Properties

Label 576.6.a.ba
Level $576$
Weight $6$
Character orbit 576.a
Self dual yes
Analytic conductor $92.381$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 38 q^{5} - 120 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 38 q^{5} - 120 q^{7} - 524 q^{11} + 962 q^{13} + 1358 q^{17} - 2284 q^{19} + 2552 q^{23} - 1681 q^{25} + 3966 q^{29} + 2992 q^{31} - 4560 q^{35} - 13206 q^{37} + 15126 q^{41} - 7316 q^{43} - 6960 q^{47} - 2407 q^{49} - 17482 q^{53} - 19912 q^{55} - 33884 q^{59} - 39118 q^{61} + 36556 q^{65} + 32996 q^{67} + 14248 q^{71} - 35990 q^{73} + 62880 q^{77} + 29888 q^{79} + 51884 q^{83} + 51604 q^{85} - 30714 q^{89} - 115440 q^{91} - 86792 q^{95} - 48478 q^{97}+O(q^{100}) \) 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 38.0000 0 −120.000 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.ba 1
3.b odd 2 1 192.6.a.j 1
4.b odd 2 1 576.6.a.bb 1
8.b even 2 1 144.6.a.d 1
8.d odd 2 1 72.6.a.b 1
12.b even 2 1 192.6.a.b 1
24.f even 2 1 24.6.a.c 1
24.h odd 2 1 48.6.a.b 1
48.i odd 4 2 768.6.d.m 2
48.k even 4 2 768.6.d.f 2
120.m even 2 1 600.6.a.a 1
120.q odd 4 2 600.6.f.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.6.a.c 1 24.f even 2 1
48.6.a.b 1 24.h odd 2 1
72.6.a.b 1 8.d odd 2 1
144.6.a.d 1 8.b even 2 1
192.6.a.b 1 12.b even 2 1
192.6.a.j 1 3.b odd 2 1
576.6.a.ba 1 1.a even 1 1 trivial
576.6.a.bb 1 4.b odd 2 1
600.6.a.a 1 120.m even 2 1
600.6.f.h 2 120.q odd 4 2
768.6.d.f 2 48.k even 4 2
768.6.d.m 2 48.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(576))\):

\( T_{5} - 38 \) Copy content Toggle raw display
\( T_{7} + 120 \) Copy content Toggle raw display
\( T_{11} + 524 \) 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
$5$ \( T - 38 \) Copy content Toggle raw display
$7$ \( T + 120 \) Copy content Toggle raw display
$11$ \( T + 524 \) Copy content Toggle raw display
$13$ \( T - 962 \) Copy content Toggle raw display
$17$ \( T - 1358 \) Copy content Toggle raw display
$19$ \( T + 2284 \) Copy content Toggle raw display
$23$ \( T - 2552 \) Copy content Toggle raw display
$29$ \( T - 3966 \) Copy content Toggle raw display
$31$ \( T - 2992 \) Copy content Toggle raw display
$37$ \( T + 13206 \) Copy content Toggle raw display
$41$ \( T - 15126 \) Copy content Toggle raw display
$43$ \( T + 7316 \) Copy content Toggle raw display
$47$ \( T + 6960 \) Copy content Toggle raw display
$53$ \( T + 17482 \) Copy content Toggle raw display
$59$ \( T + 33884 \) Copy content Toggle raw display
$61$ \( T + 39118 \) Copy content Toggle raw display
$67$ \( T - 32996 \) Copy content Toggle raw display
$71$ \( T - 14248 \) Copy content Toggle raw display
$73$ \( T + 35990 \) Copy content Toggle raw display
$79$ \( T - 29888 \) Copy content Toggle raw display
$83$ \( T - 51884 \) Copy content Toggle raw display
$89$ \( T + 30714 \) Copy content Toggle raw display
$97$ \( T + 48478 \) Copy content Toggle raw display
show more
show less