Properties

Label 1155.2.c.e
Level 1155
Weight 2
Character orbit 1155.c
Analytic conductor 9.223
Analytic rank 0
Dimension 20
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1155.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 33 x^{18} + 456 x^{16} + 3426 x^{14} + 15194 x^{12} + 40320 x^{10} + 61593 x^{8} + 48545 x^{6} + 15624 x^{4} + 2116 x^{2} + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 3 \)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{18} - 65 \nu^{16} - 883 \nu^{14} - 6504 \nu^{12} - 28141 \nu^{10} - 72149 \nu^{8} - 104134 \nu^{6} - 72770 \nu^{4} - 15852 \nu^{2} - 1000 \)\()/20\)
\(\beta_{4}\)\(=\)\((\)\(-5 \nu^{19} - 2 \nu^{18} - 167 \nu^{17} - 72 \nu^{16} - 2337 \nu^{15} - 1090 \nu^{14} - 17770 \nu^{13} - 8984 \nu^{12} - 79489 \nu^{11} - 43540 \nu^{10} - 210785 \nu^{9} - 124604 \nu^{8} - 314269 \nu^{7} - 198808 \nu^{6} - 225998 \nu^{5} - 150428 \nu^{4} - 50148 \nu^{3} - 33220 \nu^{2} - 3220 \nu - 2040\)\()/20\)
\(\beta_{5}\)\(=\)\((\)\(-10 \nu^{19} - \nu^{18} - 337 \nu^{17} - 23 \nu^{16} - 4762 \nu^{15} - 157 \nu^{14} - 36585 \nu^{13} + 198 \nu^{12} - 165399 \nu^{11} + 7591 \nu^{10} - 443130 \nu^{9} + 38433 \nu^{8} - 666509 \nu^{7} + 83507 \nu^{6} - 481448 \nu^{5} + 75668 \nu^{4} - 105408 \nu^{3} + 17612 \nu^{2} - 6620 \nu + 1100\)\()/40\)
\(\beta_{6}\)\(=\)\((\)\(6 \nu^{19} + 5 \nu^{18} + 194 \nu^{17} + 167 \nu^{16} + 2614 \nu^{15} + 2337 \nu^{14} + 19028 \nu^{13} + 17770 \nu^{12} + 81026 \nu^{11} + 79489 \nu^{10} + 203584 \nu^{9} + 210785 \nu^{8} + 286860 \nu^{7} + 314269 \nu^{6} + 195158 \nu^{5} + 225998 \nu^{4} + 41326 \nu^{3} + 50148 \nu^{2} + 2564 \nu + 3220\)\()/20\)
\(\beta_{7}\)\(=\)\((\)\(-11 \nu^{19} - 10 \nu^{18} - 364 \nu^{17} - 337 \nu^{16} - 5039 \nu^{15} - 4762 \nu^{14} - 37843 \nu^{13} - 36585 \nu^{12} - 166936 \nu^{11} - 165399 \nu^{10} - 435929 \nu^{9} - 443130 \nu^{8} - 639090 \nu^{7} - 666509 \nu^{6} - 450488 \nu^{5} - 481448 \nu^{4} - 96196 \nu^{3} - 105408 \nu^{2} - 5664 \nu - 6620\)\()/40\)
\(\beta_{8}\)\(=\)\((\)\( -9 \nu^{18} - 296 \nu^{16} - 4067 \nu^{14} - 30273 \nu^{12} - 132214 \nu^{10} - 341693 \nu^{8} - 496350 \nu^{6} - 348454 \nu^{4} - 76008 \nu^{2} - 4740 \)\()/20\)
\(\beta_{9}\)\(=\)\((\)\( 13 \nu^{18} + 413 \nu^{16} + 5445 \nu^{14} + 38586 \nu^{12} + 159025 \nu^{10} + 384351 \nu^{8} + 518257 \nu^{6} + 337002 \nu^{4} + 69240 \nu^{2} + 4100 \)\()/20\)
\(\beta_{10}\)\(=\)\((\)\(-5 \nu^{19} + 2 \nu^{18} - 167 \nu^{17} + 72 \nu^{16} - 2337 \nu^{15} + 1090 \nu^{14} - 17770 \nu^{13} + 8984 \nu^{12} - 79489 \nu^{11} + 43540 \nu^{10} - 210785 \nu^{9} + 124604 \nu^{8} - 314269 \nu^{7} + 198808 \nu^{6} - 225998 \nu^{5} + 150428 \nu^{4} - 50148 \nu^{3} + 33220 \nu^{2} - 3220 \nu + 2040\)\()/20\)
