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:

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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