Properties

Label 6020.2.a.f
Level 6020
Weight 2
Character orbit 6020.a
Self dual Yes
Analytic conductor 48.070
Analytic rank 1
Dimension 8
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \beta_{1} ) q^{3} \) \(- q^{5}\) \(+ q^{7}\) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \beta_{1} ) q^{3} \) \(- q^{5}\) \(+ q^{7}\) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{9} \) \( + ( 1 + \beta_{6} ) q^{11} \) \( + ( -1 + \beta_{4} - \beta_{5} ) q^{13} \) \( + ( 1 - \beta_{1} ) q^{15} \) \( + ( -\beta_{1} + \beta_{5} ) q^{17} \) \( + ( -2 - \beta_{2} ) q^{19} \) \( + ( -1 + \beta_{1} ) q^{21} \) \( + ( \beta_{3} - \beta_{4} - \beta_{7} ) q^{23} \) \(+ q^{25}\) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{27} \) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{29} \) \( + ( -3 + \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{31} \) \( + ( -1 + \beta_{1} - \beta_{6} - \beta_{7} ) q^{33} \) \(- q^{35}\) \( + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{37} \) \( + ( 2 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{39} \) \( + ( -3 - \beta_{2} - 2 \beta_{6} ) q^{41} \) \(- q^{43}\) \( + ( -2 + \beta_{1} - \beta_{2} ) q^{45} \) \( + ( 1 - \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{47} \) \(+ q^{49}\) \( + ( -2 + \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{51} \) \( + ( 1 - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{53} \) \( + ( -1 - \beta_{6} ) q^{55} \) \( + ( 1 - 4 \beta_{1} + \beta_{3} + \beta_{6} - \beta_{7} ) q^{57} \) \( + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{59} \) \( + ( -3 - \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{61} \) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{63} \) \( + ( 1 - \beta_{4} + \beta_{5} ) q^{65} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{67} \) \( + ( -4 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{69} \) \( + ( 3 + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{71} \) \( + ( -2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{73} \) \( + ( -1 + \beta_{1} ) q^{75} \) \( + ( 1 + \beta_{6} ) q^{77} \) \( + ( 5 - 2 \beta_{1} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} ) q^{79} \) \( + ( -2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{81} \) \( + ( -3 \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{83} \) \( + ( \beta_{1} - \beta_{5} ) q^{85} \) \( + ( 3 - 4 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{6} - 2 \beta_{7} ) q^{87} \) \( + ( -1 + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{89} \) \( + ( -1 + \beta_{4} - \beta_{5} ) q^{91} \) \( + ( -1 - \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{93} \) \( + ( 2 + \beta_{2} ) q^{95} \) \( + ( 2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{97} \) \( + ( -\beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut 8q^{5} \) \(\mathstrut +\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 11q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut 8q^{5} \) \(\mathstrut +\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 11q^{9} \) \(\mathstrut +\mathstrut 6q^{11} \) \(\mathstrut -\mathstrut 13q^{13} \) \(\mathstrut +\mathstrut 5q^{15} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut -\mathstrut 14q^{19} \) \(\mathstrut -\mathstrut 5q^{21} \) \(\mathstrut +\mathstrut 6q^{23} \) \(\mathstrut +\mathstrut 8q^{25} \) \(\mathstrut -\mathstrut 11q^{27} \) \(\mathstrut -\mathstrut 7q^{29} \) \(\mathstrut -\mathstrut 18q^{31} \) \(\mathstrut -\mathstrut q^{33} \) \(\mathstrut -\mathstrut 8q^{35} \) \(\mathstrut +\mathstrut 2q^{37} \) \(\mathstrut +\mathstrut 9q^{39} \) \(\mathstrut -\mathstrut 18q^{41} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut -\mathstrut 