# Properties

 Label 170.2.n.a Level $170$ Weight $2$ Character orbit 170.n Analytic conductor $1.357$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$170 = 2 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 170.n (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.35745683436$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$5$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 16 x^{15} + 52 x^{14} + 992 x^{13} + 6181 x^{12} + 8952 x^{11} + 6244 x^{10} - 11448 x^{9} - 14520 x^{8} + 27936 x^{7} + 27880 x^{6} - 121104 x^{5} + 187460 x^{4} - 142208 x^{3} + 73856 x^{2} - 19456 x + 2048$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 16 x^{15} + 52 x^{14} + 992 x^{13} + 6181 x^{12} + 8952 x^{11} + 6244 x^{10} - 11448 x^{9} - 14520 x^{8} + 27936 x^{7} + 27880 x^{6} - 121104 x^{5} + 187460 x^{4} - 142208 x^{3} + 73856 x^{2} - 19456 x + 2048$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$24\!\cdots\!67$$$$\nu^{19} +$$$$64\!\cdots\!72$$$$\nu^{18} +$$$$11\!\cdots\!32$$$$\nu^{17} +$$$$10\!\cdots\!52$$$$\nu^{16} +$$$$12\!\cdots\!32$$$$\nu^{15} -$$$$35\!\cdots\!00$$$$\nu^{14} +$$$$25\!\cdots\!04$$$$\nu^{13} +$$$$25\!\cdots\!68$$$$\nu^{12} +$$$$22\!\cdots\!75$$$$\nu^{11} +$$$$73\!\cdots\!24$$$$\nu^{10} +$$$$15\!\cdots\!92$$$$\nu^{9} +$$$$19\!\cdots\!76$$$$\nu^{8} +$$$$14\!\cdots\!76$$$$\nu^{7} +$$$$45\!\cdots\!88$$$$\nu^{6} +$$$$72\!\cdots\!88$$$$\nu^{5} -$$$$61\!\cdots\!20$$$$\nu^{4} +$$$$73\!\cdots\!40$$$$\nu^{3} -$$$$44\!\cdots\!36$$$$\nu^{2} +$$$$50\!\cdots\!36$$$$\nu -$$$$24\!\cdots\!36$$$$)/$$$$51\!\cdots\!40$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$47\!\cdots\!67$$$$\nu^{19} -$$$$23\!\cdots\!52$$$$\nu^{18} -$$$$34\!\cdots\!72$$$$\nu^{17} -$$$$39\!\cdots\!12$$$$\nu^{16} -$$$$14\!\cdots\!92$$$$\nu^{15} +$$$$82\!\cdots\!80$$$$\nu^{14} +$$$$14\!\cdots\!76$$$$\nu^{13} -$$$$53\!\cdots\!08$$$$\nu^{12} -$$$$53\!\cdots\!75$$$$\nu^{11} -$$$$22\!\cdots\!44$$$$\nu^{10} -$$$$48\!\cdots\!72$$$$\nu^{9} -$$$$65\!\cdots\!76$$$$\nu^{8} -$$$$31\!\cdots\!76$$$$\nu^{7} +$$$$28\!\cdots\!72$$$$\nu^{6} +$$$$25\!\cdots\!72$$$$\nu^{5} -$$$$10\!\cdots\!20$$$$\nu^{4} -$$$$30\!\cdots\!40$$$$\nu^{3} -$$$$13\!\cdots\!44$$$$\nu^{2} +$$$$63\!\cdots\!84$$$$\nu -$$$$14\!\cdots\!04$$$$)/$$$$27\!\cdots\!60$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$42\!\cdots\!91$$$$\nu^{19} -$$$$28\!\cdots\!16$$$$\nu^{18} -$$$$10\!\cdots\!56$$$$\nu^{17} +$$$$34\!\cdots\!24$$$$\nu^{16} +$$$$36\!\cdots\!24$$$$\nu^{15} +$$$$64\!\cdots\!80$$$$\nu^{14} +$$$$22\!\cdots\!88$$$$\nu^{13} -$$$$55\!\cdots\!64$$$$\nu^{12} -$$$$54\!\cdots\!55$$$$\nu^{11} -$$$$22\!\cdots\!32$$$$\nu^{10} -$$$$33\!\cdots\!16$$$$\nu^{9} -$$$$19\!\cdots\!88$$$$\nu^{8} +$$$$37\!\cdots\!72$$$$\nu^{7} +$$$$44\!\cdots\!16$$$$\nu^{6} -$$$$84\!\cdots\!44$$$$\nu^{5} -$$$$76\!\cdots\!80$$$$\nu^{4} +$$$$23\!\cdots\!60$$$$\nu^{3} -$$$$31\!\cdots\!72$$$$\nu^{2} +$$$$16\!\cdots\!92$$$$\nu -$$$$49\!\cdots\!32$$$$)/$$$$10\!\cdots\!80$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$15\!\cdots\!03$$$$\nu^{19} +$$$$15\!\cdots\!36$$$$\nu^{18} +$$$$17\!\cdots\!92$$$$\nu^{17} -$$$$19\!\cdots\!44$$$$\nu^{16} -$$$$18\!\cdots\!72$$$$\nu^{15} +$$$$24\!\cdots\!36$$$$\nu^{14} -$$$$82\!\cdots\!88$$$$\nu^{13} -$$$$15\!\cdots\!76$$$$\nu^{12} -$$$$92\!\cdots\!75$$$$\nu^{11} -$$$$12\!\cdots\!08$$$$\nu^{10} -$$$$70\!\cdots\!80$$$$\nu^{9} +$$$$19\!\cdots\!68$$$$\nu^{8} +$$$$19\!\cdots\!44$$$$\nu^{7} -$$$$51\!\cdots\!24$$$$\nu^{6} -$$$$48\!\cdots\!92$$$$\nu^{5} +$$$$18\!\cdots\!48$$$$\nu^{4} -$$$$29\!\cdots\!60$$$$\nu^{3} +$$$$22\!\cdots\!12$$$$\nu^{2} -$$$$10\!\cdots\!40$$$$\nu +$$$$17\!\cdots\!44$$$$)/$$$$15\!\cdots\!76$$ $$\beta_{6}$$ $$=$$ $$($$$$38\!\cdots\!43$$$$\nu^{19} +$$$$12\!\cdots\!98$$$$\nu^{18} +$$$$10\!\cdots\!08$$$$\nu^{17} -$$$$16\!\cdots\!52$$$$\nu^{16} -$$$$13\!\cdots\!32$$$$\nu^{15} -$$$$62\!\cdots\!80$$$$\nu^{14} +$$$$18\!\cdots\!76$$$$\nu^{13} +$$$$39\!\cdots\!92$$$$\nu^{12} +$$$$25\!\cdots\!15$$$$\nu^{11} +$$$$42\!\cdots\!26$$$$\nu^{10} +$$$$36\!\cdots\!88$$$$\nu^{9} -$$$$36\!\cdots\!36$$$$\nu^{8} -$$$$72\!\cdots\!56$$$$\nu^{7} +$$$$86\!\cdots\!32$$$$\nu^{6} +$$$$14\!\cdots\!72$$$$\nu^{5} -$$$$42\!\cdots\!40$$$$\nu^{4} +$$$$59\!\cdots\!20$$$$\nu^{3} -$$$$32\!\cdots\!24$$$$\nu^{2} +$$$$14\!\cdots\!64$$$$\nu -$$$$19\!\cdots\!44$$$$)/$$$$25\!\cdots\!20$$ $$\beta_{7}$$ $$=$$ $$($$$$11\!\cdots\!32$$$$\nu^{19} +$$$$24\!\cdots\!67$$$$\nu^{18} +$$$$64\!\cdots\!72$$$$\nu^{17} +$$$$11\!\cdots\!32$$$$\nu^{16} +$$$$10\!\cdots\!52$$$$\nu^{15} -$$$$18\!\cdots\!80$$$$\nu^{14} +$$$$60\!\cdots\!64$$$$\nu^{13} +$$$$11\!\cdots\!48$$$$\nu^{12} +$$$$73\!\cdots\!60$$$$\nu^{11} +$$$$10\!\cdots\!39$$$$\nu^{10} +$$$$81\!\cdots\!32$$$$\nu^{9} -$$$$11\!\cdots\!44$$$$\nu^{8} -$$$$15\!\cdots\!64$$$$\nu^{7} +$$$$34\!\cdots\!28$$$$\nu^{6} +$$$$33\!\cdots\!48$$$$\nu^{5} -$$$$14\!\cdots\!40$$$$\nu^{4} +$$$$22\!\cdots\!00$$$$\nu^{3} -$$$$16\!\cdots\!16$$$$\nu^{2} +$$$$87\!\cdots\!56$$$$\nu -$$$$17\!\cdots\!56$$$$)/$$$$51\!\cdots\!40$$ $$\beta_{8}$$ $$=$$ $$($$$$74\!\cdots\!99$$$$\nu^{19} +$$$$31\!\cdots\!44$$$$\nu^{18} +$$$$10\!\cdots\!84$$$$\nu^{17} +$$$$86\!\cdots\!64$$$$\nu^{16} -$$$$12\!\cdots\!16$$$$\nu^{15} -$$$$11\!\cdots\!40$$$$\nu^{14} +$$$$33\!\cdots\!08$$$$\nu^{13} +$$$$75\!\cdots\!16$$$$\nu^{12} +$$$$49\!\cdots\!55$$$$\nu^{11} +$$$$87\!\cdots\!68$$$$\nu^{10} +$$$$81\!\cdots\!64$$$$\nu^{9} -$$$$56\!\cdots\!48$$$$\nu^{8} -$$$$13\!