Properties

Label 6024.2.a.n
Level 6024
Weight 2
Character orbit 6024.a
Self dual Yes
Analytic conductor 48.102
Analytic rank 1
Dimension 14
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6024.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1018821776\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \( + ( -1 + \beta_{1} ) q^{5} \) \( -\beta_{3} q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( + ( -1 + \beta_{1} ) q^{5} \) \( -\beta_{3} q^{7} \) \(+ q^{9}\) \( + ( 1 - \beta_{12} ) q^{11} \) \( + ( -\beta_{1} + \beta_{5} - \beta_{7} - \beta_{8} ) q^{13} \) \( + ( 1 - \beta_{1} ) q^{15} \) \( + ( -3 + \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{17} \) \( + ( -1 - \beta_{7} + \beta_{12} ) q^{19} \) \( + \beta_{3} q^{21} \) \( + ( 1 + \beta_{3} - \beta_{4} + \beta_{7} ) q^{23} \) \( + ( 5 - 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{10} + \beta_{13} ) q^{25} \) \(- q^{27}\) \( + ( -3 + \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + \beta_{10} - \beta_{13} ) q^{29} \) \( + ( -2 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{12} ) q^{31} \) \( + ( -1 + \beta_{12} ) q^{33} \) \( + ( 2 + \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{35} \) \( + ( -2 \beta_{1} - \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{37} \) \( + ( \beta_{1} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{39} \) \( + ( -3 - \beta_{1} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} ) q^{41} \) \( + ( 4 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{43} \) \( + ( -1 + \beta_{1} ) q^{45} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{47} \) \( + ( 4 - 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{10} - \beta_{12} + \beta_{13} ) q^{49} \) \( + ( 3 - \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{10} ) q^{51} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{12} + 2 \beta_{13} ) q^{53} \) \( + ( -4 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{55} \) \( + ( 1 + \beta_{7} - \beta_{12} ) q^{57} \) \( + ( 1 - 2 \beta_{2} - \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{10} + \beta_{12} + \beta_{13} ) q^{59} \) \( + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{10} + \beta_{13} ) q^{61} \) \( -\beta_{3} q^{63} \) \( + ( -4 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} ) q^{65} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{67} \) \( + ( -1 - \beta_{3} + \beta_{4} - \beta_{7} ) q^{69} \) \( + ( 2 - 3 \beta_{1} + \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{71} \) \( + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} ) q^{73} \) \( + ( -5 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{6} - \beta_{8} + 2 \beta_{10} - \beta_{13} ) q^{75} \) \( + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{13} ) q^{77} \) \( + ( -2 - \beta_{1} - \beta_{3} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{12} - \beta_{13} ) q^{79} \) \(+ q^{81}\) \( + ( 2 + \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{83} \) \( + ( -3 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} + \beta_{12} - 2 \beta_{13} ) q^{85} \) \( + ( 3 - \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - \beta_{10} + \beta_{13} ) q^{87} \) \( + ( -7 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} ) q^{89} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{13} ) q^{91} \) \( + ( 2 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{12} ) q^{93} \) \( + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{95} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{97} \) \( + ( 1 - \beta_{12} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut 10q^{11} \) \(\mathstrut -\mathstrut 9q^{13} \) \(\mathstrut +\mathstrut 7q^{15} \) \(\mathstrut -\mathstrut 22q^{17} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut -\mathstrut q^{21} \) \(\mathstrut +\mathstrut 3q^{23} \) \(\mathstrut +\mathstrut 23q^{25} \) \(\mathstrut -\mathstrut 14q^{27} \) \(\mathstrut -\mathstrut 12q^{29} \) \(\mathstrut -\mathstrut 13q^{31} \) \(\mathstrut -\mathstrut 10q^{33} \) \(\mathstrut +\mathstrut 23q^{35} \) \(\mathstrut +\mathstrut 5q^{37} \) \(\mathstrut +\mathstrut 9q^{39} \) \(\mathstrut -\mathstrut 52q^{41} \) \(\mathstrut +\mathstrut 16q^{43} \) \(\mathstrut -\mathstrut 7q^{45} \) \(\mathstrut +\mathstrut q^{47} \) \(\mathstrut +\mathstrut 9q^{49} \) \(\mathstrut +\mathstrut 22q^{51} \) \(\mathstrut -\mathstrut 13q^{53} \) \(\mathstrut -\mathstrut 12q^{55} \) \(\mathstrut +\mathstrut 6q^{57} \) \(\mathstrut +\mathstrut 12q^{59} \) \(\mathstrut -\mathstrut 20q^{61} \) \(\mathstrut +\mathstrut q^{63} \) \(\mathstrut -\mathstrut 40q^{65} \) \(\mathstrut +\mathstrut 21q^{67} \) \(\mathstrut -\mathstrut 3q^{69} \) \(\mathstrut -\mathstrut 5q^{71} \) \(\mathstrut -\mathstrut 14q^{73} \) \(\mathstrut -\mathstrut 23q^{75} \) \(\mathstrut -\mathstrut 14q^{77} \) \(\mathstrut -\mathstrut 23q^{79} \) \(\mathstrut +\mathstrut 14q^{81} \) \(\mathstrut +\mathstrut 25q^{83} \) \(\mathstrut -\mathstrut 11q^{85} \) \(\mathstrut +\mathstrut 12q^{87} \) \(\mathstrut -\mathstrut 79q^{89} \) \(\mathstrut +\mathstrut 6q^{91} \) \(\mathstrut +\mathstrut 13q^{93} \) \(\mathstrut +\mathstrut 3q^{95} \) \(\mathstrut -\mathstrut 17q^{97} \) \(\mathstrut +\mathstrut 10q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(7\) \(x^{13}\mathstrut -\mathstrut \) \(22\) \(x^{12}\mathstrut +\mathstrut \) \(214\) \(x^{11}\mathstrut +\mathstrut \) \(91\) \(x^{10}\mathstrut -\mathstrut \) \(2481\) \(x^{9}\mathstrut +\mathstrut \) \(1285\) \(x^{8}\mathstrut +\mathstrut \) \(13253\) \(x^{7}\mathstrut -\mathstrut \) \(14287\) \(x^{6}\mathstrut -\mathstrut \) \(29907\) \(x^{5}\mathstrut +\mathstrut \) \(49572\) \(x^{4}\mathstrut +\mathstrut \) \(10044\) \(x^{3}\mathstrut -\mathstrut \) \(53784\) \(x^{2}\mathstrut +\mathstrut \) \(32076\) \(x\mathstrut -\mathstrut \) \(5832\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(132354\) \(\nu^{13}\mathstrut +\mathstrut \) \(252317\) \(\nu^{12}\mathstrut -\mathstrut \) \(9737975\) \(\nu^{11}\mathstrut -\mathstrut \) \(2149364\) \(\nu^{10}\mathstrut +\mathstrut \) \(195557078\) \(\nu^{9}\mathstrut -\mathstrut \) \(19621651\) \(\nu^{8}\mathstrut -\mathstrut \) \(1651422087\) \(\nu^{7}\mathstrut +\mathstrut \) \(220560557\) \(\nu^{6}\mathstrut +\mathstrut \) \(6250763011\) \(\nu^{5}\mathstrut -\mathstrut \) \(497128331\) \(\nu^{4}\mathstrut -\mathstrut \) \(9359381937\) \(\nu^{3}\mathstrut +\mathstrut \) \(400473558\) \(\nu^{2}\mathstrut +\mathstrut \) \(3612420558\) \(\nu\mathstrut -\mathstrut \) \(778401900\)\()/\)\(258641748\)
\(\beta_{3}\)\(=\)\((\)\(465848\) \(\nu^{13}\mathstrut -\mathstrut \) \(1692092\) \(\nu^{12}\mathstrut -\mathstrut \) \(14853797\) \(\nu^{11}\mathstrut +\mathstrut \) \(23529578\) \(\nu^{10}\mathstrut +\mathstrut \) \(242839136\) \(\nu^{9}\mathstrut +\mathstrut \) \(87322476\) \(\nu^{8}\mathstrut -\mathstrut \) \(2398790377\) \(\nu^{7}\mathstrut -\mathstrut \) \(2557728392\) \(\nu^{6}\mathstrut +\mathstrut \) \(13271128399\) \(\nu^{5}\mathstrut +\mathstrut \) \(11051495994\) \(\nu^{4}\mathstrut -\mathstrut \) \(36133898595\) \(\nu^{3}\mathstrut -\mathstrut \) \(9029265516\) \(\nu^{2}\mathstrut +\mathstrut \) \(36399541374\) \(\nu\mathstrut -\mathstrut \) \(12313868832\)\()/\)\(258641748\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(2419411\) \(\nu^{13}\mathstrut +\mathstrut \) \(8486962\) \(\nu^{12}\mathstrut +\mathstrut \) \(97888897\) \(\nu^{11}\mathstrut -\mathstrut \) \(257866054\) \(\nu^{10}\mathstrut -\mathstrut \) \(1545725563\) \(\nu^{9}\mathstrut +\mathstrut \) \(2859777120\) \(\nu^{8}\mathstrut +\mathstrut \) \(11807999612\) \(\nu^{7}\mathstrut -\mathstrut \) \(14253051902\) \(\nu^{6}\mathstrut -\mathstrut \) \(44180354618\) \(\nu^{5}\mathstrut +\mathstrut \) \(32206178676\) \(\nu^{4}\mathstrut +\mathstrut \) \(72522077535\) \(\nu^{3}\mathstrut -\mathstrut \) \(31979787552\) \(\nu^{2}\mathstrut -\mathstrut \) \(37256227506\) \(\nu\mathstrut +\mathstrut \) \(16494766128\)\()/\)\(775925244\)
