Properties

Label 531.5.c.d.235.10
Level $531$
Weight $5$
Character 531.235
Analytic conductor $54.889$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 531.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(54.8894503975\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: no (minimal twist has level 177)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 235.10
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.31

$q$-expansion

\(f(q)\) \(=\) \(q-4.64719i q^{2} -5.59638 q^{4} +0.691812 q^{5} -76.1022 q^{7} -48.3476i q^{8} +O(q^{10})\) \(q-4.64719i q^{2} -5.59638 q^{4} +0.691812 q^{5} -76.1022 q^{7} -48.3476i q^{8} -3.21498i q^{10} -74.8765i q^{11} -105.330i q^{13} +353.662i q^{14} -314.223 q^{16} +141.930 q^{17} -170.820 q^{19} -3.87164 q^{20} -347.966 q^{22} -226.464i q^{23} -624.521 q^{25} -489.487 q^{26} +425.897 q^{28} +677.129 q^{29} -114.554i q^{31} +686.691i q^{32} -659.575i q^{34} -52.6484 q^{35} +488.383i q^{37} +793.831i q^{38} -33.4475i q^{40} +1825.63 q^{41} +527.862i q^{43} +419.037i q^{44} -1052.42 q^{46} +1903.37i q^{47} +3390.55 q^{49} +2902.27i q^{50} +589.465i q^{52} -4683.03 q^{53} -51.8005i q^{55} +3679.36i q^{56} -3146.75i q^{58} +(-2331.20 + 2585.12i) q^{59} +6414.67i q^{61} -532.355 q^{62} -1836.38 q^{64} -72.8684i q^{65} -1772.67i q^{67} -794.294 q^{68} +244.667i q^{70} -170.581 q^{71} +8999.83i q^{73} +2269.61 q^{74} +955.971 q^{76} +5698.27i q^{77} -204.796 q^{79} -217.383 q^{80} -8484.05i q^{82} -1391.38i q^{83} +98.1888 q^{85} +2453.07 q^{86} -3620.10 q^{88} -3001.60i q^{89} +8015.83i q^{91} +1267.38i q^{92} +8845.32 q^{94} -118.175 q^{95} +9224.90i q^{97} -15756.5i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/531\mathbb{Z}\right)^\times\).

\(n\) \(119\) \(415\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.64719i 1.16180i −0.813976 0.580899i \(-0.802701\pi\)
0.813976 0.580899i \(-0.197299\pi\)
\(3\) 0 0
\(4\) −5.59638 −0.349774
\(5\) 0.691812 0.0276725 0.0138362 0.999904i \(-0.495596\pi\)
0.0138362 + 0.999904i \(0.495596\pi\)
\(6\) 0 0
\(7\) −76.1022 −1.55311 −0.776553 0.630051i \(-0.783034\pi\)
−0.776553 + 0.630051i \(0.783034\pi\)
\(8\) 48.3476i 0.755431i
\(9\) 0 0
\(10\) 3.21498i 0.0321498i
\(11\) 74.8765i 0.618814i −0.950930 0.309407i \(-0.899869\pi\)
0.950930 0.309407i \(-0.100131\pi\)
\(12\) 0 0
\(13\) 105.330i 0.623253i −0.950205 0.311626i \(-0.899126\pi\)
0.950205 0.311626i \(-0.100874\pi\)
\(14\) 353.662i 1.80440i
\(15\) 0 0
\(16\) −314.223 −1.22743
\(17\) 141.930 0.491107 0.245553 0.969383i \(-0.421030\pi\)
0.245553 + 0.969383i \(0.421030\pi\)
\(18\) 0 0
\(19\) −170.820 −0.473184 −0.236592 0.971609i \(-0.576031\pi\)
−0.236592 + 0.971609i \(0.576031\pi\)
\(20\) −3.87164 −0.00967911
\(21\) 0 0
\(22\) −347.966 −0.718937
\(23\) 226.464i 0.428098i −0.976823 0.214049i \(-0.931335\pi\)
0.976823 0.214049i \(-0.0686653\pi\)
\(24\) 0 0
\(25\) −624.521 −0.999234
\(26\) −489.487 −0.724093
\(27\) 0 0
\(28\) 425.897 0.543236
\(29\) 677.129 0.805148 0.402574 0.915388i \(-0.368116\pi\)
0.402574 + 0.915388i \(0.368116\pi\)
\(30\) 0 0
\(31\) 114.554i 0.119203i −0.998222 0.0596016i \(-0.981017\pi\)
0.998222 0.0596016i \(-0.0189830\pi\)
\(32\) 686.691i 0.670596i
\(33\) 0 0
\(34\) 659.575i 0.570567i
\(35\) −52.6484 −0.0429783
\(36\) 0 0
\(37\) 488.383i 0.356745i 0.983963 + 0.178372i \(0.0570831\pi\)
−0.983963 + 0.178372i \(0.942917\pi\)
\(38\) 793.831i 0.549744i
\(39\) 0 0
\(40\) 33.4475i 0.0209047i
\(41\) 1825.63 1.08604 0.543019 0.839720i \(-0.317281\pi\)
0.543019 + 0.839720i \(0.317281\pi\)
\(42\) 0 0
\(43\) 527.862i 0.285485i 0.989760 + 0.142742i \(0.0455921\pi\)
−0.989760 + 0.142742i \(0.954408\pi\)
\(44\) 419.037i 0.216445i
\(45\) 0 0
