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.
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) |
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 \(S_{2}^{\mathrm{new}}(585, [\chi])\).