Properties

Label 177.6.a.c.1.8
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3879361069\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + 15565376 x^{4} + 6775664 x^{3} - 75006848 x^{2} + 21512960 x + 49172480\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-3.21799\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.21799 q^{2} +9.00000 q^{3} -4.77260 q^{4} -21.6600 q^{5} +46.9619 q^{6} +150.168 q^{7} -191.879 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+5.21799 q^{2} +9.00000 q^{3} -4.77260 q^{4} -21.6600 q^{5} +46.9619 q^{6} +150.168 q^{7} -191.879 q^{8} +81.0000 q^{9} -113.022 q^{10} +527.794 q^{11} -42.9534 q^{12} +51.3772 q^{13} +783.574 q^{14} -194.940 q^{15} -848.499 q^{16} -968.342 q^{17} +422.657 q^{18} +2271.01 q^{19} +103.375 q^{20} +1351.51 q^{21} +2754.02 q^{22} +3803.24 q^{23} -1726.91 q^{24} -2655.84 q^{25} +268.086 q^{26} +729.000 q^{27} -716.691 q^{28} +7264.82 q^{29} -1017.20 q^{30} -513.041 q^{31} +1712.67 q^{32} +4750.15 q^{33} -5052.79 q^{34} -3252.64 q^{35} -386.581 q^{36} +11311.7 q^{37} +11850.1 q^{38} +462.395 q^{39} +4156.10 q^{40} +17019.0 q^{41} +7052.17 q^{42} -7359.17 q^{43} -2518.95 q^{44} -1754.46 q^{45} +19845.3 q^{46} -15367.8 q^{47} -7636.49 q^{48} +5743.39 q^{49} -13858.2 q^{50} -8715.07 q^{51} -245.203 q^{52} -32078.1 q^{53} +3803.91 q^{54} -11432.0 q^{55} -28814.1 q^{56} +20439.1 q^{57} +37907.7 q^{58} -3481.00 q^{59} +930.371 q^{60} +13858.8 q^{61} -2677.04 q^{62} +12163.6 q^{63} +36088.7 q^{64} -1112.83 q^{65} +24786.2 q^{66} -26206.8 q^{67} +4621.51 q^{68} +34229.2 q^{69} -16972.2 q^{70} -49223.6 q^{71} -15542.2 q^{72} -5312.00 q^{73} +59024.5 q^{74} -23902.6 q^{75} -10838.6 q^{76} +79257.7 q^{77} +2412.77 q^{78} -41133.2 q^{79} +18378.5 q^{80} +6561.00 q^{81} +88805.2 q^{82} +60927.7 q^{83} -6450.22 q^{84} +20974.3 q^{85} -38400.0 q^{86} +65383.3 q^{87} -101273. q^{88} +4240.18 q^{89} -9154.76 q^{90} +7715.21 q^{91} -18151.3 q^{92} -4617.37 q^{93} -80188.8 q^{94} -49190.1 q^{95} +15414.0 q^{96} -4918.71 q^{97} +29968.9 q^{98} +42751.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + O(q^{10}) \) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + 601q^{10} + 1480q^{11} + 1782q^{12} + 472q^{13} + 1065q^{14} + 1422q^{15} + 6370q^{16} + 1565q^{17} + 1782q^{18} + 3939q^{19} + 8033q^{20} + 3717q^{21} - 1738q^{22} + 7245q^{23} + 6507q^{24} + 9690q^{25} + 3764q^{26} + 8748q^{27} + 12154q^{28} + 10003q^{29} + 5409q^{30} + 7295q^{31} + 11628q^{32} + 13320q^{33} - 16344q^{34} + 11015q^{35} + 16038q^{36} + 6741q^{37} + 3035q^{38} + 4248q^{39} + 5572q^{40} + 34025q^{41} + 9585q^{42} - 6336q^{43} + 41168q^{44} + 12798q^{45} + 2345q^{46} + 66167q^{47} + 57330q^{48} + 28319q^{49} + 31173q^{50} + 14085q^{51} + 16440q^{52} + 62290q^{53} + 16038q^{54} + 55764q^{55} + 107306q^{56} + 35451q^{57} + 37952q^{58} - 41772q^{59} + 72297q^{60} + 68469q^{61} + 99190q^{62} + 33453q^{63} + 68525q^{64} + 80156q^{65} - 15642q^{66} + 113310q^{67} + 33887q^{68} + 65205q^{69} + 32034q^{70} + 84520q^{71} + 58563q^{72} + 135895q^{73} - 31962q^{74} + 87210q^{75} - 61848q^{76} - 3799q^{77} + 33876q^{78} + 14122q^{79} + 77609q^{80} + 78732q^{81} - 1501q^{82} + 114463q^{83} + 109386q^{84} - 101097q^{85} - 203536q^{86} + 90027q^{87} - 244967q^{88} + 189109q^{89} + 48681q^{90} - 168249q^{91} - 71946q^{92} + 65655q^{93} - 472284q^{94} + 21923q^{95} + 104652q^{96} - 76192q^{97} - 17544q^{98} + 119880q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.21799 0.922419 0.461209 0.887291i \(-0.347416\pi\)
0.461209 + 0.887291i \(0.347416\pi\)
\(3\) 9.00000 0.577350
\(4\) −4.77260 −0.149144
\(5\) −21.6600 −0.387466 −0.193733 0.981054i \(-0.562060\pi\)
−0.193733 + 0.981054i \(0.562060\pi\)
\(6\) 46.9619 0.532559
\(7\) 150.168 1.15833 0.579164 0.815211i \(-0.303379\pi\)
0.579164 + 0.815211i \(0.303379\pi\)
\(8\) −191.879 −1.05999
\(9\) 81.0000 0.333333
\(10\) −113.022 −0.357406
\(11\) 527.794 1.31517 0.657587 0.753379i \(-0.271577\pi\)
0.657587 + 0.753379i \(0.271577\pi\)
\(12\) −42.9534 −0.0861082
\(13\) 51.3772 0.0843165 0.0421582 0.999111i \(-0.486577\pi\)
0.0421582 + 0.999111i \(0.486577\pi\)
\(14\) 783.574 1.06846
\(15\) −194.940 −0.223704
\(16\) −848.499 −0.828612
\(17\) −968.342 −0.812655 −0.406328 0.913727i \(-0.633191\pi\)
−0.406328 + 0.913727i \(0.633191\pi\)
\(18\) 422.657 0.307473
\(19\) 2271.01 1.44323 0.721614 0.692296i \(-0.243401\pi\)
0.721614 + 0.692296i \(0.243401\pi\)
\(20\) 103.375 0.0577881
\(21\) 1351.51 0.668762
\(22\) 2754.02 1.21314
\(23\) 3803.24 1.49911 0.749556 0.661941i \(-0.230267\pi\)
0.749556 + 0.661941i \(0.230267\pi\)
\(24\) −1726.91 −0.611986
\(25\) −2655.84 −0.849870
\(26\) 268.086 0.0777751
\(27\) 729.000 0.192450
\(28\) −716.691 −0.172757
\(29\) 7264.82 1.60409 0.802047 0.597261i \(-0.203744\pi\)
0.802047 + 0.597261i \(0.203744\pi\)
\(30\) −1017.20 −0.206348
\(31\) −513.041 −0.0958843 −0.0479422 0.998850i \(-0.515266\pi\)
−0.0479422 + 0.998850i \(0.515266\pi\)
\(32\) 1712.67 0.295664
\(33\) 4750.15 0.759316
\(34\) −5052.79 −0.749608
\(35\) −3252.64 −0.448813
\(36\) −386.581 −0.0497146
\(37\) 11311.7 1.35839 0.679196 0.733957i \(-0.262329\pi\)
0.679196 + 0.733957i \(0.262329\pi\)
\(38\) 11850.1 1.33126
\(39\) 462.395 0.0486801
\(40\) 4156.10 0.410711
\(41\) 17019.0 1.58116 0.790579 0.612360i \(-0.209780\pi\)
0.790579 + 0.612360i \(0.209780\pi\)
\(42\) 7052.17 0.616878
\(43\) −7359.17 −0.606956 −0.303478 0.952838i \(-0.598148\pi\)
