Properties

Label 210.3.l.b
Level 210
Weight 3
Character orbit 210.l
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.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.72208555157\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{2} + \beta_{3} q^{3} + 2 \beta_{2} q^{4} + ( -1 + \beta_{2} + \beta_{10} ) q^{5} + ( -\beta_{3} - \beta_{4} ) q^{6} -\beta_{6} q^{7} + ( 2 - 2 \beta_{2} ) q^{8} -3 \beta_{2} q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{2} + \beta_{3} q^{3} + 2 \beta_{2} q^{4} + ( -1 + \beta_{2} + \beta_{10} ) q^{5} + ( -\beta_{3} - \beta_{4} ) q^{6} -\beta_{6} q^{7} + ( 2 - 2 \beta_{2} ) q^{8} -3 \beta_{2} q^{9} + ( 2 - \beta_{10} - \beta_{12} ) q^{10} + ( 2 \beta_{5} + 2 \beta_{6} - \beta_{14} - \beta_{15} ) q^{11} + 2 \beta_{4} q^{12} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{7} - \beta_{10} + \beta_{13} - \beta_{15} ) q^{13} + ( -\beta_{5} + \beta_{6} ) q^{14} + ( -1 - \beta_{2} - \beta_{3} - \beta_{15} ) q^{15} -4 q^{16} + ( 2 + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + \beta_{10} + \beta_{11} - \beta_{14} - 2 \beta_{15} ) q^{17} + ( -3 + 3 \beta_{2} ) q^{18} + ( -1 - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + 2 \beta_{11} + \beta_{13} - 2 \beta_{15} ) q^{19} + ( -2 - 2 \beta_{2} + 2 \beta_{12} ) q^{20} + \beta_{1} q^{21} + ( -\beta_{5} - 3 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{14} + \beta_{15} ) q^{22} + ( 1 - 4 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 4 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{13} - \beta_{14} ) q^{23} + ( 2 \beta_{3} - 2 \beta_{4} ) q^{24} + ( 1 - 3 \beta_{2} + \beta_{3} - 7 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{25} + ( 3 + 4 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{26} -3 \beta_{4} q^{27} + 2 \beta_{5} q^{28} + ( -1 + 7 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + \beta_{9} + 3 \beta_{10} - \beta_{11} + 3 \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{29} + ( 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{9} + \beta_{15} ) q^{30} + ( -6 + 4 \beta_{1} - \beta_{2} + 5 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{15} ) q^{31} + ( 4 + 4 \beta_{2} ) q^{32} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{7} + \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{33} + ( 1 - 5 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{34} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{35} + 6 q^{36} + ( -11 - 11 \beta_{2} - 4 \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{8} + \beta_{9} - 4 \beta_{10} + 2 \beta_{11} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{37} + ( -1 + 3 \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - 3 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{38} + ( 1 - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{39} + ( 4 \beta_{2} + 2 \beta_{10} - 2 \beta_{12} ) q^{40} + ( -3 + 4 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + \beta_{9} + 3 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{41} + ( -\beta_{1} + \beta_{7} ) q^{42} + ( -4 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} + 8 \beta_{5} - 4 \beta_{7} - 2 \beta_{9} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{43} + ( -2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} ) q^{44} + ( 3 + 3 \beta_{2} - 3 \beta_{12} ) q^{45} + ( -1 + 8 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{46} + ( 4 - 4 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} + 4 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} ) q^{47} -4 \beta_{3} q^{48} + 7 \beta_{2} q^{49} + ( -4 + 2 \beta_{2} - 8 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} + 9 \beta_{6} + \beta_{8} + 3 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{50} + ( -1 - 2 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 4 \beta_{12} + 3 \beta_{13} + \beta_{14} - \beta_{15} ) q^{51} + ( -2 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{7} - 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} ) q^{52} + ( 2 - 4 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - 4 \beta_{5} - 4 \beta_{7} - \beta_{10} - 2 \beta_{11} + 4 \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{53} + ( -3 \beta_{3} + 3 \beta_{4} ) q^{54} + ( 2 + 2 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} + 16 \beta_{4} - 13 \beta_{5} - 3 \beta_{6} + 6 \beta_{7} + \beta_{8} - 3 \beta_{9} - 3 \beta_{11} + \beta_{13} + 4 \beta_{14} + 2 \beta_{15} ) q^{55} + ( -2 \beta_{5} - 2 \beta_{6} ) q^{56} + ( 6 + 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{8} - 3 \beta_{11} - 3 \beta_{12} + 3 \beta_{14} + 3 \beta_{15} ) q^{57} + ( 8 - 6 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + 10 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{11} - 6 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{58} + ( 2 - 8 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 4 \beta_{8} - 6 \beta_{9} - 2 \beta_{10} - 4 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{59} + ( 2 - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{9} ) q^{60} + ( 5 + 4 \beta_{1} + \beta_{2} - 6 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 5 \beta_{12} - \beta_{13} + 2 \beta_{14} + 3 \beta_{15} ) q^{61} + ( 5 - 4 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} - \beta_{5} + \beta_{6} + 4 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + 5 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{62} -3 \beta_{5} q^{63} -8 \beta_{2} q^{64} + ( -1 + 4 \beta_{1} + 12 \beta_{2} + 9 \beta_{3} + 5 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} + 12 