Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [588,2,Mod(19,588)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(588, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("588.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 588.o (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: | \(4.69520363885\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −1.41214 | − | 0.0764704i | −0.500000 | − | 0.866025i | 1.98830 | + | 0.215975i | −2.68862 | − | 1.55228i | 0.639847 | + | 1.26119i | 0 | −2.79126 | − | 0.457034i | −0.500000 | + | 0.866025i | 3.67802 | + | 2.39764i | ||
19.2 | −1.33068 | − | 0.478844i | −0.500000 | − | 0.866025i | 1.54142 | + | 1.27438i | 2.93763 | + | 1.69604i | 0.250649 | + | 1.39182i | 0 | −1.44090 | − | 2.43388i | −0.500000 | + | 0.866025i | −3.09691 | − | 3.66356i | ||
19.3 | −1.22355 | + | 0.709180i | −0.500000 | − | 0.866025i | 0.994127 | − | 1.73543i | −3.15890 | − | 1.82379i | 1.22594 | + | 0.705031i | 0 | 0.0143727 | + | 2.82839i | −0.500000 | + | 0.866025i | 5.15845 | − | 0.00873760i | ||
19.4 | −1.04209 | + | 0.956063i | −0.500000 | − | 0.866025i | 0.171888 | − | 1.99260i | −0.110790 | − | 0.0639645i | 1.34902 | + | 0.424442i | 0 | 1.72593 | + | 2.24080i | −0.500000 | + | 0.866025i | 0.176607 | − | 0.0392655i | ||
19.5 | −0.914808 | − | 1.07848i | −0.500000 | − | 0.866025i | −0.326251 | + | 1.97321i | 0.441565 | + | 0.254938i | −0.476589 | + | 1.33149i | 0 | 2.42653 | − | 1.45325i | −0.500000 | + | 0.866025i | −0.129001 | − | 0.709440i | ||
19.6 | −0.381130 | − | 1.36189i | −0.500000 | − | 0.866025i | −1.70948 | + | 1.03811i | 0.977624 | + | 0.564431i | −0.988865 | + | 1.01101i | 0 | 2.06533 | + | 1.93246i | −0.500000 | + | 0.866025i | 0.396091 | − | 1.54654i | ||
19.7 | −0.306932 | + | 1.38050i | −0.500000 | − | 0.866025i | −1.81159 | − | 0.847441i | 0.110790 | + | 0.0639645i | 1.34902 | − | 0.424442i | 0 | 1.72593 | − | 2.24080i | −0.500000 | + | 0.866025i | −0.122308 | + | 0.133313i | ||
19.8 | −0.00239545 | + | 1.41421i | −0.500000 | − | 0.866025i | −1.99999 | − | 0.00677535i | 3.15890 | + | 1.82379i | 1.22594 | − | 0.705031i | 0 | 0.0143727 | − | 2.82839i | −0.500000 | + | 0.866025i | −2.58679 | + | 4.46298i | ||
19.9 | 0.772298 | + | 1.18472i | −0.500000 | − | 0.866025i | −0.807113 | + | 1.82991i | 2.68862 | + | 1.55228i | 0.639847 | − | 1.26119i | 0 | −2.79126 | + | 0.457034i | −0.500000 | + | 0.866025i | 0.237406 | + | 4.38407i | ||
19.10 | 1.08003 | + | 0.912980i | −0.500000 | − | 0.866025i | 0.332934 | + | 1.97209i | −2.93763 | − | 1.69604i | 0.250649 | − | 1.39182i | 0 | −1.44090 | + | 2.43388i | −0.500000 | + | 0.866025i | −1.62428 | − | 4.51378i | ||
19.11 | 1.36999 | − | 0.350876i | −0.500000 | − | 0.866025i | 1.75377 | − | 0.961396i | −0.977624 | − | 0.564431i | −0.988865 | − | 1.01101i | 0 | 2.06533 | − | 1.93246i | −0.500000 | + | 0.866025i | −1.53738 | − | 0.430244i | ||
19.12 | 1.39140 | + | 0.253006i | −0.500000 | − | 0.866025i | 1.87198 | + | 0.704063i | −0.441565 | − | 0.254938i | −0.476589 | − | 1.33149i | 0 | 2.42653 | + | 1.45325i | −0.500000 | + | 0.866025i | −0.549892 | − | 0.466438i | ||
31.1 | −1.41214 | + | 0.0764704i | −0.500000 | + | 0.866025i | 1.98830 | − | 0.215975i | −2.68862 | + | 1.55228i | 0.639847 | − | 1.26119i | 0 | −2.79126 | + | 0.457034i | −0.500000 | − | 0.866025i | 3.67802 | − | 2.39764i | ||
31.2 | −1.33068 | + | 0.478844i | −0.500000 | + | 0.866025i | 1.54142 | − | 1.27438i | 2.93763 | − | 1.69604i | 0.250649 | − | 1.39182i | 0 | −1.44090 | + | 2.43388i | −0.500000 | − | 0.866025i | −3.09691 | + | 3.66356i | ||
