# Properties

 Label 370.2.h.e Level $370$ Weight $2$ Character orbit 370.h Analytic conductor $2.954$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.h (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 4 x^{19} + 8 x^{18} + 4 x^{17} + 103 x^{16} - 394 x^{15} + 760 x^{14} + 278 x^{13} + 2009 x^{12} - 7362 x^{11} + 13826 x^{10} + 4848 x^{9} + 13544 x^{8} - 44248 x^{7} + 76384 x^{6} + 24512 x^{5} + 28432 x^{4} - 61952 x^{3} + 61952 x^{2} - 5632 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 4 x^{19} + 8 x^{18} + 4 x^{17} + 103 x^{16} - 394 x^{15} + 760 x^{14} + 278 x^{13} + 2009 x^{12} - 7362 x^{11} + 13826 x^{10} + 4848 x^{9} + 13544 x^{8} - 44248 x^{7} + 76384 x^{6} + 24512 x^{5} + 28432 x^{4} - 61952 x^{3} + 61952 x^{2} - 5632 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$58\!\cdots\!94$$$$\nu^{19} -$$$$16\!\cdots\!17$$$$\nu^{18} +$$$$19\!\cdots\!00$$$$\nu^{17} +$$$$79\!\cdots\!76$$$$\nu^{16} +$$$$62\!\cdots\!18$$$$\nu^{15} -$$$$15\!\cdots\!23$$$$\nu^{14} +$$$$17\!\cdots\!02$$$$\nu^{13} +$$$$70\!\cdots\!04$$$$\nu^{12} +$$$$13\!\cdots\!20$$$$\nu^{11} -$$$$29\!\cdots\!33$$$$\nu^{10} +$$$$31\!\cdots\!06$$$$\nu^{9} +$$$$13\!\cdots\!58$$$$\nu^{8} +$$$$10\!\cdots\!04$$$$\nu^{7} -$$$$16\!\cdots\!08$$$$\nu^{6} +$$$$15\!\cdots\!48$$$$\nu^{5} +$$$$93\!\cdots\!56$$$$\nu^{4} +$$$$30\!\cdots\!44$$$$\nu^{3} -$$$$16\!\cdots\!04$$$$\nu^{2} +$$$$14\!\cdots\!04$$$$\nu +$$$$27\!\cdots\!56$$$$)/$$$$69\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$30\!\cdots\!87$$$$\nu^{19} +$$$$18\!\cdots\!86$$$$\nu^{18} -$$$$44\!\cdots\!50$$$$\nu^{17} +$$$$40\!\cdots\!52$$$$\nu^{16} +$$$$13\!\cdots\!91$$$$\nu^{15} +$$$$22\!\cdots\!84$$$$\nu^{14} -$$$$42\!\cdots\!46$$$$\nu^{13} +$$$$28\!\cdots\!18$$$$\nu^{12} +$$$$10\!\cdots\!65$$$$\nu^{11} +$$$$58\!\cdots\!84$$$$\nu^{10} -$$$$80\!\cdots\!28$$$$\nu^{9} +$$$$75\!\cdots\!76$$$$\nu^{8} +$$$$94\!\cdots\!28$$$$\nu^{7} +$$$$45\!\cdots\!64$$$$\nu^{6} -$$$$45\!\cdots\!04$$$$\nu^{5} -$$$$20\!\cdots\!48$$$$\nu^{4} -$$$$11\!\cdots\!12$$$$\nu^{3} +$$$$54\!\cdots\!12$$$$\nu^{2} -$$$$30\!\cdots\!52$$$$\nu +$$$$38\!\cdots\!52$$$$)/$$$$27\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$58\!\cdots\!71$$$$\nu^{19} -$$$$21\!\cdots\!08$$$$\nu^{18} +$$$$40\!\cdots\!00$$$$\nu^{17} +$$$$31\!\cdots\!84$$$$\nu^{16} +$$$$63\!\cdots\!17$$$$\nu^{15} -$$$$20\!\cdots\!02$$$$\nu^{14} +$$$$38\!\cdots\!68$$$$\nu^{13} +$$$$23\!\cdots\!46$$$$\nu^{12} +$$$$14\!\cdots\!55$$$$\nu^{11} -$$$$38\!\cdots\!22$$$$\nu^{10} +$$$$69\!\cdots\!14$$$$\nu^{9} +$$$$41\!\cdots\!32$$$$\nu^{8} +$$$$13\!\cdots\!56$$$$\nu^{7} -$$$$21\!\cdots\!92$$$$\nu^{6} +$$$$38\!\cdots\!32$$$$\nu^{5} +$$$$20\!\cdots\!44$$$$\nu^{4} +$$$$54\!\cdots\!96$$$$\nu^{3} -$$$$24\!\cdots\!16$$$$\nu^{2} +$$$$29\!\cdots\!76$$$$\nu -$$$$27\!\cdots\!56$$$$)/$$$$27\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$18\!\cdots\!01$$$$\nu^{19} -$$$$21\!\cdots\!40$$$$\nu^{18} +$$$$53\!\cdots\!60$$$$\nu^{17} -$$$$43\!\cdots\!16$$$$\nu^{16} +$$$$73\!\cdots\!39$$$$\nu^{15} -$$$$23\!\cdots\!50$$$$\nu^{14} +$$$$51\!\cdots\!88$$$$\nu^{13} -$$$$35\!\cdots\!10$$$$\nu^{12} -$$$$41\!\cdots\!75$$$$\nu^{11} -$$$$56\!\cdots\!22$$$$\nu^{10} +$$$$93\!\cdots\!78$$$$\nu^{9} -$$$$27\!\cdots\!24$$$$\nu^{8} -$$$$56\!\cdots\!76$$$$\nu^{7} -$$$$41\!\cdots\!40$$$$\nu^{6} +$$$$51\!\cdots\!92$$$$\nu^{5} +$$$$75\!\cdots\!20$$$$\nu^{4} -$$$$20\!\cdots\!04$$$$\nu^{3} -$$$$55\!\cdots\!68$$$$\nu^{2} +$$$$45\!\cdots\!72$$$$\nu -$$$$40\!\cdots\!60$$$$)/$$$$37\!\cdots\!40$$ $$\beta_{6}$$ $$=$$ $$($$$$18\!\cdots\!82$$$$\nu^{19} -$$$$72\!\cdots\!52$$$$\nu^{18} +$$$$22\!\cdots\!65$$$$\nu^{17} +$$$$20\!\cdots\!08$$$$\nu^{16} +$$$$19\!\cdots\!70$$$$\nu^{15} -$$$$62\!\cdots\!28$$$$\nu^{14} +$$$$25\!\cdots\!11$$$$\nu^{13} +$$$$41\!