Properties

Label 531.5.c.d.235.12
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 235.12
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.29

$q$-expansion

\(f(q)\) \(=\) \(q-4.05068i q^{2} -0.408022 q^{4} +16.4107 q^{5} +47.3137 q^{7} -63.1581i q^{8} +O(q^{10})\) \(q-4.05068i q^{2} -0.408022 q^{4} +16.4107 q^{5} +47.3137 q^{7} -63.1581i q^{8} -66.4744i q^{10} -25.6339i q^{11} +105.588i q^{13} -191.653i q^{14} -262.362 q^{16} -441.346 q^{17} -560.726 q^{19} -6.69591 q^{20} -103.835 q^{22} -764.114i q^{23} -355.690 q^{25} +427.703 q^{26} -19.3050 q^{28} +1381.28 q^{29} -950.873i q^{31} +52.2142i q^{32} +1787.75i q^{34} +776.449 q^{35} +632.407i q^{37} +2271.32i q^{38} -1036.47i q^{40} -1029.59 q^{41} -2959.97i q^{43} +10.4592i q^{44} -3095.18 q^{46} -4322.20i q^{47} -162.413 q^{49} +1440.79i q^{50} -43.0822i q^{52} +2110.03 q^{53} -420.669i q^{55} -2988.25i q^{56} -5595.14i q^{58} +(3445.38 - 496.740i) q^{59} +4274.01i q^{61} -3851.68 q^{62} -3986.29 q^{64} +1732.77i q^{65} -867.430i q^{67} +180.079 q^{68} -3145.15i q^{70} +236.548 q^{71} -6211.26i q^{73} +2561.68 q^{74} +228.788 q^{76} -1212.84i q^{77} +8916.22 q^{79} -4305.53 q^{80} +4170.56i q^{82} -10864.4i q^{83} -7242.78 q^{85} -11989.9 q^{86} -1618.99 q^{88} -6262.72i q^{89} +4995.75i q^{91} +311.775i q^{92} -17507.9 q^{94} -9201.88 q^{95} +7894.97i q^{97} +657.884i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.05068i 1.01267i −0.862337 0.506335i \(-0.831000\pi\)
0.862337 0.506335i \(-0.169000\pi\)
\(3\) 0 0
\(4\) −0.408022 −0.0255014
\(5\) 16.4107 0.656426 0.328213 0.944604i \(-0.393554\pi\)
0.328213 + 0.944604i \(0.393554\pi\)
\(6\) 0 0
\(7\) 47.3137 0.965586 0.482793 0.875735i \(-0.339622\pi\)
0.482793 + 0.875735i \(0.339622\pi\)
\(8\) 63.1581i 0.986846i
\(9\) 0 0
\(10\) 66.4744i 0.664744i
\(11\) 25.6339i 0.211850i −0.994374 0.105925i \(-0.966220\pi\)
0.994374 0.105925i \(-0.0337804\pi\)
\(12\) 0 0
\(13\) 105.588i 0.624780i 0.949954 + 0.312390i \(0.101130\pi\)
−0.949954 + 0.312390i \(0.898870\pi\)
\(14\) 191.653i 0.977820i
\(15\) 0 0
\(16\) −262.362 −1.02485
\(17\) −441.346 −1.52715 −0.763575 0.645719i \(-0.776558\pi\)
−0.763575 + 0.645719i \(0.776558\pi\)
\(18\) 0 0
\(19\) −560.726 −1.55326 −0.776629 0.629959i \(-0.783072\pi\)
−0.776629 + 0.629959i \(0.783072\pi\)
\(20\) −6.69591 −0.0167398
\(21\) 0 0
\(22\) −103.835 −0.214535
\(23\) 764.114i 1.44445i −0.691658 0.722225i \(-0.743120\pi\)
0.691658 0.722225i \(-0.256880\pi\)
\(24\) 0 0
\(25\) −355.690 −0.569104
\(26\) 427.703 0.632696
\(27\) 0 0
\(28\) −19.3050 −0.0246238
\(29\) 1381.28 1.64243 0.821216 0.570618i \(-0.193296\pi\)
0.821216 + 0.570618i \(0.193296\pi\)
\(30\) 0 0
\(31\) 950.873i 0.989462i −0.869046 0.494731i \(-0.835267\pi\)
0.869046 0.494731i \(-0.164733\pi\)
\(32\) 52.2142i 0.0509904i
\(33\) 0 0
\(34\) 1787.75i 1.54650i
\(35\) 776.449 0.633836
\(36\) 0 0
\(37\) 632.407i 0.461948i 0.972960 + 0.230974i \(0.0741913\pi\)
−0.972960 + 0.230974i \(0.925809\pi\)
\(38\) 2271.32i 1.57294i
\(39\) 0 0
\(40\) 1036.47i 0.647792i
\(41\) −1029.59 −0.612489 −0.306245 0.951953i \(-0.599073\pi\)
−0.306245 + 0.951953i \(0.599073\pi\)
\(42\) 0 0
\(43\) 2959.97i 1.60085i −0.599434 0.800424i \(-0.704608\pi\)
0.599434 0.800424i \(-0.295392\pi\)
\(44\) 10.4592i 0.00540248i
\(45\) 0 0
\(46\) −3095.18 −1.46275
\(47\) 4322.20i 1.95663i −0.207119 0.978316i \(-0.566409\pi\)
0.207119 0.978316i \(-0.433591\pi\)
\(48\) 0 0
\(49\) −162.413 −0.0676439
\(50\) 1440.79i 0.576315i
\(51\) 0 0
\(52\) 43.0822i 0.0159327i
\(53\) 2110.03 0.751168 0.375584 0.926788i \(-0.377442\pi\)
0.375584 + 0.926788i \(0.377442\pi\)
\(54\) 0 0
