Properties

Label 177.6.a.b.1.5
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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 1.5
Root \(2.87969\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.87969 q^{2} -9.00000 q^{3} -23.7074 q^{4} -77.5334 q^{5} +25.9172 q^{6} +45.2937 q^{7} +160.420 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-2.87969 q^{2} -9.00000 q^{3} -23.7074 q^{4} -77.5334 q^{5} +25.9172 q^{6} +45.2937 q^{7} +160.420 q^{8} +81.0000 q^{9} +223.272 q^{10} +155.365 q^{11} +213.367 q^{12} +523.303 q^{13} -130.432 q^{14} +697.801 q^{15} +296.678 q^{16} +1585.45 q^{17} -233.255 q^{18} -2043.33 q^{19} +1838.12 q^{20} -407.643 q^{21} -447.403 q^{22} -1063.14 q^{23} -1443.78 q^{24} +2886.43 q^{25} -1506.95 q^{26} -729.000 q^{27} -1073.80 q^{28} +2231.64 q^{29} -2009.45 q^{30} -8697.85 q^{31} -5987.78 q^{32} -1398.29 q^{33} -4565.61 q^{34} -3511.78 q^{35} -1920.30 q^{36} +15093.5 q^{37} +5884.16 q^{38} -4709.73 q^{39} -12437.9 q^{40} -864.137 q^{41} +1173.88 q^{42} +11392.8 q^{43} -3683.30 q^{44} -6280.21 q^{45} +3061.51 q^{46} +5535.68 q^{47} -2670.11 q^{48} -14755.5 q^{49} -8312.01 q^{50} -14269.1 q^{51} -12406.2 q^{52} +18099.6 q^{53} +2099.29 q^{54} -12046.0 q^{55} +7266.01 q^{56} +18390.0 q^{57} -6426.42 q^{58} -3481.00 q^{59} -16543.0 q^{60} -44361.9 q^{61} +25047.1 q^{62} +3668.79 q^{63} +7749.20 q^{64} -40573.5 q^{65} +4026.62 q^{66} -33357.0 q^{67} -37587.0 q^{68} +9568.25 q^{69} +10112.8 q^{70} +2860.19 q^{71} +12994.0 q^{72} -63785.8 q^{73} -43464.5 q^{74} -25977.9 q^{75} +48442.2 q^{76} +7037.06 q^{77} +13562.5 q^{78} +82957.4 q^{79} -23002.5 q^{80} +6561.00 q^{81} +2488.44 q^{82} -67941.5 q^{83} +9664.17 q^{84} -122926. q^{85} -32807.6 q^{86} -20084.8 q^{87} +24923.6 q^{88} +22542.4 q^{89} +18085.0 q^{90} +23702.3 q^{91} +25204.3 q^{92} +78280.6 q^{93} -15941.0 q^{94} +158427. q^{95} +53890.0 q^{96} +26284.2 q^{97} +42491.1 q^{98} +12584.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10}) \) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.87969 −0.509061 −0.254531 0.967065i \(-0.581921\pi\)
−0.254531 + 0.967065i \(0.581921\pi\)
\(3\) −9.00000 −0.577350
\(4\) −23.7074 −0.740857
\(5\) −77.5334 −1.38696 −0.693480 0.720476i \(-0.743923\pi\)
−0.693480 + 0.720476i \(0.743923\pi\)
\(6\) 25.9172 0.293907
\(7\) 45.2937 0.349376 0.174688 0.984624i \(-0.444108\pi\)
0.174688 + 0.984624i \(0.444108\pi\)
\(8\) 160.420 0.886203
\(9\) 81.0000 0.333333
\(10\) 223.272 0.706048
\(11\) 155.365 0.387143 0.193572 0.981086i \(-0.437993\pi\)
0.193572 + 0.981086i \(0.437993\pi\)
\(12\) 213.367 0.427734
\(13\) 523.303 0.858806 0.429403 0.903113i \(-0.358724\pi\)
0.429403 + 0.903113i \(0.358724\pi\)
\(14\) −130.432 −0.177854
\(15\) 697.801 0.800762
\(16\) 296.678 0.289725
\(17\) 1585.45 1.33055 0.665275 0.746598i \(-0.268314\pi\)
0.665275 + 0.746598i \(0.268314\pi\)
\(18\) −233.255 −0.169687
\(19\) −2043.33 −1.29854 −0.649270 0.760558i \(-0.724926\pi\)
−0.649270 + 0.760558i \(0.724926\pi\)
\(20\) 1838.12 1.02754
\(21\) −407.643 −0.201712
\(22\) −447.403 −0.197080
\(23\) −1063.14 −0.419055 −0.209527 0.977803i \(-0.567193\pi\)
−0.209527 + 0.977803i \(0.567193\pi\)
\(24\) −1443.78 −0.511649
\(25\) 2886.43 0.923657
\(26\) −1506.95 −0.437185
\(27\) −729.000 −0.192450
\(28\) −1073.80 −0.258837
\(29\) 2231.64 0.492753 0.246376 0.969174i \(-0.420760\pi\)
0.246376 + 0.969174i \(0.420760\pi\)
\(30\) −2009.45 −0.407637
\(31\) −8697.85 −1.62558 −0.812788 0.582559i \(-0.802051\pi\)
−0.812788 + 0.582559i \(0.802051\pi\)
\(32\) −5987.78 −1.03369
\(33\) −1398.29 −0.223517
\(34\) −4565.61 −0.677332
\(35\) −3511.78 −0.484570
\(36\) −1920.30 −0.246952
\(37\) 15093.5 1.81253 0.906265 0.422711i \(-0.138921\pi\)
0.906265 + 0.422711i \(0.138921\pi\)
\(38\) 5884.16 0.661037
\(39\) −4709.73 −0.495832
\(40\) −12437.9 −1.22913
\(41\) −864.137 −0.0802829 −0.0401414 0.999194i \(-0.512781\pi\)
−0.0401414 + 0.999194i \(0.512781\pi\)
\(42\) 1173.88 0.102684
\(43\) 11392.8 0.939633 0.469817 0.882764i \(-0.344320\pi\)
0.469817 + 0.882764i \(0.344320\pi\)
\(44\) −3683.30 −0.286818
\(45\) −6280.21 −0.462320
\(46\) 3061.51 0.213324
\(47\) 5535.68 0.365533 0.182767 0.983156i \(-0.441495\pi\)
0.182767 + 0.983156i \(0.441495\pi\)
\(48\) −2670.11 −0.167273
\(49\) −14755.5 −0.877937
\(50\) −8312.01 −0.470198
\(51\) −14269.1 −0.768194
\(52\) −12406.2 −0.636252
\(53\) 18099.6 0.885071 0.442536 0.896751i \(-0.354079\pi\)
0.442536 + 0.896751i \(0.354079\pi\)
\(54\) 2099.29 0.0979689
\(55\) −12046.0 −0.536952
\(56\) 7266.01 0.309618
\(57\) 18390.0 0.749713
\(58\) −6426.42 −0.250841
\(59\) −3481.00 −0.130189
\(60\) −16543.0 −0.593249
\(61\) −44361.9 −1.52646 −0.763230 0.646127i \(-0.776387\pi\)
−0.763230 + 0.646127i \(0.776387\pi\)
\(62\) 25047.1 0.827518
\(63\) 3668.79 0.116459
\(64\) 7749.20 0.236487
\(65\) −40573.5 −1.19113
\(66\) 4026.62 0.113784
\(67\) −33357.0 −0.907821 −0.453910 0.891047i \(-0.649971\pi\)
−0.453910 + 0.891047i \(0.649971\pi\)
\(68\) −37587.0 −0.985747
\(69\) 9568.25 0.241941
\(70\) 10112.8 0.246676
\(71\) 2860.19 0.0673363 0.0336681 0.999433i \(-0.489281\pi\)
0.0336681 + 0.999433i \(0.489281\pi\)
\(72\) 12994.0 0.295401
\(73\) −63785.8 −1.40093 −0.700466 0.713686i \(-0.747024\pi\)
