Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [108,3,Mod(7,108)]
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, 16]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("108.7");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 108 = 2^{2} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 108.j (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: | \(2.94278685509\) |
Analytic rank: | \(0\) |
Dimension: | \(204\) |
Relative dimension: | \(34\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | −1.95448 | + | 0.424266i | 0.0189322 | + | 2.99994i | 3.64000 | − | 1.65844i | −1.07944 | − | 6.12179i | −1.30977 | − | 5.85530i | 6.40210 | − | 7.62973i | −6.41069 | + | 4.78571i | −8.99928 | + | 0.113591i | 4.70701 | + | 11.5070i |
7.2 | −1.95123 | − | 0.438997i | −1.49386 | − | 2.60161i | 3.61456 | + | 1.71316i | 0.0372686 | + | 0.211361i | 1.77277 | + | 5.73213i | 7.43930 | − | 8.86581i | −6.30076 | − | 4.92955i | −4.53674 | + | 7.77290i | 0.0200672 | − | 0.428773i |
7.3 | −1.91270 | − | 0.584441i | 2.24560 | + | 1.98929i | 3.31686 | + | 2.23572i | 0.641948 | + | 3.64067i | −3.13254 | − | 5.11734i | −2.33709 | + | 2.78523i | −5.03752 | − | 6.21478i | 1.08543 | + | 8.93431i | 0.899899 | − | 7.33869i |
7.4 | −1.90734 | − | 0.601703i | −2.99807 | + | 0.107643i | 3.27591 | + | 2.29531i | −0.755508 | − | 4.28470i | 5.78311 | + | 1.59863i | −6.44162 | + | 7.67682i | −4.86718 | − | 6.34906i | 8.97683 | − | 0.645444i | −1.13710 | + | 8.62697i |
7.5 | −1.89957 | + | 0.625818i | 2.48687 | − | 1.67794i | 3.21670 | − | 2.37757i | 1.04974 | + | 5.95339i | −3.67389 | + | 4.74368i | 1.18627 | − | 1.41374i | −4.62241 | + | 6.52942i | 3.36905 | − | 8.34563i | −5.71979 | − | 10.6519i |
7.6 | −1.85742 | + | 0.741613i | −2.48687 | + | 1.67794i | 2.90002 | − | 2.75497i | 1.04974 | + | 5.95339i | 3.37478 | − | 4.96093i | −1.18627 | + | 1.41374i | −3.34343 | + | 7.26784i | 3.36905 | − | 8.34563i | −6.36492 | − | 10.2794i |
7.7 | −1.76993 | + | 0.931310i | −0.0189322 | − | 2.99994i | 2.26532 | − | 3.29671i | −1.07944 | − | 6.12179i | 2.82738 | + | 5.29206i | −6.40210 | + | 7.62973i | −0.939205 | + | 7.94468i | −8.99928 | + | 0.113591i | 7.61182 | + | 9.82987i |
7.8 | −1.59321 | − | 1.20900i | 2.60238 | − | 1.49253i | 1.07665 | + | 3.85238i | −0.890579 | − | 5.05073i | −5.95060 | − | 0.768354i | 0.0265129 | − | 0.0315969i | 2.94218 | − | 7.43932i | 4.54473 | − | 7.76823i | −4.68743 | + | 9.12359i |
7.9 | −1.21254 | + | 1.59051i | 1.49386 | + | 2.60161i | −1.05947 | − | 3.85714i | 0.0372686 | + | 0.211361i | −5.94927 | − | 0.778554i | −7.43930 | + | 8.86581i | 7.41949 | + | 2.99184i | −4.53674 | + | 7.77290i | −0.381362 | − | 0.197008i |
7.10 | −1.20427 | − | 1.59678i | −0.204519 | − | 2.99302i | −1.09944 | + | 3.84594i | 1.50670 | + | 8.54492i | −4.53291 | + | 3.93099i | −7.53806 | + | 8.98351i | 7.46516 | − | 2.87599i | −8.91634 | + | 1.22426i | 11.8299 | − | 12.6963i |
7.11 | −1.08954 | + | 1.67717i | −2.24560 | − | 1.98929i | −1.62579 | − | 3.65470i | 0.641948 | + | 3.64067i | 5.78306 | − | 1.59883i | 2.33709 | − | 2.78523i | 7.90091 | + | 1.25523i | 1.08543 | + | 8.93431i | −6.80544 | − | 2.89001i |
