Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [64,2,Mod(5,64)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(64, base_ring=CyclotomicField(16))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("64.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 64.i (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: | \(0.511042572936\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(7\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −1.33595 | + | 0.463928i | −1.31138 | − | 0.876237i | 1.56954 | − | 1.23957i | 3.52249 | + | 0.700667i | 2.15846 | + | 0.562226i | 1.02503 | − | 2.47464i | −1.52176 | + | 2.38416i | −0.196120 | − | 0.473476i | −5.03094 | + | 0.698122i |
5.2 | −1.20609 | + | 0.738481i | 2.51381 | + | 1.67967i | 0.909293 | − | 1.78134i | −2.28487 | − | 0.454489i | −4.27227 | − | 0.169433i | 0.303950 | − | 0.733799i | 0.218802 | + | 2.81995i | 2.34987 | + | 5.67309i | 3.09139 | − | 1.13918i |
5.3 | −0.599818 | − | 1.28071i | −2.03902 | − | 1.36243i | −1.28044 | + | 1.53639i | −1.53851 | − | 0.306028i | −0.521836 | + | 3.42860i | 1.01301 | − | 2.44562i | 2.73569 | + | 0.718315i | 1.15334 | + | 2.78440i | 0.530891 | + | 2.15394i |
5.4 | 0.0973444 | + | 1.41086i | 0.306211 | + | 0.204603i | −1.98105 | + | 0.274678i | 1.42470 | + | 0.283390i | −0.258859 | + | 0.451937i | −0.666723 | + | 1.60961i | −0.580376 | − | 2.76824i | −1.09615 | − | 2.64634i | −0.261137 | + | 2.03763i |
5.5 | 0.406933 | − | 1.35440i | 1.06920 | + | 0.714416i | −1.66881 | − | 1.10230i | −0.330507 | − | 0.0657419i | 1.40270 | − | 1.15741i | −0.739314 | + | 1.78486i | −2.17206 | + | 1.81168i | −0.515254 | − | 1.24393i | −0.223535 | + | 0.420887i |
5.6 | 1.19265 | + | 0.759993i | 0.0799701 | + | 0.0534343i | 0.844823 | + | 1.81281i | −3.47403 | − | 0.691028i | 0.0547666 | + | 0.124505i | 1.22800 | − | 2.96465i | −0.370144 | + | 2.80410i | −1.14451 | − | 2.76309i | −3.61812 | − | 3.46439i |
5.7 | 1.36881 | − | 0.355465i | −2.00147 | − | 1.33734i | 1.74729 | − | 0.973128i | 0.756852 | + | 0.150547i | −3.21501 | − | 1.11911i | −1.69148 | + | 4.08359i | 2.04580 | − | 1.95313i | 1.06935 | + | 2.58163i | 1.08950 | − | 0.0629634i |
13.1 | −1.33595 | − | 0.463928i | −1.31138 | + | 0.876237i | 1.56954 | + | 1.23957i | 3.52249 | − | 0.700667i | 2.15846 | − | 0.562226i | 1.02503 | + | 2.47464i | −1.52176 | − | 2.38416i | −0.196120 | + | 0.473476i | −5.03094 | − | 0.698122i |
13.2 | −1.20609 | − | 0.738481i | 2.51381 | − | 1.67967i | 0.909293 | + | 1.78134i | −2.28487 | + | 0.454489i | −4.27227 | + | 0.169433i | 0.303950 | + | 0.733799i | 0.218802 | − | 2.81995i | 2.34987 | − | 5.67309i | 3.09139 | + | 1.13918i |
13.3 | −0.599818 | + | 1.28071i | −2.03902 | + | 1.36243i | −1.28044 | − | 1.53639i | −1.53851 | + | 0.306028i | −0.521836 | − | 3.42860i | 1.01301 | + | 2.44562i | 2.73569 | − | 0.718315i | 1.15334 | − | 2.78440i | 0.530891 | − | 2.15394i |
13.4 | 0.0973444 | − | 1.41086i | 0.306211 | − | 0.204603i | −1.98105 | − | 0.274678i | 1.42470 | − | 0.283390i | −0.258859 | − | 0.451937i | −0.666723 | − | 1.60961i | −0.580376 | + | 2.76824i | −1.09615 | + | 2.64634i | −0.261137 | − | 2.03763i |
