Properties

Label 585.2.a.m.1.1
Level $585$
Weight $2$
Character 585.1
Self dual yes
Analytic conductor $4.671$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 585.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.414214 q^{2} -1.82843 q^{4} -1.00000 q^{5} -0.828427 q^{7} +1.58579 q^{8} +O(q^{10})\) \(q-0.414214 q^{2} -1.82843 q^{4} -1.00000 q^{5} -0.828427 q^{7} +1.58579 q^{8} +0.414214 q^{10} -0.585786 q^{11} -1.00000 q^{13} +0.343146 q^{14} +3.00000 q^{16} +4.82843 q^{17} +3.41421 q^{19} +1.82843 q^{20} +0.242641 q^{22} +1.41421 q^{23} +1.00000 q^{25} +0.414214 q^{26} +1.51472 q^{28} -5.65685 q^{29} +10.2426 q^{31} -4.41421 q^{32} -2.00000 q^{34} +0.828427 q^{35} +8.48528 q^{37} -1.41421 q^{38} -1.58579 q^{40} +8.82843 q^{41} +3.07107 q^{43} +1.07107 q^{44} -0.585786 q^{46} -0.828427 q^{47} -6.31371 q^{49} -0.414214 q^{50} +1.82843 q^{52} +14.4853 q^{53} +0.585786 q^{55} -1.31371 q^{56} +2.34315 q^{58} -10.2426 q^{59} -8.00000 q^{61} -4.24264 q^{62} -4.17157 q^{64} +1.00000 q^{65} -2.00000 q^{67} -8.82843 q^{68} -0.343146 q^{70} +7.89949 q^{71} -8.48528 q^{73} -3.51472 q^{74} -6.24264 q^{76} +0.485281 q^{77} +8.48528 q^{79} -3.00000 q^{80} -3.65685 q^{82} +8.82843 q^{83} -4.82843 q^{85} -1.27208 q^{86} -0.928932 q^{88} -6.00000 q^{89} +0.828427 q^{91} -2.58579 q^{92} +0.343146 q^{94} -3.41421 q^{95} +3.65685 q^{97} +2.61522 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8} - 2 q^{10} - 4 q^{11} - 2 q^{13} + 12 q^{14} + 6 q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} - 8 q^{22} + 2 q^{25} - 2 q^{26} + 20 q^{28} + 12 q^{31} - 6 q^{32} - 4 q^{34} - 4 q^{35} - 6 q^{40} + 12 q^{41} - 8 q^{43} - 12 q^{44} - 4 q^{46} + 4 q^{47} + 10 q^{49} + 2 q^{50} - 2 q^{52} + 12 q^{53} + 4 q^{55} + 20 q^{56} + 16 q^{58} - 12 q^{59} - 16 q^{61} - 14 q^{64} + 2 q^{65} - 4 q^{67} - 12 q^{68} - 12 q^{70} - 4 q^{71} - 24 q^{74} - 4 q^{76} - 16 q^{77} - 6 q^{80} + 4 q^{82} + 12 q^{83} - 4 q^{85} - 28 q^{86} - 16 q^{88} - 12 q^{89} - 4 q^{91} - 8 q^{92} + 12 q^{94} - 4 q^{95} - 4 q^{97} + 42 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.414214 −0.292893 −0.146447 0.989219i \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) 0 0
\(4\) −1.82843 −0.914214
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.828427 −0.313116 −0.156558 0.987669i \(-0.550040\pi\)
−0.156558 + 0.987669i \(0.550040\pi\)
\(8\) 1.58579 0.560660
\(9\) 0 0
\(10\) 0.414214 0.130986
\(11\) −0.585786 −0.176621 −0.0883106 0.996093i \(-0.528147\pi\)
−0.0883106 + 0.996093i \(0.528147\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0.343146 0.0917096
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 4.82843 1.17107 0.585533 0.810649i \(-0.300885\pi\)
0.585533 + 0.810649i \(0.300885\pi\)
\(18\) 0 0
\(19\) 3.41421 0.783274 0.391637 0.920120i \(-0.371909\pi\)
0.391637 + 0.920120i \(0.371909\pi\)
\(20\) 1.82843 0.408849
\(21\) 0 0
\(22\) 0.242641 0.0517312
\(23\) 1.41421 0.294884 0.147442 0.989071i \(-0.452896\pi\)
0.147442 + 0.989071i \(0.452896\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.414214 0.0812340
\(27\) 0 0
\(28\) 1.51472 0.286255
\(29\) −5.65685 −1.05045 −0.525226 0.850963i \(-0.676019\pi\)
−0.525226 + 0.850963i \(0.676019\pi\)
\(30\) 0 0
\(31\) 10.2426 1.83963 0.919816 0.392349i \(-0.128338\pi\)
0.919816 + 0.392349i \(0.128338\pi\)
\(32\) −4.41421 −0.780330
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0.828427 0.140030
\(36\) 0 0
\(37\) 8.48528 1.39497 0.697486 0.716599i \(-0.254302\pi\)
0.697486 + 0.716599i \(0.254302\pi\)
\(38\) −1.41421 −0.229416
\(39\) 0 0
\(40\) −1.58579 −0.250735
\(41\) 8.82843 1.37877 0.689384 0.724396i \(-0.257881\pi\)
0.689384 + 0.724396i \(0.257881\pi\)
\(42\) 0 0
\(43\) 3.07107 0.468333 0.234167 0.972196i \(-0.424764\pi\)
0.234167 + 0.972196i \(0.424764\pi\)
\(44\) 1.07107 0.161470
\(45\) 0 0
\(46\) −0.585786 −0.0863695
\(47\) −0.828427 −0.120839 −0.0604193 0.998173i \(-0.519244\pi\)
−0.0604193 + 0.998173i \(0.519244\pi\)
\(48\) 0 0
\(49\) −6.31371 −0.901958
\(50\) −0.414214 −0.0585786
\(51\) 0 0
\(52\) 1.82843 0.253557
\(53\) 14.4853 1.98971 0.994853 0.101327i \(-0.0323087\pi\)
0.994853 + 0.101327i \(0.0323087\pi\)
\(54\) 0 0
\(55\) 0.585786 0.0789874
\(56\) −1.31371 −0.175552
\(57\) 0 0
\(58\) 2.34315 0.307670
\(59\) −10.2426 −1.33348 −0.666739 0.745291i \(-0.732310\pi\)
−0.666739 + 0.745291i \(0.732310\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −4.24264 −0.538816
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −8.82843 −1.07060
\(69\) 0 0
\(70\) −0.343146 −0.0410138
\(71\) 7.89949 0.937498 0.468749 0.883332i \(-0.344705\pi\)
0.468749 + 0.883332i \(0.344705\pi\)
