Properties

Label 8034.2.a.w.1.1
Level 8034
Weight 2
Character 8034.1
Self dual yes
Analytic conductor 64.152
Analytic rank 1
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} - 2790 x^{3} - 352 x^{2} + 1768 x + 768\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.37328\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.83399 q^{5} -1.00000 q^{6} -3.29790 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.83399 q^{5} -1.00000 q^{6} -3.29790 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.83399 q^{10} -1.08694 q^{11} +1.00000 q^{12} +1.00000 q^{13} +3.29790 q^{14} -3.83399 q^{15} +1.00000 q^{16} -5.28528 q^{17} -1.00000 q^{18} +7.52849 q^{19} -3.83399 q^{20} -3.29790 q^{21} +1.08694 q^{22} -4.86940 q^{23} -1.00000 q^{24} +9.69950 q^{25} -1.00000 q^{26} +1.00000 q^{27} -3.29790 q^{28} -8.02922 q^{29} +3.83399 q^{30} +3.95486 q^{31} -1.00000 q^{32} -1.08694 q^{33} +5.28528 q^{34} +12.6441 q^{35} +1.00000 q^{36} +3.95146 q^{37} -7.52849 q^{38} +1.00000 q^{39} +3.83399 q^{40} +5.35805 q^{41} +3.29790 q^{42} +11.4696 q^{43} -1.08694 q^{44} -3.83399 q^{45} +4.86940 q^{46} +11.5592 q^{47} +1.00000 q^{48} +3.87612 q^{49} -9.69950 q^{50} -5.28528 q^{51} +1.00000 q^{52} +1.82123 q^{53} -1.00000 q^{54} +4.16732 q^{55} +3.29790 q^{56} +7.52849 q^{57} +8.02922 q^{58} +10.3370 q^{59} -3.83399 q^{60} -14.8682 q^{61} -3.95486 q^{62} -3.29790 q^{63} +1.00000 q^{64} -3.83399 q^{65} +1.08694 q^{66} +9.36012 q^{67} -5.28528 q^{68} -4.86940 q^{69} -12.6441 q^{70} -9.48805 q^{71} -1.00000 q^{72} -5.79522 q^{73} -3.95146 q^{74} +9.69950 q^{75} +7.52849 q^{76} +3.58461 q^{77} -1.00000 q^{78} +6.22766 q^{79} -3.83399 q^{80} +1.00000 q^{81} -5.35805 q^{82} +2.49468 q^{83} -3.29790 q^{84} +20.2637 q^{85} -11.4696 q^{86} -8.02922 q^{87} +1.08694 q^{88} +5.79834 q^{89} +3.83399 q^{90} -3.29790 q^{91} -4.86940 q^{92} +3.95486 q^{93} -11.5592 q^{94} -28.8642 q^{95} -1.00000 q^{96} -5.65476 q^{97} -3.87612 q^{98} -1.08694 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + 4q^{10} - 9q^{11} + 12q^{12} + 12q^{13} - 4q^{15} + 12q^{16} - 20q^{17} - 12q^{18} + 4q^{19} - 4q^{20} + 9q^{22} - 30q^{23} - 12q^{24} + 14q^{25} - 12q^{26} + 12q^{27} - 29q^{29} + 4q^{30} + 6q^{31} - 12q^{32} - 9q^{33} + 20q^{34} - 22q^{35} + 12q^{36} + 7q^{37} - 4q^{38} + 12q^{39} + 4q^{40} - 8q^{41} - 8q^{43} - 9q^{44} - 4q^{45} + 30q^{46} - 16q^{47} + 12q^{48} + 10q^{49} - 14q^{50} - 20q^{51} + 12q^{52} - 9q^{53} - 12q^{54} - 20q^{55} + 4q^{57} + 29q^{58} - 29q^{59} - 4q^{60} - 26q^{61} - 6q^{62} + 12q^{64} - 4q^{65} + 9q^{66} + 12q^{67} - 20q^{68} - 30q^{69} + 22q^{70} - 35q^{71} - 12q^{72} + 18q^{73} - 7q^{74} + 14q^{75} + 4q^{76} - 25q^{77} - 12q^{78} - 37q^{79} - 4q^{80} + 12q^{81} + 8q^{82} - 24q^{83} - 17q^{85} + 8q^{86} - 29q^{87} + 9q^{88} + 15q^{89} + 4q^{90} - 30q^{92} + 6q^{93} + 16q^{94} - 54q^{95} - 12q^{96} - 11q^{97} - 10q^{98} - 9q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −3.83399 −1.71461 −0.857307 0.514806i \(-0.827864\pi\)
−0.857307 + 0.514806i \(0.827864\pi\)
\(6\) −1.00000 −0.408248
\(7\) −3.29790 −1.24649 −0.623244 0.782028i \(-0.714186\pi\)
−0.623244 + 0.782028i \(0.714186\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.83399 1.21241
\(11\) −1.08694 −0.327724 −0.163862 0.986483i \(-0.552395\pi\)
−0.163862 + 0.986483i \(0.552395\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 3.29790 0.881400
\(15\) −3.83399 −0.989933
\(16\) 1.00000 0.250000
\(17\) −5.28528 −1.28187 −0.640934 0.767596i \(-0.721453\pi\)
−0.640934 + 0.767596i \(0.721453\pi\)
\(18\) −1.00000 −0.235702
\(19\) 7.52849 1.72715 0.863577 0.504218i \(-0.168219\pi\)
0.863577 + 0.504218i \(0.168219\pi\)
\(20\) −3.83399 −0.857307
\(21\) −3.29790 −0.719660
\(22\) 1.08694 0.231736
\(23\) −4.86940 −1.01534 −0.507670 0.861551i \(-0.669493\pi\)
−0.507670 + 0.861551i \(0.669493\pi\)
\(24\) −1.00000 −0.204124
\(25\) 9.69950 1.93990
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) −3.29790 −0.623244
\(29\) −8.02922 −1.49099 −0.745494 0.666512i \(-0.767787\pi\)
−0.745494 + 0.666512i \(0.767787\pi\)
\(30\) 3.83399 0.699988
\(31\) 3.95486 0.710314 0.355157 0.934807i \(-0.384427\pi\)
0.355157 + 0.934807i \(0.384427\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.08694 −0.189212
\(34\) 5.28528 0.906418
\(35\) 12.6441 2.13724
\(36\) 1.00000 0.166667
\(37\) 3.95146 0.649616 0.324808 0.945780i \(-0.394700\pi\)
0.324808 + 0.945780i \(0.394700\pi\)
\(38\) −7.52849 −1.22128
\(39\) 1.00000 0.160128
\(40\) 3.83399 0.606207
\(41\) 5.35805 0.836787 0.418393 0.908266i \(-0.362593\pi\)
0.418393 + 0.908266i \(0.362593\pi\)
\(42\) 3.29790 0.508876
\(43\) 11.4696 1.74910 0.874549 0.484938i \(-0.161158\pi\)
0.874549 + 0.484938i \(0.161158\pi\)
\(44\) −1.08694 −0.163862
\(45\) −3.83399 −0.571538
\(46\) 4.86940 0.717954
\(47\) 11.5592 1.68608 0.843038 0.537854i \(-0.180765\pi\)
0.843038 + 0.537854i \(0.180765\pi\)
\(48\) 1.00000 0.144338
\(49\) 3.87612 0.553731
\(50\) −9.69950 −1.37172
\(51\) −5.28528 −0.740087
\(52\) 1.00000 0.138675
\(53\) 1.82123 0.250166 0.125083 0.992146i \(-0.460080\pi\)
0.125083 + 0.992146i \(0.460080\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.16732 0.561921
