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.

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

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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

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