Properties

Label 3696.2.a.bo.1.3
Level 3696
Weight 2
Character 3696.1
Self dual yes
Analytic conductor 29.513
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3696.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.5127085871\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.254102\)
Character \(\chi\) = 3696.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +4.18953 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +4.18953 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -3.17313 q^{13} -4.18953 q^{15} +6.85446 q^{17} +0.318669 q^{19} -1.00000 q^{21} +1.87086 q^{23} +12.5522 q^{25} -1.00000 q^{27} -3.17313 q^{29} -9.23353 q^{31} -1.00000 q^{33} +4.18953 q^{35} -7.55220 q^{37} +3.17313 q^{39} +9.36266 q^{41} +10.8873 q^{43} +4.18953 q^{45} +8.06040 q^{47} +1.00000 q^{49} -6.85446 q^{51} +0.508203 q^{53} +4.18953 q^{55} -0.318669 q^{57} +7.04399 q^{59} -2.00000 q^{61} +1.00000 q^{63} -13.2939 q^{65} +2.66492 q^{67} -1.87086 q^{69} +5.01641 q^{71} -4.82687 q^{73} -12.5522 q^{75} +1.00000 q^{77} -5.01641 q^{79} +1.00000 q^{81} -3.52461 q^{83} +28.7170 q^{85} +3.17313 q^{87} -1.74173 q^{89} -3.17313 q^{91} +9.23353 q^{93} +1.33508 q^{95} -12.2499 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 4q^{5} + 3q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 4q^{5} + 3q^{7} + 3q^{9} + 3q^{11} - 4q^{13} - 4q^{15} + 8q^{17} + 8q^{19} - 3q^{21} - 10q^{23} + 15q^{25} - 3q^{27} - 4q^{29} + 2q^{31} - 3q^{33} + 4q^{35} + 4q^{39} + 14q^{41} + 14q^{43} + 4q^{45} + 3q^{49} - 8q^{51} + 4q^{55} - 8q^{57} - 6q^{61} + 3q^{63} + 14q^{65} + 4q^{67} + 10q^{69} + 12q^{71} - 20q^{73} - 15q^{75} + 3q^{77} - 12q^{79} + 3q^{81} - 6q^{83} - 6q^{85} + 4q^{87} + 26q^{89} - 4q^{91} - 2q^{93} + 8q^{95} - 4q^{97} + 3q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 4.18953 1.87362 0.936808 0.349843i \(-0.113765\pi\)
0.936808 + 0.349843i \(0.113765\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −3.17313 −0.880067 −0.440034 0.897981i \(-0.645033\pi\)
−0.440034 + 0.897981i \(0.645033\pi\)
\(14\) 0 0
\(15\) −4.18953 −1.08173
\(16\) 0 0
\(17\) 6.85446 1.66245 0.831225 0.555936i \(-0.187640\pi\)
0.831225 + 0.555936i \(0.187640\pi\)
\(18\) 0 0
\(19\) 0.318669 0.0731078 0.0365539 0.999332i \(-0.488362\pi\)
0.0365539 + 0.999332i \(0.488362\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.87086 0.390102 0.195051 0.980793i \(-0.437513\pi\)
0.195051 + 0.980793i \(0.437513\pi\)
\(24\) 0 0
\(25\) 12.5522 2.51044
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.17313 −0.589235 −0.294617 0.955615i \(-0.595192\pi\)
−0.294617 + 0.955615i \(0.595192\pi\)
\(30\) 0 0
\(31\) −9.23353 −1.65839 −0.829195 0.558959i \(-0.811201\pi\)
−0.829195 + 0.558959i \(0.811201\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 4.18953 0.708161
\(36\) 0 0
\(37\) −7.55220 −1.24157 −0.620787 0.783980i \(-0.713187\pi\)
−0.620787 + 0.783980i \(0.713187\pi\)
\(38\) 0 0
\(39\) 3.17313 0.508107
\(40\) 0 0
\(41\) 9.36266 1.46220 0.731101 0.682269i \(-0.239007\pi\)
0.731101 + 0.682269i \(0.239007\pi\)
\(42\) 0 0
\(43\) 10.8873 1.66029 0.830147 0.557545i \(-0.188257\pi\)
0.830147 + 0.557545i \(0.188257\pi\)
\(44\) 0 0
\(45\) 4.18953 0.624539
\(46\) 0 0
\(47\) 8.06040 1.17573 0.587865 0.808959i \(-0.299969\pi\)
0.587865 + 0.808959i \(0.299969\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −6.85446 −0.959816
\(52\) 0 0
\(53\) 0.508203 0.0698071 0.0349036 0.999391i \(-0.488888\pi\)
0.0349036 + 0.999391i \(0.488888\pi\)
\(54\) 0 0
\(55\) 4.18953 0.564917
\(56\) 0 0
\(57\) −0.318669 −0.0422088
\(58\) 0 0
\(59\) 7.04399 0.917050 0.458525 0.888682i \(-0.348378\pi\)
0.458525 + 0.888682i \(0.348378\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −13.2939 −1.64891
\(66\) 0 0
\(67\) 2.66492 0.325572 0.162786 0.986661i \(-0.447952\pi\)
0.162786 + 0.986661i \(0.447952\pi\)
\(68\) 0 0
\(69\) −1.87086 −0.225226
\(70\) 0 0
\(71\) 5.01641 0.595338 0.297669 0.954669i \(-0.403791\pi\)
0.297669 + 0.954669i \(0.403791\pi\)
\(72\) 0 0
\(73\) −4.82687 −0.564943 −0.282471 0.959276i \(-0.591154\pi\)
−0.282471 + 0.959276i \(0.591154\pi\)
\(74\) 0 0
