Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [135,4,Mod(4,135)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(135, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([2, 9]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("135.4");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 135.p (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: | \(7.96525785077\) |
Analytic rank: | \(0\) |
Dimension: | \(312\) |
Relative dimension: | \(52\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −1.89118 | + | 5.19596i | −0.0220314 | − | 5.19611i | −17.2931 | − | 14.5107i | −9.65682 | − | 5.63435i | 27.0404 | + | 9.71227i | 13.7470 | + | 16.3830i | 69.7922 | − | 40.2945i | −26.9990 | + | 0.228955i | 47.5386 | − | 39.5209i |
4.2 | −1.85666 | + | 5.10114i | −2.41243 | + | 4.60219i | −16.4461 | − | 13.7999i | 4.19064 | + | 10.3653i | −18.9974 | − | 20.8509i | −9.50379 | − | 11.3262i | 63.3201 | − | 36.5579i | −15.3604 | − | 22.2049i | −60.6552 | + | 2.13223i |
4.3 | −1.76130 | + | 4.83912i | 5.11954 | − | 0.888993i | −14.1866 | − | 11.9040i | 10.8512 | − | 2.69293i | −4.71508 | + | 26.3399i | −7.10582 | − | 8.46839i | 46.9135 | − | 27.0855i | 25.4194 | − | 9.10247i | −6.08071 | + | 57.2533i |
4.4 | −1.65382 | + | 4.54382i | 4.52413 | + | 2.55583i | −11.7829 | − | 9.88700i | −5.35022 | + | 9.81709i | −19.0953 | + | 16.3299i | 22.7072 | + | 27.0614i | 30.9106 | − | 17.8462i | 13.9354 | + | 23.1258i | −35.7588 | − | 40.5461i |
4.5 | −1.63162 | + | 4.48285i | 2.53614 | + | 4.53520i | −11.3054 | − | 9.48636i | −0.966721 | − | 11.1385i | −24.4686 | + | 3.96938i | −4.14355 | − | 4.93809i | 27.9208 | − | 16.1201i | −14.1360 | + | 23.0038i | 51.5094 | + | 13.8401i |
4.6 | −1.62821 | + | 4.47348i | −4.69383 | + | 2.22890i | −11.2326 | − | 9.42528i | −2.71524 | − | 10.8456i | −2.32838 | − | 24.6269i | 0.578750 | + | 0.689728i | 27.4707 | − | 15.8602i | 17.0640 | − | 20.9241i | 52.9387 | + | 5.51241i |
4.7 | −1.53007 | + | 4.20384i | −0.242589 | − | 5.19049i | −9.20281 | − | 7.72208i | 4.66423 | + | 10.1610i | 22.1912 | + | 6.92202i | −7.23087 | − | 8.61742i | 15.5491 | − | 8.97729i | −26.8823 | + | 2.51831i | −49.8517 | + | 4.06069i |
4.8 | −1.49986 | + | 4.12082i | −4.97861 | − | 1.48778i | −8.60322 | − | 7.21896i | 10.8774 | + | 2.58511i | 13.5981 | − | 18.2845i | 20.6597 | + | 24.6212i | 12.2695 | − | 7.08378i | 22.5730 | + | 14.8141i | −26.9672 | + | 40.9464i |
4.9 | −1.47218 | + | 4.04479i | −4.79721 | − | 1.99668i | −8.06463 | − | 6.76703i | −9.33180 | + | 6.15772i | 15.1385 | − | 16.4642i | −11.5687 | − | 13.7870i | 9.42223 | − | 5.43993i | 19.0265 | + | 19.1570i | −11.1686 | − | 46.8104i |
4.10 | −1.37958 | + | 3.79035i | 4.90110 | − | 1.72604i | −6.33518 | − | 5.31585i | −11.0884 | − | 1.43121i | −0.219118 | + | 20.9581i | −13.3794 | − | 15.9450i | 0.943144 | − | 0.544524i | 21.0415 | − | 16.9190i | 20.7220 | − | 40.0543i |