\(46\) −1052.42 −0.497364
\(47\) 1903.37i 0.861643i 0.902437 + 0.430822i \(0.141776\pi\)
−0.902437 + 0.430822i \(0.858224\pi\)
\(48\) 0 0
\(49\) 3390.55 1.41214
\(50\) 2902.27i 1.16091i
\(51\) 0 0
\(52\) 589.465i 0.217997i
\(53\) −4683.03 −1.66715 −0.833576 0.552404i \(-0.813710\pi\)
−0.833576 + 0.552404i \(0.813710\pi\)
\(54\) 0 0
\(55\) 51.8005i 0.0171241i
\(56\) 3679.36i 1.17327i
\(57\) 0 0
\(58\) 3146.75i 0.935419i
\(59\) −2331.20 + 2585.12i −0.669693 + 0.742638i
\(60\) 0 0
\(61\) 6414.67i 1.72391i 0.506985 + 0.861955i \(0.330760\pi\)
−0.506985 + 0.861955i \(0.669240\pi\)
\(62\) −532.355 −0.138490
\(63\) 0 0
\(64\) −1836.38 −0.448335
\(65\) 72.8684i 0.0172469i
\(66\) 0 0
\(67\) 1772.67i 0.394892i −0.980314 0.197446i \(-0.936735\pi\)
0.980314 0.197446i \(-0.0632647\pi\)
\(68\) −794.294 −0.171776
\(69\) 0 0
\(70\) 244.667i 0.0499321i
\(71\) −170.581 −0.0338387 −0.0169193 0.999857i \(-0.505386\pi\)
−0.0169193 + 0.999857i \(0.505386\pi\)
\(72\) 0 0
\(73\) 8999.83i 1.68884i 0.535682 + 0.844420i \(0.320055\pi\)
−0.535682 + 0.844420i \(0.679945\pi\)
\(74\) 2269.61 0.414465
\(75\) 0 0
\(76\) 955.971 0.165507
\(77\) 5698.27i 0.961085i
\(78\) 0 0
\(79\) −204.796 −0.0328146 −0.0164073 0.999865i \(-0.505223\pi\)
−0.0164073 + 0.999865i \(0.505223\pi\)
\(80\) −217.383 −0.0339661
\(81\) 0 0
\(82\) 8484.05i 1.26176i
\(83\) 1391.38i 0.201971i −0.994888 0.100986i \(-0.967800\pi\)
0.994888 0.100986i \(-0.0321996\pi\)
\(84\) 0 0
\(85\) 98.1888 0.0135901
\(86\) 2453.07 0.331676
\(87\) 0 0
\(88\) −3620.10 −0.467472
\(89\) 3001.60i 0.378942i −0.981886 0.189471i \(-0.939323\pi\)
0.981886 0.189471i \(-0.0606773\pi\)
\(90\) 0 0
\(91\) 8015.83i 0.967978i
\(92\) 1267.38i 0.149738i
\(93\) 0 0
\(94\) 8845.32 1.00105
\(95\) −118.175 −0.0130942
\(96\) 0 0
\(97\) 9224.90i 0.980434i 0.871600 + 0.490217i \(0.163082\pi\)
−0.871600 + 0.490217i \(0.836918\pi\)
\(98\) 15756.5i 1.64062i
\(99\) 0 0
\(100\) 3495.06 0.349506
\(101\) 15773.4i 1.54626i −0.634250 0.773128i \(-0.718691\pi\)
0.634250 0.773128i \(-0.281309\pi\)
\(102\) 0 0
\(103\) 12464.1i 1.17486i 0.809273 + 0.587432i \(0.199861\pi\)
−0.809273 + 0.587432i \(0.800139\pi\)
\(104\) −5092.44 −0.470825
\(105\) 0 0
\(106\) 21762.9i 1.93689i
\(107\) 5179.20 0.452372 0.226186 0.974084i \(-0.427374\pi\)
0.226186 + 0.974084i \(0.427374\pi\)
\(108\) 0 0
\(109\) 653.130i 0.0549727i 0.999622 + 0.0274863i \(0.00875028\pi\)
−0.999622 + 0.0274863i \(0.991250\pi\)
\(110\) −240.727 −0.0198948
\(111\) 0 0
\(112\) 23913.0 1.90633
\(113\) 12708.3i 0.995247i 0.867393 + 0.497623i \(0.165794\pi\)
−0.867393 + 0.497623i \(0.834206\pi\)
\(114\) 0 0
\(115\) 156.671i 0.0118465i
\(116\) −3789.47 −0.281619
\(117\) 0 0
\(118\) 12013.6 + 10833.5i 0.862795 + 0.778048i
\(119\) −10801.2 −0.762742
\(120\) 0 0
\(121\) 9034.50 0.617069
\(122\) 29810.2 2.00283
\(123\) 0 0
\(124\) 641.089i 0.0416941i
\(125\) −864.434 −0.0553238
\(126\) 0 0
\(127\) −3763.18 −0.233318 −0.116659 0.993172i \(-0.537218\pi\)
−0.116659 + 0.993172i \(0.537218\pi\)
\(128\) 19521.1i 1.19147i
\(129\) 0 0
\(130\) −338.633 −0.0200375
\(131\) 1872.87i 0.109135i 0.998510 + 0.0545677i \(0.0173781\pi\)
−0.998510 + 0.0545677i \(0.982622\pi\)
\(132\) 0 0
\(133\) 12999.7 0.734906
\(134\) −8237.93 −0.458784
\(135\) 0 0
\(136\) 6861.97i 0.370998i
\(137\) 7217.96 0.384568 0.192284 0.981339i \(-0.438410\pi\)
0.192284 + 0.981339i \(0.438410\pi\)
\(138\) 0 0
\(139\) −4580.22 −0.237059 −0.118530 0.992951i \(-0.537818\pi\)
−0.118530 + 0.992951i \(0.537818\pi\)
\(140\) 294.641 0.0150327
\(141\) 0 0
\(142\) 792.721i 0.0393137i
\(143\) −7886.72 −0.385678
\(144\) 0 0
\(145\) 468.446 0.0222804
\(146\) 41823.9 1.96209
\(147\) 0 0
\(148\) 2733.18i 0.124780i