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} - 4 \beta_{13} - 2 \beta_{14} ) q^{65} + ( -4 + 4 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{66} + ( -2 - 8 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} + 8 \beta_{6} + 8 \beta_{7} - 4 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} + 2 \beta_{13} + 2 \beta_{15} ) q^{67} + ( -6 + 4 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - 4 \beta_{9} + 2 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} ) q^{68} + ( -1 - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + 13 \beta_{5} - 13 \beta_{6} - \beta_{8} - \beta_{10} + 2 \beta_{11} + \beta_{13} - 2 \beta_{15} ) q^{69} + ( -2 \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} + \beta_{9} - 2 \beta_{13} + \beta_{15} ) q^{70} + ( 35 + 8 \beta_{1} - \beta_{2} + 10 \beta_{3} + 8 \beta_{4} - 10 \beta_{5} - 2 \beta_{6} + 4 \beta_{8} - 5 \beta_{9} - \beta_{10} - \beta_{11} + 7 \beta_{12} + \beta_{13} + 5 \beta_{14} + 4 \beta_{15} ) q^{71} + ( -6 - 6 \beta_{2} ) q^{72} + ( 2 - 6 \beta_{1} - \beta_{2} + 23 \beta_{3} + \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} - 4 \beta_{8} - 3 \beta_{10} - 4 \beta_{12} - \beta_{13} + 5 \beta_{15} ) q^{73} + ( 22 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{10} - 4 \beta_{11} + 4 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} ) q^{74} + ( -16 + 2 \beta_{1} - 7 \beta_{2} + 4 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 6 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{75} + ( 4 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + 4 \beta_{13} + 2 \beta_{14} ) q^{76} + ( -10 - 10 \beta_{2} + 4 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{11} + 2 \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{77} + ( -3 + \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 11 \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} + 3 \beta_{12} + \beta_{13} + 3 \beta_{14} ) q^{78} + ( 2 + 40 \beta_{2} + 14 \beta_{3} - 10 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} - 2 \beta_{13} + 4 \beta_{15} ) q^{79} + ( 4 - 4 \beta_{2} - 4 \beta_{10} ) q^{80} -9 q^{81} + ( 5 - 4 \beta_{1} + \beta_{2} + 4 \beta_{3} + 10 \beta_{4} - 3 \beta_{5} - 5 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} - \beta_{10} - 3 \beta_{11} - 5 \beta_{12} - \beta_{13} + 3 \beta_{14} + 2 \beta_{15} ) q^{82} + ( -14 - 4 \beta_{1} + 16 \beta_{2} - 10 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} - 6 \beta_{9} + 6 \beta_{10} + 2 \beta_{11} + 4 \beta_{14} + 2 \beta_{15} ) q^{83} -2 \beta_{7} q^{84} + ( 1 + 4 \beta_{1} + 14 \beta_{2} - 27 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 14 \beta_{6} + 12 \beta_{7} + \beta_{9} + 2 \beta_{10} + \beta_{11} + 5 \beta_{12} + 4 \beta_{13} + \beta_{14} + \beta_{15} ) q^{85} + ( -2 + 8 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 8 \beta_{4} - 6 \beta_{5} - 10 \beta_{6} - 2 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{86} + ( -7 - 4 \beta_{1} - 11 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} - \beta_{5} + \beta_{6} + 4 \beta_{7} + \beta_{8} - 2 \beta_{9} - 5 \beta_{10} - \beta_{11} - 5 \beta_{12} - 3 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{87} + ( 6 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{14} - 2 \beta_{15} ) q^{88} + ( -53 \beta_{2} - 17 \beta_{3} + 17 \beta_{4} - 10 \beta_{5} + 10 \beta_{6} - 8 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} + \beta_{14} + 2 \beta_{15} ) q^{89} + ( -6 \beta_{2} - 3 \beta_{10} + 3 \beta_{12} ) q^{90} + ( -\beta_{2} - 9 \beta_{3} - 11 \beta_{4} - \beta_{5} + 3 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} + 3 \beta_{12} + \beta_{14} ) q^{91} + ( -8 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} - 2 \beta_{8} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{92} + ( 13 - 14 \beta_{2} - 7 \beta_{3} - \beta_{4} - 10 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - 5 \beta_{10} - 4 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{93} + ( 2 - 10 \beta_{2} + 8 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 8 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} - 6 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{94} + ( 26 + 4 \beta_{1} - 11 \beta_{2} + \beta_{3} - 27 \beta_{4} + 16 \beta_{5} + 8 \beta_{6} + 12 \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 3 \beta_{11} + \beta_{12} + 2 \beta_{13} + 5 \beta_{14} + 3 \beta_{15} ) q^{95} + ( 4 \beta_{3} + 4 \beta_{4} ) q^{96} + ( 41 + 6 \beta_{1} + 44 \beta_{2} - 3 \beta_{3} + 13 \beta_{4} + 2 \beta_{5} + 22 \beta_{6} - 6 \beta_{7} - 2 \beta_{8} + 5 \beta_{9} + 4 \beta_{10} + 3 \beta_{11} - \beta_{12} + 2 \beta_{14} - 2 \beta_{15} ) q^{97} + ( 7 - 7 \beta_{2} ) q^{98} + ( 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{8} + 3 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{2} - 16q^{5} + 32q^{8} + O(q^{10}) \) \( 16q - 16q^{2} - 16q^{5} + 32q^{8} + 24q^{10} + 8q^{11} - 32q^{13} - 12q^{15} - 64q^{16} + 56q^{17} - 48q^{18} - 16q^{20} - 8q^{22} + 24q^{23} + 40q^{25} + 64q^{26} - 112q^{31} + 64q^{32} + 24q^{33} + 28q^{35} + 96q^{36} - 152q^{37} - 16q^{40} + 24q^{45} - 48q^{46} + 80q^{47} - 72q^{50} - 72q^{51} - 64q^{52} + 48q^{53} - 24q^{55} + 24q^{57} + 96q^{58} + 24q^{60} + 96q^{61} + 112q^{62} + 16q^{65} - 48q^{66} - 80q^{67} - 112q^{68} + 536q^{71} - 96q^{72} - 288q^{75} - 168q^{77} - 48q^{78} + 64q^{80} - 144q^{81} - 256q^{83} + 40q^{85} - 144q^{87} + 16q^{88} + 24q^{90} + 48q^{92} + 192q^{93} + 360q^{95} + 688q^{97} + 112q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{15} + 32 x^{14} + 152 x^{13} + 1954 