Properties

Label 6018.2.a.bb
Level 6018
Weight 2
Character orbit 6018.a
Self dual Yes
Analytic conductor 48.054
Analytic rank 0
Dimension 13
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6018.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.0539719364\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13}\mathstrut -\mathstrut \) \(4\) \(x^{12}\mathstrut -\mathstrut \) \(41\) \(x^{11}\mathstrut +\mathstrut \) \(179\) \(x^{10}\mathstrut +\mathstrut \) \(540\) \(x^{9}\mathstrut -\mathstrut \) \(2773\) \(x^{8}\mathstrut -\mathstrut \) \(2260\) \(x^{7}\mathstrut +\mathstrut \) \(17621\) \(x^{6}\mathstrut -\mathstrut \) \(838\) \(x^{5}\mathstrut -\mathstrut \) \(44478\) \(x^{4}\mathstrut +\mathstrut \) \(16472\) \(x^{3}\mathstrut +\mathstrut \) \(29944\) \(x^{2}\mathstrut -\mathstrut \) \(6856\) \(x\mathstrut +\mathstrut \) \(128\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(1796388205553\) \(\nu^{12}\mathstrut -\mathstrut \) \(1489108919712\) \(\nu^{11}\mathstrut +\mathstrut \) \(81660319298773\) \(\nu^{10}\mathstrut +\mathstrut \) \(60295853640717\) \(\nu^{9}\mathstrut -\mathstrut \) \(1346976232371876\) \(\nu^{8}\mathstrut -\mathstrut \) \(922973188703411\) \(\nu^{7}\mathstrut +\mathstrut \) \(9899380051264956\) \(\nu^{6}\mathstrut +\mathstrut \) \(6716883469612343\) \(\nu^{5}\mathstrut -\mathstrut \) \(32386874868109258\) \(\nu^{4}\mathstrut -\mathstrut \) \(21507116682985566\) \(\nu^{3}\mathstrut +\mathstrut \) \(37358751314234452\) \(\nu^{2}\mathstrut +\mathstrut \) \(20860161005825920\) \(\nu\mathstrut -\mathstrut \) \(1768227501996184\)\()/\)\(689049844074136\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(10253913982431\) \(\nu^{12}\mathstrut +\mathstrut \) \(12143449436408\) \(\nu^{11}\mathstrut +\mathstrut \) \(459309915620523\) \(\nu^{10}\mathstrut -\mathstrut \) \(550253809975141\) \(\nu^{9}\mathstrut -\mathstrut \) \(7291775225900956\) \(\nu^{8}\mathstrut +\mathstrut \) \(8262582263045907\) \(\nu^{7}\mathstrut +\mathstrut \) \(49539905362356236\) \(\nu^{6}\mathstrut -\mathstrut \) \(46501901494687967\) \(\nu^{5}\mathstrut -\mathstrut \) \(141537061577422086\) \(\nu^{4}\mathstrut +\mathstrut \) \(86711893064448398\) \(\nu^{3}\mathstrut +\mathstrut \) \(122348683101464684\) \(\nu^{2}\mathstrut -\mathstrut \) \(15147039293522512\) \(\nu\mathstrut -\mathstrut \) \(2559670711426040\)\()/\)\(689049844074136\)
\(\beta_{4}\)\(=\)\((\)\(2643124952245\) \(\nu^{12}\mathstrut -\mathstrut \) \(3844423118131\) \(\nu^{11}\mathstrut -\mathstrut \) \(118025134245614\) \(\nu^{10}\mathstrut +\mathstrut \) \(172279045954540\) \(\nu^{9}\mathstrut +\mathstrut \) \(1859746846457838\) \(\nu^{8}\mathstrut -\mathstrut \) \(2577999496689658\) \(\nu^{7}\mathstrut -\mathstrut \) \(12435310500957227\) \(\nu^{6}\mathstrut +\mathstrut \) \(14674795233528832\) \(\nu^{5}\mathstrut +\mathstrut \) \(34437813698810893\) \(\nu^{4}\mathstrut -\mathstrut \) \(28641901734087195\) \(\nu^{3}\mathstrut -\mathstrut \) \(27309473007286086\) \(\nu^{2}\mathstrut +\mathstrut \) \(7953201992223538\) \(\nu\mathstrut +\mathstrut \) \(191018910747970\)\()/\)\(172262461018534\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(2806630645981\) \(\nu^{12}\mathstrut +\mathstrut \) \(3962699821337\) \(\nu^{11}\mathstrut +\mathstrut \) \(124624713157090\) \(\nu^{10}\mathstrut -\mathstrut \) \(179172235132576\) \(\nu^{9}\mathstrut -\mathstrut \) \(1946700518875936\) \(\nu^{8}\mathstrut +\mathstrut \) \(2711523524200234\) \(\nu^{7}\mathstrut +\mathstrut \) \(12817031913020937\) \(\nu^{6}\mathstrut -\mathstrut \) \(15728312750603394\) \(\nu^{5}\mathstrut -\mathstrut \) \(34425778503897719\) \(\nu^{4}\mathstrut +\mathstrut \) \(31982336352673793\) \(\nu^{3}\mathstrut +\mathstrut \) \(25008787374956530\) \(\nu^{2}\mathstrut -\mathstrut \) \(10497946895775678\) \(\nu\mathstrut +\mathstrut \) \(558668664048170\)\()/\)\(172262461018534\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(11419786805749\) \(\nu^{12}\mathstrut +\mathstrut \) \(16698434980110\) \(\nu^{11}\mathstrut +\mathstrut \) \(504336575792465\) \(\nu^{10}\mathstrut -\mathstrut \) \(755378470856885\) \(\nu^{9}\mathstrut -\mathstrut \) \(7810944784378346\) \(\nu^{8}\mathstrut +\mathstrut \) \(11452358170544105\) \(\nu^{7}\mathstrut +\mathstrut \) \(50713678365610526\) \(\nu^{6}\mathstrut -\mathstrut \) \(66799597776713869\) \(\nu^{5}\mathstrut -\mathstrut \) \(133463695266617588\) \(\nu^{4}\mathstrut +\mathstrut \) \(138168015452056174\) \(\nu^{3}\mathstrut +\mathstrut \) \(94078149732007496\) \(\nu^{2}\mathstrut -\mathstrut \) \(48634450370623920\) \(\nu\mathstrut +\mathstrut \) \(1161596938398896\)\()/\)\(689049844074136\)
\(\beta_{7}\)\(=\)\((\)\(5836486040157\) \(\nu^{12}\mathstrut -\mathstrut \) \(6831410905550\) \(\nu^{11}\mathstrut -\mathstrut \) \(260080147076221\) \(\nu^{10}\mathstrut +\mathstrut \) \(311196222902321\) \(\nu^{9}\mathstrut +\mathstrut \) \(4094961906303710\) \(\nu^{8}\mathstrut -\mathstrut \) \(4701200261840201\) \(\nu^{7}\mathstrut -\mathstrut \) \(27429895945466362\) \(\nu^{6}\mathstrut +\mathstrut \) \(26705214956593781\) \(\nu^{5}\mathstrut +\mathstrut \) \(76513736980135184\) \(\nu^{4}\mathstrut -\mathstrut \) \(50745708591403802\) \(\nu^{3}\mathstrut -\mathstrut \) \(63294691908648740\) \(\nu^{2}\mathstrut +\mathstrut \) \(9206122997578932\) \(\nu\mathstrut +\mathstrut \) \(990949223000480\)\()/\)\(344524922037068\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(3800074510933\) \(\nu^{12}\mathstrut +\mathstrut \) \(4801952864360\) \(\nu^{11}\mathstrut +\mathstrut \) \(168177451801958\) \(\nu^{10}\mathstrut -\mathstrut \) \(219681987395845\) \(\nu^{9}\mathstrut -\mathstrut \) \(2617125025528437\) \(\nu^{8}\mathstrut +\mathstrut \) \(3357195618452128\) \(\nu^{7}\mathstrut +\mathstrut \) \(17159458829299488\) \(\nu^{6}\mathstrut -\mathstrut \) \(19652933906266972\) \(\nu^{5}\mathstrut -\mathstrut \) \(45999555735978216\) \(\nu^{4}\mathstrut +\mathstrut \) \(40631636419194187\) \(\nu^{3}\mathstrut +\mathstrut \) \(34418056162936816\) \(\nu^{2}\mathstrut -\mathstrut \) \(14039929218321888\) \(\nu\mathstrut +\mathstrut \) \(228430125895582\)\()/\)\(172262461018534\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(15713129420919\) \(\nu^{12}\mathstrut +\mathstrut \) \(20121588852528\) \(\nu^{11}\mathstrut +\mathstrut \) \(698648560224535\) \(\nu^{10}\mathstrut -\mathstrut \) \(915337629159613\) \(\nu^{9}\mathstrut -\mathstrut \) \(10951008089524496\) \(\nu^{8}\mathstrut +\mathstrut \) \(13885279897829999\) \(\nu^{7}\mathstrut +\mathstrut \) \(72691678822176548\) \(\nu^{6}\mathstrut -\mathstrut \) \(80202504730163387\) \(\nu^{5}\mathstrut -\mathstrut \) \(199066645616036446\) \(\nu^{4}\mathstrut +\mathstrut \) \(160388343465178226\) \(\nu^{3}\mathstrut +\mathstrut \) \(155400440504302388\) \(\nu^{2}\mathstrut -\mathstrut \) \(46826258636980632\) \(\nu\mathstrut +\mathstrut \) \(927871747678256\)\()/\)\(689049844074136\)
\(\beta_{10}\)\(=\)\((\)\(48613352793023\) \(\nu^{12}\mathstrut -\mathstrut \) \(64814272554394\) \(\nu^{11}\mathstrut -\mathstrut \) \(2160793904265883\) \(\nu^{10}\mathstrut +\mathstrut \) \(2937658850521943\) \(\nu^{9}\mathstrut +\mathstrut \) \(33840530175307542\) \(\nu^{8}\mathstrut -\mathstrut \) \(44446314428220947\) \(\nu^{7}\mathstrut -\mathstrut \) \(224213390202227058\) \(\nu^{6}\mathstrut +\mathstrut \) \(256417255003978799\) \(\nu^{5}\mathstrut +\mathstrut \) \(611770516475666996\) \(\nu^{4}\mathstrut -\mathstrut \) \(513197597392377970\) \(\nu^{3}\mathstrut -\mathstrut \) \(471173372993526552\) \(\nu^{2}\mathstrut +\mathstrut \) \(152599535884229776\) \(\nu\mathstrut -\mathstrut \) \(7276871383436880\)\()/\)\(689049844074136\)
\(\beta_{11}\)\(=\)\((\)\(98658980574257\) \(\nu^{12}\mathstrut -\mathstrut \) \(121059374370454\) \(\nu^{11}\mathstrut -\mathstrut \) \(4393285532037869\) \(\nu^{10}\mathstrut +\mathstrut \) \(5495632108851737\) \(\nu^{9}\mathstrut +\mathstrut \) \(69064662601312658\) \(\nu^{8}\mathstrut -\mathstrut \) \(82857036982229621\) \(\nu^{7}\mathstrut -\mathstrut \) \(461116336742527902\) \(\nu^{6}\mathstrut +\mathstrut \) \(471078241980980913\) \(\nu^{5}\mathstrut +\mathstrut \) \(1277245628332544140\) \(\nu^{4}\mathstrut -\mathstrut \) \(904590081886303550\) \(\nu^{3}\mathstrut -\mathstrut \) \(1027333801147935896\) \(\nu^{2}\mathstrut +\mathstrut \) \(200744255133819480\) \(\nu\mathstrut +\mathstrut \) \(497584607825056\)\()/\)\(689049844074136\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(114490409117923\) \(\nu^{12}\mathstrut +\mathstrut \) \(149510059488566\) \(\nu^{11}\mathstrut +\mathstrut \) \(5090085024310823\) \(\nu^{10}\mathstrut -\mathstrut \) \(6771601944160655\) \(\nu^{9}\mathstrut -\mathstrut \) \(79766880804874594\) \(\nu^{8}\mathstrut +\mathstrut \) \(102183142140723143\) \(\nu^{7}\mathstrut +\mathstrut \) \(529363065829535614\) \(\nu^{6}\mathstrut -\mathstrut \) \(585411706046996443\) \(\nu^{5}\mathstrut -\mathstrut \) \(1450584604586481880\) \(\nu^{4}\mathstrut +\mathstrut \) \(1152338156086936082\) \(\nu^{3}\mathstrut +\mathstrut \) \(1135181228111546144\) \(\nu^{2}\mathstrut -\mathstrut \) \(313510477401767040\) \(\nu\mathstrut +\mathstrut \) \(7990135291149864\)\()/\)\(689049844074136\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut -\mathstrut \) \(4\)
