Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1050,3,Mod(199,1050)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1050, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1050.199");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1050.q (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: | \(28.6104277578\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 210) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
199.1 | −1.22474 | + | 0.707107i | −0.866025 | + | 1.50000i | 1.00000 | − | 1.73205i | 0 | − | 2.44949i | −2.86123 | − | 6.38854i | 2.82843i | −1.50000 | − | 2.59808i | 0 | |||||||
199.2 | −1.22474 | + | 0.707107i | 0.866025 | − | 1.50000i | 1.00000 | − | 1.73205i | 0 | 2.44949i | 4.01037 | − | 5.73733i | 2.82843i | −1.50000 | − | 2.59808i | 0 | ||||||||
199.3 | −1.22474 | + | 0.707107i | 0.866025 | − | 1.50000i | 1.00000 | − | 1.73205i | 0 | 2.44949i | −6.50174 | + | 2.59373i | 2.82843i | −1.50000 | − | 2.59808i | 0 | ||||||||
199.4 | −1.22474 | + | 0.707107i | 0.866025 | − | 1.50000i | 1.00000 | − | 1.73205i | 0 | 2.44949i | −6.72455 | − | 1.94434i | 2.82843i | −1.50000 | − | 2.59808i | 0 | ||||||||
199.5 | −1.22474 | + | 0.707107i | −0.866025 | + | 1.50000i | 1.00000 | − | 1.73205i | 0 | − | 2.44949i | 5.56601 | + | 4.24494i | 2.82843i | −1.50000 | − | 2.59808i | 0 | |||||||
199.6 | −1.22474 | + | 0.707107i | 0.866025 | − | 1.50000i | 1.00000 | − | 1.73205i | 0 | 2.44949i | 6.46925 | + | 2.67372i | 2.82843i | −1.50000 | − | 2.59808i | 0 | ||||||||
199.7 | −1.22474 | + | 0.707107i | −0.866025 | + | 1.50000i | 1.00000 | − | 1.73205i | 0 | − | 2.44949i | 4.61524 | − | 5.26304i | 2.82843i | −1.50000 | − | 2.59808i | 0 | |||||||
199.8 | −1.22474 | + | 0.707107i | −0.866025 | + | 1.50000i | 1.00000 | − | 1.73205i | 0 | − | 2.44949i | 0.325616 | + | 6.99242i | 2.82843i | −1.50000 | − | 2.59808i | 0 | |||||||
199.9 | 1.22474 | − | 0.707107i | 0.866025 | − | 1.50000i | 1.00000 | − | 1.73205i | 0 | − | 2.44949i | −0.325616 | − | 6.99242i | − | 2.82843i | −1.50000 | − | 2.59808i | 0 | ||||||
199.10 | 1.22474 | − | 0.707107i | 0.866025 | − | 1.50000i | 1.00000 | − | 1.73205i | 0 | − | 2.44949i | 2.86123 | + | 6.38854i | − | 2.82843i | −1.50000 | − | 2.59808i | 0 | ||||||
199.11 | 1.22474 | − | 0.707107i | 0.866025 | − | 1.50000i | 1.00000 | − | 1.73205i | 0 | − | 2.44949i | −5.56601 | − | 4.24494i | − | 2.82843i | −1.50000 | − | 2.59808i | 0 | ||||||
199.12 | 1.22474 | − | 0.707107i | −0.866025 | + | 1.50000i | 1.00000 | − | 1.73205i | 0 | 2.44949i | 6.72455 | + | 1.94434i | − | 2.82843i | −1.50000 | − | 2.59808i | 0 | |||||||
199.13 | 1.22474 | − | 0.707107i | −0.866025 | + | 1.50000i | 1.00000 | − | 1.73205i | 0 | 2.44949i | −6.46925 | − | 2.67372i | − | 2.82843i | −1.50000 | − | 2.59808i | 0 | |||||||
