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.

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

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

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