Properties

Label 1155.2.c.f
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 33 x^{18} + 456 x^{16} + 3426 x^{14} + 15210 x^{12} + 40640 x^{10} + 63865 x^{8} + 55281 x^{6} + 22984 x^{4} + 3428 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 3 \)
\(\beta_{3}\)\(=\)\((\)\( -932 \nu^{18} - 29599 \nu^{16} - 388661 \nu^{14} - 2720792 \nu^{12} - 10903803 \nu^{10} - 24948019 \nu^{8} - 30691092 \nu^{6} - 17834990 \nu^{4} - 3622164 \nu^{2} - 10112 \)\()/171356\)
\(\beta_{4}\)\(=\)\((\)\(-21445 \nu^{19} - 701609 \nu^{17} - 9437765 \nu^{15} - 67184034 \nu^{13} - 271650511 \nu^{11} - 625930321 \nu^{9} - 801263181 \nu^{7} - 580092520 \nu^{5} - 263850094 \nu^{3} - 52934572 \nu\)\()/3255764\)
\(\beta_{5}\)\(=\)\((\)\(-17223 \nu^{19} - 39658 \nu^{18} - 539302 \nu^{17} - 1304803 \nu^{16} - 7239623 \nu^{15} - 17627746 \nu^{14} - 55067041 \nu^{13} - 125789839 \nu^{12} - 262962518 \nu^{11} - 508016561 \nu^{10} - 813882325 \nu^{9} - 1157693438 \nu^{8} - 1595603488 \nu^{7} - 1413013347 \nu^{6} - 1831293380 \nu^{5} - 842531572 \nu^{4} - 1064523572 \nu^{3} - 207237480 \nu^{2} - 233638456 \nu - 12649508\)\()/6511528\)
\(\beta_{6}\)\(=\)\((\)\( 28357 \nu^{18} + 807316 \nu^{16} + 9309882 \nu^{14} + 55840854 \nu^{12} + 186103148 \nu^{10} + 341926714 \nu^{8} + 328984885 \nu^{6} + 163946632 \nu^{4} + 46688632 \nu^{2} + 1859432 \)\()/3255764\)
\(\beta_{7}\)\(=\)\((\)\(11377 \nu^{19} - 77206 \nu^{18} + 402088 \nu^{17} - 2366551 \nu^{16} + 6106853 \nu^{15} - 29875558 \nu^{14} + 52047565 \nu^{13} - 200454623 \nu^{12} + 271905052 \nu^{11} - 769151417 \nu^{10} + 890119595 \nu^{9} - 1690185486 \nu^{8} + 1787344850 \nu^{7} - 2021456739 \nu^{6} + 2060015128 \nu^{5} - 1184482080 \nu^{4} + 1186256172 \nu^{3} - 280996784 \nu^{2} + 242033984 \nu - 12324404\)\()/6511528\)
\(\beta_{8}\)\(=\)\((\)\(-40638 \nu^{19} + 2923 \nu^{18} - 1359828 \nu^{17} + 68607 \nu^{16} - 19061802 \nu^{15} + 566385 \nu^{14} - 145349694 \nu^{13} + 1509738 \nu^{12} - 655436372 \nu^{11} - 4471267 \nu^{10} - 1782095748 \nu^{9} - 38118635 \nu^{8} - 2861591894 \nu^{7} - 95870681 \nu^{6} - 2550730974 \nu^{5} - 114360874 \nu^{4} - 1104999046 \nu^{3} - 60866300 \nu^{2} - 176186716 \nu - 4197764\)\()/3255764\)
\(\beta_{9}\)\(=\)\((\)\(40638 \nu^{19} + 2923 \nu^{18} + 1359828 \nu^{17} + 68607 \nu^{16} + 19061802 \nu^{15} + 566385 \nu^{14} + 145349694 \nu^{13} + 1509738 \nu^{12} + 655436372 \nu^{11} - 4471267 \nu^{10} + 1782095748 \nu^{9} - 38118635 \nu^{8} + 2861591894 \nu^{7} - 95870681 \nu^{6} + 2550730974 \nu^{5} - 114360874 \nu^{4} + 1104999046 \nu^{3} - 60866300 \nu^{2} + 176186716 \nu - 4197764\)\()/3255764\)
\(\beta_{10}\)\(=\)\((\)\(-11377 \nu^{19} - 77206 \nu^{18} - 402088 \nu^{17} - 2366551 \nu^{16} - 6106853 \nu^{15} - 29875558 \nu^{14} - 52047565 \nu^{13} - 200454623 \nu^{12} - 271905052 \nu^{11} - 769151417 \nu^{10} - 890119595 \nu^{9} - 1690185486 \nu^{8} - 1787344850 \nu^{7} - 2021456739 \nu^{6} - 2060015128 \nu^{5} - 1184482080 \nu^{4} - 1186256172 \nu^{3} - 280996784 \nu^{2} - 242033984 \nu - 12324404\)\()/6511528\)
