Properties

Label 5445.2.a.bx.1.2
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.27433728.1
Defining polynomial: \(x^{6} - 9 x^{4} + 15 x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 605)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.37268\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.37268 q^{2} -0.115749 q^{4} -1.00000 q^{5} -1.37268 q^{7} +2.90425 q^{8} +O(q^{10})\) \(q-1.37268 q^{2} -0.115749 q^{4} -1.00000 q^{5} -1.37268 q^{7} +2.90425 q^{8} +1.37268 q^{10} -1.93253 q^{13} +1.88425 q^{14} -3.75510 q^{16} -6.20946 q^{17} +0.812826 q^{19} +0.115749 q^{20} +3.63935 q^{23} +1.00000 q^{25} +2.65275 q^{26} +0.158887 q^{28} +7.83511 q^{29} -3.40786 q^{31} -0.653939 q^{32} +8.52360 q^{34} +1.37268 q^{35} -4.52360 q^{37} -1.11575 q^{38} -2.90425 q^{40} -1.82613 q^{41} -6.46243 q^{43} -4.99567 q^{46} -4.73820 q^{47} -5.11575 q^{49} -1.37268 q^{50} +0.223690 q^{52} +8.39446 q^{53} -3.98660 q^{56} -10.7551 q^{58} -1.11575 q^{59} -2.54488 q^{61} +4.67789 q^{62} +8.40786 q^{64} +1.93253 q^{65} +2.73820 q^{67} +0.718741 q^{68} -1.88425 q^{70} +1.11575 q^{71} -12.4189 q^{73} +6.20946 q^{74} -0.0940841 q^{76} +5.80849 q^{79} +3.75510 q^{80} +2.50670 q^{82} -15.9771 q^{83} +6.20946 q^{85} +8.87085 q^{86} +2.70789 q^{89} +2.65275 q^{91} -0.421253 q^{92} +6.50403 q^{94} -0.812826 q^{95} +14.1630 q^{97} +7.02229 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{4} - 6q^{5} + O(q^{10}) \) \( 6q + 6q^{4} - 6q^{5} + 18q^{14} + 18q^{16} - 6q^{20} - 12q^{23} + 6q^{25} + 36q^{26} + 24q^{34} - 42q^{47} - 24q^{49} - 24q^{53} + 30q^{56} - 24q^{58} + 30q^{64} + 30q^{67} - 18q^{70} - 18q^{80} + 42q^{82} + 6q^{86} + 30q^{89} + 36q^{91} - 36q^{92} + 24q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.37268 −0.970631 −0.485316 0.874339i \(-0.661295\pi\)
−0.485316 + 0.874339i \(0.661295\pi\)
\(3\) 0 0
\(4\) −0.115749 −0.0578747
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.37268 −0.518824 −0.259412 0.965767i \(-0.583529\pi\)
−0.259412 + 0.965767i \(0.583529\pi\)
\(8\) 2.90425 1.02681
\(9\) 0 0
\(10\) 1.37268 0.434080
\(11\) 0 0
\(12\) 0 0
\(13\) −1.93253 −0.535989 −0.267994 0.963420i \(-0.586361\pi\)
−0.267994 + 0.963420i \(0.586361\pi\)
\(14\) 1.88425 0.503587
\(15\) 0 0
\(16\) −3.75510 −0.938776
\(17\) −6.20946 −1.50602 −0.753008 0.658012i \(-0.771398\pi\)
−0.753008 + 0.658012i \(0.771398\pi\)
\(18\) 0 0
\(19\) 0.812826 0.186475 0.0932375 0.995644i \(-0.470278\pi\)
0.0932375 + 0.995644i \(0.470278\pi\)
\(20\) 0.115749 0.0258824
\(21\) 0 0
\(22\) 0 0
\(23\) 3.63935 0.758858 0.379429 0.925221i \(-0.376120\pi\)
0.379429 + 0.925221i \(0.376120\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.65275 0.520247
\(27\) 0 0
\(28\) 0.158887 0.0300268
\(29\) 7.83511 1.45494 0.727472 0.686137i \(-0.240695\pi\)
0.727472 + 0.686137i \(0.240695\pi\)
\(30\) 0 0
\(31\) −3.40786 −0.612069 −0.306034 0.952020i \(-0.599002\pi\)
−0.306034 + 0.952020i \(0.599002\pi\)
\(32\) −0.653939 −0.115601
\(33\) 0 0
\(34\) 8.52360 1.46179
\(35\) 1.37268 0.232025
\(36\) 0 0
\(37\) −4.52360 −0.743676 −0.371838 0.928298i \(-0.621272\pi\)
−0.371838 + 0.928298i \(0.621272\pi\)
\(38\) −1.11575 −0.180998
\(39\) 0 0
\(40\) −2.90425 −0.459202
\(41\) −1.82613 −0.285194 −0.142597 0.989781i \(-0.545545\pi\)
−0.142597 + 0.989781i \(0.545545\pi\)
\(42\) 0 0
\(43\) −6.46243 −0.985512 −0.492756 0.870168i \(-0.664010\pi\)
−0.492756 + 0.870168i \(0.664010\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −4.99567 −0.736571
\(47\) −4.73820 −0.691137 −0.345569 0.938393i \(-0.612314\pi\)
−0.345569 + 0.938393i \(0.612314\pi\)
\(48\) 0 0
\(49\) −5.11575 −0.730821
\(50\) −1.37268 −0.194126
\(51\) 0 0
\(52\) 0.223690 0.0310202
\(53\) 8.39446 1.15307 0.576534 0.817073i \(-0.304405\pi\)
0.576534 + 0.817073i \(0.304405\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.98660 −0.532732
\(57\) 0 0
\(58\) −10.7551 −1.41221
\(59\) −1.11575 −0.145258 −0.0726291 0.997359i \(-0.523139\pi\)
−0.0726291 + 0.997359i \(0.523139\pi\)
\(60\) 0 0
\(61\) −2.54488 −0.325838 −0.162919 0.986639i \(-0.552091\pi\)
−0.162919 + 0.986639i \(0.552091\pi\)
\(62\) 4.67789 0.594093
\(63\) 0 0
\(64\) 8.40786 1.05098
\(65\) 1.93253 0.239701
\(66\) 0 0
\(67\) 2.73820 0.334524 0.167262 0.985912i \(-0.446507\pi\)
0.167262 + 0.985912i \(0.446507\pi\)
\(68\) 0.718741 0.0871602
\(69\) 0 0
\(70\) −1.88425 −0.225211
\(71\) 1.11575 0.132415 0.0662075 0.997806i \(-0.478910\pi\)
0.0662075 + 0.997806i \(0.478910\pi\)
\(72\) 0 0
\(73\) −12.4189 −1.45353 −0.726763 0.686889i \(-0.758976\pi\)
