Properties

Label 360.6.a.b
Level 360
Weight 6
Character orbit 360.a
Self dual yes
Analytic conductor 57.738
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.7381751327\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 25q^{5} - 108q^{7} + O(q^{10}) \) \( q - 25q^{5} - 108q^{7} + 604q^{11} - 306q^{13} - 930q^{17} - 1324q^{19} + 852q^{23} + 625q^{25} - 5902q^{29} - 3320q^{31} + 2700q^{35} + 10774q^{37} + 17958q^{41} + 9264q^{43} + 9796q^{47} - 5143q^{49} + 31434q^{53} - 15100q^{55} - 33228q^{59} - 40210q^{61} + 7650q^{65} + 58864q^{67} + 55312q^{71} + 27258q^{73} - 65232q^{77} + 31456q^{79} - 24552q^{83} + 23250q^{85} + 90854q^{89} + 33048q^{91} + 33100q^{95} + 154706q^{97} + O(q^{100}) \)

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 −25.0000 0 −108.000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 360.6.a.b 1
3.b odd 2 1 40.6.a.b 1
4.b odd 2 1 720.6.a.h 1
12.b even 2 1 80.6.a.f 1
15.d odd 2 1 200.6.a.c 1
15.e even 4 2 200.6.c.c 2
24.f even 2 1 320.6.a.e 1
24.h odd 2 1 320.6.a.l 1
60.h even 2 1 400.6.a.f 1
60.l odd 4 2 400.6.c.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.b 1 3.b odd 2 1
80.6.a.f 1 12.b even 2 1
200.6.a.c 1 15.d odd 2 1
200.6.c.c 2 15.e even 4 2
320.6.a.e 1 24.f even 2 1
320.6.a.l 1 24.h odd 2 1
360.6.a.b 1 1.a even 1 1 trivial
400.6.a.f 1 60.h even 2 1
400.6.c.h 2 60.l odd 4 2
720.6.a.h 1 4.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(360))\):

\( T_{7} + 108 \)
\( T_{11} - 604 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 25 T \)
$7$ \( 1 + 108 T + 16807 T^{2} \)
$11$ \( 1 - 604 T + 161051 T^{2} \)
$13$ \( 1 + 306 T + 371293 T^{2} \)
$17$ \( 1 + 930 T + 1419857 T^{2} \)
$19$ \( 1 + 1324 T + 2476099 T^{2} \)
$23$ \( 1 - 852 T + 6436343 T^{2} \)
$29$ \( 1 + 5902 T + 20511149 T^{2} \)
$31$ \( 1 + 3320 T + 28629151 T^{2} \)
$37$ \( 1 - 10774 T + 69343957 T^{2} \)
$41$ \( 1 - 17958 T + 115856201 T^{2} \)
$43$ \( 1 - 9264 T + 147008443 T^{2} \)
$47$ \( 1 - 9796 T + 229345007 T^{2} \)
$53$ \( 1 - 31434 T + 418195493 T^{2} \)
$59$ \( 1 + 33228 T + 714924299 T^{2} \)
$61$ \( 1 + 40210 T + 844596301 T^{2} \)
$67$ \( 1 - 58864 T + 1350125107 T^{2} \)
$71$ \( 1 - 55312 T + 1804229351 T^{2} \)
$73$ \( 1 - 27258 T + 2073071593 T^{2} \)
$79$ \( 1 - 31456 T + 3077056399 T^{2} \)
$83$ \( 1 + 24552 T + 3939040643 T^{2} \)
$89$ \( 1 - 90854 T + 5584059449 T^{2} \)
$97$ \( 1 - 154706 T + 8587340257 T^{2} \)
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