\(\beta_{11}\)\(=\)\((\)\( 2528 \nu^{19} + 82492 \nu^{17} + 1123169 \nu^{15} + 8272267 \nu^{13} + 35730088 \nu^{11} + 91834117 \nu^{9} + 136502701 \nu^{7} + 109059276 \nu^{5} + 40268562 \nu^{3} + 5043820 \nu \)\()/171356\)
\(\beta_{12}\)\(=\)\((\)\(-3503 \nu^{18} - 111664 \nu^{16} - 1471065 \nu^{14} - 10332043 \nu^{12} - 41590554 \nu^{10} - 95909007 \nu^{8} - 119768558 \nu^{6} - 71376026 \nu^{4} - 14754680 \nu^{2} + 467236\)\()/171356\)
\(\beta_{13}\)\(=\)\((\)\(158482 \nu^{19} - 20801 \nu^{18} + 5086207 \nu^{17} - 781727 \nu^{16} + 67999162 \nu^{15} - 11937193 \nu^{14} + 491154011 \nu^{13} - 95841406 \nu^{12} + 2080024161 \nu^{11} - 436700849 \nu^{10} + 5254376982 \nu^{9} - 1136990015 \nu^{8} + 7744640527 \nu^{7} - 1622824553 \nu^{6} + 6285944028 \nu^{5} - 1153488952 \nu^{4} + 2490994876 \nu^{3} - 324766228 \nu^{2} + 364697836 \nu - 8350020\)\()/6511528\)
\(\beta_{14}\)\(=\)\((\)\(-158482 \nu^{19} - 20801 \nu^{18} - 5086207 \nu^{17} - 781727 \nu^{16} - 67999162 \nu^{15} - 11937193 \nu^{14} - 491154011 \nu^{13} - 95841406 \nu^{12} - 2080024161 \nu^{11} - 436700849 \nu^{10} - 5254376982 \nu^{9} - 1136990015 \nu^{8} - 7744640527 \nu^{7} - 1622824553 \nu^{6} - 6285944028 \nu^{5} - 1153488952 \nu^{4} - 2490994876 \nu^{3} - 324766228 \nu^{2} - 364697836 \nu - 8350020\)\()/6511528\)
\(\beta_{15}\)\(=\)\((\)\(-192793 \nu^{19} + 21163 \nu^{18} - 6452077 \nu^{17} + 777136 \nu^{16} - 90636213 \nu^{15} + 12008451 \nu^{14} - 694260760 \nu^{13} + 101264279 \nu^{12} - 3152696619 \nu^{11} + 504609204 \nu^{10} - 8644761455 \nu^{9} + 1498448529 \nu^{8} - 13969193167 \nu^{7} + 2525053938 \nu^{6} - 12401839144 \nu^{5} + 2109983528 \nu^{4} - 5229989096 \nu^{3} + 617774332 \nu^{2} - 769879164 \nu + 12865832\)\()/6511528\)
\(\beta_{16}\)\(=\)\((\)\(-110783 \nu^{19} - 3732736 \nu^{17} - 52853791 \nu^{15} - 408736717 \nu^{13} - 1878395562 \nu^{11} - 5231616809 \nu^{9} - 8634683824 \nu^{7} - 7886314178 \nu^{5} - 3430562652 \nu^{3} - 500823852 \nu\)\()/3255764\)
\(\beta_{17}\)\(=\)\((\)\( 6652 \nu^{19} + 217877 \nu^{17} + 2980846 \nu^{15} + 22096009 \nu^{13} + 96286461 \nu^{11} + 250554332 \nu^{9} + 378817011 \nu^{7} + 309342838 \nu^{5} + 117183522 \nu^{3} + 15121348 \nu \)\()/171356\)
\(\beta_{18}\)\(=\)\((\)\(-6652 \nu^{19} - 217877 \nu^{17} - 2980846 \nu^{15} - 22096009 \nu^{13} - 96286461 \nu^{11} - 250554332 \nu^{9} - 378817011 \nu^{7} - 309342838 \nu^{5} - 117012166 \nu^{3} - 14264568 \nu\)\()/171356\)
