Properties

Label 8008.2.a.x
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 0
Dimension 12
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{4} q^{5} + q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{4} q^{5} + q^{7} + ( 1 + \beta_{2} ) q^{9} + q^{11} + q^{13} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{8} - \beta_{9} ) q^{15} + ( 1 - \beta_{7} ) q^{17} -\beta_{11} q^{19} + \beta_{1} q^{21} + ( 1 + \beta_{2} + \beta_{10} - \beta_{11} ) q^{23} + ( 1 + \beta_{4} + \beta_{6} ) q^{25} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{8} - \beta_{9} ) q^{27} + ( 1 - \beta_{10} ) q^{29} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{31} + \beta_{1} q^{33} + \beta_{4} q^{35} + ( 1 + \beta_{5} - \beta_{10} ) q^{37} + \beta_{1} q^{39} + ( 3 - \beta_{1} - \beta_{3} - \beta_{8} ) q^{41} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{9} + \beta_{10} ) q^{43} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{8} + \beta_{10} ) q^{45} + ( \beta_{1} + \beta_{3} + \beta_{5} + \beta_{7} ) q^{47} + q^{49} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{51} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{11} ) q^{53} + \beta_{4} q^{55} + ( -\beta_{1} + \beta_{2} + \beta_{5} + \beta_{9} - \beta_{11} ) q^{57} + ( 2 - \beta_{2} - \beta_{4} + \beta_{9} + \beta_{11} ) q^{59} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{61} + ( 1 + \beta_{2} ) q^{63} + \beta_{4} q^{65} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{67} + ( -2 + 2 \beta_{1} + \beta_{4} - \beta_{6} + \beta_{8} ) q^{69} + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{71} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{9} + \beta_{11} ) q^{73} + ( -3 + 3 \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{75} + q^{77} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{79} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{9} + \beta_{10} ) q^{81} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} ) q^{83} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{85} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{11} ) q^{87} + ( 2 - \beta_{1} + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{89} + q^{91} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{11} ) q^{93} + ( 2 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{10} ) q^{95} + ( -1 + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} ) q^{97} + ( 1 + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 4q^{3} + 6q^{5} + 12q^{7} + 14q^{9} + O(q^{10}) \) \( 12q + 4q^{3} + 6q^{5} + 12q^{7} + 14q^{9} + 12q^{11} + 12q^{13} - 3q^{15} + 16q^{17} - 2q^{19} + 4q^{21} + 9q^{23} + 14q^{25} + 7q^{27} + 15q^{29} + 10q^{31} + 4q^{33} + 6q^{35} + 18q^{37} + 4q^{39} + 24q^{41} + 15q^{45} + 5q^{47} + 12q^{49} + 4q^{51} + 15q^{53} + 6q^{55} - 4q^{57} + 15q^{59} + 17q^{61} + 14q^{63} + 6q^{65} - 7q^{67} + 9q^{71} + 32q^{73} - 8q^{75} + 12q^{77} + 20q^{79} - 4q^{81} - 5q^{83} + 25q^{85} + 19q^{87} + 16q^{89} + 12q^{91} + 21q^{93} + 8q^{95} + 10q^{97} + 14q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} - 17 x^{10} + 79 x^{9} + 80 x^{8} - 536 x^{7} - 4 x^{6} + 1484 x^{5} - 682 x^{4} - 1431 x^{3} + 1069 x^{2} - 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( 77 \nu^{11} + 116 \nu^{10} - 2017 \nu^{9} - 1221 \nu^{8} + 18608 \nu^{7} - 2976 \nu^{6} - 71120 \nu^{5} + 55928 \nu^{4} + 100022 \nu^{3} - 130123 \nu^{2} - 30863 \nu + 42444 \)\()/6100\)
\(\beta_{4}\)\(=\)\((\)\( 113 \nu^{11} - 416 \nu^{10} - 2453 \nu^{9} + 8491 \nu^{8} + 18752 \nu^{7} - 60424 \nu^{6} - 57900 \nu^{5} + 180912 \nu^{4} + 47918 \nu^{3} - 207707 \nu^{2} + 37113 \nu + 31376 \)\()/6100\)
\(\beta_{5}\)\(=\)\((\)\( 46 \nu^{11} - 2 \nu^{10} - 896 \nu^{9} - 468 \nu^{8} + 6284 \nu^{7} + 8097 \nu^{6} - 22080 \nu^{5} - 40751 \nu^{4} + 48306 \nu^{3} + 60651 \nu^{2} - 52764 \nu - 248 \)\()/1525\)