\(55\) 420.669i 0.139064i
\(56\) 2988.25i 0.952884i
\(57\) 0 0
\(58\) 5595.14i 1.66324i
\(59\) 3445.38 496.740i 0.989766 0.142700i
\(60\) 0 0
\(61\) 4274.01i 1.14862i 0.818638 + 0.574309i \(0.194729\pi\)
−0.818638 + 0.574309i \(0.805271\pi\)
\(62\) −3851.68 −1.00200
\(63\) 0 0
\(64\) −3986.29 −0.973215
\(65\) 1732.77i 0.410122i
\(66\) 0 0
\(67\) 867.430i 0.193234i −0.995322 0.0966172i \(-0.969198\pi\)
0.995322 0.0966172i \(-0.0308023\pi\)
\(68\) 180.079 0.0389444
\(69\) 0 0
\(70\) 3145.15i 0.641867i
\(71\) 236.548 0.0469249 0.0234624 0.999725i \(-0.492531\pi\)
0.0234624 + 0.999725i \(0.492531\pi\)
\(72\) 0 0
\(73\) 6211.26i 1.16556i −0.812631 0.582779i \(-0.801965\pi\)
0.812631 0.582779i \(-0.198035\pi\)
\(74\) 2561.68 0.467801
\(75\) 0 0
\(76\) 228.788 0.0396102
\(77\) 1212.84i 0.204560i
\(78\) 0 0
\(79\) 8916.22 1.42865 0.714326 0.699813i \(-0.246733\pi\)
0.714326 + 0.699813i \(0.246733\pi\)
\(80\) −4305.53 −0.672739
\(81\) 0 0
\(82\) 4170.56i 0.620250i
\(83\) 10864.4i 1.57706i −0.614994 0.788532i \(-0.710842\pi\)
0.614994 0.788532i \(-0.289158\pi\)
\(84\) 0 0
\(85\) −7242.78 −1.00246
\(86\) −11989.9 −1.62113
\(87\) 0 0
\(88\) −1618.99 −0.209064
\(89\) 6262.72i 0.790647i −0.918542 0.395324i \(-0.870632\pi\)
0.918542 0.395324i \(-0.129368\pi\)
\(90\) 0 0
\(91\) 4995.75i 0.603279i
\(92\) 311.775i 0.0368355i
\(93\) 0 0
\(94\) −17507.9 −1.98142
\(95\) −9201.88 −1.01960
\(96\) 0 0
\(97\) 7894.97i 0.839087i 0.907735 + 0.419543i \(0.137810\pi\)
−0.907735 + 0.419543i \(0.862190\pi\)
\(98\) 657.884i 0.0685010i
\(99\) 0 0
\(100\) 145.129 0.0145129
\(101\) 6756.86i 0.662372i −0.943566 0.331186i \(-0.892551\pi\)
0.943566 0.331186i \(-0.107449\pi\)
\(102\) 0 0
\(103\) 19989.4i 1.88420i 0.335338 + 0.942098i \(0.391149\pi\)
−0.335338 + 0.942098i \(0.608851\pi\)
\(104\) 6668.73 0.616562
\(105\) 0 0
\(106\) 8547.07i 0.760686i
\(107\) −14259.8 −1.24550 −0.622751 0.782420i \(-0.713985\pi\)
−0.622751 + 0.782420i \(0.713985\pi\)
\(108\) 0 0
\(109\) 219.395i 0.0184660i 0.999957 + 0.00923302i \(0.00293900\pi\)
−0.999957 + 0.00923302i \(0.997061\pi\)
\(110\) −1704.00 −0.140826
\(111\) 0 0
\(112\) −12413.3 −0.989582
\(113\) 12616.9i 0.988089i 0.869437 + 0.494044i \(0.164482\pi\)
−0.869437 + 0.494044i \(0.835518\pi\)
\(114\) 0 0
\(115\) 12539.6i 0.948175i
\(116\) −563.595 −0.0418843
\(117\) 0 0
\(118\) −2012.14 13956.1i −0.144508 1.00231i
\(119\) −20881.7 −1.47459
\(120\) 0 0
\(121\) 13983.9 0.955119
\(122\) 17312.7 1.16317
\(123\) 0 0
\(124\) 387.977i 0.0252326i
\(125\) −16093.8 −1.03000
\(126\) 0 0
\(127\) 3740.54 0.231914 0.115957 0.993254i \(-0.463007\pi\)
0.115957 + 0.993254i \(0.463007\pi\)
\(128\) 16982.6i 1.03654i
\(129\) 0 0
\(130\) 7018.88 0.415318
\(131\) 21977.1i 1.28064i 0.768108 + 0.640320i \(0.221198\pi\)
−0.768108 + 0.640320i \(0.778802\pi\)
\(132\) 0 0
\(133\) −26530.0 −1.49980
\(134\) −3513.68 −0.195683
\(135\) 0 0
\(136\) 27874.6i 1.50706i
\(137\) 8143.37 0.433873 0.216937 0.976186i \(-0.430394\pi\)
0.216937 + 0.976186i \(0.430394\pi\)
\(138\) 0 0
\(139\) 13508.7 0.699172 0.349586 0.936904i \(-0.386322\pi\)
0.349586 + 0.936904i \(0.386322\pi\)
\(140\) −316.808 −0.0161637
\(141\) 0 0
\(142\) 958.182i 0.0475194i
\(143\) 2706.63 0.132360
\(144\) 0 0
\(145\) 22667.8 1.07814
\(146\) −25159.8 −1.18033
\(147\) 0 0
\(148\) 258.036i 0.0117803i
\(149\) 5748.17i 0.258915i −0.991585 0.129457i \(-0.958676\pi\)
0.991585 0.129457i \(-0.0413235\pi\)
\(150\) 0 0
\(151\) 39105.9i 1.71510i −0.514404 0.857548i \(-0.671987\pi\)
0.514404 0.857548i \(-0.328013\pi\)
\(152\) 35414.4i 1.53283i
\(153\) 0 0
\(154\) −4912.81 −0.207152
\(155\) 15604.5i 0.649509i
\(156\) 0 0
