Properties

Label 2842.2.a.p.1.4
Level $2842$
Weight $2$
Character 2842.1
Self dual yes
Analytic conductor $22.693$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2842.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.6934842544\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.9248.1
Defining polynomial: \( x^{4} - 5x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.662153\) of defining polynomial
Character \(\chi\) \(=\) 2842.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.35829 q^{3} +1.00000 q^{4} -0.662153 q^{5} -2.35829 q^{6} -1.00000 q^{8} +2.56155 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.35829 q^{3} +1.00000 q^{4} -0.662153 q^{5} -2.35829 q^{6} -1.00000 q^{8} +2.56155 q^{9} +0.662153 q^{10} -0.438447 q^{11} +2.35829 q^{12} -4.05444 q^{13} -1.56155 q^{15} +1.00000 q^{16} -4.34475 q^{17} -2.56155 q^{18} +7.36520 q^{19} -0.662153 q^{20} +0.438447 q^{22} -8.24621 q^{23} -2.35829 q^{24} -4.56155 q^{25} +4.05444 q^{26} -1.03399 q^{27} +1.00000 q^{29} +1.56155 q^{30} -4.05444 q^{31} -1.00000 q^{32} -1.03399 q^{33} +4.34475 q^{34} +2.56155 q^{36} +7.12311 q^{37} -7.36520 q^{38} -9.56155 q^{39} +0.662153 q^{40} -4.34475 q^{41} -4.68466 q^{43} -0.438447 q^{44} -1.69614 q^{45} +8.24621 q^{46} -0.662153 q^{47} +2.35829 q^{48} +4.56155 q^{50} -10.2462 q^{51} -4.05444 q^{52} -11.5616 q^{53} +1.03399 q^{54} +0.290319 q^{55} +17.3693 q^{57} -1.00000 q^{58} +4.34475 q^{59} -1.56155 q^{60} +1.32431 q^{61} +4.05444 q^{62} +1.00000 q^{64} +2.68466 q^{65} +1.03399 q^{66} -4.87689 q^{67} -4.34475 q^{68} -19.4470 q^{69} +11.3693 q^{71} -2.56155 q^{72} -8.48071 q^{73} -7.12311 q^{74} -10.7575 q^{75} +7.36520 q^{76} +9.56155 q^{78} +7.80776 q^{79} -0.662153 q^{80} -10.1231 q^{81} +4.34475 q^{82} -5.08842 q^{83} +2.87689 q^{85} +4.68466 q^{86} +2.35829 q^{87} +0.438447 q^{88} -9.06134 q^{89} +1.69614 q^{90} -8.24621 q^{92} -9.56155 q^{93} +0.662153 q^{94} -4.87689 q^{95} -2.35829 q^{96} -6.99337 q^{97} -1.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 2 q^{9} - 10 q^{11} + 2 q^{15} + 4 q^{16} - 2 q^{18} + 10 q^{22} - 10 q^{25} + 4 q^{29} - 2 q^{30} - 4 q^{32} + 2 q^{36} + 12 q^{37} - 30 q^{39} + 6 q^{43} - 10 q^{44} + 10 q^{50} - 8 q^{51} - 38 q^{53} + 20 q^{57} - 4 q^{58} + 2 q^{60} + 4 q^{64} - 14 q^{65} - 36 q^{67} - 4 q^{71} - 2 q^{72} - 12 q^{74} + 30 q^{78} - 10 q^{79} - 24 q^{81} + 28 q^{85} - 6 q^{86} + 10 q^{88} - 30 q^{93} - 36 q^{95} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.35829 1.36156 0.680781 0.732487i \(-0.261641\pi\)
0.680781 + 0.732487i \(0.261641\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.662153 −0.296124 −0.148062 0.988978i \(-0.547304\pi\)
−0.148062 + 0.988978i \(0.547304\pi\)
\(6\) −2.35829 −0.962770
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 2.56155 0.853851
\(10\) 0.662153 0.209391
\(11\) −0.438447 −0.132197 −0.0660984 0.997813i \(-0.521055\pi\)
−0.0660984 + 0.997813i \(0.521055\pi\)
\(12\) 2.35829 0.680781
\(13\) −4.05444 −1.12450 −0.562249 0.826968i \(-0.690064\pi\)
−0.562249 + 0.826968i \(0.690064\pi\)
\(14\) 0 0
\(15\) −1.56155 −0.403191
\(16\) 1.00000 0.250000
\(17\) −4.34475 −1.05376 −0.526879 0.849940i \(-0.676638\pi\)
−0.526879 + 0.849940i \(0.676638\pi\)
\(18\) −2.56155 −0.603764
\(19\) 7.36520 1.68969 0.844847 0.535008i \(-0.179692\pi\)
0.844847 + 0.535008i \(0.179692\pi\)
\(20\) −0.662153 −0.148062
\(21\) 0 0
\(22\) 0.438447 0.0934773
\(23\) −8.24621 −1.71945 −0.859727 0.510754i \(-0.829366\pi\)
−0.859727 + 0.510754i \(0.829366\pi\)
\(24\) −2.35829 −0.481385
\(25\) −4.56155 −0.912311
\(26\) 4.05444 0.795140
\(27\) −1.03399 −0.198991
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 1.56155 0.285099
\(31\) −4.05444 −0.728198 −0.364099 0.931360i \(-0.618623\pi\)
−0.364099 + 0.931360i \(0.618623\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.03399 −0.179994
\(34\) 4.34475 0.745119
\(35\) 0 0
\(36\) 2.56155 0.426925
\(37\) 7.12311 1.17103 0.585516 0.810661i \(-0.300892\pi\)
0.585516 + 0.810661i \(0.300892\pi\)
\(38\) −7.36520 −1.19479
\(39\) −9.56155 −1.53107
\(40\) 0.662153 0.104696
\(41\) −4.34475 −0.678537 −0.339268 0.940690i \(-0.610179\pi\)
−0.339268 + 0.940690i \(0.610179\pi\)
\(42\) 0 0
\(43\) −4.68466 −0.714404 −0.357202 0.934027i \(-0.616269\pi\)
−0.357202 + 0.934027i \(0.616269\pi\)
\(44\) −0.438447 −0.0660984
\(45\) −1.69614 −0.252846
\(46\) 8.24621 1.21584
\(47\) −0.662153 −0.0965850 −0.0482925 0.998833i \(-0.515378\pi\)
−0.0482925 + 0.998833i \(0.515378\pi\)
\(48\) 2.35829 0.340390
\(49\) 0 0
\(50\) 4.56155 0.645101
\(51\) −10.2462 −1.43476
\(52\) −4.05444 −0.562249
\(53\) −11.5616 −1.58810 −0.794051 0.607852i \(-0.792032\pi\)
−0.794051 + 0.607852i \(0.792032\pi\)
\(54\) 1.03399 0.140708
\(55\) 0.290319 0.0391466
\(56\) 0 0
\(57\) 17.3693 2.30062
\(58\) −1.00000 −0.131306
\(59\) 4.34475 0.565639 0.282819 0.959173i \(-0.408730\pi\)
