# Properties

 Label 354.3.d.a Level 354 Weight 3 Character orbit 354.d Analytic conductor 9.646 Analytic rank 0 Dimension 20 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$354 = 2 \cdot 3 \cdot 59$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 354.d (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$9.64580135835$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{11}\cdot 3^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} -\beta_{7} q^{3} -2 q^{4} + \beta_{5} q^{5} + \beta_{6} q^{6} + \beta_{9} q^{7} -2 \beta_{4} q^{8} + 3 q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{2} -\beta_{7} q^{3} -2 q^{4} + \beta_{5} q^{5} + \beta_{6} q^{6} + \beta_{9} q^{7} -2 \beta_{4} q^{8} + 3 q^{9} + \beta_{3} q^{10} + ( -\beta_{6} - \beta_{8} ) q^{11} + 2 \beta_{7} q^{12} + ( -\beta_{4} - 2 \beta_{6} + \beta_{16} + \beta_{18} ) q^{13} + \beta_{10} q^{14} + ( -1 + \beta_{1} ) q^{15} + 4 q^{16} + ( 1 + \beta_{1} + \beta_{5} - \beta_{9} - \beta_{11} + \beta_{14} - \beta_{15} ) q^{17} + 3 \beta_{4} q^{18} + ( 3 - \beta_{2} - \beta_{5} + \beta_{13} + \beta_{15} ) q^{19} -2 \beta_{5} q^{20} + ( \beta_{5} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{21} + ( 1 - 2 \beta_{7} + \beta_{13} ) q^{22} + ( -2 \beta_{6} - \beta_{12} + 2 \beta_{18} ) q^{23} -2 \beta_{6} q^{24} + ( 7 + \beta_{2} + \beta_{5} + 2 \beta_{7} + \beta_{9} + 2 \beta_{14} - \beta_{15} ) q^{25} + ( 3 + \beta_{1} - 4 \beta_{7} + \beta_{11} + \beta_{13} + \beta_{15} ) q^{26} -3 \beta_{7} q^{27} -2 \beta_{9} q^{28} + ( 1 - \beta_{1} - 2 \beta_{2} + 4 \beta_{7} + \beta_{9} + \beta_{11} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{29} + ( -\beta_{4} - \beta_{8} - \beta_{16} + \beta_{17} ) q^{30} + ( 2 \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{12} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{31} + 4 \beta_{4} q^{32} + ( \beta_{3} - 2 \beta_{4} - \beta_{8} - \beta_{17} ) q^{33} + ( \beta_{8} - \beta_{10} - \beta_{12} + 2 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{34} + ( -4 + 3 \beta_{1} - \beta_{2} + 3 \beta_{5} - 6 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} - \beta_{13} + \beta_{15} ) q^{35} -6 q^{36} + ( -\beta_{3} + 6 \beta_{4} + \beta_{6} + 3 \beta_{10} - 3 \beta_{12} + 2 \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{37} + ( -\beta_{3} + 3 \beta_{4} + 2 \beta_{8} - \beta_{10} + \beta_{12} - \beta_{16} - \beta_{17} + \beta_{19} ) q^{38} + ( -5 \beta_{4} - \beta_{6} - \beta_{8} + \beta_{12} - \beta_{16} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{39} -2 \beta_{3} q^{40} + ( -3 + \beta_{1} + 2 \beta_{5} - 4 \beta_{7} + 3 \beta_{9} + \beta_{11} + \beta_{14} + 3 \beta_{15} ) q^{41} + ( -\beta_{4} + \beta_{10} - \beta_{12} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{42} + ( 3 \beta_{3} + 11 \beta_{4} - \beta_{6} - \beta_{8} + \beta_{12} - \beta_{16} + 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{43} + ( 2 \beta_{6} + 2 \beta_{8} ) q^{44} + 3 \beta_{5} q^{45} + ( 1 - \beta_{2} - \beta_{5} - 4 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} - \beta_{14} ) q^{46} + ( 4 \beta_{3} + 14 \beta_{4} - 3 \beta_{6} - \beta_{8} - 4 \beta_{10} - \beta_{12} - 2 \beta_{16} - 2 \beta_{17} - 2 \beta_{19} ) q^{47} -4 \beta_{7} q^{48} + ( 5 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{5} - 2 \beta_{11} + 5 \beta_{13} - \beta_{14} + \beta_{15} ) q^{49} + ( -\beta_{3} + 4 \beta_{4} - 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{10} - 3 \beta_{12} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{50} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{5} - 2 \beta_{7} + \beta_{9} + \beta_{11} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{51} + ( 2 \beta_{4} + 4 \beta_{6} - 2 \beta_{16} - 2 \beta_{18} ) q^{52} + ( -2 \beta_{1} + 3 \beta_{5} + 8 \beta_{7} - 2 \beta_{13} + 2 \beta_{15} ) q^{53} + 3 \beta_{6} q^{54} + ( -3 \beta_{3} - 7 \beta_{4} + 6 \beta_{6} + 4 \beta_{8} + 2 \beta_{10} + 3 \beta_{16} + \beta_{18} ) q^{55} -2 \beta_{10} q^{56} + ( -1 - \beta_{1} - \beta_{2} - 3 \beta_{5} - 2 \beta_{7} - 2 \beta_{9} - \beta_{11} - \beta_{14} - 