Properties

Label 6018.2.a.ba
Level 6018
Weight 2
Character orbit 6018.a
Self dual Yes
Analytic conductor 48.054
Analytic rank 0
Dimension 12
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(4\) \(x^{11}\mathstrut -\mathstrut \) \(31\) \(x^{10}\mathstrut +\mathstrut \) \(111\) \(x^{9}\mathstrut +\mathstrut \) \(381\) \(x^{8}\mathstrut -\mathstrut \) \(1101\) \(x^{7}\mathstrut -\mathstrut \) \(2301\) \(x^{6}\mathstrut +\mathstrut \) \(4690\) \(x^{5}\mathstrut +\mathstrut \) \(6915\) \(x^{4}\mathstrut -\mathstrut \) \(8325\) \(x^{3}\mathstrut -\mathstrut \) \(10082\) \(x^{2}\mathstrut +\mathstrut \) \(5149\) \(x\mathstrut +\mathstrut \) \(5653\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(37967\) \(\nu^{11}\mathstrut +\mathstrut \) \(239615\) \(\nu^{10}\mathstrut +\mathstrut \) \(833951\) \(\nu^{9}\mathstrut -\mathstrut \) \(6728760\) \(\nu^{8}\mathstrut -\mathstrut \) \(5503599\) \(\nu^{7}\mathstrut +\mathstrut \) \(69490919\) \(\nu^{6}\mathstrut +\mathstrut \) \(4825579\) \(\nu^{5}\mathstrut -\mathstrut \) \(319841161\) \(\nu^{4}\mathstrut +\mathstrut \) \(53318216\) \(\nu^{3}\mathstrut +\mathstrut \) \(631646387\) \(\nu^{2}\mathstrut -\mathstrut \) \(84801718\) \(\nu\mathstrut -\mathstrut \) \(435582988\)\()/1323382\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(118047\) \(\nu^{11}\mathstrut +\mathstrut \) \(575314\) \(\nu^{10}\mathstrut +\mathstrut \) \(3132406\) \(\nu^{9}\mathstrut -\mathstrut \) \(15664289\) \(\nu^{8}\mathstrut -\mathstrut \) \(30426407\) \(\nu^{7}\mathstrut +\mathstrut \) \(151503348\) \(\nu^{6}\mathstrut +\mathstrut \) \(126683562\) \(\nu^{5}\mathstrut -\mathstrut \) \(617251320\) \(\nu^{4}\mathstrut -\mathstrut \) \(192373155\) \(\nu^{3}\mathstrut +\mathstrut \) \(993409765\) \(\nu^{2}\mathstrut +\mathstrut \) \(94398553\) \(\nu\mathstrut -\mathstrut \) \(541927834\)\()/2646764\)
\(\beta_{4}\)\(=\)\((\)\(127547\) \(\nu^{11}\mathstrut -\mathstrut \) \(600100\) \(\nu^{10}\mathstrut -\mathstrut \) \(3402596\) \(\nu^{9}\mathstrut +\mathstrut \) \(16221775\) \(\nu^{8}\mathstrut +\mathstrut \) \(33453103\) \(\nu^{7}\mathstrut -\mathstrut \) \(156149770\) \(\nu^{6}\mathstrut -\mathstrut \) \(143333204\) \(\nu^{5}\mathstrut +\mathstrut \) \(634988602\) \(\nu^{4}\mathstrut +\mathstrut \) \(237183129\) \(\nu^{3}\mathstrut -\mathstrut \) \(1020839309\) \(\nu^{2}\mathstrut -\mathstrut \) \(139596031\) \(\nu\mathstrut +\mathstrut \) \(546225598\)\()/2646764\)
\(\beta_{5}\)\(=\)\((\)\(154847\) \(\nu^{11}\mathstrut -\mathstrut \) \(789735\) \(\nu^{10}\mathstrut -\mathstrut \) \(3900430\) \(\nu^{9}\mathstrut +\mathstrut \) \(21598935\) \(\nu^{8}\mathstrut +\mathstrut \) \(34189684\) \(\nu^{7}\mathstrut -\mathstrut \) \(211704075\) \(\nu^{6}\mathstrut -\mathstrut \) \(113127340\) \(\nu^{5}\mathstrut +\mathstrut \) \(886236402\) \(\nu^{4}\mathstrut +\mathstrut \) \(66506323\) \(\nu^{3}\mathstrut -\mathstrut \) \(1492728506\) \(\nu^{2}\mathstrut +\mathstrut \) \(64214944\) \(\nu\mathstrut +\mathstrut \) \(870706107\)\()/2646764\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(107852\) \(\nu^{11}\mathstrut +\mathstrut \) \(581451\) \(\nu^{10}\mathstrut +\mathstrut \) \(2636977\) \(\nu^{9}\mathstrut -\mathstrut \) \(15934575\) \(\nu^{8}\mathstrut -\mathstrut \) \(21914010\) \(\nu^{7}\mathstrut +\mathstrut \) \(156316299\) \(\nu^{6}\mathstrut +\mathstrut \) \(63583367\) \(\nu^{5}\mathstrut -\mathstrut \) \(652711197\) \(\nu^{4}\mathstrut -\mathstrut \) \(1932955\) \(\nu^{3}\mathstrut +\mathstrut \) \(1085960100\) \(\nu^{2}\mathstrut -\mathstrut \) \(73257917\) \(\nu\mathstrut -\mathstrut \) \(622275386\)\()/1323382\)
\(\beta_{7}\)\(=\)\((\)\(237037\) \(\nu^{11}\mathstrut -\mathstrut \) \(1126760\) \(\nu^{10}\mathstrut -\mathstrut \) \(6439990\) \(\nu^{9}\mathstrut +\mathstrut \) \(30758583\) \(\nu^{8}\mathstrut +\mathstrut \) \(65032295\) \(\nu^{7}\mathstrut -\mathstrut \) \(298396854\) \(\nu^{6}\mathstrut -\mathstrut \) \(289297782\) \(\nu^{5}\mathstrut +\mathstrut \) \(1220063292\) \(\nu^{4}\mathstrut +\mathstrut \) \(503450773\) \(\nu^{3}\mathstrut -\mathstrut \) \(1967030013\) \(\nu^{2}\mathstrut -\mathstrut \) \(295295825\) \(\nu\mathstrut +\mathstrut \) \(1047731024\)\()/2646764\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(128019\) \(\nu^{11}\mathstrut +\mathstrut \) \(656217\) \(\nu^{10}\mathstrut +\mathstrut \) \(3243284\) \(\nu^{9}\mathstrut -\mathstrut \) \(17843661\) \(\nu^{8}\mathstrut -\mathstrut \) \(29046033\) \(\nu^{7}\mathstrut +\mathstrut \) \(172914262\) \(\nu^{6}\mathstrut +\mathstrut \) \(102692323\) \(\nu^{5}\mathstrut -\mathstrut \) \(707942363\) \(\nu^{4}\mathstrut -\mathstrut \) \(93300582\) \(\nu^{3}\mathstrut +\mathstrut \) \(1141470325\) \(\nu^{2}\mathstrut -\mathstrut \) \(3011304\) \(\nu\mathstrut -\mathstrut \) \(625188159\)\()/661691\)
\(\beta_{9}\)\(=\)\((\)\(310088\) \(\nu^{11}\mathstrut -\mathstrut \) \(1606687\) \(\nu^{10}\mathstrut -\mathstrut \) \(7780026\) \(\nu^{9}\mathstrut +\mathstrut \) \(43783498\) \(\nu^{8}\mathstrut +\mathstrut \) \(68395961\) \(\nu^{7}\mathstrut -\mathstrut \) \(426411439\) \(\nu^{6}\mathstrut -\mathstrut \) \(231916382\) \(\nu^{5}\mathstrut +\mathstrut \) \(1762942612\) \(\nu^{4}\mathstrut +\mathstrut \) \(171457996\) \(\nu^{3}\mathstrut -\mathstrut \) \(2893219847\) \(\nu^{2}\mathstrut +\mathstrut \) \(65188905\) \(\nu\mathstrut +\mathstrut \) \(1621505661\)\()/1323382\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(324090\) \(\nu^{11}\mathstrut +\mathstrut \) \(1636811\) \(\nu^{10}\mathstrut +\mathstrut \) \(8235372\) \(\nu^{9}\mathstrut -\mathstrut \) \(44361950\) \(\nu^{8}\mathstrut -\mathstrut \) \(74242225\) \(\nu^{7}\mathstrut +\mathstrut \) \(428531541\) \(\nu^{6}\mathstrut +\mathstrut \) \(267410740\) \(\nu^{5}\mathstrut -\mathstrut \) \(1749467342\) \(\nu^{4}\mathstrut -\mathstrut \) \(266922102\) \(\nu^{3}\mathstrut +\mathstrut \) \(2814323071\) \(\nu^{2}\mathstrut +\mathstrut \) \(22537731\) \(\nu\mathstrut -\mathstrut \) \(1533113265\)\()/1323382\)
