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: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

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

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

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

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:

Inner twists

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

Atkin-Lehner signs

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

Hecke kernels

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

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