Properties

Label 8034.2.a.x
Level 8034
Weight 2
Character orbit 8034.a
Self dual Yes
Analytic conductor 64.152
Analytic rank 0
Dimension 13
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8034.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(- q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + \beta_{1} q^{5} \) \(- q^{6}\) \( + \beta_{3} q^{7} \) \(- q^{8}\) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + \beta_{1} q^{5} \) \(- q^{6}\) \( + \beta_{3} q^{7} \) \(- q^{8}\) \(+ q^{9}\) \( -\beta_{1} q^{10} \) \( + \beta_{4} q^{11} \) \(+ q^{12}\) \(+ q^{13}\) \( -\beta_{3} q^{14} \) \( + \beta_{1} q^{15} \) \(+ q^{16}\) \( + ( 2 - \beta_{1} + \beta_{3} - \beta_{11} ) q^{17} \) \(- q^{18}\) \( + ( \beta_{1} - \beta_{5} + \beta_{12} ) q^{19} \) \( + \beta_{1} q^{20} \) \( + \beta_{3} q^{21} \) \( -\beta_{4} q^{22} \) \( + ( 3 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{23} \) \(- q^{24}\) \( + ( 2 + \beta_{2} + \beta_{6} + \beta_{7} ) q^{25} \) \(- q^{26}\) \(+ q^{27}\) \( + \beta_{3} q^{28} \) \( + ( 1 + \beta_{3} + \beta_{6} - \beta_{8} + \beta_{11} ) q^{29} \) \( -\beta_{1} q^{30} \) \( + ( \beta_{1} - \beta_{5} - \beta_{12} ) q^{31} \) \(- q^{32}\) \( + \beta_{4} q^{33} \) \( + ( -2 + \beta_{1} - \beta_{3} + \beta_{11} ) q^{34} \) \( + ( 3 + \beta_{2} + \beta_{8} - \beta_{12} ) q^{35} \) \(+ q^{36}\) \( + ( -\beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{37} \) \( + ( -\beta_{1} + \beta_{5} - \beta_{12} ) q^{38} \) \(+ q^{39}\) \( -\beta_{1} q^{40} \) \( + ( 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{41} \) \( -\beta_{3} q^{42} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{43} \) \( + \beta_{4} q^{44} \) \( + \beta_{1} q^{45} \) \( + ( -3 + \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{46} \) \( + ( 1 - \beta_{1} + \beta_{4} - \beta_{6} + \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{47} \) \(+ q^{48}\) \( + ( -1 + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{10} ) q^{49} \) \( + ( -2 - \beta_{2} - \beta_{6} - \beta_{7} ) q^{50} \) \( + ( 2 - \beta_{1} + \beta_{3} - \beta_{11} ) q^{51} \) \(+ q^{52}\) \( + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} + \beta_{10} + 2 \beta_{12} ) q^{53} \) \(- q^{54}\) \( + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{55} \) \( -\beta_{3} q^{56} \) \( + ( \beta_{1} - \beta_{5} + \beta_{12} ) q^{57} \) \( + ( -1 - \beta_{3} - \beta_{6} + \beta_{8} - \beta_{11} ) q^{58} \) \( + ( 2 + \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{9} + 2 \beta_{12} ) q^{59} \) \( + \beta_{1} q^{60} \) \( + ( 1 - \beta_{1} + \beta_{4} - \beta_{6} - \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{61} \) \( + ( -\beta_{1} + \beta_{5} + \beta_{12} ) q^{62} \) \( + \beta_{3} q^{63} \) \(+ q^{64}\) \( + \beta_{1} q^{65} \) \( -\beta_{4} q^{66} \) \( + ( -2 - 2 \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{12} ) q^{67} \) \( + ( 