Properties

Label 531.5.c.d.235.11
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 235.11
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.30

$q$-expansion

\(f(q)\) \(=\) \(q-4.10319i q^{2} -0.836195 q^{4} +39.6276 q^{5} -85.8400 q^{7} -62.2200i q^{8} +O(q^{10})\) \(q-4.10319i q^{2} -0.836195 q^{4} +39.6276 q^{5} -85.8400 q^{7} -62.2200i q^{8} -162.600i q^{10} +68.0678i q^{11} +254.341i q^{13} +352.218i q^{14} -268.680 q^{16} -229.593 q^{17} +425.861 q^{19} -33.1364 q^{20} +279.295 q^{22} +954.505i q^{23} +945.343 q^{25} +1043.61 q^{26} +71.7790 q^{28} +396.928 q^{29} -1421.82i q^{31} +106.925i q^{32} +942.065i q^{34} -3401.63 q^{35} +996.969i q^{37} -1747.39i q^{38} -2465.63i q^{40} +1020.06 q^{41} +2587.59i q^{43} -56.9180i q^{44} +3916.52 q^{46} +1477.61i q^{47} +4967.50 q^{49} -3878.93i q^{50} -212.679i q^{52} +3156.03 q^{53} +2697.36i q^{55} +5340.97i q^{56} -1628.67i q^{58} +(-3480.96 - 16.5981i) q^{59} +2170.20i q^{61} -5834.00 q^{62} -3860.14 q^{64} +10078.9i q^{65} +667.653i q^{67} +191.985 q^{68} +13957.5i q^{70} +2586.30 q^{71} +1200.66i q^{73} +4090.76 q^{74} -356.103 q^{76} -5842.94i q^{77} -1243.99 q^{79} -10647.1 q^{80} -4185.50i q^{82} +6111.54i q^{83} -9098.21 q^{85} +10617.4 q^{86} +4235.18 q^{88} +12034.3i q^{89} -21832.7i q^{91} -798.152i q^{92} +6062.92 q^{94} +16875.8 q^{95} -16943.5i q^{97} -20382.6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.10319i 1.02580i −0.858449 0.512899i \(-0.828571\pi\)
0.858449 0.512899i \(-0.171429\pi\)
\(3\) 0 0
\(4\) −0.836195 −0.0522622
\(5\) 39.6276 1.58510 0.792551 0.609805i \(-0.208752\pi\)
0.792551 + 0.609805i \(0.208752\pi\)
\(6\) 0 0
\(7\) −85.8400 −1.75184 −0.875918 0.482459i \(-0.839744\pi\)
−0.875918 + 0.482459i \(0.839744\pi\)
\(8\) 62.2200i 0.972188i
\(9\) 0 0
\(10\) 162.600i 1.62600i
\(11\) 68.0678i 0.562544i 0.959628 + 0.281272i \(0.0907563\pi\)
−0.959628 + 0.281272i \(0.909244\pi\)
\(12\) 0 0
\(13\) 254.341i 1.50498i 0.658605 + 0.752489i \(0.271147\pi\)
−0.658605 + 0.752489i \(0.728853\pi\)
\(14\) 352.218i 1.79703i
\(15\) 0 0
\(16\) −268.680 −1.04953
\(17\) −229.593 −0.794440 −0.397220 0.917724i \(-0.630025\pi\)
−0.397220 + 0.917724i \(0.630025\pi\)
\(18\) 0 0
\(19\) 425.861 1.17967 0.589836 0.807523i \(-0.299193\pi\)
0.589836 + 0.807523i \(0.299193\pi\)
\(20\) −33.1364 −0.0828409
\(21\) 0 0
\(22\) 279.295 0.577056
\(23\) 954.505i 1.80436i 0.431363 + 0.902178i \(0.358033\pi\)
−0.431363 + 0.902178i \(0.641967\pi\)
\(24\) 0 0
\(25\) 945.343 1.51255
\(26\) 1043.61 1.54380
\(27\) 0 0
\(28\) 71.7790 0.0915548
\(29\) 396.928 0.471971 0.235986 0.971757i \(-0.424168\pi\)
0.235986 + 0.971757i \(0.424168\pi\)
\(30\) 0 0
\(31\) 1421.82i 1.47952i −0.672871 0.739760i \(-0.734939\pi\)
0.672871 0.739760i \(-0.265061\pi\)
\(32\) 106.925i 0.104419i
\(33\) 0 0
\(34\) 942.065i 0.814935i
\(35\) −3401.63 −2.77684
\(36\) 0 0
\(37\) 996.969i 0.728246i 0.931351 + 0.364123i \(0.118631\pi\)
−0.931351 + 0.364123i \(0.881369\pi\)
\(38\) 1747.39i 1.21010i
\(39\) 0 0
\(40\) 2465.63i 1.54102i
\(41\) 1020.06 0.606817 0.303408 0.952861i \(-0.401875\pi\)
0.303408 + 0.952861i \(0.401875\pi\)
\(42\) 0 0
\(43\) 2587.59i 1.39945i 0.714410 + 0.699727i \(0.246695\pi\)
−0.714410 + 0.699727i \(0.753305\pi\)
\(44\) 56.9180i 0.0293998i
\(45\) 0 0
\(46\) 3916.52 1.85091
\(47\) 1477.61i 0.668905i 0.942413 + 0.334452i \(0.108551\pi\)
−0.942413 + 0.334452i \(0.891449\pi\)
\(48\) 0 0
\(49\) 4967.50 2.06893
\(50\) 3878.93i 1.55157i
\(51\) 0 0
\(52\) 212.679i 0.0786535i
\(53\) 3156.03 1.12354 0.561770 0.827293i \(-0.310120\pi\)
0.561770 + 0.827293i \(0.310120\pi\)
\(54\) 0 0
\(55\) 2697.36i 0.891689i
