Properties

Label 6039.2.a.i.1.8
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2013)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.805107\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.805107 q^{2} -1.35180 q^{4} -1.06503 q^{5} -0.203035 q^{7} -2.69856 q^{8} +O(q^{10})\) \(q+0.805107 q^{2} -1.35180 q^{4} -1.06503 q^{5} -0.203035 q^{7} -2.69856 q^{8} -0.857467 q^{10} -1.00000 q^{11} +0.801491 q^{13} -0.163465 q^{14} +0.530979 q^{16} +0.376525 q^{17} +2.62261 q^{19} +1.43972 q^{20} -0.805107 q^{22} -2.93099 q^{23} -3.86570 q^{25} +0.645286 q^{26} +0.274464 q^{28} -0.907071 q^{29} +2.07822 q^{31} +5.82461 q^{32} +0.303143 q^{34} +0.216240 q^{35} -2.14816 q^{37} +2.11148 q^{38} +2.87406 q^{40} +1.83276 q^{41} -2.25276 q^{43} +1.35180 q^{44} -2.35976 q^{46} +0.685433 q^{47} -6.95878 q^{49} -3.11230 q^{50} -1.08346 q^{52} -14.0454 q^{53} +1.06503 q^{55} +0.547903 q^{56} -0.730289 q^{58} -7.73715 q^{59} -1.00000 q^{61} +1.67319 q^{62} +3.62748 q^{64} -0.853616 q^{65} -7.59429 q^{67} -0.508987 q^{68} +0.174096 q^{70} +9.23954 q^{71} +7.49398 q^{73} -1.72950 q^{74} -3.54525 q^{76} +0.203035 q^{77} +7.41703 q^{79} -0.565511 q^{80} +1.47557 q^{82} +8.55136 q^{83} -0.401012 q^{85} -1.81371 q^{86} +2.69856 q^{88} +2.97849 q^{89} -0.162731 q^{91} +3.96212 q^{92} +0.551846 q^{94} -2.79317 q^{95} +17.8193 q^{97} -5.60256 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + O(q^{10}) \) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + 6q^{10} - 13q^{11} + 13q^{13} - q^{14} + 18q^{16} - 17q^{17} + 14q^{19} + 7q^{20} + 2q^{22} - 7q^{23} + 18q^{25} + 10q^{26} + 19q^{28} + 6q^{29} + 27q^{31} - 5q^{32} + 6q^{34} - 14q^{35} + 10q^{37} - 2q^{38} + 8q^{40} - 3q^{41} + 29q^{43} - 16q^{44} - 24q^{46} - 8q^{47} + 8q^{49} + 27q^{50} + 37q^{52} + 24q^{53} + 3q^{55} - 24q^{56} - 5q^{58} - 13q^{59} - 13q^{61} - 39q^{62} + 47q^{64} + 11q^{65} + 44q^{67} + 8q^{68} - 12q^{70} - 3q^{71} + 48q^{73} + 22q^{74} + 47q^{76} - 11q^{77} - 17q^{79} + 26q^{80} + 56q^{82} - 50q^{83} + 8q^{85} - 18q^{86} + 9q^{88} + 15q^{89} + 47q^{91} - 14q^{92} + 45q^{94} + q^{95} + 27q^{97} - 47q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.805107 0.569296 0.284648 0.958632i \(-0.408123\pi\)
0.284648 + 0.958632i \(0.408123\pi\)
\(3\) 0 0
\(4\) −1.35180 −0.675902
\(5\) −1.06503 −0.476298 −0.238149 0.971229i \(-0.576541\pi\)
−0.238149 + 0.971229i \(0.576541\pi\)
\(6\) 0 0
\(7\) −0.203035 −0.0767401 −0.0383701 0.999264i \(-0.512217\pi\)
−0.0383701 + 0.999264i \(0.512217\pi\)
\(8\) −2.69856 −0.954085
\(9\) 0 0
\(10\) −0.857467 −0.271155
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.801491 0.222294 0.111147 0.993804i \(-0.464548\pi\)
0.111147 + 0.993804i \(0.464548\pi\)
\(14\) −0.163465 −0.0436879
\(15\) 0 0
\(16\) 0.530979 0.132745
\(17\) 0.376525 0.0913207 0.0456603 0.998957i \(-0.485461\pi\)
0.0456603 + 0.998957i \(0.485461\pi\)
\(18\) 0 0
\(19\) 2.62261 0.601667 0.300833 0.953677i \(-0.402735\pi\)
0.300833 + 0.953677i \(0.402735\pi\)
\(20\) 1.43972 0.321931
\(21\) 0 0
\(22\) −0.805107 −0.171649
\(23\) −2.93099 −0.611154 −0.305577 0.952167i \(-0.598849\pi\)
−0.305577 + 0.952167i \(0.598849\pi\)
\(24\) 0 0
\(25\) −3.86570 −0.773140
\(26\) 0.645286 0.126551
\(27\) 0 0
\(28\) 0.274464 0.0518688
\(29\) −0.907071 −0.168439 −0.0842195 0.996447i \(-0.526840\pi\)
−0.0842195 + 0.996447i \(0.526840\pi\)
\(30\) 0 0
\(31\) 2.07822 0.373260 0.186630 0.982430i \(-0.440243\pi\)
0.186630 + 0.982430i \(0.440243\pi\)
\(32\) 5.82461 1.02966
\(33\) 0 0
\(34\) 0.303143 0.0519885
\(35\) 0.216240 0.0365512
\(36\) 0 0
\(37\) −2.14816 −0.353155 −0.176578 0.984287i \(-0.556503\pi\)
−0.176578 + 0.984287i \(0.556503\pi\)
\(38\) 2.11148 0.342527
\(39\) 0 0
\(40\) 2.87406 0.454429
\(41\) 1.83276 0.286229 0.143115 0.989706i \(-0.454288\pi\)
0.143115 + 0.989706i \(0.454288\pi\)
\(42\) 0 0
\(43\) −2.25276 −0.343542 −0.171771 0.985137i \(-0.554949\pi\)
−0.171771 + 0.985137i \(0.554949\pi\)
\(44\) 1.35180 0.203792
\(45\) 0 0
\(46\) −2.35976 −0.347928
\(47\) 0.685433 0.0999806 0.0499903 0.998750i \(-0.484081\pi\)
0.0499903 + 0.998750i \(0.484081\pi\)
\(48\) 0 0
\(49\) −6.95878 −0.994111
\(50\) −3.11230 −0.440146
\(51\) 0 0
\(52\) −1.08346 −0.150249
\(53\) −14.0454 −1.92928 −0.964640 0.263570i \(-0.915100\pi\)
−0.964640 + 0.263570i \(0.915100\pi\)
\(54\) 0 0
\(55\) 1.06503 0.143609
\(56\) 0.547903 0.0732166
\(57\) 0 0
\(58\) −0.730289 −0.0958917
\(59\) −7.73715 −1.00729 −0.503646 0.863910i \(-0.668008\pi\)
−0.503646 + 0.863910i \(0.668008\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) 1.67319 0.212495