\(157\) 995.408i 0.0403833i 0.999796 + 0.0201917i \(0.00642764\pi\)
−0.999796 + 0.0201917i \(0.993572\pi\)
\(158\) 36116.8i 1.44675i
\(159\) 0 0
\(160\) 856.869i 0.0334715i
\(161\) 36153.1i 1.39474i
\(162\) 0 0
\(163\) −51373.5 −1.93359 −0.966794 0.255559i \(-0.917741\pi\)
−0.966794 + 0.255559i \(0.917741\pi\)
\(164\) 420.097 0.0156193
\(165\) 0 0
\(166\) −44008.2 −1.59705
\(167\) 17563.4 0.629760 0.314880 0.949132i \(-0.398036\pi\)
0.314880 + 0.949132i \(0.398036\pi\)
\(168\) 0 0
\(169\) 17412.2 0.609650
\(170\) 29338.2i 1.01516i
\(171\) 0 0
\(172\) 1207.73i 0.0408238i
\(173\) 59732.1i 1.99579i 0.0648356 + 0.997896i \(0.479348\pi\)
−0.0648356 + 0.997896i \(0.520652\pi\)
\(174\) 0 0
\(175\) −16829.0 −0.549519
\(176\) 6725.36i 0.217115i
\(177\) 0 0
\(178\) −25368.3 −0.800665
\(179\) 52499.1i 1.63850i 0.573439 + 0.819248i \(0.305609\pi\)
−0.573439 + 0.819248i \(0.694391\pi\)
\(180\) 0 0
\(181\) −1001.39 −0.0305665 −0.0152832 0.999883i \(-0.504865\pi\)
−0.0152832 + 0.999883i \(0.504865\pi\)
\(182\) 20236.2 0.610923
\(183\) 0 0
\(184\) −48260.0 −1.42545
\(185\) 10378.2i 0.303235i
\(186\) 0 0
\(187\) 11313.4i 0.323527i
\(188\) 1763.55i 0.0498968i
\(189\) 0 0
\(190\) 37273.9i 1.03252i
\(191\) 65871.0i 1.80563i −0.430034 0.902813i \(-0.641498\pi\)
0.430034 0.902813i \(-0.358502\pi\)
\(192\) 0 0
\(193\) 46752.7 1.25514 0.627570 0.778560i \(-0.284050\pi\)
0.627570 + 0.778560i \(0.284050\pi\)
\(194\) 31980.0 0.849718
\(195\) 0 0
\(196\) 66.2681 0.00172501
\(197\) −56429.6 −1.45403 −0.727017 0.686619i \(-0.759094\pi\)
−0.727017 + 0.686619i \(0.759094\pi\)
\(198\) 0 0
\(199\) 4451.64 0.112412 0.0562061 0.998419i \(-0.482100\pi\)
0.0562061 + 0.998419i \(0.482100\pi\)
\(200\) 22464.7i 0.561618i
\(201\) 0 0
\(202\) −27369.9 −0.670764
\(203\) 65353.7 1.58591
\(204\) 0 0
\(205\) −16896.3 −0.402054
\(206\) 80970.8 1.90807
\(207\) 0 0
\(208\) 27702.2i 0.640306i
\(209\) 14373.6i 0.329058i
\(210\) 0 0
\(211\) 42832.4i 0.962071i 0.876701 + 0.481036i \(0.159739\pi\)
−0.876701 + 0.481036i \(0.840261\pi\)
\(212\) −860.939 −0.0191558
\(213\) 0 0
\(214\) 57761.7i 1.26128i
\(215\) 48575.0i 1.05084i
\(216\) 0 0
\(217\) 44989.3i 0.955411i
\(218\) 888.699 0.0187000
\(219\) 0 0
\(220\) 171.642i 0.00354633i
\(221\) 46600.8i 0.954132i
\(222\) 0 0
\(223\) −14776.1 −0.297132 −0.148566 0.988902i \(-0.547466\pi\)
−0.148566 + 0.988902i \(0.547466\pi\)
\(224\) 2470.45i 0.0492356i
\(225\) 0 0
\(226\) 51107.1 1.00061
\(227\) 37257.5i 0.723040i 0.932364 + 0.361520i \(0.117742\pi\)
−0.932364 + 0.361520i \(0.882258\pi\)
\(228\) 0 0
\(229\) 36389.1i 0.693906i 0.937883 + 0.346953i \(0.112784\pi\)
−0.937883 + 0.346953i \(0.887216\pi\)
\(230\) −50794.0 −0.960189
\(231\) 0 0
\(232\) 87239.4i 1.62083i
\(233\) 20634.6i 0.380088i −0.981776 0.190044i \(-0.939137\pi\)
0.981776 0.190044i \(-0.0608630\pi\)
\(234\) 0 0
\(235\) 70930.1i 1.28438i
\(236\) −1405.79 + 202.681i −0.0252404 + 0.00363906i
\(237\) 0 0
\(238\) 84585.2i 1.49328i
\(239\) 18302.2 0.320411 0.160205 0.987084i \(-0.448784\pi\)
0.160205 + 0.987084i \(0.448784\pi\)
\(240\) 0 0
\(241\) 73211.9 1.26051 0.630257 0.776386i \(-0.282949\pi\)
0.630257 + 0.776386i \(0.282949\pi\)
\(242\) 56644.3i 0.967221i
\(243\) 0 0
\(244\) 1743.89i 0.0292914i
\(245\) −2665.31 −0.0444033
\(246\) 0 0
\(247\) 59205.8i 0.970444i
\(248\) −60055.4 −0.976447
\(249\) 0 0
\(250\) 65190.8i 1.04305i
\(251\) −66492.8 −1.05542 −0.527712 0.849423i \(-0.676950\pi\)
−0.527712 + 0.849423i \(0.676950\pi\)
\(252\) 0 0
\(253\) −19587.2 −0.306007
\(254\) 15151.7i 0.234852i
\(255\) 0 0
\(256\) 5010.54 0.0764548
\(257\) 48002.7 0.726774 0.363387 0.931638i \(-0.381620\pi\)
