Properties

Label 531.5.c.d.235.14
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 235.14
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.27

$q$-expansion

\(f(q)\) \(=\) \(q-2.43119i q^{2} +10.0893 q^{4} -25.9319 q^{5} -17.4595 q^{7} -63.4281i q^{8} +O(q^{10})\) \(q-2.43119i q^{2} +10.0893 q^{4} -25.9319 q^{5} -17.4595 q^{7} -63.4281i q^{8} +63.0454i q^{10} -44.7152i q^{11} +157.364i q^{13} +42.4475i q^{14} +7.22282 q^{16} +328.532 q^{17} +678.379 q^{19} -261.635 q^{20} -108.711 q^{22} +424.331i q^{23} +47.4624 q^{25} +382.582 q^{26} -176.154 q^{28} -870.549 q^{29} -1380.41i q^{31} -1032.41i q^{32} -798.725i q^{34} +452.758 q^{35} +2568.09i q^{37} -1649.27i q^{38} +1644.81i q^{40} -503.324 q^{41} -2003.40i q^{43} -451.145i q^{44} +1031.63 q^{46} -2422.01i q^{47} -2096.17 q^{49} -115.390i q^{50} +1587.69i q^{52} +1526.26 q^{53} +1159.55i q^{55} +1107.42i q^{56} +2116.47i q^{58} +(1105.23 - 3300.88i) q^{59} -5708.35i q^{61} -3356.05 q^{62} -2394.42 q^{64} -4080.74i q^{65} -4773.92i q^{67} +3314.66 q^{68} -1100.74i q^{70} +2795.15 q^{71} -4719.76i q^{73} +6243.53 q^{74} +6844.37 q^{76} +780.705i q^{77} +8669.75 q^{79} -187.301 q^{80} +1223.68i q^{82} -7458.93i q^{83} -8519.45 q^{85} -4870.65 q^{86} -2836.20 q^{88} +4001.99i q^{89} -2747.50i q^{91} +4281.21i q^{92} -5888.37 q^{94} -17591.6 q^{95} -5912.30i q^{97} +5096.18i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.43119i 0.607798i −0.952704 0.303899i \(-0.901711\pi\)
0.952704 0.303899i \(-0.0982886\pi\)
\(3\) 0 0
\(4\) 10.0893 0.630581
\(5\) −25.9319 −1.03728 −0.518638 0.854994i \(-0.673561\pi\)
−0.518638 + 0.854994i \(0.673561\pi\)
\(6\) 0 0
\(7\) −17.4595 −0.356317 −0.178158 0.984002i \(-0.557014\pi\)
−0.178158 + 0.984002i \(0.557014\pi\)
\(8\) 63.4281i 0.991064i
\(9\) 0 0
\(10\) 63.0454i 0.630454i
\(11\) 44.7152i 0.369547i −0.982781 0.184773i \(-0.940845\pi\)
0.982781 0.184773i \(-0.0591551\pi\)
\(12\) 0 0
\(13\) 157.364i 0.931148i 0.885009 + 0.465574i \(0.154152\pi\)
−0.885009 + 0.465574i \(0.845848\pi\)
\(14\) 42.4475i 0.216569i
\(15\) 0 0
\(16\) 7.22282 0.0282141
\(17\) 328.532 1.13679 0.568395 0.822756i \(-0.307565\pi\)
0.568395 + 0.822756i \(0.307565\pi\)
\(18\) 0 0
\(19\) 678.379 1.87917 0.939583 0.342322i \(-0.111213\pi\)
0.939583 + 0.342322i \(0.111213\pi\)
\(20\) −261.635 −0.654086
\(21\) 0 0
\(22\) −108.711 −0.224610
\(23\) 424.331i 0.802139i 0.916048 + 0.401069i \(0.131361\pi\)
−0.916048 + 0.401069i \(0.868639\pi\)
\(24\) 0 0
\(25\) 47.4624 0.0759398
\(26\) 382.582 0.565950
\(27\) 0 0
\(28\) −176.154 −0.224687
\(29\) −870.549 −1.03514 −0.517568 0.855642i \(-0.673163\pi\)
−0.517568 + 0.855642i \(0.673163\pi\)
\(30\) 0 0
\(31\) 1380.41i 1.43643i −0.695820 0.718216i \(-0.744959\pi\)
0.695820 0.718216i \(-0.255041\pi\)
\(32\) 1032.41i 1.00821i
\(33\) 0 0
\(34\) 798.725i 0.690938i
\(35\) 452.758 0.369599
\(36\) 0 0
\(37\) 2568.09i 1.87589i 0.346784 + 0.937945i \(0.387274\pi\)
−0.346784 + 0.937945i \(0.612726\pi\)
\(38\) 1649.27i 1.14215i
\(39\) 0 0
\(40\) 1644.81i 1.02801i
\(41\) −503.324 −0.299419 −0.149710 0.988730i \(-0.547834\pi\)
−0.149710 + 0.988730i \(0.547834\pi\)
\(42\) 0 0
\(43\) 2003.40i 1.08350i −0.840538 0.541752i \(-0.817761\pi\)
0.840538 0.541752i \(-0.182239\pi\)
\(44\) 451.145i 0.233029i
\(45\) 0 0
\(46\) 1031.63 0.487538
\(47\) 2422.01i 1.09643i −0.836338 0.548214i \(-0.815308\pi\)
0.836338 0.548214i \(-0.184692\pi\)
\(48\) 0 0
\(49\) −2096.17 −0.873038
\(50\) 115.390i 0.0461561i
\(51\) 0 0
\(52\) 1587.69i 0.587164i
\(53\) 1526.26 0.543347 0.271674 0.962389i \(-0.412423\pi\)
0.271674 + 0.962389i \(0.412423\pi\)
\(54\) 0 0