\cdots\!94$$$$\nu^{12} +$$$$39\!\cdots\!10$$$$\nu^{11} -$$$$74\!\cdots\!94$$$$\nu^{10} +$$$$64\!\cdots\!45$$$$\nu^{9} +$$$$23\!\cdots\!46$$$$\nu^{8} +$$$$29\!\cdots\!06$$$$\nu^{7} -$$$$33\!\cdots\!28$$$$\nu^{6} +$$$$45\!\cdots\!44$$$$\nu^{5} +$$$$22\!\cdots\!36$$$$\nu^{4} +$$$$77\!\cdots\!52$$$$\nu^{3} -$$$$20\!\cdots\!28$$$$\nu^{2} +$$$$19\!\cdots\!40$$$$\nu -$$$$91\!\cdots\!64$$$$)/$$$$27\!\cdots\!80$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!49$$$$\nu^{19} +$$$$18\!\cdots\!42$$$$\nu^{18} -$$$$10\!\cdots\!00$$$$\nu^{17} -$$$$14\!\cdots\!46$$$$\nu^{16} -$$$$14\!\cdots\!63$$$$\nu^{15} +$$$$17\!\cdots\!48$$$$\nu^{14} +$$$$88\!\cdots\!08$$$$\nu^{13} +$$$$43\!\cdots\!96$$$$\nu^{12} -$$$$50\!\cdots\!45$$$$\nu^{11} +$$$$30\!\cdots\!68$$$$\nu^{10} +$$$$11\!\cdots\!54$$$$\nu^{9} +$$$$26\!\cdots\!62$$$$\nu^{8} -$$$$33\!\cdots\!24$$$$\nu^{7} +$$$$15\!\cdots\!08$$$$\nu^{6} +$$$$11\!\cdots\!92$$$$\nu^{5} +$$$$24\!\cdots\!44$$$$\nu^{4} +$$$$58\!\cdots\!76$$$$\nu^{3} +$$$$16\!\cdots\!44$$$$\nu^{2} -$$$$42\!\cdots\!64$$$$\nu +$$$$59\!\cdots\!44$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$74\!\cdots\!19$$$$\nu^{19} -$$$$52\!\cdots\!62$$$$\nu^{18} +$$$$11\!\cdots\!00$$$$\nu^{17} -$$$$21\!\cdots\!24$$$$\nu^{16} +$$$$51\!\cdots\!13$$$$\nu^{15} -$$$$56\!\cdots\!28$$$$\nu^{14} +$$$$10\!\cdots\!52$$$$\nu^{13} -$$$$19\!\cdots\!06$$$$\nu^{12} -$$$$34\!\cdots\!05$$$$\nu^{11} -$$$$12\!\cdots\!08$$$$\nu^{10} +$$$$16\!\cdots\!46$$$$\nu^{9} -$$$$37\!\cdots\!52$$$$\nu^{8} -$$$$86\!\cdots\!16$$$$\nu^{7} -$$$$85\!\cdots\!88$$$$\nu^{6} +$$$$83\!\cdots\!48$$$$\nu^{5} +$$$$15\!\cdots\!16$$$$\nu^{4} +$$$$69\!\cdots\!44$$$$\nu^{3} -$$$$95\!\cdots\!24$$$$\nu^{2} +$$$$71\!\cdots\!64$$$$\nu -$$$$15\!\cdots\!84$$$$)/$$$$55\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$23\!\cdots\!89$$$$\nu^{19} -$$$$11\!\cdots\!02$$$$\nu^{18} +$$$$14\!\cdots\!00$$$$\nu^{17} +$$$$19\!\cdots\!56$$$$\nu^{16} +$$$$21\!\cdots\!83$$$$\nu^{15} -$$$$12\!\cdots\!88$$$$\nu^{14} +$$$$10\!\cdots\!12$$$$\nu^{13} +$$$$14\!\cdots\!74$$$$\nu^{12} +$$$$28\!\cdots\!45$$$$\nu^{11} -$$$$29\!\cdots\!48$$$$\nu^{10} +$$$$89\!\cdots\!86$$$$\nu^{9} +$$$$17\!\cdots\!48$$$$\nu^{8} +$$$$16\!\cdots\!24$$$$\nu^{7} -$$$$21\!\cdots\!48$$$$\nu^{6} -$$$$12\!\cdots\!12$$$$\nu^{5} +$$$$42\!\cdots\!36$$$$\nu^{4} +$$$$88\!\cdots\!64$$$$\nu^{3} -$$$$17\!\cdots\!24$$$$\nu^{2} +$$$$50\!\cdots\!24$$$$\nu +$$$$34\!\cdots\!36$$$$)/$$$$92\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$10\!\cdots\!51$$$$\nu^{19} -$$$$42\!\cdots\!33$$$$\nu^{18} +$$$$83\!\cdots\!00$$$$\nu^{17} +$$$$46\!\cdots\!04$$$$\nu^{16} +$$$$11\!\cdots\!37$$$$\nu^{15} -$$$$41\!\cdots\!77$$$$\nu^{14} +$$$$79\!\cdots\!58$$$$\nu^{13} +$$$$33\!\cdots\!46$$$$\nu^{12} +$$$$21\!\cdots\!05$$$$\nu^{11} -$$$$77\!\cdots\!07$$$$\nu^{10} +$$$$14\!\cdots\!04$$$$\nu^{9} +$$$$58\!\cdots\!62$$$$\nu^{8} +$$$$14\!\cdots\!76$$$$\nu^{7} -$$$$46\!\cdots\!92$$$$\nu^{6} +$$$$79\!\cdots\!92$$$$\nu^{5} +$$$$30\!\cdots\!44$$$$\nu^{4} +$$$$32\!\cdots\!76$$$$\nu^{3} -$$$$61\!\cdots\!56$$$$\nu^{2} +$$$$63\!\cdots\!36$$$$\nu -$$$$30\!\cdots\!56$$$$)/$$$$27\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!77$$$$\nu^{19} +$$$$47\!\cdots\!41$$$$\nu^{18} -$$$$80\!\cdots\!50$$$$\nu^{17} -$$$$68\!\cdots\!08$$$$\nu^{16} -$$$$11\!\cdots\!99$$$$\nu^{15} +$$$$48\!\cdots\!29$$$$\nu^{14} -$$$$71\!\cdots\!16$$$$\nu^{13} -$$$$49\!\cdots\!42$$$$\nu^{12} -$$$$19\!\cdots\!35$$$$\nu^{11} +$$$$99\!\cdots\!39$$$$\nu^{10} -$$$$10\!\cdots\!58$$$$\nu^{9} -$$$$71\!\cdots\!74$$$$\nu^{8} -$$$$12\!\cdots\!52$$$$\nu^{7} +$$$$63\!\cdots\!84$$$$\nu^{6} -$$$$45\!\cdots\!84$$$$\nu^{5} -$$$$28\!\cdots\!88$$$$\nu^{4} -$$$$31\!\cdots\!52$$$$\nu^{3} +$$$$67\!\cdots\!12$$$$\nu^{2} -$$$$37\!\cdots\!72$$$$\nu -$$$$42\!