Properties

Label 8008.2.a.k
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 0
Dimension 6
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: \(6\)
Coefficient field: 6.6.244558277.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -\beta_{5} q^{5} + q^{7} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -\beta_{5} q^{5} + q^{7} + ( 2 + \beta_{2} ) q^{9} - q^{11} - q^{13} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{15} + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{17} + ( 1 + \beta_{2} - \beta_{4} + \beta_{5} ) q^{19} -\beta_{1} q^{21} + ( 2 - 2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{23} + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} ) q^{25} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{27} + ( 2 - \beta_{3} - \beta_{5} ) q^{29} + ( 4 - 2 \beta_{1} - \beta_{4} + \beta_{5} ) q^{31} + \beta_{1} q^{33} -\beta_{5} q^{35} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{37} + \beta_{1} q^{39} + ( -2 \beta_{1} - \beta_{3} - \beta_{5} ) q^{41} + ( 2 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{43} + ( 3 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{45} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{47} + q^{49} + ( -3 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{51} + ( 3 \beta_{1} + \beta_{3} - \beta_{4} ) q^{53} + \beta_{5} q^{55} + ( -1 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{57} + ( 2 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{59} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{61} + ( 2 + \beta_{2} ) q^{63} + \beta_{5} q^{65} + ( -5 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{67} + ( 5 + \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{69} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{71} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{73} + ( -8 - \beta_{1} - 2 \beta_{2} - \beta_{5} ) q^{75} - q^{77} + ( 4 - 2 \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{79} + ( 8 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{81} + ( 3 + 4 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{83} + ( -7 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{85} + ( -3 - 2 \beta_{1} - \beta_{2} - \beta_{5} ) q^{87} + ( 9 - \beta_{1} - \beta_{2} + \beta_{5} ) q^{89} - q^{91} + ( 8 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{93} + ( -5 + \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{95} + ( 5 + 2 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{97} + ( -2 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{3} + q^{5} + 6q^{7} + 9q^{9} + O(q^{10}) \) \( 6q - q^{3} + q^{5} + 6q^{7} + 9q^{9} - 6q^{11} - 6q^{13} - q^{15} + 2q^{19} - q^{21} + 11q^{23} + 9q^{25} - q^{27} + 14q^{29} + 21q^{31} + q^{33} + q^{35} - 5q^{37} + q^{39} + 12q^{43} + 2q^{45} + 2q^{47} + 6q^{49} - 16q^{51} + 2q^{53} - q^{55} - 12q^{57} + 3q^{59} - 18q^{61} + 9q^{63} - q^{65} - 25q^{67} + 29q^{69} - 11q^{71} + 14q^{73} - 42q^{75} - 6q^{77} + 24q^{79} + 42q^{81} + 24q^{83} - 46q^{85} - 16q^{87} + 55q^{89} - 6q^{91} + 41q^{93} - 30q^{95} + 37q^{97} - 9q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 13 x^{4} + 11 x^{3} + 29 x^{2} + 5 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - \nu^{4} - 12 \nu^{3} + 12 \nu^{2} + 19 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 11 \nu^{3} + 23 \nu^{2} + 7 \nu - 10 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 12 \nu^{3} + 22 \nu^{2} + 16 \nu - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + \beta_{4} - \beta_{2} + 9 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-\beta_{5} + \beta_{3} + 10 \beta_{2} - 3 \beta_{1} + 44\)
\(\nu^{5}\)\(=\)\(-13 \beta_{5} + 12 \beta_{4} + 2 \beta_{3} - 14 \beta_{2} + 86 \beta_{1} - 28\)

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
2.98317
2.50454
0.287253
−0.676795
−0.847024
−3.25115
0 −2.98317 0 3.19751 0 1.00000 0 5.89931 0
1.2 0 −2.50454 0 −3.40126 0 1.00000 0 3.27272 0
1.3 0 −0.287253 0 −0.115270 0 1.00000 0 −2.91749 0
1.4 0 0.676795 0 3.59312 0 1.00000 0 −2.54195 0
1.5 0 0.847024 0 −2.05840 0 1.00000 0 −2.28255 0
1.6 0 3.25115 0 −0.215704 0 1.00000 0 7.56995 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{6} + T_{3}^{5} - 13 T_{3}^{4} - 11 T_{3}^{3} + 29 T_{3}^{2} - 5 T_{3} - 4 \)
\( T_{5}^{6} - T_{5}^{5} - 19 T_{5}^{4} + 9 T_{5}^{3} + 85 T_{5}^{2} + 27 T_{5} + 2 \)
\( T_{17}^{6} - 71 T_{17}^{4} - 20 T_{17}^{3} + 1225 T_{17}^{2} + 200 T_{17} - 1792 \)