\(149\) 26501.4i 1.19370i −0.802352 0.596851i \(-0.796418\pi\)
0.802352 0.596851i \(-0.203582\pi\)
\(150\) 0 0
\(151\) 28179.8i 1.23590i −0.786217 0.617950i \(-0.787963\pi\)
0.786217 0.617950i \(-0.212037\pi\)
\(152\) 8258.72i 0.357458i
\(153\) 0 0
\(154\) 26481.0 1.11659
\(155\) 79.2500i 0.00329865i
\(156\) 0 0
\(157\) 12508.2i 0.507452i −0.967276 0.253726i \(-0.918344\pi\)
0.967276 0.253726i \(-0.0816562\pi\)
\(158\) 951.725i 0.0381239i
\(159\) 0 0
\(160\) 475.061i 0.0185571i
\(161\) 17234.4i 0.664882i
\(162\) 0 0
\(163\) −17112.4 −0.644075 −0.322038 0.946727i \(-0.604368\pi\)
−0.322038 + 0.946727i \(0.604368\pi\)
\(164\) −10216.9 −0.379867
\(165\) 0 0
\(166\) −6466.00 −0.234650
\(167\) 5275.71 0.189168 0.0945841 0.995517i \(-0.469848\pi\)
0.0945841 + 0.995517i \(0.469848\pi\)
\(168\) 0 0
\(169\) 17466.7 0.611556
\(170\) 456.302i 0.0157890i
\(171\) 0 0
\(172\) 2954.11i 0.0998551i
\(173\) 19148.2i 0.639786i 0.947454 + 0.319893i \(0.103647\pi\)
−0.947454 + 0.319893i \(0.896353\pi\)
\(174\) 0 0
\(175\) 47527.5 1.55192
\(176\) 23527.9i 0.759553i
\(177\) 0 0
\(178\) −13949.0 −0.440254
\(179\) 59089.1i 1.84417i −0.386987 0.922085i \(-0.626484\pi\)
0.386987 0.922085i \(-0.373516\pi\)
\(180\) 0 0
\(181\) 21278.1 0.649494 0.324747 0.945801i \(-0.394721\pi\)
0.324747 + 0.945801i \(0.394721\pi\)
\(182\) 37251.1 1.12459
\(183\) 0 0
\(184\) −10949.0 −0.323399
\(185\) 337.869i 0.00987201i
\(186\) 0 0
\(187\) 10627.2i 0.303904i
\(188\) 10652.0i 0.301380i
\(189\) 0 0
\(190\) 549.182i 0.0152128i
\(191\) 12072.7i 0.330932i −0.986215 0.165466i \(-0.947087\pi\)
0.986215 0.165466i \(-0.0529129\pi\)
\(192\) 0 0
\(193\) −43480.2 −1.16728 −0.583642 0.812011i \(-0.698373\pi\)
−0.583642 + 0.812011i \(0.698373\pi\)
\(194\) 42869.9 1.13907
\(195\) 0 0
\(196\) −18974.8 −0.493930
\(197\) −28225.1 −0.727283 −0.363642 0.931539i \(-0.618467\pi\)
−0.363642 + 0.931539i \(0.618467\pi\)
\(198\) 0 0
\(199\) −46347.4 −1.17036 −0.585180 0.810903i \(-0.698976\pi\)
−0.585180 + 0.810903i \(0.698976\pi\)
\(200\) 30194.1i 0.754853i
\(201\) 0 0
\(202\) −73301.8 −1.79644
\(203\) −51531.0 −1.25048
\(204\) 0 0
\(205\) 1262.99 0.0300534
\(206\) 57923.2 1.36495
\(207\) 0 0
\(208\) 33097.0i 0.765000i
\(209\) 12790.4i 0.292813i
\(210\) 0 0
\(211\) 49595.9i 1.11399i −0.830516 0.556994i \(-0.811955\pi\)
0.830516 0.556994i \(-0.188045\pi\)
\(212\) 26208.0 0.583126
\(213\) 0 0
\(214\) 24068.7i 0.525564i
\(215\) 365.181i 0.00790008i
\(216\) 0 0
\(217\) 8717.83i 0.185135i
\(218\) 3035.22 0.0638671
\(219\) 0 0
\(220\) 289.895i 0.00598957i
\(221\) 14949.4i 0.306084i
\(222\) 0 0
\(223\) 27897.9 0.560998 0.280499 0.959854i \(-0.409500\pi\)
0.280499 + 0.959854i \(0.409500\pi\)
\(224\) 52258.7i 1.04151i
\(225\) 0 0
\(226\) 59057.9 1.15628
\(227\) 34726.4i 0.673919i 0.941519 + 0.336960i \(0.109399\pi\)
−0.941519 + 0.336960i \(0.890601\pi\)
\(228\) 0 0
\(229\) 57357.6i 1.09376i 0.837212 + 0.546878i \(0.184184\pi\)
−0.837212 + 0.546878i \(0.815816\pi\)
\(230\) −728.078 −0.0137633
\(231\) 0 0
\(232\) 32737.6i 0.608234i
\(233\) 64148.1i 1.18160i 0.806816 + 0.590802i \(0.201189\pi\)
−0.806816 + 0.590802i \(0.798811\pi\)
\(234\) 0 0
\(235\) 1316.77i 0.0238438i
\(236\) 13046.3 14467.3i 0.234241 0.259755i
\(237\) 0 0
\(238\) 50195.2i 0.886151i
\(239\) 33929.5 0.593994 0.296997 0.954878i \(-0.404015\pi\)
0.296997 + 0.954878i \(0.404015\pi\)
\(240\) 0 0
\(241\) −35255.6 −0.607007 −0.303503 0.952830i \(-0.598156\pi\)
−0.303503 + 0.952830i \(0.598156\pi\)
\(242\) 41985.1i 0.716909i
\(243\) 0 0
\(244\) 35898.9i 0.602978i
\(245\) 2345.62 0.0390775
\(246\) 0 0
\(247\) 17992.4i 0.294913i
\(248\) −5538.42 −0.0900498
\(249\) 0 0
\(250\) 4017.19i 0.0642750i
