Properties

Label 177.6.a.c.1.5
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 1.5
Root \(1.70991\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.290087 q^{2} +9.00000 q^{3} -31.9158 q^{4} +87.1048 q^{5} +2.61079 q^{6} +167.610 q^{7} -18.5412 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+0.290087 q^{2} +9.00000 q^{3} -31.9158 q^{4} +87.1048 q^{5} +2.61079 q^{6} +167.610 q^{7} -18.5412 q^{8} +81.0000 q^{9} +25.2680 q^{10} +252.804 q^{11} -287.243 q^{12} -289.403 q^{13} +48.6216 q^{14} +783.943 q^{15} +1015.93 q^{16} -227.568 q^{17} +23.4971 q^{18} -1162.71 q^{19} -2780.02 q^{20} +1508.49 q^{21} +73.3353 q^{22} -1231.89 q^{23} -166.871 q^{24} +4462.24 q^{25} -83.9520 q^{26} +729.000 q^{27} -5349.42 q^{28} +1975.28 q^{29} +227.412 q^{30} +7791.87 q^{31} +888.026 q^{32} +2275.24 q^{33} -66.0147 q^{34} +14599.6 q^{35} -2585.18 q^{36} +7046.70 q^{37} -337.287 q^{38} -2604.62 q^{39} -1615.03 q^{40} -11251.3 q^{41} +437.594 q^{42} +18519.7 q^{43} -8068.46 q^{44} +7055.49 q^{45} -357.355 q^{46} -20101.5 q^{47} +9143.36 q^{48} +11286.1 q^{49} +1294.44 q^{50} -2048.11 q^{51} +9236.53 q^{52} +37357.8 q^{53} +211.474 q^{54} +22020.5 q^{55} -3107.69 q^{56} -10464.4 q^{57} +573.003 q^{58} -3481.00 q^{59} -25020.2 q^{60} -998.231 q^{61} +2260.32 q^{62} +13576.4 q^{63} -32252.1 q^{64} -25208.3 q^{65} +660.018 q^{66} -19664.7 q^{67} +7263.03 q^{68} -11087.0 q^{69} +4235.17 q^{70} +21610.0 q^{71} -1501.84 q^{72} +46265.2 q^{73} +2044.16 q^{74} +40160.2 q^{75} +37108.8 q^{76} +42372.5 q^{77} -755.568 q^{78} -85014.5 q^{79} +88492.2 q^{80} +6561.00 q^{81} -3263.86 q^{82} +75468.7 q^{83} -48144.8 q^{84} -19822.3 q^{85} +5372.32 q^{86} +17777.5 q^{87} -4687.29 q^{88} +105603. q^{89} +2046.71 q^{90} -48506.8 q^{91} +39316.7 q^{92} +70126.8 q^{93} -5831.19 q^{94} -101277. q^{95} +7992.23 q^{96} -109794. q^{97} +3273.97 q^{98} +20477.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + O(q^{10}) \) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + 601q^{10} + 1480q^{11} + 1782q^{12} + 472q^{13} + 1065q^{14} + 1422q^{15} + 6370q^{16} + 1565q^{17} + 1782q^{18} + 3939q^{19} + 8033q^{20} + 3717q^{21} - 1738q^{22} + 7245q^{23} + 6507q^{24} + 9690q^{25} + 3764q^{26} + 8748q^{27} + 12154q^{28} + 10003q^{29} + 5409q^{30} + 7295q^{31} + 11628q^{32} + 13320q^{33} - 16344q^{34} + 11015q^{35} + 16038q^{36} + 6741q^{37} + 3035q^{38} + 4248q^{39} + 5572q^{40} + 34025q^{41} + 9585q^{42} - 6336q^{43} + 41168q^{44} + 12798q^{45} + 2345q^{46} + 66167q^{47} + 57330q^{48} + 28319q^{49} + 31173q^{50} + 14085q^{51} + 16440q^{52} + 62290q^{53} + 16038q^{54} + 55764q^{55} + 107306q^{56} + 35451q^{57} + 37952q^{58} - 41772q^{59} + 72297q^{60} + 68469q^{61} + 99190q^{62} + 33453q^{63} + 68525q^{64} + 80156q^{65} - 15642q^{66} + 113310q^{67} + 33887q^{68} + 65205q^{69} + 32034q^{70} + 84520q^{71} + 58563q^{72} + 135895q^{73} - 31962q^{74} + 87210q^{75} - 61848q^{76} - 3799q^{77} + 33876q^{78} + 14122q^{79} + 77609q^{80} + 78732q^{81} - 1501q^{82} + 114463q^{83} + 109386q^{84} - 101097q^{85} - 203536q^{86} + 90027q^{87} - 244967q^{88} + 189109q^{89} + 48681q^{90} - 168249q^{91} - 71946q^{92} + 65655q^{93} - 472284q^{94} + 21923q^{95} + 104652q^{96} - 76192q^{97} - 17544q^{98} + 119880q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.290087 0.0512807 0.0256403 0.999671i \(-0.491838\pi\)
0.0256403 + 0.999671i \(0.491838\pi\)
\(3\) 9.00000 0.577350
\(4\) −31.9158 −0.997370
\(5\) 87.1048 1.55818 0.779089 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(6\) 2.61079 0.0296069
\(7\) 167.610 1.29287 0.646435 0.762969i \(-0.276259\pi\)
0.646435 + 0.762969i \(0.276259\pi\)
\(8\) −18.5412 −0.102427
\(9\) 81.0000 0.333333
\(10\) 25.2680 0.0799044
\(11\) 252.804 0.629945 0.314972 0.949101i \(-0.398005\pi\)
0.314972 + 0.949101i \(0.398005\pi\)
\(12\) −287.243 −0.575832
\(13\) −289.403 −0.474946 −0.237473 0.971394i \(-0.576319\pi\)
−0.237473 + 0.971394i \(0.576319\pi\)
\(14\) 48.6216 0.0662993
\(15\) 783.943 0.899614
\(16\) 1015.93 0.992118
\(17\) −227.568 −0.190981 −0.0954903 0.995430i \(-0.530442\pi\)
−0.0954903 + 0.995430i \(0.530442\pi\)
\(18\) 23.4971 0.0170936
\(19\) −1162.71 −0.738902 −0.369451 0.929250i \(-0.620454\pi\)
−0.369451 + 0.929250i \(0.620454\pi\)
\(20\) −2780.02 −1.55408
\(21\) 1508.49 0.746439
\(22\) 73.3353 0.0323040
\(23\) −1231.89 −0.485570 −0.242785 0.970080i \(-0.578061\pi\)
−0.242785 + 0.970080i \(0.578061\pi\)
\(24\) −166.871 −0.0591360
\(25\) 4462.24 1.42792
\(26\) −83.9520 −0.0243555
\(27\) 729.000 0.192450
\(28\) −5349.42 −1.28947
\(29\) 1975.28 0.436147 0.218073 0.975932i \(-0.430023\pi\)
0.218073 + 0.975932i \(0.430023\pi\)
\(30\) 227.412 0.0461328
\(31\) 7791.87 1.45626 0.728128 0.685441i \(-0.240391\pi\)
0.728128 + 0.685441i \(0.240391\pi\)
\(32\) 888.026 0.153303
\(33\) 2275.24 0.363699
\(34\) −66.0147 −0.00979362
\(35\) 14599.6 2.01452
\(36\) −2585.18 −0.332457
\(37\) 7046.70 0.846216 0.423108 0.906079i \(-0.360939\pi\)
0.423108 + 0.906079i \(0.360939\pi\)
\(38\) −337.287 −0.0378914
\(39\) −2604.62 −0.274210
\(40\) −1615.03 −0.159599
\(41\) −11251.3 −1.04530 −0.522652 0.852546i \(-0.675057\pi\)
−0.522652 + 0.852546i \(0.675057\pi\)
\(42\) 437.594 0.0382779
\(43\) 18519.7 1.52743 0.763716 0.645552i \(-0.223373\pi\)
0.763716 + 0.645552i \(0.223373\pi\)
\(44\) −8068.46 −0.628288
\(45\) 7055.49 0.519393
\(46\) −357.355 −0.0249003