\(72\) 0 0
\(73\) −8.48528 −0.993127 −0.496564 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) −3.51472 −0.408578
\(75\) 0 0
\(76\) −6.24264 −0.716080
\(77\) 0.485281 0.0553029
\(78\) 0 0
\(79\) 8.48528 0.954669 0.477334 0.878722i \(-0.341603\pi\)
0.477334 + 0.878722i \(0.341603\pi\)
\(80\) −3.00000 −0.335410
\(81\) 0 0
\(82\) −3.65685 −0.403832
\(83\) 8.82843 0.969046 0.484523 0.874779i \(-0.338993\pi\)
0.484523 + 0.874779i \(0.338993\pi\)
\(84\) 0 0
\(85\) −4.82843 −0.523716
\(86\) −1.27208 −0.137172
\(87\) 0 0
\(88\) −0.928932 −0.0990245
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0.828427 0.0868428
\(92\) −2.58579 −0.269587
\(93\) 0 0
\(94\) 0.343146 0.0353928
\(95\) −3.41421 −0.350291
\(96\) 0 0
\(97\) 3.65685 0.371297 0.185649 0.982616i \(-0.440561\pi\)
0.185649 + 0.982616i \(0.440561\pi\)
\(98\) 2.61522 0.264177
\(99\) 0 0
\(100\) −1.82843 −0.182843
\(101\) −7.65685 −0.761885 −0.380943 0.924599i \(-0.624401\pi\)
−0.380943 + 0.924599i \(0.624401\pi\)
\(102\) 0 0
\(103\) 17.4142 1.71587 0.857937 0.513755i \(-0.171746\pi\)
0.857937 + 0.513755i \(0.171746\pi\)
\(104\) −1.58579 −0.155499
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 6.58579 0.636672 0.318336 0.947978i \(-0.396876\pi\)
0.318336 + 0.947978i \(0.396876\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −0.242641 −0.0231349
\(111\) 0 0
\(112\) −2.48528 −0.234837
\(113\) 3.17157 0.298356 0.149178 0.988810i \(-0.452337\pi\)
0.149178 + 0.988810i \(0.452337\pi\)
\(114\) 0 0
\(115\) −1.41421 −0.131876
\(116\) 10.3431 0.960337
\(117\) 0 0
\(118\) 4.24264 0.390567
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −10.6569 −0.968805
\(122\) 3.31371 0.300009
\(123\) 0 0
\(124\) −18.7279 −1.68182
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −9.41421 −0.835376 −0.417688 0.908590i \(-0.637160\pi\)
−0.417688 + 0.908590i \(0.637160\pi\)
\(128\) 10.5563 0.933058
\(129\) 0 0
\(130\) −0.414214 −0.0363289
\(131\) −16.9706 −1.48272 −0.741362 0.671105i \(-0.765820\pi\)
−0.741362 + 0.671105i \(0.765820\pi\)
\(132\) 0 0
\(133\) −2.82843 −0.245256
\(134\) 0.828427 0.0715652
\(135\) 0 0
\(136\) 7.65685 0.656570
\(137\) −5.31371 −0.453981 −0.226990 0.973897i \(-0.572889\pi\)
−0.226990 + 0.973897i \(0.572889\pi\)
\(138\) 0 0
\(139\) −12.4853 −1.05899 −0.529494 0.848314i \(-0.677618\pi\)
−0.529494 + 0.848314i \(0.677618\pi\)
\(140\) −1.51472 −0.128017
\(141\) 0 0
\(142\) −3.27208 −0.274587
\(143\) 0.585786 0.0489859
\(144\) 0 0
\(145\) 5.65685 0.469776
\(146\) 3.51472 0.290880
\(147\) 0 0
\(148\) −15.5147 −1.27530
\(149\) 0.343146 0.0281116 0.0140558 0.999901i \(-0.495526\pi\)
0.0140558 + 0.999901i \(0.495526\pi\)
\(150\) 0 0
\(151\) 18.2426 1.48457 0.742283 0.670087i \(-0.233743\pi\)
0.742283 + 0.670087i \(0.233743\pi\)
\(152\) 5.41421 0.439151
\(153\) 0 0
\(154\) −0.201010 −0.0161979
\(155\) −10.2426 −0.822709
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −3.51472 −0.279616
\(159\) 0 0
\(160\) 4.41421 0.348974
\(161\) −1.17157 −0.0923329
\(162\) 0 0
\(163\) −14.9706 −1.17258 −0.586292 0.810099i \(-0.699413\pi\)
−0.586292 + 0.810099i \(0.699413\pi\)
\(164\) −16.1421 −1.26049
\(165\) 0 0
\(166\) −3.65685 −0.283827
\(167\) 8.82843 0.683164 0.341582 0.939852i \(-0.389037\pi\)
0.341582 + 0.939852i \(0.389037\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) −5.61522 −0.428157
\(173\) −11.1716 −0.849359 −0.424679 0.905344i \(-0.639613\pi\)
−0.424679 + 0.905344i \(0.639613\pi\)
\(174\) 0 0
\(175\) −0.828427 −0.0626232
\(176\) −1.75736 −0.132466
\(177\) 0 0
\(178\) 2.48528 0.186280
\(179\) 5.65685 0.422813 0.211407 0.977398i \(-0.432196\pi\)
0.211407 + 0.977398i \(0.432196\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) −0.343146 −0.0254357
\(183\) 0 0
\(184\) 2.24264 0.165330
\(185\) −8.48528 −0.623850
\(186\) 0 0
\(187\) −2.82843 −0.206835
\(188\) 1.51472 0.110472
\(189\) 0 0
\(190\) 1.41421 0.102598
\(191\) −13.6569 −0.988175 −0.494088 0.869412i \(-0.664498\pi\)
−0.494088 + 0.869412i \(0.664498\pi\)
\(192\) 0 0
\(193\) 15.6569 1.12701 0.563503 0.826114i \(-0.309454\pi\)
0.563503 + 0.826114i \(0.309454\pi\)
\(194\) −1.51472 −0.108750
\(195\) 0 0
\(196\) 11.5442 0.824583
\(197\) 22.9706 1.63658 0.818292 0.574802i \(-0.194921\pi\)
0.818292 + 0.574802i \(0.194921\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 1.58579 0.112132
\(201\) 0 0
\(202\) 3.17157 0.223151
\(203\) 4.68629 0.328913
\(204\) 0 0
\(205\) −8.82843 −0.616604
\(206\) −7.21320 −0.502568
\(207\) 0 0
\(208\) −3.00000 −0.208013
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −19.3137 −1.32961 −0.664805 0.747017i \(-0.731485\pi\)