0.363387 + 0.931638i \(0.381620\pi\)
\(258\) 0 0
\(259\) 29921.5i 0.446051i
\(260\) 707.007i 0.0104587i
\(261\) 0 0
\(262\) 89022.1 1.29687
\(263\) −23476.9 −0.339413 −0.169707 0.985495i \(-0.554282\pi\)
−0.169707 + 0.985495i \(0.554282\pi\)
\(264\) 0 0
\(265\) 34627.0 0.493087
\(266\) 107465.i 1.51881i
\(267\) 0 0
\(268\) 353.930i 0.00492775i
\(269\) 86186.2i 1.19106i −0.803334 0.595529i \(-0.796942\pi\)
0.803334 0.595529i \(-0.203058\pi\)
\(270\) 0 0
\(271\) −18673.8 −0.254270 −0.127135 0.991885i \(-0.540578\pi\)
−0.127135 + 0.991885i \(0.540578\pi\)
\(272\) 115792. 1.56510
\(273\) 0 0
\(274\) 32986.2i 0.439371i
\(275\) 9117.73i 0.120565i
\(276\) 0 0
\(277\) 132710. 1.72959 0.864797 0.502122i \(-0.167447\pi\)
0.864797 + 0.502122i \(0.167447\pi\)
\(278\) 54719.4i 0.708030i
\(279\) 0 0
\(280\) 49039.1i 0.625499i
\(281\) −42447.9 −0.537581 −0.268790 0.963199i \(-0.586624\pi\)
−0.268790 + 0.963199i \(0.586624\pi\)
\(282\) 0 0
\(283\) 1417.90i 0.0177041i −0.999961 0.00885205i \(-0.997182\pi\)
0.999961 0.00885205i \(-0.00281773\pi\)
\(284\) −96.5169 −0.00119665
\(285\) 0 0
\(286\) 10963.7i 0.134037i
\(287\) −48713.9 −0.591411
\(288\) 0 0
\(289\) 111265. 1.33219
\(290\) 91820.0i 1.09180i
\(291\) 0 0
\(292\) 2534.33i 0.0297233i
\(293\) −44835.8 −0.522264 −0.261132 0.965303i \(-0.584096\pi\)
−0.261132 + 0.965303i \(0.584096\pi\)
\(294\) 0 0
\(295\) 56540.9 8151.83i 0.649708 0.0936723i
\(296\) 39941.7 0.455872
\(297\) 0 0
\(298\) −23284.0 −0.262195
\(299\) 80681.1 0.902463
\(300\) 0 0
\(301\) 140047.i 1.54576i
\(302\) −158406. −1.73683
\(303\) 0 0
\(304\) 147113. 1.59186
\(305\) 70139.3i 0.753984i
\(306\) 0 0
\(307\) −15561.6 −0.165112 −0.0825559 0.996586i \(-0.526308\pi\)
−0.0825559 + 0.996586i \(0.526308\pi\)
\(308\) 494.863i 0.00521656i
\(309\) 0 0
\(310\) −63208.7 −0.657739
\(311\) 160952. 1.66409 0.832043 0.554711i \(-0.187171\pi\)
0.832043 + 0.554711i \(0.187171\pi\)
\(312\) 0 0
\(313\) 79828.7i 0.814836i 0.913242 + 0.407418i \(0.133571\pi\)
−0.913242 + 0.407418i \(0.866429\pi\)
\(314\) 4032.08 0.0408950
\(315\) 0 0
\(316\) −3638.01 −0.0364326
\(317\) −108281. −1.07754 −0.538770 0.842453i \(-0.681111\pi\)
−0.538770 + 0.842453i \(0.681111\pi\)
\(318\) 0 0
\(319\) 35407.7i 0.347950i
\(320\) −65417.6 −0.638844
\(321\) 0 0
\(322\) −146445. −1.41241
\(323\) 247474. 2.37206
\(324\) 0 0
\(325\) 37556.6i 0.355565i
\(326\) 208098.i 1.95809i
\(327\) 0 0
\(328\) 65027.2i 0.604432i
\(329\) 204499.i 1.88930i
\(330\) 0 0
\(331\) 124431. 1.13573 0.567864 0.823123i \(-0.307770\pi\)
0.567864 + 0.823123i \(0.307770\pi\)
\(332\) 4432.91i 0.0402173i
\(333\) 0 0
\(334\) 71143.6i 0.637739i
\(335\) 14235.1i 0.126844i
\(336\) 0 0
\(337\) 88676.9i 0.780820i 0.920641 + 0.390410i \(0.127667\pi\)
−0.920641 + 0.390410i \(0.872333\pi\)
\(338\) 70531.3i 0.617374i
\(339\) 0 0
\(340\) 2955.21 0.0255641
\(341\) −24374.6 −0.209618
\(342\) 0 0
\(343\) −121285. −1.03090
\(344\) −186946. −1.57979
\(345\) 0 0
\(346\) 241956. 2.02108
\(347\) 55831.8i 0.463685i −0.972753 0.231842i \(-0.925525\pi\)
0.972753 0.231842i \(-0.0744754\pi\)
\(348\) 0 0
\(349\) 66212.9i 0.543616i 0.962352 + 0.271808i \(0.0876215\pi\)
−0.962352 + 0.271808i \(0.912378\pi\)
\(350\) 68169.0i 0.556482i
\(351\) 0 0
\(352\) 1338.45 0.0108023
\(353\) 119306.i 0.957439i −0.877968 0.478719i \(-0.841101\pi\)
0.877968 0.478719i \(-0.158899\pi\)
\(354\) 0 0
\(355\) 3881.91 0.0308027
\(356\) 2555.33i 0.0201626i
\(357\) 0 0
\(358\) 212657. 1.65926
\(359\) 164492. 1.27631 0.638155 0.769908i \(-0.279698\pi\)
0.638155 + 0.769908i \(0.279698\pi\)
\(360\) 0 0
\(361\) 184092. 1.41261
\(362\) 4056.30i 0.0309538i
\(363\) 0 0
