Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1764,2,Mod(1685,1764)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 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("1764.1685");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1764.bm (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: | \(14.0856109166\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1685.1 | 0 | −1.67980 | − | 0.422207i | 0 | 1.76434 | 0 | 0 | 0 | 2.64348 | + | 1.41845i | 0 | ||||||||||||||
1685.2 | 0 | −1.61111 | + | 0.635870i | 0 | −4.19356 | 0 | 0 | 0 | 2.19134 | − | 2.04891i | 0 | ||||||||||||||
1685.3 | 0 | −1.60072 | − | 0.661583i | 0 | −1.32437 | 0 | 0 | 0 | 2.12462 | + | 2.11802i | 0 | ||||||||||||||
1685.4 | 0 | −1.53224 | + | 0.807605i | 0 | 1.49089 | 0 | 0 | 0 | 1.69555 | − | 2.47490i | 0 | ||||||||||||||
1685.5 | 0 | −1.34187 | + | 1.09516i | 0 | 0.113102 | 0 | 0 | 0 | 0.601240 | − | 2.93913i | 0 | ||||||||||||||
1685.6 | 0 | −1.28699 | − | 1.15916i | 0 | −3.28666 | 0 | 0 | 0 | 0.312692 | + | 2.98366i | 0 | ||||||||||||||
1685.7 | 0 | −1.00057 | − | 1.41381i | 0 | 2.62774 | 0 | 0 | 0 | −0.997726 | + | 2.82923i | 0 | ||||||||||||||
1685.8 | 0 | −0.942894 | + | 1.45291i | 0 | 1.64101 | 0 | 0 | 0 | −1.22190 | − | 2.73988i | 0 | ||||||||||||||
1685.9 | 0 | −0.855097 | − | 1.50626i | 0 | −3.29237 | 0 | 0 | 0 | −1.53762 | + | 2.57599i | 0 | ||||||||||||||
1685.10 | 0 | −0.602975 | − | 1.62371i | 0 | 0.184109 | 0 | 0 | 0 | −2.27284 | + | 1.95811i | 0 | ||||||||||||||
1685.11 | 0 | −0.371109 | + | 1.69183i | 0 | −2.14608 | 0 | 0 | 0 | −2.72456 | − | 1.25571i | 0 | ||||||||||||||
1685.12 | 0 | −0.304737 | − | 1.70503i | 0 | 3.38118 | 0 | 0 | 0 | −2.81427 | + | 1.03917i | 0 | ||||||||||||||
1685.13 | 0 | 0.304737 | + | 1.70503i | 0 | −3.38118 | 0 | 0 | 0 | −2.81427 | + | 1.03917i | 0 | ||||||||||||||
1685.14 | 0 | 0.371109 | − | 1.69183i | 0 | 2.14608 | 0 | 0 | 0 | −2.72456 | − | 1.25571i | 0 | ||||||||||||||
1685.15 | 0 | 0.602975 | + | 1.62371i | 0 | −0.184109 | 0 | 0 | 0 | −2.27284 | + | 1.95811i | 0 | ||||||||||||||
1685.16 | 0 | 0.855097 | + | 1.50626i | 0 | 3.29237 | 0 | 0 | 0 | −1.53762 | + | 2.57599i | 0 | ||||||||||||||
1685.17 | 0 | 0.942894 | − | 1.45291i | 0 | −1.64101 | 0 | 0 | 0 | −1.22190 | − | 2.73988i | 0 | ||||||||||||||
1685.18 | 0 | 1.00057 | + | 1.41381i | 0 | −2.62774 | 0 | 0 | 0 | −0.997726 | + | 2.82923i | 0 | ||||||||||||||
1685.19 | 0 | 1.28699 | + | 1.15916i | 0 | 3.28666 | 0 | 0 | 0 | 0.312692 | + | 2.98366i | 0 | ||||||||||||||