\(47\) −20101.5 −1.32735 −0.663673 0.748023i \(-0.731003\pi\)
−0.663673 + 0.748023i \(0.731003\pi\)
\(48\) 9143.36 0.572799
\(49\) 11286.1 0.671514
\(50\) 1294.44 0.0732246
\(51\) −2048.11 −0.110263
\(52\) 9236.53 0.473697
\(53\) 37357.8 1.82681 0.913403 0.407058i \(-0.133445\pi\)
0.913403 + 0.407058i \(0.133445\pi\)
\(54\) 211.474 0.00986897
\(55\) 22020.5 0.981566
\(56\) −3107.69 −0.132424
\(57\) −10464.4 −0.426605
\(58\) 573.003 0.0223659
\(59\) −3481.00 −0.130189
\(60\) −25020.2 −0.897249
\(61\) −998.231 −0.0343484 −0.0171742 0.999853i \(-0.505467\pi\)
−0.0171742 + 0.999853i \(0.505467\pi\)
\(62\) 2260.32 0.0746778
\(63\) 13576.4 0.430957
\(64\) −32252.1 −0.984256
\(65\) −25208.3 −0.740050
\(66\) 660.018 0.0186507
\(67\) −19664.7 −0.535181 −0.267590 0.963533i \(-0.586227\pi\)
−0.267590 + 0.963533i \(0.586227\pi\)
\(68\) 7263.03 0.190478
\(69\) −11087.0 −0.280344
\(70\) 4235.17 0.103306
\(71\) 21610.0 0.508755 0.254378 0.967105i \(-0.418129\pi\)
0.254378 + 0.967105i \(0.418129\pi\)
\(72\) −1501.84 −0.0341422
\(73\) 46265.2 1.01613 0.508063 0.861320i \(-0.330362\pi\)
0.508063 + 0.861320i \(0.330362\pi\)
\(74\) 2044.16 0.0433945
\(75\) 40160.2 0.824409
\(76\) 37108.8 0.736958
\(77\) 42372.5 0.814437
\(78\) −755.568 −0.0140617
\(79\) −85014.5 −1.53259 −0.766294 0.642490i \(-0.777901\pi\)
−0.766294 + 0.642490i \(0.777901\pi\)
\(80\) 88492.2 1.54590
\(81\) 6561.00 0.111111
\(82\) −3263.86 −0.0536039
\(83\) 75468.7 1.20246 0.601231 0.799075i \(-0.294677\pi\)
0.601231 + 0.799075i \(0.294677\pi\)
\(84\) −48144.8 −0.744476
\(85\) −19822.3 −0.297582
\(86\) 5372.32 0.0783278
\(87\) 17777.5 0.251810
\(88\) −4687.29 −0.0645231
\(89\) 105603. 1.41319 0.706596 0.707617i \(-0.250230\pi\)
0.706596 + 0.707617i \(0.250230\pi\)
\(90\) 2046.71 0.0266348
\(91\) −48506.8 −0.614043
\(92\) 39316.7 0.484293
\(93\) 70126.8 0.840769
\(94\) −5831.19 −0.0680672
\(95\) −101277. −1.15134
\(96\) 7992.23 0.0885095
\(97\) −109794. −1.18481 −0.592404 0.805641i \(-0.701821\pi\)
−0.592404 + 0.805641i \(0.701821\pi\)
\(98\) 3273.97 0.0344357
\(99\) 20477.1 0.209982
\(100\) −142416. −1.42416
\(101\) 96278.1 0.939126 0.469563 0.882899i \(-0.344411\pi\)
0.469563 + 0.882899i \(0.344411\pi\)
\(102\) −594.132 −0.00565435
\(103\) −120043. −1.11492 −0.557459 0.830205i \(-0.688223\pi\)
−0.557459 + 0.830205i \(0.688223\pi\)
\(104\) 5365.87 0.0486470
\(105\) 131397. 1.16308
\(106\) 10837.0 0.0936798
\(107\) −37527.7 −0.316878 −0.158439 0.987369i \(-0.550646\pi\)
−0.158439 + 0.987369i \(0.550646\pi\)
\(108\) −23266.7 −0.191944
\(109\) −100498. −0.810201 −0.405101 0.914272i \(-0.632763\pi\)
−0.405101 + 0.914272i \(0.632763\pi\)
\(110\) 6387.85 0.0503354
\(111\) 63420.3 0.488563
\(112\) 170280. 1.28268
\(113\) −96992.3 −0.714564 −0.357282 0.933997i \(-0.616297\pi\)
−0.357282 + 0.933997i \(0.616297\pi\)
\(114\) −3035.58 −0.0218766
\(115\) −107303. −0.756604
\(116\) −63042.6 −0.435000
\(117\) −23441.6 −0.158315
\(118\) −1009.79 −0.00667618
\(119\) −38142.7 −0.246913
\(120\) −14535.2 −0.0921444
\(121\) −97141.1 −0.603170
\(122\) −289.574 −0.00176141
\(123\) −101262. −0.603506
\(124\) −248684. −1.45243
\(125\) 116480. 0.666772
\(126\) 3938.35 0.0220998
\(127\) 22759.5 0.125214 0.0626071 0.998038i \(-0.480058\pi\)
0.0626071 + 0.998038i \(0.480058\pi\)
\(128\) −37772.8 −0.203776
\(129\) 166677. 0.881863
\(130\) −7312.62 −0.0379503
\(131\) −218805. −1.11398 −0.556992 0.830518i \(-0.688045\pi\)
−0.556992 + 0.830518i \(0.688045\pi\)
\(132\) −72616.1 −0.362742
\(133\) −194882. −0.955304
\(134\) −5704.48 −0.0274444
\(135\) 63499.4 0.299871
\(136\) 4219.38 0.0195615
\(137\) −377313. −1.71752 −0.858758 0.512382i \(-0.828763\pi\)
−0.858758 + 0.512382i \(0.828763\pi\)
\(138\) −3216.20 −0.0143762
\(139\) −5026.98 −0.0220683 −0.0110342 0.999939i \(-0.503512\pi\)
−0.0110342 + 0.999939i \(0.503512\pi\)
\(140\) −465960. −2.00922
\(141\) −180914. −0.766343
\(142\) 6268.79 0.0260893
\(143\) −73162.2 −0.299190
\(144\) 82290.2 0.330706
\(145\) 172056. 0.679594
\(146\) 13421.0 0.0521076
\(147\) 101575. 0.387699
\(148\) −224901. −0.843991
\(149\) −165819. −0.611884 −0.305942 0.952050i \(-0.598971\pi\)
−0.305942 + 0.952050i \(0.598971\pi\)
\(150\) 11650.0 0.0422763
\(151\) 305526. 1.09045 0.545225 0.838290i \(-0.316444\pi\)
0.545225 + 0.838290i \(0.316444\pi\)
\(152\) 21558.0 0.0756831
\(153\) −18433.0 −0.0636602
\(154\) 12291.7 0.0417649
\(155\) 678709. 2.26910
\(156\) 83128.8 0.273489
\(157\) −412996. −1.33720 −0.668600 0.743622i \(-0.733106\pi\)
−0.668600 + 0.743622i \(0.733106\pi\)
\(158\) −24661.6 −0.0785921
\(159\) 336221. 1.05471
\(160\) 77351.3 0.238873
\(161\) −206477. −0.627779
\(162\) 1903.26 0.00569785
\(163\) 234544. 0.691441 0.345721 0.938337i \(-0.387634\pi\)
0.345721 + 0.938337i \(0.387634\pi\)
\(164\) 359094. 1.04255
\(165\) 198184. 0.566707
\(166\) 21892.5 0.0616631
\(167\) 138449. 0.384148 0.192074 0.981380i \(-0.438479\pi\)
0.192074 + 0.981380i \(0.438479\pi\)
\(168\) −27969.2 −0.0764552
\(169\) −287539. −0.774427
\(170\) −5750.19 −0.0152602
\(171\) −94179.3 −0.246301
\(172\) −591071. −1.52342
\(173\) 150940. 0.383433 0.191717 0.981450i \(-0.438595\pi\)
0.191717 + 0.981450i \(0.438595\pi\)
\(174\) 5157.02 0.0129130
\(175\) 747917. 1.84611
\(176\) 256831. 0.624979
\(177\) −31329.0 −0.0751646
\(178\) 30634.1 0.0724694
\(179\) 2851.21 0.00665114 0.00332557 0.999994i \(-0.498941\pi\)
0.00332557 + 0.999994i \(0.498941\pi\)
