Properties

Label 1340.3.g.a
Level $1340$
Weight $3$
Character orbit 1340.g
Analytic conductor $36.512$
Analytic rank $0$
Dimension $68$
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] = [1340,3,Mod(669,1340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1340, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1340.669");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1340.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: \(36.5123554243\)
Analytic rank: \(0\)
Dimension: \(68\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 208 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 208 q^{9} - 18 q^{15} + 48 q^{19} - 48 q^{21} + 34 q^{25} + 20 q^{29} + 22 q^{35} - 24 q^{39} + 656 q^{49} + 44 q^{55} + 124 q^{59} - 106 q^{65} - 28 q^{71} + 660 q^{81} + 12 q^{89} - 684 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
669.1 0 −5.92367 0 2.53183 + 4.31160i 0 −4.10255 0 26.0899 0
669.2 0 −5.92367 0 2.53183 4.31160i 0 −4.10255 0 26.0899 0
669.3 0 −5.25359 0 −4.88691 + 1.05742i 0 4.79083 0 18.6003 0
669.4 0 −5.25359 0 −4.88691 1.05742i 0 4.79083 0 18.6003 0
669.5 0 −4.80687 0 −0.0438558 4.99981i 0 7.91446 0 14.1060 0
669.6 0 −4.80687 0 −0.0438558 + 4.99981i 0 7.91446 0 14.1060 0
669.7 0 −4.51228 0 3.78719 3.26453i 0 13.1927 0 11.3607 0
669.8 0 −4.51228 0 3.78719 + 3.26453i 0 13.1927 0 11.3607 0
669.9 0 −4.43361 0 −4.17384 2.75300i 0 −7.13124 0 10.6569 0
669.10 0 −4.43361 0 −4.17384 + 2.75300i 0 −7.13124 0 10.6569 0
669.11 0 −4.21016 0 −2.49779 + 4.33140i 0 −11.5732 0 8.72545 0
669.12 0 −4.21016 0 −2.49779 4.33140i 0 −11.5732 0 8.72545 0
669.13 0 −3.95913 0 4.75828 + 1.53581i 0 −6.32946 0 6.67473 0
669.14 0 −3.95913 0 4.75828 1.53581i 0 −6.32946 0 6.67473 0
669.15 0 −3.94848 0 4.78326 + 1.45617i 0 0.855885 0 6.59049 0
669.16 0 −3.94848 0 4.78326 1.45617i 0 0.855885 0 6.59049 0
669.17 0 −3.05585 0 −3.31835 + 3.74012i 0 8.59815 0 0.338244 0
669.18 0 −3.05585 0 −3.31835 3.74012i 0 8.59815 0 0.338244 0
669.19 0 −2.66092 0 0.602251 + 4.96360i 0 −3.70312 0 −1.91951 0
669.20 0 −2.66092 0 0.602251 4.96360i 0 −3.70312 0 −1.91951 0
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 669.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
67.b odd 2 1 inner
335.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1340.3.g.a 68
5.b even 2 1 inner 1340.3.g.a 68
67.b odd 2 1 inner 1340.3.g.a 68
335.d odd 2 1 inner 1340.3.g.a 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1340.3.g.a 68 1.a even 1 1 trivial
1340.3.g.a 68 5.b even 2 1 inner
1340.3.g.a 68 67.b odd 2 1 inner
1340.3.g.a 68 335.d odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1340, [\chi])\).