2 - \beta_{1} + \beta_{3} - \beta_{11} ) q^{68} \) \( + ( 3 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{69} \) \( + ( -3 - \beta_{2} - \beta_{8} + \beta_{12} ) q^{70} \) \( + ( \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{71} \) \(- q^{72}\) \( + ( -3 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{9} + \beta_{10} ) q^{73} \) \( + ( \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{74} \) \( + ( 2 + \beta_{2} + \beta_{6} + \beta_{7} ) q^{75} \) \( + ( \beta_{1} - \beta_{5} + \beta_{12} ) q^{76} \) \( + ( 3 + 2 \beta_{3} + \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{77} \) \(- q^{78}\) \( + ( -2 - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{8} + 2 \beta_{10} + \beta_{12} ) q^{79} \) \( + \beta_{1} q^{80} \) \(+ q^{81}\) \( + ( -2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{82} \) \( + ( 1 + \beta_{3} + 2 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{12} ) q^{83} \) \( + \beta_{3} q^{84} \) \( + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{12} ) q^{85} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{86} \) \( + ( 1 + \beta_{3} + \beta_{6} - \beta_{8} + \beta_{11} ) q^{87} \) \( -\beta_{4} q^{88} \) \( + ( 1 - 2 \beta_{1} + \beta_{3} - 2 \beta_{5} - 3 \beta_{6} + \beta_{8} - 3 \beta_{10} - \beta_{11} - 3 \beta_{12} ) q^{89} \) \( -\beta_{1} q^{90} \) \( + \beta_{3} q^{91} \) \( + ( 3 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{92} \) \( + ( \beta_{1} - \beta_{5} - \beta_{12} ) q^{93} \) \( + ( -1 + \beta_{1} - \beta_{4} + \beta_{6} - \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{94} \) \( + ( 1 + \beta_{1} - 3 \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} - 4 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{95} \) \(- q^{96}\) \( + ( -4 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{97} \) \( + ( 1 - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{10} ) q^{98} \) \( + \beta_{4} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(13q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut +\mathstrut 13q^{3} \) \(\mathstrut +\mathstrut 13q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut 13q^{6} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut -\mathstrut 13q^{8} \) \(\mathstrut +\mathstrut 13q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(13q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut +\mathstrut 13q^{3} \) \(\mathstrut +\mathstrut 13q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut 13q^{6} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut -\mathstrut 13q^{8} \) \(\mathstrut +\mathstrut 13q^{9} \) \(\mathstrut -\mathstrut q^{10} \) \(\mathstrut +\mathstrut 6q^{11} \) \(\mathstrut +\mathstrut 13q^{12} \) \(\mathstrut +\mathstrut 13q^{13} \) \(\mathstrut +\mathstrut q^{14} \) \(\mathstrut +\mathstrut q^{15} \) \(\mathstrut +\mathstrut 