\(251\) −57278.6 −0.909170 −0.454585 0.890703i \(-0.650212\pi\)
−0.454585 + 0.890703i \(0.650212\pi\)
\(252\) 0 0
\(253\) −16956.8 −0.264913
\(254\) 17488.2i 0.271068i
\(255\) 0 0
\(256\) 61336.0 0.935913
\(257\) 15290.7 0.231506 0.115753 0.993278i \(-0.463072\pi\)
0.115753 + 0.993278i \(0.463072\pi\)
\(258\) 0 0
\(259\) 37167.1i 0.554062i
\(260\) 407.799i 0.00603253i
\(261\) 0 0
\(262\) 8703.60 0.126793
\(263\) −90982.9 −1.31537 −0.657685 0.753293i \(-0.728464\pi\)
−0.657685 + 0.753293i \(0.728464\pi\)
\(264\) 0 0
\(265\) −3239.78 −0.0461342
\(266\) 60412.3i 0.853812i
\(267\) 0 0
\(268\) 9920.53i 0.138123i
\(269\) 68865.4i 0.951692i 0.879529 + 0.475846i \(0.157858\pi\)
−0.879529 + 0.475846i \(0.842142\pi\)
\(270\) 0 0
\(271\) −120932. −1.64666 −0.823330 0.567563i \(-0.807886\pi\)
−0.823330 + 0.567563i \(0.807886\pi\)
\(272\) −44597.6 −0.602800
\(273\) 0 0
\(274\) 33543.2i 0.446790i
\(275\) 46762.0i 0.618340i
\(276\) 0 0
\(277\) 81260.6 1.05906 0.529530 0.848291i \(-0.322368\pi\)
0.529530 + 0.848291i \(0.322368\pi\)
\(278\) 21285.1i 0.275415i
\(279\) 0 0
\(280\) 2545.43i 0.0324672i
\(281\) 58788.8 0.744530 0.372265 0.928127i \(-0.378581\pi\)
0.372265 + 0.928127i \(0.378581\pi\)
\(282\) 0 0
\(283\) 74223.1i 0.926757i 0.886160 + 0.463379i \(0.153363\pi\)
−0.886160 + 0.463379i \(0.846637\pi\)
\(284\) 954.634 0.0118359
\(285\) 0 0
\(286\) 36651.1i 0.448079i
\(287\) −138934. −1.68673
\(288\) 0 0
\(289\) −63376.9 −0.758814
\(290\) 2176.96i 0.0258854i
\(291\) 0 0
\(292\) 50366.4i 0.590712i
\(293\) −105265. −1.22617 −0.613083 0.790019i \(-0.710071\pi\)
−0.613083 + 0.790019i \(0.710071\pi\)
\(294\) 0 0
\(295\) −1612.75 + 1788.42i −0.0185321 + 0.0205506i
\(296\) 23612.2 0.269496
\(297\) 0 0
\(298\) −123157. −1.38684
\(299\) −23853.4 −0.266813
\(300\) 0 0
\(301\) 40171.5i 0.443389i
\(302\) −130957. −1.43587
\(303\) 0 0
\(304\) 53675.4 0.580802
\(305\) 4437.75i 0.0477049i
\(306\) 0 0
\(307\) 3547.26 0.0376371 0.0188185 0.999823i \(-0.494010\pi\)
0.0188185 + 0.999823i \(0.494010\pi\)
\(308\) 31889.7i 0.336162i
\(309\) 0 0
\(310\) −368.290 −0.00383236
\(311\) −151291. −1.56420 −0.782102 0.623150i \(-0.785853\pi\)
−0.782102 + 0.623150i \(0.785853\pi\)
\(312\) 0 0
\(313\) 25650.3i 0.261821i −0.991394 0.130911i \(-0.958210\pi\)
0.991394 0.130911i \(-0.0417901\pi\)
\(314\) −58128.0 −0.589557
\(315\) 0 0
\(316\) 1146.11 0.0114777
\(317\) −129307. −1.28677 −0.643387 0.765541i \(-0.722471\pi\)
−0.643387 + 0.765541i \(0.722471\pi\)
\(318\) 0 0
\(319\) 50701.1i 0.498237i
\(320\) −1270.43 −0.0124065
\(321\) 0 0
\(322\) 80091.6 0.772459
\(323\) −24244.4 −0.232384
\(324\) 0 0
\(325\) 65780.6i 0.622775i
\(326\) 79524.7i 0.748285i
\(327\) 0 0
\(328\) 88264.8i 0.820427i
\(329\) 144851.i 1.33822i
\(330\) 0 0
\(331\) −132570. −1.21001 −0.605004 0.796223i \(-0.706828\pi\)
−0.605004 + 0.796223i \(0.706828\pi\)
\(332\) 7786.68i 0.0706442i
\(333\) 0 0
\(334\) 24517.2i 0.219775i
\(335\) 1226.35i 0.0109276i
\(336\) 0 0
\(337\) 182026.i 1.60277i −0.598146 0.801387i \(-0.704096\pi\)
0.598146 0.801387i \(-0.295904\pi\)
\(338\) 81170.9i 0.710505i
\(339\) 0 0
\(340\) −549.502 −0.00475348
\(341\) −8577.42 −0.0737646
\(342\) 0 0
\(343\) −75307.0 −0.640099
\(344\) 25520.8 0.215664
\(345\) 0 0
\(346\) 88985.1 0.743302
\(347\) 36084.6i 0.299684i 0.988710 + 0.149842i \(0.0478765\pi\)
−0.988710 + 0.149842i \(0.952124\pi\)
\(348\) 0 0
\(349\) 155958.i 1.28043i 0.768194 + 0.640217i \(0.221156\pi\)
−0.768194 + 0.640217i \(0.778844\pi\)
\(350\) 220869.i 1.80301i
\(351\) 0 0
\(352\) 51417.0 0.414975
\(353\) 106349.i 0.853464i −0.904378 0.426732i \(-0.859665\pi\)
0.904378 0.426732i \(-0.140335\pi\)
\(354\) 0 0
\(355\) −118.010 −0.000936400
\(356\) 16798.1i 0.132544i
\(357\) 0 0