\(\beta_{5}\)\(=\)\((\)\(3693740\) \(\nu^{13}\mathstrut -\mathstrut \) \(15082412\) \(\nu^{12}\mathstrut -\mathstrut \) \(138107111\) \(\nu^{11}\mathstrut +\mathstrut \) \(463087526\) \(\nu^{10}\mathstrut +\mathstrut \) \(2008311974\) \(\nu^{9}\mathstrut -\mathstrut \) \(5321129922\) \(\nu^{8}\mathstrut -\mathstrut \) \(13750748749\) \(\nu^{7}\mathstrut +\mathstrut \) \(28346301064\) \(\nu^{6}\mathstrut +\mathstrut \) \(42287226997\) \(\nu^{5}\mathstrut -\mathstrut \) \(69327164388\) \(\nu^{4}\mathstrut -\mathstrut \) \(40972975929\) \(\nu^{3}\mathstrut +\mathstrut \) \(60089181264\) \(\nu^{2}\mathstrut -\mathstrut \) \(12096974646\) \(\nu\mathstrut +\mathstrut \) \(2650201740\)\()/\)\(775925244\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(1660386\) \(\nu^{13}\mathstrut +\mathstrut \) \(8973449\) \(\nu^{12}\mathstrut +\mathstrut \) \(51206689\) \(\nu^{11}\mathstrut -\mathstrut \) \(276220658\) \(\nu^{10}\mathstrut -\mathstrut \) \(597576178\) \(\nu^{9}\mathstrut +\mathstrut \) \(3226181717\) \(\nu^{8}\mathstrut +\mathstrut \) \(3061351197\) \(\nu^{7}\mathstrut -\mathstrut \) \(17712804031\) \(\nu^{6}\mathstrut -\mathstrut \) \(4901897117\) \(\nu^{5}\mathstrut +\mathstrut \) \(45182693341\) \(\nu^{4}\mathstrut -\mathstrut \) \(8895945651\) \(\nu^{3}\mathstrut -\mathstrut \) \(41643741978\) \(\nu^{2}\mathstrut +\mathstrut \) \(23572139238\) \(\nu\mathstrut -\mathstrut \) \(1852305084\)\()/\)\(258641748\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(5125025\) \(\nu^{13}\mathstrut +\mathstrut \) \(32681531\) \(\nu^{12}\mathstrut +\mathstrut \) \(129487169\) \(\nu^{11}\mathstrut -\mathstrut \) \(984047240\) \(\nu^{10}\mathstrut -\mathstrut \) \(1068758501\) \(\nu^{9}\mathstrut +\mathstrut \) \(11343125943\) \(\nu^{8}\mathstrut +\mathstrut \) \(1499471806\) \(\nu^{7}\mathstrut -\mathstrut \) \(61843890199\) \(\nu^{6}\mathstrut +\mathstrut \) \(22856343572\) \(\nu^{5}\mathstrut +\mathstrut \) \(155885343771\) \(\nu^{4}\mathstrut -\mathstrut \) \(111236499315\) \(\nu^{3}\mathstrut -\mathstrut \) \(132695842734\) \(\nu^{2}\mathstrut +\mathstrut \) \(140697751842\) \(\nu\mathstrut -\mathstrut \) \(29622972348\)\()/\)\(775925244\)
\(\beta_{8}\)\(=\)\((\)\(5665553\) \(\nu^{13}\mathstrut -\mathstrut \) \(31990091\) \(\nu^{12}\mathstrut -\mathstrut \) \(166113722\) \(\nu^{11}\mathstrut +\mathstrut \) \(976567838\) \(\nu^{10}\mathstrut +\mathstrut \) \(1795427681\) \(\nu^{9}\mathstrut -\mathstrut \) \(11351117373\) \(\nu^{8}\mathstrut -\mathstrut \) \(7704484477\) \(\nu^{7}\mathstrut +\mathstrut \) \(62129643733\) \(\nu^{6}\mathstrut +\mathstrut \) \(1545769951\) \(\nu^{5}\mathstrut -\mathstrut \) \(157239528657\) \(\nu^{4}\mathstrut +\mathstrut \) \(70041696066\) \(\nu^{3}\mathstrut +\mathstrut \) \(137751377382\) \(\nu^{2}\mathstrut -\mathstrut \) \(115939491588\) \(\nu\mathstrut +\mathstrut \) \(20026158768\)\()/\)\(775925244\)
