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.

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

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

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

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

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