Properties

Label 1216.2.a.x.1.1
Level $1216$
Weight $2$
Character 1216.1
Self dual yes
Analytic conductor $9.710$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.329727\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155 q^{3} -3.40617 q^{5} -2.50407 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q-1.56155 q^{3} -3.40617 q^{5} -2.50407 q^{7} -0.561553 q^{9} -1.40617 q^{11} -6.22101 q^{13} +5.31891 q^{15} -3.16352 q^{17} +1.00000 q^{19} +3.91023 q^{21} +7.91023 q^{23} +6.60197 q^{25} +5.56155 q^{27} -6.22101 q^{29} -8.13124 q^{31} +2.19580 q^{33} +8.52927 q^{35} +6.13124 q^{37} +9.71443 q^{39} +1.68923 q^{41} +1.71694 q^{43} +1.91274 q^{45} -9.84818 q^{47} -0.729644 q^{49} +4.94001 q^{51} -2.78713 q^{53} +4.78964 q^{55} -1.56155 q^{57} +9.25078 q^{59} -6.52927 q^{61} +1.40617 q^{63} +21.1898 q^{65} +1.87233 q^{67} -12.3522 q^{69} +13.0081 q^{71} -6.28663 q^{73} -10.3093 q^{75} +3.52114 q^{77} +11.2543 q^{79} -7.00000 q^{81} +1.80420 q^{83} +10.7755 q^{85} +9.71443 q^{87} -7.57426 q^{89} +15.5778 q^{91} +12.6974 q^{93} -3.40617 q^{95} +4.81233 q^{97} +0.789637 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - q^{5} + q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - q^{5} + q^{7} + 6 q^{9} + 7 q^{11} - 10 q^{13} + 8 q^{15} + 5 q^{17} + 4 q^{19} - 8 q^{21} + 8 q^{23} + 17 q^{25} + 14 q^{27} - 10 q^{29} + 6 q^{31} + 12 q^{33} + 5 q^{35} - 14 q^{37} + 12 q^{39} - 2 q^{41} + 3 q^{43} + 7 q^{45} + 3 q^{47} + 7 q^{49} - 6 q^{51} - 4 q^{53} + 35 q^{55} + 2 q^{57} + 20 q^{59} + 3 q^{61} - 7 q^{63} - 12 q^{65} + 8 q^{67} + 4 q^{69} + 30 q^{71} + 9 q^{73} + 34 q^{75} + 7 q^{77} - 10 q^{79} - 28 q^{81} + 4 q^{83} - 19 q^{85} + 12 q^{87} - 16 q^{89} + 10 q^{91} + 20 q^{93} - q^{95} - 6 q^{97} + 19 q^{99}+O(q^{100}) \) 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 0
\(3\) −1.56155 −0.901563 −0.450781 0.892634i \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 0 0
\(5\) −3.40617 −1.52328 −0.761642 0.647998i \(-0.775606\pi\)
−0.761642 + 0.647998i \(0.775606\pi\)
\(6\) 0 0
\(7\) −2.50407 −0.946449 −0.473224 0.880942i \(-0.656910\pi\)
−0.473224 + 0.880942i \(0.656910\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −1.40617 −0.423975 −0.211988 0.977272i \(-0.567994\pi\)
−0.211988 + 0.977272i \(0.567994\pi\)
\(12\) 0 0
\(13\) −6.22101 −1.72540 −0.862698 0.505719i \(-0.831227\pi\)
−0.862698 + 0.505719i \(0.831227\pi\)
\(14\) 0 0
\(15\) 5.31891 1.37334
\(16\) 0 0
\(17\) −3.16352 −0.767267 −0.383633 0.923485i \(-0.625327\pi\)
−0.383633 + 0.923485i \(0.625327\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 3.91023 0.853283
\(22\) 0 0
\(23\) 7.91023 1.64940 0.824699 0.565572i \(-0.191345\pi\)
0.824699 + 0.565572i \(0.191345\pi\)
\(24\) 0 0
\(25\) 6.60197 1.32039
\(26\) 0 0
\(27\) 5.56155 1.07032
\(28\) 0 0
\(29\) −6.22101 −1.15521 −0.577606 0.816316i \(-0.696013\pi\)
−0.577606 + 0.816316i \(0.696013\pi\)
\(30\) 0 0
\(31\) −8.13124 −1.46041 −0.730207 0.683226i \(-0.760576\pi\)
−0.730207 + 0.683226i \(0.760576\pi\)
\(32\) 0 0
\(33\) 2.19580 0.382240
\(34\) 0 0
\(35\) 8.52927 1.44171
\(36\) 0 0
\(37\) 6.13124 1.00797 0.503985 0.863712i \(-0.331867\pi\)
0.503985 + 0.863712i \(0.331867\pi\)
\(38\) 0 0
\(39\) 9.71443 1.55555
\(40\) 0 0
\(41\) 1.68923 0.263813 0.131906 0.991262i \(-0.457890\pi\)
0.131906 + 0.991262i \(0.457890\pi\)
\(42\) 0 0
\(43\) 1.71694 0.261831 0.130915 0.991394i \(-0.458208\pi\)
0.130915 + 0.991394i \(0.458208\pi\)
\(44\) 0 0
\(45\) 1.91274 0.285135
\(46\) 0 0
\(47\) −9.84818 −1.43650 −0.718252 0.695783i \(-0.755058\pi\)
−0.718252 + 0.695783i \(0.755058\pi\)
\(48\) 0 0
\(49\) −0.729644 −0.104235
\(50\) 0 0
\(51\) 4.94001 0.691739
\(52\) 0 0
\(53\) −2.78713 −0.382842 −0.191421 0.981508i \(-0.561310\pi\)
−0.191421 + 0.981508i \(0.561310\pi\)
\(54\) 0 0
\(55\) 4.78964 0.645834
\(56\) 0 0
\(57\) −1.56155 −0.206833
\(58\) 0 0
\(59\) 9.25078 1.20435 0.602174 0.798365i \(-0.294301\pi\)
0.602174 + 0.798365i \(0.294301\pi\)
\(60\) 0 0
\(61\) −6.52927 −0.835988 −0.417994 0.908450i \(-0.637267\pi\)
−0.417994 + 0.908450i \(0.637267\pi\)
\(62\) 0 0
\(63\) 1.40617 0.177160
\(64\) 0 0
\(65\) 21.1898 2.62827
\(66\) 0 0
\(67\) 1.87233 0.228741 0.114370 0.993438i \(-0.463515\pi\)
