Properties

Label 531.5.c.d.235.5
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 235.5
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.36

$q$-expansion

\(f(q)\) \(=\) \(q-6.70986i q^{2} -29.0222 q^{4} +41.0870 q^{5} -6.70931 q^{7} +87.3773i q^{8} +O(q^{10})\) \(q-6.70986i q^{2} -29.0222 q^{4} +41.0870 q^{5} -6.70931 q^{7} +87.3773i q^{8} -275.688i q^{10} +21.2822i q^{11} -73.2439i q^{13} +45.0185i q^{14} +121.934 q^{16} -76.4439 q^{17} +439.767 q^{19} -1192.43 q^{20} +142.801 q^{22} +164.608i q^{23} +1063.14 q^{25} -491.456 q^{26} +194.719 q^{28} +788.819 q^{29} -754.698i q^{31} +579.878i q^{32} +512.928i q^{34} -275.665 q^{35} -1700.99i q^{37} -2950.77i q^{38} +3590.07i q^{40} -72.3910 q^{41} -2692.60i q^{43} -617.657i q^{44} +1104.50 q^{46} -2903.21i q^{47} -2355.99 q^{49} -7133.51i q^{50} +2125.70i q^{52} -1176.18 q^{53} +874.422i q^{55} -586.241i q^{56} -5292.86i q^{58} +(3479.47 - 103.350i) q^{59} -2041.78i q^{61} -5063.92 q^{62} +5841.84 q^{64} -3009.37i q^{65} -796.587i q^{67} +2218.57 q^{68} +1849.67i q^{70} +4861.25 q^{71} -1824.82i q^{73} -11413.4 q^{74} -12763.0 q^{76} -142.789i q^{77} -7086.59 q^{79} +5009.89 q^{80} +485.734i q^{82} +4425.07i q^{83} -3140.85 q^{85} -18067.0 q^{86} -1859.58 q^{88} +8041.54i q^{89} +491.415i q^{91} -4777.30i q^{92} -19480.1 q^{94} +18068.7 q^{95} +2354.04i q^{97} +15808.3i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.70986i 1.67747i −0.544544 0.838733i \(-0.683297\pi\)
0.544544 0.838733i \(-0.316703\pi\)
\(3\) 0 0
\(4\) −29.0222 −1.81389
\(5\) 41.0870 1.64348 0.821739 0.569864i \(-0.193004\pi\)
0.821739 + 0.569864i \(0.193004\pi\)
\(6\) 0 0
\(7\) −6.70931 −0.136925 −0.0684623 0.997654i \(-0.521809\pi\)
−0.0684623 + 0.997654i \(0.521809\pi\)
\(8\) 87.3773i 1.36527i
\(9\) 0 0
\(10\) 275.688i 2.75688i
\(11\) 21.2822i 0.175886i 0.996126 + 0.0879430i \(0.0280293\pi\)
−0.996126 + 0.0879430i \(0.971971\pi\)
\(12\) 0 0
\(13\) 73.2439i 0.433396i −0.976239 0.216698i \(-0.930471\pi\)
0.976239 0.216698i \(-0.0695286\pi\)
\(14\) 45.0185i 0.229686i
\(15\) 0 0
\(16\) 121.934 0.476304
\(17\) −76.4439 −0.264512 −0.132256 0.991216i \(-0.542222\pi\)
−0.132256 + 0.991216i \(0.542222\pi\)
\(18\) 0 0
\(19\) 439.767 1.21819 0.609095 0.793097i \(-0.291533\pi\)
0.609095 + 0.793097i \(0.291533\pi\)
\(20\) −1192.43 −2.98109
\(21\) 0 0
\(22\) 142.801 0.295043
\(23\) 164.608i 0.311169i 0.987823 + 0.155584i \(0.0497260\pi\)
−0.987823 + 0.155584i \(0.950274\pi\)
\(24\) 0 0
\(25\) 1063.14 1.70102
\(26\) −491.456 −0.727006
\(27\) 0 0
\(28\) 194.719 0.248366
\(29\) 788.819 0.937954 0.468977 0.883211i \(-0.344623\pi\)
0.468977 + 0.883211i \(0.344623\pi\)
\(30\) 0 0
\(31\) 754.698i 0.785325i −0.919683 0.392663i \(-0.871554\pi\)
0.919683 0.392663i \(-0.128446\pi\)
\(32\) 579.878i 0.566287i
\(33\) 0 0
\(34\) 512.928i 0.443709i
\(35\) −275.665 −0.225033
\(36\) 0 0
\(37\) 1700.99i 1.24251i −0.783610 0.621253i \(-0.786624\pi\)
0.783610 0.621253i \(-0.213376\pi\)
\(38\) 2950.77i 2.04347i
\(39\) 0 0
\(40\) 3590.07i 2.24379i
\(41\) −72.3910 −0.0430643 −0.0215321 0.999768i \(-0.506854\pi\)
−0.0215321 + 0.999768i \(0.506854\pi\)
\(42\) 0 0
\(43\) 2692.60i 1.45625i −0.685447 0.728123i \(-0.740393\pi\)
0.685447 0.728123i \(-0.259607\pi\)
\(44\) 617.657i 0.319038i
\(45\) 0 0
\(46\) 1104.50 0.521975
\(47\) 2903.21i 1.31426i −0.753775 0.657132i \(-0.771769\pi\)
0.753775 0.657132i \(-0.228231\pi\)
\(48\) 0 0
\(49\) −2355.99 −0.981252
\(50\) 7133.51i 2.85340i
\(51\) 0 0
\(52\) 2125.70i 0.786131i
\(53\) −1176.18 −0.418719 −0.209360 0.977839i \(-0.567138\pi\)
−0.209360 + 0.977839i \(0.567138\pi\)
\(54\) 0 0
