Properties

Label 966.2.q.e
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: \(x^{20} - 9 x^{19} + 42 x^{18} - 126 x^{17} + 277 x^{16} - 538 x^{15} + 1046 x^{14} - 1170 x^{13} - 619 x^{12} + 2218 x^{11} + 1660 x^{10} - 3225 x^{9} + 2719 x^{8} + 8761 x^{7} + 14705 x^{6} + 21691 x^{5} + 21163 x^{4} + 14656 x^{3} + 7846 x^{2} + 2622 x + 529\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 9 x^{19} + 42 x^{18} - 126 x^{17} + 277 x^{16} - 538 x^{15} + 1046 x^{14} - 1170 x^{13} - 619 x^{12} + 2218 x^{11} + 1660 x^{10} - 3225 x^{9} + 2719 x^{8} + 8761 x^{7} + 14705 x^{6} + 21691 x^{5} + 21163 x^{4} + 14656 x^{3} + 7846 x^{2} + 2622 x + 529\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(22\!\cdots\!16\)\( \nu^{19} - \)\(25\!\cdots\!51\)\( \nu^{18} + \)\(14\!\cdots\!15\)\( \nu^{17} - \)\(53\!\cdots\!15\)\( \nu^{16} + \)\(14\!\cdots\!70\)\( \nu^{15} - \)\(30\!\cdots\!74\)\( \nu^{14} + \)\(61\!\cdots\!18\)\( \nu^{13} - \)\(99\!\cdots\!06\)\( \nu^{12} + \)\(83\!\cdots\!58\)\( \nu^{11} + \)\(35\!\cdots\!44\)\( \nu^{10} - \)\(72\!\cdots\!80\)\( \nu^{9} - \)\(12\!\cdots\!29\)\( \nu^{8} + \)\(29\!\cdots\!28\)\( \nu^{7} - \)\(44\!\cdots\!23\)\( \nu^{6} - \)\(42\!\cdots\!35\)\( \nu^{5} - \)\(76\!\cdots\!93\)\( \nu^{4} - \)\(24\!\cdots\!67\)\( \nu^{3} - \)\(25\!\cdots\!31\)\( \nu^{2} - \)\(13\!\cdots\!06\)\( \nu - \)\(56\!\cdots\!30\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(23\!\cdots\!45\)\( \nu^{19} + \)\(25\!\cdots\!60\)\( \nu^{18} - \)\(13\!\cdots\!65\)\( \nu^{17} + \)\(47\!\cdots\!64\)\( \nu^{16} - \)\(12\!\cdots\!96\)\( \nu^{15} + \)\(24\!\cdots\!88\)\( \nu^{14} - \)\(47\!\cdots\!65\)\( \nu^{13} + \)\(70\!\cdots\!62\)\( \nu^{12} - \)\(31\!\cdots\!94\)\( \nu^{11} - \)\(93\!\cdots\!22\)\( \nu^{10} + \)\(10\!\cdots\!19\)\( \nu^{9} + \)\(85\!\cdots\!72\)\( \nu^{8} - \)\(23\!\cdots\!27\)\( \nu^{7} - \)\(54\!\cdots\!25\)\( \nu^{6} + \)\(43\!\cdots\!47\)\( \nu^{5} - \)\(84\!\cdots\!98\)\( \nu^{4} + \)\(15\!\cdots\!67\)\( \nu^{3} + \)\(11\!\cdots\!88\)\( \nu^{2} - \)\(24\!\cdots\!24\)\( \nu + \)\(21\!\cdots\!76\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(23\!\cdots\!45\)\( \nu^{19} + \)\(25\!\cdots\!60\)\( \nu^{18} - \)\(13\!\cdots\!65\)\( \nu^{17} + \)\(47\!\cdots\!64\)\( \nu^{16} - \)\(12\!\cdots\!96\)\( \nu^{15} + \)\(24\!\cdots\!88\)\( \nu^{14} - \)\(47\!\cdots\!65\)\( \nu^{13} + \)\(70\!\cdots\!62\)\( \nu^{12} - \)\(31\!\cdots\!94\)\( \nu^{11} - \)\(93\!\cdots\!22\)\( \nu^{10} + \)\(10\!\cdots\!19\)\( \nu^{9} + \)\(85\!\cdots\!72\)\( \nu^{8} - \)\(23\!\cdots\!27\)\( \nu^{7} - \)\(54\!\cdots\!25\)\( \nu^{6} + \)\(43\!\cdots\!47\)\( \nu^{5} - \)\(84\!\cdots\!98\)\( \nu^{4} + \)\(15\!\cdots\!67\)\( \nu^{3} + \)\(11\!\cdots\!88\)\( \nu^{2} + \)\(33\!\cdots\!25\)\( \nu + \)\(21\!\cdots\!76\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(23\!\cdots\!28\)\( \nu^{19} - \)\(21\!\cdots\!34\)\( \nu^{18} + \)\(10\!\cdots\!67\)\( \nu^{17} - \)\(33\!\cdots\!34\)\( \nu^{16} + \)\(75\!\cdots\!24\)\( \nu^{15} - \)\(14\!\cdots\!78\)\( \nu^{14} + \)\(28\!\cdots\!93\)\( \nu^{13} - \)\(34\!\cdots\!94\)\( \nu^{12} - \)\(79\!\cdots\!10\)\( \nu^{11} + \)\(63\!\cdots\!40\)\( \nu^{10} - \)\(14\!\cdots\!00\)\( \nu^{9} - \)\(66\!\cdots\!27\)\( \nu^{8} + \)\(14\!\cdots\!39\)\( \nu^{7} + \)\(11\!\cdots\!06\)\( \nu^{6} + \)\(28\!\cdots\!08\)\( \nu^{5} + \)\(55\!\cdots\!36\)\( \nu^{4} + \)\(47\!\cdots\!25\)\( \nu^{3} + \)\(30\!\cdots\!54\)\( \nu^{2} + \)\(22\!\cdots\!02\)\( \nu + \)\(23\!\cdots\!62\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(40\!\cdots\!44\)\( \nu^{19} - \)\(34\!\cdots\!51\)\( \nu^{18} + \)\(14\!\cdots\!88\)\( \nu^{17} - \)\(37\!\cdots\!79\)\( \nu^{16} + \)\(64\!\cdots\!24\)\( \nu^{15} - \)\(96\!\cdots\!76\)\( \nu^{14} + \)\(17\!\cdots\!36\)\( \nu^{13} + \)\(39\!\cdots\!85\)\( \nu^{12} - \)\(95\!\cdots\!98\)\( \nu^{11} + \)\(12\!\cdots\!86\)\( \nu^{10} + \)\(16\!\cdots\!62\)\( \nu^{9} - \)\(23\!\cdots\!19\)\( \nu^{8} + \)\(24\!\cdots\!64\)\( \nu^{7} + \)\(58\!\cdots\!11\)\( \nu^{6} + \)\(64\!\cdots\!45\)\( \nu^{5} + \)\(83\!\cdots\!57\)\( \nu^{4} + \)\(93\!\cdots\!70\)\( \nu^{3} + \)\(44\!\cdots\!97\)\( \nu^{2} + \)\(19\!\cdots\!36\)\( \nu + \)\(72\!