Properties

Label 5054.2.a.bi.1.5
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.19520000000.1
Defining polynomial: \( x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 \) 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 1.5
Root \(2.35877\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.717767 q^{3} +1.00000 q^{4} +2.91631 q^{5} +0.717767 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.48481 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.717767 q^{3} +1.00000 q^{4} +2.91631 q^{5} +0.717767 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.48481 q^{9} +2.91631 q^{10} +2.56382 q^{11} +0.717767 q^{12} +1.75896 q^{13} -1.00000 q^{14} +2.09323 q^{15} +1.00000 q^{16} +1.39477 q^{17} -2.48481 q^{18} +2.91631 q^{20} -0.717767 q^{21} +2.56382 q^{22} -3.44122 q^{23} +0.717767 q^{24} +3.50487 q^{25} +1.75896 q^{26} -3.93682 q^{27} -1.00000 q^{28} -0.789597 q^{29} +2.09323 q^{30} +4.99865 q^{31} +1.00000 q^{32} +1.84023 q^{33} +1.39477 q^{34} -2.91631 q^{35} -2.48481 q^{36} +7.90419 q^{37} +1.26252 q^{39} +2.91631 q^{40} +6.60934 q^{41} -0.717767 q^{42} +12.2294 q^{43} +2.56382 q^{44} -7.24648 q^{45} -3.44122 q^{46} -1.23536 q^{47} +0.717767 q^{48} +1.00000 q^{49} +3.50487 q^{50} +1.00112 q^{51} +1.75896 q^{52} -7.30992 q^{53} -3.93682 q^{54} +7.47689 q^{55} -1.00000 q^{56} -0.789597 q^{58} +1.68722 q^{59} +2.09323 q^{60} +4.80597 q^{61} +4.99865 q^{62} +2.48481 q^{63} +1.00000 q^{64} +5.12966 q^{65} +1.84023 q^{66} +2.63849 q^{67} +1.39477 q^{68} -2.46999 q^{69} -2.91631 q^{70} +8.70323 q^{71} -2.48481 q^{72} -4.32196 q^{73} +7.90419 q^{74} +2.51568 q^{75} -2.56382 q^{77} +1.26252 q^{78} -4.23183 q^{79} +2.91631 q^{80} +4.62871 q^{81} +6.60934 q^{82} -13.6474 q^{83} -0.717767 q^{84} +4.06760 q^{85} +12.2294 q^{86} -0.566747 q^{87} +2.56382 q^{88} +1.50287 q^{89} -7.24648 q^{90} -1.75896 q^{91} -3.44122 q^{92} +3.58787 q^{93} -1.23536 q^{94} +0.717767 q^{96} -11.0606 q^{97} +1.00000 q^{98} -6.37060 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} + 8 q^{8} + 8 q^{9} + 6 q^{10} + 4 q^{11} + 4 q^{12} + 6 q^{13} - 8 q^{14} + 8 q^{15} + 8 q^{16} + 2 q^{17} + 8 q^{18} + 6 q^{20} - 4 q^{21} + 4 q^{22} + 20 q^{23} + 4 q^{24} - 8 q^{25} + 6 q^{26} + 22 q^{27} - 8 q^{28} + 16 q^{29} + 8 q^{30} + 22 q^{31} + 8 q^{32} - 8 q^{33} + 2 q^{34} - 6 q^{35} + 8 q^{36} - 12 q^{37} + 8 q^{39} + 6 q^{40} + 36 q^{41} - 4 q^{42} + 4 q^{44} - 4 q^{45} + 20 q^{46} + 6 q^{47} + 4 q^{48} + 8 q^{49} - 8 q^{50} - 4 q^{51} + 6 q^{52} + 16 q^{53} + 22 q^{54} + 18 q^{55} - 8 q^{56} + 16 q^{58} + 14 q^{59} + 8 q^{60} + 10 q^{61} + 22 q^{62} - 8 q^{63} + 8 q^{64} + 32 q^{65} - 8 q^{66} - 20 q^{67} + 2 q^{68} + 40 q^{69} - 6 q^{70} + 8 q^{72} - 18 q^{73} - 12 q^{74} + 16 q^{75} - 4 q^{77} + 8 q^{78} + 4 q^{79} + 6 q^{80} + 12 q^{81} + 36 q^{82} + 36 q^{83} - 4 q^{84} - 16 q^{85} + 28 q^{87} + 4 q^{88} + 18 q^{89} - 4 q^{90} - 6 q^{91} + 20 q^{92} + 16 q^{93} + 6 q^{94} + 4 q^{96} + 26 q^{97} + 8 q^{98} + 4 q^{99}+O(q^{100}) \) 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
\(3\) 0.717767 0.414403 0.207202 0.978298i \(-0.433564\pi\)
0.207202 + 0.978298i \(0.433564\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.91631 1.30421 0.652107 0.758127i \(-0.273885\pi\)
0.652107 + 0.758127i \(0.273885\pi\)
\(6\) 0.717767 0.293027
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.48481 −0.828270
\(10\) 2.91631 0.922219
\(11\) 2.56382 0.773020 0.386510 0.922285i \(-0.373680\pi\)
0.386510 + 0.922285i \(0.373680\pi\)
\(12\) 0.717767 0.207202
\(13\) 1.75896 0.487847 0.243923 0.969795i \(-0.421565\pi\)
0.243923 + 0.969795i \(0.421565\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.09323 0.540471
\(16\) 1.00000 0.250000
\(17\) 1.39477 0.338283 0.169141 0.985592i \(-0.445901\pi\)
0.169141 + 0.985592i \(0.445901\pi\)
\(18\) −2.48481 −0.585675
\(19\) 0 0
\(20\) 2.91631 0.652107
\(21\) −0.717767 −0.156630
\(22\) 2.56382 0.546608
\(23\) −3.44122 −0.717543 −0.358772 0.933425i \(-0.616804\pi\)
−0.358772 + 0.933425i \(0.616804\pi\)
\(24\) 0.717767 0.146514
\(25\) 3.50487 0.700974
\(26\) 1.75896 0.344960
\(27\) −3.93682 −0.757641
\(28\) −1.00000 −0.188982
\(29\) −0.789597 −0.146625 −0.0733123 0.997309i \(-0.523357\pi\)
−0.0733123 + 0.997309i \(0.523357\pi\)
\(30\) 2.09323 0.382170
\(31\) 4.99865 0.897784 0.448892 0.893586i \(-0.351819\pi\)
0.448892 + 0.893586i \(0.351819\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.84023 0.320342
\(34\) 1.39477 0.239202
\(35\) −2.91631 −0.492947
\(36\) −2.48481 −0.414135
\(37\) 7.90419 1.29944 0.649721 0.760173i \(-0.274886\pi\)
0.649721 + 0.760173i \(0.274886\pi\)
\(38\) 0 0
\(39\) 1.26252 0.202165
\(40\) 2.91631 0.461109
\(41\) 6.60934 1.03221 0.516103 0.856527i \(-0.327382\pi\)
0.516103 + 0.856527i \(0.327382\pi\)
\(42\) −0.717767 −0.110754
\(43\) 12.2294 1.86497 0.932486 0.361205i \(-0.117635\pi\)
0.932486 + 0.361205i \(0.117635\pi\)
\(44\) 2.56382 0.386510
\(45\) −7.24648 −1.08024
