Properties

Label 8008.2.a.f
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: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
0 −0.618034 0 1.61803 0 −1.00000 0 −2.61803 0
1.2 0 1.61803 0 −0.618034 0 −1.00000 0 −0.381966 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

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