\(55\) 1159.55i 0.383322i
\(56\) 1107.42i 0.353133i
\(57\) 0 0
\(58\) 2116.47i 0.629154i
\(59\) 1105.23 3300.88i 0.317504 0.948257i
\(60\) 0 0
\(61\) 5708.35i 1.53409i −0.641594 0.767045i \(-0.721726\pi\)
0.641594 0.767045i \(-0.278274\pi\)
\(62\) −3356.05 −0.873061
\(63\) 0 0
\(64\) −2394.42 −0.584576
\(65\) 4080.74i 0.965857i
\(66\) 0 0
\(67\) 4773.92i 1.06347i −0.846911 0.531735i \(-0.821540\pi\)
0.846911 0.531735i \(-0.178460\pi\)
\(68\) 3314.66 0.716838
\(69\) 0 0
\(70\) 1100.74i 0.224641i
\(71\) 2795.15 0.554484 0.277242 0.960800i \(-0.410580\pi\)
0.277242 + 0.960800i \(0.410580\pi\)
\(72\) 0 0
\(73\) 4719.76i 0.885674i −0.896602 0.442837i \(-0.853972\pi\)
0.896602 0.442837i \(-0.146028\pi\)
\(74\) 6243.53 1.14016
\(75\) 0 0
\(76\) 6844.37 1.18497
\(77\) 780.705i 0.131676i
\(78\) 0 0
\(79\) 8669.75 1.38916 0.694580 0.719415i \(-0.255590\pi\)
0.694580 + 0.719415i \(0.255590\pi\)
\(80\) −187.301 −0.0292658
\(81\) 0 0
\(82\) 1223.68i 0.181987i
\(83\) 7458.93i 1.08273i −0.840787 0.541365i \(-0.817908\pi\)
0.840787 0.541365i \(-0.182092\pi\)
\(84\) 0 0
\(85\) −8519.45 −1.17916
\(86\) −4870.65 −0.658552
\(87\) 0 0
\(88\) −2836.20 −0.366245
\(89\) 4001.99i 0.505238i 0.967566 + 0.252619i \(0.0812920\pi\)
−0.967566 + 0.252619i \(0.918708\pi\)
\(90\) 0 0
\(91\) 2747.50i 0.331784i
\(92\) 4281.21i 0.505814i
\(93\) 0 0
\(94\) −5888.37 −0.666407
\(95\) −17591.6 −1.94921
\(96\) 0 0
\(97\) 5912.30i 0.628367i −0.949362 0.314183i \(-0.898269\pi\)
0.949362 0.314183i \(-0.101731\pi\)
\(98\) 5096.18i 0.530631i
\(99\) 0 0
\(100\) 478.862 0.0478862
\(101\) 14483.7i 1.41983i 0.704286 + 0.709916i \(0.251267\pi\)
−0.704286 + 0.709916i \(0.748733\pi\)
\(102\) 0 0
\(103\) 211.537i 0.0199394i 0.999950 + 0.00996968i \(0.00317350\pi\)
−0.999950 + 0.00996968i \(0.996826\pi\)
\(104\) 9981.30 0.922828
\(105\) 0 0
\(106\) 3710.64i 0.330246i
\(107\) 12751.0 1.11373 0.556863 0.830605i \(-0.312005\pi\)
0.556863 + 0.830605i \(0.312005\pi\)
\(108\) 0 0
\(109\) 18050.8i 1.51930i −0.650334 0.759648i \(-0.725371\pi\)
0.650334 0.759648i \(-0.274629\pi\)
\(110\) 2819.09 0.232982
\(111\) 0 0
\(112\) −126.107 −0.0100532
\(113\) 9631.35i 0.754276i 0.926157 + 0.377138i \(0.123092\pi\)
−0.926157 + 0.377138i \(0.876908\pi\)
\(114\) 0 0
\(115\) 11003.7i 0.832039i
\(116\) −8783.24 −0.652737
\(117\) 0 0
\(118\) −8025.08 2687.03i −0.576349 0.192979i
\(119\) −5736.01 −0.405057
\(120\) 0 0
\(121\) 12641.6 0.863435
\(122\) −13878.1 −0.932417
\(123\) 0 0
\(124\) 13927.4i 0.905788i
\(125\) 14976.6 0.958505
\(126\) 0 0
\(127\) −25023.8 −1.55148 −0.775741 0.631052i \(-0.782624\pi\)
−0.775741 + 0.631052i \(0.782624\pi\)
\(128\) 10697.3i 0.652909i
\(129\) 0 0
\(130\) −9921.08 −0.587046
\(131\) 2066.15i 0.120398i −0.998186 0.0601990i \(-0.980826\pi\)
0.998186 0.0601990i \(-0.0191735\pi\)
\(132\) 0 0
\(133\) −11844.2 −0.669578
\(134\) −11606.3 −0.646375
\(135\) 0 0
\(136\) 20838.2i 1.12663i
\(137\) −8060.06 −0.429434 −0.214717 0.976676i \(-0.568883\pi\)
−0.214717 + 0.976676i \(0.568883\pi\)
\(138\) 0 0
\(139\) −8208.19 −0.424833 −0.212416 0.977179i \(-0.568133\pi\)
−0.212416 + 0.977179i \(0.568133\pi\)
\(140\) 4568.01 0.233062
\(141\) 0 0
\(142\) 6795.56i 0.337014i
\(143\) 7036.56 0.344103
\(144\) 0 0
\(145\) 22575.0 1.07372
\(146\) −11474.6 −0.538311
\(147\) 0 0
\(148\) 25910.3i 1.18290i
\(149\) 37339.6i 1.68189i −0.541122 0.840944i \(-0.682000\pi\)
0.541122 0.840944i \(-0.318000\pi\)
\(150\) 0 0
\(151\) 10313.8i 0.452340i −0.974088 0.226170i \(-0.927379\pi\)
0.974088 0.226170i \(-0.0726205\pi\)
\(152\) 43028.3i 1.86237i
\(153\) 0 0
\(154\) 1898.04 0.0800323
\(155\) 35796.7i 1.48998i
\(156\) 0 0
