Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2100,2,Mod(157,2100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2100, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2100.157");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2100.ce (of order \(12\), degree \(4\), 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: | \(16.7685844245\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
157.1 | 0 | −0.965926 | + | 0.258819i | 0 | 0 | 0 | 2.25190 | + | 1.38886i | 0 | 0.866025 | − | 0.500000i | 0 | ||||||||||||
157.2 | 0 | −0.965926 | + | 0.258819i | 0 | 0 | 0 | −0.728357 | − | 2.54352i | 0 | 0.866025 | − | 0.500000i | 0 | ||||||||||||
157.3 | 0 | −0.965926 | + | 0.258819i | 0 | 0 | 0 | −2.48947 | + | 0.895840i | 0 | 0.866025 | − | 0.500000i | 0 | ||||||||||||
157.4 | 0 | 0.965926 | − | 0.258819i | 0 | 0 | 0 | 0.728357 | + | 2.54352i | 0 | 0.866025 | − | 0.500000i | 0 | ||||||||||||
157.5 | 0 | 0.965926 | − | 0.258819i | 0 | 0 | 0 | 2.48947 | − | 0.895840i | 0 | 0.866025 | − | 0.500000i | 0 | ||||||||||||
157.6 | 0 | 0.965926 | − | 0.258819i | 0 | 0 | 0 | −2.25190 | − | 1.38886i | 0 | 0.866025 | − | 0.500000i | 0 | ||||||||||||
493.1 | 0 | −0.258819 | − | 0.965926i | 0 | 0 | 0 | 1.38886 | − | 2.25190i | 0 | −0.866025 | + | 0.500000i | 0 | ||||||||||||
493.2 | 0 | −0.258819 | − | 0.965926i | 0 | 0 | 0 | −2.54352 | + | 0.728357i | 0 | −0.866025 | + | 0.500000i | 0 | ||||||||||||
493.3 | 0 | −0.258819 | − | 0.965926i | 0 | 0 | 0 | 0.895840 | + | 2.48947i | 0 | −0.866025 | + | 0.500000i | 0 | ||||||||||||
493.4 | 0 | 0.258819 | + | 0.965926i | 0 | 0 | 0 | 2.54352 | − | 0.728357i | 0 | −0.866025 | + | 0.500000i | 0 | ||||||||||||
493.5 | 0 | 0.258819 | + | 0.965926i | 0 | 0 | 0 | −1.38886 | + | 2.25190i | 0 | −0.866025 | + | 0.500000i | 0 | ||||||||||||
493.6 | 0 | 0.258819 | + | 0.965926i | 0 | 0 | 0 | −0.895840 | − | 2.48947i | 0 | −0.866025 | + | 0.500000i | 0 | ||||||||||||
1657.1 | 0 | −0.258819 | + | 0.965926i | 0 | 0 | 0 | 1.38886 | + | 2.25190i | 0 | −0.866025 | − | 0.500000i | 0 | ||||||||||||
1657.2 | 0 | −0.258819 | + | 0.965926i | 0 | 0 | 0 | −2.54352 | − | 0.728357i | 0 | −0.866025 | − | 0.500000i | 0 | ||||||||||||
1657.3 | 0 | −0.258819 | + | 0.965926i | 0 | 0 | 0 | 0.895840 | − | 2.48947i | 0 | −0.866025 | − | 0.500000i | 0 | ||||||||||||
1657.4 | 0 | 0.258819 | − | 0.965926i | 0 | 0 | 0 | 2.54352 | + | 0.728357i | 0 | −0.866025 | − | 0.500000i | 0 | ||||||||||||
1657.5 | 0 | 0.258819 | − | 0.965926i | 0 | 0 | 0 | −1.38886 | − | 2.25190i | 0 | −0.866025 | − | 0.500000i | 0 | ||||||||||||
1657.6 | 0 | 0.258819 | − | 0.965926i | 0 | 0 | 0 | −0.895840 | + | 2.48947i | 0 | −0.866025 | − | 0.500000i | 0 | ||||||||||||
1993.1 | 0 | −0.965926 | − | 0.258819i | 0 | 0 | 0 | 2.25190 | − | 1.38886i | 0 | 0.866025 | + | 0.500000i | 0 | ||||||||||||
1993.2 | 0 | −0.965926 | − | 0.258819i | 0 | 0 | 0 | −0.728357 | + | 2.54352i | 0 | 0.866025 | + | 0.500000i | 0 | ||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
7.d | odd | 6 | 1 | inner |
35.i | odd | 6 | 1 | inner |
35.k | even | 12 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2100.2.ce.d | ✓ | 24 |
5.b | even | 2 | 1 | inner | 2100.2.ce.d | ✓ | 24 |
5.c | odd | 4 | 2 | inner | 2100.2.ce.d | ✓ | 24 |
7.d | odd | 6 | 1 | inner | 2100.2.ce.d | ✓ | 24 |
35.i | odd | 6 | 1 | inner | 2100.2.ce.d | ✓ | 24 |
35.k | even | 12 | 2 | inner | 2100.2.ce.d | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2100.2.ce.d | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
2100.2.ce.d | ✓ | 24 | 5.b | even | 2 | 1 | inner |
2100.2.ce.d | ✓ | 24 | 5.c | odd | 4 | 2 | inner |
2100.2.ce.d | ✓ | 24 | 7.d | odd | 6 | 1 | inner |
2100.2.ce.d | ✓ | 24 | 35.i | odd | 6 | 1 | inner |
2100.2.ce.d | ✓ | 24 | 35.k | even | 12 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\):
\( T_{11}^{6} + 2T_{11}^{5} + 24T_{11}^{4} + 20T_{11}^{3} + 460T_{11}^{2} + 600T_{11} + 900 \) |
\( T_{17}^{24} - 392 T_{17}^{20} + 111408 T_{17}^{16} - 14944352 T_{17}^{12} + 1468049536 T_{17}^{8} + \cdots + 656100000000 \) |