Properties

Label 49.4.a.e.1.1
Level $49$
Weight $4$
Character 49.1
Self dual yes
Analytic conductor $2.891$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.89109359028\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{65})\)
Defining polynomial: \(x^{4} - 2 x^{3} - 35 x^{2} + 36 x + 194\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.11692\) of defining polynomial
Character \(\chi\) \(=\) 49.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.53113 q^{2} -7.82220 q^{3} +4.46887 q^{4} -2.07730 q^{5} +27.6212 q^{6} +12.4689 q^{8} +34.1868 q^{9} +O(q^{10})\) \(q-3.53113 q^{2} -7.82220 q^{3} +4.46887 q^{4} -2.07730 q^{5} +27.6212 q^{6} +12.4689 q^{8} +34.1868 q^{9} +7.33521 q^{10} +49.1868 q^{11} -34.9564 q^{12} -44.8559 q^{13} +16.2490 q^{15} -79.7802 q^{16} +26.5179 q^{17} -120.718 q^{18} +77.7350 q^{19} -9.28317 q^{20} -173.685 q^{22} +55.7510 q^{23} -97.5340 q^{24} -120.685 q^{25} +158.392 q^{26} -56.2164 q^{27} +121.436 q^{29} -57.3774 q^{30} +305.553 q^{31} +181.963 q^{32} -384.749 q^{33} -93.6380 q^{34} +152.776 q^{36} +77.1868 q^{37} -274.492 q^{38} +350.872 q^{39} -25.9016 q^{40} +248.720 q^{41} -147.179 q^{43} +219.809 q^{44} -71.0161 q^{45} -196.864 q^{46} +269.851 q^{47} +624.056 q^{48} +426.154 q^{50} -207.428 q^{51} -200.455 q^{52} -141.121 q^{53} +198.507 q^{54} -102.176 q^{55} -608.058 q^{57} -428.805 q^{58} -424.834 q^{59} +72.6148 q^{60} -587.996 q^{61} -1078.95 q^{62} -4.29373 q^{64} +93.1790 q^{65} +1358.60 q^{66} -179.634 q^{67} +118.505 q^{68} -436.095 q^{69} +674.872 q^{71} +426.270 q^{72} +237.489 q^{73} -272.556 q^{74} +944.021 q^{75} +347.388 q^{76} -1238.97 q^{78} +495.852 q^{79} +165.727 q^{80} -483.307 q^{81} -878.262 q^{82} -24.4406 q^{83} -55.0855 q^{85} +519.708 q^{86} -949.895 q^{87} +613.304 q^{88} +1072.29 q^{89} +250.767 q^{90} +249.144 q^{92} -2390.09 q^{93} -952.877 q^{94} -161.479 q^{95} -1423.35 q^{96} +1667.43 q^{97} +1681.54 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 34q^{4} + 66q^{8} + 40q^{9} + O(q^{10}) \) \( 4q + 2q^{2} + 34q^{4} + 66q^{8} + 40q^{9} + 100q^{11} - 64q^{15} - 174q^{16} - 370q^{18} - 340q^{22} + 352q^{23} - 128q^{25} + 260q^{29} - 552q^{30} - 30q^{32} - 50q^{36} + 212q^{37} + 952q^{39} + 540q^{43} + 460q^{44} + 696q^{46} + 1366q^{50} + 428q^{51} + 16q^{53} - 1884q^{57} - 780q^{58} - 1064q^{60} - 1678q^{64} - 756q^{65} - 1944q^{67} + 2248q^{71} + 270q^{72} - 284q^{74} - 1344q^{78} - 1048q^{79} - 1256q^{81} - 3284q^{85} + 4820q^{86} + 1260q^{88} + 3512q^{92} - 5368q^{93} + 2192q^{95} + 3340q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.53113 −1.24844 −0.624221 0.781248i \(-0.714584\pi\)
−0.624221 + 0.781248i \(0.714584\pi\)
\(3\) −7.82220 −1.50538 −0.752691 0.658374i \(-0.771245\pi\)
−0.752691 + 0.658374i \(0.771245\pi\)
\(4\) 4.46887 0.558609
\(5\) −2.07730 −0.185799 −0.0928996 0.995675i \(-0.529614\pi\)
−0.0928996 + 0.995675i \(0.529614\pi\)
\(6\) 27.6212 1.87938
\(7\) 0 0
\(8\) 12.4689 0.551051
\(9\) 34.1868 1.26618
\(10\) 7.33521 0.231960
\(11\) 49.1868 1.34822 0.674108 0.738633i \(-0.264528\pi\)
0.674108 + 0.738633i \(0.264528\pi\)
\(12\) −34.9564 −0.840920
\(13\) −44.8559 −0.956983 −0.478492 0.878092i \(-0.658816\pi\)
−0.478492 + 0.878092i \(0.658816\pi\)
\(14\) 0 0
\(15\) 16.2490 0.279699
\(16\) −79.7802 −1.24656
\(17\) 26.5179 0.378325 0.189163 0.981946i \(-0.439423\pi\)
0.189163 + 0.981946i \(0.439423\pi\)
\(18\) −120.718 −1.58075
\(19\) 77.7350 0.938612 0.469306 0.883036i \(-0.344504\pi\)
0.469306 + 0.883036i \(0.344504\pi\)
\(20\) −9.28317 −0.103789
\(21\) 0 0
\(22\) −173.685 −1.68317
\(23\) 55.7510 0.505430 0.252715 0.967541i \(-0.418677\pi\)
0.252715 + 0.967541i \(0.418677\pi\)
\(24\) −97.5340 −0.829543
\(25\) −120.685 −0.965479
\(26\) 158.392 1.19474
\(27\) −56.2164 −0.400698
\(28\) 0 0
\(29\) 121.436 0.777588 0.388794 0.921325i \(-0.372892\pi\)
0.388794 + 0.921325i \(0.372892\pi\)
\(30\) −57.3774 −0.349188
\(31\) 305.553 1.77029 0.885143 0.465319i \(-0.154060\pi\)
0.885143 + 0.465319i \(0.154060\pi\)
\(32\) 181.963 1.00521
\(33\) −384.749 −2.02958
\(34\) −93.6380 −0.472317
\(35\) 0 0
\(36\) 152.776 0.707298
\(37\) 77.1868 0.342957 0.171479 0.985188i \(-0.445146\pi\)
0.171479 + 0.985188i \(0.445146\pi\)
\(38\) −274.492 −1.17180
\(39\) 350.872 1.44063
\(40\) −25.9016 −0.102385
\(41\) 248.720 0.947403 0.473702 0.880685i \(-0.342917\pi\)
0.473702 + 0.880685i \(0.342917\pi\)
\(42\) 0 0
\(43\) −147.179 −0.521967 −0.260984 0.965343i \(-0.584047\pi\)
−0.260984 + 0.965343i \(0.584047\pi\)
\(44\) 219.809 0.753125
\(45\) −71.0161 −0.235255
\(46\) −196.864 −0.631000
\(47\) 269.851 0.837484 0.418742 0.908105i \(-0.362471\pi\)
0.418742 + 0.908105i \(0.362471\pi\)
\(48\) 624.056 1.87656
\(49\) 0 0
\(50\) 426.154 1.20534
\(51\) −207.428 −0.569524
\(52\) −200.455 −0.534579
\(53\) −141.121 −0.365744 −0.182872 0.983137i \(-0.558539\pi\)
−0.182872 + 0.983137i \(0.558539\pi\)
\(54\) 198.507 0.500248
\(55\) −102.176 −0.250497
\(56\) 0 0
\(57\) −608.058 −1.41297
\(58\) −428.805 −0.970774