−0.700466 + 0.713686i \(0.747024\pi\)
\(74\) −43464.5 −0.922689
\(75\) −25977.9 −0.533274
\(76\) 48442.2 0.962032
\(77\) 7037.06 0.135258
\(78\) 13562.5 0.252409
\(79\) 82957.4 1.49550 0.747751 0.663979i \(-0.231133\pi\)
0.747751 + 0.663979i \(0.231133\pi\)
\(80\) −23002.5 −0.401837
\(81\) 6561.00 0.111111
\(82\) 2488.44 0.0408689
\(83\) −67941.5 −1.08253 −0.541265 0.840852i \(-0.682054\pi\)
−0.541265 + 0.840852i \(0.682054\pi\)
\(84\) 9664.17 0.149440
\(85\) −122926. −1.84542
\(86\) −32807.6 −0.478331
\(87\) −20084.8 −0.284491
\(88\) 24923.6 0.343087
\(89\) 22542.4 0.301665 0.150833 0.988559i \(-0.451805\pi\)
0.150833 + 0.988559i \(0.451805\pi\)
\(90\) 18085.0 0.235349
\(91\) 23702.3 0.300046
\(92\) 25204.3 0.310459
\(93\) 78280.6 0.938527
\(94\) −15941.0 −0.186079
\(95\) 158427. 1.80102
\(96\) 53890.0 0.596802
\(97\) 26284.2 0.283639 0.141820 0.989893i \(-0.454705\pi\)
0.141820 + 0.989893i \(0.454705\pi\)
\(98\) 42491.1 0.446924
\(99\) 12584.6 0.129048
\(100\) −68429.7 −0.684297
\(101\) −61695.6 −0.601798 −0.300899 0.953656i \(-0.597287\pi\)
−0.300899 + 0.953656i \(0.597287\pi\)
\(102\) 41090.5 0.391058
\(103\) −65169.6 −0.605274 −0.302637 0.953106i \(-0.597867\pi\)
−0.302637 + 0.953106i \(0.597867\pi\)
\(104\) 83948.2 0.761076
\(105\) 31606.0 0.279767
\(106\) −52121.0 −0.450556
\(107\) 153050. 1.29233 0.646167 0.763196i \(-0.276371\pi\)
0.646167 + 0.763196i \(0.276371\pi\)
\(108\) 17282.7 0.142578
\(109\) −203743. −1.64254 −0.821271 0.570539i \(-0.806734\pi\)
−0.821271 + 0.570539i \(0.806734\pi\)
\(110\) 34688.6 0.273342
\(111\) −135841. −1.04646
\(112\) 13437.7 0.101223
\(113\) 240950. 1.77514 0.887568 0.460678i \(-0.152394\pi\)
0.887568 + 0.460678i \(0.152394\pi\)
\(114\) −52957.5 −0.381650
\(115\) 82428.8 0.581212
\(116\) −52906.4 −0.365059
\(117\) 42387.5 0.286269
\(118\) 10024.2 0.0662741
\(119\) 71811.1 0.464862
\(120\) 111941. 0.709637
\(121\) −136913. −0.850120
\(122\) 127748. 0.777061
\(123\) 7777.23 0.0463513
\(124\) 206203. 1.20432
\(125\) 18497.3 0.105885
\(126\) −10565.0 −0.0592846
\(127\) −150990. −0.830689 −0.415344 0.909664i \(-0.636339\pi\)
−0.415344 + 0.909664i \(0.636339\pi\)
\(128\) 169294. 0.913304
\(129\) −102535. −0.542498
\(130\) 116839. 0.606358
\(131\) −250601. −1.27586 −0.637932 0.770092i \(-0.720210\pi\)
−0.637932 + 0.770092i \(0.720210\pi\)
\(132\) 33149.7 0.165594
\(133\) −92550.2 −0.453679
\(134\) 96057.7 0.462136
\(135\) 56521.8 0.266921
\(136\) 254338. 1.17914
\(137\) −15685.5 −0.0713996 −0.0356998 0.999363i \(-0.511366\pi\)
−0.0356998 + 0.999363i \(0.511366\pi\)
\(138\) −27553.6 −0.123163
\(139\) 67099.0 0.294563 0.147282 0.989095i \(-0.452948\pi\)
0.147282 + 0.989095i \(0.452948\pi\)
\(140\) 83255.1 0.358997
\(141\) −49821.2 −0.211041
\(142\) −8236.45 −0.0342783
\(143\) 81303.0 0.332481
\(144\) 24030.9 0.0965750
\(145\) −173027. −0.683428
\(146\) 183683. 0.713160
\(147\) 132799. 0.506877
\(148\) −357827. −1.34282
\(149\) 109381. 0.403623 0.201811 0.979424i \(-0.435317\pi\)
0.201811 + 0.979424i \(0.435317\pi\)
\(150\) 74808.1 0.271469
\(151\) −325298. −1.16102 −0.580509 0.814254i \(-0.697146\pi\)
−0.580509 + 0.814254i \(0.697146\pi\)
\(152\) −327791. −1.15077
\(153\) 128422. 0.443517
\(154\) −20264.5 −0.0688549
\(155\) 674374. 2.25461
\(156\) 111655. 0.367340
\(157\) −244345. −0.791142 −0.395571 0.918435i \(-0.629453\pi\)
−0.395571 + 0.918435i \(0.629453\pi\)
\(158\) −238891. −0.761303
\(159\) −162896. −0.510996
\(160\) 464253. 1.43369
\(161\) −48153.5 −0.146408
\(162\) −18893.6 −0.0565624
\(163\) 86224.9 0.254193 0.127097 0.991890i \(-0.459434\pi\)
0.127097 + 0.991890i \(0.459434\pi\)
\(164\) 20486.5 0.0594781
\(165\) 108414. 0.310009
\(166\) 195650. 0.551074
\(167\) 161244. 0.447396 0.223698 0.974659i \(-0.428187\pi\)
0.223698 + 0.974659i \(0.428187\pi\)
\(168\) −65394.1 −0.178758
\(169\) −97447.0 −0.262453
\(170\) 353987. 0.939432
\(171\) −165510. −0.432847
\(172\) −270093. −0.696133
\(173\) 348734. 0.885887 0.442944 0.896549i \(-0.353934\pi\)
0.442944 + 0.896549i \(0.353934\pi\)
\(174\) 57837.8 0.144823
\(175\) 130737. 0.322703
\(176\) 46093.5 0.112165
\(177\) 31329.0 0.0751646
\(178\) −64915.0 −0.153566
\(179\) −306324. −0.714577 −0.357288 0.933994i \(-0.616299\pi\)
−0.357288 + 0.933994i \(0.616299\pi\)
\(180\) 148887. 0.342513
\(181\) 525696. 1.19272 0.596360 0.802717i \(-0.296613\pi\)
0.596360 + 0.802717i \(0.296613\pi\)
\(182\) −68255.3 −0.152742
\(183\) 399257. 0.881302
\(184\) −170549. −0.371367
\(185\) −1.17025e6 −2.51391
\(186\) −225424. −0.477768
\(187\) 246324. 0.515114
\(188\) −131237. −0.270808
\(189\) −33019.1 −0.0672374
\(190\) −456219. −0.916831
\(191\) −540574. −1.07219 −0.536095 0.844158i \(-0.680101\pi\)
−0.536095 + 0.844158i \(0.680101\pi\)
\(192\) −69742.8 −0.136536
\(193\) −646535. −1.24939 −0.624696 0.780868i \(-0.714777\pi\)
−0.624696 + 0.780868i \(0.714777\pi\)
\(194\) −75690.4 −0.144390
\(195\) 365161. 0.687698
\(196\) 349814. 0.650425
\(197\) −628338. −1.15353 −0.576764 0.816911i \(-0.695685\pi\)
−0.576764 + 0.816911i \(0.695685\pi\)
\(198\) −36239.6 −0.0656932
\(199\) −196577. −0.351884 −0.175942 0.984401i \(-0.556297\pi\)
−0.175942 + 0.984401i \(0.556297\pi\)
\(200\) 463040. 0.818547
\(201\) 300213. 0.524130
\(202\) 177664. 0.306352
