Properties

Label 6018.2.a.n
Level 6018
Weight 2
Character orbit 6018.a
Self dual Yes
Analytic conductor 48.054
Analytic rank 1
Dimension 4
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: \(4\)
Coefficient field: 4.4.2525.1
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 3\)

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
2.46673
−0.777484
1.77748
−1.46673
1.00000 −1.00000 1.00000 −3.37322 −1.00000 3.14256 1.00000 1.00000 −3.37322
1.2 1.00000 −1.00000 1.00000 −2.09855 −1.00000 0.639962 1.00000 1.00000 −2.09855
1.3 1.00000 −1.00000 1.00000 −0.519488 −1.00000 −3.49406 1.00000 1.00000 −0.519488
1.4 1.00000 −1.00000 1.00000 2.99126 −1.00000 0.711544 1.00000 1.00000 2.99126
\(n\): e.g. 2-40 or 990-1000
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\( T_{5}^{4} + 3 T_{5}^{3} - 8 T_{5}^{2} - 26 T_{5} - 11 \)
\( T_{7}^{4} - T_{7}^{3} - 11 T_{7}^{2} + 15 T_{7} - 5 \)