\(56\) 5340.97i 1.70311i
\(57\) 0 0
\(58\) 1628.67i 0.484147i
\(59\) −3480.96 16.5981i −0.999989 0.00476820i
\(60\) 0 0
\(61\) 2170.20i 0.583230i 0.956536 + 0.291615i \(0.0941927\pi\)
−0.956536 + 0.291615i \(0.905807\pi\)
\(62\) −5834.00 −1.51769
\(63\) 0 0
\(64\) −3860.14 −0.942418
\(65\) 10078.9i 2.38554i
\(66\) 0 0
\(67\) 667.653i 0.148731i 0.997231 + 0.0743655i \(0.0236931\pi\)
−0.997231 + 0.0743655i \(0.976307\pi\)
\(68\) 191.985 0.0415192
\(69\) 0 0
\(70\) 13957.5i 2.84848i
\(71\) 2586.30 0.513054 0.256527 0.966537i \(-0.417422\pi\)
0.256527 + 0.966537i \(0.417422\pi\)
\(72\) 0 0
\(73\) 1200.66i 0.225306i 0.993634 + 0.112653i \(0.0359349\pi\)
−0.993634 + 0.112653i \(0.964065\pi\)
\(74\) 4090.76 0.747034
\(75\) 0 0
\(76\) −356.103 −0.0616522
\(77\) 5842.94i 0.985485i
\(78\) 0 0
\(79\) −1243.99 −0.199325 −0.0996626 0.995021i \(-0.531776\pi\)
−0.0996626 + 0.995021i \(0.531776\pi\)
\(80\) −10647.1 −1.66361
\(81\) 0 0
\(82\) 4185.50i 0.622471i
\(83\) 6111.54i 0.887145i 0.896239 + 0.443572i \(0.146289\pi\)
−0.896239 + 0.443572i \(0.853711\pi\)
\(84\) 0 0
\(85\) −9098.21 −1.25927
\(86\) 10617.4 1.43556
\(87\) 0 0
\(88\) 4235.18 0.546898
\(89\) 12034.3i 1.51929i 0.650340 + 0.759643i \(0.274626\pi\)
−0.650340 + 0.759643i \(0.725374\pi\)
\(90\) 0 0
\(91\) 21832.7i 2.63648i
\(92\) 798.152i 0.0942997i
\(93\) 0 0
\(94\) 6062.92 0.686161
\(95\) 16875.8 1.86990
\(96\) 0 0
\(97\) 16943.5i 1.80078i −0.435085 0.900389i \(-0.643282\pi\)
0.435085 0.900389i \(-0.356718\pi\)
\(98\) 20382.6i 2.12231i
\(99\) 0 0
\(100\) −790.492 −0.0790492
\(101\) 8659.11i 0.848849i 0.905463 + 0.424424i \(0.139524\pi\)
−0.905463 + 0.424424i \(0.860476\pi\)
\(102\) 0 0
\(103\) 1183.70i 0.111575i 0.998443 + 0.0557873i \(0.0177669\pi\)
−0.998443 + 0.0557873i \(0.982233\pi\)
\(104\) 15825.1 1.46312
\(105\) 0 0
\(106\) 12949.8i 1.15253i
\(107\) −11910.3 −1.04029 −0.520145 0.854078i \(-0.674122\pi\)
−0.520145 + 0.854078i \(0.674122\pi\)
\(108\) 0 0
\(109\) 5882.90i 0.495152i −0.968868 0.247576i \(-0.920366\pi\)
0.968868 0.247576i \(-0.0796340\pi\)
\(110\) 11067.8 0.914693
\(111\) 0 0
\(112\) 23063.5 1.83861
\(113\) 21194.2i 1.65982i −0.557900 0.829908i \(-0.688393\pi\)
0.557900 0.829908i \(-0.311607\pi\)
\(114\) 0 0
\(115\) 37824.7i 2.86009i
\(116\) −331.909 −0.0246663
\(117\) 0 0
\(118\) −68.1052 + 14283.1i −0.00489121 + 1.02579i
\(119\) 19708.3 1.39173
\(120\) 0 0
\(121\) 10007.8 0.683545
\(122\) 8904.75 0.598277
\(123\) 0 0
\(124\) 1188.92i 0.0773230i
\(125\) 12694.4 0.812443
\(126\) 0 0
\(127\) 26800.6 1.66164 0.830820 0.556541i \(-0.187872\pi\)
0.830820 + 0.556541i \(0.187872\pi\)
\(128\) 17549.7i 1.07115i
\(129\) 0 0
\(130\) 41355.8 2.44709
\(131\) 1959.64i 0.114192i 0.998369 + 0.0570958i \(0.0181840\pi\)
−0.998369 + 0.0570958i \(0.981816\pi\)
\(132\) 0 0
\(133\) −36555.9 −2.06659
\(134\) 2739.51 0.152568
\(135\) 0 0
\(136\) 14285.3i 0.772345i
\(137\) 3510.97 0.187062 0.0935310 0.995616i \(-0.470185\pi\)
0.0935310 + 0.995616i \(0.470185\pi\)
\(138\) 0 0
\(139\) −8435.85 −0.436616 −0.218308 0.975880i \(-0.570054\pi\)
−0.218308 + 0.975880i \(0.570054\pi\)
\(140\) 2844.43 0.145124
\(141\) 0 0
\(142\) 10612.1i 0.526290i
\(143\) −17312.4 −0.846616
\(144\) 0 0
\(145\) 15729.3 0.748123
\(146\) 4926.52 0.231119
\(147\) 0 0
\(148\) 833.661i 0.0380597i
\(149\) 25735.7i 1.15921i 0.814896 + 0.579607i \(0.196794\pi\)
−0.814896 + 0.579607i \(0.803206\pi\)
\(150\) 0 0
\(151\) 13763.3i 0.603628i −0.953367 0.301814i \(-0.902408\pi\)
0.953367 0.301814i \(-0.0975921\pi\)
\(152\) 26497.1i 1.14686i
\(153\) 0 0
\(154\) −23974.7 −1.01091
\(155\) 56343.2i 2.34519i
\(156\) 0 0