−0.303478 + 0.952838i \(0.598148\pi\)
\(44\) −2518.95 −0.196150
\(45\) −1754.46 −0.129155
\(46\) 19845.3 1.38281
\(47\) −15367.8 −1.01477 −0.507383 0.861720i \(-0.669387\pi\)
−0.507383 + 0.861720i \(0.669387\pi\)
\(48\) −7636.49 −0.478400
\(49\) 5743.39 0.341726
\(50\) −13858.2 −0.783936
\(51\) −8715.07 −0.469187
\(52\) −245.203 −0.0125753
\(53\) −32078.1 −1.56862 −0.784312 0.620367i \(-0.786984\pi\)
−0.784312 + 0.620367i \(0.786984\pi\)
\(54\) 3803.91 0.177520
\(55\) −11432.0 −0.509585
\(56\) −28814.1 −1.22782
\(57\) 20439.1 0.833248
\(58\) 37907.7 1.47965
\(59\) −3481.00 −0.130189
\(60\) 930.371 0.0333640
\(61\) 13858.8 0.476872 0.238436 0.971158i \(-0.423365\pi\)
0.238436 + 0.971158i \(0.423365\pi\)
\(62\) −2677.04 −0.0884455
\(63\) 12163.6 0.386110
\(64\) 36088.7 1.10134
\(65\) −1112.83 −0.0326698
\(66\) 24786.2 0.700407
\(67\) −26206.8 −0.713224 −0.356612 0.934252i \(-0.616068\pi\)
−0.356612 + 0.934252i \(0.616068\pi\)
\(68\) 4621.51 0.121202
\(69\) 34229.2 0.865513
\(70\) −16972.2 −0.413994
\(71\) −49223.6 −1.15885 −0.579426 0.815025i \(-0.696723\pi\)
−0.579426 + 0.815025i \(0.696723\pi\)
\(72\) −15542.2 −0.353331
\(73\) −5312.00 −0.116668 −0.0583339 0.998297i \(-0.518579\pi\)
−0.0583339 + 0.998297i \(0.518579\pi\)
\(74\) 59024.5 1.25301
\(75\) −23902.6 −0.490673
\(76\) −10838.6 −0.215248
\(77\) 79257.7 1.52340
\(78\) 2412.77 0.0449035
\(79\) −41133.2 −0.741524 −0.370762 0.928728i \(-0.620903\pi\)
−0.370762 + 0.928728i \(0.620903\pi\)
\(80\) 18378.5 0.321059
\(81\) 6561.00 0.111111
\(82\) 88805.2 1.45849
\(83\) 60927.7 0.970777 0.485388 0.874299i \(-0.338678\pi\)
0.485388 + 0.874299i \(0.338678\pi\)
\(84\) −6450.22 −0.0997416
\(85\) 20974.3 0.314876
\(86\) −38400.0 −0.559868
\(87\) 65383.3 0.926124
\(88\) −101273. −1.39407
\(89\) 4240.18 0.0567426 0.0283713 0.999597i \(-0.490968\pi\)
0.0283713 + 0.999597i \(0.490968\pi\)
\(90\) −9154.76 −0.119135
\(91\) 7715.21 0.0976662
\(92\) −18151.3 −0.223583
\(93\) −4617.37 −0.0553588
\(94\) −80188.8 −0.936040
\(95\) −49190.1 −0.559202
\(96\) 15414.0 0.170702
\(97\) −4918.71 −0.0530789 −0.0265395 0.999648i \(-0.508449\pi\)
−0.0265395 + 0.999648i \(0.508449\pi\)
\(98\) 29968.9 0.315214
\(99\) 42751.3 0.438391
\(100\) 12675.3 0.126753
\(101\) −152708. −1.48956 −0.744780 0.667310i \(-0.767446\pi\)
−0.744780 + 0.667310i \(0.767446\pi\)
\(102\) −45475.2 −0.432787
\(103\) 104142. 0.967237 0.483619 0.875279i \(-0.339322\pi\)
0.483619 + 0.875279i \(0.339322\pi\)
\(104\) −9858.21 −0.0893747
\(105\) −29273.7 −0.259122
\(106\) −167383. −1.44693
\(107\) 148892. 1.25722 0.628611 0.777720i \(-0.283624\pi\)
0.628611 + 0.777720i \(0.283624\pi\)
\(108\) −3479.22 −0.0287027
\(109\) −115623. −0.932135 −0.466067 0.884749i \(-0.654330\pi\)
−0.466067 + 0.884749i \(0.654330\pi\)
\(110\) −59652.2 −0.470051
\(111\) 101806. 0.784268
\(112\) −127417. −0.959806
\(113\) −6712.64 −0.0494535 −0.0247268 0.999694i \(-0.507872\pi\)
−0.0247268 + 0.999694i \(0.507872\pi\)
\(114\) 106651. 0.768603
\(115\) −82378.2 −0.580855
\(116\) −34672.1 −0.239240
\(117\) 4161.56 0.0281055
\(118\) −18163.8 −0.120089
\(119\) −145414. −0.941322
\(120\) 37404.9 0.237124
\(121\) 117516. 0.729680
\(122\) 72315.2 0.439876
\(123\) 153171. 0.912882
\(124\) 2448.54 0.0143005
\(125\) 125213. 0.716762
\(126\) 63469.5 0.356155
\(127\) −254609. −1.40076 −0.700381 0.713769i \(-0.746986\pi\)
−0.700381 + 0.713769i \(0.746986\pi\)
\(128\) 133505. 0.720231
\(129\) −66232.5 −0.350426
\(130\) −5806.74 −0.0301352
\(131\) −246062. −1.25276 −0.626379 0.779519i \(-0.715464\pi\)
−0.626379 + 0.779519i \(0.715464\pi\)
\(132\) −22670.6 −0.113247
\(133\) 341032. 1.67173
\(134\) −136747. −0.657892
\(135\) −15790.2 −0.0745679
\(136\) 185804. 0.861408
\(137\) 171097. 0.778827 0.389413 0.921063i \(-0.372678\pi\)
0.389413 + 0.921063i \(0.372678\pi\)
\(138\) 178607. 0.798365
\(139\) −256672. −1.12678 −0.563392 0.826189i \(-0.690504\pi\)
−0.563392 + 0.826189i \(0.690504\pi\)
\(140\) 15523.5 0.0669377
\(141\) −138310. −0.585876
\(142\) −256848. −1.06895
\(143\) 27116.6 0.110891
\(144\) −68728.4 −0.276204
\(145\) −157356. −0.621532
\(146\) −27718.0 −0.107617
\(147\) 51690.5 0.197296
\(148\) −53986.4 −0.202596
\(149\) 122191. 0.450894 0.225447 0.974255i \(-0.427616\pi\)
0.225447 + 0.974255i \(0.427616\pi\)
\(150\) −124723. −0.452606
\(151\) −485698. −1.73350 −0.866750 0.498743i \(-0.833795\pi\)
−0.866750 + 0.498743i \(0.833795\pi\)
\(152\) −435759. −1.52981
\(153\) −78435.7 −0.270885
\(154\) 413566. 1.40522
\(155\) 11112.5 0.0371519
\(156\) −2206.83 −0.00726034
\(157\) −472070. −1.52847 −0.764235 0.644938i \(-0.776883\pi\)
−0.764235 + 0.644938i \(0.776883\pi\)
\(158\) −214633. −0.683995
\(159\) −288703. −0.905645
\(160\) −37096.4 −0.114560
\(161\) 571124. 1.73646
\(162\) 34235.2 0.102491
\(163\) 253916. 0.748549 0.374275 0.927318i \(-0.377892\pi\)
0.374275 + 0.927318i \(0.377892\pi\)
\(164\) −81225.1 −0.235820
\(165\) −102888. −0.294209
\(166\) 317920. 0.895463
\(167\) −79311.3 −0.220061 −0.110031 0.993928i \(-0.535095\pi\)
−0.110031 + 0.993928i \(0.535095\pi\)
\(168\) −259327. −0.708882
\(169\) −368653. −0.992891
\(170\) 109444. 0.290448
\(171\) 183952. 0.481076
\(172\) 35122.3 0.0905237
\(173\) 551760. 1.40163 0.700817 0.713341i \(-0.252819\pi\)
0.700817 + 0.713341i \(0.252819\pi\)
\(174\) 341170. 0.854274
\(175\) −398822. −0.984429
\(176\) −447833. −1.08977
\(177\) −31329.0 −0.0751646