\(\beta_{19}\)\(=\)\((\)\(-8594 \nu^{19} - 286263 \nu^{17} - 3995054 \nu^{15} - 30327549 \nu^{13} - 136061779 \nu^{11} - 367228218 \nu^{9} - 581983849 \nu^{7} - 506157206 \nu^{5} - 211022344 \nu^{3} - 32364064 \nu\)\()/171356\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{18} + \beta_{17} - 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{12} - \beta_{10} - \beta_{7} - \beta_{6} - \beta_{3} - 7 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(-9 \beta_{18} - 10 \beta_{17} + \beta_{16} + 2 \beta_{11} + 2 \beta_{9} - 2 \beta_{8} + \beta_{4} + 28 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-\beta_{19} + 3 \beta_{18} - \beta_{16} - \beta_{15} - 3 \beta_{14} - 10 \beta_{12} + 9 \beta_{10} - 3 \beta_{9} + 2 \beta_{8} + 9 \beta_{7} + 10 \beta_{6} + 3 \beta_{5} + 12 \beta_{3} + 48 \beta_{2} - \beta_{1} - 85\)
\(\nu^{7}\)\(=\)\(-3 \beta_{19} + 71 \beta_{18} + 83 \beta_{17} - 13 \beta_{16} - \beta_{14} + \beta_{13} - 32 \beta_{11} + \beta_{10} - 29 \beta_{9} + 29 \beta_{8} - \beta_{7} - 14 \beta_{4} - 165 \beta_{1}\)
\(\nu^{8}\)\(=\)\(14 \beta_{19} - 42 \beta_{18} + 14 \beta_{16} + 14 \beta_{15} + 44 \beta_{14} + 2 \beta_{13} + 81 \beta_{12} - 67 \beta_{10} + 37 \beta_{9} - 33 \beta_{8} - 67 \beta_{7} - 81 \beta_{6} - 42 \beta_{5} - 113 \beta_{3} - 331 \beta_{2} + 14 \beta_{1} + 513\)
\(\nu^{9}\)\(=\)\(49 \beta_{19} - 545 \beta_{18} - 648 \beta_{17} + 130 \beta_{16} + 19 \beta_{14} - 19 \beta_{13} + 350 \beta_{11} - 23 \beta_{10} + 300 \beta_{9} - 300 \beta_{8} + 23 \beta_{7} + 141 \beta_{4} + 1006 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-140 \beta_{19} + 421 \beta_{18} - 140 \beta_{16} - 141 \beta_{15} - 464 \beta_{14} - 43 \beta_{13} - 610 \beta_{12} + 478 \beta_{10} - 330 \beta_{9} + 371 \beta_{8} + 478 \beta_{7} + 626 \beta_{6} + 421 \beta_{5} + 960 \beta_{3} + 2302 \beta_{2} - 141 \beta_{1} - 3235\)
\(\nu^{11}\)\(=\)\(-555 \beta_{19} + 4157 \beta_{18} + 4919 \beta_{17} - 1181 \beta_{16} - 239 \beta_{14} + 239 \beta_{13} - 3268 \beta_{11} + 317 \beta_{10} - 2732 \beta_{9} + 2732 \beta_{8} - 317 \beta_{7} - 1258 \beta_{4} - 6301 \beta_{1}\)
\(\nu^{12}\)\(=\)\(1235 \beta_{19} - 3728 \beta_{18} + 1235 \beta_{16} + 1258 \beta_{15} + 4307 \beta_{14} + 579 \beta_{13} + 4441 \beta_{12} - 3386 \beta_{10} + 2617 \beta_{9} - 3581 \beta_{8} - 3386 \beta_{7} - 4783 \beta_{6} - 3728 \beta_{5} - 7691 \beta_{3} - 16157 \beta_{2} + 1258 \beta_{1} + 21117\)
\(\nu^{13}\)\(=\)\(5418 \beta_{19} - 31673 \beta_{18} - 36800 \beta_{17} + 10201 \beta_{16} + 2508 \beta_{14} - 2508 \beta_{13} + 28062 \beta_{11} - 3504 \beta_{10} + 23371 \beta_{9} - 23371 \beta_{8} + 3504 \beta_{7} + 10585 \beta_{4} + 40376 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-10278 \beta_{19} + 31141 \beta_{18} - 10278 \beta_{16} - 10585 \beta_{15} - 37478 \beta_{14} - 6337 \beta_{13} - 31766 \beta_{12} + 24072 \beta_{10} - 19711 \beta_{9} + 31986 \beta_{8} + 24072 \beta_{7} + 36456 \beta_{6} + 31141 \beta_{5} + 59386 \beta_{3} + 114418 \beta_{2} - 10585 \beta_{1} - 141781\)
\(\nu^{15}\)\(=\)\(-48908 \beta_{19} + 241331 \beta_{18} + 273167 \beta_{17} - 85364 \beta_{16} - 23814 \beta_{14} + 23814 \beta_{13} - 229332 \beta_{11} + 34382 \beta_{10} - 192948 \beta_{9} + 192948 \beta_{8} - 34382 \beta_{7} - 86166 \beta_{4} - 263895 \beta_{1}\)
