Properties

Label 2013.2.a.h
Level 2013
Weight 2
Character orbit 2013.a
Self dual yes
Analytic conductor 16.074
Analytic rank 0
Dimension 14
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0738859269\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} - 745 x^{5} - 802 x^{4} + 386 x^{3} + 74 x^{2} - 15 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} - 745 x^{5} - 802 x^{4} + 386 x^{3} + 74 x^{2} - 15 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{13} - 71 \nu^{12} - 586 \nu^{11} + 2838 \nu^{10} + 10454 \nu^{9} - 35539 \nu^{8} - 62411 \nu^{7} + 186606 \nu^{6} + 140254 \nu^{5} - 411132 \nu^{4} - 83942 \nu^{3} + 307840 \nu^{2} - 27224 \nu - 13953 \)\()/4505\)
\(\beta_{5}\)\(=\)\((\)\( -82 \nu^{13} - 1317 \nu^{12} + 1503 \nu^{11} + 25486 \nu^{10} - 7732 \nu^{9} - 179663 \nu^{8} - 4527 \nu^{7} + 556827 \nu^{6} + 125703 \nu^{5} - 713699 \nu^{4} - 233864 \nu^{3} + 253645 \nu^{2} + 24637 \nu - 22401 \)\()/4505\)
\(\beta_{6}\)\(=\)\((\)\( -311 \nu^{13} + 444 \nu^{12} + 6964 \nu^{11} - 9372 \nu^{10} - 59981 \nu^{9} + 74721 \nu^{8} + 245494 \nu^{7} - 273749 \nu^{6} - 466936 \nu^{5} + 435843 \nu^{4} + 306853 \nu^{3} - 209250 \nu^{2} + 43281 \nu - 1068 \)\()/4505\)
\(\beta_{7}\)\(=\)\((\)\(-349 \nu^{13} - 2254 \nu^{12} + 7221 \nu^{11} + 44412 \nu^{10} - 54664 \nu^{9} - 320701 \nu^{8} + 180436 \nu^{7} + 1032859 \nu^{6} - 232439 \nu^{5} - 1428903 \nu^{4} + 76842 \nu^{3} + 618105 \nu^{2} - 67706 \nu - 17712\)\()/4505\)
\(\beta_{8}\)\(=\)\((\)\( -251 \nu^{13} + 199 \nu^{12} + 5183 \nu^{11} - 3957 \nu^{10} - 40303 \nu^{9} + 29344 \nu^{8} + 146488 \nu^{7} - 100390 \nu^{6} - 248794 \nu^{5} + 158677 \nu^{4} + 158118 \nu^{3} - 92020 \nu^{2} - 2743 \nu + 885 \)\()/901\)
\(\beta_{9}\)\(=\)\((\)\(1423 \nu^{13} - 2582 \nu^{12} - 31082 \nu^{11} + 52066 \nu^{10} + 260758 \nu^{9} - 388563 \nu^{8} - 1050382 \nu^{7} + 1308832 \nu^{6} + 2048843 \nu^{5} - 1915614 \nu^{4} - 1595379 \nu^{3} + 883850 \nu^{2} + 113882 \nu - 20941\)\()/4505\)
\(\beta_{10}\)\(=\)\((\)\(1433 \nu^{13} - 1872 \nu^{12} - 29727 \nu^{11} + 37201 \nu^{10} + 232803 \nu^{9} - 271938 \nu^{8} - 858752 \nu^{7} + 893382 \nu^{6} + 1511263 \nu^{5} - 1286659 \nu^{4} - 1048784 \nu^{3} + 607560 \nu^{2} + 52752 \nu - 12056\)\()/4505\)
\(\beta_{11}\)\(=\)\((\)\( -303 \nu^{13} + 111 \nu^{12} + 6246 \nu^{11} - 2343 \nu^{10} - 48107 \nu^{9} + 18455 \nu^{8} + 170845 \nu^{7} - 67311 \nu^{6} - 277112 \nu^{5} + 115493 \nu^{4} + 162984 \nu^{3} - 77090 \nu^{2} - 5623 \nu + 3337 \)\()/901\)
\(\beta_{12}\)\(=\)\((\)\(-1576 \nu^{13} + 729 \nu^{12} + 31524 \nu^{11} - 14292 \nu^{10} - 233541 \nu^{9} + 104816 \nu^{8} + 786304 \nu^{7} - 360249 \nu^{6} - 1177831 \nu^{5} + 604778 \nu^{4} + 591393 \nu^{3} - 424595 \nu^{2} + 14111 \nu + 21502\)\()/4505\)