\(178\) 22125.2 0.0523405
\(179\) 738237. 1.72212 0.861060 0.508504i \(-0.169801\pi\)
0.861060 + 0.508504i \(0.169801\pi\)
\(180\) 8373.34 0.0192627
\(181\) 663540. 1.50546 0.752732 0.658327i \(-0.228736\pi\)
0.752732 + 0.658327i \(0.228736\pi\)
\(182\) 40257.9 0.0900891
\(183\) 124729. 0.275322
\(184\) −729762. −1.58905
\(185\) −245013. −0.526331
\(186\) −24093.4 −0.0510640
\(187\) −511085. −1.06878
\(188\) 73344.2 0.151346
\(189\) 109472. 0.222921
\(190\) −256673. −0.515818
\(191\) −825937. −1.63819 −0.819093 0.573660i \(-0.805523\pi\)
−0.819093 + 0.573660i \(0.805523\pi\)
\(192\) 324798. 0.635858
\(193\) −596260. −1.15224 −0.576120 0.817365i \(-0.695434\pi\)
−0.576120 + 0.817365i \(0.695434\pi\)
\(194\) −25665.8 −0.0489610
\(195\) −10015.5 −0.0188619
\(196\) −27410.9 −0.0509663
\(197\) 847138. 1.55521 0.777604 0.628754i \(-0.216435\pi\)
0.777604 + 0.628754i \(0.216435\pi\)
\(198\) 223076. 0.404380
\(199\) −481833. −0.862509 −0.431254 0.902230i \(-0.641929\pi\)
−0.431254 + 0.902230i \(0.641929\pi\)
\(200\) 509601. 0.900855
\(201\) −235861. −0.411780
\(202\) −796828. −1.37400
\(203\) 1.09094e6 1.85807
\(204\) 41593.6 0.0699762
\(205\) −368633. −0.612645
\(206\) 543412. 0.892198
\(207\) 308062. 0.499704
\(208\) −43593.5 −0.0698657
\(209\) 1.19863e6 1.89809
\(210\) −152750. −0.239019
\(211\) 612379. 0.946922 0.473461 0.880815i \(-0.343005\pi\)
0.473461 + 0.880815i \(0.343005\pi\)
\(212\) 153096. 0.233950
\(213\) −443013. −0.669063
\(214\) 776917. 1.15969
\(215\) 159400. 0.235175
\(216\) −139880. −0.203995
\(217\) −77042.2 −0.111066
\(218\) −603320. −0.859818
\(219\) −47808.0 −0.0673582
\(220\) 54560.5 0.0760014
\(221\) −49750.7 −0.0685202
\(222\) 531221. 0.723424
\(223\) 1.24590e6 1.67773 0.838865 0.544340i \(-0.183220\pi\)
0.838865 + 0.544340i \(0.183220\pi\)
\(224\) 257188. 0.342476
\(225\) −215123. −0.283290
\(226\) −35026.5 −0.0456169
\(227\) −1.24352e6 −1.60173 −0.800864 0.598846i \(-0.795626\pi\)
−0.800864 + 0.598846i \(0.795626\pi\)
\(228\) −97547.5 −0.124274
\(229\) 524991. 0.661550 0.330775 0.943710i \(-0.392690\pi\)
0.330775 + 0.943710i \(0.392690\pi\)
\(230\) −429849. −0.535792
\(231\) 713320. 0.879537
\(232\) −1.39397e6 −1.70033
\(233\) 529442. 0.638894 0.319447 0.947604i \(-0.396503\pi\)
0.319447 + 0.947604i \(0.396503\pi\)
\(234\) 21714.9 0.0259250
\(235\) 332866. 0.393188
\(236\) 16613.4 0.0194169
\(237\) −370199. −0.428119
\(238\) −758767. −0.868293
\(239\) −443263. −0.501958 −0.250979 0.967993i \(-0.580752\pi\)
−0.250979 + 0.967993i \(0.580752\pi\)
\(240\) 165407. 0.185364
\(241\) −54132.2 −0.0600362 −0.0300181 0.999549i \(-0.509556\pi\)
−0.0300181 + 0.999549i \(0.509556\pi\)
\(242\) 613196. 0.673071
\(243\) 59049.0 0.0641500
\(244\) −66142.6 −0.0711225
\(245\) −124402. −0.132407
\(246\) 799246. 0.842060
\(247\) 116678. 0.121688
\(248\) 98441.7 0.101637
\(249\) 548349. 0.560478
\(250\) 653361. 0.661155
\(251\) 841171. 0.842753 0.421376 0.906886i \(-0.361547\pi\)
0.421376 + 0.906886i \(0.361547\pi\)
\(252\) −58052.0 −0.0575858
\(253\) 2.00733e6 1.97159
\(254\) −1.32855e6 −1.29209
\(255\) 188769. 0.181794
\(256\) −458211. −0.436984
\(257\) −338918. −0.320083 −0.160041 0.987110i \(-0.551163\pi\)
−0.160041 + 0.987110i \(0.551163\pi\)
\(258\) −345600. −0.323240
\(259\) 1.69866e6 1.57347
\(260\) 5311.10 0.00487249
\(261\) 588450. 0.534698
\(262\) −1.28395e6 −1.15557
\(263\) 2.18386e6 1.94686 0.973431 0.228981i \(-0.0735395\pi\)
0.973431 + 0.228981i \(0.0735395\pi\)
\(264\) −911454. −0.804868
\(265\) 694812. 0.607789
\(266\) 1.77950e6 1.54204
\(267\) 38161.6 0.0327604
\(268\) 125074. 0.106373
\(269\) −104452. −0.0880108 −0.0440054 0.999031i \(-0.514012\pi\)
−0.0440054 + 0.999031i \(0.514012\pi\)
\(270\) −82392.8 −0.0687828
\(271\) −263637. −0.218063 −0.109032 0.994038i \(-0.534775\pi\)
−0.109032 + 0.994038i \(0.534775\pi\)
\(272\) 821637. 0.673376
\(273\) 69436.9 0.0563876
\(274\) 892782. 0.718405
\(275\) −1.40174e6 −1.11773
\(276\) −163362. −0.129086
\(277\) 677687. 0.530677 0.265338 0.964155i \(-0.414516\pi\)
0.265338 + 0.964155i \(0.414516\pi\)
\(278\) −1.33931e6 −1.03937
\(279\) −41556.3 −0.0319614
\(280\) 624113. 0.475738
\(281\) 31956.5 0.0241431 0.0120716 0.999927i \(-0.496157\pi\)
0.0120716 + 0.999927i \(0.496157\pi\)
\(282\) −721700. −0.540423
\(283\) 395858. 0.293815 0.146907 0.989150i \(-0.453068\pi\)
0.146907 + 0.989150i \(0.453068\pi\)
\(284\) 234925. 0.172835
\(285\) −442711. −0.322855
\(286\) 141494. 0.102288
\(287\) 2.55571e6 1.83150
\(288\) 138726. 0.0985547
\(289\) −482172. −0.339592
\(290\) −821082. −0.573313
\(291\) −44268.4 −0.0306451
\(292\) 25352.0 0.0174003
\(293\) 1.30735e6 0.889658 0.444829 0.895616i \(-0.353264\pi\)
0.444829 + 0.895616i \(0.353264\pi\)
\(294\) 269720. 0.181989
\(295\) 75398.5 0.0504438
\(296\) −2.17049e6 −1.43988
\(297\) 384762. 0.253105
\(298\) 637593. 0.415913
\(299\) 195400. 0.126400
\(300\) 114077. 0.0731807
\(301\) −1.10511e6 −0.703055
\(302\) −2.53437e6 −1.59901
\(303\) −1.37437e6 −0.859998
\(304\) −1.92695e6 −1.19588
\(305\) −300182. −0.184772
\(306\) −409276. −0.249869
\(307\) −2.11977e6 −1.28364 −0.641820 0.766856i \(-0.721820\pi\)
−0.641820 + 0.766856i \(0.721820\pi\)
\(308\) −378265. −0.227206
\(309\) 937278. 0.558435
\(310\) 57984.7 0.0342696
\(311\) 2.50058e6 1.46602 0.733009 0.680219i \(-0.238115\pi\)
0.733009 + 0.680219i \(0.238115\pi\)
\(312\) −88723.9 −0.0516005
