↑

Properties

Label 756.2.e.b
Level $756$
Weight $2$
Character orbit 756.e
Analytic conductor $6.037$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [756,2,Mod(323,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.e (of order \(2\), degree \(1\), 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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{4} + 20 q^{10} + 20 q^{16} - 8 q^{22} - 24 q^{25} - 8 q^{28} - 20 q^{34} + 16 q^{37} - 32 q^{40} + 36 q^{46} - 24 q^{49} + 16 q^{52} - 52 q^{58} + 16 q^{61} + 4 q^{64} + 12 q^{70} + 4 q^{82} - 64 q^{85} - 16 q^{88} + 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1 −1.40500 − 0.161188i 0 1.94804 + 0.452937i − 4.14952i 0 − 1.00000i −2.66398 − 0.950375i 0 −0.668852 + 5.83006i
323.2 −1.40500 + 0.161188i 0 1.94804 − 0.452937i 4.14952i 0 1.00000i −2.66398 + 0.950375i 0 −0.668852 − 5.83006i
323.3 −1.38801 − 0.270968i 0 1.85315 + 0.752213i − 1.41649i 0 1.00000i −2.36837 − 1.54622i 0 −0.383823 + 1.96611i
323.4 −1.38801 + 0.270968i 0 1.85315 − 0.752213i 1.41649i 0 − 1.00000i −2.36837 + 1.54622i 0 −0.383823 − 1.96611i
323.5 −1.21808 − 0.718530i 0 0.967429 + 1.75045i 1.77738i 0 − 1.00000i 0.0793484 − 2.82731i 0 1.27710 − 2.16499i
323.6 −1.21808 + 0.718530i 0 0.967429 − 1.75045i − 1.77738i 0 1.00000i 0.0793484 + 2.82731i 0 1.27710 + 2.16499i
323.7 −0.932512 − 1.06321i 0 −0.260844 + 1.98292i − 0.546170i 0 1.00000i 2.35150 − 1.57176i 0 −0.580695 + 0.509310i
323.8 −0.932512 + 1.06321i 0 −0.260844 − 1.98292i 0.546170i 0 − 1.00000i 2.35150 + 1.57176i 0 −0.580695 − 0.509310i
323.9 −0.492782 − 1.32558i 0 −1.51433 + 1.30645i 3.62883i 0 1.00000i 2.47803 + 1.36358i 0 4.81031 − 1.78822i
323.10 −0.492782 + 1.32558i 0 −1.51433 − 1.30645i − 3.62883i 0 − 1.00000i 2.47803 − 1.36358i 0 4.81031 + 1.78822i
323.11 −0.0572568 − 1.41305i 0 −1.99344 + 0.161814i 0.386370i 0 1.00000i 0.342790 + 2.80758i 0 0.545962 − 0.0221223i
323.12 −0.0572568 + 1.41305i 0 −1.99344 − 0.161814i − 0.386370i 0 − 1.00000i 0.342790 − 2.80758i 0 0.545962 + 0.0221223i
323.13 0.0572568 − 1.41305i 0 −1.99344 − 0.161814i 0.386370i 0 − 1.00000i −0.342790 + 2.80758i 0 0.545962 + 0.0221223i
323.14 0.0572568 + 1.41305i 0 −1.99344 + 0.161814i − 0.386370i 0 1.00000i −0.342790 − 2.80758i 0 0.545962 − 0.0221223i
323.15 0.492782 − 1.32558i 0 −1.51433 − 1.30645i 3.62883i 0 − 1.00000i −2.47803 + 1.36358i 0 4.81031 + 1.78822i
323.16 0.492782 + 1.32558i 0 −1.51433 + 1.30645i − 3.62883i 0 1.00000i −2.47803 − 1.36358i 0 4.81031 − 1.78822i
323.17 0.932512 − 1.06321i 0 −0.260844 − 1.98292i − 0.546170i 0 − 1.00000i −2.35150 − 1.57176i 0 −0.580695 − 0.509310i
323.18 0.932512 + 1.06321i 0 −0.260844 + 1.98292i 0.546170i 0 1.00000i −2.35150 + 1.57176i 0 −0.580695 + 0.509310i
323.19 1.21808 − 0.718530i 0 0.967429 − 1.75045i 1.77738i 0 1.00000i −0.0793484 − 2.82731i 0 1.27710 + 2.16499i
323.20 1.21808 + 0.718530i 0 0.967429 + 1.75045i − 1.77738i 0 − 1.00000i −0.0793484 + 2.82731i 0 1.27710 − 2.16499i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.e.b ✓ 24
3.b odd 2 1 inner 756.2.e.b ✓ 24
4.b odd 2 1 inner 756.2.e.b ✓ 24
12.b even 2 1 inner 756.2.e.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.e.b ✓ 24 1.a even 1 1 trivial
756.2.e.b ✓ 24 3.b odd 2 1 inner
756.2.e.b ✓ 24 4.b odd 2 1 inner
756.2.e.b ✓ 24 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 36T_{5}^{10} + 406T_{5}^{8} + 1540T_{5}^{6} + 2065T_{5}^{4} + 704T_{5}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\). Copy content Toggle raw display