Properties

Label 2160.4.a.d
Level $2160$
Weight $4$
Character orbit 2160.a
Self dual yes
Analytic conductor $127.444$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - 5 q^{5} + O(q^{10}) \) \( q - 5 q^{5} + 10 q^{11} - 80 q^{13} - 7 q^{17} + 113 q^{19} - 81 q^{23} + 25 q^{25} + 220 q^{29} + 189 q^{31} + 170 q^{37} + 130 q^{41} - 10 q^{43} + 160 q^{47} - 343 q^{49} - 631 q^{53} - 50 q^{55} - 560 q^{59} + 229 q^{61} + 400 q^{65} - 750 q^{67} + 890 q^{71} - 890 q^{73} + 27 q^{79} + 429 q^{83} + 35 q^{85} + 750 q^{89} - 565 q^{95} - 1480 q^{97} + O(q^{100}) \)

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

\( T_{7} \)
\( T_{11} - 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 5 + T \)
$7$ \( T \)
$11$ \( -10 + T \)
$13$ \( 80 + T \)
$17$ \( 7 + T \)
$19$ \( -113 + T \)
$23$ \( 81 + T \)
$29$ \( -220 + T \)
$31$ \( -189 + T \)
$37$ \( -170 + T \)
$41$ \( -130 + T \)
$43$ \( 10 + T \)
$47$ \( -160 + T \)
$53$ \( 631 + T \)
$59$ \( 560 + T \)
$61$ \( -229 + T \)
$67$ \( 750 + T \)
$71$ \( -890 + T \)
$73$ \( 890 + T \)
$79$ \( -27 + T \)
$83$ \( -429 + T \)
$89$ \( -750 + T \)
$97$ \( 1480 + T \)
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