x^{12} - 12664 x^{11} + 50336 x^{10} + 231896 x^{9} + 1093889 x^{8} - 4595248 x^{7} + 18837632 x^{6} + 86081152 x^{5} + 178889856 x^{4} + 70149120 x^{3} + 10035200 x^{2} - 7168000 x + 2560000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-124150896048631 \nu^{15} + 759860784649304 \nu^{14} - 962633185803656 \nu^{13} - 37961089316763336 \nu^{12} - 220441925294230158 \nu^{11} + 1204837597044011592 \nu^{10} - 1461099351530560688 \nu^{9} - 59153440044822509928 \nu^{8} - 100458761515785846391 \nu^{7} + 446250282498400102704 \nu^{6} - 588890574148529033256 \nu^{5} - 22232452750211023274336 \nu^{4} - 9966351473395582185920 \nu^{3} - 2157494988382877363200 \nu^{2} + 1309863686202312128000 \nu - 125597356615831969536000\)\()/ \)\(27\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(50521933035234406935387 \nu^{15} - 355849584339626868196590 \nu^{14} + 1220781585234851432658840 \nu^{13} + 9304805374903308743399240 \nu^{12} + 105717930910748956540229574 \nu^{11} - 546511703953657264363821140 \nu^{10} + 1912893592429770663178189120 \nu^{9} + 14274226782583577122771384520 \nu^{8} + 65924256015119669450932994891 \nu^{7} - 180976850729426810190271470950 \nu^{6} + 719449018130531781232088723240 \nu^{5} + 5305696558618628641218886482240 \nu^{4} + 12993474700473951273694265843648 \nu^{3} + 11620918919670154992391536887680 \nu^{2} + 2225350312372339004103034892800 \nu - 232431648484874621115141632000\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(7312607874029475001394987 \nu^{15} - 50322989299182724918321074 \nu^{14} + 168283183559866103821772176 \nu^{13} + 1371720702266455093133696536 \nu^{12} + 15547180873839092314908209030 \nu^{11} - 76809414399688904270132589212 \nu^{10} + 263505718309346300935072397648 \nu^{9} + 2102021021189220528568128879528 \nu^{8} + 9911131083816128297130191166779 \nu^{7} - 25009453348432333318581627650154 \nu^{6} + 99192306063654602485356687469376 \nu^{5} + 780014500024526499493845235763536 \nu^{4} + 2014834486401151120674936909338304 \nu^{3} + 1818182355719495270833534626247040 \nu^{2} + 351325748136599941624346108518400 \nu - 222122905273351925296690427776000\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-3829248573711961810228154 \nu^{15} + 26239924034908349561047733 \nu^{14} - 85716107970375280109650292 \nu^{13} - 738472474668082305176147912 \nu^{12} - 8080492117822719953566872060 \nu^{11} + 40115168057085959901727168954 \nu^{10} - 134228933858105791931167618416 \nu^{9} - 1135855513549067340147092753376 \nu^{8} - 5097784662125274247991681038018 \nu^{7} + 13137999125753012940783686579493 \nu^{6} - 50574570308491286816955297317692 \nu^{5} - 422996013884134839010228418101912 \nu^{4} - 1022192686228881977575596028483968 \nu^{3} - 912283111994794320793370813810880 \nu^{2} - 173723596390269875933568240172800 \nu - 74392129685396065048852419008000\)\()/ \)\(75\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(12085404159758350350699247 \nu^{15} - 85224520217749198675066170 \nu^{14} + 292450754027165513539470160 \nu^{13} + 2226878906398720288454917560 \nu^{12} + 25263266575828111291938782414 \nu^{11} - 130955780952902562464699541580 \nu^{10} + 458255913561199101206385102480 \nu^{9} + 3418446210754313761500284959880 \nu^{8} + 15752394612797340867688025357631 \nu^{7} - 43372335081959687277592960733330 \nu^{6} + 172352038909657772404504958906560 \nu^{5} + 1270855705906394283493640257820560 \nu^{4} + 3111633157131511050847624678326208 \nu^{3} + 2782839408275349352230347459422080 \nu^{2} + 532880132548126797451977997516800 \nu - 338043110964360494658210332032000\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-6056238176411623354901114 \nu^{15} + 42688660290736661449235625 \nu^{14} - 146396752377212370392985060 \nu^{13} - 1114365932830071889909269160 \nu^{12} - 12669223252126716516697758108 \nu^{11} + 65598741434497573017700291490 \nu^{10} - 228974356306765774837195962480 \nu^{9} - 1708385418054973711004999115680 \nu^{8} - 7897245961507910001212281287842 \nu^{7} + 21731443531600977047188683957145 \nu^{6} - 85927199122479636045696950022860 \nu^{5} - 634888487783576647750492215095160 \nu^{4} - 1555603783724061364253191658007936 \nu^{3} - 1391388873664253031217367420877760 \nu^{2} - 266461998719592764984523512185600 \nu - 113359430152712771622478808256000\)\()/ \)\(75\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(2097730881 \nu^{15} - 14409577452 \nu^{14} + 47557870248 \nu^{13} + 399345069528 \nu^{12} + 4443245195650 \nu^{11} - 22040222469376 \nu^{10} + 74450035079504 \nu^{9} + 613147040366744 \nu^{8} + 2818574049615457 \nu^{7} - 7210077963870292 \nu^{6} + 28041792915591848 \nu^{5} + 227948096200534528 \nu^{4} + 569226084296757312 \nu^{3} + 510847011841073920 \nu^{2} + 97996854480243200 \nu - 10285863897088000\)\()/ 11314720808960000 \)
