Properties

Label 6036.2.a.f
Level 6036
Weight 2
Character orbit 6036.a
Self dual Yes
Analytic conductor 48.198
Analytic rank 1
Dimension 14
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6036.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.19770266\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(6\) \(x^{13}\mathstrut -\mathstrut \) \(12\) \(x^{12}\mathstrut +\mathstrut \) \(112\) \(x^{11}\mathstrut +\mathstrut \) \(7\) \(x^{10}\mathstrut -\mathstrut \) \(710\) \(x^{9}\mathstrut +\mathstrut \) \(281\) \(x^{8}\mathstrut +\mathstrut \) \(1850\) \(x^{7}\mathstrut -\mathstrut \) \(830\) \(x^{6}\mathstrut -\mathstrut \) \(1918\) \(x^{5}\mathstrut +\mathstrut \) \(623\) \(x^{4}\mathstrut +\mathstrut \) \(480\) \(x^{3}\mathstrut -\mathstrut \) \(29\) \(x^{2}\mathstrut -\mathstrut \) \(17\) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(7952245\) \(\nu^{13}\mathstrut +\mathstrut \) \(3028708645\) \(\nu^{12}\mathstrut -\mathstrut \) \(18771968543\) \(\nu^{11}\mathstrut -\mathstrut \) \(30186210676\) \(\nu^{10}\mathstrut +\mathstrut \) \(340361208007\) \(\nu^{9}\mathstrut -\mathstrut \) \(80387457108\) \(\nu^{8}\mathstrut -\mathstrut \) \(2044944803430\) \(\nu^{7}\mathstrut +\mathstrut \) \(1364696185591\) \(\nu^{6}\mathstrut +\mathstrut \) \(4893856962964\) \(\nu^{5}\mathstrut -\mathstrut \) \(3375561656235\) \(\nu^{4}\mathstrut -\mathstrut \) \(4540099016410\) \(\nu^{3}\mathstrut +\mathstrut \) \(2294729878823\) \(\nu^{2}\mathstrut +\mathstrut \) \(889051980100\) \(\nu\mathstrut -\mathstrut \) \(90550432649\)\()/\)\(27475685729\)
\(\beta_{3}\)\(=\)\((\)\(322100129\) \(\nu^{13}\mathstrut -\mathstrut \) \(2221309782\) \(\nu^{12}\mathstrut -\mathstrut \) \(1385989722\) \(\nu^{11}\mathstrut +\mathstrut \) \(33379320665\) \(\nu^{10}\mathstrut -\mathstrut \) \(26925118609\) \(\nu^{9}\mathstrut -\mathstrut \) \(141373459510\) \(\nu^{8}\mathstrut +\mathstrut \) \(108603473694\) \(\nu^{7}\mathstrut +\mathstrut \) \(189473079974\) \(\nu^{6}\mathstrut +\mathstrut \) \(294369123865\) \(\nu^{5}\mathstrut -\mathstrut \) \(351047224710\) \(\nu^{4}\mathstrut -\mathstrut \) \(960664064613\) \(\nu^{3}\mathstrut +\mathstrut \) \(605489812423\) \(\nu^{2}\mathstrut +\mathstrut \) \(293945891681\) \(\nu\mathstrut -\mathstrut \) \(60056755878\)\()/\)\(27475685729\)
\(\beta_{4}\)\(=\)\((\)\(2007881369\) \(\nu^{13}\mathstrut -\mathstrut \) \(11372279696\) \(\nu^{12}\mathstrut -\mathstrut \) \(27653630613\) \(\nu^{11}\mathstrut +\mathstrut \) \(212995039129\) \(\nu^{10}\mathstrut +\mathstrut \) \(90447398538\) \(\nu^{9}\mathstrut -\mathstrut \) \(1366694869203\) \(\nu^{8}\mathstrut -\mathstrut \) \(13103462483\) \(\nu^{7}\mathstrut +\mathstrut \) \(3716454436865\) \(\nu^{6}\mathstrut +\mathstrut \) \(140577439895\) \(\nu^{5}\mathstrut -\mathstrut \) \(4444849279339\) \(\nu^{4}\mathstrut -\mathstrut \) \(761666677339\) \(\nu^{3}\mathstrut +\mathstrut \) \(1787405514567\) \(\nu^{2}\mathstrut +\mathstrut \) \(248097278534\) \(\nu\mathstrut -\mathstrut \) \(56412665837\)\()/\)\(27475685729\)
