Properties

Label 8046.2.a.i
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: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(- q^{2}\) \(+ q^{4}\) \( -\beta_{1} q^{5} \) \( + ( 1 - \beta_{5} ) q^{7} \) \(- q^{8}\) \(+O(q^{10})\) \( q\) \(- q^{2}\) \(+ q^{4}\) \( -\beta_{1} q^{5} \) \( + ( 1 - \beta_{5} ) q^{7} \) \(- q^{8}\) \( + \beta_{1} q^{10} \) \( + ( -1 + \beta_{2} + \beta_{3} + \beta_{8} ) q^{11} \) \( + ( 1 - \beta_{5} - \beta_{6} + \beta_{11} ) q^{13} \) \( + ( -1 + \beta_{5} ) q^{14} \) \(+ q^{16}\) \( + ( -\beta_{4} + \beta_{6} - \beta_{7} ) q^{17} \) \( + ( \beta_{4} - \beta_{10} ) q^{19} \) \( -\beta_{1} q^{20} \) \( + ( 1 - \beta_{2} - \beta_{3} - \beta_{8} ) q^{22} \) \( + ( -1 + \beta_{5} + \beta_{6} + \beta_{10} ) q^{23} \) \( + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{25} \) \( + ( -1 + \beta_{5} + \beta_{6} - \beta_{11} ) q^{26} \) \( + ( 1 - \beta_{5} ) q^{28} \) \( + ( -3 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{29} \) \( + ( \beta_{1} + \beta_{10} ) q^{31} \) \(- q^{32}\) \( + ( \beta_{4} - \beta_{6} + \beta_{7} ) q^{34} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{35} \) \( + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{37} \) \( + ( -\beta_{4} + \beta_{10} ) q^{38} \) \( + \beta_{1} q^{40} \) \( + ( -3 + \beta_{1} - \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{41} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{43} \) \( + ( -1 + \beta_{2} + \beta_{3} + \beta_{8} ) q^{44} \) \( + ( 1 - \beta_{5} - \beta_{6} - \beta_{10} ) q^{46} \) \( + ( -2 - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{47} \) \( + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} - 2 \beta_{11} ) q^{49} \) \( + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{50} \) \( + ( 1 - \beta_{5} - \beta_{6} + \beta_{11} ) q^{52} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{53} \) \( + ( \beta_{1} - 2 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{55} \) \( + ( -1 + \beta_{5} ) q^{56} \) \( + ( 3 + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{58} \) \( + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{10} + \beta_{11} ) q^{59} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{61} \) \( + ( -\beta_{1} - \beta_{10} ) q^{62} \) \(+ q^{64}\) \( + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{9} ) q^{65} \) \( + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{67} \) \( + ( -\beta_{4} + \beta_{6} - \beta_{7} ) q^{68} \) \( + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{70} \) \( + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{71} \) \( + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{73} \) \( + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{74} \) \( + ( \beta_{4} - \beta_{10} ) q^{76} \) \( + ( -6 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} ) q^{77} \) \( + ( 4 - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{79} \) \( -\beta_{1} q^{80} \) \( + ( 3 - \beta_{1} + \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} ) q^{82} \) \( + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{83} \) \( + ( -3 + 2 \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} - \beta_{11} ) q^{85} \) \( + ( \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{86} \) \( + ( 1 - \beta_{2} - \beta_{3} - \beta_{8} ) q^{88} \) \( + ( -\beta_{2} - 2 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{10} ) q^{89} \) \( + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 4 \beta_{9} ) q^{91} \) \( + ( -1 + \beta_{5} + \beta_{6} + \beta_{10} ) q^{92} \) \( + ( 2 + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{94} \) \( + ( -6 - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{95} \) \( + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{97} \) \( + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + 2 \beta_{11} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut -\mathstrut 12q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 12q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 5q^{10} \) \(\mathstrut -\mathstrut 6q^{11} \) \(\mathstrut +\mathstrut 3q^{13} \) \(\mathstrut -\mathstrut 6q^{14} \) \(\mathstrut +\mathstrut 12q^{16} \) \(\mathstrut -\mathstrut 6q^{17} \) \(\mathstrut +\mathstrut 8q^{19} \) \(\mathstrut -\mathstrut 5q^{20} \) \(\mathstrut +\mathstrut 6q^{22} \) \(\mathstrut -\mathstrut 11q^{23} \) \(\mathstrut +\mathstrut 11q^{25} \) \(\mathstrut -\mathstrut 3q^{26} \) \(\mathstrut +\mathstrut 6q^{28} \) \(\mathstrut -\mathstrut 29q^{29} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 12q^{32} \) \(\mathstrut +\mathstrut 6q^{34} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 5q^{37} \) \(\mathstrut -\mathstrut 8q^{38} \) \(\mathstrut +\mathstrut 5q^{40} \) \(\mathstrut -\mathstrut 22q^{41} \) \(\mathstrut +\mathstrut 9q^{43} \) \(\mathstrut -\mathstrut 6q^{44} \) \(\mathstrut +\mathstrut 11q^{46} \) \(\mathstrut -\mathstrut 15q^{47} \) \(\mathstrut +\mathstrut 14q^{49} \) \(\mathstrut -\mathstrut 11q^{50} \) \(\mathstrut +\mathstrut 3q^{52} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 13q^{55} \) \(\mathstrut -\mathstrut 6q^{56} \) \(\mathstrut +\mathstrut 29q^{58} \) \(\mathstrut -\mathstrut 34q^{59} \) \(\mathstrut -\mathstrut 4q^{61} \) \(\mathstrut -\mathstrut 2q^{62} \) \(\mathstrut +\mathstrut 12q^{64} \) \(\mathstrut -\mathstrut 12q^{65} \) \(\mathstrut +\mathstrut q^{67} \) \(\mathstrut -\mathstrut 6q^{68} \) \(\mathstrut +\mathstrut 4q^{70} \) \(\mathstrut -\mathstrut 21q^{71} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut 5q^{74} \) \(\mathstrut +\mathstrut 8q^{76} \) \(\mathstrut -\mathstrut 34q^{77} \) \(\mathstrut +\mathstrut 9q^{79} \) \(\mathstrut -\mathstrut 5q^{80} \) \(\mathstrut +\mathstrut 22q^{82} \) \(\mathstrut -\mathstrut 10q^{83} \) \(\mathstrut +\mathstrut 5q^{85} \) \(\mathstrut -\mathstrut 9q^{86} \) \(\mathstrut +\mathstrut 6q^{88} \) \(\mathstrut +\mathstrut 2q^{89} \) \(\mathstrut +\mathstrut 17q^{91} \) \(\mathstrut -\mathstrut 11q^{92} \) \(\mathstrut +\mathstrut 15q^{94} \) \(\mathstrut -\mathstrut 69q^{95} \) \(\mathstrut -\mathstrut 