Properties

Label 4029.2.a.k.1.1
Level 4029
Weight 2
Character 4029.1
Self dual yes
Analytic conductor 32.172
Analytic rank 0
Dimension 31
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4029.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1717269744\)
Analytic rank: \(0\)
Dimension: \(31\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) = 4029.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.61491 q^{2} +1.00000 q^{3} +4.83775 q^{4} -2.62328 q^{5} -2.61491 q^{6} -4.80522 q^{7} -7.42046 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.61491 q^{2} +1.00000 q^{3} +4.83775 q^{4} -2.62328 q^{5} -2.61491 q^{6} -4.80522 q^{7} -7.42046 q^{8} +1.00000 q^{9} +6.85964 q^{10} +5.03885 q^{11} +4.83775 q^{12} -0.424255 q^{13} +12.5652 q^{14} -2.62328 q^{15} +9.72832 q^{16} +1.00000 q^{17} -2.61491 q^{18} -1.69266 q^{19} -12.6908 q^{20} -4.80522 q^{21} -13.1761 q^{22} +2.20587 q^{23} -7.42046 q^{24} +1.88160 q^{25} +1.10939 q^{26} +1.00000 q^{27} -23.2464 q^{28} +0.0142232 q^{29} +6.85964 q^{30} +4.78194 q^{31} -10.5978 q^{32} +5.03885 q^{33} -2.61491 q^{34} +12.6054 q^{35} +4.83775 q^{36} -11.1161 q^{37} +4.42615 q^{38} -0.424255 q^{39} +19.4659 q^{40} +10.7076 q^{41} +12.5652 q^{42} -2.19176 q^{43} +24.3767 q^{44} -2.62328 q^{45} -5.76814 q^{46} -11.5926 q^{47} +9.72832 q^{48} +16.0901 q^{49} -4.92021 q^{50} +1.00000 q^{51} -2.05244 q^{52} -1.42050 q^{53} -2.61491 q^{54} -13.2183 q^{55} +35.6569 q^{56} -1.69266 q^{57} -0.0371923 q^{58} -3.70537 q^{59} -12.6908 q^{60} -12.6621 q^{61} -12.5043 q^{62} -4.80522 q^{63} +8.25553 q^{64} +1.11294 q^{65} -13.1761 q^{66} -4.61390 q^{67} +4.83775 q^{68} +2.20587 q^{69} -32.9621 q^{70} -7.41269 q^{71} -7.42046 q^{72} -4.98159 q^{73} +29.0675 q^{74} +1.88160 q^{75} -8.18866 q^{76} -24.2128 q^{77} +1.10939 q^{78} +1.00000 q^{79} -25.5201 q^{80} +1.00000 q^{81} -27.9994 q^{82} +4.67216 q^{83} -23.2464 q^{84} -2.62328 q^{85} +5.73126 q^{86} +0.0142232 q^{87} -37.3906 q^{88} -1.95995 q^{89} +6.85964 q^{90} +2.03864 q^{91} +10.6714 q^{92} +4.78194 q^{93} +30.3135 q^{94} +4.44032 q^{95} -10.5978 q^{96} -13.9016 q^{97} -42.0742 q^{98} +5.03885 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + O(q^{10}) \) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + 5q^{10} + 26q^{11} + 34q^{12} + 7q^{13} + 19q^{14} + 11q^{15} + 40q^{16} + 31q^{17} + 4q^{18} + 32q^{19} + 23q^{20} + 4q^{21} + 2q^{22} + 29q^{23} + 12q^{24} + 32q^{25} + 13q^{26} + 31q^{27} - 13q^{28} + 25q^{29} + 5q^{30} + 22q^{31} + 28q^{32} + 26q^{33} + 4q^{34} + 20q^{35} + 34q^{36} - 4q^{37} + 19q^{38} + 7q^{39} - 3q^{40} + 33q^{41} + 19q^{42} + 6q^{43} + 30q^{44} + 11q^{45} - 11q^{46} + 23q^{47} + 40q^{48} + 31q^{49} + 6q^{50} + 31q^{51} - 7q^{52} + 12q^{53} + 4q^{54} + 40q^{56} + 32q^{57} + 9q^{58} + 27q^{59} + 23q^{60} - 4q^{61} + 25q^{62} + 4q^{63} + 10q^{64} + 54q^{65} + 2q^{66} + 34q^{68} + 29q^{69} - 59q^{70} + 35q^{71} + 12q^{72} + 5q^{73} + 48q^{74} + 32q^{75} + 32q^{76} + 42q^{77} + 13q^{78} + 31q^{79} + 24q^{80} + 31q^{81} + 5q^{82} + 67q^{83} - 13q^{84} + 11q^{85} - 20q^{86} + 25q^{87} - 7q^{88} + 22q^{89} + 5q^{90} + 16q^{91} + 57q^{92} + 22q^{93} + 45q^{94} + 73q^{95} + 28q^{96} - 13q^{97} - 19q^{98} + 26q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.61491 −1.84902 −0.924510 0.381158i \(-0.875525\pi\)
−0.924510 + 0.381158i \(0.875525\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.83775 2.41887
\(5\) −2.62328 −1.17317 −0.586583 0.809889i \(-0.699527\pi\)
−0.586583 + 0.809889i \(0.699527\pi\)
\(6\) −2.61491 −1.06753
\(7\) −4.80522 −1.81620 −0.908101 0.418751i \(-0.862468\pi\)
−0.908101 + 0.418751i \(0.862468\pi\)
\(8\) −7.42046 −2.62353
\(9\) 1.00000 0.333333
\(10\) 6.85964 2.16921
\(11\) 5.03885 1.51927 0.759636 0.650349i \(-0.225377\pi\)
0.759636 + 0.650349i \(0.225377\pi\)
\(12\) 4.83775 1.39654
\(13\) −0.424255 −0.117667 −0.0588336 0.998268i \(-0.518738\pi\)
−0.0588336 + 0.998268i \(0.518738\pi\)
\(14\) 12.5652 3.35819
\(15\) −2.62328 −0.677328
\(16\) 9.72832 2.43208
\(17\) 1.00000 0.242536
\(18\) −2.61491 −0.616340
\(19\) −1.69266 −0.388322 −0.194161 0.980970i \(-0.562198\pi\)
−0.194161 + 0.980970i \(0.562198\pi\)
\(20\) −12.6908 −2.83774
\(21\) −4.80522 −1.04858
\(22\) −13.1761 −2.80916
\(23\) 2.20587 0.459955 0.229978 0.973196i \(-0.426135\pi\)
0.229978 + 0.973196i \(0.426135\pi\)
\(24\) −7.42046 −1.51469
\(25\) 1.88160 0.376319
\(26\) 1.10939 0.217569
\(27\) 1.00000 0.192450
\(28\) −23.2464 −4.39316
\(29\) 0.0142232 0.00264118 0.00132059 0.999999i \(-0.499580\pi\)
0.00132059 + 0.999999i \(0.499580\pi\)
\(30\) 6.85964 1.25239
\(31\) 4.78194 0.858861 0.429431 0.903100i \(-0.358714\pi\)
0.429431 + 0.903100i \(0.358714\pi\)
\(32\) −10.5978 −1.87344
\(33\) 5.03885 0.877152
\(34\) −2.61491 −0.448453
\(35\) 12.6054 2.13071
\(36\) 4.83775 0.806292
\(37\) −11.1161 −1.82747 −0.913736 0.406309i \(-0.866816\pi\)
−0.913736 + 0.406309i \(0.866816\pi\)
\(38\) 4.42615 0.718016
\(39\) −0.424255 −0.0679352
\(40\) 19.4659 3.07783
\(41\) 10.7076 1.67225 0.836124 0.548540i \(-0.184816\pi\)
0.836124 + 0.548540i \(0.184816\pi\)
\(42\) 12.5652 1.93885
\(43\) −2.19176 −0.334241 −0.167120 0.985936i \(-0.553447\pi\)
−0.167120 + 0.985936i \(0.553447\pi\)
