Properties

Label 531.5.c.d.235.1
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 235.1
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.40

$q$-expansion

\(f(q)\) \(=\) \(q-7.81880i q^{2} -45.1337 q^{4} -21.3172 q^{5} -30.7266 q^{7} +227.790i q^{8} +O(q^{10})\) \(q-7.81880i q^{2} -45.1337 q^{4} -21.3172 q^{5} -30.7266 q^{7} +227.790i q^{8} +166.675i q^{10} +207.480i q^{11} -206.020i q^{13} +240.245i q^{14} +1058.91 q^{16} +391.490 q^{17} -321.627 q^{19} +962.125 q^{20} +1622.24 q^{22} +287.967i q^{23} -170.575 q^{25} -1610.83 q^{26} +1386.80 q^{28} -1020.53 q^{29} -560.747i q^{31} -4634.75i q^{32} -3060.98i q^{34} +655.006 q^{35} +1257.89i q^{37} +2514.74i q^{38} -4855.86i q^{40} +1524.79 q^{41} +2155.53i q^{43} -9364.31i q^{44} +2251.55 q^{46} -245.854i q^{47} -1456.88 q^{49} +1333.69i q^{50} +9298.43i q^{52} +1142.98 q^{53} -4422.89i q^{55} -6999.22i q^{56} +7979.34i q^{58} +(3465.52 - 327.946i) q^{59} +518.066i q^{61} -4384.37 q^{62} -19295.6 q^{64} +4391.77i q^{65} -2495.71i q^{67} -17669.4 q^{68} -5121.36i q^{70} -9311.79 q^{71} -8005.70i q^{73} +9835.18 q^{74} +14516.2 q^{76} -6375.14i q^{77} +5130.68 q^{79} -22573.0 q^{80} -11922.1i q^{82} +13428.1i q^{83} -8345.49 q^{85} +16853.7 q^{86} -47261.8 q^{88} -3141.70i q^{89} +6330.29i q^{91} -12997.0i q^{92} -1922.28 q^{94} +6856.20 q^{95} -5322.39i q^{97} +11391.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.81880i 1.95470i −0.211629 0.977350i \(-0.567877\pi\)
0.211629 0.977350i \(-0.432123\pi\)
\(3\) 0 0
\(4\) −45.1337 −2.82085
\(5\) −21.3172 −0.852690 −0.426345 0.904561i \(-0.640199\pi\)
−0.426345 + 0.904561i \(0.640199\pi\)
\(6\) 0 0
\(7\) −30.7266 −0.627073 −0.313537 0.949576i \(-0.601514\pi\)
−0.313537 + 0.949576i \(0.601514\pi\)
\(8\) 227.790i 3.55922i
\(9\) 0 0
\(10\) 166.675i 1.66675i
\(11\) 207.480i 1.71471i 0.514728 + 0.857353i \(0.327893\pi\)
−0.514728 + 0.857353i \(0.672107\pi\)
\(12\) 0 0
\(13\) 206.020i 1.21905i −0.792766 0.609526i \(-0.791360\pi\)
0.792766 0.609526i \(-0.208640\pi\)
\(14\) 240.245i 1.22574i
\(15\) 0 0
\(16\) 1058.91 4.13636
\(17\) 391.490 1.35464 0.677318 0.735690i \(-0.263142\pi\)
0.677318 + 0.735690i \(0.263142\pi\)
\(18\) 0 0
\(19\) −321.627 −0.890934 −0.445467 0.895298i \(-0.646962\pi\)
−0.445467 + 0.895298i \(0.646962\pi\)
\(20\) 962.125 2.40531
\(21\) 0 0
\(22\) 1622.24 3.35174
\(23\) 287.967i 0.544360i 0.962246 + 0.272180i \(0.0877447\pi\)
−0.962246 + 0.272180i \(0.912255\pi\)
\(24\) 0 0
\(25\) −170.575 −0.272921
\(26\) −1610.83 −2.38288
\(27\) 0 0
\(28\) 1386.80 1.76888
\(29\) −1020.53 −1.21347 −0.606737 0.794902i \(-0.707522\pi\)
−0.606737 + 0.794902i \(0.707522\pi\)
\(30\) 0 0
\(31\) 560.747i 0.583504i −0.956494 0.291752i \(-0.905762\pi\)
0.956494 0.291752i \(-0.0942381\pi\)
\(32\) 4634.75i 4.52612i
\(33\) 0 0
\(34\) 3060.98i 2.64791i
\(35\) 655.006 0.534699
\(36\) 0 0
\(37\) 1257.89i 0.918838i 0.888220 + 0.459419i \(0.151942\pi\)
−0.888220 + 0.459419i \(0.848058\pi\)
\(38\) 2514.74i 1.74151i
\(39\) 0 0
\(40\) 4855.86i 3.03491i
\(41\) 1524.79 0.907076 0.453538 0.891237i \(-0.350162\pi\)
0.453538 + 0.891237i \(0.350162\pi\)
\(42\) 0 0
\(43\) 2155.53i 1.16578i 0.812550 + 0.582892i \(0.198079\pi\)
−0.812550 + 0.582892i \(0.801921\pi\)
\(44\) 9364.31i 4.83694i
\(45\) 0 0
\(46\) 2251.55 1.06406
\(47\) 245.854i 0.111296i −0.998450 0.0556482i \(-0.982277\pi\)
0.998450 0.0556482i \(-0.0177225\pi\)
\(48\) 0 0
\(49\) −1456.88 −0.606779
\(50\) 1333.69i 0.533478i
\(51\) 0 0
\(52\) 9298.43i 3.43877i
\(53\) 1142.98 0.406899 0.203449 0.979085i \(-0.434785\pi\)
0.203449 + 0.979085i \(0.434785\pi\)
\(54\) 0 0