0.114370 + 0.993438i \(0.463515\pi\)
\(68\) 0 0
\(69\) −12.3522 −1.48704
\(70\) 0 0
\(71\) 13.0081 1.54378 0.771891 0.635755i \(-0.219311\pi\)
0.771891 + 0.635755i \(0.219311\pi\)
\(72\) 0 0
\(73\) −6.28663 −0.735794 −0.367897 0.929867i \(-0.619922\pi\)
−0.367897 + 0.929867i \(0.619922\pi\)
\(74\) 0 0
\(75\) −10.3093 −1.19042
\(76\) 0 0
\(77\) 3.52114 0.401271
\(78\) 0 0
\(79\) 11.2543 1.26621 0.633106 0.774065i \(-0.281780\pi\)
0.633106 + 0.774065i \(0.281780\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 1.80420 0.198036 0.0990182 0.995086i \(-0.468430\pi\)
0.0990182 + 0.995086i \(0.468430\pi\)
\(84\) 0 0
\(85\) 10.7755 1.16877
\(86\) 0 0
\(87\) 9.71443 1.04150
\(88\) 0 0
\(89\) −7.57426 −0.802870 −0.401435 0.915888i \(-0.631488\pi\)
−0.401435 + 0.915888i \(0.631488\pi\)
\(90\) 0 0
\(91\) 15.5778 1.63300
\(92\) 0 0
\(93\) 12.6974 1.31666
\(94\) 0 0
\(95\) −3.40617 −0.349465
\(96\) 0 0
\(97\) 4.81233 0.488618 0.244309 0.969697i \(-0.421439\pi\)
0.244309 + 0.969697i \(0.421439\pi\)
\(98\) 0 0
\(99\) 0.789637 0.0793615
\(100\) 0 0
\(101\) 16.2462 1.61656 0.808279 0.588799i \(-0.200399\pi\)
0.808279 + 0.588799i \(0.200399\pi\)
\(102\) 0 0
\(103\) −7.68923 −0.757642 −0.378821 0.925470i \(-0.623670\pi\)
−0.378821 + 0.925470i \(0.623670\pi\)
\(104\) 0 0
\(105\) −13.3189 −1.29979
\(106\) 0 0
\(107\) −11.8886 −1.14931 −0.574657 0.818394i \(-0.694865\pi\)
−0.574657 + 0.818394i \(0.694865\pi\)
\(108\) 0 0
\(109\) 5.65489 0.541640 0.270820 0.962630i \(-0.412705\pi\)
0.270820 + 0.962630i \(0.412705\pi\)
\(110\) 0 0
\(111\) −9.57426 −0.908748
\(112\) 0 0
\(113\) −8.81233 −0.828995 −0.414497 0.910051i \(-0.636043\pi\)
−0.414497 + 0.910051i \(0.636043\pi\)
\(114\) 0 0
\(115\) −26.9436 −2.51250
\(116\) 0 0
\(117\) 3.49342 0.322967
\(118\) 0 0
\(119\) 7.92167 0.726179
\(120\) 0 0
\(121\) −9.02270 −0.820245
\(122\) 0 0
\(123\) −2.63782 −0.237844
\(124\) 0 0
\(125\) −5.45657 −0.488051
\(126\) 0 0
\(127\) −12.0504 −1.06930 −0.534650 0.845073i \(-0.679557\pi\)
−0.534650 + 0.845073i \(0.679557\pi\)
\(128\) 0 0
\(129\) −2.68109 −0.236057
\(130\) 0 0
\(131\) 0.920878 0.0804575 0.0402287 0.999190i \(-0.487191\pi\)
0.0402287 + 0.999190i \(0.487191\pi\)
\(132\) 0 0
\(133\) −2.50407 −0.217130
\(134\) 0 0
\(135\) −18.9436 −1.63040
\(136\) 0 0
\(137\) −0.525705 −0.0449140 −0.0224570 0.999748i \(-0.507149\pi\)
−0.0224570 + 0.999748i \(0.507149\pi\)
\(138\) 0 0
\(139\) −4.52927 −0.384168 −0.192084 0.981379i \(-0.561525\pi\)
−0.192084 + 0.981379i \(0.561525\pi\)
\(140\) 0 0
\(141\) 15.3785 1.29510
\(142\) 0 0
\(143\) 8.74777 0.731525
\(144\) 0 0
\(145\) 21.1898 1.75972
\(146\) 0 0
\(147\) 1.13938 0.0939743
\(148\) 0 0
\(149\) 18.9713 1.55419 0.777094 0.629384i \(-0.216693\pi\)
0.777094 + 0.629384i \(0.216693\pi\)
\(150\) 0 0
\(151\) −6.55698 −0.533600 −0.266800 0.963752i \(-0.585966\pi\)
−0.266800 + 0.963752i \(0.585966\pi\)
\(152\) 0 0
\(153\) 1.77648 0.143620
\(154\) 0 0
\(155\) 27.6964 2.22463
\(156\) 0 0
\(157\) −17.5651 −1.40185 −0.700925 0.713235i \(-0.747229\pi\)
−0.700925 + 0.713235i \(0.747229\pi\)
\(158\) 0 0
\(159\) 4.35225 0.345156
\(160\) 0 0
\(161\) −19.8078 −1.56107
\(162\) 0 0
\(163\) −22.9436 −1.79708 −0.898540 0.438892i \(-0.855371\pi\)
−0.898540 + 0.438892i \(0.855371\pi\)
\(164\) 0 0
\(165\) −7.47927 −0.582260
\(166\) 0 0
\(167\) −6.24621 −0.483346 −0.241673 0.970358i \(-0.577696\pi\)
−0.241673 + 0.970358i \(0.577696\pi\)
\(168\) 0 0
\(169\) 25.7009 1.97699
\(170\) 0 0
\(171\) −0.561553 −0.0429430
\(172\) 0 0
\(173\) 16.9436 1.28820 0.644098 0.764943i \(-0.277233\pi\)
0.644098 + 0.764943i \(0.277233\pi\)
\(174\) 0 0
\(175\) −16.5318 −1.24969
\(176\) 0 0
\(177\) −14.4456 −1.08580
\(178\) 0 0
\(179\) 18.2462 1.36379 0.681893 0.731452i \(-0.261157\pi\)
0.681893 + 0.731452i \(0.261157\pi\)
\(180\) 0 0
\(181\) −10.9273 −0.812220 −0.406110 0.913824i \(-0.633115\pi\)
−0.406110 + 0.913824i \(0.633115\pi\)
\(182\) 0 0
\(183\) 10.1958 0.753695
\(184\) 0 0
\(185\) −20.8840 −1.53542
\(186\) 0 0
\(187\) 4.44844 0.325302
\(188\) 0 0
\(189\) −13.9265 −1.01300
\(190\) 0 0
\(191\) 15.5122 1.12242 0.561212 0.827672i \(-0.310335\pi\)
0.561212 + 0.827672i \(0.310335\pi\)
\(192\) 0 0