\cdots\!68$$$$\nu^{7} +$$$$15\!\cdots\!16$$$$\nu^{6} +$$$$27\!\cdots\!76$$$$\nu^{5} -$$$$79\!\cdots\!20$$$$\nu^{4} +$$$$10\!\cdots\!20$$$$\nu^{3} -$$$$59\!\cdots\!32$$$$\nu^{2} +$$$$29\!\cdots\!52$$$$\nu -$$$$33\!\cdots\!32$$$$)/$$$$20\!\cdots\!60$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!25$$$$\nu^{19} -$$$$70\!\cdots\!88$$$$\nu^{18} -$$$$14\!\cdots\!52$$$$\nu^{17} +$$$$33\!\cdots\!72$$$$\nu^{16} -$$$$61\!\cdots\!28$$$$\nu^{15} +$$$$17\!\cdots\!60$$$$\nu^{14} -$$$$47\!\cdots\!36$$$$\nu^{13} -$$$$11\!\cdots\!00$$$$\nu^{12} -$$$$76\!\cdots\!77$$$$\nu^{11} -$$$$14\!\cdots\!84$$$$\nu^{10} -$$$$14\!\cdots\!04$$$$\nu^{9} +$$$$73\!\cdots\!68$$$$\nu^{8} +$$$$23\!\cdots\!68$$$$\nu^{7} -$$$$20\!\cdots\!20$$$$\nu^{6} -$$$$50\!\cdots\!28$$$$\nu^{5} +$$$$11\!\cdots\!56$$$$\nu^{4} -$$$$12\!\cdots\!96$$$$\nu^{3} +$$$$48\!\cdots\!84$$$$\nu^{2} -$$$$17\!\cdots\!00$$$$\nu -$$$$17\!\cdots\!00$$$$)/$$$$20\!\cdots\!16$$ $$\beta_{10}$$ $$=$$ $$($$$$17\!\cdots\!31$$$$\nu^{19} +$$$$30\!\cdots\!06$$$$\nu^{18} -$$$$31\!\cdots\!72$$$$\nu^{17} -$$$$35\!\cdots\!84$$$$\nu^{16} +$$$$38\!\cdots\!88$$$$\nu^{15} -$$$$27\!\cdots\!52$$$$\nu^{14} +$$$$84\!\cdots\!40$$$$\nu^{13} +$$$$17\!\cdots\!28$$$$\nu^{12} +$$$$10\!\cdots\!63$$$$\nu^{11} +$$$$17\!\cdots\!62$$$$\nu^{10} +$$$$13\!\cdots\!80$$$$\nu^{9} -$$$$18\!\cdots\!28$$$$\nu^{8} -$$$$28\!\cdots\!56$$$$\nu^{7} +$$$$44\!\cdots\!28$$$$\nu^{6} +$$$$58\!\cdots\!28$$$$\nu^{5} -$$$$19\!\cdots\!40$$$$\nu^{4} +$$$$28\!\cdots\!64$$$$\nu^{3} -$$$$18\!\cdots\!28$$$$\nu^{2} +$$$$81\!\cdots\!12$$$$\nu -$$$$12\!\cdots\!56$$$$)/$$$$30\!\cdots\!52$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$48\!\cdots\!23$$$$\nu^{19} -$$$$24\!\cdots\!14$$$$\nu^{18} +$$$$87\!\cdots\!88$$$$\nu^{17} +$$$$54\!\cdots\!72$$$$\nu^{16} +$$$$27\!\cdots\!68$$$$\nu^{15} +$$$$77\!\cdots\!64$$$$\nu^{14} -$$$$24\!\cdots\!20$$$$\nu^{13} -$$$$48\!\cdots\!72$$$$\nu^{12} -$$$$30\!\cdots\!63$$$$\nu^{11} -$$$$44\!\cdots\!90$$$$\nu^{10} -$$$$26\!\cdots\!48$$$$\nu^{9} +$$$$65\!\cdots\!04$$$$\nu^{8} +$$$$83\!\cdots\!56$$$$\nu^{7} -$$$$13\!\cdots\!80$$$$\nu^{6} -$$$$15\!\cdots\!24$$$$\nu^{5} +$$$$59\!\cdots\!16$$$$\nu^{4} -$$$$83\!\cdots\!68$$$$\nu^{3} +$$$$54\!\cdots\!00$$$$\nu^{2} -$$$$23\!\cdots\!80$$$$\nu +$$$$48\!\cdots\!24$$$$)/$$$$68\!\cdots\!72$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!29$$$$\nu^{19} -$$$$15\!\cdots\!04$$$$\nu^{18} +$$$$19\!\cdots\!12$$$$\nu^{17} -$$$$73\!\cdots\!52$$$$\nu^{16} -$$$$12\!\cdots\!52$$$$\nu^{15} +$$$$16\!\cdots\!32$$$$\nu^{14} -$$$$50\!\cdots\!76$$$$\nu^{13} -$$$$10\!\cdots\!68$$$$\nu^{12} -$$$$64\!\cdots\!97$$$$\nu^{11} -$$$$10\!\cdots\!84$$$$\nu^{10} -$$$$77\!\cdots\!28$$$$\nu^{9} +$$$$10\!\cdots\!84$$$$\nu^{8} +$$$$15\!\cdots\!24$$$$\nu^{7} -$$$$28\!\cdots\!20$$$$\nu^{6} -$$$$33\!\cdots\!80$$$$\nu^{5} +$$$$12\!\cdots\!44$$$$\nu^{4} -$$$$16\!\cdots\!72$$$$\nu^{3} +$$$$10\!\cdots\!16$$$$\nu^{2} -$$$$46\!\cdots\!60$$$$\nu +$$$$44\!\cdots\!56$$$$)/$$$$13\!\cdots\!44$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$96\!\cdots\!58$$$$\nu^{19} -$$$$29\!\cdots\!53$$$$\nu^{18} -$$$$11\!\cdots\!16$$$$\nu^{17} +$$$$27\!\cdots\!04$$$$\nu^{16} +$$$$71\!\cdots\!44$$$$\nu^{15} +$$$$15\!\cdots\!68$$$$\nu^{14} -$$$$45\!\cdots\!32$$$$\nu^{13} -$$$$96\!\cdots\!72$$$$\nu^{12} -$$$$62\!\cdots\!46$$$$\nu^{11} -$$$$10\!\cdots\!05$$$$\nu^{10} -$$$$93\!\cdots\!16$$$$\nu^{9} +$$$$84\!\cdots\!96$$$$\nu^{8} +$$$$17\!\cdots\!56$$$$\nu^{7} -$$$$20\!\cdots\!96$$$$\nu^{6} -$$$$33\!\cdots\!48$$$$\nu^{5} +$$$$10\!\cdots\!24$$$$\nu^{4} -$$$$14\!\cdots\!16$$$$\nu^{3} +$$$$97\!\cdots\!32$$$$\nu^{2} -$$$$49\!\cdots\!52$$$$\nu +$$$$87\!\cdots\!56$$$$)/$$$$10\!\cdots\!08$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$48\!\cdots\!91$$$$\nu^{19} -$$$$18\!\cdots\!21$$$$\nu^{18} -$$$$11\!\cdots\!96$$$$\nu^{17} -$$$$48\!\cdots\!76$$$$\nu^{16} +$$$$11\!\cdots\!44$$$$\nu^{15} +$$$$77\!\cdots\!80$$$$\nu^{14} -$$$$22\!\cdots\!12$$$$\nu^{13} -$$$$48\!\cdots\!64$$$$\nu^{12} -$$$$31\!\cdots\!15$$$$\nu^{11} -$$$$55\!\cdots\!17$$$$\nu^{10} -$$$$53\!\cdots\!56$$$$\nu^{9} +$$$$31\!\cdots\!52$$$$\nu^{8} +$$$$80\!\cdots\!12$$$$\nu^{7} -$$$$95\!\cdots\!84$$$$\nu^{6} -$$$$15\!\cdots\!24$$$$\nu^{5} +$$$$51\!\cdots\!60$$$$\nu^{4} -$$$$73\!\cdots\!40$$$$\nu^{3} +$$$$46\!\cdots\!08$$$$\nu^{2} -$$$$23\!\cdots\!48$$$$\nu +$$$$37\!\cdots\!48$$$$)/$$$$51\!\cdots\!40$$ $$\beta_{15}$$ $$=$$ $$($$$$42\!\cdots\!55$$$$\nu^{19} +$$$$73\!\cdots\!04$$$$\nu^{18} +$$$$18\!\cdots\!24$$$$\nu^{17} +$$$$26\!\cdots\!36$$$$\nu^{16} +$$$$90\!\cdots\!60$$$$\nu^{15} -$$$$68\!\cdots\!84$$$$\nu^{14} +$$$$21\!\cdots\!84$$$$\nu^{13} +$$$$42\!\cdots\!36$$$$\nu^{12} +$$$$27\!\cdots\!23$$$$\nu^{11} +$$$$43\!\cdots\!44$$$$\nu^{10} +$$$$34\!\cdots\!48$$$$\nu^{9} -$$$$41\!\cdots\!64$$$$\nu^{8} -$$$$66\!\cdots\!80$$$$\nu^{7} +$$$$10\!\cdots\!28$$$$\nu^{6} +$$$$13\!\cdots\!80$$$$\nu^{5} -$$$$50\!\cdots\!12$$$$\nu^{4} +$$$$72\!\cdots\!08$$$$\nu^{3} -$$$$47\!\cdots\!20$$$$\nu^{2} +$$$$21\!\cdots\!84$$$$\nu -$$$$27\!\cdots\!84$$$$)/$$$$41\!\cdots\!32$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$24\!\cdots\!51$$$$\nu^{19} -$$$$52\!\cdots\!70$$$$\nu^{18} +$$$$12\!\cdots\!84$$$$\nu^{17} +$$$$23\!\cdots\!72$$$$\nu^{16} +$$$$13\!\cdots\!44$$$$\nu^{15} +$$$$39\!\cdots\!04$$$$\nu^{14} -$$$$11\!\cdots\!44$$$$\nu^{13} -$$$$24\!\cdots\!24$$$$\nu^{12} -$$$$15\!\cdots\!31$$$$\nu^{11} -$$$$25\!\cdots\!98$$$$\nu^{10} -$$$$18\!\cdots\!40$$$$\nu^{9} +$$$$27\!