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Properties

Label 1336.3.g.a
Level $1336$
Weight $3$
Character orbit 1336.g
Analytic conductor $36.403$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1336,3,Mod(1001,1336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1336.1001");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1336.g (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: \(36.4033633185\)
Analytic rank: \(0\)
Dimension: \(84\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 268 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q + 268 q^{9} - 8 q^{11} - 24 q^{19} - 88 q^{21} - 420 q^{25} + 24 q^{27} - 104 q^{29} - 16 q^{31} + 64 q^{33} + 32 q^{47} + 516 q^{49} + 80 q^{57} - 168 q^{61} - 144 q^{63} - 120 q^{65} + 80 q^{75} + 120 q^{77} + 1004 q^{81} + 128 q^{85} - 440 q^{87} - 176 q^{89} + 240 q^{93} - 416 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1001.1 0 −5.64246 0 9.24025i 0 5.39801 0 22.8373 0
1001.2 0 −5.64246 0 − 9.24025i 0 5.39801 0 22.8373 0
1001.3 0 −5.24337 0 − 4.44251i 0 1.55134 0 18.4929 0
1001.4 0 −5.24337 0 4.44251i 0 1.55134 0 18.4929 0
1001.5 0 −5.23740 0 − 3.42645i 0 5.52165 0 18.4303 0
1001.6 0 −5.23740 0 3.42645i 0 5.52165 0 18.4303 0
1001.7 0 −5.16982 0 4.36182i 0 −9.63872 0 17.7271 0
1001.8 0 −5.16982 0 − 4.36182i 0 −9.63872 0 17.7271 0
1001.9 0 −4.98229 0 5.47329i 0 7.97632 0 15.8232 0
1001.10 0 −4.98229 0 − 5.47329i 0 7.97632 0 15.8232 0
1001.11 0 −4.96209 0 2.24534i 0 −9.38817 0 15.6223 0
1001.12 0 −4.96209 0 − 2.24534i 0 −9.38817 0 15.6223 0
1001.13 0 −3.99434 0 − 2.69255i 0 9.38175 0 6.95473 0
1001.14 0 −3.99434 0 2.69255i 0 9.38175 0 6.95473 0
1001.15 0 −3.69298 0 − 6.42967i 0 −8.46692 0 4.63808 0
1001.16 0 −3.69298 0 6.42967i 0 −8.46692 0 4.63808 0
1001.17 0 −3.38555 0 − 3.02144i 0 0.290107 0 2.46197 0
1001.18 0 −3.38555 0 3.02144i 0 0.290107 0 2.46197 0
1001.19 0 −3.19880 0 9.03358i 0 −5.90111 0 1.23235 0
1001.20 0 −3.19880 0 − 9.03358i 0 −5.90111 0 1.23235 0
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1001.84
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
167.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1336.3.g.a ✓ 84
167.b odd 2 1 inner 1336.3.g.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1336.3.g.a ✓ 84 1.a even 1 1 trivial
1336.3.g.a ✓ 84 167.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1336, [\chi])\).