Properties

Label 3640.2.a.a
Level $3640$
Weight $2$
Character orbit 3640.a
Self dual yes
Analytic conductor $29.066$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.0655463357\)
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

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

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 3640.2.a.a 1
4.b odd 2 1 7280.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.a 1 1.a even 1 1 trivial
7280.2.a.u 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3640))\):

\( T_{3} + 2 \)
\( T_{11} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 2 + T \)
$5$ \( -1 + T \)
$7$ \( -1 + T \)
$11$ \( 2 + T \)
$13$ \( -1 + T \)
$17$ \( 6 + T \)
$19$ \( -6 + T \)
$23$ \( -6 + T \)
$29$ \( 6 + T \)
$31$ \( 10 + T \)
$37$ \( 6 + T \)
$41$ \( -10 + T \)
$43$ \( -2 + T \)
$47$ \( T \)
$53$ \( 10 + T \)
$59$ \( 14 + T \)
$61$ \( -2 + T \)
$67$ \( -16 + T \)
$71$ \( -6 + T \)
$73$ \( 2 + T \)
$79$ \( -4 + T \)
$83$ \( -4 + T \)
$89$ \( -2 + T \)
$97$ \( 14 + T \)
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