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.

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

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

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

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

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