Properties

Label 531.5.c.d.235.18
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 235.18
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.23

$q$-expansion

\(f(q)\) \(=\) \(q-1.07792i q^{2} +14.8381 q^{4} -26.1400 q^{5} +49.1298 q^{7} -33.2410i q^{8} +O(q^{10})\) \(q-1.07792i q^{2} +14.8381 q^{4} -26.1400 q^{5} +49.1298 q^{7} -33.2410i q^{8} +28.1769i q^{10} -190.415i q^{11} -198.277i q^{13} -52.9580i q^{14} +201.578 q^{16} +81.6050 q^{17} -590.362 q^{19} -387.868 q^{20} -205.252 q^{22} +808.730i q^{23} +58.3017 q^{25} -213.726 q^{26} +728.992 q^{28} +714.690 q^{29} +1370.46i q^{31} -749.141i q^{32} -87.9637i q^{34} -1284.26 q^{35} -2106.01i q^{37} +636.363i q^{38} +868.921i q^{40} -1132.48 q^{41} -349.669i q^{43} -2825.40i q^{44} +871.746 q^{46} -2257.08i q^{47} +12.7381 q^{49} -62.8446i q^{50} -2942.05i q^{52} -4733.28 q^{53} +4977.46i q^{55} -1633.12i q^{56} -770.378i q^{58} +(3019.48 - 1732.08i) q^{59} +1500.86i q^{61} +1477.24 q^{62} +2417.74 q^{64} +5182.96i q^{65} +233.568i q^{67} +1210.86 q^{68} +1384.32i q^{70} -2664.24 q^{71} -9123.79i q^{73} -2270.11 q^{74} -8759.85 q^{76} -9355.06i q^{77} -8286.52 q^{79} -5269.26 q^{80} +1220.72i q^{82} +7478.73i q^{83} -2133.16 q^{85} -376.915 q^{86} -6329.59 q^{88} -10590.0i q^{89} -9741.29i q^{91} +12000.0i q^{92} -2432.95 q^{94} +15432.1 q^{95} -15056.8i q^{97} -13.7306i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.07792i 0.269480i −0.990881 0.134740i \(-0.956980\pi\)
0.990881 0.134740i \(-0.0430199\pi\)
\(3\) 0 0
\(4\) 14.8381 0.927380
\(5\) −26.1400 −1.04560 −0.522801 0.852455i \(-0.675113\pi\)
−0.522801 + 0.852455i \(0.675113\pi\)
\(6\) 0 0
\(7\) 49.1298 1.00265 0.501325 0.865259i \(-0.332846\pi\)
0.501325 + 0.865259i \(0.332846\pi\)
\(8\) 33.2410i 0.519391i
\(9\) 0 0
\(10\) 28.1769i 0.281769i
\(11\) 190.415i 1.57368i −0.617158 0.786839i \(-0.711716\pi\)
0.617158 0.786839i \(-0.288284\pi\)
\(12\) 0 0
\(13\) 198.277i 1.17323i −0.809864 0.586617i \(-0.800459\pi\)
0.809864 0.586617i \(-0.199541\pi\)
\(14\) 52.9580i 0.270194i
\(15\) 0 0
\(16\) 201.578 0.787415
\(17\) 81.6050 0.282370 0.141185 0.989983i \(-0.454909\pi\)
0.141185 + 0.989983i \(0.454909\pi\)
\(18\) 0 0
\(19\) −590.362 −1.63535 −0.817676 0.575678i \(-0.804738\pi\)
−0.817676 + 0.575678i \(0.804738\pi\)
\(20\) −387.868 −0.969671
\(21\) 0 0
\(22\) −205.252 −0.424075
\(23\) 808.730i 1.52879i 0.644748 + 0.764395i \(0.276962\pi\)
−0.644748 + 0.764395i \(0.723038\pi\)
\(24\) 0 0
\(25\) 58.3017 0.0932828
\(26\) −213.726 −0.316163
\(27\) 0 0
\(28\) 728.992 0.929837
\(29\) 714.690 0.849809 0.424905 0.905238i \(-0.360308\pi\)
0.424905 + 0.905238i \(0.360308\pi\)
\(30\) 0 0
\(31\) 1370.46i 1.42607i 0.701126 + 0.713037i \(0.252681\pi\)
−0.701126 + 0.713037i \(0.747319\pi\)
\(32\) 749.141i 0.731583i
\(33\) 0 0
\(34\) 87.9637i 0.0760932i
\(35\) −1284.26 −1.04837
\(36\) 0 0
\(37\) 2106.01i 1.53835i −0.639036 0.769177i \(-0.720666\pi\)
0.639036 0.769177i \(-0.279334\pi\)
\(38\) 636.363i 0.440695i
\(39\) 0 0
\(40\) 868.921i 0.543076i
\(41\) −1132.48 −0.673694 −0.336847 0.941559i \(-0.609360\pi\)
−0.336847 + 0.941559i \(0.609360\pi\)
\(42\) 0 0
\(43\) 349.669i 0.189113i −0.995520 0.0945563i \(-0.969857\pi\)
0.995520 0.0945563i \(-0.0301432\pi\)
\(44\) 2825.40i 1.45940i
\(45\) 0 0
\(46\) 871.746 0.411978
\(47\) 2257.08i 1.02176i −0.859651 0.510882i \(-0.829319\pi\)
0.859651 0.510882i \(-0.170681\pi\)
\(48\) 0 0
\(49\) 12.7381 0.00530533
\(50\) 62.8446i 0.0251378i
\(51\) 0 0
\(52\) 2942.05i 1.08803i
\(53\) −4733.28 −1.68504 −0.842520 0.538665i \(-0.818929\pi\)
−0.842520 + 0.538665i \(0.818929\pi\)
\(54\) 0 0
