Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [276,2,Mod(7,276)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(276, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 0, 19]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("276.7");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 276 = 2^{2} \cdot 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 276.m (of order \(22\), degree \(10\), 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: | \(2.20387109579\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | −1.40160 | + | 0.188444i | 0.540641 | + | 0.841254i | 1.92898 | − | 0.528247i | −1.29084 | + | 0.589505i | −0.916292 | − | 1.07722i | 1.16908 | − | 0.343272i | −2.60411 | + | 1.10390i | −0.415415 | + | 0.909632i | 1.69815 | − | 1.06950i |
7.2 | −1.39487 | − | 0.233083i | −0.540641 | − | 0.841254i | 1.89135 | + | 0.650241i | −1.34962 | + | 0.616349i | 0.558044 | + | 1.29946i | 3.09157 | − | 0.907767i | −2.48663 | − | 1.34784i | −0.415415 | + | 0.909632i | 2.02620 | − | 0.545157i |
7.3 | −1.35202 | + | 0.414766i | −0.540641 | − | 0.841254i | 1.65594 | − | 1.12155i | 3.59483 | − | 1.64171i | 1.07988 | + | 0.913155i | 0.696898 | − | 0.204628i | −1.77369 | + | 2.20319i | −0.415415 | + | 0.909632i | −4.17938 | + | 3.71064i |
7.4 | −1.24424 | − | 0.672203i | −0.540641 | − | 0.841254i | 1.09629 | + | 1.67277i | 1.03264 | − | 0.471590i | 0.107195 | + | 1.41015i | −4.96264 | + | 1.45716i | −0.239604 | − | 2.81826i | −0.415415 | + | 0.909632i | −1.60186 | − | 0.107370i |
7.5 | −1.14571 | − | 0.829070i | 0.540641 | + | 0.841254i | 0.625286 | + | 1.89974i | −3.51516 | + | 1.60532i | 0.0780426 | − | 1.41206i | 0.708179 | − | 0.207940i | 0.858625 | − | 2.69495i | −0.415415 | + | 0.909632i | 5.35826 | + | 1.07509i |
7.6 | −1.03590 | + | 0.962761i | 0.540641 | + | 0.841254i | 0.146182 | − | 1.99465i | 0.0885609 | − | 0.0404444i | −1.36998 | − | 0.350947i | 3.48963 | − | 1.02465i | 1.76894 | + | 2.20700i | −0.415415 | + | 0.909632i | −0.0528020 | + | 0.127159i |
7.7 | −0.916755 | − | 1.07683i | 0.540641 | + | 0.841254i | −0.319120 | + | 1.97438i | 1.92635 | − | 0.879733i | 0.410251 | − | 1.35340i | −0.906673 | + | 0.266223i | 2.41862 | − | 1.46638i | −0.415415 | + | 0.909632i | −2.71331 | − | 1.26785i |
7.8 | −0.805537 | + | 1.16237i | −0.540641 | − | 0.841254i | −0.702219 | − | 1.87267i | 0.0885609 | − | 0.0404444i | 1.41336 | + | 0.0492354i | −3.48963 | + | 1.02465i | 2.74240 | + | 0.692265i | −0.415415 | + | 0.909632i | −0.0243277 | + | 0.135520i |
7.9 | −0.723978 | − | 1.21485i | −0.540641 | − | 0.841254i | −0.951711 | + | 1.75905i | −2.68981 | + | 1.22840i | −0.630583 | + | 1.26585i | 0.0149354 | − | 0.00438542i | 2.82599 | − | 0.117329i | −0.415415 | + | 0.909632i | 3.43968 | + | 2.37838i |
7.10 | −0.416968 | − | 1.35135i | −0.540641 | − | 0.841254i | −1.65228 | + | 1.12694i | 3.16176 | − | 1.44393i | −0.911395 | + | 1.08137i | 3.54109 | − | 1.03976i | 2.21183 | + | 1.76290i | −0.415415 | + | 0.909632i | −3.26960 | − | 3.67057i |
7.11 | −0.218131 | + | 1.39729i | 0.540641 | + | 0.841254i | −1.90484 | − | 0.609585i | 3.59483 | − | 1.64171i | −1.29341 | + | 0.571928i | −0.696898 | + | 0.204628i | 1.26727 | − | 2.52864i | −0.415415 | + | 0.909632i | 1.50979 | + | 5.38113i |
