Properties

Label 3.8.a.a
Level $3$
Weight $8$
Character orbit 3.a
Self dual yes
Analytic conductor $0.937$
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] = [3,8,Mod(1,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, 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("3.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 3.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: \(0.937155076452\)
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)
 
\(f(q)\) \(=\) \( q + 6 q^{2} - 27 q^{3} - 92 q^{4} + 390 q^{5} - 162 q^{6} - 64 q^{7} - 1320 q^{8} + 729 q^{9} + 2340 q^{10} - 948 q^{11} + 2484 q^{12} - 5098 q^{13} - 384 q^{14} - 10530 q^{15} + 3856 q^{16} + 28386 q^{17}+ \cdots - 691092 q^{99}+O(q^{100}) \) 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
6.00000 −27.0000 −92.0000 390.000 −162.000 −64.0000 −1320.00 729.000 2340.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)

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 3.8.a.a 1
3.b odd 2 1 9.8.a.a 1
4.b odd 2 1 48.8.a.g 1
5.b even 2 1 75.8.a.a 1
5.c odd 4 2 75.8.b.c 2
7.b odd 2 1 147.8.a.b 1
7.c even 3 2 147.8.e.b 2
7.d odd 6 2 147.8.e.a 2
8.b even 2 1 192.8.a.i 1
8.d odd 2 1 192.8.a.a 1
9.c even 3 2 81.8.c.a 2
9.d odd 6 2 81.8.c.c 2
11.b odd 2 1 363.8.a.b 1
12.b even 2 1 144.8.a.b 1
13.b even 2 1 507.8.a.a 1
15.d odd 2 1 225.8.a.i 1
15.e even 4 2 225.8.b.f 2
21.c even 2 1 441.8.a.a 1
24.f even 2 1 576.8.a.x 1
24.h odd 2 1 576.8.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.8.a.a 1 1.a even 1 1 trivial
9.8.a.a 1 3.b odd 2 1
48.8.a.g 1 4.b odd 2 1
75.8.a.a 1 5.b even 2 1
75.8.b.c 2 5.c odd 4 2
81.8.c.a 2 9.c even 3 2
81.8.c.c 2 9.d odd 6 2
144.8.a.b 1 12.b even 2 1
147.8.a.b 1 7.b odd 2 1
147.8.e.a 2 7.d odd 6 2
147.8.e.b 2 7.c even 3 2
192.8.a.a 1 8.d odd 2 1
192.8.a.i 1 8.b even 2 1
225.8.a.i 1 15.d odd 2 1
225.8.b.f 2 15.e even 4 2
363.8.a.b 1 11.b odd 2 1
441.8.a.a 1 21.c even 2 1
507.8.a.a 1 13.b even 2 1
576.8.a.w 1 24.h odd 2 1
576.8.a.x 1 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 6 \) Copy content Toggle raw display
$3$ \( T + 27 \) Copy content Toggle raw display
$5$ \( T - 390 \) Copy content Toggle raw display
$7$ \( T + 64 \) Copy content Toggle raw display
$11$ \( T + 948 \) Copy content Toggle raw display
$13$ \( T + 5098 \) Copy content Toggle raw display
$17$ \( T - 28386 \) Copy content Toggle raw display
$19$ \( T + 8620 \) Copy content Toggle raw display
$23$ \( T + 15288 \) Copy content Toggle raw display
$29$ \( T - 36510 \) Copy content Toggle raw display
$31$ \( T + 276808 \) Copy content Toggle raw display
$37$ \( T - 268526 \) Copy content Toggle raw display
$41$ \( T + 629718 \) Copy content Toggle raw display
$43$ \( T - 685772 \) Copy content Toggle raw display
$47$ \( T - 583296 \) Copy content Toggle raw display
$53$ \( T + 428058 \) Copy content Toggle raw display
$59$ \( T - 1306380 \) Copy content Toggle raw display
$61$ \( T - 300662 \) Copy content Toggle raw display
$67$ \( T + 507244 \) Copy content Toggle raw display
$71$ \( T - 5560632 \) Copy content Toggle raw display
$73$ \( T - 1369082 \) Copy content Toggle raw display
$79$ \( T + 6913720 \) Copy content Toggle raw display
$83$ \( T + 4376748 \) Copy content Toggle raw display
$89$ \( T + 8528310 \) Copy content Toggle raw display
$97$ \( T + 8826814 \) Copy content Toggle raw display
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