Properties

Label 8008.2.a.t
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 0
Dimension 10
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{6} q^{5} + q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{6} q^{5} + q^{7} + ( 1 + \beta_{2} ) q^{9} - q^{11} + q^{13} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{15} + ( 1 - \beta_{6} + \beta_{8} - \beta_{9} ) q^{17} + ( 2 + \beta_{2} + \beta_{8} ) q^{19} + \beta_{1} q^{21} + ( 2 + \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{23} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} ) q^{25} + ( \beta_{3} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{27} + ( -\beta_{3} - \beta_{4} - \beta_{6} + \beta_{9} ) q^{29} + ( -1 - \beta_{4} + \beta_{7} - \beta_{9} ) q^{31} -\beta_{1} q^{33} + \beta_{6} q^{35} + ( -1 - \beta_{1} - \beta_{3} - \beta_{5} + \beta_{7} ) q^{37} + \beta_{1} q^{39} + ( 1 - \beta_{1} + 2 \beta_{4} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{41} + ( 2 - \beta_{1} + \beta_{4} + \beta_{8} ) q^{43} + ( -1 + \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{45} + ( 2 + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{47} + q^{49} + ( -1 + \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} ) q^{51} + ( 3 + \beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{8} - \beta_{9} ) q^{53} -\beta_{6} q^{55} + ( 4 \beta_{1} + 3 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{57} + ( 1 - \beta_{1} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{59} + ( 1 + 2 \beta_{1} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{61} + ( 1 + \beta_{2} ) q^{63} + \beta_{6} q^{65} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{67} + ( 3 + 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{69} + ( \beta_{2} + 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{71} + ( -2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{9} ) q^{73} + ( 6 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{75} - q^{77} + ( 1 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{79} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} ) q^{81} + ( 4 + \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{83} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{9} ) q^{85} + ( -3 - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{9} ) q^{87} + ( -3 - \beta_{1} + 3 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{89} + q^{91} + ( 1 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{8} - 2 \beta_{9} ) q^{93} + ( 1 - \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{95} + ( -\beta_{1} + \beta_{2} - 5 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{97} + ( -1 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + q^{3} + 3q^{5} + 10q^{7} + 5q^{9} + O(q^{10}) \) \( 10q + q^{3} + 3q^{5} + 10q^{7} + 5q^{9} - 10q^{11} + 10q^{13} + 9q^{15} + 7q^{17} + 17q^{19} + q^{21} + 15q^{23} + 13q^{25} + 7q^{27} - q^{29} - 6q^{31} - q^{33} + 3q^{35} - 12q^{37} + q^{39} + 4q^{41} + 17q^{43} - 15q^{45} + 21q^{47} + 10q^{49} + 3q^{51} + 20q^{53} - 3q^{55} + 12q^{57} + 9q^{59} + 6q^{61} + 5q^{63} + 3q^{65} + 26q^{67} + 30q^{69} + 18q^{71} - 4q^{73} + 48q^{75} - 10q^{77} + 19q^{79} - 14q^{81} + 40q^{83} - 25q^{85} - 25q^{87} - 19q^{89} + 10q^{91} + q^{93} + 11q^{95} - 22q^{97} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} - 17 x^{8} + 13 x^{7} + 96 x^{6} - 49 x^{5} - 207 x^{4} + 58 x^{3} + 169 x^{2} - 12 x - 32\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( 15 \nu^{9} + \nu^{8} - 247 \nu^{7} - 65 \nu^{6} + 1258 \nu^{5} + 541 \nu^{4} - 1949 \nu^{3} - 864 \nu^{2} + 641 \nu + 138 \)\()/26\)