\cdots\!88$$$$)/$$$$27\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$24\!\cdots\!73$$$$\nu^{19} -$$$$92\!\cdots\!84$$$$\nu^{18} +$$$$13\!\cdots\!00$$$$\nu^{17} +$$$$17\!\cdots\!92$$$$\nu^{16} +$$$$24\!\cdots\!51$$$$\nu^{15} -$$$$97\!\cdots\!46$$$$\nu^{14} +$$$$11\!\cdots\!84$$$$\nu^{13} +$$$$12\!\cdots\!58$$$$\nu^{12} +$$$$48\!\cdots\!65$$$$\nu^{11} -$$$$21\!\cdots\!86$$$$\nu^{10} +$$$$13\!\cdots\!42$$$$\nu^{9} +$$$$12\!\cdots\!76$$$$\nu^{8} +$$$$31\!\cdots\!48$$$$\nu^{7} -$$$$14\!\cdots\!16$$$$\nu^{6} +$$$$40\!\cdots\!16$$$$\nu^{5} +$$$$16\!\cdots\!12$$$$\nu^{4} +$$$$51\!\cdots\!48$$$$\nu^{3} -$$$$15\!\cdots\!88$$$$\nu^{2} +$$$$65\!\cdots\!28$$$$\nu -$$$$52\!\cdots\!88$$$$)/$$$$55\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$33\!\cdots\!27$$$$\nu^{19} +$$$$15\!\cdots\!66$$$$\nu^{18} -$$$$33\!\cdots\!00$$$$\nu^{17} -$$$$81\!\cdots\!08$$$$\nu^{16} -$$$$32\!\cdots\!49$$$$\nu^{15} +$$$$15\!\cdots\!04$$$$\nu^{14} -$$$$31\!\cdots\!16$$$$\nu^{13} -$$$$58\!\cdots\!42$$$$\nu^{12} -$$$$50\!\cdots\!35$$$$\nu^{11} +$$$$30\!\cdots\!64$$$$\nu^{10} -$$$$59\!\cdots\!58$$$$\nu^{9} -$$$$16\!\cdots\!24$$$$\nu^{8} -$$$$28\!\cdots\!52$$$$\nu^{7} +$$$$18\!\cdots\!84$$$$\nu^{6} -$$$$34\!\cdots\!84$$$$\nu^{5} -$$$$12\!\cdots\!88$$$$\nu^{4} -$$$$91\!\cdots\!52$$$$\nu^{3} +$$$$26\!\cdots\!12$$$$\nu^{2} -$$$$27\!\cdots\!72$$$$\nu +$$$$16\!\cdots\!12$$$$)/$$$$55\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$54\!\cdots\!63$$$$\nu^{19} +$$$$21\!\cdots\!03$$$$\nu^{18} -$$$$44\!\cdots\!70$$$$\nu^{17} -$$$$21\!\cdots\!12$$$$\nu^{16} -$$$$56\!\cdots\!05$$$$\nu^{15} +$$$$20\!\cdots\!67$$$$\nu^{14} -$$$$43\!\cdots\!44$$$$\nu^{13} -$$$$16\!\cdots\!06$$$$\nu^{12} -$$$$11\!\cdots\!05$$$$\nu^{11} +$$$$37\!\cdots\!81$$$$\nu^{10} -$$$$83\!\cdots\!50$$$$\nu^{9} -$$$$34\!\cdots\!34$$$$\nu^{8} -$$$$77\!\cdots\!24$$$$\nu^{7} +$$$$21\!\cdots\!52$$$$\nu^{6} -$$$$47\!\cdots\!96$$$$\nu^{5} -$$$$21\!\cdots\!44$$$$\nu^{4} -$$$$17\!\cdots\!08$$$$\nu^{3} +$$$$34\!\cdots\!92$$$$\nu^{2} -$$$$35\!\cdots\!20$$$$\nu +$$$$15\!\cdots\!16$$$$)/$$$$55\!\cdots\!60$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$31\!\cdots\!11$$$$\nu^{19} +$$$$11\!\cdots\!53$$$$\nu^{18} -$$$$21\!\cdots\!50$$$$\nu^{17} -$$$$16\!\cdots\!44$$$$\nu^{16} -$$$$32\!\cdots\!97$$$$\nu^{15} +$$$$11\!\cdots\!57$$$$\nu^{14} -$$$$20\!\cdots\!88$$$$\nu^{13} -$$$$11\!\cdots\!86$$$$\nu^{12} -$$$$67\!\cdots\!05$$$$\nu^{11} +$$$$22\!\cdots\!27$$$$\nu^{10} -$$$$34\!\cdots\!74$$$$\nu^{9} -$$$$17\!\cdots\!62$$$$\nu^{8} -$$$$47\!\cdots\!96$$$$\nu^{7} +$$$$13\!\cdots\!72$$$$\nu^{6} -$$$$17\!\cdots\!12$$$$\nu^{5} -$$$$72\!\cdots\!04$$$$\nu^{4} -$$$$98\!\cdots\!36$$$$\nu^{3} +$$$$16\!\cdots\!56$$$$\nu^{2} -$$$$14\!\cdots\!16$$$$\nu +$$$$52\!\cdots\!96$$$$)/$$$$27\!\cdots\!00$$ $$\beta_{16}$$ $$=$$ $$($$$$80\!\cdots\!83$$$$\nu^{19} -$$$$36\!\cdots\!29$$$$\nu^{18} +$$$$78\!\cdots\!25$$$$\nu^{17} +$$$$16\!\cdots\!82$$$$\nu^{16} +$$$$78\!\cdots\!61$$$$\nu^{15} -$$$$36\!\cdots\!01$$$$\nu^{14} +$$$$75\!\cdots\!89$$$$\nu^{13} +$$$$11\!\cdots\!98$$$$\nu^{12} +$$$$13\!\cdots\!65$$$$\nu^{11} -$$$$70\!\cdots\!81$$$$\nu^{10} +$$$$14\!\cdots\!37$$$$\nu^{9} +$$$$36\!\cdots\!26$$$$\nu^{8} +$$$$87\!\cdots\!18$$$$\nu^{7} -$$$$43\!\cdots\!96$$$$\nu^{6} +$$$$80\!\cdots\!36$$$$\nu^{5} +$$$$28\!\cdots\!72$$$$\nu^{4} +$$$$27\!\cdots\!08$$$$\nu^{3} -$$$$59\!\cdots\!88$$$$\nu^{2} +$$$$63\!\cdots\!08$$$$\nu -$$$$31\!\cdots\!28$$$$)/$$$$69\!\cdots\!00$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$15\!\cdots\!09$$$$\nu^{19} +$$$$57\!\cdots\!62$$$$\nu^{18} -$$$$11\!\cdots\!00$$$$\nu^{17} -$$$$76\!\cdots\!36$$$$\nu^{16} -$$$$16\!\cdots\!23$$$$\nu^{15} +$$$$55\!\cdots\!28$$$$\nu^{14} -$$$$10\!\cdots\!72$$$$\nu^{13} -$$$$53\!\cdots\!94$$$$\nu^{12} -$$$$35\!