−0.726763 + 0.686889i \(0.758976\pi\)
\(74\) 6.20946 0.721835
\(75\) 0 0
\(76\) −0.0940841 −0.0107922
\(77\) 0 0
\(78\) 0 0
\(79\) 5.80849 0.653507 0.326753 0.945110i \(-0.394045\pi\)
0.326753 + 0.945110i \(0.394045\pi\)
\(80\) 3.75510 0.419833
\(81\) 0 0
\(82\) 2.50670 0.276819
\(83\) −15.9771 −1.75372 −0.876858 0.480750i \(-0.840365\pi\)
−0.876858 + 0.480750i \(0.840365\pi\)
\(84\) 0 0
\(85\) 6.20946 0.673511
\(86\) 8.87085 0.956569
\(87\) 0 0
\(88\) 0 0
\(89\) 2.70789 0.287036 0.143518 0.989648i \(-0.454158\pi\)
0.143518 + 0.989648i \(0.454158\pi\)
\(90\) 0 0
\(91\) 2.65275 0.278084
\(92\) −0.421253 −0.0439187
\(93\) 0 0
\(94\) 6.50403 0.670839
\(95\) −0.812826 −0.0833941
\(96\) 0 0
\(97\) 14.1630 1.43803 0.719015 0.694994i \(-0.244593\pi\)
0.719015 + 0.694994i \(0.244593\pi\)
\(98\) 7.02229 0.709358
\(99\) 0 0
\(100\) −0.115749 −0.0115749
\(101\) −11.4056 −1.13490 −0.567451 0.823408i \(-0.692070\pi\)
−0.567451 + 0.823408i \(0.692070\pi\)
\(102\) 0 0
\(103\) 13.8709 1.36674 0.683368 0.730074i \(-0.260515\pi\)
0.683368 + 0.730074i \(0.260515\pi\)
\(104\) −5.61256 −0.550357
\(105\) 0 0
\(106\) −11.5229 −1.11920
\(107\) 1.06580 0.103034 0.0515172 0.998672i \(-0.483594\pi\)
0.0515172 + 0.998672i \(0.483594\pi\)
\(108\) 0 0
\(109\) 11.3115 1.08345 0.541724 0.840556i \(-0.317772\pi\)
0.541724 + 0.840556i \(0.317772\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 5.15456 0.487060
\(113\) 3.76850 0.354511 0.177255 0.984165i \(-0.443278\pi\)
0.177255 + 0.984165i \(0.443278\pi\)
\(114\) 0 0
\(115\) −3.63935 −0.339371
\(116\) −0.906910 −0.0842044
\(117\) 0 0
\(118\) 1.53157 0.140992
\(119\) 8.52360 0.781358
\(120\) 0 0
\(121\) 0 0
\(122\) 3.49330 0.316269
\(123\) 0 0
\(124\) 0.394457 0.0354233
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −1.67956 −0.149037 −0.0745186 0.997220i \(-0.523742\pi\)
−0.0745186 + 0.997220i \(0.523742\pi\)
\(128\) −10.2334 −0.904515
\(129\) 0 0
\(130\) −2.65275 −0.232662
\(131\) −11.7943 −1.03047 −0.515235 0.857049i \(-0.672295\pi\)
−0.515235 + 0.857049i \(0.672295\pi\)
\(132\) 0 0
\(133\) −1.11575 −0.0967477
\(134\) −3.75867 −0.324700
\(135\) 0 0
\(136\) −18.0338 −1.54639
\(137\) 7.40786 0.632896 0.316448 0.948610i \(-0.397510\pi\)
0.316448 + 0.948610i \(0.397510\pi\)
\(138\) 0 0
\(139\) −2.65128 −0.224878 −0.112439 0.993659i \(-0.535866\pi\)
−0.112439 + 0.993659i \(0.535866\pi\)
\(140\) −0.158887 −0.0134284
\(141\) 0 0
\(142\) −1.53157 −0.128526
\(143\) 0 0
\(144\) 0 0
\(145\) −7.83511 −0.650671
\(146\) 17.0472 1.41084
\(147\) 0 0
\(148\) 0.523604 0.0430400
\(149\) −8.35337 −0.684335 −0.342167 0.939639i \(-0.611161\pi\)
−0.342167 + 0.939639i \(0.611161\pi\)
\(150\) 0 0
\(151\) 7.42325 0.604096 0.302048 0.953293i \(-0.402330\pi\)
0.302048 + 0.953293i \(0.402330\pi\)
\(152\) 2.36065 0.191474
\(153\) 0 0
\(154\) 0 0
\(155\) 3.40786 0.273726
\(156\) 0 0
\(157\) −8.58421 −0.685095 −0.342547 0.939501i \(-0.611290\pi\)
−0.342547 + 0.939501i \(0.611290\pi\)
\(158\) −7.97320 −0.634314
\(159\) 0 0
\(160\) 0.653939 0.0516984
\(161\) −4.99567 −0.393714
\(162\) 0 0
\(163\) 16.3776 1.28279 0.641394 0.767211i \(-0.278356\pi\)
0.641394 + 0.767211i \(0.278356\pi\)
\(164\) 0.211374 0.0165055
\(165\) 0 0
\(166\) 21.9315 1.70221
\(167\) −21.4385 −1.65896 −0.829482 0.558533i \(-0.811364\pi\)
−0.829482 + 0.558533i \(0.811364\pi\)
\(168\) 0 0
\(169\) −9.26531 −0.712716
\(170\) −8.52360 −0.653731
\(171\) 0 0
\(172\) 0.748023 0.0570362
\(173\) −13.7377 −1.04446 −0.522229 0.852806i \(-0.674899\pi\)
−0.522229 + 0.852806i \(0.674899\pi\)
\(174\) 0 0
\(175\) −1.37268 −0.103765
\(176\) 0 0
\(177\) 0 0
\(178\) −3.71707 −0.278606
\(179\) 3.76850 0.281671 0.140836 0.990033i \(-0.455021\pi\)
0.140836 + 0.990033i \(0.455021\pi\)
\(180\) 0 0
\(181\) −0.0133979 −0.000995860 0 −0.000497930 1.00000i \(-0.500158\pi\)
−0.000497930 1.00000i \(0.500158\pi\)
\(182\) −3.64138 −0.269917
\(183\) 0 0
\(184\) 10.5696 0.779200
\(185\) 4.52360 0.332582
\(186\) 0 0
\(187\) 0 0
\(188\) 0.548444 0.0399994
\(189\) 0 0
\(190\) 1.11575 0.0809450
\(191\) 16.6260 1.20301 0.601506 0.798868i \(-0.294568\pi\)
0.601506 + 0.798868i \(0.294568\pi\)
\(192\) 0 0
\(193\) −26.7813 −1.92776 −0.963879 0.266340i \(-0.914186\pi\)
−0.963879 + 0.266340i \(0.914186\pi\)
\(194\) −19.4412 −1.39580
\(195\) 0 0
\(196\) 0.592145 0.0422961
\(197\) 2.55719 0.182192 0.0910962 0.995842i \(-0.470963\pi\)
0.0910962 + 0.995842i \(0.470963\pi\)
\(198\) 0 0
\(199\) −21.8629 −1.54982 −0.774911 0.632071i \(-0.782205\pi\)
