Properties

Label 531.5.c.d.235.4
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 235.4
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.37

$q$-expansion

\(f(q)\) \(=\) \(q-6.76718i q^{2} -29.7948 q^{4} -6.77685 q^{5} +45.8846 q^{7} +93.3518i q^{8} +O(q^{10})\) \(q-6.76718i q^{2} -29.7948 q^{4} -6.77685 q^{5} +45.8846 q^{7} +93.3518i q^{8} +45.8602i q^{10} -152.372i q^{11} +138.574i q^{13} -310.510i q^{14} +155.012 q^{16} -61.1978 q^{17} -184.090 q^{19} +201.915 q^{20} -1031.13 q^{22} -55.4703i q^{23} -579.074 q^{25} +937.758 q^{26} -1367.12 q^{28} -290.483 q^{29} +681.108i q^{31} +444.632i q^{32} +414.137i q^{34} -310.953 q^{35} +1422.17i q^{37} +1245.77i q^{38} -632.631i q^{40} -1532.90 q^{41} -404.708i q^{43} +4539.89i q^{44} -375.377 q^{46} +1601.56i q^{47} -295.602 q^{49} +3918.70i q^{50} -4128.79i q^{52} +1325.56 q^{53} +1032.60i q^{55} +4283.41i q^{56} +1965.75i q^{58} +(-3430.65 + 589.891i) q^{59} -2732.09i q^{61} +4609.19 q^{62} +5489.10 q^{64} -939.097i q^{65} +2225.52i q^{67} +1823.37 q^{68} +2104.28i q^{70} +6157.04 q^{71} +2816.89i q^{73} +9624.09 q^{74} +5484.93 q^{76} -6991.53i q^{77} +7747.95 q^{79} -1050.49 q^{80} +10373.4i q^{82} +648.806i q^{83} +414.728 q^{85} -2738.73 q^{86} +14224.2 q^{88} +5751.21i q^{89} +6358.43i q^{91} +1652.72i q^{92} +10838.1 q^{94} +1247.55 q^{95} -1510.67i q^{97} +2000.39i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.76718i 1.69180i −0.533345 0.845898i \(-0.679065\pi\)
0.533345 0.845898i \(-0.320935\pi\)
\(3\) 0 0
\(4\) −29.7948 −1.86217
\(5\) −6.77685 −0.271074 −0.135537 0.990772i \(-0.543276\pi\)
−0.135537 + 0.990772i \(0.543276\pi\)
\(6\) 0 0
\(7\) 45.8846 0.936421 0.468210 0.883617i \(-0.344899\pi\)
0.468210 + 0.883617i \(0.344899\pi\)
\(8\) 93.3518i 1.45862i
\(9\) 0 0
\(10\) 45.8602i 0.458602i
\(11\) 152.372i 1.25927i −0.776890 0.629636i \(-0.783204\pi\)
0.776890 0.629636i \(-0.216796\pi\)
\(12\) 0 0
\(13\) 138.574i 0.819966i 0.912093 + 0.409983i \(0.134465\pi\)
−0.912093 + 0.409983i \(0.865535\pi\)
\(14\) 310.510i 1.58423i
\(15\) 0 0
\(16\) 155.012 0.605517
\(17\) −61.1978 −0.211757 −0.105878 0.994379i \(-0.533765\pi\)
−0.105878 + 0.994379i \(0.533765\pi\)
\(18\) 0 0
\(19\) −184.090 −0.509946 −0.254973 0.966948i \(-0.582067\pi\)
−0.254973 + 0.966948i \(0.582067\pi\)
\(20\) 201.915 0.504787
\(21\) 0 0
\(22\) −1031.13 −2.13043
\(23\) 55.4703i 0.104859i −0.998625 0.0524294i \(-0.983304\pi\)
0.998625 0.0524294i \(-0.0166964\pi\)
\(24\) 0 0
\(25\) −579.074 −0.926519
\(26\) 937.758 1.38722
\(27\) 0 0
\(28\) −1367.12 −1.74378
\(29\) −290.483 −0.345402 −0.172701 0.984974i \(-0.555250\pi\)
−0.172701 + 0.984974i \(0.555250\pi\)
\(30\) 0 0
\(31\) 681.108i 0.708750i 0.935103 + 0.354375i \(0.115306\pi\)
−0.935103 + 0.354375i \(0.884694\pi\)
\(32\) 444.632i 0.434211i
\(33\) 0 0
\(34\) 414.137i 0.358250i
\(35\) −310.953 −0.253839
\(36\) 0 0
\(37\) 1422.17i 1.03884i 0.854519 + 0.519420i \(0.173852\pi\)
−0.854519 + 0.519420i \(0.826148\pi\)
\(38\) 1245.77i 0.862725i
\(39\) 0 0
\(40\) 632.631i 0.395394i
\(41\) −1532.90 −0.911895 −0.455948 0.890007i \(-0.650700\pi\)
−0.455948 + 0.890007i \(0.650700\pi\)
\(42\) 0 0
\(43\) 404.708i 0.218879i −0.993993 0.109440i \(-0.965094\pi\)
0.993993 0.109440i \(-0.0349056\pi\)
\(44\) 4539.89i 2.34498i
\(45\) 0 0
\(46\) −375.377 −0.177400
\(47\) 1601.56i 0.725016i 0.931981 + 0.362508i \(0.118079\pi\)
−0.931981 + 0.362508i \(0.881921\pi\)
\(48\) 0 0
\(49\) −295.602 −0.123116
\(50\) 3918.70i 1.56748i
\(51\) 0 0
\(52\) 4128.79i 1.52692i
\(53\) 1325.56 0.471897 0.235949 0.971766i \(-0.424180\pi\)
0.235949 + 0.971766i \(0.424180\pi\)
\(54\) 0 0