\cdots\!45$$$$\nu^{11} +$$$$99\!\cdots\!88$$$$\nu^{10} -$$$$19\!\cdots\!66$$$$\nu^{9} -$$$$77\!\cdots\!88$$$$\nu^{8} -$$$$24\!\cdots\!44$$$$\nu^{7} +$$$$57\!\cdots\!88$$$$\nu^{6} -$$$$10\!\cdots\!28$$$$\nu^{5} -$$$$30\!\cdots\!16$$$$\nu^{4} -$$$$42\!\cdots\!84$$$$\nu^{3} +$$$$88\!\cdots\!44$$$$\nu^{2} -$$$$87\!\cdots\!44$$$$\nu +$$$$79\!\cdots\!84$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{18}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!51$$$$\nu^{19} +$$$$42\!\cdots\!33$$$$\nu^{18} -$$$$83\!\cdots\!00$$$$\nu^{17} -$$$$46\!\cdots\!04$$$$\nu^{16} -$$$$11\!\cdots\!37$$$$\nu^{15} +$$$$41\!\cdots\!77$$$$\nu^{14} -$$$$79\!\cdots\!58$$$$\nu^{13} -$$$$33\!\cdots\!46$$$$\nu^{12} -$$$$21\!\cdots\!05$$$$\nu^{11} +$$$$77\!\cdots\!07$$$$\nu^{10} -$$$$14\!\cdots\!04$$$$\nu^{9} -$$$$58\!\cdots\!62$$$$\nu^{8} -$$$$14\!\cdots\!76$$$$\nu^{7} +$$$$46\!\cdots\!92$$$$\nu^{6} -$$$$79\!\cdots\!92$$$$\nu^{5} -$$$$30\!\cdots\!44$$$$\nu^{4} -$$$$32\!\cdots\!76$$$$\nu^{3} +$$$$61\!\cdots\!56$$$$\nu^{2} -$$$$63\!\cdots\!36$$$$\nu +$$$$30\!\cdots\!56$$$$)/$$$$69\!\cdots\!00$$ $$\beta_{19}$$ $$=$$ $$($$$$-$$$$45\!\cdots\!13$$$$\nu^{19} +$$$$17\!\cdots\!39$$$$\nu^{18} -$$$$34\!\cdots\!00$$$$\nu^{17} -$$$$20\!\cdots\!52$$$$\nu^{16} -$$$$48\!\cdots\!91$$$$\nu^{15} +$$$$16\!\cdots\!91$$$$\nu^{14} -$$$$33\!\cdots\!54$$$$\nu^{13} -$$$$14\!\cdots\!18$$$$\nu^{12} -$$$$99\!\cdots\!15$$$$\nu^{11} +$$$$30\!\cdots\!41$$$$\nu^{10} -$$$$62\!\cdots\!72$$$$\nu^{9} -$$$$25\!\cdots\!26$$$$\nu^{8} -$$$$69\!\cdots\!28$$$$\nu^{7} +$$$$18\!\cdots\!36$$$$\nu^{6} -$$$$35\!\cdots\!96$$$$\nu^{5} -$$$$13\!\cdots\!52$$$$\nu^{4} -$$$$12\!\cdots\!88$$$$\nu^{3} +$$$$28\!\cdots\!88$$$$\nu^{2} -$$$$27\!\cdots\!48$$$$\nu +$$$$13\!\cdots\!48$$$$)/$$$$27\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{18} + 4 \beta_{10}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{19} + \beta_{17} + 2 \beta_{15} - 2 \beta_{11} + 2 \beta_{10} - \beta_{7} - \beta_{5} + 7 \beta_{4} - 2$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{18} + \beta_{17} - 2 \beta_{16} + \beta_{15} - \beta_{14} - \beta_{13} - \beta_{9} - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} + 9 \beta_{2} - \beta_{1} - 30$$ $$\nu^{5}$$ $$=$$ $$13 \beta_{19} - 15 \beta_{18} - 16 \beta_{16} - 2 \beta_{15} - 12 \beta_{14} - 27 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 29 \beta_{10} - 2 \beta_{9} - 27 \beta_{8} - 13 \beta_{7} + 2 \beta_{5} + 16 \beta_{4} - 16 \beta_{3} + \beta_{2} - 62 \beta_{1} - 29$$ $$\nu^{6}$$ $$=$$ $$30 \beta_{19} - 113 \beta_{18} - 15 \beta_{17} - 47 \beta_{16} - 12 \beta_{15} - 15 \beta_{14} - 12 \beta_{13} - 18 \beta_{12} + 20 \beta_{11} - 280 \beta_{10} + \beta_{9} - 20 \beta_{8} - \beta_{6} + 47 \beta_{5} + 9 \beta_{4} - 30 \beta_{3} - \beta_{2} - 39 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$143 \beta_{19} - 26 \beta_{18} - 124 \beta_{17} - 37 \beta_{16} - 312 \beta_{15} - 33 \beta_{13} - 12 \beta_{12} + 312 \beta_{11} - 363 \beta_{10} + 12 \beta_{9} + 33 \beta_{8} + 143 \beta_{7} + 37 \beta_{6} + 202 \beta_{5} - 605 \beta_{4} + 37 \beta_{3} - \beta_{2} - 12 \beta_{1} + 363$$ $$\nu^{8}$$ $$=$$ $$367 \beta_{18} - 179 \beta_{17} + 367 \beta_{16} - 303 \beta_{15} + 179 \beta_{14} + 303 \beta_{13} - 15 \beta_{12} + 272 \beta_{11} + 247 \beta_{9} + 272 \beta_{8} + 382 \beta_{7} + 247 \beta_{6} + 367 \beta_{5} - 938 \beta_{4} + 599 \beta_{3} - 821 \beta_{2} + 324 \beta_{1} + 2878$$ $$\nu^{9}$$ $$=$$ $$-1540 \beta_{19} + 2345 \beta_{18} + 2411 \beta_{16} + 420 \beta_{15} + 1278 \beta_{14} + 3492 \beta_{13} + 520 \beta_{12} + 420 \beta_{11} + 4428 \beta_{10} + 520 \beta_{9} + 3492 \beta_{8} + 1540 \beta_{7} + 281 \beta_{6} - 520 \beta_{5} - 2411 \beta_{4} + 2411 \beta_{3} - 454 \beta_{2} + 6206 \beta_{1} + 4428$$ $$\nu^{10}$$ $$=$$ $$-4690 \beta_{19} + 12450 \beta_{18} + 2064 \beta_{17} + 7228 \beta_{16} + 4345 \beta_{15} + 2064 \beta_{14} + 4345 \beta_{13} + 3085 \beta_{12} - 4120 \beta_{11} + 30984 \beta_{10} - 162 \beta_{9} + 4120 \beta_{8} + 162 \beta_{6} - 7228 \beta_{5} + 585 \beta_{4} + 4305 \beta_{3} + 162 \beta_{2} + 7975 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$-16755 \beta_{19} + 6769 \beta_{18} + 13508 \beta_{17} + 6658 \beta_{16} + 38934 \beta_{15} + 4941 \beta_{13} + 4522 \beta_{12} - 38934 \beta_{11} + 53553 \beta_{10} - 4522 \beta_{9} - 4941 \beta_{8} - 16755 \beta_{7} - 6658 \beta_{6} - 28328 \beta_{5} + 65694 \beta_{4} - 6658 \beta_{3} + 4633 \beta_{2} + 4522 \beta_{1} - 53553$$ $$\nu^{12}$$ $$=$$ $$-50135 \beta_{18} + 23917 \beta_{17} - 50135 \beta_{16} + 53046 \beta_{15} - 23917 \beta_{14} - 53046 \beta_{13} + 1530 \beta_{12} - 60548 \beta_{11} - 36999 \beta_{9} - 60548 \beta_{8} - 56878 \beta_{7} - 36999 \beta_{6} - 50135 \beta_{5} + 151663 \beta_{4} - 85604 \beta_{3} + 86397 \beta_{2} - 64529 \beta_{1} - 342236$$ $$\nu^{13}$$ $$=$$ $$185137 \beta_{19} - 342404 \beta_{18} - 331289 \beta_{16} - 56474 \beta_{15} - 146608 \beta_{14} - 435945 \beta_{13} - 81772 \beta_{12} - 56474 \beta_{11} - 643707 \beta_{10} - 81772 \beta_{9} - 435945 \beta_{8} - 185137 \beta_{7} - 62849 \beta_{6} + 81772 \beta_{5} + 331289 \beta_{4} - 331289 \beta_{3} + 92887 \beta_{2} - 711316 \beta_{1} - 643707$$ $$\nu^{14}$$ $$=$$ $$684808 \beta_{19} - 1490121 \beta_{18} - 279555 \beta_{17} - 1006273 \beta_{16} - 788457 \beta_{15} - 279555 \beta_{14} - 788457 \beta_{13} - 435489 \beta_{12} + 661684 \beta_{11} - 3841544 \beta_{10} + 13231 \beta_{9} - 661684 \beta_{8} - 13231 \beta_{6} + 1006273 \beta_{5} - 271248 \beta_{4} - 584015 \beta_{3} - 13231 \beta_{2} - 1290752 \beta_{1}$$ $$\nu^{15}$$ $$=$$ $$2074136 \beta_{19} - 1212487 \beta_{18} - 1625416 \beta_{17} - 983149 \beta_{16} - 4912634 \beta_{15} - 639156 \beta_{13} - 813362 \beta_{12} + 4912634 \beta_{11} - 7694530 \beta_{10} + 813362 \beta_{9} + 639156 \beta_{8} + 2074136 \beta_{7} + 983149 \beta_{6} + 3868866 \beta_{5} - 7836393 \beta_{4} + 983149 \beta_{3} - 1042700 \beta_{2} - 813362 \beta_{1} + 7694530$$ $$\nu^{16}$$ $$=$$ $$6811496 \beta_{18} - 3284842 \beta_{17} + 6811496 \beta_{16} - 8090351 \beta_{15} + 3284842 \beta_{14} + 8090351 \beta_{13} - 104727 \beta_{12} + 9884608 \beta_{11} + 5081948 \beta_{9} + 9884608 \beta_{8} + 8196408 \beta_{7} + 5081948 \beta_{6} + 6811496 \beta_{5} - 22546309 \beta_{4} + 11788717 \beta_{3} - 9971148 \beta_{2} + 10652865 \beta_{1} + 43593678$$ $$\nu^{17}$$ $$=$$ $$-23489413 \beta_{19} + 48801367 \beta_{18} + 45162738 \beta_{16} + 7223690 \beta_{15} + 18302738 \beta_{14} + 55720641 \beta_{13} + 11682660 \beta_{12} + 7223690 \beta_{11} + 91527319 \beta_{10} + 11682660 \beta_{9} + 55720641 \beta_{8} + 23489413 \beta_{7} + 10126420 \beta_{6} - 11682660 \beta_{5} - 45162738 \beta_{4} + 45162738 \beta_{3} - 15321289 \beta_{2} + 87504566 \beta_{1} + 91527319$$ $$\nu^{18}$$ $$=$$ $$-97602734 \beta_{19} + 188655987 \beta_{18} + 38674947 \beta_{17} + 137877993 \beta_{16} + 121102068 \beta_{15} + 38674947 \beta_{14} + 121102068 \beta_{13} + 59091228 \beta_{12} - 97614396 \beta_{11} + 498516416 \beta_{10} - 727705 \beta_{9} + 97614396 \beta_{8} + 727705 \beta_{6} - 137877993 \beta_{5} + 53086263 \beta_{4} + 79514470 \beta_{3} + 727705 \beta_{2} + 191691961 \beta_{1}$$ $$\nu^{19}$$ $$=$$ $$-268170457 \beta_{19} + 189375606 \beta_{18} + 208351524 \beta_{17} + 137889655 \beta_{16} + 635700440 \beta_{15} + 81842487 \beta_{13} + 123266644 \beta_{12} - 635700440 \beta_{11} + 1084260885 \beta_{10} - 123266644 \beta_{9} - 81842487 \beta_{8} - 268170457 \beta_{7} - 137889655 \beta_{6} - 527150518 \beta_{5} + 987449703 \beta_{4} - 137889655 \beta_{3} + 174752595 \beta_{2} + 123266644 \beta_{1} - 1084260885$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$\beta_{10}$$ $$-\beta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
