# Properties

 Label 966.2.q.d Level $966$ Weight $2$ Character orbit 966.q Analytic conductor $7.714$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.q (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$2$$ over $$\Q(\zeta_{11})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 6 x^{19} + 10 x^{18} - 14 x^{17} + 77 x^{16} + 12 x^{15} - 226 x^{14} - 793 x^{13} + 690 x^{12} + 6105 x^{11} + 13883 x^{10} + 24365 x^{9} + 36461 x^{8} + 41912 x^{7} + 45709 x^{6} + 33023 x^{5} + 25949 x^{4} + 12217 x^{3} + 2895 x^{2} - 172 x + 1849$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 6 x^{19} + 10 x^{18} - 14 x^{17} + 77 x^{16} + 12 x^{15} - 226 x^{14} - 793 x^{13} + 690 x^{12} + 6105 x^{11} + 13883 x^{10} + 24365 x^{9} + 36461 x^{8} + 41912 x^{7} + 45709 x^{6} + 33023 x^{5} + 25949 x^{4} + 12217 x^{3} + 2895 x^{2} - 172 x + 1849$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$68\!\cdots\!47$$$$\nu^{19} -$$$$10\!\cdots\!82$$$$\nu^{18} +$$$$50\!\cdots\!32$$$$\nu^{17} -$$$$10\!\cdots\!18$$$$\nu^{16} +$$$$19\!\cdots\!82$$$$\nu^{15} -$$$$56\!\cdots\!21$$$$\nu^{14} +$$$$10\!\cdots\!77$$$$\nu^{13} +$$$$10\!\cdots\!15$$$$\nu^{12} +$$$$45\!\cdots\!94$$$$\nu^{11} -$$$$37\!\cdots\!34$$$$\nu^{10} -$$$$27\!\cdots\!21$$$$\nu^{9} -$$$$47\!\cdots\!01$$$$\nu^{8} -$$$$71\!\cdots\!10$$$$\nu^{7} -$$$$94\!\cdots\!47$$$$\nu^{6} -$$$$75\!\cdots\!30$$$$\nu^{5} -$$$$10\!\cdots\!54$$$$\nu^{4} +$$$$84\!\cdots\!37$$$$\nu^{3} +$$$$26\!\cdots\!46$$$$\nu^{2} +$$$$37\!\cdots\!36$$$$\nu +$$$$41\!\cdots\!66$$$$)/$$$$57\!\cdots\!53$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$23\!\cdots\!05$$$$\nu^{19} +$$$$12\!\cdots\!11$$$$\nu^{18} -$$$$15\!\cdots\!01$$$$\nu^{17} +$$$$16\!\cdots\!68$$$$\nu^{16} -$$$$14\!\cdots\!14$$$$\nu^{15} -$$$$18\!\cdots\!94$$$$\nu^{14} +$$$$66\!\cdots\!47$$$$\nu^{13} +$$$$18\!\cdots\!71$$$$\nu^{12} -$$$$97\!\cdots\!40$$$$\nu^{11} -$$$$15\!\cdots\!85$$$$\nu^{10} -$$$$39\!\cdots\!90$$$$\nu^{9} -$$$$77\!\cdots\!31$$$$\nu^{8} -$$$$11\!\cdots\!64$$$$\nu^{7} -$$$$10\!\cdots\!74$$$$\nu^{6} -$$$$84\!\cdots\!12$$$$\nu^{5} -$$$$21\!\cdots\!41$$$$\nu^{4} +$$$$22\!\cdots\!04$$$$\nu^{3} +$$$$38\!\cdots\!56$$$$\nu^{2} +$$$$99\!\cdots\!55$$$$\nu +$$$$11\!\cdots\!87$$$$)/$$$$57\!\cdots\!53$$ $$\beta_{4}$$ $$=$$ $$($$$$60\!\cdots\!14$$$$\nu^{19} -$$$$35\!\cdots\!31$$$$\nu^{18} +$$$$59\!\cdots\!96$$$$\nu^{17} -$$$$12\!\cdots\!37$$$$\nu^{16} +$$$$62\!\cdots\!37$$$$\nu^{15} -$$$$15\!\cdots\!99$$$$\nu^{14} -$$$$69\!\cdots\!11$$$$\nu^{13} -$$$$62\!\cdots\!74$$$$\nu^{12} +$$$$25\!\cdots\!82$$$$\nu^{11} +$$$$39\!\cdots\!30$$$$\nu^{10} +$$$$10\!\cdots\!98$$$$\nu^{9} +$$$$16\!\cdots\!08$$$$\nu^{8} +$$$$20\!\cdots\!17$$$$\nu^{7} +$$$$27\!\cdots\!39$$$$\nu^{6} +$$$$33\!\cdots\!73$$$$\nu^{5} +$$$$20\!\cdots\!38$$$$\nu^{4} +$$$$17\!\cdots\!86$$$$\nu^{3} +$$$$33\!\cdots\!00$$$$\nu^{2} +$$$$50\!\cdots\!75$$$$\nu -$$$$50\!\cdots\!27$$$$)/$$$$13\!\cdots\!71$$ $$\beta_{5}$$ $$=$$ $$($$$$29\!\cdots\!75$$$$\nu^{19} -$$$$14\!\cdots\!77$$$$\nu^{18} +$$$$54\!\cdots\!14$$$$\nu^{17} +$$$$55\!\cdots\!28$$$$\nu^{16} +$$$$52\!\cdots\!88$$$$\nu^{15} +$$$$43\!\cdots\!96$$$$\nu^{14} -$$$$13\!\cdots\!45$$$$\nu^{13} -$$$$27\!\cdots\!25$$$$\nu^{12} +$$$$22\!\cdots\!40$$$$\nu^{11} +$$$$26\!\cdots\!31$$$$\nu^{10} +$$$$46\!\cdots\!30$$$$\nu^{9} +$$$$53\!\cdots\!24$$$$\nu^{8} +$$$$85\!\cdots\!17$$$$\nu^{7} +$$$$93\!\cdots\!60$$$$\nu^{6} +$$$$55\!\cdots\!74$$$$\nu^{5} +$$$$22\!\cdots\!64$$$$\nu^{4} -$$$$59\!\cdots\!75$$$$\nu^{3} -$$$$12\!\cdots\!43$$$$\nu^{2} -$$$$73\!\cdots\!93$$$$\nu -$$$$31\!\cdots\!72$$$$)/$$$$57\!\cdots\!53$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$30\!\cdots\!48$$$$\nu^{19} +$$$$78\!\cdots\!01$$$$\nu^{18} +$$$$46\!\cdots\!06$$$$\nu^{17} -$$$$16\!\cdots\!49$$$$\nu^{16} +$$$$19\!\cdots\!87$$$$\nu^{15} -$$$$13\!\cdots\!41$$$$\nu^{14} +$$$$20\!\cdots\!72$$$$\nu^{13} +$$$$30\!\cdots\!09$$$$\nu^{12} +$$$$37\!\cdots\!52$$$$\nu^{11} -$$$$31\!\cdots\!35$$$$\nu^{10} -$$$$80\!