\(364\) 2038.38i 0.0153844i
\(365\) 101931.i 0.765103i
\(366\) 0 0
\(367\) 10621.8i 0.0788613i −0.999222 0.0394307i \(-0.987446\pi\)
0.999222 0.0394307i \(-0.0125544\pi\)
\(368\) 200474.i 1.48035i
\(369\) 0 0
\(370\) 42038.9 0.307077
\(371\) 99833.4 0.725318
\(372\) 0 0
\(373\) 49988.6 0.359297 0.179648 0.983731i \(-0.442504\pi\)
0.179648 + 0.983731i \(0.442504\pi\)
\(374\) 45827.1 0.327627
\(375\) 0 0
\(376\) −272982. −1.93089
\(377\) 145847.i 1.02616i
\(378\) 0 0
\(379\) −23033.2 −0.160352 −0.0801762 0.996781i \(-0.525548\pi\)
−0.0801762 + 0.996781i \(0.525548\pi\)
\(380\) 3754.57 0.0260012
\(381\) 0 0
\(382\) −266823. −1.82850
\(383\) 35518.2 0.242133 0.121067 0.992644i \(-0.461369\pi\)
0.121067 + 0.992644i \(0.461369\pi\)
\(384\) 0 0
\(385\) 19903.4i 0.134278i
\(386\) 189380.i 1.27104i
\(387\) 0 0
\(388\) 3221.32i 0.0213979i
\(389\) −157436. −1.04041 −0.520206 0.854041i \(-0.674145\pi\)
−0.520206 + 0.854041i \(0.674145\pi\)
\(390\) 0 0
\(391\) 337239.i 2.20589i
\(392\) 10257.7i 0.0667541i
\(393\) 0 0
\(394\) 228578.i 1.47246i
\(395\) 146321. 0.937805
\(396\) 0 0
\(397\) 163490.i 1.03731i −0.854983 0.518656i \(-0.826432\pi\)
0.854983 0.518656i \(-0.173568\pi\)
\(398\) 18032.2i 0.113837i
\(399\) 0 0
\(400\) 93319.6 0.583247
\(401\) 74973.1i 0.466248i 0.972447 + 0.233124i \(0.0748948\pi\)
−0.972447 + 0.233124i \(0.925105\pi\)
\(402\) 0 0
\(403\) 100401. 0.618196
\(404\) 2756.95i 0.0168914i
\(405\) 0 0
\(406\) 264727.i 1.60600i
\(407\) 16211.1 0.0978640
\(408\) 0 0
\(409\) 216548.i 1.29452i 0.762271 + 0.647258i \(0.224084\pi\)
−0.762271 + 0.647258i \(0.775916\pi\)
\(410\) 68441.6i 0.407148i
\(411\) 0 0
\(412\) 8156.13i 0.0480496i
\(413\) 163013. 23502.6i 0.955704 0.137789i
\(414\) 0 0
\(415\) 178292.i 1.03523i
\(416\) −5513.18 −0.0318578
\(417\) 0 0
\(418\) 58222.8 0.333228
\(419\) 221474.i 1.26152i 0.775977 + 0.630761i \(0.217257\pi\)
−0.775977 + 0.630761i \(0.782743\pi\)
\(420\) 0 0
\(421\) 315604.i 1.78065i 0.455329 + 0.890323i \(0.349522\pi\)
−0.455329 + 0.890323i \(0.650478\pi\)
\(422\) 173500. 0.974261
\(423\) 0 0
\(424\) 133266.i 0.741287i
\(425\) 156983. 0.869107
\(426\) 0 0
\(427\) 202219.i 1.10909i
\(428\) 5818.29 0.0317620
\(429\) 0 0
\(430\) −196762. −1.06415
\(431\) 37329.5i 0.200954i −0.994939 0.100477i \(-0.967963\pi\)
0.994939 0.100477i \(-0.0320369\pi\)
\(432\) 0 0
\(433\) 60211.0 0.321144 0.160572 0.987024i \(-0.448666\pi\)
0.160572 + 0.987024i \(0.448666\pi\)
\(434\) −182237. −0.967516
\(435\) 0 0
\(436\) 89.5180i 0.000470909i
\(437\) 428458.i 2.24360i
\(438\) 0 0
\(439\) −185108. −0.960495 −0.480248 0.877133i \(-0.659453\pi\)
−0.480248 + 0.877133i \(0.659453\pi\)
\(440\) −26568.7 −0.137235
\(441\) 0 0
\(442\) −188765. −0.966222
\(443\) 234937.i 1.19714i −0.801071 0.598569i \(-0.795736\pi\)
0.801071 0.598569i \(-0.204264\pi\)
\(444\) 0 0
\(445\) 102775.i 0.519002i
\(446\) 59853.3i 0.300897i
\(447\) 0 0
\(448\) −188606. −0.939722
\(449\) 69120.8 0.342859 0.171430 0.985196i \(-0.445161\pi\)
0.171430 + 0.985196i \(0.445161\pi\)
\(450\) 0 0
\(451\) 26392.5i 0.129756i
\(452\) 5147.98i 0.0251976i
\(453\) 0 0
\(454\) 150918. 0.732201
\(455\) 81983.6i 0.396008i
\(456\) 0 0
\(457\) 307630.i 1.47298i −0.676449 0.736489i \(-0.736482\pi\)
0.676449 0.736489i \(-0.263518\pi\)
\(458\) 147401. 0.702698
\(459\) 0 0
\(460\) 5116.44i 0.0241798i
\(461\) −20386.5 −0.0959272 −0.0479636 0.998849i \(-0.515273\pi\)
−0.0479636 + 0.998849i \(0.515273\pi\)
\(462\) 0 0
\(463\) 50749.8i 0.236740i −0.992970 0.118370i \(-0.962233\pi\)
0.992970 0.118370i \(-0.0377669\pi\)
\(464\) −362396. −1.68325
\(465\) 0 0
\(466\) −83584.1 −0.384904