\(157\) 5484.96i 0.222523i −0.993791 0.111261i \(-0.964511\pi\)
0.993791 0.111261i \(-0.0354891\pi\)
\(158\) 5104.33i 0.204467i
\(159\) 0 0
\(160\) 4237.19i 0.165515i
\(161\) 81934.7i 3.16094i
\(162\) 0 0
\(163\) 137.440 0.00517295 0.00258648 0.999997i \(-0.499177\pi\)
0.00258648 + 0.999997i \(0.499177\pi\)
\(164\) −852.968 −0.0317136
\(165\) 0 0
\(166\) 25076.8 0.910032
\(167\) 4707.17 0.168782 0.0843911 0.996433i \(-0.473105\pi\)
0.0843911 + 0.996433i \(0.473105\pi\)
\(168\) 0 0
\(169\) −36128.5 −1.26496
\(170\) 37331.7i 1.29176i
\(171\) 0 0
\(172\) 2163.73i 0.0731386i
\(173\) 25132.5i 0.839737i 0.907585 + 0.419869i \(0.137924\pi\)
−0.907585 + 0.419869i \(0.862076\pi\)
\(174\) 0 0
\(175\) −81148.3 −2.64974
\(176\) 18288.4i 0.590407i
\(177\) 0 0
\(178\) 49378.9 1.55848
\(179\) 16585.2i 0.517623i 0.965928 + 0.258811i \(0.0833308\pi\)
−0.965928 + 0.258811i \(0.916669\pi\)
\(180\) 0 0
\(181\) 51985.4 1.58681 0.793404 0.608695i \(-0.208307\pi\)
0.793404 + 0.608695i \(0.208307\pi\)
\(182\) −89583.6 −2.70449
\(183\) 0 0
\(184\) 59389.3 1.75417
\(185\) 39507.4i 1.15434i
\(186\) 0 0
\(187\) 15627.9i 0.446907i
\(188\) 1235.57i 0.0349584i
\(189\) 0 0
\(190\) 69244.8i 1.91814i
\(191\) 39194.8i 1.07439i −0.843458 0.537195i \(-0.819484\pi\)
0.843458 0.537195i \(-0.180516\pi\)
\(192\) 0 0
\(193\) 36220.3 0.972384 0.486192 0.873852i \(-0.338386\pi\)
0.486192 + 0.873852i \(0.338386\pi\)
\(194\) −69522.6 −1.84724
\(195\) 0 0
\(196\) −4153.80 −0.108127
\(197\) −37466.1 −0.965396 −0.482698 0.875787i \(-0.660343\pi\)
−0.482698 + 0.875787i \(0.660343\pi\)
\(198\) 0 0
\(199\) −76403.0 −1.92932 −0.964660 0.263497i \(-0.915124\pi\)
−0.964660 + 0.263497i \(0.915124\pi\)
\(200\) 58819.3i 1.47048i
\(201\) 0 0
\(202\) 35530.0 0.870748
\(203\) −34072.3 −0.826817
\(204\) 0 0
\(205\) 40422.4 0.961866
\(206\) 4856.93 0.114453
\(207\) 0 0
\(208\) 68336.4i 1.57952i
\(209\) 28987.4i 0.663616i
\(210\) 0 0
\(211\) 24399.1i 0.548036i 0.961725 + 0.274018i \(0.0883527\pi\)
−0.961725 + 0.274018i \(0.911647\pi\)
\(212\) −2639.05 −0.0587187
\(213\) 0 0
\(214\) 48870.2i 1.06713i
\(215\) 102540.i 2.21828i
\(216\) 0 0
\(217\) 122049.i 2.59188i
\(218\) −24138.7 −0.507926
\(219\) 0 0
\(220\) 2255.52i 0.0466016i
\(221\) 58395.0i 1.19561i
\(222\) 0 0
\(223\) −26520.0 −0.533290 −0.266645 0.963795i \(-0.585915\pi\)
−0.266645 + 0.963795i \(0.585915\pi\)
\(224\) 9178.46i 0.182925i
\(225\) 0 0
\(226\) −86963.9 −1.70264
\(227\) 71618.1i 1.38986i 0.719077 + 0.694930i \(0.244565\pi\)
−0.719077 + 0.694930i \(0.755435\pi\)
\(228\) 0 0
\(229\) 42669.2i 0.813660i −0.913504 0.406830i \(-0.866634\pi\)
0.913504 0.406830i \(-0.133366\pi\)
\(230\) 155202. 2.93388
\(231\) 0 0
\(232\) 24696.9i 0.458845i
\(233\) 68586.1i 1.26335i 0.775233 + 0.631676i \(0.217633\pi\)
−0.775233 + 0.631676i \(0.782367\pi\)
\(234\) 0 0
\(235\) 58554.1i 1.06028i
\(236\) 2910.76 + 13.8793i 0.0522616 + 0.000249197i
\(237\) 0 0
\(238\) 80866.8i 1.42763i
\(239\) −963.908 −0.0168749 −0.00843743 0.999964i \(-0.502686\pi\)
−0.00843743 + 0.999964i \(0.502686\pi\)
\(240\) 0 0
\(241\) 16329.4 0.281149 0.140574 0.990070i \(-0.455105\pi\)
0.140574 + 0.990070i \(0.455105\pi\)
\(242\) 41063.8i 0.701179i
\(243\) 0 0
\(244\) 1814.71i 0.0304809i
\(245\) 196850. 3.27947
\(246\) 0 0
\(247\) 108314.i 1.77538i
\(248\) −88465.6 −1.43837
\(249\) 0 0
\(250\) 52087.7i 0.833403i
\(251\) −57856.7 −0.918346 −0.459173 0.888347i \(-0.651854\pi\)
−0.459173 + 0.888347i \(0.651854\pi\)
\(252\) 0 0
\(253\) −64971.0 −1.01503
\(254\) 109968.i 1.70451i
\(255\) 0 0
\(256\) 10247.6 0.156366
\(257\) 94257.4 1.42708 0.713542 0.700613i \(-0.247090\pi\)
