Properties

Label 4008.2.a.d
Level $4008$
Weight $2$
Character orbit 4008.a
Self dual yes
Analytic conductor $32.004$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0040411301\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + q^{3} + 4q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} + 4q^{7} + q^{9} - 4q^{11} + 2q^{13} - 4q^{19} + 4q^{21} - 5q^{25} + q^{27} + 6q^{29} + 8q^{31} - 4q^{33} + 6q^{37} + 2q^{39} + 2q^{43} + 8q^{47} + 9q^{49} + 8q^{53} - 4q^{57} + 10q^{61} + 4q^{63} + 14q^{67} - 12q^{71} - 2q^{73} - 5q^{75} - 16q^{77} + 2q^{79} + q^{81} - 12q^{83} + 6q^{87} - 6q^{89} + 8q^{91} + 8q^{93} - 6q^{97} - 4q^{99} + O(q^{100}) \)

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 0 0 4.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(167\) \(1\)

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 4008.2.a.d 1
4.b odd 2 1 8016.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.d 1 1.a even 1 1 trivial
8016.2.a.d 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\( T_{5} \)
\( T_{7} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( -4 + T \)
$11$ \( 4 + T \)
$13$ \( -2 + T \)
$17$ \( T \)
$19$ \( 4 + T \)
$23$ \( T \)
$29$ \( -6 + T \)
$31$ \( -8 + T \)
$37$ \( -6 + T \)
$41$ \( T \)
$43$ \( -2 + T \)
$47$ \( -8 + T \)
$53$ \( -8 + T \)
$59$ \( T \)
$61$ \( -10 + T \)
$67$ \( -14 + T \)
$71$ \( 12 + T \)
$73$ \( 2 + T \)
$79$ \( -2 + T \)
$83$ \( 12 + T \)
$89$ \( 6 + T \)
$97$ \( 6 + T \)
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