Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [363,3,Mod(245,363)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 8]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("363.245");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 363.h (of order \(10\), degree \(4\), 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: | \(9.89103359628\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
245.1 | −2.05778 | − | 2.83229i | −2.95837 | − | 0.498071i | −2.55135 | + | 7.85225i | −4.64306 | + | 6.39063i | 4.67699 | + | 9.40388i | 0.560737 | − | 1.72577i | 14.1718 | − | 4.60468i | 8.50385 | + | 2.94695i | 27.6545 | ||
245.2 | −1.69141 | − | 2.32802i | 2.68453 | + | 1.33914i | −1.32276 | + | 4.07105i | 2.90829 | − | 4.00292i | −1.42308 | − | 8.51468i | −3.45484 | + | 10.6329i | 0.767802 | − | 0.249474i | 5.41340 | + | 7.18993i | −14.2380 | ||
245.3 | −0.399998 | − | 0.550550i | −2.18131 | + | 2.05958i | 1.09296 | − | 3.36379i | 3.81474 | − | 5.25053i | 2.00642 | + | 0.377094i | 2.89410 | − | 8.90713i | −4.87795 | + | 1.58494i | 0.516261 | − | 8.98518i | −4.41657 | ||
245.4 | 0.399998 | + | 0.550550i | 2.97531 | + | 0.384091i | 1.09296 | − | 3.36379i | −3.81474 | + | 5.25053i | 0.978656 | + | 1.79169i | 2.89410 | − | 8.90713i | 4.87795 | − | 1.58494i | 8.70495 | + | 2.28558i | −4.41657 | ||
245.5 | 1.69141 | + | 2.32802i | −1.38470 | + | 2.66132i | −1.32276 | + | 4.07105i | −2.90829 | + | 4.00292i | −8.53770 | + | 1.27775i | −3.45484 | + | 10.6329i | −0.767802 | + | 0.249474i | −5.16520 | − | 7.37026i | −14.2380 | ||
245.6 | 2.05778 | + | 2.83229i | 2.10061 | − | 2.14183i | −2.55135 | + | 7.85225i | 4.64306 | − | 6.39063i | 10.3889 | + | 1.54212i | 0.560737 | − | 1.72577i | −14.1718 | + | 4.60468i | −0.174883 | − | 8.99830i | 27.6545 | ||
251.1 | −3.32956 | − | 1.08184i | 2.68613 | + | 1.33594i | 6.67952 | + | 4.85296i | −7.51264 | + | 2.44100i | −7.49835 | − | 7.35404i | −1.46803 | − | 1.06658i | −8.75863 | − | 12.0552i | 5.43055 | + | 7.17698i | 27.6545 | ||
251.2 | −2.73676 | − | 0.889226i | −2.95896 | − | 0.494538i | 3.46304 | + | 2.51605i | 4.70571 | − | 1.52898i | 7.65819 | + | 3.98461i | 9.04489 | + | 6.57149i | −0.474528 | − | 0.653131i | 8.51086 | + | 2.92664i | −14.2380 | ||
251.3 | −0.647210 | − | 0.210291i | 0.554129 | + | 2.94838i | −2.86141 | − | 2.07894i | 6.17237 | − | 2.00553i | 0.261380 | − | 2.02475i | −7.57686 | − | 5.50491i | 3.01474 | + | 4.14944i | −8.38588 | + | 3.26757i | −4.41657 | ||
251.4 | 0.647210 | + | 0.210291i | −2.63284 | − | 1.43811i | −2.86141 | − | 2.07894i | −6.17237 | + | 2.00553i | −1.40158 | − | 1.48442i | −7.57686 | − | 5.50491i | −3.01474 | − | 4.14944i | 4.86369 | + | 7.57261i | −4.41657 | ||
251.5 | 2.73676 | + | 0.889226i | −0.444035 | + | 2.96696i | 3.46304 | + | 2.51605i | −4.70571 | + | 1.52898i | −3.85351 | + | 7.72499i | 9.04489 | + | 6.57149i | 0.474528 | + | 0.653131i | −8.60567 | − | 2.63486i | −14.2380 | ||
251.6 | 3.32956 | + | 1.08184i | −0.440492 | − | 2.96748i | 6.67952 | + | 4.85296i | 7.51264 | − | 2.44100i | 1.74370 | − | 10.3570i | −1.46803 | − | 1.06658i | 8.75863 | + | 12.0552i | −8.61193 | + | 2.61430i | 27.6545 | ||
269.1 | −3.32956 | + | 1.08184i | 2.68613 | − | 1.33594i | 6.67952 | − | 4.85296i | −7.51264 | − | 2.44100i | −7.49835 | + | 7.35404i | −1.46803 | + | 1.06658i | −8.75863 | + | 12.0552i | 5.43055 | − | 7.17698i | 27.6545 | ||
