Properties

Label 27.10.e.a
Level 27
Weight 10
Character orbit 27.e
Analytic conductor 13.906
Analytic rank 0
Dimension 156
CM No

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 27.e (of order \(9\) and degree \(6\))

Newform invariants

Self dual: No
Analytic conductor: \(13.9059675764\)
Analytic rank: \(0\)
Dimension: \(156\)
Relative dimension: \(26\) 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.

\(\operatorname{Tr}(f)(q) = \) \(156q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut +\mathstrut 2382q^{5} \) \(\mathstrut -\mathstrut 6822q^{6} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 36861q^{8} \) \(\mathstrut +\mathstrut 2538q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(156q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut +\mathstrut 2382q^{5} \) \(\mathstrut -\mathstrut 6822q^{6} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 36861q^{8} \) \(\mathstrut +\mathstrut 2538q^{9} \) \(\mathstrut -\mathstrut 3q^{10} \) \(\mathstrut -\mathstrut 121767q^{11} \) \(\mathstrut +\mathstrut 319935q^{12} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 720417q^{14} \) \(\mathstrut -\mathstrut 706356q^{15} \) \(\mathstrut +\mathstrut 1530q^{16} \) \(\mathstrut +\mathstrut 1002249q^{17} \) \(\mathstrut +\mathstrut 922437q^{18} \) \(\mathstrut -\mathstrut 3q^{19} \) \(\mathstrut -\mathstrut 8448573q^{20} \) \(\mathstrut -\mathstrut 3497394q^{21} \) \(\mathstrut -\mathstrut 1585014q^{22} \) \(\mathstrut +\mathstrut 9316482q^{23} \) \(\mathstrut +\mathstrut 7959258q^{24} \) \(\mathstrut +\mathstrut 2192178q^{25} \) \(\mathstrut -\mathstrut 33129678q^{26} \) \(\mathstrut -\mathstrut 3148155q^{27} \) \(\mathstrut -\mathstrut 12q^{28} \) \(\mathstrut +\mathstrut 19657236q^{29} \) \(\mathstrut +\mathstrut 21174507q^{30} \) \(\mathstrut -\mathstrut 4633692q^{31} \) \(\mathstrut -\mathstrut 37550952q^{32} \) \(\mathstrut -\mathstrut 19055889q^{33} \) \(\mathstrut +\mathstrut 14341770q^{34} \) \(\mathstrut +\mathstrut 34449663q^{35} \) \(\mathstrut +\mathstrut 137733552q^{36} \) \(\mathstrut -\mathstrut 3q^{37} \) \(\mathstrut -\mathstrut 92022060q^{38} \) \(\mathstrut -\mathstrut 89330160q^{39} \) \(\mathstrut +\mathstrut 6645801q^{40} \) \(\mathstrut -\mathstrut 33127221q^{41} \) \(\mathstrut +\mathstrut 227141658q^{42} \) \(\mathstrut +\mathstrut 2356203q^{43} \) \(\mathstrut +\mathstrut 1001541q^{44} \) \(\mathstrut -\mathstrut 78992532q^{45} \) \(\mathstrut -\mathstrut 3q^{46} \) \(\mathstrut -\mathstrut 33488952q^{47} \) \(\mathstrut +\mathstrut 248865393q^{48} \) \(\mathstrut -\mathstrut 93214644q^{49} \) \(\mathstrut -\mathstrut 72027753q^{50} \) \(\mathstrut -\mathstrut 101541537q^{51} \) \(\mathstrut +\mathstrut 160531713q^{52} \) \(\mathstrut +\mathstrut 4841352q^{53} \) \(\mathstrut -\mathstrut 315431874q^{54} \) \(\mathstrut -\mathstrut 12q^{55} \) \(\mathstrut -\mathstrut 91770981q^{56} \) \(\mathstrut -\mathstrut 103852992q^{57} \) \(\mathstrut -\mathstrut 320682489q^{58} \) \(\mathstrut +\mathstrut 538090854q^{59} \) \(\mathstrut +\mathstrut 1127571210q^{60} \) \(\mathstrut +\mathstrut 36683436q^{61} \) \(\mathstrut -\mathstrut 55862958q^{62} \) \(\mathstrut -\mathstrut 419972706q^{63} \) \(\mathstrut -\mathstrut 956301315q^{64} \) \(\mathstrut -\mathstrut 852347310q^{65} \) \(\mathstrut +\mathstrut 472205277q^{66} \) \(\mathstrut -\mathstrut 109026897q^{67} \) \(\mathstrut -\mathstrut 508886217q^{68} \) \(\mathstrut -\mathstrut 1234413090q^{69} \) \(\mathstrut -\mathstrut 815297649q^{70} \) \(\mathstrut +\mathstrut 940878639q^{71} \) \(\mathstrut +\mathstrut 2831326452q^{72} \) \(\mathstrut -\mathstrut 221849931q^{73} \) \(\mathstrut +\mathstrut 838884579q^{74} \) \(\mathstrut -\mathstrut 226667796q^{75} \) \(\mathstrut +\mathstrut 908401146q^{76} \) \(\mathstrut -\mathstrut 967811358q^{77} \) \(\mathstrut -\mathstrut 1247881230q^{78} \) \(\mathstrut -\mathstrut 1228059150q^{79} \) \(\mathstrut -\mathstrut 2497115154q^{80} \) \(\mathstrut -\mathstrut 392600970q^{81} \) \(\mathstrut -\mathstrut 12q^{82} \) \(\mathstrut -\mathstrut 297703284q^{83} \) \(\mathstrut +\mathstrut 1027656618q^{84} \) \(\mathstrut +\mathstrut 1535949369q^{85} \) \(\mathstrut +\mathstrut 2930201574q^{86} \) \(\mathstrut +\mathstrut 2687612040q^{87} \) \(\mathstrut +\mathstrut 2837227002q^{88} \) \(\mathstrut +\mathstrut 1774997208q^{89} \) \(\mathstrut +\mathstrut 7127326206q^{90} \) \(\mathstrut +\mathstrut 551343507q^{91} \) \(\mathstrut -\mathstrut 6896645517q^{92} \) \(\mathstrut -\mathstrut 7299151842q^{93} \) \(\mathstrut +\mathstrut 414852915q^{94} \) \(\mathstrut -\mathstrut 9000269169q^{95} \) \(\mathstrut -\mathstrut 2644638174q^{96} \) \(\mathstrut +\mathstrut 3371772333q^{97} \) \(\mathstrut +\mathstrut 6901039926q^{98} \) \(\mathstrut +\mathstrut 5248107342q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −40.0613 14.5811i −125.031 + 63.6419i 1000.08 + 839.170i −306.059 + 1735.74i 5936.87 726.489i 764.835 641.773i −16914.7 29297.2i 11582.4 15914.4i 37570.2 65073.5i
4.2 −39.3878 14.3360i 29.9043 137.072i 953.662 + 800.217i 27.4469 155.659i −3142.92 + 4970.25i 1724.19 1446.77i −15360.3 26604.9i −17894.5 8198.08i −3312.60 + 5737.59i
4.3 −34.7751 12.6571i 37.7394 + 135.125i 656.889 + 551.196i 233.333 1323.30i 397.899 5176.65i −1098.75 + 921.961i −6393.09 11073.2i −16834.5 + 10199.1i −24863.3 + 43064.5i
4.4 −31.5968 11.5003i 132.307 + 46.6679i 473.884 + 397.636i −451.726 + 2561.87i −3643.77 2996.12i −2730.60 + 2291.24i −1792.39 3104.51i 15327.2 + 12349.0i 43735.3 75751.7i
4.5 −29.1224 10.5997i −114.519 81.0463i 343.545 + 288.269i 182.505 1035.04i 2475.99 + 3574.12i −6317.76 + 5301.23i 984.496 + 1705.20i 6546.00 + 18562.6i −16286.1 + 28208.3i
