Properties

Label 177.6.a.b.1.3
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $1$
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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + 3566264 x^{5} + 15496192 x^{4} - 53008480 x^{3} - 16576192 x^{2} + 120303168 x - 50564480\)
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 1.3
Root \(8.80731\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.80731 q^{2} -9.00000 q^{3} +45.5687 q^{4} -88.4760 q^{5} +79.2658 q^{6} -134.279 q^{7} -119.504 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-8.80731 q^{2} -9.00000 q^{3} +45.5687 q^{4} -88.4760 q^{5} +79.2658 q^{6} -134.279 q^{7} -119.504 q^{8} +81.0000 q^{9} +779.236 q^{10} +423.269 q^{11} -410.118 q^{12} -713.936 q^{13} +1182.64 q^{14} +796.284 q^{15} -405.693 q^{16} +924.109 q^{17} -713.392 q^{18} -64.3204 q^{19} -4031.74 q^{20} +1208.51 q^{21} -3727.86 q^{22} +3703.37 q^{23} +1075.53 q^{24} +4703.01 q^{25} +6287.86 q^{26} -729.000 q^{27} -6118.92 q^{28} +4990.17 q^{29} -7013.12 q^{30} +3097.37 q^{31} +7397.18 q^{32} -3809.42 q^{33} -8138.92 q^{34} +11880.5 q^{35} +3691.06 q^{36} -4978.34 q^{37} +566.489 q^{38} +6425.43 q^{39} +10573.2 q^{40} +996.199 q^{41} -10643.7 q^{42} +7715.17 q^{43} +19287.8 q^{44} -7166.56 q^{45} -32616.7 q^{46} -17274.4 q^{47} +3651.24 q^{48} +1223.84 q^{49} -41420.8 q^{50} -8316.98 q^{51} -32533.1 q^{52} -34756.3 q^{53} +6420.53 q^{54} -37449.1 q^{55} +16046.8 q^{56} +578.883 q^{57} -43950.0 q^{58} -3481.00 q^{59} +36285.6 q^{60} +33316.0 q^{61} -27279.5 q^{62} -10876.6 q^{63} -52167.1 q^{64} +63166.3 q^{65} +33550.7 q^{66} +50498.8 q^{67} +42110.4 q^{68} -33330.3 q^{69} -104635. q^{70} -16903.5 q^{71} -9679.79 q^{72} -84572.2 q^{73} +43845.8 q^{74} -42327.1 q^{75} -2930.99 q^{76} -56836.1 q^{77} -56590.7 q^{78} +9984.38 q^{79} +35894.1 q^{80} +6561.00 q^{81} -8773.83 q^{82} +11880.4 q^{83} +55070.2 q^{84} -81761.5 q^{85} -67949.9 q^{86} -44911.6 q^{87} -50582.1 q^{88} -8140.62 q^{89} +63118.1 q^{90} +95866.6 q^{91} +168758. q^{92} -27876.3 q^{93} +152141. q^{94} +5690.81 q^{95} -66574.6 q^{96} -137348. q^{97} -10778.7 q^{98} +34284.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10}) \) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.80731 −1.55693 −0.778463 0.627690i \(-0.784001\pi\)
−0.778463 + 0.627690i \(0.784001\pi\)
\(3\) −9.00000 −0.577350
\(4\) 45.5687 1.42402
\(5\) −88.4760 −1.58271 −0.791354 0.611359i \(-0.790623\pi\)
−0.791354 + 0.611359i \(0.790623\pi\)
\(6\) 79.2658 0.898892
\(7\) −134.279 −1.03577 −0.517884 0.855451i \(-0.673280\pi\)
−0.517884 + 0.855451i \(0.673280\pi\)
\(8\) −119.504 −0.660170
\(9\) 81.0000 0.333333
\(10\) 779.236 2.46416
\(11\) 423.269 1.05471 0.527357 0.849644i \(-0.323183\pi\)
0.527357 + 0.849644i \(0.323183\pi\)
\(12\) −410.118 −0.822159
\(13\) −713.936 −1.17166 −0.585829 0.810434i \(-0.699231\pi\)
−0.585829 + 0.810434i \(0.699231\pi\)
\(14\) 1182.64 1.61262
\(15\) 796.284 0.913776
\(16\) −405.693 −0.396184
\(17\) 924.109 0.775534 0.387767 0.921757i \(-0.373246\pi\)
0.387767 + 0.921757i \(0.373246\pi\)
\(18\) −713.392 −0.518976
\(19\) −64.3204 −0.0408756 −0.0204378 0.999791i \(-0.506506\pi\)
−0.0204378 + 0.999791i \(0.506506\pi\)
\(20\) −4031.74 −2.25381
\(21\) 1208.51 0.598001
\(22\) −3727.86 −1.64211
\(23\) 3703.37 1.45975 0.729873 0.683583i \(-0.239579\pi\)
0.729873 + 0.683583i \(0.239579\pi\)
\(24\) 1075.53 0.381150
\(25\) 4703.01 1.50496
\(26\) 6287.86 1.82419
\(27\) −729.000 −0.192450
\(28\) −6118.92 −1.47496
\(29\) 4990.17 1.10185 0.550923 0.834556i \(-0.314276\pi\)
0.550923 + 0.834556i \(0.314276\pi\)
\(30\) −7013.12 −1.42268
\(31\) 3097.37 0.578880 0.289440 0.957196i \(-0.406531\pi\)
0.289440 + 0.957196i \(0.406531\pi\)
\(32\) 7397.18 1.27700
\(33\) −3809.42 −0.608939
\(34\) −8138.92 −1.20745
\(35\) 11880.5 1.63932
\(36\) 3691.06 0.474674
\(37\) −4978.34 −0.597834 −0.298917 0.954279i \(-0.596625\pi\)
−0.298917 + 0.954279i \(0.596625\pi\)
\(38\) 566.489 0.0636404
\(39\) 6425.43 0.676458
\(40\) 10573.2 1.04486
\(41\) 996.199 0.0925522 0.0462761 0.998929i \(-0.485265\pi\)
0.0462761 + 0.998929i \(0.485265\pi\)
\(42\) −10643.7 −0.931044
\(43\) 7715.17 0.636318 0.318159 0.948037i \(-0.396935\pi\)
0.318159 + 0.948037i \(0.396935\pi\)
\(44\) 19287.8 1.50193
\(45\) −7166.56 −0.527569
\(46\) −32616.7 −2.27272
\(47\) −17274.4 −1.14067 −0.570333 0.821414i \(-0.693186\pi\)
−0.570333 + 0.821414i \(0.693186\pi\)
\(48\) 3651.24 0.228737
\(49\) 1223.84 0.0728171
\(50\) −41420.8 −2.34312
\(51\) −8316.98 −0.447755
\(52\) −32533.1 −1.66847
\(53\) −34756.3 −1.69959 −0.849796 0.527112i \(-0.823275\pi\)
−0.849796 + 0.527112i \(0.823275\pi\)
\(54\) 6420.53 0.299631
\(55\) −37449.1 −1.66930
\(56\) 16046.8 0.683784
\(57\) 578.883 0.0235996
\(58\) −43950.0 −1.71549
\(59\) −3481.00 −0.130189
\(60\) 36285.6 1.30124
\(61\) 33316.0 1.14638 0.573190 0.819422i \(-0.305706\pi\)
0.573190 + 0.819422i \(0.305706\pi\)
\(62\) −27279.5 −0.901274
\(63\) −10876.6 −0.345256
\(64\) −52167.1 −1.59201
\(65\) 63166.3 1.85439
\(66\) 33550.7 0.948073
\(67\) 50498.8 1.37434 0.687169 0.726497i \(-0.258853\pi\)
0.687169 + 0.726497i \(0.258853\pi\)
\(68\) 42110.4 1.10438
\(69\) −33330.3 −0.842784
\(70\) −104635. −2.55230
\(71\) −16903.5 −0.397951 −0.198975 0.980004i \(-0.563761\pi\)
