Properties

Label 8008.2.a.n
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 0
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8008.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9442019386\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -\beta_{6} q^{5} - q^{7} + ( 1 + \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -\beta_{6} q^{5} - q^{7} + ( 1 + \beta_{4} - \beta_{5} ) q^{9} + q^{11} - q^{13} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{8} ) q^{15} + ( 1 - \beta_{3} + \beta_{8} ) q^{17} + ( -2 - \beta_{2} ) q^{19} + \beta_{1} q^{21} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} ) q^{23} + ( 2 - \beta_{2} + \beta_{4} + \beta_{7} ) q^{25} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} ) q^{27} + ( \beta_{3} - \beta_{5} - \beta_{6} + \beta_{8} ) q^{29} + ( -1 - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{31} -\beta_{1} q^{33} + \beta_{6} q^{35} + ( -\beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{37} + \beta_{1} q^{39} + ( 2 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{41} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{43} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{8} ) q^{45} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} ) q^{47} + q^{49} + ( -\beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{51} + ( -1 - \beta_{5} - \beta_{6} ) q^{53} -\beta_{6} q^{55} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{57} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{59} + ( 2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{61} + ( -1 - \beta_{4} + \beta_{5} ) q^{63} + \beta_{6} q^{65} + ( -4 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{67} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} ) q^{69} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{71} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{73} + ( 2 - 4 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{75} - q^{77} + ( 3 - 2 \beta_{1} + \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{79} + ( -1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{5} - \beta_{8} ) q^{81} + ( -3 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} ) q^{83} + ( 7 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{85} + ( 6 - 4 \beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{87} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{89} + q^{91} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{93} + ( 4 - 3 \beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{95} + ( \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{97} + ( 1 + \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 3q^{3} + q^{5} - 9q^{7} + 12q^{9} + O(q^{10}) \) \( 9q - 3q^{3} + q^{5} - 9q^{7} + 12q^{9} + 9q^{11} - 9q^{13} + 11q^{15} + 7q^{17} - 17q^{19} + 3q^{21} + 11q^{23} + 18q^{25} - 9q^{27} + 9q^{29} - 8q^{31} - 3q^{33} - q^{35} + 2q^{37} + 3q^{39} + 18q^{41} + 7q^{43} + 5q^{45} + 15q^{47} + 9q^{49} - 7q^{51} - 4q^{53} + q^{55} + 22q^{57} - 23q^{59} + 12q^{61} - 12q^{63} - q^{65} - 16q^{67} - 32q^{69} - 6q^{71} + 4q^{73} - 14q^{75} - 9q^{77} + 21q^{79} + 5q^{81} - 16q^{83} + 53q^{85} + 41q^{87} + 5q^{89} + 9q^{91} + 29q^{93} + 19q^{95} + 18q^{97} + 12q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 15 x^{7} + 45 x^{6} + 64 x^{5} - 201 x^{4} - 63 x^{3} + 282 x^{2} + 3 x - 116\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{8} - 14 \nu^{7} - 53 \nu^{6} + 159 \nu^{5} + 129 \nu^{4} - 346 \nu^{3} + 491 \nu^{2} - 20 \nu - 738 \)\()/95\)
\(\beta_{3}\)\(=\)\((\)\( 8 \nu^{8} - 28 \nu^{7} - 106 \nu^{6} + 413 \nu^{5} + 353 \nu^{4} - 1737 \nu^{3} - 158 \nu^{2} + 1670 \nu + 139 \)\()/95\)
\(\beta_{4}\)\(=\)\((\)\( 29 \nu^{8} - 54 \nu^{7} - 503 \nu^{6} + 749 \nu^{5} + 2764 \nu^{4} - 2841 \nu^{3} - 5014 \nu^{2} + 2705 \nu + 2677 \)\()/95\)
\(\beta_{5}\)\(=\)\((\)\( 29 \nu^{8} - 54 \nu^{7} - 503 \nu^{6} + 749 \nu^{5} + 2764 \nu^{4} - 2841 \nu^{3} - 5109 \nu^{2} + 2705 \nu + 3057 \)\()/95\)
\(\beta_{6}\)\(=\)\((\)\( 56 \nu^{8} - 101 \nu^{7} - 932 \nu^{6} + 1371 \nu^{5} + 4846 \nu^{4} - 5034 \nu^{3} - 8326 \nu^{2} + 4375 \nu + 4678 \)\()/95\)
\(\beta_{7}\)\(=\)\((\)\( -10 \nu^{8} + 16 \nu^{7} + 180 \nu^{6} - 217 \nu^{5} - 1054 \nu^{4} + 808 \nu^{3} + 2164 \nu^{2} - 824 \nu - 1328 \)\()/19\)
\(\beta_{8}\)\(=\)\((\)\( -62 \nu^{8} + 122 \nu^{7} + 1059 \nu^{6} - 1657 \nu^{5} - 5752 \nu^{4} + 6028 \nu^{3} + 10582 \nu^{2} - 5200 \nu - 6326 \)\()/95\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 6 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{8} - 11 \beta_{5} + 9 \beta_{4} - \beta_{2} + 2 \beta_{1} + 26\)
\(\nu^{5}\)\(=\)\(-10 \beta_{8} + 11 \beta_{7} - 12 \beta_{5} + 14 \beta_{4} - 10 \beta_{3} - 12 \beta_{2} + 46 \beta_{1} + 16\)
\(\nu^{6}\)\(=\)\(-13 \beta_{8} + \beta_{7} + 2 \beta_{6} - 104 \beta_{5} + 76 \beta_{4} - 14 \beta_{2} + 34 \beta_{1} + 202\)
\(\nu^{7}\)\(=\)\(-86 \beta_{8} + 103 \beta_{7} + 5 \beta_{6} - 126 \beta_{5} + 152 \beta_{4} - 92 \beta_{3} - 120 \beta_{2} + 381 \beta_{1} + 200\)
\(\nu^{8}\)\(=\)\(-130 \beta_{8} + 23 \beta_{7} + 44 \beta_{6} - 951 \beta_{5} + 656 \beta_{4} - 11 \beta_{3} - 159 \beta_{2} + 415 \beta_{1} + 1682\)

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
3.07040
3.00639
1.63046
1.21838
1.10969
−0.763644
−1.15169
−2.29224
−2.82774
0 −3.07040 0 −3.53994 0 −1.00000 0 6.42735 0
1.2 0 −3.00639 0 2.62191 0 −1.00000 0 6.03838 0
1.3 0 −1.63046 0 −1.01052 0 −1.00000 0 −0.341611 0
1.4 0 −1.21838 0 2.90030 0 −1.00000 0 −1.51556 0
1.5 0 −1.10969 0 −2.48203 0 −1.00000 0 −1.76859 0
1.6 0 0.763644 0 1.55421 0 −1.00000 0 −2.41685 0
1.7 0 1.15169 0 −3.19313 0 −1.00000 0 −1.67362 0
1.8 0 2.29224 0 3.91599 0 −1.00000 0 2.25436 0
1.9 0 2.82774 0 0.233208 0 −1.00000 0 4.99612 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
\(2\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)
\(13\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\(T_{3}^{9} + \cdots\)
\(T_{5}^{9} - \cdots\)
\(T_{17}^{9} - \cdots\)