0.713542 + 0.700613i \(0.247090\pi\)
\(258\) 0 0
\(259\) 85579.8i 1.27577i
\(260\) 8427.95i 0.124674i
\(261\) 0 0
\(262\) 8040.79 0.117137
\(263\) −6388.77 −0.0923646 −0.0461823 0.998933i \(-0.514706\pi\)
−0.0461823 + 0.998933i \(0.514706\pi\)
\(264\) 0 0
\(265\) 125066. 1.78093
\(266\) 149996.i 2.11991i
\(267\) 0 0
\(268\) 558.288i 0.00777301i
\(269\) 50975.4i 0.704460i −0.935914 0.352230i \(-0.885424\pi\)
0.935914 0.352230i \(-0.114576\pi\)
\(270\) 0 0
\(271\) 103419. 1.40820 0.704098 0.710103i \(-0.251352\pi\)
0.704098 + 0.710103i \(0.251352\pi\)
\(272\) 61687.0 0.833789
\(273\) 0 0
\(274\) 14406.2i 0.191888i
\(275\) 64347.4i 0.850875i
\(276\) 0 0
\(277\) −118682. −1.54676 −0.773381 0.633941i \(-0.781436\pi\)
−0.773381 + 0.633941i \(0.781436\pi\)
\(278\) 34613.9i 0.447879i
\(279\) 0 0
\(280\) 211649.i 2.69961i
\(281\) 63690.0 0.806600 0.403300 0.915068i \(-0.367863\pi\)
0.403300 + 0.915068i \(0.367863\pi\)
\(282\) 0 0
\(283\) 7522.96i 0.0939325i −0.998896 0.0469662i \(-0.985045\pi\)
0.998896 0.0469662i \(-0.0149553\pi\)
\(284\) −2162.66 −0.0268133
\(285\) 0 0
\(286\) 71036.3i 0.868457i
\(287\) −87561.8 −1.06304
\(288\) 0 0
\(289\) −30808.0 −0.368866
\(290\) 64540.3i 0.767423i
\(291\) 0 0
\(292\) 1003.98i 0.0117750i
\(293\) 32964.4 0.383981 0.191990 0.981397i \(-0.438506\pi\)
0.191990 + 0.981397i \(0.438506\pi\)
\(294\) 0 0
\(295\) −137942. 657.742i −1.58508 0.00755809i
\(296\) 62031.4 0.707992
\(297\) 0 0
\(298\) 105599. 1.18912
\(299\) −242770. −2.71552
\(300\) 0 0
\(301\) 222119.i 2.45162i
\(302\) −56473.5 −0.619200
\(303\) 0 0
\(304\) −114420. −1.23810
\(305\) 85999.7i 0.924480i
\(306\) 0 0
\(307\) −85864.5 −0.911039 −0.455520 0.890226i \(-0.650547\pi\)
−0.455520 + 0.890226i \(0.650547\pi\)
\(308\) 4885.84i 0.0515036i
\(309\) 0 0
\(310\) −231187. −2.40569
\(311\) 20927.4 0.216368 0.108184 0.994131i \(-0.465496\pi\)
0.108184 + 0.994131i \(0.465496\pi\)
\(312\) 0 0
\(313\) 132516.i 1.35264i 0.736610 + 0.676318i \(0.236425\pi\)
−0.736610 + 0.676318i \(0.763575\pi\)
\(314\) −22505.9 −0.228263
\(315\) 0 0
\(316\) 1040.22 0.0104172
\(317\) 108659. 1.08130 0.540649 0.841248i \(-0.318179\pi\)
0.540649 + 0.841248i \(0.318179\pi\)
\(318\) 0 0
\(319\) 27018.0i 0.265504i
\(320\) −152968. −1.49383
\(321\) 0 0
\(322\) −336194. −3.24249
\(323\) −97774.8 −0.937177
\(324\) 0 0
\(325\) 240440.i 2.27635i
\(326\) 563.944i 0.00530641i
\(327\) 0 0
\(328\) 63468.1i 0.589940i
\(329\) 126838.i 1.17181i
\(330\) 0 0
\(331\) −101850. −0.929619 −0.464809 0.885411i \(-0.653877\pi\)
−0.464809 + 0.885411i \(0.653877\pi\)
\(332\) 5110.44i 0.0463641i
\(333\) 0 0
\(334\) 19314.4i 0.173137i
\(335\) 26457.5i 0.235754i
\(336\) 0 0
\(337\) 74871.2i 0.659258i 0.944111 + 0.329629i \(0.106924\pi\)
−0.944111 + 0.329629i \(0.893076\pi\)
\(338\) 148242.i 1.29759i
\(339\) 0 0
\(340\) 7607.88 0.0658121
\(341\) 96780.0 0.832294
\(342\) 0 0
\(343\) −220309. −1.87259
\(344\) 161000. 1.36053
\(345\) 0 0
\(346\) 103123. 0.861401
\(347\) 90583.5i 0.752298i −0.926559 0.376149i \(-0.877248\pi\)
0.926559 0.376149i \(-0.122752\pi\)
\(348\) 0 0
\(349\) 134492.i 1.10420i −0.833779 0.552098i \(-0.813828\pi\)
0.833779 0.552098i \(-0.186172\pi\)
\(350\) 332967.i 2.71810i
\(351\) 0 0
\(352\) −7278.16 −0.0587403
\(353\) 81414.3i 0.653358i 0.945135 + 0.326679i \(0.105930\pi\)
−0.945135 + 0.326679i \(0.894070\pi\)
\(354\) 0 0
\(355\) 102489. 0.813243
\(356\) 10063.0i 0.0794013i
\(357\) 0 0
\(358\) 68052.1 0.530977
\(359\) −64385.0 −0.499570 −0.249785 0.968301i \(-0.580360\pi\)
−0.249785 + 0.968301i \(0.580360\pi\)
\(360\) 0 0
\(361\) 51036.8 0.391624
\(362\) 213306.i 1.62775i
\(363\) 0 0
\(364\) 18256.4i 0.137788i