−0.774911 + 0.632071i \(0.782205\pi\)
\(200\) 2.90425 0.205361
\(201\) 0 0
\(202\) 15.6563 1.10157
\(203\) −10.7551 −0.754860
\(204\) 0 0
\(205\) 1.82613 0.127543
\(206\) −19.0402 −1.32660
\(207\) 0 0
\(208\) 7.25687 0.503173
\(209\) 0 0
\(210\) 0 0
\(211\) −12.8308 −0.883307 −0.441654 0.897186i \(-0.645608\pi\)
−0.441654 + 0.897186i \(0.645608\pi\)
\(212\) −0.971653 −0.0667334
\(213\) 0 0
\(214\) −1.46300 −0.100008
\(215\) 6.46243 0.440734
\(216\) 0 0
\(217\) 4.67789 0.317556
\(218\) −15.5271 −1.05163
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −7.36415 −0.493140 −0.246570 0.969125i \(-0.579304\pi\)
−0.246570 + 0.969125i \(0.579304\pi\)
\(224\) 0.897649 0.0599767
\(225\) 0 0
\(226\) −5.17295 −0.344099
\(227\) 1.37268 0.0911080 0.0455540 0.998962i \(-0.485495\pi\)
0.0455540 + 0.998962i \(0.485495\pi\)
\(228\) 0 0
\(229\) 29.4968 1.94920 0.974602 0.223944i \(-0.0718933\pi\)
0.974602 + 0.223944i \(0.0718933\pi\)
\(230\) 4.99567 0.329405
\(231\) 0 0
\(232\) 22.7551 1.49395
\(233\) 21.9984 1.44116 0.720582 0.693370i \(-0.243875\pi\)
0.720582 + 0.693370i \(0.243875\pi\)
\(234\) 0 0
\(235\) 4.73820 0.309086
\(236\) 0.129147 0.00840677
\(237\) 0 0
\(238\) −11.7002 −0.758410
\(239\) 18.7225 1.21106 0.605528 0.795824i \(-0.292962\pi\)
0.605528 + 0.795824i \(0.292962\pi\)
\(240\) 0 0
\(241\) 26.2630 1.69175 0.845875 0.533382i \(-0.179079\pi\)
0.845875 + 0.533382i \(0.179079\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0.294568 0.0188578
\(245\) 5.11575 0.326833
\(246\) 0 0
\(247\) −1.57081 −0.0999485
\(248\) −9.89725 −0.628476
\(249\) 0 0
\(250\) 1.37268 0.0868159
\(251\) 22.6260 1.42814 0.714069 0.700075i \(-0.246850\pi\)
0.714069 + 0.700075i \(0.246850\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 2.30550 0.144660
\(255\) 0 0
\(256\) −2.76850 −0.173031
\(257\) −17.2102 −1.07354 −0.536770 0.843728i \(-0.680356\pi\)
−0.536770 + 0.843728i \(0.680356\pi\)
\(258\) 0 0
\(259\) 6.20946 0.385837
\(260\) −0.223690 −0.0138726
\(261\) 0 0
\(262\) 16.1898 1.00021
\(263\) 5.71441 0.352366 0.176183 0.984357i \(-0.443625\pi\)
0.176183 + 0.984357i \(0.443625\pi\)
\(264\) 0 0
\(265\) −8.39446 −0.515667
\(266\) 1.53157 0.0939064
\(267\) 0 0
\(268\) −0.316945 −0.0193605
\(269\) 2.87085 0.175039 0.0875195 0.996163i \(-0.472106\pi\)
0.0875195 + 0.996163i \(0.472106\pi\)
\(270\) 0 0
\(271\) 7.64694 0.464519 0.232259 0.972654i \(-0.425388\pi\)
0.232259 + 0.972654i \(0.425388\pi\)
\(272\) 23.3172 1.41381
\(273\) 0 0
\(274\) −10.1686 −0.614309
\(275\) 0 0
\(276\) 0 0
\(277\) 29.3970 1.76630 0.883148 0.469094i \(-0.155420\pi\)
0.883148 + 0.469094i \(0.155420\pi\)
\(278\) 3.63935 0.218274
\(279\) 0 0
\(280\) 3.98660 0.238245
\(281\) 24.2241 1.44509 0.722544 0.691325i \(-0.242973\pi\)
0.722544 + 0.691325i \(0.242973\pi\)
\(282\) 0 0
\(283\) −5.11903 −0.304295 −0.152147 0.988358i \(-0.548619\pi\)
−0.152147 + 0.988358i \(0.548619\pi\)
\(284\) −0.129147 −0.00766348
\(285\) 0 0
\(286\) 0 0
\(287\) 2.50670 0.147966
\(288\) 0 0
\(289\) 21.5574 1.26808
\(290\) 10.7551 0.631561
\(291\) 0 0
\(292\) 1.43748 0.0841223
\(293\) 10.9923 0.642179 0.321089 0.947049i \(-0.395951\pi\)
0.321089 + 0.947049i \(0.395951\pi\)
\(294\) 0 0
\(295\) 1.11575 0.0649614
\(296\) −13.1377 −0.763611
\(297\) 0 0
\(298\) 11.4665 0.664237
\(299\) −7.03318 −0.406739
\(300\) 0 0
\(301\) 8.87085 0.511307
\(302\) −10.1898 −0.586354
\(303\) 0 0
\(304\) −3.05224 −0.175058
\(305\) 2.54488 0.145719
\(306\) 0 0
\(307\) 14.3515 0.819081 0.409540 0.912292i \(-0.365689\pi\)
0.409540 + 0.912292i \(0.365689\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.67789 −0.265687
\(311\) −27.3811 −1.55264 −0.776319 0.630341i \(-0.782915\pi\)
−0.776319 + 0.630341i \(0.782915\pi\)
\(312\) 0 0
\(313\) 3.87085 0.218794 0.109397 0.993998i \(-0.465108\pi\)
0.109397 + 0.993998i \(0.465108\pi\)
\(314\) 11.7834 0.664974
\(315\) 0 0
\(316\) −0.672330 −0.0378215
\(317\) 15.5102 0.871140 0.435570 0.900155i \(-0.356547\pi\)
0.435570 + 0.900155i \(0.356547\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −8.40786 −0.470013
\(321\) 0 0
\(322\) 6.85745 0.382151
\(323\) −5.04721 −0.280834
\(324\) 0 0
\(325\) −1.93253 −0.107198
\(326\) −22.4811 −1.24512
\(327\) 0 0
\(328\) −5.30355 −0.292839
\(329\) 6.50403 0.358579
\(330\) 0 0
\(331\) 26.8575 1.47622 0.738110 0.674681i \(-0.235719\pi\)
0.738110 + 0.674681i \(0.235719\pi\)
\(332\) 1.84934 0.101496
\(333\) 0 0
\(334\) 29.4283 1.61024
\(335\) −2.73820 −0.149604
\(336\) 0 0
\(337\) 1.93253 0.105272 0.0526359 0.998614i \(-0.483238\pi\)