\(180\) −225182. −0.518027
\(181\) 340696. 0.772985 0.386492 0.922293i \(-0.373687\pi\)
0.386492 + 0.922293i \(0.373687\pi\)
\(182\) −14071.2 −0.0314886
\(183\) −8984.08 −0.0198311
\(184\) 22840.7 0.0497352
\(185\) 613801. 1.31856
\(186\) 20342.9 0.0431152
\(187\) −57530.2 −0.120307
\(188\) 641557. 1.32386
\(189\) 122188. 0.248813
\(190\) −29379.3 −0.0590415
\(191\) −101124. −0.200571 −0.100286 0.994959i \(-0.531976\pi\)
−0.100286 + 0.994959i \(0.531976\pi\)
\(192\) −290269. −0.568261
\(193\) 344995. 0.666683 0.333342 0.942806i \(-0.391824\pi\)
0.333342 + 0.942806i \(0.391824\pi\)
\(194\) −31849.8 −0.0607578
\(195\) −226875. −0.427268
\(196\) −360207. −0.669748
\(197\) −506599. −0.930034 −0.465017 0.885302i \(-0.653952\pi\)
−0.465017 + 0.885302i \(0.653952\pi\)
\(198\) 5940.16 0.0107680
\(199\) −221500. −0.396498 −0.198249 0.980152i \(-0.563525\pi\)
−0.198249 + 0.980152i \(0.563525\pi\)
\(200\) −82735.3 −0.146257
\(201\) −176982. −0.308987
\(202\) 27929.1 0.0481591
\(203\) 331076. 0.563882
\(204\) 65367.3 0.109973
\(205\) −980041. −1.62877
\(206\) −34822.9 −0.0571737
\(207\) −99782.9 −0.161857
\(208\) −294012. −0.471202
\(209\) −293937. −0.465467
\(210\) 38116.5 0.0596438
\(211\) −969455. −1.49907 −0.749534 0.661966i \(-0.769723\pi\)
−0.749534 + 0.661966i \(0.769723\pi\)
\(212\) −1.19231e6 −1.82200
\(213\) 194490. 0.293730
\(214\) −10886.3 −0.0162497
\(215\) 1.61315e6 2.38001
\(216\) −13516.5 −0.0197120
\(217\) 1.30600e6 1.88275
\(218\) −29153.3 −0.0415477
\(219\) 416387. 0.586660
\(220\) −702801. −0.978985
\(221\) 65858.8 0.0907054
\(222\) 18397.4 0.0250539
\(223\) −1.33305e6 −1.79509 −0.897544 0.440925i \(-0.854650\pi\)
−0.897544 + 0.440925i \(0.854650\pi\)
\(224\) 148842. 0.198201
\(225\) 361442. 0.475973
\(226\) −28136.2 −0.0366433
\(227\) 734891. 0.946582 0.473291 0.880906i \(-0.343066\pi\)
0.473291 + 0.880906i \(0.343066\pi\)
\(228\) 333979. 0.425483
\(229\) 315332. 0.397355 0.198677 0.980065i \(-0.436335\pi\)
0.198677 + 0.980065i \(0.436335\pi\)
\(230\) −31127.3 −0.0387992
\(231\) 381353. 0.470215
\(232\) −36624.0 −0.0446730
\(233\) 447622. 0.540159 0.270079 0.962838i \(-0.412950\pi\)
0.270079 + 0.962838i \(0.412950\pi\)
\(234\) −6800.12 −0.00811852
\(235\) −1.75094e6 −2.06824
\(236\) 111099. 0.129847
\(237\) −765131. −0.884840
\(238\) −11064.7 −0.0126619
\(239\) 220701. 0.249925 0.124962 0.992161i \(-0.460119\pi\)
0.124962 + 0.992161i \(0.460119\pi\)
\(240\) 796430. 0.892523
\(241\) 643890. 0.714117 0.357058 0.934082i \(-0.383780\pi\)
0.357058 + 0.934082i \(0.383780\pi\)
\(242\) −28179.4 −0.0309310
\(243\) 59049.0 0.0641500
\(244\) 31859.4 0.0342581
\(245\) 983077. 1.04634
\(246\) −29374.7 −0.0309482
\(247\) 336491. 0.350938
\(248\) −144471. −0.149159
\(249\) 679218. 0.694242
\(250\) 33789.5 0.0341925
\(251\) 1.13484e6 1.13697 0.568485 0.822693i \(-0.307530\pi\)
0.568485 + 0.822693i \(0.307530\pi\)
\(252\) −433303. −0.429824
\(253\) −311426. −0.305882
\(254\) 6602.25 0.00642107
\(255\) −178401. −0.171809
\(256\) 1.02111e6 0.973807
\(257\) 157266. 0.148526 0.0742630 0.997239i \(-0.476340\pi\)
0.0742630 + 0.997239i \(0.476340\pi\)
\(258\) 48350.9 0.0452226
\(259\) 1.18110e6 1.09405
\(260\) 804546. 0.738104
\(261\) 159997. 0.145382
\(262\) −63472.6 −0.0571259
\(263\) −943147. −0.840794 −0.420397 0.907340i \(-0.638109\pi\)
−0.420397 + 0.907340i \(0.638109\pi\)
\(264\) −42185.6 −0.0372524
\(265\) 3.25405e6 2.84649
\(266\) −56532.7 −0.0489886
\(267\) 950427. 0.815907
\(268\) 627616. 0.533773
\(269\) 528620. 0.445413 0.222707 0.974885i \(-0.428511\pi\)
0.222707 + 0.974885i \(0.428511\pi\)
\(270\) 18420.4 0.0153776
\(271\) 1.81630e6 1.50233 0.751163 0.660117i \(-0.229493\pi\)
0.751163 + 0.660117i \(0.229493\pi\)
\(272\) −231193. −0.189475
\(273\) −436561. −0.354518
\(274\) −109454. −0.0880754
\(275\) 1.12807e6 0.899509
\(276\) 353851. 0.279607
\(277\) −1.67266e6 −1.30981 −0.654905 0.755711i \(-0.727291\pi\)
−0.654905 + 0.755711i \(0.727291\pi\)
\(278\) −1458.26 −0.00113168
\(279\) 631142. 0.485418
\(280\) −270695. −0.206340
\(281\) 461678. 0.348798 0.174399 0.984675i \(-0.444202\pi\)
0.174399 + 0.984675i \(0.444202\pi\)
\(282\) −52480.7 −0.0392986
\(283\) 880352. 0.653417 0.326708 0.945125i \(-0.394061\pi\)
0.326708 + 0.945125i \(0.394061\pi\)
\(284\) −689702. −0.507418
\(285\) −911497. −0.664726
\(286\) −21223.4 −0.0153426
\(287\) −1.88583e6 −1.35144
\(288\) 71930.1 0.0511010
\(289\) −1.36807e6 −0.963526
\(290\) 49911.3 0.0348501
\(291\) −988143. −0.684049
\(292\) −1.47659e6 −1.01345
\(293\) −2.64738e6 −1.80155 −0.900777 0.434281i \(-0.857002\pi\)
−0.900777 + 0.434281i \(0.857002\pi\)
\(294\) 29465.7 0.0198815
\(295\) −303212. −0.202857
\(296\) −130654. −0.0866750
\(297\) 184294. 0.121233
\(298\) −48102.0 −0.0313778
\(299\) 356512. 0.230619
\(300\) −1.28175e6 −0.822241
\(301\) 3.10408e6 1.97477
\(302\) 88629.2 0.0559190
\(303\) 866503. 0.542205
\(304\) −1.18123e6 −0.733077
\(305\) −86950.7 −0.0535209
\(306\) −5347.19 −0.00326454
\(307\) −2.30973e6 −1.39867 −0.699336 0.714794i \(-0.746521\pi\)
−0.699336 + 0.714794i \(0.746521\pi\)
\(308\) −1.35236e6 −0.812295
\(309\) −1.08038e6 −0.643698
\(310\) 196885. 0.116361
\(311\) −2.74862e6 −1.61144 −0.805719 0.592298i \(-0.798221\pi\)
−0.805719 + 0.592298i \(0.798221\pi\)
\(312\) 48292.8 0.0280864
\(313\) −2.11318e6 −1.21920 −0.609601 0.792708i \(-0.708670\pi\)
−0.609601 + 0.792708i \(0.708670\pi\)
