Properties

Label 98.4.a.g.1.2
Level $98$
Weight $4$
Character 98.1
Self dual yes
Analytic conductor $5.782$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.78218718056\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 98.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +7.07107 q^{3} +4.00000 q^{4} +19.7990 q^{5} -14.1421 q^{6} -8.00000 q^{8} +23.0000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +7.07107 q^{3} +4.00000 q^{4} +19.7990 q^{5} -14.1421 q^{6} -8.00000 q^{8} +23.0000 q^{9} -39.5980 q^{10} -14.0000 q^{11} +28.2843 q^{12} -50.9117 q^{13} +140.000 q^{15} +16.0000 q^{16} -1.41421 q^{17} -46.0000 q^{18} +1.41421 q^{19} +79.1960 q^{20} +28.0000 q^{22} +140.000 q^{23} -56.5685 q^{24} +267.000 q^{25} +101.823 q^{26} -28.2843 q^{27} -286.000 q^{29} -280.000 q^{30} +93.3381 q^{31} -32.0000 q^{32} -98.9949 q^{33} +2.82843 q^{34} +92.0000 q^{36} -38.0000 q^{37} -2.82843 q^{38} -360.000 q^{39} -158.392 q^{40} +125.865 q^{41} -34.0000 q^{43} -56.0000 q^{44} +455.377 q^{45} -280.000 q^{46} -523.259 q^{47} +113.137 q^{48} -534.000 q^{50} -10.0000 q^{51} -203.647 q^{52} -74.0000 q^{53} +56.5685 q^{54} -277.186 q^{55} +10.0000 q^{57} +572.000 q^{58} -434.164 q^{59} +560.000 q^{60} -14.1421 q^{61} -186.676 q^{62} +64.0000 q^{64} -1008.00 q^{65} +197.990 q^{66} +684.000 q^{67} -5.65685 q^{68} +989.949 q^{69} +588.000 q^{71} -184.000 q^{72} +270.115 q^{73} +76.0000 q^{74} +1887.98 q^{75} +5.65685 q^{76} +720.000 q^{78} +1220.00 q^{79} +316.784 q^{80} -821.000 q^{81} -251.730 q^{82} -422.850 q^{83} -28.0000 q^{85} +68.0000 q^{86} -2022.33 q^{87} +112.000 q^{88} -618.011 q^{89} -910.754 q^{90} +560.000 q^{92} +660.000 q^{93} +1046.52 q^{94} +28.0000 q^{95} -226.274 q^{96} -1483.51 q^{97} -322.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 8q^{4} - 16q^{8} + 46q^{9} + O(q^{10}) \) \( 2q - 4q^{2} + 8q^{4} - 16q^{8} + 46q^{9} - 28q^{11} + 280q^{15} + 32q^{16} - 92q^{18} + 56q^{22} + 280q^{23} + 534q^{25} - 572q^{29} - 560q^{30} - 64q^{32} + 184q^{36} - 76q^{37} - 720q^{39} - 68q^{43} - 112q^{44} - 560q^{46} - 1068q^{50} - 20q^{51} - 148q^{53} + 20q^{57} + 1144q^{58} + 1120q^{60} + 128q^{64} - 2016q^{65} + 1368q^{67} + 1176q^{71} - 368q^{72} + 152q^{74} + 1440q^{78} + 2440q^{79} - 1642q^{81} - 56q^{85} + 136q^{86} + 224q^{88} + 1120q^{92} + 1320q^{93} + 56q^{95} - 644q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 7.07107 1.36083 0.680414 0.732828i \(-0.261800\pi\)
0.680414 + 0.732828i \(0.261800\pi\)
\(4\) 4.00000 0.500000
\(5\) 19.7990 1.77088 0.885438 0.464758i \(-0.153859\pi\)
0.885438 + 0.464758i \(0.153859\pi\)
\(6\) −14.1421 −0.962250
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 23.0000 0.851852
\(10\) −39.5980 −1.25220
\(11\) −14.0000 −0.383742 −0.191871 0.981420i \(-0.561455\pi\)
−0.191871 + 0.981420i \(0.561455\pi\)
\(12\) 28.2843 0.680414
\(13\) −50.9117 −1.08618 −0.543091 0.839674i \(-0.682746\pi\)
−0.543091 + 0.839674i \(0.682746\pi\)
\(14\) 0 0
\(15\) 140.000 2.40986
\(16\) 16.0000 0.250000
\(17\) −1.41421 −0.0201763 −0.0100882 0.999949i \(-0.503211\pi\)
−0.0100882 + 0.999949i \(0.503211\pi\)
\(18\) −46.0000 −0.602350
\(19\) 1.41421 0.0170759 0.00853797 0.999964i \(-0.497282\pi\)
0.00853797 + 0.999964i \(0.497282\pi\)
\(20\) 79.1960 0.885438
\(21\) 0 0
\(22\) 28.0000 0.271346
\(23\) 140.000 1.26922 0.634609 0.772833i \(-0.281161\pi\)
0.634609 + 0.772833i \(0.281161\pi\)
\(24\) −56.5685 −0.481125
\(25\) 267.000 2.13600
\(26\) 101.823 0.768046
\(27\) −28.2843 −0.201604
\(28\) 0 0
\(29\) −286.000 −1.83134 −0.915670 0.401931i \(-0.868339\pi\)
−0.915670 + 0.401931i \(0.868339\pi\)
\(30\) −280.000 −1.70403
\(31\) 93.3381 0.540775 0.270387 0.962752i \(-0.412848\pi\)
0.270387 + 0.962752i \(0.412848\pi\)
\(32\) −32.0000 −0.176777
\(33\) −98.9949 −0.522206
\(34\) 2.82843 0.0142668
\(35\) 0 0
\(36\) 92.0000 0.425926
\(37\) −38.0000 −0.168842 −0.0844211 0.996430i \(-0.526904\pi\)
−0.0844211 + 0.996430i \(0.526904\pi\)
\(38\) −2.82843 −0.0120745
\(39\) −360.000 −1.47811
\(40\) −158.392 −0.626099
\(41\) 125.865 0.479434 0.239717 0.970843i \(-0.422945\pi\)
0.239717 + 0.970843i \(0.422945\pi\)
\(42\) 0 0
\(43\) −34.0000 −0.120580 −0.0602901 0.998181i \(-0.519203\pi\)
−0.0602901 + 0.998181i \(0.519203\pi\)
\(44\) −56.0000 −0.191871
\(45\) 455.377 1.50852
\(46\) −280.000 −0.897473
\(47\) −523.259 −1.62394 −0.811970 0.583699i \(-0.801605\pi\)
−0.811970 + 0.583699i \(0.801605\pi\)
\(48\) 113.137 0.340207
\(49\) 0 0
\(50\) −534.000 −1.51038
\(51\) −10.0000 −0.0274565
\(52\) −203.647 −0.543091
\(53\) −74.0000 −0.191786 −0.0958932 0.995392i \(-0.530571\pi\)
−0.0958932 + 0.995392i \(0.530571\pi\)
\(54\) 56.5685 0.142556
\(55\) −277.186 −0.679559
\(56\) 0 0
\(57\) 10.0000 0.0232374
\(58\) 572.000 1.29495
\(59\) −434.164 −0.958022 −0.479011 0.877809i \(-0.659005\pi\)
−0.479011 + 0.877809i \(0.659005\pi\)
\(60\) 560.000 1.20493
\(61\) −14.1421 −0.0296839 −0.0148419 0.999890i \(-0.504725\pi\)
−0.0148419 + 0.999890i \(0.504725\pi\)