\cdots\!43\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(46\!\cdots\!83\)\( \nu^{19} + \)\(48\!\cdots\!48\)\( \nu^{18} - \)\(25\!\cdots\!12\)\( \nu^{17} + \)\(87\!\cdots\!38\)\( \nu^{16} - \)\(21\!\cdots\!54\)\( \nu^{15} + \)\(45\!\cdots\!96\)\( \nu^{14} - \)\(87\!\cdots\!83\)\( \nu^{13} + \)\(12\!\cdots\!33\)\( \nu^{12} - \)\(59\!\cdots\!49\)\( \nu^{11} - \)\(13\!\cdots\!83\)\( \nu^{10} + \)\(11\!\cdots\!75\)\( \nu^{9} + \)\(16\!\cdots\!68\)\( \nu^{8} - \)\(38\!\cdots\!71\)\( \nu^{7} - \)\(13\!\cdots\!76\)\( \nu^{6} - \)\(15\!\cdots\!06\)\( \nu^{5} - \)\(44\!\cdots\!30\)\( \nu^{4} + \)\(13\!\cdots\!03\)\( \nu^{3} + \)\(89\!\cdots\!31\)\( \nu^{2} + \)\(17\!\cdots\!54\)\( \nu + \)\(41\!\cdots\!40\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(47\!\cdots\!93\)\( \nu^{19} + \)\(52\!\cdots\!22\)\( \nu^{18} - \)\(28\!\cdots\!67\)\( \nu^{17} + \)\(10\!\cdots\!16\)\( \nu^{16} - \)\(26\!\cdots\!87\)\( \nu^{15} + \)\(57\!\cdots\!96\)\( \nu^{14} - \)\(11\!\cdots\!74\)\( \nu^{13} + \)\(18\!\cdots\!67\)\( \nu^{12} - \)\(13\!\cdots\!08\)\( \nu^{11} - \)\(79\!\cdots\!61\)\( \nu^{10} + \)\(10\!\cdots\!89\)\( \nu^{9} + \)\(22\!\cdots\!91\)\( \nu^{8} - \)\(45\!\cdots\!73\)\( \nu^{7} - \)\(36\!\cdots\!46\)\( \nu^{6} - \)\(89\!\cdots\!61\)\( \nu^{5} + \)\(16\!\cdots\!42\)\( \nu^{4} + \)\(40\!\cdots\!07\)\( \nu^{3} + \)\(46\!\cdots\!26\)\( \nu^{2} + \)\(26\!\cdots\!49\)\( \nu + \)\(11\!\cdots\!19\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(48\!\cdots\!91\)\( \nu^{19} + \)\(31\!\cdots\!53\)\( \nu^{18} - \)\(91\!\cdots\!74\)\( \nu^{17} + \)\(59\!\cdots\!51\)\( \nu^{16} + \)\(39\!\cdots\!50\)\( \nu^{15} - \)\(13\!\cdots\!73\)\( \nu^{14} + \)\(27\!\cdots\!47\)\( \nu^{13} - \)\(92\!\cdots\!69\)\( \nu^{12} + \)\(21\!\cdots\!66\)\( \nu^{11} - \)\(73\!\cdots\!18\)\( \nu^{10} - \)\(44\!\cdots\!01\)\( \nu^{9} + \)\(15\!\cdots\!60\)\( \nu^{8} + \)\(39\!\cdots\!07\)\( \nu^{7} - \)\(95\!\cdots\!96\)\( \nu^{6} - \)\(16\!\cdots\!43\)\( \nu^{5} - \)\(19\!\cdots\!92\)\( \nu^{4} - \)\(24\!\cdots\!97\)\( \nu^{3} - \)\(15\!\cdots\!91\)\( \nu^{2} - \)\(68\!\cdots\!59\)\( \nu - \)\(12\!\cdots\!09\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(49\!\cdots\!74\)\( \nu^{19} + \)\(45\!\cdots\!37\)\( \nu^{18} - \)\(21\!\cdots\!11\)\( \nu^{17} + \)\(66\!\cdots\!54\)\( \nu^{16} - \)\(15\!\cdots\!82\)\( \nu^{15} + \)\(30\!\cdots\!25\)\( \nu^{14} - \)\(61\!\cdots\!19\)\( \nu^{13} + \)\(76\!\cdots\!60\)\( \nu^{12} - \)\(27\!\cdots\!33\)\( \nu^{11} - \)\(73\!\cdots\!75\)\( \nu^{10} - \)\(83\!\cdots\!72\)\( \nu^{9} + \)\(12\!\cdots\!98\)\( \nu^{8} - \)\(16\!\cdots\!72\)\( \nu^{7} - \)\(34\!\cdots\!34\)\( \nu^{6} - \)\(75\!\cdots\!94\)\( \nu^{5} - \)\(11\!\cdots\!19\)\( \nu^{4} - \)\(11\!\cdots\!36\)\( \nu^{3} - \)\(81\!\cdots\!09\)\( \nu^{2} - \)\(43\!\cdots\!86\)\( \nu - \)\(14\!\cdots\!26\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(54\!\cdots\!21\)\( \nu^{19} + \)\(50\!\cdots\!54\)\( \nu^{18} - \)\(23\!\cdots\!98\)\( \nu^{17} + \)\(73\!\cdots\!26\)\( \nu^{16} - \)\(16\!\cdots\!21\)\( \nu^{15} + \)\(31\!\cdots\!04\)\( \nu^{14} - \)\(60\!\cdots\!42\)\( \nu^{13} + \)\(70\!\cdots\!96\)\( \nu^{12} + \)\(32\!\cdots\!21\)\( \nu^{11} - \)\(15\!\cdots\!66\)\( \nu^{10} - \)\(46\!\cdots\!09\)\( \nu^{9} + \)\(22\!\cdots\!89\)\( \nu^{8} - \)\(22\!\cdots\!72\)\( \nu^{7} - \)\(45\!\cdots\!46\)\( \nu^{6} - \)\(59\!\cdots\!77\)\( \nu^{5} - \)\(94\!\cdots\!87\)\( \nu^{4} - \)\(82\!\cdots\!61\)\( \nu^{3} - \)\(37\!\cdots\!32\)\( \nu^{2} - \)\(24\!\cdots\!11\)\( \nu - \)\(61\!\cdots\!61\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(62\!\cdots\!20\)\( \nu^{19} + \)\(52\!\cdots\!08\)\( \nu^{18} - \)\(22\!\cdots\!33\)\( \nu^{17} + \)\(60\!\cdots\!42\)\( \nu^{16} - \)\(11\!\cdots\!73\)\( \nu^{15} + \)\(22\!\cdots\!16\)\( \nu^{14} - \)\(44\!\cdots\!73\)\( \nu^{13} + \)\(34\!\cdots\!31\)\( \nu^{12} + \)\(79\!\cdots\!30\)\( \nu^{11} - \)\(86\!\cdots\!11\)\( \nu^{10} - \)\(25\!\cdots\!88\)\( \nu^{9} + \)\(12\!\cdots\!23\)\( \nu^{8} + \)\(91\!\cdots\!11\)\( \nu^{7} - \)\(66\!\cdots\!11\)\( \nu^{6} - \)\(14\!\cdots\!75\)\( \nu^{5} - \)\(17\!\cdots\!15\)\( \nu^{4} - \)\(17\!\cdots\!81\)\( \nu^{3} - \)\(12\!