Properties

Label 882.5.c.c.685.2
Level $882$
Weight $5$
Character 882.685
Analytic conductor $91.172$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 882.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(91.1723074400\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
Defining polynomial: \(x^{4} + 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 685.2
Root \(1.84776i\) of defining polynomial
Character \(\chi\) \(=\) 882.685
Dual form 882.5.c.c.685.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843 q^{2} +8.00000 q^{4} +0.710974i q^{5} -22.6274 q^{8} +O(q^{10})\) \(q-2.82843 q^{2} +8.00000 q^{4} +0.710974i q^{5} -22.6274 q^{8} -2.01094i q^{10} -151.664 q^{11} -260.864i q^{13} +64.0000 q^{16} -385.598i q^{17} -390.140i q^{19} +5.68779i q^{20} +428.971 q^{22} +177.647 q^{23} +624.495 q^{25} +737.836i q^{26} +320.887 q^{29} +1346.37i q^{31} -181.019 q^{32} +1090.64i q^{34} -797.088 q^{37} +1103.48i q^{38} -16.0875i q^{40} +815.856i q^{41} -2167.70 q^{43} -1213.31 q^{44} -502.461 q^{46} -4285.61i q^{47} -1766.34 q^{50} -2086.91i q^{52} +3171.57 q^{53} -107.829i q^{55} -907.606 q^{58} +4706.39i q^{59} +2534.78i q^{61} -3808.12i q^{62} +512.000 q^{64} +185.468 q^{65} -4092.43 q^{67} -3084.78i q^{68} +2255.28 q^{71} +4653.19i q^{73} +2254.51 q^{74} -3121.12i q^{76} -4193.74 q^{79} +45.5023i q^{80} -2307.59i q^{82} -7799.12i q^{83} +274.150 q^{85} +6131.18 q^{86} +3431.76 q^{88} +9469.11i q^{89} +1421.17 q^{92} +12121.5i q^{94} +277.379 q^{95} -9945.39i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{4} + O(q^{10}) \) \( 4q + 32q^{4} - 24q^{11} + 256q^{16} + 1648q^{22} - 104q^{23} + 948q^{25} + 1408q^{29} - 3392q^{37} - 2024q^{43} - 192q^{44} - 2304q^{46} - 4384q^{50} + 16680q^{53} + 352q^{58} + 2048q^{64} + 6048q^{65} - 20816q^{67} + 1984q^{71} - 576q^{74} - 29616q^{79} - 29688q^{85} + 18800q^{86} + 13184q^{88} - 832q^{92} - 7240q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.82843 −0.707107
\(3\) 0 0
\(4\) 8.00000 0.500000
\(5\) 0.710974i 0.0284390i 0.999899 + 0.0142195i \(0.00452635\pi\)
−0.999899 + 0.0142195i \(0.995474\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −22.6274 −0.353553
\(9\) 0 0
\(10\) − 2.01094i − 0.0201094i
\(11\) −151.664 −1.25342 −0.626711 0.779252i \(-0.715599\pi\)
−0.626711 + 0.779252i \(0.715599\pi\)
\(12\) 0 0
\(13\) − 260.864i − 1.54358i −0.635880 0.771788i \(-0.719363\pi\)
0.635880 0.771788i \(-0.280637\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) − 385.598i − 1.33425i −0.744946 0.667125i \(-0.767525\pi\)
0.744946 0.667125i \(-0.232475\pi\)
\(18\) 0 0
\(19\) − 390.140i − 1.08072i −0.841434 0.540360i \(-0.818288\pi\)
0.841434 0.540360i \(-0.181712\pi\)
\(20\) 5.68779i 0.0142195i
\(21\) 0 0
\(22\) 428.971 0.886303
\(23\) 177.647 0.335816 0.167908 0.985803i \(-0.446299\pi\)
0.167908 + 0.985803i \(0.446299\pi\)
\(24\) 0 0
\(25\) 624.495 0.999191
\(26\) 737.836i 1.09147i
\(27\) 0 0
\(28\) 0 0
\(29\) 320.887 0.381554 0.190777 0.981633i \(-0.438899\pi\)
0.190777 + 0.981633i \(0.438899\pi\)
\(30\) 0 0
\(31\) 1346.37i 1.40101i 0.713646 + 0.700506i \(0.247042\pi\)
−0.713646 + 0.700506i \(0.752958\pi\)
\(32\) −181.019 −0.176777
\(33\) 0 0
\(34\) 1090.64i 0.943457i
\(35\) 0 0
\(36\) 0 0
\(37\) −797.088 −0.582241 −0.291121 0.956686i \(-0.594028\pi\)
−0.291121 + 0.956686i \(0.594028\pi\)
\(38\) 1103.48i 0.764184i
\(39\) 0 0
\(40\) − 16.0875i − 0.0100547i
\(41\) 815.856i 0.485340i 0.970109 + 0.242670i \(0.0780232\pi\)
−0.970109 + 0.242670i \(0.921977\pi\)
\(42\) 0 0
\(43\) −2167.70 −1.17236 −0.586182 0.810179i \(-0.699370\pi\)
−0.586182 + 0.810179i \(0.699370\pi\)
\(44\) −1213.31 −0.626711
\(45\) 0 0
\(46\) −502.461 −0.237458
\(47\) − 4285.61i − 1.94007i −0.242970 0.970034i \(-0.578122\pi\)
0.242970 0.970034i \(-0.421878\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1766.34 −0.706535
\(51\) 0 0
\(52\) − 2086.91i − 0.771788i
\(53\) 3171.57 1.12907 0.564536 0.825408i \(-0.309055\pi\)
0.564536 + 0.825408i \(0.309055\pi\)
\(54\) 0 0
\(55\) − 107.829i − 0.0356460i
\(56\) 0 0
\(57\) 0 0
\(58\) −907.606 −0.269800
\(59\) 4706.39i 1.35202i 0.736891 + 0.676012i \(0.236293\pi\)
−0.736891 + 0.676012i \(0.763707\pi\)
\(60\) 0 0
\(61\) 2534.78i 0.681209i 0.940207 + 0.340604i \(0.110632\pi\)
−0.940207 + 0.340604i \(0.889368\pi\)
\(62\) − 3808.12i − 0.990665i
\(63\) 0 0
\(64\) 512.000 0.125000
\(65\) 185.468 0.0438977
\(66\) 0 0
\(67\) −4092.43 −0.911657 −0.455828 0.890068i \(-0.650657\pi\)
−0.455828 + 0.890068i \(0.650657\pi\)
\(68\) − 3084.78i − 0.667125i
\(69\) 0 0
\(70\) 0 0