\(75\) −12.5522 −1.44940
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −5.01641 −0.564390 −0.282195 0.959357i \(-0.591062\pi\)
−0.282195 + 0.959357i \(0.591062\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.52461 −0.386876 −0.193438 0.981112i \(-0.561964\pi\)
−0.193438 + 0.981112i \(0.561964\pi\)
\(84\) 0 0
\(85\) 28.7170 3.11479
\(86\) 0 0
\(87\) 3.17313 0.340195
\(88\) 0 0
\(89\) −1.74173 −0.184623 −0.0923115 0.995730i \(-0.529426\pi\)
−0.0923115 + 0.995730i \(0.529426\pi\)
\(90\) 0 0
\(91\) −3.17313 −0.332634
\(92\) 0 0
\(93\) 9.23353 0.957472
\(94\) 0 0
\(95\) 1.33508 0.136976
\(96\) 0 0
\(97\) −12.2499 −1.24379 −0.621896 0.783100i \(-0.713637\pi\)
−0.621896 + 0.783100i \(0.713637\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 4.88727 0.486302 0.243151 0.969988i \(-0.421819\pi\)
0.243151 + 0.969988i \(0.421819\pi\)
\(102\) 0 0
\(103\) 0.637339 0.0627988 0.0313994 0.999507i \(-0.490004\pi\)
0.0313994 + 0.999507i \(0.490004\pi\)
\(104\) 0 0
\(105\) −4.18953 −0.408857
\(106\) 0 0
\(107\) −0.956008 −0.0924208 −0.0462104 0.998932i \(-0.514714\pi\)
−0.0462104 + 0.998932i \(0.514714\pi\)
\(108\) 0 0
\(109\) −7.61259 −0.729154 −0.364577 0.931173i \(-0.618786\pi\)
−0.364577 + 0.931173i \(0.618786\pi\)
\(110\) 0 0
\(111\) 7.55220 0.716823
\(112\) 0 0
\(113\) −7.70892 −0.725194 −0.362597 0.931946i \(-0.618110\pi\)
−0.362597 + 0.931946i \(0.618110\pi\)
\(114\) 0 0
\(115\) 7.83805 0.730902
\(116\) 0 0
\(117\) −3.17313 −0.293356
\(118\) 0 0
\(119\) 6.85446 0.628347
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −9.36266 −0.844203
\(124\) 0 0
\(125\) 31.6402 2.82998
\(126\) 0 0
\(127\) 5.49180 0.487318 0.243659 0.969861i \(-0.421652\pi\)
0.243659 + 0.969861i \(0.421652\pi\)
\(128\) 0 0
\(129\) −10.8873 −0.958571
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 0.318669 0.0276321
\(134\) 0 0
\(135\) −4.18953 −0.360578
\(136\) 0 0
\(137\) 15.6126 1.33387 0.666937 0.745114i \(-0.267605\pi\)
0.666937 + 0.745114i \(0.267605\pi\)
\(138\) 0 0
\(139\) 9.01641 0.764762 0.382381 0.924005i \(-0.375104\pi\)
0.382381 + 0.924005i \(0.375104\pi\)
\(140\) 0 0
\(141\) −8.06040 −0.678808
\(142\) 0 0
\(143\) −3.17313 −0.265350
\(144\) 0 0
\(145\) −13.2939 −1.10400
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −5.20594 −0.426487 −0.213244 0.976999i \(-0.568403\pi\)
−0.213244 + 0.976999i \(0.568403\pi\)
\(150\) 0 0
\(151\) −6.24993 −0.508612 −0.254306 0.967124i \(-0.581847\pi\)
−0.254306 + 0.967124i \(0.581847\pi\)
\(152\) 0 0
\(153\) 6.85446 0.554150
\(154\) 0 0
\(155\) −38.6842 −3.10719
\(156\) 0 0
\(157\) −18.1208 −1.44620 −0.723099 0.690745i \(-0.757283\pi\)
−0.723099 + 0.690745i \(0.757283\pi\)
\(158\) 0 0
\(159\) −0.508203 −0.0403031
\(160\) 0 0
\(161\) 1.87086 0.147445
\(162\) 0 0
\(163\) 2.66492 0.208733 0.104366 0.994539i \(-0.466719\pi\)
0.104366 + 0.994539i \(0.466719\pi\)
\(164\) 0 0
\(165\) −4.18953 −0.326155
\(166\) 0 0
\(167\) 11.1455 0.862468 0.431234 0.902240i \(-0.358078\pi\)
0.431234 + 0.902240i \(0.358078\pi\)
\(168\) 0 0
\(169\) −2.93126 −0.225482
\(170\) 0 0
\(171\) 0.318669 0.0243693
\(172\) 0 0
\(173\) −24.8461 −1.88902 −0.944508 0.328489i \(-0.893461\pi\)
−0.944508 + 0.328489i \(0.893461\pi\)
\(174\) 0 0
\(175\) 12.5522 0.948857
\(176\) 0 0
\(177\) −7.04399 −0.529459
\(178\) 0 0
\(179\) 12.7581 0.953588 0.476794 0.879015i \(-0.341799\pi\)
0.476794 + 0.879015i \(0.341799\pi\)
\(180\) 0 0
\(181\) −3.23353 −0.240346 −0.120173 0.992753i \(-0.538345\pi\)
−0.120173 + 0.992753i \(0.538345\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −31.6402 −2.32623
\(186\) 0 0
\(187\) 6.85446 0.501248
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −20.9753 −1.51772 −0.758858 0.651256i \(-0.774242\pi\)
−0.758858 + 0.651256i \(0.774242\pi\)
\(192\) 0 0
\(193\) 0.249933 0.0179905 0.00899527 0.999960i \(-0.497137\pi\)
0.00899527 + 0.999960i \(0.497137\pi\)
\(194\) 0 0
\(195\) 13.2939 0.951998
\(196\) 0 0
\(197\) 18.4999 1.31806 0.659030 0.752116i \(-0.270967\pi\)
0.659030 + 0.752116i \(0.270967\pi\)
\(198\) 0 0
\(199\) −9.87086 −0.699727 −0.349864 0.936801i \(-0.613772\pi\)
−0.349864 + 0.936801i \(0.613772\pi\)
