Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,2,Mod(22,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([14, 24]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.22");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 441.ba (of order \(21\), degree \(12\), 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: | \(3.52140272914\) |
Analytic rank: | \(0\) |
Dimension: | \(648\) |
Relative dimension: | \(54\) over \(\Q(\zeta_{21})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{21}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
22.1 | −2.28558 | − | 1.55828i | −0.781588 | − | 1.54568i | 2.06496 | + | 5.26143i | −0.806726 | − | 0.748533i | −0.622222 | + | 4.75071i | 1.94410 | + | 1.79457i | 2.24807 | − | 9.84945i | −1.77824 | + | 2.41617i | 0.677414 | + | 2.96794i |
22.2 | −2.20796 | − | 1.50536i | −1.57686 | + | 0.716598i | 1.87830 | + | 4.78582i | 2.79913 | + | 2.59721i | 4.56039 | + | 0.791525i | −2.53111 | − | 0.770385i | 1.86790 | − | 8.18382i | 1.97297 | − | 2.25995i | −2.27063 | − | 9.94826i |
22.3 | −2.13496 | − | 1.45559i | 1.68474 | + | 0.402062i | 1.70863 | + | 4.35352i | 2.61229 | + | 2.42385i | −3.01161 | − | 3.31068i | 2.64469 | + | 0.0747901i | 1.53912 | − | 6.74332i | 2.67669 | + | 1.35474i | −2.04900 | − | 8.97724i |
22.4 | −2.12732 | − | 1.45038i | 1.48498 | + | 0.891527i | 1.69120 | + | 4.30910i | −2.50301 | − | 2.32245i | −1.86598 | − | 4.05036i | −2.17833 | + | 1.50162i | 1.50627 | − | 6.59941i | 1.41036 | + | 2.64781i | 1.95625 | + | 8.57090i |
22.5 | −2.11562 | − | 1.44241i | −0.314751 | + | 1.70321i | 1.66464 | + | 4.24143i | −1.01272 | − | 0.939663i | 3.12262 | − | 3.14936i | 1.62237 | − | 2.08995i | 1.45657 | − | 6.38167i | −2.80186 | − | 1.07218i | 0.787149 | + | 3.44872i |
22.6 | −2.08565 | − | 1.42197i | 1.06450 | − | 1.36632i | 1.59724 | + | 4.06970i | 0.525543 | + | 0.487633i | −4.16304 | + | 1.33598i | −2.64570 | − | 0.0162007i | 1.33231 | − | 5.83723i | −0.733680 | − | 2.90890i | −0.402698 | − | 1.76434i |
22.7 | −1.80236 | − | 1.22883i | −1.63261 | + | 0.578434i | 1.00781 | + | 2.56785i | −1.67523 | − | 1.55438i | 3.65335 | + | 0.963652i | −0.0954394 | + | 2.64403i | 0.368203 | − | 1.61320i | 2.33083 | − | 1.88871i | 1.10930 | + | 4.86014i |
22.8 | −1.76857 | − | 1.20579i | −1.66628 | − | 0.472785i | 0.943225 | + | 2.40330i | −0.0377034 | − | 0.0349836i | 2.37684 | + | 2.84533i | 1.13677 | − | 2.38909i | 0.277100 | − | 1.21405i | 2.55295 | + | 1.57558i | 0.0244981 | + | 0.107333i |
22.9 | −1.71062 | − | 1.16628i | −0.227483 | − | 1.71705i | 0.835328 | + | 2.12838i | 1.88227 | + | 1.74649i | −1.61342 | + | 3.20252i | 1.73079 | − | 2.00109i | 0.131960 | − | 0.578153i | −2.89650 | + | 0.781197i | −1.18295 | − | 5.18284i |
22.10 | −1.58020 | − | 1.07736i | −0.0140747 | − | 1.73199i | 0.605641 | + | 1.54315i | −3.06138 | − | 2.84054i | −1.84374 | + | 2.75206i | −1.13241 | − | 2.39116i | −0.145657 | + | 0.638167i | −2.99960 | + | 0.0487547i | 1.77730 | + | 7.78684i |