\(\beta_{13}\)\(=\)\((\)\(-3393 \nu^{13} + 2367 \nu^{12} + 70487 \nu^{11} - 47406 \nu^{10} - 551558 \nu^{9} + 352898 \nu^{8} + 2019747 \nu^{7} - 1203902 \nu^{6} - 3477208 \nu^{5} + 1876454 \nu^{4} + 2324484 \nu^{3} - 1064330 \nu^{2} - 176207 \nu + 36556\)\()/4505\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{11} + \beta_{10} - \beta_{5} + 7 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(-\beta_{12} + \beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + 9 \beta_{3} + 28 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{13} + 9 \beta_{11} + 9 \beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{6} - 10 \beta_{5} + \beta_{3} + 48 \beta_{2} - \beta_{1} + 84\)
\(\nu^{7}\)\(=\)\(2 \beta_{13} - 12 \beta_{12} + 10 \beta_{11} - \beta_{10} + 13 \beta_{9} + 9 \beta_{8} + 12 \beta_{6} - 12 \beta_{5} - \beta_{4} + 67 \beta_{3} + 2 \beta_{2} + 168 \beta_{1} - 1\)
\(\nu^{8}\)\(=\)\(16 \beta_{13} - 2 \beta_{12} + 64 \beta_{11} + 66 \beta_{10} + 16 \beta_{9} - 26 \beta_{8} - \beta_{7} + 28 \beta_{6} - 79 \beta_{5} - \beta_{4} + 14 \beta_{3} + 326 \beta_{2} - 13 \beta_{1} + 504\)
\(\nu^{9}\)\(=\)\(35 \beta_{13} - 109 \beta_{12} + 79 \beta_{11} - 10 \beta_{10} + 125 \beta_{9} + 57 \beta_{8} - 4 \beta_{7} + 114 \beta_{6} - 106 \beta_{5} - 12 \beta_{4} + 472 \beta_{3} + 33 \beta_{2} + 1046 \beta_{1} - 12\)
\(\nu^{10}\)\(=\)\(176 \beta_{13} - 37 \beta_{12} + 426 \beta_{11} + 464 \beta_{10} + 174 \beta_{9} - 243 \beta_{8} - 20 \beta_{7} + 282 \beta_{6} - 579 \beta_{5} - 11 \beta_{4} + 141 \beta_{3} + 2204 \beta_{2} - 118 \beta_{1} + 3147\)
\(\nu^{11}\)\(=\)\(405 \beta_{13} - 898 \beta_{12} + 586 \beta_{11} - 56 \beta_{10} + 1066 \beta_{9} + 302 \beta_{8} - 75 \beta_{7} + 984 \beta_{6} - 839 \beta_{5} - 98 \beta_{4} + 3258 \beta_{3} + 370 \beta_{2} + 6644 \beta_{1} - 89\)
\(\nu^{12}\)\(=\)\(1652 \beta_{13} - 443 \beta_{12} + 2777 \beta_{11} + 3245 \beta_{10} + 1609 \beta_{9} - 2010 \beta_{8} - 256 \beta_{7} + 2491 \beta_{6} - 4113 \beta_{5} - 72 \beta_{4} + 1251 \beta_{3} + 14874 \beta_{2} - 923 \beta_{1} + 20093\)
\(\nu^{13}\)\(=\)\(3916 \beta_{13} - 7055 \beta_{12} + 4261 \beta_{11} - 128 \beta_{10} + 8540 \beta_{9} + 1339 \beta_{8} - 911 \beta_{7} + 8029 \beta_{6} - 6318 \beta_{5} - 681 \beta_{4} + 22304 \beta_{3} + 3528 \beta_{2} + 42680 \beta_{1} - 465\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.63401
2.45909
1.93923
1.67203
1.61392
0.546298
0.179763
−0.0561655
−0.231279
−1.18140
−1.69494
−1.76637
−2.54331
−2.57087