\(358\) −274598. −2.14255
\(359\) 221323. 1.71726 0.858632 0.512592i \(-0.171315\pi\)
0.858632 + 0.512592i \(0.171315\pi\)
\(360\) 0 0
\(361\) −101142. −0.776097
\(362\) 98883.2i 0.754580i
\(363\) 0 0
\(364\) 44859.6i 0.338573i
\(365\) 6226.19i 0.0467344i
\(366\) 0 0
\(367\) 231374.i 1.71784i −0.512108 0.858921i \(-0.671135\pi\)
0.512108 0.858921i \(-0.328865\pi\)
\(368\) 71160.1i 0.525462i
\(369\) 0 0
\(370\) 1570.14 0.0114693
\(371\) 356389. 2.58927
\(372\) 0 0
\(373\) 129853. 0.933327 0.466663 0.884435i \(-0.345456\pi\)
0.466663 + 0.884435i \(0.345456\pi\)
\(374\) −49386.7 −0.353075
\(375\) 0 0
\(376\) 92023.4 0.650912
\(377\) 71321.8i 0.501810i
\(378\) 0 0
\(379\) −271372. −1.88924 −0.944618 0.328173i \(-0.893567\pi\)
−0.944618 + 0.328173i \(0.893567\pi\)
\(380\) 661.352 0.00458000
\(381\) 0 0
\(382\) −56104.4 −0.384477
\(383\) 228592. 1.55834 0.779171 0.626811i \(-0.215640\pi\)
0.779171 + 0.626811i \(0.215640\pi\)
\(384\) 0 0
\(385\) 3942.13i 0.0265956i
\(386\) 202061.i 1.35615i
\(387\) 0 0
\(388\) 51626.1i 0.342930i
\(389\) −19842.8 −0.131131 −0.0655654 0.997848i \(-0.520885\pi\)
−0.0655654 + 0.997848i \(0.520885\pi\)
\(390\) 0 0
\(391\) 32142.0i 0.210242i
\(392\) 163925.i 1.06678i
\(393\) 0 0
\(394\) 131168.i 0.844956i
\(395\) −141.680 −0.000908061
\(396\) 0 0
\(397\) 286794.i 1.81966i 0.414986 + 0.909828i \(0.363787\pi\)
−0.414986 + 0.909828i \(0.636213\pi\)
\(398\) 215385.i 1.35972i
\(399\) 0 0
\(400\) 196239. 1.22649
\(401\) 227378.i 1.41403i 0.707198 + 0.707016i \(0.249959\pi\)
−0.707198 + 0.707016i \(0.750041\pi\)
\(402\) 0 0
\(403\) −12066.0 −0.0742937
\(404\) 88273.7i 0.540839i
\(405\) 0 0
\(406\) 239475.i 1.45281i
\(407\) 36568.4 0.220759
\(408\) 0 0
\(409\) 158393.i 0.946870i −0.880829 0.473435i \(-0.843014\pi\)
0.880829 0.473435i \(-0.156986\pi\)
\(410\) 5869.37i 0.0349159i
\(411\) 0 0
\(412\) 69754.0i 0.410937i
\(413\) 177410. 196734.i 1.04011 1.15340i
\(414\) 0 0
\(415\) 962.573i 0.00558904i
\(416\) 72328.9 0.417951
\(417\) 0 0
\(418\) 59439.3 0.340190
\(419\) 122537.i 0.697976i −0.937127 0.348988i \(-0.886525\pi\)
0.937127 0.348988i \(-0.113475\pi\)
\(420\) 0 0
\(421\) 46500.7i 0.262359i −0.991359 0.131179i \(-0.958124\pi\)
0.991359 0.131179i \(-0.0418764\pi\)
\(422\) −230481. −1.29423
\(423\) 0 0
\(424\) 226413.i 1.25942i
\(425\) −88638.3 −0.490731
\(426\) 0 0
\(427\) 488171.i 2.67742i
\(428\) −28984.8 −0.158228
\(429\) 0 0
\(430\) 1697.07 0.00917829
\(431\) 17223.7i 0.0927198i 0.998925 + 0.0463599i \(0.0147621\pi\)
−0.998925 + 0.0463599i \(0.985238\pi\)
\(432\) 0 0
\(433\) 28363.6 0.151281 0.0756407 0.997135i \(-0.475900\pi\)
0.0756407 + 0.997135i \(0.475900\pi\)
\(434\) 40513.4 0.215090
\(435\) 0 0
\(436\) 3655.17i 0.0192280i
\(437\) 38684.5i 0.202569i
\(438\) 0 0
\(439\) −222696. −1.15554 −0.577769 0.816200i \(-0.696077\pi\)
−0.577769 + 0.816200i \(0.696077\pi\)
\(440\) −2504.43 −0.0129361
\(441\) 0 0
\(442\) −69472.9 −0.355607
\(443\) 174192.i 0.887609i 0.896124 + 0.443804i \(0.146371\pi\)
−0.896124 + 0.443804i \(0.853629\pi\)
\(444\) 0 0
\(445\) 2076.54i 0.0104863i
\(446\) 129647.i 0.651766i
\(447\) 0 0
\(448\) 139753. 0.696312
\(449\) 24381.9 0.120941 0.0604706 0.998170i \(-0.480740\pi\)
0.0604706 + 0.998170i \(0.480740\pi\)
\(450\) 0 0
\(451\) 136697.i 0.672056i
\(452\) 71120.5i 0.348111i
\(453\) 0 0
\(454\) 161380. 0.782958
\(455\) 5545.44i 0.0267864i
\(456\) 0 0
\(457\) 278285.i 1.33247i −0.745741 0.666236i \(-0.767904\pi\)
0.745741 0.666236i \(-0.232096\pi\)
\(458\) 266552. 1.27072
\(459\) 0 0
\(460\) 876.788i 0.00414361i
\(461\) −13805.0 −0.0649581 −0.0324791 0.999472i \(-0.510340\pi\)
−0.0324791 + 0.999472i \(0.510340\pi\)
\(462\) 0 0