\(59\) −424.834 −0.937434 −0.468717 0.883348i \(-0.655284\pi\)
−0.468717 + 0.883348i \(0.655284\pi\)
\(60\) 72.6148 0.156242
\(61\) −587.996 −1.23418 −0.617092 0.786891i \(-0.711689\pi\)
−0.617092 + 0.786891i \(0.711689\pi\)
\(62\) −1078.95 −2.21010
\(63\) 0 0
\(64\) −4.29373 −0.00838618
\(65\) 93.1790 0.177807
\(66\) 1358.60 2.53381
\(67\) −179.634 −0.327549 −0.163775 0.986498i \(-0.552367\pi\)
−0.163775 + 0.986498i \(0.552367\pi\)
\(68\) 118.505 0.211336
\(69\) −436.095 −0.760865
\(70\) 0 0
\(71\) 674.872 1.12806 0.564032 0.825753i \(-0.309250\pi\)
0.564032 + 0.825753i \(0.309250\pi\)
\(72\) 426.270 0.697729
\(73\) 237.489 0.380767 0.190383 0.981710i \(-0.439027\pi\)
0.190383 + 0.981710i \(0.439027\pi\)
\(74\) −272.556 −0.428163
\(75\) 944.021 1.45341
\(76\) 347.388 0.524317
\(77\) 0 0
\(78\) −1238.97 −1.79854
\(79\) 495.852 0.706174 0.353087 0.935591i \(-0.385132\pi\)
0.353087 + 0.935591i \(0.385132\pi\)
\(80\) 165.727 0.231611
\(81\) −483.307 −0.662973
\(82\) −878.262 −1.18278
\(83\) −24.4406 −0.0323217 −0.0161609 0.999869i \(-0.505144\pi\)
−0.0161609 + 0.999869i \(0.505144\pi\)
\(84\) 0 0
\(85\) −55.0855 −0.0702925
\(86\) 519.708 0.651646
\(87\) −949.895 −1.17057
\(88\) 613.304 0.742936
\(89\) 1072.29 1.27710 0.638552 0.769579i \(-0.279534\pi\)
0.638552 + 0.769579i \(0.279534\pi\)
\(90\) 250.767 0.293702
\(91\) 0 0
\(92\) 249.144 0.282337
\(93\) −2390.09 −2.66496
\(94\) −952.877 −1.04555
\(95\) −161.479 −0.174393
\(96\) −1423.35 −1.51323
\(97\) 1667.43 1.74538 0.872690 0.488275i \(-0.162374\pi\)
0.872690 + 0.488275i \(0.162374\pi\)
\(98\) 0 0
\(99\) 1681.54 1.70708
\(100\) −539.325 −0.539325
\(101\) −77.1187 −0.0759762 −0.0379881 0.999278i \(-0.512095\pi\)
−0.0379881 + 0.999278i \(0.512095\pi\)
\(102\) 732.455 0.711018
\(103\) 164.693 0.157550 0.0787749 0.996892i \(-0.474899\pi\)
0.0787749 + 0.996892i \(0.474899\pi\)
\(104\) −559.302 −0.527347
\(105\) 0 0
\(106\) 498.315 0.456610
\(107\) 1022.62 0.923931 0.461966 0.886898i \(-0.347144\pi\)
0.461966 + 0.886898i \(0.347144\pi\)
\(108\) −251.224 −0.223833
\(109\) 1362.52 1.19730 0.598649 0.801011i \(-0.295704\pi\)
0.598649 + 0.801011i \(0.295704\pi\)
\(110\) 360.795 0.312731
\(111\) −603.770 −0.516282
\(112\) 0 0
\(113\) −1538.41 −1.28072 −0.640360 0.768075i \(-0.721215\pi\)
−0.640360 + 0.768075i \(0.721215\pi\)
\(114\) 2147.13 1.76401
\(115\) −115.811 −0.0939084
\(116\) 542.681 0.434368
\(117\) −1533.48 −1.21171
\(118\) 1500.14 1.17033
\(119\) 0 0
\(120\) 202.607 0.154128
\(121\) 1088.34 0.817685
\(122\) 2076.29 1.54081
\(123\) −1945.54 −1.42620
\(124\) 1365.48 0.988898
\(125\) 510.360 0.365184
\(126\) 0 0
\(127\) −170.358 −0.119030 −0.0595151 0.998227i \(-0.518955\pi\)
−0.0595151 + 0.998227i \(0.518955\pi\)
\(128\) −1440.54 −0.994744
\(129\) 1151.26 0.785760
\(130\) −329.027 −0.221981
\(131\) −751.935 −0.501503 −0.250751 0.968051i \(-0.580678\pi\)
−0.250751 + 0.968051i \(0.580678\pi\)
\(132\) −1719.39 −1.13374
\(133\) 0 0
\(134\) 634.312 0.408927
\(135\) 116.778 0.0744493
\(136\) 330.648 0.208477
\(137\) 518.623 0.323423 0.161711 0.986838i \(-0.448299\pi\)
0.161711 + 0.986838i \(0.448299\pi\)
\(138\) 1539.91 0.949896
\(139\) −2975.72 −1.81581 −0.907905 0.419177i \(-0.862319\pi\)
−0.907905 + 0.419177i \(0.862319\pi\)
\(140\) 0 0
\(141\) −2110.83 −1.26073
\(142\) −2383.06 −1.40832
\(143\) −2206.32 −1.29022
\(144\) −2727.43 −1.57837
\(145\) −252.258 −0.144475
\(146\) −838.604 −0.475365
\(147\) 0 0
\(148\) 344.938 0.191579
\(149\) −2717.94 −1.49438 −0.747188 0.664612i \(-0.768597\pi\)
−0.747188 + 0.664612i \(0.768597\pi\)
\(150\) −3333.46 −1.81450
\(151\) 707.650 0.381376 0.190688 0.981651i \(-0.438928\pi\)
0.190688 + 0.981651i \(0.438928\pi\)
\(152\) 969.267 0.517224
\(153\) 906.561 0.479027
\(154\) 0 0
\(155\) −634.724 −0.328918
\(156\) 1568.00 0.804747
\(157\) 3117.91 1.58495 0.792473 0.609906i \(-0.208793\pi\)
0.792473 + 0.609906i \(0.208793\pi\)
\(158\) −1750.92 −0.881618
\(159\) 1103.87 0.550584
\(160\) −377.991 −0.186768
\(161\) 0 0
\(162\) 1706.62 0.827684
\(163\) 1808.77 0.869167 0.434583 0.900632i \(-0.356896\pi\)
0.434583 + 0.900632i \(0.356896\pi\)
\(164\) 1111.50 0.529228
\(165\) 799.237 0.377094
\(166\) 86.3028 0.0403518
\(167\) 3147.38 1.45839 0.729197 0.684303i \(-0.239894\pi\)
0.729197 + 0.684303i \(0.239894\pi\)
\(168\) 0 0
\(169\) −184.949 −0.0841827
\(170\) 194.514 0.0877562
\(171\) 2657.51 1.18845
\(172\) −657.724 −0.291576
\(173\) −3284.36 −1.44338 −0.721691 0.692215i \(-0.756635\pi\)
−0.721691 + 0.692215i \(0.756635\pi\)
\(174\) 3354.20 1.46139
\(175\) 0 0
\(176\) −3924.13 −1.68064
\(177\) 3323.13 1.41120
\(178\) −3786.39 −1.59439
\(179\) 2798.83 1.16868 0.584341 0.811508i \(-0.301353\pi\)
0.584341 + 0.811508i \(0.301353\pi\)
\(180\) −317.362 −0.131415
\(181\) −3723.04 −1.52890 −0.764451 0.644682i \(-0.776990\pi\)
−0.764451 + 0.644682i \(0.776990\pi\)
\(182\) 0 0
\(183\) 4599.42 1.85792
\(184\) 695.152 0.278518
\(185\) −160.340 −0.0637212
\(186\) 8439.73 3.32705
\(187\) 1304.33 0.510064
