Properties

Label 6018.2.a.y
Level 6018
Weight 2
Character orbit 6018.a
Self dual Yes
Analytic conductor 48.054
Analytic rank 1
Dimension 10
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: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(2\) \(x^{9}\mathstrut -\mathstrut \) \(33\) \(x^{8}\mathstrut +\mathstrut \) \(53\) \(x^{7}\mathstrut +\mathstrut \) \(356\) \(x^{6}\mathstrut -\mathstrut \) \(433\) \(x^{5}\mathstrut -\mathstrut \) \(1296\) \(x^{4}\mathstrut +\mathstrut \) \(1135\) \(x^{3}\mathstrut +\mathstrut \) \(930\) \(x^{2}\mathstrut -\mathstrut \) \(186\) \(x\mathstrut -\mathstrut \) \(104\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(43951\) \(\nu^{9}\mathstrut -\mathstrut \) \(76913\) \(\nu^{8}\mathstrut +\mathstrut \) \(988626\) \(\nu^{7}\mathstrut +\mathstrut \) \(8141512\) \(\nu^{6}\mathstrut -\mathstrut \) \(20812677\) \(\nu^{5}\mathstrut -\mathstrut \) \(127966190\) \(\nu^{4}\mathstrut +\mathstrut \) \(261039766\) \(\nu^{3}\mathstrut +\mathstrut \) \(516802760\) \(\nu^{2}\mathstrut -\mathstrut \) \(917083841\) \(\nu\mathstrut -\mathstrut \) \(232805875\)\()/51474769\)
\(\beta_{3}\)\(=\)\((\)\(214570\) \(\nu^{9}\mathstrut -\mathstrut \) \(1275880\) \(\nu^{8}\mathstrut -\mathstrut \) \(6472015\) \(\nu^{7}\mathstrut +\mathstrut \) \(39620633\) \(\nu^{6}\mathstrut +\mathstrut \) \(61025321\) \(\nu^{5}\mathstrut -\mathstrut \) \(392668405\) \(\nu^{4}\mathstrut -\mathstrut \) \(154873115\) \(\nu^{3}\mathstrut +\mathstrut \) \(1263523452\) \(\nu^{2}\mathstrut -\mathstrut \) \(135541142\) \(\nu\mathstrut -\mathstrut \) \(459680139\)\()/51474769\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(249465\) \(\nu^{9}\mathstrut +\mathstrut \) \(1418601\) \(\nu^{8}\mathstrut +\mathstrut \) \(4174377\) \(\nu^{7}\mathstrut -\mathstrut \) \(28056148\) \(\nu^{6}\mathstrut -\mathstrut \) \(13181246\) \(\nu^{5}\mathstrut +\mathstrut \) \(145009646\) \(\nu^{4}\mathstrut -\mathstrut \) \(29021559\) \(\nu^{3}\mathstrut -\mathstrut \) \(194173638\) \(\nu^{2}\mathstrut +\mathstrut \) \(117499426\) \(\nu\mathstrut -\mathstrut \) \(47082704\)\()/51474769\)
\(\beta_{5}\)\(=\)\((\)\(414280\) \(\nu^{9}\mathstrut -\mathstrut \) \(956844\) \(\nu^{8}\mathstrut -\mathstrut \) \(14645292\) \(\nu^{7}\mathstrut +\mathstrut \) \(33222269\) \(\nu^{6}\mathstrut +\mathstrut \) \(160178219\) \(\nu^{5}\mathstrut -\mathstrut \) \(349088916\) \(\nu^{4}\mathstrut -\mathstrut \) \(537665586\) \(\nu^{3}\mathstrut +\mathstrut \) \(1070383049\) \(\nu^{2}\mathstrut +\mathstrut \) \(174956104\) \(\nu\mathstrut -\mathstrut \) \(154245468\)\()/51474769\)
