Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [19,4,Mod(4,19)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("19.4");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 19 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 19.e (of order \(9\), degree \(6\), 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: | \(1.12103629011\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −5.03941 | − | 1.83419i | 0.944789 | + | 5.35816i | 15.9030 | + | 13.3442i | −7.64679 | + | 6.41642i | 5.06673 | − | 28.7349i | −9.43070 | + | 16.3344i | −34.2145 | − | 59.2612i | −2.44557 | + | 0.890116i | 50.3042 | − | 18.3092i |
4.2 | −2.16911 | − | 0.789492i | −0.930433 | − | 5.27675i | −2.04660 | − | 1.71731i | 6.28846 | − | 5.27664i | −2.14774 | + | 12.1804i | −1.44586 | + | 2.50430i | 12.3168 | + | 21.3333i | −1.60664 | + | 0.584768i | −17.8062 | + | 6.48094i |
4.3 | 0.579114 | + | 0.210780i | 1.63522 | + | 9.27381i | −5.83741 | − | 4.89817i | 9.88078 | − | 8.29096i | −1.00776 | + | 5.71527i | 6.05695 | − | 10.4909i | −4.81321 | − | 8.33672i | −57.9579 | + | 21.0950i | 7.46967 | − | 2.71874i |
4.4 | 2.98397 | + | 1.08608i | −0.341646 | − | 1.93757i | 1.59618 | + | 1.33936i | −6.63193 | + | 5.56485i | 1.08489 | − | 6.15271i | −4.36312 | + | 7.55715i | −9.39359 | − | 16.2702i | 21.7342 | − | 7.91062i | −25.8334 | + | 9.40257i |
5.1 | −5.03941 | + | 1.83419i | 0.944789 | − | 5.35816i | 15.9030 | − | 13.3442i | −7.64679 | − | 6.41642i | 5.06673 | + | 28.7349i | −9.43070 | − | 16.3344i | −34.2145 | + | 59.2612i | −2.44557 | − | 0.890116i | 50.3042 | + | 18.3092i |
5.2 | −2.16911 | + | 0.789492i | −0.930433 | + | 5.27675i | −2.04660 | + | 1.71731i | 6.28846 | + | 5.27664i | −2.14774 | − | 12.1804i | −1.44586 | − | 2.50430i | 12.3168 | − | 21.3333i | −1.60664 | − | 0.584768i | −17.8062 | − | 6.48094i |
5.3 | 0.579114 | − | 0.210780i | 1.63522 | − | 9.27381i | −5.83741 | + | 4.89817i | 9.88078 | + | 8.29096i | −1.00776 | − | 5.71527i | 6.05695 | + | 10.4909i | −4.81321 | + | 8.33672i | −57.9579 | − | 21.0950i | 7.46967 | + | 2.71874i |
5.4 | 2.98397 | − | 1.08608i | −0.341646 | + | 1.93757i | 1.59618 | − | 1.33936i | −6.63193 | − | 5.56485i | 1.08489 | + | 6.15271i | −4.36312 | − | 7.55715i | −9.39359 | + | 16.2702i | 21.7342 | + | 7.91062i | −25.8334 | − | 9.40257i |
6.1 | −2.26193 | + | 1.89799i | −8.82799 | − | 3.21312i | 0.124797 | − | 0.707761i | −0.0925462 | − | 0.524856i | 26.0668 | − | 9.48752i | −11.8949 | + | 20.6025i | −10.7499 | − | 18.6194i | 46.9260 | + | 39.3756i | 1.20550 | + | 1.01154i |
6.2 | −1.74757 | + | 1.46638i | 4.00345 | + | 1.45714i | −0.485473 | + | 2.75325i | 1.27381 | + | 7.22413i | −9.13303 | + | 3.32415i | 13.1019 | − | 22.6932i | −12.3141 | − | 21.3286i | −6.77881 | − | 5.68809i | −12.8194 | − | 10.7568i |
6.3 | 1.38101 | − | 1.15881i | 2.58415 | + | 0.940555i | −0.824823 | + | 4.67780i | −3.13553 | − | 17.7825i | 4.65867 | − | 1.69562i | −14.1277 | + | 24.4699i | 11.4927 | + | 19.9060i | −14.8900 | − | 12.4942i | −24.9367 | − | 20.9244i |
