Properties

Label 210.3.e.a
Level 210
Weight 3
Character orbit 210.e
Analytic conductor 5.722
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 210.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.72208555157\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{4} q^{2} -\beta_{1} q^{3} -2 q^{4} -\beta_{9} q^{5} + ( 1 - \beta_{13} ) q^{6} -\beta_{5} q^{7} + 2 \beta_{4} q^{8} + ( -1 + \beta_{1} - \beta_{5} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} ) q^{9} +O(q^{10})\) \( q -\beta_{4} q^{2} -\beta_{1} q^{3} -2 q^{4} -\beta_{9} q^{5} + ( 1 - \beta_{13} ) q^{6} -\beta_{5} q^{7} + 2 \beta_{4} q^{8} + ( -1 + \beta_{1} - \beta_{5} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} ) q^{9} -\beta_{10} q^{10} + ( -\beta_{1} - 2 \beta_{2} - 3 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{11} + \beta_{14} ) q^{11} + 2 \beta_{1} q^{12} + ( -\beta_{1} - 2 \beta_{4} + \beta_{6} - 2 \beta_{8} + \beta_{10} - \beta_{13} + 2 \beta_{14} ) q^{13} + \beta_{11} q^{14} + ( -2 + \beta_{6} + \beta_{9} + \beta_{10} + \beta_{14} ) q^{15} + 4 q^{16} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 6 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{14} ) q^{17} + ( -1 - 2 \beta_{1} + 3 \beta_{5} + \beta_{7} + 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{18} + ( 2 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{10} + \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{19} + 2 \beta_{9} q^{20} + ( 2 + \beta_{5} - \beta_{6} - \beta_{11} + \beta_{14} ) q^{21} + ( -6 - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{22} + ( -2 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{23} + ( -2 + 2 \beta_{13} ) q^{24} -5 q^{25} + ( -1 + 2 \beta_{1} - \beta_{3} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{11} - \beta_{13} + 2 \beta_{15} ) q^{26} + ( 4 + \beta_{2} - \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} + 4 \beta_{10} - 4 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{27} + 2 \beta_{5} q^{28} + ( 1 - \beta_{1} - 4 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{11} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{29} + ( -\beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{30} + ( -\beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - 2 \beta_{10} + 3 \beta_{12} + \beta_{13} + \beta_{15} ) q^{31} -4 \beta_{4} q^{32} + ( -8 + 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + \beta_{6} - \beta_{7} - 3 \beta_{8} - 5 \beta_{9} - \beta_{10} + 2 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} + \beta_{14} + 3 \beta_{15} ) q^{33} + ( 11 + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} + \beta_{13} - 2 \beta_{15} ) q^{34} + \beta_{2} q^{35} + ( 2 - 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} ) q^{36} + ( 2 + 4 \beta_{1} + 2 \beta_{4} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 8 \beta_{10} - 2 \beta_{14} + 2 \beta_{15} ) q^{37} + ( -2 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{38} + ( 11 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 8 \beta_{4} - 6 \beta_{5} + \beta_{8} + 9 \beta_{9} - 3 \beta_{10} + 4 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{39} + 2 \beta_{10} q^{40} + ( 2 + 2 \beta_{3} - 20 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} + 8 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{41} + ( \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{42} + ( -16 - 8 \beta_{1} - 4 \beta_{3} - 6 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{43} + ( 2 \beta_{1} + 4 \beta_{2} + 6 \beta_{4} - 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{11} - 2 \beta_{14} ) q^{44} + ( -5 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 7 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{10} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{45} + ( 6 - 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{12} - 4 \beta_{14} - 2 \beta_{15} ) q^{46} + ( 1 + 5 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} - 5 \beta_{6} - \beta_{8} - 12 \beta_{9} - 6 \beta_{13} - 2 \beta_{14} ) q^{47} -4 \beta_{1} q^{48} + 7 q^{49} + 5 \beta_{4} q^{50} + ( 6 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} + 