Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1890,2,Mod(89,1890)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1890, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1890.89");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1890.r (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: | \(15.0917259820\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 630) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
89.1 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −2.23138 | − | 0.144779i | 0 | 2.37108 | + | 1.17388i | −1.00000 | 0 | −0.990306 | − | 2.00482i | ||||||||
89.2 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −2.20770 | − | 0.355056i | 0 | 0.692019 | + | 2.55365i | −1.00000 | 0 | −0.796362 | − | 2.08945i | ||||||||
89.3 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −2.19316 | + | 0.435950i | 0 | −2.64572 | + | 0.0128895i | −1.00000 | 0 | −1.47412 | − | 1.68136i | ||||||||
89.4 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −2.07866 | + | 0.824119i | 0 | 0.837534 | − | 2.50969i | −1.00000 | 0 | −1.75304 | − | 1.38811i | ||||||||
89.5 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.96632 | + | 1.06470i | 0 | −0.733422 | + | 2.54206i | −1.00000 | 0 | −1.90522 | − | 1.17053i | ||||||||
89.6 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.76507 | − | 1.37278i | 0 | 1.05699 | − | 2.42544i | −1.00000 | 0 | 0.306329 | − | 2.21499i | ||||||||
89.7 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.29544 | + | 1.82259i | 0 | −1.77597 | − | 1.96110i | −1.00000 | 0 | −2.22613 | − | 0.210587i | ||||||||
89.8 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.28213 | − | 1.83198i | 0 | −2.37211 | − | 1.17179i | −1.00000 | 0 | 0.945479 | − | 2.02634i | ||||||||
89.9 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.782489 | − | 2.09469i | 0 | 2.24311 | + | 1.40302i | −1.00000 | 0 | 1.42281 | − | 1.72500i | ||||||||
89.10 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.619562 | + | 2.14852i | 0 | 2.06156 | − | 1.65830i | −1.00000 | 0 | −2.17045 | + | 0.537704i | ||||||||
89.11 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.390088 | + | 2.20178i | 0 | −0.160573 | + | 2.64087i | −1.00000 | 0 | −2.10184 | + | 0.763064i | ||||||||
89.12 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.0986685 | − | 2.23389i | 0 | −1.62563 | + | 2.08742i | −1.00000 | 0 | 1.88527 | − | 1.20239i | ||||||||
89.13 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0.00346114 | + | 2.23607i | 0 | 2.57793 | + | 0.595221i | −1.00000 | 0 | −1.93476 | + | 1.12103i | ||||||||
89.14 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0.438968 | − | 2.19256i | 0 | −0.272771 | − | 2.63165i | −1.00000 | 0 | 2.11829 | − | 0.716121i | ||||||||
89.15 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0.717414 | + | 2.11786i | 0 | −0.647443 | + | 2.56531i | −1.00000 | 0 | −1.47541 | + | 1.68023i | ||||||||
89.16 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.00280 | − | 1.99860i | 0 | −2.55747 | + | 0.677764i | −1.00000 | 0 | 2.23224 | − | 0.130844i | ||||||||
89.17 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.26026 | + | 1.84709i | 0 | −2.59865 | + | 0.497024i | −1.00000 | 0 | −0.969500 | + | 2.01496i | ||||||||
89.18 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.60670 | − | 1.55516i | 0 | 1.24799 | − | 2.33292i | −1.00000 | 0 | 2.15016 | + | 0.613862i | ||||||||
89.19 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.76448 | − | 1.37353i | 0 | 2.04479 | − | 1.67894i | −1.00000 | 0 | 2.07176 | + | 0.841321i | ||||||||
89.20 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.83728 | + | 1.27452i | 0 | 2.46948 | − | 0.949557i | −1.00000 | 0 | −0.185127 | + | 2.22839i | ||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
315.u | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1890.2.r.b | 48 | |
3.b | odd | 2 | 1 | 630.2.r.a | ✓ | 48 | |
5.b | even | 2 | 1 | 1890.2.r.a | 48 | ||
7.d | odd | 6 | 1 | 1890.2.bi.a | 48 | ||
9.c | even | 3 | 1 | 630.2.bi.a | yes | 48 | |
9.d | odd | 6 | 1 | 1890.2.bi.b | 48 | ||
15.d | odd | 2 | 1 | 630.2.r.b | yes | 48 | |
21.g | even | 6 | 1 | 630.2.bi.b | yes | 48 | |
35.i | odd | 6 | 1 | 1890.2.bi.b | 48 | ||
45.h | odd | 6 | 1 | 1890.2.bi.a | 48 | ||
45.j | even | 6 | 1 | 630.2.bi.b | yes | 48 | |
63.k | odd | 6 | 1 | 630.2.r.b | yes | 48 | |
63.s | even | 6 | 1 | 1890.2.r.a | 48 | ||
105.p | even | 6 | 1 | 630.2.bi.a | yes | 48 | |
315.u | even | 6 | 1 | inner | 1890.2.r.b | 48 | |
315.bn | odd | 6 | 1 | 630.2.r.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
630.2.r.a | ✓ | 48 | 3.b | odd | 2 | 1 | |
630.2.r.a | ✓ | 48 | 315.bn | odd | 6 | 1 | |
630.2.r.b | yes | 48 | 15.d | odd | 2 | 1 | |
630.2.r.b | yes | 48 | 63.k | odd | 6 | 1 | |
630.2.bi.a | yes | 48 | 9.c | even | 3 | 1 | |
630.2.bi.a | yes | 48 | 105.p | even | 6 | 1 | |
630.2.bi.b | yes | 48 | 21.g | even | 6 | 1 | |
630.2.bi.b | yes | 48 | 45.j | even | 6 | 1 | |
1890.2.r.a | 48 | 5.b | even | 2 | 1 | ||
1890.2.r.a | 48 | 63.s | even | 6 | 1 | ||
1890.2.r.b | 48 | 1.a | even | 1 | 1 | trivial | |
1890.2.r.b | 48 | 315.u | even | 6 | 1 | inner | |
1890.2.bi.a | 48 | 7.d | odd | 6 | 1 | ||
1890.2.bi.a | 48 | 45.h | odd | 6 | 1 | ||
1890.2.bi.b | 48 | 9.d | odd | 6 | 1 | ||
1890.2.bi.b | 48 | 35.i | odd | 6 | 1 |