\(55\) 874.422i 0.289065i
\(56\) 586.241i 0.186939i
\(57\) 0 0
\(58\) 5292.86i 1.57338i
\(59\) 3479.47 103.350i 0.999559 0.0296897i
\(60\) 0 0
\(61\) 2041.78i 0.548718i −0.961627 0.274359i \(-0.911534\pi\)
0.961627 0.274359i \(-0.0884657\pi\)
\(62\) −5063.92 −1.31736
\(63\) 0 0
\(64\) 5841.84 1.42623
\(65\) 3009.37i 0.712276i
\(66\) 0 0
\(67\) 796.587i 0.177453i −0.996056 0.0887266i \(-0.971720\pi\)
0.996056 0.0887266i \(-0.0282797\pi\)
\(68\) 2218.57 0.479795
\(69\) 0 0
\(70\) 1849.67i 0.377484i
\(71\) 4861.25 0.964343 0.482171 0.876077i \(-0.339848\pi\)
0.482171 + 0.876077i \(0.339848\pi\)
\(72\) 0 0
\(73\) 1824.82i 0.342433i −0.985233 0.171216i \(-0.945230\pi\)
0.985233 0.171216i \(-0.0547697\pi\)
\(74\) −11413.4 −2.08426
\(75\) 0 0
\(76\) −12763.0 −2.20966
\(77\) 142.789i 0.0240831i
\(78\) 0 0
\(79\) −7086.59 −1.13549 −0.567745 0.823204i \(-0.692184\pi\)
−0.567745 + 0.823204i \(0.692184\pi\)
\(80\) 5009.89 0.782795
\(81\) 0 0
\(82\) 485.734i 0.0722388i
\(83\) 4425.07i 0.642338i 0.947022 + 0.321169i \(0.104076\pi\)
−0.947022 + 0.321169i \(0.895924\pi\)
\(84\) 0 0
\(85\) −3140.85 −0.434719
\(86\) −18067.0 −2.44280
\(87\) 0 0
\(88\) −1859.58 −0.240132
\(89\) 8041.54i 1.01522i 0.861588 + 0.507609i \(0.169471\pi\)
−0.861588 + 0.507609i \(0.830529\pi\)
\(90\) 0 0
\(91\) 491.415i 0.0593425i
\(92\) 4777.30i 0.564425i
\(93\) 0 0
\(94\) −19480.1 −2.20463
\(95\) 18068.7 2.00207
\(96\) 0 0
\(97\) 2354.04i 0.250190i 0.992145 + 0.125095i \(0.0399235\pi\)
−0.992145 + 0.125095i \(0.960076\pi\)
\(98\) 15808.3i 1.64602i
\(99\) 0 0
\(100\) −30854.6 −3.08546
\(101\) 15071.0i 1.47741i −0.674032 0.738703i \(-0.735439\pi\)
0.674032 0.738703i \(-0.264561\pi\)
\(102\) 0 0
\(103\) 13089.2i 1.23378i −0.787048 0.616892i \(-0.788392\pi\)
0.787048 0.616892i \(-0.211608\pi\)
\(104\) 6399.85 0.591702
\(105\) 0 0
\(106\) 7892.02i 0.702387i
\(107\) 264.977 0.0231441 0.0115721 0.999933i \(-0.496316\pi\)
0.0115721 + 0.999933i \(0.496316\pi\)
\(108\) 0 0
\(109\) 13760.9i 1.15823i −0.815247 0.579113i \(-0.803399\pi\)
0.815247 0.579113i \(-0.196601\pi\)
\(110\) 5867.25 0.484896
\(111\) 0 0
\(112\) −818.091 −0.0652177
\(113\) 8838.66i 0.692197i 0.938198 + 0.346098i \(0.112494\pi\)
−0.938198 + 0.346098i \(0.887506\pi\)
\(114\) 0 0
\(115\) 6763.25i 0.511399i
\(116\) −22893.3 −1.70134
\(117\) 0 0
\(118\) −693.464 23346.7i −0.0498035 1.67673i
\(119\) 512.886 0.0362182
\(120\) 0 0
\(121\) 14188.1 0.969064
\(122\) −13700.1 −0.920456
\(123\) 0 0
\(124\) 21903.0i 1.42449i
\(125\) 18001.8 1.15211
\(126\) 0 0
\(127\) −12508.4 −0.775524 −0.387762 0.921759i \(-0.626752\pi\)
−0.387762 + 0.921759i \(0.626752\pi\)
\(128\) 29919.9i 1.82616i
\(129\) 0 0
\(130\) −20192.4 −1.19482
\(131\) 30033.4i 1.75010i 0.484036 + 0.875048i \(0.339170\pi\)
−0.484036 + 0.875048i \(0.660830\pi\)
\(132\) 0 0
\(133\) −2950.53 −0.166800
\(134\) −5344.99 −0.297672
\(135\) 0 0
\(136\) 6679.46i 0.361130i
\(137\) −30546.6 −1.62750 −0.813751 0.581213i \(-0.802578\pi\)
−0.813751 + 0.581213i \(0.802578\pi\)
\(138\) 0 0
\(139\) 13641.7 0.706056 0.353028 0.935613i \(-0.385152\pi\)
0.353028 + 0.935613i \(0.385152\pi\)
\(140\) 8000.41 0.408184
\(141\) 0 0
\(142\) 32618.3i 1.61765i
\(143\) 1558.79 0.0762283
\(144\) 0 0
\(145\) 32410.2 1.54151
\(146\) −12244.3 −0.574419
\(147\) 0 0
\(148\) 49366.5i 2.25377i
\(149\) 18026.1i 0.811949i −0.913884 0.405975i \(-0.866932\pi\)
0.913884 0.405975i \(-0.133068\pi\)
\(150\) 0 0
\(151\) 39094.7i 1.71460i 0.514814 + 0.857302i \(0.327861\pi\)
−0.514814 + 0.857302i \(0.672139\pi\)
\(152\) 38425.6i 1.66316i
\(153\) 0 0
\(154\) −958.094 −0.0403986
\(155\) 31008.2i 1.29067i
\(156\) 0 0