13q^{16} \) \(\mathstrut +\mathstrut 18q^{17} \) \(\mathstrut -\mathstrut 13q^{18} \) \(\mathstrut -\mathstrut q^{19} \) \(\mathstrut +\mathstrut q^{20} \) \(\mathstrut -\mathstrut q^{21} \) \(\mathstrut -\mathstrut 6q^{22} \) \(\mathstrut +\mathstrut 20q^{23} \) \(\mathstrut -\mathstrut 13q^{24} \) \(\mathstrut +\mathstrut 16q^{25} \) \(\mathstrut -\mathstrut 13q^{26} \) \(\mathstrut +\mathstrut 13q^{27} \) \(\mathstrut -\mathstrut q^{28} \) \(\mathstrut +\mathstrut 25q^{29} \) \(\mathstrut -\mathstrut q^{30} \) \(\mathstrut -\mathstrut 5q^{31} \) \(\mathstrut -\mathstrut 13q^{32} \) \(\mathstrut +\mathstrut 6q^{33} \) \(\mathstrut -\mathstrut 18q^{34} \) \(\mathstrut +\mathstrut 26q^{35} \) \(\mathstrut +\mathstrut 13q^{36} \) \(\mathstrut -\mathstrut 6q^{37} \) \(\mathstrut +\mathstrut q^{38} \) \(\mathstrut +\mathstrut 13q^{39} \) \(\mathstrut -\mathstrut q^{40} \) \(\mathstrut +\mathstrut 13q^{41} \) \(\mathstrut +\mathstrut q^{42} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut 6q^{44} \) \(\mathstrut +\mathstrut q^{45} \) \(\mathstrut -\mathstrut 20q^{46} \) \(\mathstrut +\mathstrut 5q^{47} \) \(\mathstrut +\mathstrut 13q^{48} \) \(\mathstrut -\mathstrut 16q^{50} \) \(\mathstrut +\mathstrut 18q^{51} \) \(\mathstrut +\mathstrut 13q^{52} \) \(\mathstrut +\mathstrut 21q^{53} \) \(\mathstrut -\mathstrut 13q^{54} \) \(\mathstrut +\mathstrut 2q^{55} \) \(\mathstrut +\mathstrut q^{56} \) \(\mathstrut -\mathstrut q^{57} \) \(\mathstrut -\mathstrut 25q^{58} \) \(\mathstrut +\mathstrut 22q^{59} \) \(\mathstrut +\mathstrut q^{60} \) \(\mathstrut +\mathstrut 2q^{61} \) \(\mathstrut +\mathstrut 5q^{62} \) \(\mathstrut -\mathstrut q^{63} \) \(\mathstrut +\mathstrut 13q^{64} \) \(\mathstrut +\mathstrut q^{65} \) \(\mathstrut -\mathstrut 6q^{66} \) \(\mathstrut -\mathstrut q^{67} \) \(\mathstrut +\mathstrut 18q^{68} \) \(\mathstrut +\mathstrut 20q^{69} \) \(\mathstrut -\mathstrut 26q^{70} \) \(\mathstrut +\mathstrut 32q^{71} \) \(\mathstrut -\mathstrut 13q^{72} \) \(\mathstrut -\mathstrut 13q^{73} \) \(\mathstrut +\mathstrut 6q^{74} \) \(\mathstrut +\mathstrut 16q^{75} \) \(\mathstrut -\mathstrut q^{76} \) \(\mathstrut +\mathstrut 33q^{77} \) \(\mathstrut -\mathstrut 13q^{78} \) \(\mathstrut -\mathstrut 7q^{79} \) \(\mathstrut +\mathstrut q^{80} \) \(\mathstrut +\mathstrut 13q^{81} \) \(\mathstrut -\mathstrut 13q^{82} \) \(\mathstrut +\mathstrut 17q^{83} \) \(\mathstrut -\mathstrut q^{84} \) \(\mathstrut -\mathstrut 25q^{85} \) \(\mathstrut +\mathstrut 2q^{86} \) \(\mathstrut +\mathstrut 25q^{87} \) \(\mathstrut -\mathstrut 6q^{88} \) \(\mathstrut -\mathstrut 12q^{89} \) \(\mathstrut -\mathstrut q^{90} \) \(\mathstrut -\mathstrut q^{91} \) \(\mathstrut +\mathstrut 20q^{92} \) \(\mathstrut -\mathstrut 5q^{93} \) \(\mathstrut -\mathstrut 5q^{94} \) \(\mathstrut +\mathstrut 36q^{95} \) \(\mathstrut -\mathstrut 13q^{96} \) \(\mathstrut -\mathstrut 