Properties

Label 531.5.c.d.235.17
Level $531$
Weight $5$
Character 531.235
Analytic conductor $54.889$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

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 235.17
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.24

$q$-expansion

\(f(q)\) \(=\) \(q-2.15560i q^{2} +11.3534 q^{4} -30.9385 q^{5} +73.3307 q^{7} -58.9629i q^{8} +O(q^{10})\) \(q-2.15560i q^{2} +11.3534 q^{4} -30.9385 q^{5} +73.3307 q^{7} -58.9629i q^{8} +66.6909i q^{10} +203.451i q^{11} -115.092i q^{13} -158.071i q^{14} +54.5542 q^{16} +149.563 q^{17} -97.5337 q^{19} -351.257 q^{20} +438.557 q^{22} +785.535i q^{23} +332.189 q^{25} -248.092 q^{26} +832.553 q^{28} -279.591 q^{29} +520.224i q^{31} -1061.00i q^{32} -322.398i q^{34} -2268.74 q^{35} +737.478i q^{37} +210.243i q^{38} +1824.22i q^{40} -540.849 q^{41} +1171.64i q^{43} +2309.86i q^{44} +1693.30 q^{46} -1505.45i q^{47} +2976.40 q^{49} -716.067i q^{50} -1306.69i q^{52} +3630.24 q^{53} -6294.45i q^{55} -4323.79i q^{56} +602.686i q^{58} +(-3344.70 + 964.542i) q^{59} +5340.44i q^{61} +1121.39 q^{62} -1414.23 q^{64} +3560.77i q^{65} +1694.69i q^{67} +1698.05 q^{68} +4890.49i q^{70} +9066.29 q^{71} -876.744i q^{73} +1589.71 q^{74} -1107.34 q^{76} +14919.2i q^{77} +7895.46 q^{79} -1687.82 q^{80} +1165.85i q^{82} -3535.87i q^{83} -4627.26 q^{85} +2525.58 q^{86} +11996.0 q^{88} +11418.1i q^{89} -8439.78i q^{91} +8918.49i q^{92} -3245.15 q^{94} +3017.55 q^{95} +15153.6i q^{97} -6415.91i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.15560i 0.538899i −0.963014 0.269450i \(-0.913158\pi\)
0.963014 0.269450i \(-0.0868418\pi\)
\(3\) 0 0
\(4\) 11.3534 0.709588
\(5\) −30.9385 −1.23754 −0.618770 0.785573i \(-0.712369\pi\)
−0.618770 + 0.785573i \(0.712369\pi\)
\(6\) 0 0
\(7\) 73.3307 1.49655 0.748273 0.663391i \(-0.230883\pi\)
0.748273 + 0.663391i \(0.230883\pi\)
\(8\) 58.9629i 0.921295i
\(9\) 0 0
\(10\) 66.6909i 0.666909i
\(11\) 203.451i 1.68141i 0.541494 + 0.840705i \(0.317859\pi\)
−0.541494 + 0.840705i \(0.682141\pi\)
\(12\) 0 0
\(13\) 115.092i 0.681018i −0.940241 0.340509i \(-0.889401\pi\)
0.940241 0.340509i \(-0.110599\pi\)
\(14\) 158.071i 0.806487i
\(15\) 0 0
\(16\) 54.5542 0.213102
\(17\) 149.563 0.517520 0.258760 0.965942i \(-0.416686\pi\)
0.258760 + 0.965942i \(0.416686\pi\)
\(18\) 0 0
\(19\) −97.5337 −0.270177 −0.135088 0.990834i \(-0.543132\pi\)
−0.135088 + 0.990834i \(0.543132\pi\)
\(20\) −351.257 −0.878143
\(21\) 0 0
\(22\) 438.557 0.906110
\(23\) 785.535i 1.48494i 0.669878 + 0.742472i \(0.266347\pi\)
−0.669878 + 0.742472i \(0.733653\pi\)
\(24\) 0 0
\(25\) 332.189 0.531503
\(26\) −248.092 −0.367000
\(27\) 0 0
\(28\) 832.553 1.06193
\(29\) −279.591 −0.332451 −0.166226 0.986088i \(-0.553158\pi\)
−0.166226 + 0.986088i \(0.553158\pi\)
\(30\) 0 0
\(31\) 520.224i 0.541336i 0.962673 + 0.270668i \(0.0872445\pi\)
−0.962673 + 0.270668i \(0.912755\pi\)
\(32\) 1061.00i 1.03614i
\(33\) 0 0
\(34\) 322.398i 0.278891i
\(35\) −2268.74 −1.85203
\(36\) 0 0
\(37\) 737.478i 0.538699i 0.963043 + 0.269349i \(0.0868086\pi\)
−0.963043 + 0.269349i \(0.913191\pi\)
\(38\) 210.243i 0.145598i
\(39\) 0 0
\(40\) 1824.22i 1.14014i
\(41\) −540.849 −0.321743 −0.160871 0.986975i \(-0.551430\pi\)
−0.160871 + 0.986975i \(0.551430\pi\)
\(42\) 0 0
\(43\) 1171.64i 0.633660i 0.948482 + 0.316830i \(0.102618\pi\)
−0.948482 + 0.316830i \(0.897382\pi\)
\(44\) 2309.86i 1.19311i
\(45\) 0 0
\(46\) 1693.30 0.800235
\(47\) 1505.45i 0.681509i −0.940152 0.340755i \(-0.889318\pi\)
0.940152 0.340755i \(-0.110682\pi\)
\(48\) 0 0
\(49\) 2976.40 1.23965
\(50\) 716.067i 0.286427i
\(51\) 0 0
\(52\) 1306.69i 0.483242i