\(203\) 101079. 0.172156
\(204\) 338283. 0.569121
\(205\) 66999.5 0.111349
\(206\) 187668. 0.308121
\(207\) −86114.3 −0.139685
\(208\) 155253. 0.248817
\(209\) −317463. −0.502721
\(210\) −91015.3 −0.142418
\(211\) −869514. −1.34453 −0.672265 0.740310i \(-0.734679\pi\)
−0.672265 + 0.740310i \(0.734679\pi\)
\(212\) −429094. −0.655711
\(213\) −25741.7 −0.0388766
\(214\) −440737. −0.657877
\(215\) −883321. −1.30323
\(216\) −116946. −0.170550
\(217\) −393958. −0.567937
\(218\) 586716. 0.836154
\(219\) 574072. 0.808828
\(220\) 285579. 0.397804
\(221\) 829673. 1.14268
\(222\) 391180. 0.532715
\(223\) −108640. −0.146294 −0.0731472 0.997321i \(-0.523304\pi\)
−0.0731472 + 0.997321i \(0.523304\pi\)
\(224\) −271209. −0.361146
\(225\) 233801. 0.307886
\(226\) −693861. −0.903653
\(227\) −1.10451e6 −1.42268 −0.711338 0.702850i \(-0.751910\pi\)
−0.711338 + 0.702850i \(0.751910\pi\)
\(228\) −435980. −0.555430
\(229\) 113945. 0.143584 0.0717920 0.997420i \(-0.477128\pi\)
0.0717920 + 0.997420i \(0.477128\pi\)
\(230\) −237369. −0.295872
\(231\) −63333.5 −0.0780915
\(232\) 357999. 0.436679
\(233\) −374241. −0.451608 −0.225804 0.974173i \(-0.572501\pi\)
−0.225804 + 0.974173i \(0.572501\pi\)
\(234\) −122063. −0.145728
\(235\) −429200. −0.506980
\(236\) 82525.5 0.0964513
\(237\) −746616. −0.863429
\(238\) −206793. −0.236643
\(239\) −1.06804e6 −1.20946 −0.604731 0.796430i \(-0.706719\pi\)
−0.604731 + 0.796430i \(0.706719\pi\)
\(240\) 207022. 0.232001
\(241\) −330951. −0.367046 −0.183523 0.983015i \(-0.558750\pi\)
−0.183523 + 0.983015i \(0.558750\pi\)
\(242\) 394266. 0.432763
\(243\) −59049.0 −0.0641500
\(244\) 1.05170e6 1.13089
\(245\) 1.14404e6 1.21766
\(246\) −22396.0 −0.0235957
\(247\) −1.06928e6 −1.11519
\(248\) −1.39531e6 −1.44059
\(249\) 611474. 0.624999
\(250\) −53266.3 −0.0539017
\(251\) 903692. 0.905391 0.452695 0.891665i \(-0.350463\pi\)
0.452695 + 0.891665i \(0.350463\pi\)
\(252\) −86977.5 −0.0862791
\(253\) −165175. −0.162234
\(254\) 434803. 0.422872
\(255\) 1.10633e6 1.06545
\(256\) −735487. −0.701415
\(257\) 721353. 0.681264 0.340632 0.940197i \(-0.389359\pi\)
0.340632 + 0.940197i \(0.389359\pi\)
\(258\) 295269. 0.276165
\(259\) 683640. 0.633254
\(260\) 961892. 0.882455
\(261\) 180763. 0.164251
\(262\) 721652. 0.649493
\(263\) 601446. 0.536176 0.268088 0.963394i \(-0.413608\pi\)
0.268088 + 0.963394i \(0.413608\pi\)
\(264\) −224313. −0.198082
\(265\) −1.40332e6 −1.22756
\(266\) 266516. 0.230950
\(267\) −202882. −0.174166
\(268\) 790808. 0.672565
\(269\) 1.22319e6 1.03065 0.515326 0.856994i \(-0.327671\pi\)
0.515326 + 0.856994i \(0.327671\pi\)
\(270\) −162765. −0.135879
\(271\) −1.75129e6 −1.44855 −0.724277 0.689509i \(-0.757826\pi\)
−0.724277 + 0.689509i \(0.757826\pi\)
\(272\) 470370. 0.385494
\(273\) −213321. −0.173232
\(274\) 45169.2 0.0363468
\(275\) 448450. 0.357588
\(276\) −226838. −0.179244
\(277\) 638822. 0.500243 0.250121 0.968214i \(-0.419529\pi\)
0.250121 + 0.968214i \(0.419529\pi\)
\(278\) −193224. −0.149951
\(279\) −704526. −0.541859
\(280\) −563358. −0.429427
\(281\) −1.72109e6 −1.30028 −0.650141 0.759814i \(-0.725290\pi\)
−0.650141 + 0.759814i \(0.725290\pi\)
\(282\) 143469. 0.107433
\(283\) −480652. −0.356750 −0.178375 0.983963i \(-0.557084\pi\)
−0.178375 + 0.983963i \(0.557084\pi\)
\(284\) −67807.7 −0.0498865
\(285\) −1.42584e6 −1.03982
\(286\) −234127. −0.169253
\(287\) −39140.0 −0.0280489
\(288\) −485010. −0.344564
\(289\) 1.09381e6 0.770364
\(290\) 498262. 0.347907
\(291\) −236558. −0.163759
\(292\) 1.51220e6 1.03789
\(293\) −2.08942e6 −1.42186 −0.710931 0.703261i \(-0.751726\pi\)
−0.710931 + 0.703261i \(0.751726\pi\)
\(294\) −382420. −0.258031
\(295\) 269894. 0.180567
\(296\) 2.42129e6 1.60627
\(297\) −113261. −0.0745057
\(298\) −314982. −0.205469
\(299\) −556344. −0.359886
\(300\) 615868. 0.395079
\(301\) 516021. 0.328285
\(302\) 936756. 0.591029
\(303\) 555260. 0.347448
\(304\) −606213. −0.376220
\(305\) 3.43953e6 2.11714
\(306\) −369814. −0.225777
\(307\) 2.82729e6 1.71208 0.856040 0.516910i \(-0.172918\pi\)
0.856040 + 0.516910i \(0.172918\pi\)
\(308\) −166830. −0.100207
\(309\) 586526. 0.349455
\(310\) −1.94198e6 −1.14773
\(311\) −1.90959e6 −1.11954 −0.559769 0.828648i \(-0.689110\pi\)
−0.559769 + 0.828648i \(0.689110\pi\)
\(312\) −755534. −0.439407
\(313\) −2.15361e6 −1.24253 −0.621264 0.783602i \(-0.713380\pi\)
−0.621264 + 0.783602i \(0.713380\pi\)
\(314\) 703637. 0.402740
\(315\) −284454. −0.161523
\(316\) −1.96670e6 −1.10795
\(317\) −2.27397e6 −1.27097 −0.635486 0.772113i \(-0.719200\pi\)
−0.635486 + 0.772113i \(0.719200\pi\)
\(318\) 469089. 0.260128
\(319\) 346719. 0.190766
\(320\) −600822. −0.327998
\(321\) −1.37745e6 −0.746129
\(322\) 138667. 0.0745304
\(323\) −3.23961e6 −1.72777
\(324\) −155544. −0.0823174
\(325\) 1.51048e6 0.793242
\(326\) −248301. −0.129400
\(327\) 1.83369e6 0.948322
\(328\) −138625. −0.0711469
\(329\) 250732. 0.127708
\(330\) −312198. −0.157814
\(331\) 3.45401e6 1.73282 0.866410 0.499334i \(-0.166422\pi\)
0.866410 + 0.499334i \(0.166422\pi\)
\(332\) 1.61072e6 0.801999
\(333\) 1.22257e6 0.604176
\(334\) −464332. −0.227752
\(335\) 2.58628e6 1.25911
\(336\) −120939. −0.0584411
\(337\) 2.83240e6 1.35856 0.679281 0.733878i \(-0.262292\pi\)
