Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [320,3,Mod(19,320)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(320, base_ring=CyclotomicField(16))
chi = DirichletCharacter(H, H._module([8, 7, 8]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("320.19");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 320.bh (of order \(16\), degree \(8\), 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: | \(8.71936845953\) |
Analytic rank: | \(0\) |
Dimension: | \(752\) |
Relative dimension: | \(94\) over \(\Q(\zeta_{16})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{16}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −1.99980 | + | 0.0281854i | 2.55665 | + | 3.82630i | 3.99841 | − | 0.112730i | −4.32079 | − | 2.51611i | −5.22064 | − | 7.57978i | −1.57357 | − | 3.79893i | −7.99285 | + | 0.338135i | −4.65995 | + | 11.2501i | 8.71164 | + | 4.90993i |
19.2 | −1.99976 | + | 0.0312769i | −0.222450 | − | 0.332920i | 3.99804 | − | 0.125092i | −2.11099 | − | 4.53252i | 0.455259 | + | 0.658801i | 3.34064 | + | 8.06503i | −7.99120 | + | 0.375200i | 3.38280 | − | 8.16680i | 4.36322 | + | 8.99791i |
19.3 | −1.99936 | − | 0.0504630i | −1.71781 | − | 2.57088i | 3.99491 | + | 0.201788i | 3.28925 | + | 3.76575i | 3.30479 | + | 5.22681i | 1.24880 | + | 3.01488i | −7.97709 | − | 0.605042i | −0.214417 | + | 0.517649i | −6.38637 | − | 7.69508i |
19.4 | −1.99775 | − | 0.0947843i | 0.132332 | + | 0.198048i | 3.98203 | + | 0.378711i | 3.01166 | − | 3.99123i | −0.245594 | − | 0.408194i | −1.16058 | − | 2.80188i | −7.91922 | − | 1.13401i | 3.42244 | − | 8.26250i | −6.39485 | + | 7.68804i |
19.5 | −1.99464 | − | 0.146372i | 2.90773 | + | 4.35173i | 3.95715 | + | 0.583917i | 3.94474 | + | 3.07230i | −5.16290 | − | 9.10572i | −4.28936 | − | 10.3554i | −7.80761 | − | 1.74392i | −7.03847 | + | 16.9924i | −7.41863 | − | 6.70551i |
19.6 | −1.98966 | + | 0.203107i | −2.41543 | − | 3.61494i | 3.91750 | − | 0.808226i | 4.97198 | + | 0.528631i | 5.54010 | + | 6.70192i | −3.66182 | − | 8.84042i | −7.63033 | + | 2.40376i | −3.78936 | + | 9.14833i | −9.99991 | − | 0.0419546i |
19.7 | −1.94208 | − | 0.477854i | −0.278697 | − | 0.417100i | 3.54331 | + | 1.85606i | −0.765800 | + | 4.94101i | 0.341938 | + | 0.943216i | −0.143376 | − | 0.346141i | −5.99445 | − | 5.29779i | 3.34785 | − | 8.08243i | 3.84832 | − | 9.22987i |
19.8 | −1.93592 | + | 0.502193i | 2.05543 | + | 3.07617i | 3.49560 | − | 1.94442i | −2.13995 | + | 4.51892i | −5.52399 | − | 4.92300i | 3.65857 | + | 8.83258i | −5.79075 | + | 5.51971i | −1.79387 | + | 4.33078i | 1.87341 | − | 9.82295i |
19.9 | −1.92098 | + | 0.556639i | −3.12248 | − | 4.67312i | 3.38031 | − | 2.13858i | −2.04138 | − | 4.56429i | 8.59945 | + | 7.23886i | 0.911578 | + | 2.20074i | −5.30307 | + | 5.98978i | −8.64401 | + | 20.8685i | 6.46211 | + | 7.63159i |
19.10 | −1.89436 | + | 0.641417i | −0.368816 | − | 0.551972i | 3.17717 | − | 2.43014i | −4.94122 | − | 0.764408i | 1.05271 | + | 0.809067i | −4.69689 | − | 11.3393i | −4.45996 | + | 6.64144i | 3.27550 | − | 7.90776i | 9.85074 | − | 1.72132i |