\(53\) 3630.24 1.29236 0.646179 0.763185i \(-0.276366\pi\)
0.646179 + 0.763185i \(0.276366\pi\)
\(54\) 0 0
\(55\) 6294.45i 2.08081i
\(56\) 4323.79i 1.37876i
\(57\) 0 0
\(58\) 602.686i 0.179158i
\(59\) −3344.70 + 964.542i −0.960845 + 0.277088i
\(60\) 0 0
\(61\) 5340.44i 1.43522i 0.696447 + 0.717608i \(0.254763\pi\)
−0.696447 + 0.717608i \(0.745237\pi\)
\(62\) 1121.39 0.291725
\(63\) 0 0
\(64\) −1414.23 −0.345270
\(65\) 3560.77i 0.842786i
\(66\) 0 0
\(67\) 1694.69i 0.377520i 0.982023 + 0.188760i \(0.0604469\pi\)
−0.982023 + 0.188760i \(0.939553\pi\)
\(68\) 1698.05 0.367226
\(69\) 0 0
\(70\) 4890.49i 0.998059i
\(71\) 9066.29 1.79851 0.899255 0.437425i \(-0.144109\pi\)
0.899255 + 0.437425i \(0.144109\pi\)
\(72\) 0 0
\(73\) 876.744i 0.164523i −0.996611 0.0822616i \(-0.973786\pi\)
0.996611 0.0822616i \(-0.0262143\pi\)
\(74\) 1589.71 0.290304
\(75\) 0 0
\(76\) −1107.34 −0.191714
\(77\) 14919.2i 2.51631i
\(78\) 0 0
\(79\) 7895.46 1.26509 0.632547 0.774522i \(-0.282009\pi\)
0.632547 + 0.774522i \(0.282009\pi\)
\(80\) −1687.82 −0.263723
\(81\) 0 0
\(82\) 1165.85i 0.173387i
\(83\) 3535.87i 0.513264i −0.966509 0.256632i \(-0.917387\pi\)
0.966509 0.256632i \(-0.0826128\pi\)
\(84\) 0 0
\(85\) −4627.26 −0.640451
\(86\) 2525.58 0.341479
\(87\) 0 0
\(88\) 11996.0 1.54907
\(89\) 11418.1i 1.44150i 0.693195 + 0.720750i \(0.256203\pi\)
−0.693195 + 0.720750i \(0.743797\pi\)
\(90\) 0 0
\(91\) 8439.78i 1.01917i
\(92\) 8918.49i 1.05370i
\(93\) 0 0
\(94\) −3245.15 −0.367265
\(95\) 3017.55 0.334354
\(96\) 0 0
\(97\) 15153.6i 1.61054i 0.592909 + 0.805269i \(0.297979\pi\)
−0.592909 + 0.805269i \(0.702021\pi\)
\(98\) 6415.91i 0.668046i
\(99\) 0 0
\(100\) 3771.48 0.377148
\(101\) 10320.2i 1.01168i −0.862626 0.505842i \(-0.831182\pi\)
0.862626 0.505842i \(-0.168818\pi\)
\(102\) 0 0
\(103\) 13724.8i 1.29369i 0.762620 + 0.646847i \(0.223913\pi\)
−0.762620 + 0.646847i \(0.776087\pi\)
\(104\) −6786.16 −0.627419
\(105\) 0 0
\(106\) 7825.33i 0.696451i
\(107\) 9382.60 0.819513 0.409756 0.912195i \(-0.365614\pi\)
0.409756 + 0.912195i \(0.365614\pi\)
\(108\) 0 0
\(109\) 15647.4i 1.31701i −0.752575 0.658507i \(-0.771188\pi\)
0.752575 0.658507i \(-0.228812\pi\)
\(110\) −13568.3 −1.12135
\(111\) 0 0
\(112\) 4000.50 0.318917
\(113\) 16451.0i 1.28836i −0.764875 0.644179i \(-0.777199\pi\)
0.764875 0.644179i \(-0.222801\pi\)
\(114\) 0 0
\(115\) 24303.3i 1.83768i
\(116\) −3174.31 −0.235903
\(117\) 0 0
\(118\) 2079.16 + 7209.82i 0.149322 + 0.517798i
\(119\) 10967.6 0.774492
\(120\) 0 0
\(121\) −26751.1 −1.82714
\(122\) 11511.8 0.773437
\(123\) 0 0
\(124\) 5906.31i 0.384125i
\(125\) 9059.11 0.579783
\(126\) 0 0
\(127\) 15007.8 0.930484 0.465242 0.885183i \(-0.345967\pi\)
0.465242 + 0.885183i \(0.345967\pi\)
\(128\) 13927.5i 0.850070i
\(129\) 0 0
\(130\) 7675.59 0.454177
\(131\) 521.489i 0.0303880i 0.999885 + 0.0151940i \(0.00483659\pi\)
−0.999885 + 0.0151940i \(0.995163\pi\)
\(132\) 0 0
\(133\) −7152.22 −0.404332
\(134\) 3653.07 0.203445
\(135\) 0 0
\(136\) 8818.68i 0.476789i
\(137\) 29732.3 1.58411 0.792057 0.610447i \(-0.209010\pi\)
0.792057 + 0.610447i \(0.209010\pi\)
\(138\) 0 0
\(139\) −5711.68 −0.295620 −0.147810 0.989016i \(-0.547222\pi\)
−0.147810 + 0.989016i \(0.547222\pi\)
\(140\) −25757.9 −1.31418
\(141\) 0 0
\(142\) 19543.3i 0.969216i
\(143\) 23415.5 1.14507
\(144\) 0 0
\(145\) 8650.13 0.411421
\(146\) −1889.91 −0.0886614
\(147\) 0 0
\(148\) 8372.89i 0.382254i
\(149\) 6283.15i 0.283012i 0.989937 + 0.141506i \(0.0451944\pi\)
−0.989937 + 0.141506i \(0.954806\pi\)
\(150\) 0 0
\(151\) 39803.8i 1.74571i 0.487983 + 0.872853i \(0.337733\pi\)
−0.487983 + 0.872853i \(0.662267\pi\)