\(467\) 9160.46i 0.0420033i −0.999779 0.0210017i \(-0.993314\pi\)
0.999779 0.0210017i \(-0.00668553\pi\)
\(468\) 0 0
\(469\) 41041.3i 0.186584i
\(470\) −287315. −1.30066
\(471\) 0 0
\(472\) −31373.2 217603.i −0.140823 0.976746i
\(473\) −75875.5 −0.339140
\(474\) 0 0
\(475\) 199445. 0.883965
\(476\) 8520.20 0.0376042
\(477\) 0 0
\(478\) 74136.3i 0.324471i
\(479\) −281255. −1.22583 −0.612913 0.790150i \(-0.710003\pi\)
−0.612913 + 0.790150i \(0.710003\pi\)
\(480\) 0 0
\(481\) −66774.5 −0.288616
\(482\) 296558.i 1.27649i
\(483\) 0 0
\(484\) −5705.74 −0.0243569
\(485\) 129562.i 0.550799i
\(486\) 0 0
\(487\) −12464.0 −0.0525534 −0.0262767 0.999655i \(-0.508365\pi\)
−0.0262767 + 0.999655i \(0.508365\pi\)
\(488\) 269939. 1.13351
\(489\) 0 0
\(490\) 10796.3i 0.0449659i
\(491\) 267488. 1.10954 0.554768 0.832005i \(-0.312807\pi\)
0.554768 + 0.832005i \(0.312807\pi\)
\(492\) 0 0
\(493\) −609625. −2.50824
\(494\) −239824. −0.982740
\(495\) 0 0
\(496\) 249473.i 1.01405i
\(497\) 11192.0 0.0453100
\(498\) 0 0
\(499\) −211721. −0.850283 −0.425142 0.905127i \(-0.639776\pi\)
−0.425142 + 0.905127i \(0.639776\pi\)
\(500\) 6566.61 0.0262665
\(501\) 0 0
\(502\) 269341.i 1.06880i
\(503\) 46231.2i 0.182725i −0.995818 0.0913627i \(-0.970878\pi\)
0.995818 0.0913627i \(-0.0291223\pi\)
\(504\) 0 0
\(505\) 110884.i 0.434798i
\(506\) 79341.6i 0.309885i
\(507\) 0 0
\(508\) −1526.22 −0.00591412
\(509\) 118612.i 0.457819i 0.973448 + 0.228909i \(0.0735159\pi\)
−0.973448 + 0.228909i \(0.926484\pi\)
\(510\) 0 0
\(511\) 293878.i 1.12545i
\(512\) 251426.i 0.959113i
\(513\) 0 0
\(514\) 194444.i 0.735982i
\(515\) 328040.i 1.23684i
\(516\) 0 0
\(517\) −110795. −0.414513
\(518\) 121203. 0.451702
\(519\) 0 0
\(520\) 109438. 0.404727
\(521\) 20361.9 0.0750140 0.0375070 0.999296i \(-0.488058\pi\)
0.0375070 + 0.999296i \(0.488058\pi\)
\(522\) 0 0
\(523\) 36769.4 0.134426 0.0672130 0.997739i \(-0.478589\pi\)
0.0672130 + 0.997739i \(0.478589\pi\)
\(524\) 8967.13i 0.0326581i
\(525\) 0 0
\(526\) 95097.4i 0.343714i
\(527\) 419664.i 1.51106i
\(528\) 0 0
\(529\) −304029. −1.08643
\(530\) 140263.i 0.499334i
\(531\) 0 0
\(532\) 10824.8 0.0382470
\(533\) 108713.i 0.382671i
\(534\) 0 0
\(535\) −234012. −0.817581
\(536\) −54785.2 −0.190693
\(537\) 0 0
\(538\) −349113. −1.20615
\(539\) 4163.28i 0.0143304i
\(540\) 0 0
\(541\) 136111.i 0.465049i 0.972590 + 0.232525i \(0.0746986\pi\)
−0.972590 + 0.232525i \(0.925301\pi\)
\(542\) 75641.7i 0.257491i
\(543\) 0 0
\(544\) 23044.5i 0.0778700i
\(545\) 3600.42i 0.0121216i
\(546\) 0 0
\(547\) 532541. 1.77983 0.889915 0.456125i \(-0.150763\pi\)
0.889915 + 0.456125i \(0.150763\pi\)
\(548\) −3322.67 −0.0110644
\(549\) 0 0
\(550\) 36933.0 0.122093
\(551\) −774522. −2.55112
\(552\) 0 0
\(553\) 421859. 1.37949
\(554\) 537566.i 1.75151i
\(555\) 0 0
\(556\) −5511.84 −0.0178298
\(557\) 166657. 0.537172 0.268586 0.963256i \(-0.413444\pi\)
0.268586 + 0.963256i \(0.413444\pi\)
\(558\) 0 0
\(559\) 312537. 1.00018
\(560\) −203711. −0.649588
\(561\) 0 0
\(562\) 171943.i 0.544392i
\(563\) 111610.i 0.352116i −0.984380 0.176058i \(-0.943665\pi\)
0.984380 0.176058i \(-0.0563347\pi\)
\(564\) 0 0
\(565\) 207052.i 0.648608i
\(566\) −5743.47 −0.0179284
\(567\) 0 0
\(568\) 14939.9i 0.0463076i
\(569\) 112850.i 0.348560i 0.984696 + 0.174280i \(0.0557598\pi\)
−0.984696 + 0.174280i \(0.944240\pi\)
\(570\) 0 0
\(571\) 441479.i 1.35406i 0.735956 + 0.677029i \(0.236733\pi\)
−0.735956 + 0.677029i \(0.763267\pi\)
\(572\) −1104.36 −0.00337536
\(573\) 0 0
\(574\) 197325.i 0.598904i
\(575\) 271788.i 0.822043i
\(576\) 0 0
\(577\) 214151. 0.643234 0.321617 0.946870i \(-0.395774\pi\)
0.321617 + 0.946870i \(0.395774\pi\)