\(46\) −3.44122 −0.507380
\(47\) −1.23536 −0.180196 −0.0900981 0.995933i \(-0.528718\pi\)
−0.0900981 + 0.995933i \(0.528718\pi\)
\(48\) 0.717767 0.103601
\(49\) 1.00000 0.142857
\(50\) 3.50487 0.495664
\(51\) 1.00112 0.140185
\(52\) 1.75896 0.243923
\(53\) −7.30992 −1.00409 −0.502047 0.864840i \(-0.667420\pi\)
−0.502047 + 0.864840i \(0.667420\pi\)
\(54\) −3.93682 −0.535733
\(55\) 7.47689 1.00818
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −0.789597 −0.103679
\(59\) 1.68722 0.219657 0.109829 0.993951i \(-0.464970\pi\)
0.109829 + 0.993951i \(0.464970\pi\)
\(60\) 2.09323 0.270235
\(61\) 4.80597 0.615342 0.307671 0.951493i \(-0.400450\pi\)
0.307671 + 0.951493i \(0.400450\pi\)
\(62\) 4.99865 0.634829
\(63\) 2.48481 0.313057
\(64\) 1.00000 0.125000
\(65\) 5.12966 0.636257
\(66\) 1.84023 0.226516
\(67\) 2.63849 0.322343 0.161172 0.986926i \(-0.448473\pi\)
0.161172 + 0.986926i \(0.448473\pi\)
\(68\) 1.39477 0.169141
\(69\) −2.46999 −0.297352
\(70\) −2.91631 −0.348566
\(71\) 8.70323 1.03288 0.516442 0.856322i \(-0.327256\pi\)
0.516442 + 0.856322i \(0.327256\pi\)
\(72\) −2.48481 −0.292838
\(73\) −4.32196 −0.505847 −0.252923 0.967486i \(-0.581392\pi\)
−0.252923 + 0.967486i \(0.581392\pi\)
\(74\) 7.90419 0.918844
\(75\) 2.51568 0.290486
\(76\) 0 0
\(77\) −2.56382 −0.292174
\(78\) 1.26252 0.142952
\(79\) −4.23183 −0.476118 −0.238059 0.971251i \(-0.576511\pi\)
−0.238059 + 0.971251i \(0.576511\pi\)
\(80\) 2.91631 0.326054
\(81\) 4.62871 0.514301
\(82\) 6.60934 0.729879
\(83\) −13.6474 −1.49800 −0.748999 0.662571i \(-0.769465\pi\)
−0.748999 + 0.662571i \(0.769465\pi\)
\(84\) −0.717767 −0.0783149
\(85\) 4.06760 0.441193
\(86\) 12.2294 1.31873
\(87\) −0.566747 −0.0607617
\(88\) 2.56382 0.273304
\(89\) 1.50287 0.159304 0.0796521 0.996823i \(-0.474619\pi\)
0.0796521 + 0.996823i \(0.474619\pi\)
\(90\) −7.24648 −0.763846
\(91\) −1.75896 −0.184389
\(92\) −3.44122 −0.358772
\(93\) 3.58787 0.372045
\(94\) −1.23536 −0.127418
\(95\) 0 0
\(96\) 0.717767 0.0732568
\(97\) −11.0606 −1.12303 −0.561516 0.827466i \(-0.689782\pi\)
−0.561516 + 0.827466i \(0.689782\pi\)
\(98\) 1.00000 0.101015
\(99\) −6.37060 −0.640270
\(100\) 3.50487 0.350487
\(101\) −15.0746 −1.49998 −0.749989 0.661450i \(-0.769941\pi\)
−0.749989 + 0.661450i \(0.769941\pi\)
\(102\) 1.00112 0.0991260
\(103\) 13.0346 1.28434 0.642169 0.766563i \(-0.278035\pi\)
0.642169 + 0.766563i \(0.278035\pi\)
\(104\) 1.75896 0.172480
\(105\) −2.09323 −0.204279
\(106\) −7.30992 −0.710002
\(107\) 15.3752 1.48638 0.743188 0.669082i \(-0.233313\pi\)
0.743188 + 0.669082i \(0.233313\pi\)
\(108\) −3.93682 −0.378821
\(109\) 19.9059 1.90664 0.953322 0.301957i \(-0.0976400\pi\)
0.953322 + 0.301957i \(0.0976400\pi\)
\(110\) 7.47689 0.712894
\(111\) 5.67337 0.538493
\(112\) −1.00000 −0.0944911
\(113\) −2.98600 −0.280899 −0.140450 0.990088i \(-0.544855\pi\)
−0.140450 + 0.990088i \(0.544855\pi\)
\(114\) 0 0
\(115\) −10.0357 −0.935830
\(116\) −0.789597 −0.0733123
\(117\) −4.37067 −0.404069
\(118\) 1.68722 0.155321
\(119\) −1.39477 −0.127859
\(120\) 2.09323 0.191085
\(121\) −4.42683 −0.402439
\(122\) 4.80597 0.435112
\(123\) 4.74397 0.427749
\(124\) 4.99865 0.448892
\(125\) −4.36026 −0.389993
\(126\) 2.48481 0.221364
\(127\) 4.15241 0.368467 0.184233 0.982883i \(-0.441020\pi\)
0.184233 + 0.982883i \(0.441020\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.77790 0.772851
\(130\) 5.12966 0.449901
\(131\) 18.9379 1.65461 0.827305 0.561753i \(-0.189873\pi\)
0.827305 + 0.561753i \(0.189873\pi\)
\(132\) 1.84023 0.160171
\(133\) 0 0
\(134\) 2.63849 0.227931
\(135\) −11.4810 −0.988126
\(136\) 1.39477 0.119601
\(137\) −16.3088 −1.39336 −0.696678 0.717384i \(-0.745340\pi\)
−0.696678 + 0.717384i \(0.745340\pi\)
\(138\) −2.46999 −0.210260
\(139\) −3.62222 −0.307233 −0.153616 0.988131i \(-0.549092\pi\)
−0.153616 + 0.988131i \(0.549092\pi\)
\(140\) −2.91631 −0.246473
\(141\) −0.886703 −0.0746739
\(142\) 8.70323 0.730359
\(143\) 4.50965 0.377115
\(144\) −2.48481 −0.207067
\(145\) −2.30271 −0.191230
\(146\) −4.32196 −0.357688
\(147\) 0.717767 0.0592005
\(148\) 7.90419 0.649721
\(149\) −10.5028 −0.860420 −0.430210 0.902729i \(-0.641561\pi\)
−0.430210 + 0.902729i \(0.641561\pi\)
\(150\) 2.51568 0.205405
\(151\) −20.8828 −1.69942 −0.849708 0.527253i \(-0.823222\pi\)
−0.849708 + 0.527253i \(0.823222\pi\)
\(152\) 0 0
\(153\) −3.46575 −0.280189
\(154\) −2.56382 −0.206598
\(155\) 14.5776 1.17090
\(156\) 1.26252 0.101083
\(157\) 14.4157 1.15049 0.575247 0.817980i \(-0.304906\pi\)
0.575247 + 0.817980i \(0.304906\pi\)
\(158\) −4.23183 −0.336666
\(159\) −5.24682 −0.416100
\(160\) 2.91631 0.230555
\(161\) 3.44122 0.271206
\(162\) 4.62871 0.363666
\(163\) 0.718973 0.0563143 0.0281572 0.999604i \(-0.491036\pi\)
0.0281572 + 0.999604i \(0.491036\pi\)
\(164\) 6.60934 0.516103
\(165\) 5.36667 0.417795
\(166\) −13.6474 −1.05924
\(167\) −14.5419 −1.12528 −0.562642 0.826701i \(-0.690215\pi\)
−0.562642 + 0.826701i \(0.690215\pi\)
\(168\) −0.717767 −0.0553770