\(\beta_{11}\)\(=\)\((\)\(6 \nu^{19} - 5 \nu^{18} + 194 \nu^{17} - 167 \nu^{16} + 2614 \nu^{15} - 2337 \nu^{14} + 19028 \nu^{13} - 17770 \nu^{12} + 81026 \nu^{11} - 79489 \nu^{10} + 203584 \nu^{9} - 210785 \nu^{8} + 286860 \nu^{7} - 314269 \nu^{6} + 195158 \nu^{5} - 225998 \nu^{4} + 41326 \nu^{3} - 50148 \nu^{2} + 2564 \nu - 3220\)\()/20\)
\(\beta_{12}\)\(=\)\((\)\(-11 \nu^{19} + 10 \nu^{18} - 364 \nu^{17} + 337 \nu^{16} - 5039 \nu^{15} + 4762 \nu^{14} - 37843 \nu^{13} + 36585 \nu^{12} - 166936 \nu^{11} + 165399 \nu^{10} - 435929 \nu^{9} + 443130 \nu^{8} - 639090 \nu^{7} + 666509 \nu^{6} - 450488 \nu^{5} + 481448 \nu^{4} - 96196 \nu^{3} + 105408 \nu^{2} - 5664 \nu + 6620\)\()/40\)
\(\beta_{13}\)\(=\)\((\)\( -10 \nu^{19} - 328 \nu^{17} - 4495 \nu^{15} - 33377 \nu^{13} - 145436 \nu^{11} - 375059 \nu^{9} - 543781 \nu^{7} - 381316 \nu^{5} - 83470 \nu^{3} - 5308 \nu \)\()/20\)
\(\beta_{14}\)\(=\)\((\)\(-11 \nu^{19} - 32 \nu^{18} - 364 \nu^{17} - 1023 \nu^{16} - 5039 \nu^{15} - 13596 \nu^{14} - 37843 \nu^{13} - 97349 \nu^{12} - 166936 \nu^{11} - 406577 \nu^{10} - 435929 \nu^{9} - 999584 \nu^{8} - 639090 \nu^{7} - 1377875 \nu^{6} - 450488 \nu^{5} - 923432 \nu^{4} - 96196 \nu^{3} - 201024 \nu^{2} - 5664 \nu - 12860\)\()/40\)
\(\beta_{15}\)\(=\)\((\)\(24 \nu^{19} - \nu^{18} + 795 \nu^{17} - 23 \nu^{16} + 11018 \nu^{15} - 157 \nu^{14} + 82855 \nu^{13} + 198 \nu^{12} + 366117 \nu^{11} + 7591 \nu^{10} + 958442 \nu^{9} + 38433 \nu^{8} + 1411273 \nu^{7} + 83507 \nu^{6} + 1004904 \nu^{5} + 75668 \nu^{4} + 223552 \nu^{3} + 17612 \nu^{2} + 14468 \nu + 1100\)\()/40\)
\(\beta_{16}\)\(=\)\((\)\(-24 \nu^{19} - \nu^{18} - 795 \nu^{17} - 23 \nu^{16} - 11018 \nu^{15} - 157 \nu^{14} - 82855 \nu^{13} + 198 \nu^{12} - 366117 \nu^{11} + 7591 \nu^{10} - 958442 \nu^{9} + 38433 \nu^{8} - 1411273 \nu^{7} + 83507 \nu^{6} - 1004904 \nu^{5} + 75668 \nu^{4} - 223552 \nu^{3} + 17612 \nu^{2} - 14468 \nu + 1100\)\()/40\)
\(\beta_{17}\)\(=\)\((\)\( 28 \nu^{19} + 919 \nu^{17} + 12602 \nu^{15} + 93627 \nu^{13} + 408167 \nu^{11} + 1053028 \nu^{9} + 1527209 \nu^{7} + 1071178 \nu^{5} + 234558 \nu^{3} + 14924 \nu \)\()/20\)
\(\beta_{18}\)\(=\)\((\)\( -28 \nu^{19} - 919 \nu^{17} - 12602 \nu^{15} - 93627 \nu^{13} - 408167 \nu^{11} - 1053028 \nu^{9} - 1527209 \nu^{7} - 1071178 \nu^{5} - 234538 \nu^{3} - 14824 \nu \)\()/20\)
\(\beta_{19}\)\(=\)\((\)\( 33 \nu^{19} + 1090 \nu^{17} + 15057 \nu^{15} + 112801 \nu^{13} + 496244 \nu^{11} + 1292171 \nu^{9} + 1889226 \nu^{7} + 1329130 \nu^{5} + 284588 \nu^{3} + 17340 \nu \)\()/20\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{18} + \beta_{17} - 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{16} - \beta_{15} + \beta_{14} - \beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{3} - 7 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(\beta_{19} - 9 \beta_{18} - 10 \beta_{17} - \beta_{16} - \beta_{13} + 2 \beta_{12} + 2 \beta_{7} + \beta_{5} + 29 \beta_{1}\)