\cdots\!68$$$$\nu^{9} -$$$$16\!\cdots\!78$$$$\nu^{8} -$$$$32\!\cdots\!52$$$$\nu^{7} -$$$$47\!\cdots\!44$$$$\nu^{6} -$$$$54\!\cdots\!32$$$$\nu^{5} -$$$$59\!\cdots\!32$$$$\nu^{4} -$$$$38\!\cdots\!75$$$$\nu^{3} -$$$$45\!\cdots\!95$$$$\nu^{2} -$$$$53\!\cdots\!92$$$$\nu -$$$$32\!\cdots\!11$$$$)/$$$$57\!\cdots\!53$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$34\!\cdots\!90$$$$\nu^{19} +$$$$35\!\cdots\!10$$$$\nu^{18} -$$$$12\!\cdots\!00$$$$\nu^{17} +$$$$21\!\cdots\!26$$$$\nu^{16} -$$$$48\!\cdots\!02$$$$\nu^{15} +$$$$10\!\cdots\!70$$$$\nu^{14} +$$$$73\!\cdots\!52$$$$\nu^{13} -$$$$79\!\cdots\!37$$$$\nu^{12} -$$$$13\!\cdots\!70$$$$\nu^{11} -$$$$74\!\cdots\!97$$$$\nu^{10} +$$$$40\!\cdots\!87$$$$\nu^{9} +$$$$95\!\cdots\!06$$$$\nu^{8} +$$$$16\!\cdots\!83$$$$\nu^{7} +$$$$27\!\cdots\!17$$$$\nu^{6} +$$$$28\!\cdots\!02$$$$\nu^{5} +$$$$34\!\cdots\!44$$$$\nu^{4} +$$$$18\!\cdots\!45$$$$\nu^{3} +$$$$17\!\cdots\!08$$$$\nu^{2} +$$$$74\!\cdots\!21$$$$\nu +$$$$17\!\cdots\!49$$$$)/$$$$57\!\cdots\!53$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$83\!\cdots\!51$$$$\nu^{19} +$$$$42\!\cdots\!31$$$$\nu^{18} -$$$$12\!\cdots\!72$$$$\nu^{17} -$$$$11\!\cdots\!75$$$$\nu^{16} -$$$$25\!\cdots\!24$$$$\nu^{15} -$$$$10\!\cdots\!65$$$$\nu^{14} +$$$$37\!\cdots\!92$$$$\nu^{13} +$$$$83\!\cdots\!32$$$$\nu^{12} -$$$$63\!\cdots\!37$$$$\nu^{11} -$$$$75\!\cdots\!19$$$$\nu^{10} -$$$$13\!\cdots\!67$$$$\nu^{9} -$$$$15\!\cdots\!21$$$$\nu^{8} -$$$$18\!\cdots\!90$$$$\nu^{7} -$$$$17\!\cdots\!60$$$$\nu^{6} -$$$$65\!\cdots\!05$$$$\nu^{5} -$$$$13\!\cdots\!54$$$$\nu^{4} +$$$$11\!\cdots\!27$$$$\nu^{3} -$$$$20\!\cdots\!68$$$$\nu^{2} +$$$$85\!\cdots\!97$$$$\nu +$$$$33\!\cdots\!22$$$$)/$$$$13\!\cdots\!71$$ $$\beta_{9}$$ $$=$$ $$($$$$36\!\cdots\!04$$$$\nu^{19} -$$$$17\!\cdots\!39$$$$\nu^{18} +$$$$59\!\cdots\!64$$$$\nu^{17} +$$$$12\!\cdots\!45$$$$\nu^{16} +$$$$17\!\cdots\!25$$$$\nu^{15} +$$$$49\!\cdots\!46$$$$\nu^{14} -$$$$10\!\cdots\!06$$$$\nu^{13} -$$$$37\!\cdots\!11$$$$\nu^{12} -$$$$90\!\cdots\!75$$$$\nu^{11} +$$$$26\!\cdots\!21$$$$\nu^{10} +$$$$77\!\cdots\!69$$$$\nu^{9} +$$$$14\!\cdots\!90$$$$\nu^{8} +$$$$24\!\cdots\!15$$$$\nu^{7} +$$$$31\!\cdots\!21$$$$\nu^{6} +$$$$34\!\cdots\!34$$$$\nu^{5} +$$$$32\!\cdots\!93$$$$\nu^{4} +$$$$23\!\cdots\!31$$$$\nu^{3} +$$$$21\!\cdots\!78$$$$\nu^{2} +$$$$48\!\cdots\!36$$$$\nu +$$$$12\!\cdots\!09$$$$)/$$$$57\!\cdots\!53$$ $$\beta_{10}$$ $$=$$ $$($$$$10\!\cdots\!03$$$$\nu^{19} -$$$$59\!\cdots\!87$$$$\nu^{18} +$$$$69\!\cdots\!46$$$$\nu^{17} -$$$$55\!\cdots\!43$$$$\nu^{16} +$$$$70\!\cdots\!60$$$$\nu^{15} +$$$$56\!\cdots\!97$$$$\nu^{14} -$$$$30\!\cdots\!44$$$$\nu^{13} -$$$$97\!\cdots\!85$$$$\nu^{12} +$$$$50\!\cdots\!17$$$$\nu^{11} +$$$$78\!\cdots\!92$$$$\nu^{10} +$$$$17\!\cdots\!60$$$$\nu^{9} +$$$$26\!\cdots\!28$$$$\nu^{8} +$$$$32\!\cdots\!99$$$$\nu^{7} +$$$$32\!\cdots\!33$$$$\nu^{6} +$$$$30\!\cdots\!25$$$$\nu^{5} +$$$$18\!\cdots\!90$$$$\nu^{4} +$$$$12\!\cdots\!63$$$$\nu^{3} +$$$$41\!\cdots\!63$$$$\nu^{2} -$$$$14\!\cdots\!49$$$$\nu +$$$$79\!\cdots\!84$$$$)/$$$$13\!\cdots\!71$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$53\!\cdots\!45$$$$\nu^{19} +$$$$21\!\cdots\!57$$$$\nu^{18} +$$$$27\!\cdots\!29$$$$\nu^{17} -$$$$11\!\cdots\!61$$$$\nu^{16} -$$$$15\!\cdots\!06$$$$\nu^{15} -$$$$10\!\cdots\!92$$$$\nu^{14} +$$$$19\!\cdots\!05$$$$\nu^{13} +$$$$73\!\cdots\!09$$$$\nu^{12} +$$$$15\!\cdots\!76$$$$\nu^{11} -$$$$52\!\cdots\!41$$$$\nu^{10} -$$$$13\!\cdots\!58$$$$\nu^{9} -$$$$19\!\cdots\!88$$$$\nu^{8} -$$$$27\!\cdots\!47$$$$\nu^{7} -$$$$33\!\cdots\!11$$$$\nu^{6} -$$$$31\!\cdots\!93$$$$\nu^{5} -$$$$28\!\cdots\!63$$$$\nu^{4} -$$$$17\!\cdots\!03$$$$\nu^{3} -$$$$13\!\cdots\!82$$$$\nu^{2} -$$$$72\!\cdots\!82$$$$\nu +$$$$15\!\cdots\!03$$$$)/$$$$58\!\cdots\!77$$ $$\beta_{12}$$ $$=$$ $$($$$$15\!\cdots\!75$$$$\nu^{19} -$$$$88\!\cdots\!10$$$$\nu^{18} +$$$$14\!\cdots\!98$$$$\nu^{17} -$$$$23\!\cdots\!82$$$$\nu^{16} +$$$$11\!\cdots\!77$$$$\nu^{15} +$$$$50\!