0.679281 + 0.733878i \(0.262292\pi\)
\(338\) 280617. 0.133605
\(339\) −2.16855e6 −1.02487
\(340\) 2.91425e6 1.36719
\(341\) −1.35134e6 −0.629331
\(342\) 476617. 0.220346
\(343\) −1.42958e6 −0.656106
\(344\) 1.82763e6 0.832706
\(345\) −741859. −0.335563
\(346\) −1.00424e6 −0.450971
\(347\) −152475. −0.0679791 −0.0339895 0.999422i \(-0.510821\pi\)
−0.0339895 + 0.999422i \(0.510821\pi\)
\(348\) 476158. 0.210767
\(349\) 3.07524e6 1.35150 0.675748 0.737132i \(-0.263821\pi\)
0.675748 + 0.737132i \(0.263821\pi\)
\(350\) −376482. −0.164276
\(351\) −381488. −0.165277
\(352\) −930291. −0.400186
\(353\) 2.89392e6 1.23609 0.618045 0.786142i \(-0.287925\pi\)
0.618045 + 0.786142i \(0.287925\pi\)
\(354\) −90217.7 −0.0382634
\(355\) −221760. −0.0933927
\(356\) −534422. −0.223491
\(357\) −646300. −0.268388
\(358\) 882118. 0.363764
\(359\) −3.98179e6 −1.63058 −0.815291 0.579052i \(-0.803423\pi\)
−0.815291 + 0.579052i \(0.803423\pi\)
\(360\) −1.00747e6 −0.409709
\(361\) 1.69912e6 0.686207
\(362\) −1.51384e6 −0.607167
\(363\) 1.23221e6 0.490817
\(364\) −561921. −0.222291
\(365\) 4.94553e6 1.94304
\(366\) −1.14973e6 −0.448637
\(367\) 1.99313e6 0.772451 0.386225 0.922404i \(-0.373779\pi\)
0.386225 + 0.922404i \(0.373779\pi\)
\(368\) −315410. −0.121411
\(369\) −69995.1 −0.0267610
\(370\) 3.36995e6 1.27973
\(371\) 819796. 0.309223
\(372\) −1.85583e6 −0.695314
\(373\) 1.35506e6 0.504298 0.252149 0.967688i \(-0.418863\pi\)
0.252149 + 0.967688i \(0.418863\pi\)
\(374\) −709336. −0.262224
\(375\) −166475. −0.0611325
\(376\) 888034. 0.323936
\(377\) 1.16782e6 0.423179
\(378\) 95084.7 0.0342280
\(379\) 54757.5 0.0195815 0.00979074 0.999952i \(-0.496883\pi\)
0.00979074 + 0.999952i \(0.496883\pi\)
\(380\) −3.75589e6 −1.33430
\(381\) 1.35891e6 0.479598
\(382\) 1.55668e6 0.545810
\(383\) −2.22595e6 −0.775388 −0.387694 0.921788i \(-0.626728\pi\)
−0.387694 + 0.921788i \(0.626728\pi\)
\(384\) −1.52364e6 −0.527296
\(385\) −545607. −0.187598
\(386\) 1.86182e6 0.636017
\(387\) 922815. 0.313211
\(388\) −623131. −0.210136
\(389\) 3.54979e6 1.18940 0.594702 0.803947i \(-0.297270\pi\)
0.594702 + 0.803947i \(0.297270\pi\)
\(390\) −1.05155e6 −0.350081
\(391\) −1.68556e6 −0.557573
\(392\) −2.36707e6 −0.778030
\(393\) 2.25541e6 0.736621
\(394\) 1.80942e6 0.587216
\(395\) −6.43197e6 −2.07420
\(396\) −298348. −0.0956059
\(397\) −3.38798e6 −1.07886 −0.539430 0.842031i \(-0.681360\pi\)
−0.539430 + 0.842031i \(0.681360\pi\)
\(398\) 566079. 0.179130
\(399\) 832952. 0.261931
\(400\) 856341. 0.267607
\(401\) 1.54937e6 0.481165 0.240583 0.970629i \(-0.422661\pi\)
0.240583 + 0.970629i \(0.422661\pi\)
\(402\) −864519. −0.266815
\(403\) −4.55161e6 −1.39605
\(404\) 1.46264e6 0.445846
\(405\) −508697. −0.154107
\(406\) −291077. −0.0876379
\(407\) 2.34500e6 0.701708
\(408\) −2.28904e6 −0.680775
\(409\) −1.40903e6 −0.416497 −0.208248 0.978076i \(-0.566776\pi\)
−0.208248 + 0.978076i \(0.566776\pi\)
\(410\) −192937. −0.0566835
\(411\) 141169. 0.0412226
\(412\) 1.54500e6 0.448421
\(413\) −157667. −0.0454849
\(414\) 247982. 0.0711082
\(415\) 5.26774e6 1.50143
\(416\) −3.13342e6 −0.887739
\(417\) −603891. −0.170066
\(418\) 914193. 0.255916
\(419\) −1.78943e6 −0.497942 −0.248971 0.968511i \(-0.580092\pi\)
−0.248971 + 0.968511i \(0.580092\pi\)
\(420\) −749296. −0.207267
\(421\) 4.50624e6 1.23911 0.619554 0.784954i \(-0.287314\pi\)
0.619554 + 0.784954i \(0.287314\pi\)
\(422\) 2.50393e6 0.684449
\(423\) 448390. 0.121844
\(424\) 2.90353e6 0.784353
\(425\) 4.57630e6 1.22897
\(426\) 74128.0 0.0197906
\(427\) −2.00931e6 −0.533308
\(428\) −3.62843e6 −0.957434
\(429\) −731727. −0.191958
\(430\) 2.54369e6 0.663426
\(431\) −2.68824e6 −0.697067 −0.348534 0.937296i \(-0.613320\pi\)
−0.348534 + 0.937296i \(0.613320\pi\)
\(432\) −216279. −0.0557576
\(433\) 1.53010e6 0.392194 0.196097 0.980584i \(-0.437173\pi\)
0.196097 + 0.980584i \(0.437173\pi\)
\(434\) 1.13447e6 0.289115
\(435\) 1.55724e6 0.394578
\(436\) 4.83022e6 1.21689
\(437\) 2.17235e6 0.544159
\(438\) −1.65315e6 −0.411743
\(439\) −698534. −0.172992 −0.0864960 0.996252i \(-0.527567\pi\)
−0.0864960 + 0.996252i \(0.527567\pi\)
\(440\) −1.93241e6 −0.475848
\(441\) −1.19519e6 −0.292646
\(442\) −2.38920e6 −0.581696
\(443\) −1.73221e6 −0.419365 −0.209682 0.977770i \(-0.567243\pi\)
−0.209682 + 0.977770i \(0.567243\pi\)
\(444\) 3.22045e6 0.775280
\(445\) −1.74779e6 −0.418397
\(446\) 312849. 0.0744728
\(447\) −984427. −0.233032
\(448\) 350990. 0.0826228
\(449\) 2.98288e6 0.698265 0.349133 0.937073i \(-0.386476\pi\)
0.349133 + 0.937073i \(0.386476\pi\)
\(450\) −673273. −0.156733
\(451\) −134257. −0.0310810
\(452\) −5.71231e6 −1.31512
\(453\) 2.92768e6 0.670314
\(454\) 3.18065e6 0.724229
\(455\) −1.83772e6 −0.416152
\(456\) 2.95012e6 0.664397
\(457\) −3.19426e6 −0.715451 −0.357725 0.933827i \(-0.616448\pi\)
−0.357725 + 0.933827i \(0.616448\pi\)
\(458\) −328125. −0.0730931
\(459\) −1.15580e6 −0.256065
\(460\) −1.95417e6 −0.430595
\(461\) −7.83598e6 −1.71728 −0.858640 0.512580i \(-0.828690\pi\)
−0.858640 + 0.512580i \(0.828690\pi\)
\(462\) 182381. 0.0397534
\(463\) −763425. −0.165506 −0.0827530 0.996570i \(-0.526371\pi\)
−0.0827530 + 0.996570i \(0.526371\pi\)
\(464\) 662079. 0.142763
\(465\) −6.06936e6 −1.30170
\(466\) 1.07770e6 0.229896
