Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [224,4,Mod(29,224)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(224, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 3, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("224.29");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 224 = 2^{5} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 224.u (of order \(8\), degree \(4\), 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: | \(13.2164278413\) |
Analytic rank: | \(0\) |
Dimension: | \(140\) |
Relative dimension: | \(35\) over \(\Q(\zeta_{8})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
29.1 | −2.82701 | − | 0.0894526i | 2.36840 | + | 0.981025i | 7.98400 | + | 0.505767i | 2.98710 | + | 7.21150i | −6.60775 | − | 2.98523i | 4.94975 | − | 4.94975i | −22.5256 | − | 2.14400i | −14.4450 | − | 14.4450i | −7.79949 | − | 20.6542i |
29.2 | −2.82267 | − | 0.180425i | 8.94755 | + | 3.70620i | 7.93489 | + | 1.01856i | −5.50821 | − | 13.2980i | −24.5873 | − | 12.0757i | 4.94975 | − | 4.94975i | −22.2138 | − | 4.30670i | 47.2309 | + | 47.2309i | 13.1486 | + | 38.5296i |
29.3 | −2.81375 | − | 0.287765i | −8.01711 | − | 3.32080i | 7.83438 | + | 1.61940i | −2.55421 | − | 6.16641i | 21.6025 | + | 11.6509i | 4.94975 | − | 4.94975i | −21.5780 | − | 6.81103i | 34.1545 | + | 34.1545i | 5.41244 | + | 18.0857i |
29.4 | −2.67774 | − | 0.910887i | 0.836532 | + | 0.346503i | 6.34057 | + | 4.87824i | 2.18623 | + | 5.27803i | −1.92439 | − | 1.68983i | 4.94975 | − | 4.94975i | −12.5349 | − | 18.8382i | −18.5122 | − | 18.5122i | −1.04647 | − | 16.1246i |
29.5 | −2.62799 | + | 1.04577i | −6.58597 | − | 2.72800i | 5.81271 | − | 5.49658i | 7.52198 | + | 18.1597i | 20.1608 | + | 0.281723i | 4.94975 | − | 4.94975i | −9.52759 | + | 20.5238i | 16.8412 | + | 16.8412i | −38.7586 | − | 39.8572i |
29.6 | −2.56089 | + | 1.20077i | 7.06644 | + | 2.92701i | 5.11632 | − | 6.15006i | 3.96866 | + | 9.58119i | −21.6110 | + | 0.989380i | 4.94975 | − | 4.94975i | −5.71754 | + | 21.8931i | 22.2752 | + | 22.2752i | −21.6681 | − | 19.7709i |
29.7 | −2.33082 | − | 1.60227i | 2.54429 | + | 1.05388i | 2.86545 | + | 7.46922i | −6.41104 | − | 15.4776i | −4.24168 | − | 6.53304i | 4.94975 | − | 4.94975i | 5.28887 | − | 22.0006i | −13.7292 | − | 13.7292i | −9.85637 | + | 46.3478i |
29.8 | −2.27517 | + | 1.68036i | 0.692049 | + | 0.286656i | 2.35276 | − | 7.64621i | −5.63604 | − | 13.6066i | −2.05621 | + | 0.510703i | 4.94975 | − | 4.94975i | 7.49548 | + | 21.3499i | −18.6951 | − | 18.6951i | 35.6870 | + | 21.4867i |
29.9 | −2.14397 | − | 1.84483i | −6.50322 | − | 2.69372i | 1.19322 | + | 7.91051i | 0.166115 | + | 0.401037i | 8.97325 | + | 17.7726i | 4.94975 | − | 4.94975i | 12.0353 | − | 19.1612i | 15.9438 | + | 15.9438i | 0.383699 | − | 1.16627i |
29.10 | −1.91298 | + | 2.08338i | −8.05349 | − | 3.33586i | −0.680979 | − | 7.97096i | −5.74420 | − | 13.8677i | 22.3561 | − | 10.3971i | 4.94975 | − | 4.94975i | 17.9093 | + | 13.8296i | 34.6388 | + | 34.6388i | 39.8804 | + | 14.5614i |