0.282819 + 0.959173i \(0.408730\pi\)
\(60\) −1.56155 −0.201596
\(61\) 1.32431 0.169560 0.0847801 0.996400i \(-0.472981\pi\)
0.0847801 + 0.996400i \(0.472981\pi\)
\(62\) 4.05444 0.514914
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.68466 0.332991
\(66\) 1.03399 0.127275
\(67\) −4.87689 −0.595807 −0.297904 0.954596i \(-0.596287\pi\)
−0.297904 + 0.954596i \(0.596287\pi\)
\(68\) −4.34475 −0.526879
\(69\) −19.4470 −2.34114
\(70\) 0 0
\(71\) 11.3693 1.34929 0.674645 0.738142i \(-0.264297\pi\)
0.674645 + 0.738142i \(0.264297\pi\)
\(72\) −2.56155 −0.301882
\(73\) −8.48071 −0.992591 −0.496296 0.868154i \(-0.665307\pi\)
−0.496296 + 0.868154i \(0.665307\pi\)
\(74\) −7.12311 −0.828044
\(75\) −10.7575 −1.24217
\(76\) 7.36520 0.844847
\(77\) 0 0
\(78\) 9.56155 1.08263
\(79\) 7.80776 0.878442 0.439221 0.898379i \(-0.355254\pi\)
0.439221 + 0.898379i \(0.355254\pi\)
\(80\) −0.662153 −0.0740310
\(81\) −10.1231 −1.12479
\(82\) 4.34475 0.479798
\(83\) −5.08842 −0.558527 −0.279263 0.960215i \(-0.590090\pi\)
−0.279263 + 0.960215i \(0.590090\pi\)
\(84\) 0 0
\(85\) 2.87689 0.312043
\(86\) 4.68466 0.505160
\(87\) 2.35829 0.252836
\(88\) 0.438447 0.0467386
\(89\) −9.06134 −0.960501 −0.480250 0.877132i \(-0.659454\pi\)
−0.480250 + 0.877132i \(0.659454\pi\)
\(90\) 1.69614 0.178789
\(91\) 0 0
\(92\) −8.24621 −0.859727
\(93\) −9.56155 −0.991487
\(94\) 0.662153 0.0682959
\(95\) −4.87689 −0.500359
\(96\) −2.35829 −0.240692
\(97\) −6.99337 −0.710069 −0.355034 0.934853i \(-0.615531\pi\)
−0.355034 + 0.934853i \(0.615531\pi\)
\(98\) 0 0
\(99\) −1.12311 −0.112876
\(100\) −4.56155 −0.456155
\(101\) 11.3381 1.12819 0.564093 0.825711i \(-0.309226\pi\)
0.564093 + 0.825711i \(0.309226\pi\)
\(102\) 10.2462 1.01453
\(103\) 12.8255 1.26373 0.631865 0.775078i \(-0.282290\pi\)
0.631865 + 0.775078i \(0.282290\pi\)
\(104\) 4.05444 0.397570
\(105\) 0 0
\(106\) 11.5616 1.12296
\(107\) −0.876894 −0.0847726 −0.0423863 0.999101i \(-0.513496\pi\)
−0.0423863 + 0.999101i \(0.513496\pi\)
\(108\) −1.03399 −0.0994955
\(109\) 10.6847 1.02340 0.511702 0.859163i \(-0.329015\pi\)
0.511702 + 0.859163i \(0.329015\pi\)
\(110\) −0.290319 −0.0276809
\(111\) 16.7984 1.59443
\(112\) 0 0
\(113\) −19.3693 −1.82211 −0.911056 0.412283i \(-0.864732\pi\)
−0.911056 + 0.412283i \(0.864732\pi\)
\(114\) −17.3693 −1.62679
\(115\) 5.46026 0.509172
\(116\) 1.00000 0.0928477
\(117\) −10.3857 −0.960154
\(118\) −4.34475 −0.399967
\(119\) 0 0
\(120\) 1.56155 0.142550
\(121\) −10.8078 −0.982524
\(122\) −1.32431 −0.119897
\(123\) −10.2462 −0.923870
\(124\) −4.05444 −0.364099
\(125\) 6.33122 0.566281
\(126\) 0 0
\(127\) −2.24621 −0.199319 −0.0996595 0.995022i \(-0.531775\pi\)
−0.0996595 + 0.995022i \(0.531775\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −11.0478 −0.972705
\(130\) −2.68466 −0.235460
\(131\) −2.06798 −0.180680 −0.0903399 0.995911i \(-0.528795\pi\)
−0.0903399 + 0.995911i \(0.528795\pi\)
\(132\) −1.03399 −0.0899971
\(133\) 0 0
\(134\) 4.87689 0.421300
\(135\) 0.684658 0.0589260
\(136\) 4.34475 0.372560
\(137\) −8.24621 −0.704521 −0.352261 0.935902i \(-0.614587\pi\)
−0.352261 + 0.935902i \(0.614587\pi\)
\(138\) 19.4470 1.65544
\(139\) −8.31768 −0.705496 −0.352748 0.935718i \(-0.614753\pi\)
−0.352748 + 0.935718i \(0.614753\pi\)
\(140\) 0 0
\(141\) −1.56155 −0.131506
\(142\) −11.3693 −0.954092
\(143\) 1.77766 0.148655
\(144\) 2.56155 0.213463
\(145\) −0.662153 −0.0549889
\(146\) 8.48071 0.701868
\(147\) 0 0
\(148\) 7.12311 0.585516
\(149\) 2.68466 0.219936 0.109968 0.993935i \(-0.464925\pi\)
0.109968 + 0.993935i \(0.464925\pi\)
\(150\) 10.7575 0.878345
\(151\) 4.24621 0.345552 0.172776 0.984961i \(-0.444726\pi\)
0.172776 + 0.984961i \(0.444726\pi\)
\(152\) −7.36520 −0.597397
\(153\) −11.1293 −0.899752
\(154\) 0 0
\(155\) 2.68466 0.215637
\(156\) −9.56155 −0.765537
\(157\) −17.3790 −1.38700 −0.693498 0.720458i \(-0.743932\pi\)
−0.693498 + 0.720458i \(0.743932\pi\)
\(158\) −7.80776 −0.621152
\(159\) −27.2655 −2.16230
\(160\) 0.662153 0.0523478
\(161\) 0 0
\(162\) 10.1231 0.795346
\(163\) −4.43845 −0.347646 −0.173823 0.984777i \(-0.555612\pi\)
−0.173823 + 0.984777i \(0.555612\pi\)
\(164\) −4.34475 −0.339268
\(165\) 0.684658 0.0533006
\(166\) 5.08842 0.394938
\(167\) 9.43318 0.729961 0.364981 0.931015i \(-0.381076\pi\)
0.364981 + 0.931015i \(0.381076\pi\)
\(168\) 0 0
\(169\) 3.43845 0.264496
\(170\) −2.87689 −0.220648
\(171\) 18.8664 1.44275
\(172\) −4.68466 −0.357202
\(173\) 11.1293 0.846146 0.423073 0.906095i \(-0.360951\pi\)
0.423073 + 0.906095i \(0.360951\pi\)
\(174\) −2.35829 −0.178782
\(175\) 0 0
\(176\) −0.438447 −0.0330492
\(177\) 10.2462 0.770152
\(178\) 9.06134 0.679176
\(179\) −1.75379 −0.131084 −0.0655422 0.997850i \(-0.520878\pi\)
−0.0655422 + 0.997850i \(0.520878\pi\)
\(180\) −1.69614 −0.126423
\(181\) 22.1771 1.64841 0.824206 0.566290i \(-0.191622\pi\)