\(467\) 2.85947e6 0.606727 0.303363 0.952875i \(-0.401890\pi\)
0.303363 + 0.952875i \(0.401890\pi\)
\(468\) −1.00490e6 −0.212084
\(469\) −1.51086e6 −0.317171
\(470\) 1.23596e6 0.258084
\(471\) 2.19911e6 0.456766
\(472\) −558421. −0.115374
\(473\) 1.77004e6 0.363773
\(474\) 2.15002e6 0.439538
\(475\) −5.89794e6 −1.19941
\(476\) −1.70246e6 −0.344396
\(477\) 1.46606e6 0.295024
\(478\) 3.07562e6 0.615690
\(479\) 5.27842e6 1.05115 0.525575 0.850747i \(-0.323850\pi\)
0.525575 + 0.850747i \(0.323850\pi\)
\(480\) −4.17827e6 −0.827740
\(481\) 7.89846e6 1.55661
\(482\) 953034. 0.186849
\(483\) 433382. 0.0845284
\(484\) 3.24585e6 0.629817
\(485\) −2.03791e6 −0.393396
\(486\) 170043. 0.0326563
\(487\) −5.59895e6 −1.06975 −0.534877 0.844930i \(-0.679642\pi\)
−0.534877 + 0.844930i \(0.679642\pi\)
\(488\) −7.11652e6 −1.35275
\(489\) −776024. −0.146758
\(490\) −3.29448e6 −0.619865
\(491\) 4.25242e6 0.796035 0.398018 0.917378i \(-0.369698\pi\)
0.398018 + 0.917378i \(0.369698\pi\)
\(492\) −184378. −0.0343397
\(493\) 3.53816e6 0.655633
\(494\) 3.07920e6 0.567702
\(495\) −975725. −0.178984
\(496\) −2.58046e6 −0.470970
\(497\) 129549. 0.0235257
\(498\) −1.76085e6 −0.318163
\(499\) −45082.1 −0.00810500 −0.00405250 0.999992i \(-0.501290\pi\)
−0.00405250 + 0.999992i \(0.501290\pi\)
\(500\) −438522. −0.0784453
\(501\) −1.45119e6 −0.258304
\(502\) −2.60235e6 −0.460899
\(503\) −5.54321e6 −0.976881 −0.488440 0.872597i \(-0.662434\pi\)
−0.488440 + 0.872597i \(0.662434\pi\)
\(504\) 588547. 0.103206
\(505\) 4.78347e6 0.834669
\(506\) 475651. 0.0825871
\(507\) 877023. 0.151527
\(508\) 3.57958e6 0.615421
\(509\) −8.38202e6 −1.43402 −0.717008 0.697065i \(-0.754489\pi\)
−0.717008 + 0.697065i \(0.754489\pi\)
\(510\) −3.18588e6 −0.542381
\(511\) −2.88910e6 −0.489452
\(512\) −3.29942e6 −0.556241
\(513\) 1.48959e6 0.249904
\(514\) −2.07727e6 −0.346805
\(515\) 5.05282e6 0.839490
\(516\) 2.43084e6 0.401913
\(517\) 860052. 0.141514
\(518\) −1.96867e6 −0.322365
\(519\) −3.13860e6 −0.511467
\(520\) −6.50879e6 −1.05558
\(521\) −6.69577e6 −1.08070 −0.540351 0.841440i \(-0.681709\pi\)
−0.540351 + 0.841440i \(0.681709\pi\)
\(522\) −520540. −0.0836138
\(523\) −4.06164e6 −0.649303 −0.324651 0.945834i \(-0.605247\pi\)
−0.324651 + 0.945834i \(0.605247\pi\)
\(524\) 5.94110e6 0.945233
\(525\) −1.17663e6 −0.186313
\(526\) −1.73198e6 −0.272946
\(527\) −1.37900e7 −2.16291
\(528\) −414841. −0.0647585
\(529\) −5.30608e6 −0.824393
\(530\) 4.04112e6 0.624903
\(531\) −281961. −0.0433963
\(532\) 2.19413e6 0.336111
\(533\) −452205. −0.0689474
\(534\) 584235. 0.0886614
\(535\) −1.18665e7 −1.79242
\(536\) −5.35113e6 −0.804513
\(537\) 2.75692e6 0.412561
\(538\) −3.52239e6 −0.524665
\(539\) −2.29249e6 −0.339887
\(540\) −1.33999e6 −0.197750
\(541\) −2.10875e6 −0.309765 −0.154883 0.987933i \(-0.549500\pi\)
−0.154883 + 0.987933i \(0.549500\pi\)
\(542\) 5.04317e6 0.737403
\(543\) −4.73127e6 −0.688617
\(544\) −9.49334e6 −1.37538
\(545\) 1.57969e7 2.27814
\(546\) 614298. 0.0881855
\(547\) −7.48493e6 −1.06960 −0.534798 0.844980i \(-0.679612\pi\)
−0.534798 + 0.844980i \(0.679612\pi\)
\(548\) 371862. 0.0528969
\(549\) −3.59331e6 −0.508820
\(550\) −1.29140e6 −0.182034
\(551\) −4.55999e6 −0.639860
\(552\) 1.53494e6 0.214409
\(553\) 3.75745e6 0.522493
\(554\) −1.83961e6 −0.254654
\(555\) 1.05322e7 1.45140
\(556\) −1.59074e6 −0.218229
\(557\) 7.42234e6 1.01368 0.506842 0.862039i \(-0.330813\pi\)
0.506842 + 0.862039i \(0.330813\pi\)
\(558\) 2.02881e6 0.275839
\(559\) 5.96187e6 0.806962
\(560\) −1.04187e6 −0.140392
\(561\) −2.21692e6 −0.297401
\(562\) 4.95619e6 0.661923
\(563\) −818283. −0.108801 −0.0544005 0.998519i \(-0.517325\pi\)
−0.0544005 + 0.998519i \(0.517325\pi\)
\(564\) 1.18113e6 0.156351
\(565\) −1.86817e7 −2.46204
\(566\) 1.38413e6 0.181608
\(567\) 297172. 0.0388195
\(568\) 458831. 0.0596736
\(569\) 1.44362e7 1.86928 0.934638 0.355599i \(-0.115723\pi\)
0.934638 + 0.355599i \(0.115723\pi\)
\(570\) 4.10597e6 0.529333
\(571\) 7.48418e6 0.960625 0.480312 0.877098i \(-0.340523\pi\)
0.480312 + 0.877098i \(0.340523\pi\)
\(572\) −1.92748e6 −0.246321
\(573\) 4.86516e6 0.619029
\(574\) 112711. 0.0142786
\(575\) −3.06868e6 −0.387063
\(576\) 627686. 0.0788290
\(577\) −1.08673e7 −1.35888 −0.679441 0.733730i \(-0.737778\pi\)
−0.679441 + 0.733730i \(0.737778\pi\)
\(578\) −3.14982e6 −0.392163
\(579\) 5.81881e6 0.721336
\(580\) 4.10201e6 0.506322
\(581\) −3.07732e6 −0.378210
\(582\) 681213. 0.0833634
\(583\) 2.81204e6 0.342649
\(584\) −1.02325e7 −1.24151
\(585\) −3.28645e6 −0.397043
\(586\) 6.01689e6 0.723815
\(587\) 1.25762e7 1.50644 0.753222 0.657766i \(-0.228499\pi\)
0.753222 + 0.657766i \(0.228499\pi\)
\(588\) −3.14833e6 −0.375523
\(589\) 1.77726e7 2.11088
\(590\) −777209. −0.0919196
\(591\) 5.65504e6 0.665989
\(592\) 4.47791e6 0.525135
\(593\) −1.44755e7 −1.69042 −0.845212 0.534430i \(-0.820526\pi\)
−0.845212 + 0.534430i \(0.820526\pi\)
\(594\) 326156. 0.0379280
\(595\) −5.56776e6 −0.644745
\(596\) −2.59314e6 −0.299026
\(597\) 1.76919e6 0.203160
\(598\) 1.60210e6 0.183204
\(599\) 898434. 0.102310 0.0511551 0.998691i \(-0.483710\pi\)
0.0511551 + 0.998691i \(0.483710\pi\)
\(600\) −4.16736e6 −0.472589
\(601\) 1.02932e7 1.16242 0.581210 0.813754i \(-0.302580\pi\)
0.581210 + 0.813754i \(0.302580\pi\)
