Properties

Label 3136.1.bc.a
Level $3136$
Weight $1$
Character orbit 3136.bc
Analytic conductor $1.565$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3136,1,Mod(913,3136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3136, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3136.913");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3136.bc (of order \(12\), degree \(4\), 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: no
Analytic conductor: \(1.56506787962\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.14336.1
Artin image: $C_4\wr C_2\times C_6$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{48} + \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{5} q^{9} + (\zeta_{12}^{4} - \zeta_{12}) q^{11} - \zeta_{12} q^{25} + (\zeta_{12}^{3} - 1) q^{29} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{37} + ( - \zeta_{12}^{3} - 1) q^{43} + (\zeta_{12}^{4} - \zeta_{12}) q^{53} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{67} + \zeta_{12}^{3} q^{71} - \zeta_{12}^{4} q^{81} + ( - \zeta_{12}^{3} + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{11} - 4 q^{29} + 2 q^{37} - 4 q^{43} - 2 q^{53} + 2 q^{67} + 2 q^{81} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(1\) \(\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
913.1
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 0 0 0 0 0 0 0.866025 0.500000i 0
1489.1 0 0 0 0 0 0 0 −0.866025 0.500000i 0
2481.1 0 0 0 0 0 0 0 −0.866025 + 0.500000i 0
3057.1 0 0 0 0 0 0 0 0.866025 + 0.500000i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
7.c even 3 1 inner
7.d odd 6 1 inner
16.e even 4 1 inner
112.l odd 4 1 inner
112.w even 12 1 inner
112.x odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.1.bc.a 4
4.b odd 2 1 784.1.y.a 4
7.b odd 2 1 CM 3136.1.bc.a 4
7.c even 3 1 448.1.l.a 2
7.c even 3 1 inner 3136.1.bc.a 4
7.d odd 6 1 448.1.l.a 2
7.d odd 6 1 inner 3136.1.bc.a 4
16.e even 4 1 inner 3136.1.bc.a 4
16.f odd 4 1 784.1.y.a 4
28.d even 2 1 784.1.y.a 4
28.f even 6 1 112.1.l.a 2
28.f even 6 1 784.1.y.a 4
28.g odd 6 1 112.1.l.a 2
28.g odd 6 1 784.1.y.a 4
56.j odd 6 1 896.1.l.a 2
56.k odd 6 1 896.1.l.b 2
56.m even 6 1 896.1.l.b 2
56.p even 6 1 896.1.l.a 2
84.j odd 6 1 1008.1.u.b 2
84.n even 6 1 1008.1.u.b 2
112.j even 4 1 784.1.y.a 4
112.l odd 4 1 inner 3136.1.bc.a 4
112.u odd 12 1 112.1.l.a 2
112.u odd 12 1 784.1.y.a 4
112.u odd 12 1 896.1.l.b 2
112.v even 12 1 112.1.l.a 2
112.v even 12 1 784.1.y.a 4
112.v even 12 1 896.1.l.b 2
112.w even 12 1 448.1.l.a 2
112.w even 12 1 896.1.l.a 2
112.w even 12 1 inner 3136.1.bc.a 4
112.x odd 12 1 448.1.l.a 2
112.x odd 12 1 896.1.l.a 2
112.x odd 12 1 inner 3136.1.bc.a 4
140.p odd 6 1 2800.1.z.a 2
140.s even 6 1 2800.1.z.a 2
140.w even 12 1 2800.1.bf.a 2
140.w even 12 1 2800.1.bf.b 2
140.x odd 12 1 2800.1.bf.a 2
140.x odd 12 1 2800.1.bf.b 2
336.br odd 12 1 1008.1.u.b 2
336.bu even 12 1 1008.1.u.b 2
560.ce odd 12 1 2800.1.bf.b 2
560.cf even 12 1 2800.1.bf.a 2
560.co even 12 1 2800.1.z.a 2
560.cs odd 12 1 2800.1.z.a 2
560.da odd 12 1 2800.1.bf.a 2
560.db even 12 1 2800.1.bf.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.1.l.a 2 28.f even 6 1
112.1.l.a 2 28.g odd 6 1
112.1.l.a 2 112.u odd 12 1
112.1.l.a 2 112.v even 12 1
448.1.l.a 2 7.c even 3 1
448.1.l.a 2 7.d odd 6 1
448.1.l.a 2 112.w even 12 1
448.1.l.a 2 112.x odd 12 1
784.1.y.a 4 4.b odd 2 1
784.1.y.a 4 16.f odd 4 1
784.1.y.a 4 28.d even 2 1
784.1.y.a 4 28.f even 6 1
784.1.y.a 4 28.g odd 6 1
784.1.y.a 4 112.j even 4 1
784.1.y.a 4 112.u odd 12 1
784.1.y.a 4 112.v even 12 1
896.1.l.a 2 56.j odd 6 1
896.1.l.a 2 56.p even 6 1
896.1.l.a 2 112.w even 12 1
896.1.l.a 2 112.x odd 12 1
896.1.l.b 2 56.k odd 6 1
896.1.l.b 2 56.m even 6 1
896.1.l.b 2 112.u odd 12 1
896.1.l.b 2 112.v even 12 1
1008.1.u.b 2 84.j odd 6 1
1008.1.u.b 2 84.n even 6 1
1008.1.u.b 2 336.br odd 12 1
1008.1.u.b 2 336.bu even 12 1
2800.1.z.a 2 140.p odd 6 1
2800.1.z.a 2 140.s even 6 1
2800.1.z.a 2 560.co even 12 1
2800.1.z.a 2 560.cs odd 12 1
2800.1.bf.a 2 140.w even 12 1
2800.1.bf.a 2 140.x odd 12 1
2800.1.bf.a 2 560.cf even 12 1
2800.1.bf.a 2 560.da odd 12 1
2800.1.bf.b 2 140.w even 12 1
2800.1.bf.b 2 140.x odd 12 1
2800.1.bf.b 2 560.ce odd 12 1
2800.1.bf.b 2 560.db even 12 1
3136.1.bc.a 4 1.a even 1 1 trivial
3136.1.bc.a 4 7.b odd 2 1 CM
3136.1.bc.a 4 7.c even 3 1 inner
3136.1.bc.a 4 7.d odd 6 1 inner
3136.1.bc.a 4 16.e even 4 1 inner
3136.1.bc.a 4 112.l odd 4 1 inner
3136.1.bc.a 4 112.w even 12 1 inner
3136.1.bc.a 4 112.x odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3136, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$71$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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