1685.20 | 0 | 1.34187 | − | 1.09516i | 0 | −0.113102 | 0 | 0 | 0 | 0.601240 | − | 2.93913i | 0 | ||||||||||||||
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 | 1764.2.bm.c | 48 | |
3.b | odd | 2 | 1 | 5292.2.bm.c | 48 | ||
7.b | odd | 2 | 1 | inner | 1764.2.bm.c | 48 | |
7.c | even | 3 | 1 | 1764.2.w.c | 48 | ||
7.c | even | 3 | 1 | 1764.2.x.c | ✓ | 48 | |
7.d | odd | 6 | 1 | 1764.2.w.c | 48 | ||
7.d | odd | 6 | 1 | 1764.2.x.c | ✓ | 48 | |
9.c | even | 3 | 1 | 5292.2.w.c | 48 | ||
9.d | odd | 6 | 1 | 1764.2.w.c | 48 | ||
21.c | even | 2 | 1 | 5292.2.bm.c | 48 | ||
21.g | even | 6 | 1 | 5292.2.w.c | 48 | ||
21.g | even | 6 | 1 | 5292.2.x.c | 48 | ||
21.h | odd | 6 | 1 | 5292.2.w.c | 48 | ||
21.h | odd | 6 | 1 | 5292.2.x.c | 48 | ||
63.g | even | 3 | 1 | 5292.2.bm.c | 48 | ||
63.h | even | 3 | 1 | 5292.2.x.c | 48 | ||
63.i | even | 6 | 1 | 1764.2.x.c | ✓ | 48 | |
63.j | odd | 6 | 1 | 1764.2.x.c | ✓ | 48 | |
63.k | odd | 6 | 1 | 5292.2.bm.c | 48 | ||
63.l | odd | 6 | 1 | 5292.2.w.c | 48 | ||
63.n | odd | 6 | 1 | inner | 1764.2.bm.c | 48 | |
63.o | even | 6 | 1 | 1764.2.w.c | 48 | ||
63.s | even | 6 | 1 | inner | 1764.2.bm.c | 48 | |
63.t | odd | 6 | 1 | 5292.2.x.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1764.2.w.c | 48 | 7.c | even | 3 | 1 | ||
1764.2.w.c | 48 | 7.d | odd | 6 | 1 | ||
1764.2.w.c | 48 | 9.d | odd | 6 | 1 | ||
1764.2.w.c | 48 | 63.o | even | 6 | 1 | ||
1764.2.x.c | ✓ | 48 | 7.c | even | 3 | 1 | |
1764.2.x.c | ✓ | 48 | 7.d | odd | 6 | 1 | |
1764.2.x.c | ✓ | 48 | 63.i | even | 6 | 1 | |
1764.2.x.c | ✓ | 48 | 63.j | odd | 6 | 1 | |
1764.2.bm.c | 48 | 1.a | even | 1 | 1 | trivial | |
1764.2.bm.c | 48 | 7.b | odd | 2 | 1 | inner | |
1764.2.bm.c | 48 | 63.n | odd | 6 | 1 | inner | |
1764.2.bm.c | 48 | 63.s | even | 6 | 1 | inner | |
5292.2.w.c | 48 | 9.c | even | 3 | 1 | ||
5292.2.w.c | 48 | 21.g | even | 6 | 1 | ||
5292.2.w.c | 48 | 21.h | odd | 6 | 1 | ||
5292.2.w.c | 48 | 63.l | odd | 6 | 1 | ||
5292.2.x.c | 48 | 21.g | even | 6 | 1 | ||
5292.2.x.c | 48 | 21.h | odd | 6 | 1 | ||
5292.2.x.c | 48 | 63.h | even | 3 | 1 | ||
5292.2.x.c | 48 | 63.t | odd | 6 | 1 | ||
5292.2.bm.c | 48 | 3.b | odd | 2 | 1 | ||
5292.2.bm.c | 48 | 21.c | even | 2 | 1 | ||
5292.2.bm.c | 48 | 63.g | even | 3 | 1 | ||
5292.2.bm.c | 48 | 63.k | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 72 T_{5}^{22} + 2208 T_{5}^{20} - 37880 T_{5}^{18} + 401421 T_{5}^{16} - 2738940 T_{5}^{14} + \cdots + 10609 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).