Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [552,2,Mod(25,552)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(552, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("552.25");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 552 = 2^{3} \cdot 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 552.q (of order \(11\), 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: | \(4.40774219157\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{11})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25.1 | 0 | −0.142315 | − | 0.989821i | 0 | −2.98780 | − | 0.877298i | 0 | 0.0902354 | − | 0.104137i | 0 | −0.959493 | + | 0.281733i | 0 | ||||||||||
25.2 | 0 | −0.142315 | − | 0.989821i | 0 | −0.768760 | − | 0.225728i | 0 | −0.186229 | + | 0.214919i | 0 | −0.959493 | + | 0.281733i | 0 | ||||||||||
25.3 | 0 | −0.142315 | − | 0.989821i | 0 | 2.63475 | + | 0.773633i | 0 | −2.50193 | + | 2.88738i | 0 | −0.959493 | + | 0.281733i | 0 | ||||||||||
49.1 | 0 | 0.415415 | − | 0.909632i | 0 | −1.96611 | + | 2.26901i | 0 | 3.59431 | − | 2.30992i | 0 | −0.654861 | − | 0.755750i | 0 | ||||||||||
49.2 | 0 | 0.415415 | − | 0.909632i | 0 | −1.13129 | + | 1.30558i | 0 | −2.90759 | + | 1.86859i | 0 | −0.654861 | − | 0.755750i | 0 | ||||||||||
49.3 | 0 | 0.415415 | − | 0.909632i | 0 | 0.531006 | − | 0.612813i | 0 | 0.0574024 | − | 0.0368903i | 0 | −0.654861 | − | 0.755750i | 0 | ||||||||||
73.1 | 0 | −0.959493 | + | 0.281733i | 0 | −1.03173 | − | 0.663053i | 0 | −0.0639556 | − | 0.444821i | 0 | 0.841254 | − | 0.540641i | 0 | ||||||||||
73.2 | 0 | −0.959493 | + | 0.281733i | 0 | 0.191143 | + | 0.122840i | 0 | 0.0348746 | + | 0.242558i | 0 | 0.841254 | − | 0.540641i | 0 | ||||||||||
73.3 | 0 | −0.959493 | + | 0.281733i | 0 | 3.62490 | + | 2.32958i | 0 | −0.646888 | − | 4.49921i | 0 | 0.841254 | − | 0.540641i | 0 | ||||||||||
121.1 | 0 | −0.959493 | − | 0.281733i | 0 | −1.03173 | + | 0.663053i | 0 | −0.0639556 | + | 0.444821i | 0 | 0.841254 | + | 0.540641i | 0 | ||||||||||
121.2 | 0 | −0.959493 | − | 0.281733i | 0 | 0.191143 | − | 0.122840i | 0 | 0.0348746 | − | 0.242558i | 0 | 0.841254 | + | 0.540641i | 0 | ||||||||||
121.3 | 0 | −0.959493 | − | 0.281733i | 0 | 3.62490 | − | 2.32958i | 0 | −0.646888 | + | 4.49921i | 0 | 0.841254 | + | 0.540641i | 0 | ||||||||||
169.1 | 0 | 0.415415 | + | 0.909632i | 0 | −1.96611 | − | 2.26901i | 0 | 3.59431 | + | 2.30992i | 0 | −0.654861 | + | 0.755750i | 0 | ||||||||||
169.2 | 0 | 0.415415 | + | 0.909632i | 0 | −1.13129 | − | 1.30558i | 0 | −2.90759 | − | 1.86859i | 0 | −0.654861 | + | 0.755750i | 0 | ||||||||||
169.3 | 0 | 0.415415 | + | 0.909632i | 0 | 0.531006 | + | 0.612813i | 0 | 0.0574024 | + | 0.0368903i | 0 | −0.654861 | + | 0.755750i | 0 | ||||||||||
193.1 | 0 | −0.654861 | − | 0.755750i | 0 | −0.494929 | − | 3.44231i | 0 | 0.813772 | − | 1.78191i | 0 | −0.142315 | + | 0.989821i | 0 | ||||||||||
193.2 | 0 | −0.654861 | − | 0.755750i | 0 | 0.140637 | + | 0.978155i | 0 | −1.46137 | + | 3.19994i | 0 | −0.142315 | + | 0.989821i | 0 | ||||||||||
193.3 | 0 | −0.654861 | − | 0.755750i | 0 | 0.309108 | + | 2.14989i | 0 | 1.22533 | − | 2.68309i | 0 | −0.142315 | + | 0.989821i | 0 | ||||||||||
265.1 | 0 | −0.142315 | + | 0.989821i | 0 | −2.98780 | + | 0.877298i | 0 | 0.0902354 | + | 0.104137i | 0 | −0.959493 | − | 0.281733i | 0 | ||||||||||
265.2 | 0 | −0.142315 | + | 0.989821i | 0 | −0.768760 | + | 0.225728i | 0 | −0.186229 | − | 0.214919i | 0 | −0.959493 | − | 0.281733i | 0 | ||||||||||
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 552.2.q.a | ✓ | 30 |
23.c | even | 11 | 1 | inner | 552.2.q.a | ✓ | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
552.2.q.a | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
552.2.q.a | ✓ | 30 | 23.c | even | 11 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{30} + 5 T_{5}^{28} + 17 T_{5}^{27} + 80 T_{5}^{26} + 511 T_{5}^{25} + 2218 T_{5}^{24} + \cdots + 214369 \) acting on \(S_{2}^{\mathrm{new}}(552, [\chi])\).