−0.198975 + 0.980004i \(0.563761\pi\)
\(72\) −9679.79 −0.220057
\(73\) −84572.2 −1.85746 −0.928732 0.370752i \(-0.879100\pi\)
−0.928732 + 0.370752i \(0.879100\pi\)
\(74\) 43845.8 0.930783
\(75\) −42327.1 −0.868890
\(76\) −2930.99 −0.0582078
\(77\) −56836.1 −1.09244
\(78\) −56590.7 −1.05320
\(79\) 9984.38 0.179992 0.0899960 0.995942i \(-0.471315\pi\)
0.0899960 + 0.995942i \(0.471315\pi\)
\(80\) 35894.1 0.627044
\(81\) 6561.00 0.111111
\(82\) −8773.83 −0.144097
\(83\) 11880.4 0.189293 0.0946467 0.995511i \(-0.469828\pi\)
0.0946467 + 0.995511i \(0.469828\pi\)
\(84\) 55070.2 0.851567
\(85\) −81761.5 −1.22744
\(86\) −67949.9 −0.990701
\(87\) −44911.6 −0.636151
\(88\) −50582.1 −0.696290
\(89\) −8140.62 −0.108939 −0.0544694 0.998515i \(-0.517347\pi\)
−0.0544694 + 0.998515i \(0.517347\pi\)
\(90\) 63118.1 0.821387
\(91\) 95866.6 1.21357
\(92\) 168758. 2.07871
\(93\) −27876.3 −0.334216
\(94\) 152141. 1.77593
\(95\) 5690.81 0.0646942
\(96\) −66574.6 −0.737277
\(97\) −137348. −1.48215 −0.741077 0.671420i \(-0.765685\pi\)
−0.741077 + 0.671420i \(0.765685\pi\)
\(98\) −10778.7 −0.113371
\(99\) 34284.8 0.351571
\(100\) 214310. 2.14310
\(101\) −32965.3 −0.321553 −0.160777 0.986991i \(-0.551400\pi\)
−0.160777 + 0.986991i \(0.551400\pi\)
\(102\) 73250.2 0.697122
\(103\) 168560. 1.56553 0.782764 0.622319i \(-0.213809\pi\)
0.782764 + 0.622319i \(0.213809\pi\)
\(104\) 85318.0 0.773494
\(105\) −106924. −0.946461
\(106\) 306110. 2.64614
\(107\) −36729.3 −0.310137 −0.155069 0.987904i \(-0.549560\pi\)
−0.155069 + 0.987904i \(0.549560\pi\)
\(108\) −33219.6 −0.274053
\(109\) 171480. 1.38244 0.691222 0.722642i \(-0.257073\pi\)
0.691222 + 0.722642i \(0.257073\pi\)
\(110\) 329826. 2.59898
\(111\) 44805.1 0.345159
\(112\) 54476.0 0.410356
\(113\) 10840.2 0.0798624 0.0399312 0.999202i \(-0.487286\pi\)
0.0399312 + 0.999202i \(0.487286\pi\)
\(114\) −5098.40 −0.0367428
\(115\) −327659. −2.31035
\(116\) 227396. 1.56905
\(117\) −57828.9 −0.390553
\(118\) 30658.2 0.202695
\(119\) −124088. −0.803274
\(120\) −95158.8 −0.603248
\(121\) 18105.3 0.112420
\(122\) −293425. −1.78483
\(123\) −8965.79 −0.0534350
\(124\) 141143. 0.824337
\(125\) −139616. −0.799208
\(126\) 95793.5 0.537539
\(127\) −266071. −1.46382 −0.731911 0.681400i \(-0.761371\pi\)
−0.731911 + 0.681400i \(0.761371\pi\)
\(128\) 222742. 1.20165
\(129\) −69436.5 −0.367379
\(130\) −556325. −2.88715
\(131\) 344872. 1.75582 0.877910 0.478826i \(-0.158938\pi\)
0.877910 + 0.478826i \(0.158938\pi\)
\(132\) −173590. −0.867142
\(133\) 8636.87 0.0423377
\(134\) −444758. −2.13975
\(135\) 64499.0 0.304592
\(136\) −110434. −0.511985
\(137\) 299475. 1.36320 0.681600 0.731725i \(-0.261284\pi\)
0.681600 + 0.731725i \(0.261284\pi\)
\(138\) 293550. 1.31215
\(139\) −279442. −1.22675 −0.613373 0.789793i \(-0.710188\pi\)
−0.613373 + 0.789793i \(0.710188\pi\)
\(140\) 541377. 2.33443
\(141\) 155470. 0.658563
\(142\) 148874. 0.619581
\(143\) −302187. −1.23576
\(144\) −32861.1 −0.132061
\(145\) −441511. −1.74390
\(146\) 744853. 2.89193
\(147\) −11014.5 −0.0420410
\(148\) −226856. −0.851328
\(149\) 522957. 1.92974 0.964872 0.262719i \(-0.0846191\pi\)
0.964872 + 0.262719i \(0.0846191\pi\)
\(150\) 372788. 1.35280
\(151\) −225544. −0.804988 −0.402494 0.915423i \(-0.631857\pi\)
−0.402494 + 0.915423i \(0.631857\pi\)
\(152\) 7686.52 0.0269849
\(153\) 74852.9 0.258511
\(154\) 500573. 1.70085
\(155\) −274043. −0.916197
\(156\) 292798. 0.963290
\(157\) 169574. 0.549047 0.274524 0.961580i \(-0.411480\pi\)
0.274524 + 0.961580i \(0.411480\pi\)
\(158\) −87935.5 −0.280234
\(159\) 312807. 0.981259
\(160\) −654473. −2.02112
\(161\) −497284. −1.51196
\(162\) −57784.8 −0.172992
\(163\) 247010. 0.728192 0.364096 0.931361i \(-0.381378\pi\)
0.364096 + 0.931361i \(0.381378\pi\)
\(164\) 45395.5 0.131796
\(165\) 337042. 0.963772
\(166\) −104634. −0.294716
\(167\) −37137.6 −0.103044 −0.0515220 0.998672i \(-0.516407\pi\)
−0.0515220 + 0.998672i \(0.516407\pi\)
\(168\) −144421. −0.394783
\(169\) 138412. 0.372785
\(170\) 720099. 1.91104
\(171\) −5209.95 −0.0136252
\(172\) 351570. 0.906131
\(173\) −49966.1 −0.126929 −0.0634644 0.997984i \(-0.520215\pi\)
−0.0634644 + 0.997984i \(0.520215\pi\)
\(174\) 395550. 0.990440
\(175\) −631515. −1.55879
\(176\) −171717. −0.417861
\(177\) 31329.0 0.0751646
\(178\) 71697.0 0.169610
\(179\) −53182.0 −0.124060 −0.0620300 0.998074i \(-0.519757\pi\)
−0.0620300 + 0.998074i \(0.519757\pi\)
\(180\) −326571. −0.751270
\(181\) −228956. −0.519463 −0.259732 0.965681i \(-0.583634\pi\)
−0.259732 + 0.965681i \(0.583634\pi\)
\(182\) −844327. −1.88944
\(183\) −299844. −0.661863
\(184\) −442566. −0.963681
\(185\) 440464. 0.946195
\(186\) 245515. 0.520351
\(187\) 391146. 0.817966
\(188\) −787171. −1.62433
\(189\) 97889.4 0.199334
\(190\) −50120.7 −0.100724
\(191\) 383018. 0.759689 0.379845 0.925050i \(-0.375977\pi\)
0.379845 + 0.925050i \(0.375977\pi\)
\(192\) 469503. 0.919149
\(193\) −671956. −1.29852 −0.649258 0.760568i \(-0.724921\pi\)
−0.649258 + 0.760568i \(0.724921\pi\)
\(194\) 1.20967e6 2.30761
\(195\) −568496. −1.07063
\(196\) 55768.6 0.103693
\(197\) −517980. −0.950927 −0.475464 0.879735i \(-0.657720\pi\)
−0.475464 + 0.879735i \(0.657720\pi\)
\(198\) −301956. −0.547370
