Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [368,2,Mod(93,368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("368.93");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 368 = 2^{4} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 368.j (of order \(4\), 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: | \(2.93849479438\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
93.1 | −1.41334 | − | 0.0497254i | 0.387219 | + | 0.387219i | 1.99505 | + | 0.140558i | 1.36361 | − | 1.36361i | −0.528018 | − | 0.566527i | − | 1.65176i | −2.81270 | − | 0.297860i | − | 2.70012i | −1.99505 | + | 1.85944i | ||
93.2 | −1.27953 | + | 0.602332i | 0.203642 | + | 0.203642i | 1.27439 | − | 1.54140i | 1.88186 | − | 1.88186i | −0.383226 | − | 0.137906i | 4.71330i | −0.702189 | + | 2.73988i | − | 2.91706i | −1.27439 | + | 3.54140i | |||
93.3 | −1.20316 | + | 0.743237i | 2.30229 | + | 2.30229i | 0.895196 | − | 1.78847i | 1.94640 | − | 1.94640i | −4.48117 | − | 1.05888i | − | 3.32801i | 0.252192 | + | 2.81716i | 7.60105i | −0.895196 | + | 3.78847i | |||
93.4 | −1.11221 | − | 0.873496i | −1.75354 | − | 1.75354i | 0.474011 | + | 1.94302i | 0.238712 | − | 0.238712i | 0.418590 | + | 3.48200i | − | 4.31299i | 1.17002 | − | 2.57508i | 3.14978i | −0.474011 | + | 0.0569833i | |||
93.5 | −0.741286 | − | 1.20436i | 1.96106 | + | 1.96106i | −0.900989 | + | 1.78556i | −0.463078 | + | 0.463078i | 0.908123 | − | 3.81553i | 1.33926i | 2.81835 | − | 0.238491i | 4.69148i | 0.900989 | + | 0.214442i | ||||
93.6 | −0.408999 | − | 1.35378i | −1.35709 | − | 1.35709i | −1.66544 | + | 1.10739i | −0.944781 | + | 0.944781i | −1.28215 | + | 2.39224i | 1.03914i | 2.18032 | + | 1.80172i | 0.683369i | 1.66544 | + | 0.892612i | ||||
93.7 | −0.340155 | + | 1.37270i | −1.72800 | − | 1.72800i | −1.76859 | − | 0.933858i | 1.71285 | − | 1.71285i | 2.95981 | − | 1.78424i | 1.69946i | 1.88350 | − | 2.11008i | 2.97199i | 1.76859 | + | 2.93386i | ||||
93.8 | 0.326476 | + | 1.37601i | −0.248916 | − | 0.248916i | −1.78683 | + | 0.898472i | 1.04954 | − | 1.04954i | 0.261246 | − | 0.423776i | − | 3.80824i | −1.81967 | − | 2.16537i | − | 2.87608i | 1.78683 | + | 1.10153i | ||
93.9 | 0.730764 | + | 1.21078i | −2.15319 | − | 2.15319i | −0.931969 | + | 1.76959i | 0.480015 | − | 0.480015i | 1.03356 | − | 4.18052i | 2.39661i | −2.82363 | + | 0.164741i | 6.27249i | 0.931969 | + | 0.230414i | ||||
93.10 | 0.737124 | − | 1.20692i | −0.383765 | − | 0.383765i | −0.913296 | − | 1.77929i | −1.94404 | + | 1.94404i | −0.746054 | + | 0.180290i | − | 1.91069i | −2.82067 | − | 0.209289i | − | 2.70545i | 0.913296 | + | 3.77929i | ||
93.11 | 1.29063 | − | 0.578171i | 1.73006 | + | 1.73006i | 1.33144 | − | 1.49241i | −1.86880 | + | 1.86880i | 3.23314 | + | 1.23259i | 1.07441i | 0.855521 | − | 2.69594i | 2.98625i | −1.33144 | + | 3.49241i | ||||