\(55\) 4977.46i 1.64544i
\(56\) 1633.12i 0.520767i
\(57\) 0 0
\(58\) 770.378i 0.229007i
\(59\) 3019.48 1732.08i 0.867417 0.497582i
\(60\) 0 0
\(61\) 1500.86i 0.403349i 0.979453 + 0.201675i \(0.0646384\pi\)
−0.979453 + 0.201675i \(0.935362\pi\)
\(62\) 1477.24 0.384298
\(63\) 0 0
\(64\) 2417.74 0.590268
\(65\) 5182.96i 1.22674i
\(66\) 0 0
\(67\) 233.568i 0.0520312i 0.999662 + 0.0260156i \(0.00828195\pi\)
−0.999662 + 0.0260156i \(0.991718\pi\)
\(68\) 1210.86 0.261865
\(69\) 0 0
\(70\) 1384.32i 0.282515i
\(71\) −2664.24 −0.528515 −0.264258 0.964452i \(-0.585127\pi\)
−0.264258 + 0.964452i \(0.585127\pi\)
\(72\) 0 0
\(73\) 9123.79i 1.71210i −0.516892 0.856051i \(-0.672911\pi\)
0.516892 0.856051i \(-0.327089\pi\)
\(74\) −2270.11 −0.414556
\(75\) 0 0
\(76\) −8759.85 −1.51659
\(77\) 9355.06i 1.57785i
\(78\) 0 0
\(79\) −8286.52 −1.32776 −0.663878 0.747841i \(-0.731090\pi\)
−0.663878 + 0.747841i \(0.731090\pi\)
\(80\) −5269.26 −0.823323
\(81\) 0 0
\(82\) 1220.72i 0.181547i
\(83\) 7478.73i 1.08560i 0.839860 + 0.542802i \(0.182637\pi\)
−0.839860 + 0.542802i \(0.817363\pi\)
\(84\) 0 0
\(85\) −2133.16 −0.295247
\(86\) −376.915 −0.0509621
\(87\) 0 0
\(88\) −6329.59 −0.817354
\(89\) 10590.0i 1.33696i −0.743732 0.668478i \(-0.766946\pi\)
0.743732 0.668478i \(-0.233054\pi\)
\(90\) 0 0
\(91\) 9741.29i 1.17634i
\(92\) 12000.0i 1.41777i
\(93\) 0 0
\(94\) −2432.95 −0.275345
\(95\) 15432.1 1.70993
\(96\) 0 0
\(97\) 15056.8i 1.60025i −0.599833 0.800125i \(-0.704766\pi\)
0.599833 0.800125i \(-0.295234\pi\)
\(98\) 13.7306i 0.00142968i
\(99\) 0 0
\(100\) 865.086 0.0865086
\(101\) 7157.42i 0.701639i 0.936443 + 0.350819i \(0.114097\pi\)
−0.936443 + 0.350819i \(0.885903\pi\)
\(102\) 0 0
\(103\) 10141.1i 0.955896i 0.878388 + 0.477948i \(0.158619\pi\)
−0.878388 + 0.477948i \(0.841381\pi\)
\(104\) −6590.91 −0.609367
\(105\) 0 0
\(106\) 5102.10i 0.454085i
\(107\) −3578.48 −0.312558 −0.156279 0.987713i \(-0.549950\pi\)
−0.156279 + 0.987713i \(0.549950\pi\)
\(108\) 0 0
\(109\) 2420.34i 0.203716i −0.994799 0.101858i \(-0.967521\pi\)
0.994799 0.101858i \(-0.0324787\pi\)
\(110\) 5365.30 0.443413
\(111\) 0 0
\(112\) 9903.50 0.789501
\(113\) 1468.79i 0.115028i 0.998345 + 0.0575140i \(0.0183174\pi\)
−0.998345 + 0.0575140i \(0.981683\pi\)
\(114\) 0 0
\(115\) 21140.2i 1.59851i
\(116\) 10604.6 0.788097
\(117\) 0 0
\(118\) −1867.05 3254.76i −0.134089 0.233752i
\(119\) 4009.24 0.283118
\(120\) 0 0
\(121\) −21616.9 −1.47646
\(122\) 1617.81 0.108695
\(123\) 0 0
\(124\) 20335.0i 1.32251i
\(125\) 14813.5 0.948065
\(126\) 0 0
\(127\) −23012.3 −1.42677 −0.713383 0.700774i \(-0.752838\pi\)
−0.713383 + 0.700774i \(0.752838\pi\)
\(128\) 14592.4i 0.890649i
\(129\) 0 0
\(130\) 5586.82 0.330581
\(131\) 16089.2i 0.937543i 0.883319 + 0.468772i \(0.155303\pi\)
−0.883319 + 0.468772i \(0.844697\pi\)
\(132\) 0 0
\(133\) −29004.4 −1.63968
\(134\) 251.768 0.0140214
\(135\) 0 0
\(136\) 2712.63i 0.146661i
\(137\) 4170.90 0.222223 0.111111 0.993808i \(-0.464559\pi\)
0.111111 + 0.993808i \(0.464559\pi\)
\(138\) 0 0
\(139\) 22317.2 1.15508 0.577538 0.816364i \(-0.304014\pi\)
0.577538 + 0.816364i \(0.304014\pi\)
\(140\) −19055.9 −0.972239
\(141\) 0 0
\(142\) 2871.84i 0.142424i
\(143\) −37754.9 −1.84629
\(144\) 0 0
\(145\) −18682.0 −0.888562
\(146\) −9834.72 −0.461377
\(147\) 0 0
\(148\) 31249.1i 1.42664i
\(149\) 20688.1i 0.931853i −0.884823 0.465927i \(-0.845721\pi\)
0.884823 0.465927i \(-0.154279\pi\)
\(150\) 0 0
\(151\) 30416.0i 1.33397i −0.745069 0.666987i \(-0.767583\pi\)
0.745069 0.666987i \(-0.232417\pi\)
\(152\) 19624.2i 0.849387i
\(153\) 0 0
\(154\) −10084.0 −0.425198
\(155\) 35823.8i 1.49111i
