Properties

Label 2496.1.bi.a
Level $2496$
Weight $1$
Character orbit 2496.bi
Analytic conductor $1.246$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -39
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(1169,2496)]
 
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([0, 3, 2, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2496.1169");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2496.bi (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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 624)
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.3234424946688.6

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{16}^{2} q^{3} + (\zeta_{16}^{3} - \zeta_{16}) q^{5} + \zeta_{16}^{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{16}^{2} q^{3} + (\zeta_{16}^{3} - \zeta_{16}) q^{5} + \zeta_{16}^{4} q^{9} + ( - \zeta_{16}^{7} - \zeta_{16}^{5}) q^{11} + \zeta_{16}^{2} q^{13} + (\zeta_{16}^{5} - \zeta_{16}^{3}) q^{15} + (\zeta_{16}^{6} - \zeta_{16}^{4} + \zeta_{16}^{2}) q^{25} + \zeta_{16}^{6} q^{27} + ( - \zeta_{16}^{7} + \zeta_{16}) q^{33} + \zeta_{16}^{4} q^{39} + (\zeta_{16}^{7} + \zeta_{16}) q^{41} + ( - \zeta_{16}^{4} + 1) q^{43} + (\zeta_{16}^{7} - \zeta_{16}^{5}) q^{45} + (\zeta_{16}^{7} - \zeta_{16}) q^{47} - q^{49} + (\zeta_{16}^{6} + \zeta_{16}^{2}) q^{55} + ( - \zeta_{16}^{3} + \zeta_{16}) q^{59} + (\zeta_{16}^{5} - \zeta_{16}^{3}) q^{65} + (\zeta_{16}^{5} + \zeta_{16}^{3}) q^{71} + ( - \zeta_{16}^{6} + \zeta_{16}^{4} - 1) q^{75} + ( - \zeta_{16}^{6} + \zeta_{16}^{2}) q^{79} - q^{81} + (\zeta_{16}^{7} - \zeta_{16}^{5}) q^{83} + (\zeta_{16}^{5} + \zeta_{16}^{3}) q^{89} + (\zeta_{16}^{3} + \zeta_{16}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{43} - 8 q^{49} - 8 q^{75} - 8 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(703\) \(769\) \(833\) \(1093\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-\zeta_{16}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1169.1
0.382683 0.923880i
−0.382683 + 0.923880i
0.923880 + 0.382683i
−0.923880 0.382683i
0.382683 + 0.923880i
−0.382683 0.923880i
0.923880 0.382683i
−0.923880 + 0.382683i
0 −0.707107 0.707107i 0 −1.30656 + 1.30656i 0 0 0 1.00000i 0
1169.2 0 −0.707107 0.707107i 0 1.30656 1.30656i 0 0 0 1.00000i 0
1169.3 0 0.707107 + 0.707107i 0 −0.541196 + 0.541196i 0 0 0 1.00000i 0
1169.4 0 0.707107 + 0.707107i 0 0.541196 0.541196i 0 0 0 1.00000i 0
2417.1 0 −0.707107 + 0.707107i 0 −1.30656 1.30656i 0 0 0 1.00000i 0
2417.2 0 −0.707107 + 0.707107i 0 1.30656 + 1.30656i 0 0 0 1.00000i 0
2417.3 0 0.707107 0.707107i 0 −0.541196 0.541196i 0 0 0 1.00000i 0
2417.4 0 0.707107 0.707107i 0 0.541196 + 0.541196i 0 0 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1169.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
13.b even 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner
208.p even 4 1 inner
624.bi 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.bi.a 8
3.b odd 2 1 inner 2496.1.bi.a 8
4.b odd 2 1 624.1.bi.a 8
12.b even 2 1 624.1.bi.a 8
13.b even 2 1 inner 2496.1.bi.a 8
16.e even 4 1 inner 2496.1.bi.a 8
16.f odd 4 1 624.1.bi.a 8
39.d odd 2 1 CM 2496.1.bi.a 8
48.i odd 4 1 inner 2496.1.bi.a 8
48.k even 4 1 624.1.bi.a 8
52.b odd 2 1 624.1.bi.a 8
156.h even 2 1 624.1.bi.a 8
208.o odd 4 1 624.1.bi.a 8
208.p even 4 1 inner 2496.1.bi.a 8
624.v even 4 1 624.1.bi.a 8
624.bi odd 4 1 inner 2496.1.bi.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.1.bi.a 8 4.b odd 2 1
624.1.bi.a 8 12.b even 2 1
624.1.bi.a 8 16.f odd 4 1
624.1.bi.a 8 48.k even 4 1
624.1.bi.a 8 52.b odd 2 1
624.1.bi.a 8 156.h even 2 1
624.1.bi.a 8 208.o odd 4 1
624.1.bi.a 8 624.v even 4 1
2496.1.bi.a 8 1.a even 1 1 trivial
2496.1.bi.a 8 3.b odd 2 1 inner
2496.1.bi.a 8 13.b even 2 1 inner
2496.1.bi.a 8 16.e even 4 1 inner
2496.1.bi.a 8 39.d odd 2 1 CM
2496.1.bi.a 8 48.i odd 4 1 inner
2496.1.bi.a 8 208.p even 4 1 inner
2496.1.bi.a 8 624.bi odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2496, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$13$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$89$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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