Properties

Label 27.6.e.a
Level 27
Weight 6
Character orbit 27.e
Analytic conductor 4.330
Analytic rank 0
Dimension 84
CM No

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(4.33036313495\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \(84q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 93q^{5} \) \(\mathstrut -\mathstrut 126q^{6} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 573q^{8} \) \(\mathstrut +\mathstrut 324q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(84q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 93q^{5} \) \(\mathstrut -\mathstrut 126q^{6} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 573q^{8} \) \(\mathstrut +\mathstrut 324q^{9} \) \(\mathstrut -\mathstrut 3q^{10} \) \(\mathstrut +\mathstrut 111q^{11} \) \(\mathstrut -\mathstrut 2769q^{12} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 1641q^{14} \) \(\mathstrut +\mathstrut 1989q^{15} \) \(\mathstrut +\mathstrut 90q^{16} \) \(\mathstrut +\mathstrut 3465q^{17} \) \(\mathstrut -\mathstrut 99q^{18} \) \(\mathstrut -\mathstrut 3q^{19} \) \(\mathstrut +\mathstrut 9987q^{20} \) \(\mathstrut +\mathstrut 5424q^{21} \) \(\mathstrut -\mathstrut 2850q^{22} \) \(\mathstrut -\mathstrut 7716q^{23} \) \(\mathstrut -\mathstrut 18486q^{24} \) \(\mathstrut +\mathstrut 4953q^{25} \) \(\mathstrut -\mathstrut 7806q^{26} \) \(\mathstrut -\mathstrut 8109q^{27} \) \(\mathstrut -\mathstrut 12q^{28} \) \(\mathstrut -\mathstrut 20418q^{29} \) \(\mathstrut -\mathstrut 6453q^{30} \) \(\mathstrut -\mathstrut 6657q^{31} \) \(\mathstrut +\mathstrut 51192q^{32} \) \(\mathstrut +\mathstrut 50634q^{33} \) \(\mathstrut -\mathstrut 11394q^{34} \) \(\mathstrut +\mathstrut 35868q^{35} \) \(\mathstrut -\mathstrut 3600q^{36} \) \(\mathstrut -\mathstrut 3q^{37} \) \(\mathstrut -\mathstrut 44076q^{38} \) \(\mathstrut -\mathstrut 13971q^{39} \) \(\mathstrut +\mathstrut 12441q^{40} \) \(\mathstrut -\mathstrut 79077q^{41} \) \(\mathstrut -\mathstrut 64422q^{42} \) \(\mathstrut -\mathstrut 9465q^{43} \) \(\mathstrut +\mathstrut 110757q^{44} \) \(\mathstrut +\mathstrut 92493q^{45} \) \(\mathstrut -\mathstrut 3q^{46} \) \(\mathstrut +\mathstrut 103557q^{47} \) \(\mathstrut +\mathstrut 134049q^{48} \) \(\mathstrut +\mathstrut 5484q^{49} \) \(\mathstrut -\mathstrut 105513q^{50} \) \(\mathstrut -\mathstrut 124866q^{51} \) \(\mathstrut +\mathstrut 68625q^{52} \) \(\mathstrut -\mathstrut 206406q^{53} \) \(\mathstrut -\mathstrut 350946q^{54} \) \(\mathstrut -\mathstrut 12q^{55} \) \(\mathstrut -\mathstrut 147237q^{56} \) \(\mathstrut +\mathstrut 67767q^{57} \) \(\mathstrut -\mathstrut 9753q^{58} \) \(\mathstrut +\mathstrut 116484q^{59} \) \(\mathstrut +\mathstrut 241290q^{60} \) \(\mathstrut -\mathstrut 70116q^{61} \) \(\mathstrut +\mathstrut 246066q^{62} \) \(\mathstrut +\mathstrut 135981q^{63} \) \(\mathstrut -\mathstrut 86019q^{64} \) \(\mathstrut -\mathstrut 19815q^{65} \) \(\mathstrut -\mathstrut 32931q^{66} \) \(\mathstrut +\mathstrut 48117q^{67} \) \(\mathstrut -\mathstrut 48105q^{68} \) \(\mathstrut +\mathstrut 25263q^{69} \) \(\mathstrut +\mathstrut 270111q^{70} \) \(\mathstrut +\mathstrut 279531q^{71} \) \(\mathstrut +\mathstrut 441684q^{72} \) \(\mathstrut -\mathstrut 27012q^{73} \) \(\mathstrut +\mathstrut 233691q^{74} \) \(\mathstrut +\mathstrut 18399q^{75} \) \(\mathstrut -\mathstrut 125670q^{76} \) \(\mathstrut -\mathstrut 345135q^{77} \) \(\mathstrut -\mathstrut 520974q^{78} \) \(\mathstrut -\mathstrut 216186q^{79} \) \(\mathstrut -\mathstrut 924114q^{80} \) \(\mathstrut -\mathstrut 428616q^{81} \) \(\mathstrut -\mathstrut 12q^{82} \) \(\mathstrut -\mathstrut 370401q^{83} \) \(\mathstrut -\mathstrut 522390q^{84} \) \(\mathstrut -\mathstrut 43731q^{85} \) \(\mathstrut +\mathstrut 116682q^{86} \) \(\mathstrut +\mathstrut 304173q^{87} \) \(\mathstrut +\mathstrut 371418q^{88} \) \(\mathstrut +\mathstrut 154827q^{89} \) \(\mathstrut +\mathstrut 327366q^{90} \) \(\mathstrut -\mathstrut 91002q^{91} \) \(\mathstrut +\mathstrut 1279059q^{92} \) \(\mathstrut +\mathstrut 890079q^{93} \) \(\mathstrut +\mathstrut 11667q^{94} \) \(\mathstrut +\mathstrut 1087671q^{95} \) \(\mathstrut +\mathstrut 1165698q^{96} \) \(\mathstrut -\mathstrut 420621q^{97} \) \(\mathstrut +\mathstrut 463410q^{98} \) \(\mathstrut -\mathstrut 28323q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.14
Significant digits:
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Inner twists

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

Hecke kernels

There are no other newforms in \(S_{6}^{\mathrm{new}}(27, [\chi])\).