\(63\) 0 0
\(64\) 3.62748 0.453435
\(65\) −0.853616 −0.105878
\(66\) 0 0
\(67\) −7.59429 −0.927790 −0.463895 0.885890i \(-0.653549\pi\)
−0.463895 + 0.885890i \(0.653549\pi\)
\(68\) −0.508987 −0.0617238
\(69\) 0 0
\(70\) 0.174096 0.0208085
\(71\) 9.23954 1.09653 0.548266 0.836304i \(-0.315288\pi\)
0.548266 + 0.836304i \(0.315288\pi\)
\(72\) 0 0
\(73\) 7.49398 0.877104 0.438552 0.898706i \(-0.355491\pi\)
0.438552 + 0.898706i \(0.355491\pi\)
\(74\) −1.72950 −0.201050
\(75\) 0 0
\(76\) −3.54525 −0.406668
\(77\) 0.203035 0.0231380
\(78\) 0 0
\(79\) 7.41703 0.834481 0.417240 0.908796i \(-0.362997\pi\)
0.417240 + 0.908796i \(0.362997\pi\)
\(80\) −0.565511 −0.0632260
\(81\) 0 0
\(82\) 1.47557 0.162949
\(83\) 8.55136 0.938634 0.469317 0.883030i \(-0.344500\pi\)
0.469317 + 0.883030i \(0.344500\pi\)
\(84\) 0 0
\(85\) −0.401012 −0.0434959
\(86\) −1.81371 −0.195577
\(87\) 0 0
\(88\) 2.69856 0.287667
\(89\) 2.97849 0.315719 0.157860 0.987462i \(-0.449541\pi\)
0.157860 + 0.987462i \(0.449541\pi\)
\(90\) 0 0
\(91\) −0.162731 −0.0170588
\(92\) 3.96212 0.413080
\(93\) 0 0
\(94\) 0.551846 0.0569186
\(95\) −2.79317 −0.286573
\(96\) 0 0
\(97\) 17.8193 1.80928 0.904638 0.426181i \(-0.140141\pi\)
0.904638 + 0.426181i \(0.140141\pi\)
\(98\) −5.60256 −0.565944
\(99\) 0 0
\(100\) 5.22567 0.522567
\(101\) 4.50884 0.448646 0.224323 0.974515i \(-0.427983\pi\)
0.224323 + 0.974515i \(0.427983\pi\)
\(102\) 0 0
\(103\) 19.8815 1.95898 0.979489 0.201499i \(-0.0645812\pi\)
0.979489 + 0.201499i \(0.0645812\pi\)
\(104\) −2.16287 −0.212087
\(105\) 0 0
\(106\) −11.3080 −1.09833
\(107\) −6.58861 −0.636945 −0.318472 0.947932i \(-0.603170\pi\)
−0.318472 + 0.947932i \(0.603170\pi\)
\(108\) 0 0
\(109\) 19.1473 1.83398 0.916990 0.398910i \(-0.130612\pi\)
0.916990 + 0.398910i \(0.130612\pi\)
\(110\) 0.857467 0.0817562
\(111\) 0 0
\(112\) −0.107807 −0.0101868
\(113\) 2.84072 0.267232 0.133616 0.991033i \(-0.457341\pi\)
0.133616 + 0.991033i \(0.457341\pi\)
\(114\) 0 0
\(115\) 3.12161 0.291091
\(116\) 1.22618 0.113848
\(117\) 0 0
\(118\) −6.22923 −0.573447
\(119\) −0.0764478 −0.00700796
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −0.805107 −0.0728909
\(123\) 0 0
\(124\) −2.80935 −0.252287
\(125\) 9.44228 0.844543
\(126\) 0 0
\(127\) 5.00348 0.443987 0.221993 0.975048i \(-0.428744\pi\)
0.221993 + 0.975048i \(0.428744\pi\)
\(128\) −8.72872 −0.771517
\(129\) 0 0
\(130\) −0.687252 −0.0602760
\(131\) 19.0550 1.66485 0.832423 0.554140i \(-0.186953\pi\)
0.832423 + 0.554140i \(0.186953\pi\)
\(132\) 0 0
\(133\) −0.532481 −0.0461720
\(134\) −6.11422 −0.528188
\(135\) 0 0
\(136\) −1.01607 −0.0871277
\(137\) −18.4521 −1.57647 −0.788237 0.615372i \(-0.789006\pi\)
−0.788237 + 0.615372i \(0.789006\pi\)
\(138\) 0 0
\(139\) 14.6161 1.23972 0.619861 0.784711i \(-0.287189\pi\)
0.619861 + 0.784711i \(0.287189\pi\)
\(140\) −0.292313 −0.0247050
\(141\) 0 0
\(142\) 7.43882 0.624252
\(143\) −0.801491 −0.0670241
\(144\) 0 0
\(145\) 0.966063 0.0802271
\(146\) 6.03345 0.499332
\(147\) 0 0
\(148\) 2.90389 0.238698
\(149\) 4.26078 0.349057 0.174528 0.984652i \(-0.444160\pi\)
0.174528 + 0.984652i \(0.444160\pi\)
\(150\) 0 0
\(151\) 19.9980 1.62741 0.813707 0.581276i \(-0.197446\pi\)
0.813707 + 0.581276i \(0.197446\pi\)
\(152\) −7.07726 −0.574041
\(153\) 0 0
\(154\) 0.163465 0.0131724
\(155\) −2.21338 −0.177783
\(156\) 0 0
\(157\) 0.492192 0.0392812 0.0196406 0.999807i \(-0.493748\pi\)
0.0196406 + 0.999807i \(0.493748\pi\)
\(158\) 5.97150 0.475067
\(159\) 0 0
\(160\) −6.20342 −0.490423
\(161\) 0.595095 0.0469000
\(162\) 0 0
\(163\) −1.93731 −0.151742 −0.0758708 0.997118i \(-0.524174\pi\)
−0.0758708 + 0.997118i \(0.524174\pi\)
\(164\) −2.47753 −0.193463
\(165\) 0 0
\(166\) 6.88476 0.534361
\(167\) −14.8338 −1.14788 −0.573938 0.818899i \(-0.694585\pi\)
−0.573938 + 0.818899i \(0.694585\pi\)
\(168\) 0 0
\(169\) −12.3576 −0.950586
\(170\) −0.322857 −0.0247620
\(171\) 0 0
\(172\) 3.04529 0.232201
\(173\) 10.3859 0.789623 0.394812 0.918762i \(-0.370810\pi\)
0.394812 + 0.918762i \(0.370810\pi\)
\(174\) 0 0
\(175\) 0.784874 0.0593309
\(176\) −0.530979 −0.0400240
\(177\) 0 0
\(178\) 2.39800 0.179738
\(179\) −16.9049 −1.26353 −0.631765 0.775160i \(-0.717669\pi\)
−0.631765 + 0.775160i \(0.717669\pi\)
\(180\) 0 0
\(181\) 2.91712 0.216828 0.108414 0.994106i \(-0.465423\pi\)
0.108414 + 0.994106i \(0.465423\pi\)
\(182\) −0.131016 −0.00971154
\(183\) 0 0
\(184\) 7.90945 0.583093
\(185\) 2.28787 0.168207
\(186\) 0 0
\(187\) −0.376525 −0.0275342
\(188\) −0.926570 −0.0675771
\(189\) 0 0
