Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1440,2,Mod(847,1440)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1440, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1440.847");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1440.bi (of order \(4\), degree \(2\), 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: | \(11.4984578911\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 120) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
847.1 | 0 | 0 | 0 | −2.22965 | + | 0.169312i | 0 | 0.645414 | − | 0.645414i | 0 | 0 | 0 | ||||||||||||||
847.2 | 0 | 0 | 0 | −2.11218 | + | 0.733965i | 0 | −1.93078 | + | 1.93078i | 0 | 0 | 0 | ||||||||||||||
847.3 | 0 | 0 | 0 | −1.51371 | − | 1.64581i | 0 | 3.43671 | − | 3.43671i | 0 | 0 | 0 | ||||||||||||||
847.4 | 0 | 0 | 0 | −1.28903 | + | 1.82713i | 0 | 1.45533 | − | 1.45533i | 0 | 0 | 0 | ||||||||||||||
847.5 | 0 | 0 | 0 | −0.780766 | − | 2.09533i | 0 | 2.10796 | − | 2.10796i | 0 | 0 | 0 | ||||||||||||||
847.6 | 0 | 0 | 0 | −0.0696909 | − | 2.23498i | 0 | −1.21782 | + | 1.21782i | 0 | 0 | 0 | ||||||||||||||
847.7 | 0 | 0 | 0 | 0.0696909 | + | 2.23498i | 0 | 1.21782 | − | 1.21782i | 0 | 0 | 0 | ||||||||||||||
847.8 | 0 | 0 | 0 | 0.780766 | + | 2.09533i | 0 | −2.10796 | + | 2.10796i | 0 | 0 | 0 | ||||||||||||||
847.9 | 0 | 0 | 0 | 1.28903 | − | 1.82713i | 0 | −1.45533 | + | 1.45533i | 0 | 0 | 0 | ||||||||||||||
847.10 | 0 | 0 | 0 | 1.51371 | + | 1.64581i | 0 | −3.43671 | + | 3.43671i | 0 | 0 | 0 | ||||||||||||||
847.11 | 0 | 0 | 0 | 2.11218 | − | 0.733965i | 0 | 1.93078 | − | 1.93078i | 0 | 0 | 0 | ||||||||||||||
847.12 | 0 | 0 | 0 | 2.22965 | − | 0.169312i | 0 | −0.645414 | + | 0.645414i | 0 | 0 | 0 | ||||||||||||||
1423.1 | 0 | 0 | 0 | −2.22965 | − | 0.169312i | 0 | 0.645414 | + | 0.645414i | 0 | 0 | 0 | ||||||||||||||
1423.2 | 0 | 0 | 0 | −2.11218 | − | 0.733965i | 0 | −1.93078 | − | 1.93078i | 0 | 0 | 0 | ||||||||||||||
1423.3 | 0 | 0 | 0 | −1.51371 | + | 1.64581i | 0 | 3.43671 | + | 3.43671i | 0 | 0 | 0 | ||||||||||||||
1423.4 | 0 | 0 | 0 | −1.28903 | − | 1.82713i | 0 | 1.45533 | + | 1.45533i | 0 | 0 | 0 | ||||||||||||||
1423.5 | 0 | 0 | 0 | −0.780766 | + | 2.09533i | 0 | 2.10796 | + | 2.10796i | 0 | 0 | 0 | ||||||||||||||
1423.6 | 0 | 0 | 0 | −0.0696909 | + | 2.23498i | 0 | −1.21782 | − | 1.21782i | 0 | 0 | 0 | ||||||||||||||
1423.7 | 0 | 0 | 0 | 0.0696909 | − | 2.23498i | 0 | 1.21782 | + | 1.21782i | 0 | 0 | 0 | ||||||||||||||