\(\nu^{6}\)\(=\)\(10 \beta_{16} + 10 \beta_{15} - 12 \beta_{14} + 3 \beta_{12} + 13 \beta_{11} + 11 \beta_{10} - \beta_{9} + \beta_{8} + 9 \beta_{7} - 13 \beta_{6} - 11 \beta_{4} + 22 \beta_{3} + 48 \beta_{2} - 75\)
\(\nu^{7}\)\(=\)\(-12 \beta_{19} + 69 \beta_{18} + 81 \beta_{17} + 14 \beta_{16} + 2 \beta_{15} + 12 \beta_{13} - 22 \beta_{12} + \beta_{11} + \beta_{10} - 22 \beta_{7} + \beta_{6} - 16 \beta_{5} + \beta_{4} - 183 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-77 \beta_{16} - 77 \beta_{15} + 111 \beta_{14} - 48 \beta_{12} - 128 \beta_{11} - 97 \beta_{10} + 16 \beta_{9} - 16 \beta_{8} - 63 \beta_{7} + 128 \beta_{6} + 97 \beta_{4} - 190 \beta_{3} - 333 \beta_{2} + 438\)
\(\nu^{9}\)\(=\)\(111 \beta_{19} - 503 \beta_{18} - 618 \beta_{17} - 149 \beta_{16} - 30 \beta_{15} - 111 \beta_{13} + 188 \beta_{12} - 14 \beta_{11} - 15 \beta_{10} + 188 \beta_{7} - 14 \beta_{6} + 179 \beta_{5} - 15 \beta_{4} + 1215 \beta_{1}\)
\(\nu^{10}\)\(=\)\(542 \beta_{16} + 542 \beta_{15} - 940 \beta_{14} + 535 \beta_{12} + 1134 \beta_{11} + 800 \beta_{10} - 179 \beta_{9} + 181 \beta_{8} + 405 \beta_{7} - 1134 \beta_{6} - 800 \beta_{4} + 1526 \beta_{3} + 2336 \beta_{2} - 2703\)
\(\nu^{11}\)\(=\)\(-940 \beta_{19} + 3599 \beta_{18} + 4621 \beta_{17} + 1412 \beta_{16} + 316 \beta_{15} + 956 \beta_{13} - 1482 \beta_{12} + 142 \beta_{11} + 153 \beta_{10} - 1482 \beta_{7} + 142 \beta_{6} - 1728 \beta_{5} + 153 \beta_{4} - 8315 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-3669 \beta_{16} - 3669 \beta_{15} + 7647 \beta_{14} - 5152 \beta_{12} - 9528 \beta_{11} - 6412 \beta_{10} + 1728 \beta_{9} - 1780 \beta_{8} - 2495 \beta_{7} + 9528 \beta_{6} + 6412 \beta_{4} - 11926 \beta_{3} - 16535 \beta_{2} + 17276\)
\(\nu^{13}\)\(=\)\(7647 \beta_{19} - 25601 \beta_{18} - 34328 \beta_{17} - 12527 \beta_{16} - 2902 \beta_{15} - 8021 \beta_{13} + 11316 \beta_{12} - 1287 \beta_{11} - 1336 \beta_{10} + 11316 \beta_{7} - 1287 \beta_{6} + 15429 \beta_{5} - 1336 \beta_{4} + 57993 \beta_{1}\)
\(\nu^{14}\)\(=\)\(24390 \beta_{16} + 24390 \beta_{15} - 60876 \beta_{14} + 45959 \beta_{12} + 77641 \beta_{11} + 50606 \beta_{10} - 15429 \beta_{9} + 16229 \beta_{8} + 14917 \beta_{7} - 77641 \beta_{6} - 50606 \beta_{4} + 92106 \beta_{3} + 117922 \beta_{2} - 113005\)
\(\nu^{15}\)\(=\)\(-60876 \beta_{19} + 182131 \beta_{18} + 254675 \beta_{17} + 106552 \beta_{16} + 24902 \beta_{15} + 66250 \beta_{13} - 85266 \beta_{12} + 11070 \beta_{11} + 10806 \beta_{10} - 85266 \beta_{7} + 11070 \beta_{6} - 131454 \beta_{5} + 10806 \beta_{4} - 409725 \beta_{1}\)
\(\nu^{16}\)\(=\)\(-160845 \beta_{16} - 160845 \beta_{15} + 478251 \beta_{14} - 391600 \beta_{12} - 620511 \beta_{11} - 395499 \beta_{10} + 131454 \beta_{9} - 141088 \beta_{8} - 86651 \beta_{7} + 620511 \beta_{6} + 395499 \beta_{4} - 707054 \beta_{3} - 846531 \beta_{2} + 751452\)