\(55\) 1032.60i 0.341356i
\(56\) 4283.41i 1.36588i
\(57\) 0 0
\(58\) 1965.75i 0.584350i
\(59\) −3430.65 + 589.891i −0.985537 + 0.169460i
\(60\) 0 0
\(61\) 2732.09i 0.734236i −0.930174 0.367118i \(-0.880344\pi\)
0.930174 0.367118i \(-0.119656\pi\)
\(62\) 4609.19 1.19906
\(63\) 0 0
\(64\) 5489.10 1.34011
\(65\) 939.097i 0.222271i
\(66\) 0 0
\(67\) 2225.52i 0.495771i 0.968789 + 0.247885i \(0.0797357\pi\)
−0.968789 + 0.247885i \(0.920264\pi\)
\(68\) 1823.37 0.394328
\(69\) 0 0
\(70\) 2104.28i 0.429444i
\(71\) 6157.04 1.22139 0.610696 0.791865i \(-0.290890\pi\)
0.610696 + 0.791865i \(0.290890\pi\)
\(72\) 0 0
\(73\) 2816.89i 0.528597i 0.964441 + 0.264298i \(0.0851404\pi\)
−0.964441 + 0.264298i \(0.914860\pi\)
\(74\) 9624.09 1.75750
\(75\) 0 0
\(76\) 5484.93 0.949608
\(77\) 6991.53i 1.17921i
\(78\) 0 0
\(79\) 7747.95 1.24146 0.620730 0.784025i \(-0.286836\pi\)
0.620730 + 0.784025i \(0.286836\pi\)
\(80\) −1050.49 −0.164140
\(81\) 0 0
\(82\) 10373.4i 1.54274i
\(83\) 648.806i 0.0941800i 0.998891 + 0.0470900i \(0.0149948\pi\)
−0.998891 + 0.0470900i \(0.985005\pi\)
\(84\) 0 0
\(85\) 414.728 0.0574018
\(86\) −2738.73 −0.370299
\(87\) 0 0
\(88\) 14224.2 1.83680
\(89\) 5751.21i 0.726071i 0.931775 + 0.363036i \(0.118260\pi\)
−0.931775 + 0.363036i \(0.881740\pi\)
\(90\) 0 0
\(91\) 6358.43i 0.767833i
\(92\) 1652.72i 0.195265i
\(93\) 0 0
\(94\) 10838.1 1.22658
\(95\) 1247.55 0.138233
\(96\) 0 0
\(97\) 1510.67i 0.160556i −0.996773 0.0802779i \(-0.974419\pi\)
0.996773 0.0802779i \(-0.0255808\pi\)
\(98\) 2000.39i 0.208287i
\(99\) 0 0
\(100\) 17253.4 1.72534
\(101\) 2457.45i 0.240903i 0.992719 + 0.120451i \(0.0384342\pi\)
−0.992719 + 0.120451i \(0.961566\pi\)
\(102\) 0 0
\(103\) 8751.92i 0.824952i 0.910968 + 0.412476i \(0.135336\pi\)
−0.910968 + 0.412476i \(0.864664\pi\)
\(104\) −12936.2 −1.19602
\(105\) 0 0
\(106\) 8970.31i 0.798354i
\(107\) −21034.2 −1.83721 −0.918603 0.395182i \(-0.870681\pi\)
−0.918603 + 0.395182i \(0.870681\pi\)
\(108\) 0 0
\(109\) 9627.62i 0.810338i 0.914242 + 0.405169i \(0.132787\pi\)
−0.914242 + 0.405169i \(0.867213\pi\)
\(110\) 6987.81 0.577505
\(111\) 0 0
\(112\) 7112.68 0.567018
\(113\) 23222.8i 1.81869i −0.416044 0.909344i \(-0.636584\pi\)
0.416044 0.909344i \(-0.363416\pi\)
\(114\) 0 0
\(115\) 375.914i 0.0284245i
\(116\) 8654.89 0.643199
\(117\) 0 0
\(118\) 3991.90 + 23215.9i 0.286692 + 1.66733i
\(119\) −2808.04 −0.198294
\(120\) 0 0
\(121\) −8576.22 −0.585768
\(122\) −18488.6 −1.24218
\(123\) 0 0
\(124\) 20293.5i 1.31981i
\(125\) 8159.83 0.522229
\(126\) 0 0
\(127\) 3258.32 0.202016 0.101008 0.994886i \(-0.467793\pi\)
0.101008 + 0.994886i \(0.467793\pi\)
\(128\) 30031.7i 1.83299i
\(129\) 0 0
\(130\) −6355.04 −0.376038
\(131\) 23879.6i 1.39151i 0.718281 + 0.695753i \(0.244929\pi\)
−0.718281 + 0.695753i \(0.755071\pi\)
\(132\) 0 0
\(133\) −8446.92 −0.477524
\(134\) 15060.5 0.838743
\(135\) 0 0
\(136\) 5712.92i 0.308873i
\(137\) −25992.0 −1.38483 −0.692417 0.721497i \(-0.743454\pi\)
−0.692417 + 0.721497i \(0.743454\pi\)
\(138\) 0 0
\(139\) 10296.8 0.532931 0.266466 0.963844i \(-0.414144\pi\)
0.266466 + 0.963844i \(0.414144\pi\)
\(140\) 9264.78 0.472693
\(141\) 0 0
\(142\) 41665.8i 2.06635i
\(143\) 21114.8 1.03256
\(144\) 0 0
\(145\) 1968.56 0.0936296
\(146\) 19062.4 0.894278
\(147\) 0 0
\(148\) 42373.3i 1.93450i
\(149\) 34672.4i 1.56175i 0.624687 + 0.780875i \(0.285227\pi\)
−0.624687 + 0.780875i \(0.714773\pi\)
\(150\) 0 0
\(151\) 8100.65i 0.355276i 0.984096 + 0.177638i \(0.0568456\pi\)
−0.984096 + 0.177638i \(0.943154\pi\)
\(152\) 17185.2i 0.743818i
\(153\) 0 0
\(154\) −47313.0 −1.99498
\(155\) 4615.77i 0.192124i
