Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [403,2,Mod(87,403)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(403, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("403.87");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 403 = 13 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 403.g (of order \(3\), degree \(2\), 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: | \(3.21797120146\) |
Analytic rank: | \(0\) |
Dimension: | \(70\) |
Relative dimension: | \(35\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
87.1 | −1.34742 | + | 2.33379i | 1.77488 | −2.63106 | − | 4.55713i | 0.115559 | − | 0.200154i | −2.39150 | + | 4.14220i | −0.966983 | + | 1.67486i | 8.79087 | 0.150199 | 0.311412 | + | 0.539381i | ||||||
87.2 | −1.27542 | + | 2.20909i | −1.13859 | −2.25340 | − | 3.90300i | −0.675176 | + | 1.16944i | 1.45218 | − | 2.51524i | −0.952417 | + | 1.64964i | 6.39445 | −1.70362 | −1.72227 | − | 2.98305i | ||||||
87.3 | −1.25452 | + | 2.17290i | −3.37142 | −2.14766 | − | 3.71985i | 0.669409 | − | 1.15945i | 4.22952 | − | 7.32574i | −0.576214 | + | 0.998031i | 5.75904 | 8.36645 | 1.67958 | + | 2.90912i | ||||||
87.4 | −1.18452 | + | 2.05166i | 0.601104 | −1.80620 | − | 3.12843i | 2.14607 | − | 3.71711i | −0.712022 | + | 1.23326i | −0.998859 | + | 1.73007i | 3.81984 | −2.63867 | 5.08415 | + | 8.80601i | ||||||
87.5 | −1.06665 | + | 1.84749i | −1.21169 | −1.27547 | − | 2.20918i | −1.78766 | + | 3.09632i | 1.29245 | − | 2.23859i | −0.455966 | + | 0.789757i | 1.17532 | −1.53180 | −3.81360 | − | 6.60535i | ||||||
87.6 | −1.05012 | + | 1.81887i | 2.74399 | −1.20552 | − | 2.08802i | 0.498066 | − | 0.862675i | −2.88153 | + | 4.99096i | 1.76422 | − | 3.05572i | 0.863293 | 4.52949 | 1.04606 | + | 1.81183i | ||||||
87.7 | −0.948089 | + | 1.64214i | 0.407364 | −0.797747 | − | 1.38174i | 0.840173 | − | 1.45522i | −0.386218 | + | 0.668949i | 1.87077 | − | 3.24028i | −0.767016 | −2.83405 | 1.59312 | + | 2.75936i | ||||||
87.8 | −0.937381 | + | 1.62359i | −2.30722 | −0.757366 | − | 1.31180i | −1.23736 | + | 2.14316i | 2.16274 | − | 3.74598i | 2.24919 | − | 3.89572i | −0.909763 | 2.32325 | −2.31975 | − | 4.01792i | ||||||
87.9 | −0.925450 | + | 1.60293i | 3.11839 | −0.712914 | − | 1.23480i | −0.0505045 | + | 0.0874763i | −2.88591 | + | 4.99854i | −2.21347 | + | 3.83384i | −1.06274 | 6.72433 | −0.0934787 | − | 0.161910i | ||||||
87.10 | −0.852641 | + | 1.47682i | −1.87577 | −0.453992 | − | 0.786337i | 1.48433 | − | 2.57094i | 1.59936 | − | 2.77017i | −0.0391156 | + | 0.0677502i | −1.86219 | 0.518520 | 2.53120 | + | 4.38417i | ||||||
87.11 | −0.770644 | + | 1.33479i | 0.658984 | −0.187783 | − | 0.325250i | −0.614094 | + | 1.06364i | −0.507842 | + | 0.879608i | −0.291650 | + | 0.505152i | −2.50372 | −2.56574 | −0.946495 | − | 1.63938i | ||||||
87.12 | −0.447963 | + | 0.775894i | −0.172885 | 0.598659 | + | 1.03691i | −0.364951 | + | 0.632113i | 0.0774459 | − | 0.134140i | −1.61455 | + | 2.79649i | −2.86456 | −2.97011 | −0.326969 | − | 0.566326i | ||||||
87.13 | −0.381984 | + | 0.661616i | −2.36827 | 0.708176 | + | 1.22660i | 0.878528 | − | 1.52165i | 0.904643 | − | 1.56689i | 0.396513 | − | 0.686780i | −2.60998 | 2.60872 | 0.671167 | + | 1.16250i | ||||||
87.14 | −0.356919 | + | 0.618202i | 1.24248 | 0.745217 | + | 1.29075i | −0.797665 | + | 1.38160i | −0.443464 | + | 0.768103i | 0.782978 | − | 1.35616i | −2.49161 | −1.45625 | −0.569404 | − | 0.986237i | ||||||
87.15 | −0.351546 | + | 0.608895i | −3.24498 | 0.752831 | + | 1.30394i | −1.57412 | + | 2.72646i | 1.14076 | − | 1.97585i | −2.53334 | + | 4.38787i | −2.46480 | 7.52991 | −1.10675 | − | 1.91695i | ||||||
87.16 | −0.333798 | + | 0.578156i | 2.80149 | 0.777157 | + | 1.34608i | −1.97182 | + | 3.41529i | −0.935132 | + | 1.61970i | 0.768201 | − | 1.33056i | −2.37285 | 4.84832 | −1.31638 | − | 2.28004i | ||||||
87.17 | −0.254102 | + | 0.440118i | 1.55532 | 0.870864 | + | 1.50838i | 1.96757 | − | 3.40793i | −0.395209 | + | 0.684522i | −1.78085 | + | 3.08453i | −1.90156 | −0.580989 | 0.999926 | + | 1.73192i | ||||||
87.18 | −0.0861040 | + | 0.149137i | −2.47814 | 0.985172 | + | 1.70637i | −0.342759 | + | 0.593675i | 0.213378 | − | 0.369581i | 1.94989 | − | 3.37730i | −0.683725 | 3.14119 | −0.0590258 | − | 0.102236i | ||||||
87.19 | 0.0840732 | − | 0.145619i | −0.346311 | 0.985863 | + | 1.70757i | 1.49511 | − | 2.58961i | −0.0291155 | + | 0.0504294i | 1.68076 | − | 2.91117i | 0.667832 | −2.88007 | −0.251398 | − | 0.435434i | ||||||
87.20 | 0.170223 | − | 0.294835i | 2.25499 | 0.942048 | + | 1.63168i | 0.624468 | − | 1.08161i | 0.383851 | − | 0.664850i | 0.451761 | − | 0.782473i | 1.32232 | 2.08499 | −0.212598 | − | 0.368230i | ||||||
See all 70 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
403.g | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 403.2.g.a | yes | 70 |
13.c | even | 3 | 1 | 403.2.e.a | ✓ | 70 | |
31.c | even | 3 | 1 | 403.2.e.a | ✓ | 70 | |
403.g | even | 3 | 1 | inner | 403.2.g.a | yes | 70 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
403.2.e.a | ✓ | 70 | 13.c | even | 3 | 1 | |
403.2.e.a | ✓ | 70 | 31.c | even | 3 | 1 | |
403.2.g.a | yes | 70 | 1.a | even | 1 | 1 | trivial |
403.2.g.a | yes | 70 | 403.g | even | 3 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(403, [\chi])\).