\(56\) 3.29790 0.440700
\(57\) 7.52849 0.997172
\(58\) 8.02922 1.05429
\(59\) 10.3370 1.34577 0.672883 0.739749i \(-0.265056\pi\)
0.672883 + 0.739749i \(0.265056\pi\)
\(60\) −3.83399 −0.494966
\(61\) −14.8682 −1.90368 −0.951839 0.306597i \(-0.900810\pi\)
−0.951839 + 0.306597i \(0.900810\pi\)
\(62\) −3.95486 −0.502268
\(63\) −3.29790 −0.415496
\(64\) 1.00000 0.125000
\(65\) −3.83399 −0.475548
\(66\) 1.08694 0.133793
\(67\) 9.36012 1.14352 0.571761 0.820421i \(-0.306261\pi\)
0.571761 + 0.820421i \(0.306261\pi\)
\(68\) −5.28528 −0.640934
\(69\) −4.86940 −0.586207
\(70\) −12.6441 −1.51126
\(71\) −9.48805 −1.12602 −0.563012 0.826449i \(-0.690358\pi\)
−0.563012 + 0.826449i \(0.690358\pi\)
\(72\) −1.00000 −0.117851
\(73\) −5.79522 −0.678279 −0.339140 0.940736i \(-0.610136\pi\)
−0.339140 + 0.940736i \(0.610136\pi\)
\(74\) −3.95146 −0.459348
\(75\) 9.69950 1.12000
\(76\) 7.52849 0.863577
\(77\) 3.58461 0.408504
\(78\) −1.00000 −0.113228
\(79\) 6.22766 0.700666 0.350333 0.936625i \(-0.386068\pi\)
0.350333 + 0.936625i \(0.386068\pi\)
\(80\) −3.83399 −0.428653
\(81\) 1.00000 0.111111
\(82\) −5.35805 −0.591697
\(83\) 2.49468 0.273827 0.136913 0.990583i \(-0.456282\pi\)
0.136913 + 0.990583i \(0.456282\pi\)
\(84\) −3.29790 −0.359830
\(85\) 20.2637 2.19791
\(86\) −11.4696 −1.23680
\(87\) −8.02922 −0.860823
\(88\) 1.08694 0.115868
\(89\) 5.79834 0.614623 0.307311 0.951609i \(-0.400571\pi\)
0.307311 + 0.951609i \(0.400571\pi\)
\(90\) 3.83399 0.404138
\(91\) −3.29790 −0.345713
\(92\) −4.86940 −0.507670
\(93\) 3.95486 0.410100
\(94\) −11.5592 −1.19224
\(95\) −28.8642 −2.96140
\(96\) −1.00000 −0.102062
\(97\) −5.65476 −0.574153 −0.287077 0.957908i \(-0.592683\pi\)
−0.287077 + 0.957908i \(0.592683\pi\)
\(98\) −3.87612 −0.391547
\(99\) −1.08694 −0.109241
\(100\) 9.69950 0.969950
\(101\) −9.00487 −0.896018 −0.448009 0.894029i \(-0.647867\pi\)
−0.448009 + 0.894029i \(0.647867\pi\)
\(102\) 5.28528 0.523320
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 12.6441 1.23394
\(106\) −1.82123 −0.176894
\(107\) 12.8287 1.24019 0.620097 0.784525i \(-0.287093\pi\)
0.620097 + 0.784525i \(0.287093\pi\)
\(108\) 1.00000 0.0962250
\(109\) −18.0578 −1.72962 −0.864812 0.502095i \(-0.832563\pi\)
−0.864812 + 0.502095i \(0.832563\pi\)
\(110\) −4.16732 −0.397338
\(111\) 3.95146 0.375056
\(112\) −3.29790 −0.311622
\(113\) −10.4337 −0.981521 −0.490761 0.871294i \(-0.663281\pi\)
−0.490761 + 0.871294i \(0.663281\pi\)
\(114\) −7.52849 −0.705107
\(115\) 18.6693 1.74092
\(116\) −8.02922 −0.745494
\(117\) 1.00000 0.0924500
\(118\) −10.3370 −0.951600
\(119\) 17.4303 1.59783
\(120\) 3.83399 0.349994
\(121\) −9.81856 −0.892597
\(122\) 14.8682 1.34610
\(123\) 5.35805 0.483119
\(124\) 3.95486 0.355157
\(125\) −18.0179 −1.61157
\(126\) 3.29790 0.293800
\(127\) −9.93819 −0.881871 −0.440936 0.897539i \(-0.645353\pi\)
−0.440936 + 0.897539i \(0.645353\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.4696 1.00984
\(130\) 3.83399 0.336263
\(131\) −13.4253 −1.17297 −0.586486 0.809959i \(-0.699489\pi\)
−0.586486 + 0.809959i \(0.699489\pi\)
\(132\) −1.08694 −0.0946059
\(133\) −24.8282 −2.15287
\(134\) −9.36012 −0.808591
\(135\) −3.83399 −0.329978
\(136\) 5.28528 0.453209
\(137\) 18.7763 1.60416 0.802082 0.597214i \(-0.203726\pi\)
0.802082 + 0.597214i \(0.203726\pi\)
\(138\) 4.86940 0.414511
\(139\) −0.112175 −0.00951457 −0.00475728 0.999989i \(-0.501514\pi\)
−0.00475728 + 0.999989i \(0.501514\pi\)
\(140\) 12.6441 1.06862
\(141\) 11.5592 0.973457
\(142\) 9.48805 0.796219
\(143\) −1.08694 −0.0908944
\(144\) 1.00000 0.0833333
\(145\) 30.7840 2.55647
\(146\) 5.79522 0.479616
\(147\) 3.87612 0.319697
\(148\) 3.95146 0.324808
\(149\) −3.20558 −0.262612 −0.131306 0.991342i \(-0.541917\pi\)
−0.131306 + 0.991342i \(0.541917\pi\)
\(150\) −9.69950 −0.791961
\(151\) 9.10079 0.740612 0.370306 0.928910i \(-0.379253\pi\)
0.370306 + 0.928910i \(0.379253\pi\)
\(152\) −7.52849 −0.610641
\(153\) −5.28528 −0.427289
\(154\) −3.58461 −0.288856
\(155\) −15.1629 −1.21791
\(156\) 1.00000 0.0800641
\(157\) −14.3195 −1.14282 −0.571410 0.820665i \(-0.693603\pi\)
−0.571410 + 0.820665i \(0.693603\pi\)
\(158\) −6.22766 −0.495446
\(159\) 1.82123 0.144433
\(160\) 3.83399 0.303104
\(161\) 16.0588 1.26561
\(162\) −1.00000 −0.0785674
\(163\) −14.6517 −1.14761 −0.573804 0.818993i \(-0.694533\pi\)
−0.573804 + 0.818993i \(0.694533\pi\)
\(164\) 5.35805 0.418393
\(165\) 4.16732 0.324425
\(166\) −2.49468 −0.193625
\(167\) 3.44383 0.266492 0.133246 0.991083i \(-0.457460\pi\)
0.133246 + 0.991083i \(0.457460\pi\)
\(168\) 3.29790 0.254438
\(169\) 1.00000 0.0769231
\(170\) −20.2637 −1.55416
\(171\) 7.52849 0.575718
\(172\) 11.4696 0.874549
\(173\) −4.14641 −0.315246 −0.157623 0.987499i \(-0.550383\pi\)
−0.157623 + 0.987499i \(0.550383\pi\)
\(174\) 8.02922 0.608694
\(175\) −31.9879 −2.41806
\(176\) −1.08694 −0.0819311
\(177\) 10.3370 0.776978
\(178\) −5.79834 −0.434604
\(179\) 7.84629 0.586459 0.293230 0.956042i \(-0.405270\pi\)
0.293230 + 0.956042i \(0.405270\pi\)
\(180\) −3.83399 −0.285769
\(181\) 4.55624 0.338663 0.169331 0.985559i \(-0.445839\pi\)
0.169331 + 0.985559i \(0.445839\pi\)
\(182\) 3.29790 0.244456
\(183\) −14.8682 −1.09909