\(313\) −3.29921e6 −1.90349 −0.951743 0.306897i \(-0.900709\pi\)
−0.951743 + 0.306897i \(0.900709\pi\)
\(314\) −2.46325e6 −1.40989
\(315\) −263464. −0.149604
\(316\) 196312. 0.110594
\(317\) −275609. −0.154044 −0.0770220 0.997029i \(-0.524541\pi\)
−0.0770220 + 0.997029i \(0.524541\pi\)
\(318\) −1.50645e6 −0.835384
\(319\) 3.83433e6 2.10966
\(320\) −781681. −0.426731
\(321\) 1.34003e6 0.725858
\(322\) 2.98012e6 1.60175
\(323\) −2.19911e6 −1.17285
\(324\) −31313.0 −0.0165715
\(325\) −136450. −0.0716580
\(326\) 1.32493e6 0.690476
\(327\) −1.04061e6 −0.538168
\(328\) −3.26560e6 −1.67601
\(329\) −2.30775e6 −1.17543
\(330\) −536870. −0.271384
\(331\) −2.58017e6 −1.29443 −0.647215 0.762307i \(-0.724067\pi\)
−0.647215 + 0.762307i \(0.724067\pi\)
\(332\) −290783. −0.144785
\(333\) 916251. 0.452797
\(334\) −413846. −0.202989
\(335\) 567639. 0.276350
\(336\) −1.14676e6 −0.554144
\(337\) 223512. 0.107208 0.0536039 0.998562i \(-0.482929\pi\)
0.0536039 + 0.998562i \(0.482929\pi\)
\(338\) −1.92363e6 −0.915861
\(339\) −60413.8 −0.0285520
\(340\) −100102. −0.0469618
\(341\) −270780. −0.126104
\(342\) 959858. 0.443753
\(343\) −1.66140e6 −0.762498
\(344\) 1.41207e6 0.643369
\(345\) −741404. −0.335357
\(346\) 2.87908e6 1.29289
\(347\) −2.39743e6 −1.06886 −0.534431 0.845212i \(-0.679474\pi\)
−0.534431 + 0.845212i \(0.679474\pi\)
\(348\) −312049. −0.138126
\(349\) 1.79102e6 0.787113 0.393557 0.919300i \(-0.371245\pi\)
0.393557 + 0.919300i \(0.371245\pi\)
\(350\) −2.08105e6 −0.908056
\(351\) 37454.0 0.0162267
\(352\) 903937. 0.388849
\(353\) −883662. −0.377441 −0.188721 0.982031i \(-0.560434\pi\)
−0.188721 + 0.982031i \(0.560434\pi\)
\(354\) −163474. −0.0693332
\(355\) 1.06618e6 0.449016
\(356\) −20236.7 −0.00846281
\(357\) −1.30872e6 −0.543472
\(358\) 3.85211e6 1.58851
\(359\) −597411. −0.244645 −0.122323 0.992490i \(-0.539034\pi\)
−0.122323 + 0.992490i \(0.539034\pi\)
\(360\) 336644. 0.136904
\(361\) 2.68138e6 1.08291
\(362\) 3.46234e6 1.38867
\(363\) 1.05764e6 0.421281
\(364\) −36821.6 −0.0145663
\(365\) 115058. 0.0452048
\(366\) 650837. 0.253962
\(367\) −2.44464e6 −0.947437 −0.473719 0.880676i \(-0.657089\pi\)
−0.473719 + 0.880676i \(0.657089\pi\)
\(368\) −3.22705e6 −1.24218
\(369\) 1.37854e6 0.527053
\(370\) −1.27847e6 −0.485498
\(371\) −4.81710e6 −1.81698
\(372\) 22036.8 0.00825642
\(373\) −3.44062e6 −1.28045 −0.640227 0.768186i \(-0.721160\pi\)
−0.640227 + 0.768186i \(0.721160\pi\)
\(374\) −2.66684e6 −0.985865
\(375\) 1.12692e6 0.413823
\(376\) 2.94875e6 1.07564
\(377\) 373246. 0.135251
\(378\) 571226. 0.205626
\(379\) −695527. −0.248723 −0.124361 0.992237i \(-0.539688\pi\)
−0.124361 + 0.992237i \(0.539688\pi\)
\(380\) 234765. 0.0834014
\(381\) −2.29148e6 −0.808730
\(382\) −4.30973e6 −1.51109
\(383\) 453295. 0.157901 0.0789503 0.996879i \(-0.474843\pi\)
0.0789503 + 0.996879i \(0.474843\pi\)
\(384\) 1.20154e6 0.415826
\(385\) −1.71672e6 −0.590267
\(386\) −3.11128e6 −1.06285
\(387\) −596092. −0.202319
\(388\) 23475.1 0.00791639
\(389\) −5.66314e6 −1.89751 −0.948753 0.316017i \(-0.897654\pi\)
−0.948753 + 0.316017i \(0.897654\pi\)
\(390\) −52260.7 −0.0173986
\(391\) −3.68283e6 −1.21826
\(392\) −1.10204e6 −0.362227
\(393\) −2.21456e6 −0.723280
\(394\) 4.42036e6 1.43455
\(395\) 890946. 0.287315
\(396\) −204035. −0.0653833
\(397\) 2.54465e6 0.810312 0.405156 0.914248i \(-0.367217\pi\)
0.405156 + 0.914248i \(0.367217\pi\)
\(398\) −2.51420e6 −0.795594
\(399\) 3.06929e6 0.965175
\(400\) 2.25348e6 0.704213
\(401\) −168328. −0.0522753 −0.0261376 0.999658i \(-0.508321\pi\)
−0.0261376 + 0.999658i \(0.508321\pi\)
\(402\) −1.23072e6 −0.379834
\(403\) −26358.6 −0.00808462
\(404\) 728814. 0.222159
\(405\) −142111. −0.0430518
\(406\) 5.69252e6 1.71392
\(407\) 5.97027e6 1.78652
\(408\) 1.67224e6 0.497334
\(409\) 2.01773e6 0.596423 0.298212 0.954500i \(-0.403610\pi\)
0.298212 + 0.954500i \(0.403610\pi\)
\(410\) −1.92352e6 −0.565116
\(411\) 1.53987e6 0.449656
\(412\) −497028. −0.144257
\(413\) −522734. −0.150802
\(414\) 1.60747e6 0.460936
\(415\) −1.31969e6 −0.376143
\(416\) 87992.2 0.0249293
\(417\) −2.31005e6 −0.650550
\(418\) 6.25441e6 1.75084
\(419\) 3.20396e6 0.891563 0.445782 0.895142i \(-0.352926\pi\)
0.445782 + 0.895142i \(0.352926\pi\)
\(420\) 139712. 0.0386465
\(421\) 993569. 0.273208 0.136604 0.990626i \(-0.456381\pi\)
0.136604 + 0.990626i \(0.456381\pi\)
\(422\) 3.19539e6 0.873458
\(423\) −1.24479e6 −0.338256
\(424\) 6.15511e6 1.66273
\(425\) 2.57176e6 0.690651
\(426\) −2.31163e6 −0.617156
\(427\) 2.08115e6 0.552375
\(428\) −710602. −0.187507
\(429\) 244049. 0.0640228
\(430\) 831746. 0.216930
\(431\) −7.34140e6 −1.90365 −0.951823 0.306649i \(-0.900792\pi\)
−0.951823 + 0.306649i \(0.900792\pi\)
\(432\) −618556. −0.159467
\(433\) −5.16376e6 −1.32357 −0.661784 0.749695i \(-0.730200\pi\)
−0.661784 + 0.749695i \(0.730200\pi\)
\(434\) −402005. −0.102449
\(435\) −1.41620e6 −0.358842
\(436\) 551823. 0.139022
\(437\) 8.63719e6 2.16356
\(438\) −249462. −0.0621324
\(439\) 335620. 0.0831163 0.0415582 0.999136i \(-0.486768\pi\)
0.0415582 + 0.999136i \(0.486768\pi\)
\(440\) 2.19357e6 0.540156
\(441\) 465214. 0.113909
\(442\) −259599. −0.0632043
\(443\) 6.81650e6 1.65026 0.825129 0.564944i \(-0.191102\pi\)
0.825129 + 0.564944i \(0.191102\pi\)
\(444\) −485878. −0.116969
\(445\) −91842.4 −0.0219859
\(446\) 6.50111e6 1.54757
\(447\) 1.09972e6 0.260324
\(448\) 5.41936e6 1.27571
\(449\) 1.94054e6 0.454263 0.227132 0.973864i \(-0.427065\pi\)