\(314\) −119805. −0.0685725
\(315\) 1.18257e6 0.671507
\(316\) 2.71331e6 1.52856
\(317\) 1.75231e6 0.979407 0.489703 0.871889i \(-0.337105\pi\)
0.489703 + 0.871889i \(0.337105\pi\)
\(318\) 97533.4 0.0540861
\(319\) 499358. 0.274748
\(320\) −2.80931e6 −1.53365
\(321\) −337749. −0.182950
\(322\) −59896.3 −0.0321929
\(323\) 264595. 0.141116
\(324\) −209400. −0.110819
\(325\) −1.29138e6 −0.678184
\(326\) 68038.2 0.0354576
\(327\) −904485. −0.467770
\(328\) 208612. 0.107067
\(329\) −3.36922e6 −1.71609
\(330\) 57490.7 0.0290611
\(331\) −971356. −0.487313 −0.243657 0.969862i \(-0.578347\pi\)
−0.243657 + 0.969862i \(0.578347\pi\)
\(332\) −2.40865e6 −1.19930
\(333\) 570782. 0.282072
\(334\) 40162.3 0.0196994
\(335\) −1.71289e6 −0.833907
\(336\) 1.53252e6 0.740556
\(337\) 1.02817e6 0.493164 0.246582 0.969122i \(-0.420693\pi\)
0.246582 + 0.969122i \(0.420693\pi\)
\(338\) −83411.5 −0.0397131
\(339\) −872931. −0.412554
\(340\) 632645. 0.296799
\(341\) 1.96982e6 0.917360
\(342\) −27320.2 −0.0126305
\(343\) −925352. −0.424690
\(344\) −343376. −0.156450
\(345\) −965730. −0.436825
\(346\) 43785.9 0.0196627
\(347\) 2.24381e6 1.00038 0.500188 0.865917i \(-0.333264\pi\)
0.500188 + 0.865917i \(0.333264\pi\)
\(348\) −567384. −0.251147
\(349\) 2.31221e6 1.01616 0.508081 0.861309i \(-0.330355\pi\)
0.508081 + 0.861309i \(0.330355\pi\)
\(350\) 216961. 0.0946700
\(351\) −210974. −0.0914034
\(352\) 224497. 0.0965724
\(353\) −749517. −0.320144 −0.160072 0.987105i \(-0.551173\pi\)
−0.160072 + 0.987105i \(0.551173\pi\)
\(354\) −9088.15 −0.00385449
\(355\) 1.88234e6 0.792731
\(356\) −3.37041e6 −1.40948
\(357\) −343285. −0.142555
\(358\) 827.099 0.000341075 0
\(359\) 2.49839e6 1.02312 0.511558 0.859249i \(-0.329069\pi\)
0.511558 + 0.859249i \(0.329069\pi\)
\(360\) −130817. −0.0531996
\(361\) −1.12421e6 −0.454024
\(362\) 98831.7 0.0396392
\(363\) −874270. −0.348240
\(364\) 1.54814e6 0.612429
\(365\) 4.02992e6 1.58330
\(366\) −2606.17 −0.00101695
\(367\) −2.57748e6 −0.998920 −0.499460 0.866337i \(-0.666468\pi\)
−0.499460 + 0.866337i \(0.666468\pi\)
\(368\) −1.25151e6 −0.481742
\(369\) −911354. −0.348435
\(370\) 178056. 0.0676164
\(371\) 6.26155e6 2.36182
\(372\) −2.23816e6 −0.838558
\(373\) −999728. −0.372057 −0.186029 0.982544i \(-0.559562\pi\)
−0.186029 + 0.982544i \(0.559562\pi\)
\(374\) −16688.8 −0.00616944
\(375\) 1.04832e6 0.384961
\(376\) 372706. 0.135955
\(377\) −571650. −0.207146
\(378\) 35445.1 0.0127593
\(379\) −3.57477e6 −1.27835 −0.639175 0.769062i \(-0.720724\pi\)
−0.639175 + 0.769062i \(0.720724\pi\)
\(380\) 3.23235e6 1.14831
\(381\) 204836. 0.0722925
\(382\) −29334.7 −0.0102854
\(383\) −47574.2 −0.0165720 −0.00828599 0.999966i \(-0.502638\pi\)
−0.00828599 + 0.999966i \(0.502638\pi\)
\(384\) −339955. −0.117650
\(385\) 3.69085e6 1.26904
\(386\) 100079. 0.0341880
\(387\) 1.50009e6 0.509144
\(388\) 3.50416e6 1.18169
\(389\) −4.27552e6 −1.43257 −0.716284 0.697809i \(-0.754158\pi\)
−0.716284 + 0.697809i \(0.754158\pi\)
\(390\) −65813.6 −0.0219106
\(391\) 280339. 0.0927344
\(392\) −209258. −0.0687809
\(393\) −1.96925e6 −0.643159
\(394\) −146958. −0.0476928
\(395\) −7.40517e6 −2.38804
\(396\) −653545. −0.209429
\(397\) −4.92533e6 −1.56841 −0.784204 0.620504i \(-0.786928\pi\)
−0.784204 + 0.620504i \(0.786928\pi\)
\(398\) −64254.3 −0.0203327
\(399\) −1.75393e6 −0.551545
\(400\) 4.53332e6 1.41666
\(401\) −3.08195e6 −0.957117 −0.478559 0.878056i \(-0.658841\pi\)
−0.478559 + 0.878056i \(0.658841\pi\)
\(402\) −51340.4 −0.0158451
\(403\) −2.25499e6 −0.691642
\(404\) −3.07280e6 −0.936657
\(405\) 571495. 0.173131
\(406\) 96041.0 0.0289162
\(407\) 1.78143e6 0.533069
\(408\) 37974.5 0.0112938
\(409\) −4.03756e6 −1.19347 −0.596734 0.802439i \(-0.703535\pi\)
−0.596734 + 0.802439i \(0.703535\pi\)
\(410\) −284297. −0.0835244
\(411\) −3.39582e6 −0.991608
\(412\) 3.83126e6 1.11199
\(413\) −583451. −0.168317
\(414\) −28945.8 −0.00830012
\(415\) 6.57369e6 1.87365
\(416\) −256997. −0.0728106
\(417\) −45242.8 −0.0127412
\(418\) −85267.5 −0.0238695
\(419\) 3.20507e6 0.891873 0.445936 0.895065i \(-0.352871\pi\)
0.445936 + 0.895065i \(0.352871\pi\)
\(420\) −4.19364e6 −1.16003
\(421\) 3.86174e6 1.06189 0.530943 0.847408i \(-0.321838\pi\)
0.530943 + 0.847408i \(0.321838\pi\)
\(422\) −281227. −0.0768733
\(423\) −1.62822e6 −0.442449
\(424\) −692659. −0.187113
\(425\) −1.01546e6 −0.272705
\(426\) 56419.1 0.0150627
\(427\) −167314. −0.0444080
\(428\) 1.19773e6 0.316045
\(429\) −658460. −0.172737
\(430\) 467955. 0.122049
\(431\) 4.57174e6 1.18547 0.592733 0.805399i \(-0.298049\pi\)
0.592733 + 0.805399i \(0.298049\pi\)
\(432\) 740612. 0.190933
\(433\) 6.35435e6 1.62874 0.814370 0.580346i \(-0.197083\pi\)
0.814370 + 0.580346i \(0.197083\pi\)
\(434\) 378853. 0.0965487
\(435\) 1.54850e6 0.392364
\(436\) 3.20749e6 0.808070
\(437\) 1.43233e6 0.358788
\(438\) 120789. 0.0300843
\(439\) −5.50122e6 −1.36238 −0.681189 0.732108i \(-0.738537\pi\)
−0.681189 + 0.732108i \(0.738537\pi\)
\(440\) −408285. −0.100538
\(441\) 914177. 0.223838
\(442\) 19104.8 0.00465144
\(443\) 3.19936e6 0.774558 0.387279 0.921963i \(-0.373415\pi\)
0.387279 + 0.921963i \(0.373415\pi\)
\(444\) −2.02411e6 −0.487278
\(445\) 9.19852e6 2.20200
\(446\) −386702. −0.0920533
\(447\) −1.49237e6 −0.353271
\(448\) −5.40578e6 −1.27252
\(449\) −4.16986e6 −0.976124 −0.488062 0.872809i \(-0.662296\pi\)
−0.488062 + 0.872809i \(0.662296\pi\)
