### Section 16: Radius of convergence of power expansions around singular points of algebraic functions, Part II


### Work in progress . . .

This is a continuation of Part 1. In this section, we study the fourth test case in the reference: Determining the radii of convergence of power expansions around singular points of algebraic functions which involved the following function: $$$$f_4(z,w)=\left(z^{30}+z^{32}\right)+\left(z^{14}+z^{20}\right) w^5+\left(z^5+z^9\right)w^9+\left(z+z^3\right) w^{12}+6 w^{14}+\left(2+z^2\right) w^{15}=0$$$$ Readers should review the paper and Part I first. The branch summary table over the set of finite singular points for this function was too long to include in the paper and is shown in the table below.

#### Table 1: Branch types and CLSPs

$s_n$$s_n Value(Branch Type,CLSP) s_{1} 0$$(T,s_{118})$, $(F_{3}^{4},s_{2})$, $(F_{4}^{9},s_{7})$, $(F_{5}^{16},s_{27})$, $(V_{2},s_{2})$
$s_{2}$ $-0.0511 - 0.1588 i$$(T,s_{118}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (V_{2},s_{1}) s_{3} -0.0511 + 0.1588 i$$(T,s_{119})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(V_{2},s_{1})$
$s_{4}$ $0.1431 - 0.1047 i$$(T,s_{122}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (V_{2},s_{1}) s_{5} 0.1431 + 0.1047 i$$(T,s_{123})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(T,s_{1})$, $(V_{2},s_{1})$
$s_{6}$ $-0.1855$$(T,s_{118}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (T,s_{1}), (V_{2},s_{1}) s_{7} 0.3839 - 0.3280 i$$(T,s_{122})$, $(T,s_{19})$, $(T,s_{19})$, $(T,s_{79})$, $(T,s_{53})$, $(T,s_{71})$, $(T,s_{23})$, $(T,s_{41})$, $(T,s_{55})$, $(T,s_{41})$, $(T,s_{35})$, $(T,s_{55})$, $(T,s_{57})$, $(V_{2},s_{53})$
$s_{8}$ $0.3839 + 0.3280 i$$(T,s_{123}), (T,s_{20}), (T,s_{20}), (T,s_{80}), (T,s_{54}), (T,s_{72}), (T,s_{24}), (T,s_{42}), (T,s_{56}), (T,s_{42}), (T,s_{36}), (T,s_{56}), (T,s_{58}), (V_{2},s_{54}) s_{9} -0.2792 - 0.4284 i$$(T,s_{118})$, $(T,s_{25})$, $(T,s_{25})$, $(T,s_{61})$, $(T,s_{45})$, $(T,s_{85})$, $(T,s_{69})$, $(T,s_{59})$, $(T,s_{65})$, $(T,s_{21})$, $(T,s_{59})$, $(T,s_{21})$, $(T,s_{65})$, $(V_{2},s_{61})$
$s_{10}$ $-0.2792 + 0.4284 i$$(T,s_{119}), (T,s_{26}), (T,s_{26}), (T,s_{62}), (T,s_{46}), (T,s_{86}), (T,s_{70}), (T,s_{60}), (T,s_{66}), (T,s_{22}), (T,s_{60}), (T,s_{22}), (T,s_{66}), (V_{2},s_{62}) s_{11} -0.4834 - 0.1976 i$$(T,s_{118})$, $(T,s_{25})$, $(T,s_{17})$, $(T,s_{45})$, $(T,s_{17})$, $(T,s_{88})$, $(T,s_{45})$, $(T,s_{67})$, $(T,s_{73})$, $(T,s_{31})$, $(T,s_{43})$, $(T,s_{31})$, $(T,s_{40})$, $(V_{2},s_{43})$
$s_{12}$ $-0.4834 + 0.1976 i$$(T,s_{119}), (T,s_{26}), (T,s_{18}), (T,s_{46}), (T,s_{18}), (T,s_{89}), (T,s_{46}), (T,s_{68}), (T,s_{74}), (T,s_{32}), (T,s_{44}), (T,s_{32}), (T,s_{40}), (V_{2},s_{44}) s_{13} 0.1227 - 0.5252 i$$(T,s_{138})$, $(T,s_{19})$, $(T,s_{47})$, $(T,s_{23})$, $(T,s_{47})$, $(T,s_{23})$, $(T,s_{7})$, $(T,s_{63})$, $(T,s_{75})$, $(T,s_{81})$, $(T,s_{29})$, $(T,s_{98})$, $(T,s_{29})$, $(V_{2},s_{81})$
$s_{14}$ $0.1227 + 0.5252 i$$(T,s_{139}), (T,s_{20}), (T,s_{48}), (T,s_{24}), (T,s_{48}), (T,s_{24}), (T,s_{8}), (T,s_{64}), (T,s_{76}), (T,s_{82}), (T,s_{30}), (T,s_{99}), (T,s_{30}), (V_{2},s_{82}) s_{15} 0.5618 - 0.0005 i$$(T,s_{124})$, $(T,s_{87})$, $(T,s_{27})$, $(T,s_{28})$, $(T,s_{87})$, $(T,s_{36})$, $(T,s_{35})$, $(T,s_{27})$, $(T,s_{28})$, $(T,s_{50})$, $(T,s_{49})$, $(T,s_{16})$, $(T,s_{16})$, $(V_{2},s_{49})$
$s_{16}$ $0.5618 + 0.0005 i$$(T,s_{125}), (T,s_{87}), (T,s_{28}), (T,s_{27}), (T,s_{87}), (T,s_{35}), (T,s_{36}), (T,s_{28}), (T,s_{27}), (T,s_{49}), (T,s_{50}), (T,s_{15}), (T,s_{15}), (V_{2},s_{50}) s_{17} -0.5476 - 0.2293 i$$(T,s_{118})$, $(T,s_{88})$, $(T,s_{45})$, $(T,s_{88})$, $(T,s_{73})$, $(T,s_{67})$, $(T,s_{73})$, $(T,s_{31})$, $(T,s_{43})$, $(T,s_{31})$, $(T,s_{11})$, $(T,s_{11})$, $(T,s_{83})$, $(V_{2},s_{67})$
$s_{18}$ $-0.5476 + 0.2293 i$$(T,s_{119}), (T,s_{89}), (T,s_{46}), (T,s_{89}), (T,s_{74}), (T,s_{68}), (T,s_{74}), (T,s_{32}), (T,s_{44}), (T,s_{32}), (T,s_{12}), (T,s_{12}), (T,s_{84}), (V_{2},s_{68}) s_{19} 0.4473 - 0.4023 i$$(T,s_{122})$, $(T,s_{7})$, $(T,s_{79})$, $(T,s_{7})$, $(T,s_{53})$, $(T,s_{71})$, $(T,s_{71})$, $(T,s_{41})$, $(T,s_{51})$, $(T,s_{55})$, $(T,s_{41})$, $(T,s_{55})$, $(T,s_{57})$, $(V_{2},s_{79})$
$s_{20}$ $0.4473 + 0.4023 i$$(T,s_{123}), (T,s_{8}), (T,s_{80}), (T,s_{8}), (T,s_{54}), (T,s_{72}), (T,s_{72}), (T,s_{42}), (T,s_{52}), (T,s_{56}), (T,s_{42}), (T,s_{56}), (T,s_{58}), (V_{2},s_{80}) s_{21} -0.2736 - 0.5458 i$$(T,s_{136})$, $(T,s_{25})$, $(T,s_{9})$, $(T,s_{9})$, $(T,s_{61})$, $(T,s_{25})$, $(T,s_{85})$, $(T,s_{77})$, $(T,s_{69})$, $(T,s_{65})$, $(T,s_{59})$, $(T,s_{59})$, $(T,s_{65})$, $(V_{2},s_{85})$
$s_{22}$ $-0.2736 + 0.5458 i$$(T,s_{137}), (T,s_{26}), (T,s_{10}), (T,s_{10}), (T,s_{62}), (T,s_{26}), (T,s_{86}), (T,s_{78}), (T,s_{70}), (T,s_{66}), (T,s_{60}), (T,s_{60}), (T,s_{66}), (V_{2},s_{86}) s_{23} 0.2482 - 0.5617 i$$(T,s_{138})$, $(T,s_{19})$, $(T,s_{47})$, $(T,s_{47})$, $(T,s_{7})$, $(T,s_{90})$, $(T,s_{63})$, $(T,s_{29})$, $(T,s_{98})$, $(T,s_{81})$, $(T,s_{29})$, $(T,s_{13})$, $(T,s_{13})$, $(V_{2},s_{63})$
$s_{24}$ $0.2482 + 0.5617 i$$(T,s_{139}), (T,s_{20}), (T,s_{48}), (T,s_{48}), (T,s_{8}), (T,s_{91}), (T,s_{64}), (T,s_{30}), (T,s_{99}), (T,s_{82}), (T,s_{30}), (T,s_{14}), (T,s_{14}), (V_{2},s_{64}) s_{25} -0.3983 - 0.4778 i$$(T,s_{136})$, $(T,s_{9})$, $(T,s_{45})$, $(T,s_{61})$, $(T,s_{9})$, $(T,s_{69})$, $(T,s_{69})$, $(T,s_{85})$, $(T,s_{59})$, $(T,s_{65})$, $(T,s_{21})$, $(T,s_{21})$, $(T,s_{65})$, $(V_{2},s_{45})$