\(190\) −2.24880 −0.163145
\(191\) −10.8546 −0.785410 −0.392705 0.919664i \(-0.628461\pi\)
−0.392705 + 0.919664i \(0.628461\pi\)
\(192\) 0 0
\(193\) 9.85294 0.709230 0.354615 0.935012i \(-0.384612\pi\)
0.354615 + 0.935012i \(0.384612\pi\)
\(194\) 14.3464 1.03001
\(195\) 0 0
\(196\) 9.40690 0.671921
\(197\) −1.58403 −0.112857 −0.0564286 0.998407i \(-0.517971\pi\)
−0.0564286 + 0.998407i \(0.517971\pi\)
\(198\) 0 0
\(199\) 25.5545 1.81151 0.905756 0.423799i \(-0.139304\pi\)
0.905756 + 0.423799i \(0.139304\pi\)
\(200\) 10.4318 0.737641
\(201\) 0 0
\(202\) 3.63010 0.255413
\(203\) 0.184168 0.0129260
\(204\) 0 0
\(205\) −1.95196 −0.136330
\(206\) 16.0067 1.11524
\(207\) 0 0
\(208\) 0.425575 0.0295083
\(209\) −2.62261 −0.181409
\(210\) 0 0
\(211\) 16.1904 1.11460 0.557298 0.830313i \(-0.311838\pi\)
0.557298 + 0.830313i \(0.311838\pi\)
\(212\) 18.9866 1.30400
\(213\) 0 0
\(214\) −5.30453 −0.362610
\(215\) 2.39927 0.163629
\(216\) 0 0
\(217\) −0.421952 −0.0286440
\(218\) 15.4156 1.04408
\(219\) 0 0
\(220\) −1.43972 −0.0970657
\(221\) 0.301781 0.0203000
\(222\) 0 0
\(223\) −4.93169 −0.330250 −0.165125 0.986273i \(-0.552803\pi\)
−0.165125 + 0.986273i \(0.552803\pi\)
\(224\) −1.18260 −0.0790159
\(225\) 0 0
\(226\) 2.28708 0.152134
\(227\) 2.59452 0.172204 0.0861022 0.996286i \(-0.472559\pi\)
0.0861022 + 0.996286i \(0.472559\pi\)
\(228\) 0 0
\(229\) −6.75414 −0.446326 −0.223163 0.974781i \(-0.571638\pi\)
−0.223163 + 0.974781i \(0.571638\pi\)
\(230\) 2.51323 0.165717
\(231\) 0 0
\(232\) 2.44779 0.160705
\(233\) 27.8481 1.82439 0.912195 0.409757i \(-0.134386\pi\)
0.912195 + 0.409757i \(0.134386\pi\)
\(234\) 0 0
\(235\) −0.730010 −0.0476206
\(236\) 10.4591 0.680830
\(237\) 0 0
\(238\) −0.0615487 −0.00398961
\(239\) −16.9202 −1.09448 −0.547240 0.836976i \(-0.684321\pi\)
−0.547240 + 0.836976i \(0.684321\pi\)
\(240\) 0 0
\(241\) −1.06455 −0.0685739 −0.0342869 0.999412i \(-0.510916\pi\)
−0.0342869 + 0.999412i \(0.510916\pi\)
\(242\) 0.805107 0.0517542
\(243\) 0 0
\(244\) 1.35180 0.0865403
\(245\) 7.41134 0.473493
\(246\) 0 0
\(247\) 2.10199 0.133747
\(248\) −5.60821 −0.356121
\(249\) 0 0
\(250\) 7.60204 0.480795
\(251\) 3.19791 0.201850 0.100925 0.994894i \(-0.467820\pi\)
0.100925 + 0.994894i \(0.467820\pi\)
\(252\) 0 0
\(253\) 2.93099 0.184270
\(254\) 4.02833 0.252760
\(255\) 0 0
\(256\) −14.2825 −0.892657
\(257\) 12.0296 0.750388 0.375194 0.926946i \(-0.377576\pi\)
0.375194 + 0.926946i \(0.377576\pi\)
\(258\) 0 0
\(259\) 0.436152 0.0271012
\(260\) 1.15392 0.0715631
\(261\) 0 0
\(262\) 15.3413 0.947791
\(263\) 3.98169 0.245521 0.122761 0.992436i \(-0.460825\pi\)
0.122761 + 0.992436i \(0.460825\pi\)
\(264\) 0 0
\(265\) 14.9588 0.918912
\(266\) −0.428704 −0.0262855
\(267\) 0 0
\(268\) 10.2660 0.627095
\(269\) −2.20633 −0.134522 −0.0672611 0.997735i \(-0.521426\pi\)
−0.0672611 + 0.997735i \(0.521426\pi\)
\(270\) 0 0
\(271\) 10.9268 0.663754 0.331877 0.943323i \(-0.392318\pi\)
0.331877 + 0.943323i \(0.392318\pi\)
\(272\) 0.199927 0.0121223
\(273\) 0 0
\(274\) −14.8559 −0.897480
\(275\) 3.86570 0.233111
\(276\) 0 0
\(277\) 7.40923 0.445177 0.222589 0.974912i \(-0.428549\pi\)
0.222589 + 0.974912i \(0.428549\pi\)
\(278\) 11.7675 0.705770
\(279\) 0 0
\(280\) −0.583535 −0.0348729
\(281\) 0.155459 0.00927391 0.00463696 0.999989i \(-0.498524\pi\)
0.00463696 + 0.999989i \(0.498524\pi\)
\(282\) 0 0
\(283\) −11.9494 −0.710316 −0.355158 0.934806i \(-0.615573\pi\)
−0.355158 + 0.934806i \(0.615573\pi\)
\(284\) −12.4900 −0.741148
\(285\) 0 0
\(286\) −0.645286 −0.0381565
\(287\) −0.372115 −0.0219653
\(288\) 0 0
\(289\) −16.8582 −0.991661
\(290\) 0.777784 0.0456730
\(291\) 0 0
\(292\) −10.1304 −0.592836
\(293\) −16.9701 −0.991404 −0.495702 0.868493i \(-0.665089\pi\)
−0.495702 + 0.868493i \(0.665089\pi\)
\(294\) 0 0
\(295\) 8.24034 0.479771
\(296\) 5.79694 0.336940
\(297\) 0 0
\(298\) 3.43038 0.198717
\(299\) −2.34916 −0.135856
\(300\) 0 0
\(301\) 0.457390 0.0263635
\(302\) 16.1005 0.926480
\(303\) 0 0
\(304\) 1.39255 0.0798681
\(305\) 1.06503 0.0609837
\(306\) 0 0
\(307\) 13.3520 0.762041 0.381021 0.924567i \(-0.375573\pi\)
0.381021 + 0.924567i \(0.375573\pi\)
\(308\) −0.274464 −0.0156390
\(309\) 0 0
\(310\) −1.78201 −0.101211
\(311\) 12.6506 0.717352 0.358676 0.933462i \(-0.383228\pi\)
0.358676 + 0.933462i \(0.383228\pi\)
\(312\) 0 0
\(313\) 8.15405 0.460894 0.230447 0.973085i \(-0.425981\pi\)
0.230447 + 0.973085i \(0.425981\pi\)
\(314\) 0.396267 0.0223627
\(315\) 0 0
\(316\) −10.0264 −0.564027
\(317\) 22.7512 1.27784 0.638918 0.769274i \(-0.279382\pi\)