\(62\) −186.676 −0.382385
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −1008.00 −1.92349
\(66\) 197.990 0.369256
\(67\) 684.000 1.24722 0.623611 0.781735i \(-0.285665\pi\)
0.623611 + 0.781735i \(0.285665\pi\)
\(68\) −5.65685 −0.0100882
\(69\) 989.949 1.72719
\(70\) 0 0
\(71\) 588.000 0.982856 0.491428 0.870918i \(-0.336475\pi\)
0.491428 + 0.870918i \(0.336475\pi\)
\(72\) −184.000 −0.301175
\(73\) 270.115 0.433076 0.216538 0.976274i \(-0.430523\pi\)
0.216538 + 0.976274i \(0.430523\pi\)
\(74\) 76.0000 0.119389
\(75\) 1887.98 2.90673
\(76\) 5.65685 0.00853797
\(77\) 0 0
\(78\) 720.000 1.04518
\(79\) 1220.00 1.73748 0.868739 0.495271i \(-0.164931\pi\)
0.868739 + 0.495271i \(0.164931\pi\)
\(80\) 316.784 0.442719
\(81\) −821.000 −1.12620
\(82\) −251.730 −0.339011
\(83\) −422.850 −0.559202 −0.279601 0.960116i \(-0.590202\pi\)
−0.279601 + 0.960116i \(0.590202\pi\)
\(84\) 0 0
\(85\) −28.0000 −0.0357297
\(86\) 68.0000 0.0852631
\(87\) −2022.33 −2.49214
\(88\) 112.000 0.135673
\(89\) −618.011 −0.736057 −0.368028 0.929815i \(-0.619967\pi\)
−0.368028 + 0.929815i \(0.619967\pi\)
\(90\) −910.754 −1.06669
\(91\) 0 0
\(92\) 560.000 0.634609
\(93\) 660.000 0.735901
\(94\) 1046.52 1.14830
\(95\) 28.0000 0.0302394
\(96\) −226.274 −0.240563
\(97\) −1483.51 −1.55286 −0.776431 0.630202i \(-0.782972\pi\)
−0.776431 + 0.630202i \(0.782972\pi\)
\(98\) 0 0
\(99\) −322.000 −0.326891
\(100\) 1068.00 1.06800
\(101\) −1128.54 −1.11182 −0.555912 0.831241i \(-0.687631\pi\)
−0.555912 + 0.831241i \(0.687631\pi\)
\(102\) 20.0000 0.0194147
\(103\) 868.327 0.830668 0.415334 0.909669i \(-0.363665\pi\)
0.415334 + 0.909669i \(0.363665\pi\)
\(104\) 407.294 0.384023
\(105\) 0 0
\(106\) 148.000 0.135613
\(107\) −1684.00 −1.52148 −0.760740 0.649056i \(-0.775164\pi\)
−0.760740 + 0.649056i \(0.775164\pi\)
\(108\) −113.137 −0.100802
\(109\) −818.000 −0.718809 −0.359405 0.933182i \(-0.617020\pi\)
−0.359405 + 0.933182i \(0.617020\pi\)
\(110\) 554.372 0.480521
\(111\) −268.701 −0.229765
\(112\) 0 0
\(113\) −540.000 −0.449548 −0.224774 0.974411i \(-0.572164\pi\)
−0.224774 + 0.974411i \(0.572164\pi\)
\(114\) −20.0000 −0.0164313
\(115\) 2771.86 2.24763
\(116\) −1144.00 −0.915670
\(117\) −1170.97 −0.925266
\(118\) 868.327 0.677424
\(119\) 0 0
\(120\) −1120.00 −0.852013
\(121\) −1135.00 −0.852742
\(122\) 28.2843 0.0209897
\(123\) 890.000 0.652428
\(124\) 373.352 0.270387
\(125\) 2811.46 2.01171
\(126\) 0 0
\(127\) 1720.00 1.20177 0.600887 0.799334i \(-0.294814\pi\)
0.600887 + 0.799334i \(0.294814\pi\)
\(128\) −128.000 −0.0883883
\(129\) −240.416 −0.164089
\(130\) 2016.00 1.36011
\(131\) 1735.24 1.15732 0.578659 0.815570i \(-0.303576\pi\)
0.578659 + 0.815570i \(0.303576\pi\)
\(132\) −395.980 −0.261103
\(133\) 0 0
\(134\) −1368.00 −0.881919
\(135\) −560.000 −0.357016
\(136\) 11.3137 0.00713340
\(137\) 828.000 0.516356 0.258178 0.966097i \(-0.416878\pi\)
0.258178 + 0.966097i \(0.416878\pi\)
\(138\) −1979.90 −1.22131
\(139\) 425.678 0.259752 0.129876 0.991530i \(-0.458542\pi\)
0.129876 + 0.991530i \(0.458542\pi\)
\(140\) 0 0
\(141\) −3700.00 −2.20990
\(142\) −1176.00 −0.694984
\(143\) 712.764 0.416813
\(144\) 368.000 0.212963
\(145\) −5662.51 −3.24308
\(146\) −540.230 −0.306231
\(147\) 0 0
\(148\) −152.000 −0.0844211
\(149\) 2050.00 1.12713 0.563566 0.826071i \(-0.309429\pi\)
0.563566 + 0.826071i \(0.309429\pi\)
\(150\) −3775.95 −2.05537
\(151\) −472.000 −0.254376 −0.127188 0.991879i \(-0.540595\pi\)
−0.127188 + 0.991879i \(0.540595\pi\)
\(152\) −11.3137 −0.00603726
\(153\) −32.5269 −0.0171872
\(154\) 0 0
\(155\) 1848.00 0.957645
\(156\) −1440.00 −0.739053
\(157\) 2211.83 1.12435 0.562176 0.827018i \(-0.309964\pi\)
0.562176 + 0.827018i \(0.309964\pi\)
\(158\) −2440.00 −1.22858
\(159\) −523.259 −0.260988
\(160\) −633.568 −0.313050
\(161\) 0 0
\(162\) 1642.00 0.796344
\(163\) 3286.00 1.57901 0.789507 0.613741i \(-0.210336\pi\)
0.789507 + 0.613741i \(0.210336\pi\)
\(164\) 503.460 0.239717
\(165\) −1960.00 −0.924762
\(166\) 845.700 0.395416
\(167\) 1490.58 0.690686 0.345343 0.938476i \(-0.387763\pi\)
0.345343 + 0.938476i \(0.387763\pi\)
\(168\) 0 0
\(169\) 395.000 0.179791
\(170\) 56.0000 0.0252647
\(171\) 32.5269 0.0145462
\(172\) −136.000 −0.0602901
\(173\) 2070.41 0.909886 0.454943 0.890521i \(-0.349660\pi\)
0.454943 + 0.890521i \(0.349660\pi\)
\(174\) 4044.65 1.76221
\(175\) 0 0
\(176\) −224.000 −0.0959354
\(177\) −3070.00 −1.30370
\(178\) 1236.02 0.520471
\(179\) 540.000 0.225483 0.112742 0.993624i \(-0.464037\pi\)
0.112742 + 0.993624i \(0.464037\pi\)
\(180\) 1821.51 0.754262
\(181\) −3784.44 −1.55412 −0.777058 0.629429i \(-0.783289\pi\)
−0.777058 + 0.629429i \(0.783289\pi\)
\(182\) 0 0
\(183\) −100.000 −0.0403946
\(184\) −1120.00 −0.448736
\(185\) −752.362 −0.298999
\(186\) −1320.00 −0.520361
\(187\) 19.7990 0.00774249
\(188\) −2093.04 −0.811970
\(189\) 0 0
\(190\) −56.0000 −0.0213825
\(191\) 1028.00 0.389442 0.194721 0.980859i \(-0.437620\pi\)
0.194721 + 0.980859i \(0.437620\pi\)
\(192\) 452.548 0.170103