\(602\) −1.48598e6 −0.167117
\(603\) −2.70192e6 −0.302607
\(604\) 7.71197e6 0.860148
\(605\) 1.06153e7 1.17908
\(606\) −1.59898e6 −0.176872
\(607\) 1.82491e6 0.201034 0.100517 0.994935i \(-0.467950\pi\)
0.100517 + 0.994935i \(0.467950\pi\)
\(608\) 1.22350e7 1.34229
\(609\) −909714. −0.0993943
\(610\) −9.90476e6 −1.07775
\(611\) 2.89684e6 0.313922
\(612\) −3.04455e6 −0.328582
\(613\) −621392. −0.0667905 −0.0333953 0.999442i \(-0.510632\pi\)
−0.0333953 + 0.999442i \(0.510632\pi\)
\(614\) −8.14170e6 −0.871554
\(615\) −602995. −0.0642874
\(616\) 1.12888e6 0.119866
\(617\) 5.04840e6 0.533877 0.266938 0.963714i \(-0.413988\pi\)
0.266938 + 0.963714i \(0.413988\pi\)
\(618\) −1.68901e6 −0.177894
\(619\) 1.49642e7 1.56974 0.784869 0.619662i \(-0.212730\pi\)
0.784869 + 0.619662i \(0.212730\pi\)
\(620\) −1.59877e7 −1.67034
\(621\) 775028. 0.0806471
\(622\) 5.49902e6 0.569914
\(623\) 1.02103e6 0.105395
\(624\) −1.39727e6 −0.143655
\(625\) −1.04542e7 −1.07051
\(626\) 6.20172e6 0.632523
\(627\) 2.85717e6 0.290246
\(628\) 5.79279e6 0.586123
\(629\) 2.39300e7 2.41166
\(630\) 819138. 0.0822253
\(631\) 1.72533e7 1.72504 0.862519 0.506024i \(-0.168885\pi\)
0.862519 + 0.506024i \(0.168885\pi\)
\(632\) 1.33080e7 1.32532
\(633\) 7.82563e6 0.776265
\(634\) 6.54831e6 0.647002
\(635\) 1.17068e7 1.15213
\(636\) 3.86184e6 0.378575
\(637\) −7.72159e6 −0.753977
\(638\) −998442. −0.0971116
\(639\) 231675. 0.0224454
\(640\) −1.31259e7 −1.26672
\(641\) −1.83730e7 −1.76618 −0.883092 0.469200i \(-0.844543\pi\)
−0.883092 + 0.469200i \(0.844543\pi\)
\(642\) 3.96663e6 0.379826
\(643\) 1.64115e7 1.56538 0.782690 0.622411i \(-0.213847\pi\)
0.782690 + 0.622411i \(0.213847\pi\)
\(644\) 1.14160e6 0.108467
\(645\) 7.94989e6 0.752422
\(646\) 9.32907e6 0.879543
\(647\) −6.16024e6 −0.578545 −0.289272 0.957247i \(-0.593413\pi\)
−0.289272 + 0.957247i \(0.593413\pi\)
\(648\) 1.05251e6 0.0984670
\(649\) −540826. −0.0504018
\(650\) −4.34970e6 −0.403809
\(651\) 3.54562e6 0.327899
\(652\) −2.04417e6 −0.188321
\(653\) 1.38748e7 1.27334 0.636671 0.771135i \(-0.280311\pi\)
0.636671 + 0.771135i \(0.280311\pi\)
\(654\) −5.28044e6 −0.482754
\(655\) 1.94300e7 1.76957
\(656\) −256371. −0.0232600
\(657\) −5.16665e6 −0.466977
\(658\) −722029. −0.0650114
\(659\) 1.76364e7 1.58196 0.790982 0.611839i \(-0.209570\pi\)
0.790982 + 0.611839i \(0.209570\pi\)
\(660\) −2.57021e6 −0.229672
\(661\) 6.64665e6 0.591697 0.295849 0.955235i \(-0.404398\pi\)
0.295849 + 0.955235i \(0.404398\pi\)
\(662\) −9.94646e6 −0.882111
\(663\) −7.46706e6 −0.659729
\(664\) −1.08992e7 −0.959341
\(665\) 7.17573e6 0.629234
\(666\) −3.52062e6 −0.307563
\(667\) −2.37254e6 −0.206490
\(668\) −3.82267e6 −0.331456
\(669\) 977760. 0.0844631
\(670\) −7.44768e6 −0.640965
\(671\) −6.89228e6 −0.590958
\(672\) 2.44088e6 0.208508
\(673\) −1.52418e7 −1.29717 −0.648586 0.761141i \(-0.724639\pi\)
−0.648586 + 0.761141i \(0.724639\pi\)
\(674\) −8.15642e6 −0.691591
\(675\) −2.10421e6 −0.177758
\(676\) 2.31022e6 0.194440
\(677\) −4.32639e6 −0.362789 −0.181394 0.983410i \(-0.558061\pi\)
−0.181394 + 0.983410i \(0.558061\pi\)
\(678\) 6.24475e6 0.521724
\(679\) 1.19051e6 0.0990967
\(680\) −1.97197e7 −1.63542
\(681\) 9.94061e6 0.821382
\(682\) 3.89144e6 0.320368
\(683\) 6.89334e6 0.565428 0.282714 0.959204i \(-0.408765\pi\)
0.282714 + 0.959204i \(0.408765\pi\)
\(684\) 3.92382e6 0.320677
\(685\) 1.21615e6 0.0990284
\(686\) 4.11675e6 0.333998
\(687\) −1.02550e6 −0.0828983
\(688\) 3.37999e6 0.272235
\(689\) 9.47155e6 0.760104
\(690\) 2.13632e6 0.170822
\(691\) −1.77702e7 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(692\) −8.26757e6 −0.656315
\(693\) 570002. 0.0450862
\(694\) 439080. 0.0346055
\(695\) −5.20241e6 −0.408548
\(696\) −3.22199e6 −0.252117
\(697\) −1.37005e6 −0.106820
\(698\) −8.85571e6 −0.687995
\(699\) 3.36817e6 0.260736
\(700\) −3.09944e6 −0.239077
\(701\) −2.94835e6 −0.226613 −0.113306 0.993560i \(-0.536144\pi\)
−0.113306 + 0.993560i \(0.536144\pi\)
\(702\) 1.09857e6 0.0841362
\(703\) −3.08410e7 −2.35364
\(704\) 1.20396e6 0.0915543
\(705\) 3.86280e6 0.292705
\(706\) −8.33359e6 −0.629246
\(707\) −2.79442e6 −0.210254
\(708\) −742729. −0.0556862
\(709\) −1.10740e7 −0.827346 −0.413673 0.910425i \(-0.635754\pi\)
−0.413673 + 0.910425i \(0.635754\pi\)
\(710\) 638600. 0.0475426
\(711\) 6.71955e6 0.498501
\(712\) 3.61625e6 0.267336
\(713\) 9.24702e6 0.681205
\(714\) 1.86114e6 0.136626
\(715\) −6.30370e6 −0.461137
\(716\) 7.26216e6 0.529399
\(717\) 9.61235e6 0.698283
\(718\) 1.14663e7 0.830066
\(719\) −5.21161e6 −0.375967 −0.187984 0.982172i \(-0.560195\pi\)
−0.187984 + 0.982172i \(0.560195\pi\)
\(720\) −1.86320e6 −0.133946
\(721\) −2.95177e6 −0.211468
\(722\) −4.89292e6 −0.349322
\(723\) 2.97856e6 0.211914
\(724\) −1.24629e7 −0.883634
\(725\) 6.44147e6 0.455135
\(726\) −3.54839e6 −0.249856
\(727\) −6.91096e6 −0.484956 −0.242478 0.970157i \(-0.577960\pi\)
−0.242478 + 0.970157i \(0.577960\pi\)
\(728\) 3.80232e6 0.265901
\(729\) 531441. 0.0370370
\(730\) −1.42416e7 −0.989124
\(731\) 1.80627e7 1.25023
\(732\) −9.46534e6 −0.652918
\(733\) 9.42723e6 0.648073 0.324037 0.946045i \(-0.394960\pi\)
0.324037 + 0.946045i \(0.394960\pi\)
\(734\) −5.73959e6 −0.393225
\(735\) −1.02964e7 −0.703018
