Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1575,2,Mod(899,1575)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1575, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1575.899");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1575.bc (of order \(6\), 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: | \(12.5764383184\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 315) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
899.1 | −1.37068 | − | 2.37409i | 0 | −2.75754 | + | 4.77621i | 0 | 0 | 1.65853 | + | 2.06138i | 9.63615 | 0 | 0 | ||||||||||||
899.2 | −1.23340 | − | 2.13631i | 0 | −2.04255 | + | 3.53780i | 0 | 0 | 1.88881 | − | 1.85267i | 5.14351 | 0 | 0 | ||||||||||||
899.3 | −0.997835 | − | 1.72830i | 0 | −0.991350 | + | 1.71707i | 0 | 0 | 1.95727 | + | 1.78020i | −0.0345244 | 0 | 0 | ||||||||||||
899.4 | −0.569823 | − | 0.986962i | 0 | 0.350603 | − | 0.607263i | 0 | 0 | 1.48558 | − | 2.18931i | −3.07842 | 0 | 0 | ||||||||||||
899.5 | −0.489834 | − | 0.848417i | 0 | 0.520126 | − | 0.900884i | 0 | 0 | −2.33346 | − | 1.24698i | −2.97844 | 0 | 0 | ||||||||||||
899.6 | −0.199105 | − | 0.344861i | 0 | 0.920714 | − | 1.59472i | 0 | 0 | −1.30338 | − | 2.30243i | −1.52970 | 0 | 0 | ||||||||||||
899.7 | 0.199105 | + | 0.344861i | 0 | 0.920714 | − | 1.59472i | 0 | 0 | 1.30338 | + | 2.30243i | 1.52970 | 0 | 0 | ||||||||||||
899.8 | 0.489834 | + | 0.848417i | 0 | 0.520126 | − | 0.900884i | 0 | 0 | 2.33346 | + | 1.24698i | 2.97844 | 0 | 0 | ||||||||||||
899.9 | 0.569823 | + | 0.986962i | 0 | 0.350603 | − | 0.607263i | 0 | 0 | −1.48558 | + | 2.18931i | 3.07842 | 0 | 0 | ||||||||||||
899.10 | 0.997835 | + | 1.72830i | 0 | −0.991350 | + | 1.71707i | 0 | 0 | −1.95727 | − | 1.78020i | 0.0345244 | 0 | 0 | ||||||||||||
899.11 | 1.23340 | + | 2.13631i | 0 | −2.04255 | + | 3.53780i | 0 | 0 | −1.88881 | + | 1.85267i | −5.14351 | 0 | 0 | ||||||||||||
899.12 | 1.37068 | + | 2.37409i | 0 | −2.75754 | + | 4.77621i | 0 | 0 | −1.65853 | − | 2.06138i | −9.63615 | 0 | 0 | ||||||||||||
1349.1 | −1.37068 | + | 2.37409i | 0 | −2.75754 | − | 4.77621i | 0 | 0 | 1.65853 | − | 2.06138i | 9.63615 | 0 | 0 | ||||||||||||
1349.2 | −1.23340 | + | 2.13631i | 0 | −2.04255 | − | 3.53780i | 0 | 0 | 1.88881 | + | 1.85267i | 5.14351 | 0 | 0 | ||||||||||||
1349.3 | −0.997835 | + | 1.72830i | 0 | −0.991350 | − | 1.71707i | 0 | 0 | 1.95727 | − | 1.78020i | −0.0345244 | 0 | 0 | ||||||||||||
1349.4 | −0.569823 | + | 0.986962i | 0 | 0.350603 | + | 0.607263i | 0 | 0 | 1.48558 | + | 2.18931i | −3.07842 | 0 | 0 | ||||||||||||
1349.5 | −0.489834 | + | 0.848417i | 0 | 0.520126 | + | 0.900884i | 0 | 0 | −2.33346 | + | 1.24698i | −2.97844 | 0 | 0 | ||||||||||||