−2.63401 −1.00000 4.93799 2.62502 2.63401 5.18852 −7.73867 1.00000 −6.91432
1.2 −2.45909 −1.00000 4.04715 −1.85442 2.45909 2.32343 −5.03413 1.00000 4.56020
1.3 −1.93923 −1.00000 1.76061 3.07794 1.93923 −3.15900 0.464237 1.00000 −5.96883
1.4 −1.67203 −1.00000 0.795679 −1.45953 1.67203 −0.113904 2.01366 1.00000 2.44037
1.5 −1.61392 −1.00000 0.604750 −3.65776 1.61392 0.985739 2.25183 1.00000 5.90335
1.6 −0.546298 −1.00000 −1.70156 −0.842631 0.546298 −4.19208 2.02216 1.00000 0.460328
1.7 −0.179763 −1.00000 −1.96769 2.20477 0.179763 3.17177 0.713244 1.00000 −0.396336
1.8 0.0561655 −1.00000 −1.99685 −2.87016 −0.0561655 3.53988 −0.224485 1.00000 −0.161204
1.9 0.231279 −1.00000 −1.94651 3.70694 −0.231279 −0.911108 −0.912746 1.00000 0.857338
1.10 1.18140 −1.00000 −0.604292 −2.90081 −1.18140 −0.528840 −3.07671 1.00000 −3.42702
1.11 1.69494 −1.00000 0.872835 1.13650 −1.69494 4.14659 −1.91048 1.00000 1.92630
1.12 1.76637 −1.00000 1.12006 −2.50284 −1.76637 −4.08919 −1.55430 1.00000 −4.42094
1.13 2.54331 −1.00000 4.46845 0.329781 −2.54331 0.595463 6.27804 1.00000 0.838736
1.14 2.57087 −1.00000 4.60938 4.00721 −2.57087 2.04273 6.70837 1.00000 10.3020
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(1\)
\(61\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2013.2.a.h 14
3.b odd 2 1 6039.2.a.j 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.h 14 1.a even 1 1 trivial
6039.2.a.j 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{14} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 7 T^{2} + 6 T^{3} + 27 T^{4} + 20 T^{5} + 81 T^{6} + 55 T^{7} + 207 T^{8} + 139 T^{9} + 514 T^{10} + 356 T^{11} + 1230 T^{12} + 851 T^{13} + 2615 T^{14} + 1702 T^{15} + 4920 T^{16} + 2848 T^{17} + 8224 T^{18} + 4448 T^{19} + 13248 T^{20} + 7040 T^{21} + 20736 T^{22} + 10240 T^{23} + 27648 T^{24} + 12288 T^{25} + 28672 T^{26} + 8192 T^{27} + 16384 T^{28} \)
$3$ \( ( 1 + T )^{14} \)
$5$ \( 1 - T + 23 T^{2} - 35 T^{3} + 331 T^{4} - 561 T^{5} + 3552 T^{6} - 6250 T^{7} + 30710 T^{8} - 53589 T^{9} + 222534 T^{10} - 374432 T^{11} + 1380392 T^{12} - 2199712 T^{13} + 7403730 T^{14} - 10998560 T^{15} + 34509800 T^{16} - 46804000 T^{17} + 139083750 T^{18} - 167465625 T^{19} + 479843750 T^{20} - 488281250 T^{21} + 1387500000 T^{22} - 1095703125 T^{23} + 3232421875 T^{24} - 1708984375 T^{25} + 5615234375 T^{26} - 1220703125 T^{27} + 6103515625 T^{28} \)
$7$ \( 1 - 9 T + 77 T^{2} - 432 T^{3} + 2270 T^{4} - 9803 T^{5} + 40388 T^{6} - 149410 T^{7} + 533718 T^{8} - 1782087 T^{9} + 5737179 T^{10} - 17535815 T^{11} + 51464307 T^{12} - 144627270 T^{13} + 390279048 T^{14} - 1012390890 T^{15} + 2521751043 T^{16} - 6014784545 T^{17} + 13774966779 T^{18} - 29951536209 T^{19} + 62791388982 T^{20} - 123045559630 T^{21} + 232828782788 T^{22} - 395586409421 T^{23} + 641218815230 T^{24} - 854205152976 T^{25} + 1065779114477 T^{26} - 872001093663 T^{27} + 678223072849 T^{28} \)