\(157\) 27675.1i 1.12277i 0.827556 + 0.561383i \(0.189731\pi\)
−0.827556 + 0.561383i \(0.810269\pi\)
\(158\) 47550.1i 1.90475i
\(159\) 0 0
\(160\) 23825.4i 0.930680i
\(161\) 1104.41i 0.0426067i
\(162\) 0 0
\(163\) −28910.4 −1.08813 −0.544063 0.839044i \(-0.683115\pi\)
−0.544063 + 0.839044i \(0.683115\pi\)
\(164\) 2100.95 0.0781138
\(165\) 0 0
\(166\) 29691.6 1.07750
\(167\) 44721.0 1.60354 0.801768 0.597635i \(-0.203893\pi\)
0.801768 + 0.597635i \(0.203893\pi\)
\(168\) 0 0
\(169\) 23196.3 0.812168
\(170\) 21074.6i 0.729227i
\(171\) 0 0
\(172\) 78145.2i 2.64147i
\(173\) 37239.4i 1.24426i 0.782914 + 0.622129i \(0.213732\pi\)
−0.782914 + 0.622129i \(0.786268\pi\)
\(174\) 0 0
\(175\) −7132.92 −0.232912
\(176\) 2595.02i 0.0837752i
\(177\) 0 0
\(178\) 53957.6 1.70299
\(179\) 23586.9i 0.736147i −0.929797 0.368073i \(-0.880017\pi\)
0.929797 0.368073i \(-0.119983\pi\)
\(180\) 0 0
\(181\) −1350.22 −0.0412144 −0.0206072 0.999788i \(-0.506560\pi\)
−0.0206072 + 0.999788i \(0.506560\pi\)
\(182\) 3297.33 0.0995450
\(183\) 0 0
\(184\) −14383.0 −0.424829
\(185\) 69888.5i 2.04203i
\(186\) 0 0
\(187\) 1626.90i 0.0465239i
\(188\) 84257.6i 2.38393i
\(189\) 0 0
\(190\) 121238.i 3.35840i
\(191\) 34964.8i 0.958438i 0.877695 + 0.479219i \(0.159080\pi\)
−0.877695 + 0.479219i \(0.840920\pi\)
\(192\) 0 0
\(193\) −28062.0 −0.753362 −0.376681 0.926343i \(-0.622935\pi\)
−0.376681 + 0.926343i \(0.622935\pi\)
\(194\) 15795.3 0.419685
\(195\) 0 0
\(196\) 68375.9 1.77988
\(197\) 8658.99 0.223118 0.111559 0.993758i \(-0.464416\pi\)
0.111559 + 0.993758i \(0.464416\pi\)
\(198\) 0 0
\(199\) −31291.6 −0.790173 −0.395087 0.918644i \(-0.629285\pi\)
−0.395087 + 0.918644i \(0.629285\pi\)
\(200\) 92894.1i 2.32235i
\(201\) 0 0
\(202\) −101124. −2.47830
\(203\) −5292.43 −0.128429
\(204\) 0 0
\(205\) −2974.33 −0.0707752
\(206\) −87826.8 −2.06963
\(207\) 0 0
\(208\) 8930.90i 0.206428i
\(209\) 9359.21i 0.214263i
\(210\) 0 0
\(211\) 1277.25i 0.0286888i −0.999897 0.0143444i \(-0.995434\pi\)
0.999897 0.0143444i \(-0.00456612\pi\)
\(212\) 34135.4 0.759510
\(213\) 0 0
\(214\) 1777.96i 0.0388235i
\(215\) 110631.i 2.39331i
\(216\) 0 0
\(217\) 5063.50i 0.107530i
\(218\) −92333.7 −1.94288
\(219\) 0 0
\(220\) 25377.7i 0.524332i
\(221\) 5599.05i 0.114638i
\(222\) 0 0
\(223\) 18556.2 0.373146 0.186573 0.982441i \(-0.440262\pi\)
0.186573 + 0.982441i \(0.440262\pi\)
\(224\) 3890.58i 0.0775386i
\(225\) 0 0
\(226\) 59306.2 1.16114
\(227\) 61320.0i 1.19001i −0.803722 0.595005i \(-0.797150\pi\)
0.803722 0.595005i \(-0.202850\pi\)
\(228\) 0 0
\(229\) 86222.5i 1.64418i −0.569356 0.822091i \(-0.692807\pi\)
0.569356 0.822091i \(-0.307193\pi\)
\(230\) 45380.5 0.857854
\(231\) 0 0
\(232\) 68924.9i 1.28056i
\(233\) 40899.5i 0.753367i −0.926342 0.376683i \(-0.877064\pi\)
0.926342 0.376683i \(-0.122936\pi\)
\(234\) 0 0
\(235\) 119284.i 2.15997i
\(236\) −100982. + 2999.45i −1.81309 + 0.0538539i
\(237\) 0 0
\(238\) 3441.39i 0.0607547i
\(239\) −4438.81 −0.0777089 −0.0388544 0.999245i \(-0.512371\pi\)
−0.0388544 + 0.999245i \(0.512371\pi\)
\(240\) 0 0
\(241\) −65511.7 −1.12794 −0.563968 0.825797i \(-0.690726\pi\)
−0.563968 + 0.825797i \(0.690726\pi\)
\(242\) 95199.9i 1.62557i
\(243\) 0 0
\(244\) 59257.0i 0.995314i
\(245\) −96800.3 −1.61267
\(246\) 0 0
\(247\) 32210.2i 0.527959i
\(248\) 65943.4 1.07218
\(249\) 0 0
\(250\) 120789.i 1.93263i
\(251\) 26481.1 0.420328 0.210164 0.977666i \(-0.432600\pi\)
0.210164 + 0.977666i \(0.432600\pi\)
\(252\) 0 0
\(253\) −3503.23 −0.0547302
\(254\) 83929.8i 1.30091i
\(255\) 0 0
\(256\) −107289. −1.63710
\(257\) 73530.4 1.11327 0.556635 0.830757i \(-0.312092\pi\)
