Properties

Label 8034.2.a.z
Level 8034
Weight 2
Character orbit 8034.a
Self dual Yes
Analytic conductor 64.152
Analytic rank 0
Dimension 14
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(3\) \(x^{13}\mathstrut -\mathstrut \) \(36\) \(x^{12}\mathstrut +\mathstrut \) \(108\) \(x^{11}\mathstrut +\mathstrut \) \(434\) \(x^{10}\mathstrut -\mathstrut \) \(1239\) \(x^{9}\mathstrut -\mathstrut \) \(2404\) \(x^{8}\mathstrut +\mathstrut \) \(6204\) \(x^{7}\mathstrut +\mathstrut \) \(6361\) \(x^{6}\mathstrut -\mathstrut \) \(14618\) \(x^{5}\mathstrut -\mathstrut \) \(7015\) \(x^{4}\mathstrut +\mathstrut \) \(15763\) \(x^{3}\mathstrut +\mathstrut \) \(1118\) \(x^{2}\mathstrut -\mathstrut \) \(6316\) \(x\mathstrut +\mathstrut \) \(1552\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(3341759347\) \(\nu^{13}\mathstrut +\mathstrut \) \(518140367105\) \(\nu^{12}\mathstrut -\mathstrut \) \(1741789540612\) \(\nu^{11}\mathstrut -\mathstrut \) \(17247628357060\) \(\nu^{10}\mathstrut +\mathstrut \) \(62687524315698\) \(\nu^{9}\mathstrut +\mathstrut \) \(176417731545861\) \(\nu^{8}\mathstrut -\mathstrut \) \(670560099753944\) \(\nu^{7}\mathstrut -\mathstrut \) \(742563932810444\) \(\nu^{6}\mathstrut +\mathstrut \) \(2912651300509245\) \(\nu^{5}\mathstrut +\mathstrut \) \(1186311372117602\) \(\nu^{4}\mathstrut -\mathstrut \) \(5100601073725863\) \(\nu^{3}\mathstrut -\mathstrut \) \(235038449926693\) \(\nu^{2}\mathstrut +\mathstrut \) \(2735906855139534\) \(\nu\mathstrut -\mathstrut \) \(592135085106752\)\()/\)\(30448521494580\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(12670008395\) \(\nu^{13}\mathstrut +\mathstrut \) \(121522864993\) \(\nu^{12}\mathstrut +\mathstrut \) \(241219668796\) \(\nu^{11}\mathstrut -\mathstrut \) \(4283106154742\) \(\nu^{10}\mathstrut +\mathstrut \) \(1787398276572\) \(\nu^{9}\mathstrut +\mathstrut \) \(49208409954315\) \(\nu^{8}\mathstrut -\mathstrut \) \(43074872724922\) \(\nu^{7}\mathstrut -\mathstrut \) \(252692469808438\) \(\nu^{6}\mathstrut +\mathstrut \) \(223588882118493\) \(\nu^{5}\mathstrut +\mathstrut \) \(575804728969528\) \(\nu^{4}\mathstrut -\mathstrut \) \(446376121324599\) \(\nu^{3}\mathstrut -\mathstrut \) \(429175712907053\) \(\nu^{2}\mathstrut +\mathstrut \) \(305776841980152\) \(\nu\mathstrut -\mathstrut \) \(7510828872730\)\()/\)\(15224260747290\)
\(\beta_{4}\)\(=\)\((\)\(7437129421\) \(\nu^{13}\mathstrut +\mathstrut \) \(8250683722\) \(\nu^{12}\mathstrut -\mathstrut \) \(328982202812\) \(\nu^{11}\mathstrut -\mathstrut \) \(355914392252\) \(\nu^{10}\mathstrut +\mathstrut \) \(5237503644027\) \(\nu^{9}\mathstrut +\mathstrut \) \(5955785366799\) \(\nu^{8}\mathstrut -\mathstrut \) \(36307339325161\) \(\nu^{7}\mathstrut -\mathstrut \) \(45172405213102\) \(\nu^{6}\mathstrut +\mathstrut \) \(103511839862979\) \(\nu^{5}\mathstrut +\mathstrut \) \(143064053475349\) \(\nu^{4}\mathstrut -\mathstrut \) \(94789427122899\) \(\nu^{3}\mathstrut -\mathstrut \) \(160392768597467\) \(\nu^{2}\mathstrut +\mathstrut \) \(16523568578403\) \(\nu\mathstrut +\mathstrut \) \(41261138165888\)\()/\)\(7612130373645\)