0.824206 + 0.566290i \(0.191622\pi\)
\(182\) 0 0
\(183\) 3.12311 0.230867
\(184\) 8.24621 0.607919
\(185\) −4.71659 −0.346771
\(186\) 9.56155 0.701087
\(187\) 1.90495 0.139303
\(188\) −0.662153 −0.0482925
\(189\) 0 0
\(190\) 4.87689 0.353807
\(191\) −10.2462 −0.741390 −0.370695 0.928755i \(-0.620880\pi\)
−0.370695 + 0.928755i \(0.620880\pi\)
\(192\) 2.35829 0.170195
\(193\) −3.36932 −0.242529 −0.121264 0.992620i \(-0.538695\pi\)
−0.121264 + 0.992620i \(0.538695\pi\)
\(194\) 6.99337 0.502095
\(195\) 6.33122 0.453388
\(196\) 0 0
\(197\) −3.75379 −0.267446 −0.133723 0.991019i \(-0.542693\pi\)
−0.133723 + 0.991019i \(0.542693\pi\)
\(198\) 1.12311 0.0798156
\(199\) 20.7713 1.47244 0.736219 0.676743i \(-0.236609\pi\)
0.736219 + 0.676743i \(0.236609\pi\)
\(200\) 4.56155 0.322550
\(201\) −11.5012 −0.811229
\(202\) −11.3381 −0.797748
\(203\) 0 0
\(204\) −10.2462 −0.717378
\(205\) 2.87689 0.200931
\(206\) −12.8255 −0.893592
\(207\) −21.1231 −1.46816
\(208\) −4.05444 −0.281125
\(209\) −3.22925 −0.223372
\(210\) 0 0
\(211\) 4.68466 0.322505 0.161253 0.986913i \(-0.448447\pi\)
0.161253 + 0.986913i \(0.448447\pi\)
\(212\) −11.5616 −0.794051
\(213\) 26.8122 1.83714
\(214\) 0.876894 0.0599433
\(215\) 3.10196 0.211552
\(216\) 1.03399 0.0703539
\(217\) 0 0
\(218\) −10.6847 −0.723656
\(219\) −20.0000 −1.35147
\(220\) 0.290319 0.0195733
\(221\) 17.6155 1.18495
\(222\) −16.7984 −1.12743
\(223\) 3.39228 0.227164 0.113582 0.993529i \(-0.463768\pi\)
0.113582 + 0.993529i \(0.463768\pi\)
\(224\) 0 0
\(225\) −11.6847 −0.778977
\(226\) 19.3693 1.28843
\(227\) 12.4536 0.826576 0.413288 0.910600i \(-0.364380\pi\)
0.413288 + 0.910600i \(0.364380\pi\)
\(228\) 17.3693 1.15031
\(229\) 23.5829 1.55840 0.779202 0.626772i \(-0.215624\pi\)
0.779202 + 0.626772i \(0.215624\pi\)
\(230\) −5.46026 −0.360039
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) −12.9309 −0.847129 −0.423565 0.905866i \(-0.639221\pi\)
−0.423565 + 0.905866i \(0.639221\pi\)
\(234\) 10.3857 0.678931
\(235\) 0.438447 0.0286011
\(236\) 4.34475 0.282819
\(237\) 18.4130 1.19605
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.56155 −0.100798
\(241\) −24.6169 −1.58572 −0.792858 0.609406i \(-0.791408\pi\)
−0.792858 + 0.609406i \(0.791408\pi\)
\(242\) 10.8078 0.694749
\(243\) −20.7713 −1.33248
\(244\) 1.32431 0.0847801
\(245\) 0 0
\(246\) 10.2462 0.653275
\(247\) −29.8617 −1.90006
\(248\) 4.05444 0.257457
\(249\) −12.0000 −0.760469
\(250\) −6.33122 −0.400421
\(251\) 22.5490 1.42328 0.711639 0.702546i \(-0.247953\pi\)
0.711639 + 0.702546i \(0.247953\pi\)
\(252\) 0 0
\(253\) 3.61553 0.227306
\(254\) 2.24621 0.140940
\(255\) 6.78456 0.424866
\(256\) 1.00000 0.0625000
\(257\) 13.8594 0.864529 0.432264 0.901747i \(-0.357715\pi\)
0.432264 + 0.901747i \(0.357715\pi\)
\(258\) 11.0478 0.687806
\(259\) 0 0
\(260\) 2.68466 0.166495
\(261\) 2.56155 0.158556
\(262\) 2.06798 0.127760
\(263\) −15.8078 −0.974748 −0.487374 0.873193i \(-0.662045\pi\)
−0.487374 + 0.873193i \(0.662045\pi\)
\(264\) 1.03399 0.0636375
\(265\) 7.65552 0.470275
\(266\) 0 0
\(267\) −21.3693 −1.30778
\(268\) −4.87689 −0.297904
\(269\) 24.9073 1.51862 0.759311 0.650728i \(-0.225536\pi\)
0.759311 + 0.650728i \(0.225536\pi\)
\(270\) −0.684658 −0.0416670
\(271\) −12.0003 −0.728965 −0.364482 0.931210i \(-0.618754\pi\)
−0.364482 + 0.931210i \(0.618754\pi\)
\(272\) −4.34475 −0.263439
\(273\) 0 0
\(274\) 8.24621 0.498172
\(275\) 2.00000 0.120605
\(276\) −19.4470 −1.17057
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 8.31768 0.498861
\(279\) −10.3857 −0.621773
\(280\) 0 0
\(281\) −29.1771 −1.74056 −0.870279 0.492558i \(-0.836062\pi\)
−0.870279 + 0.492558i \(0.836062\pi\)
\(282\) 1.56155 0.0929891
\(283\) 10.5487 0.627054 0.313527 0.949579i \(-0.398489\pi\)
0.313527 + 0.949579i \(0.398489\pi\)
\(284\) 11.3693 0.674645
\(285\) −11.5012 −0.681270
\(286\) −1.77766 −0.105115
\(287\) 0 0
\(288\) −2.56155 −0.150941
\(289\) 1.87689 0.110406
\(290\) 0.662153 0.0388830
\(291\) −16.4924 −0.966803
\(292\) −8.48071 −0.496296
\(293\) −11.5012 −0.671905 −0.335952 0.941879i \(-0.609058\pi\)
−0.335952 + 0.941879i \(0.609058\pi\)
\(294\) 0 0
\(295\) −2.87689 −0.167499
\(296\) −7.12311 −0.414022
\(297\) 0.453349 0.0263060
\(298\) −2.68466 −0.155518
\(299\) 33.4337 1.93352
\(300\) −10.7575 −0.621084
\(301\) 0 0
\(302\) −4.24621 −0.244342
\(303\) 26.7386 1.53609
\(304\) 7.36520 0.422423
\(305\) −0.876894 −0.0502108
\(306\) 11.1293 0.636221
\(307\) 11.2108 0.639836 0.319918 0.947445i \(-0.396345\pi\)
0.319918 + 0.947445i \(0.396345\pi\)
\(308\) 0 0
\(309\) 30.2462 1.72065
\(310\) −2.68466 −0.152478
\(311\) −9.80501 −0.555991 −0.277996 0.960582i \(-0.589670\pi\)
−0.277996 + 0.960582i \(0.589670\pi\)
\(312\) 9.56155 0.541316
\(313\) 27.2655 1.54114 0.770570 0.637356i \(-0.219972\pi\)
0.770570 + 0.637356i \(0.219972\pi\)
\(314\) 17.3790 0.980755