\(450\) 104850. 0.0244082
\(451\) −2.84437e6 −0.658484
\(452\) 3.09559e6 0.712685
\(453\) 2.74973e6 0.629571
\(454\) 213183. 0.0485414
\(455\) −4.22517e6 −0.956789
\(456\) 194022. 0.0436957
\(457\) 5.33516e6 1.19497 0.597485 0.801880i \(-0.296167\pi\)
0.597485 + 0.801880i \(0.296167\pi\)
\(458\) 91473.7 0.0203766
\(459\) −165897. −0.0367542
\(460\) 3.42468e6 0.754614
\(461\) −5.91111e6 −1.29544 −0.647719 0.761879i \(-0.724277\pi\)
−0.647719 + 0.761879i \(0.724277\pi\)
\(462\) 110626. 0.0241130
\(463\) −7.25730e6 −1.57334 −0.786670 0.617373i \(-0.788197\pi\)
−0.786670 + 0.617373i \(0.788197\pi\)
\(464\) 2.00674e6 0.432709
\(465\) 6.10838e6 1.31007
\(466\) 129849. 0.0276997
\(467\) 8.09273e6 1.71713 0.858565 0.512706i \(-0.171357\pi\)
0.858565 + 0.512706i \(0.171357\pi\)
\(468\) 748159. 0.157899
\(469\) −3.29600e6 −0.691919
\(470\) −507925. −0.106061
\(471\) −3.71696e6 −0.772033
\(472\) 64541.9 0.0133348
\(473\) 4.68185e6 0.962198
\(474\) −221955. −0.0453752
\(475\) −5.18829e6 −1.05509
\(476\) 1.21736e6 0.246264
\(477\) 3.02599e6 0.608935
\(478\) 64022.6 0.0128163
\(479\) 4.31869e6 0.860030 0.430015 0.902822i \(-0.358508\pi\)
0.430015 + 0.902822i \(0.358508\pi\)
\(480\) 696162. 0.137914
\(481\) −2.03933e6 −0.401907
\(482\) 186784. 0.0366204
\(483\) −1.85829e6 −0.362448
\(484\) 3.10034e6 0.601583
\(485\) −9.56356e6 −1.84614
\(486\) 17129.4 0.00328966
\(487\) −134182. −0.0256372 −0.0128186 0.999918i \(-0.504080\pi\)
−0.0128186 + 0.999918i \(0.504080\pi\)
\(488\) 18508.4 0.00351819
\(489\) 2.11090e6 0.399204
\(490\) 285178. 0.0536570
\(491\) 3.28974e6 0.615825 0.307913 0.951415i \(-0.400370\pi\)
0.307913 + 0.951415i \(0.400370\pi\)
\(492\) 3.23185e6 0.601919
\(493\) −449510. −0.0832956
\(494\) 97611.7 0.0179964
\(495\) 1.78366e6 0.327189
\(496\) 7.91599e6 1.44478
\(497\) 3.62205e6 0.657755
\(498\) 197033. 0.0356012
\(499\) 9.24373e6 1.66187 0.830933 0.556373i \(-0.187807\pi\)
0.830933 + 0.556373i \(0.187807\pi\)
\(500\) −3.71757e6 −0.665019
\(501\) 1.24604e6 0.221788
\(502\) 329202. 0.0583046
\(503\) −5.84762e6 −1.03053 −0.515263 0.857032i \(-0.672306\pi\)
−0.515263 + 0.857032i \(0.672306\pi\)
\(504\) −251723. −0.0441414
\(505\) 8.38628e6 1.46333
\(506\) −90340.9 −0.0156858
\(507\) −2.58785e6 −0.447115
\(508\) −726389. −0.124885
\(509\) −9.97745e6 −1.70697 −0.853483 0.521120i \(-0.825514\pi\)
−0.853483 + 0.521120i \(0.825514\pi\)
\(510\) −51751.7 −0.00881048
\(511\) 7.75451e6 1.31372
\(512\) 1.50494e6 0.253714
\(513\) −847614. −0.142202
\(514\) 45620.9 0.00761652
\(515\) −1.04563e7 −1.73724
\(516\) −5.31964e6 −0.879544
\(517\) −5.08174e6 −0.836154
\(518\) 342621. 0.0561035
\(519\) 1.35846e6 0.221375
\(520\) 467393. 0.0758007
\(521\) 6.29007e6 1.01522 0.507611 0.861586i \(-0.330529\pi\)
0.507611 + 0.861586i \(0.330529\pi\)
\(522\) 46413.2 0.00745531
\(523\) 777756. 0.124334 0.0621669 0.998066i \(-0.480199\pi\)
0.0621669 + 0.998066i \(0.480199\pi\)
\(524\) 6.98335e6 1.11106
\(525\) 6.73125e6 1.06585
\(526\) −273595. −0.0431165
\(527\) −1.77318e6 −0.278117
\(528\) 2.31148e6 0.360832
\(529\) −4.91880e6 −0.764222
\(530\) 943958. 0.145970
\(531\) −281961. −0.0433963
\(532\) 6.21981e6 0.952792
\(533\) 3.25615e6 0.496463
\(534\) 275707. 0.0418403
\(535\) −3.26884e6 −0.493753
\(536\) 364607. 0.0548167
\(537\) 25660.9 0.00384004
\(538\) 153346. 0.0228411
\(539\) 2.85318e6 0.423017
\(540\) −2.02664e6 −0.299083
\(541\) 3.53668e6 0.519520 0.259760 0.965673i \(-0.416357\pi\)
0.259760 + 0.965673i \(0.416357\pi\)
\(542\) 526885. 0.0770403
\(543\) 3.06627e6 0.446283
\(544\) −202086. −0.0292779
\(545\) −8.75389e6 −1.26244
\(546\) −126641. −0.0181799
\(547\) −7.93013e6 −1.13321 −0.566607 0.823988i \(-0.691744\pi\)
−0.566607 + 0.823988i \(0.691744\pi\)
\(548\) 1.20423e7 1.71300
\(549\) −80856.7 −0.0114495
\(550\) 327240. 0.0461275
\(551\) −2.29667e6 −0.322270
\(552\) 205566. 0.0287146
\(553\) −1.42493e7 −1.98144
\(554\) −485217. −0.0671679
\(555\) 5.52421e6 0.761268
\(556\) 160440. 0.0220103
\(557\) −1.08864e7 −1.48678 −0.743392 0.668857i \(-0.766784\pi\)
−0.743392 + 0.668857i \(0.766784\pi\)
\(558\) 183086. 0.0248926
\(559\) −5.35964e6 −0.725447
\(560\) 1.48322e7 1.99864
\(561\) −517772. −0.0694594
\(562\) 133927. 0.0178866
\(563\) 3.05550e6 0.406267 0.203133 0.979151i \(-0.434887\pi\)
0.203133 + 0.979151i \(0.434887\pi\)
\(564\) 5.77401e6 0.764328
\(565\) −8.44850e6 −1.11342
\(566\) 255379. 0.0335077
\(567\) 1.09969e6 0.143652
\(568\) −400675. −0.0521100
\(569\) 7.58838e6 0.982581 0.491291 0.870996i \(-0.336525\pi\)
0.491291 + 0.870996i \(0.336525\pi\)
\(570\) −264414. −0.0340876
\(571\) −6.32584e6 −0.811948 −0.405974 0.913885i \(-0.633068\pi\)
−0.405974 + 0.913885i \(0.633068\pi\)
\(572\) 2.33503e6 0.298403
\(573\) −910112. −0.115800
\(574\) −547055. −0.0693029
\(575\) −5.49698e6 −0.693354
\(576\) −2.61242e6 −0.328085
\(577\) −2.66897e6 −0.333737 −0.166868 0.985979i \(-0.553365\pi\)
−0.166868 + 0.985979i \(0.553365\pi\)
\(578\) −396860. −0.0494103
\(579\) 3.10495e6 0.384910
\(580\) −5.49131e6 −0.677807
\(581\) 1.26493e7 1.55463
\(582\) −286648. −0.0350785
\(583\) 9.44422e6 1.15079
\(584\) −857811. −0.104078
\(585\) −2.04188e6 −0.246683
\(586\) −767972. −0.0923850
\(587\) 5.82526e6 0.697782 0.348891 0.937163i \(-0.386558\pi\)
0.348891 + 0.937163i \(0.386558\pi\)
\(588\) −3.24186e6 −0.386679
\(589\) −9.05967e6 −1.07603
\(590\) −87957.9 −0.0104027
\(591\) −4.55939e6 −0.536955