0.0526359 + 0.998614i \(0.483238\pi\)
\(338\) 12.7183 0.691785
\(339\) 0 0
\(340\) −0.718741 −0.0389792
\(341\) 0 0
\(342\) 0 0
\(343\) 16.6310 0.897992
\(344\) −18.7685 −1.01193
\(345\) 0 0
\(346\) 18.8575 1.01378
\(347\) −3.80027 −0.204009 −0.102004 0.994784i \(-0.532526\pi\)
−0.102004 + 0.994784i \(0.532526\pi\)
\(348\) 0 0
\(349\) 17.1909 0.920208 0.460104 0.887865i \(-0.347812\pi\)
0.460104 + 0.887865i \(0.347812\pi\)
\(350\) 1.88425 0.100717
\(351\) 0 0
\(352\) 0 0
\(353\) 1.99207 0.106027 0.0530135 0.998594i \(-0.483117\pi\)
0.0530135 + 0.998594i \(0.483117\pi\)
\(354\) 0 0
\(355\) −1.11575 −0.0592178
\(356\) −0.313437 −0.0166121
\(357\) 0 0
\(358\) −5.17295 −0.273399
\(359\) 15.5761 0.822077 0.411039 0.911618i \(-0.365166\pi\)
0.411039 + 0.911618i \(0.365166\pi\)
\(360\) 0 0
\(361\) −18.3393 −0.965227
\(362\) 0.0183911 0.000966613 0
\(363\) 0 0
\(364\) −0.307054 −0.0160940
\(365\) 12.4189 0.650036
\(366\) 0 0
\(367\) 18.7720 0.979891 0.489945 0.871753i \(-0.337017\pi\)
0.489945 + 0.871753i \(0.337017\pi\)
\(368\) −13.6661 −0.712397
\(369\) 0 0
\(370\) −6.20946 −0.322815
\(371\) −11.5229 −0.598239
\(372\) 0 0
\(373\) −2.62664 −0.136003 −0.0680013 0.997685i \(-0.521662\pi\)
−0.0680013 + 0.997685i \(0.521662\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −13.7609 −0.709664
\(377\) −15.1416 −0.779833
\(378\) 0 0
\(379\) 2.66069 0.136670 0.0683351 0.997662i \(-0.478231\pi\)
0.0683351 + 0.997662i \(0.478231\pi\)
\(380\) 0.0940841 0.00482641
\(381\) 0 0
\(382\) −22.8221 −1.16768
\(383\) −21.9315 −1.12065 −0.560323 0.828274i \(-0.689323\pi\)
−0.560323 + 0.828274i \(0.689323\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 36.7621 1.87114
\(387\) 0 0
\(388\) −1.63935 −0.0832256
\(389\) 3.42125 0.173464 0.0867322 0.996232i \(-0.472358\pi\)
0.0867322 + 0.996232i \(0.472358\pi\)
\(390\) 0 0
\(391\) −22.5984 −1.14285
\(392\) −14.8574 −0.750412
\(393\) 0 0
\(394\) −3.51021 −0.176842
\(395\) −5.80849 −0.292257
\(396\) 0 0
\(397\) 5.51021 0.276549 0.138275 0.990394i \(-0.455844\pi\)
0.138275 + 0.990394i \(0.455844\pi\)
\(398\) 30.0108 1.50431
\(399\) 0 0
\(400\) −3.75510 −0.187755
\(401\) 17.9260 0.895181 0.447591 0.894239i \(-0.352282\pi\)
0.447591 + 0.894239i \(0.352282\pi\)
\(402\) 0 0
\(403\) 6.58580 0.328062
\(404\) 1.32019 0.0656821
\(405\) 0 0
\(406\) 14.7633 0.732691
\(407\) 0 0
\(408\) 0 0
\(409\) −29.3261 −1.45008 −0.725042 0.688704i \(-0.758180\pi\)
−0.725042 + 0.688704i \(0.758180\pi\)
\(410\) −2.50670 −0.123797
\(411\) 0 0
\(412\) −1.60554 −0.0790994
\(413\) 1.53157 0.0753635
\(414\) 0 0
\(415\) 15.9771 0.784285
\(416\) 1.26376 0.0619609
\(417\) 0 0
\(418\) 0 0
\(419\) −2.65275 −0.129595 −0.0647977 0.997898i \(-0.520640\pi\)
−0.0647977 + 0.997898i \(0.520640\pi\)
\(420\) 0 0
\(421\) −11.1630 −0.544049 −0.272025 0.962290i \(-0.587693\pi\)
−0.272025 + 0.962290i \(0.587693\pi\)
\(422\) 17.6126 0.857366
\(423\) 0 0
\(424\) 24.3796 1.18398
\(425\) −6.20946 −0.301203
\(426\) 0 0
\(427\) 3.49330 0.169053
\(428\) −0.123365 −0.00596309
\(429\) 0 0
\(430\) −8.87085 −0.427791
\(431\) 4.55918 0.219608 0.109804 0.993953i \(-0.464978\pi\)
0.109804 + 0.993953i \(0.464978\pi\)
\(432\) 0 0
\(433\) 20.1630 0.968970 0.484485 0.874800i \(-0.339007\pi\)
0.484485 + 0.874800i \(0.339007\pi\)
\(434\) −6.42125 −0.308230
\(435\) 0 0
\(436\) −1.30930 −0.0627042
\(437\) 2.95816 0.141508
\(438\) 0 0
\(439\) 34.4868 1.64596 0.822982 0.568067i \(-0.192309\pi\)
0.822982 + 0.568067i \(0.192309\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −16.4722 −0.783501
\(443\) −31.4586 −1.49464 −0.747321 0.664463i \(-0.768660\pi\)
−0.747321 + 0.664463i \(0.768660\pi\)
\(444\) 0 0
\(445\) −2.70789 −0.128367
\(446\) 10.1086 0.478657
\(447\) 0 0
\(448\) −11.5413 −0.545275
\(449\) 4.77643 0.225414 0.112707 0.993628i \(-0.464048\pi\)
0.112707 + 0.993628i \(0.464048\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −0.436202 −0.0205172
\(453\) 0 0
\(454\) −1.88425 −0.0884323
\(455\) −2.65275 −0.124363
\(456\) 0 0
\(457\) −1.53157 −0.0716437 −0.0358218 0.999358i \(-0.511405\pi\)
−0.0358218 + 0.999358i \(0.511405\pi\)
\(458\) −40.4897 −1.89196
\(459\) 0 0
\(460\) 0.421253 0.0196410
\(461\) 18.5220 0.862655 0.431327 0.902195i \(-0.358045\pi\)
0.431327 + 0.902195i \(0.358045\pi\)
\(462\) 0 0
\(463\) 9.68306 0.450010 0.225005 0.974358i \(-0.427760\pi\)
0.225005 + 0.974358i \(0.427760\pi\)
\(464\) −29.4217 −1.36587
\(465\) 0 0
\(466\) −30.1968 −1.39884
\(467\) 31.6980 1.46681 0.733404 0.679793i \(-0.237930\pi\)