0.227132 + 0.973864i \(0.427065\pi\)
\(450\) −1.12251e6 −0.261312
\(451\) 8.98255e6 2.07950
\(452\) 32036.7 0.00737568
\(453\) −4.37128e6 −1.00084
\(454\) −6.48868e6 −1.47746
\(455\) −167112. −0.0378423
\(456\) −3.92183e6 −0.883236
\(457\) 4.70547e6 1.05393 0.526966 0.849887i \(-0.323330\pi\)
0.526966 + 0.849887i \(0.323330\pi\)
\(458\) 2.73939e6 0.610226
\(459\) −705921. −0.156396
\(460\) 393158. 0.0866309
\(461\) 1.04225e6 0.228413 0.114206 0.993457i \(-0.463567\pi\)
0.114206 + 0.993457i \(0.463567\pi\)
\(462\) 3.72209e6 0.811302
\(463\) 670929. 0.145454 0.0727268 0.997352i \(-0.476830\pi\)
0.0727268 + 0.997352i \(0.476830\pi\)
\(464\) −6.16419e6 −1.32917
\(465\) 100012. 0.0214497
\(466\) 2.76262e6 0.589328
\(467\) −2.18853e6 −0.464367 −0.232183 0.972672i \(-0.574587\pi\)
−0.232183 + 0.972672i \(0.574587\pi\)
\(468\) −19861.4 −0.00419176
\(469\) −3.93541e6 −0.826149
\(470\) 1.73689e6 0.362684
\(471\) −4.24863e6 −0.882463
\(472\) 667931. 0.137999
\(473\) −3.88413e6 −0.798253
\(474\) −1.93169e6 −0.394905
\(475\) −6.03144e6 −1.22656
\(476\) 694002. 0.140392
\(477\) −2.59832e6 −0.522875
\(478\) −2.31294e6 −0.463015
\(479\) −4.83496e6 −0.962840 −0.481420 0.876490i \(-0.659879\pi\)
−0.481420 + 0.876490i \(0.659879\pi\)
\(480\) −333868. −0.0661411
\(481\) 581166. 0.114535
\(482\) −282461. −0.0553785
\(483\) 5.14012e6 1.00255
\(484\) −560855. −0.108827
\(485\) 106539. 0.0205663
\(486\) 308117. 0.0591732
\(487\) 7.05912e6 1.34874 0.674370 0.738394i \(-0.264415\pi\)
0.674370 + 0.738394i \(0.264415\pi\)
\(488\) −2.65922e6 −0.505480
\(489\) 2.28524e6 0.432175
\(490\) −649128. −0.122135
\(491\) 2.74825e6 0.514462 0.257231 0.966350i \(-0.417190\pi\)
0.257231 + 0.966350i \(0.417190\pi\)
\(492\) −731025. −0.136151
\(493\) −7.03482e6 −1.30357
\(494\) 608825. 0.112247
\(495\) −925995. −0.169862
\(496\) 435315. 0.0794509
\(497\) −7.39181e6 −1.34233
\(498\) 2.86128e6 0.516996
\(499\) −1.00805e7 −1.81230 −0.906149 0.422958i \(-0.860992\pi\)
−0.906149 + 0.422958i \(0.860992\pi\)
\(500\) −597592. −0.106901
\(501\) −713802. −0.127053
\(502\) 4.38922e6 0.777371
\(503\) 8.49561e6 1.49718 0.748591 0.663032i \(-0.230731\pi\)
0.748591 + 0.663032i \(0.230731\pi\)
\(504\) −2.33394e6 −0.409273
\(505\) 3.30766e6 0.577154
\(506\) 1.04742e7 1.81863
\(507\) −3.31788e6 −0.573246
\(508\) 1.21515e6 0.208915
\(509\) −6.16900e6 −1.05541 −0.527704 0.849428i \(-0.676947\pi\)
−0.527704 + 0.849428i \(0.676947\pi\)
\(510\) 984993. 0.167690
\(511\) −797692. −0.135140
\(512\) −6.66309e6 −1.12331
\(513\) 1.65557e6 0.277749
\(514\) −1.76847e6 −0.295250
\(515\) −2.25572e6 −0.374772
\(516\) 316101. 0.0522639
\(517\) −8.11102e6 −1.33459
\(518\) 8.86359e6 1.45139
\(519\) 4.96584e6 0.809234
\(520\) 213529. 0.0346297
\(521\) −4.98898e6 −0.805225 −0.402612 0.915371i \(-0.631898\pi\)
−0.402612 + 0.915371i \(0.631898\pi\)
\(522\) 3.07053e6 0.493215
\(523\) −2.82384e6 −0.451425 −0.225712 0.974194i \(-0.572471\pi\)
−0.225712 + 0.974194i \(0.572471\pi\)
\(524\) 1.17436e6 0.186841
\(525\) −3.58940e6 −0.568360
\(526\) 1.13953e7 1.79582
\(527\) 496799. 0.0779209
\(528\) −4.03050e6 −0.629178
\(529\) 8.02829e6 1.24734
\(530\) 3.62552e6 0.560636
\(531\) −281961. −0.0433963
\(532\) −1.62761e6 −0.249328
\(533\) 874391. 0.133318
\(534\) 199127. 0.0302188
\(535\) −3.22501e6 −0.487131
\(536\) 5.02853e6 0.756012
\(537\) 6.64413e6 0.994266
\(538\) −545029. −0.0811829
\(539\) 3.03133e6 0.449429
\(540\) 75360.1 0.0111213
\(541\) −1.48508e6 −0.218151 −0.109075 0.994033i \(-0.534789\pi\)
−0.109075 + 0.994033i \(0.534789\pi\)
\(542\) −1.37565e6 −0.201146
\(543\) 5.97186e6 0.869180
\(544\) −1.65845e6 −0.240273
\(545\) 2.50440e6 0.361171
\(546\) 362321. 0.0520130
\(547\) −1.20672e7 −1.72440 −0.862198 0.506571i \(-0.830913\pi\)
−0.862198 + 0.506571i \(0.830913\pi\)
\(548\) −816577. −0.116157
\(549\) 1.12256e6 0.158957
\(550\) −7.31426e6 −1.03101
\(551\) 1.64985e7 2.31507
\(552\) −6.56786e6 −0.917436
\(553\) −6.17689e6 −0.858928
\(554\) 3.53616e6 0.489506
\(555\) −2.20511e6 −0.303877
\(556\) 1.22499e6 0.168053
\(557\) −1.54643e6 −0.211199 −0.105600 0.994409i \(-0.533676\pi\)
−0.105600 + 0.994409i \(0.533676\pi\)
\(558\) −216840. −0.0294818
\(559\) −378094. −0.0511764
\(560\) 2.75986e6 0.371892
\(561\) −4.59977e6 −0.617062
\(562\) 166749. 0.0222701
\(563\) 1.88354e6 0.250439 0.125220 0.992129i \(-0.460036\pi\)
0.125220 + 0.992129i \(0.460036\pi\)
\(564\) 660098. 0.0873797
\(565\) 145396. 0.0191616
\(566\) 2.06558e6 0.271020
\(567\) 985251. 0.128703
\(568\) 9.44498e6 1.22837
\(569\) 1.00339e6 0.129924 0.0649622 0.997888i \(-0.479307\pi\)
0.0649622 + 0.997888i \(0.479307\pi\)
\(570\) −2.31006e6 −0.297808
\(571\) 7.14637e6 0.917266 0.458633 0.888626i \(-0.348339\pi\)
0.458633 + 0.888626i \(0.348339\pi\)
\(572\) −129417. −0.0165387
\(573\) −7.43343e6 −0.945808
\(574\) 1.33357e7 1.68941
\(575\) −1.01008e7 −1.27405
\(576\) 2.92318e6 0.367113
\(577\) −4.39566e6 −0.549649 −0.274824 0.961494i \(-0.588620\pi\)
−0.274824 + 0.961494i \(0.588620\pi\)
\(578\) −2.51597e6 −0.313246
\(579\) −5.36634e6 −0.665246
\(580\) 750997. 0.0926976
\(581\) 9.14938e6 1.12448
\(582\) −230992. −0.0282677
\(583\) −1.69306e7 −2.06301
\(584\) 1.01926e6 0.123667
\(585\) −90139.4 −0.0108899
\(586\) 6.82174e6 0.820637
\(587\) −9.07708e6 −1.08730 −0.543652 0.839311i \(-0.682959\pi\)
−0.543652 + 0.839311i \(0.682959\pi\)
\(588\) −246698. −0.0294254
\(589\) −1.16512e6 −0.138383