\(152\) 5750.87i 0.248912i
\(153\) 0 0
\(154\) 32159.7 1.35604
\(155\) 16094.9i 0.669924i
\(156\) 0 0
\(157\) 2933.82i 0.119024i −0.998228 0.0595119i \(-0.981046\pi\)
0.998228 0.0595119i \(-0.0189544\pi\)
\(158\) 17019.4i 0.681758i
\(159\) 0 0
\(160\) 32825.8i 1.28226i
\(161\) 57603.8i 2.22228i
\(162\) 0 0
\(163\) 44939.2 1.69142 0.845708 0.533645i \(-0.179178\pi\)
0.845708 + 0.533645i \(0.179178\pi\)
\(164\) −6140.48 −0.228305
\(165\) 0 0
\(166\) −7621.92 −0.276597
\(167\) −5237.32 −0.187791 −0.0938957 0.995582i \(-0.529932\pi\)
−0.0938957 + 0.995582i \(0.529932\pi\)
\(168\) 0 0
\(169\) 15314.8 0.536215
\(170\) 9974.50i 0.345138i
\(171\) 0 0
\(172\) 13302.1i 0.449637i
\(173\) 23988.2i 0.801503i 0.916187 + 0.400751i \(0.131251\pi\)
−0.916187 + 0.400751i \(0.868749\pi\)
\(174\) 0 0
\(175\) 24359.7 0.795419
\(176\) 11099.1i 0.358312i
\(177\) 0 0
\(178\) 24612.9 0.776824
\(179\) 48379.7i 1.50993i −0.655765 0.754965i \(-0.727654\pi\)
0.655765 0.754965i \(-0.272346\pi\)
\(180\) 0 0
\(181\) −9403.26 −0.287026 −0.143513 0.989648i \(-0.545840\pi\)
−0.143513 + 0.989648i \(0.545840\pi\)
\(182\) −18192.8 −0.549232
\(183\) 0 0
\(184\) 46317.4 1.36807
\(185\) 22816.5i 0.666661i
\(186\) 0 0
\(187\) 30428.7i 0.870163i
\(188\) 17092.0i 0.483590i
\(189\) 0 0
\(190\) 6504.61i 0.180183i
\(191\) 26293.8i 0.720754i 0.932807 + 0.360377i \(0.117352\pi\)
−0.932807 + 0.360377i \(0.882648\pi\)
\(192\) 0 0
\(193\) −35955.0 −0.965260 −0.482630 0.875824i \(-0.660318\pi\)
−0.482630 + 0.875824i \(0.660318\pi\)
\(194\) 32665.0 0.867918
\(195\) 0 0
\(196\) 33792.2 0.879639
\(197\) −16157.0 −0.416321 −0.208160 0.978095i \(-0.566748\pi\)
−0.208160 + 0.978095i \(0.566748\pi\)
\(198\) 0 0
\(199\) −8326.75 −0.210266 −0.105133 0.994458i \(-0.533527\pi\)
−0.105133 + 0.994458i \(0.533527\pi\)
\(200\) 19586.9i 0.489671i
\(201\) 0 0
\(202\) −22246.2 −0.545195
\(203\) −20502.6 −0.497528
\(204\) 0 0
\(205\) 16733.1 0.398169
\(206\) 29585.1 0.697171
\(207\) 0 0
\(208\) 6278.75i 0.145127i
\(209\) 19843.3i 0.454277i
\(210\) 0 0
\(211\) 75444.4i 1.69458i −0.531130 0.847290i \(-0.678232\pi\)
0.531130 0.847290i \(-0.321768\pi\)
\(212\) 41215.5 0.917042
\(213\) 0 0
\(214\) 20225.1i 0.441635i
\(215\) 36248.7i 0.784179i
\(216\) 0 0
\(217\) 38148.4i 0.810134i
\(218\) −33729.6 −0.709738
\(219\) 0 0
\(220\) 71463.4i 1.47652i
\(221\) 17213.5i 0.352440i
\(222\) 0 0
\(223\) 80926.3 1.62735 0.813673 0.581322i \(-0.197464\pi\)
0.813673 + 0.581322i \(0.197464\pi\)
\(224\) 77804.2i 1.55062i
\(225\) 0 0
\(226\) −35461.8 −0.694295
\(227\) 50468.1i 0.979411i 0.871888 + 0.489706i \(0.162896\pi\)
−0.871888 + 0.489706i \(0.837104\pi\)
\(228\) 0 0
\(229\) 88590.6i 1.68934i −0.535288 0.844669i \(-0.679797\pi\)
0.535288 0.844669i \(-0.320203\pi\)
\(230\) −52388.0 −0.990322
\(231\) 0 0
\(232\) 16485.5i 0.306286i
\(233\) 21367.1i 0.393580i 0.980446 + 0.196790i \(0.0630518\pi\)
−0.980446 + 0.196790i \(0.936948\pi\)
\(234\) 0 0
\(235\) 46576.4i 0.843394i
\(236\) −37973.7 + 10950.8i −0.681804 + 0.196618i
\(237\) 0 0
\(238\) 23641.7i 0.417373i
\(239\) −27789.3 −0.486498 −0.243249 0.969964i \(-0.578213\pi\)
−0.243249 + 0.969964i \(0.578213\pi\)
\(240\) 0 0
\(241\) −114284. −1.96767 −0.983836 0.179071i \(-0.942691\pi\)
−0.983836 + 0.179071i \(0.942691\pi\)
\(242\) 57664.6i 0.984643i
\(243\) 0 0
\(244\) 60632.1i 1.01841i
\(245\) −92085.2 −1.53411
\(246\) 0 0
\(247\) 11225.4i 0.183995i
\(248\) 30673.9 0.498730
\(249\) 0 0
\(250\) 19527.8i 0.312445i
\(251\) −76189.7 −1.20934 −0.604671 0.796475i \(-0.706695\pi\)
−0.604671 + 0.796475i \(0.706695\pi\)
\(252\) 0 0
\(253\) −159818. −2.49680
\(254\) 32350.7i 0.501437i
\(255\) 0 0
