Properties

Label 6020.2.a.g
Level 6020
Weight 2
Character orbit 6020.a
Self dual Yes
Analytic conductor 48.070
Analytic rank 1
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6020.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.0699420168\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \(+ q^{5}\) \(- q^{7}\) \( + ( 1 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \(+ q^{5}\) \(- q^{7}\) \( + ( 1 + \beta_{2} ) q^{9} \) \( + ( -\beta_{3} - \beta_{4} ) q^{11} \) \( + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{13} \) \( + \beta_{1} q^{15} \) \( + ( -2 - \beta_{2} - \beta_{3} ) q^{17} \) \( + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{19} \) \( -\beta_{1} q^{21} \) \( + ( -1 + \beta_{2} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{23} \) \(+ q^{25}\) \( + ( \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{27} \) \( + ( -1 - \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{29} \) \( + ( -2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{31} \) \( + ( -2 - \beta_{2} + \beta_{5} - 2 \beta_{6} ) q^{33} \) \(- q^{35}\) \( + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{37} \) \( + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} ) q^{39} \) \( + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{41} \) \(- q^{43}\) \( + ( 1 + \beta_{2} ) q^{45} \) \( + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{47} \) \(+ q^{49}\) \( + ( -1 - 4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{51} \) \( + ( -3 + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{53} \) \( + ( -\beta_{3} - \beta_{4} ) q^{55} \) \( + ( -2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{57} \) \( + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{59} \) \( + ( -2 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{61} \) \( + ( -1 - \beta_{2} ) q^{63} \) \( + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{65} \) \( + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{67} \) \( + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{8} ) q^{69} \) \( + ( 1 - 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} ) q^{71} \) \( + ( -4 - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} ) q^{73} \) \( + \beta_{1} q^{75} \) \( + ( \beta_{3} + \beta_{4} ) q^{77} \) \( + ( 1 - 3 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{79} \) \( + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{81} \) \( + ( -2 - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{7} + 2 \beta_{8} ) q^{83} \) \( + ( -2 - \beta_{2} - \beta_{3} ) q^{85} \) \( + ( 1 - \beta_{1} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{87} \) \( + ( \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} - 3 \beta_{8} ) q^{89} \) \( + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{91} \) \( + ( -5 - \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{8} ) q^{93} \) \( + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{95} \) \( + ( -3 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{7} - 2 \beta_{8} ) q^{97} \) \( + ( -1 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(9q \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 5q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 5q^{9} \) \(\mathstrut +\mathstrut q^{11} \) \(\mathstrut -\mathstrut 14q^{13} \) \(\mathstrut -\mathstrut 11q^{17} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 6q^{23} \) \(\mathstrut +\mathstrut 9q^{25} \) \(\mathstrut -\mathstrut 6q^{29} \) \(\mathstrut +\mathstrut 6q^{31} \) \(\mathstrut -\mathstrut 14q^{33} \) \(\mathstrut -\mathstrut 9q^{35} \) \(\mathstrut -\mathstrut 20q^{37} \) \(\mathstrut -\mathstrut 6q^{39} \) \(\mathstrut -\mathstrut 6q^{41} \) \(\mathstrut -\mathstrut 9q^{43} \) \(\mathstrut +\mathstrut 5q^{45} \) \(\mathstrut +\mathstrut 9q^{49} \) \(\mathstrut -\mathstrut 8q^{51} \) \(\mathstrut -\mathstrut 31q^{53} \) \(\mathstrut +\mathstrut q^{55} \) \(\mathstrut -\mathstrut 16q^{57} \) \(\mathstrut +\mathstrut 2q^{59} \) \(\mathstrut -\mathstrut 13q^{61} \) \(\mathstrut -\mathstrut 5q^{63} \) \(\mathstrut -\mathstrut 14q^{65} \) \(\mathstrut -\mathstrut 10q^{67} \) \(\mathstrut -\mathstrut 18q^{69} \) \(\mathstrut +\mathstrut 12q^{71} \) \(\mathstrut -\mathstrut 32q^{73} \) \(\mathstrut -\mathstrut q^{77} \) \(\mathstrut +\mathstrut q^{79} \) \(\mathstrut -\mathstrut 27q^{81} \) \(\mathstrut -\mathstrut 10q^{83} \) \(\mathstrut -\mathstrut 11q^{85} \) \(\mathstrut -\mathstrut 5q^{87} \) \(\mathstrut -\mathstrut q^{89} \) \(\mathstrut +\mathstrut 14q^{91} \) \(\mathstrut -\mathstrut 49q^{93} \) \(\mathstrut -\mathstrut 2q^{95} \) \(\mathstrut -\mathstrut 28q^{97} \) \(\mathstrut -\mathstrut 14q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(16\) \(x^{7}\mathstrut +\mathstrut \) \(83\) \(x^{5}\mathstrut -\mathstrut \) \(9\) \(x^{4}\mathstrut -\mathstrut \) \(160\) \(x^{3}\mathstrut +\mathstrut \) \(32\) \(x^{2}\mathstrut +\mathstrut \) \(77\) \(x\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{8} - 12 \nu^{7} - 7 \nu^{6} + 136 \nu^{5} - 86 \nu^{4} - 442 \nu^{3} + 358 \nu^{2} + 360 \nu - 173 \)\()/47\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{8} - 18 \nu^{7} - 34 \nu^{6} + 204 \nu^{5} + 153 \nu^{4} - 616 \nu^{3} - 309 \nu^{2} + 446 \nu + 234 \)\()/47\)
\(\beta_{5}\)\(=\)\((\)\( 6 \nu^{8} + 11 \nu^{7} - 68 \nu^{6} - 156 \nu^{5} + 165 \nu^{4} + 507 \nu^{3} - 7 \nu^{2} - 330 \nu - 190 \)\()/47\)
\(\beta_{6}\)\(=\)\((\)\( -12 \nu^{8} + 25 \nu^{7} + 136 \nu^{6} - 252 \nu^{5} - 424 \nu^{4} + 678 \nu^{3} + 296 \nu^{2} - 327 \nu - 43 \)\()/47\)
\(\beta_{7}\)\(=\)\((\)\( 12 \nu^{8} + 22 \nu^{7} - 183 \nu^{6} - 265 \nu^{5} + 800 \nu^{4} + 732 \nu^{3} - 1142 \nu^{2} - 331 \nu + 184 \)\()/47\)
\(\beta_{8}\)\(=\)\((\)\( 20 \nu^{8} + 21 \nu^{7} - 258 \nu^{6} - 285 \nu^{5} + 879 \nu^{4} + 844 \nu^{3} - 838 \nu^{2} - 583 \nu + 9 \)\()/47\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(23\)
\(\nu^{5}\)\(=\)\(10\) \(\beta_{8}\mathstrut -\mathstrut \) \(9\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(39\) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{6}\)\(=\)\(14\) \(\beta_{8}\mathstrut -\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(22\) \(\beta_{4}\mathstrut -\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(62\) \(\beta_{2}\mathstrut +\mathstrut \) \(20\) \(\beta_{1}\mathstrut +\mathstrut \) \(153\)
\(\nu^{7}\)\(=\)\(86\) \(\beta_{8}\mathstrut -\mathstrut \) \(74\) \(\beta_{7}\mathstrut +\mathstrut \) \(27\) \(\beta_{6}\mathstrut -\mathstrut \) \(82\) \(\beta_{5}\mathstrut +\mathstrut \) \(28\) \(\beta_{4}\mathstrut -\mathstrut \) \(50\) \(\beta_{3}\mathstrut +\mathstrut \) \(118\) \(\beta_{2}\mathstrut +\mathstrut \) \(275\) \(\beta_{1}\mathstrut +\mathstrut \) \(115\)
\(\nu^{8}\)\(=\)\(149\) \(\beta_{8}\mathstrut -\mathstrut \) \(145\) \(\beta_{7}\mathstrut +\mathstrut \) \(111\) \(\beta_{6}\mathstrut -\mathstrut \) \(40\) \(\beta_{5}\mathstrut +\mathstrut \) \(195\) \(\beta_{4}\mathstrut -\mathstrut \) \(103\) \(\beta_{3}\mathstrut +\mathstrut \) \(495\) \(\beta_{2}\mathstrut +\mathstrut \) \(257\) \(\beta_{1}\mathstrut +\mathstrut \) \(1109\)