\(193\) 4592.00 1.71264 0.856320 0.516446i \(-0.172745\pi\)
0.856320 + 0.516446i \(0.172745\pi\)
\(194\) 2967.02 1.09804
\(195\) −7127.64 −2.61754
\(196\) 0 0
\(197\) 794.000 0.287158 0.143579 0.989639i \(-0.454139\pi\)
0.143579 + 0.989639i \(0.454139\pi\)
\(198\) 644.000 0.231147
\(199\) −2486.19 −0.885634 −0.442817 0.896612i \(-0.646021\pi\)
−0.442817 + 0.896612i \(0.646021\pi\)
\(200\) −2136.00 −0.755190
\(201\) 4836.61 1.69725
\(202\) 2257.08 0.786178
\(203\) 0 0
\(204\) −40.0000 −0.0137282
\(205\) 2492.00 0.849019
\(206\) −1736.65 −0.587371
\(207\) 3220.00 1.08119
\(208\) −814.587 −0.271545
\(209\) −19.7990 −0.00655275
\(210\) 0 0
\(211\) −2748.00 −0.896588 −0.448294 0.893886i \(-0.647968\pi\)
−0.448294 + 0.893886i \(0.647968\pi\)
\(212\) −296.000 −0.0958932
\(213\) 4157.79 1.33750
\(214\) 3368.00 1.07585
\(215\) −673.166 −0.213533
\(216\) 226.274 0.0712778
\(217\) 0 0
\(218\) 1636.00 0.508275
\(219\) 1910.00 0.589342
\(220\) −1108.74 −0.339779
\(221\) 72.0000 0.0219151
\(222\) 537.401 0.162468
\(223\) 3428.05 1.02941 0.514707 0.857366i \(-0.327901\pi\)
0.514707 + 0.857366i \(0.327901\pi\)
\(224\) 0 0
\(225\) 6141.00 1.81956
\(226\) 1080.00 0.317878
\(227\) −5290.57 −1.54691 −0.773453 0.633854i \(-0.781472\pi\)
−0.773453 + 0.633854i \(0.781472\pi\)
\(228\) 40.0000 0.0116187
\(229\) 2749.23 0.793338 0.396669 0.917962i \(-0.370166\pi\)
0.396669 + 0.917962i \(0.370166\pi\)
\(230\) −5543.72 −1.58931
\(231\) 0 0
\(232\) 2288.00 0.647477
\(233\) 72.0000 0.0202441 0.0101221 0.999949i \(-0.496778\pi\)
0.0101221 + 0.999949i \(0.496778\pi\)
\(234\) 2341.94 0.654262
\(235\) −10360.0 −2.87580
\(236\) −1736.65 −0.479011
\(237\) 8626.70 2.36441
\(238\) 0 0
\(239\) 4308.00 1.16595 0.582974 0.812491i \(-0.301889\pi\)
0.582974 + 0.812491i \(0.301889\pi\)
\(240\) 2240.00 0.602464
\(241\) 1540.08 0.411640 0.205820 0.978590i \(-0.434014\pi\)
0.205820 + 0.978590i \(0.434014\pi\)
\(242\) 2270.00 0.602980
\(243\) −5041.67 −1.33096
\(244\) −56.5685 −0.0148419
\(245\) 0 0
\(246\) −1780.00 −0.461336
\(247\) −72.0000 −0.0185476
\(248\) −746.705 −0.191193
\(249\) −2990.00 −0.760978
\(250\) −5622.91 −1.42250
\(251\) −931.967 −0.234363 −0.117182 0.993110i \(-0.537386\pi\)
−0.117182 + 0.993110i \(0.537386\pi\)
\(252\) 0 0
\(253\) −1960.00 −0.487052
\(254\) −3440.00 −0.849783
\(255\) −197.990 −0.0486220
\(256\) 256.000 0.0625000
\(257\) −937.624 −0.227577 −0.113789 0.993505i \(-0.536299\pi\)
−0.113789 + 0.993505i \(0.536299\pi\)
\(258\) 480.833 0.116028
\(259\) 0 0
\(260\) −4032.00 −0.961746
\(261\) −6578.00 −1.56003
\(262\) −3470.48 −0.818347
\(263\) 7140.00 1.67404 0.837018 0.547176i \(-0.184297\pi\)
0.837018 + 0.547176i \(0.184297\pi\)
\(264\) 791.960 0.184628
\(265\) −1465.13 −0.339630
\(266\) 0 0
\(267\) −4370.00 −1.00165
\(268\) 2736.00 0.623611
\(269\) −4610.34 −1.04497 −0.522485 0.852648i \(-0.674995\pi\)
−0.522485 + 0.852648i \(0.674995\pi\)
\(270\) 1120.00 0.252448
\(271\) 2364.57 0.530026 0.265013 0.964245i \(-0.414624\pi\)
0.265013 + 0.964245i \(0.414624\pi\)
\(272\) −22.6274 −0.00504408
\(273\) 0 0
\(274\) −1656.00 −0.365119
\(275\) −3738.00 −0.819672
\(276\) 3959.80 0.863594
\(277\) 4006.00 0.868943 0.434472 0.900686i \(-0.356935\pi\)
0.434472 + 0.900686i \(0.356935\pi\)
\(278\) −851.357 −0.183673
\(279\) 2146.78 0.460660
\(280\) 0 0
\(281\) −5984.00 −1.27038 −0.635188 0.772358i \(-0.719077\pi\)
−0.635188 + 0.772358i \(0.719077\pi\)
\(282\) 7400.00 1.56264
\(283\) 4928.53 1.03523 0.517617 0.855613i \(-0.326819\pi\)
0.517617 + 0.855613i \(0.326819\pi\)
\(284\) 2352.00 0.491428
\(285\) 197.990 0.0411506
\(286\) −1425.53 −0.294731
\(287\) 0 0
\(288\) −736.000 −0.150588
\(289\) −4911.00 −0.999593
\(290\) 11325.0 2.29320
\(291\) −10490.0 −2.11318
\(292\) 1080.46 0.216538
\(293\) −1971.41 −0.393076 −0.196538 0.980496i \(-0.562970\pi\)
−0.196538 + 0.980496i \(0.562970\pi\)
\(294\) 0 0
\(295\) −8596.00 −1.69654
\(296\) 304.000 0.0596947
\(297\) 395.980 0.0773639
\(298\) −4100.00 −0.797002
\(299\) −7127.64 −1.37860
\(300\) 7551.90 1.45336
\(301\) 0 0
\(302\) 944.000 0.179871
\(303\) −7980.00 −1.51300
\(304\) 22.6274 0.00426898
\(305\) −280.000 −0.0525664
\(306\) 65.0538 0.0121532
\(307\) 4767.31 0.886270 0.443135 0.896455i \(-0.353866\pi\)
0.443135 + 0.896455i \(0.353866\pi\)
\(308\) 0 0
\(309\) 6140.00 1.13040
\(310\) −3696.00 −0.677157
\(311\) 6776.91 1.23564 0.617819 0.786320i \(-0.288016\pi\)
0.617819 + 0.786320i \(0.288016\pi\)
\(312\) 2880.00 0.522589
\(313\) 6190.01 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(314\) −4423.66 −0.795037
\(315\) 0 0
\(316\) 4880.00 0.868739
\(317\) 9826.00 1.74096 0.870478 0.492207i \(-0.163810\pi\)
0.870478 + 0.492207i \(0.163810\pi\)
\(318\) 1046.52 0.184547
\(319\) 4004.00 0.702762
\(320\) 1267.14 0.221359
\(321\) −11907.7 −2.07047
\(322\) 0 0
\(323\) −2.00000 −0.000344529 0
\(324\) −3284.00 −0.563100
\(325\) −13593.4 −2.32008
\(326\) −6572.00 −1.11653
\(327\) −5784.13 −0.978175
\(328\) −1006.92 −0.169506
\(329\) 0 0