$s_{26}$ $-0.3983 + 0.4778 i$$(T,s_{137}), (T,s_{10}), (T,s_{46}), (T,s_{62}), (T,s_{10}), (T,s_{70}), (T,s_{70}), (T,s_{86}), (T,s_{60}), (T,s_{66}), (T,s_{22}), (T,s_{22}), (T,s_{66}), (V_{2},s_{46}) s_{27} 0.5986 - 0.2303 i$$(T,s_{122})$, $(T,s_{53})$, $(T,s_{79})$, $(T,s_{51})$, $(T,s_{38})$, $(T,s_{35})$, $(T,s_{38})$, $(T,s_{35})$, $(T,s_{57})$, $(T,s_{71})$, $(T,s_{71})$, $(T,s_{15})$, $(T,s_{57})$, $(V_{2},s_{53})$
$s_{28}$ $0.5986 + 0.2303 i$$(T,s_{123}), (T,s_{54}), (T,s_{80}), (T,s_{52}), (T,s_{39}), (T,s_{36}), (T,s_{39}), (T,s_{36}), (T,s_{58}), (T,s_{72}), (T,s_{72}), (T,s_{16}), (T,s_{58}), (V_{2},s_{54}) s_{29} -0.0048 - 0.6440 i$$(T,s_{144})$, $(T,s_{9})$, $(T,s_{63})$, $(T,s_{23})$, $(T,s_{63})$, $(T,s_{47})$, $(T,s_{47})$, $(T,s_{102})$, $(T,s_{21})$, $(T,s_{13})$, $(T,s_{75})$, $(T,s_{92})$, $(T,s_{13})$, $(V_{2},s_{75})$
$s_{30}$ $-0.0048 + 0.6440 i$$(T,s_{145}), (T,s_{10}), (T,s_{64}), (T,s_{24}), (T,s_{64}), (T,s_{48}), (T,s_{48}), (T,s_{103}), (T,s_{22}), (T,s_{14}), (T,s_{76}), (T,s_{93}), (T,s_{14}), (V_{2},s_{76}) s_{31} -0.5584 - 0.3214 i$$(T,s_{118})$, $(T,s_{25})$, $(T,s_{45})$, $(T,s_{17})$, $(T,s_{69})$, $(T,s_{45})$, $(T,s_{73})$, $(T,s_{17})$, $(T,s_{67})$, $(T,s_{43})$, $(T,s_{11})$, $(T,s_{43})$, $(T,s_{83})$, $(V_{2},s_{83})$
$s_{32}$ $-0.5584 + 0.3214 i$$(T,s_{119}), (T,s_{26}), (T,s_{46}), (T,s_{18}), (T,s_{70}), (T,s_{46}), (T,s_{74}), (T,s_{18}), (T,s_{68}), (T,s_{44}), (T,s_{12}), (T,s_{44}), (T,s_{84}), (V_{2},s_{84}) s_{33} -0.6477 - 0.1319 i$$(T,s_{118})$, $(T,s_{88})$, $(T,s_{67})$, $(T,s_{37})$, $(T,s_{88})$, $(T,s_{67})$, $(T,s_{34})$, $(T,s_{73})$, $(T,s_{73})$, $(T,s_{83})$, $(T,s_{43})$, $(T,s_{40})$, $(T,s_{40})$, $(V_{2},s_{43})$
$s_{34}$ $-0.6477 + 0.1319 i$$(T,s_{119}), (T,s_{89}), (T,s_{68}), (T,s_{37}), (T,s_{89}), (T,s_{68}), (T,s_{33}), (T,s_{74}), (T,s_{74}), (T,s_{84}), (T,s_{44}), (T,s_{40}), (T,s_{40}), (V_{2},s_{44}) s_{35} 0.5839 - 0.3279 i$$(T,s_{122})$, $(T,s_{27})$, $(T,s_{79})$, $(T,s_{53})$, $(T,s_{27})$, $(T,s_{79})$, $(T,s_{19})$, $(T,s_{51})$, $(T,s_{71})$, $(T,s_{71})$, $(T,s_{57})$, $(T,s_{55})$, $(T,s_{57})$, $(V_{2},s_{51})$
$s_{36}$ $0.5839 + 0.3279 i$$(T,s_{123}), (T,s_{28}), (T,s_{80}), (T,s_{54}), (T,s_{28}), (T,s_{80}), (T,s_{20}), (T,s_{52}), (T,s_{72}), (T,s_{72}), (T,s_{58}), (T,s_{56}), (T,s_{58}), (V_{2},s_{52}) s_{37} -0.6788$$(T,s_{118})$, $(T,s_{67})$, $(T,s_{68})$, $(T,s_{33})$, $(T,s_{34})$, $(T,s_{33})$, $(T,s_{34})$, $(T,s_{74})$, $(T,s_{73})$, $(T,s_{74})$, $(T,s_{73})$, $(T,s_{40})$, $(T,s_{40})$, $(V_{2},s_{67})$
$s_{38}$ $0.6664 - 0.1297 i$$(T,s_{124}), (T,s_{87}), (T,s_{27}), (T,s_{87}), (T,s_{27}), (T,s_{79}), (T,s_{35}), (T,s_{35}), (T,s_{96}), (T,s_{50}), (T,s_{49}), (T,s_{15}), (T,s_{49}), (V_{2},s_{50}) s_{39} 0.6664 + 0.1297 i$$(T,s_{125})$, $(T,s_{87})$, $(T,s_{28})$, $(T,s_{87})$, $(T,s_{28})$, $(T,s_{80})$, $(T,s_{36})$, $(T,s_{36})$, $(T,s_{97})$, $(T,s_{49})$, $(T,s_{50})$, $(T,s_{16})$, $(T,s_{50})$, $(V_{2},s_{49})$
$s_{40}$ $-0.6814$$(T,s_{118}), (T,s_{67}), (T,s_{68}), (T,s_{37}), (T,s_{37}), (T,s_{33}), (T,s_{34}), (T,s_{33}), (T,s_{34}), (T,s_{74}), (T,s_{73}), (T,s_{74}), (T,s_{73}), (V_{2},s_{83}) s_{41} 0.5416 - 0.4400 i$$(T,s_{122})$, $(T,s_{19})$, $(T,s_{7})$, $(T,s_{79})$, $(T,s_{19})$, $(T,s_{53})$, $(T,s_{53})$, $(T,s_{71})$, $(T,s_{71})$, $(T,s_{51})$, $(T,s_{55})$, $(T,s_{57})$, $(T,s_{57})$, $(V_{2},s_{51})$
$s_{42}$ $0.5416 + 0.4400 i$$(T,s_{123}), (T,s_{20}), (T,s_{8}), (T,s_{80}), (T,s_{20}), (T,s_{54}), (T,s_{54}), (T,s_{72}), (T,s_{72}), (T,s_{52}), (T,s_{56}), (T,s_{58}), (T,s_{58}), (V_{2},s_{52}) s_{43} -0.6397 - 0.2804 i$$(T,s_{118})$, $(T,s_{88})$, $(T,s_{17})$, $(T,s_{45})$, $(T,s_{88})$, $(T,s_{17})$, $(T,s_{73})$, $(T,s_{73})$, $(T,s_{67})$, $(T,s_{31})$, $(T,s_{31})$, $(T,s_{33})$, $(T,s_{83})$, $(V_{2},s_{33})$
$s_{44}$ $-0.6397 + 0.2804 i$$(T,s_{119}), (T,s_{89}), (T,s_{18}), (T,s_{46}), (T,s_{89}), (T,s_{18}), (T,s_{74}), (T,s_{74}), (T,s_{68}), (T,s_{32}), (T,s_{32}), (T,s_{34}), (T,s_{84}), (V_{2},s_{34}) s_{45} -0.5220 - 0.4699 i$$(T,s_{136})$, $(T,s_{25})$, $(T,s_{61})$, $(T,s_{69})$, $(T,s_{61})$, $(T,s_{69})$, $(T,s_{31})$, $(T,s_{17})$, $(T,s_{85})$, $(T,s_{65})$, $(T,s_{85})$, $(T,s_{65})$, $(T,s_{43})$, $(V_{2},s_{25})$
$s_{46}$ $-0.5220 + 0.4699 i$$(T,s_{137}), (T,s_{26}), (T,s_{62}), (T,s_{70}), (T,s_{62}), (T,s_{70}), (T,s_{32}), (T,s_{18}), (T,s_{86}), (T,s_{66}), (T,s_{86}), (T,s_{66}), (T,s_{44}), (V_{2},s_{26}) s_{47} 0.0644 - 0.7098 i$$(T,s_{144})$, $(T,s_{112})$, $(T,s_{63})$, $(T,s_{63})$, $(T,s_{23})$, $(T,s_{108})$, $(T,s_{81})$, $(T,s_{75})$, $(T,s_{13})$, $(T,s_{92})$, $(T,s_{29})$, $(T,s_{13})$, $(T,s_{29})$, $(V_{2},s_{108})$
$s_{48}$ $0.0644 + 0.7098 i$$(T,s_{145}), (T,s_{113}), (T,s_{64}), (T,s_{64}), (T,s_{24}), (T,s_{109}), (T,s_{82}), (T,s_{76}), (T,s_{14}), (T,s_{93}), (T,s_{30}), (T,s_{14}), (T,s_{30}), (V_{2},s_{109}) s_{49} 0.7173 - 0.0097 i$$(T,s_{124})$, $(T,s_{87})$, $(T,s_{87})$, $(T,s_{27})$, $(T,s_{28})$, $(T,s_{97})$, $(T,s_{97})$, $(T,s_{96})$, $(T,s_{96})$, $(T,s_{38})$, $(T,s_{39})$, $(T,s_{50})$, $(T,s_{50})$, $(V_{2},s_{39})$
