Properties

Label 8001.2.a.s
Level 8001
Weight 2
Character orbit 8001.a
Self dual yes
Analytic conductor 63.888
Analytic rank 0
Dimension 16
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} - 298 x^{8} - 5422 x^{7} + 2075 x^{6} + 5385 x^{5} - 3163 x^{4} - 1882 x^{3} + 1614 x^{2} - 200 x - 20\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} - 298 x^{8} - 5422 x^{7} + 2075 x^{6} + 5385 x^{5} - 3163 x^{4} - 1882 x^{3} + 1614 x^{2} - 200 x - 20\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 3 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 7 \nu^{2} - \nu + 6 \)
\(\beta_{5}\)\(=\)\((\)\(16172 \nu^{15} - 233047 \nu^{14} + 545740 \nu^{13} + 3569234 \nu^{12} - 13969268 \nu^{11} - 15371453 \nu^{10} + 106160280 \nu^{9} - 7403894 \nu^{8} - 351255852 \nu^{7} + 189480617 \nu^{6} + 508824053 \nu^{5} - 399632645 \nu^{4} - 240231533 \nu^{3} + 249231930 \nu^{2} - 32996882 \nu - 1833530\)\()/739814\)
\(\beta_{6}\)\(=\)\((\)\(-55506 \nu^{15} + 171907 \nu^{14} + 1127781 \nu^{13} - 3471014 \nu^{12} - 8968081 \nu^{11} + 26857942 \nu^{10} + 35796817 \nu^{9} - 100836497 \nu^{8} - 75361440 \nu^{7} + 193857030 \nu^{6} + 74881691 \nu^{5} - 187228846 \nu^{4} - 20078726 \nu^{3} + 77060622 \nu^{2} - 7448006 \nu - 3035278\)\()/739814\)
\(\beta_{7}\)\(=\)\((\)\(-79649 \nu^{15} + 179031 \nu^{14} + 1937788 \nu^{13} - 4128253 \nu^{12} - 18731714 \nu^{11} + 37509563 \nu^{10} + 90669787 \nu^{9} - 170332316 \nu^{8} - 224913536 \nu^{7} + 404529319 \nu^{6} + 250701988 \nu^{5} - 476307526 \nu^{4} - 57970538 \nu^{3} + 219774482 \nu^{2} - 52297458 \nu - 1805692\)\()/739814\)
\(\beta_{8}\)\(=\)\((\)\(-49011 \nu^{15} + 202993 \nu^{14} + 777233 \nu^{13} - 3856011 \nu^{12} - 3798508 \nu^{11} + 27243628 \nu^{10} + 3013336 \nu^{9} - 88425697 \nu^{8} + 22495878 \nu^{7} + 131065711 \nu^{6} - 53417264 \nu^{5} - 71773574 \nu^{4} + 32187666 \nu^{3} + 1013322 \nu^{2} + 932298 \nu + 482636\)\()/369907\)
\(\beta_{9}\)\(=\)\((\)\(123393 \nu^{15} - 271059 \nu^{14} - 3028354 \nu^{13} + 6241792 \nu^{12} + 29665237 \nu^{11} - 56580819 \nu^{10} - 146738516 \nu^{9} + 256052832 \nu^{8} + 378583411 \nu^{7} - 605454280 \nu^{6} - 461426392 \nu^{5} + 709532970 \nu^{4} + 170356469 \nu^{3} - 325487958 \nu^{2} + 50732374 \nu + 3426510\)\()/739814\)
\(\beta_{10}\)\(=\)\((\)\(125049 \nu^{15} - 467479 \nu^{14} - 2288559 \nu^{13} + 9361224 \nu^{12} + 15315132 \nu^{11} - 71647986 \nu^{10} - 45326027 \nu^{9} + 264664955 \nu^{8} + 52335901 \nu^{7} - 493686395 \nu^{6} + 12589986 \nu^{5} + 444594657 \nu^{4} - 76420466 \nu^{3} - 158630492 \nu^{2} + 46767048 \nu + 4758460\)\()/739814\)
\(\beta_{11}\)\(=\)\((\)\(129018 \nu^{15} - 383386 \nu^{14} - 2688667 \nu^{13} + 7911672 \nu^{12} + 21971299 \nu^{11} - 62975173 \nu^{10} - 89947843 \nu^{9} + 244582021 \nu^{8} + 193199050 \nu^{7} - 484210085 \nu^{6} - 196229912 \nu^{5} + 461790873 \nu^{4} + 51084111 \nu^{3} - 166221504 \nu^{2} + 26121758 \nu + 725548\)\()/739814\)
\(\beta_{12}\)\(=\)\((\)\(-137239 \nu^{15} + 515922 \nu^{14} + 2401726 \nu^{13} - 10047088 \nu^{12} - 14807929 \nu^{11} + 73837244 \nu^{10} + 36812718 \nu^{9} - 256493020 \nu^{8} - 24053391 \nu^{7} + 434181743 \nu^{6} - 30567173 \nu^{5} - 332083099 \nu^{4} + 31220698 \nu^{3} + 88655714 \nu^{2} + 2540988 \nu - 4842758\)\()/739814\)