7 \beta_{5} - 7 \beta_{9} - \beta_{10} - \beta_{11} - 4 \beta_{12} + 5 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{51} + ( 2 \beta_{1} + 4 \beta_{4} - 2 \beta_{6} + 4 \beta_{8} - 2 \beta_{10} + 2 \beta_{13} - 4 \beta_{14} ) q^{52} + ( 6 - 12 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 6 \beta_{8} + 5 \beta_{9} - 6 \beta_{11} - 3 \beta_{13} - 3 \beta_{15} ) q^{53} + ( -3 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 8 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 7 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{54} + ( -7 + 3 \beta_{1} + \beta_{3} + 2 \beta_{4} - 8 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{55} -2 \beta_{11} q^{56} + ( -14 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 11 \beta_{4} - 7 \beta_{5} - \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 7 \beta_{9} - 2 \beta_{10} - 6 \beta_{11} - 4 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{57} + ( 6 - 2 \beta_{1} - 4 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - 4 \beta_{8} - \beta_{9} + 2 \beta_{10} + 4 \beta_{12} - \beta_{13} + 4 \beta_{14} + \beta_{15} ) q^{58} + ( -4 + 8 \beta_{1} - 4 \beta_{3} + 18 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} - 6 \beta_{8} - 16 \beta_{9} + 4 \beta_{11} + 6 \beta_{13} - 2 \beta_{14} ) q^{59} + ( 4 - 2 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{14} ) q^{60} + ( 6 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 16 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} + 10 \beta_{10} - 2 \beta_{13} + 6 \beta_{14} + 4 \beta_{15} ) q^{61} + ( -2 \beta_{1} + 6 \beta_{2} + 2 \beta_{5} + 2 \beta_{8} + 4 \beta_{9} + 2 \beta_{14} ) q^{62} + ( 9 - 4 \beta_{1} + \beta_{2} - \beta_{3} - 9 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{63} -8 q^{64} + ( -3 + 5 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{65} + ( -5 + 6 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} + 10 \beta_{4} + 7 \beta_{5} - 3 \beta_{6} + \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} ) q^{66} + ( 20 - 8 \beta_{1} - 4 \beta_{3} - 6 \beta_{4} + 10 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 16 \beta_{10} + 4 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{67} + ( 2 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 12 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - 4 \beta_{9} - 4 \beta_{14} ) q^{68} + ( 4 \beta_{1} + 4 \beta_{3} + 8 \beta_{4} - 4 \beta_{7} - 2 \beta_{8} - 14 \beta_{9} + 8 \beta_{10} + 6 \beta_{11} - 6 \beta_{12} ) q^{69} -\beta_{12} q^{70} + ( 5 - 12 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + \beta_{5} + 6 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + 4 \beta_{9} + \beta_{11} + \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{71} + ( 2 + 4 \beta_{1} - 6 \beta_{5} - 2 \beta_{7} - 6 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{13} ) q^{72} + ( -1 + 4 \beta_{1} - \beta_{3} + 3 \beta_{5} - 4 \beta_{6} - 8 \beta_{7} + \beta_{8} - 8 \beta_{9} - 9 \beta_{10} - 2 \beta_{12} + 3 \beta_{13} - 2 \beta_{14} + 8 \beta_{15} ) q^{73} + ( -2 - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 14 \beta_{9} + 4 \beta_{13} + 4 \beta_{14} - 2 \beta_{15} ) q^{74} + 5 \beta_{1} q^{75} + ( -4 - 4 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 4 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} - 4 \beta_{14} + 2 \beta_{15} ) q^{76} + ( 1 - 3 \beta_{1} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{7} + 3 \beta_{8} + 13 \beta_{9} + 2 \beta_{11} - \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{77} + ( 17 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 13 \beta_{4} + 5 \beta_{5} - \beta_{7} - 2 \beta_{8} + 5 \beta_{9} + 9 \beta_{10} + 7 \beta_{11} - 2 \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{78} + ( 1 + 7 \beta_{1} + 5 \beta_{3} + 15 \beta_{4} - 8 \beta_{6} + 10 \beta_{8} - 14 \beta_{10} - 2 \beta_{12} + 13 \beta_{13} - 5 \beta_{14} ) q^{79} -4 \beta_{9} q^{80} + ( -4 - 8 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} + 11 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} + 5 \beta_{9} - 7 \beta_{10} + \beta_{11} - 2 \beta_{13} + 4 \beta_{14} - 6 \beta_{15} ) q^{81} + ( -42 + 4 \beta_{1} + 2 \beta_{3} + 12 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} + 4 \beta_{14} + 2 \beta_{15} ) q^{82} + ( 2 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 20 \beta_{4} + 2 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} + 12 \beta_{9} - 20 \beta_{11} - 8 \beta_{13} - 4 \beta_{14} + 4 \beta_{15} ) q^{83} + ( -4 - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{11} - 2 \beta_{14} ) q^{84} + ( 7 + 3 \beta_{1} + \beta_{3} + 5 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} + \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{85} + ( 2 + 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 20 \beta_{4} - 6 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 6 \beta_{11} - 8 \beta_{13} - 2 \beta_{15} ) q^{86} + ( -15 + 8 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} + 11 \beta_{4} - 17 \beta_{5} - \beta_{6} + \beta_{7} - 8 \beta_{8} - \beta_{9} - 8 \beta_{10} - 8 \beta_{11} - \beta_{12} + 4 \beta_{13} - \beta_{14} + 11 \beta_{15} ) q^{87} + ( 12 + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 4 \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{88} + ( -4 + 14 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 24 \beta_{4} - 8 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - 4 \beta_{11} + 2 \beta_{13} - 4 \beta_{15} ) q^{89} + ( 9 - 2 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} + 3 \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{90} + ( 3 - 7 \beta_{1} - \beta_{3} - \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} + 3 \beta_{9} + \beta_{12} + 2 \beta_{13} - \beta_{14} - 3 \beta_{15} ) q^{91} + ( 4 + 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{7} - 4 \beta_{9} + 4 \beta_{11} - 4 \beta_{13} - 4 \beta_{14} - 4 \beta_{15} ) q^{92} + ( 4 - 4 \beta_{2} - 8 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} - 9 \beta_{6} + \beta_{7} + 6 \beta_{8} - 5 \beta_{9} + 6 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} + 7 \beta_{15} ) q^{93} + ( -5 + 12 \beta_{1} + 5 \beta_{3} + 10 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} - 12 \beta_{10} + 2 \beta_{12} + 5 \beta_{13} + 2 \beta_{15} ) q^{94} + ( 1 + 4 \beta_{1} + \beta_{3} - 4 \beta_{4} - 5 \beta_{5} - 4 \beta_{7} - \beta_{8} - 6 \beta_{9} + 5 \beta_{11} - \beta_{13} - 2 \beta_{14} - 4 \beta_{15} ) q^{95} + ( 4 - 4 \beta_{13} ) q^{96} + ( -14 - \beta_{1} - 2 \beta_{3} - 6 \beta_{4} + 10 \beta_{5} + 5 \beta_{6} - 4 \beta_{8} - \beta_{10} - 4 \beta_{12} - 7 \beta_{13} + 2 \beta_{14} ) q^{97} -7 \beta_{4} q^{98} + ( 2 + 6 \beta_{2} + 6 \beta_{3} + 9 \beta_{4} + 4 \beta_{6} + 6 \beta_{7} + 3 \beta_{8} + 25 \beta_{9} + 13 \beta_{10} + 12 \beta_{11} + 6 \beta_{12} - \beta_{13} - 2 \beta_{14} - 12 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 8q^{3} - 32q^{4} + 16q^{6} - 4q^{9} + O(q^{10}) \) \( 16q - 8q^{3} - 32q^{4} + 16q^{6} - 4q^{9} + 16q^{12} - 20q^{15} + 64q^{16} - 32q^{18} + 48q^{19} + 28q^{21} - 96q^{22} - 32q^{24} - 80q^{25} + 64q^{27} - 88q^{33} + 160q^{34} + 8q^{36} + 80q^{37} + 156q^{39} - 336q^{43} - 80q^{45} + 32q^{46} - 32q^{48} + 112q^{49} + 84q^{51} - 32q^{54} - 80q^{55} - 264q^{57} + 96q^{58} + 40q^{60} + 112q^{61} + 112q^{63} - 128q^{64} + 240q^{67} + 8q^{69} + 64q^{72} + 48q^{73} + 40q^{75} - 96q^{76} + 208q^{78} + 8q^{79} - 124q^{81} - 608q^{82} - 56q^{84} + 120q^{85} - 120q^{87} + 192q^{88} + 160q^{90} - 56q^{91} + 104q^{93} + 32q^{94} + 64q^{96} - 192q^{97} - 52q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{15} + 34 x^{14} - 80 x^{13} + 97 x^{12} - 80 x^{11} + 498 x^{10} - 3288 x^{9} + 11844 x^{8} - 29592 x^{7} + 40338 x^{6} - 58320 x^{5} + 636417 x^{4} - 4723920 x^{3} + 18068994 x^{2} - 38263752 x + 43046721\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(52123 \nu^{15} - 1700114 \nu^{14} + 6007501 \nu^{13} - 12749018 \nu^{12} - 1811744 \nu^{11} + 7604824 \nu^{10} + 87057894 \nu^{9} - 295348116 \nu^{8} + 1956320442 \nu^{7} - 6251244444 \nu^{6} + 1557455040 \nu^{5} + 20625223752 \nu^{4} + 31798457307 \nu^{3} - 822534737202 \nu^{2} + 3281727359709 \nu - 4939018146594\)\()/ 586124153136 \)
