Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [880,2,Mod(3,880)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(880, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([10, 15, 15, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("880.3");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 880 = 2^{4} \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 880.ch (of order \(20\), degree \(8\), 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: | \(7.02683537787\) |
Analytic rank: | \(0\) |
Dimension: | \(1120\) |
Relative dimension: | \(140\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −1.41419 | − | 0.00742043i | −0.420156 | − | 1.29311i | 1.99989 | + | 0.0209879i | −1.45621 | + | 1.69690i | 0.584587 | + | 1.83182i | −2.03662 | − | 3.99710i | −2.82808 | − | 0.0445209i | 0.931452 | − | 0.676740i | 2.07195 | − | 2.38894i |
3.2 | −1.41413 | + | 0.0154771i | 0.615081 | + | 1.89302i | 1.99952 | − | 0.0437732i | 2.17850 | + | 0.504120i | −0.899102 | − | 2.66746i | 1.39271 | + | 2.73334i | −2.82690 | + | 0.0928478i | −0.778167 | + | 0.565372i | −3.08848 | − | 0.679174i |
3.3 | −1.41368 | + | 0.0389587i | −0.937940 | − | 2.88668i | 1.99696 | − | 0.110150i | 1.44999 | + | 1.70221i | 1.43841 | + | 4.04430i | 0.678210 | + | 1.33106i | −2.81877 | + | 0.233516i | −5.02616 | + | 3.65172i | −2.11613 | − | 2.34989i |
3.4 | −1.41234 | − | 0.0728289i | 0.576842 | + | 1.77534i | 1.98939 | + | 0.205718i | −2.23607 | + | 0.00245174i | −0.685400 | − | 2.54939i | −0.104289 | − | 0.204678i | −2.79471 | − | 0.435428i | −0.392025 | + | 0.284823i | 3.15826 | + | 0.159388i |
3.5 | −1.41156 | − | 0.0865730i | 0.181125 | + | 0.557445i | 1.98501 | + | 0.244406i | 1.50243 | − | 1.65611i | −0.207409 | − | 0.802548i | −0.999464 | − | 1.96156i | −2.78080 | − | 0.516843i | 2.14911 | − | 1.56142i | −2.26414 | + | 2.20763i |
3.6 | −1.40992 | + | 0.110113i | −0.245249 | − | 0.754798i | 1.97575 | − | 0.310501i | −0.273486 | + | 2.21928i | 0.428894 | + | 1.03720i | 1.26650 | + | 2.48564i | −2.75146 | + | 0.655336i | 1.91748 | − | 1.39313i | 0.141222 | − | 3.15912i |
3.7 | −1.40476 | − | 0.163251i | −0.00637238 | − | 0.0196122i | 1.94670 | + | 0.458656i | 2.20539 | − | 0.369125i | 0.00574996 | + | 0.0285907i | 0.841532 | + | 1.65160i | −2.65977 | − | 0.962102i | 2.42671 | − | 1.76311i | −3.15830 | + | 0.158501i |
3.8 | −1.39587 | + | 0.227068i | −0.278371 | − | 0.856738i | 1.89688 | − | 0.633912i | −1.64351 | − | 1.51620i | 0.583106 | + | 1.13268i | 0.635022 | + | 1.24630i | −2.50385 | + | 1.31558i | 1.77054 | − | 1.28637i | 2.63840 | + | 1.74322i |
3.9 | −1.38155 | + | 0.302196i | 0.944887 | + | 2.90806i | 1.81736 | − | 0.834996i | −0.886842 | − | 2.05268i | −2.18421 | − | 3.73209i | 2.37324 | + | 4.65774i | −2.25843 | + | 1.70279i | −5.13697 | + | 3.73223i | 1.84553 | + | 2.56788i |
3.10 | −1.38081 | − | 0.305559i | 0.672650 | + | 2.07020i | 1.81327 | + | 0.843837i | −1.89509 | − | 1.18685i | −0.296232 | − | 3.06409i | −1.34718 | − | 2.64400i | −2.24593 | − | 1.71924i | −1.40623 | + | 1.02169i | 2.25411 | + | 2.21788i |
3.11 | −1.37207 | + | 0.342675i | 0.979144 | + | 3.01349i | 1.76515 | − | 0.940347i | 1.59160 | − | 1.57061i | −2.37610 | − | 3.79919i | −1.58669 | − | 3.11406i | −2.09967 | + | 1.89509i | −5.69537 | + | 4.13793i | −1.64557 | + | 2.70039i |
