Properties

Label 8046.2.a.m
Level 8046
Weight 2
Character orbit 8046.a
Self dual Yes
Analytic conductor 64.248
Analytic rank 1
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8046.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2476334663\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \(+ q^{4}\) \( -\beta_{1} q^{5} \) \( -\beta_{2} q^{7} \) \(+ q^{8}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(+ q^{4}\) \( -\beta_{1} q^{5} \) \( -\beta_{2} q^{7} \) \(+ q^{8}\) \( -\beta_{1} q^{10} \) \( + ( \beta_{1} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{11} \) \( + ( -1 - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{13} \) \( -\beta_{2} q^{14} \) \(+ q^{16}\) \( + ( -1 + 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{17} \) \( + ( -2 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{19} \) \( -\beta_{1} q^{20} \) \( + ( \beta_{1} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{22} \) \( + ( -2 + \beta_{1} + \beta_{3} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{23} \) \( + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{25} \) \( + ( -1 - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{26} \) \( -\beta_{2} q^{28} \) \( + ( -3 - 2 \beta_{1} - 2 \beta_{4} - \beta_{6} - \beta_{10} - \beta_{11} ) q^{29} \) \( + ( -1 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{31} \) \(+ q^{32}\) \( + ( -1 + 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{34} \) \( + ( -3 + 2 \beta_{1} + \beta_{3} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{35} \) \( + ( -2 + \beta_{1} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{37} \) \( + ( -2 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{38} \) \( -\beta_{1} q^{40} \) \( + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{41} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{43} \) \( + ( \beta_{1} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{44} \) \( + ( -2 + \beta_{1} + \beta_{3} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{46} \) \( + ( \beta_{1} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{47} \) \( + ( -\beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{49} \) \( + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{50} \) \( + ( -1 - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{52} \) \( + ( -2 - \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{53} \) \( + ( -3 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{55} \) \( -\beta_{2} q^{56} \) \( + ( -3 - 2 \beta_{1} - 2 \beta_{4} - \beta_{6} - \beta_{10} - \beta_{11} ) q^{58} \) \( + ( \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{59} \) \( + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{61} \) \( + ( -1 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{62} \) \(+ q^{64}\) \( + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{65} \) \( + ( -3 + \beta_{3} - \beta_{5} - \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{67} \) \( + ( -1 + 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{68} \) \( + ( -3 + 2 \beta_{1} + \beta_{3} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{70} \) \( + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{71} \) \( + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{73} \) \( + ( -2 + \beta_{1} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{74} \) \( + ( -2 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{76} \) \( + ( -1 + \beta_{2} - \beta_{4} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{77} \) \( + ( -\beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{79} \) \( -\beta_{1} q^{80} \) \( + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{82} \) \( + ( -\beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{83} \) \( + ( -2 - 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{85} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{86} \) \( + ( \beta_{1} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{88} \) \( + ( -2 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} - \beta_{8} - 3 \beta_{11} ) q^{89} \) \( + ( 1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} ) q^{91} \) \( + ( -2 + \beta_{1} + \beta_{3} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{92} \) \( + ( \beta_{1} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{94} \) \( + ( 4 \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 4 \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{95} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{97} \) \( + ( -\beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut 12q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 12q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut -\mathstrut 5q^{10} \) \(\mathstrut -\mathstrut 10q^{11} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut -\mathstrut 6q^{14} \) \(\mathstrut +\mathstrut 12q^{16} \) \(\mathstrut -\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 10q^{19} \) \(\mathstrut -\mathstrut 5q^{20} \) \(\mathstrut -\mathstrut 10q^{22} \) \(\mathstrut -\mathstrut 15q^{23} \) \(\mathstrut +\mathstrut 7q^{25} \) \(\mathstrut -\mathstrut q^{26} \) \(\mathstrut -\mathstrut 6q^{28} \) \(\mathstrut -\mathstrut 33q^{29} \) \(\mathstrut -\mathstrut 6q^{31} \) \(\mathstrut +\mathstrut 12q^{32} \) \(\mathstrut -\mathstrut 6q^{34} \) \(\mathstrut -\mathstrut 16q^{35} \) \(\mathstrut -\mathstrut 13q^{37} \) \(\mathstrut -\mathstrut 10q^{38} \) \(\mathstrut -\mathstrut 5q^{40} \) \(\mathstrut -\mathstrut 20q^{41} \) \(\mathstrut -\mathstrut 11q^{43} \) \(\mathstrut -\mathstrut 10q^{44} \) \(\mathstrut -\mathstrut 15q^{46} \) \(\mathstrut -\mathstrut 15q^{47} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 7q^{50} \) \(\mathstrut -\mathstrut q^{52} \) \(\mathstrut -\mathstrut 4q^{53} \) \(\mathstrut -\mathstrut 17q^{55} \) \(\mathstrut -\mathstrut 6q^{56} \) \(\mathstrut -\mathstrut 33q^{58} \) \(\mathstrut -\mathstrut 10q^{59} \) \(\mathstrut -\mathstrut 12q^{61} \) \(\mathstrut -\mathstrut 6q^{62} \) \(\mathstrut +\mathstrut 12q^{64} \) \(\mathstrut -\mathstrut 40q^{65} \) \(\mathstrut -\mathstrut 19q^{67} \) \(\mathstrut -\mathstrut 6q^{68} \) \(\mathstrut -\mathstrut 16q^{70} \) \(\mathstrut -\mathstrut 47q^{71} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut 13q^{74} \) \(\mathstrut -\mathstrut 10q^{76} \) \(\mathstrut +\mathstrut 6q^{77} \) \(\mathstrut -\mathstrut 15q^{79} \) \(\mathstrut -\mathstrut 5q^{80} \) \(\mathstrut -\mathstrut 20q^{82} \) \(\mathstrut -\mathstrut 18q^{83} \) \(\mathstrut -\mathstrut 25q^{85} \) \(\mathstrut -\mathstrut 11q^{86} \) \(\mathstrut -\mathstrut 10q^{88} \) \(\mathstrut -\mathstrut 24q^{89} \) \(\mathstrut -\mathstrut 3q^{91} \) \(\mathstrut -\mathstrut 15q^{92} \) \(\mathstrut -\mathstrut 15q^{94} \) \(\mathstrut +\mathstrut 3q^{95} \) \(\mathstrut -\mathstrut 25q^{97} \) \(\mathstrut +\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(5\) \(x^{11}\mathstrut -\mathstrut \) \(21\) \(x^{10}\mathstrut +\mathstrut \) \(116\) \(x^{9}\mathstrut +\mathstrut \) \(106\) \(x^{8}\mathstrut -\mathstrut \) \(774\) \(x^{7}\mathstrut -\mathstrut \) \(63\) \(x^{6}\mathstrut +\mathstrut \) \(2013\) \(x^{5}\mathstrut -\mathstrut \) \(417\) \(x^{4}\mathstrut -\mathstrut \) \(2249\) \(x^{3}\mathstrut +\mathstrut \) \(761\) \(x^{2}\mathstrut +\mathstrut \) \(910\) \(x\mathstrut -\mathstrut \) \(375\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(32197\) \(\nu^{11}\mathstrut +\mathstrut \) \(5673372\) \(\nu^{10}\mathstrut -\mathstrut \) \(27408721\) \(\nu^{9}\mathstrut -\mathstrut \) \(109537587\) \(\nu^{8}\mathstrut +\mathstrut \) \(609086107\) \(\nu^{7}\mathstrut +\mathstrut \) \(410288818\) \(\nu^{6}\mathstrut -\mathstrut \) \(3495027303\) \(\nu^{5}\mathstrut +\mathstrut \) \(447932378\) \(\nu^{4}\mathstrut +\mathstrut \) \(6463643167\) \(\nu^{3}\mathstrut -\mathstrut \) \(2256579470\) \(\nu^{2}\mathstrut -\mathstrut \) \(3729321842\) \(\nu\mathstrut +\mathstrut \) \(1592878652\)\()/5855999\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(337779\) \(\nu^{11}\mathstrut -\mathstrut \) \(12051765\) \(\nu^{10}\mathstrut +\mathstrut \) \(77892894\) \(\nu^{9}\mathstrut +\mathstrut \) \(222792211\) \(\nu^{8}\mathstrut -\mathstrut \) \(1586491494\) \(\nu^{7}\mathstrut -\mathstrut \) \(664646379\) \(\nu^{6}\mathstrut +\mathstrut \) \(8970420017\) \(\nu^{5}\mathstrut -\mathstrut \) \(2097844062\) \(\nu^{4}\mathstrut -\mathstrut \) \(16685502287\) \(\nu^{3}\mathstrut +\mathstrut \) \(6698494966\) \(\nu^{2}\mathstrut +\mathstrut \) \(9660687071\) \(\nu\mathstrut -\mathstrut \) \(4301878920\)\()/29279995\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(4626967\) \(\nu^{11}\mathstrut +\mathstrut \) \(27982820\) \(\nu^{10}\mathstrut +\mathstrut \) \(68317507\) \(\nu^{9}\mathstrut -\mathstrut \) \(610246892\) \(\nu^{8}\mathstrut +\mathstrut \) \(137663433\) \(\nu^{7}\mathstrut +\mathstrut \) \(3479105328\) \(\nu^{6}\mathstrut -\mathstrut \) \(3268300794\) \(\nu^{5}\mathstrut -\mathstrut \) \(6124911496\) \(\nu^{4}\mathstrut +\mathstrut \) \(8040696879\) \(\nu^{3}\mathstrut +\mathstrut \) \(2382541943\) \(\nu^{2}\mathstrut -\mathstrut \) \(5435421687\) \(\nu\mathstrut +\mathstrut \) \(1309434950\)\()/29279995\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(5834388\) \(\nu^{11}\mathstrut +\mathstrut \) \(38266875\) \(\nu^{10}\mathstrut +\mathstrut \) \(69519048\) \(\nu^{9}\mathstrut -\mathstrut \) \(818338273\) \(\nu^{8}\mathstrut +\mathstrut \) \(526825367\) \(\nu^{7}\mathstrut +\mathstrut \) \(4419720832\) \(\nu^{6}\mathstrut -\mathstrut \) \(6005098721\) \(\nu^{5}\mathstrut -\mathstrut \) \(6551037399\) \(\nu^{4}\mathstrut +\mathstrut \) \(13075477756\) \(\nu^{3}\mathstrut +\mathstrut \) \(594551152\) \(\nu^{2}\mathstrut -\mathstrut \) \(8170221548\) \(\nu\mathstrut +\mathstrut \) \(2800717475\)\()/29279995\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(11071809\) \(\nu^{11}\mathstrut +\mathstrut \) \(33647510\) \(\nu^{10}\mathstrut +\mathstrut \) \(329349654\) \(\nu^{9}\mathstrut -\mathstrut \) \(795142464\) \(\nu^{8}\mathstrut -\mathstrut \) \(3330885579\) \(\nu^{7}\mathstrut +\mathstrut \) \(5470397091\) \(\nu^{6}\mathstrut +\mathstrut \) \(13696849342\) \(\nu^{5}\mathstrut -\mathstrut \) \(15262968602\) \(\nu^{4}\mathstrut -\mathstrut \) \(22956311877\) \(\nu^{3}\mathstrut +\mathstrut \) \(17416329301\) \(\nu^{2}\mathstrut +\mathstrut \) \(12966000711\) \(\nu\mathstrut -\mathstrut \) \(6906022130\)\()/29279995\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(15600107\) \(\nu^{11}\mathstrut +\mathstrut \) \(74241930\) \(\nu^{10}\mathstrut +\mathstrut \) \(327537312\) \(\nu^{9}\mathstrut -\mathstrut \) \(1640249962\) \(\nu^{8}\mathstrut -\mathstrut \) \(1697104197\) \(\nu^{7}\mathstrut +\mathstrut \) \(9679352383\) \(\nu^{6}\mathstrut +\mathstrut \) \(1916697621\) \(\nu^{5}\mathstrut -\mathstrut \) \(19399390581\) \(\nu^{4}\mathstrut +\mathstrut \) \(804575664\) \(\nu^{3}\mathstrut +\mathstrut \) \(13154737748\) \(\nu^{2}\mathstrut -\mathstrut \) \(1591143977\) \(\nu\mathstrut -\mathstrut \) \(1289750025\)\()/29279995\)
\(\beta_{8}\)\(=\)\((\)\(15915726\) \(\nu^{11}\mathstrut -\mathstrut \) \(100688050\) \(\nu^{10}\mathstrut -\mathstrut \) \(207068641\) \(\nu^{9}\mathstrut +\mathstrut \) \(2153825106\) \(\nu^{8}\mathstrut -\mathstrut \) \(1047579084\) \(\nu^{7}\mathstrut -\mathstrut \) \(11652307114\) \(\nu^{6}\mathstrut +\mathstrut \) \(14014610012\) \(\nu^{5}\mathstrut +\mathstrut \) \(17633361118\) \(\nu^{4}\mathstrut -\mathstrut \) \(30654436132\) \(\nu^{3}\mathstrut -\mathstrut \) \(3005054784\) \(\nu^{2}\mathstrut +\mathstrut \) \(18959758726\) \(\nu\mathstrut -\mathstrut \) \(6112777815\)\()/29279995\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(3317302\) \(\nu^{11}\mathstrut +\mathstrut \) \(16732001\) \(\nu^{10}\mathstrut +\mathstrut \) \(64707528\) \(\nu^{9}\mathstrut -\mathstrut \) \(366093260\) \(\nu^{8}\mathstrut -\mathstrut \) \(254489192\) \(\nu^{7}\mathstrut +\mathstrut \) \(2107854700\) \(\nu^{6}\mathstrut -\mathstrut \) \(179845605\) \(\nu^{5}\mathstrut -\mathstrut \) \(3967835268\) \(\nu^{4}\mathstrut +\mathstrut \) \(1171309316\) \(\nu^{3}\mathstrut +\mathstrut \) \(2366997820\) \(\nu^{2}\mathstrut -\mathstrut \) \(842422122\) \(\nu\mathstrut -\mathstrut \) \(38973036\)\()/5855999\)
