Properties

Label 531.5.c.d.235.8
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 235.8
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.33

$q$-expansion

\(f(q)\) \(=\) \(q-4.96663i q^{2} -8.66741 q^{4} -41.3974 q^{5} +1.08721 q^{7} -36.4182i q^{8} +O(q^{10})\) \(q-4.96663i q^{2} -8.66741 q^{4} -41.3974 q^{5} +1.08721 q^{7} -36.4182i q^{8} +205.606i q^{10} +142.708i q^{11} +35.2627i q^{13} -5.39978i q^{14} -319.555 q^{16} -527.661 q^{17} +76.0375 q^{19} +358.808 q^{20} +708.780 q^{22} -138.991i q^{23} +1088.75 q^{25} +175.137 q^{26} -9.42332 q^{28} +1044.28 q^{29} +1364.14i q^{31} +1004.42i q^{32} +2620.69i q^{34} -45.0078 q^{35} -2076.33i q^{37} -377.650i q^{38} +1507.62i q^{40} +1604.49 q^{41} +746.077i q^{43} -1236.91i q^{44} -690.318 q^{46} +613.862i q^{47} -2399.82 q^{49} -5407.39i q^{50} -305.636i q^{52} +3306.58 q^{53} -5907.75i q^{55} -39.5944i q^{56} -5186.56i q^{58} +(-330.884 - 3465.24i) q^{59} -3021.56i q^{61} +6775.17 q^{62} -124.304 q^{64} -1459.78i q^{65} +3679.74i q^{67} +4573.45 q^{68} +223.537i q^{70} -1827.67 q^{71} +2692.51i q^{73} -10312.4 q^{74} -659.048 q^{76} +155.154i q^{77} +3087.31 q^{79} +13228.7 q^{80} -7968.90i q^{82} -7955.09i q^{83} +21843.8 q^{85} +3705.49 q^{86} +5197.19 q^{88} +5141.10i q^{89} +38.3380i q^{91} +1204.70i q^{92} +3048.82 q^{94} -3147.75 q^{95} -13063.6i q^{97} +11919.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.96663i 1.24166i −0.783946 0.620829i \(-0.786796\pi\)
0.783946 0.620829i \(-0.213204\pi\)
\(3\) 0 0
\(4\) −8.66741 −0.541713
\(5\) −41.3974 −1.65590 −0.827948 0.560805i \(-0.810492\pi\)
−0.827948 + 0.560805i \(0.810492\pi\)
\(6\) 0 0
\(7\) 1.08721 0.0221880 0.0110940 0.999938i \(-0.496469\pi\)
0.0110940 + 0.999938i \(0.496469\pi\)
\(8\) 36.4182i 0.569035i
\(9\) 0 0
\(10\) 205.606i 2.05606i
\(11\) 142.708i 1.17941i 0.807620 + 0.589704i \(0.200756\pi\)
−0.807620 + 0.589704i \(0.799244\pi\)
\(12\) 0 0
\(13\) 35.2627i 0.208655i 0.994543 + 0.104327i \(0.0332690\pi\)
−0.994543 + 0.104327i \(0.966731\pi\)
\(14\) 5.39978i 0.0275499i
\(15\) 0 0
\(16\) −319.555 −1.24826
\(17\) −527.661 −1.82581 −0.912907 0.408167i \(-0.866168\pi\)
−0.912907 + 0.408167i \(0.866168\pi\)
\(18\) 0 0
\(19\) 76.0375 0.210630 0.105315 0.994439i \(-0.466415\pi\)
0.105315 + 0.994439i \(0.466415\pi\)
\(20\) 358.808 0.897021
\(21\) 0 0
\(22\) 708.780 1.46442
\(23\) 138.991i 0.262744i −0.991333 0.131372i \(-0.958062\pi\)
0.991333 0.131372i \(-0.0419382\pi\)
\(24\) 0 0
\(25\) 1088.75 1.74199
\(26\) 175.137 0.259078
\(27\) 0 0
\(28\) −9.42332 −0.0120195
\(29\) 1044.28 1.24171 0.620857 0.783924i \(-0.286785\pi\)
0.620857 + 0.783924i \(0.286785\pi\)
\(30\) 0 0
\(31\) 1364.14i 1.41950i 0.704454 + 0.709749i \(0.251192\pi\)
−0.704454 + 0.709749i \(0.748808\pi\)
\(32\) 1004.42i 0.980876i
\(33\) 0 0
\(34\) 2620.69i 2.26704i
\(35\) −45.0078 −0.0367410
\(36\) 0 0
\(37\) 2076.33i 1.51668i −0.651860 0.758339i \(-0.726011\pi\)
0.651860 0.758339i \(-0.273989\pi\)
\(38\) 377.650i 0.261530i
\(39\) 0 0
\(40\) 1507.62i 0.942263i
\(41\) 1604.49 0.954485 0.477242 0.878772i \(-0.341636\pi\)
0.477242 + 0.878772i \(0.341636\pi\)
\(42\) 0 0
\(43\) 746.077i 0.403503i 0.979437 + 0.201751i \(0.0646633\pi\)
−0.979437 + 0.201751i \(0.935337\pi\)
\(44\) 1236.91i 0.638901i
\(45\) 0 0
\(46\) −690.318 −0.326237
\(47\) 613.862i 0.277891i 0.990300 + 0.138946i \(0.0443713\pi\)
−0.990300 + 0.138946i \(0.955629\pi\)
\(48\) 0 0
\(49\) −2399.82 −0.999508
\(50\) 5407.39i 2.16296i
\(51\) 0 0
\(52\) 305.636i 0.113031i
\(53\) 3306.58 1.17714 0.588569 0.808447i \(-0.299692\pi\)
0.588569 + 0.808447i \(0.299692\pi\)
\(54\) 0 0