\(156\) 0 0
\(157\) 35250.4i 1.43009i −0.699077 0.715046i \(-0.746406\pi\)
0.699077 0.715046i \(-0.253594\pi\)
\(158\) 52431.8i 2.10030i
\(159\) 0 0
\(160\) 3013.21i 0.117703i
\(161\) 2545.23i 0.0981919i
\(162\) 0 0
\(163\) −12118.8 −0.456126 −0.228063 0.973646i \(-0.573239\pi\)
−0.228063 + 0.973646i \(0.573239\pi\)
\(164\) 45672.3 1.69811
\(165\) 0 0
\(166\) 4390.59 0.159333
\(167\) −24887.2 −0.892365 −0.446182 0.894942i \(-0.647217\pi\)
−0.446182 + 0.894942i \(0.647217\pi\)
\(168\) 0 0
\(169\) 9358.17 0.327656
\(170\) 2806.54i 0.0971121i
\(171\) 0 0
\(172\) 12058.2i 0.407591i
\(173\) 6149.40i 0.205466i 0.994709 + 0.102733i \(0.0327588\pi\)
−0.994709 + 0.102733i \(0.967241\pi\)
\(174\) 0 0
\(175\) −26570.6 −0.867612
\(176\) 23619.5i 0.762510i
\(177\) 0 0
\(178\) 38919.5 1.22836
\(179\) 17529.1i 0.547082i 0.961860 + 0.273541i \(0.0881949\pi\)
−0.961860 + 0.273541i \(0.911805\pi\)
\(180\) 0 0
\(181\) 20780.3 0.634299 0.317150 0.948376i \(-0.397274\pi\)
0.317150 + 0.948376i \(0.397274\pi\)
\(182\) 43028.6 1.29902
\(183\) 0 0
\(184\) 5178.25 0.152949
\(185\) 9637.84i 0.281602i
\(186\) 0 0
\(187\) 9324.83i 0.266660i
\(188\) 47718.1i 1.35011i
\(189\) 0 0
\(190\) 8442.42i 0.233862i
\(191\) 13668.6i 0.374676i 0.982296 + 0.187338i \(0.0599860\pi\)
−0.982296 + 0.187338i \(0.940014\pi\)
\(192\) 0 0
\(193\) 24606.0 0.660582 0.330291 0.943879i \(-0.392853\pi\)
0.330291 + 0.943879i \(0.392853\pi\)
\(194\) −10223.0 −0.271628
\(195\) 0 0
\(196\) 8807.38 0.229263
\(197\) −7657.45 −0.197311 −0.0986556 0.995122i \(-0.531454\pi\)
−0.0986556 + 0.995122i \(0.531454\pi\)
\(198\) 0 0
\(199\) 50335.0 1.27105 0.635527 0.772079i \(-0.280783\pi\)
0.635527 + 0.772079i \(0.280783\pi\)
\(200\) 54057.6i 1.35144i
\(201\) 0 0
\(202\) 16630.0 0.407558
\(203\) −13328.7 −0.323442
\(204\) 0 0
\(205\) 10388.2 0.247191
\(206\) 59225.8 1.39565
\(207\) 0 0
\(208\) 21480.7i 0.496503i
\(209\) 28050.2i 0.642161i
\(210\) 0 0
\(211\) 34304.2i 0.770518i 0.922809 + 0.385259i \(0.125888\pi\)
−0.922809 + 0.385259i \(0.874112\pi\)
\(212\) −39494.7 −0.878755
\(213\) 0 0
\(214\) 142342.i 3.10818i
\(215\) 2742.64i 0.0593324i
\(216\) 0 0
\(217\) 31252.4i 0.663688i
\(218\) 65151.9 1.37093
\(219\) 0 0
\(220\) 30766.1i 0.635664i
\(221\) 8480.44i 0.173634i
\(222\) 0 0
\(223\) 12252.7 0.246390 0.123195 0.992383i \(-0.460686\pi\)
0.123195 + 0.992383i \(0.460686\pi\)
\(224\) 20401.8i 0.406604i
\(225\) 0 0
\(226\) −157153. −3.07685
\(227\) 83145.6i 1.61357i 0.590846 + 0.806784i \(0.298794\pi\)
−0.590846 + 0.806784i \(0.701206\pi\)
\(228\) 0 0
\(229\) 84147.1i 1.60460i −0.596918 0.802302i \(-0.703608\pi\)
0.596918 0.802302i \(-0.296392\pi\)
\(230\) 2543.88 0.0480884
\(231\) 0 0
\(232\) 27117.1i 0.503811i
\(233\) 10889.2i 0.200578i −0.994958 0.100289i \(-0.968023\pi\)
0.994958 0.100289i \(-0.0319767\pi\)
\(234\) 0 0
\(235\) 10853.5i 0.196533i
\(236\) 102216. 17575.7i 1.83524 0.315564i
\(237\) 0 0
\(238\) 19002.5i 0.335472i
\(239\) −62954.4 −1.10212 −0.551061 0.834465i \(-0.685777\pi\)
−0.551061 + 0.834465i \(0.685777\pi\)
\(240\) 0 0
\(241\) −63118.5 −1.08673 −0.543366 0.839496i \(-0.682851\pi\)
−0.543366 + 0.839496i \(0.682851\pi\)
\(242\) 58036.9i 0.990999i
\(243\) 0 0
\(244\) 81402.1i 1.36728i
\(245\) 2003.25 0.0333736
\(246\) 0 0
\(247\) 25510.2i 0.418138i
\(248\) −63582.7 −1.03380
\(249\) 0 0
\(250\) 55219.1i 0.883505i
\(251\) 66048.7 1.04838 0.524188 0.851603i \(-0.324369\pi\)
0.524188 + 0.851603i \(0.324369\pi\)
\(252\) 0 0
\(253\) −8452.12 −0.132046
\(254\) 22049.6i 0.341770i
\(255\) 0 0
\(256\) −115404. −1.76093
\(257\) −91076.3 −1.37892 −0.689460 0.724323i \(-0.742152\pi\)