\(330\) 3920.00 0.653906
\(331\) −5738.00 −0.952837 −0.476418 0.879219i \(-0.658065\pi\)
−0.476418 + 0.879219i \(0.658065\pi\)
\(332\) −1691.40 −0.279601
\(333\) −874.000 −0.143829
\(334\) −2981.16 −0.488389
\(335\) 13542.5 2.20868
\(336\) 0 0
\(337\) −2254.00 −0.364342 −0.182171 0.983267i \(-0.558312\pi\)
−0.182171 + 0.983267i \(0.558312\pi\)
\(338\) −790.000 −0.127131
\(339\) −3818.38 −0.611757
\(340\) −112.000 −0.0178649
\(341\) −1306.73 −0.207518
\(342\) −65.0538 −0.0102857
\(343\) 0 0
\(344\) 272.000 0.0426316
\(345\) 19600.0 3.05863
\(346\) −4140.82 −0.643386
\(347\) −1986.00 −0.307245 −0.153623 0.988130i \(-0.549094\pi\)
−0.153623 + 0.988130i \(0.549094\pi\)
\(348\) −8089.30 −1.24607
\(349\) −6771.25 −1.03856 −0.519279 0.854605i \(-0.673800\pi\)
−0.519279 + 0.854605i \(0.673800\pi\)
\(350\) 0 0
\(351\) 1440.00 0.218979
\(352\) 448.000 0.0678366
\(353\) −6993.29 −1.05443 −0.527217 0.849731i \(-0.676764\pi\)
−0.527217 + 0.849731i \(0.676764\pi\)
\(354\) 6140.00 0.921857
\(355\) 11641.8 1.74052
\(356\) −2472.05 −0.368028
\(357\) 0 0
\(358\) −1080.00 −0.159441
\(359\) 5944.00 0.873850 0.436925 0.899498i \(-0.356067\pi\)
0.436925 + 0.899498i \(0.356067\pi\)
\(360\) −3643.01 −0.533344
\(361\) −6857.00 −0.999708
\(362\) 7568.87 1.09893
\(363\) −8025.66 −1.16044
\(364\) 0 0
\(365\) 5348.00 0.766924
\(366\) 200.000 0.0285633
\(367\) 842.871 0.119884 0.0599421 0.998202i \(-0.480908\pi\)
0.0599421 + 0.998202i \(0.480908\pi\)
\(368\) 2240.00 0.317305
\(369\) 2894.90 0.408407
\(370\) 1504.72 0.211424
\(371\) 0 0
\(372\) 2640.00 0.367951
\(373\) −5726.00 −0.794855 −0.397428 0.917634i \(-0.630097\pi\)
−0.397428 + 0.917634i \(0.630097\pi\)
\(374\) −39.5980 −0.00547477
\(375\) 19880.0 2.73760
\(376\) 4186.07 0.574149
\(377\) 14560.7 1.98917
\(378\) 0 0
\(379\) 10330.0 1.40004 0.700022 0.714122i \(-0.253174\pi\)
0.700022 + 0.714122i \(0.253174\pi\)
\(380\) 112.000 0.0151197
\(381\) 12162.2 1.63541
\(382\) −2056.00 −0.275377
\(383\) −1004.09 −0.133960 −0.0669800 0.997754i \(-0.521336\pi\)
−0.0669800 + 0.997754i \(0.521336\pi\)
\(384\) −905.097 −0.120281
\(385\) 0 0
\(386\) −9184.00 −1.21102
\(387\) −782.000 −0.102717
\(388\) −5934.04 −0.776431
\(389\) 5210.00 0.679068 0.339534 0.940594i \(-0.389731\pi\)
0.339534 + 0.940594i \(0.389731\pi\)
\(390\) 14255.3 1.85088
\(391\) −197.990 −0.0256081
\(392\) 0 0
\(393\) 12270.0 1.57491
\(394\) −1588.00 −0.203051
\(395\) 24154.8 3.07686
\(396\) −1288.00 −0.163446
\(397\) 73.5391 0.00929678 0.00464839 0.999989i \(-0.498520\pi\)
0.00464839 + 0.999989i \(0.498520\pi\)
\(398\) 4972.37 0.626238
\(399\) 0 0
\(400\) 4272.00 0.534000
\(401\) −498.000 −0.0620173 −0.0310086 0.999519i \(-0.509872\pi\)
−0.0310086 + 0.999519i \(0.509872\pi\)
\(402\) −9673.22 −1.20014
\(403\) −4752.00 −0.587380
\(404\) −4514.17 −0.555912
\(405\) −16255.0 −1.99436
\(406\) 0 0
\(407\) 532.000 0.0647918
\(408\) 80.0000 0.00970733
\(409\) −3355.93 −0.405721 −0.202861 0.979208i \(-0.565024\pi\)
−0.202861 + 0.979208i \(0.565024\pi\)
\(410\) −4984.00 −0.600347
\(411\) 5854.84 0.702672
\(412\) 3473.31 0.415334
\(413\) 0 0
\(414\) −6440.00 −0.764514
\(415\) −8372.00 −0.990278
\(416\) 1629.17 0.192012
\(417\) 3010.00 0.353478
\(418\) 39.5980 0.00463349
\(419\) −14545.2 −1.69589 −0.847946 0.530082i \(-0.822161\pi\)
−0.847946 + 0.530082i \(0.822161\pi\)
\(420\) 0 0
\(421\) 10854.0 1.25651 0.628256 0.778007i \(-0.283769\pi\)
0.628256 + 0.778007i \(0.283769\pi\)
\(422\) 5496.00 0.633984
\(423\) −12035.0 −1.38336
\(424\) 592.000 0.0678067
\(425\) −377.595 −0.0430966
\(426\) −8315.58 −0.945753
\(427\) 0 0
\(428\) −6736.00 −0.760740
\(429\) 5040.00 0.567211
\(430\) 1346.33 0.150990
\(431\) −5364.00 −0.599477 −0.299739 0.954021i \(-0.596900\pi\)
−0.299739 + 0.954021i \(0.596900\pi\)
\(432\) −452.548 −0.0504010
\(433\) 6487.00 0.719966 0.359983 0.932959i \(-0.382783\pi\)
0.359983 + 0.932959i \(0.382783\pi\)
\(434\) 0 0
\(435\) −40040.0 −4.41327
\(436\) −3272.00 −0.359405
\(437\) 197.990 0.0216731
\(438\) −3820.00 −0.416728
\(439\) −13932.8 −1.51476 −0.757378 0.652977i \(-0.773520\pi\)
−0.757378 + 0.652977i \(0.773520\pi\)
\(440\) 2217.49 0.240260
\(441\) 0 0
\(442\) −144.000 −0.0154963
\(443\) 5996.00 0.643067 0.321533 0.946898i \(-0.395802\pi\)
0.321533 + 0.946898i \(0.395802\pi\)
\(444\) −1074.80 −0.114883
\(445\) −12236.0 −1.30347
\(446\) −6856.11 −0.727906
\(447\) 14495.7 1.53383
\(448\) 0 0
\(449\) 2622.00 0.275590 0.137795 0.990461i \(-0.455999\pi\)
0.137795 + 0.990461i \(0.455999\pi\)
\(450\) −12282.0 −1.28662
\(451\) −1762.11 −0.183979
\(452\) −2160.00 −0.224774
\(453\) −3337.54 −0.346162
\(454\) 10581.1 1.09383
\(455\) 0 0
\(456\) −80.0000 −0.00821567
\(457\) 11208.0 1.14724 0.573619 0.819122i \(-0.305539\pi\)
0.573619 + 0.819122i \(0.305539\pi\)
\(458\) −5498.46 −0.560974
\(459\) 40.0000 0.00406763
\(460\) 11087.4 1.12381
\(461\) −9786.36 −0.988712 −0.494356 0.869260i \(-0.664596\pi\)
−0.494356 + 0.869260i \(0.664596\pi\)
\(462\) 0 0