\(\beta_{5}\)\(=\)\((\)\(16735095548\) \(\nu^{13}\mathstrut -\mathstrut \) \(59503252771\) \(\nu^{12}\mathstrut -\mathstrut \) \(534709503235\) \(\nu^{11}\mathstrut +\mathstrut \) \(2013117619607\) \(\nu^{10}\mathstrut +\mathstrut \) \(4893999455973\) \(\nu^{9}\mathstrut -\mathstrut \) \(20391279195918\) \(\nu^{8}\mathstrut -\mathstrut \) \(13884105134675\) \(\nu^{7}\mathstrut +\mathstrut \) \(81401298589306\) \(\nu^{6}\mathstrut -\mathstrut \) \(20121761021580\) \(\nu^{5}\mathstrut -\mathstrut \) \(121000025836105\) \(\nu^{4}\mathstrut +\mathstrut \) \(146667384042234\) \(\nu^{3}\mathstrut +\mathstrut \) \(22338499957991\) \(\nu^{2}\mathstrut -\mathstrut \) \(164021431732794\) \(\nu\mathstrut +\mathstrut \) \(67234006456384\)\()/\)\(7612130373645\)
\(\beta_{6}\)\(=\)\((\)\(6829796404\) \(\nu^{13}\mathstrut -\mathstrut \) \(24088984468\) \(\nu^{12}\mathstrut -\mathstrut \) \(220735533930\) \(\nu^{11}\mathstrut +\mathstrut \) \(818192200816\) \(\nu^{10}\mathstrut +\mathstrut \) \(2167385042979\) \(\nu^{9}\mathstrut -\mathstrut \) \(8331580582394\) \(\nu^{8}\mathstrut -\mathstrut \) \(9159648834785\) \(\nu^{7}\mathstrut +\mathstrut \) \(33626274944798\) \(\nu^{6}\mathstrut +\mathstrut \) \(19719493652720\) \(\nu^{5}\mathstrut -\mathstrut \) \(50057127693710\) \(\nu^{4}\mathstrut -\mathstrut \) \(27525433033153\) \(\nu^{3}\mathstrut +\mathstrut \) \(16363934034728\) \(\nu^{2}\mathstrut +\mathstrut \) \(18990965483608\) \(\nu\mathstrut -\mathstrut \) \(696096892188\)\()/\)\(2537376791215\)
\(\beta_{7}\)\(=\)\((\)\(101168043581\) \(\nu^{13}\mathstrut -\mathstrut \) \(428736219223\) \(\nu^{12}\mathstrut -\mathstrut \) \(3096334884004\) \(\nu^{11}\mathstrut +\mathstrut \) \(14443924013984\) \(\nu^{10}\mathstrut +\mathstrut \) \(24846996819930\) \(\nu^{9}\mathstrut -\mathstrut \) \(146139006849039\) \(\nu^{8}\mathstrut -\mathstrut \) \(37437704187596\) \(\nu^{7}\mathstrut +\mathstrut \) \(574313684971036\) \(\nu^{6}\mathstrut -\mathstrut \) \(284435428081239\) \(\nu^{5}\mathstrut -\mathstrut \) \(772353790449958\) \(\nu^{4}\mathstrut +\mathstrut \) \(1021694063716269\) \(\nu^{3}\mathstrut +\mathstrut \) \(18632735153039\) \(\nu^{2}\mathstrut -\mathstrut \) \(873824979116130\) \(\nu\mathstrut +\mathstrut \) \(321698456894260\)\()/\)\(30448521494580\)
\(\beta_{8}\)\(=\)\((\)\(132195076697\) \(\nu^{13}\mathstrut -\mathstrut \) \(378058949095\) \(\nu^{12}\mathstrut -\mathstrut \) \(4698358621144\) \(\nu^{11}\mathstrut +\mathstrut \) \(13089742387748\) \(\nu^{10}\mathstrut +\mathstrut \) \(54759751083666\) \(\nu^{9}\mathstrut -\mathstrut \) \(137259949634175\) \(\nu^{8}\mathstrut -\mathstrut \) \(280935521431316\) \(\nu^{7}\mathstrut +\mathstrut \) \(563752133186128\) \(\nu^{6}\mathstrut +\mathstrut \) \(636877173580161\) \(\nu^{5}\mathstrut -\mathstrut \) \(853332240720538\) \(\nu^{4}\mathstrut -\mathstrut \) \(538687991427303\) \(\nu^{3}\mathstrut +\mathstrut \) \(267983097437951\) \(\nu^{2}\mathstrut +\mathstrut \) \(72415118017302\) \(\nu\mathstrut +\mathstrut \) \(93265012848748\)\()/\)\(30448521494580\)