\(\beta_{8}\)\(=\)\((\)\(32855433884161413600786413 \nu^{15} - 242881987450695476744051722 \nu^{14} + 871161927604647418785675808 \nu^{13} + 5788268119260471302117910088 \nu^{12} + 66736303643423945685902433034 \nu^{11} - 380988323317994855078296121756 \nu^{10} + 1361636072012469463115585731184 \nu^{9} + 8878699531067865896278523313624 \nu^{8} + 39855087018970249650230574815293 \nu^{7} - 134880513368049375947794748088722 \nu^{6} + 505538527532165038949999971833808 \nu^{5} + 3301443704469897880689965824795088 \nu^{4} + 7356143154379500356313529867082560 \nu^{3} + 3865220491343500644870997547440000 \nu^{2} - 1782015691578969602114305069440000 \nu - 578511245920333805845166998400000\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(18157761202582385835845453 \nu^{15} - 138201480217931814755258697 \nu^{14} + 531672511457828758625255308 \nu^{13} + 2927634693131037963229766288 \nu^{12} + 36775410834714584892565399714 \nu^{11} - 215117302827581639464603206106 \nu^{10} + 838646339177981324328878961184 \nu^{9} + 4469526381715517371670585549224 \nu^{8} + 21815504759964780226944741365813 \nu^{7} - 74159946440587864531526315498097 \nu^{6} + 317790730961322578382490691364308 \nu^{5} + 1658880350791282506855334044563288 \nu^{4} + 3960864469926590399395433656416320 \nu^{3} + 3113414347872909791204126651723200 \nu^{2} + 2177788264840499473721106893600000 \nu + 156991172636117816978821586880000\)\()/ \)\(75\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-25619847127364202804004088 \nu^{15} + 209013915909942067698997291 \nu^{14} - 846389187876494638451214684 \nu^{13} - 3816030313296174660555045224 \nu^{12} - 49206426328983244912405952680 \nu^{11} + 333019027745361728377740895958 \nu^{10} - 1329910328587199825907171723632 \nu^{9} - 5817938692905048834300855007952 \nu^{8} - 26702026038063337195485867742576 \nu^{7} + 123101801538077153612327213798411 \nu^{6} - 495777422475835530711960706934484 \nu^{5} - 2158391084077825233059704640933224 \nu^{4} - 4087003797708202574724196686285056 \nu^{3} - 741923545412565971612274577738560 \nu^{2} + 662781347076851192596360652614400 \nu + 277280066820098054074805262784000\)\()/ \)\(75\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-30210166485329928547596197 \nu^{15} + 251256414977023779662258048 \nu^{14} - 1041640945465414842749367672 \nu^{13} - 4304316034535906379155885592 \nu^{12} - 57453883747226921850999013866 \nu^{11} + 401278250762337189387697730504 \nu^{10} - 1638578943248015095930652851856 \nu^{9} - 6554476762032694134934175321016 \nu^{8} - 30595634419827574462518605423077 \nu^{7} + 149411967066120270337621889373448 \nu^{6} - 610747435633717741112331950811672 \nu^{5} - 2432295735985634937265953883482592 \nu^{4} - 4480165239972381135404607380362560 \nu^{3} - 301899735384081685505650056947200 \nu^{2} + 916995256244655122978920939584000 \nu + 351229866651616794027057921280000\)\()/ \)\(75\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(79423963299802415213428159 \nu^{15} - 637398970433385122773090418 \nu^{14} + 2570420193123899633387122032 \nu^{13} + 11904051310699587852608181752 \nu^{12} + 155290462324251649821365419310 \nu^{11} - 1007748229131034253939180973884 \nu^{10} + 4047505737212729675542314305936 \nu^{9} + 18148342974333468833428203106696 \nu^{8} + 87027351311464961269957603831503 \nu^{7} - 364198561763488524906836092405578 \nu^{6} + 1517842937803649758010961151510432 \nu^{5} + 6735333608399852024063411905308752 \nu^{4} + 14259969796714783357035631787169728 \nu^{3} + 6275709390438039035024865942686080 \nu^{2} + 2189604048871786919170488226764800 \nu - 268229090643105600095955394432000\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(48008523463796959406891937 \nu^{15} - 383734193142080325729672276 \nu^{14} + 1542876160727974109994264584 \nu^{13} + 7230013894824407560407961624 \nu^{12} + 94222818806802536818828103298 \nu^{11} - 606296556504396943109256018128 \nu^{10} + 2430444413972650654024369636432 \nu^{9} + 11030722642356792956362503088152 \nu^{8} + 53179278687780696517455621864193 \nu^{7} - 218680193033144990980038171773116 \nu^{6} + 911940339331313184019076750490184 \nu^{5} + 4096984927306688652604273117730624 \nu^{4} + 8837192332068518501535894696081472 \nu^{3} + 4022959077777966547452123543537920 \nu^{2} + 1412561912246498912690027617907200 \nu - 233753043249498438207760168448000\)\()/ \)\(75\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(123585303480524997884648893 \nu^{15} - 989933054674581662528277490 \nu^{14} + 3954839715704469569811065280 \nu^{13} + 18836138570789250849481747080 \nu^{12} + 240876834315876628200248272106 \nu^{11} - 1568530855388415159639096171500 \nu^{10} + 6219828013733831984160550086640 \nu^{9} + 28737598450342389474404480794840 \nu^{8} + 134271957971637412609448544632909 \nu^{7} - 570568583838898452656409028449530 \nu^{6} + 2327946541006283012588389143592880 \nu^{5} + 10665221561761649876802701931304080 \nu^{4} + 21780767738934702147402476774774592 \nu^{3} + 8046252818223053116061802256731520 \nu^{2} + 1682261296482405187622675467443200 \nu - 606508668662217530050672969088000\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-154588593766939683177084097 \nu^{15} + 1245088980323407888951858514 \nu^{14} - 5012937608173823679310184736 \nu^{13} - 23221279964821955496818728296 \nu^{12} - 300895852204709338451466564210 \nu^{11} + 1974677569417393936516357056332 \nu^{10} - 7883590770144275742813070676528 \nu^{9} - 35412348382788580638042115262008 \nu^{8} - 167334587478993405604532163851089 \nu^{7} + 720255158235518105142458800089594 \nu^{6} - 2948223419297075651954316089953936 \nu^{5} - 13144333292294696935240264533345296 \nu^{4} - 27003671336103006834157847486320704 \nu^{3} - 9157328852483389193017946181447040 \nu^{2} - 1003618701632613008603638676198400 \nu + 1038375690517642357720280308096000\)\()/ \)\(15\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{14} + \beta_{13} - 5 \beta_{12} - 3 \beta_{11} + 5 \beta_{10} - \beta_{8} - \beta_{6} + \beta_{5} + 6 \beta_{4} + 2 \beta_{3} + 7 \beta_{2} + 9\)\()/10\)
\(\nu^{2}\)\(=\)\((\)\(-5 \beta_{15} + 6 \beta_{14} - 3 \beta_{13} - 5 \beta_{12} - \beta_{11} + 15 \beta_{10} - 5 \beta_{9} - 2 \beta_{8} + 38 \beta_{6} - 38 \beta_{5} + 32 \beta_{4} - 26 \beta_{3} + 229 \beta_{2} + 3\)\()/10\)