6.4 | 2.98693 | − | 2.50633i | −3.72413 | − | 1.35547i | 1.25086 | − | 7.09397i | 2.58856 | + | 14.6804i | −14.5210 | + | 5.28519i | 5.35146 | − | 9.26900i | 1.55302 | + | 2.68992i | −8.65137 | − | 7.25936i | 44.5258 | + | 37.3616i |
9.1 | −0.851642 | − | 4.82990i | 0.458552 | − | 0.384771i | −15.0851 | + | 5.49052i | 6.94640 | + | 2.52828i | −2.24893 | − | 1.88707i | 14.5751 | − | 25.2448i | 19.7481 | + | 34.2048i | −4.62628 | + | 26.2369i | 6.29551 | − | 35.7036i |
9.2 | −0.163781 | − | 0.928850i | 1.56059 | − | 1.30949i | 6.68160 | − | 2.43190i | −3.55727 | − | 1.29474i | −1.47192 | − | 1.23509i | −11.7083 | + | 20.2794i | −7.12592 | − | 12.3424i | −3.96782 | + | 22.5026i | −0.620006 | + | 3.51623i |
9.3 | 0.477474 | + | 2.70789i | −4.62264 | + | 3.87886i | 0.412858 | − | 0.150268i | 5.11043 | + | 1.86004i | −12.7107 | − | 10.6656i | 11.7392 | − | 20.3328i | 11.6027 | + | 20.0964i | 1.63479 | − | 9.27135i | −2.59670 | + | 14.7266i |
9.4 | 0.824938 | + | 4.67846i | 5.76007 | − | 4.83328i | −13.6899 | + | 4.98271i | −14.0244 | − | 5.10445i | 27.3640 | + | 22.9611i | 3.64598 | − | 6.31503i | −15.6022 | − | 27.0238i | 5.12940 | − | 29.0903i | 12.3117 | − | 69.8233i |
16.1 | −2.26193 | − | 1.89799i | −8.82799 | + | 3.21312i | 0.124797 | + | 0.707761i | −0.0925462 | + | 0.524856i | 26.0668 | + | 9.48752i | −11.8949 | − | 20.6025i | −10.7499 | + | 18.6194i | 46.9260 | − | 39.3756i | 1.20550 | − | 1.01154i |
16.2 | −1.74757 | − | 1.46638i | 4.00345 | − | 1.45714i | −0.485473 | − | 2.75325i | 1.27381 | − | 7.22413i | −9.13303 | − | 3.32415i | 13.1019 | + | 22.6932i | −12.3141 | + | 21.3286i | −6.77881 | + | 5.68809i | −12.8194 | + | 10.7568i |
16.3 | 1.38101 | + | 1.15881i | 2.58415 | − | 0.940555i | −0.824823 | − | 4.67780i | −3.13553 | + | 17.7825i | 4.65867 | + | 1.69562i | −14.1277 | − | 24.4699i | 11.4927 | − | 19.9060i | −14.8900 | + | 12.4942i | −24.9367 | + | 20.9244i |
16.4 | 2.98693 | + | 2.50633i | −3.72413 | + | 1.35547i | 1.25086 | + | 7.09397i | 2.58856 | − | 14.6804i | −14.5210 | − | 5.28519i | 5.35146 | + | 9.26900i | 1.55302 | − | 2.68992i | −8.65137 | + | 7.25936i | 44.5258 | − | 37.3616i |
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 19.4.e.a | ✓ | 24 |
3.b | odd | 2 | 1 | 171.4.u.b | 24 | ||
19.e | even | 9 | 1 | inner | 19.4.e.a | ✓ | 24 |
19.e | even | 9 | 1 | 361.4.a.n | 12 | ||
19.f | odd | 18 | 1 | 361.4.a.m | 12 | ||
57.l | odd | 18 | 1 | 171.4.u.b | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
19.4.e.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
19.4.e.a | ✓ | 24 | 19.e | even | 9 | 1 | inner |
171.4.u.b | 24 | 3.b | odd | 2 | 1 | ||
171.4.u.b | 24 | 57.l | odd | 18 | 1 | ||
361.4.a.m | 12 | 19.f | odd | 18 | 1 | ||
361.4.a.n | 12 | 19.e | even | 9 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(19, [\chi])\).