\(\beta_{9}\)\(=\)\((\)\(37454660035\) \(\nu^{13}\mathstrut -\mathstrut \) \(65377606061\) \(\nu^{12}\mathstrut -\mathstrut \) \(1543899870542\) \(\nu^{11}\mathstrut +\mathstrut \) \(2517262146709\) \(\nu^{10}\mathstrut +\mathstrut \) \(22989677068086\) \(\nu^{9}\mathstrut -\mathstrut \) \(31415177667555\) \(\nu^{8}\mathstrut -\mathstrut \) \(162281355426796\) \(\nu^{7}\mathstrut +\mathstrut \) \(170536386071396\) \(\nu^{6}\mathstrut +\mathstrut \) \(566623348152159\) \(\nu^{5}\mathstrut -\mathstrut \) \(440040849178856\) \(\nu^{4}\mathstrut -\mathstrut \) \(910338490594662\) \(\nu^{3}\mathstrut +\mathstrut \) \(529906004653921\) \(\nu^{2}\mathstrut +\mathstrut \) \(498400052484456\) \(\nu\mathstrut -\mathstrut \) \(240724229362315\)\()/\)\(7612130373645\)
\(\beta_{10}\)\(=\)\((\)\(61385752497\) \(\nu^{13}\mathstrut -\mathstrut \) \(224068680263\) \(\nu^{12}\mathstrut -\mathstrut \) \(2016688298880\) \(\nu^{11}\mathstrut +\mathstrut \) \(7666899083920\) \(\nu^{10}\mathstrut +\mathstrut \) \(20353862368234\) \(\nu^{9}\mathstrut -\mathstrut \) \(80361026660595\) \(\nu^{8}\mathstrut -\mathstrut \) \(88069692253324\) \(\nu^{7}\mathstrut +\mathstrut \) \(350770178370856\) \(\nu^{6}\mathstrut +\mathstrut \) \(171611111684913\) \(\nu^{5}\mathstrut -\mathstrut \) \(659138629686466\) \(\nu^{4}\mathstrut -\mathstrut \) \(144254361560619\) \(\nu^{3}\mathstrut +\mathstrut \) \(482165035857659\) \(\nu^{2}\mathstrut +\mathstrut \) \(23740354322750\) \(\nu\mathstrut -\mathstrut \) \(113491089496292\)\()/\)\(10149507164860\)
\(\beta_{11}\)\(=\)\((\)\(61385752497\) \(\nu^{13}\mathstrut -\mathstrut \) \(224068680263\) \(\nu^{12}\mathstrut -\mathstrut \) \(2016688298880\) \(\nu^{11}\mathstrut +\mathstrut \) \(7666899083920\) \(\nu^{10}\mathstrut +\mathstrut \) \(20353862368234\) \(\nu^{9}\mathstrut -\mathstrut \) \(80361026660595\) \(\nu^{8}\mathstrut -\mathstrut \) \(88069692253324\) \(\nu^{7}\mathstrut +\mathstrut \) \(350770178370856\) \(\nu^{6}\mathstrut +\mathstrut \) \(171611111684913\) \(\nu^{5}\mathstrut -\mathstrut \) \(659138629686466\) \(\nu^{4}\mathstrut -\mathstrut \) \(144254361560619\) \(\nu^{3}\mathstrut +\mathstrut \) \(472015528692799\) \(\nu^{2}\mathstrut +\mathstrut \) \(23740354322750\) \(\nu\mathstrut -\mathstrut \) \(52594046507132\)\()/\)\(10149507164860\)