\(\beta_{5}\)\(=\)\((\)\(3152210972\) \(\nu^{13}\mathstrut -\mathstrut \) \(20759780039\) \(\nu^{12}\mathstrut -\mathstrut \) \(24374867588\) \(\nu^{11}\mathstrut +\mathstrut \) \(357762732425\) \(\nu^{10}\mathstrut -\mathstrut \) \(190637263243\) \(\nu^{9}\mathstrut -\mathstrut \) \(1975075093259\) \(\nu^{8}\mathstrut +\mathstrut \) \(1863340381380\) \(\nu^{7}\mathstrut +\mathstrut \) \(4044469895292\) \(\nu^{6}\mathstrut -\mathstrut \) \(3826100152528\) \(\nu^{5}\mathstrut -\mathstrut \) \(2938481582867\) \(\nu^{4}\mathstrut +\mathstrut \) \(1894425132298\) \(\nu^{3}\mathstrut +\mathstrut \) \(318919205885\) \(\nu^{2}\mathstrut +\mathstrut \) \(143731895017\) \(\nu\mathstrut +\mathstrut \) \(3219219179\)\()/\)\(27475685729\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(4165887700\) \(\nu^{13}\mathstrut +\mathstrut \) \(24780738655\) \(\nu^{12}\mathstrut +\mathstrut \) \(51258093215\) \(\nu^{11}\mathstrut -\mathstrut \) \(465960412783\) \(\nu^{10}\mathstrut -\mathstrut \) \(40614512801\) \(\nu^{9}\mathstrut +\mathstrut \) \(2972385841950\) \(\nu^{8}\mathstrut -\mathstrut \) \(1214872247346\) \(\nu^{7}\mathstrut -\mathstrut \) \(7679322381514\) \(\nu^{6}\mathstrut +\mathstrut \) \(3968036094497\) \(\nu^{5}\mathstrut +\mathstrut \) \(7304963685496\) \(\nu^{4}\mathstrut -\mathstrut \) \(3328805948384\) \(\nu^{3}\mathstrut -\mathstrut \) \(784133180722\) \(\nu^{2}\mathstrut +\mathstrut \) \(26632417227\) \(\nu\mathstrut -\mathstrut \) \(119337497143\)\()/\)\(27475685729\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(4212643376\) \(\nu^{13}\mathstrut +\mathstrut \) \(23573883354\) \(\nu^{12}\mathstrut +\mathstrut \) \(58572637137\) \(\nu^{11}\mathstrut -\mathstrut \) \(436114908343\) \(\nu^{10}\mathstrut -\mathstrut \) \(211668844498\) \(\nu^{9}\mathstrut +\mathstrut \) \(2742795204711\) \(\nu^{8}\mathstrut +\mathstrut \) \(273075199697\) \(\nu^{7}\mathstrut -\mathstrut \) \(7251378248568\) \(\nu^{6}\mathstrut -\mathstrut \) \(1288294144460\) \(\nu^{5}\mathstrut +\mathstrut \) \(8566098010242\) \(\nu^{4}\mathstrut +\mathstrut \) \(2908141882017\) \(\nu^{3}\mathstrut -\mathstrut \) \(3695267323010\) \(\nu^{2}\mathstrut -\mathstrut \) \(796815072181\) \(\nu\mathstrut +\mathstrut \) \(176609367247\)\()/\)\(27475685729\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(4905000419\) \(\nu^{13}\mathstrut +\mathstrut \) \(31618495960\) \(\nu^{12}\mathstrut +\mathstrut \) \(44138491000\) \(\nu^{11}\mathstrut -\mathstrut \) \(564543134100\) \(\nu^{10}\mathstrut +\mathstrut \) \(217855681025\) \(\nu^{9}\mathstrut +\mathstrut \) \(3320329763540\) \(\nu^{8}\mathstrut -\mathstrut \) \(2757968746381\) \(\nu^{7}\mathstrut -\mathstrut \) \(7619196456661\) \(\nu^{6}\mathstrut +\mathstrut \) \(6870213497832\) \(\nu^{5}\mathstrut +\mathstrut \) \(6442164287423\) \(\nu^{4}\mathstrut -\mathstrut \) \(5143052518658\) \(\nu^{3}\mathstrut -\mathstrut \) \(805240090037\) \(\nu^{2}\mathstrut +\mathstrut \) \(458125175223\) \(\nu\mathstrut +\mathstrut \) \(2484730916\)\()/\)\(27475685729\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(5439301880\) \(\nu^{13}\mathstrut +\mathstrut \) \(31641095191\) \(\nu^{12}\mathstrut +\mathstrut \) \(72188909248\) \(\nu^{11}\mathstrut -\mathstrut \) \(605726146495\) \(\nu^{10}\mathstrut -\mathstrut \) \(142566061996\) \(\nu^{9}\mathstrut +\mathstrut \) \(3977671773047\) \(\nu^{8}\mathstrut -\mathstrut \) \(1104496521124\) \(\nu^{7}\mathstrut -\mathstrut \) \(10794875184923\) \(\nu^{6}\mathstrut +\mathstrut \) \(4255215809352\) \(\nu^{5}\mathstrut +\mathstrut \) \(11365564243223\) \(\nu^{4}\mathstrut -\mathstrut \) \(3793738276432\) \(\nu^{3}\mathstrut -\mathstrut \) \(2568116450402\) \(\nu^{2}\mathstrut +\mathstrut \) \(61012448226\) \(\nu\mathstrut +\mathstrut \) \(82980122956\)\()/\)\(27475685729\)
\(\beta_{10}\)\(=\)\((\)\(5590186212\) \(\nu^{13}\mathstrut -\mathstrut \) \(31970020417\) \(\nu^{12}\mathstrut -\mathstrut \) \(78495590618\) \(\nu^{11}\mathstrut +\mathstrut \) \(621759608457\) \(\nu^{10}\mathstrut +\mathstrut \) \(220244655572\) \(\nu^{9}\mathstrut -\mathstrut \) \(4186680361883\) \(\nu^{8}\mathstrut +\mathstrut \) \(729167452279\) \(\nu^{7}\mathstrut +\mathstrut \) \(11827134969219\) \(\nu^{6}\mathstrut -\mathstrut \) \(3525560595104\) \(\nu^{5}\mathstrut -\mathstrut \) \(13313274652841\) \(\nu^{4}\mathstrut +\mathstrut \) \(3237196035944\) \(\nu^{3}\mathstrut +\mathstrut \) \(3743508421935\) \(\nu^{2}\mathstrut +\mathstrut \) \(53467299082\) \(\nu\mathstrut -\mathstrut \) \(146248145768\)\()/\)\(27475685729\)
\(\beta_{11}\)\(=\)\((\)\(6957537558\) \(\nu^{13}\mathstrut -\mathstrut \) \(41934570665\) \(\nu^{12}\mathstrut -\mathstrut \) \(81776887223\) \(\nu^{11}\mathstrut +\mathstrut \) \(778020971724\) \(\nu^{10}\mathstrut +\mathstrut \) \(20346448125\) \(\nu^{9}\mathstrut -\mathstrut \) \(4874086179798\) \(\nu^{8}\mathstrut +\mathstrut \) \(2096077938535\) \(\nu^{7}\mathstrut +\mathstrut \) \(12373242334648\) \(\nu^{6}\mathstrut -\mathstrut \) \(5968092819620\) \(\nu^{5}\mathstrut -\mathstrut \) \(11953799699222\) \(\nu^{4}\mathstrut +\mathstrut \) \(4220854652683\) \(\nu^{3}\mathstrut +\mathstrut \) \(1933119775716\) \(\nu^{2}\mathstrut +\mathstrut \) \(30920955088\) \(\nu\mathstrut +\mathstrut \) \(29960854643\)\()/\)\(27475685729\)