\(184\) 4.86940 0.358977
\(185\) −15.1499 −1.11384
\(186\) −3.95486 −0.289985
\(187\) 5.74477 0.420099
\(188\) 11.5592 0.843038
\(189\) −3.29790 −0.239887
\(190\) 28.8642 2.09403
\(191\) 5.88806 0.426045 0.213023 0.977047i \(-0.431669\pi\)
0.213023 + 0.977047i \(0.431669\pi\)
\(192\) 1.00000 0.0721688
\(193\) 3.69894 0.266256 0.133128 0.991099i \(-0.457498\pi\)
0.133128 + 0.991099i \(0.457498\pi\)
\(194\) 5.65476 0.405988
\(195\) −3.83399 −0.274558
\(196\) 3.87612 0.276865
\(197\) 5.26650 0.375223 0.187611 0.982243i \(-0.439925\pi\)
0.187611 + 0.982243i \(0.439925\pi\)
\(198\) 1.08694 0.0772454
\(199\) 14.3971 1.02059 0.510293 0.860001i \(-0.329537\pi\)
0.510293 + 0.860001i \(0.329537\pi\)
\(200\) −9.69950 −0.685858
\(201\) 9.36012 0.660212
\(202\) 9.00487 0.633581
\(203\) 26.4795 1.85850
\(204\) −5.28528 −0.370043
\(205\) −20.5427 −1.43477
\(206\) 1.00000 0.0696733
\(207\) −4.86940 −0.338447
\(208\) 1.00000 0.0693375
\(209\) −8.18301 −0.566030
\(210\) −12.6441 −0.872526
\(211\) 10.4702 0.720795 0.360398 0.932799i \(-0.382641\pi\)
0.360398 + 0.932799i \(0.382641\pi\)
\(212\) 1.82123 0.125083
\(213\) −9.48805 −0.650110
\(214\) −12.8287 −0.876950
\(215\) −43.9744 −2.99903
\(216\) −1.00000 −0.0680414
\(217\) −13.0427 −0.885398
\(218\) 18.0578 1.22303
\(219\) −5.79522 −0.391605
\(220\) 4.16732 0.280960
\(221\) −5.28528 −0.355526
\(222\) −3.95146 −0.265205
\(223\) 15.9747 1.06975 0.534873 0.844932i \(-0.320359\pi\)
0.534873 + 0.844932i \(0.320359\pi\)
\(224\) 3.29790 0.220350
\(225\) 9.69950 0.646633
\(226\) 10.4337 0.694040
\(227\) 4.41065 0.292745 0.146373 0.989230i \(-0.453240\pi\)
0.146373 + 0.989230i \(0.453240\pi\)
\(228\) 7.52849 0.498586
\(229\) −7.15661 −0.472922 −0.236461 0.971641i \(-0.575988\pi\)
−0.236461 + 0.971641i \(0.575988\pi\)
\(230\) −18.6693 −1.23101
\(231\) 3.58461 0.235850
\(232\) 8.02922 0.527144
\(233\) 24.7436 1.62101 0.810503 0.585735i \(-0.199194\pi\)
0.810503 + 0.585735i \(0.199194\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −44.3177 −2.89097
\(236\) 10.3370 0.672883
\(237\) 6.22766 0.404530
\(238\) −17.4303 −1.12984
\(239\) −3.79838 −0.245697 −0.122848 0.992425i \(-0.539203\pi\)
−0.122848 + 0.992425i \(0.539203\pi\)
\(240\) −3.83399 −0.247483
\(241\) −21.6703 −1.39591 −0.697954 0.716143i \(-0.745906\pi\)
−0.697954 + 0.716143i \(0.745906\pi\)
\(242\) 9.81856 0.631161
\(243\) 1.00000 0.0641500
\(244\) −14.8682 −0.951839
\(245\) −14.8610 −0.949435
\(246\) −5.35805 −0.341617
\(247\) 7.52849 0.479026
\(248\) −3.95486 −0.251134
\(249\) 2.49468 0.158094
\(250\) 18.0179 1.13955
\(251\) −15.8714 −1.00179 −0.500897 0.865507i \(-0.666996\pi\)
−0.500897 + 0.865507i \(0.666996\pi\)
\(252\) −3.29790 −0.207748
\(253\) 5.29275 0.332752
\(254\) 9.93819 0.623577
\(255\) 20.2637 1.26896
\(256\) 1.00000 0.0625000
\(257\) −19.3680 −1.20814 −0.604072 0.796930i \(-0.706456\pi\)
−0.604072 + 0.796930i \(0.706456\pi\)
\(258\) −11.4696 −0.714066
\(259\) −13.0315 −0.809738
\(260\) −3.83399 −0.237774
\(261\) −8.02922 −0.496996
\(262\) 13.4253 0.829417
\(263\) 2.61834 0.161454 0.0807269 0.996736i \(-0.474276\pi\)
0.0807269 + 0.996736i \(0.474276\pi\)
\(264\) 1.08694 0.0668965
\(265\) −6.98260 −0.428938
\(266\) 24.8282 1.52231
\(267\) 5.79834 0.354853
\(268\) 9.36012 0.571761
\(269\) 22.0924 1.34700 0.673498 0.739189i \(-0.264791\pi\)
0.673498 + 0.739189i \(0.264791\pi\)
\(270\) 3.83399 0.233329
\(271\) −13.7745 −0.836741 −0.418370 0.908277i \(-0.637399\pi\)
−0.418370 + 0.908277i \(0.637399\pi\)
\(272\) −5.28528 −0.320467
\(273\) −3.29790 −0.199598
\(274\) −18.7763 −1.13432
\(275\) −10.5428 −0.635753
\(276\) −4.86940 −0.293104
\(277\) 5.22578 0.313987 0.156993 0.987600i \(-0.449820\pi\)
0.156993 + 0.987600i \(0.449820\pi\)
\(278\) 0.112175 0.00672782
\(279\) 3.95486 0.236771
\(280\) −12.6441 −0.755630
\(281\) −10.2219 −0.609784 −0.304892 0.952387i \(-0.598620\pi\)
−0.304892 + 0.952387i \(0.598620\pi\)
\(282\) −11.5592 −0.688338
\(283\) −12.3497 −0.734116 −0.367058 0.930198i \(-0.619635\pi\)
−0.367058 + 0.930198i \(0.619635\pi\)
\(284\) −9.48805 −0.563012
\(285\) −28.8642 −1.70977
\(286\) 1.08694 0.0642721
\(287\) −17.6703 −1.04304
\(288\) −1.00000 −0.0589256
\(289\) 10.9342 0.643185
\(290\) −30.7840 −1.80770
\(291\) −5.65476 −0.331488
\(292\) −5.79522 −0.339140
\(293\) −6.42954 −0.375618 −0.187809 0.982206i \(-0.560139\pi\)
−0.187809 + 0.982206i \(0.560139\pi\)
\(294\) −3.87612 −0.226060
\(295\) −39.6321 −2.30747
\(296\) −3.95146 −0.229674
\(297\) −1.08694 −0.0630706
\(298\) 3.20558 0.185694
\(299\) −4.86940 −0.281605
\(300\) 9.69950 0.560001
\(301\) −37.8255 −2.18023
\(302\) −9.10079 −0.523692
\(303\) −9.00487 −0.517316
\(304\) 7.52849 0.431788
\(305\) 57.0046 3.26407
\(306\) 5.28528 0.302139
\(307\) 14.6133 0.834026 0.417013 0.908901i \(-0.363077\pi\)
0.417013 + 0.908901i \(0.363077\pi\)
\(308\) 3.58461 0.204252
\(309\) −1.00000 −0.0568880
\(310\) 15.1629 0.861195
\(311\) −11.9468 −0.677438 −0.338719 0.940888i \(-0.609994\pi\)
−0.338719 + 0.940888i \(0.609994\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −7.08127 −0.400257 −0.200128 0.979770i \(-0.564136\pi\)
−0.200128 + 0.979770i \(0.564136\pi\)