42q^{97} \) \(\mathstrut +\mathstrut 6q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13}\mathstrut -\mathstrut \) \(x^{12}\mathstrut -\mathstrut \) \(40\) \(x^{11}\mathstrut +\mathstrut \) \(34\) \(x^{10}\mathstrut +\mathstrut \) \(604\) \(x^{9}\mathstrut -\mathstrut \) \(381\) \(x^{8}\mathstrut -\mathstrut \) \(4352\) \(x^{7}\mathstrut +\mathstrut \) \(1474\) \(x^{6}\mathstrut +\mathstrut \) \(15809\) \(x^{5}\mathstrut -\mathstrut \) \(368\) \(x^{4}\mathstrut -\mathstrut \) \(27369\) \(x^{3}\mathstrut -\mathstrut \) \(7621\) \(x^{2}\mathstrut +\mathstrut \) \(17300\) \(x\mathstrut +\mathstrut \) \(8832\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(82571179\) \(\nu^{12}\mathstrut -\mathstrut \) \(4674469543\) \(\nu^{11}\mathstrut +\mathstrut \) \(4045229380\) \(\nu^{10}\mathstrut +\mathstrut \) \(153227692402\) \(\nu^{9}\mathstrut -\mathstrut \) \(190916406966\) \(\nu^{8}\mathstrut -\mathstrut \) \(1693362162947\) \(\nu^{7}\mathstrut +\mathstrut \) \(2186335491570\) \(\nu^{6}\mathstrut +\mathstrut \) \(7311077583168\) \(\nu^{5}\mathstrut -\mathstrut \) \(7974918999565\) \(\nu^{4}\mathstrut -\mathstrut \) \(12395566278704\) \(\nu^{3}\mathstrut +\mathstrut \) \(7496600830547\) \(\nu^{2}\mathstrut +\mathstrut \) \(7178855402017\) \(\nu\mathstrut -\mathstrut \) \(94191942330\)\()/\)\(316533270094\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(821635533\) \(\nu^{12}\mathstrut -\mathstrut \) \(14688528363\) \(\nu^{11}\mathstrut +\mathstrut \) \(23628366040\) \(\nu^{10}\mathstrut +\mathstrut \) \(567415004562\) \(\nu^{9}\mathstrut -\mathstrut \) \(132389839580\) \(\nu^{8}\mathstrut -\mathstrut \) \(8016782345595\) \(\nu^{7}\mathstrut -\mathstrut \) \(1886872567880\) \(\nu^{6}\mathstrut +\mathstrut \) \(50022172152038\) \(\nu^{5}\mathstrut +\mathstrut \) \(24369898063691\) \(\nu^{4}\mathstrut -\mathstrut \) \(130289551131540\) \(\nu^{3}\mathstrut -\mathstrut \) \(81913590142811\) \(\nu^{2}\mathstrut +\mathstrut \) \(104592523119481\) \(\nu\mathstrut +\mathstrut \) \(74380301896924\)\()/\)\(633066540188\)
\(\beta_{4}\)\(=\)\((\)\(464768053\) \(\nu^{12}\mathstrut -\mathstrut \) \(3020129323\) \(\nu^{11}\mathstrut -\mathstrut \) \(17668924332\) \(\nu^{10}\mathstrut +\mathstrut \) \(110747092838\) \(\nu^{9}\mathstrut +\mathstrut \) \(253508520438\) \(\nu^{8}\mathstrut -\mathstrut \) \(1447775318393\) \(\nu^{7}\mathstrut -\mathstrut \) \(1820042919050\) \(\nu^{6}\mathstrut +\mathstrut \) \(8096109949528\) \(\nu^{5}\mathstrut +\mathstrut \) \(7507442330311\) \(\nu^{4}\mathstrut -\mathstrut \) \(18589696680318\) \(\nu^{3}\mathstrut -\mathstrut \) \(15726961057905\) \(\nu^{2}\mathstrut +\mathstrut \) \(13031570002261\) \(\nu\mathstrut +\mathstrut \) \(10932434824650\)\()/\)\(316533270094\)