\cdots\!51$$$$\nu^{14} -$$$$30\!\cdots\!46$$$$\nu^{13} -$$$$11\!\cdots\!67$$$$\nu^{12} +$$$$42\!\cdots\!64$$$$\nu^{11} +$$$$85\!\cdots\!04$$$$\nu^{10} +$$$$23\!\cdots\!39$$$$\nu^{9} +$$$$49\!\cdots\!73$$$$\nu^{8} +$$$$76\!\cdots\!49$$$$\nu^{7} +$$$$89\!\cdots\!50$$$$\nu^{6} +$$$$99\!\cdots\!53$$$$\nu^{5} +$$$$80\!\cdots\!34$$$$\nu^{4} +$$$$54\!\cdots\!40$$$$\nu^{3} +$$$$27\!\cdots\!26$$$$\nu^{2} +$$$$18\!\cdots\!10$$$$\nu +$$$$46\!\cdots\!54$$$$)/$$$$13\!\cdots\!71$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$15\!\cdots\!00$$$$\nu^{19} +$$$$10\!\cdots\!34$$$$\nu^{18} -$$$$21\!\cdots\!20$$$$\nu^{17} +$$$$32\!\cdots\!41$$$$\nu^{16} -$$$$13\!\cdots\!95$$$$\nu^{15} +$$$$60\!\cdots\!93$$$$\nu^{14} +$$$$37\!\cdots\!02$$$$\nu^{13} +$$$$95\!\cdots\!48$$$$\nu^{12} -$$$$18\!\cdots\!83$$$$\nu^{11} -$$$$87\!\cdots\!54$$$$\nu^{10} -$$$$14\!\cdots\!92$$$$\nu^{9} -$$$$22\!\cdots\!39$$$$\nu^{8} -$$$$28\!\cdots\!77$$$$\nu^{7} -$$$$24\!\cdots\!71$$$$\nu^{6} -$$$$30\!\cdots\!45$$$$\nu^{5} -$$$$21\!\cdots\!62$$$$\nu^{4} -$$$$13\!\cdots\!71$$$$\nu^{3} +$$$$83\!\cdots\!97$$$$\nu^{2} +$$$$97\!\cdots\!50$$$$\nu -$$$$29\!\cdots\!21$$$$)/$$$$13\!\cdots\!71$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$17\!\cdots\!99$$$$\nu^{19} +$$$$98\!\cdots\!31$$$$\nu^{18} -$$$$11\!\cdots\!72$$$$\nu^{17} +$$$$48\!\cdots\!64$$$$\nu^{16} -$$$$11\!\cdots\!37$$$$\nu^{15} -$$$$70\!\cdots\!43$$$$\nu^{14} +$$$$52\!\cdots\!14$$$$\nu^{13} +$$$$16\!\cdots\!64$$$$\nu^{12} -$$$$14\!\cdots\!03$$$$\nu^{11} -$$$$13\!\cdots\!38$$$$\nu^{10} -$$$$26\!\cdots\!77$$$$\nu^{9} -$$$$33\!\cdots\!22$$$$\nu^{8} -$$$$43\!\cdots\!71$$$$\nu^{7} -$$$$53\!\cdots\!66$$$$\nu^{6} -$$$$53\!\cdots\!16$$$$\nu^{5} -$$$$39\!\cdots\!60$$$$\nu^{4} -$$$$40\!\cdots\!91$$$$\nu^{3} -$$$$29\!\cdots\!41$$$$\nu^{2} -$$$$46\!\cdots\!67$$$$\nu -$$$$41\!\cdots\!67$$$$)/$$$$13\!\cdots\!71$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$76\!\cdots\!54$$$$\nu^{19} +$$$$42\!\cdots\!31$$$$\nu^{18} -$$$$58\!\cdots\!07$$$$\nu^{17} +$$$$10\!\cdots\!60$$$$\nu^{16} -$$$$64\!\cdots\!83$$$$\nu^{15} -$$$$20\!\cdots\!80$$$$\nu^{14} +$$$$12\!\cdots\!09$$$$\nu^{13} +$$$$76\!\cdots\!78$$$$\nu^{12} -$$$$17\!\cdots\!84$$$$\nu^{11} -$$$$49\!\cdots\!61$$$$\nu^{10} -$$$$13\!\cdots\!99$$$$\nu^{9} -$$$$24\!\cdots\!91$$$$\nu^{8} -$$$$34\!\cdots\!97$$$$\nu^{7} -$$$$40\!\cdots\!18$$$$\nu^{6} -$$$$42\!\cdots\!66$$$$\nu^{5} -$$$$28\!\cdots\!57$$$$\nu^{4} -$$$$20\!\cdots\!68$$$$\nu^{3} -$$$$44\!\cdots\!57$$$$\nu^{2} -$$$$31\!\cdots\!54$$$$\nu +$$$$38\!\cdots\!59$$$$)/$$$$57\!\cdots\!53$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$94\!\cdots\!52$$$$\nu^{19} +$$$$58\!\cdots\!64$$$$\nu^{18} -$$$$11\!\cdots\!80$$$$\nu^{17} +$$$$20\!\cdots\!05$$$$\nu^{16} -$$$$85\!\cdots\!85$$$$\nu^{15} +$$$$21\!\cdots\!33$$$$\nu^{14} +$$$$14\!\cdots\!13$$$$\nu^{13} +$$$$71\!\cdots\!79$$$$\nu^{12} -$$$$62\!\cdots\!31$$$$\nu^{11} -$$$$49\!\cdots\!67$$$$\nu^{10} -$$$$12\!\cdots\!31$$$$\nu^{9} -$$$$25\!\cdots\!31$$$$\nu^{8} -$$$$39\!\cdots\!58$$$$\nu^{7} -$$$$47\!\cdots\!54$$$$\nu^{6} -$$$$53\!\cdots\!09$$$$\nu^{5} -$$$$38\!\cdots\!85$$$$\nu^{4} -$$$$35\!\cdots\!75$$$$\nu^{3} -$$$$10\!\cdots\!38$$$$\nu^{2} -$$$$28\!\cdots\!75$$$$\nu +$$$$14\!\cdots\!27$$$$)/$$$$57\!\cdots\!53$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$22\!\cdots\!79$$$$\nu^{19} +$$$$13\!\cdots\!06$$$$\nu^{18} -$$$$19\!\cdots\!93$$$$\nu^{17} +$$$$22\!\cdots\!03$$$$\nu^{16} -$$$$16\!\cdots\!48$$$$\nu^{15} -$$$$31\!\cdots\!32$$$$\nu^{14} +$$$$56\!\cdots\!07$$$$\nu^{13} +$$$$19\!\cdots\!47$$$$\nu^{12} -$$$$18\!\cdots\!64$$$$\nu^{11} -$$$$15\!\cdots\!03$$$$\nu^{10} -$$$$31\!\cdots\!19$$$$\nu^{9} -$$$$46\!\cdots\!31$$$$\nu^{8} -$$$$58\!\cdots\!99$$$$\nu^{7} -$$$$54\!\cdots\!15$$$$\nu^{6} -$$$$41\!\cdots\!94$$$$\nu^{5} -$$$$13\!\cdots\!56$$$$\nu^{4} -$$$$12\!\cdots\!56$$$$\nu^{3} +$$$$13\!\cdots\!50$$$$\nu^{2} +$$$$81\!\cdots\!07$$$$\nu +$$$$58\!\cdots\!09$$$$)/$$$$13\!\cdots\!71$$ $$\beta_{18}$$ $$=$$ $$($$$$10\!