2 \beta_{15} ) q^{57} + ( -\beta_{3} - 4 \beta_{6} + \beta_{8} + 3 \beta_{10} + \beta_{12} + 2 \beta_{16} - 2 \beta_{18} + 3 \beta_{19} ) q^{58} + ( 11 - \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 5 \beta_{7} - 3 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} + 3 \beta_{13} + \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{59} + ( 2 - 2 \beta_{1} ) q^{60} + ( \beta_{3} - \beta_{4} + 5 \beta_{6} + \beta_{10} - \beta_{12} + 3 \beta_{16} + \beta_{17} + 3 \beta_{18} - 2 \beta_{19} ) q^{61} + ( 1 + \beta_{2} - 3 \beta_{5} - 4 \beta_{7} - 3 \beta_{9} + \beta_{11} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{62} + 3 \beta_{9} q^{63} -8 q^{64} + ( -7 \beta_{3} + 3 \beta_{4} - 9 \beta_{6} + 5 \beta_{8} - 2 \beta_{10} - 3 \beta_{12} + 7 \beta_{16} + \beta_{17} + 6 \beta_{18} + \beta_{19} ) q^{65} + ( 6 + \beta_{1} - 2 \beta_{5} - \beta_{9} + \beta_{13} - \beta_{15} ) q^{66} + ( \beta_{3} - 11 \beta_{4} + 3 \beta_{6} + \beta_{8} + 6 \beta_{10} + \beta_{12} + \beta_{16} - 3 \beta_{18} + 3 \beta_{19} ) q^{67} + ( -2 - 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{9} + 2 \beta_{11} - 2 \beta_{14} + 2 \beta_{15} ) q^{68} + ( 2 \beta_{3} - 6 \beta_{4} - 3 \beta_{8} + \beta_{10} + 2 \beta_{12} - 3 \beta_{18} + \beta_{19} ) q^{69} + ( 3 \beta_{3} - 4 \beta_{4} + 6 \beta_{6} - 5 \beta_{8} - \beta_{10} + \beta_{12} - 4 \beta_{16} + 2 \beta_{17} - 4 \beta_{18} + \beta_{19} ) q^{70} + ( -29 + \beta_{1} - 2 \beta_{2} - 3 \beta_{5} + 10 \beta_{7} - 2 \beta_{9} - \beta_{11} - 5 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{71} -6 \beta_{4} q^{72} + ( -\beta_{3} - 9 \beta_{4} - \beta_{6} + \beta_{10} + 3 \beta_{12} - 3 \beta_{16} + 3 \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{73} + ( -14 + \beta_{1} - 3 \beta_{2} - \beta_{5} + 2 \beta_{7} - 5 \beta_{9} + 2 \beta_{11} + 2 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{74} + ( -5 + \beta_{2} - 2 \beta_{5} - 5 \beta_{7} + 3 \beta_{9} + \beta_{11} - 2 \beta_{15} ) q^{75} + ( -6 + 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{13} - 2 \beta_{15} ) q^{76} + ( 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} + 3 \beta_{8} - 5 \beta_{10} + 3 \beta_{12} - 2 \beta_{16} + 2 \beta_{18} - 2 \beta_{19} ) q^{77} + ( 12 + 2 \beta_{2} - 2 \beta_{7} - \beta_{9} - \beta_{11} + \beta_{13} - 2 \beta_{15} ) q^{78} + ( 33 + 4 \beta_{1} + 3 \beta_{2} + \beta_{5} + 8 \beta_{7} - 5 \beta_{9} - 2 \beta_{11} + 3 \beta_{13} - 2 \beta_{14} - 3 \beta_{15} ) q^{79} + 4 \beta_{5} q^{80} + 9 q^{81} + ( \beta_{3} - 2 \beta_{4} + 4 \beta_{6} - 3 \beta_{8} + \beta_{10} - \beta_{12} - 4 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{82} + ( -4 \beta_{3} - 6 \beta_{4} + 5 \beta_{6} - 5 \beta_{8} - 5 \beta_{12} - 6 \beta_{16} + 4 \beta_{17} - 2 \beta_{18} - 2 \beta_{19} ) q^{83} + ( -2 \beta_{5} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{84} + ( 32 - 6 \beta_{1} + 6 \beta_{2} + 2 \beta_{5} - 12 \beta_{7} + 2 \beta_{9} - 6 \beta_{11} - \beta_{13} + 3 \beta_{14} - 7 \beta_{15} ) q^{85} + ( -23 - 3 \beta_{1} - 4 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} - \beta_{11} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{86} + ( -12 - 2 \beta_{1} - 3 \beta_{5} + 5 \beta_{9} - 3 \beta_{11} - \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{87} + ( -2 + 4 \beta_{7} - 2 \beta_{13} ) q^{88} + ( -\beta_{3} + 10 \beta_{4} - 11 \beta_{6} + 5 \beta_{8} + 3 \beta_{10} + 4 \beta_{12} + \beta_{16} - 5 \beta_{17} - 4 \beta_{18} - 3 \beta_{19} ) q^{89} + 3 \beta_{3} q^{90} + ( -4 \beta_{3} - 5 \beta_{4} + 3 \beta_{6} - \beta_{8} - 6 \beta_{10} + 6 \beta_{12} - 5 \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{91} + ( 4 \beta_{6} + 2 \beta_{12} - 4 \beta_{18} ) q^{92} + ( \beta_{3} - 9 \beta_{4} - 3 \beta_{8} - \beta_{10} + \beta_{12} + 3 \beta_{17} + 2 \beta_{19} ) q^{93} + ( -26 - 3 \beta_{2} - 7 \beta_{5} - 6 \beta_{7} + 4 \beta_{9} - 3 \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{94} + ( -29 - 6 \beta_{1} - 5 \beta_{2} + 2 \beta_{5} + 14 \beta_{7} + 3 \beta_{11} - 4 \beta_{14} + 8 \beta_{15} ) q^{95} + 4 \beta_{6} q^{96} + ( 2 \beta_{3} + 15 \beta_{4} + 6 \beta_{6} - 2 \beta_{8} - 4 \beta_{10} + 6 \beta_{12} + \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{97} + ( -3 \beta_{3} + 4 \beta_{4} + 4 \beta_{8} - 3 \beta_{10} - \beta_{12} - 3 \beta_{16} + \beta_{17} + 4 \beta_{18} - 3 \beta_{19} ) q^{98} + ( -3 \beta_{6} - 3 \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 40q^{4} + 8q^{7} + 60q^{9} + O(q^{10})$$ $$20q - 40q^{4} + 8q^{7} + 60q^{9} - 24q^{15} + 80q^{16} + 72q^{19} + 16q^{22} + 140q^{25} + 64q^{26} - 16q^{28} + 56q^{29} - 80q^{35} - 120q^{36} - 8q^{41} + 16q^{46} + 52q^{49} + 32q^{53} - 48q^{57} + 192q^{59} + 48q^{60} - 16q^{62} + 24q^{63} - 160q^{64} + 96q^{66} - 568q^{71} - 288q^{74} - 96q^{75} - 144q^{76} + 192q^{78} + 528q^{79} + 180q^{81} + 568q^{85} - 416q^{86} - 216q^{87} - 32q^{88} - 480q^{94} - 456q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 300 x^{18} - 88 x^{17} + 37952 x^{16} + 27000 x^{15} - 2574056 x^{14} - 3295208 x^{13} + 98138134 x^{12} + 195465720 x^{11} - 1935828112 x^{10} - 5590976152 x^{9} + 12661641204 x^{8} + 55962519336 x^{7} + 75365256136 x^{6} + 219349710792 x^{5} + 570681930321 x^{4} + 614778274152 x^{3} + 1559028888924 x^{2} + 534075251472 x + 2455573689828$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!96$$$$\nu^{19} +$$$$15\!\cdots\!35$$$$\nu^{18} +$$$$20\!\cdots\!22$$$$\nu^{17} -$$$$36\!\cdots\!53$$$$\nu^{16} -$$$$48\!\cdots\!84$$$$\nu^{15} +$$$$30\!\cdots\!71$$$$\nu^{14} +$$$$48\!\cdots\!86$$$$\nu^{13} -$$$$88\!\cdots\!53$$$$\nu^{12} -$$$$24\!\cdots\!00$$$$\nu^{11} -$$$$18\!\cdots\!55$$$$\nu^{10} +$$$$55\!\cdots\!82$$$$\nu^{9} +$$$$15\!\cdots\!25$$$$\nu^{8} -$$$$21\!\cdots\!64$$$$\nu^{7} -$$$$18\!\cdots\!67$$$$\nu^{6} -$$$$76\!\cdots\!58$$$$\nu^{5} -$$$$20\!\cdots\!71$$$$\nu^{4} -$$$$36\!\cdots\!08$$$$\nu^{3} -$$$$57\!\cdots\!28$$$$\nu^{2} -$$$$46\!\cdots\!12$$$$\nu -$$$$49\!\cdots\!00$$$$)/$$$$78\!\cdots\!56$$ $$\beta_{2}$$ $$=$$ $$($$$$30\!\cdots\!89$$$$\nu^{19} +$$$$11\!\cdots\!07$$$$\nu^{18} -$$$$35\!\cdots\!01$$$$\nu^{17} -$$$$34\!\cdots\!95$$$$\nu^{16} -$$$$56\!\cdots\!71$$$$\nu^{15} +$$$$42\!\cdots\!11$$$$\nu^{14} +$$$$15\!\cdots\!91$$$$\nu^{13} -$$$$27\!\cdots\!75$$$$\nu^{12} -$$$$13\!\cdots\!13$$$$\nu^{11} +$$$$91\!\cdots\!53$$$$\nu^{10} +$$$$64\!\cdots\!93$$$$\nu^{9} -$$$$12\!\cdots\!41$$$$\nu^{8} -$$$$15\!\cdots\!25$$$$\nu^{7} -$$$$11\!\cdots\!71$$$$\nu^{6} +$$$$13\!\cdots\!01$$$$\nu^{5} +$$$$34\!\cdots\!15$$$$\nu^{4} +$$$$39\!\cdots\!44$$$$\nu^{3} +$$$$95\!\cdots\!92$$$$\nu^{2} +$$$$81\!\cdots\!00$$$$\nu +$$$$67\!\cdots\!28$$$$)/$$$$18\!\cdots\!64$$ $$\beta_{3}$$ $$=$$ $$($$$$10\!\cdots\!82$$$$\nu^{19} -$$$$39\!\cdots\!27$$$$\nu^{18} -$$$$31\!\cdots\!04$$$$\nu^{17} +$$$$10\!\cdots\!41$$$$\nu^{16} +$$$$38\!\cdots\!98$$$$\nu^{15} -$$$$10\!\cdots\!55$$$$\nu^{14} -$$$$25\!\cdots\!16$$$$\nu^{13} +$$$$54\!\cdots\!45$$$$\nu^{12} +$$$$99\!\cdots\!02$$$$\nu^{11} -$$$$11\!\cdots\!69$$$$\nu^{10} -$$$$21\!\cdots\!20$$$$\nu^{9} +$$$$26\!\cdots\!95$$$$\nu^{8} +$$$$20\!\cdots\!86$$$$\nu^{7} +$$$$16\!\cdots\!19$$$$\nu^{6} -$$$$14\!\cdots\!72$$$$\nu^{5} +$$$$67\!\cdots\!11$$$$\nu^{4} -$$$$24\!\cdots\!28$$$$\nu^{3} -$$$$13\!\cdots\!36$$$$\nu^{2} -$$$$25\!\cdots\!48$$$$\nu -$$$$22\!\cdots\!64$$$$)/$$$$31\!\cdots\!72$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!45$$$$\nu^{19} +$$$$74\!\cdots\!74$$$$\nu^{18} +$$$$35\!\cdots\!11$$$$\nu^{17} -$$$$11\!\cdots\!68$$$$\nu^{16} -$$$$44\!\cdots\!53$$$$\nu^{15} -$$$$49\!\cdots\!14$$$$\nu^{14} +$$$$29\!\cdots\!35$$$$\nu^{13} +$$$$20\!\cdots\!48$$$$\nu^{12} -$$$$11\!\cdots\!15$$$$\nu^{11} -$$$$16\!\cdots\!86$$$$\nu^{10} +$$$$22\!\cdots\!57$$$$\nu^{9} +$$$$51\!\cdots\!00$$$$\nu^{8} -$$$$14\!\cdots\!15$$$$\nu^{7} -$$$$51\!\cdots\!18$$$$\nu^{6} -$$$$77\!\cdots\!87$$$$\nu^{5} -$$$$25\!\cdots\!44$$$$\nu^{4} -$$$$62\!\cdots\!68$$$$\nu^{3} -$$$$74\!\cdots\!36$$$$\nu^{2} -$$$$27\!\cdots\!32$$$$\nu -$$$$80\!\cdots\!76$$$$)/$$$$22\!\cdots\!04$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!45$$$$\nu^{19} +$$$$74\!\cdots\!74$$$$\nu^{18} +$$$$35\!\cdots\!11$$$$\nu^{17} -$$$$11\!