\(463\) 3952.00 0.396685 0.198342 0.980133i \(-0.436444\pi\)
0.198342 + 0.980133i \(0.436444\pi\)
\(464\) −4576.00 −0.457835
\(465\) 13067.3 1.30319
\(466\) −144.000 −0.0143147
\(467\) −17506.5 −1.73470 −0.867352 0.497696i \(-0.834180\pi\)
−0.867352 + 0.497696i \(0.834180\pi\)
\(468\) −4683.88 −0.462633
\(469\) 0 0
\(470\) 20720.0 2.03349
\(471\) 15640.0 1.53005
\(472\) 3473.31 0.338712
\(473\) 476.000 0.0462717
\(474\) −17253.4 −1.67189
\(475\) 377.595 0.0364742
\(476\) 0 0
\(477\) −1702.00 −0.163374
\(478\) −8616.00 −0.824449
\(479\) −2288.20 −0.218268 −0.109134 0.994027i \(-0.534808\pi\)
−0.109134 + 0.994027i \(0.534808\pi\)
\(480\) −4480.00 −0.426006
\(481\) 1934.64 0.183393
\(482\) −3080.16 −0.291073
\(483\) 0 0
\(484\) −4540.00 −0.426371
\(485\) −29372.0 −2.74993
\(486\) 10083.3 0.941131
\(487\) 972.000 0.0904426 0.0452213 0.998977i \(-0.485601\pi\)
0.0452213 + 0.998977i \(0.485601\pi\)
\(488\) 113.137 0.0104948
\(489\) 23235.5 2.14877
\(490\) 0 0
\(491\) −7404.00 −0.680525 −0.340263 0.940330i \(-0.610516\pi\)
−0.340263 + 0.940330i \(0.610516\pi\)
\(492\) 3560.00 0.326214
\(493\) 404.465 0.0369497
\(494\) 144.000 0.0131151
\(495\) −6375.27 −0.578883
\(496\) 1493.41 0.135194
\(497\) 0 0
\(498\) 5980.00 0.538093
\(499\) −12244.0 −1.09843 −0.549215 0.835681i \(-0.685073\pi\)
−0.549215 + 0.835681i \(0.685073\pi\)
\(500\) 11245.8 1.00586
\(501\) 10540.0 0.939905
\(502\) 1863.93 0.165720
\(503\) 2415.48 0.214117 0.107058 0.994253i \(-0.465857\pi\)
0.107058 + 0.994253i \(0.465857\pi\)
\(504\) 0 0
\(505\) −22344.0 −1.96890
\(506\) 3920.00 0.344398
\(507\) 2793.07 0.244664
\(508\) 6880.00 0.600887
\(509\) 5707.77 0.497038 0.248519 0.968627i \(-0.420056\pi\)
0.248519 + 0.968627i \(0.420056\pi\)
\(510\) 395.980 0.0343809
\(511\) 0 0
\(512\) −512.000 −0.0441942
\(513\) −40.0000 −0.00344258
\(514\) 1875.25 0.160921
\(515\) 17192.0 1.47101
\(516\) −961.665 −0.0820445
\(517\) 7325.63 0.623173
\(518\) 0 0
\(519\) 14640.0 1.23820
\(520\) 8064.00 0.680057
\(521\) −1.41421 −0.000118921 0 −5.94605e−5 1.00000i \(-0.500019\pi\)
−5.94605e−5 1.00000i \(0.500019\pi\)
\(522\) 13156.0 1.10311
\(523\) 12257.0 1.02478 0.512391 0.858752i \(-0.328760\pi\)
0.512391 + 0.858752i \(0.328760\pi\)
\(524\) 6940.96 0.578659
\(525\) 0 0
\(526\) −14280.0 −1.18372
\(527\) −132.000 −0.0109108
\(528\) −1583.92 −0.130552
\(529\) 7433.00 0.610915
\(530\) 2930.25 0.240155
\(531\) −9985.76 −0.816093
\(532\) 0 0
\(533\) −6408.00 −0.520753
\(534\) 8740.00 0.708271
\(535\) −33341.5 −2.69435
\(536\) −5472.00 −0.440960
\(537\) 3818.38 0.306844
\(538\) 9220.67 0.738906
\(539\) 0 0
\(540\) −2240.00 −0.178508
\(541\) 2050.00 0.162914 0.0814569 0.996677i \(-0.474043\pi\)
0.0814569 + 0.996677i \(0.474043\pi\)
\(542\) −4729.13 −0.374785
\(543\) −26760.0 −2.11488
\(544\) 45.2548 0.00356670
\(545\) −16195.6 −1.27292
\(546\) 0 0
\(547\) 14554.0 1.13763 0.568815 0.822465i \(-0.307402\pi\)
0.568815 + 0.822465i \(0.307402\pi\)
\(548\) 3312.00 0.258178
\(549\) −325.269 −0.0252862
\(550\) 7476.00 0.579596
\(551\) −404.465 −0.0312719
\(552\) −7919.60 −0.610653
\(553\) 0 0
\(554\) −8012.00 −0.614435
\(555\) −5320.00 −0.406885
\(556\) 1702.71 0.129876
\(557\) 6954.00 0.528995 0.264498 0.964386i \(-0.414794\pi\)
0.264498 + 0.964386i \(0.414794\pi\)
\(558\) −4293.55 −0.325736
\(559\) 1731.00 0.130972
\(560\) 0 0
\(561\) 140.000 0.0105362
\(562\) 11968.0 0.898291
\(563\) 1636.25 0.122486 0.0612429 0.998123i \(-0.480494\pi\)
0.0612429 + 0.998123i \(0.480494\pi\)
\(564\) −14800.0 −1.10495
\(565\) −10691.5 −0.796094
\(566\) −9857.07 −0.732020
\(567\) 0 0
\(568\) −4704.00 −0.347492
\(569\) −7142.00 −0.526201 −0.263100 0.964768i \(-0.584745\pi\)
−0.263100 + 0.964768i \(0.584745\pi\)
\(570\) −395.980 −0.0290978
\(571\) −20606.0 −1.51022 −0.755109 0.655599i \(-0.772416\pi\)
−0.755109 + 0.655599i \(0.772416\pi\)
\(572\) 2851.05 0.208407
\(573\) 7269.06 0.529964
\(574\) 0 0
\(575\) 37380.0 2.71105
\(576\) 1472.00 0.106481
\(577\) 8803.48 0.635171 0.317585 0.948230i \(-0.397128\pi\)
0.317585 + 0.948230i \(0.397128\pi\)
\(578\) 9822.00 0.706819
\(579\) 32470.3 2.33061
\(580\) −22650.0 −1.62154
\(581\) 0 0
\(582\) 20980.0 1.49424
\(583\) 1036.00 0.0735965
\(584\) −2160.92 −0.153115
\(585\) −23184.0 −1.63853
\(586\) 3942.83 0.277947
\(587\) −6503.97 −0.457321 −0.228661 0.973506i \(-0.573435\pi\)
−0.228661 + 0.973506i \(0.573435\pi\)
\(588\) 0 0
\(589\) 132.000 0.00923424
\(590\) 17192.0 1.19963
\(591\) 5614.43 0.390773
\(592\) −608.000 −0.0422106
\(593\) 23140.8 1.60249 0.801246 0.598335i \(-0.204171\pi\)
0.801246 + 0.598335i \(0.204171\pi\)
\(594\) −791.960 −0.0547045
\(595\) 0 0
\(596\) 8200.00 0.563566
\(597\) −17580.0 −1.20520
\(598\) 14255.3 0.974818
\(599\) −11296.0 −0.770521 −0.385260 0.922808i \(-0.625888\pi\)
−0.385260 + 0.922808i \(0.625888\pi\)
\(600\) −15103.8 −1.02768
\(601\) −8727.11 −0.592323 −0.296162 0.955138i \(-0.595707\pi\)
−0.296162 + 0.955138i \(0.595707\pi\)
\(602\) 0 0
\(603\) 15732.0 1.06245
\(604\) −1888.00 −0.127188
