Properties

Label 8001.2.a.q
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $15$
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: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + 726 x^{3} + 145 x^{2} - 83 x - 13\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 889)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{8} q^{5} - q^{7} + ( -\beta_{4} + \beta_{6} - \beta_{9} + \beta_{12} + \beta_{14} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{8} q^{5} - q^{7} + ( -\beta_{4} + \beta_{6} - \beta_{9} + \beta_{12} + \beta_{14} ) q^{8} + ( 1 - \beta_{1} - \beta_{3} - \beta_{5} + \beta_{7} ) q^{10} + ( \beta_{4} - \beta_{8} - \beta_{12} - \beta_{14} ) q^{11} + ( -\beta_{1} - \beta_{4} ) q^{13} -\beta_{1} q^{14} + ( 2 + 2 \beta_{2} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{16} + ( -1 - \beta_{6} - \beta_{12} ) q^{17} + ( 1 - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{14} ) q^{19} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{20} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{14} ) q^{22} + ( -1 + \beta_{4} - \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{23} + ( -\beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{8} - \beta_{10} - \beta_{14} ) q^{25} + ( -3 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{26} + ( -1 - \beta_{2} ) q^{28} + ( -2 - \beta_{3} - 2 \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{29} + ( 2 + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{31} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{14} ) q^{32} + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{34} + \beta_{8} q^{35} + ( -1 + \beta_{2} + \beta_{3} + 3 \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{37} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{38} + ( 3 - 2 \beta_{1} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{40} + ( -3 - \beta_{3} - 2 \beta_{4} + 2 \beta_{8} + \beta_{11} + \beta_{12} ) q^{41} + ( \beta_{1} + \beta_{4} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{14} ) q^{43} + ( 1 - 2 \beta_{1} - \beta_{2} + 4 \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 3 \beta_{14} ) q^{44} + ( -3 - \beta_{1} - 2 \beta_{2} - \beta_{6} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{46} + ( -3 - \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{11} + \beta_{12} + 2 \beta_{14} ) q^{47} + q^{49} + ( -3 - \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - \beta_{7} + 2 \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{50} + ( -3 + 2 \beta_{1} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{52} + ( 1 + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{11} - \beta_{14} ) q^{53} + ( 2 - \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{55} + ( \beta_{4} - \beta_{6} + \beta_{9} - \beta_{12} - \beta_{14} ) q^{56} + ( -\beta_{1} + \beta_{4} + \beta_{6} - 