0.733404 + 0.679793i \(0.237930\pi\)
\(468\) 0 0
\(469\) −3.75867 −0.173559
\(470\) −6.50403 −0.300009
\(471\) 0 0
\(472\) −3.24041 −0.149152
\(473\) 0 0
\(474\) 0 0
\(475\) 0.812826 0.0372950
\(476\) −0.986602 −0.0452208
\(477\) 0 0
\(478\) −25.7000 −1.17549
\(479\) 30.8345 1.40886 0.704432 0.709771i \(-0.251202\pi\)
0.704432 + 0.709771i \(0.251202\pi\)
\(480\) 0 0
\(481\) 8.74202 0.398602
\(482\) −36.0507 −1.64207
\(483\) 0 0
\(484\) 0 0
\(485\) −14.1630 −0.643107
\(486\) 0 0
\(487\) 29.9921 1.35907 0.679535 0.733643i \(-0.262182\pi\)
0.679535 + 0.733643i \(0.262182\pi\)
\(488\) −7.39095 −0.334573
\(489\) 0 0
\(490\) −7.02229 −0.317235
\(491\) 34.5114 1.55748 0.778739 0.627348i \(-0.215860\pi\)
0.778739 + 0.627348i \(0.215860\pi\)
\(492\) 0 0
\(493\) −48.6518 −2.19117
\(494\) 2.15622 0.0970131
\(495\) 0 0
\(496\) 12.7968 0.574595
\(497\) −1.53157 −0.0687002
\(498\) 0 0
\(499\) 17.8629 0.799654 0.399827 0.916591i \(-0.369070\pi\)
0.399827 + 0.916591i \(0.369070\pi\)
\(500\) 0.115749 0.00517647
\(501\) 0 0
\(502\) −31.0582 −1.38620
\(503\) 5.11903 0.228246 0.114123 0.993467i \(-0.463594\pi\)
0.114123 + 0.993467i \(0.463594\pi\)
\(504\) 0 0
\(505\) 11.4056 0.507543
\(506\) 0 0
\(507\) 0 0
\(508\) 0.194408 0.00862548
\(509\) 18.1078 0.802615 0.401307 0.915943i \(-0.368556\pi\)
0.401307 + 0.915943i \(0.368556\pi\)
\(510\) 0 0
\(511\) 17.0472 0.754124
\(512\) 24.2671 1.07246
\(513\) 0 0
\(514\) 23.6241 1.04201
\(515\) −13.8709 −0.611223
\(516\) 0 0
\(517\) 0 0
\(518\) −8.52360 −0.374506
\(519\) 0 0
\(520\) 5.61256 0.246127
\(521\) 34.5708 1.51457 0.757287 0.653082i \(-0.226524\pi\)
0.757287 + 0.653082i \(0.226524\pi\)
\(522\) 0 0
\(523\) −32.7917 −1.43388 −0.716940 0.697135i \(-0.754458\pi\)
−0.716940 + 0.697135i \(0.754458\pi\)
\(524\) 1.36518 0.0596381
\(525\) 0 0
\(526\) −7.84406 −0.342017
\(527\) 21.1609 0.921785
\(528\) 0 0
\(529\) −9.75510 −0.424135
\(530\) 11.5229 0.500523
\(531\) 0 0
\(532\) 0.129147 0.00559925
\(533\) 3.52907 0.152861
\(534\) 0 0
\(535\) −1.06580 −0.0460784
\(536\) 7.95240 0.343491
\(537\) 0 0
\(538\) −3.94076 −0.169898
\(539\) 0 0
\(540\) 0 0
\(541\) −4.37244 −0.187986 −0.0939929 0.995573i \(-0.529963\pi\)
−0.0939929 + 0.995573i \(0.529963\pi\)
\(542\) −10.4968 −0.450877
\(543\) 0 0
\(544\) 4.06061 0.174097
\(545\) −11.3115 −0.484533
\(546\) 0 0
\(547\) −31.6473 −1.35314 −0.676571 0.736377i \(-0.736535\pi\)
−0.676571 + 0.736377i \(0.736535\pi\)
\(548\) −0.857455 −0.0366287
\(549\) 0 0
\(550\) 0 0
\(551\) 6.36858 0.271311
\(552\) 0 0
\(553\) −7.97320 −0.339055
\(554\) −40.3527 −1.71442
\(555\) 0 0
\(556\) 0.306884 0.0130148
\(557\) 1.52068 0.0644331 0.0322166 0.999481i \(-0.489743\pi\)
0.0322166 + 0.999481i \(0.489743\pi\)
\(558\) 0 0
\(559\) 12.4889 0.528223
\(560\) −5.15456 −0.217820
\(561\) 0 0
\(562\) −33.2519 −1.40265
\(563\) 0.653939 0.0275602 0.0137801 0.999905i \(-0.495614\pi\)
0.0137801 + 0.999905i \(0.495614\pi\)
\(564\) 0 0
\(565\) −3.76850 −0.158542
\(566\) 7.02680 0.295358
\(567\) 0 0
\(568\) 3.24041 0.135965
\(569\) 32.6606 1.36921 0.684603 0.728916i \(-0.259976\pi\)
0.684603 + 0.728916i \(0.259976\pi\)
\(570\) 0 0
\(571\) −26.1457 −1.09416 −0.547082 0.837079i \(-0.684262\pi\)
−0.547082 + 0.837079i \(0.684262\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −3.44090 −0.143620
\(575\) 3.63935 0.151772
\(576\) 0 0
\(577\) 5.73377 0.238700 0.119350 0.992852i \(-0.461919\pi\)
0.119350 + 0.992852i \(0.461919\pi\)
\(578\) −29.5914 −1.23084
\(579\) 0 0
\(580\) 0.906910 0.0376574
\(581\) 21.9315 0.909870
\(582\) 0 0
\(583\) 0 0
\(584\) −36.0676 −1.49249
\(585\) 0 0
\(586\) −15.0890 −0.623319
\(587\) −43.4586 −1.79373 −0.896864 0.442307i \(-0.854160\pi\)
−0.896864 + 0.442307i \(0.854160\pi\)
\(588\) 0 0
\(589\) −2.76999 −0.114136
\(590\) −1.53157 −0.0630536
\(591\) 0 0
\(592\) 16.9866 0.698145
\(593\) 9.23707 0.379321 0.189661 0.981850i \(-0.439261\pi\)
0.189661 + 0.981850i \(0.439261\pi\)
\(594\) 0 0
\(595\) −8.52360 −0.349434
\(596\) 0.966898 0.0396057
\(597\) 0 0
\(598\) 9.65430 0.394794
\(599\) 36.0865 1.47445 0.737227 0.675645i \(-0.236135\pi\)
0.737227 + 0.675645i \(0.236135\pi\)
\(600\) 0 0
\(601\) 23.1290 0.943452 0.471726 0.881745i \(-0.343631\pi\)
0.471726 + 0.881745i \(0.343631\pi\)
\(602\) −12.1768 −0.496291
\(603\) 0 0
\(604\) −0.859237 −0.0349619
\(605\) 0 0
\(606\) 0 0
\(607\) −21.9984 −0.892888 −0.446444 0.894812i \(-0.647310\pi\)
−0.446444 + 0.894812i \(0.647310\pi\)
\(608\) −0.531538 −0.0215567
\(609\) 0 0
\(610\) −3.49330 −0.141440
\(611\) 9.15673 0.370442