\cdots\!78$$$$\nu^{19} -$$$$71\!\cdots\!93$$$$\nu^{18} +$$$$14\!\cdots\!10$$$$\nu^{17} -$$$$21\!\cdots\!06$$$$\nu^{16} +$$$$93\!\cdots\!32$$$$\nu^{15} -$$$$36\!\cdots\!75$$$$\nu^{14} -$$$$26\!\cdots\!21$$$$\nu^{13} -$$$$73\!\cdots\!76$$$$\nu^{12} +$$$$12\!\cdots\!01$$$$\nu^{11} +$$$$64\!\cdots\!38$$$$\nu^{10} +$$$$11\!\cdots\!02$$$$\nu^{9} +$$$$16\!\cdots\!93$$$$\nu^{8} +$$$$18\!\cdots\!19$$$$\nu^{7} +$$$$12\!\cdots\!29$$$$\nu^{6} +$$$$11\!\cdots\!52$$$$\nu^{5} -$$$$69\!\cdots\!85$$$$\nu^{4} -$$$$65\!\cdots\!40$$$$\nu^{3} -$$$$10\!\cdots\!94$$$$\nu^{2} -$$$$86\!\cdots\!08$$$$\nu -$$$$96\!\cdots\!46$$$$)/$$$$57\!\cdots\!53$$ $$\beta_{19}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!89$$$$\nu^{19} +$$$$68\!\cdots\!32$$$$\nu^{18} -$$$$10\!\cdots\!57$$$$\nu^{17} +$$$$13\!\cdots\!18$$$$\nu^{16} -$$$$85\!\cdots\!62$$$$\nu^{15} -$$$$41\!\cdots\!59$$$$\nu^{14} +$$$$27\!\cdots\!71$$$$\nu^{13} +$$$$96\!\cdots\!50$$$$\nu^{12} -$$$$54\!\cdots\!28$$$$\nu^{11} -$$$$73\!\cdots\!71$$$$\nu^{10} -$$$$18\!\cdots\!77$$$$\nu^{9} -$$$$33\!\cdots\!99$$$$\nu^{8} -$$$$50\!\cdots\!73$$$$\nu^{7} -$$$$58\!\cdots\!99$$$$\nu^{6} -$$$$65\!\cdots\!78$$$$\nu^{5} -$$$$53\!\cdots\!86$$$$\nu^{4} -$$$$39\!\cdots\!95$$$$\nu^{3} -$$$$22\!\cdots\!11$$$$\nu^{2} -$$$$35\!\cdots\!55$$$$\nu -$$$$19\!\cdots\!17$$$$)/$$$$57\!\cdots\!53$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{19} - \beta_{17} - \beta_{15} + \beta_{14} - \beta_{12} + \beta_{10} + 3 \beta_{9} + \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{2} + 2$$ $$\nu^{3}$$ $$=$$ $$11 \beta_{19} + \beta_{18} - 7 \beta_{17} + \beta_{16} - 8 \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + 4 \beta_{9} + 7 \beta_{8} - 8 \beta_{7} - 8 \beta_{6} - 8 \beta_{5} + \beta_{3} - 12 \beta_{2} + \beta_{1} + 11$$ $$\nu^{4}$$ $$=$$ $$29 \beta_{19} + \beta_{18} - 14 \beta_{17} - \beta_{16} - 15 \beta_{15} - 6 \beta_{14} - 14 \beta_{13} - \beta_{12} + 6 \beta_{11} + 14 \beta_{9} + 14 \beta_{8} - 15 \beta_{7} - 15 \beta_{6} - 25 \beta_{5} - 10 \beta_{4} - 39 \beta_{2} + 10 \beta_{1} + 25$$ $$\nu^{5}$$ $$=$$ $$64 \beta_{19} + 7 \beta_{18} - 17 \beta_{17} - 21 \beta_{16} + 10 \beta_{15} - 8 \beta_{14} - 69 \beta_{13} + 8 \beta_{12} - 17 \beta_{11} + 35 \beta_{9} + 36 \beta_{8} - 3 \beta_{7} - 11 \beta_{6} - 46 \beta_{5} - 42 \beta_{4} + 8 \beta_{3} - 64 \beta_{2} + 25 \beta_{1} + 11$$ $$\nu^{6}$$ $$=$$ $$149 \beta_{19} + 148 \beta_{18} - 133 \beta_{16} + 148 \beta_{15} + 13 \beta_{14} - 183 \beta_{13} + 34 \beta_{12} - 134 \beta_{11} - 17 \beta_{10} + 134 \beta_{8} + 133 \beta_{7} - 70 \beta_{5} - 183 \beta_{4} + 111 \beta_{3} - 70 \beta_{2} + 17 \beta_{1} - 111$$ $$\nu^{7}$$ $$=$$ $$747 \beta_{18} + 240 \beta_{17} - 587 \beta_{16} + 915 \beta_{15} - 535 \beta_{13} + 17 \beta_{12} - 535 \beta_{11} - 295 \beta_{10} - 404 \beta_{9} + 278 \beta_{8} + 992 \beta_{7} + 245 \beta_{6} - 765 \beta_{4} + 404 \beta_{3} + 245 \beta_{2} + 5 \beta_{1} - 832$$ $$\nu^{8}$$ $$=$$ $$-2565 \beta_{19} + 2871 \beta_{18} + 1956 \beta_{17} - 2343 \beta_{16} + 5697 \beta_{15} - 198 \beta_{14} - 1773 \beta_{13} - 2367 \beta_{11} - 1773 \beta_{10} - 2343 \beta_{9} + 307 \beta_{8} + 5697 \beta_{7} + 2565 \beta_{6} + 1815 \beta_{5} - 2565 \beta_{4} + 1015 \beta_{3} + 3580 \beta_{2} + 198 \beta_{1} - 5436$$ $$\nu^{9}$$ $$=$$ $$-16436 \beta_{19} + 11137 \beta_{18} + 8882 \beta_{17} - 8052 \beta_{16} + 26583 \beta_{15} - 913 \beta_{14} - 4395 \beta_{13} + 78 \beta_{12} - 8882 \beta_{11} - 7579 \beta_{10} - 11137 \beta_{9} + 913 \beta_{8} + 26390 \beta_{7} + 13277 \beta_{6} + 13277 \beta_{5} - 7579 \beta_{4} + 3159 \beta_{3} + 21329 \beta_{2} + 78 \beta_{1} - 26583$$ $$\nu^{10}$$ $$=$$ $$-75250 \beta_{19} + 38542 \beta_{18} + 30704 \beta_{17} - 21783 \beta_{16} + 101258 \beta_{15} - 8988 \beta_{14} - 8988 \beta_{13} + 1806 \beta_{12} - 23522 \beta_{11} - 32510 \beta_{10} - 47791 \beta_{9} + 3380 \beta_{8} + 105731 \beta_{7} + 53467 \beta_{6} + 