4.6 −28.3865 10.3319i 136.675 31.6686i 306.833 + 257.464i 180.775 1025.22i −4206.93 513.146i 4261.26 3575.62i 1683.47 + 2915.86i 17677.2 8656.62i −15724.0 + 27234.8i
4.7 −22.2970 8.11544i −117.506 + 76.6512i 39.0810 + 32.7929i 151.895 861.438i 3242.08 755.482i 4408.97 3699.56i 5469.10 + 9472.76i 7932.18 18013.9i −10377.8 + 17974.8i
4.8 −20.2233 7.36069i −95.1961 103.057i −37.4118 31.3922i −358.438 + 2032.80i 1166.61 + 2784.86i 8116.71 6810.73i 6034.95 + 10452.8i −1558.40 + 19621.2i 22211.6 38471.7i
4.9 −15.0855 5.49066i 69.4935 121.876i −194.791 163.449i −120.371 + 682.656i −1717.52 + 1456.99i −4765.46 + 3998.70i 6150.79 + 10653.5i −10024.3 16939.1i 5564.08 9637.27i
4.10 −12.9570 4.71595i −37.0275 + 135.322i −246.572 206.898i −222.415 + 1261.38i 1117.94 1578.74i −4998.24 + 4194.02i 5748.96 + 9957.50i −16940.9 10021.3i 8830.42 15294.7i
4.11 −6.47952 2.35835i 83.5586 + 112.699i −355.792 298.545i −63.8763 + 362.261i −275.637 927.294i 6593.75 5532.82i 3366.50 + 5830.96i −5718.93 + 18833.9i 1268.23 2196.63i
4.12 −4.86289 1.76995i 133.111 + 44.3232i −371.700 311.893i 375.168 2127.68i −568.853 451.138i −8670.18 + 7275.14i 2580.30 + 4469.20i 15753.9 + 11799.8i −5590.29 + 9682.67i
4.13 −4.57613 1.66557i −24.3571 138.166i −374.048 313.863i 461.629 2618.03i −118.664 + 672.832i 5320.32 4464.28i 2435.60 + 4218.58i −18496.5 + 6730.62i −6472.99 + 11211.5i
4.14 3.93000 + 1.43040i −134.904 38.5223i −378.816 317.864i −245.629 + 1393.03i −475.070 344.359i −4348.21 + 3648.58i −2104.72 3645.48i 16715.1 + 10393.6i −2957.92 + 5123.27i
4.15 4.82322 + 1.75551i −137.543 + 27.6597i −372.033 312.173i 203.603 1154.69i −711.954 108.048i −440.948 + 369.999i −2560.36 4434.68i 18152.9 7608.76i 3009.09 5211.90i
4.16 8.90982 + 3.24291i 132.389 46.4332i −323.346 271.320i −128.517 + 728.857i 1330.14 + 15.6152i 4124.79 3461.11i −4428.39 7670.20i 15370.9 12294.5i −3508.68 + 6077.21i
4.17 13.5174 + 4.91993i −10.0180 139.938i −233.700 196.098i −142.171 + 806.289i 553.068 1940.89i −461.968 + 387.637i −5876.77 10178.9i −19482.3 + 2803.80i −5888.66 + 10199.5i
4.18 17.4636 + 6.35623i −17.1474 + 139.244i −127.639 107.102i 280.318 1589.76i −1184.52 + 2322.71i 610.601 512.355i −6305.87 10922.1i −19094.9 4775.35i 15000.2 25981.2i
4.19 23.5230 + 8.56168i −83.7573 + 112.551i 87.8153 + 73.6858i −425.267 + 2411.81i −2933.85 + 1930.43i 3548.49 2977.54i −4973.56 8614.45i −5652.42 18853.9i −30652.7 + 53092.0i
4.20 23.7357 + 8.63908i 115.700 + 79.3502i 96.5341 + 81.0017i −237.448 + 1346.63i 2060.71 + 2882.98i −5786.38 + 4855.35i −4874.78 8443.36i 7090.08 + 18361.7i −17269.7 + 29911.9i
See next 80 embeddings (of 156 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.26
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 \(S_{10}^{\mathrm{new}}(27, [\chi])\).