\(463\) 129315.i 0.603234i 0.953429 + 0.301617i \(0.0975264\pi\)
−0.953429 + 0.301617i \(0.902474\pi\)
\(464\) −212769. −0.988264
\(465\) 0 0
\(466\) 298109. 1.37279
\(467\) 294700.i 1.35128i 0.737230 + 0.675641i \(0.236133\pi\)
−0.737230 + 0.675641i \(0.763867\pi\)
\(468\) 0 0
\(469\) 134904.i 0.613309i
\(470\) 6119.30 0.0277017
\(471\) 0 0
\(472\) 124984. + 112708.i 0.561012 + 0.505907i
\(473\) 39524.5 0.176662
\(474\) 0 0
\(475\) 106680. 0.472822
\(476\) 60447.5 0.266787
\(477\) 0 0
\(478\) 157677.i 0.690100i
\(479\) 144629. 0.630355 0.315177 0.949033i \(-0.397936\pi\)
0.315177 + 0.949033i \(0.397936\pi\)
\(480\) 0 0
\(481\) 51441.3 0.222342
\(482\) 163839.i 0.705219i
\(483\) 0 0
\(484\) −50560.5 −0.215834
\(485\) 6381.90i 0.0271310i
\(486\) 0 0
\(487\) −275005. −1.15953 −0.579766 0.814783i \(-0.696856\pi\)
−0.579766 + 0.814783i \(0.696856\pi\)
\(488\) 310134. 1.30230
\(489\) 0 0
\(490\) 10900.6i 0.0454001i
\(491\) −32951.9 −0.136684 −0.0683419 0.997662i \(-0.521771\pi\)
−0.0683419 + 0.997662i \(0.521771\pi\)
\(492\) 0 0
\(493\) 96104.9 0.395414
\(494\) 83614.0 0.342630
\(495\) 0 0
\(496\) 35995.5i 0.146314i
\(497\) 12981.6 0.0525551
\(498\) 0 0
\(499\) −17305.7 −0.0695005 −0.0347503 0.999396i \(-0.511064\pi\)
−0.0347503 + 0.999396i \(0.511064\pi\)
\(500\) 4837.70 0.0193508
\(501\) 0 0
\(502\) 266185.i 1.05627i
\(503\) 264972.i 1.04728i −0.851938 0.523642i \(-0.824573\pi\)
0.851938 0.523642i \(-0.175427\pi\)
\(504\) 0 0
\(505\) 10912.2i 0.0427887i
\(506\) 78801.7i 0.307776i
\(507\) 0 0
\(508\) 21060.2 0.0816084
\(509\) 446664.i 1.72403i −0.506882 0.862015i \(-0.669202\pi\)
0.506882 0.862015i \(-0.330798\pi\)
\(510\) 0 0
\(511\) 684907.i 2.62295i
\(512\) 27296.9i 0.104129i
\(513\) 0 0
\(514\) 71059.0i 0.268963i
\(515\) 8622.84i 0.0325114i
\(516\) 0 0
\(517\) 142518. 0.533197
\(518\) −172722. −0.643708
\(519\) 0 0
\(520\) −3523.01 −0.0130289
\(521\) −14595.5 −0.0537703 −0.0268852 0.999639i \(-0.508559\pi\)
−0.0268852 + 0.999639i \(0.508559\pi\)
\(522\) 0 0
\(523\) −40396.8 −0.147687 −0.0738437 0.997270i \(-0.523527\pi\)
−0.0738437 + 0.997270i \(0.523527\pi\)
\(524\) 10481.3i 0.0381727i
\(525\) 0 0
\(526\) 422815.i 1.52819i
\(527\) 16258.7i 0.0585415i
\(528\) 0 0
\(529\) 228555. 0.816732
\(530\) 15055.9i 0.0535987i
\(531\) 0 0
\(532\) −72751.5 −0.257051
\(533\) 192293.i 0.676876i
\(534\) 0 0
\(535\) 3583.03 0.0125182
\(536\) −85704.3 −0.298314
\(537\) 0 0
\(538\) 320030. 1.10567
\(539\) 253873.i 0.873853i
\(540\) 0 0
\(541\) 528895.i 1.80707i 0.428513 + 0.903536i \(0.359038\pi\)
−0.428513 + 0.903536i \(0.640962\pi\)
\(542\) 561996.i 1.91309i
\(543\) 0 0
\(544\) 97461.9i 0.329334i
\(545\) 451.843i 0.00152123i
\(546\) 0 0
\(547\) −223190. −0.745935 −0.372967 0.927844i \(-0.621660\pi\)
−0.372967 + 0.927844i \(0.621660\pi\)
\(548\) −40394.4 −0.134512
\(549\) 0 0
\(550\) 217312. 0.718386
\(551\) −115667. −0.380983
\(552\) 0 0
\(553\) 15585.4 0.0509645
\(554\) 377634.i 1.23041i
\(555\) 0 0
\(556\) 25632.6 0.0829170
\(557\) −497882. −1.60478 −0.802392 0.596798i \(-0.796439\pi\)
−0.802392 + 0.596798i \(0.796439\pi\)
\(558\) 0 0
\(559\) 55599.5 0.177929
\(560\) 16543.3 0.0527530
\(561\) 0 0
\(562\) 273203.i 0.864993i
\(563\) 268686.i 0.847671i 0.905739 + 0.423836i \(0.139317\pi\)
−0.905739 + 0.423836i \(0.860683\pi\)
\(564\) 0 0
\(565\) 8791.76i 0.0275409i
\(566\) 344929. 1.07670
\(567\) 0 0
\(568\) 8247.17i 0.0255628i
\(569\) 122187.i 0.377398i −0.982035 0.188699i \(-0.939573\pi\)
0.982035 0.188699i \(-0.0604271\pi\)
\(570\) 0 0
\(571\) 20881.3i 0.0640449i 0.999487 + 0.0320224i \(0.0101948\pi\)
−0.999487 + 0.0320224i \(0.989805\pi\)
\(572\) 44137.1 0.134900
\(573\) 0 0
\(574\) 645655.i 1.95964i
\(575\) 141432.i 0.427770i
\(576\) 0 0