\(44\) 24.3767 3.67493
\(45\) −2.62328 −0.391055
\(46\) −5.76814 −0.850466
\(47\) −11.5926 −1.69095 −0.845475 0.534014i \(-0.820683\pi\)
−0.845475 + 0.534014i \(0.820683\pi\)
\(48\) 9.72832 1.40416
\(49\) 16.0901 2.29859
\(50\) −4.92021 −0.695822
\(51\) 1.00000 0.140028
\(52\) −2.05244 −0.284622
\(53\) −1.42050 −0.195120 −0.0975600 0.995230i \(-0.531104\pi\)
−0.0975600 + 0.995230i \(0.531104\pi\)
\(54\) −2.61491 −0.355844
\(55\) −13.2183 −1.78236
\(56\) 35.6569 4.76485
\(57\) −1.69266 −0.224198
\(58\) −0.0371923 −0.00488359
\(59\) −3.70537 −0.482398 −0.241199 0.970476i \(-0.577541\pi\)
−0.241199 + 0.970476i \(0.577541\pi\)
\(60\) −12.6908 −1.63837
\(61\) −12.6621 −1.62122 −0.810609 0.585588i \(-0.800864\pi\)
−0.810609 + 0.585588i \(0.800864\pi\)
\(62\) −12.5043 −1.58805
\(63\) −4.80522 −0.605401
\(64\) 8.25553 1.03194
\(65\) 1.11294 0.138043
\(66\) −13.1761 −1.62187
\(67\) −4.61390 −0.563678 −0.281839 0.959462i \(-0.590944\pi\)
−0.281839 + 0.959462i \(0.590944\pi\)
\(68\) 4.83775 0.586663
\(69\) 2.20587 0.265555
\(70\) −32.9621 −3.93972
\(71\) −7.41269 −0.879724 −0.439862 0.898065i \(-0.644973\pi\)
−0.439862 + 0.898065i \(0.644973\pi\)
\(72\) −7.42046 −0.874509
\(73\) −4.98159 −0.583051 −0.291526 0.956563i \(-0.594163\pi\)
−0.291526 + 0.956563i \(0.594163\pi\)
\(74\) 29.0675 3.37903
\(75\) 1.88160 0.217268
\(76\) −8.18866 −0.939303
\(77\) −24.2128 −2.75930
\(78\) 1.10939 0.125614
\(79\) 1.00000 0.112509
\(80\) −25.5201 −2.85323
\(81\) 1.00000 0.111111
\(82\) −27.9994 −3.09202
\(83\) 4.67216 0.512837 0.256418 0.966566i \(-0.417458\pi\)
0.256418 + 0.966566i \(0.417458\pi\)
\(84\) −23.2464 −2.53639
\(85\) −2.62328 −0.284535
\(86\) 5.73126 0.618018
\(87\) 0.0142232 0.00152488
\(88\) −37.3906 −3.98585
\(89\) −1.95995 −0.207755 −0.103877 0.994590i \(-0.533125\pi\)
−0.103877 + 0.994590i \(0.533125\pi\)
\(90\) 6.85964 0.723069
\(91\) 2.03864 0.213708
\(92\) 10.6714 1.11257
\(93\) 4.78194 0.495864
\(94\) 30.3135 3.12660
\(95\) 4.44032 0.455567
\(96\) −10.5978 −1.08163
\(97\) −13.9016 −1.41150 −0.705748 0.708463i \(-0.749389\pi\)
−0.705748 + 0.708463i \(0.749389\pi\)
\(98\) −42.0742 −4.25014
\(99\) 5.03885 0.506424
\(100\) 9.10270 0.910270
\(101\) 17.9380 1.78490 0.892448 0.451151i \(-0.148986\pi\)
0.892448 + 0.451151i \(0.148986\pi\)
\(102\) −2.61491 −0.258915
\(103\) 6.26519 0.617328 0.308664 0.951171i \(-0.400118\pi\)
0.308664 + 0.951171i \(0.400118\pi\)
\(104\) 3.14817 0.308703
\(105\) 12.6054 1.23016
\(106\) 3.71447 0.360781
\(107\) 15.0928 1.45908 0.729538 0.683941i \(-0.239735\pi\)
0.729538 + 0.683941i \(0.239735\pi\)
\(108\) 4.83775 0.465513
\(109\) −1.33822 −0.128179 −0.0640893 0.997944i \(-0.520414\pi\)
−0.0640893 + 0.997944i \(0.520414\pi\)
\(110\) 34.5647 3.29561
\(111\) −11.1161 −1.05509
\(112\) −46.7467 −4.41715
\(113\) −11.1926 −1.05291 −0.526456 0.850202i \(-0.676480\pi\)
−0.526456 + 0.850202i \(0.676480\pi\)
\(114\) 4.42615 0.414547
\(115\) −5.78661 −0.539604
\(116\) 0.0688081 0.00638868
\(117\) −0.424255 −0.0392224
\(118\) 9.68921 0.891964
\(119\) −4.80522 −0.440494
\(120\) 19.4659 1.77699
\(121\) 14.3900 1.30818
\(122\) 33.1103 2.99766
\(123\) 10.7076 0.965473
\(124\) 23.1338 2.07748
\(125\) 8.18044 0.731681
\(126\) 12.5652 1.11940
\(127\) 7.49201 0.664809 0.332404 0.943137i \(-0.392140\pi\)
0.332404 + 0.943137i \(0.392140\pi\)
\(128\) −0.391945 −0.0346433
\(129\) −2.19176 −0.192974
\(130\) −2.91024 −0.255245
\(131\) −9.03147 −0.789083 −0.394542 0.918878i \(-0.629097\pi\)
−0.394542 + 0.918878i \(0.629097\pi\)
\(132\) 24.3767 2.12172
\(133\) 8.13359 0.705272
\(134\) 12.0649 1.04225
\(135\) −2.62328 −0.225776
\(136\) −7.42046 −0.636299
\(137\) 17.5708 1.50117 0.750587 0.660771i \(-0.229771\pi\)
0.750587 + 0.660771i \(0.229771\pi\)
\(138\) −5.76814 −0.491017
\(139\) −7.16731 −0.607923 −0.303962 0.952684i \(-0.598309\pi\)
−0.303962 + 0.952684i \(0.598309\pi\)
\(140\) 60.9819 5.15391
\(141\) −11.5926 −0.976271
\(142\) 19.3835 1.62663
\(143\) −2.13776 −0.178768
\(144\) 9.72832 0.810693
\(145\) −0.0373114 −0.00309854
\(146\) 13.0264 1.07807
\(147\) 16.0901 1.32709
\(148\) −53.7768 −4.42042
\(149\) −3.43129 −0.281102 −0.140551 0.990073i \(-0.544887\pi\)
−0.140551 + 0.990073i \(0.544887\pi\)
\(150\) −4.92021 −0.401733
\(151\) 12.2495 0.996848 0.498424 0.866933i \(-0.333912\pi\)
0.498424 + 0.866933i \(0.333912\pi\)
\(152\) 12.5603 1.01877
\(153\) 1.00000 0.0808452
\(154\) 63.3142 5.10201
\(155\) −12.5444 −1.00759
\(156\) −2.05244 −0.164327
\(157\) 3.22816 0.257635 0.128818 0.991668i \(-0.458882\pi\)
0.128818 + 0.991668i \(0.458882\pi\)
\(158\) −2.61491 −0.208031
\(159\) −1.42050 −0.112653
\(160\) 27.8009 2.19785
\(161\) −10.5997 −0.835372
\(162\) −2.61491 −0.205447
\(163\) 12.3203 0.964997 0.482498 0.875897i \(-0.339729\pi\)
0.482498 + 0.875897i \(0.339729\pi\)
\(164\) 51.8007 4.04496
\(165\) −13.2183 −1.02904
\(166\) −12.2173 −0.948245
\(167\) −12.1321 −0.938811 −0.469405 0.882983i \(-0.655532\pi\)
−0.469405 + 0.882983i \(0.655532\pi\)
\(168\) 35.6569 2.75099
\(169\) −12.8200 −0.986154
\(170\) 6.85964 0.526110
\(171\) −1.69266 −0.129441
\(172\) −10.6032 −0.808486
\(173\) 21.7686 1.65504 0.827519 0.561438i \(-0.189752\pi\)
0.827519 + 0.561438i \(0.189752\pi\)