−0.664805 + 0.747017i \(0.731485\pi\)
\(212\) −26.4853 −1.81902
\(213\) 0 0
\(214\) −2.72792 −0.186477
\(215\) −3.07107 −0.209445
\(216\) 0 0
\(217\) −8.48528 −0.576018
\(218\) 0.828427 0.0561082
\(219\) 0 0
\(220\) −1.07107 −0.0722114
\(221\) −4.82843 −0.324795
\(222\) 0 0
\(223\) 26.4853 1.77359 0.886793 0.462167i \(-0.152928\pi\)
0.886793 + 0.462167i \(0.152928\pi\)
\(224\) 3.65685 0.244334
\(225\) 0 0
\(226\) −1.31371 −0.0873866
\(227\) 27.6569 1.83565 0.917825 0.396985i \(-0.129944\pi\)
0.917825 + 0.396985i \(0.129944\pi\)
\(228\) 0 0
\(229\) 0.828427 0.0547440 0.0273720 0.999625i \(-0.491286\pi\)
0.0273720 + 0.999625i \(0.491286\pi\)
\(230\) 0.585786 0.0386256
\(231\) 0 0
\(232\) −8.97056 −0.588946
\(233\) −24.6274 −1.61340 −0.806698 0.590964i \(-0.798747\pi\)
−0.806698 + 0.590964i \(0.798747\pi\)
\(234\) 0 0
\(235\) 0.828427 0.0540406
\(236\) 18.7279 1.21908
\(237\) 0 0
\(238\) 1.65685 0.107398
\(239\) 0.585786 0.0378914 0.0189457 0.999821i \(-0.493969\pi\)
0.0189457 + 0.999821i \(0.493969\pi\)
\(240\) 0 0
\(241\) 2.48528 0.160091 0.0800455 0.996791i \(-0.474493\pi\)
0.0800455 + 0.996791i \(0.474493\pi\)
\(242\) 4.41421 0.283756
\(243\) 0 0
\(244\) 14.6274 0.936424
\(245\) 6.31371 0.403368
\(246\) 0 0
\(247\) −3.41421 −0.217241
\(248\) 16.2426 1.03141
\(249\) 0 0
\(250\) 0.414214 0.0261972
\(251\) 19.7990 1.24970 0.624851 0.780744i \(-0.285160\pi\)
0.624851 + 0.780744i \(0.285160\pi\)
\(252\) 0 0
\(253\) −0.828427 −0.0520828
\(254\) 3.89949 0.244676
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) −16.3431 −1.01946 −0.509729 0.860335i \(-0.670254\pi\)
−0.509729 + 0.860335i \(0.670254\pi\)
\(258\) 0 0
\(259\) −7.02944 −0.436788
\(260\) −1.82843 −0.113394
\(261\) 0 0
\(262\) 7.02944 0.434280
\(263\) 13.4142 0.827156 0.413578 0.910469i \(-0.364279\pi\)
0.413578 + 0.910469i \(0.364279\pi\)
\(264\) 0 0
\(265\) −14.4853 −0.889824
\(266\) 1.17157 0.0718337
\(267\) 0 0
\(268\) 3.65685 0.223378
\(269\) 2.68629 0.163786 0.0818930 0.996641i \(-0.473903\pi\)
0.0818930 + 0.996641i \(0.473903\pi\)
\(270\) 0 0
\(271\) 1.27208 0.0772732 0.0386366 0.999253i \(-0.487699\pi\)
0.0386366 + 0.999253i \(0.487699\pi\)
\(272\) 14.4853 0.878299
\(273\) 0 0
\(274\) 2.20101 0.132968
\(275\) −0.585786 −0.0353243
\(276\) 0 0
\(277\) −7.17157 −0.430898 −0.215449 0.976515i \(-0.569122\pi\)
−0.215449 + 0.976515i \(0.569122\pi\)
\(278\) 5.17157 0.310170
\(279\) 0 0
\(280\) 1.31371 0.0785091
\(281\) 17.7990 1.06180 0.530899 0.847435i \(-0.321854\pi\)
0.530899 + 0.847435i \(0.321854\pi\)
\(282\) 0 0
\(283\) 8.72792 0.518821 0.259411 0.965767i \(-0.416472\pi\)
0.259411 + 0.965767i \(0.416472\pi\)
\(284\) −14.4437 −0.857073
\(285\) 0 0
\(286\) −0.242641 −0.0143476
\(287\) −7.31371 −0.431715
\(288\) 0 0
\(289\) 6.31371 0.371395
\(290\) −2.34315 −0.137594
\(291\) 0 0
\(292\) 15.5147 0.907930
\(293\) 2.14214 0.125145 0.0625724 0.998040i \(-0.480070\pi\)
0.0625724 + 0.998040i \(0.480070\pi\)
\(294\) 0 0
\(295\) 10.2426 0.596350
\(296\) 13.4558 0.782105
\(297\) 0 0
\(298\) −0.142136 −0.00823370
\(299\) −1.41421 −0.0817861
\(300\) 0 0
\(301\) −2.54416 −0.146643
\(302\) −7.55635 −0.434819
\(303\) 0 0
\(304\) 10.2426 0.587456
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) 19.1716 1.09418 0.547090 0.837074i \(-0.315736\pi\)
0.547090 + 0.837074i \(0.315736\pi\)
\(308\) −0.887302 −0.0505587
\(309\) 0 0
\(310\) 4.24264 0.240966
\(311\) 8.48528 0.481156 0.240578 0.970630i \(-0.422663\pi\)
0.240578 + 0.970630i \(0.422663\pi\)
\(312\) 0 0
\(313\) 0.828427 0.0468255 0.0234127 0.999726i \(-0.492547\pi\)
0.0234127 + 0.999726i \(0.492547\pi\)
\(314\) −7.45584 −0.420758
\(315\) 0 0
\(316\) −15.5147 −0.872771
\(317\) −26.1421 −1.46829 −0.734144 0.678993i \(-0.762416\pi\)
−0.734144 + 0.678993i \(0.762416\pi\)
\(318\) 0 0
\(319\) 3.31371 0.185532
\(320\) 4.17157 0.233198
\(321\) 0 0
\(322\) 0.485281 0.0270437
\(323\) 16.4853 0.917266
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 6.20101 0.343442
\(327\) 0 0
\(328\) 14.0000 0.773021
\(329\) 0.686292 0.0378365
\(330\) 0 0
\(331\) 22.0416 1.21152 0.605759 0.795648i \(-0.292870\pi\)
0.605759 + 0.795648i \(0.292870\pi\)
\(332\) −16.1421 −0.885915
\(333\) 0 0
\(334\) −3.65685 −0.200094
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) 7.17157 0.390660 0.195330 0.980738i \(-0.437422\pi\)
0.195330 + 0.980738i \(0.437422\pi\)
\(338\) −0.414214 −0.0225302
\(339\) 0 0
\(340\) 8.82843 0.478789
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 11.0294 0.595534
\(344\) 4.87006 0.262576
\(345\) 0 0
\(346\) 4.62742 0.248771
\(347\) −4.24264 −0.227757 −0.113878 0.993495i \(-0.536327\pi\)
