Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [473,2,Mod(4,473)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(473, base_ring=CyclotomicField(70))
chi = DirichletCharacter(H, H._module([14, 20]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("473.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 473 = 11 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 473.v (of order \(35\), degree \(24\), 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: | \(3.77692401561\) |
Analytic rank: | \(0\) |
Dimension: | \(1008\) |
Relative dimension: | \(42\) over \(\Q(\zeta_{35})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{35}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −2.64958 | + | 0.731237i | −2.23456 | + | 0.838643i | 4.76865 | − | 2.84914i | 1.60552 | − | 0.291358i | 5.30738 | − | 3.85604i | 1.23939 | + | 3.81445i | −6.75256 | + | 7.06263i | 2.03070 | − | 1.77417i | −4.04088 | + | 1.94599i |
4.2 | −2.60508 | + | 0.718957i | 2.83141 | − | 1.06265i | 4.55265 | − | 2.72008i | 0.299230 | − | 0.0543022i | −6.61205 | + | 4.80394i | −0.899463 | − | 2.76826i | −6.16924 | + | 6.45252i | 4.62844 | − | 4.04375i | −0.740476 | + | 0.356595i |
4.3 | −2.50139 | + | 0.690340i | −0.708636 | + | 0.265955i | 4.06349 | − | 2.42782i | −3.57232 | + | 0.648280i | 1.58898 | − | 1.15446i | −0.217631 | − | 0.669799i | −4.90186 | + | 5.12695i | −1.82778 | + | 1.59688i | 8.48822 | − | 4.08771i |
4.4 | −2.35605 | + | 0.650227i | 0.242385 | − | 0.0909685i | 3.41126 | − | 2.03813i | 0.0975855 | − | 0.0177092i | −0.511919 | + | 0.371931i | −0.573559 | − | 1.76523i | −3.33375 | + | 3.48683i | −2.20874 | + | 1.92972i | −0.218401 | + | 0.105176i |
4.5 | −2.29877 | + | 0.634421i | 0.881746 | − | 0.330925i | 3.16497 | − | 1.89098i | 3.39980 | − | 0.616973i | −1.81699 | + | 1.32012i | −0.233542 | − | 0.718767i | −2.77989 | + | 2.90754i | −1.59125 | + | 1.39023i | −7.42395 | + | 3.57518i |
4.6 | −2.20423 | + | 0.608330i | −2.68549 | + | 1.00788i | 2.77168 | − | 1.65600i | 1.15096 | − | 0.208869i | 5.30633 | − | 3.85527i | −1.45526 | − | 4.47882i | −1.94162 | + | 2.03078i | 3.93682 | − | 3.43950i | −2.40993 | + | 1.16056i |
4.7 | −2.15618 | + | 0.595068i | 2.20094 | − | 0.826027i | 2.57811 | − | 1.54035i | −2.29238 | + | 0.416005i | −4.25409 | + | 3.09078i | 0.928872 | + | 2.85877i | −1.55075 | + | 1.62195i | 1.90261 | − | 1.66226i | 4.69523 | − | 2.26110i |
4.8 | −1.92141 | + | 0.530276i | 0.216379 | − | 0.0812083i | 1.69374 | − | 1.01196i | 1.47051 | − | 0.266858i | −0.372690 | + | 0.270775i | 0.937210 | + | 2.88444i | 0.0371585 | − | 0.0388648i | −2.21899 | + | 1.93867i | −2.68395 | + | 1.29252i |
4.9 | −1.66037 | + | 0.458232i | −1.88126 | + | 0.706049i | 0.829940 | − | 0.495866i | 3.17696 | − | 0.576534i | 2.80005 | − | 2.03435i | −0.186938 | − | 0.575337i | 1.22984 | − | 1.28631i | 0.781425 | − | 0.682711i | −5.01074 | + | 2.41304i |
4.10 | −1.55494 | + | 0.429136i | −2.73531 | + | 1.02658i | 0.516785 | − | 0.308764i | −2.57482 | + | 0.467261i | 3.81270 | − | 2.77009i | 0.285364 | + | 0.878262i | 1.55840 | − | 1.62996i | 4.16883 | − | 3.64219i | 3.80317 | − | 1.83151i |