\(\beta_{13}\)\(=\)\((\)\(145145 \nu^{15} - 426142 \nu^{14} - 3134036 \nu^{13} + 9144531 \nu^{12} + 26680262 \nu^{11} - 76732984 \nu^{10} - 113889853 \nu^{9} + 320134980 \nu^{8} + 252475352 \nu^{7} - 697758738 \nu^{6} - 255070845 \nu^{5} + 754990315 \nu^{4} + 44791941 \nu^{3} - 321397232 \nu^{2} + 60680346 \nu + 5228678\)\()/739814\)
\(\beta_{14}\)\(=\)\((\)\(158942 \nu^{15} - 398038 \nu^{14} - 3767459 \nu^{13} + 8967247 \nu^{12} + 35638598 \nu^{11} - 79277367 \nu^{10} - 170414762 \nu^{9} + 348630065 \nu^{8} + 425502097 \nu^{7} - 797772424 \nu^{6} - 500952602 \nu^{5} + 901431563 \nu^{4} + 177265028 \nu^{3} - 399924308 \nu^{2} + 51868034 \nu + 7379728\)\()/739814\)
\(\beta_{15}\)\(=\)\((\)\(322032 \nu^{15} - 1082571 \nu^{14} - 6442221 \nu^{13} + 22367158 \nu^{12} + 49725923 \nu^{11} - 178585472 \nu^{10} - 187902897 \nu^{9} + 698572039 \nu^{8} + 358046232 \nu^{7} - 1405569986 \nu^{6} - 279881591 \nu^{5} + 1389436646 \nu^{4} - 43714386 \nu^{3} - 540456742 \nu^{2} + 124105592 \nu + 7468534\)\()/739814\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 7 \beta_{2} + \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(\beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{4} + 8 \beta_{3} + 9 \beta_{2} + 31 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(2 \beta_{15} - \beta_{14} + \beta_{12} - \beta_{11} - 4 \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} + 11 \beta_{4} + \beta_{3} + 45 \beta_{2} + 16 \beta_{1} + 86\)
\(\nu^{7}\)\(=\)\(15 \beta_{15} - 14 \beta_{14} - 13 \beta_{13} - 14 \beta_{12} - 17 \beta_{10} + 13 \beta_{9} + 14 \beta_{8} + \beta_{7} + 12 \beta_{6} + 15 \beta_{4} + 55 \beta_{3} + 69 \beta_{2} + 209 \beta_{1} + 23\)
\(\nu^{8}\)\(=\)\(33 \beta_{15} - 19 \beta_{14} - 3 \beta_{13} + 10 \beta_{12} - 14 \beta_{11} - 61 \beta_{10} + 34 \beta_{9} - 7 \beta_{8} + 18 \beta_{7} + \beta_{6} + 15 \beta_{5} + 99 \beta_{4} + 16 \beta_{3} + 292 \beta_{2} + 175 \beta_{1} + 520\)
\(\nu^{9}\)\(=\)\(162 \beta_{15} - 146 \beta_{14} - 125 \beta_{13} - 139 \beta_{12} + 2 \beta_{11} - 196 \beta_{10} + 127 \beta_{9} + 142 \beta_{8} + 19 \beta_{7} + 106 \beta_{6} + 163 \beta_{4} + 370 \beta_{3} + 510 \beta_{2} + 1466 \beta_{1} + 203\)
\(\nu^{10}\)\(=\)\(378 \beta_{15} - 239 \beta_{14} - 59 \beta_{13} + 62 \beta_{12} - 129 \beta_{11} - 657 \beta_{10} + 388 \beta_{9} - 3 \beta_{8} + 218 \beta_{7} + 21 \beta_{6} + 154 \beta_{5} + 837 \beta_{4} + 174 \beta_{3} + 1942 \beta_{2} + 1657 \beta_{1} + 3232\)
\(\nu^{11}\)\(=\)\(1538 \beta_{15} - 1365 \beta_{14} - 1076 \beta_{13} - 1217 \beta_{12} + 47 \beta_{11} - 1930 \beta_{10} + 1130 \beta_{9} + 1286 \beta_{8} + 259 \beta_{7} + 840 \beta_{6} - \beta_{5} + 1560 \beta_{4} + 2500 \beta_{3} + 3747 \beta_{2} + 10526 \beta_{1} + 1641\)
\(\nu^{12}\)\(=\)\(3733 \beta_{15} - 2530 \beta_{14} - 752 \beta_{13} + 223 \beta_{12} - 982 \beta_{11} - 6181 \beta_{10} + 3756 \beta_{9} + 537 \beta_{8} + 2236 \beta_{7} + 274 \beta_{6} + 1350 \beta_{5} + 6877 \beta_{4} + 1629 \beta_{3} + 13248 \beta_{2} + 14610 \beta_{1} + 20471\)
\(\nu^{13}\)\(=\)\(13671 \beta_{15} - 12084 \beta_{14} - 8781 \beta_{13} - 10051 \beta_{12} + 703 \beta_{11} - 17520 \beta_{10} + 9659 \beta_{9} + 11081 \beta_{8} + 3015 \beta_{7} + 6340 \beta_{6} - 11 \beta_{5} + 13979 \beta_{4} + 17072 \beta_{3} + 27638 \beta_{2} + 76824 \beta_{1} + 12767\)
\(\nu^{14}\)\(=\)\(34127 \beta_{15} - 24451 \beta_{14} - 7956 \beta_{13} - 756 \beta_{12} - 6564 \beta_{11} - 54356 \beta_{10} + 33417 \beta_{9} + 8925 \beta_{8} + 20997 \beta_{7} + 2898 \beta_{6} + 10885 \beta_{5} + 55614 \beta_{4} + 14172 \beta_{3} + 92494 \beta_{2} + 123744 \beta_{1} + 131697\)