−0.689460 + 0.724323i \(0.742152\pi\)
\(258\) 0 0
\(259\) 65255.8i 0.972791i
\(260\) 27980.2i 0.413908i
\(261\) 0 0
\(262\) 161598. 2.35414
\(263\) 83369.8 1.20531 0.602653 0.798003i \(-0.294110\pi\)
0.602653 + 0.798003i \(0.294110\pi\)
\(264\) 0 0
\(265\) −8983.12 −0.127919
\(266\) 57161.9i 0.807873i
\(267\) 0 0
\(268\) 66308.7i 0.923211i
\(269\) 13600.6i 0.187955i 0.995574 + 0.0939775i \(0.0299582\pi\)
−0.995574 + 0.0939775i \(0.970042\pi\)
\(270\) 0 0
\(271\) −374.789 −0.00510327 −0.00255164 0.999997i \(-0.500812\pi\)
−0.00255164 + 0.999997i \(0.500812\pi\)
\(272\) −9486.40 −0.128222
\(273\) 0 0
\(274\) 175892.i 2.34286i
\(275\) 88234.7i 1.16674i
\(276\) 0 0
\(277\) −34517.8 −0.449866 −0.224933 0.974374i \(-0.572216\pi\)
−0.224933 + 0.974374i \(0.572216\pi\)
\(278\) 69680.1i 0.901611i
\(279\) 0 0
\(280\) 29028.0i 0.370255i
\(281\) −100630. −1.27442 −0.637210 0.770690i \(-0.719912\pi\)
−0.637210 + 0.770690i \(0.719912\pi\)
\(282\) 0 0
\(283\) 85403.6i 1.06636i −0.846002 0.533179i \(-0.820997\pi\)
0.846002 0.533179i \(-0.179003\pi\)
\(284\) −183448. −2.27444
\(285\) 0 0
\(286\) 142888.i 1.74688i
\(287\) −70336.4 −0.853918
\(288\) 0 0
\(289\) −79775.8 −0.955159
\(290\) 13321.6i 0.158402i
\(291\) 0 0
\(292\) 83928.7i 0.984339i
\(293\) −133357. −1.55339 −0.776696 0.629876i \(-0.783106\pi\)
−0.776696 + 0.629876i \(0.783106\pi\)
\(294\) 0 0
\(295\) 23249.0 3997.60i 0.267153 0.0459362i
\(296\) −132762. −1.51527
\(297\) 0 0
\(298\) 234635. 2.64216
\(299\) 7686.75 0.0859806
\(300\) 0 0
\(301\) 18569.9i 0.204963i
\(302\) 54818.6 0.601055
\(303\) 0 0
\(304\) −28536.3 −0.308781
\(305\) 18515.0i 0.199032i
\(306\) 0 0
\(307\) 20011.7 0.212328 0.106164 0.994349i \(-0.466143\pi\)
0.106164 + 0.994349i \(0.466143\pi\)
\(308\) 208311.i 2.19589i
\(309\) 0 0
\(310\) −31235.8 −0.325034
\(311\) −32402.1 −0.335006 −0.167503 0.985872i \(-0.553570\pi\)
−0.167503 + 0.985872i \(0.553570\pi\)
\(312\) 0 0
\(313\) 191288.i 1.95254i 0.216568 + 0.976268i \(0.430514\pi\)
−0.216568 + 0.976268i \(0.569486\pi\)
\(314\) −238546. −2.41943
\(315\) 0 0
\(316\) −230848. −2.31181
\(317\) 127089. 1.26470 0.632351 0.774682i \(-0.282090\pi\)
0.632351 + 0.774682i \(0.282090\pi\)
\(318\) 0 0
\(319\) 44261.5i 0.434956i
\(320\) −37198.8 −0.363270
\(321\) 0 0
\(322\) −17224.1 −0.166121
\(323\) 11265.9 0.107985
\(324\) 0 0
\(325\) 80244.8i 0.759714i
\(326\) 82010.3i 0.771673i
\(327\) 0 0
\(328\) 143099.i 1.33011i
\(329\) 73487.0i 0.678920i
\(330\) 0 0
\(331\) −190707. −1.74064 −0.870321 0.492484i \(-0.836089\pi\)
−0.870321 + 0.492484i \(0.836089\pi\)
\(332\) 19331.0i 0.175379i
\(333\) 0 0
\(334\) 168416.i 1.50970i
\(335\) 15082.0i 0.134391i
\(336\) 0 0
\(337\) 121592.i 1.07065i 0.844648 + 0.535323i \(0.179810\pi\)
−0.844648 + 0.535323i \(0.820190\pi\)
\(338\) 63328.5i 0.554326i
\(339\) 0 0
\(340\) −12356.7 −0.106892
\(341\) 103782. 0.892509
\(342\) 0 0
\(343\) −123733. −1.05171
\(344\) 37780.2 0.319262
\(345\) 0 0
\(346\) 41614.1 0.347607
\(347\) 11383.8i 0.0945426i 0.998882 + 0.0472713i \(0.0150525\pi\)
−0.998882 + 0.0472713i \(0.984947\pi\)
\(348\) 0 0
\(349\) 133595.i 1.09683i 0.836207 + 0.548414i \(0.184768\pi\)
−0.836207 + 0.548414i \(0.815232\pi\)
\(350\) 179808.i 1.46782i
\(351\) 0 0
\(352\) 67749.5 0.546790
\(353\) 152806.i 1.22628i 0.789973 + 0.613141i \(0.210094\pi\)
−0.789973 + 0.613141i \(0.789906\pi\)
\(354\) 0 0
\(355\) −41725.3 −0.331088
\(356\) 171356.i 1.35207i
\(357\) 0 0
\(358\) 118622. 0.925551
\(359\) 84352.3 0.654497 0.327249 0.944938i \(-0.393879\pi\)
0.327249 + 0.944938i \(0.393879\pi\)
\(360\) 0 0
\(361\) −96431.7 −0.739955
\(362\) 140624.i 1.07310i
\(363\) 0 0