\(188\) 1205.93 0.467826
\(189\) 0 0
\(190\) 570.202 0.217720
\(191\) 959.650 0.363549 0.181774 0.983340i \(-0.441816\pi\)
0.181774 + 0.983340i \(0.441816\pi\)
\(192\) 33.5864 0.0126244
\(193\) −3790.25 −1.41362 −0.706808 0.707406i \(-0.749865\pi\)
−0.706808 + 0.707406i \(0.749865\pi\)
\(194\) −5887.91 −2.17901
\(195\) −728.865 −0.267667
\(196\) 0 0
\(197\) 5117.99 1.85097 0.925487 0.378779i \(-0.123656\pi\)
0.925487 + 0.378779i \(0.123656\pi\)
\(198\) −5937.72 −2.13119
\(199\) 864.855 0.308080 0.154040 0.988065i \(-0.450772\pi\)
0.154040 + 0.988065i \(0.450772\pi\)
\(200\) −1504.80 −0.532028
\(201\) 1405.13 0.493087
\(202\) 272.316 0.0948519
\(203\) 0 0
\(204\) −926.969 −0.318141
\(205\) −516.665 −0.176027
\(206\) −581.550 −0.196692
\(207\) 1905.95 0.639963
\(208\) 3578.61 1.19294
\(209\) 3823.53 1.26545
\(210\) 0 0
\(211\) −1344.61 −0.438707 −0.219353 0.975645i \(-0.570395\pi\)
−0.219353 + 0.975645i \(0.570395\pi\)
\(212\) −630.650 −0.204308
\(213\) −5278.98 −1.69817
\(214\) −3611.01 −1.15348
\(215\) 305.735 0.0969811
\(216\) −700.955 −0.220805
\(217\) 0 0
\(218\) −4811.23 −1.49476
\(219\) −1857.68 −0.573200
\(220\) −456.609 −0.139930
\(221\) −1189.48 −0.362051
\(222\) 2131.99 0.644549
\(223\) −864.916 −0.259727 −0.129863 0.991532i \(-0.541454\pi\)
−0.129863 + 0.991532i \(0.541454\pi\)
\(224\) 0 0
\(225\) −4125.83 −1.22247
\(226\) 5432.32 1.59890
\(227\) 1715.34 0.501548 0.250774 0.968046i \(-0.419315\pi\)
0.250774 + 0.968046i \(0.419315\pi\)
\(228\) −2717.33 −0.789298
\(229\) 1045.46 0.301685 0.150842 0.988558i \(-0.451801\pi\)
0.150842 + 0.988558i \(0.451801\pi\)
\(230\) 408.945 0.117239
\(231\) 0 0
\(232\) 1514.17 0.428491
\(233\) 1448.67 0.407320 0.203660 0.979042i \(-0.434716\pi\)
0.203660 + 0.979042i \(0.434716\pi\)
\(234\) 5414.91 1.51275
\(235\) −560.560 −0.155604
\(236\) −1898.53 −0.523659
\(237\) −3878.65 −1.06306
\(238\) 0 0
\(239\) −3153.12 −0.853383 −0.426691 0.904397i \(-0.640321\pi\)
−0.426691 + 0.904397i \(0.640321\pi\)
\(240\) −1296.35 −0.348663
\(241\) 381.012 0.101839 0.0509194 0.998703i \(-0.483785\pi\)
0.0509194 + 0.998703i \(0.483785\pi\)
\(242\) −3843.06 −1.02083
\(243\) 5298.37 1.39873
\(244\) −2627.68 −0.689426
\(245\) 0 0
\(246\) 6869.94 1.78053
\(247\) −3486.87 −0.898236
\(248\) 3809.90 0.975519
\(249\) 191.179 0.0486565
\(250\) −1802.15 −0.455912
\(251\) −3776.23 −0.949617 −0.474808 0.880089i \(-0.657483\pi\)
−0.474808 + 0.880089i \(0.657483\pi\)
\(252\) 0 0
\(253\) 2742.21 0.681428
\(254\) 601.556 0.148602
\(255\) 430.890 0.105817
\(256\) 5121.09 1.25027
\(257\) −4258.42 −1.03359 −0.516795 0.856109i \(-0.672875\pi\)
−0.516795 + 0.856109i \(0.672875\pi\)
\(258\) −4065.26 −0.980977
\(259\) 0 0
\(260\) 416.405 0.0993244
\(261\) 4151.50 0.984564
\(262\) 2655.18 0.626098
\(263\) 4198.83 0.984451 0.492226 0.870468i \(-0.336184\pi\)
0.492226 + 0.870468i \(0.336184\pi\)
\(264\) −4797.38 −1.11840
\(265\) 293.150 0.0679548
\(266\) 0 0
\(267\) −8387.65 −1.92253
\(268\) −802.762 −0.182972
\(269\) −3740.59 −0.847835 −0.423917 0.905701i \(-0.639345\pi\)
−0.423917 + 0.905701i \(0.639345\pi\)
\(270\) −412.359 −0.0929457
\(271\) −4356.30 −0.976480 −0.488240 0.872709i \(-0.662361\pi\)
−0.488240 + 0.872709i \(0.662361\pi\)
\(272\) −2115.60 −0.471607
\(273\) 0 0
\(274\) −1831.32 −0.403775
\(275\) −5936.10 −1.30167
\(276\) −1948.85 −0.425026
\(277\) −1344.30 −0.291593 −0.145797 0.989315i \(-0.546575\pi\)
−0.145797 + 0.989315i \(0.546575\pi\)
\(278\) 10507.7 2.26693
\(279\) 10445.9 2.24150
\(280\) 0 0
\(281\) 4205.54 0.892817 0.446408 0.894829i \(-0.352703\pi\)
0.446408 + 0.894829i \(0.352703\pi\)
\(282\) 7453.60 1.57395
\(283\) −4752.03 −0.998159 −0.499079 0.866556i \(-0.666328\pi\)
−0.499079 + 0.866556i \(0.666328\pi\)
\(284\) 3015.91 0.630146
\(285\) 1263.12 0.262529
\(286\) 7790.79 1.61077
\(287\) 0 0
\(288\) 6220.73 1.27278
\(289\) −4209.80 −0.856870
\(290\) 890.757 0.180369
\(291\) −13043.0 −2.62746
\(292\) 1061.31 0.212700
\(293\) 4961.17 0.989196 0.494598 0.869122i \(-0.335315\pi\)
0.494598 + 0.869122i \(0.335315\pi\)
\(294\) 0 0
\(295\) 882.506 0.174174
\(296\) 962.432 0.188987
\(297\) −2765.10 −0.540227
\(298\) 9597.39 1.86564
\(299\) −2500.76 −0.483688
\(300\) 4218.71 0.811890
\(301\) 0 0
\(302\) −2498.80 −0.476126
\(303\) 603.237 0.114373
\(304\) −6201.71 −1.17004
\(305\) 1221.44 0.229310
\(306\) −3201.18 −0.598037
\(307\) 4234.00 0.787124 0.393562 0.919298i \(-0.371243\pi\)
0.393562 + 0.919298i \(0.371243\pi\)
\(308\) 0 0
\(309\) −1288.26 −0.237173
\(310\) 2241.29 0.410635
\(311\) 684.700 0.124842 0.0624209 0.998050i \(-0.480118\pi\)
0.0624209 + 0.998050i \(0.480118\pi\)
\(312\) 4374.97 0.793859
\(313\) 5944.07 1.07341 0.536707 0.843768i \(-0.319668\pi\)
0.536707 + 0.843768i \(0.319668\pi\)
\(314\) −11009.8 −1.97872
\(315\) 0 0
\(316\) 2215.90 0.394475
\(317\) −2823.89 −0.500333 −0.250166 0.968203i \(-0.580485\pi\)
−0.250166 + 0.968203i \(0.580485\pi\)
\(318\) −3897.92 −0.687373
\(319\) 5973.04 1.04836
\(320\) 8.91935 0.00155815
\(321\) −7999.16 −1.39087