\(\nu^{16}\)\(=\)\(82968 \beta_{19} - 252102 \beta_{18} + 82968 \beta_{16} + 86166 \beta_{15} + 313974 \beta_{14} + 61872 \beta_{13} + 225061 \beta_{12} - 172285 \beta_{10} + 144998 \beta_{9} - 273040 \beta_{8} - 172285 \beta_{7} - 277787 \beta_{6} - 252102 \beta_{5} - 447691 \beta_{3} - 816931 \beta_{2} + 86166 \beta_{1} + 974063\)
\(\nu^{17}\)\(=\)\(421078 \beta_{19} - 1838815 \beta_{18} - 2019572 \beta_{17} + 698865 \beta_{16} + 212606 \beta_{14} - 212606 \beta_{13} + 1816978 \beta_{11} - 313816 \beta_{10} + 1557768 \beta_{9} - 1557768 \beta_{8} + 313816 \beta_{7} + 687073 \beta_{4} + 1754916 \beta_{1}\)
\(\nu^{18}\)\(=\)\(-658089 \beta_{19} + 2003251 \beta_{18} - 658089 \beta_{16} - 687073 \beta_{15} - 2566315 \beta_{14} - 563064 \beta_{13} - 1586702 \beta_{12} + 1241773 \beta_{10} - 1055751 \beta_{9} + 2263678 \beta_{8} + 1241773 \beta_{7} + 2116602 \beta_{6} + 2003251 \beta_{5} + 3322576 \beta_{3} + 5875164 \beta_{2} - 687073 \beta_{1} - 6817561\)
\(\nu^{19}\)\(=\)\(-3513815 \beta_{19} + 14006619 \beta_{18} + 14904295 \beta_{17} - 5630417 \beta_{16} - 1822329 \beta_{14} + 1822329 \beta_{13} - 14109884 \beta_{11} + 2731717 \beta_{10} - 12387861 \beta_{9} + 12387861 \beta_{8} - 2731717 \beta_{7} - 5403154 \beta_{4} - 11848557 \beta_{1}\)

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
2.74150i
2.55795i
2.32980i
2.28323i
1.93467i
1.40171i
1.28094i
0.792050i
0.568210i
0.0342944i
0.0342944i
0.568210i
0.792050i
1.28094i
1.40171i
1.93467i
2.28323i
2.32980i
2.55795i
2.74150i
2.74150i 1.00000i −5.51584 −0.739422 2.11027i 2.74150 1.00000i 9.63869i −1.00000 −5.78532 + 2.02713i
694.2 2.55795i 1.00000i −4.54312 0.0738813 + 2.23485i 2.55795 1.00000i 6.50518i −1.00000 5.71663 0.188985i
694.3 2.32980i 1.00000i −3.42795 1.29334 + 1.82408i −2.32980 1.00000i 3.32682i −1.00000 4.24974 3.01322i
694.4 2.28323i 1.00000i −3.21315 −2.23363 0.104355i −2.28323 1.00000i 2.76990i −1.00000 −0.238267 + 5.09990i
694.5 1.93467i 1.00000i −1.74296 1.72871 1.41829i 1.93467 1.00000i 0.497289i −1.00000 −2.74393 3.34450i
694.6 1.40171i 1.00000i 0.0352144 −2.22487 + 0.223473i 1.40171 1.00000i 2.85278i −1.00000 0.313244 + 3.11862i
694.7 1.28094i 1.00000i 0.359182 2.05729 0.876110i −1.28094 1.00000i 3.02198i −1.00000 −1.12225 2.63527i
694.8 0.792050i 1.00000i 1.37266 −0.199975 + 2.22711i 0.792050 1.00000i 2.67131i −1.00000 1.76398 + 0.158390i
694.9 0.568210i 1.00000i 1.67714 1.08688 1.95415i −0.568210 1.00000i 2.08939i −1.00000 −1.11037 0.617578i
694.10 0.0342944i 1.00000i 1.99882 −1.84220 1.26739i 0.0342944 1.00000i 0.137137i −1.00000 −0.0434646 + 0.0631773i
694.11 0.0342944i 1.00000i 1.99882 −1.84220 + 1.26739i 0.0342944 1.00000i 0.137137i −1.00000 −0.0434646 0.0631773i
694.12 0.568210i 1.00000i 1.67714 1.08688 + 1.95415i −0.568210 1.00000i 2.08939i −1.00000 −1.11037 + 0.617578i
694.13 0.792050i 1.00000i 1.37266 −0.199975 2.22711i 0.792050 1.00000i 2.67131i −1.00000 1.76398 0.158390i