\(\nu^{17}\)\(=\)\(478251 \beta_{19} - 1299675 \beta_{18} - 1890748 \beta_{17} - 880665 \beta_{16} - 205648 \beta_{15} - 539857 \beta_{13} + 639096 \beta_{12} - 92386 \beta_{11} - 83924 \beta_{10} + 639096 \beta_{7} - 92386 \beta_{6} + 1086313 \beta_{5} - 83924 \beta_{4} + 2922765 \beta_{1}\)
\(\nu^{18}\)\(=\)\(1058106 \beta_{16} + 1058106 \beta_{15} - 3724028 \beta_{14} + 3238063 \beta_{12} + 4894265 \beta_{11} + 3069467 \beta_{10} - 1086313 \beta_{9} + 1187181 \beta_{8} + 485965 \beta_{7} - 4894265 \beta_{6} - 3069467 \beta_{4} + 5407666 \beta_{3} + 6113188 \beta_{2} - 5061191\)
\(\nu^{19}\)\(=\)\(-3724028 \beta_{19} + 9315713 \beta_{18} + 14057701 \beta_{17} + 7131710 \beta_{16} + 1658454 \beta_{15} + 4344692 \beta_{13} - 4782134 \beta_{12} + 755429 \beta_{11} + 637617 \beta_{10} - 4782134 \beta_{7} + 755429 \beta_{6} - 8790164 \beta_{5} + 637617 \beta_{4} - 21011595 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
694.1
2.74619i
2.51841i
2.51027i
2.02181i
1.76958i
1.76011i
1.41336i
0.445112i
0.437361i
0.332438i
0.332438i
0.437361i
0.445112i
1.41336i
1.76011i
1.76958i
2.02181i
2.51027i
2.51841i
2.74619i
2.74619i 1.00000i −5.54154 2.23561 + 0.0452669i 2.74619 1.00000i 9.72574i −1.00000 0.124311 6.13940i
694.2 2.51841i 1.00000i −4.34238 −2.17741 0.508790i 2.51841 1.00000i 5.89907i −1.00000 −1.28134 + 5.48362i
694.3 2.51027i 1.00000i −4.30144 −1.80593 1.31856i −2.51027 1.00000i 5.77723i −1.00000 −3.30994 + 4.53337i
694.4 2.02181i 1.00000i −2.08773 −0.549616 + 2.16747i 2.02181 1.00000i 0.177380i −1.00000 4.38222 + 1.11122i
694.5 1.76958i 1.00000i −1.13141 −1.09533 + 1.94942i −1.76958 1.00000i 1.53703i −1.00000 3.44966 + 1.93828i
694.6 1.76011i 1.00000i −1.09799 2.22415 0.230563i −1.76011 1.00000i 1.58764i −1.00000 −0.405817 3.91475i
694.7 1.41336i 1.00000i 0.00241967 1.69889 1.45388i 1.41336 1.00000i 2.83014i −1.00000 −2.05485 2.40114i
694.8 0.445112i 1.00000i 1.80187 −2.13786 0.655413i 0.445112 1.00000i 1.69226i −1.00000 −0.291732 + 0.951587i
694.9 0.437361i 1.00000i 1.80872 0.978157 2.01077i −0.437361 1.00000i 1.66578i −1.00000 −0.879433 0.427807i
694.10 0.332438i 1.00000i 1.88948 −0.370655 2.20513i 0.332438 1.00000i 1.29301i −1.00000 −0.733071 + 0.123220i
694.11 0.332438i 1.00000i 1.88948 −0.370655 + 2.20513i 0.332438 1.00000i 1.29301i −1.00000 −0.733071 0.123220i
694.12 0.437361i 1.00000i 1.80872 0.978157 + 2.01077i −0.437361 1.00000i 1.66578i −1.00000 −0.879433 + 0.427807i
694.13 0.445112i 1.00000i 1.80187 −2.13786 + 0.655413i 0.445112 1.00000i 1.69226i −1.00000 −0.291732 0.951587i
694.14 1.41336i 1.00000i 0.00241967 1.69889 + 1.45388i 1.41336 1.00000i 2.83014i −1.00000 −2.05485 + 2.40114i
694.15 1.76011i 1.00000i −1.09799 2.22415 + 0.230563i −1.76011 1.00000i 1.58764i −1.00000 −0.405817 + 3.91475i
694.16 1.76958i 1.00000i −1.13141 −1.09533 1.94942i −1.76958 1.00000i 1.53703i −1.00000 3.44966 1.93828i