\(156\) 0 0
\(157\) 37967.4i 1.54032i 0.637849 + 0.770162i \(0.279825\pi\)
−0.637849 + 0.770162i \(0.720175\pi\)
\(158\) 8932.21i 0.357804i
\(159\) 0 0
\(160\) 19582.6i 0.764945i
\(161\) 39732.7i 1.53284i
\(162\) 0 0
\(163\) −25366.6 −0.954745 −0.477372 0.878701i \(-0.658411\pi\)
−0.477372 + 0.878701i \(0.658411\pi\)
\(164\) −16803.8 −0.624770
\(165\) 0 0
\(166\) 8061.48 0.292549
\(167\) −33472.9 −1.20022 −0.600110 0.799917i \(-0.704877\pi\)
−0.600110 + 0.799917i \(0.704877\pi\)
\(168\) 0 0
\(169\) −10752.6 −0.376479
\(170\) 2299.38i 0.0795632i
\(171\) 0 0
\(172\) 5188.42i 0.175379i
\(173\) 23093.0i 0.771594i −0.922584 0.385797i \(-0.873927\pi\)
0.922584 0.385797i \(-0.126073\pi\)
\(174\) 0 0
\(175\) 2864.35 0.0935299
\(176\) 38383.5i 1.23914i
\(177\) 0 0
\(178\) −11415.2 −0.360283
\(179\) 49281.4i 1.53807i −0.639205 0.769036i \(-0.720736\pi\)
0.639205 0.769036i \(-0.279264\pi\)
\(180\) 0 0
\(181\) 52984.7 1.61731 0.808656 0.588282i \(-0.200196\pi\)
0.808656 + 0.588282i \(0.200196\pi\)
\(182\) −10500.3 −0.317001
\(183\) 0 0
\(184\) 26883.0 0.794039
\(185\) 55051.1i 1.60851i
\(186\) 0 0
\(187\) 15538.8i 0.444360i
\(188\) 33490.7i 0.947565i
\(189\) 0 0
\(190\) 16634.6i 0.460791i
\(191\) 31600.3i 0.866213i −0.901343 0.433106i \(-0.857417\pi\)
0.901343 0.433106i \(-0.142583\pi\)
\(192\) 0 0
\(193\) −6662.77 −0.178871 −0.0894355 0.995993i \(-0.528506\pi\)
−0.0894355 + 0.995993i \(0.528506\pi\)
\(194\) −16230.0 −0.431236
\(195\) 0 0
\(196\) 189.009 0.00492006
\(197\) 3249.83 0.0837391 0.0418696 0.999123i \(-0.486669\pi\)
0.0418696 + 0.999123i \(0.486669\pi\)
\(198\) 0 0
\(199\) −39419.7 −0.995421 −0.497710 0.867343i \(-0.665826\pi\)
−0.497710 + 0.867343i \(0.665826\pi\)
\(200\) 1938.01i 0.0484502i
\(201\) 0 0
\(202\) 7715.12 0.189078
\(203\) 35112.6 0.852061
\(204\) 0 0
\(205\) 29603.0 0.704415
\(206\) 10931.3 0.257595
\(207\) 0 0
\(208\) 39968.3i 0.923822i
\(209\) 112414.i 2.57352i
\(210\) 0 0
\(211\) 8215.58i 0.184533i −0.995734 0.0922663i \(-0.970589\pi\)
0.995734 0.0922663i \(-0.0294111\pi\)
\(212\) −70232.8 −1.56267
\(213\) 0 0
\(214\) 3857.31i 0.0842281i
\(215\) 9140.37i 0.197736i
\(216\) 0 0
\(217\) 67330.3i 1.42985i
\(218\) −2608.94 −0.0548973
\(219\) 0 0
\(220\) 73856.0i 1.52595i
\(221\) 16180.4i 0.331287i
\(222\) 0 0
\(223\) 24717.2 0.497037 0.248519 0.968627i \(-0.420056\pi\)
0.248519 + 0.968627i \(0.420056\pi\)
\(224\) 36805.2i 0.733521i
\(225\) 0 0
\(226\) 1583.24 0.0309978
\(227\) 22950.1i 0.445382i 0.974889 + 0.222691i \(0.0714842\pi\)
−0.974889 + 0.222691i \(0.928516\pi\)
\(228\) 0 0
\(229\) 26740.2i 0.509911i −0.966953 0.254955i \(-0.917939\pi\)
0.966953 0.254955i \(-0.0820608\pi\)
\(230\) −22787.5 −0.430765
\(231\) 0 0
\(232\) 23757.0i 0.441383i
\(233\) 29499.8i 0.543385i −0.962384 0.271693i \(-0.912417\pi\)
0.962384 0.271693i \(-0.0875834\pi\)
\(234\) 0 0
\(235\) 59000.1i 1.06836i
\(236\) 44803.3 25700.8i 0.804425 0.461448i
\(237\) 0 0
\(238\) 4321.64i 0.0762948i
\(239\) −37689.9 −0.659825 −0.329913 0.944011i \(-0.607019\pi\)
−0.329913 + 0.944011i \(0.607019\pi\)
\(240\) 0 0
\(241\) 49193.4 0.846979 0.423489 0.905901i \(-0.360805\pi\)
0.423489 + 0.905901i \(0.360805\pi\)
\(242\) 23301.3i 0.397877i
\(243\) 0 0
\(244\) 22269.9i 0.374058i
\(245\) −332.974 −0.00554726
\(246\) 0 0
\(247\) 117055.i 1.91865i
\(248\) 45555.4 0.740689
\(249\) 0 0
\(250\) 15967.8i 0.255485i
\(251\) 67494.9 1.07133 0.535666 0.844430i \(-0.320061\pi\)
0.535666 + 0.844430i \(0.320061\pi\)
\(252\) 0 0
\(253\) 153994. 2.40582
\(254\) 24805.4i 0.384485i
\(255\) 0 0
\(256\) 22954.4 0.350256
\(257\) 94378.9 1.42892 0.714461 0.699675i \(-0.246672\pi\)