\(199\) 437764. 0.783624 0.391812 0.920045i \(-0.371848\pi\)
0.391812 + 0.920045i \(0.371848\pi\)
\(200\) −562026. −0.993531
\(201\) −454489. −0.793475
\(202\) 290335. 0.500635
\(203\) −670075. −1.14126
\(204\) −378994. −0.637613
\(205\) −88139.8 −0.146483
\(206\) −1.48456e6 −2.43741
\(207\) 299973. 0.486582
\(208\) 289639. 0.464193
\(209\) −27224.8 −0.0431121
\(210\) 941715. 1.47357
\(211\) 795724. 1.23043 0.615214 0.788360i \(-0.289070\pi\)
0.615214 + 0.788360i \(0.289070\pi\)
\(212\) −1.58380e6 −2.42025
\(213\) 152131. 0.229757
\(214\) 323487. 0.482861
\(215\) −682608. −1.00711
\(216\) 87118.1 0.127050
\(217\) −415911. −0.599586
\(218\) −1.51028e6 −2.15237
\(219\) 761149. 1.07241
\(220\) −1.70651e6 −2.37712
\(221\) −659755. −0.908662
\(222\) −394612. −0.537388
\(223\) −1.44232e6 −1.94222 −0.971110 0.238632i \(-0.923301\pi\)
−0.971110 + 0.238632i \(0.923301\pi\)
\(224\) −993285. −1.32268
\(225\) 380944. 0.501654
\(226\) −95473.3 −0.124340
\(227\) −421642. −0.543099 −0.271549 0.962425i \(-0.587536\pi\)
−0.271549 + 0.962425i \(0.587536\pi\)
\(228\) 26379.0 0.0336063
\(229\) −1.41705e6 −1.78565 −0.892823 0.450407i \(-0.851279\pi\)
−0.892823 + 0.450407i \(0.851279\pi\)
\(230\) 2.88580e6 3.59705
\(231\) 511525. 0.630720
\(232\) −596344. −0.727406
\(233\) 673154. 0.812315 0.406158 0.913803i \(-0.366868\pi\)
0.406158 + 0.913803i \(0.366868\pi\)
\(234\) 509317. 0.608062
\(235\) 1.52837e6 1.80534
\(236\) −158625. −0.185392
\(237\) −89859.4 −0.103918
\(238\) 1.09289e6 1.25064
\(239\) −760337. −0.861016 −0.430508 0.902587i \(-0.641666\pi\)
−0.430508 + 0.902587i \(0.641666\pi\)
\(240\) −323047. −0.362024
\(241\) 470539. 0.521859 0.260929 0.965358i \(-0.415971\pi\)
0.260929 + 0.965358i \(0.415971\pi\)
\(242\) −159459. −0.175029
\(243\) −59049.0 −0.0641500
\(244\) 1.51817e6 1.63247
\(245\) −108280. −0.115248
\(246\) 78964.5 0.0831944
\(247\) 45920.7 0.0478923
\(248\) −370147. −0.382159
\(249\) −106923. −0.109289
\(250\) 1.22964e6 1.24431
\(251\) 962815. 0.964625 0.482313 0.875999i \(-0.339797\pi\)
0.482313 + 0.875999i \(0.339797\pi\)
\(252\) −495632. −0.491652
\(253\) 1.56752e6 1.53961
\(254\) 2.34337e6 2.27906
\(255\) 735854. 0.708665
\(256\) −292409. −0.278863
\(257\) −931917. −0.880125 −0.440062 0.897967i \(-0.645044\pi\)
−0.440062 + 0.897967i \(0.645044\pi\)
\(258\) 611549. 0.571982
\(259\) 668486. 0.619217
\(260\) 2.87840e6 2.64070
\(261\) 404204. 0.367282
\(262\) −3.03740e6 −2.73368
\(263\) −1.26124e6 −1.12437 −0.562184 0.827012i \(-0.690039\pi\)
−0.562184 + 0.827012i \(0.690039\pi\)
\(264\) 455239. 0.402003
\(265\) 3.07510e6 2.68996
\(266\) −76067.6 −0.0659167
\(267\) 73265.6 0.0628958
\(268\) 2.30116e6 1.95709
\(269\) −688569. −0.580185 −0.290093 0.956999i \(-0.593686\pi\)
−0.290093 + 0.956999i \(0.593686\pi\)
\(270\) −568063. −0.474228
\(271\) −733019. −0.606306 −0.303153 0.952942i \(-0.598039\pi\)
−0.303153 + 0.952942i \(0.598039\pi\)
\(272\) −374905. −0.307255
\(273\) −862800. −0.700654
\(274\) −2.63757e6 −2.12240
\(275\) 1.99064e6 1.58730
\(276\) −1.51882e6 −1.20014
\(277\) −176065. −0.137871 −0.0689356 0.997621i \(-0.521960\pi\)
−0.0689356 + 0.997621i \(0.521960\pi\)
\(278\) 2.46113e6 1.90995
\(279\) 250887. 0.192960
\(280\) −1.41976e6 −1.08223
\(281\) −1.04068e6 −0.786232 −0.393116 0.919489i \(-0.628603\pi\)
−0.393116 + 0.919489i \(0.628603\pi\)
\(282\) −1.36927e6 −1.02534
\(283\) 715219. 0.530851 0.265426 0.964131i \(-0.414488\pi\)
0.265426 + 0.964131i \(0.414488\pi\)
\(284\) −770268. −0.566691
\(285\) −51217.3 −0.0373512
\(286\) 2.66145e6 1.92399
\(287\) −133769. −0.0958626
\(288\) 599171. 0.425667
\(289\) −565879. −0.398546
\(290\) 3.88852e6 2.71512
\(291\) 1.23613e6 0.855722
\(292\) −3.85384e6 −2.64507
\(293\) 2.03867e6 1.38732 0.693662 0.720301i \(-0.255996\pi\)
0.693662 + 0.720301i \(0.255996\pi\)
\(294\) 97008.4 0.0654547
\(295\) 307985. 0.206051
\(296\) 594930. 0.394672
\(297\) −308563. −0.202980
\(298\) −4.60584e6 −3.00447
\(299\) −2.64397e6 −1.71032
\(300\) −1.92879e6 −1.23732
\(301\) −1.03599e6 −0.659079
\(302\) 1.98644e6 1.25331
\(303\) 296687. 0.185649
\(304\) 26094.3 0.0161943
\(305\) −2.94767e6 −1.81439
\(306\) −659252. −0.402483
\(307\) −1.35132e6 −0.818301 −0.409151 0.912467i \(-0.634175\pi\)
−0.409151 + 0.912467i \(0.634175\pi\)
\(308\) −2.58994e6 −1.55566
\(309\) −1.51704e6 −0.903858
\(310\) 2.41358e6 1.42645
\(311\) −1.87621e6 −1.09997 −0.549985 0.835175i \(-0.685367\pi\)
−0.549985 + 0.835175i \(0.685367\pi\)
\(312\) −767862. −0.446577
\(313\) 2.45653e6 1.41730 0.708649 0.705561i \(-0.249305\pi\)
0.708649 + 0.705561i \(0.249305\pi\)
\(314\) −1.49349e6 −0.854826
\(315\) 962318. 0.546440
\(316\) 454975. 0.256312
\(317\) −1.75984e6 −0.983615 −0.491808 0.870704i \(-0.663664\pi\)
−0.491808 + 0.870704i \(0.663664\pi\)
\(318\) −2.75499e6 −1.52775
\(319\) 2.11218e6 1.16213
\(320\) 4.61553e6 2.51969
\(321\) 330564. 0.179058
\(322\) 4.37974e6 2.35401
\(323\) −59439.1 −0.0317005
\(324\) 298976. 0.158225
\(325\) −3.35765e6 −1.76330
\(326\) −2.17549e6 −1.13374
\(327\) −1.54332e6 −0.798155
\(328\) −119049. −0.0611002
\(329\) 2.31959e6 1.18147
\(330\) −2.96843e6 −1.50052
\(331\) −2.62035e6 −1.31459 −0.657293 0.753635i \(-0.728299\pi\)
−0.657293 + 0.753635i \(0.728299\pi\)
\(332\) 541374. 0.269558
\(333\) −403246. −0.199278