\(\nu^{15}\)\(=\)\(116930 \beta_{15} - 103619 \beta_{14} - 69592 \beta_{13} - 80533 \beta_{12} + 8580 \beta_{11} - 151698 \beta_{10} + 80862 \beta_{9} + 93120 \beta_{8} + 31789 \beta_{7} + 46689 \beta_{6} - 12 \beta_{5} + 120476 \beta_{4} + 117968 \beta_{3} + 205325 \beta_{2} + 567801 \beta_{1} + 97481\)

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
2.80317
2.63870
2.45059
2.35655
1.50110
1.17466
0.825012
0.716430
0.284398
−0.0643619
−1.14773
−1.35902
−1.40936
−1.84433
−2.35145
−2.57436
−2.80317 0 5.85777 −1.45069 0 −1.00000 −10.8140 0 4.06653
1.2 −2.63870 0 4.96274 −0.422965 0 −1.00000 −7.81779 0 1.11608
1.3 −2.45059 0 4.00539 2.70231 0 −1.00000 −4.91439 0 −6.62224
1.4 −2.35655 0 3.55331 −2.88159 0 −1.00000 −3.66045 0 6.79060
1.5 −1.50110 0 0.253312 2.96917 0 −1.00000 2.62196 0 −4.45703
1.6 −1.17466 0 −0.620175 −3.75278 0 −1.00000 3.07781 0 4.40823
1.7 −0.825012 0 −1.31936 1.66882 0 −1.00000 2.73851 0 −1.37680
1.8 −0.716430 0 −1.48673 −0.617976 0 −1.00000 2.49800 0 0.442737
1.9 −0.284398 0 −1.91912 3.66581 0 −1.00000 1.11459 0 −1.04255
1.10 0.0643619 0 −1.99586 −3.34910 0 −1.00000 −0.257181 0 −0.215554
1.11 1.14773 0 −0.682718 −1.07917 0 −1.00000 −3.07903 0 −1.23860
1.12 1.35902 0 −0.153057 2.54404 0 −1.00000 −2.92605 0 3.45741
1.13 1.40936 0 −0.0136985 −2.06019 0 −1.00000 −2.83803 0 −2.90356
1.14 1.84433 0 1.40154 −2.16633 0 −1.00000 −1.10376 0 −3.99542
1.15 2.35145 0 3.52933 2.01529 0 −1.00000 3.59614 0 4.73886
1.16 2.57436 0 4.62732 −2.78465 0 −1.00000 6.76367 0 −7.16870
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

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 8001.2.a.s 16
3.b odd 2 1 2667.2.a.n 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.n 16 3.b odd 2 1
8001.2.a.s 16 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(127\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{16} + \cdots\)
\(T_{5}^{16} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 14 T^{2} + 37 T^{3} + 88 T^{4} + 190 T^{5} + 381 T^{6} + 712 T^{7} + 1274 T^{8} + 2174 T^{9} + 3615 T^{10} + 5835 T^{11} + 9161 T^{12} + 14072 T^{13} + 21098 T^{14} + 30924 T^{15} + 44348 T^{16} + 61848 T^{17} + 84392 T^{18} + 112576 T^{19} + 146576 T^{20} + 186720 T^{21} + 231360 T^{22} + 278272 T^{23} + 326144 T^{24} + 364544 T^{25} + 390144 T^{26} + 389120 T^{27} + 360448 T^{28} + 303104 T^{29} + 229376 T^{30} + 131072 T^{31} + 65536 T^{32} \)
$3$ 1
$5$ \( 1 + 5 T + 44 T^{2} + 165 T^{3} + 915 T^{4} + 2942 T^{5} + 12907 T^{6} + 37018 T^{7} + 138404 T^{8} + 361252 T^{9} + 1199295 T^{10} + 2886522 T^{11} + 8700809 T^{12} + 19438343 T^{13} + 53957730 T^{14} + 112352057 T^{15} + 289570702 T^{16} + 561760285 T^{17} + 1348943250 T^{18} + 2429792875 T^{19} + 5438005625 T^{20} + 9020381250 T^{21} + 18738984375 T^{22} + 28222812500 T^{23} + 54064062500 T^{24} + 72300781250 T^{25} + 126044921875 T^{26} + 143652343750 T^{27} + 223388671875 T^{28} + 201416015625 T^{29} + 268554687500 T^{30} + 152587890625 T^{31} + 152587890625 T^{32} \)
$7$ \( ( 1 + T )^{16} \)