11q^{45} \) \(\mathstrut -\mathstrut q^{47} \) \(\mathstrut +\mathstrut 8q^{49} \) \(\mathstrut -\mathstrut 19q^{51} \) \(\mathstrut +\mathstrut 5q^{53} \) \(\mathstrut -\mathstrut 6q^{55} \) \(\mathstrut -\mathstrut 4q^{57} \) \(\mathstrut -\mathstrut 12q^{59} \) \(\mathstrut -\mathstrut 23q^{61} \) \(\mathstrut +\mathstrut 11q^{63} \) \(\mathstrut +\mathstrut 13q^{65} \) \(\mathstrut +\mathstrut 8q^{67} \) \(\mathstrut -\mathstrut 18q^{69} \) \(\mathstrut +\mathstrut 20q^{71} \) \(\mathstrut -\mathstrut 4q^{73} \) \(\mathstrut -\mathstrut 5q^{75} \) \(\mathstrut +\mathstrut 6q^{77} \) \(\mathstrut +\mathstrut 24q^{79} \) \(\mathstrut -\mathstrut 8q^{81} \) \(\mathstrut -\mathstrut 14q^{83} \) \(\mathstrut +\mathstrut 2q^{85} \) \(\mathstrut +\mathstrut 10q^{87} \) \(\mathstrut -\mathstrut 21q^{89} \) \(\mathstrut -\mathstrut 13q^{91} \) \(\mathstrut +\mathstrut q^{93} \) \(\mathstrut +\mathstrut 14q^{95} \) \(\mathstrut +\mathstrut 7q^{97} \) \(\mathstrut +\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(3\) \(x^{7}\mathstrut -\mathstrut \) \(12\) \(x^{6}\mathstrut +\mathstrut \) \(26\) \(x^{5}\mathstrut +\mathstrut \) \(55\) \(x^{4}\mathstrut -\mathstrut \) \(52\) \(x^{3}\mathstrut -\mathstrut \) \(82\) \(x^{2}\mathstrut +\mathstrut \) \(22\) \(x\mathstrut +\mathstrut \) \(27\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\( -7 \nu^{7} + 23 \nu^{6} + 65 \nu^{5} - 126 \nu^{4} - 349 \nu^{3} + 66 \nu^{2} + 580 \nu - 9 \)\()/87\)
\(\beta_{4}\)\(=\)\((\)\( 14 \nu^{7} - 46 \nu^{6} - 130 \nu^{5} + 339 \nu^{4} + 437 \nu^{3} - 480 \nu^{2} - 290 \nu + 18 \)\()/87\)
\(\beta_{5}\)\(=\)\((\)\( 4 \nu^{7} - 9 \nu^{6} - 62 \nu^{5} + 101 \nu^{4} + 274 \nu^{3} - 220 \nu^{2} - 261 \nu + 59 \)\()/29\)
\(\beta_{6}\)\(=\)\((\)\( -25 \nu^{7} + 107 \nu^{6} + 170 \nu^{5} - 885 \nu^{4} - 364 \nu^{3} + 1926 \nu^{2} + 232 \nu - 927 \)\()/87\)
\(\beta_{7}\)\(=\)\((\)\( -32 \nu^{7} + 130 \nu^{6} + 235 \nu^{5} - 1011 \nu^{4} - 626 \nu^{3} + 1818 \nu^{2} + 377 \nu - 675 \)\()/87\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{4}\)\(=\)\(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\) \(\beta_{1}\mathstrut +\mathstrut \) \(31\)
\(\nu^{5}\)\(=\)\(16\) \(\beta_{7}\mathstrut -\mathstrut \) \(15\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut -\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(28\) \(\beta_{2}\mathstrut +\mathstrut \) \(64\) \(\beta_{1}\mathstrut +\mathstrut \) \(78\)
\(\nu^{6}\)\(=\)\(57\) \(\beta_{7}\mathstrut -\mathstrut \) \(51\) \(\beta_{6}\mathstrut -\mathstrut \) \(5\) \(\beta_{5}\mathstrut +\mathstrut \) \(35\) \(\beta_{4}\mathstrut -\mathstrut \) \(17\) \(\beta_{3}\mathstrut +\mathstrut \) \(106\) \(\beta_{2}\mathstrut +\mathstrut \) \(180\) \(\beta_{1}\mathstrut +\mathstrut \) \(324\)
\(\nu^{7}\)\(=\)\(232\) \(\beta_{7}\mathstrut -\mathstrut \) \(203\) \(\beta_{6}\mathstrut -\mathstrut \) \(35\) \(\beta_{5}\mathstrut +\mathstrut \) \(162\) \(\beta_{4}\mathstrut -\mathstrut \) \(84\) \(\beta_{3}\mathstrut +\mathstrut \) \(338\) \(\beta_{2}\mathstrut +\mathstrut \) \(659\) \(\beta_{1}\mathstrut +\mathstrut \) \(1018\)

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
−2.11466
−1.73402
−1.16692
−0.604219
0.704685
1.41294
2.96620
3.53600
0 −3.11466 0 −1.00000 0 1.00000 0 6.70113 0
1.2 0 −2.73402 0 −1.00000 0 1.00000 0 4.47486 0
1.3 0 −2.16692 0 −1.00000 0 1.00000 0 1.69555 0
1.4 0 −1.60422 0 −1.00000 0 1.00000 0 −0.426481 0
1.5 0 −0.295315 0 −1.00000 0 1.00000 0 −2.91279 0
1.6 0 0.412945 0 −1.00000 0 1.00000 0 −2.82948 0
1.7 0 1.96620 0 −1.00000 0 1.00000 0 0.865935 0
1.8 0 2.53600 0 −1.00000 0 1.00000 0 3.43127 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6020))\):

\(T_{3}^{8} + \cdots\)
\(T_{11}^{8} - \cdots\)