\(71\) 2255.28 0.447388 0.223694 0.974659i \(-0.428188\pi\)
0.223694 + 0.974659i \(0.428188\pi\)
\(72\) 0 0
\(73\) 4653.19i 0.873183i 0.899660 + 0.436591i \(0.143814\pi\)
−0.899660 + 0.436591i \(0.856186\pi\)
\(74\) 2254.51 0.411707
\(75\) 0 0
\(76\) − 3121.12i − 0.540360i
\(77\) 0 0
\(78\) 0 0
\(79\) −4193.74 −0.671965 −0.335983 0.941868i \(-0.609068\pi\)
−0.335983 + 0.941868i \(0.609068\pi\)
\(80\) 45.5023i 0.00710974i
\(81\) 0 0
\(82\) − 2307.59i − 0.343187i
\(83\) − 7799.12i − 1.13211i −0.824367 0.566056i \(-0.808468\pi\)
0.824367 0.566056i \(-0.191532\pi\)
\(84\) 0 0
\(85\) 274.150 0.0379447
\(86\) 6131.18 0.828986
\(87\) 0 0
\(88\) 3431.76 0.443151
\(89\) 9469.11i 1.19544i 0.801704 + 0.597722i \(0.203927\pi\)
−0.801704 + 0.597722i \(0.796073\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1421.17 0.167908
\(93\) 0 0
\(94\) 12121.5i 1.37183i
\(95\) 277.379 0.0307345
\(96\) 0 0
\(97\) − 9945.39i − 1.05701i −0.848931 0.528504i \(-0.822753\pi\)
0.848931 0.528504i \(-0.177247\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 4995.96 0.499596
\(101\) 2068.32i 0.202756i 0.994848 + 0.101378i \(0.0323252\pi\)
−0.994848 + 0.101378i \(0.967675\pi\)
\(102\) 0 0
\(103\) − 2504.76i − 0.236098i −0.993008 0.118049i \(-0.962336\pi\)
0.993008 0.118049i \(-0.0376639\pi\)
\(104\) 5902.68i 0.545736i
\(105\) 0 0
\(106\) −8970.54 −0.798375
\(107\) −6443.39 −0.562791 −0.281395 0.959592i \(-0.590797\pi\)
−0.281395 + 0.959592i \(0.590797\pi\)
\(108\) 0 0
\(109\) 23546.0 1.98182 0.990912 0.134515i \(-0.0429477\pi\)
0.990912 + 0.134515i \(0.0429477\pi\)
\(110\) 304.987i 0.0252055i
\(111\) 0 0
\(112\) 0 0
\(113\) −19692.2 −1.54219 −0.771096 0.636719i \(-0.780291\pi\)
−0.771096 + 0.636719i \(0.780291\pi\)
\(114\) 0 0
\(115\) 126.302i 0.00955026i
\(116\) 2567.10 0.190777
\(117\) 0 0
\(118\) − 13311.7i − 0.956025i
\(119\) 0 0
\(120\) 0 0
\(121\) 8360.97 0.571065
\(122\) − 7169.44i − 0.481687i
\(123\) 0 0
\(124\) 10771.0i 0.700506i
\(125\) 888.358i 0.0568549i
\(126\) 0 0
\(127\) −12550.5 −0.778135 −0.389067 0.921209i \(-0.627203\pi\)
−0.389067 + 0.921209i \(0.627203\pi\)
\(128\) −1448.15 −0.0883883
\(129\) 0 0
\(130\) −524.582 −0.0310403
\(131\) − 19323.3i − 1.12600i −0.826457 0.562999i \(-0.809647\pi\)
0.826457 0.562999i \(-0.190353\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 11575.1 0.644639
\(135\) 0 0
\(136\) 8725.09i 0.471728i
\(137\) −3708.59 −0.197591 −0.0987955 0.995108i \(-0.531499\pi\)
−0.0987955 + 0.995108i \(0.531499\pi\)
\(138\) 0 0
\(139\) − 1096.09i − 0.0567307i −0.999598 0.0283653i \(-0.990970\pi\)
0.999598 0.0283653i \(-0.00903018\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6378.90 −0.316351
\(143\) 39563.7i 1.93475i
\(144\) 0 0
\(145\) 228.143i 0.0108510i
\(146\) − 13161.2i − 0.617433i
\(147\) 0 0
\(148\) −6376.71 −0.291121
\(149\) 1233.82 0.0555748 0.0277874 0.999614i \(-0.491154\pi\)
0.0277874 + 0.999614i \(0.491154\pi\)
\(150\) 0 0
\(151\) −42672.8 −1.87153 −0.935766 0.352622i \(-0.885290\pi\)
−0.935766 + 0.352622i \(0.885290\pi\)
\(152\) 8827.86i 0.382092i
\(153\) 0 0
\(154\) 0 0
\(155\) −957.236 −0.0398433
\(156\) 0 0
\(157\) 29884.9i 1.21242i 0.795305 + 0.606209i \(0.207311\pi\)
−0.795305 + 0.606209i \(0.792689\pi\)
\(158\) 11861.7 0.475151
\(159\) 0 0
\(160\) − 128.700i − 0.00502734i
\(161\) 0 0
\(162\) 0 0
\(163\) −26205.2 −0.986308 −0.493154 0.869942i \(-0.664156\pi\)
−0.493154 + 0.869942i \(0.664156\pi\)
\(164\) 6526.85i 0.242670i
\(165\) 0 0
\(166\) 22059.3i 0.800525i
\(167\) − 35887.5i − 1.28680i −0.765532 0.643398i \(-0.777524\pi\)
0.765532 0.643398i \(-0.222476\pi\)
\(168\) 0 0
\(169\) −39489.2 −1.38263
\(170\) −775.414 −0.0268309
\(171\) 0 0
\(172\) −17341.6 −0.586182
\(173\) 10972.7i 0.366625i 0.983055 + 0.183312i \(0.0586820\pi\)
−0.983055 + 0.183312i \(0.941318\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −9706.50 −0.313355
\(177\) 0 0
\(178\) − 26782.7i − 0.845306i
\(179\) 33844.1 1.05628 0.528138 0.849159i \(-0.322891\pi\)
0.528138 + 0.849159i \(0.322891\pi\)
\(180\) 0 0
\(181\) − 50094.5i − 1.52909i −0.644571 0.764545i \(-0.722964\pi\)
0.644571 0.764545i \(-0.277036\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4019.69 −0.118729
\(185\) − 566.709i − 0.0165583i
\(186\) 0 0
\(187\) 58481.4i 1.67238i
\(188\) − 34284.9i − 0.970034i
\(189\) 0 0
\(190\) −784.547 −0.0217326
\(191\) −17612.9 −0.482798 −0.241399 0.970426i \(-0.577606\pi\)
−0.241399 + 0.970426i \(0.577606\pi\)
\(192\) 0 0
\(193\) 13264.2 0.356097 0.178048 0.984022i \(-0.443022\pi\)
0.178048 + 0.984022i \(0.443022\pi\)
\(194\) 28129.8i 0.747417i
