Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [27,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("27.4");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 27 = 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
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: | \(8.43439568807\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(20\) 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 | −20.1276 | − | 7.32584i | 46.7116 | − | 2.24141i | 253.398 | + | 212.626i | 26.6624 | − | 151.210i | −956.612 | − | 297.088i | −1123.74 | + | 942.932i | −2171.79 | − | 3761.65i | 2176.95 | − | 209.400i | −1644.39 | + | 2848.17i |
4.2 | −19.1819 | − | 6.98163i | −45.3621 | − | 11.3703i | 221.148 | + | 185.565i | 73.2555 | − | 415.452i | 790.746 | + | 534.805i | 1090.38 | − | 914.934i | −1640.05 | − | 2840.66i | 1928.43 | + | 1031.56i | −4305.71 | + | 7457.72i |
4.3 | −16.4594 | − | 5.99073i | 7.02376 | + | 46.2349i | 136.969 | + | 114.931i | −40.0449 | + | 227.106i | 161.374 | − | 803.077i | 882.649 | − | 740.631i | −444.907 | − | 770.602i | −2088.33 | + | 649.486i | 2019.65 | − | 3498.13i |
4.4 | −15.5171 | − | 5.64776i | −4.90318 | − | 46.5076i | 110.829 | + | 92.9968i | −48.1497 | + | 273.071i | −186.581 | + | 749.355i | −92.9707 | + | 78.0117i | −137.696 | − | 238.496i | −2138.92 | + | 456.070i | 2289.38 | − | 3965.33i |
4.5 | −13.9790 | − | 5.08793i | −44.6310 | + | 13.9670i | 71.4707 | + | 59.9711i | −48.7897 | + | 276.700i | 694.958 | + | 31.8356i | −1113.53 | + | 934.359i | 258.112 | + | 447.063i | 1796.85 | − | 1246.72i | 2089.86 | − | 3619.74i |
4.6 | −9.62864 | − | 3.50454i | 31.7200 | − | 34.3634i | −17.6248 | − | 14.7890i | 40.7285 | − | 230.983i | −425.848 | + | 219.708i | 500.665 | − | 420.108i | 773.656 | + | 1340.01i | −174.680 | − | 2180.01i | −1201.65 | + | 2081.32i |
4.7 | −9.32840 | − | 3.39526i | 39.8042 | + | 24.5484i | −22.5624 | − | 18.9321i | 12.2615 | − | 69.5386i | −287.961 | − | 364.143i | 48.0590 | − | 40.3263i | 781.525 | + | 1353.64i | 981.750 | + | 1954.26i | −350.482 | + | 607.053i |
4.8 | −9.04432 | − | 3.29186i | −20.3312 | + | 42.1146i | −27.0904 | − | 22.7315i | 82.6399 | − | 468.674i | 322.518 | − | 313.970i | −558.283 | + | 468.455i | 786.170 | + | 1361.69i | −1360.28 | − | 1712.48i | −2290.23 | + | 3966.80i |
4.9 | −2.79327 | − | 1.01667i | −34.1687 | − | 31.9296i | −91.2849 | − | 76.5971i | 17.7337 | − | 100.573i | 62.9806 | + | 123.926i | −71.4548 | + | 59.9577i | 367.352 | + | 636.273i | 147.997 | + | 2181.99i | −151.785 | + | 262.899i |
4.10 | −2.03404 | − | 0.740329i | 45.5454 | − | 10.6122i | −94.4645 | − | 79.2651i | −90.6912 | + | 514.336i | −100.497 | − | 12.1330i | −162.436 | + | 136.300i | 271.995 | + | 471.109i | 1961.76 | − | 966.672i | 565.246 | − | 979.036i |
