Properties

Label 6024.2.a.p
Level 6024
Weight 2
Character orbit 6024.a
Self dual Yes
Analytic conductor 48.102
Analytic rank 0
Dimension 14
CM No
Inner twists 1

Related objects

Downloads

Learn more about

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: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \( + \beta_{1} q^{5} \) \( + \beta_{10} q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( + \beta_{1} q^{5} \) \( + \beta_{10} q^{7} \) \(+ q^{9}\) \( -\beta_{8} q^{11} \) \( + \beta_{4} q^{13} \) \( -\beta_{1} q^{15} \) \( + ( 2 + \beta_{4} + \beta_{7} + \beta_{8} ) q^{17} \) \( + ( -\beta_{5} + \beta_{6} + \beta_{9} ) q^{19} \) \( -\beta_{10} q^{21} \) \( + ( \beta_{7} + \beta_{12} ) q^{23} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{25} \) \(- q^{27}\) \( + ( 1 + \beta_{3} + \beta_{9} ) q^{29} \) \( + ( -\beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{31} \) \( + \beta_{8} q^{33} \) \( + ( -2 + \beta_{4} - \beta_{7} + 2 \beta_{10} - \beta_{12} ) q^{35} \) \( + ( \beta_{1} - \beta_{2} + \beta_{7} + \beta_{11} ) q^{37} \) \( -\beta_{4} q^{39} \) \( + ( 2 - \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{41} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{11} ) q^{43} \) \( + \beta_{1} q^{45} \) \( + ( 1 - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{10} + \beta_{13} ) q^{47} \) \( + ( 2 + \beta_{3} + \beta_{5} - \beta_{11} + \beta_{12} ) q^{49} \) \( + ( -2 - \beta_{4} - \beta_{7} - \beta_{8} ) q^{51} \) \( + ( -\beta_{3} + \beta_{4} + \beta_{10} - \beta_{12} ) q^{53} \) \( + ( \beta_{2} + 2 \beta_{5} + \beta_{11} - \beta_{13} ) q^{55} \) \( + ( \beta_{5} - \beta_{6} - \beta_{9} ) q^{57} \) \( + ( -\beta_{4} + \beta_{9} - \beta_{13} ) q^{59} \) \( + ( 1 + \beta_{1} - \beta_{11} - \beta_{12} ) q^{61} \) \( + \beta_{10} q^{63} \) \( + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{10} - \beta_{13} ) q^{65} \) \( + ( -3 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{13} ) q^{67} \) \( + ( -\beta_{7} - \beta_{12} ) q^{69} \) \( + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{71} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{11} + \beta_{13} ) q^{73} \) \( + ( -1 - \beta_{1} - \beta_{2} ) q^{75} \) \( + ( 1 - \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{12} ) q^{77} \) \( + ( -1 + 2 \beta_{3} - \beta_{4} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{12} + \beta_{13} ) q^{79} \) \(+ q^{81}\) \( + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{83} \) \( + ( 4 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{85} \) \( + ( -1 - \beta_{3} - \beta_{9} ) q^{87} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{89} \) \( + ( -4 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{91} \) \( + ( \beta_{6} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{93} \) \( + ( 