Properties

Label 585.2.dn.a
Level $585$
Weight $2$
Character orbit 585.dn
Analytic conductor $4.671$
Analytic rank $0$
Dimension $112$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [585,2,Mod(107,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.dn (of order \(12\), degree \(4\), 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: \(112\)
Relative dimension: \(28\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 16 q^{10} + 20 q^{13} + 56 q^{16} - 16 q^{22} - 96 q^{31} - 12 q^{37} + 88 q^{40} + 24 q^{43} - 16 q^{46} + 16 q^{52} + 32 q^{55} + 52 q^{58} - 16 q^{61} + 96 q^{67} - 144 q^{70} - 136 q^{73} + 64 q^{76} + 124 q^{82} + 76 q^{85} - 96 q^{88} - 160 q^{91} + 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −2.55314 0.684111i 0 4.31845 + 2.49326i −2.20022 + 0.398805i 0 −1.65127 + 0.442458i −5.58188 5.58188i 0 5.89028 + 0.486988i
107.2 −2.51369 0.673541i 0 4.13292 + 2.38614i 1.63719 + 1.52302i 0 3.60055 0.964763i −5.10141 5.10141i 0 −3.08957 4.93112i
107.3 −2.49890 0.669579i 0 4.06414 + 2.34643i 0.377855 2.20391i 0 −4.99557 + 1.33856i −4.92612 4.92612i 0 −2.41992 + 5.25436i
107.4 −2.05417 0.550414i 0 2.18462 + 1.26129i −1.97089 1.05622i 0 2.41543 0.647213i −0.785838 0.785838i 0 3.46719 + 3.25446i
107.5 −1.99171 0.533677i 0 1.95005 + 1.12586i −0.717853 2.11771i 0 1.51879 0.406959i −0.367024 0.367024i 0 0.299584 + 4.60096i
107.6 −1.94192 0.520335i 0 1.76824 + 1.02090i 0.268423 + 2.21990i 0 1.87543 0.502520i −0.0594080 0.0594080i 0 0.633836 4.45053i
107.7 −1.46053 0.391348i 0 0.247943 + 0.143150i 1.96992 + 1.05802i 0 −2.73983 + 0.734135i 1.83226 + 1.83226i 0 −2.46308 2.31619i
107.8 −1.45612 0.390166i 0 0.236005 + 0.136258i −1.32895 + 1.79830i 0 −0.804080 + 0.215453i 1.84142 + 1.84142i 0 2.63675 2.10004i
107.9 −1.27023 0.340357i 0 −0.234407 0.135335i 0.422581 2.19577i 0 −2.52240 + 0.675874i 2.11144 + 2.11144i 0 −1.28412 + 2.64531i
107.10 −1.04244 0.279321i 0 −0.723390 0.417649i 1.45730 1.69596i 0 3.97020 1.06381i 2.16367 + 2.16367i 0 −1.99286 + 1.36088i
107.11 −0.907772 0.243237i 0 −0.967166 0.558393i 2.23601 0.0161462i 0 4.20400 1.12646i 2.07121 + 2.07121i 0 −2.03371 0.529222i
107.12 −0.595543 0.159575i 0 −1.40284 0.809932i −2.22145 0.255304i 0 −0.564606 + 0.151286i 1.57814 + 1.57814i 0 1.28223 + 0.506532i
107.13 −0.107004 0.0286717i 0 −1.72142 0.993864i −2.23171 + 0.139478i 0 −3.18110 + 0.852373i 0.312369 + 0.312369i 0 0.242802 + 0.0490623i
107.14 −0.0593036 0.0158904i 0 −1.72879 0.998115i −0.00190283 2.23607i 0 −1.12555 + 0.301591i 0.173489 + 0.173489i 0 −0.0354191 + 0.132637i
107.15 0.0593036 + 0.0158904i 0 −1.72879 0.998115i 0.00190283 + 2.23607i 0 −1.12555 + 0.301591i −0.173489 0.173489i 0 −0.0354191 + 0.132637i
107.16 0.107004 + 0.0286717i 0 −1.72142 0.993864i 2.23171 0.139478i 0 −3.18110 + 0.852373i −0.312369 0.312369i 0 0.242802 + 0.0490623i
107.17 0.595543 + 0.159575i 0 −1.40284 0.809932i 2.22145 + 0.255304i 0 −0.564606 + 0.151286i −1.57814 1.57814i 0 1.28223 + 0.506532i
107.18 0.907772 + 0.243237i 0 −0.967166 0.558393i −2.23601 + 0.0161462i 0 4.20400 1.12646i −2.07121 2.07121i 0 −2.03371 0.529222i
107.19 1.04244 + 0.279321i 0 −0.723390 0.417649i −1.45730 + 1.69596i 0 3.97020 1.06381i −2.16367 2.16367i 0 −1.99286 + 1.36088i
107.20 1.27023 + 0.340357i 0 −0.234407 0.135335i −0.422581 + 2.19577i 0 −2.52240 + 0.675874i −2.11144 2.11144i 0 −1.28412 + 2.64531i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
13.c even 3 1 inner
15.e even 4 1 inner
39.i odd 6 1 inner
65.q odd 12 1 inner
195.bl even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.dn.a 112
3.b odd 2 1 inner 585.2.dn.a 112
5.c odd 4 1 inner 585.2.dn.a 112
13.c even 3 1 inner 585.2.dn.a 112
15.e even 4 1 inner 585.2.dn.a 112
39.i odd 6 1 inner 585.2.dn.a 112
65.q odd 12 1 inner 585.2.dn.a 112
195.bl even 12 1 inner 585.2.dn.a 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.dn.a 112 1.a even 1 1 trivial
585.2.dn.a 112 3.b odd 2 1 inner
585.2.dn.a 112 5.c odd 4 1 inner
585.2.dn.a 112 13.c even 3 1 inner
585.2.dn.a 112 15.e even 4 1 inner
585.2.dn.a 112 39.i odd 6 1 inner
585.2.dn.a 112 65.q odd 12 1 inner
585.2.dn.a 112 195.bl even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(585, [\chi])\).