61381 \beta_{5} - 22763 \beta_{4} + 7914 \beta_{3} + 92009 \beta_{2} - 1574 \beta_{1} - 105731$$ $$\nu^{11}$$ $$=$$ $$-313555 \beta_{19} + 102085 \beta_{18} + 103091 \beta_{17} - 40451 \beta_{16} + 360429 \beta_{15} - 67186 \beta_{14} - 23525 \beta_{13} + 23525 \beta_{12} - 41193 \beta_{11} - 131642 \beta_{10} - 180056 \beta_{9} + 391526 \beta_{7} + 211470 \beta_{6} + 251921 \beta_{5} - 67186 \beta_{4} + 360429 \beta_{2} - 395560$$ $$\nu^{12}$$ $$=$$ $$-1201324 \beta_{19} + 185477 \beta_{18} + 338433 \beta_{17} + 1201324 \beta_{15} - 338433 \beta_{14} - 57090 \beta_{13} + 168682 \beta_{12} - 458450 \beta_{10} - 621768 \beta_{9} - 57090 \beta_{8} + 1346177 \beta_{7} + 796883 \beta_{6} + 982360 \beta_{5} - 168682 \beta_{4} - 90187 \beta_{3} + 1346177 \beta_{2} + 992 \beta_{1} - 1418651$$ $$\nu^{13}$$ $$=$$ $$-4147140 \beta_{19} + 999504 \beta_{17} + 521377 \beta_{16} + 3466498 \beta_{15} - 1408112 \beta_{14} + 878365 \beta_{12} + 529747 \beta_{11} - 1408112 \beta_{10} - 2040815 \beta_{9} - 433682 \beta_{8} + 4147140 \beta_{7} + 2668546 \beta_{6} + 3466498 \beta_{5} - 312543 \beta_{4} - 521377 \beta_{3} + 4599753 \beta_{2} - 121139 \beta_{1} - 4599753$$ $$\nu^{14}$$ $$=$$ $$-13098153 \beta_{19} - 2467606 \beta_{18} + 2727662 \beta_{17} + 3555288 \beta_{16} + 8021991 \beta_{15} - 5294329 \beta_{14} + 922370 \beta_{13} + 3798522 \beta_{12} + 3650258 \beta_{11} - 3798522 \beta_{10} - 6196805 \beta_{9} - 2727662 \beta_{8} + 10943816 \beta_{7} + 8021991 \beta_{6} + 10943816 \beta_{5} - 2467606 \beta_{3} + 14218796 \beta_{2} - 922370 \beta_{1} - 13098153$$ $$\nu^{15}$$ $$=$$ $$-37638988 \beta_{19} - 17315737 \beta_{18} + 7073276 \beta_{17} + 17315737 \beta_{16} + 10881693 \beta_{15} - 17414930 \beta_{14} + 7073276 \beta_{13} + 14334018 \beta_{12} + 17414930 \beta_{11} - 7494453 \beta_{10} - 16231694 \beta_{9} - 14567729 \beta_{8} + 21407294 \beta_{7} + 21407294 \beta_{6} + 31073356 \beta_{5} + 4881999 \beta_{4} - 10525601 \beta_{3} + 39605501 \beta_{2} - 4881999 \beta_{1} - 31073356$$ $$\nu^{16}$$ $$=$$ $$-90164625 \beta_{19} - 86028536 \beta_{18} + 14293796 \beta_{17} + 72372259 \beta_{16} - 30137932 \beta_{15} - 46873408 \beta_{14} + 41640299 \beta_{13} + 46873408 \beta_{12} + 70452311 \beta_{11} - 31171292 \beta_{9} - 65813230 \beta_{8} + 2757215 \beta_{7} + 42234327 \beta_{6} + 73405619 \beta_{5} + 37872699 \beta_{4} - 39477112 \beta_{3} + 90164625 \beta_{2} - 23578903 \beta_{1} - 42234327$$ $$\nu^{17}$$ $$=$$ $$-125514925 \beta_{19} - 365683870 \beta_{18} + 276298682 \beta_{16} - 365683870 \beta_{15} - 89053662 \beta_{14} + 209154735 \beta_{13} + 126882870 \beta_{12} + 262509569 \beta_{11} + 100644316 \beta_{10} - 262509569 \beta_{8} - 276298682 \beta_{7} + 94400822 \beta_{5} + 209154735 \beta_{4} - 135671513 \beta_{3} + 94400822 \beta_{2} - 100644316 \beta_{1} + 135671513$$ $$\nu^{18}$$ $$=$$ $$337802687 \beta_{19} - 1421774709 \beta_{18} - 214501895 \beta_{17} + 981157816 \beta_{16} - 2180733691 \beta_{15} + 915891844 \beta_{13} + 243384338 \beta_{12} + 915891844 \beta_{11} + 701389949 \beta_{10} + 443831579 \beta_{9} - 944774287 \beta_{8} - 2004430959 \beta_{7} - 582656250 \beta_{6} - 337802687 \beta_{5} + 981988330 \beta_{4} - 443831579 \beta_{3} - 582656250 \beta_{2} - 368154355 \beta_{1} + 1563814066$$ $$\nu^{19}$$ $$=$$ $$4141128378 \beta_{19} - 5132324382 \beta_{18} - 1606872268 \beta_{17} + 3236739490 \beta_{16} - 10461611385 \beta_{15} + 1170889285 \beta_{14} + 3591921664 \beta_{13} + 2970239093 \beta_{11} + 3591921664 \beta_{10} + 3236739490 \beta_{9} - 3012616711 \beta_{8} - 10461611385 \beta_{7} - 4141128378 \beta_{6} - 3675699812 \beta_{5} + 4141128378 \beta_{4} - 1344895634 \beta_{3} - 5486024012 \beta_{2} - 1170889285 \beta_{1} + 9273452760$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/966\mathbb{Z}\right)^\times$$.

 $$n$$ $$323$$ $$829$$ $$925$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
85.1