\(\beta_{6}\)\(=\)\((\)\(417990\) \(\nu^{9}\mathstrut -\mathstrut \) \(662240\) \(\nu^{8}\mathstrut -\mathstrut \) \(15536862\) \(\nu^{7}\mathstrut +\mathstrut \) \(20221183\) \(\nu^{6}\mathstrut +\mathstrut \) \(181938909\) \(\nu^{5}\mathstrut -\mathstrut \) \(182686833\) \(\nu^{4}\mathstrut -\mathstrut \) \(684929521\) \(\nu^{3}\mathstrut +\mathstrut \) \(504776496\) \(\nu^{2}\mathstrut +\mathstrut \) \(448652800\) \(\nu\mathstrut -\mathstrut \) \(87407927\)\()/51474769\)
\(\beta_{7}\)\(=\)\((\)\(743180\) \(\nu^{9}\mathstrut -\mathstrut \) \(1756250\) \(\nu^{8}\mathstrut -\mathstrut \) \(21953329\) \(\nu^{7}\mathstrut +\mathstrut \) \(43199010\) \(\nu^{6}\mathstrut +\mathstrut \) \(209998564\) \(\nu^{5}\mathstrut -\mathstrut \) \(329491644\) \(\nu^{4}\mathstrut -\mathstrut \) \(657244249\) \(\nu^{3}\mathstrut +\mathstrut \) \(833201672\) \(\nu^{2}\mathstrut +\mathstrut \) \(274790504\) \(\nu\mathstrut -\mathstrut \) \(191137786\)\()/51474769\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(1112920\) \(\nu^{9}\mathstrut +\mathstrut \) \(3475010\) \(\nu^{8}\mathstrut +\mathstrut \) \(31167323\) \(\nu^{7}\mathstrut -\mathstrut \) \(82936201\) \(\nu^{6}\mathstrut -\mathstrut \) \(285693609\) \(\nu^{5}\mathstrut +\mathstrut \) \(595528719\) \(\nu^{4}\mathstrut +\mathstrut \) \(862414939\) \(\nu^{3}\mathstrut -\mathstrut \) \(1317064691\) \(\nu^{2}\mathstrut -\mathstrut \) \(250454498\) \(\nu\mathstrut +\mathstrut \) \(114968772\)\()/51474769\)
\(\beta_{9}\)\(=\)\((\)\(1849040\) \(\nu^{9}\mathstrut -\mathstrut \) \(4265675\) \(\nu^{8}\mathstrut -\mathstrut \) \(59471294\) \(\nu^{7}\mathstrut +\mathstrut \) \(114940606\) \(\nu^{6}\mathstrut +\mathstrut \) \(618418870\) \(\nu^{5}\mathstrut -\mathstrut \) \(964601918\) \(\nu^{4}\mathstrut -\mathstrut \) \(2064959828\) \(\nu^{3}\mathstrut +\mathstrut \) \(2587254209\) \(\nu^{2}\mathstrut +\mathstrut \) \(760020540\) \(\nu\mathstrut -\mathstrut \) \(338810025\)\()/51474769\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(-\)\(21\) \(\beta_{8}\mathstrut -\mathstrut \) \(5\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(31\) \(\beta_{5}\mathstrut +\mathstrut \) \(36\) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(90\)
\(\nu^{5}\)\(=\)\(38\) \(\beta_{9}\mathstrut -\mathstrut \) \(46\) \(\beta_{8}\mathstrut -\mathstrut \) \(59\) \(\beta_{7}\mathstrut -\mathstrut \) \(60\) \(\beta_{6}\mathstrut -\mathstrut \) \(86\) \(\beta_{5}\mathstrut +\mathstrut \) \(91\) \(\beta_{4}\mathstrut +\mathstrut \) \(26\) \(\beta_{3}\mathstrut -\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(117\) \(\beta_{1}\mathstrut +\mathstrut \) \(67\)