\(256\) −52649.8 −0.803373
\(257\) 9932.87 0.150386 0.0751932 0.997169i \(-0.476043\pi\)
0.0751932 + 0.997169i \(0.476043\pi\)
\(258\) 0 0
\(259\) 54079.8i 0.806187i
\(260\) 40426.9i 0.598031i
\(261\) 0 0
\(262\) 1124.12 0.0163761
\(263\) −34983.1 −0.505762 −0.252881 0.967497i \(-0.581378\pi\)
−0.252881 + 0.967497i \(0.581378\pi\)
\(264\) 0 0
\(265\) −112314. −1.59934
\(266\) 15417.3i 0.217894i
\(267\) 0 0
\(268\) 19240.5i 0.267884i
\(269\) 61848.8i 0.854725i 0.904080 + 0.427363i \(0.140557\pi\)
−0.904080 + 0.427363i \(0.859443\pi\)
\(270\) 0 0
\(271\) −91570.5 −1.24686 −0.623429 0.781880i \(-0.714261\pi\)
−0.623429 + 0.781880i \(0.714261\pi\)
\(272\) 8159.30 0.110285
\(273\) 0 0
\(274\) 64090.7i 0.853678i
\(275\) 67584.1i 0.893675i
\(276\) 0 0
\(277\) −88687.7 −1.15586 −0.577928 0.816088i \(-0.696139\pi\)
−0.577928 + 0.816088i \(0.696139\pi\)
\(278\) 12312.1i 0.159309i
\(279\) 0 0
\(280\) 133772.i 1.70627i
\(281\) 54231.2 0.686809 0.343405 0.939188i \(-0.388420\pi\)
0.343405 + 0.939188i \(0.388420\pi\)
\(282\) 0 0
\(283\) 19900.8i 0.248483i −0.992252 0.124242i \(-0.960350\pi\)
0.992252 0.124242i \(-0.0396498\pi\)
\(284\) 102933. 1.27620
\(285\) 0 0
\(286\) 50474.5i 0.617077i
\(287\) −39660.9 −0.481503
\(288\) 0 0
\(289\) −61151.9 −0.732173
\(290\) 18646.2i 0.221715i
\(291\) 0 0
\(292\) 9954.03i 0.116744i
\(293\) −88852.6 −1.03499 −0.517494 0.855687i \(-0.673135\pi\)
−0.517494 + 0.855687i \(0.673135\pi\)
\(294\) 0 0
\(295\) 103480. 29841.5i 1.18908 0.342907i
\(296\) 43483.9 0.496301
\(297\) 0 0
\(298\) 13543.9 0.152515
\(299\) 90408.8 1.01127
\(300\) 0 0
\(301\) 85917.0i 0.948301i
\(302\) 85801.0 0.940760
\(303\) 0 0
\(304\) −5320.88 −0.0575753
\(305\) 165225.i 1.77614i
\(306\) 0 0
\(307\) −5886.60 −0.0624579 −0.0312290 0.999512i \(-0.509942\pi\)
−0.0312290 + 0.999512i \(0.509942\pi\)
\(308\) 169383.i 1.78554i
\(309\) 0 0
\(310\) −34694.2 −0.361022
\(311\) 94490.4 0.976938 0.488469 0.872581i \(-0.337556\pi\)
0.488469 + 0.872581i \(0.337556\pi\)
\(312\) 0 0
\(313\) 90255.1i 0.921262i −0.887592 0.460631i \(-0.847623\pi\)
0.887592 0.460631i \(-0.152377\pi\)
\(314\) −6324.13 −0.0641418
\(315\) 0 0
\(316\) 89640.3 0.897696
\(317\) −159546. −1.58770 −0.793849 0.608115i \(-0.791926\pi\)
−0.793849 + 0.608115i \(0.791926\pi\)
\(318\) 0 0
\(319\) 56883.0i 0.558987i
\(320\) 43754.1 0.427286
\(321\) 0 0
\(322\) 124171. 1.19759
\(323\) −14587.5 −0.139822
\(324\) 0 0
\(325\) 38232.4i 0.361963i
\(326\) 96870.9i 0.911503i
\(327\) 0 0
\(328\) 31890.1i 0.296420i
\(329\) 110396.i 1.01991i
\(330\) 0 0
\(331\) 20831.2 0.190134 0.0950669 0.995471i \(-0.469694\pi\)
0.0950669 + 0.995471i \(0.469694\pi\)
\(332\) 40144.2i 0.364206i
\(333\) 0 0
\(334\) 11289.5i 0.101201i
\(335\) 52431.1i 0.467196i
\(336\) 0 0
\(337\) 189217.i 1.66610i 0.553199 + 0.833049i \(0.313407\pi\)
−0.553199 + 0.833049i \(0.686593\pi\)
\(338\) 33012.6i 0.288966i
\(339\) 0 0
\(340\) −52535.1 −0.454456
\(341\) −105840. −0.910207
\(342\) 0 0
\(343\) 42194.2 0.358645
\(344\) 69083.1 0.583788
\(345\) 0 0
\(346\) 51708.8 0.431929
\(347\) 150636.i 1.25104i 0.780209 + 0.625519i \(0.215113\pi\)
−0.780209 + 0.625519i \(0.784887\pi\)
\(348\) 0 0
\(349\) 80454.6i 0.660541i −0.943886 0.330271i \(-0.892860\pi\)
0.943886 0.330271i \(-0.107140\pi\)
\(350\) 52509.7i 0.428651i
\(351\) 0 0
\(352\) 215862. 1.74217
\(353\) 93087.5i 0.747037i 0.927623 + 0.373518i \(0.121849\pi\)
−0.927623 + 0.373518i \(0.878151\pi\)
\(354\) 0 0
\(355\) −280497. −2.22573
\(356\) 129635.i 1.02287i
\(357\) 0 0
\(358\) −104287. −0.813700
\(359\) 36181.0 0.280732 0.140366 0.990100i \(-0.455172\pi\)
0.140366 + 0.990100i \(0.455172\pi\)
\(360\) 0 0