0.714461 + 0.699675i \(0.246672\pi\)
\(258\) 0 0
\(259\) 103468.i 1.54243i
\(260\) 76905.2i 1.13765i
\(261\) 0 0
\(262\) 17342.8 0.252649
\(263\) 96817.8 1.39973 0.699864 0.714276i \(-0.253244\pi\)
0.699864 + 0.714276i \(0.253244\pi\)
\(264\) 0 0
\(265\) 123728. 1.76188
\(266\) 31264.4i 0.441862i
\(267\) 0 0
\(268\) 3465.70i 0.0482527i
\(269\) 56566.9i 0.781732i 0.920448 + 0.390866i \(0.127824\pi\)
−0.920448 + 0.390866i \(0.872176\pi\)
\(270\) 0 0
\(271\) −17865.0 −0.243257 −0.121628 0.992576i \(-0.538812\pi\)
−0.121628 + 0.992576i \(0.538812\pi\)
\(272\) 16449.8 0.222343
\(273\) 0 0
\(274\) 4495.90i 0.0598846i
\(275\) 11101.5i 0.146797i
\(276\) 0 0
\(277\) −86629.0 −1.12902 −0.564512 0.825425i \(-0.690936\pi\)
−0.564512 + 0.825425i \(0.690936\pi\)
\(278\) 24056.2i 0.311270i
\(279\) 0 0
\(280\) 42689.9i 0.544514i
\(281\) 84063.8 1.06462 0.532312 0.846548i \(-0.321323\pi\)
0.532312 + 0.846548i \(0.321323\pi\)
\(282\) 0 0
\(283\) 10026.8i 0.125195i 0.998039 + 0.0625977i \(0.0199385\pi\)
−0.998039 + 0.0625977i \(0.980062\pi\)
\(284\) −39532.3 −0.490135
\(285\) 0 0
\(286\) 40696.7i 0.497539i
\(287\) −55638.5 −0.675478
\(288\) 0 0
\(289\) −76861.6 −0.920267
\(290\) 20137.7i 0.239450i
\(291\) 0 0
\(292\) 135380.i 1.58777i
\(293\) 68620.8 0.799319 0.399660 0.916664i \(-0.369128\pi\)
0.399660 + 0.916664i \(0.369128\pi\)
\(294\) 0 0
\(295\) −78929.3 + 45276.8i −0.906972 + 0.520273i
\(296\) −70005.8 −0.799007
\(297\) 0 0
\(298\) −22300.1 −0.251116
\(299\) 160352. 1.79363
\(300\) 0 0
\(301\) 17179.2i 0.189614i
\(302\) −32786.0 −0.359480
\(303\) 0 0
\(304\) −119004. −1.28770
\(305\) 39232.6i 0.421743i
\(306\) 0 0
\(307\) 158352. 1.68014 0.840072 0.542475i \(-0.182513\pi\)
0.840072 + 0.542475i \(0.182513\pi\)
\(308\) 138811.i 1.46326i
\(309\) 0 0
\(310\) −38615.2 −0.401823
\(311\) −58229.0 −0.602030 −0.301015 0.953619i \(-0.597325\pi\)
−0.301015 + 0.953619i \(0.597325\pi\)
\(312\) 0 0
\(313\) 38385.1i 0.391808i −0.980623 0.195904i \(-0.937236\pi\)
0.980623 0.195904i \(-0.0627642\pi\)
\(314\) 40925.9 0.415086
\(315\) 0 0
\(316\) −122956. −1.23133
\(317\) −25846.8 −0.257210 −0.128605 0.991696i \(-0.541050\pi\)
−0.128605 + 0.991696i \(0.541050\pi\)
\(318\) 0 0
\(319\) 136088.i 1.33733i
\(320\) −63199.8 −0.617185
\(321\) 0 0
\(322\) 42828.7 0.413070
\(323\) −48176.5 −0.461775
\(324\) 0 0
\(325\) 11559.9i 0.109443i
\(326\) 27343.2i 0.257285i
\(327\) 0 0
\(328\) 37644.7i 0.349910i
\(329\) 110890.i 1.02447i
\(330\) 0 0
\(331\) 127603. 1.16467 0.582337 0.812947i \(-0.302138\pi\)
0.582337 + 0.812947i \(0.302138\pi\)
\(332\) 110970.i 1.00677i
\(333\) 0 0
\(334\) 36081.2i 0.323435i
\(335\) 6105.48i 0.0544039i
\(336\) 0 0
\(337\) 55504.5i 0.488730i 0.969683 + 0.244365i \(0.0785794\pi\)
−0.969683 + 0.244365i \(0.921421\pi\)
\(338\) 11590.5i 0.101454i
\(339\) 0 0
\(340\) −31652.0 −0.273806
\(341\) 260956. 2.24418
\(342\) 0 0
\(343\) −117335. −0.997330
\(344\) −11623.4 −0.0982233
\(345\) 0 0
\(346\) −24892.5 −0.207929
\(347\) 84690.6i 0.703358i −0.936121 0.351679i \(-0.885611\pi\)
0.936121 0.351679i \(-0.114389\pi\)
\(348\) 0 0
\(349\) 74255.3i 0.609645i 0.952409 + 0.304822i \(0.0985971\pi\)
−0.952409 + 0.304822i \(0.901403\pi\)
\(350\) 3087.54i 0.0252044i
\(351\) 0 0
\(352\) −142648. −1.15128
\(353\) 221926.i 1.78098i 0.455001 + 0.890491i \(0.349639\pi\)
−0.455001 + 0.890491i \(0.650361\pi\)
\(354\) 0 0
\(355\) 69643.5 0.552616
\(356\) 157136.i 1.23987i
\(357\) 0 0
\(358\) −53121.4 −0.414480
\(359\) −55048.5 −0.427127 −0.213563 0.976929i \(-0.568507\pi\)
−0.213563 + 0.976929i \(0.568507\pi\)
\(360\) 0 0
\(361\) 218207. 1.67438
\(362\) 57113.3i 0.435833i