\(364\) 189448.i 1.42984i
\(365\) 19089.6i 0.143289i
\(366\) 0 0
\(367\) 22612.6i 0.167888i −0.996470 0.0839439i \(-0.973248\pi\)
0.996470 0.0839439i \(-0.0267517\pi\)
\(368\) 8598.57i 0.0634937i
\(369\) 0 0
\(370\) −65221.0 −0.476414
\(371\) 60822.8 0.441894
\(372\) 0 0
\(373\) −142008. −1.02069 −0.510347 0.859968i \(-0.670483\pi\)
−0.510347 + 0.859968i \(0.670483\pi\)
\(374\) 63102.8 0.451134
\(375\) 0 0
\(376\) −149509. −1.05752
\(377\) 40253.5i 0.283218i
\(378\) 0 0
\(379\) −71134.7 −0.495226 −0.247613 0.968859i \(-0.579646\pi\)
−0.247613 + 0.968859i \(0.579646\pi\)
\(380\) −37170.6 −0.257414
\(381\) 0 0
\(382\) 92497.6 0.633875
\(383\) −5389.15 −0.0367386 −0.0183693 0.999831i \(-0.505847\pi\)
−0.0183693 + 0.999831i \(0.505847\pi\)
\(384\) 0 0
\(385\) 47380.5i 0.319653i
\(386\) 166514.i 1.11757i
\(387\) 0 0
\(388\) 45010.1i 0.298983i
\(389\) −227686. −1.50466 −0.752329 0.658787i \(-0.771070\pi\)
−0.752329 + 0.658787i \(0.771070\pi\)
\(390\) 0 0
\(391\) 3394.66i 0.0222046i
\(392\) 27594.9i 0.179580i
\(393\) 0 0
\(394\) 51819.4i 0.333810i
\(395\) −52506.7 −0.336527
\(396\) 0 0
\(397\) 66414.0i 0.421385i 0.977552 + 0.210692i \(0.0675718\pi\)
−0.977552 + 0.210692i \(0.932428\pi\)
\(398\) 340626.i 2.15036i
\(399\) 0 0
\(400\) −89763.6 −0.561022
\(401\) 185618.i 1.15434i −0.816625 0.577168i \(-0.804158\pi\)
0.816625 0.577168i \(-0.195842\pi\)
\(402\) 0 0
\(403\) −94384.1 −0.581151
\(404\) 73219.1i 0.448602i
\(405\) 0 0
\(406\) 90197.9i 0.547198i
\(407\) 216699. 1.30818
\(408\) 0 0
\(409\) 69799.6i 0.417260i −0.977995 0.208630i \(-0.933100\pi\)
0.977995 0.208630i \(-0.0669004\pi\)
\(410\) 70298.9i 0.418197i
\(411\) 0 0
\(412\) 260761.i 1.53620i
\(413\) −157414. + 27066.9i −0.922877 + 0.158686i
\(414\) 0 0
\(415\) 4396.86i 0.0255297i
\(416\) −61614.6 −0.356038
\(417\) 0 0
\(418\) 189821. 1.08641
\(419\) 68851.9i 0.392182i −0.980586 0.196091i \(-0.937175\pi\)
0.980586 0.196091i \(-0.0628248\pi\)
\(420\) 0 0
\(421\) 52611.8i 0.296837i 0.988925 + 0.148419i \(0.0474183\pi\)
−0.988925 + 0.148419i \(0.952582\pi\)
\(422\) 232143. 1.30356
\(423\) 0 0
\(424\) 123743.i 0.688320i
\(425\) 35438.1 0.196197
\(426\) 0 0
\(427\) 125361.i 0.687554i
\(428\) 626708. 3.42120
\(429\) 0 0
\(430\) 18560.0 0.100378
\(431\) 127282.i 0.685193i −0.939483 0.342596i \(-0.888694\pi\)
0.939483 0.342596i \(-0.111306\pi\)
\(432\) 0 0
\(433\) 67016.8 0.357444 0.178722 0.983900i \(-0.442804\pi\)
0.178722 + 0.983900i \(0.442804\pi\)
\(434\) 211491. 1.12282
\(435\) 0 0
\(436\) 286853.i 1.50899i
\(437\) 10211.5i 0.0534723i
\(438\) 0 0
\(439\) 131134. 0.680434 0.340217 0.940347i \(-0.389499\pi\)
0.340217 + 0.940347i \(0.389499\pi\)
\(440\) −96395.2 −0.497909
\(441\) 0 0
\(442\) −57388.7 −0.293753
\(443\) 238014.i 1.21282i −0.795154 0.606408i \(-0.792610\pi\)
0.795154 0.606408i \(-0.207390\pi\)
\(444\) 0 0
\(445\) 38975.1i 0.196819i
\(446\) 82916.3i 0.416841i
\(447\) 0 0
\(448\) 251865. 1.25491
\(449\) 140599. 0.697411 0.348706 0.937232i \(-0.386621\pi\)
0.348706 + 0.937232i \(0.386621\pi\)
\(450\) 0 0
\(451\) 233570.i 1.14832i
\(452\) 691919.i 3.38671i
\(453\) 0 0
\(454\) 562661. 2.72983
\(455\) 43090.1i 0.208140i
\(456\) 0 0
\(457\) 381250.i 1.82548i −0.408539 0.912741i \(-0.633962\pi\)
0.408539 0.912741i \(-0.366038\pi\)
\(458\) −569439. −2.71466
\(459\) 0 0
\(460\) 11200.3i 0.0529313i
\(461\) 66518.7 0.312998 0.156499 0.987678i \(-0.449979\pi\)
0.156499 + 0.987678i \(0.449979\pi\)
\(462\) 0 0
\(463\) 362433.i 1.69070i −0.534216 0.845348i \(-0.679393\pi\)
0.534216 0.845348i \(-0.320607\pi\)
\(464\) −45028.5 −0.209147
\(465\) 0 0
\(466\) −73689.1 −0.339337
