Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3360,2,Mod(911,3360)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3360, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3360.911");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3360.e (of order \(2\), degree \(1\), not 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: | \(26.8297350792\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Twist minimal: | no (minimal twist has level 840) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
911.1 | 0 | −1.72227 | − | 0.183849i | 0 | −1.00000 | 0 | − | 1.00000i | 0 | 2.93240 | + | 0.633274i | 0 | |||||||||||||
911.2 | 0 | −1.72227 | + | 0.183849i | 0 | −1.00000 | 0 | 1.00000i | 0 | 2.93240 | − | 0.633274i | 0 | ||||||||||||||
911.3 | 0 | −1.70474 | − | 0.306378i | 0 | −1.00000 | 0 | 1.00000i | 0 | 2.81226 | + | 1.04459i | 0 | ||||||||||||||
911.4 | 0 | −1.70474 | + | 0.306378i | 0 | −1.00000 | 0 | − | 1.00000i | 0 | 2.81226 | − | 1.04459i | 0 | |||||||||||||
911.5 | 0 | −1.67255 | − | 0.450097i | 0 | −1.00000 | 0 | − | 1.00000i | 0 | 2.59483 | + | 1.50562i | 0 | |||||||||||||
911.6 | 0 | −1.67255 | + | 0.450097i | 0 | −1.00000 | 0 | 1.00000i | 0 | 2.59483 | − | 1.50562i | 0 | ||||||||||||||
911.7 | 0 | −1.52280 | − | 0.825275i | 0 | −1.00000 | 0 | − | 1.00000i | 0 | 1.63784 | + | 2.51346i | 0 | |||||||||||||
911.8 | 0 | −1.52280 | + | 0.825275i | 0 | −1.00000 | 0 | 1.00000i | 0 | 1.63784 | − | 2.51346i | 0 | ||||||||||||||
911.9 | 0 | −1.51567 | − | 0.838298i | 0 | −1.00000 | 0 | − | 1.00000i | 0 | 1.59451 | + | 2.54117i | 0 | |||||||||||||
911.10 | 0 | −1.51567 | + | 0.838298i | 0 | −1.00000 | 0 | 1.00000i | 0 | 1.59451 | − | 2.54117i | 0 | ||||||||||||||
911.11 | 0 | −1.39892 | − | 1.02128i | 0 | −1.00000 | 0 | 1.00000i | 0 | 0.913962 | + | 2.85739i | 0 | ||||||||||||||
911.12 | 0 | −1.39892 | + | 1.02128i | 0 | −1.00000 | 0 | − | 1.00000i | 0 | 0.913962 | − | 2.85739i | 0 | |||||||||||||
911.13 | 0 | −1.20864 | − | 1.24064i | 0 | −1.00000 | 0 | 1.00000i | 0 | −0.0783833 | + | 2.99898i | 0 | ||||||||||||||
911.14 | 0 | −1.20864 | + | 1.24064i | 0 | −1.00000 | 0 | − | 1.00000i | 0 | −0.0783833 | − | 2.99898i | 0 | |||||||||||||
911.15 | 0 | −1.08178 | − | 1.35269i | 0 | −1.00000 | 0 | 1.00000i | 0 | −0.659524 | + | 2.92661i | 0 | ||||||||||||||
911.16 | 0 | −1.08178 | + | 1.35269i | 0 | −1.00000 | 0 | − | 1.00000i | 0 | −0.659524 | − | 2.92661i | 0 | |||||||||||||
911.17 | 0 | −0.663478 | − | 1.59994i | 0 | −1.00000 | 0 | − | 1.00000i | 0 | −2.11959 | + | 2.12305i | 0 | |||||||||||||
911.18 | 0 | −0.663478 | + | 1.59994i | 0 | −1.00000 | 0 | 1.00000i | 0 | −2.11959 | − | 2.12305i | 0 | ||||||||||||||
911.19 | 0 | −0.467013 | − | 1.66790i | 0 | −1.00000 | 0 | 1.00000i | 0 | −2.56380 | + | 1.55787i | 0 | ||||||||||||||
911.20 | 0 | −0.467013 | + | 1.66790i | 0 | −1.00000 | 0 | − | 1.00000i | 0 | −2.56380 | − | 1.55787i | 0 | |||||||||||||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
24.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3360.2.e.c | 44 | |
3.b | odd | 2 | 1 | 3360.2.e.d | 44 | ||
4.b | odd | 2 | 1 | 840.2.e.d | yes | 44 | |
8.b | even | 2 | 1 | 840.2.e.c | ✓ | 44 | |
8.d | odd | 2 | 1 | 3360.2.e.d | 44 | ||
12.b | even | 2 | 1 | 840.2.e.c | ✓ | 44 | |
24.f | even | 2 | 1 | inner | 3360.2.e.c | 44 | |
24.h | odd | 2 | 1 | 840.2.e.d | yes | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
840.2.e.c | ✓ | 44 | 8.b | even | 2 | 1 | |
840.2.e.c | ✓ | 44 | 12.b | even | 2 | 1 | |
840.2.e.d | yes | 44 | 4.b | odd | 2 | 1 | |
840.2.e.d | yes | 44 | 24.h | odd | 2 | 1 | |
3360.2.e.c | 44 | 1.a | even | 1 | 1 | trivial | |
3360.2.e.c | 44 | 24.f | even | 2 | 1 | inner | |
3360.2.e.d | 44 | 3.b | odd | 2 | 1 | ||
3360.2.e.d | 44 | 8.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3360, [\chi])\):
\( T_{11}^{44} + 264 T_{11}^{42} + 31844 T_{11}^{40} + 2327928 T_{11}^{38} + 115382518 T_{11}^{36} + \cdots + 202012780134400 \) |
\( T_{23}^{22} + 8 T_{23}^{21} - 216 T_{23}^{20} - 1776 T_{23}^{19} + 18240 T_{23}^{18} + 153728 T_{23}^{17} + \cdots - 527433728 \) |