\(361\) −120808. −0.927005
\(362\) 20269.6i 0.154678i
\(363\) 0 0
\(364\) 95820.3i 0.723194i
\(365\) 27125.1i 0.203604i
\(366\) 0 0
\(367\) 147497.i 1.09509i −0.836775 0.547547i \(-0.815562\pi\)
0.836775 0.547547i \(-0.184438\pi\)
\(368\) 42854.2i 0.316445i
\(369\) 0 0
\(370\) −49183.1 −0.359263
\(371\) 266208. 1.93407
\(372\) 0 0
\(373\) 93006.2 0.668489 0.334244 0.942486i \(-0.391519\pi\)
0.334244 + 0.942486i \(0.391519\pi\)
\(374\) 65592.0 0.468930
\(375\) 0 0
\(376\) −88765.9 −0.627871
\(377\) 32178.7i 0.226405i
\(378\) 0 0
\(379\) −197981. −1.37831 −0.689153 0.724615i \(-0.742017\pi\)
−0.689153 + 0.724615i \(0.742017\pi\)
\(380\) 34259.4 0.237254
\(381\) 0 0
\(382\) 56678.9 0.388414
\(383\) −130011. −0.886307 −0.443153 0.896446i \(-0.646140\pi\)
−0.443153 + 0.896446i \(0.646140\pi\)
\(384\) 0 0
\(385\) 461577.i 3.11403i
\(386\) 77504.4i 0.520178i
\(387\) 0 0
\(388\) 172044.i 1.14282i
\(389\) 97356.2 0.643375 0.321688 0.946846i \(-0.395750\pi\)
0.321688 + 0.946846i \(0.395750\pi\)
\(390\) 0 0
\(391\) 117487.i 0.768487i
\(392\) 175497.i 1.14208i
\(393\) 0 0
\(394\) 34827.9i 0.224355i
\(395\) −244273. −1.56560
\(396\) 0 0
\(397\) 21590.6i 0.136988i 0.997652 + 0.0684942i \(0.0218195\pi\)
−0.997652 + 0.0684942i \(0.978181\pi\)
\(398\) 17949.1i 0.113312i
\(399\) 0 0
\(400\) 18122.3 0.113265
\(401\) 14919.3i 0.0927809i −0.998923 0.0463905i \(-0.985228\pi\)
0.998923 0.0463905i \(-0.0147718\pi\)
\(402\) 0 0
\(403\) 59873.6 0.368659
\(404\) 117169.i 0.717878i
\(405\) 0 0
\(406\) 44195.4i 0.268118i
\(407\) −150040. −0.905773
\(408\) 0 0
\(409\) 90196.9i 0.539194i −0.962973 0.269597i \(-0.913110\pi\)
0.962973 0.269597i \(-0.0868905\pi\)
\(410\) 36069.7i 0.214573i
\(411\) 0 0
\(412\) 155823.i 0.917989i
\(413\) −245269. + 70730.5i −1.43795 + 0.414674i
\(414\) 0 0
\(415\) 109395.i 0.635184i
\(416\) −122113. −0.705627
\(417\) 0 0
\(418\) −42774.1 −0.244810
\(419\) 141820.i 0.807808i 0.914801 + 0.403904i \(0.132347\pi\)
−0.914801 + 0.403904i \(0.867653\pi\)
\(420\) 0 0
\(421\) 328314.i 1.85236i 0.377084 + 0.926179i \(0.376927\pi\)
−0.377084 + 0.926179i \(0.623073\pi\)
\(422\) −162628. −0.913208
\(423\) 0 0
\(424\) 214049.i 1.19064i
\(425\) 49683.3 0.275063
\(426\) 0 0
\(427\) 391618.i 2.14787i
\(428\) 106524. 0.581516
\(429\) 0 0
\(430\) −78137.5 −0.422593
\(431\) 256724.i 1.38201i −0.722849 0.691006i \(-0.757168\pi\)
0.722849 0.691006i \(-0.242832\pi\)
\(432\) 0 0
\(433\) 63696.1 0.339733 0.169866 0.985467i \(-0.445666\pi\)
0.169866 + 0.985467i \(0.445666\pi\)
\(434\) 82232.5 0.436580
\(435\) 0 0
\(436\) 177652.i 0.934537i
\(437\) 76616.2i 0.401197i
\(438\) 0 0
\(439\) 301024. 1.56197 0.780985 0.624550i \(-0.214717\pi\)
0.780985 + 0.624550i \(0.214717\pi\)
\(440\) −371139. −1.91704
\(441\) 0 0
\(442\) −37105.4 −0.189930
\(443\) 214549.i 1.09325i −0.837378 0.546625i \(-0.815912\pi\)
0.837378 0.546625i \(-0.184088\pi\)
\(444\) 0 0
\(445\) 353260.i 1.78391i
\(446\) 174445.i 0.876976i
\(447\) 0 0
\(448\) −103706. −0.516713
\(449\) −20938.9 −0.103863 −0.0519315 0.998651i \(-0.516538\pi\)
−0.0519315 + 0.998651i \(0.516538\pi\)
\(450\) 0 0
\(451\) 110036.i 0.540981i
\(452\) 186775.i 0.914203i
\(453\) 0 0
\(454\) 108789. 0.527804
\(455\) 261114.i 1.26127i
\(456\) 0 0
\(457\) 155521.i 0.744659i −0.928101 0.372330i \(-0.878559\pi\)
0.928101 0.372330i \(-0.121441\pi\)
\(458\) −190966. −0.910383
\(459\) 0 0
\(460\) 275925.i 1.30399i
\(461\) −424691. −1.99835 −0.999175 0.0406056i \(-0.987071\pi\)
−0.999175 + 0.0406056i \(0.987071\pi\)
\(462\) 0 0
\(463\) 51304.1i 0.239326i −0.992815 0.119663i \(-0.961819\pi\)
0.992815 0.119663i \(-0.0381814\pi\)
\(464\) −15252.9 −0.0708461
