Properties

Label 1216.2.a.t.1.1
Level $1216$
Weight $2$
Character 1216.1
Self dual yes
Analytic conductor $9.710$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.70980888579\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155 q^{3} +2.56155 q^{5} -3.00000 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q-1.56155 q^{3} +2.56155 q^{5} -3.00000 q^{7} -0.561553 q^{9} +0.561553 q^{11} +1.56155 q^{13} -4.00000 q^{15} +0.123106 q^{17} +1.00000 q^{19} +4.68466 q^{21} -5.56155 q^{23} +1.56155 q^{25} +5.56155 q^{27} -4.68466 q^{29} -9.12311 q^{31} -0.876894 q^{33} -7.68466 q^{35} -7.12311 q^{37} -2.43845 q^{39} +4.00000 q^{41} +5.43845 q^{43} -1.43845 q^{45} +3.68466 q^{47} +2.00000 q^{49} -0.192236 q^{51} +4.43845 q^{53} +1.43845 q^{55} -1.56155 q^{57} -4.43845 q^{59} -14.8078 q^{61} +1.68466 q^{63} +4.00000 q^{65} -0.438447 q^{67} +8.68466 q^{69} +4.24621 q^{71} -9.24621 q^{73} -2.43845 q^{75} -1.68466 q^{77} -10.0000 q^{79} -7.00000 q^{81} -17.3693 q^{83} +0.315342 q^{85} +7.31534 q^{87} -15.3693 q^{89} -4.68466 q^{91} +14.2462 q^{93} +2.56155 q^{95} +7.12311 q^{97} -0.315342 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + q^{5} - 6q^{7} + 3q^{9} + O(q^{10}) \) \( 2q + q^{3} + q^{5} - 6q^{7} + 3q^{9} - 3q^{11} - q^{13} - 8q^{15} - 8q^{17} + 2q^{19} - 3q^{21} - 7q^{23} - q^{25} + 7q^{27} + 3q^{29} - 10q^{31} - 10q^{33} - 3q^{35} - 6q^{37} - 9q^{39} + 8q^{41} + 15q^{43} - 7q^{45} - 5q^{47} + 4q^{49} - 21q^{51} + 13q^{53} + 7q^{55} + q^{57} - 13q^{59} - 9q^{61} - 9q^{63} + 8q^{65} - 5q^{67} + 5q^{69} - 8q^{71} - 2q^{73} - 9q^{75} + 9q^{77} - 20q^{79} - 14q^{81} - 10q^{83} + 13q^{85} + 27q^{87} - 6q^{89} + 3q^{91} + 12q^{93} + q^{95} + 6q^{97} - 13q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.56155 −0.901563 −0.450781 0.892634i \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 0 0
\(5\) 2.56155 1.14556 0.572781 0.819709i \(-0.305865\pi\)
0.572781 + 0.819709i \(0.305865\pi\)
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) 0.561553 0.169315 0.0846573 0.996410i \(-0.473020\pi\)
0.0846573 + 0.996410i \(0.473020\pi\)
\(12\) 0 0
\(13\) 1.56155 0.433097 0.216548 0.976272i \(-0.430520\pi\)
0.216548 + 0.976272i \(0.430520\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) 0.123106 0.0298575 0.0149287 0.999889i \(-0.495248\pi\)
0.0149287 + 0.999889i \(0.495248\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 4.68466 1.02228
\(22\) 0 0
\(23\) −5.56155 −1.15966 −0.579832 0.814736i \(-0.696882\pi\)
−0.579832 + 0.814736i \(0.696882\pi\)
\(24\) 0 0
\(25\) 1.56155 0.312311
\(26\) 0 0
\(27\) 5.56155 1.07032
\(28\) 0 0
\(29\) −4.68466 −0.869919 −0.434960 0.900450i \(-0.643237\pi\)
−0.434960 + 0.900450i \(0.643237\pi\)
\(30\) 0 0
\(31\) −9.12311 −1.63856 −0.819279 0.573395i \(-0.805626\pi\)
−0.819279 + 0.573395i \(0.805626\pi\)
\(32\) 0 0
\(33\) −0.876894 −0.152648
\(34\) 0 0
\(35\) −7.68466 −1.29894
\(36\) 0 0
\(37\) −7.12311 −1.17103 −0.585516 0.810661i \(-0.699108\pi\)
−0.585516 + 0.810661i \(0.699108\pi\)
\(38\) 0 0
\(39\) −2.43845 −0.390464
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 5.43845 0.829355 0.414678 0.909968i \(-0.363894\pi\)
0.414678 + 0.909968i \(0.363894\pi\)
\(44\) 0 0
\(45\) −1.43845 −0.214431
\(46\) 0 0
\(47\) 3.68466 0.537463 0.268731 0.963215i \(-0.413396\pi\)
0.268731 + 0.963215i \(0.413396\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −0.192236 −0.0269184
\(52\) 0 0
\(53\) 4.43845 0.609668 0.304834 0.952406i \(-0.401399\pi\)
0.304834 + 0.952406i \(0.401399\pi\)
\(54\) 0 0
\(55\) 1.43845 0.193960
\(56\) 0 0
\(57\) −1.56155 −0.206833
\(58\) 0 0
\(59\) −4.43845 −0.577837 −0.288918 0.957354i \(-0.593296\pi\)
−0.288918 + 0.957354i \(0.593296\pi\)
\(60\) 0 0
\(61\) −14.8078 −1.89594 −0.947970 0.318360i \(-0.896868\pi\)
−0.947970 + 0.318360i \(0.896868\pi\)
\(62\) 0 0
\(63\) 1.68466 0.212247
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −0.438447 −0.0535648 −0.0267824 0.999641i \(-0.508526\pi\)
−0.0267824 + 0.999641i \(0.508526\pi\)
\(68\) 0 0
\(69\) 8.68466 1.04551
\(70\) 0 0
\(71\) 4.24621 0.503933 0.251966 0.967736i \(-0.418923\pi\)