\cdots\!40$$$$\nu^{8} +$$$$45\!\cdots\!56$$$$\nu^{7} -$$$$59\!\cdots\!88$$$$\nu^{6} -$$$$86\!\cdots\!92$$$$\nu^{5} +$$$$27\!\cdots\!84$$$$\nu^{4} -$$$$38\!\cdots\!36$$$$\nu^{3} +$$$$24\!\cdots\!08$$$$\nu^{2} -$$$$10\!\cdots\!76$$$$\nu +$$$$18\!\cdots\!16$$$$)/$$$$20\!\cdots\!16$$ $$\beta_{17}$$ $$=$$ $$($$$$10\!\cdots\!13$$$$\nu^{19} +$$$$82\!\cdots\!48$$$$\nu^{18} +$$$$93\!\cdots\!38$$$$\nu^{17} +$$$$74\!\cdots\!08$$$$\nu^{16} -$$$$55\!\cdots\!32$$$$\nu^{15} -$$$$16\!\cdots\!60$$$$\nu^{14} +$$$$53\!\cdots\!96$$$$\nu^{13} +$$$$10\!\cdots\!92$$$$\nu^{12} +$$$$65\!\cdots\!25$$$$\nu^{11} +$$$$10\!\cdots\!96$$$$\nu^{10} +$$$$79\!\cdots\!78$$$$\nu^{9} -$$$$10\!\cdots\!76$$$$\nu^{8} -$$$$15\!\cdots\!76$$$$\nu^{7} +$$$$27\!\cdots\!32$$$$\nu^{6} +$$$$28\!\cdots\!72$$$$\nu^{5} -$$$$12\!\cdots\!60$$$$\nu^{4} +$$$$19\!\cdots\!60$$$$\nu^{3} -$$$$14\!\cdots\!64$$$$\nu^{2} +$$$$72\!\cdots\!24$$$$\nu -$$$$14\!\cdots\!64$$$$)/$$$$85\!\cdots\!40$$ $$\beta_{18}$$ $$=$$ $$($$$$-$$$$49\!\cdots\!77$$$$\nu^{19} -$$$$21\!\cdots\!32$$$$\nu^{18} -$$$$68\!\cdots\!42$$$$\nu^{17} +$$$$48\!\cdots\!88$$$$\nu^{16} +$$$$21\!\cdots\!08$$$$\nu^{15} +$$$$79\!\cdots\!60$$$$\nu^{14} -$$$$22\!\cdots\!84$$$$\nu^{13} -$$$$50\!\cdots\!68$$$$\nu^{12} -$$$$32\!\cdots\!05$$$$\nu^{11} -$$$$58\!\cdots\!44$$$$\nu^{10} -$$$$54\!\cdots\!02$$$$\nu^{9} +$$$$38\!\cdots\!84$$$$\nu^{8} +$$$$94\!\cdots\!44$$$$\nu^{7} -$$$$98\!\cdots\!68$$$$\nu^{6} -$$$$18\!\cdots\!88$$$$\nu^{5} +$$$$52\!\cdots\!80$$$$\nu^{4} -$$$$69\!\cdots\!00$$$$\nu^{3} +$$$$40\!\cdots\!96$$$$\nu^{2} -$$$$19\!\cdots\!96$$$$\nu +$$$$22\!\cdots\!96$$$$)/$$$$25\!\cdots\!20$$ $$\beta_{19}$$ $$=$$ $$($$$$-$$$$13\!\cdots\!01$$$$\nu^{19} -$$$$23\!\cdots\!76$$$$\nu^{18} +$$$$33\!\cdots\!04$$$$\nu^{17} +$$$$12\!\cdots\!24$$$$\nu^{16} -$$$$37\!\cdots\!56$$$$\nu^{15} +$$$$22\!\cdots\!00$$$$\nu^{14} -$$$$68\!\cdots\!12$$$$\nu^{13} -$$$$13\!\cdots\!44$$$$\nu^{12} -$$$$88\!\cdots\!25$$$$\nu^{11} -$$$$13\!\cdots\!12$$$$\nu^{10} -$$$$10\!\cdots\!96$$$$\nu^{9} +$$$$14\!\cdots\!12$$$$\nu^{8} +$$$$23\!\cdots\!32$$$$\nu^{7} -$$$$36\!\cdots\!84$$$$\nu^{6} -$$$$46\!\cdots\!24$$$$\nu^{5} +$$$$16\!\cdots\!60$$$$\nu^{4} -$$$$23\!\cdots\!00$$$$\nu^{3} +$$$$15\!\cdots\!48$$$$\nu^{2} -$$$$66\!\cdots\!28$$$$\nu +$$$$10\!\cdots\!68$$$$)/$$$$51\!\cdots\!40$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{18} - \beta_{16} + \beta_{11} + 4 \beta_{8}$$ $$\nu^{3}$$ $$=$$ $$\beta_{19} + \beta_{18} - \beta_{16} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 7 \beta_{6} + \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{19} - \beta_{16} + \beta_{15} + 28 \beta_{10} - 10 \beta_{9} + \beta_{8} + \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 10 \beta_{4} - \beta_{2} - \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$13 \beta_{19} - 13 \beta_{17} + 13 \beta_{15} - 11 \beta_{14} + 13 \beta_{12} - 11 \beta_{11} + 23 \beta_{10} - 13 \beta_{9} - 4 \beta_{8} + 23 \beta_{7} + \beta_{6} - 57 \beta_{5} + 13 \beta_{4} + \beta_{2} + 4$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{19} - 83 \beta_{17} - 2 \beta_{16} + 94 \beta_{15} + \beta_{14} - 14 \beta_{13} + 94 \beta_{12} - 2 \beta_{11} + 17 \beta_{10} - \beta_{9} + 232 \beta_{7} - 15 \beta_{6} - 51 \beta_{5} + 14 \beta_{4} - 3 \beta_{3} + 51 \beta_{2} + 15 \beta_{1} - 17$$ $$\nu^{7}$$ $$=$$ $$-3 \beta_{19} + 3 \beta_{18} - 146 \beta_{17} - 104 \beta_{16} + 147 \beta_{15} + 147 \beta_{14} - 150 \beta_{13} + 150 \beta_{12} - 57 \beta_{10} + 104 \beta_{9} + 57 \beta_{8} + 308 \beta_{7} - 22 \beta_{5} - 146 \beta_{3} + 499 \beta_{2} - 22 \beta_{1} - 308$$ $$\nu^{8}$$ $$=$$ $$-68 \beta_{18} - 68 \beta_{17} + 21 \beta_{16} + 21 \beta_{15} + 899 \beta_{14} - 899 \beta_{13} + 166 \beta_{12} - 166 \beta_{11} + 49 \beta_{9} - 233 \beta_{8} + 233 \beta_{7} + 167 \beta_{6} + 167 \beta_{5} + 49 \beta_{4} - 793 \beta_{3} + 647 \beta_{2} - 647 \beta_{1} - 2094$$ $$\nu^{9}$$ $$=$$ $$50 \beta_{19} - 1567 \beta_{18} + 50 \beta_{17} + 1585 \beta_{16} - 949 \beta_{15} + 1663 \beta_{14} - 1585 \beta_{13} + 20 \beta_{12} - 1663 \beta_{11} + 590 \beta_{10} + 20 \beta_{9} - 3557 \beta_{8} - 590 \beta_{7} - 343 \beta_{6} + 949 \beta_{4} - 1567 \beta_{3} + 343 \beta_{2} - 4587 \beta_{1} - 3557$$ $$\nu^{10}$$ $$=$$ $$-1057 \beta_{19} - 7757 \beta_{18} + 8766 \beta_{16} - 764 \beta_{15} + 1898 \beta_{14} - 345 \beta_{13} + 764 \beta_{12} - 8766 \beta_{11} - 2925 \beta_{10} + 1898 \beta_{9} - 19876 \beta_{8} - 7403 \beta_{6} - 1647 \beta_{5} - 345 \beta_{4} - 1057 \beta_{3} - 1647 \beta_{2} - 7403 \beta_{1} - 2925$$ $$\nu^{11}$$ $$=$$ $$-16514 \beta_{19} - 16514 \beta_{18} - 513 \beta_{17} + 17990 \beta_{16} - 596 \beta_{15} + 596 \beta_{14} + 8592 \beta_{13} + 8592 \beta_{12} - 16713 \beta_{11} - 38800 \beta_{10} + 17990 \beta_{9} - 38800 \beta_{8} + 5311 \beta_{7} - 43637 \beta_{6} + 4672 \beta_{5} - 16713 \beta_{4} + 513 \beta_{3} - 4672 \beta_{1} + 5311$$ $$\nu^{12}$$ $$=$$ $$-77063 \beta_{19} - 14070 \beta_{18} + 14070 \beta_{17} + 21468 \beta_{16} - 21468 \beta_{15} + 9911 \beta_{14} + 9911 \beta_{13} - 5069 \beta_{12} - 5069 \beta_{11} - 194370 \beta_{10} + 86733 \beta_{9} - 35087 \beta_{8} - 35087 \beta_{7} - 80829 \beta_{6} + 80829 \beta_{5} - 86733 \beta_{4} + 15097 \beta_{2} + 15097 \beta_{1}$$ $$\nu^{13}$$ $$=$$ $$-172631 \beta_{19} + 3540 \beta_{18} + 172631 \beta_{17} + 11440 \beta_{16} - 191543 \beta_{15} + 77849 \beta_{14} + 