−0.113878 + 0.993495i \(0.536327\pi\)
\(348\) 0 0
\(349\) 1.51472 0.0810810 0.0405405 0.999178i \(-0.487092\pi\)
0.0405405 + 0.999178i \(0.487092\pi\)
\(350\) 0.343146 0.0183419
\(351\) 0 0
\(352\) 2.58579 0.137823
\(353\) −9.17157 −0.488154 −0.244077 0.969756i \(-0.578485\pi\)
−0.244077 + 0.969756i \(0.578485\pi\)
\(354\) 0 0
\(355\) −7.89949 −0.419262
\(356\) 10.9706 0.581439
\(357\) 0 0
\(358\) −2.34315 −0.123839
\(359\) 27.8995 1.47248 0.736240 0.676721i \(-0.236600\pi\)
0.736240 + 0.676721i \(0.236600\pi\)
\(360\) 0 0
\(361\) −7.34315 −0.386481
\(362\) 0 0
\(363\) 0 0
\(364\) −1.51472 −0.0793928
\(365\) 8.48528 0.444140
\(366\) 0 0
\(367\) 4.44365 0.231957 0.115978 0.993252i \(-0.463000\pi\)
0.115978 + 0.993252i \(0.463000\pi\)
\(368\) 4.24264 0.221163
\(369\) 0 0
\(370\) 3.51472 0.182722
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −25.3137 −1.31069 −0.655347 0.755328i \(-0.727478\pi\)
−0.655347 + 0.755328i \(0.727478\pi\)
\(374\) 1.17157 0.0605806
\(375\) 0 0
\(376\) −1.31371 −0.0677493
\(377\) 5.65685 0.291343
\(378\) 0 0
\(379\) 14.9289 0.766848 0.383424 0.923572i \(-0.374745\pi\)
0.383424 + 0.923572i \(0.374745\pi\)
\(380\) 6.24264 0.320241
\(381\) 0 0
\(382\) 5.65685 0.289430
\(383\) −33.1127 −1.69198 −0.845990 0.533199i \(-0.820990\pi\)
−0.845990 + 0.533199i \(0.820990\pi\)
\(384\) 0 0
\(385\) −0.485281 −0.0247322
\(386\) −6.48528 −0.330092
\(387\) 0 0
\(388\) −6.68629 −0.339445
\(389\) 16.6274 0.843044 0.421522 0.906818i \(-0.361496\pi\)
0.421522 + 0.906818i \(0.361496\pi\)
\(390\) 0 0
\(391\) 6.82843 0.345328
\(392\) −10.0122 −0.505692
\(393\) 0 0
\(394\) −9.51472 −0.479345
\(395\) −8.48528 −0.426941
\(396\) 0 0
\(397\) −27.7990 −1.39519 −0.697596 0.716492i \(-0.745747\pi\)
−0.697596 + 0.716492i \(0.745747\pi\)
\(398\) −1.65685 −0.0830506
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) −17.3137 −0.864605 −0.432303 0.901729i \(-0.642299\pi\)
−0.432303 + 0.901729i \(0.642299\pi\)
\(402\) 0 0
\(403\) −10.2426 −0.510222
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) −1.94113 −0.0963364
\(407\) −4.97056 −0.246382
\(408\) 0 0
\(409\) 12.8284 0.634325 0.317162 0.948371i \(-0.397270\pi\)
0.317162 + 0.948371i \(0.397270\pi\)
\(410\) 3.65685 0.180599
\(411\) 0 0
\(412\) −31.8406 −1.56867
\(413\) 8.48528 0.417533
\(414\) 0 0
\(415\) −8.82843 −0.433370
\(416\) 4.41421 0.216425
\(417\) 0 0
\(418\) 0.828427 0.0405197
\(419\) −5.17157 −0.252648 −0.126324 0.991989i \(-0.540318\pi\)
−0.126324 + 0.991989i \(0.540318\pi\)
\(420\) 0 0
\(421\) −1.02944 −0.0501717 −0.0250859 0.999685i \(-0.507986\pi\)
−0.0250859 + 0.999685i \(0.507986\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) 22.9706 1.11555
\(425\) 4.82843 0.234213
\(426\) 0 0
\(427\) 6.62742 0.320723
\(428\) −12.0416 −0.582054
\(429\) 0 0
\(430\) 1.27208 0.0613450
\(431\) −3.61522 −0.174139 −0.0870696 0.996202i \(-0.527750\pi\)
−0.0870696 + 0.996202i \(0.527750\pi\)
\(432\) 0 0
\(433\) −3.65685 −0.175737 −0.0878686 0.996132i \(-0.528006\pi\)
−0.0878686 + 0.996132i \(0.528006\pi\)
\(434\) 3.51472 0.168712
\(435\) 0 0
\(436\) 3.65685 0.175132
\(437\) 4.82843 0.230975
\(438\) 0 0
\(439\) −32.9706 −1.57360 −0.786800 0.617209i \(-0.788263\pi\)
−0.786800 + 0.617209i \(0.788263\pi\)
\(440\) 0.928932 0.0442851
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) 6.58579 0.312900 0.156450 0.987686i \(-0.449995\pi\)
0.156450 + 0.987686i \(0.449995\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) −10.9706 −0.519471
\(447\) 0 0
\(448\) 3.45584 0.163273
\(449\) −29.1127 −1.37391 −0.686957 0.726698i \(-0.741054\pi\)
−0.686957 + 0.726698i \(0.741054\pi\)
\(450\) 0 0
\(451\) −5.17157 −0.243520
\(452\) −5.79899 −0.272762
\(453\) 0 0
\(454\) −11.4558 −0.537649
\(455\) −0.828427 −0.0388373
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) −0.343146 −0.0160341
\(459\) 0 0
\(460\) 2.58579 0.120563
\(461\) −26.4853 −1.23354 −0.616771 0.787142i \(-0.711560\pi\)
−0.616771 + 0.787142i \(0.711560\pi\)
\(462\) 0 0
\(463\) −15.6569 −0.727636 −0.363818 0.931470i \(-0.618527\pi\)
−0.363818 + 0.931470i \(0.618527\pi\)
\(464\) −16.9706 −0.787839
\(465\) 0 0
\(466\) 10.2010 0.472553
\(467\) 10.5858 0.489852 0.244926 0.969542i \(-0.421236\pi\)
0.244926 + 0.969542i \(0.421236\pi\)
\(468\) 0 0
\(469\) 1.65685 0.0765064
\(470\) −0.343146 −0.0158281
\(471\) 0 0
\(472\) −16.2426 −0.747628
\(473\) −1.79899 −0.0827176
\(474\) 0 0
\(475\) 3.41421 0.156655
\(476\) 7.31371 0.335223
\(477\) 0 0
\(478\) −0.242641 −0.0110981
\(479\) 5.27208 0.240887 0.120444 0.992720i \(-0.461568\pi\)
0.120444 + 0.992720i \(0.461568\pi\)
\(480\) 0 0
\(481\) −8.48528 −0.386896
\(482\) −1.02944 −0.0468896
