Properties

Label 2160.4.a.g
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 270)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{5} + 13 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} + 13 q^{7} + 30 q^{11} - 61 q^{13} + 12 q^{17} + 49 q^{19} - 18 q^{23} + 25 q^{25} - 186 q^{29} + 160 q^{31} - 65 q^{35} - 91 q^{37} + 378 q^{41} + 268 q^{43} - 144 q^{47} - 174 q^{49} + 570 q^{53} - 150 q^{55} - 204 q^{59} - 877 q^{61} + 305 q^{65} + 187 q^{67} + 606 q^{71} + 431 q^{73} + 390 q^{77} - 1151 q^{79} - 102 q^{83} - 60 q^{85} + 984 q^{89} - 793 q^{91} - 245 q^{95} - 265 q^{97}+O(q^{100}) \) 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 13.0000 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.g 1
3.b odd 2 1 2160.4.a.q 1
4.b odd 2 1 270.4.a.h yes 1
12.b even 2 1 270.4.a.d 1
20.d odd 2 1 1350.4.a.i 1
20.e even 4 2 1350.4.c.f 2
36.f odd 6 2 810.4.e.j 2
36.h even 6 2 810.4.e.r 2
60.h even 2 1 1350.4.a.w 1
60.l odd 4 2 1350.4.c.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.d 1 12.b even 2 1
270.4.a.h yes 1 4.b odd 2 1
810.4.e.j 2 36.f odd 6 2
810.4.e.r 2 36.h even 6 2
1350.4.a.i 1 20.d odd 2 1
1350.4.a.w 1 60.h even 2 1
1350.4.c.f 2 20.e even 4 2
1350.4.c.o 2 60.l odd 4 2
2160.4.a.g 1 1.a even 1 1 trivial
2160.4.a.q 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7} - 13 \) Copy content Toggle raw display
\( T_{11} - 30 \) 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
$7$ \( T - 13 \) Copy content Toggle raw display
$11$ \( T - 30 \) Copy content Toggle raw display
$13$ \( T + 61 \) Copy content Toggle raw display
$17$ \( T - 12 \) Copy content Toggle raw display
$19$ \( T - 49 \) Copy content Toggle raw display
$23$ \( T + 18 \) Copy content Toggle raw display
$29$ \( T + 186 \) Copy content Toggle raw display
$31$ \( T - 160 \) Copy content Toggle raw display
$37$ \( T + 91 \) Copy content Toggle raw display
$41$ \( T - 378 \) Copy content Toggle raw display
$43$ \( T - 268 \) Copy content Toggle raw display
$47$ \( T + 144 \) Copy content Toggle raw display
$53$ \( T - 570 \) Copy content Toggle raw display
$59$ \( T + 204 \) Copy content Toggle raw display
$61$ \( T + 877 \) Copy content Toggle raw display
$67$ \( T - 187 \) Copy content Toggle raw display
$71$ \( T - 606 \) Copy content Toggle raw display
$73$ \( T - 431 \) Copy content Toggle raw display
$79$ \( T + 1151 \) Copy content Toggle raw display
$83$ \( T + 102 \) Copy content Toggle raw display
$89$ \( T - 984 \) Copy content Toggle raw display
$97$ \( T + 265 \) Copy content Toggle raw display
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