13q^{97} \) \(\mathstrut -\mathstrut 14q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(5\) \(x^{11}\mathstrut -\mathstrut \) \(23\) \(x^{10}\mathstrut +\mathstrut \) \(142\) \(x^{9}\mathstrut +\mathstrut \) \(104\) \(x^{8}\mathstrut -\mathstrut \) \(1302\) \(x^{7}\mathstrut +\mathstrut \) \(607\) \(x^{6}\mathstrut +\mathstrut \) \(4323\) \(x^{5}\mathstrut -\mathstrut \) \(4461\) \(x^{4}\mathstrut -\mathstrut \) \(3333\) \(x^{3}\mathstrut +\mathstrut \) \(4805\) \(x^{2}\mathstrut -\mathstrut \) \(224\) \(x\mathstrut -\mathstrut \) \(553\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(39753790711\) \(\nu^{11}\mathstrut -\mathstrut \) \(198200510956\) \(\nu^{10}\mathstrut -\mathstrut \) \(967064971419\) \(\nu^{9}\mathstrut +\mathstrut \) \(5714838266656\) \(\nu^{8}\mathstrut +\mathstrut \) \(5759628424968\) \(\nu^{7}\mathstrut -\mathstrut \) \(53850580778217\) \(\nu^{6}\mathstrut +\mathstrut \) \(7072339723081\) \(\nu^{5}\mathstrut +\mathstrut \) \(189124537678876\) \(\nu^{4}\mathstrut -\mathstrut \) \(106414471211396\) \(\nu^{3}\mathstrut -\mathstrut \) \(177474875785820\) \(\nu^{2}\mathstrut +\mathstrut \) \(99233618007315\) \(\nu\mathstrut +\mathstrut \) \(17827377683725\)\()/\)\(7247830497003\)
\(\beta_{3}\)\(=\)\((\)\(80534503868\) \(\nu^{11}\mathstrut -\mathstrut \) \(351696237482\) \(\nu^{10}\mathstrut -\mathstrut \) \(2108871784872\) \(\nu^{9}\mathstrut +\mathstrut \) \(10131881046614\) \(\nu^{8}\mathstrut +\mathstrut \) \(15896674895505\) \(\nu^{7}\mathstrut -\mathstrut \) \(95422133728626\) \(\nu^{6}\mathstrut -\mathstrut \) \(23898794925802\) \(\nu^{5}\mathstrut +\mathstrut \) \(335975682458783\) \(\nu^{4}\mathstrut -\mathstrut \) \(90437371366366\) \(\nu^{3}\mathstrut -\mathstrut \) \(327059747358370\) \(\nu^{2}\mathstrut +\mathstrut \) \(92295543950502\) \(\nu\mathstrut +\mathstrut \) \(42821455278854\)\()/\)\(7247830497003\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(82493402629\) \(\nu^{11}\mathstrut +\mathstrut \) \(263830494832\) \(\nu^{10}\mathstrut +\mathstrut \) \(2411694496521\) \(\nu^{9}\mathstrut -\mathstrut \) \(7581331666612\) \(\nu^{8}\mathstrut -\mathstrut \) \(23166629157747\) \(\nu^{7}\mathstrut +\mathstrut \) \(71744854896003\) \(\nu^{6}\mathstrut +\mathstrut \) \(84556908978098\) \(\nu^{5}\mathstrut -\mathstrut \) \(260250839987092\) \(\nu^{4}\mathstrut -\mathstrut \) \(94448906762320\) \(\nu^{3}\mathstrut +\mathstrut \) \(286297569815276\) \(\nu^{2}\mathstrut +\mathstrut \) \(36333927662739\) \(\nu\mathstrut -\mathstrut \) \(37905558611308\)\()/\)\(7247830497003\)
\(\beta_{5}\)\(=\)\((\)\(111684601157\) \(\nu^{11}\mathstrut -\mathstrut \) \(439380667640\) \(\nu^{10}\mathstrut -\mathstrut \) \(3018114108333\) \(\nu^{9}\mathstrut +\mathstrut \) \(12565700557217\) \(\nu^{8}\mathstrut +\mathstrut \) \(24513574501275\) \(\nu^{7}\mathstrut -\mathstrut \) \(117331699982658\) \(\nu^{6}\mathstrut -\mathstrut \) \(53450947539877\) \(\nu^{5}\mathstrut +\mathstrut \) \(411530092181318\) \(\nu^{4}\mathstrut -\mathstrut \) \(68568106482415\) \(\nu^{3}\mathstrut -\mathstrut \) \(416256290933614\) \(\nu^{2}\mathstrut +\mathstrut \) \(100786856841096\) \(\nu\mathstrut +\mathstrut \) \(78764454042977\)\()/\)\(7247830497003\)