\(169\) −9.90607 −0.762006
\(170\) 4.06760 0.311970
\(171\) 0 0
\(172\) 12.2294 0.932486
\(173\) −24.2394 −1.84289 −0.921444 0.388510i \(-0.872990\pi\)
−0.921444 + 0.388510i \(0.872990\pi\)
\(174\) −0.566747 −0.0429650
\(175\) −3.50487 −0.264943
\(176\) 2.56382 0.193255
\(177\) 1.21103 0.0910267
\(178\) 1.50287 0.112645
\(179\) 9.38770 0.701670 0.350835 0.936437i \(-0.385898\pi\)
0.350835 + 0.936437i \(0.385898\pi\)
\(180\) −7.24648 −0.540121
\(181\) 23.7807 1.76760 0.883801 0.467862i \(-0.154976\pi\)
0.883801 + 0.467862i \(0.154976\pi\)
\(182\) −1.75896 −0.130383
\(183\) 3.44957 0.255000
\(184\) −3.44122 −0.253690
\(185\) 23.0511 1.69475
\(186\) 3.58787 0.263075
\(187\) 3.57595 0.261499
\(188\) −1.23536 −0.0900981
\(189\) 3.93682 0.286361
\(190\) 0 0
\(191\) −0.561930 −0.0406599 −0.0203299 0.999793i \(-0.506472\pi\)
−0.0203299 + 0.999793i \(0.506472\pi\)
\(192\) 0.717767 0.0518004
\(193\) −17.3688 −1.25024 −0.625118 0.780530i \(-0.714949\pi\)
−0.625118 + 0.780530i \(0.714949\pi\)
\(194\) −11.0606 −0.794103
\(195\) 3.68191 0.263667
\(196\) 1.00000 0.0714286
\(197\) 20.4849 1.45949 0.729746 0.683719i \(-0.239638\pi\)
0.729746 + 0.683719i \(0.239638\pi\)
\(198\) −6.37060 −0.452739
\(199\) 13.4529 0.953650 0.476825 0.878998i \(-0.341787\pi\)
0.476825 + 0.878998i \(0.341787\pi\)
\(200\) 3.50487 0.247832
\(201\) 1.89382 0.133580
\(202\) −15.0746 −1.06064
\(203\) 0.789597 0.0554189
\(204\) 1.00112 0.0700927
\(205\) 19.2749 1.34622
\(206\) 13.0346 0.908164
\(207\) 8.55077 0.594319
\(208\) 1.75896 0.121962
\(209\) 0 0
\(210\) −2.09323 −0.144447
\(211\) −5.21924 −0.359308 −0.179654 0.983730i \(-0.557498\pi\)
−0.179654 + 0.983730i \(0.557498\pi\)
\(212\) −7.30992 −0.502047
\(213\) 6.24689 0.428030
\(214\) 15.3752 1.05103
\(215\) 35.6649 2.43232
\(216\) −3.93682 −0.267867
\(217\) −4.99865 −0.339330
\(218\) 19.9059 1.34820
\(219\) −3.10216 −0.209625
\(220\) 7.47689 0.504092
\(221\) 2.45335 0.165030
\(222\) 5.67337 0.380772
\(223\) −18.2290 −1.22070 −0.610352 0.792130i \(-0.708972\pi\)
−0.610352 + 0.792130i \(0.708972\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −8.70894 −0.580596
\(226\) −2.98600 −0.198626
\(227\) −10.7390 −0.712774 −0.356387 0.934338i \(-0.615992\pi\)
−0.356387 + 0.934338i \(0.615992\pi\)
\(228\) 0 0
\(229\) −16.6977 −1.10341 −0.551707 0.834038i \(-0.686023\pi\)
−0.551707 + 0.834038i \(0.686023\pi\)
\(230\) −10.0357 −0.661732
\(231\) −1.84023 −0.121078
\(232\) −0.789597 −0.0518396
\(233\) −16.8460 −1.10362 −0.551810 0.833970i \(-0.686063\pi\)
−0.551810 + 0.833970i \(0.686063\pi\)
\(234\) −4.37067 −0.285720
\(235\) −3.60270 −0.235014
\(236\) 1.68722 0.109829
\(237\) −3.03747 −0.197305
\(238\) −1.39477 −0.0904098
\(239\) −23.2019 −1.50081 −0.750403 0.660981i \(-0.770141\pi\)
−0.750403 + 0.660981i \(0.770141\pi\)
\(240\) 2.09323 0.135118
\(241\) 1.76578 0.113744 0.0568720 0.998381i \(-0.481887\pi\)
0.0568720 + 0.998381i \(0.481887\pi\)
\(242\) −4.42683 −0.284568
\(243\) 15.1328 0.970769
\(244\) 4.80597 0.307671
\(245\) 2.91631 0.186316
\(246\) 4.74397 0.302464
\(247\) 0 0
\(248\) 4.99865 0.317414
\(249\) −9.79567 −0.620775
\(250\) −4.36026 −0.275767
\(251\) 4.47898 0.282711 0.141355 0.989959i \(-0.454854\pi\)
0.141355 + 0.989959i \(0.454854\pi\)
\(252\) 2.48481 0.156528
\(253\) −8.82265 −0.554676
\(254\) 4.15241 0.260545
\(255\) 2.91959 0.182832
\(256\) 1.00000 0.0625000
\(257\) 24.0895 1.50266 0.751330 0.659926i \(-0.229413\pi\)
0.751330 + 0.659926i \(0.229413\pi\)
\(258\) 8.77790 0.546488
\(259\) −7.90419 −0.491143
\(260\) 5.12966 0.318128
\(261\) 1.96200 0.121445
\(262\) 18.9379 1.16999
\(263\) 8.88831 0.548077 0.274038 0.961719i \(-0.411640\pi\)
0.274038 + 0.961719i \(0.411640\pi\)
\(264\) 1.84023 0.113258
\(265\) −21.3180 −1.30955
\(266\) 0 0
\(267\) 1.07871 0.0660162
\(268\) 2.63849 0.161172
\(269\) −12.1748 −0.742313 −0.371156 0.928570i \(-0.621039\pi\)
−0.371156 + 0.928570i \(0.621039\pi\)
\(270\) −11.4810 −0.698711
\(271\) 10.8375 0.658332 0.329166 0.944272i \(-0.393232\pi\)
0.329166 + 0.944272i \(0.393232\pi\)
\(272\) 1.39477 0.0845706
\(273\) −1.26252 −0.0764113
\(274\) −16.3088 −0.985252
\(275\) 8.98586 0.541868
\(276\) −2.46999 −0.148676
\(277\) −11.3275 −0.680602 −0.340301 0.940317i \(-0.610529\pi\)
−0.340301 + 0.940317i \(0.610529\pi\)
\(278\) −3.62222 −0.217247
\(279\) −12.4207 −0.743607
\(280\) −2.91631 −0.174283
\(281\) 16.8113 1.00288 0.501438 0.865194i \(-0.332805\pi\)
0.501438 + 0.865194i \(0.332805\pi\)
\(282\) −0.886703 −0.0528024
\(283\) −32.7713 −1.94805 −0.974025 0.226442i \(-0.927291\pi\)
−0.974025 + 0.226442i \(0.927291\pi\)
\(284\) 8.70323 0.516442
\(285\) 0 0
\(286\) 4.50965 0.266661
\(287\) −6.60934 −0.390137
\(288\) −2.48481 −0.146419
\(289\) −15.0546 −0.885565
\(290\) −2.30271 −0.135220
\(291\) −7.93892 −0.465388
\(292\) −4.32196 −0.252923
\(293\) 16.5383 0.966176 0.483088 0.875572i \(-0.339515\pi\)
0.483088 + 0.875572i \(0.339515\pi\)
\(294\) 0.717767 0.0418610
\(295\) 4.92046 0.286480
