Properties

Label 2160.2.a.x
Level $2160$
Weight $2$
Character orbit 2160.a
Self dual yes
Analytic conductor $17.248$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.2476868366\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 540)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} + 4 q^{7} + O(q^{10}) \) \( q + q^{5} + 4 q^{7} + 6 q^{11} - 4 q^{13} + 3 q^{17} + 7 q^{19} - 9 q^{23} + q^{25} + 7 q^{31} + 4 q^{35} + 2 q^{37} - 6 q^{41} - 2 q^{43} + 9 q^{49} - 9 q^{53} + 6 q^{55} - 12 q^{59} - 7 q^{61} - 4 q^{65} - 2 q^{67} + 6 q^{71} + 2 q^{73} + 24 q^{77} + q^{79} + 9 q^{83} + 3 q^{85} + 6 q^{89} - 16 q^{91} + 7 q^{95} + 8 q^{97} + O(q^{100}) \)

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

Atkin-Lehner signs

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

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 2160.2.a.x 1
3.b odd 2 1 2160.2.a.k 1
4.b odd 2 1 540.2.a.d yes 1
8.b even 2 1 8640.2.a.bc 1
8.d odd 2 1 8640.2.a.b 1
12.b even 2 1 540.2.a.a 1
20.d odd 2 1 2700.2.a.r 1
20.e even 4 2 2700.2.d.b 2
24.f even 2 1 8640.2.a.be 1
24.h odd 2 1 8640.2.a.ch 1
36.f odd 6 2 1620.2.i.f 2
36.h even 6 2 1620.2.i.k 2
60.h even 2 1 2700.2.a.t 1
60.l odd 4 2 2700.2.d.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.a.a 1 12.b even 2 1
540.2.a.d yes 1 4.b odd 2 1
1620.2.i.f 2 36.f odd 6 2
1620.2.i.k 2 36.h even 6 2
2160.2.a.k 1 3.b odd 2 1
2160.2.a.x 1 1.a even 1 1 trivial
2700.2.a.r 1 20.d odd 2 1
2700.2.a.t 1 60.h even 2 1
2700.2.d.b 2 20.e even 4 2
2700.2.d.l 2 60.l odd 4 2
8640.2.a.b 1 8.d odd 2 1
8640.2.a.bc 1 8.b even 2 1
8640.2.a.be 1 24.f even 2 1
8640.2.a.ch 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7} - 4 \)
\( T_{11} - 6 \)
\( T_{13} + 4 \)
\( T_{17} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -1 + T \)
$7$ \( -4 + T \)
$11$ \( -6 + T \)
$13$ \( 4 + T \)
$17$ \( -3 + T \)
$19$ \( -7 + T \)
$23$ \( 9 + T \)
$29$ \( T \)
$31$ \( -7 + T \)
$37$ \( -2 + T \)
$41$ \( 6 + T \)
$43$ \( 2 + T \)
$47$ \( T \)
$53$ \( 9 + T \)
$59$ \( 12 + T \)
$61$ \( 7 + T \)
$67$ \( 2 + T \)
$71$ \( -6 + T \)
$73$ \( -2 + T \)
$79$ \( -1 + T \)
$83$ \( -9 + T \)
$89$ \( -6 + T \)
$97$ \( -8 + T \)
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