\(200\) 0 0
\(201\) −2.66492 −0.187969
\(202\) 0 0
\(203\) −3.17313 −0.222710
\(204\) 0 0
\(205\) 39.2252 2.73961
\(206\) 0 0
\(207\) 1.87086 0.130034
\(208\) 0 0
\(209\) 0.318669 0.0220428
\(210\) 0 0
\(211\) 4.63734 0.319248 0.159624 0.987178i \(-0.448972\pi\)
0.159624 + 0.987178i \(0.448972\pi\)
\(212\) 0 0
\(213\) −5.01641 −0.343719
\(214\) 0 0
\(215\) 45.6126 3.11075
\(216\) 0 0
\(217\) −9.23353 −0.626813
\(218\) 0 0
\(219\) 4.82687 0.326170
\(220\) 0 0
\(221\) −21.7501 −1.46307
\(222\) 0 0
\(223\) 18.3463 1.22856 0.614278 0.789090i \(-0.289447\pi\)
0.614278 + 0.789090i \(0.289447\pi\)
\(224\) 0 0
\(225\) 12.5522 0.836813
\(226\) 0 0
\(227\) −0.379068 −0.0251596 −0.0125798 0.999921i \(-0.504004\pi\)
−0.0125798 + 0.999921i \(0.504004\pi\)
\(228\) 0 0
\(229\) 24.9424 1.64824 0.824121 0.566413i \(-0.191669\pi\)
0.824121 + 0.566413i \(0.191669\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) 23.4506 1.53630 0.768151 0.640268i \(-0.221177\pi\)
0.768151 + 0.640268i \(0.221177\pi\)
\(234\) 0 0
\(235\) 33.7693 2.20287
\(236\) 0 0
\(237\) 5.01641 0.325851
\(238\) 0 0
\(239\) −5.07681 −0.328391 −0.164196 0.986428i \(-0.552503\pi\)
−0.164196 + 0.986428i \(0.552503\pi\)
\(240\) 0 0
\(241\) 19.2939 1.24283 0.621415 0.783481i \(-0.286558\pi\)
0.621415 + 0.783481i \(0.286558\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 4.18953 0.267660
\(246\) 0 0
\(247\) −1.01118 −0.0643397
\(248\) 0 0
\(249\) 3.52461 0.223363
\(250\) 0 0
\(251\) −23.8021 −1.50238 −0.751188 0.660088i \(-0.770519\pi\)
−0.751188 + 0.660088i \(0.770519\pi\)
\(252\) 0 0
\(253\) 1.87086 0.117620
\(254\) 0 0
\(255\) −28.7170 −1.79833
\(256\) 0 0
\(257\) 14.9149 0.930363 0.465182 0.885215i \(-0.345989\pi\)
0.465182 + 0.885215i \(0.345989\pi\)
\(258\) 0 0
\(259\) −7.55220 −0.469271
\(260\) 0 0
\(261\) −3.17313 −0.196412
\(262\) 0 0
\(263\) −2.92319 −0.180252 −0.0901259 0.995930i \(-0.528727\pi\)
−0.0901259 + 0.995930i \(0.528727\pi\)
\(264\) 0 0
\(265\) 2.12914 0.130792
\(266\) 0 0
\(267\) 1.74173 0.106592
\(268\) 0 0
\(269\) 11.9672 0.729652 0.364826 0.931076i \(-0.381128\pi\)
0.364826 + 0.931076i \(0.381128\pi\)
\(270\) 0 0
\(271\) 20.3187 1.23427 0.617136 0.786857i \(-0.288293\pi\)
0.617136 + 0.786857i \(0.288293\pi\)
\(272\) 0 0
\(273\) 3.17313 0.192046
\(274\) 0 0
\(275\) 12.5522 0.756926
\(276\) 0 0
\(277\) 18.0552 1.08483 0.542415 0.840111i \(-0.317510\pi\)
0.542415 + 0.840111i \(0.317510\pi\)
\(278\) 0 0
\(279\) −9.23353 −0.552797
\(280\) 0 0
\(281\) 27.2939 1.62822 0.814110 0.580711i \(-0.197226\pi\)
0.814110 + 0.580711i \(0.197226\pi\)
\(282\) 0 0
\(283\) −29.8901 −1.77678 −0.888391 0.459087i \(-0.848177\pi\)
−0.888391 + 0.459087i \(0.848177\pi\)
\(284\) 0 0
\(285\) −1.33508 −0.0790831
\(286\) 0 0
\(287\) 9.36266 0.552660
\(288\) 0 0
\(289\) 29.9836 1.76374
\(290\) 0 0
\(291\) 12.2499 0.718104
\(292\) 0 0
\(293\) −4.34625 −0.253911 −0.126955 0.991908i \(-0.540521\pi\)
−0.126955 + 0.991908i \(0.540521\pi\)
\(294\) 0 0
\(295\) 29.5110 1.71820
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −5.93649 −0.343316
\(300\) 0 0
\(301\) 10.8873 0.627532
\(302\) 0 0
\(303\) −4.88727 −0.280766
\(304\) 0 0
\(305\) −8.37907 −0.479784
\(306\) 0 0
\(307\) 20.7581 1.18473 0.592365 0.805670i \(-0.298194\pi\)
0.592365 + 0.805670i \(0.298194\pi\)
\(308\) 0 0
\(309\) −0.637339 −0.0362569
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 8.12914 0.459486 0.229743 0.973251i \(-0.426211\pi\)
0.229743 + 0.973251i \(0.426211\pi\)
\(314\) 0 0
\(315\) 4.18953 0.236054
\(316\) 0 0
\(317\) −19.9917 −1.12284 −0.561422 0.827530i \(-0.689745\pi\)
−0.561422 + 0.827530i \(0.689745\pi\)
\(318\) 0 0
\(319\) −3.17313 −0.177661
\(320\) 0 0
\(321\) 0.956008 0.0533592
\(322\) 0 0
\(323\) 2.18431 0.121538
\(324\) 0 0
\(325\) −39.8297 −2.20935
\(326\) 0 0
\(327\) 7.61259 0.420977
\(328\) 0 0
\(329\) 8.06040 0.444384
\(330\) 0 0
\(331\) −17.0164 −0.935306 −0.467653 0.883912i \(-0.654900\pi\)
−0.467653 + 0.883912i \(0.654900\pi\)
\(332\) 0 0
\(333\) −7.55220 −0.413858
\(334\) 0 0
\(335\) 11.1648 0.609998
\(336\) 0 0
\(337\) 1.52461 0.0830508 0.0415254 0.999137i \(-0.486778\pi\)