\(590\) 393429. 0.0465303
\(591\) 7.62424e6 0.897900
\(592\) −9.59800e6 −1.12558
\(593\) −2.24665e6 −0.262360 −0.131180 0.991359i \(-0.541877\pi\)
−0.131180 + 0.991359i \(0.541877\pi\)
\(594\) 2.00768e6 0.233469
\(595\) 3.14967e6 0.364730
\(596\) −583170. −0.0672481
\(597\) −4.33649e6 −0.497970
\(598\) 1.01959e6 0.116594
\(599\) 2.81454e6 0.320509 0.160255 0.987076i \(-0.448768\pi\)
0.160255 + 0.987076i \(0.448768\pi\)
\(600\) 4.58641e6 0.520109
\(601\) −7.44310e6 −0.840557 −0.420279 0.907395i \(-0.638068\pi\)
−0.420279 + 0.907395i \(0.638068\pi\)
\(602\) −5.76645e6 −0.648511
\(603\) −2.12275e6 −0.237741
\(604\) 2.31804e6 0.258541
\(605\) −2.54539e6 −0.282726
\(606\) −7.17145e6 −0.793278
\(607\) −1.48571e7 −1.63667 −0.818335 0.574742i \(-0.805102\pi\)
−0.818335 + 0.574742i \(0.805102\pi\)
\(608\) 3.88949e6 0.426710
\(609\) 9.81848e6 1.07276
\(610\) −1.56635e6 −0.170437
\(611\) −789554. −0.0855615
\(612\) 374342. 0.0404008
\(613\) −8.75960e6 −0.941528 −0.470764 0.882259i \(-0.656022\pi\)
−0.470764 + 0.882259i \(0.656022\pi\)
\(614\) −1.10609e7 −1.18405
\(615\) −3.31769e6 −0.353711
\(616\) −1.52079e7 −1.61479
\(617\) 1.22453e6 0.129496 0.0647478 0.997902i \(-0.479376\pi\)
0.0647478 + 0.997902i \(0.479376\pi\)
\(618\) 4.89071e6 0.515111
\(619\) 1.13512e7 1.19074 0.595369 0.803452i \(-0.297006\pi\)
0.595369 + 0.803452i \(0.297006\pi\)
\(620\) −53035.4 −0.00554098
\(621\) 2.77256e6 0.288504
\(622\) 1.30480e7 1.35228
\(623\) 636739. 0.0657266
\(624\) −392342. −0.0403370
\(625\) 5.58739e6 0.572149
\(626\) −1.72153e7 −1.75581
\(627\) 1.07876e7 1.09586
\(628\) 2.25300e6 0.227962
\(629\) −1.09536e7 −1.10390
\(630\) −1.37475e6 −0.137998
\(631\) 8.37476e6 0.837334 0.418667 0.908140i \(-0.362497\pi\)
0.418667 + 0.908140i \(0.362497\pi\)
\(632\) 7.89260e6 0.786009
\(633\) 5.51141e6 0.546706
\(634\) −1.43812e6 −0.142093
\(635\) 5.51483e6 0.542748
\(636\) 1.37786e6 0.135071
\(637\) 295079. 0.0288131
\(638\) 2.00075e7 1.94599
\(639\) −3.98711e6 −0.386284
\(640\) −2.89172e6 −0.279065
\(641\) 3.09372e6 0.297396 0.148698 0.988883i \(-0.452492\pi\)
0.148698 + 0.988883i \(0.452492\pi\)
\(642\) 6.99225e6 0.669545
\(643\) 3.68773e6 0.351748 0.175874 0.984413i \(-0.443725\pi\)
0.175874 + 0.984413i \(0.443725\pi\)
\(644\) −2.72575e6 −0.258983
\(645\) 1.43460e6 0.135778
\(646\) −1.14749e7 −1.08186
\(647\) 1.12690e7 1.05834 0.529171 0.848516i \(-0.322503\pi\)
0.529171 + 0.848516i \(0.322503\pi\)
\(648\) −1.25892e6 −0.117777
\(649\) −1.83725e6 −0.171221
\(650\) −711994. −0.0660987
\(651\) −693380. −0.0641237
\(652\) −1.21184e6 −0.111641
\(653\) 9.48265e6 0.870255 0.435128 0.900369i \(-0.356703\pi\)
0.435128 + 0.900369i \(0.356703\pi\)
\(654\) −5.42988e6 −0.496416
\(655\) 5.32972e6 0.485401
\(656\) −1.44406e7 −1.31017
\(657\) −430272. −0.0388893
\(658\) −1.20418e7 −1.08424
\(659\) 1.67506e6 0.150251 0.0751255 0.997174i \(-0.476064\pi\)
0.0751255 + 0.997174i \(0.476064\pi\)
\(660\) 491045. 0.0438794
\(661\) −1.47650e7 −1.31441 −0.657203 0.753714i \(-0.728260\pi\)
−0.657203 + 0.753714i \(0.728260\pi\)
\(662\) −1.34633e7 −1.19401
\(663\) −447756. −0.0395602
\(664\) −1.16907e7 −1.02902
\(665\) −7.38677e6 −0.647740
\(666\) 4.78099e6 0.417669
\(667\) 2.76298e7 2.40472
\(668\) 378521. 0.0328208
\(669\) 1.12131e7 0.968638
\(670\) 2.96193e6 0.254911
\(671\) 7.31461e6 0.627169
\(672\) 2.31469e6 0.197729
\(673\) 4.22348e6 0.359445 0.179723 0.983717i \(-0.442480\pi\)
0.179723 + 0.983717i \(0.442480\pi\)
\(674\) 1.16628e6 0.0988904
\(675\) −1.93611e6 −0.163558
\(676\) 1.75943e6 0.148083
\(677\) −5.35812e6 −0.449304 −0.224652 0.974439i \(-0.572125\pi\)
−0.224652 + 0.974439i \(0.572125\pi\)
\(678\) −315238. −0.0263369
\(679\) −738633. −0.0614829
\(680\) −4.02453e6 −0.333766
\(681\) −1.11917e7 −0.924758
\(682\) −1.41293e6 −0.116321
\(683\) −3.41248e6 −0.279910 −0.139955 0.990158i \(-0.544696\pi\)
−0.139955 + 0.990158i \(0.544696\pi\)
\(684\) −877928. −0.0717494
\(685\) −3.70596e6 −0.301769
\(686\) −8.66916e6 −0.703342
\(687\) 4.72492e6 0.381946
\(688\) 6.24425e6 0.502932
\(689\) −1.64808e6 −0.132261
\(690\) −3.86864e6 −0.309339
\(691\) −1.38499e7 −1.10345 −0.551724 0.834027i \(-0.686030\pi\)
−0.551724 + 0.834027i \(0.686030\pi\)
\(692\) −2.63333e6 −0.209045
\(693\) 6.41988e6 0.507801
\(694\) −1.25097e7 −0.985938
\(695\) 5.55951e6 0.436591
\(696\) −1.25457e7 −0.981683
\(697\) −1.64802e7 −1.28494
\(698\) 9.34553e6 0.726048
\(699\) 4.76498e6 0.368866
\(700\) 1.90342e6 0.146821
\(701\) −232405. −0.0178628 −0.00893142 0.999960i \(-0.502843\pi\)
−0.00893142 + 0.999960i \(0.502843\pi\)
\(702\) 195435. 0.0149678
\(703\) 2.56891e7 1.96047
\(704\) 1.90474e7 1.44845
\(705\) 2.99580e6 0.227007
\(706\) −4.61094e6 −0.348159
\(707\) −2.29318e7 −1.72540
\(708\) 149521. 0.0112103
\(709\) −3.14519e6 −0.234981 −0.117490 0.993074i \(-0.537485\pi\)
−0.117490 + 0.993074i \(0.537485\pi\)
\(710\) 5.56334e6 0.414181
\(711\) −3.33179e6 −0.247175
\(712\) −813602. −0.0601467
\(713\) −1.95122e6 −0.143741
\(714\) −6.82891e6 −0.501309
\(715\) −587346. −0.0429664
\(716\) −3.52331e6 −0.256843
\(717\) −3.98937e6 −0.289805
\(718\) −3.11728e6 −0.225665
\(719\) −8.47359e6 −0.611287 −0.305644 0.952146i \(-0.598872\pi\)
−0.305644 + 0.952146i \(0.598872\pi\)
\(720\) 1.48866e6 0.107020
\(721\) 1.56388e7 1.12038
\(722\) 1.39914e7 0.998892
\(723\) −487190. −0.0346619
\(724\) −3.16681e6 −0.224531
\(725\) −1.92942e7 −1.36327
\(726\) 5.51876e6 0.388598