\(296\) 7.90419 0.459422
\(297\) −10.0933 −0.585672
\(298\) −10.5028 −0.608409
\(299\) −6.05295 −0.350051
\(300\) 2.51568 0.145243
\(301\) −12.2294 −0.704893
\(302\) −20.8828 −1.20167
\(303\) −10.8201 −0.621596
\(304\) 0 0
\(305\) 14.0157 0.802537
\(306\) −3.46575 −0.198124
\(307\) 10.7452 0.613262 0.306631 0.951828i \(-0.400798\pi\)
0.306631 + 0.951828i \(0.400798\pi\)
\(308\) −2.56382 −0.146087
\(309\) 9.35582 0.532234
\(310\) 14.5776 0.827953
\(311\) −32.3009 −1.83161 −0.915807 0.401619i \(-0.868448\pi\)
−0.915807 + 0.401619i \(0.868448\pi\)
\(312\) 1.26252 0.0714762
\(313\) −11.5120 −0.650697 −0.325349 0.945594i \(-0.605482\pi\)
−0.325349 + 0.945594i \(0.605482\pi\)
\(314\) 14.4157 0.813522
\(315\) 7.24648 0.408293
\(316\) −4.23183 −0.238059
\(317\) −12.4053 −0.696751 −0.348376 0.937355i \(-0.613267\pi\)
−0.348376 + 0.937355i \(0.613267\pi\)
\(318\) −5.24682 −0.294227
\(319\) −2.02438 −0.113344
\(320\) 2.91631 0.163027
\(321\) 11.0358 0.615959
\(322\) 3.44122 0.191771
\(323\) 0 0
\(324\) 4.62871 0.257151
\(325\) 6.16492 0.341968
\(326\) 0.718973 0.0398202
\(327\) 14.2878 0.790119
\(328\) 6.60934 0.364940
\(329\) 1.23536 0.0681077
\(330\) 5.36667 0.295426
\(331\) 4.28173 0.235345 0.117673 0.993052i \(-0.462457\pi\)
0.117673 + 0.993052i \(0.462457\pi\)
\(332\) −13.6474 −0.748999
\(333\) −19.6404 −1.07629
\(334\) −14.5419 −0.795696
\(335\) 7.69467 0.420404
\(336\) −0.717767 −0.0391574
\(337\) 34.6005 1.88481 0.942404 0.334477i \(-0.108560\pi\)
0.942404 + 0.334477i \(0.108560\pi\)
\(338\) −9.90607 −0.538819
\(339\) −2.14325 −0.116405
\(340\) 4.06760 0.220596
\(341\) 12.8156 0.694005
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 12.2294 0.659367
\(345\) −7.20327 −0.387811
\(346\) −24.2394 −1.30312
\(347\) −8.61568 −0.462514 −0.231257 0.972893i \(-0.574284\pi\)
−0.231257 + 0.972893i \(0.574284\pi\)
\(348\) −0.566747 −0.0303808
\(349\) −30.1120 −1.61186 −0.805930 0.592011i \(-0.798334\pi\)
−0.805930 + 0.592011i \(0.798334\pi\)
\(350\) −3.50487 −0.187343
\(351\) −6.92469 −0.369613
\(352\) 2.56382 0.136652
\(353\) 4.02781 0.214379 0.107189 0.994239i \(-0.465815\pi\)
0.107189 + 0.994239i \(0.465815\pi\)
\(354\) 1.21103 0.0643656
\(355\) 25.3813 1.34710
\(356\) 1.50287 0.0796521
\(357\) −1.00112 −0.0529851
\(358\) 9.38770 0.496156
\(359\) 33.6389 1.77539 0.887696 0.460429i \(-0.152305\pi\)
0.887696 + 0.460429i \(0.152305\pi\)
\(360\) −7.24648 −0.381923
\(361\) 0 0
\(362\) 23.7807 1.24988
\(363\) −3.17744 −0.166772
\(364\) −1.75896 −0.0921944
\(365\) −12.6042 −0.659732
\(366\) 3.44957 0.180312
\(367\) 10.8822 0.568047 0.284024 0.958817i \(-0.408331\pi\)
0.284024 + 0.958817i \(0.408331\pi\)
\(368\) −3.44122 −0.179386
\(369\) −16.4229 −0.854944
\(370\) 23.0511 1.19837
\(371\) 7.30992 0.379512
\(372\) 3.58787 0.186022
\(373\) −10.9132 −0.565062 −0.282531 0.959258i \(-0.591174\pi\)
−0.282531 + 0.959258i \(0.591174\pi\)
\(374\) 3.57595 0.184908
\(375\) −3.12965 −0.161615
\(376\) −1.23536 −0.0637090
\(377\) −1.38887 −0.0715303
\(378\) 3.93682 0.202488
\(379\) −30.4270 −1.56293 −0.781465 0.623949i \(-0.785527\pi\)
−0.781465 + 0.623949i \(0.785527\pi\)
\(380\) 0 0
\(381\) 2.98046 0.152694
\(382\) −0.561930 −0.0287509
\(383\) −3.04363 −0.155522 −0.0777612 0.996972i \(-0.524777\pi\)
−0.0777612 + 0.996972i \(0.524777\pi\)
\(384\) 0.717767 0.0366284
\(385\) −7.47689 −0.381058
\(386\) −17.3688 −0.884051
\(387\) −30.3878 −1.54470
\(388\) −11.0606 −0.561516
\(389\) 15.7728 0.799712 0.399856 0.916578i \(-0.369060\pi\)
0.399856 + 0.916578i \(0.369060\pi\)
\(390\) 3.68191 0.186441
\(391\) −4.79972 −0.242732
\(392\) 1.00000 0.0505076
\(393\) 13.5930 0.685676
\(394\) 20.4849 1.03202
\(395\) −12.3413 −0.620959
\(396\) −6.37060 −0.320135
\(397\) −29.8753 −1.49940 −0.749700 0.661778i \(-0.769802\pi\)
−0.749700 + 0.661778i \(0.769802\pi\)
\(398\) 13.4529 0.674333
\(399\) 0 0
\(400\) 3.50487 0.175244
\(401\) −10.0388 −0.501316 −0.250658 0.968076i \(-0.580647\pi\)
−0.250658 + 0.968076i \(0.580647\pi\)
\(402\) 1.89382 0.0944554
\(403\) 8.79240 0.437981
\(404\) −15.0746 −0.749989
\(405\) 13.4988 0.670759
\(406\) 0.789597 0.0391871
\(407\) 20.2649 1.00449
\(408\) 1.00112 0.0495630
\(409\) −10.0191 −0.495412 −0.247706 0.968835i \(-0.579677\pi\)
−0.247706 + 0.968835i \(0.579677\pi\)
\(410\) 19.2749 0.951919
\(411\) −11.7059 −0.577412
\(412\) 13.0346 0.642169
\(413\) −1.68722 −0.0830227
\(414\) 8.55077 0.420247
\(415\) −39.8001 −1.95371
\(416\) 1.75896 0.0862399
\(417\) −2.59991 −0.127318
\(418\) 0 0
\(419\) 2.90178 0.141761 0.0708807 0.997485i \(-0.477419\pi\)
0.0708807 + 0.997485i \(0.477419\pi\)
\(420\) −2.09323 −0.102139
\(421\) 19.6306 0.956738 0.478369 0.878159i \(-0.341228\pi\)
0.478369 + 0.878159i \(0.341228\pi\)
\(422\) −5.21924 −0.254069
\(423\) 3.06964 0.149251
\(424\) −7.30992 −0.355001
\(425\) 4.88851 0.237127
\(426\) 6.24689 0.302663
\(427\) −4.80597 −0.232577
\(428\) 15.3752 0.743188
\(429\) 3.23688 0.156278
\(430\) 35.6649 1.71991
