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:

Atkin-Lehner signs

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

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 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

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

$p$ $F_p(T)$
$2$ \( -20 + 200 T + 1614 T^{2} + 1882 T^{3} - 3163 T^{4} - 5385 T^{5} + 2075 T^{6} + 5422 T^{7} - 298 T^{8} - 2648 T^{9} - 235 T^{10} + 668 T^{11} + 112 T^{12} - 83 T^{13} - 18 T^{14} + 4 T^{15} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( 46352 + 247912 T + 396800 T^{2} + 42868 T^{3} - 397916 T^{4} - 224398 T^{5} + 131760 T^{6} + 120392 T^{7} - 14196 T^{8} - 28542 T^{9} - 1293 T^{10} + 3467 T^{11} + 435 T^{12} - 210 T^{13} - 36 T^{14} + 5 T^{15} + T^{16} \)
$7$ \( ( 1 + T )^{16} \)
$11$ \( -4096 - 11264 T + 286720 T^{2} - 862080 T^{3} + 561600 T^{4} + 668304 T^{5} - 726544 T^{6} - 128752 T^{7} + 253096 T^{8} + 192 T^{9} - 37043 T^{10} + 1262 T^{11} + 2597 T^{12} - 69 T^{13} - 84 T^{14} + T^{15} + T^{16} \)
$13$ \( -143360 + 1585152 T + 2738176 T^{2} - 15951104 T^{3} + 17946368 T^{4} - 3490400 T^{5} - 5878320 T^{6} + 3538768 T^{7} + 21216 T^{8} - 529930 T^{9} + 124987 T^{10} + 13322 T^{11} - 8753 T^{12} + 874 T^{13} + 82 T^{14} - 20 T^{15} + T^{16} \)
$17$ \( -27287872 + 7666368 T + 276752368 T^{2} + 148829152 T^{3} - 204008404 T^{4} - 155860720 T^{5} + 2122349 T^{6} + 22540878 T^{7} + 2728431 T^{8} - 1354498 T^{9} - 244130 T^{10} + 40033 T^{11} + 8958 T^{12} - 568 T^{13} - 153 T^{14} + 3 T^{15} + T^{16} \)
$19$ \( -59975840 - 126001240 T + 387923792 T^{2} - 152497356 T^{3} - 148431844 T^{4} + 86231342 T^{5} + 22158002 T^{6} - 16272394 T^{7} - 1529644 T^{8} + 1442632 T^{9} + 30943 T^{10} - 65654 T^{11} + 1516 T^{12} + 1478 T^{13} - 80 T^{14} - 13 T^{15} + T^{16} \)
$23$ \( -92323840 + 74818752 T + 524148032 T^{2} + 330790768 T^{3} - 205430368 T^{4} - 218588028 T^{5} - 7829180 T^{6} + 32643952 T^{7} + 4413556 T^{8} - 2100942 T^{9} - 355035 T^{10} + 66044 T^{11} + 12139 T^{12} - 969 T^{13} - 186 T^{14} + 5 T^{15} + T^{16} \)
$29$ \( -265579040 + 310167920 T + 387045072 T^{2} - 292116496 T^{3} - 224099764 T^{4} + 86325472 T^{5} + 61134227 T^{6} - 8239912 T^{7} - 7900481 T^{8} - 43746 T^{9} + 475519 T^{10} + 41756 T^{11} - 11776 T^{12} - 1804 T^{13} + 58 T^{14} + 22 T^{15} + T^{16} \)
$31$ \( -2220064768 - 3714553984 T + 2884376320 T^{2} + 4693594224 T^{3} - 1651682632 T^{4} - 1441946074 T^{5} + 557645614 T^{6} + 96703040 T^{7} - 55543448 T^{8} + 448452 T^{9} + 2038323 T^{10} - 165346 T^{11} - 27762 T^{12} + 3885 T^{13} + 48 T^{14} - 26 T^{15} + T^{16} \)
$37$ \( 2281961720 - 2457379980 T - 5440093684 T^{2} + 2738674004 T^{3} + 1798707752 T^{4} - 1067836704 T^{5} - 97855071 T^{6} + 137744598 T^{7} - 13400566 T^{8} - 5969555 T^{9} + 1256269 T^{10} + 48938 T^{11} - 32507 T^{12} + 1977 T^{13} + 214 T^{14} - 30 T^{15} + T^{16} \)
$41$ \( 6198211456 + 5061204736 T - 4583186720 T^{2} - 3158258896 T^{3} + 1560932864 T^{4} + 684255130 T^{5} - 280575735 T^{6} - 63674447 T^{7} + 25776102 T^{8} + 2664910 T^{9} - 1202560 T^{10} - 44056 T^{11} + 27323 T^{12} + 138 T^{13} - 275 T^{14} + T^{15} + T^{16} \)
$43$ \( -270850329856 + 796337972672 T - 549726886976 T^{2} + 79090202464 T^{3} + 50367532336 T^{4} - 19026830772 T^{5} + 23557740 T^{6} + 969715232 T^{7} - 122134260 T^{8} - 16227304 T^{9} + 4373947 T^{10} - 65142 T^{11} - 55787 T^{12} + 4325 T^{13} + 164 T^{14} - 31 T^{15} + T^{16} \)
