Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [800,2,Mod(107,800)]
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, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("800.107");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 800.bb (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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
107.1 | −1.41220 | − | 0.0754956i | −1.68650 | − | 0.698571i | 1.98860 | + | 0.213229i | 0 | 2.32893 | + | 1.11384i | 2.70081 | −2.79220 | − | 0.451252i | 0.234961 | + | 0.234961i | 0 | ||||||
107.2 | −1.39973 | + | 0.201889i | 2.09546 | + | 0.867966i | 1.91848 | − | 0.565181i | 0 | −3.10830 | − | 0.791867i | 1.82364 | −2.57125 | + | 1.17842i | 1.51625 | + | 1.51625i | 0 | ||||||
107.3 | −1.35631 | − | 0.400521i | 1.51557 | + | 0.627770i | 1.67917 | + | 1.08646i | 0 | −1.80415 | − | 1.45847i | −4.80429 | −1.84232 | − | 2.14613i | −0.218458 | − | 0.218458i | 0 | ||||||
107.4 | −1.19311 | + | 0.759267i | −0.608697 | − | 0.252131i | 0.847028 | − | 1.81178i | 0 | 0.917677 | − | 0.161344i | 1.49067 | 0.365026 | + | 2.80477i | −1.81438 | − | 1.81438i | 0 | ||||||
107.5 | −1.00742 | − | 0.992527i | −0.673021 | − | 0.278775i | 0.0297801 | + | 1.99978i | 0 | 0.401322 | + | 0.948834i | −0.467309 | 1.95483 | − | 2.04417i | −1.74608 | − | 1.74608i | 0 | ||||||
107.6 | −1.00177 | + | 0.998231i | 1.22899 | + | 0.509063i | 0.00706901 | − | 1.99999i | 0 | −1.73932 | + | 0.716851i | −2.73471 | 1.98937 | + | 2.01058i | −0.870059 | − | 0.870059i | 0 | ||||||
107.7 | −0.551886 | − | 1.30208i | −1.39485 | − | 0.577765i | −1.39084 | + | 1.43720i | 0 | 0.0174990 | + | 2.13507i | −1.62907 | 2.63895 | + | 1.01782i | −0.509534 | − | 0.509534i | 0 | ||||||
107.8 | −0.521383 | − | 1.31460i | 2.23011 | + | 0.923741i | −1.45632 | + | 1.37081i | 0 | 0.0516058 | − | 3.41331i | 3.63945 | 2.56137 | + | 1.19975i | 1.99876 | + | 1.99876i | 0 | ||||||
107.9 | −0.500260 | + | 1.32278i | −2.50226 | − | 1.03647i | −1.49948 | − | 1.32347i | 0 | 2.62280 | − | 2.79143i | 2.65674 | 2.50078 | − | 1.32140i | 3.06572 | + | 3.06572i | 0 | ||||||
107.10 | −0.184269 | + | 1.40216i | 1.10776 | + | 0.458849i | −1.93209 | − | 0.516749i | 0 | −0.847505 | + | 1.46870i | 4.27741 | 1.08059 | − | 2.61387i | −1.10473 | − | 1.10473i | 0 | ||||||
107.11 | −0.0556644 | + | 1.41312i | −0.532554 | − | 0.220591i | −1.99380 | − | 0.157321i | 0 | 0.341365 | − | 0.740282i | −3.48272 | 0.333297 | − | 2.80872i | −1.88637 | − | 1.88637i | 0 | ||||||
107.12 | 0.178125 | − | 1.40295i | −2.54622 | − | 1.05468i | −1.93654 | − | 0.499802i | 0 | −1.93321 | + | 3.38436i | 3.70855 | −1.04614 | + | 2.62785i | 3.24959 | + | 3.24959i | 0 | ||||||
107.13 | 0.343520 | + | 1.37186i | 2.69126 | + | 1.11476i | −1.76399 | + | 0.942521i | 0 | −0.604785 | + | 4.07496i | 0.518179 | −1.89897 | − | 2.09617i | 3.87887 | + | 3.87887i | 0 | ||||||