$11$ \( 1 + T + 92 T^{2} + 96 T^{3} + 4181 T^{4} + 4100 T^{5} + 126197 T^{6} + 107277 T^{7} + 2867972 T^{8} + 2007877 T^{9} + 52697233 T^{10} + 30067300 T^{11} + 817852019 T^{12} + 392443888 T^{13} + 10985297878 T^{14} + 4676327073 T^{15} + 129047873478 T^{16} + 51439597803 T^{17} + 1329221043238 T^{18} + 522342814928 T^{19} + 11974171410179 T^{20} + 4842368732300 T^{21} + 93356362790713 T^{22} + 39127842445967 T^{23} + 614775268659332 T^{24} + 252953554447407 T^{25} + 3273225172372397 T^{26} + 1169777849505100 T^{27} + 13121769043070501 T^{28} + 3314180365817376 T^{29} + 34936984689658172 T^{30} + 4177248169415651 T^{31} + 45949729863572161 T^{32} \)
$13$ \( 1 - 20 T + 290 T^{2} - 3026 T^{3} + 26451 T^{4} - 193872 T^{5} + 1250917 T^{6} - 7096516 T^{7} + 36195640 T^{8} - 165179444 T^{9} + 682331085 T^{10} - 2528068092 T^{11} + 8470190389 T^{12} - 25438784618 T^{13} + 70775694820 T^{14} - 192799829316 T^{15} + 616167289950 T^{16} - 2506397781108 T^{17} + 11961092424580 T^{18} - 55889009805746 T^{19} + 241917107700229 T^{20} - 938653986082956 T^{21} + 3293481822057765 T^{22} - 10364765149884548 T^{23} + 29525895514256440 T^{24} - 75254999472484468 T^{25} + 172449531048275533 T^{26} - 347449719912741264 T^{27} + 616257649574744931 T^{28} - 916500072548157578 T^{29} + 1141839151852793810 T^{30} - 1023717860281815140 T^{31} + 665416609183179841 T^{32} \)
$17$ \( 1 + 3 T + 119 T^{2} + 197 T^{3} + 7224 T^{4} + 5540 T^{5} + 310835 T^{6} + 34062 T^{7} + 10484647 T^{8} - 4461310 T^{9} + 291065858 T^{10} - 241198952 T^{11} + 6894239766 T^{12} - 7415074569 T^{13} + 141937841380 T^{14} - 165486582283 T^{15} + 2567277444404 T^{16} - 2813271898811 T^{17} + 41020036158820 T^{18} - 36430261357497 T^{19} + 575813799496086 T^{20} - 342468020389864 T^{21} + 7025622231019202 T^{22} - 1830648025241630 T^{23} + 73138354326508327 T^{24} + 4039340249240814 T^{25} + 626641464046064915 T^{26} + 189866305544286820 T^{27} + 4208863041747793464 T^{28} + 1951201872482469589 T^{29} + 20036961360568710551 T^{30} + 8587269154529447379 T^{31} + 48661191875666868481 T^{32} \)
$19$ \( 1 - 13 T + 224 T^{2} - 2227 T^{3} + 23556 T^{4} - 193353 T^{5} + 1589551 T^{6} - 11232515 T^{7} + 77919882 T^{8} - 486334465 T^{9} + 2959429581 T^{10} - 16577586969 T^{11} + 90351387300 T^{12} - 458691179259 T^{13} + 2264953054380 T^{14} - 10477514679859 T^{15} + 47144548718330 T^{16} - 199072778917321 T^{17} + 817648052631180 T^{18} - 3146162798537481 T^{19} + 11774683144323300 T^{20} - 41047746516353931 T^{21} + 139228971895605861 T^{22} - 434720633965184635 T^{23} + 1323357228094281162 T^{24} - 3624594405618084185 T^{25} + 9745642501153837351 T^{26} - 22523741028747338307 T^{27} + 52136846233522488516 T^{28} - 93651994170446470393 T^{29} + \)\(17\!\cdots\!04\)\( T^{30} - \)\(19\!\cdots\!87\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( 1 + 5 T + 182 T^{2} + 756 T^{3} + 15727 T^{4} + 54038 T^{5} + 854995 T^{6} + 2305237 T^{7} + 32132604 T^{8} + 57324585 T^{9} + 857073607 T^{10} + 371762126 T^{11} + 15989916385 T^{12} - 32570990844 T^{13} + 206622863040 T^{14} - 1527911913935 T^{15} + 2891712727590 T^{16} - 35141974020505 T^{17} + 109303494548160 T^{18} - 396291245598948 T^{19} + 4474634191094785 T^{20} + 2392788557345218 T^{21} + 126877653350681623 T^{22} + 195180205746714495 T^{23} + 2516335878884201724 T^{24} + 4152083757852981731 T^{25} + 35419459955113826755 T^{26} + 51487933698152787226 T^{27} + \)\(34\!\cdots\!67\)\( T^{28} + \)\(38\!\cdots\!48\)\( T^{29} + \)\(21\!\cdots\!38\)\( T^{30} + \)\(13\!