\(\beta_{12}\)\(=\)\((\)\(47799908567\) \(\nu^{13}\mathstrut -\mathstrut \) \(114887884225\) \(\nu^{12}\mathstrut -\mathstrut \) \(1783176710104\) \(\nu^{11}\mathstrut +\mathstrut \) \(4177793518358\) \(\nu^{10}\mathstrut +\mathstrut \) \(22844790904641\) \(\nu^{9}\mathstrut -\mathstrut \) \(47813470495515\) \(\nu^{8}\mathstrut -\mathstrut \) \(135044548880321\) \(\nu^{7}\mathstrut +\mathstrut \) \(231962318420908\) \(\nu^{6}\mathstrut +\mathstrut \) \(378111654620451\) \(\nu^{5}\mathstrut -\mathstrut \) \(496358883141133\) \(\nu^{4}\mathstrut -\mathstrut \) \(434168909972403\) \(\nu^{3}\mathstrut +\mathstrut \) \(413593508460356\) \(\nu^{2}\mathstrut +\mathstrut \) \(106258840381992\) \(\nu\mathstrut -\mathstrut \) \(91513305290492\)\()/\)\(7612130373645\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(120974609621\) \(\nu^{13}\mathstrut +\mathstrut \) \(303334971739\) \(\nu^{12}\mathstrut +\mathstrut \) \(4434352237832\) \(\nu^{11}\mathstrut -\mathstrut \) \(10647593900116\) \(\nu^{10}\mathstrut -\mathstrut \) \(55483675391710\) \(\nu^{9}\mathstrut +\mathstrut \) \(114757728296943\) \(\nu^{8}\mathstrut +\mathstrut \) \(325219663165232\) \(\nu^{7}\mathstrut -\mathstrut \) \(513376507176776\) \(\nu^{6}\mathstrut -\mathstrut \) \(932125820605101\) \(\nu^{5}\mathstrut +\mathstrut \) \(1007892134519590\) \(\nu^{4}\mathstrut +\mathstrut \) \(1197390391324439\) \(\nu^{3}\mathstrut -\mathstrut \) \(853786741692003\) \(\nu^{2}\mathstrut -\mathstrut \) \(506871319390622\) \(\nu\mathstrut +\mathstrut \) \(280944669298364\)\()/\)\(10149507164860\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(6\)
\(\nu^{3}\)\(=\)\(3\) \(\beta_{13}\mathstrut +\mathstrut \) \(3\) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{13}\mathstrut -\mathstrut \) \(4\) \(\beta_{12}\mathstrut -\mathstrut \) \(19\) \(\beta_{11}\mathstrut +\mathstrut \) \(16\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(4\) \(\beta_{8}\mathstrut -\mathstrut \) \(6\) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(8\) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(71\)
\(\nu^{5}\)\(=\)\(60\) \(\beta_{13}\mathstrut +\mathstrut \) \(61\) \(\beta_{12}\mathstrut +\mathstrut \) \(53\) \(\beta_{11}\mathstrut +\mathstrut \) \(7\) \(\beta_{10}\mathstrut +\mathstrut \) \(21\) \(\beta_{9}\mathstrut -\mathstrut \) \(41\) \(\beta_{8}\mathstrut +\mathstrut \) \(41\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(49\) \(\beta_{5}\mathstrut +\mathstrut \) \(46\) \(\beta_{4}\mathstrut +\mathstrut \) \(39\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(137\) \(\beta_{1}\mathstrut -\mathstrut \) \(54\)
\(\nu^{6}\)\(=\)\(-\)\(59\) \(\beta_{13}\mathstrut -\mathstrut \) \(126\) \(\beta_{12}\mathstrut -\mathstrut \) \(354\) \(\beta_{11}\mathstrut +\mathstrut \) \(250\) \(\beta_{10}\mathstrut +\mathstrut \) \(15\) \(\beta_{9}\mathstrut +\mathstrut \) \(101\) \(\beta_{8}\mathstrut -\mathstrut \) \(148\) \(\beta_{7}\mathstrut +\mathstrut \) \(64\) \(\beta_{6}\mathstrut +\mathstrut \) \(212\) \(\beta_{5}\mathstrut -\mathstrut \) \(30\) \(\beta_{4}\mathstrut -\mathstrut \) \(105\) \(\beta_{3}\mathstrut -\mathstrut \) \(151\) \(\beta_{1}\mathstrut +\mathstrut \) \(1064\)
