Properties

Label 6018.2.a.s
Level 6018
Weight 2
Character orbit 6018.a
Self dual Yes
Analytic conductor 48.054
Analytic rank 0
Dimension 8
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(3\) \(x^{7}\mathstrut -\mathstrut \) \(17\) \(x^{6}\mathstrut +\mathstrut \) \(37\) \(x^{5}\mathstrut +\mathstrut \) \(105\) \(x^{4}\mathstrut -\mathstrut \) \(117\) \(x^{3}\mathstrut -\mathstrut \) \(238\) \(x^{2}\mathstrut +\mathstrut \) \(42\) \(x\mathstrut +\mathstrut \) \(90\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 23 \nu^{7} - 126 \nu^{6} - 82 \nu^{5} + 1079 \nu^{4} - 276 \nu^{3} - 2202 \nu^{2} + 184 \nu + 810 \)\()/30\)
\(\beta_{3}\)\(=\)\((\)\( 34 \nu^{7} - 183 \nu^{6} - 146 \nu^{5} + 1612 \nu^{4} - 213 \nu^{3} - 3456 \nu^{2} - 58 \nu + 1200 \)\()/30\)
\(\beta_{4}\)\(=\)\((\)\( 34 \nu^{7} - 183 \nu^{6} - 146 \nu^{5} + 1612 \nu^{4} - 213 \nu^{3} - 3486 \nu^{2} - 28 \nu + 1350 \)\()/30\)
\(\beta_{5}\)\(=\)\((\)\( 43 \nu^{7} - 231 \nu^{6} - 182 \nu^{5} + 2029 \nu^{4} - 321 \nu^{3} - 4362 \nu^{2} + 164 \nu + 1710 \)\()/30\)
\(\beta_{6}\)\(=\)\((\)\( -77 \nu^{7} + 414 \nu^{6} + 328 \nu^{5} - 3641 \nu^{4} + 549 \nu^{3} + 7803 \nu^{2} - 196 \nu - 2925 \)\()/15\)
\(\beta_{7}\)\(=\)\((\)\( 169 \nu^{7} - 903 \nu^{6} - 746 \nu^{5} + 7957 \nu^{4} - 993 \nu^{3} - 17106 \nu^{2} + 92 \nu + 6420 \)\()/30\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut -\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(16\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(45\)
\(\nu^{5}\)\(=\)\(-\)\(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(17\) \(\beta_{6}\mathstrut +\mathstrut \) \(49\) \(\beta_{5}\mathstrut -\mathstrut \) \(31\) \(\beta_{4}\mathstrut +\mathstrut \) \(58\) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(62\) \(\beta_{1}\mathstrut +\mathstrut \) \(106\)
\(\nu^{6}\)\(=\)\(-\)\(10\) \(\beta_{7}\mathstrut +\mathstrut \) \(59\) \(\beta_{6}\mathstrut +\mathstrut \) \(206\) \(\beta_{5}\mathstrut -\mathstrut \) \(175\) \(\beta_{4}\mathstrut +\mathstrut \) \(249\) \(\beta_{3}\mathstrut -\mathstrut \) \(26\) \(\beta_{2}\mathstrut +\mathstrut \) \(153\) \(\beta_{1}\mathstrut +\mathstrut \) \(520\)
\(\nu^{7}\)\(=\)\(-\)\(15\) \(\beta_{7}\mathstrut +\mathstrut \) \(302\) \(\beta_{6}\mathstrut +\mathstrut \) \(905\) \(\beta_{5}\mathstrut -\mathstrut \) \(614\) \(\beta_{4}\mathstrut +\mathstrut \) \(952\) \(\beta_{3}\mathstrut -\mathstrut \) \(58\) \(\beta_{2}\mathstrut +\mathstrut \) \(668\) \(\beta_{1}\mathstrut +\mathstrut \) \(1631\)

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
−2.36472
2.14759
−2.36258
3.92135
0.672052
3.02133
−1.32287
−0.712152
−1.00000 −1.00000 1.00000 −3.85158 1.00000 4.10228 −1.00000 1.00000 3.85158
1.2 −1.00000 −1.00000 1.00000 −2.25098 1.00000 0.744783 −1.00000 1.00000 2.25098
1.3 −1.00000 −1.00000 1.00000 −1.89711 1.00000 1.64158 −1.00000 1.00000 1.89711
1.4 −1.00000 −1.00000 1.00000 −1.03120 1.00000 −2.85327 −1.00000 1.00000 1.03120
1.5 −1.00000 −1.00000 1.00000 −0.462452 1.00000 −3.09978 −1.00000 1.00000 0.462452
1.6 −1.00000 −1.00000 1.00000 2.51738 1.00000 4.42797 −1.00000 1.00000 −2.51738
1.7 −1.00000 −1.00000 1.00000 2.86769 1.00000 0.636539 −1.00000 1.00000 −2.86769
1.8 −1.00000 −1.00000 1.00000 3.10826 1.00000 0.399898 −1.00000 1.00000 −3.10826
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{8} + \cdots\)
\(T_{7}^{8} - \cdots\)