0.0415254 + 0.999137i \(0.486778\pi\)
\(338\) 0 0
\(339\) 7.70892 0.418691
\(340\) 0 0
\(341\) −9.23353 −0.500023
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −7.83805 −0.421986
\(346\) 0 0
\(347\) −17.4506 −0.936800 −0.468400 0.883517i \(-0.655169\pi\)
−0.468400 + 0.883517i \(0.655169\pi\)
\(348\) 0 0
\(349\) 6.85969 0.367191 0.183595 0.983002i \(-0.441226\pi\)
0.183595 + 0.983002i \(0.441226\pi\)
\(350\) 0 0
\(351\) 3.17313 0.169369
\(352\) 0 0
\(353\) −31.9313 −1.69953 −0.849765 0.527162i \(-0.823256\pi\)
−0.849765 + 0.527162i \(0.823256\pi\)
\(354\) 0 0
\(355\) 21.0164 1.11544
\(356\) 0 0
\(357\) −6.85446 −0.362776
\(358\) 0 0
\(359\) 24.4342 1.28959 0.644795 0.764356i \(-0.276943\pi\)
0.644795 + 0.764356i \(0.276943\pi\)
\(360\) 0 0
\(361\) −18.8984 −0.994655
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −20.2223 −1.05849
\(366\) 0 0
\(367\) −17.0716 −0.891129 −0.445565 0.895250i \(-0.646997\pi\)
−0.445565 + 0.895250i \(0.646997\pi\)
\(368\) 0 0
\(369\) 9.36266 0.487401
\(370\) 0 0
\(371\) 0.508203 0.0263846
\(372\) 0 0
\(373\) 3.17836 0.164569 0.0822845 0.996609i \(-0.473778\pi\)
0.0822845 + 0.996609i \(0.473778\pi\)
\(374\) 0 0
\(375\) −31.6402 −1.63389
\(376\) 0 0
\(377\) 10.0687 0.518566
\(378\) 0 0
\(379\) −3.93960 −0.202364 −0.101182 0.994868i \(-0.532262\pi\)
−0.101182 + 0.994868i \(0.532262\pi\)
\(380\) 0 0
\(381\) −5.49180 −0.281353
\(382\) 0 0
\(383\) 24.7581 1.26508 0.632541 0.774527i \(-0.282012\pi\)
0.632541 + 0.774527i \(0.282012\pi\)
\(384\) 0 0
\(385\) 4.18953 0.213518
\(386\) 0 0
\(387\) 10.8873 0.553431
\(388\) 0 0
\(389\) 3.65375 0.185252 0.0926261 0.995701i \(-0.470474\pi\)
0.0926261 + 0.995701i \(0.470474\pi\)
\(390\) 0 0
\(391\) 12.8238 0.648526
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) −21.0164 −1.05745
\(396\) 0 0
\(397\) 4.56337 0.229029 0.114515 0.993422i \(-0.463469\pi\)
0.114515 + 0.993422i \(0.463469\pi\)
\(398\) 0 0
\(399\) −0.318669 −0.0159534
\(400\) 0 0
\(401\) −7.23353 −0.361225 −0.180613 0.983554i \(-0.557808\pi\)
−0.180613 + 0.983554i \(0.557808\pi\)
\(402\) 0 0
\(403\) 29.2992 1.45949
\(404\) 0 0
\(405\) 4.18953 0.208180
\(406\) 0 0
\(407\) −7.55220 −0.374348
\(408\) 0 0
\(409\) 5.30749 0.262439 0.131219 0.991353i \(-0.458111\pi\)
0.131219 + 0.991353i \(0.458111\pi\)
\(410\) 0 0
\(411\) −15.6126 −0.770112
\(412\) 0 0
\(413\) 7.04399 0.346612
\(414\) 0 0
\(415\) −14.7665 −0.724858
\(416\) 0 0
\(417\) −9.01641 −0.441535
\(418\) 0 0
\(419\) 11.4231 0.558053 0.279026 0.960283i \(-0.409988\pi\)
0.279026 + 0.960283i \(0.409988\pi\)
\(420\) 0 0
\(421\) −27.4147 −1.33611 −0.668056 0.744111i \(-0.732873\pi\)
−0.668056 + 0.744111i \(0.732873\pi\)
\(422\) 0 0
\(423\) 8.06040 0.391910
\(424\) 0 0
\(425\) 86.0385 4.17348
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 0 0
\(429\) 3.17313 0.153200
\(430\) 0 0
\(431\) 2.28586 0.110106 0.0550529 0.998483i \(-0.482467\pi\)
0.0550529 + 0.998483i \(0.482467\pi\)
\(432\) 0 0
\(433\) 31.5714 1.51723 0.758613 0.651541i \(-0.225877\pi\)
0.758613 + 0.651541i \(0.225877\pi\)
\(434\) 0 0
\(435\) 13.2939 0.637395
\(436\) 0 0
\(437\) 0.596187 0.0285195
\(438\) 0 0
\(439\) 18.5275 0.884267 0.442133 0.896949i \(-0.354222\pi\)
0.442133 + 0.896949i \(0.354222\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −28.4342 −1.35095 −0.675476 0.737382i \(-0.736062\pi\)
−0.675476 + 0.737382i \(0.736062\pi\)
\(444\) 0 0
\(445\) −7.29703 −0.345913
\(446\) 0 0
\(447\) 5.20594 0.246233
\(448\) 0 0
\(449\) −5.68656 −0.268365 −0.134183 0.990957i \(-0.542841\pi\)
−0.134183 + 0.990957i \(0.542841\pi\)
\(450\) 0 0
\(451\) 9.36266 0.440871
\(452\) 0 0
\(453\) 6.24993 0.293647
\(454\) 0 0
\(455\) −13.2939 −0.623229
\(456\) 0 0
\(457\) −13.0081 −0.608492 −0.304246 0.952594i \(-0.598404\pi\)
−0.304246 + 0.952594i \(0.598404\pi\)
\(458\) 0 0
\(459\) −6.85446 −0.319939
\(460\) 0 0
\(461\) −6.37907 −0.297103 −0.148551 0.988905i \(-0.547461\pi\)
−0.148551 + 0.988905i \(0.547461\pi\)
\(462\) 0 0
\(463\) −34.0932 −1.58445 −0.792223 0.610232i \(-0.791076\pi\)
−0.792223 + 0.610232i \(0.791076\pi\)
\(464\) 0 0
\(465\) 38.6842 1.79394