\(483\) 0 0
\(484\) 19.4853 0.885695
\(485\) −3.65685 −0.166049
\(486\) 0 0
\(487\) 22.9706 1.04090 0.520448 0.853894i \(-0.325765\pi\)
0.520448 + 0.853894i \(0.325765\pi\)
\(488\) −12.6863 −0.574281
\(489\) 0 0
\(490\) −2.61522 −0.118144
\(491\) −10.8284 −0.488680 −0.244340 0.969690i \(-0.578571\pi\)
−0.244340 + 0.969690i \(0.578571\pi\)
\(492\) 0 0
\(493\) −27.3137 −1.23015
\(494\) 1.41421 0.0636285
\(495\) 0 0
\(496\) 30.7279 1.37972
\(497\) −6.54416 −0.293546
\(498\) 0 0
\(499\) 10.4437 0.467522 0.233761 0.972294i \(-0.424897\pi\)
0.233761 + 0.972294i \(0.424897\pi\)
\(500\) 1.82843 0.0817697
\(501\) 0 0
\(502\) −8.20101 −0.366029
\(503\) −18.1005 −0.807062 −0.403531 0.914966i \(-0.632217\pi\)
−0.403531 + 0.914966i \(0.632217\pi\)
\(504\) 0 0
\(505\) 7.65685 0.340726
\(506\) 0.343146 0.0152547
\(507\) 0 0
\(508\) 17.2132 0.763712
\(509\) 21.1127 0.935804 0.467902 0.883780i \(-0.345010\pi\)
0.467902 + 0.883780i \(0.345010\pi\)
\(510\) 0 0
\(511\) 7.02944 0.310964
\(512\) −22.7574 −1.00574
\(513\) 0 0
\(514\) 6.76955 0.298592
\(515\) −17.4142 −0.767362
\(516\) 0 0
\(517\) 0.485281 0.0213427
\(518\) 2.91169 0.127932
\(519\) 0 0
\(520\) 1.58579 0.0695413
\(521\) 6.34315 0.277898 0.138949 0.990300i \(-0.455628\pi\)
0.138949 + 0.990300i \(0.455628\pi\)
\(522\) 0 0
\(523\) −28.2426 −1.23496 −0.617482 0.786585i \(-0.711847\pi\)
−0.617482 + 0.786585i \(0.711847\pi\)
\(524\) 31.0294 1.35553
\(525\) 0 0
\(526\) −5.55635 −0.242268
\(527\) 49.4558 2.15433
\(528\) 0 0
\(529\) −21.0000 −0.913043
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 5.17157 0.224216
\(533\) −8.82843 −0.382402
\(534\) 0 0
\(535\) −6.58579 −0.284728
\(536\) −3.17157 −0.136991
\(537\) 0 0
\(538\) −1.11270 −0.0479718
\(539\) 3.69848 0.159305
\(540\) 0 0
\(541\) −12.8284 −0.551537 −0.275769 0.961224i \(-0.588932\pi\)
−0.275769 + 0.961224i \(0.588932\pi\)
\(542\) −0.526912 −0.0226328
\(543\) 0 0
\(544\) −21.3137 −0.913818
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −29.2132 −1.24907 −0.624533 0.780998i \(-0.714711\pi\)
−0.624533 + 0.780998i \(0.714711\pi\)
\(548\) 9.71573 0.415035
\(549\) 0 0
\(550\) 0.242641 0.0103462
\(551\) −19.3137 −0.822792
\(552\) 0 0
\(553\) −7.02944 −0.298922
\(554\) 2.97056 0.126207
\(555\) 0 0
\(556\) 22.8284 0.968141
\(557\) −3.79899 −0.160968 −0.0804842 0.996756i \(-0.525647\pi\)
−0.0804842 + 0.996756i \(0.525647\pi\)
\(558\) 0 0
\(559\) −3.07107 −0.129892
\(560\) 2.48528 0.105022
\(561\) 0 0
\(562\) −7.37258 −0.310994
\(563\) 16.2426 0.684546 0.342273 0.939601i \(-0.388803\pi\)
0.342273 + 0.939601i \(0.388803\pi\)
\(564\) 0 0
\(565\) −3.17157 −0.133429
\(566\) −3.61522 −0.151959
\(567\) 0 0
\(568\) 12.5269 0.525618
\(569\) 21.6569 0.907903 0.453951 0.891027i \(-0.350014\pi\)
0.453951 + 0.891027i \(0.350014\pi\)
\(570\) 0 0
\(571\) −28.4853 −1.19207 −0.596036 0.802958i \(-0.703258\pi\)
−0.596036 + 0.802958i \(0.703258\pi\)
\(572\) −1.07107 −0.0447836
\(573\) 0 0
\(574\) 3.02944 0.126446
\(575\) 1.41421 0.0589768
\(576\) 0 0
\(577\) −29.1716 −1.21443 −0.607214 0.794538i \(-0.707713\pi\)
−0.607214 + 0.794538i \(0.707713\pi\)
\(578\) −2.61522 −0.108779
\(579\) 0 0
\(580\) −10.3431 −0.429476
\(581\) −7.31371 −0.303424
\(582\) 0 0
\(583\) −8.48528 −0.351424
\(584\) −13.4558 −0.556807
\(585\) 0 0
\(586\) −0.887302 −0.0366541
\(587\) −31.6569 −1.30662 −0.653309 0.757091i \(-0.726620\pi\)
−0.653309 + 0.757091i \(0.726620\pi\)
\(588\) 0 0
\(589\) 34.9706 1.44094
\(590\) −4.24264 −0.174667
\(591\) 0 0
\(592\) 25.4558 1.04623
\(593\) −20.6274 −0.847066 −0.423533 0.905881i \(-0.639210\pi\)
−0.423533 + 0.905881i \(0.639210\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) −0.627417 −0.0257000
\(597\) 0 0
\(598\) 0.585786 0.0239546
\(599\) 25.4558 1.04010 0.520049 0.854137i \(-0.325914\pi\)
0.520049 + 0.854137i \(0.325914\pi\)
\(600\) 0 0
\(601\) 0.627417 0.0255929 0.0127964 0.999918i \(-0.495927\pi\)
0.0127964 + 0.999918i \(0.495927\pi\)
\(602\) 1.05382 0.0429507
\(603\) 0 0
\(604\) −33.3553 −1.35721
\(605\) 10.6569 0.433263
\(606\) 0 0
\(607\) 40.2426 1.63340 0.816699 0.577064i \(-0.195802\pi\)
0.816699 + 0.577064i \(0.195802\pi\)
\(608\) −15.0711 −0.611213
\(609\) 0 0
\(610\) −3.31371 −0.134168
\(611\) 0.828427 0.0335146
\(612\) 0 0
\(613\) −37.3137 −1.50709 −0.753543 0.657398i \(-0.771657\pi\)
−0.753543 + 0.657398i \(0.771657\pi\)
\(614\) −7.94113 −0.320478
\(615\) 0 0
\(616\) 0.769553 0.0310062
\(617\) 22.9706 0.924760 0.462380 0.886682i \(-0.346996\pi\)
0.462380 + 0.886682i \(0.346996\pi\)
\(618\) 0 0
\(619\) 10.2426 0.411686 0.205843 0.978585i \(-0.434006\pi\)
0.205843 + 0.978585i \(0.434006\pi\)
