Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [27,6,Mod(4,27)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(27, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("27.4");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 27 = 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 27.e (of order \(9\), 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: | \(4.33036313495\) |
Analytic rank: | \(0\) |
Dimension: | \(84\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | −9.69504 | − | 3.52870i | −5.72592 | + | 14.4988i | 57.0285 | + | 47.8526i | 2.39953 | − | 13.6084i | 106.675 | − | 120.361i | −83.8831 | + | 70.3863i | −218.960 | − | 379.250i | −177.428 | − | 166.037i | −71.2837 | + | 123.467i |
4.2 | −8.79489 | − | 3.20108i | 13.9939 | − | 6.86812i | 42.5898 | + | 35.7371i | −11.0981 | + | 62.9404i | −145.060 | + | 15.6088i | 158.627 | − | 133.104i | −110.426 | − | 191.263i | 148.658 | − | 192.223i | 299.084 | − | 518.028i |
4.3 | −7.30121 | − | 2.65742i | −0.728957 | − | 15.5714i | 21.7323 | + | 18.2356i | 15.2097 | − | 86.2586i | −36.0575 | + | 115.627i | −90.4944 | + | 75.9338i | 14.1040 | + | 24.4288i | −241.937 | + | 22.7018i | −340.275 | + | 589.373i |
4.4 | −5.55790 | − | 2.02291i | −15.5209 | − | 1.44989i | 2.28464 | + | 1.91704i | −3.90711 | + | 22.1583i | 83.3305 | + | 39.4557i | 69.5798 | − | 58.3844i | 85.8137 | + | 148.634i | 238.796 | + | 45.0070i | 66.5396 | − | 115.250i |
4.5 | −4.77365 | − | 1.73746i | 11.5896 | + | 10.4250i | −4.74451 | − | 3.98112i | −2.77896 | + | 15.7603i | −37.2117 | − | 69.9018i | −95.3445 | + | 80.0035i | 97.0117 | + | 168.029i | 25.6389 | + | 241.644i | 40.6487 | − | 70.4057i |
4.6 | −1.14127 | − | 0.415390i | −3.98171 | + | 15.0714i | −23.3835 | − | 19.6211i | 10.2948 | − | 58.3849i | 10.8047 | − | 15.5466i | 144.334 | − | 121.111i | 37.9689 | + | 65.7640i | −211.292 | − | 120.020i | −36.0017 | + | 62.3568i |
4.7 | −0.752106 | − | 0.273744i | −2.15220 | − | 15.4392i | −24.0227 | − | 20.1574i | −15.6402 | + | 88.6998i | −2.60770 | + | 12.2010i | −70.2017 | + | 58.9062i | 25.3556 | + | 43.9172i | −233.736 | + | 66.4565i | 36.0441 | − | 62.4302i |
4.8 | 0.224938 | + | 0.0818707i | 14.5788 | − | 5.51901i | −24.4695 | − | 20.5324i | 7.42390 | − | 42.1031i | 3.73116 | − | 0.0478599i | 7.97190 | − | 6.68922i | −7.65311 | − | 13.2556i | 182.081 | − | 160.921i | 5.11693 | − | 8.86277i |
4.9 | 3.25396 | + | 1.18435i | −12.0549 | + | 9.88326i | −15.3278 | − | 12.8616i | −5.36783 | + | 30.4425i | −50.9315 | + | 17.8826i | −155.790 | + | 130.724i | −90.0483 | − | 155.968i | 47.6424 | − | 238.284i | −53.5211 | + | 92.7013i |
4.10 | 4.64663 | + | 1.69124i | −11.1566 | − | 10.8872i | −5.78253 | − | 4.85212i | 9.66519 | − | 54.8140i | −33.4278 | − | 69.4570i | 60.6928 | − | 50.9273i | −97.7806 | − | 169.361i | 5.93943 | + | 242.927i | 137.614 | − | 238.354i |