\(578\) 450701.i 1.34906i
\(579\) 0 0
\(580\) −9248.96 −0.0274939
\(581\) 514034.i 1.52279i
\(582\) 0 0
\(583\) 54088.4i 0.159135i
\(584\) −392291. −1.15023
\(585\) 0 0
\(586\) 181616.i 0.528881i
\(587\) 229548.i 0.666189i −0.942893 0.333094i \(-0.891907\pi\)
0.942893 0.333094i \(-0.108093\pi\)
\(588\) 0 0
\(589\) 533179.i 1.53689i
\(590\) −33020.5 229029.i −0.0948592 0.657941i
\(591\) 0 0
\(592\) 165920.i 0.473428i
\(593\) −161537. −0.459371 −0.229686 0.973265i \(-0.573770\pi\)
−0.229686 + 0.973265i \(0.573770\pi\)
\(594\) 0 0
\(595\) −342683. −0.967962
\(596\) 2345.38i 0.00660268i
\(597\) 0 0
\(598\) 326813.i 0.913898i
\(599\) 259778. 0.724017 0.362008 0.932175i \(-0.382091\pi\)
0.362008 + 0.932175i \(0.382091\pi\)
\(600\) 0 0
\(601\) 609538.i 1.68753i −0.536713 0.843765i \(-0.680334\pi\)
0.536713 0.843765i \(-0.319666\pi\)
\(602\) −567286. −1.56534
\(603\) 0 0
\(604\) 15956.1i 0.0437373i
\(605\) 229485. 0.626966
\(606\) 0 0
\(607\) 123432. 0.335005 0.167502 0.985872i \(-0.446430\pi\)
0.167502 + 0.985872i \(0.446430\pi\)
\(608\) 29277.8i 0.0792012i
\(609\) 0 0
\(610\) 284112. 0.763537
\(611\) 456372. 1.22246
\(612\) 0 0
\(613\) 182402.i 0.485409i −0.970100 0.242705i \(-0.921965\pi\)
0.970100 0.242705i \(-0.0780346\pi\)
\(614\) 63035.2i 0.167204i
\(615\) 0 0
\(616\) −76600.4 −0.201869
\(617\) −550891. −1.44709 −0.723544 0.690278i \(-0.757488\pi\)
−0.723544 + 0.690278i \(0.757488\pi\)
\(618\) 0 0
\(619\) 172751. 0.450856 0.225428 0.974260i \(-0.427622\pi\)
0.225428 + 0.974260i \(0.427622\pi\)
\(620\) 6366.96i 0.0165634i
\(621\) 0 0
\(622\) 651966.i 1.68517i
\(623\) 296312.i 0.763438i
\(624\) 0 0
\(625\) −41803.1 −0.107016
\(626\) 323361. 0.825160
\(627\) 0 0
\(628\) 406.149i 0.00102983i
\(629\) 279111.i 0.705464i
\(630\) 0 0
\(631\) −6939.21 −0.0174282 −0.00871408 0.999962i \(-0.502774\pi\)
−0.00871408 + 0.999962i \(0.502774\pi\)
\(632\) 563132.i 1.40986i
\(633\) 0 0
\(634\) 438611.i 1.09119i
\(635\) 61384.7 0.152234
\(636\) 0 0
\(637\) 17148.8i 0.0422626i
\(638\) −143425. −0.352358
\(639\) 0 0
\(640\) 278696.i 0.680410i
\(641\) 213083. 0.518601 0.259300 0.965797i \(-0.416508\pi\)
0.259300 + 0.965797i \(0.416508\pi\)
\(642\) 0 0
\(643\) −323868. −0.783333 −0.391666 0.920107i \(-0.628101\pi\)
−0.391666 + 0.920107i \(0.628101\pi\)
\(644\) 14751.2i 0.0355678i
\(645\) 0 0
\(646\) 1.00244e6i 2.40211i
\(647\) 268558. 0.641548 0.320774 0.947156i \(-0.396057\pi\)
0.320774 + 0.947156i \(0.396057\pi\)
\(648\) 0 0
\(649\) −12733.4 88318.4i −0.0302311 0.209682i
\(650\) −152130. −0.360070
\(651\) 0 0
\(652\) 20961.5 0.0493091
\(653\) 43918.8 0.102997 0.0514985 0.998673i \(-0.483600\pi\)
0.0514985 + 0.998673i \(0.483600\pi\)
\(654\) 0 0
\(655\) 360658.i 0.840646i
\(656\) 270126. 0.627710
\(657\) 0 0
\(658\) −828361. −1.91323
\(659\) 246753.i 0.568187i 0.958797 + 0.284093i \(0.0916925\pi\)
−0.958797 + 0.284093i \(0.908307\pi\)
\(660\) 0 0
\(661\) 91190.3 0.208711 0.104356 0.994540i \(-0.466722\pi\)
0.104356 + 0.994540i \(0.466722\pi\)
\(662\) 504032.i 1.15012i
\(663\) 0 0
\(664\) −686175. −1.55632
\(665\) −435375. −0.984510
\(666\) 0 0
\(667\) 1.05546e6i 2.37241i
\(668\) −7166.24 −0.0160597
\(669\) 0 0
\(670\) −57661.8 −0.128451
\(671\) 109560. 0.243335
\(672\) 0 0
\(673\) 377278.i 0.832973i −0.909142 0.416486i \(-0.863261\pi\)
0.909142 0.416486i \(-0.136739\pi\)
\(674\) 359202. 0.790713
\(675\) 0 0
\(676\) −7104.57 −0.0155469
\(677\) −301726. −0.658317 −0.329159 0.944275i \(-0.606765\pi\)
−0.329159 + 0.944275i \(0.606765\pi\)
\(678\) 0 0
\(679\) 373540.i 0.810210i
\(680\) 457441.i 0.989275i
\(681\) 0 0
\(682\) 98733.7i 0.212274i