\(\beta_{4}\)\(=\)\((\)\( -25 \nu^{9} + 7 \nu^{8} + 403 \nu^{7} - 13 \nu^{6} - 1984 \nu^{5} - 425 \nu^{4} + 2841 \nu^{3} + 1050 \nu^{2} - 739 \nu - 308 \)\()/26\)
\(\beta_{5}\)\(=\)\( \nu^{9} - 16 \nu^{7} - 4 \nu^{6} + 77 \nu^{5} + 39 \nu^{4} - 101 \nu^{3} - 72 \nu^{2} + 16 \nu + 18 \)
\(\beta_{6}\)\(=\)\((\)\( 23 \nu^{9} + 5 \nu^{8} - 377 \nu^{7} - 169 \nu^{6} + 1870 \nu^{5} + 1249 \nu^{4} - 2569 \nu^{3} - 2058 \nu^{2} + 449 \nu + 456 \)\()/26\)
\(\beta_{7}\)\(=\)\((\)\( -21 \nu^{9} - 17 \nu^{8} + 351 \nu^{7} + 351 \nu^{6} - 1782 \nu^{5} - 2099 \nu^{4} + 2531 \nu^{3} + 3300 \nu^{2} - 445 \nu - 812 \)\()/26\)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{9} + \nu^{8} - 49 \nu^{7} - 27 \nu^{6} + 242 \nu^{5} + 185 \nu^{4} - 331 \nu^{3} - 304 \nu^{2} + 57 \nu + 72 \)\()/2\)
\(\beta_{9}\)\(=\)\((\)\( -59 \nu^{9} - 23 \nu^{8} + 975 \nu^{7} + 585 \nu^{6} - 4910 \nu^{5} - 3889 \nu^{4} + 7075 \nu^{3} + 6222 \nu^{2} - 1691 \nu - 1406 \)\()/26\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{9} - \beta_{7} + \beta_{6} + \beta_{3} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 8 \beta_{2} + \beta_{1} + 25\)
\(\nu^{5}\)\(=\)\(9 \beta_{9} - \beta_{8} - 10 \beta_{7} + 8 \beta_{6} + \beta_{5} + 10 \beta_{3} + \beta_{2} + 42 \beta_{1} + 3\)
\(\nu^{6}\)\(=\)\(\beta_{9} + 13 \beta_{8} + \beta_{7} - 10 \beta_{6} - 12 \beta_{5} - 10 \beta_{4} - 9 \beta_{3} + 59 \beta_{2} + 14 \beta_{1} + 174\)
\(\nu^{7}\)\(=\)\(72 \beta_{9} - 12 \beta_{8} - 84 \beta_{7} + 57 \beta_{6} + 16 \beta_{5} + 2 \beta_{4} + 85 \beta_{3} + 15 \beta_{2} + 308 \beta_{1} + 47\)
\(\nu^{8}\)\(=\)\(16 \beta_{9} + 128 \beta_{8} + 13 \beta_{7} - 85 \beta_{6} - 110 \beta_{5} - 80 \beta_{4} - 64 \beta_{3} + 433 \beta_{2} + 147 \beta_{1} + 1258\)
\(\nu^{9}\)\(=\)\(564 \beta_{9} - 102 \beta_{8} - 671 \beta_{7} + 396 \beta_{6} + 171 \beta_{5} + 31 \beta_{4} + 694 \beta_{3} + 159 \beta_{2} + 2301 \beta_{1} + 512\)

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
−2.72305
−2.25881
−1.29966
−1.02605
−0.516933
0.514400
1.37400
1.38780
2.73324
2.81506
0 −2.72305 0 −1.62788 0 1.00000 0 4.41500 0
1.2 0 −2.25881 0 −2.11934 0 1.00000 0 2.10224 0
1.3 0 −1.29966 0 0.367835 0 1.00000 0 −1.31089 0
1.4 0 −1.02605 0 3.56808 0 1.00000 0 −1.94722 0
1.5 0 −0.516933 0 1.90177 0 1.00000 0 −2.73278 0
1.6 0 0.514400 0 −2.27351 0 1.00000 0 −2.73539 0
1.7 0 1.37400 0 −0.949904 0 1.00000 0 −1.11212 0
1.8 0 1.38780 0 2.74127 0 1.00000 0 −1.07400 0
1.9 0 2.73324 0 4.21870 0 1.00000 0 4.47060 0
1.10 0 2.81506 0 −2.82702 0 1.00000 0 4.92456 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\(T_{3}^{10} - \cdots\)
\(T_{5}^{10} - \cdots\)
\(T_{17}^{10} - \cdots\)