0.251966 + 0.967736i \(0.418923\pi\)
\(72\) 0 0
\(73\) −9.24621 −1.08219 −0.541094 0.840962i \(-0.681990\pi\)
−0.541094 + 0.840962i \(0.681990\pi\)
\(74\) 0 0
\(75\) −2.43845 −0.281568
\(76\) 0 0
\(77\) −1.68466 −0.191985
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −17.3693 −1.90653 −0.953265 0.302135i \(-0.902301\pi\)
−0.953265 + 0.302135i \(0.902301\pi\)
\(84\) 0 0
\(85\) 0.315342 0.0342036
\(86\) 0 0
\(87\) 7.31534 0.784287
\(88\) 0 0
\(89\) −15.3693 −1.62914 −0.814572 0.580062i \(-0.803028\pi\)
−0.814572 + 0.580062i \(0.803028\pi\)
\(90\) 0 0
\(91\) −4.68466 −0.491086
\(92\) 0 0
\(93\) 14.2462 1.47726
\(94\) 0 0
\(95\) 2.56155 0.262810
\(96\) 0 0
\(97\) 7.12311 0.723242 0.361621 0.932325i \(-0.382223\pi\)
0.361621 + 0.932325i \(0.382223\pi\)
\(98\) 0 0
\(99\) −0.315342 −0.0316930
\(100\) 0 0
\(101\) 0.246211 0.0244989 0.0122495 0.999925i \(-0.496101\pi\)
0.0122495 + 0.999925i \(0.496101\pi\)
\(102\) 0 0
\(103\) −14.4924 −1.42798 −0.713990 0.700155i \(-0.753114\pi\)
−0.713990 + 0.700155i \(0.753114\pi\)
\(104\) 0 0
\(105\) 12.0000 1.17108
\(106\) 0 0
\(107\) 16.9309 1.63677 0.818384 0.574671i \(-0.194870\pi\)
0.818384 + 0.574671i \(0.194870\pi\)
\(108\) 0 0
\(109\) 3.56155 0.341135 0.170567 0.985346i \(-0.445440\pi\)
0.170567 + 0.985346i \(0.445440\pi\)
\(110\) 0 0
\(111\) 11.1231 1.05576
\(112\) 0 0
\(113\) 4.87689 0.458780 0.229390 0.973335i \(-0.426327\pi\)
0.229390 + 0.973335i \(0.426327\pi\)
\(114\) 0 0
\(115\) −14.2462 −1.32847
\(116\) 0 0
\(117\) −0.876894 −0.0810689
\(118\) 0 0
\(119\) −0.369317 −0.0338552
\(120\) 0 0
\(121\) −10.6847 −0.971333
\(122\) 0 0
\(123\) −6.24621 −0.563202
\(124\) 0 0
\(125\) −8.80776 −0.787790
\(126\) 0 0
\(127\) −15.1231 −1.34196 −0.670979 0.741476i \(-0.734126\pi\)
−0.670979 + 0.741476i \(0.734126\pi\)
\(128\) 0 0
\(129\) −8.49242 −0.747716
\(130\) 0 0
\(131\) −13.4384 −1.17412 −0.587061 0.809542i \(-0.699715\pi\)
−0.587061 + 0.809542i \(0.699715\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) 0 0
\(135\) 14.2462 1.22612
\(136\) 0 0
\(137\) 14.3693 1.22765 0.613827 0.789441i \(-0.289629\pi\)
0.613827 + 0.789441i \(0.289629\pi\)
\(138\) 0 0
\(139\) 5.43845 0.461283 0.230642 0.973039i \(-0.425918\pi\)
0.230642 + 0.973039i \(0.425918\pi\)
\(140\) 0 0
\(141\) −5.75379 −0.484556
\(142\) 0 0
\(143\) 0.876894 0.0733296
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) −3.12311 −0.257589
\(148\) 0 0
\(149\) −10.5616 −0.865236 −0.432618 0.901577i \(-0.642410\pi\)
−0.432618 + 0.901577i \(0.642410\pi\)
\(150\) 0 0
\(151\) 22.4924 1.83041 0.915204 0.402992i \(-0.132030\pi\)
0.915204 + 0.402992i \(0.132030\pi\)
\(152\) 0 0
\(153\) −0.0691303 −0.00558885
\(154\) 0 0
\(155\) −23.3693 −1.87707
\(156\) 0 0
\(157\) 7.75379 0.618820 0.309410 0.950929i \(-0.399869\pi\)
0.309410 + 0.950929i \(0.399869\pi\)
\(158\) 0 0
\(159\) −6.93087 −0.549654
\(160\) 0 0
\(161\) 16.6847 1.31494
\(162\) 0 0
\(163\) 16.4924 1.29179 0.645893 0.763428i \(-0.276485\pi\)
0.645893 + 0.763428i \(0.276485\pi\)
\(164\) 0 0
\(165\) −2.24621 −0.174867
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −10.5616 −0.812427
\(170\) 0 0
\(171\) −0.561553 −0.0429430
\(172\) 0 0
\(173\) 20.2462 1.53929 0.769645 0.638471i \(-0.220433\pi\)
0.769645 + 0.638471i \(0.220433\pi\)
\(174\) 0 0
\(175\) −4.68466 −0.354127
\(176\) 0 0
\(177\) 6.93087 0.520956
\(178\) 0 0
\(179\) 2.24621 0.167890 0.0839449 0.996470i \(-0.473248\pi\)
0.0839449 + 0.996470i \(0.473248\pi\)
\(180\) 0 0
\(181\) 0.246211 0.0183007 0.00915037 0.999958i \(-0.497087\pi\)
0.00915037 + 0.999958i \(0.497087\pi\)
\(182\) 0 0
\(183\) 23.1231 1.70931
\(184\) 0 0
\(185\) −18.2462 −1.34149
\(186\) 0 0
\(187\) 0.0691303 0.00505531
\(188\) 0 0
\(189\) −16.6847 −1.21363
\(190\) 0 0
\(191\) 1.00000 0.0723575 0.0361787 0.999345i \(-0.488481\pi\)
0.0361787 + 0.999345i \(0.488481\pi\)
\(192\) 0 0
\(193\) 18.4924 1.33111 0.665557 0.746347i \(-0.268194\pi\)
0.665557 + 0.746347i \(0.268194\pi\)
\(194\) 0 0
\(195\) −6.24621 −0.447300
\(196\) 0 0
\(197\) −11.3693 −0.810030 −0.405015 0.914310i \(-0.632734\pi\)
−0.405015 + 0.914310i \(0.632734\pi\)
