Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [512,2,Mod(5,512)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(512, base_ring=CyclotomicField(128))
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("512.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 512 = 2^{9} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 512.o (of order \(128\), degree \(64\), 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: | \(4.08834058349\) |
Analytic rank: | \(0\) |
Dimension: | \(4032\) |
Relative dimension: | \(63\) over \(\Q(\zeta_{128})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{128}]$ |
$q$-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 | −1.41230 | − | 0.0736323i | 0.668640 | + | 0.775139i | 1.98916 | + | 0.207981i | −0.455024 | − | 0.0111702i | −0.887242 | − | 1.14396i | −0.579250 | + | 2.31250i | −2.79396 | − | 0.440197i | 0.286430 | − | 1.93095i | 0.641806 | + | 0.0492801i |
5.2 | −1.40353 | + | 0.173508i | 1.19537 | + | 1.38577i | 1.93979 | − | 0.487048i | 2.41010 | + | 0.0591645i | −1.91818 | − | 1.73756i | −0.464653 | + | 1.85500i | −2.63804 | + | 1.02016i | −0.0512479 | + | 0.345485i | −3.39291 | + | 0.335133i |
5.3 | −1.38795 | − | 0.271261i | −1.88159 | − | 2.18129i | 1.85283 | + | 0.752997i | −2.67923 | − | 0.0657714i | 2.01986 | + | 3.53793i | −0.581717 | + | 2.32235i | −2.36739 | − | 1.54773i | −0.777434 | + | 5.24104i | 3.70081 | + | 0.818059i |
5.4 | −1.37702 | + | 0.322217i | −0.556414 | − | 0.645038i | 1.79235 | − | 0.887397i | −2.40397 | − | 0.0590141i | 0.974033 | + | 0.708942i | 1.06676 | − | 4.25872i | −2.18216 | + | 1.79949i | 0.333714 | − | 2.24971i | 3.32932 | − | 0.693336i |
5.5 | −1.37343 | − | 0.337181i | −0.833389 | − | 0.966130i | 1.77262 | + | 0.926190i | 3.44138 | + | 0.0844812i | 0.818841 | + | 1.60791i | 0.359378 | − | 1.43472i | −2.12227 | − | 1.86975i | 0.201322 | − | 1.35721i | −4.69801 | − | 1.27640i |
5.6 | −1.37064 | − | 0.348339i | 0.943297 | + | 1.09354i | 1.75732 | + | 0.954897i | −1.67661 | − | 0.0411585i | −0.911998 | − | 1.82744i | 0.698599 | − | 2.78896i | −2.07603 | − | 1.92096i | 0.134164 | − | 0.904459i | 2.28370 | + | 0.640444i |
5.7 | −1.34552 | + | 0.435416i | −0.955526 | − | 1.10772i | 1.62083 | − | 1.17172i | −0.885780 | − | 0.0217447i | 1.76799 | + | 1.07440i | −0.000328152 | 0.00131006i | −1.67066 | + | 2.28230i | 0.126178 | − | 0.850621i | 1.20130 | − | 0.356425i | |
5.8 | −1.34363 | + | 0.441201i | 1.62403 | + | 1.88270i | 1.61068 | − | 1.18562i | −3.80814 | − | 0.0934845i | −3.01274 | − | 1.81313i | −0.545363 | + | 2.17721i | −1.64106 | + | 2.30367i | −0.466898 | + | 3.14757i | 5.15797 | − | 1.55455i |
5.9 | −1.21995 | − | 0.715356i | −0.595169 | − | 0.689967i | 0.976532 | + | 1.74539i | 0.263295 | + | 0.00646353i | 0.232502 | + | 1.26748i | −0.849867 | + | 3.39286i | 0.0572585 | − | 2.82785i | 0.318364 | − | 2.14624i | −0.316582 | − | 0.196235i |
5.10 | −1.19094 | + | 0.762675i | 0.173232 | + | 0.200824i | 0.836653 | − | 1.81659i | 2.81168 | + | 0.0690230i | −0.359472 | − | 0.107049i | 0.330864 | − | 1.32088i | 0.389072 | + | 2.80154i | 0.429870 | − | 2.89795i | −3.40118 | + | 2.06220i |
