Properties

Label 8008.2.a.m
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: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \beta_{1} ) q^{3} \) \( + ( -1 + \beta_{6} ) q^{5} \) \(- q^{7}\) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \beta_{1} ) q^{3} \) \( + ( -1 + \beta_{6} ) q^{5} \) \(- q^{7}\) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{9} \) \(- q^{11}\) \(+ q^{13}\) \( + ( \beta_{5} - \beta_{6} - \beta_{7} ) q^{15} \) \( + ( 2 - \beta_{3} - \beta_{8} ) q^{17} \) \( + ( -1 + \beta_{8} ) q^{19} \) \( + ( 1 - \beta_{1} ) q^{21} \) \( + ( -1 - \beta_{2} + \beta_{7} ) q^{23} \) \( + ( 2 - \beta_{1} + 2 \beta_{7} ) q^{25} \) \( + ( -4 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{27} \) \( + ( -3 + \beta_{1} - \beta_{7} + \beta_{8} ) q^{29} \) \( + ( \beta_{1} + \beta_{3} - \beta_{5} ) q^{31} \) \( + ( 1 - \beta_{1} ) q^{33} \) \( + ( 1 - \beta_{6} ) q^{35} \) \( + ( 3 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{8} ) q^{37} \) \( + ( -1 + \beta_{1} ) q^{39} \) \( + ( 3 - 2 \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{41} \) \( + ( -3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{43} \) \( + ( \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{45} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{6} - \beta_{7} ) q^{47} \) \(+ q^{49}\) \( + ( -2 + \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{8} ) q^{51} \) \( + ( -3 + \beta_{1} - \beta_{3} - \beta_{7} ) q^{53} \) \( + ( 1 - \beta_{6} ) q^{55} \) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{8} ) q^{57} \) \( + ( -1 - \beta_{3} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - \beta_{8} ) q^{59} \) \( + ( 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{61} \) \( + ( -2 + \beta_{1} - \beta_{2} ) q^{63} \) \( + ( -1 + \beta_{6} ) q^{65} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{67} \) \( + ( 3 - 5 \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{69} \) \( + ( 1 + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{71} \) \( + ( 5 + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{73} \) \( + ( -4 - \beta_{2} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{75} \) \(+ q^{77}\) \( + ( -1 + \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{79} \) \( + ( 3 - \beta_{1} + \beta_{4} + 2 \beta_{6} + \beta_{8} ) q^{81} \) \( + ( -3 - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} ) q^{83} \) \( + ( 2 \beta_{3} + \beta_{5} + 4 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} ) q^{85} \) \( + ( 6 - 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{7} - 2 \beta_{8} ) q^{87} \) \( + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{89} \) \(- q^{91}\) \( + ( 7 + \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{93} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{6} - 2 \beta_{8} ) q^{95} \) \( + ( -2 + \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{97} \) \( + ( -2 + \beta_{1} - \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(9q \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 12q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 12q^{9} \) \(\mathstrut -\mathstrut 9q^{11} \) \(\mathstrut +\mathstrut 9q^{13} \) \(\mathstrut -\mathstrut q^{15} \) \(\mathstrut +\mathstrut 13q^{17} \) \(\mathstrut -\mathstrut 5q^{19} \) \(\mathstrut +\mathstrut 5q^{21} \) \(\mathstrut -\mathstrut 7q^{23} \) \(\mathstrut +\mathstrut 14q^{25} \) \(\mathstrut -\mathstrut 23q^{27} \) \(\mathstrut -\mathstrut 19q^{29} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut +\mathstrut 5q^{33} \) \(\mathstrut +\mathstrut 5q^{35} \) \(\mathstrut +\mathstrut 4q^{37} \) \(\mathstrut -\mathstrut 5q^{39} \) \(\mathstrut +\mathstrut 10q^{41} \) \(\mathstrut -\mathstrut 17q^{43} \) \(\mathstrut +\mathstrut 11q^{45} \) \(\mathstrut -\mathstrut 5q^{47} \) \(\mathstrut +\mathstrut 9q^{49} \) \(\mathstrut -\mathstrut 9q^{51} \) \(\mathstrut -\mathstrut 24q^{53} \) \(\mathstrut +\mathstrut 5q^{55} \) \(\mathstrut -\mathstrut 4q^{57} \) \(\mathstrut -\mathstrut 19q^{59} \) \(\mathstrut -\mathstrut 12q^{63} \) \(\mathstrut -\mathstrut 5q^{65} \) \(\mathstrut +\mathstrut 2q^{67} \) \(\mathstrut -\mathstrut 2q^{69} \) \(\mathstrut -\mathstrut 2q^{71} \) \(\mathstrut +\mathstrut 34q^{73} \) \(\mathstrut -\mathstrut 46q^{75} \) \(\mathstrut +\mathstrut 9q^{77} \) \(\mathstrut +\mathstrut 5q^{79} \) \(\mathstrut +\mathstrut 37q^{81} \) \(\mathstrut -\mathstrut 24q^{83} \) \(\mathstrut +\mathstrut 33q^{85} \) \(\mathstrut +\mathstrut 41q^{87} \) \(\mathstrut -\mathstrut 11q^{89} \) \(\mathstrut -\mathstrut 9q^{91} \) \(\mathstrut +\mathstrut 53q^{93} \) \(\mathstrut -\mathstrut 23q^{95} \) \(\mathstrut -\mathstrut 12q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(4\) \(x^{8}\mathstrut -\mathstrut \) \(11\) \(x^{7}\mathstrut +\mathstrut \) \(46\) \(x^{6}\mathstrut +\mathstrut \) \(37\) \(x^{5}\mathstrut -\mathstrut \) \(169\) \(x^{4}\mathstrut -\mathstrut \) \(18\) \(x^{3}\mathstrut +\mathstrut \) \(195\) \(x^{2}\mathstrut -\mathstrut \) \(72\) \(x\mathstrut +\mathstrut \) \(7\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{8} - 11 \nu^{7} - 37 \nu^{6} + 128 \nu^{5} + 152 \nu^{4} - 473 \nu^{3} - 185 \nu^{2} + 549 \nu - 92 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( 8 \nu^{8} - 31 \nu^{7} - 91 \nu^{6} + 351 \nu^{5} + 342 \nu^{4} - 1266 \nu^{3} - 359 \nu^{2} + 1437 \nu - 267 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( 9 \nu^{8} - 34 \nu^{7} - 106 \nu^{6} + 389 \nu^{5} + 412 \nu^{4} - 1415 \nu^{3} - 452 \nu^{2} + 1622 \nu - 297 \)\()/2\)
\(\beta_{6}\)\(=\)\( -5 \nu^{8} + 19 \nu^{7} + 59 \nu^{6} - 219 \nu^{5} - 230 \nu^{4} + 804 \nu^{3} + 253 \nu^{2} - 931 \nu + 173 \)
\(\beta_{7}\)\(=\)\((\)\( 11 \nu^{8} - 42 \nu^{7} - 128 \nu^{6} + 479 \nu^{5} + 494 \nu^{4} - 1741 \nu^{3} - 540 \nu^{2} + 1998 \nu - 369 \)\()/2\)
\(\beta_{8}\)\(=\)\((\)\( 12 \nu^{8} - 45 \nu^{7} - 145 \nu^{6} + 525 \nu^{5} + 580 \nu^{4} - 1958 \nu^{3} - 659 \nu^{2} + 2317 \nu - 429 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(20\) \(\beta_{1}\mathstrut +\mathstrut \) \(26\)
\(\nu^{5}\)\(=\)\(4\) \(\beta_{8}\mathstrut -\mathstrut \) \(23\) \(\beta_{7}\mathstrut +\mathstrut \) \(7\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(25\) \(\beta_{4}\mathstrut +\mathstrut \) \(19\) \(\beta_{3}\mathstrut +\mathstrut \) \(34\) \(\beta_{2}\mathstrut +\mathstrut \) \(94\) \(\beta_{1}\mathstrut +\mathstrut \) \(48\)
\(\nu^{6}\)\(=\)\(23\) \(\beta_{8}\mathstrut -\mathstrut \) \(89\) \(\beta_{7}\mathstrut +\mathstrut \) \(44\) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(105\) \(\beta_{4}\mathstrut +\mathstrut \) \(74\) \(\beta_{3}\mathstrut +\mathstrut \) \(143\) \(\beta_{2}\mathstrut +\mathstrut \) \(318\) \(\beta_{1}\mathstrut +\mathstrut \) \(231\)
\(\nu^{7}\)\(=\)\(91\) \(\beta_{8}\mathstrut -\mathstrut \) \(388\) \(\beta_{7}\mathstrut +\mathstrut \) \(167\) \(\beta_{6}\mathstrut +\mathstrut \) \(53\) \(\beta_{5}\mathstrut +\mathstrut \) \(434\) \(\beta_{4}\mathstrut +\mathstrut \) \(299\) \(\beta_{3}\mathstrut +\mathstrut \) \(512\) \(\beta_{2}\mathstrut +\mathstrut \) \(1290\) \(\beta_{1}\mathstrut +\mathstrut \) \(654\)
\(\nu^{8}\)\(=\)\(396\) \(\beta_{8}\mathstrut -\mathstrut \) \(1494\) \(\beta_{7}\mathstrut +\mathstrut \) \(755\) \(\beta_{6}\mathstrut +\mathstrut \) \(220\) \(\beta_{5}\mathstrut +\mathstrut \) \(1724\) \(\beta_{4}\mathstrut +\mathstrut \) \(1154\) \(\beta_{3}\mathstrut +\mathstrut \) \(2010\) \(\beta_{2}\mathstrut +\mathstrut \) \(4768\) \(\beta_{1}\mathstrut +\mathstrut \) \(2632\)

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
−2.16724
−1.94872
−1.78308
0.200201
0.207911
1.17419
1.98056
2.50934
3.82684
0 −3.16724 0 3.81158 0 −1.00000 0 7.03141 0
1.2 0 −2.94872 0 −1.59937 0 −1.00000 0 5.69497 0
1.3 0 −2.78308 0 −2.69929 0 −1.00000 0 4.74553 0
1.4 0 −0.799799 0 1.76846 0 −1.00000 0 −2.36032 0
1.5 0 −0.792089 0 −3.91230 0 −1.00000 0 −2.37259 0
1.6 0 0.174190 0 −1.75196 0 −1.00000 0 −2.96966 0
1.7 0 0.980556 0 2.37283 0 −1.00000 0 −2.03851 0
1.8 0 1.50934 0 −2.72474 0 −1.00000 0 −0.721886 0
1.9 0 2.82684 0 −0.265205 0 −1.00000 0 4.99105 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

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