\(\beta_{3}\)\(=\)\((\)\(399407 \nu^{15} - 1791193 \nu^{14} - 7693021 \nu^{13} + 32003807 \nu^{12} - 38173168 \nu^{11} - 127960096 \nu^{10} + 562590078 \nu^{9} + 393216582 \nu^{8} - 1125862434 \nu^{7} + 7487765982 \nu^{6} + 14965942320 \nu^{5} - 51909908832 \nu^{4} + 494622551007 \nu^{3} - 356475092913 \nu^{2} - 3998139588405 \nu + 15882398605935\)\()/ 2344496612544 \)
\(\beta_{4}\)\(=\)\((\)\(-242026 \nu^{15} + 942563 \nu^{14} - 1277806 \nu^{13} + 701903 \nu^{12} + 2378336 \nu^{11} + 16831352 \nu^{10} - 49349244 \nu^{9} + 356817486 \nu^{8} - 697331988 \nu^{7} + 1884212550 \nu^{6} + 35940672 \nu^{5} + 12489525432 \nu^{4} - 125302411026 \nu^{3} + 434417529051 \nu^{2} - 924863583654 \nu - 348051871161\)\()/ 1172248306272 \)
\(\beta_{5}\)\(=\)\((\)\(13307 \nu^{15} - 179536 \nu^{14} + 640907 \nu^{13} - 1208056 \nu^{12} + 207224 \nu^{11} + 1262120 \nu^{10} + 11237550 \nu^{9} - 68579808 \nu^{8} + 228169854 \nu^{7} - 402475608 \nu^{6} + 480193272 \nu^{5} + 1548121896 \nu^{4} + 13093466211 \nu^{3} - 104554521360 \nu^{2} + 364144967523 \nu - 532478372832\)\()/ 63364773312 \)
\(\beta_{6}\)\(=\)\((\)\(-317857 \nu^{15} + 4721090 \nu^{14} - 19290205 \nu^{13} + 36928814 \nu^{12} - 37149256 \nu^{11} + 4023536 \nu^{10} - 309774954 \nu^{9} + 1489257012 \nu^{8} - 6976055682 \nu^{7} + 15682012236 \nu^{6} - 29779628616 \nu^{5} + 18213954192 \nu^{4} - 314695327257 \nu^{3} + 2629252738674 \nu^{2} - 9653113987317 \nu + 19313925366078\)\()/ 1172248306272 \)
\(\beta_{7}\)\(=\)\((\)\(220079 \nu^{15} - 1436965 \nu^{14} + 2437835 \nu^{13} - 629917 \nu^{12} - 8683744 \nu^{11} - 35001376 \nu^{10} + 210536430 \nu^{9} - 840997122 \nu^{8} + 2272572270 \nu^{7} - 738874602 \nu^{6} - 7635580992 \nu^{5} - 34803999648 \nu^{4} + 119307211983 \nu^{3} - 608915472813 \nu^{2} + 1223438297715 \nu - 111447960669\)\()/ 781498870848 \)
\(\beta_{8}\)\(=\)\((\)\(-758476 \nu^{15} + 6985817 \nu^{14} - 22326856 \nu^{13} + 53671265 \nu^{12} - 47350168 \nu^{11} + 123531416 \nu^{10} - 893755488 \nu^{9} + 2573548410 \nu^{8} - 7991916480 \nu^{7} + 12018821418 \nu^{6} - 32635757304 \nu^{5} - 5995418472 \nu^{4} - 784349520372 \nu^{3} + 4175473419441 \nu^{2} - 13040503509528 \nu + 20507080029849\)\()/ 2344496612544 \)
\(\beta_{9}\)\(=\)\((\)\(-33544 \nu^{15} + 152423 \nu^{14} - 301840 \nu^{13} + 248939 \nu^{12} + 144488 \nu^{11} - 671464 \nu^{10} - 13125864 \nu^{9} + 43804902 \nu^{8} - 133470144 \nu^{7} + 43469406 \nu^{6} + 380897640 \nu^{5} + 1588257720 \nu^{4} - 15922129824 \nu^{3} + 70616403855 \nu^{2} - 133285402800 \nu + 49412852739\)\()/ 101934635328 \)
\(\beta_{10}\)\(=\)\((\)\(16127 \nu^{15} - 95230 \nu^{14} + 302735 \nu^{13} - 423640 \nu^{12} - 101032 \nu^{11} - 568144 \nu^{10} + 6767574 \nu^{9} - 30501636 \nu^{8} + 82349334 \nu^{7} - 156899160 \nu^{6} + 237488760 \nu^{5} - 891222912 \nu^{4} + 8587712583 \nu^{3} - 57280718646 \nu^{2} + 147616772247 \nu - 173181741552\)\()/ 31682386656 \)
\(\beta_{11}\)\(=\)\((\)\(29989 \nu^{15} - 210221 \nu^{14} + 754963 \nu^{13} - 1055177 \nu^{12} - 61616 \nu^{11} - 4040 \nu^{10} + 18376986 \nu^{9} - 82340922 \nu^{8} + 256812894 \nu^{7} - 544203954 \nu^{6} + 349811136 \nu^{5} - 705934440 \nu^{4} + 21657106485 \nu^{3} - 118977889149 \nu^{2} + 363162864555 \nu - 467281722393\)\()/ 50967317664 \)
\(\beta_{12}\)\(=\)\((\)\(-343 \nu^{15} + 3788 \nu^{14} - 14263 \nu^{13} + 22760 \nu^{12} - 13840 \nu^{11} - 1864 \nu^{10} - 366438 \nu^{9} + 1648344 \nu^{8} - 3942990 \nu^{7} + 10656792 \nu^{6} - 8269776 \nu^{5} - 1358856 \nu^{4} - 213514623 \nu^{3} + 2149147404 \nu^{2} - 7273832967 \nu + 11096488080\)\()/ 459165024 \)
\(\beta_{13}\)\(=\)\((\)\(110405 \nu^{15} - 772342 \nu^{14} + 2073353 \nu^{13} - 2872762 \nu^{12} + 281192 \nu^{11} - 7908856 \nu^{10} + 48773778 \nu^{9} - 241024884 \nu^{8} + 586424538 \nu^{7} - 1088753940 \nu^{6} + 180603432 \nu^{5} - 3191894424 \nu^{4} + 78765992541 \nu^{3} - 383144750910 \nu^{2} + 1067652745857 \nu - 1027353043386\)\()/ 130249811808 \)