$11$ \( ( 1 + T )^{14} \)
$13$ \( 1 - T + 81 T^{2} - 98 T^{3} + 3422 T^{4} - 4866 T^{5} + 100104 T^{6} - 160773 T^{7} + 2269434 T^{8} - 3949833 T^{9} + 42329042 T^{10} - 76600455 T^{11} + 673466313 T^{12} - 1208879886 T^{13} + 9341312614 T^{14} - 15715438518 T^{15} + 113815806897 T^{16} - 168291199635 T^{17} + 1208959768562 T^{18} - 1466545344069 T^{19} + 10954124456106 T^{20} - 10088267323641 T^{21} + 81657908094984 T^{22} - 51601493949018 T^{23} + 471751759107278 T^{24} - 175631718615626 T^{25} + 1887144894920961 T^{26} - 302875106592253 T^{27} + 3937376385699289 T^{28} \)
$17$ \( 1 + 9 T + 161 T^{2} + 1229 T^{3} + 12271 T^{4} + 79788 T^{5} + 588756 T^{6} + 3306669 T^{7} + 20089312 T^{8} + 99501923 T^{9} + 526646471 T^{10} + 2355773010 T^{11} + 11265239146 T^{12} + 46479022192 T^{13} + 205561578844 T^{14} + 790143377264 T^{15} + 3255654113194 T^{16} + 11573912798130 T^{17} + 43986039904391 T^{18} + 141278501885011 T^{19} + 484907154562528 T^{20} + 1356854169510237 T^{21} + 4107019047933396 T^{22} + 9461889489942636 T^{23} + 24738261152409679 T^{24} + 42120160562080957 T^{25} + 93802180193991521 T^{26} + 89141202296153433 T^{27} + 168377826559400929 T^{28} \)
$19$ \( 1 - 22 T + 405 T^{2} - 5162 T^{3} + 58169 T^{4} - 545208 T^{5} + 4656573 T^{6} - 35148668 T^{7} + 245940070 T^{8} - 1563436698 T^{9} + 9305286079 T^{10} - 51028457017 T^{11} + 263455290000 T^{12} - 1261886979845 T^{13} + 5704736463782 T^{14} - 23975852617055 T^{15} + 95107359690000 T^{16} - 350004186679603 T^{17} + 1212674187101359 T^{18} - 3871224044481102 T^{19} + 11570467266351670 T^{20} - 31418400988693652 T^{21} + 79085201100518493 T^{22} - 175931914330693032 T^{23} + 356637993150026369 T^{24} - 601322716432606478 T^{25} + 896392542221795205 T^{26} - 925165636169655298 T^{27} + 799006685782884121 T^{28} \)
$23$ \( 1 - T + 162 T^{2} - 129 T^{3} + 13736 T^{4} - 8073 T^{5} + 796428 T^{6} - 295782 T^{7} + 35119436 T^{8} - 6199464 T^{9} + 1245290749 T^{10} - 41635747 T^{11} + 36690460719 T^{12} + 1399840196 T^{13} + 914656857314 T^{14} + 32196324508 T^{15} + 19409253720351 T^{16} - 506582133749 T^{17} + 348483408490909 T^{18} - 39901876720152 T^{19} + 5198936929438604 T^{20} - 1007086080364554 T^{21} + 62369061385376268 T^{22} - 14540705435990799 T^{23} + 569034558030682664 T^{24} - 122912458770896583 T^{25} + 3550169157987292002 T^{26} - 504036361936467383 T^{27} + 11592836324538749809 T^{28} \)