\(\beta_{11}\)\(=\)\((\)\(1137721\) \(\nu^{11}\mathstrut -\mathstrut \) \(5777705\) \(\nu^{10}\mathstrut -\mathstrut \) \(28952712\) \(\nu^{9}\mathstrut +\mathstrut \) \(157286279\) \(\nu^{8}\mathstrut +\mathstrut \) \(261108042\) \(\nu^{7}\mathstrut -\mathstrut \) \(1528293261\) \(\nu^{6}\mathstrut -\mathstrut \) \(937729774\) \(\nu^{5}\mathstrut +\mathstrut \) \(6293405452\) \(\nu^{4}\mathstrut +\mathstrut \) \(915294055\) \(\nu^{3}\mathstrut -\mathstrut \) \(10269561860\) \(\nu^{2}\mathstrut -\mathstrut \) \(32595192\) \(\nu\mathstrut +\mathstrut \) \(5703167183\)\()/2646764\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{3}\)\(=\)\(-\)\(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{4}\)\(=\)\(-\)\(17\) \(\beta_{11}\mathstrut -\mathstrut \) \(15\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut -\mathstrut \) \(16\) \(\beta_{6}\mathstrut +\mathstrut \) \(17\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(22\) \(\beta_{1}\mathstrut +\mathstrut \) \(59\)
\(\nu^{5}\)\(=\)\(-\)\(48\) \(\beta_{11}\mathstrut -\mathstrut \) \(27\) \(\beta_{10}\mathstrut +\mathstrut \) \(7\) \(\beta_{9}\mathstrut -\mathstrut \) \(16\) \(\beta_{8}\mathstrut +\mathstrut \) \(26\) \(\beta_{7}\mathstrut -\mathstrut \) \(41\) \(\beta_{6}\mathstrut +\mathstrut \) \(48\) \(\beta_{5}\mathstrut +\mathstrut \) \(34\) \(\beta_{4}\mathstrut +\mathstrut \) \(18\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(133\) \(\beta_{1}\mathstrut +\mathstrut \) \(92\)
\(\nu^{6}\)\(=\)\(-\)\(268\) \(\beta_{11}\mathstrut -\mathstrut \) \(212\) \(\beta_{10}\mathstrut +\mathstrut \) \(35\) \(\beta_{9}\mathstrut -\mathstrut \) \(23\) \(\beta_{8}\mathstrut +\mathstrut \) \(191\) \(\beta_{7}\mathstrut -\mathstrut \) \(240\) \(\beta_{6}\mathstrut +\mathstrut \) \(264\) \(\beta_{5}\mathstrut +\mathstrut \) \(80\) \(\beta_{4}\mathstrut +\mathstrut \) \(117\) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(403\) \(\beta_{1}\mathstrut +\mathstrut \) \(719\)
\(\nu^{7}\)\(=\)\(-\)\(902\) \(\beta_{11}\mathstrut -\mathstrut \) \(536\) \(\beta_{10}\mathstrut +\mathstrut \) \(191\) \(\beta_{9}\mathstrut -\mathstrut \) \(218\) \(\beta_{8}\mathstrut +\mathstrut \) \(501\) \(\beta_{7}\mathstrut -\mathstrut \) \(734\) \(\beta_{6}\mathstrut +\mathstrut \) \(886\) \(\beta_{5}\mathstrut +\mathstrut \) \(539\) \(\beta_{4}\mathstrut +\mathstrut \) \(272\) \(\beta_{3}\mathstrut +\mathstrut \) \(29\) \(\beta_{2}\mathstrut +\mathstrut \) \(1973\) \(\beta_{1}\mathstrut +\mathstrut \) \(1728\)
\(\nu^{8}\)\(=\)\(-\)\(4271\) \(\beta_{11}\mathstrut -\mathstrut \) \(3111\) \(\beta_{10}\mathstrut +\mathstrut \) \(801\) \(\beta_{9}\mathstrut -\mathstrut \) \(422\) \(\beta_{8}\mathstrut +\mathstrut \) \(2767\) \(\beta_{7}\mathstrut -\mathstrut \) \(3690\) \(\beta_{6}\mathstrut +\mathstrut \) \(4153\) \(\beta_{5}\mathstrut +\mathstrut \) \(1618\) \(\beta_{4}\mathstrut +\mathstrut \) \(1373\) \(\beta_{3}\mathstrut +\mathstrut \) \(123\) \(\beta_{2}\mathstrut +\mathstrut \) \(6942\) \(\beta_{1}\mathstrut +\mathstrut \) \(9879\)
