Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [108,2,Mod(11,108)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(108, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 13]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("108.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 108 = 2^{2} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 108.l (of order \(18\), degree \(6\), 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: | \(0.862384341830\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −1.37686 | − | 0.322907i | 1.59825 | + | 0.667539i | 1.79146 | + | 0.889191i | −1.27150 | + | 3.49343i | −1.98500 | − | 1.43519i | −1.63150 | − | 0.287677i | −2.17946 | − | 1.80276i | 2.10878 | + | 2.13378i | 2.87873 | − | 4.39937i |
11.2 | −1.36287 | − | 0.377622i | −1.69735 | + | 0.344959i | 1.71480 | + | 1.02930i | 0.420820 | − | 1.15619i | 2.44353 | + | 0.170825i | 1.81474 | + | 0.319988i | −1.94836 | − | 2.05034i | 2.76201 | − | 1.17103i | −1.01013 | + | 1.41682i |
11.3 | −1.27725 | + | 0.607153i | 0.971698 | + | 1.43381i | 1.26273 | − | 1.55097i | 1.29726 | − | 3.56419i | −2.11164 | − | 1.24136i | 2.58045 | + | 0.455003i | −0.671144 | + | 2.74765i | −1.11160 | + | 2.78646i | 0.507086 | + | 5.33999i |
11.4 | −1.06666 | − | 0.928568i | 0.745195 | − | 1.56355i | 0.275524 | + | 1.98093i | 0.605021 | − | 1.66228i | −2.24673 | + | 0.975810i | 0.748045 | + | 0.131901i | 1.54554 | − | 2.36882i | −1.88937 | − | 2.33030i | −2.18889 | + | 1.21129i |
11.5 | −1.00492 | + | 0.995056i | −1.23220 | + | 1.21724i | 0.0197253 | − | 1.99990i | −0.710267 | + | 1.95144i | 0.0270364 | − | 2.44934i | −3.83975 | − | 0.677051i | 1.97019 | + | 2.02937i | 0.0366374 | − | 2.99978i | −1.22804 | − | 2.66780i |
11.6 | −0.805437 | + | 1.16244i | 1.23220 | − | 1.21724i | −0.702543 | − | 1.87255i | −0.710267 | + | 1.95144i | 0.422515 | + | 2.41277i | 3.83975 | + | 0.677051i | 2.74258 | + | 0.691554i | 0.0366374 | − | 2.99978i | −1.69636 | − | 2.39741i |
11.7 | −0.419899 | − | 1.35044i | −0.537194 | + | 1.64664i | −1.64737 | + | 1.13410i | −0.847966 | + | 2.32977i | 2.44925 | + | 0.0340252i | 4.59553 | + | 0.810316i | 2.22326 | + | 1.74846i | −2.42284 | − | 1.76913i | 3.50227 | + | 0.166858i |
11.8 | −0.376137 | + | 1.36328i | −0.971698 | − | 1.43381i | −1.71704 | − | 1.02556i | 1.29726 | − | 3.56419i | 2.32017 | − | 0.785385i | −2.58045 | − | 0.455003i | 2.04396 | − | 1.95505i | −1.11160 | + | 2.78646i | 4.37103 | + | 3.10915i |
11.9 | 0.0793838 | − | 1.41198i | 1.70299 | + | 0.315971i | −1.98740 | − | 0.224177i | 0.470103 | − | 1.29160i | 0.581335 | − | 2.37951i | −1.57428 | − | 0.277589i | −0.474302 | + | 2.78838i | 2.80032 | + | 1.07619i | −1.78640 | − | 0.766310i |
11.10 | 0.554410 | − | 1.30101i | −0.490974 | − | 1.66101i | −1.38526 | − | 1.44259i | −0.197421 | + | 0.542409i | −2.43319 | − | 0.282117i | 1.70706 | + | 0.301001i | −2.64482 | + | 1.00245i | −2.51789 | + | 1.63102i | 0.596228 | + | 0.557564i |
11.11 | 0.557089 | + | 1.29987i | −1.59825 | − | 0.667539i | −1.37930 | + | 1.44828i | −1.27150 | + | 3.49343i | −0.0226541 | − | 2.44938i | 1.63150 | + | 0.287677i | −2.65097 | − | 0.986086i | 2.10878 | + | 2.13378i | −5.24933 | + | 0.293367i |
11.12 | 0.608544 | + | 1.27659i | 1.69735 | − | 0.344959i | −1.25935 | + | 1.55372i | 0.420820 | − | 1.15619i | 1.47328 | + | 1.95689i | −1.81474 | − | 0.319988i | −2.74983 | − | 0.662160i | 2.76201 | − | 1.17103i | 1.73207 | − | 0.166382i |
11.13 | 1.09968 | + | 0.889210i | −0.745195 | + | 1.56355i | 0.418610 | + | 1.95570i | 0.605021 | − | 1.66228i | −2.20980 | + | 1.05677i | −0.748045 | − | 0.131901i | −1.27869 | + | 2.52289i | −1.88937 | − | 2.33030i | 2.14345 | − | 1.28999i |
11.14 | 1.18497 | − | 0.771906i | 0.490974 | + | 1.66101i | 0.808323 | − | 1.82938i | −0.197421 | + | 0.542409i | 1.86393 | + | 1.58926i | −1.70706 | − | 0.301001i | −0.454264 | − | 2.79171i | −2.51789 | + | 1.63102i | 0.184750 | + | 0.795131i |