\cdots\!35\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( 1 + 22 T + 522 T^{2} + 7766 T^{3} + 112692 T^{4} + 1304358 T^{5} + 14474109 T^{6} + 139071516 T^{7} + 1278513361 T^{8} + 10568056376 T^{9} + 83695188240 T^{10} + 607919137474 T^{11} + 4238256585198 T^{12} + 27397319711554 T^{13} + 170225839422745 T^{14} + 986146653004046 T^{15} + 5495004250987864 T^{16} + 28598252937117334 T^{17} + 143159930954528545 T^{18} + 668193230445090506 T^{19} + 2997638355835426638 T^{20} + 12469120008680697626 T^{21} + 49783849820636945040 T^{22} + \)\(18\!\cdots\!84\)\( T^{23} + \)\(63\!\cdots\!21\)\( T^{24} + \)\(20\!\cdots\!04\)\( T^{25} + \)\(60\!\cdots\!09\)\( T^{26} + \)\(15\!\cdots\!82\)\( T^{27} + \)\(39\!\cdots\!72\)\( T^{28} + \)\(79\!\cdots\!74\)\( T^{29} + \)\(15\!\cdots\!82\)\( T^{30} + \)\(18\!\cdots\!78\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( 1 - 26 T + 544 T^{2} - 8205 T^{3} + 108390 T^{4} - 1223221 T^{5} + 12591467 T^{6} - 117150234 T^{7} + 1016820642 T^{8} - 8192223962 T^{9} + 62402913981 T^{10} - 447676790195 T^{11} + 3061424282902 T^{12} - 19881426212383 T^{13} + 123673849035384 T^{14} - 733836777735502 T^{15} + 4180499851186114 T^{16} - 22748940109800562 T^{17} + 118850568923004024 T^{18} - 592287568293101953 T^{19} + 2827289615169937942 T^{20} - 12816606425687974445 T^{21} + 55382815863263864061 T^{22} - \)\(22\!\cdots\!82\)\( T^{23} + \)\(86\!\cdots\!22\)\( T^{24} - \)\(30\!\cdots\!14\)\( T^{25} + \)\(10\!\cdots\!67\)\( T^{26} - \)\(31\!\cdots\!51\)\( T^{27} + \)\(85\!\cdots\!90\)\( T^{28} - \)\(20\!\cdots\!55\)\( T^{29} + \)\(41\!\cdots\!24\)\( T^{30} - \)\(61\!\cdots\!26\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( 1 - 30 T + 806 T^{2} - 14673 T^{3} + 242625 T^{4} - 3312475 T^{5} + 41848747 T^{6} - 466357225 T^{7} + 4870920594 T^{8} - 46271902891 T^{9} + 415447002148 T^{10} - 3448551643407 T^{11} + 27200630462421 T^{12} - 200172652127761 T^{13} + 1404240604044519 T^{14} - 9234259977455106 T^{15} + 57972877864016510 T^{16} - 341667619165838922 T^{17} + 1922405386936946511 T^{18} - 10139345348227477933 T^{19} + 50978360788081403781 T^{20} - \)\(23\!\cdots\!99\)\( T^{21} + \)\(10\!\cdots\!32\)\( T^{22} - \)\(43\!\cdots\!03\)\( T^{23} + \)\(17\!\cdots\!74\)\( T^{24} - \)\(60\!\cdots\!25\)\( T^{25} + \)\(20\!\cdots\!03\)\( T^{26} - \)\(58\!\cdots\!75\)\( T^{27} + \)\(15\!\cdots\!25\)\( T^{28} - \)\(35\!\cdots\!81\)\( T^{29} + \)\(72\!\cdots\!34\)\( T^{30} - \)\(10\!\cdots\!90\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( 1 + T + 381 T^{2} + 753 T^{3} + 71193 T^{4} + 206003 T^{5} + 8769091 T^{6} + 32248993 T^{7} + 807996980 T^{8} + 3423817656 T^{9} + 59693127474 T^{10} + 269410101834 T^{11} + 3672268203653 T^{12} + 16586234784178 T^{13} + 191391546657934 T^{14} + 826335588463734 T^{15} + 8497746395094842 T^{16} + 33879759127013094 T^{17} + 321729189931987054 T^{18} + 1143139887560331938 T^{19} + 10376952271422704933 T^{20} + 31212830909510372634 T^{21} + \)\(28\!\cdots\!34\)\( T^{22} + \)\(66\!\cdots\!36\)\( T^{23} + \)\(64\!\cdots\!80\)\( T^{24} + \)\(10\!\cdots\!73\)\( T^{25} + \)\(11\!\cdots\!91\)\( T^{26} + \)\(11\!\cdots\!23\)\( T^{27} + \)\(16\!\cdots\!33\)\( T^{28} + \)\(69\!\cdots\!13\)\( T^{29} + \)\(14\!\cdots\!41\)\( T^{30} + \)\(15\!