0.556635 + 0.830757i \(0.312092\pi\)
\(258\) 0 0
\(259\) 11412.5i 0.170130i
\(260\) 87338.5i 1.29199i
\(261\) 0 0
\(262\) 201520. 2.93573
\(263\) 58037.1 0.839062 0.419531 0.907741i \(-0.362194\pi\)
0.419531 + 0.907741i \(0.362194\pi\)
\(264\) 0 0
\(265\) −48325.8 −0.688156
\(266\) 19797.6i 0.279802i
\(267\) 0 0
\(268\) 23118.7i 0.321880i
\(269\) 119064.i 1.64542i −0.568464 0.822708i \(-0.692462\pi\)
0.568464 0.822708i \(-0.307538\pi\)
\(270\) 0 0
\(271\) 95279.2 1.29736 0.648679 0.761062i \(-0.275322\pi\)
0.648679 + 0.761062i \(0.275322\pi\)
\(272\) −9321.10 −0.125988
\(273\) 0 0
\(274\) 204963.i 2.73008i
\(275\) 22625.9i 0.299186i
\(276\) 0 0
\(277\) −70994.3 −0.925260 −0.462630 0.886551i \(-0.653094\pi\)
−0.462630 + 0.886551i \(0.653094\pi\)
\(278\) 91534.0i 1.18438i
\(279\) 0 0
\(280\) 24086.9i 0.307230i
\(281\) −89157.2 −1.12913 −0.564565 0.825389i \(-0.690956\pi\)
−0.564565 + 0.825389i \(0.690956\pi\)
\(282\) 0 0
\(283\) 121771.i 1.52044i 0.649664 + 0.760222i \(0.274910\pi\)
−0.649664 + 0.760222i \(0.725090\pi\)
\(284\) −141084. −1.74921
\(285\) 0 0
\(286\) 10459.3i 0.127870i
\(287\) 485.693 0.00589656
\(288\) 0 0
\(289\) −77677.3 −0.930034
\(290\) 217468.i 2.58582i
\(291\) 0 0
\(292\) 52960.5i 0.621135i
\(293\) 14072.9 0.163927 0.0819633 0.996635i \(-0.473881\pi\)
0.0819633 + 0.996635i \(0.473881\pi\)
\(294\) 0 0
\(295\) 142961. 4246.34i 1.64275 0.0487944i
\(296\) 148628. 1.69636
\(297\) 0 0
\(298\) −120953. −1.36202
\(299\) 12056.5 0.134859
\(300\) 0 0
\(301\) 18065.5i 0.199396i
\(302\) 262320. 2.87619
\(303\) 0 0
\(304\) 53622.4 0.580229
\(305\) 83890.6i 0.901807i
\(306\) 0 0
\(307\) 84061.4 0.891908 0.445954 0.895056i \(-0.352865\pi\)
0.445954 + 0.895056i \(0.352865\pi\)
\(308\) 4144.05i 0.0436841i
\(309\) 0 0
\(310\) −208061. −2.16505
\(311\) 27808.7 0.287514 0.143757 0.989613i \(-0.454082\pi\)
0.143757 + 0.989613i \(0.454082\pi\)
\(312\) 0 0
\(313\) 3675.80i 0.0375200i 0.999824 + 0.0187600i \(0.00597184\pi\)
−0.999824 + 0.0187600i \(0.994028\pi\)
\(314\) 185696. 1.88340
\(315\) 0 0
\(316\) 205669. 2.05965
\(317\) −30461.2 −0.303129 −0.151565 0.988447i \(-0.548431\pi\)
−0.151565 + 0.988447i \(0.548431\pi\)
\(318\) 0 0
\(319\) 16787.8i 0.164973i
\(320\) 240023. 2.34398
\(321\) 0 0
\(322\) −7410.42 −0.0714712
\(323\) −33617.5 −0.322226
\(324\) 0 0
\(325\) 77868.4i 0.737215i
\(326\) 193985.i 1.82529i
\(327\) 0 0
\(328\) 6325.33i 0.0587943i
\(329\) 19478.5i 0.179955i
\(330\) 0 0
\(331\) 206607. 1.88577 0.942884 0.333120i \(-0.108102\pi\)
0.942884 + 0.333120i \(0.108102\pi\)
\(332\) 128425.i 1.16513i
\(333\) 0 0
\(334\) 300072.i 2.68988i
\(335\) 32729.4i 0.291641i
\(336\) 0 0
\(337\) 145823.i 1.28401i 0.766702 + 0.642004i \(0.221897\pi\)
−0.766702 + 0.642004i \(0.778103\pi\)
\(338\) 155644.i 1.36238i
\(339\) 0 0
\(340\) 91154.4 0.788533
\(341\) 16061.6 0.138128
\(342\) 0 0
\(343\) 31916.1 0.271282
\(344\) 235272. 1.98817
\(345\) 0 0
\(346\) 249871. 2.08720
\(347\) 32401.9i 0.269098i 0.990907 + 0.134549i \(0.0429586\pi\)
−0.990907 + 0.134549i \(0.957041\pi\)
\(348\) 0 0
\(349\) 165183.i 1.35617i −0.734982 0.678086i \(-0.762809\pi\)
0.734982 0.678086i \(-0.237191\pi\)
\(350\) 47860.9i 0.390701i
\(351\) 0 0
\(352\) −12341.1 −0.0996020
\(353\) 193446.i 1.55242i 0.630472 + 0.776212i \(0.282861\pi\)
−0.630472 + 0.776212i \(0.717139\pi\)
\(354\) 0 0
\(355\) 199734. 1.58488
\(356\) 233383.i 1.84149i
\(357\) 0 0
\(358\) −158265. −1.23486
\(359\) 94638.7 0.734311 0.367155 0.930160i \(-0.380332\pi\)
0.367155 + 0.930160i \(0.380332\pi\)
\(360\) 0 0
\(361\) 63073.9 0.483989
\(362\) 9059.82i 0.0691357i
\(363\) 0 0
\(364\) 14262.0i 0.107641i