199.14 | 1.22474 | − | 0.707107i | −0.866025 | + | 1.50000i | 1.00000 | − | 1.73205i | 0 | 2.44949i | 6.50174 | − | 2.59373i | − | 2.82843i | −1.50000 | − | 2.59808i | 0 | |||||||
199.15 | 1.22474 | − | 0.707107i | −0.866025 | + | 1.50000i | 1.00000 | − | 1.73205i | 0 | 2.44949i | −4.01037 | + | 5.73733i | − | 2.82843i | −1.50000 | − | 2.59808i | 0 | |||||||
199.16 | 1.22474 | − | 0.707107i | 0.866025 | − | 1.50000i | 1.00000 | − | 1.73205i | 0 | − | 2.44949i | −4.61524 | + | 5.26304i | − | 2.82843i | −1.50000 | − | 2.59808i | 0 | ||||||
649.1 | −1.22474 | − | 0.707107i | −0.866025 | − | 1.50000i | 1.00000 | + | 1.73205i | 0 | 2.44949i | −2.86123 | + | 6.38854i | − | 2.82843i | −1.50000 | + | 2.59808i | 0 | |||||||
649.2 | −1.22474 | − | 0.707107i | 0.866025 | + | 1.50000i | 1.00000 | + | 1.73205i | 0 | − | 2.44949i | 4.01037 | + | 5.73733i | − | 2.82843i | −1.50000 | + | 2.59808i | 0 | ||||||
649.3 | −1.22474 | − | 0.707107i | 0.866025 | + | 1.50000i | 1.00000 | + | 1.73205i | 0 | − | 2.44949i | −6.50174 | − | 2.59373i | − | 2.82843i | −1.50000 | + | 2.59808i | 0 | ||||||
649.4 | −1.22474 | − | 0.707107i | 0.866025 | + | 1.50000i | 1.00000 | + | 1.73205i | 0 | − | 2.44949i | −6.72455 | + | 1.94434i | − | 2.82843i | −1.50000 | + | 2.59808i | 0 | ||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
35.i | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1050.3.q.e | 32 | |
5.b | even | 2 | 1 | inner | 1050.3.q.e | 32 | |
5.c | odd | 4 | 1 | 210.3.o.b | ✓ | 16 | |
5.c | odd | 4 | 1 | 1050.3.p.i | 16 | ||
7.d | odd | 6 | 1 | inner | 1050.3.q.e | 32 | |
15.e | even | 4 | 1 | 630.3.v.c | 16 | ||
35.i | odd | 6 | 1 | inner | 1050.3.q.e | 32 | |
35.k | even | 12 | 1 | 210.3.o.b | ✓ | 16 | |
35.k | even | 12 | 1 | 1050.3.p.i | 16 | ||
35.k | even | 12 | 1 | 1470.3.f.d | 16 | ||
35.l | odd | 12 | 1 | 1470.3.f.d | 16 | ||
105.w | odd | 12 | 1 | 630.3.v.c | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
210.3.o.b | ✓ | 16 | 5.c | odd | 4 | 1 | |
210.3.o.b | ✓ | 16 | 35.k | even | 12 | 1 | |
630.3.v.c | 16 | 15.e | even | 4 | 1 | ||
630.3.v.c | 16 | 105.w | odd | 12 | 1 | ||
1050.3.p.i | 16 | 5.c | odd | 4 | 1 | ||
1050.3.p.i | 16 | 35.k | even | 12 | 1 | ||
1050.3.q.e | 32 | 1.a | even | 1 | 1 | trivial | |
1050.3.q.e | 32 | 5.b | even | 2 | 1 | inner | |
1050.3.q.e | 32 | 7.d | odd | 6 | 1 | inner | |
1050.3.q.e | 32 | 35.i | odd | 6 | 1 | inner | |
1470.3.f.d | 16 | 35.k | even | 12 | 1 | ||
1470.3.f.d | 16 | 35.l | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{16} + 4 T_{11}^{15} + 604 T_{11}^{14} + 352 T_{11}^{13} + 252064 T_{11}^{12} + \cdots + 9682651996416 \) acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\).