Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2880,2,Mod(737,2880)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2880, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 2, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2880.737");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2880.bj (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: | \(22.9969157821\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
737.1 | 0 | 0 | 0 | −2.11477 | + | 0.726469i | 0 | −3.61002 | − | 3.61002i | 0 | 0 | 0 | ||||||||||||||
737.2 | 0 | 0 | 0 | −2.11477 | + | 0.726469i | 0 | 3.61002 | + | 3.61002i | 0 | 0 | 0 | ||||||||||||||
737.3 | 0 | 0 | 0 | −1.94535 | − | 1.10255i | 0 | −0.880206 | − | 0.880206i | 0 | 0 | 0 | ||||||||||||||
737.4 | 0 | 0 | 0 | −1.94535 | − | 1.10255i | 0 | 0.880206 | + | 0.880206i | 0 | 0 | 0 | ||||||||||||||
737.5 | 0 | 0 | 0 | −1.27770 | + | 1.83507i | 0 | −0.254252 | − | 0.254252i | 0 | 0 | 0 | ||||||||||||||
737.6 | 0 | 0 | 0 | −1.27770 | + | 1.83507i | 0 | 0.254252 | + | 0.254252i | 0 | 0 | 0 | ||||||||||||||
737.7 | 0 | 0 | 0 | −0.332926 | − | 2.21114i | 0 | −2.47556 | − | 2.47556i | 0 | 0 | 0 | ||||||||||||||
737.8 | 0 | 0 | 0 | −0.332926 | − | 2.21114i | 0 | 2.47556 | + | 2.47556i | 0 | 0 | 0 | ||||||||||||||
737.9 | 0 | 0 | 0 | 0.332926 | + | 2.21114i | 0 | −2.47556 | − | 2.47556i | 0 | 0 | 0 | ||||||||||||||
737.10 | 0 | 0 | 0 | 0.332926 | + | 2.21114i | 0 | 2.47556 | + | 2.47556i | 0 | 0 | 0 | ||||||||||||||
737.11 | 0 | 0 | 0 | 1.27770 | − | 1.83507i | 0 | −0.254252 | − | 0.254252i | 0 | 0 | 0 | ||||||||||||||
737.12 | 0 | 0 | 0 | 1.27770 | − | 1.83507i | 0 | 0.254252 | + | 0.254252i | 0 | 0 | 0 | ||||||||||||||
737.13 | 0 | 0 | 0 | 1.94535 | + | 1.10255i | 0 | −0.880206 | − | 0.880206i | 0 | 0 | 0 | ||||||||||||||
737.14 | 0 | 0 | 0 | 1.94535 | + | 1.10255i | 0 | 0.880206 | + | 0.880206i | 0 | 0 | 0 | ||||||||||||||
737.15 | 0 | 0 | 0 | 2.11477 | − | 0.726469i | 0 | −3.61002 | − | 3.61002i | 0 | 0 | 0 | ||||||||||||||
737.16 | 0 | 0 | 0 | 2.11477 | − | 0.726469i | 0 | 3.61002 | + | 3.61002i | 0 | 0 | 0 | ||||||||||||||
1313.1 | 0 | 0 | 0 | −2.11477 | − | 0.726469i | 0 | −3.61002 | + | 3.61002i | 0 | 0 | 0 | ||||||||||||||
1313.2 | 0 | 0 | 0 | −2.11477 | − | 0.726469i | 0 | 3.61002 | − | 3.61002i | 0 | 0 | 0 | ||||||||||||||
1313.3 | 0 | 0 | 0 | −1.94535 | + | 1.10255i | 0 | −0.880206 | + | 0.880206i | 0 | 0 | 0 | ||||||||||||||
1313.4 | 0 | 0 | 0 | −1.94535 | + | 1.10255i | 0 | 0.880206 | − | 0.880206i | 0 | 0 | 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 |
20.e | even | 4 | 1 | inner |
24.f | even | 2 | 1 | inner |
40.i | odd | 4 | 1 | inner |
60.l | odd | 4 | 1 | inner |
120.w | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2880.2.bj.e | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 2880.2.bj.e | ✓ | 32 |
4.b | odd | 2 | 1 | 2880.2.bj.f | yes | 32 | |
5.c | odd | 4 | 1 | 2880.2.bj.f | yes | 32 | |
8.b | even | 2 | 1 | 2880.2.bj.f | yes | 32 | |
8.d | odd | 2 | 1 | inner | 2880.2.bj.e | ✓ | 32 |
12.b | even | 2 | 1 | 2880.2.bj.f | yes | 32 | |
15.e | even | 4 | 1 | 2880.2.bj.f | yes | 32 | |
20.e | even | 4 | 1 | inner | 2880.2.bj.e | ✓ | 32 |
24.f | even | 2 | 1 | inner | 2880.2.bj.e | ✓ | 32 |
24.h | odd | 2 | 1 | 2880.2.bj.f | yes | 32 | |
40.i | odd | 4 | 1 | inner | 2880.2.bj.e | ✓ | 32 |
40.k | even | 4 | 1 | 2880.2.bj.f | yes | 32 | |
60.l | odd | 4 | 1 | inner | 2880.2.bj.e | ✓ | 32 |
120.q | odd | 4 | 1 | 2880.2.bj.f | yes | 32 | |
120.w | even | 4 | 1 | inner | 2880.2.bj.e | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2880.2.bj.e | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
2880.2.bj.e | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
2880.2.bj.e | ✓ | 32 | 8.d | odd | 2 | 1 | inner |
2880.2.bj.e | ✓ | 32 | 20.e | even | 4 | 1 | inner |
2880.2.bj.e | ✓ | 32 | 24.f | even | 2 | 1 | inner |
2880.2.bj.e | ✓ | 32 | 40.i | odd | 4 | 1 | inner |
2880.2.bj.e | ✓ | 32 | 60.l | odd | 4 | 1 | inner |
2880.2.bj.e | ✓ | 32 | 120.w | even | 4 | 1 | inner |
2880.2.bj.f | yes | 32 | 4.b | odd | 2 | 1 | |
2880.2.bj.f | yes | 32 | 5.c | odd | 4 | 1 | |
2880.2.bj.f | yes | 32 | 8.b | even | 2 | 1 | |
2880.2.bj.f | yes | 32 | 12.b | even | 2 | 1 | |
2880.2.bj.f | yes | 32 | 15.e | even | 4 | 1 | |
2880.2.bj.f | yes | 32 | 24.h | odd | 2 | 1 | |
2880.2.bj.f | yes | 32 | 40.k | even | 4 | 1 | |
2880.2.bj.f | yes | 32 | 120.q | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2880, [\chi])\):
\( T_{7}^{16} + 832T_{7}^{12} + 104064T_{7}^{8} + 246784T_{7}^{4} + 4096 \) |
\( T_{19}^{4} + 4T_{19}^{3} - 24T_{19}^{2} - 80T_{19} - 32 \) |
\( T_{23}^{16} + 2560T_{23}^{12} + 1400832T_{23}^{8} + 189792256T_{23}^{4} + 16777216 \) |