Properties

Label 6018.2.a.v
Level 6018
Weight 2
Character orbit 6018.a
Self dual Yes
Analytic conductor 48.054
Analytic rank 1
Dimension 9
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(3\) \(x^{8}\mathstrut -\mathstrut \) \(21\) \(x^{7}\mathstrut +\mathstrut \) \(42\) \(x^{6}\mathstrut +\mathstrut \) \(121\) \(x^{5}\mathstrut -\mathstrut \) \(127\) \(x^{4}\mathstrut -\mathstrut \) \(141\) \(x^{3}\mathstrut +\mathstrut \) \(27\) \(x^{2}\mathstrut +\mathstrut \) \(26\) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 5907 \nu^{8} - 25576 \nu^{7} - 96263 \nu^{6} + 398353 \nu^{5} + 298486 \nu^{4} - 1475547 \nu^{3} + 637856 \nu^{2} + 499745 \nu - 289639 \)\()/61504\)
\(\beta_{3}\)\(=\)\((\)\( 9203 \nu^{8} - 31080 \nu^{7} - 182951 \nu^{6} + 461489 \nu^{5} + 961014 \nu^{4} - 1597499 \nu^{3} - 788640 \nu^{2} + 560961 \nu + 75129 \)\()/61504\)
\(\beta_{4}\)\(=\)\((\)\( -6401 \nu^{8} + 21400 \nu^{7} + 124669 \nu^{6} - 306155 \nu^{5} - 617266 \nu^{4} + 963353 \nu^{3} + 294624 \nu^{2} - 161915 \nu + 20861 \)\()/30752\)
\(\beta_{5}\)\(=\)\((\)\( 12987 \nu^{8} - 49640 \nu^{7} - 236943 \nu^{6} + 752521 \nu^{5} + 1066438 \nu^{4} - 2690819 \nu^{3} - 286688 \nu^{2} + 1014297 \nu + 60241 \)\()/61504\)
\(\beta_{6}\)\(=\)\((\)\( -4883 \nu^{8} + 14312 \nu^{7} + 103367 \nu^{6} - 196913 \nu^{5} - 602150 \nu^{4} + 561227 \nu^{3} + 720192 \nu^{2} - 19745 \nu - 125641 \)\()/15376\)
\(\beta_{7}\)\(=\)\((\)\( -17371 \nu^{8} + 67112 \nu^{7} + 312239 \nu^{6} - 1016233 \nu^{5} - 1332326 \nu^{4} + 3591075 \nu^{3} - 85312 \nu^{2} - 1078105 \nu + 120975 \)\()/30752\)
\(\beta_{8}\)\(=\)\((\)\( 42985 \nu^{8} - 154744 \nu^{7} - 810245 \nu^{6} + 2291651 \nu^{5} + 3836354 \nu^{4} - 7760065 \nu^{3} - 1443360 \nu^{2} + 2036531 \nu - 159013 \)\()/61504\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(4\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(5\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut -\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{4}\)\(=\)\(38\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(21\) \(\beta_{6}\mathstrut -\mathstrut \) \(23\) \(\beta_{5}\mathstrut +\mathstrut \) \(19\) \(\beta_{4}\mathstrut -\mathstrut \) \(33\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(69\)
\(\nu^{5}\)\(=\)\(119\) \(\beta_{8}\mathstrut +\mathstrut \) \(16\) \(\beta_{7}\mathstrut +\mathstrut \) \(70\) \(\beta_{6}\mathstrut -\mathstrut \) \(118\) \(\beta_{5}\mathstrut +\mathstrut \) \(69\) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut -\mathstrut \) \(147\) \(\beta_{2}\mathstrut +\mathstrut \) \(123\) \(\beta_{1}\mathstrut +\mathstrut \) \(181\)
\(\nu^{6}\)\(=\)\(762\) \(\beta_{8}\mathstrut +\mathstrut \) \(225\) \(\beta_{7}\mathstrut +\mathstrut \) \(452\) \(\beta_{6}\mathstrut -\mathstrut \) \(528\) \(\beta_{5}\mathstrut +\mathstrut \) \(364\) \(\beta_{4}\mathstrut +\mathstrut \) \(19\) \(\beta_{3}\mathstrut -\mathstrut \) \(807\) \(\beta_{2}\mathstrut +\mathstrut \) \(245\) \(\beta_{1}\mathstrut +\mathstrut \) \(1225\)
\(\nu^{7}\)\(=\)\(2963\) \(\beta_{8}\mathstrut +\mathstrut \) \(548\) \(\beta_{7}\mathstrut +\mathstrut \) \(1826\) \(\beta_{6}\mathstrut -\mathstrut \) \(2630\) \(\beta_{5}\mathstrut +\mathstrut \) \(1510\) \(\beta_{4}\mathstrut +\mathstrut \) \(260\) \(\beta_{3}\mathstrut -\mathstrut \) \(3651\) \(\beta_{2}\mathstrut +\mathstrut \) \(2063\) \(\beta_{1}\mathstrut +\mathstrut \) \(4466\)
\(\nu^{8}\)\(=\)\(16085\) \(\beta_{8}\mathstrut +\mathstrut \) \(4145\) \(\beta_{7}\mathstrut +\mathstrut \) \(9882\) \(\beta_{6}\mathstrut -\mathstrut \) \(12013\) \(\beta_{5}\mathstrut +\mathstrut \) \(7498\) \(\beta_{4}\mathstrut +\mathstrut \) \(761\) \(\beta_{3}\mathstrut -\mathstrut \) \(18509\) \(\beta_{2}\mathstrut +\mathstrut \) \(6440\) \(\beta_{1}\mathstrut +\mathstrut \) \(24615\)

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
4.72920
3.03878
1.53312
0.460972
0.0373069
−0.542287
−0.625332
−2.43329
−3.19847
−1.00000 −1.00000 1.00000 −3.72920 1.00000 −3.41012 −1.00000 1.00000 3.72920
1.2 −1.00000 −1.00000 1.00000 −2.03878 1.00000 −3.87855 −1.00000 1.00000 2.03878
1.3 −1.00000 −1.00000 1.00000 −0.533116 1.00000 0.275545 −1.00000 1.00000 0.533116
1.4 −1.00000 −1.00000 1.00000 0.539028 1.00000 0.211506 −1.00000 1.00000 −0.539028
1.5 −1.00000 −1.00000 1.00000 0.962693 1.00000 0.389187 −1.00000 1.00000 −0.962693
1.6 −1.00000 −1.00000 1.00000 1.54229 1.00000 2.97024 −1.00000 1.00000 −1.54229
1.7 −1.00000 −1.00000 1.00000 1.62533 1.00000 −4.68222 −1.00000 1.00000 −1.62533
1.8 −1.00000 −1.00000 1.00000 3.43329 1.00000 −0.177683 −1.00000 1.00000 −3.43329
1.9 −1.00000 −1.00000 1.00000 4.19847 1.00000 −2.69790 −1.00000 1.00000 −4.19847
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{9} - \cdots\)
\(T_{7}^{9} + \cdots\)