\(\beta_{5}\)\(=\)\((\)\(1800405167\) \(\nu^{12}\mathstrut -\mathstrut \) \(9821990403\) \(\nu^{11}\mathstrut -\mathstrut \) \(66844233676\) \(\nu^{10}\mathstrut +\mathstrut \) \(372531445822\) \(\nu^{9}\mathstrut +\mathstrut \) \(923116143424\) \(\nu^{8}\mathstrut -\mathstrut \) \(5111459259651\) \(\nu^{7}\mathstrut -\mathstrut \) \(6174254472800\) \(\nu^{6}\mathstrut +\mathstrut \) \(30716279416750\) \(\nu^{5}\mathstrut +\mathstrut \) \(22995984560079\) \(\nu^{4}\mathstrut -\mathstrut \) \(78136989988656\) \(\nu^{3}\mathstrut -\mathstrut \) \(46139543962307\) \(\nu^{2}\mathstrut +\mathstrut \) \(63313812889133\) \(\nu\mathstrut +\mathstrut \) \(37558928545592\)\()/\)\(633066540188\)
\(\beta_{6}\)\(=\)\((\)\(459562957\) \(\nu^{12}\mathstrut +\mathstrut \) \(518271920\) \(\nu^{11}\mathstrut -\mathstrut \) \(16642314467\) \(\nu^{10}\mathstrut -\mathstrut \) \(17632719959\) \(\nu^{9}\mathstrut +\mathstrut \) \(219058129306\) \(\nu^{8}\mathstrut +\mathstrut \) \(220995431293\) \(\nu^{7}\mathstrut -\mathstrut \) \(1289495659648\) \(\nu^{6}\mathstrut -\mathstrut \) \(1257189213960\) \(\nu^{5}\mathstrut +\mathstrut \) \(3402241451437\) \(\nu^{4}\mathstrut +\mathstrut \) \(3081177289065\) \(\nu^{3}\mathstrut -\mathstrut \) \(3076261637442\) \(\nu^{2}\mathstrut -\mathstrut \) \(2594200123208\) \(\nu\mathstrut -\mathstrut \) \(261207897518\)\()/\)\(158266635047\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(1001697093\) \(\nu^{12}\mathstrut +\mathstrut \) \(3637925703\) \(\nu^{11}\mathstrut +\mathstrut \) \(29239399554\) \(\nu^{10}\mathstrut -\mathstrut \) \(117962252484\) \(\nu^{9}\mathstrut -\mathstrut \) \(247199851646\) \(\nu^{8}\mathstrut +\mathstrut \) \(1251371300361\) \(\nu^{7}\mathstrut +\mathstrut \) \(392655827726\) \(\nu^{6}\mathstrut -\mathstrut \) \(4796699155248\) \(\nu^{5}\mathstrut +\mathstrut \) \(1170436096691\) \(\nu^{4}\mathstrut +\mathstrut \) \(6233211700574\) \(\nu^{3}\mathstrut -\mathstrut \) \(1027544285569\) \(\nu^{2}\mathstrut -\mathstrut \) \(1990455155601\) \(\nu\mathstrut -\mathstrut \) \(1599125153292\)\()/\)\(316533270094\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(1301185969\) \(\nu^{12}\mathstrut -\mathstrut \) \(3095768311\) \(\nu^{11}\mathstrut +\mathstrut \) \(48687375005\) \(\nu^{10}\mathstrut +\mathstrut \) \(117305988087\) \(\nu^{9}\mathstrut -\mathstrut \) \(652269947584\) \(\nu^{8}\mathstrut -\mathstrut \) \(1636981333955\) \(\nu^{7}\mathstrut +\mathstrut \) \(3626438951305\) \(\nu^{6}\mathstrut +\mathstrut \) \(10185117990171\) \(\nu^{5}\mathstrut -\mathstrut \) \(6405537841610\) \(\nu^{4}\mathstrut -\mathstrut \) \(26666364418514\) \(\nu^{3}\mathstrut -\mathstrut \) \(4445159036711\) \(\nu^{2}\mathstrut +\mathstrut \) \(21974554085695\) \(\nu\mathstrut +\mathstrut \) \(11198108957142\)\()/\)\(158266635047\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(5709491399\) \(\nu^{12}\mathstrut +\mathstrut \) \(1884380107\) \(\nu^{11}\mathstrut +\mathstrut \) \(220983218852\) \(\nu^{10}\mathstrut -\mathstrut \) \(60804918910\) \(\nu^{9}\mathstrut -\mathstrut \) \(3140305882384\) \(\nu^{8}\mathstrut +\mathstrut \) \(596077416591\) \(\nu^{7}\mathstrut +\mathstrut \) \(20057540387488\) \(\nu^{6}\mathstrut -\mathstrut \) \(1615844172274\) \(\nu^{5}\mathstrut -\mathstrut \) \(57139879804783\) \(\nu^{4}\mathstrut -\mathstrut \) \(114622375696\) \(\nu^{3}\mathstrut +\mathstrut \) \(63720036244919\) \(\nu^{2}\mathstrut +\mathstrut \) \(3045455679311\) \(\nu\mathstrut -\mathstrut \) \(22176589418488\)\()/\)\(633066540188\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(1572124490\) \(\nu^{12}\mathstrut +\mathstrut \) \(1168195578\) \(\nu^{11}\mathstrut +\mathstrut \) \(59893192506\) \(\nu^{10}\mathstrut -\mathstrut \) \(46050449196\) \(\nu^{9}\mathstrut -\mathstrut \) \(844559885012\) \(\nu^{8}\mathstrut +\mathstrut \) \(638669608161\) \(\nu^{7}\mathstrut +\mathstrut \) \(5500753705133\) \(\nu^{6}\mathstrut -\mathstrut \) \(3769562907595\) \(\nu^{5}\mathstrut -\mathstrut \) \(17246842712312\) \(\nu^{4}\mathstrut +\mathstrut \) \(9535362819479\) \(\nu^{3}\mathstrut +\mathstrut \) \(24610988707373\) \(\nu^{2}\mathstrut -\mathstrut \) \(8054888236900\) \(\nu\mathstrut -\mathstrut \) \(13075070948772\)\()/\)\(158266635047\)
\(\beta_{11}\)\(=\)\((\)\(4905042985\) \(\nu^{12}\mathstrut -\mathstrut \) \(8698615733\) \(\nu^{11}\mathstrut -\mathstrut \) \(187812103994\) \(\nu^{10}\mathstrut +\mathstrut \) \(306055612136\) \(\nu^{9}\mathstrut +\mathstrut \) \(2649761260912\) \(\nu^{8}\mathstrut -\mathstrut \) \(3752712058411\) \(\nu^{7}\mathstrut -\mathstrut \) \(17188892321224\) \(\nu^{6}\mathstrut +\mathstrut \) \(19215174884610\) \(\nu^{5}\mathstrut +\mathstrut \) \(53780502305993\) \(\nu^{4}\mathstrut -\mathstrut \) \(41767633025424\) \(\nu^{3}\mathstrut -\mathstrut \) \(76738490139713\) \(\nu^{2}\mathstrut +\mathstrut \) \(30387722952273\) \(\nu\mathstrut +\mathstrut \) \(40291809896428\)\()/\)\(316533270094\)
\(\beta_{12}\)\(=\)\((\)\(5235281189\) \(\nu^{12}\mathstrut -\mathstrut \) \(6247478525\) \(\nu^{11}\mathstrut -\mathstrut \) \(196255326952\) \(\nu^{10}\mathstrut +\mathstrut \) \(205900657400\) \(\nu^{9}\mathstrut +\mathstrut \) \(2669456439700\) \(\nu^{8}\mathstrut -\mathstrut \) \(2236009627109\) \(\nu^{7}\mathstrut -\mathstrut \) \(16177418069660\) \(\nu^{6}\mathstrut +\mathstrut \) \(9001746461066\) \(\nu^{5}\mathstrut +\mathstrut \) \(44509961821057\) \(\nu^{4}\mathstrut -\mathstrut \) \(13527828592988\) \(\nu^{3}\mathstrut -\mathstrut \) \(50559136604119\) \(\nu^{2}\mathstrut +\mathstrut \) \(6830665264495\) \(\nu\mathstrut +\mathstrut \) \(19623283268508\)\()/\)\(316533270094\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(15\) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(16\) \(\beta_{2}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(75\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(16\) \(\beta_{10}\mathstrut +\mathstrut \) \(7\) \(\beta_{9}\mathstrut -\mathstrut \) \(33\) \(\beta_{8}\mathstrut +\mathstrut \) \(16\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut -\mathstrut \) \(37\) \(\beta_{4}\mathstrut +\mathstrut \) \(49\) \(\beta_{3}\mathstrut +\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(100\) \(\beta_{1}\mathstrut +\mathstrut \) \(24\)