\(\nu^{3}\)\(=\)\((\)\(40 \beta_{15} + 81 \beta_{14} - 123 \beta_{13} + 205 \beta_{12} - 41 \beta_{11} + 205 \beta_{10} - 40 \beta_{9} + 83 \beta_{8} + 80 \beta_{7} + 83 \beta_{6} - 243 \beta_{5} + 82 \beta_{4} - 406 \beta_{3} + 589 \beta_{2} + 80 \beta_{1} - 507\)\()/10\)
\(\nu^{4}\)\(=\)\((\)\(121 \beta_{15} + 78 \beta_{14} - 45 \beta_{13} + 247 \beta_{12} + 43 \beta_{11} + 41 \beta_{10} - 57 \beta_{9} + 102 \beta_{8} - 338 \beta_{6} - 542 \beta_{5} - 298 \beta_{4} - 384 \beta_{3} + 43 \beta_{2} + 160 \beta_{1} - 1897\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(4100 \beta_{15} - 2547 \beta_{14} - 1749 \beta_{13} + 10445 \beta_{12} + 5547 \beta_{11} - 10145 \beta_{10} + 3000 \beta_{9} + 5849 \beta_{8} - 7200 \beta_{7} - 10151 \beta_{6} - 5849 \beta_{5} - 24594 \beta_{4} - 3798 \beta_{3} - 31743 \beta_{2} + 7200 \beta_{1} - 35541\)\()/10\)
\(\nu^{6}\)\(=\)\((\)\(16245 \beta_{15} - 36214 \beta_{14} + 14507 \beta_{13} + 9045 \beta_{12} + 19969 \beta_{11} - 86935 \beta_{10} + 46645 \beta_{9} + 32138 \beta_{8} - 75200 \beta_{7} - 97022 \beta_{6} + 97022 \beta_{5} - 128608 \beta_{4} + 99594 \beta_{3} - 476101 \beta_{2} - 14507\)\()/10\)
\(\nu^{7}\)\(=\)\((\)\(-204360 \beta_{15} - 400809 \beta_{14} + 288947 \beta_{13} - 569245 \beta_{12} + 83849 \beta_{11} - 606645 \beta_{10} + 316960 \beta_{9} - 84587 \beta_{8} - 516720 \beta_{7} - 84587 \beta_{6} + 1284427 \beta_{5} - 205098 \beta_{4} + 1521534 \beta_{3} - 2238821 \beta_{2} - 516720 \beta_{1} + 2033723\)\()/10\)
\(\nu^{8}\)\(=\)\((\)\(-641969 \beta_{15} - 449350 \beta_{14} + 293269 \beta_{13} - 1169783 \beta_{12} - 192619 \beta_{11} - 91969 \beta_{10} + 195313 \beta_{9} - 488582 \beta_{8} + 1105618 \beta_{6} + 2082782 \beta_{5} + 1347946 \beta_{4} + 1733184 \beta_{3} - 192619 \beta_{2} - 1117280 \beta_{1} + 5425889\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-22295100 \beta_{15} + 3142203 \beta_{14} + 4516101 \beta_{13} - 37620805 \beta_{12} - 16699603 \beta_{11} + 34469505 \beta_{10} - 13557400 \beta_{9} - 26811201 \beta_{8} + 34769600 \beta_{7} + 55868799 \beta_{6} + 26811201 \beta_{5} + 96239706 \beta_{4} + 12183502 \beta_{3} + 132013007 \beta_{2} - 34769600 \beta_{1} + 144196509\)\()/10\)
\(\nu^{10}\)\(=\)\((\)\(-60895005 \beta_{15} + 161696566 \beta_{14} - 63463483 \beta_{13} - 26125405 \beta_{12} - 100801561 \beta_{11} + 386856615 \beta_{10} - 213114205 \beta_{9} - 149650722 \beta_{8} + 384809600 \beta_{7} + 356150918 \beta_{6} - 356150918 \beta_{5} + 578206752 \beta_{4} - 451279786 \beta_{3} + 1660369069 \beta_{2} + 63463483\)\()/10\)
\(\nu^{11}\)\(=\)\((\)\(890611240 \beta_{15} + 1771701601 \beta_{14} - 1025775883 \beta_{13} + 2171613805 \beta_{12} - 264747561 \beta_{11} + 2403147005 \beta_{10} - 1506954040 \beta_{9} + 135164643 \beta_{8} + 2291086480 \beta_{7} + 135164643 \beta_{6} - 5656844803 \beta_{5} + 761028322 \beta_{4} - 6174792326 \beta_{3} + 9371249469 \beta_{2} + 2291086480 \beta_{1} - 8610221147\)\()/10\)
\(\nu^{12}\)\(=\)\((\)\(2796620041 \beta_{15} + 1963647742 \beta_{14} - 1348312477 \beta_{13} + 5081296087 \beta_{12} + 832972299 \beta_{11} + 317632121 \beta_{10} - 775849417 \beta_{9} + 2124161894 \beta_{8} - 4387973266 \beta_{6} - 8636297054 \beta_{5} - 5981748970 \beta_{4} - 7647693568 \beta_{3} + 832972299 \beta_{2} + 5144386080 \beta_{1} - 21052502265\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(100017523700 \beta_{15} - 6752694947 \beta_{14} - 16340771349 \beta_{13} + 155464748445 \beta_{12} + 65035301147 \beta_{11} - 139451761345 \beta_{10} + 58282606200 \beta_{9} + 116358295049 \beta_{8} - 149939701600 \beta_{7} - 247726472951 \beta_{6} - 116358295049 \beta_{5} - 399436669794 \beta_{4} - 48694529798 \beta_{3} - 562349185343 \beta_{2} + 149939701600 \beta_{1} - 611043715141\)\()/10\)
\(\nu^{14}\)\(=\)\((\)\(250038606245 \beta_{15} - 695324153654 \beta_{14} + 272692226027 \beta_{13} + 100098904645 \beta_{12} + 445285547409 \beta_{11} - 1663340533335 \beta_{10} + 914002777445 \beta_{9} + 641310551418 \beta_{8} - 1697234785600 \beta_{7} - 1469893976142 \beta_{6} + 1469893976142 \beta_{5} - 2514309521888 \beta_{4} + 1968925069834 \beta_{3} - 6782611526261 \beta_{2} - 272692226027\)\()/10\)
\(\nu^{15}\)\(=\)\((\)\(-3808439313160 \beta_{15} - 7618714957689 \beta_{14} + 4187571520387 \beta_{13} - 9035112495645 \beta_{12} + 1037265330729 \beta_{11} - 10110888023845 \beta_{10} + 6581449626960 \beta_{9} - 379132207227 \beta_{8} - 9792533190320 \beta_{7} - 379132207227 \beta_{6} + 24255276527067 \beta_{5} - 3150306189658 \beta_{4} + 25954079128014 \beta_{3} - 39885777879541 \beta_{2} - 9792533190320 \beta_{1} + 36735471689883\)\()/10\)

Character values

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

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−0.394902 + 0.394902i
−3.99135 + 3.99135i
3.76660 3.76660i
0.170157 0.170157i
3.60306 3.60306i
5.71348 5.71348i
−3.48873 + 3.48873i
−1.37832 + 1.37832i
−0.394902 0.394902i
−3.99135 3.99135i
3.76660 + 3.76660i
0.170157 + 0.170157i
3.60306 + 3.60306i
5.71348 + 5.71348i
−3.48873 3.48873i
−1.37832 1.37832i
−1.00000 + 1.00000i −1.22474 1.22474i 2.00000i −4.93066 0.829843i 2.44949 1.87083 1.87083i 2.00000 + 2.00000i 3.00000i 5.76050 4.10081i
43.2 −1.00000 + 1.00000i −1.22474 1.22474i 2.00000i −4.16484 + 2.76660i 2.44949 −1.87083 + 1.87083i 2.00000 + 2.00000i 3.00000i 1.39824 6.93144i
43.3 −1.00000 + 1.00000i −1.22474 1.22474i 2.00000i 0.294013 4.99135i 2.44949 −1.87083 + 1.87083i 2.00000 + 2.00000i 3.00000i 4.69734 + 5.28536i
43.4 −1.00000 + 1.00000i −1.22474 1.22474i 2.00000i 4.80148 1.39490i 2.44949 1.87083 1.87083i 2.00000 + 2.00000i 3.00000i −3.40658 + 6.19639i
