Properties

Label 2160.4.a.f
Level $2160$
Weight $4$
Character orbit 2160.a
Self dual yes
Analytic conductor $127.444$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(127.444125612\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 135)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} + 6 q^{7} - 47 q^{11} - 5 q^{13} + 131 q^{17} + 56 q^{19} + 3 q^{23} + 25 q^{25} + 157 q^{29} - 225 q^{31} - 30 q^{35} - 70 q^{37} - 140 q^{41} - 397 q^{43} - 347 q^{47} - 307 q^{49} - 4 q^{53} + 235 q^{55} + 748 q^{59} - 338 q^{61} + 25 q^{65} - 492 q^{67} + 32 q^{71} + 970 q^{73} - 282 q^{77} + 1257 q^{79} - 102 q^{83} - 655 q^{85} + 1488 q^{89} - 30 q^{91} - 280 q^{95} + 974 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −5.00000 0 6.00000 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 2160.4.a.f 1
3.b odd 2 1 2160.4.a.p 1
4.b odd 2 1 135.4.a.b 1
12.b even 2 1 135.4.a.c yes 1
20.d odd 2 1 675.4.a.h 1
20.e even 4 2 675.4.b.f 2
36.f odd 6 2 405.4.e.h 2
36.h even 6 2 405.4.e.f 2
60.h even 2 1 675.4.a.c 1
60.l odd 4 2 675.4.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.b 1 4.b odd 2 1
135.4.a.c yes 1 12.b even 2 1
405.4.e.f 2 36.h even 6 2
405.4.e.h 2 36.f odd 6 2
675.4.a.c 1 60.h even 2 1
675.4.a.h 1 20.d odd 2 1
675.4.b.e 2 60.l odd 4 2
675.4.b.f 2 20.e even 4 2
2160.4.a.f 1 1.a even 1 1 trivial
2160.4.a.p 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7} - 6 \) Copy content Toggle raw display
\( T_{11} + 47 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T - 6 \) Copy content Toggle raw display
$11$ \( T + 47 \) Copy content Toggle raw display
$13$ \( T + 5 \) Copy content Toggle raw display
$17$ \( T - 131 \) Copy content Toggle raw display
$19$ \( T - 56 \) Copy content Toggle raw display
$23$ \( T - 3 \) Copy content Toggle raw display
$29$ \( T - 157 \) Copy content Toggle raw display
$31$ \( T + 225 \) Copy content Toggle raw display
$37$ \( T + 70 \) Copy content Toggle raw display
$41$ \( T + 140 \) Copy content Toggle raw display
$43$ \( T + 397 \) Copy content Toggle raw display
$47$ \( T + 347 \) Copy content Toggle raw display
$53$ \( T + 4 \) Copy content Toggle raw display
$59$ \( T - 748 \) Copy content Toggle raw display
$61$ \( T + 338 \) Copy content Toggle raw display
$67$ \( T + 492 \) Copy content Toggle raw display
$71$ \( T - 32 \) Copy content Toggle raw display
$73$ \( T - 970 \) Copy content Toggle raw display
$79$ \( T - 1257 \) Copy content Toggle raw display
$83$ \( T + 102 \) Copy content Toggle raw display
$89$ \( T - 1488 \) Copy content Toggle raw display
$97$ \( T - 974 \) Copy content Toggle raw display
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