\(727\) −5.10460e6 −0.358200 −0.179100 0.983831i \(-0.557319\pi\)
−0.179100 + 0.983831i \(0.557319\pi\)
\(728\) −1.48039e6 −0.103525
\(729\) 531441. 0.0370370
\(730\) 600371. 0.0416978
\(731\) 7.12619e6 0.493246
\(732\) −595284. −0.0410626
\(733\) −3.23461e6 −0.222363 −0.111181 0.993800i \(-0.535463\pi\)
−0.111181 + 0.993800i \(0.535463\pi\)
\(734\) −1.27561e7 −0.873934
\(735\) −1.11962e6 −0.0764454
\(736\) 6.51369e6 0.443234
\(737\) −1.38318e7 −0.938014
\(738\) 7.19322e6 0.486163
\(739\) −2.43694e7 −1.64147 −0.820735 0.571309i \(-0.806436\pi\)
−0.820735 + 0.571309i \(0.806436\pi\)
\(740\) 1.16935e6 0.0784990
\(741\) 1.05010e6 0.0702565
\(742\) −2.51356e7 −1.67602
\(743\) −1.35628e7 −0.901315 −0.450657 0.892697i \(-0.648810\pi\)
−0.450657 + 0.892697i \(0.648810\pi\)
\(744\) 885975. 0.0586799
\(745\) −2.64667e6 −0.174706
\(746\) −1.79531e7 −1.18112
\(747\) 4.93514e6 0.323592
\(748\) 2.43920e6 0.159402
\(749\) 2.23588e7 1.45628
\(750\) 5.88025e6 0.381718
\(751\) −3.68893e6 −0.238671 −0.119336 0.992854i \(-0.538076\pi\)
−0.119336 + 0.992854i \(0.538076\pi\)
\(752\) 1.30395e7 0.840848
\(753\) 7.57054e6 0.486564
\(754\) 1.94759e6 0.124758
\(755\) 1.05202e7 0.671673
\(756\) −522468. −0.0332472
\(757\) 4.79135e6 0.303892 0.151946 0.988389i \(-0.451446\pi\)
0.151946 + 0.988389i \(0.451446\pi\)
\(758\) −3.62925e6 −0.229427
\(759\) 1.80660e7 1.13830
\(760\) 9.43854e6 0.592749
\(761\) 7.35535e6 0.460407 0.230203 0.973143i \(-0.426061\pi\)
0.230203 + 0.973143i \(0.426061\pi\)
\(762\) −1.19569e7 −0.745988
\(763\) −1.73629e7 −1.07972
\(764\) 3.94187e6 0.244325
\(765\) 1.69892e6 0.104959
\(766\) 2.36529e6 0.145650
\(767\) −178844. −0.0109771
\(768\) −4.12390e6 −0.252293
\(769\) −2.77997e7 −1.69521 −0.847607 0.530625i \(-0.821957\pi\)
−0.847607 + 0.530625i \(0.821957\pi\)
\(770\) −8.95784e6 −0.544473
\(771\) −3.05026e6 −0.184800
\(772\) 2.84571e6 0.171849
\(773\) 9.35408e6 0.563057 0.281529 0.959553i \(-0.409159\pi\)
0.281529 + 0.959553i \(0.409159\pi\)
\(774\) −3.11040e6 −0.186623
\(775\) 1.36256e6 0.0814892
\(776\) 943798. 0.0562632
\(777\) 1.52879e7 0.908441
\(778\) −2.95502e7 −1.75030
\(779\) 3.86504e7 2.28197
\(780\) 47799.9 0.00281313
\(781\) −2.59799e7 −1.52409
\(782\) −1.92170e7 −1.12375
\(783\) 5.29605e6 0.308708
\(784\) −4.87326e6 −0.283158
\(785\) 1.02250e7 0.592231
\(786\) −1.15556e7 −0.667167
\(787\) 8.96042e6 0.515693 0.257847 0.966186i \(-0.416987\pi\)
0.257847 + 0.966186i \(0.416987\pi\)
\(788\) −4.04305e6 −0.231950
\(789\) 1.96547e7 1.12402
\(790\) 4.64895e6 0.265025
\(791\) −1.00802e6 −0.0572835
\(792\) −8.20308e6 −0.464691
\(793\) 712028. 0.0402082
\(794\) 1.32780e7 0.747447
\(795\) 6.25331e6 0.350907
\(796\) 2.29959e6 0.128638
\(797\) −1.17914e7 −0.657536 −0.328768 0.944411i \(-0.606633\pi\)
−0.328768 + 0.944411i \(0.606633\pi\)
\(798\) 1.60155e7 0.890295
\(799\) 1.48812e7 0.824655
\(800\) −4.54858e6 −0.251276
\(801\) 343455. 0.0189142
\(802\) −878336. −0.0482197
\(803\) −2.80364e6 −0.153438
\(804\) 1.12567e6 0.0614145
\(805\) −1.23706e7 −0.672821
\(806\) −137539. −0.00745741
\(807\) −940068. −0.0508131
\(808\) 2.93014e7 1.57892
\(809\) 2.15179e7 1.15592 0.577960 0.816065i \(-0.303849\pi\)
0.577960 + 0.816065i \(0.303849\pi\)
\(810\) −741535. −0.0397118
\(811\) 1.33906e7 0.714906 0.357453 0.933931i \(-0.383645\pi\)
0.357453 + 0.933931i \(0.383645\pi\)
\(812\) −5.20663e6 −0.277119
\(813\) −2.37273e6 −0.125899
\(814\) 3.11528e7 1.64792
\(815\) −5.49982e6 −0.290038
\(816\) 7.39473e6 0.388774
\(817\) −1.67127e7 −0.875976
\(818\) 1.05285e7 0.550152
\(819\) 624932. 0.0325554
\(820\) 1.75934e6 0.0913722
\(821\) −8.26050e6 −0.427709 −0.213854 0.976866i \(-0.568602\pi\)
−0.213854 + 0.976866i \(0.568602\pi\)
\(822\) 8.03504e6 0.414771
\(823\) 3.28873e7 1.69250 0.846251 0.532784i \(-0.178854\pi\)
0.846251 + 0.532784i \(0.178854\pi\)
\(824\) −1.99827e7 −1.02526
\(825\) −1.26156e7 −0.645319
\(826\) −2.72762e6 −0.139102
\(827\) −2.91579e6 −0.148249 −0.0741247 0.997249i \(-0.523616\pi\)
−0.0741247 + 0.997249i \(0.523616\pi\)
\(828\) −1.47026e6 −0.0745277
\(829\) 1.16799e6 0.0590274 0.0295137 0.999564i \(-0.490604\pi\)
0.0295137 + 0.999564i \(0.490604\pi\)
\(830\) −6.88615e6 −0.346961
\(831\) 6.09919e6 0.306386
\(832\) 1.85414e6 0.0928610
\(833\) −5.56156e6 −0.277705
\(834\) −1.20538e7 −0.600079
\(835\) 1.71788e6 0.0852664
\(836\) −5.72056e6 −0.283089
\(837\) −374007. −0.0184529
\(838\) 1.67182e7 0.822395
\(839\) 4.82533e6 0.236659 0.118329 0.992974i \(-0.462246\pi\)
0.118329 + 0.992974i \(0.462246\pi\)
\(840\) 5.61702e6 0.274668
\(841\) 3.22664e7 1.57312
\(842\) 5.18443e6 0.252012
\(843\) 287609. 0.0139391
\(844\) −2.92264e6 −0.141227
\(845\) 7.98504e6 0.384712
\(846\) −6.49530e6 −0.312013
\(847\) 1.76471e7 0.845210
\(848\) 2.72182e7 1.29978
\(849\) 3.56273e6 0.169634
\(850\) 1.34194e7 0.637070
\(851\) 4.30213e7 2.03638
\(852\) 2.11432e6 0.0997866
\(853\) 2.55798e7 1.20372 0.601859 0.798603i \(-0.294427\pi\)
0.601859 + 0.798603i \(0.294427\pi\)
\(854\) 1.08594e7 0.509521
\(855\) −3.98440e6 −0.186401
\(856\) −2.85693e7 −1.33265
\(857\) −1.16798e7 −0.543232 −0.271616 0.962406i \(-0.587558\pi\)
−0.271616 + 0.962406i \(0.587558\pi\)
\(858\) 1.27345e6 0.0590558
\(859\) 1.36617e6 0.0631717 0.0315858 0.999501i \(-0.489944\pi\)
0.0315858 + 0.999501i \(0.489944\pi\)
\(860\) −760751. −0.0350749
\(861\) 2.30014e7 1.05742
\(862\) −3.83074e7 −1.75596
\(863\) −2.59879e7 −1.18780 −0.593901 0.804538i \(-0.702413\pi\)