\(365\) 47579.1i 0.357133i
\(366\) 0 0
\(367\) 129178.i 0.959083i −0.877519 0.479541i \(-0.840803\pi\)
0.877519 0.479541i \(-0.159197\pi\)
\(368\) 256456.i 1.89373i
\(369\) 0 0
\(370\) 162107. 1.18412
\(371\) −270913. −1.96826
\(372\) 0 0
\(373\) −71866.8 −0.516548 −0.258274 0.966072i \(-0.583154\pi\)
−0.258274 + 0.966072i \(0.583154\pi\)
\(374\) −64124.3 −0.458436
\(375\) 0 0
\(376\) 91936.9 0.650301
\(377\) 100955.i 0.710306i
\(378\) 0 0
\(379\) 85467.7 0.595009 0.297504 0.954720i \(-0.403846\pi\)
0.297504 + 0.954720i \(0.403846\pi\)
\(380\) −14111.5 −0.0977251
\(381\) 0 0
\(382\) −160824. −1.10211
\(383\) −278371. −1.89769 −0.948846 0.315739i \(-0.897747\pi\)
−0.948846 + 0.315739i \(0.897747\pi\)
\(384\) 0 0
\(385\) 231541.i 1.56209i
\(386\) 148619.i 0.997470i
\(387\) 0 0
\(388\) 14168.1i 0.0941127i
\(389\) 21226.2 0.140273 0.0701364 0.997537i \(-0.477657\pi\)
0.0701364 + 0.997537i \(0.477657\pi\)
\(390\) 0 0
\(391\) 219148.i 1.43345i
\(392\) 309078.i 2.01139i
\(393\) 0 0
\(394\) 153731.i 0.990302i
\(395\) −49296.2 −0.315951
\(396\) 0 0
\(397\) 45611.1i 0.289394i −0.989476 0.144697i \(-0.953779\pi\)
0.989476 0.144697i \(-0.0462208\pi\)
\(398\) 313496.i 1.97909i
\(399\) 0 0
\(400\) −253995. −1.58747
\(401\) 56817.4i 0.353340i 0.984270 + 0.176670i \(0.0565325\pi\)
−0.984270 + 0.176670i \(0.943468\pi\)
\(402\) 0 0
\(403\) 361627. 2.22664
\(404\) 7240.70i 0.0443627i
\(405\) 0 0
\(406\) 139805.i 0.848147i
\(407\) −67861.5 −0.409670
\(408\) 0 0
\(409\) 152246.i 0.910123i −0.890460 0.455061i \(-0.849617\pi\)
0.890460 0.455061i \(-0.150383\pi\)
\(410\) 165861.i 0.986681i
\(411\) 0 0
\(412\) 989.801i 0.00583114i
\(413\) 298806. + 1424.78i 1.75182 + 0.00835311i
\(414\) 0 0
\(415\) 242185.i 1.40622i
\(416\) −27195.5 −0.157149
\(417\) 0 0
\(418\) 118941. 0.680737
\(419\) 14956.8i 0.0851945i 0.999092 + 0.0425972i \(0.0135632\pi\)
−0.999092 + 0.0425972i \(0.986437\pi\)
\(420\) 0 0
\(421\) 192146.i 1.08410i −0.840347 0.542048i \(-0.817649\pi\)
0.840347 0.542048i \(-0.182351\pi\)
\(422\) 100114. 0.562174
\(423\) 0 0
\(424\) 196368.i 1.09229i
\(425\) −217044. −1.20163
\(426\) 0 0
\(427\) 186290.i 1.02172i
\(428\) 9959.31 0.0543678
\(429\) 0 0
\(430\) 420741. 2.27551
\(431\) 134989.i 0.726679i 0.931657 + 0.363340i \(0.118364\pi\)
−0.931657 + 0.363340i \(0.881636\pi\)
\(432\) 0 0
\(433\) −27732.6 −0.147916 −0.0739580 0.997261i \(-0.523563\pi\)
−0.0739580 + 0.997261i \(0.523563\pi\)
\(434\) 500790. 2.65874
\(435\) 0 0
\(436\) 4919.25i 0.0258777i
\(437\) 406487.i 2.12855i
\(438\) 0 0
\(439\) −256732. −1.33215 −0.666073 0.745887i \(-0.732026\pi\)
−0.666073 + 0.745887i \(0.732026\pi\)
\(440\) 167830. 0.866889
\(441\) 0 0
\(442\) −239606. −1.22646
\(443\) 104213.i 0.531022i 0.964108 + 0.265511i \(0.0855407\pi\)
−0.964108 + 0.265511i \(0.914459\pi\)
\(444\) 0 0
\(445\) 476889.i 2.40822i
\(446\) 108817.i 0.547048i
\(447\) 0 0
\(448\) 331355. 1.65096
\(449\) −15237.3 −0.0755814 −0.0377907 0.999286i \(-0.512032\pi\)
−0.0377907 + 0.999286i \(0.512032\pi\)
\(450\) 0 0
\(451\) 69433.1i 0.341361i
\(452\) 17722.5i 0.0867457i
\(453\) 0 0
\(454\) 293863. 1.42572
\(455\) 865175.i 4.17908i
\(456\) 0 0
\(457\) 351744.i 1.68420i 0.539318 + 0.842102i \(0.318682\pi\)
−0.539318 + 0.842102i \(0.681318\pi\)
\(458\) −175080. −0.834651
\(459\) 0 0
\(460\) 31628.8i 0.149475i
\(461\) 91900.9 0.432432 0.216216 0.976346i \(-0.430628\pi\)
0.216216 + 0.976346i \(0.430628\pi\)
\(462\) 0 0
\(463\) 284633.i 1.32777i −0.747835 0.663885i \(-0.768907\pi\)
0.747835 0.663885i \(-0.231093\pi\)
\(464\) −106647. −0.495348
\(465\) 0 0
\(466\) 281422. 1.29594
