Properties

Label 8004.2.a.c
Level $8004$
Weight $2$
Character orbit 8004.a
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0 1.00000 0 −2.00000 0 3.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8004.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.c 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\( T_{5} + 2 \)
\( T_{7} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( 2 + T \)
$7$ \( -3 + T \)
$11$ \( -3 + T \)
$13$ \( -1 + T \)
$17$ \( 5 + T \)
$19$ \( 2 + T \)
$23$ \( 1 + T \)
$29$ \( 1 + T \)
$31$ \( 6 + T \)
$37$ \( 10 + T \)
$41$ \( 6 + T \)
$43$ \( 2 + T \)
$47$ \( -9 + T \)
$53$ \( -8 + T \)
$59$ \( 4 + T \)
$61$ \( 6 + T \)
$67$ \( 7 + T \)
$71$ \( 12 + T \)
$73$ \( 6 + T \)
$79$ \( 8 + T \)
$83$ \( -2 + T \)
$89$ \( -9 + T \)
$97$ \( -6 + T \)
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