Properties

Label 1216.2.a.w.1.3
Level $1216$
Weight $2$
Character 1216.1
Self dual yes
Analytic conductor $9.710$
Analytic rank $1$
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: \(1\)
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 1.3
Root \(1.32973\) 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 \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 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 \(-0.343090\pi\)
0.473224 + 0.880942i \(0.343090\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) 1.40617 0.423975 0.211988 0.977272i \(-0.432006\pi\)
0.211988 + 0.977272i \(0.432006\pi\)
\(12\) 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 \(-0.808655\pi\)
−0.824699 + 0.565572i \(0.808655\pi\)
\(24\) 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 \(-0.239424\pi\)
0.730207 + 0.683226i \(0.239424\pi\)
\(32\) 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 \(-0.541792\pi\)
−0.130915 + 0.991394i \(0.541792\pi\)
\(44\) 0 0
\(45\) 1.91274 0.285135
\(46\) 0 0
\(47\) 9.84818 1.43650 0.718252 0.695783i \(-0.244942\pi\)
0.718252 + 0.695783i \(0.244942\pi\)
\(48\) 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 \(-0.705699\pi\)
−0.602174 + 0.798365i \(0.705699\pi\)
\(60\) 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 \(-0.536485\pi\)
−0.114370 + 0.993438i \(0.536485\pi\)
\(68\) 0 0
\(69\) −12.3522 −1.48704
\(70\) 0 0
\(71\) −13.0081 −1.54378 −0.771891 0.635755i \(-0.780689\pi\)
−0.771891 + 0.635755i \(0.780689\pi\)
\(72\) 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 \(-0.718220\pi\)
−0.633106 + 0.774065i \(0.718220\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −1.80420 −0.198036 −0.0990182 0.995086i \(-0.531570\pi\)
−0.0990182 + 0.995086i \(0.531570\pi\)
\(84\) 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 \(-0.376330\pi\)
0.378821 + 0.925470i \(0.376330\pi\)
\(104\) 0 0
\(105\) −13.3189 −1.29979
\(106\) 0 0
\(107\) 11.8886 1.14931 0.574657 0.818394i \(-0.305135\pi\)
0.574657 + 0.818394i \(0.305135\pi\)
\(108\) 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 \(-0.320443\pi\)
0.534650 + 0.845073i \(0.320443\pi\)
\(128\) 0 0
\(129\) −2.68109 −0.236057
\(130\) 0 0
\(131\) −0.920878 −0.0804575 −0.0402287 0.999190i \(-0.512809\pi\)
−0.0402287 + 0.999190i \(0.512809\pi\)
\(132\) 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 \(-0.438475\pi\)
0.192084 + 0.981379i \(0.438475\pi\)
\(140\) 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 \(-0.414034\pi\)
0.266800 + 0.963752i \(0.414034\pi\)
\(152\) 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 \(-0.144629\pi\)
0.898540 + 0.438892i \(0.144629\pi\)
\(164\) 0 0
\(165\) −7.47927 −0.582260
\(166\) 0 0
\(167\) 6.24621 0.483346 0.241673 0.970358i \(-0.422304\pi\)
0.241673 + 0.970358i \(0.422304\pi\)
\(168\) 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 \(-0.738843\pi\)
−0.681893 + 0.731452i \(0.738843\pi\)
\(180\) 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 \(-0.689665\pi\)
−0.561212 + 0.827672i \(0.689665\pi\)
\(192\) 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 \(-0.501509\pi\)
−0.00474062 + 0.999989i \(0.501509\pi\)
\(200\) 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 \(-0.208990\pi\)
0.792095 + 0.610397i \(0.208990\pi\)
\(212\) 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 \(-0.419040\pi\)
0.251611 + 0.967828i \(0.419040\pi\)
\(224\) 0 0
\(225\) −3.70735 −0.247157
\(226\) 0 0
\(227\) −5.81690 −0.386081 −0.193041 0.981191i \(-0.561835\pi\)
−0.193041 + 0.981191i \(0.561835\pi\)
\(228\) 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 \(-0.415320\pi\)
0.262902 + 0.964823i \(0.415320\pi\)
\(240\) 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 \(-0.484549\pi\)
0.0485202 + 0.998822i \(0.484549\pi\)