\(174\) −0.0371923 −0.00281954
\(175\) −9.04149 −0.683472
\(176\) 49.0196 3.69499
\(177\) −3.70537 −0.278513
\(178\) 5.12510 0.384143
\(179\) −14.3372 −1.07161 −0.535806 0.844341i \(-0.679992\pi\)
−0.535806 + 0.844341i \(0.679992\pi\)
\(180\) −12.6908 −0.945914
\(181\) 18.8588 1.40177 0.700883 0.713276i \(-0.252789\pi\)
0.700883 + 0.713276i \(0.252789\pi\)
\(182\) −5.33086 −0.395149
\(183\) −12.6621 −0.936011
\(184\) −16.3685 −1.20671
\(185\) 29.1606 2.14393
\(186\) −12.5043 −0.916862
\(187\) 5.03885 0.368477
\(188\) −56.0820 −4.09020
\(189\) −4.80522 −0.349528
\(190\) −11.6110 −0.842352
\(191\) 0.813947 0.0588951 0.0294476 0.999566i \(-0.490625\pi\)
0.0294476 + 0.999566i \(0.490625\pi\)
\(192\) 8.25553 0.595791
\(193\) 15.2769 1.09966 0.549828 0.835278i \(-0.314693\pi\)
0.549828 + 0.835278i \(0.314693\pi\)
\(194\) 36.3515 2.60988
\(195\) 1.11294 0.0796993
\(196\) 77.8400 5.56000
\(197\) 18.6764 1.33064 0.665320 0.746558i \(-0.268295\pi\)
0.665320 + 0.746558i \(0.268295\pi\)
\(198\) −13.1761 −0.936387
\(199\) −0.597834 −0.0423793 −0.0211897 0.999775i \(-0.506745\pi\)
−0.0211897 + 0.999775i \(0.506745\pi\)
\(200\) −13.9623 −0.987284
\(201\) −4.61390 −0.325439
\(202\) −46.9062 −3.30031
\(203\) −0.0683454 −0.00479691
\(204\) 4.83775 0.338710
\(205\) −28.0891 −1.96183
\(206\) −16.3829 −1.14145
\(207\) 2.20587 0.153318
\(208\) −4.12729 −0.286176
\(209\) −8.52906 −0.589967
\(210\) −32.9621 −2.27460
\(211\) 20.2479 1.39392 0.696962 0.717108i \(-0.254535\pi\)
0.696962 + 0.717108i \(0.254535\pi\)
\(212\) −6.87200 −0.471971
\(213\) −7.41269 −0.507909
\(214\) −39.4663 −2.69786
\(215\) 5.74961 0.392120
\(216\) −7.42046 −0.504898
\(217\) −22.9782 −1.55986
\(218\) 3.49933 0.237005
\(219\) −4.98159 −0.336625
\(220\) −63.9469 −4.31130
\(221\) −0.424255 −0.0285385
\(222\) 29.0675 1.95088
\(223\) −9.04780 −0.605886 −0.302943 0.953009i \(-0.597969\pi\)
−0.302943 + 0.953009i \(0.597969\pi\)
\(224\) 50.9245 3.40254
\(225\) 1.88160 0.125440
\(226\) 29.2677 1.94686
\(227\) 15.1259 1.00394 0.501971 0.864885i \(-0.332608\pi\)
0.501971 + 0.864885i \(0.332608\pi\)
\(228\) −8.18866 −0.542307
\(229\) 11.4726 0.758128 0.379064 0.925370i \(-0.376246\pi\)
0.379064 + 0.925370i \(0.376246\pi\)
\(230\) 15.1315 0.997739
\(231\) −24.2128 −1.59308
\(232\) −0.105542 −0.00692920
\(233\) 6.68129 0.437706 0.218853 0.975758i \(-0.429769\pi\)
0.218853 + 0.975758i \(0.429769\pi\)
\(234\) 1.10939 0.0725230
\(235\) 30.4106 1.98377
\(236\) −17.9257 −1.16686
\(237\) 1.00000 0.0649570
\(238\) 12.5652 0.814481
\(239\) −23.0293 −1.48964 −0.744819 0.667266i \(-0.767464\pi\)
−0.744819 + 0.667266i \(0.767464\pi\)
\(240\) −25.5201 −1.64732
\(241\) 8.26243 0.532230 0.266115 0.963941i \(-0.414260\pi\)
0.266115 + 0.963941i \(0.414260\pi\)
\(242\) −37.6286 −2.41886
\(243\) 1.00000 0.0641500
\(244\) −61.2562 −3.92152
\(245\) −42.2089 −2.69663
\(246\) −27.9994 −1.78518
\(247\) 0.718119 0.0456928
\(248\) −35.4841 −2.25325
\(249\) 4.67216 0.296086
\(250\) −21.3911 −1.35289
\(251\) 11.8050 0.745128 0.372564 0.928007i \(-0.378479\pi\)
0.372564 + 0.928007i \(0.378479\pi\)
\(252\) −23.2464 −1.46439
\(253\) 11.1150 0.698797
\(254\) −19.5909 −1.22924
\(255\) −2.62328 −0.164276
\(256\) −15.4862 −0.967885
\(257\) 23.7690 1.48267 0.741335 0.671135i \(-0.234193\pi\)
0.741335 + 0.671135i \(0.234193\pi\)
\(258\) 5.73126 0.356813
\(259\) 53.4152 3.31906
\(260\) 5.38413 0.333909
\(261\) 0.0142232 0.000880392 0
\(262\) 23.6165 1.45903
\(263\) 27.3452 1.68618 0.843089 0.537774i \(-0.180735\pi\)
0.843089 + 0.537774i \(0.180735\pi\)
\(264\) −37.3906 −2.30123
\(265\) 3.72636 0.228908
\(266\) −21.2686 −1.30406
\(267\) −1.95995 −0.119947
\(268\) −22.3209 −1.36347
\(269\) −21.3647 −1.30263 −0.651314 0.758808i \(-0.725782\pi\)
−0.651314 + 0.758808i \(0.725782\pi\)
\(270\) 6.85964 0.417464
\(271\) −10.5082 −0.638330 −0.319165 0.947699i \(-0.603402\pi\)
−0.319165 + 0.947699i \(0.603402\pi\)
\(272\) 9.72832 0.589866
\(273\) 2.03864 0.123384
\(274\) −45.9460 −2.77570
\(275\) 9.48109 0.571731
\(276\) 10.6714 0.642345
\(277\) −1.59108 −0.0955988 −0.0477994 0.998857i \(-0.515221\pi\)
−0.0477994 + 0.998857i \(0.515221\pi\)
\(278\) 18.7419 1.12406
\(279\) 4.78194 0.286287
\(280\) −93.5381 −5.58997
\(281\) −26.3797 −1.57368 −0.786842 0.617155i \(-0.788285\pi\)
−0.786842 + 0.617155i \(0.788285\pi\)
\(282\) 30.3135 1.80514
\(283\) 17.1406 1.01890 0.509451 0.860500i \(-0.329848\pi\)
0.509451 + 0.860500i \(0.329848\pi\)
\(284\) −35.8607 −2.12794
\(285\) 4.44032 0.263022
\(286\) 5.59005 0.330546
\(287\) −51.4524 −3.03714
\(288\) −10.5978 −0.624479
\(289\) 1.00000 0.0588235
\(290\) 0.0975658 0.00572926
\(291\) −13.9016 −0.814927
\(292\) −24.0997 −1.41033
\(293\) 15.7659 0.921052 0.460526 0.887646i \(-0.347661\pi\)
0.460526 + 0.887646i \(0.347661\pi\)
\(294\) −42.0742 −2.45382
\(295\) 9.72023 0.565934
\(296\) 82.4863 4.79442
\(297\) 5.03885 0.292384
\(298\) 8.97251 0.519764
\(299\) −0.935851 −0.0541217
\(300\) 9.10270 0.525544
\(301\) 10.5319 0.607049
\(302\) −32.0313 −1.84319
\(303\) 17.9380 1.03051
\(304\) −16.4667 −0.944431
\(305\) 33.2163 1.90196
\(306\) −2.61491 −0.149484
\(307\) −2.76806 −0.157981 −0.0789906 0.996875i \(-0.525170\pi\)
−0.0789906 + 0.996875i \(0.525170\pi\)
