Properties

Label 3120.2.a.q
Level $3120$
Weight $2$
Character orbit 3120.a
Self dual yes
Analytic conductor $24.913$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.9133254306\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + q^{9} + O(q^{10}) \) \( q + q^{3} - q^{5} + q^{9} - q^{13} - q^{15} - 6q^{17} + 4q^{23} + q^{25} + q^{27} - 10q^{29} - 6q^{37} - q^{39} + 2q^{41} + 4q^{43} - q^{45} - 7q^{49} - 6q^{51} - 6q^{53} + 6q^{61} + q^{65} - 4q^{67} + 4q^{69} - 16q^{71} - 2q^{73} + q^{75} + q^{81} - 4q^{83} + 6q^{85} - 10q^{87} - 6q^{89} + 14q^{97} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(1\)

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 3120.2.a.q 1
3.b odd 2 1 9360.2.a.bn 1
4.b odd 2 1 390.2.a.a 1
12.b even 2 1 1170.2.a.m 1
20.d odd 2 1 1950.2.a.y 1
20.e even 4 2 1950.2.e.l 2
52.b odd 2 1 5070.2.a.s 1
52.f even 4 2 5070.2.b.c 2
60.h even 2 1 5850.2.a.m 1
60.l odd 4 2 5850.2.e.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.a 1 4.b odd 2 1
1170.2.a.m 1 12.b even 2 1
1950.2.a.y 1 20.d odd 2 1
1950.2.e.l 2 20.e even 4 2
3120.2.a.q 1 1.a even 1 1 trivial
5070.2.a.s 1 52.b odd 2 1
5070.2.b.c 2 52.f even 4 2
5850.2.a.m 1 60.h even 2 1
5850.2.e.p 2 60.l odd 4 2
9360.2.a.bn 1 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3120))\):

\( T_{7} \)
\( T_{11} \)
\( T_{17} + 6 \)
\( T_{19} \)
\( T_{31} \)

Hecke characteristic polynomials

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