\(198\) 0 0
\(199\) 23.7386 1.68279 0.841394 0.540423i \(-0.181736\pi\)
0.841394 + 0.540423i \(0.181736\pi\)
\(200\) 0 0
\(201\) 0.684658 0.0482921
\(202\) 0 0
\(203\) 14.0540 0.986396
\(204\) 0 0
\(205\) 10.2462 0.715626
\(206\) 0 0
\(207\) 3.12311 0.217071
\(208\) 0 0
\(209\) 0.561553 0.0388434
\(210\) 0 0
\(211\) 15.5616 1.07130 0.535651 0.844440i \(-0.320066\pi\)
0.535651 + 0.844440i \(0.320066\pi\)
\(212\) 0 0
\(213\) −6.63068 −0.454327
\(214\) 0 0
\(215\) 13.9309 0.950077
\(216\) 0 0
\(217\) 27.3693 1.85795
\(218\) 0 0
\(219\) 14.4384 0.975660
\(220\) 0 0
\(221\) 0.192236 0.0129312
\(222\) 0 0
\(223\) 1.36932 0.0916962 0.0458481 0.998948i \(-0.485401\pi\)
0.0458481 + 0.998948i \(0.485401\pi\)
\(224\) 0 0
\(225\) −0.876894 −0.0584596
\(226\) 0 0
\(227\) 16.6847 1.10740 0.553700 0.832716i \(-0.313216\pi\)
0.553700 + 0.832716i \(0.313216\pi\)
\(228\) 0 0
\(229\) 12.5616 0.830091 0.415045 0.909801i \(-0.363766\pi\)
0.415045 + 0.909801i \(0.363766\pi\)
\(230\) 0 0
\(231\) 2.63068 0.173086
\(232\) 0 0
\(233\) −19.9309 −1.30571 −0.652857 0.757481i \(-0.726430\pi\)
−0.652857 + 0.757481i \(0.726430\pi\)
\(234\) 0 0
\(235\) 9.43845 0.615696
\(236\) 0 0
\(237\) 15.6155 1.01434
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) −14.2462 −0.917679 −0.458840 0.888519i \(-0.651735\pi\)
−0.458840 + 0.888519i \(0.651735\pi\)
\(242\) 0 0
\(243\) −5.75379 −0.369106
\(244\) 0 0
\(245\) 5.12311 0.327303
\(246\) 0 0
\(247\) 1.56155 0.0993592
\(248\) 0 0
\(249\) 27.1231 1.71886
\(250\) 0 0
\(251\) 5.68466 0.358812 0.179406 0.983775i \(-0.442582\pi\)
0.179406 + 0.983775i \(0.442582\pi\)
\(252\) 0 0
\(253\) −3.12311 −0.196348
\(254\) 0 0
\(255\) −0.492423 −0.0308367
\(256\) 0 0
\(257\) 26.2462 1.63719 0.818597 0.574369i \(-0.194752\pi\)
0.818597 + 0.574369i \(0.194752\pi\)
\(258\) 0 0
\(259\) 21.3693 1.32782
\(260\) 0 0
\(261\) 2.63068 0.162835
\(262\) 0 0
\(263\) −11.0540 −0.681617 −0.340809 0.940133i \(-0.610701\pi\)
−0.340809 + 0.940133i \(0.610701\pi\)
\(264\) 0 0
\(265\) 11.3693 0.698412
\(266\) 0 0
\(267\) 24.0000 1.46878
\(268\) 0 0
\(269\) 18.4924 1.12750 0.563751 0.825944i \(-0.309358\pi\)
0.563751 + 0.825944i \(0.309358\pi\)
\(270\) 0 0
\(271\) −10.0540 −0.610736 −0.305368 0.952234i \(-0.598779\pi\)
−0.305368 + 0.952234i \(0.598779\pi\)
\(272\) 0 0
\(273\) 7.31534 0.442745
\(274\) 0 0
\(275\) 0.876894 0.0528787
\(276\) 0 0
\(277\) 11.6847 0.702063 0.351032 0.936364i \(-0.385831\pi\)
0.351032 + 0.936364i \(0.385831\pi\)
\(278\) 0 0
\(279\) 5.12311 0.306712
\(280\) 0 0
\(281\) −0.630683 −0.0376234 −0.0188117 0.999823i \(-0.505988\pi\)
−0.0188117 + 0.999823i \(0.505988\pi\)
\(282\) 0 0
\(283\) 28.5616 1.69781 0.848904 0.528547i \(-0.177263\pi\)
0.848904 + 0.528547i \(0.177263\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −16.9848 −0.999109
\(290\) 0 0
\(291\) −11.1231 −0.652048
\(292\) 0 0
\(293\) −15.3153 −0.894732 −0.447366 0.894351i \(-0.647638\pi\)
−0.447366 + 0.894351i \(0.647638\pi\)
\(294\) 0 0
\(295\) −11.3693 −0.661947
\(296\) 0 0
\(297\) 3.12311 0.181221
\(298\) 0 0
\(299\) −8.68466 −0.502247
\(300\) 0 0
\(301\) −16.3153 −0.940401
\(302\) 0 0
\(303\) −0.384472 −0.0220873
\(304\) 0 0
\(305\) −37.9309 −2.17192
\(306\) 0 0
\(307\) 4.63068 0.264287 0.132144 0.991231i \(-0.457814\pi\)
0.132144 + 0.991231i \(0.457814\pi\)
\(308\) 0 0
\(309\) 22.6307 1.28741
\(310\) 0 0
\(311\) 6.12311 0.347209 0.173605 0.984815i \(-0.444458\pi\)
0.173605 + 0.984815i \(0.444458\pi\)
\(312\) 0 0
\(313\) −0.930870 −0.0526159 −0.0263079 0.999654i \(-0.508375\pi\)
−0.0263079 + 0.999654i \(0.508375\pi\)
\(314\) 0 0
\(315\) 4.31534 0.243142
\(316\) 0 0
\(317\) −16.4384 −0.923275 −0.461638 0.887069i \(-0.652738\pi\)
−0.461638 + 0.887069i \(0.652738\pi\)
\(318\) 0 0
\(319\) −2.63068 −0.147290
\(320\) 0 0
\(321\) −26.4384 −1.47565
\(322\) 0 0
\(323\) 0.123106 0.00684978
\(324\) 0 0
\(325\) 2.43845 0.135261
\(326\) 0 0
\(327\) −5.56155 −0.307555
\(328\) 0 0
\(329\) −11.0540 −0.609425
\(330\) 0 0
\(331\) 4.43845 0.243959 0.121980 0.992533i \(-0.461076\pi\)
0.121980 + 0.992533i \(0.461076\pi\)
\(332\) 0 0