2 \beta_{9} - \beta_{12} - \beta_{13} ) q^{58} + ( -4 - 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{59} + ( 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{14} ) q^{61} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{11} + \beta_{13} ) q^{62} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{64} + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} ) q^{65} + ( -2 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{67} + ( -2 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} - \beta_{12} - 2 \beta_{14} ) q^{68} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} - \beta_{7} ) q^{70} + ( -6 - \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} + 3 \beta_{14} ) q^{71} + ( 1 - \beta_{1} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + 2 \beta_{7} - \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} ) q^{73} + ( 7 - 4 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{74} + ( -1 - 2 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - 3 \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{76} + ( -\beta_{4} + \beta_{8} + \beta_{12} + \beta_{14} ) q^{77} + ( 2 + 4 \beta_{1} + 4 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{9} - \beta_{11} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{79} + ( -2 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{80} + ( 2 \beta_{2} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{14} ) q^{82} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{9} - \beta_{10} - \beta_{12} - 3 \beta_{13} - \beta_{14} ) q^{83} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{85} + ( -2 - 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{13} + 2 \beta_{14} ) q^{86} + ( -6 - 3 \beta_{1} - 4 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{11} + 2 \beta_{13} + 2 \beta_{14} ) q^{88} + ( -4 - 4 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{89} + ( \beta_{1} + \beta_{4} ) q^{91} + ( -3 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - 3 \beta_{14} ) q^{92} + ( 4 - 5 \beta_{1} - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + \beta_{10} - 4 \beta_{11} - 4 \beta_{12} + 3 \beta_{13} - 2 \beta_{14} ) q^{94} + ( -2 - \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{95} + ( -1 - \beta_{1} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{9} - \beta_{10} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 14q^{4} - 7q^{5} - 15q^{7} + O(q^{10}) \) \( 15q + 14q^{4} - 7q^{5} - 15q^{7} + 10q^{10} - 14q^{11} + 6q^{13} + 20q^{16} - 10q^{17} + 13q^{19} - 8q^{20} - 11q^{22} - 15q^{23} - 22q^{26} - 14q^{28} - 16q^{29} + 22q^{31} - 15q^{34} + 7q^{35} - 14q^{37} + 6q^{38} + 22q^{40} - 19q^{41} - q^{43} - 25q^{44} - 28q^{46} - 49q^{47} + 15q^{49} - 24q^{50} - 17q^{52} + 28q^{53} + 39q^{55} - 10q^{58} - 43q^{59} + 27q^{61} - 