\(322\) 0 0
\(323\) 2061.37 0.355101
\(324\) −2159.84 −0.370343
\(325\) 5413.43 0.923947
\(326\) −6387.02 −1.08510
\(327\) −10657.9 −1.80239
\(328\) 3101.26 0.522068
\(329\) 0 0
\(330\) −2822.21 −0.470780
\(331\) −2812.97 −0.467114 −0.233557 0.972343i \(-0.575037\pi\)
−0.233557 + 0.972343i \(0.575037\pi\)
\(332\) −109.222 −0.0180552
\(333\) 2638.77 0.434245
\(334\) −11113.8 −1.82072
\(335\) 373.154 0.0608584
\(336\) 0 0
\(337\) 4260.10 0.688612 0.344306 0.938857i \(-0.388114\pi\)
0.344306 + 0.938857i \(0.388114\pi\)
\(338\) 653.080 0.105097
\(339\) 12033.7 1.92797
\(340\) −246.170 −0.0392660
\(341\) 15029.2 2.38673
\(342\) −9384.00 −1.48371
\(343\) 0 0
\(344\) −1835.16 −0.287631
\(345\) 905.899 0.141368
\(346\) 11597.5 1.80198
\(347\) 36.0584 0.00557843 0.00278922 0.999996i \(-0.499112\pi\)
0.00278922 + 0.999996i \(0.499112\pi\)
\(348\) −4244.96 −0.653890
\(349\) 242.692 0.0372236 0.0186118 0.999827i \(-0.494075\pi\)
0.0186118 + 0.999827i \(0.494075\pi\)
\(350\) 0 0
\(351\) 2521.63 0.383461
\(352\) 8950.18 1.35524
\(353\) 109.990 0.0165840 0.00829201 0.999966i \(-0.497361\pi\)
0.00829201 + 0.999966i \(0.497361\pi\)
\(354\) −11734.4 −1.76180
\(355\) −1401.91 −0.209593
\(356\) 4791.92 0.713402
\(357\) 0 0
\(358\) −9883.01 −1.45903
\(359\) 12404.5 1.82363 0.911814 0.410604i \(-0.134682\pi\)
0.911814 + 0.410604i \(0.134682\pi\)
\(360\) −885.491 −0.129637
\(361\) −816.273 −0.119008
\(362\) 13146.5 1.90875
\(363\) −8513.20 −1.23093
\(364\) 0 0
\(365\) −493.335 −0.0707461
\(366\) −16241.2 −2.31951
\(367\) −13859.6 −1.97130 −0.985649 0.168807i \(-0.946009\pi\)
−0.985649 + 0.168807i \(0.946009\pi\)
\(368\) −4447.82 −0.630051
\(369\) 8502.93 1.19958
\(370\) 566.181 0.0795523
\(371\) 0 0
\(372\) −10681.0 −1.48867
\(373\) 4898.06 0.679925 0.339963 0.940439i \(-0.389586\pi\)
0.339963 + 0.940439i \(0.389586\pi\)
\(374\) −4605.75 −0.636786
\(375\) −3992.14 −0.549742
\(376\) 3364.73 0.461497
\(377\) −5447.11 −0.744139
\(378\) 0 0
\(379\) −9806.25 −1.32906 −0.664530 0.747262i \(-0.731368\pi\)
−0.664530 + 0.747262i \(0.731368\pi\)
\(380\) −721.627 −0.0974176
\(381\) 1332.57 0.179186
\(382\) −3388.65 −0.453870
\(383\) −10729.7 −1.43149 −0.715746 0.698361i \(-0.753913\pi\)
−0.715746 + 0.698361i \(0.753913\pi\)
\(384\) 11268.2 1.49747
\(385\) 0 0
\(386\) 13383.8 1.76482
\(387\) −5031.58 −0.660903
\(388\) 7451.53 0.974984
\(389\) 5264.05 0.686113 0.343057 0.939315i \(-0.388538\pi\)
0.343057 + 0.939315i \(0.388538\pi\)
\(390\) 2573.72 0.334167
\(391\) 1478.40 0.191217
\(392\) 0 0
\(393\) 5881.79 0.754954
\(394\) −18072.3 −2.31083
\(395\) −1030.03 −0.131206
\(396\) 7514.57 0.953590
\(397\) −1214.90 −0.153587 −0.0767935 0.997047i \(-0.524468\pi\)
−0.0767935 + 0.997047i \(0.524468\pi\)
\(398\) −3053.91 −0.384620
\(399\) 0 0
\(400\) 9628.26 1.20353
\(401\) 2295.45 0.285859 0.142929 0.989733i \(-0.454348\pi\)
0.142929 + 0.989733i \(0.454348\pi\)
\(402\) −4961.71 −0.615591
\(403\) −13705.8 −1.69413
\(404\) −344.633 −0.0424410
\(405\) 1003.97 0.123180
\(406\) 0 0
\(407\) 3796.57 0.462381
\(408\) −2586.39 −0.313837
\(409\) −4646.54 −0.561753 −0.280876 0.959744i \(-0.590625\pi\)
−0.280876 + 0.959744i \(0.590625\pi\)
\(410\) 1824.41 0.219759
\(411\) −4056.77 −0.486875
\(412\) 735.990 0.0880087
\(413\) 0 0
\(414\) −6730.14 −0.798957
\(415\) 50.7703 0.00600535
\(416\) −8162.11 −0.961973
\(417\) 23276.7 2.73349
\(418\) −13501.4 −1.57984
\(419\) −7541.24 −0.879269 −0.439634 0.898177i \(-0.644892\pi\)
−0.439634 + 0.898177i \(0.644892\pi\)
\(420\) 0 0
\(421\) −6243.63 −0.722794 −0.361397 0.932412i \(-0.617700\pi\)
−0.361397 + 0.932412i \(0.617700\pi\)
\(422\) 4748.01 0.547700
\(423\) 9225.32 1.06040
\(424\) −1759.62 −0.201544
\(425\) −3200.31 −0.365265
\(426\) 18640.8 2.12006
\(427\) 0 0
\(428\) 4569.97 0.516116
\(429\) 17258.2 1.94227
\(430\) −1079.59 −0.121075
\(431\) 11465.8 1.28141 0.640706 0.767786i \(-0.278642\pi\)
0.640706 + 0.767786i \(0.278642\pi\)
\(432\) 4484.95 0.499496
\(433\) −5156.40 −0.572289 −0.286144 0.958187i \(-0.592374\pi\)
−0.286144 + 0.958187i \(0.592374\pi\)
\(434\) 0 0
\(435\) 1973.21 0.217491
\(436\) 6088.92 0.668822
\(437\) 4333.80 0.474402
\(438\) 6559.72 0.715607
\(439\) 5064.25 0.550577 0.275289 0.961362i \(-0.411227\pi\)
0.275289 + 0.961362i \(0.411227\pi\)
\(440\) −1274.01 −0.138037
\(441\) 0 0
\(442\) 4200.22 0.452000
\(443\) 12703.6 1.36246 0.681228 0.732071i \(-0.261446\pi\)
0.681228 + 0.732071i \(0.261446\pi\)
\(444\) −2698.17 −0.288400
\(445\) −2227.46 −0.237285
\(446\) 3054.13 0.324254
\(447\) 21260.2 2.24961
\(448\) 0 0
\(449\) 13942.2 1.46542 0.732709 0.680542i \(-0.238256\pi\)
0.732709 + 0.680542i \(0.238256\pi\)
\(450\) 14568.8 1.52618
\(451\) 12233.7 1.27730
\(452\) −6874.95 −0.715421
\(453\) −5535.38 −0.574116
\(454\) −6057.10 −0.626154
\(455\) 0 0
\(456\) −7581.80 −0.778619
\(457\) −15214.0 −1.55729 −0.778646 0.627464i \(-0.784093\pi\)
−0.778646 + 0.627464i \(0.784093\pi\)
\(458\) −3691.65 −0.376636
\(459\) −1490.74 −0.151594
\(460\) −517.546 −0.0524581