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) −1.12311 −0.0613618
\(336\) 0 0
\(337\) −20.4924 −1.11629 −0.558147 0.829742i \(-0.688487\pi\)
−0.558147 + 0.829742i \(0.688487\pi\)
\(338\) 0 0
\(339\) −7.61553 −0.413619
\(340\) 0 0
\(341\) −5.12311 −0.277432
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) 22.2462 1.19770
\(346\) 0 0
\(347\) −4.56155 −0.244877 −0.122438 0.992476i \(-0.539071\pi\)
−0.122438 + 0.992476i \(0.539071\pi\)
\(348\) 0 0
\(349\) −13.6847 −0.732523 −0.366261 0.930512i \(-0.619362\pi\)
−0.366261 + 0.930512i \(0.619362\pi\)
\(350\) 0 0
\(351\) 8.68466 0.463553
\(352\) 0 0
\(353\) −28.9309 −1.53983 −0.769917 0.638144i \(-0.779703\pi\)
−0.769917 + 0.638144i \(0.779703\pi\)
\(354\) 0 0
\(355\) 10.8769 0.577286
\(356\) 0 0
\(357\) 0.576708 0.0305226
\(358\) 0 0
\(359\) −27.8769 −1.47129 −0.735643 0.677369i \(-0.763120\pi\)
−0.735643 + 0.677369i \(0.763120\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 16.6847 0.875717
\(364\) 0 0
\(365\) −23.6847 −1.23971
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 0 0
\(369\) −2.24621 −0.116933
\(370\) 0 0
\(371\) −13.3153 −0.691298
\(372\) 0 0
\(373\) 14.9309 0.773091 0.386546 0.922270i \(-0.373668\pi\)
0.386546 + 0.922270i \(0.373668\pi\)
\(374\) 0 0
\(375\) 13.7538 0.710243
\(376\) 0 0
\(377\) −7.31534 −0.376759
\(378\) 0 0
\(379\) 16.0540 0.824637 0.412319 0.911040i \(-0.364719\pi\)
0.412319 + 0.911040i \(0.364719\pi\)
\(380\) 0 0
\(381\) 23.6155 1.20986
\(382\) 0 0
\(383\) −24.2462 −1.23892 −0.619462 0.785027i \(-0.712649\pi\)
−0.619462 + 0.785027i \(0.712649\pi\)
\(384\) 0 0
\(385\) −4.31534 −0.219930
\(386\) 0 0
\(387\) −3.05398 −0.155242
\(388\) 0 0
\(389\) −26.8078 −1.35921 −0.679604 0.733579i \(-0.737848\pi\)
−0.679604 + 0.733579i \(0.737848\pi\)
\(390\) 0 0
\(391\) −0.684658 −0.0346247
\(392\) 0 0
\(393\) 20.9848 1.05855
\(394\) 0 0
\(395\) −25.6155 −1.28886
\(396\) 0 0
\(397\) 5.43845 0.272948 0.136474 0.990644i \(-0.456423\pi\)
0.136474 + 0.990644i \(0.456423\pi\)
\(398\) 0 0
\(399\) 4.68466 0.234526
\(400\) 0 0
\(401\) 23.3693 1.16701 0.583504 0.812110i \(-0.301681\pi\)
0.583504 + 0.812110i \(0.301681\pi\)
\(402\) 0 0
\(403\) −14.2462 −0.709654
\(404\) 0 0
\(405\) −17.9309 −0.890992
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) 18.8769 0.933402 0.466701 0.884415i \(-0.345442\pi\)
0.466701 + 0.884415i \(0.345442\pi\)
\(410\) 0 0
\(411\) −22.4384 −1.10681
\(412\) 0 0
\(413\) 13.3153 0.655205
\(414\) 0 0
\(415\) −44.4924 −2.18405
\(416\) 0 0
\(417\) −8.49242 −0.415876
\(418\) 0 0
\(419\) −26.2462 −1.28221 −0.641106 0.767453i \(-0.721524\pi\)
−0.641106 + 0.767453i \(0.721524\pi\)
\(420\) 0 0
\(421\) −20.0540 −0.977371 −0.488685 0.872460i \(-0.662523\pi\)
−0.488685 + 0.872460i \(0.662523\pi\)
\(422\) 0 0
\(423\) −2.06913 −0.100605
\(424\) 0 0
\(425\) 0.192236 0.00932481
\(426\) 0 0
\(427\) 44.4233 2.14979
\(428\) 0 0
\(429\) −1.36932 −0.0661112
\(430\) 0 0
\(431\) 31.1231 1.49915 0.749574 0.661921i \(-0.230259\pi\)
0.749574 + 0.661921i \(0.230259\pi\)
\(432\) 0 0
\(433\) 39.6155 1.90380 0.951900 0.306408i \(-0.0991271\pi\)
0.951900 + 0.306408i \(0.0991271\pi\)
\(434\) 0 0
\(435\) 18.7386 0.898449
\(436\) 0 0
\(437\) −5.56155 −0.266045
\(438\) 0 0
\(439\) −1.75379 −0.0837038 −0.0418519 0.999124i \(-0.513326\pi\)
−0.0418519 + 0.999124i \(0.513326\pi\)
\(440\) 0 0
\(441\) −1.12311 −0.0534812
\(442\) 0 0
\(443\) 1.68466 0.0800405 0.0400203 0.999199i \(-0.487258\pi\)
0.0400203 + 0.999199i \(0.487258\pi\)
\(444\) 0 0
\(445\) −39.3693 −1.86628
\(446\) 0 0
\(447\) 16.4924 0.780065
\(448\) 0 0
\(449\) −15.1231 −0.713703 −0.356852 0.934161i \(-0.616150\pi\)
−0.356852 + 0.934161i \(0.616150\pi\)
\(450\) 0 0
\(451\) 2.24621 0.105770
\(452\) 0 0
\(453\) −35.1231 −1.65023
\(454\) 0 0
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) −36.1231 −1.68977 −0.844884 0.534950i \(-0.820330\pi\)
−0.844884 + 0.534950i \(0.820330\pi\)
\(458\) 0 0
\(459\) 0.684658 0.0319571
\(460\) 0 0
\(461\) 33.9309 1.58032 0.790159 0.612902i \(-0.209998\pi\)
0.790159 + 0.612902i \(0.209998\pi\)
\(462\) 0 0
\(463\) −15.0540 −0.699618 −0.349809 0.936821i \(-0.613753\pi\)