\cdots\!68$$$$\nu^{16} -$$$$44\!\cdots\!53$$$$\nu^{15} -$$$$49\!\cdots\!14$$$$\nu^{14} +$$$$29\!\cdots\!35$$$$\nu^{13} +$$$$20\!\cdots\!48$$$$\nu^{12} -$$$$11\!\cdots\!15$$$$\nu^{11} -$$$$16\!\cdots\!86$$$$\nu^{10} +$$$$22\!\cdots\!57$$$$\nu^{9} +$$$$51\!\cdots\!00$$$$\nu^{8} -$$$$14\!\cdots\!15$$$$\nu^{7} -$$$$51\!\cdots\!18$$$$\nu^{6} -$$$$77\!\cdots\!87$$$$\nu^{5} -$$$$25\!\cdots\!44$$$$\nu^{4} -$$$$62\!\cdots\!68$$$$\nu^{3} -$$$$74\!\cdots\!36$$$$\nu^{2} -$$$$54\!\cdots\!28$$$$\nu -$$$$80\!\cdots\!76$$$$)/$$$$22\!\cdots\!04$$ $$\beta_{6}$$ $$=$$ $$($$$$23\!\cdots\!82$$$$\nu^{19} +$$$$80\!\cdots\!67$$$$\nu^{18} -$$$$71\!\cdots\!88$$$$\nu^{17} -$$$$26\!\cdots\!65$$$$\nu^{16} +$$$$90\!\cdots\!30$$$$\nu^{15} +$$$$36\!\cdots\!87$$$$\nu^{14} -$$$$61\!\cdots\!16$$$$\nu^{13} -$$$$28\!\cdots\!57$$$$\nu^{12} +$$$$22\!\cdots\!38$$$$\nu^{11} +$$$$12\!\cdots\!05$$$$\nu^{10} -$$$$38\!\cdots\!72$$$$\nu^{9} -$$$$31\!\cdots\!63$$$$\nu^{8} -$$$$49\!\cdots\!26$$$$\nu^{7} +$$$$30\!\cdots\!53$$$$\nu^{6} +$$$$61\!\cdots\!96$$$$\nu^{5} +$$$$44\!\cdots\!05$$$$\nu^{4} +$$$$11\!\cdots\!12$$$$\nu^{3} +$$$$28\!\cdots\!80$$$$\nu^{2} +$$$$18\!\cdots\!12$$$$\nu +$$$$17\!\cdots\!28$$$$)/$$$$25\!\cdots\!48$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$30\!\cdots\!13$$$$\nu^{19} +$$$$30\!\cdots\!25$$$$\nu^{18} +$$$$93\!\cdots\!93$$$$\nu^{17} -$$$$73\!\cdots\!27$$$$\nu^{16} -$$$$11\!\cdots\!25$$$$\nu^{15} +$$$$53\!\cdots\!53$$$$\nu^{14} +$$$$83\!\cdots\!69$$$$\nu^{13} +$$$$25\!\cdots\!73$$$$\nu^{12} -$$$$33\!\cdots\!79$$$$\nu^{11} -$$$$21\!\cdots\!53$$$$\nu^{10} +$$$$77\!\cdots\!11$$$$\nu^{9} +$$$$10\!\cdots\!59$$$$\nu^{8} -$$$$86\!\cdots\!59$$$$\nu^{7} -$$$$16\!\cdots\!89$$$$\nu^{6} +$$$$24\!\cdots\!03$$$$\nu^{5} +$$$$43\!\cdots\!19$$$$\nu^{4} -$$$$32\!\cdots\!76$$$$\nu^{3} +$$$$11\!\cdots\!88$$$$\nu^{2} +$$$$12\!\cdots\!44$$$$\nu +$$$$27\!\cdots\!76$$$$)/$$$$32\!\cdots\!72$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$50\!\cdots\!80$$$$\nu^{19} +$$$$16\!\cdots\!84$$$$\nu^{18} +$$$$15\!\cdots\!45$$$$\nu^{17} -$$$$47\!\cdots\!87$$$$\nu^{16} -$$$$20\!\cdots\!78$$$$\nu^{15} +$$$$54\!\cdots\!02$$$$\nu^{14} +$$$$14\!\cdots\!19$$$$\nu^{13} -$$$$32\!\cdots\!61$$$$\nu^{12} -$$$$62\!\cdots\!92$$$$\nu^{11} +$$$$97\!\cdots\!48$$$$\nu^{10} +$$$$15\!\cdots\!43$$$$\nu^{9} -$$$$14\!\cdots\!85$$$$\nu^{8} -$$$$21\!\cdots\!74$$$$\nu^{7} +$$$$84\!\cdots\!14$$$$\nu^{6} +$$$$10\!\cdots\!61$$$$\nu^{5} -$$$$10\!\cdots\!59$$$$\nu^{4} +$$$$13\!\cdots\!56$$$$\nu^{3} +$$$$13\!\cdots\!88$$$$\nu^{2} -$$$$55\!\cdots\!08$$$$\nu +$$$$20\!\cdots\!36$$$$)/$$$$39\!\cdots\!28$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$17\!\cdots\!88$$$$\nu^{19} +$$$$63\!\cdots\!53$$$$\nu^{18} +$$$$54\!\cdots\!78$$$$\nu^{17} +$$$$19\!\cdots\!61$$$$\nu^{16} -$$$$70\!\cdots\!32$$$$\nu^{15} -$$$$58\!\cdots\!07$$$$\nu^{14} +$$$$49\!\cdots\!26$$$$\nu^{13} +$$$$69\!\cdots\!17$$$$\nu^{12} -$$$$19\!\cdots\!36$$$$\nu^{11} -$$$$41\!\cdots\!05$$$$\nu^{10} +$$$$40\!\cdots\!70$$$$\nu^{9} +$$$$12\!\cdots\!31$$$$\nu^{8} -$$$$30\!\cdots\!92$$$$\nu^{7} -$$$$12\!\cdots\!61$$$$\nu^{6} -$$$$99\!\cdots\!90$$$$\nu^{5} -$$$$34\!\cdots\!33$$$$\nu^{4} -$$$$13\!\cdots\!80$$$$\nu^{3} -$$$$92\!\cdots\!12$$$$\nu^{2} -$$$$10\!\cdots\!96$$$$\nu -$$$$26\!\cdots\!64$$$$)/$$$$54\!\cdots\!92$$ $$\beta_{10}$$ $$=$$ $$($$$$18\!\cdots\!37$$$$\nu^{19} -$$$$45\!\cdots\!45$$$$\nu^{18} -$$$$54\!\cdots\!43$$$$\nu^{17} +$$$$12\!\cdots\!15$$$$\nu^{16} +$$$$68\!\cdots\!73$$$$\nu^{15} -$$$$13\!\cdots\!57$$$$\nu^{14} -$$$$47\!\cdots\!63$$$$\nu^{13} +$$$$66\!\cdots\!27$$$$\nu^{12} +$$$$18\!\cdots\!99$$$$\nu^{11} -$$$$13\!\cdots\!75$$$$\nu^{10} -$$$$42\!\cdots\!61$$$$\nu^{9} -$$$$34\!\cdots\!23$$$$\nu^{8} +$$$$44\!\cdots\!47$$$$\nu^{7} +$$$$19\!\cdots\!77$$$$\nu^{6} -$$$$88\!\cdots\!37$$$$\nu^{5} +$$$$31\!\cdots\!05$$$$\nu^{4} +$$$$22\!\cdots\!00$$$$\nu^{3} -$$$$17\!\cdots\!76$$$$\nu^{2} +$$$$17\!\cdots\!24$$$$\nu -$$$$28\!\cdots\!04$$$$)/$$$$54\!\cdots\!92$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$60\!\cdots\!59$$$$\nu^{19} +$$$$25\!\cdots\!58$$$$\nu^{18} +$$$$18\!\cdots\!89$$$$\nu^{17} -$$$$68\!