\(55\) 5907.75i 1.95298i
\(56\) 39.5944i 0.0126258i
\(57\) 0 0
\(58\) 5186.56i 1.54178i
\(59\) −330.884 3465.24i −0.0950543 0.995472i
\(60\) 0 0
\(61\) 3021.56i 0.812028i −0.913867 0.406014i \(-0.866918\pi\)
0.913867 0.406014i \(-0.133082\pi\)
\(62\) 6775.17 1.76253
\(63\) 0 0
\(64\) −124.304 −0.0303476
\(65\) 1459.78i 0.345511i
\(66\) 0 0
\(67\) 3679.74i 0.819724i 0.912148 + 0.409862i \(0.134423\pi\)
−0.912148 + 0.409862i \(0.865577\pi\)
\(68\) 4573.45 0.989068
\(69\) 0 0
\(70\) 223.537i 0.0456198i
\(71\) −1827.67 −0.362560 −0.181280 0.983431i \(-0.558024\pi\)
−0.181280 + 0.983431i \(0.558024\pi\)
\(72\) 0 0
\(73\) 2692.51i 0.505256i 0.967563 + 0.252628i \(0.0812949\pi\)
−0.967563 + 0.252628i \(0.918705\pi\)
\(74\) −10312.4 −1.88320
\(75\) 0 0
\(76\) −659.048 −0.114101
\(77\) 155.154i 0.0261687i
\(78\) 0 0
\(79\) 3087.31 0.494682 0.247341 0.968928i \(-0.420443\pi\)
0.247341 + 0.968928i \(0.420443\pi\)
\(80\) 13228.7 2.06699
\(81\) 0 0
\(82\) 7968.90i 1.18514i
\(83\) 7955.09i 1.15475i −0.816478 0.577377i \(-0.804076\pi\)
0.816478 0.577377i \(-0.195924\pi\)
\(84\) 0 0
\(85\) 21843.8 3.02336
\(86\) 3705.49 0.501012
\(87\) 0 0
\(88\) 5197.19 0.671124
\(89\) 5141.10i 0.649047i 0.945878 + 0.324524i \(0.105204\pi\)
−0.945878 + 0.324524i \(0.894796\pi\)
\(90\) 0 0
\(91\) 38.3380i 0.00462964i
\(92\) 1204.70i 0.142332i
\(93\) 0 0
\(94\) 3048.82 0.345046
\(95\) −3147.75 −0.348782
\(96\) 0 0
\(97\) 13063.6i 1.38842i −0.719772 0.694210i \(-0.755754\pi\)
0.719772 0.694210i \(-0.244246\pi\)
\(98\) 11919.0i 1.24105i
\(99\) 0 0
\(100\) −9436.60 −0.943660
\(101\) 6212.97i 0.609055i −0.952504 0.304527i \(-0.901502\pi\)
0.952504 0.304527i \(-0.0984985\pi\)
\(102\) 0 0
\(103\) 12027.0i 1.13366i −0.823834 0.566831i \(-0.808169\pi\)
0.823834 0.566831i \(-0.191831\pi\)
\(104\) 1284.21 0.118732
\(105\) 0 0
\(106\) 16422.6i 1.46160i
\(107\) 5915.89 0.516717 0.258359 0.966049i \(-0.416818\pi\)
0.258359 + 0.966049i \(0.416818\pi\)
\(108\) 0 0
\(109\) 17887.0i 1.50552i −0.658298 0.752758i \(-0.728723\pi\)
0.658298 0.752758i \(-0.271277\pi\)
\(110\) −29341.6 −2.42493
\(111\) 0 0
\(112\) −347.424 −0.0276964
\(113\) 24933.1i 1.95262i 0.216368 + 0.976312i \(0.430579\pi\)
−0.216368 + 0.976312i \(0.569421\pi\)
\(114\) 0 0
\(115\) 5753.88i 0.435076i
\(116\) −9051.23 −0.672653
\(117\) 0 0
\(118\) −17210.6 + 1643.38i −1.23604 + 0.118025i
\(119\) −573.679 −0.0405112
\(120\) 0 0
\(121\) −5724.67 −0.391003
\(122\) −15007.0 −1.00826
\(123\) 0 0
\(124\) 11823.6i 0.768962i
\(125\) −19197.8 −1.22866
\(126\) 0 0
\(127\) 22413.1 1.38961 0.694806 0.719197i \(-0.255490\pi\)
0.694806 + 0.719197i \(0.255490\pi\)
\(128\) 16688.0i 1.01856i
\(129\) 0 0
\(130\) −7250.21 −0.429006
\(131\) 17802.3i 1.03737i −0.854965 0.518685i \(-0.826422\pi\)
0.854965 0.518685i \(-0.173578\pi\)
\(132\) 0 0
\(133\) 82.6689 0.00467346
\(134\) 18275.9 1.01782
\(135\) 0 0
\(136\) 19216.5i 1.03895i
\(137\) 11891.9 0.633595 0.316797 0.948493i \(-0.397393\pi\)
0.316797 + 0.948493i \(0.397393\pi\)
\(138\) 0 0
\(139\) −24841.5 −1.28572 −0.642862 0.765982i \(-0.722253\pi\)
−0.642862 + 0.765982i \(0.722253\pi\)
\(140\) 390.101 0.0199031
\(141\) 0 0
\(142\) 9077.34i 0.450176i
\(143\) −5032.28 −0.246089
\(144\) 0 0
\(145\) −43230.6 −2.05615
\(146\) 13372.7 0.627355
\(147\) 0 0
\(148\) 17996.4i 0.821605i
\(149\) 26032.6i 1.17259i 0.810099 + 0.586293i \(0.199413\pi\)
−0.810099 + 0.586293i \(0.800587\pi\)
\(150\) 0 0
\(151\) 12278.3i 0.538497i −0.963071 0.269249i \(-0.913225\pi\)
0.963071 0.269249i \(-0.0867753\pi\)
\(152\) 2769.15i 0.119856i
\(153\) 0 0
\(154\) 770.594 0.0324926
\(155\) 56471.8i 2.35054i
\(156\) 0 0