 0.474546 − 1.03911i −0.511167 + 1.11930i 3.56300 + 1.04619i −1.42457 − 0.418291i 0.287778 − 0.332113i −1.18676 + 1.36960i 2.68238 − 1.72386i −0.665158 + 0.427471i 0.287778 + 0.332113i −1.18676 − 1.36960i 0.150891 + 1.04947i −0.370938 − 2.57993i 2.68238 + 1.72386i −0.665158 − 0.427471i 3.56300 − 1.04619i −1.42457 + 0.418291i 0.474546 + 1.03911i −0.511167 − 1.11930i 0.150891 − 1.04947i −0.370938 + 2.57993i
−0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i −0.129407 0.283361i 0.142315 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i 0.298893 + 0.0877630i
85.2 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i 0.856306 + 1.87505i 0.142315 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −1.97783 0.580743i
127.1 0.841254 0.540641i 0.142315 0.989821i 0.415415 0.909632i −1.72174 + 0.505549i −0.415415 0.909632i −0.654861 0.755750i −0.142315 0.989821i −0.959493 0.281733i −1.17510 + 1.35614i
127.2 0.841254 0.540641i 0.142315 0.989821i 0.415415 0.909632i 3.26582 0.958931i −0.415415 0.909632i −0.654861 0.755750i −0.142315 0.989821i −0.959493 0.281733i 2.22895 2.57234i
169.1 −0.142315 + 0.989821i −0.415415 0.909632i −0.959493 0.281733i 0.569907 + 0.657708i 0.959493 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i −0.732120 + 0.470505i
169.2 −0.142315 + 0.989821i −0.415415 0.909632i −0.959493 0.281733i 2.04445 + 2.35942i 0.959493 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i −2.62636 + 1.68786i
211.1 0.415415 + 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i −1.26697 0.814230i 0.654861 + 0.755750i −0.142315 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0.214333 1.49072i
211.2 0.415415 + 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i 2.08057 + 1.33710i 0.654861 + 0.755750i −0.142315 0.989821i −0.959493 0.281733i 0.841254 0.540641i −0.351971 + 2.44801i
463.1 −0.142315 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.569907 0.657708i 0.959493 + 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i −0.732120 0.470505i
463.2 −0.142315 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 2.04445 2.35942i 0.959493 + 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i −2.62636 1.68786i
547.1 −0.959493 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i −0.110384 + 0.767734i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i −0.142315 0.989821i 0.322208 0.705537i
547.2 −0.959493 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i 0.411445 2.86166i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i −0.142315 0.989821i −1.20100 + 2.62983i
673.1 0.415415 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i −1.26697 + 0.814230i 0.654861 0.755750i −0.142315 + 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.214333 + 1.49072i
673.2 0.415415 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i 2.08057 1.33710i 0.654861 0.755750i −0.142315 + 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.351971 2.44801i
715.1 0.841254 + 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i −1.72174 0.505549i −0.415415 + 0.909632i −0.654861 + 0.755750i −0.142315 + 0.989821i −0.959493 + 0.281733i −1.17510 1.35614i
715.2 0.841254 + 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i 3.26582 + 0.958931i −0.415415 + 0.909632i −0.654861 + 0.755750i −0.142315 + 0.989821i −0.959493 + 0.281733i 2.22895 + 2.57234i
841.1 −0.654861 0.755750i −0.841254 0.540641i −0.142315 + 0.989821i −0.129407 + 0.283361i 0.142315 + 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i 0.298893 0.0877630i
841.2 −0.654861 0.755750i −0.841254 0.540641i −0.142315 + 0.989821i 0.856306 1.87505i 0.142315 + 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i −1.97783 + 0.580743i
883.1 −0.959493 + 0.281733i 0.654861 + 0.755750i 0.841254 0.540641i −0.110384 0.767734i −0.841254 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.322208 + 0.705537i
