Properties

Label 6018.2.a.p
Level 6018
Weight 2
Character orbit 6018.a
Self dual Yes
Analytic conductor 48.054
Analytic rank 0
Dimension 5
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: \(5\)
Coefficient field: 5.5.1668357.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 11 x^{3} + x^{2} + 17 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 3 \nu^{2} - 6 \nu + 10 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 9 \nu^{2} - 2 \nu + 5 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{4} + 2 \nu^{3} + 9 \nu^{2} - 7 \nu - 10 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(3 \beta_{4} + 3 \beta_{3} + 9 \beta_{1} + 5\)
\(\nu^{4}\)\(=\)\(12 \beta_{4} + 15 \beta_{3} - 9 \beta_{2} + 20 \beta_{1} + 45\)

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
0.117900
−2.44651
1.24777
3.61712
−1.53628
−1.00000 1.00000 1.00000 −3.56570 −1.00000 −0.117900 −1.00000 1.00000 3.56570
1.2 −1.00000 1.00000 1.00000 −1.37256 −1.00000 2.44651 −1.00000 1.00000 1.37256
1.3 −1.00000 1.00000 1.00000 −1.08688 −1.00000 −1.24777 −1.00000 1.00000 1.08688
1.4 −1.00000 1.00000 1.00000 1.96737 −1.00000 −3.61712 −1.00000 1.00000 −1.96737
1.5 −1.00000 1.00000 1.00000 3.05778 −1.00000 1.53628 −1.00000 1.00000 −3.05778
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\( T_{5}^{5} + T_{5}^{4} - 14 T_{5}^{3} - 10 T_{5}^{2} + 35 T_{5} + 32 \)
\( T_{7}^{5} + T_{7}^{4} - 11 T_{7}^{3} - T_{7}^{2} + 17 T_{7} + 2 \)