7.12 | −1.07434 | + | 1.68695i | 2.99807 | − | 0.107643i | −1.69158 | − | 3.62471i | −0.755508 | − | 4.28470i | −3.03936 | + | 5.17323i | 6.44162 | − | 7.67682i | 7.93204 | + | 1.04057i | 8.97683 | − | 0.645444i | 8.03973 | + | 3.32873i |
7.13 | −1.00307 | − | 1.73027i | −2.96394 | + | 0.463733i | −1.98770 | + | 3.47117i | 0.608002 | + | 3.44815i | 3.77543 | + | 4.66328i | 5.71739 | − | 6.81372i | 7.99989 | − | 0.0425704i | 8.56990 | − | 2.74895i | 5.35638 | − | 4.51075i |
7.14 | −0.974385 | − | 1.74659i | −0.586935 | + | 2.94202i | −2.10115 | + | 3.40370i | −1.26933 | − | 7.19876i | 5.71041 | − | 1.84153i | −2.67622 | + | 3.18940i | 7.99219 | + | 0.353324i | −8.31101 | − | 3.45355i | −11.3364 | + | 9.23137i |
7.15 | −0.619770 | − | 1.90155i | 2.41946 | + | 1.77376i | −3.23177 | + | 2.35705i | 0.463207 | + | 2.62698i | 1.87339 | − | 5.70004i | 4.34885 | − | 5.18275i | 6.48499 | + | 4.68454i | 2.70753 | + | 8.58308i | 4.70825 | − | 2.50894i |
7.16 | −0.443343 | + | 1.95024i | −2.60238 | + | 1.49253i | −3.60689 | − | 1.72925i | −0.890579 | − | 5.05073i | −1.75705 | − | 5.73697i | −0.0265129 | + | 0.0315969i | 4.97155 | − | 6.26767i | 4.54473 | − | 7.76823i | 10.2450 | + | 0.502357i |
7.17 | 0.103865 | + | 1.99730i | 0.204519 | + | 2.99302i | −3.97842 | + | 0.414901i | 1.50670 | + | 8.54492i | −5.95672 | + | 0.719357i | 7.53806 | − | 8.98351i | −1.24190 | − | 7.90302i | −8.91634 | + | 1.22426i | −16.9103 | + | 3.89685i |
7.18 | 0.210380 | − | 1.98890i | 1.64282 | − | 2.51021i | −3.91148 | − | 0.836851i | −0.133306 | − | 0.756014i | −4.64695 | − | 3.79550i | 3.08121 | − | 3.67204i | −2.48731 | + | 7.60350i | −3.60231 | − | 8.24763i | −1.53168 | + | 0.106082i |
7.19 | 0.277531 | − | 1.98065i | −2.64062 | − | 1.42377i | −3.84595 | − | 1.09938i | −0.542224 | − | 3.07510i | −3.55285 | + | 4.83500i | −2.98812 | + | 3.56110i | −3.24486 | + | 7.31238i | 4.94574 | + | 7.51929i | −6.24119 | + | 0.220520i |
7.20 | 0.343802 | + | 1.97023i | 2.96394 | − | 0.463733i | −3.76360 | + | 1.35474i | 0.608002 | + | 3.44815i | 1.93267 | + | 5.68021i | −5.71739 | + | 6.81372i | −3.96308 | − | 6.94939i | 8.56990 | − | 2.74895i | −6.58462 | + | 2.38339i |
See next 80 embeddings (of 204 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
27.e | even | 9 | 1 | inner |
108.j | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 108.3.j.a | ✓ | 204 |
3.b | odd | 2 | 1 | 324.3.j.a | 204 | ||
4.b | odd | 2 | 1 | inner | 108.3.j.a | ✓ | 204 |
12.b | even | 2 | 1 | 324.3.j.a | 204 | ||
27.e | even | 9 | 1 | inner | 108.3.j.a | ✓ | 204 |
27.f | odd | 18 | 1 | 324.3.j.a | 204 | ||
108.j | odd | 18 | 1 | inner | 108.3.j.a | ✓ | 204 |
108.l | even | 18 | 1 | 324.3.j.a | 204 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
108.3.j.a | ✓ | 204 | 1.a | even | 1 | 1 | trivial |
108.3.j.a | ✓ | 204 | 4.b | odd | 2 | 1 | inner |
108.3.j.a | ✓ | 204 | 27.e | even | 9 | 1 | inner |
108.3.j.a | ✓ | 204 | 108.j | odd | 18 | 1 | inner |
324.3.j.a | 204 | 3.b | odd | 2 | 1 | ||
324.3.j.a | 204 | 12.b | even | 2 | 1 | ||
324.3.j.a | 204 | 27.f | odd | 18 | 1 | ||
324.3.j.a | 204 | 108.l | even | 18 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(108, [\chi])\).