\(157\) 10989.2i 0.445826i −0.974838 0.222913i \(-0.928443\pi\)
0.974838 0.222913i \(-0.0715566\pi\)
\(158\) 21077.8i 0.844329i
\(159\) 0 0
\(160\) 26772.3i 1.04579i
\(161\) 7408.62i 0.285815i
\(162\) 0 0
\(163\) 1659.70 0.0624676 0.0312338 0.999512i \(-0.490056\pi\)
0.0312338 + 0.999512i \(0.490056\pi\)
\(164\) −5078.19 −0.188808
\(165\) 0 0
\(166\) −18134.1 −0.658082
\(167\) 5744.24 0.205968 0.102984 0.994683i \(-0.467161\pi\)
0.102984 + 0.994683i \(0.467161\pi\)
\(168\) 0 0
\(169\) 3797.58 0.132964
\(170\) 20712.4i 0.716693i
\(171\) 0 0
\(172\) 20212.9i 0.683238i
\(173\) 48337.4i 1.61507i 0.589821 + 0.807534i \(0.299198\pi\)
−0.589821 + 0.807534i \(0.700802\pi\)
\(174\) 0 0
\(175\) −828.671 −0.0270586
\(176\) 322.969i 0.0104264i
\(177\) 0 0
\(178\) 9729.62 0.307083
\(179\) 8489.09i 0.264945i −0.991187 0.132472i \(-0.957708\pi\)
0.991187 0.132472i \(-0.0422916\pi\)
\(180\) 0 0
\(181\) −63579.4 −1.94070 −0.970352 0.241696i \(-0.922296\pi\)
−0.970352 + 0.241696i \(0.922296\pi\)
\(182\) −6679.70 −0.201657
\(183\) 0 0
\(184\) 26914.5 0.794971
\(185\) 66595.5i 1.94581i
\(186\) 0 0
\(187\) 14690.4i 0.420097i
\(188\) 24436.4i 0.691387i
\(189\) 0 0
\(190\) 42768.7i 1.18473i
\(191\) 36006.4i 0.986991i −0.869748 0.493495i \(-0.835719\pi\)
0.869748 0.493495i \(-0.164281\pi\)
\(192\) 0 0
\(193\) 7371.06 0.197886 0.0989431 0.995093i \(-0.468454\pi\)
0.0989431 + 0.995093i \(0.468454\pi\)
\(194\) −14373.9 −0.381920
\(195\) 0 0
\(196\) −21148.8 −0.550522
\(197\) −62886.8 −1.62042 −0.810209 0.586141i \(-0.800647\pi\)
−0.810209 + 0.586141i \(0.800647\pi\)
\(198\) 0 0
\(199\) 24801.5 0.626285 0.313143 0.949706i \(-0.398618\pi\)
0.313143 + 0.949706i \(0.398618\pi\)
\(200\) 3010.45i 0.0752613i
\(201\) 0 0
\(202\) 35212.7 0.862972
\(203\) 15199.4 0.368836
\(204\) 0 0
\(205\) 13052.1 0.310580
\(206\) 514.286 0.0121191
\(207\) 0 0
\(208\) 1136.61i 0.0262715i
\(209\) 30333.8i 0.694440i
\(210\) 0 0
\(211\) 17502.7i 0.393134i −0.980490 0.196567i \(-0.937021\pi\)
0.980490 0.196567i \(-0.0629794\pi\)
\(212\) 15398.9 0.342625
\(213\) 0 0
\(214\) 31000.2i 0.676920i
\(215\) 51951.9i 1.12389i
\(216\) 0 0
\(217\) 24101.3i 0.511825i
\(218\) −43884.9 −0.923426
\(219\) 0 0
\(220\) 11699.0i 0.241716i
\(221\) 51699.1i 1.05852i
\(222\) 0 0
\(223\) 54042.6 1.08674 0.543371 0.839493i \(-0.317148\pi\)
0.543371 + 0.839493i \(0.317148\pi\)
\(224\) 18025.4i 0.359243i
\(225\) 0 0
\(226\) 23415.7 0.458448
\(227\) 24892.4i 0.483076i −0.970391 0.241538i \(-0.922348\pi\)
0.970391 0.241538i \(-0.0776518\pi\)
\(228\) 0 0
\(229\) 81045.6i 1.54546i 0.634733 + 0.772731i \(0.281110\pi\)
−0.634733 + 0.772731i \(0.718890\pi\)
\(230\) −26752.1 −0.505712
\(231\) 0 0
\(232\) 55217.3i 1.02589i
\(233\) 92463.9i 1.70318i 0.524209 + 0.851589i \(0.324361\pi\)
−0.524209 + 0.851589i \(0.675639\pi\)
\(234\) 0 0
\(235\) 62807.2i 1.13730i
\(236\) 11151.0 33303.6i 0.200212 0.597953i
\(237\) 0 0
\(238\) 13945.4i 0.246193i
\(239\) 66752.2 1.16861 0.584305 0.811534i \(-0.301367\pi\)
0.584305 + 0.811534i \(0.301367\pi\)
\(240\) 0 0
\(241\) −57886.0 −0.996643 −0.498322 0.866992i \(-0.666050\pi\)
−0.498322 + 0.866992i \(0.666050\pi\)
\(242\) 30734.1i 0.524794i
\(243\) 0 0
\(244\) 57593.2i 0.967368i
\(245\) 54357.5 0.905581
\(246\) 0 0
\(247\) 106752.i 1.74978i
\(248\) −87556.9 −1.42360
\(249\) 0 0
\(250\) 36411.1i 0.582577i
\(251\) 1482.64 0.0235336 0.0117668 0.999931i \(-0.496254\pi\)
0.0117668 + 0.999931i \(0.496254\pi\)
\(252\) 0 0
\(253\) 18974.0 0.296428
\(254\) 60837.8i 0.942988i
\(255\) 0 0
\(256\) −64317.9 −0.981413
\(257\) 63595.3 0.962850 0.481425 0.876487i \(-0.340119\pi\)
0.481425 + 0.876487i \(0.340119\pi\)
