Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [277,2,Mod(16,277)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(277, base_ring=CyclotomicField(46))
chi = DirichletCharacter(H, H._module([6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("277.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 277 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 277.g (of order \(23\), degree \(22\), 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: | \(2.21185613599\) |
Analytic rank: | \(0\) |
Dimension: | \(462\) |
Relative dimension: | \(21\) over \(\Q(\zeta_{23})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{23}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −2.26253 | − | 1.37587i | 0.191052 | − | 2.79307i | 2.30588 | + | 4.45014i | −2.28039 | − | 0.990512i | −4.27518 | + | 6.05655i | 3.28175 | − | 1.42546i | 0.544304 | − | 7.95744i | −4.79271 | − | 0.658743i | 3.79662 | + | 5.37858i |
16.2 | −2.13644 | − | 1.29920i | −0.142346 | + | 2.08102i | 1.95633 | + | 3.77554i | 1.88109 | + | 0.817073i | 3.00777 | − | 4.26104i | 1.95720 | − | 0.850132i | 0.384323 | − | 5.61861i | −1.33832 | − | 0.183948i | −2.95730 | − | 4.18953i |
16.3 | −2.04323 | − | 1.24251i | −0.0903377 | + | 1.32069i | 1.71080 | + | 3.30170i | −1.31876 | − | 0.572817i | 1.82556 | − | 2.58622i | −1.60050 | + | 0.695194i | 0.280465 | − | 4.10025i | 1.23600 | + | 0.169884i | 1.98279 | + | 2.80897i |
16.4 | −1.63953 | − | 0.997023i | 0.0265916 | − | 0.388755i | 0.773888 | + | 1.49354i | −3.28121 | − | 1.42523i | −0.431196 | + | 0.610865i | −1.93539 | + | 0.840657i | −0.0416251 | + | 0.608538i | 2.82163 | + | 0.387825i | 3.95867 | + | 5.60815i |
16.5 | −1.61130 | − | 0.979854i | 0.106997 | − | 1.56423i | 0.716048 | + | 1.38191i | 1.44267 | + | 0.626638i | −1.70513 | + | 2.41561i | 2.54898 | − | 1.10718i | −0.0570881 | + | 0.834598i | 0.536677 | + | 0.0737645i | −1.71056 | − | 2.42331i |
16.6 | −1.29591 | − | 0.788063i | 0.228574 | − | 3.34164i | 0.138222 | + | 0.266756i | 1.36313 | + | 0.592092i | −2.92964 | + | 4.15035i | −3.34717 | + | 1.45388i | −0.175912 | + | 2.57175i | −8.14226 | − | 1.11913i | −1.29990 | − | 1.84154i |
16.7 | −1.20024 | − | 0.729879i | −0.171528 | + | 2.50765i | −0.0122887 | − | 0.0237160i | −3.60151 | − | 1.56436i | 2.03616 | − | 2.88458i | 3.47311 | − | 1.50859i | −0.194286 | + | 2.84036i | −3.28684 | − | 0.451766i | 3.18087 | + | 4.50627i |
16.8 | −1.09201 | − | 0.664069i | −0.00466840 | + | 0.0682496i | −0.168621 | − | 0.325424i | 0.936249 | + | 0.406670i | 0.0504204 | − | 0.0714294i | 1.89133 | − | 0.821521i | −0.206405 | + | 3.01754i | 2.96742 | + | 0.407863i | −0.752341 | − | 1.06582i |
16.9 | −0.965710 | − | 0.587261i | −0.0593620 | + | 0.867841i | −0.332411 | − | 0.641523i | 3.35404 | + | 1.45687i | 0.566976 | − | 0.803221i | −4.23678 | + | 1.84029i | −0.209992 | + | 3.06997i | 2.22243 | + | 0.305466i | −2.38347 | − | 3.37661i |
16.10 | −0.331159 | − | 0.201382i | 0.0927188 | − | 1.35550i | −0.851019 | − | 1.64239i | −2.09811 | − | 0.911339i | −0.303679 | + | 0.430215i | −1.62492 | + | 0.705800i | −0.101825 | + | 1.48863i | 1.14327 | + | 0.157139i | 0.511282 | + | 0.724321i |