\(\beta_{10}\)\(=\)\((\)\(17078923\) \(\nu^{11}\mathstrut -\mathstrut \) \(69687500\) \(\nu^{10}\mathstrut -\mathstrut \) \(418005468\) \(\nu^{9}\mathstrut +\mathstrut \) \(1573561298\) \(\nu^{8}\mathstrut +\mathstrut \) \(3159684708\) \(\nu^{7}\mathstrut -\mathstrut \) \(9796137762\) \(\nu^{6}\mathstrut -\mathstrut \) \(9615020874\) \(\nu^{5}\mathstrut +\mathstrut \) \(22410652829\) \(\nu^{4}\mathstrut +\mathstrut \) \(13297311074\) \(\nu^{3}\mathstrut -\mathstrut \) \(19695978447\) \(\nu^{2}\mathstrut -\mathstrut \) \(6697290927\) \(\nu\mathstrut +\mathstrut \) \(5205009495\)\()/29279995\)
\(\beta_{11}\)\(=\)\((\)\(5445786\) \(\nu^{11}\mathstrut -\mathstrut \) \(29080114\) \(\nu^{10}\mathstrut -\mathstrut \) \(98087514\) \(\nu^{9}\mathstrut +\mathstrut \) \(632612308\) \(\nu^{8}\mathstrut +\mathstrut \) \(238661630\) \(\nu^{7}\mathstrut -\mathstrut \) \(3586733161\) \(\nu^{6}\mathstrut +\mathstrut \) \(1341737333\) \(\nu^{5}\mathstrut +\mathstrut \) \(6428622786\) \(\nu^{4}\mathstrut -\mathstrut \) \(3951467976\) \(\nu^{3}\mathstrut -\mathstrut \) \(3278532055\) \(\nu^{2}\mathstrut +\mathstrut \) \(2627858133\) \(\nu\mathstrut -\mathstrut \) \(395658693\)\()/5855999\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(13\) \(\beta_{10}\mathstrut -\mathstrut \) \(11\) \(\beta_{9}\mathstrut +\mathstrut \) \(19\) \(\beta_{8}\mathstrut +\mathstrut \) \(21\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(16\) \(\beta_{3}\mathstrut +\mathstrut \) \(17\) \(\beta_{2}\mathstrut +\mathstrut \) \(25\) \(\beta_{1}\mathstrut +\mathstrut \) \(60\)
\(\nu^{5}\)\(=\)\(-\)\(15\) \(\beta_{11}\mathstrut +\mathstrut \) \(51\) \(\beta_{10}\mathstrut -\mathstrut \) \(23\) \(\beta_{9}\mathstrut +\mathstrut \) \(44\) \(\beta_{8}\mathstrut +\mathstrut \) \(79\) \(\beta_{7}\mathstrut +\mathstrut \) \(27\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(23\) \(\beta_{4}\mathstrut -\mathstrut \) \(13\) \(\beta_{3}\mathstrut +\mathstrut \) \(27\) \(\beta_{2}\mathstrut +\mathstrut \) \(222\) \(\beta_{1}\mathstrut +\mathstrut \) \(49\)
\(\nu^{6}\)\(=\)\(34\) \(\beta_{11}\mathstrut -\mathstrut \) \(162\) \(\beta_{10}\mathstrut -\mathstrut \) \(138\) \(\beta_{9}\mathstrut +\mathstrut \) \(341\) \(\beta_{8}\mathstrut +\mathstrut \) \(378\) \(\beta_{7}\mathstrut -\mathstrut \) \(11\) \(\beta_{6}\mathstrut -\mathstrut \) \(24\) \(\beta_{5}\mathstrut +\mathstrut \) \(61\) \(\beta_{4}\mathstrut -\mathstrut \) \(258\) \(\beta_{3}\mathstrut +\mathstrut \) \(280\) \(\beta_{2}\mathstrut +\mathstrut \) \(506\) \(\beta_{1}\mathstrut +\mathstrut \) \(923\)
\(\nu^{7}\)\(=\)\(-\)\(223\) \(\beta_{11}\mathstrut +\mathstrut \) \(787\) \(\beta_{10}\mathstrut -\mathstrut \) \(406\) \(\beta_{9}\mathstrut +\mathstrut \) \(914\) \(\beta_{8}\mathstrut +\mathstrut \) \(1435\) \(\beta_{7}\mathstrut +\mathstrut \) \(498\) \(\beta_{6}\mathstrut -\mathstrut \) \(222\) \(\beta_{5}\mathstrut +\mathstrut \) \(450\) \(\beta_{4}\mathstrut -\mathstrut \) \(384\) \(\beta_{3}\mathstrut +\mathstrut \) \(578\) \(\beta_{2}\mathstrut +\mathstrut \) \(3665\) \(\beta_{1}\mathstrut +\mathstrut \) \(1285\)