11440 \beta_{13} - 174291 \beta_{12} + 77849 \beta_{11} - 412605 \beta_{10} + 174291 \beta_{9} + 43136 \beta_{8} - 412605 \beta_{7} - 59295 \beta_{6} + 425077 \beta_{5} - 191543 \beta_{4} - 3540 \beta_{3} - 59295 \beta_{2} - 43136$$ $$\nu^{14}$$ $$=$$ $$-172989 \beta_{19} + 773659 \beta_{17} + 117234 \beta_{16} - 867100 \beta_{15} - 69075 \beta_{14} + 241486 \beta_{13} - 867100 \beta_{12} + 117234 \beta_{11} - 409591 \beta_{10} + 69075 \beta_{9} - 1934776 \beta_{7} + 130337 \beta_{6} + 861781 \beta_{5} - 241486 \beta_{4} + 172989 \beta_{3} - 861781 \beta_{2} - 130337 \beta_{1} + 409591$$ $$\nu^{15}$$ $$=$$ $$6085 \beta_{19} - 6085 \beta_{18} + 1797956 \beta_{17} + 707214 \beta_{16} - 1807851 \beta_{15} - 1807851 \beta_{14} + 2019104 \beta_{13} - 2019104 \beta_{12} + 180224 \beta_{11} + 310095 \beta_{10} - 707214 \beta_{9} - 310095 \beta_{8} - 4334562 \beta_{7} + 720730 \beta_{5} - 180224 \beta_{4} + 1797956 \beta_{3} - 4208167 \beta_{2} + 720730 \beta_{1} + 4334562$$ $$\nu^{16}$$ $$=$$ $$2032620 \beta_{18} + 2032620 \beta_{17} - 891059 \beta_{16} - 891059 \beta_{15} - 8732441 \beta_{14} + 8732441 \beta_{13} - 2702720 \beta_{12} + 2702720 \beta_{11} - 1317975 \beta_{9} + 4694581 \beta_{8} - 4694581 \beta_{7} - 1054423 \beta_{6} - 1054423 \beta_{5} - 1317975 \beta_{4} + 7823869 \beta_{3} - 9072035 \beta_{2} + 9072035 \beta_{1} + 19468618$$ $$\nu^{17}$$ $$=$$ $$330322 \beta_{19} + 18695535 \beta_{18} + 330322 \beta_{17} - 18707565 \beta_{16} + 6436269 \beta_{15} - 21155071 \beta_{14} + 18707565 \beta_{13} - 2539356 \beta_{12} + 21155071 \beta_{11} - 1770186 \beta_{10} - 2539356 \beta_{9} + 45263765 \beta_{8} + 1770186 \beta_{7} + 8509319 \beta_{6} - 6436269 \beta_{4} + 18695535 \beta_{3} - 8509319 \beta_{2} + 42121419 \beta_{1} + 45263765$$ $$\nu^{18}$$ $$=$$ $$23228121 \beta_{19} + 79536549 \beta_{18} - 88408294 \beta_{16} + 14388480 \beta_{15} - 30086562 \beta_{14} + 11036645 \beta_{13} - 14388480 \beta_{12} + 88408294 \beta_{11} + 53087193 \beta_{10} - 30086562 \beta_{9} + 197262636 \beta_{8} + 94846311 \beta_{6} + 7784283 \beta_{5} + 11036645 \beta_{4} + 23228121 \beta_{3} + 7784283 \beta_{2} + 94846311 \beta_{1} + 53087193$$ $$\nu^{19}$$ $$=$$ $$194291250 \beta_{19} + 194291250 \beta_{18} - 7913799 \beta_{17} - 220871206 \beta_{16} + 33338924 \beta_{15} - 33338924 \beta_{14} - 58575976 \beta_{13} - 58575976 \beta_{12} + 193432777 \beta_{11} + 471300144 \beta_{10} - 220871206 \beta_{9} + 471300144 \beta_{8} - 3698703 \beta_{7} + 424838813 \beta_{6} - 98388604 \beta_{5} + 193432777 \beta_{4} + 7913799 \beta_{3} + 98388604 \beta_{1} - 3698703$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/170\mathbb{Z}\right)^\times$$.

 $$n$$ $$71$$ $$137$$ $$\chi(n)$$ $$-\beta_{7}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 2.99334 − 1.23988i 0.953222 − 0.394838i 0.236338 − 0.0978946i −1.86202 + 0.771273i −2.32088 + 0.961341i 2.99334 + 1.23988i 0.953222 + 0.394838i 0.236338 + 0.0978946i −1.86202 − 0.771273i −2.32088 − 0.961341i −0.969032 + 2.33945i −0.826884 + 1.99627i 0.254075 − 0.613391i 0.355063 − 0.857197i 1.18678 − 2.86514i −0.969032 − 2.33945i −0.826884 − 1.99627i 0.254075 + 0.613391i 0.355063 + 0.857197i 1.18678 + 2.86514i
−0.707107 0.707107i −1.23988 + 2.99334i 1.00000i −2.18859 0.458323i 2.99334 1.23988i 1.49921 0.620992i 0.707107 0.707107i −5.30145 5.30145i 1.22349 + 1.87165i
9.2 −0.707107 0.707107i −0.394838 + 0.953222i 1.00000i 1.21037 1.88016i 0.953222 0.394838i −0.363965 + 0.150759i 0.707107 0.707107i 1.36858 + 1.36858i −2.18534 + 0.473610i
9.3 −0.707107 0.707107i −0.0978946 + 0.236338i 1.00000i −0.496623 + 2.18022i 0.236338 0.0978946i −3.48449 + 1.44332i 0.707107 0.707107i 2.07505 + 2.07505i 1.89281 1.19048i
9.4 −0.707107 0.707107i 0.771273 1.86202i 1.00000i 1.71825 + 1.43095i −1.86202 + 0.771273i 2.47378 1.02468i 0.707107 0.707107i −0.750924 0.750924i −0.203147 2.22682i
9.5 −0.707107 0.707107i 0.961341 2.32088i 1.00000i −1.24340 1.85848i −2.32088 + 0.961341i −0.124542 + 0.0515871i 0.707107 0.707107i −2.34100 2.34100i −0.434925 + 2.19336i
19.1 −0.707107 + 0.707107i −1.23988 2.99334i 1.00000i −2.18859 + 0.458323i 2.99334 + 1.23988i 1.49921 + 0.620992i 0.707107 + 0.707107i −5.30145 + 5.30145i 1.22349 1.87165i
19.2 −0.707107 + 0.707107i −0.394838 0.953222i 1.00000i 1.21037 + 1.88016i 0.953222 + 0.394838i −0.363965 0.150759i 0.707107 + 0.707107i 1.36858 1.36858i −2.18534 0.473610i
19.3 −0.707107 + 0.707107i −0.0978946 0.236338i 1.00000i −0.496623 2.18022i 0.236338 + 0.0978946i −3.48449 1.44332i 0.707107 + 0.707107i 2.07505 2.07505i 1.89281 + 1.19048i
19.4 −0.707107 + 0.707107i 0.771273 + 1.86202i 1.00000i 1.71825 1.43095i −1.86202 0.771273i 2.47378 + 1.02468i 0.707107 + 0.707107i −0.750924 + 0.750924i −0.203147 + 2.22682i
19.5 −0.707107 + 0.707107i 0.961341 + 2.32088i 1.00000i −1.24340 + 1.85848i −2.32088 0.961341i −0.124542 0.0515871i 0.707107 + 0.707107i −2.34100 + 2.34100i −0.434925 2.19336i
49.1 0.707107 0.707107i −2.33945 + 0.969032i 1.00000i −0.309710 + 2.21452i −0.969032 + 2.33945i −1.26758 + 3.06021i −0.707107 0.707107i 2.41268 2.41268i 1.34690 + 1.78490i
49.2 0.707107 0.707107i −1.99627 + 0.826884i 1.00000i −1.27343 1.83804i −0.826884 + 1.99627i 1.32795 3.20595i −0.707107 0.707107i 1.18005 1.18005i −2.20014 0.399242i
49.3 0.707107 0.707107i 0.613391 0.254075i 1.00000i 0.384345 + 2.20279i 0.254075 0.613391i 1.97319 4.76369i −0.707107 0.707107i −1.80963 + 1.80963i 1.82938 + 1.28583i