$s_{50}$ $0.7173 + 0.0097 i$$(T,s_{125}), (T,s_{87}), (T,s_{87}), (T,s_{28}), (T,s_{27}), (T,s_{96}), (T,s_{96}), (T,s_{97}), (T,s_{97}), (T,s_{39}), (T,s_{38}), (T,s_{49}), (T,s_{49}), (V_{2},s_{38}) s_{51} 0.5998 - 0.3948 i$$(T,s_{122})$, $(T,s_{27})$, $(T,s_{79})$, $(T,s_{79})$, $(T,s_{53})$, $(T,s_{53})$, $(T,s_{19})$, $(T,s_{71})$, $(T,s_{71})$, $(T,s_{35})$, $(T,s_{55})$, $(T,s_{57})$, $(T,s_{57})$, $(V_{2},s_{35})$
$s_{52}$ $0.5998 + 0.3948 i$$(T,s_{123}), (T,s_{28}), (T,s_{80}), (T,s_{80}), (T,s_{54}), (T,s_{54}), (T,s_{20}), (T,s_{72}), (T,s_{72}), (T,s_{36}), (T,s_{56}), (T,s_{58}), (T,s_{58}), (V_{2},s_{36}) s_{53} 0.5889 - 0.4371 i$$(T,s_{122})$, $(T,s_{27})$, $(T,s_{79})$, $(T,s_{79})$, $(T,s_{19})$, $(T,s_{71})$, $(T,s_{51})$, $(T,s_{71})$, $(T,s_{51})$, $(T,s_{41})$, $(T,s_{55})$, $(T,s_{57})$, $(T,s_{57})$, $(V_{2},s_{27})$
$s_{54}$ $0.5889 + 0.4371 i$$(T,s_{123}), (T,s_{28}), (T,s_{80}), (T,s_{80}), (T,s_{20}), (T,s_{72}), (T,s_{52}), (T,s_{72}), (T,s_{52}), (T,s_{42}), (T,s_{56}), (T,s_{58}), (T,s_{58}), (V_{2},s_{28}) s_{55} 0.4492 - 0.5842 i$$(T,s_{126})$, $(T,s_{19})$, $(T,s_{94})$, $(T,s_{90})$, $(T,s_{90})$, $(T,s_{94})$, $(T,s_{53})$, $(T,s_{23})$, $(T,s_{98})$, $(T,s_{98})$, $(T,s_{41})$, $(T,s_{41})$, $(T,s_{81})$, $(V_{2},s_{23})$
$s_{56}$ $0.4492 + 0.5842 i$$(T,s_{127}), (T,s_{20}), (T,s_{95}), (T,s_{91}), (T,s_{91}), (T,s_{95}), (T,s_{54}), (T,s_{24}), (T,s_{99}), (T,s_{99}), (T,s_{42}), (T,s_{42}), (T,s_{82}), (V_{2},s_{24}) s_{57} 0.6600 - 0.3677 i$$(T,s_{122})$, $(T,s_{27})$, $(T,s_{79})$, $(T,s_{53})$, $(T,s_{53})$, $(T,s_{79})$, $(T,s_{19})$, $(T,s_{51})$, $(T,s_{51})$, $(T,s_{71})$, $(T,s_{71})$, $(T,s_{35})$, $(T,s_{55})$, $(V_{2},s_{55})$
$s_{58}$ $0.6600 + 0.3677 i$$(T,s_{123}), (T,s_{28}), (T,s_{80}), (T,s_{54}), (T,s_{54}), (T,s_{80}), (T,s_{20}), (T,s_{52}), (T,s_{52}), (T,s_{72}), (T,s_{72}), (T,s_{36}), (T,s_{56}), (V_{2},s_{56}) s_{59} -0.2558 - 0.7249 i$$(T,s_{150})$, $(T,s_{61})$, $(T,s_{102})$, $(T,s_{104})$, $(T,s_{102})$, $(T,s_{104})$, $(T,s_{77})$, $(T,s_{21})$, $(T,s_{92})$, $(T,s_{100})$, $(T,s_{92})$, $(T,s_{21})$, $(T,s_{75})$, $(V_{2},s_{75})$
$s_{60}$ $-0.2558 + 0.7249 i$$(T,s_{151}), (T,s_{62}), (T,s_{103}), (T,s_{105}), (T,s_{103}), (T,s_{105}), (T,s_{78}), (T,s_{22}), (T,s_{93}), (T,s_{101}), (T,s_{93}), (T,s_{22}), (T,s_{76}), (V_{2},s_{76}) s_{61} -0.5103 - 0.5840 i$$(T,s_{136})$, $(T,s_{45})$, $(T,s_{77})$, $(T,s_{77})$, $(T,s_{69})$, $(T,s_{31})$, $(T,s_{69})$, $(T,s_{85})$, $(T,s_{59})$, $(T,s_{85})$, $(T,s_{65})$, $(T,s_{65})$, $(T,s_{21})$, $(V_{2},s_{31})$
$s_{62}$ $-0.5103 + 0.5840 i$$(T,s_{137}), (T,s_{46}), (T,s_{78}), (T,s_{78}), (T,s_{70}), (T,s_{32}), (T,s_{70}), (T,s_{86}), (T,s_{60}), (T,s_{86}), (T,s_{66}), (T,s_{66}), (T,s_{22}), (V_{2},s_{32}) s_{63} 0.2395 - 0.7416 i$$(T,s_{138})$, $(T,s_{94})$, $(T,s_{47})$, $(T,s_{47})$, $(T,s_{90})$, $(T,s_{90})$, $(T,s_{23})$, $(T,s_{29})$, $(T,s_{81})$, $(T,s_{98})$, $(T,s_{81})$, $(T,s_{98})$, $(T,s_{13})$, $(V_{2},s_{23})$
$s_{64}$ $0.2395 + 0.7416 i$$(T,s_{139}), (T,s_{95}), (T,s_{48}), (T,s_{48}), (T,s_{91}), (T,s_{91}), (T,s_{24}), (T,s_{30}), (T,s_{82}), (T,s_{99}), (T,s_{82}), (T,s_{99}), (T,s_{14}), (V_{2},s_{24}) s_{65} -0.5522 - 0.5537 i$$(T,s_{136})$, $(T,s_{45})$, $(T,s_{77})$, $(T,s_{45})$, $(T,s_{61})$, $(T,s_{61})$, $(T,s_{69})$, $(T,s_{69})$, $(T,s_{31})$, $(T,s_{17})$, $(T,s_{85})$, $(T,s_{85})$, $(T,s_{21})$, $(V_{2},s_{11})$
$s_{66}$ $-0.5522 + 0.5537 i$$(T,s_{137}), (T,s_{46}), (T,s_{78}), (T,s_{46}), (T,s_{62}), (T,s_{62}), (T,s_{70}), (T,s_{70}), (T,s_{32}), (T,s_{18}), (T,s_{86}), (T,s_{86}), (T,s_{22}), (V_{2},s_{12}) s_{67} -0.7538 - 0.2106 i$$(T,s_{118})$, $(T,s_{88})$, $(T,s_{17})$, $(T,s_{88})$, $(T,s_{45})$, $(T,s_{73})$, $(T,s_{73})$, $(T,s_{33})$, $(T,s_{33})$, $(T,s_{40})$, $(T,s_{43})$, $(T,s_{83})$, $(T,s_{83})$, $(V_{2},s_{17})$
$s_{68}$ $-0.7538 + 0.2106 i$$(T,s_{119}), (T,s_{89}), (T,s_{18}), (T,s_{89}), (T,s_{46}), (T,s_{74}), (T,s_{74}), (T,s_{34}), (T,s_{34}), (T,s_{40}), (T,s_{44}), (T,s_{84}), (T,s_{84}), (V_{2},s_{18}) s_{69} -0.5939 - 0.5122 i$$(T,s_{136})$, $(T,s_{25})$, $(T,s_{45})$, $(T,s_{45})$, $(T,s_{61})$, $(T,s_{61})$, $(T,s_{31})$, $(T,s_{17})$, $(T,s_{65})$, $(T,s_{85})$, $(T,s_{85})$, $(T,s_{65})$, $(T,s_{43})$, $(V_{2},s_{88})$
$s_{70}$ $-0.5939 + 0.5122 i$$(T,s_{137}), (T,s_{26}), (T,s_{46}), (T,s_{46}), (T,s_{62}), (T,s_{62}), (T,s_{32}), (T,s_{18}), (T,s_{66}), (T,s_{86}), (T,s_{86}), (T,s_{66}), (T,s_{44}), (V_{2},s_{89}) s_{71} 0.6542 - 0.4333 i$$(T,s_{122})$, $(T,s_{27})$, $(T,s_{79})$, $(T,s_{79})$, $(T,s_{53})$, $(T,s_{53})$, $(T,s_{19})$, $(T,s_{51})$, $(T,s_{51})$, $(T,s_{41})$, $(T,s_{55})$, $(T,s_{57})$, $(T,s_{57})$, $(V_{2},s_{94})$
$s_{72}$ $0.6542 + 0.4333 i$$(T,s_{123}), (T,s_{28}), (T,s_{80}), (T,s_{80}), (T,s_{54}), (T,s_{54}), (T,s_{20}), (T,s_{52}), (T,s_{52}), (T,s_{42}), (T,s_{56}), (T,s_{58}), (T,s_{58}), (V_{2},s_{95}) s_{73} -0.7620 - 0.2140 i$$(T,s_{118})$, $(T,s_{88})$, $(T,s_{17})$, $(T,s_{88})$, $(T,s_{67})$, $(T,s_{45})$, $(T,s_{67})$, $(T,s_{33})$, $(T,s_{43})$, $(T,s_{40})$, $(T,s_{43})$, $(T,s_{83})$, $(T,s_{83})$, $(V_{2},s_{45})$
$s_{74}$ $-0.7620 + 0.2140 i$$(T,s_{119}), (T,s_{89}), (T,s_{18}), (T,s_{89}), (T,s_{68}), (T,s_{46}), (T,s_{68}), (T,s_{34}), (T,s_{44}), (T,s_{40}), (T,s_{44}), (T,s_{84}), (T,s_{84}), (V_{2},s_{46}) s_{75} -0.0873 - 0.7871 i$$(T,s_{144})$, $(T,s_{104})$, $(T,s_{110})$, $(T,s_{102})$, $(T,s_{104})$, $(T,s_{47})$, $(T,s_{102})$, $(T,s_{47})$, $(T,s_{110})$, $(T,s_{100})$, $(T,s_{92})$, $(T,s_{92})$, $(T,s_{29})$, $(V_{2},s_{29})$