\(467\) 304437.i 1.39593i −0.716132 0.697965i \(-0.754089\pi\)
0.716132 0.697965i \(-0.245911\pi\)
\(468\) 0 0
\(469\) 57311.3i 0.260552i
\(470\) 240259. 1.08764
\(471\) 0 0
\(472\) −1032.73 + 216585.i −0.00463559 + 0.972177i
\(473\) −176132. −0.787254
\(474\) 0 0
\(475\) 402585. 1.78431
\(476\) −16480.0 −0.0727348
\(477\) 0 0
\(478\) 3955.10i 0.0173102i
\(479\) −397464. −1.73231 −0.866157 0.499772i \(-0.833417\pi\)
−0.866157 + 0.499772i \(0.833417\pi\)
\(480\) 0 0
\(481\) −253570. −1.09599
\(482\) 67002.7i 0.288402i
\(483\) 0 0
\(484\) −8368.46 −0.0357235
\(485\) 671431.i 2.85442i
\(486\) 0 0
\(487\) −157762. −0.665187 −0.332593 0.943070i \(-0.607924\pi\)
−0.332593 + 0.943070i \(0.607924\pi\)
\(488\) 135030. 0.567010
\(489\) 0 0
\(490\) 807714.i 3.36407i
\(491\) −76943.9 −0.319162 −0.159581 0.987185i \(-0.551014\pi\)
−0.159581 + 0.987185i \(0.551014\pi\)
\(492\) 0 0
\(493\) −91131.9 −0.374953
\(494\) 444434. 1.82118
\(495\) 0 0
\(496\) 382014.i 1.55280i
\(497\) −222008. −0.898787
\(498\) 0 0
\(499\) 1118.26 0.00449100 0.00224550 0.999997i \(-0.499285\pi\)
0.00224550 + 0.999997i \(0.499285\pi\)
\(500\) −10615.0 −0.0424601
\(501\) 0 0
\(502\) 237397.i 0.942038i
\(503\) 143292.i 0.566351i −0.959068 0.283176i \(-0.908612\pi\)
0.959068 0.283176i \(-0.0913879\pi\)
\(504\) 0 0
\(505\) 343139.i 1.34551i
\(506\) 266589.i 1.04122i
\(507\) 0 0
\(508\) −22410.5 −0.0868410
\(509\) 39802.5i 0.153630i 0.997045 + 0.0768148i \(0.0244750\pi\)
−0.997045 + 0.0768148i \(0.975525\pi\)
\(510\) 0 0
\(511\) 103064.i 0.394699i
\(512\) 238748.i 0.910750i
\(513\) 0 0
\(514\) 386756.i 1.46390i
\(515\) 46907.0i 0.176857i
\(516\) 0 0
\(517\) −100578. −0.376288
\(518\) −351150. −1.30868
\(519\) 0 0
\(520\) 627111. 2.31920
\(521\) 228531. 0.841917 0.420958 0.907080i \(-0.361694\pi\)
0.420958 + 0.907080i \(0.361694\pi\)
\(522\) 0 0
\(523\) 508793. 1.86011 0.930054 0.367423i \(-0.119760\pi\)
0.930054 + 0.367423i \(0.119760\pi\)
\(524\) 1638.64i 0.00596790i
\(525\) 0 0
\(526\) 26214.4i 0.0947475i
\(527\) 326440.i 1.17539i
\(528\) 0 0
\(529\) −631238. −2.25570
\(530\) 513168.i 1.82687i
\(531\) 0 0
\(532\) 30567.9 0.108005
\(533\) 259443.i 0.913246i
\(534\) 0 0
\(535\) −471975. −1.64897
\(536\) 41541.4 0.144594
\(537\) 0 0
\(538\) −209162. −0.722634
\(539\) 338127.i 1.16386i
\(540\) 0 0
\(541\) 560174.i 1.91394i 0.290187 + 0.956970i \(0.406282\pi\)
−0.290187 + 0.956970i \(0.593718\pi\)
\(542\) 424349.i 1.44452i
\(543\) 0 0
\(544\) 24549.3i 0.0829547i
\(545\) 233125.i 0.784867i
\(546\) 0 0
\(547\) −273979. −0.915679 −0.457839 0.889035i \(-0.651377\pi\)
−0.457839 + 0.889035i \(0.651377\pi\)
\(548\) −2935.85 −0.00977627
\(549\) 0 0
\(550\) 264030. 0.872826
\(551\) 169036. 0.556771
\(552\) 0 0
\(553\) 106784. 0.349185
\(554\) 486973.i 1.58667i
\(555\) 0 0
\(556\) 7054.02 0.0228185
\(557\) −339532. −1.09439 −0.547193 0.837006i \(-0.684304\pi\)
−0.547193 + 0.837006i \(0.684304\pi\)
\(558\) 0 0
\(559\) −658131. −2.10615
\(560\) 913949. 2.91438
\(561\) 0 0
\(562\) 261332.i 0.827409i
\(563\) 92097.7i 0.290558i 0.989391 + 0.145279i \(0.0464079\pi\)
−0.989391 + 0.145279i \(0.953592\pi\)
\(564\) 0 0
\(565\) 839874.i 2.63098i
\(566\) −30868.1 −0.0963558
\(567\) 0 0
\(568\) 160920.i 0.498785i
\(569\) 443913.i 1.37111i −0.728019 0.685557i \(-0.759559\pi\)
0.728019 0.685557i \(-0.240441\pi\)
\(570\) 0 0
\(571\) 254417.i 0.780321i 0.920747 + 0.390161i \(0.127581\pi\)
−0.920747 + 0.390161i \(0.872419\pi\)
\(572\) 14476.6 0.0442460
\(573\) 0 0
\(574\) 359283.i 1.09047i
\(575\) 902335.i 2.72918i
\(576\) 0 0
\(577\) 311437. 0.935445 0.467723 0.883875i \(-0.345075\pi\)
0.467723 + 0.883875i \(0.345075\pi\)