\cdots\!01\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( 1 - 31 T + 852 T^{2} - 15670 T^{3} + 264821 T^{4} - 3665962 T^{5} + 47706251 T^{6} - 544725555 T^{7} + 5919219884 T^{8} - 58255247771 T^{9} + 549520016843 T^{10} - 4775114475230 T^{11} + 39944199949147 T^{12} - 310729232709570 T^{13} + 2332751593885182 T^{14} - 16365338610668583 T^{15} + 110929007559371878 T^{16} - 703709560258749069 T^{17} + 4313257697093701518 T^{18} - 24705149105039781990 T^{19} + \)\(13\!\cdots\!47\)\( T^{20} - \)\(70\!\cdots\!90\)\( T^{21} + \)\(34\!\cdots\!07\)\( T^{22} - \)\(15\!\cdots\!97\)\( T^{23} + \)\(69\!\cdots\!84\)\( T^{24} - \)\(27\!\cdots\!65\)\( T^{25} + \)\(10\!\cdots\!99\)\( T^{26} - \)\(34\!\cdots\!34\)\( T^{27} + \)\(10\!\cdots\!21\)\( T^{28} - \)\(26\!\cdots\!10\)\( T^{29} + \)\(62\!\cdots\!48\)\( T^{30} - \)\(98\!\cdots\!17\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 - T + 360 T^{2} - 1350 T^{3} + 63404 T^{4} - 418163 T^{5} + 7649067 T^{6} - 68548326 T^{7} + 747449622 T^{8} - 7437562764 T^{9} + 63784570875 T^{10} - 597692191691 T^{11} + 4745753222012 T^{12} - 38339036949818 T^{13} + 298426805613138 T^{14} - 2072618878066403 T^{15} + 15494086707737218 T^{16} - 97413087269120941 T^{17} + 659224813599421842 T^{18} - 3980473833240954214 T^{19} + 23157761828140738172 T^{20} - \)\(13\!\cdots\!37\)\( T^{21} + \)\(68\!\cdots\!75\)\( T^{22} - \)\(37\!\cdots\!32\)\( T^{23} + \)\(17\!\cdots\!42\)\( T^{24} - \)\(76\!\cdots\!42\)\( T^{25} + \)\(40\!\cdots\!83\)\( T^{26} - \)\(10\!\cdots\!89\)\( T^{27} + \)\(73\!\cdots\!64\)\( T^{28} - \)\(73\!\cdots\!50\)\( T^{29} + \)\(92\!\cdots\!40\)\( T^{30} - \)\(12\!\cdots\!43\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 + 24 T + 623 T^{2} + 9286 T^{3} + 145196 T^{4} + 1667275 T^{5} + 20422321 T^{6} + 202557693 T^{7} + 2163294243 T^{8} + 19651949700 T^{9} + 190486009842 T^{10} + 1611744454580 T^{11} + 14312697141274 T^{12} + 112880703254671 T^{13} + 924871830380006 T^{14} + 6823886144051675 T^{15} + 52193981804810860 T^{16} + 361665965634738775 T^{17} + 2597964971537436854 T^{18} + 16805340458445654467 T^{19} + \)\(11\!\cdots\!94\)\( T^{20} + \)\(67\!\cdots\!40\)\( T^{21} + \)\(42\!\cdots\!18\)\( T^{22} + \)\(23\!\cdots\!00\)\( T^{23} + \)\(13\!\cdots\!23\)\( T^{24} + \)\(66\!\cdots\!69\)\( T^{25} + \)\(35\!\cdots\!29\)\( T^{26} + \)\(15\!\cdots\!75\)\( T^{27} + \)\(71\!\cdots\!36\)\( T^{28} + \)\(24\!\cdots\!78\)\( T^{29} + \)\(85\!\cdots\!87\)\( T^{30} + \)\(17\!\cdots\!68\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 - 17 T + 846 T^{2} - 11802 T^{3} + 329143 T^{4} - 3917144 T^{5} + 79697417 T^{6} - 829139959 T^{7} + 13603939474 T^{8} - 125686534583 T^{9} + 1749118455509 T^{10} - 14494136574176 T^{11} + 176087374935777 T^{12} - 1316023362978522 T^{13} + 14200806947115068 T^{14} - 95904511241320065 T^{15} + 928657148255566346 T^{16} - 5658366163237883835 T^{17} + 49433008982907551708 T^{18} - \)\(27\!\cdots\!38\)\( T^{19} + \)\(21\!\cdots\!97\)\( T^{20} - \)\(10\!\cdots\!24\)\( T^{21} + \)\(73\!\cdots\!69\)\( T^{22} - \)\(31\!\cdots\!77\)\( T^{23} + \)\(19\!\cdots\!54\)\( T^{24} - \)\(71\!\cdots\!01\)\( T^{25} + \)\(40\!\cdots\!17\)\( T^{26} - \)\(11\!\cdots\!96\)\( T^{27} + \)\(58\!\cdots\!83\)\( T^{28} - \)\(12\!\cdots\!58\)\( T^{29} + \)\(52\!\cdots\!06\)\( T^{30} - \)\(62\!