\(577\) −309645. −0.930063 −0.465032 0.885294i \(-0.653957\pi\)
−0.465032 + 0.885294i \(0.653957\pi\)
\(578\) 294525.i 0.881588i
\(579\) 0 0
\(580\) −2621.60 −0.00779311
\(581\) 105887.i 0.313683i
\(582\) 0 0
\(583\) 350649.i 1.03166i
\(584\) 435120. 1.27580
\(585\) 0 0
\(586\) 489187.i 1.42456i
\(587\) 245888.i 0.713611i −0.934179 0.356806i \(-0.883866\pi\)
0.934179 0.356806i \(-0.116134\pi\)
\(588\) 0 0
\(589\) 19568.1i 0.0564050i
\(590\) 8311.12 + 7494.77i 0.0238757 + 0.0215305i
\(591\) 0 0
\(592\) 153461.i 0.437880i
\(593\) −396294. −1.12696 −0.563480 0.826130i \(-0.690538\pi\)
−0.563480 + 0.826130i \(0.690538\pi\)
\(594\) 0 0
\(595\) −7472.39 −0.0211070
\(596\) 148312.i 0.417526i
\(597\) 0 0
\(598\) 110851.i 0.309983i
\(599\) 107130. 0.298579 0.149289 0.988794i \(-0.452301\pi\)
0.149289 + 0.988794i \(0.452301\pi\)
\(600\) 0 0
\(601\) 479681.i 1.32802i 0.747725 + 0.664008i \(0.231146\pi\)
−0.747725 + 0.664008i \(0.768854\pi\)
\(602\) −186684. −0.515128
\(603\) 0 0
\(604\) 157705.i 0.432285i
\(605\) 6250.18 0.0170758
\(606\) 0 0
\(607\) −182930. −0.496486 −0.248243 0.968698i \(-0.579853\pi\)
−0.248243 + 0.968698i \(0.579853\pi\)
\(608\) 117300.i 0.317316i
\(609\) 0 0
\(610\) 20623.1 0.0554234
\(611\) 200481. 0.537021
\(612\) 0 0
\(613\) 474523.i 1.26281i −0.775455 0.631403i \(-0.782479\pi\)
0.775455 0.631403i \(-0.217521\pi\)
\(614\) 16484.8i 0.0437267i
\(615\) 0 0
\(616\) 275498. 0.726034
\(617\) 339241. 0.891123 0.445562 0.895251i \(-0.353004\pi\)
0.445562 + 0.895251i \(0.353004\pi\)
\(618\) 0 0
\(619\) 307348. 0.802137 0.401069 0.916048i \(-0.368639\pi\)
0.401069 + 0.916048i \(0.368639\pi\)
\(620\) 443.513i 0.00115378i
\(621\) 0 0
\(622\) 703080.i 1.81729i
\(623\) 228429.i 0.588538i
\(624\) 0 0
\(625\) 389728. 0.997703
\(626\) −119202. −0.304183
\(627\) 0 0
\(628\) 70000.6i 0.177493i
\(629\) 69316.2i 0.175200i
\(630\) 0 0
\(631\) 478230. 1.20110 0.600548 0.799589i \(-0.294949\pi\)
0.600548 + 0.799589i \(0.294949\pi\)
\(632\) 9901.38i 0.0247892i
\(633\) 0 0
\(634\) 600913.i 1.49497i
\(635\) −2603.41 −0.00645648
\(636\) 0 0
\(637\) 357126.i 0.880121i
\(638\) −235618. −0.578850
\(639\) 0 0
\(640\) 13504.9i 0.0329710i
\(641\) −84112.5 −0.204713 −0.102356 0.994748i \(-0.532638\pi\)
−0.102356 + 0.994748i \(0.532638\pi\)
\(642\) 0 0
\(643\) 202387. 0.489508 0.244754 0.969585i \(-0.421293\pi\)
0.244754 + 0.969585i \(0.421293\pi\)
\(644\) 96450.3i 0.232558i
\(645\) 0 0
\(646\) 112668.i 0.269983i
\(647\) −729383. −1.74240 −0.871199 0.490930i \(-0.836657\pi\)
−0.871199 + 0.490930i \(0.836657\pi\)
\(648\) 0 0
\(649\) 193565. + 174552.i 0.459555 + 0.414416i
\(650\) 305695. 0.723539
\(651\) 0 0
\(652\) 95767.6 0.225280
\(653\) 33872.3 0.0794362 0.0397181 0.999211i \(-0.487354\pi\)
0.0397181 + 0.999211i \(0.487354\pi\)
\(654\) 0 0
\(655\) 1295.68i 0.00302005i
\(656\) −573654. −1.33304
\(657\) 0 0
\(658\) −673149. −1.55475
\(659\) 12252.6i 0.0282134i 0.999900 + 0.0141067i \(0.00449046\pi\)
−0.999900 + 0.0141067i \(0.995510\pi\)
\(660\) 0 0
\(661\) −193919. −0.443832 −0.221916 0.975066i \(-0.571231\pi\)
−0.221916 + 0.975066i \(0.571231\pi\)
\(662\) 616076.i 1.40578i
\(663\) 0 0
\(664\) −67269.8 −0.152575
\(665\) 8993.38 0.0203367
\(666\) 0 0
\(667\) 153345.i 0.344682i
\(668\) −29524.9 −0.0661660
\(669\) 0 0
\(670\) −5699.10 −0.0126957
\(671\) 480308. 1.06678
\(672\) 0 0
\(673\) 527119.i 1.16380i 0.813260 + 0.581900i \(0.197690\pi\)
−0.813260 + 0.581900i \(0.802310\pi\)
\(674\) −845907. −1.86210
\(675\) 0 0
\(676\) −97750.0 −0.213906
\(677\) −160122. −0.349359 −0.174680 0.984625i \(-0.555889\pi\)
−0.174680 + 0.984625i \(0.555889\pi\)
\(678\) 0 0
\(679\) 702036.i 1.52272i
\(680\) 4747.19i 0.0102664i
\(681\) 0 0