\(\nu^{7}\)\(=\)\(1048\) \(\beta_{13}\mathstrut +\mathstrut \) \(1103\) \(\beta_{12}\mathstrut +\mathstrut \) \(1135\) \(\beta_{11}\mathstrut -\mathstrut \) \(76\) \(\beta_{10}\mathstrut +\mathstrut \) \(365\) \(\beta_{9}\mathstrut -\mathstrut \) \(772\) \(\beta_{8}\mathstrut +\mathstrut \) \(788\) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(981\) \(\beta_{5}\mathstrut +\mathstrut \) \(844\) \(\beta_{4}\mathstrut +\mathstrut \) \(701\) \(\beta_{3}\mathstrut -\mathstrut \) \(47\) \(\beta_{2}\mathstrut +\mathstrut \) \(2167\) \(\beta_{1}\mathstrut -\mathstrut \) \(1613\)
\(\nu^{8}\)\(=\)\(-\)\(1728\) \(\beta_{13}\mathstrut -\mathstrut \) \(2966\) \(\beta_{12}\mathstrut -\mathstrut \) \(6601\) \(\beta_{11}\mathstrut +\mathstrut \) \(4024\) \(\beta_{10}\mathstrut +\mathstrut \) \(101\) \(\beta_{9}\mathstrut +\mathstrut \) \(2173\) \(\beta_{8}\mathstrut -\mathstrut \) \(3046\) \(\beta_{7}\mathstrut +\mathstrut \) \(1168\) \(\beta_{6}\mathstrut +\mathstrut \) \(4462\) \(\beta_{5}\mathstrut -\mathstrut \) \(1070\) \(\beta_{4}\mathstrut -\mathstrut \) \(2280\) \(\beta_{3}\mathstrut +\mathstrut \) \(16\) \(\beta_{2}\mathstrut -\mathstrut \) \(3982\) \(\beta_{1}\mathstrut +\mathstrut \) \(17581\)
\(\nu^{9}\)\(=\)\(18247\) \(\beta_{13}\mathstrut +\mathstrut \) \(19912\) \(\beta_{12}\mathstrut +\mathstrut \) \(23085\) \(\beta_{11}\mathstrut -\mathstrut \) \(4313\) \(\beta_{10}\mathstrut +\mathstrut \) \(6128\) \(\beta_{9}\mathstrut -\mathstrut \) \(14338\) \(\beta_{8}\mathstrut +\mathstrut \) \(15014\) \(\beta_{7}\mathstrut -\mathstrut \) \(420\) \(\beta_{6}\mathstrut -\mathstrut \) \(18991\) \(\beta_{5}\mathstrut +\mathstrut \) \(14884\) \(\beta_{4}\mathstrut +\mathstrut \) \(12767\) \(\beta_{3}\mathstrut -\mathstrut \) \(873\) \(\beta_{2}\mathstrut +\mathstrut \) \(36799\) \(\beta_{1}\mathstrut -\mathstrut \) \(38962\)
\(\nu^{10}\)\(=\)\(-\)\(41245\) \(\beta_{13}\mathstrut -\mathstrut \) \(63259\) \(\beta_{12}\mathstrut -\mathstrut \) \(123644\) \(\beta_{11}\mathstrut +\mathstrut \) \(67177\) \(\beta_{10}\mathstrut -\mathstrut \) \(1642\) \(\beta_{9}\mathstrut +\mathstrut \) \(44844\) \(\beta_{8}\mathstrut -\mathstrut \) \(60274\) \(\beta_{7}\mathstrut +\mathstrut \) \(20510\) \(\beta_{6}\mathstrut +\mathstrut \) \(88164\) \(\beta_{5}\mathstrut -\mathstrut \) \(27624\) \(\beta_{4}\mathstrut -\mathstrut \) \(46650\) \(\beta_{3}\mathstrut +\mathstrut \) \(676\) \(\beta_{2}\mathstrut -\mathstrut \) \(91210\) \(\beta_{1}\mathstrut +\mathstrut \) \(305954\)
\(\nu^{11}\)\(=\)\(323677\) \(\beta_{13}\mathstrut +\mathstrut \) \(365063\) \(\beta_{12}\mathstrut +\mathstrut \) \(460293\) \(\beta_{11}\mathstrut -\mathstrut \) \(120204\) \(\beta_{10}\mathstrut +\mathstrut \) \(102976\) \(\beta_{9}\mathstrut -\mathstrut \) \(266446\) \(\beta_{8}\mathstrut +\mathstrut \) \(285976\) \(\beta_{7}\mathstrut -\mathstrut \) \(17338\) \(\beta_{6}\mathstrut -\mathstrut \) \(366787\) \(\beta_{5}\mathstrut +\mathstrut \) \(263343\) \(\beta_{4}\mathstrut +\mathstrut \) \(237094\) \(\beta_{3}\mathstrut -\mathstrut \) \(15430\) \(\beta_{2}\mathstrut +\mathstrut \) \(650857\) \(\beta_{1}\mathstrut -\mathstrut \) \(854144\)