\(308\) −117.135 −6.67441
\(309\) 6.26519 0.356414
\(310\) 32.8023 1.86305
\(311\) 19.7793 1.12158 0.560790 0.827958i \(-0.310498\pi\)
0.560790 + 0.827958i \(0.310498\pi\)
\(312\) 3.14817 0.178230
\(313\) 20.6041 1.16462 0.582308 0.812969i \(-0.302150\pi\)
0.582308 + 0.812969i \(0.302150\pi\)
\(314\) −8.44134 −0.476373
\(315\) 12.6054 0.710236
\(316\) 4.83775 0.272145
\(317\) −7.73561 −0.434475 −0.217237 0.976119i \(-0.569705\pi\)
−0.217237 + 0.976119i \(0.569705\pi\)
\(318\) 3.71447 0.208297
\(319\) 0.0716685 0.00401266
\(320\) −21.6566 −1.21064
\(321\) 15.0928 0.842397
\(322\) 27.7172 1.54462
\(323\) −1.69266 −0.0941820
\(324\) 4.83775 0.268764
\(325\) −0.798278 −0.0442805
\(326\) −32.2163 −1.78430
\(327\) −1.33822 −0.0740039
\(328\) −79.4554 −4.38719
\(329\) 55.7049 3.07111
\(330\) 34.5647 1.90272
\(331\) −13.1774 −0.724296 −0.362148 0.932121i \(-0.617956\pi\)
−0.362148 + 0.932121i \(0.617956\pi\)
\(332\) 22.6028 1.24049
\(333\) −11.1161 −0.609157
\(334\) 31.7244 1.73588
\(335\) 12.1036 0.661288
\(336\) −46.7467 −2.55024
\(337\) −15.7916 −0.860225 −0.430112 0.902775i \(-0.641526\pi\)
−0.430112 + 0.902775i \(0.641526\pi\)
\(338\) 33.5232 1.82342
\(339\) −11.1926 −0.607899
\(340\) −12.6908 −0.688254
\(341\) 24.0955 1.30484
\(342\) 4.42615 0.239339
\(343\) −43.6800 −2.35850
\(344\) 16.2639 0.876890
\(345\) −5.78661 −0.311541
\(346\) −56.9230 −3.06020
\(347\) −34.0392 −1.82732 −0.913661 0.406478i \(-0.866757\pi\)
−0.913661 + 0.406478i \(0.866757\pi\)
\(348\) 0.0688081 0.00368850
\(349\) −13.3243 −0.713234 −0.356617 0.934251i \(-0.616070\pi\)
−0.356617 + 0.934251i \(0.616070\pi\)
\(350\) 23.6427 1.26375
\(351\) −0.424255 −0.0226451
\(352\) −53.4005 −2.84626
\(353\) 5.72620 0.304775 0.152387 0.988321i \(-0.451304\pi\)
0.152387 + 0.988321i \(0.451304\pi\)
\(354\) 9.68921 0.514976
\(355\) 19.4456 1.03206
\(356\) −9.48177 −0.502533
\(357\) −4.80522 −0.254319
\(358\) 37.4904 1.98143
\(359\) −0.808049 −0.0426472 −0.0213236 0.999773i \(-0.506788\pi\)
−0.0213236 + 0.999773i \(0.506788\pi\)
\(360\) 19.4659 1.02594
\(361\) −16.1349 −0.849206
\(362\) −49.3141 −2.59189
\(363\) 14.3900 0.755281
\(364\) 9.86243 0.516932
\(365\) 13.0681 0.684016
\(366\) 33.1103 1.73070
\(367\) 5.29883 0.276597 0.138298 0.990391i \(-0.455837\pi\)
0.138298 + 0.990391i \(0.455837\pi\)
\(368\) 21.4594 1.11865
\(369\) 10.7076 0.557416
\(370\) −76.2523 −3.96417
\(371\) 6.82579 0.354377
\(372\) 23.1338 1.19943
\(373\) −2.49991 −0.129440 −0.0647202 0.997903i \(-0.520616\pi\)
−0.0647202 + 0.997903i \(0.520616\pi\)
\(374\) −13.1761 −0.681322
\(375\) 8.18044 0.422436
\(376\) 86.0222 4.43626
\(377\) −0.00603426 −0.000310780 0
\(378\) 12.5652 0.646285
\(379\) 19.6511 1.00941 0.504705 0.863292i \(-0.331601\pi\)
0.504705 + 0.863292i \(0.331601\pi\)
\(380\) 21.4811 1.10196
\(381\) 7.49201 0.383827
\(382\) −2.12840 −0.108898
\(383\) 23.5058 1.20109 0.600545 0.799591i \(-0.294950\pi\)
0.600545 + 0.799591i \(0.294950\pi\)
\(384\) −0.391945 −0.0200013
\(385\) 63.5169 3.23712
\(386\) −39.9478 −2.03329
\(387\) −2.19176 −0.111414
\(388\) −67.2525 −3.41423
\(389\) −18.5680 −0.941437 −0.470718 0.882284i \(-0.656005\pi\)
−0.470718 + 0.882284i \(0.656005\pi\)
\(390\) −2.91024 −0.147366
\(391\) 2.20587 0.111556
\(392\) −119.396 −6.03041
\(393\) −9.03147 −0.455577
\(394\) −48.8371 −2.46038
\(395\) −2.62328 −0.131992
\(396\) 24.3767 1.22498
\(397\) −11.2993 −0.567096 −0.283548 0.958958i \(-0.591512\pi\)
−0.283548 + 0.958958i \(0.591512\pi\)
\(398\) 1.56328 0.0783602
\(399\) 8.13359 0.407189
\(400\) 18.3048 0.915239
\(401\) 12.9628 0.647331 0.323666 0.946172i \(-0.395085\pi\)
0.323666 + 0.946172i \(0.395085\pi\)
\(402\) 12.0649 0.601744
\(403\) −2.02876 −0.101060
\(404\) 86.7794 4.31744
\(405\) −2.62328 −0.130352
\(406\) 0.178717 0.00886958
\(407\) −56.0123 −2.77642
\(408\) −7.42046 −0.367367
\(409\) 32.4363 1.60387 0.801936 0.597410i \(-0.203803\pi\)
0.801936 + 0.597410i \(0.203803\pi\)
\(410\) 73.4504 3.62745
\(411\) 17.5708 0.866704
\(412\) 30.3094 1.49324
\(413\) 17.8051 0.876133
\(414\) −5.76814 −0.283489
\(415\) −12.2564 −0.601643
\(416\) 4.49615 0.220442
\(417\) −7.16731 −0.350985
\(418\) 22.3027 1.09086
\(419\) 9.92000 0.484624 0.242312 0.970198i \(-0.422094\pi\)
0.242312 + 0.970198i \(0.422094\pi\)
\(420\) 60.9819 2.97561
\(421\) 7.37803 0.359583 0.179792 0.983705i \(-0.442458\pi\)
0.179792 + 0.983705i \(0.442458\pi\)
\(422\) −52.9465 −2.57739
\(423\) −11.5926 −0.563650
\(424\) 10.5407 0.511903
\(425\) 1.88160 0.0912709
\(426\) 19.3835 0.939134
\(427\) 60.8442 2.94446
\(428\) 73.0151 3.52932
\(429\) −2.13776 −0.103212
\(430\) −15.0347 −0.725038
\(431\) 31.6327 1.52369 0.761847 0.647757i \(-0.224293\pi\)
0.761847 + 0.647757i \(0.224293\pi\)
\(432\) 9.72832 0.468054
\(433\) 21.2106 1.01932 0.509658 0.860377i \(-0.329772\pi\)
0.509658 + 0.860377i \(0.329772\pi\)
\(434\) 60.0860 2.88422
\(435\) −0.0373114 −0.00178894
\(436\) −6.47399 −0.310048
\(437\) −3.73378 −0.178611
\(438\) 13.0264 0.622426
\(439\) −12.1855 −0.581581 −0.290791 0.956787i \(-0.593918\pi\)
−0.290791 + 0.956787i \(0.593918\pi\)
\(440\) 98.0859 4.67606
\(441\) 16.0901 0.766196
\(442\) 1.10939 0.0527683
\(443\) 27.0387 1.28465 0.642323 0.766434i \(-0.277971\pi\)