\(55\) 4422.89i 1.46211i
\(56\) 6999.22i 2.23189i
\(57\) 0 0
\(58\) 7979.34i 2.37198i
\(59\) 3465.52 327.946i 0.995552 0.0942101i
\(60\) 0 0
\(61\) 518.066i 0.139228i 0.997574 + 0.0696138i \(0.0221767\pi\)
−0.997574 + 0.0696138i \(0.977823\pi\)
\(62\) −4384.37 −1.14057
\(63\) 0 0
\(64\) −19295.6 −4.71085
\(65\) 4391.77i 1.03947i
\(66\) 0 0
\(67\) 2495.71i 0.555961i −0.960587 0.277980i \(-0.910335\pi\)
0.960587 0.277980i \(-0.0896650\pi\)
\(68\) −17669.4 −3.82123
\(69\) 0 0
\(70\) 5121.36i 1.04518i
\(71\) −9311.79 −1.84721 −0.923606 0.383344i \(-0.874772\pi\)
−0.923606 + 0.383344i \(0.874772\pi\)
\(72\) 0 0
\(73\) 8005.70i 1.50229i −0.660138 0.751144i \(-0.729502\pi\)
0.660138 0.751144i \(-0.270498\pi\)
\(74\) 9835.18 1.79605
\(75\) 0 0
\(76\) 14516.2 2.51319
\(77\) 6375.14i 1.07525i
\(78\) 0 0
\(79\) 5130.68 0.822093 0.411047 0.911614i \(-0.365163\pi\)
0.411047 + 0.911614i \(0.365163\pi\)
\(80\) −22573.0 −3.52703
\(81\) 0 0
\(82\) 11922.1i 1.77306i
\(83\) 13428.1i 1.94921i 0.223939 + 0.974603i \(0.428108\pi\)
−0.223939 + 0.974603i \(0.571892\pi\)
\(84\) 0 0
\(85\) −8345.49 −1.15508
\(86\) 16853.7 2.27876
\(87\) 0 0
\(88\) −47261.8 −6.10302
\(89\) 3141.70i 0.396630i −0.980138 0.198315i \(-0.936453\pi\)
0.980138 0.198315i \(-0.0635468\pi\)
\(90\) 0 0
\(91\) 6330.29i 0.764435i
\(92\) 12997.0i 1.53556i
\(93\) 0 0
\(94\) −1922.28 −0.217551
\(95\) 6856.20 0.759690
\(96\) 0 0
\(97\) 5322.39i 0.565670i −0.959169 0.282835i \(-0.908725\pi\)
0.959169 0.282835i \(-0.0912749\pi\)
\(98\) 11391.0i 1.18607i
\(99\) 0 0
\(100\) 7698.69 0.769869
\(101\) 6022.91i 0.590424i −0.955432 0.295212i \(-0.904610\pi\)
0.955432 0.295212i \(-0.0953902\pi\)
\(102\) 0 0
\(103\) 3738.09i 0.352350i 0.984359 + 0.176175i \(0.0563725\pi\)
−0.984359 + 0.176175i \(0.943627\pi\)
\(104\) 46929.3 4.33888
\(105\) 0 0
\(106\) 8936.72i 0.795365i
\(107\) −10702.1 −0.934764 −0.467382 0.884056i \(-0.654803\pi\)
−0.467382 + 0.884056i \(0.654803\pi\)
\(108\) 0 0
\(109\) 21298.0i 1.79261i −0.443442 0.896303i \(-0.646243\pi\)
0.443442 0.896303i \(-0.353757\pi\)
\(110\) −34581.7 −2.85799
\(111\) 0 0
\(112\) −32536.6 −2.59380
\(113\) 16771.3i 1.31344i −0.754135 0.656720i \(-0.771943\pi\)
0.754135 0.656720i \(-0.228057\pi\)
\(114\) 0 0
\(115\) 6138.65i 0.464170i
\(116\) 46060.4 3.42303
\(117\) 0 0
\(118\) −2564.14 27096.2i −0.184153 1.94601i
\(119\) −12029.2 −0.849456
\(120\) 0 0
\(121\) −28406.8 −1.94022
\(122\) 4050.65 0.272148
\(123\) 0 0
\(124\) 25308.6i 1.64598i
\(125\) 16959.5 1.08541
\(126\) 0 0
\(127\) 27235.6 1.68861 0.844305 0.535863i \(-0.180014\pi\)
0.844305 + 0.535863i \(0.180014\pi\)
\(128\) 76712.8i 4.68218i
\(129\) 0 0
\(130\) 34338.4 2.03186
\(131\) 1718.72i 0.100153i 0.998745 + 0.0500763i \(0.0159465\pi\)
−0.998745 + 0.0500763i \(0.984054\pi\)
\(132\) 0 0
\(133\) 9882.51 0.558681
\(134\) −19513.4 −1.08674
\(135\) 0 0
\(136\) 89177.6i 4.82145i
\(137\) −8888.26 −0.473560 −0.236780 0.971563i \(-0.576092\pi\)
−0.236780 + 0.971563i \(0.576092\pi\)
\(138\) 0 0
\(139\) 2771.10 0.143424 0.0717121 0.997425i \(-0.477154\pi\)
0.0717121 + 0.997425i \(0.477154\pi\)
\(140\) −29562.8 −1.50831
\(141\) 0 0
\(142\) 72807.1i 3.61074i
\(143\) 42744.9 2.09032
\(144\) 0 0
\(145\) 21754.9 1.03472
\(146\) −62594.9 −2.93652
\(147\) 0 0
\(148\) 56773.1i 2.59191i
\(149\) 8123.08i 0.365888i 0.983123 + 0.182944i \(0.0585627\pi\)
−0.983123 + 0.182944i \(0.941437\pi\)
\(150\) 0 0
\(151\) 2438.11i 0.106930i 0.998570 + 0.0534651i \(0.0170266\pi\)
−0.998570 + 0.0534651i \(0.982973\pi\)
\(152\) 73263.5i 3.17103i
\(153\) 0 0
\(154\) −49845.9 −2.10179
\(155\) 11953.6i 0.497547i
\(156\) 0 0