\(252\) 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 \(-0.698090\pi\)
−0.582919 + 0.812530i \(0.698090\pi\)
\(264\) 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 \(-0.831209\pi\)
−0.862669 + 0.505768i \(0.831209\pi\)
\(272\) 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 \(-0.204903\pi\)
0.799868 + 0.600175i \(0.204903\pi\)
\(284\) 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 \(-0.546110\pi\)
−0.144353 + 0.989526i \(0.546110\pi\)
\(308\) 0 0
\(309\) 12.0071 0.683062
\(310\) 0 0
\(311\) −13.4222 −0.761105 −0.380553 0.924759i \(-0.624266\pi\)
−0.380553 + 0.924759i \(0.624266\pi\)
\(312\) 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 \(-0.536043\pi\)
−0.112992 + 0.993596i \(0.536043\pi\)
\(332\) 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 \(-0.474094\pi\)
0.0812949 + 0.996690i \(0.474094\pi\)
\(348\) 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 \(-0.873495\pi\)
−0.922059 + 0.387048i \(0.873495\pi\)
\(360\) 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 \(-0.436368\pi\)
0.198577 + 0.980085i \(0.436368\pi\)
\(368\) 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 \(-0.591208\pi\)
−0.282633 + 0.959228i \(0.591208\pi\)
\(380\) 0 0
\(381\) 18.8173 0.964042
\(382\) 0 0
\(383\) 20.6923 1.05733 0.528665 0.848831i \(-0.322693\pi\)
0.528665 + 0.848831i \(0.322693\pi\)
\(384\) 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 \(-0.533540\pi\)
−0.105175 + 0.994454i \(0.533540\pi\)
\(420\) 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 \(-0.644395\pi\)
−0.438230 + 0.898863i \(0.644395\pi\)
\(432\) 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 \(-0.589347\pi\)
−0.277019 + 0.960864i \(0.589347\pi\)
\(440\) 0 0
\(441\) 0.409734 0.0195111
\(442\) 0 0
\(443\) −23.8140 −1.13144 −0.565720 0.824598i \(-0.691402\pi\)
−0.565720 + 0.824598i \(0.691402\pi\)
\(444\) 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 \(-0.603852\pi\)
−0.320502 + 0.947248i \(0.603852\pi\)
\(464\) 0 0
\(465\) −43.2493 −2.00564
\(466\) 0 0
\(467\) −27.7282 −1.28311 −0.641554 0.767078i \(-0.721710\pi\)
−0.641554 + 0.767078i \(0.721710\pi\)
\(468\) 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 \(-0.396247\pi\)
0.320210 + 0.947347i \(0.396247\pi\)
\(480\) 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 \(-0.824230\pi\)
−0.851374 + 0.524559i \(0.824230\pi\)
\(488\) 0 0
\(489\) 35.8276 1.62018
\(490\) 0 0
\(491\) −5.24309 −0.236617 −0.118309 0.992977i \(-0.537747\pi\)
−0.118309 + 0.992977i \(0.537747\pi\)
\(492\) 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 \(-0.725367\pi\)
−0.650323 + 0.759658i \(0.725367\pi\)
\(500\) 0 0
\(501\) 9.75379 0.435767
\(502\) 0 0
\(503\) −31.5512 −1.40680 −0.703399 0.710796i \(-0.748335\pi\)
−0.703399 + 0.710796i \(0.748335\pi\)
\(504\) 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 \(-0.713010\pi\)
−0.620352 + 0.784324i \(0.713010\pi\)
\(524\) 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 \(-0.407555\pi\)
0.286359 + 0.958123i \(0.407555\pi\)
\(548\) 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 \(-0.345672\pi\)
0.466062 + 0.884752i \(0.345672\pi\)
\(564\) 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 \(-0.457411\pi\)
0.133400 + 0.991062i \(0.457411\pi\)
\(572\) 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 \(-0.290565\pi\)
0.611503 + 0.791242i \(0.290565\pi\)
\(588\) 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 \(-0.582641\pi\)
−0.256719 + 0.966486i \(0.582641\pi\)
\(600\) 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 \(-0.487508\pi\)
0.0392355 + 0.999230i \(0.487508\pi\)
\(608\) 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 \(-0.697332\pi\)
−0.580984 + 0.813915i \(0.697332\pi\)
\(620\) 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 \(-0.275803\pi\)
0.647527 + 0.762042i \(0.275803\pi\)