\cdots\!36\)\( \nu^{2} - \)\(56\!\cdots\!83\)\( \nu - \)\(12\!\cdots\!32\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(70\!\cdots\!54\)\( \nu^{19} - \)\(62\!\cdots\!76\)\( \nu^{18} + \)\(28\!\cdots\!61\)\( \nu^{17} - \)\(80\!\cdots\!81\)\( \nu^{16} + \)\(16\!\cdots\!62\)\( \nu^{15} - \)\(28\!\cdots\!59\)\( \nu^{14} + \)\(52\!\cdots\!63\)\( \nu^{13} - \)\(40\!\cdots\!31\)\( \nu^{12} - \)\(11\!\cdots\!44\)\( \nu^{11} + \)\(22\!\cdots\!06\)\( \nu^{10} + \)\(14\!\cdots\!89\)\( \nu^{9} - \)\(38\!\cdots\!53\)\( \nu^{8} + \)\(25\!\cdots\!05\)\( \nu^{7} + \)\(76\!\cdots\!49\)\( \nu^{6} + \)\(88\!\cdots\!65\)\( \nu^{5} + \)\(14\!\cdots\!32\)\( \nu^{4} + \)\(14\!\cdots\!09\)\( \nu^{3} + \)\(70\!\cdots\!37\)\( \nu^{2} + \)\(46\!\cdots\!92\)\( \nu + \)\(10\!\cdots\!17\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(91\!\cdots\!85\)\( \nu^{19} + \)\(86\!\cdots\!61\)\( \nu^{18} - \)\(42\!\cdots\!98\)\( \nu^{17} + \)\(13\!\cdots\!26\)\( \nu^{16} - \)\(31\!\cdots\!62\)\( \nu^{15} + \)\(64\!\cdots\!96\)\( \nu^{14} - \)\(12\!\cdots\!57\)\( \nu^{13} + \)\(16\!\cdots\!75\)\( \nu^{12} - \)\(27\!\cdots\!13\)\( \nu^{11} - \)\(18\!\cdots\!69\)\( \nu^{10} - \)\(69\!\cdots\!66\)\( \nu^{9} + \)\(32\!\cdots\!06\)\( \nu^{8} - \)\(38\!\cdots\!27\)\( \nu^{7} - \)\(61\!\cdots\!04\)\( \nu^{6} - \)\(10\!\cdots\!05\)\( \nu^{5} - \)\(14\!\cdots\!66\)\( \nu^{4} - \)\(11\!\cdots\!34\)\( \nu^{3} - \)\(63\!\cdots\!27\)\( \nu^{2} - \)\(24\!\cdots\!65\)\( \nu - \)\(25\!\cdots\!97\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(10\!\cdots\!52\)\( \nu^{19} - \)\(82\!\cdots\!00\)\( \nu^{18} + \)\(33\!\cdots\!87\)\( \nu^{17} - \)\(81\!\cdots\!33\)\( \nu^{16} + \)\(12\!\cdots\!78\)\( \nu^{15} - \)\(17\!\cdots\!38\)\( \nu^{14} + \)\(30\!\cdots\!81\)\( \nu^{13} + \)\(32\!\cdots\!01\)\( \nu^{12} - \)\(27\!\cdots\!62\)\( \nu^{11} + \)\(29\!\cdots\!95\)\( \nu^{10} + \)\(35\!\cdots\!04\)\( \nu^{9} - \)\(23\!\cdots\!31\)\( \nu^{8} - \)\(17\!\cdots\!40\)\( \nu^{7} + \)\(14\!\cdots\!74\)\( \nu^{6} + \)\(21\!\cdots\!34\)\( \nu^{5} + \)\(32\!\cdots\!84\)\( \nu^{4} + \)\(34\!\cdots\!34\)\( \nu^{3} + \)\(25\!\cdots\!64\)\( \nu^{2} + \)\(12\!\cdots\!02\)\( \nu + \)\(45\!\cdots\!18\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(10\!\cdots\!70\)\( \nu^{19} + \)\(93\!\cdots\!14\)\( \nu^{18} - \)\(42\!\cdots\!89\)\( \nu^{17} + \)\(11\!\cdots\!05\)\( \nu^{16} - \)\(24\!\cdots\!75\)\( \nu^{15} + \)\(43\!\cdots\!90\)\( \nu^{14} - \)\(80\!\cdots\!46\)\( \nu^{13} + \)\(63\!\cdots\!82\)\( \nu^{12} + \)\(16\!\cdots\!36\)\( \nu^{11} - \)\(32\!\cdots\!18\)\( \nu^{10} - \)\(21\!\cdots\!44\)\( \nu^{9} + \)\(41\!\cdots\!30\)\( \nu^{8} - \)\(16\!\cdots\!01\)\( \nu^{7} - \)\(12\!\cdots\!98\)\( \nu^{6} - \)\(15\!\cdots\!27\)\( \nu^{5} - \)\(22\!\cdots\!35\)\( \nu^{4} - \)\(21\!\cdots\!17\)\( \nu^{3} - \)\(13\!\cdots\!53\)\( \nu^{2} - \)\(58\!\cdots\!89\)\( \nu - \)\(14\!\cdots\!34\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(14\!\cdots\!89\)\( \nu^{19} - \)\(13\!\cdots\!38\)\( \nu^{18} + \)\(65\!\cdots\!02\)\( \nu^{17} - \)\(20\!\cdots\!52\)\( \nu^{16} + \)\(45\!\cdots\!38\)\( \nu^{15} - \)\(87\!\cdots\!56\)\( \nu^{14} + \)\(16\!\cdots\!19\)\( \nu^{13} - \)\(19\!\cdots\!17\)\( \nu^{12} - \)\(81\!\cdots\!45\)\( \nu^{11} + \)\(43\!\cdots\!41\)\( \nu^{10} + \)\(66\!\cdots\!06\)\( \nu^{9} - \)\(58\!\cdots\!21\)\( \nu^{8} + \)\(57\!\cdots\!59\)\( \nu^{7} + \)\(12\!\cdots\!97\)\( \nu^{6} + \)\(14\!\cdots\!21\)\( \nu^{5} + \)\(22\!\cdots\!24\)\( \nu^{4} + \)\(17\!\cdots\!63\)\( \nu^{3} + \)\(87\!\cdots\!23\)\( \nu^{2} + \)\(37\!\cdots\!44\)\( \nu + \)\(57\!\cdots\!63\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(15\!\cdots\!30\)\( \nu^{19} - \)\(15\!\cdots\!36\)\( \nu^{18} + \)\(73\!\cdots\!21\)\( \nu^{17} - \)\(23\!\cdots\!61\)\( \nu^{16} + \)\(54\!\cdots\!98\)\( \nu^{15} - \)\(10\!\cdots\!95\)\( \nu^{14} + \)\(21\!\cdots\!41\)\( \nu^{13} - \)\(28\!\cdots\!43\)\( \nu^{12} + \)\(22\!\cdots\!63\)\( \nu^{11} + \)\(35\!\cdots\!56\)\( \nu^{10} + \)\(11\!\cdots\!88\)\( \nu^{9} - \)\(64\!\cdots\!62\)\( \nu^{8} + \)\(70\!\cdots\!26\)\( \nu^{7} + \)\(11\!\cdots\!84\)\( \nu^{6} + \)\(17\!\cdots\!22\)\( \nu^{5} + \)\(23\!