694.14 1.28094i 1.00000i 0.359182 2.05729 + 0.876110i −1.28094 1.00000i 3.02198i −1.00000 −1.12225 + 2.63527i
694.15 1.40171i 1.00000i 0.0352144 −2.22487 0.223473i 1.40171 1.00000i 2.85278i −1.00000 0.313244 3.11862i
694.16 1.93467i 1.00000i −1.74296 1.72871 + 1.41829i 1.93467 1.00000i 0.497289i −1.00000 −2.74393 + 3.34450i
694.17 2.28323i 1.00000i −3.21315 −2.23363 + 0.104355i −2.28323 1.00000i 2.76990i −1.00000 −0.238267 5.09990i
694.18 2.32980i 1.00000i −3.42795 1.29334 1.82408i −2.32980 1.00000i 3.32682i −1.00000 4.24974 + 3.01322i
694.19 2.55795i 1.00000i −4.54312 0.0738813 2.23485i 2.55795 1.00000i 6.50518i −1.00000 5.71663 + 0.188985i
694.20 2.74150i 1.00000i −5.51584 −0.739422 + 2.11027i 2.74150 1.00000i 9.63869i −1.00000 −5.78532 2.02713i
\(n\): 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.f 20
5.b even 2 1 inner 1155.2.c.f 20
5.c odd 4 1 5775.2.a.cn 10
5.c odd 4 1 5775.2.a.co 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.c.f 20 1.a even 1 1 trivial
1155.2.c.f 20 5.b even 2 1 inner
5775.2.a.cn 10 5.c odd 4 1
5775.2.a.co 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

$p$ $F_p(T)$
$2$ \( 1 - 7 T^{2} + 28 T^{4} - 90 T^{6} + 258 T^{8} - 664 T^{10} + 1545 T^{12} - 3391 T^{14} + 7244 T^{16} - 15008 T^{18} + 30196 T^{20} - 60032 T^{22} + 115904 T^{24} - 217024 T^{26} + 395520 T^{28} - 679936 T^{30} + 1056768 T^{32} - 1474560 T^{34} + 1835008 T^{36} - 1835008 T^{38} + 1048576 T^{40} \)
$3$ \( ( 1 + T^{2} )^{10} \)
$5$ \( 1 + 2 T + 4 T^{2} + 26 T^{3} + 34 T^{4} + 126 T^{5} + 402 T^{6} + 438 T^{7} + 1757 T^{8} + 4496 T^{9} + 6708 T^{10} + 22480 T^{11} + 43925 T^{12} + 54750 T^{13} + 251250 T^{14} + 393750 T^{15} + 531250 T^{16} + 2031250 T^{17} + 1562500 T^{18} + 3906250 T^{19} + 9765625 T^{20} \)
$7$ \( ( 1 + T^{2} )^{10} \)
$11$ \( ( 1 - T )^{20} \)
$13$ \( 1 - 108 T^{2} + 5944 T^{4} - 219580 T^{6} + 6036678 T^{8} - 129984768 T^{10} + 2257177886 T^{12} - 32314565340 T^{14} + 394348032353 T^{16} - 4437894622204 T^{18} + 53274540958148 T^{20} - 750004191152476 T^{22} + 11262974152034033 T^{24} - 155976234814200060 T^{26} + 1841249344372035806 T^{28} - 17919504079822156032 T^{30} + \)\(14\!\cdots\!18\)\( T^{32} - \)\(86\!\cdots\!20\)\( T^{34} + \)\(39\!\cdots\!04\)\( T^{36} - \)\(12\!\cdots\!32\)\( T^{38} + \)\(19\!\cdots\!01\)\( T^{40} \)
$17$ \( 1 - 149 T^{2} + 11639 T^{4} - 631904 T^{6} + 26638931 T^{8} - 923975797 T^{10} + 27284514928 T^{12} - 700708863983 T^{14} + 15863803860108 T^{16} - 319282468644871 T^{18} + 5739174583030002 T^{20} - 92272633438367719 T^{22} + 1324960762200080268 T^{24} - 16913408553301277327 T^{26} + \)\(19\!\cdots\!48\)\( T^{28} - \)\(18\!\cdots\!53\)\( T^{30} + \)\(15\!\cdots\!91\)\( T^{32} - \)\(10\!\cdots\!16\)\( T^{34} + \)\(56\!\cdots\!59\)\( T^{36} - \)\(20\!\cdots\!41\)\( T^{38} + \)\(40\!\cdots\!01\)\( T^{40} \)