$29$ \( 1 + 6 T + 164 T^{2} + 548 T^{3} + 11705 T^{4} + 21353 T^{5} + 580119 T^{6} + 778214 T^{7} + 25224091 T^{8} + 43384579 T^{9} + 965373316 T^{10} + 1935780164 T^{11} + 30569274113 T^{12} + 61567478972 T^{13} + 883259702894 T^{14} + 1785456890188 T^{15} + 25708759529033 T^{16} + 47211742419796 T^{17} + 682790204313796 T^{18} + 889867564171271 T^{19} + 15003877577826211 T^{20} + 13424095241932126 T^{21} + 290202448840522359 T^{22} + 309771088022730757 T^{23} + 4924378165778852705 T^{24} + 6685879351606794292 T^{25} + 58025624445696922724 T^{26} + 61563772277751613134 T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \)
$31$ \( 1 - 9 T + 239 T^{2} - 1595 T^{3} + 24505 T^{4} - 122870 T^{5} + 1461485 T^{6} - 5328640 T^{7} + 58869289 T^{8} - 145579448 T^{9} + 1870221962 T^{10} - 3013751093 T^{11} + 56112840139 T^{12} - 69208980055 T^{13} + 1726770502432 T^{14} - 2145478381705 T^{15} + 53924439373579 T^{16} - 89782658811563 T^{17} + 1727189256568202 T^{18} - 4167815999288648 T^{19} + 52246710685352809 T^{20} - 146604816056439040 T^{21} + 1246487457854459885 T^{22} - 3248636374881645770 T^{23} + 20084991172464528505 T^{24} - 40526520649765705445 T^{25} + \)\(18\!\cdots\!79\)\( T^{26} - \)\(21\!\cdots\!19\)\( T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \)
$37$ \( 1 - 18 T + 432 T^{2} - 5905 T^{3} + 85438 T^{4} - 946870 T^{5} + 10472125 T^{6} - 98228108 T^{7} + 903726227 T^{8} - 7378741926 T^{9} + 58846420685 T^{10} - 425956868704 T^{11} + 3011460506632 T^{12} - 19535248148689 T^{13} + 123848656581976 T^{14} - 722804181501493 T^{15} + 4122689433579208 T^{16} - 21575993270463712 T^{17} + 110287666637420285 T^{18} - 511671162830641182 T^{19} + 2318714247119828843 T^{20} - 9324978679663054364 T^{21} + 36783123901392452125 T^{22} - \)\(12\!\cdots\!90\)\( T^{23} + \)\(41\!\cdots\!62\)\( T^{24} - \)\(10\!\cdots\!65\)\( T^{25} + \)\(28\!\cdots\!92\)\( T^{26} - \)\(43\!\cdots\!46\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \)
$41$ \( 1 + 25 T + 555 T^{2} + 8481 T^{3} + 115073 T^{4} + 1302241 T^{5} + 13419824 T^{6} + 122695156 T^{7} + 1044133400 T^{8} + 8136457371 T^{9} + 60228332496 T^{10} + 418792302802 T^{11} + 2832480690002 T^{12} + 18464723912586 T^{13} + 119639509456198 T^{14} + 757053680416026 T^{15} + 4761400039893362 T^{16} + 28863584301416642 T^{17} + 170190873062229456 T^{18} + 942659040602507571 T^{19} + 4959742491509749400 T^{20} + 23895406015496020436 T^{21} + \)\(10\!\cdots\!04\)\( T^{22} + \)\(42\!\cdots\!01\)\( T^{23} + \)\(15\!\cdots\!73\)\( T^{24} + \)\(46\!\cdots\!21\)\( T^{25} + \)\(12\!\cdots\!55\)\( T^{26} + \)\(23\!\cdots\!25\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} \)