\(315\) 0 0
\(316\) 7.80776 0.439221
\(317\) 14.4924 0.813976 0.406988 0.913434i \(-0.366579\pi\)
0.406988 + 0.913434i \(0.366579\pi\)
\(318\) 27.2655 1.52898
\(319\) −0.438447 −0.0245483
\(320\) −0.662153 −0.0370155
\(321\) −2.06798 −0.115423
\(322\) 0 0
\(323\) −32.0000 −1.78053
\(324\) −10.1231 −0.562395
\(325\) 18.4945 1.02589
\(326\) 4.43845 0.245823
\(327\) 25.1976 1.39343
\(328\) 4.34475 0.239899
\(329\) 0 0
\(330\) −0.684658 −0.0376892
\(331\) 36.3002 1.99524 0.997619 0.0689611i \(-0.0219684\pi\)
0.997619 + 0.0689611i \(0.0219684\pi\)
\(332\) −5.08842 −0.279263
\(333\) 18.2462 0.999886
\(334\) −9.43318 −0.516161
\(335\) 3.22925 0.176433
\(336\) 0 0
\(337\) −24.2462 −1.32078 −0.660388 0.750925i \(-0.729608\pi\)
−0.660388 + 0.750925i \(0.729608\pi\)
\(338\) −3.43845 −0.187027
\(339\) −45.6786 −2.48092
\(340\) 2.87689 0.156022
\(341\) 1.77766 0.0962655
\(342\) −18.8664 −1.02018
\(343\) 0 0
\(344\) 4.68466 0.252580
\(345\) 12.8769 0.693269
\(346\) −11.1293 −0.598316
\(347\) −31.1231 −1.67078 −0.835388 0.549661i \(-0.814757\pi\)
−0.835388 + 0.549661i \(0.814757\pi\)
\(348\) 2.35829 0.126418
\(349\) −22.9208 −1.22692 −0.613461 0.789725i \(-0.710223\pi\)
−0.613461 + 0.789725i \(0.710223\pi\)
\(350\) 0 0
\(351\) 4.19224 0.223765
\(352\) 0.438447 0.0233693
\(353\) −9.43318 −0.502077 −0.251039 0.967977i \(-0.580772\pi\)
−0.251039 + 0.967977i \(0.580772\pi\)
\(354\) −10.2462 −0.544580
\(355\) −7.52823 −0.399557
\(356\) −9.06134 −0.480250
\(357\) 0 0
\(358\) 1.75379 0.0926906
\(359\) −28.6847 −1.51392 −0.756959 0.653462i \(-0.773316\pi\)
−0.756959 + 0.653462i \(0.773316\pi\)
\(360\) 1.69614 0.0893945
\(361\) 35.2462 1.85506
\(362\) −22.1771 −1.16560
\(363\) −25.4879 −1.33777
\(364\) 0 0
\(365\) 5.61553 0.293930
\(366\) −3.12311 −0.163247
\(367\) −0.371834 −0.0194096 −0.00970479 0.999953i \(-0.503089\pi\)
−0.00970479 + 0.999953i \(0.503089\pi\)
\(368\) −8.24621 −0.429863
\(369\) −11.1293 −0.579369
\(370\) 4.71659 0.245204
\(371\) 0 0
\(372\) −9.56155 −0.495743
\(373\) −21.8078 −1.12916 −0.564582 0.825377i \(-0.690962\pi\)
−0.564582 + 0.825377i \(0.690962\pi\)
\(374\) −1.90495 −0.0985024
\(375\) 14.9309 0.771027
\(376\) 0.662153 0.0341480
\(377\) −4.05444 −0.208814
\(378\) 0 0
\(379\) 32.4924 1.66902 0.834512 0.550990i \(-0.185750\pi\)
0.834512 + 0.550990i \(0.185750\pi\)
\(380\) −4.87689 −0.250179
\(381\) −5.29723 −0.271385
\(382\) 10.2462 0.524242
\(383\) −7.36520 −0.376344 −0.188172 0.982136i \(-0.560256\pi\)
−0.188172 + 0.982136i \(0.560256\pi\)
\(384\) −2.35829 −0.120346
\(385\) 0 0
\(386\) 3.36932 0.171494
\(387\) −12.0000 −0.609994
\(388\) −6.99337 −0.355034
\(389\) −32.9848 −1.67240 −0.836199 0.548426i \(-0.815227\pi\)
−0.836199 + 0.548426i \(0.815227\pi\)
\(390\) −6.33122 −0.320594
\(391\) 35.8278 1.81189
\(392\) 0 0
\(393\) −4.87689 −0.246007
\(394\) 3.75379 0.189113
\(395\) −5.16994 −0.260128
\(396\) −1.12311 −0.0564382
\(397\) 10.2584 0.514852 0.257426 0.966298i \(-0.417126\pi\)
0.257426 + 0.966298i \(0.417126\pi\)
\(398\) −20.7713 −1.04117
\(399\) 0 0
\(400\) −4.56155 −0.228078
\(401\) 14.1922 0.708726 0.354363 0.935108i \(-0.384698\pi\)
0.354363 + 0.935108i \(0.384698\pi\)
\(402\) 11.5012 0.573625
\(403\) 16.4384 0.818857
\(404\) 11.3381 0.564093
\(405\) 6.70305 0.333077
\(406\) 0 0
\(407\) −3.12311 −0.154807
\(408\) 10.2462 0.507263
\(409\) −29.9957 −1.48319 −0.741595 0.670848i \(-0.765931\pi\)
−0.741595 + 0.670848i \(0.765931\pi\)
\(410\) −2.87689 −0.142080
\(411\) −19.4470 −0.959249
\(412\) 12.8255 0.631865
\(413\) 0 0
\(414\) 21.1231 1.03814
\(415\) 3.36932 0.165393
\(416\) 4.05444 0.198785
\(417\) −19.6155 −0.960577
\(418\) 3.22925 0.157948
\(419\) −5.83209 −0.284916 −0.142458 0.989801i \(-0.545501\pi\)
−0.142458 + 0.989801i \(0.545501\pi\)
\(420\) 0 0
\(421\) −31.1231 −1.51685 −0.758424 0.651762i \(-0.774030\pi\)
−0.758424 + 0.651762i \(0.774030\pi\)
\(422\) −4.68466 −0.228046
\(423\) −1.69614 −0.0824692
\(424\) 11.5616 0.561479
\(425\) 19.8188 0.961354
\(426\) −26.8122 −1.29906
\(427\) 0 0
\(428\) −0.876894 −0.0423863
\(429\) 4.19224 0.202403
\(430\) −3.10196 −0.149590
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) −1.03399 −0.0497478
\(433\) 16.5896 0.797244 0.398622 0.917115i \(-0.369489\pi\)
0.398622 + 0.917115i \(0.369489\pi\)
\(434\) 0 0
\(435\) −1.56155 −0.0748707
\(436\) 10.6847 0.511702
\(437\) −60.7350 −2.90535
\(438\) 20.0000 0.955637
\(439\) −35.6647 −1.70218 −0.851092 0.525016i \(-0.824059\pi\)
−0.851092 + 0.525016i \(0.824059\pi\)
\(440\) −0.290319 −0.0138404
\(441\) 0 0
\(442\) −17.6155 −0.837885
\(443\) 33.1231 1.57373 0.786863 0.617128i \(-0.211704\pi\)
0.786863 + 0.617128i \(0.211704\pi\)
\(444\) 16.7984 0.797216
\(445\) 6.00000 0.284427
\(446\) −3.39228 −0.160629
\(447\) 6.33122 0.299456
\(448\) 0 0
\(449\) −19.3693 −0.914095 −0.457047 0.889442i \(-0.651093\pi\)
