Properties

Label 2496.1.bd.a
Level $2496$
Weight $1$
Character orbit 2496.bd
Analytic conductor $1.246$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
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(1919,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 2, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2496.1919");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2496.bd (of order \(4\), degree \(2\), 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: \(1.24566627153\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 624)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.105456.1
Artin image: $C_4^2:C_2^2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-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.

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

Character values

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

\(n\) \(703\) \(769\) \(833\) \(1093\)
\(\chi(n)\) \(-1\) \(i\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1919.1
1.00000i
1.00000i
0 1.00000i 0 0 0 −1.00000 1.00000i 0 −1.00000 0
2111.1 0 1.00000i 0 0 0 −1.00000 + 1.00000i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
52.f even 4 1 inner
156.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.1.bd.a 2
3.b odd 2 1 CM 2496.1.bd.a 2
4.b odd 2 1 2496.1.bd.b 2
8.b even 2 1 624.1.bd.a 2
8.d odd 2 1 624.1.bd.b yes 2
12.b even 2 1 2496.1.bd.b 2
13.d odd 4 1 2496.1.bd.b 2
24.f even 2 1 624.1.bd.b yes 2
24.h odd 2 1 624.1.bd.a 2
39.f even 4 1 2496.1.bd.b 2
52.f even 4 1 inner 2496.1.bd.a 2
104.j odd 4 1 624.1.bd.b yes 2
104.m even 4 1 624.1.bd.a 2
156.l odd 4 1 inner 2496.1.bd.a 2
312.w odd 4 1 624.1.bd.a 2
312.y even 4 1 624.1.bd.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.1.bd.a 2 8.b even 2 1
624.1.bd.a 2 24.h odd 2 1
624.1.bd.a 2 104.m even 4 1
624.1.bd.a 2 312.w odd 4 1
624.1.bd.b yes 2 8.d odd 2 1
624.1.bd.b yes 2 24.f even 2 1
624.1.bd.b yes 2 104.j odd 4 1
624.1.bd.b yes 2 312.y even 4 1
2496.1.bd.a 2 1.a even 1 1 trivial
2496.1.bd.a 2 3.b odd 2 1 CM
2496.1.bd.a 2 52.f even 4 1 inner
2496.1.bd.a 2 156.l odd 4 1 inner
2496.1.bd.b 2 4.b odd 2 1
2496.1.bd.b 2 12.b even 2 1
2496.1.bd.b 2 13.d odd 4 1
2496.1.bd.b 2 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 2T_{7} + 2 \) acting on \(S_{1}^{\mathrm{new}}(2496, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$79$ \( T^{2} + 4 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
show more
show less