\(258\) 0 0
\(259\) 44837.7i 0.668411i
\(260\) 41171.9i 0.609051i
\(261\) 0 0
\(262\) −5023.21 −0.0731777
\(263\) 79407.3 1.14802 0.574010 0.818849i \(-0.305387\pi\)
0.574010 + 0.818849i \(0.305387\pi\)
\(264\) 0 0
\(265\) −39578.9 −0.563601
\(266\) 28795.5i 0.406968i
\(267\) 0 0
\(268\) 48165.5i 0.670605i
\(269\) 103489.i 1.43018i 0.699031 + 0.715091i \(0.253615\pi\)
−0.699031 + 0.715091i \(0.746385\pi\)
\(270\) 0 0
\(271\) 144360. 1.96566 0.982830 0.184513i \(-0.0590709\pi\)
0.982830 + 0.184513i \(0.0590709\pi\)
\(272\) 2372.93 0.0320735
\(273\) 0 0
\(274\) 19595.5i 0.261010i
\(275\) 2122.29i 0.0280633i
\(276\) 0 0
\(277\) 47254.9 0.615867 0.307934 0.951408i \(-0.400363\pi\)
0.307934 + 0.951408i \(0.400363\pi\)
\(278\) 19955.7i 0.258213i
\(279\) 0 0
\(280\) 28717.6i 0.366296i
\(281\) 31731.5 0.401863 0.200931 0.979605i \(-0.435603\pi\)
0.200931 + 0.979605i \(0.435603\pi\)
\(282\) 0 0
\(283\) 12999.5i 0.162313i −0.996701 0.0811563i \(-0.974139\pi\)
0.996701 0.0811563i \(-0.0258613\pi\)
\(284\) 28201.1 0.349647
\(285\) 0 0
\(286\) 17107.2i 0.209145i
\(287\) 8787.80 0.106688
\(288\) 0 0
\(289\) 24412.3 0.292289
\(290\) 54884.1i 0.652606i
\(291\) 0 0
\(292\) 47619.0i 0.558489i
\(293\) 27627.7 0.321817 0.160909 0.986969i \(-0.448558\pi\)
0.160909 + 0.986969i \(0.448558\pi\)
\(294\) 0 0
\(295\) −28660.8 + 85598.1i −0.329339 + 0.983603i
\(296\) 162889. 1.85913
\(297\) 0 0
\(298\) −90779.8 −1.02225
\(299\) −66774.5 −0.746910
\(300\) 0 0
\(301\) 34978.4i 0.386071i
\(302\) −25074.9 −0.274932
\(303\) 0 0
\(304\) 4899.81 0.0530190
\(305\) 148028.i 1.59127i
\(306\) 0 0
\(307\) −25754.0 −0.273255 −0.136627 0.990623i \(-0.543626\pi\)
−0.136627 + 0.990623i \(0.543626\pi\)
\(308\) 7876.77i 0.0830322i
\(309\) 0 0
\(310\) 87028.6 0.905605
\(311\) −123728. −1.27923 −0.639613 0.768697i \(-0.720905\pi\)
−0.639613 + 0.768697i \(0.720905\pi\)
\(312\) 0 0
\(313\) 12824.2i 0.130900i −0.997856 0.0654502i \(-0.979152\pi\)
0.997856 0.0654502i \(-0.0208484\pi\)
\(314\) −26716.8 −0.270972
\(315\) 0 0
\(316\) 87471.7 0.875978
\(317\) 136329. 1.35666 0.678329 0.734759i \(-0.262705\pi\)
0.678329 + 0.734759i \(0.262705\pi\)
\(318\) 0 0
\(319\) 38926.8i 0.382531i
\(320\) 62091.9 0.606366
\(321\) 0 0
\(322\) −18011.8 −0.173718
\(323\) 222869. 2.13622
\(324\) 0 0
\(325\) 7468.87i 0.0707112i
\(326\) 4035.05i 0.0379677i
\(327\) 0 0
\(328\) 31924.9i 0.296744i
\(329\) 42287.1i 0.390676i
\(330\) 0 0
\(331\) −153145. −1.39781 −0.698904 0.715216i \(-0.746328\pi\)
−0.698904 + 0.715216i \(0.746328\pi\)
\(332\) 75255.4i 0.682750i
\(333\) 0 0
\(334\) 13965.4i 0.125187i
\(335\) 123797.i 1.10311i
\(336\) 0 0
\(337\) 60949.0i 0.536669i −0.963326 0.268335i \(-0.913527\pi\)
0.963326 0.268335i \(-0.0864733\pi\)
\(338\) 9232.64i 0.0808151i
\(339\) 0 0
\(340\) −85955.3 −0.743558
\(341\) −61725.3 −0.530829
\(342\) 0 0
\(343\) 78518.3 0.667395
\(344\) −127072. −1.07382
\(345\) 0 0
\(346\) 117517. 0.981635
\(347\) 150274.i 1.24803i −0.781412 0.624015i \(-0.785500\pi\)
0.781412 0.624015i \(-0.214500\pi\)
\(348\) 0 0
\(349\) 4128.34i 0.0338942i 0.999856 + 0.0169471i \(0.00539468\pi\)
−0.999856 + 0.0169471i \(0.994605\pi\)
\(350\) 2014.66i 0.0164462i
\(351\) 0 0
\(352\) −46164.4 −0.372582
\(353\) 126346.i 1.01394i −0.861964 0.506969i \(-0.830766\pi\)
0.861964 0.506969i \(-0.169234\pi\)
\(354\) 0 0
\(355\) −72483.6 −0.575152
\(356\) 40377.3i 0.318594i
\(357\) 0 0
\(358\) −20638.6 −0.161033
\(359\) −182628. −1.41703 −0.708514 0.705697i \(-0.750634\pi\)
−0.708514 + 0.705697i \(0.750634\pi\)
\(360\) 0 0
\(361\) 329877. 2.53126
\(362\) 154574.i 1.17956i
\(363\) 0 0
\(364\) 27720.4i 0.209217i