1349.6 | −0.199105 | + | 0.344861i | 0 | 0.920714 | + | 1.59472i | 0 | 0 | −1.30338 | + | 2.30243i | −1.52970 | 0 | 0 | ||||||||||||
1349.7 | 0.199105 | − | 0.344861i | 0 | 0.920714 | + | 1.59472i | 0 | 0 | 1.30338 | − | 2.30243i | 1.52970 | 0 | 0 | ||||||||||||
1349.8 | 0.489834 | − | 0.848417i | 0 | 0.520126 | + | 0.900884i | 0 | 0 | 2.33346 | − | 1.24698i | 2.97844 | 0 | 0 | ||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
21.g | even | 6 | 1 | inner |
105.p | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1575.2.bc.c | 24 | |
3.b | odd | 2 | 1 | 1575.2.bc.d | 24 | ||
5.b | even | 2 | 1 | inner | 1575.2.bc.c | 24 | |
5.c | odd | 4 | 1 | 315.2.bj.b | yes | 12 | |
5.c | odd | 4 | 1 | 1575.2.bk.e | 12 | ||
7.d | odd | 6 | 1 | 1575.2.bc.d | 24 | ||
15.d | odd | 2 | 1 | 1575.2.bc.d | 24 | ||
15.e | even | 4 | 1 | 315.2.bj.a | ✓ | 12 | |
15.e | even | 4 | 1 | 1575.2.bk.f | 12 | ||
21.g | even | 6 | 1 | inner | 1575.2.bc.c | 24 | |
35.i | odd | 6 | 1 | 1575.2.bc.d | 24 | ||
35.k | even | 12 | 1 | 315.2.bj.a | ✓ | 12 | |
35.k | even | 12 | 1 | 1575.2.bk.f | 12 | ||
35.k | even | 12 | 1 | 2205.2.b.b | 12 | ||
35.l | odd | 12 | 1 | 2205.2.b.a | 12 | ||
105.p | even | 6 | 1 | inner | 1575.2.bc.c | 24 | |
105.w | odd | 12 | 1 | 315.2.bj.b | yes | 12 | |
105.w | odd | 12 | 1 | 1575.2.bk.e | 12 | ||
105.w | odd | 12 | 1 | 2205.2.b.a | 12 | ||
105.x | even | 12 | 1 | 2205.2.b.b | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.bj.a | ✓ | 12 | 15.e | even | 4 | 1 | |
315.2.bj.a | ✓ | 12 | 35.k | even | 12 | 1 | |
315.2.bj.b | yes | 12 | 5.c | odd | 4 | 1 | |
315.2.bj.b | yes | 12 | 105.w | odd | 12 | 1 | |
1575.2.bc.c | 24 | 1.a | even | 1 | 1 | trivial | |
1575.2.bc.c | 24 | 5.b | even | 2 | 1 | inner | |
1575.2.bc.c | 24 | 21.g | even | 6 | 1 | inner | |
1575.2.bc.c | 24 | 105.p | even | 6 | 1 | inner | |
1575.2.bc.d | 24 | 3.b | odd | 2 | 1 | ||
1575.2.bc.d | 24 | 7.d | odd | 6 | 1 | ||
1575.2.bc.d | 24 | 15.d | odd | 2 | 1 | ||
1575.2.bc.d | 24 | 35.i | odd | 6 | 1 | ||
1575.2.bk.e | 12 | 5.c | odd | 4 | 1 | ||
1575.2.bk.e | 12 | 105.w | odd | 12 | 1 | ||
1575.2.bk.f | 12 | 15.e | even | 4 | 1 | ||
1575.2.bk.f | 12 | 35.k | even | 12 | 1 | ||
2205.2.b.a | 12 | 35.l | odd | 12 | 1 | ||
2205.2.b.a | 12 | 105.w | odd | 12 | 1 | ||
2205.2.b.b | 12 | 35.k | even | 12 | 1 | ||
2205.2.b.b | 12 | 105.x | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1575, [\chi])\):
\( T_{2}^{24} + 20 T_{2}^{22} + 256 T_{2}^{20} + 1976 T_{2}^{18} + 11092 T_{2}^{16} + 41240 T_{2}^{14} + \cdots + 1296 \) |
\( T_{11}^{12} + 12 T_{11}^{11} + 38 T_{11}^{10} - 120 T_{11}^{9} - 660 T_{11}^{8} + 3000 T_{11}^{7} + \cdots + 19044 \) |