1423.8 | 0 | 0 | 0 | 0.780766 | − | 2.09533i | 0 | −2.10796 | − | 2.10796i | 0 | 0 | 0 | ||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
8.d | odd | 2 | 1 | inner |
40.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1440.2.bi.e | 24 | |
3.b | odd | 2 | 1 | 480.2.bh.a | 24 | ||
4.b | odd | 2 | 1 | 360.2.w.e | 24 | ||
5.c | odd | 4 | 1 | inner | 1440.2.bi.e | 24 | |
8.b | even | 2 | 1 | 360.2.w.e | 24 | ||
8.d | odd | 2 | 1 | inner | 1440.2.bi.e | 24 | |
12.b | even | 2 | 1 | 120.2.v.a | ✓ | 24 | |
15.d | odd | 2 | 1 | 2400.2.bh.b | 24 | ||
15.e | even | 4 | 1 | 480.2.bh.a | 24 | ||
15.e | even | 4 | 1 | 2400.2.bh.b | 24 | ||
20.e | even | 4 | 1 | 360.2.w.e | 24 | ||
24.f | even | 2 | 1 | 480.2.bh.a | 24 | ||
24.h | odd | 2 | 1 | 120.2.v.a | ✓ | 24 | |
40.i | odd | 4 | 1 | 360.2.w.e | 24 | ||
40.k | even | 4 | 1 | inner | 1440.2.bi.e | 24 | |
60.h | even | 2 | 1 | 600.2.v.b | 24 | ||
60.l | odd | 4 | 1 | 120.2.v.a | ✓ | 24 | |
60.l | odd | 4 | 1 | 600.2.v.b | 24 | ||
120.i | odd | 2 | 1 | 600.2.v.b | 24 | ||
120.m | even | 2 | 1 | 2400.2.bh.b | 24 | ||
120.q | odd | 4 | 1 | 480.2.bh.a | 24 | ||
120.q | odd | 4 | 1 | 2400.2.bh.b | 24 | ||
120.w | even | 4 | 1 | 120.2.v.a | ✓ | 24 | |
120.w | even | 4 | 1 | 600.2.v.b | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
120.2.v.a | ✓ | 24 | 12.b | even | 2 | 1 | |
120.2.v.a | ✓ | 24 | 24.h | odd | 2 | 1 | |
120.2.v.a | ✓ | 24 | 60.l | odd | 4 | 1 | |
120.2.v.a | ✓ | 24 | 120.w | even | 4 | 1 | |
360.2.w.e | 24 | 4.b | odd | 2 | 1 | ||
360.2.w.e | 24 | 8.b | even | 2 | 1 | ||
360.2.w.e | 24 | 20.e | even | 4 | 1 | ||
360.2.w.e | 24 | 40.i | odd | 4 | 1 | ||
480.2.bh.a | 24 | 3.b | odd | 2 | 1 | ||
480.2.bh.a | 24 | 15.e | even | 4 | 1 | ||
480.2.bh.a | 24 | 24.f | even | 2 | 1 | ||
480.2.bh.a | 24 | 120.q | odd | 4 | 1 | ||
600.2.v.b | 24 | 60.h | even | 2 | 1 | ||
600.2.v.b | 24 | 60.l | odd | 4 | 1 | ||
600.2.v.b | 24 | 120.i | odd | 2 | 1 | ||
600.2.v.b | 24 | 120.w | even | 4 | 1 | ||
1440.2.bi.e | 24 | 1.a | even | 1 | 1 | trivial | |
1440.2.bi.e | 24 | 5.c | odd | 4 | 1 | inner | |
1440.2.bi.e | 24 | 8.d | odd | 2 | 1 | inner | |
1440.2.bi.e | 24 | 40.k | even | 4 | 1 | inner | |
2400.2.bh.b | 24 | 15.d | odd | 2 | 1 | ||
2400.2.bh.b | 24 | 15.e | even | 4 | 1 | ||
2400.2.bh.b | 24 | 120.m | even | 2 | 1 | ||
2400.2.bh.b | 24 | 120.q | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{24} + 720T_{7}^{20} + 98656T_{7}^{16} + 4752640T_{7}^{12} + 81309952T_{7}^{8} + 440926208T_{7}^{4} + 268435456 \)
acting on \(S_{2}^{\mathrm{new}}(1440, [\chi])\).