Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [800,2,Mod(43,800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(800, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 5, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("800.43");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 800.v (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: | \(6.38803216170\) |
Analytic rank: | \(0\) |
Dimension: | \(88\) |
Relative dimension: | \(22\) over \(\Q(\zeta_{8})\) |
Twist minimal: | no (minimal twist has level 160) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 | −1.40295 | − | 0.178125i | 1.05468 | − | 2.54622i | 1.93654 | + | 0.499802i | 0 | −1.93321 | + | 3.38436i | − | 3.70855i | −2.62785 | − | 1.04614i | −3.24959 | − | 3.24959i | 0 | |||||
43.2 | −1.31460 | + | 0.521383i | −0.923741 | + | 2.23011i | 1.45632 | − | 1.37081i | 0 | 0.0516058 | − | 3.41331i | − | 3.63945i | −1.19975 | + | 2.56137i | −1.99876 | − | 1.99876i | 0 | |||||
43.3 | −1.30876 | − | 0.535863i | −0.461737 | + | 1.11473i | 1.42570 | + | 1.40263i | 0 | 1.20165 | − | 1.21149i | 2.85280i | −1.11428 | − | 2.59969i | 1.09189 | + | 1.09189i | 0 | ||||||
43.4 | −1.30208 | + | 0.551886i | 0.577765 | − | 1.39485i | 1.39084 | − | 1.43720i | 0 | 0.0174990 | + | 2.13507i | 1.62907i | −1.01782 | + | 2.63895i | 0.509534 | + | 0.509534i | 0 | ||||||
43.5 | −1.02775 | − | 0.971453i | 1.05354 | − | 2.54348i | 0.112559 | + | 1.99683i | 0 | −3.55365 | + | 1.59060i | 4.43630i | 1.82414 | − | 2.16160i | −3.23800 | − | 3.23800i | 0 | ||||||
43.6 | −0.992527 | + | 1.00742i | 0.278775 | − | 0.673021i | −0.0297801 | − | 1.99978i | 0 | 0.401322 | + | 0.948834i | 0.467309i | 2.04417 | + | 1.95483i | 1.74608 | + | 1.74608i | 0 | ||||||
43.7 | −0.816290 | − | 1.15485i | −1.28377 | + | 3.09930i | −0.667342 | + | 1.88538i | 0 | 4.62714 | − | 1.04737i | 0.906290i | 2.72207 | − | 0.768338i | −5.83625 | − | 5.83625i | 0 | ||||||
43.8 | −0.640905 | − | 1.26065i | −0.0983610 | + | 0.237464i | −1.17848 | + | 1.61591i | 0 | 0.362400 | − | 0.0281932i | − | 4.12414i | 2.79240 | + | 0.450008i | 2.07461 | + | 2.07461i | 0 | |||||
43.9 | −0.400521 | + | 1.35631i | −0.627770 | + | 1.51557i | −1.67917 | − | 1.08646i | 0 | −1.80415 | − | 1.45847i | 4.80429i | 2.14613 | − | 1.84232i | 0.218458 | + | 0.218458i | 0 | ||||||
43.10 | −0.375898 | − | 1.36334i | 0.508197 | − | 1.22690i | −1.71740 | + | 1.02496i | 0 | −1.86371 | − | 0.231658i | 0.810621i | 2.04293 | + | 1.95613i | 0.874309 | + | 0.874309i | 0 | ||||||
43.11 | −0.0754956 | + | 1.41220i | 0.698571 | − | 1.68650i | −1.98860 | − | 0.213229i | 0 | 2.32893 | + | 1.11384i | − | 2.70081i | 0.451252 | − | 2.79220i | −0.234961 | − | 0.234961i | 0 | |||||
43.12 | 0.201889 | + | 1.39973i | −0.867966 | + | 2.09546i | −1.91848 | + | 0.565181i | 0 | −3.10830 | − | 0.791867i | − | 1.82364i | −1.17842 | − | 2.57125i | −1.51625 | − | 1.51625i | 0 | |||||
