Properties

Label 2496.1.de.c
Level $2496$
Weight $1$
Character orbit 2496.de
Analytic conductor $1.246$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2496,1,Mod(479,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 6, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2496.479");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2496.de (of order \(12\), degree \(4\), 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: \(1.24566627153\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{2} q^{3} + (\zeta_{12}^{5} - 1) q^{7} + \zeta_{12}^{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{2} q^{3} + (\zeta_{12}^{5} - 1) q^{7} + \zeta_{12}^{4} q^{9} - \zeta_{12} q^{13} + (\zeta_{12}^{4} - \zeta_{12}) q^{19} + ( - \zeta_{12}^{2} - \zeta_{12}) q^{21} - \zeta_{12}^{3} q^{25} - q^{27} + ( - \zeta_{12}^{2} + \zeta_{12}) q^{31} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{37} - \zeta_{12}^{3} q^{39} - \zeta_{12} q^{43} + ( - \zeta_{12}^{5} - \zeta_{12}^{4} + 1) q^{49} + ( - \zeta_{12}^{3} - 1) q^{57} - \zeta_{12} q^{61} + ( - \zeta_{12}^{4} - \zeta_{12}^{3}) q^{63} + (\zeta_{12}^{4} + \zeta_{12}^{3}) q^{67} + (\zeta_{12}^{5} + \zeta_{12}^{4}) q^{73} - \zeta_{12}^{5} q^{75} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{79} - \zeta_{12}^{2} q^{81} + (\zeta_{12} + 1) q^{91} + ( - \zeta_{12}^{4} + \zeta_{12}^{3}) q^{93} + (\zeta_{12}^{3} - \zeta_{12}^{2}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 4 q^{7} - 2 q^{9} - 2 q^{19} - 2 q^{21} - 4 q^{27} - 2 q^{31} + 2 q^{37} + 6 q^{49} - 4 q^{57} + 2 q^{63} - 2 q^{67} - 2 q^{73} - 2 q^{81} + 4 q^{91} + 2 q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(769\) \(833\) \(1093\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
479.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0.500000 + 0.866025i 0 0 0 −1.86603 + 0.500000i 0 −0.500000 + 0.866025i 0
1055.1 0 0.500000 + 0.866025i 0 0 0 −0.133975 0.500000i 0 −0.500000 + 0.866025i 0
1631.1 0 0.500000 0.866025i 0 0 0 −1.86603 0.500000i 0 −0.500000 0.866025i 0
2399.1 0 0.500000 0.866025i 0 0 0 −0.133975 + 0.500000i 0 −0.500000 0.866025i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
104.u even 12 1 inner
312.bq odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.1.de.c yes 4
3.b odd 2 1 CM 2496.1.de.c yes 4
4.b odd 2 1 2496.1.de.b yes 4
8.b even 2 1 2496.1.de.a 4
8.d odd 2 1 2496.1.de.d yes 4
12.b even 2 1 2496.1.de.b yes 4
13.f odd 12 1 2496.1.de.d yes 4
24.f even 2 1 2496.1.de.d yes 4
24.h odd 2 1 2496.1.de.a 4
39.k even 12 1 2496.1.de.d yes 4
52.l even 12 1 2496.1.de.a 4
104.u even 12 1 inner 2496.1.de.c yes 4
104.x odd 12 1 2496.1.de.b yes 4
156.v odd 12 1 2496.1.de.a 4
312.bo even 12 1 2496.1.de.b yes 4
312.bq odd 12 1 inner 2496.1.de.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2496.1.de.a 4 8.b even 2 1
2496.1.de.a 4 24.h odd 2 1
2496.1.de.a 4 52.l even 12 1
2496.1.de.a 4 156.v odd 12 1
2496.1.de.b yes 4 4.b odd 2 1
2496.1.de.b yes 4 12.b even 2 1
2496.1.de.b yes 4 104.x odd 12 1
2496.1.de.b yes 4 312.bo even 12 1
2496.1.de.c yes 4 1.a even 1 1 trivial
2496.1.de.c yes 4 3.b odd 2 1 CM
2496.1.de.c yes 4 104.u even 12 1 inner
2496.1.de.c yes 4 312.bq odd 12 1 inner
2496.1.de.d yes 4 8.d odd 2 1
2496.1.de.d yes 4 13.f odd 12 1
2496.1.de.d yes 4 24.f even 2 1
2496.1.de.d yes 4 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2496, [\chi])\):

\( T_{7}^{4} + 4T_{7}^{3} + 5T_{7}^{2} + 2T_{7} + 1 \) Copy content Toggle raw display
\( T_{19}^{4} + 2T_{19}^{3} + 2T_{19}^{2} + 4T_{19} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} + 2 T^{2} - 2 T + 1 \) 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
$43$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} + 2 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$79$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1 \) Copy content Toggle raw display
show more
show less