49.4 0.707107 0.707107i 0.857197 0.355063i 1.00000i 2.23422 0.0908004i 0.355063 0.857197i −0.939960 + 2.26926i −0.707107 0.707107i −1.51260 + 1.51260i 1.51563 1.64404i
49.5 0.707107 0.707107i 2.86514 1.18678i 1.00000i −2.03543 + 0.925748i 1.18678 2.86514i −1.09360 + 2.64018i −0.707107 0.707107i 4.67924 4.67924i −0.784666 + 2.09387i
59.1 0.707107 + 0.707107i −2.33945 0.969032i 1.00000i −0.309710 2.21452i −0.969032 2.33945i −1.26758 3.06021i −0.707107 + 0.707107i 2.41268 + 2.41268i 1.34690 1.78490i
59.2 0.707107 + 0.707107i −1.99627 0.826884i 1.00000i −1.27343 + 1.83804i −0.826884 1.99627i 1.32795 + 3.20595i −0.707107 + 0.707107i 1.18005 + 1.18005i −2.20014 + 0.399242i
59.3 0.707107 + 0.707107i 0.613391 + 0.254075i 1.00000i 0.384345 2.20279i 0.254075 + 0.613391i 1.97319 + 4.76369i −0.707107 + 0.707107i −1.80963 1.80963i 1.82938 1.28583i
59.4 0.707107 + 0.707107i 0.857197 + 0.355063i 1.00000i 2.23422 + 0.0908004i 0.355063 + 0.857197i −0.939960 2.26926i −0.707107 + 0.707107i −1.51260 1.51260i 1.51563 + 1.64404i
59.5 0.707107 + 0.707107i 2.86514 + 1.18678i 1.00000i −2.03543 0.925748i 1.18678 + 2.86514i −1.09360 2.64018i −0.707107 + 0.707107i 4.67924 + 4.67924i −0.784666 2.09387i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 59.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.m even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.n.a 20
5.b even 2 1 170.2.n.b yes 20
5.c odd 4 1 850.2.l.h 20
5.c odd 4 1 850.2.l.i 20
17.d even 8 1 170.2.n.b yes 20
85.k odd 8 1 850.2.l.i 20
85.m even 8 1 inner 170.2.n.a 20
85.n odd 8 1 850.2.l.h 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.n.a 20 1.a even 1 1 trivial
170.2.n.a 20 85.m even 8 1 inner
170.2.n.b yes 20 5.b even 2 1
170.2.n.b yes 20 17.d even 8 1
850.2.l.h 20 5.c odd 4 1
850.2.l.h 20 85.n odd 8 1
850.2.l.i 20 5.c odd 4 1
850.2.l.i 20 85.k odd 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{20} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(170, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{5}$$
$3$ $$1 - 8 T^{3} + 32 T^{5} + 28 T^{6} - 72 T^{7} - 234 T^{8} - 24 T^{9} + 1224 T^{10} + 728 T^{11} - 2488 T^{12} - 6728 T^{13} + 1256 T^{14} + 27240 T^{15} + 14713 T^{16} - 60616 T^{17} - 122316 T^{18} + 66920 T^{19} + 485264 T^{20} + 200760 T^{21} - 1100844 T^{22} - 1636632 T^{23} + 1191753 T^{24} + 6619320 T^{25} + 915624 T^{26} - 14714136 T^{27} - 16323768 T^{28} + 14329224 T^{29} + 72275976 T^{30} - 4251528 T^{31} - 124357194 T^{32} - 114791256 T^{33} + 133923132 T^{34} + 459165024 T^{35} - 1033121304 T^{37} + 3486784401 T^{40}$$
$5$ $$1 + 4 T + 14 T^{2} + 28 T^{3} + 41 T^{4} + 32 T^{5} - 72 T^{6} - 416 T^{7} - 1550 T^{8} - 4944 T^{9} - 10916 T^{10} - 24720 T^{11} - 38750 T^{12} - 52000 T^{13} - 45000 T^{14} + 100000 T^{15} + 640625 T^{16} + 2187500 T^{17} + 5468750 T^{18} + 7812500 T^{19} + 9765625 T^{20}$$
$7$ $$1 + 28 T^{2} + 72 T^{3} + 392 T^{4} + 2152 T^{5} + 5636 T^{6} + 31696 T^{7} + 93057 T^{8} + 321232 T^{9} + 1251096 T^{10} + 3073136 T^{11} + 11857952 T^{12} + 32094064 T^{13} + 87528552 T^{14} + 298482128 T^{15} + 656393822 T^{16} + 2152770768 T^{17} + 5629095584 T^{18} + 13238676192 T^{19} + 44114872624 T^{20} + 92670733344 T^{21} + 275825683616 T^{22} + 738400373424 T^{23} + 1576001566622 T^{24} + 5016589125296 T^{25} + 10297646614248 T^{26} + 26430841748752 T^{27} + 68358733547552 T^{28} + 124012122401552 T^{29} + 353403654122904 T^{30} + 635180624307376 T^{31} + 1288028663063457 T^{32} + 3070994073860272 T^{33} + 3822465238576964 T^{34} + 10216752369397336 T^{35} + 13027308783283592 T^{36} + 16749397007078904 T^{37} + 45595580741492572 T^{38} + 79792266297612001 T^{40}$$
$11$ $$1 + 8 T + 32 T^{2} + 88 T^{3} + 192 T^{4} + 1336 T^{5} + 8272 T^{6} + 31784 T^{7} + 74909 T^{8} + 75696 T^{9} + 299120 T^{10} + 2946256 T^{11} + 14548096 T^{12} + 38626064 T^{13} + 40847888 T^{14} + 70277616 T^{15} + 1208780434 T^{16} + 7630937792 T^{17} + 25235168784 T^{18} + 46671777024 T^{19} + 75581355904 T^{20} + 513389547264 T^{21} + 3053455422864 T^{22} + 10156778201152 T^{23} + 17697754334194 T^{24} + 11318280334416 T^{25} + 72364525313168 T^{26} + 752712714224944 T^{27} + 3118513579240576 T^{28} + 6947117532294896 T^{29} + 7758402446651120 T^{30} + 21596952218570256 T^{31} + 235096531271793389 T^{32} + 1097269882782702904 T^{33} + 3141290623400569552 T^{34} + 5580803554339309736 T^{35} + 8822348133805854912 T^{36} + 44479338507937851848 T^{37} +$$$$17\!\cdots\!92$$$$T^{38} +$$$$48\!\cdots\!28$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$
$13$ $$( 1 + 12 T + 126 T^{2} + 944 T^{3} + 6370 T^{4} + 36804 T^{5} + 195036 T^{6} + 930636 T^{7} + 4101389 T^{8} + 16591780 T^{9} + 62248284 T^{10} + 215693140 T^{11} + 693134741 T^{12} + 2044607292 T^{13} + 5570423196 T^{14} + 13665067572 T^{15} + 30746773330 T^{16} + 59234600048 T^{17} + 102782070846 T^{18} + 127253992476 T^{19} + 137858491849 T^{20} )^{2}$$
$17$ $$1 + 18 T^{2} - 16 T^{3} + 803 T^{4} - 1696 T^{5} + 11588 T^{6} - 48032 T^{7} + 238596 T^{8} - 1229744 T^{9} + 3632100 T^{10} - 20905648 T^{11} + 68954244 T^{12} - 235981216 T^{13} + 967841348 T^{14} - 2408077472 T^{15} + 19382467907 T^{16} - 6565418768 T^{17} + 125563633938 T^{18} + 2015993900449 T^{20}$$
