Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2496,2,Mod(287,2496)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2496, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2496.287");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2496 = 2^{6} \cdot 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2496.j (of order \(2\), degree \(1\), 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: | \(19.9306603445\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
287.1 | 0 | −1.62133 | − | 0.609324i | 0 | −1.44117 | 0 | 3.46791i | 0 | 2.25745 | + | 1.97583i | 0 | ||||||||||||||
287.2 | 0 | −1.62133 | − | 0.609324i | 0 | 1.44117 | 0 | − | 3.46791i | 0 | 2.25745 | + | 1.97583i | 0 | |||||||||||||
287.3 | 0 | −1.62133 | + | 0.609324i | 0 | −1.44117 | 0 | − | 3.46791i | 0 | 2.25745 | − | 1.97583i | 0 | |||||||||||||
287.4 | 0 | −1.62133 | + | 0.609324i | 0 | 1.44117 | 0 | 3.46791i | 0 | 2.25745 | − | 1.97583i | 0 | ||||||||||||||
287.5 | 0 | −1.34468 | − | 1.09171i | 0 | −0.320469 | 0 | 1.42033i | 0 | 0.616340 | + | 2.93600i | 0 | ||||||||||||||
287.6 | 0 | −1.34468 | − | 1.09171i | 0 | 0.320469 | 0 | − | 1.42033i | 0 | 0.616340 | + | 2.93600i | 0 | |||||||||||||
287.7 | 0 | −1.34468 | + | 1.09171i | 0 | −0.320469 | 0 | − | 1.42033i | 0 | 0.616340 | − | 2.93600i | 0 | |||||||||||||
287.8 | 0 | −1.34468 | + | 1.09171i | 0 | 0.320469 | 0 | 1.42033i | 0 | 0.616340 | − | 2.93600i | 0 | ||||||||||||||
287.9 | 0 | −0.840553 | − | 1.51442i | 0 | −3.32809 | 0 | 3.31275i | 0 | −1.58694 | + | 2.54590i | 0 | ||||||||||||||
287.10 | 0 | −0.840553 | − | 1.51442i | 0 | 3.32809 | 0 | − | 3.31275i | 0 | −1.58694 | + | 2.54590i | 0 | |||||||||||||
287.11 | 0 | −0.840553 | + | 1.51442i | 0 | −3.32809 | 0 | − | 3.31275i | 0 | −1.58694 | − | 2.54590i | 0 | |||||||||||||
287.12 | 0 | −0.840553 | + | 1.51442i | 0 | 3.32809 | 0 | 3.31275i | 0 | −1.58694 | − | 2.54590i | 0 | ||||||||||||||
287.13 | 0 | −0.517777 | − | 1.65285i | 0 | −0.379917 | 0 | − | 2.38318i | 0 | −2.46381 | + | 1.71161i | 0 | |||||||||||||
287.14 | 0 | −0.517777 | − | 1.65285i | 0 | 0.379917 | 0 | 2.38318i | 0 | −2.46381 | + | 1.71161i | 0 | ||||||||||||||
287.15 | 0 | −0.517777 | + | 1.65285i | 0 | −0.379917 | 0 | 2.38318i | 0 | −2.46381 | − | 1.71161i | 0 | ||||||||||||||
287.16 | 0 | −0.517777 | + | 1.65285i | 0 | 0.379917 | 0 | − | 2.38318i | 0 | −2.46381 | − | 1.71161i | 0 | |||||||||||||
287.17 | 0 | 0.0969806 | − | 1.72933i | 0 | −3.03737 | 0 | 2.31810i | 0 | −2.98119 | − | 0.335424i | 0 | ||||||||||||||
287.18 | 0 | 0.0969806 | − | 1.72933i | 0 | 3.03737 | 0 | − | 2.31810i | 0 | −2.98119 | − | 0.335424i | 0 | |||||||||||||
287.19 | 0 | 0.0969806 | + | 1.72933i | 0 | −3.03737 | 0 | − | 2.31810i | 0 | −2.98119 | + | 0.335424i | 0 | |||||||||||||
287.20 | 0 | 0.0969806 | + | 1.72933i | 0 | 3.03737 | 0 | 2.31810i | 0 | −2.98119 | + | 0.335424i | 0 | ||||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
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 | 2496.2.j.e | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 2496.2.j.e | ✓ | 32 |
4.b | odd | 2 | 1 | 2496.2.j.f | yes | 32 | |
8.b | even | 2 | 1 | 2496.2.j.f | yes | 32 | |
8.d | odd | 2 | 1 | inner | 2496.2.j.e | ✓ | 32 |
12.b | even | 2 | 1 | 2496.2.j.f | yes | 32 | |
24.f | even | 2 | 1 | inner | 2496.2.j.e | ✓ | 32 |
24.h | odd | 2 | 1 | 2496.2.j.f | yes | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2496.2.j.e | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
2496.2.j.e | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
2496.2.j.e | ✓ | 32 | 8.d | odd | 2 | 1 | inner |
2496.2.j.e | ✓ | 32 | 24.f | even | 2 | 1 | inner |
2496.2.j.f | yes | 32 | 4.b | odd | 2 | 1 | |
2496.2.j.f | yes | 32 | 8.b | even | 2 | 1 | |
2496.2.j.f | yes | 32 | 12.b | even | 2 | 1 | |
2496.2.j.f | yes | 32 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2496, [\chi])\):
\( T_{5}^{16} - 54 T_{5}^{14} + 1149 T_{5}^{12} - 12248 T_{5}^{10} + 68228 T_{5}^{8} - 186336 T_{5}^{6} + \cdots + 2368 \) |
\( T_{19}^{8} + 8T_{19}^{7} - 68T_{19}^{6} - 588T_{19}^{5} + 708T_{19}^{4} + 10440T_{19}^{3} + 16920T_{19}^{2} + 1728T_{19} - 6912 \) |
\( T_{43}^{8} + 4T_{43}^{7} - 163T_{43}^{6} - 808T_{43}^{5} + 5772T_{43}^{4} + 33760T_{43}^{3} + 22816T_{43}^{2} - 17472T_{43} - 11456 \) |