Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [930,2,Mod(107,930)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, base_ring=CyclotomicField(60))
chi = DirichletCharacter(H, H._module([30, 15, 44]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("930.107");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 930.bs (of order \(60\), degree \(16\), 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: | \(7.42608738798\) |
Analytic rank: | \(0\) |
Dimension: | \(1024\) |
Relative dimension: | \(64\) over \(\Q(\zeta_{60})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{60}]$ |
$q$-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 | −0.453990 | − | 0.891007i | −1.71996 | − | 0.204302i | −0.587785 | + | 0.809017i | −1.77453 | − | 1.36053i | 0.598811 | + | 1.62525i | −3.08239 | + | 2.49607i | 0.987688 | + | 0.156434i | 2.91652 | + | 0.702783i | −0.406621 | + | 2.19879i |
107.2 | −0.453990 | − | 0.891007i | −1.70228 | − | 0.319750i | −0.587785 | + | 0.809017i | 1.94583 | + | 1.10171i | 0.487920 | + | 1.66191i | 0.0931562 | − | 0.0754364i | 0.987688 | + | 0.156434i | 2.79552 | + | 1.08861i | 0.0982390 | − | 2.23391i |
107.3 | −0.453990 | − | 0.891007i | −1.69507 | + | 0.355982i | −0.587785 | + | 0.809017i | 2.01411 | − | 0.971260i | 1.08673 | + | 1.34871i | 2.26525 | − | 1.83436i | 0.987688 | + | 0.156434i | 2.74655 | − | 1.20683i | −1.77979 | − | 1.35365i |
107.4 | −0.453990 | − | 0.891007i | −1.67186 | + | 0.452624i | −0.587785 | + | 0.809017i | 2.07413 | + | 0.835445i | 1.16230 | + | 1.28416i | −1.71190 | + | 1.38627i | 0.987688 | + | 0.156434i | 2.59026 | − | 1.51345i | −0.197250 | − | 2.22735i |
107.5 | −0.453990 | − | 0.891007i | −1.61622 | − | 0.622763i | −0.587785 | + | 0.809017i | −0.949618 | + | 2.02441i | 0.178863 | + | 1.72279i | 0.463591 | − | 0.375409i | 0.987688 | + | 0.156434i | 2.22433 | + | 2.01304i | 2.23488 | − | 0.0729461i |
107.6 | −0.453990 | − | 0.891007i | −1.52373 | − | 0.823562i | −0.587785 | + | 0.809017i | −1.29986 | − | 1.81944i | −0.0420417 | + | 1.73154i | 2.38916 | − | 1.93471i | 0.987688 | + | 0.156434i | 1.64349 | + | 2.50977i | −1.03101 | + | 1.98419i |
107.7 | −0.453990 | − | 0.891007i | −1.49231 | + | 0.879203i | −0.587785 | + | 0.809017i | −2.08977 | + | 0.795522i | 1.46087 | + | 0.930512i | 3.09036 | − | 2.50253i | 0.987688 | + | 0.156434i | 1.45400 | − | 2.62409i | 1.65755 | + | 1.50084i |
107.8 | −0.453990 | − | 0.891007i | −1.19166 | − | 1.25696i | −0.587785 | + | 0.809017i | 1.48838 | − | 1.66875i | −0.578959 | + | 1.63242i | −2.67101 | + | 2.16294i | 0.987688 | + | 0.156434i | −0.159900 | + | 2.99574i | −2.16258 | − | 0.568562i |
107.9 | −0.453990 | − | 0.891007i | −1.02426 | + | 1.39674i | −0.587785 | + | 0.809017i | −2.05905 | − | 0.871953i | 1.70951 | + | 0.278513i | −0.725093 | + | 0.587168i | 0.987688 | + | 0.156434i | −0.901788 | − | 2.86125i | 0.157874 | + | 2.23049i |
107.10 | −0.453990 | − | 0.891007i | −1.00357 | + | 1.41168i | −0.587785 | + | 0.809017i | 0.618932 | − | 2.14870i | 1.71343 | + | 0.253300i | −0.574024 | + | 0.464836i | 0.987688 | + | 0.156434i | −0.985684 | − | 2.83345i | −2.19550 | + | 0.424018i |
107.11 | −0.453990 | − | 0.891007i | −0.827104 | + | 1.52181i | −0.587785 | + | 0.809017i | 0.791841 | + | 2.09117i | 1.73144 | + | 0.0460690i | 2.74072 | − | 2.21939i | 0.987688 | + | 0.156434i | −1.63180 | − | 2.51739i | 1.50376 | − | 1.65491i |