\(365\) 122392.i 0.918688i
\(366\) 0 0
\(367\) 8500.43i 0.0631116i 0.999502 + 0.0315558i \(0.0100462\pi\)
−0.999502 + 0.0315558i \(0.989954\pi\)
\(368\) 3064.87i 0.0226316i
\(369\) 0 0
\(370\) −161907. −1.18266
\(371\) −26647.8 −0.193604
\(372\) 0 0
\(373\) 82283.7 0.591420 0.295710 0.955278i \(-0.404444\pi\)
0.295710 + 0.955278i \(0.404444\pi\)
\(374\) −35715.1 −0.255334
\(375\) 0 0
\(376\) −153623. −1.08663
\(377\) 136993.i 0.963865i
\(378\) 0 0
\(379\) 2637.31 0.0183604 0.00918022 0.999958i \(-0.497078\pi\)
0.00918022 + 0.999958i \(0.497078\pi\)
\(380\) −177487. −1.22914
\(381\) 0 0
\(382\) −87538.5 −0.599891
\(383\) −45438.5 −0.309761 −0.154880 0.987933i \(-0.549499\pi\)
−0.154880 + 0.987933i \(0.549499\pi\)
\(384\) 0 0
\(385\) 20245.2i 0.136584i
\(386\) 17920.5i 0.120275i
\(387\) 0 0
\(388\) 59651.0i 0.396236i
\(389\) 36809.8 0.243257 0.121628 0.992576i \(-0.461188\pi\)
0.121628 + 0.992576i \(0.461188\pi\)
\(390\) 0 0
\(391\) 139406.i 0.911862i
\(392\) 132956.i 0.865237i
\(393\) 0 0
\(394\) 152890.i 0.984888i
\(395\) −224823. −1.44094
\(396\) 0 0
\(397\) 184237.i 1.16895i 0.811412 + 0.584475i \(0.198700\pi\)
−0.811412 + 0.584475i \(0.801300\pi\)
\(398\) 60297.3i 0.380655i
\(399\) 0 0
\(400\) 342.812 0.00214258
\(401\) 243995.i 1.51738i −0.651455 0.758688i \(-0.725841\pi\)
0.651455 0.758688i \(-0.274159\pi\)
\(402\) 0 0
\(403\) 217227. 1.33753
\(404\) 146131.i 0.895320i
\(405\) 0 0
\(406\) 36952.6i 0.224178i
\(407\) 114833. 0.693229
\(408\) 0 0
\(409\) 39142.7i 0.233994i 0.993132 + 0.116997i \(0.0373268\pi\)
−0.993132 + 0.116997i \(0.962673\pi\)
\(410\) 31732.3i 0.188770i
\(411\) 0 0
\(412\) 2134.26i 0.0125734i
\(413\) −19296.8 + 57631.8i −0.113132 + 0.337880i
\(414\) 0 0
\(415\) 193424.i 1.12309i
\(416\) 162464. 0.938795
\(417\) 0 0
\(418\) −73747.4 −0.422079
\(419\) 148939.i 0.848362i −0.905577 0.424181i \(-0.860562\pi\)
0.905577 0.424181i \(-0.139438\pi\)
\(420\) 0 0
\(421\) 258237.i 1.45698i 0.685055 + 0.728492i \(0.259778\pi\)
−0.685055 + 0.728492i \(0.740222\pi\)
\(422\) −42552.5 −0.238946
\(423\) 0 0
\(424\) 96808.0i 0.538492i
\(425\) 15592.9 0.0863276
\(426\) 0 0
\(427\) 99665.0i 0.546622i
\(428\) 128649. 0.702295
\(429\) 0 0
\(430\) 126305. 0.683100
\(431\) 260135.i 1.40037i −0.713960 0.700187i \(-0.753100\pi\)
0.713960 0.700187i \(-0.246900\pi\)
\(432\) 0 0
\(433\) 281301. 1.50036 0.750181 0.661232i \(-0.229966\pi\)
0.750181 + 0.661232i \(0.229966\pi\)
\(434\) 58595.0 0.311086
\(435\) 0 0
\(436\) 182120.i 0.958040i
\(437\) 287857.i 1.50735i
\(438\) 0 0
\(439\) −226376. −1.17463 −0.587316 0.809357i \(-0.699816\pi\)
−0.587316 + 0.809357i \(0.699816\pi\)
\(440\) 73548.0 0.379897
\(441\) 0 0
\(442\) 125691. 0.643366
\(443\) 191214.i 0.974346i −0.873305 0.487173i \(-0.838028\pi\)
0.873305 0.487173i \(-0.161972\pi\)
\(444\) 0 0
\(445\) 103779.i 0.524071i
\(446\) 131388.i 0.660519i
\(447\) 0 0
\(448\) 41805.5 0.208294
\(449\) −205380. −1.01874 −0.509372 0.860546i \(-0.670122\pi\)
−0.509372 + 0.860546i \(0.670122\pi\)
\(450\) 0 0
\(451\) 22506.2i 0.110649i
\(452\) 97173.6i 0.475633i
\(453\) 0 0
\(454\) −60518.3 −0.293613
\(455\) 71247.8i 0.344151i
\(456\) 0 0
\(457\) 145715.i 0.697703i 0.937178 + 0.348852i \(0.113428\pi\)
−0.937178 + 0.348852i \(0.886572\pi\)
\(458\) 197038. 0.939329
\(459\) 0 0
\(460\) 111020.i 0.524668i
\(461\) 91731.3 0.431634 0.215817 0.976434i \(-0.430759\pi\)
0.215817 + 0.976434i \(0.430759\pi\)
\(462\) 0 0
\(463\) 363576.i 1.69603i 0.529974 + 0.848014i \(0.322202\pi\)
−0.529974 + 0.848014i \(0.677798\pi\)
\(464\) −6287.82 −0.0292055
\(465\) 0 0
\(466\) 224797. 1.03519
