Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2016,2,Mod(431,2016)]
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, 3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2016.431");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2016.bu (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: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 504) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
431.1 | 0 | 0 | 0 | −1.88256 | − | 3.26069i | 0 | 2.23426 | + | 1.41706i | 0 | 0 | 0 | ||||||||||||||
431.2 | 0 | 0 | 0 | −1.88256 | − | 3.26069i | 0 | −2.23426 | − | 1.41706i | 0 | 0 | 0 | ||||||||||||||
431.3 | 0 | 0 | 0 | −1.51483 | − | 2.62375i | 0 | −1.48957 | + | 2.18659i | 0 | 0 | 0 | ||||||||||||||
431.4 | 0 | 0 | 0 | −1.51483 | − | 2.62375i | 0 | 1.48957 | − | 2.18659i | 0 | 0 | 0 | ||||||||||||||
431.5 | 0 | 0 | 0 | −1.02787 | − | 1.78033i | 0 | 1.24680 | + | 2.33356i | 0 | 0 | 0 | ||||||||||||||
431.6 | 0 | 0 | 0 | −1.02787 | − | 1.78033i | 0 | −1.24680 | − | 2.33356i | 0 | 0 | 0 | ||||||||||||||
431.7 | 0 | 0 | 0 | −0.635051 | − | 1.09994i | 0 | 2.64362 | − | 0.106220i | 0 | 0 | 0 | ||||||||||||||
431.8 | 0 | 0 | 0 | −0.635051 | − | 1.09994i | 0 | −2.64362 | + | 0.106220i | 0 | 0 | 0 | ||||||||||||||
431.9 | 0 | 0 | 0 | −0.317795 | − | 0.550436i | 0 | 2.11900 | − | 1.58425i | 0 | 0 | 0 | ||||||||||||||
431.10 | 0 | 0 | 0 | −0.317795 | − | 0.550436i | 0 | −2.11900 | + | 1.58425i | 0 | 0 | 0 | ||||||||||||||
431.11 | 0 | 0 | 0 | −0.316953 | − | 0.548978i | 0 | −2.06297 | − | 1.65655i | 0 | 0 | 0 | ||||||||||||||
431.12 | 0 | 0 | 0 | −0.316953 | − | 0.548978i | 0 | 2.06297 | + | 1.65655i | 0 | 0 | 0 | ||||||||||||||
431.13 | 0 | 0 | 0 | 0.316953 | + | 0.548978i | 0 | 2.06297 | + | 1.65655i | 0 | 0 | 0 | ||||||||||||||
431.14 | 0 | 0 | 0 | 0.316953 | + | 0.548978i | 0 | −2.06297 | − | 1.65655i | 0 | 0 | 0 | ||||||||||||||
431.15 | 0 | 0 | 0 | 0.317795 | + | 0.550436i | 0 | −2.11900 | + | 1.58425i | 0 | 0 | 0 | ||||||||||||||
431.16 | 0 | 0 | 0 | 0.317795 | + | 0.550436i | 0 | 2.11900 | − | 1.58425i | 0 | 0 | 0 | ||||||||||||||
431.17 | 0 | 0 | 0 | 0.635051 | + | 1.09994i | 0 | −2.64362 | + | 0.106220i | 0 | 0 | 0 | ||||||||||||||
431.18 | 0 | 0 | 0 | 0.635051 | + | 1.09994i | 0 | 2.64362 | − | 0.106220i | 0 | 0 | 0 | ||||||||||||||
431.19 | 0 | 0 | 0 | 1.02787 | + | 1.78033i | 0 | −1.24680 | − | 2.33356i | 0 | 0 | 0 | ||||||||||||||
431.20 | 0 | 0 | 0 | 1.02787 | + | 1.78033i | 0 | 1.24680 | + | 2.33356i | 0 | 0 | 0 | ||||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
8.d | odd | 2 | 1 | inner |
21.h | odd | 6 | 1 | inner |
24.f | even | 2 | 1 | inner |
56.k | odd | 6 | 1 | inner |
168.v | 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.bu.c | 48 | |
3.b | odd | 2 | 1 | inner | 2016.2.bu.c | 48 | |
4.b | odd | 2 | 1 | 504.2.bm.c | ✓ | 48 | |
7.c | even | 3 | 1 | inner | 2016.2.bu.c | 48 | |
8.b | even | 2 | 1 | 504.2.bm.c | ✓ | 48 | |
8.d | odd | 2 | 1 | inner | 2016.2.bu.c | 48 | |
12.b | even | 2 | 1 | 504.2.bm.c | ✓ | 48 | |
21.h | odd | 6 | 1 | inner | 2016.2.bu.c | 48 | |
24.f | even | 2 | 1 | inner | 2016.2.bu.c | 48 | |
24.h | odd | 2 | 1 | 504.2.bm.c | ✓ | 48 | |
28.g | odd | 6 | 1 | 504.2.bm.c | ✓ | 48 | |
56.k | odd | 6 | 1 | inner | 2016.2.bu.c | 48 | |
56.p | even | 6 | 1 | 504.2.bm.c | ✓ | 48 | |
84.n | even | 6 | 1 | 504.2.bm.c | ✓ | 48 | |
168.s | odd | 6 | 1 | 504.2.bm.c | ✓ | 48 | |
168.v | even | 6 | 1 | inner | 2016.2.bu.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
504.2.bm.c | ✓ | 48 | 4.b | odd | 2 | 1 | |
504.2.bm.c | ✓ | 48 | 8.b | even | 2 | 1 | |
504.2.bm.c | ✓ | 48 | 12.b | even | 2 | 1 | |
504.2.bm.c | ✓ | 48 | 24.h | odd | 2 | 1 | |
504.2.bm.c | ✓ | 48 | 28.g | odd | 6 | 1 | |
504.2.bm.c | ✓ | 48 | 56.p | even | 6 | 1 | |
504.2.bm.c | ✓ | 48 | 84.n | even | 6 | 1 | |
504.2.bm.c | ✓ | 48 | 168.s | odd | 6 | 1 | |
2016.2.bu.c | 48 | 1.a | even | 1 | 1 | trivial | |
2016.2.bu.c | 48 | 3.b | odd | 2 | 1 | inner | |
2016.2.bu.c | 48 | 7.c | even | 3 | 1 | inner | |
2016.2.bu.c | 48 | 8.d | odd | 2 | 1 | inner | |
2016.2.bu.c | 48 | 21.h | odd | 6 | 1 | inner | |
2016.2.bu.c | 48 | 24.f | even | 2 | 1 | inner | |
2016.2.bu.c | 48 | 56.k | odd | 6 | 1 | inner | |
2016.2.bu.c | 48 | 168.v | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 30 T_{5}^{22} + 603 T_{5}^{20} + 6622 T_{5}^{18} + 52217 T_{5}^{16} + 240312 T_{5}^{14} + \cdots + 20736 \) acting on \(S_{2}^{\mathrm{new}}(2016, [\chi])\).