Properties

Label 8008.2.a.j
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 0
Dimension 5
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: \(5\)
Coefficient field: 5.5.668973.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{3} q^{5} \) \(+ q^{7}\) \( + ( 1 + \beta_{2} - \beta_{4} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{3} q^{5} \) \(+ q^{7}\) \( + ( 1 + \beta_{2} - \beta_{4} ) q^{9} \) \(- q^{11}\) \(- q^{13}\) \( + ( -1 + \beta_{2} ) q^{15} \) \( + ( 4 + \beta_{3} ) q^{17} \) \( + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} \) \( -\beta_{1} q^{21} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{23} \) \( + ( 3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{25} \) \( + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{27} \) \( + ( -4 - 2 \beta_{1} - \beta_{2} - \beta_{4} ) q^{29} \) \( + ( 1 + 2 \beta_{1} - \beta_{4} ) q^{31} \) \( + \beta_{1} q^{33} \) \( + \beta_{3} q^{35} \) \( + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{37} \) \( + \beta_{1} q^{39} \) \( + ( 2 + \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{41} \) \( + ( -5 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{43} \) \( + ( \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{45} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{47} \) \(+ q^{49}\) \( + ( -1 - 4 \beta_{1} + \beta_{2} ) q^{51} \) \( + ( 4 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{53} \) \( -\beta_{3} q^{55} \) \( + ( 3 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{57} \) \( + ( 3 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{59} \) \( + ( 5 - \beta_{1} + \beta_{2} ) q^{61} \) \( + ( 1 + \beta_{2} - \beta_{4} ) q^{63} \) \( -\beta_{3} q^{65} \) \( + ( 5 - 2 \beta_{3} - \beta_{4} ) q^{67} \) \( + ( 3 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{69} \) \( + ( 6 + 2 \beta_{4} ) q^{71} \) \( + ( -1 - \beta_{1} + 3 \beta_{2} ) q^{73} \) \( + ( 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{75} \) \(- q^{77}\) \( + ( -3 - \beta_{3} - 3 \beta_{4} ) q^{79} \) \( + ( 2 + 5 \beta_{1} - \beta_{3} + \beta_{4} ) q^{81} \) \( + ( -1 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} ) q^{83} \) \( + ( 8 + \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} ) q^{85} \) \( + ( 6 + 3 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{87} \) \( + ( -7 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{89} \) \(- q^{91}\) \( + ( -10 - 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} ) q^{93} \) \( + ( 7 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} ) q^{95} \) \( + ( 3 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{97} \) \( + ( -1 - \beta_{2} + \beta_{4} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(5q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut +\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 6q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut +\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 6q^{9} \) \(\mathstrut -\mathstrut 5q^{11} \) \(\mathstrut -\mathstrut 5q^{13} \) \(\mathstrut -\mathstrut 5q^{15} \) \(\mathstrut +\mathstrut 19q^{17} \) \(\mathstrut -\mathstrut 