Properties

Label 8034.2.a.y
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

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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: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13}\mathstrut -\mathstrut \) \(5\) \(x^{12}\mathstrut -\mathstrut \) \(25\) \(x^{11}\mathstrut +\mathstrut \) \(134\) \(x^{10}\mathstrut +\mathstrut \) \(196\) \(x^{9}\mathstrut -\mathstrut \) \(1151\) \(x^{8}\mathstrut -\mathstrut \) \(801\) \(x^{7}\mathstrut +\mathstrut \) \(4263\) \(x^{6}\mathstrut +\mathstrut \) \(2205\) \(x^{5}\mathstrut -\mathstrut \) \(6840\) \(x^{4}\mathstrut -\mathstrut \) \(3579\) \(x^{3}\mathstrut +\mathstrut \) \(3559\) \(x^{2}\mathstrut +\mathstrut \) \(1839\) \(x\mathstrut -\mathstrut \) \(180\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(29990952597\) \(\nu^{12}\mathstrut -\mathstrut \) \(43375192180\) \(\nu^{11}\mathstrut -\mathstrut \) \(1136475186639\) \(\nu^{10}\mathstrut +\mathstrut \) \(1007708200373\) \(\nu^{9}\mathstrut +\mathstrut \) \(15167079562047\) \(\nu^{8}\mathstrut -\mathstrut \) \(5134111470190\) \(\nu^{7}\mathstrut -\mathstrut \) \(86013366708565\) \(\nu^{6}\mathstrut -\mathstrut \) \(14172806973364\) \(\nu^{5}\mathstrut +\mathstrut \) \(192287295025937\) \(\nu^{4}\mathstrut +\mathstrut \) \(102799778649133\) \(\nu^{3}\mathstrut -\mathstrut \) \(112700868033808\) \(\nu^{2}\mathstrut -\mathstrut \) \(76422756434293\) \(\nu\mathstrut +\mathstrut \) \(4629967598166\)\()/\)\(1742391215594\)
\(\beta_{3}\)\(=\)\((\)\(65603934565\) \(\nu^{12}\mathstrut -\mathstrut \) \(181553890487\) \(\nu^{11}\mathstrut -\mathstrut \) \(2150123291131\) \(\nu^{10}\mathstrut +\mathstrut \) \(4562233057208\) \(\nu^{9}\mathstrut +\mathstrut \) \(25073320454392\) \(\nu^{8}\mathstrut -\mathstrut \) \(32931751958387\) \(\nu^{7}\mathstrut -\mathstrut \) \(132998549746986\) \(\nu^{6}\mathstrut +\mathstrut \) \(69118693945305\) \(\nu^{5}\mathstrut +\mathstrut \) \(295215629097864\) \(\nu^{4}\mathstrut +\mathstrut \) \(15139766651310\) \(\nu^{3}\mathstrut -\mathstrut \) \(186156246848460\) \(\nu^{2}\mathstrut -\mathstrut \) \(51907864728215\) \(\nu\mathstrut +\mathstrut \) \(10324372473579\)\()/\)\(2613586823391\)
\(\beta_{4}\)\(=\)\((\)\(162857412779\) \(\nu^{12}\mathstrut -\mathstrut \) \(677670898672\) \(\nu^{11}\mathstrut -\mathstrut \) \(4278067082381\) \(\nu^{10}\mathstrut +\mathstrut \) \(16667349248383\) \(\nu^{9}\mathstrut +\mathstrut \) \(36727857551753\) \(\nu^{8}\mathstrut -\mathstrut \) \(118345177553344\) \(\nu^{7}\mathstrut -\mathstrut \) \(156262962783561\) \(\nu^{6}\mathstrut +\mathstrut \) \(294974331366942\) \(\nu^{5}\mathstrut +\mathstrut \) \(316924790401641\) \(\nu^{4}\mathstrut -\mathstrut \) \(194513299987389\) \(\nu^{3}\mathstrut -\mathstrut \) \(188494938439110\) \(\nu^{2}\mathstrut +\mathstrut \) \(13543461373865\) \(\nu\mathstrut +\mathstrut \) \(13092136637310\)\()/\)\(5227173646782\)