\(467\) 107740.i 0.494019i 0.969013 + 0.247009i \(0.0794478\pi\)
−0.969013 + 0.247009i \(0.920552\pi\)
\(468\) 0 0
\(469\) 83350.3i 0.378932i
\(470\) 152697. 0.691247
\(471\) 0 0
\(472\) −209369. 70102.9i −0.939784 0.314667i
\(473\) −89582.3 −0.400406
\(474\) 0 0
\(475\) 32197.5 0.142704
\(476\) −57872.4 −0.255421
\(477\) 0 0
\(478\) 162288.i 0.710280i
\(479\) −148486. −0.647165 −0.323582 0.946200i \(-0.604887\pi\)
−0.323582 + 0.946200i \(0.604887\pi\)
\(480\) 0 0
\(481\) −404125. −1.74673
\(482\) 140732.i 0.605758i
\(483\) 0 0
\(484\) 127544. 0.544466
\(485\) 153317.i 0.651789i
\(486\) 0 0
\(487\) −3404.16 −0.0143533 −0.00717666 0.999974i \(-0.502284\pi\)
−0.00717666 + 0.999974i \(0.502284\pi\)
\(488\) −362070. −1.52038
\(489\) 0 0
\(490\) 132154.i 0.550411i
\(491\) −326718. −1.35522 −0.677611 0.735420i \(-0.736985\pi\)
−0.677611 + 0.735420i \(0.736985\pi\)
\(492\) 0 0
\(493\) −286003. −1.17673
\(494\) 259536. 1.06351
\(495\) 0 0
\(496\) 9970.46i 0.0405277i
\(497\) −48802.0 −0.197572
\(498\) 0 0
\(499\) 170255. 0.683754 0.341877 0.939745i \(-0.388937\pi\)
0.341877 + 0.939745i \(0.388937\pi\)
\(500\) 151104. 0.604415
\(501\) 0 0
\(502\) 3604.58i 0.0143037i
\(503\) 198224.i 0.783466i −0.920079 0.391733i \(-0.871876\pi\)
0.920079 0.391733i \(-0.128124\pi\)
\(504\) 0 0
\(505\) 375590.i 1.47276i
\(506\) 46129.6i 0.180168i
\(507\) 0 0
\(508\) −252473. −0.978335
\(509\) 252370.i 0.974099i 0.873374 + 0.487049i \(0.161927\pi\)
−0.873374 + 0.487049i \(0.838073\pi\)
\(510\) 0 0
\(511\) 82404.7i 0.315580i
\(512\) 14787.0i 0.0564079i
\(513\) 0 0
\(514\) 154612.i 0.585219i
\(515\) 5485.54i 0.0206826i
\(516\) 0 0
\(517\) −108301. −0.405181
\(518\) −109009. −0.406259
\(519\) 0 0
\(520\) −258834. −0.957226
\(521\) 372104. 1.37085 0.685424 0.728144i \(-0.259617\pi\)
0.685424 + 0.728144i \(0.259617\pi\)
\(522\) 0 0
\(523\) −68686.7 −0.251113 −0.125557 0.992086i \(-0.540072\pi\)
−0.125557 + 0.992086i \(0.540072\pi\)
\(524\) 20846.0i 0.0759207i
\(525\) 0 0
\(526\) 193055.i 0.697764i
\(527\) 453510.i 1.63292i
\(528\) 0 0
\(529\) 99783.9 0.356574
\(530\) 96223.8i 0.342556i
\(531\) 0 0
\(532\) −119499. −0.422223
\(533\) 79205.1i 0.278804i
\(534\) 0 0
\(535\) −330659. −1.15524
\(536\) −302801. −1.05397
\(537\) 0 0
\(538\) 251603. 0.869262
\(539\) 93730.4i 0.322629i
\(540\) 0 0
\(541\) 51022.3i 0.174327i −0.996194 0.0871637i \(-0.972220\pi\)
0.996194 0.0871637i \(-0.0277803\pi\)
\(542\) 350967.i 1.19472i
\(543\) 0 0
\(544\) 339180.i 1.14613i
\(545\) 468090.i 1.57593i
\(546\) 0 0
\(547\) 116201. 0.388362 0.194181 0.980966i \(-0.437795\pi\)
0.194181 + 0.980966i \(0.437795\pi\)
\(548\) −81320.3 −0.270793
\(549\) 0 0
\(550\) −5159.69 −0.0170568
\(551\) −590562. −1.94519
\(552\) 0 0
\(553\) −151370. −0.494981
\(554\) 114886.i 0.374323i
\(555\) 0 0
\(556\) −82814.9 −0.267892
\(557\) 428570. 1.38137 0.690687 0.723154i \(-0.257308\pi\)
0.690687 + 0.723154i \(0.257308\pi\)
\(558\) 0 0
\(559\) 315263. 1.00890
\(560\) 3270.19 0.0104279
\(561\) 0 0
\(562\) 77145.4i 0.244251i
\(563\) 131577.i 0.415110i 0.978223 + 0.207555i \(0.0665506\pi\)
−0.978223 + 0.207555i \(0.933449\pi\)
\(564\) 0 0
\(565\) 249759.i 0.782392i
\(566\) −31604.2 −0.0986533
\(567\) 0 0
\(568\) 177291.i 0.549529i
\(569\) 399439.i 1.23375i −0.787063 0.616873i \(-0.788399\pi\)
0.787063 0.616873i \(-0.211601\pi\)
\(570\) 0 0
\(571\) 131744.i 0.404072i −0.979378 0.202036i \(-0.935244\pi\)
0.979378 0.202036i \(-0.0647558\pi\)
\(572\) 70993.9 0.216985
\(573\) 0 0
\(574\) 21364.8i 0.0648449i
\(575\) 20139.8i 0.0609143i
\(576\) 0 0
\(577\) −160918. −0.483339 −0.241670 0.970359i \(-0.577695\pi\)
−0.241670 + 0.970359i \(0.577695\pi\)
