Properties

Label 8004.2.a.d
Level 8004
Weight 2
Character orbit 8004.a
Self dual Yes
Analytic conductor 63.912
Analytic rank 1
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8004.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(3\) \(x^{7}\mathstrut -\mathstrut \) \(8\) \(x^{6}\mathstrut +\mathstrut \) \(19\) \(x^{5}\mathstrut +\mathstrut \) \(19\) \(x^{4}\mathstrut -\mathstrut \) \(35\) \(x^{3}\mathstrut -\mathstrut \) \(10\) \(x^{2}\mathstrut +\mathstrut \) \(18\) \(x\mathstrut -\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{7} - 13 \nu^{6} - 46 \nu^{5} + 79 \nu^{4} + 131 \nu^{3} - 131 \nu^{2} - 108 \nu + 50 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{7} + 13 \nu^{6} + 46 \nu^{5} - 79 \nu^{4} - 131 \nu^{3} + 133 \nu^{2} + 106 \nu - 56 \)\()/2\)
\(\beta_{4}\)\(=\)\( 5 \nu^{7} - 13 \nu^{6} - 45 \nu^{5} + 76 \nu^{4} + 126 \nu^{3} - 122 \nu^{2} - 102 \nu + 50 \)
\(\beta_{5}\)\(=\)\((\)\( -11 \nu^{7} + 29 \nu^{6} + 98 \nu^{5} - 171 \nu^{4} - 269 \nu^{3} + 273 \nu^{2} + 210 \nu - 104 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -13 \nu^{7} + 33 \nu^{6} + 120 \nu^{5} - 195 \nu^{4} - 339 \nu^{3} + 317 \nu^{2} + 272 \nu - 132 \)\()/2\)
\(\beta_{7}\)\(=\)\( 7 \nu^{7} - 18 \nu^{6} - 64 \nu^{5} + 107 \nu^{4} + 179 \nu^{3} - 175 \nu^{2} - 143 \nu + 71 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(5\) \(\beta_{7}\mathstrut +\mathstrut \) \(4\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(16\) \(\beta_{1}\mathstrut +\mathstrut \) \(20\)
\(\nu^{5}\)\(=\)\(20\) \(\beta_{7}\mathstrut +\mathstrut \) \(17\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(31\) \(\beta_{3}\mathstrut +\mathstrut \) \(26\) \(\beta_{2}\mathstrut +\mathstrut \) \(63\) \(\beta_{1}\mathstrut +\mathstrut \) \(48\)
\(\nu^{6}\)\(=\)\(77\) \(\beta_{7}\mathstrut +\mathstrut \) \(60\) \(\beta_{6}\mathstrut +\mathstrut \) \(48\) \(\beta_{5}\mathstrut +\mathstrut \) \(34\) \(\beta_{4}\mathstrut +\mathstrut \) \(120\) \(\beta_{3}\mathstrut +\mathstrut \) \(98\) \(\beta_{2}\mathstrut +\mathstrut \) \(211\) \(\beta_{1}\mathstrut +\mathstrut \) \(199\)
\(\nu^{7}\)\(=\)\(279\) \(\beta_{7}\mathstrut +\mathstrut \) \(223\) \(\beta_{6}\mathstrut +\mathstrut \) \(180\) \(\beta_{5}\mathstrut +\mathstrut \) \(141\) \(\beta_{4}\mathstrut +\mathstrut \) \(413\) \(\beta_{3}\mathstrut +\mathstrut \) \(326\) \(\beta_{2}\mathstrut +\mathstrut \) \(766\) \(\beta_{1}\mathstrut +\mathstrut \) \(633\)

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
−1.85152
0.386133
−1.03502
3.55127
−1.58311
1.24887
0.411238
1.87214
0 1.00000 0 −3.41642 0 2.05098 0 1.00000 0
1.2 0 1.00000 0 −2.77471 0 0.556399 0 1.00000 0
1.3 0 1.00000 0 −1.84132 0 −2.79101 0 1.00000 0
1.4 0 1.00000 0 −1.29538 0 2.07218 0 1.00000 0
1.5 0 1.00000 0 −0.270840 0 0.409136 0 1.00000 0
1.6 0 1.00000 0 0.138622 0 −3.33662 0 1.00000 0
1.7 0 1.00000 0 1.71910 0 0.210007 0 1.00000 0
1.8 0 1.00000 0 2.74094 0 −3.17108 0 1.00000 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
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(-1\)
\(29\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\(T_{5}^{8} + \cdots\)
\(T_{7}^{8} + \cdots\)