\cdots\!95\)\( \nu^{4} + \)\(18\!\cdots\!48\)\( \nu^{3} + \)\(97\!\cdots\!67\)\( \nu^{2} + \)\(37\!\cdots\!36\)\( \nu + \)\(29\!\cdots\!01\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(55\!\cdots\!48\)\( \nu^{19} + \)\(46\!\cdots\!86\)\( \nu^{18} - \)\(19\!\cdots\!77\)\( \nu^{17} + \)\(48\!\cdots\!59\)\( \nu^{16} - \)\(81\!\cdots\!14\)\( \nu^{15} + \)\(11\!\cdots\!67\)\( \nu^{14} - \)\(19\!\cdots\!31\)\( \nu^{13} - \)\(11\!\cdots\!09\)\( \nu^{12} + \)\(15\!\cdots\!50\)\( \nu^{11} - \)\(19\!\cdots\!64\)\( \nu^{10} - \)\(16\!\cdots\!77\)\( \nu^{9} + \)\(22\!\cdots\!56\)\( \nu^{8} - \)\(45\!\cdots\!22\)\( \nu^{7} - \)\(76\!\cdots\!39\)\( \nu^{6} - \)\(95\!\cdots\!90\)\( \nu^{5} - \)\(14\!\cdots\!47\)\( \nu^{4} - \)\(15\!\cdots\!77\)\( \nu^{3} - \)\(89\!\cdots\!31\)\( \nu^{2} - \)\(42\!\cdots\!22\)\( \nu - \)\(12\!\cdots\!39\)\(\)\()/ \)\(18\!\cdots\!07\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(28\!\cdots\!08\)\( \nu^{19} + \)\(26\!\cdots\!64\)\( \nu^{18} - \)\(12\!\cdots\!42\)\( \nu^{17} + \)\(37\!\cdots\!87\)\( \nu^{16} - \)\(82\!\cdots\!33\)\( \nu^{15} + \)\(15\!\cdots\!10\)\( \nu^{14} - \)\(29\!\cdots\!15\)\( \nu^{13} + \)\(32\!\cdots\!72\)\( \nu^{12} + \)\(25\!\cdots\!20\)\( \nu^{11} - \)\(87\!\cdots\!62\)\( \nu^{10} - \)\(26\!\cdots\!51\)\( \nu^{9} + \)\(12\!\cdots\!45\)\( \nu^{8} - \)\(10\!\cdots\!64\)\( \nu^{7} - \)\(26\!\cdots\!32\)\( \nu^{6} - \)\(31\!\cdots\!21\)\( \nu^{5} - \)\(51\!\cdots\!29\)\( \nu^{4} - \)\(44\!\cdots\!02\)\( \nu^{3} - \)\(23\!\cdots\!31\)\( \nu^{2} - \)\(11\!\cdots\!93\)\( \nu - \)\(23\!\cdots\!18\)\(\)\()/ \)\(58\!\cdots\!49\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{3} - \beta_{2}\)
\(\nu^{2}\)\(=\)\(-\beta_{17} + \beta_{14} + \beta_{11} + 5 \beta_{10} - \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{18} - 4 \beta_{16} + \beta_{15} + 7 \beta_{14} + 2 \beta_{13} + 6 \beta_{11} + 15 \beta_{10} + 6 \beta_{9} - 8 \beta_{8} - 4 \beta_{7} - 15 \beta_{6} + 8 \beta_{5} + 6 \beta_{4} - 2 \beta_{3} + 8 \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(7 \beta_{19} + 2 \beta_{18} + 9 \beta_{17} - 6 \beta_{16} + 3 \beta_{15} + 9 \beta_{14} + 9 \beta_{12} + 9 \beta_{11} + 8 \beta_{10} + 9 \beta_{9} - 11 \beta_{8} + 11 \beta_{7} - 36 \beta_{6} + 6 \beta_{5} - 11 \beta_{4} - 5 \beta_{3} + \beta_{2} + 10 \beta_{1} + 2\)
\(\nu^{5}\)\(=\)\(35 \beta_{19} - 9 \beta_{18} + 5 \beta_{17} - 27 \beta_{16} - 65 \beta_{15} - 11 \beta_{14} - 20 \beta_{13} + 47 \beta_{12} - 3 \beta_{11} - 18 \beta_{10} + 6 \beta_{9} + 11 \beta_{8} + 5 \beta_{7} - 52 \beta_{6} - 26 \beta_{5} - 23 \beta_{4} - \beta_{3} - 13 \beta_{1} + 6\)
\(\nu^{6}\)\(=\)\(70 \beta_{19} - 46 \beta_{18} - 65 \beta_{17} - 154 \beta_{16} - 312 \beta_{15} - 41 \beta_{14} - 31 \beta_{13} + 75 \beta_{12} - 46 \beta_{11} + 16 \beta_{10} + 40 \beta_{9} + 70 \beta_{8} - 154 \beta_{7} - 70 \beta_{6} - 46 \beta_{5} + 115 \beta_{4} + 55 \beta_{3} - 24 \beta_{2} + 5 \beta_{1} + 91\)
\(\nu^{7}\)\(=\)\(-53 \beta_{18} - 330 \beta_{17} - 117 \beta_{16} - 330 \beta_{15} + 64 \beta_{14} + 83 \beta_{13} - 45 \beta_{12} - 83 \beta_{11} + 328 \beta_{10} + 25 \beta_{9} + 53 \beta_{8} - 493 \beta_{7} - 25 \beta_{6} + 166 \beta_{5} + 373 \beta_{4} + 162 \beta_{3} - 45 \beta_{2} + 341 \beta_{1} + 4\)
\(\nu^{8}\)\(=\)\(-646 \beta_{19} + 671 \beta_{18} - 780 \beta_{17} + 988 \beta_{16} + 1466 \beta_{15} + 1044 \beta_{14} + 551 \beta_{13} - 348 \beta_{12} + 373 \beta_{11} + 1538 \beta_{10} + 224 \beta_{9} - 1019 \beta_{8} - 373 \beta_{7} - 133 \beta_{6} + 892 \beta_{5} + 133 \beta_{4} - 646 \beta_{3} + 622 \beta_{2} + 1690 \beta_{1} - 342\)
\(\nu^{9}\)\(=\)\(-2337 \beta_{19} + 3434 \beta_{18} + 804 \beta_{17} + 3278 \beta_{16} + 6903 \beta_{15} + 3426 \beta_{14} + 700 \beta_{13} + 1880 \beta_{11} + 848 \beta_{10} + 2622 \beta_{9} - 4126 \beta_{8} + 4048 \beta_{7} - 778 \beta_{6} + 86 \beta_{5} - 3477 \beta_{4} - 4126 \beta_{3} + 3434 \beta_{2} + 3426 \beta_{1} + 86\)
\(\nu^{10}\)\(=\)\(-1844 \beta_{19} + 5208 \beta_{18} + 8197 \beta_{17} + 3393 \beta_{16} + 7560 \beta_{15} + 42 \beta_{14} - 5970 \beta_{13} + 4255 \beta_{12} - 17985 \beta_{10} + 6902 \beta_{9} + 17985 \beta_{7} + 5237 \beta_{6} - 11773 \beta_{5} - 14254 \beta_{4} - 4126 \beta_{3} + 5970 \beta_{2} - 6057 \beta_{1}\)
