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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,6,Mod(1,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 360.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 25 q^{5} - 108 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 25 q^{5} - 108 q^{7} + 604 q^{11} - 306 q^{13} - 930 q^{17} - 1324 q^{19} + 852 q^{23} + 625 q^{25} - 5902 q^{29} - 3320 q^{31} + 2700 q^{35} + 10774 q^{37} + 17958 q^{41} + 9264 q^{43} + 9796 q^{47} - 5143 q^{49} + 31434 q^{53} - 15100 q^{55} - 33228 q^{59} - 40210 q^{61} + 7650 q^{65} + 58864 q^{67} + 55312 q^{71} + 27258 q^{73} - 65232 q^{77} + 31456 q^{79} - 24552 q^{83} + 23250 q^{85} + 90854 q^{89} + 33048 q^{91} + 33100 q^{95} + 154706 q^{97}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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:

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 360.6.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

Hecke kernels

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

\( T_{7} + 108 \) Copy content Toggle raw display
\( T_{11} - 604 \) 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
$5$ \( T + 25 \) Copy content Toggle raw display
$7$ \( T + 108 \) Copy content Toggle raw display
$11$ \( T - 604 \) Copy content Toggle raw display
$13$ \( T + 306 \) Copy content Toggle raw display
$17$ \( T + 930 \) Copy content Toggle raw display
$19$ \( T + 1324 \) Copy content Toggle raw display
$23$ \( T - 852 \) Copy content Toggle raw display
$29$ \( T + 5902 \) Copy content Toggle raw display
$31$ \( T + 3320 \) Copy content Toggle raw display
$37$ \( T - 10774 \) Copy content Toggle raw display
$41$ \( T - 17958 \) Copy content Toggle raw display
$43$ \( T - 9264 \) Copy content Toggle raw display
$47$ \( T - 9796 \) Copy content Toggle raw display
$53$ \( T - 31434 \) Copy content Toggle raw display
$59$ \( T + 33228 \) Copy content Toggle raw display
$61$ \( T + 40210 \) Copy content Toggle raw display
$67$ \( T - 58864 \) Copy content Toggle raw display
$71$ \( T - 55312 \) Copy content Toggle raw display
$73$ \( T - 27258 \) Copy content Toggle raw display
$79$ \( T - 31456 \) Copy content Toggle raw display
$83$ \( T + 24552 \) Copy content Toggle raw display
$89$ \( T - 90854 \) Copy content Toggle raw display
$97$ \( T - 154706 \) Copy content Toggle raw display
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