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

Related objects

Downloads

Learn more about

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.

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

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

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

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

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