\(\nu^{11}\)\(=\)\(9017 \beta_{19} - 9017 \beta_{18} + 6186 \beta_{17} - 14930 \beta_{16} - 41932 \beta_{15} - 27002 \beta_{14} - 30857 \beta_{13} + 9017 \beta_{12} - 18718 \beta_{11} - 57859 \beta_{10} + 4584 \beta_{9} + 39874 \beta_{8} + 8842 \beta_{7} + 39699 \beta_{6} - 33188 \beta_{5} + 4584 \beta_{4} + 28803 \beta_{3} - 8284 \beta_{2} - 47521 \beta_{1} + 1982\)
\(\nu^{12}\)\(=\)\(14008 \beta_{19} - 48118 \beta_{18} - 68930 \beta_{17} - 68930 \beta_{16} - 158194 \beta_{15} - 48118 \beta_{14} - 44283 \beta_{13} - 44283 \beta_{12} - 43986 \beta_{11} + 21088 \beta_{10} - 7560 \beta_{9} + 102277 \beta_{8} - 144186 \beta_{7} + 88269 \beta_{6} + 50731 \beta_{5} + 195985 \beta_{4} + 122264 \beta_{3} - 63973 \beta_{2} - 58291 \beta_{1} + 21088\)
\(\nu^{13}\)\(=\)\(-98908 \beta_{19} - 244346 \beta_{17} - 3245 \beta_{16} + 98908 \beta_{15} + 173330 \beta_{14} + 107463 \beta_{13} - 284254 \beta_{12} + 74422 \beta_{11} + 590422 \beta_{10} + 3245 \beta_{9} - 92621 \beta_{8} - 417676 \beta_{7} - 65063 \beta_{6} + 565374 \beta_{5} + 565374 \beta_{4} + 74422 \beta_{3} - 74422 \beta_{2} + 280793 \beta_{1} - 9359\)
\(\nu^{14}\)\(=\)\(-460076 \beta_{19} + 521894 \beta_{18} + 592649 \beta_{16} + 1900647 \beta_{15} + 1004853 \beta_{14} + 521894 \beta_{13} - 500311 \beta_{12} + 661599 \beta_{11} + 1430397 \beta_{10} + 243079 \beta_{9} - 1183493 \beta_{8} + 278815 \beta_{7} - 970321 \beta_{6} + 1392383 \beta_{5} - 30520 \beta_{4} - 1022205 \beta_{3} + 460076 \beta_{2} + 1183493 \beta_{1} - 257078\)
\(\nu^{15}\)\(=\)\(-213172 \beta_{19} + 1214013 \beta_{18} + 2487134 \beta_{17} + 771418 \beta_{16} + 3799098 \beta_{15} + 1000841 \beta_{14} + 1000841 \beta_{12} + 1000841 \beta_{11} - 1857843 \beta_{10} + 1000841 \beta_{9} - 1941255 \beta_{8} + 4522515 \beta_{7} - 2239628 \beta_{6} - 771418 \beta_{5} - 4522515 \beta_{4} - 3612080 \beta_{3} + 1900647 \beta_{2} + 40608 \beta_{1} - 545879\)
\(\nu^{16}\)\(=\)\(4686154 \beta_{19} - 3386541 \beta_{18} + 6746559 \beta_{17} - 5924164 \beta_{16} - 9813145 \beta_{15} - 8145872 \beta_{14} - 5523356 \beta_{13} + 6762143 \beta_{12} - 4698505 \beta_{11} - 19651762 \beta_{10} - 418291 \beta_{9} + 8145872 \beta_{8} + 8147590 \beta_{7} + 3092058 \beta_{6} - 13610120 \beta_{5} - 10221861 \beta_{4} + 60826 \beta_{3} - 12905203 \beta_{1} - 418291\)
\(\nu^{17}\)\(=\)\(16559704 \beta_{19} - 25515100 \beta_{18} - 7526168 \beta_{17} - 27924648 \beta_{16} - 67499365 \beta_{15} - 34945001 \beta_{14} - 13252655 \beta_{13} + 7129803 \beta_{12} - 25515100 \beta_{11} - 35057653 \beta_{10} - 15824182 \beta_{9} + 44748571 \beta_{8} - 27924648 \beta_{7} + 30092600 \beta_{6} - 25515100 \beta_{5} + 18796421 \beta_{4} + 32486126 \beta_{3} - 19233471 \beta_{2} - 37618768 \beta_{1} - 437050\)
\(\nu^{18}\)\(=\)\(-44764593 \beta_{18} - 86288805 \beta_{17} - 22943573 \beta_{16} - 86288805 \beta_{15} - 21821020 \beta_{14} + 30496175 \beta_{13} - 48889021 \beta_{12} - 30496175 \beta_{11} + 92497875 \beta_{10} - 53316936 \beta_{9} + 44764593 \beta_{8} - 162936345 \beta_{7} + 53316936 \beta_{6} + 63003212 \beta_{5} + 141386896 \beta_{4} + 71832594 \beta_{3} - 48889021 \beta_{2} + 23883759 \beta_{1} - 8829382\)
\(\nu^{19}\)\(=\)\(-163489500 \beta_{19} + 110172564 \beta_{18} - 141421132 \beta_{17} + 231063521 \beta_{16} + 458616410 \beta_{15} + 251559460 \beta_{14} + 246749456 \beta_{13} - 194703832 \beta_{12} + 141386896 \beta_{11} + 526344742 \beta_{10} - 43567450 \beta_{9} - 304876396 \beta_{8} - 141386896 \beta_{7} - 108368497 \beta_{6} + 362855242 \beta_{5} + 108368497 \beta_{4} - 163489500 \beta_{3} + 70112925 \beta_{2} + 415048960 \beta_{1} - 67574021\)

Character values

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

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{17}\)

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
−0.748652 0.219824i
2.58991 + 0.760465i
−0.308405 + 0.355918i
1.16609 1.34574i
2.33806 1.50258i
−0.922645 + 0.592948i
−0.0895295 + 0.622691i
0.130037 0.904424i
2.33806 + 1.50258i
−0.922645 0.592948i
1.31816 2.88636i
−0.973019 + 2.13061i
−0.0895295 0.622691i
