Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [100,2,Mod(3,100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(100, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([10, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("100.3");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 100 = 2^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 100.l (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: | \(0.798504020213\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(12\) 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 | −1.41421 | 0.000293203i | −1.30110 | + | 0.206075i | 2.00000 | 0.000829304i | −2.22277 | + | 0.243473i | 1.84010 | − | 0.291052i | −1.76525 | − | 1.76525i | −2.82843 | − | 0.00175922i | −1.20277 | + | 0.390802i | 3.14355 | − | 0.343670i | ||
3.2 | −1.16222 | + | 0.805750i | 1.77314 | − | 0.280838i | 0.701533 | − | 1.87293i | 1.24326 | − | 1.85858i | −1.83450 | + | 1.75511i | −2.22547 | − | 2.22547i | 0.693772 | + | 2.74202i | 0.211989 | − | 0.0688795i | 0.0526065 | + | 3.16184i |
3.3 | −1.15706 | − | 0.813156i | 1.79631 | − | 0.284507i | 0.677554 | + | 1.88173i | −0.456295 | − | 2.18902i | −2.30978 | − | 1.13149i | 2.23536 | + | 2.23536i | 0.746176 | − | 2.72823i | 0.292613 | − | 0.0950758i | −1.25205 | + | 2.90385i |
3.4 | −1.06026 | − | 0.935866i | −1.97842 | + | 0.313350i | 0.248311 | + | 1.98453i | 1.82539 | + | 1.29150i | 2.39089 | + | 1.51930i | 1.00369 | + | 1.00369i | 1.59398 | − | 2.33650i | 0.962775 | − | 0.312824i | −0.726717 | − | 3.07764i |
3.5 | −0.934812 | + | 1.06119i | 0.173581 | − | 0.0274925i | −0.252253 | − | 1.98403i | −0.138125 | + | 2.23180i | −0.133091 | + | 0.209903i | 2.95323 | + | 2.95323i | 2.34124 | + | 1.58700i | −2.82379 | + | 0.917507i | −2.23924 | − | 2.23289i |
3.6 | −0.133925 | − | 1.40786i | 1.97842 | − | 0.313350i | −1.96413 | + | 0.377095i | 1.82539 | + | 1.29150i | −0.706112 | − | 2.74336i | −1.00369 | − | 1.00369i | 0.793941 | + | 2.71471i | 0.962775 | − | 0.312824i | 1.57378 | − | 2.74285i |
3.7 | 0.0222429 | − | 1.41404i | −1.79631 | + | 0.284507i | −1.99901 | − | 0.0629046i | −0.456295 | − | 2.18902i | 0.362349 | + | 2.54638i | −2.23536 | − | 2.23536i | −0.133413 | + | 2.82528i | 0.292613 | − | 0.0950758i | −3.10550 | + | 0.596529i |
3.8 | 0.189713 | + | 1.40143i | 3.09018 | − | 0.489437i | −1.92802 | + | 0.531739i | −2.06046 | + | 0.868611i | 1.27216 | + | 4.23782i | −1.45172 | − | 1.45172i | −1.11097 | − | 2.60111i | 6.45651 | − | 2.09785i | −1.60820 | − | 2.72281i |
3.9 | 0.831017 | − | 1.14430i | 1.30110 | − | 0.206075i | −0.618823 | − | 1.90186i | −2.22277 | + | 0.243473i | 0.845429 | − | 1.66010i | 1.76525 | + | 1.76525i | −2.69054 | − | 0.872359i | −1.20277 | + | 0.390802i | −1.56856 | + | 2.74584i |
3.10 | 1.02227 | + | 0.977222i | −3.09018 | + | 0.489437i | 0.0900763 | + | 1.99797i | −2.06046 | + | 0.868611i | −3.63729 | − | 2.51946i | 1.45172 | + | 1.45172i | −1.86038 | + | 2.13049i | 6.45651 | − | 2.09785i | −2.95518 | − | 1.12557i |
3.11 | 1.33500 | − | 0.466651i | −1.77314 | + | 0.280838i | 1.56447 | − | 1.24596i | 1.24326 | − | 1.85858i | −2.23610 | + | 1.20236i | 2.22547 | + | 2.22547i | 1.50715 | − | 2.39343i | 0.211989 | − | 0.0688795i | 0.792444 | − | 3.06138i |
3.12 | 1.40799 | − | 0.132526i | −0.173581 | + | 0.0274925i | 1.96487 | − | 0.373191i | −0.138125 | + | 2.23180i | −0.240757 | + | 0.0617133i | −2.95323 | − | 2.95323i | 2.71707 | − | 0.785847i | −2.82379 | + | 0.917507i | 0.101293 | + | 3.16065i |
23.1 | −1.41083 | − | 0.0976968i | 0.0867473 | + | 0.170251i | 1.98091 | + | 0.275668i | −1.18066 | + | 1.89896i | −0.105753 | − | 0.248671i | 1.89427 | + | 1.89427i | −2.76781 | − | 0.582451i | 1.74190 | − | 2.39751i | 1.85124 | − | 2.56377i |