\(334\) 327082. 0.160432
\(335\) −4.46793e6 −2.17518
\(336\) −490284. −0.236919
\(337\) 2.56955e6 1.23249 0.616244 0.787555i \(-0.288653\pi\)
0.616244 + 0.787555i \(0.288653\pi\)
\(338\) −1.21904e6 −0.580398
\(339\) −97562.1 −0.0461086
\(340\) −3.72577e6 −1.74791
\(341\) 1.31102e6 0.610552
\(342\) 45885.6 0.0212135
\(343\) 2.09249e6 0.960347
\(344\) −921991. −0.420079
\(345\) 2.94893e6 1.33388
\(346\) 440067. 0.197619
\(347\) 1.69640e6 0.756318 0.378159 0.925741i \(-0.376557\pi\)
0.378159 + 0.925741i \(0.376557\pi\)
\(348\) −2.04656e6 −0.905892
\(349\) −2.36923e6 −1.04122 −0.520612 0.853794i \(-0.674296\pi\)
−0.520612 + 0.853794i \(0.674296\pi\)
\(350\) 5.56195e6 2.42693
\(351\) 520460. 0.225486
\(352\) 3.13099e6 1.34687
\(353\) 4.33080e6 1.84983 0.924914 0.380175i \(-0.124136\pi\)
0.924914 + 0.380175i \(0.124136\pi\)
\(354\) −275924. −0.117026
\(355\) 1.49555e6 0.629840
\(356\) −370957. −0.155131
\(357\) 1.11680e6 0.463771
\(358\) 468390. 0.193152
\(359\) −1.01517e6 −0.415723 −0.207862 0.978158i \(-0.566650\pi\)
−0.207862 + 0.978158i \(0.566650\pi\)
\(360\) 856429. 0.348285
\(361\) −2.47196e6 −0.998329
\(362\) 2.01648e6 0.808766
\(363\) −162948. −0.0649055
\(364\) 4.36852e6 1.72815
\(365\) 7.48261e6 2.93982
\(366\) 2.64082e6 1.03047
\(367\) −2.21426e6 −0.858151 −0.429076 0.903269i \(-0.641161\pi\)
−0.429076 + 0.903269i \(0.641161\pi\)
\(368\) −1.50243e6 −0.578328
\(369\) 80692.1 0.0308507
\(370\) −3.87930e6 −1.47316
\(371\) 4.66705e6 1.76038
\(372\) −1.27029e6 −0.475931
\(373\) −2.01840e6 −0.751166 −0.375583 0.926789i \(-0.622557\pi\)
−0.375583 + 0.926789i \(0.622557\pi\)
\(374\) −3.44495e6 −1.27351
\(375\) 1.25654e6 0.461423
\(376\) 2.06435e6 0.753033
\(377\) −3.56267e6 −1.29099
\(378\) −862142. −0.310348
\(379\) 4.76169e6 1.70280 0.851398 0.524520i \(-0.175755\pi\)
0.851398 + 0.524520i \(0.175755\pi\)
\(380\) 259323. 0.0921259
\(381\) 2.39464e6 0.845138
\(382\) −3.37336e6 −1.18278
\(383\) 1.00203e6 0.349046 0.174523 0.984653i \(-0.444162\pi\)
0.174523 + 0.984653i \(0.444162\pi\)
\(384\) −2.00467e6 −0.693771
\(385\) 5.02863e6 1.72901
\(386\) 5.91812e6 2.02170
\(387\) 624929. 0.212106
\(388\) −6.25877e6 −2.11062
\(389\) −5.08858e6 −1.70499 −0.852496 0.522734i \(-0.824912\pi\)
−0.852496 + 0.522734i \(0.824912\pi\)
\(390\) 5.00692e6 1.66690
\(391\) 3.42232e6 1.13208
\(392\) −146253. −0.0480717
\(393\) −3.10385e6 −1.01372
\(394\) 4.56201e6 1.48052
\(395\) −883378. −0.284875
\(396\) 1.56231e6 0.500645
\(397\) −4.16172e6 −1.32525 −0.662624 0.748953i \(-0.730557\pi\)
−0.662624 + 0.748953i \(0.730557\pi\)
\(398\) −3.85552e6 −1.22004
\(399\) −77731.9 −0.0244437
\(400\) −1.90798e6 −0.596243
\(401\) 6.12721e6 1.90284 0.951420 0.307897i \(-0.0996252\pi\)
0.951420 + 0.307897i \(0.0996252\pi\)
\(402\) 4.00282e6 1.23538
\(403\) −2.21132e6 −0.678250
\(404\) −1.50218e6 −0.457899
\(405\) −580491. −0.175856
\(406\) 5.90156e6 1.77685
\(407\) −2.10718e6 −0.630543
\(408\) 993910. 0.295595
\(409\) −1.39272e6 −0.411675 −0.205837 0.978586i \(-0.565992\pi\)
−0.205837 + 0.978586i \(0.565992\pi\)
\(410\) 776274. 0.228063
\(411\) −2.69528e6 −0.787044
\(412\) 7.68104e6 2.22934
\(413\) 467425. 0.134846
\(414\) −2.64195e6 −0.757572
\(415\) −1.05113e6 −0.299596
\(416\) −5.28111e6 −1.49621
\(417\) 2.51498e6 0.708262
\(418\) 239777. 0.0671223
\(419\) −4.89970e6 −1.36343 −0.681717 0.731616i \(-0.738767\pi\)
−0.681717 + 0.731616i \(0.738767\pi\)
\(420\) −4.87240e6 −1.34778
\(421\) −1.47517e6 −0.405637 −0.202818 0.979216i \(-0.565010\pi\)
−0.202818 + 0.979216i \(0.565010\pi\)
\(422\) −7.00819e6 −1.91569
\(423\) −1.39923e6 −0.380222
\(424\) 4.15351e6 1.12202
\(425\) 4.34609e6 1.16715
\(426\) −1.33987e6 −0.357715
\(427\) −4.47364e6 −1.18739
\(428\) −1.67371e6 −0.441642
\(429\) 2.71968e6 0.713469
\(430\) 6.01194e6 1.56799
\(431\) −6.48538e6 −1.68168 −0.840838 0.541286i \(-0.817937\pi\)
−0.840838 + 0.541286i \(0.817937\pi\)
\(432\) 295750. 0.0762457
\(433\) 5.86229e6 1.50262 0.751308 0.659952i \(-0.229423\pi\)
0.751308 + 0.659952i \(0.229423\pi\)
\(434\) 3.66306e6 0.933511
\(435\) 3.97360e6 1.00684
\(436\) 7.81413e6 1.96863
\(437\) −238202. −0.0596680
\(438\) −6.70368e6 −1.66966
\(439\) −4.42815e6 −1.09663 −0.548316 0.836271i \(-0.684731\pi\)
−0.548316 + 0.836271i \(0.684731\pi\)
\(440\) 4.47530e6 1.10202
\(441\) 99130.8 0.0242724
\(442\) 5.81067e6 1.41472
\(443\) −7.50913e6 −1.81794 −0.908971 0.416859i \(-0.863131\pi\)
−0.908971 + 0.416859i \(0.863131\pi\)
\(444\) 2.04171e6 0.491514
\(445\) 720250. 0.172418
\(446\) 1.27029e7 3.02389
\(447\) −4.70661e6 −1.11414
\(448\) 7.00494e6 1.64896
\(449\) 1.95240e6 0.457040 0.228520 0.973539i \(-0.426611\pi\)
0.228520 + 0.973539i \(0.426611\pi\)
\(450\) −3.35509e6 −0.781039
\(451\) 421660. 0.0976160
\(452\) 493975. 0.113726
\(453\) 2.02990e6 0.464760
\(454\) 3.71353e6 0.845565
\(455\) −8.48190e6 −1.92072
\(456\) −69178.6 −0.0155797
\(457\) −5.94530e6 −1.33163 −0.665814 0.746118i \(-0.731916\pi\)
−0.665814 + 0.746118i \(0.731916\pi\)
\(458\) 1.24804e7 2.78012
\(459\) −673676. −0.149252
\(460\) −1.49310e7 −3.28999
\(461\) 6.52618e6 1.43023 0.715116 0.699006i \(-0.246374\pi\)
0.715116 + 0.699006i \(0.246374\pi\)
\(462\) −4.50515e6 −0.981985
\(463\) −1.21024e6 −0.262373 −0.131187 0.991358i \(-0.541879\pi\)