694.17 2.02181i 1.00000i −2.08773 −0.549616 2.16747i 2.02181 1.00000i 0.177380i −1.00000 4.38222 1.11122i
694.18 2.51027i 1.00000i −4.30144 −1.80593 + 1.31856i −2.51027 1.00000i 5.77723i −1.00000 −3.30994 4.53337i
694.19 2.51841i 1.00000i −4.34238 −2.17741 + 0.508790i 2.51841 1.00000i 5.89907i −1.00000 −1.28134 5.48362i
694.20 2.74619i 1.00000i −5.54154 2.23561 0.0452669i 2.74619 1.00000i 9.72574i −1.00000 0.124311 + 6.13940i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 694.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.2.c.e 20
5.b even 2 1 inner 1155.2.c.e 20
5.c odd 4 1 5775.2.a.cm 10
5.c odd 4 1 5775.2.a.cp 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.c.e 20 1.a even 1 1 trivial
1155.2.c.e 20 5.b even 2 1 inner
5775.2.a.cm 10 5.c odd 4 1
5775.2.a.cp 10 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\(T_{2}^{20} + \cdots\)
\(T_{13}^{20} + \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 7 T^{2} + 28 T^{4} - 90 T^{6} + 242 T^{8} - 600 T^{10} + 1449 T^{12} - 3215 T^{14} + 6732 T^{16} - 13440 T^{18} + 26324 T^{20} - 53760 T^{22} + 107712 T^{24} - 205760 T^{26} + 370944 T^{28} - 614400 T^{30} + 991232 T^{32} - 1474560 T^{34} + 1835008 T^{36} - 1835008 T^{38} + 1048576 T^{40} \)
$3$ \( ( 1 + T^{2} )^{10} \)
$5$ \( 1 + 2 T - 4 T^{2} - 22 T^{3} - 38 T^{4} + 30 T^{5} + 194 T^{6} - 10 T^{7} - 411 T^{8} + 544 T^{9} + 3812 T^{10} + 2720 T^{11} - 10275 T^{12} - 1250 T^{13} + 121250 T^{14} + 93750 T^{15} - 593750 T^{16} - 1718750 T^{17} - 1562500 T^{18} + 3906250 T^{19} + 9765625 T^{20} \)
$7$ \( ( 1 + T^{2} )^{10} \)
$11$ \( ( 1 + T )^{20} \)
$13$ \( 1 - 76 T^{2} + 3016 T^{4} - 84092 T^{6} + 1823558 T^{8} - 31925088 T^{10} + 456227726 T^{12} - 5292844156 T^{14} + 49221283041 T^{16} - 383795453276 T^{18} + 3610174593988 T^{20} - 64861431603644 T^{22} + 1405809064934001 T^{24} - 25547547807778204 T^{26} + 372158971870170446 T^{28} - 4401144483826607712 T^{30} + 42485409509781207398 T^{32} - \)\(33\!\cdots\!88\)\( T^{34} + \)\(20\!\cdots\!56\)\( T^{36} - \)\(85\!\cdots\!04\)\( T^{38} + \)\(19\!\cdots\!01\)\( T^{40} \)
$17$ \( 1 - 125 T^{2} + 8183 T^{4} - 379280 T^{6} + 14024259 T^{8} - 437409389 T^{10} + 11869036496 T^{12} - 285904405719 T^{14} + 6190841630620 T^{16} - 121353224519439 T^{18} + 2162205885093682 T^{20} - 35071081886117871 T^{22} + 517065283831013020 T^{24} - 6901037320446357111 T^{26} + 82795519654472566736 T^{28} - \)\(88\!\cdots\!61\)\( T^{30} + \)\(81\!\cdots\!99\)\( T^{32} - \)\(63\!\cdots\!20\)\( T^{34} + \)\(39\!\cdots\!23\)\( T^{36} - \)\(17\!\cdots\!25\)\( T^{38} + \)\(40\!\cdots\!01\)\( T^{40} \)
$19$ \( ( 1 - T + 67 T^{2} - 28 T^{3} + 2388 T^{4} - 404 T^{5} + 64277 T^{6} + 16491 T^{7} + 1449423 T^{8} + 928264 T^{9} + 29194968 T^{10} + 17637016 T^{11} + 523241703 T^{12} + 113111769 T^{13} + 8376642917 T^{14} - 1000343996 T^{15} + 112345563828 T^{16} - 25028408692 T^{17} + 1137898723747 T^{18} - 322687697779 T^{19} + 6131066257801 T^{20} )^{2} \)