0.638918 + 0.769274i \(0.279382\pi\)
\(318\) 0 0
\(319\) 0.907071 0.0507863
\(320\) −3.86339 −0.215970
\(321\) 0 0
\(322\) 0.479115 0.0267000
\(323\) 0.987476 0.0549446
\(324\) 0 0
\(325\) −3.09832 −0.171864
\(326\) −1.55974 −0.0863860
\(327\) 0 0
\(328\) −4.94582 −0.273087
\(329\) −0.139167 −0.00767253
\(330\) 0 0
\(331\) 2.47607 0.136097 0.0680486 0.997682i \(-0.478323\pi\)
0.0680486 + 0.997682i \(0.478323\pi\)
\(332\) −11.5598 −0.634424
\(333\) 0 0
\(334\) −11.9428 −0.653481
\(335\) 8.08819 0.441905
\(336\) 0 0
\(337\) −11.8667 −0.646419 −0.323210 0.946327i \(-0.604762\pi\)
−0.323210 + 0.946327i \(0.604762\pi\)
\(338\) −9.94920 −0.541165
\(339\) 0 0
\(340\) 0.542089 0.0293989
\(341\) −2.07822 −0.112542
\(342\) 0 0
\(343\) 2.83412 0.153028
\(344\) 6.07920 0.327769
\(345\) 0 0
\(346\) 8.36174 0.449530
\(347\) −33.4479 −1.79558 −0.897789 0.440426i \(-0.854827\pi\)
−0.897789 + 0.440426i \(0.854827\pi\)
\(348\) 0 0
\(349\) −19.7785 −1.05872 −0.529359 0.848398i \(-0.677567\pi\)
−0.529359 + 0.848398i \(0.677567\pi\)
\(350\) 0.631907 0.0337768
\(351\) 0 0
\(352\) −5.82461 −0.310453
\(353\) 15.2833 0.813448 0.406724 0.913551i \(-0.366671\pi\)
0.406724 + 0.913551i \(0.366671\pi\)
\(354\) 0 0
\(355\) −9.84043 −0.522276
\(356\) −4.02633 −0.213395
\(357\) 0 0
\(358\) −13.6102 −0.719324
\(359\) −10.2459 −0.540759 −0.270380 0.962754i \(-0.587149\pi\)
−0.270380 + 0.962754i \(0.587149\pi\)
\(360\) 0 0
\(361\) −12.1219 −0.637997
\(362\) 2.34859 0.123439
\(363\) 0 0
\(364\) 0.219980 0.0115301
\(365\) −7.98135 −0.417763
\(366\) 0 0
\(367\) 11.5481 0.602807 0.301403 0.953497i \(-0.402545\pi\)
0.301403 + 0.953497i \(0.402545\pi\)
\(368\) −1.55629 −0.0811274
\(369\) 0 0
\(370\) 1.84198 0.0957598
\(371\) 2.85171 0.148053
\(372\) 0 0
\(373\) −26.2319 −1.35824 −0.679119 0.734028i \(-0.737638\pi\)
−0.679119 + 0.734028i \(0.737638\pi\)
\(374\) −0.303143 −0.0156751
\(375\) 0 0
\(376\) −1.84968 −0.0953900
\(377\) −0.727010 −0.0374429
\(378\) 0 0
\(379\) 11.1404 0.572245 0.286123 0.958193i \(-0.407634\pi\)
0.286123 + 0.958193i \(0.407634\pi\)
\(380\) 3.77581 0.193695
\(381\) 0 0
\(382\) −8.73910 −0.447131
\(383\) −37.6696 −1.92483 −0.962413 0.271590i \(-0.912451\pi\)
−0.962413 + 0.271590i \(0.912451\pi\)
\(384\) 0 0
\(385\) −0.216240 −0.0110206
\(386\) 7.93266 0.403762
\(387\) 0 0
\(388\) −24.0882 −1.22289
\(389\) 10.9370 0.554527 0.277263 0.960794i \(-0.410573\pi\)
0.277263 + 0.960794i \(0.410573\pi\)
\(390\) 0 0
\(391\) −1.10359 −0.0558110
\(392\) 18.7787 0.948466
\(393\) 0 0
\(394\) −1.27531 −0.0642492
\(395\) −7.89939 −0.397461
\(396\) 0 0
\(397\) 33.2094 1.66673 0.833365 0.552722i \(-0.186411\pi\)
0.833365 + 0.552722i \(0.186411\pi\)
\(398\) 20.5741 1.03129
\(399\) 0 0
\(400\) −2.05260 −0.102630
\(401\) 18.7562 0.936640 0.468320 0.883559i \(-0.344859\pi\)
0.468320 + 0.883559i \(0.344859\pi\)
\(402\) 0 0
\(403\) 1.66568 0.0829733
\(404\) −6.09506 −0.303241
\(405\) 0 0
\(406\) 0.148274 0.00735874
\(407\) 2.14816 0.106480
\(408\) 0 0
\(409\) 5.97380 0.295385 0.147693 0.989033i \(-0.452815\pi\)
0.147693 + 0.989033i \(0.452815\pi\)
\(410\) −1.57153 −0.0776125
\(411\) 0 0
\(412\) −26.8758 −1.32408
\(413\) 1.57091 0.0772997
\(414\) 0 0
\(415\) −9.10750 −0.447069
\(416\) 4.66838 0.228886
\(417\) 0 0
\(418\) −2.11148 −0.103276
\(419\) 21.7376 1.06195 0.530976 0.847387i \(-0.321826\pi\)
0.530976 + 0.847387i \(0.321826\pi\)
\(420\) 0 0
\(421\) −16.2177 −0.790402 −0.395201 0.918595i \(-0.629325\pi\)
−0.395201 + 0.918595i \(0.629325\pi\)
\(422\) 13.0350 0.634535
\(423\) 0 0
\(424\) 37.9023 1.84070
\(425\) −1.45553 −0.0706037
\(426\) 0 0
\(427\) 0.203035 0.00982557
\(428\) 8.90650 0.430512
\(429\) 0 0
\(430\) 1.93167 0.0931532
\(431\) 31.0756 1.49686 0.748430 0.663214i \(-0.230808\pi\)
0.748430 + 0.663214i \(0.230808\pi\)
\(432\) 0 0
\(433\) 37.4627 1.80034 0.900171 0.435536i \(-0.143441\pi\)
0.900171 + 0.435536i \(0.143441\pi\)
\(434\) −0.339717 −0.0163069
\(435\) 0 0
\(436\) −25.8834 −1.23959
\(437\) −7.68683 −0.367711
\(438\) 0 0
\(439\) 15.0607 0.718809 0.359405 0.933182i \(-0.382980\pi\)
0.359405 + 0.933182i \(0.382980\pi\)
\(440\) −2.87406 −0.137015
\(441\) 0 0
\(442\) 0.242966 0.0115567
\(443\) −7.50403 −0.356527 −0.178264 0.983983i \(-0.557048\pi\)
−0.178264 + 0.983983i \(0.557048\pi\)
\(444\) 0 0
\(445\) −3.17220 −0.150376
\(446\) −3.97054 −0.188010
\(447\) 0 0
\(448\) −0.736506 −0.0347966
\(449\) −13.5424 −0.639103 −0.319552 0.947569i \(-0.603532\pi\)
−0.319552 + 0.947569i \(0.603532\pi\)
\(450\) 0 0
\(451\) −1.83276 −0.0863014
\(452\) −3.84009 −0.180623