4.11 | 5.36004 | + | 1.95090i | 9.76327 | + | 12.1523i | 0.410645 | + | 0.344572i | −14.4659 | + | 82.0402i | 28.6236 | + | 84.1840i | 131.197 | − | 110.088i | −89.7358 | − | 155.427i | −52.3572 | + | 237.292i | −237.590 | + | 411.517i |
4.12 | 8.01235 | + | 2.91626i | 10.1819 | − | 11.8038i | 31.1798 | + | 26.1630i | −3.30454 | + | 18.7410i | 116.004 | − | 64.8829i | −6.74337 | + | 5.65836i | 37.1007 | + | 64.2603i | −35.6577 | − | 240.370i | −81.1307 | + | 140.523i |
4.13 | 8.47173 | + | 3.08346i | 8.38757 | + | 13.1396i | 37.7491 | + | 31.6753i | 17.5895 | − | 99.7549i | 30.5419 | + | 137.178i | −143.917 | + | 120.761i | 77.8841 | + | 134.899i | −102.297 | + | 220.418i | 456.604 | − | 790.861i |
4.14 | 10.0391 | + | 3.65394i | −15.3240 | + | 2.85912i | 62.9193 | + | 52.7956i | −7.04748 | + | 39.9683i | −164.287 | − | 27.2900i | 72.7973 | − | 61.0842i | 267.808 | + | 463.857i | 226.651 | − | 87.6263i | −216.792 | + | 375.495i |
7.1 | −9.69504 | + | 3.52870i | −5.72592 | − | 14.4988i | 57.0285 | − | 47.8526i | 2.39953 | + | 13.6084i | 106.675 | + | 120.361i | −83.8831 | − | 70.3863i | −218.960 | + | 379.250i | −177.428 | + | 166.037i | −71.2837 | − | 123.467i |
7.2 | −8.79489 | + | 3.20108i | 13.9939 | + | 6.86812i | 42.5898 | − | 35.7371i | −11.0981 | − | 62.9404i | −145.060 | − | 15.6088i | 158.627 | + | 133.104i | −110.426 | + | 191.263i | 148.658 | + | 192.223i | 299.084 | + | 518.028i |
7.3 | −7.30121 | + | 2.65742i | −0.728957 | + | 15.5714i | 21.7323 | − | 18.2356i | 15.2097 | + | 86.2586i | −36.0575 | − | 115.627i | −90.4944 | − | 75.9338i | 14.1040 | − | 24.4288i | −241.937 | − | 22.7018i | −340.275 | − | 589.373i |
7.4 | −5.55790 | + | 2.02291i | −15.5209 | + | 1.44989i | 2.28464 | − | 1.91704i | −3.90711 | − | 22.1583i | 83.3305 | − | 39.4557i | 69.5798 | + | 58.3844i | 85.8137 | − | 148.634i | 238.796 | − | 45.0070i | 66.5396 | + | 115.250i |
7.5 | −4.77365 | + | 1.73746i | 11.5896 | − | 10.4250i | −4.74451 | + | 3.98112i | −2.77896 | − | 15.7603i | −37.2117 | + | 69.9018i | −95.3445 | − | 80.0035i | 97.0117 | − | 168.029i | 25.6389 | − | 241.644i | 40.6487 | + | 70.4057i |
7.6 | −1.14127 | + | 0.415390i | −3.98171 | − | 15.0714i | −23.3835 | + | 19.6211i | 10.2948 | + | 58.3849i | 10.8047 | + | 15.5466i | 144.334 | + | 121.111i | 37.9689 | − | 65.7640i | −211.292 | + | 120.020i | −36.0017 | − | 62.3568i |
See all 84 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 27.6.e.a | ✓ | 84 |
3.b | odd | 2 | 1 | 81.6.e.a | 84 | ||
27.e | even | 9 | 1 | inner | 27.6.e.a | ✓ | 84 |
27.e | even | 9 | 1 | 729.6.a.c | 42 | ||
27.f | odd | 18 | 1 | 81.6.e.a | 84 | ||
27.f | odd | 18 | 1 | 729.6.a.e | 42 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
27.6.e.a | ✓ | 84 | 1.a | even | 1 | 1 | trivial |
27.6.e.a | ✓ | 84 | 27.e | even | 9 | 1 | inner |
81.6.e.a | 84 | 3.b | odd | 2 | 1 | ||
81.6.e.a | 84 | 27.f | odd | 18 | 1 | ||
729.6.a.c | 42 | 27.e | even | 9 | 1 | ||
729.6.a.e | 42 | 27.f | odd | 18 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(27, [\chi])\).