43.13 | 0.458371 | − | 1.33787i | −0.218753 | + | 0.528116i | −1.57979 | − | 1.22648i | 0 | 0.606280 | + | 0.534736i | − | 0.814088i | −2.36500 | + | 1.55137i | 1.89027 | + | 1.89027i | 0 | |||||
43.14 | 0.491080 | − | 1.32621i | −0.896482 | + | 2.16430i | −1.51768 | − | 1.30255i | 0 | 2.43008 | + | 2.25177i | 0.225996i | −2.47277 | + | 1.37311i | −1.75919 | − | 1.75919i | 0 | ||||||
43.15 | 0.759267 | + | 1.19311i | 0.252131 | − | 0.608697i | −0.847028 | + | 1.81178i | 0 | 0.917677 | − | 0.161344i | − | 1.49067i | −2.80477 | + | 0.365026i | 1.81438 | + | 1.81438i | 0 | |||||
43.16 | 0.998231 | + | 1.00177i | −0.509063 | + | 1.22899i | −0.00706901 | + | 1.99999i | 0 | −1.73932 | + | 0.716851i | 2.73471i | −2.01058 | + | 1.98937i | 0.870059 | + | 0.870059i | 0 | ||||||
43.17 | 1.04802 | − | 0.949552i | 0.255424 | − | 0.616647i | 0.196703 | − | 1.99030i | 0 | −0.317849 | − | 0.888798i | 2.27809i | −1.68375 | − | 2.27266i | 1.80631 | + | 1.80631i | 0 | ||||||
43.18 | 1.19101 | − | 0.762560i | 1.11089 | − | 2.68192i | 0.837005 | − | 1.81643i | 0 | −0.722048 | − | 4.04131i | − | 0.874514i | −0.388258 | − | 2.80165i | −3.83732 | − | 3.83732i | 0 | |||||
43.19 | 1.32278 | + | 0.500260i | 1.03647 | − | 2.50226i | 1.49948 | + | 1.32347i | 0 | 2.62280 | − | 2.79143i | − | 2.65674i | 1.32140 | + | 2.50078i | −3.06572 | − | 3.06572i | 0 | |||||
43.20 | 1.37186 | − | 0.343520i | −1.11476 | + | 2.69126i | 1.76399 | − | 0.942521i | 0 | −0.604785 | + | 4.07496i | − | 0.518179i | 2.09617 | − | 1.89897i | −3.87887 | − | 3.87887i | 0 | |||||
See all 88 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
160.u | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 800.2.v.b | 88 | |
5.b | even | 2 | 1 | 160.2.u.a | ✓ | 88 | |
5.c | odd | 4 | 1 | 160.2.ba.a | yes | 88 | |
5.c | odd | 4 | 1 | 800.2.bb.b | 88 | ||
20.d | odd | 2 | 1 | 640.2.u.a | 88 | ||
20.e | even | 4 | 1 | 640.2.ba.a | 88 | ||
32.h | odd | 8 | 1 | 800.2.bb.b | 88 | ||
160.u | even | 8 | 1 | inner | 800.2.v.b | 88 | |
160.v | odd | 8 | 1 | 640.2.u.a | 88 | ||
160.y | odd | 8 | 1 | 160.2.ba.a | yes | 88 | |
160.z | even | 8 | 1 | 640.2.ba.a | 88 | ||
160.ba | even | 8 | 1 | 160.2.u.a | ✓ | 88 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
160.2.u.a | ✓ | 88 | 5.b | even | 2 | 1 | |
160.2.u.a | ✓ | 88 | 160.ba | even | 8 | 1 | |
160.2.ba.a | yes | 88 | 5.c | odd | 4 | 1 | |
160.2.ba.a | yes | 88 | 160.y | odd | 8 | 1 | |
640.2.u.a | 88 | 20.d | odd | 2 | 1 | ||
640.2.u.a | 88 | 160.v | odd | 8 | 1 | ||
640.2.ba.a | 88 | 20.e | even | 4 | 1 | ||
640.2.ba.a | 88 | 160.z | even | 8 | 1 | ||
800.2.v.b | 88 | 1.a | even | 1 | 1 | trivial | |
800.2.v.b | 88 | 160.u | even | 8 | 1 | inner | |
800.2.bb.b | 88 | 5.c | odd | 4 | 1 | ||
800.2.bb.b | 88 | 32.h | odd | 8 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{88} - 4 T_{3}^{87} + 8 T_{3}^{86} - 24 T_{3}^{85} + 64 T_{3}^{84} - 8 T_{3}^{83} + \cdots + 154719502336 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\).