\(\nu^{6}\)\(=\)\(14\) \(\beta_{9}\mathstrut -\mathstrut \) \(349\) \(\beta_{8}\mathstrut -\mathstrut \) \(117\) \(\beta_{7}\mathstrut -\mathstrut \) \(70\) \(\beta_{6}\mathstrut -\mathstrut \) \(475\) \(\beta_{5}\mathstrut +\mathstrut \) \(585\) \(\beta_{4}\mathstrut +\mathstrut \) \(207\) \(\beta_{3}\mathstrut -\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(65\) \(\beta_{1}\mathstrut +\mathstrut \) \(1256\)
\(\nu^{7}\)\(=\)\(596\) \(\beta_{9}\mathstrut -\mathstrut \) \(866\) \(\beta_{8}\mathstrut -\mathstrut \) \(952\) \(\beta_{7}\mathstrut -\mathstrut \) \(1005\) \(\beta_{6}\mathstrut -\mathstrut \) \(1522\) \(\beta_{5}\mathstrut +\mathstrut \) \(1682\) \(\beta_{4}\mathstrut +\mathstrut \) \(499\) \(\beta_{3}\mathstrut -\mathstrut \) \(110\) \(\beta_{2}\mathstrut +\mathstrut \) \(1516\) \(\beta_{1}\mathstrut +\mathstrut \) \(1552\)
\(\nu^{8}\)\(=\)\(501\) \(\beta_{9}\mathstrut -\mathstrut \) \(5491\) \(\beta_{8}\mathstrut -\mathstrut \) \(2178\) \(\beta_{7}\mathstrut -\mathstrut \) \(1690\) \(\beta_{6}\mathstrut -\mathstrut \) \(7398\) \(\beta_{5}\mathstrut +\mathstrut \) \(9322\) \(\beta_{4}\mathstrut +\mathstrut \) \(3127\) \(\beta_{3}\mathstrut -\mathstrut \) \(160\) \(\beta_{2}\mathstrut +\mathstrut \) \(1492\) \(\beta_{1}\mathstrut +\mathstrut \) \(18164\)
\(\nu^{9}\)\(=\)\(9007\) \(\beta_{9}\mathstrut -\mathstrut \) \(15231\) \(\beta_{8}\mathstrut -\mathstrut \) \(14597\) \(\beta_{7}\mathstrut -\mathstrut \) \(16198\) \(\beta_{6}\mathstrut -\mathstrut \) \(25570\) \(\beta_{5}\mathstrut +\mathstrut \) \(29253\) \(\beta_{4}\mathstrut +\mathstrut \) \(8721\) \(\beta_{3}\mathstrut -\mathstrut \) \(1924\) \(\beta_{2}\mathstrut +\mathstrut \) \(20830\) \(\beta_{1}\mathstrut +\mathstrut \) \(30910\)

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
4.05993
3.94189
2.38622
1.40255
0.390474
−0.313984
−0.556815
−2.34533
−3.46791
−3.49703
−1.00000 1.00000 1.00000 −4.05993 −1.00000 4.32251 −1.00000 1.00000 4.05993
1.2 −1.00000 1.00000 1.00000 −3.94189 −1.00000 −4.15692 −1.00000 1.00000 3.94189
1.3 −1.00000 1.00000 1.00000 −2.38622 −1.00000 0.0691424 −1.00000 1.00000 2.38622
1.4 −1.00000 1.00000 1.00000 −1.40255 −1.00000 0.699671 −1.00000 1.00000 1.40255
1.5 −1.00000 1.00000 1.00000 −0.390474 −1.00000 −3.35881 −1.00000 1.00000 0.390474
1.6 −1.00000 1.00000 1.00000 0.313984 −1.00000 2.10092 −1.00000 1.00000 −0.313984
1.7 −1.00000 1.00000 1.00000 0.556815 −1.00000 0.727556 −1.00000 1.00000 −0.556815
1.8 −1.00000 1.00000 1.00000 2.34533 −1.00000 −0.244087 −1.00000 1.00000 −2.34533
1.9 −1.00000 1.00000 1.00000 3.46791 −1.00000 −1.61634 −1.00000 1.00000 −3.46791
1.10 −1.00000 1.00000 1.00000 3.49703 −1.00000 −4.54363 −1.00000 1.00000 −3.49703
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{10} + \cdots\)
\(T_{7}^{10} + \cdots\)