\(157\) 13811.6i 0.560332i −0.959952 0.280166i \(-0.909611\pi\)
0.959952 0.280166i \(-0.0903895\pi\)
\(158\) 40115.8i 1.60695i
\(159\) 0 0
\(160\) 98800.0i 3.85938i
\(161\) 8848.23i 0.341354i
\(162\) 0 0
\(163\) −14070.7 −0.529593 −0.264796 0.964304i \(-0.585305\pi\)
−0.264796 + 0.964304i \(0.585305\pi\)
\(164\) −68819.5 −2.55873
\(165\) 0 0
\(166\) 104991. 3.81011
\(167\) 37863.4 1.35765 0.678823 0.734302i \(-0.262490\pi\)
0.678823 + 0.734302i \(0.262490\pi\)
\(168\) 0 0
\(169\) −13883.2 −0.486089
\(170\) 65251.7i 2.25784i
\(171\) 0 0
\(172\) 97287.2i 3.28851i
\(173\) 13221.6i 0.441765i 0.975300 + 0.220883i \(0.0708938\pi\)
−0.975300 + 0.220883i \(0.929106\pi\)
\(174\) 0 0
\(175\) 5241.20 0.171141
\(176\) 219702.i 7.09264i
\(177\) 0 0
\(178\) −24564.3 −0.775292
\(179\) 398.257i 0.0124296i −0.999981 0.00621480i \(-0.998022\pi\)
0.999981 0.00621480i \(-0.00197825\pi\)
\(180\) 0 0
\(181\) 46013.5 1.40452 0.702260 0.711920i \(-0.252174\pi\)
0.702260 + 0.711920i \(0.252174\pi\)
\(182\) 49495.3 1.49424
\(183\) 0 0
\(184\) −65596.0 −1.93750
\(185\) 26814.7i 0.783483i
\(186\) 0 0
\(187\) 81226.2i 2.32281i
\(188\) 11096.3i 0.313951i
\(189\) 0 0
\(190\) 53607.3i 1.48497i
\(191\) 60179.4i 1.64961i −0.565418 0.824804i \(-0.691285\pi\)
0.565418 0.824804i \(-0.308715\pi\)
\(192\) 0 0
\(193\) −31955.1 −0.857879 −0.428939 0.903333i \(-0.641113\pi\)
−0.428939 + 0.903333i \(0.641113\pi\)
\(194\) −41614.7 −1.10572
\(195\) 0 0
\(196\) 65754.2 1.71163
\(197\) 34993.6 0.901689 0.450844 0.892603i \(-0.351123\pi\)
0.450844 + 0.892603i \(0.351123\pi\)
\(198\) 0 0
\(199\) 3731.67 0.0942317 0.0471159 0.998889i \(-0.484997\pi\)
0.0471159 + 0.998889i \(0.484997\pi\)
\(200\) 38855.4i 0.971385i
\(201\) 0 0
\(202\) −47092.0 −1.15410
\(203\) 31357.5 0.760938
\(204\) 0 0
\(205\) −32504.4 −0.773454
\(206\) 29227.4 0.688740
\(207\) 0 0
\(208\) 218156.i 5.04244i
\(209\) 66731.1i 1.52769i
\(210\) 0 0
\(211\) 13733.7i 0.308476i −0.988034 0.154238i \(-0.950708\pi\)
0.988034 0.154238i \(-0.0492923\pi\)
\(212\) −51586.8 −1.14780
\(213\) 0 0
\(214\) 83677.7i 1.82718i
\(215\) 45950.0i 0.994052i
\(216\) 0 0
\(217\) 17229.8i 0.365899i
\(218\) −166524. −3.50401
\(219\) 0 0
\(220\) 199621.i 4.12441i
\(221\) 80654.7i 1.65137i
\(222\) 0 0
\(223\) −62492.9 −1.25667 −0.628335 0.777943i \(-0.716263\pi\)
−0.628335 + 0.777943i \(0.716263\pi\)
\(224\) 142410.i 2.83821i
\(225\) 0 0
\(226\) −131132. −2.56738
\(227\) 15746.2i 0.305579i 0.988259 + 0.152790i \(0.0488257\pi\)
−0.988259 + 0.152790i \(0.951174\pi\)
\(228\) 0 0
\(229\) 50347.4i 0.960078i −0.877247 0.480039i \(-0.840623\pi\)
0.877247 0.480039i \(-0.159377\pi\)
\(230\) −47996.9 −0.907314
\(231\) 0 0
\(232\) 232467.i 4.31903i
\(233\) 54304.7i 1.00029i −0.865942 0.500144i \(-0.833280\pi\)
0.865942 0.500144i \(-0.166720\pi\)
\(234\) 0 0
\(235\) 5240.92i 0.0949012i
\(236\) −156411. + 14801.4i −2.80831 + 0.265753i
\(237\) 0 0
\(238\) 94053.6i 1.66043i
\(239\) −44247.5 −0.774628 −0.387314 0.921948i \(-0.626597\pi\)
−0.387314 + 0.921948i \(0.626597\pi\)
\(240\) 0 0
\(241\) 93615.1 1.61180 0.805902 0.592050i \(-0.201681\pi\)
0.805902 + 0.592050i \(0.201681\pi\)
\(242\) 222107.i 3.79255i
\(243\) 0 0
\(244\) 23382.2i 0.392741i
\(245\) 31056.6 0.517394
\(246\) 0 0
\(247\) 66261.6i 1.08610i
\(248\) 127733. 2.07682
\(249\) 0 0
\(250\) 132603.i 2.12164i
\(251\) 31025.2 0.492455 0.246228 0.969212i \(-0.420809\pi\)
0.246228 + 0.969212i \(0.420809\pi\)
\(252\) 0 0
\(253\) −59747.2 −0.933419
\(254\) 212950.i 3.30073i
\(255\) 0 0
\(256\) 291072. 4.44141
\(257\) 51107.2 0.773777 0.386889 0.922126i \(-0.373550\pi\)
0.386889 + 0.922126i \(0.373550\pi\)
