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Properties

Label 816.2.bf.e
Level $816$
Weight $2$
Character orbit 816.bf
Analytic conductor $6.516$
Analytic rank $0$
Dimension $24$
CM no
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{13} - 24 q^{21} + 16 q^{33} + 8 q^{37} - 24 q^{45} - 8 q^{57} - 56 q^{61} - 16 q^{69} - 64 q^{73} + 8 q^{81} - 40 q^{85} + 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −1.68082 + 0.418161i 0 −2.43510 + 2.43510i 0 3.07469 + 3.07469i 0 2.65028 − 1.40570i 0
47.2 0 −1.61358 − 0.629575i 0 0.583363 − 0.583363i 0 −1.30857 − 1.30857i 0 2.20727 + 2.03174i 0
47.3 0 −1.49173 + 0.880198i 0 1.49331 − 1.49331i 0 −0.913206 − 0.913206i 0 1.45050 − 2.62603i 0
47.4 0 −0.880198 + 1.49173i 0 −1.49331 + 1.49331i 0 −0.913206 − 0.913206i 0 −1.45050 − 2.62603i 0
47.5 0 −0.629575 − 1.61358i 0 −0.583363 + 0.583363i 0 1.30857 + 1.30857i 0 −2.20727 + 2.03174i 0
47.6 0 −0.418161 + 1.68082i 0 2.43510 − 2.43510i 0 3.07469 + 3.07469i 0 −2.65028 − 1.40570i 0
47.7 0 0.418161 − 1.68082i 0 2.43510 − 2.43510i 0 −3.07469 − 3.07469i 0 −2.65028 − 1.40570i 0
47.8 0 0.629575 + 1.61358i 0 −0.583363 + 0.583363i 0 −1.30857 − 1.30857i 0 −2.20727 + 2.03174i 0
47.9 0 0.880198 − 1.49173i 0 −1.49331 + 1.49331i 0 0.913206 + 0.913206i 0 −1.45050 − 2.62603i 0
47.10 0 1.49173 − 0.880198i 0 1.49331 − 1.49331i 0 0.913206 + 0.913206i 0 1.45050 − 2.62603i 0
47.11 0 1.61358 + 0.629575i 0 0.583363 − 0.583363i 0 1.30857 + 1.30857i 0 2.20727 + 2.03174i 0
47.12 0 1.68082 − 0.418161i 0 −2.43510 + 2.43510i 0 −3.07469 − 3.07469i 0 2.65028 − 1.40570i 0
191.1 0 −1.68082 − 0.418161i 0 −2.43510 − 2.43510i 0 3.07469 − 3.07469i 0 2.65028 + 1.40570i 0
191.2 0 −1.61358 + 0.629575i 0 0.583363 + 0.583363i 0 −1.30857 + 1.30857i 0 2.20727 − 2.03174i 0
191.3 0 −1.49173 − 0.880198i 0 1.49331 + 1.49331i 0 −0.913206 + 0.913206i 0 1.45050 + 2.62603i 0
191.4 0 −0.880198 − 1.49173i 0 −1.49331 − 1.49331i 0 −0.913206 + 0.913206i 0 −1.45050 + 2.62603i 0
191.5 0 −0.629575 + 1.61358i 0 −0.583363 − 0.583363i 0 1.30857 − 1.30857i 0 −2.20727 − 2.03174i 0
191.6 0 −0.418161 − 1.68082i 0 2.43510 + 2.43510i 0 3.07469 − 3.07469i 0 −2.65028 + 1.40570i 0
191.7 0 0.418161 + 1.68082i 0 2.43510 + 2.43510i 0 −3.07469 + 3.07469i 0 −2.65028 + 1.40570i 0
191.8 0 0.629575 − 1.61358i 0 −0.583363 − 0.583363i 0 −1.30857 + 1.30857i 0 −2.20727 − 2.03174i 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.e ✓ 24
3.b odd 2 1 inner 816.2.bf.e ✓ 24
4.b odd 2 1 inner 816.2.bf.e ✓ 24
12.b even 2 1 inner 816.2.bf.e ✓ 24
17.c even 4 1 inner 816.2.bf.e ✓ 24
51.f odd 4 1 inner 816.2.bf.e ✓ 24
68.f odd 4 1 inner 816.2.bf.e ✓ 24
204.l even 4 1 inner 816.2.bf.e ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
816.2.bf.e ✓ 24 1.a even 1 1 trivial
816.2.bf.e ✓ 24 3.b odd 2 1 inner
816.2.bf.e ✓ 24 4.b odd 2 1 inner
816.2.bf.e ✓ 24 12.b even 2 1 inner
816.2.bf.e ✓ 24 17.c even 4 1 inner
816.2.bf.e ✓ 24 51.f odd 4 1 inner
816.2.bf.e ✓ 24 68.f odd 4 1 inner
816.2.bf.e ✓ 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 \(S_{2}^{\mathrm{new}}(816, [\chi])\):

\( T_{5}^{12} + 161T_{5}^{8} + 2872T_{5}^{4} + 1296 \) Copy content Toggle raw display
\( T_{11}^{12} + 1369T_{11}^{8} + 12036T_{11}^{4} + 22500 \) Copy content Toggle raw display