\(195\) 0 0
\(196\) 0 0
\(197\) −5362.20 −0.138169 −0.0690844 0.997611i \(-0.522008\pi\)
−0.0690844 + 0.997611i \(0.522008\pi\)
\(198\) 0 0
\(199\) 42230.6i 1.06640i 0.845988 + 0.533202i \(0.179011\pi\)
−0.845988 + 0.533202i \(0.820989\pi\)
\(200\) −14130.7 −0.353267
\(201\) 0 0
\(202\) − 5850.08i − 0.143370i
\(203\) 0 0
\(204\) 0 0
\(205\) −580.053 −0.0138026
\(206\) 7084.53i 0.166946i
\(207\) 0 0
\(208\) − 16695.3i − 0.385894i
\(209\) 59170.2i 1.35460i
\(210\) 0 0
\(211\) −77049.0 −1.73062 −0.865311 0.501236i \(-0.832879\pi\)
−0.865311 + 0.501236i \(0.832879\pi\)
\(212\) 25372.5 0.564536
\(213\) 0 0
\(214\) 18224.7 0.397953
\(215\) − 1541.18i − 0.0333408i
\(216\) 0 0
\(217\) 0 0
\(218\) −66598.3 −1.40136
\(219\) 0 0
\(220\) − 862.633i − 0.0178230i
\(221\) −100589. −2.05951
\(222\) 0 0
\(223\) − 3702.60i − 0.0744555i −0.999307 0.0372278i \(-0.988147\pi\)
0.999307 0.0372278i \(-0.0118527\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 55698.1 1.09049
\(227\) − 12018.8i − 0.233243i −0.993176 0.116621i \(-0.962794\pi\)
0.993176 0.116621i \(-0.0372064\pi\)
\(228\) 0 0
\(229\) − 33360.5i − 0.636153i −0.948065 0.318076i \(-0.896963\pi\)
0.948065 0.318076i \(-0.103037\pi\)
\(230\) − 357.237i − 0.00675305i
\(231\) 0 0
\(232\) −7260.85 −0.134900
\(233\) −54138.9 −0.997235 −0.498618 0.866822i \(-0.666159\pi\)
−0.498618 + 0.866822i \(0.666159\pi\)
\(234\) 0 0
\(235\) 3046.96 0.0551735
\(236\) 37651.2i 0.676012i
\(237\) 0 0
\(238\) 0 0
\(239\) −29165.4 −0.510590 −0.255295 0.966863i \(-0.582173\pi\)
−0.255295 + 0.966863i \(0.582173\pi\)
\(240\) 0 0
\(241\) 100130.i 1.72398i 0.506929 + 0.861988i \(0.330781\pi\)
−0.506929 + 0.861988i \(0.669219\pi\)
\(242\) −23648.4 −0.403804
\(243\) 0 0
\(244\) 20278.2i 0.340604i
\(245\) 0 0
\(246\) 0 0
\(247\) −101774. −1.66817
\(248\) − 30464.9i − 0.495333i
\(249\) 0 0
\(250\) − 2512.66i − 0.0402025i
\(251\) 81988.2i 1.30138i 0.759344 + 0.650690i \(0.225520\pi\)
−0.759344 + 0.650690i \(0.774480\pi\)
\(252\) 0 0
\(253\) −26942.6 −0.420919
\(254\) 35498.3 0.550224
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) 73774.8i 1.11697i 0.829515 + 0.558485i \(0.188617\pi\)
−0.829515 + 0.558485i \(0.811383\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1483.74 0.0219488
\(261\) 0 0
\(262\) 54654.4i 0.796201i
\(263\) −39243.6 −0.567358 −0.283679 0.958919i \(-0.591555\pi\)
−0.283679 + 0.958919i \(0.591555\pi\)
\(264\) 0 0
\(265\) 2254.90i 0.0321096i
\(266\) 0 0
\(267\) 0 0
\(268\) −32739.4 −0.455828
\(269\) − 10311.6i − 0.142502i −0.997458 0.0712509i \(-0.977301\pi\)
0.997458 0.0712509i \(-0.0226991\pi\)
\(270\) 0 0
\(271\) 36969.3i 0.503388i 0.967807 + 0.251694i \(0.0809876\pi\)
−0.967807 + 0.251694i \(0.919012\pi\)
\(272\) − 24678.3i − 0.333562i
\(273\) 0 0
\(274\) 10489.5 0.139718
\(275\) −94713.3 −1.25241
\(276\) 0 0
\(277\) −92405.8 −1.20431 −0.602157 0.798378i \(-0.705692\pi\)
−0.602157 + 0.798378i \(0.705692\pi\)
\(278\) 3100.22i 0.0401146i
\(279\) 0 0
\(280\) 0 0
\(281\) −21171.6 −0.268128 −0.134064 0.990973i \(-0.542803\pi\)
−0.134064 + 0.990973i \(0.542803\pi\)
\(282\) 0 0
\(283\) − 48065.2i − 0.600147i −0.953916 0.300074i \(-0.902989\pi\)
0.953916 0.300074i \(-0.0970112\pi\)
\(284\) 18042.3 0.223694
\(285\) 0 0
\(286\) − 111903.i − 1.36808i
\(287\) 0 0
\(288\) 0 0
\(289\) −65164.9 −0.780222
\(290\) − 645.284i − 0.00767282i
\(291\) 0 0
\(292\) 37225.5i 0.436591i
\(293\) − 26781.1i − 0.311956i −0.987761 0.155978i \(-0.950147\pi\)
0.987761 0.155978i \(-0.0498528\pi\)
\(294\) 0 0
\(295\) −3346.12 −0.0384501
\(296\) 18036.0 0.205853
\(297\) 0 0
\(298\) −3489.76 −0.0392973
\(299\) − 46341.7i − 0.518358i
\(300\) 0 0
\(301\) 0 0
\(302\) 120697. 1.32337
\(303\) 0 0
\(304\) − 24968.9i − 0.270180i
\(305\) −1802.16 −0.0193729
\(306\) 0 0
\(307\) 69996.0i 0.742671i 0.928499 + 0.371336i \(0.121100\pi\)
−0.928499 + 0.371336i \(0.878900\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2707.47 0.0281735
\(311\) 86961.6i 0.899097i 0.893256 + 0.449549i \(0.148415\pi\)
−0.893256 + 0.449549i \(0.851585\pi\)
\(312\) 0 0
\(313\) 33163.2i 0.338507i 0.985573 + 0.169253i \(0.0541357\pi\)
−0.985573 + 0.169253i \(0.945864\pi\)
\(314\) − 84527.3i − 0.857310i
\(315\) 0 0
\(316\) −33549.9 −0.335983
\(317\) −29206.0 −0.290639 −0.145319 0.989385i \(-0.546421\pi\)
−0.145319 + 0.989385i \(0.546421\pi\)
\(318\) 0 0
\(319\) −48667.1 −0.478249
\(320\) 364.019i 0.00355487i
\(321\) 0 0
\(322\) 0 0
\(323\) −150437. −1.44195
\(324\) 0 0
\(325\) − 162908.i − 1.54233i
\(326\) 74119.5 0.697425
\(327\) 0 0
\(328\) − 18460.7i − 0.171594i
