Properties

Label 8008.2.a.p
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 1
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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{3} q^{5} - q^{7} + ( -\beta_{3} + \beta_{5} + \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{3} q^{5} - q^{7} + ( -\beta_{3} + \beta_{5} + \beta_{7} ) q^{9} + q^{11} + q^{13} + ( -1 + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{8} ) q^{15} + ( -1 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{17} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{19} -\beta_{1} q^{21} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} ) q^{23} + ( -\beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{25} + ( -\beta_{1} - \beta_{2} + \beta_{6} + \beta_{8} ) q^{27} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{29} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{31} + \beta_{1} q^{33} -\beta_{3} q^{35} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{8} ) q^{37} + \beta_{1} q^{39} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{41} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{8} ) q^{43} + ( -4 + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{45} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} ) q^{47} + q^{49} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{51} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{53} + \beta_{3} q^{55} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} ) q^{57} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{59} + ( -2 + 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{61} + ( \beta_{3} - \beta_{5} - \beta_{7} ) q^{63} + \beta_{3} q^{65} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{67} + ( -2 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{7} - 2 \beta_{8} ) q^{69} + ( -1 - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{71} + ( -1 + \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{7} + \beta_{8} ) q^{73} + ( 3 - 3 \beta_{1} - \beta_{2} - 4 \beta_{3} + 3 \beta_{6} + \beta_{7} + 3 \beta_{8} ) q^{75} - q^{77} + ( 1 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{79} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} - 4 \beta_{7} - \beta_{8} ) q^{81} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{83} + ( -2 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} + \beta_{8} ) q^{85} + ( -1 - 4 \beta_{1} - 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{87} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{89} - q^{91} + ( -1 - \beta_{1} + \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} ) q^{93} + ( 1 + \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{95} + ( 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{97} + ( -\beta_{3} + \beta_{5} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + q^{3} - 4q^{5} - 9q^{7} + 4q^{9} + O(q^{10}) \) \( 9q + q^{3} - 4q^{5} - 9q^{7} + 4q^{9} + 9q^{11} + 9q^{13} - 9q^{15} - 11q^{17} + 10q^{19} - q^{21} - 14q^{23} - q^{25} - 5q^{27} - 10q^{29} + 5q^{31} + q^{33} + 4q^{35} - 16q^{37} + q^{39} + 2q^{41} + 4q^{43} - 30q^{45} + 9q^{49} + 3q^{51} - 23q^{53} - 4q^{55} + 14q^{57} + 9q^{59} - 14q^{61} - 4q^{63} - 4q^{65} + 8q^{67} - 26q^{69} - 20q^{71} - 23q^{73} + 32q^{75} - 9q^{77} + 2q^{79} - 11q^{81} - 9q^{83} - 3q^{85} - 7q^{87} - 6q^{89} - 9q^{91} - 19q^{93} - 4q^{95} - 3q^{97} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - x^{8} - 15 x^{7} + 15 x^{6} + 66 x^{5} - 59 x^{4} - 77 x^{3} + 34 x^{2} + 11 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{7} + 2 \nu^{6} - 25 \nu^{5} - 19 \nu^{4} + 77 \nu^{3} + 40 \nu^{2} - 13 \nu - 10 \)\()/9\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{8} - 15 \nu^{6} + 66 \nu^{4} + 7 \nu^{3} - 79 \nu^{2} - 45 \nu + 11 \)\()/9\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{8} - 3 \nu^{7} - 27 \nu^{6} + 45 \nu^{5} + 93 \nu^{4} - 181 \nu^{3} - 26 \nu^{2} + 129 \nu - 14 \)\()/9\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{8} + \nu^{7} + 31 \nu^{6} - 17 \nu^{5} - 146 \nu^{4} + 74 \nu^{3} + 205 \nu^{2} - 47 \nu - 45 \)\()/9\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{8} - 2 \nu^{7} - 44 \nu^{6} + 31 \nu^{5} + 190 \nu^{4} - 116 \nu^{3} - 217 \nu^{2} + 25 \nu + 13 \)\()/9\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{8} - \nu^{7} - 46 \nu^{6} + 17 \nu^{5} + 212 \nu^{4} - 67 \nu^{3} - 275 \nu^{2} + 2 \nu + 29 \)\()/9\)
\(\beta_{8}\)\(=\)\((\)\( -3 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 56 \nu^{5} - 209 \nu^{4} + 202 \nu^{3} + 257 \nu^{2} - 83 \nu - 23 \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{5} - \beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{8} + \beta_{6} - \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{8} + 5 \beta_{7} + \beta_{6} + 6 \beta_{5} - \beta_{4} - 7 \beta_{3} + \beta_{2} - \beta_{1} + 18\)
\(\nu^{5}\)\(=\)\(12 \beta_{8} + \beta_{7} + 10 \beta_{6} - 2 \beta_{5} + \beta_{4} - 3 \beta_{3} - 11 \beta_{2} + 27 \beta_{1} - 4\)
\(\nu^{6}\)\(=\)\(-13 \beta_{8} + 23 \beta_{7} + 12 \beta_{6} + 38 \beta_{5} - 11 \beta_{4} - 46 \beta_{3} + 14 \beta_{2} - 12 \beta_{1} + 120\)
\(\nu^{7}\)\(=\)\(115 \beta_{8} + 17 \beta_{7} + 84 \beta_{6} - 26 \beta_{5} + 14 \beta_{4} - 38 \beta_{3} - 99 \beta_{2} + 154 \beta_{1} - 54\)
\(\nu^{8}\)\(=\)\(-136 \beta_{8} + 94 \beta_{7} + 107 \beta_{6} + 253 \beta_{5} - 99 \beta_{4} - 298 \beta_{3} + 151 \beta_{2} - 104 \beta_{1} + 838\)

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
−2.82075
−2.37534
−1.00974
−0.333725
0.141545
0.536705
1.94061
2.24883
2.67186
0 −2.82075 0 −1.92109 0 −1.00000 0 4.95665 0
1.2 0 −2.37534 0 −0.155697 0 −1.00000 0 2.64224 0
1.3 0 −1.00974 0 2.49743 0 −1.00000 0 −1.98043 0
1.4 0 −0.333725 0 1.97301 0 −1.00000 0 −2.88863 0
1.5 0 0.141545 0 0.343768 0 −1.00000 0 −2.97996 0
1.6 0 0.536705 0 −3.30011 0 −1.00000 0 −2.71195 0
1.7 0 1.94061 0 1.48281 0 −1.00000 0 0.765982 0
1.8 0 2.24883 0 −0.903831 0 −1.00000 0 2.05724 0
1.9 0 2.67186 0 −4.01631 0 −1.00000 0 4.13885 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

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