\(605\) −22471.9 −1.51010
\(606\) 15960.0 1.06985
\(607\) −19736.8 −1.31975 −0.659877 0.751374i \(-0.729392\pi\)
−0.659877 + 0.751374i \(0.729392\pi\)
\(608\) −45.2548 −0.00301863
\(609\) 0 0
\(610\) 560.000 0.0371701
\(611\) 26640.0 1.76389
\(612\) −130.108 −0.00859361
\(613\) 16962.0 1.11760 0.558800 0.829302i \(-0.311262\pi\)
0.558800 + 0.829302i \(0.311262\pi\)
\(614\) −9534.63 −0.626688
\(615\) 17621.1 1.15537
\(616\) 0 0
\(617\) −19034.0 −1.24194 −0.620972 0.783832i \(-0.713262\pi\)
−0.620972 + 0.783832i \(0.713262\pi\)
\(618\) −12280.0 −0.799311
\(619\) 18677.5 1.21278 0.606392 0.795166i \(-0.292616\pi\)
0.606392 + 0.795166i \(0.292616\pi\)
\(620\) 7392.00 0.478822
\(621\) −3959.80 −0.255880
\(622\) −13553.8 −0.873728
\(623\) 0 0
\(624\) −5760.00 −0.369527
\(625\) 22289.0 1.42650
\(626\) −12380.0 −0.790424
\(627\) −140.000 −0.00891716
\(628\) 8847.32 0.562176
\(629\) 53.7401 0.00340661
\(630\) 0 0
\(631\) −14716.0 −0.928423 −0.464211 0.885724i \(-0.653662\pi\)
−0.464211 + 0.885724i \(0.653662\pi\)
\(632\) −9760.00 −0.614291
\(633\) −19431.3 −1.22010
\(634\) −19652.0 −1.23104
\(635\) 34054.3 2.12819
\(636\) −2093.04 −0.130494
\(637\) 0 0
\(638\) −8008.00 −0.496928
\(639\) 13524.0 0.837248
\(640\) −2534.27 −0.156525
\(641\) 4730.00 0.291457 0.145728 0.989325i \(-0.453447\pi\)
0.145728 + 0.989325i \(0.453447\pi\)
\(642\) 23815.4 1.46405
\(643\) −19056.5 −1.16877 −0.584383 0.811478i \(-0.698663\pi\)
−0.584383 + 0.811478i \(0.698663\pi\)
\(644\) 0 0
\(645\) −4760.00 −0.290581
\(646\) 4.00000 0.000243619 0
\(647\) −9342.29 −0.567672 −0.283836 0.958873i \(-0.591607\pi\)
−0.283836 + 0.958873i \(0.591607\pi\)
\(648\) 6568.00 0.398172
\(649\) 6078.29 0.367633
\(650\) 27186.8 1.64055
\(651\) 0 0
\(652\) 13144.0 0.789507
\(653\) 3774.00 0.226169 0.113084 0.993585i \(-0.463927\pi\)
0.113084 + 0.993585i \(0.463927\pi\)
\(654\) 11568.3 0.691674
\(655\) 34356.0 2.04947
\(656\) 2013.84 0.119859
\(657\) 6212.64 0.368917
\(658\) 0 0
\(659\) −21150.0 −1.25021 −0.625104 0.780541i \(-0.714943\pi\)
−0.625104 + 0.780541i \(0.714943\pi\)
\(660\) −7840.00 −0.462381
\(661\) 10377.5 0.610647 0.305324 0.952249i \(-0.401235\pi\)
0.305324 + 0.952249i \(0.401235\pi\)
\(662\) 11476.0 0.673757
\(663\) 509.117 0.0298227
\(664\) 3382.80 0.197708
\(665\) 0 0
\(666\) 1748.00 0.101702
\(667\) −40040.0 −2.32437
\(668\) 5962.32 0.345343
\(669\) 24240.0 1.40086
\(670\) −27085.0 −1.56177
\(671\) 197.990 0.0113909
\(672\) 0 0
\(673\) −1164.00 −0.0666700 −0.0333350 0.999444i \(-0.510613\pi\)
−0.0333350 + 0.999444i \(0.510613\pi\)
\(674\) 4508.00 0.257629
\(675\) −7551.90 −0.430626
\(676\) 1580.00 0.0898953
\(677\) −27152.9 −1.54146 −0.770732 0.637160i \(-0.780109\pi\)
−0.770732 + 0.637160i \(0.780109\pi\)
\(678\) 7636.75 0.432578
\(679\) 0 0
\(680\) 224.000 0.0126324
\(681\) −37410.0 −2.10507
\(682\) 2613.47 0.146737
\(683\) −16596.0 −0.929763 −0.464882 0.885373i \(-0.653903\pi\)
−0.464882 + 0.885373i \(0.653903\pi\)
\(684\) 130.108 0.00727309
\(685\) 16393.6 0.914403
\(686\) 0 0
\(687\) 19440.0 1.07960
\(688\) −544.000 −0.0301451
\(689\) 3767.46 0.208315
\(690\) −39200.0 −2.16278
\(691\) −11298.2 −0.622000 −0.311000 0.950410i \(-0.600664\pi\)
−0.311000 + 0.950410i \(0.600664\pi\)
\(692\) 8281.63 0.454943
\(693\) 0 0
\(694\) 3972.00 0.217255
\(695\) 8428.00 0.459989
\(696\) 16178.6 0.881104
\(697\) −178.000 −0.00967321
\(698\) 13542.5 0.734372
\(699\) 509.117 0.0275487
\(700\) 0 0
\(701\) −2754.00 −0.148384 −0.0741920 0.997244i \(-0.523638\pi\)
−0.0741920 + 0.997244i \(0.523638\pi\)
\(702\) −2880.00 −0.154841
\(703\) −53.7401 −0.00288314
\(704\) −896.000 −0.0479677
\(705\) −73256.3 −3.91346
\(706\) 13986.6 0.745597
\(707\) 0 0
\(708\) −12280.0 −0.651851
\(709\) 29434.0 1.55912 0.779561 0.626327i \(-0.215442\pi\)
0.779561 + 0.626327i \(0.215442\pi\)
\(710\) −23283.6 −1.23073
\(711\) 28060.0 1.48007
\(712\) 4944.09 0.260235
\(713\) 13067.3 0.686361
\(714\) 0 0
\(715\) 14112.0 0.738124
\(716\) 2160.00 0.112742
\(717\) 30462.2 1.58665
\(718\) −11888.0 −0.617906
\(719\) 17669.2 0.916480 0.458240 0.888828i \(-0.348480\pi\)
0.458240 + 0.888828i \(0.348480\pi\)
\(720\) 7286.03 0.377131
\(721\) 0 0
\(722\) 13714.0 0.706901
\(723\) 10890.0 0.560171
\(724\) −15137.7 −0.777058
\(725\) −76362.0 −3.91174
\(726\) 16051.3 0.820552
\(727\) −28445.5 −1.45115 −0.725574 0.688144i \(-0.758426\pi\)
−0.725574 + 0.688144i \(0.758426\pi\)
\(728\) 0 0
\(729\) −13483.0 −0.685007
\(730\) −10696.0 −0.542297
\(731\) 48.0833 0.00243286
\(732\) −400.000 −0.0201973
\(733\) −22341.7 −1.12580 −0.562900 0.826525i \(-0.690314\pi\)
−0.562900 + 0.826525i \(0.690314\pi\)
\(734\) −1685.74 −0.0847710
\(735\) 0 0
\(736\) −4480.00 −0.224368
\(737\) −9576.00 −0.478611
\(738\) −5789.79 −0.288787
\(739\) 20670.0 1.02890 0.514451 0.857520i \(-0.327996\pi\)
0.514451 + 0.857520i \(0.327996\pi\)
\(740\) −3009.45 −0.149499
\(741\) −509.117 −0.0252400
\(742\) 0 0
\(743\) −25400.0 −1.25415 −0.627076 0.778958i \(-0.715749\pi\)
−0.627076 + 0.778958i \(0.715749\pi\)