883.2 −0.959493 + 0.281733i 0.654861 + 0.755750i 0.841254 0.540641i 0.411445 + 2.86166i −0.841254 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i −0.142315 + 0.989821i −1.20100 2.62983i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 883.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.q.d 20
23.c even 11 1 inner 966.2.q.d 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.d 20 1.a even 1 1 trivial
966.2.q.d 20 23.c even 11 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{20} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
$3$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
$5$ $$7921 + 12282 T + 74710 T^{2} - 58419 T^{3} + 151272 T^{4} - 86030 T^{5} + 123272 T^{6} + 50829 T^{7} - 46066 T^{8} - 2937 T^{9} + 30119 T^{10} - 14652 T^{11} - 1380 T^{12} + 4623 T^{13} - 1695 T^{14} - 186 T^{15} + 462 T^{16} - 236 T^{17} + 69 T^{18} - 12 T^{19} + T^{20}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
$11$ $$121 - 2057 T + 18150 T^{2} - 57112 T^{3} + 35090 T^{4} + 163570 T^{5} - 12870 T^{6} - 174702 T^{7} - 94688 T^{8} + 31735 T^{9} + 96647 T^{10} + 77657 T^{11} + 50136 T^{12} + 24096 T^{13} + 10652 T^{14} + 3662 T^{15} + 1125 T^{16} + 277 T^{17} + 60 T^{18} + 7 T^{19} + T^{20}$$
$13$ $$529 + 5911 T + 48784 T^{2} + 212800 T^{3} + 1049404 T^{4} + 2699914 T^{5} + 6158919 T^{6} + 6167387 T^{7} + 3859689 T^{8} + 2090033 T^{9} + 1180114 T^{10} + 655677 T^{11} + 333756 T^{12} + 125310 T^{13} + 23986 T^{14} - 844 T^{15} - 1300 T^{16} - 246 T^{17} + 7 T^{18} + 8 T^{19} + T^{20}$$
$17$ $$19562929 + 53960600 T + 119397241 T^{2} + 130865946 T^{3} + 219335941 T^{4} + 188359238 T^{5} + 187173537 T^{6} + 119900524 T^{7} + 57551760 T^{8} + 20677613 T^{9} + 1226446 T^{10} - 1471041 T^{11} - 116561 T^{12} - 9410 T^{13} + 64810 T^{14} - 10680 T^{15} - 880 T^{16} + 237 T^{17} + 39 T^{18} - 13 T^{19} + T^{20}$$
$19$ $$591361 - 7541583 T + 25172064 T^{2} + 3961021 T^{3} + 27918735 T^{4} + 4978463 T^{5} - 4161279 T^{6} + 6458447 T^{7} + 10148645 T^{8} + 6182579 T^{9} + 3360544 T^{10} + 2198017 T^{11} + 1137789 T^{12} + 269430 T^{13} + 71055 T^{14} + 8718 T^{15} + 3425 T^{16} + 608 T^{17} + 81 T^{18} + 8 T^{19} + T^{20}$$
$23$ $$41426511213649 - 2505951528992 T^{2} - 1123592397510 T^{3} + 92966538292 T^{4} + 60958604553 T^{5} + 10472489743 T^{6} - 2332377399 T^{7} - 591966341 T^{8} + 11896566 T^{9} + 29715973 T^{10} + 517242 T^{11} - 1119029 T^{12} - 191697 T^{13} + 37423 T^{14} + 9471 T^{15} + 628 T^{16} - 330 T^{17} - 32 T^{18} + T^{20}$$
$29$ $$4280761 - 14489207 T + 16461807 T^{2} - 9643431 T^{3} + 9586009 T^{4} - 4756271 T^{5} + 754971 T^{6} + 557255 T^{7} + 256925 T^{8} + 1061874 T^{9} + 286551 T^{10} + 64856 T^{11} + 105176 T^{12} - 43716 T^{13} + 15098 T^{14} + 2258 T^{15} + 2110 T^{16} + 84 T^{17} + 29 T^{18} - 3 T^{19} + T^{20}$$
$31$ $$113401201 - 382192610 T + 649069676 T^{2} - 564855207 T^{3} + 279499480 T^{4} - 108953653 T^{5} + 55674728 T^{6} - 36016777 T^{7} + 24673454 T^{8} - 14149091 T^{9} + 6256207 T^{10} - 2458467 T^{11} + 921694 T^{12} - 273184 T^{13} + 61443 T^{14} - 13018 T^{15} + 2503 T^{16} - 409 T^{17} + 95 T^{18} - 9 T^{19} + T^{20}$$
$37$ $$73462139521 + 133435752168 T + 46726794247 T^{2} - 82460746501 T^{3} + 214804395446 T^{4} - 132723526700 T^{5} + 36698021093 T^{6} - 17506861206 T^{7} + 6521551822 T^{8} - 686486752 T^{9} + 330052074 T^{10} + 25170330 T^{11} + 2379358 T^{12} + 544531 T^{13} + 150746 T^{14} + 21422 T^{15} + 1464 T^{16} + 39 T^{17} + 9 T^{18} + T^{19} + T^{20}$$
$41$ $$529 + 20378 T + 302750 T^{2} + 1119504 T^{3} + 1718895 T^{4} - 9961429 T^{5} + 9167072 T^{6} - 3853742 T^{7} + 6401602 T^{8} - 8144301 T^{9} + 5910928 T^{10} - 3001306 T^{11} + 1311579 T^{12} - 436021 T^{13} + 117938 T^{14} - 26043 T^{15} + 4685 T^{16} - 740 T^{17} + 98 T^{18} - 10 T^{19} + T^{20}$$
$43$ $$982081 - 5904378 T + 1290866801 T^{2} + 5859702157 T^{3} + 9484425533 T^{4} + 848870912 T^{5} + 246120574 T^{6} + 703434816 T^{7} + 221180574 T^{8} - 338711241 T^{9} + 32811329 T^{10} - 1435467 T^{11} + 5802103 T^{12} - 221324 T^{13} - 14227 T^{14} - 40052 T^{15} - 682 T^{16} + 166 T^{17} + 158 T^{18} + 15 T^{19} + T^{20}$$