14q^{62} + 18q^{64} + 8q^{65} + 3q^{67} - 13q^{68} - 10q^{70} - 55q^{71} - 3q^{73} + 12q^{74} - 20q^{76} + 14q^{77} + 18q^{79} - 29q^{80} + 14q^{82} - 17q^{83} + 7q^{85} - 4q^{86} - 114q^{88} - 36q^{89} - 6q^{91} - 45q^{92} - 15q^{94} - 59q^{95} - 2q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + 726 x^{3} + 145 x^{2} - 83 x - 13\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(5995 \nu^{14} - 50387 \nu^{13} - 70689 \nu^{12} + 1136059 \nu^{11} - 104425 \nu^{10} - 9736369 \nu^{9} + 4215784 \nu^{8} + 39113971 \nu^{7} - 17615205 \nu^{6} - 72393647 \nu^{5} + 21852674 \nu^{4} + 49208323 \nu^{3} - 1669103 \nu^{2} - 5838776 \nu - 1488901\)\()/624662\)
\(\beta_{4}\)\(=\)\((\)\(13615 \nu^{14} - 201176 \nu^{13} - 414259 \nu^{12} + 4188080 \nu^{11} + 4802080 \nu^{10} - 32567535 \nu^{9} - 26615464 \nu^{8} + 116731019 \nu^{7} + 72850775 \nu^{6} - 191755763 \nu^{5} - 91354475 \nu^{4} + 124577258 \nu^{3} + 45768926 \nu^{2} - 18038688 \nu - 2677270\)\()/312331\)
\(\beta_{5}\)\(=\)\((\)\(46461 \nu^{14} + 346281 \nu^{13} - 856573 \nu^{12} - 7272379 \nu^{11} + 5275825 \nu^{10} + 57143503 \nu^{9} - 10437030 \nu^{8} - 207601489 \nu^{7} - 10268173 \nu^{6} + 347242687 \nu^{5} + 51693498 \nu^{4} - 231453269 \nu^{3} - 46110397 \nu^{2} + 36069156 \nu + 3303757\)\()/624662\)
\(\beta_{6}\)\(=\)\((\)\(61025 \nu^{14} + 251381 \nu^{13} - 1185327 \nu^{12} - 5203625 \nu^{11} + 8156911 \nu^{10} + 39981133 \nu^{9} - 22898694 \nu^{8} - 140135849 \nu^{7} + 18310123 \nu^{6} + 220387379 \nu^{5} + 16351100 \nu^{4} - 130941925 \nu^{3} - 25148965 \nu^{2} + 12786392 \nu + 1161803\)\()/624662\)
\(\beta_{7}\)\(=\)\((\)\(89367 \nu^{14} + 251991 \nu^{13} - 1775425 \nu^{12} - 5236867 \nu^{11} + 12760035 \nu^{10} + 40507123 \nu^{9} - 39479230 \nu^{8} - 144087755 \nu^{7} + 45457397 \nu^{6} + 235076729 \nu^{5} - 555434 \nu^{4} - 155145979 \nu^{3} - 20742165 \nu^{2} + 24822268 \nu - 424393\)\()/624662\)
\(\beta_{8}\)\(=\)\((\)\(124199 \nu^{14} - 36911 \nu^{13} - 2688475 \nu^{12} + 848163 \nu^{11} + 22201561 \nu^{10} - 7588635 \nu^{9} - 87863826 \nu^{8} + 32388591 \nu^{7} + 172828249 \nu^{6} - 65392039 \nu^{5} - 162051064 \nu^{4} + 51124791 \nu^{3} + 63069507 \nu^{2} - 9028480 \nu - 4275743\)\()/624662\)
\(\beta_{9}\)\(=\)\((\)\(-71314 \nu^{14} + 21453 \nu^{13} + 1521763 \nu^{12} - 459426 \nu^{11} - 12246648 \nu^{10} + 3742105 \nu^{9} + 46094392 \nu^{8} - 14021902 \nu^{7} - 81489765 \nu^{6} + 23298689 \nu^{5} + 59802271 \nu^{4} - 12931376 \nu^{3} - 13620319 \nu^{2} + 2031255 \nu + 295618\)\()/312331\)
\(\beta_{10}\)\(=\)\((\)\(124624 \nu^{14} + 196305 \nu^{13} - 2558030 \nu^{12} - 4051663 \nu^{11} + 19403758 \nu^{10} + 31050350 \nu^{9} - 66378289 \nu^{8} - 109257358 \nu^{7} + 97930304 \nu^{6} + 176047189 \nu^{5} - 43222236 \nu^{4} - 114555237 \nu^{3} - 13192721 \nu^{2} + 15841558 \nu + 2534792\)\()/312331\)
