Properties

Label 576.6.a.bg
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 24)
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)
 
\(f(q)\) \(=\) \( q + 94 q^{5} + 144 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 94 q^{5} + 144 q^{7} - 380 q^{11} - 814 q^{13} + 862 q^{17} + 1156 q^{19} + 488 q^{23} + 5711 q^{25} - 5466 q^{29} + 9560 q^{31} + 13536 q^{35} + 10506 q^{37} + 5190 q^{41} + 17084 q^{43} - 3168 q^{47} + 3929 q^{49} - 24770 q^{53} - 35720 q^{55} + 17380 q^{59} - 4366 q^{61} - 76516 q^{65} + 5284 q^{67} - 8360 q^{71} + 39466 q^{73} - 54720 q^{77} + 42376 q^{79} - 61828 q^{83} + 81028 q^{85} + 63078 q^{89} - 117216 q^{91} + 108664 q^{95} - 16318 q^{97}+O(q^{100}) \) 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 94.0000 0 144.000 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.bg 1
3.b odd 2 1 192.6.a.i 1
4.b odd 2 1 576.6.a.bf 1
8.b even 2 1 72.6.a.a 1
8.d odd 2 1 144.6.a.b 1
12.b even 2 1 192.6.a.a 1
24.f even 2 1 48.6.a.e 1
24.h odd 2 1 24.6.a.b 1
48.i odd 4 2 768.6.d.d 2
48.k even 4 2 768.6.d.o 2
120.i odd 2 1 600.6.a.d 1
120.w even 4 2 600.6.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.6.a.b 1 24.h odd 2 1
48.6.a.e 1 24.f even 2 1
72.6.a.a 1 8.b even 2 1
144.6.a.b 1 8.d odd 2 1
192.6.a.a 1 12.b even 2 1
192.6.a.i 1 3.b odd 2 1
576.6.a.bf 1 4.b odd 2 1
576.6.a.bg 1 1.a even 1 1 trivial
600.6.a.d 1 120.i odd 2 1
600.6.f.b 2 120.w even 4 2
768.6.d.d 2 48.i odd 4 2
768.6.d.o 2 48.k even 4 2

Hecke kernels

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

\( T_{5} - 94 \) Copy content Toggle raw display
\( T_{7} - 144 \) Copy content Toggle raw display
\( T_{11} + 380 \) 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
$5$ \( T - 94 \) Copy content Toggle raw display
$7$ \( T - 144 \) Copy content Toggle raw display
$11$ \( T + 380 \) Copy content Toggle raw display
$13$ \( T + 814 \) Copy content Toggle raw display
$17$ \( T - 862 \) Copy content Toggle raw display
$19$ \( T - 1156 \) Copy content Toggle raw display
$23$ \( T - 488 \) Copy content Toggle raw display
$29$ \( T + 5466 \) Copy content Toggle raw display
$31$ \( T - 9560 \) Copy content Toggle raw display
$37$ \( T - 10506 \) Copy content Toggle raw display
$41$ \( T - 5190 \) Copy content Toggle raw display
$43$ \( T - 17084 \) Copy content Toggle raw display
$47$ \( T + 3168 \) Copy content Toggle raw display
$53$ \( T + 24770 \) Copy content Toggle raw display
$59$ \( T - 17380 \) Copy content Toggle raw display
$61$ \( T + 4366 \) Copy content Toggle raw display
$67$ \( T - 5284 \) Copy content Toggle raw display
$71$ \( T + 8360 \) Copy content Toggle raw display
$73$ \( T - 39466 \) Copy content Toggle raw display
$79$ \( T - 42376 \) Copy content Toggle raw display
$83$ \( T + 61828 \) Copy content Toggle raw display
$89$ \( T - 63078 \) Copy content Toggle raw display
$97$ \( T + 16318 \) Copy content Toggle raw display
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