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Properties

Label 1216.3.f.a
Level $1216$
Weight $3$
Character orbit 1216.f
Analytic conductor $33.134$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,3,Mod(799,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.799");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1216.f (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: \(33.1336001462\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{9} - 52 q^{17} - 12 q^{25} + 8 q^{33} - 44 q^{49} - 16 q^{65} - 12 q^{73} + 72 q^{81} + 120 q^{89} + 248 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1 0 −5.25058 0 − 0.631961i 0 8.44898i 0 18.5686 0
799.2 0 −5.25058 0 0.631961i 0 − 8.44898i 0 18.5686 0
799.3 0 −4.45856 0 7.02316i 0 − 1.07537i 0 10.8788 0
799.4 0 −4.45856 0 − 7.02316i 0 1.07537i 0 10.8788 0
799.5 0 −3.34620 0 − 3.40348i 0 − 3.74141i 0 2.19704 0
799.6 0 −3.34620 0 3.40348i 0 3.74141i 0 2.19704 0
799.7 0 −2.36495 0 3.34112i 0 − 11.6288i 0 −3.40699 0
799.8 0 −2.36495 0 − 3.34112i 0 11.6288i 0 −3.40699 0
799.9 0 −1.19515 0 − 1.13969i 0 8.87243i 0 −7.57162 0
799.10 0 −1.19515 0 1.13969i 0 − 8.87243i 0 −7.57162 0
799.11 0 −0.578112 0 8.90113i 0 − 2.12414i 0 −8.66579 0
799.12 0 −0.578112 0 − 8.90113i 0 2.12414i 0 −8.66579 0
799.13 0 0.578112 0 8.90113i 0 2.12414i 0 −8.66579 0
799.14 0 0.578112 0 − 8.90113i 0 − 2.12414i 0 −8.66579 0
799.15 0 1.19515 0 − 1.13969i 0 − 8.87243i 0 −7.57162 0
799.16 0 1.19515 0 1.13969i 0 8.87243i 0 −7.57162 0
799.17 0 2.36495 0 3.34112i 0 11.6288i 0 −3.40699 0
799.18 0 2.36495 0 − 3.34112i 0 − 11.6288i 0 −3.40699 0
799.19 0 3.34620 0 − 3.40348i 0 3.74141i 0 2.19704 0
799.20 0 3.34620 0 3.40348i 0 − 3.74141i 0 2.19704 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 799.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.3.f.a ✓ 24
4.b odd 2 1 inner 1216.3.f.a ✓ 24
8.b even 2 1 inner 1216.3.f.a ✓ 24
8.d odd 2 1 inner 1216.3.f.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.3.f.a ✓ 24 1.a even 1 1 trivial
1216.3.f.a ✓ 24 4.b odd 2 1 inner
1216.3.f.a ✓ 24 8.b even 2 1 inner
1216.3.f.a ✓ 24 8.d odd 2 1 inner