\(\beta_{11}\)\(=\)\((\)\(165788 \nu^{14} + 202150 \nu^{13} - 3394240 \nu^{12} - 4150019 \nu^{11} + 25626527 \nu^{10} + 31492447 \nu^{9} - 86766924 \nu^{8} - 108999529 \nu^{7} + 124533409 \nu^{6} + 170598430 \nu^{5} - 48501611 \nu^{4} - 105475921 \nu^{3} - 20545614 \nu^{2} + 11029016 \nu + 2727176\)\()/312331\)
\(\beta_{12}\)\(=\)\((\)\(457793 \nu^{14} - 140549 \nu^{13} - 9783541 \nu^{12} + 3108413 \nu^{11} + 79045069 \nu^{10} - 26346187 \nu^{9} - 300616102 \nu^{8} + 104110845 \nu^{7} + 546854897 \nu^{6} - 187710855 \nu^{5} - 436619790 \nu^{4} + 122931523 \nu^{3} + 127323163 \nu^{2} - 22007624 \nu - 6116335\)\()/624662\)
\(\beta_{13}\)\(=\)\((\)\(243430 \nu^{14} + 83019 \nu^{13} - 5093930 \nu^{12} - 1622009 \nu^{11} + 39878673 \nu^{10} + 11264670 \nu^{9} - 144341909 \nu^{8} - 33752528 \nu^{7} + 241269487 \nu^{6} + 41532867 \nu^{5} - 163752385 \nu^{4} - 19959288 \nu^{3} + 29818639 \nu^{2} - 1373965 \nu - 283714\)\()/312331\)
\(\beta_{14}\)\(=\)\((\)\(-317108 \nu^{14} - 235139 \nu^{13} + 6591938 \nu^{12} + 4776260 \nu^{11} - 51045558 \nu^{10} - 35642903 \nu^{9} + 181236326 \nu^{8} + 120721619 \nu^{7} - 291221500 \nu^{6} - 184795336 \nu^{5} + 178582141 \nu^{4} + 115963414 \nu^{3} - 18938492 \nu^{2} - 12646141 \nu + 95614\)\()/312331\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{14} + \beta_{12} - \beta_{9} + \beta_{6} - \beta_{4} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + 8 \beta_{2} + 16\)
\(\nu^{5}\)\(=\)\(10 \beta_{14} + 9 \beta_{12} + \beta_{11} - 10 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 8 \beta_{6} + 3 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} + \beta_{2} + 22 \beta_{1} - 2\)
\(\nu^{6}\)\(=\)\(-\beta_{14} + 9 \beta_{13} - 11 \beta_{12} - 10 \beta_{11} + 9 \beta_{10} - 13 \beta_{9} + 2 \beta_{8} - 11 \beta_{7} - \beta_{4} + 57 \beta_{2} + \beta_{1} + 96\)
\(\nu^{7}\)\(=\)\(79 \beta_{14} - \beta_{13} + 68 \beta_{12} + 13 \beta_{11} - 2 \beta_{10} - 81 \beta_{9} + 26 \beta_{8} - 28 \beta_{7} + 56 \beta_{6} + 37 \beta_{5} - 70 \beta_{4} + 23 \beta_{3} + 14 \beta_{2} + 137 \beta_{1} - 24\)
\(\nu^{8}\)\(=\)\(-8 \beta_{14} + 66 \beta_{13} - 91 \beta_{12} - 77 \beta_{11} + 67 \beta_{10} - 128 \beta_{9} + 32 \beta_{8} - 97 \beta_{7} + \beta_{6} + 3 \beta_{5} - 18 \beta_{4} + 2 \beta_{3} + 398 \beta_{2} + 16 \beta_{1} + 601\)
\(\nu^{9}\)\(=\)\(580 \beta_{14} - 13 \beta_{13} + 486 \beta_{12} + 117 \beta_{11} - 25 \beta_{10} - 616 \beta_{9} + 245 \beta_{8} - 280 \beta_{7} + 380 \beta_{6} + 337 \beta_{5} - 521 \beta_{4} + 200 \beta_{3} + 150 \beta_{2} + 901 \beta_{1} - 208\)
\(\nu^{10}\)\(=\)\(-24 \beta_{14} + 461 \beta_{13} - 673 \beta_{12} - 545 \beta_{11} + 473 \beta_{10} - 1123 \beta_{9} + 352 \beta_{8} - 792 \beta_{7} + 21 \beta_{6} + 58 \beta_{5} - 222 \beta_{4} + 34 \beta_{3} + 2775 \beta_{2} + 180 \beta_{1} + 3848\)