\cdots\!96$$$$\nu^{16} -$$$$24\!\cdots\!99$$$$\nu^{15} +$$$$73\!\cdots\!06$$$$\nu^{14} +$$$$18\!\cdots\!73$$$$\nu^{13} -$$$$37\!\cdots\!68$$$$\nu^{12} -$$$$80\!\cdots\!85$$$$\nu^{11} +$$$$78\!\cdots\!66$$$$\nu^{10} +$$$$20\!\cdots\!19$$$$\nu^{9} +$$$$31\!\cdots\!68$$$$\nu^{8} -$$$$28\!\cdots\!57$$$$\nu^{7} -$$$$34\!\cdots\!62$$$$\nu^{6} +$$$$12\!\cdots\!91$$$$\nu^{5} +$$$$28\!\cdots\!36$$$$\nu^{4} +$$$$24\!\cdots\!72$$$$\nu^{3} +$$$$76\!\cdots\!60$$$$\nu^{2} +$$$$82\!\cdots\!48$$$$\nu +$$$$17\!\cdots\!32$$$$)/$$$$14\!\cdots\!28$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$19\!\cdots\!79$$$$\nu^{19} -$$$$19\!\cdots\!70$$$$\nu^{18} +$$$$58\!\cdots\!49$$$$\nu^{17} +$$$$76\!\cdots\!98$$$$\nu^{16} -$$$$74\!\cdots\!75$$$$\nu^{15} -$$$$12\!\cdots\!22$$$$\nu^{14} +$$$$50\!\cdots\!97$$$$\nu^{13} +$$$$11\!\cdots\!06$$$$\nu^{12} -$$$$18\!\cdots\!93$$$$\nu^{11} -$$$$57\!\cdots\!58$$$$\nu^{10} +$$$$35\!\cdots\!51$$$$\nu^{9} +$$$$14\!\cdots\!62$$$$\nu^{8} -$$$$16\!\cdots\!85$$$$\nu^{7} -$$$$13\!\cdots\!10$$$$\nu^{6} -$$$$23\!\cdots\!45$$$$\nu^{5} -$$$$52\!\cdots\!22$$$$\nu^{4} -$$$$13\!\cdots\!24$$$$\nu^{3} -$$$$25\!\cdots\!92$$$$\nu^{2} -$$$$55\!\cdots\!00$$$$\nu -$$$$29\!\cdots\!56$$$$)/$$$$42\!\cdots\!84$$ $$\beta_{13}$$ $$=$$ $$($$$$26\!\cdots\!84$$$$\nu^{19} +$$$$22\!\cdots\!73$$$$\nu^{18} -$$$$79\!\cdots\!08$$$$\nu^{17} -$$$$92\!\cdots\!49$$$$\nu^{16} +$$$$10\!\cdots\!92$$$$\nu^{15} +$$$$16\!\cdots\!65$$$$\nu^{14} -$$$$70\!\cdots\!76$$$$\nu^{13} -$$$$14\!\cdots\!29$$$$\nu^{12} +$$$$27\!\cdots\!04$$$$\nu^{11} +$$$$77\!\cdots\!55$$$$\nu^{10} -$$$$56\!\cdots\!64$$$$\nu^{9} -$$$$20\!\cdots\!51$$$$\nu^{8} +$$$$41\!\cdots\!40$$$$\nu^{7} +$$$$21\!\cdots\!59$$$$\nu^{6} +$$$$13\!\cdots\!96$$$$\nu^{5} +$$$$43\!\cdots\!85$$$$\nu^{4} +$$$$22\!\cdots\!20$$$$\nu^{3} +$$$$10\!\cdots\!28$$$$\nu^{2} +$$$$17\!\cdots\!00$$$$\nu +$$$$11\!\cdots\!48$$$$)/$$$$54\!\cdots\!92$$ $$\beta_{14}$$ $$=$$ $$($$$$34\!\cdots\!88$$$$\nu^{19} -$$$$66\!\cdots\!51$$$$\nu^{18} -$$$$10\!\cdots\!84$$$$\nu^{17} +$$$$16\!\cdots\!71$$$$\nu^{16} +$$$$13\!\cdots\!72$$$$\nu^{15} -$$$$15\!\cdots\!75$$$$\nu^{14} -$$$$96\!\cdots\!68$$$$\nu^{13} +$$$$45\!\cdots\!87$$$$\nu^{12} +$$$$39\!\cdots\!00$$$$\nu^{11} +$$$$11\!\cdots\!79$$$$\nu^{10} -$$$$93\!\cdots\!52$$$$\nu^{9} -$$$$10\!\cdots\!63$$$$\nu^{8} +$$$$10\!\cdots\!84$$$$\nu^{7} +$$$$19\!\cdots\!27$$$$\nu^{6} -$$$$30\!\cdots\!48$$$$\nu^{5} -$$$$57\!\cdots\!83$$$$\nu^{4} +$$$$27\!\cdots\!76$$$$\nu^{3} -$$$$15\!\cdots\!76$$$$\nu^{2} -$$$$16\!\cdots\!68$$$$\nu -$$$$12\!\cdots\!08$$$$)/$$$$54\!\cdots\!92$$ $$\beta_{15}$$ $$=$$ $$($$$$85\!\cdots\!45$$$$\nu^{19} -$$$$18\!\cdots\!33$$$$\nu^{18} -$$$$26\!\cdots\!14$$$$\nu^{17} +$$$$46\!\cdots\!77$$$$\nu^{16} +$$$$33\!\cdots\!81$$$$\nu^{15} -$$$$44\!\cdots\!55$$$$\nu^{14} -$$$$23\!\cdots\!78$$$$\nu^{13} +$$$$16\!\cdots\!59$$$$\nu^{12} +$$$$96\!\cdots\!87$$$$\nu^{11} +$$$$38\!\cdots\!09$$$$\nu^{10} -$$$$22\!\cdots\!78$$$$\nu^{9} -$$$$19\!\cdots\!65$$$$\nu^{8} +$$$$24\!\cdots\!79$$$$\nu^{7} +$$$$39\!\cdots\!35$$$$\nu^{6} -$$$$62\!\cdots\!46$$$$\nu^{5} -$$$$10\!\cdots\!35$$$$\nu^{4} +$$$$27\!\cdots\!96$$$$\nu^{3} -$$$$24\!\cdots\!20$$$$\nu^{2} -$$$$39\!\cdots\!88$$$$\nu -$$$$18\!\cdots\!76$$$$)/$$$$13\!\cdots\!48$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$26\!\cdots\!13$$$$\nu^{19} -$$$$40\!\cdots\!24$$$$\nu^{18} +$$$$79\!\cdots\!77$$$$\nu^{17} +$$$$14\!\cdots\!96$$$$\nu^{16} -$$$$10\!\cdots\!09$$$$\nu^{15} -$$$$22\!\cdots\!32$$$$\nu^{14} +$$$$67\!\cdots\!57$$$$\nu^{13} +$$$$19\!\cdots\!84$$$$\nu^{12} -$$$$25\!\cdots\!27$$$$\nu^{11} -$$$$94\!\cdots\!44$$$$\nu^{10} +$$$$45\!\cdots\!23$$$$\nu^{9} +$$$$24\!\cdots\!56$$$$\nu^{8} -$$$$12\!\cdots\!79$$$$\nu^{7} -$$$$24\!\cdots\!72$$$$\nu^{6} -$$$$47\!\cdots\!01$$$$\nu^{5} -$$$$53\!\cdots\!16$$$$\nu^{4} -$$$$14\!\cdots\!92$$$$\nu^{3} -$$$$27\!\cdots\!96$$$$\nu^{2} -$$$$41\!\cdots\!08$$$$\nu -$$$$23\!\cdots\!68$$$$)/$$$$42\!\cdots\!84$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$37\!\cdots\!17$$$$\nu^{19} -$$$$29\!\cdots\!86$$$$\nu^{18} +$$$$11\!\cdots\!59$$$$\nu^{17} +$$$$12\!\cdots\!36$$$$\nu^{16} -$$$$14\!