\(431\) −0.757246 −0.0364752 −0.0182376 0.999834i \(-0.505806\pi\)
−0.0182376 + 0.999834i \(0.505806\pi\)
\(432\) −3.93682 −0.189410
\(433\) −2.51099 −0.120670 −0.0603351 0.998178i \(-0.519217\pi\)
−0.0603351 + 0.998178i \(0.519217\pi\)
\(434\) −4.99865 −0.239943
\(435\) −1.65281 −0.0792462
\(436\) 19.9059 0.953322
\(437\) 0 0
\(438\) −3.10216 −0.148227
\(439\) 22.3934 1.06878 0.534389 0.845239i \(-0.320542\pi\)
0.534389 + 0.845239i \(0.320542\pi\)
\(440\) 7.47689 0.356447
\(441\) −2.48481 −0.118324
\(442\) 2.45335 0.116694
\(443\) 2.45606 0.116691 0.0583455 0.998296i \(-0.481418\pi\)
0.0583455 + 0.998296i \(0.481418\pi\)
\(444\) 5.67337 0.269246
\(445\) 4.38285 0.207767
\(446\) −18.2290 −0.863169
\(447\) −7.53855 −0.356561
\(448\) −1.00000 −0.0472456
\(449\) 4.08691 0.192873 0.0964367 0.995339i \(-0.469255\pi\)
0.0964367 + 0.995339i \(0.469255\pi\)
\(450\) −8.70894 −0.410543
\(451\) 16.9451 0.797916
\(452\) −2.98600 −0.140450
\(453\) −14.9890 −0.704244
\(454\) −10.7390 −0.504008
\(455\) −5.12966 −0.240482
\(456\) 0 0
\(457\) 10.5514 0.493576 0.246788 0.969069i \(-0.420625\pi\)
0.246788 + 0.969069i \(0.420625\pi\)
\(458\) −16.6977 −0.780232
\(459\) −5.49097 −0.256297
\(460\) −10.0357 −0.467915
\(461\) −17.6728 −0.823105 −0.411552 0.911386i \(-0.635013\pi\)
−0.411552 + 0.911386i \(0.635013\pi\)
\(462\) −1.84023 −0.0856150
\(463\) 27.5515 1.28043 0.640214 0.768196i \(-0.278846\pi\)
0.640214 + 0.768196i \(0.278846\pi\)
\(464\) −0.789597 −0.0366561
\(465\) 10.4633 0.485226
\(466\) −16.8460 −0.780378
\(467\) −34.2363 −1.58427 −0.792133 0.610349i \(-0.791029\pi\)
−0.792133 + 0.610349i \(0.791029\pi\)
\(468\) −4.37067 −0.202034
\(469\) −2.63849 −0.121834
\(470\) −3.60270 −0.166180
\(471\) 10.3471 0.476769
\(472\) 1.68722 0.0776606
\(473\) 31.3541 1.44166
\(474\) −3.03747 −0.139516
\(475\) 0 0
\(476\) −1.39477 −0.0639294
\(477\) 18.1638 0.831661
\(478\) −23.2019 −1.06123
\(479\) 24.7601 1.13132 0.565660 0.824639i \(-0.308622\pi\)
0.565660 + 0.824639i \(0.308622\pi\)
\(480\) 2.09323 0.0955426
\(481\) 13.9031 0.633928
\(482\) 1.76578 0.0804292
\(483\) 2.46999 0.112389
\(484\) −4.42683 −0.201220
\(485\) −32.2561 −1.46467
\(486\) 15.1328 0.686437
\(487\) −22.2019 −1.00606 −0.503032 0.864268i \(-0.667782\pi\)
−0.503032 + 0.864268i \(0.667782\pi\)
\(488\) 4.80597 0.217556
\(489\) 0.516056 0.0233368
\(490\) 2.91631 0.131746
\(491\) −31.0334 −1.40052 −0.700259 0.713888i \(-0.746932\pi\)
−0.700259 + 0.713888i \(0.746932\pi\)
\(492\) 4.74397 0.213875
\(493\) −1.10131 −0.0496005
\(494\) 0 0
\(495\) −18.5787 −0.835049
\(496\) 4.99865 0.224446
\(497\) −8.70323 −0.390393
\(498\) −9.79567 −0.438954
\(499\) −24.0668 −1.07738 −0.538690 0.842504i \(-0.681080\pi\)
−0.538690 + 0.842504i \(0.681080\pi\)
\(500\) −4.36026 −0.194997
\(501\) −10.4377 −0.466321
\(502\) 4.47898 0.199907
\(503\) −6.87199 −0.306407 −0.153203 0.988195i \(-0.548959\pi\)
−0.153203 + 0.988195i \(0.548959\pi\)
\(504\) 2.48481 0.110682
\(505\) −43.9622 −1.95629
\(506\) −8.82265 −0.392215
\(507\) −7.11026 −0.315778
\(508\) 4.15241 0.184233
\(509\) 4.26876 0.189210 0.0946048 0.995515i \(-0.469841\pi\)
0.0946048 + 0.995515i \(0.469841\pi\)
\(510\) 2.91959 0.129282
\(511\) 4.32196 0.191192
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 24.0895 1.06254
\(515\) 38.0130 1.67505
\(516\) 8.77790 0.386425
\(517\) −3.16725 −0.139295
\(518\) −7.90419 −0.347290
\(519\) −17.3983 −0.763699
\(520\) 5.12966 0.224951
\(521\) 23.3507 1.02301 0.511506 0.859280i \(-0.329088\pi\)
0.511506 + 0.859280i \(0.329088\pi\)
\(522\) 1.96200 0.0858744
\(523\) 16.6780 0.729278 0.364639 0.931149i \(-0.381192\pi\)
0.364639 + 0.931149i \(0.381192\pi\)
\(524\) 18.9379 0.827305
\(525\) −2.51568 −0.109793
\(526\) 8.88831 0.387549
\(527\) 6.97199 0.303705
\(528\) 1.84023 0.0800855
\(529\) −11.1580 −0.485132
\(530\) −21.3180 −0.925995
\(531\) −4.19242 −0.181936
\(532\) 0 0
\(533\) 11.6255 0.503558
\(534\) 1.07871 0.0466805
\(535\) 44.8389 1.93855
\(536\) 2.63849 0.113966
\(537\) 6.73819 0.290774
\(538\) −12.1748 −0.524894
\(539\) 2.56382 0.110431
\(540\) −11.4810 −0.494063
\(541\) −30.8053 −1.32442 −0.662211 0.749318i \(-0.730382\pi\)
−0.662211 + 0.749318i \(0.730382\pi\)
\(542\) 10.8375 0.465511
\(543\) 17.0690 0.732500
\(544\) 1.39477 0.0598005
\(545\) 58.0519 2.48667
\(546\) −1.26252 −0.0540309
\(547\) 5.43027 0.232182 0.116091 0.993239i \(-0.462964\pi\)
0.116091 + 0.993239i \(0.462964\pi\)
\(548\) −16.3088 −0.696678
\(549\) −11.9419 −0.509669
\(550\) 8.98586 0.383158
\(551\) 0 0
\(552\) −2.46999 −0.105130
\(553\) 4.23183 0.179956
\(554\) −11.3275 −0.481258
\(555\) 16.5453 0.702310
\(556\) −3.62222 −0.153616
\(557\) −10.0307 −0.425013 −0.212507 0.977160i \(-0.568163\pi\)
−0.212507 + 0.977160i \(0.568163\pi\)
\(558\) −12.4207 −0.525810
\(559\) 21.5111 0.909821
\(560\) −2.91631 −0.123237
\(561\) 2.56670 0.108366
\(562\) 16.8113 0.709140
\(563\) −1.13981 −0.0480373 −0.0240186 0.999712i \(-0.507646\pi\)
−0.0240186 + 0.999712i \(0.507646\pi\)