\(592\) 7.15894e6 0.839546
\(593\) −1.39791e7 −1.63246 −0.816232 0.577724i \(-0.803941\pi\)
−0.816232 + 0.577724i \(0.803941\pi\)
\(594\) 53461.4 0.00621691
\(595\) −3.32241e6 −0.384735
\(596\) 5.29226e6 0.610275
\(597\) −1.99350e6 −0.228918
\(598\) 103419. 0.0118263
\(599\) −1.42274e7 −1.62016 −0.810081 0.586319i \(-0.800577\pi\)
−0.810081 + 0.586319i \(0.800577\pi\)
\(600\) −744617. −0.0844413
\(601\) 8.29425e6 0.936679 0.468340 0.883549i \(-0.344852\pi\)
0.468340 + 0.883549i \(0.344852\pi\)
\(602\) 900455. 0.101268
\(603\) −1.59284e6 −0.178394
\(604\) −9.75111e6 −1.08758
\(605\) −8.46145e6 −0.939845
\(606\) 251362. 0.0278046
\(607\) 6.09959e6 0.671938 0.335969 0.941873i \(-0.390936\pi\)
0.335969 + 0.941873i \(0.390936\pi\)
\(608\) −1.03251e6 −0.113276
\(609\) 2.97969e6 0.325557
\(610\) −25223.3 −0.00274459
\(611\) 5.81743e6 0.630417
\(612\) 588306. 0.0634928
\(613\) −9.06279e6 −0.974116 −0.487058 0.873370i \(-0.661930\pi\)
−0.487058 + 0.873370i \(0.661930\pi\)
\(614\) −670024. −0.0717248
\(615\) −8.82037e6 −0.940370
\(616\) −785637. −0.0834200
\(617\) 1.70732e7 1.80552 0.902760 0.430145i \(-0.141537\pi\)
0.902760 + 0.430145i \(0.141537\pi\)
\(618\) −313406. −0.0330093
\(619\) 8.30455e6 0.871143 0.435572 0.900154i \(-0.356546\pi\)
0.435572 + 0.900154i \(0.356546\pi\)
\(620\) −2.16616e7 −2.26314
\(621\) −898046. −0.0934479
\(622\) −797340. −0.0826357
\(623\) 1.77001e7 1.82707
\(624\) −2.64611e6 −0.272049
\(625\) −3.79852e6 −0.388968
\(626\) −613007. −0.0625216
\(627\) −2.64544e6 −0.268738
\(628\) 1.31811e7 1.33368
\(629\) −1.60360e6 −0.161611
\(630\) 343049. 0.0344354
\(631\) 1.07615e7 1.07597 0.537983 0.842956i \(-0.319186\pi\)
0.537983 + 0.842956i \(0.319186\pi\)
\(632\) 1.57627e6 0.156978
\(633\) −8.72509e6 −0.865488
\(634\) 508323. 0.0502246
\(635\) 1.98246e6 0.195106
\(636\) −1.07308e7 −1.05193
\(637\) −3.26624e6 −0.318933
\(638\) 144857. 0.0140893
\(639\) 1.75041e6 0.169585
\(640\) −3.29019e6 −0.317520
\(641\) −1.33893e7 −1.28711 −0.643553 0.765402i \(-0.722540\pi\)
−0.643553 + 0.765402i \(0.722540\pi\)
\(642\) −97976.8 −0.00938179
\(643\) −2.78332e6 −0.265482 −0.132741 0.991151i \(-0.542378\pi\)
−0.132741 + 0.991151i \(0.542378\pi\)
\(644\) 6.58988e6 0.626128
\(645\) 1.45184e7 1.37410
\(646\) 76755.8 0.00723652
\(647\) 2.39129e6 0.224580 0.112290 0.993675i \(-0.464181\pi\)
0.112290 + 0.993675i \(0.464181\pi\)
\(648\) −121649. −0.0113807
\(649\) −880011. −0.0820118
\(650\) −374614. −0.0347777
\(651\) 1.17540e7 1.08701
\(652\) −7.48567e6 −0.689623
\(653\) 1.02862e7 0.943997 0.471998 0.881599i \(-0.343533\pi\)
0.471998 + 0.881599i \(0.343533\pi\)
\(654\) −262380. −0.0239876
\(655\) −1.90590e7 −1.73579
\(656\) −1.14305e7 −1.03706
\(657\) 3.74748e6 0.338708
\(658\) −977367. −0.0880021
\(659\) 1.52742e7 1.37007 0.685036 0.728509i \(-0.259786\pi\)
0.685036 + 0.728509i \(0.259786\pi\)
\(660\) −6.32521e6 −0.565217
\(661\) −2.48992e6 −0.221657 −0.110829 0.993840i \(-0.535350\pi\)
−0.110829 + 0.993840i \(0.535350\pi\)
\(662\) −281778. −0.0249898
\(663\) 592729. 0.0523688
\(664\) −1.39928e6 −0.123164
\(665\) −1.69751e7 −1.48853
\(666\) 165577. 0.0144648
\(667\) −2.43332e6 −0.211780
\(668\) −4.41872e6 −0.383138
\(669\) −1.19975e7 −1.03639
\(670\) −496888. −0.0427633
\(671\) −252357. −0.0216376
\(672\) 1.33958e6 0.114431
\(673\) 1.62042e7 1.37908 0.689542 0.724246i \(-0.257812\pi\)
0.689542 + 0.724246i \(0.257812\pi\)
\(674\) 298260. 0.0252898
\(675\) 3.25298e6 0.274803
\(676\) 9.17706e6 0.772390
\(677\) 8.11683e6 0.680636 0.340318 0.940310i \(-0.389465\pi\)
0.340318 + 0.940310i \(0.389465\pi\)
\(678\) −253226. −0.0211560
\(679\) −1.84025e7 −1.53180
\(680\) 367529. 0.0304803
\(681\) 6.61402e6 0.546510
\(682\) 571419. 0.0470429
\(683\) −1.39257e7 −1.14226 −0.571131 0.820859i \(-0.693495\pi\)
−0.571131 + 0.820859i \(0.693495\pi\)
\(684\) 3.00581e6 0.245653
\(685\) −3.28658e7 −2.67619
\(686\) −268433. −0.0217784
\(687\) 2.83798e6 0.229413
\(688\) 1.88147e7 1.51539
\(689\) −1.08115e7 −0.867633
\(690\) −280146. −0.0224007
\(691\) 2.03394e7 1.62048 0.810238 0.586101i \(-0.199338\pi\)
0.810238 + 0.586101i \(0.199338\pi\)
\(692\) −4.81739e6 −0.382425
\(693\) 3.43217e6 0.271479
\(694\) 650902. 0.0513000
\(695\) −437874. −0.0343864
\(696\) −329616. −0.0257920
\(697\) 2.56043e6 0.199633
\(698\) 670742. 0.0521095
\(699\) 4.02859e6 0.311861
\(700\) −2.38704e7 −1.84126
\(701\) 2.46892e6 0.189763 0.0948815 0.995489i \(-0.469753\pi\)
0.0948815 + 0.995489i \(0.469753\pi\)
\(702\) −61201.0 −0.00468723
\(703\) −8.19325e6 −0.625270
\(704\) −8.15347e6 −0.620027
\(705\) −1.57584e7 −1.19410
\(706\) −217426. −0.0164172
\(707\) 1.61372e7 1.21417
\(708\) 999892. 0.0749669
\(709\) 1.05339e7 0.786994 0.393497 0.919326i \(-0.371265\pi\)
0.393497 + 0.919326i \(0.371265\pi\)
\(710\) 546042. 0.0406518
\(711\) −6.88617e6 −0.510862
\(712\) −1.95800e6 −0.144748
\(713\) −9.59871e6 −0.707113
\(714\) −99582.5 −0.00731034
\(715\) −6.37278e6 −0.466191
\(716\) −90998.7 −0.00663365
\(717\) 1.98631e6 0.144294
\(718\) 724752. 0.0524660
\(719\) 1.89639e7 1.36806 0.684029 0.729455i \(-0.260226\pi\)
0.684029 + 0.729455i \(0.260226\pi\)
\(720\) 7.16787e6 0.515299
\(721\) −2.01204e7 −1.44144
\(722\) −326119. −0.0232827
\(723\) 5.79501e6 0.412296
\(724\) −1.08736e7 −0.770952
\(725\) 8.81416e6 0.622782
\(726\) −253615. −0.0178580
\(727\) −579638. −0.0406744 −0.0203372 0.999793i \(-0.506474\pi\)
