Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1890,2,Mod(629,1890)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1890, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 3, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1890.629");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1890.bf (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: | \(15.0917259820\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 630) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
629.1 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −2.23546 | − | 0.0519524i | 0 | 2.59656 | − | 0.507832i | 1.00000 | 0 | 1.16272 | − | 1.90999i | ||||||||
629.2 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −2.18778 | − | 0.462197i | 0 | 1.99922 | + | 1.73295i | 1.00000 | 0 | 1.49416 | − | 1.66357i | ||||||||
629.3 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −2.16779 | + | 0.548354i | 0 | −1.29573 | − | 2.30675i | 1.00000 | 0 | 0.609005 | − | 2.15154i | ||||||||
629.4 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −2.12268 | − | 0.703023i | 0 | −2.48907 | − | 0.896960i | 1.00000 | 0 | 1.67017 | − | 1.48678i | ||||||||
629.5 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −2.00747 | − | 0.984910i | 0 | 2.28974 | − | 1.32556i | 1.00000 | 0 | 1.85669 | − | 1.24607i | ||||||||
629.6 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −1.25628 | + | 1.84980i | 0 | 1.02580 | + | 2.43880i | 1.00000 | 0 | −0.973830 | − | 2.01287i | ||||||||
629.7 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −1.00815 | − | 1.99591i | 0 | −2.45843 | − | 0.977807i | 1.00000 | 0 | 2.23258 | + | 0.124872i | ||||||||
629.8 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −0.797934 | − | 2.08885i | 0 | 0.843817 | − | 2.50758i | 1.00000 | 0 | 2.20797 | + | 0.353395i | ||||||||
629.9 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 0.797934 | + | 2.08885i | 0 | −1.74972 | + | 1.98456i | 1.00000 | 0 | −2.20797 | − | 0.353395i | ||||||||
629.10 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 1.00815 | + | 1.99591i | 0 | −2.07602 | − | 1.64016i | 1.00000 | 0 | −2.23258 | − | 0.124872i | ||||||||
629.11 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 1.25628 | − | 1.84980i | 0 | 2.62496 | − | 0.331028i | 1.00000 | 0 | 0.973830 | + | 2.01287i | ||||||||
629.12 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 2.00747 | + | 0.984910i | 0 | −0.00309790 | + | 2.64575i | 1.00000 | 0 | −1.85669 | + | 1.24607i | ||||||||
629.13 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 2.12268 | + | 0.703023i | 0 | −2.02132 | − | 1.70712i | 1.00000 | 0 | −1.67017 | + | 1.48678i | ||||||||
629.14 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 2.16779 | − | 0.548354i | 0 | −2.64557 | + | 0.0312377i | 1.00000 | 0 | −0.609005 | + | 2.15154i | ||||||||
629.15 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 2.18778 | + | 0.462197i | 0 | 2.50039 | + | 0.864900i | 1.00000 | 0 | −1.49416 | + | 1.66357i | ||||||||
629.16 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 2.23546 | + | 0.0519524i | 0 | 0.858483 | + | 2.50260i | 1.00000 | 0 | −1.16272 | + | 1.90999i | ||||||||
1259.1 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | −2.23546 | + | 0.0519524i | 0 | 2.59656 | + | 0.507832i | 1.00000 | 0 | 1.16272 | + | 1.90999i | ||||||||
1259.2 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | −2.18778 | + | 0.462197i | 0 | 1.99922 | − | 1.73295i | 1.00000 | 0 | 1.49416 | + | 1.66357i | ||||||||
1259.3 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | −2.16779 | − | 0.548354i | 0 | −1.29573 | + | 2.30675i | 1.00000 | 0 | 0.609005 | + | 2.15154i | ||||||||
1259.4 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | −2.12268 | + | 0.703023i | 0 | −2.48907 | + | 0.896960i | 1.00000 | 0 | 1.67017 | + | 1.48678i | ||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
45.h | odd | 6 | 1 | inner |
315.z | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1890.2.bf.e | 32 | |
3.b | odd | 2 | 1 | 630.2.bf.f | yes | 32 | |
5.b | even | 2 | 1 | 1890.2.bf.f | 32 | ||
7.b | odd | 2 | 1 | inner | 1890.2.bf.e | 32 | |
9.c | even | 3 | 1 | 630.2.bf.e | ✓ | 32 | |
9.d | odd | 6 | 1 | 1890.2.bf.f | 32 | ||
15.d | odd | 2 | 1 | 630.2.bf.e | ✓ | 32 | |
21.c | even | 2 | 1 | 630.2.bf.f | yes | 32 | |
35.c | odd | 2 | 1 | 1890.2.bf.f | 32 | ||
45.h | odd | 6 | 1 | inner | 1890.2.bf.e | 32 | |
45.j | even | 6 | 1 | 630.2.bf.f | yes | 32 | |
63.l | odd | 6 | 1 | 630.2.bf.e | ✓ | 32 | |
63.o | even | 6 | 1 | 1890.2.bf.f | 32 | ||
105.g | even | 2 | 1 | 630.2.bf.e | ✓ | 32 | |
315.z | even | 6 | 1 | inner | 1890.2.bf.e | 32 | |
315.bg | odd | 6 | 1 | 630.2.bf.f | yes | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
630.2.bf.e | ✓ | 32 | 9.c | even | 3 | 1 | |
630.2.bf.e | ✓ | 32 | 15.d | odd | 2 | 1 | |
630.2.bf.e | ✓ | 32 | 63.l | odd | 6 | 1 | |
630.2.bf.e | ✓ | 32 | 105.g | even | 2 | 1 | |
630.2.bf.f | yes | 32 | 3.b | odd | 2 | 1 | |
630.2.bf.f | yes | 32 | 21.c | even | 2 | 1 | |
630.2.bf.f | yes | 32 | 45.j | even | 6 | 1 | |
630.2.bf.f | yes | 32 | 315.bg | odd | 6 | 1 | |
1890.2.bf.e | 32 | 1.a | even | 1 | 1 | trivial | |
1890.2.bf.e | 32 | 7.b | odd | 2 | 1 | inner | |
1890.2.bf.e | 32 | 45.h | odd | 6 | 1 | inner | |
1890.2.bf.e | 32 | 315.z | even | 6 | 1 | inner | |
1890.2.bf.f | 32 | 5.b | even | 2 | 1 | ||
1890.2.bf.f | 32 | 9.d | odd | 6 | 1 | ||
1890.2.bf.f | 32 | 35.c | odd | 2 | 1 | ||
1890.2.bf.f | 32 | 63.o | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1890, [\chi])\):
\( T_{11}^{16} - 12 T_{11}^{15} + 7 T_{11}^{14} + 492 T_{11}^{13} - 830 T_{11}^{12} - 21498 T_{11}^{11} + \cdots + 239754256 \) |
\( T_{13}^{32} + 131 T_{13}^{30} + 10509 T_{13}^{28} + 533244 T_{13}^{26} + 19752852 T_{13}^{24} + \cdots + 28\!\cdots\!76 \) |
\( T_{23}^{16} + 12 T_{23}^{15} + 177 T_{23}^{14} + 1156 T_{23}^{13} + 11169 T_{23}^{12} + \cdots + 2958924816 \) |