\cdots\!83\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 - 32 T + 1206 T^{2} - 25791 T^{3} + 574786 T^{4} - 9432859 T^{5} + 157027093 T^{6} - 2115073800 T^{7} + 28728344596 T^{8} - 330500301176 T^{9} + 3837456872265 T^{10} - 38710131992009 T^{11} + 395349964688070 T^{12} - 3560087269732381 T^{13} + 32564374072402884 T^{14} - 264721387143865088 T^{15} + 2190402127678037446 T^{16} - 16148004615775770368 T^{17} + \)\(12\!\cdots\!64\)\( T^{18} - \)\(80\!\cdots\!61\)\( T^{19} + \)\(54\!\cdots\!70\)\( T^{20} - \)\(32\!\cdots\!09\)\( T^{21} + \)\(19\!\cdots\!65\)\( T^{22} - \)\(10\!\cdots\!96\)\( T^{23} + \)\(55\!\cdots\!76\)\( T^{24} - \)\(24\!\cdots\!00\)\( T^{25} + \)\(11\!\cdots\!93\)\( T^{26} - \)\(41\!\cdots\!99\)\( T^{27} + \)\(15\!\cdots\!06\)\( T^{28} - \)\(41\!\cdots\!71\)\( T^{29} + \)\(11\!\cdots\!46\)\( T^{30} - \)\(19\!\cdots\!32\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( 1 - 16 T + 628 T^{2} - 8280 T^{3} + 190794 T^{4} - 2210435 T^{5} + 38583413 T^{6} - 404285065 T^{7} + 5872118516 T^{8} - 56444576871 T^{9} + 715144787325 T^{10} - 6351231831685 T^{11} + 72089223934430 T^{12} - 593244848334966 T^{13} + 6130469576154946 T^{14} - 46757932432905050 T^{15} + 444257383187169606 T^{16} - 3132781473004638350 T^{17} + 27519677927359552594 T^{18} - \)\(17\!\cdots\!58\)\( T^{19} + \)\(14\!\cdots\!30\)\( T^{20} - \)\(85\!\cdots\!95\)\( T^{21} + \)\(64\!\cdots\!25\)\( T^{22} - \)\(34\!\cdots\!33\)\( T^{23} + \)\(23\!\cdots\!56\)\( T^{24} - \)\(10\!\cdots\!55\)\( T^{25} + \)\(70\!\cdots\!37\)\( T^{26} - \)\(26\!\cdots\!05\)\( T^{27} + \)\(15\!\cdots\!34\)\( T^{28} - \)\(45\!\cdots\!60\)\( T^{29} + \)\(23\!\cdots\!12\)\( T^{30} - \)\(39\!\cdots\!88\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( 1 - 10 T + 733 T^{2} - 8415 T^{3} + 271870 T^{4} - 3298766 T^{5} + 67832304 T^{6} - 816827387 T^{7} + 12630578324 T^{8} - 144866921557 T^{9} + 1840550492694 T^{10} - 19647637055160 T^{11} + 215273356920058 T^{12} - 2113451975430609 T^{13} + 20509996777358637 T^{14} - 183899987372631804 T^{15} + 1604770401332365030 T^{16} - 13056899103456858084 T^{17} + \)\(10\!\cdots\!17\)\( T^{18} - \)\(75\!\cdots\!99\)\( T^{19} + \)\(54\!\cdots\!98\)\( T^{20} - \)\(35\!\cdots\!60\)\( T^{21} + \)\(23\!\cdots\!74\)\( T^{22} - \)\(13\!\cdots\!87\)\( T^{23} + \)\(81\!\cdots\!64\)\( T^{24} - \)\(37\!\cdots\!97\)\( T^{25} + \)\(22\!\cdots\!04\)\( T^{26} - \)\(76\!\cdots\!86\)\( T^{27} + \)\(44\!\cdots\!70\)\( T^{28} - \)\(98\!\cdots\!65\)\( T^{29} + \)\(60\!\cdots\!73\)\( T^{30} - \)\(58\!\cdots\!10\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \)
$73$ \( 1 - 23 T + 844 T^{2} - 14258 T^{3} + 313221 T^{4} - 4311528 T^{5} + 72723743 T^{6} - 863577203 T^{7} + 12288571822 T^{8} - 130212669665 T^{9} + 1636466187575 T^{10} - 15774708137328 T^{11} + 179347554825123 T^{12} - 1589309353137782 T^{13} + 16565296817973582 T^{14} - 135691770432833349 T^{15} + 1306081775474694290 T^{16} - 9905499241596834477 T^{17} + 88276466742981218478 T^{18} - \)\(61\!\cdots\!94\)\( T^{19} + \)\(50\!\cdots\!43\)\( T^{20} - \)\(32\!\cdots\!04\)\( T^{21} + \)\(24\!\cdots\!75\)\( T^{22} - \)\(14\!\cdots\!05\)\( T^{23} + \)\(99\!\cdots\!82\)\( T^{24} - \)\(50\!\cdots\!39\)\( T^{25} + \)\(31\!\cdots\!07\)\( T^{26} - \)\(13\!\cdots\!56\)\( T^{27} + \)\(71\!\cdots\!41\)\( T^{28} - \)\(23\!\cdots\!14\)\( T^{29} + \)\(10\!\cdots\!96\)\( T^{30} - \)\(20\!