\(564\) −0.886703 −0.0373369
\(565\) −8.70810 −0.366353
\(566\) −32.7713 −1.37748
\(567\) −4.62871 −0.194388
\(568\) 8.70323 0.365179
\(569\) 16.1500 0.677041 0.338521 0.940959i \(-0.390074\pi\)
0.338521 + 0.940959i \(0.390074\pi\)
\(570\) 0 0
\(571\) −13.3743 −0.559698 −0.279849 0.960044i \(-0.590284\pi\)
−0.279849 + 0.960044i \(0.590284\pi\)
\(572\) 4.50965 0.188558
\(573\) −0.403335 −0.0168496
\(574\) −6.60934 −0.275868
\(575\) −12.0610 −0.502979
\(576\) −2.48481 −0.103534
\(577\) 8.76918 0.365066 0.182533 0.983200i \(-0.441570\pi\)
0.182533 + 0.983200i \(0.441570\pi\)
\(578\) −15.0546 −0.626189
\(579\) −12.4668 −0.518102
\(580\) −2.30271 −0.0956149
\(581\) 13.6474 0.566190
\(582\) −7.93892 −0.329079
\(583\) −18.7413 −0.776185
\(584\) −4.32196 −0.178844
\(585\) −12.7462 −0.526992
\(586\) 16.5383 0.683190
\(587\) 30.1409 1.24405 0.622024 0.782998i \(-0.286311\pi\)
0.622024 + 0.782998i \(0.286311\pi\)
\(588\) 0.717767 0.0296002
\(589\) 0 0
\(590\) 4.92046 0.202572
\(591\) 14.7034 0.604818
\(592\) 7.90419 0.324860
\(593\) 17.0917 0.701871 0.350935 0.936400i \(-0.385864\pi\)
0.350935 + 0.936400i \(0.385864\pi\)
\(594\) −10.0933 −0.414133
\(595\) −4.06760 −0.166755
\(596\) −10.5028 −0.430210
\(597\) 9.65605 0.395196
\(598\) −6.05295 −0.247524
\(599\) −3.36233 −0.137381 −0.0686906 0.997638i \(-0.521882\pi\)
−0.0686906 + 0.997638i \(0.521882\pi\)
\(600\) 2.51568 0.102702
\(601\) 32.7613 1.33636 0.668181 0.743999i \(-0.267073\pi\)
0.668181 + 0.743999i \(0.267073\pi\)
\(602\) −12.2294 −0.498435
\(603\) −6.55615 −0.266987
\(604\) −20.8828 −0.849708
\(605\) −12.9100 −0.524867
\(606\) −10.8201 −0.439534
\(607\) 41.7569 1.69486 0.847431 0.530906i \(-0.178148\pi\)
0.847431 + 0.530906i \(0.178148\pi\)
\(608\) 0 0
\(609\) 0.566747 0.0229658
\(610\) 14.0157 0.567480
\(611\) −2.17295 −0.0879081
\(612\) −3.46575 −0.140095
\(613\) 22.3997 0.904716 0.452358 0.891836i \(-0.350583\pi\)
0.452358 + 0.891836i \(0.350583\pi\)
\(614\) 10.7452 0.433642
\(615\) 13.8349 0.557876
\(616\) −2.56382 −0.103299
\(617\) −37.3977 −1.50558 −0.752788 0.658263i \(-0.771292\pi\)
−0.752788 + 0.658263i \(0.771292\pi\)
\(618\) 9.35582 0.376346
\(619\) 21.9167 0.880906 0.440453 0.897776i \(-0.354818\pi\)
0.440453 + 0.897776i \(0.354818\pi\)
\(620\) 14.5776 0.585451
\(621\) 13.5474 0.543640
\(622\) −32.3009 −1.29515
\(623\) −1.50287 −0.0602113
\(624\) 1.26252 0.0505413
\(625\) −30.2402 −1.20961
\(626\) −11.5120 −0.460113
\(627\) 0 0
\(628\) 14.4157 0.575247
\(629\) 11.0246 0.439578
\(630\) 7.24648 0.288707
\(631\) 17.8262 0.709650 0.354825 0.934933i \(-0.384541\pi\)
0.354825 + 0.934933i \(0.384541\pi\)
\(632\) −4.23183 −0.168333
\(633\) −3.74620 −0.148898
\(634\) −12.4053 −0.492677
\(635\) 12.1097 0.480559
\(636\) −5.24682 −0.208050
\(637\) 1.75896 0.0696924
\(638\) −2.02438 −0.0801461
\(639\) −21.6259 −0.855506
\(640\) 2.91631 0.115277
\(641\) 9.79762 0.386983 0.193491 0.981102i \(-0.438019\pi\)
0.193491 + 0.981102i \(0.438019\pi\)
\(642\) 11.0358 0.435549
\(643\) 27.1491 1.07066 0.535328 0.844645i \(-0.320188\pi\)
0.535328 + 0.844645i \(0.320188\pi\)
\(644\) 3.44122 0.135603
\(645\) 25.5991 1.00796
\(646\) 0 0
\(647\) 25.7144 1.01094 0.505469 0.862845i \(-0.331320\pi\)
0.505469 + 0.862845i \(0.331320\pi\)
\(648\) 4.62871 0.181833
\(649\) 4.32573 0.169800
\(650\) 6.16492 0.241808
\(651\) −3.58787 −0.140620
\(652\) 0.718973 0.0281572
\(653\) −5.60452 −0.219322 −0.109661 0.993969i \(-0.534976\pi\)
−0.109661 + 0.993969i \(0.534976\pi\)
\(654\) 14.2878 0.558699
\(655\) 55.2288 2.15797
\(656\) 6.60934 0.258051
\(657\) 10.7392 0.418978
\(658\) 1.23536 0.0481595
\(659\) 24.3115 0.947040 0.473520 0.880783i \(-0.342983\pi\)
0.473520 + 0.880783i \(0.342983\pi\)
\(660\) 5.36667 0.208897
\(661\) −10.6106 −0.412705 −0.206352 0.978478i \(-0.566159\pi\)
−0.206352 + 0.978478i \(0.566159\pi\)
\(662\) 4.28173 0.166414
\(663\) 1.76093 0.0683890
\(664\) −13.6474 −0.529622
\(665\) 0 0
\(666\) −19.6404 −0.761051
\(667\) 2.71717 0.105209
\(668\) −14.5419 −0.562642
\(669\) −13.0842 −0.505864
\(670\) 7.69467 0.297271
\(671\) 12.3216 0.475672
\(672\) −0.717767 −0.0276885
\(673\) −33.2603 −1.28209 −0.641045 0.767503i \(-0.721499\pi\)
−0.641045 + 0.767503i \(0.721499\pi\)
\(674\) 34.6005 1.33276
\(675\) −13.7980 −0.531087
\(676\) −9.90607 −0.381003
\(677\) −51.6253 −1.98412 −0.992061 0.125758i \(-0.959864\pi\)
−0.992061 + 0.125758i \(0.959864\pi\)
\(678\) −2.14325 −0.0823111
\(679\) 11.0606 0.424466
\(680\) 4.06760 0.155985
\(681\) −7.70813 −0.295376
\(682\) 12.8156 0.490736
\(683\) −10.1408 −0.388026 −0.194013 0.980999i \(-0.562150\pi\)
−0.194013 + 0.980999i \(0.562150\pi\)
\(684\) 0 0
\(685\) −47.5616 −1.81724
\(686\) −1.00000 −0.0381802
\(687\) −11.9851 −0.457259
\(688\) 12.2294 0.466243
\(689\) −12.8578 −0.489844
\(690\) −7.20327 −0.274224
\(691\) 7.25018 0.275810 0.137905 0.990445i \(-0.455963\pi\)
0.137905 + 0.990445i \(0.455963\pi\)
\(692\) −24.2394 −0.921444
\(693\) 6.37060 0.241999
