Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [128,2,Mod(5,128)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(128, base_ring=CyclotomicField(32))
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("128.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 128.k (of order \(32\), 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: | \(1.02208514587\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(15\) over \(\Q(\zeta_{32})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{32}]$ |
$q$-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 | −1.41035 | − | 0.104463i | 1.66739 | + | 0.505799i | 1.97817 | + | 0.294660i | 0.127039 | + | 0.0125123i | −2.29877 | − | 0.887535i | 1.18113 | + | 1.76769i | −2.75914 | − | 0.622220i | 0.0299647 | + | 0.0200218i | −0.177863 | − | 0.0309177i |
5.2 | −1.31795 | − | 0.512840i | −1.88141 | − | 0.570718i | 1.47399 | + | 1.35180i | 2.61618 | + | 0.257671i | 2.18691 | + | 1.71704i | −1.78728 | − | 2.67485i | −1.24939 | − | 2.53752i | 0.719558 | + | 0.480794i | −3.31585 | − | 1.68128i |
5.3 | −1.26050 | + | 0.641195i | −0.417839 | − | 0.126750i | 1.17774 | − | 1.61646i | −2.96993 | − | 0.292513i | 0.607959 | − | 0.108147i | −1.08102 | − | 1.61787i | −0.448081 | + | 2.79271i | −2.33589 | − | 1.56079i | 3.93117 | − | 1.53559i |
5.4 | −0.950249 | − | 1.04739i | −1.73829 | − | 0.527306i | −0.194054 | + | 1.99056i | −2.46085 | − | 0.242372i | 1.09952 | + | 2.32174i | 2.82715 | + | 4.23114i | 2.26930 | − | 1.68828i | 0.249204 | + | 0.166513i | 2.08456 | + | 2.80778i |
5.5 | −0.919458 | + | 1.07452i | −2.80857 | − | 0.851969i | −0.309193 | − | 1.97596i | 1.35768 | + | 0.133720i | 3.49782 | − | 2.23451i | 1.67452 | + | 2.50610i | 2.40750 | + | 1.48457i | 4.66778 | + | 3.11891i | −1.39201 | + | 1.33591i |
5.6 | −0.891064 | + | 1.09818i | 1.53052 | + | 0.464277i | −0.412009 | − | 1.95710i | 3.76242 | + | 0.370566i | −1.87365 | + | 1.26709i | −2.04991 | − | 3.06791i | 2.51638 | + | 1.29144i | −0.367482 | − | 0.245544i | −3.75950 | + | 3.80162i |
5.7 | −0.652002 | − | 1.25495i | 2.77794 | + | 0.842679i | −1.14979 | + | 1.63646i | 0.0628209 | + | 0.00618731i | −0.753703 | − | 4.03560i | −0.619337 | − | 0.926904i | 2.80333 | + | 0.375951i | 4.51243 | + | 3.01511i | −0.0331945 | − | 0.0828710i |
5.8 | −0.163059 | − | 1.40478i | −0.757526 | − | 0.229793i | −1.94682 | + | 0.458124i | −1.57990 | − | 0.155607i | −0.199287 | + | 1.10163i | −2.26604 | − | 3.39136i | 0.961012 | + | 2.66016i | −1.97337 | − | 1.31856i | 0.0390236 | + | 2.24479i |
5.9 | 0.432421 | + | 1.34648i | 2.51607 | + | 0.763243i | −1.62602 | + | 1.16449i | −1.42404 | − | 0.140256i | 0.0603114 | + | 3.71789i | −0.954234 | − | 1.42811i | −2.27110 | − | 1.68586i | 3.25368 | + | 2.17404i | −0.426934 | − | 1.97810i |
5.10 | 0.441120 | + | 1.34366i | −0.451169 | − | 0.136861i | −1.61083 | + | 1.18543i | 2.96377 | + | 0.291905i | −0.0151258 | − | 0.666588i | 1.40525 | + | 2.10310i | −2.30337 | − | 1.64148i | −2.30959 | − | 1.54322i | 0.915154 | + | 4.11105i |