\(193\) −9.56512 −0.688512 −0.344256 0.938876i \(-0.611869\pi\)
−0.344256 + 0.938876i \(0.611869\pi\)
\(194\) 0 0
\(195\) −33.0890 −2.36955
\(196\) 0 0
\(197\) −19.1394 −1.36362 −0.681812 0.731527i \(-0.738808\pi\)
−0.681812 + 0.731527i \(0.738808\pi\)
\(198\) 0 0
\(199\) 0.133749 0.00948125 0.00474062 0.999989i \(-0.498491\pi\)
0.00474062 + 0.999989i \(0.498491\pi\)
\(200\) 0 0
\(201\) −2.92374 −0.206224
\(202\) 0 0
\(203\) 15.5778 1.09335
\(204\) 0 0
\(205\) −5.75379 −0.401862
\(206\) 0 0
\(207\) −4.44201 −0.308741
\(208\) 0 0
\(209\) −1.40617 −0.0972666
\(210\) 0 0
\(211\) −23.0117 −1.58419 −0.792095 0.610397i \(-0.791010\pi\)
−0.792095 + 0.610397i \(0.791010\pi\)
\(212\) 0 0
\(213\) −20.3129 −1.39182
\(214\) 0 0
\(215\) −5.84818 −0.398843
\(216\) 0 0
\(217\) 20.3612 1.38221
\(218\) 0 0
\(219\) 9.81690 0.663365
\(220\) 0 0
\(221\) 19.6803 1.32384
\(222\) 0 0
\(223\) −7.51471 −0.503222 −0.251611 0.967828i \(-0.580960\pi\)
−0.251611 + 0.967828i \(0.580960\pi\)
\(224\) 0 0
\(225\) −3.70735 −0.247157
\(226\) 0 0
\(227\) 5.81690 0.386081 0.193041 0.981191i \(-0.438165\pi\)
0.193041 + 0.981191i \(0.438165\pi\)
\(228\) 0 0
\(229\) 26.1702 1.72938 0.864688 0.502309i \(-0.167516\pi\)
0.864688 + 0.502309i \(0.167516\pi\)
\(230\) 0 0
\(231\) −5.49844 −0.361771
\(232\) 0 0
\(233\) 2.22352 0.145667 0.0728337 0.997344i \(-0.476796\pi\)
0.0728337 + 0.997344i \(0.476796\pi\)
\(234\) 0 0
\(235\) 33.5445 2.18820
\(236\) 0 0
\(237\) −17.5743 −1.14157
\(238\) 0 0
\(239\) −8.12873 −0.525804 −0.262902 0.964823i \(-0.584680\pi\)
−0.262902 + 0.964823i \(0.584680\pi\)
\(240\) 0 0
\(241\) 25.9517 1.67170 0.835848 0.548960i \(-0.184976\pi\)
0.835848 + 0.548960i \(0.184976\pi\)
\(242\) 0 0
\(243\) −5.75379 −0.369106
\(244\) 0 0
\(245\) 2.48529 0.158779
\(246\) 0 0
\(247\) −6.22101 −0.395833
\(248\) 0 0
\(249\) −2.81735 −0.178542
\(250\) 0 0
\(251\) −1.53741 −0.0970403 −0.0485202 0.998822i \(-0.515451\pi\)
−0.0485202 + 0.998822i \(0.515451\pi\)
\(252\) 0 0
\(253\) −11.1231 −0.699304
\(254\) 0 0
\(255\) −16.8265 −1.05372
\(256\) 0 0
\(257\) −0.370318 −0.0230998 −0.0115499 0.999933i \(-0.503677\pi\)
−0.0115499 + 0.999933i \(0.503677\pi\)
\(258\) 0 0
\(259\) −15.3530 −0.953992
\(260\) 0 0
\(261\) 3.49342 0.216238
\(262\) 0 0
\(263\) 18.9067 1.16584 0.582919 0.812530i \(-0.301910\pi\)
0.582919 + 0.812530i \(0.301910\pi\)
\(264\) 0 0
\(265\) 9.49342 0.583176
\(266\) 0 0
\(267\) 11.8276 0.723838
\(268\) 0 0
\(269\) 15.1898 0.926138 0.463069 0.886322i \(-0.346748\pi\)
0.463069 + 0.886322i \(0.346748\pi\)
\(270\) 0 0
\(271\) 28.4027 1.72534 0.862669 0.505768i \(-0.168791\pi\)
0.862669 + 0.505768i \(0.168791\pi\)
\(272\) 0 0
\(273\) −24.3256 −1.47225
\(274\) 0 0
\(275\) −9.28347 −0.559814
\(276\) 0 0
\(277\) −28.4647 −1.71028 −0.855139 0.518398i \(-0.826529\pi\)
−0.855139 + 0.518398i \(0.826529\pi\)
\(278\) 0 0
\(279\) 4.56612 0.273367
\(280\) 0 0
\(281\) −27.1394 −1.61900 −0.809500 0.587120i \(-0.800262\pi\)
−0.809500 + 0.587120i \(0.800262\pi\)
\(282\) 0 0
\(283\) −26.9117 −1.59974 −0.799868 0.600175i \(-0.795097\pi\)
−0.799868 + 0.600175i \(0.795097\pi\)
\(284\) 0 0
\(285\) 5.31891 0.315065
\(286\) 0 0
\(287\) −4.22994 −0.249685
\(288\) 0 0
\(289\) −6.99213 −0.411302
\(290\) 0 0
\(291\) −7.51471 −0.440520
\(292\) 0 0
\(293\) −10.2714 −0.600062 −0.300031 0.953929i \(-0.596997\pi\)
−0.300031 + 0.953929i \(0.596997\pi\)
\(294\) 0 0
\(295\) −31.5097 −1.83457
\(296\) 0 0
\(297\) −7.82047 −0.453790
\(298\) 0 0
\(299\) −49.2096 −2.84587
\(300\) 0 0
\(301\) −4.29933 −0.247809
\(302\) 0 0
\(303\) −25.3693 −1.45743
\(304\) 0 0
\(305\) 22.2398 1.27345
\(306\) 0 0
\(307\) 5.05854 0.288706 0.144353 0.989526i \(-0.453890\pi\)
0.144353 + 0.989526i \(0.453890\pi\)
\(308\) 0 0
\(309\) 12.0071 0.683062
\(310\) 0 0
\(311\) 13.4222 0.761105 0.380553 0.924759i \(-0.375734\pi\)
0.380553 + 0.924759i \(0.375734\pi\)
\(312\) 0 0
\(313\) 6.94001 0.392272 0.196136 0.980577i \(-0.437161\pi\)
0.196136 + 0.980577i \(0.437161\pi\)
\(314\) 0 0
\(315\) −4.78964 −0.269865
\(316\) 0 0
\(317\) −5.60448 −0.314779 −0.157389 0.987537i \(-0.550308\pi\)
−0.157389 + 0.987537i \(0.550308\pi\)
\(318\) 0 0
\(319\) 8.74777 0.489781
\(320\) 0 0
\(321\) 18.5647 1.03618