\(465\) 0 0
\(466\) 46058.8 0.212100
\(467\) 18311.1i 0.0839614i 0.999118 + 0.0419807i \(0.0133668\pi\)
−0.999118 + 0.0419807i \(0.986633\pi\)
\(468\) 0 0
\(469\) 124273.i 0.564976i
\(470\) 100400. 0.454504
\(471\) 0 0
\(472\) 56872.2 + 197213.i 0.255279 + 0.885222i
\(473\) −238370. −1.06544
\(474\) 0 0
\(475\) −32399.7 −0.143600
\(476\) 124519. 0.549570
\(477\) 0 0
\(478\) 59902.4i 0.262173i
\(479\) 117396. 0.511659 0.255830 0.966722i \(-0.417651\pi\)
0.255830 + 0.966722i \(0.417651\pi\)
\(480\) 0 0
\(481\) 84877.9 0.366863
\(482\) 246351.i 1.06038i
\(483\) 0 0
\(484\) −303716. −1.29651
\(485\) 468828.i 1.99310i
\(486\) 0 0
\(487\) 276423. 1.16551 0.582756 0.812647i \(-0.301974\pi\)
0.582756 + 0.812647i \(0.301974\pi\)
\(488\) 314888. 1.32226
\(489\) 0 0
\(490\) 198499.i 0.826733i
\(491\) 95438.0 0.395875 0.197938 0.980215i \(-0.436576\pi\)
0.197938 + 0.980215i \(0.436576\pi\)
\(492\) 0 0
\(493\) −41816.6 −0.172050
\(494\) 24197.3 0.0991548
\(495\) 0 0
\(496\) 28380.4i 0.115360i
\(497\) 664838. 2.69155
\(498\) 0 0
\(499\) 296293. 1.18993 0.594964 0.803752i \(-0.297166\pi\)
0.594964 + 0.803752i \(0.297166\pi\)
\(500\) 102852. 0.411407
\(501\) 0 0
\(502\) 164234.i 0.651713i
\(503\) 38668.3i 0.152834i 0.997076 + 0.0764169i \(0.0243480\pi\)
−0.997076 + 0.0764169i \(0.975652\pi\)
\(504\) 0 0
\(505\) 319291.i 1.25200i
\(506\) 344502.i 1.34552i
\(507\) 0 0
\(508\) 170389. 0.660260
\(509\) 382145.i 1.47500i −0.675347 0.737500i \(-0.736006\pi\)
0.675347 0.737500i \(-0.263994\pi\)
\(510\) 0 0
\(511\) 64292.3i 0.246217i
\(512\) 109349.i 0.417133i
\(513\) 0 0
\(514\) 21411.3i 0.0810431i
\(515\) 424624.i 1.60100i
\(516\) 0 0
\(517\) 306285. 1.14590
\(518\) 116574. 0.434454
\(519\) 0 0
\(520\) 209953. 0.776455
\(521\) 193927. 0.714437 0.357218 0.934021i \(-0.383725\pi\)
0.357218 + 0.934021i \(0.383725\pi\)
\(522\) 0 0
\(523\) 242685. 0.887238 0.443619 0.896216i \(-0.353694\pi\)
0.443619 + 0.896216i \(0.353694\pi\)
\(524\) 5920.68i 0.0215630i
\(525\) 0 0
\(526\) 75409.4i 0.272555i
\(527\) 77806.3i 0.280152i
\(528\) 0 0
\(529\) −337224. −1.20506
\(530\) 242104.i 0.861886i
\(531\) 0 0
\(532\) −81202.0 −0.286909
\(533\) 62247.5i 0.219113i
\(534\) 0 0
\(535\) −290283. −1.01418
\(536\) 99923.8 0.347808
\(537\) 0 0
\(538\) 133321. 0.460611
\(539\) 605550.i 2.08436i
\(540\) 0 0
\(541\) 87558.4i 0.299160i −0.988750 0.149580i \(-0.952208\pi\)
0.988750 0.149580i \(-0.0477922\pi\)
\(542\) 197389.i 0.671931i
\(543\) 0 0
\(544\) 158687.i 0.536221i
\(545\) 484108.i 1.62986i
\(546\) 0 0
\(547\) −169774. −0.567411 −0.283705 0.958912i \(-0.591564\pi\)
−0.283705 + 0.958912i \(0.591564\pi\)
\(548\) 337562. 1.12407
\(549\) 0 0
\(550\) 145684. 0.481600
\(551\) 27269.6 0.0898205
\(552\) 0 0
\(553\) 578980. 1.89327
\(554\) 191175.i 0.622890i
\(555\) 0 0
\(556\) −64847.0 −0.209768
\(557\) 76223.9 0.245686 0.122843 0.992426i \(-0.460799\pi\)
0.122843 + 0.992426i \(0.460799\pi\)
\(558\) 0 0
\(559\) 134846. 0.431534
\(560\) −123769. −0.394673
\(561\) 0 0
\(562\) 116900.i 0.370121i
\(563\) 589979.i 1.86131i −0.365892 0.930657i \(-0.619236\pi\)
0.365892 0.930657i \(-0.380764\pi\)
\(564\) 0 0
\(565\) 508970.i 1.59439i
\(566\) −42898.0 −0.133907
\(567\) 0 0
\(568\) 534575.i 1.65696i
\(569\) 312707.i 0.965856i 0.875660 + 0.482928i \(0.160427\pi\)
−0.875660 + 0.482928i \(0.839573\pi\)
\(570\) 0 0
\(571\) 614610.i 1.88507i −0.334110 0.942534i \(-0.608436\pi\)
0.334110 0.942534i \(-0.391564\pi\)
\(572\) 265846. 0.812528
\(573\) 0 0
\(574\) 85492.9i 0.259481i
\(575\) 260946.i 0.789252i
\(576\) 0 0
\(577\) −170463. −0.512010 −0.256005 0.966675i \(-0.582406\pi\)
