Properties

Label 6004.2.a.d
Level 6004
Weight 2
Character orbit 6004.a
Self dual Yes
Analytic conductor 47.942
Analytic rank 0
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6004.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9421813736\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{5} \) \( -\beta_{4} q^{7} \) \( -3 q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{5} \) \( -\beta_{4} q^{7} \) \( -3 q^{9} \) \( + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{11} \) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} ) q^{13} \) \( + ( -1 - \beta_{5} + \beta_{6} ) q^{17} \) \(+ q^{19}\) \( + ( 2 + \beta_{2} - \beta_{3} + \beta_{7} ) q^{23} \) \( + ( \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{25} \) \( + ( 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{29} \) \( + ( -1 + 2 \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{31} \) \( + ( -2 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{35} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} + 2 \beta_{7} ) q^{37} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{41} \) \( + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{43} \) \( -3 \beta_{1} q^{45} \) \( + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{47} \) \( + ( 4 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{49} \) \( + ( -\beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{5} ) q^{53} \) \( + ( -1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{55} \) \( + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{59} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{61} \) \( + 3 \beta_{4} q^{63} \) \( + ( 2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{65} \) \( + ( 3 - \beta_{1} - \beta_{5} + \beta_{6} ) q^{67} \) \( + ( 6 - \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{71} \) \( + ( -3 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{73} \) \( + ( 1 + 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{77} \) \(+ q^{79}\) \( + 9 q^{81} \) \( + ( \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{83} \) \( + ( -3 + 2 \beta_{3} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{85} \) \( + ( -2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{89} \) \( + ( 6 + 5 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{91} \) \( + \beta_{1} q^{95} \) \( + ( -4 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{97} \) \( + ( 3 - 3 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut -\mathstrut 24q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut -\mathstrut 24q^{9} \) \(\mathstrut -\mathstrut 7q^{11} \) \(\mathstrut -\mathstrut 12q^{13} \) \(\mathstrut -\mathstrut 7q^{17} \) \(\mathstrut +\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 10q^{23} \) \(\mathstrut +\mathstrut 6q^{25} \) \(\mathstrut +\mathstrut 5q^{29} \) \(\mathstrut +\mathstrut 3q^{31} \) \(\mathstrut -\mathstrut 11q^{35} \) \(\mathstrut -\mathstrut 15q^{37} \) \(\mathstrut -\mathstrut 5q^{41} \) \(\mathstrut +\mathstrut 10q^{43} \) \(\mathstrut -\mathstrut 12q^{45} \) \(\mathstrut +\mathstrut 18q^{47} \) \(\mathstrut +\mathstrut 31q^{49} \) \(\mathstrut +\mathstrut 9q^{53} \) \(\mathstrut -\mathstrut 17q^{55} \) \(\mathstrut +\mathstrut 8q^{59} \) \(\mathstrut +\mathstrut 13q^{61} \) \(\mathstrut -\mathstrut 3q^{63} \) \(\mathstrut +\mathstrut 4q^{65} \) \(\mathstrut +\mathstrut 21q^{67} \) \(\mathstrut +\mathstrut 