$19$ \( ( 1 + 17 T + 219 T^{2} + 1940 T^{3} + 15300 T^{4} + 101292 T^{5} + 632573 T^{6} + 3500645 T^{7} + 18436271 T^{8} + 87566008 T^{9} + 399711704 T^{10} + 1663754152 T^{11} + 6655493831 T^{12} + 24010924055 T^{13} + 82437545933 T^{14} + 250809019908 T^{15} + 719801979300 T^{16} + 1734111173660 T^{17} + 3719400305979 T^{18} + 5485690862243 T^{19} + 6131066257801 T^{20} )^{2} \)
$23$ \( 1 - 149 T^{2} + 11627 T^{4} - 634984 T^{6} + 27788887 T^{8} - 1051120093 T^{10} + 35693788136 T^{12} - 1101953725015 T^{14} + 31049241038440 T^{16} - 802588589114367 T^{18} + 19170409876986906 T^{20} - 424569363641500143 T^{22} + 8688850661438088040 T^{24} - \)\(16\!\cdots\!35\)\( T^{26} + \)\(27\!\cdots\!16\)\( T^{28} - \)\(43\!\cdots\!57\)\( T^{30} + \)\(60\!\cdots\!27\)\( T^{32} - \)\(73\!\cdots\!56\)\( T^{34} + \)\(71\!\cdots\!47\)\( T^{36} - \)\(48\!\cdots\!81\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \)
$29$ \( ( 1 + 5 T + 117 T^{2} + 478 T^{3} + 7888 T^{4} + 28558 T^{5} + 387835 T^{6} + 1242511 T^{7} + 14971543 T^{8} + 43150668 T^{9} + 474100176 T^{10} + 1251369372 T^{11} + 12591067663 T^{12} + 30303600779 T^{13} + 274308326635 T^{14} + 585757393142 T^{15} + 4691966356048 T^{16} + 8245440875702 T^{17} + 58528830316437 T^{18} + 72535729879345 T^{19} + 420707233300201 T^{20} )^{2} \)
$31$ \( ( 1 - 6 T + 74 T^{2} - 420 T^{3} + 5569 T^{4} - 26250 T^{5} + 244816 T^{6} - 1152066 T^{7} + 10772798 T^{8} - 42870934 T^{9} + 332523084 T^{10} - 1328998954 T^{11} + 10352658878 T^{12} - 34321198206 T^{13} + 226092717136 T^{14} - 751515213750 T^{15} + 4942507999489 T^{16} - 11555297926620 T^{17} + 63113936770634 T^{18} - 158637732964026 T^{19} + 819628286980801 T^{20} )^{2} \)
$37$ \( 1 - 276 T^{2} + 39800 T^{4} - 3928044 T^{6} + 298871830 T^{8} - 18845095064 T^{10} + 1035557317166 T^{12} - 51263841660388 T^{14} + 2326196427291569 T^{16} - 97297827407840132 T^{18} + 3750960894212410852 T^{20} - \)\(13\!\cdots\!08\)\( T^{22} + \)\(43\!\cdots\!09\)\( T^{24} - \)\(13\!\cdots\!92\)\( T^{26} + \)\(36\!\cdots\!86\)\( T^{28} - \)\(90\!\cdots\!36\)\( T^{30} + \)\(19\!\cdots\!30\)\( T^{32} - \)\(35\!\cdots\!16\)\( T^{34} + \)\(49\!\cdots\!00\)\( T^{36} - \)\(46\!\cdots\!04\)\( T^{38} + \)\(23\!\cdots\!01\)\( T^{40} \)
$41$ \( ( 1 - 26 T + 544 T^{2} - 7724 T^{3} + 95777 T^{4} - 968190 T^{5} + 8934600 T^{6} - 71969834 T^{7} + 547213230 T^{8} - 3776336450 T^{9} + 25186272816 T^{10} - 154829794450 T^{11} + 919865439630 T^{12} - 4960232929114 T^{13} + 25247044230600 T^{14} - 112170815246190 T^{15} + 454950733890257 T^{16} - 1504282011456844 T^{17} + 4343799324641824 T^{18} - 8511930294242986 T^{19} + 13422659310152401 T^{20} )^{2} \)
$43$ \( 1 - 445 T^{2} + 101583 T^{4} - 15733964 T^{6} + 1850110019 T^{8} - 175376856657 T^{10} + 13900353862768 T^{12} - 943206133698667 T^{14} + 55645680002705772 T^{16} - 2882364628830608611 T^{18} + \)\(13\!\cdots\!38\)\( T^{20} - \)\(53\!\cdots\!39\)\( T^{22} + \)\(19\!\cdots\!72\)\( T^{24} - \)\(59\!\cdots\!83\)\( T^{26} + \)\(16\!\cdots\!68\)\( T^{28} - \)\(37\!\cdots\!93\)\( T^{30} + \)\(73\!\cdots\!19\)\( T^{32} - \)\(11\!\cdots\!36\)\( T^{34} + \)\(13\!\cdots\!83\)\( T^{36} - \)\(11\!\cdots\!05\)\( T^{38} + \)\(46\!\cdots\!01\)\( T^{40} \)