0.130037 + 0.904424i
−0.308405 0.355918i
1.16609 + 1.34574i
−0.748652 + 0.219824i
2.58991 0.760465i
1.31816 + 2.88636i
−0.973019 2.13061i
0.654861 0.755750i 0.841254 0.540641i −0.142315 0.989821i −1.38365 3.02977i 0.142315 0.989821i −0.959493 + 0.281733i −0.841254 0.540641i 0.415415 0.909632i −3.19585 0.938386i
85.2 0.654861 0.755750i 0.841254 0.540641i −0.142315 0.989821i −0.182739 0.400142i 0.142315 0.989821i −0.959493 + 0.281733i −0.841254 0.540641i 0.415415 0.909632i −0.422076 0.123933i
127.1 −0.841254 + 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i −0.180741 + 0.0530703i −0.415415 0.909632i −0.654861 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 0.123357 0.142361i
127.2 −0.841254 + 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i 3.96506 1.16425i −0.415415 0.909632i −0.654861 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i −2.70618 + 3.12310i
169.1 0.142315 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −1.18479 1.36732i 0.959493 0.281733i 0.841254 + 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i −1.52202 + 0.978140i
169.2 0.142315 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i 2.13961 + 2.46924i 0.959493 0.281733i 0.841254 + 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i 2.74860 1.76642i
211.1 −0.415415 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −0.117330 0.0754032i 0.654861 + 0.755750i −0.142315 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i −0.0198486 + 0.138050i
211.2 −0.415415 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 2.06640 + 1.32799i 0.654861 + 0.755750i −0.142315 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i 0.349573 2.43133i
463.1 0.142315 + 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −1.18479 + 1.36732i 0.959493 + 0.281733i 0.841254 0.540641i −0.415415 0.909632i −0.654861 0.755750i −1.52202 0.978140i
463.2 0.142315 + 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i 2.13961 2.46924i 0.959493 + 0.281733i 0.841254 0.540641i −0.415415 0.909632i −0.654861 0.755750i 2.74860 + 1.76642i
547.1 0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.172611 + 1.20054i −0.841254 + 0.540641i 0.415415 + 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −0.503850 + 1.10328i
547.2 0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.0508009 0.353328i −0.841254 + 0.540641i 0.415415 + 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i 0.148287 0.324704i
673.1 −0.415415 + 0.909632i −0.959493 0.281733i −0.654861 0.755750i −0.117330 + 0.0754032i 0.654861 0.755750i −0.142315 + 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i −0.0198486 0.138050i
673.2 −0.415415 + 0.909632i −0.959493 0.281733i −0.654861 0.755750i 2.06640 1.32799i 0.654861 0.755750i −0.142315 + 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i 0.349573 + 2.43133i
715.1 −0.841254 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i −0.180741 0.0530703i −0.415415 + 0.909632i −0.654861 + 0.755750i 0.142315 0.989821i −0.959493 + 0.281733i 0.123357 + 0.142361i
715.2 −0.841254 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i 3.96506 + 1.16425i −0.415415 + 0.909632i −0.654861 + 0.755750i 0.142315 0.989821i −0.959493 + 0.281733i −2.70618 3.12310i
841.1 0.654861 + 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −1.38365 + 3.02977i 0.142315 + 0.989821i −0.959493 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −3.19585 + 0.938386i
841.2 0.654861 + 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −0.182739 + 0.400142i 0.142315 + 0.989821i −0.959493 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.422076 + 0.123933i
883.1 0.959493 0.281733i −0.654861 0.755750i 0.841254 0.540641i −0.172611 1.20054i −0.841254 0.540641i 0.415415 0.909632i 0.654861 0.755750i −0.142315 + 0.989821i −0.503850 1.10328i
883.2 0.959493 0.281733i −0.654861 0.755750i 0.841254 0.540641i 0.0508009 + 0.353328i −0.841254 0.540641i 0.415415 0.909632i 0.654861 0.755750i −0.142315 + 0.989821i 0.148287 + 0.324704i
\(n\): 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.e 20
23.c even 11 1 inner 966.2.q.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.e 20 1.a even 1 1 trivial
