Properties

Label 3960.2.a.k
Level $3960$
Weight $2$
Character orbit 3960.a
Self dual yes
Analytic conductor $31.621$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.6207592004\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1320)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} - 2 q^{7} + O(q^{10}) \) \( q + q^{5} - 2 q^{7} - q^{11} + 8 q^{17} - 8 q^{19} - 4 q^{23} + q^{25} + 6 q^{29} - 2 q^{35} + 6 q^{37} + 2 q^{41} + 2 q^{43} + 4 q^{47} - 3 q^{49} + 2 q^{53} - q^{55} + 12 q^{59} - 6 q^{61} + 8 q^{67} - 8 q^{73} + 2 q^{77} + 4 q^{79} + 6 q^{83} + 8 q^{85} + 10 q^{89} - 8 q^{95} - 10 q^{97} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)

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 3960.2.a.k 1
3.b odd 2 1 1320.2.a.k 1
4.b odd 2 1 7920.2.a.bj 1
12.b even 2 1 2640.2.a.c 1
15.d odd 2 1 6600.2.a.n 1
15.e even 4 2 6600.2.d.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.a.k 1 3.b odd 2 1
2640.2.a.c 1 12.b even 2 1
3960.2.a.k 1 1.a even 1 1 trivial
6600.2.a.n 1 15.d odd 2 1
6600.2.d.y 2 15.e even 4 2
7920.2.a.bj 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3960))\):

\( T_{7} + 2 \)
\( T_{13} \)
\( T_{17} - 8 \)
\( T_{19} + 8 \)
\( T_{23} + 4 \)
\( T_{29} - 6 \)

Hecke characteristic polynomials

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