\(365\) 74976.5i 0.562781i
\(366\) 0 0
\(367\) 250626.i 1.86078i −0.366576 0.930388i \(-0.619470\pi\)
0.366576 0.930388i \(-0.380530\pi\)
\(368\) 20071.3i 0.148211i
\(369\) 0 0
\(370\) −468942. −3.42544
\(371\) 7891.37 0.0573330
\(372\) 0 0
\(373\) 65456.3 0.470472 0.235236 0.971938i \(-0.424414\pi\)
0.235236 + 0.971938i \(0.424414\pi\)
\(374\) −10916.2 −0.0780423
\(375\) 0 0
\(376\) 253675. 1.79433
\(377\) 57776.1i 0.406505i
\(378\) 0 0
\(379\) 60322.5 0.419953 0.209977 0.977706i \(-0.432661\pi\)
0.209977 + 0.977706i \(0.432661\pi\)
\(380\) −524393. −3.63153
\(381\) 0 0
\(382\) 234609. 1.60775
\(383\) 53450.5 0.364380 0.182190 0.983263i \(-0.441681\pi\)
0.182190 + 0.983263i \(0.441681\pi\)
\(384\) 0 0
\(385\) 5866.76i 0.0395801i
\(386\) 188292.i 1.26374i
\(387\) 0 0
\(388\) 68319.4i 0.453817i
\(389\) −49209.3 −0.325198 −0.162599 0.986692i \(-0.551988\pi\)
−0.162599 + 0.986692i \(0.551988\pi\)
\(390\) 0 0
\(391\) 12583.3i 0.0823078i
\(392\) 205860.i 1.33967i
\(393\) 0 0
\(394\) 58100.6i 0.374273i
\(395\) −291167. −1.86615
\(396\) 0 0
\(397\) 75598.4i 0.479658i 0.970815 + 0.239829i \(0.0770914\pi\)
−0.970815 + 0.239829i \(0.922909\pi\)
\(398\) 209963.i 1.32549i
\(399\) 0 0
\(400\) 129633. 0.810203
\(401\) 289441.i 1.80000i 0.435893 + 0.899999i \(0.356433\pi\)
−0.435893 + 0.899999i \(0.643567\pi\)
\(402\) 0 0
\(403\) −55277.0 −0.340357
\(404\) 437394.i 2.67985i
\(405\) 0 0
\(406\) 35511.4i 0.215435i
\(407\) 36200.9 0.218540
\(408\) 0 0
\(409\) 157956.i 0.944256i −0.881530 0.472128i \(-0.843486\pi\)
0.881530 0.472128i \(-0.156514\pi\)
\(410\) 19957.3i 0.118723i
\(411\) 0 0
\(412\) 379878.i 2.23795i
\(413\) −23344.8 + 693.407i −0.136864 + 0.00406526i
\(414\) 0 0
\(415\) 181813.i 1.05567i
\(416\) 42472.5 0.245426
\(417\) 0 0
\(418\) 62799.0 0.359418
\(419\) 280260.i 1.59637i 0.602412 + 0.798185i \(0.294206\pi\)
−0.602412 + 0.798185i \(0.705794\pi\)
\(420\) 0 0
\(421\) 123028.i 0.694126i 0.937842 + 0.347063i \(0.112821\pi\)
−0.937842 + 0.347063i \(0.887179\pi\)
\(422\) −8570.20 −0.0481245
\(423\) 0 0
\(424\) 102772.i 0.571665i
\(425\) −81270.4 −0.449940
\(426\) 0 0
\(427\) 13698.9i 0.0751331i
\(428\) −7690.23 −0.0419809
\(429\) 0 0
\(430\) −742317. −4.01469
\(431\) 118678.i 0.638877i 0.947607 + 0.319438i \(0.103494\pi\)
−0.947607 + 0.319438i \(0.896506\pi\)
\(432\) 0 0
\(433\) −336890. −1.79685 −0.898425 0.439127i \(-0.855288\pi\)
−0.898425 + 0.439127i \(0.855288\pi\)
\(434\) 33975.4 0.180378
\(435\) 0 0
\(436\) 399372.i 2.10089i
\(437\) 72389.3i 0.379063i
\(438\) 0 0
\(439\) 333202. 1.72893 0.864466 0.502691i \(-0.167657\pi\)
0.864466 + 0.502691i \(0.167657\pi\)
\(440\) −76404.6 −0.394652
\(441\) 0 0
\(442\) 37568.8 0.192302
\(443\) 79729.2i 0.406265i 0.979151 + 0.203133i \(0.0651123\pi\)
−0.979151 + 0.203133i \(0.934888\pi\)
\(444\) 0 0
\(445\) 330403.i 1.66849i
\(446\) 124509.i 0.625940i
\(447\) 0 0
\(448\) −39194.7 −0.195286
\(449\) 278079. 1.37935 0.689676 0.724118i \(-0.257753\pi\)
0.689676 + 0.724118i \(0.257753\pi\)
\(450\) 0 0
\(451\) 1540.64i 0.00757440i
\(452\) 256518.i 1.25557i
\(453\) 0 0
\(454\) −411449. −1.99620
\(455\) 20190.8i 0.0975282i
\(456\) 0 0
\(457\) 175865.i 0.842067i 0.907045 + 0.421033i \(0.138332\pi\)
−0.907045 + 0.421033i \(0.861668\pi\)
\(458\) −578541. −2.75806
\(459\) 0 0
\(460\) 196285.i 0.927621i
\(461\) 158569. 0.746134 0.373067 0.927804i \(-0.378306\pi\)
0.373067 + 0.927804i \(0.378306\pi\)
\(462\) 0 0
\(463\) 77926.4i 0.363515i −0.983343 0.181758i \(-0.941821\pi\)
0.983343 0.181758i \(-0.0581786\pi\)
\(464\) 96183.7 0.446751
\(465\) 0 0
\(466\) −274430. −1.26375
\(467\) 408542.i 1.87328i −0.350291 0.936641i \(-0.613917\pi\)