\(\nu^{9}\)\(=\)\(-\)\(15745\) \(\beta_{11}\mathstrut -\mathstrut \) \(9639\) \(\beta_{10}\mathstrut +\mathstrut \) \(3856\) \(\beta_{9}\mathstrut -\mathstrut \) \(2964\) \(\beta_{8}\mathstrut +\mathstrut \) \(8818\) \(\beta_{7}\mathstrut -\mathstrut \) \(12646\) \(\beta_{6}\mathstrut +\mathstrut \) \(15255\) \(\beta_{5}\mathstrut +\mathstrut \) \(8628\) \(\beta_{4}\mathstrut +\mathstrut \) \(4044\) \(\beta_{3}\mathstrut +\mathstrut \) \(617\) \(\beta_{2}\mathstrut +\mathstrut \) \(30847\) \(\beta_{1}\mathstrut +\mathstrut \) \(30171\)
\(\nu^{10}\)\(=\)\(-\)\(69069\) \(\beta_{11}\mathstrut -\mathstrut \) \(47589\) \(\beta_{10}\mathstrut +\mathstrut \) \(15600\) \(\beta_{9}\mathstrut -\mathstrut \) \(7283\) \(\beta_{8}\mathstrut +\mathstrut \) \(42177\) \(\beta_{7}\mathstrut -\mathstrut \) \(58223\) \(\beta_{6}\mathstrut +\mathstrut \) \(66633\) \(\beta_{5}\mathstrut +\mathstrut \) \(29649\) \(\beta_{4}\mathstrut +\mathstrut \) \(17988\) \(\beta_{3}\mathstrut +\mathstrut \) \(2657\) \(\beta_{2}\mathstrut +\mathstrut \) \(116710\) \(\beta_{1}\mathstrut +\mathstrut \) \(146554\)
\(\nu^{11}\)\(=\)\(-\)\(266914\) \(\beta_{11}\mathstrut -\mathstrut \) \(166147\) \(\beta_{10}\mathstrut +\mathstrut \) \(70032\) \(\beta_{9}\mathstrut -\mathstrut \) \(41789\) \(\beta_{8}\mathstrut +\mathstrut \) \(149858\) \(\beta_{7}\mathstrut -\mathstrut \) \(214001\) \(\beta_{6}\mathstrut +\mathstrut \) \(256688\) \(\beta_{5}\mathstrut +\mathstrut \) \(140031\) \(\beta_{4}\mathstrut +\mathstrut \) \(61399\) \(\beta_{3}\mathstrut +\mathstrut \) \(11732\) \(\beta_{2}\mathstrut +\mathstrut \) \(495453\) \(\beta_{1}\mathstrut +\mathstrut \) \(510941\)

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
4.07242
4.06325
3.06946
2.01103
1.37577
1.26686
−1.07790
−1.11808
−1.17036
−2.56401
−2.88446
−3.04399
1.00000 1.00000 1.00000 −3.07242 1.00000 −1.48607 1.00000 1.00000 −3.07242
1.2 1.00000 1.00000 1.00000 −3.06325 1.00000 1.90874 1.00000 1.00000 −3.06325
1.3 1.00000 1.00000 1.00000 −2.06946 1.00000 −4.76338 1.00000 1.00000 −2.06946
1.4 1.00000 1.00000 1.00000 −1.01103 1.00000 −2.46588 1.00000 1.00000 −1.01103
1.5 1.00000 1.00000 1.00000 −0.375767 1.00000 4.48414 1.00000 1.00000 −0.375767
1.6 1.00000 1.00000 1.00000 −0.266858 1.00000 0.661205 1.00000 1.00000 −0.266858
1.7 1.00000 1.00000 1.00000 2.07790 1.00000 3.88310 1.00000 1.00000 2.07790
1.8 1.00000 1.00000 1.00000 2.11808 1.00000 1.85275 1.00000 1.00000 2.11808
1.9 1.00000 1.00000 1.00000 2.17036 1.00000 1.24708 1.00000 1.00000 2.17036
1.10 1.00000 1.00000 1.00000 3.56401 1.00000 −2.46585 1.00000 1.00000 3.56401
1.11 1.00000 1.00000 1.00000 3.88446 1.00000 −1.31412 1.00000 1.00000 3.88446
1.12 1.00000 1.00000 1.00000 4.04399 1.00000 3.45827 1.00000 1.00000 4.04399
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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

Hecke kernels

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

\(T_{5}^{12} - \cdots\)
\(T_{7}^{12} - \cdots\)