\(467\) 184210.i 0.844656i −0.906443 0.422328i \(-0.861213\pi\)
0.906443 0.422328i \(-0.138787\pi\)
\(468\) 0 0
\(469\) 102117.i 0.464250i
\(470\) −73447.9 −0.332494
\(471\) 0 0
\(472\) −55067.4 320258.i −0.247178 1.43753i
\(473\) −61666.1 −0.275629
\(474\) 0 0
\(475\) 106602. 0.472475
\(476\) 83664.8 0.369257
\(477\) 0 0
\(478\) 426024.i 1.86457i
\(479\) 348835. 1.52037 0.760185 0.649707i \(-0.225108\pi\)
0.760185 + 0.649707i \(0.225108\pi\)
\(480\) 0 0
\(481\) −197076. −0.851813
\(482\) 427135.i 1.83853i
\(483\) 0 0
\(484\) 255527. 1.09080
\(485\) 10237.6i 0.0435225i
\(486\) 0 0
\(487\) −451419. −1.90336 −0.951681 0.307087i \(-0.900646\pi\)
−0.951681 + 0.307087i \(0.900646\pi\)
\(488\) 255046. 1.07097
\(489\) 0 0
\(490\) 13556.3i 0.0564612i
\(491\) 64824.7 0.268892 0.134446 0.990921i \(-0.457075\pi\)
0.134446 + 0.990921i \(0.457075\pi\)
\(492\) 0 0
\(493\) 17776.9 0.0731414
\(494\) −172632. −0.707405
\(495\) 0 0
\(496\) 105580.i 0.429160i
\(497\) 282513. 1.14374
\(498\) 0 0
\(499\) −151617. −0.608900 −0.304450 0.952528i \(-0.598473\pi\)
−0.304450 + 0.952528i \(0.598473\pi\)
\(500\) −243120. −0.972481
\(501\) 0 0
\(502\) 446964.i 1.77364i
\(503\) 154559.i 0.610883i −0.952211 0.305442i \(-0.901196\pi\)
0.952211 0.305442i \(-0.0988040\pi\)
\(504\) 0 0
\(505\) 16653.8i 0.0653024i
\(506\) 57197.0i 0.223394i
\(507\) 0 0
\(508\) −97080.8 −0.376189
\(509\) 6251.16i 0.0241282i 0.999927 + 0.0120641i \(0.00384022\pi\)
−0.999927 + 0.0120641i \(0.996160\pi\)
\(510\) 0 0
\(511\) 129252.i 0.494989i
\(512\) 300454.i 1.14614i
\(513\) 0 0
\(514\) 616330.i 2.33285i
\(515\) 59310.4i 0.223623i
\(516\) 0 0
\(517\) 244033. 0.912993
\(518\) 441598. 1.64576
\(519\) 0 0
\(520\) 87666.4 0.324210
\(521\) −351023. −1.29318 −0.646591 0.762836i \(-0.723806\pi\)
−0.646591 + 0.762836i \(0.723806\pi\)
\(522\) 0 0
\(523\) −58841.2 −0.215119 −0.107559 0.994199i \(-0.534304\pi\)
−0.107559 + 0.994199i \(0.534304\pi\)
\(524\) 711489.i 2.59123i
\(525\) 0 0
\(526\) 564179.i 2.03913i
\(527\) 41682.3i 0.150083i
\(528\) 0 0
\(529\) 276764. 0.989005
\(530\) 60790.4i 0.216413i
\(531\) 0 0
\(532\) 251674. 0.889233
\(533\) 212420.i 0.747723i
\(534\) 0 0
\(535\) 142545. 0.498019
\(536\) −207756. −0.723142
\(537\) 0 0
\(538\) 92037.8 0.317981
\(539\) 45041.4i 0.155037i
\(540\) 0 0
\(541\) 451192.i 1.54158i −0.637088 0.770791i \(-0.719861\pi\)
0.637088 0.770791i \(-0.280139\pi\)
\(542\) 2536.27i 0.00863369i
\(543\) 0 0
\(544\) 27210.5i 0.0919472i
\(545\) 65244.9i 0.219661i
\(546\) 0 0
\(547\) −341064. −1.13988 −0.569942 0.821685i \(-0.693034\pi\)
−0.569942 + 0.821685i \(0.693034\pi\)
\(548\) 774425. 2.57880
\(549\) 0 0
\(550\) 597100. 1.97389
\(551\) 53475.2 0.176137
\(552\) 0 0
\(553\) 355512. 1.16253
\(554\) 233588.i 0.761082i
\(555\) 0 0
\(556\) −306790. −0.992410
\(557\) −333403. −1.07463 −0.537315 0.843382i \(-0.680561\pi\)
−0.537315 + 0.843382i \(0.680561\pi\)
\(558\) 0 0
\(559\) 56082.1 0.179474
\(560\) −48201.5 −0.153704
\(561\) 0 0
\(562\) 680978.i 2.15606i
\(563\) 48980.2i 0.154527i 0.997011 + 0.0772634i \(0.0246182\pi\)
−0.997011 + 0.0772634i \(0.975382\pi\)
\(564\) 0 0
\(565\) 157378.i 0.492999i
\(566\) −577942. −1.80406
\(567\) 0 0
\(568\) 574771.i 1.78155i
\(569\) 47923.5i 0.148021i 0.997257 + 0.0740106i \(0.0235799\pi\)
−0.997257 + 0.0740106i \(0.976420\pi\)
\(570\) 0 0
\(571\) 518303.i 1.58969i −0.606815 0.794843i \(-0.707553\pi\)
0.606815 0.794843i \(-0.292447\pi\)
\(572\) −629112. −1.92281
\(573\) 0 0
\(574\) 475979.i 1.44465i
\(575\) 32121.4i 0.0971536i
\(576\) 0 0
\(577\) 92997.1 0.279330 0.139665 0.990199i \(-0.455397\pi\)
0.139665 + 0.990199i \(0.455397\pi\)