7.12 | 0.0129431 | + | 1.41415i | −0.540641 | − | 0.841254i | −1.99966 | + | 0.0366070i | −1.29084 | + | 0.589505i | 1.18266 | − | 0.775438i | −1.16908 | + | 0.343272i | −0.0776497 | − | 2.82736i | −0.415415 | + | 0.909632i | −0.850358 | − | 1.81781i |
7.13 | 0.101854 | − | 1.41054i | 0.540641 | + | 0.841254i | −1.97925 | − | 0.287338i | −1.98780 | + | 0.907796i | 1.24169 | − | 0.676911i | −4.66467 | + | 1.36967i | −0.606897 | + | 2.76255i | −0.415415 | + | 0.909632i | 1.07802 | + | 2.89633i |
7.14 | 0.190297 | − | 1.40135i | −0.540641 | − | 0.841254i | −1.92757 | − | 0.533346i | −0.953526 | + | 0.435461i | −1.28177 | + | 0.597540i | −1.56668 | + | 0.460018i | −1.11422 | + | 2.59972i | −0.415415 | + | 0.909632i | 0.428780 | + | 1.41909i |
7.15 | 0.429221 | + | 1.34750i | 0.540641 | + | 0.841254i | −1.63154 | + | 1.15676i | −1.34962 | + | 0.616349i | −0.901539 | + | 1.08960i | −3.09157 | + | 0.907767i | −2.25902 | − | 1.70200i | −0.415415 | + | 0.909632i | −1.40982 | − | 1.55406i |
7.16 | 0.740078 | − | 1.20511i | 0.540641 | + | 0.841254i | −0.904569 | − | 1.78375i | 1.98260 | − | 0.905423i | 1.41392 | − | 0.0289372i | 1.01964 | − | 0.299394i | −2.81906 | − | 0.230009i | −0.415415 | + | 0.909632i | 0.376146 | − | 3.05933i |
7.17 | 0.842436 | + | 1.13591i | 0.540641 | + | 0.841254i | −0.580604 | + | 1.91387i | 1.03264 | − | 0.471590i | −0.500137 | + | 1.32282i | 4.96264 | − | 1.45716i | −2.66311 | + | 0.952796i | −0.415415 | + | 0.909632i | 1.40562 | + | 0.775704i |
7.18 | 0.983682 | + | 1.01606i | −0.540641 | − | 0.841254i | −0.0647382 | + | 1.99895i | −3.51516 | + | 1.60532i | 0.322942 | − | 1.37685i | −0.708179 | + | 0.207940i | −2.09473 | + | 1.90056i | −0.415415 | + | 0.909632i | −5.08889 | − | 1.99247i |
7.19 | 1.08752 | − | 0.904050i | −0.540641 | − | 0.841254i | 0.365388 | − | 1.96634i | 1.98260 | − | 0.905423i | −1.34849 | − | 0.426112i | −1.01964 | + | 0.299394i | −1.38030 | − | 2.46876i | −0.415415 | + | 0.909632i | 1.33756 | − | 2.77703i |
7.20 | 1.19634 | + | 0.754175i | −0.540641 | − | 0.841254i | 0.862440 | + | 1.80449i | 1.92635 | − | 0.879733i | −0.0123356 | − | 1.41416i | 0.906673 | − | 0.266223i | −0.329137 | + | 2.80921i | −0.415415 | + | 0.909632i | 2.96803 | + | 0.400347i |
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
92.h | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 276.2.m.a | ✓ | 240 |
3.b | odd | 2 | 1 | 828.2.u.c | 240 | ||
4.b | odd | 2 | 1 | inner | 276.2.m.a | ✓ | 240 |
12.b | even | 2 | 1 | 828.2.u.c | 240 | ||
23.d | odd | 22 | 1 | inner | 276.2.m.a | ✓ | 240 |
69.g | even | 22 | 1 | 828.2.u.c | 240 | ||
92.h | even | 22 | 1 | inner | 276.2.m.a | ✓ | 240 |
276.j | odd | 22 | 1 | 828.2.u.c | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
276.2.m.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
276.2.m.a | ✓ | 240 | 4.b | odd | 2 | 1 | inner |
276.2.m.a | ✓ | 240 | 23.d | odd | 22 | 1 | inner |
276.2.m.a | ✓ | 240 | 92.h | even | 22 | 1 | inner |
828.2.u.c | 240 | 3.b | odd | 2 | 1 | ||
828.2.u.c | 240 | 12.b | even | 2 | 1 | ||
828.2.u.c | 240 | 69.g | even | 22 | 1 | ||
828.2.u.c | 240 | 276.j | odd | 22 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(276, [\chi])\).