$23$ \( 1 - 141 T^{2} + 10587 T^{4} - 559880 T^{6} + 23898567 T^{8} - 889773365 T^{10} + 29999892104 T^{12} - 922885123935 T^{14} + 25903837523960 T^{16} - 666701419393415 T^{18} + 15893936460155130 T^{20} - 352685050859116535 T^{22} + 7248955796542490360 T^{24} - \)\(13\!\cdots\!15\)\( T^{26} + \)\(23\!\cdots\!24\)\( T^{28} - \)\(36\!\cdots\!85\)\( T^{30} + \)\(52\!\cdots\!07\)\( T^{32} - \)\(64\!\cdots\!20\)\( T^{34} + \)\(64\!\cdots\!07\)\( T^{36} - \)\(45\!\cdots\!29\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \)
$29$ \( ( 1 + 19 T + 285 T^{2} + 2902 T^{3} + 26028 T^{4} + 186854 T^{5} + 1240867 T^{6} + 7049545 T^{7} + 39524327 T^{8} + 202341500 T^{9} + 1112908184 T^{10} + 5867903500 T^{11} + 33239959007 T^{12} + 171931353005 T^{13} + 877641652627 T^{14} + 3832590235246 T^{15} + 15482061398988 T^{16} + 50059141048718 T^{17} + 142570227693885 T^{18} + 275635773541511 T^{19} + 420707233300201 T^{20} )^{2} \)
$31$ \( ( 1 - 10 T + 222 T^{2} - 1764 T^{3} + 20825 T^{4} - 135630 T^{5} + 1119824 T^{6} - 6212150 T^{7} + 41423910 T^{8} - 210761730 T^{9} + 1303997476 T^{10} - 6533613630 T^{11} + 39808377510 T^{12} - 185066160650 T^{13} + 1034180980304 T^{14} - 3882971750130 T^{15} + 18482264156825 T^{16} - 48532251291804 T^{17} + 189341810311902 T^{18} - 264396221606710 T^{19} + 819628286980801 T^{20} )^{2} \)
$37$ \( 1 - 420 T^{2} + 85048 T^{4} - 11129532 T^{6} + 1067670262 T^{8} - 80952785416 T^{10} + 5110273334094 T^{12} - 278612539606740 T^{14} + 13434614767508177 T^{16} - 580509426620078532 T^{18} + 22607554396001093156 T^{20} - \)\(79\!\cdots\!08\)\( T^{22} + \)\(25\!\cdots\!97\)\( T^{24} - \)\(71\!\cdots\!60\)\( T^{26} + \)\(17\!\cdots\!74\)\( T^{28} - \)\(38\!\cdots\!84\)\( T^{30} + \)\(70\!\cdots\!22\)\( T^{32} - \)\(10\!\cdots\!48\)\( T^{34} + \)\(10\!\cdots\!68\)\( T^{36} - \)\(70\!\cdots\!80\)\( T^{38} + \)\(23\!\cdots\!01\)\( T^{40} \)
$41$ \( ( 1 - 6 T + 188 T^{2} - 520 T^{3} + 16457 T^{4} - 14494 T^{5} + 1058088 T^{6} + 286678 T^{7} + 55605510 T^{8} + 45683142 T^{9} + 2463539512 T^{10} + 1873008822 T^{11} + 93472862310 T^{12} + 19758134438 T^{13} + 2989903804968 T^{14} - 1679219777294 T^{15} + 78172465494137 T^{16} - 101272222418120 T^{17} + 1501165943074748 T^{18} - 1964291606363766 T^{19} + 13422659310152401 T^{20} )^{2} \)
$43$ \( 1 - 301 T^{2} + 40567 T^{4} - 3114932 T^{6} + 139359283 T^{8} - 2731659641 T^{10} - 55736331088 T^{12} + 3220097377053 T^{14} + 225705463472508 T^{16} - 31610273187179755 T^{18} + 1798746642937635378 T^{20} - 58447395123095366995 T^{22} + \)\(77\!\cdots\!08\)\( T^{24} + \)\(20\!\cdots\!97\)\( T^{26} - \)\(65\!\cdots\!88\)\( T^{28} - \)\(59\!\cdots\!09\)\( T^{30} + \)\(55\!\cdots\!83\)\( T^{32} - \)\(23\!\cdots\!68\)\( T^{34} + \)\(55\!\cdots\!67\)\( T^{36} - \)\(76\!\cdots\!49\)\( T^{38} + \)\(46\!\cdots\!01\)\( T^{40} \)