\(329\) 0 0
\(330\) 0 0
\(331\) 166993. 1.52420 0.762102 0.647457i \(-0.224168\pi\)
0.762102 + 0.647457i \(0.224168\pi\)
\(332\) − 62393.0i − 0.566056i
\(333\) 0 0
\(334\) 101505.i 0.909902i
\(335\) − 2909.61i − 0.0259266i
\(336\) 0 0
\(337\) 16126.0 0.141993 0.0709964 0.997477i \(-0.477382\pi\)
0.0709964 + 0.997477i \(0.477382\pi\)
\(338\) 111692. 0.977664
\(339\) 0 0
\(340\) 2193.20 0.0189723
\(341\) − 204196.i − 1.75606i
\(342\) 0 0
\(343\) 0 0
\(344\) 49049.5 0.414493
\(345\) 0 0
\(346\) − 31035.5i − 0.259243i
\(347\) −10071.3 −0.0836428 −0.0418214 0.999125i \(-0.513316\pi\)
−0.0418214 + 0.999125i \(0.513316\pi\)
\(348\) 0 0
\(349\) 97674.1i 0.801915i 0.916097 + 0.400958i \(0.131323\pi\)
−0.916097 + 0.400958i \(0.868677\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 27454.1 0.221576
\(353\) 85255.3i 0.684182i 0.939667 + 0.342091i \(0.111135\pi\)
−0.939667 + 0.342091i \(0.888865\pi\)
\(354\) 0 0
\(355\) 1603.45i 0.0127232i
\(356\) 75752.9i 0.597722i
\(357\) 0 0
\(358\) −95725.6 −0.746899
\(359\) 142240. 1.10365 0.551826 0.833959i \(-0.313931\pi\)
0.551826 + 0.833959i \(0.313931\pi\)
\(360\) 0 0
\(361\) −21888.1 −0.167955
\(362\) 141689.i 1.08123i
\(363\) 0 0
\(364\) 0 0
\(365\) −3308.30 −0.0248324
\(366\) 0 0
\(367\) 177841.i 1.32038i 0.751098 + 0.660191i \(0.229525\pi\)
−0.751098 + 0.660191i \(0.770475\pi\)
\(368\) 11369.4 0.0839540
\(369\) 0 0
\(370\) 1602.90i 0.0117085i
\(371\) 0 0
\(372\) 0 0
\(373\) −21670.8 −0.155761 −0.0778803 0.996963i \(-0.524815\pi\)
−0.0778803 + 0.996963i \(0.524815\pi\)
\(374\) − 165410.i − 1.18255i
\(375\) 0 0
\(376\) 96972.3i 0.685917i
\(377\) − 83708.0i − 0.588958i
\(378\) 0 0
\(379\) 125199. 0.871608 0.435804 0.900042i \(-0.356464\pi\)
0.435804 + 0.900042i \(0.356464\pi\)
\(380\) 2219.03 0.0153673
\(381\) 0 0
\(382\) 49816.9 0.341390
\(383\) 211410.i 1.44121i 0.693345 + 0.720605i \(0.256136\pi\)
−0.693345 + 0.720605i \(0.743864\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −37516.9 −0.251798
\(387\) 0 0
\(388\) − 79563.1i − 0.528504i
\(389\) 39160.8 0.258793 0.129396 0.991593i \(-0.458696\pi\)
0.129396 + 0.991593i \(0.458696\pi\)
\(390\) 0 0
\(391\) − 68500.3i − 0.448063i
\(392\) 0 0
\(393\) 0 0
\(394\) 15166.6 0.0977001
\(395\) − 2981.64i − 0.0191100i
\(396\) 0 0
\(397\) − 196525.i − 1.24691i −0.781858 0.623456i \(-0.785728\pi\)
0.781858 0.623456i \(-0.214272\pi\)
\(398\) − 119446.i − 0.754061i
\(399\) 0 0
\(400\) 39967.6 0.249798
\(401\) −213650. −1.32866 −0.664330 0.747439i \(-0.731283\pi\)
−0.664330 + 0.747439i \(0.731283\pi\)
\(402\) 0 0
\(403\) 351221. 2.16257
\(404\) 16546.5i 0.101378i
\(405\) 0 0
\(406\) 0 0
\(407\) 120890. 0.729794
\(408\) 0 0
\(409\) 35604.1i 0.212840i 0.994321 + 0.106420i \(0.0339388\pi\)
−0.994321 + 0.106420i \(0.966061\pi\)
\(410\) 1640.64 0.00975988
\(411\) 0 0
\(412\) − 20038.1i − 0.118049i
\(413\) 0 0
\(414\) 0 0
\(415\) 5544.97 0.0321961
\(416\) 47221.5i 0.272868i
\(417\) 0 0
\(418\) − 167358.i − 0.957845i
\(419\) 254947.i 1.45219i 0.687597 + 0.726093i \(0.258666\pi\)
−0.687597 + 0.726093i \(0.741334\pi\)
\(420\) 0 0
\(421\) 107186. 0.604748 0.302374 0.953189i \(-0.402221\pi\)
0.302374 + 0.953189i \(0.402221\pi\)
\(422\) 217927. 1.22373
\(423\) 0 0
\(424\) −71764.3 −0.399187
\(425\) − 240804.i − 1.33317i
\(426\) 0 0
\(427\) 0 0
\(428\) −51547.1 −0.281395
\(429\) 0 0
\(430\) 4359.11i 0.0235755i
\(431\) −119958. −0.645768 −0.322884 0.946439i \(-0.604652\pi\)
−0.322884 + 0.946439i \(0.604652\pi\)
\(432\) 0 0
\(433\) 110159.i 0.587547i 0.955875 + 0.293773i \(0.0949111\pi\)
−0.955875 + 0.293773i \(0.905089\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 188368. 0.990912
\(437\) − 69307.1i − 0.362923i
\(438\) 0 0
\(439\) − 352295.i − 1.82801i −0.405708 0.914003i \(-0.632975\pi\)
0.405708 0.914003i \(-0.367025\pi\)
\(440\) 2439.90i 0.0126028i
\(441\) 0 0
\(442\) 284508. 1.45630
\(443\) −285732. −1.45597 −0.727984 0.685594i \(-0.759543\pi\)
−0.727984 + 0.685594i \(0.759543\pi\)
\(444\) 0 0
\(445\) −6732.29 −0.0339972
\(446\) 10472.5i 0.0526480i
\(447\) 0 0
\(448\) 0 0
\(449\) 342184. 1.69733 0.848667 0.528927i \(-0.177406\pi\)
0.848667 + 0.528927i \(0.177406\pi\)
\(450\) 0 0
\(451\) − 123736.i − 0.608335i
\(452\) −157538. −0.771096
\(453\) 0 0
\(454\) 33994.2i 0.164927i
\(455\) 0 0
\(456\) 0 0
\(457\) −277624. −1.32931 −0.664653 0.747152i \(-0.731421\pi\)
−0.664653 + 0.747152i \(0.731421\pi\)
\(458\) 94357.7i 0.449828i
\(459\) 0 0
\(460\) 1010.42i 0.00477513i
\(461\) 361934.i 1.70305i 0.524315 + 0.851524i \(0.324322\pi\)
−0.524315 + 0.851524i \(0.675678\pi\)