\(\beta_{5}\)\(=\)\((\)\(102100965329\) \(\nu^{12}\mathstrut -\mathstrut \) \(304209647056\) \(\nu^{11}\mathstrut -\mathstrut \) \(3169887944810\) \(\nu^{10}\mathstrut +\mathstrut \) \(7262339101606\) \(\nu^{9}\mathstrut +\mathstrut \) \(34848573902027\) \(\nu^{8}\mathstrut -\mathstrut \) \(46654184977558\) \(\nu^{7}\mathstrut -\mathstrut \) \(178805884240980\) \(\nu^{6}\mathstrut +\mathstrut \) \(68608084577061\) \(\nu^{5}\mathstrut +\mathstrut \) \(379650761799534\) \(\nu^{4}\mathstrut +\mathstrut \) \(101715657395343\) \(\nu^{3}\mathstrut -\mathstrut \) \(187075360189239\) \(\nu^{2}\mathstrut -\mathstrut \) \(98831180192488\) \(\nu\mathstrut -\mathstrut \) \(14347124251584\)\()/\)\(2613586823391\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(217714208045\) \(\nu^{12}\mathstrut +\mathstrut \) \(1005449553016\) \(\nu^{11}\mathstrut +\mathstrut \) \(5411101570403\) \(\nu^{10}\mathstrut -\mathstrut \) \(25448761107733\) \(\nu^{9}\mathstrut -\mathstrut \) \(41088867689801\) \(\nu^{8}\mathstrut +\mathstrut \) \(193780513946266\) \(\nu^{7}\mathstrut +\mathstrut \) \(145599003207645\) \(\nu^{6}\mathstrut -\mathstrut \) \(575826243150648\) \(\nu^{5}\mathstrut -\mathstrut \) \(243229487220105\) \(\nu^{4}\mathstrut +\mathstrut \) \(638472984040929\) \(\nu^{3}\mathstrut +\mathstrut \) \(104596802213862\) \(\nu^{2}\mathstrut -\mathstrut \) \(206073950699651\) \(\nu\mathstrut +\mathstrut \) \(13131093814746\)\()/\)\(5227173646782\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(255095470579\) \(\nu^{12}\mathstrut +\mathstrut \) \(661810602824\) \(\nu^{11}\mathstrut +\mathstrut \) \(8285169875425\) \(\nu^{10}\mathstrut -\mathstrut \) \(15232827751271\) \(\nu^{9}\mathstrut -\mathstrut \) \(96662482099681\) \(\nu^{8}\mathstrut +\mathstrut \) \(86623598556140\) \(\nu^{7}\mathstrut +\mathstrut \) \(524052689971935\) \(\nu^{6}\mathstrut -\mathstrut \) \(27010984297314\) \(\nu^{5}\mathstrut -\mathstrut \) \(1182427264108857\) \(\nu^{4}\mathstrut -\mathstrut \) \(557297580549693\) \(\nu^{3}\mathstrut +\mathstrut \) \(699141028801284\) \(\nu^{2}\mathstrut +\mathstrut \) \(417558588046067\) \(\nu\mathstrut -\mathstrut \) \(13275752967450\)\()/\)\(5227173646782\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(87548046093\) \(\nu^{12}\mathstrut +\mathstrut \) \(264959999010\) \(\nu^{11}\mathstrut +\mathstrut \) \(2727333681067\) \(\nu^{10}\mathstrut -\mathstrut \) \(6425835183533\) \(\nu^{9}\mathstrut -\mathstrut \) \(30213793968393\) \(\nu^{8}\mathstrut +\mathstrut \) \(43083002164404\) \(\nu^{7}\mathstrut +\mathstrut \) \(157724847260109\) \(\nu^{6}\mathstrut -\mathstrut \) \(76859446109462\) \(\nu^{5}\mathstrut -\mathstrut \) \(352937772776757\) \(\nu^{4}\mathstrut -\mathstrut \) \(51039017330433\) \(\nu^{3}\mathstrut +\mathstrut \) \(224469599916190\) \(\nu^{2}\mathstrut +\mathstrut \) \(76919571886465\) \(\nu\mathstrut -\mathstrut \) \(15396019776588\)\()/\)\(1742391215594\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(137156022856\) \(\nu^{12}\mathstrut +\mathstrut \) \(292035064997\) \(\nu^{11}\mathstrut +\mathstrut \) \(4686634469818\) \(\nu^{10}\mathstrut -\mathstrut \) \(6331690706375\) \(\nu^{9}\mathstrut -\mathstrut \) \(57883623377098\) \(\nu^{8}\mathstrut +\mathstrut \) \(27934282538669\) \(\nu^{7}\mathstrut +\mathstrut \) \(327020126294178\) \(\nu^{6}\mathstrut +\mathstrut \) \(73722367174863\) \(\nu^{5}\mathstrut -\mathstrut \) \(764557275936216\) \(\nu^{4}\mathstrut -\mathstrut \) \(483649010501916\) \(\nu^{3}\mathstrut +\mathstrut \) \(497688206439438\) \(\nu^{2}\mathstrut +\mathstrut \) \(341001851855144\) \(\nu\mathstrut -\mathstrut \) \(33755880545082\)\()/\)\(2613586823391\)