\(157\) 21189.6i 0.859655i 0.902911 + 0.429827i \(0.141426\pi\)
−0.902911 + 0.429827i \(0.858574\pi\)
\(158\) 15333.5i 0.614226i
\(159\) 0 0
\(160\) 41580.3i 1.62423i
\(161\) 151.113i 0.00582976i
\(162\) 0 0
\(163\) 22822.0 0.858970 0.429485 0.903074i \(-0.358695\pi\)
0.429485 + 0.903074i \(0.358695\pi\)
\(164\) −13906.8 −0.517057
\(165\) 0 0
\(166\) −39510.0 −1.43381
\(167\) 28368.2 1.01718 0.508592 0.861008i \(-0.330166\pi\)
0.508592 + 0.861008i \(0.330166\pi\)
\(168\) 0 0
\(169\) 27317.5 0.956463
\(170\) 108490.i 3.75398i
\(171\) 0 0
\(172\) 6466.56i 0.218583i
\(173\) 7580.94i 0.253297i −0.991948 0.126649i \(-0.959578\pi\)
0.991948 0.126649i \(-0.0404221\pi\)
\(174\) 0 0
\(175\) 1183.70 0.0386513
\(176\) 45603.1i 1.47221i
\(177\) 0 0
\(178\) 25533.9 0.805894
\(179\) 34939.2i 1.09045i 0.838289 + 0.545226i \(0.183556\pi\)
−0.838289 + 0.545226i \(0.816444\pi\)
\(180\) 0 0
\(181\) 37840.7 1.15505 0.577527 0.816372i \(-0.304018\pi\)
0.577527 + 0.816372i \(0.304018\pi\)
\(182\) 190.411 0.00574843
\(183\) 0 0
\(184\) −5061.82 −0.149510
\(185\) 85954.8i 2.51146i
\(186\) 0 0
\(187\) 75301.6i 2.15338i
\(188\) 5320.59i 0.150537i
\(189\) 0 0
\(190\) 15633.7i 0.433067i
\(191\) 3569.74i 0.0978521i 0.998802 + 0.0489261i \(0.0155799\pi\)
−0.998802 + 0.0489261i \(0.984420\pi\)
\(192\) 0 0
\(193\) 24451.4 0.656430 0.328215 0.944603i \(-0.393553\pi\)
0.328215 + 0.944603i \(0.393553\pi\)
\(194\) −64882.3 −1.72394
\(195\) 0 0
\(196\) 20800.2 0.541447
\(197\) 29706.3 0.765449 0.382724 0.923863i \(-0.374986\pi\)
0.382724 + 0.923863i \(0.374986\pi\)
\(198\) 0 0
\(199\) −22181.5 −0.560125 −0.280062 0.959982i \(-0.590355\pi\)
−0.280062 + 0.959982i \(0.590355\pi\)
\(200\) 39650.2i 0.991255i
\(201\) 0 0
\(202\) −30857.5 −0.756237
\(203\) 1135.36 0.0275512
\(204\) 0 0
\(205\) −66421.7 −1.58053
\(206\) −59733.8 −1.40762
\(207\) 0 0
\(208\) 11268.4i 0.260456i
\(209\) 10851.2i 0.248419i
\(210\) 0 0
\(211\) 81724.8i 1.83565i 0.396989 + 0.917824i \(0.370055\pi\)
−0.396989 + 0.917824i \(0.629945\pi\)
\(212\) −28659.5 −0.637671
\(213\) 0 0
\(214\) 29382.1i 0.641586i
\(215\) 30885.6i 0.668159i
\(216\) 0 0
\(217\) 1483.11i 0.0314959i
\(218\) −88838.3 −1.86933
\(219\) 0 0
\(220\) 51205.0i 1.05795i
\(221\) 18606.7i 0.380965i
\(222\) 0 0
\(223\) −54373.6 −1.09340 −0.546699 0.837329i \(-0.684116\pi\)
−0.546699 + 0.837329i \(0.684116\pi\)
\(224\) 1092.02i 0.0217637i
\(225\) 0 0
\(226\) 123833. 2.42449
\(227\) 55801.8i 1.08292i 0.840726 + 0.541460i \(0.182128\pi\)
−0.840726 + 0.541460i \(0.817872\pi\)
\(228\) 0 0
\(229\) 102771.i 1.95975i 0.199624 + 0.979873i \(0.436028\pi\)
−0.199624 + 0.979873i \(0.563972\pi\)
\(230\) 28577.4 0.540215
\(231\) 0 0
\(232\) 38030.9i 0.706579i
\(233\) 60636.1i 1.11691i −0.829534 0.558457i \(-0.811394\pi\)
0.829534 0.558457i \(-0.188606\pi\)
\(234\) 0 0
\(235\) 25412.3i 0.460159i
\(236\) 2867.91 + 30034.7i 0.0514922 + 0.539261i
\(237\) 0 0
\(238\) 2849.25i 0.0503010i
\(239\) −12138.8 −0.212510 −0.106255 0.994339i \(-0.533886\pi\)
−0.106255 + 0.994339i \(0.533886\pi\)
\(240\) 0 0
\(241\) −13471.2 −0.231938 −0.115969 0.993253i \(-0.536997\pi\)
−0.115969 + 0.993253i \(0.536997\pi\)
\(242\) 28432.3i 0.485491i
\(243\) 0 0
\(244\) 26189.1i 0.439886i
\(245\) 99346.2 1.65508
\(246\) 0 0
\(247\) 2681.29i 0.0439490i
\(248\) 49679.5 0.807745
\(249\) 0 0
\(250\) 95348.6i 1.52558i
\(251\) 20999.2 0.333315 0.166657 0.986015i \(-0.446703\pi\)
0.166657 + 0.986015i \(0.446703\pi\)
\(252\) 0 0
\(253\) 19835.2 0.309882
\(254\) 111317.i 1.72542i
\(255\) 0 0
\(256\) 80894.5 1.23435
\(257\) 39315.0 0.595240 0.297620 0.954684i \(-0.403807\pi\)
0.297620 + 0.954684i \(0.403807\pi\)