\(\beta_{14}\)\(=\)\((\)\(-817513 \nu^{15} + 4462733 \nu^{14} - 10958341 \nu^{13} + 9975341 \nu^{12} + 429008 \nu^{11} + 43919120 \nu^{10} - 369524802 \nu^{9} + 1446356370 \nu^{8} - 3634237602 \nu^{7} + 6376204602 \nu^{6} + 8253263664 \nu^{5} + 39262342032 \nu^{4} - 553424923017 \nu^{3} + 2743657282773 \nu^{2} - 5645432375757 \nu + 3650385855645\)\()/ 781498870848 \)
\(\beta_{15}\)\(=\)\((\)\(-2027164 \nu^{15} + 11413589 \nu^{14} - 32738950 \nu^{13} + 39772031 \nu^{12} - 22732624 \nu^{11} + 144647024 \nu^{10} - 942441384 \nu^{9} + 3934891362 \nu^{8} - 11821849740 \nu^{7} + 18815275134 \nu^{6} + 4134880224 \nu^{5} + 8153999136 \nu^{4} - 1093256569404 \nu^{3} + 6276525412941 \nu^{2} - 16343497543734 \nu + 17674749277119\)\()/ 1172248306272 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{5} + \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{15} - \beta_{14} + 2 \beta_{13} + \beta_{12} + 4 \beta_{11} - 4 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{4} - \beta_{2} - 4\)
\(\nu^{4}\)\(=\)\(-6 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} + \beta_{11} - 7 \beta_{10} + 5 \beta_{9} - 6 \beta_{7} + 6 \beta_{6} + 11 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 8 \beta_{1} - 4\)
\(\nu^{5}\)\(=\)\(-2 \beta_{15} + 20 \beta_{14} - 10 \beta_{13} - 2 \beta_{12} - 8 \beta_{11} + 32 \beta_{10} + 10 \beta_{9} + 5 \beta_{8} - 10 \beta_{7} - 9 \beta_{6} + 27 \beta_{5} - 50 \beta_{4} + 21 \beta_{3} - 16 \beta_{2} - 12 \beta_{1} + 41\)
\(\nu^{6}\)\(=\)\(-18 \beta_{15} + 40 \beta_{14} + 46 \beta_{13} + 24 \beta_{12} - 56 \beta_{11} + 80 \beta_{10} - 70 \beta_{9} + 30 \beta_{7} + 6 \beta_{6} + 86 \beta_{5} + 294 \beta_{4} - 60 \beta_{2} + 52 \beta_{1} + 35\)
\(\nu^{7}\)\(=\)\(-98 \beta_{15} + 8 \beta_{14} - 190 \beta_{13} + 154 \beta_{12} + 40 \beta_{11} - 64 \beta_{10} - 38 \beta_{9} + 56 \beta_{8} + 182 \beta_{7} - 174 \beta_{6} + 222 \beta_{5} + 88 \beta_{4} + 12 \beta_{3} + 8 \beta_{2} + 15 \beta_{1} + 536\)
\(\nu^{8}\)\(=\)\(-102 \beta_{15} - 221 \beta_{14} - 167 \beta_{13} + 336 \beta_{12} + 175 \beta_{11} + 185 \beta_{10} - 649 \beta_{9} - 72 \beta_{8} - 294 \beta_{7} - 318 \beta_{6} - 829 \beta_{5} + 1806 \beta_{4} + 204 \beta_{3} + 672 \beta_{2} + 931 \beta_{1} - 259\)
\(\nu^{9}\)\(=\)\(157 \beta_{15} + 113 \beta_{14} - 1540 \beta_{13} - 851 \beta_{12} + 1708 \beta_{11} - 628 \beta_{10} - 2975 \beta_{9} - 2071 \beta_{8} - 157 \beta_{7} + 1146 \beta_{6} - 2778 \beta_{5} + 2863 \beta_{4} + 984 \beta_{3} + 977 \beta_{2} + 1608 \beta_{1} - 2524\)
\(\nu^{10}\)\(=\)\(3612 \beta_{15} - 1760 \beta_{14} - 3188 \beta_{13} - 1872 \beta_{12} + 1855 \beta_{11} - 3985 \beta_{10} - 19291 \beta_{9} + 1296 \beta_{8} - 3876 \beta_{7} - 444 \beta_{6} + 4763 \beta_{5} - 3603 \beta_{4} + 273 \beta_{3} - 1152 \beta_{2} + 568 \beta_{1} + 1256\)
\(\nu^{11}\)\(=\)\(-4196 \beta_{15} - 7504 \beta_{14} + 548 \beta_{13} - 6356 \beta_{12} - 18944 \beta_{11} - 27616 \beta_{10} - 20348 \beta_{9} + 1427 \beta_{8} - 9028 \beta_{7} - 16425 \beta_{6} - 6561 \beta_{5} + 2470 \beta_{4} - 2373 \beta_{3} - 10888 \beta_{2} + 2544 \beta_{1} + 68243\)
\(\nu^{12}\)\(=\)\(-25596 \beta_{15} - 7736 \beta_{14} - 2996 \beta_{13} - 12000 \beta_{12} - 4304 \beta_{11} - 29440 \beta_{10} + 45740 \beta_{9} + 79920 \beta_{8} - 8124 \beta_{7} - 28236 \beta_{6} + 22100 \beta_{5} + 71196 \beta_{4} + 43272 \beta_{3} - 34152 \beta_{2} + 68272 \beta_{1} - 143191\)
\(\nu^{13}\)\(=\)\(-60164 \beta_{15} + 72440 \beta_{14} + 121148 \beta_{13} + 6292 \beta_{12} + 187456 \beta_{11} + 35936 \beta_{10} + 120292 \beta_{9} + 114608 \beta_{8} - 28132 \beta_{7} - 156 \beta_{6} - 156036 \beta_{5} + 643048 \beta_{4} + 6816 \beta_{3} - 66664 \beta_{2} - 85575 \beta_{1} + 496904\)
\(\nu^{14}\)\(=\)\(-101604 \beta_{15} - 179663 \beta_{14} - 520475 \beta_{13} + 117264 \beta_{12} + 646033 \beta_{11} + 329279 \beta_{10} + 697427 \beta_{9} - 250848 \beta_{8} - 30372 \beta_{7} + 203532 \beta_{6} - 144325 \beta_{5} + 410316 \beta_{4} - 230304 \beta_{3} - 182784 \beta_{2} + 296113 \beta_{1} + 582167\)
\(\nu^{15}\)\(=\)\(-810131 \beta_{15} + 1160807 \beta_{14} - 1263466 \beta_{13} + 90013 \beta_{12} + 589588 \beta_{11} + 1586252 \beta_{10} - 2166683 \beta_{9} - 572833 \beta_{8} - 837229 \beta_{7} + 673140 \beta_{6} - 1256532 \beta_{5} - 3226247 \beta_{4} - 588264 \beta_{3} + 170663 \beta_{2} + 832776 \beta_{1} - 2515036\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