\(258\) 0 0
\(259\) 38650.6i 0.576179i
\(260\) 198217.i 2.93220i
\(261\) 0 0
\(262\) 13438.3 0.195768
\(263\) −83386.8 −1.20555 −0.602776 0.797911i \(-0.705939\pi\)
−0.602776 + 0.797911i \(0.705939\pi\)
\(264\) 0 0
\(265\) −24365.1 −0.346958
\(266\) 77269.3i 1.09205i
\(267\) 0 0
\(268\) 112640.i 1.56828i
\(269\) 55398.4i 0.765584i 0.923835 + 0.382792i \(0.125037\pi\)
−0.923835 + 0.382792i \(0.874963\pi\)
\(270\) 0 0
\(271\) 33799.6 0.460228 0.230114 0.973164i \(-0.426090\pi\)
0.230114 + 0.973164i \(0.426090\pi\)
\(272\) 414552. 5.60326
\(273\) 0 0
\(274\) 69495.5i 0.925669i
\(275\) 35390.9i 0.467979i
\(276\) 0 0
\(277\) 71567.4 0.932730 0.466365 0.884592i \(-0.345563\pi\)
0.466365 + 0.884592i \(0.345563\pi\)
\(278\) 21666.7i 0.280351i
\(279\) 0 0
\(280\) 149204.i 1.90311i
\(281\) −121564. −1.53955 −0.769774 0.638316i \(-0.779631\pi\)
−0.769774 + 0.638316i \(0.779631\pi\)
\(282\) 0 0
\(283\) 142151.i 1.77491i −0.460890 0.887457i \(-0.652470\pi\)
0.460890 0.887457i \(-0.347530\pi\)
\(284\) 420275. 5.21071
\(285\) 0 0
\(286\) 334214.i 4.08594i
\(287\) −46851.7 −0.568803
\(288\) 0 0
\(289\) 69743.4 0.835041
\(290\) 170097.i 2.02256i
\(291\) 0 0
\(292\) 361326.i 4.23774i
\(293\) 96135.1 1.11982 0.559908 0.828555i \(-0.310836\pi\)
0.559908 + 0.828555i \(0.310836\pi\)
\(294\) 0 0
\(295\) −73875.3 + 6990.89i −0.848897 + 0.0803320i
\(296\) −286535. −3.27035
\(297\) 0 0
\(298\) 63512.8 0.715202
\(299\) 59326.9 0.663604
\(300\) 0 0
\(301\) 66232.2i 0.731032i
\(302\) 19063.1 0.209016
\(303\) 0 0
\(304\) −340574. −3.68522
\(305\) 11043.7i 0.118718i
\(306\) 0 0
\(307\) 35837.3 0.380240 0.190120 0.981761i \(-0.439112\pi\)
0.190120 + 0.981761i \(0.439112\pi\)
\(308\) 287733.i 3.03311i
\(309\) 0 0
\(310\) 93462.6 0.972556
\(311\) 55265.8 0.571394 0.285697 0.958320i \(-0.407775\pi\)
0.285697 + 0.958320i \(0.407775\pi\)
\(312\) 0 0
\(313\) 96714.2i 0.987192i 0.869691 + 0.493596i \(0.164318\pi\)
−0.869691 + 0.493596i \(0.835682\pi\)
\(314\) −107990. −1.09528
\(315\) 0 0
\(316\) −231566. −2.31900
\(317\) 132005. 1.31363 0.656813 0.754053i \(-0.271904\pi\)
0.656813 + 0.754053i \(0.271904\pi\)
\(318\) 0 0
\(319\) 211740.i 2.08075i
\(320\) 411330. 4.01689
\(321\) 0 0
\(322\) −69182.6 −0.667245
\(323\) −125914. −1.20689
\(324\) 0 0
\(325\) 35141.9i 0.332705i
\(326\) 110016.i 1.03519i
\(327\) 0 0
\(328\) 347333.i 3.22848i
\(329\) 7554.24i 0.0697909i
\(330\) 0 0
\(331\) 53983.3 0.492724 0.246362 0.969178i \(-0.420765\pi\)
0.246362 + 0.969178i \(0.420765\pi\)
\(332\) 606058.i 5.49842i
\(333\) 0 0
\(334\) 296046.i 2.65379i
\(335\) 53201.6i 0.474062i
\(336\) 0 0
\(337\) 87020.3i 0.766233i 0.923700 + 0.383116i \(0.125149\pi\)
−0.923700 + 0.383116i \(0.874851\pi\)
\(338\) 108550.i 0.950158i
\(339\) 0 0
\(340\) 376662. 3.25832
\(341\) 116344. 1.00054
\(342\) 0 0
\(343\) 118539. 1.00757
\(344\) −491010. −4.14928
\(345\) 0 0
\(346\) 103377. 0.863519
\(347\) 74094.0i 0.615353i −0.951491 0.307677i \(-0.900449\pi\)
0.951491 0.307677i \(-0.0995515\pi\)
\(348\) 0 0
\(349\) 27934.6i 0.229347i 0.993403 + 0.114673i \(0.0365821\pi\)
−0.993403 + 0.114673i \(0.963418\pi\)
\(350\) 40979.9i 0.334530i
\(351\) 0 0
\(352\) 961615. 7.76097
\(353\) 140969.i 1.13129i −0.824648 0.565647i \(-0.808627\pi\)
0.824648 0.565647i \(-0.191373\pi\)
\(354\) 0 0
\(355\) 198502. 1.57510
\(356\) 141797.i 1.11883i
\(357\) 0 0
\(358\) −3113.89 −0.0242961
\(359\) 29736.8 0.230731 0.115365 0.993323i \(-0.463196\pi\)
0.115365 + 0.993323i \(0.463196\pi\)
\(360\) 0 0
\(361\) −26877.0 −0.206237
\(362\) 359770.i 2.74542i
\(363\) 0 0
\(364\) 285709.i 2.15636i
\(365\) 170659.i 1.28099i
