Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1764,2,Mod(949,1764)]
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, 2, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1764.949");
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.l (of order \(3\), 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: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
949.1 | 0 | −1.73160 | + | 0.0393104i | 0 | 2.38486 | 0 | 0 | 0 | 2.99691 | − | 0.136140i | 0 | ||||||||||||||
949.2 | 0 | −1.52658 | + | 0.818256i | 0 | 3.89246 | 0 | 0 | 0 | 1.66092 | − | 2.49827i | 0 | ||||||||||||||
949.3 | 0 | −1.36474 | − | 1.06653i | 0 | 0.0223826 | 0 | 0 | 0 | 0.725035 | + | 2.91107i | 0 | ||||||||||||||
949.4 | 0 | −0.755620 | + | 1.55854i | 0 | −3.47961 | 0 | 0 | 0 | −1.85808 | − | 2.35532i | 0 | ||||||||||||||
949.5 | 0 | −0.666148 | − | 1.59883i | 0 | −1.47387 | 0 | 0 | 0 | −2.11249 | + | 2.13011i | 0 | ||||||||||||||
949.6 | 0 | −0.542084 | + | 1.64504i | 0 | 0.938454 | 0 | 0 | 0 | −2.41229 | − | 1.78350i | 0 | ||||||||||||||
949.7 | 0 | 0.542084 | − | 1.64504i | 0 | −0.938454 | 0 | 0 | 0 | −2.41229 | − | 1.78350i | 0 | ||||||||||||||
949.8 | 0 | 0.666148 | + | 1.59883i | 0 | 1.47387 | 0 | 0 | 0 | −2.11249 | + | 2.13011i | 0 | ||||||||||||||
949.9 | 0 | 0.755620 | − | 1.55854i | 0 | 3.47961 | 0 | 0 | 0 | −1.85808 | − | 2.35532i | 0 | ||||||||||||||
949.10 | 0 | 1.36474 | + | 1.06653i | 0 | −0.0223826 | 0 | 0 | 0 | 0.725035 | + | 2.91107i | 0 | ||||||||||||||
949.11 | 0 | 1.52658 | − | 0.818256i | 0 | −3.89246 | 0 | 0 | 0 | 1.66092 | − | 2.49827i | 0 | ||||||||||||||
949.12 | 0 | 1.73160 | − | 0.0393104i | 0 | −2.38486 | 0 | 0 | 0 | 2.99691 | − | 0.136140i | 0 | ||||||||||||||
961.1 | 0 | −1.73160 | − | 0.0393104i | 0 | 2.38486 | 0 | 0 | 0 | 2.99691 | + | 0.136140i | 0 | ||||||||||||||
961.2 | 0 | −1.52658 | − | 0.818256i | 0 | 3.89246 | 0 | 0 | 0 | 1.66092 | + | 2.49827i | 0 | ||||||||||||||
961.3 | 0 | −1.36474 | + | 1.06653i | 0 | 0.0223826 | 0 | 0 | 0 | 0.725035 | − | 2.91107i | 0 | ||||||||||||||
961.4 | 0 | −0.755620 | − | 1.55854i | 0 | −3.47961 | 0 | 0 | 0 | −1.85808 | + | 2.35532i | 0 | ||||||||||||||
961.5 | 0 | −0.666148 | + | 1.59883i | 0 | −1.47387 | 0 | 0 | 0 | −2.11249 | − | 2.13011i | 0 | ||||||||||||||
961.6 | 0 | −0.542084 | − | 1.64504i | 0 | 0.938454 | 0 | 0 | 0 | −2.41229 | + | 1.78350i | 0 | ||||||||||||||
961.7 | 0 | 0.542084 | + | 1.64504i | 0 | −0.938454 | 0 | 0 | 0 | −2.41229 | + | 1.78350i | 0 | ||||||||||||||