\(578\) 539858.i 1.61593i
\(579\) 0 0
\(580\) −58652.9 −0.174355
\(581\) 29770.2i 0.0881921i
\(582\) 0 0
\(583\) 201978.i 0.594247i
\(584\) −262962. −0.771022
\(585\) 0 0
\(586\) 902452.i 2.62802i
\(587\) 333846.i 0.968881i 0.874824 + 0.484440i \(0.160977\pi\)
−0.874824 + 0.484440i \(0.839023\pi\)
\(588\) 0 0
\(589\) 125386.i 0.361424i
\(590\) −27052.5 157330.i −0.0777147 0.451969i
\(591\) 0 0
\(592\) 220454.i 0.629034i
\(593\) −122051. −0.347081 −0.173541 0.984827i \(-0.555521\pi\)
−0.173541 + 0.984827i \(0.555521\pi\)
\(594\) 0 0
\(595\) 19029.6 0.0537522
\(596\) 1.03306e6i 2.90825i
\(597\) 0 0
\(598\) 52017.7i 0.145462i
\(599\) 118410. 0.330015 0.165008 0.986292i \(-0.447235\pi\)
0.165008 + 0.986292i \(0.447235\pi\)
\(600\) 0 0
\(601\) 316761.i 0.876966i −0.898740 0.438483i \(-0.855516\pi\)
0.898740 0.438483i \(-0.144484\pi\)
\(602\) −125666. −0.346756
\(603\) 0 0
\(604\) 241357.i 0.661586i
\(605\) 58119.8 0.158786
\(606\) 0 0
\(607\) −635965. −1.72606 −0.863030 0.505153i \(-0.831436\pi\)
−0.863030 + 0.505153i \(0.831436\pi\)
\(608\) 81852.6i 0.221424i
\(609\) 0 0
\(610\) 125294. 0.336722
\(611\) −221935. −0.594489
\(612\) 0 0
\(613\) 664506.i 1.76839i 0.467119 + 0.884195i \(0.345292\pi\)
−0.467119 + 0.884195i \(0.654708\pi\)
\(614\) 135423.i 0.359215i
\(615\) 0 0
\(616\) 652672. 1.72002
\(617\) −214690. −0.563952 −0.281976 0.959422i \(-0.590990\pi\)
−0.281976 + 0.959422i \(0.590990\pi\)
\(618\) 0 0
\(619\) −638321. −1.66594 −0.832968 0.553322i \(-0.813360\pi\)
−0.832968 + 0.553322i \(0.813360\pi\)
\(620\) 137526.i 0.357767i
\(621\) 0 0
\(622\) 219271.i 0.566761i
\(623\) 263892.i 0.679908i
\(624\) 0 0
\(625\) 306624. 0.784956
\(626\) 1.29448e6 3.30329
\(627\) 0 0
\(628\) 1.05028e6i 2.66308i
\(629\) 87033.7i 0.219981i
\(630\) 0 0
\(631\) −375421. −0.942888 −0.471444 0.881896i \(-0.656267\pi\)
−0.471444 + 0.881896i \(0.656267\pi\)
\(632\) 723285.i 1.81082i
\(633\) 0 0
\(634\) 860032.i 2.13962i
\(635\) −22081.1 −0.0547613
\(636\) 0 0
\(637\) 40962.8i 0.100951i
\(638\) 299526. 0.735857
\(639\) 0 0
\(640\) 203520.i 0.496875i
\(641\) −704608. −1.71487 −0.857435 0.514592i \(-0.827943\pi\)
−0.857435 + 0.514592i \(0.827943\pi\)
\(642\) 0 0
\(643\) −448900. −1.08574 −0.542872 0.839816i \(-0.682663\pi\)
−0.542872 + 0.839816i \(0.682663\pi\)
\(644\) 75834.6i 0.182850i
\(645\) 0 0
\(646\) 76238.6i 0.182688i
\(647\) 211597. 0.505477 0.252738 0.967535i \(-0.418669\pi\)
0.252738 + 0.967535i \(0.418669\pi\)
\(648\) 0 0
\(649\) 89882.8 + 522736.i 0.213397 + 1.24106i
\(650\) −543031. −1.28528
\(651\) 0 0
\(652\) 361078. 0.849387
\(653\) 171313. 0.401757 0.200878 0.979616i \(-0.435620\pi\)
0.200878 + 0.979616i \(0.435620\pi\)
\(654\) 0 0
\(655\) 161829.i 0.377201i
\(656\) −237618. −0.552168
\(657\) 0 0
\(658\) 497300. 1.14859
\(659\) 123614.i 0.284641i −0.989821 0.142321i \(-0.954544\pi\)
0.989821 0.142321i \(-0.0454564\pi\)
\(660\) 0 0
\(661\) 759419. 1.73811 0.869057 0.494712i \(-0.164726\pi\)
0.869057 + 0.494712i \(0.164726\pi\)
\(662\) 1.29055e6i 2.94481i
\(663\) 0 0
\(664\) −60567.2 −0.137373
\(665\) 57243.5 0.129444
\(666\) 0 0
\(667\) 16113.2i 0.0362185i
\(668\) 741507. 1.66174
\(669\) 0 0
\(670\) −102063. −0.227361
\(671\) −416295. −0.924604
\(672\) 0 0
\(673\) 366990.i 0.810260i −0.914259 0.405130i \(-0.867226\pi\)
0.914259 0.405130i \(-0.132774\pi\)
\(674\) 822836. 1.81131
\(675\) 0 0
\(676\) −278825. −0.610152
\(677\) −411628. −0.898105 −0.449053 0.893505i \(-0.648238\pi\)
−0.449053 + 0.893505i \(0.648238\pi\)
\(678\) 0 0
\(679\) 69316.5i 0.150348i
\(680\) 38715.6i 0.0837275i
\(681\) 0 0
\(682\) 702311.i 1.50994i