\(\nu^{6}\)\(=\)\(-\)\(4\) \(\beta_{12}\mathstrut -\mathstrut \) \(12\) \(\beta_{11}\mathstrut -\mathstrut \) \(36\) \(\beta_{10}\mathstrut +\mathstrut \) \(7\) \(\beta_{9}\mathstrut -\mathstrut \) \(22\) \(\beta_{8}\mathstrut +\mathstrut \) \(198\) \(\beta_{7}\mathstrut +\mathstrut \) \(139\) \(\beta_{6}\mathstrut -\mathstrut \) \(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(15\) \(\beta_{3}\mathstrut +\mathstrut \) \(215\) \(\beta_{2}\mathstrut -\mathstrut \) \(86\) \(\beta_{1}\mathstrut +\mathstrut \) \(896\)
\(\nu^{7}\)\(=\)\(52\) \(\beta_{12}\mathstrut +\mathstrut \) \(42\) \(\beta_{11}\mathstrut +\mathstrut \) \(216\) \(\beta_{10}\mathstrut +\mathstrut \) \(178\) \(\beta_{9}\mathstrut -\mathstrut \) \(477\) \(\beta_{8}\mathstrut -\mathstrut \) \(9\) \(\beta_{7}\mathstrut +\mathstrut \) \(201\) \(\beta_{6}\mathstrut -\mathstrut \) \(229\) \(\beta_{5}\mathstrut -\mathstrut \) \(535\) \(\beta_{4}\mathstrut +\mathstrut \) \(693\) \(\beta_{3}\mathstrut +\mathstrut \) \(163\) \(\beta_{2}\mathstrut +\mathstrut \) \(1187\) \(\beta_{1}\mathstrut +\mathstrut \) \(237\)
\(\nu^{8}\)\(=\)\(-\)\(136\) \(\beta_{12}\mathstrut -\mathstrut \) \(70\) \(\beta_{11}\mathstrut -\mathstrut \) \(520\) \(\beta_{10}\mathstrut +\mathstrut \) \(190\) \(\beta_{9}\mathstrut -\mathstrut \) \(367\) \(\beta_{8}\mathstrut +\mathstrut \) \(2549\) \(\beta_{7}\mathstrut +\mathstrut \) \(1644\) \(\beta_{6}\mathstrut -\mathstrut \) \(154\) \(\beta_{5}\mathstrut +\mathstrut \) \(57\) \(\beta_{4}\mathstrut +\mathstrut \) \(166\) \(\beta_{3}\mathstrut +\mathstrut \) \(2789\) \(\beta_{2}\mathstrut -\mathstrut \) \(1416\) \(\beta_{1}\mathstrut +\mathstrut \) \(11090\)
\(\nu^{9}\)\(=\)\(1011\) \(\beta_{12}\mathstrut +\mathstrut \) \(650\) \(\beta_{11}\mathstrut +\mathstrut \) \(2812\) \(\beta_{10}\mathstrut +\mathstrut \) \(3318\) \(\beta_{9}\mathstrut -\mathstrut \) \(6682\) \(\beta_{8}\mathstrut -\mathstrut \) \(241\) \(\beta_{7}\mathstrut +\mathstrut \) \(2403\) \(\beta_{6}\mathstrut -\mathstrut \) \(3190\) \(\beta_{5}\mathstrut -\mathstrut \) \(7268\) \(\beta_{4}\mathstrut +\mathstrut \) \(9510\) \(\beta_{3}\mathstrut +\mathstrut \) \(1815\) \(\beta_{2}\mathstrut +\mathstrut \) \(14464\) \(\beta_{1}\mathstrut +\mathstrut \) \(2129\)