2 + \beta_{1} - \beta_{2} - 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} + 2 \beta_{9} + 2 \beta_{11} ) q^{95} \) \( + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{12} - 2 \beta_{13} ) q^{97} \) \( -\beta_{8} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut 7q^{11} \) \(\mathstrut +\mathstrut 5q^{13} \) \(\mathstrut -\mathstrut 6q^{15} \) \(\mathstrut +\mathstrut 28q^{17} \) \(\mathstrut -\mathstrut q^{19} \) \(\mathstrut -\mathstrut 3q^{21} \) \(\mathstrut -\mathstrut 3q^{23} \) \(\mathstrut +\mathstrut 24q^{25} \) \(\mathstrut -\mathstrut 14q^{27} \) \(\mathstrut +\mathstrut 15q^{29} \) \(\mathstrut +\mathstrut 9q^{31} \) \(\mathstrut -\mathstrut 7q^{33} \) \(\mathstrut -\mathstrut 14q^{35} \) \(\mathstrut +\mathstrut 4q^{37} \) \(\mathstrut -\mathstrut 5q^{39} \) \(\mathstrut +\mathstrut 32q^{41} \) \(\mathstrut -\mathstrut 3q^{43} \) \(\mathstrut +\mathstrut 6q^{45} \) \(\mathstrut +\mathstrut 15q^{47} \) \(\mathstrut +\mathstrut 27q^{49} \) \(\mathstrut -\mathstrut 28q^{51} \) \(\mathstrut +\mathstrut 8q^{53} \) \(\mathstrut +\mathstrut q^{57} \) \(\mathstrut -\mathstrut 11q^{59} \) \(\mathstrut +\mathstrut 25q^{61} \) \(\mathstrut +\mathstrut 3q^{63} \) \(\mathstrut +\mathstrut 18q^{65} \) \(\mathstrut -\mathstrut 20q^{67} \) \(\mathstrut +\mathstrut 3q^{69} \) \(\mathstrut +\mathstrut 33q^{71} \) \(\mathstrut +\mathstrut 8q^{73} \) \(\mathstrut -\mathstrut 24q^{75} \) \(\mathstrut +\mathstrut 14q^{77} \) \(\mathstrut -\mathstrut 17q^{79} \) \(\mathstrut +\mathstrut 14q^{81} \) \(\mathstrut -\mathstrut 4q^{83} \) \(\mathstrut +\mathstrut 16q^{85} \) \(\mathstrut -\mathstrut 15q^{87} \) \(\mathstrut +\mathstrut 14q^{89} \) \(\mathstrut -\mathstrut 28q^{91} \) \(\mathstrut -\mathstrut 9q^{93} \) \(\mathstrut +\mathstrut 3q^{95} \) \(\mathstrut +\mathstrut 9q^{97} \) \(\mathstrut +\mathstrut 7q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(6\) \(x^{13}\mathstrut -\mathstrut \) \(29\) \(x^{12}\mathstrut +\mathstrut \) \(210\) \(x^{11}\mathstrut +\mathstrut \) \(280\) \(x^{10}\mathstrut -\mathstrut \) \(2796\) \(x^{9}\mathstrut -\mathstrut \) \(863\) \(x^{8}\mathstrut +\mathstrut \) \(17652\) \(x^{7}\mathstrut -\mathstrut \) \(1300\) \(x^{6}\mathstrut -\mathstrut \) \(53458\) \(x^{5}\mathstrut +\mathstrut \) \(10892\) \(x^{4}\mathstrut +\mathstrut \) \(67298\) \(x^{3}\mathstrut -\mathstrut \) \(13761\) \(x^{2}\mathstrut -\mathstrut \) \(22140\) \(x\mathstrut -\mathstrut \) \(3364\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 6 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(34385\) \(\nu^{13}\mathstrut +\mathstrut \) \(26892\) \(\nu^{12}\mathstrut +\mathstrut \) \(1553117\) \(\nu^{11}\mathstrut -\mathstrut \) \(869644\) \(\nu^{10}\mathstrut -\mathstrut \) \(26472460\) \(\nu^{9}\mathstrut +\mathstrut \) \(9438328\) \(\nu^{8}\mathstrut +\mathstrut \) \(207665199\) \(\nu^{7}\mathstrut -\mathstrut \) \(38974970\) \(\nu^{6}\mathstrut -\mathstrut \) \(726563800\) \(\nu^{5}\mathstrut +\mathstrut \) \(59731310\) \(\nu^{4}\mathstrut +\mathstrut \) \(933220260\) \(\nu^{3}\mathstrut -\mathstrut \) \(52788358\) \(\nu^{2}\mathstrut -\mathstrut \) \(253959291\) \(\nu\mathstrut -\mathstrut \) \(56474950\)\()/10615128\)
\(\beta_{4}\)\(=\)\((\)\(891500\) \(\nu^{13}\mathstrut -\mathstrut \) \(2582249\) \(\nu^{12}\mathstrut -\mathstrut \) \(32877251\) \(\nu^{11}\mathstrut +\mathstrut \) \(81418106\) \(\nu^{10}\mathstrut +\mathstrut \) \(470527080\) \(\nu^{9}\mathstrut -\mathstrut \) \(921011508\) \(\nu^{8}\mathstrut -\mathstrut \) \(3245410102\) \(\nu^{7}\mathstrut +\mathstrut \) \(4526703895\) \(\nu^{6}\mathstrut +\mathstrut \) \(10740423537\) \(\nu^{5}\mathstrut -\mathstrut \) \(9653543531\) \(\nu^{4}\mathstrut -\mathstrut \) \(14580166089\) \(\nu^{3}\mathstrut +\mathstrut \) \(7501423177\) \(\nu^{2}\mathstrut +\mathstrut \) \(5250877601\) \(\nu\mathstrut +\mathstrut \) \(543511258\)\()/47768076\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(2285707\) \(\nu^{13}\mathstrut +\mathstrut \) \(5763184\) \(\nu^{12}\mathstrut +\mathstrut \) \(85005175\) \(\nu^{11}\mathstrut -\mathstrut \) \(177040372\) \(\nu^{10}\mathstrut -\mathstrut \) \(1224526764\) \(\nu^{9}\mathstrut +\mathstrut \) \(1920505680\) \(\nu^{8}\mathstrut +\mathstrut \) \(8476711133\) \(\nu^{7}\mathstrut -\mathstrut \) \(8770496906\) \(\nu^{6}\mathstrut -\mathstrut \) \(28100462208\) \(\nu^{5}\mathstrut +\mathstrut \) \(16436289310\) \(\nu^{4}\mathstrut +\mathstrut \) \(38536651212\) \(\nu^{3}\mathstrut -\mathstrut \) \(10547722286\) \(\nu^{2}\mathstrut -\mathstrut \) \(15515835721\) \(\nu\mathstrut -\mathstrut \) \(2060172338\)\()/95536152\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(510731\) \(\nu^{13}\mathstrut +\mathstrut \) \(1185871\) \(\nu^{12}\mathstrut +\mathstrut \) \(19439080\) \(\nu^{11}\mathstrut -\mathstrut \) \(36738588\) \(\nu^{10}\mathstrut -\mathstrut \) \(286607520\) \(\nu^{9}\mathstrut +\mathstrut \) \(402727860\) \(\nu^{8}\mathstrut +\mathstrut \) \(2026352809\) \(\nu^{7}\mathstrut -\mathstrut \) \(1865943851\) \(\nu^{6}\mathstrut -\mathstrut \) \(6827363977\) \(\nu^{5}\mathstrut +\mathstrut \) \(3566414523\) \(\nu^{4}\mathstrut +\mathstrut \) \(9440736513\) \(\nu^{3}\mathstrut -\mathstrut \) \(2259965299\) \(\nu^{2}\mathstrut -\mathstrut \) \(3691152442\) \(\nu\mathstrut -\mathstrut \) \(677427348\)\()/15922692\)