0.350291 0.936641i \(-0.386083\pi\)
\(468\) 0 0
\(469\) 5344.55i 0.0242977i
\(470\) −800380. −3.62327
\(471\) 0 0
\(472\) 9030.44 + 304026.i 0.0405345 + 1.36467i
\(473\) 57304.5 0.256133
\(474\) 0 0
\(475\) 467533. 2.07217
\(476\) −14885.1 −0.0656957
\(477\) 0 0
\(478\) 29783.8i 0.130354i
\(479\) 161846. 0.705393 0.352697 0.935738i \(-0.385265\pi\)
0.352697 + 0.935738i \(0.385265\pi\)
\(480\) 0 0
\(481\) −124587. −0.538497
\(482\) 439574.i 1.89207i
\(483\) 0 0
\(484\) −411769. −1.75777
\(485\) 96720.2i 0.411182i
\(486\) 0 0
\(487\) −156630. −0.660414 −0.330207 0.943909i \(-0.607119\pi\)
−0.330207 + 0.943909i \(0.607119\pi\)
\(488\) 178405. 0.749149
\(489\) 0 0
\(490\) 649516.i 2.70519i
\(491\) −289340. −1.20018 −0.600089 0.799933i \(-0.704868\pi\)
−0.600089 + 0.799933i \(0.704868\pi\)
\(492\) 0 0
\(493\) −60300.4 −0.248100
\(494\) −216126. −0.885632
\(495\) 0 0
\(496\) 92023.2i 0.374054i
\(497\) −32615.6 −0.132042
\(498\) 0 0
\(499\) 366469. 1.47176 0.735878 0.677114i \(-0.236769\pi\)
0.735878 + 0.677114i \(0.236769\pi\)
\(500\) −522452. −2.08981
\(501\) 0 0
\(502\) 177684.i 0.705085i
\(503\) 160477.i 0.634274i 0.948380 + 0.317137i \(0.102722\pi\)
−0.948380 + 0.317137i \(0.897278\pi\)
\(504\) 0 0
\(505\) 619222.i 2.42808i
\(506\) 23506.2i 0.0918081i
\(507\) 0 0
\(508\) 363022. 1.40671
\(509\) 244953.i 0.945468i 0.881205 + 0.472734i \(0.156733\pi\)
−0.881205 + 0.472734i \(0.843267\pi\)
\(510\) 0 0
\(511\) 12243.3i 0.0468875i
\(512\) 241175.i 0.920008i
\(513\) 0 0
\(514\) 493379.i 1.86747i
\(515\) 537796.i 2.02770i
\(516\) 0 0
\(517\) 61786.8 0.231161
\(518\) 76576.1 0.285387
\(519\) 0 0
\(520\) 262950. 0.972450
\(521\) 366691. 1.35090 0.675452 0.737404i \(-0.263948\pi\)
0.675452 + 0.737404i \(0.263948\pi\)
\(522\) 0 0
\(523\) 97476.4 0.356366 0.178183 0.983997i \(-0.442978\pi\)
0.178183 + 0.983997i \(0.442978\pi\)
\(524\) 871636.i 3.17448i
\(525\) 0 0
\(526\) 389421.i 1.40750i
\(527\) 57692.0i 0.207728i
\(528\) 0 0
\(529\) 252745. 0.903174
\(530\) 324259.i 1.15436i
\(531\) 0 0
\(532\) 85631.0 0.302557
\(533\) 5302.20i 0.0186639i
\(534\) 0 0
\(535\) 10887.1 0.0380369
\(536\) 69603.6 0.242272
\(537\) 0 0
\(538\) −798903. −2.76013
\(539\) 50140.6i 0.172589i
\(540\) 0 0
\(541\) 321027.i 1.09685i −0.836199 0.548425i \(-0.815227\pi\)
0.836199 0.548425i \(-0.184773\pi\)
\(542\) 639310.i 2.17627i
\(543\) 0 0
\(544\) 44328.1i 0.149790i
\(545\) 565393.i 1.90352i
\(546\) 0 0
\(547\) −82148.9 −0.274554 −0.137277 0.990533i \(-0.543835\pi\)
−0.137277 + 0.990533i \(0.543835\pi\)
\(548\) 886530. 2.95211
\(549\) 0 0
\(550\) 151817. 0.501874
\(551\) 346896. 1.14261
\(552\) 0 0
\(553\) 47546.1 0.155477
\(554\) 476362.i 1.55209i
\(555\) 0 0
\(556\) −395913. −1.28071
\(557\) 92439.5 0.297953 0.148976 0.988841i \(-0.452402\pi\)
0.148976 + 0.988841i \(0.452402\pi\)
\(558\) 0 0
\(559\) −197216. −0.631131
\(560\) −33612.9 −0.107184
\(561\) 0 0
\(562\) 598232.i 1.89408i
\(563\) 332807.i 1.04997i 0.851112 + 0.524984i \(0.175929\pi\)
−0.851112 + 0.524984i \(0.824071\pi\)
\(564\) 0 0
\(565\) 363154.i 1.13761i
\(566\) 817065. 2.55049
\(567\) 0 0
\(568\) 424763.i 1.31659i
\(569\) 71655.2i 0.221321i 0.993858 + 0.110661i \(0.0352966\pi\)
−0.993858 + 0.110661i \(0.964703\pi\)
\(570\) 0 0
\(571\) 235330.i 0.721780i 0.932608 + 0.360890i \(0.117527\pi\)
−0.932608 + 0.360890i \(0.882473\pi\)
\(572\) −45239.6 −0.138270
\(573\) 0 0
\(574\) 3258.94i 0.00989127i
\(575\) 175001.i 0.529305i
\(576\) 0 0
\(577\) 582269. 1.74893 0.874465 0.485089i \(-0.161213\pi\)
0.874465 + 0.485089i \(0.161213\pi\)
\(578\) 521204.i 1.56010i
\(579\) 0 0
