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.

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

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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

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