4.11 | −1.53787 | + | 0.424427i | 0.517766 | − | 0.194321i | 0.468024 | − | 0.279631i | −3.90061 | + | 0.707856i | −0.713785 | + | 0.518595i | −0.197839 | − | 0.608886i | 1.60392 | − | 1.67757i | −2.02889 | + | 1.77259i | 5.69821 | − | 2.74411i |
4.12 | −1.39350 | + | 0.384582i | 3.03112 | − | 1.13760i | 0.0770500 | − | 0.0460352i | 2.34716 | − | 0.425946i | −3.78637 | + | 2.75096i | 1.13017 | + | 3.47830i | 1.90833 | − | 1.99596i | 5.63434 | − | 4.92257i | −3.10696 | + | 1.49623i |
4.13 | −1.30207 | + | 0.359348i | −1.43507 | + | 0.538592i | −0.150647 | + | 0.0900074i | −1.18882 | + | 0.215738i | 1.67502 | − | 1.21697i | 1.09100 | + | 3.35776i | 2.03071 | − | 2.12395i | −0.489863 | + | 0.427980i | 1.47039 | − | 0.708105i |
4.14 | −1.26189 | + | 0.348258i | 1.92898 | − | 0.723959i | −0.245827 | + | 0.146875i | 2.19533 | − | 0.398394i | −2.18203 | + | 1.58534i | −0.592754 | − | 1.82431i | 2.06834 | − | 2.16331i | 0.937643 | − | 0.819194i | −2.63152 | + | 1.26727i |
4.15 | −1.22164 | + | 0.337151i | −0.665373 | + | 0.249719i | −0.338163 | + | 0.202043i | −0.914590 | + | 0.165974i | 0.728654 | − | 0.529398i | −1.37743 | − | 4.23930i | 2.09658 | − | 2.19285i | −1.87885 | + | 1.64150i | 1.06134 | − | 0.511115i |
4.16 | −0.739140 | + | 0.203990i | 1.12386 | − | 0.421792i | −1.21218 | + | 0.724245i | −2.05879 | + | 0.373616i | −0.744649 | + | 0.541019i | 1.02720 | + | 3.16140i | 1.80801 | − | 1.89103i | −1.17406 | + | 1.02575i | 1.44552 | − | 0.696126i |
4.17 | −0.669119 | + | 0.184665i | −1.55099 | + | 0.582096i | −1.30328 | + | 0.778672i | −0.0591621 | + | 0.0107363i | 0.930304 | − | 0.675905i | −0.562464 | − | 1.73109i | 1.68763 | − | 1.76513i | −0.192484 | + | 0.168168i | 0.0376039 | − | 0.0181091i |
4.18 | −0.410819 | + | 0.113379i | −3.17237 | + | 1.19061i | −1.56098 | + | 0.932642i | 2.69162 | − | 0.488457i | 1.16828 | − | 0.848805i | 0.728507 | + | 2.24211i | 1.12457 | − | 1.17621i | 6.38717 | − | 5.58030i | −1.05039 | + | 0.505840i |
4.19 | −0.261439 | + | 0.0721526i | 1.82397 | − | 0.684548i | −1.65375 | + | 0.988072i | 1.99651 | − | 0.362314i | −0.427466 | + | 0.310572i | −0.337623 | − | 1.03910i | 0.735914 | − | 0.769706i | 0.599052 | − | 0.523376i | −0.495825 | + | 0.238777i |
4.20 | −0.179531 | + | 0.0495474i | 2.12349 | − | 0.796961i | −1.68712 | + | 1.00801i | −1.95877 | + | 0.355463i | −0.341746 | + | 0.248293i | 0.490319 | + | 1.50905i | 0.510357 | − | 0.533792i | 1.61487 | − | 1.41087i | 0.334047 | − | 0.160869i |
See next 80 embeddings (of 1008 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
43.e | even | 7 | 1 | inner |
473.v | even | 35 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 473.2.v.a | ✓ | 1008 |
11.c | even | 5 | 1 | inner | 473.2.v.a | ✓ | 1008 |
43.e | even | 7 | 1 | inner | 473.2.v.a | ✓ | 1008 |
473.v | even | 35 | 1 | inner | 473.2.v.a | ✓ | 1008 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
473.2.v.a | ✓ | 1008 | 1.a | even | 1 | 1 | trivial |
473.2.v.a | ✓ | 1008 | 11.c | even | 5 | 1 | inner |
473.2.v.a | ✓ | 1008 | 43.e | even | 7 | 1 | inner |
473.2.v.a | ✓ | 1008 | 473.v | even | 35 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(473, [\chi])\).