\(\beta_{9}\)\(=\)\((\)\(3057855\) \(\nu^{13}\mathstrut -\mathstrut \) \(16874308\) \(\nu^{12}\mathstrut -\mathstrut \) \(90044327\) \(\nu^{11}\mathstrut +\mathstrut \) \(504753364\) \(\nu^{10}\mathstrut +\mathstrut \) \(991939655\) \(\nu^{9}\mathstrut -\mathstrut \) \(5711937544\) \(\nu^{8}\mathstrut -\mathstrut \) \(4557204534\) \(\nu^{7}\mathstrut +\mathstrut \) \(30194899850\) \(\nu^{6}\mathstrut +\mathstrut \) \(3727777090\) \(\nu^{5}\mathstrut -\mathstrut \) \(72960617120\) \(\nu^{4}\mathstrut +\mathstrut \) \(29261077233\) \(\nu^{3}\mathstrut +\mathstrut \) \(58554025560\) \(\nu^{2}\mathstrut -\mathstrut \) \(56200870818\) \(\nu\mathstrut +\mathstrut \) \(14162096700\)\()/\)\(258641748\)
\(\beta_{10}\)\(=\)\((\)\(4582742\) \(\nu^{13}\mathstrut -\mathstrut \) \(25547763\) \(\nu^{12}\mathstrut -\mathstrut \) \(138217810\) \(\nu^{11}\mathstrut +\mathstrut \) \(794697624\) \(\nu^{10}\mathstrut +\mathstrut \) \(1534796044\) \(\nu^{9}\mathstrut -\mathstrut \) \(9401210821\) \(\nu^{8}\mathstrut -\mathstrut \) \(6805424140\) \(\nu^{7}\mathstrut +\mathstrut \) \(52256162589\) \(\nu^{6}\mathstrut +\mathstrut \) \(2231205644\) \(\nu^{5}\mathstrut -\mathstrut \) \(133525058219\) \(\nu^{4}\mathstrut +\mathstrut \) \(60758574672\) \(\nu^{3}\mathstrut +\mathstrut \) \(114854769702\) \(\nu^{2}\mathstrut -\mathstrut \) \(106760112048\) \(\nu\mathstrut +\mathstrut \) \(25073033220\)\()/\)\(258641748\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(5062862\) \(\nu^{13}\mathstrut +\mathstrut \) \(32973361\) \(\nu^{12}\mathstrut +\mathstrut \) \(122994141\) \(\nu^{11}\mathstrut -\mathstrut \) \(984692176\) \(\nu^{10}\mathstrut -\mathstrut \) \(920360232\) \(\nu^{9}\mathstrut +\mathstrut \) \(11225429261\) \(\nu^{8}\mathstrut +\mathstrut \) \(110845633\) \(\nu^{7}\mathstrut -\mathstrut \) \(60252839573\) \(\nu^{6}\mathstrut +\mathstrut \) \(28915531203\) \(\nu^{5}\mathstrut +\mathstrut \) \(148046812711\) \(\nu^{4}\mathstrut -\mathstrut \) \(123505523355\) \(\nu^{3}\mathstrut -\mathstrut \) \(117321201774\) \(\nu^{2}\mathstrut +\mathstrut \) \(150642394002\) \(\nu\mathstrut -\mathstrut \) \(38528861760\)\()/\)\(258641748\)
\(\beta_{12}\)\(=\)\((\)\(5104653\) \(\nu^{13}\mathstrut -\mathstrut \) \(28639646\) \(\nu^{12}\mathstrut -\mathstrut \) \(147986245\) \(\nu^{11}\mathstrut +\mathstrut \) \(854007062\) \(\nu^{10}\mathstrut +\mathstrut \) \(1612330183\) \(\nu^{9}\mathstrut -\mathstrut \) \(9668793260\) \(\nu^{8}\mathstrut -\mathstrut \) \(7393707270\) \(\nu^{7}\mathstrut +\mathstrut \) \(51387448984\) \(\nu^{6}\mathstrut +\mathstrut \) \(6364019144\) \(\nu^{5}\mathstrut -\mathstrut \) \(125935699258\) \(\nu^{4}\mathstrut +\mathstrut \) \(46461515751\) \(\nu^{3}\mathstrut +\mathstrut \) \(105859625616\) \(\nu^{2}\mathstrut -\mathstrut \) \(90920242962\) \(\nu\mathstrut +\mathstrut \) \(19653622272\)\()/\)\(258641748\)