$43$ \( 1 - 25 T + 532 T^{2} - 8239 T^{3} + 114958 T^{4} - 1374000 T^{5} + 15215678 T^{6} - 152200843 T^{7} + 1433865593 T^{8} - 12506182861 T^{9} + 103731364030 T^{10} - 807115680118 T^{11} + 6006992048132 T^{12} - 42217165145506 T^{13} + 284592836335592 T^{14} - 1815338101256758 T^{15} + 11106928296996068 T^{16} - 64171346379141826 T^{17} + 354636891077128030 T^{18} - 1838514470268895423 T^{19} + 9063984976822673057 T^{20} - 41371021753574563201 T^{21} + \)\(17\!\cdots\!78\)\( T^{22} - \)\(69\!\cdots\!00\)\( T^{23} + \)\(24\!\cdots\!42\)\( T^{24} - \)\(76\!\cdots\!73\)\( T^{25} + \)\(21\!\cdots\!32\)\( T^{26} - \)\(42\!\cdots\!75\)\( T^{27} + \)\(73\!\cdots\!49\)\( T^{28} \)
$47$ \( 1 - 36 T + 781 T^{2} - 11882 T^{3} + 146937 T^{4} - 1545655 T^{5} + 14728490 T^{6} - 128324787 T^{7} + 1055895172 T^{8} - 8241399586 T^{9} + 63121300759 T^{10} - 471780048623 T^{11} + 3479881314414 T^{12} - 24801469326359 T^{13} + 173052355308364 T^{14} - 1165669058338873 T^{15} + 7687057823540526 T^{16} - 48981619988185729 T^{17} + 308011812008977879 T^{18} - 1890123845740967102 T^{19} + 11381721423839491588 T^{20} - 65012304022689816381 T^{21} + \)\(35\!\cdots\!90\)\( T^{22} - \)\(17\!\cdots\!85\)\( T^{23} + \)\(77\!\cdots\!13\)\( T^{24} - \)\(29\!\cdots\!46\)\( T^{25} + \)\(90\!\cdots\!21\)\( T^{26} - \)\(19\!\cdots\!72\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \)
$53$ \( 1 + 402 T^{2} + 227 T^{3} + 83634 T^{4} + 79457 T^{5} + 11882258 T^{6} + 14130355 T^{7} + 1278036905 T^{8} + 1675943848 T^{9} + 109466085477 T^{10} + 146445928242 T^{11} + 7669739305332 T^{12} + 9873375239069 T^{13} + 445518985784738 T^{14} + 523288887670657 T^{15} + 21544297708677588 T^{16} + 21802430458884234 T^{17} + 863740067600644437 T^{18} + 700872163754677064 T^{19} + 28326871498609465745 T^{20} + 16599085428351452135 T^{21} + \)\(73\!\cdots\!38\)\( T^{22} + \)\(26\!\cdots\!81\)\( T^{23} + \)\(14\!\cdots\!66\)\( T^{24} + \)\(21\!\cdots\!19\)\( T^{25} + \)\(19\!\cdots\!82\)\( T^{26} + \)\(13\!\cdots\!69\)\( T^{28} \)
$59$ \( 1 - 17 T + 389 T^{2} - 4740 T^{3} + 64287 T^{4} - 608577 T^{5} + 6371355 T^{6} - 49592532 T^{7} + 442577701 T^{8} - 2858040770 T^{9} + 22812298546 T^{10} - 115897267986 T^{11} + 906553113883 T^{12} - 3650539556082 T^{13} + 39630010779372 T^{14} - 215381833808838 T^{15} + 3155711389426723 T^{16} - 23802865001696694 T^{17} + 276424856721657106 T^{18} - 2043282794005670230 T^{19} + 18668163605786939341 T^{20} - \)\(12\!\cdots\!08\)\( T^{21} + \)\(93\!\cdots\!55\)\( T^{22} - \)\(52\!\cdots\!03\)\( T^{23} + \)\(32\!\cdots\!87\)\( T^{24} - \)\(14\!\cdots\!60\)\( T^{25} + \)\(69\!\cdots\!09\)\( T^{26} - \)\(17\!\cdots\!43\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \)