\(366\) 0 0
\(367\) 52026.5i 0.386272i −0.981172 0.193136i \(-0.938134\pi\)
0.981172 0.193136i \(-0.0618658\pi\)
\(368\) 304930.i 2.25167i
\(369\) 0 0
\(370\) −209659. −1.53148
\(371\) −35119.8 −0.255155
\(372\) 0 0
\(373\) 70424.5 0.506181 0.253091 0.967443i \(-0.418553\pi\)
0.253091 + 0.967443i \(0.418553\pi\)
\(374\) 635091. 4.54039
\(375\) 0 0
\(376\) 56003.0 0.396128
\(377\) 210250.i 1.47929i
\(378\) 0 0
\(379\) −39937.6 −0.278038 −0.139019 0.990290i \(-0.544395\pi\)
−0.139019 + 0.990290i \(0.544395\pi\)
\(380\) −309445. −2.14297
\(381\) 0 0
\(382\) −470531. −3.22449
\(383\) −155977. −1.06332 −0.531659 0.846959i \(-0.678431\pi\)
−0.531659 + 0.846959i \(0.678431\pi\)
\(384\) 0 0
\(385\) 135900.i 0.916852i
\(386\) 249851.i 1.67690i
\(387\) 0 0
\(388\) 240219.i 1.59567i
\(389\) 184572. 1.21974 0.609870 0.792501i \(-0.291222\pi\)
0.609870 + 0.792501i \(0.291222\pi\)
\(390\) 0 0
\(391\) 112736.i 0.737411i
\(392\) 331862.i 2.15966i
\(393\) 0 0
\(394\) 273608.i 1.76253i
\(395\) −109372. −0.700990
\(396\) 0 0
\(397\) 216054.i 1.37082i −0.728155 0.685412i \(-0.759622\pi\)
0.728155 0.685412i \(-0.240378\pi\)
\(398\) 29177.2i 0.184195i
\(399\) 0 0
\(400\) −180624. −1.12890
\(401\) 186079.i 1.15720i 0.815611 + 0.578601i \(0.196401\pi\)
−0.815611 + 0.578601i \(0.803599\pi\)
\(402\) 0 0
\(403\) −115525. −0.711321
\(404\) 271836.i 1.66550i
\(405\) 0 0
\(406\) 245178.i 1.48741i
\(407\) −260986. −1.57554
\(408\) 0 0
\(409\) 142032.i 0.849064i −0.905413 0.424532i \(-0.860439\pi\)
0.905413 0.424532i \(-0.139561\pi\)
\(410\) 254145.i 1.51187i
\(411\) 0 0
\(412\) 168713.i 0.993929i
\(413\) −106484. + 10076.6i −0.624284 + 0.0590767i
\(414\) 0 0
\(415\) 286250.i 1.66207i
\(416\) −954850. −5.51758
\(417\) 0 0
\(418\) −521757. −2.98618
\(419\) 305281.i 1.73889i −0.494032 0.869444i \(-0.664477\pi\)
0.494032 0.869444i \(-0.335523\pi\)
\(420\) 0 0
\(421\) 40020.9i 0.225799i −0.993606 0.112900i \(-0.963986\pi\)
0.993606 0.112900i \(-0.0360139\pi\)
\(422\) −107381. −0.602978
\(423\) 0 0
\(424\) 260359.i 1.44824i
\(425\) −66778.6 −0.369708
\(426\) 0 0
\(427\) 15918.4i 0.0873059i
\(428\) 483025. 2.63683
\(429\) 0 0
\(430\) −359274. −1.94307
\(431\) 199954.i 1.07641i −0.842815 0.538203i \(-0.819103\pi\)
0.842815 0.538203i \(-0.180897\pi\)
\(432\) 0 0
\(433\) −173422. −0.924972 −0.462486 0.886627i \(-0.653042\pi\)
−0.462486 + 0.886627i \(0.653042\pi\)
\(434\) 134717. 0.715224
\(435\) 0 0
\(436\) 961254.i 5.05668i
\(437\) 92617.9i 0.484989i
\(438\) 0 0
\(439\) 236050. 1.22483 0.612415 0.790536i \(-0.290198\pi\)
0.612415 + 0.790536i \(0.290198\pi\)
\(440\) 1.00749e6 5.20398
\(441\) 0 0
\(442\) −630623. −3.22794
\(443\) 390579.i 1.99022i 0.0987629 + 0.995111i \(0.468511\pi\)
−0.0987629 + 0.995111i \(0.531489\pi\)
\(444\) 0 0
\(445\) 66972.4i 0.338202i
\(446\) 488620.i 2.45641i
\(447\) 0 0
\(448\) 592889. 2.95405
\(449\) −297658. −1.47647 −0.738236 0.674543i \(-0.764341\pi\)
−0.738236 + 0.674543i \(0.764341\pi\)
\(450\) 0 0
\(451\) 316364.i 1.55537i
\(452\) 756951.i 3.70502i
\(453\) 0 0
\(454\) 123116. 0.597316
\(455\) 134944.i 0.651826i
\(456\) 0 0
\(457\) 138561.i 0.663449i −0.943376 0.331725i \(-0.892370\pi\)
0.943376 0.331725i \(-0.107630\pi\)
\(458\) −393657. −1.87666
\(459\) 0 0
\(460\) 277060.i 1.30936i
\(461\) 249929. 1.17602 0.588010 0.808853i \(-0.299911\pi\)
0.588010 + 0.808853i \(0.299911\pi\)
\(462\) 0 0
\(463\) 101021.i 0.471248i 0.971844 + 0.235624i \(0.0757134\pi\)
−0.971844 + 0.235624i \(0.924287\pi\)
\(464\) −1.08065e6 −5.01937
\(465\) 0 0
\(466\) −424597. −1.95526
\(467\) 290164.i 1.33048i 0.746628 + 0.665242i \(0.231672\pi\)