\(\beta_{13}\)\(=\)\((\)\(22370297\) \(\nu^{13}\mathstrut -\mathstrut \) \(130485959\) \(\nu^{12}\mathstrut -\mathstrut \) \(650249336\) \(\nu^{11}\mathstrut +\mathstrut \) \(4098099680\) \(\nu^{10}\mathstrut +\mathstrut \) \(6701297225\) \(\nu^{9}\mathstrut -\mathstrut \) \(49074003417\) \(\nu^{8}\mathstrut -\mathstrut \) \(23650380991\) \(\nu^{7}\mathstrut +\mathstrut \) \(276701321761\) \(\nu^{6}\mathstrut -\mathstrut \) \(43061264939\) \(\nu^{5}\mathstrut -\mathstrut \) \(716537928321\) \(\nu^{4}\mathstrut +\mathstrut \) \(438163598178\) \(\nu^{3}\mathstrut +\mathstrut \) \(614326237578\) \(\nu^{2}\mathstrut -\mathstrut \) \(659012791752\) \(\nu\mathstrut +\mathstrut \) \(159639955164\)\()/\)\(775925244\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{13}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(9\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(5\) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(7\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{4}\)\(=\)\(18\) \(\beta_{13}\mathstrut +\mathstrut \) \(3\) \(\beta_{12}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(40\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(25\) \(\beta_{8}\mathstrut +\mathstrut \) \(9\) \(\beta_{7}\mathstrut +\mathstrut \) \(22\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(49\) \(\beta_{4}\mathstrut -\mathstrut \) \(8\) \(\beta_{3}\mathstrut -\mathstrut \) \(38\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(150\)
\(\nu^{5}\)\(=\)\(59\) \(\beta_{13}\mathstrut +\mathstrut \) \(33\) \(\beta_{12}\mathstrut -\mathstrut \) \(10\) \(\beta_{11}\mathstrut -\mathstrut \) \(158\) \(\beta_{10}\mathstrut -\mathstrut \) \(12\) \(\beta_{9}\mathstrut +\mathstrut \) \(117\) \(\beta_{8}\mathstrut +\mathstrut \) \(61\) \(\beta_{7}\mathstrut +\mathstrut \) \(91\) \(\beta_{6}\mathstrut +\mathstrut \) \(11\) \(\beta_{5}\mathstrut -\mathstrut \) \(228\) \(\beta_{4}\mathstrut -\mathstrut \) \(72\) \(\beta_{3}\mathstrut -\mathstrut \) \(210\) \(\beta_{2}\mathstrut +\mathstrut \) \(127\) \(\beta_{1}\mathstrut +\mathstrut \) \(548\)
\(\nu^{6}\)\(=\)\(376\) \(\beta_{13}\mathstrut +\mathstrut \) \(157\) \(\beta_{12}\mathstrut -\mathstrut \) \(77\) \(\beta_{11}\mathstrut -\mathstrut \) \(947\) \(\beta_{10}\mathstrut -\mathstrut \) \(92\) \(\beta_{9}\mathstrut +\mathstrut \) \(709\) \(\beta_{8}\mathstrut +\mathstrut \) \(390\) \(\beta_{7}\mathstrut +\mathstrut \) \(529\) \(\beta_{6}\mathstrut +\mathstrut \) \(85\) \(\beta_{5}\mathstrut -\mathstrut \) \(1296\) \(\beta_{4}\mathstrut -\mathstrut \) \(352\) \(\beta_{3}\mathstrut -\mathstrut \) \(1164\) \(\beta_{2}\mathstrut +\mathstrut \) \(400\) \(\beta_{1}\mathstrut +\mathstrut \) \(3459\)