\(258\) 0 0
\(259\) 2257.42i 0.0336521i
\(260\) 12652.5i 0.187168i
\(261\) 0 0
\(262\) −88417.5 −1.28806
\(263\) −37861.6 −0.547378 −0.273689 0.961818i \(-0.588244\pi\)
−0.273689 + 0.961818i \(0.588244\pi\)
\(264\) 0 0
\(265\) −136884. −1.94922
\(266\) 410.586i 0.00580284i
\(267\) 0 0
\(268\) 31893.8i 0.444055i
\(269\) 123756.i 1.71026i 0.518413 + 0.855130i \(0.326523\pi\)
−0.518413 + 0.855130i \(0.673477\pi\)
\(270\) 0 0
\(271\) 4855.26 0.0661110 0.0330555 0.999454i \(-0.489476\pi\)
0.0330555 + 0.999454i \(0.489476\pi\)
\(272\) 168616. 2.27909
\(273\) 0 0
\(274\) 59062.9i 0.786707i
\(275\) 155373.i 2.05452i
\(276\) 0 0
\(277\) 97089.7 1.26536 0.632680 0.774414i \(-0.281955\pi\)
0.632680 + 0.774414i \(0.281955\pi\)
\(278\) 123378.i 1.59643i
\(279\) 0 0
\(280\) 1639.10i 0.0209069i
\(281\) 108342. 1.37209 0.686045 0.727559i \(-0.259345\pi\)
0.686045 + 0.727559i \(0.259345\pi\)
\(282\) 0 0
\(283\) 67893.8i 0.847729i −0.905726 0.423864i \(-0.860673\pi\)
0.905726 0.423864i \(-0.139327\pi\)
\(284\) 15841.1 0.196404
\(285\) 0 0
\(286\) 24993.5i 0.305559i
\(287\) 1744.42 0.0211781
\(288\) 0 0
\(289\) 194905. 2.33360
\(290\) 214710.i 2.55304i
\(291\) 0 0
\(292\) 23337.1i 0.273704i
\(293\) −51965.3 −0.605311 −0.302655 0.953100i \(-0.597873\pi\)
−0.302655 + 0.953100i \(0.597873\pi\)
\(294\) 0 0
\(295\) 13697.7 + 143452.i 0.157400 + 1.64840i
\(296\) −75616.4 −0.863043
\(297\) 0 0
\(298\) 129294. 1.45595
\(299\) 4901.21 0.0548227
\(300\) 0 0
\(301\) 811.144i 0.00895293i
\(302\) −60981.7 −0.668629
\(303\) 0 0
\(304\) −24298.1 −0.262921
\(305\) 125085.i 1.34463i
\(306\) 0 0
\(307\) 122608. 1.30089 0.650445 0.759553i \(-0.274582\pi\)
0.650445 + 0.759553i \(0.274582\pi\)
\(308\) 1344.79i 0.0141759i
\(309\) 0 0
\(310\) −280475. −2.91857
\(311\) 159419. 1.64823 0.824117 0.566420i \(-0.191672\pi\)
0.824117 + 0.566420i \(0.191672\pi\)
\(312\) 0 0
\(313\) 17723.3i 0.180907i −0.995901 0.0904534i \(-0.971168\pi\)
0.995901 0.0904534i \(-0.0288316\pi\)
\(314\) 105241. 1.06740
\(315\) 0 0
\(316\) −26759.0 −0.267976
\(317\) 112577. 1.12029 0.560145 0.828394i \(-0.310745\pi\)
0.560145 + 0.828394i \(0.310745\pi\)
\(318\) 0 0
\(319\) 149028.i 1.46449i
\(320\) 5145.85 0.0502524
\(321\) 0 0
\(322\) −750.523 −0.00723856
\(323\) −40122.0 −0.384572
\(324\) 0 0
\(325\) 38392.1i 0.363475i
\(326\) 113348.i 1.06655i
\(327\) 0 0
\(328\) 58432.7i 0.543135i
\(329\) 667.398i 0.00616585i
\(330\) 0 0
\(331\) −126890. −1.15817 −0.579083 0.815268i \(-0.696589\pi\)
−0.579083 + 0.815268i \(0.696589\pi\)
\(332\) 68950.1i 0.625545i
\(333\) 0 0
\(334\) 140895.i 1.26299i
\(335\) 152332.i 1.35738i
\(336\) 0 0
\(337\) 145883.i 1.28454i 0.766480 + 0.642268i \(0.222006\pi\)
−0.766480 + 0.642268i \(0.777994\pi\)
\(338\) 135676.i 1.18760i
\(339\) 0 0
\(340\) −189329. −1.63779
\(341\) −194674. −1.67417
\(342\) 0 0
\(343\) −5219.51 −0.0443651
\(344\) 27170.8 0.229607
\(345\) 0 0
\(346\) −37651.7 −0.314509
\(347\) 118230.i 0.981905i 0.871186 + 0.490953i \(0.163351\pi\)
−0.871186 + 0.490953i \(0.836649\pi\)
\(348\) 0 0
\(349\) 24509.0i 0.201222i 0.994926 + 0.100611i \(0.0320797\pi\)
−0.994926 + 0.100611i \(0.967920\pi\)
\(350\) 5878.99i 0.0479917i
\(351\) 0 0
\(352\) −143339. −1.15685
\(353\) 38007.7i 0.305015i −0.988302 0.152508i \(-0.951265\pi\)
0.988302 0.152508i \(-0.0487349\pi\)
\(354\) 0 0
\(355\) 75660.7 0.600362
\(356\) 44560.1i 0.351597i
\(357\) 0 0
\(358\) 173530. 1.35397
\(359\) −192503. −1.49365 −0.746826 0.665020i \(-0.768423\pi\)
−0.746826 + 0.665020i \(0.768423\pi\)
\(360\) 0 0
\(361\) −124539. −0.955635
\(362\) 187941.i 1.43418i
\(363\) 0 0
\(364\) 332.292i 0.00250794i
