Properties

Label 6040.2.a.l
Level 6040
Weight 2
Character orbit 6040.a
Self dual Yes
Analytic conductor 48.230
Analytic rank 1
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6040.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2296428209\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{7} q^{3} + q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} -\beta_{5} q^{9} +O(q^{10})\) \( q -\beta_{7} q^{3} + q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} -\beta_{5} q^{9} + ( -1 - \beta_{1} - \beta_{3} + \beta_{7} ) q^{11} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} ) q^{13} -\beta_{7} q^{15} + ( -1 - \beta_{3} - \beta_{6} ) q^{17} + ( -1 + \beta_{3} + \beta_{5} ) q^{19} + ( -1 - \beta_{2} - \beta_{8} ) q^{21} + ( -1 - \beta_{1} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{23} + q^{25} + ( 2 + \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{27} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{6} + \beta_{8} ) q^{29} + ( 1 + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} ) q^{31} + ( -1 + \beta_{1} + \beta_{3} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{33} + ( \beta_{1} + \beta_{4} ) q^{35} + ( -1 - \beta_{3} + \beta_{4} - 2 \beta_{5} + 4 \beta_{7} + \beta_{8} ) q^{37} + ( -1 - \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{39} + ( -1 + \beta_{1} - \beta_{2} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{41} + ( -1 + 2 \beta_{1} - 3 \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{8} ) q^{43} -\beta_{5} q^{45} + ( 4 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{47} + ( -4 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} ) q^{49} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{51} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} ) q^{53} + ( -1 - \beta_{1} - \beta_{3} + \beta_{7} ) q^{55} + ( -3 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{7} - \beta_{8} ) q^{57} + ( 2 - 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{8} ) q^{59} + ( -2 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{61} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{63} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} ) q^{65} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} - 3 \beta_{7} - 3 \beta_{8} ) q^{67} + ( -3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{6} + 4 \beta_{7} + 4 \beta_{8} ) q^{69} + ( -4 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{71} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} - 3 \beta_{7} - 3 \beta_{8} ) q^{73} -\beta_{7} q^{75} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{77} + ( 1 - \beta_{1} + 5 \beta_{2} + \beta_{3} - \beta_{4} + 4 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{79} + ( -4 - \beta_{2} + \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{81} + ( -1 + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{83} + ( -1 - \beta_{3} - \beta_{6} ) q^{85} + ( -1 - \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{87} + ( -1 - \beta_{1} - 4 \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{89} + ( -2 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{91} + ( -3 + \beta_{1} + 2 \beta_{2} + 3 \beta_{5} - 2 \beta_{6} - 5 \beta_{7} - 3 \beta_{8} ) q^{93} + ( -1 + \beta_{3} + \beta_{5} ) q^{95} + ( -4 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - 3 