↑

Properties

Label 3264.2.l.f
Level $3264$
Weight $2$
Character orbit 3264.l
Analytic conductor $26.063$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3264,2,Mod(2209,3264)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3264, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3264.2209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3264 = 2^{6} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3264.l (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: \(26.0631712197\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{3} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{3} + 24 q^{9} + 12 q^{11} + 12 q^{17} + 12 q^{25} + 24 q^{27} + 12 q^{33} - 24 q^{49} + 12 q^{51} + 12 q^{75} + 24 q^{81} - 96 q^{89} - 24 q^{91} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2209.1 0 1.00000 0 −3.70688 0 − 2.81500i 0 1.00000 0
2209.2 0 1.00000 0 −3.70688 0 2.81500i 0 1.00000 0
2209.3 0 1.00000 0 −3.56097 0 − 2.04494i 0 1.00000 0
2209.4 0 1.00000 0 −3.56097 0 2.04494i 0 1.00000 0
2209.5 0 1.00000 0 −1.69618 0 − 3.76398i 0 1.00000 0
2209.6 0 1.00000 0 −1.69618 0 3.76398i 0 1.00000 0
2209.7 0 1.00000 0 −1.33914 0 − 1.04644i 0 1.00000 0
2209.8 0 1.00000 0 −1.33914 0 1.04644i 0 1.00000 0
2209.9 0 1.00000 0 −1.07218 0 4.44099i 0 1.00000 0
2209.10 0 1.00000 0 −1.07218 0 − 4.44099i 0 1.00000 0
2209.11 0 1.00000 0 −0.870997 0 − 0.953387i 0 1.00000 0
2209.12 0 1.00000 0 −0.870997 0 0.953387i 0 1.00000 0
2209.13 0 1.00000 0 0.870997 0 − 0.953387i 0 1.00000 0
2209.14 0 1.00000 0 0.870997 0 0.953387i 0 1.00000 0
2209.15 0 1.00000 0 1.07218 0 4.44099i 0 1.00000 0
2209.16 0 1.00000 0 1.07218 0 − 4.44099i 0 1.00000 0
2209.17 0 1.00000 0 1.33914 0 − 1.04644i 0 1.00000 0
2209.18 0 1.00000 0 1.33914 0 1.04644i 0 1.00000 0
2209.19 0 1.00000 0 1.69618 0 − 3.76398i 0 1.00000 0
2209.20 0 1.00000 0 1.69618 0 3.76398i 0 1.00000 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2209.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
68.d odd 2 1 inner
136.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3264.2.l.f yes 24
4.b odd 2 1 3264.2.l.e ✓ 24
8.b even 2 1 3264.2.l.e ✓ 24
8.d odd 2 1 inner 3264.2.l.f yes 24
17.b even 2 1 3264.2.l.e ✓ 24
68.d odd 2 1 inner 3264.2.l.f yes 24
136.e odd 2 1 3264.2.l.e ✓ 24
136.h even 2 1 inner 3264.2.l.f yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3264.2.l.e ✓ 24 4.b odd 2 1
3264.2.l.e ✓ 24 8.b even 2 1
3264.2.l.e ✓ 24 17.b even 2 1
3264.2.l.e ✓ 24 136.e odd 2 1
3264.2.l.f yes 24 1.a even 1 1 trivial
3264.2.l.f yes 24 8.d odd 2 1 inner
3264.2.l.f yes 24 68.d odd 2 1 inner
3264.2.l.f yes 24 136.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3264, [\chi])\):

\( T_{5}^{12} - 33T_{5}^{10} + 363T_{5}^{8} - 1555T_{5}^{6} + 2976T_{5}^{4} - 2544T_{5}^{2} + 784 \) Copy content Toggle raw display
\( T_{11}^{6} - 3T_{11}^{5} - 21T_{11}^{4} + 43T_{11}^{3} + 36T_{11}^{2} - 48T_{11} - 32 \) Copy content Toggle raw display