93.12 | 1.41369 | − | 0.0386035i | −0.959768 | − | 0.959768i | 1.99702 | − | 0.109146i | −1.45229 | + | 1.45229i | −1.39386 | − | 1.31976i | − | 3.25050i | 2.81895 | − | 0.231391i | − | 1.15769i | −1.99702 | + | 2.10915i | ||
277.1 | −1.41334 | + | 0.0497254i | 0.387219 | − | 0.387219i | 1.99505 | − | 0.140558i | 1.36361 | + | 1.36361i | −0.528018 | + | 0.566527i | 1.65176i | −2.81270 | + | 0.297860i | 2.70012i | −1.99505 | − | 1.85944i | ||||
277.2 | −1.27953 | − | 0.602332i | 0.203642 | − | 0.203642i | 1.27439 | + | 1.54140i | 1.88186 | + | 1.88186i | −0.383226 | + | 0.137906i | − | 4.71330i | −0.702189 | − | 2.73988i | 2.91706i | −1.27439 | − | 3.54140i | |||
277.3 | −1.20316 | − | 0.743237i | 2.30229 | − | 2.30229i | 0.895196 | + | 1.78847i | 1.94640 | + | 1.94640i | −4.48117 | + | 1.05888i | 3.32801i | 0.252192 | − | 2.81716i | − | 7.60105i | −0.895196 | − | 3.78847i | |||
277.4 | −1.11221 | + | 0.873496i | −1.75354 | + | 1.75354i | 0.474011 | − | 1.94302i | 0.238712 | + | 0.238712i | 0.418590 | − | 3.48200i | 4.31299i | 1.17002 | + | 2.57508i | − | 3.14978i | −0.474011 | − | 0.0569833i | |||
277.5 | −0.741286 | + | 1.20436i | 1.96106 | − | 1.96106i | −0.900989 | − | 1.78556i | −0.463078 | − | 0.463078i | 0.908123 | + | 3.81553i | − | 1.33926i | 2.81835 | + | 0.238491i | − | 4.69148i | 0.900989 | − | 0.214442i | ||
277.6 | −0.408999 | + | 1.35378i | −1.35709 | + | 1.35709i | −1.66544 | − | 1.10739i | −0.944781 | − | 0.944781i | −1.28215 | − | 2.39224i | − | 1.03914i | 2.18032 | − | 1.80172i | − | 0.683369i | 1.66544 | − | 0.892612i | ||
277.7 | −0.340155 | − | 1.37270i | −1.72800 | + | 1.72800i | −1.76859 | + | 0.933858i | 1.71285 | + | 1.71285i | 2.95981 | + | 1.78424i | − | 1.69946i | 1.88350 | + | 2.11008i | − | 2.97199i | 1.76859 | − | 2.93386i | ||
277.8 | 0.326476 | − | 1.37601i | −0.248916 | + | 0.248916i | −1.78683 | − | 0.898472i | 1.04954 | + | 1.04954i | 0.261246 | + | 0.423776i | 3.80824i | −1.81967 | + | 2.16537i | 2.87608i | 1.78683 | − | 1.10153i | ||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 368.2.j.d | ✓ | 24 |
4.b | odd | 2 | 1 | 1472.2.j.d | 24 | ||
16.e | even | 4 | 1 | inner | 368.2.j.d | ✓ | 24 |
16.f | odd | 4 | 1 | 1472.2.j.d | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
368.2.j.d | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
368.2.j.d | ✓ | 24 | 16.e | even | 4 | 1 | inner |
1472.2.j.d | 24 | 4.b | odd | 2 | 1 | ||
1472.2.j.d | 24 | 16.f | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 4 T_{3}^{23} + 8 T_{3}^{22} + 12 T_{3}^{21} + 208 T_{3}^{20} + 828 T_{3}^{19} + 1720 T_{3}^{18} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(368, [\chi])\).