\(736\) 6.36584e6 0.433173
\(737\) −5.18251e6 −0.351457
\(738\) 201564. 0.0136230
\(739\) −1.88305e7 −1.26839 −0.634193 0.773175i \(-0.718668\pi\)
−0.634193 + 0.773175i \(0.718668\pi\)
\(740\) 2.77436e7 1.86244
\(741\) 9.62355e6 0.643857
\(742\) −2.36076e6 −0.157413
\(743\) 2.36519e6 0.157179 0.0785893 0.996907i \(-0.474958\pi\)
0.0785893 + 0.996907i \(0.474958\pi\)
\(744\) 1.25578e7 0.831725
\(745\) −8.48067e6 −0.559808
\(746\) −3.90216e6 −0.256719
\(747\) −5.50326e6 −0.360843
\(748\) −5.83971e6 −0.381625
\(749\) 6.93222e6 0.451510
\(750\) 479397. 0.0311202
\(751\) −1.67747e7 −1.08531 −0.542657 0.839955i \(-0.682581\pi\)
−0.542657 + 0.839955i \(0.682581\pi\)
\(752\) 1.64232e6 0.105904
\(753\) −8.13323e6 −0.522728
\(754\) −3.36297e6 −0.215424
\(755\) 2.52215e7 1.61029
\(756\) 782798. 0.0498133
\(757\) −9.79306e6 −0.621124 −0.310562 0.950553i \(-0.600517\pi\)
−0.310562 + 0.950553i \(0.600517\pi\)
\(758\) −157684. −0.00996817
\(759\) 1.48657e6 0.0936659
\(760\) 2.54148e7 1.59607
\(761\) 1.83642e7 1.14950 0.574751 0.818328i \(-0.305099\pi\)
0.574751 + 0.818328i \(0.305099\pi\)
\(762\) −3.91323e6 −0.244145
\(763\) −9.22827e6 −0.573864
\(764\) 1.28156e7 0.794338
\(765\) −9.95698e6 −0.615140
\(766\) 6.41005e6 0.394720
\(767\) −1.82162e6 −0.111807
\(768\) 6.61938e6 0.404962
\(769\) −1.44796e6 −0.0882962 −0.0441481 0.999025i \(-0.514057\pi\)
−0.0441481 + 0.999025i \(0.514057\pi\)
\(770\) 1.57118e6 0.0954989
\(771\) −6.49218e6 −0.393328
\(772\) 1.53277e7 0.925620
\(773\) −136665. −0.00822637 −0.00411318 0.999992i \(-0.501309\pi\)
−0.00411318 + 0.999992i \(0.501309\pi\)
\(774\) −2.65742e6 −0.159444
\(775\) −2.51057e7 −1.50148
\(776\) 4.21651e6 0.251362
\(777\) −6.15276e6 −0.365609
\(778\) −1.02223e7 −0.605479
\(779\) 1.76572e6 0.104251
\(780\) −8.65702e6 −0.509486
\(781\) 444374. 0.0260688
\(782\) 4.85388e6 0.283839
\(783\) −1.62687e6 −0.0948303
\(784\) −4.37763e6 −0.254360
\(785\) 1.89449e7 1.09728
\(786\) −6.49487e6 −0.374985
\(787\) 2.64152e7 1.52026 0.760129 0.649772i \(-0.225136\pi\)
0.760129 + 0.649772i \(0.225136\pi\)
\(788\) 1.48963e7 0.854598
\(789\) −5.41301e6 −0.309561
\(790\) 1.85220e7 1.05590
\(791\) 1.09135e7 0.620189
\(792\) 2.01881e6 0.114362
\(793\) −2.32147e7 −1.31093
\(794\) 9.75633e6 0.549206
\(795\) 1.26299e7 0.708731
\(796\) 4.66032e6 0.260695
\(797\) −1.25477e7 −0.699708 −0.349854 0.936804i \(-0.613769\pi\)
−0.349854 + 0.936804i \(0.613769\pi\)
\(798\) −2.39864e6 −0.133339
\(799\) 8.77657e6 0.486360
\(800\) −1.72833e7 −0.954776
\(801\) 1.82593e6 0.100555
\(802\) −4.46170e6 −0.244943
\(803\) −9.91009e6 −0.542361
\(804\) −7.11727e6 −0.388305
\(805\) 3.73351e6 0.203061
\(806\) 1.31072e7 0.710677
\(807\) −1.10087e7 −0.595047
\(808\) −9.89720e6 −0.533315
\(809\) 2.17849e7 1.17027 0.585134 0.810937i \(-0.301042\pi\)
0.585134 + 0.810937i \(0.301042\pi\)
\(810\) 1.46489e6 0.0784497
\(811\) −2.63202e7 −1.40520 −0.702599 0.711586i \(-0.747977\pi\)
−0.702599 + 0.711586i \(0.747977\pi\)
\(812\) −2.39633e6 −0.127543
\(813\) 1.57616e7 0.836323
\(814\) −6.75286e6 −0.357213
\(815\) −6.68531e6 −0.352556
\(816\) −4.23333e6 −0.222565
\(817\) −2.32793e7 −1.22015
\(818\) 4.05756e6 0.212022
\(819\) 1.91989e6 0.100015
\(820\) −1.58838e6 −0.0824937
\(821\) −3.00385e7 −1.55532 −0.777661 0.628684i \(-0.783594\pi\)
−0.777661 + 0.628684i \(0.783594\pi\)
\(822\) −406523. −0.0209848
\(823\) 8.67425e6 0.446408 0.223204 0.974772i \(-0.428348\pi\)
0.223204 + 0.974772i \(0.428348\pi\)
\(824\) −1.04545e7 −0.536395
\(825\) −4.03605e6 −0.206453
\(826\) 454033. 0.0231546
\(827\) −5.30575e6 −0.269763 −0.134882 0.990862i \(-0.543065\pi\)
−0.134882 + 0.990862i \(0.543065\pi\)
\(828\) 2.04155e6 0.103486
\(829\) −3221.11 −0.000162787 0 −8.13934e−5 1.00000i \(-0.500026\pi\)
−8.13934e−5 1.00000i \(0.500026\pi\)
\(830\) −1.51694e7 −0.764318
\(831\) −5.74940e6 −0.288815
\(832\) 4.05518e6 0.203096
\(833\) −2.33941e7 −1.16814
\(834\) 1.73902e6 0.0865742
\(835\) −1.25018e7 −0.620520
\(836\) 7.52622e6 0.372444
\(837\) 6.34073e6 0.312842
\(838\) 5.15298e6 0.253483
\(839\) −1.48233e7 −0.727008 −0.363504 0.931593i \(-0.618420\pi\)
−0.363504 + 0.931593i \(0.618420\pi\)
\(840\) 5.07023e6 0.247930
\(841\) −1.55309e7 −0.757195
\(842\) −1.29766e7 −0.630782
\(843\) 1.54898e7 0.750718
\(844\) 2.06139e7 0.996104
\(845\) 7.55540e6 0.364012
\(846\) −1.29122e6 −0.0620263
\(847\) −6.20128e6 −0.297011
\(848\) 5.36975e6 0.256427
\(849\) 4.32586e6 0.205970
\(850\) −1.31783e7 −0.625622
\(851\) −1.60465e7 −0.759549
\(852\) 610269. 0.0288020
\(853\) −1.50069e7 −0.706183 −0.353092 0.935589i \(-0.614870\pi\)
−0.353092 + 0.935589i \(0.614870\pi\)
\(854\) 5.78619e6 0.271486
\(855\) 1.28326e7 0.600341
\(856\) 2.45523e7 1.14527
\(857\) 9.04584e6 0.420724 0.210362 0.977624i \(-0.432536\pi\)
0.210362 + 0.977624i \(0.432536\pi\)
\(858\) 2.10714e6 0.0977183
\(859\) 5.42003e6 0.250622 0.125311 0.992118i \(-0.460007\pi\)
0.125311 + 0.992118i \(0.460007\pi\)
\(860\) 2.09412e7 0.965509
\(861\) 352260. 0.0161940
\(862\) 7.74129e6 0.354850
\(863\) −2.45589e7 −1.12249 −0.561245 0.827650i \(-0.689678\pi\)
−0.561245 + 0.827650i \(0.689678\pi\)
\(864\) 4.36509e6 0.198934
\(865\) −2.70385e7 −1.22869
\(866\) −4.40622e6 −0.199651
\(867\) −9.84426e6 −0.444770
\(868\) 9.33972e6 0.420760