\(462\) 0 0
\(463\) 146876. 0.685153 0.342577 0.939490i \(-0.388700\pi\)
0.342577 + 0.939490i \(0.388700\pi\)
\(464\) 20536.8 0.0953886
\(465\) 0 0
\(466\) 153128. 0.705152
\(467\) − 268410.i − 1.23073i −0.788241 0.615367i \(-0.789008\pi\)
0.788241 0.615367i \(-0.210992\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −8618.09 −0.0390136
\(471\) 0 0
\(472\) − 106494.i − 0.478013i
\(473\) 328762. 1.46947
\(474\) 0 0
\(475\) − 243640.i − 1.07985i
\(476\) 0 0
\(477\) 0 0
\(478\) 82492.2 0.361042
\(479\) − 155285.i − 0.676797i −0.941003 0.338399i \(-0.890115\pi\)
0.941003 0.338399i \(-0.109885\pi\)
\(480\) 0 0
\(481\) 207932.i 0.898733i
\(482\) − 283211.i − 1.21903i
\(483\) 0 0
\(484\) 66887.7 0.285533
\(485\) 7070.91 0.0300602
\(486\) 0 0
\(487\) −155628. −0.656190 −0.328095 0.944645i \(-0.606407\pi\)
−0.328095 + 0.944645i \(0.606407\pi\)
\(488\) − 57355.5i − 0.240844i
\(489\) 0 0
\(490\) 0 0
\(491\) −37550.7 −0.155760 −0.0778798 0.996963i \(-0.524815\pi\)
−0.0778798 + 0.996963i \(0.524815\pi\)
\(492\) 0 0
\(493\) − 123734.i − 0.509089i
\(494\) 287859. 1.17958
\(495\) 0 0
\(496\) 86167.9i 0.350253i
\(497\) 0 0
\(498\) 0 0
\(499\) 167820. 0.673975 0.336987 0.941509i \(-0.390592\pi\)
0.336987 + 0.941509i \(0.390592\pi\)
\(500\) 7106.86i 0.0284275i
\(501\) 0 0
\(502\) − 231898.i − 0.920214i
\(503\) − 133035.i − 0.525810i −0.964822 0.262905i \(-0.915319\pi\)
0.964822 0.262905i \(-0.0846806\pi\)
\(504\) 0 0
\(505\) −1470.52 −0.00576617
\(506\) 76205.2 0.297635
\(507\) 0 0
\(508\) −100404. −0.389067
\(509\) − 227994.i − 0.880011i −0.897995 0.440005i \(-0.854976\pi\)
0.897995 0.440005i \(-0.145024\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11585.2 −0.0441942
\(513\) 0 0
\(514\) − 208667.i − 0.789817i
\(515\) 1780.82 0.00671437
\(516\) 0 0
\(517\) 649973.i 2.43172i
\(518\) 0 0
\(519\) 0 0
\(520\) −4196.65 −0.0155202
\(521\) 341411.i 1.25777i 0.777498 + 0.628885i \(0.216489\pi\)
−0.777498 + 0.628885i \(0.783511\pi\)
\(522\) 0 0
\(523\) − 62912.0i − 0.230001i −0.993365 0.115001i \(-0.963313\pi\)
0.993365 0.115001i \(-0.0366870\pi\)
\(524\) − 154586.i − 0.562999i
\(525\) 0 0
\(526\) 110998. 0.401182
\(527\) 519159. 1.86930
\(528\) 0 0
\(529\) −248283. −0.887228
\(530\) − 6377.82i − 0.0227050i
\(531\) 0 0
\(532\) 0 0
\(533\) 212828. 0.749159
\(534\) 0 0
\(535\) − 4581.08i − 0.0160052i
\(536\) 92601.1 0.322319
\(537\) 0 0
\(538\) 29165.5i 0.100764i
\(539\) 0 0
\(540\) 0 0
\(541\) 251203. 0.858282 0.429141 0.903237i \(-0.358816\pi\)
0.429141 + 0.903237i \(0.358816\pi\)
\(542\) − 104565.i − 0.355949i
\(543\) 0 0
\(544\) 69800.7i 0.235864i
\(545\) 16740.6i 0.0563610i
\(546\) 0 0
\(547\) −153431. −0.512789 −0.256395 0.966572i \(-0.582535\pi\)
−0.256395 + 0.966572i \(0.582535\pi\)
\(548\) −29668.7 −0.0987955
\(549\) 0 0
\(550\) 267890. 0.885586
\(551\) − 125191.i − 0.412353i
\(552\) 0 0
\(553\) 0 0
\(554\) 261363. 0.851579
\(555\) 0 0
\(556\) − 8768.75i − 0.0283653i
\(557\) 475151. 1.53152 0.765758 0.643129i \(-0.222364\pi\)
0.765758 + 0.643129i \(0.222364\pi\)
\(558\) 0 0
\(559\) 565476.i 1.80963i
\(560\) 0 0
\(561\) 0 0
\(562\) 59882.5 0.189595
\(563\) − 39487.0i − 0.124577i −0.998058 0.0622884i \(-0.980160\pi\)
0.998058 0.0622884i \(-0.0198398\pi\)
\(564\) 0 0
\(565\) − 14000.7i − 0.0438583i
\(566\) 135949.i 0.424368i
\(567\) 0 0
\(568\) −51031.2 −0.158175
\(569\) 85307.7 0.263490 0.131745 0.991284i \(-0.457942\pi\)
0.131745 + 0.991284i \(0.457942\pi\)
\(570\) 0 0
\(571\) 89099.0 0.273276 0.136638 0.990621i \(-0.456370\pi\)
0.136638 + 0.990621i \(0.456370\pi\)
\(572\) 316510.i 0.967375i
\(573\) 0 0
\(574\) 0 0
\(575\) 110939. 0.335545
\(576\) 0 0
\(577\) − 311076.i − 0.934362i −0.884162 0.467181i \(-0.845270\pi\)
0.884162 0.467181i \(-0.154730\pi\)
\(578\) 184314. 0.551700
\(579\) 0 0
\(580\) 1825.14i 0.00542551i
\(581\) 0 0
\(582\) 0 0
\(583\) −481012. −1.41520
\(584\) − 105290.i − 0.308717i
\(585\) 0 0
\(586\) 75748.3i 0.220586i
\(587\) 62466.3i 0.181288i 0.995883 + 0.0906440i \(0.0288926\pi\)
−0.995883 + 0.0906440i \(0.971107\pi\)
\(588\) 0 0
\(589\) 525274. 1.51410
\(590\) 9464.27 0.0271884
\(591\) 0 0
\(592\) −51013.7 −0.145560
\(593\) − 47567.3i − 0.135269i −0.997710 0.0676346i \(-0.978455\pi\)
0.997710 0.0676346i \(-0.0215452\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 9870.52 0.0277874
\(597\) 0 0
\(598\) 131074.i 0.366534i
\(599\) −564035. −1.57200 −0.785999 0.618227i \(-0.787851\pi\)
−0.785999 + 0.618227i \(0.787851\pi\)
\(600\) 0 0
\(601\) − 255637.i − 0.707740i −0.935295 0.353870i \(-0.884865\pi\)