$19$ $$1 - 104 T^{3} - 440 T^{4} - 1768 T^{5} + 5408 T^{6} - 5432 T^{7} + 242710 T^{8} + 738648 T^{9} + 4507360 T^{10} + 7934112 T^{11} + 6690854 T^{12} - 462658112 T^{13} - 1445661280 T^{14} - 8971090888 T^{15} - 40777542279 T^{16} - 43430694488 T^{17} + 214418884704 T^{18} + 3590696280576 T^{19} + 25541206222212 T^{20} + 68223229330944 T^{21} + 77405217378144 T^{22} - 297891133493192 T^{23} - 5314170087341559 T^{24} - 22213309176685912 T^{25} - 68012408545187680 T^{26} - 413557011135896768 T^{27} + 113634540707127014 T^{28} + 2560240335200737248 T^{29} + 27634922807761915360 T^{30} + 86045296754651667912 T^{31} +$$$$53\!\cdots\!10$$$$T^{32} -$$$$22\!\cdots\!88$$$$T^{33} +$$$$43\!\cdots\!68$$$$T^{34} -$$$$26\!\cdots\!32$$$$T^{35} -$$$$12\!\cdots\!40$$$$T^{36} -$$$$56\!\cdots\!56$$$$T^{37} +$$$$37\!\cdots\!01$$$$T^{40}$$
$23$ $$1 - 16 T + 56 T^{2} + 376 T^{3} - 2400 T^{4} - 7256 T^{5} + 72440 T^{6} + 121072 T^{7} - 2621531 T^{8} + 4314208 T^{9} + 70866208 T^{10} - 388785408 T^{11} - 779461504 T^{12} + 12408165568 T^{13} - 8863476320 T^{14} - 287308681888 T^{15} + 756303717514 T^{16} + 5606916061312 T^{17} - 30507258885872 T^{18} - 66054575985616 T^{19} + 960733835992000 T^{20} - 1519255247669168 T^{21} - 16138339950626288 T^{22} + 68219347717983104 T^{23} + 211644788612835274 T^{24} - 1849217223509055584 T^{25} - 1312112596661648480 T^{26} + 42247637876515608896 T^{27} - 61040398366850122624 T^{28} -$$$$70\!\cdots\!04$$$$T^{29} +$$$$29\!\cdots\!92$$$$T^{30} +$$$$41\!\cdots\!16$$$$T^{31} -$$$$57\!\cdots\!51$$$$T^{32} +$$$$61\!\cdots\!76$$$$T^{33} +$$$$83\!\cdots\!60$$$$T^{34} -$$$$19\!\cdots\!92$$$$T^{35} -$$$$14\!\cdots\!00$$$$T^{36} +$$$$53\!\cdots\!28$$$$T^{37} +$$$$18\!\cdots\!64$$$$T^{38} -$$$$11\!\cdots\!92$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$
$29$ $$1 + 12 T + 38 T^{2} + 212 T^{3} + 2978 T^{4} + 17140 T^{5} + 92422 T^{6} + 571116 T^{7} + 2331938 T^{8} + 14922092 T^{9} + 130048102 T^{10} + 770925860 T^{11} + 3851687522 T^{12} + 19564235252 T^{13} + 133504284918 T^{14} + 966299457452 T^{15} + 4642708862081 T^{16} + 19643962031712 T^{17} + 123559244554608 T^{18} + 710977606960016 T^{19} + 3560114011124736 T^{20} + 20618350601840464 T^{21} + 103913324670425328 T^{22} + 479096589991423968 T^{23} + 3283699766681511761 T^{24} + 19819912150417132348 T^{25} + 79411462122654972678 T^{26} +$$$$33\!\cdots\!68$$$$T^{27} +$$$$19\!\cdots\!42$$$$T^{28} +$$$$11\!\cdots\!40$$$$T^{29} +$$$$54\!\cdots\!02$$$$T^{30} +$$$$18\!\cdots\!68$$$$T^{31} +$$$$82\!\cdots\!58$$$$T^{32} +$$$$58\!\cdots\!24$$$$T^{33} +$$$$27\!\cdots\!82$$$$T^{34} +$$$$14\!\cdots\!60$$$$T^{35} +$$$$74\!\cdots\!38$$$$T^{36} +$$$$15\!\cdots\!08$$$$T^{37} +$$$$79\!\cdots\!18$$$$T^{38} +$$$$73\!\cdots\!28$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$
$31$ $$1 - 8 T + 4 T^{2} + 384 T^{3} - 2296 T^{4} + 6096 T^{5} + 40040 T^{6} - 153440 T^{7} + 1584090 T^{8} - 14476232 T^{9} + 89328068 T^{10} + 315411224 T^{11} - 3760050272 T^{12} + 22889881832 T^{13} + 17254141740 T^{14} - 332815622952 T^{15} + 2990654769257 T^{16} - 6114910723832 T^{17} + 38570348356996 T^{18} + 367744608569728 T^{19} - 1791803809774288 T^{20} + 11400082865661568 T^{21} + 37066104771073156 T^{22} - 182169305373679112 T^{23} + 2761932483158993897 T^{24} - 9528228724651873752 T^{25} + 15313114306745744940 T^{26} +$$$$62\!\cdots\!52$$$$T^{27} -$$$$32\!\cdots\!52$$$$T^{28} +$$$$83\!\cdots\!04$$$$T^{29} +$$$$73\!\cdots\!68$$$$T^{30} -$$$$36\!\cdots\!92$$$$T^{31} +$$$$12\!\cdots\!90$$$$T^{32} -$$$$37\!\cdots\!40$$$$T^{33} +$$$$30\!\cdots\!40$$$$T^{34} +$$$$14\!\cdots\!96$$$$T^{35} -$$$$16\!\cdots\!76$$$$T^{36} +$$$$86\!\cdots\!24$$$$T^{37} +$$$$27\!\cdots\!64$$$$T^{38} -$$$$17\!\cdots\!68$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$
$37$ $$1 + 8 T + 118 T^{2} + 960 T^{3} + 8114 T^{4} + 50464 T^{5} + 338982 T^{6} + 1440600 T^{7} + 4335441 T^{8} - 9800336 T^{9} - 346829856 T^{10} - 3402505696 T^{11} - 26710359232 T^{12} - 167815912224 T^{13} - 904985189600 T^{14} - 3602804409392 T^{15} - 5581747209506 T^{16} + 63717621771584 T^{17} + 1054915539916916 T^{18} + 8107624670209056 T^{19} + 59840742028993692 T^{20} + 299982112797735072 T^{21} + 1444179374146258004 T^{22} + 3227488695596044352 T^{23} - 10461092931914974466 T^{24} -$$$$24\!\cdots\!44$$$$T^{25} -$$$$23\!\cdots\!00$$$$T^{26} -$$$$15\!\cdots\!92$$$$T^{27} -$$$$93\!\cdots\!72$$$$T^{28} -$$$$44\!\cdots\!92$$$$T^{29} -$$$$16\!\cdots\!44$$$$T^{30} -$$$$17\!\cdots\!68$$$$T^{31} +$$$$28\!\cdots\!21$$$$T^{32} +$$$$35\!\cdots\!00$$$$T^{33} +$$$$30\!\cdots\!98$$$$T^{34} +$$$$16\!\cdots\!52$$$$T^{35} +$$$$10\!\cdots\!74$$$$T^{36} +$$$$43\!\cdots\!20$$$$T^{37} +$$$$19\!\cdots\!22$$$$T^{38} +$$$$49\!\cdots\!84$$$$T^{39} +$$$$23\!\cdots\!01$$$$T^{40}$$
$41$ $$1 - 4 T + 30 T^{2} + 92 T^{3} - 334 T^{4} - 10116 T^{5} + 85854 T^{6} - 842916 T^{7} - 2808099 T^{8} + 21400784 T^{9} - 13842264 T^{10} - 1079831088 T^{11} + 13460410040 T^{12} + 2388153040 T^{13} - 195580560792 T^{14} + 2644046148048 T^{15} + 10500427183938 T^{16} - 192141210610008 T^{17} + 664554433081812 T^{18} - 1101475111191192 T^{19} - 30298126262993364 T^{20} - 45160479558838872 T^{21} + 1117116002010525972 T^{22} - 13242564376452361368 T^{23} + 29671697619711826818 T^{24} +$$$$30\!\cdots\!48$$$$T^{25} -$$$$92\!\cdots\!72$$$$T^{26} +$$$$46\!\cdots\!40$$$$T^{27} +$$$$10\!\cdots\!40$$$$T^{28} -$$$$35\!\cdots\!68$$$$T^{29} -$$$$18\!\cdots\!64$$$$T^{30} +$$$$11\!\cdots\!44$$$$T^{31} -$$$$63\!\cdots\!19$$$$T^{32} -$$$$77\!\cdots\!36$$$$T^{33} +$$$$32\!\cdots\!94$$$$T^{34} -$$$$15\!\cdots\!16$$$$T^{35} -$$$$21\!\cdots\!94$$$$T^{36} +$$$$24\!\cdots\!52$$$$T^{37} +$$$$32\!\cdots\!30$$$$T^{38} -$$$$17\!\cdots\!44$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$