−0.457047 + 0.889442i \(0.651093\pi\)
\(450\) 11.6847 0.550820
\(451\) 1.90495 0.0897004
\(452\) −19.3693 −0.911056
\(453\) 10.0138 0.470490
\(454\) −12.4536 −0.584478
\(455\) 0 0
\(456\) −17.3693 −0.813393
\(457\) 29.3693 1.37384 0.686919 0.726734i \(-0.258963\pi\)
0.686919 + 0.726734i \(0.258963\pi\)
\(458\) −23.5829 −1.10196
\(459\) 4.49242 0.209688
\(460\) 5.46026 0.254586
\(461\) −23.4199 −1.09077 −0.545387 0.838184i \(-0.683617\pi\)
−0.545387 + 0.838184i \(0.683617\pi\)
\(462\) 0 0
\(463\) −23.3693 −1.08606 −0.543032 0.839712i \(-0.682724\pi\)
−0.543032 + 0.839712i \(0.682724\pi\)
\(464\) 1.00000 0.0464238
\(465\) 6.33122 0.293603
\(466\) 12.9309 0.599011
\(467\) −6.33122 −0.292974 −0.146487 0.989213i \(-0.546797\pi\)
−0.146487 + 0.989213i \(0.546797\pi\)
\(468\) −10.3857 −0.480077
\(469\) 0 0
\(470\) −0.438447 −0.0202241
\(471\) −40.9848 −1.88848
\(472\) −4.34475 −0.199984
\(473\) 2.05398 0.0944419
\(474\) −18.4130 −0.845737
\(475\) −33.5968 −1.54153
\(476\) 0 0
\(477\) −29.6155 −1.35600
\(478\) 0 0
\(479\) −40.2998 −1.84135 −0.920673 0.390336i \(-0.872359\pi\)
−0.920673 + 0.390336i \(0.872359\pi\)
\(480\) 1.56155 0.0712748
\(481\) −28.8802 −1.31682
\(482\) 24.6169 1.12127
\(483\) 0 0
\(484\) −10.8078 −0.491262
\(485\) 4.63068 0.210268
\(486\) 20.7713 0.942205
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −1.32431 −0.0599486
\(489\) −10.4672 −0.473342
\(490\) 0 0
\(491\) −25.1771 −1.13623 −0.568113 0.822951i \(-0.692326\pi\)
−0.568113 + 0.822951i \(0.692326\pi\)
\(492\) −10.2462 −0.461935
\(493\) −4.34475 −0.195678
\(494\) 29.8617 1.34354
\(495\) 0.743668 0.0334254
\(496\) −4.05444 −0.182050
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 3.12311 0.139809 0.0699047 0.997554i \(-0.477730\pi\)
0.0699047 + 0.997554i \(0.477730\pi\)
\(500\) 6.33122 0.283141
\(501\) 22.2462 0.993887
\(502\) −22.5490 −1.00641
\(503\) 33.6783 1.50164 0.750820 0.660507i \(-0.229659\pi\)
0.750820 + 0.660507i \(0.229659\pi\)
\(504\) 0 0
\(505\) −7.50758 −0.334083
\(506\) −3.61553 −0.160730
\(507\) 8.10887 0.360128
\(508\) −2.24621 −0.0996595
\(509\) 34.4219 1.52573 0.762863 0.646560i \(-0.223793\pi\)
0.762863 + 0.646560i \(0.223793\pi\)
\(510\) −6.78456 −0.300426
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −7.61553 −0.336234
\(514\) −13.8594 −0.611314
\(515\) −8.49242 −0.374221
\(516\) −11.0478 −0.486352
\(517\) 0.290319 0.0127682
\(518\) 0 0
\(519\) 26.2462 1.15208
\(520\) −2.68466 −0.117730
\(521\) −24.4539 −1.07134 −0.535672 0.844426i \(-0.679942\pi\)
−0.535672 + 0.844426i \(0.679942\pi\)
\(522\) −2.56155 −0.112116
\(523\) 43.8194 1.91609 0.958044 0.286621i \(-0.0925322\pi\)
0.958044 + 0.286621i \(0.0925322\pi\)
\(524\) −2.06798 −0.0903399
\(525\) 0 0
\(526\) 15.8078 0.689251
\(527\) 17.6155 0.767344
\(528\) −1.03399 −0.0449985
\(529\) 45.0000 1.95652
\(530\) −7.65552 −0.332535
\(531\) 11.1293 0.482971
\(532\) 0 0
\(533\) 17.6155 0.763013
\(534\) 21.3693 0.924741
\(535\) 0.580639 0.0251032
\(536\) 4.87689 0.210650
\(537\) −4.13595 −0.178479
\(538\) −24.9073 −1.07383
\(539\) 0 0
\(540\) 0.684658 0.0294630
\(541\) 9.12311 0.392233 0.196116 0.980581i \(-0.437167\pi\)
0.196116 + 0.980581i \(0.437167\pi\)
\(542\) 12.0003 0.515456
\(543\) 52.3002 2.24442
\(544\) 4.34475 0.186280
\(545\) −7.07488 −0.303055
\(546\) 0 0
\(547\) 7.61553 0.325616 0.162808 0.986658i \(-0.447945\pi\)
0.162808 + 0.986658i \(0.447945\pi\)
\(548\) −8.24621 −0.352261
\(549\) 3.39228 0.144779
\(550\) −2.00000 −0.0852803
\(551\) 7.36520 0.313768
\(552\) 19.4470 0.827719
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) −11.1231 −0.472150
\(556\) −8.31768 −0.352748
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 10.3857 0.439660
\(559\) 18.9936 0.803346
\(560\) 0 0
\(561\) 4.49242 0.189670
\(562\) 29.1771 1.23076
\(563\) 18.4130 0.776016 0.388008 0.921656i \(-0.373163\pi\)
0.388008 + 0.921656i \(0.373163\pi\)
\(564\) −1.56155 −0.0657532
\(565\) 12.8255 0.539571
\(566\) −10.5487 −0.443394
\(567\) 0 0
\(568\) −11.3693 −0.477046
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 11.5012 0.481730
\(571\) 16.8769 0.706276 0.353138 0.935571i \(-0.385115\pi\)
0.353138 + 0.935571i \(0.385115\pi\)
\(572\) 1.77766 0.0743275
\(573\) −24.1636 −1.00945
\(574\) 0 0
\(575\) 37.6155 1.56868
\(576\) 2.56155 0.106731
\(577\) 4.18173 0.174087 0.0870437 0.996204i \(-0.472258\pi\)
0.0870437 + 0.996204i \(0.472258\pi\)
\(578\) −1.87689 −0.0780685
\(579\) −7.94584 −0.330218
\(580\) −0.662153 −0.0274944
\(581\) 0 0
\(582\) 16.4924 0.683633
\(583\) 5.06913 0.209942
\(584\) 8.48071 0.350934
\(585\) 6.87689 0.284325
\(586\) 11.5012 0.475108
\(587\) 35.8735 1.48066 0.740330 0.672244i \(-0.234669\pi\)
0.740330 + 0.672244i \(0.234669\pi\)
\(588\) 0 0
\(589\) −29.8617 −1.23043
\(590\) 2.87689 0.118440
\(591\) −8.85254 −0.364145
\(592\) 7.12311 0.292758
\(593\) −11.6284 −0.477523 −0.238761 0.971078i \(-0.576741\pi\)