43.5 −1.00000 + 1.00000i 1.22474 + 1.22474i 2.00000i −4.39814 2.37832i −2.44949 1.87083 1.87083i 2.00000 + 2.00000i 3.00000i 6.77646 2.01982i
43.6 −1.00000 + 1.00000i 1.22474 + 1.22474i 2.00000i −2.20256 4.48873i −2.44949 −1.87083 + 1.87083i 2.00000 + 2.00000i 3.00000i 6.69129 + 2.28617i
43.7 −1.00000 + 1.00000i 1.22474 + 1.22474i 2.00000i −1.66827 + 4.71348i −2.44949 −1.87083 + 1.87083i 2.00000 + 2.00000i 3.00000i −3.04521 6.38175i
43.8 −1.00000 + 1.00000i 1.22474 + 1.22474i 2.00000i 4.26896 + 2.60306i −2.44949 1.87083 1.87083i 2.00000 + 2.00000i 3.00000i −6.87203 + 1.66590i
127.1 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i −4.93066 + 0.829843i 2.44949 1.87083 + 1.87083i 2.00000 2.00000i 3.00000i 5.76050 + 4.10081i
127.2 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i −4.16484 2.76660i 2.44949 −1.87083 1.87083i 2.00000 2.00000i 3.00000i 1.39824 + 6.93144i
127.3 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 0.294013 + 4.99135i 2.44949 −1.87083 1.87083i 2.00000 2.00000i 3.00000i 4.69734 5.28536i
127.4 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 4.80148 + 1.39490i 2.44949 1.87083 + 1.87083i 2.00000 2.00000i 3.00000i −3.40658 6.19639i
127.5 −1.00000 1.00000i 1.22474 1.22474i 2.00000i −4.39814 + 2.37832i −2.44949 1.87083 + 1.87083i 2.00000 2.00000i 3.00000i 6.77646 + 2.01982i
127.6 −1.00000 1.00000i 1.22474 1.22474i 2.00000i −2.20256 + 4.48873i −2.44949 −1.87083 1.87083i 2.00000 2.00000i 3.00000i 6.69129 2.28617i
127.7 −1.00000 1.00000i 1.22474 1.22474i 2.00000i −1.66827 4.71348i −2.44949 −1.87083 1.87083i 2.00000 2.00000i 3.00000i −3.04521 + 6.38175i
127.8 −1.00000 1.00000i 1.22474 1.22474i 2.00000i 4.26896 2.60306i −2.44949 1.87083 + 1.87083i 2.00000 2.00000i 3.00000i −6.87203 1.66590i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.3.l.b 16
3.b odd 2 1 630.3.o.f 16
5.b even 2 1 1050.3.l.h 16
5.c odd 4 1 inner 210.3.l.b 16
5.c odd 4 1 1050.3.l.h 16
15.e even 4 1 630.3.o.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.l.b 16 1.a even 1 1 trivial
210.3.l.b 16 5.c odd 4 1 inner
630.3.o.f 16 3.b odd 2 1
630.3.o.f 16 15.e even 4 1
1050.3.l.h 16 5.b even 2 1
1050.3.l.h 16 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{11}^{8} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(210, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T + 2 T^{2} )^{8} \)
$3$ \( ( 1 + 9 T^{4} )^{4} \)
$5$ \( 1 + 16 T + 108 T^{2} + 240 T^{3} - 1736 T^{4} - 15856 T^{5} - 26268 T^{6} + 353520 T^{7} + 2938350 T^{8} + 8838000 T^{9} - 16417500 T^{10} - 247750000 T^{11} - 678125000 T^{12} + 2343750000 T^{13} + 26367187500 T^{14} + 97656250000 T^{15} + 152587890625 T^{16} \)
$7$ \( ( 1 + 49 T^{4} )^{4} \)
$11$ \( ( 1 - 4 T + 428 T^{2} - 3020 T^{3} + 112132 T^{4} - 750180 T^{5} + 21255892 T^{6} - 132232716 T^{7} + 2897491254 T^{8} - 16000158636 T^{9} + 311207514772 T^{10} - 1328989630980 T^{11} + 24036490044292 T^{12} - 78331022295020 T^{13} + 1343247345236588 T^{14} - 1518999334332964 T^{15} + 45949729863572161 T^{16} )^{2} \)
$13$ \( 1 + 32 T + 512 T^{2} + 12624 T^{3} + 210400 T^{4} + 1218608 T^{5} + 10953344 T^{6} - 12361888 T^{7} - 9595251012 T^{8} - 144909422048 T^{9} - 1442470310272 T^{10} - 30698489798640 T^{11} - 199614992290272 T^{12} + 2586969858854768 T^{13} + 29532641854346496 T^{14} + 799237423444942944 T^{15} + 19007129532539001606 T^{16} + \)\(13\!\cdots\!36\)\( T^{17} + \)\(84\!\cdots\!56\)\( T^{18} + \)\(12\!\cdots\!12\)\( T^{19} - \)\(16\!\cdots\!12\)\( T^{20} - \)\(42\!\cdots\!60\)\( T^{21} - \)\(33\!\cdots\!32\)\( T^{22} - \)\(57\!\cdots\!72\)\( T^{23} - \)\(63\!\cdots\!92\)\( T^{24} - \)\(13\!\cdots\!52\)\( T^{25} + \)\(20\!\cdots\!44\)\( T^{26} + \)\(39\!\cdots\!52\)\( T^{27} + \)\(11\!\cdots\!00\)\( T^{28} + \)\(11\!\cdots\!16\)\( T^{29} + \)\(79\!\cdots\!52\)\( T^{30} + \)\(83\!\cdots\!68\)\( T^{31} + \)\(44\!\cdots\!81\)\( T^{32} \)
$17$ \( 1 - 56 T + 1568 T^{2} - 29832 T^{3} + 267808 T^{4} + 3397960 T^{5} - 165234592 T^{6} + 3465012472 T^{7} - 48572233284 T^{8} + 491667950696 T^{9} - 4716183819232 T^{10} + 39445096533336 T^{11} + 280624802792928 T^{12} - 26208629063564312 T^{13} + 761979769352889696 T^{14} - 15987258252632453928 T^{15} + \)\(28\!\cdots\!74\)\( T^{16} - \)\(46\!\cdots\!92\)\( T^{17} + \)\(63\!\cdots\!16\)\( T^{18} - \)\(63\!\cdots\!28\)\( T^{19} + \)\(19\!\cdots\!48\)\( T^{20} + \)\(79\!\cdots\!64\)\( T^{21} - \)\(27\!\cdots\!52\)\( T^{22} + \)\(82\!\cdots\!84\)\( T^{23} - \)\(23\!\cdots\!04\)\( T^{24} + \)\(48\!\cdots\!48\)\( T^{25} - \)\(67\!\cdots\!92\)\( T^{26} + \)\(39\!\cdots\!40\)\( T^{27} + \)\(90\!\cdots\!68\)\( T^{28} - \)\(29\!\cdots\!08\)\( T^{29} + \)\(44\!\cdots\!88\)\( T^{30} - \)\(45\!\cdots\!44\)\( T^{31} + \)\(23\!\cdots\!61\)\( T^{32} \)
$19$ \( 1 - 2056 T^{2} + 2230944 T^{4} - 1603740248 T^{6} + 819184531004 T^{8} - 297045269538504 T^{10} + 69987022118810464 T^{12} - 7175141044809308632 T^{14} - \)\(43\!\cdots\!46\)\( T^{16} - \)\(93\!\cdots\!72\)\( T^{18} + \)\(11\!\cdots\!24\)\( T^{20} - \)\(65\!\cdots\!44\)\( T^{22} + \)\(23\!\cdots\!24\)\( T^{24} - \)\(60\!\cdots\!48\)\( T^{26} + \)\(10\!\cdots\!24\)\( T^{28} - \)\(13\!\cdots\!96\)\( T^{30} + \)\(83\!\cdots\!61\)\( T^{32} \)
$23$ \( 1 - 24 T + 288 T^{2} - 17192 T^{3} + 221472 T^{4} - 2391768 T^{5} + 141400928 T^{6} - 3172207336 T^{7} + 137674628156 T^{8} - 274022502200 T^{9} - 6993233578720 T^{10} - 103772124341000 T^{11} - 60508914798085408 T^{12} + 933592691352233992 T^{13} - 15427273581773601184 T^{14} + \)\(57\!\cdots\!36\)\( T^{15} - \)\(64\!\cdots\!26\)\( T^{16} + \)\(30\!\cdots\!44\)\( T^{17} - \)\(43\!\cdots\!44\)\( T^{18} + \)\(13\!\cdots\!88\)\( T^{19} - \)\(47\!\cdots\!48\)\( T^{20} - \)\(42\!\cdots\!00\)\( T^{21} - \)\(15\!\cdots\!20\)\( T^{22} - \)\(31\!\cdots\!00\)\( T^{23} + \)\(84\!\cdots\!16\)\( T^{24} - \)\(10\!\cdots\!84\)\( T^{25} + \)\(24\!\cdots\!28\)\( T^{26} - \)\(21\!\cdots\!72\)\( T^{27} + \)\(10\!\cdots\!52\)\( T^{28} - \)\(43\!\cdots\!88\)\( T^{29} + \)\(38\!\cdots\!28\)\( T^{30} - \)\(17\!\cdots\!76\)\( T^{31} + \)\(37\!\cdots\!21\)\( T^{32} \)