\(\beta_{12}\)\(=\)\((\)\(9079109911\) \(\nu^{13}\mathstrut -\mathstrut \) \(55197598148\) \(\nu^{12}\mathstrut -\mathstrut \) \(103938633805\) \(\nu^{11}\mathstrut +\mathstrut \) \(1019970626770\) \(\nu^{10}\mathstrut -\mathstrut \) \(15375661377\) \(\nu^{9}\mathstrut -\mathstrut \) \(6366193544144\) \(\nu^{8}\mathstrut +\mathstrut \) \(2906083325035\) \(\nu^{7}\mathstrut +\mathstrut \) \(16231443971053\) \(\nu^{6}\mathstrut -\mathstrut \) \(7907306353990\) \(\nu^{5}\mathstrut -\mathstrut \) \(16474469130588\) \(\nu^{4}\mathstrut +\mathstrut \) \(5441706575046\) \(\nu^{3}\mathstrut +\mathstrut \) \(4026197625691\) \(\nu^{2}\mathstrut -\mathstrut \) \(114830934896\) \(\nu\mathstrut -\mathstrut \) \(77123318663\)\()/\)\(27475685729\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(21130993205\) \(\nu^{13}\mathstrut +\mathstrut \) \(130464228246\) \(\nu^{12}\mathstrut +\mathstrut \) \(228630777256\) \(\nu^{11}\mathstrut -\mathstrut \) \(2390891047375\) \(\nu^{10}\mathstrut +\mathstrut \) \(278650380659\) \(\nu^{9}\mathstrut +\mathstrut \) \(14701619372219\) \(\nu^{8}\mathstrut -\mathstrut \) \(8258332157603\) \(\nu^{7}\mathstrut -\mathstrut \) \(36417219408842\) \(\nu^{6}\mathstrut +\mathstrut \) \(22136370008464\) \(\nu^{5}\mathstrut +\mathstrut \) \(34756741871282\) \(\nu^{4}\mathstrut -\mathstrut \) \(16212111998511\) \(\nu^{3}\mathstrut -\mathstrut \) \(6531867947174\) \(\nu^{2}\mathstrut +\mathstrut \) \(662798462431\) \(\nu\mathstrut +\mathstrut \) \(70165343107\)\()/\)\(27475685729\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(2\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(4\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{4}\)\(=\)\(-\)\(19\) \(\beta_{13}\mathstrut -\mathstrut \) \(13\) \(\beta_{12}\mathstrut -\mathstrut \) \(7\) \(\beta_{11}\mathstrut -\mathstrut \) \(17\) \(\beta_{10}\mathstrut -\mathstrut \) \(11\) \(\beta_{9}\mathstrut +\mathstrut \) \(21\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(11\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\) \(\beta_{1}\mathstrut +\mathstrut \) \(39\)
\(\nu^{5}\)\(=\)\(-\)\(61\) \(\beta_{13}\mathstrut -\mathstrut \) \(21\) \(\beta_{12}\mathstrut -\mathstrut \) \(34\) \(\beta_{11}\mathstrut -\mathstrut \) \(48\) \(\beta_{10}\mathstrut -\mathstrut \) \(17\) \(\beta_{9}\mathstrut +\mathstrut \) \(87\) \(\beta_{8}\mathstrut -\mathstrut \) \(18\) \(\beta_{7}\mathstrut +\mathstrut \) \(22\) \(\beta_{6}\mathstrut -\mathstrut \) \(51\) \(\beta_{5}\mathstrut -\mathstrut \) \(45\) \(\beta_{4}\mathstrut +\mathstrut \) \(24\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(105\) \(\beta_{1}\mathstrut +\mathstrut \) \(86\)
\(\nu^{6}\)\(=\)\(-\)\(351\) \(\beta_{13}\mathstrut -\mathstrut \) \(201\) \(\beta_{12}\mathstrut -\mathstrut \) \(176\) \(\beta_{11}\mathstrut -\mathstrut \) \(283\) \(\beta_{10}\mathstrut -\mathstrut \) \(125\) \(\beta_{9}\mathstrut +\mathstrut \) \(384\) \(\beta_{8}\mathstrut -\mathstrut \) \(43\) \(\beta_{7}\mathstrut +\mathstrut \) \(146\) \(\beta_{6}\mathstrut -\mathstrut \) \(286\) \(\beta_{5}\mathstrut -\mathstrut \) \(150\) \(\beta_{4}\mathstrut +\mathstrut \) \(150\) \(\beta_{3}\mathstrut +\mathstrut \) \(56\) \(\beta_{2}\mathstrut +\mathstrut \) \(296\) \(\beta_{1}\mathstrut +\mathstrut \) \(509\)
