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Properties

Label 816.2.bf.d
Level $816$
Weight $2$
Character orbit 816.bf
Analytic conductor $6.516$
Analytic rank $0$
Dimension $24$
Inner twists $8$

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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: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.bf (of order \(4\), degree \(2\), 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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

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

Embeddings

For each embedding \(\iota_m\) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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
\(n\): 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 \(S_{2}^{\mathrm{new}}(816, [\chi])\):

\( T_{5}^{12} + 360T_{5}^{8} + 5136T_{5}^{4} + 18496 \) Copy content Toggle raw display
\( T_{11}^{12} + 216T_{11}^{8} + 12816T_{11}^{4} + 166464 \) Copy content Toggle raw display