Properties

Label 27.8.e.a
Level 27
Weight 8
Character orbit 27.e
Analytic conductor 8.434
Analytic rank 0
Dimension 120
CM No

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 27.e (of order \(9\) and degree \(6\))

Newform invariants

Self dual: No
Analytic conductor: \(8.43439568807\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{9})\)
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.

\(\operatorname{Tr}(f)(q) = \) \(120q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 219q^{5} \) \(\mathstrut +\mathstrut 1386q^{6} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 4611q^{8} \) \(\mathstrut -\mathstrut 1728q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(120q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 219q^{5} \) \(\mathstrut +\mathstrut 1386q^{6} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 4611q^{8} \) \(\mathstrut -\mathstrut 1728q^{9} \) \(\mathstrut -\mathstrut 3q^{10} \) \(\mathstrut +\mathstrut 9399q^{11} \) \(\mathstrut -\mathstrut 10545q^{12} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut 16647q^{14} \) \(\mathstrut +\mathstrut 11709q^{15} \) \(\mathstrut +\mathstrut 378q^{16} \) \(\mathstrut -\mathstrut 58959q^{17} \) \(\mathstrut -\mathstrut 156051q^{18} \) \(\mathstrut -\mathstrut 3q^{19} \) \(\mathstrut +\mathstrut 240243q^{20} \) \(\mathstrut -\mathstrut 516q^{21} \) \(\mathstrut +\mathstrut 105762q^{22} \) \(\mathstrut -\mathstrut 144084q^{23} \) \(\mathstrut -\mathstrut 150894q^{24} \) \(\mathstrut -\mathstrut 107997q^{25} \) \(\mathstrut +\mathstrut 443202q^{26} \) \(\mathstrut +\mathstrut 114363q^{27} \) \(\mathstrut -\mathstrut 12q^{28} \) \(\mathstrut +\mathstrut 107886q^{29} \) \(\mathstrut -\mathstrut 179901q^{30} \) \(\mathstrut +\mathstrut 297975q^{31} \) \(\mathstrut -\mathstrut 639576q^{32} \) \(\mathstrut -\mathstrut 31338q^{33} \) \(\mathstrut -\mathstrut 265806q^{34} \) \(\mathstrut -\mathstrut 450672q^{35} \) \(\mathstrut +\mathstrut 51264q^{36} \) \(\mathstrut -\mathstrut 3q^{37} \) \(\mathstrut -\mathstrut 145596q^{38} \) \(\mathstrut +\mathstrut 1919661q^{39} \) \(\mathstrut +\mathstrut 283521q^{40} \) \(\mathstrut +\mathstrut 227085q^{41} \) \(\mathstrut +\mathstrut 3792042q^{42} \) \(\mathstrut -\mathstrut 1080969q^{43} \) \(\mathstrut -\mathstrut 1205115q^{44} \) \(\mathstrut -\mathstrut 4430925q^{45} \) \(\mathstrut -\mathstrut 3q^{46} \) \(\mathstrut -\mathstrut 5184339q^{47} \) \(\mathstrut -\mathstrut 2424039q^{48} \) \(\mathstrut +\mathstrut 976008q^{49} \) \(\mathstrut +\mathstrut 2324559q^{50} \) \(\mathstrut +\mathstrut 2918034q^{51} \) \(\mathstrut -\mathstrut 3366135q^{52} \) \(\mathstrut +\mathstrut 5631750q^{53} \) \(\mathstrut +\mathstrut 11971422q^{54} \) \(\mathstrut -\mathstrut 12q^{55} \) \(\mathstrut +\mathstrut 5254923q^{56} \) \(\mathstrut -\mathstrut 284637q^{57} \) \(\mathstrut +\mathstrut 3884295q^{58} \) \(\mathstrut -\mathstrut 12534492q^{59} \) \(\mathstrut -\mathstrut 29390886q^{60} \) \(\mathstrut +\mathstrut 3865440q^{61} \) \(\mathstrut -\mathstrut 9923646q^{62} \) \(\mathstrut +\mathstrut 3292389q^{63} \) \(\mathstrut -\mathstrut 10223619q^{64} \) \(\mathstrut +\mathstrut 17158395q^{65} \) \(\mathstrut +\mathstrut 39100653q^{66} \) \(\mathstrut +\mathstrut 5567061q^{67} \) \(\mathstrut +\mathstrut 19858455q^{68} \) \(\mathstrut -\mathstrut 12682935q^{69} \) \(\mathstrut -\mathstrut 1020993q^{70} \) \(\mathstrut -\mathstrut 18563877q^{71} \) \(\mathstrut -\mathstrut 33138540q^{72} \) \(\mathstrut +\mathstrut 3770934q^{73} \) \(\mathstrut -\mathstrut 36278085q^{74} \) \(\mathstrut -\mathstrut 26317941q^{75} \) \(\mathstrut -\mathstrut 1156998q^{76} \) \(\mathstrut +\mathstrut 8471751q^{77} \) \(\mathstrut +\mathstrut 36593658q^{78} \) \(\mathstrut +\mathstrut 16415526q^{79} \) \(\mathstrut +\mathstrut 101080590q^{80} \) \(\mathstrut +\mathstrut 45736524q^{81} \) \(\mathstrut -\mathstrut 12q^{82} \) \(\mathstrut +\mathstrut 3147699q^{83} \) \(\mathstrut -\mathstrut 27588054q^{84} \) \(\mathstrut -\mathstrut 7672131q^{85} \) \(\mathstrut -\mathstrut 70854834q^{86} \) \(\mathstrut -\mathstrut 55384731q^{87} \) \(\mathstrut -\mathstrut 34323078q^{88} \) \(\mathstrut -\mathstrut 54647613q^{89} \) \(\mathstrut -\mathstrut 25256970q^{90} \) \(\mathstrut +\mathstrut 3315066q^{91} \) \(\mathstrut +\mathstrut 73858515q^{92} \) \(\mathstrut +\mathstrut 69808065q^{93} \) \(\mathstrut +\mathstrut 22126539q^{94} \) \(\mathstrut +\mathstrut 32541447q^{95} \) \(\mathstrut +\mathstrut 33014034q^{96} \) \(\mathstrut +\mathstrut 41250117q^{97} \) \(\mathstrut -\mathstrut 88493274q^{98} \) \(\mathstrut -\mathstrut 111262167q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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.

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 990-1000
Embeddings: e.g. 1-3 or 25.20
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

There are no other newforms in \(S_{8}^{\mathrm{new}}(27, [\chi])\).