Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [99,2,Mod(4,99)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(99, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([10, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("99.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 99.m (of order \(15\), 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: | \(0.790518980011\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −0.231548 | + | 2.20304i | −0.679127 | + | 1.59336i | −2.84345 | − | 0.604395i | 0.0650661 | + | 0.619063i | −3.35297 | − | 1.86508i | 1.72584 | − | 1.91674i | 0.620850 | − | 1.91078i | −2.07757 | − | 2.16418i | −1.37888 | ||
4.2 | −0.161615 | + | 1.53767i | 1.18421 | − | 1.26398i | −0.382004 | − | 0.0811974i | −0.0618842 | − | 0.588789i | 1.75219 | + | 2.02520i | 0.483079 | − | 0.536514i | −0.768973 | + | 2.36665i | −0.195290 | − | 2.99364i | 0.915362 | ||
4.3 | −0.150377 | + | 1.43074i | −1.39014 | − | 1.03320i | −0.0681222 | − | 0.0144798i | 0.364652 | + | 3.46943i | 1.68730 | − | 1.83357i | −0.226061 | + | 0.251066i | −0.858159 | + | 2.64114i | 0.864983 | + | 2.87260i | −5.01870 | ||
4.4 | −0.0603978 | + | 0.574647i | 0.641057 | + | 1.60905i | 1.62972 | + | 0.346409i | −0.314048 | − | 2.98797i | −0.963355 | + | 0.271198i | −0.779082 | + | 0.865259i | −0.654602 | + | 2.01466i | −2.17809 | + | 2.06299i | 1.73599 | ||
4.5 | 0.0512706 | − | 0.487807i | −1.72269 | − | 0.179809i | 1.72097 | + | 0.365803i | −0.178553 | − | 1.69882i | −0.176036 | + | 0.831122i | 2.28209 | − | 2.53451i | 0.569818 | − | 1.75372i | 2.93534 | + | 0.619512i | −0.837851 | ||
4.6 | 0.0694404 | − | 0.660681i | −0.106925 | − | 1.72875i | 1.52462 | + | 0.324067i | −0.116719 | − | 1.11051i | −1.14958 | − | 0.0494014i | −2.61265 | + | 2.90164i | 0.730548 | − | 2.24840i | −2.97713 | + | 0.369694i | −0.741796 | ||
4.7 | 0.122179 | − | 1.16246i | −0.613763 | + | 1.61966i | 0.619915 | + | 0.131767i | 0.331034 | + | 3.14958i | 1.80780 | + | 0.911362i | 0.166529 | − | 0.184949i | 0.951310 | − | 2.92783i | −2.24659 | − | 1.98817i | 3.70170 | ||
4.8 | 0.209960 | − | 1.99763i | 1.72741 | + | 0.126717i | −1.99016 | − | 0.423021i | −0.000569867 | − | 0.00542192i | 0.615820 | − | 3.42412i | −2.56130 | + | 2.84461i | −0.0214877 | + | 0.0661323i | 2.96789 | + | 0.437784i | −0.0109507 | ||
4.9 | 0.255617 | − | 2.43204i | −0.195654 | − | 1.72096i | −3.89317 | − | 0.827519i | 0.313886 | + | 2.98642i | −4.23546 | + | 0.0359288i | 2.60423 | − | 2.89229i | −1.49636 | + | 4.60531i | −2.92344 | + | 0.673427i | 7.34333 | ||
16.1 | −2.27811 | − | 0.484227i | 1.65119 | − | 0.523037i | 3.12821 | + | 1.39277i | 0.310092 | − | 0.0659121i | −4.01486 | + | 0.391985i | 0.148273 | + | 1.41072i | −2.68358 | − | 1.94973i | 2.45286 | − | 1.72727i | −0.738340 | ||
16.2 | −2.09212 | − | 0.444695i | −1.47961 | − | 0.900424i | 2.35214 | + | 1.04724i | −1.63936 | + | 0.348457i | 2.69511 | + | 2.54177i | 0.135431 | + | 1.28854i | −0.994512 | − | 0.722555i | 1.37847 | + | 2.66455i | 3.58471 | ||
16.3 | −0.984579 | − | 0.209279i | −1.59971 | + | 0.664019i | −0.901493 | − | 0.401371i | 3.90391 | − | 0.829803i | 1.71401 | − | 0.318993i | −0.288110 | − | 2.74118i | 2.43227 | + | 1.76714i | 2.11816 | − | 2.12448i | −4.01737 | ||
16.4 | −0.662860 | − | 0.140895i | 0.607840 | − | 1.62189i | −1.40756 | − | 0.626686i | −0.284440 | + | 0.0604596i | −0.631430 | + | 0.989445i | −0.350744 | − | 3.33711i | 1.94121 | + | 1.41037i | −2.26106 | − | 1.97170i | 0.197063 | ||
16.5 | −0.278917 | − | 0.0592857i | −1.04736 | + | 1.37950i | −1.75281 | − | 0.780402i | −2.64954 | + | 0.563177i | 0.373912 | − | 0.322674i | 0.425779 | + | 4.05102i | 0.904002 | + | 0.656796i | −0.806061 | − | 2.88968i | 0.772391 | ||