\(\nu^{7}\)\(=\)\(-\)\(1313\) \(\beta_{13}\mathstrut -\mathstrut \) \(577\) \(\beta_{12}\mathstrut -\mathstrut \) \(758\) \(\beta_{11}\mathstrut -\mathstrut \) \(956\) \(\beta_{10}\mathstrut -\mathstrut \) \(256\) \(\beta_{9}\mathstrut +\mathstrut \) \(1607\) \(\beta_{8}\mathstrut -\mathstrut \) \(316\) \(\beta_{7}\mathstrut +\mathstrut \) \(415\) \(\beta_{6}\mathstrut -\mathstrut \) \(1063\) \(\beta_{5}\mathstrut -\mathstrut \) \(894\) \(\beta_{4}\mathstrut +\mathstrut \) \(453\) \(\beta_{3}\mathstrut +\mathstrut \) \(108\) \(\beta_{2}\mathstrut +\mathstrut \) \(1475\) \(\beta_{1}\mathstrut +\mathstrut \) \(1568\)
\(\nu^{8}\)\(=\)\(-\)\(6332\) \(\beta_{13}\mathstrut -\mathstrut \) \(3363\) \(\beta_{12}\mathstrut -\mathstrut \) \(3506\) \(\beta_{11}\mathstrut -\mathstrut \) \(4769\) \(\beta_{10}\mathstrut -\mathstrut \) \(1530\) \(\beta_{9}\mathstrut +\mathstrut \) \(6821\) \(\beta_{8}\mathstrut -\mathstrut \) \(1028\) \(\beta_{7}\mathstrut +\mathstrut \) \(2178\) \(\beta_{6}\mathstrut -\mathstrut \) \(5123\) \(\beta_{5}\mathstrut -\mathstrut \) \(3447\) \(\beta_{4}\mathstrut +\mathstrut \) \(2301\) \(\beta_{3}\mathstrut +\mathstrut \) \(1170\) \(\beta_{2}\mathstrut +\mathstrut \) \(5109\) \(\beta_{1}\mathstrut +\mathstrut \) \(7649\)
\(\nu^{9}\)\(=\)\(-\)\(25174\) \(\beta_{13}\mathstrut -\mathstrut \) \(11936\) \(\beta_{12}\mathstrut -\mathstrut \) \(14774\) \(\beta_{11}\mathstrut -\mathstrut \) \(17822\) \(\beta_{10}\mathstrut -\mathstrut \) \(3989\) \(\beta_{9}\mathstrut +\mathstrut \) \(28577\) \(\beta_{8}\mathstrut -\mathstrut \) \(5537\) \(\beta_{7}\mathstrut +\mathstrut \) \(7455\) \(\beta_{6}\mathstrut -\mathstrut \) \(20278\) \(\beta_{5}\mathstrut -\mathstrut \) \(16756\) \(\beta_{4}\mathstrut +\mathstrut \) \(8177\) \(\beta_{3}\mathstrut +\mathstrut \) \(3712\) \(\beta_{2}\mathstrut +\mathstrut \) \(23052\) \(\beta_{1}\mathstrut +\mathstrut \) \(27544\)
\(\nu^{10}\)\(=\)\(-\)\(112550\) \(\beta_{13}\mathstrut -\mathstrut \) \(57882\) \(\beta_{12}\mathstrut -\mathstrut \) \(64910\) \(\beta_{11}\mathstrut -\mathstrut \) \(81496\) \(\beta_{10}\mathstrut -\mathstrut \) \(20608\) \(\beta_{9}\mathstrut +\mathstrut \) \(119964\) \(\beta_{8}\mathstrut -\mathstrut \) \(20571\) \(\beta_{7}\mathstrut +\mathstrut \) \(34817\) \(\beta_{6}\mathstrut -\mathstrut \) \(90916\) \(\beta_{5}\mathstrut -\mathstrut \) \(67847\) \(\beta_{4}\mathstrut +\mathstrut \) \(37636\) \(\beta_{3}\mathstrut +\mathstrut \) \(22173\) \(\beta_{2}\mathstrut +\mathstrut \) \(88090\) \(\beta_{1}\mathstrut +\mathstrut \) \(123472\)