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} ) q^{97} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 9q^{5} - 2q^{7} - 3q^{9} + O(q^{10}) \) \( 9q + 9q^{5} - 2q^{7} - 3q^{9} - 6q^{11} - 9q^{13} - 2q^{17} - 10q^{19} - 9q^{21} - 6q^{23} + 9q^{25} + 12q^{27} - 6q^{29} + 9q^{31} - 11q^{33} - 2q^{35} - 12q^{37} - 3q^{39} - 20q^{41} + q^{43} - 3q^{45} + 22q^{47} - 29q^{49} + 2q^{51} - 35q^{53} - 6q^{55} - 20q^{57} + 14q^{59} - 22q^{61} - 12q^{63} - 9q^{65} + 4q^{67} + 5q^{69} - 22q^{71} - 34q^{73} - 5q^{77} + 8q^{79} - 31q^{81} - 3q^{83} - 2q^{85} - 5q^{89} - 7q^{91} - 21q^{93} - 10q^{95} - 33q^{97} - 15q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - x^{8} - 11 x^{7} + 9 x^{6} + 32 x^{5} - 17 x^{4} - 27 x^{3} + 10 x^{2} + 3 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{8} + 4 \nu^{7} - 41 \nu^{6} - 47 \nu^{5} + 164 \nu^{4} + 141 \nu^{3} - 181 \nu^{2} - 89 \nu + 31 \)\()/13\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{8} - 8 \nu^{7} - 74 \nu^{6} + 68 \nu^{5} + 192 \nu^{4} - 100 \nu^{3} - 106 \nu^{2} + 22 \nu - 23 \)\()/13\)
\(\beta_{4}\)\(=\)\( -\nu^{8} + \nu^{7} + 11 \nu^{6} - 9 \nu^{5} - 32 \nu^{4} + 17 \nu^{3} + 27 \nu^{2} - 10 \nu - 3 \)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{8} - 9 \nu^{7} - 184 \nu^{6} + 70 \nu^{5} + 580 \nu^{4} - 67 \nu^{3} - 532 \nu^{2} - 24 \nu + 70 \)\()/13\)
\(\beta_{6}\)\(=\)\((\)\( -25 \nu^{8} + 23 \nu^{7} + 268 \nu^{6} - 189 \nu^{5} - 734 \nu^{4} + 242 \nu^{3} + 555 \nu^{2} - 34 \nu - 59 \)\()/13\)
\(\beta_{7}\)\(=\)\((\)\( -27 \nu^{8} + 16 \nu^{7} + 304 \nu^{6} - 123 \nu^{5} - 917 \nu^{4} + 122 \nu^{3} + 771 \nu^{2} - 18 \nu - 71 \)\()/13\)
\(\beta_{8}\)\(=\)\((\)\( 57 \nu^{8} - 41 \nu^{7} - 636 \nu^{6} + 329 \nu^{5} + 1894 \nu^{4} - 389 \nu^{3} - 1606 \nu^{2} + 38 \nu + 160 \)\()/13\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} - \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(8 \beta_{8} + \beta_{7} + 7 \beta_{6} - 8 \beta_{5} + 7 \beta_{4} - 2 \beta_{3} - 7 \beta_{2} + 7 \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(-\beta_{8} - 2 \beta_{7} + 9 \beta_{5} + 8 \beta_{4} - \beta_{3} - 10 \beta_{2} + 30 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(56 \beta_{8} + 6 \beta_{7} + 47 \beta_{6} - 57 \beta_{5} + 50 \beta_{4} - 20 \beta_{3} - 51 \beta_{2} + 47 \beta_{1} + 100\)
\(\nu^{7}\)\(=\)\(-10 \beta_{8} - 24 \beta_{7} + 66 \beta_{5} + 61 \beta_{4} - 14 \beta_{3} - 81 \beta_{2} + 197 \beta_{1} - 12\)
\(\nu^{8}\)\(=\)\(386 \beta_{8} + 28 \beta_{7} + 320 \beta_{6} - 396 \beta_{5} + 358 \beta_{4} - 161 \beta_{3} - 372 \beta_{2} + 322 \beta_{1} + 663\)

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
0.390645
2.02556
−1.10984
−0.357360
−1.41589
2.68790
0.289144
1.10659
−2.61675
0 −1.96251 0 1.00000 0 −2.16922 0 0.851433 0
1.2 0 −1.75822 0 1.00000 0 1.53187 0 0.0913304 0
1.3 0 −1.57672 0 1.00000 0 −0.208807 0 −0.513952 0
1.4 0 −1.13091 0 1.00000 0 2.44094 0 −1.72105 0
1.5 0 0.0722232 0 1.00000 0 −0.709626 0 −2.99478 0
1.6 0 0.325171 0 1.00000 0 2.31586 0 −2.89426 0
1.7 0 1.17504 0 1.00000 0 −3.16934 0 −1.61928 0
1.8 0 2.35518 0 1.00000 0 0.202920 0 2.54689 0
1.9 0 2.50073 0 1.00000 0 −2.23459 0 3.25367 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(151\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\(T_{3}^{9} - \cdots\)
\(T_{7}^{9} + \cdots\)
\(T_{11}^{9} + \cdots\)