Properties

Label 54.8.a.c
Level $54$
Weight $8$
Character orbit 54.a
Self dual yes
Analytic conductor $16.869$
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] = [54,8,Mod(1,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 54.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: \(16.8687913761\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 8 q^{2} + 64 q^{4} + 312 q^{5} + 323 q^{7} - 512 q^{8} - 2496 q^{10} + 3720 q^{11} - 14179 q^{13} - 2584 q^{14} + 4096 q^{16} + 15912 q^{17} + 22421 q^{19} + 19968 q^{20} - 29760 q^{22} + 57768 q^{23}+ \cdots + 5753712 q^{98}+O(q^{100}) \) 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
−8.00000 0 64.0000 312.000 0 323.000 −512.000 0 −2496.00
\(n\): 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 54.8.a.c 1
3.b odd 2 1 54.8.a.d yes 1
4.b odd 2 1 432.8.a.h 1
9.c even 3 2 162.8.c.g 2
9.d odd 6 2 162.8.c.f 2
12.b even 2 1 432.8.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.8.a.c 1 1.a even 1 1 trivial
54.8.a.d yes 1 3.b odd 2 1
162.8.c.f 2 9.d odd 6 2
162.8.c.g 2 9.c even 3 2
432.8.a.a 1 12.b even 2 1
432.8.a.h 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 312 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(54))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 8 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 312 \) Copy content Toggle raw display
$7$ \( T - 323 \) Copy content Toggle raw display
$11$ \( T - 3720 \) Copy content Toggle raw display
$13$ \( T + 14179 \) Copy content Toggle raw display
$17$ \( T - 15912 \) Copy content Toggle raw display
$19$ \( T - 22421 \) Copy content Toggle raw display
$23$ \( T - 57768 \) Copy content Toggle raw display
$29$ \( T - 166656 \) Copy content Toggle raw display
$31$ \( T - 94820 \) Copy content Toggle raw display
$37$ \( T - 453971 \) Copy content Toggle raw display
$41$ \( T - 627072 \) Copy content Toggle raw display
$43$ \( T + 42472 \) Copy content Toggle raw display
$47$ \( T + 1235256 \) Copy content Toggle raw display
$53$ \( T - 107280 \) Copy content Toggle raw display
$59$ \( T + 2479224 \) Copy content Toggle raw display
$61$ \( T - 2874383 \) Copy content Toggle raw display
$67$ \( T - 1501097 \) Copy content Toggle raw display
$71$ \( T - 4733136 \) Copy content Toggle raw display
$73$ \( T + 85111 \) Copy content Toggle raw display
$79$ \( T + 1180819 \) Copy content Toggle raw display
$83$ \( T + 1116528 \) Copy content Toggle raw display
$89$ \( T - 9368136 \) Copy content Toggle raw display
$97$ \( T + 2039995 \) Copy content Toggle raw display
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