\(\nu^{4}\)\(=\)\(3\) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(19\) \(\beta_{10}\mathstrut +\mathstrut \) \(20\) \(\beta_{9}\mathstrut -\mathstrut \) \(5\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{6}\mathstrut +\mathstrut \) \(24\) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(20\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(107\)
\(\nu^{5}\)\(=\)\(4\) \(\beta_{12}\mathstrut -\mathstrut \) \(20\) \(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(71\) \(\beta_{9}\mathstrut +\mathstrut \) \(61\) \(\beta_{8}\mathstrut +\mathstrut \) \(53\) \(\beta_{7}\mathstrut -\mathstrut \) \(73\) \(\beta_{6}\mathstrut -\mathstrut \) \(34\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(75\) \(\beta_{2}\mathstrut +\mathstrut \) \(108\) \(\beta_{1}\mathstrut -\mathstrut \) \(100\)
\(\nu^{6}\)\(=\)\(83\) \(\beta_{12}\mathstrut +\mathstrut \) \(57\) \(\beta_{11}\mathstrut +\mathstrut \) \(343\) \(\beta_{10}\mathstrut +\mathstrut \) \(370\) \(\beta_{9}\mathstrut -\mathstrut \) \(121\) \(\beta_{8}\mathstrut -\mathstrut \) \(20\) \(\beta_{7}\mathstrut +\mathstrut \) \(197\) \(\beta_{6}\mathstrut +\mathstrut \) \(449\) \(\beta_{5}\mathstrut +\mathstrut \) \(220\) \(\beta_{4}\mathstrut +\mathstrut \) \(354\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(26\) \(\beta_{1}\mathstrut +\mathstrut \) \(1611\)
\(\nu^{7}\)\(=\)\(120\) \(\beta_{12}\mathstrut -\mathstrut \) \(324\) \(\beta_{11}\mathstrut +\mathstrut \) \(63\) \(\beta_{10}\mathstrut -\mathstrut \) \(1398\) \(\beta_{9}\mathstrut +\mathstrut \) \(1079\) \(\beta_{8}\mathstrut +\mathstrut \) \(814\) \(\beta_{7}\mathstrut -\mathstrut \) \(1413\) \(\beta_{6}\mathstrut -\mathstrut \) \(497\) \(\beta_{5}\mathstrut +\mathstrut \) \(434\) \(\beta_{4}\mathstrut +\mathstrut \) \(123\) \(\beta_{3}\mathstrut -\mathstrut \) \(1409\) \(\beta_{2}\mathstrut +\mathstrut \) \(1527\) \(\beta_{1}\mathstrut -\mathstrut \) \(2028\)
\(\nu^{8}\)\(=\)\(1747\) \(\beta_{12}\mathstrut +\mathstrut \) \(1234\) \(\beta_{11}\mathstrut +\mathstrut \) \(6054\) \(\beta_{10}\mathstrut +\mathstrut \) \(6709\) \(\beta_{9}\mathstrut -\mathstrut \) \(2374\) \(\beta_{8}\mathstrut -\mathstrut \) \(717\) \(\beta_{7}\mathstrut +\mathstrut \) \(2841\) \(\beta_{6}\mathstrut +\mathstrut \) \(7974\) \(\beta_{5}\mathstrut +\mathstrut \) \(4195\) \(\beta_{4}\mathstrut +\mathstrut \) \(5929\) \(\beta_{3}\mathstrut -\mathstrut \) \(42\) \(\beta_{2}\mathstrut +\mathstrut \) \(615\) \(\beta_{1}\mathstrut +\mathstrut \) \(25577\)
\(\nu^{9}\)\(=\)\(2658\) \(\beta_{12}\mathstrut -\mathstrut \) \(5011\) \(\beta_{11}\mathstrut +\mathstrut \) \(949\) \(\beta_{10}\mathstrut -\mathstrut \) \(26224\) \(\beta_{9}\mathstrut +\mathstrut \) \(18554\) \(\beta_{8}\mathstrut +\mathstrut \) \(12170\) \(\beta_{7}\mathstrut -\mathstrut \) \(25538\) \(\beta_{6}\mathstrut -\mathstrut \) \(7021\) \(\beta_{5}\mathstrut +\mathstrut \) \(8879\) \(\beta_{4}\mathstrut +\mathstrut \) \(2710\) \(\beta_{3}\mathstrut -\mathstrut \) \(24450\) \(\beta_{2}\mathstrut +\mathstrut \) \(23401\) \(\beta_{1}\mathstrut -\mathstrut \) \(38176\)