$47$ \( -80836384768 - 76373551264 T + 43441489056 T^{2} + 49679000768 T^{3} - 1307082184 T^{4} - 8456898534 T^{5} - 796150660 T^{6} + 583875536 T^{7} + 87168336 T^{8} - 17646480 T^{9} - 3423005 T^{10} + 207877 T^{11} + 56260 T^{12} - 645 T^{13} - 392 T^{14} - T^{15} + T^{16} \)
$53$ \( 3330453274336 + 266174831760 T - 2732186959424 T^{2} - 1440170393912 T^{3} - 71508188512 T^{4} + 112052595120 T^{5} + 24248021647 T^{6} - 1347958393 T^{7} - 906195881 T^{8} - 56567522 T^{9} + 10423500 T^{10} + 1336661 T^{11} - 24934 T^{12} - 9794 T^{13} - 225 T^{14} + 24 T^{15} + T^{16} \)
$59$ \( 2877741056 - 2104027648 T - 3575985408 T^{2} + 3941108224 T^{3} - 132775872 T^{4} - 1094722208 T^{5} + 348450348 T^{6} + 49269828 T^{7} - 37385154 T^{8} + 2853792 T^{9} + 1130067 T^{10} - 190940 T^{11} - 7629 T^{12} + 3243 T^{13} - 98 T^{14} - 17 T^{15} + T^{16} \)
$61$ \( 9236458496 - 33411940352 T + 38587323392 T^{2} - 15145636224 T^{3} - 4090670400 T^{4} + 5904678680 T^{5} - 2019899068 T^{6} + 144145184 T^{7} + 88914990 T^{8} - 26132826 T^{9} + 1925931 T^{10} + 302924 T^{11} - 68154 T^{12} + 3489 T^{13} + 230 T^{14} - 32 T^{15} + T^{16} \)
$67$ \( 5716113605888 - 9577606936128 T + 4994099126336 T^{2} - 263776173632 T^{3} - 518652263552 T^{4} + 110249099280 T^{5} + 15997004748 T^{6} - 5855481496 T^{7} - 94042410 T^{8} + 132182930 T^{9} - 3613455 T^{10} - 1462715 T^{11} + 68586 T^{12} + 7800 T^{13} - 444 T^{14} - 16 T^{15} + T^{16} \)
$71$ \( 4708249600 + 4130234560 T - 26016842560 T^{2} - 53854787232 T^{3} - 40221276768 T^{4} - 12289946288 T^{5} + 10996052 T^{6} + 897806512 T^{7} + 154931894 T^{8} - 13536866 T^{9} - 5266527 T^{10} - 68621 T^{11} + 67532 T^{12} + 2235 T^{13} - 403 T^{14} - 10 T^{15} + T^{16} \)
$73$ \( -30744464128 - 2070684395520 T + 1292312728000 T^{2} + 478025811712 T^{3} - 367590595440 T^{4} - 10172106464 T^{5} + 30522086668 T^{6} - 2443796756 T^{7} - 871940302 T^{8} + 120354976 T^{9} + 7729215 T^{10} - 1811716 T^{11} + 4869 T^{12} + 10927 T^{13} - 324 T^{14} - 23 T^{15} + T^{16} \)
$79$ \( 1732480000 + 45047124160 T - 74333528480 T^{2} - 1434337488 T^{3} + 40044706824 T^{4} - 8439587464 T^{5} - 6119774643 T^{6} + 2108505257 T^{7} + 167319982 T^{8} - 142258268 T^{9} + 15614977 T^{10} + 689437 T^{11} - 214427 T^{12} + 8582 T^{13} + 526 T^{14} - 48 T^{15} + T^{16} \)
$83$ \( 6896650240 - 61449116160 T + 126597341952 T^{2} - 86983008256 T^{3} + 1327247360 T^{4} + 18997302208 T^{5} - 4701066912 T^{6} - 620007744 T^{7} + 279325048 T^{8} + 194766 T^{9} - 6305681 T^{10} + 198040 T^{11} + 70366 T^{12} - 2454 T^{13} - 406 T^{14} + 9 T^{15} + T^{16} \)
$89$ \( 550072064 - 1213654784 T - 8870724832 T^{2} + 1430513264 T^{3} + 11233982796 T^{4} + 6251380018 T^{5} + 202721414 T^{6} - 816888652 T^{7} - 263197438 T^{8} - 20368336 T^{9} + 4121769 T^{10} + 793480 T^{11} + 5853 T^{12} - 6712 T^{13} - 297 T^{14} + 17 T^{15} + T^{16} \)
$97$ \( -24693460838972 + 10074310186226 T + 11369341994086 T^{2} - 5106204045108 T^{3} - 906256851328 T^{4} + 497966894564 T^{5} + 17097439689 T^{6} - 19028628880 T^{7} + 347594659 T^{8} + 332573188 T^{9} - 13732100 T^{10} - 2814943 T^{11} + 146220 T^{12} + 11246 T^{13} - 639 T^{14} - 17 T^{15} + T^{16} \)
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