$s_{76}$ $-0.0873 + 0.7871 i$$(T,s_{145}), (T,s_{105}), (T,s_{111}), (T,s_{103}), (T,s_{105}), (T,s_{48}), (T,s_{103}), (T,s_{48}), (T,s_{111}), (T,s_{101}), (T,s_{93}), (T,s_{93}), (T,s_{30}), (V_{2},s_{30}) s_{77} -0.4366 - 0.6651 i$$(T,s_{136})$, $(T,s_{25})$, $(T,s_{61})$, $(T,s_{61})$, $(T,s_{9})$, $(T,s_{69})$, $(T,s_{85})$, $(T,s_{59})$, $(T,s_{85})$, $(T,s_{59})$, $(T,s_{65})$, $(T,s_{21})$, $(T,s_{65})$, $(V_{2},s_{25})$
$s_{78}$ $-0.4366 + 0.6651 i$$(T,s_{137}), (T,s_{26}), (T,s_{62}), (T,s_{62}), (T,s_{10}), (T,s_{70}), (T,s_{86}), (T,s_{60}), (T,s_{86}), (T,s_{60}), (T,s_{66}), (T,s_{22}), (T,s_{66}), (V_{2},s_{26}) s_{79} 0.7129 - 0.3861 i$$(T,s_{122})$, $(T,s_{27})$, $(T,s_{53})$, $(T,s_{53})$, $(T,s_{38})$, $(T,s_{51})$, $(T,s_{51})$, $(T,s_{71})$, $(T,s_{71})$, $(T,s_{35})$, $(T,s_{57})$, $(T,s_{55})$, $(T,s_{57})$, $(V_{2},s_{38})$
$s_{80}$ $0.7129 + 0.3861 i$$(T,s_{123}), (T,s_{28}), (T,s_{54}), (T,s_{54}), (T,s_{39}), (T,s_{52}), (T,s_{52}), (T,s_{72}), (T,s_{72}), (T,s_{36}), (T,s_{58}), (T,s_{56}), (T,s_{58}), (V_{2},s_{39}) s_{81} 0.3594 - 0.7350 i$$(T,s_{138})$, $(T,s_{94})$, $(T,s_{47})$, $(T,s_{94})$, $(T,s_{90})$, $(T,s_{63})$, $(T,s_{90})$, $(T,s_{63})$, $(T,s_{29})$, $(T,s_{55})$, $(T,s_{98})$, $(T,s_{98})$, $(T,s_{13})$, $(V_{2},s_{13})$
$s_{82}$ $0.3594 + 0.7350 i$$(T,s_{139}), (T,s_{95}), (T,s_{48}), (T,s_{95}), (T,s_{91}), (T,s_{64}), (T,s_{91}), (T,s_{64}), (T,s_{30}), (T,s_{56}), (T,s_{99}), (T,s_{99}), (T,s_{14}), (V_{2},s_{14}) s_{83} -0.7782 - 0.2551 i$$(T,s_{118})$, $(T,s_{88})$, $(T,s_{17})$, $(T,s_{88})$, $(T,s_{45})$, $(T,s_{67})$, $(T,s_{67})$, $(T,s_{73})$, $(T,s_{73})$, $(T,s_{33})$, $(T,s_{43})$, $(T,s_{31})$, $(T,s_{43})$, $(V_{2},s_{31})$
$s_{84}$ $-0.7782 + 0.2551 i$$(T,s_{119}), (T,s_{89}), (T,s_{18}), (T,s_{89}), (T,s_{46}), (T,s_{68}), (T,s_{68}), (T,s_{74}), (T,s_{74}), (T,s_{34}), (T,s_{44}), (T,s_{32}), (T,s_{44}), (V_{2},s_{32}) s_{85} -0.6011 - 0.5711 i$$(T,s_{136})$, $(T,s_{45})$, $(T,s_{77})$, $(T,s_{45})$, $(T,s_{61})$, $(T,s_{69})$, $(T,s_{61})$, $(T,s_{69})$, $(T,s_{31})$, $(T,s_{17})$, $(T,s_{65})$, $(T,s_{65})$, $(T,s_{43})$, $(V_{2},s_{43})$
$s_{86}$ $-0.6011 + 0.5711 i$$(T,s_{137}), (T,s_{46}), (T,s_{78}), (T,s_{46}), (T,s_{62}), (T,s_{70}), (T,s_{62}), (T,s_{70}), (T,s_{32}), (T,s_{18}), (T,s_{66}), (T,s_{66}), (T,s_{44}), (V_{2},s_{44}) s_{87} 0.8419$$(T,s_{124})$, $(T,s_{27})$, $(T,s_{28})$, $(T,s_{97})$, $(T,s_{96})$, $(T,s_{97})$, $(T,s_{96})$, $(T,s_{38})$, $(T,s_{39})$, $(T,s_{50})$, $(T,s_{49})$, $(T,s_{50})$, $(T,s_{49})$, $(V_{2},s_{124})$
$s_{88}$ $-0.7854 - 0.3371 i$$(T,s_{118}), (T,s_{45}), (T,s_{69}), (T,s_{67}), (T,s_{73}), (T,s_{73}), (T,s_{67}), (T,s_{31}), (T,s_{33}), (T,s_{83}), (T,s_{43}), (T,s_{43}), (T,s_{83}), (V_{2},s_{69}) s_{89} -0.7854 + 0.3371 i$$(T,s_{119})$, $(T,s_{46})$, $(T,s_{70})$, $(T,s_{68})$, $(T,s_{74})$, $(T,s_{74})$, $(T,s_{68})$, $(T,s_{32})$, $(T,s_{34})$, $(T,s_{84})$, $(T,s_{44})$, $(T,s_{44})$, $(T,s_{84})$, $(V_{2},s_{70})$
$s_{90}$ $0.4578 - 0.7352 i$$(T,s_{138}), (T,s_{19}), (T,s_{94}), (T,s_{94}), (T,s_{63}), (T,s_{53}), (T,s_{55}), (T,s_{55}), (T,s_{98}), (T,s_{98}), (T,s_{81}), (T,s_{81}), (T,s_{41}), (V_{2},s_{7}) s_{91} 0.4578 + 0.7352 i$$(T,s_{139})$, $(T,s_{20})$, $(T,s_{95})$, $(T,s_{95})$, $(T,s_{64})$, $(T,s_{54})$, $(T,s_{56})$, $(T,s_{56})$, $(T,s_{99})$, $(T,s_{99})$, $(T,s_{82})$, $(T,s_{82})$, $(T,s_{42})$, $(V_{2},s_{8})$
$s_{92}$ $-0.1489 - 0.8630 i$$(T,s_{144}), (T,s_{104}), (T,s_{110}), (T,s_{102}), (T,s_{104}), (T,s_{102}), (T,s_{110}), (T,s_{114}), (T,s_{100}), (T,s_{100}), (T,s_{75}), (T,s_{75}), (T,s_{59}), (V_{2},s_{114}) s_{93} -0.1489 + 0.8630 i$$(T,s_{145})$, $(T,s_{105})$, $(T,s_{111})$, $(T,s_{103})$, $(T,s_{105})$, $(T,s_{103})$, $(T,s_{111})$, $(T,s_{115})$, $(T,s_{101})$, $(T,s_{101})$, $(T,s_{76})$, $(T,s_{76})$, $(T,s_{60})$, $(V_{2},s_{115})$
$s_{94}$ $0.5415 - 0.6974 i$$(T,s_{126}), (T,s_{19}), (T,s_{90}), (T,s_{90}), (T,s_{53}), (T,s_{71}), (T,s_{98}), (T,s_{55}), (T,s_{98}), (T,s_{55}), (T,s_{81}), (T,s_{41}), (T,s_{81}), (V_{2},s_{71}) s_{95} 0.5415 + 0.6974 i$$(T,s_{127})$, $(T,s_{20})$, $(T,s_{91})$, $(T,s_{91})$, $(T,s_{54})$, $(T,s_{72})$, $(T,s_{99})$, $(T,s_{56})$, $(T,s_{99})$, $(T,s_{56})$, $(T,s_{82})$, $(T,s_{42})$, $(T,s_{82})$, $(V_{2},s_{72})$
$s_{96}$ $0.8863 - 0.0763 i$$(T,s_{124}), (T,s_{87}), (T,s_{87}), (T,s_{27}), (T,s_{35}), (T,s_{97}), (T,s_{97}), (T,s_{38}), (T,s_{50}), (T,s_{39}), (T,s_{49}), (T,s_{50}), (T,s_{49}), (V_{2},s_{57}) s_{97} 0.8863 + 0.0763 i$$(T,s_{125})$, $(T,s_{87})$, $(T,s_{87})$, $(T,s_{28})$, $(T,s_{36})$, $(T,s_{96})$, $(T,s_{96})$, $(T,s_{39})$, $(T,s_{49})$, $(T,s_{38})$, $(T,s_{50})$, $(T,s_{49})$, $(T,s_{50})$, $(V_{2},s_{58})$
$s_{98}$ $0.4373 - 0.7850 i$$(T,s_{138}), (T,s_{47}), (T,s_{94}), (T,s_{94}), (T,s_{90}), (T,s_{90}), (T,s_{63}), (T,s_{55}), (T,s_{53}), (T,s_{55}), (T,s_{81}), (T,s_{81}), (T,s_{41}), (V_{2},s_{53}) s_{99} 0.4373 + 0.7850 i$$(T,s_{139})$, $(T,s_{48})$, $(T,s_{95})$, $(T,s_{95})$, $(T,s_{91})$, $(T,s_{91})$, $(T,s_{64})$, $(T,s_{56})$, $(T,s_{54})$, $(T,s_{56})$, $(T,s_{82})$, $(T,s_{82})$, $(T,s_{42})$, $(V_{2},s_{54})$
$s_{100}$ $-0.2518 - 0.9314 i$$(T,s_{144}), (T,s_{110}), (T,s_{104}), (T,s_{102}), (T,s_{102}), (T,s_{104}), (T,s_{110}), (T,s_{114}), (T,s_{92}), (T,s_{92}), (T,s_{59}), (T,s_{59}), (T,s_{75}), (V_{2},s_{110}) s_{101} -0.2518 + 0.9314 i$$(T,s_{145})$, $(T,s_{111})$, $(T,s_{105})$, $(T,s_{103})$, $(T,s_{103})$, $(T,s_{105})$, $(T,s_{111})$, $(T,s_{115})$, $(T,s_{93})$, $(T,s_{93})$, $(T,s_{60})$, $(T,s_{60})$, $(T,s_{76})$, $(V_{2},s_{111})$