0.642323 + 0.766434i \(0.277971\pi\)
\(444\) −53.7768 −2.55213
\(445\) 5.14151 0.243731
\(446\) 23.6592 1.12029
\(447\) −3.43129 −0.162294
\(448\) −39.6696 −1.87421
\(449\) −36.3117 −1.71365 −0.856827 0.515604i \(-0.827568\pi\)
−0.856827 + 0.515604i \(0.827568\pi\)
\(450\) −4.92021 −0.231941
\(451\) 53.9541 2.54060
\(452\) −54.1470 −2.54686
\(453\) 12.2495 0.575530
\(454\) −39.5529 −1.85631
\(455\) −5.34792 −0.250714
\(456\) 12.5603 0.588190
\(457\) 29.0585 1.35930 0.679650 0.733537i \(-0.262132\pi\)
0.679650 + 0.733537i \(0.262132\pi\)
\(458\) −29.9997 −1.40179
\(459\) 1.00000 0.0466760
\(460\) −27.9942 −1.30523
\(461\) 29.6091 1.37903 0.689517 0.724270i \(-0.257823\pi\)
0.689517 + 0.724270i \(0.257823\pi\)
\(462\) 63.3142 2.94564
\(463\) −6.29861 −0.292721 −0.146361 0.989231i \(-0.546756\pi\)
−0.146361 + 0.989231i \(0.546756\pi\)
\(464\) 0.138368 0.00642355
\(465\) −12.5444 −0.581731
\(466\) −17.4710 −0.809326
\(467\) −17.7787 −0.822702 −0.411351 0.911477i \(-0.634943\pi\)
−0.411351 + 0.911477i \(0.634943\pi\)
\(468\) −2.05244 −0.0948741
\(469\) 22.1708 1.02375
\(470\) −79.5209 −3.66802
\(471\) 3.22816 0.148746
\(472\) 27.4956 1.26559
\(473\) −11.0440 −0.507802
\(474\) −2.61491 −0.120107
\(475\) −3.18490 −0.146133
\(476\) −23.2464 −1.06550
\(477\) −1.42050 −0.0650400
\(478\) 60.2194 2.75437
\(479\) −5.08581 −0.232377 −0.116188 0.993227i \(-0.537068\pi\)
−0.116188 + 0.993227i \(0.537068\pi\)
\(480\) 27.8009 1.26893
\(481\) 4.71605 0.215034
\(482\) −21.6055 −0.984104
\(483\) −10.5997 −0.482302
\(484\) 69.6153 3.16433
\(485\) 36.4678 1.65592
\(486\) −2.61491 −0.118615
\(487\) −16.2181 −0.734912 −0.367456 0.930041i \(-0.619771\pi\)
−0.367456 + 0.930041i \(0.619771\pi\)
\(488\) 93.9587 4.25331
\(489\) 12.3203 0.557141
\(490\) 110.372 4.98612
\(491\) 18.6235 0.840468 0.420234 0.907416i \(-0.361948\pi\)
0.420234 + 0.907416i \(0.361948\pi\)
\(492\) 51.8007 2.33536
\(493\) 0.0142232 0.000640579 0
\(494\) −1.87782 −0.0844870
\(495\) −13.2183 −0.594119
\(496\) 46.5202 2.08882
\(497\) 35.6196 1.59776
\(498\) −12.2173 −0.547470
\(499\) 42.0101 1.88063 0.940315 0.340306i \(-0.110531\pi\)
0.940315 + 0.340306i \(0.110531\pi\)
\(500\) 39.5749 1.76984
\(501\) −12.1321 −0.542023
\(502\) −30.8691 −1.37776
\(503\) −2.25841 −0.100698 −0.0503488 0.998732i \(-0.516033\pi\)
−0.0503488 + 0.998732i \(0.516033\pi\)
\(504\) 35.6569 1.58828
\(505\) −47.0563 −2.09398
\(506\) −29.0648 −1.29209
\(507\) −12.8200 −0.569357
\(508\) 36.2445 1.60809
\(509\) −38.1268 −1.68994 −0.844970 0.534814i \(-0.820382\pi\)
−0.844970 + 0.534814i \(0.820382\pi\)
\(510\) 6.85964 0.303750
\(511\) 23.9376 1.05894
\(512\) 41.2788 1.82428
\(513\) −1.69266 −0.0747327
\(514\) −62.1538 −2.74149
\(515\) −16.4354 −0.724228
\(516\) −10.6032 −0.466780
\(517\) −58.4133 −2.56901
\(518\) −139.676 −6.13700
\(519\) 21.7686 0.955536
\(520\) −8.25853 −0.362160
\(521\) −21.4571 −0.940053 −0.470026 0.882652i \(-0.655756\pi\)
−0.470026 + 0.882652i \(0.655756\pi\)
\(522\) −0.0371923 −0.00162786
\(523\) −4.99511 −0.218421 −0.109210 0.994019i \(-0.534832\pi\)
−0.109210 + 0.994019i \(0.534832\pi\)
\(524\) −43.6920 −1.90869
\(525\) −9.04149 −0.394603
\(526\) −71.5052 −3.11778
\(527\) 4.78194 0.208304
\(528\) 49.0196 2.13330
\(529\) −18.1341 −0.788441
\(530\) −9.74409 −0.423256
\(531\) −3.70537 −0.160799
\(532\) 39.3483 1.70596
\(533\) −4.54276 −0.196769
\(534\) 5.12510 0.221785
\(535\) −39.5926 −1.71174
\(536\) 34.2372 1.47882
\(537\) −14.3372 −0.618695
\(538\) 55.8668 2.40859
\(539\) 81.0757 3.49218
\(540\) −12.6908 −0.546124
\(541\) 12.2190 0.525336 0.262668 0.964886i \(-0.415398\pi\)
0.262668 + 0.964886i \(0.415398\pi\)
\(542\) 27.4781 1.18028
\(543\) 18.8588 0.809310
\(544\) −10.5978 −0.454375
\(545\) 3.51053 0.150375
\(546\) −5.33086 −0.228140
\(547\) 38.3094 1.63799 0.818995 0.573800i \(-0.194531\pi\)
0.818995 + 0.573800i \(0.194531\pi\)
\(548\) 85.0031 3.63115
\(549\) −12.6621 −0.540406
\(550\) −24.7922 −1.05714
\(551\) −0.0240750 −0.00102563
\(552\) −16.3685 −0.696692
\(553\) −4.80522 −0.204339
\(554\) 4.16053 0.176764
\(555\) 29.1606 1.23780
\(556\) −34.6736 −1.47049
\(557\) 14.7175 0.623601 0.311801 0.950148i \(-0.399068\pi\)
0.311801 + 0.950148i \(0.399068\pi\)
\(558\) −12.5043 −0.529350
\(559\) 0.929867 0.0393292
\(560\) 122.630 5.18205
\(561\) 5.03885 0.212740
\(562\) 68.9806 2.90977
\(563\) −28.2021 −1.18858 −0.594289 0.804252i \(-0.702566\pi\)
−0.594289 + 0.804252i \(0.702566\pi\)
\(564\) −56.0820 −2.36148
\(565\) 29.3614 1.23524
\(566\) −44.8211 −1.88397
\(567\) −4.80522 −0.201800
\(568\) 55.0055 2.30798
\(569\) 3.98075 0.166882 0.0834408 0.996513i \(-0.473409\pi\)
0.0834408 + 0.996513i \(0.473409\pi\)
\(570\) −11.6110 −0.486332
\(571\) 4.41742 0.184863 0.0924317 0.995719i \(-0.470536\pi\)
0.0924317 + 0.995719i \(0.470536\pi\)
\(572\) −10.3419 −0.432418
\(573\) 0.813947 0.0340031
\(574\) 134.543 5.61573
\(575\) 4.15056 0.173090
\(576\) 8.25553 0.343980
\(577\) 12.5308 0.521665 0.260832 0.965384i \(-0.416003\pi\)
0.260832 + 0.965384i \(0.416003\pi\)
\(578\) −2.61491 −0.108766
\(579\) 15.2769 0.634887
\(580\) −0.180503 −0.00749498
\(581\) −22.4508 −0.931415
\(582\) 36.3515 1.50682
\(583\) −7.15767 −0.296440
\(584\) 36.9657 1.52965