19.11 | −1.88434 | − | 0.670266i | −2.53228 | − | 3.78983i | 3.10149 | + | 2.52602i | −4.77579 | + | 1.48047i | 2.23149 | + | 8.83864i | 1.82456 | + | 4.40488i | −4.15115 | − | 6.83871i | −4.50621 | + | 10.8789i | 9.99154 | + | 0.411344i |
19.12 | −1.83592 | − | 0.793342i | 1.45741 | + | 2.18116i | 2.74122 | + | 2.91303i | 4.71962 | + | 1.65082i | −0.945275 | − | 5.16066i | 3.79732 | + | 9.16754i | −2.72163 | − | 7.52281i | 0.810715 | − | 1.95724i | −7.35518 | − | 6.77506i |
19.13 | −1.80115 | + | 0.869401i | −1.52115 | − | 2.27657i | 2.48828 | − | 3.13184i | −3.98138 | + | 3.02466i | 4.71908 | + | 2.77795i | 1.81341 | + | 4.37797i | −1.75895 | + | 7.80424i | 0.575297 | − | 1.38889i | 4.54143 | − | 8.90929i |
19.14 | −1.78511 | + | 0.901887i | 1.57535 | + | 2.35768i | 2.37320 | − | 3.21993i | 4.15406 | − | 2.78277i | −4.93853 | − | 2.78792i | −0.244084 | − | 0.589270i | −1.33240 | + | 7.88826i | 0.367225 | − | 0.886559i | −4.90569 | + | 8.71403i |
19.15 | −1.78397 | − | 0.904121i | 3.00507 | + | 4.49741i | 2.36513 | + | 3.22586i | 1.29786 | − | 4.82862i | −1.29477 | − | 10.7402i | 3.88041 | + | 9.36813i | −1.30277 | − | 7.89321i | −7.75206 | + | 18.7151i | −6.68101 | + | 7.44071i |
19.16 | −1.77717 | − | 0.917423i | 1.50868 | + | 2.25790i | 2.31667 | + | 3.26084i | −4.54717 | + | 2.07923i | −0.609733 | − | 5.39678i | −1.69013 | − | 4.08034i | −1.12555 | − | 7.92043i | 0.622145 | − | 1.50199i | 9.98864 | + | 0.476530i |
19.17 | −1.70689 | − | 1.04237i | −1.63844 | − | 2.45209i | 1.82692 | + | 3.55842i | −2.16920 | − | 4.50495i | 0.240632 | + | 5.89330i | −4.03762 | − | 9.74768i | 0.590847 | − | 7.97815i | 0.115865 | − | 0.279723i | −0.993257 | + | 9.95055i |
19.18 | −1.68479 | + | 1.07772i | 0.795457 | + | 1.19049i | 1.67702 | − | 3.63147i | 1.75943 | + | 4.68021i | −2.62319 | − | 1.14843i | −3.20466 | − | 7.73674i | 1.08829 | + | 7.92563i | 2.65965 | − | 6.42096i | −8.00825 | − | 5.98899i |
19.19 | −1.66317 | − | 1.11079i | −2.54647 | − | 3.81106i | 1.53229 | + | 3.69487i | 3.64742 | − | 3.41999i | 0.00193507 | + | 9.16703i | 1.01842 | + | 2.45869i | 1.55575 | − | 7.84727i | −4.59550 | + | 11.0945i | −9.86518 | + | 1.63651i |
19.20 | −1.63387 | + | 1.15346i | −1.28831 | − | 1.92809i | 1.33908 | − | 3.76920i | 4.99552 | + | 0.211605i | 4.32891 | + | 1.66425i | 4.87580 | + | 11.7712i | 2.15971 | + | 7.70296i | 1.38635 | − | 3.34695i | −8.40612 | + | 5.41637i |
See next 80 embeddings (of 752 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
64.j | odd | 16 | 1 | inner |
320.bh | odd | 16 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 320.3.bh.a | ✓ | 752 |
5.b | even | 2 | 1 | inner | 320.3.bh.a | ✓ | 752 |
64.j | odd | 16 | 1 | inner | 320.3.bh.a | ✓ | 752 |
320.bh | odd | 16 | 1 | inner | 320.3.bh.a | ✓ | 752 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
320.3.bh.a | ✓ | 752 | 1.a | even | 1 | 1 | trivial |
320.3.bh.a | ✓ | 752 | 5.b | even | 2 | 1 | inner |
320.3.bh.a | ✓ | 752 | 64.j | odd | 16 | 1 | inner |
320.3.bh.a | ✓ | 752 | 320.bh | odd | 16 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(320, [\chi])\).