Properties

Label 4004.2.a.c
Level $4004$
Weight $2$
Character orbit 4004.a
Self dual yes
Analytic conductor $31.972$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9721009693\)
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

\(f(q)\) \(=\) \( q + 2q^{3} + 4q^{5} + q^{7} + q^{9} + O(q^{10}) \) \( q + 2q^{3} + 4q^{5} + q^{7} + q^{9} + q^{11} - q^{13} + 8q^{15} + 2q^{17} + 4q^{19} + 2q^{21} + 8q^{23} + 11q^{25} - 4q^{27} - 10q^{29} + 8q^{31} + 2q^{33} + 4q^{35} + 2q^{37} - 2q^{39} - 10q^{41} - 10q^{43} + 4q^{45} - 12q^{47} + q^{49} + 4q^{51} - 10q^{53} + 4q^{55} + 8q^{57} + 12q^{59} + 2q^{61} + q^{63} - 4q^{65} - 4q^{67} + 16q^{69} + 2q^{73} + 22q^{75} + q^{77} + 10q^{79} - 11q^{81} - 4q^{83} + 8q^{85} - 20q^{87} + 12q^{89} - q^{91} + 16q^{93} + 16q^{95} - 8q^{97} + q^{99} + O(q^{100}) \)

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 2.00000 0 4.00000 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)
\(13\) \(1\)

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 4004.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.a.c 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).

Hecke characteristic polynomials

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