16.6 | 1.42554 | + | 0.303007i | −1.06518 | − | 1.36579i | 0.113254 | + | 0.0504238i | 2.94034 | − | 0.624989i | −1.10462 | − | 2.26974i | 0.172491 | + | 1.64114i | −2.21193 | − | 1.60706i | −0.730763 | + | 2.90964i | 4.38094 | ||
16.7 | 1.45109 | + | 0.308439i | 0.266086 | + | 1.71149i | 0.183441 | + | 0.0816732i | 0.0707467 | − | 0.0150377i | −0.141775 | + | 2.56560i | −0.203853 | − | 1.93954i | −2.15937 | − | 1.56887i | −2.85840 | + | 0.910807i | 0.107298 | ||
16.8 | 1.85327 | + | 0.393925i | 0.886115 | − | 1.48822i | 1.45235 | + | 0.646630i | −3.43625 | + | 0.730397i | 2.22846 | − | 2.40902i | 0.358742 | + | 3.41320i | −0.628765 | − | 0.456825i | −1.42960 | − | 2.63747i | −6.65603 | ||
16.9 | 2.54483 | + | 0.540921i | −1.71102 | + | 0.269100i | 4.35649 | + | 1.93963i | −2.00724 | + | 0.426651i | −4.49982 | − | 0.240711i | −0.333407 | − | 3.17215i | 5.82773 | + | 4.23409i | 2.85517 | − | 0.920871i | −5.33887 | ||
25.1 | −0.231548 | − | 2.20304i | −0.679127 | − | 1.59336i | −2.84345 | + | 0.604395i | 0.0650661 | − | 0.619063i | −3.35297 | + | 1.86508i | 1.72584 | + | 1.91674i | 0.620850 | + | 1.91078i | −2.07757 | + | 2.16418i | −1.37888 | ||
25.2 | −0.161615 | − | 1.53767i | 1.18421 | + | 1.26398i | −0.382004 | + | 0.0811974i | −0.0618842 | + | 0.588789i | 1.75219 | − | 2.02520i | 0.483079 | + | 0.536514i | −0.768973 | − | 2.36665i | −0.195290 | + | 2.99364i | 0.915362 | ||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
11.c | even | 5 | 1 | inner |
99.m | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 99.2.m.b | ✓ | 72 |
3.b | odd | 2 | 1 | 297.2.n.b | 72 | ||
9.c | even | 3 | 1 | inner | 99.2.m.b | ✓ | 72 |
9.c | even | 3 | 1 | 891.2.f.f | 36 | ||
9.d | odd | 6 | 1 | 297.2.n.b | 72 | ||
9.d | odd | 6 | 1 | 891.2.f.e | 36 | ||
11.c | even | 5 | 1 | inner | 99.2.m.b | ✓ | 72 |
11.c | even | 5 | 1 | 1089.2.e.p | 36 | ||
11.d | odd | 10 | 1 | 1089.2.e.o | 36 | ||
33.h | odd | 10 | 1 | 297.2.n.b | 72 | ||
99.m | even | 15 | 1 | inner | 99.2.m.b | ✓ | 72 |
99.m | even | 15 | 1 | 891.2.f.f | 36 | ||
99.m | even | 15 | 1 | 1089.2.e.p | 36 | ||
99.m | even | 15 | 1 | 9801.2.a.cm | 18 | ||
99.n | odd | 30 | 1 | 297.2.n.b | 72 | ||
99.n | odd | 30 | 1 | 891.2.f.e | 36 | ||
99.n | odd | 30 | 1 | 9801.2.a.cp | 18 | ||
99.o | odd | 30 | 1 | 1089.2.e.o | 36 | ||
99.o | odd | 30 | 1 | 9801.2.a.co | 18 | ||
99.p | even | 30 | 1 | 9801.2.a.cn | 18 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
99.2.m.b | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
99.2.m.b | ✓ | 72 | 9.c | even | 3 | 1 | inner |
99.2.m.b | ✓ | 72 | 11.c | even | 5 | 1 | inner |
99.2.m.b | ✓ | 72 | 99.m | even | 15 | 1 | inner |
297.2.n.b | 72 | 3.b | odd | 2 | 1 | ||
297.2.n.b | 72 | 9.d | odd | 6 | 1 | ||
297.2.n.b | 72 | 33.h | odd | 10 | 1 | ||
297.2.n.b | 72 | 99.n | odd | 30 | 1 | ||
891.2.f.e | 36 | 9.d | odd | 6 | 1 | ||
891.2.f.e | 36 | 99.n | odd | 30 | 1 | ||
891.2.f.f | 36 | 9.c | even | 3 | 1 | ||
891.2.f.f | 36 | 99.m | even | 15 | 1 | ||
1089.2.e.o | 36 | 11.d | odd | 10 | 1 | ||
1089.2.e.o | 36 | 99.o | odd | 30 | 1 | ||
1089.2.e.p | 36 | 11.c | even | 5 | 1 | ||
1089.2.e.p | 36 | 99.m | even | 15 | 1 | ||
9801.2.a.cm | 18 | 99.m | even | 15 | 1 | ||
9801.2.a.cn | 18 | 99.p | even | 30 | 1 | ||
9801.2.a.co | 18 | 99.o | odd | 30 | 1 | ||
9801.2.a.cp | 18 | 99.n | odd | 30 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{72} + T_{2}^{71} - 14 T_{2}^{70} - 19 T_{2}^{69} + 76 T_{2}^{68} + 112 T_{2}^{67} + \cdots + 9150625 \)
acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\).