117.1
 −2.09082 + 2.09082i −1.28900 + 1.28900i −1.23675 + 1.23675i −0.794932 + 0.794932i 0.0477388 − 0.0477388i 0.536506 − 0.536506i 1.28931 − 1.28931i 1.29397 − 1.29397i 1.82785 − 1.82785i 2.41612 − 2.41612i −2.09082 − 2.09082i −1.28900 − 1.28900i −1.23675 − 1.23675i −0.794932 − 0.794932i 0.0477388 + 0.0477388i 0.536506 + 0.536506i 1.28931 + 1.28931i 1.29397 + 1.29397i 1.82785 + 1.82785i 2.41612 + 2.41612i
−1.00000 −2.09082 2.09082i 1.00000 −1.81614 + 1.30447i 2.09082 + 2.09082i −0.643605 0.643605i −1.00000 5.74304i 1.81614 1.30447i
117.2 −1.00000 −1.28900 1.28900i 1.00000 1.63786 1.52231i 1.28900 + 1.28900i −3.01566 3.01566i −1.00000 0.323040i −1.63786 + 1.52231i
117.3 −1.00000 −1.23675 1.23675i 1.00000 −1.26849 1.84145i 1.23675 + 1.23675i 1.28708 + 1.28708i −1.00000 0.0591090i 1.26849 + 1.84145i
117.4 −1.00000 −0.794932 0.794932i 1.00000 2.22550 + 0.217179i 0.794932 + 0.794932i 2.93770 + 2.93770i −1.00000 1.73617i −2.22550 0.217179i
117.5 −1.00000 0.0477388 + 0.0477388i 1.00000 0.531109 + 2.17208i −0.0477388 0.0477388i −2.77997 2.77997i −1.00000 2.99544i −0.531109 2.17208i
117.6 −1.00000 0.536506 + 0.536506i 1.00000 −2.23241 0.127776i −0.536506 0.536506i 0.767774 + 0.767774i −1.00000 2.42432i 2.23241 + 0.127776i
117.7 −1.00000 1.28931 + 1.28931i 1.00000 1.69364 + 1.45999i −1.28931 1.28931i 0.579841 + 0.579841i −1.00000 0.324646i −1.69364 1.45999i
117.8 −1.00000 1.29397 + 1.29397i 1.00000 −2.20164 0.390851i −1.29397 1.29397i −2.67087 2.67087i −1.00000 0.348729i 2.20164 + 0.390851i
117.9 −1.00000 1.82785 + 1.82785i 1.00000 −1.23698 + 1.86276i −1.82785 1.82785i 3.41332 + 3.41332i −1.00000 3.68208i 1.23698 1.86276i
117.10 −1.00000 2.41612 + 2.41612i 1.00000 0.667560 2.13410i −2.41612 2.41612i −0.875609 0.875609i −1.00000 8.67529i −0.667560 + 2.13410i
253.1 −1.00000 −2.09082 + 2.09082i 1.00000 −1.81614 1.30447i 2.09082 2.09082i −0.643605 + 0.643605i −1.00000 5.74304i 1.81614 + 1.30447i
253.2 −1.00000 −1.28900 + 1.28900i 1.00000 1.63786 + 1.52231i 1.28900 1.28900i −3.01566 + 3.01566i −1.00000 0.323040i −1.63786 1.52231i
253.3 −1.00000 −1.23675 + 1.23675i 1.00000 −1.26849 + 1.84145i 1.23675 1.23675i 1.28708 1.28708i −1.00000 0.0591090i 1.26849 1.84145i
253.4 −1.00000 −0.794932 + 0.794932i 1.00000 2.22550 0.217179i 0.794932 0.794932i 2.93770 2.93770i −1.00000 1.73617i −2.22550 + 0.217179i
253.5 −1.00000 0.0477388 0.0477388i 1.00000 0.531109 2.17208i −0.0477388 + 0.0477388i −2.77997 + 2.77997i −1.00000 2.99544i −0.531109 + 2.17208i
253.6 −1.00000 0.536506 0.536506i 1.00000 −2.23241 + 0.127776i −0.536506 + 0.536506i 0.767774 0.767774i −1.00000 2.42432i 2.23241 0.127776i
253.7 −1.00000 1.28931 1.28931i 1.00000 1.69364 1.45999i −1.28931 + 1.28931i 0.579841 0.579841i −1.00000 0.324646i −1.69364 + 1.45999i
253.8 −1.00000 1.29397 1.29397i 1.00000 −2.20164 + 0.390851i −1.29397 + 1.29397i −2.67087 + 2.67087i −1.00000 0.348729i 2.20164 0.390851i
253.9 −1.00000 1.82785 1.82785i 1.00000 −1.23698 1.86276i −1.82785 + 1.82785i 3.41332 3.41332i −1.00000 3.68208i 1.23698 + 1.86276i
253.10 −1.00000 2.41612 2.41612i 1.00000 0.667560 + 2.13410i −2.41612 + 2.41612i −0.875609 + 0.875609i −1.00000 8.67529i −0.667560 2.13410i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 253.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.h.e yes 20
5.c odd 4 1 370.2.g.e 20
37.d odd 4 1 370.2.g.e 20
185.k even 4 1 inner 370.2.h.e yes 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.g.e 20 5.c odd 4 1
370.2.g.e 20 37.d odd 4 1