\(365\) 111463.i 0.836652i
\(366\) 0 0
\(367\) 18295.0i 0.135831i 0.997691 + 0.0679157i \(0.0216349\pi\)
−0.997691 + 0.0679157i \(0.978365\pi\)
\(368\) 44415.3i 0.327972i
\(369\) 0 0
\(370\) 426906. 3.11838
\(371\) 3594.96 0.0261183
\(372\) 0 0
\(373\) 57630.7 0.414225 0.207112 0.978317i \(-0.433593\pi\)
0.207112 + 0.978317i \(0.433593\pi\)
\(374\) −373995. −2.67376
\(375\) 0 0
\(376\) 22355.8 0.158130
\(377\) 36824.2i 0.259090i
\(378\) 0 0
\(379\) 170818. 1.18920 0.594599 0.804023i \(-0.297311\pi\)
0.594599 + 0.804023i \(0.297311\pi\)
\(380\) 27282.9 0.188940
\(381\) 0 0
\(382\) 17729.6 0.121499
\(383\) 221122. 1.50742 0.753709 0.657208i \(-0.228263\pi\)
0.753709 + 0.657208i \(0.228263\pi\)
\(384\) 0 0
\(385\) 6422.99i 0.0433327i
\(386\) 121441.i 0.815061i
\(387\) 0 0
\(388\) 113228.i 0.752126i
\(389\) −87151.4 −0.575937 −0.287969 0.957640i \(-0.592980\pi\)
−0.287969 + 0.957640i \(0.592980\pi\)
\(390\) 0 0
\(391\) 73340.2i 0.479721i
\(392\) 87397.2i 0.568755i
\(393\) 0 0
\(394\) 147540.i 0.950425i
\(395\) −127807. −0.819143
\(396\) 0 0
\(397\) 126155.i 0.800431i 0.916421 + 0.400216i \(0.131065\pi\)
−0.916421 + 0.400216i \(0.868935\pi\)
\(398\) 110167.i 0.695483i
\(399\) 0 0
\(400\) −347913. −2.17446
\(401\) 27343.9i 0.170048i 0.996379 + 0.0850242i \(0.0270967\pi\)
−0.996379 + 0.0850242i \(0.972903\pi\)
\(402\) 0 0
\(403\) −48103.2 −0.296185
\(404\) 53850.4i 0.329933i
\(405\) 0 0
\(406\) 5638.90i 0.0342091i
\(407\) 296310. 1.78878
\(408\) 0 0
\(409\) 104932.i 0.627282i −0.949542 0.313641i \(-0.898451\pi\)
0.949542 0.313641i \(-0.101549\pi\)
\(410\) 329892.i 1.96247i
\(411\) 0 0
\(412\) 104243.i 0.614120i
\(413\) −359.741 3767.45i −0.00210907 0.0220875i
\(414\) 0 0
\(415\) 329320.i 1.91215i
\(416\) −35418.5 −0.204665
\(417\) 0 0
\(418\) 53893.8 0.308451
\(419\) 203320.i 1.15812i −0.815286 0.579058i \(-0.803420\pi\)
0.815286 0.579058i \(-0.196580\pi\)
\(420\) 0 0
\(421\) 7915.89i 0.0446617i −0.999751 0.0223309i \(-0.992891\pi\)
0.999751 0.0223309i \(-0.00710872\pi\)
\(422\) 405897. 2.27924
\(423\) 0 0
\(424\) 120420.i 0.669833i
\(425\) −574488. −3.18056
\(426\) 0 0
\(427\) 3285.07i 0.0180173i
\(428\) −51275.5 −0.279913
\(429\) 0 0
\(430\) −153398. −0.829625
\(431\) 166950.i 0.898734i −0.893347 0.449367i \(-0.851650\pi\)
0.893347 0.449367i \(-0.148350\pi\)
\(432\) 0 0
\(433\) −47184.7 −0.251666 −0.125833 0.992051i \(-0.540160\pi\)
−0.125833 + 0.992051i \(0.540160\pi\)
\(434\) 7366.05 0.0391071
\(435\) 0 0
\(436\) 155034.i 0.815558i
\(437\) 10568.5i 0.0553417i
\(438\) 0 0
\(439\) 168825. 0.876010 0.438005 0.898973i \(-0.355685\pi\)
0.438005 + 0.898973i \(0.355685\pi\)
\(440\) −215150. −1.11131
\(441\) 0 0
\(442\) −92412.7 −0.473028
\(443\) 307500.i 1.56689i −0.621462 0.783444i \(-0.713461\pi\)
0.621462 0.783444i \(-0.286539\pi\)
\(444\) 0 0
\(445\) 212828.i 1.07475i
\(446\) 270053.i 1.35763i
\(447\) 0 0
\(448\) −135.145 −0.000673352
\(449\) 243616. 1.20841 0.604203 0.796830i \(-0.293491\pi\)
0.604203 + 0.796830i \(0.293491\pi\)
\(450\) 0 0
\(451\) 228974.i 1.12573i
\(452\) 216105.i 1.05776i
\(453\) 0 0
\(454\) 277147. 1.34462
\(455\) 1587.10i 0.00766620i
\(456\) 0 0
\(457\) 240817.i 1.15307i 0.817073 + 0.576535i \(0.195595\pi\)
−0.817073 + 0.576535i \(0.804405\pi\)
\(458\) 510426. 2.43333
\(459\) 0 0
\(460\) 49871.3i 0.235686i
\(461\) −105419. −0.496043 −0.248021 0.968755i \(-0.579780\pi\)
−0.248021 + 0.968755i \(0.579780\pi\)
\(462\) 0 0
\(463\) 177261.i 0.826895i −0.910528 0.413447i \(-0.864325\pi\)
0.910528 0.413447i \(-0.135675\pi\)
\(464\) −333705. −1.54998
\(465\) 0 0
\(466\) −301157. −1.38682
