Properties

Label 2160.4.a.s
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 540)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 5 q^{5} + 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + 22 q^{7} - 9 q^{11} + 17 q^{13} + 75 q^{17} + 4 q^{19} + 183 q^{23} + 25 q^{25} - 129 q^{29} + 187 q^{31} + 110 q^{35} - 34 q^{37} - 264 q^{41} - 443 q^{43} + 609 q^{47} + 141 q^{49} + 228 q^{53} - 45 q^{55} + 60 q^{59} - 454 q^{61} + 85 q^{65} + 244 q^{67} + 444 q^{71} + 398 q^{73} - 198 q^{77} + 349 q^{79} + 1038 q^{83} + 375 q^{85} - 852 q^{89} + 374 q^{91} + 20 q^{95} + 914 q^{97}+O(q^{100}) \) 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 22.0000 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.s 1
3.b odd 2 1 2160.4.a.h 1
4.b odd 2 1 540.4.a.c yes 1
12.b even 2 1 540.4.a.a 1
36.f odd 6 2 1620.4.i.e 2
36.h even 6 2 1620.4.i.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.a 1 12.b even 2 1
540.4.a.c yes 1 4.b odd 2 1
1620.4.i.e 2 36.f odd 6 2
1620.4.i.k 2 36.h even 6 2
2160.4.a.h 1 3.b odd 2 1
2160.4.a.s 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7} - 22 \) Copy content Toggle raw display
\( T_{11} + 9 \) 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
$7$ \( T - 22 \) Copy content Toggle raw display
$11$ \( T + 9 \) Copy content Toggle raw display
$13$ \( T - 17 \) Copy content Toggle raw display
$17$ \( T - 75 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T - 183 \) Copy content Toggle raw display
$29$ \( T + 129 \) Copy content Toggle raw display
$31$ \( T - 187 \) Copy content Toggle raw display
$37$ \( T + 34 \) Copy content Toggle raw display
$41$ \( T + 264 \) Copy content Toggle raw display
$43$ \( T + 443 \) Copy content Toggle raw display
$47$ \( T - 609 \) Copy content Toggle raw display
$53$ \( T - 228 \) Copy content Toggle raw display
$59$ \( T - 60 \) Copy content Toggle raw display
$61$ \( T + 454 \) Copy content Toggle raw display
$67$ \( T - 244 \) Copy content Toggle raw display
$71$ \( T - 444 \) Copy content Toggle raw display
$73$ \( T - 398 \) Copy content Toggle raw display
$79$ \( T - 349 \) Copy content Toggle raw display
$83$ \( T - 1038 \) Copy content Toggle raw display
$89$ \( T + 852 \) Copy content Toggle raw display
$97$ \( T - 914 \) Copy content Toggle raw display
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