23.2 | −1.35937 | + | 0.390031i | 1.41506 | + | 2.77721i | 1.69575 | − | 1.06039i | 1.36636 | − | 1.77005i | −3.00678 | − | 3.22332i | 0.467533 | + | 0.467533i | −1.89156 | + | 2.10285i | −3.94714 | + | 5.43277i | −1.16701 | + | 2.93906i |
23.3 | −1.06970 | + | 0.925063i | −0.458745 | − | 0.900337i | 0.288515 | − | 1.97908i | −1.53117 | − | 1.62957i | 1.32359 | + | 0.538722i | −2.42430 | − | 2.42430i | 1.52215 | + | 2.38392i | 1.16320 | − | 1.60100i | 3.14535 | + | 0.326722i |
23.4 | −0.776413 | − | 1.18202i | −1.25043 | − | 2.45410i | −0.794365 | + | 1.83548i | −2.21556 | − | 0.302124i | −1.92996 | + | 3.38344i | 1.05978 | + | 1.05978i | 2.78634 | − | 0.486130i | −2.69570 | + | 3.71031i | 1.36307 | + | 2.85342i |
23.5 | −0.643767 | + | 1.25919i | −0.700193 | − | 1.37421i | −1.17113 | − | 1.62125i | 2.14271 | + | 0.639373i | 2.18115 | + | 0.00299155i | 3.31824 | + | 3.31824i | 2.79540 | − | 0.430965i | 0.365182 | − | 0.502631i | −2.18450 | + | 2.28647i |
23.6 | −0.481662 | − | 1.32966i | 0.956383 | + | 1.87701i | −1.53600 | + | 1.28090i | 0.727343 | + | 2.11447i | 2.03513 | − | 2.17575i | 0.0806175 | + | 0.0806175i | 2.44299 | + | 1.42541i | −0.845131 | + | 1.16322i | 2.46119 | − | 1.98558i |
23.7 | 0.223147 | + | 1.39650i | 0.700193 | + | 1.37421i | −1.90041 | + | 0.623249i | 2.14271 | + | 0.639373i | −1.76283 | + | 1.28447i | −3.31824 | − | 3.31824i | −1.29444 | − | 2.51484i | 0.365182 | − | 0.502631i | −0.414743 | + | 3.13496i |
23.8 | 0.731485 | + | 1.21034i | 0.458745 | + | 0.900337i | −0.929861 | + | 1.77069i | −1.53117 | − | 1.62957i | −0.754152 | + | 1.21382i | 2.42430 | + | 2.42430i | −2.82333 | + | 0.169786i | 1.16320 | − | 1.60100i | 0.852314 | − | 3.04525i |
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
25.f | odd | 20 | 1 | inner |
100.l | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 100.2.l.b | ✓ | 96 |
3.b | odd | 2 | 1 | 900.2.bj.d | 96 | ||
4.b | odd | 2 | 1 | inner | 100.2.l.b | ✓ | 96 |
5.b | even | 2 | 1 | 500.2.l.f | 96 | ||
5.c | odd | 4 | 1 | 500.2.l.d | 96 | ||
5.c | odd | 4 | 1 | 500.2.l.e | 96 | ||
12.b | even | 2 | 1 | 900.2.bj.d | 96 | ||
20.d | odd | 2 | 1 | 500.2.l.f | 96 | ||
20.e | even | 4 | 1 | 500.2.l.d | 96 | ||
20.e | even | 4 | 1 | 500.2.l.e | 96 | ||
25.d | even | 5 | 1 | 500.2.l.e | 96 | ||
25.e | even | 10 | 1 | 500.2.l.d | 96 | ||
25.f | odd | 20 | 1 | inner | 100.2.l.b | ✓ | 96 |
25.f | odd | 20 | 1 | 500.2.l.f | 96 | ||
75.l | even | 20 | 1 | 900.2.bj.d | 96 | ||
100.h | odd | 10 | 1 | 500.2.l.d | 96 | ||
100.j | odd | 10 | 1 | 500.2.l.e | 96 | ||
100.l | even | 20 | 1 | inner | 100.2.l.b | ✓ | 96 |
100.l | even | 20 | 1 | 500.2.l.f | 96 | ||
300.u | odd | 20 | 1 | 900.2.bj.d | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
100.2.l.b | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
100.2.l.b | ✓ | 96 | 4.b | odd | 2 | 1 | inner |
100.2.l.b | ✓ | 96 | 25.f | odd | 20 | 1 | inner |
100.2.l.b | ✓ | 96 | 100.l | even | 20 | 1 | inner |
500.2.l.d | 96 | 5.c | odd | 4 | 1 | ||
500.2.l.d | 96 | 20.e | even | 4 | 1 | ||
500.2.l.d | 96 | 25.e | even | 10 | 1 | ||
500.2.l.d | 96 | 100.h | odd | 10 | 1 | ||
500.2.l.e | 96 | 5.c | odd | 4 | 1 | ||
500.2.l.e | 96 | 20.e | even | 4 | 1 | ||
500.2.l.e | 96 | 25.d | even | 5 | 1 | ||
500.2.l.e | 96 | 100.j | odd | 10 | 1 | ||
500.2.l.f | 96 | 5.b | even | 2 | 1 | ||
500.2.l.f | 96 | 20.d | odd | 2 | 1 | ||
500.2.l.f | 96 | 25.f | odd | 20 | 1 | ||
500.2.l.f | 96 | 100.l | even | 20 | 1 | ||
900.2.bj.d | 96 | 3.b | odd | 2 | 1 | ||
900.2.bj.d | 96 | 12.b | even | 2 | 1 | ||
900.2.bj.d | 96 | 75.l | even | 20 | 1 | ||
900.2.bj.d | 96 | 300.u | odd | 20 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{96} + 10 T_{3}^{94} - 103 T_{3}^{92} - 1600 T_{3}^{90} + 7218 T_{3}^{88} + 167890 T_{3}^{86} - 499696 T_{3}^{84} - 13339180 T_{3}^{82} + 61015935 T_{3}^{80} + 1192607560 T_{3}^{78} + \cdots + 75\!\cdots\!00 \)
acting on \(S_{2}^{\mathrm{new}}(100, [\chi])\).