5.11 | 0.674225 | − | 1.24315i | 0.865940 | + | 0.262680i | −1.09084 | − | 1.67632i | 1.24951 | + | 0.123066i | 0.910389 | − | 0.899388i | 0.832309 | + | 1.24564i | −2.81939 | + | 0.225861i | −1.81356 | − | 1.21178i | 0.995438 | − | 1.47035i |
5.12 | 0.825274 | + | 1.14844i | −3.11090 | − | 0.943682i | −0.637847 | + | 1.89556i | −2.52957 | − | 0.249141i | −1.48358 | − | 4.35149i | −1.75170 | − | 2.62161i | −2.70334 | + | 0.831825i | 6.29277 | + | 4.20469i | −1.80147 | − | 3.11068i |
5.13 | 1.26626 | + | 0.629746i | −0.156077 | − | 0.0473453i | 1.20684 | + | 1.59485i | −0.851642 | − | 0.0838794i | −0.167818 | − | 0.158240i | 1.43739 | + | 2.15120i | 0.523829 | + | 2.77950i | −2.47229 | − | 1.65193i | −1.02558 | − | 0.642531i |
5.14 | 1.35411 | − | 0.407902i | 1.87695 | + | 0.569366i | 1.66723 | − | 1.10469i | −4.38038 | − | 0.431430i | 2.77384 | + | 0.00537365i | 0.0783406 | + | 0.117245i | 1.80701 | − | 2.17594i | 0.704347 | + | 0.470630i | −6.10750 | + | 1.20256i |
5.15 | 1.37613 | − | 0.325966i | −1.74451 | − | 0.529190i | 1.78749 | − | 0.897147i | 2.07613 | + | 0.204481i | −2.57317 | − | 0.159587i | −0.655035 | − | 0.980329i | 2.16739 | − | 1.81726i | 0.268850 | + | 0.179640i | 2.92369 | − | 0.395355i |
13.1 | −1.41410 | + | 0.0181162i | 0.180024 | + | 0.593459i | 1.99934 | − | 0.0512363i | −0.372310 | − | 3.78013i | −0.265323 | − | 0.835948i | −0.885307 | + | 1.32496i | −2.82634 | + | 0.108674i | 2.17462 | − | 1.45304i | 0.594964 | + | 5.33872i |
13.2 | −1.36942 | − | 0.353095i | −0.855442 | − | 2.82001i | 1.75065 | + | 0.967074i | −0.0181846 | − | 0.184631i | 0.175731 | + | 4.16385i | 1.98359 | − | 2.96866i | −2.05591 | − | 1.94248i | −4.72629 | + | 3.15801i | −0.0402899 | + | 0.259259i |
13.3 | −1.31306 | + | 0.525230i | 0.580122 | + | 1.91241i | 1.44827 | − | 1.37932i | 0.217075 | + | 2.20400i | −1.76619 | − | 2.20641i | 0.814297 | − | 1.21868i | −1.17721 | + | 2.57181i | −0.826349 | + | 0.552149i | −1.44264 | − | 2.77998i |
13.4 | −1.10268 | − | 0.885498i | −0.152748 | − | 0.503541i | 0.431788 | + | 1.95283i | 0.326146 | + | 3.31141i | −0.277453 | + | 0.690500i | −2.79715 | + | 4.18623i | 1.25311 | − | 2.53569i | 2.26419 | − | 1.51288i | 2.57262 | − | 3.94022i |
13.5 | −0.836598 | − | 1.14022i | 0.187193 | + | 0.617093i | −0.600207 | + | 1.90781i | −0.112545 | − | 1.14269i | 0.547017 | − | 0.729700i | 1.96260 | − | 2.93724i | 2.67746 | − | 0.911705i | 2.14865 | − | 1.43568i | −1.20876 | + | 1.08430i |
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
128.k | even | 32 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 128.2.k.a | ✓ | 240 |
4.b | odd | 2 | 1 | 512.2.k.a | 240 | ||
128.k | even | 32 | 1 | inner | 128.2.k.a | ✓ | 240 |
128.l | odd | 32 | 1 | 512.2.k.a | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
128.2.k.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
128.2.k.a | ✓ | 240 | 128.k | even | 32 | 1 | inner |
512.2.k.a | 240 | 4.b | odd | 2 | 1 | ||
512.2.k.a | 240 | 128.l | odd | 32 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(128, [\chi])\).