\(\beta_{7}\)\(=\)\((\)\(1750919\) \(\nu^{13}\mathstrut -\mathstrut \) \(4072364\) \(\nu^{12}\mathstrut -\mathstrut \) \(65938241\) \(\nu^{11}\mathstrut +\mathstrut \) \(124543394\) \(\nu^{10}\mathstrut +\mathstrut \) \(961333458\) \(\nu^{9}\mathstrut -\mathstrut \) \(1341558738\) \(\nu^{8}\mathstrut -\mathstrut \) \(6714256819\) \(\nu^{7}\mathstrut +\mathstrut \) \(6055282684\) \(\nu^{6}\mathstrut +\mathstrut \) \(22294893624\) \(\nu^{5}\mathstrut -\mathstrut \) \(11158864646\) \(\nu^{4}\mathstrut -\mathstrut \) \(30232129854\) \(\nu^{3}\mathstrut +\mathstrut \) \(7029931060\) \(\nu^{2}\mathstrut +\mathstrut \) \(11780644049\) \(\nu\mathstrut +\mathstrut \) \(1930586878\)\()/47768076\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(799400\) \(\nu^{13}\mathstrut +\mathstrut \) \(2007980\) \(\nu^{12}\mathstrut +\mathstrut \) \(29958653\) \(\nu^{11}\mathstrut -\mathstrut \) \(62535530\) \(\nu^{10}\mathstrut -\mathstrut \) \(434576058\) \(\nu^{9}\mathstrut +\mathstrut \) \(693826704\) \(\nu^{8}\mathstrut +\mathstrut \) \(3020215300\) \(\nu^{7}\mathstrut -\mathstrut \) \(3303220780\) \(\nu^{6}\mathstrut -\mathstrut \) \(9973980471\) \(\nu^{5}\mathstrut +\mathstrut \) \(6716846546\) \(\nu^{4}\mathstrut +\mathstrut \) \(13368961119\) \(\nu^{3}\mathstrut -\mathstrut \) \(4982795950\) \(\nu^{2}\mathstrut -\mathstrut \) \(4904604035\) \(\nu\mathstrut -\mathstrut \) \(636577414\)\()/15922692\)
\(\beta_{9}\)\(=\)\((\)\(2690857\) \(\nu^{13}\mathstrut -\mathstrut \) \(6588652\) \(\nu^{12}\mathstrut -\mathstrut \) \(101245663\) \(\nu^{11}\mathstrut +\mathstrut \) \(205940440\) \(\nu^{10}\mathstrut +\mathstrut \) \(1473998214\) \(\nu^{9}\mathstrut -\mathstrut \) \(2300026932\) \(\nu^{8}\mathstrut -\mathstrut \) \(10278303353\) \(\nu^{7}\mathstrut +\mathstrut \) \(11083245044\) \(\nu^{6}\mathstrut +\mathstrut \) \(34072151028\) \(\nu^{5}\mathstrut -\mathstrut \) \(22949713318\) \(\nu^{4}\mathstrut -\mathstrut \) \(45888598104\) \(\nu^{3}\mathstrut +\mathstrut \) \(17130259784\) \(\nu^{2}\mathstrut +\mathstrut \) \(16722285277\) \(\nu\mathstrut +\mathstrut \) \(2458464146\)\()/47768076\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(1085369\) \(\nu^{13}\mathstrut +\mathstrut \) \(2654776\) \(\nu^{12}\mathstrut +\mathstrut \) \(40709620\) \(\nu^{11}\mathstrut -\mathstrut \) \(82526094\) \(\nu^{10}\mathstrut -\mathstrut \) \(590411514\) \(\nu^{9}\mathstrut +\mathstrut \) \(913528686\) \(\nu^{8}\mathstrut +\mathstrut \) \(4096485739\) \(\nu^{7}\mathstrut -\mathstrut \) \(4337434292\) \(\nu^{6}\mathstrut -\mathstrut \) \(13473230383\) \(\nu^{5}\mathstrut +\mathstrut \) \(8801355990\) \(\nu^{4}\mathstrut +\mathstrut \) \(17847143889\) \(\nu^{3}\mathstrut -\mathstrut \) \(6578394766\) \(\nu^{2}\mathstrut -\mathstrut \) \(6272928610\) \(\nu\mathstrut -\mathstrut \) \(798333708\)\()/15922692\)