107.12 | −0.453990 | − | 0.891007i | −0.695665 | − | 1.58621i | −0.587785 | + | 0.809017i | 1.24127 | + | 1.85990i | −1.09749 | + | 1.33996i | 3.78072 | − | 3.06157i | 0.987688 | + | 0.156434i | −2.03210 | + | 2.20694i | 1.09366 | − | 1.95036i |
107.13 | −0.453990 | − | 0.891007i | −0.602074 | − | 1.62404i | −0.587785 | + | 0.809017i | −1.44079 | − | 1.71001i | −1.17369 | + | 1.27375i | 0.478316 | − | 0.387333i | 0.987688 | + | 0.156434i | −2.27501 | + | 1.95559i | −0.869520 | + | 2.06008i |
107.14 | −0.453990 | − | 0.891007i | −0.510421 | − | 1.65513i | −0.587785 | + | 0.809017i | 1.93109 | + | 1.12734i | −1.24301 | + | 1.20620i | −2.49173 | + | 2.01776i | 0.987688 | + | 0.156434i | −2.47894 | + | 1.68963i | 0.127771 | − | 2.23241i |
107.15 | −0.453990 | − | 0.891007i | −0.205138 | − | 1.71986i | −0.587785 | + | 0.809017i | 1.37425 | − | 1.76393i | −1.43928 | + | 0.963580i | 1.73979 | − | 1.40885i | 0.987688 | + | 0.156434i | −2.91584 | + | 0.705618i | −2.19557 | − | 0.423656i |
107.16 | −0.453990 | − | 0.891007i | −0.148049 | − | 1.72571i | −0.587785 | + | 0.809017i | −1.13774 | + | 1.92498i | −1.47041 | + | 0.915369i | −1.33909 | + | 1.08437i | 0.987688 | + | 0.156434i | −2.95616 | + | 0.510979i | 2.23169 | + | 0.139806i |
107.17 | −0.453990 | − | 0.891007i | 0.0494968 | + | 1.73134i | −0.587785 | + | 0.809017i | 0.307868 | + | 2.21477i | 1.52017 | − | 0.830115i | −0.186966 | + | 0.151402i | 0.987688 | + | 0.156434i | −2.99510 | + | 0.171392i | 1.83361 | − | 1.27980i |
107.18 | −0.453990 | − | 0.891007i | 0.0504550 | + | 1.73132i | −0.587785 | + | 0.809017i | −2.07141 | + | 0.842184i | 1.51971 | − | 0.830957i | −3.23873 | + | 2.62267i | 0.987688 | + | 0.156434i | −2.99491 | + | 0.174707i | 1.69079 | + | 1.46329i |
107.19 | −0.453990 | − | 0.891007i | 0.546386 | + | 1.64361i | −0.587785 | + | 0.809017i | 0.869573 | − | 2.06006i | 1.21642 | − | 1.23302i | −1.46148 | + | 1.18348i | 0.987688 | + | 0.156434i | −2.40292 | + | 1.79610i | −2.23030 | + | 0.160452i |
107.20 | −0.453990 | − | 0.891007i | 0.559329 | − | 1.63925i | −0.587785 | + | 0.809017i | −2.23601 | + | 0.0166336i | −1.71452 | + | 0.245839i | −0.828800 | + | 0.671149i | 0.987688 | + | 0.156434i | −2.37430 | − | 1.83376i | 1.02995 | + | 1.98474i |
See next 80 embeddings (of 1024 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 |
15.e | even | 4 | 1 | inner |
31.g | even | 15 | 1 | inner |
93.o | odd | 30 | 1 | inner |
155.w | odd | 60 | 1 | inner |
465.bt | even | 60 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 930.2.bs.a | ✓ | 1024 |
3.b | odd | 2 | 1 | inner | 930.2.bs.a | ✓ | 1024 |
5.c | odd | 4 | 1 | inner | 930.2.bs.a | ✓ | 1024 |
15.e | even | 4 | 1 | inner | 930.2.bs.a | ✓ | 1024 |
31.g | even | 15 | 1 | inner | 930.2.bs.a | ✓ | 1024 |
93.o | odd | 30 | 1 | inner | 930.2.bs.a | ✓ | 1024 |
155.w | odd | 60 | 1 | inner | 930.2.bs.a | ✓ | 1024 |
465.bt | even | 60 | 1 | inner | 930.2.bs.a | ✓ | 1024 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
930.2.bs.a | ✓ | 1024 | 1.a | even | 1 | 1 | trivial |
930.2.bs.a | ✓ | 1024 | 3.b | odd | 2 | 1 | inner |
930.2.bs.a | ✓ | 1024 | 5.c | odd | 4 | 1 | inner |
930.2.bs.a | ✓ | 1024 | 15.e | even | 4 | 1 | inner |
930.2.bs.a | ✓ | 1024 | 31.g | even | 15 | 1 | inner |
930.2.bs.a | ✓ | 1024 | 93.o | odd | 30 | 1 | inner |
930.2.bs.a | ✓ | 1024 | 155.w | odd | 60 | 1 | inner |
930.2.bs.a | ✓ | 1024 | 465.bt | even | 60 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(930, [\chi])\).