4.11 | −1.23285 | + | 3.38722i | 1.73354 | − | 4.89845i | −3.82500 | − | 3.20956i | 7.45058 | − | 8.33600i | 14.4550 | + | 11.9109i | 1.63240 | + | 1.94541i | −9.38631 | + | 5.41919i | −20.9897 | − | 16.9833i | 19.0505 | + | 35.5138i |
4.12 | −1.15191 | + | 3.16486i | 0.627092 | + | 5.15817i | −2.56107 | − | 2.14899i | 10.8432 | − | 2.72480i | −17.0472 | − | 3.95712i | 6.28747 | + | 7.49311i | −13.5826 | + | 7.84192i | −26.2135 | + | 6.46930i | −3.86685 | + | 37.4560i |
4.13 | −1.11091 | + | 3.05220i | −2.06434 | + | 4.76849i | −1.95347 | − | 1.63916i | −10.7613 | + | 3.03229i | −12.2611 | − | 11.5982i | 8.11002 | + | 9.66515i | −15.3302 | + | 8.85092i | −18.4770 | − | 19.6876i | 2.69965 | − | 36.2142i |
4.14 | −1.00430 | + | 2.75929i | −2.84251 | − | 4.34973i | −0.476691 | − | 0.399991i | 1.21373 | − | 11.1143i | 14.8569 | − | 3.47489i | −10.7265 | − | 12.7834i | −18.7613 | + | 10.8319i | −10.8402 | + | 24.7283i | 29.4485 | + | 14.5111i |
4.15 | −0.944176 | + | 2.59410i | 3.52324 | + | 3.81927i | 0.290454 | + | 0.243719i | 0.662990 | + | 11.1607i | −13.2341 | + | 5.53358i | −19.8929 | − | 23.7074i | −20.0324 | + | 11.5657i | −2.17360 | + | 26.9124i | −29.5779 | − | 8.81777i |
4.16 | −0.935901 | + | 2.57137i | 4.16251 | − | 3.11022i | 0.392344 | + | 0.329216i | 3.40677 | + | 10.6487i | 4.10182 | + | 13.6142i | 10.1992 | + | 12.1549i | −20.1720 | + | 11.6463i | 7.65302 | − | 25.8927i | −30.5700 | − | 1.20603i |
4.17 | −0.842463 | + | 2.31465i | −4.78143 | + | 2.03418i | 1.48050 | + | 1.24229i | 10.7613 | − | 3.03227i | −0.680240 | − | 12.7811i | −19.1625 | − | 22.8370i | −21.1883 | + | 12.2330i | 18.7242 | − | 19.4526i | −2.04734 | + | 27.4632i |
4.18 | −0.716355 | + | 1.96817i | 4.76638 | + | 2.06920i | 2.76783 | + | 2.32248i | −4.23077 | − | 10.3489i | −7.48696 | + | 7.89877i | 8.31506 | + | 9.90951i | −21.0648 | + | 12.1618i | 18.4368 | + | 19.7252i | 23.3992 | − | 0.913357i |
4.19 | −0.627147 | + | 1.72307i | −4.15670 | − | 3.11798i | 3.55269 | + | 2.98106i | −8.04407 | − | 7.76486i | 7.97938 | − | 5.20687i | 15.6915 | + | 18.7004i | −20.0686 | + | 11.5866i | 7.55637 | + | 25.9211i | 18.4242 | − | 8.99080i |
4.20 | −0.617815 | + | 1.69743i | 1.85019 | − | 4.85559i | 3.62877 | + | 3.04490i | −10.9792 | + | 2.11101i | 7.09897 | + | 6.14044i | 10.3906 | + | 12.3830i | −19.9253 | + | 11.5039i | −20.1536 | − | 17.9676i | 3.19985 | − | 19.9407i |
See next 80 embeddings (of 312 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
27.e | even | 9 | 1 | inner |
135.p | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 135.4.p.a | ✓ | 312 |
5.b | even | 2 | 1 | inner | 135.4.p.a | ✓ | 312 |
27.e | even | 9 | 1 | inner | 135.4.p.a | ✓ | 312 |
135.p | even | 18 | 1 | inner | 135.4.p.a | ✓ | 312 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
135.4.p.a | ✓ | 312 | 1.a | even | 1 | 1 | trivial |
135.4.p.a | ✓ | 312 | 5.b | even | 2 | 1 | inner |
135.4.p.a | ✓ | 312 | 27.e | even | 9 | 1 | inner |
135.4.p.a | ✓ | 312 | 135.p | even | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(135, [\chi])\).