\(612\) 0 0
\(613\) −8.42425 −0.340252 −0.170126 0.985422i \(-0.554418\pi\)
−0.170126 + 0.985422i \(0.554418\pi\)
\(614\) −19.7000 −0.795026
\(615\) 0 0
\(616\) 0 0
\(617\) 17.8709 0.719453 0.359727 0.933058i \(-0.382870\pi\)
0.359727 + 0.933058i \(0.382870\pi\)
\(618\) 0 0
\(619\) 29.5102 1.18612 0.593058 0.805160i \(-0.297921\pi\)
0.593058 + 0.805160i \(0.297921\pi\)
\(620\) −0.394457 −0.0158418
\(621\) 0 0
\(622\) 37.5854 1.50704
\(623\) −3.71707 −0.148921
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −5.31344 −0.212368
\(627\) 0 0
\(628\) 0.993617 0.0396496
\(629\) 28.0891 1.11999
\(630\) 0 0
\(631\) 23.0204 0.916428 0.458214 0.888842i \(-0.348489\pi\)
0.458214 + 0.888842i \(0.348489\pi\)
\(632\) 16.8693 0.671025
\(633\) 0 0
\(634\) −21.2906 −0.845556
\(635\) 1.67956 0.0666515
\(636\) 0 0
\(637\) 9.88636 0.391712
\(638\) 0 0
\(639\) 0 0
\(640\) 10.2334 0.404511
\(641\) 3.76850 0.148847 0.0744234 0.997227i \(-0.476288\pi\)
0.0744234 + 0.997227i \(0.476288\pi\)
\(642\) 0 0
\(643\) 28.3011 1.11609 0.558043 0.829812i \(-0.311552\pi\)
0.558043 + 0.829812i \(0.311552\pi\)
\(644\) 0.578246 0.0227861
\(645\) 0 0
\(646\) 6.92820 0.272587
\(647\) −11.7775 −0.463020 −0.231510 0.972832i \(-0.574367\pi\)
−0.231510 + 0.972832i \(0.574367\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 2.65275 0.104049
\(651\) 0 0
\(652\) −1.89569 −0.0742410
\(653\) 46.1282 1.80514 0.902569 0.430546i \(-0.141679\pi\)
0.902569 + 0.430546i \(0.141679\pi\)
\(654\) 0 0
\(655\) 11.7943 0.460840
\(656\) 6.85733 0.267734
\(657\) 0 0
\(658\) −8.92795 −0.348048
\(659\) 15.7781 0.614626 0.307313 0.951609i \(-0.400570\pi\)
0.307313 + 0.951609i \(0.400570\pi\)
\(660\) 0 0
\(661\) −38.6732 −1.50421 −0.752106 0.659042i \(-0.770962\pi\)
−0.752106 + 0.659042i \(0.770962\pi\)
\(662\) −36.8667 −1.43286
\(663\) 0 0
\(664\) −46.4015 −1.80073
\(665\) 1.11575 0.0432669
\(666\) 0 0
\(667\) 28.5147 1.10410
\(668\) 2.48150 0.0960121
\(669\) 0 0
\(670\) 3.75867 0.145210
\(671\) 0 0
\(672\) 0 0
\(673\) 44.2791 1.70683 0.853416 0.521230i \(-0.174527\pi\)
0.853416 + 0.521230i \(0.174527\pi\)
\(674\) −2.65275 −0.102180
\(675\) 0 0
\(676\) 1.07245 0.0412482
\(677\) −6.74004 −0.259041 −0.129520 0.991577i \(-0.541344\pi\)
−0.129520 + 0.991577i \(0.541344\pi\)
\(678\) 0 0
\(679\) −19.4412 −0.746085
\(680\) 18.0338 0.691565
\(681\) 0 0
\(682\) 0 0
\(683\) −33.3303 −1.27535 −0.637675 0.770305i \(-0.720104\pi\)
−0.637675 + 0.770305i \(0.720104\pi\)
\(684\) 0 0
\(685\) −7.40786 −0.283040
\(686\) −22.8291 −0.871619
\(687\) 0 0
\(688\) 24.2671 0.925175
\(689\) −16.2226 −0.618031
\(690\) 0 0
\(691\) 9.00546 0.342584 0.171292 0.985220i \(-0.445206\pi\)
0.171292 + 0.985220i \(0.445206\pi\)
\(692\) 1.59013 0.0604477
\(693\) 0 0
\(694\) 5.21655 0.198018
\(695\) 2.65128 0.100569
\(696\) 0 0
\(697\) 11.3393 0.429507
\(698\) −23.5976 −0.893183
\(699\) 0 0
\(700\) 0.158887 0.00600536
\(701\) −45.8938 −1.73339 −0.866693 0.498842i \(-0.833759\pi\)
−0.866693 + 0.498842i \(0.833759\pi\)
\(702\) 0 0
\(703\) −3.67690 −0.138677
\(704\) 0 0
\(705\) 0 0
\(706\) −2.73447 −0.102913
\(707\) 15.6563 0.588814
\(708\) 0 0
\(709\) −8.51021 −0.319608 −0.159804 0.987149i \(-0.551086\pi\)
−0.159804 + 0.987149i \(0.551086\pi\)
\(710\) 1.53157 0.0574787
\(711\) 0 0
\(712\) 7.86439 0.294731
\(713\) −12.4024 −0.464473
\(714\) 0 0
\(715\) 0 0
\(716\) −0.436202 −0.0163016
\(717\) 0 0
\(718\) −21.3811 −0.797934
\(719\) 9.60554 0.358226 0.179113 0.983828i \(-0.442677\pi\)
0.179113 + 0.983828i \(0.442677\pi\)
\(720\) 0 0
\(721\) −19.0402 −0.709096
\(722\) 25.1740 0.936880
\(723\) 0 0
\(724\) 0.00155080 5.76351e−5 0
\(725\) 7.83511 0.290989
\(726\) 0 0
\(727\) 32.9688 1.22274 0.611372 0.791343i \(-0.290618\pi\)
0.611372 + 0.791343i \(0.290618\pi\)
\(728\) 7.70425 0.285538
\(729\) 0 0
\(730\) −17.0472 −0.630946
\(731\) 40.1282 1.48420
\(732\) 0 0
\(733\) −27.3704 −1.01095 −0.505475 0.862842i \(-0.668683\pi\)
−0.505475 + 0.862842i \(0.668683\pi\)
\(734\) −25.7680 −0.951113
\(735\) 0 0
\(736\) −2.37991 −0.0877248
\(737\) 0 0
\(738\) 0 0
\(739\) 38.5646 1.41862 0.709312 0.704895i \(-0.249006\pi\)
0.709312 + 0.704895i \(0.249006\pi\)
\(740\) −0.523604 −0.0192481
\(741\) 0 0
\(742\) 15.8173 0.580670
\(743\) −34.1644 −1.25337 −0.626684 0.779273i \(-0.715588\pi\)
−0.626684 + 0.779273i \(0.715588\pi\)
\(744\) 0 0
\(745\) 8.35337 0.306044
\(746\) 3.60554 0.132008
\(747\) 0 0
\(748\) 0 0
\(749\) −1.46300 −0.0534568
\(750\) 0 0
\(751\) −45.9305 −1.67603 −0.838015 0.545648i \(-0.816284\pi\)