\(\nu^{11}\)\(=\)\(4133 \beta_{14} - 121 \beta_{13} + 3390 \beta_{12} + 919 \beta_{11} - 212 \beta_{10} - 4567 \beta_{9} + 2051 \beta_{8} - 2454 \beta_{7} + 2562 \beta_{6} + 2738 \beta_{5} - 3801 \beta_{4} + 1579 \beta_{3} + 1430 \beta_{2} + 6099 \beta_{1} - 1566\)
\(\nu^{12}\)\(=\)\(235 \beta_{14} + 3186 \beta_{13} - 4698 \beta_{12} - 3722 \beta_{11} + 3269 \beta_{10} - 9269 \beta_{9} + 3314 \beta_{8} - 6231 \beta_{7} + 282 \beta_{6} + 738 \beta_{5} - 2302 \beta_{4} + 395 \beta_{3} + 19409 \beta_{2} + 1776 \beta_{1} + 25023\)
\(\nu^{13}\)\(=\)\(29066 \beta_{14} - 987 \beta_{13} + 23369 \beta_{12} + 6765 \beta_{11} - 1512 \beta_{10} - 33474 \beta_{9} + 16229 \beta_{8} - 20119 \beta_{7} + 17306 \beta_{6} + 21050 \beta_{5} - 27447 \beta_{4} + 11933 \beta_{3} + 12736 \beta_{2} + 42013 \beta_{1} - 10818\)
\(\nu^{14}\)\(=\)\(5213 \beta_{14} + 22030 \beta_{13} - 31683 \beta_{12} - 24988 \beta_{11} + 22382 \beta_{10} - 73792 \beta_{9} + 28733 \beta_{8} - 48003 \beta_{7} + 3120 \beta_{6} + 7809 \beta_{5} - 21594 \beta_{4} + 3926 \beta_{3} + 136319 \beta_{2} + 16365 \beta_{1} + 164749\)

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
−2.59170
−2.51564
−2.07483
−1.64397
−0.955258
−0.735585
−0.601235
−0.146355
0.371530
1.15799
1.22206
1.32572
2.16969
2.30266
2.71493
−2.59170 0 4.71692 −3.45466 0 −1.00000 −7.04143 0 8.95345
1.2 −2.51564 0 4.32847 1.19101 0 −1.00000 −5.85760 0 −2.99615
1.3 −2.07483 0 2.30492 −1.11193 0 −1.00000 −0.632654 0 2.30706
1.4 −1.64397 0 0.702628 −1.50234 0 −1.00000 2.13284 0 2.46980
1.5 −0.955258 0 −1.08748 2.56419 0 −1.00000 2.94934 0 −2.44946
1.6 −0.735585 0 −1.45891 −0.430466 0 −1.00000 2.54433 0 0.316644
1.7 −0.601235 0 −1.63852 −4.40463 0 −1.00000 2.18760 0 2.64822
1.8 −0.146355 0 −1.97858 2.93285 0 −1.00000 0.582285 0 −0.429238
1.9 0.371530 0 −1.86197 −0.962757 0 −1.00000 −1.43484 0 −0.357693
1.10 1.15799 0 −0.659052 −3.10556 0 −1.00000 −3.07916 0 −3.59621
1.11 1.22206 0 −0.506578 −1.90746 0 −1.00000 −3.06318 0 −2.33103
1.12 1.32572 0 −0.242472 1.07970 0 −1.00000 −2.97289 0 1.43138
1.13 2.16969 0 2.70756 2.09371 0 −1.00000 1.53520 0 4.54272
1.14 2.30266 0 3.30224 1.35662 0 −1.00000 2.99860 0 3.12384
1.15 2.71493 0 5.37083 −1.33828 0 −1.00000 9.15156 0 −3.63333
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.q 15
3.b odd 2 1 889.2.a.b 15
21.c even 2 1 6223.2.a.j 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
889.2.a.b 15 3.b odd 2 1
6223.2.a.j 15 21.c even 2 1
8001.2.a.q 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{15} - \cdots\)
\(T_{5}^{15} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -13 - 83 T + 145 T^{2} + 726 T^{3} - 185 T^{4} - 1625 T^{5} + 64 T^{6} + 1588 T^{7} - 7 T^{8} - 763 T^{9} + 186 T^{11} - 22 T^{13} + T^{15} \)
$3$ \( T^{15} \)