\(314\) 14.3195 0.808096
\(315\) 12.6441 0.712415
\(316\) 6.22766 0.350333
\(317\) −23.8099 −1.33730 −0.668648 0.743579i \(-0.733127\pi\)
−0.668648 + 0.743579i \(0.733127\pi\)
\(318\) −1.82123 −0.102130
\(319\) 8.72727 0.488634
\(320\) −3.83399 −0.214327
\(321\) 12.8287 0.716027
\(322\) −16.0588 −0.894921
\(323\) −39.7901 −2.21398
\(324\) 1.00000 0.0555556
\(325\) 9.69950 0.538032
\(326\) 14.6517 0.811481
\(327\) −18.0578 −0.998599
\(328\) −5.35805 −0.295849
\(329\) −38.1209 −2.10167
\(330\) −4.16732 −0.229403
\(331\) −35.9491 −1.97594 −0.987969 0.154650i \(-0.950575\pi\)
−0.987969 + 0.154650i \(0.950575\pi\)
\(332\) 2.49468 0.136913
\(333\) 3.95146 0.216539
\(334\) −3.44383 −0.188438
\(335\) −35.8866 −1.96070
\(336\) −3.29790 −0.179915
\(337\) 0.991712 0.0540220 0.0270110 0.999635i \(-0.491401\pi\)
0.0270110 + 0.999635i \(0.491401\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −10.4337 −0.566681
\(340\) 20.2637 1.09895
\(341\) −4.29869 −0.232787
\(342\) −7.52849 −0.407094
\(343\) 10.3022 0.556269
\(344\) −11.4696 −0.618399
\(345\) 18.6693 1.00512
\(346\) 4.14641 0.222912
\(347\) −18.9111 −1.01520 −0.507600 0.861593i \(-0.669467\pi\)
−0.507600 + 0.861593i \(0.669467\pi\)
\(348\) −8.02922 −0.430411
\(349\) −28.2832 −1.51397 −0.756984 0.653434i \(-0.773328\pi\)
−0.756984 + 0.653434i \(0.773328\pi\)
\(350\) 31.9879 1.70983
\(351\) 1.00000 0.0533761
\(352\) 1.08694 0.0579340
\(353\) 15.5041 0.825198 0.412599 0.910913i \(-0.364621\pi\)
0.412599 + 0.910913i \(0.364621\pi\)
\(354\) −10.3370 −0.549407
\(355\) 36.3771 1.93070
\(356\) 5.79834 0.307311
\(357\) 17.4303 0.922509
\(358\) −7.84629 −0.414689
\(359\) 9.62071 0.507761 0.253881 0.967236i \(-0.418293\pi\)
0.253881 + 0.967236i \(0.418293\pi\)
\(360\) 3.83399 0.202069
\(361\) 37.6781 1.98306
\(362\) −4.55624 −0.239471
\(363\) −9.81856 −0.515341
\(364\) −3.29790 −0.172857
\(365\) 22.2188 1.16299
\(366\) 14.8682 0.777174
\(367\) 24.2108 1.26379 0.631896 0.775053i \(-0.282277\pi\)
0.631896 + 0.775053i \(0.282277\pi\)
\(368\) −4.86940 −0.253835
\(369\) 5.35805 0.278929
\(370\) 15.1499 0.787604
\(371\) −6.00624 −0.311828
\(372\) 3.95486 0.205050
\(373\) −1.14257 −0.0591598 −0.0295799 0.999562i \(-0.509417\pi\)
−0.0295799 + 0.999562i \(0.509417\pi\)
\(374\) −5.74477 −0.297055
\(375\) −18.0179 −0.930438
\(376\) −11.5592 −0.596118
\(377\) −8.02922 −0.413526
\(378\) 3.29790 0.169625
\(379\) −8.84686 −0.454433 −0.227216 0.973844i \(-0.572962\pi\)
−0.227216 + 0.973844i \(0.572962\pi\)
\(380\) −28.8642 −1.48070
\(381\) −9.93819 −0.509149
\(382\) −5.88806 −0.301259
\(383\) −3.54043 −0.180908 −0.0904538 0.995901i \(-0.528832\pi\)
−0.0904538 + 0.995901i \(0.528832\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −13.7434 −0.700427
\(386\) −3.69894 −0.188271
\(387\) 11.4696 0.583032
\(388\) −5.65476 −0.287077
\(389\) 15.8087 0.801534 0.400767 0.916180i \(-0.368744\pi\)
0.400767 + 0.916180i \(0.368744\pi\)
\(390\) 3.83399 0.194142
\(391\) 25.7361 1.30153
\(392\) −3.87612 −0.195773
\(393\) −13.4253 −0.677216
\(394\) −5.26650 −0.265323
\(395\) −23.8768 −1.20137
\(396\) −1.08694 −0.0546207
\(397\) 36.8850 1.85120 0.925602 0.378499i \(-0.123560\pi\)
0.925602 + 0.378499i \(0.123560\pi\)
\(398\) −14.3971 −0.721663
\(399\) −24.8282 −1.24296
\(400\) 9.69950 0.484975
\(401\) 21.5806 1.07769 0.538843 0.842406i \(-0.318862\pi\)
0.538843 + 0.842406i \(0.318862\pi\)
\(402\) −9.36012 −0.466840
\(403\) 3.95486 0.197006
\(404\) −9.00487 −0.448009
\(405\) −3.83399 −0.190513
\(406\) −26.4795 −1.31416
\(407\) −4.29500 −0.212895
\(408\) 5.28528 0.261660
\(409\) −39.4851 −1.95241 −0.976207 0.216840i \(-0.930425\pi\)
−0.976207 + 0.216840i \(0.930425\pi\)
\(410\) 20.5427 1.01453
\(411\) 18.7763 0.926165
\(412\) −1.00000 −0.0492665
\(413\) −34.0904 −1.67748
\(414\) 4.86940 0.239318
\(415\) −9.56459 −0.469507
\(416\) −1.00000 −0.0490290
\(417\) −0.112175 −0.00549324
\(418\) 8.18301 0.400244
\(419\) −16.1944 −0.791147 −0.395574 0.918434i \(-0.629454\pi\)
−0.395574 + 0.918434i \(0.629454\pi\)
\(420\) 12.6441 0.616969
\(421\) −16.4463 −0.801544 −0.400772 0.916178i \(-0.631258\pi\)
−0.400772 + 0.916178i \(0.631258\pi\)
\(422\) −10.4702 −0.509679
\(423\) 11.5592 0.562025
\(424\) −1.82123 −0.0884469
\(425\) −51.2646 −2.48670
\(426\) 9.48805 0.459697
\(427\) 49.0338 2.37291
\(428\) 12.8287 0.620097
\(429\) −1.08694 −0.0524779
\(430\) 43.9744 2.12063
\(431\) 7.93853 0.382386 0.191193 0.981553i \(-0.438764\pi\)
0.191193 + 0.981553i \(0.438764\pi\)
\(432\) 1.00000 0.0481125
\(433\) 40.9214 1.96656 0.983278 0.182108i \(-0.0582921\pi\)
0.983278 + 0.182108i \(0.0582921\pi\)
\(434\) 13.0427 0.626071
\(435\) 30.7840 1.47598
\(436\) −18.0578 −0.864812
\(437\) −36.6592 −1.75365
\(438\) 5.79522 0.276906
\(439\) −26.1722 −1.24913 −0.624565 0.780973i \(-0.714724\pi\)
−0.624565 + 0.780973i \(0.714724\pi\)
\(440\) −4.16732 −0.198669
\(441\) 3.87612 0.184577
\(442\) 5.28528 0.251395
\(443\) −8.73740 −0.415126 −0.207563 0.978222i \(-0.566553\pi\)
−0.207563 + 0.978222i \(0.566553\pi\)
\(444\) 3.95146 0.187528
\(445\) −22.2308 −1.05384
\(446\) −15.9747 −0.756425
\(447\) −3.20558 −0.151619
\(448\) −3.29790 −0.155811
\(449\) −27.0573 −1.27692 −0.638458 0.769657i \(-0.720427\pi\)