$47$ \( 1 - 348 T^{2} + 69124 T^{4} - 9929156 T^{6} + 1132382918 T^{8} - 107583204096 T^{10} + 8759246940354 T^{12} - 622164180171628 T^{14} + 38998757285971457 T^{16} - 2172877670496645428 T^{18} + \)\(10\!\cdots\!52\)\( T^{20} - \)\(47\!\cdots\!52\)\( T^{22} + \)\(19\!\cdots\!17\)\( T^{24} - \)\(67\!\cdots\!12\)\( T^{26} + \)\(20\!\cdots\!94\)\( T^{28} - \)\(56\!\cdots\!04\)\( T^{30} + \)\(13\!\cdots\!38\)\( T^{32} - \)\(25\!\cdots\!64\)\( T^{34} + \)\(39\!\cdots\!04\)\( T^{36} - \)\(43\!\cdots\!72\)\( T^{38} + \)\(27\!\cdots\!01\)\( T^{40} \)
$53$ \( 1 - 397 T^{2} + 83271 T^{4} - 11925936 T^{6} + 1290772875 T^{8} - 111126345933 T^{10} + 7857223936736 T^{12} - 468699015269855 T^{14} + 24497573777265572 T^{16} - 1201050518125355583 T^{18} + 60909706273895107890 T^{20} - \)\(33\!\cdots\!47\)\( T^{22} + \)\(19\!\cdots\!32\)\( T^{24} - \)\(10\!\cdots\!95\)\( T^{26} + \)\(48\!\cdots\!96\)\( T^{28} - \)\(19\!\cdots\!17\)\( T^{30} + \)\(63\!\cdots\!75\)\( T^{32} - \)\(16\!\cdots\!84\)\( T^{34} + \)\(32\!\cdots\!91\)\( T^{36} - \)\(43\!\cdots\!33\)\( T^{38} + \)\(30\!\cdots\!01\)\( T^{40} \)
$59$ \( ( 1 + 11 T + 407 T^{2} + 3330 T^{3} + 73908 T^{4} + 470630 T^{5} + 8334465 T^{6} + 42784951 T^{7} + 681366495 T^{8} + 2966908468 T^{9} + 44302162232 T^{10} + 175047599612 T^{11} + 2371836769095 T^{12} + 8787130451429 T^{13} + 100991721146865 T^{14} + 336464822838370 T^{15} + 3117478880339028 T^{16} + 8287209444447270 T^{17} + 59759988104958647 T^{18} + 95292954005204329 T^{19} + 511116753300641401 T^{20} )^{2} \)
$61$ \( ( 1 - 17 T + 335 T^{2} - 4264 T^{3} + 49055 T^{4} - 484205 T^{5} + 4426960 T^{6} - 37564131 T^{7} + 310404408 T^{8} - 2529217579 T^{9} + 19598560162 T^{10} - 154282272319 T^{11} + 1155014802168 T^{12} - 8526344018511 T^{13} + 61294984273360 T^{14} - 408957751925705 T^{15} + 2527331964278855 T^{16} - 13400655452793544 T^{17} + 64221949854089135 T^{18} - 198800483578180397 T^{19} + 713342911662882601 T^{20} )^{2} \)
$67$ \( 1 - 428 T^{2} + 99524 T^{4} - 16563640 T^{6} + 2222713174 T^{8} - 254734787908 T^{10} + 25671863713850 T^{12} - 2317293714434892 T^{14} + 190211097756591737 T^{16} - 14350217250745009068 T^{18} + \)\(99\!\cdots\!24\)\( T^{20} - \)\(64\!\cdots\!52\)\( T^{22} + \)\(38\!\cdots\!77\)\( T^{24} - \)\(20\!\cdots\!48\)\( T^{26} + \)\(10\!\cdots\!50\)\( T^{28} - \)\(46\!\cdots\!92\)\( T^{30} + \)\(18\!\cdots\!14\)\( T^{32} - \)\(60\!\cdots\!60\)\( T^{34} + \)\(16\!\cdots\!44\)\( T^{36} - \)\(31\!\cdots\!52\)\( T^{38} + \)\(33\!\cdots\!01\)\( T^{40} \)
$71$ \( ( 1 - 36 T + 976 T^{2} - 19438 T^{3} + 331117 T^{4} - 4790434 T^{5} + 62063776 T^{6} - 716228202 T^{7} + 7550418114 T^{8} - 72276130322 T^{9} + 637462169120 T^{10} - 5131605252862 T^{11} + 38061657712674 T^{12} - 256345952006022 T^{13} + 1577144877367456 T^{14} - 8643041626828334 T^{15} + 42416181711069757 T^{16} - 176790945638804258 T^{17} + 630255446495862736 T^{18} - 1650546025864165116 T^{19} + 3255243551009881201 T^{20} )^{2} \)