\(620\) 18.7279 0.752131
\(621\) 0 0
\(622\) −3.51472 −0.140927
\(623\) 4.97056 0.199141
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −0.343146 −0.0137149
\(627\) 0 0
\(628\) −32.9117 −1.31332
\(629\) 40.9706 1.63360
\(630\) 0 0
\(631\) −18.2426 −0.726228 −0.363114 0.931745i \(-0.618286\pi\)
−0.363114 + 0.931745i \(0.618286\pi\)
\(632\) 13.4558 0.535245
\(633\) 0 0
\(634\) 10.8284 0.430052
\(635\) 9.41421 0.373592
\(636\) 0 0
\(637\) 6.31371 0.250158
\(638\) −1.37258 −0.0543411
\(639\) 0 0
\(640\) −10.5563 −0.417276
\(641\) −36.3431 −1.43547 −0.717734 0.696317i \(-0.754821\pi\)
−0.717734 + 0.696317i \(0.754821\pi\)
\(642\) 0 0
\(643\) 26.4853 1.04448 0.522239 0.852799i \(-0.325097\pi\)
0.522239 + 0.852799i \(0.325097\pi\)
\(644\) 2.14214 0.0844120
\(645\) 0 0
\(646\) −6.82843 −0.268661
\(647\) −6.58579 −0.258914 −0.129457 0.991585i \(-0.541323\pi\)
−0.129457 + 0.991585i \(0.541323\pi\)
\(648\) 0 0
\(649\) 6.00000 0.235521
\(650\) 0.414214 0.0162468
\(651\) 0 0
\(652\) 27.3726 1.07199
\(653\) −13.0294 −0.509881 −0.254941 0.966957i \(-0.582056\pi\)
−0.254941 + 0.966957i \(0.582056\pi\)
\(654\) 0 0
\(655\) 16.9706 0.663095
\(656\) 26.4853 1.03408
\(657\) 0 0
\(658\) −0.284271 −0.0110820
\(659\) −46.1421 −1.79744 −0.898721 0.438520i \(-0.855503\pi\)
−0.898721 + 0.438520i \(0.855503\pi\)
\(660\) 0 0
\(661\) −49.5980 −1.92914 −0.964569 0.263831i \(-0.915014\pi\)
−0.964569 + 0.263831i \(0.915014\pi\)
\(662\) −9.12994 −0.354845
\(663\) 0 0
\(664\) 14.0000 0.543305
\(665\) 2.82843 0.109682
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) −16.1421 −0.624558
\(669\) 0 0
\(670\) −0.828427 −0.0320049
\(671\) 4.68629 0.180912
\(672\) 0 0
\(673\) −10.4853 −0.404178 −0.202089 0.979367i \(-0.564773\pi\)
−0.202089 + 0.979367i \(0.564773\pi\)
\(674\) −2.97056 −0.114422
\(675\) 0 0
\(676\) −1.82843 −0.0703241
\(677\) −8.14214 −0.312928 −0.156464 0.987684i \(-0.550009\pi\)
−0.156464 + 0.987684i \(0.550009\pi\)
\(678\) 0 0
\(679\) −3.02944 −0.116259
\(680\) −7.65685 −0.293627
\(681\) 0 0
\(682\) 2.48528 0.0951663
\(683\) −33.3137 −1.27471 −0.637357 0.770569i \(-0.719972\pi\)
−0.637357 + 0.770569i \(0.719972\pi\)
\(684\) 0 0
\(685\) 5.31371 0.203026
\(686\) −4.56854 −0.174428
\(687\) 0 0
\(688\) 9.21320 0.351250
\(689\) −14.4853 −0.551845
\(690\) 0 0
\(691\) 21.0711 0.801581 0.400791 0.916170i \(-0.368735\pi\)
0.400791 + 0.916170i \(0.368735\pi\)
\(692\) 20.4264 0.776495
\(693\) 0 0
\(694\) 1.75736 0.0667084
\(695\) 12.4853 0.473594
\(696\) 0 0
\(697\) 42.6274 1.61463
\(698\) −0.627417 −0.0237481
\(699\) 0 0
\(700\) 1.51472 0.0572510
\(701\) −37.3137 −1.40932 −0.704660 0.709545i \(-0.748900\pi\)
−0.704660 + 0.709545i \(0.748900\pi\)
\(702\) 0 0
\(703\) 28.9706 1.09265
\(704\) 2.44365 0.0920986
\(705\) 0 0
\(706\) 3.79899 0.142977
\(707\) 6.34315 0.238559
\(708\) 0 0
\(709\) −17.1127 −0.642681 −0.321340 0.946964i \(-0.604133\pi\)
−0.321340 + 0.946964i \(0.604133\pi\)
\(710\) 3.27208 0.122799
\(711\) 0 0
\(712\) −9.51472 −0.356579
\(713\) 14.4853 0.542478
\(714\) 0 0
\(715\) −0.585786 −0.0219072
\(716\) −10.3431 −0.386542
\(717\) 0 0
\(718\) −11.5563 −0.431279
\(719\) 4.97056 0.185371 0.0926854 0.995695i \(-0.470455\pi\)
0.0926854 + 0.995695i \(0.470455\pi\)
\(720\) 0 0
\(721\) −14.4264 −0.537267
\(722\) 3.04163 0.113198
\(723\) 0 0
\(724\) 0 0
\(725\) −5.65685 −0.210090
\(726\) 0 0
\(727\) 19.3553 0.717850 0.358925 0.933366i \(-0.383143\pi\)
0.358925 + 0.933366i \(0.383143\pi\)
\(728\) 1.31371 0.0486893
\(729\) 0 0
\(730\) −3.51472 −0.130086
\(731\) 14.8284 0.548449
\(732\) 0 0
\(733\) −1.31371 −0.0485229 −0.0242615 0.999706i \(-0.507723\pi\)
−0.0242615 + 0.999706i \(0.507723\pi\)
\(734\) −1.84062 −0.0679385
\(735\) 0 0
\(736\) −6.24264 −0.230107
\(737\) 1.17157 0.0431554
\(738\) 0 0
\(739\) 30.7279 1.13034 0.565172 0.824973i \(-0.308810\pi\)
0.565172 + 0.824973i \(0.308810\pi\)
\(740\) 15.5147 0.570332
\(741\) 0 0
\(742\) 4.97056 0.182475
\(743\) 38.4853 1.41189 0.705944 0.708268i \(-0.250523\pi\)
0.705944 + 0.708268i \(0.250523\pi\)
\(744\) 0 0
\(745\) −0.343146 −0.0125719
\(746\) 10.4853 0.383893
\(747\) 0 0
\(748\) 5.17157 0.189091
\(749\) −5.45584 −0.199352
\(750\) 0 0
\(751\) −44.4853 −1.62329 −0.811645 0.584150i \(-0.801428\pi\)
−0.811645 + 0.584150i \(0.801428\pi\)
\(752\) −2.48528 −0.0906289
\(753\) 0 0
\(754\) −2.34315 −0.0853323
\(755\) −18.2426 −0.663918
\(756\) 0 0
\(757\) 4.14214 0.150548 0.0752742 0.997163i \(-0.476017\pi\)
0.0752742 + 0.997163i \(0.476017\pi\)
\(758\) −6.18377 −0.224605
\(759\) 0 0
\(760\) −5.41421 −0.196394
\(761\) 36.6274 1.32774 0.663871 0.747847i \(-0.268912\pi\)
0.663871 + 0.747847i \(0.268912\pi\)