\(363\) 0 0
\(364\) 144542.i 1.09092i
\(365\) 238496.i 1.79018i
\(366\) 0 0
\(367\) 146662.i 1.08889i 0.838796 + 0.544446i \(0.183260\pi\)
−0.838796 + 0.544446i \(0.816740\pi\)
\(368\) 163022.i 1.20379i
\(369\) 0 0
\(370\) 59340.7 0.433460
\(371\) −232545. −1.68950
\(372\) 0 0
\(373\) 166279. 1.19515 0.597573 0.801815i \(-0.296132\pi\)
0.597573 + 0.801815i \(0.296132\pi\)
\(374\) −16749.6 −0.119746
\(375\) 0 0
\(376\) −75027.5 −0.530695
\(377\) 141706.i 0.997025i
\(378\) 0 0
\(379\) 60758.9 0.422991 0.211496 0.977379i \(-0.432167\pi\)
0.211496 + 0.977379i \(0.432167\pi\)
\(380\) 228983. 1.58575
\(381\) 0 0
\(382\) −34062.6 −0.233427
\(383\) −120080. −0.818601 −0.409300 0.912400i \(-0.634227\pi\)
−0.409300 + 0.912400i \(0.634227\pi\)
\(384\) 0 0
\(385\) 244542.i 1.64980i
\(386\) 7181.93i 0.0482022i
\(387\) 0 0
\(388\) 223413.i 1.48404i
\(389\) −184834. −1.22147 −0.610735 0.791835i \(-0.709126\pi\)
−0.610735 + 0.791835i \(0.709126\pi\)
\(390\) 0 0
\(391\) 65996.4i 0.431685i
\(392\) 423.427i 0.00275554i
\(393\) 0 0
\(394\) 3503.06i 0.0225660i
\(395\) 216610. 1.38830
\(396\) 0 0
\(397\) 170314.i 1.08061i −0.841469 0.540306i \(-0.818308\pi\)
0.841469 0.540306i \(-0.181692\pi\)
\(398\) 42491.2i 0.268246i
\(399\) 0 0
\(400\) 11752.4 0.0734523
\(401\) 32629.9i 0.202921i −0.994840 0.101461i \(-0.967648\pi\)
0.994840 0.101461i \(-0.0323516\pi\)
\(402\) 0 0
\(403\) 271730. 1.67312
\(404\) 106202.i 0.650686i
\(405\) 0 0
\(406\) 37848.5i 0.229613i
\(407\) −401015. −2.42087
\(408\) 0 0
\(409\) 278424.i 1.66441i 0.554468 + 0.832205i \(0.312922\pi\)
−0.554468 + 0.832205i \(0.687078\pi\)
\(410\) 31909.7i 0.189826i
\(411\) 0 0
\(412\) 150475.i 0.886480i
\(413\) 148346. 85097.0i 0.869715 0.498901i
\(414\) 0 0
\(415\) 195494.i 1.13511i
\(416\) −148537. −0.858319
\(417\) 0 0
\(418\) 121173. 0.693512
\(419\) 99149.5i 0.564758i 0.959303 + 0.282379i \(0.0911236\pi\)
−0.959303 + 0.282379i \(0.908876\pi\)
\(420\) 0 0
\(421\) 124249.i 0.701015i 0.936560 + 0.350508i \(0.113991\pi\)
−0.936560 + 0.350508i \(0.886009\pi\)
\(422\) −8855.74 −0.0497279
\(423\) 0 0
\(424\) 157339.i 0.875194i
\(425\) 4757.72 0.0263403
\(426\) 0 0
\(427\) 73737.1i 0.404418i
\(428\) −53097.7 −0.289860
\(429\) 0 0
\(430\) 9852.59 0.0532860
\(431\) 80751.0i 0.434704i 0.976093 + 0.217352i \(0.0697419\pi\)
−0.976093 + 0.217352i \(0.930258\pi\)
\(432\) 0 0
\(433\) 267478. 1.42663 0.713317 0.700841i \(-0.247192\pi\)
0.713317 + 0.700841i \(0.247192\pi\)
\(434\) 72576.7 0.385317
\(435\) 0 0
\(436\) 35913.3i 0.188922i
\(437\) 477444.i 2.50011i
\(438\) 0 0
\(439\) 244540. 1.26888 0.634441 0.772971i \(-0.281230\pi\)
0.634441 + 0.772971i \(0.281230\pi\)
\(440\) 165456. 0.854626
\(441\) 0 0
\(442\) −17441.1 −0.0892751
\(443\) 303375.i 1.54587i −0.634487 0.772933i \(-0.718789\pi\)
0.634487 0.772933i \(-0.281211\pi\)
\(444\) 0 0
\(445\) 276824.i 1.39792i
\(446\) 26643.1i 0.133942i
\(447\) 0 0
\(448\) 118783. 0.591832
\(449\) 265110. 1.31503 0.657513 0.753444i \(-0.271609\pi\)
0.657513 + 0.753444i \(0.271609\pi\)
\(450\) 0 0
\(451\) 215641.i 1.06018i
\(452\) 21794.1i 0.106675i
\(453\) 0 0
\(454\) 24738.4 0.120022
\(455\) 254638.i 1.22999i
\(456\) 0 0
\(457\) 351393.i 1.68252i −0.540628 0.841262i \(-0.681813\pi\)
0.540628 0.841262i \(-0.318187\pi\)
\(458\) −28823.8 −0.137411
\(459\) 0 0
\(460\) 313681.i 1.48242i
\(461\) −96578.1 −0.454440 −0.227220 0.973843i \(-0.572964\pi\)
−0.227220 + 0.973843i \(0.572964\pi\)
\(462\) 0 0
\(463\) 247348.i 1.15384i 0.816800 + 0.576922i \(0.195746\pi\)
−0.816800 + 0.576922i \(0.804254\pi\)
\(464\) 144066. 0.669153
\(465\) 0 0
\(466\) −31798.5 −0.146432