$29$ \( 1 - 4032 T^{2} + 8105976 T^{4} - 11130637632 T^{6} + 11901856158364 T^{8} - 11208853722734016 T^{10} + 10516466734892486856 T^{12} - \)\(99\!\cdots\!40\)\( T^{14} + \)\(87\!\cdots\!46\)\( T^{16} - \)\(70\!\cdots\!40\)\( T^{18} + \)\(52\!\cdots\!16\)\( T^{20} - \)\(39\!\cdots\!56\)\( T^{22} + \)\(29\!\cdots\!44\)\( T^{24} - \)\(19\!\cdots\!32\)\( T^{26} + \)\(10\!\cdots\!56\)\( T^{28} - \)\(35\!\cdots\!52\)\( T^{30} + \)\(62\!\cdots\!41\)\( T^{32} \)
$31$ \( ( 1 + 56 T + 4484 T^{2} + 186776 T^{3} + 8992264 T^{4} + 304202984 T^{5} + 11800345036 T^{6} + 348556649544 T^{7} + 12266333612430 T^{8} + 334962940211784 T^{9} + 10897866447991756 T^{10} + 269981268071184104 T^{11} + 7669421371903356424 T^{12} + \)\(15\!\cdots\!76\)\( T^{13} + \)\(35\!\cdots\!24\)\( T^{14} + \)\(42\!\cdots\!76\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} )^{2} \)
$37$ \( 1 + 152 T + 11552 T^{2} + 598344 T^{3} + 25962168 T^{4} + 1167158904 T^{5} + 56500959840 T^{6} + 2639490056104 T^{7} + 111109511071516 T^{8} + 4242418231122872 T^{9} + 154876079427922976 T^{10} + 5762441784032775208 T^{11} + \)\(22\!\cdots\!72\)\( T^{12} + \)\(89\!\cdots\!08\)\( T^{13} + \)\(32\!\cdots\!12\)\( T^{14} + \)\(10\!\cdots\!40\)\( T^{15} + \)\(37\!\cdots\!46\)\( T^{16} + \)\(14\!\cdots\!60\)\( T^{17} + \)\(60\!\cdots\!32\)\( T^{18} + \)\(22\!\cdots\!72\)\( T^{19} + \)\(79\!\cdots\!12\)\( T^{20} + \)\(27\!\cdots\!92\)\( T^{21} + \)\(10\!\cdots\!56\)\( T^{22} + \)\(38\!\cdots\!08\)\( T^{23} + \)\(13\!\cdots\!56\)\( T^{24} + \)\(44\!\cdots\!16\)\( T^{25} + \)\(13\!\cdots\!40\)\( T^{26} + \)\(36\!\cdots\!76\)\( T^{27} + \)\(11\!\cdots\!48\)\( T^{28} + \)\(35\!\cdots\!96\)\( T^{29} + \)\(93\!\cdots\!92\)\( T^{30} + \)\(16\!\cdots\!48\)\( T^{31} + \)\(15\!\cdots\!81\)\( T^{32} \)
$41$ \( ( 1 + 8252 T^{2} - 19936 T^{3} + 34068728 T^{4} - 127251488 T^{5} + 92291784532 T^{6} - 367862933760 T^{7} + 180432835211182 T^{8} - 618377591650560 T^{9} + 260794525350928852 T^{10} - 604457832822360608 T^{11} + \)\(27\!\cdots\!88\)\( T^{12} - \)\(26\!\cdots\!36\)\( T^{13} + \)\(18\!\cdots\!12\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} )^{2} \)
$43$ \( 1 - 158592 T^{3} + 2725896 T^{4} - 61004160 T^{5} + 12575711232 T^{6} - 618250768128 T^{7} + 18017190018460 T^{8} - 968512496926464 T^{9} + 65630298643144704 T^{10} - 3092793950726971008 T^{11} + \)\(10\!\cdots\!80\)\( T^{12} - \)\(37\!\cdots\!52\)\( T^{13} + \)\(33\!\cdots\!64\)\( T^{14} - \)\(12\!\cdots\!48\)\( T^{15} + \)\(20\!\cdots\!26\)\( T^{16} - \)\(22\!\cdots\!52\)\( T^{17} + \)\(11\!\cdots\!64\)\( T^{18} - \)\(23\!\cdots\!48\)\( T^{19} + \)\(11\!\cdots\!80\)\( T^{20} - \)\(66\!\cdots\!92\)\( T^{21} + \)\(26\!\cdots\!04\)\( T^{22} - \)\(71\!\cdots\!36\)\( T^{23} + \)\(24\!\cdots\!60\)\( T^{24} - \)\(15\!\cdots\!72\)\( T^{25} + \)\(58\!\cdots\!32\)\( T^{26} - \)\(52\!\cdots\!40\)\( T^{27} + \)\(43\!\cdots\!96\)\( T^{28} - \)\(46\!\cdots\!08\)\( T^{29} + \)\(18\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 - 80 T + 3200 T^{2} - 185424 T^{3} + 10961416 T^{4} - 416599088 T^{5} + 15442425728 T^{6} - 918151881776 T^{7} + 40326873674268 T^{8} - 452413353633424 T^{9} - 5098985333045632 T^{10} + 1563817770500267376 T^{11} - \)\(31\!\cdots\!76\)\( T^{12} + \)\(17\!\cdots\!60\)\( T^{13} - \)\(66\!\cdots\!16\)\( T^{14} + \)\(36\!\cdots\!52\)\( T^{15} - \)\(19\!\cdots\!38\)\( T^{16} + \)\(79\!\cdots\!68\)\( T^{17} - \)\(32\!\cdots\!96\)\( T^{18} + \)\(19\!\cdots\!40\)\( T^{19} - \)\(75\!\cdots\!36\)\( T^{20} + \)\(82\!\cdots\!24\)\( T^{21} - \)\(59\!\cdots\!12\)\( T^{22} - \)\(11\!\cdots\!56\)\( T^{23} + \)\(22\!\cdots\!28\)\( T^{24} - \)\(11\!\cdots\!64\)\( T^{25} + \)\(42\!\cdots\!28\)\( T^{26} - \)\(25\!\cdots\!92\)\( T^{27} + \)\(14\!\cdots\!96\)\( T^{28} - \)\(55\!\cdots\!96\)\( T^{29} + \)\(21\!\cdots\!00\)\( T^{30} - \)\(11\!\cdots\!20\)\( T^{31} + \)\(32\!\cdots\!41\)\( T^{32} \)
$53$ \( 1 - 48 T + 1152 T^{2} - 172480 T^{3} + 14460640 T^{4} - 33746752 T^{5} - 164137984 T^{6} + 7272556080 T^{7} - 144052676687940 T^{8} + 6754246786116880 T^{9} - 156566570729557248 T^{10} + 21067810577807322688 T^{11} - \)\(28\!\cdots\!60\)\( T^{12} - \)\(50\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!80\)\( T^{14} - \)\(15\!\cdots\!88\)\( T^{15} + \)\(20\!\cdots\!78\)\( T^{16} - \)\(43\!\cdots\!92\)\( T^{17} + \)\(90\!\cdots\!80\)\( T^{18} - \)\(11\!\cdots\!00\)\( T^{19} - \)\(17\!\cdots\!60\)\( T^{20} + \)\(36\!\cdots\!12\)\( T^{21} - \)\(76\!\cdots\!68\)\( T^{22} + \)\(93\!\cdots\!20\)\( T^{23} - \)\(55\!\cdots\!40\)\( T^{24} + \)\(79\!\cdots\!20\)\( T^{25} - \)\(50\!\cdots\!84\)\( T^{26} - \)\(28\!\cdots\!68\)\( T^{27} + \)\(34\!\cdots\!40\)\( T^{28} - \)\(11\!\cdots\!20\)\( T^{29} + \)\(21\!\cdots\!72\)\( T^{30} - \)\(25\!\cdots\!52\)\( T^{31} + \)\(15\!\cdots\!41\)\( T^{32} \)
$59$ \( 1 - 14320 T^{2} + 157767608 T^{4} - 1244890527696 T^{6} + 8207935216533276 T^{8} - 45520264315988706288 T^{10} + \)\(21\!\cdots\!36\)\( T^{12} - \)\(91\!\cdots\!96\)\( T^{14} + \)\(33\!\cdots\!38\)\( T^{16} - \)\(11\!\cdots\!56\)\( T^{18} + \)\(32\!\cdots\!56\)\( T^{20} - \)\(80\!\cdots\!28\)\( T^{22} + \)\(17\!\cdots\!16\)\( T^{24} - \)\(32\!\cdots\!96\)\( T^{26} + \)\(49\!\cdots\!88\)\( T^{28} - \)\(54\!\cdots\!20\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \)