\(467\) 387563.i 1.77709i −0.458792 0.888544i \(-0.651718\pi\)
0.458792 0.888544i \(-0.348282\pi\)
\(468\) 0 0
\(469\) 4000.66i 0.0181880i
\(470\) −126213. −0.571360
\(471\) 0 0
\(472\) −126198. + 12050.2i −0.566459 + 0.0540892i
\(473\) −106471. −0.475894
\(474\) 0 0
\(475\) 82785.4 0.366916
\(476\) 4972.31 0.0219455
\(477\) 0 0
\(478\) 60288.8i 0.263865i
\(479\) −346182. −1.50881 −0.754404 0.656411i \(-0.772074\pi\)
−0.754404 + 0.656411i \(0.772074\pi\)
\(480\) 0 0
\(481\) 73217.1 0.316463
\(482\) 66906.3i 0.287987i
\(483\) 0 0
\(484\) 49618.1 0.211811
\(485\) 540801.i 2.29908i
\(486\) 0 0
\(487\) −204643. −0.862855 −0.431428 0.902148i \(-0.641990\pi\)
−0.431428 + 0.902148i \(0.641990\pi\)
\(488\) −110040. −0.462072
\(489\) 0 0
\(490\) 493416.i 2.05504i
\(491\) 124429. 0.516128 0.258064 0.966128i \(-0.416915\pi\)
0.258064 + 0.966128i \(0.416915\pi\)
\(492\) 0 0
\(493\) −551026. −2.26714
\(494\) 13317.0 0.0545696
\(495\) 0 0
\(496\) 435917.i 1.77190i
\(497\) −1987.06 −0.00804449
\(498\) 0 0
\(499\) −410778. −1.64970 −0.824851 0.565350i \(-0.808741\pi\)
−0.824851 + 0.565350i \(0.808741\pi\)
\(500\) 166396. 0.665583
\(501\) 0 0
\(502\) 104295.i 0.413863i
\(503\) 245979.i 0.972213i −0.873899 0.486107i \(-0.838417\pi\)
0.873899 0.486107i \(-0.161583\pi\)
\(504\) 0 0
\(505\) 257201.i 1.00853i
\(506\) 98514.2i 0.384767i
\(507\) 0 0
\(508\) −194263. −0.752771
\(509\) 32679.0i 0.126134i −0.998009 0.0630672i \(-0.979912\pi\)
0.998009 0.0630672i \(-0.0200882\pi\)
\(510\) 0 0
\(511\) 2927.33i 0.0112106i
\(512\) 134764.i 0.514085i
\(513\) 0 0
\(514\) 195263.i 0.739084i
\(515\) 497887.i 1.87723i
\(516\) 0 0
\(517\) −87603.2 −0.327747
\(518\) −11211.7 −0.0417844
\(519\) 0 0
\(520\) −53162.8 −0.196608
\(521\) 113050. 0.416479 0.208240 0.978078i \(-0.433227\pi\)
0.208240 + 0.978078i \(0.433227\pi\)
\(522\) 0 0
\(523\) 237542. 0.868435 0.434218 0.900808i \(-0.357025\pi\)
0.434218 + 0.900808i \(0.357025\pi\)
\(524\) 154300.i 0.561957i
\(525\) 0 0
\(526\) 188044.i 0.679656i
\(527\) 719802.i 2.59174i
\(528\) 0 0
\(529\) 260522. 0.930966
\(530\) 679851.i 2.42026i
\(531\) 0 0
\(532\) −716.525 −0.00253168
\(533\) 56578.6i 0.199158i
\(534\) 0 0
\(535\) −244903. −0.855630
\(536\) 134010. 0.466452
\(537\) 0 0
\(538\) 614651. 2.12356
\(539\) 342474.i 1.17883i
\(540\) 0 0
\(541\) 203519.i 0.695360i 0.937613 + 0.347680i \(0.113030\pi\)
−0.937613 + 0.347680i \(0.886970\pi\)
\(542\) 24114.3i 0.0820873i
\(543\) 0 0
\(544\) 529991.i 1.79090i
\(545\) 740477.i 2.49298i
\(546\) 0 0
\(547\) −388812. −1.29947 −0.649733 0.760163i \(-0.725119\pi\)
−0.649733 + 0.760163i \(0.725119\pi\)
\(548\) −103072. −0.343227
\(549\) 0 0
\(550\) 771680. 2.55101
\(551\) 79404.6 0.261543
\(552\) 0 0
\(553\) 3356.57 0.0109760
\(554\) 482209.i 1.57114i
\(555\) 0 0
\(556\) 215311. 0.696493
\(557\) −307150. −0.990011 −0.495006 0.868890i \(-0.664834\pi\)
−0.495006 + 0.868890i \(0.664834\pi\)
\(558\) 0 0
\(559\) −26308.7 −0.0841929
\(560\) 14382.4 0.0458624
\(561\) 0 0
\(562\) 538093.i 1.70367i
\(563\) 224936.i 0.709647i −0.934933 0.354824i \(-0.884541\pi\)
0.934933 0.354824i \(-0.115459\pi\)
\(564\) 0 0
\(565\) 1.03216e6i 3.23334i
\(566\) −337203. −1.05259
\(567\) 0 0
\(568\) 66560.4i 0.206310i
\(569\) 48390.1i 0.149463i 0.997204 + 0.0747313i \(0.0238099\pi\)
−0.997204 + 0.0747313i \(0.976190\pi\)
\(570\) 0 0
\(571\) 344185.i 1.05565i −0.849353 0.527825i \(-0.823008\pi\)
0.849353 0.527825i \(-0.176992\pi\)
\(572\) 43616.8 0.133310
\(573\) 0 0
\(574\) 8663.89i 0.0262960i
\(575\) 151326.i 0.457697i
\(576\) 0 0
\(577\) 68280.6 0.205091 0.102545 0.994728i \(-0.467301\pi\)
0.102545 + 0.994728i \(0.467301\pi\)