0.935295 0.353870i \(-0.115135\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −341382. −0.935766
\(605\) 5944.43i 0.0162405i
\(606\) 0 0
\(607\) − 572699.i − 1.55435i −0.629284 0.777176i \(-0.716652\pi\)
0.629284 0.777176i \(-0.283348\pi\)
\(608\) 70622.8i 0.191046i
\(609\) 0 0
\(610\) 5097.28 0.0136987
\(611\) −1.11796e6 −2.99464
\(612\) 0 0
\(613\) 283002. 0.753127 0.376563 0.926391i \(-0.377106\pi\)
0.376563 + 0.926391i \(0.377106\pi\)
\(614\) − 197979.i − 0.525148i
\(615\) 0 0
\(616\) 0 0
\(617\) −248170. −0.651896 −0.325948 0.945388i \(-0.605683\pi\)
−0.325948 + 0.945388i \(0.605683\pi\)
\(618\) 0 0
\(619\) − 56588.1i − 0.147688i −0.997270 0.0738438i \(-0.976473\pi\)
0.997270 0.0738438i \(-0.0235266\pi\)
\(620\) −7657.89 −0.0199217
\(621\) 0 0
\(622\) − 245964.i − 0.635758i
\(623\) 0 0
\(624\) 0 0
\(625\) 389677. 0.997574
\(626\) − 93799.7i − 0.239361i
\(627\) 0 0
\(628\) 239079.i 0.606209i
\(629\) 307356.i 0.776855i
\(630\) 0 0
\(631\) 681448. 1.71149 0.855744 0.517399i \(-0.173100\pi\)
0.855744 + 0.517399i \(0.173100\pi\)
\(632\) 94893.4 0.237576
\(633\) 0 0
\(634\) 82607.0 0.205513
\(635\) − 8923.10i − 0.0221293i
\(636\) 0 0
\(637\) 0 0
\(638\) 137651. 0.338173
\(639\) 0 0
\(640\) − 1029.60i − 0.00251367i
\(641\) −678114. −1.65039 −0.825195 0.564848i \(-0.808935\pi\)
−0.825195 + 0.564848i \(0.808935\pi\)
\(642\) 0 0
\(643\) − 345520.i − 0.835701i −0.908516 0.417851i \(-0.862784\pi\)
0.908516 0.417851i \(-0.137216\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 425501. 1.01961
\(647\) − 7757.50i − 0.0185316i −0.999957 0.00926581i \(-0.997051\pi\)
0.999957 0.00926581i \(-0.00294944\pi\)
\(648\) 0 0
\(649\) − 713791.i − 1.69466i
\(650\) 460774.i 1.09059i
\(651\) 0 0
\(652\) −209642. −0.493154
\(653\) 448473. 1.05174 0.525872 0.850564i \(-0.323739\pi\)
0.525872 + 0.850564i \(0.323739\pi\)
\(654\) 0 0
\(655\) 13738.3 0.0320222
\(656\) 52214.8i 0.121335i
\(657\) 0 0
\(658\) 0 0
\(659\) −583660. −1.34397 −0.671984 0.740565i \(-0.734558\pi\)
−0.671984 + 0.740565i \(0.734558\pi\)
\(660\) 0 0
\(661\) − 382886.i − 0.876327i −0.898895 0.438164i \(-0.855629\pi\)
0.898895 0.438164i \(-0.144371\pi\)
\(662\) −472328. −1.07777
\(663\) 0 0
\(664\) 176474.i 0.400262i
\(665\) 0 0
\(666\) 0 0
\(667\) 57004.6 0.128132
\(668\) − 287100.i − 0.643398i
\(669\) 0 0
\(670\) 8229.62i 0.0183329i
\(671\) − 384435.i − 0.853842i
\(672\) 0 0
\(673\) −323802. −0.714907 −0.357453 0.933931i \(-0.616355\pi\)
−0.357453 + 0.933931i \(0.616355\pi\)
\(674\) −45611.2 −0.100404
\(675\) 0 0
\(676\) −315913. −0.691313
\(677\) 374803.i 0.817760i 0.912588 + 0.408880i \(0.134081\pi\)
−0.912588 + 0.408880i \(0.865919\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −6203.31 −0.0134155
\(681\) 0 0
\(682\) 577554.i 1.24172i
\(683\) −97364.5 −0.208718 −0.104359 0.994540i \(-0.533279\pi\)
−0.104359 + 0.994540i \(0.533279\pi\)
\(684\) 0 0
\(685\) − 2636.71i − 0.00561928i
\(686\) 0 0
\(687\) 0 0
\(688\) −138733. −0.293091
\(689\) − 827348.i − 1.74281i
\(690\) 0 0
\(691\) 787116.i 1.64848i 0.566244 + 0.824238i \(0.308396\pi\)
−0.566244 + 0.824238i \(0.691604\pi\)
\(692\) 87781.7i 0.183312i
\(693\) 0 0
\(694\) 28486.1 0.0591444
\(695\) 779.294 0.00161336
\(696\) 0 0
\(697\) 314593. 0.647564
\(698\) − 276264.i − 0.567040i
\(699\) 0 0
\(700\) 0 0
\(701\) 232522. 0.473182 0.236591 0.971609i \(-0.423970\pi\)
0.236591 + 0.971609i \(0.423970\pi\)
\(702\) 0 0
\(703\) 310976.i 0.629240i
\(704\) −77652.0 −0.156678
\(705\) 0 0
\(706\) − 241138.i − 0.483790i
\(707\) 0 0
\(708\) 0 0
\(709\) 151049. 0.300487 0.150243 0.988649i \(-0.451994\pi\)
0.150243 + 0.988649i \(0.451994\pi\)
\(710\) − 4535.23i − 0.00899669i
\(711\) 0 0
\(712\) − 214261.i − 0.422653i
\(713\) 239179.i 0.470483i
\(714\) 0 0
\(715\) −28128.8 −0.0550223
\(716\) 270753. 0.528138
\(717\) 0 0
\(718\) −402315. −0.780400
\(719\) 712110.i 1.37749i 0.725002 + 0.688746i \(0.241839\pi\)
−0.725002 + 0.688746i \(0.758161\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 61908.8 0.118762
\(723\) 0 0
\(724\) − 400756.i − 0.764545i
\(725\) 200392. 0.381246
\(726\) 0 0
\(727\) 693160.i 1.31149i 0.754983 + 0.655744i \(0.227645\pi\)
−0.754983 + 0.655744i \(0.772355\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 9357.28 0.0175592
\(731\) 835861.i 1.56423i
\(732\) 0 0
\(733\) 269107.i 0.500861i 0.968135 + 0.250431i \(0.0805722\pi\)
−0.968135 + 0.250431i \(0.919428\pi\)
\(734\) − 503010.i − 0.933651i
\(735\) 0 0
\(736\) −32157.5 −0.0593645
\(737\) 620674. 1.14269
\(738\) 0 0
\(739\) 86185.1 0.157813 0.0789066 0.996882i \(-0.474857\pi\)
0.0789066 + 0.996882i \(0.474857\pi\)