−0.638458 + 0.769657i \(0.720427\pi\)
\(450\) −9.69950 −0.457239
\(451\) −5.82387 −0.274235
\(452\) −10.4337 −0.490761
\(453\) 9.10079 0.427593
\(454\) −4.41065 −0.207002
\(455\) 12.6441 0.592765
\(456\) −7.52849 −0.352554
\(457\) −12.4155 −0.580771 −0.290385 0.956910i \(-0.593783\pi\)
−0.290385 + 0.956910i \(0.593783\pi\)
\(458\) 7.15661 0.334406
\(459\) −5.28528 −0.246696
\(460\) 18.6693 0.870459
\(461\) −25.3222 −1.17937 −0.589687 0.807632i \(-0.700749\pi\)
−0.589687 + 0.807632i \(0.700749\pi\)
\(462\) −3.58461 −0.166771
\(463\) −11.0274 −0.512487 −0.256244 0.966612i \(-0.582485\pi\)
−0.256244 + 0.966612i \(0.582485\pi\)
\(464\) −8.02922 −0.372747
\(465\) −15.1629 −0.703163
\(466\) −24.7436 −1.14622
\(467\) −0.314734 −0.0145641 −0.00728207 0.999973i \(-0.502318\pi\)
−0.00728207 + 0.999973i \(0.502318\pi\)
\(468\) 1.00000 0.0462250
\(469\) −30.8687 −1.42538
\(470\) 44.3177 2.04422
\(471\) −14.3195 −0.659808
\(472\) −10.3370 −0.475800
\(473\) −12.4668 −0.573222
\(474\) −6.22766 −0.286046
\(475\) 73.0226 3.35050
\(476\) 17.4303 0.798916
\(477\) 1.82123 0.0833886
\(478\) 3.79838 0.173734
\(479\) −30.3516 −1.38680 −0.693401 0.720552i \(-0.743888\pi\)
−0.693401 + 0.720552i \(0.743888\pi\)
\(480\) 3.83399 0.174997
\(481\) 3.95146 0.180171
\(482\) 21.6703 0.987056
\(483\) 16.0588 0.730700
\(484\) −9.81856 −0.446298
\(485\) 21.6803 0.984451
\(486\) −1.00000 −0.0453609
\(487\) 22.7338 1.03017 0.515083 0.857140i \(-0.327761\pi\)
0.515083 + 0.857140i \(0.327761\pi\)
\(488\) 14.8682 0.673052
\(489\) −14.6517 −0.662571
\(490\) 14.8610 0.671352
\(491\) −37.6519 −1.69921 −0.849604 0.527422i \(-0.823159\pi\)
−0.849604 + 0.527422i \(0.823159\pi\)
\(492\) 5.35805 0.241559
\(493\) 42.4367 1.91125
\(494\) −7.52849 −0.338723
\(495\) 4.16732 0.187307
\(496\) 3.95486 0.177579
\(497\) 31.2906 1.40357
\(498\) −2.49468 −0.111789
\(499\) −37.8361 −1.69378 −0.846889 0.531770i \(-0.821527\pi\)
−0.846889 + 0.531770i \(0.821527\pi\)
\(500\) −18.0179 −0.805783
\(501\) 3.44383 0.153859
\(502\) 15.8714 0.708375
\(503\) 18.3213 0.816905 0.408452 0.912780i \(-0.366069\pi\)
0.408452 + 0.912780i \(0.366069\pi\)
\(504\) 3.29790 0.146900
\(505\) 34.5246 1.53633
\(506\) −5.29275 −0.235291
\(507\) 1.00000 0.0444116
\(508\) −9.93819 −0.440936
\(509\) 31.1958 1.38273 0.691364 0.722507i \(-0.257010\pi\)
0.691364 + 0.722507i \(0.257010\pi\)
\(510\) −20.2637 −0.897292
\(511\) 19.1120 0.845467
\(512\) −1.00000 −0.0441942
\(513\) 7.52849 0.332391
\(514\) 19.3680 0.854286
\(515\) 3.83399 0.168946
\(516\) 11.4696 0.504921
\(517\) −12.5641 −0.552568
\(518\) 13.0315 0.572571
\(519\) −4.14641 −0.182007
\(520\) 3.83399 0.168132
\(521\) −39.7366 −1.74089 −0.870446 0.492264i \(-0.836170\pi\)
−0.870446 + 0.492264i \(0.836170\pi\)
\(522\) 8.02922 0.351429
\(523\) −15.5654 −0.680626 −0.340313 0.940312i \(-0.610533\pi\)
−0.340313 + 0.940312i \(0.610533\pi\)
\(524\) −13.4253 −0.586486
\(525\) −31.9879 −1.39607
\(526\) −2.61834 −0.114165
\(527\) −20.9025 −0.910529
\(528\) −1.08694 −0.0473030
\(529\) 0.711092 0.0309170
\(530\) 6.98260 0.303305
\(531\) 10.3370 0.448589
\(532\) −24.8282 −1.07644
\(533\) 5.35805 0.232083
\(534\) −5.79834 −0.250919
\(535\) −49.1850 −2.12645
\(536\) −9.36012 −0.404296
\(537\) 7.84629 0.338592
\(538\) −22.0924 −0.952469
\(539\) −4.21310 −0.181471
\(540\) −3.83399 −0.164989
\(541\) 36.4563 1.56738 0.783689 0.621153i \(-0.213336\pi\)
0.783689 + 0.621153i \(0.213336\pi\)
\(542\) 13.7745 0.591665
\(543\) 4.55624 0.195527
\(544\) 5.28528 0.226604
\(545\) 69.2335 2.96564
\(546\) 3.29790 0.141137
\(547\) 23.1673 0.990562 0.495281 0.868733i \(-0.335065\pi\)
0.495281 + 0.868733i \(0.335065\pi\)
\(548\) 18.7763 0.802082
\(549\) −14.8682 −0.634560
\(550\) 10.5428 0.449545
\(551\) −60.4479 −2.57517
\(552\) 4.86940 0.207256
\(553\) −20.5382 −0.873372
\(554\) −5.22578 −0.222022
\(555\) −15.1499 −0.643076
\(556\) −0.112175 −0.00475728
\(557\) −9.59679 −0.406629 −0.203315 0.979113i \(-0.565171\pi\)
−0.203315 + 0.979113i \(0.565171\pi\)
\(558\) −3.95486 −0.167423
\(559\) 11.4696 0.485112
\(560\) 12.6441 0.534311
\(561\) 5.74477 0.242545
\(562\) 10.2219 0.431183
\(563\) −23.4170 −0.986911 −0.493455 0.869771i \(-0.664266\pi\)
−0.493455 + 0.869771i \(0.664266\pi\)
\(564\) 11.5592 0.486728
\(565\) 40.0028 1.68293
\(566\) 12.3497 0.519098
\(567\) −3.29790 −0.138499
\(568\) 9.48805 0.398110
\(569\) −26.6193 −1.11594 −0.557969 0.829862i \(-0.688419\pi\)
−0.557969 + 0.829862i \(0.688419\pi\)
\(570\) 28.8642 1.20899
\(571\) 0.0404241 0.00169170 0.000845848 1.00000i \(-0.499731\pi\)
0.000845848 1.00000i \(0.499731\pi\)
\(572\) −1.08694 −0.0454472
\(573\) 5.88806 0.245977
\(574\) 17.6703 0.737543
\(575\) −47.2308 −1.96966
\(576\) 1.00000 0.0416667
\(577\) −38.2566 −1.59264 −0.796321 0.604875i \(-0.793223\pi\)
−0.796321 + 0.604875i \(0.793223\pi\)
\(578\) −10.9342 −0.454801
\(579\) 3.69894 0.153723
\(580\) 30.7840 1.27823
\(581\) −8.22719 −0.341322
\(582\) 5.65476 0.234397
\(583\) −1.97957 −0.0819854
\(584\) 5.79522 0.239808
\(585\) −3.83399 −0.158516
\(586\) 6.42954 0.265602
\(587\) −16.9963 −0.701512 −0.350756 0.936467i \(-0.614075\pi\)
−0.350756 + 0.936467i \(0.614075\pi\)