−0.256005 + 0.966675i \(0.582406\pi\)
\(578\) 131819.i 0.394568i
\(579\) 0 0
\(580\) 98208.5 0.291940
\(581\) 259288.i 0.768123i
\(582\) 0 0
\(583\) 738574.i 2.17298i
\(584\) −51695.4 −0.151575
\(585\) 0 0
\(586\) 191530.i 0.557754i
\(587\) 315184.i 0.914721i −0.889281 0.457360i \(-0.848795\pi\)
0.889281 0.457360i \(-0.151205\pi\)
\(588\) 0 0
\(589\) 50739.4i 0.146256i
\(590\) −64326.1 223061.i −0.184792 0.640796i
\(591\) 0 0
\(592\) 40232.6i 0.114798i
\(593\) 93311.9 0.265355 0.132678 0.991159i \(-0.457643\pi\)
0.132678 + 0.991159i \(0.457643\pi\)
\(594\) 0 0
\(595\) −339320. −0.958464
\(596\) 71335.1i 0.200822i
\(597\) 0 0
\(598\) 194885.i 0.544974i
\(599\) 70659.2 0.196932 0.0984658 0.995140i \(-0.468607\pi\)
0.0984658 + 0.995140i \(0.468607\pi\)
\(600\) 0 0
\(601\) 624282.i 1.72835i −0.503190 0.864176i \(-0.667841\pi\)
0.503190 0.864176i \(-0.332159\pi\)
\(602\) 185202. 0.511039
\(603\) 0 0
\(604\) 451909.i 1.23873i
\(605\) 827639. 2.26115
\(606\) 0 0
\(607\) 72992.7 0.198108 0.0990540 0.995082i \(-0.468418\pi\)
0.0990540 + 0.995082i \(0.468418\pi\)
\(608\) 103484.i 0.279940i
\(609\) 0 0
\(610\) −356158. −0.957158
\(611\) −173266. −0.464120
\(612\) 0 0
\(613\) 446331.i 1.18778i 0.804546 + 0.593890i \(0.202409\pi\)
−0.804546 + 0.593890i \(0.797591\pi\)
\(614\) 12689.1i 0.0336585i
\(615\) 0 0
\(616\) 879678. 2.31826
\(617\) 676732. 1.77765 0.888825 0.458246i \(-0.151522\pi\)
0.888825 + 0.458246i \(0.151522\pi\)
\(618\) 0 0
\(619\) 217204. 0.566874 0.283437 0.958991i \(-0.408525\pi\)
0.283437 + 0.958991i \(0.408525\pi\)
\(620\) 182732.i 0.475370i
\(621\) 0 0
\(622\) 203683.i 0.526471i
\(623\) 837300.i 2.15727i
\(624\) 0 0
\(625\) −487894. −1.24901
\(626\) −194554. −0.496467
\(627\) 0 0
\(628\) 33308.8i 0.0844578i
\(629\) 110300.i 0.278787i
\(630\) 0 0
\(631\) 74809.2 0.187887 0.0939434 0.995578i \(-0.470053\pi\)
0.0939434 + 0.995578i \(0.470053\pi\)
\(632\) 465539.i 1.16553i
\(633\) 0 0
\(634\) 343917.i 0.855609i
\(635\) −464318. −1.15151
\(636\) 0 0
\(637\) 342559.i 0.844223i
\(638\) −122617. −0.301237
\(639\) 0 0
\(640\) 430897.i 1.05200i
\(641\) 461489. 1.12317 0.561585 0.827419i \(-0.310192\pi\)
0.561585 + 0.827419i \(0.310192\pi\)
\(642\) 0 0
\(643\) −363629. −0.879501 −0.439751 0.898120i \(-0.644933\pi\)
−0.439751 + 0.898120i \(0.644933\pi\)
\(644\) 654000.i 1.57691i
\(645\) 0 0
\(646\) 31444.7i 0.0753498i
\(647\) −641950. −1.53353 −0.766766 0.641926i \(-0.778135\pi\)
−0.766766 + 0.641926i \(0.778135\pi\)
\(648\) 0 0
\(649\) −196237. 680481.i −0.465898 1.61557i
\(650\) −82413.6 −0.195062
\(651\) 0 0
\(652\) 510213. 1.20021
\(653\) −426122. −0.999327 −0.499664 0.866220i \(-0.666543\pi\)
−0.499664 + 0.866220i \(0.666543\pi\)
\(654\) 0 0
\(655\) 16134.1i 0.0376064i
\(656\) −29505.6 −0.0685641
\(657\) 0 0
\(658\) −237969. −0.549628
\(659\) 468334.i 1.07841i 0.842174 + 0.539206i \(0.181276\pi\)
−0.842174 + 0.539206i \(0.818724\pi\)
\(660\) 0 0
\(661\) 485472. 1.11112 0.555561 0.831476i \(-0.312504\pi\)
0.555561 + 0.831476i \(0.312504\pi\)
\(662\) 44903.8i 0.102463i
\(663\) 0 0
\(664\) −208485. −0.472868
\(665\) 221279. 0.500376
\(666\) 0 0
\(667\) 219629.i 0.493671i
\(668\) −59461.4 −0.133255
\(669\) 0 0
\(670\) −113020. −0.251772
\(671\) −1.08651e6 −2.41319
\(672\) 0 0
\(673\) 153856.i 0.339692i 0.985471 + 0.169846i \(0.0543270\pi\)
−0.985471 + 0.169846i \(0.945673\pi\)
\(674\) 407876. 0.897859
\(675\) 0 0
\(676\) 173875. 0.380491
\(677\) 624029. 1.36153 0.680766 0.732501i \(-0.261647\pi\)
0.680766 + 0.732501i \(0.261647\pi\)
\(678\) 0 0
\(679\) 1.11122e6i 2.41024i
\(680\) 272837.i 0.590044i
\(681\) 0 0
\(682\) 228148.i 0.490510i