370.2.h.e yes 20 1.a even 1 1 trivial
370.2.h.e yes 20 185.k even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{20} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{20}$$
$3$ $$256 - 5632 T + 61952 T^{2} - 61952 T^{3} + 28432 T^{4} + 24512 T^{5} + 76384 T^{6} - 44248 T^{7} + 13544 T^{8} + 4848 T^{9} + 13826 T^{10} - 7362 T^{11} + 2009 T^{12} + 278 T^{13} + 760 T^{14} - 394 T^{15} + 103 T^{16} + 4 T^{17} + 8 T^{18} - 4 T^{19} + T^{20}$$
$5$ $$9765625 + 7812500 T + 1171875 T^{2} + 437500 T^{4} + 187500 T^{5} - 72500 T^{6} - 25500 T^{7} + 11875 T^{8} - 9780 T^{9} - 11342 T^{10} - 1956 T^{11} + 475 T^{12} - 204 T^{13} - 116 T^{14} + 60 T^{15} + 28 T^{16} + 3 T^{18} + 4 T^{19} + T^{20}$$
$7$ $$5382400 - 3637760 T + 1229312 T^{2} + 1962624 T^{3} + 11778624 T^{4} - 6218880 T^{5} + 1870752 T^{6} + 2399040 T^{7} + 5002176 T^{8} - 1792960 T^{9} + 442904 T^{10} + 379696 T^{11} + 169908 T^{12} - 5904 T^{13} + 2058 T^{14} + 1842 T^{15} + 813 T^{16} + 2 T^{18} + 2 T^{19} + T^{20}$$
$11$ $$2130745600 + 3626288896 T^{2} + 2486816096 T^{4} + 916014116 T^{6} + 202392569 T^{8} + 28283876 T^{10} + 2558072 T^{12} + 149330 T^{14} + 5432 T^{16} + 112 T^{18} + T^{20}$$
$13$ $$( -32672 - 43872 T + 49696 T^{2} + 23612 T^{3} - 19622 T^{4} - 1822 T^{5} + 2095 T^{6} + 32 T^{7} - 83 T^{8} + T^{10} )^{2}$$
$17$ $$6206918656 + 60758539264 T^{2} + 49655772416 T^{4} + 17101066688 T^{6} + 3143031632 T^{8} + 335841008 T^{10} + 21545900 T^{12} + 828332 T^{14} + 18389 T^{16} + 214 T^{18} + T^{20}$$
$19$ $$36191257600 + 65454735360 T + 59190018048 T^{2} - 67804944896 T^{3} + 45401601536 T^{4} + 6667364352 T^{5} + 1321983104 T^{6} - 1095675520 T^{7} + 741580096 T^{8} + 122763008 T^{9} + 17689984 T^{10} - 7342976 T^{11} + 2563408 T^{12} + 308128 T^{13} + 43616 T^{14} - 15144 T^{15} + 3080 T^{16} + 128 T^{17} + 18 T^{18} - 6 T^{19} + T^{20}$$
$23$ $$( -23040 - 123264 T + 91008 T^{2} + 67248 T^{3} - 33282 T^{4} - 7932 T^{5} + 4119 T^{6} + 30 T^{7} - 115 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$29$ $$651474576 - 3125362752 T + 7496756352 T^{2} - 6999318672 T^{3} + 3804990184 T^{4} - 1518057144 T^{5} + 1096935912 T^{6} - 842633364 T^{7} + 438476161 T^{8} - 139267218 T^{9} + 37437282 T^{10} - 13699140 T^{11} + 5897580 T^{12} - 1776426 T^{13} + 347292 T^{14} - 43146 T^{15} + 4828 T^{16} - 840 T^{17} + 162 T^{18} - 18 T^{19} + T^{20}$$
$31$ $$11580342544 + 72564923840 T + 227353731200 T^{2} - 131903946664 T^{3} + 50910065528 T^{4} + 43890601504 T^{5} + 26739310482 T^{6} - 3487136522 T^{7} + 313303225 T^{8} + 148228426 T^{9} + 119599090 T^{10} - 16084752 T^{11} + 1090176 T^{12} + 118154 T^{13} + 168056 T^{14} - 25370 T^{15} + 1896 T^{16} + 16 T^{17} + 72 T^{18} - 12 T^{19} + T^{20}$$
$37$ $$4808584372417849 + 4158775673442464 T + 1728139891329132 T^{2} + 407067889146304 T^{3} + 40510254271701 T^{4} - 7600097687200 T^{5} - 3826047204992 T^{6} - 679828095840 T^{7} - 28597366822 T^{8} + 15076294688 T^{9} + 4017596840 T^{10} + 407467424 T^{11} - 20889238 T^{12} - 13421280 T^{13} - 2041472 T^{14} - 109600 T^{15} + 15789 T^{16} + 4288 T^{17} + 492 T^{18} + 32 T^{19} + T^{20}$$
$41$ $$3430496665600 + 48342471806976 T^{2} + 21964817089024 T^{4} + 3885073610368 T^{6} + 343867731145 T^{8} + 16995731836 T^{10} + 495661480 T^{12} + 8645170 T^{14} + 87568 T^{16} + 468 T^{18} + T^{20}$$
$43$ $$( -1401856 + 4148736 T + 1991488 T^{2} - 1119712 T^{3} - 612788 T^{4} - 21916 T^{5} + 18706 T^{6} + 1070 T^{7} - 221 T^{8} - 8 T^{9} + T^{10} )^{2}$$
$47$ $$7191040000 + 57419776000 T + 229245747200 T^{2} + 186603550720 T^{3} + 77400617216 T^{4} + 9419001856 T^{5} + 28857747968 T^{6} + 19000936576 T^{7} + 6260258560 T^{8} + 1454404992 T^{9} + 662975520 T^{10} + 334318368 T^{11} + 113496928 T^{12} + 24622528 T^{13} + 3516464 T^{14} + 326008 T^{15} + 21920 T^{16} + 1816 T^{17} + 242 T^{18} + 22 T^{19} + T^{20}$$