\(\beta_{10}\)\(=\)\((\)\(48721679165\) \(\nu^{12}\mathstrut -\mathstrut \) \(146572031608\) \(\nu^{11}\mathstrut -\mathstrut \) \(1525594606540\) \(\nu^{10}\mathstrut +\mathstrut \) \(3583008630877\) \(\nu^{9}\mathstrut +\mathstrut \) \(16951906999904\) \(\nu^{8}\mathstrut -\mathstrut \) \(24498849631117\) \(\nu^{7}\mathstrut -\mathstrut \) \(87888224114164\) \(\nu^{6}\mathstrut +\mathstrut \) \(46835853163763\) \(\nu^{5}\mathstrut +\mathstrut \) \(192397099265341\) \(\nu^{4}\mathstrut +\mathstrut \) \(17354205928509\) \(\nu^{3}\mathstrut -\mathstrut \) \(113951919651177\) \(\nu^{2}\mathstrut -\mathstrut \) \(30844919826139\) \(\nu\mathstrut +\mathstrut \) \(4959261111660\)\()/\)\(871195607797\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(389665509547\) \(\nu^{12}\mathstrut +\mathstrut \) \(1140098743058\) \(\nu^{11}\mathstrut +\mathstrut \) \(12323418899743\) \(\nu^{10}\mathstrut -\mathstrut \) \(27777758676353\) \(\nu^{9}\mathstrut -\mathstrut \) \(138454266189829\) \(\nu^{8}\mathstrut +\mathstrut \) \(187683228295316\) \(\nu^{7}\mathstrut +\mathstrut \) \(721098516997287\) \(\nu^{6}\mathstrut -\mathstrut \) \(334211798029062\) \(\nu^{5}\mathstrut -\mathstrut \) \(1567675103232729\) \(\nu^{4}\mathstrut -\mathstrut \) \(233571225428787\) \(\nu^{3}\mathstrut +\mathstrut \) \(874832876252430\) \(\nu^{2}\mathstrut +\mathstrut \) \(315223219608983\) \(\nu\mathstrut +\mathstrut \) \(1172172013944\)\()/\)\(5227173646782\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(114093379497\) \(\nu^{12}\mathstrut +\mathstrut \) \(401458140693\) \(\nu^{11}\mathstrut +\mathstrut \) \(3322146417204\) \(\nu^{10}\mathstrut -\mathstrut \) \(9875157476924\) \(\nu^{9}\mathstrut -\mathstrut \) \(33571232899839\) \(\nu^{8}\mathstrut +\mathstrut \) \(69389648399099\) \(\nu^{7}\mathstrut +\mathstrut \) \(162730885578322\) \(\nu^{6}\mathstrut -\mathstrut \) \(157270125105526\) \(\nu^{5}\mathstrut -\mathstrut \) \(343557388191574\) \(\nu^{4}\mathstrut +\mathstrut \) \(46987436209824\) \(\nu^{3}\mathstrut +\mathstrut \) \(192897249331282\) \(\nu^{2}\mathstrut +\mathstrut \) \(35315463008526\) \(\nu\mathstrut -\mathstrut \) \(5083097540960\)\()/\)\(871195607797\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(6\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(16\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(12\) \(\beta_{12}\mathstrut +\mathstrut \) \(9\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(4\) \(\beta_{9}\mathstrut -\mathstrut \) \(22\) \(\beta_{8}\mathstrut -\mathstrut \) \(5\) \(\beta_{7}\mathstrut -\mathstrut \) \(14\) \(\beta_{6}\mathstrut +\mathstrut \) \(18\) \(\beta_{5}\mathstrut -\mathstrut \) \(6\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(36\) \(\beta_{1}\mathstrut +\mathstrut \) \(58\)