$s_{102}$ $-0.1680 - 0.9516 i$$(T,s_{144}), (T,s_{110}), (T,s_{104}), (T,s_{104}), (T,s_{110}), (T,s_{114}), (T,s_{100}), (T,s_{100}), (T,s_{92}), (T,s_{92}), (T,s_{110}), (T,s_{106}), (T,s_{106}), (V_{2},s_{110}) s_{103} -0.1680 + 0.9516 i$$(T,s_{145})$, $(T,s_{111})$, $(T,s_{105})$, $(T,s_{105})$, $(T,s_{111})$, $(T,s_{115})$, $(T,s_{101})$, $(T,s_{101})$, $(T,s_{93})$, $(T,s_{93})$, $(T,s_{111})$, $(T,s_{107})$, $(T,s_{107})$, $(V_{2},s_{111})$
$s_{104}$ $-0.2304 - 0.9598 i$$(T,s_{144}), (T,s_{110}), (T,s_{102}), (T,s_{102}), (T,s_{110}), (T,s_{114}), (T,s_{92}), (T,s_{100}), (T,s_{100}), (T,s_{92}), (T,s_{75}), (T,s_{106}), (T,s_{106}), (V_{2},s_{110}) s_{105} -0.2304 + 0.9598 i$$(T,s_{145})$, $(T,s_{111})$, $(T,s_{103})$, $(T,s_{103})$, $(T,s_{111})$, $(T,s_{115})$, $(T,s_{93})$, $(T,s_{101})$, $(T,s_{101})$, $(T,s_{93})$, $(T,s_{76})$, $(T,s_{107})$, $(T,s_{107})$, $(V_{2},s_{111})$
$s_{106}$ $-0.0130 - 0.9876 i$$(T,s_{144}), (T,s_{112}), (T,s_{110}), (T,s_{110}), (T,s_{116}), (T,s_{110}), (T,s_{110}), (T,s_{110}), (T,s_{108}), (T,s_{110}), (T,s_{110}), (T,s_{114}), (T,s_{110}), (V_{2},s_{110}) s_{107} -0.0130 + 0.9876 i$$(T,s_{145})$, $(T,s_{113})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{117})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{109})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{115})$, $(T,s_{111})$, $(V_{2},s_{111})$
$s_{108}$ $0.0140 - 0.9884 i$$(T,s_{144}), (T,s_{112}), (T,s_{110}), (T,s_{110}), (T,s_{116}), (T,s_{110}), (T,s_{110}), (T,s_{110}), (T,s_{110}), (T,s_{114}), (T,s_{110}), (T,s_{110}), (T,s_{106}), (V_{2},s_{110}) s_{109} 0.0140 + 0.9884 i$$(T,s_{145})$, $(T,s_{113})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{117})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{115})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{107})$, $(V_{2},s_{111})$
$s_{110}$ $- 1.0000 i$$(T,s_{144}), (T,s_{112}), (T,s_{116}), (T,s_{108}), (T,s_{114}), (T,s_{106}), (V_{9},s_{114}) s_{111} 1.000 i$$(T,s_{145})$, $(T,s_{113})$, $(T,s_{117})$, $(T,s_{109})$, $(T,s_{115})$, $(T,s_{107})$, $(V_{9},s_{115})$
$s_{112}$ $0.0174 - 1.0090 i$$(T,s_{144}), (T,s_{110}), (T,s_{116}), (T,s_{110}), (T,s_{116}), (T,s_{108}), (T,s_{110}), (T,s_{110}), (T,s_{110}), (T,s_{114}), (T,s_{110}), (T,s_{110}), (T,s_{106}), (V_{2},s_{110}) s_{113} 0.0174 + 1.0090 i$$(T,s_{145})$, $(T,s_{111})$, $(T,s_{117})$, $(T,s_{111})$, $(T,s_{117})$, $(T,s_{109})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{115})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{107})$, $(V_{2},s_{111})$
$s_{114}$ $-0.0125 - 1.0105 i$$(T,s_{144}), (T,s_{112}), (T,s_{110}), (T,s_{110}), (T,s_{110}), (T,s_{116}), (T,s_{110}), (T,s_{110}), (T,s_{108}), (T,s_{110}), (T,s_{110}), (T,s_{110}), (T,s_{106}), (V_{2},s_{110}) s_{115} -0.0125 + 1.0105 i$$(T,s_{145})$, $(T,s_{113})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{117})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{109})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{107})$, $(V_{2},s_{111})$
$s_{116}$ $0.0041 - 1.0174 i$$(T,s_{144}), (T,s_{112}), (T,s_{112}), (T,s_{110}), (T,s_{110}), (T,s_{108}), (T,s_{110}), (T,s_{110}), (T,s_{114}), (T,s_{110}), (T,s_{110}), (T,s_{110}), (T,s_{106}), (V_{2},s_{110}) s_{117} 0.0041 + 1.0174 i$$(T,s_{145})$, $(T,s_{113})$, $(T,s_{113})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{109})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{115})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{107})$, $(V_{2},s_{111})$
$s_{118}$ $-1.0731 - 0.2103 i$ $(V_{2},s_{120})$, $(T,s_{120})$, $(T,s_{128})$, $(T,s_{121})$, $(T,s_{67})$, $(T,s_{134})$, $(T,s_{33})$, $(T,s_{73})$, $(T,s_{73})$, $(T,s_{33})$, $(T,s_{43})$, $(T,s_{40})$, $(T,s_{83})$, $(T,s_{83})$
$s_{119}$ $-1.0731 + 0.2103 i$ $(V_{2},s_{121})$, $(T,s_{121})$, $(T,s_{129})$, $(T,s_{120})$, $(T,s_{68})$, $(T,s_{135})$, $(T,s_{34})$, $(T,s_{74})$, $(T,s_{74})$, $(T,s_{34})$, $(T,s_{44})$, $(T,s_{40})$, $(T,s_{84})$, $(T,s_{84})$
$s_{120}$ $-1.1459 - 0.0600 i$ $(V_{2},s_{121})$, $(T,s_{121})$, $(T,s_{118})$, $(T,s_{119})$, $(T,s_{128})$, $(T,s_{129})$, $(T,s_{134})$, $(T,s_{33})$, $(T,s_{73})$, $(T,s_{74})$, $(T,s_{73})$, $(T,s_{83})$, $(T,s_{40})$, $(T,s_{83})$
$s_{121}$ $-1.1459 + 0.0600 i$ $(V_{2},s_{120})$, $(T,s_{120})$, $(T,s_{119})$, $(T,s_{118})$, $(T,s_{129})$, $(T,s_{128})$, $(T,s_{135})$, $(T,s_{34})$, $(T,s_{74})$, $(T,s_{73})$, $(T,s_{74})$, $(T,s_{84})$, $(T,s_{40})$, $(T,s_{84})$
$s_{122}$ $1.0345 - 0.6353 i$ $(V_{2},s_{132})$, $(T,s_{132})$, $(T,s_{130})$, $(T,s_{126})$, $(T,s_{53})$, $(T,s_{140})$, $(T,s_{51})$, $(T,s_{71})$, $(T,s_{51})$, $(T,s_{71})$, $(T,s_{41})$, $(T,s_{55})$, $(T,s_{57})$, $(T,s_{57})$
$s_{123}$ $1.0345 + 0.6353 i$ $(V_{2},s_{133})$, $(T,s_{133})$, $(T,s_{131})$, $(T,s_{127})$, $(T,s_{54})$, $(T,s_{141})$, $(T,s_{52})$, $(T,s_{72})$, $(T,s_{52})$, $(T,s_{72})$, $(T,s_{42})$, $(T,s_{56})$, $(T,s_{58})$, $(T,s_{58})$
$s_{124}$ $1.2292 - 0.0476 i$ $(V_{2},s_{125})$, $(T,s_{125})$, $(T,s_{130})$, $(T,s_{131})$, $(T,s_{97})$, $(T,s_{96})$, $(T,s_{97})$, $(T,s_{96})$, $(T,s_{38})$, $(T,s_{39})$, $(T,s_{50})$, $(T,s_{49})$, $(T,s_{50})$, $(T,s_{49})$
$s_{125}$ $1.2292 + 0.0476 i$ $(V_{2},s_{124})$, $(T,s_{124})$, $(T,s_{131})$, $(T,s_{130})$, $(T,s_{96})$, $(T,s_{97})$, $(T,s_{96})$, $(T,s_{97})$, $(T,s_{39})$, $(T,s_{38})$, $(T,s_{49})$, $(T,s_{50})$, $(T,s_{49})$, $(T,s_{50})$