\(322\) 0 0
\(323\) −3.16352 −0.176023
\(324\) 0 0
\(325\) −41.0709 −2.27820
\(326\) 0 0
\(327\) −8.83040 −0.488322
\(328\) 0 0
\(329\) 24.6605 1.35958
\(330\) 0 0
\(331\) 4.11140 0.225983 0.112992 0.993596i \(-0.463957\pi\)
0.112992 + 0.993596i \(0.463957\pi\)
\(332\) 0 0
\(333\) −3.44302 −0.188676
\(334\) 0 0
\(335\) −6.37745 −0.348437
\(336\) 0 0
\(337\) −0.632801 −0.0344709 −0.0172354 0.999851i \(-0.505486\pi\)
−0.0172354 + 0.999851i \(0.505486\pi\)
\(338\) 0 0
\(339\) 13.7609 0.747391
\(340\) 0 0
\(341\) 11.4339 0.619179
\(342\) 0 0
\(343\) 19.3556 1.04510
\(344\) 0 0
\(345\) 42.0738 2.26518
\(346\) 0 0
\(347\) −3.02871 −0.162590 −0.0812949 0.996690i \(-0.525906\pi\)
−0.0812949 + 0.996690i \(0.525906\pi\)
\(348\) 0 0
\(349\) −19.0954 −1.02215 −0.511076 0.859535i \(-0.670753\pi\)
−0.511076 + 0.859535i \(0.670753\pi\)
\(350\) 0 0
\(351\) −34.5985 −1.84673
\(352\) 0 0
\(353\) 15.2025 0.809147 0.404573 0.914506i \(-0.367420\pi\)
0.404573 + 0.914506i \(0.367420\pi\)
\(354\) 0 0
\(355\) −44.3079 −2.35162
\(356\) 0 0
\(357\) −12.3701 −0.654696
\(358\) 0 0
\(359\) 34.9411 1.84412 0.922059 0.387048i \(-0.126505\pi\)
0.922059 + 0.387048i \(0.126505\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 14.0894 0.739503
\(364\) 0 0
\(365\) 21.4133 1.12082
\(366\) 0 0
\(367\) −7.60839 −0.397155 −0.198577 0.980085i \(-0.563632\pi\)
−0.198577 + 0.980085i \(0.563632\pi\)
\(368\) 0 0
\(369\) −0.948590 −0.0493816
\(370\) 0 0
\(371\) 6.97916 0.362340
\(372\) 0 0
\(373\) −9.38739 −0.486060 −0.243030 0.970019i \(-0.578141\pi\)
−0.243030 + 0.970019i \(0.578141\pi\)
\(374\) 0 0
\(375\) 8.52073 0.440009
\(376\) 0 0
\(377\) 38.7009 1.99320
\(378\) 0 0
\(379\) 11.0046 0.565267 0.282633 0.959228i \(-0.408792\pi\)
0.282633 + 0.959228i \(0.408792\pi\)
\(380\) 0 0
\(381\) 18.8173 0.964042
\(382\) 0 0
\(383\) −20.6923 −1.05733 −0.528665 0.848831i \(-0.677307\pi\)
−0.528665 + 0.848831i \(0.677307\pi\)
\(384\) 0 0
\(385\) −11.9936 −0.611249
\(386\) 0 0
\(387\) −0.964152 −0.0490106
\(388\) 0 0
\(389\) 14.4306 0.731659 0.365830 0.930682i \(-0.380785\pi\)
0.365830 + 0.930682i \(0.380785\pi\)
\(390\) 0 0
\(391\) −25.0242 −1.26553
\(392\) 0 0
\(393\) −1.43800 −0.0725375
\(394\) 0 0
\(395\) −38.3342 −1.92880
\(396\) 0 0
\(397\) 14.2739 0.716388 0.358194 0.933647i \(-0.383393\pi\)
0.358194 + 0.933647i \(0.383393\pi\)
\(398\) 0 0
\(399\) 3.91023 0.195757
\(400\) 0 0
\(401\) 5.38559 0.268943 0.134472 0.990917i \(-0.457066\pi\)
0.134472 + 0.990917i \(0.457066\pi\)
\(402\) 0 0
\(403\) 50.5845 2.51979
\(404\) 0 0
\(405\) 23.8432 1.18478
\(406\) 0 0
\(407\) −8.62155 −0.427354
\(408\) 0 0
\(409\) 5.07983 0.251182 0.125591 0.992082i \(-0.459917\pi\)
0.125591 + 0.992082i \(0.459917\pi\)
\(410\) 0 0
\(411\) 0.820916 0.0404928
\(412\) 0 0
\(413\) −23.1646 −1.13985
\(414\) 0 0
\(415\) −6.14539 −0.301666
\(416\) 0 0
\(417\) 7.07270 0.346351
\(418\) 0 0
\(419\) 4.30576 0.210350 0.105175 0.994454i \(-0.466460\pi\)
0.105175 + 0.994454i \(0.466460\pi\)
\(420\) 0 0
\(421\) 17.8670 0.870782 0.435391 0.900241i \(-0.356610\pi\)
0.435391 + 0.900241i \(0.356610\pi\)
\(422\) 0 0
\(423\) 5.53027 0.268891
\(424\) 0 0
\(425\) −20.8855 −1.01309
\(426\) 0 0
\(427\) 16.3497 0.791219
\(428\) 0 0
\(429\) −13.6601 −0.659516
\(430\) 0 0
\(431\) 18.1958 0.876461 0.438230 0.898863i \(-0.355605\pi\)
0.438230 + 0.898863i \(0.355605\pi\)
\(432\) 0 0
\(433\) −16.6074 −0.798100 −0.399050 0.916929i \(-0.630660\pi\)
−0.399050 + 0.916929i \(0.630660\pi\)
\(434\) 0 0
\(435\) −33.0890 −1.58649
\(436\) 0 0
\(437\) 7.91023 0.378398
\(438\) 0 0
\(439\) 11.6084 0.554038 0.277019 0.960864i \(-0.410653\pi\)
0.277019 + 0.960864i \(0.410653\pi\)
\(440\) 0 0
\(441\) 0.409734 0.0195111
\(442\) 0 0
\(443\) 23.8140 1.13144 0.565720 0.824598i \(-0.308598\pi\)
0.565720 + 0.824598i \(0.308598\pi\)
\(444\) 0 0
\(445\) 25.7992 1.22300
\(446\) 0 0
\(447\) −29.6247 −1.40120
\(448\) 0 0
\(449\) 3.72336 0.175716 0.0878582 0.996133i \(-0.471998\pi\)
0.0878582 + 0.996133i \(0.471998\pi\)
\(450\) 0 0
\(451\) −2.37533 −0.111850
\(452\) 0 0
\(453\) 10.2391 0.481074
\(454\) 0 0
\(455\) −53.0607 −2.48752
\(456\) 0 0
\(457\) 26.7719 1.25234 0.626169 0.779688i \(-0.284622\pi\)