$73$ \( 1 - 936 T^{2} + 438364 T^{4} - 136338996 T^{6} + 31533022886 T^{8} - 5757351991336 T^{10} + 860098254780642 T^{12} - 107563201317401920 T^{14} + 11429183850756931913 T^{16} - \)\(10\!\cdots\!68\)\( T^{18} + \)\(81\!\cdots\!68\)\( T^{20} - \)\(55\!\cdots\!72\)\( T^{22} + \)\(32\!\cdots\!33\)\( T^{24} - \)\(16\!\cdots\!80\)\( T^{26} + \)\(69\!\cdots\!02\)\( T^{28} - \)\(24\!\cdots\!64\)\( T^{30} + \)\(72\!\cdots\!06\)\( T^{32} - \)\(16\!\cdots\!64\)\( T^{34} + \)\(28\!\cdots\!04\)\( T^{36} - \)\(32\!\cdots\!84\)\( T^{38} + \)\(18\!\cdots\!01\)\( T^{40} \)
$79$ \( ( 1 - 2 T + 382 T^{2} - 236 T^{3} + 75401 T^{4} + 1610 T^{5} + 10724928 T^{6} + 1802770 T^{7} + 1182764310 T^{8} + 260660158 T^{9} + 103760069636 T^{10} + 20592152482 T^{11} + 7381632058710 T^{12} + 888835918030 T^{13} + 417736814319168 T^{14} + 4954060802390 T^{15} + 18329037233738921 T^{16} - 4532122520733524 T^{17} + 579535565384306302 T^{18} - 239703191965236638 T^{19} + 9468276082626847201 T^{20} )^{2} \)
$83$ \( 1 - 685 T^{2} + 254131 T^{4} - 65345072 T^{6} + 12947358623 T^{8} - 2092210467133 T^{10} + 286385049310552 T^{12} - 34110090479625375 T^{14} + 3608900681760218544 T^{16} - \)\(34\!\cdots\!51\)\( T^{18} + \)\(29\!\cdots\!22\)\( T^{20} - \)\(23\!\cdots\!39\)\( T^{22} + \)\(17\!\cdots\!24\)\( T^{24} - \)\(11\!\cdots\!75\)\( T^{26} + \)\(64\!\cdots\!32\)\( T^{28} - \)\(32\!\cdots\!17\)\( T^{30} + \)\(13\!\cdots\!03\)\( T^{32} - \)\(48\!\cdots\!88\)\( T^{34} + \)\(12\!\cdots\!11\)\( T^{36} - \)\(23\!\cdots\!65\)\( T^{38} + \)\(24\!\cdots\!01\)\( T^{40} \)
$89$ \( ( 1 - 3 T + 471 T^{2} - 572 T^{3} + 110521 T^{4} + 34189 T^{5} + 17467996 T^{6} + 25520951 T^{7} + 2095249526 T^{8} + 4296264099 T^{9} + 204133231578 T^{10} + 382367504811 T^{11} + 16596471495446 T^{12} + 17991479305519 T^{13} + 1095981214819036 T^{14} + 190913408501861 T^{15} + 54926869258300681 T^{16} - 25300323560242588 T^{17} + 1854133327485680151 T^{18} - 1051069211122455627 T^{19} + 31181719929966183601 T^{20} )^{2} \)
$97$ \( 1 - 553 T^{2} + 200359 T^{4} - 53877996 T^{6} + 11779364323 T^{8} - 2181747162701 T^{10} + 350569578802512 T^{12} - 49724314262059831 T^{14} + 6285659438414427580 T^{16} - \)\(71\!\cdots\!19\)\( T^{18} + \)\(72\!\cdots\!26\)\( T^{20} - \)\(67\!\cdots\!71\)\( T^{22} + \)\(55\!\cdots\!80\)\( T^{24} - \)\(41\!\cdots\!99\)\( T^{26} + \)\(27\!\cdots\!32\)\( T^{28} - \)\(16\!\cdots\!49\)\( T^{30} + \)\(81\!\cdots\!43\)\( T^{32} - \)\(35\!\cdots\!24\)\( T^{34} + \)\(12\!\cdots\!39\)\( T^{36} - \)\(31\!\cdots\!17\)\( T^{38} + \)\(54\!\cdots\!01\)\( T^{40} \)
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