Properties

Label 392.2.a.d
Level $392$
Weight $2$
Character orbit 392.a
Self dual yes
Analytic conductor $3.130$
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] = [392,2,Mod(1,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.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: \(3.13013575923\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{5} - 3 q^{9} - 4 q^{11} - 2 q^{13} + 6 q^{17} - 8 q^{19} - q^{25} + 6 q^{29} - 8 q^{31} - 2 q^{37} - 2 q^{41} - 4 q^{43} + 6 q^{45} + 8 q^{47} + 6 q^{53} + 8 q^{55} + 6 q^{61} + 4 q^{65} - 4 q^{67} - 8 q^{71} - 10 q^{73} + 16 q^{79} + 9 q^{81} - 8 q^{83} - 12 q^{85} + 6 q^{89} + 16 q^{95} + 6 q^{97} + 12 q^{99}+O(q^{100}) \) 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
0 0 0 −2.00000 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)

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 392.2.a.d 1
3.b odd 2 1 3528.2.a.x 1
4.b odd 2 1 784.2.a.e 1
5.b even 2 1 9800.2.a.u 1
7.b odd 2 1 56.2.a.a 1
7.c even 3 2 392.2.i.d 2
7.d odd 6 2 392.2.i.c 2
8.b even 2 1 3136.2.a.q 1
8.d odd 2 1 3136.2.a.p 1
12.b even 2 1 7056.2.a.bo 1
21.c even 2 1 504.2.a.c 1
21.g even 6 2 3528.2.s.t 2
21.h odd 6 2 3528.2.s.e 2
28.d even 2 1 112.2.a.b 1
28.f even 6 2 784.2.i.e 2
28.g odd 6 2 784.2.i.g 2
35.c odd 2 1 1400.2.a.g 1
35.f even 4 2 1400.2.g.g 2
56.e even 2 1 448.2.a.e 1
56.h odd 2 1 448.2.a.d 1
77.b even 2 1 6776.2.a.g 1
84.h odd 2 1 1008.2.a.d 1
91.b odd 2 1 9464.2.a.c 1
112.j even 4 2 1792.2.b.d 2
112.l odd 4 2 1792.2.b.i 2
140.c even 2 1 2800.2.a.p 1
140.j odd 4 2 2800.2.g.p 2
168.e odd 2 1 4032.2.a.bk 1
168.i even 2 1 4032.2.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.a 1 7.b odd 2 1
112.2.a.b 1 28.d even 2 1
392.2.a.d 1 1.a even 1 1 trivial
392.2.i.c 2 7.d odd 6 2
392.2.i.d 2 7.c even 3 2
448.2.a.d 1 56.h odd 2 1
448.2.a.e 1 56.e even 2 1
504.2.a.c 1 21.c even 2 1
784.2.a.e 1 4.b odd 2 1
784.2.i.e 2 28.f even 6 2
784.2.i.g 2 28.g odd 6 2
1008.2.a.d 1 84.h odd 2 1
1400.2.a.g 1 35.c odd 2 1
1400.2.g.g 2 35.f even 4 2
1792.2.b.d 2 112.j even 4 2
1792.2.b.i 2 112.l odd 4 2
2800.2.a.p 1 140.c even 2 1
2800.2.g.p 2 140.j odd 4 2
3136.2.a.p 1 8.d odd 2 1
3136.2.a.q 1 8.b even 2 1
3528.2.a.x 1 3.b odd 2 1
3528.2.s.e 2 21.h odd 6 2
3528.2.s.t 2 21.g even 6 2
4032.2.a.bb 1 168.i even 2 1
4032.2.a.bk 1 168.e odd 2 1
6776.2.a.g 1 77.b even 2 1
7056.2.a.bo 1 12.b even 2 1
9464.2.a.c 1 91.b odd 2 1
9800.2.a.u 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(392))\). 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
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T + 8 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 6 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T - 16 \) Copy content Toggle raw display
$83$ \( T + 8 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T - 6 \) Copy content Toggle raw display
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