\(683\) 717942.i 1.53903i −0.638627 0.769516i \(-0.720497\pi\)
0.638627 0.769516i \(-0.279503\pi\)
\(684\) 0 0
\(685\) −919871. −1.96040
\(686\) 90953.8i 0.193274i
\(687\) 0 0
\(688\) 63917.7i 0.135034i
\(689\) 417811.i 0.880120i
\(690\) 0 0
\(691\) 647155.i 1.35535i −0.735361 0.677676i \(-0.762987\pi\)
0.735361 0.677676i \(-0.237013\pi\)
\(692\) 272347.i 0.568737i
\(693\) 0 0
\(694\) 324711. 0.674184
\(695\) 176711. 0.365841
\(696\) 0 0
\(697\) −80891.2 −0.166508
\(698\) −173428. −0.355965
\(699\) 0 0
\(700\) 276565. 0.564419
\(701\) 394820.i 0.803457i 0.915759 + 0.401729i \(0.131591\pi\)
−0.915759 + 0.401729i \(0.868409\pi\)
\(702\) 0 0
\(703\) 71929.0i 0.145544i
\(704\) 287725.i 0.580541i
\(705\) 0 0
\(706\) 200659. 0.402578
\(707\) 756787.i 1.51403i
\(708\) 0 0
\(709\) −404978. −0.805637 −0.402818 0.915280i \(-0.631969\pi\)
−0.402818 + 0.915280i \(0.631969\pi\)
\(710\) 604639.i 1.19944i
\(711\) 0 0
\(712\) 673246. 1.32805
\(713\) −408654. −0.803853
\(714\) 0 0
\(715\) −724441. −1.41707
\(716\) 549274.i 1.07143i
\(717\) 0 0
\(718\) 77991.6i 0.151286i
\(719\) 54443.6i 0.105315i −0.998613 0.0526574i \(-0.983231\pi\)
0.998613 0.0526574i \(-0.0167691\pi\)
\(720\) 0 0
\(721\) 1.00645e6i 1.93607i
\(722\) 260414.i 0.499562i
\(723\) 0 0
\(724\) −106759. −0.203670
\(725\) −92877.3 −0.176699
\(726\) 0 0
\(727\) 611738. 1.15743 0.578717 0.815528i \(-0.303553\pi\)
0.578717 + 0.815528i \(0.303553\pi\)
\(728\) −497634. −0.938961
\(729\) 0 0
\(730\) 58470.9 0.109722
\(731\) 175234.i 0.327931i
\(732\) 0 0
\(733\) 523681. 0.974674 0.487337 0.873214i \(-0.337968\pi\)
0.487337 + 0.873214i \(0.337968\pi\)
\(734\) −317944. −0.590146
\(735\) 0 0
\(736\) 833455. 1.53860
\(737\) −344785. −0.634766
\(738\) 0 0
\(739\) 304017.i 0.556684i 0.960482 + 0.278342i \(0.0897848\pi\)
−0.960482 + 0.278342i \(0.910215\pi\)
\(740\) 259044.i 0.473054i
\(741\) 0 0
\(742\) 573837.i 1.04227i
\(743\) −678235. −1.22858 −0.614289 0.789081i \(-0.710557\pi\)
−0.614289 + 0.789081i \(0.710557\pi\)
\(744\) 0 0
\(745\) 194391.i 0.350238i
\(746\) 200484.i 0.360248i
\(747\) 0 0
\(748\) 345469.i 0.617457i
\(749\) 688033. 1.22644
\(750\) 0 0
\(751\) 908208.i 1.61030i −0.593074 0.805148i \(-0.702086\pi\)
0.593074 0.805148i \(-0.297914\pi\)
\(752\) 82128.8i 0.145231i
\(753\) 0 0
\(754\) 69364.4 0.122010
\(755\) 1.23147e6i 2.16038i
\(756\) 0 0
\(757\) −761493. −1.32885 −0.664423 0.747357i \(-0.731322\pi\)
−0.664423 + 0.747357i \(0.731322\pi\)
\(758\) 426768.i 0.742768i
\(759\) 0 0
\(760\) 177923.i 0.308039i
\(761\) −631628. −1.09067 −0.545334 0.838219i \(-0.683597\pi\)
−0.545334 + 0.838219i \(0.683597\pi\)
\(762\) 0 0
\(763\) 1.14744e6i 1.97097i
\(764\) 298525.i 0.511438i
\(765\) 0 0
\(766\) 280252.i 0.477630i
\(767\) 111011. + 384948.i 0.188702 + 0.654352i
\(768\) 0 0
\(769\) 361776.i 0.611769i 0.952069 + 0.305885i \(0.0989521\pi\)
−0.952069 + 0.305885i \(0.901048\pi\)
\(770\) −994973. −1.67815
\(771\) 0 0
\(772\) −408211. −0.684937
\(773\) 300146.i 0.502312i −0.967947 0.251156i \(-0.919189\pi\)
0.967947 0.251156i \(-0.0808108\pi\)
\(774\) 0 0
\(775\) 172813.i 0.287722i
\(776\) 893498. 1.48378
\(777\) 0 0
\(778\) 209861.i 0.346714i
\(779\) 52751.1 0.0869273
\(780\) 0 0
\(781\) 1.84454e6i 3.02403i
\(782\) 253255. 0.414137
\(783\) 0 0
\(784\) 162375. 0.264172
\(785\) 90767.8i 0.147297i
\(786\) 0 0
\(787\) −174818. −0.282253 −0.141126 0.989992i \(-0.545072\pi\)
−0.141126 + 0.989992i \(0.545072\pi\)
\(788\) −183437. −0.295416
\(789\) 0 0
\(790\) 526555.i 0.843703i
\(791\) 1.20637e6i 1.92809i
\(792\) 0 0
\(793\) 614642. 0.977408
\(794\) 46540.6 0.0738229
\(795\) 0 0
\(796\) −94537.0 −0.149202
\(797\) 142726.i 0.224691i