Properties

Label 768.1.e.a
Level $768$
Weight $1$
Character orbit 768.e
Self dual yes
Analytic conductor $0.383$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -3, -8, 24
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,1,Mod(257,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.257");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 768.e (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: yes
Analytic conductor: \(0.383281929702\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 192)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{-3})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.6144.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{3} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{9} + 2 q^{19} + q^{25} - q^{27} + 2 q^{43} - q^{49} - 2 q^{57} - 2 q^{67} - 2 q^{73} - q^{75} + q^{81} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(0\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0
0 −1.00000 0 0 0 0 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 \(\Q(\sqrt{-3}) \)
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
24.f even 2 1 RM by \(\Q(\sqrt{6}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.1.e.a 1
3.b odd 2 1 CM 768.1.e.a 1
4.b odd 2 1 768.1.e.b 1
8.b even 2 1 768.1.e.b 1
8.d odd 2 1 CM 768.1.e.a 1
12.b even 2 1 768.1.e.b 1
16.e even 4 2 192.1.h.a 2
16.f odd 4 2 192.1.h.a 2
24.f even 2 1 RM 768.1.e.a 1
24.h odd 2 1 768.1.e.b 1
32.g even 8 4 3072.1.i.g 4
32.h odd 8 4 3072.1.i.g 4
48.i odd 4 2 192.1.h.a 2
48.k even 4 2 192.1.h.a 2
96.o even 8 4 3072.1.i.g 4
96.p odd 8 4 3072.1.i.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.1.h.a 2 16.e even 4 2
192.1.h.a 2 16.f odd 4 2
192.1.h.a 2 48.i odd 4 2
192.1.h.a 2 48.k even 4 2
768.1.e.a 1 1.a even 1 1 trivial
768.1.e.a 1 3.b odd 2 1 CM
768.1.e.a 1 8.d odd 2 1 CM
768.1.e.a 1 24.f even 2 1 RM
768.1.e.b 1 4.b odd 2 1
768.1.e.b 1 8.b even 2 1
768.1.e.b 1 12.b even 2 1
768.1.e.b 1 24.h odd 2 1
3072.1.i.g 4 32.g even 8 4
3072.1.i.g 4 32.h odd 8 4
3072.1.i.g 4 96.o even 8 4
3072.1.i.g 4 96.p odd 8 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(768, [\chi])\):

\( T_{11} \) Copy content Toggle raw display
\( T_{19} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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