966.2.q.e 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

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$3$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$5$ \( 1 + 23 T + 231 T^{2} + 1286 T^{3} + 4696 T^{4} + 13728 T^{5} + 32077 T^{6} + 47037 T^{7} + 59510 T^{8} + 1770 T^{9} + 22923 T^{10} - 20879 T^{11} + 8459 T^{12} - 3409 T^{13} + 3389 T^{14} - 1683 T^{15} + 527 T^{16} - 144 T^{17} + 44 T^{18} - 10 T^{19} + T^{20} \)
$7$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$11$ \( 279841 + 2068390 T + 11681378 T^{2} + 30550808 T^{3} + 84459956 T^{4} + 46342631 T^{5} + 49410492 T^{6} + 38029381 T^{7} + 21513251 T^{8} + 6532585 T^{9} + 3166581 T^{10} + 105262 T^{11} - 87215 T^{12} - 64313 T^{13} + 3801 T^{14} + 4405 T^{15} + 620 T^{16} - 157 T^{17} - 40 T^{18} + 3 T^{19} + T^{20} \)
$13$ \( 1024 - 4096 T + 185600 T^{2} + 59136 T^{3} - 2407744 T^{4} + 3015136 T^{5} + 17850464 T^{6} - 13308240 T^{7} + 9403684 T^{8} + 1987706 T^{9} + 1789259 T^{10} + 669431 T^{11} + 146710 T^{12} + 36971 T^{13} - 398 T^{14} - 377 T^{15} + 754 T^{16} - 55 T^{17} + 29 T^{18} - 6 T^{19} + T^{20} \)
$17$ \( 529 + 21896 T + 168569 T^{2} - 7653274 T^{3} + 47579376 T^{4} - 59213214 T^{5} + 54316888 T^{6} - 33290680 T^{7} + 21753359 T^{8} - 10919440 T^{9} + 5099072 T^{10} - 1991278 T^{11} + 724662 T^{12} - 245502 T^{13} + 75378 T^{14} - 20553 T^{15} + 5262 T^{16} - 1075 T^{17} + 147 T^{18} - 14 T^{19} + T^{20} \)
$19$ \( 12499910809 - 27172601120 T + 22304839975 T^{2} - 2888821008 T^{3} - 2999979970 T^{4} + 1561925772 T^{5} + 235302457 T^{6} - 103131845 T^{7} + 623898 T^{8} - 8321414 T^{9} + 776544 T^{10} - 856829 T^{11} + 394273 T^{12} + 284170 T^{13} + 129721 T^{14} + 33649 T^{15} + 7464 T^{16} + 1147 T^{17} + 154 T^{18} + 13 T^{19} + T^{20} \)
$23$ \( 41426511213649 + 3602305322926 T - 4072171234612 T^{2} - 211099177714 T^{3} + 367129004720 T^{4} + 17950960627 T^{5} - 21182564495 T^{6} - 366482207 T^{7} + 1189268705 T^{8} + 17753240 T^{9} - 51933563 T^{10} + 771880 T^{11} + 2248145 T^{12} - 30121 T^{13} - 75695 T^{14} + 2789 T^{15} + 2480 T^{16} - 62 T^{17} - 52 T^{18} + 2 T^{19} + T^{20} \)
$29$ \( 41729946256384 + 61352970233344 T - 11153368697088 T^{2} - 9838217765888 T^{3} + 27423922623616 T^{4} - 21186510161312 T^{5} + 10691891893200 T^{6} - 3504906714624 T^{7} + 818671484628 T^{8} - 139161280408 T^{9} + 18941233761 T^{10} - 2220233552 T^{11} + 249881986 T^{12} - 27148907 T^{13} + 3070934 T^{14} - 329592 T^{15} + 37289 T^{16} - 4019 T^{17} + 366 T^{18} - 26 T^{19} + T^{20} \)
$31$ \( 6404133748321 - 13121320194137 T + 14461653268852 T^{2} - 9504791908684 T^{3} + 4651662553297 T^{4} - 1895526773024 T^{5} + 580092440193 T^{6} - 141247836739 T^{7} + 32994433365 T^{8} - 3610588122 T^{9} + 945439682 T^{10} - 142505984 T^{11} + 22440874 T^{12} - 3256474 T^{13} + 569247 T^{14} - 123436 T^{15} + 14116 T^{16} - 549 T^{17} + 67 T^{18} - 15 T^{19} + T^{20} \)
$37$ \( 744266818681 - 2971633933442 T + 7954765902312 T^{2} - 2402101588804 T^{3} + 10115742367381 T^{4} - 930317586296 T^{5} + 469330154954 T^{6} + 179000465200 T^{7} - 3594541701 T^{8} - 3736716493 T^{9} - 197543358 T^{10} + 3195102 T^{11} + 5019071 T^{12} + 1289953 T^{13} + 324465 T^{14} + 60444 T^{15} + 11538 T^{16} + 2116 T^{17} + 311 T^{18} + 26 T^{19} + T^{20} \)
$41$ \( 55504332715129 - 104748923083198 T + 139758764248112 T^{2} - 107669364047030 T^{3} + 65302004499195 T^{4} - 26793099234417 T^{5} + 10151758877206 T^{6} - 2285132266894 T^{7} + 273931822524 T^{8} - 30684017687 T^{9} + 12160613816 T^{10} - 4032185896 T^{11} + 841508351 T^{12} - 130454605 T^{13} + 16881214 T^{14} - 1830093 T^{15} + 159873 T^{16} - 11766 T^{17} + 784 T^{18} - 40 T^{19} + T^{20} \)
$43$ \( 41787170080162816 + 4312337108931072 T + 14731349819109888 T^{2} + 2735270468658688 T^{3} + 1249104173794240 T^{4} + 102008223751104 T^{5} + 11443107946096 T^{6} - 3965585600656 T^{7} + 284086101304 T^{8} + 53714032688 T^{9} + 2792983929 T^{10} - 1098153585 T^{11} + 22309543 T^{12} + 6157279 T^{13} + 1095804 T^{14} - 146289 T^{15} - 5106 T^{16} + 1813 T^{17} - 60 T^{18} - 8 T^{19} + T^{20} \)
$47$ \( ( 2073248 - 1847936 T - 1369216 T^{2} + 760436 T^{3} + 281944 T^{4} - 78379 T^{5} - 11029 T^{6} + 3417 T^{7} - 107 T^{8} - 18 T^{9} + T^{10} )^{2} \)