\(467\) 233363.i 1.07004i 0.844841 + 0.535018i \(0.179695\pi\)
−0.844841 + 0.535018i \(0.820305\pi\)
\(468\) 0 0
\(469\) 11475.1i 0.0521690i
\(470\) 63597.4 0.287901
\(471\) 0 0
\(472\) −57576.2 100370.i −0.258440 0.450528i
\(473\) −66582.3 −0.297602
\(474\) 0 0
\(475\) −34419.1 −0.152550
\(476\) 59489.5 0.262559
\(477\) 0 0
\(478\) 40626.7i 0.177810i
\(479\) −254275. −1.10824 −0.554119 0.832438i \(-0.686945\pi\)
−0.554119 + 0.832438i \(0.686945\pi\)
\(480\) 0 0
\(481\) −417572. −1.80485
\(482\) 53026.5i 0.228244i
\(483\) 0 0
\(484\) −320753. −1.36924
\(485\) 393584.i 1.67322i
\(486\) 0 0
\(487\) 221313. 0.933146 0.466573 0.884483i \(-0.345488\pi\)
0.466573 + 0.884483i \(0.345488\pi\)
\(488\) 49890.2 0.209496
\(489\) 0 0
\(490\) 358.920i 0.00149488i
\(491\) −20143.2 −0.0835537 −0.0417769 0.999127i \(-0.513302\pi\)
−0.0417769 + 0.999127i \(0.513302\pi\)
\(492\) 0 0
\(493\) 58322.3 0.239961
\(494\) 126176. 0.517038
\(495\) 0 0
\(496\) 276254.i 1.12291i
\(497\) −130894. −0.529915
\(498\) 0 0
\(499\) −142382. −0.571813 −0.285907 0.958258i \(-0.592295\pi\)
−0.285907 + 0.958258i \(0.592295\pi\)
\(500\) 219804. 0.879217
\(501\) 0 0
\(502\) 72754.1i 0.288702i
\(503\) 61029.3i 0.241214i 0.992700 + 0.120607i \(0.0384841\pi\)
−0.992700 + 0.120607i \(0.961516\pi\)
\(504\) 0 0
\(505\) 187095.i 0.733635i
\(506\) 165994.i 0.648321i
\(507\) 0 0
\(508\) −341459. −1.32315
\(509\) 119724.i 0.462111i −0.972941 0.231056i \(-0.925782\pi\)
0.972941 0.231056i \(-0.0742179\pi\)
\(510\) 0 0
\(511\) 448250.i 1.71664i
\(512\) 258221.i 0.985036i
\(513\) 0 0
\(514\) 101733.i 0.385066i
\(515\) 265089.i 0.999487i
\(516\) 0 0
\(517\) −429782. −1.60793
\(518\) −111530. −0.415654
\(519\) 0 0
\(520\) 172287. 0.637155
\(521\) −526245. −1.93871 −0.969355 0.245663i \(-0.920994\pi\)
−0.969355 + 0.245663i \(0.920994\pi\)
\(522\) 0 0
\(523\) 43580.1 0.159325 0.0796626 0.996822i \(-0.474616\pi\)
0.0796626 + 0.996822i \(0.474616\pi\)
\(524\) 238733.i 0.869459i
\(525\) 0 0
\(526\) 104362.i 0.377199i
\(527\) 111836.i 0.402681i
\(528\) 0 0
\(529\) −374203. −1.33720
\(530\) 133369.i 0.474792i
\(531\) 0 0
\(532\) −430370. −1.52061
\(533\) 224544.i 0.790400i
\(534\) 0 0
\(535\) 93541.5 0.326811
\(536\) 7764.03 0.0270245
\(537\) 0 0
\(538\) 60974.6 0.210661
\(539\) 2425.52i 0.00834888i
\(540\) 0 0
\(541\) 318207.i 1.08721i −0.839340 0.543606i \(-0.817058\pi\)
0.839340 0.543606i \(-0.182942\pi\)
\(542\) 19257.1i 0.0655529i
\(543\) 0 0
\(544\) 61133.7i 0.206577i
\(545\) 63267.9i 0.213005i
\(546\) 0 0
\(547\) −151057. −0.504856 −0.252428 0.967616i \(-0.581229\pi\)
−0.252428 + 0.967616i \(0.581229\pi\)
\(548\) 61888.2 0.206085
\(549\) 0 0
\(550\) −11966.6 −0.0395589
\(551\) −421926. −1.38974
\(552\) 0 0
\(553\) −407115. −1.33127
\(554\) 93379.1i 0.304250i
\(555\) 0 0
\(556\) 331145. 1.07119
\(557\) 439612. 1.41697 0.708483 0.705728i \(-0.249380\pi\)
0.708483 + 0.705728i \(0.249380\pi\)
\(558\) 0 0
\(559\) −69331.2 −0.221873
\(560\) −258878. −0.825504
\(561\) 0 0
\(562\) 90614.0i 0.286895i
\(563\) 468132.i 1.47690i −0.674308 0.738450i \(-0.735558\pi\)
0.674308 0.738450i \(-0.264442\pi\)
\(564\) 0 0
\(565\) 38394.3i 0.120274i
\(566\) 10808.1 0.0337376
\(567\) 0 0
\(568\) 88562.2i 0.274506i
\(569\) 156283.i 0.482711i −0.970437 0.241356i \(-0.922408\pi\)
0.970437 0.241356i \(-0.0775920\pi\)
\(570\) 0 0
\(571\) 214333.i 0.657379i 0.944438 + 0.328690i \(0.106607\pi\)
−0.944438 + 0.328690i \(0.893393\pi\)
\(572\) −560210. −1.71222
\(573\) 0 0
\(574\) 59973.8i 0.182028i
\(575\) 47150.4i 0.142610i
\(576\) 0 0
\(577\) −83530.0 −0.250894 −0.125447 0.992100i \(-0.540037\pi\)
−0.125447 + 0.992100i \(0.540037\pi\)
