Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2646,2,Mod(1979,2646)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2646, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2646.1979");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2646.t (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: | \(21.1284163748\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 882) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1979.1 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | −1.44900 | 0 | 0 | 1.00000i | 0 | 1.25487 | − | 0.724499i | ||||||||||||
1979.2 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 1.44900 | 0 | 0 | 1.00000i | 0 | −1.25487 | + | 0.724499i | ||||||||||||
1979.3 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | −1.42597 | 0 | 0 | 1.00000i | 0 | 1.23492 | − | 0.712984i | ||||||||||||
1979.4 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 1.42597 | 0 | 0 | 1.00000i | 0 | −1.23492 | + | 0.712984i | ||||||||||||
1979.5 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | −1.16972 | 0 | 0 | 1.00000i | 0 | 1.01301 | − | 0.584859i | ||||||||||||
1979.6 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 1.16972 | 0 | 0 | 1.00000i | 0 | −1.01301 | + | 0.584859i | ||||||||||||
1979.7 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | −0.949113 | 0 | 0 | 1.00000i | 0 | 0.821956 | − | 0.474556i | ||||||||||||
1979.8 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.949113 | 0 | 0 | 1.00000i | 0 | −0.821956 | + | 0.474556i | ||||||||||||
1979.9 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | −2.70053 | 0 | 0 | 1.00000i | 0 | 2.33872 | − | 1.35026i | ||||||||||||
1979.10 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 2.70053 | 0 | 0 | 1.00000i | 0 | −2.33872 | + | 1.35026i | ||||||||||||
1979.11 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | −3.92134 | 0 | 0 | 1.00000i | 0 | 3.39598 | − | 1.96067i | ||||||||||||
1979.12 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 3.92134 | 0 | 0 | 1.00000i | 0 | −3.39598 | + | 1.96067i | ||||||||||||
1979.13 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | −3.98682 | 0 | 0 | − | 1.00000i | 0 | −3.45268 | + | 1.99341i | |||||||||||
1979.14 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 3.98682 | 0 | 0 | − | 1.00000i | 0 | 3.45268 | − | 1.99341i | |||||||||||
1979.15 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | −0.440174 | 0 | 0 | − | 1.00000i | 0 | −0.381202 | + | 0.220087i | |||||||||||
1979.16 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.440174 | 0 | 0 | − | 1.00000i | 0 | 0.381202 | − | 0.220087i | |||||||||||
1979.17 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | −1.89776 | 0 | 0 | − | 1.00000i | 0 | −1.64351 | + | 0.948881i | |||||||||||
1979.18 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 1.89776 | 0 | 0 | − | 1.00000i | 0 | 1.64351 | − | 0.948881i | |||||||||||
1979.19 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | −1.99040 | 0 | 0 | − | 1.00000i | 0 | −1.72374 | + | 0.995200i | |||||||||||
1979.20 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 1.99040 | 0 | 0 | − | 1.00000i | 0 | 1.72374 | − | 0.995200i | |||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
63.n | odd | 6 | 1 | inner |
63.s | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2646.2.t.c | 48 | |
3.b | odd | 2 | 1 | 882.2.t.c | 48 | ||
7.b | odd | 2 | 1 | inner | 2646.2.t.c | 48 | |
7.c | even | 3 | 1 | 2646.2.l.c | 48 | ||
7.c | even | 3 | 1 | 2646.2.m.c | 48 | ||
7.d | odd | 6 | 1 | 2646.2.l.c | 48 | ||
7.d | odd | 6 | 1 | 2646.2.m.c | 48 | ||
9.c | even | 3 | 1 | 882.2.l.c | 48 | ||
9.d | odd | 6 | 1 | 2646.2.l.c | 48 | ||
21.c | even | 2 | 1 | 882.2.t.c | 48 | ||
21.g | even | 6 | 1 | 882.2.l.c | 48 | ||
21.g | even | 6 | 1 | 882.2.m.c | ✓ | 48 | |
21.h | odd | 6 | 1 | 882.2.l.c | 48 | ||
21.h | odd | 6 | 1 | 882.2.m.c | ✓ | 48 | |
63.g | even | 3 | 1 | 882.2.t.c | 48 | ||
63.h | even | 3 | 1 | 882.2.m.c | ✓ | 48 | |
63.i | even | 6 | 1 | 2646.2.m.c | 48 | ||
63.j | odd | 6 | 1 | 2646.2.m.c | 48 | ||
63.k | odd | 6 | 1 | 882.2.t.c | 48 | ||
63.l | odd | 6 | 1 | 882.2.l.c | 48 | ||
63.n | odd | 6 | 1 | inner | 2646.2.t.c | 48 | |
63.o | even | 6 | 1 | 2646.2.l.c | 48 | ||
63.s | even | 6 | 1 | inner | 2646.2.t.c | 48 | |
63.t | odd | 6 | 1 | 882.2.m.c | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
882.2.l.c | 48 | 9.c | even | 3 | 1 | ||
882.2.l.c | 48 | 21.g | even | 6 | 1 | ||
882.2.l.c | 48 | 21.h | odd | 6 | 1 | ||
882.2.l.c | 48 | 63.l | odd | 6 | 1 | ||
882.2.m.c | ✓ | 48 | 21.g | even | 6 | 1 | |
882.2.m.c | ✓ | 48 | 21.h | odd | 6 | 1 | |
882.2.m.c | ✓ | 48 | 63.h | even | 3 | 1 | |
882.2.m.c | ✓ | 48 | 63.t | odd | 6 | 1 | |
882.2.t.c | 48 | 3.b | odd | 2 | 1 | ||
882.2.t.c | 48 | 21.c | even | 2 | 1 | ||
882.2.t.c | 48 | 63.g | even | 3 | 1 | ||
882.2.t.c | 48 | 63.k | odd | 6 | 1 | ||
2646.2.l.c | 48 | 7.c | even | 3 | 1 | ||
2646.2.l.c | 48 | 7.d | odd | 6 | 1 | ||
2646.2.l.c | 48 | 9.d | odd | 6 | 1 | ||
2646.2.l.c | 48 | 63.o | even | 6 | 1 | ||
2646.2.m.c | 48 | 7.c | even | 3 | 1 | ||
2646.2.m.c | 48 | 7.d | odd | 6 | 1 | ||
2646.2.m.c | 48 | 63.i | even | 6 | 1 | ||
2646.2.m.c | 48 | 63.j | odd | 6 | 1 | ||
2646.2.t.c | 48 | 1.a | even | 1 | 1 | trivial | |
2646.2.t.c | 48 | 7.b | odd | 2 | 1 | inner | |
2646.2.t.c | 48 | 63.n | odd | 6 | 1 | inner | |
2646.2.t.c | 48 | 63.s | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 72 T_{5}^{22} + 2208 T_{5}^{20} - 37880 T_{5}^{18} + 402114 T_{5}^{16} - 2762952 T_{5}^{14} + \cdots + 2408704 \) acting on \(S_{2}^{\mathrm{new}}(2646, [\chi])\).