\(466\) 0 0
\(467\) −18.1484 −0.839807 −0.419903 0.907569i \(-0.637936\pi\)
−0.419903 + 0.907569i \(0.637936\pi\)
\(468\) 0 0
\(469\) 2.66492 0.123055
\(470\) 0 0
\(471\) 18.1208 0.834962
\(472\) 0 0
\(473\) 10.8873 0.500597
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 0.508203 0.0232690
\(478\) 0 0
\(479\) 42.7498 1.95329 0.976644 0.214864i \(-0.0689307\pi\)
0.976644 + 0.214864i \(0.0689307\pi\)
\(480\) 0 0
\(481\) 23.9641 1.09267
\(482\) 0 0
\(483\) −1.87086 −0.0851273
\(484\) 0 0
\(485\) −51.3215 −2.33039
\(486\) 0 0
\(487\) −26.7909 −1.21401 −0.607007 0.794697i \(-0.707630\pi\)
−0.607007 + 0.794697i \(0.707630\pi\)
\(488\) 0 0
\(489\) −2.66492 −0.120512
\(490\) 0 0
\(491\) −31.6813 −1.42976 −0.714879 0.699248i \(-0.753518\pi\)
−0.714879 + 0.699248i \(0.753518\pi\)
\(492\) 0 0
\(493\) −21.7501 −0.979574
\(494\) 0 0
\(495\) 4.18953 0.188306
\(496\) 0 0
\(497\) 5.01641 0.225017
\(498\) 0 0
\(499\) 24.1260 1.08003 0.540015 0.841656i \(-0.318419\pi\)
0.540015 + 0.841656i \(0.318419\pi\)
\(500\) 0 0
\(501\) −11.1455 −0.497946
\(502\) 0 0
\(503\) −30.8873 −1.37720 −0.688598 0.725144i \(-0.741773\pi\)
−0.688598 + 0.725144i \(0.741773\pi\)
\(504\) 0 0
\(505\) 20.4754 0.911143
\(506\) 0 0
\(507\) 2.93126 0.130182
\(508\) 0 0
\(509\) 28.2088 1.25033 0.625166 0.780492i \(-0.285031\pi\)
0.625166 + 0.780492i \(0.285031\pi\)
\(510\) 0 0
\(511\) −4.82687 −0.213528
\(512\) 0 0
\(513\) −0.318669 −0.0140696
\(514\) 0 0
\(515\) 2.67015 0.117661
\(516\) 0 0
\(517\) 8.06040 0.354496
\(518\) 0 0
\(519\) 24.8461 1.09062
\(520\) 0 0
\(521\) −33.7610 −1.47910 −0.739548 0.673104i \(-0.764961\pi\)
−0.739548 + 0.673104i \(0.764961\pi\)
\(522\) 0 0
\(523\) −12.8185 −0.560515 −0.280258 0.959925i \(-0.590420\pi\)
−0.280258 + 0.959925i \(0.590420\pi\)
\(524\) 0 0
\(525\) −12.5522 −0.547823
\(526\) 0 0
\(527\) −63.2908 −2.75699
\(528\) 0 0
\(529\) −19.4999 −0.847820
\(530\) 0 0
\(531\) 7.04399 0.305683
\(532\) 0 0
\(533\) −29.7089 −1.28684
\(534\) 0 0
\(535\) −4.00523 −0.173161
\(536\) 0 0
\(537\) −12.7581 −0.550554
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −21.8625 −0.939943 −0.469972 0.882681i \(-0.655736\pi\)
−0.469972 + 0.882681i \(0.655736\pi\)
\(542\) 0 0
\(543\) 3.23353 0.138764
\(544\) 0 0
\(545\) −31.8932 −1.36616
\(546\) 0 0
\(547\) −23.4178 −1.00127 −0.500637 0.865657i \(-0.666901\pi\)
−0.500637 + 0.865657i \(0.666901\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −1.01118 −0.0430776
\(552\) 0 0
\(553\) −5.01641 −0.213319
\(554\) 0 0
\(555\) 31.6402 1.34305
\(556\) 0 0
\(557\) −6.91486 −0.292992 −0.146496 0.989211i \(-0.546800\pi\)
−0.146496 + 0.989211i \(0.546800\pi\)
\(558\) 0 0
\(559\) −34.5467 −1.46117
\(560\) 0 0
\(561\) −6.85446 −0.289395
\(562\) 0 0
\(563\) −11.8381 −0.498914 −0.249457 0.968386i \(-0.580252\pi\)
−0.249457 + 0.968386i \(0.580252\pi\)
\(564\) 0 0
\(565\) −32.2968 −1.35874
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 15.2252 0.637154 0.318577 0.947897i \(-0.396795\pi\)
0.318577 + 0.947897i \(0.396795\pi\)
\(572\) 0 0
\(573\) 20.9753 0.876254
\(574\) 0 0
\(575\) 23.4835 0.979328
\(576\) 0 0
\(577\) 17.3215 0.721104 0.360552 0.932739i \(-0.382588\pi\)
0.360552 + 0.932739i \(0.382588\pi\)
\(578\) 0 0
\(579\) −0.249933 −0.0103868
\(580\) 0 0
\(581\) −3.52461 −0.146225
\(582\) 0 0
\(583\) 0.508203 0.0210476
\(584\) 0 0
\(585\) −13.2939 −0.549636
\(586\) 0 0
\(587\) −27.8678 −1.15023 −0.575113 0.818074i \(-0.695042\pi\)
−0.575113 + 0.818074i \(0.695042\pi\)
\(588\) 0 0
\(589\) −2.94244 −0.121241
\(590\) 0 0
\(591\) −18.4999 −0.760983
\(592\) 0 0
\(593\) −24.3463 −0.999781 −0.499890 0.866089i \(-0.666626\pi\)
−0.499890 + 0.866089i \(0.666626\pi\)
\(594\) 0 0
\(595\) 28.7170 1.17728
\(596\) 0 0
\(597\) 9.87086 0.403988
\(598\) 0 0
\(599\) −21.0164 −0.858707 −0.429354 0.903136i \(-0.641259\pi\)
−0.429354 + 0.903136i \(0.641259\pi\)
\(600\) 0 0
\(601\) 40.9477 1.67029 0.835145 0.550030i \(-0.185384\pi\)
0.835145 + 0.550030i \(0.185384\pi\)
\(602\) 0 0
\(603\) 2.66492 0.108524
\(604\) 0 0
\(605\) 4.18953 0.170329
\(606\) 0 0
\(607\) 14.0276 0.569362 0.284681 0.958622i \(-0.408112\pi\)