\(453\) 0 0
\(454\) 2.08887 0.0980353
\(455\) 0.173314 0.00812509
\(456\) 0 0
\(457\) 10.1840 0.476388 0.238194 0.971218i \(-0.423445\pi\)
0.238194 + 0.971218i \(0.423445\pi\)
\(458\) −5.43780 −0.254092
\(459\) 0 0
\(460\) −4.21980 −0.196749
\(461\) 14.0649 0.655067 0.327533 0.944840i \(-0.393783\pi\)
0.327533 + 0.944840i \(0.393783\pi\)
\(462\) 0 0
\(463\) −27.3090 −1.26916 −0.634578 0.772859i \(-0.718826\pi\)
−0.634578 + 0.772859i \(0.718826\pi\)
\(464\) −0.481636 −0.0223594
\(465\) 0 0
\(466\) 22.4207 1.03862
\(467\) −16.1833 −0.748875 −0.374437 0.927252i \(-0.622164\pi\)
−0.374437 + 0.927252i \(0.622164\pi\)
\(468\) 0 0
\(469\) 1.54191 0.0711988
\(470\) −0.587736 −0.0271102
\(471\) 0 0
\(472\) 20.8792 0.961041
\(473\) 2.25276 0.103582
\(474\) 0 0
\(475\) −10.1382 −0.465173
\(476\) 0.103342 0.00473669
\(477\) 0 0
\(478\) −13.6226 −0.623083
\(479\) 19.5798 0.894623 0.447311 0.894378i \(-0.352382\pi\)
0.447311 + 0.894378i \(0.352382\pi\)
\(480\) 0 0
\(481\) −1.72173 −0.0785042
\(482\) −0.857078 −0.0390389
\(483\) 0 0
\(484\) −1.35180 −0.0614456
\(485\) −18.9782 −0.861754
\(486\) 0 0
\(487\) −25.6857 −1.16393 −0.581966 0.813213i \(-0.697716\pi\)
−0.581966 + 0.813213i \(0.697716\pi\)
\(488\) 2.69856 0.122158
\(489\) 0 0
\(490\) 5.96692 0.269558
\(491\) −30.0246 −1.35499 −0.677497 0.735526i \(-0.736935\pi\)
−0.677497 + 0.735526i \(0.736935\pi\)
\(492\) 0 0
\(493\) −0.341535 −0.0153820
\(494\) 1.69233 0.0761415
\(495\) 0 0
\(496\) 1.10349 0.0495482
\(497\) −1.87595 −0.0841480
\(498\) 0 0
\(499\) 16.6668 0.746109 0.373054 0.927810i \(-0.378311\pi\)
0.373054 + 0.927810i \(0.378311\pi\)
\(500\) −12.7641 −0.570828
\(501\) 0 0
\(502\) 2.57466 0.114912
\(503\) 27.2800 1.21635 0.608176 0.793802i \(-0.291901\pi\)
0.608176 + 0.793802i \(0.291901\pi\)
\(504\) 0 0
\(505\) −4.80207 −0.213689
\(506\) 2.35976 0.104904
\(507\) 0 0
\(508\) −6.76372 −0.300092
\(509\) 21.2805 0.943243 0.471622 0.881801i \(-0.343669\pi\)
0.471622 + 0.881801i \(0.343669\pi\)
\(510\) 0 0
\(511\) −1.52154 −0.0673091
\(512\) 5.95850 0.263331
\(513\) 0 0
\(514\) 9.68514 0.427193
\(515\) −21.1744 −0.933057
\(516\) 0 0
\(517\) −0.685433 −0.0301453
\(518\) 0.351149 0.0154286
\(519\) 0 0
\(520\) 2.30353 0.101017
\(521\) 25.1670 1.10258 0.551292 0.834312i \(-0.314135\pi\)
0.551292 + 0.834312i \(0.314135\pi\)
\(522\) 0 0
\(523\) 20.0188 0.875362 0.437681 0.899130i \(-0.355800\pi\)
0.437681 + 0.899130i \(0.355800\pi\)
\(524\) −25.7587 −1.12527
\(525\) 0 0
\(526\) 3.20568 0.139774
\(527\) 0.782502 0.0340863
\(528\) 0 0
\(529\) −14.4093 −0.626491
\(530\) 12.0434 0.523134
\(531\) 0 0
\(532\) 0.719810 0.0312077
\(533\) 1.46894 0.0636270
\(534\) 0 0
\(535\) 7.01710 0.303376
\(536\) 20.4936 0.885191
\(537\) 0 0
\(538\) −1.77633 −0.0765830
\(539\) 6.95878 0.299736
\(540\) 0 0
\(541\) 33.9981 1.46169 0.730847 0.682541i \(-0.239125\pi\)
0.730847 + 0.682541i \(0.239125\pi\)
\(542\) 8.79722 0.377873
\(543\) 0 0
\(544\) 2.19311 0.0940289
\(545\) −20.3926 −0.873521
\(546\) 0 0
\(547\) −17.6728 −0.755636 −0.377818 0.925880i \(-0.623325\pi\)
−0.377818 + 0.925880i \(0.623325\pi\)
\(548\) 24.9437 1.06554
\(549\) 0 0
\(550\) 3.11230 0.132709
\(551\) −2.37889 −0.101344
\(552\) 0 0
\(553\) −1.50592 −0.0640381
\(554\) 5.96522 0.253438
\(555\) 0 0
\(556\) −19.7581 −0.837931
\(557\) −40.6753 −1.72347 −0.861734 0.507360i \(-0.830621\pi\)
−0.861734 + 0.507360i \(0.830621\pi\)
\(558\) 0 0
\(559\) −1.80557 −0.0763673
\(560\) 0.114819 0.00485197
\(561\) 0 0
\(562\) 0.125161 0.00527961
\(563\) −43.3214 −1.82578 −0.912890 0.408207i \(-0.866154\pi\)
−0.912890 + 0.408207i \(0.866154\pi\)
\(564\) 0 0
\(565\) −3.02546 −0.127282
\(566\) −9.62051 −0.404380
\(567\) 0 0
\(568\) −24.9335 −1.04618
\(569\) 26.7783 1.12260 0.561302 0.827611i \(-0.310301\pi\)
0.561302 + 0.827611i \(0.310301\pi\)
\(570\) 0 0
\(571\) 29.9130 1.25182 0.625910 0.779896i \(-0.284728\pi\)
0.625910 + 0.779896i \(0.284728\pi\)
\(572\) 1.08346 0.0453017
\(573\) 0 0
\(574\) −0.299593 −0.0125048
\(575\) 11.3303 0.472508
\(576\) 0 0
\(577\) −19.3583 −0.805897 −0.402948 0.915223i \(-0.632015\pi\)
−0.402948 + 0.915223i \(0.632015\pi\)
\(578\) −13.5727 −0.564549
\(579\) 0 0
\(580\) −1.30593 −0.0542257
\(581\) −1.73623 −0.0720309
\(582\) 0 0
\(583\) 14.0454 0.581700
\(584\) −20.2230 −0.836832
\(585\) 0 0
\(586\) −13.6627 −0.564402
\(587\) 33.1146 1.36679 0.683394 0.730050i \(-0.260503\pi\)
0.683394 + 0.730050i \(0.260503\pi\)
\(588\) 0 0
\(589\) 5.45036 0.224578
\(590\) 6.63435 0.273132
\(591\) 0 0
\(592\) −1.14063 −0.0468795