−0.0203372 + 0.999793i \(0.506474\pi\)
\(728\) 899373. 0.0628943
\(729\) 531441. 0.0370370
\(730\) 1.16903e6 0.0811929
\(731\) −4.21449e6 −0.291710
\(732\) 286735. 0.0197789
\(733\) −1.22334e7 −0.840980 −0.420490 0.907297i \(-0.638142\pi\)
−0.420490 + 0.907297i \(0.638142\pi\)
\(734\) −747696. −0.0512253
\(735\) 8.84769e6 0.604104
\(736\) −1.09395e6 −0.0744393
\(737\) −4.97132e6 −0.337134
\(738\) −264372. −0.0178680
\(739\) 1.16099e7 0.782017 0.391008 0.920387i \(-0.372126\pi\)
0.391008 + 0.920387i \(0.372126\pi\)
\(740\) −1.95900e7 −1.31509
\(741\) 3.02842e6 0.202614
\(742\) 1.81640e6 0.121116
\(743\) 2.39450e7 1.59127 0.795634 0.605777i \(-0.207138\pi\)
0.795634 + 0.605777i \(0.207138\pi\)
\(744\) −1.30023e6 −0.0861171
\(745\) −1.44436e7 −0.953424
\(746\) −290008. −0.0190793
\(747\) 6.11297e6 0.400821
\(748\) 1.83612e6 0.119991
\(749\) −6.29002e6 −0.409683
\(750\) 304105. 0.0197411
\(751\) 1.54209e6 0.0997720 0.0498860 0.998755i \(-0.484114\pi\)
0.0498860 + 0.998755i \(0.484114\pi\)
\(752\) −2.04217e7 −1.31688
\(753\) 1.02135e7 0.656430
\(754\) −165828. −0.0106226
\(755\) 2.66128e7 1.69911
\(756\) −3.89973e6 −0.248159
\(757\) −2.03724e7 −1.29212 −0.646059 0.763288i \(-0.723584\pi\)
−0.646059 + 0.763288i \(0.723584\pi\)
\(758\) −1.03699e6 −0.0655546
\(759\) −2.80284e6 −0.176601
\(760\) 1.87780e6 0.117928
\(761\) −6.63887e6 −0.415559 −0.207779 0.978176i \(-0.566624\pi\)
−0.207779 + 0.978176i \(0.566624\pi\)
\(762\) 59420.3 0.00370721
\(763\) −1.68445e7 −1.04749
\(764\) 3.22744e6 0.200044
\(765\) −1.60560e6 −0.0991939
\(766\) −13800.7 −0.000849822 0
\(767\) 1.00741e6 0.0618327
\(768\) 9.18999e6 0.562227
\(769\) −1.86945e7 −1.13998 −0.569992 0.821650i \(-0.693054\pi\)
−0.569992 + 0.821650i \(0.693054\pi\)
\(770\) 1.07067e6 0.0650771
\(771\) 1.41540e6 0.0857515
\(772\) −1.10108e7 −0.664930
\(773\) 2.02459e7 1.21867 0.609337 0.792911i \(-0.291436\pi\)
0.609337 + 0.792911i \(0.291436\pi\)
\(774\) 435158. 0.0261093
\(775\) 3.47692e7 2.07941
\(776\) 2.03570e6 0.121356
\(777\) 1.06299e7 0.631649
\(778\) −1.24027e6 −0.0734630
\(779\) 1.30820e7 0.772377
\(780\) 7.24091e6 0.426144
\(781\) 5.46310e6 0.320488
\(782\) 81322.7 0.00475548
\(783\) 1.43998e6 0.0839365
\(784\) 1.14659e7 0.666221
\(785\) −3.59739e7 −2.08359
\(786\) −571253. −0.0329816
\(787\) 1.82754e7 1.05179 0.525895 0.850550i \(-0.323730\pi\)
0.525895 + 0.850550i \(0.323730\pi\)
\(788\) 1.61685e7 0.927588
\(789\) −8.48832e6 −0.485433
\(790\) −2.14815e6 −0.122461
\(791\) −1.62569e7 −0.923839
\(792\) −379670. −0.0215077
\(793\) 288891. 0.0163136
\(794\) −1.42878e6 −0.0804290
\(795\) 2.92864e7 1.64342
\(796\) 7.06935e6 0.395455
\(797\) −2.12442e7 −1.18466 −0.592331 0.805695i \(-0.701792\pi\)
−0.592331 + 0.805695i \(0.701792\pi\)
\(798\) −508794. −0.0282836
\(799\) 4.57446e6 0.253497
\(800\) 3.96259e6 0.218904
\(801\) 8.55384e6 0.471064
\(802\) −894036. −0.0490816
\(803\) 1.16960e7 0.640103
\(804\) 5.64854e6 0.308174
\(805\) −1.79851e7 −0.978191
\(806\) −654144. −0.0354679
\(807\) 4.75758e6 0.257160
\(808\) −1.78511e6 −0.0961915
\(809\) 1.25471e7 0.674020 0.337010 0.941501i \(-0.390584\pi\)
0.337010 + 0.941501i \(0.390584\pi\)
\(810\) 165783. 0.00887827
\(811\) 727251. 0.0388269 0.0194134 0.999812i \(-0.493820\pi\)
0.0194134 + 0.999812i \(0.493820\pi\)
\(812\) −1.05666e7 −0.562399
\(813\) 1.63467e7 0.867368
\(814\) 516772. 0.0273362
\(815\) 2.04299e7 1.07739
\(816\) −2.08074e6 −0.109394
\(817\) −2.15330e7 −1.12862
\(818\) −1.17125e6 −0.0612019
\(819\) −3.92905e6 −0.204681
\(820\) 3.12788e7 1.62449
\(821\) 1.33577e7 0.691629 0.345815 0.938303i \(-0.387603\pi\)
0.345815 + 0.938303i \(0.387603\pi\)
\(822\) −985084. −0.0508503
\(823\) −1.38719e7 −0.713896 −0.356948 0.934124i \(-0.616183\pi\)
−0.356948 + 0.934124i \(0.616183\pi\)
\(824\) 2.22573e6 0.114197
\(825\) 1.01527e7 0.519332
\(826\) −169252. −0.00863143
\(827\) 3.82559e7 1.94507 0.972533 0.232765i \(-0.0747773\pi\)
0.972533 + 0.232765i \(0.0747773\pi\)
\(828\) 3.18466e6 0.161431
\(829\) 1.78067e7 0.899907 0.449954 0.893052i \(-0.351441\pi\)
0.449954 + 0.893052i \(0.351441\pi\)
\(830\) 1.90694e6 0.0960821
\(831\) −1.50539e7 −0.756219
\(832\) 9.33384e6 0.467468
\(833\) −2.56837e6 −0.128246
\(834\) −13124.4 −0.000653376 0
\(835\) 1.20596e7 0.598571
\(836\) 9.38126e6 0.464243
\(837\) 5.68027e6 0.280256
\(838\) 929751. 0.0457359
\(839\) 9.51395e6 0.466612 0.233306 0.972403i \(-0.425046\pi\)
0.233306 + 0.972403i \(0.425046\pi\)
\(840\) −2.43625e6 −0.119131
\(841\) −1.66094e7 −0.809776
\(842\) 1.12024e6 0.0544542
\(843\) 4.15510e6 0.201378
\(844\) 3.09410e7 1.49513
\(845\) −2.50460e7 −1.20669
\(846\) −472327. −0.0226891
\(847\) −1.62818e7 −0.779820
\(848\) 3.79529e7 1.81241
\(849\) 7.92317e6 0.377250
\(850\) −294574. −0.0139845
\(851\) −8.68074e6 −0.410897
\(852\) −6.20732e6 −0.292958
\(853\) −1.04920e7 −0.493727 −0.246864 0.969050i \(-0.579400\pi\)
−0.246864 + 0.969050i \(0.579400\pi\)
\(854\) −48535.6 −0.00227727
\(855\) −8.20347e6 −0.383780
\(856\) 695808. 0.0324567
\(857\) 1.41436e7 0.657821 0.328911 0.944361i \(-0.393318\pi\)
0.328911 + 0.944361i \(0.393318\pi\)
\(858\) −191011. −0.00885808
\(859\) 3.56902e7 1.65031 0.825155 0.564906i \(-0.191088\pi\)
0.825155 + 0.564906i \(0.191088\pi\)
\(860\) −5.14851e7 −2.37375
\(861\) −1.69725e7 −0.780256
\(862\) 1.32621e6 0.0607915
\(863\) 1.25188e7 0.572183 0.286091 0.958202i \(-0.407644\pi\)