\(\nu^{5}\)\(=\)\(\beta_{12}\mathstrut +\mathstrut \) \(37\) \(\beta_{11}\mathstrut -\mathstrut \) \(62\) \(\beta_{10}\mathstrut +\mathstrut \) \(34\) \(\beta_{9}\mathstrut -\mathstrut \) \(144\) \(\beta_{8}\mathstrut -\mathstrut \) \(13\) \(\beta_{7}\mathstrut -\mathstrut \) \(16\) \(\beta_{6}\mathstrut +\mathstrut \) \(54\) \(\beta_{5}\mathstrut -\mathstrut \) \(63\) \(\beta_{4}\mathstrut -\mathstrut \) \(36\) \(\beta_{3}\mathstrut +\mathstrut \) \(21\) \(\beta_{2}\mathstrut +\mathstrut \) \(273\) \(\beta_{1}\mathstrut +\mathstrut \) \(65\)
\(\nu^{6}\)\(=\)\(161\) \(\beta_{12}\mathstrut +\mathstrut \) \(243\) \(\beta_{11}\mathstrut -\mathstrut \) \(67\) \(\beta_{10}\mathstrut +\mathstrut \) \(127\) \(\beta_{9}\mathstrut -\mathstrut \) \(479\) \(\beta_{8}\mathstrut -\mathstrut \) \(145\) \(\beta_{7}\mathstrut -\mathstrut \) \(206\) \(\beta_{6}\mathstrut +\mathstrut \) \(347\) \(\beta_{5}\mathstrut -\mathstrut \) \(170\) \(\beta_{4}\mathstrut +\mathstrut \) \(34\) \(\beta_{3}\mathstrut +\mathstrut \) \(45\) \(\beta_{2}\mathstrut +\mathstrut \) \(696\) \(\beta_{1}\mathstrut +\mathstrut \) \(900\)
\(\nu^{7}\)\(=\)\(35\) \(\beta_{12}\mathstrut +\mathstrut \) \(951\) \(\beta_{11}\mathstrut -\mathstrut \) \(1180\) \(\beta_{10}\mathstrut +\mathstrut \) \(801\) \(\beta_{9}\mathstrut -\mathstrut \) \(2920\) \(\beta_{8}\mathstrut -\mathstrut \) \(447\) \(\beta_{7}\mathstrut -\mathstrut \) \(267\) \(\beta_{6}\mathstrut +\mathstrut \) \(1218\) \(\beta_{5}\mathstrut -\mathstrut \) \(1212\) \(\beta_{4}\mathstrut -\mathstrut \) \(495\) \(\beta_{3}\mathstrut +\mathstrut \) \(375\) \(\beta_{2}\mathstrut +\mathstrut \) \(4847\) \(\beta_{1}\mathstrut +\mathstrut \) \(1468\)
\(\nu^{8}\)\(=\)\(2393\) \(\beta_{12}\mathstrut +\mathstrut \) \(5389\) \(\beta_{11}\mathstrut -\mathstrut \) \(2073\) \(\beta_{10}\mathstrut +\mathstrut \) \(3071\) \(\beta_{9}\mathstrut -\mathstrut \) \(10213\) \(\beta_{8}\mathstrut -\mathstrut \) \(3254\) \(\beta_{7}\mathstrut -\mathstrut \) \(3203\) \(\beta_{6}\mathstrut +\mathstrut \) \(6899\) \(\beta_{5}\mathstrut -\mathstrut \) \(3815\) \(\beta_{4}\mathstrut +\mathstrut \) \(848\) \(\beta_{3}\mathstrut +\mathstrut \) \(866\) \(\beta_{2}\mathstrut +\mathstrut \) \(14054\) \(\beta_{1}\mathstrut +\mathstrut \) \(15532\)
\(\nu^{9}\)\(=\)\(1150\) \(\beta_{12}\mathstrut +\mathstrut \) \(21643\) \(\beta_{11}\mathstrut -\mathstrut \) \(22293\) \(\beta_{10}\mathstrut +\mathstrut \) \(17056\) \(\beta_{9}\mathstrut -\mathstrut \) \(57599\) \(\beta_{8}\mathstrut -\mathstrut \) \(11144\) \(\beta_{7}\mathstrut -\mathstrut \) \(4950\) \(\beta_{6}\mathstrut +\mathstrut \) \(26262\) \(\beta_{5}\mathstrut -\mathstrut \) \(23245\) \(\beta_{4}\mathstrut -\mathstrut \) \(6042\) \(\beta_{3}\mathstrut +\mathstrut \) \(6485\) \(\beta_{2}\mathstrut +\mathstrut \) \(89148\) \(\beta_{1}\mathstrut +\mathstrut \) \(33244\)