$5$ \( 2294 + 7628 T - 396 T^{2} - 19249 T^{3} - 8499 T^{4} + 18028 T^{5} + 11384 T^{6} - 7622 T^{7} - 6009 T^{8} + 1357 T^{9} + 1477 T^{10} - 42 T^{11} - 167 T^{12} - 13 T^{13} + 7 T^{14} + T^{15} \)
$7$ \( ( 1 + T )^{15} \)
$11$ \( 302112 + 1156336 T - 2325180 T^{2} - 2545671 T^{3} + 1930756 T^{4} + 2002646 T^{5} - 336541 T^{6} - 596721 T^{7} - 50769 T^{8} + 63092 T^{9} + 12380 T^{10} - 2345 T^{11} - 743 T^{12} + 2 T^{13} + 14 T^{14} + T^{15} \)
$13$ \( -296 - 3548 T - 6494 T^{2} + 31135 T^{3} + 43063 T^{4} - 70298 T^{5} - 54785 T^{6} + 45717 T^{7} + 25733 T^{8} - 11495 T^{9} - 4700 T^{10} + 1294 T^{11} + 318 T^{12} - 62 T^{13} - 6 T^{14} + T^{15} \)
$17$ \( -13032 + 9188 T + 201246 T^{2} - 133265 T^{3} - 371016 T^{4} + 156683 T^{5} + 263389 T^{6} - 52135 T^{7} - 83564 T^{8} + 3151 T^{9} + 11245 T^{10} + 471 T^{11} - 605 T^{12} - 48 T^{13} + 10 T^{14} + T^{15} \)
$19$ \( -2261282 + 7583414 T - 2237368 T^{2} - 10412999 T^{3} + 6214108 T^{4} + 3528781 T^{5} - 2526558 T^{6} - 489036 T^{7} + 406142 T^{8} + 27952 T^{9} - 29691 T^{10} - 87 T^{11} + 1008 T^{12} - 43 T^{13} - 13 T^{14} + T^{15} \)
$23$ \( -159348 + 938896 T - 347488 T^{2} - 2726729 T^{3} + 987376 T^{4} + 2690496 T^{5} - 26537 T^{6} - 848611 T^{7} - 103897 T^{8} + 105137 T^{9} + 21034 T^{10} - 4312 T^{11} - 1316 T^{12} - 26 T^{13} + 15 T^{14} + T^{15} \)
$29$ \( 24350536 + 14523124 T - 77370370 T^{2} - 88356051 T^{3} + 3498790 T^{4} + 33033627 T^{5} + 5564005 T^{6} - 4288104 T^{7} - 1061097 T^{8} + 215665 T^{9} + 70623 T^{10} - 2886 T^{11} - 1882 T^{12} - 56 T^{13} + 16 T^{14} + T^{15} \)
$31$ \( -508362642 + 1106349290 T - 310644668 T^{2} - 492781149 T^{3} + 180867980 T^{4} + 90468548 T^{5} - 31279304 T^{6} - 8374648 T^{7} + 2558037 T^{8} + 389654 T^{9} - 109645 T^{10} - 7618 T^{11} + 2428 T^{12} + 9 T^{13} - 22 T^{14} + T^{15} \)
$37$ \( -40940005196 + 38041799844 T + 15390327322 T^{2} - 15372197233 T^{3} - 3583743265 T^{4} + 2025308739 T^{5} + 537702802 T^{6} - 69180164 T^{7} - 26692292 T^{8} + 173338 T^{9} + 526954 T^{10} + 20006 T^{11} - 4504 T^{12} - 267 T^{13} + 14 T^{14} + T^{15} \)
$41$ \( -14583848 - 230325324 T - 608078046 T^{2} + 181282831 T^{3} + 658940872 T^{4} + 251299276 T^{5} - 35643900 T^{6} - 33413800 T^{7} - 2798248 T^{8} + 1169333 T^{9} + 194439 T^{10} - 10621 T^{11} - 3568 T^{12} - 83 T^{13} + 19 T^{14} + T^{15} \)
$43$ \( 2881799908 + 11534877440 T + 8894092804 T^{2} - 347835737 T^{3} - 2129048098 T^{4} - 328092937 T^{5} + 175436696 T^{6} + 39674340 T^{7} - 6247768 T^{8} - 1786407 T^{9} + 94517 T^{10} + 35660 T^{11} - 564 T^{12} - 315 T^{13} + T^{14} + T^{15} \)
$47$ \( 127047598350 + 441253093900 T + 511021588630 T^{2} + 214902228235 T^{3} + 12795824849 T^{4} - 13764122681 T^{5} - 2971844939 T^{6} + 131885457 T^{7} + 95895638 T^{8} + 6135322 T^{9} - 959163 T^{10} - 139778 T^{11} - 1496 T^{12} + 729 T^{13} + 49 T^{14} + T^{15} \)