\cdots\!05$$$$\nu^{15} -$$$$21\!\cdots\!90$$$$\nu^{14} +$$$$98\!\cdots\!51$$$$\nu^{13} +$$$$20\!\cdots\!48$$$$\nu^{12} -$$$$37\!\cdots\!87$$$$\nu^{11} -$$$$10\!\cdots\!46$$$$\nu^{10} +$$$$72\!\cdots\!33$$$$\nu^{9} +$$$$27\!\cdots\!20$$$$\nu^{8} -$$$$40\!\cdots\!27$$$$\nu^{7} -$$$$27\!\cdots\!86$$$$\nu^{6} -$$$$42\!\cdots\!75$$$$\nu^{5} -$$$$80\!\cdots\!68$$$$\nu^{4} -$$$$16\!\cdots\!84$$$$\nu^{3} -$$$$18\!\cdots\!32$$$$\nu^{2} -$$$$56\!\cdots\!88$$$$\nu -$$$$69\!\cdots\!40$$$$)/$$$$54\!\cdots\!92$$ $$\beta_{18}$$ $$=$$ $$($$$$45\!\cdots\!07$$$$\nu^{19} +$$$$30\!\cdots\!32$$$$\nu^{18} -$$$$13\!\cdots\!91$$$$\nu^{17} -$$$$13\!\cdots\!04$$$$\nu^{16} +$$$$17\!\cdots\!95$$$$\nu^{15} +$$$$24\!\cdots\!92$$$$\nu^{14} -$$$$11\!\cdots\!71$$$$\nu^{13} -$$$$23\!\cdots\!60$$$$\nu^{12} +$$$$44\!\cdots\!81$$$$\nu^{11} +$$$$12\!\cdots\!96$$$$\nu^{10} -$$$$86\!\cdots\!93$$$$\nu^{9} -$$$$33\!\cdots\!08$$$$\nu^{8} +$$$$45\!\cdots\!21$$$$\nu^{7} +$$$$34\!\cdots\!44$$$$\nu^{6} +$$$$57\!\cdots\!07$$$$\nu^{5} +$$$$78\!\cdots\!80$$$$\nu^{4} +$$$$14\!\cdots\!48$$$$\nu^{3} +$$$$11\!\cdots\!12$$$$\nu^{2} +$$$$36\!\cdots\!80$$$$\nu -$$$$12\!\cdots\!84$$$$)/$$$$54\!\cdots\!92$$ $$\beta_{19}$$ $$=$$ $$($$$$32\!\cdots\!27$$$$\nu^{19} +$$$$16\!\cdots\!59$$$$\nu^{18} -$$$$97\!\cdots\!97$$$$\nu^{17} -$$$$80\!\cdots\!27$$$$\nu^{16} +$$$$12\!\cdots\!47$$$$\nu^{15} +$$$$15\!\cdots\!15$$$$\nu^{14} -$$$$84\!\cdots\!53$$$$\nu^{13} -$$$$15\!\cdots\!23$$$$\nu^{12} +$$$$31\!\cdots\!41$$$$\nu^{11} +$$$$85\!\cdots\!65$$$$\nu^{10} -$$$$61\!\cdots\!03$$$$\nu^{9} -$$$$23\!\cdots\!25$$$$\nu^{8} +$$$$33\!\cdots\!97$$$$\nu^{7} +$$$$24\!\cdots\!09$$$$\nu^{6} +$$$$39\!\cdots\!33$$$$\nu^{5} +$$$$56\!\cdots\!83$$$$\nu^{4} +$$$$10\!\cdots\!88$$$$\nu^{3} +$$$$84\!\cdots\!96$$$$\nu^{2} +$$$$29\!\cdots\!20$$$$\nu -$$$$75\!\cdots\!48$$$$)/$$$$26\!\cdots\!52$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{5} - \beta_{4}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{15} + 2 \beta_{14} + \beta_{9} + 2 \beta_{7} + \beta_{5} - 2 \beta_{3} + \beta_{2} + 30$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{19} - 3 \beta_{17} - 3 \beta_{16} + 4 \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + 9 \beta_{12} - 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} + 12 \beta_{7} + 6 \beta_{6} + 49 \beta_{5} - 85 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + 4 \beta_{1} + 16$$ $$\nu^{4}$$ $$=$$ $$24 \beta_{19} - 16 \beta_{18} + 32 \beta_{16} - 85 \beta_{15} + 108 \beta_{14} - 6 \beta_{13} + 8 \beta_{12} - 2 \beta_{11} + 32 \beta_{10} + 51 \beta_{9} + 40 \beta_{8} - 54 \beta_{7} + 48 \beta_{6} + 29 \beta_{5} - 88 \beta_{4} - 220 \beta_{3} + 51 \beta_{2} - 26 \beta_{1} + 1416$$ $$\nu^{5}$$ $$=$$ $$-325 \beta_{19} - 20 \beta_{18} - 335 \beta_{17} - 595 \beta_{16} + 238 \beta_{15} - 140 \beta_{14} + 64 \beta_{13} + 915 \beta_{12} - 164 \beta_{11} - 750 \beta_{10} - 242 \beta_{9} - 860 \beta_{8} + 1304 \beta_{7} - 190 \beta_{6} + 2271 \beta_{5} - 7299 \beta_{4} + 435 \beta_{3} + 194 \beta_{2} + 436 \beta_{1} + 90$$ $$\nu^{6}$$ $$=$$ $$2484 \beta_{19} - 2288 \beta_{18} - 1188 \beta_{17} + 4684 \beta_{16} - 4747 \beta_{15} + 4766 \beta_{14} - 296 \beta_{13} + 484 \beta_{12} - 272 \beta_{11} + 4504 \beta_{10} + 2995 \beta_{9} + 6080 \beta_{8} - 9306 \beta_{7} + 8784 \beta_{6} - 1141 \beta_{5} - 5168 \beta_{4} - 18930 \beta_{3} + 2435 \beta_{2} - 2616 \beta_{1} + 57892$$ $$\nu^{7}$$ $$=$$ $$-25641 \beta_{19} - 4368 \beta_{18} - 24521 \beta_{17} - 68425 \beta_{16} + 12404 \beta_{15} - 8622 \beta_{14} + 182 \beta_{13} + 76699 \beta_{12} - 7398 \beta_{11} - 73738 \beta_{10} - 19142 \beta_{9} - 88242 \beta_{8} + 79108 \beta_{7} - 70014 \beta_{6} + 81853 \beta_{5} - 553433 \beta_{4} + 53753 \beta_{3} + 4916 \beta_{2} + 27524 \beta_{1} - 65192$$ $$\nu^{8}$$ $$=$$ $$204624 \beta_{19} - 206176 \beta_{18} - 165312 \beta_{17} + 499072 \beta_{16} - 168489 \beta_{15} + 126656 \beta_{14} - 2250 \beta_{13} - 10384 \beta_{12} - 12294 \beta_{11} + 484480 \beta_{10} + 132523 \beta_{9} + 657776 \beta_{8} - 560270 \beta_{7} + 967968 \beta_{6} - 194335 \beta_{5} + 