\(694\) −8.61568 −0.327047
\(695\) −10.5635 −0.400698
\(696\) −0.566747 −0.0214825
\(697\) 9.21853 0.349177
\(698\) −30.1120 −1.13976
\(699\) −12.0915 −0.457344
\(700\) −3.50487 −0.132472
\(701\) −13.1064 −0.495022 −0.247511 0.968885i \(-0.579613\pi\)
−0.247511 + 0.968885i \(0.579613\pi\)
\(702\) −6.92469 −0.261356
\(703\) 0 0
\(704\) 2.56382 0.0966276
\(705\) −2.58590 −0.0973907
\(706\) 4.02781 0.151589
\(707\) 15.0746 0.566938
\(708\) 1.21103 0.0455134
\(709\) −47.4592 −1.78237 −0.891184 0.453642i \(-0.850125\pi\)
−0.891184 + 0.453642i \(0.850125\pi\)
\(710\) 25.3813 0.952544
\(711\) 10.5153 0.394354
\(712\) 1.50287 0.0563225
\(713\) −17.2014 −0.644199
\(714\) −1.00112 −0.0374661
\(715\) 13.1515 0.491839
\(716\) 9.38770 0.350835
\(717\) −16.6536 −0.621939
\(718\) 33.6389 1.25539
\(719\) 35.7207 1.33216 0.666080 0.745881i \(-0.267971\pi\)
0.666080 + 0.745881i \(0.267971\pi\)
\(720\) −7.24648 −0.270060
\(721\) −13.0346 −0.485434
\(722\) 0 0
\(723\) 1.26742 0.0471359
\(724\) 23.7807 0.883801
\(725\) −2.76744 −0.102780
\(726\) −3.17744 −0.117926
\(727\) 22.2094 0.823701 0.411851 0.911251i \(-0.364883\pi\)
0.411851 + 0.911251i \(0.364883\pi\)
\(728\) −1.75896 −0.0651913
\(729\) −3.02430 −0.112011
\(730\) −12.6042 −0.466501
\(731\) 17.0573 0.630888
\(732\) 3.44957 0.127500
\(733\) −35.2245 −1.30105 −0.650523 0.759486i \(-0.725450\pi\)
−0.650523 + 0.759486i \(0.725450\pi\)
\(734\) 10.8822 0.401670
\(735\) 2.09323 0.0772101
\(736\) −3.44122 −0.126845
\(737\) 6.76462 0.249178
\(738\) −16.4229 −0.604537
\(739\) 3.67115 0.135045 0.0675227 0.997718i \(-0.478490\pi\)
0.0675227 + 0.997718i \(0.478490\pi\)
\(740\) 23.0511 0.847375
\(741\) 0 0
\(742\) 7.30992 0.268356
\(743\) −36.1232 −1.32523 −0.662615 0.748960i \(-0.730554\pi\)
−0.662615 + 0.748960i \(0.730554\pi\)
\(744\) 3.58787 0.131538
\(745\) −30.6293 −1.12217
\(746\) −10.9132 −0.399559
\(747\) 33.9112 1.24075
\(748\) 3.57595 0.130750
\(749\) −15.3752 −0.561798
\(750\) −3.12965 −0.114279
\(751\) −30.9862 −1.13070 −0.565351 0.824851i \(-0.691259\pi\)
−0.565351 + 0.824851i \(0.691259\pi\)
\(752\) −1.23536 −0.0450490
\(753\) 3.21487 0.117156
\(754\) −1.38887 −0.0505796
\(755\) −60.9007 −2.21640
\(756\) 3.93682 0.143181
\(757\) 53.0879 1.92951 0.964757 0.263143i \(-0.0847592\pi\)
0.964757 + 0.263143i \(0.0847592\pi\)
\(758\) −30.4270 −1.10516
\(759\) −6.33261 −0.229859
\(760\) 0 0
\(761\) −4.65429 −0.168718 −0.0843589 0.996435i \(-0.526884\pi\)
−0.0843589 + 0.996435i \(0.526884\pi\)
\(762\) 2.98046 0.107971
\(763\) −19.9059 −0.720643
\(764\) −0.561930 −0.0203299
\(765\) −10.1072 −0.365427
\(766\) −3.04363 −0.109971
\(767\) 2.96775 0.107159
\(768\) 0.717767 0.0259002
\(769\) −23.8596 −0.860399 −0.430200 0.902734i \(-0.641557\pi\)
−0.430200 + 0.902734i \(0.641557\pi\)
\(770\) −7.47689 −0.269449
\(771\) 17.2906 0.622707
\(772\) −17.3688 −0.625118
\(773\) −30.2120 −1.08665 −0.543324 0.839523i \(-0.682835\pi\)
−0.543324 + 0.839523i \(0.682835\pi\)
\(774\) −30.3878 −1.09227
\(775\) 17.5196 0.629323
\(776\) −11.0606 −0.397051
\(777\) −5.67337 −0.203531
\(778\) 15.7728 0.565482
\(779\) 0 0
\(780\) 3.68191 0.131833
\(781\) 22.3135 0.798440
\(782\) −4.79972 −0.171638
\(783\) 3.10850 0.111089
\(784\) 1.00000 0.0357143
\(785\) 42.0405 1.50049
\(786\) 13.5930 0.484846
\(787\) 2.65870 0.0947723 0.0473862 0.998877i \(-0.484911\pi\)
0.0473862 + 0.998877i \(0.484911\pi\)
\(788\) 20.4849 0.729746
\(789\) 6.37974 0.227125
\(790\) −12.3413 −0.439085
\(791\) 2.98600 0.106170
\(792\) −6.37060 −0.226369
\(793\) 8.45350 0.300192
\(794\) −29.8753 −1.06024
\(795\) −15.3014 −0.542683
\(796\) 13.4529 0.476825
\(797\) −12.1882 −0.431729 −0.215864 0.976423i \(-0.569257\pi\)
−0.215864 + 0.976423i \(0.569257\pi\)
\(798\) 0 0
\(799\) −1.72305 −0.0609572
\(800\) 3.50487 0.123916
\(801\) −3.73435 −0.131947
\(802\) −10.0388 −0.354484
\(803\) −11.0807 −0.391030
\(804\) 1.89382 0.0667900
\(805\) 10.0357 0.353710
\(806\) 8.79240 0.309699
\(807\) −8.73870 −0.307617
\(808\) −15.0746 −0.530322
\(809\) 17.3289 0.609253 0.304626 0.952472i \(-0.401468\pi\)
0.304626 + 0.952472i \(0.401468\pi\)
\(810\) 13.4988 0.474298
\(811\) 24.6678 0.866203 0.433102 0.901345i \(-0.357419\pi\)
0.433102 + 0.901345i \(0.357419\pi\)
\(812\) 0.789597 0.0277094
\(813\) 7.77881 0.272815
\(814\) 20.2649 0.710285
\(815\) 2.09675 0.0734459
\(816\) 1.00112 0.0350463
\(817\) 0 0
\(818\) −10.0191 −0.350309
\(819\) 4.37067 0.152724
\(820\) 19.2749 0.673108
\(821\) −27.8333 −0.971389 −0.485694 0.874129i \(-0.661433\pi\)
−0.485694 + 0.874129i \(0.661433\pi\)
\(822\) −11.7059 −0.408292
\(823\) −19.0395 −0.663676 −0.331838 0.943336i \(-0.607669\pi\)
−0.331838 + 0.943336i \(0.607669\pi\)
\(824\) 13.0346 0.454082
\(825\) 6.44976 0.224552
\(826\) −1.68722 −0.0587059
\(827\) 21.3800 0.743455 0.371728 0.928342i \(-0.378766\pi\)
0.371728 + 0.928342i \(0.378766\pi\)
\(828\) 8.55077 0.297160
\(829\) 8.19845 0.284744 0.142372 0.989813i \(-0.454527\pi\)
