Properties

Label 576.6.a.bi
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 18)
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)
 
\(f(q)\) \(=\) \( q + 96 q^{5} + 148 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 96 q^{5} + 148 q^{7} + 384 q^{11} + 334 q^{13} + 576 q^{17} - 664 q^{19} + 3840 q^{23} + 6091 q^{25} - 96 q^{29} + 4564 q^{31} + 14208 q^{35} - 5798 q^{37} - 6720 q^{41} - 14872 q^{43} + 19200 q^{47} + 5097 q^{49} - 7776 q^{53} + 36864 q^{55} - 13056 q^{59} - 42782 q^{61} + 32064 q^{65} + 36656 q^{67} - 64512 q^{71} - 16810 q^{73} + 56832 q^{77} - 28076 q^{79} - 66432 q^{83} + 55296 q^{85} - 81792 q^{89} + 49432 q^{91} - 63744 q^{95} - 29938 q^{97}+O(q^{100}) \) 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 96.0000 0 148.000 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.bi 1
3.b odd 2 1 576.6.a.b 1
4.b odd 2 1 576.6.a.bh 1
8.b even 2 1 144.6.a.a 1
8.d odd 2 1 18.6.a.a 1
12.b even 2 1 576.6.a.a 1
24.f even 2 1 18.6.a.c yes 1
24.h odd 2 1 144.6.a.l 1
40.e odd 2 1 450.6.a.v 1
40.k even 4 2 450.6.c.m 2
56.e even 2 1 882.6.a.k 1
72.l even 6 2 162.6.c.a 2
72.p odd 6 2 162.6.c.l 2
120.m even 2 1 450.6.a.k 1
120.q odd 4 2 450.6.c.c 2
168.e odd 2 1 882.6.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.a.a 1 8.d odd 2 1
18.6.a.c yes 1 24.f even 2 1
144.6.a.a 1 8.b even 2 1
144.6.a.l 1 24.h odd 2 1
162.6.c.a 2 72.l even 6 2
162.6.c.l 2 72.p odd 6 2
450.6.a.k 1 120.m even 2 1
450.6.a.v 1 40.e odd 2 1
450.6.c.c 2 120.q odd 4 2
450.6.c.m 2 40.k even 4 2
576.6.a.a 1 12.b even 2 1
576.6.a.b 1 3.b odd 2 1
576.6.a.bh 1 4.b odd 2 1
576.6.a.bi 1 1.a even 1 1 trivial
882.6.a.k 1 56.e even 2 1
882.6.a.l 1 168.e odd 2 1

Hecke kernels

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

\( T_{5} - 96 \) Copy content Toggle raw display
\( T_{7} - 148 \) Copy content Toggle raw display
\( T_{11} - 384 \) 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
$5$ \( T - 96 \) Copy content Toggle raw display
$7$ \( T - 148 \) Copy content Toggle raw display
$11$ \( T - 384 \) Copy content Toggle raw display
$13$ \( T - 334 \) Copy content Toggle raw display
$17$ \( T - 576 \) Copy content Toggle raw display
$19$ \( T + 664 \) Copy content Toggle raw display
$23$ \( T - 3840 \) Copy content Toggle raw display
$29$ \( T + 96 \) Copy content Toggle raw display
$31$ \( T - 4564 \) Copy content Toggle raw display
$37$ \( T + 5798 \) Copy content Toggle raw display
$41$ \( T + 6720 \) Copy content Toggle raw display
$43$ \( T + 14872 \) Copy content Toggle raw display
$47$ \( T - 19200 \) Copy content Toggle raw display
$53$ \( T + 7776 \) Copy content Toggle raw display
$59$ \( T + 13056 \) Copy content Toggle raw display
$61$ \( T + 42782 \) Copy content Toggle raw display
$67$ \( T - 36656 \) Copy content Toggle raw display
$71$ \( T + 64512 \) Copy content Toggle raw display
$73$ \( T + 16810 \) Copy content Toggle raw display
$79$ \( T + 28076 \) Copy content Toggle raw display
$83$ \( T + 66432 \) Copy content Toggle raw display
$89$ \( T + 81792 \) Copy content Toggle raw display
$97$ \( T + 29938 \) Copy content Toggle raw display
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