89520 \beta_{4} - 1464056 \beta_{3} + 69883 \beta_{2} - 147062 \beta_{1} + 1398812$$ $$\nu^{9}$$ $$=$$ $$-1832157 \beta_{19} - 436332 \beta_{18} - 1462971 \beta_{17} - 6287367 \beta_{16} + 421506 \beta_{15} - 364688 \beta_{14} - 128252 \beta_{13} + 5705571 \beta_{12} - 80000 \beta_{11} - 6345630 \beta_{10} - 784130 \beta_{9} - 7783344 \beta_{8} + 2424768 \beta_{7} - 8448030 \beta_{6} + 482815 \beta_{5} - 38678923 \beta_{4} + 5589063 \beta_{3} - 179122 \beta_{2} + 860028 \beta_{1} - 5685682$$ $$\nu^{10}$$ $$=$$ $$15307372 \beta_{19} - 15036544 \beta_{18} - 14836164 \beta_{17} + 44940012 \beta_{16} + 3076317 \beta_{15} - 5516278 \beta_{14} + 700692 \beta_{13} - 5823828 \beta_{12} + 325164 \beta_{11} + 44024872 \beta_{10} - 72145 \beta_{9} + 59528840 \beta_{8} - 8348554 \beta_{7} + 86383872 \beta_{6} - 10071725 \beta_{5} + 49743016 \beta_{4} - 105374518 \beta_{3} - 2867697 \beta_{2} - 1975028 \beta_{1} - 75783484$$ $$\nu^{11}$$ $$=$$ $$-125173037 \beta_{19} - 28031784 \beta_{18} - 70936833 \beta_{17} - 500549225 \beta_{16} - 22370312 \beta_{15} + 10500534 \beta_{14} - 8327838 \beta_{13} + 385536943 \beta_{12} + 26053702 \beta_{11} - 490560642 \beta_{10} + 30134994 \beta_{9} - 612046270 \beta_{8} - 168278084 \beta_{7} - 761581062 \beta_{6} - 321838299 \beta_{5} - 2518737145 \beta_{4} + 500334153 \beta_{3} - 33671344 \beta_{2} - 59193348 \beta_{1} - 112108372$$ $$\nu^{12}$$ $$=$$ $$1062670936 \beta_{19} - 942502256 \beta_{18} - 1051657104 \beta_{17} + 3557900432 \beta_{16} + 1375925959 \beta_{15} - 1357445676 \beta_{14} + 68660698 \beta_{13} - 795519640 \beta_{12} + 99742190 \beta_{11} + 3495987264 \beta_{10} - 903107305 \beta_{9} + 4695379512 \beta_{8} + 2778423378 \beta_{7} + 6706804944 \beta_{6} + 393564993 \beta_{5} + 6360219768 \beta_{4} - 7059358404 \beta_{3} - 720186313 \beta_{2} + 777702278 \beta_{1} - 16622556444$$ $$\nu^{13}$$ $$=$$ $$-8123893089 \beta_{19} - 1082448276 \beta_{18} - 2405988091 \beta_{17} - 35188337951 \beta_{16} - 7003180210 \beta_{15} + 4878316204 \beta_{14} - 130484072 \beta_{13} + 23561151367 \beta_{12} + 3735435140 \beta_{11} - 34045287094 \beta_{10} + 10352613110 \beta_{9} - 43029944932 \beta_{8} - 41520682456 \beta_{7} - 57464727814 \beta_{6} - 46105055481 \beta_{5} - 151119238347 \beta_{4} + 38999816063 \beta_{3} - 2484743454 \beta_{2} - 14479302604 \beta_{1} + 37434050762$$ $$\nu^{14}$$ $$=$$ $$67111931348 \beta_{19} - 50400346448 \beta_{18} - 61156315220 \beta_{17} + 247175087068 \beta_{16} + 178219432237 \beta_{15} - 158441331498 \beta_{14} + 4019914688 \beta_{13} - 75611435292 \beta_{12} + 8987440280 \beta_{11} + 242578280344 \beta_{10} - 137002913133 \beta_{9} + 323315978832 \beta_{8} + 478585788726 \beta_{7} + 456095497072 \beta_{6} + 155129182515 \beta_{5} + 571604318592 \beta_{4} - 429708452810 \beta_{3} - 81072791101 \beta_{2} + 134709421344 \beta_{1} - 1883906818388$$ $$\nu^{15}$$ $$=$$ $$-478148148233 \beta_{19} + 5843040736 \beta_{18} - 4148820705 \beta_{17} - 2138290857937 \beta_{16} - 982072045748 \beta_{15} + 730210382842 \beta_{14} + 30311048942 \beta_{13} + 1251128744555 \beta_{12} + 359819731890 \beta_{11} - 2059055043114 \beta_{10} + 1379993530746 \beta_{9} - 2631616237578 \beta_{8} - 5209399366668 \beta_{7} - 3662190581598 \beta_{6} - 4723058319875 \beta_{5} - 7990641491561 \beta_{4} + 2604957285729 \beta_{3} - 92230866020 \beta_{2} - 1788182253580 \beta_{1} + 7169280884976$$ $$\nu^{16}$$ $$=$$ $$3611012467360 \beta_{19} - 2086838653888 \beta_{18} - 2732907357952 \beta_{17} + 14399130551680 \beta_{16} + 17721292301927 \beta_{15} - 14989765340664 \beta_{14} + 132326104510 \beta_{13} - 5555083772960 \beta_{12} + 455680449218 \beta_{11} + 14078600855808 \beta_{10} - 14940285253805 \beta_{9} + 18600882290272 \beta_{8} + 54146776113490 \beta_{7} + 25979400195392 \beta_{6} + 22038723810737 \beta_{5} + 39981341159904 \beta_{4} - 22173074382960 \beta_{3} - 7244352072957 \beta_{2} + 15504106523778 \beta_{1} - 175295813267812$$ $$\nu^{17}$$ $$=$$ $$-22622149643317 \beta_{19} + 5160254586820 \beta_{18} + 7839490976085 \beta_{17} - 101513362710887 \beta_{16} - 