$53$ \( 1938817024 + 1262485504 T - 5148704768 T^{2} - 7461437440 T^{3} + 26021249024 T^{4} - 39537969152 T^{5} + 43559556864 T^{6} - 25332639040 T^{7} + 10509561456 T^{8} - 2625282236 T^{9} + 388966181 T^{10} + 8061367 T^{11} - 13025366 T^{12} + 2492221 T^{13} - 10374 T^{14} - 31911 T^{15} + 10033 T^{16} - 523 T^{17} + 57 T^{18} + 5 T^{19} + T^{20} \)
$59$ \( 6045905961419776 - 3319601742094336 T + 970315669278208 T^{2} - 59528593919744 T^{3} + 66099889727104 T^{4} - 27906821726784 T^{5} + 1748001148624 T^{6} - 333946992728 T^{7} + 298976617412 T^{8} - 38343433280 T^{9} + 1571242441 T^{10} - 1236127326 T^{11} + 316614210 T^{12} - 33510242 T^{13} + 5137669 T^{14} - 758307 T^{15} + 66094 T^{16} - 5756 T^{17} + 418 T^{18} - 13 T^{19} + T^{20} \)
$61$ \( 76535691699119104 - 400685086758151680 T + 1213520461487114240 T^{2} - 196898525299952768 T^{3} + 32582794390038080 T^{4} + 2583401289509376 T^{5} - 814293398719744 T^{6} + 124462684299208 T^{7} + 935250426740 T^{8} - 2345015027232 T^{9} + 388967521789 T^{10} - 29787843783 T^{11} + 751860268 T^{12} + 14318633 T^{13} + 13929594 T^{14} - 3594651 T^{15} + 435537 T^{16} - 33571 T^{17} + 1761 T^{18} - 59 T^{19} + T^{20} \)
$67$ \( 682426593903616 - 65317435400192 T + 512518814596096 T^{2} - 116209182132224 T^{3} + 159417620567872 T^{4} - 40445549344992 T^{5} + 26388974406336 T^{6} - 6274816409248 T^{7} + 1041222821496 T^{8} + 25965845462 T^{9} - 1991737485 T^{10} - 946654459 T^{11} + 134523631 T^{12} + 41724017 T^{13} + 5260666 T^{14} + 130020 T^{15} - 48744 T^{16} - 4772 T^{17} + 66 T^{18} + 26 T^{19} + T^{20} \)
$71$ \( 9548694366891241 - 6506681769509752 T + 3991162438081357 T^{2} - 1490560015902544 T^{3} + 414523631006124 T^{4} - 80300161197916 T^{5} + 8875304059231 T^{6} - 128593342303 T^{7} - 16207515368 T^{8} - 30357054088 T^{9} + 9341546202 T^{10} - 1126142011 T^{11} + 37804767 T^{12} + 5979428 T^{13} - 722669 T^{14} + 33231 T^{15} - 1138 T^{16} + 63 T^{17} + 88 T^{18} - 13 T^{19} + T^{20} \)
$73$ \( 53223169182966784 - 58450602461614592 T + 52983355484946944 T^{2} - 19011150434053888 T^{3} + 3936908892635264 T^{4} - 230883850421408 T^{5} - 66794857418352 T^{6} - 25695919661104 T^{7} + 5027437201880 T^{8} + 1650820197696 T^{9} + 295297675441 T^{10} + 25005653385 T^{11} + 688878793 T^{12} - 153827971 T^{13} - 17679867 T^{14} - 741194 T^{15} + 38310 T^{16} + 6424 T^{17} + 611 T^{18} + 34 T^{19} + T^{20} \)
$79$ \( 3072419271230464 + 6868085892397568 T + 7820649934415872 T^{2} + 5917967692711808 T^{3} + 3300978533721664 T^{4} + 1405655151016000 T^{5} + 462356730588704 T^{6} + 118127339126504 T^{7} + 23557130936068 T^{8} + 3681134374668 T^{9} + 448443956653 T^{10} + 41607199935 T^{11} + 3182304401 T^{12} + 225667425 T^{13} + 16341170 T^{14} + 1048140 T^{15} + 60564 T^{16} + 4556 T^{17} + 401 T^{18} + 27 T^{19} + T^{20} \)
$83$ \( 24050157577216 - 39380910931968 T + 32885753063168 T^{2} - 24516471601664 T^{3} + 14596196839616 T^{4} - 5689505702880 T^{5} + 2155621656160 T^{6} - 531799515248 T^{7} + 125956492012 T^{8} - 17349860190 T^{9} + 3296722681 T^{10} - 788033740 T^{11} + 273097886 T^{12} - 61977445 T^{13} + 9428478 T^{14} - 851973 T^{15} + 53686 T^{16} - 4861 T^{17} + 617 T^{18} - 41 T^{19} + T^{20} \)
$89$ \( 2037500161310329 - 3591960326206470 T + 6096056693656239 T^{2} - 5883929588844769 T^{3} + 3575611414516225 T^{4} - 1397621497714362 T^{5} + 375519691320612 T^{6} - 73564524298331 T^{7} + 11750702511355 T^{8} - 1203935290845 T^{9} + 32770126212 T^{10} + 7080073130 T^{11} - 912999870 T^{12} + 51616433 T^{13} - 214628 T^{14} - 469791 T^{15} + 102638 T^{16} - 12507 T^{17} + 963 T^{18} - 43 T^{19} + T^{20} \)
$97$ \( 834755060368384 + 1393645771222528 T + 1592003300743424 T^{2} - 1171059800893184 T^{3} + 573679015637568 T^{4} - 168069668245600 T^{5} + 37760079762352 T^{6} - 5758680483800 T^{7} + 647033141936 T^{8} - 30117226290 T^{9} - 525560553 T^{10} + 427797217 T^{11} + 8824742 T^{12} - 889181 T^{13} + 381871 T^{14} + 33440 T^{15} + 3364 T^{16} - 1175 T^{17} - 93 T^{18} + 8 T^{19} + T^{20} \)
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