3.12 | −1.36438 | − | 0.372123i | −0.640082 | − | 1.96997i | 1.72305 | + | 1.01543i | −2.10934 | − | 0.742086i | 0.140241 | + | 2.92597i | 1.78514 | + | 3.50354i | −1.97302 | − | 2.02662i | −1.04403 | + | 0.758530i | 2.60178 | + | 1.79742i |
3.13 | −1.36287 | − | 0.377605i | −0.495583 | − | 1.52525i | 1.71483 | + | 1.02925i | −0.149487 | − | 2.23107i | 0.0994734 | + | 2.26585i | −1.50066 | − | 2.94521i | −1.94844 | − | 2.05027i | 0.346271 | − | 0.251580i | −0.638731 | + | 3.09710i |
3.14 | −1.35013 | + | 0.420899i | −0.200873 | − | 0.618224i | 1.64569 | − | 1.13653i | 0.939878 | + | 2.02895i | 0.531414 | + | 0.750134i | −1.18591 | − | 2.32748i | −1.74352 | + | 2.22713i | 2.08520 | − | 1.51499i | −2.12294 | − | 2.34374i |
3.15 | −1.33212 | + | 0.474831i | −0.880415 | − | 2.70964i | 1.54907 | − | 1.26506i | 0.453949 | − | 2.18950i | 2.45944 | + | 3.19151i | −1.13236 | − | 2.22239i | −1.46286 | + | 2.42075i | −4.13996 | + | 3.00786i | 0.434930 | + | 3.13223i |
3.16 | −1.32110 | − | 0.504675i | −0.718182 | − | 2.21034i | 1.49061 | + | 1.33345i | 2.20754 | − | 0.356039i | −0.166712 | + | 3.28252i | −0.492736 | − | 0.967049i | −1.29628 | − | 2.51389i | −1.94275 | + | 1.41149i | −3.09606 | − | 0.643729i |
3.17 | −1.29404 | + | 0.570493i | 0.316697 | + | 0.974695i | 1.34907 | − | 1.47648i | −2.08879 | + | 0.798093i | −0.965876 | − | 1.08062i | 0.205045 | + | 0.402424i | −0.903433 | + | 2.68026i | 1.57732 | − | 1.14599i | 2.24767 | − | 2.22441i |
3.18 | −1.27747 | − | 0.606694i | 0.250972 | + | 0.772414i | 1.26385 | + | 1.55006i | −1.43109 | + | 1.71813i | 0.148009 | − | 1.13900i | 1.73454 | + | 3.40422i | −0.674108 | − | 2.74692i | 1.89342 | − | 1.37565i | 2.87055 | − | 1.32662i |
3.19 | −1.27435 | + | 0.613223i | −0.0348820 | − | 0.107356i | 1.24791 | − | 1.56292i | −0.0740804 | − | 2.23484i | 0.110285 | + | 0.115418i | 0.817935 | + | 1.60529i | −0.631859 | + | 2.75695i | 2.41674 | − | 1.75587i | 1.46486 | + | 2.80253i |
3.20 | −1.27188 | − | 0.618313i | 0.560562 | + | 1.72523i | 1.23538 | + | 1.57285i | 1.60613 | + | 1.55575i | 0.353764 | − | 2.54090i | −2.02820 | − | 3.98057i | −0.598746 | − | 2.76433i | −0.235147 | + | 0.170845i | −1.08087 | − | 2.97182i |
See next 80 embeddings (of 1120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
80.s | even | 4 | 1 | inner |
880.ch | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 880.2.ch.a | ✓ | 1120 |
5.c | odd | 4 | 1 | 880.2.cz.a | yes | 1120 | |
11.c | even | 5 | 1 | inner | 880.2.ch.a | ✓ | 1120 |
16.f | odd | 4 | 1 | 880.2.cz.a | yes | 1120 | |
55.k | odd | 20 | 1 | 880.2.cz.a | yes | 1120 | |
80.s | even | 4 | 1 | inner | 880.2.ch.a | ✓ | 1120 |
176.v | odd | 20 | 1 | 880.2.cz.a | yes | 1120 | |
880.ch | even | 20 | 1 | inner | 880.2.ch.a | ✓ | 1120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
880.2.ch.a | ✓ | 1120 | 1.a | even | 1 | 1 | trivial |
880.2.ch.a | ✓ | 1120 | 11.c | even | 5 | 1 | inner |
880.2.ch.a | ✓ | 1120 | 80.s | even | 4 | 1 | inner |
880.2.ch.a | ✓ | 1120 | 880.ch | even | 20 | 1 | inner |
880.2.cz.a | yes | 1120 | 5.c | odd | 4 | 1 | |
880.2.cz.a | yes | 1120 | 16.f | odd | 4 | 1 | |
880.2.cz.a | yes | 1120 | 55.k | odd | 20 | 1 | |
880.2.cz.a | yes | 1120 | 176.v | odd | 20 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(880, [\chi])\).