22.11 | −1.53495 | − | 1.04651i | 1.72901 | − | 0.102560i | 0.530198 | + | 1.35092i | −0.256118 | − | 0.237642i | −2.76127 | − | 1.65200i | −0.648127 | − | 2.56514i | −0.226850 | + | 0.993895i | 2.97896 | − | 0.354656i | 0.144432 | + | 0.632798i |
22.12 | −1.50607 | − | 1.02682i | −0.181074 | + | 1.72256i | 0.483202 | + | 1.23118i | 0.989195 | + | 0.917838i | 2.04147 | − | 2.40836i | −2.54327 | + | 0.729243i | −0.274759 | + | 1.20380i | −2.93442 | − | 0.623821i | −0.547340 | − | 2.39805i |
22.13 | −1.48693 | − | 1.01377i | 1.13536 | − | 1.30804i | 0.452539 | + | 1.15305i | 1.39969 | + | 1.29873i | −3.01424 | + | 0.793970i | −0.229184 | + | 2.63581i | −0.304877 | + | 1.33575i | −0.421935 | − | 2.97018i | −0.764633 | − | 3.35008i |
22.14 | −1.31403 | − | 0.895892i | 1.71871 | + | 0.214588i | 0.193375 | + | 0.492712i | −1.33434 | − | 1.23808i | −2.06619 | − | 1.82175i | 2.50568 | + | 0.849447i | −0.520469 | + | 2.28032i | 2.90790 | + | 0.737628i | 0.644172 | + | 2.82230i |
22.15 | −1.21486 | − | 0.828276i | −1.14380 | + | 1.30067i | 0.0591562 | + | 0.150728i | 2.23394 | + | 2.07279i | 2.46686 | − | 0.632745i | 2.42421 | + | 1.05980i | −0.601388 | + | 2.63485i | −0.383462 | − | 2.97539i | −0.997073 | − | 4.36846i |
22.16 | −1.20704 | − | 0.822949i | −1.50963 | − | 0.849130i | 0.0490286 | + | 0.124923i | −0.415450 | − | 0.385481i | 1.12340 | + | 2.26728i | −2.62355 | + | 0.341999i | −0.606532 | + | 2.65739i | 1.55796 | + | 2.56374i | 0.184235 | + | 0.807187i |
22.17 | −1.11972 | − | 0.763412i | 0.735246 | + | 1.56825i | −0.0597084 | − | 0.152135i | −1.91052 | − | 1.77271i | 0.373953 | − | 2.31730i | 2.60358 | + | 0.470493i | −0.652406 | + | 2.85838i | −1.91883 | + | 2.30610i | 0.785945 | + | 3.44345i |
22.18 | −0.946246 | − | 0.645140i | 1.51934 | + | 0.831632i | −0.251505 | − | 0.640825i | 1.33307 | + | 1.23691i | −0.901149 | − | 1.76711i | −1.78159 | + | 1.95600i | −0.685118 | + | 3.00170i | 1.61678 | + | 2.52706i | −0.463433 | − | 2.03043i |
22.19 | −0.912794 | − | 0.622332i | −0.609005 | + | 1.62145i | −0.284787 | − | 0.725626i | −1.69424 | − | 1.57203i | 1.56498 | − | 1.10105i | −1.84008 | − | 1.90108i | −0.683292 | + | 2.99370i | −2.25823 | − | 1.97495i | 0.568171 | + | 2.48932i |
22.20 | −0.819051 | − | 0.558419i | −0.438134 | − | 1.67572i | −0.371670 | − | 0.947000i | −1.06857 | − | 0.991491i | −0.576900 | + | 1.61716i | 0.639861 | + | 2.56721i | −0.665577 | + | 2.91608i | −2.61608 | + | 1.46838i | 0.321548 | + | 1.40879i |
See next 80 embeddings (of 648 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
49.e | even | 7 | 1 | inner |
441.ba | even | 21 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 441.2.ba.a | ✓ | 648 |
9.c | even | 3 | 1 | inner | 441.2.ba.a | ✓ | 648 |
49.e | even | 7 | 1 | inner | 441.2.ba.a | ✓ | 648 |
441.ba | even | 21 | 1 | inner | 441.2.ba.a | ✓ | 648 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
441.2.ba.a | ✓ | 648 | 1.a | even | 1 | 1 | trivial |
441.2.ba.a | ✓ | 648 | 9.c | even | 3 | 1 | inner |
441.2.ba.a | ✓ | 648 | 49.e | even | 7 | 1 | inner |
441.2.ba.a | ✓ | 648 | 441.ba | even | 21 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(441, [\chi])\).