−0.746628 + 0.665242i \(0.768328\pi\)
\(468\) 0 0
\(469\) 76684.6i 0.348628i
\(470\) 40977.7 0.185503
\(471\) 0 0
\(472\) 74702.8 + 789411.i 0.335315 + 3.54339i
\(473\) −447229. −1.99898
\(474\) 0 0
\(475\) 54861.7 0.243154
\(476\) 542920. 2.39619
\(477\) 0 0
\(478\) 345963.i 1.51417i
\(479\) 90383.7 0.393930 0.196965 0.980411i \(-0.436891\pi\)
0.196965 + 0.980411i \(0.436891\pi\)
\(480\) 0 0
\(481\) 259150. 1.12011
\(482\) 731958.i 3.15059i
\(483\) 0 0
\(484\) 1.28210e6 5.47308
\(485\) 113459.i 0.482341i
\(486\) 0 0
\(487\) 151820. 0.640132 0.320066 0.947395i \(-0.396295\pi\)
0.320066 + 0.947395i \(0.396295\pi\)
\(488\) −118010. −0.495542
\(489\) 0 0
\(490\) 242825.i 1.01135i
\(491\) 103785. 0.430497 0.215249 0.976559i \(-0.430944\pi\)
0.215249 + 0.976559i \(0.430944\pi\)
\(492\) 0 0
\(493\) −399528. −1.64382
\(494\) 518086. 2.12299
\(495\) 0 0
\(496\) 593779.i 2.41358i
\(497\) 286120. 1.15834
\(498\) 0 0
\(499\) 469746. 1.88652 0.943262 0.332049i \(-0.107740\pi\)
0.943262 + 0.332049i \(0.107740\pi\)
\(500\) −765443. −3.06177
\(501\) 0 0
\(502\) 242580.i 0.962602i
\(503\) 138093.i 0.545801i 0.962042 + 0.272901i \(0.0879831\pi\)
−0.962042 + 0.272901i \(0.912017\pi\)
\(504\) 0 0
\(505\) 128392.i 0.503448i
\(506\) 467151.i 1.82455i
\(507\) 0 0
\(508\) −1.22924e6 −4.76332
\(509\) 123336.i 0.476051i 0.971259 + 0.238026i \(0.0765002\pi\)
−0.971259 + 0.238026i \(0.923500\pi\)
\(510\) 0 0
\(511\) 245988.i 0.942045i
\(512\) 1.04843e6i 3.99944i
\(513\) 0 0
\(514\) 399597.i 1.51250i
\(515\) 79685.7i 0.300446i
\(516\) 0 0
\(517\) 51009.6 0.190841
\(518\) −302202. −1.12626
\(519\) 0 0
\(520\) −1.00040e6 −3.69972
\(521\) 68858.9 0.253679 0.126840 0.991923i \(-0.459517\pi\)
0.126840 + 0.991923i \(0.459517\pi\)
\(522\) 0 0
\(523\) 269285. 0.984482 0.492241 0.870459i \(-0.336178\pi\)
0.492241 + 0.870459i \(0.336178\pi\)
\(524\) 77572.1i 0.282516i
\(525\) 0 0
\(526\) 651985.i 2.35649i
\(527\) 219527.i 0.790435i
\(528\) 0 0
\(529\) 196916. 0.703672
\(530\) 190506.i 0.678199i
\(531\) 0 0
\(532\) −446034. −1.57596
\(533\) 314138.i 1.10577i
\(534\) 0 0
\(535\) 228139. 0.797063
\(536\) 568498. 1.97879
\(537\) 0 0
\(538\) 433149. 1.49649
\(539\) 302272.i 1.04045i
\(540\) 0 0
\(541\) 110046.i 0.375993i −0.982170 0.187997i \(-0.939801\pi\)
0.982170 0.187997i \(-0.0601994\pi\)
\(542\) 264272.i 0.899607i
\(543\) 0 0
\(544\) 1.81446e6i 6.13125i
\(545\) 454014.i 1.52854i
\(546\) 0 0
\(547\) 163036. 0.544889 0.272444 0.962172i \(-0.412168\pi\)
0.272444 + 0.962172i \(0.412168\pi\)
\(548\) 401159. 1.33584
\(549\) 0 0
\(550\) −276714. −0.914758
\(551\) 328231. 1.08113
\(552\) 0 0
\(553\) −157648. −0.515513
\(554\) 559572.i 1.82321i
\(555\) 0 0
\(556\) −125070. −0.404578
\(557\) 447290. 1.44171 0.720857 0.693084i \(-0.243749\pi\)
0.720857 + 0.693084i \(0.243749\pi\)
\(558\) 0 0
\(559\) 444083. 1.42115
\(560\) 693591. 2.21171
\(561\) 0 0
\(562\) 950487.i 3.00936i
\(563\) 50595.3i 0.159622i 0.996810 + 0.0798111i \(0.0254317\pi\)
−0.996810 + 0.0798111i \(0.974568\pi\)
\(564\) 0 0
\(565\) 357518.i 1.11996i
\(566\) −1.11145e6 −3.46943
\(567\) 0 0
\(568\) 2.12114e6i 6.57464i
\(569\) 596739.i 1.84315i 0.388204 + 0.921573i \(0.373096\pi\)
−0.388204 + 0.921573i \(0.626904\pi\)
\(570\) 0 0
\(571\) 2297.80i 0.00704759i 0.999994 + 0.00352379i \(0.00112166\pi\)
−0.999994 + 0.00352379i \(0.998878\pi\)
\(572\) −1.92923e6 −5.89648
\(573\) 0 0
\(574\) 366324.i 1.11184i
\(575\) 49120.0i 0.148567i
\(576\) 0 0
\(577\) −635182. −1.90786 −0.953931 0.300027i \(-0.903004\pi\)
−0.953931 + 0.300027i \(0.903004\pi\)
\(578\) 545310.i 1.63225i
\(579\) 0 0
\(580\) −981880. −2.91879