\(\beta_{6}\)\(=\)\((\)\( 163 \nu^{11} - 816 \nu^{10} - 1703 \nu^{9} + 15541 \nu^{8} - 5448 \nu^{7} - 100224 \nu^{6} + 95000 \nu^{5} + 259612 \nu^{4} - 246082 \nu^{3} - 231757 \nu^{2} + 144063 \nu + 18376 \)\()/6100\)
\(\beta_{7}\)\(=\)\((\)\( -19 \nu^{11} + 91 \nu^{10} + 386 \nu^{9} - 1764 \nu^{8} - 2821 \nu^{7} + 11464 \nu^{6} + 9303 \nu^{5} - 29113 \nu^{4} - 13574 \nu^{3} + 25304 \nu^{2} + 6817 \nu - 3234 \)\()/610\)
\(\beta_{8}\)\(=\)\((\)\( 124 \nu^{11} - 443 \nu^{10} - 2044 \nu^{9} + 7968 \nu^{8} + 9646 \nu^{7} - 46427 \nu^{6} - 7975 \nu^{5} + 100626 \nu^{4} - 29511 \nu^{3} - 68611 \nu^{2} + 32399 \nu + 4848 \)\()/3050\)
\(\beta_{9}\)\(=\)\((\)\( 65 \nu^{11} - 154 \nu^{10} - 1221 \nu^{9} + 2943 \nu^{8} + 7580 \nu^{7} - 19166 \nu^{6} - 17414 \nu^{5} + 51436 \nu^{4} + 6980 \nu^{3} - 53469 \nu^{2} + 16547 \nu + 9208 \)\()/1220\)
\(\beta_{10}\)\(=\)\((\)\( -103 \nu^{11} + 214 \nu^{10} + 2115 \nu^{9} - 4397 \nu^{8} - 15052 \nu^{7} + 31846 \nu^{6} + 43218 \nu^{5} - 97584 \nu^{4} - 36568 \nu^{3} + 108835 \nu^{2} - 19017 \nu - 6892 \)\()/1220\)
\(\beta_{11}\)\(=\)\((\)\( -679 \nu^{11} + 1528 \nu^{10} + 13849 \nu^{9} - 30103 \nu^{8} - 97266 \nu^{7} + 203642 \nu^{6} + 269800 \nu^{5} - 562446 \nu^{4} - 193544 \nu^{3} + 549981 \nu^{2} - 157229 \nu - 22808 \)\()/6100\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{9} + \beta_{8} + \beta_{3} + 8 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{10} + \beta_{9} + \beta_{4} + \beta_{3} + 10 \beta_{2} + \beta_{1} + 26\)
\(\nu^{5}\)\(=\)\(-10 \beta_{9} + 12 \beta_{8} - \beta_{6} - \beta_{5} - 2 \beta_{4} + 11 \beta_{3} + 65 \beta_{1} - 7\)
\(\nu^{6}\)\(=\)\(-2 \beta_{11} + 15 \beta_{10} + 15 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - \beta_{5} + 7 \beta_{4} + 13 \beta_{3} + 93 \beta_{2} + 11 \beta_{1} + 202\)
\(\nu^{7}\)\(=\)\(3 \beta_{11} - 3 \beta_{10} - 84 \beta_{9} + 118 \beta_{8} - 3 \beta_{7} - 17 \beta_{6} - 16 \beta_{5} - 37 \beta_{4} + 102 \beta_{3} + \beta_{2} + 539 \beta_{1} - 42\)
\(\nu^{8}\)\(=\)\(-32 \beta_{11} + 165 \beta_{10} + 168 \beta_{9} - 33 \beta_{8} - 36 \beta_{7} - 19 \beta_{5} + 26 \beta_{4} + 137 \beta_{3} + 843 \beta_{2} + 99 \beta_{1} + 1688\)
\(\nu^{9}\)\(=\)\(53 \beta_{11} - 50 \beta_{10} - 681 \beta_{9} + 1094 \beta_{8} - 55 \beta_{7} - 201 \beta_{6} - 188 \beta_{5} - 471 \beta_{4} + 914 \beta_{3} + 27 \beta_{2} + 4549 \beta_{1} - 229\)
\(\nu^{10}\)\(=\)\(-368 \beta_{11} + 1629 \beta_{10} + 1694 \beta_{9} - 381 \beta_{8} - 444 \beta_{7} - 5 \beta_{6} - 246 \beta_{5} - 81 \beta_{4} + 1349 \beta_{3} + 7560 \beta_{2} + 865 \beta_{1} + 14538\)
\(\nu^{11}\)\(=\)\(633 \beta_{11} - 569 \beta_{10} - 5511 \beta_{9} + 9899 \beta_{8} - 695 \beta_{7} - 2073 \beta_{6} - 1951 \beta_{5} - 5165 \beta_{4} + 8149 \beta_{3} + 465 \beta_{2} + 38915 \beta_{1} - 1019\)

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
−2.95543
−2.33021
−1.78254
−1.54910
−0.219004
0.336827
0.818045
1.39079
1.75763
2.54627
2.98199
3.00474
0 −2.95543 0 3.71005 0 1.00000 0 5.73458 0
1.2 0 −2.33021 0 −0.450718 0 1.00000 0 2.42987 0
1.3 0 −1.78254 0 0.474905 0 1.00000 0 0.177459 0
1.4 0 −1.54910 0 −3.31877 0 1.00000 0 −0.600281 0
1.5 0 −0.219004 0 2.16734 0 1.00000 0 −2.95204 0
1.6 0 0.336827 0 3.95783 0 1.00000 0 −2.88655 0
1.7 0 0.818045 0 −0.570018 0 1.00000 0 −2.33080 0
1.8 0 1.39079 0 0.208204 0 1.00000 0 −1.06569 0
1.9 0 1.75763 0 −3.24204 0 1.00000 0 0.0892594 0
1.10 0 2.54627 0 3.33743 0 1.00000 0 3.48349 0
1.11 0 2.98199 0 1.64834 0 1.00000 0 5.89225 0
1.12 0 3.00474 0 −1.92256 0 1.00000 0 6.02845 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\(T_{3}^{12} - \cdots\)
\(T_{5}^{12} - \cdots\)
\(T_{17}^{12} - \cdots\)