\(593\) 11.3857 0.467555 0.233778 0.972290i \(-0.424891\pi\)
0.233778 + 0.972290i \(0.424891\pi\)
\(594\) 0 0
\(595\) 0.0814196 0.00333788
\(596\) −5.75974 −0.235928
\(597\) 0 0
\(598\) −1.89133 −0.0773421
\(599\) −14.0221 −0.572927 −0.286463 0.958091i \(-0.592480\pi\)
−0.286463 + 0.958091i \(0.592480\pi\)
\(600\) 0 0
\(601\) 1.14648 0.0467660 0.0233830 0.999727i \(-0.492556\pi\)
0.0233830 + 0.999727i \(0.492556\pi\)
\(602\) 0.368247 0.0150086
\(603\) 0 0
\(604\) −27.0333 −1.09997
\(605\) −1.06503 −0.0432998
\(606\) 0 0
\(607\) −15.6771 −0.636315 −0.318157 0.948038i \(-0.603064\pi\)
−0.318157 + 0.948038i \(0.603064\pi\)
\(608\) 15.2757 0.619510
\(609\) 0 0
\(610\) 0.857467 0.0347178
\(611\) 0.549368 0.0222251
\(612\) 0 0
\(613\) 18.6826 0.754584 0.377292 0.926094i \(-0.376855\pi\)
0.377292 + 0.926094i \(0.376855\pi\)
\(614\) 10.7498 0.433827
\(615\) 0 0
\(616\) −0.547903 −0.0220756
\(617\) −3.28234 −0.132142 −0.0660709 0.997815i \(-0.521046\pi\)
−0.0660709 + 0.997815i \(0.521046\pi\)
\(618\) 0 0
\(619\) −20.4789 −0.823116 −0.411558 0.911383i \(-0.635015\pi\)
−0.411558 + 0.911383i \(0.635015\pi\)
\(620\) 2.99205 0.120164
\(621\) 0 0
\(622\) 10.1851 0.408386
\(623\) −0.604738 −0.0242283
\(624\) 0 0
\(625\) 9.27215 0.370886
\(626\) 6.56488 0.262385
\(627\) 0 0
\(628\) −0.665347 −0.0265502
\(629\) −0.808836 −0.0322504
\(630\) 0 0
\(631\) −21.1120 −0.840454 −0.420227 0.907419i \(-0.638050\pi\)
−0.420227 + 0.907419i \(0.638050\pi\)
\(632\) −20.0153 −0.796165
\(633\) 0 0
\(634\) 18.3172 0.727468
\(635\) −5.32888 −0.211470
\(636\) 0 0
\(637\) −5.57740 −0.220985
\(638\) 0.730289 0.0289124
\(639\) 0 0
\(640\) 9.29639 0.367472
\(641\) 28.2912 1.11743 0.558717 0.829358i \(-0.311294\pi\)
0.558717 + 0.829358i \(0.311294\pi\)
\(642\) 0 0
\(643\) −20.9767 −0.827238 −0.413619 0.910450i \(-0.635736\pi\)
−0.413619 + 0.910450i \(0.635736\pi\)
\(644\) −0.804451 −0.0316998
\(645\) 0 0
\(646\) 0.795024 0.0312798
\(647\) −20.5640 −0.808456 −0.404228 0.914658i \(-0.632460\pi\)
−0.404228 + 0.914658i \(0.632460\pi\)
\(648\) 0 0
\(649\) 7.73715 0.303710
\(650\) −2.49448 −0.0978416
\(651\) 0 0
\(652\) 2.61886 0.102562
\(653\) −3.95699 −0.154849 −0.0774246 0.996998i \(-0.524670\pi\)
−0.0774246 + 0.996998i \(0.524670\pi\)
\(654\) 0 0
\(655\) −20.2943 −0.792963
\(656\) 0.973158 0.0379954
\(657\) 0 0
\(658\) −0.112044 −0.00436794
\(659\) 13.5419 0.527519 0.263759 0.964589i \(-0.415038\pi\)
0.263759 + 0.964589i \(0.415038\pi\)
\(660\) 0 0
\(661\) −37.6849 −1.46577 −0.732886 0.680351i \(-0.761827\pi\)
−0.732886 + 0.680351i \(0.761827\pi\)
\(662\) 1.99350 0.0774797
\(663\) 0 0
\(664\) −23.0764 −0.895536
\(665\) 0.567111 0.0219916
\(666\) 0 0
\(667\) 2.65862 0.102942
\(668\) 20.0524 0.775851
\(669\) 0 0
\(670\) 6.51185 0.251575
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 46.2878 1.78427 0.892133 0.451774i \(-0.149209\pi\)
0.892133 + 0.451774i \(0.149209\pi\)
\(674\) −9.55394 −0.368004
\(675\) 0 0
\(676\) 16.7051 0.642502
\(677\) −26.6561 −1.02448 −0.512239 0.858843i \(-0.671184\pi\)
−0.512239 + 0.858843i \(0.671184\pi\)
\(678\) 0 0
\(679\) −3.61795 −0.138844
\(680\) 1.08215 0.0414987
\(681\) 0 0
\(682\) −1.67319 −0.0640698
\(683\) −27.7294 −1.06104 −0.530519 0.847673i \(-0.678003\pi\)
−0.530519 + 0.847673i \(0.678003\pi\)
\(684\) 0 0
\(685\) 19.6522 0.750871
\(686\) 2.28177 0.0871185
\(687\) 0 0
\(688\) −1.19617 −0.0456034
\(689\) −11.2572 −0.428867
\(690\) 0 0
\(691\) −14.4148 −0.548365 −0.274183 0.961678i \(-0.588407\pi\)
−0.274183 + 0.961678i \(0.588407\pi\)
\(692\) −14.0397 −0.533708
\(693\) 0 0
\(694\) −26.9291 −1.02222
\(695\) −15.5667 −0.590477
\(696\) 0 0
\(697\) 0.690080 0.0261387
\(698\) −15.9238 −0.602724
\(699\) 0 0
\(700\) −1.06099 −0.0401018
\(701\) 21.9567 0.829295 0.414647 0.909982i \(-0.363905\pi\)
0.414647 + 0.909982i \(0.363905\pi\)
\(702\) 0 0
\(703\) −5.63378 −0.212482
\(704\) −3.62748 −0.136716
\(705\) 0 0
\(706\) 12.3047 0.463093
\(707\) −0.915453 −0.0344292
\(708\) 0 0
\(709\) −5.41410 −0.203331 −0.101665 0.994819i \(-0.532417\pi\)
−0.101665 + 0.994819i \(0.532417\pi\)
\(710\) −7.92260 −0.297330
\(711\) 0 0
\(712\) −8.03763 −0.301223
\(713\) −6.09125 −0.228119
\(714\) 0 0
\(715\) 0.853616 0.0319234
\(716\) 22.8521 0.854023
\(717\) 0 0
\(718\) −8.24906 −0.307852
\(719\) −34.7613 −1.29638 −0.648188 0.761480i \(-0.724473\pi\)
−0.648188 + 0.761480i \(0.724473\pi\)
\(720\) 0 0
\(721\) −4.03664 −0.150332
\(722\) −9.75946 −0.363209
\(723\) 0 0
\(724\) −3.94337 −0.146554
\(725\) 3.50647 0.130227
\(726\) 0 0
\(727\) 34.1852 1.26786 0.633930 0.773390i \(-0.281441\pi\)