$47$ \( 1 - 444 T^{2} + 92196 T^{4} - 12149780 T^{6} + 1184517974 T^{8} - 95625758080 T^{10} + 6880096091314 T^{12} - 450201056694860 T^{14} + 26553630005312817 T^{16} - 1412555151749395236 T^{18} + 68938276638065329860 T^{20} - \)\(31\!\cdots\!24\)\( T^{22} + \)\(12\!\cdots\!77\)\( T^{24} - \)\(48\!\cdots\!40\)\( T^{26} + \)\(16\!\cdots\!54\)\( T^{28} - \)\(50\!\cdots\!20\)\( T^{30} + \)\(13\!\cdots\!34\)\( T^{32} - \)\(31\!\cdots\!20\)\( T^{34} + \)\(52\!\cdots\!16\)\( T^{36} - \)\(55\!\cdots\!16\)\( T^{38} + \)\(27\!\cdots\!01\)\( T^{40} \)
$53$ \( 1 - 389 T^{2} + 81639 T^{4} - 12035744 T^{6} + 1395484923 T^{8} - 135312803605 T^{10} + 11385806864320 T^{12} - 849942472919799 T^{14} + 57052023207879220 T^{16} - 3472559969098943159 T^{18} + \)\(19\!\cdots\!66\)\( T^{20} - \)\(97\!\cdots\!31\)\( T^{22} + \)\(45\!\cdots\!20\)\( T^{24} - \)\(18\!\cdots\!71\)\( T^{26} + \)\(70\!\cdots\!20\)\( T^{28} - \)\(23\!\cdots\!45\)\( T^{30} + \)\(68\!\cdots\!43\)\( T^{32} - \)\(16\!\cdots\!36\)\( T^{34} + \)\(31\!\cdots\!19\)\( T^{36} - \)\(42\!\cdots\!21\)\( T^{38} + \)\(30\!\cdots\!01\)\( T^{40} \)
$59$ \( ( 1 + 7 T + 291 T^{2} + 818 T^{3} + 36608 T^{4} - 9834 T^{5} + 3271101 T^{6} - 6599101 T^{7} + 259218183 T^{8} - 554218956 T^{9} + 17333140048 T^{10} - 32698918404 T^{11} + 902338495023 T^{12} - 1355316764279 T^{13} + 39637111684461 T^{14} - 7030565556366 T^{15} + 1544144975529728 T^{16} + 2035716914581942 T^{17} + 42727657342857411 T^{18} + 60640970730584573 T^{19} + 511116753300641401 T^{20} )^{2} \)
$61$ \( ( 1 - 39 T + 951 T^{2} - 17248 T^{3} + 259735 T^{4} - 3372403 T^{5} + 39049984 T^{6} - 409043933 T^{7} + 3923831552 T^{8} - 34592326741 T^{9} + 281355699058 T^{10} - 2110131931201 T^{11} + 14600577204992 T^{12} - 92845200956273 T^{13} + 540679869516544 T^{14} - 2848319099281303 T^{15} + 13381644434654335 T^{16} - 54206028435690208 T^{17} + 182313654660414231 T^{18} - 456071697620531499 T^{19} + 713342911662882601 T^{20} )^{2} \)
$67$ \( 1 - 860 T^{2} + 368036 T^{4} - 104191816 T^{6} + 21885547798 T^{8} - 3626233790356 T^{10} + 491852500080410 T^{12} - 55930669460610204 T^{14} + 5415764148185626937 T^{16} - \)\(45\!\cdots\!56\)\( T^{18} + \)\(32\!\cdots\!44\)\( T^{20} - \)\(20\!\cdots\!84\)\( T^{22} + \)\(10\!\cdots\!77\)\( T^{24} - \)\(50\!\cdots\!76\)\( T^{26} + \)\(19\!\cdots\!10\)\( T^{28} - \)\(66\!\cdots\!44\)\( T^{30} + \)\(17\!\cdots\!78\)\( T^{32} - \)\(38\!\cdots\!64\)\( T^{34} + \)\(60\!\cdots\!16\)\( T^{36} - \)\(63\!\cdots\!40\)\( T^{38} + \)\(33\!\cdots\!01\)\( T^{40} \)