269.2 | −2.73676 | + | 0.889226i | −2.95896 | + | 0.494538i | 3.46304 | − | 2.51605i | 4.70571 | + | 1.52898i | 7.65819 | − | 3.98461i | 9.04489 | − | 6.57149i | −0.474528 | + | 0.653131i | 8.51086 | − | 2.92664i | −14.2380 | ||
269.3 | −0.647210 | + | 0.210291i | 0.554129 | − | 2.94838i | −2.86141 | + | 2.07894i | 6.17237 | + | 2.00553i | 0.261380 | + | 2.02475i | −7.57686 | + | 5.50491i | 3.01474 | − | 4.14944i | −8.38588 | − | 3.26757i | −4.41657 | ||
269.4 | 0.647210 | − | 0.210291i | −2.63284 | + | 1.43811i | −2.86141 | + | 2.07894i | −6.17237 | − | 2.00553i | −1.40158 | + | 1.48442i | −7.57686 | + | 5.50491i | −3.01474 | + | 4.14944i | 4.86369 | − | 7.57261i | −4.41657 | ||
269.5 | 2.73676 | − | 0.889226i | −0.444035 | − | 2.96696i | 3.46304 | − | 2.51605i | −4.70571 | − | 1.52898i | −3.85351 | − | 7.72499i | 9.04489 | − | 6.57149i | 0.474528 | − | 0.653131i | −8.60567 | + | 2.63486i | −14.2380 | ||
269.6 | 3.32956 | − | 1.08184i | −0.440492 | + | 2.96748i | 6.67952 | − | 4.85296i | 7.51264 | + | 2.44100i | 1.74370 | + | 10.3570i | −1.46803 | + | 1.06658i | 8.75863 | − | 12.0552i | −8.61193 | − | 2.61430i | 27.6545 | ||
323.1 | −2.05778 | + | 2.83229i | −2.95837 | + | 0.498071i | −2.55135 | − | 7.85225i | −4.64306 | − | 6.39063i | 4.67699 | − | 9.40388i | 0.560737 | + | 1.72577i | 14.1718 | + | 4.60468i | 8.50385 | − | 2.94695i | 27.6545 | ||
323.2 | −1.69141 | + | 2.32802i | 2.68453 | − | 1.33914i | −1.32276 | − | 4.07105i | 2.90829 | + | 4.00292i | −1.42308 | + | 8.51468i | −3.45484 | − | 10.6329i | 0.767802 | + | 0.249474i | 5.41340 | − | 7.18993i | −14.2380 | ||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
11.c | even | 5 | 3 | inner |
33.h | odd | 10 | 3 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 363.3.h.q | 24 | |
3.b | odd | 2 | 1 | inner | 363.3.h.q | 24 | |
11.b | odd | 2 | 1 | 363.3.h.p | 24 | ||
11.c | even | 5 | 1 | 363.3.b.j | ✓ | 6 | |
11.c | even | 5 | 3 | inner | 363.3.h.q | 24 | |
11.d | odd | 10 | 1 | 363.3.b.k | yes | 6 | |
11.d | odd | 10 | 3 | 363.3.h.p | 24 | ||
33.d | even | 2 | 1 | 363.3.h.p | 24 | ||
33.f | even | 10 | 1 | 363.3.b.k | yes | 6 | |
33.f | even | 10 | 3 | 363.3.h.p | 24 | ||
33.h | odd | 10 | 1 | 363.3.b.j | ✓ | 6 | |
33.h | odd | 10 | 3 | inner | 363.3.h.q | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
363.3.b.j | ✓ | 6 | 11.c | even | 5 | 1 | |
363.3.b.j | ✓ | 6 | 33.h | odd | 10 | 1 | |
363.3.b.k | yes | 6 | 11.d | odd | 10 | 1 | |
363.3.b.k | yes | 6 | 33.f | even | 10 | 1 | |
363.3.h.p | 24 | 11.b | odd | 2 | 1 | ||
363.3.h.p | 24 | 11.d | odd | 10 | 3 | ||
363.3.h.p | 24 | 33.d | even | 2 | 1 | ||
363.3.h.p | 24 | 33.f | even | 10 | 3 | ||
363.3.h.q | 24 | 1.a | even | 1 | 1 | trivial | |
363.3.h.q | 24 | 3.b | odd | 2 | 1 | inner | |
363.3.h.q | 24 | 11.c | even | 5 | 3 | inner | |
363.3.h.q | 24 | 33.h | odd | 10 | 3 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):
\( T_{2}^{24} - 21 T_{2}^{22} + 330 T_{2}^{20} - 4646 T_{2}^{18} + 61923 T_{2}^{16} - 484686 T_{2}^{14} + \cdots + 4879681 \) |
\( T_{5}^{24} - 129 T_{5}^{22} + 11454 T_{5}^{20} - 872786 T_{5}^{18} + 61477743 T_{5}^{16} + \cdots + 17\!\cdots\!01 \) |
\( T_{7}^{12} + 108 T_{7}^{10} - 190 T_{7}^{9} + 11664 T_{7}^{8} + 61560 T_{7}^{7} + 1295812 T_{7}^{6} + \cdots + 1303210000 \) |