\(\beta_{11}\)\(=\)\((\)\(1359385\) \(\nu^{13}\mathstrut -\mathstrut \) \(3285059\) \(\nu^{12}\mathstrut -\mathstrut \) \(51337298\) \(\nu^{11}\mathstrut +\mathstrut \) \(101829564\) \(\nu^{10}\mathstrut +\mathstrut \) \(751000860\) \(\nu^{9}\mathstrut -\mathstrut \) \(1119757014\) \(\nu^{8}\mathstrut -\mathstrut \) \(5266901315\) \(\nu^{7}\mathstrut +\mathstrut \) \(5240205949\) \(\nu^{6}\mathstrut +\mathstrut \) \(17571182555\) \(\nu^{5}\mathstrut -\mathstrut \) \(10334058561\) \(\nu^{4}\mathstrut -\mathstrut \) \(23897723349\) \(\nu^{3}\mathstrut +\mathstrut \) \(7318586177\) \(\nu^{2}\mathstrut +\mathstrut \) \(9075081374\) \(\nu\mathstrut +\mathstrut \) \(1300892340\)\()/15922692\)
\(\beta_{12}\)\(=\)\((\)\(1400227\) \(\nu^{13}\mathstrut -\mathstrut \) \(3427662\) \(\nu^{12}\mathstrut -\mathstrut \) \(52961826\) \(\nu^{11}\mathstrut +\mathstrut \) \(107080910\) \(\nu^{10}\mathstrut +\mathstrut \) \(776737884\) \(\nu^{9}\mathstrut -\mathstrut \) \(1192572510\) \(\nu^{8}\mathstrut -\mathstrut \) \(5472245579\) \(\nu^{7}\mathstrut +\mathstrut \) \(5699815236\) \(\nu^{6}\mathstrut +\mathstrut \) \(18422349217\) \(\nu^{5}\mathstrut -\mathstrut \) \(11564497352\) \(\nu^{4}\mathstrut -\mathstrut \) \(25553159163\) \(\nu^{3}\mathstrut +\mathstrut \) \(8209065926\) \(\nu^{2}\mathstrut +\mathstrut \) \(10105957332\) \(\nu\mathstrut +\mathstrut \) \(1560309976\)\()/15922692\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(6572812\) \(\nu^{13}\mathstrut +\mathstrut \) \(16236166\) \(\nu^{12}\mathstrut +\mathstrut \) \(247008691\) \(\nu^{11}\mathstrut -\mathstrut \) \(503783854\) \(\nu^{10}\mathstrut -\mathstrut \) \(3595411272\) \(\nu^{9}\mathstrut +\mathstrut \) \(5552233938\) \(\nu^{8}\mathstrut +\mathstrut \) \(25099371248\) \(\nu^{7}\mathstrut -\mathstrut \) \(26089480472\) \(\nu^{6}\mathstrut -\mathstrut \) \(83439350505\) \(\nu^{5}\mathstrut +\mathstrut \) \(51662191384\) \(\nu^{4}\mathstrut +\mathstrut \) \(113289156915\) \(\nu^{3}\mathstrut -\mathstrut \) \(36259637984\) \(\nu^{2}\mathstrut -\mathstrut \) \(43344239917\) \(\nu\mathstrut -\mathstrut \) \(6511648574\)\()/47768076\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{3}\)\(=\)\(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(16\) \(\beta_{1}\mathstrut +\mathstrut \) \(60\)
\(\nu^{5}\)\(=\)\(15\) \(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(19\) \(\beta_{8}\mathstrut +\mathstrut \) \(15\) \(\beta_{7}\mathstrut -\mathstrut \) \(15\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(13\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(131\) \(\beta_{1}\mathstrut +\mathstrut \) \(44\)
\(\nu^{6}\)\(=\)\(-\)\(12\) \(\beta_{13}\mathstrut -\mathstrut \) \(16\) \(\beta_{12}\mathstrut +\mathstrut \) \(3\) \(\beta_{11}\mathstrut +\mathstrut \) \(17\) \(\beta_{10}\mathstrut +\mathstrut \) \(14\) \(\beta_{9}\mathstrut +\mathstrut \) \(9\) \(\beta_{8}\mathstrut +\mathstrut \) \(20\) \(\beta_{7}\mathstrut -\mathstrut \) \(19\) \(\beta_{6}\mathstrut +\mathstrut \) \(54\) \(\beta_{5}\mathstrut +\mathstrut \) \(18\) \(\beta_{4}\mathstrut +\mathstrut \) \(20\) \(\beta_{3}\mathstrut +\mathstrut \) \(175\) \(\beta_{2}\mathstrut +\mathstrut \) \(233\) \(\beta_{1}\mathstrut +\mathstrut \) \(678\)