\(869\) 1.28887e7 0.578974
\(870\) −4.48436e6 −0.200864
\(871\) −1.74558e7 −0.779641
\(872\) −3.26844e7 −1.45562
\(873\) 2.12902e6 0.0945464
\(874\) −6.25568e6 −0.277010
\(875\) 837810. 0.0369935
\(876\) −1.36098e7 −0.599226
\(877\) 1.94355e7 0.853289 0.426644 0.904420i \(-0.359696\pi\)
0.426644 + 0.904420i \(0.359696\pi\)
\(878\) 2.01156e6 0.0880636
\(879\) 1.88048e7 0.820913
\(880\) −3.57378e6 −0.155568
\(881\) −4.44437e7 −1.92917 −0.964584 0.263775i \(-0.915032\pi\)
−0.964584 + 0.263775i \(0.915032\pi\)
\(882\) 3.44178e6 0.148975
\(883\) −1.10995e7 −0.479072 −0.239536 0.970888i \(-0.576995\pi\)
−0.239536 + 0.970888i \(0.576995\pi\)
\(884\) −1.96694e7 −0.846565
\(885\) −2.42904e6 −0.104250
\(886\) 4.98823e6 0.213482
\(887\) −1.92347e7 −0.820872 −0.410436 0.911889i \(-0.634623\pi\)
−0.410436 + 0.911889i \(0.634623\pi\)
\(888\) −2.17916e7 −0.927380
\(889\) −6.83889e6 −0.290223
\(890\) 5.03308e6 0.212990
\(891\) 1.01935e6 0.0430159
\(892\) 2.57557e6 0.108383
\(893\) −1.13113e7 −0.474659
\(894\) 2.83484e6 0.118627
\(895\) 2.37504e7 0.991089
\(896\) 7.66793e6 0.319086
\(897\) 5.00710e6 0.207780
\(898\) −8.58977e6 −0.355460
\(899\) −1.94105e7 −0.801008
\(900\) −5.54281e6 −0.228099
\(901\) 2.86960e7 1.17763
\(902\) 386617. 0.0158221
\(903\) −4.64419e6 −0.189536
\(904\) 3.86532e7 1.57313
\(905\) −4.07590e7 −1.65425
\(906\) −8.43080e6 −0.341231
\(907\) −1.01314e6 −0.0408931 −0.0204466 0.999791i \(-0.506509\pi\)
−0.0204466 + 0.999791i \(0.506509\pi\)
\(908\) 2.61851e7 1.05400
\(909\) −4.99734e6 −0.200599
\(910\) 5.29206e6 0.211847
\(911\) −3.28323e7 −1.31071 −0.655354 0.755322i \(-0.727480\pi\)
−0.655354 + 0.755322i \(0.727480\pi\)
\(912\) 5.45592e6 0.217210
\(913\) −1.05557e7 −0.419094
\(914\) 9.19846e6 0.364208
\(915\) −3.09557e7 −1.22233
\(916\) −2.70134e6 −0.106375
\(917\) −1.13507e7 −0.445756
\(918\) 3.32833e6 0.130353
\(919\) 6.66545e6 0.260340 0.130170 0.991492i \(-0.458448\pi\)
0.130170 + 0.991492i \(0.458448\pi\)
\(920\) 1.32232e7 0.515071
\(921\) −2.54456e7 −0.988470
\(922\) 2.25652e7 0.874201
\(923\) 1.49675e6 0.0578288
\(924\) 1.50147e6 0.0578546
\(925\) 4.35662e7 1.67416
\(926\) 2.19842e6 0.0842528
\(927\) −5.27873e6 −0.201758
\(928\) −1.33626e7 −0.509354
\(929\) 1.25443e7 0.476877 0.238439 0.971158i \(-0.423364\pi\)
0.238439 + 0.971158i \(0.423364\pi\)
\(930\) 1.74779e7 0.662645
\(931\) 3.01504e7 1.14004
\(932\) 8.87229e6 0.334577
\(933\) 1.71863e7 0.646366
\(934\) −8.23437e6 −0.308861
\(935\) −1.90983e7 −0.714442
\(936\) 6.79980e6 0.253692
\(937\) −3.57395e7 −1.32984 −0.664921 0.746914i \(-0.731535\pi\)
−0.664921 + 0.746914i \(0.731535\pi\)
\(938\) 4.35081e6 0.161459
\(939\) 1.93825e7 0.717373
\(940\) 1.01752e7 0.375599
\(941\) 2.36120e7 0.869277 0.434639 0.900605i \(-0.356876\pi\)
0.434639 + 0.900605i \(0.356876\pi\)
\(942\) −6.33274e6 −0.232522
\(943\) 918698. 0.0336429
\(944\) −1.03274e6 −0.0377190
\(945\) 2.56008e6 0.0932556
\(946\) −5.09716e6 −0.185183
\(947\) −5.01198e7 −1.81608 −0.908038 0.418887i \(-0.862420\pi\)
−0.908038 + 0.418887i \(0.862420\pi\)
\(948\) 1.77003e7 0.639677
\(949\) −3.33793e7 −1.20313
\(950\) 1.69842e7 0.610571
\(951\) 2.04657e7 0.733795
\(952\) 1.15199e7 0.411962
\(953\) 3.11140e7 1.10975 0.554874 0.831935i \(-0.312767\pi\)
0.554874 + 0.831935i \(0.312767\pi\)
\(954\) −4.22180e6 −0.150185
\(955\) 4.19125e7 1.48708
\(956\) 2.53204e7 0.896038
\(957\) −3.12047e6 −0.110139
\(958\) −1.52002e7 −0.535100
\(959\) −710453. −0.0249453
\(960\) 5.40740e6 0.189370
\(961\) 4.70234e7 1.64250
\(962\) −2.27451e7 −0.792410
\(963\) 1.23971e7 0.430778
\(964\) 7.84598e6 0.271928
\(965\) 5.01280e7 1.73286
\(966\) −1.24800e6 −0.0430302
\(967\) −1.64097e7 −0.564331 −0.282166 0.959366i \(-0.591053\pi\)
−0.282166 + 0.959366i \(0.591053\pi\)
\(968\) −2.19635e7 −0.753379
\(969\) 2.91565e7 0.997531
\(970\) 5.86853e6 0.200263
\(971\) −2.88792e7 −0.982962 −0.491481 0.870888i \(-0.663544\pi\)
−0.491481 + 0.870888i \(0.663544\pi\)
\(972\) 1.39990e6 0.0475260
\(973\) 3.03916e6 0.102913
\(974\) 1.61232e7 0.544571
\(975\) −1.35943e7 −0.457978
\(976\) −1.31612e7 −0.442253
\(977\) 2.68826e7 0.901021 0.450510 0.892771i \(-0.351242\pi\)
0.450510 + 0.892771i \(0.351242\pi\)
\(978\) 2.23471e6 0.0747091
\(979\) 3.50230e6 0.116788
\(980\) −2.71223e7 −0.902113
\(981\) −1.65032e7 −0.547514
\(982\) −1.22456e7 −0.405231
\(983\) −1.32072e7 −0.435942 −0.217971 0.975955i \(-0.569944\pi\)
−0.217971 + 0.975955i \(0.569944\pi\)
\(984\) 1.24762e6 0.0410767
\(985\) 4.87172e7 1.59990
\(986\) −1.01888e7 −0.333757
\(987\) −2.25659e6 −0.0737325
\(988\) 2.53499e7 0.826199
\(989\) −1.21121e7 −0.393758
\(990\) 2.80978e6 0.0911138
\(991\) 2.08572e6 0.0674639 0.0337320 0.999431i \(-0.489261\pi\)
0.0337320 + 0.999431i \(0.489261\pi\)
\(992\) 5.20807e7 1.68034
\(993\) −3.10861e7 −1.00044
\(994\) −373059. −0.0119760
\(995\) 1.52413e7 0.488048
\(996\) −1.44965e7 −0.463035
\(997\) 3.49916e6 0.111487 0.0557437 0.998445i \(-0.482247\pi\)
0.0557437 + 0.998445i \(0.482247\pi\)
\(998\) 129822. 0.00412594
\(999\) −1.10031e7 −0.348821
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.1.5 12
3.2 odd 2 531.6.a.d.1.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.b.1.5 12 1.1 even 1 trivial
531.6.a.d.1.8 12 3.2 odd 2