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
−2.49127
−2.42779
−1.89794
−0.701670
0.0257343
1.10254
1.66233
1.79476
2.93329
0 −2.49127 0 1.00000 0 −1.00000 0 3.20641 0
1.2 0 −2.42779 0 1.00000 0 −1.00000 0 2.89414 0
1.3 0 −1.89794 0 1.00000 0 −1.00000 0 0.602159 0
1.4 0 −0.701670 0 1.00000 0 −1.00000 0 −2.50766 0
1.5 0 0.0257343 0 1.00000 0 −1.00000 0 −2.99934 0
1.6 0 1.10254 0 1.00000 0 −1.00000 0 −1.78440 0
1.7 0 1.66233 0 1.00000 0 −1.00000 0 −0.236670 0
1.8 0 1.79476 0 1.00000 0 −1.00000 0 0.221173 0
1.9 0 2.93329 0 1.00000 0 −1.00000 0 5.60419 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
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)
\(43\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6020))\):

\(T_{3}^{9} \) \(\mathstrut -\mathstrut 16 T_{3}^{7} \) \(\mathstrut +\mathstrut 83 T_{3}^{5} \) \(\mathstrut -\mathstrut 9 T_{3}^{4} \) \(\mathstrut -\mathstrut 160 T_{3}^{3} \) \(\mathstrut +\mathstrut 32 T_{3}^{2} \) \(\mathstrut +\mathstrut 77 T_{3} \) \(\mathstrut -\mathstrut 2 \)
\(T_{11}^{9} - \cdots\)