Properties

Label 816.2.bf.d
Level 816816
Weight 22
Character orbit 816.bf
Analytic conductor 6.5166.516
Analytic rank 00
Dimension 2424
Inner twists 88

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [816,2,Mod(47,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 816=24317 816 = 2^{4} \cdot 3 \cdot 17
Weight: k k == 2 2
Character orbit: [χ][\chi] == 816.bf (of order 44, degree 22, 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: 6.515792804946.51579280494
Analytic rank: 00
Dimension: 2424
Relative dimension: 1212 over Q(i)\Q(i)
Twist minimal: yes
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

qq-expansion

The algebraic qq-expansion of this newform has not been computed, but we have computed the trace expansion.

Tr(f)(q)=\operatorname{Tr}(f)(q) = 24q+24q2196q33+24q4548q5748q61+72q69120q73+72q81+192q85+24q97+O(q100) 24 q + 24 q^{21} - 96 q^{33} + 24 q^{45} - 48 q^{57} - 48 q^{61} + 72 q^{69} - 120 q^{73} + 72 q^{81} + 192 q^{85} + 24 q^{97}+O(q^{100}) Copy content Toggle raw display

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
47.1 0 −1.71292 + 0.256691i 0 −1.13275 + 1.13275i 0 −2.41609 2.41609i 0 2.86822 0.879385i 0
47.2 0 −1.68530 0.399721i 0 3.04809 3.04809i 0 −0.237565 0.237565i 0 2.68045 + 1.34730i 0
47.3 0 −1.51804 0.834000i 0 −1.19416 + 1.19416i 0 3.01763 + 3.01763i 0 1.60889 + 2.53209i 0
47.4 0 −0.834000 1.51804i 0 1.19416 1.19416i 0 −3.01763 3.01763i 0 −1.60889 + 2.53209i 0
47.5 0 −0.399721 1.68530i 0 −3.04809 + 3.04809i 0 0.237565 + 0.237565i 0 −2.68045 + 1.34730i 0
47.6 0 −0.256691 + 1.71292i 0 1.13275 1.13275i 0 −2.41609 2.41609i 0 −2.86822 0.879385i 0
47.7 0 0.256691 1.71292i 0 1.13275 1.13275i 0 2.41609 + 2.41609i 0 −2.86822 0.879385i 0
47.8 0 0.399721 + 1.68530i 0 −3.04809 + 3.04809i 0 −0.237565 0.237565i 0 −2.68045 + 1.34730i 0
47.9 0 0.834000 + 1.51804i 0 1.19416 1.19416i 0 3.01763 + 3.01763i 0 −1.60889 + 2.53209i 0
47.10 0 1.51804 + 0.834000i 0 −1.19416 + 1.19416i 0 −3.01763 3.01763i 0 1.60889 + 2.53209i 0
47.11 0 1.68530 + 0.399721i 0 3.04809 3.04809i 0 0.237565 + 0.237565i 0 2.68045 + 1.34730i 0
47.12 0 1.71292 0.256691i 0 −1.13275 + 1.13275i 0 2.41609 + 2.41609i 0 2.86822 0.879385i 0
191.1 0 −1.71292 0.256691i 0 −1.13275 1.13275i 0 −2.41609 + 2.41609i 0 2.86822 + 0.879385i 0
191.2 0 −1.68530 + 0.399721i 0 3.04809 + 3.04809i 0 −0.237565 + 0.237565i 0 2.68045 1.34730i 0
191.3 0 −1.51804 + 0.834000i 0 −1.19416 1.19416i 0 3.01763 3.01763i 0 1.60889 2.53209i 0
191.4 0 −0.834000 + 1.51804i 0 1.19416 + 1.19416i 0 −3.01763 + 3.01763i 0 −1.60889 2.53209i 0
191.5 0 −0.399721 + 1.68530i 0 −3.04809 3.04809i 0 0.237565 0.237565i 0 −2.68045 1.34730i 0
191.6 0 −0.256691 1.71292i 0 1.13275 + 1.13275i 0 −2.41609 + 2.41609i 0 −2.86822 + 0.879385i 0
191.7 0 0.256691 + 1.71292i 0 1.13275 + 1.13275i 0 2.41609 2.41609i 0 −2.86822 + 0.879385i 0
191.8 0 0.399721 1.68530i 0 −3.04809 3.04809i 0 −0.237565 + 0.237565i 0 −2.68045 1.34730i 0
See all 24 embeddings
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.12
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
17.c even 4 1 inner
51.f odd 4 1 inner
68.f odd 4 1 inner
204.l even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.bf.d 24
3.b odd 2 1 inner 816.2.bf.d 24
4.b odd 2 1 inner 816.2.bf.d 24
12.b even 2 1 inner 816.2.bf.d 24
17.c even 4 1 inner 816.2.bf.d 24
51.f odd 4 1 inner 816.2.bf.d 24
68.f odd 4 1 inner 816.2.bf.d 24
204.l even 4 1 inner 816.2.bf.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
816.2.bf.d 24 1.a even 1 1 trivial
816.2.bf.d 24 3.b odd 2 1 inner
816.2.bf.d 24 4.b odd 2 1 inner
816.2.bf.d 24 12.b even 2 1 inner
816.2.bf.d 24 17.c even 4 1 inner
816.2.bf.d 24 51.f odd 4 1 inner
816.2.bf.d 24 68.f odd 4 1 inner
816.2.bf.d 24 204.l even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(816,[χ])S_{2}^{\mathrm{new}}(816, [\chi]):

T512+360T58+5136T54+18496 T_{5}^{12} + 360T_{5}^{8} + 5136T_{5}^{4} + 18496 Copy content Toggle raw display
T1112+216T118+12816T114+166464 T_{11}^{12} + 216T_{11}^{8} + 12816T_{11}^{4} + 166464 Copy content Toggle raw display