−0.238761 + 0.971078i \(0.576741\pi\)
\(594\) −0.453349 −0.0186011
\(595\) 0 0
\(596\) 2.68466 0.109968
\(597\) 48.9848 2.00482
\(598\) −33.4337 −1.36721
\(599\) −24.6847 −1.00859 −0.504294 0.863532i \(-0.668247\pi\)
−0.504294 + 0.863532i \(0.668247\pi\)
\(600\) 10.7575 0.439172
\(601\) 5.83209 0.237896 0.118948 0.992900i \(-0.462048\pi\)
0.118948 + 0.992900i \(0.462048\pi\)
\(602\) 0 0
\(603\) −12.4924 −0.508731
\(604\) 4.24621 0.172776
\(605\) 7.15640 0.290949
\(606\) −26.7386 −1.08618
\(607\) 32.9346 1.33677 0.668387 0.743813i \(-0.266985\pi\)
0.668387 + 0.743813i \(0.266985\pi\)
\(608\) −7.36520 −0.298698
\(609\) 0 0
\(610\) 0.876894 0.0355044
\(611\) 2.68466 0.108610
\(612\) −11.1293 −0.449876
\(613\) 20.0540 0.809972 0.404986 0.914323i \(-0.367276\pi\)
0.404986 + 0.914323i \(0.367276\pi\)
\(614\) −11.2108 −0.452432
\(615\) 6.78456 0.273580
\(616\) 0 0
\(617\) 7.36932 0.296678 0.148339 0.988937i \(-0.452607\pi\)
0.148339 + 0.988937i \(0.452607\pi\)
\(618\) −30.2462 −1.21668
\(619\) 31.4015 1.26213 0.631066 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(620\) 2.68466 0.107818
\(621\) 8.52648 0.342156
\(622\) 9.80501 0.393145
\(623\) 0 0
\(624\) −9.56155 −0.382768
\(625\) 18.6155 0.744621
\(626\) −27.2655 −1.08975
\(627\) −7.61553 −0.304135
\(628\) −17.3790 −0.693498
\(629\) −30.9481 −1.23398
\(630\) 0 0
\(631\) 22.2462 0.885608 0.442804 0.896619i \(-0.353984\pi\)
0.442804 + 0.896619i \(0.353984\pi\)
\(632\) −7.80776 −0.310576
\(633\) 11.0478 0.439111
\(634\) −14.4924 −0.575568
\(635\) 1.48734 0.0590231
\(636\) −27.2655 −1.08115
\(637\) 0 0
\(638\) 0.438447 0.0173583
\(639\) 29.1231 1.15209
\(640\) 0.662153 0.0261739
\(641\) 7.36932 0.291071 0.145535 0.989353i \(-0.453510\pi\)
0.145535 + 0.989353i \(0.453510\pi\)
\(642\) 2.06798 0.0816165
\(643\) −13.0343 −0.514021 −0.257011 0.966409i \(-0.582738\pi\)
−0.257011 + 0.966409i \(0.582738\pi\)
\(644\) 0 0
\(645\) 7.31534 0.288041
\(646\) 32.0000 1.25902
\(647\) 0.743668 0.0292366 0.0146183 0.999893i \(-0.495347\pi\)
0.0146183 + 0.999893i \(0.495347\pi\)
\(648\) 10.1231 0.397673
\(649\) −1.90495 −0.0747757
\(650\) −18.4945 −0.725415
\(651\) 0 0
\(652\) −4.43845 −0.173823
\(653\) 47.8617 1.87297 0.936487 0.350701i \(-0.114057\pi\)
0.936487 + 0.350701i \(0.114057\pi\)
\(654\) −25.1976 −0.985303
\(655\) 1.36932 0.0535036
\(656\) −4.34475 −0.169634
\(657\) −21.7238 −0.847525
\(658\) 0 0
\(659\) −34.3002 −1.33615 −0.668073 0.744096i \(-0.732881\pi\)
−0.668073 + 0.744096i \(0.732881\pi\)
\(660\) 0.684658 0.0266503
\(661\) −25.8597 −1.00583 −0.502913 0.864337i \(-0.667739\pi\)
−0.502913 + 0.864337i \(0.667739\pi\)
\(662\) −36.3002 −1.41085
\(663\) 41.5426 1.61338
\(664\) 5.08842 0.197469
\(665\) 0 0
\(666\) −18.2462 −0.707026
\(667\) −8.24621 −0.319295
\(668\) 9.43318 0.364981
\(669\) 8.00000 0.309298
\(670\) −3.22925 −0.124757
\(671\) −0.580639 −0.0224153
\(672\) 0 0
\(673\) 41.6695 1.60624 0.803121 0.595816i \(-0.203171\pi\)
0.803121 + 0.595816i \(0.203171\pi\)
\(674\) 24.2462 0.933929
\(675\) 4.71659 0.181542
\(676\) 3.43845 0.132248
\(677\) 1.16128 0.0446315 0.0223158 0.999751i \(-0.492896\pi\)
0.0223158 + 0.999751i \(0.492896\pi\)
\(678\) 45.6786 1.75427
\(679\) 0 0
\(680\) −2.87689 −0.110324
\(681\) 29.3693 1.12543
\(682\) −1.77766 −0.0680700
\(683\) −22.2462 −0.851228 −0.425614 0.904905i \(-0.639942\pi\)
−0.425614 + 0.904905i \(0.639942\pi\)
\(684\) 18.8664 0.721373
\(685\) 5.46026 0.208626
\(686\) 0 0
\(687\) 55.6155 2.12186
\(688\) −4.68466 −0.178601
\(689\) 46.8756 1.78582
\(690\) −12.8769 −0.490215
\(691\) −29.9957 −1.14109 −0.570545 0.821267i \(-0.693268\pi\)
−0.570545 + 0.821267i \(0.693268\pi\)
\(692\) 11.1293 0.423073
\(693\) 0 0
\(694\) 31.1231 1.18142
\(695\) 5.50758 0.208914
\(696\) −2.35829 −0.0893909
\(697\) 18.8769 0.715013
\(698\) 22.9208 0.867565
\(699\) −30.4948 −1.15342
\(700\) 0 0
\(701\) −12.0540 −0.455272 −0.227636 0.973746i \(-0.573100\pi\)
−0.227636 + 0.973746i \(0.573100\pi\)
\(702\) −4.19224 −0.158226
\(703\) 52.4631 1.97868
\(704\) −0.438447 −0.0165246
\(705\) 1.03399 0.0389422
\(706\) 9.43318 0.355022
\(707\) 0 0
\(708\) 10.2462 0.385076
\(709\) 3.06913 0.115264 0.0576318 0.998338i \(-0.481645\pi\)
0.0576318 + 0.998338i \(0.481645\pi\)
\(710\) 7.52823 0.282530
\(711\) 20.0000 0.750059
\(712\) 9.06134 0.339588
\(713\) 33.4337 1.25210
\(714\) 0 0
\(715\) −1.17708 −0.0440203
\(716\) −1.75379 −0.0655422
\(717\) 0 0
\(718\) 28.6847 1.07050
\(719\) −29.6238 −1.10478 −0.552391 0.833585i \(-0.686285\pi\)
−0.552391 + 0.833585i \(0.686285\pi\)
\(720\) −1.69614 −0.0632114
\(721\) 0 0
\(722\) −35.2462 −1.31173
\(723\) −58.0540 −2.15905
\(724\) 22.1771 0.824206
\(725\) −4.56155 −0.169412
\(726\) 25.4879 0.945944
\(727\) −34.7123 −1.28741 −0.643703 0.765275i \(-0.722603\pi\)
−0.643703 + 0.765275i \(0.722603\pi\)
\(728\) 0 0