\(632\) 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 \(-0.630232\pi\)
−0.397817 + 0.917465i \(0.630232\pi\)
\(644\) 0 0
\(645\) 9.13224 0.359582
\(646\) 0 0
\(647\) 14.2741 0.561174 0.280587 0.959829i \(-0.409471\pi\)
0.280587 + 0.959829i \(0.409471\pi\)
\(648\) 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 \(-0.423944\pi\)
0.236669 + 0.971590i \(0.423944\pi\)
\(660\) 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 \(-0.821491\pi\)
−0.846828 + 0.531866i \(0.821491\pi\)
\(684\) 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 \(-0.286577\pi\)
0.621368 + 0.783519i \(0.286577\pi\)
\(692\) 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 \(-0.631279\pi\)
−0.400831 + 0.916152i \(0.631279\pi\)
\(720\) 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 \(-0.844274\pi\)
−0.882696 + 0.469944i \(0.844274\pi\)
\(728\) 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 \(-0.726685\pi\)
−0.653464 + 0.756957i \(0.726685\pi\)
\(740\) 0 0
\(741\) 9.71443 0.356869
\(742\) 0 0
\(743\) 20.3612 0.746979 0.373490 0.927634i \(-0.378161\pi\)
0.373490 + 0.927634i \(0.378161\pi\)
\(744\) 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 \(-0.387397\pi\)
0.346420 + 0.938080i \(0.387397\pi\)
\(752\) 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 \(-0.572591\pi\)
−0.226080 + 0.974109i \(0.572591\pi\)
\(788\) 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 \(-0.721872\pi\)
−0.641944 + 0.766751i \(0.721872\pi\)
\(812\) 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 \(-0.546283\pi\)
−0.144891 + 0.989448i \(0.546283\pi\)
\(824\) 0 0
\(825\) 14.4966 0.504708
\(826\) 0 0
\(827\) −0.280204 −0.00974363 −0.00487182 0.999988i \(-0.501551\pi\)
−0.00487182 + 0.999988i \(0.501551\pi\)
\(828\) 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 \(-0.329330\pi\)
0.510853 + 0.859668i \(0.329330\pi\)
\(840\) 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 \(-0.489589\pi\)
0.0327013 + 0.999465i \(0.489589\pi\)
\(860\) 0 0
\(861\) 6.60527 0.225107
\(862\) 0 0
\(863\) −35.8530 −1.22045 −0.610225 0.792228i \(-0.708921\pi\)
−0.610225 + 0.792228i \(0.708921\pi\)
\(864\) 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 \(-0.378059\pi\)
0.373787 + 0.927515i \(0.378059\pi\)
\(884\) 0 0
\(885\) 49.2041 1.65398
\(886\) 0 0
\(887\) 42.1729 1.41603 0.708014 0.706198i \(-0.249591\pi\)
0.708014 + 0.706198i \(0.249591\pi\)
\(888\) 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 \(-0.353483\pi\)
0.444215 + 0.895920i \(0.353483\pi\)
\(908\) 0 0
\(909\) −9.12311 −0.302594
\(910\) 0 0
\(911\) −11.1281 −0.368691 −0.184346 0.982861i \(-0.559017\pi\)
−0.184346 + 0.982861i \(0.559017\pi\)
\(912\) 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 \(-0.528662\pi\)
−0.0899216 + 0.995949i \(0.528662\pi\)
\(920\) 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 \(-0.489963\pi\)
0.0315281 + 0.999503i \(0.489963\pi\)
\(948\) 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 \(-0.102130\pi\)
0.948968 + 0.315373i \(0.102130\pi\)
\(968\) 0 0
\(969\) 4.94001 0.158696
\(970\) 0 0
\(971\) −39.7743 −1.27642 −0.638209 0.769863i \(-0.720324\pi\)
−0.638209 + 0.769863i \(0.720324\pi\)
\(972\) 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 \(-0.459630\pi\)
0.126485 + 0.991969i \(0.459630\pi\)
\(984\) 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 \(-0.370902\pi\)
0.394546 + 0.918876i \(0.370902\pi\)
\(992\) 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.w.1.3 4
4.3 odd 2 1216.2.a.x.1.1 4
8.3 odd 2 608.2.a.i.1.4 4
8.5 even 2 608.2.a.j.1.2 yes 4
24.5 odd 2 5472.2.a.bs.1.1 4
24.11 even 2 5472.2.a.bt.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.i.1.4 4 8.3 odd 2
608.2.a.j.1.2 yes 4 8.5 even 2
1216.2.a.w.1.3 4 1.1 even 1 trivial
1216.2.a.x.1.1 4 4.3 odd 2
5472.2.a.bs.1.1 4 24.5 odd 2
5472.2.a.bt.1.1 4 24.11 even 2