−0.131187 + 0.991358i \(0.541879\pi\)
\(464\) −2.02448e6 −0.436534
\(465\) 2.46638e6 0.528967
\(466\) −5.92868e6 −1.26472
\(467\) −3.93565e6 −0.835073 −0.417536 0.908660i \(-0.637106\pi\)
−0.417536 + 0.908660i \(0.637106\pi\)
\(468\) −2.63518e6 −0.556156
\(469\) −6.78092e6 −1.42350
\(470\) −1.34608e7 −2.81078
\(471\) −1.52616e6 −0.316993
\(472\) 415992. 0.0859469
\(473\) 3.26559e6 0.671133
\(474\) 791419. 0.161793
\(475\) −302499. −0.0615163
\(476\) −5.65455e6 −1.14388
\(477\) −2.81526e6 −0.566530
\(478\) 6.69652e6 1.34054
\(479\) −242633. −0.0483183 −0.0241592 0.999708i \(-0.507691\pi\)
−0.0241592 + 0.999708i \(0.507691\pi\)
\(480\) 5.89026e6 1.16689
\(481\) 3.55422e6 0.700457
\(482\) −4.14418e6 −0.812496
\(483\) 4.47556e6 0.872930
\(484\) 825035. 0.160088
\(485\) 1.21520e7 2.34582
\(486\) 520063. 0.0998769
\(487\) −2.57797e6 −0.492556 −0.246278 0.969199i \(-0.579208\pi\)
−0.246278 + 0.969199i \(0.579208\pi\)
\(488\) −3.98139e6 −0.756807
\(489\) −2.22309e6 −0.420422
\(490\) 953657. 0.179433
\(491\) −287405. −0.0538010 −0.0269005 0.999638i \(-0.508564\pi\)
−0.0269005 + 0.999638i \(0.508564\pi\)
\(492\) −408559. −0.0760926
\(493\) 4.61147e6 0.854519
\(494\) −404437. −0.0745648
\(495\) −3.03338e6 −0.556434
\(496\) −1.25658e6 −0.229343
\(497\) 2.26978e6 0.412185
\(498\) 941708. 0.170154
\(499\) 4.52048e6 0.812706 0.406353 0.913716i \(-0.366800\pi\)
0.406353 + 0.913716i \(0.366800\pi\)
\(500\) −6.36211e6 −1.13809
\(501\) 334238. 0.0594925
\(502\) −8.47981e6 −1.50185
\(503\) −2.75026e6 −0.484679 −0.242339 0.970192i \(-0.577915\pi\)
−0.242339 + 0.970192i \(0.577915\pi\)
\(504\) 1.29979e6 0.227928
\(505\) 2.91664e6 0.508925
\(506\) −1.38056e7 −2.39706
\(507\) −1.24571e6 −0.215227
\(508\) −1.21245e7 −2.08451
\(509\) 2.32843e6 0.398354 0.199177 0.979964i \(-0.436173\pi\)
0.199177 + 0.979964i \(0.436173\pi\)
\(510\) −6.48089e6 −1.10334
\(511\) 1.13563e7 1.92390
\(512\) −4.55240e6 −0.767477
\(513\) 46889.6 0.00786652
\(514\) 8.20768e6 1.37029
\(515\) −1.49135e7 −2.47777
\(516\) −3.16413e6 −0.523155
\(517\) −7.31171e6 −1.20307
\(518\) −5.88757e6 −0.964076
\(519\) 449695. 0.0732823
\(520\) −7.54860e6 −1.22422
\(521\) 1.70375e6 0.274986 0.137493 0.990503i \(-0.456096\pi\)
0.137493 + 0.990503i \(0.456096\pi\)
\(522\) −3.55995e6 −0.571831
\(523\) −4.39690e6 −0.702898 −0.351449 0.936207i \(-0.614311\pi\)
−0.351449 + 0.936207i \(0.614311\pi\)
\(524\) 1.57154e7 2.50032
\(525\) 5.68363e6 0.899970
\(526\) 1.11081e7 1.75056
\(527\) 2.86231e6 0.448941
\(528\) 1.54545e6 0.241252
\(529\) 7.27858e6 1.13086
\(530\) −2.70834e7 −4.18806
\(531\) −281961. −0.0433963
\(532\) 393571. 0.0602898
\(533\) −711223. −0.108440
\(534\) −645273. −0.0979242
\(535\) 3.24967e6 0.490856
\(536\) −6.03478e6 −0.907298
\(537\) 478638. 0.0716261
\(538\) 6.06444e6 0.903306
\(539\) 518012. 0.0768011
\(540\) 2.93914e6 0.433746
\(541\) −2.96004e6 −0.434815 −0.217407 0.976081i \(-0.569760\pi\)
−0.217407 + 0.976081i \(0.569760\pi\)
\(542\) 6.45593e6 0.943975
\(543\) 2.06060e6 0.299912
\(544\) 6.83580e6 0.990358
\(545\) −1.51719e7 −2.18801
\(546\) 7.59894e6 1.09087
\(547\) −3.64363e6 −0.520673 −0.260337 0.965518i \(-0.583834\pi\)
−0.260337 + 0.965518i \(0.583834\pi\)
\(548\) 1.36467e7 1.94123
\(549\) 2.69860e6 0.382127
\(550\) −1.75321e7 −2.47132
\(551\) −320970. −0.0450386
\(552\) 3.98309e6 0.556381
\(553\) −1.34069e6 −0.186430
\(554\) 1.55066e6 0.214655
\(555\) −3.96417e6 −0.546286
\(556\) −1.27338e7 −1.74691
\(557\) −5.39423e6 −0.736701 −0.368350 0.929687i \(-0.620077\pi\)
−0.368350 + 0.929687i \(0.620077\pi\)
\(558\) −2.20964e6 −0.300425
\(559\) −5.50814e6 −0.745548
\(560\) −4.81982e6 −0.649473
\(561\) −3.52032e6 −0.472253
\(562\) 9.16558e6 1.22411
\(563\) −8.18538e6 −1.08835 −0.544174 0.838973i \(-0.683157\pi\)
−0.544174 + 0.838973i \(0.683157\pi\)
\(564\) 7.08454e6 0.937808
\(565\) −959101. −0.126399
\(566\) −6.29915e6 −0.826496
\(567\) −881004. −0.115085
\(568\) 2.02002e6 0.262715
\(569\) 1.23194e7 1.59518 0.797590 0.603201i \(-0.206108\pi\)
0.797590 + 0.603201i \(0.206108\pi\)
\(570\) 451087. 0.0581531
\(571\) 1.09533e7 1.40590 0.702948 0.711241i \(-0.251866\pi\)
0.702948 + 0.711241i \(0.251866\pi\)
\(572\) −1.37703e7 −1.75975
\(573\) −3.44716e6 −0.438607
\(574\) 1.17814e6 0.149251
\(575\) 1.74170e7 2.19686
\(576\) −4.22553e6 −0.530671
\(577\) −803138. −0.100427 −0.0502135 0.998739i \(-0.515990\pi\)
−0.0502135 + 0.998739i \(0.515990\pi\)
\(578\) 4.98387e6 0.620508
\(579\) 6.04760e6 0.749699
\(580\) −2.01191e7 −2.48335
\(581\) −1.59529e6 −0.196064
\(582\) −1.08870e7 −1.33230
\(583\) −1.47113e7 −1.79258
\(584\) 1.01067e7 1.22624
\(585\) 5.11647e6 0.618131
\(586\) −1.79552e7 −2.15996
\(587\) 6.51982e6 0.780980 0.390490 0.920607i \(-0.372306\pi\)
0.390490 + 0.920607i \(0.372306\pi\)
\(588\) −501918. −0.0598672
\(589\) −199224. −0.0236621
\(590\) −2.71252e6 −0.320806
\(591\) 4.66182e6 0.549018
\(592\) 2.01968e6 0.236852
\(593\) −2.02927e6 −0.236975 −0.118488 0.992956i \(-0.537805\pi\)
−0.118488 + 0.992956i \(0.537805\pi\)
\(594\) 2.71761e6 0.316024
\(595\) 1.09789e7 1.27135
\(596\) 2.38304e7 2.74800
\(597\) −3.93988e6 −0.452425
\(598\) 2.32862e7 2.66285
\(599\) 667254. 0.0759843 0.0379922 0.999278i \(-0.487904\pi\)
0.0379922 + 0.999278i \(0.487904\pi\)