−0.838015 + 0.545648i \(0.816284\pi\)
\(752\) 17.7924 0.648823
\(753\) 0 0
\(754\) 20.7846 0.756931
\(755\) −7.42325 −0.270160
\(756\) 0 0
\(757\) −30.2236 −1.09849 −0.549247 0.835660i \(-0.685085\pi\)
−0.549247 + 0.835660i \(0.685085\pi\)
\(758\) −3.65227 −0.132656
\(759\) 0 0
\(760\) −2.36065 −0.0856296
\(761\) 13.4308 0.486866 0.243433 0.969918i \(-0.421726\pi\)
0.243433 + 0.969918i \(0.421726\pi\)
\(762\) 0 0
\(763\) −15.5271 −0.562119
\(764\) −1.92444 −0.0696240
\(765\) 0 0
\(766\) 30.1049 1.08773
\(767\) 2.15622 0.0778567
\(768\) 0 0
\(769\) 10.8982 0.393001 0.196500 0.980504i \(-0.437042\pi\)
0.196500 + 0.980504i \(0.437042\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.09992 0.111568
\(773\) −13.1496 −0.472957 −0.236478 0.971637i \(-0.575993\pi\)
−0.236478 + 0.971637i \(0.575993\pi\)
\(774\) 0 0
\(775\) −3.40786 −0.122414
\(776\) 41.1327 1.47658
\(777\) 0 0
\(778\) −4.69629 −0.168370
\(779\) −1.48433 −0.0531816
\(780\) 0 0
\(781\) 0 0
\(782\) 31.0204 1.10929
\(783\) 0 0
\(784\) 19.2102 0.686077
\(785\) 8.58421 0.306384
\(786\) 0 0
\(787\) 11.5413 0.411403 0.205701 0.978615i \(-0.434052\pi\)
0.205701 + 0.978615i \(0.434052\pi\)
\(788\) −0.295993 −0.0105443
\(789\) 0 0
\(790\) 7.97320 0.283674
\(791\) −5.17295 −0.183929
\(792\) 0 0
\(793\) 4.91806 0.174645
\(794\) −7.56375 −0.268427
\(795\) 0 0
\(796\) 2.53062 0.0896954
\(797\) 5.54494 0.196412 0.0982059 0.995166i \(-0.468690\pi\)
0.0982059 + 0.995166i \(0.468690\pi\)
\(798\) 0 0
\(799\) 29.4217 1.04086
\(800\) −0.653939 −0.0231202
\(801\) 0 0
\(802\) −24.6067 −0.868891
\(803\) 0 0
\(804\) 0 0
\(805\) 4.99567 0.176074
\(806\) −9.04019 −0.318427
\(807\) 0 0
\(808\) −33.1247 −1.16532
\(809\) 18.4402 0.648324 0.324162 0.946002i \(-0.394918\pi\)
0.324162 + 0.946002i \(0.394918\pi\)
\(810\) 0 0
\(811\) −6.32818 −0.222212 −0.111106 0.993809i \(-0.535439\pi\)
−0.111106 + 0.993809i \(0.535439\pi\)
\(812\) 1.24490 0.0436873
\(813\) 0 0
\(814\) 0 0
\(815\) −16.3776 −0.573681
\(816\) 0 0
\(817\) −5.25283 −0.183773
\(818\) 40.2554 1.40750
\(819\) 0 0
\(820\) −0.211374 −0.00738150
\(821\) 7.95240 0.277541 0.138770 0.990325i \(-0.455685\pi\)
0.138770 + 0.990325i \(0.455685\pi\)
\(822\) 0 0
\(823\) 1.32241 0.0460963 0.0230481 0.999734i \(-0.492663\pi\)
0.0230481 + 0.999734i \(0.492663\pi\)
\(824\) 40.2844 1.40337
\(825\) 0 0
\(826\) −2.10235 −0.0731502
\(827\) −4.52990 −0.157520 −0.0787600 0.996894i \(-0.525096\pi\)
−0.0787600 + 0.996894i \(0.525096\pi\)
\(828\) 0 0
\(829\) −37.8148 −1.31336 −0.656681 0.754168i \(-0.728040\pi\)
−0.656681 + 0.754168i \(0.728040\pi\)
\(830\) −21.9315 −0.761252
\(831\) 0 0
\(832\) −16.2485 −0.563314
\(833\) 31.7661 1.10063
\(834\) 0 0
\(835\) 21.4385 0.741912
\(836\) 0 0
\(837\) 0 0
\(838\) 3.64138 0.125789
\(839\) 39.2519 1.35513 0.677563 0.735465i \(-0.263036\pi\)
0.677563 + 0.735465i \(0.263036\pi\)
\(840\) 0 0
\(841\) 32.3890 1.11686
\(842\) 15.3232 0.528071
\(843\) 0 0
\(844\) 1.48516 0.0511212
\(845\) 9.26531 0.318736
\(846\) 0 0
\(847\) 0 0
\(848\) −31.5221 −1.08247
\(849\) 0 0
\(850\) 8.52360 0.292357
\(851\) −16.4630 −0.564344
\(852\) 0 0
\(853\) −33.3917 −1.14331 −0.571655 0.820494i \(-0.693698\pi\)
−0.571655 + 0.820494i \(0.693698\pi\)
\(854\) −4.79518 −0.164088
\(855\) 0 0
\(856\) 3.09534 0.105796
\(857\) −16.9781 −0.579961 −0.289980 0.957033i \(-0.593649\pi\)
−0.289980 + 0.957033i \(0.593649\pi\)
\(858\) 0 0
\(859\) −40.8148 −1.39258 −0.696291 0.717759i \(-0.745168\pi\)
−0.696291 + 0.717759i \(0.745168\pi\)
\(860\) −0.748023 −0.0255074
\(861\) 0 0
\(862\) −6.25829 −0.213158
\(863\) −35.0303 −1.19245 −0.596223 0.802819i \(-0.703332\pi\)
−0.596223 + 0.802819i \(0.703332\pi\)
\(864\) 0 0
\(865\) 13.7377 0.467096
\(866\) −27.6773 −0.940513
\(867\) 0 0
\(868\) −0.541464 −0.0183785
\(869\) 0 0
\(870\) 0 0
\(871\) −5.29166 −0.179301
\(872\) 32.8515 1.11249
\(873\) 0 0
\(874\) −4.06061 −0.137352
\(875\) 1.37268 0.0464051
\(876\) 0 0
\(877\) 26.6626 0.900331 0.450165 0.892945i \(-0.351365\pi\)
0.450165 + 0.892945i \(0.351365\pi\)
\(878\) −47.3393 −1.59762
\(879\) 0 0
\(880\) 0 0
\(881\) 4.66615 0.157207 0.0786033 0.996906i \(-0.474954\pi\)
0.0786033 + 0.996906i \(0.474954\pi\)
\(882\) 0 0
\(883\) 6.47093 0.217764 0.108882 0.994055i \(-0.465273\pi\)
0.108882 + 0.994055i \(0.465273\pi\)
\(884\) −1.38899 −0.0467169
\(885\) 0 0
\(886\) 43.1826 1.45075
\(887\) 18.3508 0.616159 0.308079 0.951361i \(-0.400314\pi\)
0.308079 + 0.951361i \(0.400314\pi\)
\(888\) 0 0
\(889\) 2.30550 0.0773241
\(890\) 3.71707 0.124597