$71$ \( ( 1 + 276 T^{2} + 778 T^{3} + 36429 T^{4} + 196426 T^{5} + 3736592 T^{6} + 23142226 T^{7} + 350659810 T^{8} + 1917129650 T^{9} + 27778410936 T^{10} + 136116205150 T^{11} + 1767676102210 T^{12} + 8282857249886 T^{13} + 94953083931152 T^{14} + 354397554499526 T^{15} + 4666565242958109 T^{16} + 7076003483228198 T^{17} + 178227974623830036 T^{18} + 3255243551009881201 T^{20} )^{2} \)
$73$ \( 1 - 840 T^{2} + 338748 T^{4} - 87754740 T^{6} + 16504326726 T^{8} - 2418252763080 T^{10} + 289904194790306 T^{12} - 29570586397109664 T^{14} + 2655840494932708265 T^{16} - \)\(21\!\cdots\!04\)\( T^{18} + \)\(16\!\cdots\!00\)\( T^{20} - \)\(11\!\cdots\!16\)\( T^{22} + \)\(75\!\cdots\!65\)\( T^{24} - \)\(44\!\cdots\!96\)\( T^{26} + \)\(23\!\cdots\!86\)\( T^{28} - \)\(10\!\cdots\!20\)\( T^{30} + \)\(37\!\cdots\!46\)\( T^{32} - \)\(10\!\cdots\!60\)\( T^{34} + \)\(22\!\cdots\!28\)\( T^{36} - \)\(29\!\cdots\!60\)\( T^{38} + \)\(18\!\cdots\!01\)\( T^{40} \)
$79$ \( ( 1 + 26 T + 722 T^{2} + 12736 T^{3} + 221769 T^{4} + 3052578 T^{5} + 41043696 T^{6} + 467927082 T^{7} + 5206423382 T^{8} + 50541132238 T^{9} + 478939578876 T^{10} + 3992749446802 T^{11} + 32493288327062 T^{12} + 230706300582198 T^{13} + 1598655283739376 T^{14} + 9392954668346622 T^{15} + 53909261923436649 T^{16} + 244580984847721024 T^{17} + 1095352560752537042 T^{18} + 3116141495548076294 T^{19} + 9468276082626847201 T^{20} )^{2} \)
$83$ \( 1 - 565 T^{2} + 146323 T^{4} - 22297440 T^{6} + 2107834031 T^{8} - 114807579093 T^{10} + 2731866902904 T^{12} - 212169470725031 T^{14} + 77833540515327648 T^{16} - 12865723835530791335 T^{18} + \)\(13\!\cdots\!22\)\( T^{20} - \)\(88\!\cdots\!15\)\( T^{22} + \)\(36\!\cdots\!08\)\( T^{24} - \)\(69\!\cdots\!39\)\( T^{26} + \)\(61\!\cdots\!64\)\( T^{28} - \)\(17\!\cdots\!57\)\( T^{30} + \)\(22\!\cdots\!91\)\( T^{32} - \)\(16\!\cdots\!60\)\( T^{34} + \)\(74\!\cdots\!63\)\( T^{36} - \)\(19\!\cdots\!85\)\( T^{38} + \)\(24\!\cdots\!01\)\( T^{40} \)
$89$ \( ( 1 + 5 T + 423 T^{2} + 2300 T^{3} + 89801 T^{4} + 562621 T^{5} + 12317564 T^{6} + 91880199 T^{7} + 1283985254 T^{8} + 10908120867 T^{9} + 117187677178 T^{10} + 970822757163 T^{11} + 10170447196934 T^{12} + 64772692008831 T^{13} + 772831569020924 T^{14} + 3141709111255829 T^{15} + 44629416909588761 T^{16} + 101732070259716700 T^{17} + 1665177064811980263 T^{18} + 1751782018537426045 T^{19} + 31181719929966183601 T^{20} )^{2} \)
$97$ \( 1 - 745 T^{2} + 269023 T^{4} - 62727220 T^{6} + 10674371027 T^{8} - 1429985200421 T^{10} + 159494324073040 T^{12} - 15457224318890367 T^{14} + 1353013921718174412 T^{16} - \)\(11\!\cdots\!15\)\( T^{18} + \)\(10\!\cdots\!66\)\( T^{20} - \)\(10\!\cdots\!35\)\( T^{22} + \)\(11\!\cdots\!72\)\( T^{24} - \)\(12\!\cdots\!43\)\( T^{26} + \)\(12\!\cdots\!40\)\( T^{28} - \)\(10\!\cdots\!29\)\( T^{30} + \)\(74\!\cdots\!07\)\( T^{32} - \)\(40\!\cdots\!80\)\( T^{34} + \)\(16\!\cdots\!83\)\( T^{36} - \)\(43\!\cdots\!05\)\( T^{38} + \)\(54\!\cdots\!01\)\( T^{40} \)
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