\(600\) 5.05824e6 0.573616
\(601\) −7.42028e6 −0.837981 −0.418991 0.907991i \(-0.637616\pi\)
−0.418991 + 0.907991i \(0.637616\pi\)
\(602\) 9.12424e6 1.02614
\(603\) 4.09040e6 0.458113
\(604\) −1.02778e7 −1.14632
\(605\) −1.60189e6 −0.177927
\(606\) −2.61302e6 −0.289042
\(607\) −2.14891e6 −0.236727 −0.118363 0.992970i \(-0.537765\pi\)
−0.118363 + 0.992970i \(0.537765\pi\)
\(608\) −475789. −0.0521982
\(609\) 6.03068e6 0.658905
\(610\) 2.59611e7 2.82487
\(611\) 1.23328e7 1.33647
\(612\) 3.41095e6 0.368126
\(613\) 1.69964e7 1.82686 0.913429 0.406999i \(-0.133425\pi\)
0.913429 + 0.406999i \(0.133425\pi\)
\(614\) 1.19015e7 1.27404
\(615\) 793258. 0.0845720
\(616\) 6.79211e6 0.721196
\(617\) −3.94765e6 −0.417470 −0.208735 0.977972i \(-0.566935\pi\)
−0.208735 + 0.977972i \(0.566935\pi\)
\(618\) 1.33610e7 1.40724
\(619\) −4.73731e6 −0.496942 −0.248471 0.968639i \(-0.579928\pi\)
−0.248471 + 0.968639i \(0.579928\pi\)
\(620\) −1.24878e7 −1.30468
\(621\) −2.69975e6 −0.280928
\(622\) 1.65244e7 1.71257
\(623\) 1.09311e6 0.112835
\(624\) −2.60675e6 −0.268002
\(625\) −2.34425e6 −0.240051
\(626\) −2.16354e7 −2.20663
\(627\) 245023. 0.0248908
\(628\) 7.72726e6 0.781855
\(629\) −4.60053e6 −0.463640
\(630\) −8.47543e6 −0.850767
\(631\) −1.34194e7 −1.34172 −0.670859 0.741585i \(-0.734074\pi\)
−0.670859 + 0.741585i \(0.734074\pi\)
\(632\) −1.19317e6 −0.118825
\(633\) −7.16152e6 −0.710388
\(634\) 1.54995e7 1.53142
\(635\) 2.35409e7 2.31680
\(636\) 1.42542e7 1.39733
\(637\) −873742. −0.0853168
\(638\) −1.86027e7 −1.80935
\(639\) −1.36918e6 −0.132650
\(640\) −1.97073e7 −1.90185
\(641\) 1.36524e7 1.31239 0.656196 0.754591i \(-0.272165\pi\)
0.656196 + 0.754591i \(0.272165\pi\)
\(642\) −2.91138e6 −0.278780
\(643\) −918910. −0.0876487 −0.0438244 0.999039i \(-0.513954\pi\)
−0.0438244 + 0.999039i \(0.513954\pi\)
\(644\) −2.26606e7 −2.15306
\(645\) 6.14347e6 0.581453
\(646\) 523498. 0.0493553
\(647\) −1.68376e7 −1.58132 −0.790658 0.612258i \(-0.790262\pi\)
−0.790658 + 0.612258i \(0.790262\pi\)
\(648\) −784063. −0.0733523
\(649\) −1.47340e6 −0.137312
\(650\) 2.95718e7 2.74533
\(651\) 3.74320e6 0.346171
\(652\) 1.12559e7 1.03696
\(653\) 7.85260e6 0.720661 0.360330 0.932825i \(-0.382664\pi\)
0.360330 + 0.932825i \(0.382664\pi\)
\(654\) 1.35925e7 1.24267
\(655\) −3.05129e7 −2.77895
\(656\) −404151. −0.0366677
\(657\) −6.85034e6 −0.619154
\(658\) −2.04293e7 −1.83946
\(659\) 1.37264e7 1.23124 0.615622 0.788042i \(-0.288905\pi\)
0.615622 + 0.788042i \(0.288905\pi\)
\(660\) 1.53586e7 1.37243
\(661\) −1.55788e7 −1.38686 −0.693428 0.720526i \(-0.743901\pi\)
−0.693428 + 0.720526i \(0.743901\pi\)
\(662\) 2.30782e7 2.04672
\(663\) 5.93780e6 0.524616
\(664\) −1.41975e6 −0.124966
\(665\) −764156. −0.0670082
\(666\) 3.55151e6 0.310261
\(667\) 1.84804e7 1.60841
\(668\) −1.69231e6 −0.146737
\(669\) 1.29809e7 1.12134
\(670\) 3.93504e7 3.38659
\(671\) 1.41016e7 1.20910
\(672\) 8.93957e6 0.763648
\(673\) 9.65010e6 0.821285 0.410643 0.911796i \(-0.365304\pi\)
0.410643 + 0.911796i \(0.365304\pi\)
\(674\) −2.26308e7 −1.91889
\(675\) −3.42849e6 −0.289630
\(676\) 6.30727e6 0.530853
\(677\) 2.85548e6 0.239446 0.119723 0.992807i \(-0.461799\pi\)
0.119723 + 0.992807i \(0.461799\pi\)
\(678\) 859260. 0.0717877
\(679\) 1.84430e7 1.53517
\(680\) 9.77080e6 0.810322
\(681\) 3.79477e6 0.313558
\(682\) −1.15465e7 −0.950585
\(683\) −4.39043e6 −0.360127 −0.180063 0.983655i \(-0.557630\pi\)
−0.180063 + 0.983655i \(0.557630\pi\)
\(684\) −237411. −0.0194026
\(685\) −2.64964e7 −2.15755
\(686\) −1.84292e7 −1.49519
\(687\) 1.27534e7 1.03094
\(688\) −3.12999e6 −0.252099
\(689\) 2.48138e7 1.99134
\(690\) −2.59722e7 −2.07676
\(691\) −8.79380e6 −0.700618 −0.350309 0.936634i \(-0.613923\pi\)
−0.350309 + 0.936634i \(0.613923\pi\)
\(692\) −2.27689e6 −0.180749
\(693\) −4.60372e6 −0.364146
\(694\) −1.49407e7 −1.17753
\(695\) 2.47239e7 1.94158
\(696\) 5.36709e6 0.419968
\(697\) 920597. 0.0717774
\(698\) 2.08666e7 1.62111
\(699\) −6.05839e6 −0.468991
\(700\) −2.87773e7 −2.21975
\(701\) 1.57360e7 1.20948 0.604742 0.796421i \(-0.293276\pi\)
0.604742 + 0.796421i \(0.293276\pi\)
\(702\) −4.58385e6 −0.351065
\(703\) 320209. 0.0244368
\(704\) −2.20807e7 −1.67912
\(705\) −1.37553e7 −1.04231
\(706\) −3.81427e7 −2.88005
\(707\) 4.42654e6 0.333055
\(708\) 1.42762e6 0.107036
\(709\) 2.11378e7 1.57922 0.789611 0.613607i \(-0.210282\pi\)
0.789611 + 0.613607i \(0.210282\pi\)
\(710\) −1.31718e7 −0.980615
\(711\) 808734. 0.0599973
\(712\) 972834. 0.0719182
\(713\) 1.14707e7 0.845017
\(714\) −9.83597e6 −0.722057
\(715\) 2.67363e7 1.95585
\(716\) −2.42343e6 −0.176664
\(717\) 6.84303e6 0.497108
\(718\) 8.94095e6 0.647250
\(719\) −2.29229e7 −1.65366 −0.826831 0.562451i \(-0.809859\pi\)
−0.826831 + 0.562451i \(0.809859\pi\)
\(720\) 2.90742e6 0.209015
\(721\) −2.26340e7 −1.62152
\(722\) 2.17713e7 1.55433
\(723\) −4.23485e6 −0.301295
\(724\) −1.04332e7 −0.739727
\(725\) 2.34688e7 1.65824
\(726\) 1.43513e6 0.101053
\(727\) 6.81555e6 0.478261 0.239130 0.970987i \(-0.423138\pi\)
0.239130 + 0.970987i \(0.423138\pi\)
\(728\) −1.14564e7 −0.801161
\(729\) 531441. 0.0370370
\(730\) −6.59016e7 −4.57709
\(731\) 7.12966e6 0.493487
\(732\) −1.36635e7 −0.942507
\(733\) 1.68755e7 1.16011 0.580053 0.814578i \(-0.303032\pi\)