0.633930 + 0.773390i \(0.281441\pi\)
\(728\) 0.439139 0.0162756
\(729\) 0 0
\(730\) −6.42584 −0.237831
\(731\) −0.848220 −0.0313725
\(732\) 0 0
\(733\) −21.4554 −0.792474 −0.396237 0.918148i \(-0.629684\pi\)
−0.396237 + 0.918148i \(0.629684\pi\)
\(734\) 9.29746 0.343176
\(735\) 0 0
\(736\) −17.0719 −0.629278
\(737\) 7.59429 0.279739
\(738\) 0 0
\(739\) 33.6003 1.23601 0.618004 0.786175i \(-0.287942\pi\)
0.618004 + 0.786175i \(0.287942\pi\)
\(740\) −3.09274 −0.113692
\(741\) 0 0
\(742\) 2.29593 0.0842862
\(743\) −3.15148 −0.115617 −0.0578083 0.998328i \(-0.518411\pi\)
−0.0578083 + 0.998328i \(0.518411\pi\)
\(744\) 0 0
\(745\) −4.53788 −0.166255
\(746\) −21.1195 −0.773240
\(747\) 0 0
\(748\) 0.508987 0.0186104
\(749\) 1.33772 0.0488792
\(750\) 0 0
\(751\) −21.1165 −0.770551 −0.385276 0.922802i \(-0.625894\pi\)
−0.385276 + 0.922802i \(0.625894\pi\)
\(752\) 0.363950 0.0132719
\(753\) 0 0
\(754\) −0.585320 −0.0213161
\(755\) −21.2986 −0.775134
\(756\) 0 0
\(757\) 25.3065 0.919779 0.459890 0.887976i \(-0.347889\pi\)
0.459890 + 0.887976i \(0.347889\pi\)
\(758\) 8.96923 0.325777
\(759\) 0 0
\(760\) 7.53752 0.273415
\(761\) 26.3449 0.955000 0.477500 0.878632i \(-0.341543\pi\)
0.477500 + 0.878632i \(0.341543\pi\)
\(762\) 0 0
\(763\) −3.88758 −0.140740
\(764\) 14.6733 0.530860
\(765\) 0 0
\(766\) −30.3280 −1.09580
\(767\) −6.20126 −0.223914
\(768\) 0 0
\(769\) −0.854669 −0.0308202 −0.0154101 0.999881i \(-0.504905\pi\)
−0.0154101 + 0.999881i \(0.504905\pi\)
\(770\) −0.174096 −0.00627398
\(771\) 0 0
\(772\) −13.3192 −0.479370
\(773\) −14.3339 −0.515555 −0.257778 0.966204i \(-0.582990\pi\)
−0.257778 + 0.966204i \(0.582990\pi\)
\(774\) 0 0
\(775\) −8.03378 −0.288582
\(776\) −48.0864 −1.72620
\(777\) 0 0
\(778\) 8.80543 0.315690
\(779\) 4.80661 0.172215
\(780\) 0 0
\(781\) −9.23954 −0.330617
\(782\) −0.888508 −0.0317730
\(783\) 0 0
\(784\) −3.69496 −0.131963
\(785\) −0.524202 −0.0187096
\(786\) 0 0
\(787\) 46.4807 1.65686 0.828430 0.560093i \(-0.189235\pi\)
0.828430 + 0.560093i \(0.189235\pi\)
\(788\) 2.14129 0.0762804
\(789\) 0 0
\(790\) −6.35985 −0.226273
\(791\) −0.576766 −0.0205074
\(792\) 0 0
\(793\) −0.801491 −0.0284618
\(794\) 26.7371 0.948864
\(795\) 0 0
\(796\) −34.5447 −1.22440
\(797\) 13.5176 0.478817 0.239409 0.970919i \(-0.423046\pi\)
0.239409 + 0.970919i \(0.423046\pi\)
\(798\) 0 0
\(799\) 0.258082 0.00913030
\(800\) −22.5162 −0.796068
\(801\) 0 0
\(802\) 15.1007 0.533226
\(803\) −7.49398 −0.264457
\(804\) 0 0
\(805\) −0.633797 −0.0223384
\(806\) 1.34105 0.0472364
\(807\) 0 0
\(808\) −12.1674 −0.428046
\(809\) −44.3814 −1.56037 −0.780183 0.625551i \(-0.784874\pi\)
−0.780183 + 0.625551i \(0.784874\pi\)
\(810\) 0 0
\(811\) −2.02464 −0.0710946 −0.0355473 0.999368i \(-0.511317\pi\)
−0.0355473 + 0.999368i \(0.511317\pi\)
\(812\) −0.248958 −0.00873672
\(813\) 0 0
\(814\) 1.72950 0.0606189
\(815\) 2.06330 0.0722743
\(816\) 0 0
\(817\) −5.90810 −0.206698
\(818\) 4.80954 0.168162
\(819\) 0 0
\(820\) 2.63866 0.0921460
\(821\) −26.4603 −0.923472 −0.461736 0.887017i \(-0.652773\pi\)
−0.461736 + 0.887017i \(0.652773\pi\)
\(822\) 0 0
\(823\) −40.6265 −1.41615 −0.708075 0.706137i \(-0.750436\pi\)
−0.708075 + 0.706137i \(0.750436\pi\)
\(824\) −53.6513 −1.86903
\(825\) 0 0
\(826\) 1.26475 0.0440064
\(827\) 39.7573 1.38250 0.691248 0.722617i \(-0.257061\pi\)
0.691248 + 0.722617i \(0.257061\pi\)
\(828\) 0 0
\(829\) −8.76395 −0.304385 −0.152192 0.988351i \(-0.548633\pi\)
−0.152192 + 0.988351i \(0.548633\pi\)
\(830\) −7.33251 −0.254515
\(831\) 0 0
\(832\) 2.90739 0.100796
\(833\) −2.62015 −0.0907829
\(834\) 0 0
\(835\) 15.7985 0.546731
\(836\) 3.54525 0.122615
\(837\) 0 0
\(838\) 17.5011 0.604565
\(839\) 0.750852 0.0259223 0.0129611 0.999916i \(-0.495874\pi\)
0.0129611 + 0.999916i \(0.495874\pi\)
\(840\) 0 0
\(841\) −28.1772 −0.971628
\(842\) −13.0570 −0.449973
\(843\) 0 0
\(844\) −21.8863 −0.753357
\(845\) 13.1613 0.452762
\(846\) 0 0
\(847\) −0.203035 −0.00697637
\(848\) −7.45779 −0.256102
\(849\) 0 0
\(850\) −1.17186 −0.0401944
\(851\) 6.29624 0.215832
\(852\) 0 0
\(853\) 27.3127 0.935168 0.467584 0.883949i \(-0.345124\pi\)
0.467584 + 0.883949i \(0.345124\pi\)
\(854\) 0.163465 0.00559366
\(855\) 0 0
\(856\) 17.7798 0.607699
\(857\) 8.31434 0.284012 0.142006 0.989866i \(-0.454645\pi\)
0.142006 + 0.989866i \(0.454645\pi\)
\(858\) 0 0
\(859\) 27.0139 0.921703 0.460852 0.887477i \(-0.347544\pi\)
0.460852 + 0.887477i \(0.347544\pi\)
\(860\) −3.24334 −0.110597
\(861\) 0 0
\(862\) 25.0192 0.852157
\(863\) 7.54619 0.256875 0.128438 0.991718i \(-0.459004\pi\)
