Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2240,2,Mod(1569,2240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2240.1569");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2240 = 2^{6} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2240.l (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: | \(17.8864900528\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1569.1 | 0 | −3.11575 | 0 | −0.660542 | − | 2.13628i | 0 | 1.00000i | 0 | 6.70793 | 0 | ||||||||||||||||
1569.2 | 0 | −3.11575 | 0 | −0.660542 | + | 2.13628i | 0 | − | 1.00000i | 0 | 6.70793 | 0 | |||||||||||||||
1569.3 | 0 | −2.41750 | 0 | −1.94498 | − | 1.10320i | 0 | − | 1.00000i | 0 | 2.84431 | 0 | |||||||||||||||
1569.4 | 0 | −2.41750 | 0 | −1.94498 | + | 1.10320i | 0 | 1.00000i | 0 | 2.84431 | 0 | ||||||||||||||||
1569.5 | 0 | −1.58895 | 0 | 1.82232 | − | 1.29582i | 0 | 1.00000i | 0 | −0.475223 | 0 | ||||||||||||||||
1569.6 | 0 | −1.58895 | 0 | 1.82232 | + | 1.29582i | 0 | − | 1.00000i | 0 | −0.475223 | 0 | |||||||||||||||
1569.7 | 0 | −1.29255 | 0 | 0.437032 | − | 2.19294i | 0 | − | 1.00000i | 0 | −1.32931 | 0 | |||||||||||||||
1569.8 | 0 | −1.29255 | 0 | 0.437032 | + | 2.19294i | 0 | 1.00000i | 0 | −1.32931 | 0 | ||||||||||||||||
1569.9 | 0 | −1.09381 | 0 | −2.20510 | − | 0.370875i | 0 | 1.00000i | 0 | −1.80358 | 0 | ||||||||||||||||
1569.10 | 0 | −1.09381 | 0 | −2.20510 | + | 0.370875i | 0 | − | 1.00000i | 0 | −1.80358 | 0 | |||||||||||||||
1569.11 | 0 | −0.236390 | 0 | 1.55127 | + | 1.61046i | 0 | − | 1.00000i | 0 | −2.94412 | 0 | |||||||||||||||
1569.12 | 0 | −0.236390 | 0 | 1.55127 | − | 1.61046i | 0 | 1.00000i | 0 | −2.94412 | 0 | ||||||||||||||||
1569.13 | 0 | 0.236390 | 0 | 1.55127 | − | 1.61046i | 0 | − | 1.00000i | 0 | −2.94412 | 0 | |||||||||||||||
1569.14 | 0 | 0.236390 | 0 | 1.55127 | + | 1.61046i | 0 | 1.00000i | 0 | −2.94412 | 0 | ||||||||||||||||
1569.15 | 0 | 1.09381 | 0 | −2.20510 | + | 0.370875i | 0 | 1.00000i | 0 | −1.80358 | 0 | ||||||||||||||||
1569.16 | 0 | 1.09381 | 0 | −2.20510 | − | 0.370875i | 0 | − | 1.00000i | 0 | −1.80358 | 0 | |||||||||||||||
1569.17 | 0 | 1.29255 | 0 | 0.437032 | + | 2.19294i | 0 | − | 1.00000i | 0 | −1.32931 | 0 | |||||||||||||||
1569.18 | 0 | 1.29255 | 0 | 0.437032 | − | 2.19294i | 0 | 1.00000i | 0 | −1.32931 | 0 | ||||||||||||||||
1569.19 | 0 | 1.58895 | 0 | 1.82232 | + | 1.29582i | 0 | 1.00000i | 0 | −0.475223 | 0 | ||||||||||||||||
1569.20 | 0 | 1.58895 | 0 | 1.82232 | − | 1.29582i | 0 | − | 1.00000i | 0 | −0.475223 | 0 | |||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
40.e | odd | 2 | 1 | inner |
40.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2240.2.l.c | ✓ | 24 |
4.b | odd | 2 | 1 | inner | 2240.2.l.c | ✓ | 24 |
5.b | even | 2 | 1 | 2240.2.l.d | yes | 24 | |
8.b | even | 2 | 1 | 2240.2.l.d | yes | 24 | |
8.d | odd | 2 | 1 | 2240.2.l.d | yes | 24 | |
20.d | odd | 2 | 1 | 2240.2.l.d | yes | 24 | |
40.e | odd | 2 | 1 | inner | 2240.2.l.c | ✓ | 24 |
40.f | even | 2 | 1 | inner | 2240.2.l.c | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2240.2.l.c | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
2240.2.l.c | ✓ | 24 | 4.b | odd | 2 | 1 | inner |
2240.2.l.c | ✓ | 24 | 40.e | odd | 2 | 1 | inner |
2240.2.l.c | ✓ | 24 | 40.f | even | 2 | 1 | inner |
2240.2.l.d | yes | 24 | 5.b | even | 2 | 1 | |
2240.2.l.d | yes | 24 | 8.b | even | 2 | 1 | |
2240.2.l.d | yes | 24 | 8.d | odd | 2 | 1 | |
2240.2.l.d | yes | 24 | 20.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 \(S_{2}^{\mathrm{new}}(2240, [\chi])\):
\( T_{3}^{12} - 21T_{3}^{10} + 151T_{3}^{8} - 463T_{3}^{6} + 628T_{3}^{4} - 320T_{3}^{2} + 16 \) |
\( T_{13}^{6} + T_{13}^{5} - 33T_{13}^{4} + 3T_{13}^{3} + 260T_{13}^{2} - 124T_{13} - 356 \) |