Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1764,2,Mod(509,1764)]
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, 5, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1764.509");
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.w (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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
509.1 | 0 | −1.72970 | + | 0.0901145i | 0 | 0.820505 | + | 1.42116i | 0 | 0 | 0 | 2.98376 | − | 0.311743i | 0 | ||||||||||||
509.2 | 0 | −1.65072 | − | 0.524523i | 0 | −1.07304 | − | 1.85856i | 0 | 0 | 0 | 2.44975 | + | 1.73168i | 0 | ||||||||||||
509.3 | 0 | −1.61937 | + | 0.614514i | 0 | 0.0565510 | + | 0.0979493i | 0 | 0 | 0 | 2.24475 | − | 1.99026i | 0 | ||||||||||||
509.4 | 0 | −1.46553 | + | 0.923160i | 0 | 0.745447 | + | 1.29115i | 0 | 0 | 0 | 1.29555 | − | 2.70584i | 0 | ||||||||||||
509.5 | 0 | −1.35623 | + | 1.07733i | 0 | −2.09678 | − | 3.63173i | 0 | 0 | 0 | 0.678740 | − | 2.92221i | 0 | ||||||||||||
509.6 | 0 | −1.32423 | − | 1.11643i | 0 | −1.69059 | − | 2.92819i | 0 | 0 | 0 | 0.507186 | + | 2.95682i | 0 | ||||||||||||
509.7 | 0 | −1.10468 | − | 1.33404i | 0 | −0.0920545 | − | 0.159443i | 0 | 0 | 0 | −0.559350 | + | 2.94739i | 0 | ||||||||||||
509.8 | 0 | −0.876908 | − | 1.49366i | 0 | 1.64618 | + | 2.85127i | 0 | 0 | 0 | −1.46207 | + | 2.61961i | 0 | ||||||||||||
509.9 | 0 | −0.724112 | − | 1.57342i | 0 | −1.31387 | − | 2.27569i | 0 | 0 | 0 | −1.95132 | + | 2.27867i | 0 | ||||||||||||
509.10 | 0 | −0.474260 | + | 1.66586i | 0 | 0.882170 | + | 1.52796i | 0 | 0 | 0 | −2.55016 | − | 1.58010i | 0 | ||||||||||||
509.11 | 0 | −0.360367 | − | 1.69415i | 0 | 1.64333 | + | 2.84633i | 0 | 0 | 0 | −2.74027 | + | 1.22103i | 0 | ||||||||||||
509.12 | 0 | −0.227413 | + | 1.71706i | 0 | −0.662184 | − | 1.14694i | 0 | 0 | 0 | −2.89657 | − | 0.780960i | 0 | ||||||||||||
509.13 | 0 | 0.227413 | − | 1.71706i | 0 | 0.662184 | + | 1.14694i | 0 | 0 | 0 | −2.89657 | − | 0.780960i | 0 | ||||||||||||
509.14 | 0 | 0.360367 | + | 1.69415i | 0 | −1.64333 | − | 2.84633i | 0 | 0 | 0 | −2.74027 | + | 1.22103i | 0 | ||||||||||||
509.15 | 0 | 0.474260 | − | 1.66586i | 0 | −0.882170 | − | 1.52796i | 0 | 0 | 0 | −2.55016 | − | 1.58010i | 0 | ||||||||||||
509.16 | 0 | 0.724112 | + | 1.57342i | 0 | 1.31387 | + | 2.27569i | 0 | 0 | 0 | −1.95132 | + | 2.27867i | 0 | ||||||||||||
509.17 | 0 | 0.876908 | + | 1.49366i | 0 | −1.64618 | − | 2.85127i | 0 | 0 | 0 | −1.46207 | + | 2.61961i | 0 | ||||||||||||
509.18 | 0 | 1.10468 | + | 1.33404i | 0 | 0.0920545 | + | 0.159443i | 0 | 0 | 0 | −0.559350 | + | 2.94739i | 0 | ||||||||||||