\(585\) 1.11294 0.0460144
\(586\) −41.2263 −1.70304
\(587\) −1.65542 −0.0683267 −0.0341633 0.999416i \(-0.510877\pi\)
−0.0341633 + 0.999416i \(0.510877\pi\)
\(588\) 77.8400 3.21007
\(589\) −8.09418 −0.333515
\(590\) −25.4175 −1.04642
\(591\) 18.6764 0.768245
\(592\) −108.141 −4.44456
\(593\) −27.4662 −1.12790 −0.563951 0.825808i \(-0.690719\pi\)
−0.563951 + 0.825808i \(0.690719\pi\)
\(594\) −13.1761 −0.540624
\(595\) 12.6054 0.516772
\(596\) −16.5997 −0.679951
\(597\) −0.597834 −0.0244677
\(598\) 2.44717 0.100072
\(599\) −17.4117 −0.711423 −0.355712 0.934596i \(-0.615761\pi\)
−0.355712 + 0.934596i \(0.615761\pi\)
\(600\) −13.9623 −0.570009
\(601\) 6.64335 0.270988 0.135494 0.990778i \(-0.456738\pi\)
0.135494 + 0.990778i \(0.456738\pi\)
\(602\) −27.5400 −1.12245
\(603\) −4.61390 −0.187893
\(604\) 59.2599 2.41125
\(605\) −37.7491 −1.53472
\(606\) −46.9062 −1.90543
\(607\) 13.4506 0.545941 0.272971 0.962022i \(-0.411994\pi\)
0.272971 + 0.962022i \(0.411994\pi\)
\(608\) 17.9384 0.727497
\(609\) −0.0683454 −0.00276950
\(610\) −86.8576 −3.51676
\(611\) 4.91821 0.198970
\(612\) 4.83775 0.195554
\(613\) −7.45281 −0.301016 −0.150508 0.988609i \(-0.548091\pi\)
−0.150508 + 0.988609i \(0.548091\pi\)
\(614\) 7.23821 0.292110
\(615\) −28.0891 −1.13266
\(616\) 179.670 7.23911
\(617\) −14.1094 −0.568023 −0.284011 0.958821i \(-0.591665\pi\)
−0.284011 + 0.958821i \(0.591665\pi\)
\(618\) −16.3829 −0.659017
\(619\) −12.7217 −0.511329 −0.255664 0.966766i \(-0.582294\pi\)
−0.255664 + 0.966766i \(0.582294\pi\)
\(620\) −60.6864 −2.43723
\(621\) 2.20587 0.0885184
\(622\) −51.7210 −2.07382
\(623\) 9.41801 0.377324
\(624\) −4.12729 −0.165224
\(625\) −30.8676 −1.23470
\(626\) −53.8780 −2.15340
\(627\) −8.52906 −0.340618
\(628\) 15.6170 0.623187
\(629\) −11.1161 −0.443227
\(630\) −32.9621 −1.31324
\(631\) −0.578439 −0.0230273 −0.0115137 0.999934i \(-0.503665\pi\)
−0.0115137 + 0.999934i \(0.503665\pi\)
\(632\) −7.42046 −0.295170
\(633\) 20.2479 0.804782
\(634\) 20.2279 0.803353
\(635\) −19.6536 −0.779931
\(636\) −6.87200 −0.272493
\(637\) −6.82632 −0.270469
\(638\) −0.187407 −0.00741949
\(639\) −7.41269 −0.293241
\(640\) 1.02818 0.0406424
\(641\) 30.4589 1.20305 0.601527 0.798853i \(-0.294559\pi\)
0.601527 + 0.798853i \(0.294559\pi\)
\(642\) −39.4663 −1.55761
\(643\) 16.2566 0.641099 0.320549 0.947232i \(-0.396133\pi\)
0.320549 + 0.947232i \(0.396133\pi\)
\(644\) −51.2786 −2.02066
\(645\) 5.74961 0.226391
\(646\) 4.42615 0.174144
\(647\) −8.27283 −0.325238 −0.162619 0.986689i \(-0.551994\pi\)
−0.162619 + 0.986689i \(0.551994\pi\)
\(648\) −7.42046 −0.291503
\(649\) −18.6708 −0.732894
\(650\) 2.08742 0.0818755
\(651\) −22.9782 −0.900588
\(652\) 59.6023 2.33421
\(653\) −21.0857 −0.825146 −0.412573 0.910925i \(-0.635370\pi\)
−0.412573 + 0.910925i \(0.635370\pi\)
\(654\) 3.49933 0.136835
\(655\) 23.6921 0.925726
\(656\) 104.167 4.06704
\(657\) −4.98159 −0.194350
\(658\) −145.663 −5.67854
\(659\) 45.7095 1.78059 0.890295 0.455384i \(-0.150498\pi\)
0.890295 + 0.455384i \(0.150498\pi\)
\(660\) −63.9469 −2.48913
\(661\) −16.7803 −0.652676 −0.326338 0.945253i \(-0.605815\pi\)
−0.326338 + 0.945253i \(0.605815\pi\)
\(662\) 34.4577 1.33924
\(663\) −0.424255 −0.0164767
\(664\) −34.6696 −1.34544
\(665\) −21.3367 −0.827401
\(666\) 29.0675 1.12634
\(667\) 0.0313744 0.00121482
\(668\) −58.6921 −2.27087
\(669\) −9.04780 −0.349808
\(670\) −31.6497 −1.22273
\(671\) −63.8025 −2.46307
\(672\) 50.9245 1.96446
\(673\) 19.5956 0.755356 0.377678 0.925937i \(-0.376723\pi\)
0.377678 + 0.925937i \(0.376723\pi\)
\(674\) 41.2937 1.59057
\(675\) 1.88160 0.0724227
\(676\) −62.0200 −2.38538
\(677\) 19.6776 0.756272 0.378136 0.925750i \(-0.376565\pi\)
0.378136 + 0.925750i \(0.376565\pi\)
\(678\) 29.2677 1.12402
\(679\) 66.8003 2.56356
\(680\) 19.4659 0.746484
\(681\) 15.1259 0.579626
\(682\) −63.0075 −2.41268
\(683\) −39.9552 −1.52884 −0.764421 0.644718i \(-0.776975\pi\)
−0.764421 + 0.644718i \(0.776975\pi\)
\(684\) −8.18866 −0.313101
\(685\) −46.0931 −1.76113
\(686\) 114.219 4.36091
\(687\) 11.4726 0.437706
\(688\) −21.3222 −0.812900
\(689\) 0.602653 0.0229592
\(690\) 15.1315 0.576045
\(691\) 39.6863 1.50974 0.754869 0.655876i \(-0.227700\pi\)
0.754869 + 0.655876i \(0.227700\pi\)
\(692\) 105.311 4.00333
\(693\) −24.2128 −0.919768
\(694\) 89.0095 3.37875
\(695\) 18.8019 0.713195
\(696\) −0.105542 −0.00400058
\(697\) 10.7076 0.405580
\(698\) 34.8419 1.31878
\(699\) 6.68129 0.252709
\(700\) −43.7404 −1.65323
\(701\) 2.43342 0.0919089 0.0459544 0.998944i \(-0.485367\pi\)
0.0459544 + 0.998944i \(0.485367\pi\)
\(702\) 1.10939 0.0418712
\(703\) 18.8157 0.709648
\(704\) 41.5984 1.56780
\(705\) 30.4106 1.14533
\(706\) −14.9735 −0.563535
\(707\) −86.1959 −3.24173
\(708\) −17.9257 −0.673688
\(709\) −4.13449 −0.155274 −0.0776370 0.996982i \(-0.524738\pi\)
−0.0776370 + 0.996982i \(0.524738\pi\)
\(710\) −50.8484 −1.90830
\(711\) 1.00000 0.0375029
\(712\) 14.5438 0.545050
\(713\) 10.5483 0.395038
\(714\) 12.5652 0.470241
\(715\) 5.60794 0.209725
\(716\) −69.3597 −2.59209
\(717\) −23.0293 −0.860043
\(718\) 2.11297 0.0788555
\(719\) 45.6694 1.70318 0.851591 0.524207i \(-0.175638\pi\)
0.851591 + 0.524207i \(0.175638\pi\)
\(720\) −25.5201 −0.951078