\(581\) 412599.i 1.22229i
\(582\) 0 0
\(583\) 237145.i 0.697712i
\(584\) 1.82362e6 5.34698
\(585\) 0 0
\(586\) 751661.i 2.18891i
\(587\) 46453.2i 0.134815i −0.997726 0.0674077i \(-0.978527\pi\)
0.997726 0.0674077i \(-0.0214728\pi\)
\(588\) 0 0
\(589\) 180351.i 0.519863i
\(590\) 54660.4 + 577616.i 0.157025 + 1.65934i
\(591\) 0 0
\(592\) 1.33199e6i 3.80064i
\(593\) 292298. 0.831221 0.415611 0.909543i \(-0.363568\pi\)
0.415611 + 0.909543i \(0.363568\pi\)
\(594\) 0 0
\(595\) 256428. 0.724323
\(596\) 366624.i 1.03212i
\(597\) 0 0
\(598\) 463865.i 1.29715i
\(599\) −239592. −0.667758 −0.333879 0.942616i \(-0.608358\pi\)
−0.333879 + 0.942616i \(0.608358\pi\)
\(600\) 0 0
\(601\) 235688.i 0.652512i 0.945281 + 0.326256i \(0.105787\pi\)
−0.945281 + 0.326256i \(0.894213\pi\)
\(602\) −517857. −1.42895
\(603\) 0 0
\(604\) 110041.i 0.301634i
\(605\) 605554. 1.65441
\(606\) 0 0
\(607\) 423021. 1.14811 0.574056 0.818816i \(-0.305369\pi\)
0.574056 + 0.818816i \(0.305369\pi\)
\(608\) 1.49066e6i 4.03247i
\(609\) 0 0
\(610\) −86348.7 −0.232058
\(611\) −50650.7 −0.135676
\(612\) 0 0
\(613\) 210522.i 0.560243i −0.959965 0.280122i \(-0.909625\pi\)
0.959965 0.280122i \(-0.0903748\pi\)
\(614\) 280205.i 0.743256i
\(615\) 0 0
\(616\) 1.45219e6 3.82704
\(617\) 342587. 0.899914 0.449957 0.893050i \(-0.351439\pi\)
0.449957 + 0.893050i \(0.351439\pi\)
\(618\) 0 0
\(619\) −202870. −0.529463 −0.264732 0.964322i \(-0.585283\pi\)
−0.264732 + 0.964322i \(0.585283\pi\)
\(620\) 539508.i 1.40351i
\(621\) 0 0
\(622\) 432112.i 1.11690i
\(623\) 96533.8i 0.248716i
\(624\) 0 0
\(625\) −254919. −0.652594
\(626\) 756189. 1.92966
\(627\) 0 0
\(628\) 623369.i 1.58061i
\(629\) 492451.i 1.24469i
\(630\) 0 0
\(631\) −41966.6 −0.105401 −0.0527006 0.998610i \(-0.516783\pi\)
−0.0527006 + 0.998610i \(0.516783\pi\)
\(632\) 1.16872e6i 2.92601i
\(633\) 0 0
\(634\) 1.03212e6i 2.56775i
\(635\) −580587. −1.43986
\(636\) 0 0
\(637\) 300146.i 0.739696i
\(638\) −1.65555e6 −4.06725
\(639\) 0 0
\(640\) 1.63531e6i 3.99245i
\(641\) 336370. 0.818656 0.409328 0.912387i \(-0.365763\pi\)
0.409328 + 0.912387i \(0.365763\pi\)
\(642\) 0 0
\(643\) −387046. −0.936140 −0.468070 0.883691i \(-0.655051\pi\)
−0.468070 + 0.883691i \(0.655051\pi\)
\(644\) 399353.i 0.962909i
\(645\) 0 0
\(646\) 984495.i 2.35911i
\(647\) −640773. −1.53072 −0.765360 0.643602i \(-0.777439\pi\)
−0.765360 + 0.643602i \(0.777439\pi\)
\(648\) 0 0
\(649\) 68042.0 + 719024.i 0.161543 + 1.70708i
\(650\) 274768. 0.650338
\(651\) 0 0
\(652\) 635064. 1.49390
\(653\) −59695.3 −0.139995 −0.0699977 0.997547i \(-0.522299\pi\)
−0.0699977 + 0.997547i \(0.522299\pi\)
\(654\) 0 0
\(655\) 36638.4i 0.0853991i
\(656\) 1.61462e6 3.75199
\(657\) 0 0
\(658\) 59065.1 0.136420
\(659\) 277090.i 0.638043i 0.947747 + 0.319022i \(0.103354\pi\)
−0.947747 + 0.319022i \(0.896646\pi\)
\(660\) 0 0
\(661\) −170596. −0.390450 −0.195225 0.980758i \(-0.562544\pi\)
−0.195225 + 0.980758i \(0.562544\pi\)
\(662\) 422085.i 0.963128i
\(663\) 0 0
\(664\) −3.05879e6 −6.93766
\(665\) −210668. −0.476381
\(666\) 0 0
\(667\) 293879.i 0.660568i
\(668\) −1.70891e6 −3.82972
\(669\) 0 0
\(670\) 415973. 0.926649
\(671\) −107488. −0.238734
\(672\) 0 0
\(673\) 124494.i 0.274864i 0.990511 + 0.137432i \(0.0438848\pi\)
−0.990511 + 0.137432i \(0.956115\pi\)
\(674\) 680394. 1.49775
\(675\) 0 0
\(676\) 626599. 1.37119
\(677\) −470982. −1.02761 −0.513804 0.857908i \(-0.671764\pi\)
−0.513804 + 0.857908i \(0.671764\pi\)
\(678\) 0 0
\(679\) 163539.i 0.354716i
\(680\) 1.90102e6i 4.11120i
\(681\) 0 0
\(682\) 909667.i 1.95575i
\(683\) 543532.i 1.16516i 0.812775 + 0.582578i \(0.197956\pi\)