\(682\) 39860.9i 0.0856995i
\(683\) 96946.8i 0.207822i −0.994587 0.103911i \(-0.966864\pi\)
0.994587 0.103911i \(-0.0331358\pi\)
\(684\) 0 0
\(685\) 4993.47 0.0106420
\(686\) 349966.i 0.743666i
\(687\) 0 0
\(688\) 165866.i 0.350413i
\(689\) 493262.i 1.03906i
\(690\) 0 0
\(691\) 292122.i 0.611798i 0.952064 + 0.305899i \(0.0989570\pi\)
−0.952064 + 0.305899i \(0.901043\pi\)
\(692\) 107160.i 0.223780i
\(693\) 0 0
\(694\) 167692. 0.348172
\(695\) −3168.65 −0.00656001
\(696\) 0 0
\(697\) 259111. 0.533361
\(698\) 724767. 1.48761
\(699\) 0 0
\(700\) −265982. −0.542820
\(701\) 761446.i 1.54954i −0.632243 0.774770i \(-0.717866\pi\)
0.632243 0.774770i \(-0.282134\pi\)
\(702\) 0 0
\(703\) 83425.4i 0.168806i
\(704\) 137502.i 0.277436i
\(705\) 0 0
\(706\) −494225. −0.991552
\(707\) 1.20039e6i 2.40150i
\(708\) 0 0
\(709\) 9379.41 0.0186588 0.00932938 0.999956i \(-0.497030\pi\)
0.00932938 + 0.999956i \(0.497030\pi\)
\(710\) 548.414i 0.00108791i
\(711\) 0 0
\(712\) −145120. −0.286265
\(713\) −25942.4 −0.0510307
\(714\) 0 0
\(715\) −5456.13 −0.0106727
\(716\) 330685.i 0.645042i
\(717\) 0 0
\(718\) 1.02853e6i 1.99511i
\(719\) 474152.i 0.917192i −0.888645 0.458596i \(-0.848353\pi\)
0.888645 0.458596i \(-0.151647\pi\)
\(720\) 0 0
\(721\) 948549.i 1.82469i
\(722\) 470025.i 0.901667i
\(723\) 0 0
\(724\) −119080. −0.227176
\(725\) −422882. −0.804531
\(726\) 0 0
\(727\) −172753. −0.326857 −0.163429 0.986555i \(-0.552255\pi\)
−0.163429 + 0.986555i \(0.552255\pi\)
\(728\) 387546. 0.731241
\(729\) 0 0
\(730\) 28934.3 0.0542959
\(731\) 74919.4i 0.140204i
\(732\) 0 0
\(733\) 192952. 0.359122 0.179561 0.983747i \(-0.442532\pi\)
0.179561 + 0.983747i \(0.442532\pi\)
\(734\) −1.07524e6 −1.99578
\(735\) 0 0
\(736\) 155511. 0.287081
\(737\) −132731. −0.244365
\(738\) 0 0
\(739\) 894229.i 1.63742i 0.574208 + 0.818709i \(0.305310\pi\)
−0.574208 + 0.818709i \(0.694690\pi\)
\(740\) 1890.85i 0.00345297i
\(741\) 0 0
\(742\) 1.65621e6i 3.00820i
\(743\) 814173. 1.47482 0.737410 0.675445i \(-0.236048\pi\)
0.737410 + 0.675445i \(0.236048\pi\)
\(744\) 0 0
\(745\) 18334.0i 0.0330327i
\(746\) 603451.i 1.08434i
\(747\) 0 0
\(748\) 59473.9i 0.106298i
\(749\) −394149. −0.702582
\(750\) 0 0
\(751\) 417689.i 0.740581i −0.928916 0.370291i \(-0.879258\pi\)
0.928916 0.370291i \(-0.120742\pi\)
\(752\) 598082.i 1.05761i
\(753\) 0 0
\(754\) −331446. −0.583002
\(755\) 19495.1i 0.0342004i
\(756\) 0 0
\(757\) −999639. −1.74442 −0.872210 0.489131i \(-0.837314\pi\)
−0.872210 + 0.489131i \(0.837314\pi\)
\(758\) 1.26112e6i 2.19491i
\(759\) 0 0
\(760\) 5713.48i 0.00989176i
\(761\) 698337. 1.20586 0.602928 0.797795i \(-0.294001\pi\)
0.602928 + 0.797795i \(0.294001\pi\)
\(762\) 0 0
\(763\) 49704.7i 0.0853784i
\(764\) 67563.7i 0.115751i
\(765\) 0 0
\(766\) 1.06231e6i 1.81048i
\(767\) 272290. + 245545.i 0.462851 + 0.417388i
\(768\) 0 0
\(769\) 1.02416e6i 1.73187i 0.500155 + 0.865936i \(0.333276\pi\)
−0.500155 + 0.865936i \(0.666724\pi\)
\(770\) 18319.8 0.0308987
\(771\) 0 0
\(772\) 243332. 0.408286
\(773\) 907633.i 1.51898i −0.650521 0.759488i \(-0.725449\pi\)
0.650521 0.759488i \(-0.274551\pi\)
\(774\) 0 0
\(775\) 71541.5i 0.119112i
\(776\) 446002. 0.740651
\(777\) 0 0
\(778\) 92213.4i 0.152347i
\(779\) −311853. −0.513896
\(780\) 0 0
\(781\) 12772.5i 0.0209399i
\(782\) −149370. −0.244259
\(783\) 0 0
\(784\) −1.06539e6 −1.73331
\(785\) 8653.32i 0.0140425i
\(786\) 0 0
\(787\) 418956. 0.676423 0.338212 0.941070i \(-0.390178\pi\)
0.338212 + 0.941070i \(0.390178\pi\)
\(788\) 157959. 0.254385
\(789\) 0 0
\(790\) 658.415i 0.00105498i
\(791\) 967131.i 1.54572i
\(792\) 0 0
\(793\) 675655. 1.07443
\(794\) 1.33279e6 2.11407
\(795\) 0 0
\(796\) 259378. 0.409361
\(797\) 429165.i