Properties

Label 496.1.e.a
Level $496$
Weight $1$
Character orbit 496.e
Self dual yes
Analytic conductor $0.248$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -31
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [496,1,Mod(433,496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(496, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("496.433");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 496 = 2^{4} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 496.e (of order \(2\), degree \(1\), not minimal)

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: \(0.247536246266\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.31.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.61504.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{5} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + q^{7} + q^{9} + q^{19} - q^{31} - q^{35} - q^{41} - q^{45} - 2 q^{47} + q^{59} + q^{63} - 2 q^{67} + q^{71} + q^{81} - q^{95} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/496\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\) \(373\)
\(\chi(n)\) \(0\) \(1\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
0
0 0 0 −1.00000 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 496.1.e.a 1
4.b odd 2 1 31.1.b.a 1
8.b even 2 1 1984.1.e.b 1
8.d odd 2 1 1984.1.e.a 1
12.b even 2 1 279.1.d.b 1
20.d odd 2 1 775.1.d.b 1
20.e even 4 2 775.1.c.a 2
28.d even 2 1 1519.1.c.a 1
28.f even 6 2 1519.1.n.a 2
28.g odd 6 2 1519.1.n.b 2
31.b odd 2 1 CM 496.1.e.a 1
36.f odd 6 2 2511.1.m.e 2
36.h even 6 2 2511.1.m.a 2
44.c even 2 1 3751.1.d.b 1
44.g even 10 4 3751.1.t.a 4
44.h odd 10 4 3751.1.t.c 4
124.d even 2 1 31.1.b.a 1
124.g even 6 2 961.1.e.a 2
124.i odd 6 2 961.1.e.a 2
124.j even 10 4 961.1.f.a 4
124.l odd 10 4 961.1.f.a 4
124.n odd 30 8 961.1.h.a 8
124.p even 30 8 961.1.h.a 8
248.b even 2 1 1984.1.e.a 1
248.g odd 2 1 1984.1.e.b 1
372.b odd 2 1 279.1.d.b 1
620.e even 2 1 775.1.d.b 1
620.m odd 4 2 775.1.c.a 2
868.c odd 2 1 1519.1.c.a 1
868.r even 6 2 1519.1.n.b 2
868.bj odd 6 2 1519.1.n.a 2
1116.v even 6 2 2511.1.m.e 2
1116.bk odd 6 2 2511.1.m.a 2
1364.h odd 2 1 3751.1.d.b 1
1364.bc odd 10 4 3751.1.t.a 4
1364.bi even 10 4 3751.1.t.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 4.b odd 2 1
31.1.b.a 1 124.d even 2 1
279.1.d.b 1 12.b even 2 1
279.1.d.b 1 372.b odd 2 1
496.1.e.a 1 1.a even 1 1 trivial
496.1.e.a 1 31.b odd 2 1 CM
775.1.c.a 2 20.e even 4 2
775.1.c.a 2 620.m odd 4 2
775.1.d.b 1 20.d odd 2 1
775.1.d.b 1 620.e even 2 1
961.1.e.a 2 124.g even 6 2
961.1.e.a 2 124.i odd 6 2
961.1.f.a 4 124.j even 10 4
961.1.f.a 4 124.l odd 10 4
961.1.h.a 8 124.n odd 30 8
961.1.h.a 8 124.p even 30 8
1519.1.c.a 1 28.d even 2 1
1519.1.c.a 1 868.c odd 2 1
1519.1.n.a 2 28.f even 6 2
1519.1.n.a 2 868.bj odd 6 2
1519.1.n.b 2 28.g odd 6 2
1519.1.n.b 2 868.r even 6 2
1984.1.e.a 1 8.d odd 2 1
1984.1.e.a 1 248.b even 2 1
1984.1.e.b 1 8.b even 2 1
1984.1.e.b 1 248.g odd 2 1
2511.1.m.a 2 36.h even 6 2
2511.1.m.a 2 1116.bk odd 6 2
2511.1.m.e 2 36.f odd 6 2
2511.1.m.e 2 1116.v even 6 2
3751.1.d.b 1 44.c even 2 1
3751.1.d.b 1 1364.h odd 2 1
3751.1.t.a 4 44.g even 10 4
3751.1.t.a 4 1364.bc odd 10 4
3751.1.t.c 4 44.h odd 10 4
3751.1.t.c 4 1364.bi even 10 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(496, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 1 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T + 1 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 2 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T - 1 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 2 \) Copy content Toggle raw display
$71$ \( T - 1 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 1 \) Copy content Toggle raw display
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