−0.812775 + 0.582578i \(0.802044\pi\)
\(684\) 0 0
\(685\) 189473. 0.403800
\(686\) 926836.i 1.96949i
\(687\) 0 0
\(688\) 2.28251e6i 4.82210i
\(689\) 235476.i 0.496031i
\(690\) 0 0
\(691\) 466845.i 0.977725i 0.872361 + 0.488862i \(0.162588\pi\)
−0.872361 + 0.488862i \(0.837412\pi\)
\(692\) 596739.i 1.24615i
\(693\) 0 0
\(694\) −579327. −1.20283
\(695\) −59072.1 −0.122296
\(696\) 0 0
\(697\) 596942. 1.22876
\(698\) 218415. 0.448304
\(699\) 0 0
\(700\) −236554. −0.482764
\(701\) 661106.i 1.34535i −0.739939 0.672674i \(-0.765145\pi\)
0.739939 0.672674i \(-0.234855\pi\)
\(702\) 0 0
\(703\) 404571.i 0.818624i
\(704\) 4.00345e6i 8.07773i
\(705\) 0 0
\(706\) −1.10221e6 −2.21134
\(707\) 185064.i 0.370239i
\(708\) 0 0
\(709\) −64729.8 −0.128769 −0.0643845 0.997925i \(-0.520508\pi\)
−0.0643845 + 0.997925i \(0.520508\pi\)
\(710\) 1.55205e6i 3.07884i
\(711\) 0 0
\(712\) 715649. 1.41169
\(713\) 161476. 0.317636
\(714\) 0 0
\(715\) −911203. −1.78239
\(716\) 17974.8i 0.0350621i
\(717\) 0 0
\(718\) 232506.i 0.451009i
\(719\) 648569.i 1.25458i 0.778786 + 0.627290i \(0.215836\pi\)
−0.778786 + 0.627290i \(0.784164\pi\)
\(720\) 0 0
\(721\) 114859.i 0.220950i
\(722\) 210146.i 0.403131i
\(723\) 0 0
\(724\) −2.07676e6 −3.96195
\(725\) 174078. 0.331182
\(726\) 0 0
\(727\) 201809. 0.381831 0.190916 0.981606i \(-0.438854\pi\)
0.190916 + 0.981606i \(0.438854\pi\)
\(728\) −1.44198e6 −2.72079
\(729\) 0 0
\(730\) 1.33435e6 2.50394
\(731\) 843870.i 1.57921i
\(732\) 0 0
\(733\) −517718. −0.963575 −0.481787 0.876288i \(-0.660012\pi\)
−0.481787 + 0.876288i \(0.660012\pi\)
\(734\) −406785. −0.755045
\(735\) 0 0
\(736\) 1.33465e6 2.46384
\(737\) 517808. 0.953310
\(738\) 0 0
\(739\) 1.01019e6i 1.84976i 0.380263 + 0.924879i \(0.375834\pi\)
−0.380263 + 0.924879i \(0.624166\pi\)
\(740\) 1.21025e6i 2.21009i
\(741\) 0 0
\(742\) 274595.i 0.498752i
\(743\) 521556. 0.944765 0.472382 0.881394i \(-0.343394\pi\)
0.472382 + 0.881394i \(0.343394\pi\)
\(744\) 0 0
\(745\) 173162.i 0.311989i
\(746\) 550635.i 0.989432i
\(747\) 0 0
\(748\) 3.66603e6i 6.55229i
\(749\) 328839. 0.586165
\(750\) 0 0
\(751\) 879845.i 1.56001i −0.625776 0.780003i \(-0.715217\pi\)
0.625776 0.780003i \(-0.284783\pi\)
\(752\) 260336.i 0.460362i
\(753\) 0 0
\(754\) 1.64390e6 2.89157
\(755\) 51973.9i 0.0911782i
\(756\) 0 0
\(757\) −1.05871e6 −1.84750 −0.923748 0.383000i \(-0.874891\pi\)
−0.923748 + 0.383000i \(0.874891\pi\)
\(758\) 312264.i 0.543481i
\(759\) 0 0
\(760\) 1.56178e6i 2.70391i
\(761\) 319446. 0.551604 0.275802 0.961214i \(-0.411057\pi\)
0.275802 + 0.961214i \(0.411057\pi\)
\(762\) 0 0
\(763\) 654414.i 1.12410i
\(764\) 2.71612e6i 4.65330i
\(765\) 0 0
\(766\) 1.21955e6i 2.07847i
\(767\) −67563.3 713966.i −0.114847 1.21363i
\(768\) 0 0
\(769\) 547254.i 0.925414i −0.886511 0.462707i \(-0.846878\pi\)
0.886511 0.462707i \(-0.153122\pi\)
\(770\) 1.06258e6 1.79217
\(771\) 0 0
\(772\) 1.44225e6 2.41995
\(773\) 385413.i 0.645011i −0.946568 0.322505i \(-0.895475\pi\)
0.946568 0.322505i \(-0.104525\pi\)
\(774\) 0 0
\(775\) 95649.6i 0.159250i
\(776\) 1.21239e6 2.01335
\(777\) 0 0
\(778\) 1.44313e6i 2.38423i
\(779\) −490415. −0.808144
\(780\) 0 0
\(781\) 1.93201e6i 3.16743i
\(782\) 881461. 1.44142
\(783\) 0 0
\(784\) −1.54270e6 −2.50986
\(785\) 294426.i 0.477789i
\(786\) 0 0
\(787\) 1.03990e6 1.67896 0.839481 0.543389i \(-0.182859\pi\)
0.839481 + 0.543389i \(0.182859\pi\)
\(788\) −1.57939e6 −2.54353
\(789\) 0 0
\(790\) 855158.i 1.37023i
\(791\) 515325.i 0.823623i
\(792\) 0 0
\(793\) 106732. 0.169726
\(794\) −1.68929e6 −2.67955
\(795\) 0 0
\(796\) −168424. −0.265814
\(797\) 436583.i 0.687306i