$43$ $$1 - 16 T + 128 T^{2} - 1488 T^{3} + 20694 T^{4} - 176960 T^{5} + 1289600 T^{6} - 12374208 T^{7} + 114689745 T^{8} - 816890480 T^{9} + 5773220224 T^{10} - 47396668080 T^{11} + 376357329952 T^{12} - 2558318885360 T^{13} + 17028142721920 T^{14} - 127218542416432 T^{15} + 963177088145694 T^{16} - 6226032087351344 T^{17} + 38642451157210368 T^{18} - 278587214299949552 T^{19} + 2002378843515302292 T^{20} - 11979250214897830736 T^{21} + 71449892189681970432 T^{22} -$$$$49\!\cdots\!08$$$$T^{23} +$$$$32\!\cdots\!94$$$$T^{24} -$$$$18\!\cdots\!76$$$$T^{25} +$$$$10\!\cdots\!80$$$$T^{26} -$$$$69\!\cdots\!20$$$$T^{27} +$$$$43\!\cdots\!52$$$$T^{28} -$$$$23\!\cdots\!40$$$$T^{29} +$$$$12\!\cdots\!76$$$$T^{30} -$$$$75\!\cdots\!60$$$$T^{31} +$$$$45\!\cdots\!45$$$$T^{32} -$$$$21\!\cdots\!44$$$$T^{33} +$$$$95\!\cdots\!00$$$$T^{34} -$$$$56\!\cdots\!20$$$$T^{35} +$$$$28\!\cdots\!94$$$$T^{36} -$$$$87\!\cdots\!84$$$$T^{37} +$$$$32\!\cdots\!72$$$$T^{38} -$$$$17\!\cdots\!12$$$$T^{39} +$$$$46\!\cdots\!01$$$$T^{40}$$
$47$ $$( 1 - 20 T + 388 T^{2} - 4976 T^{3} + 62092 T^{4} - 635100 T^{5} + 6214238 T^{6} - 53431788 T^{7} + 439450571 T^{8} - 3289992044 T^{9} + 23554230924 T^{10} - 154629626068 T^{11} + 970746311339 T^{12} - 5547448525524 T^{13} + 30323499098078 T^{14} - 145657013945700 T^{15} + 669303038208268 T^{16} - 2520956647423888 T^{17} + 9238779224763268 T^{18} - 22382609462055340 T^{19} + 52599132235830049 T^{20} )^{2}$$
$53$ $$1 - 44 T + 968 T^{2} - 14204 T^{3} + 153360 T^{4} - 1253852 T^{5} + 7593816 T^{6} - 29525844 T^{7} + 21121854 T^{8} + 436104500 T^{9} + 1235332696 T^{10} - 80090622156 T^{11} + 970719706210 T^{12} - 6135983063340 T^{13} + 7164913588248 T^{14} + 319649478179492 T^{15} - 4261370408835583 T^{16} + 31329479689545576 T^{17} - 136710820885404048 T^{18} + 162474207458147632 T^{19} + 1532805880285790428 T^{20} + 8611132995281824496 T^{21} -$$$$38\!\cdots\!32$$$$T^{22} +$$$$46\!\cdots\!52$$$$T^{23} -$$$$33\!\cdots\!23$$$$T^{24} +$$$$13\!\cdots\!56$$$$T^{25} +$$$$15\!\cdots\!92$$$$T^{26} -$$$$72\!\cdots\!80$$$$T^{27} +$$$$60\!\cdots\!10$$$$T^{28} -$$$$26\!\cdots\!48$$$$T^{29} +$$$$21\!\cdots\!04$$$$T^{30} +$$$$40\!\cdots\!00$$$$T^{31} +$$$$10\!\cdots\!14$$$$T^{32} -$$$$76\!\cdots\!12$$$$T^{33} +$$$$10\!\cdots\!04$$$$T^{34} -$$$$91\!\cdots\!64$$$$T^{35} +$$$$59\!\cdots\!60$$$$T^{36} -$$$$29\!\cdots\!52$$$$T^{37} +$$$$10\!\cdots\!52$$$$T^{38} -$$$$25\!\cdots\!48$$$$T^{39} +$$$$30\!\cdots\!01$$$$T^{40}$$
$59$ $$1 - 16 T + 128 T^{2} - 640 T^{3} + 6032 T^{4} - 73144 T^{5} + 603008 T^{6} - 4225576 T^{7} + 33076134 T^{8} - 244311232 T^{9} + 1417281312 T^{10} - 16887395168 T^{11} + 211053947230 T^{12} - 1522774288880 T^{13} + 6460849018400 T^{14} - 56720710039568 T^{15} + 808791885028057 T^{16} - 6447237579834824 T^{17} + 35399211288359360 T^{18} - 220142009338344872 T^{19} + 1853897520463942116 T^{20} - 12988378550962347448 T^{21} +$$$$12\!\cdots\!60$$$$T^{22} -$$$$13\!\cdots\!96$$$$T^{23} +$$$$98\!\cdots\!77$$$$T^{24} -$$$$40\!\cdots\!32$$$$T^{25} +$$$$27\!\cdots\!00$$$$T^{26} -$$$$37\!\cdots\!20$$$$T^{27} +$$$$30\!\cdots\!30$$$$T^{28} -$$$$14\!\cdots\!52$$$$T^{29} +$$$$72\!\cdots\!12$$$$T^{30} -$$$$73\!\cdots\!88$$$$T^{31} +$$$$58\!\cdots\!54$$$$T^{32} -$$$$44\!\cdots\!04$$$$T^{33} +$$$$37\!\cdots\!88$$$$T^{34} -$$$$26\!\cdots\!56$$$$T^{35} +$$$$13\!\cdots\!12$$$$T^{36} -$$$$81\!\cdots\!60$$$$T^{37} +$$$$96\!\cdots\!88$$$$T^{38} -$$$$70\!\cdots\!24$$$$T^{39} +$$$$26\!\cdots\!01$$$$T^{40}$$
$61$ $$1 - 8 T + 10 T^{2} + 320 T^{3} - 2126 T^{4} - 27296 T^{5} - 72510 T^{6} + 8840 T^{7} - 3520990 T^{8} - 69652728 T^{9} + 659172698 T^{10} + 1929507872 T^{11} + 77537561570 T^{12} + 128446570208 T^{13} + 1861119756730 T^{14} + 37614663351992 T^{15} - 71694040127423 T^{16} - 1493114370031936 T^{17} + 3398743476006728 T^{18} - 110814064642641248 T^{19} - 1163583925604399136 T^{20} - 6759657943201116128 T^{21} + 12646724474221034888 T^{22} -$$$$33\!\cdots\!16$$$$T^{23} -$$$$99\!\cdots\!43$$$$T^{24} +$$$$31\!\cdots\!92$$$$T^{25} +$$$$95\!\cdots\!30$$$$T^{26} +$$$$40\!\cdots\!68$$$$T^{27} +$$$$14\!\cdots\!70$$$$T^{28} +$$$$22\!\cdots\!52$$$$T^{29} +$$$$47\!\cdots\!98$$$$T^{30} -$$$$30\!\cdots\!08$$$$T^{31} -$$$$93\!\cdots\!90$$$$T^{32} +$$$$14\!\cdots\!40$$$$T^{33} -$$$$71\!\cdots\!10$$$$T^{34} -$$$$16\!\cdots\!96$$$$T^{35} -$$$$78\!\cdots\!86$$$$T^{36} +$$$$71\!\cdots\!20$$$$T^{37} +$$$$13\!\cdots\!10$$$$T^{38} -$$$$66\!\cdots\!28$$$$T^{39} +$$$$50\!\cdots\!01$$$$T^{40}$$
$67$ $$1 - 532 T^{2} + 148646 T^{4} - 29200180 T^{6} + 4500891213 T^{8} - 575578725168 T^{10} + 63179028972648 T^{12} - 6089051557526448 T^{14} + 523435323978922754 T^{16} - 40558419543281652984 T^{18} +$$$$28\!\cdots\!68$$$$T^{20} -$$$$18\!\cdots\!76$$$$T^{22} +$$$$10\!\cdots\!34$$$$T^{24} -$$$$55\!\cdots\!12$$$$T^{26} +$$$$25\!\cdots\!68$$$$T^{28} -$$$$10\!\cdots\!32$$$$T^{30} +$$$$36\!\cdots\!93$$$$T^{32} -$$$$10\!\cdots\!20$$$$T^{34} +$$$$24\!\cdots\!26$$$$T^{36} -$$$$39\!\cdots\!88$$$$T^{38} +$$$$33\!\cdots\!01$$$$T^{40}$$