\(891\) 0 0
\(892\) 0.852396 0.0285403
\(893\) −3.85133 −0.128880
\(894\) 0 0
\(895\) −3.76850 −0.125967
\(896\) 14.0472 0.469284
\(897\) 0 0
\(898\) −6.55652 −0.218794
\(899\) −26.7009 −0.890526
\(900\) 0 0
\(901\) −52.1251 −1.73654
\(902\) 0 0
\(903\) 0 0
\(904\) 10.9447 0.364014
\(905\) 0.0133979 0.000445362 0
\(906\) 0 0
\(907\) −29.7845 −0.988978 −0.494489 0.869184i \(-0.664645\pi\)
−0.494489 + 0.869184i \(0.664645\pi\)
\(908\) −0.158887 −0.00527285
\(909\) 0 0
\(910\) 3.64138 0.120711
\(911\) −45.8699 −1.51974 −0.759869 0.650076i \(-0.774737\pi\)
−0.759869 + 0.650076i \(0.774737\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 2.10235 0.0695396
\(915\) 0 0
\(916\) −3.41424 −0.112810
\(917\) 16.1898 0.534633
\(918\) 0 0
\(919\) 58.1355 1.91771 0.958856 0.283893i \(-0.0916260\pi\)
0.958856 + 0.283893i \(0.0916260\pi\)
\(920\) −10.5696 −0.348469
\(921\) 0 0
\(922\) −25.4248 −0.837320
\(923\) −2.15622 −0.0709730
\(924\) 0 0
\(925\) −4.52360 −0.148735
\(926\) −13.2917 −0.436794
\(927\) 0 0
\(928\) −5.12368 −0.168193
\(929\) −39.8361 −1.30698 −0.653490 0.756935i \(-0.726696\pi\)
−0.653490 + 0.756935i \(0.726696\pi\)
\(930\) 0 0
\(931\) −4.15821 −0.136280
\(932\) −2.54630 −0.0834069
\(933\) 0 0
\(934\) −43.5112 −1.42373
\(935\) 0 0
\(936\) 0 0
\(937\) −9.67356 −0.316022 −0.158011 0.987437i \(-0.550508\pi\)
−0.158011 + 0.987437i \(0.550508\pi\)
\(938\) 5.15945 0.168462
\(939\) 0 0
\(940\) −0.548444 −0.0178883
\(941\) 3.97147 0.129466 0.0647331 0.997903i \(-0.479380\pi\)
0.0647331 + 0.997903i \(0.479380\pi\)
\(942\) 0 0
\(943\) −6.64595 −0.216422
\(944\) 4.18975 0.136365
\(945\) 0 0
\(946\) 0 0
\(947\) −2.65275 −0.0862029 −0.0431014 0.999071i \(-0.513724\pi\)
−0.0431014 + 0.999071i \(0.513724\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) −1.11575 −0.0361997
\(951\) 0 0
\(952\) 24.7547 0.802303
\(953\) −20.8678 −0.675974 −0.337987 0.941151i \(-0.609746\pi\)
−0.337987 + 0.941151i \(0.609746\pi\)
\(954\) 0 0
\(955\) −16.6260 −0.538003
\(956\) −2.16711 −0.0700895
\(957\) 0 0
\(958\) −42.3259 −1.36749
\(959\) −10.1686 −0.328362
\(960\) 0 0
\(961\) −19.3865 −0.625372
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) −3.03993 −0.0979095
\(965\) 26.7813 0.862120
\(966\) 0 0
\(967\) −58.5364 −1.88240 −0.941202 0.337843i \(-0.890303\pi\)
−0.941202 + 0.337843i \(0.890303\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 19.4412 0.624220
\(971\) 56.2653 1.80564 0.902820 0.430019i \(-0.141493\pi\)
0.902820 + 0.430019i \(0.141493\pi\)
\(972\) 0 0
\(973\) 3.63935 0.116672
\(974\) −41.1695 −1.31916
\(975\) 0 0
\(976\) 9.55627 0.305889
\(977\) −36.9866 −1.18331 −0.591653 0.806193i \(-0.701524\pi\)
−0.591653 + 0.806193i \(0.701524\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.592145 −0.0189154
\(981\) 0 0
\(982\) −47.3731 −1.51174
\(983\) 19.6642 0.627190 0.313595 0.949557i \(-0.398467\pi\)
0.313595 + 0.949557i \(0.398467\pi\)
\(984\) 0 0
\(985\) −2.55719 −0.0814789
\(986\) 66.7834 2.12682
\(987\) 0 0
\(988\) 0.181821 0.00578449
\(989\) −23.5191 −0.747863
\(990\) 0 0
\(991\) 18.5763 0.590095 0.295047 0.955483i \(-0.404665\pi\)
0.295047 + 0.955483i \(0.404665\pi\)
\(992\) 2.22853 0.0707558
\(993\) 0 0
\(994\) 2.10235 0.0666825
\(995\) 21.8629 0.693101
\(996\) 0 0
\(997\) 30.1049 0.953431 0.476716 0.879058i \(-0.341827\pi\)
0.476716 + 0.879058i \(0.341827\pi\)
\(998\) −24.5201 −0.776169
\(999\) 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 5445.2.a.bx.1.2 6
3.2 odd 2 605.2.a.m.1.5 yes 6
11.10 odd 2 inner 5445.2.a.bx.1.5 6
12.11 even 2 9680.2.a.cw.1.2 6
15.14 odd 2 3025.2.a.bg.1.2 6
33.2 even 10 605.2.g.q.81.5 24
33.5 odd 10 605.2.g.q.366.2 24
33.8 even 10 605.2.g.q.251.2 24
33.14 odd 10 605.2.g.q.251.5 24
33.17 even 10 605.2.g.q.366.5 24
33.20 odd 10 605.2.g.q.81.2 24
33.26 odd 10 605.2.g.q.511.5 24
33.29 even 10 605.2.g.q.511.2 24
33.32 even 2 605.2.a.m.1.2 6
132.131 odd 2 9680.2.a.cw.1.1 6
165.164 even 2 3025.2.a.bg.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
605.2.a.m.1.2 6 33.32 even 2
605.2.a.m.1.5 yes 6 3.2 odd 2
605.2.g.q.81.2 24 33.20 odd 10
605.2.g.q.81.5 24 33.2 even 10
605.2.g.q.251.2 24 33.8 even 10
605.2.g.q.251.5 24 33.14 odd 10
605.2.g.q.366.2 24 33.5 odd 10
605.2.g.q.366.5 24 33.17 even 10
605.2.g.q.511.2 24 33.29 even 10
605.2.g.q.511.5 24 33.26 odd 10
3025.2.a.bg.1.2 6 15.14 odd 2
3025.2.a.bg.1.5 6 165.164 even 2
5445.2.a.bx.1.2 6 1.1 even 1 trivial
5445.2.a.bx.1.5 6 11.10 odd 2 inner
9680.2.a.cw.1.1 6 132.131 odd 2
9680.2.a.cw.1.2 6 12.11 even 2