Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2016,2,Mod(17,2016)]
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([0, 3, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2016.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2016.cp (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: | \(56\) |
Relative dimension: | \(28\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | 0 | 0 | −3.18706 | + | 1.84005i | 0 | 0.998380 | + | 2.45015i | 0 | 0 | 0 | ||||||||||||||
17.2 | 0 | 0 | 0 | −3.18706 | + | 1.84005i | 0 | 0.998380 | + | 2.45015i | 0 | 0 | 0 | ||||||||||||||
17.3 | 0 | 0 | 0 | −3.16007 | + | 1.82447i | 0 | 1.64838 | − | 2.06951i | 0 | 0 | 0 | ||||||||||||||
17.4 | 0 | 0 | 0 | −3.16007 | + | 1.82447i | 0 | 1.64838 | − | 2.06951i | 0 | 0 | 0 | ||||||||||||||
17.5 | 0 | 0 | 0 | −1.87230 | + | 1.08097i | 0 | −2.55958 | − | 0.669737i | 0 | 0 | 0 | ||||||||||||||
17.6 | 0 | 0 | 0 | −1.87230 | + | 1.08097i | 0 | −2.55958 | − | 0.669737i | 0 | 0 | 0 | ||||||||||||||
17.7 | 0 | 0 | 0 | −1.53798 | + | 0.887954i | 0 | −0.843933 | + | 2.50754i | 0 | 0 | 0 | ||||||||||||||
17.8 | 0 | 0 | 0 | −1.53798 | + | 0.887954i | 0 | −0.843933 | + | 2.50754i | 0 | 0 | 0 | ||||||||||||||
17.9 | 0 | 0 | 0 | −1.00441 | + | 0.579896i | 0 | −1.24394 | − | 2.33508i | 0 | 0 | 0 | ||||||||||||||
17.10 | 0 | 0 | 0 | −1.00441 | + | 0.579896i | 0 | −1.24394 | − | 2.33508i | 0 | 0 | 0 | ||||||||||||||
17.11 | 0 | 0 | 0 | −0.785247 | + | 0.453362i | 0 | 2.47043 | + | 0.947077i | 0 | 0 | 0 | ||||||||||||||
17.12 | 0 | 0 | 0 | −0.785247 | + | 0.453362i | 0 | 2.47043 | + | 0.947077i | 0 | 0 | 0 | ||||||||||||||
17.13 | 0 | 0 | 0 | −0.331990 | + | 0.191675i | 0 | 2.03027 | − | 1.69647i | 0 | 0 | 0 | ||||||||||||||
17.14 | 0 | 0 | 0 | −0.331990 | + | 0.191675i | 0 | 2.03027 | − | 1.69647i | 0 | 0 | 0 | ||||||||||||||
17.15 | 0 | 0 | 0 | 0.331990 | − | 0.191675i | 0 | 2.03027 | − | 1.69647i | 0 | 0 | 0 | ||||||||||||||
17.16 | 0 | 0 | 0 | 0.331990 | − | 0.191675i | 0 | 2.03027 | − | 1.69647i | 0 | 0 | 0 | ||||||||||||||
17.17 | 0 | 0 | 0 | 0.785247 | − | 0.453362i | 0 | 2.47043 | + | 0.947077i | 0 | 0 | 0 | ||||||||||||||
17.18 | 0 | 0 | 0 | 0.785247 | − | 0.453362i | 0 | 2.47043 | + | 0.947077i | 0 | 0 | 0 | ||||||||||||||
17.19 | 0 | 0 | 0 | 1.00441 | − | 0.579896i | 0 | −1.24394 | − | 2.33508i | 0 | 0 | 0 | ||||||||||||||
17.20 | 0 | 0 | 0 | 1.00441 | − | 0.579896i | 0 | −1.24394 | − | 2.33508i | 0 | 0 | 0 | ||||||||||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
8.b | even | 2 | 1 | inner |
21.g | even | 6 | 1 | inner |
24.h | odd | 2 | 1 | inner |
56.j | odd | 6 | 1 | inner |
168.ba | 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.cp.b | 56 | |
3.b | odd | 2 | 1 | inner | 2016.2.cp.b | 56 | |
4.b | odd | 2 | 1 | 504.2.ch.b | ✓ | 56 | |
7.d | odd | 6 | 1 | inner | 2016.2.cp.b | 56 | |
8.b | even | 2 | 1 | inner | 2016.2.cp.b | 56 | |
8.d | odd | 2 | 1 | 504.2.ch.b | ✓ | 56 | |
12.b | even | 2 | 1 | 504.2.ch.b | ✓ | 56 | |
21.g | even | 6 | 1 | inner | 2016.2.cp.b | 56 | |
24.f | even | 2 | 1 | 504.2.ch.b | ✓ | 56 | |
24.h | odd | 2 | 1 | inner | 2016.2.cp.b | 56 | |
28.f | even | 6 | 1 | 504.2.ch.b | ✓ | 56 | |
56.j | odd | 6 | 1 | inner | 2016.2.cp.b | 56 | |
56.m | even | 6 | 1 | 504.2.ch.b | ✓ | 56 | |
84.j | odd | 6 | 1 | 504.2.ch.b | ✓ | 56 | |
168.ba | even | 6 | 1 | inner | 2016.2.cp.b | 56 | |
168.be | odd | 6 | 1 | 504.2.ch.b | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
504.2.ch.b | ✓ | 56 | 4.b | odd | 2 | 1 | |
504.2.ch.b | ✓ | 56 | 8.d | odd | 2 | 1 | |
504.2.ch.b | ✓ | 56 | 12.b | even | 2 | 1 | |
504.2.ch.b | ✓ | 56 | 24.f | even | 2 | 1 | |
504.2.ch.b | ✓ | 56 | 28.f | even | 6 | 1 | |
504.2.ch.b | ✓ | 56 | 56.m | even | 6 | 1 | |
504.2.ch.b | ✓ | 56 | 84.j | odd | 6 | 1 | |
504.2.ch.b | ✓ | 56 | 168.be | odd | 6 | 1 | |
2016.2.cp.b | 56 | 1.a | even | 1 | 1 | trivial | |
2016.2.cp.b | 56 | 3.b | odd | 2 | 1 | inner | |
2016.2.cp.b | 56 | 7.d | odd | 6 | 1 | inner | |
2016.2.cp.b | 56 | 8.b | even | 2 | 1 | inner | |
2016.2.cp.b | 56 | 21.g | even | 6 | 1 | inner | |
2016.2.cp.b | 56 | 24.h | odd | 2 | 1 | inner | |
2016.2.cp.b | 56 | 56.j | odd | 6 | 1 | inner | |
2016.2.cp.b | 56 | 168.ba | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{28} - 37 T_{5}^{26} + 882 T_{5}^{24} - 12429 T_{5}^{22} + 126330 T_{5}^{20} - 820581 T_{5}^{18} + 3879393 T_{5}^{16} - 12338064 T_{5}^{14} + 28587144 T_{5}^{12} - 42674992 T_{5}^{10} + \cdots + 186624 \)
acting on \(S_{2}^{\mathrm{new}}(2016, [\chi])\).