961.8 | 0 | 0.666148 | − | 1.59883i | 0 | 1.47387 | 0 | 0 | 0 | −2.11249 | − | 2.13011i | 0 | ||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
63.g | even | 3 | 1 | inner |
63.k | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1764.2.l.j | 24 | |
3.b | odd | 2 | 1 | 5292.2.l.j | 24 | ||
7.b | odd | 2 | 1 | inner | 1764.2.l.j | 24 | |
7.c | even | 3 | 1 | 1764.2.i.j | 24 | ||
7.c | even | 3 | 1 | 1764.2.j.i | ✓ | 24 | |
7.d | odd | 6 | 1 | 1764.2.i.j | 24 | ||
7.d | odd | 6 | 1 | 1764.2.j.i | ✓ | 24 | |
9.c | even | 3 | 1 | 1764.2.i.j | 24 | ||
9.d | odd | 6 | 1 | 5292.2.i.j | 24 | ||
21.c | even | 2 | 1 | 5292.2.l.j | 24 | ||
21.g | even | 6 | 1 | 5292.2.i.j | 24 | ||
21.g | even | 6 | 1 | 5292.2.j.i | 24 | ||
21.h | odd | 6 | 1 | 5292.2.i.j | 24 | ||
21.h | odd | 6 | 1 | 5292.2.j.i | 24 | ||
63.g | even | 3 | 1 | inner | 1764.2.l.j | 24 | |
63.h | even | 3 | 1 | 1764.2.j.i | ✓ | 24 | |
63.i | even | 6 | 1 | 5292.2.j.i | 24 | ||
63.j | odd | 6 | 1 | 5292.2.j.i | 24 | ||
63.k | odd | 6 | 1 | inner | 1764.2.l.j | 24 | |
63.l | odd | 6 | 1 | 1764.2.i.j | 24 | ||
63.n | odd | 6 | 1 | 5292.2.l.j | 24 | ||
63.o | even | 6 | 1 | 5292.2.i.j | 24 | ||
63.s | even | 6 | 1 | 5292.2.l.j | 24 | ||
63.t | odd | 6 | 1 | 1764.2.j.i | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1764.2.i.j | 24 | 7.c | even | 3 | 1 | ||
1764.2.i.j | 24 | 7.d | odd | 6 | 1 | ||
1764.2.i.j | 24 | 9.c | even | 3 | 1 | ||
1764.2.i.j | 24 | 63.l | odd | 6 | 1 | ||
1764.2.j.i | ✓ | 24 | 7.c | even | 3 | 1 | |
1764.2.j.i | ✓ | 24 | 7.d | odd | 6 | 1 | |
1764.2.j.i | ✓ | 24 | 63.h | even | 3 | 1 | |
1764.2.j.i | ✓ | 24 | 63.t | odd | 6 | 1 | |
1764.2.l.j | 24 | 1.a | even | 1 | 1 | trivial | |
1764.2.l.j | 24 | 7.b | odd | 2 | 1 | inner | |
1764.2.l.j | 24 | 63.g | even | 3 | 1 | inner | |
1764.2.l.j | 24 | 63.k | odd | 6 | 1 | inner | |
5292.2.i.j | 24 | 9.d | odd | 6 | 1 | ||
5292.2.i.j | 24 | 21.g | even | 6 | 1 | ||
5292.2.i.j | 24 | 21.h | odd | 6 | 1 | ||
5292.2.i.j | 24 | 63.o | even | 6 | 1 | ||
5292.2.j.i | 24 | 21.g | even | 6 | 1 | ||
5292.2.j.i | 24 | 21.h | odd | 6 | 1 | ||
5292.2.j.i | 24 | 63.i | even | 6 | 1 | ||
5292.2.j.i | 24 | 63.j | odd | 6 | 1 | ||
5292.2.l.j | 24 | 3.b | odd | 2 | 1 | ||
5292.2.l.j | 24 | 21.c | even | 2 | 1 | ||
5292.2.l.j | 24 | 63.n | odd | 6 | 1 | ||
5292.2.l.j | 24 | 63.s | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} - 36T_{5}^{10} + 441T_{5}^{8} - 2140T_{5}^{6} + 3834T_{5}^{4} - 1998T_{5}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).