\(578\) 126411.i 0.378382i
\(579\) 0 0
\(580\) −13152.8 −0.0390985
\(581\) 524615.i 1.55413i
\(582\) 0 0
\(583\) 214824.i 0.632041i
\(584\) 74704.8 0.219040
\(585\) 0 0
\(586\) 135259.i 0.393887i
\(587\) 581088.i 1.68642i 0.537584 + 0.843210i \(0.319337\pi\)
−0.537584 + 0.843210i \(0.680663\pi\)
\(588\) 0 0
\(589\) 605497.i 1.74535i
\(590\) −2698.84 + 566003.i −0.00775307 + 1.62598i
\(591\) 0 0
\(592\) 267866.i 0.764317i
\(593\) 218284. 0.620743 0.310371 0.950615i \(-0.399547\pi\)
0.310371 + 0.950615i \(0.399547\pi\)
\(594\) 0 0
\(595\) 780990. 2.20603
\(596\) 21520.1i 0.0605830i
\(597\) 0 0
\(598\) 996132.i 2.78557i
\(599\) 549532. 1.53158 0.765789 0.643091i \(-0.222349\pi\)
0.765789 + 0.643091i \(0.222349\pi\)
\(600\) 0 0
\(601\) 611697.i 1.69351i −0.531984 0.846755i \(-0.678553\pi\)
0.531984 0.846755i \(-0.321447\pi\)
\(602\) −911397. −2.51486
\(603\) 0 0
\(604\) 11508.8i 0.0315469i
\(605\) 396584. 1.08349
\(606\) 0 0
\(607\) 1086.59 0.00294908 0.00147454 0.999999i \(-0.499531\pi\)
0.00147454 + 0.999999i \(0.499531\pi\)
\(608\) 45535.3i 0.123180i
\(609\) 0 0
\(610\) 352874. 0.948330
\(611\) −375817. −1.00669
\(612\) 0 0
\(613\) 331052.i 0.880999i −0.897753 0.440500i \(-0.854801\pi\)
0.897753 0.440500i \(-0.145199\pi\)
\(614\) 352319.i 0.934543i
\(615\) 0 0
\(616\) −363548. −0.958076
\(617\) 247255. 0.649492 0.324746 0.945801i \(-0.394721\pi\)
0.324746 + 0.945801i \(0.394721\pi\)
\(618\) 0 0
\(619\) 307344. 0.802127 0.401064 0.916050i \(-0.368641\pi\)
0.401064 + 0.916050i \(0.368641\pi\)
\(620\) 47113.9i 0.122565i
\(621\) 0 0
\(622\) 85869.1i 0.221950i
\(623\) 1.03302e6i 2.66154i
\(624\) 0 0
\(625\) −87790.4 −0.224744
\(626\) 543740. 1.38753
\(627\) 0 0
\(628\) 4586.50i 0.0116295i
\(629\) 228897.i 0.578548i
\(630\) 0 0
\(631\) 469556. 1.17931 0.589657 0.807654i \(-0.299263\pi\)
0.589657 + 0.807654i \(0.299263\pi\)
\(632\) 77401.0i 0.193782i
\(633\) 0 0
\(634\) 445847.i 1.10919i
\(635\) 1.06204e6 2.63387
\(636\) 0 0
\(637\) 1.26344e6i 3.11370i
\(638\) 110860. 0.272354
\(639\) 0 0
\(640\) 695453.i 1.69788i
\(641\) −244837. −0.595882 −0.297941 0.954584i \(-0.596300\pi\)
−0.297941 + 0.954584i \(0.596300\pi\)
\(642\) 0 0
\(643\) −199751. −0.483133 −0.241567 0.970384i \(-0.577661\pi\)
−0.241567 + 0.970384i \(0.577661\pi\)
\(644\) 68513.4i 0.165198i
\(645\) 0 0
\(646\) 401189.i 0.961355i
\(647\) −406276. −0.970538 −0.485269 0.874365i \(-0.661278\pi\)
−0.485269 + 0.874365i \(0.661278\pi\)
\(648\) 0 0
\(649\) 1129.80 236941.i 0.00268232 0.562537i
\(650\) 986571. 2.33508
\(651\) 0 0
\(652\) −114.927 −0.000270350
\(653\) −334544. −0.784562 −0.392281 0.919845i \(-0.628314\pi\)
−0.392281 + 0.919845i \(0.628314\pi\)
\(654\) 0 0
\(655\) 77655.8i 0.181005i
\(656\) −274069. −0.636873
\(657\) 0 0
\(658\) −520441. −1.20204
\(659\) 286185.i 0.658986i 0.944158 + 0.329493i \(0.106878\pi\)
−0.944158 + 0.329493i \(0.893122\pi\)
\(660\) 0 0
\(661\) 759053. 1.73728 0.868639 0.495446i \(-0.164995\pi\)
0.868639 + 0.495446i \(0.164995\pi\)
\(662\) 417910.i 0.953601i
\(663\) 0 0
\(664\) 380260. 0.862471
\(665\) −1.44862e6 −3.27576
\(666\) 0 0
\(667\) 378870.i 0.851605i
\(668\) −3936.11 −0.00882093
\(669\) 0 0
\(670\) 108560. 0.241836
\(671\) −147721. −0.328093
\(672\) 0 0
\(673\) 664503.i 1.46712i −0.679622 0.733562i \(-0.737856\pi\)
0.679622 0.733562i \(-0.262144\pi\)
\(674\) 307211. 0.676265
\(675\) 0 0
\(676\) 30210.5 0.0661095
\(677\) 252653. 0.551249 0.275624 0.961265i \(-0.411115\pi\)
0.275624 + 0.961265i \(0.411115\pi\)
\(678\) 0 0
\(679\) 1.45443e6i 3.15467i
\(680\) 566091.i 1.22425i
\(681\) 0 0
\(682\) 397107.i 0.853766i
\(683\) 506823.i 1.08646i −0.839583 0.543232i \(-0.817200\pi\)
