Properties

Label 8008.2.a.u
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 1
Dimension 10
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( -\beta_{6} q^{5} \) \(- q^{7}\) \( + ( \beta_{1} + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( -\beta_{6} q^{5} \) \(- q^{7}\) \( + ( \beta_{1} + \beta_{2} ) q^{9} \) \(- q^{11}\) \(- q^{13}\) \( + ( -\beta_{1} - \beta_{2} + \beta_{9} ) q^{15} \) \( + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{17} \) \( + ( 2 - \beta_{1} - \beta_{2} - \beta_{5} ) q^{19} \) \( -\beta_{1} q^{21} \) \( + ( \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} ) q^{23} \) \( + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{25} \) \( + ( 1 + \beta_{2} + \beta_{3} ) q^{27} \) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{8} - \beta_{9} ) q^{29} \) \( + ( -\beta_{1} + 2 \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{31} \) \( -\beta_{1} q^{33} \) \( + \beta_{6} q^{35} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{37} \) \( -\beta_{1} q^{39} \) \( + ( 1 - \beta_{1} - \beta_{3} - \beta_{7} ) q^{41} \) \( + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{8} ) q^{43} \) \( + ( -2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{45} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{47} \) \(+ q^{49}\) \( + ( 1 + \beta_{2} + 2 \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{51} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{53} \) \( + \beta_{6} q^{55} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{5} + \beta_{8} ) q^{57} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{59} \) \( + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{61} \) \( + ( -\beta_{1} - \beta_{2} ) q^{63} \) \( + \beta_{6} q^{65} \) \( + ( 1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{67} \) \( + ( -1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{69} \) \( + ( -2 - \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{9} ) q^{71} \) \( + ( \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{73} \) \( + ( -1 + \beta_{2} - \beta_{4} + \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{75} \) \(+ q^{77}\) \( + ( -\beta_{1} - 2 \beta_{2} + \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} ) q^{79} \) \( + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{81} \) \( + ( \beta_{2} - 2 \beta_{4} + 2 \beta_{7} + \beta_{9} ) q^{83} \) \( + ( -4 - \beta_{3} + \beta_{4} + \beta_{8} + \beta_{9} ) q^{85} \) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - 3 \beta_{9} ) q^{87} \) \( + ( -3 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} ) q^{89} \) \(+ q^{91}\) \( + ( -5 + \beta_{1} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + \beta_{8} - \beta_{9} ) q^{93} \) \( + ( \beta_{4} - 3 \beta_{6} + \beta_{7} + \beta_{8} ) q^{95} \) \( + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{97} \) \( + ( -\beta_{1} - \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 6q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 6q^{9} \) \(\mathstrut -\mathstrut 10q^{11} \) \(\mathstrut -\mathstrut 10q^{13} \) \(\mathstrut -\mathstrut 4q^{15} \) \(\mathstrut -\mathstrut 3q^{17} \) \(\mathstrut +\mathstrut 13q^{19} \) \(\mathstrut -\mathstrut 2q^{21} \) \(\mathstrut +\mathstrut 6q^{25} \) \(\mathstrut +\mathstrut 14q^{27} \) \(\mathstrut -\mathstrut 3q^{29} \) \(\mathstrut -\mathstrut q^{31} \) \(\mathstrut -\mathstrut 2q^{33} \) \(\mathstrut +\mathstrut 4q^{35} \) \(\mathstrut -\mathstrut 5q^{37} \) \(\mathstrut -\mathstrut 2q^{39} \) \(\mathstrut +\mathstrut 4q^{41} \) \(\mathstrut +\mathstrut 19q^{43} \) \(\mathstrut -\mathstrut 11q^{45} \) \(\mathstrut +\mathstrut q^{47} \) \(\mathstrut +\mathstrut 10q^{49} \) \(\mathstrut +\mathstrut 15q^{51} \) \(\mathstrut -\mathstrut 6q^{53} \) \(\mathstrut +\mathstrut 4q^{55} \) \(\mathstrut -\mathstrut 6q^{57} \) \(\mathstrut +\mathstrut 4q^{59} \) \(\mathstrut -\mathstrut 18q^{61} \) \(\mathstrut -\mathstrut 6q^{63} \) \(\mathstrut +\mathstrut 4q^{65} \) \(\mathstrut +\mathstrut q^{67} \) \(\mathstrut -\mathstrut 11q^{69} \) \(\mathstrut -\mathstrut 21q^{71} \) \(\mathstrut -\mathstrut 10q^{73} \) \(\mathstrut +\mathstrut 10q^{77} \) \(\mathstrut +\mathstrut 3q^{79} \) \(\mathstrut -\mathstrut 30q^{81} \) \(\mathstrut +\mathstrut 6q^{83} \) \(\mathstrut -\mathstrut 33q^{85} \) \(\mathstrut -\mathstrut 9q^{87} \) \(\mathstrut -\mathstrut 22q^{89} \) \(\mathstrut +\mathstrut 10q^{91} \) \(\mathstrut -\mathstrut 34q^{93} \) \(\mathstrut -\mathstrut 3q^{95} \) \(\mathstrut -\mathstrut 9q^{97} \) \(\mathstrut -\mathstrut 6q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(2\) \(x^{9}\mathstrut -\mathstrut \) \(16\) \(x^{8}\mathstrut +\mathstrut \) \(26\) \(x^{7}\mathstrut +\mathstrut \) \(93\) \(x^{6}\mathstrut -\mathstrut \) \(113\) \(x^{5}\mathstrut -\mathstrut \) \(230\) \(x^{4}\mathstrut +\mathstrut \) \(197\) \(x^{3}\mathstrut +\mathstrut \) \(201\) \(x^{2}\mathstrut -\mathstrut \) \(115\) \(x\mathstrut +\mathstrut \) \(8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 2 \)
\(\beta_{4}\)\(=\)\((\)\( 6 \nu^{9} - 14 \nu^{8} - 76 \nu^{7} + 143 \nu^{6} + 334 \nu^{5} - 406 \nu^{4} - 616 \nu^{3} + 360 \nu^{2} + 396 \nu - 109 \)\()/23\)
\(\beta_{5}\)\(=\)\((\)\( 9 \nu^{9} - 21 \nu^{8} - 114 \nu^{7} + 226 \nu^{6} + 455 \nu^{5} - 701 \nu^{4} - 510 \nu^{3} + 655 \nu^{2} - 188 \nu + 9 \)\()/23\)
\(\beta_{6}\)\(=\)\((\)\( 8 \nu^{9} - 34 \nu^{8} - 63 \nu^{7} + 390 \nu^{6} - 7 \nu^{5} - 1354 \nu^{4} + 804 \nu^{3} + 1400 \nu^{2} - 1289 \nu + 192 \)\()/23\)
\(\beta_{7}\)\(=\)\((\)\( 13 \nu^{9} - 38 \nu^{8} - 157 \nu^{7} + 444 \nu^{6} + 601 \nu^{5} - 1585 \nu^{4} - 752 \nu^{3} + 1792 \nu^{2} + 30 \nu - 79 \)\()/23\)
\(\beta_{8}\)\(=\)\((\)\( 12 \nu^{9} - 51 \nu^{8} - 106 \nu^{7} + 608 \nu^{6} + 139 \nu^{5} - 2261 \nu^{4} + 585 \nu^{3} + 2675 \nu^{2} - 1117 \nu + 12 \)\()/23\)
\(\beta_{9}\)\(=\)\((\)\( 18 \nu^{9} - 65 \nu^{8} - 182 \nu^{7} + 751 \nu^{6} + 450 \nu^{5} - 2644 \nu^{4} + 176 \nu^{3} + 2920 \nu^{2} - 1112 \nu - 5 \)\()/23\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(42\) \(\beta_{1}\mathstrut +\mathstrut \) \(13\)
\(\nu^{6}\)\(=\)\(-\)\(4\) \(\beta_{9}\mathstrut -\mathstrut \) \(8\) \(\beta_{8}\mathstrut +\mathstrut \) \(12\) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut -\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(54\) \(\beta_{2}\mathstrut +\mathstrut \) \(90\) \(\beta_{1}\mathstrut +\mathstrut \) \(91\)
\(\nu^{7}\)\(=\)\(-\)\(21\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(17\) \(\beta_{7}\mathstrut +\mathstrut \) \(20\) \(\beta_{6}\mathstrut -\mathstrut \) \(13\) \(\beta_{5}\mathstrut +\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(80\) \(\beta_{3}\mathstrut +\mathstrut \) \(107\) \(\beta_{2}\mathstrut +\mathstrut \) \(323\) \(\beta_{1}\mathstrut +\mathstrut \) \(123\)
\(\nu^{8}\)\(=\)\(-\)\(75\) \(\beta_{9}\mathstrut -\mathstrut \) \(46\) \(\beta_{8}\mathstrut +\mathstrut \) \(116\) \(\beta_{7}\mathstrut +\mathstrut \) \(122\) \(\beta_{6}\mathstrut -\mathstrut \) \(80\) \(\beta_{5}\mathstrut +\mathstrut \) \(23\) \(\beta_{4}\mathstrut +\mathstrut \) \(114\) \(\beta_{3}\mathstrut +\mathstrut \) \(440\) \(\beta_{2}\mathstrut +\mathstrut \) \(786\) \(\beta_{1}\mathstrut +\mathstrut \) \(620\)
\(\nu^{9}\)\(=\)\(-\)\(290\) \(\beta_{9}\mathstrut +\mathstrut \) \(41\) \(\beta_{8}\mathstrut +\mathstrut \) \(212\) \(\beta_{7}\mathstrut +\mathstrut \) \(264\) \(\beta_{6}\mathstrut -\mathstrut \) \(125\) \(\beta_{5}\mathstrut +\mathstrut \) \(168\) \(\beta_{4}\mathstrut +\mathstrut \) \(607\) \(\beta_{3}\mathstrut +\mathstrut \) \(999\) \(\beta_{2}\mathstrut +\mathstrut \) \(2609\) \(\beta_{1}\mathstrut +\mathstrut \) \(1068\)

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
−2.44592
−1.80136
−1.64280
−1.55177
0.0822465
0.398142
1.54918
1.76763
2.67231
2.97233
0 −2.44592 0 0.430286 0 −1.00000 0 2.98251 0
1.2 0 −1.80136 0 −2.97899 0 −1.00000 0 0.244902 0
1.3 0 −1.64280 0 2.39966 0 −1.00000 0 −0.301198 0
1.4 0 −1.55177 0 −0.910737 0 −1.00000 0 −0.592010 0
1.5 0 0.0822465 0 −4.16696 0 −1.00000 0 −2.99324 0
1.6 0 0.398142 0 3.53033 0 −1.00000 0 −2.84148 0
1.7 0 1.54918 0 −0.356949 0 −1.00000 0 −0.600028 0
1.8 0 1.76763 0 0.187208 0 −1.00000 0 0.124526 0
1.9 0 2.67231 0 0.943895 0 −1.00000 0 4.14126 0
1.10 0 2.97233 0 −3.07774 0 −1.00000 0 5.83476 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\(T_{3}^{10} - \cdots\)
\(T_{5}^{10} + \cdots\)
\(T_{17}^{10} + \cdots\)