\(762\) 0 0
\(763\) 1.65685 0.0599822
\(764\) 24.9706 0.903403
\(765\) 0 0
\(766\) 13.7157 0.495569
\(767\) 10.2426 0.369840
\(768\) 0 0
\(769\) 10.9706 0.395609 0.197804 0.980242i \(-0.436619\pi\)
0.197804 + 0.980242i \(0.436619\pi\)
\(770\) 0.201010 0.00724390
\(771\) 0 0
\(772\) −28.6274 −1.03032
\(773\) −6.14214 −0.220917 −0.110459 0.993881i \(-0.535232\pi\)
−0.110459 + 0.993881i \(0.535232\pi\)
\(774\) 0 0
\(775\) 10.2426 0.367927
\(776\) 5.79899 0.208172
\(777\) 0 0
\(778\) −6.88730 −0.246922
\(779\) 30.1421 1.07995
\(780\) 0 0
\(781\) −4.62742 −0.165582
\(782\) −2.82843 −0.101144
\(783\) 0 0
\(784\) −18.9411 −0.676469
\(785\) −18.0000 −0.642448
\(786\) 0 0
\(787\) 5.51472 0.196578 0.0982892 0.995158i \(-0.468663\pi\)
0.0982892 + 0.995158i \(0.468663\pi\)
\(788\) −42.0000 −1.49619
\(789\) 0 0
\(790\) 3.51472 0.125048
\(791\) −2.62742 −0.0934202
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 11.5147 0.408642
\(795\) 0 0
\(796\) −7.31371 −0.259228
\(797\) −10.9706 −0.388597 −0.194299 0.980942i \(-0.562243\pi\)
−0.194299 + 0.980942i \(0.562243\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) −4.41421 −0.156066
\(801\) 0 0
\(802\) 7.17157 0.253237
\(803\) 4.97056 0.175407
\(804\) 0 0
\(805\) 1.17157 0.0412925
\(806\) 4.24264 0.149441
\(807\) 0 0
\(808\) −12.1421 −0.427159
\(809\) 45.2548 1.59108 0.795538 0.605904i \(-0.207189\pi\)
0.795538 + 0.605904i \(0.207189\pi\)
\(810\) 0 0
\(811\) 8.38478 0.294429 0.147215 0.989105i \(-0.452969\pi\)
0.147215 + 0.989105i \(0.452969\pi\)
\(812\) −8.56854 −0.300697
\(813\) 0 0
\(814\) 2.05887 0.0721635
\(815\) 14.9706 0.524396
\(816\) 0 0
\(817\) 10.4853 0.366834
\(818\) −5.31371 −0.185789
\(819\) 0 0
\(820\) 16.1421 0.563708
\(821\) −39.2548 −1.37000 −0.685002 0.728542i \(-0.740199\pi\)
−0.685002 + 0.728542i \(0.740199\pi\)
\(822\) 0 0
\(823\) 34.3848 1.19858 0.599289 0.800533i \(-0.295450\pi\)
0.599289 + 0.800533i \(0.295450\pi\)
\(824\) 27.6152 0.962022
\(825\) 0 0
\(826\) −3.51472 −0.122293
\(827\) −27.8579 −0.968713 −0.484356 0.874871i \(-0.660946\pi\)
−0.484356 + 0.874871i \(0.660946\pi\)
\(828\) 0 0
\(829\) 7.02944 0.244142 0.122071 0.992521i \(-0.461046\pi\)
0.122071 + 0.992521i \(0.461046\pi\)
\(830\) 3.65685 0.126931
\(831\) 0 0
\(832\) 4.17157 0.144623
\(833\) −30.4853 −1.05625
\(834\) 0 0
\(835\) −8.82843 −0.305520
\(836\) 3.65685 0.126475
\(837\) 0 0
\(838\) 2.14214 0.0739988
\(839\) 18.7279 0.646560 0.323280 0.946303i \(-0.395214\pi\)
0.323280 + 0.946303i \(0.395214\pi\)
\(840\) 0 0
\(841\) 3.00000 0.103448
\(842\) 0.426407 0.0146950
\(843\) 0 0
\(844\) 35.3137 1.21555
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 8.82843 0.303348
\(848\) 43.4558 1.49228
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) 37.4558 1.28246 0.641232 0.767347i \(-0.278424\pi\)
0.641232 + 0.767347i \(0.278424\pi\)
\(854\) −2.74517 −0.0939376
\(855\) 0 0
\(856\) 10.4437 0.356957
\(857\) −0.343146 −0.0117216 −0.00586082 0.999983i \(-0.501866\pi\)
−0.00586082 + 0.999983i \(0.501866\pi\)
\(858\) 0 0
\(859\) 11.7990 0.402576 0.201288 0.979532i \(-0.435487\pi\)
0.201288 + 0.979532i \(0.435487\pi\)
\(860\) 5.61522 0.191478
\(861\) 0 0
\(862\) 1.49747 0.0510042
\(863\) −19.4558 −0.662285 −0.331142 0.943581i \(-0.607434\pi\)
−0.331142 + 0.943581i \(0.607434\pi\)
\(864\) 0 0
\(865\) 11.1716 0.379845
\(866\) 1.51472 0.0514722
\(867\) 0 0
\(868\) 15.5147 0.526604
\(869\) −4.97056 −0.168615
\(870\) 0 0
\(871\) 2.00000 0.0677674
\(872\) −3.17157 −0.107403
\(873\) 0 0
\(874\) −2.00000 −0.0676510
\(875\) 0.828427 0.0280059
\(876\) 0 0
\(877\) 2.68629 0.0907096 0.0453548 0.998971i \(-0.485558\pi\)
0.0453548 + 0.998971i \(0.485558\pi\)
\(878\) 13.6569 0.460897
\(879\) 0 0
\(880\) 1.75736 0.0592406
\(881\) −52.9706 −1.78462 −0.892312 0.451420i \(-0.850918\pi\)
−0.892312 + 0.451420i \(0.850918\pi\)
\(882\) 0 0
\(883\) −32.2426 −1.08505 −0.542526 0.840039i \(-0.682532\pi\)
−0.542526 + 0.840039i \(0.682532\pi\)
\(884\) 8.82843 0.296932
\(885\) 0 0
\(886\) −2.72792 −0.0916463
\(887\) −14.3848 −0.482994 −0.241497 0.970402i \(-0.577638\pi\)
−0.241497 + 0.970402i \(0.577638\pi\)
\(888\) 0 0
\(889\) 7.79899 0.261570
\(890\) −2.48528 −0.0833068
\(891\) 0 0
\(892\) −48.4264 −1.62144
\(893\) −2.82843 −0.0946497
\(894\) 0 0
\(895\) −5.65685 −0.189088
\(896\) −8.74517 −0.292155
\(897\) 0 0
\(898\) 12.0589 0.402410
\(899\) −57.9411 −1.93244
\(900\) 0 0
\(901\) 69.9411 2.33008
\(902\) 2.14214 0.0713253
\(903\) 0 0
\(904\) 5.02944 0.167277
\(905\) 0 0
\(906\) 0 0
\(907\) −33.2132 −1.10283 −0.551413 0.834232i \(-0.685911\pi\)
−0.551413 + 0.834232i \(0.685911\pi\)