107.14 | 0.535863 | − | 1.30876i | 1.11473 | + | 0.461737i | −1.42570 | − | 1.40263i | 0 | 1.20165 | − | 1.21149i | −2.85280 | −2.59969 | + | 1.11428i | −1.09189 | − | 1.09189i | 0 | ||||||
107.15 | 0.762560 | + | 1.19101i | −2.68192 | − | 1.11089i | −0.837005 | + | 1.81643i | 0 | −0.722048 | − | 4.04131i | 0.874514 | −2.80165 | + | 0.388258i | 3.83732 | + | 3.83732i | 0 | ||||||
107.16 | 0.949552 | + | 1.04802i | −0.616647 | − | 0.255424i | −0.196703 | + | 1.99030i | 0 | −0.317849 | − | 0.888798i | −2.27809 | −2.27266 | + | 1.68375i | −1.80631 | − | 1.80631i | 0 | ||||||
107.17 | 0.971453 | − | 1.02775i | −2.54348 | − | 1.05354i | −0.112559 | − | 1.99683i | 0 | −3.55365 | + | 1.59060i | −4.43630 | −2.16160 | − | 1.82414i | 3.23800 | + | 3.23800i | 0 | ||||||
107.18 | 1.15485 | − | 0.816290i | 3.09930 | + | 1.28377i | 0.667342 | − | 1.88538i | 0 | 4.62714 | − | 1.04737i | −0.906290 | −0.768338 | − | 2.72207i | 5.83625 | + | 5.83625i | 0 | ||||||
107.19 | 1.26065 | − | 0.640905i | 0.237464 | + | 0.0983610i | 1.17848 | − | 1.61591i | 0 | 0.362400 | − | 0.0281932i | 4.12414 | 0.450008 | − | 2.79240i | −2.07461 | − | 2.07461i | 0 | ||||||
107.20 | 1.32621 | + | 0.491080i | 2.16430 | + | 0.896482i | 1.51768 | + | 1.30255i | 0 | 2.43008 | + | 2.25177i | −0.225996 | 1.37311 | + | 2.47277i | 1.75919 | + | 1.75919i | 0 | ||||||
See all 88 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
160.ba | 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.bb.b | 88 | |
5.b | even | 2 | 1 | 160.2.ba.a | yes | 88 | |
5.c | odd | 4 | 1 | 160.2.u.a | ✓ | 88 | |
5.c | odd | 4 | 1 | 800.2.v.b | 88 | ||
20.d | odd | 2 | 1 | 640.2.ba.a | 88 | ||
20.e | even | 4 | 1 | 640.2.u.a | 88 | ||
32.h | odd | 8 | 1 | 800.2.v.b | 88 | ||
160.u | even | 8 | 1 | 160.2.ba.a | yes | 88 | |
160.y | odd | 8 | 1 | 160.2.u.a | ✓ | 88 | |
160.z | even | 8 | 1 | 640.2.u.a | 88 | ||
160.ba | even | 8 | 1 | inner | 800.2.bb.b | 88 | |
160.bb | odd | 8 | 1 | 640.2.ba.a | 88 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
160.2.u.a | ✓ | 88 | 5.c | odd | 4 | 1 | |
160.2.u.a | ✓ | 88 | 160.y | odd | 8 | 1 | |
160.2.ba.a | yes | 88 | 5.b | even | 2 | 1 | |
160.2.ba.a | yes | 88 | 160.u | even | 8 | 1 | |
640.2.u.a | 88 | 20.e | even | 4 | 1 | ||
640.2.u.a | 88 | 160.z | even | 8 | 1 | ||
640.2.ba.a | 88 | 20.d | odd | 2 | 1 | ||
640.2.ba.a | 88 | 160.bb | odd | 8 | 1 | ||
800.2.v.b | 88 | 5.c | odd | 4 | 1 | ||
800.2.v.b | 88 | 32.h | odd | 8 | 1 | ||
800.2.bb.b | 88 | 1.a | even | 1 | 1 | trivial | |
800.2.bb.b | 88 | 160.ba | even | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{88} - 4 T_{3}^{87} + 8 T_{3}^{86} - 32 T_{3}^{84} - 8 T_{3}^{83} + 288 T_{3}^{82} + \cdots + 154719502336 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\).