\(729\) −18.6155 −0.689464
\(730\) −5.61553 −0.207840
\(731\) 20.3537 0.752809
\(732\) 3.12311 0.115433
\(733\) 24.3266 0.898524 0.449262 0.893400i \(-0.351687\pi\)
0.449262 + 0.893400i \(0.351687\pi\)
\(734\) 0.371834 0.0137246
\(735\) 0 0
\(736\) 8.24621 0.303959
\(737\) 2.13826 0.0787638
\(738\) 11.1293 0.409676
\(739\) −22.0540 −0.811269 −0.405634 0.914035i \(-0.632949\pi\)
−0.405634 + 0.914035i \(0.632949\pi\)
\(740\) −4.71659 −0.173385
\(741\) −70.4228 −2.58705
\(742\) 0 0
\(743\) 44.9848 1.65033 0.825167 0.564889i \(-0.191081\pi\)
0.825167 + 0.564889i \(0.191081\pi\)
\(744\) 9.56155 0.350544
\(745\) −1.77766 −0.0651283
\(746\) 21.8078 0.798439
\(747\) −13.0343 −0.476899
\(748\) 1.90495 0.0696517
\(749\) 0 0
\(750\) −14.9309 −0.545198
\(751\) 30.7386 1.12167 0.560834 0.827928i \(-0.310480\pi\)
0.560834 + 0.827928i \(0.310480\pi\)
\(752\) −0.662153 −0.0241463
\(753\) 53.1771 1.93788
\(754\) 4.05444 0.147654
\(755\) −2.81164 −0.102326
\(756\) 0 0
\(757\) −0.492423 −0.0178974 −0.00894870 0.999960i \(-0.502848\pi\)
−0.00894870 + 0.999960i \(0.502848\pi\)
\(758\) −32.4924 −1.18018
\(759\) 8.52648 0.309492
\(760\) 4.87689 0.176904
\(761\) −27.1383 −0.983761 −0.491881 0.870663i \(-0.663690\pi\)
−0.491881 + 0.870663i \(0.663690\pi\)
\(762\) 5.29723 0.191898
\(763\) 0 0
\(764\) −10.2462 −0.370695
\(765\) 7.36932 0.266438
\(766\) 7.36520 0.266116
\(767\) −17.6155 −0.636060
\(768\) 2.35829 0.0850976
\(769\) −24.6984 −0.890649 −0.445324 0.895369i \(-0.646912\pi\)
−0.445324 + 0.895369i \(0.646912\pi\)
\(770\) 0 0
\(771\) 32.6847 1.17711
\(772\) −3.36932 −0.121264
\(773\) 43.0299 1.54768 0.773840 0.633382i \(-0.218334\pi\)
0.773840 + 0.633382i \(0.218334\pi\)
\(774\) 12.0000 0.431331
\(775\) 18.4945 0.664343
\(776\) 6.99337 0.251047
\(777\) 0 0
\(778\) 32.9848 1.18256
\(779\) −32.0000 −1.14652
\(780\) 6.33122 0.226694
\(781\) −4.98485 −0.178372
\(782\) −35.8278 −1.28120
\(783\) −1.03399 −0.0369517
\(784\) 0 0
\(785\) 11.5076 0.410723
\(786\) 4.87689 0.173953
\(787\) 29.9957 1.06923 0.534615 0.845096i \(-0.320457\pi\)
0.534615 + 0.845096i \(0.320457\pi\)
\(788\) −3.75379 −0.133723
\(789\) −37.2794 −1.32718
\(790\) 5.16994 0.183938
\(791\) 0 0
\(792\) 1.12311 0.0399078
\(793\) −5.36932 −0.190670
\(794\) −10.2584 −0.364056
\(795\) 18.0540 0.640309
\(796\) 20.7713 0.736219
\(797\) 47.1659 1.67070 0.835351 0.549717i \(-0.185265\pi\)
0.835351 + 0.549717i \(0.185265\pi\)
\(798\) 0 0
\(799\) 2.87689 0.101777
\(800\) 4.56155 0.161275
\(801\) −23.2111 −0.820124
\(802\) −14.1922 −0.501145
\(803\) 3.71834 0.131217
\(804\) −11.5012 −0.405614
\(805\) 0 0
\(806\) −16.4384 −0.579020
\(807\) 58.7386 2.06770
\(808\) −11.3381 −0.398874
\(809\) 42.9848 1.51127 0.755633 0.654995i \(-0.227329\pi\)
0.755633 + 0.654995i \(0.227329\pi\)
\(810\) −6.70305 −0.235521
\(811\) 54.7399 1.92218 0.961089 0.276239i \(-0.0890882\pi\)
0.961089 + 0.276239i \(0.0890882\pi\)
\(812\) 0 0
\(813\) −28.3002 −0.992531
\(814\) 3.12311 0.109465
\(815\) 2.93893 0.102946
\(816\) −10.2462 −0.358689
\(817\) −34.5035 −1.20712
\(818\) 29.9957 1.04877
\(819\) 0 0
\(820\) 2.87689 0.100466
\(821\) −16.9309 −0.590891 −0.295446 0.955360i \(-0.595468\pi\)
−0.295446 + 0.955360i \(0.595468\pi\)
\(822\) 19.4470 0.678292
\(823\) 26.2462 0.914885 0.457443 0.889239i \(-0.348765\pi\)
0.457443 + 0.889239i \(0.348765\pi\)
\(824\) −12.8255 −0.446796
\(825\) 4.71659 0.164211
\(826\) 0 0
\(827\) −25.3153 −0.880301 −0.440150 0.897924i \(-0.645075\pi\)
−0.440150 + 0.897924i \(0.645075\pi\)
\(828\) −21.1231 −0.734079
\(829\) 22.2586 0.773074 0.386537 0.922274i \(-0.373671\pi\)
0.386537 + 0.922274i \(0.373671\pi\)
\(830\) −3.36932 −0.116951
\(831\) −23.5829 −0.818083
\(832\) −4.05444 −0.140562
\(833\) 0 0
\(834\) 19.6155 0.679230
\(835\) −6.24621 −0.216159
\(836\) −3.22925 −0.111686
\(837\) 4.19224 0.144905
\(838\) 5.83209 0.201466
\(839\) 21.4335 0.739965 0.369983 0.929039i \(-0.379364\pi\)
0.369983 + 0.929039i \(0.379364\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 31.1231 1.07257
\(843\) −68.8081 −2.36988
\(844\) 4.68466 0.161253
\(845\) −2.27678 −0.0783236
\(846\) 1.69614 0.0583145
\(847\) 0 0
\(848\) −11.5616 −0.397025
\(849\) 24.8769 0.853773
\(850\) −19.8188 −0.679780
\(851\) −58.7386 −2.01353
\(852\) 26.8122 0.918571
\(853\) 36.0823 1.23544 0.617718 0.786400i \(-0.288057\pi\)
0.617718 + 0.786400i \(0.288057\pi\)
\(854\) 0 0
\(855\) −12.4924 −0.427232
\(856\) 0.876894 0.0299716
\(857\) −32.5628 −1.11232 −0.556162 0.831074i \(-0.687726\pi\)
−0.556162 + 0.831074i \(0.687726\pi\)
\(858\) −4.19224 −0.143121
\(859\) −33.8871 −1.15621 −0.578106 0.815962i \(-0.696208\pi\)
−0.578106 + 0.815962i \(0.696208\pi\)
\(860\) 3.10196 0.105776
\(861\) 0 0
\(862\) −8.00000 −0.272481
\(863\) −14.0000 −0.476566 −0.238283 0.971196i \(-0.576585\pi\)
−0.238283 + 0.971196i \(0.576585\pi\)
\(864\) 1.03399 0.0351770