0.286091 + 0.958202i \(0.407644\pi\)
\(864\) 647371. 0.0295032
\(865\) 1.31476e7 0.597457
\(866\) 1.84332e6 0.0835229
\(867\) −1.23126e7 −0.556292
\(868\) −4.16820e7 −1.87780
\(869\) −2.14920e7 −0.965445
\(870\) 449201. 0.0201207
\(871\) 5.69102e6 0.254182
\(872\) 1.86336e6 0.0829861
\(873\) −8.89329e6 −0.394936
\(874\) 415500. 0.0183989
\(875\) 1.95233e7 0.862050
\(876\) −1.32893e7 −0.585118
\(877\) −3.08425e7 −1.35410 −0.677050 0.735937i \(-0.736742\pi\)
−0.677050 + 0.735937i \(0.736742\pi\)
\(878\) −1.59583e6 −0.0698637
\(879\) −2.38264e7 −1.04013
\(880\) 2.23712e7 0.973829
\(881\) −1.52828e6 −0.0663379 −0.0331690 0.999450i \(-0.510560\pi\)
−0.0331690 + 0.999450i \(0.510560\pi\)
\(882\) 265191. 0.0114786
\(883\) 1.20855e7 0.521632 0.260816 0.965389i \(-0.416008\pi\)
0.260816 + 0.965389i \(0.416008\pi\)
\(884\) −2.10194e6 −0.0904669
\(885\) −2.72891e6 −0.117120
\(886\) 928094. 0.0397198
\(887\) 1.34386e7 0.573516 0.286758 0.958003i \(-0.407422\pi\)
0.286758 + 0.958003i \(0.407422\pi\)
\(888\) −1.17589e6 −0.0500418
\(889\) 3.81473e6 0.161886
\(890\) 2.66838e6 0.112920
\(891\) 1.65865e6 0.0699939
\(892\) 4.25456e7 1.79037
\(893\) 2.33722e7 0.980778
\(894\) −432918. −0.0181160
\(895\) 248354. 0.0103637
\(896\) −6.33110e6 −0.263456
\(897\) 3.20860e6 0.133148
\(898\) −1.20962e6 −0.0500563
\(899\) 1.53911e7 0.635141
\(900\) −1.15357e7 −0.474721
\(901\) −8.50146e6 −0.348884
\(902\) −825116. −0.0337675
\(903\) 2.79367e7 1.14014
\(904\) 1.79835e6 0.0731903
\(905\) 2.96763e7 1.20445
\(906\) 797662. 0.0322848
\(907\) 4.92075e6 0.198616 0.0993078 0.995057i \(-0.468337\pi\)
0.0993078 + 0.995057i \(0.468337\pi\)
\(908\) −2.34547e7 −0.944093
\(909\) 7.79853e6 0.313042
\(910\) −1.22567e6 −0.0490648
\(911\) −4.20882e7 −1.68021 −0.840106 0.542422i \(-0.817507\pi\)
−0.840106 + 0.542422i \(0.817507\pi\)
\(912\) −1.06311e7 −0.423242
\(913\) 1.90788e7 0.757485
\(914\) 1.54766e6 0.0612789
\(915\) −782556. −0.0309003
\(916\) −1.00641e7 −0.396310
\(917\) −3.66739e7 −1.44024
\(918\) −48124.7 −0.00188478
\(919\) −4.32245e7 −1.68827 −0.844134 0.536133i \(-0.819885\pi\)
−0.844134 + 0.536133i \(0.819885\pi\)
\(920\) 1.98953e6 0.0774963
\(921\) −2.07876e7 −0.807523
\(922\) −1.71474e6 −0.0664310
\(923\) −6.25399e6 −0.241631
\(924\) −1.21712e7 −0.468979
\(925\) 3.14441e7 1.20833
\(926\) −2.10525e6 −0.0806820
\(927\) −9.72346e6 −0.371639
\(928\) 1.75410e6 0.0668626
\(929\) 4.09390e6 0.155632 0.0778158 0.996968i \(-0.475205\pi\)
0.0778158 + 0.996968i \(0.475205\pi\)
\(930\) 1.77197e6 0.0671812
\(931\) −1.31225e7 −0.496183
\(932\) −1.42862e7 −0.538738
\(933\) −2.47376e7 −0.930365
\(934\) 2.34760e6 0.0880556
\(935\) −5.01115e6 −0.187460
\(936\) 434635. 0.0162157
\(937\) 1.35185e6 0.0503015 0.0251507 0.999684i \(-0.491993\pi\)
0.0251507 + 0.999684i \(0.491993\pi\)
\(938\) −956129. −0.0354821
\(939\) −1.90186e7 −0.703907
\(940\) 5.58827e7 2.06280
\(941\) 3.76790e7 1.38715 0.693577 0.720382i \(-0.256034\pi\)
0.693577 + 0.720382i \(0.256034\pi\)
\(942\) −1.07824e6 −0.0395904
\(943\) 1.38603e7 0.507568
\(944\) −3.53645e6 −0.129163
\(945\) 1.06431e7 0.387695
\(946\) 1.35814e6 0.0493422
\(947\) 2.29344e7 0.831024 0.415512 0.909588i \(-0.363603\pi\)
0.415512 + 0.909588i \(0.363603\pi\)
\(948\) 2.44198e7 0.882513
\(949\) −1.33893e7 −0.482604
\(950\) −1.50506e6 −0.0541058
\(951\) 1.57708e7 0.565461
\(952\) 707211. 0.0252905
\(953\) 3.44029e7 1.22705 0.613526 0.789675i \(-0.289751\pi\)
0.613526 + 0.789675i \(0.289751\pi\)
\(954\) 877800. 0.0312266
\(955\) −8.80834e6 −0.312526
\(956\) −7.04386e6 −0.249268
\(957\) 4.49422e6 0.158626
\(958\) 1.25280e6 0.0441029
\(959\) −6.32415e7 −2.22052
\(960\) −2.52838e7 −0.885451
\(961\) 3.20841e7 1.12068
\(962\) −591585. −0.0206101
\(963\) −3.03974e6 −0.105626
\(964\) −2.05503e7 −0.712239
\(965\) 3.00507e7 1.03881
\(966\) −539067. −0.0185866
\(967\) 4.00865e7 1.37858 0.689289 0.724486i \(-0.257923\pi\)
0.689289 + 0.724486i \(0.257923\pi\)
\(968\) 1.80111e6 0.0617806
\(969\) 2.38136e6 0.0814733
\(970\) −2.77427e6 −0.0946714
\(971\) 3.52454e7 1.19965 0.599825 0.800131i \(-0.295237\pi\)
0.599825 + 0.800131i \(0.295237\pi\)
\(972\) −1.88460e6 −0.0639813
\(973\) −842572. −0.0285315
\(974\) −38924.5 −0.00131470
\(975\) −1.16225e7 −0.391549
\(976\) −1.01413e6 −0.0340777
\(977\) −1.35507e7 −0.454178 −0.227089 0.973874i \(-0.572921\pi\)
−0.227089 + 0.973874i \(0.572921\pi\)
\(978\) 612344. 0.0204714
\(979\) 2.66969e7 0.890233
\(980\) −3.13757e7 −1.04359
\(981\) −8.14037e6 −0.270067
\(982\) 954311. 0.0315799
\(983\) 4.74375e7 1.56580 0.782902 0.622145i \(-0.213738\pi\)
0.782902 + 0.622145i \(0.213738\pi\)
\(984\) 1.87751e6 0.0618151
\(985\) −4.41272e7 −1.44916
\(986\) −130397. −0.00427146
\(987\) −3.03229e7 −0.990783
\(988\) −1.07394e7 −0.350015
\(989\) −2.28141e7 −0.741675
\(990\) 517416. 0.0167785
\(991\) 4.59032e7 1.48477 0.742385 0.669974i \(-0.233695\pi\)
0.742385 + 0.669974i \(0.233695\pi\)
\(992\) 6.91938e6 0.223248
\(993\) −8.74220e6 −0.281351
\(994\) 1.05071e6 0.0337301
\(995\) −1.92937e7 −0.617814
\(996\) −2.16778e7 −0.692417
\(997\) 2.51295e6 0.0800655 0.0400327 0.999198i \(-0.487254\pi\)
0.0400327 + 0.999198i \(0.487254\pi\)
\(998\) 2.68149e6 0.0852216
\(999\) 5.13704e6 0.162854
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.1.5 12
3.2 odd 2 531.6.a.c.1.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.c.1.5 12 1.1 even 1 trivial
531.6.a.c.1.8 12 3.2 odd 2