Properties

Label 27.8.e.a
Level $27$
Weight $8$
Character orbit 27.e
Analytic conductor $8.434$
Analytic rank $0$
Dimension $120$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,8,Mod(4,27)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// 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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9} - 3 q^{10} + 9399 q^{11} - 10545 q^{12} - 6 q^{13} + 16647 q^{14} + 11709 q^{15} + 378 q^{16} - 58959 q^{17}+ \cdots - 111262167 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 4.20
Significant digits:
Format:

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 \(S_{8}^{\mathrm{new}}(27, [\chi])\).