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.

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

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

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

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

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