\(908\) −50.5685 −1.67818
\(909\) 0 0
\(910\) 0.343146 0.0113752
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) −5.17157 −0.171154
\(914\) 7.45584 0.246617
\(915\) 0 0
\(916\) −1.51472 −0.0500477
\(917\) 14.0589 0.464265
\(918\) 0 0
\(919\) −16.4853 −0.543799 −0.271900 0.962326i \(-0.587652\pi\)
−0.271900 + 0.962326i \(0.587652\pi\)
\(920\) −2.24264 −0.0739377
\(921\) 0 0
\(922\) 10.9706 0.361296
\(923\) −7.89949 −0.260015
\(924\) 0 0
\(925\) 8.48528 0.278994
\(926\) 6.48528 0.213120
\(927\) 0 0
\(928\) 24.9706 0.819699
\(929\) −11.1716 −0.366527 −0.183264 0.983064i \(-0.558666\pi\)
−0.183264 + 0.983064i \(0.558666\pi\)
\(930\) 0 0
\(931\) −21.5563 −0.706481
\(932\) 45.0294 1.47499
\(933\) 0 0
\(934\) −4.38478 −0.143474
\(935\) 2.82843 0.0924995
\(936\) 0 0
\(937\) 10.9706 0.358393 0.179196 0.983813i \(-0.442650\pi\)
0.179196 + 0.983813i \(0.442650\pi\)
\(938\) −0.686292 −0.0224082
\(939\) 0 0
\(940\) −1.51472 −0.0494047
\(941\) 54.7696 1.78544 0.892718 0.450615i \(-0.148795\pi\)
0.892718 + 0.450615i \(0.148795\pi\)
\(942\) 0 0
\(943\) 12.4853 0.406577
\(944\) −30.7279 −1.00011
\(945\) 0 0
\(946\) 0.745166 0.0242274
\(947\) 45.1127 1.46597 0.732983 0.680247i \(-0.238128\pi\)
0.732983 + 0.680247i \(0.238128\pi\)
\(948\) 0 0
\(949\) 8.48528 0.275444
\(950\) −1.41421 −0.0458831
\(951\) 0 0
\(952\) −6.34315 −0.205583
\(953\) 55.2548 1.78988 0.894940 0.446187i \(-0.147218\pi\)
0.894940 + 0.446187i \(0.147218\pi\)
\(954\) 0 0
\(955\) 13.6569 0.441925
\(956\) −1.07107 −0.0346408
\(957\) 0 0
\(958\) −2.18377 −0.0705543
\(959\) 4.40202 0.142149
\(960\) 0 0
\(961\) 73.9117 2.38425
\(962\) 3.51472 0.113319
\(963\) 0 0
\(964\) −4.54416 −0.146357
\(965\) −15.6569 −0.504012
\(966\) 0 0
\(967\) −19.9411 −0.641263 −0.320632 0.947204i \(-0.603895\pi\)
−0.320632 + 0.947204i \(0.603895\pi\)
\(968\) −16.8995 −0.543170
\(969\) 0 0
\(970\) 1.51472 0.0486347
\(971\) 12.2843 0.394221 0.197111 0.980381i \(-0.436844\pi\)
0.197111 + 0.980381i \(0.436844\pi\)
\(972\) 0 0
\(973\) 10.3431 0.331586
\(974\) −9.51472 −0.304871
\(975\) 0 0
\(976\) −24.0000 −0.768221
\(977\) 56.4853 1.80712 0.903562 0.428457i \(-0.140943\pi\)
0.903562 + 0.428457i \(0.140943\pi\)
\(978\) 0 0
\(979\) 3.51472 0.112331
\(980\) −11.5442 −0.368765
\(981\) 0 0
\(982\) 4.48528 0.143131
\(983\) 34.9706 1.11539 0.557694 0.830047i \(-0.311686\pi\)
0.557694 + 0.830047i \(0.311686\pi\)
\(984\) 0 0
\(985\) −22.9706 −0.731903
\(986\) 11.3137 0.360302
\(987\) 0 0
\(988\) 6.24264 0.198605
\(989\) 4.34315 0.138104
\(990\) 0 0
\(991\) 15.0294 0.477426 0.238713 0.971090i \(-0.423275\pi\)
0.238713 + 0.971090i \(0.423275\pi\)
\(992\) −45.2132 −1.43552
\(993\) 0 0
\(994\) 2.71068 0.0859775
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) 23.1716 0.733851 0.366926 0.930250i \(-0.380410\pi\)
0.366926 + 0.930250i \(0.380410\pi\)
\(998\) −4.32590 −0.136934
\(999\) 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 585.2.a.m.1.1 2
3.2 odd 2 65.2.a.b.1.2 2
4.3 odd 2 9360.2.a.cd.1.2 2
5.2 odd 4 2925.2.c.r.2224.2 4
5.3 odd 4 2925.2.c.r.2224.3 4
5.4 even 2 2925.2.a.u.1.2 2
12.11 even 2 1040.2.a.j.1.1 2
13.12 even 2 7605.2.a.x.1.2 2
15.2 even 4 325.2.b.f.274.3 4
15.8 even 4 325.2.b.f.274.2 4
15.14 odd 2 325.2.a.i.1.1 2
21.20 even 2 3185.2.a.j.1.2 2
24.5 odd 2 4160.2.a.bf.1.1 2
24.11 even 2 4160.2.a.z.1.2 2
33.32 even 2 7865.2.a.j.1.1 2
39.2 even 12 845.2.m.f.316.3 8
39.5 even 4 845.2.c.b.506.2 4
39.8 even 4 845.2.c.b.506.3 4
39.11 even 12 845.2.m.f.316.2 8
39.17 odd 6 845.2.e.c.146.2 4
39.20 even 12 845.2.m.f.361.3 8
39.23 odd 6 845.2.e.c.191.2 4
39.29 odd 6 845.2.e.h.191.1 4
39.32 even 12 845.2.m.f.361.2 8
39.35 odd 6 845.2.e.h.146.1 4
39.38 odd 2 845.2.a.g.1.1 2
60.59 even 2 5200.2.a.bu.1.2 2
195.194 odd 2 4225.2.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.b.1.2 2 3.2 odd 2
325.2.a.i.1.1 2 15.14 odd 2
325.2.b.f.274.2 4 15.8 even 4
325.2.b.f.274.3 4 15.2 even 4
585.2.a.m.1.1 2 1.1 even 1 trivial
845.2.a.g.1.1 2 39.38 odd 2
845.2.c.b.506.2 4 39.5 even 4
845.2.c.b.506.3 4 39.8 even 4
845.2.e.c.146.2 4 39.17 odd 6
845.2.e.c.191.2 4 39.23 odd 6
845.2.e.h.146.1 4 39.35 odd 6
845.2.e.h.191.1 4 39.29 odd 6
845.2.m.f.316.2 8 39.11 even 12
845.2.m.f.316.3 8 39.2 even 12
845.2.m.f.361.2 8 39.32 even 12
845.2.m.f.361.3 8 39.20 even 12
1040.2.a.j.1.1 2 12.11 even 2
2925.2.a.u.1.2 2 5.4 even 2
2925.2.c.r.2224.2 4 5.2 odd 4
2925.2.c.r.2224.3 4 5.3 odd 4
3185.2.a.j.1.2 2 21.20 even 2
4160.2.a.z.1.2 2 24.11 even 2
4160.2.a.bf.1.1 2 24.5 odd 2
4225.2.a.r.1.2 2 195.194 odd 2
5200.2.a.bu.1.2 2 60.59 even 2
7605.2.a.x.1.2 2 13.12 even 2
7865.2.a.j.1.1 2 33.32 even 2
9360.2.a.cd.1.2 2 4.3 odd 2