\(744\) −5280.00 −0.260180
\(745\) 40587.9 1.99601
\(746\) 11452.0 0.562048
\(747\) −9725.55 −0.476358
\(748\) 79.1960 0.00387124
\(749\) 0 0
\(750\) −39760.0 −1.93577
\(751\) 29180.0 1.41783 0.708917 0.705292i \(-0.249184\pi\)
0.708917 + 0.705292i \(0.249184\pi\)
\(752\) −8372.14 −0.405985
\(753\) −6590.00 −0.318928
\(754\) −29121.5 −1.40655
\(755\) −9345.12 −0.450469
\(756\) 0 0
\(757\) −26206.0 −1.25822 −0.629110 0.777316i \(-0.716581\pi\)
−0.629110 + 0.777316i \(0.716581\pi\)
\(758\) −20660.0 −0.989980
\(759\) −13859.3 −0.662794
\(760\) −224.000 −0.0106912
\(761\) −6863.18 −0.326925 −0.163463 0.986550i \(-0.552266\pi\)
−0.163463 + 0.986550i \(0.552266\pi\)
\(762\) −24324.5 −1.15641
\(763\) 0 0
\(764\) 4112.00 0.194721
\(765\) −644.000 −0.0304364
\(766\) 2008.18 0.0947240
\(767\) 22104.0 1.04059
\(768\) 1810.19 0.0850517
\(769\) 9058.04 0.424761 0.212380 0.977187i \(-0.431878\pi\)
0.212380 + 0.977187i \(0.431878\pi\)
\(770\) 0 0
\(771\) −6630.00 −0.309693
\(772\) 18368.0 0.856320
\(773\) 132.936 0.00618548 0.00309274 0.999995i \(-0.499016\pi\)
0.00309274 + 0.999995i \(0.499016\pi\)
\(774\) 1564.00 0.0726315
\(775\) 24921.3 1.15509
\(776\) 11868.1 0.549020
\(777\) 0 0
\(778\) −10420.0 −0.480174
\(779\) 178.000 0.00818679
\(780\) −28510.5 −1.30877
\(781\) −8232.00 −0.377163
\(782\) 395.980 0.0181077
\(783\) 8089.30 0.369206
\(784\) 0 0
\(785\) 43792.0 1.99109
\(786\) −24540.0 −1.11363
\(787\) 8729.94 0.395411 0.197706 0.980261i \(-0.436651\pi\)
0.197706 + 0.980261i \(0.436651\pi\)
\(788\) 3176.00 0.143579
\(789\) 50487.4 2.27807
\(790\) −48309.5 −2.17567
\(791\) 0 0
\(792\) 2576.00 0.115573
\(793\) 720.000 0.0322421
\(794\) −147.078 −0.00657382
\(795\) −10360.0 −0.462178
\(796\) −9944.75 −0.442817
\(797\) −7517.96 −0.334128 −0.167064 0.985946i \(-0.553429\pi\)
−0.167064 + 0.985946i \(0.553429\pi\)
\(798\) 0 0
\(799\) 740.000 0.0327651
\(800\) −8544.00 −0.377595
\(801\) −14214.3 −0.627011
\(802\) 996.000 0.0438528
\(803\) −3781.61 −0.166189
\(804\) 19346.4 0.848627
\(805\) 0 0
\(806\) 9504.00 0.415340
\(807\) −32600.0 −1.42203
\(808\) 9028.34 0.393089
\(809\) 3776.00 0.164100 0.0820501 0.996628i \(-0.473853\pi\)
0.0820501 + 0.996628i \(0.473853\pi\)
\(810\) 32509.9 1.41023
\(811\) 36227.9 1.56860 0.784300 0.620382i \(-0.213023\pi\)
0.784300 + 0.620382i \(0.213023\pi\)
\(812\) 0 0
\(813\) 16720.0 0.721274
\(814\) −1064.00 −0.0458147
\(815\) 65059.5 2.79624
\(816\) −160.000 −0.00686412
\(817\) −48.0833 −0.00205902
\(818\) 6711.86 0.286888
\(819\) 0 0
\(820\) 9968.00 0.424509
\(821\) −16410.0 −0.697580 −0.348790 0.937201i \(-0.613407\pi\)
−0.348790 + 0.937201i \(0.613407\pi\)
\(822\) −11709.7 −0.496864
\(823\) 22072.0 0.934850 0.467425 0.884033i \(-0.345182\pi\)
0.467425 + 0.884033i \(0.345182\pi\)
\(824\) −6946.62 −0.293686
\(825\) −26431.7 −1.11543
\(826\) 0 0
\(827\) −11628.0 −0.488930 −0.244465 0.969658i \(-0.578612\pi\)
−0.244465 + 0.969658i \(0.578612\pi\)
\(828\) 12880.0 0.540593
\(829\) 30906.2 1.29483 0.647417 0.762136i \(-0.275849\pi\)
0.647417 + 0.762136i \(0.275849\pi\)
\(830\) 16744.0 0.700232
\(831\) 28326.7 1.18248
\(832\) −3258.35 −0.135773
\(833\) 0 0
\(834\) −6020.00 −0.249947
\(835\) 29512.0 1.22312
\(836\) −79.1960 −0.00327638
\(837\) −2640.00 −0.109022
\(838\) 29090.4 1.19918
\(839\) −17884.1 −0.735911 −0.367955 0.929843i \(-0.619942\pi\)
−0.367955 + 0.929843i \(0.619942\pi\)
\(840\) 0 0
\(841\) 57407.0 2.35381
\(842\) −21708.0 −0.888488
\(843\) −42313.3 −1.72876
\(844\) −10992.0 −0.448294
\(845\) 7820.60 0.318387
\(846\) 24069.9 0.978181
\(847\) 0 0
\(848\) −1184.00 −0.0479466
\(849\) 34850.0 1.40877
\(850\) 755.190 0.0304739
\(851\) −5320.00 −0.214298
\(852\) 16631.2 0.668749
\(853\) 20755.0 0.833104 0.416552 0.909112i \(-0.363238\pi\)
0.416552 + 0.909112i \(0.363238\pi\)
\(854\) 0 0
\(855\) 644.000 0.0257595
\(856\) 13472.0 0.537925
\(857\) −44919.7 −1.79046 −0.895231 0.445602i \(-0.852990\pi\)
−0.895231 + 0.445602i \(0.852990\pi\)
\(858\) −10080.0 −0.401079
\(859\) −69.2965 −0.00275246 −0.00137623 0.999999i \(-0.500438\pi\)
−0.00137623 + 0.999999i \(0.500438\pi\)
\(860\) −2692.66 −0.106766
\(861\) 0 0
\(862\) 10728.0 0.423895
\(863\) −5452.00 −0.215050 −0.107525 0.994202i \(-0.534293\pi\)
−0.107525 + 0.994202i \(0.534293\pi\)
\(864\) 905.097 0.0356389
\(865\) 40992.0 1.61129
\(866\) −12974.0 −0.509093
\(867\) −34726.0 −1.36027
\(868\) 0 0
\(869\) −17080.0 −0.666743
\(870\) 80080.0 3.12065
\(871\) −34823.6 −1.35471
\(872\) 6544.00 0.254137
\(873\) −34120.7 −1.32281
\(874\) −395.980 −0.0153252
\(875\) 0 0
\(876\) 7640.00 0.294671
\(877\) 31106.0 1.19769 0.598845 0.800865i \(-0.295626\pi\)
0.598845 + 0.800865i \(0.295626\pi\)
\(878\) 27865.7 1.07109
\(879\) −13940.0 −0.534908
\(880\) −4434.97 −0.169890
\(881\) −5943.94 −0.227306 −0.113653 0.993521i \(-0.536255\pi\)
−0.113653 + 0.993521i \(0.536255\pi\)
\(882\) 0 0
\(883\) 34796.0 1.32614 0.663068 0.748559i \(-0.269254\pi\)
0.663068 + 0.748559i \(0.269254\pi\)