31.3 | −1.22355 | − | 0.709180i | −0.500000 | + | 0.866025i | 0.994127 | + | 1.73543i | −3.15890 | + | 1.82379i | 1.22594 | − | 0.705031i | 0 | 0.0143727 | − | 2.82839i | −0.500000 | − | 0.866025i | 5.15845 | + | 0.00873760i | ||
31.4 | −1.04209 | − | 0.956063i | −0.500000 | + | 0.866025i | 0.171888 | + | 1.99260i | −0.110790 | + | 0.0639645i | 1.34902 | − | 0.424442i | 0 | 1.72593 | − | 2.24080i | −0.500000 | − | 0.866025i | 0.176607 | + | 0.0392655i | ||
31.5 | −0.914808 | + | 1.07848i | −0.500000 | + | 0.866025i | −0.326251 | − | 1.97321i | 0.441565 | − | 0.254938i | −0.476589 | − | 1.33149i | 0 | 2.42653 | + | 1.45325i | −0.500000 | − | 0.866025i | −0.129001 | + | 0.709440i | ||
31.6 | −0.381130 | + | 1.36189i | −0.500000 | + | 0.866025i | −1.70948 | − | 1.03811i | 0.977624 | − | 0.564431i | −0.988865 | − | 1.01101i | 0 | 2.06533 | − | 1.93246i | −0.500000 | − | 0.866025i | 0.396091 | + | 1.54654i | ||
31.7 | −0.306932 | − | 1.38050i | −0.500000 | + | 0.866025i | −1.81159 | + | 0.847441i | 0.110790 | − | 0.0639645i | 1.34902 | + | 0.424442i | 0 | 1.72593 | + | 2.24080i | −0.500000 | − | 0.866025i | −0.122308 | − | 0.133313i | ||
31.8 | −0.00239545 | − | 1.41421i | −0.500000 | + | 0.866025i | −1.99999 | + | 0.00677535i | 3.15890 | − | 1.82379i | 1.22594 | + | 0.705031i | 0 | 0.0143727 | + | 2.82839i | −0.500000 | − | 0.866025i | −2.58679 | − | 4.46298i | ||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
28.d | even | 2 | 1 | inner |
28.f | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 588.2.o.e | 24 | |
4.b | odd | 2 | 1 | 588.2.o.f | 24 | ||
7.b | odd | 2 | 1 | 588.2.o.f | 24 | ||
7.c | even | 3 | 1 | 588.2.b.d | yes | 12 | |
7.c | even | 3 | 1 | inner | 588.2.o.e | 24 | |
7.d | odd | 6 | 1 | 588.2.b.c | ✓ | 12 | |
7.d | odd | 6 | 1 | 588.2.o.f | 24 | ||
21.g | even | 6 | 1 | 1764.2.b.l | 12 | ||
21.h | odd | 6 | 1 | 1764.2.b.m | 12 | ||
28.d | even | 2 | 1 | inner | 588.2.o.e | 24 | |
28.f | even | 6 | 1 | 588.2.b.d | yes | 12 | |
28.f | even | 6 | 1 | inner | 588.2.o.e | 24 | |
28.g | odd | 6 | 1 | 588.2.b.c | ✓ | 12 | |
28.g | odd | 6 | 1 | 588.2.o.f | 24 | ||
84.j | odd | 6 | 1 | 1764.2.b.m | 12 | ||
84.n | even | 6 | 1 | 1764.2.b.l | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
588.2.b.c | ✓ | 12 | 7.d | odd | 6 | 1 | |
588.2.b.c | ✓ | 12 | 28.g | odd | 6 | 1 | |
588.2.b.d | yes | 12 | 7.c | even | 3 | 1 | |
588.2.b.d | yes | 12 | 28.f | even | 6 | 1 | |
588.2.o.e | 24 | 1.a | even | 1 | 1 | trivial | |
588.2.o.e | 24 | 7.c | even | 3 | 1 | inner | |
588.2.o.e | 24 | 28.d | even | 2 | 1 | inner | |
588.2.o.e | 24 | 28.f | even | 6 | 1 | inner | |
588.2.o.f | 24 | 4.b | odd | 2 | 1 | ||
588.2.o.f | 24 | 7.b | odd | 2 | 1 | ||
588.2.o.f | 24 | 7.d | odd | 6 | 1 | ||
588.2.o.f | 24 | 28.g | odd | 6 | 1 | ||
1764.2.b.l | 12 | 21.g | even | 6 | 1 | ||
1764.2.b.l | 12 | 84.n | even | 6 | 1 | ||
1764.2.b.m | 12 | 21.h | odd | 6 | 1 | ||
1764.2.b.m | 12 | 84.j | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\):
\( T_{5}^{24} - 36 T_{5}^{22} + 850 T_{5}^{20} - 11864 T_{5}^{18} + 121032 T_{5}^{16} - 760528 T_{5}^{14} + \cdots + 64 \) |
\( T_{11}^{24} - 104 T_{11}^{22} + 6776 T_{11}^{20} - 274816 T_{11}^{18} + 8159792 T_{11}^{16} + \cdots + 3873527824384 \) |
\( T_{19}^{12} + 72 T_{19}^{10} - 64 T_{19}^{9} + 3896 T_{19}^{8} - 3584 T_{19}^{7} + 93248 T_{19}^{6} + \cdots + 65536 \) |