# Properties

 Label 170.2.n.b 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.

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

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 $$\left(\mathbb{Z}/170\mathbb{Z}\right)^\times$$.

 $$n$$ $$71$$ $$137$$ $$\chi(n)$$ $$\beta_{8}$$ $$-1$$

## 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
 −2.32088 − 0.961341i −1.86202 − 0.771273i 0.236338 + 0.0978946i 0.953222 + 0.394838i 2.99334 + 1.23988i −2.32088 + 0.961341i −1.86202 + 0.771273i 0.236338 − 0.0978946i 0.953222 − 0.394838i 2.99334 − 1.23988i 1.18678 + 2.86514i 0.355063 + 0.857197i 0.254075 + 0.613391i −0.826884 − 1.99627i −0.969032 − 2.33945i 1.18678 − 2.86514i 0.355063 − 0.857197i 0.254075 − 0.613391i −0.826884 + 1.99627i −0.969032 + 2.33945i
0.707107 + 0.707107i −0.961341 + 2.32088i 1.00000i 2.19336 0.434925i −2.32088 + 0.961341i 0.124542 0.0515871i −0.707107 + 0.707107i −2.34100 2.34100i 1.85848 + 1.24340i
9.2 0.707107 + 0.707107i −0.771273 + 1.86202i 1.00000i −2.22682 0.203147i −1.86202 + 0.771273i −2.47378 + 1.02468i −0.707107 + 0.707107i −0.750924 0.750924i −1.43095 1.71825i
9.3 0.707107 + 0.707107i 0.0978946 0.236338i 1.00000i −1.19048 + 1.89281i 0.236338 0.0978946i 3.48449 1.44332i −0.707107 + 0.707107i 2.07505 + 2.07505i −2.18022 + 0.496623i
9.4 0.707107 + 0.707107i 0.394838 0.953222i 1.00000i 0.473610 2.18534i 0.953222 0.394838i 0.363965 0.150759i −0.707107 + 0.707107i 1.36858 + 1.36858i 1.88016 1.21037i
9.5 0.707107 + 0.707107i 1.23988 2.99334i 1.00000i 1.87165 + 1.22349i 2.99334 1.23988i −1.49921 + 0.620992i −0.707107 + 0.707107i −5.30145 5.30145i 0.458323 + 2.18859i
19.1 0.707107 0.707107i −0.961341 2.32088i 1.00000i 2.19336 + 0.434925i −2.32088 0.961341i 0.124542 + 0.0515871i −0.707107 0.707107i −2.34100 + 2.34100i 1.85848 1.24340i
19.2 0.707107 0.707107i −0.771273 1.86202i 1.00000i −2.22682 + 0.203147i −1.86202 0.771273i −2.47378 1.02468i −0.707107 0.707107i −0.750924 + 0.750924i −1.43095 + 1.71825i
19.3 0.707107 0.707107i 0.0978946 + 0.236338i 1.00000i −1.19048 1.89281i 0.236338 + 0.0978946i 3.48449 + 1.44332i −0.707107 0.707107i 2.07505 2.07505i −2.18022 0.496623i
19.4 0.707107 0.707107i 0.394838 + 0.953222i 1.00000i 0.473610 + 2.18534i 0.953222 + 0.394838i 0.363965 + 0.150759i −0.707107 0.707107i 1.36858 1.36858i 1.88016 + 1.21037i
19.5 0.707107 0.707107i 1.23988 + 2.99334i 1.00000i 1.87165 1.22349i 2.99334 + 1.23988i −1.49921 0.620992i −0.707107 0.707107i −5.30145 + 5.30145i 0.458323 2.18859i
49.1 −0.707107 + 0.707107i −2.86514 + 1.18678i 1.00000i −2.09387 + 0.784666i 1.18678 2.86514i 1.09360 2.64018i 0.707107 + 0.707107i 4.67924 4.67924i 0.925748 2.03543i
49.2 −0.707107 + 0.707107i −0.857197 + 0.355063i 1.00000i 1.64404 1.51563i 0.355063 0.857197i 0.939960 2.26926i 0.707107 + 0.707107i −1.51260 + 1.51260i −0.0908004 + 2.23422i
49.3 −0.707107 + 0.707107i −0.613391 + 0.254075i 1.00000i −1.28583 1.82938i 0.254075 0.613391i −1.97319 + 4.76369i 0.707107 + 0.707107i −1.80963 + 1.80963i 2.20279 + 0.384345i
49.4 −0.707107 + 0.707107i 1.99627 0.826884i 1.00000i 0.399242 + 2.20014i −0.826884 + 1.99627i −1.32795 + 3.20595i 0.707107 + 0.707107i 1.18005 1.18005i −1.83804 1.27343i
49.5 −0.707107 + 0.707107i 2.33945 0.969032i 1.00000i −1.78490 1.34690i −0.969032 + 2.33945i 1.26758 3.06021i 0.707107 + 0.707107i 2.41268 2.41268i 2.21452 0.309710i
59.1 −0.707107 0.707107i −2.86514 1.18678i 1.00000i −2.09387 0.784666i 1.18678 + 2.86514i 1.09360 + 2.64018i 0.707107 0.707107i 4.67924 + 4.67924i 0.925748 + 2.03543i
59.2 −0.707107 0.707107i −0.857197 0.355063i 1.00000i 1.64404 + 1.51563i 0.355063 + 0.857197i 0.939960 + 2.26926i 0.707107 0.707107i −1.51260 1.51260i −0.0908004 2.23422i
59.3 −0.707107 0.707107i −0.613391 0.254075i 1.00000i −1.28583 + 1.82938i 0.254075 + 0.613391i −1.97319 4.76369i 0.707107 0.707107i −1.80963 1.80963i 2.20279 0.384345i
59.4 −0.707107 0.707107i 1.99627 + 0.826884i 1.00000i 0.399242 2.20014i −0.826884 1.99627i −1.32795 3.20595i 0.707107 0.707107i 1.18005 + 1.18005i −1.83804 + 1.27343i
59.5 −0.707107 0.707107i 2.33945 + 0.969032i 1.00000i −1.78490 + 1.34690i −0.969032 2.33945i 1.26758 + 3.06021i 0.707107 0.707107i 2.41268 + 2.41268i 2.21452 + 0.309710i
 $$n$$: 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.b yes 20
5.b even 2 1 170.2.n.a 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.a 20
85.k odd 8 1 850.2.l.h 20
85.m even 8 1 inner 170.2.n.b yes 20
85.n odd 8 1 850.2.l.i 20

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

## Hecke kernels

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