$61$ \( ( 1 - T )^{14} \)
$67$ \( 1 - 22 T + 681 T^{2} - 10157 T^{3} + 186696 T^{4} - 2114640 T^{5} + 29260327 T^{6} - 261335976 T^{7} + 2995953959 T^{8} - 21040316696 T^{9} + 216281497981 T^{10} - 1160965161854 T^{11} + 12304472925340 T^{12} - 52683459033475 T^{13} + 722943372345054 T^{14} - 3529791755242825 T^{15} + 55234778961851260 T^{16} - 349175364974694602 T^{17} + 4358314635876386701 T^{18} - 28407059830500886472 T^{19} + \)\(27\!\cdots\!71\)\( T^{20} - \)\(15\!\cdots\!48\)\( T^{21} + \)\(11\!\cdots\!07\)\( T^{22} - \)\(57\!\cdots\!80\)\( T^{23} + \)\(34\!\cdots\!04\)\( T^{24} - \)\(12\!\cdots\!31\)\( T^{25} + \)\(55\!\cdots\!41\)\( T^{26} - \)\(12\!\cdots\!14\)\( T^{27} + \)\(36\!\cdots\!29\)\( T^{28} \)
$71$ \( 1 + 13 T + 466 T^{2} + 4633 T^{3} + 99121 T^{4} + 892267 T^{5} + 14927077 T^{6} + 133506138 T^{7} + 1828465207 T^{8} + 15985056605 T^{9} + 184637787060 T^{10} + 1563978150040 T^{11} + 16068717568391 T^{12} + 130982142754738 T^{13} + 1226932266157194 T^{14} + 9299732135586398 T^{15} + 81002405262259031 T^{16} + 559764983658966440 T^{17} + 4691956545314647860 T^{18} + 28840708304137413355 T^{19} + \)\(23\!\cdots\!47\)\( T^{20} + \)\(12\!\cdots\!58\)\( T^{21} + \)\(96\!\cdots\!97\)\( T^{22} + \)\(40\!\cdots\!77\)\( T^{23} + \)\(32\!\cdots\!21\)\( T^{24} + \)\(10\!\cdots\!43\)\( T^{25} + \)\(76\!\cdots\!06\)\( T^{26} + \)\(15\!\cdots\!43\)\( T^{27} + \)\(82\!\cdots\!81\)\( T^{28} \)
$73$ \( 1 - 20 T + 815 T^{2} - 13122 T^{3} + 300368 T^{4} - 3986354 T^{5} + 66684005 T^{6} - 744919067 T^{7} + 10089838918 T^{8} - 96900011919 T^{9} + 1130454581090 T^{10} - 9593322829902 T^{11} + 101395062590991 T^{12} - 788574260711128 T^{13} + 7838904257943112 T^{14} - 57565921031912344 T^{15} + 540334288547391039 T^{16} - 3731965667319986334 T^{17} + 32102921633347862690 T^{18} - \)\(20\!\cdots\!67\)\( T^{19} + \)\(15\!\cdots\!02\)\( T^{20} - \)\(82\!\cdots\!99\)\( T^{21} + \)\(53\!\cdots\!05\)\( T^{22} - \)\(23\!\cdots\!02\)\( T^{23} + \)\(12\!\cdots\!32\)\( T^{24} - \)\(41\!\cdots\!94\)\( T^{25} + \)\(18\!\cdots\!15\)\( T^{26} - \)\(33\!\cdots\!60\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \)