2.97371 + 0.396304i
2.53169 1.60951i
1.86110 2.35293i
1.51079 + 2.59182i
0.812085 2.88800i
−0.650833 2.92855i
−2.18398 + 2.05675i
−2.85457 0.922735i
2.97371 0.396304i
2.53169 + 1.60951i
1.86110 + 2.35293i
1.51079 2.59182i
0.812085 + 2.88800i
−0.650833 + 2.92855i
−2.18398 2.05675i
−2.85457 + 0.922735i
1.41421i −2.97371 0.396304i −2.00000 2.23607i −0.560459 + 4.20546i −2.64575 2.82843i 8.68589 + 2.35699i 3.16228
71.2 1.41421i −2.53169 + 1.60951i −2.00000 2.23607i 2.27619 + 3.58035i 2.64575 2.82843i 3.81894 8.14958i 3.16228
71.3 1.41421i −1.86110 + 2.35293i −2.00000 2.23607i 3.32755 + 2.63200i 2.64575 2.82843i −2.07259 8.75810i −3.16228
71.4 1.41421i −1.51079 2.59182i −2.00000 2.23607i −3.66538 + 2.13658i −2.64575 2.82843i −4.43502 + 7.83139i −3.16228
71.5 1.41421i −0.812085 + 2.88800i −2.00000 2.23607i 4.08424 + 1.14846i −2.64575 2.82843i −7.68104 4.69059i −3.16228
71.6 1.41421i 0.650833 + 2.92855i −2.00000 2.23607i 4.14160 0.920417i −2.64575 2.82843i −8.15283 + 3.81200i 3.16228
71.7 1.41421i 2.18398 2.05675i −2.00000 2.23607i −2.90869 3.08861i 2.64575 2.82843i 0.539533 8.98381i −3.16228
71.8 1.41421i 2.85457 + 0.922735i −2.00000 2.23607i 1.30494 4.03697i 2.64575 2.82843i 7.29712 + 5.26802i 3.16228
71.9 1.41421i −2.97371 + 0.396304i −2.00000 2.23607i −0.560459 4.20546i −2.64575 2.82843i 8.68589 2.35699i 3.16228
71.10 1.41421i −2.53169 1.60951i −2.00000 2.23607i 2.27619 3.58035i 2.64575 2.82843i 3.81894 + 8.14958i 3.16228
71.11 1.41421i −1.86110 2.35293i −2.00000 2.23607i 3.32755 2.63200i 2.64575 2.82843i −2.07259 + 8.75810i −3.16228
71.12 1.41421i −1.51079 + 2.59182i −2.00000 2.23607i −3.66538 2.13658i −2.64575 2.82843i −4.43502 7.83139i −3.16228
71.13 1.41421i −0.812085 2.88800i −2.00000 2.23607i 4.08424 1.14846i −2.64575 2.82843i −7.68104 + 4.69059i −3.16228
71.14 1.41421i 0.650833 2.92855i −2.00000 2.23607i 4.14160 + 0.920417i −2.64575 2.82843i −8.15283 3.81200i 3.16228
71.15 1.41421i 2.18398 + 2.05675i −2.00000 2.23607i −2.90869 + 3.08861i 2.64575 2.82843i 0.539533 + 8.98381i −3.16228
71.16 1.41421i 2.85457 0.922735i −2.00000 2.23607i 1.30494 + 4.03697i 2.64575 2.82843i 7.29712 5.26802i 3.16228
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.3.e.a 16
3.b odd 2 1 inner 210.3.e.a 16
4.b odd 2 1 1680.3.l.c 16
5.b even 2 1 1050.3.e.d 16
5.c odd 4 2 1050.3.c.c 32
12.b even 2 1 1680.3.l.c 16
15.d odd 2 1 1050.3.e.d 16
15.e even 4 2 1050.3.c.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.e.a 16 1.a even 1 1 trivial
210.3.e.a 16 3.b odd 2 1 inner
1050.3.c.c 32 5.c odd 4 2
1050.3.c.c 32 15.e even 4 2
1050.3.e.d 16 5.b even 2 1
1050.3.e.d 16 15.d odd 2 1
1680.3.l.c 16 4.b odd 2 1
1680.3.l.c 16 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(210, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{8} \)
$3$ \( 1 + 8 T + 34 T^{2} + 80 T^{3} + 97 T^{4} + 80 T^{5} + 498 T^{6} + 3288 T^{7} + 11844 T^{8} + 29592 T^{9} + 40338 T^{10} + 58320 T^{11} + 636417 T^{12} + 4723920 T^{13} + 18068994 T^{14} + 38263752 T^{15} + 43046721 T^{16} \)
$5$ \( ( 1 + 5 T^{2} )^{8} \)
$7$ \( ( 1 - 7 T^{2} )^{8} \)
$11$ \( 1 - 588 T^{2} + 192110 T^{4} - 45729776 T^{6} + 8909879425 T^{8} - 1517060018584 T^{10} + 233462822472086 T^{12} - 32740893640977644 T^{14} + 4166709531307795396 T^{16} - \)\(47\!\cdots\!04\)\( T^{18} + \)\(50\!\cdots\!66\)\( T^{20} - \)\(47\!\cdots\!64\)\( T^{22} + \)\(40\!\cdots\!25\)\( T^{24} - \)\(30\!\cdots\!76\)\( T^{26} + \)\(18\!\cdots\!10\)\( T^{28} - \)\(84\!\cdots\!28\)\( T^{30} + \)\(21\!\cdots\!21\)\( T^{32} \)
$13$ \( ( 1 + 494 T^{2} - 456 T^{3} + 146293 T^{4} - 55296 T^{5} + 35813146 T^{6} - 25079880 T^{7} + 6821269256 T^{8} - 4238499720 T^{9} + 1022859262906 T^{10} - 266903230464 T^{11} + 119335694367253 T^{12} - 62863472283144 T^{13} + 11509254050505614 T^{14} + 665416609183179841 T^{16} )^{2} \)
