↑

Properties

Label 585.2.bs.c
Level $585$
Weight $2$
Character orbit 585.bs
Analytic conductor $4.671$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [585,2,Mod(289,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bs (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: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 20 q^{4} - 6 q^{10} - 28 q^{16} - 8 q^{19} + 28 q^{25} + 8 q^{31} - 8 q^{34} - 20 q^{40} - 8 q^{46} + 44 q^{49} + 20 q^{55} - 56 q^{61} - 136 q^{64} - 80 q^{70} + 88 q^{76} - 72 q^{79} - 50 q^{85} - 28 q^{91} + 48 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 −2.32723 + 1.34363i 0 2.61067 − 4.52182i −2.23433 + 0.0881535i 0 −3.14272 − 1.81445i 8.65659i 0 5.08136 − 3.20726i
289.2 −2.32723 + 1.34363i 0 2.61067 − 4.52182i 2.23433 + 0.0881535i 0 3.14272 + 1.81445i 8.65659i 0 −5.31825 + 2.79696i
289.3 −1.57126 + 0.907167i 0 0.645904 − 1.11874i −1.19685 + 1.88880i 0 4.06197 + 2.34518i − 1.28490i 0 0.167109 − 4.05353i
289.4 −1.57126 + 0.907167i 0 0.645904 − 1.11874i 1.19685 + 1.88880i 0 −4.06197 − 2.34518i − 1.28490i 0 −3.59402 − 1.88204i
289.5 −1.35625 + 0.783029i 0 0.226269 − 0.391909i −1.90777 − 1.16637i 0 0.331025 + 0.191118i − 2.42342i 0 3.50070 + 0.0880529i
289.6 −1.35625 + 0.783029i 0 0.226269 − 0.391909i 1.90777 − 1.16637i 0 −0.331025 − 0.191118i − 2.42342i 0 −1.67410 + 3.07573i
289.7 −0.160403 + 0.0926085i 0 −0.982847 + 1.70234i −1.29836 − 1.82051i 0 1.66258 + 0.959889i − 0.734514i 0 0.376855 + 0.171775i
289.8 −0.160403 + 0.0926085i 0 −0.982847 + 1.70234i 1.29836 − 1.82051i 0 −1.66258 − 0.959889i − 0.734514i 0 −0.0396663 + 0.412254i
289.9 0.160403 − 0.0926085i 0 −0.982847 + 1.70234i −1.29836 + 1.82051i 0 −1.66258 − 0.959889i 0.734514i 0 −0.0396663 + 0.412254i
289.10 0.160403 − 0.0926085i 0 −0.982847 + 1.70234i 1.29836 + 1.82051i 0 1.66258 + 0.959889i 0.734514i 0 0.376855 + 0.171775i
289.11 1.35625 − 0.783029i 0 0.226269 − 0.391909i −1.90777 + 1.16637i 0 −0.331025 − 0.191118i 2.42342i 0 −1.67410 + 3.07573i
289.12 1.35625 − 0.783029i 0 0.226269 − 0.391909i 1.90777 + 1.16637i 0 0.331025 + 0.191118i 2.42342i 0 3.50070 + 0.0880529i
289.13 1.57126 − 0.907167i 0 0.645904 − 1.11874i −1.19685 − 1.88880i 0 −4.06197 − 2.34518i 1.28490i 0 −3.59402 − 1.88204i
289.14 1.57126 − 0.907167i 0 0.645904 − 1.11874i 1.19685 − 1.88880i 0 4.06197 + 2.34518i 1.28490i 0 0.167109 − 4.05353i
289.15 2.32723 − 1.34363i 0 2.61067 − 4.52182i −2.23433 − 0.0881535i 0 3.14272 + 1.81445i − 8.65659i 0 −5.31825 + 2.79696i
289.16 2.32723 − 1.34363i 0 2.61067 − 4.52182i 2.23433 − 0.0881535i 0 −3.14272 − 1.81445i − 8.65659i 0 5.08136 − 3.20726i
334.1 −2.32723 − 1.34363i 0 2.61067 + 4.52182i −2.23433 − 0.0881535i 0 −3.14272 + 1.81445i − 8.65659i 0 5.08136 + 3.20726i
334.2 −2.32723 − 1.34363i 0 2.61067 + 4.52182i 2.23433 − 0.0881535i 0 3.14272 − 1.81445i − 8.65659i 0 −5.31825 − 2.79696i
334.3 −1.57126 − 0.907167i 0 0.645904 + 1.11874i −1.19685 − 1.88880i 0 4.06197 − 2.34518i 1.28490i 0 0.167109 + 4.05353i
334.4 −1.57126 − 0.907167i 0 0.645904 + 1.11874i 1.19685 − 1.88880i 0 −4.06197 + 2.34518i 1.28490i 0 −3.59402 + 1.88204i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
13.c even 3 1 inner
15.d odd 2 1 inner
39.i odd 6 1 inner
65.n even 6 1 inner
195.x odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.bs.c ✓ 32
3.b odd 2 1 inner 585.2.bs.c ✓ 32
5.b even 2 1 inner 585.2.bs.c ✓ 32
13.c even 3 1 inner 585.2.bs.c ✓ 32
15.d odd 2 1 inner 585.2.bs.c ✓ 32
39.i odd 6 1 inner 585.2.bs.c ✓ 32
65.n even 6 1 inner 585.2.bs.c ✓ 32
195.x odd 6 1 inner 585.2.bs.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.bs.c ✓ 32 1.a even 1 1 trivial
585.2.bs.c ✓ 32 3.b odd 2 1 inner
585.2.bs.c ✓ 32 5.b even 2 1 inner
585.2.bs.c ✓ 32 13.c even 3 1 inner
585.2.bs.c ✓ 32 15.d odd 2 1 inner
585.2.bs.c ✓ 32 39.i odd 6 1 inner
585.2.bs.c ✓ 32 65.n even 6 1 inner
585.2.bs.c ✓ 32 195.x odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 13T_{2}^{14} + 119T_{2}^{12} - 530T_{2}^{10} + 1718T_{2}^{8} - 2948T_{2}^{6} + 3500T_{2}^{4} - 120T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display