\(461\) 11430.2 1.15479 0.577394 0.816465i \(-0.304070\pi\)
0.577394 + 0.816465i \(0.304070\pi\)
\(462\) 0 0
\(463\) −9347.88 −0.938300 −0.469150 0.883119i \(-0.655440\pi\)
−0.469150 + 0.883119i \(0.655440\pi\)
\(464\) −9688.17 −0.969314
\(465\) 4964.94 0.495147
\(466\) −5115.44 −0.508515
\(467\) −3630.84 −0.359776 −0.179888 0.983687i \(-0.557573\pi\)
−0.179888 + 0.983687i \(0.557573\pi\)
\(468\) −6852.92 −0.676872
\(469\) 0 0
\(470\) 1979.41 0.194262
\(471\) −24388.9 −2.38595
\(472\) −5297.20 −0.516575
\(473\) −7239.26 −0.703724
\(474\) 13696.0 1.32717
\(475\) −9381.43 −0.906210
\(476\) 0 0
\(477\) −4824.46 −0.463096
\(478\) 11134.1 1.06540
\(479\) −6521.25 −0.622053 −0.311027 0.950401i \(-0.600673\pi\)
−0.311027 + 0.950401i \(0.600673\pi\)
\(480\) 2956.72 0.281157
\(481\) −3462.28 −0.328205
\(482\) −1345.40 −0.127140
\(483\) 0 0
\(484\) 4863.65 0.456766
\(485\) −3463.75 −0.324290
\(486\) −18709.2 −1.74623
\(487\) −3666.29 −0.341140 −0.170570 0.985346i \(-0.554561\pi\)
−0.170570 + 0.985346i \(0.554561\pi\)
\(488\) −7331.65 −0.680099
\(489\) −14148.6 −1.30843
\(490\) 0 0
\(491\) −12470.7 −1.14623 −0.573113 0.819476i \(-0.694264\pi\)
−0.573113 + 0.819476i \(0.694264\pi\)
\(492\) −8694.35 −0.796691
\(493\) 3220.22 0.294181
\(494\) 12312.6 1.12140
\(495\) −3493.05 −0.317174
\(496\) −24377.0 −2.20678
\(497\) 0 0
\(498\) −675.078 −0.0607449
\(499\) −2303.93 −0.206690 −0.103345 0.994646i \(-0.532955\pi\)
−0.103345 + 0.994646i \(0.532955\pi\)
\(500\) 2280.74 0.203995
\(501\) −24619.5 −2.19544
\(502\) 13334.4 1.18554
\(503\) 10520.4 0.932570 0.466285 0.884635i \(-0.345592\pi\)
0.466285 + 0.884635i \(0.345592\pi\)
\(504\) 0 0
\(505\) 160.198 0.0141163
\(506\) −9683.10 −0.850724
\(507\) 1446.71 0.126727
\(508\) −761.308 −0.0664913
\(509\) 9662.22 0.841395 0.420698 0.907201i \(-0.361785\pi\)
0.420698 + 0.907201i \(0.361785\pi\)
\(510\) −1521.53 −0.132107
\(511\) 0 0
\(512\) −6558.89 −0.566142
\(513\) −4369.98 −0.376100
\(514\) 15037.0 1.29038
\(515\) −342.115 −0.0292726
\(516\) 5144.85 0.438933
\(517\) 13273.1 1.12911
\(518\) 0 0
\(519\) 25690.9 2.17284
\(520\) 1161.84 0.0979806
\(521\) −8607.81 −0.723829 −0.361914 0.932211i \(-0.617877\pi\)
−0.361914 + 0.932211i \(0.617877\pi\)
\(522\) −14659.5 −1.22917
\(523\) 10482.7 0.876439 0.438219 0.898868i \(-0.355609\pi\)
0.438219 + 0.898868i \(0.355609\pi\)
\(524\) −3360.30 −0.280144
\(525\) 0 0
\(526\) −14826.6 −1.22903
\(527\) 8102.61 0.669744
\(528\) 30695.3 2.53000
\(529\) −9058.83 −0.744541
\(530\) −1035.15 −0.0848377
\(531\) −14523.7 −1.18696
\(532\) 0 0
\(533\) −11156.6 −0.906649
\(534\) 29617.9 2.40017
\(535\) −2124.29 −0.171666
\(536\) −2239.84 −0.180497
\(537\) −21893.0 −1.75931
\(538\) 13208.5 1.05847
\(539\) 0 0
\(540\) 521.866 0.0415881
\(541\) 20722.6 1.64683 0.823416 0.567438i \(-0.192066\pi\)
0.823416 + 0.567438i \(0.192066\pi\)
\(542\) 15382.6 1.21908
\(543\) 29122.3 2.30158
\(544\) 4825.27 0.380298
\(545\) −2830.35 −0.222457
\(546\) 0 0
\(547\) −4175.09 −0.326351 −0.163176 0.986597i \(-0.552174\pi\)
−0.163176 + 0.986597i \(0.552174\pi\)
\(548\) 2317.66 0.180667
\(549\) −20101.7 −1.56270
\(550\) 20961.1 1.62506
\(551\) 9439.81 0.729854
\(552\) −5437.61 −0.419276
\(553\) 0 0
\(554\) 4746.91 0.364038
\(555\) 1254.21 0.0959248
\(556\) −13298.1 −1.01433
\(557\) −10161.7 −0.773011 −0.386505 0.922287i \(-0.626318\pi\)
−0.386505 + 0.922287i \(0.626318\pi\)
\(558\) −36885.7 −2.79838
\(559\) 6601.85 0.499514
\(560\) 0 0
\(561\) −10202.7 −0.767841
\(562\) −14850.3 −1.11463
\(563\) 17104.4 1.28040 0.640201 0.768208i \(-0.278851\pi\)
0.640201 + 0.768208i \(0.278851\pi\)
\(564\) −9433.01 −0.704257
\(565\) 3195.73 0.237957
\(566\) 16780.0 1.24614
\(567\) 0 0
\(568\) 8414.89 0.621621
\(569\) −18257.6 −1.34516 −0.672581 0.740023i \(-0.734815\pi\)
−0.672581 + 0.740023i \(0.734815\pi\)
\(570\) −4460.23 −0.327752
\(571\) 13630.5 0.998982 0.499491 0.866319i \(-0.333520\pi\)
0.499491 + 0.866319i \(0.333520\pi\)
\(572\) −9859.74 −0.720728
\(573\) −7506.57 −0.547280
\(574\) 0 0
\(575\) −6728.30 −0.487981
\(576\) −146.789 −0.0106184
\(577\) −4442.08 −0.320496 −0.160248 0.987077i \(-0.551229\pi\)
−0.160248 + 0.987077i \(0.551229\pi\)
\(578\) 14865.4 1.06975
\(579\) 29648.0 2.12803
\(580\) −1127.31 −0.0807052
\(581\) 0 0
\(582\) 46056.4 3.28024
\(583\) −6941.27 −0.493101
\(584\) 2961.22 0.209822
\(585\) 3185.49 0.225135
\(586\) −17518.5 −1.23495
\(587\) −3103.38 −0.218211 −0.109106 0.994030i \(-0.534799\pi\)
−0.109106 + 0.994030i \(0.534799\pi\)
\(588\) 0 0
\(589\) 23752.1 1.66161
\(590\) −3116.24 −0.217447
\(591\) −40033.9 −2.78642
\(592\) −6157.97 −0.427519
\(593\) −5937.71 −0.411185 −0.205592 0.978638i \(-0.565912\pi\)
−0.205592 + 0.978638i \(0.565912\pi\)
\(594\) 9763.93 0.674443
\(595\) 0 0
\(596\) −12146.1 −0.834772
\(597\) −6765.07 −0.463778
\(598\) 8830.50 0.603856
\(599\) −2600.33 −0.177373 −0.0886866 0.996060i \(-0.528267\pi\)
−0.0886866 + 0.996060i \(0.528267\pi\)
\(600\) 11770.9 0.800906
\(601\) 13881.4 0.942156 0.471078 0.882092i \(-0.343865\pi\)