0.284681 + 0.958622i \(0.408112\pi\)
\(608\) 0 0
\(609\) 3.17313 0.128582
\(610\) 0 0
\(611\) −25.5767 −1.03472
\(612\) 0 0
\(613\) 18.5306 0.748442 0.374221 0.927339i \(-0.377910\pi\)
0.374221 + 0.927339i \(0.377910\pi\)
\(614\) 0 0
\(615\) −39.2252 −1.58171
\(616\) 0 0
\(617\) 2.43424 0.0979987 0.0489994 0.998799i \(-0.484397\pi\)
0.0489994 + 0.998799i \(0.484397\pi\)
\(618\) 0 0
\(619\) 30.4259 1.22292 0.611460 0.791275i \(-0.290582\pi\)
0.611460 + 0.791275i \(0.290582\pi\)
\(620\) 0 0
\(621\) −1.87086 −0.0750752
\(622\) 0 0
\(623\) −1.74173 −0.0697809
\(624\) 0 0
\(625\) 69.7966 2.79187
\(626\) 0 0
\(627\) −0.318669 −0.0127264
\(628\) 0 0
\(629\) −51.7662 −2.06405
\(630\) 0 0
\(631\) −34.9836 −1.39267 −0.696337 0.717715i \(-0.745188\pi\)
−0.696337 + 0.717715i \(0.745188\pi\)
\(632\) 0 0
\(633\) −4.63734 −0.184318
\(634\) 0 0
\(635\) 23.0081 0.913047
\(636\) 0 0
\(637\) −3.17313 −0.125724
\(638\) 0 0
\(639\) 5.01641 0.198446
\(640\) 0 0
\(641\) 8.56337 0.338233 0.169116 0.985596i \(-0.445909\pi\)
0.169116 + 0.985596i \(0.445909\pi\)
\(642\) 0 0
\(643\) 5.11273 0.201626 0.100813 0.994905i \(-0.467856\pi\)
0.100813 + 0.994905i \(0.467856\pi\)
\(644\) 0 0
\(645\) −45.6126 −1.79599
\(646\) 0 0
\(647\) −25.7693 −1.01310 −0.506548 0.862212i \(-0.669079\pi\)
−0.506548 + 0.862212i \(0.669079\pi\)
\(648\) 0 0
\(649\) 7.04399 0.276501
\(650\) 0 0
\(651\) 9.23353 0.361890
\(652\) 0 0
\(653\) −47.8953 −1.87429 −0.937145 0.348941i \(-0.886541\pi\)
−0.937145 + 0.348941i \(0.886541\pi\)
\(654\) 0 0
\(655\) −16.7581 −0.654795
\(656\) 0 0
\(657\) −4.82687 −0.188314
\(658\) 0 0
\(659\) −8.95601 −0.348877 −0.174438 0.984668i \(-0.555811\pi\)
−0.174438 + 0.984668i \(0.555811\pi\)
\(660\) 0 0
\(661\) −10.7993 −0.420044 −0.210022 0.977697i \(-0.567353\pi\)
−0.210022 + 0.977697i \(0.567353\pi\)
\(662\) 0 0
\(663\) 21.7501 0.844703
\(664\) 0 0
\(665\) 1.33508 0.0517720
\(666\) 0 0
\(667\) −5.93649 −0.229862
\(668\) 0 0
\(669\) −18.3463 −0.709307
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 32.2499 1.24314 0.621572 0.783357i \(-0.286494\pi\)
0.621572 + 0.783357i \(0.286494\pi\)
\(674\) 0 0
\(675\) −12.5522 −0.483134
\(676\) 0 0
\(677\) −27.7089 −1.06494 −0.532470 0.846449i \(-0.678736\pi\)
−0.532470 + 0.846449i \(0.678736\pi\)
\(678\) 0 0
\(679\) −12.2499 −0.470109
\(680\) 0 0
\(681\) 0.379068 0.0145259
\(682\) 0 0
\(683\) −22.3156 −0.853881 −0.426941 0.904280i \(-0.640409\pi\)
−0.426941 + 0.904280i \(0.640409\pi\)
\(684\) 0 0
\(685\) 65.4095 2.49917
\(686\) 0 0
\(687\) −24.9424 −0.951614
\(688\) 0 0
\(689\) −1.61259 −0.0614349
\(690\) 0 0
\(691\) −25.1372 −0.956264 −0.478132 0.878288i \(-0.658686\pi\)
−0.478132 + 0.878288i \(0.658686\pi\)
\(692\) 0 0
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) 37.7745 1.43287
\(696\) 0 0
\(697\) 64.1760 2.43084
\(698\) 0 0
\(699\) −23.4506 −0.886985
\(700\) 0 0
\(701\) −19.2747 −0.727995 −0.363997 0.931400i \(-0.618588\pi\)
−0.363997 + 0.931400i \(0.618588\pi\)
\(702\) 0 0
\(703\) −2.40665 −0.0907686
\(704\) 0 0
\(705\) −33.7693 −1.27183
\(706\) 0 0
\(707\) 4.88727 0.183805
\(708\) 0 0
\(709\) 9.76098 0.366581 0.183291 0.983059i \(-0.441325\pi\)
0.183291 + 0.983059i \(0.441325\pi\)
\(710\) 0 0
\(711\) −5.01641 −0.188130
\(712\) 0 0
\(713\) −17.2747 −0.646942
\(714\) 0 0
\(715\) −13.2939 −0.497165
\(716\) 0 0
\(717\) 5.07681 0.189597
\(718\) 0 0
\(719\) −14.0276 −0.523141 −0.261570 0.965184i \(-0.584240\pi\)
−0.261570 + 0.965184i \(0.584240\pi\)
\(720\) 0 0
\(721\) 0.637339 0.0237357
\(722\) 0 0
\(723\) −19.2939 −0.717549
\(724\) 0 0
\(725\) −39.8297 −1.47924
\(726\) 0 0
\(727\) 34.3051 1.27231 0.636153 0.771563i \(-0.280525\pi\)
0.636153 + 0.771563i \(0.280525\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 74.6263 2.76016
\(732\) 0 0
\(733\) −10.7581 −0.397361 −0.198680 0.980064i \(-0.563666\pi\)
−0.198680 + 0.980064i \(0.563666\pi\)
\(734\) 0 0
\(735\) −4.18953 −0.154533
\(736\) 0 0
\(737\) 2.66492 0.0981637
\(738\) 0 0
\(739\) 6.38741 0.234965 0.117482 0.993075i \(-0.462518\pi\)
0.117482 + 0.993075i \(0.462518\pi\)
\(740\) 0 0
\(741\) 1.01118 0.0371466
\(742\) 0 0