5.11 | −1.19074 | − | 0.762973i | 2.25131 | + | 2.60990i | 0.835745 | + | 1.81701i | −1.49149 | − | 0.0366139i | −0.689458 | − | 4.82541i | 0.0527871 | − | 0.210738i | 0.391171 | − | 2.80125i | −1.30296 | + | 8.78387i | 1.74804 | + | 1.18156i |
5.12 | −1.11786 | + | 0.866245i | −1.12670 | − | 1.30616i | 0.499239 | − | 1.93669i | −2.53567 | − | 0.0622471i | 2.39095 | + | 0.484108i | −0.796848 | + | 3.18120i | 1.11957 | + | 2.59742i | 0.00359615 | − | 0.0242433i | 2.88845 | − | 2.12692i |
5.13 | −1.09603 | + | 0.893716i | −1.81468 | − | 2.10372i | 0.402544 | − | 1.95907i | 2.77109 | + | 0.0680264i | 3.86907 | + | 0.683922i | −0.160347 | + | 0.640139i | 1.30965 | + | 2.50695i | −0.692379 | + | 4.66764i | −3.09798 | + | 2.40201i |
5.14 | −1.09005 | − | 0.900993i | 1.45075 | + | 1.68182i | 0.376424 | + | 1.96426i | 2.75573 | + | 0.0676494i | −0.0660818 | − | 3.14038i | −0.901721 | + | 3.59987i | 1.35946 | − | 2.48030i | −0.283655 | + | 1.91225i | −2.94294 | − | 2.55664i |
5.15 | −1.06151 | − | 0.934455i | −1.75460 | − | 2.03407i | 0.253587 | + | 1.98386i | −0.223365 | − | 0.00548331i | −0.0382295 | + | 3.79878i | 0.548061 | − | 2.18798i | 1.58464 | − | 2.34284i | −0.618627 | + | 4.17044i | 0.231980 | + | 0.214546i |
5.16 | −1.05109 | + | 0.946158i | 1.45593 | + | 1.68782i | 0.209569 | − | 1.98899i | −0.0634026 | − | 0.00155645i | −3.12725 | − | 0.396512i | 1.07059 | − | 4.27405i | 1.66162 | + | 2.28889i | −0.288833 | + | 1.94716i | 0.0681143 | − | 0.0583530i |
5.17 | −0.962532 | − | 1.03611i | 0.360620 | + | 0.418058i | −0.147066 | + | 1.99459i | −3.88959 | − | 0.0954842i | 0.0860484 | − | 0.776038i | −0.0593750 | + | 0.237038i | 2.20817 | − | 1.76748i | 0.395465 | − | 2.66601i | 3.64492 | + | 4.12197i |
5.18 | −0.901401 | + | 1.08971i | 1.88802 | + | 2.18874i | −0.374952 | − | 1.96454i | 1.75891 | + | 0.0431789i | −4.08696 | + | 0.0844684i | −0.903089 | + | 3.60533i | 2.47877 | + | 1.36225i | −0.785769 | + | 5.29723i | −1.63254 | + | 1.87779i |
5.19 | −0.890574 | − | 1.09858i | 1.01161 | + | 1.17273i | −0.413756 | + | 1.95673i | 2.43657 | + | 0.0598145i | 0.387431 | − | 2.15574i | 0.840927 | − | 3.35717i | 2.51811 | − | 1.28807i | 0.0882358 | − | 0.594837i | −2.10424 | − | 2.73004i |
5.20 | −0.744441 | − | 1.20242i | −2.03236 | − | 2.35607i | −0.891615 | + | 1.79026i | 2.68864 | + | 0.0660025i | −1.32001 | + | 4.19771i | −0.738845 | + | 2.94963i | 2.81639 | − | 0.260649i | −0.980393 | + | 6.60927i | −1.92217 | − | 3.28200i |
See next 80 embeddings (of 4032 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
512.o | even | 128 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 512.2.o.a | ✓ | 4032 |
512.o | even | 128 | 1 | inner | 512.2.o.a | ✓ | 4032 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
512.2.o.a | ✓ | 4032 | 1.a | even | 1 | 1 | trivial |
512.2.o.a | ✓ | 4032 | 512.o | even | 128 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(512, [\chi])\).