↑

Properties

Label 1264.2.n.i
Level $1264$
Weight $2$
Character orbit 1264.n
Analytic conductor $10.093$
Analytic rank $0$
Dimension $28$
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] = [1264,2,Mod(735,1264)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1264, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("1264.735");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1264 = 2^{4} \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1264.n (of order \(6\), 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: \(10.0930908155\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 20 q^{9} - 8 q^{13} + 32 q^{21} - 14 q^{25} + 30 q^{37} - 6 q^{45} - 12 q^{49} - 18 q^{53} - 48 q^{65} + 22 q^{73} - 66 q^{77} - 14 q^{81} + 60 q^{89} - 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
735.1 0 −1.60192 + 2.77460i 0 −1.45498 − 2.52010i 0 0.195200 + 0.338096i 0 −3.63228 − 6.29129i 0
735.2 0 −1.30739 + 2.26447i 0 0.0495497 + 0.0858226i 0 0.841498 + 1.45752i 0 −1.91856 − 3.32304i 0
735.3 0 −1.29992 + 2.25154i 0 2.04906 + 3.54908i 0 −1.22355 − 2.11925i 0 −1.87961 − 3.25558i 0
735.4 0 −1.03863 + 1.79896i 0 −0.901120 − 1.56079i 0 −2.58598 − 4.47905i 0 −0.657515 − 1.13885i 0
735.5 0 −0.799764 + 1.38523i 0 1.24399 + 2.15465i 0 0.627964 + 1.08766i 0 0.220755 + 0.382358i 0
735.6 0 −0.214059 + 0.370761i 0 0.323998 + 0.561181i 0 2.03778 + 3.52953i 0 1.40836 + 2.43935i 0
735.7 0 −0.143444 + 0.248452i 0 −1.31050 − 2.26985i 0 −0.522039 − 0.904199i 0 1.45885 + 2.52680i 0
735.8 0 0.143444 − 0.248452i 0 −1.31050 − 2.26985i 0 0.522039 + 0.904199i 0 1.45885 + 2.52680i 0
735.9 0 0.214059 − 0.370761i 0 0.323998 + 0.561181i 0 −2.03778 − 3.52953i 0 1.40836 + 2.43935i 0
735.10 0 0.799764 − 1.38523i 0 1.24399 + 2.15465i 0 −0.627964 − 1.08766i 0 0.220755 + 0.382358i 0
735.11 0 1.03863 − 1.79896i 0 −0.901120 − 1.56079i 0 2.58598 + 4.47905i 0 −0.657515 − 1.13885i 0
735.12 0 1.29992 − 2.25154i 0 2.04906 + 3.54908i 0 1.22355 + 2.11925i 0 −1.87961 − 3.25558i 0
735.13 0 1.30739 − 2.26447i 0 0.0495497 + 0.0858226i 0 −0.841498 − 1.45752i 0 −1.91856 − 3.32304i 0
735.14 0 1.60192 − 2.77460i 0 −1.45498 − 2.52010i 0 −0.195200 − 0.338096i 0 −3.63228 − 6.29129i 0
767.1 0 −1.60192 − 2.77460i 0 −1.45498 + 2.52010i 0 0.195200 − 0.338096i 0 −3.63228 + 6.29129i 0
767.2 0 −1.30739 − 2.26447i 0 0.0495497 − 0.0858226i 0 0.841498 − 1.45752i 0 −1.91856 + 3.32304i 0
767.3 0 −1.29992 − 2.25154i 0 2.04906 − 3.54908i 0 −1.22355 + 2.11925i 0 −1.87961 + 3.25558i 0
767.4 0 −1.03863 − 1.79896i 0 −0.901120 + 1.56079i 0 −2.58598 + 4.47905i 0 −0.657515 + 1.13885i 0
767.5 0 −0.799764 − 1.38523i 0 1.24399 − 2.15465i 0 0.627964 − 1.08766i 0 0.220755 − 0.382358i 0
767.6 0 −0.214059 − 0.370761i 0 0.323998 − 0.561181i 0 2.03778 − 3.52953i 0 1.40836 − 2.43935i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 735.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
79.d odd 6 1 inner
316.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1264.2.n.i ✓ 28
4.b odd 2 1 inner 1264.2.n.i ✓ 28
79.d odd 6 1 inner 1264.2.n.i ✓ 28
316.f even 6 1 inner 1264.2.n.i ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1264.2.n.i ✓ 28 1.a even 1 1 trivial
1264.2.n.i ✓ 28 4.b odd 2 1 inner
1264.2.n.i ✓ 28 79.d odd 6 1 inner
1264.2.n.i ✓ 28 316.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1264, [\chi])\):

\( T_{3}^{28} + 31 T_{3}^{26} + 592 T_{3}^{24} + 7217 T_{3}^{22} + 64868 T_{3}^{20} + 422813 T_{3}^{18} + \cdots + 6241 \) Copy content Toggle raw display
\( T_{11}^{28} - 96 T_{11}^{26} + 5562 T_{11}^{24} - 209898 T_{11}^{22} + 5863968 T_{11}^{20} + \cdots + 230328005625 \) Copy content Toggle raw display