107048356918598 \beta_{15} + 81901413941000 \beta_{14} + 4507501388780 \beta_{13} + 49715925451867 \beta_{12} + 29165151837768 \beta_{11} - 97684692915022 \beta_{10} + 141635314228462 \beta_{9} - 126468558449880 \beta_{8} - 522737688380016 \beta_{7} - 181737085173198 \beta_{6} - 417947329486353 \beta_{5} - 320554893267979 \beta_{4} + 137710913141415 \beta_{3} + 4040217694934 \beta_{2} - 176327636179188 \beta_{1} + 868159104771558$$ $$\nu^{18}$$ $$=$$ $$130657989007868 \beta_{19} - 33243943776992 \beta_{18} - 55766768971716 \beta_{17} + 584092890745836 \beta_{16} + 1535035139822973 \beta_{15} - 1262517147852094 \beta_{14} - 4225935236212 \beta_{13} - 295174377501156 \beta_{12} + 795706038468 \beta_{11} + 566900520486728 \beta_{10} - 1384491190844649 \beta_{9} + 738121140364216 \beta_{8} + 5088886578797046 \beta_{7} + 1018107038157600 \beta_{6} + 2354592927040595 \beta_{5} + 2007195062536568 \beta_{4} - 742315036548222 \beta_{3} - 569992058313257 \beta_{2} + 1483840998207188 \beta_{1} - 14602148259697708$$ $$\nu^{19}$$ $$=$$ $$-447769033388069 \beta_{19} + 492561905784744 \beta_{18} + 770621941544639 \beta_{17} - 1959381963676985 \beta_{16} - 10012300457668448 \beta_{15} + 7779402723366110 \beta_{14} + 424475348925562 \beta_{13} + 226232181731047 \beta_{12} + 2113808089162398 \beta_{11} - 1888761569117554 \beta_{10} + 12535713457468738 \beta_{9} - 2571229920050582 \beta_{8} - 45859720242426708 \beta_{7} - 4070921980391222 \beta_{6} - 33542732620996011 \beta_{5} - 1850210341337081 \beta_{4} + 3749442865882745 \beta_{3} + 1176591402274792 \beta_{2} - 15207549654619892 \beta_{1} + 85250807684934260$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/354\mathbb{Z}\right)^\times$$.

 $$n$$ $$61$$ $$119$$ $$\chi(n)$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
235.1
 −6.71849 + 1.41421i −4.74052 + 1.41421i 0.383479 + 1.41421i 6.82349 + 1.41421i 7.71614 + 1.41421i −8.79591 + 1.41421i −2.72105 + 1.41421i −0.662289 + 1.41421i 0.969662 + 1.41421i 7.74549 + 1.41421i −6.71849 − 1.41421i −4.74052 − 1.41421i 0.383479 − 1.41421i 6.82349 − 1.41421i 7.71614 − 1.41421i −8.79591 − 1.41421i −2.72105 − 1.41421i −0.662289 − 1.41421i 0.969662 − 1.41421i 7.74549 − 1.41421i
1.41421i −1.73205 −2.00000 −6.71849 2.44949i −6.32195 2.82843i 3.00000 9.50137i
235.2 1.41421i −1.73205 −2.00000 −4.74052 2.44949i 12.4076 2.82843i 3.00000 6.70411i
235.3 1.41421i −1.73205 −2.00000 0.383479 2.44949i 1.10699 2.82843i 3.00000 0.542321i
235.4 1.41421i −1.73205 −2.00000 6.82349 2.44949i 2.37240 2.82843i 3.00000 9.64987i
235.5 1.41421i −1.73205 −2.00000 7.71614 2.44949i −7.56506 2.82843i 3.00000 10.9123i
235.6 1.41421i 1.73205 −2.00000 −8.79591 2.44949i 6.82395 2.82843i 3.00000 12.4393i
235.7 1.41421i 1.73205 −2.00000 −2.72105 2.44949i −3.73640 2.82843i 3.00000 3.84815i
235.8 1.41421i 1.73205 −2.00000 −0.662289 2.44949i −11.5463 2.82843i 3.00000 0.936618i
235.9 1.41421i 1.73205 −2.00000 0.969662 2.44949i 3.05407 2.82843i 3.00000 1.37131i
235.10 1.41421i 1.73205 −2.00000 7.74549 2.44949i 7.40466 2.82843i 3.00000 10.9538i
235.11 1.41421i −1.73205 −2.00000 −6.71849 2.44949i −6.32195 2.82843i 3.00000 9.50137i
235.12 1.41421i −1.73205 −2.00000 −4.74052 2.44949i 12.4076 2.82843i 3.00000 6.70411i
235.13 1.41421i −1.73205 −2.00000 0.383479 2.44949i 1.10699 2.82843i 3.00000 0.542321i
235.14 1.41421i −1.73205 −2.00000 6.82349 2.44949i 2.37240 2.82843i 3.00000 9.64987i
235.15 1.41421i −1.73205 −2.00000 7.71614 2.44949i −7.56506 2.82843i 3.00000 10.9123i
235.16 1.41421i 1.73205 −2.00000 −8.79591 2.44949i 6.82395 2.82843i 3.00000 12.4393i
235.17 1.41421i 1.73205 −2.00000 −2.72105 2.44949i −3.73640 2.82843i 3.00000 3.84815i
235.18 1.41421i 1.73205 −2.00000 −0.662289 2.44949i −11.5463 2.82843i 3.00000 0.936618i
235.19 1.41421i 1.73205 −2.00000 0.969662 2.44949i 3.05407 2.82843i 3.00000 1.37131i
235.20 1.41421i 1.73205 −2.00000 7.74549 2.44949i 7.40466 2.82843i 3.00000 10.9538i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 235.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
59.b Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(354, [\chi])$$.