Properties

Label 576.4.a.x
Level $576$
Weight $4$
Character orbit 576.a
Self dual yes
Analytic conductor $33.985$
Analytic rank $1$
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] = [576,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(33.9851001633\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
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)
 
\(f(q)\) \(=\) \( q + 16 q^{5} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 q^{5} + 12 q^{7} - 64 q^{11} - 58 q^{13} - 32 q^{17} - 136 q^{19} - 128 q^{23} + 131 q^{25} - 144 q^{29} - 20 q^{31} + 192 q^{35} + 18 q^{37} + 288 q^{41} - 200 q^{43} + 384 q^{47} - 199 q^{49} + 496 q^{53} - 1024 q^{55} + 128 q^{59} + 458 q^{61} - 928 q^{65} - 496 q^{67} + 512 q^{71} - 602 q^{73} - 768 q^{77} - 1108 q^{79} - 704 q^{83} - 512 q^{85} + 960 q^{89} - 696 q^{91} - 2176 q^{95} + 206 q^{97}+O(q^{100}) \) 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 16.0000 0 12.0000 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 576.4.a.x 1
3.b odd 2 1 576.4.a.d 1
4.b odd 2 1 576.4.a.w 1
8.b even 2 1 144.4.a.a 1
8.d odd 2 1 72.4.a.a 1
12.b even 2 1 576.4.a.c 1
24.f even 2 1 72.4.a.d yes 1
24.h odd 2 1 144.4.a.f 1
40.e odd 2 1 1800.4.a.z 1
40.k even 4 2 1800.4.f.b 2
72.l even 6 2 648.4.i.a 2
72.p odd 6 2 648.4.i.l 2
120.m even 2 1 1800.4.a.ba 1
120.q odd 4 2 1800.4.f.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.a.a 1 8.d odd 2 1
72.4.a.d yes 1 24.f even 2 1
144.4.a.a 1 8.b even 2 1
144.4.a.f 1 24.h odd 2 1
576.4.a.c 1 12.b even 2 1
576.4.a.d 1 3.b odd 2 1
576.4.a.w 1 4.b odd 2 1
576.4.a.x 1 1.a even 1 1 trivial
648.4.i.a 2 72.l even 6 2
648.4.i.l 2 72.p odd 6 2
1800.4.a.z 1 40.e odd 2 1
1800.4.a.ba 1 120.m even 2 1
1800.4.f.b 2 40.k even 4 2
1800.4.f.x 2 120.q odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(576))\):

\( T_{5} - 16 \) Copy content Toggle raw display
\( T_{7} - 12 \) Copy content Toggle raw display
\( T_{11} + 64 \) 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
$5$ \( T - 16 \) Copy content Toggle raw display
$7$ \( T - 12 \) Copy content Toggle raw display
$11$ \( T + 64 \) Copy content Toggle raw display
$13$ \( T + 58 \) Copy content Toggle raw display
$17$ \( T + 32 \) Copy content Toggle raw display
$19$ \( T + 136 \) Copy content Toggle raw display
$23$ \( T + 128 \) Copy content Toggle raw display
$29$ \( T + 144 \) Copy content Toggle raw display
$31$ \( T + 20 \) Copy content Toggle raw display
$37$ \( T - 18 \) Copy content Toggle raw display
$41$ \( T - 288 \) Copy content Toggle raw display
$43$ \( T + 200 \) Copy content Toggle raw display
$47$ \( T - 384 \) Copy content Toggle raw display
$53$ \( T - 496 \) Copy content Toggle raw display
$59$ \( T - 128 \) Copy content Toggle raw display
$61$ \( T - 458 \) Copy content Toggle raw display
$67$ \( T + 496 \) Copy content Toggle raw display
$71$ \( T - 512 \) Copy content Toggle raw display
$73$ \( T + 602 \) Copy content Toggle raw display
$79$ \( T + 1108 \) Copy content Toggle raw display
$83$ \( T + 704 \) Copy content Toggle raw display
$89$ \( T - 960 \) Copy content Toggle raw display
$97$ \( T - 206 \) Copy content Toggle raw display
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