$s_{126}$ $0.9601 - 0.7750 i$ $(V_{2},s_{122})$, $(T,s_{122})$, $(T,s_{132})$, $(T,s_{53})$, $(T,s_{140})$, $(T,s_{53})$, $(T,s_{138})$, $(T,s_{98})$, $(T,s_{71})$, $(T,s_{51})$, $(T,s_{55})$, $(T,s_{41})$, $(T,s_{57})$, $(T,s_{57})$
$s_{127}$ $0.9601 + 0.7750 i$ $(V_{2},s_{123})$, $(T,s_{123})$, $(T,s_{133})$, $(T,s_{54})$, $(T,s_{141})$, $(T,s_{54})$, $(T,s_{139})$, $(T,s_{99})$, $(T,s_{72})$, $(T,s_{52})$, $(T,s_{56})$, $(T,s_{42})$, $(T,s_{58})$, $(T,s_{58})$
$s_{128}$ $-1.1894 - 0.3490 i$ $(V_{2},s_{118})$, $(T,s_{118})$, $(T,s_{120})$, $(T,s_{134})$, $(T,s_{121})$, $(T,s_{67})$, $(T,s_{73})$, $(T,s_{33})$, $(T,s_{73})$, $(T,s_{43})$, $(T,s_{40})$, $(T,s_{43})$, $(T,s_{83})$, $(T,s_{83})$
$s_{129}$ $-1.1894 + 0.3490 i$ $(V_{2},s_{119})$, $(T,s_{119})$, $(T,s_{121})$, $(T,s_{135})$, $(T,s_{120})$, $(T,s_{68})$, $(T,s_{74})$, $(T,s_{34})$, $(T,s_{74})$, $(T,s_{44})$, $(T,s_{40})$, $(T,s_{44})$, $(T,s_{84})$, $(T,s_{84})$
$s_{130}$ $1.2086 - 0.3062 i$ $(V_{2},s_{132})$, $(T,s_{124})$, $(T,s_{132})$, $(T,s_{125})$, $(T,s_{122})$, $(T,s_{131})$, $(T,s_{79})$, $(T,s_{96})$, $(T,s_{38})$, $(T,s_{96})$, $(T,s_{71})$, $(T,s_{57})$, $(T,s_{71})$, $(T,s_{49})$
$s_{131}$ $1.2086 + 0.3062 i$ $(V_{2},s_{133})$, $(T,s_{125})$, $(T,s_{133})$, $(T,s_{124})$, $(T,s_{123})$, $(T,s_{130})$, $(T,s_{80})$, $(T,s_{97})$, $(T,s_{39})$, $(T,s_{97})$, $(T,s_{72})$, $(T,s_{58})$, $(T,s_{72})$, $(T,s_{50})$
$s_{132}$ $1.1579 - 0.5443 i$ $(V_{2},s_{122})$, $(T,s_{130})$, $(T,s_{122})$, $(T,s_{124})$, $(T,s_{126})$, $(T,s_{51})$, $(T,s_{140})$, $(T,s_{51})$, $(T,s_{71})$, $(T,s_{96})$, $(T,s_{71})$, $(T,s_{57})$, $(T,s_{49})$, $(T,s_{57})$
$s_{133}$ $1.1579 + 0.5443 i$ $(V_{2},s_{123})$, $(T,s_{131})$, $(T,s_{123})$, $(T,s_{125})$, $(T,s_{127})$, $(T,s_{52})$, $(T,s_{141})$, $(T,s_{52})$, $(T,s_{72})$, $(T,s_{97})$, $(T,s_{72})$, $(T,s_{58})$, $(T,s_{50})$, $(T,s_{58})$
$s_{134}$ $-1.1534 - 0.5558 i$ $(V_{2},s_{128})$, $(T,s_{128})$, $(T,s_{118})$, $(T,s_{120})$, $(T,s_{136})$, $(T,s_{67})$, $(T,s_{142})$, $(T,s_{67})$, $(T,s_{146})$, $(T,s_{65})$, $(T,s_{83})$, $(T,s_{43})$, $(T,s_{83})$, $(T,s_{85})$
$s_{135}$ $-1.1534 + 0.5558 i$ $(V_{2},s_{129})$, $(T,s_{129})$, $(T,s_{119})$, $(T,s_{121})$, $(T,s_{137})$, $(T,s_{68})$, $(T,s_{143})$, $(T,s_{68})$, $(T,s_{147})$, $(T,s_{66})$, $(T,s_{84})$, $(T,s_{44})$, $(T,s_{84})$, $(T,s_{86})$
$s_{136}$ $-1.0363 - 0.7796 i$ $(V_{2},s_{142})$, $(T,s_{134})$, $(T,s_{128})$, $(T,s_{88})$, $(T,s_{142})$, $(T,s_{69})$, $(T,s_{146})$, $(T,s_{67})$, $(T,s_{83})$, $(T,s_{65})$, $(T,s_{85})$, $(T,s_{65})$, $(T,s_{85})$, $(T,s_{43})$
$s_{137}$ $-1.0363 + 0.7796 i$ $(V_{2},s_{143})$, $(T,s_{135})$, $(T,s_{129})$, $(T,s_{89})$, $(T,s_{143})$, $(T,s_{70})$, $(T,s_{147})$, $(T,s_{68})$, $(T,s_{84})$, $(T,s_{66})$, $(T,s_{86})$, $(T,s_{66})$, $(T,s_{86})$, $(T,s_{44})$
$s_{138}$ $0.8316 - 1.0206 i$$(T,s_{126}), (T,s_{140}), (T,s_{122}), (T,s_{90}), (T,s_{53}), (T,s_{148}), (T,s_{98}), (T,s_{154}), (T,s_{98}), (T,s_{158}), (T,s_{41}), (T,s_{81}), (T,s_{81}), (V_{2},s_{140}) s_{139} 0.8316 + 1.0206 i$$(T,s_{127})$, $(T,s_{141})$, $(T,s_{123})$, $(T,s_{91})$, $(T,s_{54})$, $(T,s_{149})$, $(T,s_{99})$, $(T,s_{155})$, $(T,s_{99})$, $(T,s_{159})$, $(T,s_{42})$, $(T,s_{82})$, $(T,s_{82})$, $(V_{2},s_{141})$
$s_{140}$ $0.9313 - 0.9617 i$$(T,s_{126}), (T,s_{122}), (T,s_{132}), (T,s_{53}), (T,s_{138}), (T,s_{98}), (T,s_{148}), (T,s_{98}), (T,s_{154}), (T,s_{41}), (T,s_{55}), (T,s_{81}), (T,s_{81}), (V_{2},s_{138}) s_{141} 0.9313 + 0.9617 i$$(T,s_{127})$, $(T,s_{123})$, $(T,s_{133})$, $(T,s_{54})$, $(T,s_{139})$, $(T,s_{99})$, $(T,s_{149})$, $(T,s_{99})$, $(T,s_{155})$, $(T,s_{42})$, $(T,s_{56})$, $(T,s_{82})$, $(T,s_{82})$, $(V_{2},s_{139})$
$s_{142}$ $-1.0129 - 0.9700 i$$(T,s_{136}), (T,s_{134}), (T,s_{128}), (T,s_{69}), (T,s_{146}), (T,s_{69}), (T,s_{150}), (T,s_{59}), (T,s_{152}), (T,s_{65}), (T,s_{85}), (T,s_{65}), (T,s_{100}), (V_{2},s_{146}) s_{143} -1.0129 + 0.9700 i$$(T,s_{137})$, $(T,s_{135})$, $(T,s_{129})$, $(T,s_{70})$, $(T,s_{147})$, $(T,s_{70})$, $(T,s_{151})$, $(T,s_{60})$, $(T,s_{153})$, $(T,s_{66})$, $(T,s_{86})$, $(T,s_{66})$, $(T,s_{101})$, $(V_{2},s_{147})$
$s_{144}$ $-1.414 i$$(T,s_{112}), (T,s_{112}), (T,s_{110}), (T,s_{116}), (T,s_{110}), (L_{1}^{-1},s_{172}), (T,s_{108}), (T,s_{116}), (T,s_{110}), (T,s_{110}), (T,s_{110}), (T,s_{114}), (T,s_{114}), (T,s_{172}), (T,s_{174}) s_{145} 1.414 i$$(T,s_{113})$, $(T,s_{113})$, $(T,s_{111})$, $(T,s_{117})$, $(T,s_{111})$, $(L_{1}^{-1},s_{173})$, $(T,s_{109})$, $(T,s_{117})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{111})$, $(T,s_{115})$, $(T,s_{115})$, $(T,s_{173})$, $(T,s_{175})$
$s_{146}$ $-0.880 - 1.108 i$$(T,s_{136}), (T,s_{142}), (T,s_{134}), (T,s_{77}), (T,s_{150}), (T,s_{69}), (T,s_{152}), (T,s_{69}), (T,s_{156}), (T,s_{59}), (T,s_{100}), (T,s_{65}), (T,s_{65}), (V_{2},s_{142}) s_{147} -0.880 + 1.108 i$$(T,s_{137})$, $(T,s_{143})$, $(T,s_{135})$, $(T,s_{78})$, $(T,s_{151})$, $(T,s_{70})$, $(T,s_{153})$, $(T,s_{70})$, $(T,s_{157})$, $(T,s_{60})$, $(T,s_{101})$, $(T,s_{66})$, $(T,s_{66})$, $(V_{2},s_{143})$
$s_{148}$ $0.781 - 1.241 i$$(T,s_{126}), (T,s_{140}), (T,s_{138}), (T,s_{90}), (T,s_{90}), (T,s_{53}), (T,s_{154}), (T,s_{98}), (T,s_{158}), (T,s_{98}), (T,s_{164}), (T,s_{166}), (T,s_{168}), (V_{2},s_{154}) s_{149} 0.781 + 1.241 i$$(T,s_{127})$, $(T,s_{141})$, $(T,s_{139})$, $(T,s_{91})$, $(T,s_{91})$, $(T,s_{54})$, $(T,s_{155})$, $(T,s_{99})$, $(T,s_{159})$, $(T,s_{99})$, $(T,s_{165})$, $(T,s_{167})$, $(T,s_{169})$, $(V_{2},s_{155})$