0.471078 + 0.882092i \(0.343865\pi\)
\(602\) 0 0
\(603\) −6141.11 −0.414735
\(604\) 3162.40 0.213040
\(605\) −2260.80 −0.151925
\(606\) −2130.11 −0.142788
\(607\) 12284.6 0.821442 0.410721 0.911761i \(-0.365277\pi\)
0.410721 + 0.911761i \(0.365277\pi\)
\(608\) 14144.9 0.943505
\(609\) 0 0
\(610\) −4313.07 −0.286281
\(611\) −12104.4 −0.801459
\(612\) 4051.30 0.267589
\(613\) 22062.0 1.45363 0.726815 0.686833i \(-0.241000\pi\)
0.726815 + 0.686833i \(0.241000\pi\)
\(614\) −14950.8 −0.982679
\(615\) 4041.46 0.264988
\(616\) 0 0
\(617\) −12182.2 −0.794871 −0.397436 0.917630i \(-0.630100\pi\)
−0.397436 + 0.917630i \(0.630100\pi\)
\(618\) 4549.00 0.296097
\(619\) −23248.6 −1.50960 −0.754799 0.655956i \(-0.772266\pi\)
−0.754799 + 0.655956i \(0.772266\pi\)
\(620\) −2836.50 −0.183736
\(621\) −3134.12 −0.202525
\(622\) −2417.76 −0.155858
\(623\) 0 0
\(624\) −27992.6 −1.79583
\(625\) 14025.4 0.897628
\(626\) −20989.3 −1.34010
\(627\) −29908.4 −1.90499
\(628\) 13933.6 0.885365
\(629\) 2046.83 0.129749
\(630\) 0 0
\(631\) 19184.4 1.21033 0.605165 0.796100i \(-0.293107\pi\)
0.605165 + 0.796100i \(0.293107\pi\)
\(632\) 6182.72 0.389138
\(633\) 10517.8 0.660421
\(634\) 9971.52 0.624637
\(635\) 353.884 0.0221157
\(636\) 4933.07 0.307561
\(637\) 0 0
\(638\) −21091.6 −1.30881
\(639\) 23071.7 1.42833
\(640\) 2992.44 0.184823
\(641\) −19433.4 −1.19746 −0.598730 0.800951i \(-0.704328\pi\)
−0.598730 + 0.800951i \(0.704328\pi\)
\(642\) 28246.0 1.73642
\(643\) −5777.47 −0.354341 −0.177170 0.984180i \(-0.556694\pi\)
−0.177170 + 0.984180i \(0.556694\pi\)
\(644\) 0 0
\(645\) −2391.52 −0.145994
\(646\) −7278.95 −0.443323
\(647\) 29231.5 1.77621 0.888106 0.459640i \(-0.152021\pi\)
0.888106 + 0.459640i \(0.152021\pi\)
\(648\) −6026.30 −0.365332
\(649\) −20896.2 −1.26386
\(650\) −19115.5 −1.15349
\(651\) 0 0
\(652\) 8083.18 0.485524
\(653\) −7093.27 −0.425086 −0.212543 0.977152i \(-0.568174\pi\)
−0.212543 + 0.977152i \(0.568174\pi\)
\(654\) 37634.4 2.25018
\(655\) 1561.99 0.0931788
\(656\) −19842.9 −1.18100
\(657\) 8118.98 0.482118
\(658\) 0 0
\(659\) 19014.2 1.12396 0.561980 0.827151i \(-0.310040\pi\)
0.561980 + 0.827151i \(0.310040\pi\)
\(660\) 3571.69 0.210648
\(661\) 21058.4 1.23915 0.619573 0.784939i \(-0.287306\pi\)
0.619573 + 0.784939i \(0.287306\pi\)
\(662\) 9932.96 0.583165
\(663\) 9304.37 0.545025
\(664\) −304.746 −0.0178109
\(665\) 0 0
\(666\) −9317.83 −0.542130
\(667\) 6770.16 0.393016
\(668\) 14065.3 0.814672
\(669\) 6765.54 0.390988
\(670\) −1317.65 −0.0759782
\(671\) −28921.6 −1.66395
\(672\) 0 0
\(673\) 9634.87 0.551853 0.275926 0.961179i \(-0.411015\pi\)
0.275926 + 0.961179i \(0.411015\pi\)
\(674\) −15043.0 −0.859693
\(675\) 6784.46 0.386865
\(676\) −826.515 −0.0470252
\(677\) −8371.31 −0.475237 −0.237619 0.971359i \(-0.576367\pi\)
−0.237619 + 0.971359i \(0.576367\pi\)
\(678\) −42492.7 −2.40696
\(679\) 0 0
\(680\) −686.854 −0.0387348
\(681\) −13417.8 −0.755022
\(682\) −53069.9 −2.97969
\(683\) −12068.8 −0.676137 −0.338069 0.941121i \(-0.609774\pi\)
−0.338069 + 0.941121i \(0.609774\pi\)
\(684\) 11876.1 0.663878
\(685\) −1077.33 −0.0600917
\(686\) 0 0
\(687\) −8177.78 −0.454151
\(688\) 11742.0 0.650666
\(689\) 6330.09 0.350011
\(690\) −3198.85 −0.176490
\(691\) −2981.29 −0.164130 −0.0820648 0.996627i \(-0.526151\pi\)
−0.0820648 + 0.996627i \(0.526151\pi\)
\(692\) −14677.4 −0.806286
\(693\) 0 0
\(694\) −127.327 −0.00696435
\(695\) 6181.46 0.337376
\(696\) −11844.1 −0.645043
\(697\) 6595.53 0.358427
\(698\) −856.978 −0.0464715
\(699\) −11331.8 −0.613172
\(700\) 0 0
\(701\) −28978.0 −1.56132 −0.780660 0.624956i \(-0.785117\pi\)
−0.780660 + 0.624956i \(0.785117\pi\)
\(702\) −8904.22 −0.478729
\(703\) 6000.11 0.321904
\(704\) −211.195 −0.0113064
\(705\) 4384.81 0.234243
\(706\) −388.388 −0.0207042
\(707\) 0 0
\(708\) 14850.7 0.788307
\(709\) −16372.4 −0.867249 −0.433625 0.901094i \(-0.642766\pi\)
−0.433625 + 0.901094i \(0.642766\pi\)
\(710\) 4950.32 0.261665
\(711\) 16951.6 0.894141
\(712\) 13370.2 0.703750
\(713\) 17034.9 0.894755
\(714\) 0 0
\(715\) 4583.18 0.239722
\(716\) 12507.6 0.652836
\(717\) 24664.3 1.28467
\(718\) −43801.7 −2.27669
\(719\) −23010.5 −1.19353 −0.596765 0.802416i \(-0.703547\pi\)
−0.596765 + 0.802416i \(0.703547\pi\)
\(720\) 5665.68 0.293260
\(721\) 0 0
\(722\) 2882.36 0.148574
\(723\) −2980.35 −0.153306
\(724\) −16637.8 −0.854058
\(725\) −14655.5 −0.750745
\(726\) 30061.2 1.53674
\(727\) 24636.8 1.25685 0.628423 0.777872i \(-0.283701\pi\)
0.628423 + 0.777872i \(0.283701\pi\)
\(728\) 0 0
\(729\) −28395.6 −1.44264
\(730\) 1742.03 0.0883225
\(731\) −3902.87 −0.197473
\(732\) 20554.2 1.03785
\(733\) −6904.76 −0.347931 −0.173965 0.984752i \(-0.555658\pi\)
−0.173965 + 0.984752i \(0.555658\pi\)
\(734\) 48940.1 2.46105
\(735\) 0 0
\(736\) 10144.6 0.508065
\(737\) −8835.63 −0.441607
\(738\) −30025.0 −1.49761
\(739\) −9234.89 −0.459690 −0.229845 0.973227i \(-0.573822\pi\)
−0.229845 + 0.973227i \(0.573822\pi\)
\(740\) −716.538 −0.0355952
\(741\) 27275.0 1.35219