4.11 | −0.493529 | − | 0.179630i | −38.9037 | + | 25.9520i | −97.8424 | − | 82.0995i | −47.2754 | + | 268.112i | 23.8618 | − | 5.81978i | 932.688 | − | 782.618i | 67.1535 | + | 116.313i | 839.991 | − | 2019.25i | 71.4927 | − | 123.829i |
4.12 | 4.29826 | + | 1.56444i | 11.1293 | + | 45.4218i | −82.0261 | − | 68.8281i | −20.3422 | + | 115.366i | −23.2230 | + | 212.646i | −610.300 | + | 512.103i | −537.635 | − | 931.211i | −1939.28 | + | 1011.02i | −267.920 | + | 464.051i |
4.13 | 6.42719 | + | 2.33931i | 21.8338 | − | 41.3556i | −62.2172 | − | 52.2064i | 40.2664 | − | 228.362i | 237.073 | − | 214.725i | −1346.32 | + | 1129.70i | −715.495 | − | 1239.27i | −1233.57 | − | 1805.90i | 793.009 | − | 1373.53i |
4.14 | 7.46251 | + | 2.71613i | 42.7022 | + | 19.0664i | −49.7420 | − | 41.7385i | 71.4552 | − | 405.243i | 266.878 | + | 258.268i | 898.235 | − | 753.709i | −766.085 | − | 1326.90i | 1459.95 | + | 1628.35i | 1633.93 | − | 2830.04i |
4.15 | 9.90100 | + | 3.60367i | 7.82827 | − | 46.1055i | −13.0104 | − | 10.9170i | −29.9915 | + | 170.090i | 243.657 | − | 428.280i | 1223.81 | − | 1026.90i | −763.805 | − | 1322.95i | −2064.44 | − | 721.852i | −909.893 | + | 1575.98i |
4.16 | 11.7328 | + | 4.27037i | −44.8117 | + | 13.3756i | 21.3678 | + | 17.9297i | 58.4278 | − | 331.360i | −582.884 | − | 34.4297i | −23.9260 | + | 20.0763i | −624.951 | − | 1082.45i | 1829.18 | − | 1198.77i | 2100.55 | − | 3638.26i |
4.17 | 14.3397 | + | 5.21922i | −40.6699 | − | 23.0860i | 80.3327 | + | 67.4072i | −74.8052 | + | 424.241i | −462.703 | − | 543.310i | −616.615 | + | 517.401i | −176.505 | − | 305.716i | 1121.08 | + | 1877.81i | −3286.89 | + | 5693.06i |
4.18 | 16.7921 | + | 6.11184i | 46.7511 | − | 1.15397i | 146.567 | + | 122.985i | −21.3907 | + | 121.313i | 792.104 | + | 266.358i | −285.632 | + | 239.674i | 565.850 | + | 980.080i | 2184.34 | − | 107.899i | −1100.64 | + | 1906.36i |
4.19 | 17.6042 | + | 6.40739i | −7.16273 | + | 46.2136i | 170.798 | + | 143.317i | −14.0863 | + | 79.8875i | −422.203 | + | 767.657i | 245.838 | − | 206.282i | 889.502 | + | 1540.66i | −2084.39 | − | 662.031i | −759.849 | + | 1316.10i |
4.20 | 20.7458 | + | 7.55086i | −11.1469 | − | 45.4175i | 275.320 | + | 231.021i | 74.3769 | − | 421.812i | 111.689 | − | 1026.39i | 181.714 | − | 152.476i | 2554.39 | + | 4424.34i | −1938.49 | + | 1012.53i | 4728.06 | − | 8189.24i |
See next 80 embeddings (of 120 total) |
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 | 27.8.e.a | ✓ | 120 |
3.b | odd | 2 | 1 | 81.8.e.a | 120 | ||
27.e | even | 9 | 1 | inner | 27.8.e.a | ✓ | 120 |
27.f | odd | 18 | 1 | 81.8.e.a | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
27.8.e.a | ✓ | 120 | 1.a | even | 1 | 1 | trivial |
27.8.e.a | ✓ | 120 | 27.e | even | 9 | 1 | inner |
81.8.e.a | 120 | 3.b | odd | 2 | 1 | ||
81.8.e.a | 120 | 27.f | odd | 18 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(27, [\chi])\).