\(\nu^{12}\)\(=\)\(-\)\(900094\) \(\beta_{13}\mathstrut -\mathstrut \) \(1291247\) \(\beta_{12}\mathstrut -\mathstrut \) \(2328995\) \(\beta_{11}\mathstrut +\mathstrut \) \(1159985\) \(\beta_{10}\mathstrut -\mathstrut \) \(89596\) \(\beta_{9}\mathstrut +\mathstrut \) \(907064\) \(\beta_{8}\mathstrut -\mathstrut \) \(1178554\) \(\beta_{7}\mathstrut +\mathstrut \) \(357528\) \(\beta_{6}\mathstrut +\mathstrut \) \(1706092\) \(\beta_{5}\mathstrut -\mathstrut \) \(629847\) \(\beta_{4}\mathstrut -\mathstrut \) \(929026\) \(\beta_{3}\mathstrut +\mathstrut \) \(19530\) \(\beta_{2}\mathstrut -\mathstrut \) \(1948105\) \(\beta_{1}\mathstrut +\mathstrut \) \(5498570\)
\(\nu^{13}\)\(=\)\(5856451\) \(\beta_{13}\mathstrut +\mathstrut \) \(6791797\) \(\beta_{12}\mathstrut +\mathstrut \) \(9079144\) \(\beta_{11}\mathstrut -\mathstrut \) \(2797826\) \(\beta_{10}\mathstrut +\mathstrut \) \(1753751\) \(\beta_{9}\mathstrut -\mathstrut \) \(4976458\) \(\beta_{8}\mathstrut +\mathstrut \) \(5457293\) \(\beta_{7}\mathstrut -\mathstrut \) \(489879\) \(\beta_{6}\mathstrut -\mathstrut \) \(7096317\) \(\beta_{5}\mathstrut +\mathstrut \) \(4725157\) \(\beta_{4}\mathstrut +\mathstrut \) \(4464975\) \(\beta_{3}\mathstrut -\mathstrut \) \(271490\) \(\beta_{2}\mathstrut +\mathstrut \) \(11817884\) \(\beta_{1}\mathstrut -\mathstrut \) \(17797008\)

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
3.68342
3.65422
3.21497
2.05100
1.52247
0.937159
0.664239
0.343018
−1.02849
−1.34598
−1.59859
−2.28948
−2.41893
−4.38902
−1.00000 −1.00000 1.00000 −3.68342 1.00000 5.03243 −1.00000 1.00000 3.68342
1.2 −1.00000 −1.00000 1.00000 −3.65422 1.00000 −2.98447 −1.00000 1.00000 3.65422
1.3 −1.00000 −1.00000 1.00000 −3.21497 1.00000 −1.40491 −1.00000 1.00000 3.21497
1.4 −1.00000 −1.00000 1.00000 −2.05100 1.00000 5.03739 −1.00000 1.00000 2.05100
1.5 −1.00000 −1.00000 1.00000 −1.52247 1.00000 −3.40750 −1.00000 1.00000 1.52247
1.6 −1.00000 −1.00000 1.00000 −0.937159 1.00000 −0.792704 −1.00000 1.00000 0.937159
1.7 −1.00000 −1.00000 1.00000 −0.664239 1.00000 2.36162 −1.00000 1.00000 0.664239
1.8 −1.00000 −1.00000 1.00000 −0.343018 1.00000 2.39676 −1.00000 1.00000 0.343018
1.9 −1.00000 −1.00000 1.00000 1.02849 1.00000 −4.96874 −1.00000 1.00000 −1.02849
1.10 −1.00000 −1.00000 1.00000 1.34598 1.00000 −3.24510 −1.00000 1.00000 −1.34598
1.11 −1.00000 −1.00000 1.00000 1.59859 1.00000 4.24832 −1.00000 1.00000 −1.59859
1.12 −1.00000 −1.00000 1.00000 2.28948 1.00000 −1.19378 −1.00000 1.00000 −2.28948
1.13 −1.00000 −1.00000 1.00000 2.41893 1.00000 −0.257176 −1.00000 1.00000 −2.41893
1.14 −1.00000 −1.00000 1.00000 4.38902 1.00000 3.17787 −1.00000 1.00000 −4.38902
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{14} + \cdots\)
\(T_{7}^{14} - \cdots\)