Properties

Label 576.6.a.bc
Level $576$
Weight $6$
Character orbit 576.a
Self dual yes
Analytic conductor $92.381$
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] = [576,6,Mod(1,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("576.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 576.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: \(92.3810802123\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4)
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)
 
\(f(q)\) \(=\) \( q + 54 q^{5} - 88 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 54 q^{5} - 88 q^{7} + 540 q^{11} + 418 q^{13} - 594 q^{17} - 836 q^{19} + 4104 q^{23} - 209 q^{25} - 594 q^{29} + 4256 q^{31} - 4752 q^{35} + 298 q^{37} - 17226 q^{41} + 12100 q^{43} + 1296 q^{47} - 9063 q^{49} + 19494 q^{53} + 29160 q^{55} - 7668 q^{59} + 34738 q^{61} + 22572 q^{65} - 21812 q^{67} + 46872 q^{71} + 67562 q^{73} - 47520 q^{77} - 76912 q^{79} + 67716 q^{83} - 32076 q^{85} - 29754 q^{89} - 36784 q^{91} - 45144 q^{95} - 122398 q^{97}+O(q^{100}) \) 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 54.0000 0 −88.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 576.6.a.bc 1
3.b odd 2 1 64.6.a.f 1
4.b odd 2 1 576.6.a.bd 1
8.b even 2 1 36.6.a.a 1
8.d odd 2 1 144.6.a.c 1
12.b even 2 1 64.6.a.b 1
24.f even 2 1 16.6.a.b 1
24.h odd 2 1 4.6.a.a 1
40.f even 2 1 900.6.a.h 1
40.i odd 4 2 900.6.d.a 2
48.i odd 4 2 256.6.b.g 2
48.k even 4 2 256.6.b.c 2
72.j odd 6 2 324.6.e.a 2
72.n even 6 2 324.6.e.d 2
120.i odd 2 1 100.6.a.b 1
120.m even 2 1 400.6.a.d 1
120.q odd 4 2 400.6.c.f 2
120.w even 4 2 100.6.c.b 2
168.e odd 2 1 784.6.a.d 1
168.i even 2 1 196.6.a.e 1
168.s odd 6 2 196.6.e.g 2
168.ba even 6 2 196.6.e.d 2
264.m even 2 1 484.6.a.a 1
312.b odd 2 1 676.6.a.a 1
312.y even 4 2 676.6.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 24.h odd 2 1
16.6.a.b 1 24.f even 2 1
36.6.a.a 1 8.b even 2 1
64.6.a.b 1 12.b even 2 1
64.6.a.f 1 3.b odd 2 1
100.6.a.b 1 120.i odd 2 1
100.6.c.b 2 120.w even 4 2
144.6.a.c 1 8.d odd 2 1
196.6.a.e 1 168.i even 2 1
196.6.e.d 2 168.ba even 6 2
196.6.e.g 2 168.s odd 6 2
256.6.b.c 2 48.k even 4 2
256.6.b.g 2 48.i odd 4 2
324.6.e.a 2 72.j odd 6 2
324.6.e.d 2 72.n even 6 2
400.6.a.d 1 120.m even 2 1
400.6.c.f 2 120.q odd 4 2
484.6.a.a 1 264.m even 2 1
576.6.a.bc 1 1.a even 1 1 trivial
576.6.a.bd 1 4.b odd 2 1
676.6.a.a 1 312.b odd 2 1
676.6.d.a 2 312.y even 4 2
784.6.a.d 1 168.e odd 2 1
900.6.a.h 1 40.f even 2 1
900.6.d.a 2 40.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(576))\):

\( T_{5} - 54 \) Copy content Toggle raw display
\( T_{7} + 88 \) Copy content Toggle raw display
\( T_{11} - 540 \) 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
$5$ \( T - 54 \) Copy content Toggle raw display
$7$ \( T + 88 \) Copy content Toggle raw display
$11$ \( T - 540 \) Copy content Toggle raw display
$13$ \( T - 418 \) Copy content Toggle raw display
$17$ \( T + 594 \) Copy content Toggle raw display
$19$ \( T + 836 \) Copy content Toggle raw display
$23$ \( T - 4104 \) Copy content Toggle raw display
$29$ \( T + 594 \) Copy content Toggle raw display
$31$ \( T - 4256 \) Copy content Toggle raw display
$37$ \( T - 298 \) Copy content Toggle raw display
$41$ \( T + 17226 \) Copy content Toggle raw display
$43$ \( T - 12100 \) Copy content Toggle raw display
$47$ \( T - 1296 \) Copy content Toggle raw display
$53$ \( T - 19494 \) Copy content Toggle raw display
$59$ \( T + 7668 \) Copy content Toggle raw display
$61$ \( T - 34738 \) Copy content Toggle raw display
$67$ \( T + 21812 \) Copy content Toggle raw display
$71$ \( T - 46872 \) Copy content Toggle raw display
$73$ \( T - 67562 \) Copy content Toggle raw display
$79$ \( T + 76912 \) Copy content Toggle raw display
$83$ \( T - 67716 \) Copy content Toggle raw display
$89$ \( T + 29754 \) Copy content Toggle raw display
$97$ \( T + 122398 \) Copy content Toggle raw display
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