Properties

Label 2160.2.a.f
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 1080)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} + O(q^{10}) \) \( q - q^{5} + 2 q^{11} - 3 q^{17} + q^{19} + 3 q^{23} + q^{25} - 4 q^{29} + 5 q^{31} + 10 q^{37} - 6 q^{41} + 6 q^{43} + 8 q^{47} - 7 q^{49} - 3 q^{53} - 2 q^{55} + 5 q^{61} + 2 q^{67} + 2 q^{71} + 6 q^{73} + 11 q^{79} + 9 q^{83} + 3 q^{85} - 10 q^{89} - q^{95} + 8 q^{97} + O(q^{100}) \)

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 0 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.f 1
3.b odd 2 1 2160.2.a.r 1
4.b odd 2 1 1080.2.a.d 1
8.b even 2 1 8640.2.a.bs 1
8.d odd 2 1 8640.2.a.bt 1
12.b even 2 1 1080.2.a.j yes 1
20.d odd 2 1 5400.2.a.w 1
20.e even 4 2 5400.2.f.j 2
24.f even 2 1 8640.2.a.o 1
24.h odd 2 1 8640.2.a.p 1
36.f odd 6 2 3240.2.q.s 2
36.h even 6 2 3240.2.q.e 2
60.h even 2 1 5400.2.a.x 1
60.l odd 4 2 5400.2.f.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.d 1 4.b odd 2 1
1080.2.a.j yes 1 12.b even 2 1
2160.2.a.f 1 1.a even 1 1 trivial
2160.2.a.r 1 3.b odd 2 1
3240.2.q.e 2 36.h even 6 2
3240.2.q.s 2 36.f odd 6 2
5400.2.a.w 1 20.d odd 2 1
5400.2.a.x 1 60.h even 2 1
5400.2.f.j 2 20.e even 4 2
5400.2.f.r 2 60.l odd 4 2
8640.2.a.o 1 24.f even 2 1
8640.2.a.p 1 24.h odd 2 1
8640.2.a.bs 1 8.b even 2 1
8640.2.a.bt 1 8.d odd 2 1

Hecke kernels

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

\( T_{7} \)
\( T_{11} - 2 \)
\( T_{13} \)
\( T_{17} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( -2 + T \)
$13$ \( T \)
$17$ \( 3 + T \)
$19$ \( -1 + T \)
$23$ \( -3 + T \)
$29$ \( 4 + T \)
$31$ \( -5 + T \)
$37$ \( -10 + T \)
$41$ \( 6 + T \)
$43$ \( -6 + T \)
$47$ \( -8 + T \)
$53$ \( 3 + T \)
$59$ \( T \)
$61$ \( -5 + T \)
$67$ \( -2 + T \)
$71$ \( -2 + T \)
$73$ \( -6 + T \)
$79$ \( -11 + T \)
$83$ \( -9 + T \)
$89$ \( 10 + T \)
$97$ \( -8 + T \)
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