\(578\) 59351.0i 0.177653i
\(579\) 0 0
\(580\) 227766. 0.677068
\(581\) 130229.i 0.385795i
\(582\) 0 0
\(583\) 68247.1i 0.200792i
\(584\) −299365. −0.877760
\(585\) 0 0
\(586\) 67168.2i 0.195600i
\(587\) 171759.i 0.498474i −0.968443 0.249237i \(-0.919820\pi\)
0.968443 0.249237i \(-0.0801798\pi\)
\(588\) 0 0
\(589\) 936442.i 2.69929i
\(590\) 208105. + 69679.9i 0.597832 + 0.200172i
\(591\) 0 0
\(592\) 18548.9i 0.0529266i
\(593\) −457675. −1.30151 −0.650755 0.759288i \(-0.725547\pi\)
−0.650755 + 0.759288i \(0.725547\pi\)
\(594\) 0 0
\(595\) 148746. 0.420156
\(596\) 376731.i 1.06057i
\(597\) 0 0
\(598\) 162342.i 0.453970i
\(599\) 450231. 1.25482 0.627410 0.778689i \(-0.284115\pi\)
0.627410 + 0.778689i \(0.284115\pi\)
\(600\) 0 0
\(601\) 188572.i 0.522071i −0.965329 0.261035i \(-0.915936\pi\)
0.965329 0.261035i \(-0.0840639\pi\)
\(602\) 85039.2 0.234653
\(603\) 0 0
\(604\) 104059.i 0.285237i
\(605\) −327819. −0.895620
\(606\) 0 0
\(607\) 266272. 0.722682 0.361341 0.932434i \(-0.382319\pi\)
0.361341 + 0.932434i \(0.382319\pi\)
\(608\) 700365.i 1.89460i
\(609\) 0 0
\(610\) 359885. 0.967173
\(611\) 381137. 1.02094
\(612\) 0 0
\(613\) 331432.i 0.882010i −0.897505 0.441005i \(-0.854622\pi\)
0.897505 0.441005i \(-0.145378\pi\)
\(614\) 62612.9i 0.166084i
\(615\) 0 0
\(616\) 49518.7 0.130499
\(617\) 341099. 0.896005 0.448002 0.894032i \(-0.352136\pi\)
0.448002 + 0.894032i \(0.352136\pi\)
\(618\) 0 0
\(619\) −101548. −0.265027 −0.132513 0.991181i \(-0.542305\pi\)
−0.132513 + 0.991181i \(0.542305\pi\)
\(620\) 361163.i 0.939551i
\(621\) 0 0
\(622\) 300807.i 0.777511i
\(623\) 69872.9i 0.180025i
\(624\) 0 0
\(625\) −418036. −1.07017
\(626\) −31178.1 −0.0795611
\(627\) 0 0
\(628\) 110873.i 0.281130i
\(629\) 843701.i 2.13249i
\(630\) 0 0
\(631\) −278409. −0.699238 −0.349619 0.936892i \(-0.613689\pi\)
−0.349619 + 0.936892i \(0.613689\pi\)
\(632\) 549906.i 1.37675i
\(633\) 0 0
\(634\) 331442.i 0.824574i
\(635\) 648915. 1.60931
\(636\) 0 0
\(637\) 329861.i 0.812928i
\(638\) 94638.5 0.232502
\(639\) 0 0
\(640\) 277400.i 0.677246i
\(641\) −267843. −0.651874 −0.325937 0.945391i \(-0.605680\pi\)
−0.325937 + 0.945391i \(0.605680\pi\)
\(642\) 0 0
\(643\) 247371. 0.598311 0.299155 0.954204i \(-0.403295\pi\)
0.299155 + 0.954204i \(0.403295\pi\)
\(644\) 74747.8i 0.180230i
\(645\) 0 0
\(646\) 541838.i 1.29839i
\(647\) 66763.5 0.159489 0.0797445 0.996815i \(-0.474590\pi\)
0.0797445 + 0.996815i \(0.474590\pi\)
\(648\) 0 0
\(649\) −147599. 49420.7i −0.350425 0.117333i
\(650\) 18158.3 0.0429781
\(651\) 0 0
\(652\) 16745.2 0.0393909
\(653\) 585647. 1.37344 0.686719 0.726923i \(-0.259050\pi\)
0.686719 + 0.726923i \(0.259050\pi\)
\(654\) 0 0
\(655\) 53579.2i 0.124886i
\(656\) −3635.42 −0.00844786
\(657\) 0 0
\(658\) 102808. 0.237452
\(659\) 34391.7i 0.0791922i −0.999216 0.0395961i \(-0.987393\pi\)
0.999216 0.0395961i \(-0.0126071\pi\)
\(660\) 0 0
\(661\) −151480. −0.346698 −0.173349 0.984860i \(-0.555459\pi\)
−0.173349 + 0.984860i \(0.555459\pi\)
\(662\) 372325.i 0.849585i
\(663\) 0 0
\(664\) −473106. −1.07306
\(665\) 307142. 0.694537
\(666\) 0 0
\(667\) 369401.i 0.830323i
\(668\) 57955.4 0.129880
\(669\) 0 0
\(670\) 300974. 0.670469
\(671\) −255250. −0.566918
\(672\) 0 0
\(673\) 355378.i 0.784622i 0.919833 + 0.392311i \(0.128324\pi\)
−0.919833 + 0.392311i \(0.871676\pi\)
\(674\) −148179. −0.326187
\(675\) 0 0
\(676\) 38314.9 0.0838444
\(677\) −670210. −1.46229 −0.731145 0.682222i \(-0.761014\pi\)
−0.731145 + 0.682222i \(0.761014\pi\)
\(678\) 0 0
\(679\) 103226.i 0.223898i
\(680\) 540373.i 1.16863i
\(681\) 0 0
\(682\) 150066.i 0.322637i
\(683\) 237295.i 0.508683i 0.967114 + 0.254341i \(0.0818587\pi\)