\(588\) 3.87612 0.159848
\(589\) 29.7741 1.22682
\(590\) 39.6321 1.63163
\(591\) 5.26650 0.216635
\(592\) 3.95146 0.162404
\(593\) −0.753800 −0.0309548 −0.0154774 0.999880i \(-0.504927\pi\)
−0.0154774 + 0.999880i \(0.504927\pi\)
\(594\) 1.08694 0.0445977
\(595\) −66.8276 −2.73967
\(596\) −3.20558 −0.131306
\(597\) 14.3971 0.589235
\(598\) 4.86940 0.199125
\(599\) 16.7987 0.686378 0.343189 0.939266i \(-0.388493\pi\)
0.343189 + 0.939266i \(0.388493\pi\)
\(600\) −9.69950 −0.395980
\(601\) 4.34433 0.177209 0.0886046 0.996067i \(-0.471759\pi\)
0.0886046 + 0.996067i \(0.471759\pi\)
\(602\) 37.8255 1.54165
\(603\) 9.36012 0.381174
\(604\) 9.10079 0.370306
\(605\) 37.6443 1.53046
\(606\) 9.00487 0.365798
\(607\) 27.4366 1.11362 0.556808 0.830641i \(-0.312026\pi\)
0.556808 + 0.830641i \(0.312026\pi\)
\(608\) −7.52849 −0.305320
\(609\) 26.4795 1.07300
\(610\) −57.0046 −2.30805
\(611\) 11.5592 0.467633
\(612\) −5.28528 −0.213645
\(613\) −31.6726 −1.27925 −0.639623 0.768689i \(-0.720910\pi\)
−0.639623 + 0.768689i \(0.720910\pi\)
\(614\) −14.6133 −0.589745
\(615\) −20.5427 −0.828362
\(616\) −3.58461 −0.144428
\(617\) −13.1275 −0.528495 −0.264247 0.964455i \(-0.585124\pi\)
−0.264247 + 0.964455i \(0.585124\pi\)
\(618\) 1.00000 0.0402259
\(619\) 17.3610 0.697799 0.348899 0.937160i \(-0.386555\pi\)
0.348899 + 0.937160i \(0.386555\pi\)
\(620\) −15.1629 −0.608957
\(621\) −4.86940 −0.195402
\(622\) 11.9468 0.479021
\(623\) −19.1223 −0.766120
\(624\) 1.00000 0.0400320
\(625\) 20.5828 0.823313
\(626\) 7.08127 0.283024
\(627\) −8.18301 −0.326798
\(628\) −14.3195 −0.571410
\(629\) −20.8846 −0.832722
\(630\) −12.6441 −0.503753
\(631\) 18.9049 0.752592 0.376296 0.926499i \(-0.377198\pi\)
0.376296 + 0.926499i \(0.377198\pi\)
\(632\) −6.22766 −0.247723
\(633\) 10.4702 0.416151
\(634\) 23.8099 0.945611
\(635\) 38.1029 1.51207
\(636\) 1.82123 0.0722166
\(637\) 3.87612 0.153577
\(638\) −8.72727 −0.345516
\(639\) −9.48805 −0.375341
\(640\) 3.83399 0.151552
\(641\) −8.94991 −0.353500 −0.176750 0.984256i \(-0.556558\pi\)
−0.176750 + 0.984256i \(0.556558\pi\)
\(642\) −12.8287 −0.506307
\(643\) 42.0008 1.65635 0.828174 0.560471i \(-0.189380\pi\)
0.828174 + 0.560471i \(0.189380\pi\)
\(644\) 16.0588 0.632805
\(645\) −43.9744 −1.73149
\(646\) 39.7901 1.56552
\(647\) −20.9806 −0.824834 −0.412417 0.910995i \(-0.635315\pi\)
−0.412417 + 0.910995i \(0.635315\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −11.2357 −0.441040
\(650\) −9.69950 −0.380446
\(651\) −13.0427 −0.511185
\(652\) −14.6517 −0.573804
\(653\) 43.9854 1.72128 0.860642 0.509211i \(-0.170063\pi\)
0.860642 + 0.509211i \(0.170063\pi\)
\(654\) 18.0578 0.706116
\(655\) 51.4724 2.01119
\(656\) 5.35805 0.209197
\(657\) −5.79522 −0.226093
\(658\) 38.1209 1.48611
\(659\) 7.26503 0.283005 0.141503 0.989938i \(-0.454807\pi\)
0.141503 + 0.989938i \(0.454807\pi\)
\(660\) 4.16732 0.162213
\(661\) −13.2990 −0.517273 −0.258636 0.965975i \(-0.583273\pi\)
−0.258636 + 0.965975i \(0.583273\pi\)
\(662\) 35.9491 1.39720
\(663\) −5.28528 −0.205263
\(664\) −2.49468 −0.0968124
\(665\) 95.1910 3.69135
\(666\) −3.95146 −0.153116
\(667\) 39.0975 1.51386
\(668\) 3.44383 0.133246
\(669\) 15.9747 0.617618
\(670\) 35.8866 1.38642
\(671\) 16.1608 0.623882
\(672\) 3.29790 0.127219
\(673\) −44.7188 −1.72378 −0.861891 0.507094i \(-0.830720\pi\)
−0.861891 + 0.507094i \(0.830720\pi\)
\(674\) −0.991712 −0.0381993
\(675\) 9.69950 0.373334
\(676\) 1.00000 0.0384615
\(677\) 12.3260 0.473728 0.236864 0.971543i \(-0.423880\pi\)
0.236864 + 0.971543i \(0.423880\pi\)
\(678\) 10.4337 0.400704
\(679\) 18.6488 0.715675
\(680\) −20.2637 −0.777078
\(681\) 4.41065 0.169016
\(682\) 4.29869 0.164605
\(683\) 28.1431 1.07687 0.538433 0.842668i \(-0.319017\pi\)
0.538433 + 0.842668i \(0.319017\pi\)
\(684\) 7.52849 0.287859
\(685\) −71.9881 −2.75052
\(686\) −10.3022 −0.393341
\(687\) −7.15661 −0.273042
\(688\) 11.4696 0.437274
\(689\) 1.82123 0.0693835
\(690\) −18.6693 −0.710727
\(691\) −8.55552 −0.325467 −0.162734 0.986670i \(-0.552031\pi\)
−0.162734 + 0.986670i \(0.552031\pi\)
\(692\) −4.14641 −0.157623
\(693\) 3.58461 0.136168
\(694\) 18.9111 0.717855
\(695\) 0.430079 0.0163138
\(696\) 8.02922 0.304347
\(697\) −28.3188 −1.07265
\(698\) 28.2832 1.07054
\(699\) 24.7436 0.935888
\(700\) −31.9879 −1.20903
\(701\) 5.05341 0.190865 0.0954323 0.995436i \(-0.469577\pi\)
0.0954323 + 0.995436i \(0.469577\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 29.7485 1.12199
\(704\) −1.08694 −0.0409656
\(705\) −44.3177 −1.66910
\(706\) −15.5041 −0.583503
\(707\) 29.6971 1.11688
\(708\) 10.3370 0.388489
\(709\) −24.1843 −0.908259 −0.454130 0.890936i \(-0.650050\pi\)
−0.454130 + 0.890936i \(0.650050\pi\)
\(710\) −36.3771 −1.36521
\(711\) 6.22766 0.233555
\(712\) −5.79834 −0.217302
\(713\) −19.2578 −0.721211
\(714\) −17.4303 −0.652312
\(715\) 4.16732 0.155849
\(716\) 7.84629 0.293230
\(717\) −3.79838 −0.141853
\(718\) −9.62071 −0.359042
\(719\) −25.7695 −0.961039 −0.480520 0.876984i \(-0.659552\pi\)
−0.480520 + 0.876984i \(0.659552\pi\)
\(720\) −3.83399 −0.142884
\(721\) 3.29790 0.122820
\(722\) −37.6781 −1.40223
\(723\) −21.6703 −0.805928
\(724\) 4.55624 0.169331