\(721\) −30.1056 −1.12119
\(722\) 42.1913 1.57020
\(723\) 8.26243 0.307283
\(724\) 91.2343 3.39070
\(725\) 0.0267623 0.000993926 0
\(726\) −37.6286 −1.39653
\(727\) 29.0454 1.07724 0.538618 0.842550i \(-0.318947\pi\)
0.538618 + 0.842550i \(0.318947\pi\)
\(728\) −15.1276 −0.560667
\(729\) 1.00000 0.0370370
\(730\) −34.1719 −1.26476
\(731\) −2.19176 −0.0810653
\(732\) −61.2562 −2.26409
\(733\) 41.0333 1.51560 0.757800 0.652487i \(-0.226274\pi\)
0.757800 + 0.652487i \(0.226274\pi\)
\(734\) −13.8560 −0.511433
\(735\) −42.2089 −1.55690
\(736\) −23.3772 −0.861697
\(737\) −23.2488 −0.856379
\(738\) −27.9994 −1.03067
\(739\) −14.5607 −0.535626 −0.267813 0.963471i \(-0.586301\pi\)
−0.267813 + 0.963471i \(0.586301\pi\)
\(740\) 141.072 5.18589
\(741\) 0.718119 0.0263808
\(742\) −17.8488 −0.655251
\(743\) 12.0311 0.441379 0.220690 0.975344i \(-0.429169\pi\)
0.220690 + 0.975344i \(0.429169\pi\)
\(744\) −35.4841 −1.30091
\(745\) 9.00124 0.329780
\(746\) 6.53704 0.239338
\(747\) 4.67216 0.170946
\(748\) 24.3767 0.891300
\(749\) −72.5242 −2.64997
\(750\) −21.3911 −0.781093
\(751\) 15.0478 0.549102 0.274551 0.961573i \(-0.411471\pi\)
0.274551 + 0.961573i \(0.411471\pi\)
\(752\) −112.776 −4.11253
\(753\) 11.8050 0.430200
\(754\) 0.0157790 0.000574639 0
\(755\) −32.1338 −1.16947
\(756\) −23.2464 −0.845465
\(757\) 42.6301 1.54942 0.774709 0.632318i \(-0.217896\pi\)
0.774709 + 0.632318i \(0.217896\pi\)
\(758\) −51.3858 −1.86642
\(759\) 11.1150 0.403450
\(760\) −32.9492 −1.19519
\(761\) 47.5086 1.72219 0.861093 0.508447i \(-0.169780\pi\)
0.861093 + 0.508447i \(0.169780\pi\)
\(762\) −19.5909 −0.709705
\(763\) 6.43046 0.232798
\(764\) 3.93767 0.142460
\(765\) −2.62328 −0.0948449
\(766\) −61.4656 −2.22084
\(767\) 1.57202 0.0567625
\(768\) −15.4862 −0.558808
\(769\) −2.73285 −0.0985492 −0.0492746 0.998785i \(-0.515691\pi\)
−0.0492746 + 0.998785i \(0.515691\pi\)
\(770\) −166.091 −5.98550
\(771\) 23.7690 0.856020
\(772\) 73.9059 2.65993
\(773\) −43.1338 −1.55142 −0.775708 0.631092i \(-0.782607\pi\)
−0.775708 + 0.631092i \(0.782607\pi\)
\(774\) 5.73126 0.206006
\(775\) 8.99768 0.323206
\(776\) 103.156 3.70310
\(777\) 53.4152 1.91626
\(778\) 48.5537 1.74073
\(779\) −18.1243 −0.649372
\(780\) 5.38413 0.192783
\(781\) −37.3514 −1.33654
\(782\) −5.76814 −0.206268
\(783\) 0.0142232 0.000508295 0
\(784\) 156.530 5.59035
\(785\) −8.46837 −0.302249
\(786\) 23.6165 0.842372
\(787\) 4.42299 0.157662 0.0788312 0.996888i \(-0.474881\pi\)
0.0788312 + 0.996888i \(0.474881\pi\)
\(788\) 90.3518 3.21865
\(789\) 27.3452 0.973515
\(790\) 6.85964 0.244055
\(791\) 53.7829 1.91230
\(792\) −37.3906 −1.32862
\(793\) 5.37197 0.190764
\(794\) 29.5467 1.04857
\(795\) 3.72636 0.132160
\(796\) −2.89217 −0.102510
\(797\) −41.7670 −1.47946 −0.739731 0.672903i \(-0.765047\pi\)
−0.739731 + 0.672903i \(0.765047\pi\)
\(798\) −21.2686 −0.752901
\(799\) −11.5926 −0.410116
\(800\) −19.9407 −0.705010
\(801\) −1.95995 −0.0692516
\(802\) −33.8965 −1.19693
\(803\) −25.1015 −0.885813
\(804\) −22.3209 −0.787197
\(805\) 27.8059 0.980030
\(806\) 5.30503 0.186862
\(807\) −21.3647 −0.752073
\(808\) −133.108 −4.68272
\(809\) 44.1029 1.55058 0.775288 0.631607i \(-0.217605\pi\)
0.775288 + 0.631607i \(0.217605\pi\)
\(810\) 6.85964 0.241023
\(811\) −31.3037 −1.09922 −0.549611 0.835421i \(-0.685224\pi\)
−0.549611 + 0.835421i \(0.685224\pi\)
\(812\) −0.330638 −0.0116031
\(813\) −10.5082 −0.368540
\(814\) 146.467 5.13366
\(815\) −32.3195 −1.13210
\(816\) 9.72832 0.340559
\(817\) 3.70991 0.129793
\(818\) −84.8180 −2.96559
\(819\) 2.03864 0.0712358
\(820\) −135.888 −4.74541
\(821\) 29.8272 1.04097 0.520487 0.853869i \(-0.325750\pi\)
0.520487 + 0.853869i \(0.325750\pi\)
\(822\) −45.9460 −1.60255
\(823\) −20.8186 −0.725690 −0.362845 0.931849i \(-0.618195\pi\)
−0.362845 + 0.931849i \(0.618195\pi\)
\(824\) −46.4906 −1.61958
\(825\) 9.48109 0.330089
\(826\) −46.5588 −1.61999
\(827\) 29.7121 1.03319 0.516595 0.856230i \(-0.327199\pi\)
0.516595 + 0.856230i \(0.327199\pi\)
\(828\) 10.6714 0.370858
\(829\) 38.3261 1.33112 0.665560 0.746345i \(-0.268193\pi\)
0.665560 + 0.746345i \(0.268193\pi\)
\(830\) 32.0494 1.11245
\(831\) −1.59108 −0.0551940
\(832\) −3.50245 −0.121426
\(833\) 16.0901 0.557490
\(834\) 18.7419 0.648977
\(835\) 31.8259 1.10138
\(836\) −41.2614 −1.42706
\(837\) 4.78194 0.165288
\(838\) −25.9399 −0.896079
\(839\) 56.9351 1.96562 0.982809 0.184625i \(-0.0591069\pi\)
0.982809 + 0.184625i \(0.0591069\pi\)
\(840\) −93.5381 −3.22737
\(841\) −28.9998 −0.999993
\(842\) −19.2929 −0.664876
\(843\) −26.3797 −0.908567
\(844\) 97.9543 3.37173
\(845\) 33.6305 1.15692
\(846\) 30.3135 1.04220
\(847\) −69.1472 −2.37593
\(848\) −13.8190 −0.474548
\(849\) 17.1406 0.588263
\(850\) −4.92021 −0.168762
\(851\) −24.5206 −0.840555
\(852\) −35.8607 −1.22857
\(853\) −23.8274 −0.815833 −0.407917 0.913019i \(-0.633745\pi\)
−0.407917 + 0.913019i \(0.633745\pi\)
\(854\) −159.102 −5.44436
\(855\) 4.44032 0.151856
\(856\) −111.995 −3.82792
\(857\) 47.9327 1.63735 0.818674 0.574258i \(-0.194710\pi\)
0.818674 + 0.574258i \(0.194710\pi\)
\(858\) 5.59005 0.190841
\(859\) −14.4560 −0.493233 −0.246616 0.969113i \(-0.579319\pi\)
−0.246616 + 0.969113i \(0.579319\pi\)
\(860\) 27.8152 0.948489