$53$ \( -33403676712 + 4044213460 T + 17998952606 T^{2} - 4772909097 T^{3} - 2439010519 T^{4} + 879489034 T^{5} + 111287168 T^{6} - 64167674 T^{7} + 184942 T^{8} + 2139384 T^{9} - 144203 T^{10} - 30801 T^{11} + 3654 T^{12} + 98 T^{13} - 28 T^{14} + T^{15} \)
$59$ \( 925256454912 - 5460769807808 T - 2132023558720 T^{2} + 513121211181 T^{3} + 292568841906 T^{4} + 6564059213 T^{5} - 11599350885 T^{6} - 1311033540 T^{7} + 134739376 T^{8} + 29851558 T^{9} + 428525 T^{10} - 213499 T^{11} - 12957 T^{12} + 271 T^{13} + 43 T^{14} + T^{15} \)
$61$ \( -26580006408 + 479550998404 T - 730785624978 T^{2} + 455689385783 T^{3} - 132830835370 T^{4} + 11201724862 T^{5} + 3452592981 T^{6} - 901720438 T^{7} + 25970152 T^{8} + 13421572 T^{9} - 1308801 T^{10} - 50907 T^{11} + 11185 T^{12} - 194 T^{13} - 27 T^{14} + T^{15} \)
$67$ \( -38013704592 - 52550554808 T + 15964626160 T^{2} + 22745354025 T^{3} - 4424502500 T^{4} - 2759803053 T^{5} + 368786488 T^{6} + 160160270 T^{7} - 11542650 T^{8} - 4778569 T^{9} + 113483 T^{10} + 68381 T^{11} + 109 T^{12} - 429 T^{13} - 3 T^{14} + T^{15} \)
$71$ \( 23410866384 + 110079067960 T + 126105696164 T^{2} - 20510965747 T^{3} - 39961546592 T^{4} - 7282488239 T^{5} + 1944600325 T^{6} + 813696536 T^{7} + 67198399 T^{8} - 11363279 T^{9} - 2799023 T^{10} - 197588 T^{11} + 1550 T^{12} + 1007 T^{13} + 55 T^{14} + T^{15} \)
$73$ \( -1903703128 + 5336675548 T + 680466986 T^{2} - 6274180307 T^{3} - 920187182 T^{4} + 2222729025 T^{5} + 712491550 T^{6} - 78035228 T^{7} - 49181910 T^{8} - 2750277 T^{9} + 753716 T^{10} + 76878 T^{11} - 3219 T^{12} - 497 T^{13} + 3 T^{14} + T^{15} \)
$79$ \( 1041479699184 - 1449151677880 T - 4051944308660 T^{2} + 3290507206821 T^{3} + 966483954027 T^{4} - 135218638871 T^{5} - 34020570543 T^{6} + 2640020147 T^{7} + 503736255 T^{8} - 30472075 T^{9} - 3702641 T^{10} + 205644 T^{11} + 13221 T^{12} - 725 T^{13} - 18 T^{14} + T^{15} \)
$83$ \( -328479194135568 + 26108183669288 T + 43188819334612 T^{2} - 719244096109 T^{3} - 2045732572104 T^{4} - 33913916623 T^{5} + 46145435712 T^{6} + 1579912188 T^{7} - 556654409 T^{8} - 24822587 T^{9} + 3680554 T^{10} + 188435 T^{11} - 12519 T^{12} - 697 T^{13} + 17 T^{14} + T^{15} \)
$89$ \( -217040265526 + 1816948721504 T + 1165595467422 T^{2} - 318862132135 T^{3} - 140628637853 T^{4} + 22313800962 T^{5} + 6434440611 T^{6} - 721951856 T^{7} - 145635646 T^{8} + 10538659 T^{9} + 1848895 T^{10} - 60226 T^{11} - 12689 T^{12} - 26 T^{13} + 36 T^{14} + T^{15} \)
$97$ \( -1094692262 + 20388908342 T - 38811049566 T^{2} - 36602944525 T^{3} + 3345558804 T^{4} + 10171027643 T^{5} + 2955118798 T^{6} + 101295113 T^{7} - 73952946 T^{8} - 8012469 T^{9} + 628152 T^{10} + 104362 T^{11} - 2081 T^{12} - 539 T^{13} + 2 T^{14} + T^{15} \)
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