Properties

Label 8008.2.a.q
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: \( 5 \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 1 - \beta_{7} ) q^{5} + q^{7} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 1 - \beta_{7} ) q^{5} + q^{7} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + q^{11} - q^{13} + ( -1 + \beta_{1} - \beta_{4} - \beta_{7} ) q^{15} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{17} + ( 2 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{19} + \beta_{1} q^{21} + ( \beta_{4} - \beta_{8} ) q^{23} + ( 1 - 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} ) q^{25} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{27} -\beta_{6} q^{29} + ( \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{31} + \beta_{1} q^{33} + ( 1 - \beta_{7} ) q^{35} + ( -1 + \beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{37} -\beta_{1} q^{39} + ( 2 - \beta_{2} - \beta_{4} ) q^{41} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{43} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{45} + ( -\beta_{4} + \beta_{8} ) q^{47} + q^{49} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} ) q^{51} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} ) q^{53} + ( 1 - \beta_{7} ) q^{55} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{57} + ( 2 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{59} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{61} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{63} + ( -1 + \beta_{7} ) q^{65} + ( 3 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} ) q^{67} + ( 1 - \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{7} ) q^{69} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} ) q^{71} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{73} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{75} + q^{77} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{7} - \beta_{8} ) q^{79} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} ) q^{81} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{83} + ( 2 - \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{85} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{87} + ( 3 - 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{89} - q^{91} + ( -3 + \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{93} + ( 4 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - 5 \beta_{7} + \beta_{8} ) q^{95} + ( 2 + 3 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{97} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 3q^{3} + 8q^{5} + 9q^{7} + 12q^{9} + O(q^{10}) \) \( 9q + 3q^{3} + 8q^{5} + 9q^{7} + 12q^{9} + 9q^{11} - 9q^{13} - 9q^{15} + q^{17} + 16q^{19} + 3q^{21} + 6q^{23} + 11q^{25} + 9q^{27} - 2q^{29} + 3q^{31} + 3q^{33} + 8q^{35} - 4q^{37} - 3q^{39} + 20q^{41} + 20q^{43} + 8q^{45} - 6q^{47} + 9q^{49} + 15q^{51} + 15q^{53} + 8q^{55} + 16q^{57} + 21q^{59} + 12q^{63} - 8q^{65} + 18q^{67} + 10q^{69} + 12q^{71} + 29q^{73} + 12q^{75} + 9q^{77} - 6q^{79} + 5q^{81} + 25q^{83} + 13q^{85} + 17q^{87} + 32q^{89} - 9q^{91} - 13q^{93} + 38q^{95} + 17q^{97} + 12q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 15 x^{7} + 45 x^{6} + 64 x^{5} - 201 x^{4} - 53 x^{3} + 252 x^{2} - 69 x - 26\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 25 \nu^{8} - 54 \nu^{7} - 419 \nu^{6} + 771 \nu^{5} + 2232 \nu^{4} - 3127 \nu^{3} - 3917 \nu^{2} + 2939 \nu + 773 \)\()/17\)
\(\beta_{3}\)\(=\)\((\)\( -25 \nu^{8} + 54 \nu^{7} + 419 \nu^{6} - 771 \nu^{5} - 2232 \nu^{4} + 3127 \nu^{3} + 3934 \nu^{2} - 2956 \nu - 841 \)\()/17\)
\(\beta_{4}\)\(=\)\((\)\( -50 \nu^{8} + 108 \nu^{7} + 838 \nu^{6} - 1542 \nu^{5} - 4464 \nu^{4} + 6271 \nu^{3} + 7834 \nu^{2} - 6014 \nu - 1563 \)\()/17\)
\(\beta_{5}\)\(=\)\((\)\( -110 \nu^{8} + 241 \nu^{7} + 1847 \nu^{6} - 3457 \nu^{5} - 9865 \nu^{4} + 14126 \nu^{3} + 17347 \nu^{2} - 13564 \nu - 3374 \)\()/17\)
\(\beta_{6}\)\(=\)\((\)\( -123 \nu^{8} + 265 \nu^{7} + 2071 \nu^{6} - 3794 \nu^{5} - 11095 \nu^{4} + 15461 \nu^{3} + 19596 \nu^{2} - 14789 \nu - 3965 \)\()/17\)
\(\beta_{7}\)\(=\)\((\)\( 287 \nu^{8} - 624 \nu^{7} - 4821 \nu^{6} + 8932 \nu^{5} + 25758 \nu^{4} - 36393 \nu^{3} - 45333 \nu^{2} + 34825 \nu + 9008 \)\()/17\)
\(\beta_{8}\)\(=\)\((\)\( -383 \nu^{8} + 830 \nu^{7} + 6432 \nu^{6} - 11877 \nu^{5} - 34335 \nu^{4} + 48383 \nu^{3} + 60309 \nu^{2} - 46310 \nu - 11960 \)\()/17\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{4} + 2 \beta_{2} + 8 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{8} - \beta_{6} - \beta_{5} - 2 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} + 11 \beta_{1} + 29\)
\(\nu^{5}\)\(=\)\(-3 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + 8 \beta_{4} + 23 \beta_{2} + 68 \beta_{1} + 19\)
\(\nu^{6}\)\(=\)\(12 \beta_{8} - 4 \beta_{7} - 13 \beta_{6} - 17 \beta_{5} - 28 \beta_{4} + 62 \beta_{3} + 97 \beta_{2} + 116 \beta_{1} + 238\)
\(\nu^{7}\)\(=\)\(\beta_{8} - 53 \beta_{7} - 38 \beta_{6} - 67 \beta_{5} + 46 \beta_{4} + 9 \beta_{3} + 243 \beta_{2} + 608 \beta_{1} + 252\)
\(\nu^{8}\)\(=\)\(114 \beta_{8} - 89 \beta_{7} - 149 \beta_{6} - 217 \beta_{5} - 313 \beta_{4} + 501 \beta_{3} + 956 \beta_{2} + 1218 \beta_{1} + 2079\)

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
−2.78620
−2.31438
−1.45026
−0.211829
0.827072
0.829765
2.27011
2.61891
3.21682
0 −2.78620 0 0.423993 0 1.00000 0 4.76292 0
1.2 0 −2.31438 0 3.17142 0 1.00000 0 2.35637 0
1.3 0 −1.45026 0 1.74448 0 1.00000 0 −0.896739 0
1.4 0 −0.211829 0 1.56322 0 1.00000 0 −2.95513 0
1.5 0 0.827072 0 −3.42555 0 1.00000 0 −2.31595 0
1.6 0 0.829765 0 4.12811 0 1.00000 0 −2.31149 0
1.7 0 2.27011 0 1.08525 0 1.00000 0 2.15339 0
1.8 0 2.61891 0 −2.59083 0 1.00000 0 3.85871 0
1.9 0 3.21682 0 1.89991 0 1.00000 0 7.34791 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

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