509.19 | 0 | 1.32423 | + | 1.11643i | 0 | 1.69059 | + | 2.92819i | 0 | 0 | 0 | 0.507186 | + | 2.95682i | 0 | ||||||||||||
509.20 | 0 | 1.35623 | − | 1.07733i | 0 | 2.09678 | + | 3.63173i | 0 | 0 | 0 | 0.678740 | − | 2.92221i | 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.i | even | 6 | 1 | inner |
63.j | 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.w.c | 48 | |
3.b | odd | 2 | 1 | 5292.2.w.c | 48 | ||
7.b | odd | 2 | 1 | inner | 1764.2.w.c | 48 | |
7.c | even | 3 | 1 | 1764.2.x.c | ✓ | 48 | |
7.c | even | 3 | 1 | 1764.2.bm.c | 48 | ||
7.d | odd | 6 | 1 | 1764.2.x.c | ✓ | 48 | |
7.d | odd | 6 | 1 | 1764.2.bm.c | 48 | ||
9.c | even | 3 | 1 | 5292.2.bm.c | 48 | ||
9.d | odd | 6 | 1 | 1764.2.bm.c | 48 | ||
21.c | even | 2 | 1 | 5292.2.w.c | 48 | ||
21.g | even | 6 | 1 | 5292.2.x.c | 48 | ||
21.g | even | 6 | 1 | 5292.2.bm.c | 48 | ||
21.h | odd | 6 | 1 | 5292.2.x.c | 48 | ||
21.h | odd | 6 | 1 | 5292.2.bm.c | 48 | ||
63.g | even | 3 | 1 | 5292.2.x.c | 48 | ||
63.h | even | 3 | 1 | 5292.2.w.c | 48 | ||
63.i | even | 6 | 1 | inner | 1764.2.w.c | 48 | |
63.j | odd | 6 | 1 | inner | 1764.2.w.c | 48 | |
63.k | odd | 6 | 1 | 5292.2.x.c | 48 | ||
63.l | odd | 6 | 1 | 5292.2.bm.c | 48 | ||
63.n | odd | 6 | 1 | 1764.2.x.c | ✓ | 48 | |
63.o | even | 6 | 1 | 1764.2.bm.c | 48 | ||
63.s | even | 6 | 1 | 1764.2.x.c | ✓ | 48 | |
63.t | odd | 6 | 1 | 5292.2.w.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1764.2.w.c | 48 | 1.a | even | 1 | 1 | trivial | |
1764.2.w.c | 48 | 7.b | odd | 2 | 1 | inner | |
1764.2.w.c | 48 | 63.i | even | 6 | 1 | inner | |
1764.2.w.c | 48 | 63.j | odd | 6 | 1 | inner | |
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.n | odd | 6 | 1 | |
1764.2.x.c | ✓ | 48 | 63.s | even | 6 | 1 | |
1764.2.bm.c | 48 | 7.c | even | 3 | 1 | ||
1764.2.bm.c | 48 | 7.d | odd | 6 | 1 | ||
1764.2.bm.c | 48 | 9.d | odd | 6 | 1 | ||
1764.2.bm.c | 48 | 63.o | even | 6 | 1 | ||
5292.2.w.c | 48 | 3.b | odd | 2 | 1 | ||
5292.2.w.c | 48 | 21.c | even | 2 | 1 | ||
5292.2.w.c | 48 | 63.h | even | 3 | 1 | ||
5292.2.w.c | 48 | 63.t | 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.g | even | 3 | 1 | ||
5292.2.x.c | 48 | 63.k | odd | 6 | 1 | ||
5292.2.bm.c | 48 | 9.c | even | 3 | 1 | ||
5292.2.bm.c | 48 | 21.g | even | 6 | 1 | ||
5292.2.bm.c | 48 | 21.h | odd | 6 | 1 | ||
5292.2.bm.c | 48 | 63.l | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} + 72 T_{5}^{46} + 2976 T_{5}^{44} + 83216 T_{5}^{42} + 1746483 T_{5}^{40} + 28573356 T_{5}^{38} + \cdots + 112550881 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).