−0.967114 + 0.254341i \(0.918141\pi\)
\(684\) 0 0
\(685\) 209012. 0.445442
\(686\) 190893.i 0.405641i
\(687\) 0 0
\(688\) 14470.2i 0.0305701i
\(689\) 240179.i 0.505937i
\(690\) 0 0
\(691\) 40265.6i 0.0843293i 0.999111 + 0.0421646i \(0.0134254\pi\)
−0.999111 + 0.0421646i \(0.986575\pi\)
\(692\) 487690.i 1.01843i
\(693\) 0 0
\(694\) −365345. −0.758551
\(695\) 212854. 0.440668
\(696\) 0 0
\(697\) −165358. −0.340377
\(698\) 10036.8 0.0206008
\(699\) 0 0
\(700\) −8360.71 −0.0170627
\(701\) 55287.6i 0.112510i −0.998416 0.0562551i \(-0.982084\pi\)
0.998416 0.0562551i \(-0.0179160\pi\)
\(702\) 0 0
\(703\) 1.74214e6i 3.52511i
\(704\) 107067.i 0.216028i
\(705\) 0 0
\(706\) −307171. −0.616270
\(707\) 252879.i 0.505910i
\(708\) 0 0
\(709\) 91346.8 0.181719 0.0908596 0.995864i \(-0.471039\pi\)
0.0908596 + 0.995864i \(0.471039\pi\)
\(710\) 176222.i 0.349577i
\(711\) 0 0
\(712\) 253839. 0.500724
\(713\) 585752. 1.15222
\(714\) 0 0
\(715\) −182471. −0.356929
\(716\) 85649.0i 0.167069i
\(717\) 0 0
\(718\) 444004.i 0.861267i
\(719\) 25854.5i 0.0500124i 0.999687 + 0.0250062i \(0.00796055\pi\)
−0.999687 + 0.0250062i \(0.992039\pi\)
\(720\) 0 0
\(721\) 3693.33i 0.00710473i
\(722\) 801994.i 1.53850i
\(723\) 0 0
\(724\) −641472. −1.22377
\(725\) −41318.4 −0.0786081
\(726\) 0 0
\(727\) 441000. 0.834391 0.417196 0.908817i \(-0.363013\pi\)
0.417196 + 0.908817i \(0.363013\pi\)
\(728\) −174269. −0.328819
\(729\) 0 0
\(730\) 297559. 0.558377
\(731\) 658181.i 1.23172i
\(732\) 0 0
\(733\) 426201. 0.793244 0.396622 0.917982i \(-0.370182\pi\)
0.396622 + 0.917982i \(0.370182\pi\)
\(734\) 20666.2 0.0383591
\(735\) 0 0
\(736\) 438084. 0.808727
\(737\) −213467. −0.393002
\(738\) 0 0
\(739\) 91383.8i 0.167333i −0.996494 0.0836663i \(-0.973337\pi\)
0.996494 0.0836663i \(-0.0266630\pi\)
\(740\) 671902.i 1.22699i
\(741\) 0 0
\(742\) 64786.0i 0.117672i
\(743\) −309686. −0.560975 −0.280488 0.959858i \(-0.590496\pi\)
−0.280488 + 0.959858i \(0.590496\pi\)
\(744\) 0 0
\(745\) 968286.i 1.74458i
\(746\) 200048.i 0.359464i
\(747\) 0 0
\(748\) 148215.i 0.264905i
\(749\) −222627. −0.396839
\(750\) 0 0
\(751\) 218166.i 0.386819i 0.981118 + 0.193409i \(0.0619545\pi\)
−0.981118 + 0.193409i \(0.938045\pi\)
\(752\) 17493.7i 0.0309348i
\(753\) 0 0
\(754\) −333057. −0.585835
\(755\) 267457.i 0.469201i
\(756\) 0 0
\(757\) 48000.1 0.0837627 0.0418813 0.999123i \(-0.486665\pi\)
0.0418813 + 0.999123i \(0.486665\pi\)
\(758\) 6411.81i 0.0111594i
\(759\) 0 0
\(760\) 1.11580e6i 1.93179i
\(761\) −236294. −0.408021 −0.204011 0.978969i \(-0.565398\pi\)
−0.204011 + 0.978969i \(0.565398\pi\)
\(762\) 0 0
\(763\) 315158.i 0.541351i
\(764\) 363279.i 0.622378i
\(765\) 0 0
\(766\) 110470.i 0.188272i
\(767\) 519440. + 173924.i 0.882967 + 0.295644i
\(768\) 0 0
\(769\) 110994.i 0.187692i −0.995587 0.0938459i \(-0.970084\pi\)
0.995587 0.0938459i \(-0.0299161\pi\)
\(770\) −49219.9 −0.0830155
\(771\) 0 0
\(772\) 74368.9 0.124783
\(773\) 164246.i 0.274875i −0.990510 0.137437i \(-0.956113\pi\)
0.990510 0.137437i \(-0.0438866\pi\)
\(774\) 0 0
\(775\) 65517.7i 0.109082i
\(776\) −375006. −0.622752
\(777\) 0 0
\(778\) 89491.8i 0.147851i
\(779\) −341444. −0.562659
\(780\) 0 0
\(781\) 124986.i 0.204908i
\(782\) 338924. 0.554228
\(783\) 0 0
\(784\) −15140.2 −0.0246320
\(785\) 284970.i 0.462445i
\(786\) 0 0
\(787\) 909477. 1.46839 0.734196 0.678937i \(-0.237559\pi\)
0.734196 + 0.678937i \(0.237559\pi\)
\(788\) −634484. −1.02181
\(789\) 0 0
\(790\) 546588.i 0.875802i
\(791\) 168159.i 0.268761i
\(792\) 0 0
\(793\) 898288. 1.42846
\(794\) 447916. 0.710486
\(795\) 0 0
\(796\) 250230. 0.394924
\(797\)