\(740\) − 4533.67i − 0.00827917i
\(741\) 0 0
\(742\) 0 0
\(743\) 457439. 0.828620 0.414310 0.910136i \(-0.364023\pi\)
0.414310 + 0.910136i \(0.364023\pi\)
\(744\) 0 0
\(745\) 877.210i 0.00158049i
\(746\) 61294.3 0.110139
\(747\) 0 0
\(748\) 467851.i 0.836189i
\(749\) 0 0
\(750\) 0 0
\(751\) −581442. −1.03092 −0.515462 0.856913i \(-0.672380\pi\)
−0.515462 + 0.856913i \(0.672380\pi\)
\(752\) − 274279.i − 0.485017i
\(753\) 0 0
\(754\) 236762.i 0.416456i
\(755\) − 30339.2i − 0.0532244i
\(756\) 0 0
\(757\) 398071. 0.694655 0.347327 0.937744i \(-0.387089\pi\)
0.347327 + 0.937744i \(0.387089\pi\)
\(758\) −354115. −0.616320
\(759\) 0 0
\(760\) −6276.38 −0.0108663
\(761\) 323965.i 0.559408i 0.960086 + 0.279704i \(0.0902363\pi\)
−0.960086 + 0.279704i \(0.909764\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −140904. −0.241399
\(765\) 0 0
\(766\) − 597957.i − 1.01909i
\(767\) 1.22773e6 2.08695
\(768\) 0 0
\(769\) − 318602.i − 0.538760i −0.963034 0.269380i \(-0.913181\pi\)
0.963034 0.269380i \(-0.0868187\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 106114. 0.178048
\(773\) − 512015.i − 0.856887i −0.903569 0.428443i \(-0.859062\pi\)
0.903569 0.428443i \(-0.140938\pi\)
\(774\) 0 0
\(775\) 840803.i 1.39988i
\(776\) 225038.i 0.373709i
\(777\) 0 0
\(778\) −110763. −0.182994
\(779\) 318298. 0.524516
\(780\) 0 0
\(781\) −342045. −0.560765
\(782\) 193748.i 0.316828i
\(783\) 0 0
\(784\) 0 0
\(785\) −21247.4 −0.0344799
\(786\) 0 0
\(787\) − 644919.i − 1.04125i −0.853785 0.520626i \(-0.825699\pi\)
0.853785 0.520626i \(-0.174301\pi\)
\(788\) −42897.6 −0.0690844
\(789\) 0 0
\(790\) 8433.34i 0.0135128i
\(791\) 0 0
\(792\) 0 0
\(793\) 661233. 1.05150
\(794\) 555856.i 0.881700i
\(795\) 0 0
\(796\) 337845.i 0.533202i
\(797\) − 278844.i − 0.438980i −0.975615 0.219490i \(-0.929561\pi\)
0.975615 0.219490i \(-0.0704393\pi\)
\(798\) 0 0
\(799\) −1.65252e6 −2.58853
\(800\) −113046. −0.176634
\(801\) 0 0
\(802\) 604293. 0.939505
\(803\) − 705721.i − 1.09447i
\(804\) 0 0
\(805\) 0 0
\(806\) −993402. −1.52917
\(807\) 0 0
\(808\) − 46800.6i − 0.0716851i
\(809\) −662520. −1.01228 −0.506142 0.862450i \(-0.668929\pi\)
−0.506142 + 0.862450i \(0.668929\pi\)
\(810\) 0 0
\(811\) − 586078.i − 0.891073i −0.895264 0.445537i \(-0.853013\pi\)
0.895264 0.445537i \(-0.146987\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −341927. −0.516042
\(815\) − 18631.2i − 0.0280496i
\(816\) 0 0
\(817\) 845706.i 1.26700i
\(818\) − 100704.i − 0.150501i
\(819\) 0 0
\(820\) −4640.42 −0.00690128
\(821\) −790534. −1.17283 −0.586414 0.810012i \(-0.699461\pi\)
−0.586414 + 0.810012i \(0.699461\pi\)
\(822\) 0 0
\(823\) −1.15645e6 −1.70737 −0.853684 0.520791i \(-0.825637\pi\)
−0.853684 + 0.520791i \(0.825637\pi\)
\(824\) 56676.3i 0.0834731i
\(825\) 0 0
\(826\) 0 0
\(827\) −390408. −0.570832 −0.285416 0.958404i \(-0.592132\pi\)
−0.285416 + 0.958404i \(0.592132\pi\)
\(828\) 0 0
\(829\) 531226.i 0.772984i 0.922293 + 0.386492i \(0.126313\pi\)
−0.922293 + 0.386492i \(0.873687\pi\)
\(830\) −15683.6 −0.0227661
\(831\) 0 0
\(832\) − 133562.i − 0.192947i
\(833\) 0 0
\(834\) 0 0
\(835\) 25515.1 0.0365951
\(836\) 473361.i 0.677299i
\(837\) 0 0
\(838\) − 721099.i − 1.02685i
\(839\) − 263842.i − 0.374817i −0.982282 0.187409i \(-0.939991\pi\)
0.982282 0.187409i \(-0.0600089\pi\)
\(840\) 0 0
\(841\) −604312. −0.854416
\(842\) −303168. −0.427622
\(843\) 0 0
\(844\) −616392. −0.865311
\(845\) − 28075.8i − 0.0393204i
\(846\) 0 0
\(847\) 0 0
\(848\) 202980. 0.282268
\(849\) 0 0
\(850\) 681096.i 0.942694i
\(851\) −141600. −0.195526
\(852\) 0 0
\(853\) − 1.17089e6i − 1.60923i −0.593798 0.804614i \(-0.702372\pi\)
0.593798 0.804614i \(-0.297628\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 145797. 0.198977
\(857\) − 424840.i − 0.578447i −0.957262 0.289223i \(-0.906603\pi\)
0.957262 0.289223i \(-0.0933970\pi\)
\(858\) 0 0
\(859\) 408175.i 0.553172i 0.960989 + 0.276586i \(0.0892030\pi\)
−0.960989 + 0.276586i \(0.910797\pi\)
\(860\) − 12329.4i − 0.0166704i
\(861\) 0 0
\(862\) 339294. 0.456627
\(863\) 882741. 1.18525 0.592627 0.805477i \(-0.298091\pi\)
0.592627 + 0.805477i \(0.298091\pi\)
\(864\) 0 0
\(865\) −7801.31 −0.0104264
\(866\) − 311575.i − 0.415458i
\(867\) 0 0
\(868\) 0 0
\(869\) 636039. 0.842256
\(870\) 0 0
\(871\) 1.06757e6i 1.40721i
\(872\) −532786. −0.700680
\(873\) 0 0
\(874\) 196030.i 0.256625i
\(875\) 0 0
\(876\) 0 0
\(877\) −595392. −0.774112 −0.387056 0.922056i \(-0.626508\pi\)
−0.387056 + 0.922056i \(0.626508\pi\)
\(878\) 996441.i 1.29260i
\(879\) 0 0
\(880\) − 6901.07i − 0.00891150i