\(\nu^{10}\)\(=\)\(33381\) \(\beta_{12}\mathstrut +\mathstrut \) \(24253\) \(\beta_{11}\mathstrut +\mathstrut \) \(105562\) \(\beta_{10}\mathstrut +\mathstrut \) \(121001\) \(\beta_{9}\mathstrut -\mathstrut \) \(43955\) \(\beta_{8}\mathstrut -\mathstrut \) \(17839\) \(\beta_{7}\mathstrut +\mathstrut \) \(42232\) \(\beta_{6}\mathstrut +\mathstrut \) \(138935\) \(\beta_{5}\mathstrut +\mathstrut \) \(73809\) \(\beta_{4}\mathstrut +\mathstrut \) \(97328\) \(\beta_{3}\mathstrut +\mathstrut \) \(891\) \(\beta_{2}\mathstrut +\mathstrut \) \(13849\) \(\beta_{1}\mathstrut +\mathstrut \) \(417089\)
\(\nu^{11}\)\(=\)\(52093\) \(\beta_{12}\mathstrut -\mathstrut \) \(77476\) \(\beta_{11}\mathstrut +\mathstrut \) \(10881\) \(\beta_{10}\mathstrut -\mathstrut \) \(483001\) \(\beta_{9}\mathstrut +\mathstrut \) \(317710\) \(\beta_{8}\mathstrut +\mathstrut \) \(182279\) \(\beta_{7}\mathstrut -\mathstrut \) \(449090\) \(\beta_{6}\mathstrut -\mathstrut \) \(99772\) \(\beta_{5}\mathstrut +\mathstrut \) \(165057\) \(\beta_{4}\mathstrut +\mathstrut \) \(53295\) \(\beta_{3}\mathstrut -\mathstrut \) \(414050\) \(\beta_{2}\mathstrut +\mathstrut \) \(371483\) \(\beta_{1}\mathstrut -\mathstrut \) \(695315\)
\(\nu^{12}\)\(=\)\(610129\) \(\beta_{12}\mathstrut +\mathstrut \) \(454952\) \(\beta_{11}\mathstrut +\mathstrut \) \(1829309\) \(\beta_{10}\mathstrut +\mathstrut \) \(2177976\) \(\beta_{9}\mathstrut -\mathstrut \) \(797491\) \(\beta_{8}\mathstrut -\mathstrut \) \(382102\) \(\beta_{7}\mathstrut +\mathstrut \) \(647576\) \(\beta_{6}\mathstrut +\mathstrut \) \(2398243\) \(\beta_{5}\mathstrut +\mathstrut \) \(1257836\) \(\beta_{4}\mathstrut +\mathstrut \) \(1588185\) \(\beta_{3}\mathstrut +\mathstrut \) \(51549\) \(\beta_{2}\mathstrut +\mathstrut \) \(294293\) \(\beta_{1}\mathstrut +\mathstrut \) \(6906799\)

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
−4.21367
−3.91861
−2.41505
−2.08866
−0.877543
0.0205303
0.197152
1.47504
2.66604
2.71857
2.75551
3.59366
4.08702
1.00000 1.00000 1.00000 −4.21367 1.00000 1.46512 1.00000 1.00000 −4.21367
1.2 1.00000 1.00000 1.00000 −3.91861 1.00000 −4.71846 1.00000 1.00000 −3.91861
1.3 1.00000 1.00000 1.00000 −2.41505 1.00000 1.11075 1.00000 1.00000 −2.41505
1.4 1.00000 1.00000 1.00000 −2.08866 1.00000 3.96496 1.00000 1.00000 −2.08866
1.5 1.00000 1.00000 1.00000 −0.877543 1.00000 −4.30430 1.00000 1.00000 −0.877543
1.6 1.00000 1.00000 1.00000 0.0205303 1.00000 5.09020 1.00000 1.00000 0.0205303
1.7 1.00000 1.00000 1.00000 0.197152 1.00000 1.50880 1.00000 1.00000 0.197152
1.8 1.00000 1.00000 1.00000 1.47504 1.00000 −0.770392 1.00000 1.00000 1.47504
1.9 1.00000 1.00000 1.00000 2.66604 1.00000 4.24276 1.00000 1.00000 2.66604
1.10 1.00000 1.00000 1.00000 2.71857 1.00000 1.56988 1.00000 1.00000 2.71857
1.11 1.00000 1.00000 1.00000 2.75551 1.00000 1.74830 1.00000 1.00000 2.75551
1.12 1.00000 1.00000 1.00000 3.59366 1.00000 2.37015 1.00000 1.00000 3.59366
1.13 1.00000 1.00000 1.00000 4.08702 1.00000 −2.27776 1.00000 1.00000 4.08702
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(17\) \(-1\)
\(59\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{13} - \cdots\)
\(T_{7}^{13} - \cdots\)