−0.349809 + 0.936821i \(0.613753\pi\)
\(464\) 0 0
\(465\) 36.4924 1.69230
\(466\) 0 0
\(467\) 17.6847 0.818348 0.409174 0.912456i \(-0.365817\pi\)
0.409174 + 0.912456i \(0.365817\pi\)
\(468\) 0 0
\(469\) 1.31534 0.0607368
\(470\) 0 0
\(471\) −12.1080 −0.557905
\(472\) 0 0
\(473\) 3.05398 0.140422
\(474\) 0 0
\(475\) 1.56155 0.0716490
\(476\) 0 0
\(477\) −2.49242 −0.114120
\(478\) 0 0
\(479\) 14.2462 0.650926 0.325463 0.945555i \(-0.394480\pi\)
0.325463 + 0.945555i \(0.394480\pi\)
\(480\) 0 0
\(481\) −11.1231 −0.507170
\(482\) 0 0
\(483\) −26.0540 −1.18550
\(484\) 0 0
\(485\) 18.2462 0.828518
\(486\) 0 0
\(487\) 6.49242 0.294200 0.147100 0.989122i \(-0.453006\pi\)
0.147100 + 0.989122i \(0.453006\pi\)
\(488\) 0 0
\(489\) −25.7538 −1.16463
\(490\) 0 0
\(491\) −32.4924 −1.46636 −0.733181 0.680033i \(-0.761965\pi\)
−0.733181 + 0.680033i \(0.761965\pi\)
\(492\) 0 0
\(493\) −0.576708 −0.0259736
\(494\) 0 0
\(495\) −0.807764 −0.0363063
\(496\) 0 0
\(497\) −12.7386 −0.571406
\(498\) 0 0
\(499\) −3.68466 −0.164948 −0.0824740 0.996593i \(-0.526282\pi\)
−0.0824740 + 0.996593i \(0.526282\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2.43845 0.108725 0.0543625 0.998521i \(-0.482687\pi\)
0.0543625 + 0.998521i \(0.482687\pi\)
\(504\) 0 0
\(505\) 0.630683 0.0280650
\(506\) 0 0
\(507\) 16.4924 0.732454
\(508\) 0 0
\(509\) 19.6155 0.869443 0.434721 0.900565i \(-0.356847\pi\)
0.434721 + 0.900565i \(0.356847\pi\)
\(510\) 0 0
\(511\) 27.7386 1.22708
\(512\) 0 0
\(513\) 5.56155 0.245549
\(514\) 0 0
\(515\) −37.1231 −1.63584
\(516\) 0 0
\(517\) 2.06913 0.0910002
\(518\) 0 0
\(519\) −31.6155 −1.38777
\(520\) 0 0
\(521\) 23.3693 1.02383 0.511914 0.859037i \(-0.328937\pi\)
0.511914 + 0.859037i \(0.328937\pi\)
\(522\) 0 0
\(523\) −32.5464 −1.42315 −0.711577 0.702608i \(-0.752019\pi\)
−0.711577 + 0.702608i \(0.752019\pi\)
\(524\) 0 0
\(525\) 7.31534 0.319268
\(526\) 0 0
\(527\) −1.12311 −0.0489232
\(528\) 0 0
\(529\) 7.93087 0.344820
\(530\) 0 0
\(531\) 2.49242 0.108162
\(532\) 0 0
\(533\) 6.24621 0.270553
\(534\) 0 0
\(535\) 43.3693 1.87502
\(536\) 0 0
\(537\) −3.50758 −0.151363
\(538\) 0 0
\(539\) 1.12311 0.0483756
\(540\) 0 0
\(541\) 35.4384 1.52362 0.761809 0.647802i \(-0.224312\pi\)
0.761809 + 0.647802i \(0.224312\pi\)
\(542\) 0 0
\(543\) −0.384472 −0.0164993
\(544\) 0 0
\(545\) 9.12311 0.390791
\(546\) 0 0
\(547\) 24.4924 1.04722 0.523610 0.851958i \(-0.324585\pi\)
0.523610 + 0.851958i \(0.324585\pi\)
\(548\) 0 0
\(549\) 8.31534 0.354890
\(550\) 0 0
\(551\) −4.68466 −0.199573
\(552\) 0 0
\(553\) 30.0000 1.27573
\(554\) 0 0
\(555\) 28.4924 1.20944
\(556\) 0 0
\(557\) 17.9309 0.759755 0.379878 0.925037i \(-0.375966\pi\)
0.379878 + 0.925037i \(0.375966\pi\)
\(558\) 0 0
\(559\) 8.49242 0.359191
\(560\) 0 0
\(561\) −0.107951 −0.00455768
\(562\) 0 0
\(563\) −30.2462 −1.27473 −0.637363 0.770564i \(-0.719975\pi\)
−0.637363 + 0.770564i \(0.719975\pi\)
\(564\) 0 0
\(565\) 12.4924 0.525560
\(566\) 0 0
\(567\) 21.0000 0.881917
\(568\) 0 0
\(569\) −26.7386 −1.12094 −0.560471 0.828174i \(-0.689380\pi\)
−0.560471 + 0.828174i \(0.689380\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) −1.56155 −0.0652348
\(574\) 0 0
\(575\) −8.68466 −0.362175
\(576\) 0 0
\(577\) 5.24621 0.218403 0.109201 0.994020i \(-0.465171\pi\)
0.109201 + 0.994020i \(0.465171\pi\)
\(578\) 0 0
\(579\) −28.8769 −1.20008
\(580\) 0 0
\(581\) 52.1080 2.16180
\(582\) 0 0
\(583\) 2.49242 0.103226
\(584\) 0 0
\(585\) −2.24621 −0.0928694
\(586\) 0 0
\(587\) −33.9309 −1.40048 −0.700238 0.713909i \(-0.746923\pi\)
−0.700238 + 0.713909i \(0.746923\pi\)
\(588\) 0 0
\(589\) −9.12311 −0.375911
\(590\) 0 0
\(591\) 17.7538 0.730293
\(592\) 0 0
\(593\) 32.7386 1.34441 0.672207 0.740363i \(-0.265346\pi\)
0.672207 + 0.740363i \(0.265346\pi\)
\(594\) 0 0
\(595\) −0.946025 −0.0387832
\(596\) 0 0
\(597\) −37.0691 −1.51714
\(598\) 0 0
\(599\) −43.8617 −1.79214 −0.896071 0.443911i \(-0.853591\pi\)
−0.896071 + 0.443911i \(0.853591\pi\)
\(600\) 0 0
\(601\) −4.63068 −0.188890 −0.0944448 0.995530i \(-0.530108\pi\)
−0.0944448 + 0.995530i \(0.530108\pi\)
\(602\) 0 0