\(578\) 968019.i 2.89753i
\(579\) 0 0
\(580\) 374697. 1.11384
\(581\) 8648.88i 0.0256217i
\(582\) 0 0
\(583\) 471876.i 1.38833i
\(584\) 98056.5 0.287509
\(585\) 0 0
\(586\) 258093.i 0.751589i
\(587\) 275118.i 0.798441i 0.916855 + 0.399220i \(0.130719\pi\)
−0.916855 + 0.399220i \(0.869281\pi\)
\(588\) 0 0
\(589\) 103726.i 0.298989i
\(590\) 712472. 68031.6i 2.04675 0.195437i
\(591\) 0 0
\(592\) 663502.i 1.89321i
\(593\) −84870.1 −0.241349 −0.120674 0.992692i \(-0.538506\pi\)
−0.120674 + 0.992692i \(0.538506\pi\)
\(594\) 0 0
\(595\) 23748.8 0.0670823
\(596\) 225635.i 0.635206i
\(597\) 0 0
\(598\) 24342.5i 0.0680711i
\(599\) 75105.1 0.209322 0.104661 0.994508i \(-0.466624\pi\)
0.104661 + 0.994508i \(0.466624\pi\)
\(600\) 0 0
\(601\) 94230.3i 0.260880i −0.991456 0.130440i \(-0.958361\pi\)
0.991456 0.130440i \(-0.0416391\pi\)
\(602\) 4028.65 0.0111165
\(603\) 0 0
\(604\) 106421.i 0.291711i
\(605\) 236986. 0.647460
\(606\) 0 0
\(607\) 247136. 0.670747 0.335374 0.942085i \(-0.391137\pi\)
0.335374 + 0.942085i \(0.391137\pi\)
\(608\) 76373.4i 0.206602i
\(609\) 0 0
\(610\) 621249. 1.66957
\(611\) −21646.4 −0.0579834
\(612\) 0 0
\(613\) 349727.i 0.930696i −0.885128 0.465348i \(-0.845929\pi\)
0.885128 0.465348i \(-0.154071\pi\)
\(614\) 608947.i 1.61526i
\(615\) 0 0
\(616\) 5650.45 0.0148909
\(617\) 645133. 1.69465 0.847323 0.531078i \(-0.178213\pi\)
0.847323 + 0.531078i \(0.178213\pi\)
\(618\) 0 0
\(619\) −473269. −1.23517 −0.617585 0.786504i \(-0.711889\pi\)
−0.617585 + 0.786504i \(0.711889\pi\)
\(620\) 489464.i 1.27332i
\(621\) 0 0
\(622\) 791774.i 2.04654i
\(623\) 5589.47i 0.0144011i
\(624\) 0 0
\(625\) 114275. 0.292544
\(626\) −88024.9 −0.224624
\(627\) 0 0
\(628\) 183659.i 0.465687i
\(629\) 1.09560e6i 2.76917i
\(630\) 0 0
\(631\) 775739. 1.94831 0.974153 0.225891i \(-0.0725294\pi\)
0.974153 + 0.225891i \(0.0725294\pi\)
\(632\) 112435.i 0.281492i
\(633\) 0 0
\(634\) 559128.i 1.39102i
\(635\) −927842. −2.30105
\(636\) 0 0
\(637\) 84624.0i 0.208552i
\(638\) 740166. 1.81839
\(639\) 0 0
\(640\) 690842.i 1.68663i
\(641\) −368716. −0.897380 −0.448690 0.893688i \(-0.648109\pi\)
−0.448690 + 0.893688i \(0.648109\pi\)
\(642\) 0 0
\(643\) −81254.3 −0.196528 −0.0982640 0.995160i \(-0.531329\pi\)
−0.0982640 + 0.995160i \(0.531329\pi\)
\(644\) 1309.76i 0.00315806i
\(645\) 0 0
\(646\) 199271.i 0.477506i
\(647\) 176909. 0.422612 0.211306 0.977420i \(-0.432228\pi\)
0.211306 + 0.977420i \(0.432228\pi\)
\(648\) 0 0
\(649\) 494518. 47219.9i 1.17407 0.112108i
\(650\) 190679. 0.451312
\(651\) 0 0
\(652\) −197807. −0.465316
\(653\) −467222. −1.09571 −0.547857 0.836572i \(-0.684556\pi\)
−0.547857 + 0.836572i \(0.684556\pi\)
\(654\) 0 0
\(655\) 736970.i 1.71778i
\(656\) −512722. −1.19145
\(657\) 0 0
\(658\) 3314.72 0.00765588
\(659\) 41300.9i 0.0951018i 0.998869 + 0.0475509i \(0.0151416\pi\)
−0.998869 + 0.0475509i \(0.984858\pi\)
\(660\) 0 0
\(661\) −298921. −0.684153 −0.342077 0.939672i \(-0.611130\pi\)
−0.342077 + 0.939672i \(0.611130\pi\)
\(662\) 630215.i 1.43805i
\(663\) 0 0
\(664\) −289711. −0.657095
\(665\) −3422.28 −0.00773877
\(666\) 0 0
\(667\) 145146.i 0.326253i
\(668\) −245879. −0.551022
\(669\) 0 0
\(670\) −756575. −1.68540
\(671\) 431201. 0.957712
\(672\) 0 0
\(673\) 814423.i 1.79812i −0.437821 0.899062i \(-0.644249\pi\)
0.437821 0.899062i \(-0.355751\pi\)
\(674\) 724549. 1.59495
\(675\) 0 0
\(676\) −236772. −0.518129
\(677\) 345729. 0.754324 0.377162 0.926147i \(-0.376900\pi\)
0.377162 + 0.926147i \(0.376900\pi\)
\(678\) 0 0
\(679\) 14203.0i 0.0308063i
\(680\) 795512.i 1.72040i
\(681\) 0 0
\(682\) 966873.i 2.07874i
\(683\) 239945.i 0.514363i 0.966363 + 0.257181i \(0.0827938\pi\)