$71$ $$1 - 8 T + 4 T^{2} + 1064 T^{3} - 7736 T^{4} + 33056 T^{5} + 411864 T^{6} - 3498384 T^{7} + 43320906 T^{8} - 175160784 T^{9} + 1711157444 T^{10} + 23184409480 T^{11} - 282606722400 T^{12} + 2190349172904 T^{13} + 5955655055116 T^{14} - 92105865793280 T^{15} + 1484388456856825 T^{16} - 5521262001586648 T^{17} + 13570667323190324 T^{18} + 605072991295826840 T^{19} - 4474880001515114064 T^{20} + 42960182382003705640 T^{21} + 68409733976202423284 T^{22} -$$$$19\!\cdots\!28$$$$T^{23} +$$$$37\!\cdots\!25$$$$T^{24} -$$$$16\!\cdots\!80$$$$T^{25} +$$$$76\!\cdots\!36$$$$T^{26} +$$$$19\!\cdots\!64$$$$T^{27} -$$$$18\!\cdots\!00$$$$T^{28} +$$$$10\!\cdots\!80$$$$T^{29} +$$$$55\!\cdots\!44$$$$T^{30} -$$$$40\!\cdots\!64$$$$T^{31} +$$$$71\!\cdots\!46$$$$T^{32} -$$$$40\!\cdots\!24$$$$T^{33} +$$$$34\!\cdots\!84$$$$T^{34} +$$$$19\!\cdots\!56$$$$T^{35} -$$$$32\!\cdots\!56$$$$T^{36} +$$$$31\!\cdots\!24$$$$T^{37} +$$$$84\!\cdots\!44$$$$T^{38} -$$$$11\!\cdots\!48$$$$T^{39} +$$$$10\!\cdots\!01$$$$T^{40}$$
$73$ $$1 + 60 T + 1682 T^{2} + 29988 T^{3} + 398642 T^{4} + 4449180 T^{5} + 44616114 T^{6} + 400755436 T^{7} + 3146402174 T^{8} + 22577269012 T^{9} + 172416257546 T^{10} + 1548661495372 T^{11} + 15112755685578 T^{12} + 149613932704668 T^{13} + 1492299482027146 T^{14} + 14514912884993716 T^{15} + 131595396462860353 T^{16} + 1114942726481802512 T^{17} + 9296763449815475432 T^{18} + 79328198294011301400 T^{19} +$$$$68\!\cdots\!92$$$$T^{20} +$$$$57\!\cdots\!00$$$$T^{21} +$$$$49\!\cdots\!28$$$$T^{22} +$$$$43\!\cdots\!04$$$$T^{23} +$$$$37\!\cdots\!73$$$$T^{24} +$$$$30\!\cdots\!88$$$$T^{25} +$$$$22\!\cdots\!94$$$$T^{26} +$$$$16\!\cdots\!96$$$$T^{27} +$$$$12\!\cdots\!18$$$$T^{28} +$$$$91\!\cdots\!36$$$$T^{29} +$$$$74\!\cdots\!54$$$$T^{30} +$$$$70\!\cdots\!24$$$$T^{31} +$$$$72\!\cdots\!54$$$$T^{32} +$$$$67\!\cdots\!88$$$$T^{33} +$$$$54\!\cdots\!26$$$$T^{34} +$$$$39\!\cdots\!60$$$$T^{35} +$$$$25\!\cdots\!62$$$$T^{36} +$$$$14\!\cdots\!64$$$$T^{37} +$$$$58\!\cdots\!58$$$$T^{38} +$$$$15\!\cdots\!20$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$
$79$ $$1 - 56 T + 1608 T^{2} - 32544 T^{3} + 531232 T^{4} - 7502656 T^{5} + 94723432 T^{6} - 1079876968 T^{7} + 11152309445 T^{8} - 104467318464 T^{9} + 885876705344 T^{10} - 6748309500960 T^{11} + 45237704790656 T^{12} - 254352077824672 T^{13} + 1043944393809024 T^{14} - 1173550884529024 T^{15} - 28330582069716470 T^{16} + 326793601144278544 T^{17} - 2173623899474329584 T^{18} + 10801150466444693600 T^{19} - 64674172953547878720 T^{20} +$$$$85\!\cdots\!00$$$$T^{21} -$$$$13\!\cdots\!44$$$$T^{22} +$$$$16\!\cdots\!16$$$$T^{23} -$$$$11\!\cdots\!70$$$$T^{24} -$$$$36\!\cdots\!76$$$$T^{25} +$$$$25\!\cdots\!04$$$$T^{26} -$$$$48\!\cdots\!48$$$$T^{27} +$$$$68\!\cdots\!16$$$$T^{28} -$$$$80\!\cdots\!40$$$$T^{29} +$$$$83\!\cdots\!44$$$$T^{30} -$$$$78\!\cdots\!56$$$$T^{31} +$$$$65\!\cdots\!45$$$$T^{32} -$$$$50\!\cdots\!52$$$$T^{33} +$$$$34\!\cdots\!92$$$$T^{34} -$$$$21\!\cdots\!44$$$$T^{35} +$$$$12\!\cdots\!72$$$$T^{36} -$$$$59\!\cdots\!96$$$$T^{37} +$$$$23\!\cdots\!88$$$$T^{38} -$$$$63\!\cdots\!64$$$$T^{39} +$$$$89\!\cdots\!01$$$$T^{40}$$
$83$ $$1 - 544 T^{3} - 1354 T^{4} + 59152 T^{5} + 147968 T^{6} + 546160 T^{7} - 31611727 T^{8} + 119552016 T^{9} + 1652717184 T^{10} - 6405201136 T^{11} + 252699874336 T^{12} + 937926276176 T^{13} + 17620618729088 T^{14} - 121733300402992 T^{15} - 1091081305080034 T^{16} + 10833819307749168 T^{17} + 164063275302237440 T^{18} + 183331162121083376 T^{19} - 10465764546854435628 T^{20} + 15216486456049920208 T^{21} +$$$$11\!\cdots\!60$$$$T^{22} +$$$$61\!\cdots\!16$$$$T^{23} -$$$$51\!\cdots\!14$$$$T^{24} -$$$$47\!\cdots\!56$$$$T^{25} +$$$$57\!\cdots\!72$$$$T^{26} +$$$$25\!\cdots\!52$$$$T^{27} +$$$$56\!\cdots\!76$$$$T^{28} -$$$$11\!\cdots\!08$$$$T^{29} +$$$$25\!\cdots\!16$$$$T^{30} +$$$$15\!\cdots\!72$$$$T^{31} -$$$$33\!\cdots\!47$$$$T^{32} +$$$$48\!\cdots\!80$$$$T^{33} +$$$$10\!\cdots\!72$$$$T^{34} +$$$$36\!\cdots\!64$$$$T^{35} -$$$$68\!\cdots\!74$$$$T^{36} -$$$$22\!\cdots\!12$$$$T^{37} +$$$$24\!\cdots\!01$$$$T^{40}$$
$89$ $$1 - 948 T^{2} + 453936 T^{4} - 146353496 T^{6} + 35701988702 T^{8} - 7013810546556 T^{10} + 1152109743195558 T^{12} - 162048052656574324 T^{14} + 19816119477140894897 T^{16} -$$$$21\!\cdots\!68$$$$T^{18} +$$$$20\!\cdots\!84$$$$T^{20} -$$$$16\!\cdots\!28$$$$T^{22} +$$$$12\!\cdots\!77$$$$T^{24} -$$$$80\!\cdots\!64$$$$T^{26} +$$$$45\!\cdots\!98$$$$T^{28} -$$$$21\!\cdots\!56$$$$T^{30} +$$$$88\!\cdots\!42$$$$T^{32} -$$$$28\!\cdots\!36$$$$T^{34} +$$$$70\!\cdots\!96$$$$T^{36} -$$$$11\!\cdots\!88$$$$T^{38} +$$$$97\!\cdots\!01$$$$T^{40}$$
$97$ $$1 - 48 T + 1142 T^{2} - 16936 T^{3} + 160946 T^{4} - 603200 T^{5} - 12193658 T^{6} + 329836064 T^{7} - 4766618146 T^{8} + 48871055912 T^{9} - 342706004482 T^{10} + 607199068352 T^{11} + 28990974681866 T^{12} - 605921738623824 T^{13} + 7806367455647038 T^{14} - 74491245978555832 T^{15} + 484575109311053409 T^{16} - 549564314450980192 T^{17} - 41722742455227040440 T^{18} +$$$$77\!\cdots\!16$$$$T^{19} -$$$$90\!\cdots\!40$$$$T^{20} +$$$$75\!\cdots\!52$$$$T^{21} -$$$$39\!\cdots\!60$$$$T^{22} -$$$$50\!\cdots\!16$$$$T^{23} +$$$$42\!\cdots\!29$$$$T^{24} -$$$$63\!\cdots\!24$$$$T^{25} +$$$$65\!\cdots\!02$$$$T^{26} -$$$$48\!\cdots\!12$$$$T^{27} +$$$$22\!\cdots\!26$$$$T^{28} +$$$$46\!\cdots\!84$$$$T^{29} -$$$$25\!\cdots\!18$$$$T^{30} +$$$$34\!\cdots\!36$$$$T^{31} -$$$$33\!\cdots\!86$$$$T^{32} +$$$$22\!\cdots\!28$$$$T^{33} -$$$$79\!\cdots\!02$$$$T^{34} -$$$$38\!\cdots\!00$$$$T^{35} +$$$$98\!\cdots\!66$$$$T^{36} -$$$$10\!\cdots\!32$$$$T^{37} +$$$$66\!\cdots\!38$$$$T^{38} -$$$$26\!\cdots\!84$$$$T^{39} +$$$$54\!\cdots\!01$$$$T^{40}$$