\(743\) 23.1096 0.847810 0.423905 0.905707i \(-0.360659\pi\)
0.423905 + 0.905707i \(0.360659\pi\)
\(744\) 0 0
\(745\) −21.8105 −0.799074
\(746\) 0 0
\(747\) −3.52461 −0.128959
\(748\) 0 0
\(749\) −0.956008 −0.0349318
\(750\) 0 0
\(751\) −31.8678 −1.16287 −0.581435 0.813593i \(-0.697509\pi\)
−0.581435 + 0.813593i \(0.697509\pi\)
\(752\) 0 0
\(753\) 23.8021 0.867398
\(754\) 0 0
\(755\) −26.1843 −0.952944
\(756\) 0 0
\(757\) −48.3103 −1.75587 −0.877934 0.478781i \(-0.841079\pi\)
−0.877934 + 0.478781i \(0.841079\pi\)
\(758\) 0 0
\(759\) −1.87086 −0.0679081
\(760\) 0 0
\(761\) −17.8625 −0.647516 −0.323758 0.946140i \(-0.604946\pi\)
−0.323758 + 0.946140i \(0.604946\pi\)
\(762\) 0 0
\(763\) −7.61259 −0.275594
\(764\) 0 0
\(765\) 28.7170 1.03826
\(766\) 0 0
\(767\) −22.3515 −0.807065
\(768\) 0 0
\(769\) −24.8820 −0.897269 −0.448635 0.893715i \(-0.648090\pi\)
−0.448635 + 0.893715i \(0.648090\pi\)
\(770\) 0 0
\(771\) −14.9149 −0.537145
\(772\) 0 0
\(773\) −34.9700 −1.25778 −0.628892 0.777492i \(-0.716491\pi\)
−0.628892 + 0.777492i \(0.716491\pi\)
\(774\) 0 0
\(775\) −115.901 −4.16329
\(776\) 0 0
\(777\) 7.55220 0.270933
\(778\) 0 0
\(779\) 2.98359 0.106898
\(780\) 0 0
\(781\) 5.01641 0.179501
\(782\) 0 0
\(783\) 3.17313 0.113398
\(784\) 0 0
\(785\) −75.9177 −2.70962
\(786\) 0 0
\(787\) 15.1648 0.540566 0.270283 0.962781i \(-0.412883\pi\)
0.270283 + 0.962781i \(0.412883\pi\)
\(788\) 0 0
\(789\) 2.92319 0.104068
\(790\) 0 0
\(791\) −7.70892 −0.274097
\(792\) 0 0
\(793\) 6.34625 0.225362
\(794\) 0 0
\(795\) −2.12914 −0.0755126
\(796\) 0 0
\(797\) 8.24470 0.292042 0.146021 0.989281i \(-0.453353\pi\)
0.146021 + 0.989281i \(0.453353\pi\)
\(798\) 0 0
\(799\) 55.2497 1.95459
\(800\) 0 0
\(801\) −1.74173 −0.0615410
\(802\) 0 0
\(803\) −4.82687 −0.170337
\(804\) 0 0
\(805\) 7.83805 0.276255
\(806\) 0 0
\(807\) −11.9672 −0.421265
\(808\) 0 0
\(809\) 29.8433 1.04923 0.524617 0.851338i \(-0.324209\pi\)
0.524617 + 0.851338i \(0.324209\pi\)
\(810\) 0 0
\(811\) 16.6321 0.584032 0.292016 0.956413i \(-0.405674\pi\)
0.292016 + 0.956413i \(0.405674\pi\)
\(812\) 0 0
\(813\) −20.3187 −0.712607
\(814\) 0 0
\(815\) 11.1648 0.391086
\(816\) 0 0
\(817\) 3.46944 0.121380
\(818\) 0 0
\(819\) −3.17313 −0.110878
\(820\) 0 0
\(821\) 30.3327 1.05862 0.529309 0.848429i \(-0.322451\pi\)
0.529309 + 0.848429i \(0.322451\pi\)
\(822\) 0 0
\(823\) 18.4067 0.641616 0.320808 0.947144i \(-0.396046\pi\)
0.320808 + 0.947144i \(0.396046\pi\)
\(824\) 0 0
\(825\) −12.5522 −0.437011
\(826\) 0 0
\(827\) −45.4559 −1.58066 −0.790328 0.612684i \(-0.790090\pi\)
−0.790328 + 0.612684i \(0.790090\pi\)
\(828\) 0 0
\(829\) −20.3463 −0.706655 −0.353327 0.935500i \(-0.614950\pi\)
−0.353327 + 0.935500i \(0.614950\pi\)
\(830\) 0 0
\(831\) −18.0552 −0.626327
\(832\) 0 0
\(833\) 6.85446 0.237493
\(834\) 0 0
\(835\) 46.6946 1.61593
\(836\) 0 0
\(837\) 9.23353 0.319157
\(838\) 0 0
\(839\) −16.1812 −0.558637 −0.279318 0.960199i \(-0.590109\pi\)
−0.279318 + 0.960199i \(0.590109\pi\)
\(840\) 0 0
\(841\) −18.9313 −0.652802
\(842\) 0 0
\(843\) −27.2939 −0.940053
\(844\) 0 0
\(845\) −12.2806 −0.422466
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) 29.8901 1.02583
\(850\) 0 0
\(851\) −14.1291 −0.484341
\(852\) 0 0
\(853\) −20.9836 −0.718465 −0.359232 0.933248i \(-0.616961\pi\)
−0.359232 + 0.933248i \(0.616961\pi\)
\(854\) 0 0
\(855\) 1.33508 0.0456586
\(856\) 0 0
\(857\) 4.79095 0.163656 0.0818279 0.996646i \(-0.473924\pi\)
0.0818279 + 0.996646i \(0.473924\pi\)
\(858\) 0 0
\(859\) 9.96719 0.340076 0.170038 0.985438i \(-0.445611\pi\)
0.170038 + 0.985438i \(0.445611\pi\)
\(860\) 0 0
\(861\) −9.36266 −0.319079
\(862\) 0 0
\(863\) −25.5470 −0.869629 −0.434814 0.900520i \(-0.643186\pi\)
−0.434814 + 0.900520i \(0.643186\pi\)
\(864\) 0 0
\(865\) −104.094 −3.53929
\(866\) 0 0
\(867\) −29.9836 −1.01830
\(868\) 0 0
\(869\) −5.01641 −0.170170
\(870\) 0 0
\(871\) −8.45614 −0.286525
\(872\) 0 0
\(873\) −12.2499 −0.414597
\(874\) 0 0
\(875\) 31.6402 1.06963
\(876\) 0 0
\(877\) −50.9893 −1.72179 −0.860893 0.508787i \(-0.830094\pi\)
−0.860893 + 0.508787i \(0.830094\pi\)
\(878\) 0 0
\(879\) 4.34625 0.146596
\(880\) 0 0