\(\nu^{7}\)\(=\)\(187\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(5\) \(\beta_{11}\mathstrut +\mathstrut \) \(36\) \(\beta_{10}\mathstrut -\mathstrut \) \(7\) \(\beta_{9}\mathstrut -\mathstrut \) \(268\) \(\beta_{8}\mathstrut +\mathstrut \) \(203\) \(\beta_{7}\mathstrut -\mathstrut \) \(198\) \(\beta_{6}\mathstrut +\mathstrut \) \(45\) \(\beta_{5}\mathstrut +\mathstrut \) \(161\) \(\beta_{4}\mathstrut +\mathstrut \) \(25\) \(\beta_{3}\mathstrut +\mathstrut \) \(109\) \(\beta_{2}\mathstrut +\mathstrut \) \(1602\) \(\beta_{1}\mathstrut +\mathstrut \) \(746\)
\(\nu^{8}\)\(=\)\(-\)\(102\) \(\beta_{13}\mathstrut -\mathstrut \) \(199\) \(\beta_{12}\mathstrut +\mathstrut \) \(45\) \(\beta_{11}\mathstrut +\mathstrut \) \(218\) \(\beta_{10}\mathstrut +\mathstrut \) \(125\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(332\) \(\beta_{7}\mathstrut -\mathstrut \) \(321\) \(\beta_{6}\mathstrut +\mathstrut \) \(800\) \(\beta_{5}\mathstrut +\mathstrut \) \(272\) \(\beta_{4}\mathstrut +\mathstrut \) \(312\) \(\beta_{3}\mathstrut +\mathstrut \) \(2154\) \(\beta_{2}\mathstrut +\mathstrut \) \(3272\) \(\beta_{1}\mathstrut +\mathstrut \) \(8042\)
\(\nu^{9}\)\(=\)\(2245\) \(\beta_{13}\mathstrut -\mathstrut \) \(3\) \(\beta_{12}\mathstrut -\mathstrut \) \(138\) \(\beta_{11}\mathstrut +\mathstrut \) \(473\) \(\beta_{10}\mathstrut -\mathstrut \) \(249\) \(\beta_{9}\mathstrut -\mathstrut \) \(3489\) \(\beta_{8}\mathstrut +\mathstrut \) \(2717\) \(\beta_{7}\mathstrut -\mathstrut \) \(2551\) \(\beta_{6}\mathstrut +\mathstrut \) \(1074\) \(\beta_{5}\mathstrut +\mathstrut \) \(2031\) \(\beta_{4}\mathstrut +\mathstrut \) \(484\) \(\beta_{3}\mathstrut +\mathstrut \) \(2089\) \(\beta_{2}\mathstrut +\mathstrut \) \(19856\) \(\beta_{1}\mathstrut +\mathstrut \) \(11464\)
\(\nu^{10}\)\(=\)\(-\)\(561\) \(\beta_{13}\mathstrut -\mathstrut \) \(2237\) \(\beta_{12}\mathstrut +\mathstrut \) \(393\) \(\beta_{11}\mathstrut +\mathstrut \) \(2508\) \(\beta_{10}\mathstrut +\mathstrut \) \(557\) \(\beta_{9}\mathstrut -\mathstrut \) \(1714\) \(\beta_{8}\mathstrut +\mathstrut \) \(5151\) \(\beta_{7}\mathstrut -\mathstrut \) \(5090\) \(\beta_{6}\mathstrut +\mathstrut \) \(11302\) \(\beta_{5}\mathstrut +\mathstrut \) \(3885\) \(\beta_{4}\mathstrut +\mathstrut \) \(4530\) \(\beta_{3}\mathstrut +\mathstrut \) \(26606\) \(\beta_{2}\mathstrut +\mathstrut \) \(45038\) \(\beta_{1}\mathstrut +\mathstrut \) \(97700\)