\(881\) 238444.i 0.307209i 0.988132 + 0.153605i \(0.0490882\pi\)
−0.988132 + 0.153605i \(0.950912\pi\)
\(882\) 0 0
\(883\) 189523. 0.243076 0.121538 0.992587i \(-0.461217\pi\)
0.121538 + 0.992587i \(0.461217\pi\)
\(884\) −804710. −1.02976
\(885\) 0 0
\(886\) 808173. 1.02953
\(887\) 819553.i 1.04167i 0.853658 + 0.520834i \(0.174379\pi\)
−0.853658 + 0.520834i \(0.825621\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 19041.8 0.0240396
\(891\) 0 0
\(892\) − 29620.8i − 0.0372278i
\(893\) −1.67199e6 −2.09667
\(894\) 0 0
\(895\) 24062.3i 0.0300394i
\(896\) 0 0
\(897\) 0 0
\(898\) −967843. −1.20020
\(899\) 432034.i 0.534563i
\(900\) 0 0
\(901\) − 1.22295e6i − 1.50646i
\(902\) 349978.i 0.430158i
\(903\) 0 0
\(904\) 445585. 0.545247
\(905\) 35615.9 0.0434857
\(906\) 0 0
\(907\) −657619. −0.799392 −0.399696 0.916648i \(-0.630884\pi\)
−0.399696 + 0.916648i \(0.630884\pi\)
\(908\) − 96150.0i − 0.116621i
\(909\) 0 0
\(910\) 0 0
\(911\) −1.32632e6 −1.59812 −0.799061 0.601250i \(-0.794670\pi\)
−0.799061 + 0.601250i \(0.794670\pi\)
\(912\) 0 0
\(913\) 1.18285e6i 1.41901i
\(914\) 785240. 0.939961
\(915\) 0 0
\(916\) − 266884.i − 0.318076i
\(917\) 0 0
\(918\) 0 0
\(919\) −236310. −0.279803 −0.139901 0.990165i \(-0.544679\pi\)
−0.139901 + 0.990165i \(0.544679\pi\)
\(920\) − 2857.89i − 0.00337653i
\(921\) 0 0
\(922\) − 1.02370e6i − 1.20424i
\(923\) − 588322.i − 0.690577i
\(924\) 0 0
\(925\) −497777. −0.581770
\(926\) −415427. −0.484476
\(927\) 0 0
\(928\) −58086.8 −0.0674499
\(929\) 352762.i 0.408743i 0.978893 + 0.204372i \(0.0655151\pi\)
−0.978893 + 0.204372i \(0.934485\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −433111. −0.498618
\(933\) 0 0
\(934\) 759177.i 0.870261i
\(935\) −41578.7 −0.0475607
\(936\) 0 0
\(937\) 113445.i 0.129213i 0.997911 + 0.0646066i \(0.0205793\pi\)
−0.997911 + 0.0646066i \(0.979421\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 24375.6 0.0275867
\(941\) − 1.17351e6i − 1.32528i −0.748937 0.662642i \(-0.769435\pi\)
0.748937 0.662642i \(-0.230565\pi\)
\(942\) 0 0
\(943\) 144934.i 0.162985i
\(944\) 301209.i 0.338006i
\(945\) 0 0
\(946\) −929880. −1.03907
\(947\) −271958. −0.303250 −0.151625 0.988438i \(-0.548451\pi\)
−0.151625 + 0.988438i \(0.548451\pi\)
\(948\) 0 0
\(949\) 1.21385e6 1.34782
\(950\) 689118.i 0.763566i
\(951\) 0 0
\(952\) 0 0
\(953\) −465476. −0.512520 −0.256260 0.966608i \(-0.582490\pi\)
−0.256260 + 0.966608i \(0.582490\pi\)
\(954\) 0 0
\(955\) − 12522.3i − 0.0137303i
\(956\) −233323. −0.255295
\(957\) 0 0
\(958\) 439212.i 0.478568i
\(959\) 0 0
\(960\) 0 0
\(961\) −889199. −0.962836
\(962\) − 588120.i − 0.635500i
\(963\) 0 0
\(964\) 801042.i 0.861988i
\(965\) 9430.53i 0.0101270i
\(966\) 0 0
\(967\) −1.49573e6 −1.59956 −0.799782 0.600291i \(-0.795051\pi\)
−0.799782 + 0.600291i \(0.795051\pi\)
\(968\) −189187. −0.201902
\(969\) 0 0
\(970\) −19999.6 −0.0212558
\(971\) 404524.i 0.429048i 0.976719 + 0.214524i \(0.0688201\pi\)
−0.976719 + 0.214524i \(0.931180\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 440182. 0.463996
\(975\) 0 0
\(976\) 162226.i 0.170302i
\(977\) 1.65194e6 1.73063 0.865315 0.501229i \(-0.167118\pi\)
0.865315 + 0.501229i \(0.167118\pi\)
\(978\) 0 0
\(979\) − 1.43612e6i − 1.49839i
\(980\) 0 0
\(981\) 0 0
\(982\) 106209. 0.110139
\(983\) 978562.i 1.01270i 0.862328 + 0.506350i \(0.169006\pi\)
−0.862328 + 0.506350i \(0.830994\pi\)
\(984\) 0 0
\(985\) − 3812.38i − 0.00392938i
\(986\) 349971.i 0.359980i
\(987\) 0 0
\(988\) −814188. −0.834086
\(989\) −385085. −0.393699
\(990\) 0 0
\(991\) 1.26854e6 1.29168 0.645841 0.763472i \(-0.276507\pi\)
0.645841 + 0.763472i \(0.276507\pi\)
\(992\) − 243720.i − 0.247666i
\(993\) 0 0
\(994\) 0 0
\(995\) −30024.9 −0.0303274
\(996\) 0 0
\(997\) − 2071.33i − 0.00208382i −0.999999 0.00104191i \(-0.999668\pi\)
0.999999 0.00104191i \(-0.000331650\pi\)
\(998\) −474668. −0.476572
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 882.5.c.c.685.2 4
3.2 odd 2 98.5.b.a.97.3 4
7.6 odd 2 inner 882.5.c.c.685.1 4
12.11 even 2 784.5.c.a.97.3 4
21.2 odd 6 98.5.d.c.31.2 8
21.5 even 6 98.5.d.c.31.1 8
21.11 odd 6 98.5.d.c.19.1 8
21.17 even 6 98.5.d.c.19.2 8
21.20 even 2 98.5.b.a.97.4 yes 4
84.83 odd 2 784.5.c.a.97.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.5.b.a.97.3 4 3.2 odd 2
98.5.b.a.97.4 yes 4 21.20 even 2
98.5.d.c.19.1 8 21.11 odd 6
98.5.d.c.19.2 8 21.17 even 6
98.5.d.c.31.1 8 21.5 even 6
98.5.d.c.31.2 8 21.2 odd 6
784.5.c.a.97.2 4 84.83 odd 2
784.5.c.a.97.3 4 12.11 even 2
882.5.c.c.685.1 4 7.6 odd 2 inner
882.5.c.c.685.2 4 1.1 even 1 trivial