\(\nu^{7}\)\(=\)\(1651\) \(\beta_{13}\mathstrut +\mathstrut \) \(1031\) \(\beta_{12}\mathstrut -\mathstrut \) \(424\) \(\beta_{11}\mathstrut -\mathstrut \) \(4593\) \(\beta_{10}\mathstrut -\mathstrut \) \(532\) \(\beta_{9}\mathstrut +\mathstrut \) \(3665\) \(\beta_{8}\mathstrut +\mathstrut \) \(2211\) \(\beta_{7}\mathstrut +\mathstrut \) \(2548\) \(\beta_{6}\mathstrut +\mathstrut \) \(438\) \(\beta_{5}\mathstrut -\mathstrut \) \(6660\) \(\beta_{4}\mathstrut -\mathstrut \) \(2224\) \(\beta_{3}\mathstrut -\mathstrut \) \(6296\) \(\beta_{2}\mathstrut +\mathstrut \) \(2617\) \(\beta_{1}\mathstrut +\mathstrut \) \(16515\)
\(\nu^{8}\)\(=\)\(9319\) \(\beta_{13}\mathstrut +\mathstrut \) \(5417\) \(\beta_{12}\mathstrut -\mathstrut \) \(2529\) \(\beta_{11}\mathstrut -\mathstrut \) \(25413\) \(\beta_{10}\mathstrut -\mathstrut \) \(3176\) \(\beta_{9}\mathstrut +\mathstrut \) \(20377\) \(\beta_{8}\mathstrut +\mathstrut \) \(12668\) \(\beta_{7}\mathstrut +\mathstrut \) \(13886\) \(\beta_{6}\mathstrut +\mathstrut \) \(2583\) \(\beta_{5}\mathstrut -\mathstrut \) \(36207\) \(\beta_{4}\mathstrut -\mathstrut \) \(11652\) \(\beta_{3}\mathstrut -\mathstrut \) \(34072\) \(\beta_{2}\mathstrut +\mathstrut \) \(11798\) \(\beta_{1}\mathstrut +\mathstrut \) \(92549\)
\(\nu^{9}\)\(=\)\(46564\) \(\beta_{13}\mathstrut +\mathstrut \) \(31061\) \(\beta_{12}\mathstrut -\mathstrut \) \(13909\) \(\beta_{11}\mathstrut -\mathstrut \) \(132398\) \(\beta_{10}\mathstrut -\mathstrut \) \(17734\) \(\beta_{9}\mathstrut +\mathstrut \) \(108231\) \(\beta_{8}\mathstrut +\mathstrut \) \(69156\) \(\beta_{7}\mathstrut +\mathstrut \) \(71468\) \(\beta_{6}\mathstrut +\mathstrut \) \(14214\) \(\beta_{5}\mathstrut -\mathstrut \) \(191962\) \(\beta_{4}\mathstrut -\mathstrut \) \(66202\) \(\beta_{3}\mathstrut -\mathstrut \) \(183448\) \(\beta_{2}\mathstrut +\mathstrut \) \(66338\) \(\beta_{1}\mathstrut +\mathstrut \) \(480402\)
\(\nu^{10}\)\(=\)\(253242\) \(\beta_{13}\mathstrut +\mathstrut \) \(166697\) \(\beta_{12}\mathstrut -\mathstrut \) \(77516\) \(\beta_{11}\mathstrut -\mathstrut \) \(717063\) \(\beta_{10}\mathstrut -\mathstrut \) \(98848\) \(\beta_{9}\mathstrut +\mathstrut \) \(587503\) \(\beta_{8}\mathstrut +\mathstrut \) \(380865\) \(\beta_{7}\mathstrut +\mathstrut \) \(383819\) \(\beta_{6}\mathstrut +\mathstrut \) \(78770\) \(\beta_{5}\mathstrut -\mathstrut \) \(1033899\) \(\beta_{4}\mathstrut -\mathstrut \) \(354158\) \(\beta_{3}\mathstrut -\mathstrut \) \(987932\) \(\beta_{2}\mathstrut +\mathstrut \) \(337509\) \(\beta_{1}\mathstrut +\mathstrut \) \(2610197\)
\(\nu^{11}\)\(=\)\(1327808\) \(\beta_{13}\mathstrut +\mathstrut \) \(916561\) \(\beta_{12}\mathstrut -\mathstrut \) \(422094\) \(\beta_{11}\mathstrut -\mathstrut \) \(3820164\) \(\beta_{10}\mathstrut -\mathstrut \) \(541036\) \(\beta_{9}\mathstrut +\mathstrut \) \(3147161\) \(\beta_{8}\mathstrut +\mathstrut \) \(2059009\) \(\beta_{7}\mathstrut +\mathstrut \) \(2029330\) \(\beta_{6}\mathstrut +\mathstrut \) \(432141\) \(\beta_{5}\mathstrut -\mathstrut \) \(5535195\) \(\beta_{4}\mathstrut -\mathstrut \) \(1942170\) \(\beta_{3}\mathstrut -\mathstrut \) \(5314844\) \(\beta_{2}\mathstrut +\mathstrut \) \(1829667\) \(\beta_{1}\mathstrut +\mathstrut \) \(13891354\)
