Properties

Label 8008.2.a.l
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 1
Dimension 8
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \beta_{3} ) q^{3} \) \( + ( -1 + \beta_{5} ) q^{5} \) \(+ q^{7}\) \( + ( 1 - \beta_{3} - \beta_{4} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \beta_{3} ) q^{3} \) \( + ( -1 + \beta_{5} ) q^{5} \) \(+ q^{7}\) \( + ( 1 - \beta_{3} - \beta_{4} ) q^{9} \) \(+ q^{11}\) \(+ q^{13}\) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{7} ) q^{15} \) \( + ( 1 + \beta_{2} - \beta_{3} ) q^{17} \) \( + ( -2 - \beta_{1} + \beta_{6} + \beta_{7} ) q^{19} \) \( + ( -1 + \beta_{3} ) q^{21} \) \( + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{23} \) \( + ( 2 - 2 \beta_{1} + \beta_{3} ) q^{25} \) \( + ( -\beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{27} \) \( + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{29} \) \( + ( 2 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{31} \) \( + ( -1 + \beta_{3} ) q^{33} \) \( + ( -1 + \beta_{5} ) q^{35} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} \) \( + ( -1 + \beta_{3} ) q^{39} \) \( + ( -3 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{41} \) \( + ( -1 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{43} \) \( + ( 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{45} \) \( + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{47} \) \(+ q^{49}\) \( + ( -4 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{6} - \beta_{7} ) q^{51} \) \( + ( -4 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{53} \) \( + ( -1 + \beta_{5} ) q^{55} \) \( + ( 2 + \beta_{1} - 4 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{57} \) \( + ( -1 - \beta_{1} + 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{59} \) \( + ( -1 + \beta_{3} - \beta_{4} + 3 \beta_{6} ) q^{61} \) \( + ( 1 - \beta_{3} - \beta_{4} ) q^{63} \) \( + ( -1 + \beta_{5} ) q^{65} \) \( + ( -2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{5} - \beta_{6} ) q^{67} \) \( + ( 5 - \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{69} \) \( + ( -3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{71} \) \( + ( -\beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{73} \) \( + ( 3 - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{75} \) \(+ q^{77}\) \( + ( 1 - \beta_{2} - \beta_{6} + \beta_{7} ) q^{79} \) \( + ( -2 + \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{81} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{83} \) \( + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{85} \) \( + ( -3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{87} \) \( + ( 2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{89} \) \(+ q^{91}\) \( + ( 1 - 5 \beta_{1} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{93} \) \( + ( 5 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{95} \) \( + ( 3 - \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{97} \) \( + ( 1 - \beta_{3} - \beta_{4} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 7q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 7q^{9} \) \(\mathstrut +\mathstrut 8q^{11} \) \(\mathstrut +\mathstrut 8q^{13} \) \(\mathstrut -\mathstrut 5q^{15} \) \(\mathstrut +\mathstrut 3q^{17} \) \(\mathstrut -\mathstrut 19q^{19} \) \(\mathstrut -\mathstrut 5q^{21} \) \(\mathstrut +\mathstrut q^{23} \) \(\mathstrut +\mathstrut 15q^{25} \) \(\mathstrut -\mathstrut 11q^{27} \) \(\mathstrut -\mathstrut 21q^{29} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 5q^{33} \) \(\mathstrut -\mathstrut 7q^{35} \) \(\mathstrut -\mathstrut 12q^{37} \) \(\mathstrut -\mathstrut 5q^{39} \) \(\mathstrut -\mathstrut 6q^{41} \) \(\mathstrut -\mathstrut 19q^{43} \) \(\mathstrut -\mathstrut 17q^{45} \) \(\mathstrut -\mathstrut 7q^{47} \) \(\mathstrut +\mathstrut 8q^{49} \) \(\mathstrut -\mathstrut 19q^{51} \) \(\mathstrut -\mathstrut 26q^{53} \) \(\mathstrut -\mathstrut 7q^{55} \) \(\mathstrut +\mathstrut 2q^{57} \) \(\mathstrut -\mathstrut 17q^{59} \) \(\mathstrut +\mathstrut 7q^{63} \) \(\mathstrut -\mathstrut 7q^{65} \) \(\mathstrut -\mathstrut 24q^{67} \) \(\mathstrut +\mathstrut 20q^{69} \) \(\mathstrut +\mathstrut 2q^{71} \) \(\mathstrut +\mathstrut 2q^{73} \) \(\mathstrut +\mathstrut 18q^{75} \) \(\mathstrut +\mathstrut 8q^{77} \) \(\mathstrut +\mathstrut 7q^{79} \) \(\mathstrut -\mathstrut 4q^{81} \) \(\mathstrut -\mathstrut 6q^{83} \) \(\mathstrut +\mathstrut 19q^{85} \) \(\mathstrut -\mathstrut 13q^{87} \) \(\mathstrut +\mathstrut 13q^{89} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 13q^{93} \) \(\mathstrut +\mathstrut 21q^{95} \) \(\mathstrut +\mathstrut 18q^{97} \) \(\mathstrut +\mathstrut 7q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(2\) \(x^{7}\mathstrut -\mathstrut \) \(9\) \(x^{6}\mathstrut +\mathstrut \) \(12\) \(x^{5}\mathstrut +\mathstrut \) \(24\) \(x^{4}\mathstrut -\mathstrut \) \(10\) \(x^{3}\mathstrut -\mathstrut \) \(18\) \(x^{2}\mathstrut -\mathstrut \) \(2\) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 7 \nu^{3} + 10 \nu^{2} + 9 \nu - 3 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{5} - 9 \nu^{4} + 12 \nu^{3} + 21 \nu^{2} - 9 \nu - 6 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} - 9 \nu^{5} + 12 \nu^{4} + 23 \nu^{3} - 9 \nu^{2} - 12 \nu - 2 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{7} - 4 \nu^{6} - 18 \nu^{5} + 25 \nu^{4} + 47 \nu^{3} - 26 \nu^{2} - 34 \nu - 1 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 8 \nu^{6} + 22 \nu^{5} - 51 \nu^{4} - 41 \nu^{3} + 59 \nu^{2} + 19 \nu - 8 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} + 8 \nu^{6} + 22 \nu^{5} - 50 \nu^{4} - 42 \nu^{3} + 51 \nu^{2} + 27 \nu - 1 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( -4 \nu^{7} + 10 \nu^{6} + 31 \nu^{5} - 64 \nu^{4} - 63 \nu^{3} + 76 \nu^{2} + 31 \nu - 15 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(9\) \(\beta_{7}\mathstrut +\mathstrut \) \(5\) \(\beta_{6}\mathstrut -\mathstrut \) \(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(17\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(17\) \(\beta_{7}\mathstrut +\mathstrut \) \(5\) \(\beta_{6}\mathstrut -\mathstrut \) \(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(17\) \(\beta_{3}\mathstrut -\mathstrut \) \(16\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(61\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(34\) \(\beta_{7}\mathstrut +\mathstrut \) \(18\) \(\beta_{6}\mathstrut -\mathstrut \) \(49\) \(\beta_{5}\mathstrut +\mathstrut \) \(21\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(72\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(148\) \(\beta_{7}\mathstrut +\mathstrut \) \(66\) \(\beta_{6}\mathstrut -\mathstrut \) \(151\) \(\beta_{5}\mathstrut +\mathstrut \) \(123\) \(\beta_{4}\mathstrut +\mathstrut \) \(91\) \(\beta_{3}\mathstrut -\mathstrut \) \(118\) \(\beta_{2}\mathstrut +\mathstrut \) \(79\) \(\beta_{1}\mathstrut +\mathstrut \) \(456\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(527\) \(\beta_{7}\mathstrut +\mathstrut \) \(293\) \(\beta_{6}\mathstrut -\mathstrut \) \(727\) \(\beta_{5}\mathstrut +\mathstrut \) \(397\) \(\beta_{4}\mathstrut +\mathstrut \) \(20\) \(\beta_{3}\mathstrut -\mathstrut \) \(170\) \(\beta_{2}\mathstrut +\mathstrut \) \(197\) \(\beta_{1}\mathstrut +\mathstrut \) \(1195\)\()/4\)

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
0.182882
−1.15747
−0.723237
−0.405405
2.95468
−2.20852
1.08830
2.26876
0 −3.17170 0 −2.42750 0 1.00000 0 7.05967 0
1.2 0 −2.59521 0 0.486965 0 1.00000 0 3.73512 0
1.3 0 −2.02769 0 2.88924 0 1.00000 0 1.11151 0
1.4 0 −0.867389 0 −3.42583 0 1.00000 0 −2.24764 0
1.5 0 −0.837498 0 −2.01769 0 1.00000 0 −2.29860 0
1.6 0 0.425730 0 2.69163 0 1.00000 0 −2.81875 0
1.7 0 1.75319 0 −1.19008 0 1.00000 0 0.0736754 0
1.8 0 2.32056 0 −4.00673 0 1.00000 0 2.38501 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\(T_{3}^{8} + \cdots\)
\(T_{5}^{8} + \cdots\)
\(T_{17}^{8} - \cdots\)