\(\beta_{6}\)\(=\)\((\)\(126477824479\) \(\nu^{11}\mathstrut -\mathstrut \) \(465007218553\) \(\nu^{10}\mathstrut -\mathstrut \) \(3547935397485\) \(\nu^{9}\mathstrut +\mathstrut \) \(13375681875472\) \(\nu^{8}\mathstrut +\mathstrut \) \(31285095157641\) \(\nu^{7}\mathstrut -\mathstrut \) \(125748469351326\) \(\nu^{6}\mathstrut -\mathstrut \) \(90255138053357\) \(\nu^{5}\mathstrut +\mathstrut \) \(441397343073052\) \(\nu^{4}\mathstrut +\mathstrut \) \(6081982929931\) \(\nu^{3}\mathstrut -\mathstrut \) \(427114040209208\) \(\nu^{2}\mathstrut +\mathstrut \) \(79483670481720\) \(\nu\mathstrut +\mathstrut \) \(52979135676436\)\()/\)\(7247830497003\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(233966606153\) \(\nu^{11}\mathstrut +\mathstrut \) \(906289859315\) \(\nu^{10}\mathstrut +\mathstrut \) \(6340426420320\) \(\nu^{9}\mathstrut -\mathstrut \) \(25919528717069\) \(\nu^{8}\mathstrut -\mathstrut \) \(51693664769592\) \(\nu^{7}\mathstrut +\mathstrut \) \(241755270906525\) \(\nu^{6}\mathstrut +\mathstrut \) \(112822364381107\) \(\nu^{5}\mathstrut -\mathstrut \) \(841128794307659\) \(\nu^{4}\mathstrut +\mathstrut \) \(152460063612613\) \(\nu^{3}\mathstrut +\mathstrut \) \(807815616826948\) \(\nu^{2}\mathstrut -\mathstrut \) \(241244419540353\) \(\nu\mathstrut -\mathstrut \) \(116176930526963\)\()/\)\(7247830497003\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(264188216893\) \(\nu^{11}\mathstrut +\mathstrut \) \(1052591859745\) \(\nu^{10}\mathstrut +\mathstrut \) \(7122140689275\) \(\nu^{9}\mathstrut -\mathstrut \) \(30110515868080\) \(\nu^{8}\mathstrut -\mathstrut \) \(57550285913988\) \(\nu^{7}\mathstrut +\mathstrut \) \(280571600276976\) \(\nu^{6}\mathstrut +\mathstrut \) \(122706115800737\) \(\nu^{5}\mathstrut -\mathstrut \) \(970382672201788\) \(\nu^{4}\mathstrut +\mathstrut \) \(176585646871547\) \(\nu^{3}\mathstrut +\mathstrut \) \(895224641034665\) \(\nu^{2}\mathstrut -\mathstrut \) \(259184076615561\) \(\nu\mathstrut -\mathstrut \) \(75969812453317\)\()/\)\(7247830497003\)
\(\beta_{9}\)\(=\)\((\)\(265780092172\) \(\nu^{11}\mathstrut -\mathstrut \) \(1045158248329\) \(\nu^{10}\mathstrut -\mathstrut \) \(7226529730506\) \(\nu^{9}\mathstrut +\mathstrut \) \(29933540825302\) \(\nu^{8}\mathstrut +\mathstrut \) \(59704300931763\) \(\nu^{7}\mathstrut -\mathstrut \) \(279842271576060\) \(\nu^{6}\mathstrut -\mathstrut \) \(140534451251591\) \(\nu^{5}\mathstrut +\mathstrut \) \(978788449854322\) \(\nu^{4}\mathstrut -\mathstrut \) \(122917256587556\) \(\nu^{3}\mathstrut -\mathstrut \) \(959282014883780\) \(\nu^{2}\mathstrut +\mathstrut \) \(236241029700939\) \(\nu\mathstrut +\mathstrut \) \(135370428068599\)\()/\)\(7247830497003\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(116959146335\) \(\nu^{11}\mathstrut +\mathstrut \) \(438173460485\) \(\nu^{10}\mathstrut +\mathstrut \) \(3253152757098\) \(\nu^{9}\mathstrut -\mathstrut \) \(12571233750104\) \(\nu^{8}\mathstrut -\mathstrut \) \(28273973320353\) \(\nu^{7}\mathstrut +\mathstrut \) \(117880451200146\) \(\nu^{6}\mathstrut +\mathstrut \) \(79382248663654\) \(\nu^{5}\mathstrut -\mathstrut \) \(414144051045197\) \(\nu^{4}\mathstrut -\mathstrut \) \(2612399983634\) \(\nu^{3}\mathstrut +\mathstrut \) \(408108251887606\) \(\nu^{2}\mathstrut -\mathstrut \) \(56374695744096\) \(\nu\mathstrut -\mathstrut \) \(57643004413031\)\()/\)\(2415943499001\)
\(\beta_{11}\)\(=\)\((\)\(376299976007\) \(\nu^{11}\mathstrut -\mathstrut \) \(1405118384054\) \(\nu^{10}\mathstrut -\mathstrut \) \(10441796596398\) \(\nu^{9}\mathstrut +\mathstrut \) \(40199942268095\) \(\nu^{8}\mathstrut +\mathstrut \) \(90410848803024\) \(\nu^{7}\mathstrut -\mathstrut \) \(375246438055287\) \(\nu^{6}\mathstrut -\mathstrut \) \(252528829047475\) \(\nu^{5}\mathstrut +\mathstrut \) \(1308736876472261\) \(\nu^{4}\mathstrut +\mathstrut \) \(10639560659018\) \(\nu^{3}\mathstrut -\mathstrut \) \(1271104499897389\) \(\nu^{2}\mathstrut +\mathstrut \) \(158243198361369\) \(\nu\mathstrut +\mathstrut \) \(171474493496885\)\()/\)\(7247830497003\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(17\) \(\beta_{10}\mathstrut -\mathstrut \) \(12\) \(\beta_{8}\mathstrut +\mathstrut \) \(15\) \(\beta_{7}\mathstrut +\mathstrut \) \(18\) \(\beta_{6}\mathstrut +\mathstrut \) \(19\) \(\beta_{5}\mathstrut -\mathstrut \) \(13\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(61\)
\(\nu^{5}\)\(=\)\(-\)\(22\) \(\beta_{11}\mathstrut -\mathstrut \) \(15\) \(\beta_{10}\mathstrut -\mathstrut \) \(6\) \(\beta_{9}\mathstrut -\mathstrut \) \(18\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(16\) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(36\) \(\beta_{3}\mathstrut -\mathstrut \) \(61\) \(\beta_{2}\mathstrut +\mathstrut \) \(86\) \(\beta_{1}\mathstrut +\mathstrut \) \(16\)
\(\nu^{6}\)\(=\)\(55\) \(\beta_{11}\mathstrut +\mathstrut \) \(270\) \(\beta_{10}\mathstrut -\mathstrut \) \(4\) \(\beta_{9}\mathstrut -\mathstrut \) \(149\) \(\beta_{8}\mathstrut +\mathstrut \) \(211\) \(\beta_{7}\mathstrut +\mathstrut \) \(292\) \(\beta_{6}\mathstrut +\mathstrut \) \(286\) \(\beta_{5}\mathstrut -\mathstrut \) \(168\) \(\beta_{4}\mathstrut +\mathstrut \) \(28\) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(190\) \(\beta_{1}\mathstrut +\mathstrut \) \(742\)
\(\nu^{7}\)\(=\)\(-\)\(334\) \(\beta_{11}\mathstrut -\mathstrut \) \(165\) \(\beta_{10}\mathstrut +\mathstrut \) \(18\) \(\beta_{9}\mathstrut -\mathstrut \) \(258\) \(\beta_{8}\mathstrut -\mathstrut \) \(31\) \(\beta_{7}\mathstrut -\mathstrut \) \(224\) \(\beta_{6}\mathstrut +\mathstrut \) \(23\) \(\beta_{5}\mathstrut -\mathstrut \) \(63\) \(\beta_{4}\mathstrut +\mathstrut \) \(588\) \(\beta_{3}\mathstrut -\mathstrut \) \(986\) \(\beta_{2}\mathstrut +\mathstrut \) \(1062\) \(\beta_{1}\mathstrut +\mathstrut \) \(437\)