\(683\) 44408.5i 0.0951974i −0.998867 0.0475987i \(-0.984843\pi\)
0.998867 0.0475987i \(-0.0151569\pi\)
\(684\) 0 0
\(685\) 133638. 0.284806
\(686\) 491285.i 1.04396i
\(687\) 0 0
\(688\) 776583.i 1.64063i
\(689\) 222794.i 0.469315i
\(690\) 0 0
\(691\) 428913.i 0.898284i 0.893460 + 0.449142i \(0.148270\pi\)
−0.893460 + 0.449142i \(0.851730\pi\)
\(692\) 24372.0i 0.0508954i
\(693\) 0 0
\(694\) −226157. −0.469560
\(695\) 221687. 0.458955
\(696\) 0 0
\(697\) 454407. 0.935362
\(698\) 268208. 0.550504
\(699\) 0 0
\(700\) 6866.61 0.0140135
\(701\) 142828.i 0.290654i 0.989384 + 0.145327i \(0.0464234\pi\)
−0.989384 + 0.145327i \(0.953577\pi\)
\(702\) 0 0
\(703\) 354607.i 0.717525i
\(704\) 102184.i 0.206176i
\(705\) 0 0
\(706\) −483269. −0.969570
\(707\) 319692.i 0.639577i
\(708\) 0 0
\(709\) 361262. 0.718670 0.359335 0.933209i \(-0.383004\pi\)
0.359335 + 0.933209i \(0.383004\pi\)
\(710\) 15724.4i 0.0311930i
\(711\) 0 0
\(712\) −395542. −0.780247
\(713\) −726575. −1.42923
\(714\) 0 0
\(715\) 44417.6 0.0868845
\(716\) 21420.8i 0.0417839i
\(717\) 0 0
\(718\) 666305.i 1.29248i
\(719\) 364020.i 0.704154i −0.935971 0.352077i \(-0.885475\pi\)
0.935971 0.352077i \(-0.114525\pi\)
\(720\) 0 0
\(721\) 945774.i 1.81935i
\(722\) 745700.i 1.43051i
\(723\) 0 0
\(724\) 408.588 0.000779487
\(725\) −491309. −0.934715
\(726\) 0 0
\(727\) 295247. 0.558619 0.279310 0.960201i \(-0.409894\pi\)
0.279310 + 0.960201i \(0.409894\pi\)
\(728\) 315522. 0.595343
\(729\) 0 0
\(730\) −412889. −0.774797
\(731\) 1.30637e6i 2.44473i
\(732\) 0 0
\(733\) 203200. 0.378195 0.189097 0.981958i \(-0.439444\pi\)
0.189097 + 0.981958i \(0.439444\pi\)
\(734\) −43025.3 −0.0798605
\(735\) 0 0
\(736\) 39897.6 0.0736531
\(737\) −22235.6 −0.0409368
\(738\) 0 0
\(739\) 1.00932e6i 1.84816i −0.382195 0.924082i \(-0.624832\pi\)
0.382195 0.924082i \(-0.375168\pi\)
\(740\) 4234.54i 0.00773291i
\(741\) 0 0
\(742\) 404393.i 0.734508i
\(743\) 535342. 0.969737 0.484869 0.874587i \(-0.338867\pi\)
0.484869 + 0.874587i \(0.338867\pi\)
\(744\) 0 0
\(745\) 94331.2i 0.169958i
\(746\) 202488.i 0.363849i
\(747\) 0 0
\(748\) 4616.13i 0.00825039i
\(749\) −674682. −1.20264
\(750\) 0 0
\(751\) 808120.i 1.43283i −0.697672 0.716417i \(-0.745781\pi\)
0.697672 0.716417i \(-0.254219\pi\)
\(752\) 1.13398e6i 2.00526i
\(753\) 0 0
\(754\) 590779. 1.03916
\(755\) 641754.i 1.12583i
\(756\) 0 0
\(757\) −597248. −1.04223 −0.521114 0.853487i \(-0.674483\pi\)
−0.521114 + 0.853487i \(0.674483\pi\)
\(758\) 93300.0i 0.162384i
\(759\) 0 0
\(760\) 581174.i 1.00619i
\(761\) −337522. −0.582818 −0.291409 0.956599i \(-0.594124\pi\)
−0.291409 + 0.956599i \(0.594124\pi\)
\(762\) 0 0
\(763\) 10380.4i 0.0178305i
\(764\) 26876.8i 0.0460459i
\(765\) 0 0
\(766\) 143873.i 0.245201i
\(767\) 52449.7 + 363790.i 0.0891564 + 0.618386i
\(768\) 0 0
\(769\) 135631.i 0.229354i −0.993403 0.114677i \(-0.963417\pi\)
0.993403 0.114677i \(-0.0365833\pi\)
\(770\) −80622.4 −0.135980
\(771\) 0 0
\(772\) −19076.1 −0.0320078
\(773\) 320257.i 0.535969i 0.963423 + 0.267985i \(0.0863576\pi\)
−0.963423 + 0.267985i \(0.913642\pi\)
\(774\) 0 0
\(775\) 338216.i 0.563107i
\(776\) 498631. 0.828049
\(777\) 0 0
\(778\) 637723.i 1.05359i
\(779\) 577320. 0.951353
\(780\) 0 0
\(781\) 6063.65i 0.00994105i
\(782\) 1.36605e6 2.23384
\(783\) 0 0
\(784\) 42611.0 0.0693249
\(785\) 16335.3i 0.0265087i
\(786\) 0 0
\(787\) −584994. −0.944500 −0.472250 0.881465i \(-0.656558\pi\)
−0.472250 + 0.881465i \(0.656558\pi\)
\(788\) 23024.5 0.0370799
\(789\) 0 0
\(790\) 592700.i 0.949687i
\(791\) 596953.i 0.954085i
\(792\) 0 0
\(793\) −451283. −0.717634
\(794\) −662245. −1.05046
\(795\) 0 0
\(796\) −1816.37 −0.00286667
\(797\)