\(683\) 654877.i 1.40384i 0.712254 + 0.701921i \(0.247674\pi\)
−0.712254 + 0.701921i \(0.752326\pi\)
\(684\) 0 0
\(685\) 176144. 0.375393
\(686\) 837321.i 1.77928i
\(687\) 0 0
\(688\) 62734.6i 0.132535i
\(689\) 183688.i 0.386940i
\(690\) 0 0
\(691\) 628200.i 1.31565i −0.753169 0.657827i \(-0.771476\pi\)
0.753169 0.657827i \(-0.228524\pi\)
\(692\) 183220.i 0.382614i
\(693\) 0 0
\(694\) 77036.1 0.159947
\(695\) −69779.6 −0.144464
\(696\) 0 0
\(697\) 93809.8 0.193100
\(698\) 904060. 1.85561
\(699\) 0 0
\(700\) 791665. 1.61564
\(701\) 517988.i 1.05410i −0.849833 0.527052i \(-0.823297\pi\)
0.849833 0.527052i \(-0.176703\pi\)
\(702\) 0 0
\(703\) 261808.i 0.529752i
\(704\) 836386.i 1.68757i
\(705\) 0 0
\(706\) 1.03407e6 2.07462
\(707\) 112759.i 0.225586i
\(708\) 0 0
\(709\) 417989. 0.831519 0.415759 0.909475i \(-0.363516\pi\)
0.415759 + 0.909475i \(0.363516\pi\)
\(710\) 282363.i 0.560133i
\(711\) 0 0
\(712\) −536886. −1.05906
\(713\) 37781.3 0.0743186
\(714\) 0 0
\(715\) −143092. −0.279900
\(716\) 522274.i 1.01876i
\(717\) 0 0
\(718\) 570827.i 1.10728i
\(719\) 88171.5i 0.170557i 0.996357 + 0.0852787i \(0.0271781\pi\)
−0.996357 + 0.0852787i \(0.972822\pi\)
\(720\) 0 0
\(721\) 401578.i 0.772502i
\(722\) 652571.i 1.25185i
\(723\) 0 0
\(724\) −619144. −1.18117
\(725\) 168212. 0.320022
\(726\) 0 0
\(727\) 765157. 1.44771 0.723855 0.689952i \(-0.242368\pi\)
0.723855 + 0.689952i \(0.242368\pi\)
\(728\) −593571. −1.11998
\(729\) 0 0
\(730\) −129183. −0.242415
\(731\) 24767.2i 0.0463492i
\(732\) 0 0
\(733\) 45126.1 0.0839885 0.0419943 0.999118i \(-0.486629\pi\)
0.0419943 + 0.999118i \(0.486629\pi\)
\(734\) −153024. −0.284032
\(735\) 0 0
\(736\) 24663.9 0.0455308
\(737\) 339106. 0.624311
\(738\) 0 0
\(739\) 1.03826e6i 1.90116i 0.310476 + 0.950581i \(0.399512\pi\)
−0.310476 + 0.950581i \(0.600488\pi\)
\(740\) 287157.i 0.524392i
\(741\) 0 0
\(742\) 411599.i 0.747595i
\(743\) 432123. 0.782763 0.391381 0.920229i \(-0.371997\pi\)
0.391381 + 0.920229i \(0.371997\pi\)
\(744\) 0 0
\(745\) 234970.i 0.423350i
\(746\) 960995.i 1.72681i
\(747\) 0 0
\(748\) 277831.i 0.496567i
\(749\) −965145. −1.72040
\(750\) 0 0
\(751\) 586786.i 1.04040i 0.854045 + 0.520200i \(0.174142\pi\)
−0.854045 + 0.520200i \(0.825858\pi\)
\(752\) 248262.i 0.439009i
\(753\) 0 0
\(754\) −272403. −0.479148
\(755\) 54896.9i 0.0963061i
\(756\) 0 0
\(757\) −69647.3 −0.121538 −0.0607690 0.998152i \(-0.519355\pi\)
−0.0607690 + 0.998152i \(0.519355\pi\)
\(758\) 481382.i 0.837821i
\(759\) 0 0
\(760\) 116461.i 0.201630i
\(761\) 533970. 0.922035 0.461017 0.887391i \(-0.347485\pi\)
0.461017 + 0.887391i \(0.347485\pi\)
\(762\) 0 0
\(763\) 441760.i 0.758817i
\(764\) 407252.i 0.697712i
\(765\) 0 0
\(766\) 36469.4i 0.0621542i
\(767\) −81743.7 475400.i −0.138952 0.808107i
\(768\) 0 0
\(769\) 316009.i 0.534376i 0.963644 + 0.267188i \(0.0860945\pi\)
−0.963644 + 0.267188i \(0.913905\pi\)
\(770\) 320633. 0.540787
\(771\) 0 0
\(772\) −733131. −1.23012
\(773\) 42404.6i 0.0709666i −0.999370 0.0354833i \(-0.988703\pi\)
0.999370 0.0354833i \(-0.0112971\pi\)
\(774\) 0 0
\(775\) 394412.i 0.656670i
\(776\) 141024. 0.234190
\(777\) 0 0
\(778\) 1.54080e6i 2.54558i
\(779\) 282192. 0.465017
\(780\) 0 0
\(781\) 938160.i 1.53807i
\(782\) 22972.3 0.0375656
\(783\) 0 0
\(784\) −45821.9 −0.0745488
\(785\) 238886.i 0.387661i
\(786\) 0 0
\(787\) −952711. −1.53820 −0.769098 0.639131i \(-0.779294\pi\)
−0.769098 + 0.639131i \(0.779294\pi\)
\(788\) 228152. 0.367428
\(789\) 0 0
\(790\) 355322.i 0.569335i
\(791\) 1.06557e6i 1.70306i
\(792\) 0 0
\(793\) 378598. 0.602049
\(794\) 449436. 0.712897
\(795\) 0 0
\(796\) −1.49972e6 −2.36692
\(797\) 821853.i