5q^{19} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut +\mathstrut 5q^{23} \) \(\mathstrut +\mathstrut 12q^{25} \) \(\mathstrut -\mathstrut 11q^{27} \) \(\mathstrut -\mathstrut 17q^{29} \) \(\mathstrut +\mathstrut 4q^{31} \) \(\mathstrut -\mathstrut q^{33} \) \(\mathstrut -\mathstrut q^{35} \) \(\mathstrut +\mathstrut 10q^{37} \) \(\mathstrut -\mathstrut q^{39} \) \(\mathstrut +\mathstrut 10q^{41} \) \(\mathstrut -\mathstrut 25q^{43} \) \(\mathstrut +\mathstrut q^{45} \) \(\mathstrut +\mathstrut 3q^{47} \) \(\mathstrut +\mathstrut 5q^{49} \) \(\mathstrut -\mathstrut q^{51} \) \(\mathstrut +\mathstrut 22q^{53} \) \(\mathstrut +\mathstrut q^{55} \) \(\mathstrut +\mathstrut 16q^{57} \) \(\mathstrut +\mathstrut 21q^{59} \) \(\mathstrut +\mathstrut 26q^{61} \) \(\mathstrut +\mathstrut 6q^{63} \) \(\mathstrut +\mathstrut q^{65} \) \(\mathstrut +\mathstrut 28q^{67} \) \(\mathstrut +\mathstrut 16q^{69} \) \(\mathstrut +\mathstrut 28q^{71} \) \(\mathstrut -\mathstrut 4q^{73} \) \(\mathstrut -\mathstrut 5q^{77} \) \(\mathstrut -\mathstrut 11q^{79} \) \(\mathstrut +\mathstrut 5q^{81} \) \(\mathstrut +\mathstrut 33q^{85} \) \(\mathstrut +\mathstrut 31q^{87} \) \(\mathstrut -\mathstrut 37q^{89} \) \(\mathstrut -\mathstrut 5q^{91} \) \(\mathstrut -\mathstrut 49q^{93} \) \(\mathstrut +\mathstrut 29q^{95} \) \(\mathstrut +\mathstrut 18q^{97} \) \(\mathstrut -\mathstrut 6q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(x^{4}\mathstrut -\mathstrut \) \(9\) \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut +\mathstrut \) \(7\) \(x\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{4} - \nu^{3} - 8 \nu^{2} - 3 \nu + 2 \)
\(\beta_{2}\)\(=\)\( \nu^{4} - 10 \nu^{2} - 10 \nu + 5 \)
\(\beta_{3}\)\(=\)\( -\nu^{4} + \nu^{3} + 9 \nu^{2} + 2 \nu - 6 \)
\(\beta_{4}\)\(=\)\( 3 \nu^{4} - \nu^{3} - 29 \nu^{2} - 19 \nu + 15 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(13\)\()/3\)
\(\nu^{3}\)\(=\)\(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)
\(\nu^{4}\)\(=\)\((\)\(20\) \(\beta_{4}\mathstrut +\mathstrut \) \(50\) \(\beta_{3}\mathstrut -\mathstrut \) \(37\) \(\beta_{2}\mathstrut +\mathstrut \) \(30\) \(\beta_{1}\mathstrut +\mathstrut \) \(125\)\()/3\)

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
−2.16366
3.50189
0.376114
0.553533
−1.26787
0 −3.08441 0 −0.239305 0 1.00000 0 6.51357 0
1.2 0 −0.830554 0 3.93077 0 1.00000 0 −2.31018 0
1.3 0 0.293231 0 −3.94142 0 1.00000 0 −2.91402 0
1.4 0 2.18752 0 −2.05962 0 1.00000 0 1.78523 0
1.5 0 2.43421 0 1.30958 0 1.00000 0 2.92540 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\(T_{3}^{5} \) \(\mathstrut -\mathstrut T_{3}^{4} \) \(\mathstrut -\mathstrut 10 T_{3}^{3} \) \(\mathstrut +\mathstrut 12 T_{3}^{2} \) \(\mathstrut +\mathstrut 11 T_{3} \) \(\mathstrut -\mathstrut 4 \)
\(T_{5}^{5} \) \(\mathstrut +\mathstrut T_{5}^{4} \) \(\mathstrut -\mathstrut 18 T_{5}^{3} \) \(\mathstrut -\mathstrut 16 T_{5}^{2} \) \(\mathstrut +\mathstrut 39 T_{5} \) \(\mathstrut +\mathstrut 10 \)
\(T_{17}^{5} \) \(\mathstrut -\mathstrut 19 T_{17}^{4} \) \(\mathstrut +\mathstrut 126 T_{17}^{3} \) \(\mathstrut -\mathstrut 344 T_{17}^{2} \) \(\mathstrut +\mathstrut 327 T_{17} \) \(\mathstrut -\mathstrut 18 \)