Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2016,2,Mod(271,2016)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2016, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2016.271");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2016.bs (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: | \(16.0978410475\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 168) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
271.1 | 0 | 0 | 0 | −2.08776 | + | 3.61611i | 0 | 2.39694 | − | 1.12013i | 0 | 0 | 0 | ||||||||||||||
271.2 | 0 | 0 | 0 | −1.61398 | + | 2.79550i | 0 | −1.82725 | − | 1.91341i | 0 | 0 | 0 | ||||||||||||||
271.3 | 0 | 0 | 0 | −1.44142 | + | 2.49662i | 0 | 2.63862 | − | 0.194181i | 0 | 0 | 0 | ||||||||||||||
271.4 | 0 | 0 | 0 | −1.25150 | + | 2.16767i | 0 | −1.36321 | + | 2.26752i | 0 | 0 | 0 | ||||||||||||||
271.5 | 0 | 0 | 0 | −1.14053 | + | 1.97545i | 0 | 1.95181 | + | 1.78618i | 0 | 0 | 0 | ||||||||||||||
271.6 | 0 | 0 | 0 | −0.225540 | + | 0.390646i | 0 | 0.458196 | + | 2.60577i | 0 | 0 | 0 | ||||||||||||||
271.7 | 0 | 0 | 0 | −0.155280 | + | 0.268953i | 0 | −2.58581 | − | 0.560001i | 0 | 0 | 0 | ||||||||||||||
271.8 | 0 | 0 | 0 | −0.128707 | + | 0.222928i | 0 | −0.623918 | + | 2.57113i | 0 | 0 | 0 | ||||||||||||||
271.9 | 0 | 0 | 0 | 0.128707 | − | 0.222928i | 0 | 0.623918 | − | 2.57113i | 0 | 0 | 0 | ||||||||||||||
271.10 | 0 | 0 | 0 | 0.155280 | − | 0.268953i | 0 | 2.58581 | + | 0.560001i | 0 | 0 | 0 | ||||||||||||||
271.11 | 0 | 0 | 0 | 0.225540 | − | 0.390646i | 0 | −0.458196 | − | 2.60577i | 0 | 0 | 0 | ||||||||||||||
271.12 | 0 | 0 | 0 | 1.14053 | − | 1.97545i | 0 | −1.95181 | − | 1.78618i | 0 | 0 | 0 | ||||||||||||||
271.13 | 0 | 0 | 0 | 1.25150 | − | 2.16767i | 0 | 1.36321 | − | 2.26752i | 0 | 0 | 0 | ||||||||||||||
271.14 | 0 | 0 | 0 | 1.44142 | − | 2.49662i | 0 | −2.63862 | + | 0.194181i | 0 | 0 | 0 | ||||||||||||||
271.15 | 0 | 0 | 0 | 1.61398 | − | 2.79550i | 0 | 1.82725 | + | 1.91341i | 0 | 0 | 0 | ||||||||||||||
271.16 | 0 | 0 | 0 | 2.08776 | − | 3.61611i | 0 | −2.39694 | + | 1.12013i | 0 | 0 | 0 | ||||||||||||||
1711.1 | 0 | 0 | 0 | −2.08776 | − | 3.61611i | 0 | 2.39694 | + | 1.12013i | 0 | 0 | 0 | ||||||||||||||
1711.2 | 0 | 0 | 0 | −1.61398 | − | 2.79550i | 0 | −1.82725 | + | 1.91341i | 0 | 0 | 0 | ||||||||||||||
1711.3 | 0 | 0 | 0 | −1.44142 | − | 2.49662i | 0 | 2.63862 | + | 0.194181i | 0 | 0 | 0 | ||||||||||||||
1711.4 | 0 | 0 | 0 | −1.25150 | − | 2.16767i | 0 | −1.36321 | − | 2.26752i | 0 | 0 | 0 | ||||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