\(725\) −77.8794 −2.89237
\(726\) 9.81856 0.364401
\(727\) 37.5268 1.39179 0.695897 0.718142i \(-0.255007\pi\)
0.695897 + 0.718142i \(0.255007\pi\)
\(728\) 3.29790 0.122228
\(729\) 1.00000 0.0370370
\(730\) −22.2188 −0.822356
\(731\) −60.6200 −2.24211
\(732\) −14.8682 −0.549545
\(733\) 29.6385 1.09472 0.547361 0.836896i \(-0.315632\pi\)
0.547361 + 0.836896i \(0.315632\pi\)
\(734\) −24.2108 −0.893636
\(735\) −14.8610 −0.548156
\(736\) 4.86940 0.179489
\(737\) −10.1739 −0.374760
\(738\) −5.35805 −0.197232
\(739\) −5.64072 −0.207497 −0.103749 0.994604i \(-0.533084\pi\)
−0.103749 + 0.994604i \(0.533084\pi\)
\(740\) −15.1499 −0.556920
\(741\) 7.52849 0.276566
\(742\) 6.00624 0.220496
\(743\) 22.3300 0.819210 0.409605 0.912263i \(-0.365667\pi\)
0.409605 + 0.912263i \(0.365667\pi\)
\(744\) −3.95486 −0.144992
\(745\) 12.2902 0.450278
\(746\) 1.14257 0.0418323
\(747\) 2.49468 0.0912756
\(748\) 5.74477 0.210050
\(749\) −42.3076 −1.54589
\(750\) 18.0179 0.657919
\(751\) −16.8612 −0.615274 −0.307637 0.951504i \(-0.599538\pi\)
−0.307637 + 0.951504i \(0.599538\pi\)
\(752\) 11.5592 0.421519
\(753\) −15.8714 −0.578386
\(754\) 8.02922 0.292407
\(755\) −34.8924 −1.26986
\(756\) −3.29790 −0.119943
\(757\) 2.53666 0.0921966 0.0460983 0.998937i \(-0.485321\pi\)
0.0460983 + 0.998937i \(0.485321\pi\)
\(758\) 8.84686 0.321332
\(759\) 5.29275 0.192114
\(760\) 28.8642 1.04701
\(761\) 18.5717 0.673224 0.336612 0.941643i \(-0.390719\pi\)
0.336612 + 0.941643i \(0.390719\pi\)
\(762\) 9.93819 0.360023
\(763\) 59.5528 2.15596
\(764\) 5.88806 0.213023
\(765\) 20.2637 0.732636
\(766\) 3.54043 0.127921
\(767\) 10.3370 0.373248
\(768\) 1.00000 0.0360844
\(769\) −13.3869 −0.482745 −0.241372 0.970433i \(-0.577597\pi\)
−0.241372 + 0.970433i \(0.577597\pi\)
\(770\) 13.7434 0.495277
\(771\) −19.3680 −0.697522
\(772\) 3.69894 0.133128
\(773\) 25.0931 0.902537 0.451268 0.892388i \(-0.350972\pi\)
0.451268 + 0.892388i \(0.350972\pi\)
\(774\) −11.4696 −0.412266
\(775\) 38.3602 1.37794
\(776\) 5.65476 0.202994
\(777\) −13.0315 −0.467503
\(778\) −15.8087 −0.566770
\(779\) 40.3380 1.44526
\(780\) −3.83399 −0.137279
\(781\) 10.3129 0.369026
\(782\) −25.7361 −0.920323
\(783\) −8.02922 −0.286941
\(784\) 3.87612 0.138433
\(785\) 54.9009 1.95950
\(786\) 13.4253 0.478864
\(787\) 19.8191 0.706476 0.353238 0.935533i \(-0.385081\pi\)
0.353238 + 0.935533i \(0.385081\pi\)
\(788\) 5.26650 0.187611
\(789\) 2.61834 0.0932154
\(790\) 23.8768 0.849498
\(791\) 34.4093 1.22345
\(792\) 1.08694 0.0386227
\(793\) −14.8682 −0.527985
\(794\) −36.8850 −1.30900
\(795\) −6.98260 −0.247647
\(796\) 14.3971 0.510293
\(797\) −53.8034 −1.90581 −0.952906 0.303265i \(-0.901923\pi\)
−0.952906 + 0.303265i \(0.901923\pi\)
\(798\) 24.8282 0.878907
\(799\) −61.0933 −2.16133
\(800\) −9.69950 −0.342929
\(801\) 5.79834 0.204874
\(802\) −21.5806 −0.762039
\(803\) 6.29905 0.222289
\(804\) 9.36012 0.330106
\(805\) −61.5693 −2.17003
\(806\) −3.95486 −0.139304
\(807\) 22.0924 0.777688
\(808\) 9.00487 0.316790
\(809\) 23.8958 0.840130 0.420065 0.907494i \(-0.362007\pi\)
0.420065 + 0.907494i \(0.362007\pi\)
\(810\) 3.83399 0.134713
\(811\) −19.3621 −0.679894 −0.339947 0.940445i \(-0.610409\pi\)
−0.339947 + 0.940445i \(0.610409\pi\)
\(812\) 26.4795 0.929249
\(813\) −13.7745 −0.483092
\(814\) 4.29500 0.150540
\(815\) 56.1744 1.96770
\(816\) −5.28528 −0.185022
\(817\) 86.3487 3.02096
\(818\) 39.4851 1.38057
\(819\) −3.29790 −0.115238
\(820\) −20.5427 −0.717383
\(821\) −55.5460 −1.93857 −0.969284 0.245943i \(-0.920902\pi\)
−0.969284 + 0.245943i \(0.920902\pi\)
\(822\) −18.7763 −0.654897
\(823\) −53.5826 −1.86777 −0.933886 0.357571i \(-0.883605\pi\)
−0.933886 + 0.357571i \(0.883605\pi\)
\(824\) 1.00000 0.0348367
\(825\) −10.5428 −0.367052
\(826\) 34.0904 1.18616
\(827\) 48.1665 1.67491 0.837457 0.546504i \(-0.184042\pi\)
0.837457 + 0.546504i \(0.184042\pi\)
\(828\) −4.86940 −0.169223
\(829\) 19.4417 0.675238 0.337619 0.941283i \(-0.390379\pi\)
0.337619 + 0.941283i \(0.390379\pi\)
\(830\) 9.56459 0.331992
\(831\) 5.22578 0.181280
\(832\) 1.00000 0.0346688
\(833\) −20.4863 −0.709810
\(834\) 0.112175 0.00388431
\(835\) −13.2036 −0.456930
\(836\) −8.18301 −0.283015
\(837\) 3.95486 0.136700
\(838\) 16.1944 0.559425
\(839\) −14.3329 −0.494827 −0.247414 0.968910i \(-0.579581\pi\)
−0.247414 + 0.968910i \(0.579581\pi\)
\(840\) −12.6441 −0.436263
\(841\) 35.4684 1.22305
\(842\) 16.4463 0.566777
\(843\) −10.2219 −0.352059
\(844\) 10.4702 0.360398
\(845\) −3.83399 −0.131893
\(846\) −11.5592 −0.397412
\(847\) 32.3806 1.11261
\(848\) 1.82123 0.0625414
\(849\) −12.3497 −0.423842
\(850\) 51.2646 1.75836
\(851\) −19.2413 −0.659582
\(852\) −9.48805 −0.325055
\(853\) 9.66768 0.331015 0.165507 0.986209i \(-0.447074\pi\)
0.165507 + 0.986209i \(0.447074\pi\)
\(854\) −49.0338 −1.67790
\(855\) −28.8642 −0.987133
\(856\) −12.8287 −0.438475
\(857\) −27.4062 −0.936179 −0.468089 0.883681i \(-0.655057\pi\)
−0.468089 + 0.883681i \(0.655057\pi\)
\(858\) 1.08694 0.0371075
\(859\) 36.9580 1.26099 0.630495 0.776193i \(-0.282852\pi\)
0.630495 + 0.776193i \(0.282852\pi\)
\(860\) −43.9744 −1.49951
\(861\) −17.6703 −0.602202
\(862\) −7.93853 −0.270387
