Properties

Label 6018.2.a.r
Level 6018
Weight 2
Character orbit 6018.a
Self dual Yes
Analytic conductor 48.054
Analytic rank 1
Dimension 6
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: \(6\)
Coefficient field: 6.6.18461324.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(3\) \(x^{5}\mathstrut -\mathstrut \) \(4\) \(x^{4}\mathstrut +\mathstrut \) \(12\) \(x^{3}\mathstrut +\mathstrut \) \(3\) \(x^{2}\mathstrut -\mathstrut \) \(6\) \(x\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 3 \nu^{4} - 3 \nu^{3} + 10 \nu^{2} - \nu - 2 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - 4 \nu^{4} - \nu^{3} + 15 \nu^{2} - 7 \nu - 5 \)
\(\beta_{5}\)\(=\)\( -2 \nu^{5} + 7 \nu^{4} + 5 \nu^{3} - 27 \nu^{2} + 5 \nu + 10 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(9\)
\(\nu^{5}\)\(=\)\(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(6\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\mathstrut +\mathstrut \) \(23\) \(\beta_{2}\mathstrut +\mathstrut \) \(33\) \(\beta_{1}\mathstrut +\mathstrut \) \(12\)

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
2.94685
1.91714
0.908132
−0.380739
−0.558656
−1.83273
1.00000 1.00000 1.00000 −2.94685 1.00000 2.59293 1.00000 1.00000 −2.94685
1.2 1.00000 1.00000 1.00000 −1.91714 1.00000 −1.73720 1.00000 1.00000 −1.91714
1.3 1.00000 1.00000 1.00000 −0.908132 1.00000 −1.54786 1.00000 1.00000 −0.908132
1.4 1.00000 1.00000 1.00000 0.380739 1.00000 1.68872 1.00000 1.00000 0.380739
1.5 1.00000 1.00000 1.00000 0.558656 1.00000 −3.85965 1.00000 1.00000 0.558656
1.6 1.00000 1.00000 1.00000 1.83273 1.00000 −4.13695 1.00000 1.00000 1.83273
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{6} \) \(\mathstrut +\mathstrut 3 T_{5}^{5} \) \(\mathstrut -\mathstrut 4 T_{5}^{4} \) \(\mathstrut -\mathstrut 12 T_{5}^{3} \) \(\mathstrut +\mathstrut 3 T_{5}^{2} \) \(\mathstrut +\mathstrut 6 T_{5} \) \(\mathstrut -\mathstrut 2 \)
\(T_{7}^{6} \) \(\mathstrut +\mathstrut 7 T_{7}^{5} \) \(\mathstrut +\mathstrut T_{7}^{4} \) \(\mathstrut -\mathstrut 69 T_{7}^{3} \) \(\mathstrut -\mathstrut 77 T_{7}^{2} \) \(\mathstrut +\mathstrut 140 T_{7} \) \(\mathstrut +\mathstrut 188 \)