\(580\) −940615. −2.79612
\(581\) 29689.1i 0.0879519i
\(582\) 0 0
\(583\) 25031.8i 0.0736469i
\(584\) 159448. 0.467513
\(585\) 0 0
\(586\) 94427.4i 0.274981i
\(587\) 535088.i 1.55292i 0.630166 + 0.776460i \(0.282987\pi\)
−0.630166 + 0.776460i \(0.717013\pi\)
\(588\) 0 0
\(589\) 331891.i 0.956676i
\(590\) −28492.3 959246.i −0.0818510 2.75566i
\(591\) 0 0
\(592\) 207408.i 0.591810i
\(593\) −439880. −1.25091 −0.625453 0.780262i \(-0.715086\pi\)
−0.625453 + 0.780262i \(0.715086\pi\)
\(594\) 0 0
\(595\) 21072.9 0.0595238
\(596\) 523157.i 1.47279i
\(597\) 0 0
\(598\) 80897.7i 0.226222i
\(599\) 177415. 0.494465 0.247233 0.968956i \(-0.420479\pi\)
0.247233 + 0.968956i \(0.420479\pi\)
\(600\) 0 0
\(601\) 602513.i 1.66808i 0.551703 + 0.834041i \(0.313978\pi\)
−0.551703 + 0.834041i \(0.686022\pi\)
\(602\) 121217. 0.334480
\(603\) 0 0
\(604\) 1.13461e6i 3.11010i
\(605\) 582945. 1.59264
\(606\) 0 0
\(607\) −235674. −0.639639 −0.319820 0.947478i \(-0.603622\pi\)
−0.319820 + 0.947478i \(0.603622\pi\)
\(608\) 255011.i 0.689846i
\(609\) 0 0
\(610\) −562894. −1.51275
\(611\) −212642. −0.569597
\(612\) 0 0
\(613\) 3306.79i 0.00880006i 0.999990 + 0.00440003i \(0.00140058\pi\)
−0.999990 + 0.00440003i \(0.998599\pi\)
\(614\) 564040.i 1.49614i
\(615\) 0 0
\(616\) 12476.5 0.0328800
\(617\) −194814. −0.511740 −0.255870 0.966711i \(-0.582362\pi\)
−0.255870 + 0.966711i \(0.582362\pi\)
\(618\) 0 0
\(619\) −346456. −0.904206 −0.452103 0.891966i \(-0.649326\pi\)
−0.452103 + 0.891966i \(0.649326\pi\)
\(620\) 899928.i 2.34112i
\(621\) 0 0
\(622\) 186592.i 0.482295i
\(623\) 53953.2i 0.139008i
\(624\) 0 0
\(625\) 75176.8 0.192453
\(626\) 24664.1 0.0629385
\(627\) 0 0
\(628\) 803192.i 2.03657i
\(629\) 130030.i 0.328657i
\(630\) 0 0
\(631\) 694008. 1.74303 0.871517 0.490365i \(-0.163136\pi\)
0.871517 + 0.490365i \(0.163136\pi\)
\(632\) 619207.i 1.55025i
\(633\) 0 0
\(634\) 204390.i 0.508489i
\(635\) −513933. −1.27456
\(636\) 0 0
\(637\) 172561.i 0.425270i
\(638\) 112644. 0.276736
\(639\) 0 0
\(640\) 1.22932e6i 3.00126i
\(641\) 97669.8 0.237708 0.118854 0.992912i \(-0.462078\pi\)
0.118854 + 0.992912i \(0.462078\pi\)
\(642\) 0 0
\(643\) −128561. −0.310947 −0.155474 0.987840i \(-0.549690\pi\)
−0.155474 + 0.987840i \(0.549690\pi\)
\(644\) 32052.3i 0.0772837i
\(645\) 0 0
\(646\) 225569.i 0.540523i
\(647\) 280631. 0.670390 0.335195 0.942149i \(-0.391198\pi\)
0.335195 + 0.942149i \(0.391198\pi\)
\(648\) 0 0
\(649\) 2199.52 + 74050.7i 0.00522201 + 0.175809i
\(650\) −522486. −1.23665
\(651\) 0 0
\(652\) 839044. 1.97374
\(653\) −380413. −0.892132 −0.446066 0.895000i \(-0.647175\pi\)
−0.446066 + 0.895000i \(0.647175\pi\)
\(654\) 0 0
\(655\) 1.23398e6i 2.87625i
\(656\) −8826.91 −0.0205117
\(657\) 0 0
\(658\) 130698. 0.301869
\(659\) 59943.3i 0.138029i −0.997616 0.0690145i \(-0.978015\pi\)
0.997616 0.0690145i \(-0.0219855\pi\)
\(660\) 0 0
\(661\) −87318.8 −0.199850 −0.0999252 0.994995i \(-0.531860\pi\)
−0.0999252 + 0.994995i \(0.531860\pi\)
\(662\) 1.38630e6i 3.16331i
\(663\) 0 0
\(664\) −386650. −0.876965
\(665\) −121228. −0.274133
\(666\) 0 0
\(667\) 129846.i 0.291862i
\(668\) −1.29790e6 −2.90864
\(669\) 0 0
\(670\) −219609. −0.489217
\(671\) 43453.6 0.0965119
\(672\) 0 0
\(673\) 598287.i 1.32093i 0.750857 + 0.660465i \(0.229641\pi\)
−0.750857 + 0.660465i \(0.770359\pi\)
\(674\) 978455. 2.15388
\(675\) 0 0
\(676\) −673209. −1.47318
\(677\) −874864. −1.90881 −0.954406 0.298512i \(-0.903510\pi\)
−0.954406 + 0.298512i \(0.903510\pi\)
\(678\) 0 0
\(679\) 15794.0i 0.0342572i
\(680\) 274439.i 0.593509i
\(681\) 0 0
\(682\) 107771.i 0.231705i
\(683\) 742238.i 1.59112i −0.605878 0.795558i \(-0.707178\pi\)