$47$ $$( 2025893 + 2511315 T - 1514838 T^{2} - 1511270 T^{3} + 226251 T^{4} + 160689 T^{5} + 4635 T^{6} - 3064 T^{7} - 189 T^{8} + 14 T^{9} + T^{10} )^{2}$$
$53$ $$47475035185369 + 49521125276997 T + 12725550394007 T^{2} - 7923063388031 T^{3} - 1126313357455 T^{4} + 5081923758285 T^{5} + 2930068649937 T^{6} + 730885691217 T^{7} + 225975833215 T^{8} + 581944264 T^{9} + 9632933311 T^{10} - 1461632588 T^{11} + 294487967 T^{12} - 45637232 T^{13} + 5583231 T^{14} - 617222 T^{15} + 66581 T^{16} - 7077 T^{17} + 652 T^{18} - 38 T^{19} + T^{20}$$
$59$ $$195345436441 + 766616111332 T + 6666962707028 T^{2} + 14363280604782 T^{3} + 16711752840532 T^{4} + 11929269153499 T^{5} + 4572618168098 T^{6} + 756909982996 T^{7} + 229059899616 T^{8} + 39308970481 T^{9} + 4761919790 T^{10} + 485737780 T^{11} + 176907725 T^{12} - 7141967 T^{13} + 913535 T^{14} + 119719 T^{15} + 8040 T^{16} - 2659 T^{17} - 100 T^{18} + 10 T^{19} + T^{20}$$
$61$ $$381272905729 + 147961967625 T - 194396433063 T^{2} - 208168934099 T^{3} - 3013353893 T^{4} + 68784591030 T^{5} + 40462491981 T^{6} + 11268223816 T^{7} + 3735146674 T^{8} + 1323597165 T^{9} + 352981564 T^{10} + 62710362 T^{11} + 24680185 T^{12} + 2575780 T^{13} + 640965 T^{14} + 92478 T^{15} + 9416 T^{16} + 1443 T^{17} + 108 T^{18} + 6 T^{19} + T^{20}$$
$67$ $$43574897108412121 - 4100237577136552 T - 7670828615341128 T^{2} - 317280008623800 T^{3} + 2350138696056644 T^{4} - 1099603174739789 T^{5} + 294979884152260 T^{6} - 58528980402392 T^{7} + 10137651978543 T^{8} - 1602143037829 T^{9} + 231153704848 T^{10} - 29613472430 T^{11} + 3517490673 T^{12} - 377135191 T^{13} + 38380513 T^{14} - 3306242 T^{15} + 256689 T^{16} - 17234 T^{17} + 884 T^{18} - 36 T^{19} + T^{20}$$
$71$ $$603311046361 + 1065802315884 T - 877743788465 T^{2} - 936824739988 T^{3} + 1689916611086 T^{4} - 1114591512759 T^{5} + 457761876504 T^{6} - 138364114223 T^{7} + 34888388002 T^{8} - 7689966680 T^{9} + 1477921490 T^{10} - 268093276 T^{11} + 45410631 T^{12} - 5826440 T^{13} + 966219 T^{14} - 107436 T^{15} + 12001 T^{16} - 1186 T^{17} + 150 T^{18} + T^{19} + T^{20}$$
$73$ $$29094577935721 - 45796160291700 T + 68690294210458 T^{2} - 42398933570421 T^{3} + 25464215507601 T^{4} - 10353045355251 T^{5} + 3633678075491 T^{6} - 861015302490 T^{7} + 182682912729 T^{8} - 34879693486 T^{9} + 10080679874 T^{10} - 3767687946 T^{11} + 1019172951 T^{12} - 188895754 T^{13} + 26703816 T^{14} - 3284848 T^{15} + 364044 T^{16} - 31826 T^{17} + 1880 T^{18} - 65 T^{19} + T^{20}$$
$79$ $$2135456521 - 1084248693 T - 14385881543 T^{2} - 12026205609 T^{3} + 44606132967 T^{4} + 85391575401 T^{5} + 67086584356 T^{6} + 10330309660 T^{7} + 1265596574 T^{8} + 2591841395 T^{9} + 659808469 T^{10} + 69225630 T^{11} + 29669883 T^{12} - 2104244 T^{13} + 912396 T^{14} - 102343 T^{15} + 19207 T^{16} - 1649 T^{17} + 210 T^{18} - 10 T^{19} + T^{20}$$
$83$ $$272358943896601 + 233528343477002 T + 382179948715506 T^{2} + 152422589044163 T^{3} - 35745505223719 T^{4} - 37091328339950 T^{5} + 1844903949952 T^{6} + 2583183983326 T^{7} + 120587037740 T^{8} - 49361345759 T^{9} - 5200696929 T^{10} + 600092097 T^{11} + 431062349 T^{12} + 58486467 T^{13} + 7730117 T^{14} + 507996 T^{15} + 54545 T^{16} + 2988 T^{17} + 354 T^{18} + 14 T^{19} + T^{20}$$
$89$ $$21708582607009 + 5704912485716 T + 58548277937936 T^{2} - 11146407965401 T^{3} + 6279870968797 T^{4} - 1494144611702 T^{5} - 406476967517 T^{6} - 91472153040 T^{7} + 28743924793 T^{8} + 13473281360 T^{9} + 2399618695 T^{10} + 207333104 T^{11} + 17343965 T^{12} + 5145463 T^{13} + 1108344 T^{14} + 100920 T^{15} + 1383 T^{16} - 160 T^{17} + 105 T^{18} + 18 T^{19} + T^{20}$$
$97$ $$( 3659569 - 1513183 T - 306536 T^{2} + 159477 T^{3} + 64461 T^{4} - 43308 T^{5} + 8086 T^{6} - 243 T^{7} + 243 T^{8} + 10 T^{9} + T^{10} )^{2}$$
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