\(\nu^{10}\)\(=\)\(38079\) \(\beta_{12}\mathstrut +\mathstrut \) \(113170\) \(\beta_{11}\mathstrut -\mathstrut \) \(52003\) \(\beta_{10}\mathstrut +\mathstrut \) \(67753\) \(\beta_{9}\mathstrut -\mathstrut \) \(214027\) \(\beta_{8}\mathstrut -\mathstrut \) \(67665\) \(\beta_{7}\mathstrut -\mathstrut \) \(52315\) \(\beta_{6}\mathstrut +\mathstrut \) \(138605\) \(\beta_{5}\mathstrut -\mathstrut \) \(80678\) \(\beta_{4}\mathstrut +\mathstrut \) \(18531\) \(\beta_{3}\mathstrut +\mathstrut \) \(16603\) \(\beta_{2}\mathstrut +\mathstrut \) \(288396\) \(\beta_{1}\mathstrut +\mathstrut \) \(281896\)
\(\nu^{11}\)\(=\)\(33272\) \(\beta_{12}\mathstrut +\mathstrut \) \(467279\) \(\beta_{11}\mathstrut -\mathstrut \) \(424267\) \(\beta_{10}\mathstrut +\mathstrut \) \(350742\) \(\beta_{9}\mathstrut -\mathstrut \) \(1135805\) \(\beta_{8}\mathstrut -\mathstrut \) \(249258\) \(\beta_{7}\mathstrut -\mathstrut \) \(98641\) \(\beta_{6}\mathstrut +\mathstrut \) \(554534\) \(\beta_{5}\mathstrut -\mathstrut \) \(449856\) \(\beta_{4}\mathstrut -\mathstrut \) \(64501\) \(\beta_{3}\mathstrut +\mathstrut \) \(113521\) \(\beta_{2}\mathstrut +\mathstrut \) \(1685330\) \(\beta_{1}\mathstrut +\mathstrut \) \(737045\)
\(\nu^{12}\)\(=\)\(637279\) \(\beta_{12}\mathstrut +\mathstrut \) \(2327795\) \(\beta_{11}\mathstrut -\mathstrut \) \(1193552\) \(\beta_{10}\mathstrut +\mathstrut \) \(1436886\) \(\beta_{9}\mathstrut -\mathstrut \) \(4436546\) \(\beta_{8}\mathstrut -\mathstrut \) \(1370991\) \(\beta_{7}\mathstrut -\mathstrut \) \(893575\) \(\beta_{6}\mathstrut +\mathstrut \) \(2793619\) \(\beta_{5}\mathstrut -\mathstrut \) \(1673509\) \(\beta_{4}\mathstrut +\mathstrut \) \(379292\) \(\beta_{3}\mathstrut +\mathstrut \) \(324468\) \(\beta_{2}\mathstrut +\mathstrut \) \(5942369\) \(\beta_{1}\mathstrut +\mathstrut \) \(5275921\)

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
−2.14079
0.0852190
4.50145
−1.72356
1.94320
−1.46489
−0.723662
−3.83598
−0.892751
3.49310
2.89547
1.94844
0.914750
1.00000 −1.00000 1.00000 −4.27728 −1.00000 −2.14079 1.00000 1.00000 −4.27728
1.2 1.00000 −1.00000 1.00000 −4.16745 −1.00000 0.0852190 1.00000 1.00000 −4.16745
1.3 1.00000 −1.00000 1.00000 −1.68854 −1.00000 4.50145 1.00000 1.00000 −1.68854
1.4 1.00000 −1.00000 1.00000 −1.45246 −1.00000 −1.72356 1.00000 1.00000 −1.45246
1.5 1.00000 −1.00000 1.00000 −0.714571 −1.00000 1.94320 1.00000 1.00000 −0.714571
1.6 1.00000 −1.00000 1.00000 −0.556182 −1.00000 −1.46489 1.00000 1.00000 −0.556182
1.7 1.00000 −1.00000 1.00000 0.370604 −1.00000 −0.723662 1.00000 1.00000 0.370604
1.8 1.00000 −1.00000 1.00000 1.02710 −1.00000 −3.83598 1.00000 1.00000 1.02710
1.9 1.00000 −1.00000 1.00000 1.04935 −1.00000 −0.892751 1.00000 1.00000 1.04935
1.10 1.00000 −1.00000 1.00000 2.90785 −1.00000 3.49310 1.00000 1.00000 2.90785
1.11 1.00000 −1.00000 1.00000 3.35434 −1.00000 2.89547 1.00000 1.00000 3.35434
1.12 1.00000 −1.00000 1.00000 3.48931 −1.00000 1.94844 1.00000 1.00000 3.48931
1.13 1.00000 −1.00000 1.00000 3.65794 −1.00000 0.914750 1.00000 1.00000 3.65794
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

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