Properties

Label 360.2.a.b
Level $360$
Weight $2$
Character orbit 360.a
Self dual yes
Analytic conductor $2.875$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} + O(q^{10}) \) \( q - q^{5} + 4q^{11} + 6q^{13} + 6q^{17} - 4q^{19} + q^{25} + 2q^{29} - 8q^{31} - 2q^{37} + 6q^{41} + 12q^{43} - 8q^{47} - 7q^{49} - 6q^{53} - 4q^{55} - 12q^{59} + 14q^{61} - 6q^{65} + 4q^{67} - 8q^{71} - 6q^{73} - 8q^{79} + 12q^{83} - 6q^{85} - 10q^{89} + 4q^{95} + 2q^{97} + O(q^{100}) \)

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 360.2.a.b 1
3.b odd 2 1 120.2.a.b 1
4.b odd 2 1 720.2.a.d 1
5.b even 2 1 1800.2.a.n 1
5.c odd 4 2 1800.2.f.j 2
8.b even 2 1 2880.2.a.x 1
8.d odd 2 1 2880.2.a.bb 1
9.c even 3 2 3240.2.q.q 2
9.d odd 6 2 3240.2.q.g 2
12.b even 2 1 240.2.a.c 1
15.d odd 2 1 600.2.a.c 1
15.e even 4 2 600.2.f.b 2
20.d odd 2 1 3600.2.a.t 1
20.e even 4 2 3600.2.f.c 2
21.c even 2 1 5880.2.a.a 1
24.f even 2 1 960.2.a.j 1
24.h odd 2 1 960.2.a.c 1
48.i odd 4 2 3840.2.k.o 2
48.k even 4 2 3840.2.k.j 2
60.h even 2 1 1200.2.a.o 1
60.l odd 4 2 1200.2.f.g 2
120.i odd 2 1 4800.2.a.cd 1
120.m even 2 1 4800.2.a.r 1
120.q odd 4 2 4800.2.f.i 2
120.w even 4 2 4800.2.f.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.a.b 1 3.b odd 2 1
240.2.a.c 1 12.b even 2 1
360.2.a.b 1 1.a even 1 1 trivial
600.2.a.c 1 15.d odd 2 1
600.2.f.b 2 15.e even 4 2
720.2.a.d 1 4.b odd 2 1
960.2.a.c 1 24.h odd 2 1
960.2.a.j 1 24.f even 2 1
1200.2.a.o 1 60.h even 2 1
1200.2.f.g 2 60.l odd 4 2
1800.2.a.n 1 5.b even 2 1
1800.2.f.j 2 5.c odd 4 2
2880.2.a.x 1 8.b even 2 1
2880.2.a.bb 1 8.d odd 2 1
3240.2.q.g 2 9.d odd 6 2
3240.2.q.q 2 9.c even 3 2
3600.2.a.t 1 20.d odd 2 1
3600.2.f.c 2 20.e even 4 2
3840.2.k.j 2 48.k even 4 2
3840.2.k.o 2 48.i odd 4 2
4800.2.a.r 1 120.m even 2 1
4800.2.a.cd 1 120.i odd 2 1
4800.2.f.i 2 120.q odd 4 2
4800.2.f.bc 2 120.w even 4 2
5880.2.a.a 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(360))\):

\( T_{7} \)
\( T_{11} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( -4 + T \)
$13$ \( -6 + T \)
$17$ \( -6 + T \)
$19$ \( 4 + T \)
$23$ \( T \)
$29$ \( -2 + T \)
$31$ \( 8 + T \)
$37$ \( 2 + T \)
$41$ \( -6 + T \)
$43$ \( -12 + T \)
$47$ \( 8 + T \)
$53$ \( 6 + T \)
$59$ \( 12 + T \)
$61$ \( -14 + T \)
$67$ \( -4 + T \)
$71$ \( 8 + T \)
$73$ \( 6 + T \)
$79$ \( 8 + T \)
$83$ \( -12 + T \)
$89$ \( 10 + T \)
$97$ \( -2 + T \)
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