$17$ \( 1 - 2012 T^{2} + 2016998 T^{4} - 1352297560 T^{6} + 695200323953 T^{8} - 299701490294440 T^{10} + 114417537477919350 T^{12} - 39314188029348430308 T^{14} + \)\(12\!\cdots\!52\)\( T^{16} - \)\(32\!\cdots\!68\)\( T^{18} + \)\(79\!\cdots\!50\)\( T^{20} - \)\(17\!\cdots\!40\)\( T^{22} + \)\(33\!\cdots\!93\)\( T^{24} - \)\(54\!\cdots\!60\)\( T^{26} + \)\(68\!\cdots\!58\)\( T^{28} - \)\(57\!\cdots\!92\)\( T^{30} + \)\(23\!\cdots\!61\)\( T^{32} \)
$19$ \( ( 1 - 24 T + 1612 T^{2} - 39608 T^{3} + 1389160 T^{4} - 32391560 T^{5} + 808980068 T^{6} - 17015640488 T^{7} + 341271444494 T^{8} - 6142646216168 T^{9} + 105427091441828 T^{10} - 1523889477164360 T^{11} + 23592886434035560 T^{12} - 242839272338982008 T^{13} + 3567863649534651532 T^{14} - 19176160458789218904 T^{15} + \)\(28\!\cdots\!81\)\( T^{16} )^{2} \)
$23$ \( 1 - 3856 T^{2} + 7749432 T^{4} - 10915385904 T^{6} + 12001409108892 T^{8} - 10839581553458448 T^{10} + 8275120566094665352 T^{12} - \)\(54\!\cdots\!60\)\( T^{14} + \)\(30\!\cdots\!06\)\( T^{16} - \)\(15\!\cdots\!60\)\( T^{18} + \)\(64\!\cdots\!12\)\( T^{20} - \)\(23\!\cdots\!08\)\( T^{22} + \)\(73\!\cdots\!12\)\( T^{24} - \)\(18\!\cdots\!04\)\( T^{26} + \)\(37\!\cdots\!12\)\( T^{28} - \)\(51\!\cdots\!36\)\( T^{30} + \)\(37\!\cdots\!21\)\( T^{32} \)
$29$ \( 1 - 3972 T^{2} + 8120294 T^{4} - 13034144872 T^{6} + 18580208769649 T^{8} - 22683669812342744 T^{10} + 24087258397916480502 T^{12} - \)\(23\!\cdots\!44\)\( T^{14} + \)\(20\!\cdots\!28\)\( T^{16} - \)\(16\!\cdots\!64\)\( T^{18} + \)\(12\!\cdots\!22\)\( T^{20} - \)\(80\!\cdots\!04\)\( T^{22} + \)\(46\!\cdots\!29\)\( T^{24} - \)\(23\!\cdots\!72\)\( T^{26} + \)\(10\!\cdots\!14\)\( T^{28} - \)\(35\!\cdots\!92\)\( T^{30} + \)\(62\!\cdots\!41\)\( T^{32} \)
$31$ \( ( 1 + 3684 T^{2} - 19472 T^{3} + 7802120 T^{4} - 58737456 T^{5} + 11372951532 T^{6} - 93476864320 T^{7} + 12493787155086 T^{8} - 89831266611520 T^{9} + 10503159571784172 T^{10} - 52129708412575536 T^{11} + 6654358221039174920 T^{12} - 15959802004090157072 T^{13} + \)\(29\!\cdots\!24\)\( T^{14} + \)\(72\!\cdots\!81\)\( T^{16} )^{2} \)
$37$ \( ( 1 - 40 T + 6968 T^{2} - 291512 T^{3} + 25042268 T^{4} - 987943976 T^{5} + 58242343432 T^{6} - 2052793579384 T^{7} + 94406736665478 T^{8} - 2810274410176696 T^{9} + 109155528608860552 T^{10} - 2534793949835662184 T^{11} + 87960451829583332828 T^{12} - \)\(14\!\cdots\!88\)\( T^{13} + \)\(45\!\cdots\!08\)\( T^{14} - \)\(36\!\cdots\!60\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} )^{2} \)
$41$ \( 1 - 10832 T^{2} + 65454200 T^{4} - 277210214128 T^{6} + 920889789641756 T^{8} - 2528312319464856400 T^{10} + \)\(59\!\cdots\!28\)\( T^{12} - \)\(12\!\cdots\!00\)\( T^{14} + \)\(21\!\cdots\!66\)\( T^{16} - \)\(34\!\cdots\!00\)\( T^{18} + \)\(47\!\cdots\!88\)\( T^{20} - \)\(57\!\cdots\!00\)\( T^{22} + \)\(58\!\cdots\!96\)\( T^{24} - \)\(49\!\cdots\!28\)\( T^{26} + \)\(33\!\cdots\!00\)\( T^{28} - \)\(15\!\cdots\!72\)\( T^{30} + \)\(40\!\cdots\!81\)\( T^{32} \)
$43$ \( ( 1 + 168 T + 21576 T^{2} + 1867768 T^{3} + 137965244 T^{4} + 8243885160 T^{5} + 447366405624 T^{6} + 21255986843000 T^{7} + 961142451546054 T^{8} + 39302319672707000 T^{9} + 1529456714913736824 T^{10} + 52112591030623452840 T^{11} + \)\(16\!\cdots\!44\)\( T^{12} + \)\(40\!\cdots\!32\)\( T^{13} + \)\(86\!\cdots\!76\)\( T^{14} + \)\(12\!\cdots\!32\)\( T^{15} + \)\(13\!\cdots\!01\)\( T^{16} )^{2} \)
$47$ \( 1 - 18860 T^{2} + 165198086 T^{4} - 880812409480 T^{6} + 3099424527294161 T^{8} - 7043887723434328360 T^{10} + \)\(79\!\cdots\!02\)\( T^{12} + \)\(68\!\cdots\!80\)\( T^{14} - \)\(40\!\cdots\!04\)\( T^{16} + \)\(33\!\cdots\!80\)\( T^{18} + \)\(18\!\cdots\!22\)\( T^{20} - \)\(81\!\cdots\!60\)\( T^{22} + \)\(17\!\cdots\!81\)\( T^{24} - \)\(24\!\cdots\!80\)\( T^{26} + \)\(22\!\cdots\!66\)\( T^{28} - \)\(12\!\cdots\!60\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \)
$53$ \( 1 - 15704 T^{2} + 138921152 T^{4} - 871766999944 T^{6} + 4342207935569660 T^{8} - 18113936181752364376 T^{10} + \)\(65\!\cdots\!52\)\( T^{12} - \)\(21\!\cdots\!64\)\( T^{14} + \)\(62\!\cdots\!50\)\( T^{16} - \)\(16\!\cdots\!84\)\( T^{18} + \)\(40\!\cdots\!72\)\( T^{20} - \)\(88\!\cdots\!16\)\( T^{22} + \)\(16\!\cdots\!60\)\( T^{24} - \)\(26\!\cdots\!44\)\( T^{26} + \)\(33\!\cdots\!12\)\( T^{28} - \)\(29\!\cdots\!44\)\( T^{30} + \)\(15\!\cdots\!41\)\( T^{32} \)