0.626169 + 0.779688i \(0.284622\pi\)
\(458\) 0 0
\(459\) −17.5941 −0.821222
\(460\) 0 0
\(461\) −7.76735 −0.361761 −0.180881 0.983505i \(-0.557895\pi\)
−0.180881 + 0.983505i \(0.557895\pi\)
\(462\) 0 0
\(463\) 13.7928 0.641004 0.320502 0.947248i \(-0.396148\pi\)
0.320502 + 0.947248i \(0.396148\pi\)
\(464\) 0 0
\(465\) −43.2493 −2.00564
\(466\) 0 0
\(467\) 27.7282 1.28311 0.641554 0.767078i \(-0.278290\pi\)
0.641554 + 0.767078i \(0.278290\pi\)
\(468\) 0 0
\(469\) −4.68843 −0.216492
\(470\) 0 0
\(471\) 27.4289 1.26386
\(472\) 0 0
\(473\) −2.41430 −0.111010
\(474\) 0 0
\(475\) 6.60197 0.302919
\(476\) 0 0
\(477\) 1.56512 0.0716619
\(478\) 0 0
\(479\) −14.0163 −0.640420 −0.320210 0.947347i \(-0.603753\pi\)
−0.320210 + 0.947347i \(0.603753\pi\)
\(480\) 0 0
\(481\) −38.1425 −1.73915
\(482\) 0 0
\(483\) 30.9309 1.40740
\(484\) 0 0
\(485\) −16.3916 −0.744304
\(486\) 0 0
\(487\) 37.5764 1.70275 0.851374 0.524559i \(-0.175770\pi\)
0.851374 + 0.524559i \(0.175770\pi\)
\(488\) 0 0
\(489\) 35.8276 1.62018
\(490\) 0 0
\(491\) 5.24309 0.236617 0.118309 0.992977i \(-0.462253\pi\)
0.118309 + 0.992977i \(0.462253\pi\)
\(492\) 0 0
\(493\) 19.6803 0.886356
\(494\) 0 0
\(495\) −2.68963 −0.120890
\(496\) 0 0
\(497\) −32.5733 −1.46111
\(498\) 0 0
\(499\) 29.0542 1.30065 0.650323 0.759658i \(-0.274633\pi\)
0.650323 + 0.759658i \(0.274633\pi\)
\(500\) 0 0
\(501\) 9.75379 0.435767
\(502\) 0 0
\(503\) 31.5512 1.40680 0.703399 0.710796i \(-0.251665\pi\)
0.703399 + 0.710796i \(0.251665\pi\)
\(504\) 0 0
\(505\) −55.3373 −2.46248
\(506\) 0 0
\(507\) −40.1334 −1.78239
\(508\) 0 0
\(509\) −17.5097 −0.776104 −0.388052 0.921638i \(-0.626852\pi\)
−0.388052 + 0.921638i \(0.626852\pi\)
\(510\) 0 0
\(511\) 15.7421 0.696391
\(512\) 0 0
\(513\) 5.56155 0.245549
\(514\) 0 0
\(515\) 26.1908 1.15410
\(516\) 0 0
\(517\) 13.8482 0.609042
\(518\) 0 0
\(519\) −26.4583 −1.16139
\(520\) 0 0
\(521\) −37.8780 −1.65947 −0.829733 0.558161i \(-0.811507\pi\)
−0.829733 + 0.558161i \(0.811507\pi\)
\(522\) 0 0
\(523\) 28.3739 1.24070 0.620352 0.784324i \(-0.286990\pi\)
0.620352 + 0.784324i \(0.286990\pi\)
\(524\) 0 0
\(525\) 25.8152 1.12667
\(526\) 0 0
\(527\) 25.7234 1.12053
\(528\) 0 0
\(529\) 39.5718 1.72051
\(530\) 0 0
\(531\) −5.19480 −0.225435
\(532\) 0 0
\(533\) −10.5087 −0.455182
\(534\) 0 0
\(535\) 40.4945 1.75073
\(536\) 0 0
\(537\) −28.4924 −1.22954
\(538\) 0 0
\(539\) 1.02600 0.0441930
\(540\) 0 0
\(541\) −38.5868 −1.65898 −0.829488 0.558524i \(-0.811368\pi\)
−0.829488 + 0.558524i \(0.811368\pi\)
\(542\) 0 0
\(543\) 17.0636 0.732267
\(544\) 0 0
\(545\) −19.2615 −0.825071
\(546\) 0 0
\(547\) −13.3947 −0.572717 −0.286359 0.958123i \(-0.592445\pi\)
−0.286359 + 0.958123i \(0.592445\pi\)
\(548\) 0 0
\(549\) 3.66653 0.156484
\(550\) 0 0
\(551\) −6.22101 −0.265024
\(552\) 0 0
\(553\) −28.1816 −1.19841
\(554\) 0 0
\(555\) 32.6115 1.38428
\(556\) 0 0
\(557\) −4.46471 −0.189176 −0.0945879 0.995517i \(-0.530153\pi\)
−0.0945879 + 0.995517i \(0.530153\pi\)
\(558\) 0 0
\(559\) −10.6811 −0.451762
\(560\) 0 0
\(561\) −6.94647 −0.293280
\(562\) 0 0
\(563\) −22.1171 −0.932124 −0.466062 0.884752i \(-0.654328\pi\)
−0.466062 + 0.884752i \(0.654328\pi\)
\(564\) 0 0
\(565\) 30.0163 1.26279
\(566\) 0 0
\(567\) 17.5285 0.736127
\(568\) 0 0
\(569\) 6.70238 0.280978 0.140489 0.990082i \(-0.455132\pi\)
0.140489 + 0.990082i \(0.455132\pi\)
\(570\) 0 0
\(571\) −6.37533 −0.266799 −0.133400 0.991062i \(-0.542589\pi\)
−0.133400 + 0.991062i \(0.542589\pi\)
\(572\) 0 0
\(573\) −24.2231 −1.01194
\(574\) 0 0
\(575\) 52.2231 2.17785
\(576\) 0 0
\(577\) 44.5654 1.85528 0.927641 0.373474i \(-0.121834\pi\)
0.927641 + 0.373474i \(0.121834\pi\)
\(578\) 0 0
\(579\) 14.9364 0.620737
\(580\) 0 0
\(581\) −4.51783 −0.187431
\(582\) 0 0
\(583\) 3.91917 0.162315
\(584\) 0 0
\(585\) −11.8992 −0.491971
\(586\) 0 0
\(587\) −29.6311 −1.22301 −0.611503 0.791242i \(-0.709435\pi\)
−0.611503 + 0.791242i \(0.709435\pi\)
\(588\) 0 0
\(589\) −8.13124 −0.335042
\(590\) 0 0
\(591\) 29.8871 1.22939
\(592\) 0 0
\(593\) 43.6572 1.79279 0.896393 0.443259i \(-0.146178\pi\)
0.896393 + 0.443259i \(0.146178\pi\)
\(594\) 0 0
\(595\) −26.9825 −1.10618
\(596\) 0 0
\(597\) −0.208857 −0.00854794
\(598\) 0 0