\(742\) 0 0
\(743\) 20216.9 0.998232 0.499116 0.866535i \(-0.333658\pi\)
0.499116 + 0.866535i \(0.333658\pi\)
\(744\) −29801.8 −1.46853
\(745\) 5645.97 0.277654
\(746\) −17295.7 −0.848847
\(747\) −835.544 −0.0409250
\(748\) 5828.88 0.284926
\(749\) 0 0
\(750\) 14096.8 0.686321
\(751\) 24054.9 1.16881 0.584405 0.811462i \(-0.301328\pi\)
0.584405 + 0.811462i \(0.301328\pi\)
\(752\) −21528.7 −1.04398
\(753\) 29538.4 1.42954
\(754\) 19234.5 0.929015
\(755\) −1470.00 −0.0708593
\(756\) 0 0
\(757\) −30328.2 −1.45614 −0.728069 0.685504i \(-0.759582\pi\)
−0.728069 + 0.685504i \(0.759582\pi\)
\(758\) 34627.1 1.65925
\(759\) −21450.1 −1.02581
\(760\) −2013.46 −0.0960997
\(761\) 33834.1 1.61168 0.805839 0.592135i \(-0.201715\pi\)
0.805839 + 0.592135i \(0.201715\pi\)
\(762\) −4705.49 −0.223703
\(763\) 0 0
\(764\) 4288.55 0.203082
\(765\) −1883.20 −0.0890027
\(766\) 37887.9 1.78713
\(767\) 19056.3 0.897109
\(768\) −40058.2 −1.88213
\(769\) −31738.1 −1.48830 −0.744151 0.668011i \(-0.767146\pi\)
−0.744151 + 0.668011i \(0.767146\pi\)
\(770\) 0 0
\(771\) 33310.2 1.55595
\(772\) −16938.1 −0.789658
\(773\) −27494.2 −1.27930 −0.639650 0.768667i \(-0.720920\pi\)
−0.639650 + 0.768667i \(0.720920\pi\)
\(774\) 17767.1 0.825099
\(775\) −36875.6 −1.70917
\(776\) 20791.0 0.961794
\(777\) 0 0
\(778\) −18588.0 −0.856573
\(779\) 19334.2 0.889244
\(780\) −3257.20 −0.149521
\(781\) 33194.8 1.52087
\(782\) −5220.41 −0.238723
\(783\) −6826.68 −0.311578
\(784\) 0 0
\(785\) −6476.84 −0.294482
\(786\) −20769.3 −0.942517
\(787\) 468.356 0.0212136 0.0106068 0.999944i \(-0.496624\pi\)
0.0106068 + 0.999944i \(0.496624\pi\)
\(788\) 22871.6 1.03397
\(789\) −32844.0 −1.48198
\(790\) 3637.18 0.163804
\(791\) 0 0
\(792\) 20966.9 0.940688
\(793\) 26375.1 1.18109
\(794\) 4289.97 0.191745
\(795\) −2293.07 −0.102298
\(796\) 3864.93 0.172096
\(797\) −37723.8 −1.67659 −0.838297 0.545214i \(-0.816449\pi\)
−0.838297 + 0.545214i \(0.816449\pi\)
\(798\) 0 0
\(799\) 7155.87 0.316841
\(800\) −21960.2 −0.970512
\(801\) 36658.1 1.61704
\(802\) −8105.53 −0.356878
\(803\) 11681.3 0.513356
\(804\) 6279.36 0.275443
\(805\) 0 0
\(806\) 48397.1 2.11503
\(807\) 29259.6 1.27632
\(808\) −961.583 −0.0418668
\(809\) −7797.13 −0.338854 −0.169427 0.985543i \(-0.554192\pi\)
−0.169427 + 0.985543i \(0.554192\pi\)
\(810\) −3545.16 −0.153783
\(811\) −16925.9 −0.732860 −0.366430 0.930446i \(-0.619420\pi\)
−0.366430 + 0.930446i \(0.619420\pi\)
\(812\) 0 0
\(813\) 34075.8 1.46998
\(814\) −13406.2 −0.577256
\(815\) −3757.36 −0.161490
\(816\) 16548.6 0.709949
\(817\) −11441.0 −0.489925
\(818\) 16407.5 0.701316
\(819\) 0 0
\(820\) −2308.91 −0.0983301
\(821\) −30009.3 −1.27568 −0.637840 0.770169i \(-0.720172\pi\)
−0.637840 + 0.770169i \(0.720172\pi\)
\(822\) 14325.0 0.607835
\(823\) −23385.6 −0.990486 −0.495243 0.868754i \(-0.664921\pi\)
−0.495243 + 0.868754i \(0.664921\pi\)
\(824\) 2053.53 0.0868181
\(825\) 46433.3 1.95952
\(826\) 0 0
\(827\) −37325.9 −1.56947 −0.784734 0.619833i \(-0.787200\pi\)
−0.784734 + 0.619833i \(0.787200\pi\)
\(828\) 8517.43 0.357489
\(829\) 24671.3 1.03362 0.516809 0.856100i \(-0.327120\pi\)
0.516809 + 0.856100i \(0.327120\pi\)
\(830\) −179.277 −0.00749733
\(831\) 10515.4 0.438960
\(832\) 192.599 0.00802544
\(833\) 0 0
\(834\) −82193.0 −3.41260
\(835\) −6538.05 −0.270969
\(836\) 17086.9 0.706892
\(837\) −17177.1 −0.709350
\(838\) 26629.1 1.09772
\(839\) −14147.4 −0.582147 −0.291074 0.956701i \(-0.594013\pi\)
−0.291074 + 0.956701i \(0.594013\pi\)
\(840\) 0 0
\(841\) −9642.35 −0.395356
\(842\) 22047.1 0.902366
\(843\) −32896.6 −1.34403
\(844\) −6008.91 −0.245065
\(845\) 384.195 0.0156411
\(846\) −32575.8 −1.32385
\(847\) 0 0
\(848\) 11258.6 0.455923
\(849\) 37171.3 1.50261
\(850\) 11300.7 0.456012
\(851\) 4303.24 0.173341
\(852\) −23591.1 −0.948611
\(853\) −27963.6 −1.12246 −0.561229 0.827661i \(-0.689671\pi\)
−0.561229 + 0.827661i \(0.689671\pi\)
\(854\) 0 0
\(855\) −5520.44 −0.220813
\(856\) 12750.9 0.509134
\(857\) −33855.8 −1.34947 −0.674734 0.738061i \(-0.735741\pi\)
−0.674734 + 0.738061i \(0.735741\pi\)
\(858\) −60941.1 −2.42482
\(859\) 24282.7 0.964511 0.482255 0.876031i \(-0.339818\pi\)
0.482255 + 0.876031i \(0.339818\pi\)
\(860\) 1366.29 0.0541745
\(861\) 0 0
\(862\) −40487.2 −1.59977
\(863\) −19667.0 −0.775750 −0.387875 0.921712i \(-0.626791\pi\)
−0.387875 + 0.921712i \(0.626791\pi\)
\(864\) −10229.3 −0.402787
\(865\) 6822.59 0.268179
\(866\) 18207.9 0.714469
\(867\) 32929.9 1.28992
\(868\) 0 0
\(869\) 24389.4 0.952075
\(870\) −6967.67 −0.271524
\(871\) 8057.65 0.313459
\(872\) 16989.1 0.659773
\(873\) 57004.0 2.20996
\(874\) −15303.2 −0.592264
\(875\) 0 0
\(876\) −8301.75 −0.320194
\(877\) −36061.0 −1.38848 −0.694238 0.719745i \(-0.744259\pi\)
−0.694238 + 0.719745i \(0.744259\pi\)
\(878\) −17882.5 −0.687364
\(879\) −38807.2 −1.48912
\(880\) 8151.58 0.312261
\(881\) 15889.7 0.607646 0.303823 0.952728i \(-0.401737\pi\)
0.303823 + 0.952728i \(0.401737\pi\)
\(882\) 0 0
\(883\) 14861.3 0.566390 0.283195 0.959062i \(-0.408606\pi\)