−0.966363 + 0.257181i \(0.917206\pi\)
\(684\) 0 0
\(685\) −492295. −1.04917
\(686\) 25923.4i 0.0550863i
\(687\) 0 0
\(688\) 238412.i 0.503677i
\(689\) 116599.i 0.245616i
\(690\) 0 0
\(691\) 695558.i 1.45672i 0.685193 + 0.728362i \(0.259718\pi\)
−0.685193 + 0.728362i \(0.740282\pi\)
\(692\) 65707.1i 0.137215i
\(693\) 0 0
\(694\) 587206. 1.21919
\(695\) 1.02837e6 2.12902
\(696\) 0 0
\(697\) −846626. −1.74271
\(698\) 121727. 0.249848
\(699\) 0 0
\(700\) −10259.6 −0.0209379
\(701\) 839756.i 1.70890i 0.519533 + 0.854451i \(0.326106\pi\)
−0.519533 + 0.854451i \(0.673894\pi\)
\(702\) 0 0
\(703\) 157879.i 0.319458i
\(704\) 17739.2i 0.0357922i
\(705\) 0 0
\(706\) −188770. −0.378725
\(707\) 6754.82i 0.0135137i
\(708\) 0 0
\(709\) 340569. 0.677505 0.338753 0.940875i \(-0.389995\pi\)
0.338753 + 0.940875i \(0.389995\pi\)
\(710\) 375778.i 0.745444i
\(711\) 0 0
\(712\) 187230. 0.369331
\(713\) 189603. 0.372964
\(714\) 0 0
\(715\) 208323. 0.407498
\(716\) 302832.i 0.590712i
\(717\) 0 0
\(718\) 956093.i 1.85460i
\(719\) 155487.i 0.300770i 0.988627 + 0.150385i \(0.0480514\pi\)
−0.988627 + 0.150385i \(0.951949\pi\)
\(720\) 0 0
\(721\) 13075.9i 0.0251537i
\(722\) 618541.i 1.18657i
\(723\) 0 0
\(724\) −327981. −0.625708
\(725\) 1.13696e6 2.16306
\(726\) 0 0
\(727\) −61349.1 −0.116075 −0.0580376 0.998314i \(-0.518484\pi\)
−0.0580376 + 0.998314i \(0.518484\pi\)
\(728\) 1396.20 0.00263443
\(729\) 0 0
\(730\) −553595. −1.03884
\(731\) 393675.i 0.736722i
\(732\) 0 0
\(733\) 39246.3 0.0730451 0.0365225 0.999333i \(-0.488372\pi\)
0.0365225 + 0.999333i \(0.488372\pi\)
\(734\) 90864.4 0.168656
\(735\) 0 0
\(736\) 139605. 0.257719
\(737\) −525130. −0.966789
\(738\) 0 0
\(739\) 856850.i 1.56898i 0.620144 + 0.784488i \(0.287074\pi\)
−0.620144 + 0.784488i \(0.712926\pi\)
\(740\) 745006.i 1.36049i
\(741\) 0 0
\(742\) 17854.8i 0.0324300i
\(743\) −598825. −1.08473 −0.542366 0.840143i \(-0.682471\pi\)
−0.542366 + 0.840143i \(0.682471\pi\)
\(744\) 0 0
\(745\) 1.07768e6i 1.94168i
\(746\) 286230.i 0.514325i
\(747\) 0 0
\(748\) 652670.i 1.16651i
\(749\) 6431.83 0.0114649
\(750\) 0 0
\(751\) 332338.i 0.589250i 0.955613 + 0.294625i \(0.0951947\pi\)
−0.955613 + 0.294625i \(0.904805\pi\)
\(752\) 196162.i 0.346880i
\(753\) 0 0
\(754\) 182892. 0.321701
\(755\) 508289.i 0.891696i
\(756\) 0 0
\(757\) 824719. 1.43918 0.719588 0.694401i \(-0.244331\pi\)
0.719588 + 0.694401i \(0.244331\pi\)
\(758\) 848387.i 1.47658i
\(759\) 0 0
\(760\) 114636.i 0.198469i
\(761\) −1.09768e6 −1.89542 −0.947711 0.319130i \(-0.896609\pi\)
−0.947711 + 0.319130i \(0.896609\pi\)
\(762\) 0 0
\(763\) 19447.0i 0.0334044i
\(764\) 30940.4i 0.0530078i
\(765\) 0 0
\(766\) 1.09823e6i 1.87170i
\(767\) 122194. 11667.9i 0.207710 0.0198336i
\(768\) 0 0
\(769\) 237713.i 0.401976i 0.979594 + 0.200988i \(0.0644153\pi\)
−0.979594 + 0.200988i \(0.935585\pi\)
\(770\) −31900.6 −0.0538043
\(771\) 0 0
\(772\) −211930. −0.355597
\(773\) 998180.i 1.67051i −0.549860 0.835257i \(-0.685319\pi\)
0.549860 0.835257i \(-0.314681\pi\)
\(774\) 0 0
\(775\) 1.48520e6i 2.47276i
\(776\) −475755. −0.790060
\(777\) 0 0
\(778\) 432849.i 0.715117i
\(779\) 122001. 0.201043
\(780\) 0 0
\(781\) 260823.i 0.427606i
\(782\) 364254. 0.595649
\(783\) 0 0
\(784\) 766873. 1.24765
\(785\) 877196.i 1.42350i
\(786\) 0 0
\(787\) −292759. −0.472673 −0.236337 0.971671i \(-0.575947\pi\)
−0.236337 + 0.971671i \(0.575947\pi\)
\(788\) −257477. −0.414654
\(789\) 0 0
\(790\) 634769.i 1.01709i
\(791\) 27107.5i 0.0433248i
\(792\) 0 0
\(793\) 106548. 0.169434
\(794\) 626566. 0.993861
\(795\) 0 0
\(796\) 192256. 0.303427
\(797\) 105074.i