44q^{71} \) \(\mathstrut -\mathstrut 20q^{73} \) \(\mathstrut +\mathstrut 15q^{77} \) \(\mathstrut +\mathstrut 8q^{79} \) \(\mathstrut +\mathstrut 72q^{81} \) \(\mathstrut -\mathstrut 4q^{83} \) \(\mathstrut -\mathstrut 9q^{85} \) \(\mathstrut -\mathstrut 10q^{89} \) \(\mathstrut +\mathstrut 48q^{91} \) \(\mathstrut +\mathstrut 4q^{95} \) \(\mathstrut -\mathstrut 22q^{97} \) \(\mathstrut +\mathstrut 21q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(4\) \(x^{7}\mathstrut -\mathstrut \) \(15\) \(x^{6}\mathstrut +\mathstrut \) \(56\) \(x^{5}\mathstrut +\mathstrut \) \(87\) \(x^{4}\mathstrut -\mathstrut \) \(248\) \(x^{3}\mathstrut -\mathstrut \) \(241\) \(x^{2}\mathstrut +\mathstrut \) \(340\) \(x\mathstrut +\mathstrut \) \(248\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 4 \nu^{5} - 10 \nu^{4} + 36 \nu^{3} + 37 \nu^{2} - 68 \nu - 56 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{6} + 14 \nu^{5} - 26 \nu^{4} - 73 \nu^{3} + 31 \nu^{2} + 140 \nu + 72 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{6} - 10 \nu^{5} + 40 \nu^{4} + 25 \nu^{3} - 104 \nu^{2} + 4 \nu + 64 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 5 \nu^{6} - 8 \nu^{5} + 56 \nu^{4} + 7 \nu^{3} - 171 \nu^{2} + 32 \nu + 112 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 6 \nu^{6} - 4 \nu^{5} + 66 \nu^{4} - 29 \nu^{3} - 200 \nu^{2} + 92 \nu + 128 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} - 7 \nu^{6} + 76 \nu^{4} - 69 \nu^{3} - 229 \nu^{2} + 188 \nu + 160 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(-\)\(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(4\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{4}\)\(=\)\(-\)\(6\) \(\beta_{7}\mathstrut +\mathstrut \) \(21\) \(\beta_{6}\mathstrut -\mathstrut \) \(15\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\) \(\beta_{1}\mathstrut +\mathstrut \) \(37\)
\(\nu^{5}\)\(=\)\(-\)\(36\) \(\beta_{7}\mathstrut +\mathstrut \) \(84\) \(\beta_{6}\mathstrut -\mathstrut \) \(52\) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(97\) \(\beta_{1}\mathstrut +\mathstrut \) \(68\)
\(\nu^{6}\)\(=\)\(-\)\(132\) \(\beta_{7}\mathstrut +\mathstrut \) \(365\) \(\beta_{6}\mathstrut -\mathstrut \) \(249\) \(\beta_{5}\mathstrut +\mathstrut \) \(60\) \(\beta_{4}\mathstrut +\mathstrut \) \(88\) \(\beta_{3}\mathstrut +\mathstrut \) \(137\) \(\beta_{2}\mathstrut +\mathstrut \) \(265\) \(\beta_{1}\mathstrut +\mathstrut \) \(369\)
\(\nu^{7}\)\(=\)\(-\)\(598\) \(\beta_{7}\mathstrut +\mathstrut \) \(1464\) \(\beta_{6}\mathstrut -\mathstrut \) \(970\) \(\beta_{5}\mathstrut +\mathstrut \) \(268\) \(\beta_{4}\mathstrut +\mathstrut \) \(312\) \(\beta_{3}\mathstrut +\mathstrut \) \(352\) \(\beta_{2}\mathstrut +\mathstrut \) \(1225\) \(\beta_{1}\mathstrut +\mathstrut \) \(1032\)

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
−2.49694
−1.94482
−1.76571
−0.654712
1.60188
2.45095
2.82169
3.98766
0 0 0 −2.49694 0 4.08498 0 −3.00000 0
1.2 0 0 0 −1.94482 0 −0.969069 0 −3.00000 0
1.3 0 0 0 −1.76571 0 2.40269 0 −3.00000 0
1.4 0 0 0 −0.654712 0 −2.24646 0 −3.00000 0
1.5 0 0 0 1.60188 0 −2.96099 0 −3.00000 0
1.6 0 0 0 2.45095 0 −5.04197 0 −3.00000 0
1.7 0 0 0 2.82169 0 4.85844 0 −3.00000 0
1.8 0 0 0 3.98766 0 0.872374 0 −3.00000 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
\(2\) \(-1\)
\(19\) \(-1\)
\(79\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6004))\):

\(T_{3} \)
\(T_{5}^{8} - \cdots\)