$s_{150}$ $-0.713 - 1.295 i$$(T,s_{142}), (T,s_{146}), (T,s_{136}), (T,s_{77}), (T,s_{77}), (T,s_{152}), (T,s_{92}), (T,s_{156}), (T,s_{59}), (T,s_{160}), (T,s_{59}), (T,s_{162}), (T,s_{170}), (V_{2},s_{152}) s_{151} -0.713 + 1.295 i$$(T,s_{143})$, $(T,s_{147})$, $(T,s_{137})$, $(T,s_{78})$, $(T,s_{78})$, $(T,s_{153})$, $(T,s_{93})$, $(T,s_{157})$, $(T,s_{60})$, $(T,s_{161})$, $(T,s_{60})$, $(T,s_{163})$, $(T,s_{171})$, $(V_{2},s_{153})$
$s_{152}$ $-0.694 - 1.366 i$$(T,s_{146}), (T,s_{142}), (T,s_{102}), (T,s_{150}), (T,s_{77}), (T,s_{92}), (T,s_{92}), (T,s_{156}), (T,s_{172}), (T,s_{160}), (T,s_{178}), (T,s_{162}), (T,s_{170}), (V_{2},s_{150}) s_{153} -0.694 + 1.366 i$$(T,s_{147})$, $(T,s_{143})$, $(T,s_{103})$, $(T,s_{151})$, $(T,s_{78})$, $(T,s_{93})$, $(T,s_{93})$, $(T,s_{157})$, $(T,s_{173})$, $(T,s_{161})$, $(T,s_{179})$, $(T,s_{163})$, $(T,s_{171})$, $(V_{2},s_{151})$
$s_{154}$ $0.748 - 1.401 i$$(T,s_{140}), (T,s_{138}), (T,s_{126}), (T,s_{148}), (T,s_{90}), (T,s_{90}), (T,s_{106}), (T,s_{158}), (T,s_{98}), (T,s_{164}), (T,s_{98}), (T,s_{166}), (T,s_{168}), (V_{2},s_{148}) s_{155} 0.748 + 1.401 i$$(T,s_{141})$, $(T,s_{139})$, $(T,s_{127})$, $(T,s_{149})$, $(T,s_{91})$, $(T,s_{91})$, $(T,s_{107})$, $(T,s_{159})$, $(T,s_{99})$, $(T,s_{165})$, $(T,s_{99})$, $(T,s_{167})$, $(T,s_{169})$, $(V_{2},s_{149})$
$s_{156}$ $-0.565 - 1.488 i$$(T,s_{146}), (T,s_{102}), (T,s_{150}), (T,s_{102}), (T,s_{152}), (T,s_{114}), (T,s_{114}), (T,s_{160}), (T,s_{174}), (T,s_{162}), (T,s_{172}), (T,s_{178}), (T,s_{170}), (V_{2},s_{152}) s_{157} -0.565 + 1.488 i$$(T,s_{147})$, $(T,s_{103})$, $(T,s_{151})$, $(T,s_{103})$, $(T,s_{153})$, $(T,s_{115})$, $(T,s_{115})$, $(T,s_{161})$, $(T,s_{175})$, $(T,s_{163})$, $(T,s_{173})$, $(T,s_{179})$, $(T,s_{171})$, $(V_{2},s_{153})$
$s_{158}$ $0.571 - 1.501 i$$(T,s_{138}), (T,s_{140}), (T,s_{148}), (T,s_{108}), (T,s_{154}), (T,s_{108}), (T,s_{110}), (T,s_{172}), (T,s_{164}), (T,s_{174}), (T,s_{166}), (T,s_{176}), (T,s_{168}), (V_{2},s_{164}) s_{159} 0.571 + 1.501 i$$(T,s_{139})$, $(T,s_{141})$, $(T,s_{149})$, $(T,s_{109})$, $(T,s_{155})$, $(T,s_{109})$, $(T,s_{111})$, $(T,s_{173})$, $(T,s_{165})$, $(T,s_{175})$, $(T,s_{167})$, $(T,s_{177})$, $(T,s_{169})$, $(V_{2},s_{165})$
$s_{160}$ $-0.462 - 1.634 i$$(T,s_{146}), (T,s_{150}), (T,s_{102}), (T,s_{152}), (T,s_{102}), (T,s_{156}), (T,s_{168}), (T,s_{176}), (T,s_{174}), (T,s_{162}), (T,s_{172}), (T,s_{170}), (T,s_{178}), (V_{2},s_{162}) s_{161} -0.462 + 1.634 i$$(T,s_{147})$, $(T,s_{151})$, $(T,s_{103})$, $(T,s_{153})$, $(T,s_{103})$, $(T,s_{157})$, $(T,s_{169})$, $(T,s_{177})$, $(T,s_{175})$, $(T,s_{163})$, $(T,s_{173})$, $(T,s_{171})$, $(T,s_{179})$, $(V_{2},s_{163})$
$s_{162}$ $-0.415 - 1.665 i$$(T,s_{146}), (T,s_{150}), (T,s_{102}), (T,s_{152}), (T,s_{102}), (T,s_{156}), (T,s_{168}), (T,s_{160}), (T,s_{176}), (T,s_{174}), (T,s_{170}), (T,s_{172}), (T,s_{178}), (V_{2},s_{160}) s_{163} -0.415 + 1.665 i$$(T,s_{147})$, $(T,s_{151})$, $(T,s_{103})$, $(T,s_{153})$, $(T,s_{103})$, $(T,s_{157})$, $(T,s_{169})$, $(T,s_{161})$, $(T,s_{177})$, $(T,s_{175})$, $(T,s_{171})$, $(T,s_{173})$, $(T,s_{179})$, $(V_{2},s_{161})$
$s_{164}$ $0.482 - 1.657 i$$(T,s_{138}), (T,s_{148}), (T,s_{110}), (T,s_{108}), (T,s_{154}), (T,s_{158}), (T,s_{170}), (T,s_{178}), (T,s_{172}), (T,s_{166}), (T,s_{174}), (T,s_{176}), (T,s_{168}), (V_{2},s_{166}) s_{165} 0.482 + 1.657 i$$(T,s_{139})$, $(T,s_{149})$, $(T,s_{111})$, $(T,s_{109})$, $(T,s_{155})$, $(T,s_{159})$, $(T,s_{171})$, $(T,s_{179})$, $(T,s_{173})$, $(T,s_{167})$, $(T,s_{175})$, $(T,s_{177})$, $(T,s_{169})$, $(V_{2},s_{167})$
$s_{166}$ $0.394 - 1.705 i$$(T,s_{112}), (T,s_{148}), (T,s_{110}), (T,s_{154}), (T,s_{162}), (T,s_{158}), (T,s_{170}), (T,s_{164}), (T,s_{178}), (T,s_{172}), (T,s_{174}), (T,s_{168}), (T,s_{176}), (V_{2},s_{164}) s_{167} 0.394 + 1.705 i$$(T,s_{113})$, $(T,s_{149})$, $(T,s_{111})$, $(T,s_{155})$, $(T,s_{163})$, $(T,s_{159})$, $(T,s_{171})$, $(T,s_{165})$, $(T,s_{179})$, $(T,s_{173})$, $(T,s_{175})$, $(T,s_{169})$, $(T,s_{177})$, $(V_{2},s_{165})$
$s_{168}$ $0.248 - 1.741 i$$(T,s_{148}), (T,s_{112}), (T,s_{154}), (T,s_{160}), (T,s_{158}), (T,s_{162}), (T,s_{164}), (T,s_{170}), (T,s_{166}), (T,s_{178}), (T,s_{172}), (T,s_{176}), (T,s_{174}), (V_{2},s_{176}) s_{169} 0.248 + 1.741 i$$(T,s_{149})$, $(T,s_{113})$, $(T,s_{155})$, $(T,s_{161})$, $(T,s_{159})$, $(T,s_{163})$, $(T,s_{165})$, $(T,s_{171})$, $(T,s_{167})$, $(T,s_{179})$, $(T,s_{173})$, $(T,s_{177})$, $(T,s_{175})$, $(V_{2},s_{177})$
$s_{170}$ $-0.278 - 1.765 i$$(T,s_{150}), (T,s_{116}), (T,s_{152}), (T,s_{164}), (T,s_{156}), (T,s_{166}), (T,s_{160}), (T,s_{168}), (T,s_{162}), (T,s_{176}), (T,s_{174}), (T,s_{178}), (T,s_{172}), (V_{2},s_{178}) s_{171} -0.278 + 1.765 i$$(T,s_{151})$, $(T,s_{117})$, $(T,s_{153})$, $(T,s_{165})$, $(T,s_{157})$, $(T,s_{167})$, $(T,s_{161})$, $(T,s_{169})$, $(T,s_{163})$, $(T,s_{177})$, $(T,s_{175})$, $(T,s_{179})$, $(T,s_{173})$, $(V_{2},s_{179})$
$s_{172}$ $-0.064 - 1.814 i$$(T,s_{110}), (T,s_{152}), (T,s_{158}), (T,s_{156}), (T,s_{164}), (T,s_{160}), (T,s_{166}), (T,s_{162}), (T,s_{168}), (T,s_{170}), (T,s_{176}), (T,s_{178}), (T,s_{174}), (V_{2},s_{178}) s_{173} -0.064 + 1.814 i$$(T,s_{111})$, $(T,s_{153})$, $(T,s_{159})$, $(T,s_{157})$, $(T,s_{165})$, $(T,s_{161})$, $(T,s_{167})$, $(T,s_{163})$, $(T,s_{169})$, $(T,s_{171})$, $(T,s_{177})$, $(T,s_{179})$, $(T,s_{175})$, $(V_{2},s_{179})$