\(\nu^{12}\)\(=\)\(7153651\) \(\beta_{13}\mathstrut +\mathstrut \) \(4936789\) \(\beta_{12}\mathstrut -\mathstrut \) \(2300768\) \(\beta_{11}\mathstrut -\mathstrut \) \(20580740\) \(\beta_{10}\mathstrut -\mathstrut \) \(2948702\) \(\beta_{9}\mathstrut +\mathstrut \) \(16971377\) \(\beta_{8}\mathstrut +\mathstrut \) \(11174643\) \(\beta_{7}\mathstrut +\mathstrut \) \(10884243\) \(\beta_{6}\mathstrut +\mathstrut \) \(2351883\) \(\beta_{5}\mathstrut -\mathstrut \) \(29767848\) \(\beta_{4}\mathstrut -\mathstrut \) \(10444748\) \(\beta_{3}\mathstrut -\mathstrut \) \(28596052\) \(\beta_{2}\mathstrut +\mathstrut \) \(9669817\) \(\beta_{1}\mathstrut +\mathstrut \) \(74897808\)
\(\nu^{13}\)\(=\)\(38143582\) \(\beta_{13}\mathstrut +\mathstrut \) \(26767311\) \(\beta_{12}\mathstrut -\mathstrut \) \(12449599\) \(\beta_{11}\mathstrut -\mathstrut \) \(110390851\) \(\beta_{10}\mathstrut -\mathstrut \) \(15984458\) \(\beta_{9}\mathstrut +\mathstrut \) \(91171475\) \(\beta_{8}\mathstrut +\mathstrut \) \(60232994\) \(\beta_{7}\mathstrut +\mathstrut \) \(58159081\) \(\beta_{6}\mathstrut +\mathstrut \) \(12786967\) \(\beta_{5}\mathstrut -\mathstrut \) \(159876692\) \(\beta_{4}\mathstrut -\mathstrut \) \(56581274\) \(\beta_{3}\mathstrut -\mathstrut \) \(153832270\) \(\beta_{2}\mathstrut +\mathstrut \) \(52036536\) \(\beta_{1}\mathstrut +\mathstrut \) \(401596603\)

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
−2.95046
−2.77887
−2.71742
−2.24852
−1.56679
0.419617
0.640622
0.804573
1.61468
1.68008
2.29813
2.93768
3.48636
5.38033
0 −1.00000 0 −3.95046 0 −4.62501 0 1.00000 0
1.2 0 −1.00000 0 −3.77887 0 −3.84186 0 1.00000 0
1.3 0 −1.00000 0 −3.71742 0 3.96243 0 1.00000 0
1.4 0 −1.00000 0 −3.24852 0 0.794629 0 1.00000 0
1.5 0 −1.00000 0 −2.56679 0 1.84647 0 1.00000 0
1.6 0 −1.00000 0 −0.580383 0 3.10706 0 1.00000 0
1.7 0 −1.00000 0 −0.359378 0 −3.15598 0 1.00000 0
1.8 0 −1.00000 0 −0.195427 0 −0.953396 0 1.00000 0
1.9 0 −1.00000 0 0.614677 0 2.26109 0 1.00000 0
1.10 0 −1.00000 0 0.680080 0 −1.01908 0 1.00000 0
1.11 0 −1.00000 0 1.29813 0 −0.843601 0 1.00000 0
1.12 0 −1.00000 0 1.93768 0 2.74501 0 1.00000 0
1.13 0 −1.00000 0 2.48636 0 −2.46076 0 1.00000 0
1.14 0 −1.00000 0 4.38033 0 3.18300 0 1.00000 0
\(n\): 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
\(2\) \(1\)
\(3\) \(1\)
\(251\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6024))\):

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