$59$ \( 1 - 26768 T^{2} + 368873336 T^{4} - 3501619664560 T^{6} + 25676127920457884 T^{8} - \)\(15\!\cdots\!96\)\( T^{10} + \)\(77\!\cdots\!60\)\( T^{12} - \)\(33\!\cdots\!08\)\( T^{14} + \)\(12\!\cdots\!58\)\( T^{16} - \)\(40\!\cdots\!88\)\( T^{18} + \)\(11\!\cdots\!60\)\( T^{20} - \)\(27\!\cdots\!76\)\( T^{22} + \)\(55\!\cdots\!44\)\( T^{24} - \)\(91\!\cdots\!60\)\( T^{26} + \)\(11\!\cdots\!96\)\( T^{28} - \)\(10\!\cdots\!28\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \)
$61$ \( ( 1 - 56 T + 11072 T^{2} - 269288 T^{3} + 50054524 T^{4} - 711106488 T^{5} + 247917300928 T^{6} - 7112512171688 T^{7} + 1188015110923334 T^{8} - 26465657790851048 T^{9} + 3432623529798240448 T^{10} - 36636472472295954168 T^{11} + \)\(95\!\cdots\!44\)\( T^{12} - \)\(19\!\cdots\!88\)\( T^{13} + \)\(29\!\cdots\!12\)\( T^{14} - \)\(55\!\cdots\!96\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \)
$67$ \( ( 1 - 120 T + 21032 T^{2} - 1363784 T^{3} + 159985756 T^{4} - 7095679000 T^{5} + 808761061016 T^{6} - 25648223522216 T^{7} + 3474267779595526 T^{8} - 115134875391227624 T^{9} + 16297442000621798936 T^{10} - \)\(64\!\cdots\!00\)\( T^{11} + \)\(64\!\cdots\!96\)\( T^{12} - \)\(24\!\cdots\!16\)\( T^{13} + \)\(17\!\cdots\!52\)\( T^{14} - \)\(44\!\cdots\!80\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} )^{2} \)
$71$ \( 1 - 58328 T^{2} + 1630496768 T^{4} - 29214173879944 T^{6} + 378588472445657852 T^{8} - \)\(37\!\cdots\!40\)\( T^{10} + \)\(30\!\cdots\!12\)\( T^{12} - \)\(20\!\cdots\!48\)\( T^{14} + \)\(11\!\cdots\!90\)\( T^{16} - \)\(51\!\cdots\!88\)\( T^{18} + \)\(19\!\cdots\!32\)\( T^{20} - \)\(62\!\cdots\!40\)\( T^{22} + \)\(15\!\cdots\!92\)\( T^{24} - \)\(30\!\cdots\!44\)\( T^{26} + \)\(43\!\cdots\!08\)\( T^{28} - \)\(39\!\cdots\!08\)\( T^{30} + \)\(17\!\cdots\!41\)\( T^{32} \)
$73$ \( ( 1 - 24 T + 15156 T^{2} + 137320 T^{3} + 133090184 T^{4} + 1190310168 T^{5} + 1034805205980 T^{6} + 7023266129048 T^{7} + 5908298409616782 T^{8} + 37426985201696792 T^{9} + 29386647627474681180 T^{10} + \)\(18\!\cdots\!52\)\( T^{11} + \)\(10\!\cdots\!04\)\( T^{12} + \)\(59\!\cdots\!80\)\( T^{13} + \)\(34\!\cdots\!76\)\( T^{14} - \)\(29\!\cdots\!16\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} )^{2} \)
$79$ \( ( 1 - 4 T + 21374 T^{2} - 1216 T^{3} + 230456449 T^{4} + 2685965400 T^{5} + 1772108711446 T^{6} + 44455008585260 T^{7} + 11672772046344164 T^{8} + 277443708580607660 T^{9} + 69023777851627327126 T^{10} + \)\(65\!\cdots\!00\)\( T^{11} + \)\(34\!\cdots\!89\)\( T^{12} - \)\(11\!\cdots\!16\)\( T^{13} + \)\(12\!\cdots\!34\)\( T^{14} - \)\(14\!\cdots\!24\)\( T^{15} + \)\(23\!\cdots\!21\)\( T^{16} )^{2} \)
$83$ \( 1 - 26752 T^{2} + 559157880 T^{4} - 7483229844864 T^{6} + 85634770716597660 T^{8} - \)\(74\!\cdots\!16\)\( T^{10} + \)\(60\!\cdots\!76\)\( T^{12} - \)\(41\!\cdots\!76\)\( T^{14} + \)\(29\!\cdots\!06\)\( T^{16} - \)\(19\!\cdots\!96\)\( T^{18} + \)\(13\!\cdots\!16\)\( T^{20} - \)\(80\!\cdots\!76\)\( T^{22} + \)\(43\!\cdots\!60\)\( T^{24} - \)\(18\!\cdots\!64\)\( T^{26} + \)\(63\!\cdots\!80\)\( T^{28} - \)\(14\!\cdots\!32\)\( T^{30} + \)\(25\!\cdots\!61\)\( T^{32} \)
$89$ \( 1 - 71136 T^{2} + 2535583160 T^{4} - 60654761408672 T^{6} + 1094469962087760028 T^{8} - \)\(15\!\cdots\!12\)\( T^{10} + \)\(18\!\cdots\!28\)\( T^{12} - \)\(19\!\cdots\!80\)\( T^{14} + \)\(16\!\cdots\!02\)\( T^{16} - \)\(11\!\cdots\!80\)\( T^{18} + \)\(74\!\cdots\!68\)\( T^{20} - \)\(39\!\cdots\!52\)\( T^{22} + \)\(16\!\cdots\!08\)\( T^{24} - \)\(58\!\cdots\!72\)\( T^{26} + \)\(15\!\cdots\!60\)\( T^{28} - \)\(27\!\cdots\!16\)\( T^{30} + \)\(24\!\cdots\!21\)\( T^{32} \)
$97$ \( ( 1 + 96 T + 62878 T^{2} + 5357144 T^{3} + 1785824677 T^{4} + 133941251648 T^{5} + 30441979716138 T^{6} + 1968996759462584 T^{7} + 345736614004039176 T^{8} + 18526290509783452856 T^{9} + \)\(26\!\cdots\!78\)\( T^{10} + \)\(11\!\cdots\!92\)\( T^{11} + \)\(13\!\cdots\!97\)\( T^{12} + \)\(39\!\cdots\!56\)\( T^{13} + \)\(43\!\cdots\!98\)\( T^{14} + \)\(62\!\cdots\!24\)\( T^{15} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \)
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