8.d | odd | 2 | 1 | inner |
56.m | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2016.2.bs.c | 32 | |
3.b | odd | 2 | 1 | 672.2.bb.a | 32 | ||
4.b | odd | 2 | 1 | 504.2.bk.c | 32 | ||
7.d | odd | 6 | 1 | inner | 2016.2.bs.c | 32 | |
8.b | even | 2 | 1 | 504.2.bk.c | 32 | ||
8.d | odd | 2 | 1 | inner | 2016.2.bs.c | 32 | |
12.b | even | 2 | 1 | 168.2.t.a | ✓ | 32 | |
21.g | even | 6 | 1 | 672.2.bb.a | 32 | ||
21.g | even | 6 | 1 | 4704.2.p.a | 32 | ||
21.h | odd | 6 | 1 | 4704.2.p.a | 32 | ||
24.f | even | 2 | 1 | 672.2.bb.a | 32 | ||
24.h | odd | 2 | 1 | 168.2.t.a | ✓ | 32 | |
28.f | even | 6 | 1 | 504.2.bk.c | 32 | ||
56.j | odd | 6 | 1 | 504.2.bk.c | 32 | ||
56.m | even | 6 | 1 | inner | 2016.2.bs.c | 32 | |
84.j | odd | 6 | 1 | 168.2.t.a | ✓ | 32 | |
84.j | odd | 6 | 1 | 1176.2.p.a | 32 | ||
84.n | even | 6 | 1 | 1176.2.p.a | 32 | ||
168.s | odd | 6 | 1 | 1176.2.p.a | 32 | ||
168.v | even | 6 | 1 | 4704.2.p.a | 32 | ||
168.ba | even | 6 | 1 | 168.2.t.a | ✓ | 32 | |
168.ba | even | 6 | 1 | 1176.2.p.a | 32 | ||
168.be | odd | 6 | 1 | 672.2.bb.a | 32 | ||
168.be | odd | 6 | 1 | 4704.2.p.a | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
168.2.t.a | ✓ | 32 | 12.b | even | 2 | 1 | |
168.2.t.a | ✓ | 32 | 24.h | odd | 2 | 1 | |
168.2.t.a | ✓ | 32 | 84.j | odd | 6 | 1 | |
168.2.t.a | ✓ | 32 | 168.ba | even | 6 | 1 | |
504.2.bk.c | 32 | 4.b | odd | 2 | 1 | ||
504.2.bk.c | 32 | 8.b | even | 2 | 1 | ||
504.2.bk.c | 32 | 28.f | even | 6 | 1 | ||
504.2.bk.c | 32 | 56.j | odd | 6 | 1 | ||
672.2.bb.a | 32 | 3.b | odd | 2 | 1 | ||
672.2.bb.a | 32 | 21.g | even | 6 | 1 | ||
672.2.bb.a | 32 | 24.f | even | 2 | 1 | ||
672.2.bb.a | 32 | 168.be | odd | 6 | 1 | ||
1176.2.p.a | 32 | 84.j | odd | 6 | 1 | ||
1176.2.p.a | 32 | 84.n | even | 6 | 1 | ||
1176.2.p.a | 32 | 168.s | odd | 6 | 1 | ||
1176.2.p.a | 32 | 168.ba | even | 6 | 1 | ||
2016.2.bs.c | 32 | 1.a | even | 1 | 1 | trivial | |
2016.2.bs.c | 32 | 7.d | odd | 6 | 1 | inner | |
2016.2.bs.c | 32 | 8.d | odd | 2 | 1 | inner | |
2016.2.bs.c | 32 | 56.m | even | 6 | 1 | inner | |
4704.2.p.a | 32 | 21.g | even | 6 | 1 | ||
4704.2.p.a | 32 | 21.h | odd | 6 | 1 | ||
4704.2.p.a | 32 | 168.v | even | 6 | 1 | ||
4704.2.p.a | 32 | 168.be | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} + 48 T_{5}^{30} + 1426 T_{5}^{28} + 26656 T_{5}^{26} + 365635 T_{5}^{24} + 3640464 T_{5}^{22} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(2016, [\chi])\).