0.605878 0.795558i \(-0.292822\pi\)
\(684\) 0 0
\(685\) −1.25507e6 −2.67477
\(686\) 214152.i 0.455066i
\(687\) 0 0
\(688\) 328319.i 0.693616i
\(689\) 86148.1i 0.181471i
\(690\) 0 0
\(691\) 339025.i 0.710028i −0.934861 0.355014i \(-0.884476\pi\)
0.934861 0.355014i \(-0.115524\pi\)
\(692\) 1.08077e6i 2.25695i
\(693\) 0 0
\(694\) 217412. 0.451403
\(695\) 560496. 1.16039
\(696\) 0 0
\(697\) 5533.85 0.0113910
\(698\) −1.10836e6 −2.27493
\(699\) 0 0
\(700\) 207013. 0.422476
\(701\) 288735.i 0.587576i −0.955871 0.293788i \(-0.905084\pi\)
0.955871 0.293788i \(-0.0949159\pi\)
\(702\) 0 0
\(703\) 748039.i 1.51361i
\(704\) 124327.i 0.250854i
\(705\) 0 0
\(706\) 1.29800e6 2.60414
\(707\) 101116.i 0.202293i
\(708\) 0 0
\(709\) −65496.9 −0.130295 −0.0651476 0.997876i \(-0.520752\pi\)
−0.0651476 + 0.997876i \(0.520752\pi\)
\(710\) 1.34019e6i 2.65858i
\(711\) 0 0
\(712\) −702648. −1.38605
\(713\) 124229. 0.244369
\(714\) 0 0
\(715\) 64046.0 0.125279
\(716\) 684544.i 1.33529i
\(717\) 0 0
\(718\) 635013.i 1.23178i
\(719\) 616651.i 1.19284i −0.802673 0.596420i \(-0.796589\pi\)
0.802673 0.596420i \(-0.203411\pi\)
\(720\) 0 0
\(721\) 87819.5i 0.168935i
\(722\) 423217.i 0.811875i
\(723\) 0 0
\(724\) 39186.5 0.0747583
\(725\) 838624. 1.59548
\(726\) 0 0
\(727\) 79727.7 0.150848 0.0754241 0.997152i \(-0.475969\pi\)
0.0754241 + 0.997152i \(0.475969\pi\)
\(728\) −42938.5 −0.0810186
\(729\) 0 0
\(730\) −503082. −0.944045
\(731\) 205833.i 0.385194i
\(732\) 0 0
\(733\) −797529. −1.48436 −0.742179 0.670202i \(-0.766207\pi\)
−0.742179 + 0.670202i \(0.766207\pi\)
\(734\) −1.68167e6 −3.12139
\(735\) 0 0
\(736\) −95452.7 −0.176211
\(737\) 16953.1 0.0312115
\(738\) 0 0
\(739\) 924588.i 1.69301i 0.532381 + 0.846505i \(0.321297\pi\)
−0.532381 + 0.846505i \(0.678703\pi\)
\(740\) 2.02832e6i 3.70402i
\(741\) 0 0
\(742\) 52950.0i 0.0961741i
\(743\) −671065. −1.21559 −0.607795 0.794094i \(-0.707946\pi\)
−0.607795 + 0.794094i \(0.707946\pi\)
\(744\) 0 0
\(745\) 740637.i 1.33442i
\(746\) 439202.i 0.789200i
\(747\) 0 0
\(748\) 47216.1i 0.0843893i
\(749\) −1777.81 −0.00316900
\(750\) 0 0
\(751\) 405680.i 0.719289i 0.933089 + 0.359645i \(0.117102\pi\)
−0.933089 + 0.359645i \(0.882898\pi\)
\(752\) 354000.i 0.625989i
\(753\) 0 0
\(754\) −387670. −0.681898
\(755\) 1.60628e6i 2.81791i
\(756\) 0 0
\(757\) −843491. −1.47193 −0.735967 0.677017i \(-0.763272\pi\)
−0.735967 + 0.677017i \(0.763272\pi\)
\(758\) 404755.i 0.704457i
\(759\) 0 0
\(760\) 1.57879e6i 2.73337i
\(761\) 953711. 1.64682 0.823412 0.567444i \(-0.192068\pi\)
0.823412 + 0.567444i \(0.192068\pi\)
\(762\) 0 0
\(763\) 92326.0i 0.158590i
\(764\) 1.01476e6i 1.73850i
\(765\) 0 0
\(766\) 358645.i 0.611234i
\(767\) −7569.75 254849.i −0.0128674 0.433205i
\(768\) 0 0
\(769\) 551211.i 0.932106i 0.884757 + 0.466053i \(0.154324\pi\)
−0.884757 + 0.466053i \(0.845676\pi\)
\(770\) −39365.2 −0.0663943
\(771\) 0 0
\(772\) 814421. 1.36651
\(773\) 98128.0i 0.164223i −0.996623 0.0821115i \(-0.973834\pi\)
0.996623 0.0821115i \(-0.0261664\pi\)
\(774\) 0 0
\(775\) 802348.i 1.33586i
\(776\) −205689. −0.341577
\(777\) 0 0
\(778\) 330187.i 0.545508i
\(779\) −31835.2 −0.0524605
\(780\) 0 0
\(781\) 103458.i 0.169614i
\(782\) −84432.2 −0.138068
\(783\) 0 0
\(784\) −287274. −0.467374
\(785\) 1.13708e6i 1.84524i
\(786\) 0 0
\(787\) 975267. 1.57461 0.787307 0.616561i \(-0.211475\pi\)
0.787307 + 0.616561i \(0.211475\pi\)
\(788\) −251303. −0.404712
\(789\) 0 0
\(790\) 1.95369e6i 3.13041i
\(791\) 59301.3i 0.0947788i
\(792\) 0 0
\(793\) −149548. −0.237812
\(794\) 507255. 0.804610
\(795\) 0 0
\(796\) 908153. 1.43329
\(797\) 838974.i