\(603\) 0.246211 0.0100265
\(604\) 0 0
\(605\) −27.3693 −1.11272
\(606\) 0 0
\(607\) 12.8769 0.522657 0.261329 0.965250i \(-0.415839\pi\)
0.261329 + 0.965250i \(0.415839\pi\)
\(608\) 0 0
\(609\) −21.9460 −0.889298
\(610\) 0 0
\(611\) 5.75379 0.232773
\(612\) 0 0
\(613\) 41.5464 1.67804 0.839022 0.544098i \(-0.183128\pi\)
0.839022 + 0.544098i \(0.183128\pi\)
\(614\) 0 0
\(615\) −16.0000 −0.645182
\(616\) 0 0
\(617\) 21.6847 0.872991 0.436496 0.899706i \(-0.356219\pi\)
0.436496 + 0.899706i \(0.356219\pi\)
\(618\) 0 0
\(619\) −47.1231 −1.89404 −0.947019 0.321178i \(-0.895921\pi\)
−0.947019 + 0.321178i \(0.895921\pi\)
\(620\) 0 0
\(621\) −30.9309 −1.24121
\(622\) 0 0
\(623\) 46.1080 1.84728
\(624\) 0 0
\(625\) −30.3693 −1.21477
\(626\) 0 0
\(627\) −0.876894 −0.0350198
\(628\) 0 0
\(629\) −0.876894 −0.0349641
\(630\) 0 0
\(631\) 45.3002 1.80337 0.901686 0.432391i \(-0.142330\pi\)
0.901686 + 0.432391i \(0.142330\pi\)
\(632\) 0 0
\(633\) −24.3002 −0.965846
\(634\) 0 0
\(635\) −38.7386 −1.53730
\(636\) 0 0
\(637\) 3.12311 0.123742
\(638\) 0 0
\(639\) −2.38447 −0.0943282
\(640\) 0 0
\(641\) 24.1080 0.952207 0.476103 0.879389i \(-0.342049\pi\)
0.476103 + 0.879389i \(0.342049\pi\)
\(642\) 0 0
\(643\) 27.3002 1.07661 0.538307 0.842749i \(-0.319064\pi\)
0.538307 + 0.842749i \(0.319064\pi\)
\(644\) 0 0
\(645\) −21.7538 −0.856555
\(646\) 0 0
\(647\) −47.0000 −1.84776 −0.923880 0.382682i \(-0.875001\pi\)
−0.923880 + 0.382682i \(0.875001\pi\)
\(648\) 0 0
\(649\) −2.49242 −0.0978361
\(650\) 0 0
\(651\) −42.7386 −1.67506
\(652\) 0 0
\(653\) −8.80776 −0.344674 −0.172337 0.985038i \(-0.555132\pi\)
−0.172337 + 0.985038i \(0.555132\pi\)
\(654\) 0 0
\(655\) −34.4233 −1.34503
\(656\) 0 0
\(657\) 5.19224 0.202568
\(658\) 0 0
\(659\) −32.5464 −1.26783 −0.633914 0.773404i \(-0.718553\pi\)
−0.633914 + 0.773404i \(0.718553\pi\)
\(660\) 0 0
\(661\) 21.1771 0.823693 0.411846 0.911253i \(-0.364884\pi\)
0.411846 + 0.911253i \(0.364884\pi\)
\(662\) 0 0
\(663\) −0.300187 −0.0116583
\(664\) 0 0
\(665\) −7.68466 −0.297998
\(666\) 0 0
\(667\) 26.0540 1.00881
\(668\) 0 0
\(669\) −2.13826 −0.0826699
\(670\) 0 0
\(671\) −8.31534 −0.321010
\(672\) 0 0
\(673\) −21.1231 −0.814236 −0.407118 0.913376i \(-0.633466\pi\)
−0.407118 + 0.913376i \(0.633466\pi\)
\(674\) 0 0
\(675\) 8.68466 0.334273
\(676\) 0 0
\(677\) 2.43845 0.0937171 0.0468586 0.998902i \(-0.485079\pi\)
0.0468586 + 0.998902i \(0.485079\pi\)
\(678\) 0 0
\(679\) −21.3693 −0.820079
\(680\) 0 0
\(681\) −26.0540 −0.998391
\(682\) 0 0
\(683\) −17.7538 −0.679330 −0.339665 0.940547i \(-0.610314\pi\)
−0.339665 + 0.940547i \(0.610314\pi\)
\(684\) 0 0
\(685\) 36.8078 1.40635
\(686\) 0 0
\(687\) −19.6155 −0.748379
\(688\) 0 0
\(689\) 6.93087 0.264045
\(690\) 0 0
\(691\) −45.0540 −1.71393 −0.856967 0.515371i \(-0.827654\pi\)
−0.856967 + 0.515371i \(0.827654\pi\)
\(692\) 0 0
\(693\) 0.946025 0.0359365
\(694\) 0 0
\(695\) 13.9309 0.528428
\(696\) 0 0
\(697\) 0.492423 0.0186518
\(698\) 0 0
\(699\) 31.1231 1.17718
\(700\) 0 0
\(701\) −13.6155 −0.514251 −0.257126 0.966378i \(-0.582775\pi\)
−0.257126 + 0.966378i \(0.582775\pi\)
\(702\) 0 0
\(703\) −7.12311 −0.268653
\(704\) 0 0
\(705\) −14.7386 −0.555089
\(706\) 0 0
\(707\) −0.738634 −0.0277792
\(708\) 0 0
\(709\) −50.4924 −1.89628 −0.948141 0.317849i \(-0.897040\pi\)
−0.948141 + 0.317849i \(0.897040\pi\)
\(710\) 0 0
\(711\) 5.61553 0.210599
\(712\) 0 0
\(713\) 50.7386 1.90018
\(714\) 0 0
\(715\) 2.24621 0.0840035
\(716\) 0 0
\(717\) 23.4233 0.874759
\(718\) 0 0
\(719\) −23.4924 −0.876120 −0.438060 0.898946i \(-0.644334\pi\)
−0.438060 + 0.898946i \(0.644334\pi\)
\(720\) 0 0
\(721\) 43.4773 1.61918
\(722\) 0 0
\(723\) 22.2462 0.827345
\(724\) 0 0
\(725\) −7.31534 −0.271685
\(726\) 0 0
\(727\) −22.3693 −0.829632 −0.414816 0.909905i \(-0.636154\pi\)
−0.414816 + 0.909905i \(0.636154\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) 0.669503 0.0247625
\(732\) 0 0
\(733\) −3.36932 −0.124449 −0.0622243 0.998062i \(-0.519819\pi\)
−0.0622243 + 0.998062i \(0.519819\pi\)
\(734\) 0 0
\(735\) −8.00000 −0.295084
\(736\) 0 0
\(737\) −0.246211 −0.00906931
\(738\) 0 0