13.5 | 0.406933 | + | 1.35440i | 1.06920 | − | 0.714416i | −1.66881 | + | 1.10230i | −0.330507 | + | 0.0657419i | 1.40270 | + | 1.15741i | −0.739314 | − | 1.78486i | −2.17206 | − | 1.81168i | −0.515254 | + | 1.24393i | −0.223535 | − | 0.420887i |
13.6 | 1.19265 | − | 0.759993i | 0.0799701 | − | 0.0534343i | 0.844823 | − | 1.81281i | −3.47403 | + | 0.691028i | 0.0547666 | − | 0.124505i | 1.22800 | + | 2.96465i | −0.370144 | − | 2.80410i | −1.14451 | + | 2.76309i | −3.61812 | + | 3.46439i |
13.7 | 1.36881 | + | 0.355465i | −2.00147 | + | 1.33734i | 1.74729 | + | 0.973128i | 0.756852 | − | 0.150547i | −3.21501 | + | 1.11911i | −1.69148 | − | 4.08359i | 2.04580 | + | 1.95313i | 1.06935 | − | 2.58163i | 1.08950 | + | 0.0629634i |
21.1 | −1.30780 | − | 0.538195i | 0.344545 | + | 1.73215i | 1.42069 | + | 1.40771i | −2.21982 | + | 1.48324i | 0.481636 | − | 2.45074i | 2.90595 | + | 1.20368i | −1.10036 | − | 2.60561i | −0.109979 | + | 0.0455548i | 3.70136 | − | 0.745083i |
21.2 | −1.27161 | + | 0.618873i | −0.216111 | − | 1.08646i | 1.23399 | − | 1.57393i | 1.50133 | − | 1.00316i | 0.947192 | + | 1.24781i | 1.15320 | + | 0.477669i | −0.595096 | + | 2.76512i | 1.63794 | − | 0.678457i | −1.28828 | + | 2.20476i |
21.3 | −0.887839 | − | 1.10079i | −0.435353 | − | 2.18867i | −0.423482 | + | 1.95465i | 0.649649 | − | 0.434082i | −2.02274 | + | 2.42242i | −3.64486 | − | 1.50975i | 2.52765 | − | 1.26925i | −1.82909 | + | 0.757635i | −1.05462 | − | 0.329733i |
21.4 | 0.0941531 | − | 1.41108i | 0.553854 | + | 2.78441i | −1.98227 | − | 0.265714i | 2.59756 | − | 1.73564i | 3.98116 | − | 0.519369i | −1.96508 | − | 0.813965i | −0.561580 | + | 2.77212i | −4.67456 | + | 1.93627i | −2.20455 | − | 3.82878i |
21.5 | 0.297937 | + | 1.38247i | 0.123576 | + | 0.621259i | −1.82247 | + | 0.823780i | 0.660623 | − | 0.441414i | −0.822057 | + | 0.355937i | −0.860072 | − | 0.356253i | −1.68183 | − | 2.27408i | 2.40095 | − | 0.994505i | 0.807067 | + | 0.781780i |
21.6 | 1.19357 | + | 0.758552i | −0.599600 | − | 3.01439i | 0.849199 | + | 1.81076i | −1.78465 | + | 1.19247i | 1.57091 | − | 4.05270i | 1.99271 | + | 0.825409i | −0.359981 | + | 2.80543i | −5.95540 | + | 2.46681i | −3.03465 | + | 0.0695368i |
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
64.i | even | 16 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 64.2.i.a | ✓ | 56 |
3.b | odd | 2 | 1 | 576.2.bd.a | 56 | ||
4.b | odd | 2 | 1 | 256.2.i.a | 56 | ||
8.b | even | 2 | 1 | 512.2.i.b | 56 | ||
8.d | odd | 2 | 1 | 512.2.i.a | 56 | ||
64.i | even | 16 | 1 | inner | 64.2.i.a | ✓ | 56 |
64.i | even | 16 | 1 | 512.2.i.b | 56 | ||
64.j | odd | 16 | 1 | 256.2.i.a | 56 | ||
64.j | odd | 16 | 1 | 512.2.i.a | 56 | ||
192.q | odd | 16 | 1 | 576.2.bd.a | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
64.2.i.a | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
64.2.i.a | ✓ | 56 | 64.i | even | 16 | 1 | inner |
256.2.i.a | 56 | 4.b | odd | 2 | 1 | ||
256.2.i.a | 56 | 64.j | odd | 16 | 1 | ||
512.2.i.a | 56 | 8.d | odd | 2 | 1 | ||
512.2.i.a | 56 | 64.j | odd | 16 | 1 | ||
512.2.i.b | 56 | 8.b | even | 2 | 1 | ||
512.2.i.b | 56 | 64.i | even | 16 | 1 | ||
576.2.bd.a | 56 | 3.b | odd | 2 | 1 | ||
576.2.bd.a | 56 | 192.q | odd | 16 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(64, [\chi])\).