\(865\) −7.36932 −0.250564
\(866\) −16.5896 −0.563737
\(867\) 4.42627 0.150324
\(868\) 0 0
\(869\) −3.42329 −0.116127
\(870\) 1.56155 0.0529416
\(871\) 19.7731 0.669984
\(872\) −10.6847 −0.361828
\(873\) −17.9139 −0.606293
\(874\) 60.7350 2.05439
\(875\) 0 0
\(876\) −20.0000 −0.675737
\(877\) −30.6847 −1.03615 −0.518074 0.855336i \(-0.673351\pi\)
−0.518074 + 0.855336i \(0.673351\pi\)
\(878\) 35.6647 1.20363
\(879\) −27.1231 −0.914840
\(880\) 0.290319 0.00978666
\(881\) 32.4813 1.09432 0.547161 0.837028i \(-0.315709\pi\)
0.547161 + 0.837028i \(0.315709\pi\)
\(882\) 0 0
\(883\) −35.1231 −1.18199 −0.590993 0.806676i \(-0.701264\pi\)
−0.590993 + 0.806676i \(0.701264\pi\)
\(884\) 17.6155 0.592474
\(885\) −6.78456 −0.228061
\(886\) −33.1231 −1.11279
\(887\) −6.28544 −0.211044 −0.105522 0.994417i \(-0.533651\pi\)
−0.105522 + 0.994417i \(0.533651\pi\)
\(888\) −16.7984 −0.563717
\(889\) 0 0
\(890\) −6.00000 −0.201120
\(891\) 4.43845 0.148694
\(892\) 3.39228 0.113582
\(893\) −4.87689 −0.163199
\(894\) −6.33122 −0.211748
\(895\) 1.16128 0.0388172
\(896\) 0 0
\(897\) 78.8466 2.63261
\(898\) 19.3693 0.646362
\(899\) −4.05444 −0.135223
\(900\) −11.6847 −0.389489
\(901\) 50.2321 1.67347
\(902\) −1.90495 −0.0634277
\(903\) 0 0
\(904\) 19.3693 0.644214
\(905\) −14.6847 −0.488135
\(906\) −10.0138 −0.332687
\(907\) 19.3693 0.643148 0.321574 0.946885i \(-0.395788\pi\)
0.321574 + 0.946885i \(0.395788\pi\)
\(908\) 12.4536 0.413288
\(909\) 29.0432 0.963302
\(910\) 0 0
\(911\) 0.300187 0.00994562 0.00497281 0.999988i \(-0.498417\pi\)
0.00497281 + 0.999988i \(0.498417\pi\)
\(912\) 17.3693 0.575156
\(913\) 2.23100 0.0738355
\(914\) −29.3693 −0.971451
\(915\) −2.06798 −0.0683651
\(916\) 23.5829 0.779202
\(917\) 0 0
\(918\) −4.49242 −0.148272
\(919\) 18.0000 0.593765 0.296883 0.954914i \(-0.404053\pi\)
0.296883 + 0.954914i \(0.404053\pi\)
\(920\) −5.46026 −0.180019
\(921\) 26.4384 0.871176
\(922\) 23.4199 0.771294
\(923\) −46.0962 −1.51727
\(924\) 0 0
\(925\) −32.4924 −1.06834
\(926\) 23.3693 0.767963
\(927\) 32.8531 1.07904
\(928\) −1.00000 −0.0328266
\(929\) −24.1636 −0.792781 −0.396391 0.918082i \(-0.629737\pi\)
−0.396391 + 0.918082i \(0.629737\pi\)
\(930\) −6.33122 −0.207609
\(931\) 0 0
\(932\) −12.9309 −0.423565
\(933\) −23.1231 −0.757016
\(934\) 6.33122 0.207164
\(935\) −1.26137 −0.0412511
\(936\) 10.3857 0.339466
\(937\) −15.0565 −0.491873 −0.245937 0.969286i \(-0.579096\pi\)
−0.245937 + 0.969286i \(0.579096\pi\)
\(938\) 0 0
\(939\) 64.3002 2.09836
\(940\) 0.438447 0.0143006
\(941\) 10.0953 0.329098 0.164549 0.986369i \(-0.447383\pi\)
0.164549 + 0.986369i \(0.447383\pi\)
\(942\) 40.9848 1.33536
\(943\) 35.8278 1.16671
\(944\) 4.34475 0.141410
\(945\) 0 0
\(946\) −2.05398 −0.0667805
\(947\) −52.6847 −1.71202 −0.856011 0.516958i \(-0.827064\pi\)
−0.856011 + 0.516958i \(0.827064\pi\)
\(948\) 18.4130 0.598027
\(949\) 34.3845 1.11617
\(950\) 33.5968 1.09002
\(951\) 34.1774 1.10828
\(952\) 0 0
\(953\) −38.0540 −1.23269 −0.616345 0.787477i \(-0.711387\pi\)
−0.616345 + 0.787477i \(0.711387\pi\)
\(954\) 29.6155 0.958838
\(955\) 6.78456 0.219543
\(956\) 0 0
\(957\) −1.03399 −0.0334241
\(958\) 40.2998 1.30203
\(959\) 0 0
\(960\) −1.56155 −0.0503989
\(961\) −14.5616 −0.469728
\(962\) 28.8802 0.931134
\(963\) −2.24621 −0.0723831
\(964\) −24.6169 −0.792858
\(965\) 2.23100 0.0718186
\(966\) 0 0
\(967\) −22.9309 −0.737407 −0.368704 0.929547i \(-0.620198\pi\)
−0.368704 + 0.929547i \(0.620198\pi\)
\(968\) 10.8078 0.347375
\(969\) −75.4654 −2.42430
\(970\) −4.63068 −0.148682
\(971\) −11.5012 −0.369090 −0.184545 0.982824i \(-0.559081\pi\)
−0.184545 + 0.982824i \(0.559081\pi\)
\(972\) −20.7713 −0.666240
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) 43.6155 1.39681
\(976\) 1.32431 0.0423900
\(977\) 17.6695 0.565297 0.282649 0.959223i \(-0.408787\pi\)
0.282649 + 0.959223i \(0.408787\pi\)
\(978\) 10.4672 0.334703
\(979\) 3.97292 0.126975
\(980\) 0 0
\(981\) 27.3693 0.873835
\(982\) 25.1771 0.803433
\(983\) 39.1385 1.24833 0.624163 0.781294i \(-0.285440\pi\)
0.624163 + 0.781294i \(0.285440\pi\)
\(984\) 10.2462 0.326637
\(985\) 2.48558 0.0791973
\(986\) 4.34475 0.138365
\(987\) 0 0
\(988\) −29.8617 −0.950028
\(989\) 38.6307 1.22838
\(990\) −0.743668 −0.0236353
\(991\) −39.6155 −1.25843 −0.629214 0.777232i \(-0.716623\pi\)
−0.629214 + 0.777232i \(0.716623\pi\)
\(992\) 4.05444 0.128728
\(993\) 85.6065 2.71664
\(994\) 0 0
\(995\) −13.7538 −0.436024
\(996\) −12.0000 −0.380235
\(997\) −25.4879 −0.807210 −0.403605 0.914933i \(-0.632243\pi\)
−0.403605 + 0.914933i \(0.632243\pi\)
\(998\) −3.12311 −0.0988602
\(999\) −7.36520 −0.233025
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 2842.2.a.p.1.4 yes 4
7.6 odd 2 inner 2842.2.a.p.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2842.2.a.p.1.1 4 7.6 odd 2 inner
2842.2.a.p.1.4 yes 4 1.1 even 1 trivial