\(739\) 1.43845 0.0529141 0.0264571 0.999650i \(-0.491577\pi\)
0.0264571 + 0.999650i \(0.491577\pi\)
\(740\) 0 0
\(741\) −2.43845 −0.0895786
\(742\) 0 0
\(743\) −22.8769 −0.839272 −0.419636 0.907693i \(-0.637842\pi\)
−0.419636 + 0.907693i \(0.637842\pi\)
\(744\) 0 0
\(745\) −27.0540 −0.991181
\(746\) 0 0
\(747\) 9.75379 0.356872
\(748\) 0 0
\(749\) −50.7926 −1.85592
\(750\) 0 0
\(751\) 12.0000 0.437886 0.218943 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 0 0
\(753\) −8.87689 −0.323492
\(754\) 0 0
\(755\) 57.6155 2.09684
\(756\) 0 0
\(757\) −21.0540 −0.765220 −0.382610 0.923910i \(-0.624975\pi\)
−0.382610 + 0.923910i \(0.624975\pi\)
\(758\) 0 0
\(759\) 4.87689 0.177020
\(760\) 0 0
\(761\) −40.8617 −1.48124 −0.740618 0.671926i \(-0.765467\pi\)
−0.740618 + 0.671926i \(0.765467\pi\)
\(762\) 0 0
\(763\) −10.6847 −0.386811
\(764\) 0 0
\(765\) −0.177081 −0.00640237
\(766\) 0 0
\(767\) −6.93087 −0.250259
\(768\) 0 0
\(769\) 12.3693 0.446049 0.223024 0.974813i \(-0.428407\pi\)
0.223024 + 0.974813i \(0.428407\pi\)
\(770\) 0 0
\(771\) −40.9848 −1.47603
\(772\) 0 0
\(773\) 32.0540 1.15290 0.576451 0.817132i \(-0.304437\pi\)
0.576451 + 0.817132i \(0.304437\pi\)
\(774\) 0 0
\(775\) −14.2462 −0.511739
\(776\) 0 0
\(777\) −33.3693 −1.19712
\(778\) 0 0
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) 2.38447 0.0853231
\(782\) 0 0
\(783\) −26.0540 −0.931093
\(784\) 0 0
\(785\) 19.8617 0.708896
\(786\) 0 0
\(787\) 22.4384 0.799844 0.399922 0.916549i \(-0.369037\pi\)
0.399922 + 0.916549i \(0.369037\pi\)
\(788\) 0 0
\(789\) 17.2614 0.614521
\(790\) 0 0
\(791\) −14.6307 −0.520207
\(792\) 0 0
\(793\) −23.1231 −0.821126
\(794\) 0 0
\(795\) −17.7538 −0.629662
\(796\) 0 0
\(797\) 2.68466 0.0950955 0.0475477 0.998869i \(-0.484859\pi\)
0.0475477 + 0.998869i \(0.484859\pi\)
\(798\) 0 0
\(799\) 0.453602 0.0160473
\(800\) 0 0
\(801\) 8.63068 0.304950
\(802\) 0 0
\(803\) −5.19224 −0.183230
\(804\) 0 0
\(805\) 42.7386 1.50634
\(806\) 0 0
\(807\) −28.8769 −1.01651
\(808\) 0 0
\(809\) 6.12311 0.215277 0.107638 0.994190i \(-0.465671\pi\)
0.107638 + 0.994190i \(0.465671\pi\)
\(810\) 0 0
\(811\) −11.4233 −0.401126 −0.200563 0.979681i \(-0.564277\pi\)
−0.200563 + 0.979681i \(0.564277\pi\)
\(812\) 0 0
\(813\) 15.6998 0.550616
\(814\) 0 0
\(815\) 42.2462 1.47982
\(816\) 0 0
\(817\) 5.43845 0.190267
\(818\) 0 0
\(819\) 2.63068 0.0919235
\(820\) 0 0
\(821\) −4.31534 −0.150606 −0.0753032 0.997161i \(-0.523992\pi\)
−0.0753032 + 0.997161i \(0.523992\pi\)
\(822\) 0 0
\(823\) −20.3693 −0.710030 −0.355015 0.934861i \(-0.615524\pi\)
−0.355015 + 0.934861i \(0.615524\pi\)
\(824\) 0 0
\(825\) −1.36932 −0.0476735
\(826\) 0 0
\(827\) 19.5616 0.680222 0.340111 0.940385i \(-0.389535\pi\)
0.340111 + 0.940385i \(0.389535\pi\)
\(828\) 0 0
\(829\) −18.5464 −0.644143 −0.322072 0.946715i \(-0.604379\pi\)
−0.322072 + 0.946715i \(0.604379\pi\)
\(830\) 0 0
\(831\) −18.2462 −0.632954
\(832\) 0 0
\(833\) 0.246211 0.00853071
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −50.7386 −1.75378
\(838\) 0 0
\(839\) −29.6155 −1.02244 −0.511221 0.859449i \(-0.670807\pi\)
−0.511221 + 0.859449i \(0.670807\pi\)
\(840\) 0 0
\(841\) −7.05398 −0.243241
\(842\) 0 0
\(843\) 0.984845 0.0339199
\(844\) 0 0
\(845\) −27.0540 −0.930685
\(846\) 0 0
\(847\) 32.0540 1.10139
\(848\) 0 0
\(849\) −44.6004 −1.53068
\(850\) 0 0
\(851\) 39.6155 1.35800
\(852\) 0 0
\(853\) −32.2462 −1.10409 −0.552045 0.833815i \(-0.686152\pi\)
−0.552045 + 0.833815i \(0.686152\pi\)
\(854\) 0 0
\(855\) −1.43845 −0.0491939
\(856\) 0 0
\(857\) 12.6307 0.431456 0.215728 0.976454i \(-0.430788\pi\)
0.215728 + 0.976454i \(0.430788\pi\)
\(858\) 0 0
\(859\) −35.9309 −1.22595 −0.612973 0.790104i \(-0.710026\pi\)
−0.612973 + 0.790104i \(0.710026\pi\)
\(860\) 0 0
\(861\) 18.7386 0.638611
\(862\) 0 0
\(863\) −22.6307 −0.770357 −0.385179 0.922842i \(-0.625860\pi\)
−0.385179 + 0.922842i \(0.625860\pi\)
\(864\) 0 0
\(865\) 51.8617 1.76335
\(866\) 0 0
\(867\) 26.5227 0.900759
\(868\) 0 0
\(869\) −5.61553 −0.190494
\(870\) 0 0
\(871\) −0.684658 −0.0231988
\(872\) 0 0
\(873\) −4.00000 −0.135379
\(874\) 0 0