\(861\) −51.4524 −1.75349
\(862\) −82.7167 −2.81734
\(863\) 20.3483 0.692664 0.346332 0.938112i \(-0.387427\pi\)
0.346332 + 0.938112i \(0.387427\pi\)
\(864\) −10.5978 −0.360543
\(865\) −57.1052 −1.94163
\(866\) −55.4638 −1.88474
\(867\) 1.00000 0.0339618
\(868\) −111.163 −3.77312
\(869\) 5.03885 0.170931
\(870\) 0.0975658 0.00330779
\(871\) 1.95747 0.0663264
\(872\) 9.93023 0.336280
\(873\) −13.9016 −0.470498
\(874\) 9.76350 0.330255
\(875\) −39.3088 −1.32888
\(876\) −24.0997 −0.814253
\(877\) −46.9534 −1.58550 −0.792751 0.609545i \(-0.791352\pi\)
−0.792751 + 0.609545i \(0.791352\pi\)
\(878\) 31.8639 1.07535
\(879\) 15.7659 0.531769
\(880\) −128.592 −4.33484
\(881\) 9.76965 0.329148 0.164574 0.986365i \(-0.447375\pi\)
0.164574 + 0.986365i \(0.447375\pi\)
\(882\) −42.0742 −1.41671
\(883\) −16.0773 −0.541043 −0.270522 0.962714i \(-0.587196\pi\)
−0.270522 + 0.962714i \(0.587196\pi\)
\(884\) −2.05244 −0.0690311
\(885\) 9.72023 0.326742
\(886\) −70.7037 −2.37534
\(887\) 27.6320 0.927791 0.463895 0.885890i \(-0.346451\pi\)
0.463895 + 0.885890i \(0.346451\pi\)
\(888\) 82.4863 2.76806
\(889\) −36.0008 −1.20743
\(890\) −13.4446 −0.450663
\(891\) 5.03885 0.168808
\(892\) −43.7710 −1.46556
\(893\) 19.6223 0.656634
\(894\) 8.97251 0.300086
\(895\) 37.6105 1.25718
\(896\) 1.88338 0.0629193
\(897\) −0.935851 −0.0312472
\(898\) 94.9517 3.16858
\(899\) 0.0680143 0.00226840
\(900\) 9.10270 0.303423
\(901\) −1.42050 −0.0473236
\(902\) −141.085 −4.69762
\(903\) 10.5319 0.350480
\(904\) 83.0543 2.76234
\(905\) −49.4720 −1.64451
\(906\) −32.0313 −1.06417
\(907\) −23.6067 −0.783847 −0.391923 0.919998i \(-0.628190\pi\)
−0.391923 + 0.919998i \(0.628190\pi\)
\(908\) 73.1753 2.42841
\(909\) 17.9380 0.594965
\(910\) 13.9843 0.463576
\(911\) 11.7648 0.389786 0.194893 0.980824i \(-0.437564\pi\)
0.194893 + 0.980824i \(0.437564\pi\)
\(912\) −16.4667 −0.545268
\(913\) 23.5423 0.779138
\(914\) −75.9853 −2.51337
\(915\) 33.2163 1.09810
\(916\) 55.5014 1.83382
\(917\) 43.3982 1.43313
\(918\) −2.61491 −0.0863049
\(919\) 21.2321 0.700381 0.350190 0.936679i \(-0.386117\pi\)
0.350190 + 0.936679i \(0.386117\pi\)
\(920\) 42.9393 1.41567
\(921\) −2.76806 −0.0912105
\(922\) −77.4251 −2.54986
\(923\) 3.14487 0.103515
\(924\) −117.135 −3.85347
\(925\) −20.9160 −0.687713
\(926\) 16.4703 0.541247
\(927\) 6.26519 0.205776
\(928\) −0.150734 −0.00494808
\(929\) 3.36978 0.110559 0.0552794 0.998471i \(-0.482395\pi\)
0.0552794 + 0.998471i \(0.482395\pi\)
\(930\) 32.8023 1.07563
\(931\) −27.2351 −0.892594
\(932\) 32.3224 1.05875
\(933\) 19.7793 0.647544
\(934\) 46.4898 1.52119
\(935\) −13.2183 −0.432285
\(936\) 3.14817 0.102901
\(937\) −42.1364 −1.37654 −0.688268 0.725457i \(-0.741629\pi\)
−0.688268 + 0.725457i \(0.741629\pi\)
\(938\) −57.9746 −1.89294
\(939\) 20.6041 0.672391
\(940\) 147.119 4.79848
\(941\) 27.4126 0.893625 0.446813 0.894628i \(-0.352559\pi\)
0.446813 + 0.894628i \(0.352559\pi\)
\(942\) −8.44134 −0.275034
\(943\) 23.6196 0.769159
\(944\) −36.0470 −1.17323
\(945\) 12.6054 0.410055
\(946\) 28.8790 0.938937
\(947\) −9.86369 −0.320527 −0.160263 0.987074i \(-0.551234\pi\)
−0.160263 + 0.987074i \(0.551234\pi\)
\(948\) 4.83775 0.157123
\(949\) 2.11347 0.0686060
\(950\) 8.32823 0.270203
\(951\) −7.73561 −0.250844
\(952\) 35.6569 1.15565
\(953\) 51.1565 1.65712 0.828561 0.559899i \(-0.189160\pi\)
0.828561 + 0.559899i \(0.189160\pi\)
\(954\) 3.71447 0.120260
\(955\) −2.13521 −0.0690938
\(956\) −111.410 −3.60325
\(957\) 0.0716685 0.00231671
\(958\) 13.2989 0.429669
\(959\) −84.4315 −2.72644
\(960\) −21.6566 −0.698962
\(961\) −8.13309 −0.262358
\(962\) −12.3321 −0.397601
\(963\) 15.0928 0.486358
\(964\) 39.9716 1.28740
\(965\) −40.0756 −1.29008
\(966\) 27.7172 0.891786
\(967\) −35.9081 −1.15473 −0.577364 0.816487i \(-0.695918\pi\)
−0.577364 + 0.816487i \(0.695918\pi\)
\(968\) −106.781 −3.43206
\(969\) −1.69266 −0.0543760
\(970\) −95.3600 −3.06183
\(971\) 1.48731 0.0477302 0.0238651 0.999715i \(-0.492403\pi\)
0.0238651 + 0.999715i \(0.492403\pi\)
\(972\) 4.83775 0.155171
\(973\) 34.4405 1.10411
\(974\) 42.4088 1.35887
\(975\) −0.798278 −0.0255654
\(976\) −123.181 −3.94293
\(977\) 56.9095 1.82070 0.910348 0.413844i \(-0.135814\pi\)
0.910348 + 0.413844i \(0.135814\pi\)
\(978\) −32.2163 −1.03017
\(979\) −9.87592 −0.315636
\(980\) −204.196 −6.52280
\(981\) −1.33822 −0.0427262
\(982\) −48.6988 −1.55404
\(983\) 27.8735 0.889026 0.444513 0.895772i \(-0.353377\pi\)
0.444513 + 0.895772i \(0.353377\pi\)
\(984\) −79.4554 −2.53294
\(985\) −48.9935 −1.56106
\(986\) −0.0371923 −0.00118444
\(987\) 55.7049 1.77311
\(988\) 3.47408 0.110525
\(989\) −4.83474 −0.153736
\(990\) 34.5647 1.09854
\(991\) −50.3605 −1.59975 −0.799877 0.600164i \(-0.795102\pi\)
−0.799877 + 0.600164i \(0.795102\pi\)
\(992\) −50.6778 −1.60902
\(993\) −13.1774 −0.418172
\(994\) −93.1420 −2.95428
\(995\) 1.56829 0.0497180
\(996\) 22.6028 0.716196
\(997\) −7.60988 −0.241007 −0.120504 0.992713i \(-0.538451\pi\)
−0.120504 + 0.992713i \(0.538451\pi\)
\(998\) −109.853 −3.47732
\(999\) −11.1161 −0.351697
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 4029.2.a.k.1.1 31
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4029.2.a.k.1.1 31 1.1 even 1 trivial