\(884\) 288.000 0.0109576
\(885\) −60782.9 −2.30869
\(886\) −11992.0 −0.454717
\(887\) −9964.55 −0.377200 −0.188600 0.982054i \(-0.560395\pi\)
−0.188600 + 0.982054i \(0.560395\pi\)
\(888\) 2149.60 0.0812342
\(889\) 0 0
\(890\) 24472.0 0.921689
\(891\) 11494.0 0.432170
\(892\) 13712.2 0.514707
\(893\) −740.000 −0.0277303
\(894\) −28991.4 −1.08458
\(895\) 10691.5 0.399303
\(896\) 0 0
\(897\) −50400.0 −1.87604
\(898\) −5244.00 −0.194871
\(899\) −26694.7 −0.990343
\(900\) 24564.0 0.909778
\(901\) 104.652 0.00386954
\(902\) 3524.22 0.130093
\(903\) 0 0
\(904\) 4320.00 0.158939
\(905\) −74928.0 −2.75214
\(906\) 6675.09 0.244774
\(907\) −29756.0 −1.08934 −0.544670 0.838650i \(-0.683345\pi\)
−0.544670 + 0.838650i \(0.683345\pi\)
\(908\) −21162.3 −0.773453
\(909\) −25956.5 −0.947109
\(910\) 0 0
\(911\) 21440.0 0.779735 0.389868 0.920871i \(-0.372521\pi\)
0.389868 + 0.920871i \(0.372521\pi\)
\(912\) 160.000 0.00580935
\(913\) 5919.90 0.214589
\(914\) −22416.0 −0.811220
\(915\) −1979.90 −0.0715338
\(916\) 10996.9 0.396669
\(917\) 0 0
\(918\) −80.0000 −0.00287625
\(919\) −8288.00 −0.297493 −0.148746 0.988875i \(-0.547524\pi\)
−0.148746 + 0.988875i \(0.547524\pi\)
\(920\) −22174.9 −0.794656
\(921\) 33710.0 1.20606
\(922\) 19572.7 0.699125
\(923\) −29936.1 −1.06756
\(924\) 0 0
\(925\) −10146.0 −0.360647
\(926\) −7904.00 −0.280498
\(927\) 19971.5 0.707606
\(928\) 9152.00 0.323738
\(929\) 45581.5 1.60978 0.804888 0.593427i \(-0.202226\pi\)
0.804888 + 0.593427i \(0.202226\pi\)
\(930\) −26134.7 −0.921494
\(931\) 0 0
\(932\) 288.000 0.0101221
\(933\) 47920.0 1.68149
\(934\) 35013.1 1.22662
\(935\) 392.000 0.0137110
\(936\) 9367.75 0.327131
\(937\) −11665.8 −0.406731 −0.203365 0.979103i \(-0.565188\pi\)
−0.203365 + 0.979103i \(0.565188\pi\)
\(938\) 0 0
\(939\) 43770.0 1.52117
\(940\) −41440.0 −1.43790
\(941\) −14.1421 −0.000489926 0 −0.000244963 1.00000i \(-0.500078\pi\)
−0.000244963 1.00000i \(0.500078\pi\)
\(942\) −31280.0 −1.08191
\(943\) 17621.1 0.608507
\(944\) −6946.62 −0.239505
\(945\) 0 0
\(946\) −952.000 −0.0327190
\(947\) 14034.0 0.481567 0.240783 0.970579i \(-0.422596\pi\)
0.240783 + 0.970579i \(0.422596\pi\)
\(948\) 34506.8 1.18220
\(949\) −13752.0 −0.470399
\(950\) −755.190 −0.0257912
\(951\) 69480.3 2.36914
\(952\) 0 0
\(953\) −42698.0 −1.45134 −0.725668 0.688045i \(-0.758469\pi\)
−0.725668 + 0.688045i \(0.758469\pi\)
\(954\) 3404.00 0.115523
\(955\) 20353.4 0.689654
\(956\) 17232.0 0.582974
\(957\) 28312.6 0.956337
\(958\) 4576.40 0.154339
\(959\) 0 0
\(960\) 8960.00 0.301232
\(961\) −21079.0 −0.707563
\(962\) −3869.29 −0.129679
\(963\) −38732.0 −1.29608
\(964\) 6160.31 0.205820
\(965\) 90917.0 3.03287
\(966\) 0 0
\(967\) −48492.0 −1.61261 −0.806307 0.591497i \(-0.798537\pi\)
−0.806307 + 0.591497i \(0.798537\pi\)
\(968\) 9080.00 0.301490
\(969\) −14.1421 −0.000468845 0
\(970\) 58744.0 1.94449
\(971\) −52669.6 −1.74073 −0.870364 0.492409i \(-0.836116\pi\)
−0.870364 + 0.492409i \(0.836116\pi\)
\(972\) −20166.7 −0.665480
\(973\) 0 0
\(974\) −1944.00 −0.0639525
\(975\) −96120.0 −3.15723
\(976\) −226.274 −0.00742096
\(977\) −55380.0 −1.81347 −0.906737 0.421698i \(-0.861434\pi\)
−0.906737 + 0.421698i \(0.861434\pi\)
\(978\) −46471.1 −1.51941
\(979\) 8652.16 0.282456
\(980\) 0 0
\(981\) −18814.0 −0.612319
\(982\) 14808.0 0.481204
\(983\) 50535.5 1.63971 0.819854 0.572573i \(-0.194055\pi\)
0.819854 + 0.572573i \(0.194055\pi\)
\(984\) −7120.00 −0.230668
\(985\) 15720.4 0.508521
\(986\) −808.930 −0.0261274
\(987\) 0 0
\(988\) −288.000 −0.00927379
\(989\) −4760.00 −0.153043
\(990\) 12750.5 0.409332
\(991\) −39712.0 −1.27295 −0.636475 0.771297i \(-0.719608\pi\)
−0.636475 + 0.771297i \(0.719608\pi\)
\(992\) −2986.82 −0.0955964
\(993\) −40573.8 −1.29665
\(994\) 0 0
\(995\) −49224.0 −1.56835
\(996\) −11960.0 −0.380489
\(997\) −2186.37 −0.0694515 −0.0347258 0.999397i \(-0.511056\pi\)
−0.0347258 + 0.999397i \(0.511056\pi\)
\(998\) 24488.0 0.776708
\(999\) 1074.80 0.0340393
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 98.4.a.g.1.2 yes 2
3.2 odd 2 882.4.a.bg.1.1 2
4.3 odd 2 784.4.a.y.1.1 2
5.4 even 2 2450.4.a.bx.1.1 2
7.2 even 3 98.4.c.h.67.1 4
7.3 odd 6 98.4.c.h.79.2 4
7.4 even 3 98.4.c.h.79.1 4
7.5 odd 6 98.4.c.h.67.2 4
7.6 odd 2 inner 98.4.a.g.1.1 2
21.2 odd 6 882.4.g.ba.361.2 4
21.5 even 6 882.4.g.ba.361.1 4
21.11 odd 6 882.4.g.ba.667.2 4
21.17 even 6 882.4.g.ba.667.1 4
21.20 even 2 882.4.a.bg.1.2 2
28.27 even 2 784.4.a.y.1.2 2
35.34 odd 2 2450.4.a.bx.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.4.a.g.1.1 2 7.6 odd 2 inner
98.4.a.g.1.2 yes 2 1.1 even 1 trivial
98.4.c.h.67.1 4 7.2 even 3
98.4.c.h.67.2 4 7.5 odd 6
98.4.c.h.79.1 4 7.4 even 3
98.4.c.h.79.2 4 7.3 odd 6
784.4.a.y.1.1 2 4.3 odd 2
784.4.a.y.1.2 2 28.27 even 2
882.4.a.bg.1.1 2 3.2 odd 2
882.4.a.bg.1.2 2 21.20 even 2
882.4.g.ba.361.1 4 21.5 even 6
882.4.g.ba.361.2 4 21.2 odd 6
882.4.g.ba.667.1 4 21.17 even 6
882.4.g.ba.667.2 4 21.11 odd 6
2450.4.a.bx.1.1 2 5.4 even 2
2450.4.a.bx.1.2 2 35.34 odd 2