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: \( 1 \)
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.

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

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

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

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.

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

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