\(599\) 12.5661 0.513438 0.256719 0.966486i \(-0.417359\pi\)
0.256719 + 0.966486i \(0.417359\pi\)
\(600\) 0 0
\(601\) −37.9213 −1.54684 −0.773421 0.633893i \(-0.781456\pi\)
−0.773421 + 0.633893i \(0.781456\pi\)
\(602\) 0 0
\(603\) −1.05141 −0.0428167
\(604\) 0 0
\(605\) 30.7328 1.24947
\(606\) 0 0
\(607\) −1.93332 −0.0784710 −0.0392355 0.999230i \(-0.512492\pi\)
−0.0392355 + 0.999230i \(0.512492\pi\)
\(608\) 0 0
\(609\) −24.3256 −0.985723
\(610\) 0 0
\(611\) 61.2656 2.47854
\(612\) 0 0
\(613\) −18.4306 −0.744404 −0.372202 0.928152i \(-0.621397\pi\)
−0.372202 + 0.928152i \(0.621397\pi\)
\(614\) 0 0
\(615\) 8.98485 0.362304
\(616\) 0 0
\(617\) 30.5314 1.22915 0.614574 0.788859i \(-0.289328\pi\)
0.614574 + 0.788859i \(0.289328\pi\)
\(618\) 0 0
\(619\) 28.9094 1.16197 0.580984 0.813915i \(-0.302668\pi\)
0.580984 + 0.813915i \(0.302668\pi\)
\(620\) 0 0
\(621\) 43.9932 1.76539
\(622\) 0 0
\(623\) 18.9665 0.759875
\(624\) 0 0
\(625\) −14.4238 −0.576954
\(626\) 0 0
\(627\) 2.19580 0.0876919
\(628\) 0 0
\(629\) −19.3963 −0.773382
\(630\) 0 0
\(631\) −32.5314 −1.29505 −0.647527 0.762042i \(-0.724197\pi\)
−0.647527 + 0.762042i \(0.724197\pi\)
\(632\) 0 0
\(633\) 35.9340 1.42825
\(634\) 0 0
\(635\) 41.0457 1.62885
\(636\) 0 0
\(637\) 4.53912 0.179846
\(638\) 0 0
\(639\) −7.30476 −0.288972
\(640\) 0 0
\(641\) −30.1312 −1.19011 −0.595056 0.803684i \(-0.702870\pi\)
−0.595056 + 0.803684i \(0.702870\pi\)
\(642\) 0 0
\(643\) 20.1752 0.795633 0.397817 0.917465i \(-0.369768\pi\)
0.397817 + 0.917465i \(0.369768\pi\)
\(644\) 0 0
\(645\) 9.13224 0.359582
\(646\) 0 0
\(647\) −14.2741 −0.561174 −0.280587 0.959829i \(-0.590529\pi\)
−0.280587 + 0.959829i \(0.590529\pi\)
\(648\) 0 0
\(649\) −13.0081 −0.510614
\(650\) 0 0
\(651\) −31.7951 −1.24615
\(652\) 0 0
\(653\) −9.21538 −0.360626 −0.180313 0.983609i \(-0.557711\pi\)
−0.180313 + 0.983609i \(0.557711\pi\)
\(654\) 0 0
\(655\) −3.13666 −0.122560
\(656\) 0 0
\(657\) 3.53027 0.137729
\(658\) 0 0
\(659\) −12.1511 −0.473339 −0.236669 0.971590i \(-0.576056\pi\)
−0.236669 + 0.971590i \(0.576056\pi\)
\(660\) 0 0
\(661\) −13.6924 −0.532574 −0.266287 0.963894i \(-0.585797\pi\)
−0.266287 + 0.963894i \(0.585797\pi\)
\(662\) 0 0
\(663\) −30.7318 −1.19352
\(664\) 0 0
\(665\) 8.52927 0.330751
\(666\) 0 0
\(667\) −49.2096 −1.90540
\(668\) 0 0
\(669\) 11.7346 0.453687
\(670\) 0 0
\(671\) 9.18124 0.354438
\(672\) 0 0
\(673\) 40.5341 1.56247 0.781237 0.624234i \(-0.214589\pi\)
0.781237 + 0.624234i \(0.214589\pi\)
\(674\) 0 0
\(675\) 36.7172 1.41325
\(676\) 0 0
\(677\) −11.5653 −0.444492 −0.222246 0.974991i \(-0.571339\pi\)
−0.222246 + 0.974991i \(0.571339\pi\)
\(678\) 0 0
\(679\) −12.0504 −0.462452
\(680\) 0 0
\(681\) −9.08340 −0.348077
\(682\) 0 0
\(683\) 44.2625 1.69366 0.846828 0.531866i \(-0.178509\pi\)
0.846828 + 0.531866i \(0.178509\pi\)
\(684\) 0 0
\(685\) 1.79064 0.0684168
\(686\) 0 0
\(687\) −40.8662 −1.55914
\(688\) 0 0
\(689\) 17.3387 0.660554
\(690\) 0 0
\(691\) −32.6676 −1.24274 −0.621368 0.783519i \(-0.713423\pi\)
−0.621368 + 0.783519i \(0.713423\pi\)
\(692\) 0 0
\(693\) −1.97730 −0.0751116
\(694\) 0 0
\(695\) 15.4275 0.585197
\(696\) 0 0
\(697\) −5.34391 −0.202415
\(698\) 0 0
\(699\) −3.47214 −0.131328
\(700\) 0 0
\(701\) 8.73150 0.329784 0.164892 0.986312i \(-0.447272\pi\)
0.164892 + 0.986312i \(0.447272\pi\)
\(702\) 0 0
\(703\) 6.13124 0.231244
\(704\) 0 0
\(705\) −52.3816 −1.97280
\(706\) 0 0
\(707\) −40.6816 −1.52999
\(708\) 0 0
\(709\) −14.3058 −0.537264 −0.268632 0.963243i \(-0.586572\pi\)
−0.268632 + 0.963243i \(0.586572\pi\)
\(710\) 0 0
\(711\) −6.31991 −0.237015
\(712\) 0 0
\(713\) −64.3200 −2.40880
\(714\) 0 0
\(715\) −29.7964 −1.11432
\(716\) 0 0
\(717\) 12.6934 0.474045
\(718\) 0 0
\(719\) 21.4959 0.801663 0.400831 0.916152i \(-0.368721\pi\)
0.400831 + 0.916152i \(0.368721\pi\)
\(720\) 0 0
\(721\) 19.2543 0.717069
\(722\) 0 0
\(723\) −40.5250 −1.50714
\(724\) 0 0
\(725\) −41.0709 −1.52533
\(726\) 0 0
\(727\) 47.6002 1.76539 0.882696 0.469944i \(-0.155726\pi\)
0.882696 + 0.469944i \(0.155726\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) −5.43158 −0.200894
\(732\) 0 0
\(733\) 49.7239 1.83659 0.918297 0.395892i \(-0.129565\pi\)
0.918297 + 0.395892i \(0.129565\pi\)
