Properties

Label 8008.2.a.h
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} + ( 2 + \beta ) q^{5} + q^{7} + ( 1 + 3 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} + ( 2 + \beta ) q^{5} + q^{7} + ( 1 + 3 \beta ) q^{9} - q^{11} - q^{13} + ( 5 + 4 \beta ) q^{15} + ( -3 + \beta ) q^{17} + ( 2 - 3 \beta ) q^{19} + ( 1 + \beta ) q^{21} -8 q^{23} + ( 2 + 5 \beta ) q^{25} + ( 7 + 4 \beta ) q^{27} + 4 q^{29} + ( -4 + 4 \beta ) q^{31} + ( -1 - \beta ) q^{33} + ( 2 + \beta ) q^{35} + ( 4 + 2 \beta ) q^{37} + ( -1 - \beta ) q^{39} -2 \beta q^{41} + ( 1 - 3 \beta ) q^{43} + ( 11 + 10 \beta ) q^{45} + ( 4 - 4 \beta ) q^{47} + q^{49} -\beta q^{51} + ( 5 + 3 \beta ) q^{53} + ( -2 - \beta ) q^{55} + ( -7 - 4 \beta ) q^{57} + ( -6 + 4 \beta ) q^{59} + ( 3 - 7 \beta ) q^{61} + ( 1 + 3 \beta ) q^{63} + ( -2 - \beta ) q^{65} + ( 7 + \beta ) q^{67} + ( -8 - 8 \beta ) q^{69} -7 \beta q^{71} + ( -6 + 2 \beta ) q^{73} + ( 17 + 12 \beta ) q^{75} - q^{77} + ( -2 - 3 \beta ) q^{79} + ( 16 + 6 \beta ) q^{81} + ( 14 - 3 \beta ) q^{83} -3 q^{85} + ( 4 + 4 \beta ) q^{87} + ( 3 + \beta ) q^{89} - q^{91} + ( 8 + 4 \beta ) q^{93} + ( -5 - 7 \beta ) q^{95} -10 q^{97} + ( -1 - 3 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} + 5q^{5} + 2q^{7} + 5q^{9} + O(q^{10}) \) \( 2q + 3q^{3} + 5q^{5} + 2q^{7} + 5q^{9} - 2q^{11} - 2q^{13} + 14q^{15} - 5q^{17} + q^{19} + 3q^{21} - 16q^{23} + 9q^{25} + 18q^{27} + 8q^{29} - 4q^{31} - 3q^{33} + 5q^{35} + 10q^{37} - 3q^{39} - 2q^{41} - q^{43} + 32q^{45} + 4q^{47} + 2q^{49} - q^{51} + 13q^{53} - 5q^{55} - 18q^{57} - 8q^{59} - q^{61} + 5q^{63} - 5q^{65} + 15q^{67} - 24q^{69} - 7q^{71} - 10q^{73} + 46q^{75} - 2q^{77} - 7q^{79} + 38q^{81} + 25q^{83} - 6q^{85} + 12q^{87} + 7q^{89} - 2q^{91} + 20q^{93} - 17q^{95} - 20q^{97} - 5q^{99} + O(q^{100}) \)

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
−1.30278
2.30278
0 −0.302776 0 0.697224 0 1.00000 0 −2.90833 0
1.2 0 3.30278 0 4.30278 0 1.00000 0 7.90833 0
\(n\): e.g. 2-40 or 990-1000
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{2} - 3 T_{3} - 1 \)
\( T_{5}^{2} - 5 T_{5} + 3 \)
\( T_{17}^{2} + 5 T_{17} + 3 \)