$61$ \( ( 1 - 48 T + 22640 T^{2} - 673936 T^{3} + 215932124 T^{4} - 3175993520 T^{5} + 1217381291920 T^{6} - 6110146782096 T^{7} + 5029652273881990 T^{8} - 22735856176179216 T^{9} + 16855667804298904720 T^{10} - \)\(16\!\cdots\!20\)\( T^{11} + \)\(41\!\cdots\!44\)\( T^{12} - \)\(48\!\cdots\!36\)\( T^{13} + \)\(60\!\cdots\!40\)\( T^{14} - \)\(47\!\cdots\!68\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \)
$67$ \( 1 + 80 T + 3200 T^{2} - 609968 T^{3} - 25803832 T^{4} + 2674348464 T^{5} + 482550620032 T^{6} + 12521071726512 T^{7} - 1529742891382500 T^{8} - 116256625005179504 T^{9} + 3984919157383710336 T^{10} + \)\(89\!\cdots\!92\)\( T^{11} + \)\(25\!\cdots\!88\)\( T^{12} - \)\(21\!\cdots\!76\)\( T^{13} - \)\(18\!\cdots\!68\)\( T^{14} + \)\(71\!\cdots\!08\)\( T^{15} + \)\(10\!\cdots\!46\)\( T^{16} + \)\(31\!\cdots\!12\)\( T^{17} - \)\(38\!\cdots\!28\)\( T^{18} - \)\(19\!\cdots\!44\)\( T^{19} + \)\(10\!\cdots\!08\)\( T^{20} + \)\(16\!\cdots\!08\)\( T^{21} + \)\(32\!\cdots\!96\)\( T^{22} - \)\(42\!\cdots\!16\)\( T^{23} - \)\(25\!\cdots\!00\)\( T^{24} + \)\(92\!\cdots\!08\)\( T^{25} + \)\(16\!\cdots\!32\)\( T^{26} + \)\(39\!\cdots\!96\)\( T^{27} - \)\(17\!\cdots\!72\)\( T^{28} - \)\(18\!\cdots\!92\)\( T^{29} + \)\(43\!\cdots\!00\)\( T^{30} + \)\(48\!\cdots\!20\)\( T^{31} + \)\(27\!\cdots\!61\)\( T^{32} \)
$71$ \( ( 1 - 268 T + 44204 T^{2} - 5071044 T^{3} + 471192708 T^{4} - 37316092140 T^{5} + 2755016710036 T^{6} - 195026045020868 T^{7} + 13901563235868662 T^{8} - 983126292950195588 T^{9} + 70009605785104330516 T^{10} - \)\(47\!\cdots\!40\)\( T^{11} + \)\(30\!\cdots\!88\)\( T^{12} - \)\(16\!\cdots\!44\)\( T^{13} + \)\(72\!\cdots\!64\)\( T^{14} - \)\(22\!\cdots\!08\)\( T^{15} + \)\(41\!\cdots\!21\)\( T^{16} )^{2} \)
$73$ \( 1 - 638960 T^{3} - 86919200 T^{4} - 369113360 T^{5} + 204134940800 T^{6} + 33110468899200 T^{7} + 5206311620706876 T^{8} + 26720496193596800 T^{9} - 3344897125184227200 T^{10} - \)\(13\!\cdots\!20\)\( T^{11} - \)\(22\!\cdots\!00\)\( T^{12} - \)\(53\!\cdots\!20\)\( T^{13} + \)\(75\!\cdots\!00\)\( T^{14} + \)\(36\!\cdots\!00\)\( T^{15} + \)\(78\!\cdots\!66\)\( T^{16} + \)\(19\!\cdots\!00\)\( T^{17} + \)\(21\!\cdots\!00\)\( T^{18} - \)\(80\!\cdots\!80\)\( T^{19} - \)\(17\!\cdots\!00\)\( T^{20} - \)\(59\!\cdots\!80\)\( T^{21} - \)\(76\!\cdots\!00\)\( T^{22} + \)\(32\!\cdots\!00\)\( T^{23} + \)\(33\!\cdots\!36\)\( T^{24} + \)\(11\!\cdots\!00\)\( T^{25} + \)\(37\!\cdots\!00\)\( T^{26} - \)\(36\!\cdots\!40\)\( T^{27} - \)\(45\!\cdots\!00\)\( T^{28} - \)\(17\!\cdots\!40\)\( T^{29} + \)\(42\!\cdots\!21\)\( T^{32} \)
$79$ \( 1 - 52208 T^{2} + 1391680568 T^{4} - 25119513250000 T^{6} + 343967562165562012 T^{8} - \)\(37\!\cdots\!72\)\( T^{10} + \)\(34\!\cdots\!24\)\( T^{12} - \)\(27\!\cdots\!40\)\( T^{14} + \)\(18\!\cdots\!30\)\( T^{16} - \)\(10\!\cdots\!40\)\( T^{18} + \)\(52\!\cdots\!64\)\( T^{20} - \)\(22\!\cdots\!52\)\( T^{22} + \)\(79\!\cdots\!52\)\( T^{24} - \)\(22\!\cdots\!00\)\( T^{26} + \)\(48\!\cdots\!08\)\( T^{28} - \)\(71\!\cdots\!88\)\( T^{30} + \)\(52\!\cdots\!41\)\( T^{32} \)
$83$ \( 1 + 256 T + 32768 T^{2} + 2489472 T^{3} + 78882184 T^{4} + 642592896 T^{5} + 678427795456 T^{6} + 150782876556800 T^{7} + 17212084280490396 T^{8} + 1275723698299503616 T^{9} + 84640034952293916672 T^{10} + \)\(62\!\cdots\!76\)\( T^{11} + \)\(45\!\cdots\!48\)\( T^{12} + \)\(40\!\cdots\!04\)\( T^{13} + \)\(47\!\cdots\!44\)\( T^{14} + \)\(59\!\cdots\!24\)\( T^{15} + \)\(59\!\cdots\!02\)\( T^{16} + \)\(41\!\cdots\!36\)\( T^{17} + \)\(22\!\cdots\!24\)\( T^{18} + \)\(13\!\cdots\!76\)\( T^{19} + \)\(10\!\cdots\!68\)\( T^{20} + \)\(96\!\cdots\!24\)\( T^{21} + \)\(90\!\cdots\!92\)\( T^{22} + \)\(93\!\cdots\!64\)\( T^{23} + \)\(87\!\cdots\!76\)\( T^{24} + \)\(52\!\cdots\!00\)\( T^{25} + \)\(16\!\cdots\!56\)\( T^{26} + \)\(10\!\cdots\!44\)\( T^{27} + \)\(90\!\cdots\!64\)\( T^{28} + \)\(19\!\cdots\!68\)\( T^{29} + \)\(17\!\cdots\!88\)\( T^{30} + \)\(95\!\cdots\!44\)\( T^{31} + \)\(25\!\cdots\!61\)\( T^{32} \)
$89$ \( 1 - 60568 T^{2} + 1632136448 T^{4} - 25689569431240 T^{6} + 260052896713761532 T^{8} - \)\(17\!\cdots\!32\)\( T^{10} + \)\(70\!\cdots\!24\)\( T^{12} - \)\(68\!\cdots\!20\)\( T^{14} - \)\(92\!\cdots\!90\)\( T^{16} - \)\(43\!\cdots\!20\)\( T^{18} + \)\(27\!\cdots\!44\)\( T^{20} - \)\(42\!\cdots\!72\)\( T^{22} + \)\(40\!\cdots\!52\)\( T^{24} - \)\(24\!\cdots\!40\)\( T^{26} + \)\(99\!\cdots\!68\)\( T^{28} - \)\(23\!\cdots\!08\)\( T^{30} + \)\(24\!\cdots\!21\)\( T^{32} \)
$97$ \( 1 - 688 T + 236672 T^{2} - 56991072 T^{3} + 10854023776 T^{4} - 1693042723360 T^{5} + 219961022412800 T^{6} - 24020408470599760 T^{7} + 2158083026478171708 T^{8} - \)\(14\!\cdots\!44\)\( T^{9} + \)\(52\!\cdots\!16\)\( T^{10} + \)\(44\!\cdots\!04\)\( T^{11} - \)\(12\!\cdots\!36\)\( T^{12} + \)\(17\!\cdots\!52\)\( T^{13} - \)\(20\!\cdots\!48\)\( T^{14} + \)\(21\!\cdots\!08\)\( T^{15} - \)\(20\!\cdots\!18\)\( T^{16} + \)\(19\!\cdots\!72\)\( T^{17} - \)\(18\!\cdots\!88\)\( T^{18} + \)\(14\!\cdots\!08\)\( T^{19} - \)\(97\!\cdots\!96\)\( T^{20} + \)\(32\!\cdots\!96\)\( T^{21} + \)\(36\!\cdots\!56\)\( T^{22} - \)\(96\!\cdots\!36\)\( T^{23} + \)\(13\!\cdots\!68\)\( T^{24} - \)\(13\!\cdots\!40\)\( T^{25} + \)\(11\!\cdots\!00\)\( T^{26} - \)\(86\!\cdots\!40\)\( T^{27} + \)\(52\!\cdots\!56\)\( T^{28} - \)\(25\!\cdots\!88\)\( T^{29} + \)\(10\!\cdots\!92\)\( T^{30} - \)\(27\!\cdots\!12\)\( T^{31} + \)\(37\!\cdots\!41\)\( T^{32} \)
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