\(\nu^{11}\)\(=\)\(-\)\(459495\) \(\beta_{13}\mathstrut -\mathstrut \) \(224498\) \(\beta_{12}\mathstrut -\mathstrut \) \(272049\) \(\beta_{11}\mathstrut -\mathstrut \) \(321593\) \(\beta_{10}\mathstrut -\mathstrut \) \(64717\) \(\beta_{9}\mathstrut +\mathstrut \) \(502084\) \(\beta_{8}\mathstrut -\mathstrut \) \(96953\) \(\beta_{7}\mathstrut +\mathstrut \) \(131347\) \(\beta_{6}\mathstrut -\mathstrut \) \(370278\) \(\beta_{5}\mathstrut -\mathstrut \) \(303928\) \(\beta_{4}\mathstrut +\mathstrut \) \(145193\) \(\beta_{3}\mathstrut +\mathstrut \) \(82164\) \(\beta_{2}\mathstrut +\mathstrut \) \(380592\) \(\beta_{1}\mathstrut +\mathstrut \) \(479390\)
\(\nu^{12}\)\(=\)\(-\)\(1983492\) \(\beta_{13}\mathstrut -\mathstrut \) \(1005713\) \(\beta_{12}\mathstrut -\mathstrut \) \(1165533\) \(\beta_{11}\mathstrut -\mathstrut \) \(1405827\) \(\beta_{10}\mathstrut -\mathstrut \) \(303735\) \(\beta_{9}\mathstrut +\mathstrut \) \(2100296\) \(\beta_{8}\mathstrut -\mathstrut \) \(382953\) \(\beta_{7}\mathstrut +\mathstrut \) \(579028\) \(\beta_{6}\mathstrut -\mathstrut \) \(1602169\) \(\beta_{5}\mathstrut -\mathstrut \) \(1253426\) \(\beta_{4}\mathstrut +\mathstrut \) \(636376\) \(\beta_{3}\mathstrut +\mathstrut \) \(403225\) \(\beta_{2}\mathstrut +\mathstrut \) \(1523144\) \(\beta_{1}\mathstrut +\mathstrut \) \(2068261\)
\(\nu^{13}\)\(=\)\(-\)\(8195838\) \(\beta_{13}\mathstrut -\mathstrut \) \(4057544\) \(\beta_{12}\mathstrut -\mathstrut \) \(4876133\) \(\beta_{11}\mathstrut -\mathstrut \) \(5706295\) \(\beta_{10}\mathstrut -\mathstrut \) \(1080214\) \(\beta_{9}\mathstrut +\mathstrut \) \(8781237\) \(\beta_{8}\mathstrut -\mathstrut \) \(1696213\) \(\beta_{7}\mathstrut +\mathstrut \) \(2296564\) \(\beta_{6}\mathstrut -\mathstrut \) \(6612632\) \(\beta_{5}\mathstrut -\mathstrut \) \(5413968\) \(\beta_{4}\mathstrut +\mathstrut \) \(2556932\) \(\beta_{3}\mathstrut +\mathstrut \) \(1593848\) \(\beta_{2}\mathstrut +\mathstrut \) \(6457536\) \(\beta_{1}\mathstrut +\mathstrut \) \(8330892\)

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
4.17501
3.05416
2.98659
1.79209
1.63333
0.736579
0.176671
0.0590520
−0.246625
−0.362365
−1.38751
−1.41050
−2.28353
−2.92296
0 −1.00000 0 −4.17501 0 −2.39518 0 1.00000 0
1.2 0 −1.00000 0 −3.05416 0 −1.32476 0 1.00000 0
1.3 0 −1.00000 0 −2.98659 0 −4.28847 0 1.00000 0
1.4 0 −1.00000 0 −1.79209 0 1.22660 0 1.00000 0
1.5 0 −1.00000 0 −1.63333 0 0.404256 0 1.00000 0
1.6 0 −1.00000 0 −0.736579 0 1.91291 0 1.00000 0
1.7 0 −1.00000 0 −0.176671 0 4.02875 0 1.00000 0
1.8 0 −1.00000 0 −0.0590520 0 −1.12902 0 1.00000 0
1.9 0 −1.00000 0 0.246625 0 −4.31906 0 1.00000 0
1.10 0 −1.00000 0 0.362365 0 0.748260 0 1.00000 0
1.11 0 −1.00000 0 1.38751 0 −0.715830 0 1.00000 0
1.12 0 −1.00000 0 1.41050 0 1.12091 0 1.00000 0
1.13 0 −1.00000 0 2.28353 0 −2.96859 0 1.00000 0
1.14 0 −1.00000 0 2.92296 0 0.699214 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
\(2\) \(-1\)
\(3\) \(1\)
\(503\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6036))\):

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