Properties

Label 2496.1.l.c
Level $2496$
Weight $1$
Character orbit 2496.l
Analytic conductor $1.246$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
RM discriminant 156
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2496,1,Mod(1793,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2496.1793");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2496.l (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: no
Analytic conductor: \(1.24566627153\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1248)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.8112.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{8}^{2} q^{3} + (\zeta_{8}^{3} - \zeta_{8}) q^{5} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8}^{2} q^{3} + (\zeta_{8}^{3} - \zeta_{8}) q^{5} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{7} - q^{9} + q^{13} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{15} + (\zeta_{8}^{3} + \zeta_{8}) q^{19} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{21} + \zeta_{8}^{2} q^{23} + q^{25} - \zeta_{8}^{2} q^{27} + (\zeta_{8}^{3} + \zeta_{8}) q^{31} + \zeta_{8}^{2} q^{35} + \zeta_{8}^{2} q^{39} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{41} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{45} - q^{49} + (\zeta_{8}^{3} - \zeta_{8}) q^{57} + (\zeta_{8}^{3} + \zeta_{8}) q^{63} + (\zeta_{8}^{3} - \zeta_{8}) q^{65} + (\zeta_{8}^{3} + \zeta_{8}) q^{67} - 2 q^{69} + \zeta_{8}^{2} q^{75} + q^{81} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{89} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{91} + (\zeta_{8}^{3} - \zeta_{8}) q^{93} - \zeta_{8}^{2} q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} + 4 q^{13} + 4 q^{25} - 4 q^{49} - 8 q^{69} + 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1793.1
0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
0 1.00000i 0 −1.41421 0 1.41421i 0 −1.00000 0
1793.2 0 1.00000i 0 1.41421 0 1.41421i 0 −1.00000 0
1793.3 0 1.00000i 0 −1.41421 0 1.41421i 0 −1.00000 0
1793.4 0 1.00000i 0 1.41421 0 1.41421i 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
156.h even 2 1 RM by \(\Q(\sqrt{39}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner
52.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.1.l.c 4
3.b odd 2 1 inner 2496.1.l.c 4
4.b odd 2 1 inner 2496.1.l.c 4
8.b even 2 1 1248.1.l.a 4
8.d odd 2 1 1248.1.l.a 4
12.b even 2 1 inner 2496.1.l.c 4
13.b even 2 1 inner 2496.1.l.c 4
24.f even 2 1 1248.1.l.a 4
24.h odd 2 1 1248.1.l.a 4
39.d odd 2 1 inner 2496.1.l.c 4
52.b odd 2 1 inner 2496.1.l.c 4
104.e even 2 1 1248.1.l.a 4
104.h odd 2 1 1248.1.l.a 4
156.h even 2 1 RM 2496.1.l.c 4
312.b odd 2 1 1248.1.l.a 4
312.h even 2 1 1248.1.l.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.1.l.a 4 8.b even 2 1
1248.1.l.a 4 8.d odd 2 1
1248.1.l.a 4 24.f even 2 1
1248.1.l.a 4 24.h odd 2 1
1248.1.l.a 4 104.e even 2 1
1248.1.l.a 4 104.h odd 2 1
1248.1.l.a 4 312.b odd 2 1
1248.1.l.a 4 312.h even 2 1
2496.1.l.c 4 1.a even 1 1 trivial
2496.1.l.c 4 3.b odd 2 1 inner
2496.1.l.c 4 4.b odd 2 1 inner
2496.1.l.c 4 12.b even 2 1 inner
2496.1.l.c 4 13.b even 2 1 inner
2496.1.l.c 4 39.d odd 2 1 inner
2496.1.l.c 4 52.b odd 2 1 inner
2496.1.l.c 4 156.h even 2 1 RM

Hecke kernels

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

\( T_{5}^{2} - 2 \) Copy content Toggle raw display
\( T_{43} \) Copy content Toggle raw display

Hecke characteristic polynomials

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