29.11 | −1.75433 | + | 2.21863i | −1.70574 | − | 0.706540i | −1.84463 | − | 7.78443i | 3.11491 | + | 7.52005i | 4.55998 | − | 2.54489i | 4.94975 | − | 4.94975i | 20.5069 | + | 9.56393i | −16.6815 | − | 16.6815i | −22.1488 | − | 6.28185i |
29.12 | −1.52380 | − | 2.38286i | 2.12512 | + | 0.880255i | −3.35607 | + | 7.26201i | 8.08907 | + | 19.5287i | −1.14074 | − | 6.40521i | 4.94975 | − | 4.94975i | 22.4184 | − | 3.06880i | −15.3506 | − | 15.3506i | 34.2082 | − | 49.0330i |
29.13 | −1.25291 | + | 2.53579i | 5.65221 | + | 2.34122i | −4.86044 | − | 6.35422i | −4.14360 | − | 10.0035i | −13.0186 | + | 11.3995i | 4.94975 | − | 4.94975i | 22.2026 | − | 4.36379i | 7.37430 | + | 7.37430i | 30.5584 | + | 2.02622i |
29.14 | −1.12766 | − | 2.59391i | −3.03773 | − | 1.25827i | −5.45677 | + | 5.85010i | −3.05117 | − | 7.36618i | 0.161680 | + | 9.29850i | 4.94975 | − | 4.94975i | 21.3280 | + | 7.55749i | −11.4473 | − | 11.4473i | −15.6666 | + | 16.2210i |
29.15 | −0.791489 | − | 2.71543i | 8.59902 | + | 3.56183i | −6.74709 | + | 4.29846i | −1.15499 | − | 2.78839i | 2.86587 | − | 26.1692i | 4.94975 | − | 4.94975i | 17.0124 | + | 14.9191i | 42.1647 | + | 42.1647i | −6.65750 | + | 5.34326i |
29.16 | −0.461524 | + | 2.79052i | −4.06210 | − | 1.68257i | −7.57399 | − | 2.57578i | 4.51885 | + | 10.9095i | 6.57001 | − | 10.5588i | 4.94975 | − | 4.94975i | 10.6834 | − | 19.9466i | −5.42232 | − | 5.42232i | −32.5286 | + | 7.57495i |
29.17 | −0.446664 | + | 2.79294i | 3.90182 | + | 1.61619i | −7.60098 | − | 2.49501i | −1.21986 | − | 2.94501i | −6.25670 | + | 10.1756i | 4.94975 | − | 4.94975i | 10.3635 | − | 20.1146i | −6.47977 | − | 6.47977i | 8.77009 | − | 2.09157i |
29.18 | 0.0170002 | − | 2.82838i | −6.98661 | − | 2.89395i | −7.99942 | − | 0.0961657i | 3.91954 | + | 9.46261i | −8.30395 | + | 19.7116i | 4.94975 | − | 4.94975i | −0.407984 | + | 22.6237i | 21.3459 | + | 21.3459i | 26.8305 | − | 10.9251i |
29.19 | 0.153695 | − | 2.82425i | 3.46993 | + | 1.43729i | −7.95276 | − | 0.868145i | 1.61982 | + | 3.91060i | 4.59258 | − | 9.57903i | 4.94975 | − | 4.94975i | −3.67415 | + | 22.3271i | −9.11729 | − | 9.11729i | 11.2935 | − | 3.97374i |
29.20 | 0.390074 | + | 2.80140i | −5.73255 | − | 2.37450i | −7.69569 | + | 2.18550i | −3.87321 | − | 9.35077i | 4.41581 | − | 16.9854i | 4.94975 | − | 4.94975i | −9.12436 | − | 20.7062i | 8.13200 | + | 8.13200i | 24.6844 | − | 14.4979i |
See next 80 embeddings (of 140 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
32.g | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 224.4.u.a | ✓ | 140 |
32.g | even | 8 | 1 | inner | 224.4.u.a | ✓ | 140 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
224.4.u.a | ✓ | 140 | 1.a | even | 1 | 1 | trivial |
224.4.u.a | ✓ | 140 | 32.g | even | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{140} - 88 T_{3}^{137} + 7128 T_{3}^{135} + 8168 T_{3}^{134} + 1462472 T_{3}^{133} + \cdots + 21\!\cdots\!52 \) acting on \(S_{4}^{\mathrm{new}}(224, [\chi])\).