11.15 | 1.37675 | − | 0.323366i | −1.70299 | − | 0.315971i | 1.79087 | − | 0.890387i | 0.470103 | − | 1.29160i | −2.44676 | + | 0.115676i | 1.57428 | + | 0.277589i | 2.17765 | − | 1.80495i | 2.80032 | + | 1.07619i | 0.229554 | − | 1.93022i |
11.16 | 1.40284 | + | 0.179019i | 0.537194 | − | 1.64664i | 1.93590 | + | 0.502269i | −0.847966 | + | 2.32977i | 1.04838 | − | 2.21380i | −4.59553 | − | 0.810316i | 2.62584 | + | 1.05116i | −2.42284 | − | 1.76913i | −1.60663 | + | 3.11648i |
23.1 | −1.40830 | − | 0.129170i | 1.33417 | − | 1.10453i | 1.96663 | + | 0.363822i | 0.297855 | + | 0.0525198i | −2.02159 | + | 1.38318i | 0.312070 | + | 0.371910i | −2.72261 | − | 0.766402i | 0.560024 | − | 2.94727i | −0.412685 | − | 0.112438i |
23.2 | −1.33631 | − | 0.462897i | −0.808457 | + | 1.53180i | 1.57145 | + | 1.23715i | −3.39370 | − | 0.598401i | 1.78941 | − | 1.67272i | −1.88319 | − | 2.24430i | −1.52728 | − | 2.38064i | −1.69280 | − | 2.47678i | 4.25804 | + | 2.37058i |
23.3 | −1.30706 | + | 0.540000i | −1.71606 | − | 0.234845i | 1.41680 | − | 1.41162i | 0.137245 | + | 0.0242000i | 2.36980 | − | 0.619715i | 2.98823 | + | 3.56123i | −1.08956 | + | 2.61015i | 2.88970 | + | 0.806015i | −0.192456 | + | 0.0424817i |
23.4 | −1.20024 | + | 0.747945i | 0.644352 | + | 1.60773i | 0.881156 | − | 1.79543i | 3.32285 | + | 0.585909i | −1.97587 | − | 1.44773i | −1.72640 | − | 2.05745i | 0.285283 | + | 2.81400i | −2.16962 | + | 2.07189i | −4.42645 | + | 1.78208i |
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
27.f | odd | 18 | 1 | inner |
108.l | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 108.2.l.a | ✓ | 96 |
3.b | odd | 2 | 1 | 324.2.l.a | 96 | ||
4.b | odd | 2 | 1 | inner | 108.2.l.a | ✓ | 96 |
9.c | even | 3 | 1 | 972.2.l.c | 96 | ||
9.c | even | 3 | 1 | 972.2.l.d | 96 | ||
9.d | odd | 6 | 1 | 972.2.l.a | 96 | ||
9.d | odd | 6 | 1 | 972.2.l.b | 96 | ||
12.b | even | 2 | 1 | 324.2.l.a | 96 | ||
27.e | even | 9 | 1 | 324.2.l.a | 96 | ||
27.e | even | 9 | 1 | 972.2.l.a | 96 | ||
27.e | even | 9 | 1 | 972.2.l.b | 96 | ||
27.f | odd | 18 | 1 | inner | 108.2.l.a | ✓ | 96 |
27.f | odd | 18 | 1 | 972.2.l.c | 96 | ||
27.f | odd | 18 | 1 | 972.2.l.d | 96 | ||
36.f | odd | 6 | 1 | 972.2.l.c | 96 | ||
36.f | odd | 6 | 1 | 972.2.l.d | 96 | ||
36.h | even | 6 | 1 | 972.2.l.a | 96 | ||
36.h | even | 6 | 1 | 972.2.l.b | 96 | ||
108.j | odd | 18 | 1 | 324.2.l.a | 96 | ||
108.j | odd | 18 | 1 | 972.2.l.a | 96 | ||
108.j | odd | 18 | 1 | 972.2.l.b | 96 | ||
108.l | even | 18 | 1 | inner | 108.2.l.a | ✓ | 96 |
108.l | even | 18 | 1 | 972.2.l.c | 96 | ||
108.l | even | 18 | 1 | 972.2.l.d | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
108.2.l.a | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
108.2.l.a | ✓ | 96 | 4.b | odd | 2 | 1 | inner |
108.2.l.a | ✓ | 96 | 27.f | odd | 18 | 1 | inner |
108.2.l.a | ✓ | 96 | 108.l | even | 18 | 1 | inner |
324.2.l.a | 96 | 3.b | odd | 2 | 1 | ||
324.2.l.a | 96 | 12.b | even | 2 | 1 | ||
324.2.l.a | 96 | 27.e | even | 9 | 1 | ||
324.2.l.a | 96 | 108.j | odd | 18 | 1 | ||
972.2.l.a | 96 | 9.d | odd | 6 | 1 | ||
972.2.l.a | 96 | 27.e | even | 9 | 1 | ||
972.2.l.a | 96 | 36.h | even | 6 | 1 | ||
972.2.l.a | 96 | 108.j | odd | 18 | 1 | ||
972.2.l.b | 96 | 9.d | odd | 6 | 1 | ||
972.2.l.b | 96 | 27.e | even | 9 | 1 | ||
972.2.l.b | 96 | 36.h | even | 6 | 1 | ||
972.2.l.b | 96 | 108.j | odd | 18 | 1 | ||
972.2.l.c | 96 | 9.c | even | 3 | 1 | ||
972.2.l.c | 96 | 27.f | odd | 18 | 1 | ||
972.2.l.c | 96 | 36.f | odd | 6 | 1 | ||
972.2.l.c | 96 | 108.l | even | 18 | 1 | ||
972.2.l.d | 96 | 9.c | even | 3 | 1 | ||
972.2.l.d | 96 | 27.f | odd | 18 | 1 | ||
972.2.l.d | 96 | 36.f | odd | 6 | 1 | ||
972.2.l.d | 96 | 108.l | even | 18 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(108, [\chi])\).