$s_{174}$ $0.077 - 1.817 i$$(T,s_{112}), (T,s_{154}), (T,s_{156}), (T,s_{158}), (T,s_{160}), (T,s_{164}), (T,s_{162}), (T,s_{166}), (T,s_{170}), (T,s_{168}), (T,s_{178}), (T,s_{176}), (T,s_{172}), (V_{2},s_{176}) s_{175} 0.077 + 1.817 i$$(T,s_{113})$, $(T,s_{155})$, $(T,s_{157})$, $(T,s_{159})$, $(T,s_{161})$, $(T,s_{165})$, $(T,s_{163})$, $(T,s_{167})$, $(T,s_{171})$, $(T,s_{169})$, $(T,s_{179})$, $(T,s_{177})$, $(T,s_{173})$, $(V_{2},s_{177})$
$s_{176}$ $0.177 - 1.821 i$$(T,s_{112}), (T,s_{156}), (T,s_{154}), (T,s_{158}), (T,s_{160}), (T,s_{162}), (T,s_{164}), (T,s_{170}), (T,s_{166}), (T,s_{178}), (T,s_{168}), (T,s_{172}), (T,s_{174}), (V_{2},s_{174}) s_{177} 0.177 + 1.821 i$$(T,s_{113})$, $(T,s_{157})$, $(T,s_{155})$, $(T,s_{159})$, $(T,s_{161})$, $(T,s_{163})$, $(T,s_{165})$, $(T,s_{171})$, $(T,s_{167})$, $(T,s_{179})$, $(T,s_{169})$, $(T,s_{173})$, $(T,s_{175})$, $(V_{2},s_{175})$
$s_{178}$ $-0.152 - 1.849 i$$(T,s_{150}), (T,s_{152}), (T,s_{158}), (T,s_{156}), (T,s_{164}), (T,s_{160}), (T,s_{166}), (T,s_{162}), (T,s_{168}), (T,s_{170}), (T,s_{176}), (T,s_{174}), (T,s_{172}), (V_{2},s_{172}) s_{179} -0.152 + 1.849 i$$(T,s_{151})$, $(T,s_{153})$, $(T,s_{159})$, $(T,s_{157})$, $(T,s_{165})$, $(T,s_{161})$, $(T,s_{167})$, $(T,s_{163})$, $(T,s_{169})$, $(T,s_{171})$, $(T,s_{177})$, $(T,s_{175})$, $(T,s_{173})$, $(V_{2},s_{173})$

The data in Table $1$ can be retrieved by executing the following Mathematica command:

#### Mathematica code to retrieve convergence data for $f_4$

  
convergenceData =Import["https://dl.dropboxusercontent.com/s/lwdia2179ybxefa/aaTest.m?dl=0"];


The convergence data of the expansion about singular point $n$ is $\texttt{convergenceData[[n]]}$. For example, convergence data for the expansions about the origin can be formated via $\texttt{convergenceData[[1,2]]//MatrixForm}$ resulting in the report: $$\begin{array}{ccccccc} \text{} & \text{} & \text{} & \text{S}_1 & \text{} & \text{} & \text{} \\ \text{Series #} & \text{Cycle} & \text{(Cycle,Order)} & \text{CLSP} & \text{Root Test} & \text{AC Test} & \text{\% error} \\ 1 & 1 & \{1,1\} & \text{S}_{118} & 1.09688 & 1.09352 & 0.307762 \\ 2 & 3 & \{3,4\} & \text{S}_2 & 0.168414 & 0.166817 & 0.957267 \\ 3 & 4 & \{4,9\} & \text{S}_7 & 0.504761 & 0.504901 & 0.0278736 \\ 6 & 5 & \{5,16\} & \text{S}_{27} & 0.646892 & 0.641328 & 0.867696 \\ 11 & 2 & \{2,1\} & \text{S}_2 & 0.167905 & 0.166817 & 0.652264 \\ \end{array}$$ Recall, the radius of convergence of a branch expansion of $(X,s_l)$ centered at $s_n$ is given by $R=\left|s_n-s_l\right|$ with $s_l=\text{CLSP}$.

The following figure shows the singular points as the black dots and the radii of convergence of the $1,2,3,4,5$-cycle branches at the origin with colored circles. Note the number of singular points encompassed by the convergence radius of the $1$-cycle shown in red. The purple circle is over-written by the yellow circle as both the $3$-cycle and $2$-cycle have the same radius of convergence.

Once we have the convergence data, we can easily plot the real or imaginary surfaces of the branch. This is useful if contour integrations over the branch surfaces are to be analyzed. Consider plotting the real surface of the $3$-cycle branch at the origin. According to the convergence summary report above, this branch is generated by Puiseux series $2$ (when sorted as described in Section $15$) although any $3$-cycle series of this conjugate set can generate the branch. The report indicates the $3$-cycle series have radii of convergence $R=|s_{2}|\approx 0.167$.

It's important to understand each series in the conjugate set of Puiseux series for an $n$-cycle branch generates a single-valued sheet of the branch and when all branch sheets are combined produce the analytically-continuous branch surface. Once the singular points have been computed and sorted and Puiseux series generated for the branches at the origin as per Section $15$, the following code can be used to generate the branch sheets of the $3$-cycle branch:

#### Mathematica code to plot $3$-cycle branch sheets at the origin

  
seriesIndex = 2;
(* sorted singular points are in theSingularList *)
rEnd = Abs@theSingularList[[2]] // N
rStart = 0.1 rEnd;
(* baseSeries are the Puiseux expansions at the origin*)
myF1[u_] = baseSeries[[seriesIndex, 1 ;; 50]];
theexp = Exponent[myF1[z], z, List];
theNum = Numerator@theexp;
clist = Coefficient[myF1[z], z, theexp];
(* conjugate the base series for this branch *)
f[z_, k_] :=
Sum[clist[[
n]] (Exp[2 k \[Pi] I/cycleSize])^(cycleSize theexp[[n]]) z^
theexp[[n]], {n, 1, Length[clist]}];
theColors = {Red, Blue, Darker@Green, Darker@Yellow, Orange, Pink};
(* generate the branch sheets *)
branchSheets = Table[
ParametricPlot3D[{Re[z] + Re[expCenter], Im[z] + Im[expCenter],
Re[f[z, i - 1]]} /. z -> r Exp[I t], {r, rStart,
rEnd}, {t, -\[Pi], \[Pi]}, BoxRatios -> {1, 1, 1},
PlotStyle -> theColors[[i]]],
{i, 1, cycleSize}];
(* plot the branch sheets separately *)
GraphicsGrid[{branchSheets}]


The code produces the following branch sheet plots:
and when combined together via $\texttt{Show[branchSheets]}$, produces a branch plot with each branch sheet spliced together in an analytically-continuous surface: