Properties

Label 392.2.a.b
Level $392$
Weight $2$
Character orbit 392.a
Self dual yes
Analytic conductor $3.130$
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] = [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: \(0\)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{3} + 4 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} + 4 q^{5} + q^{9} - 8 q^{15} + 2 q^{17} + 2 q^{19} + 8 q^{23} + 11 q^{25} + 4 q^{27} + 2 q^{29} - 4 q^{31} - 6 q^{37} + 2 q^{41} + 8 q^{43} + 4 q^{45} + 4 q^{47} - 4 q^{51} - 10 q^{53} - 4 q^{57} - 6 q^{59} - 4 q^{61} - 12 q^{67} - 16 q^{69} + 14 q^{73} - 22 q^{75} - 8 q^{79} - 11 q^{81} - 6 q^{83} + 8 q^{85} - 4 q^{87} - 10 q^{89} + 8 q^{93} + 8 q^{95} + 2 q^{97}+O(q^{100}) \) 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 −2.00000 0 4.00000 0 0 0 1.00000 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.b 1
3.b odd 2 1 3528.2.a.b 1
4.b odd 2 1 784.2.a.i 1
5.b even 2 1 9800.2.a.bj 1
7.b odd 2 1 56.2.a.b 1
7.c even 3 2 392.2.i.e 2
7.d odd 6 2 392.2.i.a 2
8.b even 2 1 3136.2.a.w 1
8.d odd 2 1 3136.2.a.c 1
12.b even 2 1 7056.2.a.c 1
21.c even 2 1 504.2.a.h 1
21.g even 6 2 3528.2.s.a 2
21.h odd 6 2 3528.2.s.ba 2
28.d even 2 1 112.2.a.a 1
28.f even 6 2 784.2.i.j 2
28.g odd 6 2 784.2.i.b 2
35.c odd 2 1 1400.2.a.a 1
35.f even 4 2 1400.2.g.b 2
56.e even 2 1 448.2.a.h 1
56.h odd 2 1 448.2.a.c 1
77.b even 2 1 6776.2.a.h 1
84.h odd 2 1 1008.2.a.m 1
91.b odd 2 1 9464.2.a.h 1
112.j even 4 2 1792.2.b.h 2
112.l odd 4 2 1792.2.b.a 2
140.c even 2 1 2800.2.a.bd 1
140.j odd 4 2 2800.2.g.g 2
168.e odd 2 1 4032.2.a.a 1
168.i even 2 1 4032.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 7.b odd 2 1
112.2.a.a 1 28.d even 2 1
392.2.a.b 1 1.a even 1 1 trivial
392.2.i.a 2 7.d odd 6 2
392.2.i.e 2 7.c even 3 2
448.2.a.c 1 56.h odd 2 1
448.2.a.h 1 56.e even 2 1
504.2.a.h 1 21.c even 2 1
784.2.a.i 1 4.b odd 2 1
784.2.i.b 2 28.g odd 6 2
784.2.i.j 2 28.f even 6 2
1008.2.a.m 1 84.h odd 2 1
1400.2.a.a 1 35.c odd 2 1
1400.2.g.b 2 35.f even 4 2
1792.2.b.a 2 112.l odd 4 2
1792.2.b.h 2 112.j even 4 2
2800.2.a.bd 1 140.c even 2 1
2800.2.g.g 2 140.j odd 4 2
3136.2.a.c 1 8.d odd 2 1
3136.2.a.w 1 8.b even 2 1
3528.2.a.b 1 3.b odd 2 1
3528.2.s.a 2 21.g even 6 2
3528.2.s.ba 2 21.h odd 6 2
4032.2.a.a 1 168.e odd 2 1
4032.2.a.d 1 168.i even 2 1
6776.2.a.h 1 77.b even 2 1
7056.2.a.c 1 12.b even 2 1
9464.2.a.h 1 91.b odd 2 1
9800.2.a.bj 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 2 \) 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
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T - 4 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T - 4 \) Copy content Toggle raw display
$53$ \( T + 10 \) Copy content Toggle raw display
$59$ \( T + 6 \) Copy content Toggle raw display
$61$ \( T + 4 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 14 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T + 10 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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