\cdots\!11\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( 1 - 48 T + 1790 T^{2} - 48298 T^{3} + 1116249 T^{4} - 21951489 T^{5} + 387171727 T^{6} - 6133408839 T^{7} + 89467750466 T^{8} - 1204253173348 T^{9} + 15175235165324 T^{10} - 179254222560342 T^{11} + 2002963195844289 T^{12} - 21170098555741523 T^{13} + 212885951986436599 T^{14} - 2034025008309474401 T^{15} + 18535129089765807910 T^{16} - \)\(16\!\cdots\!79\)\( T^{17} + \)\(13\!\cdots\!59\)\( T^{18} - \)\(10\!\cdots\!97\)\( T^{19} + \)\(78\!\cdots\!09\)\( T^{20} - \)\(55\!\cdots\!58\)\( T^{21} + \)\(36\!\cdots\!04\)\( T^{22} - \)\(23\!\cdots\!32\)\( T^{23} + \)\(13\!\cdots\!26\)\( T^{24} - \)\(73\!\cdots\!41\)\( T^{25} + \)\(36\!\cdots\!27\)\( T^{26} - \)\(16\!\cdots\!31\)\( T^{27} + \)\(65\!\cdots\!09\)\( T^{28} - \)\(22\!\cdots\!22\)\( T^{29} + \)\(66\!\cdots\!90\)\( T^{30} - \)\(13\!\cdots\!52\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( 1 + 9 T + 922 T^{2} + 8751 T^{3} + 425274 T^{4} + 4060279 T^{5} + 129458581 T^{6} + 1203835783 T^{7} + 28912374714 T^{8} + 256282046515 T^{9} + 4995921340093 T^{10} + 41580737482759 T^{11} + 688346845936766 T^{12} + 5315373706784247 T^{13} + 76961879728256332 T^{14} + 545261754693266101 T^{15} + 7047910466107914458 T^{16} + 45256725639541086383 T^{17} + \)\(53\!\cdots\!48\)\( T^{18} + \)\(30\!\cdots\!89\)\( T^{19} + \)\(32\!\cdots\!86\)\( T^{20} + \)\(16\!\cdots\!37\)\( T^{21} + \)\(16\!\cdots\!17\)\( T^{22} + \)\(69\!\cdots\!05\)\( T^{23} + \)\(65\!\cdots\!74\)\( T^{24} + \)\(22\!\cdots\!49\)\( T^{25} + \)\(20\!\cdots\!69\)\( T^{26} + \)\(52\!\cdots\!93\)\( T^{27} + \)\(45\!\cdots\!14\)\( T^{28} + \)\(77\!\cdots\!13\)\( T^{29} + \)\(67\!\cdots\!38\)\( T^{30} + \)\(55\!\cdots\!63\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( 1 + 17 T + 1127 T^{2} + 15983 T^{3} + 586311 T^{4} + 7166681 T^{5} + 191074546 T^{6} + 2062455143 T^{7} + 44443133398 T^{8} + 431204254609 T^{9} + 7927842129958 T^{10} + 69953463871825 T^{11} + 1131788728858665 T^{12} + 9138323523354515 T^{13} + 132584056451488545 T^{14} + 980924964406358011 T^{15} + 12914003614443073682 T^{16} + 87302321832165862979 T^{17} + \)\(10\!\cdots\!45\)\( T^{18} + \)\(64\!\cdots\!35\)\( T^{19} + \)\(71\!\cdots\!65\)\( T^{20} + \)\(39\!\cdots\!25\)\( T^{21} + \)\(39\!\cdots\!38\)\( T^{22} + \)\(19\!\cdots\!61\)\( T^{23} + \)\(17\!\cdots\!38\)\( T^{24} + \)\(72\!\cdots\!87\)\( T^{25} + \)\(59\!\cdots\!46\)\( T^{26} + \)\(19\!\cdots\!09\)\( T^{27} + \)\(14\!\cdots\!31\)\( T^{28} + \)\(35\!\cdots\!27\)\( T^{29} + \)\(22\!\cdots\!07\)\( T^{30} + \)\(29\!\cdots\!33\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \)
$97$ \( 1 - 17 T + 913 T^{2} - 13489 T^{3} + 407538 T^{4} - 5428802 T^{5} + 120440919 T^{6} - 1477034756 T^{7} + 26668402651 T^{8} - 304251849558 T^{9} + 4714542700790 T^{10} - 50215256728296 T^{11} + 689165255502424 T^{12} - 6848160468346513 T^{13} + 84938546496995736 T^{14} - 784702342845576967 T^{15} + 8914605800421217636 T^{16} - 76116127256020965799 T^{17} + \)\(79\!\cdots\!24\)\( T^{18} - \)\(62\!\cdots\!49\)\( T^{19} + \)\(61\!\cdots\!44\)\( T^{20} - \)\(43\!\cdots\!72\)\( T^{21} + \)\(39\!\cdots\!10\)\( T^{22} - \)\(24\!\cdots\!54\)\( T^{23} + \)\(20\!\cdots\!11\)\( T^{24} - \)\(11\!\cdots\!52\)\( T^{25} + \)\(88\!\cdots\!31\)\( T^{26} - \)\(38\!\cdots\!06\)\( T^{27} + \)\(28\!\cdots\!58\)\( T^{28} - \)\(90\!\cdots\!53\)\( T^{29} + \)\(59\!\cdots\!97\)\( T^{30} - \)\(10\!\cdots\!81\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \)
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