\(578\) 82850.7i 0.247994i
\(579\) 0 0
\(580\) −277205. −0.824035
\(581\) 367429.i 1.08848i
\(582\) 0 0
\(583\) 901288.i 2.65171i
\(584\) −303284. −0.889249
\(585\) 0 0
\(586\) 73967.7i 0.215401i
\(587\) 65804.0i 0.190975i −0.995431 0.0954873i \(-0.969559\pi\)
0.995431 0.0954873i \(-0.0304409\pi\)
\(588\) 0 0
\(589\) 809066.i 2.33213i
\(590\) 48804.7 + 85079.5i 0.140203 + 0.244411i
\(591\) 0 0
\(592\) 424525.i 1.21132i
\(593\) 180576. 0.513511 0.256756 0.966476i \(-0.417346\pi\)
0.256756 + 0.966476i \(0.417346\pi\)
\(594\) 0 0
\(595\) −104802. −0.296029
\(596\) 306971.i 0.864183i
\(597\) 0 0
\(598\) 172847.i 0.483347i
\(599\) 121500. 0.338629 0.169314 0.985562i \(-0.445845\pi\)
0.169314 + 0.985562i \(0.445845\pi\)
\(600\) 0 0
\(601\) 246152.i 0.681483i −0.940157 0.340742i \(-0.889322\pi\)
0.940157 0.340742i \(-0.110678\pi\)
\(602\) −18517.8 −0.0510971
\(603\) 0 0
\(604\) 451315.i 1.23710i
\(605\) 565067. 1.54379
\(606\) 0 0
\(607\) 91016.6 0.247026 0.123513 0.992343i \(-0.460584\pi\)
0.123513 + 0.992343i \(0.460584\pi\)
\(608\) 442265.i 1.19640i
\(609\) 0 0
\(610\) −42289.6 −0.113651
\(611\) −447526. −1.19877
\(612\) 0 0
\(613\) 165519.i 0.440481i −0.975446 0.220240i \(-0.929316\pi\)
0.975446 0.220240i \(-0.0706842\pi\)
\(614\) 170691.i 0.452765i
\(615\) 0 0
\(616\) −310971. −0.819519
\(617\) 337650. 0.886945 0.443472 0.896288i \(-0.353746\pi\)
0.443472 + 0.896288i \(0.353746\pi\)
\(618\) 0 0
\(619\) −158426. −0.413470 −0.206735 0.978397i \(-0.566284\pi\)
−0.206735 + 0.978397i \(0.566284\pi\)
\(620\) 531557.i 1.38282i
\(621\) 0 0
\(622\) 62766.2i 0.162235i
\(623\) 520286.i 1.34050i
\(624\) 0 0
\(625\) −423664. −1.08458
\(626\) −41376.1 −0.105585
\(627\) 0 0
\(628\) 563364.i 1.42847i
\(629\) 171861.i 0.434386i
\(630\) 0 0
\(631\) 563902. 1.41627 0.708133 0.706080i \(-0.249538\pi\)
0.708133 + 0.706080i \(0.249538\pi\)
\(632\) 275452.i 0.689624i
\(633\) 0 0
\(634\) 27860.7i 0.0693129i
\(635\) 601543. 1.49183
\(636\) 0 0
\(637\) 2525.66i 0.00622439i
\(638\) −146692. −0.360383
\(639\) 0 0
\(640\) 381446.i 0.931264i
\(641\) 46995.0 0.114376 0.0571881 0.998363i \(-0.481787\pi\)
0.0571881 + 0.998363i \(0.481787\pi\)
\(642\) 0 0
\(643\) 357241. 0.864050 0.432025 0.901862i \(-0.357799\pi\)
0.432025 + 0.901862i \(0.357799\pi\)
\(644\) 589558.i 1.42153i
\(645\) 0 0
\(646\) 51930.5i 0.124439i
\(647\) 146425. 0.349790 0.174895 0.984587i \(-0.444041\pi\)
0.174895 + 0.984587i \(0.444041\pi\)
\(648\) 0 0
\(649\) −329815. 574954.i −0.783035 1.36503i
\(650\) −12460.6 −0.0294926
\(651\) 0 0
\(652\) −376392. −0.885412
\(653\) 149895. 0.351529 0.175764 0.984432i \(-0.443760\pi\)
0.175764 + 0.984432i \(0.443760\pi\)
\(654\) 0 0
\(655\) 420572.i 0.980297i
\(656\) −228283. −0.530477
\(657\) 0 0
\(658\) −119530. −0.276075
\(659\) 561285.i 1.29245i −0.763148 0.646223i \(-0.776347\pi\)
0.763148 0.646223i \(-0.223653\pi\)
\(660\) 0 0
\(661\) −158811. −0.363478 −0.181739 0.983347i \(-0.558173\pi\)
−0.181739 + 0.983347i \(0.558173\pi\)
\(662\) 137546.i 0.313856i
\(663\) 0 0
\(664\) 248600. 0.563853
\(665\) 758176. 1.71446
\(666\) 0 0
\(667\) 577991.i 1.29918i
\(668\) −496674. −1.11306
\(669\) 0 0
\(670\) −6581.22 −0.0146608
\(671\) 285787. 0.634742
\(672\) 0 0
\(673\) 468994.i 1.03547i −0.855541 0.517735i \(-0.826775\pi\)
0.855541 0.517735i \(-0.173225\pi\)
\(674\) 59829.5 0.131703
\(675\) 0 0
\(676\) −159548. −0.349139
\(677\) 560613. 1.22317 0.611584 0.791180i \(-0.290533\pi\)
0.611584 + 0.791180i \(0.290533\pi\)
\(678\) 0 0
\(679\) 739736.i 1.60449i
\(680\) 70908.3i 0.153348i
\(681\) 0 0
\(682\) 281289.i 0.604762i