\(\nu^{10}\)\(=\)\(-\)\(3017\) \(\beta_{12}\mathstrut +\mathstrut \) \(419\) \(\beta_{11}\mathstrut -\mathstrut \) \(7116\) \(\beta_{10}\mathstrut +\mathstrut \) \(3536\) \(\beta_{9}\mathstrut -\mathstrut \) \(5573\) \(\beta_{8}\mathstrut +\mathstrut \) \(32694\) \(\beta_{7}\mathstrut +\mathstrut \) \(19796\) \(\beta_{6}\mathstrut -\mathstrut \) \(1499\) \(\beta_{5}\mathstrut +\mathstrut \) \(1131\) \(\beta_{4}\mathstrut +\mathstrut \) \(1557\) \(\beta_{3}\mathstrut +\mathstrut \) \(36022\) \(\beta_{2}\mathstrut -\mathstrut \) \(21290\) \(\beta_{1}\mathstrut +\mathstrut \) \(139500\)
\(\nu^{11}\)\(=\)\(17512\) \(\beta_{12}\mathstrut +\mathstrut \) \(8931\) \(\beta_{11}\mathstrut +\mathstrut \) \(36263\) \(\beta_{10}\mathstrut +\mathstrut \) \(54796\) \(\beta_{9}\mathstrut -\mathstrut \) \(92393\) \(\beta_{8}\mathstrut -\mathstrut \) \(4410\) \(\beta_{7}\mathstrut +\mathstrut \) \(28728\) \(\beta_{6}\mathstrut -\mathstrut \) \(44052\) \(\beta_{5}\mathstrut -\mathstrut \) \(97071\) \(\beta_{4}\mathstrut +\mathstrut \) \(129454\) \(\beta_{3}\mathstrut +\mathstrut \) \(20001\) \(\beta_{2}\mathstrut +\mathstrut \) \(178798\) \(\beta_{1}\mathstrut +\mathstrut \) \(17574\)
\(\nu^{12}\)\(=\)\(-\)\(56146\) \(\beta_{12}\mathstrut +\mathstrut \) \(22414\) \(\beta_{11}\mathstrut -\mathstrut \) \(95843\) \(\beta_{10}\mathstrut +\mathstrut \) \(56188\) \(\beta_{9}\mathstrut -\mathstrut \) \(80875\) \(\beta_{8}\mathstrut +\mathstrut \) \(420209\) \(\beta_{7}\mathstrut +\mathstrut \) \(241672\) \(\beta_{6}\mathstrut -\mathstrut \) \(12416\) \(\beta_{5}\mathstrut +\mathstrut \) \(19472\) \(\beta_{4}\mathstrut +\mathstrut \) \(11516\) \(\beta_{3}\mathstrut +\mathstrut \) \(466863\) \(\beta_{2}\mathstrut -\mathstrut \) \(306338\) \(\beta_{1}\mathstrut +\mathstrut \) \(1771965\)

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
−3.67523
−3.54523
−2.08649
−2.01149
−1.22001
−0.975286
−0.757993
1.20657
1.95504
1.98373
3.21537
3.25869
3.65233
−1.00000 1.00000 1.00000 −3.67523 −1.00000 −3.11602 −1.00000 1.00000 3.67523
1.2 −1.00000 1.00000 1.00000 −3.54523 −1.00000 0.166701 −1.00000 1.00000 3.54523
1.3 −1.00000 1.00000 1.00000 −2.08649 −1.00000 −3.62308 −1.00000 1.00000 2.08649
1.4 −1.00000 1.00000 1.00000 −2.01149 −1.00000 0.821182 −1.00000 1.00000 2.01149
1.5 −1.00000 1.00000 1.00000 −1.22001 −1.00000 3.96620 −1.00000 1.00000 1.22001
1.6 −1.00000 1.00000 1.00000 −0.975286 −1.00000 −3.46524 −1.00000 1.00000 0.975286
1.7 −1.00000 1.00000 1.00000 −0.757993 −1.00000 1.64575 −1.00000 1.00000 0.757993
1.8 −1.00000 1.00000 1.00000 1.20657 −1.00000 −1.77104 −1.00000 1.00000 −1.20657
1.9 −1.00000 1.00000 1.00000 1.95504 −1.00000 −2.89499 −1.00000 1.00000 −1.95504
1.10 −1.00000 1.00000 1.00000 1.98373 −1.00000 3.61896 −1.00000 1.00000 −1.98373
1.11 −1.00000 1.00000 1.00000 3.21537 −1.00000 −0.928130 −1.00000 1.00000 −3.21537
1.12 −1.00000 1.00000 1.00000 3.25869 −1.00000 3.01501 −1.00000 1.00000 −3.25869
1.13 −1.00000 1.00000 1.00000 3.65233 −1.00000 1.56471 −1.00000 1.00000 −3.65233
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(13\) \(-1\)
\(103\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{13} - \cdots\)
\(T_{7}^{13} + \cdots\)