\(875\) 26.4233 0.893270
\(876\) 0 0
\(877\) −2.68466 −0.0906545 −0.0453272 0.998972i \(-0.514433\pi\)
−0.0453272 + 0.998972i \(0.514433\pi\)
\(878\) 0 0
\(879\) 23.9157 0.806657
\(880\) 0 0
\(881\) 49.6847 1.67392 0.836959 0.547265i \(-0.184331\pi\)
0.836959 + 0.547265i \(0.184331\pi\)
\(882\) 0 0
\(883\) 0.0691303 0.00232642 0.00116321 0.999999i \(-0.499630\pi\)
0.00116321 + 0.999999i \(0.499630\pi\)
\(884\) 0 0
\(885\) 17.7538 0.596787
\(886\) 0 0
\(887\) 37.2311 1.25010 0.625048 0.780586i \(-0.285079\pi\)
0.625048 + 0.780586i \(0.285079\pi\)
\(888\) 0 0
\(889\) 45.3693 1.52164
\(890\) 0 0
\(891\) −3.93087 −0.131689
\(892\) 0 0
\(893\) 3.68466 0.123302
\(894\) 0 0
\(895\) 5.75379 0.192328
\(896\) 0 0
\(897\) 13.5616 0.452807
\(898\) 0 0
\(899\) 42.7386 1.42541
\(900\) 0 0
\(901\) 0.546398 0.0182032
\(902\) 0 0
\(903\) 25.4773 0.847830
\(904\) 0 0
\(905\) 0.630683 0.0209646
\(906\) 0 0
\(907\) −15.5616 −0.516713 −0.258356 0.966050i \(-0.583181\pi\)
−0.258356 + 0.966050i \(0.583181\pi\)
\(908\) 0 0
\(909\) −0.138261 −0.00458582
\(910\) 0 0
\(911\) 9.12311 0.302262 0.151131 0.988514i \(-0.451708\pi\)
0.151131 + 0.988514i \(0.451708\pi\)
\(912\) 0 0
\(913\) −9.75379 −0.322803
\(914\) 0 0
\(915\) 59.2311 1.95812
\(916\) 0 0
\(917\) 40.3153 1.33133
\(918\) 0 0
\(919\) 24.3002 0.801589 0.400795 0.916168i \(-0.368734\pi\)
0.400795 + 0.916168i \(0.368734\pi\)
\(920\) 0 0
\(921\) −7.23106 −0.238271
\(922\) 0 0
\(923\) 6.63068 0.218252
\(924\) 0 0
\(925\) −11.1231 −0.365725
\(926\) 0 0
\(927\) 8.13826 0.267296
\(928\) 0 0
\(929\) −55.1771 −1.81030 −0.905151 0.425091i \(-0.860242\pi\)
−0.905151 + 0.425091i \(0.860242\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) −9.56155 −0.313031
\(934\) 0 0
\(935\) 0.177081 0.00579117
\(936\) 0 0
\(937\) −12.7538 −0.416648 −0.208324 0.978060i \(-0.566801\pi\)
−0.208324 + 0.978060i \(0.566801\pi\)
\(938\) 0 0
\(939\) 1.45360 0.0474365
\(940\) 0 0
\(941\) −18.3002 −0.596569 −0.298285 0.954477i \(-0.596414\pi\)
−0.298285 + 0.954477i \(0.596414\pi\)
\(942\) 0 0
\(943\) −22.2462 −0.724436
\(944\) 0 0
\(945\) −42.7386 −1.39029
\(946\) 0 0
\(947\) 36.9848 1.20185 0.600923 0.799307i \(-0.294800\pi\)
0.600923 + 0.799307i \(0.294800\pi\)
\(948\) 0 0
\(949\) −14.4384 −0.468692
\(950\) 0 0
\(951\) 25.6695 0.832391
\(952\) 0 0
\(953\) −8.24621 −0.267121 −0.133560 0.991041i \(-0.542641\pi\)
−0.133560 + 0.991041i \(0.542641\pi\)
\(954\) 0 0
\(955\) 2.56155 0.0828899
\(956\) 0 0
\(957\) 4.10795 0.132791
\(958\) 0 0
\(959\) −43.1080 −1.39203
\(960\) 0 0
\(961\) 52.2311 1.68487
\(962\) 0 0
\(963\) −9.50758 −0.306377
\(964\) 0 0
\(965\) 47.3693 1.52487
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) 0 0
\(969\) −0.192236 −0.00617551
\(970\) 0 0
\(971\) −23.5076 −0.754394 −0.377197 0.926133i \(-0.623112\pi\)
−0.377197 + 0.926133i \(0.623112\pi\)
\(972\) 0 0
\(973\) −16.3153 −0.523046
\(974\) 0 0
\(975\) −3.80776 −0.121946
\(976\) 0 0
\(977\) −53.6155 −1.71531 −0.857656 0.514223i \(-0.828080\pi\)
−0.857656 + 0.514223i \(0.828080\pi\)
\(978\) 0 0
\(979\) −8.63068 −0.275838
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) −11.6155 −0.370478 −0.185239 0.982694i \(-0.559306\pi\)
−0.185239 + 0.982694i \(0.559306\pi\)
\(984\) 0 0
\(985\) −29.1231 −0.927939
\(986\) 0 0
\(987\) 17.2614 0.549435
\(988\) 0 0
\(989\) −30.2462 −0.961774
\(990\) 0 0
\(991\) 18.2462 0.579610 0.289805 0.957086i \(-0.406410\pi\)
0.289805 + 0.957086i \(0.406410\pi\)
\(992\) 0 0
\(993\) −6.93087 −0.219945
\(994\) 0 0
\(995\) 60.8078 1.92774
\(996\) 0 0
\(997\) 46.9157 1.48584 0.742918 0.669383i \(-0.233441\pi\)
0.742918 + 0.669383i \(0.233441\pi\)
\(998\) 0 0
\(999\) −39.6155 −1.25338
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 1216.2.a.t.1.1 2
4.3 odd 2 1216.2.a.s.1.2 2
8.3 odd 2 608.2.a.h.1.1 yes 2
8.5 even 2 608.2.a.g.1.2 2
24.5 odd 2 5472.2.a.bc.1.2 2
24.11 even 2 5472.2.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.g.1.2 2 8.5 even 2
608.2.a.h.1.1 yes 2 8.3 odd 2
1216.2.a.s.1.2 2 4.3 odd 2
1216.2.a.t.1.1 2 1.1 even 1 trivial
5472.2.a.bc.1.2 2 24.5 odd 2
5472.2.a.bf.1.2 2 24.11 even 2