\(863\) 20.0855 0.683719 0.341859 0.939751i \(-0.388943\pi\)
0.341859 + 0.939751i \(0.388943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 15.8973 0.540525
\(866\) −40.9214 −1.39057
\(867\) 10.9342 0.371343
\(868\) −13.0427 −0.442699
\(869\) −6.76908 −0.229625
\(870\) −30.7840 −1.04367
\(871\) 9.36012 0.317156
\(872\) 18.0578 0.611515
\(873\) −5.65476 −0.191384
\(874\) 36.6592 1.24002
\(875\) 59.4210 2.00880
\(876\) −5.79522 −0.195802
\(877\) 17.9521 0.606199 0.303099 0.952959i \(-0.401979\pi\)
0.303099 + 0.952959i \(0.401979\pi\)
\(878\) 26.1722 0.883268
\(879\) −6.42954 −0.216863
\(880\) 4.16732 0.140480
\(881\) 41.0166 1.38189 0.690943 0.722909i \(-0.257196\pi\)
0.690943 + 0.722909i \(0.257196\pi\)
\(882\) −3.87612 −0.130516
\(883\) −8.43264 −0.283781 −0.141890 0.989882i \(-0.545318\pi\)
−0.141890 + 0.989882i \(0.545318\pi\)
\(884\) −5.28528 −0.177763
\(885\) −39.6321 −1.33222
\(886\) 8.73740 0.293539
\(887\) 36.9741 1.24147 0.620734 0.784021i \(-0.286835\pi\)
0.620734 + 0.784021i \(0.286835\pi\)
\(888\) −3.95146 −0.132602
\(889\) 32.7751 1.09924
\(890\) 22.2308 0.745178
\(891\) −1.08694 −0.0364138
\(892\) 15.9747 0.534873
\(893\) 87.0229 2.91211
\(894\) 3.20558 0.107211
\(895\) −30.0826 −1.00555
\(896\) 3.29790 0.110175
\(897\) −4.86940 −0.162585
\(898\) 27.0573 0.902915
\(899\) −31.7545 −1.05907
\(900\) 9.69950 0.323317
\(901\) −9.62573 −0.320679
\(902\) 5.82387 0.193914
\(903\) −37.8255 −1.25875
\(904\) 10.4337 0.347020
\(905\) −17.4686 −0.580676
\(906\) −9.10079 −0.302354
\(907\) −33.5901 −1.11534 −0.557670 0.830063i \(-0.688305\pi\)
−0.557670 + 0.830063i \(0.688305\pi\)
\(908\) 4.41065 0.146373
\(909\) −9.00487 −0.298673
\(910\) −12.6441 −0.419148
\(911\) 4.69896 0.155683 0.0778417 0.996966i \(-0.475197\pi\)
0.0778417 + 0.996966i \(0.475197\pi\)
\(912\) 7.52849 0.249293
\(913\) −2.71157 −0.0897397
\(914\) 12.4155 0.410667
\(915\) 57.0046 1.88451
\(916\) −7.15661 −0.236461
\(917\) 44.2752 1.46209
\(918\) 5.28528 0.174440
\(919\) −1.61816 −0.0533781 −0.0266890 0.999644i \(-0.508496\pi\)
−0.0266890 + 0.999644i \(0.508496\pi\)
\(920\) −18.6693 −0.615507
\(921\) 14.6133 0.481525
\(922\) 25.3222 0.833943
\(923\) −9.48805 −0.312303
\(924\) 3.58461 0.117925
\(925\) 38.3272 1.26019
\(926\) 11.0274 0.362383
\(927\) −1.00000 −0.0328443
\(928\) 8.02922 0.263572
\(929\) −31.3272 −1.02781 −0.513905 0.857847i \(-0.671802\pi\)
−0.513905 + 0.857847i \(0.671802\pi\)
\(930\) 15.1629 0.497211
\(931\) 29.1813 0.956378
\(932\) 24.7436 0.810503
\(933\) −11.9468 −0.391119
\(934\) 0.314734 0.0102984
\(935\) −22.0254 −0.720308
\(936\) −1.00000 −0.0326860
\(937\) −17.5332 −0.572785 −0.286392 0.958112i \(-0.592456\pi\)
−0.286392 + 0.958112i \(0.592456\pi\)
\(938\) 30.8687 1.00790
\(939\) −7.08127 −0.231088
\(940\) −44.3177 −1.44548
\(941\) 26.9382 0.878161 0.439080 0.898448i \(-0.355304\pi\)
0.439080 + 0.898448i \(0.355304\pi\)
\(942\) 14.3195 0.466554
\(943\) −26.0905 −0.849624
\(944\) 10.3370 0.336441
\(945\) 12.6441 0.411313
\(946\) 12.4668 0.405329
\(947\) 48.8593 1.58772 0.793858 0.608104i \(-0.208069\pi\)
0.793858 + 0.608104i \(0.208069\pi\)
\(948\) 6.22766 0.202265
\(949\) −5.79522 −0.188121
\(950\) −73.0226 −2.36916
\(951\) −23.8099 −0.772088
\(952\) −17.4303 −0.564919
\(953\) −24.9828 −0.809274 −0.404637 0.914477i \(-0.632602\pi\)
−0.404637 + 0.914477i \(0.632602\pi\)
\(954\) −1.82123 −0.0589646
\(955\) −22.5748 −0.730503
\(956\) −3.79838 −0.122848
\(957\) 8.72727 0.282113
\(958\) 30.3516 0.980616
\(959\) −61.9222 −1.99957
\(960\) −3.83399 −0.123742
\(961\) −15.3591 −0.495454
\(962\) −3.95146 −0.127400
\(963\) 12.8287 0.413398
\(964\) −21.6703 −0.697954
\(965\) −14.1817 −0.456525
\(966\) −16.0588 −0.516683
\(967\) −30.0342 −0.965836 −0.482918 0.875666i \(-0.660423\pi\)
−0.482918 + 0.875666i \(0.660423\pi\)
\(968\) 9.81856 0.315581
\(969\) −39.7901 −1.27824
\(970\) −21.6803 −0.696112
\(971\) 46.2170 1.48317 0.741587 0.670857i \(-0.234073\pi\)
0.741587 + 0.670857i \(0.234073\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0.369942 0.0118598
\(974\) −22.7338 −0.728437
\(975\) 9.69950 0.310633
\(976\) −14.8682 −0.475920
\(977\) 17.5927 0.562839 0.281419 0.959585i \(-0.409195\pi\)
0.281419 + 0.959585i \(0.409195\pi\)
\(978\) 14.6517 0.468509
\(979\) −6.30244 −0.201427
\(980\) −14.8610 −0.474717
\(981\) −18.0578 −0.576542
\(982\) 37.6519 1.20152
\(983\) 17.4539 0.556694 0.278347 0.960481i \(-0.410213\pi\)
0.278347 + 0.960481i \(0.410213\pi\)
\(984\) −5.35805 −0.170808
\(985\) −20.1917 −0.643362
\(986\) −42.4367 −1.35146
\(987\) −38.1209 −1.21340
\(988\) 7.52849 0.239513
\(989\) −55.8501 −1.77593
\(990\) −4.16732 −0.132446
\(991\) −0.107141 −0.00340343 −0.00170172 0.999999i \(-0.500542\pi\)
−0.00170172 + 0.999999i \(0.500542\pi\)
\(992\) −3.95486 −0.125567
\(993\) −35.9491 −1.14081
\(994\) −31.2906 −0.992477
\(995\) −55.1985 −1.74991
\(996\) 2.49468 0.0790470
\(997\) −57.4384 −1.81909 −0.909546 0.415603i \(-0.863570\pi\)
−0.909546 + 0.415603i \(0.863570\pi\)
\(998\) 37.8361 1.19768
\(999\) 3.95146 0.125019
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 8034.2.a.w.1.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.w.1.1 12 1.1 even 1 trivial