16.11 | −0.0387400 | − | 0.0235583i | −0.207250 | + | 3.02989i | −0.919184 | − | 1.77395i | 3.60660 | + | 1.56657i | 0.0794080 | − | 0.112496i | 4.01897 | − | 1.74568i | −0.0123703 | + | 0.180847i | −6.16522 | − | 0.847390i | −0.102814 | − | 0.145654i |
16.12 | 0.0563172 | + | 0.0342472i | 0.163040 | − | 2.38357i | −0.918131 | − | 1.77191i | 1.84624 | + | 0.801934i | 0.0908125 | − | 0.128652i | 3.29494 | − | 1.43119i | 0.0179727 | − | 0.262751i | −2.68275 | − | 0.368735i | 0.0765109 | + | 0.108391i |
16.13 | 0.143933 | + | 0.0875274i | −0.0735688 | + | 1.07554i | −0.907074 | − | 1.75057i | −0.165823 | − | 0.0720272i | −0.104728 | + | 0.148366i | −0.376419 | + | 0.163502i | 0.0456574 | − | 0.667487i | 1.82069 | + | 0.250248i | −0.0175630 | − | 0.0248811i |
16.14 | 0.296974 | + | 0.180594i | −0.178174 | + | 2.60481i | −0.864551 | − | 1.66851i | −1.64277 | − | 0.713557i | −0.523326 | + | 0.741384i | −2.71762 | + | 1.18043i | 0.0920117 | − | 1.34516i | −3.78124 | − | 0.519719i | −0.358997 | − | 0.508583i |
16.15 | 0.722881 | + | 0.439593i | 0.142533 | − | 2.08375i | −0.590816 | − | 1.14022i | 2.38739 | + | 1.03699i | 1.01904 | − | 1.44365i | −1.63366 | + | 0.709598i | 0.189618 | − | 2.77212i | −1.34965 | − | 0.185505i | 1.26994 | + | 1.79910i |
16.16 | 0.948416 | + | 0.576744i | −0.00917760 | + | 0.134172i | −0.353272 | − | 0.681783i | −3.16508 | − | 1.37479i | −0.0860870 | + | 0.121958i | 3.88742 | − | 1.68854i | 0.209666 | − | 3.06521i | 2.95414 | + | 0.406037i | −2.20891 | − | 3.12931i |
16.17 | 1.19118 | + | 0.724372i | −0.0329619 | + | 0.481886i | −0.0259378 | − | 0.0500577i | 2.50640 | + | 1.08868i | −0.388328 | + | 0.550136i | −0.0270097 | + | 0.0117320i | 0.195643 | − | 2.86019i | 2.74093 | + | 0.376732i | 2.19696 | + | 3.11239i |
16.18 | 1.33922 | + | 0.814396i | 0.180529 | − | 2.63925i | 0.210133 | + | 0.405538i | −1.94291 | − | 0.843923i | 2.39116 | − | 3.38750i | −1.25200 | + | 0.543822i | 0.165071 | − | 2.41326i | −3.96098 | − | 0.544424i | −1.91469 | − | 2.71249i |
16.19 | 1.78257 | + | 1.08401i | −0.170303 | + | 2.48974i | 1.08237 | + | 2.08887i | −0.215365 | − | 0.0935460i | −3.00247 | + | 4.25353i | 0.626806 | − | 0.272260i | −0.0502065 | + | 0.733994i | −3.19774 | − | 0.439519i | −0.282499 | − | 0.400209i |
16.20 | 2.00704 | + | 1.22051i | 0.0304939 | − | 0.445804i | 1.61844 | + | 3.12346i | 0.912837 | + | 0.396501i | 0.605311 | − | 0.857530i | −4.02881 | + | 1.74996i | −0.243317 | + | 3.55716i | 2.77425 | + | 0.381311i | 1.34817 | + | 1.90992i |
See next 80 embeddings (of 462 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
277.g | even | 23 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 277.2.g.a | ✓ | 462 |
277.g | even | 23 | 1 | inner | 277.2.g.a | ✓ | 462 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
277.2.g.a | ✓ | 462 | 1.a | even | 1 | 1 | trivial |
277.2.g.a | ✓ | 462 | 277.g | even | 23 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(277, [\chi])\).