\(\nu^{8}\)\(=\)\(451\) \(\beta_{11}\mathstrut -\mathstrut \) \(1964\) \(\beta_{10}\mathstrut -\mathstrut \) \(1962\) \(\beta_{9}\mathstrut +\mathstrut \) \(6113\) \(\beta_{8}\mathstrut +\mathstrut \) \(6668\) \(\beta_{7}\mathstrut +\mathstrut \) \(77\) \(\beta_{6}\mathstrut -\mathstrut \) \(348\) \(\beta_{5}\mathstrut +\mathstrut \) \(1390\) \(\beta_{4}\mathstrut -\mathstrut \) \(4316\) \(\beta_{3}\mathstrut +\mathstrut \) \(4693\) \(\beta_{2}\mathstrut +\mathstrut \) \(9664\) \(\beta_{1}\mathstrut +\mathstrut \) \(15112\)
\(\nu^{9}\)\(=\)\(-\)\(3384\) \(\beta_{11}\mathstrut +\mathstrut \) \(12202\) \(\beta_{10}\mathstrut -\mathstrut \) \(6570\) \(\beta_{9}\mathstrut +\mathstrut \) \(18455\) \(\beta_{8}\mathstrut +\mathstrut \) \(25796\) \(\beta_{7}\mathstrut +\mathstrut \) \(8546\) \(\beta_{6}\mathstrut -\mathstrut \) \(2832\) \(\beta_{5}\mathstrut +\mathstrut \) \(8544\) \(\beta_{4}\mathstrut -\mathstrut \) \(8748\) \(\beta_{3}\mathstrut +\mathstrut \) \(11538\) \(\beta_{2}\mathstrut +\mathstrut \) \(61610\) \(\beta_{1}\mathstrut +\mathstrut \) \(28417\)
\(\nu^{10}\)\(=\)\(5285\) \(\beta_{11}\mathstrut -\mathstrut \) \(22019\) \(\beta_{10}\mathstrut -\mathstrut \) \(29192\) \(\beta_{9}\mathstrut +\mathstrut \) \(110188\) \(\beta_{8}\mathstrut +\mathstrut \) \(117443\) \(\beta_{7}\mathstrut +\mathstrut \) \(6505\) \(\beta_{6}\mathstrut -\mathstrut \) \(3650\) \(\beta_{5}\mathstrut +\mathstrut \) \(29179\) \(\beta_{4}\mathstrut -\mathstrut \) \(73887\) \(\beta_{3}\mathstrut +\mathstrut \) \(79863\) \(\beta_{2}\mathstrut +\mathstrut \) \(180002\) \(\beta_{1}\mathstrut +\mathstrut \) \(253681\)
\(\nu^{11}\)\(=\)\(-\)\(52344\) \(\beta_{11}\mathstrut +\mathstrut \) \(193515\) \(\beta_{10}\mathstrut -\mathstrut \) \(102076\) \(\beta_{9}\mathstrut +\mathstrut \) \(364000\) \(\beta_{8}\mathstrut +\mathstrut \) \(462496\) \(\beta_{7}\mathstrut +\mathstrut \) \(145903\) \(\beta_{6}\mathstrut -\mathstrut \) \(31755\) \(\beta_{5}\mathstrut +\mathstrut \) \(161388\) \(\beta_{4}\mathstrut -\mathstrut \) \(182355\) \(\beta_{3}\mathstrut +\mathstrut \) \(222808\) \(\beta_{2}\mathstrut +\mathstrut \) \(1046987\) \(\beta_{1}\mathstrut +\mathstrut \) \(583486\)

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
4.25305
4.11015
1.99250
1.53008
1.09665
1.06735
0.448758
−1.12518
−1.17050
−1.20126
−2.26764
−3.73396
1.00000 0 1.00000 −4.25305 0 2.10378 1.00000 0 −4.25305
1.2 1.00000 0 1.00000 −4.11015 0 −3.78475 1.00000 0 −4.11015
1.3 1.00000 0 1.00000 −1.99250 0 −0.0206178 1.00000 0 −1.99250
1.4 1.00000 0 1.00000 −1.53008 0 5.04291 1.00000 0 −1.53008
1.5 1.00000 0 1.00000 −1.09665 0 −2.72925 1.00000 0 −1.09665
1.6 1.00000 0 1.00000 −1.06735 0 2.41300 1.00000 0 −1.06735
1.7 1.00000 0 1.00000 −0.448758 0 −1.52942 1.00000 0 −0.448758
1.8 1.00000 0 1.00000 1.12518 0 −0.686663 1.00000 0 1.12518
1.9 1.00000 0 1.00000 1.17050 0 0.506529 1.00000 0 1.17050
1.10 1.00000 0 1.00000 1.20126 0 −4.29383 1.00000 0 1.20126
1.11 1.00000 0 1.00000 2.26764 0 −0.397853 1.00000 0 2.26764
1.12 1.00000 0 1.00000 3.73396 0 −2.62384 1.00000 0 3.73396
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(149\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\(T_{5}^{12} + \cdots\)
\(T_{11}^{12} + \cdots\)