Properties

Label 6018.2.a.m
Level 6018
Weight 2
Character orbit 6018.a
Self dual yes
Analytic conductor 48.054
Analytic rank 0
Dimension 3
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: \(3\)
Coefficient field: 3.3.148.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

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

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
−1.48119
2.17009
0.311108
−1.00000 1.00000 1.00000 −2.15633 −1.00000 −2.28726 −1.00000 1.00000 2.15633
1.2 −1.00000 1.00000 1.00000 2.63090 −1.00000 3.87936 −1.00000 1.00000 −2.63090
1.3 −1.00000 1.00000 1.00000 3.52543 −1.00000 −2.59210 −1.00000 1.00000 −3.52543
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6018.2.a.m 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.m 3 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(17\) \(1\)
\(59\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\( T_{5}^{3} - 4 T_{5}^{2} - 4 T_{5} + 20 \)
\( T_{7}^{3} + T_{7}^{2} - 13 T_{7} - 23 \)