\(734\) 0 0
\(735\) −3.88091 −0.143149
\(736\) 0 0
\(737\) −2.63280 −0.0969805
\(738\) 0 0
\(739\) 35.5283 1.30693 0.653464 0.756957i \(-0.273315\pi\)
0.653464 + 0.756957i \(0.273315\pi\)
\(740\) 0 0
\(741\) 9.71443 0.356869
\(742\) 0 0
\(743\) −20.3612 −0.746979 −0.373490 0.927634i \(-0.621839\pi\)
−0.373490 + 0.927634i \(0.621839\pi\)
\(744\) 0 0
\(745\) −64.6194 −2.36747
\(746\) 0 0
\(747\) −1.01315 −0.0370693
\(748\) 0 0
\(749\) 29.7699 1.08777
\(750\) 0 0
\(751\) −18.9868 −0.692840 −0.346420 0.938080i \(-0.612603\pi\)
−0.346420 + 0.938080i \(0.612603\pi\)
\(752\) 0 0
\(753\) 2.40074 0.0874880
\(754\) 0 0
\(755\) 22.3342 0.812824
\(756\) 0 0
\(757\) 22.4738 0.816826 0.408413 0.912797i \(-0.366082\pi\)
0.408413 + 0.912797i \(0.366082\pi\)
\(758\) 0 0
\(759\) 17.3693 0.630466
\(760\) 0 0
\(761\) −1.40973 −0.0511028 −0.0255514 0.999674i \(-0.508134\pi\)
−0.0255514 + 0.999674i \(0.508134\pi\)
\(762\) 0 0
\(763\) −14.1602 −0.512634
\(764\) 0 0
\(765\) −6.05100 −0.218774
\(766\) 0 0
\(767\) −57.5492 −2.07798
\(768\) 0 0
\(769\) −0.114116 −0.00411512 −0.00205756 0.999998i \(-0.500655\pi\)
−0.00205756 + 0.999998i \(0.500655\pi\)
\(770\) 0 0
\(771\) 0.578272 0.0208260
\(772\) 0 0
\(773\) 46.1799 1.66097 0.830487 0.557038i \(-0.188062\pi\)
0.830487 + 0.557038i \(0.188062\pi\)
\(774\) 0 0
\(775\) −53.6822 −1.92832
\(776\) 0 0
\(777\) 23.9746 0.860084
\(778\) 0 0
\(779\) 1.68923 0.0605228
\(780\) 0 0
\(781\) −18.2916 −0.654525
\(782\) 0 0
\(783\) −34.5985 −1.23645
\(784\) 0 0
\(785\) 59.8297 2.13541
\(786\) 0 0
\(787\) 12.6847 0.452159 0.226080 0.974109i \(-0.427409\pi\)
0.226080 + 0.974109i \(0.427409\pi\)
\(788\) 0 0
\(789\) −29.5238 −1.05108
\(790\) 0 0
\(791\) 22.0667 0.784601
\(792\) 0 0
\(793\) 40.6186 1.44241
\(794\) 0 0
\(795\) −14.8245 −0.525770
\(796\) 0 0
\(797\) −39.2796 −1.39135 −0.695677 0.718355i \(-0.744895\pi\)
−0.695677 + 0.718355i \(0.744895\pi\)
\(798\) 0 0
\(799\) 31.1549 1.10218
\(800\) 0 0
\(801\) 4.25335 0.150285
\(802\) 0 0
\(803\) 8.84004 0.311958
\(804\) 0 0
\(805\) 67.4685 2.37795
\(806\) 0 0
\(807\) −23.7197 −0.834971
\(808\) 0 0
\(809\) −1.09896 −0.0386374 −0.0193187 0.999813i \(-0.506150\pi\)
−0.0193187 + 0.999813i \(0.506150\pi\)
\(810\) 0 0
\(811\) 36.5627 1.28389 0.641944 0.766751i \(-0.278128\pi\)
0.641944 + 0.766751i \(0.278128\pi\)
\(812\) 0 0
\(813\) −44.3522 −1.55550
\(814\) 0 0
\(815\) 78.1496 2.73746
\(816\) 0 0
\(817\) 1.71694 0.0600681
\(818\) 0 0
\(819\) −8.74777 −0.305672
\(820\) 0 0
\(821\) 17.8823 0.624097 0.312049 0.950066i \(-0.398985\pi\)
0.312049 + 0.950066i \(0.398985\pi\)
\(822\) 0 0
\(823\) 8.31328 0.289783 0.144891 0.989448i \(-0.453717\pi\)
0.144891 + 0.989448i \(0.453717\pi\)
\(824\) 0 0
\(825\) 14.4966 0.504708
\(826\) 0 0
\(827\) 0.280204 0.00974363 0.00487182 0.999988i \(-0.498449\pi\)
0.00487182 + 0.999988i \(0.498449\pi\)
\(828\) 0 0
\(829\) −6.04148 −0.209829 −0.104915 0.994481i \(-0.533457\pi\)
−0.104915 + 0.994481i \(0.533457\pi\)
\(830\) 0 0
\(831\) 44.4491 1.54192
\(832\) 0 0
\(833\) 2.30824 0.0799759
\(834\) 0 0
\(835\) 21.2756 0.736274
\(836\) 0 0
\(837\) −45.2223 −1.56311
\(838\) 0 0
\(839\) −29.5942 −1.02171 −0.510853 0.859668i \(-0.670670\pi\)
−0.510853 + 0.859668i \(0.670670\pi\)
\(840\) 0 0
\(841\) 9.70093 0.334515
\(842\) 0 0
\(843\) 42.3796 1.45963
\(844\) 0 0
\(845\) −87.5416 −3.01152
\(846\) 0 0
\(847\) 22.5934 0.776320
\(848\) 0 0
\(849\) 42.0241 1.44226
\(850\) 0 0
\(851\) 48.4996 1.66254
\(852\) 0 0
\(853\) −15.2331 −0.521570 −0.260785 0.965397i \(-0.583981\pi\)
−0.260785 + 0.965397i \(0.583981\pi\)
\(854\) 0 0
\(855\) 1.91274 0.0654144
\(856\) 0 0
\(857\) −2.25535 −0.0770412 −0.0385206 0.999258i \(-0.512265\pi\)
−0.0385206 + 0.999258i \(0.512265\pi\)
\(858\) 0 0
\(859\) −1.91686 −0.0654025 −0.0327013 0.999465i \(-0.510411\pi\)
−0.0327013 + 0.999465i \(0.510411\pi\)
\(860\) 0 0
\(861\) 6.60527 0.225107
\(862\) 0 0
\(863\) 35.8530 1.22045 0.610225 0.792228i \(-0.291079\pi\)
0.610225 + 0.792228i \(0.291079\pi\)
\(864\) 0 0
\(865\) −57.7126 −1.96229
\(866\) 0 0
\(867\) 10.9186 0.370814
\(868\) 0 0
\(869\) −15.8255 −0.536843
\(870\) 0 0
\(871\) −11.6478 −0.394669
\(872\) 0 0
\(873\) −2.70238 −0.0914617