0.580053 + 0.814578i \(0.303032\pi\)
\(734\) 1.95017e7 1.33608
\(735\) 974522. 0.0665385
\(736\) 2.73945e7 1.86410
\(737\) 2.13745e7 1.44953
\(738\) −710681. −0.0480323
\(739\) 6.09722e6 0.410696 0.205348 0.978689i \(-0.434167\pi\)
0.205348 + 0.978689i \(0.434167\pi\)
\(740\) 2.00714e7 1.34740
\(741\) −413286. −0.0276506
\(742\) −4.11041e7 −2.74079
\(743\) 5.82391e6 0.387028 0.193514 0.981097i \(-0.438011\pi\)
0.193514 + 0.981097i \(0.438011\pi\)
\(744\) 3.33132e6 0.220640
\(745\) −4.62691e7 −3.05422
\(746\) 1.77767e7 1.16951
\(747\) 962311. 0.0630978
\(748\) 1.78240e7 1.16480
\(749\) 4.93198e6 0.321230
\(750\) −1.10668e7 −0.718401
\(751\) −1.45028e7 −0.938321 −0.469160 0.883113i \(-0.655443\pi\)
−0.469160 + 0.883113i \(0.655443\pi\)
\(752\) 7.00810e6 0.451914
\(753\) −8.66534e6 −0.556927
\(754\) 3.13775e7 2.00997
\(755\) 1.99553e7 1.27406
\(756\) 4.46069e6 0.283856
\(757\) 2.63506e7 1.67128 0.835642 0.549274i \(-0.185096\pi\)
0.835642 + 0.549274i \(0.185096\pi\)
\(758\) −4.19376e7 −2.65113
\(759\) −1.41077e7 −0.888896
\(760\) −680072. −0.0427092
\(761\) −8.31384e6 −0.520403 −0.260202 0.965554i \(-0.583789\pi\)
−0.260202 + 0.965554i \(0.583789\pi\)
\(762\) −2.10903e7 −1.31582
\(763\) −2.30262e7 −1.43189
\(764\) 1.74536e7 1.08181
\(765\) −6.62268e6 −0.409148
\(766\) −8.82517e6 −0.543439
\(767\) 2.48521e6 0.152537
\(768\) 2.63168e6 0.161001
\(769\) 1.06046e7 0.646666 0.323333 0.946285i \(-0.395197\pi\)
0.323333 + 0.946285i \(0.395197\pi\)
\(770\) −4.42887e7 −2.69194
\(771\) 8.38725e6 0.508140
\(772\) −3.06202e7 −1.84912
\(773\) −2.65938e7 −1.60078 −0.800390 0.599479i \(-0.795374\pi\)
−0.800390 + 0.599479i \(0.795374\pi\)
\(774\) −5.50394e6 −0.330234
\(775\) 1.45669e7 0.871192
\(776\) 1.64136e7 0.978474
\(777\) −6.01638e6 −0.357505
\(778\) 4.48167e7 2.65455
\(779\) −64075.9 −0.00378313
\(780\) −2.59056e7 −1.52461
\(781\) −7.15470e6 −0.419724
\(782\) −3.01414e7 −1.76257
\(783\) −3.63784e6 −0.212050
\(784\) −496502. −0.0288490
\(785\) −1.50032e7 −0.868981
\(786\) 2.73366e7 1.57829
\(787\) −3.05442e7 −1.75789 −0.878945 0.476923i \(-0.841752\pi\)
−0.878945 + 0.476923i \(0.841752\pi\)
\(788\) −2.36037e7 −1.35414
\(789\) 1.13512e7 0.649154
\(790\) 7.78018e6 0.443529
\(791\) −1.45562e6 −0.0827190
\(792\) −4.09715e6 −0.232097
\(793\) −2.37855e7 −1.34317
\(794\) 3.66536e7 2.06331
\(795\) −2.76759e7 −1.55305
\(796\) 1.99483e7 1.11590
\(797\) −2.58869e7 −1.44356 −0.721779 0.692123i \(-0.756675\pi\)
−0.721779 + 0.692123i \(0.756675\pi\)
\(798\) 684608. 0.0380570
\(799\) −1.59634e7 −0.884625
\(800\) 3.47890e7 1.92184
\(801\) −659390. −0.0363129
\(802\) −5.39643e7 −2.96258
\(803\) −3.57967e7 −1.95909
\(804\) −2.07105e7 −1.12993
\(805\) 4.39977e7 2.39299
\(806\) 1.94758e7 1.05599
\(807\) 6.19712e6 0.334970
\(808\) 3.93947e6 0.212280
\(809\) 6.74823e6 0.362509 0.181254 0.983436i \(-0.441984\pi\)
0.181254 + 0.983436i \(0.441984\pi\)
\(810\) 5.11257e6 0.273796
\(811\) −2.22287e7 −1.18676 −0.593378 0.804924i \(-0.702206\pi\)
−0.593378 + 0.804924i \(0.702206\pi\)
\(812\) −3.05345e7 −1.62517
\(813\) 6.59717e6 0.350051
\(814\) 1.85585e7 0.981709
\(815\) −2.18545e7 −1.15251
\(816\) 3.37414e6 0.177394
\(817\) −496243. −0.0260099
\(818\) 1.22661e7 0.640947
\(819\) 7.76520e6 0.404523
\(820\) −4.01641e6 −0.208595
\(821\) −1.57133e7 −0.813599 −0.406799 0.913517i \(-0.633355\pi\)
−0.406799 + 0.913517i \(0.633355\pi\)
\(822\) 2.37382e7 1.22537
\(823\) −2.22414e7 −1.14462 −0.572311 0.820036i \(-0.693953\pi\)
−0.572311 + 0.820036i \(0.693953\pi\)
\(824\) −2.01435e7 −1.03351
\(825\) −1.79157e7 −0.916430
\(826\) −4.11676e6 −0.209945
\(827\) 1.20770e7 0.614038 0.307019 0.951703i \(-0.400668\pi\)
0.307019 + 0.951703i \(0.400668\pi\)
\(828\) 1.36694e7 0.692903
\(829\) −1.91099e6 −0.0965765 −0.0482882 0.998833i \(-0.515377\pi\)
−0.0482882 + 0.998833i \(0.515377\pi\)
\(830\) 9.25762e6 0.466449
\(831\) 1.58458e6 0.0795999
\(832\) 3.72440e7 1.86530
\(833\) 1.13096e6 0.0564722
\(834\) −2.21502e7 −1.10271
\(835\) 3.28579e6 0.163088
\(836\) −1.24060e6 −0.0613925
\(837\) −2.25798e6 −0.111405
\(838\) 4.31532e7 2.12277
\(839\) 3.50301e7 1.71805 0.859025 0.511933i \(-0.171070\pi\)
0.859025 + 0.511933i \(0.171070\pi\)
\(840\) 1.27778e7 0.624826
\(841\) 4.39070e6 0.214064
\(842\) 1.29923e7 0.631547
\(843\) 9.36611e6 0.453931
\(844\) 3.62601e7 1.75216
\(845\) −1.22462e7 −0.590009
\(846\) 1.23234e7 0.591977
\(847\) −2.43116e6 −0.116441
\(848\) 1.41004e7 0.673352
\(849\) −6.43697e6 −0.306487
\(850\) −3.82774e7 −1.81717
\(851\) −1.84366e7 −0.872685
\(852\) 6.93241e6 0.327179
\(853\) −2.10044e7 −0.988412 −0.494206 0.869345i \(-0.664541\pi\)
−0.494206 + 0.869345i \(0.664541\pi\)
\(854\) 3.94008e7 1.84867
\(855\) 460956. 0.0215647
\(856\) 4.38929e6 0.204743
\(857\) 2.13586e7 0.993390 0.496695 0.867925i \(-0.334547\pi\)
0.496695 + 0.867925i \(0.334547\pi\)
\(858\) −2.39531e7 −1.11082
\(859\) 2.37853e7 1.09983 0.549915 0.835221i \(-0.314660\pi\)
0.549915 + 0.835221i \(0.314660\pi\)
\(860\) −3.11055e7 −1.43414
\(861\) 1.20392e6 0.0553463
\(862\) 5.71188e7 2.61825
\(863\) −345138. −0.0157749 −0.00788744 0.999969i \(-0.502511\pi\)
−0.00788744 + 0.999969i \(0.502511\pi\)
\(864\) −5.39254e6 −0.245759
\(865\) 4.42080e6 0.200891
\(866\) −5.16310e7 −2.33946
\(867\) 5.09291e6 0.230101