$53$ $$12096484000000 - 19848528640000 T + 16284239667200 T^{2} + 12964286049280 T^{3} + 6414155068736 T^{4} - 1258818079360 T^{5} + 377994521600 T^{6} + 188794245760 T^{7} + 54636606400 T^{8} - 4526554848 T^{9} + 1282064160 T^{10} + 573426096 T^{11} + 127994116 T^{12} - 523120 T^{13} + 347432 T^{14} + 163588 T^{15} + 38429 T^{16} + 32 T^{17} + 8 T^{18} + 4 T^{19} + T^{20}$$
$59$ $$73279477516926976 + 52075026826067968 T + 18503191553933312 T^{2} + 3422262471553024 T^{3} + 429003084906752 T^{4} + 74301449654272 T^{5} + 24389692712960 T^{6} + 4613942384512 T^{7} + 491831852032 T^{8} + 33311803008 T^{9} + 9845529120 T^{10} + 1922565792 T^{11} + 196368928 T^{12} + 4715680 T^{13} + 1250000 T^{14} + 252760 T^{15} + 25808 T^{16} + 160 T^{17} + 50 T^{18} + 10 T^{19} + T^{20}$$
$61$ $$778824091056400 - 2021050221434720 T + 2622314874736928 T^{2} - 2098627967635680 T^{3} + 1128767967221640 T^{4} - 414923232487296 T^{5} + 103637438188176 T^{6} - 16797440912472 T^{7} + 1779521139681 T^{8} - 169528141762 T^{9} + 33079889282 T^{10} - 5571391472 T^{11} + 526235316 T^{12} - 17779734 T^{13} + 2440452 T^{14} - 447162 T^{15} + 43044 T^{16} - 492 T^{17} + 50 T^{18} - 10 T^{19} + T^{20}$$
$67$ $$1304646887050816 + 1928304858712704 T + 1425044456489088 T^{2} + 569530482721184 T^{3} + 137560978369968 T^{4} + 20762293721984 T^{5} + 4743051658280 T^{6} + 1611043124560 T^{7} + 398356771740 T^{8} + 46926073416 T^{9} + 2627092642 T^{10} + 202782590 T^{11} + 113485981 T^{12} + 13316342 T^{13} + 503568 T^{14} - 55246 T^{15} + 14459 T^{16} + 1024 T^{17} + 32 T^{18} - 8 T^{19} + T^{20}$$
$71$ $$( -147456 + 97210368 T + 37988352 T^{2} - 2889216 T^{3} - 2312640 T^{4} - 49920 T^{5} + 44928 T^{6} + 1584 T^{7} - 352 T^{8} - 8 T^{9} + T^{10} )^{2}$$
$73$ $$1814896322560000 - 12402791255040000 T + 42379619431680000 T^{2} + 26408325131603200 T^{3} + 8153253450569744 T^{4} + 253736919037728 T^{5} + 11567450036768 T^{6} + 13164054589952 T^{7} + 6729134905432 T^{8} + 27674833040 T^{9} - 1062208016 T^{10} + 1292061088 T^{11} + 1391120209 T^{12} - 27815102 T^{13} + 452930 T^{14} + 195516 T^{15} + 67787 T^{16} - 1000 T^{17} + 18 T^{18} + 6 T^{19} + T^{20}$$
$79$ $$573388694327296 - 523894612365312 T + 239336393951232 T^{2} + 143130501044992 T^{3} + 41577742434320 T^{4} - 3392313569280 T^{5} + 3608927146112 T^{6} + 2154640041656 T^{7} + 572574232648 T^{8} + 46690582256 T^{9} + 4532291074 T^{10} + 1574787502 T^{11} + 459687817 T^{12} + 43029046 T^{13} + 1506920 T^{14} - 94338 T^{15} + 24263 T^{16} + 2060 T^{17} + 72 T^{18} - 12 T^{19} + T^{20}$$
$83$ $$48004038771097600 - 106812894978293760 T + 118833694265253888 T^{2} - 44384154333972480 T^{3} + 8596192490819840 T^{4} - 369938345490432 T^{5} + 61941732258816 T^{6} - 32016690381696 T^{7} + 8042041402624 T^{8} + 8200850304 T^{9} + 9507894048 T^{10} - 4859243232 T^{11} + 1214228512 T^{12} + 9167328 T^{13} + 702288 T^{14} - 290040 T^{15} + 61760 T^{16} + 384 T^{17} + 18 T^{18} - 6 T^{19} + T^{20}$$
$89$ $$236610798273433600 - 294580485099397120 T + 183376377651863552 T^{2} - 49493748713500672 T^{3} + 5766473706772736 T^{4} + 534240185325056 T^{5} + 42279506283008 T^{6} - 22268933228800 T^{7} + 4298650082560 T^{8} + 688543913088 T^{9} + 42149940096 T^{10} - 1251139776 T^{11} + 1043000032 T^{12} + 195836960 T^{13} + 16784864 T^{14} + 640592 T^{15} + 67904 T^{16} + 10424 T^{17} + 968 T^{18} + 44 T^{19} + T^{20}$$
$97$ $$33875251766901080064 + 5608633654203383808 T^{2} + 386682360414068736 T^{4} + 14460789374689536 T^{6} + 321741822503952 T^{8} + 4431604791168 T^{10} + 38477117916 T^{12} + 210023016 T^{14} + 697021 T^{16} + 1282 T^{18} + T^{20}$$