\(874\) 0 0
\(875\) 13.6636 0.461915
\(876\) 0 0
\(877\) −48.2323 −1.62869 −0.814344 0.580383i \(-0.802903\pi\)
−0.814344 + 0.580383i \(0.802903\pi\)
\(878\) 0 0
\(879\) 16.0394 0.540994
\(880\) 0 0
\(881\) 4.77336 0.160819 0.0804094 0.996762i \(-0.474377\pi\)
0.0804094 + 0.996762i \(0.474377\pi\)
\(882\) 0 0
\(883\) −22.2144 −0.747573 −0.373787 0.927515i \(-0.621941\pi\)
−0.373787 + 0.927515i \(0.621941\pi\)
\(884\) 0 0
\(885\) 49.2041 1.65398
\(886\) 0 0
\(887\) −42.1729 −1.41603 −0.708014 0.706198i \(-0.750409\pi\)
−0.708014 + 0.706198i \(0.750409\pi\)
\(888\) 0 0
\(889\) 30.1750 1.01204
\(890\) 0 0
\(891\) 9.84316 0.329758
\(892\) 0 0
\(893\) −9.84818 −0.329557
\(894\) 0 0
\(895\) −62.1496 −2.07743
\(896\) 0 0
\(897\) 76.8434 2.56573
\(898\) 0 0
\(899\) 50.5845 1.68709
\(900\) 0 0
\(901\) 8.81714 0.293742
\(902\) 0 0
\(903\) 6.71363 0.223416
\(904\) 0 0
\(905\) 37.2202 1.23724
\(906\) 0 0
\(907\) −26.7564 −0.888430 −0.444215 0.895920i \(-0.646517\pi\)
−0.444215 + 0.895920i \(0.646517\pi\)
\(908\) 0 0
\(909\) −9.12311 −0.302594
\(910\) 0 0
\(911\) 11.1281 0.368691 0.184346 0.982861i \(-0.440983\pi\)
0.184346 + 0.982861i \(0.440983\pi\)
\(912\) 0 0
\(913\) −2.53700 −0.0839625
\(914\) 0 0
\(915\) −34.7286 −1.14809
\(916\) 0 0
\(917\) −2.30594 −0.0761489
\(918\) 0 0
\(919\) 5.45195 0.179843 0.0899216 0.995949i \(-0.471338\pi\)
0.0899216 + 0.995949i \(0.471338\pi\)
\(920\) 0 0
\(921\) −7.89918 −0.260287
\(922\) 0 0
\(923\) −80.9237 −2.66364
\(924\) 0 0
\(925\) 40.4783 1.33092
\(926\) 0 0
\(927\) 4.31791 0.141819
\(928\) 0 0
\(929\) −1.94402 −0.0637813 −0.0318906 0.999491i \(-0.510153\pi\)
−0.0318906 + 0.999491i \(0.510153\pi\)
\(930\) 0 0
\(931\) −0.729644 −0.0239131
\(932\) 0 0
\(933\) −20.9595 −0.686184
\(934\) 0 0
\(935\) −15.1521 −0.495527
\(936\) 0 0
\(937\) −11.2897 −0.368820 −0.184410 0.982849i \(-0.559037\pi\)
−0.184410 + 0.982849i \(0.559037\pi\)
\(938\) 0 0
\(939\) −10.8372 −0.353658
\(940\) 0 0
\(941\) −51.4229 −1.67634 −0.838170 0.545409i \(-0.816374\pi\)
−0.838170 + 0.545409i \(0.816374\pi\)
\(942\) 0 0
\(943\) 13.3622 0.435133
\(944\) 0 0
\(945\) 47.4360 1.54309
\(946\) 0 0
\(947\) −1.94045 −0.0630563 −0.0315281 0.999503i \(-0.510037\pi\)
−0.0315281 + 0.999503i \(0.510037\pi\)
\(948\) 0 0
\(949\) 39.1092 1.26954
\(950\) 0 0
\(951\) 8.75169 0.283793
\(952\) 0 0
\(953\) −23.4523 −0.759693 −0.379847 0.925049i \(-0.624023\pi\)
−0.379847 + 0.925049i \(0.624023\pi\)
\(954\) 0 0
\(955\) −52.8371 −1.70977
\(956\) 0 0
\(957\) −13.6601 −0.441569
\(958\) 0 0
\(959\) 1.31640 0.0425088
\(960\) 0 0
\(961\) 35.1171 1.13281
\(962\) 0 0
\(963\) 6.67608 0.215134
\(964\) 0 0
\(965\) 32.5804 1.04880
\(966\) 0 0
\(967\) −59.0194 −1.89794 −0.948968 0.315373i \(-0.897870\pi\)
−0.948968 + 0.315373i \(0.897870\pi\)
\(968\) 0 0
\(969\) 4.94001 0.158696
\(970\) 0 0
\(971\) 39.7743 1.27642 0.638209 0.769863i \(-0.279676\pi\)
0.638209 + 0.769863i \(0.279676\pi\)
\(972\) 0 0
\(973\) 11.3416 0.363595
\(974\) 0 0
\(975\) 64.1344 2.05394
\(976\) 0 0
\(977\) −41.9538 −1.34222 −0.671111 0.741357i \(-0.734183\pi\)
−0.671111 + 0.741357i \(0.734183\pi\)
\(978\) 0 0
\(979\) 10.6507 0.340397
\(980\) 0 0
\(981\) −3.17552 −0.101386
\(982\) 0 0
\(983\) −7.93132 −0.252970 −0.126485 0.991969i \(-0.540370\pi\)
−0.126485 + 0.991969i \(0.540370\pi\)
\(984\) 0 0
\(985\) 65.1919 2.07719
\(986\) 0 0
\(987\) −38.5087 −1.22575
\(988\) 0 0
\(989\) 13.5814 0.431863
\(990\) 0 0
\(991\) −24.8408 −0.789093 −0.394546 0.918876i \(-0.629098\pi\)
−0.394546 + 0.918876i \(0.629098\pi\)
\(992\) 0 0
\(993\) −6.42017 −0.203738
\(994\) 0 0
\(995\) −0.455573 −0.0144426
\(996\) 0 0
\(997\) −28.1157 −0.890433 −0.445216 0.895423i \(-0.646873\pi\)
−0.445216 + 0.895423i \(0.646873\pi\)
\(998\) 0 0
\(999\) 34.0992 1.07885
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.x.1.1 4
4.3 odd 2 1216.2.a.w.1.3 4
8.3 odd 2 608.2.a.j.1.2 yes 4
8.5 even 2 608.2.a.i.1.4 4
24.5 odd 2 5472.2.a.bt.1.1 4
24.11 even 2 5472.2.a.bs.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.i.1.4 4 8.5 even 2
608.2.a.j.1.2 yes 4 8.3 odd 2
1216.2.a.w.1.3 4 4.3 odd 2
1216.2.a.x.1.1 4 1.1 even 1 trivial
5472.2.a.bs.1.1 4 24.11 even 2
5472.2.a.bt.1.1 4 24.5 odd 2