$79$ \( 1 - 31 T + 1023 T^{2} - 21323 T^{3} + 433671 T^{4} - 7070910 T^{5} + 110900735 T^{6} - 1512216248 T^{7} + 19835673362 T^{8} - 234800713163 T^{9} + 2683673218946 T^{10} - 28224264814010 T^{11} + 287742543203814 T^{12} - 2726029592624301 T^{13} + 25095925876714032 T^{14} - 215356337817319779 T^{15} + 1795801212135003174 T^{16} - 13915663299634676390 T^{17} + \)\(10\!\cdots\!26\)\( T^{18} - \)\(72\!\cdots\!37\)\( T^{19} + \)\(48\!\cdots\!02\)\( T^{20} - \)\(29\!\cdots\!32\)\( T^{21} + \)\(16\!\cdots\!35\)\( T^{22} - \)\(84\!\cdots\!90\)\( T^{23} + \)\(41\!\cdots\!71\)\( T^{24} - \)\(15\!\cdots\!17\)\( T^{25} + \)\(60\!\cdots\!43\)\( T^{26} - \)\(14\!\cdots\!09\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \)
$83$ \( 1 - 32 T + 1197 T^{2} - 27366 T^{3} + 616486 T^{4} - 11110672 T^{5} + 190171157 T^{6} - 2848502518 T^{7} + 40231880855 T^{8} - 516914651030 T^{9} + 6266878895874 T^{10} - 70420077805961 T^{11} + 748268506372730 T^{12} - 7434734385528863 T^{13} + 69953545105194072 T^{14} - 617082953998895629 T^{15} + 5154821740401736970 T^{16} - 40265285028437022307 T^{17} + \)\(29\!\cdots\!54\)\( T^{18} - \)\(20\!\cdots\!90\)\( T^{19} + \)\(13\!\cdots\!95\)\( T^{20} - \)\(77\!\cdots\!86\)\( T^{21} + \)\(42\!\cdots\!37\)\( T^{22} - \)\(20\!\cdots\!16\)\( T^{23} + \)\(95\!\cdots\!14\)\( T^{24} - \)\(35\!\cdots\!22\)\( T^{25} + \)\(12\!\cdots\!17\)\( T^{26} - \)\(28\!\cdots\!16\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \)
$89$ \( 1 + 21 T + 607 T^{2} + 8920 T^{3} + 171262 T^{4} + 2101735 T^{5} + 32830164 T^{6} + 352108686 T^{7} + 4829265654 T^{8} + 46726972789 T^{9} + 586211633725 T^{10} + 5219427228739 T^{11} + 61415310081737 T^{12} + 512935261401252 T^{13} + 5745185914786624 T^{14} + 45651238264711428 T^{15} + 486470671157438777 T^{16} + 3679534394016904091 T^{17} + 36780231600177677725 T^{18} + \)\(26\!\cdots\!61\)\( T^{19} + \)\(24\!\cdots\!94\)\( T^{20} + \)\(15\!\cdots\!94\)\( T^{21} + \)\(12\!\cdots\!84\)\( T^{22} + \)\(73\!\cdots\!15\)\( T^{23} + \)\(53\!\cdots\!62\)\( T^{24} + \)\(24\!\cdots\!80\)\( T^{25} + \)\(14\!\cdots\!47\)\( T^{26} + \)\(46\!\cdots\!49\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \)
$97$ \( 1 - 37 T + 1265 T^{2} - 30962 T^{3} + 679791 T^{4} - 12841719 T^{5} + 221642161 T^{6} - 3474922608 T^{7} + 50579995695 T^{8} - 684170650429 T^{9} + 8690333561707 T^{10} - 103729058988553 T^{11} + 1171015994366347 T^{12} - 12490999256601712 T^{13} + 126454166386797242 T^{14} - 1211626927890366064 T^{15} + 11018089490992958923 T^{16} - 94670711454259632169 T^{17} + \)\(76\!\cdots\!67\)\( T^{18} - \)\(58\!\cdots\!53\)\( T^{19} + \)\(42\!\cdots\!55\)\( T^{20} - \)\(28\!\cdots\!04\)\( T^{21} + \)\(17\!\cdots\!21\)\( T^{22} - \)\(97\!\cdots\!23\)\( T^{23} + \)\(50\!\cdots\!59\)\( T^{24} - \)\(22\!\cdots\!86\)\( T^{25} + \)\(87\!\cdots\!65\)\( T^{26} - \)\(24\!\cdots\!49\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \)
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