0.839583 0.543232i \(-0.182800\pi\)
\(684\) 0 0
\(685\) 139131. 0.296512
\(686\) 903969.i 1.92090i
\(687\) 0 0
\(688\) 695234.i 1.46877i
\(689\) 802708.i 1.69090i
\(690\) 0 0
\(691\) 789140.i 1.65271i −0.563146 0.826357i \(-0.690409\pi\)
0.563146 0.826357i \(-0.309591\pi\)
\(692\) 21015.7i 0.0438865i
\(693\) 0 0
\(694\) −371682. −0.771706
\(695\) −334292. −0.692080
\(696\) 0 0
\(697\) −234198. −0.482079
\(698\) −551847. −1.13268
\(699\) 0 0
\(700\) 67855.8 0.138481
\(701\) 259759.i 0.528608i −0.964439 0.264304i \(-0.914858\pi\)
0.964439 0.264304i \(-0.0851423\pi\)
\(702\) 0 0
\(703\) 424570.i 0.859091i
\(704\) 262751.i 0.530151i
\(705\) 0 0
\(706\) 334059. 0.670213
\(707\) 743298.i 1.48704i
\(708\) 0 0
\(709\) 171973. 0.342112 0.171056 0.985261i \(-0.445282\pi\)
0.171056 + 0.985261i \(0.445282\pi\)
\(710\) 420532.i 0.834223i
\(711\) 0 0
\(712\) 748772. 1.47703
\(713\) 1.35713e6 2.66958
\(714\) 0 0
\(715\) −686050. −1.34197
\(716\) 13868.4i 0.0270521i
\(717\) 0 0
\(718\) 264184.i 0.512458i
\(719\) 823365.i 1.59270i 0.604835 + 0.796351i \(0.293239\pi\)
−0.604835 + 0.796351i \(0.706761\pi\)
\(720\) 0 0
\(721\) 101608.i 0.195461i
\(722\) 209414.i 0.401727i
\(723\) 0 0
\(724\) −43470.0 −0.0829301
\(725\) 375233. 0.713880
\(726\) 0 0
\(727\) −403704. −0.763825 −0.381912 0.924198i \(-0.624734\pi\)
−0.381912 + 0.924198i \(0.624734\pi\)
\(728\) −1.35843e6 −2.56315
\(729\) 0 0
\(730\) 195226. 0.366347
\(731\) 594093.i 1.11178i
\(732\) 0 0
\(733\) −472977. −0.880302 −0.440151 0.897924i \(-0.645075\pi\)
−0.440151 + 0.897924i \(0.645075\pi\)
\(734\) −530042. −0.983825
\(735\) 0 0
\(736\) −102061. −0.188409
\(737\) −45445.7 −0.0836676
\(738\) 0 0
\(739\) 107013.i 0.195951i 0.995189 + 0.0979757i \(0.0312368\pi\)
−0.995189 + 0.0979757i \(0.968763\pi\)
\(740\) 33035.9i 0.0603286i
\(741\) 0 0
\(742\) 1.11161e6i 2.01904i
\(743\) 915468. 1.65831 0.829155 0.559019i \(-0.188822\pi\)
0.829155 + 0.559019i \(0.188822\pi\)
\(744\) 0 0
\(745\) 1.01984e6i 1.83747i
\(746\) 294883.i 0.529874i
\(747\) 0 0
\(748\) 13068.0i 0.0233563i
\(749\) 1.02238e6 1.82242
\(750\) 0 0
\(751\) 832564.i 1.47617i −0.674705 0.738087i \(-0.735729\pi\)
0.674705 0.738087i \(-0.264271\pi\)
\(752\) 397004.i 0.702036i
\(753\) 0 0
\(754\) 414238. 0.728631
\(755\) 545406.i 0.956811i
\(756\) 0 0
\(757\) 247540. 0.431970 0.215985 0.976397i \(-0.430704\pi\)
0.215985 + 0.976397i \(0.430704\pi\)
\(758\) 350690.i 0.610359i
\(759\) 0 0
\(760\) 1.05002e6i 1.81789i
\(761\) 539545. 0.931662 0.465831 0.884874i \(-0.345756\pi\)
0.465831 + 0.884874i \(0.345756\pi\)
\(762\) 0 0
\(763\) 504988.i 0.867425i
\(764\) 32774.5i 0.0561500i
\(765\) 0 0
\(766\) 1.14221e6i 1.94665i
\(767\) 4221.58 885352.i 0.00717604 1.50496i
\(768\) 0 0
\(769\) 97270.6i 0.164486i 0.996612 + 0.0822430i \(0.0262084\pi\)
−0.996612 + 0.0822430i \(0.973792\pi\)
\(770\) −950059. −1.60239
\(771\) 0 0
\(772\) −30287.3 −0.0508189
\(773\) 234313.i 0.392136i 0.980590 + 0.196068i \(0.0628173\pi\)
−0.980590 + 0.196068i \(0.937183\pi\)
\(774\) 0 0
\(775\) 1.34411e6i 2.23785i
\(776\) −1.05423e6 −1.75069
\(777\) 0 0
\(778\) 87095.3i 0.143892i
\(779\) 434403. 0.715844
\(780\) 0 0
\(781\) 176044.i 0.288615i
\(782\) −899205. −1.47043
\(783\) 0 0
\(784\) −1.33467e6 −2.17141
\(785\) 217356.i 0.352721i
\(786\) 0 0
\(787\) 773520. 1.24888 0.624442 0.781071i \(-0.285327\pi\)
0.624442 + 0.781071i \(0.285327\pi\)
\(788\) 31328.9 0.0504537
\(789\) 0 0
\(790\) 202272.i 0.324102i
\(791\) 1.81931e6i 2.90773i
\(792\) 0 0
\(793\) −551972. −0.877749
\(794\) −187151. −0.296860
\(795\) 0 0
\(796\) 63887.8 0.100831
\(797\) 254532.i 0.400705i