\(683\) 39053.9i 0.0837188i 0.999124 + 0.0418594i \(0.0133282\pi\)
−0.999124 + 0.0418594i \(0.986672\pi\)
\(684\) 0 0
\(685\) −109027. −0.232356
\(686\) 126478.i 0.268760i
\(687\) 0 0
\(688\) 70485.7i 0.148910i
\(689\) 938499.i 1.97695i
\(690\) 0 0
\(691\) 138810.i 0.290714i 0.989379 + 0.145357i \(0.0464330\pi\)
−0.989379 + 0.145357i \(0.953567\pi\)
\(692\) 342657.i 0.715561i
\(693\) 0 0
\(694\) −91289.7 −0.189541
\(695\) −583373. −1.20775
\(696\) 0 0
\(697\) −92416.0 −0.190231
\(698\) 80041.3 0.164287
\(699\) 0 0
\(700\) 42501.5 0.0867378
\(701\) 399850.i 0.813695i 0.913496 + 0.406847i \(0.133372\pi\)
−0.913496 + 0.406847i \(0.866628\pi\)
\(702\) 0 0
\(703\) 1.24331e6i 2.51575i
\(704\) 460374.i 0.928892i
\(705\) 0 0
\(706\) 239219. 0.479939
\(707\) 351642.i 0.703497i
\(708\) 0 0
\(709\) −572973. −1.13983 −0.569917 0.821702i \(-0.693025\pi\)
−0.569917 + 0.821702i \(0.693025\pi\)
\(710\) 75070.1i 0.148919i
\(711\) 0 0
\(712\) −352023. −0.694402
\(713\) −1.10833e6 −2.18017
\(714\) 0 0
\(715\) 986913. 1.93049
\(716\) 731242.i 1.42638i
\(717\) 0 0
\(718\) 59337.9i 0.115102i
\(719\) 655458.i 1.26791i 0.773372 + 0.633953i \(0.218569\pi\)
−0.773372 + 0.633953i \(0.781431\pi\)
\(720\) 0 0
\(721\) 498231.i 0.958429i
\(722\) 235209.i 0.451211i
\(723\) 0 0
\(724\) 786192. 1.49986
\(725\) 41667.6 0.0792726
\(726\) 0 0
\(727\) 438565. 0.829785 0.414892 0.909870i \(-0.363819\pi\)
0.414892 + 0.909870i \(0.363819\pi\)
\(728\) −323810. −0.610981
\(729\) 0 0
\(730\) 257080. 0.482417
\(731\) 28534.8i 0.0533998i
\(732\) 0 0
\(733\) 524612. 0.976406 0.488203 0.872730i \(-0.337653\pi\)
0.488203 + 0.872730i \(0.337653\pi\)
\(734\) 158090. 0.293435
\(735\) 0 0
\(736\) 605853. 1.11844
\(737\) 44474.9 0.0818803
\(738\) 0 0
\(739\) 354679.i 0.649451i 0.945808 + 0.324725i \(0.105272\pi\)
−0.945808 + 0.324725i \(0.894728\pi\)
\(740\) 816853.i 1.49170i
\(741\) 0 0
\(742\) 250665.i 0.455288i
\(743\) 9241.58 0.0167405 0.00837025 0.999965i \(-0.497336\pi\)
0.00837025 + 0.999965i \(0.497336\pi\)
\(744\) 0 0
\(745\) 540787.i 0.974347i
\(746\) 179236.i 0.322068i
\(747\) 0 0
\(748\) 230567.i 0.412091i
\(749\) −175810. −0.313386
\(750\) 0 0
\(751\) 1.07245e6i 1.90150i 0.309962 + 0.950749i \(0.399684\pi\)
−0.309962 + 0.950749i \(0.600316\pi\)
\(752\) 454978.i 0.804553i
\(753\) 0 0
\(754\) −152748. −0.268678
\(755\) 795074.i 1.39481i
\(756\) 0 0
\(757\) 513889. 0.896762 0.448381 0.893842i \(-0.352001\pi\)
0.448381 + 0.893842i \(0.352001\pi\)
\(758\) 65493.2i 0.113988i
\(759\) 0 0
\(760\) 512978.i 0.888120i
\(761\) −300411. −0.518736 −0.259368 0.965779i \(-0.583514\pi\)
−0.259368 + 0.965779i \(0.583514\pi\)
\(762\) 0 0
\(763\) 118911.i 0.204255i
\(764\) 468888.i 0.803309i
\(765\) 0 0
\(766\) 129436.i 0.220597i
\(767\) −343432. 598692.i −0.583781 1.01768i
\(768\) 0 0
\(769\) 690998.i 1.16849i 0.811579 + 0.584243i \(0.198609\pi\)
−0.811579 + 0.584243i \(0.801391\pi\)
\(770\) 263596. 0.444588
\(771\) 0 0
\(772\) −98862.7 −0.165882
\(773\) 312549.i 0.523069i −0.965194 0.261534i \(-0.915771\pi\)
0.965194 0.261534i \(-0.0842285\pi\)
\(774\) 0 0
\(775\) 79900.0i 0.133028i
\(776\) −500502. −0.831155
\(777\) 0 0
\(778\) 199236.i 0.329162i
\(779\) 668573. 1.10173
\(780\) 0 0
\(781\) 507312.i 0.831713i
\(782\) 71138.9 0.116331
\(783\) 0 0
\(784\) 2567.72 0.00417749
\(785\) 992470.i 1.61056i
\(786\) 0 0
\(787\) 858686. 1.38639 0.693194 0.720751i \(-0.256203\pi\)
0.693194 + 0.720751i \(0.256203\pi\)
\(788\) 48221.3 0.0776580
\(789\) 0 0
\(790\) 233488.i 0.374120i
\(791\) 72161.5i 0.115333i
\(792\) 0 0
\(793\) 297586. 0.473223
\(794\) −183585. −0.291203
\(795\) 0 0
\(796\) −584912. −0.923134
\(797\) 795885.i