\(881\) 32.1895 1.08449 0.542246 0.840219i \(-0.317574\pi\)
0.542246 + 0.840219i \(0.317574\pi\)
\(882\) 0 0
\(883\) −36.6154 −1.23221 −0.616104 0.787665i \(-0.711290\pi\)
−0.616104 + 0.787665i \(0.711290\pi\)
\(884\) 0 0
\(885\) −29.5110 −0.992003
\(886\) 0 0
\(887\) 48.2004 1.61841 0.809206 0.587525i \(-0.199897\pi\)
0.809206 + 0.587525i \(0.199897\pi\)
\(888\) 0 0
\(889\) 5.49180 0.184189
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 2.56860 0.0859550
\(894\) 0 0
\(895\) 53.4506 1.78666
\(896\) 0 0
\(897\) 5.93649 0.198214
\(898\) 0 0
\(899\) 29.2992 0.977181
\(900\) 0 0
\(901\) 3.48346 0.116051
\(902\) 0 0
\(903\) −10.8873 −0.362306
\(904\) 0 0
\(905\) −13.5470 −0.450316
\(906\) 0 0
\(907\) −22.9013 −0.760425 −0.380212 0.924899i \(-0.624149\pi\)
−0.380212 + 0.924899i \(0.624149\pi\)
\(908\) 0 0
\(909\) 4.88727 0.162101
\(910\) 0 0
\(911\) 36.0552 1.19456 0.597281 0.802032i \(-0.296248\pi\)
0.597281 + 0.802032i \(0.296248\pi\)
\(912\) 0 0
\(913\) −3.52461 −0.116648
\(914\) 0 0
\(915\) 8.37907 0.277003
\(916\) 0 0
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) 28.3379 0.934782 0.467391 0.884051i \(-0.345194\pi\)
0.467391 + 0.884051i \(0.345194\pi\)
\(920\) 0 0
\(921\) −20.7581 −0.684004
\(922\) 0 0
\(923\) −15.9177 −0.523937
\(924\) 0 0
\(925\) −94.7966 −3.11689
\(926\) 0 0
\(927\) 0.637339 0.0209329
\(928\) 0 0
\(929\) 5.08514 0.166838 0.0834191 0.996515i \(-0.473416\pi\)
0.0834191 + 0.996515i \(0.473416\pi\)
\(930\) 0 0
\(931\) 0.318669 0.0104440
\(932\) 0 0
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) 28.7170 0.939146
\(936\) 0 0
\(937\) 32.2088 1.05222 0.526108 0.850418i \(-0.323651\pi\)
0.526108 + 0.850418i \(0.323651\pi\)
\(938\) 0 0
\(939\) −8.12914 −0.265284
\(940\) 0 0
\(941\) 32.7805 1.06861 0.534307 0.845291i \(-0.320573\pi\)
0.534307 + 0.845291i \(0.320573\pi\)
\(942\) 0 0
\(943\) 17.5163 0.570408
\(944\) 0 0
\(945\) −4.18953 −0.136286
\(946\) 0 0
\(947\) 27.3627 0.889167 0.444584 0.895737i \(-0.353352\pi\)
0.444584 + 0.895737i \(0.353352\pi\)
\(948\) 0 0
\(949\) 15.3163 0.497188
\(950\) 0 0
\(951\) 19.9917 0.648274
\(952\) 0 0
\(953\) −24.6894 −0.799768 −0.399884 0.916566i \(-0.630950\pi\)
−0.399884 + 0.916566i \(0.630950\pi\)
\(954\) 0 0
\(955\) −87.8765 −2.84362
\(956\) 0 0
\(957\) 3.17313 0.102573
\(958\) 0 0
\(959\) 15.6126 0.504157
\(960\) 0 0
\(961\) 54.2580 1.75026
\(962\) 0 0
\(963\) −0.956008 −0.0308069
\(964\) 0 0
\(965\) 1.04710 0.0337074
\(966\) 0 0
\(967\) −21.3298 −0.685922 −0.342961 0.939350i \(-0.611430\pi\)
−0.342961 + 0.939350i \(0.611430\pi\)
\(968\) 0 0
\(969\) −2.18431 −0.0701700
\(970\) 0 0
\(971\) −28.4946 −0.914436 −0.457218 0.889355i \(-0.651154\pi\)
−0.457218 + 0.889355i \(0.651154\pi\)
\(972\) 0 0
\(973\) 9.01641 0.289053
\(974\) 0 0
\(975\) 39.8297 1.27557
\(976\) 0 0
\(977\) −19.8157 −0.633960 −0.316980 0.948432i \(-0.602669\pi\)
−0.316980 + 0.948432i \(0.602669\pi\)
\(978\) 0 0
\(979\) −1.74173 −0.0556659
\(980\) 0 0
\(981\) −7.61259 −0.243051
\(982\) 0 0
\(983\) 53.7745 1.71514 0.857571 0.514366i \(-0.171973\pi\)
0.857571 + 0.514366i \(0.171973\pi\)
\(984\) 0 0
\(985\) 77.5058 2.46954
\(986\) 0 0
\(987\) −8.06040 −0.256565
\(988\) 0 0
\(989\) 20.3686 0.647684
\(990\) 0 0
\(991\) −49.7693 −1.58097 −0.790487 0.612479i \(-0.790173\pi\)
−0.790487 + 0.612479i \(0.790173\pi\)
\(992\) 0 0
\(993\) 17.0164 0.539999
\(994\) 0 0
\(995\) −41.3543 −1.31102
\(996\) 0 0
\(997\) 0.659696 0.0208928 0.0104464 0.999945i \(-0.496675\pi\)
0.0104464 + 0.999945i \(0.496675\pi\)
\(998\) 0 0
\(999\) 7.55220 0.238941
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 3696.2.a.bo.1.3 3
4.3 odd 2 231.2.a.e.1.1 3
12.11 even 2 693.2.a.l.1.3 3
20.19 odd 2 5775.2.a.bp.1.3 3
28.27 even 2 1617.2.a.t.1.1 3
44.43 even 2 2541.2.a.bg.1.3 3
84.83 odd 2 4851.2.a.bi.1.3 3
132.131 odd 2 7623.2.a.cd.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.1 3 4.3 odd 2
693.2.a.l.1.3 3 12.11 even 2
1617.2.a.t.1.1 3 28.27 even 2
2541.2.a.bg.1.3 3 44.43 even 2
3696.2.a.bo.1.3 3 1.1 even 1 trivial
4851.2.a.bi.1.3 3 84.83 odd 2
5775.2.a.bp.1.3 3 20.19 odd 2
7623.2.a.cd.1.1 3 132.131 odd 2