\(\nu^{8}\)\(=\)\(1032\) \(\beta_{11}\mathstrut +\mathstrut \) \(4150\) \(\beta_{10}\mathstrut -\mathstrut \) \(61\) \(\beta_{9}\mathstrut -\mathstrut \) \(1902\) \(\beta_{8}\mathstrut +\mathstrut \) \(2902\) \(\beta_{7}\mathstrut +\mathstrut \) \(4487\) \(\beta_{6}\mathstrut +\mathstrut \) \(4003\) \(\beta_{5}\mathstrut -\mathstrut \) \(2315\) \(\beta_{4}\mathstrut +\mathstrut \) \(664\) \(\beta_{3}\mathstrut +\mathstrut \) \(37\) \(\beta_{2}\mathstrut +\mathstrut \) \(2643\) \(\beta_{1}\mathstrut +\mathstrut \) \(9899\)
\(\nu^{9}\)\(=\)\(-\)\(4527\) \(\beta_{11}\mathstrut -\mathstrut \) \(1300\) \(\beta_{10}\mathstrut +\mathstrut \) \(1303\) \(\beta_{9}\mathstrut -\mathstrut \) \(3491\) \(\beta_{8}\mathstrut -\mathstrut \) \(706\) \(\beta_{7}\mathstrut -\mathstrut \) \(2872\) \(\beta_{6}\mathstrut -\mathstrut \) \(945\) \(\beta_{5}\mathstrut -\mathstrut \) \(1461\) \(\beta_{4}\mathstrut +\mathstrut \) \(9351\) \(\beta_{3}\mathstrut -\mathstrut \) \(14872\) \(\beta_{2}\mathstrut +\mathstrut \) \(14198\) \(\beta_{1}\mathstrut +\mathstrut \) \(8796\)
\(\nu^{10}\)\(=\)\(17114\) \(\beta_{11}\mathstrut +\mathstrut \) \(62852\) \(\beta_{10}\mathstrut -\mathstrut \) \(375\) \(\beta_{9}\mathstrut -\mathstrut \) \(24898\) \(\beta_{8}\mathstrut +\mathstrut \) \(39584\) \(\beta_{7}\mathstrut +\mathstrut \) \(67054\) \(\beta_{6}\mathstrut +\mathstrut \) \(54592\) \(\beta_{5}\mathstrut -\mathstrut \) \(33297\) \(\beta_{4}\mathstrut +\mathstrut \) \(13472\) \(\beta_{3}\mathstrut -\mathstrut \) \(329\) \(\beta_{2}\mathstrut +\mathstrut \) \(37757\) \(\beta_{1}\mathstrut +\mathstrut \) \(138910\)
\(\nu^{11}\)\(=\)\(-\)\(58540\) \(\beta_{11}\mathstrut -\mathstrut \) \(1621\) \(\beta_{10}\mathstrut +\mathstrut \) \(28961\) \(\beta_{9}\mathstrut -\mathstrut \) \(46940\) \(\beta_{8}\mathstrut -\mathstrut \) \(13406\) \(\beta_{7}\mathstrut -\mathstrut \) \(33588\) \(\beta_{6}\mathstrut -\mathstrut \) \(27030\) \(\beta_{5}\mathstrut -\mathstrut \) \(29463\) \(\beta_{4}\mathstrut +\mathstrut \) \(146119\) \(\beta_{3}\mathstrut -\mathstrut \) \(218819\) \(\beta_{2}\mathstrut +\mathstrut \) \(198673\) \(\beta_{1}\mathstrut +\mathstrut \) \(158591\)

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
3.89381
3.31930
2.74383
2.27289
1.33540
1.20508
0.521020
−0.302744
−1.09224
−2.38357
−2.78478
−3.72799
−1.00000 0 1.00000 −3.89381 0 0.649105 −1.00000 0 3.89381
1.2 −1.00000 0 1.00000 −3.31930 0 −3.41918 −1.00000 0 3.31930
1.3 −1.00000 0 1.00000 −2.74383 0 0.881706 −1.00000 0 2.74383
1.4 −1.00000 0 1.00000 −2.27289 0 3.85060 −1.00000 0 2.27289
1.5 −1.00000 0 1.00000 −1.33540 0 2.26685 −1.00000 0 1.33540
1.6 −1.00000 0 1.00000 −1.20508 0 4.22185 −1.00000 0 1.20508
1.7 −1.00000 0 1.00000 −0.521020 0 −3.80450 −1.00000 0 0.521020
1.8 −1.00000 0 1.00000 0.302744 0 −1.13869 −1.00000 0 −0.302744
1.9 −1.00000 0 1.00000 1.09224 0 −1.25253 −1.00000 0 −1.09224
1.10 −1.00000 0 1.00000 2.38357 0 4.24377 −1.00000 0 −2.38357
1.11 −1.00000 0 1.00000 2.78478 0 2.18432 −1.00000 0 −2.78478
1.12 −1.00000 0 1.00000 3.72799 0 −2.68331 −1.00000 0 −3.72799
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

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