0.128438 + 0.991718i \(0.459004\pi\)
\(864\) 0 0
\(865\) −11.0613 −0.376096
\(866\) 30.1615 1.02493
\(867\) 0 0
\(868\) 0.570397 0.0193605
\(869\) −7.41703 −0.251605
\(870\) 0 0
\(871\) −6.08676 −0.206242
\(872\) −51.6702 −1.74977
\(873\) 0 0
\(874\) −6.18872 −0.209337
\(875\) −1.91712 −0.0648104
\(876\) 0 0
\(877\) −40.4243 −1.36503 −0.682515 0.730871i \(-0.739114\pi\)
−0.682515 + 0.730871i \(0.739114\pi\)
\(878\) 12.1255 0.409215
\(879\) 0 0
\(880\) 0.565511 0.0190634
\(881\) 24.8846 0.838383 0.419191 0.907898i \(-0.362314\pi\)
0.419191 + 0.907898i \(0.362314\pi\)
\(882\) 0 0
\(883\) 43.2827 1.45658 0.728289 0.685270i \(-0.240316\pi\)
0.728289 + 0.685270i \(0.240316\pi\)
\(884\) −0.407949 −0.0137208
\(885\) 0 0
\(886\) −6.04154 −0.202970
\(887\) −10.2272 −0.343396 −0.171698 0.985150i \(-0.554925\pi\)
−0.171698 + 0.985150i \(0.554925\pi\)
\(888\) 0 0
\(889\) −1.01588 −0.0340716
\(890\) −2.55396 −0.0856088
\(891\) 0 0
\(892\) 6.66667 0.223217
\(893\) 1.79762 0.0601550
\(894\) 0 0
\(895\) 18.0043 0.601817
\(896\) 1.77224 0.0592063
\(897\) 0 0
\(898\) −10.9030 −0.363839
\(899\) −1.88510 −0.0628715
\(900\) 0 0
\(901\) −5.28843 −0.176183
\(902\) −1.47557 −0.0491311
\(903\) 0 0
\(904\) −7.66584 −0.254962
\(905\) −3.10683 −0.103275
\(906\) 0 0
\(907\) −34.3774 −1.14148 −0.570741 0.821130i \(-0.693344\pi\)
−0.570741 + 0.821130i \(0.693344\pi\)
\(908\) −3.50728 −0.116393
\(909\) 0 0
\(910\) 0.139536 0.00462559
\(911\) 20.2548 0.671072 0.335536 0.942027i \(-0.391083\pi\)
0.335536 + 0.942027i \(0.391083\pi\)
\(912\) 0 0
\(913\) −8.55136 −0.283009
\(914\) 8.19921 0.271206
\(915\) 0 0
\(916\) 9.13027 0.301673
\(917\) −3.86885 −0.127761
\(918\) 0 0
\(919\) −26.9266 −0.888227 −0.444114 0.895970i \(-0.646481\pi\)
−0.444114 + 0.895970i \(0.646481\pi\)
\(920\) −8.42384 −0.277726
\(921\) 0 0
\(922\) 11.3237 0.372927
\(923\) 7.40541 0.243752
\(924\) 0 0
\(925\) 8.30415 0.273039
\(926\) −21.9866 −0.722526
\(927\) 0 0
\(928\) −5.28334 −0.173434
\(929\) −13.1495 −0.431421 −0.215711 0.976457i \(-0.569207\pi\)
−0.215711 + 0.976457i \(0.569207\pi\)
\(930\) 0 0
\(931\) −18.2501 −0.598124
\(932\) −37.6452 −1.23311
\(933\) 0 0
\(934\) −13.0293 −0.426332
\(935\) 0.401012 0.0131145
\(936\) 0 0
\(937\) 19.9881 0.652981 0.326491 0.945200i \(-0.394134\pi\)
0.326491 + 0.945200i \(0.394134\pi\)
\(938\) 1.24140 0.0405332
\(939\) 0 0
\(940\) 0.986829 0.0321868
\(941\) −24.3479 −0.793720 −0.396860 0.917879i \(-0.629900\pi\)
−0.396860 + 0.917879i \(0.629900\pi\)
\(942\) 0 0
\(943\) −5.37181 −0.174930
\(944\) −4.10826 −0.133713
\(945\) 0 0
\(946\) 1.81371 0.0589688
\(947\) 33.4192 1.08598 0.542989 0.839740i \(-0.317293\pi\)
0.542989 + 0.839740i \(0.317293\pi\)
\(948\) 0 0
\(949\) 6.00636 0.194975
\(950\) −8.16234 −0.264821
\(951\) 0 0
\(952\) 0.206299 0.00668619
\(953\) 1.05530 0.0341847 0.0170923 0.999854i \(-0.494559\pi\)
0.0170923 + 0.999854i \(0.494559\pi\)
\(954\) 0 0
\(955\) 11.5605 0.374089
\(956\) 22.8728 0.739761
\(957\) 0 0
\(958\) 15.7638 0.509305
\(959\) 3.74644 0.120979
\(960\) 0 0
\(961\) −26.6810 −0.860677
\(962\) −1.38618 −0.0446922
\(963\) 0 0
\(964\) 1.43907 0.0463492
\(965\) −10.4937 −0.337805
\(966\) 0 0
\(967\) 15.5418 0.499791 0.249895 0.968273i \(-0.419604\pi\)
0.249895 + 0.968273i \(0.419604\pi\)
\(968\) −2.69856 −0.0867350
\(969\) 0 0
\(970\) −15.2795 −0.490594
\(971\) 27.9120 0.895738 0.447869 0.894099i \(-0.352183\pi\)
0.447869 + 0.894099i \(0.352183\pi\)
\(972\) 0 0
\(973\) −2.96759 −0.0951365
\(974\) −20.6797 −0.662622
\(975\) 0 0
\(976\) −0.530979 −0.0169962
\(977\) −10.2968 −0.329423 −0.164712 0.986342i \(-0.552669\pi\)
−0.164712 + 0.986342i \(0.552669\pi\)
\(978\) 0 0
\(979\) −2.97849 −0.0951929
\(980\) −10.0187 −0.320035
\(981\) 0 0
\(982\) −24.1730 −0.771393
\(983\) 53.4728 1.70552 0.852759 0.522304i \(-0.174927\pi\)
0.852759 + 0.522304i \(0.174927\pi\)
\(984\) 0 0
\(985\) 1.68704 0.0537537
\(986\) −0.274972 −0.00875689
\(987\) 0 0
\(988\) −2.84148 −0.0903996
\(989\) 6.60282 0.209957
\(990\) 0 0
\(991\) −22.7926 −0.724031 −0.362015 0.932172i \(-0.617911\pi\)
−0.362015 + 0.932172i \(0.617911\pi\)
\(992\) 12.1048 0.384329
\(993\) 0 0
\(994\) −1.51034 −0.0479051
\(995\) −27.2165 −0.862820
\(996\) 0 0
\(997\) 53.2933 1.68782 0.843908 0.536488i \(-0.180249\pi\)
0.843908 + 0.536488i \(0.180249\pi\)
\(998\) 13.4186 0.424757
\(999\) 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 6039.2.a.i.1.8 13
3.2 odd 2 2013.2.a.e.1.6 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.6 13 3.2 odd 2
6039.2.a.i.1.8 13 1.1 even 1 trivial