\(\nu^{11}\)\(=\)\(26820\) \(\beta_{13}\mathstrut +\mathstrut \) \(346\) \(\beta_{12}\mathstrut -\mathstrut \) \(2758\) \(\beta_{11}\mathstrut +\mathstrut \) \(5426\) \(\beta_{10}\mathstrut -\mathstrut \) \(5826\) \(\beta_{9}\mathstrut -\mathstrut \) \(44574\) \(\beta_{8}\mathstrut +\mathstrut \) \(36376\) \(\beta_{7}\mathstrut -\mathstrut \) \(32878\) \(\beta_{6}\mathstrut +\mathstrut \) \(20445\) \(\beta_{5}\mathstrut +\mathstrut \) \(25946\) \(\beta_{4}\mathstrut +\mathstrut \) \(8453\) \(\beta_{3}\mathstrut +\mathstrut \) \(34772\) \(\beta_{2}\mathstrut +\mathstrut \) \(248237\) \(\beta_{1}\mathstrut +\mathstrut \) \(167648\)
\(\nu^{12}\)\(=\)\(1937\) \(\beta_{13}\mathstrut -\mathstrut \) \(23447\) \(\beta_{12}\mathstrut +\mathstrut \) \(1039\) \(\beta_{11}\mathstrut +\mathstrut \) \(26896\) \(\beta_{10}\mathstrut -\mathstrut \) \(7911\) \(\beta_{9}\mathstrut -\mathstrut \) \(43524\) \(\beta_{8}\mathstrut +\mathstrut \) \(77287\) \(\beta_{7}\mathstrut -\mathstrut \) \(77082\) \(\beta_{6}\mathstrut +\mathstrut \) \(157729\) \(\beta_{5}\mathstrut +\mathstrut \) \(54109\) \(\beta_{4}\mathstrut +\mathstrut \) \(64193\) \(\beta_{3}\mathstrut +\mathstrut \) \(331250\) \(\beta_{2}\mathstrut +\mathstrut \) \(611874\) \(\beta_{1}\mathstrut +\mathstrut \) \(1204036\)
\(\nu^{13}\)\(=\)\(322155\) \(\beta_{13}\mathstrut +\mathstrut \) \(11913\) \(\beta_{12}\mathstrut -\mathstrut \) \(48703\) \(\beta_{11}\mathstrut +\mathstrut \) \(55998\) \(\beta_{10}\mathstrut -\mathstrut \) \(113821\) \(\beta_{9}\mathstrut -\mathstrut \) \(570870\) \(\beta_{8}\mathstrut +\mathstrut \) \(487827\) \(\beta_{7}\mathstrut -\mathstrut \) \(426928\) \(\beta_{6}\mathstrut +\mathstrut \) \(348366\) \(\beta_{5}\mathstrut +\mathstrut \) \(333155\) \(\beta_{4}\mathstrut +\mathstrut \) \(139074\) \(\beta_{3}\mathstrut +\mathstrut \) \(537923\) \(\beta_{2}\mathstrut +\mathstrut \) \(3123648\) \(\beta_{1}\mathstrut +\mathstrut \) \(2379408\)

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
−3.30404
−3.23770
−2.26645
−2.15802
−1.55864
−0.379598
−0.198997
1.10854
1.26787
2.29106
3.55671
3.57026
3.63608
3.67292
0 −1.00000 0 −3.30404 0 −0.450333 0 1.00000 0
1.2 0 −1.00000 0 −3.23770 0 2.16912 0 1.00000 0
1.3 0 −1.00000 0 −2.26645 0 2.39900 0 1.00000 0
1.4 0 −1.00000 0 −2.15802 0 −0.0890340 0 1.00000 0
1.5 0 −1.00000 0 −1.55864 0 −1.20577 0 1.00000 0
1.6 0 −1.00000 0 −0.379598 0 −4.36268 0 1.00000 0
1.7 0 −1.00000 0 −0.198997 0 4.17683 0 1.00000 0
1.8 0 −1.00000 0 1.10854 0 5.12441 0 1.00000 0
1.9 0 −1.00000 0 1.26787 0 −0.0906464 0 1.00000 0
1.10 0 −1.00000 0 2.29106 0 −4.25127 0 1.00000 0
1.11 0 −1.00000 0 3.55671 0 2.50656 0 1.00000 0
1.12 0 −1.00000 0 3.57026 0 −1.78948 0 1.00000 0
1.13 0 −1.00000 0 3.63608 0 2.74254 0 1.00000 0
1.14 0 −1.00000 0 3.67292 0 −3.87925 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6024))\):

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