\(868\) −1.89525e7 −0.853823
\(869\) 4.22607e6 0.189840
\(870\) −3.49967e7 −1.56758
\(871\) −3.60529e7 −1.61026
\(872\) −2.04925e7 −0.912649
\(873\) −1.11252e7 −0.494051
\(874\) 2.09792e6 0.0928988
\(875\) 1.87475e7 0.827794
\(876\) 3.46846e7 1.52713
\(877\) −2.10330e7 −0.923425 −0.461712 0.887030i \(-0.652765\pi\)
−0.461712 + 0.887030i \(0.652765\pi\)
\(878\) 3.90001e7 1.70738
\(879\) −1.83480e7 −0.800971
\(880\) 1.51928e7 0.661352
\(881\) −3.40176e7 −1.47660 −0.738302 0.674470i \(-0.764372\pi\)
−0.738302 + 0.674470i \(0.764372\pi\)
\(882\) −873075. −0.0377903
\(883\) 9.95646e6 0.429737 0.214869 0.976643i \(-0.431068\pi\)
0.214869 + 0.976643i \(0.431068\pi\)
\(884\) −3.00642e7 −1.29395
\(885\) −2.77187e6 −0.118964
\(886\) 6.61352e7 2.83040
\(887\) −1.73362e7 −0.739850 −0.369925 0.929062i \(-0.620617\pi\)
−0.369925 + 0.929062i \(0.620617\pi\)
\(888\) −5.35437e6 −0.227864
\(889\) 3.57277e7 1.51618
\(890\) −6.34346e6 −0.268443
\(891\) 2.77707e6 0.117190
\(892\) −6.57245e7 −2.76576
\(893\) 1.11110e6 0.0466254
\(894\) 4.14526e7 1.73463
\(895\) 4.70533e6 0.196351
\(896\) −2.99095e7 −1.24463
\(897\) 2.37957e7 0.987456
\(898\) −1.71954e7 −0.711577
\(899\) 1.54564e7 0.637836
\(900\) 1.73591e7 0.714366
\(901\) −3.21187e7 −1.31809
\(902\) −3.71369e6 −0.151981
\(903\) 9.32387e6 0.380519
\(904\) −1.29545e6 −0.0527228
\(905\) 2.02571e7 0.822158
\(906\) −1.78779e7 −0.723598
\(907\) 7.69488e6 0.310587 0.155294 0.987868i \(-0.450368\pi\)
0.155294 + 0.987868i \(0.450368\pi\)
\(908\) −1.92137e7 −0.773384
\(909\) −2.67019e6 −0.107184
\(910\) 7.47027e7 2.99042
\(911\) −1.61662e7 −0.645376 −0.322688 0.946505i \(-0.604587\pi\)
−0.322688 + 0.946505i \(0.604587\pi\)
\(912\) −234849. −0.00934978
\(913\) 5.02859e6 0.199650
\(914\) 5.23621e7 2.07325
\(915\) 2.65290e7 1.04754
\(916\) −6.45730e7 −2.54280
\(917\) −4.63091e7 −1.81862
\(918\) 5.93327e6 0.232374
\(919\) −3.62199e7 −1.41468 −0.707341 0.706873i \(-0.750105\pi\)
−0.707341 + 0.706873i \(0.750105\pi\)
\(920\) 3.91564e7 1.52522
\(921\) 1.21619e7 0.472447
\(922\) −5.74780e7 −2.22677
\(923\) 1.20680e7 0.466263
\(924\) 2.33095e7 0.898159
\(925\) −2.34132e7 −0.899717
\(926\) 1.06590e7 0.408496
\(927\) 1.36533e7 0.521842
\(928\) 3.69132e7 1.40706
\(929\) −2.11322e6 −0.0803350 −0.0401675 0.999193i \(-0.512789\pi\)
−0.0401675 + 0.999193i \(0.512789\pi\)
\(930\) −2.17222e7 −0.823563
\(931\) −78717.6 −0.00297644
\(932\) 3.06747e7 1.15675
\(933\) 1.68859e7 0.635068
\(934\) 3.46625e7 1.30015
\(935\) −3.46071e7 −1.29460
\(936\) 6.91076e6 0.257831
\(937\) −1.46913e6 −0.0546652 −0.0273326 0.999626i \(-0.508701\pi\)
−0.0273326 + 0.999626i \(0.508701\pi\)
\(938\) 5.97217e7 2.21628
\(939\) −2.21088e7 −0.818278
\(940\) 6.96458e7 2.57084
\(941\) −3.58757e7 −1.32077 −0.660384 0.750928i \(-0.729607\pi\)
−0.660384 + 0.750928i \(0.729607\pi\)
\(942\) 1.34414e7 0.493534
\(943\) 3.68929e6 0.135103
\(944\) 1.41222e6 0.0515788
\(945\) −8.66086e6 −0.315487
\(946\) −2.87611e7 −1.04491
\(947\) −2.42642e7 −0.879208 −0.439604 0.898192i \(-0.644881\pi\)
−0.439604 + 0.898192i \(0.644881\pi\)
\(948\) −4.09477e6 −0.147982
\(949\) 6.03792e7 2.17631
\(950\) 2.66420e6 0.0957764
\(951\) 1.58386e7 0.567890
\(952\) 1.48290e7 0.530298
\(953\) −3.58333e7 −1.27807 −0.639036 0.769177i \(-0.720666\pi\)
−0.639036 + 0.769177i \(0.720666\pi\)
\(954\) 2.47949e7 0.882046
\(955\) −3.38879e7 −1.20237
\(956\) −3.46475e7 −1.22611
\(957\) −1.90097e7 −0.670957
\(958\) 2.13695e6 0.0752281
\(959\) −4.02132e7 −1.41196
\(960\) −4.15398e7 −1.45474
\(961\) −1.90355e7 −0.664898
\(962\) −3.13031e7 −1.09056
\(963\) −2.97508e6 −0.103379
\(964\) 2.14418e7 0.743138
\(965\) 5.94520e7 2.05517
\(966\) −3.94176e7 −1.35909
\(967\) −3.27974e7 −1.12791 −0.563953 0.825807i \(-0.690720\pi\)
−0.563953 + 0.825807i \(0.690720\pi\)
\(968\) −2.16365e6 −0.0742161
\(969\) 534952. 0.0183023
\(970\) −1.07027e8 −3.65226
\(971\) 5.64198e7 1.92036 0.960182 0.279376i \(-0.0901275\pi\)
0.960182 + 0.279376i \(0.0901275\pi\)
\(972\) −2.69079e6 −0.0913510
\(973\) 3.75232e7 1.27063
\(974\) 2.27050e7 0.766874
\(975\) 3.02188e7 1.01804
\(976\) −1.35161e7 −0.454178
\(977\) −3.59696e7 −1.20559 −0.602794 0.797897i \(-0.705946\pi\)
−0.602794 + 0.797897i \(0.705946\pi\)
\(978\) 1.95794e7 0.654566
\(979\) −3.44567e6 −0.114899
\(980\) −4.93419e6 −0.164116
\(981\) 1.38899e7 0.460815
\(982\) 2.53126e6 0.0837642
\(983\) −119682. −0.00395044 −0.00197522 0.999998i \(-0.500629\pi\)
−0.00197522 + 0.999998i \(0.500629\pi\)
\(984\) 1.07144e6 0.0352762
\(985\) 4.58288e7 1.50504
\(986\) −4.06146e7 −1.33042
\(987\) −2.08763e7 −0.682119
\(988\) 2.09254e6 0.0681997
\(989\) 2.85721e7 0.928863
\(990\) 2.67159e7 0.866327
\(991\) −7.51751e6 −0.243159 −0.121579 0.992582i \(-0.538796\pi\)
−0.121579 + 0.992582i \(0.538796\pi\)
\(992\) 2.29118e7 0.739230
\(993\) 2.35831e7 0.758977
\(994\) −1.99906e7 −0.641742
\(995\) −3.87316e7 −1.24025
\(996\) −4.87236e6 −0.155629
\(997\) 3.03530e7 0.967084 0.483542 0.875321i \(-0.339350\pi\)
0.483542 + 0.875321i \(0.339350\pi\)
\(998\) −3.98133e7 −1.26532
\(999\) 3.62921e6 0.115053
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.b.1.3 12
3.2 odd 2 531.6.a.d.1.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.b.1.3 12 1.1 even 1 trivial
531.6.a.d.1.10 12 3.2 odd 2