0.283195 + 0.959062i \(0.408606\pi\)
\(884\) −5315.65 −0.202245
\(885\) −6903.13 −0.262199
\(886\) −44858.2 −1.70095
\(887\) 38189.9 1.44565 0.722824 0.691032i \(-0.242844\pi\)
0.722824 + 0.691032i \(0.242844\pi\)
\(888\) −7528.33 −0.284498
\(889\) 0 0
\(890\) 7865.45 0.296237
\(891\) −23772.3 −0.893831
\(892\) −3865.20 −0.145086
\(893\) 20976.8 0.786073
\(894\) −75072.7 −2.80851
\(895\) −5813.99 −0.217140
\(896\) 0 0
\(897\) 19561.4 0.728135
\(898\) −49231.7 −1.82949
\(899\) 37105.0 1.37655
\(900\) −18437.8 −0.682881
\(901\) −3742.22 −0.138370
\(902\) −43198.9 −1.59464
\(903\) 0 0
\(904\) −19182.2 −0.705742
\(905\) 7733.86 0.284069
\(906\) 19546.1 0.716751
\(907\) 16865.5 0.617431 0.308715 0.951154i \(-0.400101\pi\)
0.308715 + 0.951154i \(0.400101\pi\)
\(908\) 7665.65 0.280169
\(909\) −2636.44 −0.0961993
\(910\) 0 0
\(911\) 26754.1 0.973000 0.486500 0.873681i \(-0.338273\pi\)
0.486500 + 0.873681i \(0.338273\pi\)
\(912\) 48511.0 1.76136
\(913\) −1202.15 −0.0435766
\(914\) 53722.7 1.94419
\(915\) −9554.37 −0.345200
\(916\) 4672.02 0.168524
\(917\) 0 0
\(918\) 5263.99 0.189257
\(919\) −41527.5 −1.49061 −0.745303 0.666726i \(-0.767695\pi\)
−0.745303 + 0.666726i \(0.767695\pi\)
\(920\) −1444.04 −0.0517484
\(921\) −33119.2 −1.18492
\(922\) −40361.5 −1.44169
\(923\) −30272.0 −1.07954
\(924\) 0 0
\(925\) −9315.27 −0.331118
\(926\) 33008.6 1.17141
\(927\) 5630.31 0.199486
\(928\) 22096.8 0.781642
\(929\) −4584.68 −0.161914 −0.0809572 0.996718i \(-0.525798\pi\)
−0.0809572 + 0.996718i \(0.525798\pi\)
\(930\) −17531.8 −0.618163
\(931\) 0 0
\(932\) 6473.92 0.227532
\(933\) −5355.86 −0.187935
\(934\) 12821.0 0.449159
\(935\) −2709.48 −0.0947694
\(936\) −19120.7 −0.667715
\(937\) 6928.18 0.241552 0.120776 0.992680i \(-0.461462\pi\)
0.120776 + 0.992680i \(0.461462\pi\)
\(938\) 0 0
\(939\) −46495.7 −1.61590
\(940\) −2505.07 −0.0869217
\(941\) −20944.9 −0.725596 −0.362798 0.931868i \(-0.618178\pi\)
−0.362798 + 0.931868i \(0.618178\pi\)
\(942\) 86120.5 2.97872
\(943\) 13866.4 0.478846
\(944\) 33893.3 1.16857
\(945\) 0 0
\(946\) 25562.8 0.878559
\(947\) 29278.9 1.00468 0.502342 0.864669i \(-0.332472\pi\)
0.502342 + 0.864669i \(0.332472\pi\)
\(948\) −17333.2 −0.593836
\(949\) −10652.8 −0.364387
\(950\) 33127.1 1.13135
\(951\) 22089.0 0.753192
\(952\) 0 0
\(953\) 2136.81 0.0726316 0.0363158 0.999340i \(-0.488438\pi\)
0.0363158 + 0.999340i \(0.488438\pi\)
\(954\) 17035.8 0.578149
\(955\) −1993.48 −0.0675470
\(956\) −14090.9 −0.476707
\(957\) −46722.3 −1.57818
\(958\) 23027.4 0.776597
\(959\) 0 0
\(960\) −69.7689 −0.00234561
\(961\) 63571.4 2.13391
\(962\) 12225.8 0.409745
\(963\) 34960.2 1.16986
\(964\) 1702.70 0.0568881
\(965\) 7873.47 0.262649
\(966\) 0 0
\(967\) −3921.32 −0.130405 −0.0652023 0.997872i \(-0.520769\pi\)
−0.0652023 + 0.997872i \(0.520769\pi\)
\(968\) 13570.4 0.450586
\(969\) −16124.4 −0.534562
\(970\) 12230.9 0.404857
\(971\) −47809.1 −1.58009 −0.790045 0.613049i \(-0.789943\pi\)
−0.790045 + 0.613049i \(0.789943\pi\)
\(972\) 23677.7 0.781341
\(973\) 0 0
\(974\) 12946.1 0.425894
\(975\) −42344.9 −1.39089
\(976\) 46910.5 1.53849
\(977\) −51329.1 −1.68082 −0.840411 0.541949i \(-0.817686\pi\)
−0.840411 + 0.541949i \(0.817686\pi\)
\(978\) 49960.5 1.63350
\(979\) 52742.4 1.72181
\(980\) 0 0
\(981\) 46580.1 1.51599
\(982\) 44035.8 1.43100
\(983\) 16326.3 0.529734 0.264867 0.964285i \(-0.414672\pi\)
0.264867 + 0.964285i \(0.414672\pi\)
\(984\) −24258.7 −0.785912
\(985\) −10631.6 −0.343909
\(986\) −11371.0 −0.367268
\(987\) 0 0
\(988\) −15582.4 −0.501763
\(989\) −8205.37 −0.263818
\(990\) 12334.4 0.395973
\(991\) 33770.0 1.08248 0.541242 0.840867i \(-0.317954\pi\)
0.541242 + 0.840867i \(0.317954\pi\)
\(992\) 55599.3 1.77952
\(993\) 22003.6 0.703185
\(994\) 0 0
\(995\) −1796.56 −0.0572410
\(996\) 854.354 0.0271800
\(997\) −50695.4 −1.61037 −0.805185 0.593023i \(-0.797934\pi\)
−0.805185 + 0.593023i \(0.797934\pi\)
\(998\) 8135.48 0.258040
\(999\) −4339.16 −0.137422
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 49.4.a.e.1.1 4
3.2 odd 2 441.4.a.u.1.4 4
4.3 odd 2 784.4.a.bf.1.4 4
5.4 even 2 1225.4.a.bb.1.4 4
7.2 even 3 49.4.c.e.18.4 8
7.3 odd 6 49.4.c.e.30.3 8
7.4 even 3 49.4.c.e.30.4 8
7.5 odd 6 49.4.c.e.18.3 8
7.6 odd 2 inner 49.4.a.e.1.2 yes 4
21.2 odd 6 441.4.e.y.361.1 8
21.5 even 6 441.4.e.y.361.2 8
21.11 odd 6 441.4.e.y.226.1 8
21.17 even 6 441.4.e.y.226.2 8
21.20 even 2 441.4.a.u.1.3 4
28.27 even 2 784.4.a.bf.1.1 4
35.34 odd 2 1225.4.a.bb.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.4.a.e.1.1 4 1.1 even 1 trivial
49.4.a.e.1.2 yes 4 7.6 odd 2 inner
49.4.c.e.18.3 8 7.5 odd 6
49.4.c.e.18.4 8 7.2 even 3
49.4.c.e.30.3 8 7.3 odd 6
49.4.c.e.30.4 8 7.4 even 3
441.4.a.u.1.3 4 21.20 even 2
441.4.a.u.1.4 4 3.2 odd 2
441.4.e.y.226.1 8 21.11 odd 6
441.4.e.y.226.2 8 21.17 even 6
441.4.e.y.361.1 8 21.2 odd 6
441.4.e.y.361.2 8 21.5 even 6
784.4.a.bf.1.1 4 28.27 even 2
784.4.a.bf.1.4 4 4.3 odd 2
1225.4.a.bb.1.3 4 35.34 odd 2
1225.4.a.bb.1.4 4 5.4 even 2