−0.593901 + 0.804538i \(0.702413\pi\)
\(864\) 1.24854e6 0.0569006
\(865\) −1.19511e7 −0.543086
\(866\) −2.69444e7 −1.22088
\(867\) −4.33955e6 −0.196063
\(868\) 367692. 0.0165647
\(869\) −2.17099e7 −0.975232
\(870\) −7.38974e6 −0.331002
\(871\) −1.34643e6 −0.0601366
\(872\) 2.21857e7 0.988055
\(873\) −398416. −0.0176930
\(874\) 4.50688e7 1.99571
\(875\) 1.88030e7 0.830246
\(876\) 228168. 0.0100460
\(877\) −3.18973e7 −1.40041 −0.700204 0.713943i \(-0.746907\pi\)
−0.700204 + 0.713943i \(0.746907\pi\)
\(878\) 1.75126e6 0.0766681
\(879\) 1.17662e7 0.513644
\(880\) 9.70007e6 0.422249
\(881\) 1.48478e7 0.644497 0.322249 0.946655i \(-0.395561\pi\)
0.322249 + 0.946655i \(0.395561\pi\)
\(882\) 2.42748e6 0.105071
\(883\) 2.11125e7 0.911249 0.455625 0.890172i \(-0.349416\pi\)
0.455625 + 0.890172i \(0.349416\pi\)
\(884\) 237440. 0.0102194
\(885\) 678587. 0.0291237
\(886\) 3.55684e7 1.52223
\(887\) −1.01713e7 −0.434076 −0.217038 0.976163i \(-0.569640\pi\)
−0.217038 + 0.976163i \(0.569640\pi\)
\(888\) −1.95344e7 −0.831318
\(889\) −3.82341e7 −1.62254
\(890\) −479233. −0.0202802
\(891\) 3.46286e6 0.146130
\(892\) −5.94619e6 −0.250223
\(893\) −3.49003e7 −1.46454
\(894\) 5.73834e6 0.240128
\(895\) −1.59902e7 −0.667263
\(896\) 2.00481e7 0.834265
\(897\) 1.75860e6 0.0729770
\(898\) 1.01257e7 0.419021
\(899\) −3.72715e6 −0.153807
\(900\) 1.02670e6 0.0422509
\(901\) 3.10625e7 1.27475
\(902\) 4.68708e7 1.91817
\(903\) −9.94599e6 −0.405909
\(904\) 1.28801e6 0.0524203
\(905\) −1.43723e7 −0.583316
\(906\) −2.28093e7 −0.923191
\(907\) −3.32335e7 −1.34140 −0.670700 0.741729i \(-0.734006\pi\)
−0.670700 + 0.741729i \(0.734006\pi\)
\(908\) 5.93483e6 0.238888
\(909\) −1.23693e7 −0.496520
\(910\) −871986. −0.0349065
\(911\) 701228. 0.0279939 0.0139970 0.999902i \(-0.495544\pi\)
0.0139970 + 0.999902i \(0.495544\pi\)
\(912\) −1.73425e7 −0.690439
\(913\) 3.21573e7 1.27674
\(914\) 2.45531e7 0.972166
\(915\) −2.70164e6 −0.106678
\(916\) −2.50557e6 −0.0986661
\(917\) −3.69507e7 −1.45111
\(918\) −3.68349e6 −0.144262
\(919\) 1.45028e7 0.566451 0.283226 0.959053i \(-0.408595\pi\)
0.283226 + 0.959053i \(0.408595\pi\)
\(920\) 1.58067e7 0.615702
\(921\) −1.90779e7 −0.741109
\(922\) 5.43846e6 0.210692
\(923\) −2.52897e6 −0.0977102
\(924\) −3.40439e6 −0.131177
\(925\) −3.00422e7 −1.15446
\(926\) 3.50090e6 0.134169
\(927\) 8.43550e6 0.322412
\(928\) 1.24422e7 0.474273
\(929\) −1.70226e7 −0.647124 −0.323562 0.946207i \(-0.604880\pi\)
−0.323562 + 0.946207i \(0.604880\pi\)
\(930\) 521863. 0.0197856
\(931\) 1.30433e7 0.493188
\(932\) −2.52682e6 −0.0952870
\(933\) 2.25052e7 0.846406
\(934\) −1.14197e7 −0.428340
\(935\) 1.10701e7 0.414117
\(936\) −798515. −0.0297916
\(937\) −2.02964e7 −0.755214 −0.377607 0.925966i \(-0.623253\pi\)
−0.377607 + 0.925966i \(0.623253\pi\)
\(938\) −2.05349e7 −0.762055
\(939\) −2.96929e7 −1.09898
\(940\) −1.58864e6 −0.0586415
\(941\) 4.68227e7 1.72378 0.861891 0.507093i \(-0.169280\pi\)
0.861891 + 0.507093i \(0.169280\pi\)
\(942\) −2.21693e7 −0.814000
\(943\) 6.47275e7 2.37033
\(944\) 2.95363e6 0.107876
\(945\) −2.37117e6 −0.0863742
\(946\) −2.02673e7 −0.736323
\(947\) 1.05919e7 0.383793 0.191897 0.981415i \(-0.438536\pi\)
0.191897 + 0.981415i \(0.438536\pi\)
\(948\) 1.76681e6 0.0638512
\(949\) −272916. −0.00983701
\(950\) −3.14720e7 −1.13140
\(951\) −2.48048e6 −0.0889373
\(952\) 2.79018e7 0.997793
\(953\) 2.94030e7 1.04872 0.524360 0.851497i \(-0.324305\pi\)
0.524360 + 0.851497i \(0.324305\pi\)
\(954\) −1.35580e7 −0.482309
\(955\) 1.78898e7 0.634742
\(956\) 2.11552e6 0.0748638
\(957\) 3.45090e7 1.21801
\(958\) −2.52288e7 −0.888141
\(959\) 2.56933e7 0.902138
\(960\) −7.03513e6 −0.246373
\(961\) −2.83659e7 −0.990806
\(962\) 3.03252e6 0.105649
\(963\) 1.20603e7 0.419074
\(964\) 258351. 0.00895402
\(965\) 1.29150e7 0.446454
\(966\) 2.68211e7 0.924770
\(967\) 4.00215e7 1.37635 0.688173 0.725547i \(-0.258413\pi\)
0.688173 + 0.725547i \(0.258413\pi\)
\(968\) −2.25488e7 −0.773455
\(969\) −1.97920e7 −0.677143
\(970\) 555922. 0.0189707
\(971\) 3.43019e7 1.16753 0.583767 0.811921i \(-0.301578\pi\)
0.583767 + 0.811921i \(0.301578\pi\)
\(972\) −281817. −0.00956757
\(973\) −3.85438e7 −1.30519
\(974\) 3.68344e7 1.24410
\(975\) −1.22805e6 −0.0413718
\(976\) −1.17592e7 −0.395142
\(977\) −5.48788e7 −1.83937 −0.919684 0.392660i \(-0.871555\pi\)
−0.919684 + 0.392660i \(0.871555\pi\)
\(978\) 1.19244e7 0.398646
\(979\) 2.23794e6 0.0746264
\(980\) 593720. 0.0197477
\(981\) −9.36548e6 −0.310712
\(982\) 1.43404e7 0.474549
\(983\) 4.78986e7 1.58102 0.790512 0.612446i \(-0.209814\pi\)
0.790512 + 0.612446i \(0.209814\pi\)
\(984\) −2.93904e7 −0.967648
\(985\) −1.83490e7 −0.602591
\(986\) −3.67076e7 −1.20244
\(987\) −2.07697e7 −0.678637
\(988\) −556858. −0.0181490
\(989\) −2.79887e7 −0.909896
\(990\) −4.83183e6 −0.156684
\(991\) 3.75136e7 1.21340 0.606702 0.794930i \(-0.292492\pi\)
0.606702 + 0.794930i \(0.292492\pi\)
\(992\) −878669. −0.0283495
\(993\) −2.32216e7 −0.747340
\(994\) −3.85704e7 −1.23819
\(995\) 1.04365e7 0.334193
\(996\) −2.61705e6 −0.0835918
\(997\) 3.98840e7 1.27075 0.635375 0.772203i \(-0.280845\pi\)
0.635375 + 0.772203i \(0.280845\pi\)
\(998\) −5.25998e7 −1.67170
\(999\) 8.24626e6 0.261423
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.6.a.c.1.8 12
3.2 odd 2 531.6.a.c.1.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.c.1.8 12 1.1 even 1 trivial
531.6.a.c.1.5 12 3.2 odd 2