0.142372 + 0.989813i \(0.454527\pi\)
\(830\) −39.8001 −1.38148
\(831\) −8.13050 −0.282044
\(832\) 1.75896 0.0609808
\(833\) 1.39477 0.0483261
\(834\) −2.59991 −0.0900277
\(835\) −42.4086 −1.46761
\(836\) 0 0
\(837\) −19.6788 −0.680198
\(838\) 2.90178 0.100240
\(839\) 8.97637 0.309899 0.154949 0.987922i \(-0.450479\pi\)
0.154949 + 0.987922i \(0.450479\pi\)
\(840\) −2.09323 −0.0722234
\(841\) −28.3765 −0.978501
\(842\) 19.6306 0.676516
\(843\) 12.0666 0.415595
\(844\) −5.21924 −0.179654
\(845\) −28.8892 −0.993818
\(846\) 3.06964 0.105536
\(847\) 4.42683 0.152108
\(848\) −7.30992 −0.251024
\(849\) −23.5221 −0.807278
\(850\) 4.88851 0.167674
\(851\) −27.2000 −0.932405
\(852\) 6.24689 0.214015
\(853\) −14.6720 −0.502360 −0.251180 0.967940i \(-0.580819\pi\)
−0.251180 + 0.967940i \(0.580819\pi\)
\(854\) −4.80597 −0.164457
\(855\) 0 0
\(856\) 15.3752 0.525513
\(857\) −1.20492 −0.0411594 −0.0205797 0.999788i \(-0.506551\pi\)
−0.0205797 + 0.999788i \(0.506551\pi\)
\(858\) 3.23688 0.110505
\(859\) 6.65515 0.227071 0.113535 0.993534i \(-0.463782\pi\)
0.113535 + 0.993534i \(0.463782\pi\)
\(860\) 35.6649 1.21616
\(861\) −4.74397 −0.161674
\(862\) −0.757246 −0.0257919
\(863\) −12.4694 −0.424462 −0.212231 0.977220i \(-0.568073\pi\)
−0.212231 + 0.977220i \(0.568073\pi\)
\(864\) −3.93682 −0.133933
\(865\) −70.6897 −2.40352
\(866\) −2.51099 −0.0853267
\(867\) −10.8057 −0.366981
\(868\) −4.99865 −0.169665
\(869\) −10.8496 −0.368049
\(870\) −1.65281 −0.0560356
\(871\) 4.64099 0.157254
\(872\) 19.9059 0.674100
\(873\) 27.4834 0.930173
\(874\) 0 0
\(875\) 4.36026 0.147404
\(876\) −3.10216 −0.104812
\(877\) 32.2864 1.09023 0.545117 0.838360i \(-0.316485\pi\)
0.545117 + 0.838360i \(0.316485\pi\)
\(878\) 22.3934 0.755740
\(879\) 11.8706 0.400387
\(880\) 7.47689 0.252046
\(881\) 44.5135 1.49970 0.749850 0.661608i \(-0.230126\pi\)
0.749850 + 0.661608i \(0.230126\pi\)
\(882\) −2.48481 −0.0836679
\(883\) −35.0278 −1.17878 −0.589389 0.807849i \(-0.700631\pi\)
−0.589389 + 0.807849i \(0.700631\pi\)
\(884\) 2.45335 0.0825150
\(885\) 3.53175 0.118718
\(886\) 2.45606 0.0825129
\(887\) 12.1340 0.407421 0.203710 0.979031i \(-0.434700\pi\)
0.203710 + 0.979031i \(0.434700\pi\)
\(888\) 5.67337 0.190386
\(889\) −4.15241 −0.139267
\(890\) 4.38285 0.146913
\(891\) 11.8672 0.397565
\(892\) −18.2290 −0.610352
\(893\) 0 0
\(894\) −7.53855 −0.252127
\(895\) 27.3775 0.915128
\(896\) −1.00000 −0.0334077
\(897\) −4.34461 −0.145062
\(898\) 4.08691 0.136382
\(899\) −3.94692 −0.131637
\(900\) −8.70894 −0.290298
\(901\) −10.1957 −0.339668
\(902\) 16.9451 0.564212
\(903\) −8.77790 −0.292110
\(904\) −2.98600 −0.0993128
\(905\) 69.3518 2.30533
\(906\) −14.9890 −0.497976
\(907\) −8.50269 −0.282327 −0.141164 0.989986i \(-0.545084\pi\)
−0.141164 + 0.989986i \(0.545084\pi\)
\(908\) −10.7390 −0.356387
\(909\) 37.4575 1.24239
\(910\) −5.12966 −0.170047
\(911\) −20.9608 −0.694462 −0.347231 0.937780i \(-0.612878\pi\)
−0.347231 + 0.937780i \(0.612878\pi\)
\(912\) 0 0
\(913\) −34.9895 −1.15798
\(914\) 10.5514 0.349011
\(915\) 10.0600 0.332574
\(916\) −16.6977 −0.551707
\(917\) −18.9379 −0.625384
\(918\) −5.49097 −0.181229
\(919\) 4.33153 0.142884 0.0714420 0.997445i \(-0.477240\pi\)
0.0714420 + 0.997445i \(0.477240\pi\)
\(920\) −10.0357 −0.330866
\(921\) 7.71257 0.254138
\(922\) −17.6728 −0.582023
\(923\) 15.3086 0.503889
\(924\) −1.84023 −0.0605390
\(925\) 27.7032 0.910875
\(926\) 27.5515 0.905400
\(927\) −32.3885 −1.06378
\(928\) −0.789597 −0.0259198
\(929\) −33.7598 −1.10762 −0.553811 0.832642i \(-0.686827\pi\)
−0.553811 + 0.832642i \(0.686827\pi\)
\(930\) 10.4633 0.343106
\(931\) 0 0
\(932\) −16.8460 −0.551810
\(933\) −23.1845 −0.759027
\(934\) −34.2363 −1.12024
\(935\) 10.4286 0.341051
\(936\) −4.37067 −0.142860
\(937\) −43.5661 −1.42324 −0.711621 0.702564i \(-0.752039\pi\)
−0.711621 + 0.702564i \(0.752039\pi\)
\(938\) −2.63849 −0.0861498
\(939\) −8.26295 −0.269651
\(940\) −3.60270 −0.117507
\(941\) −30.6024 −0.997609 −0.498805 0.866714i \(-0.666227\pi\)
−0.498805 + 0.866714i \(0.666227\pi\)
\(942\) 10.3471 0.337126
\(943\) −22.7442 −0.740652
\(944\) 1.68722 0.0549143
\(945\) 11.4810 0.373477
\(946\) 31.3541 1.01941
\(947\) 7.73067 0.251213 0.125606 0.992080i \(-0.459912\pi\)
0.125606 + 0.992080i \(0.459912\pi\)
\(948\) −3.03747 −0.0986524
\(949\) −7.60213 −0.246776
\(950\) 0 0
\(951\) −8.90412 −0.288736
\(952\) −1.39477 −0.0452049
\(953\) −55.9729 −1.81314 −0.906571 0.422054i \(-0.861309\pi\)
−0.906571 + 0.422054i \(0.861309\pi\)
\(954\) 18.1638 0.588073
\(955\) −1.63876 −0.0530292
\(956\) −23.2019 −0.750403
\(957\) −1.45304 −0.0469700
\(958\) 24.7601 0.799963
\(959\) 16.3088 0.526639
\(960\) 2.09323 0.0675588
\(961\) −6.01351 −0.193984
\(962\) 13.9031 0.448255
\(963\) −38.2044 −1.23112
\(964\) 1.76578 0.0568720
\(965\) −50.6530 −1.63058
\(966\) 2.46999 0.0794707
\(967\) −35.4513 −1.14004 −0.570019 0.821631i \(-0.693064\pi\)
−0.570019 + 0.821631i \(0.693064\pi\)
\(968\) −4.42683 −0.142284
\(969\) 0 0
\(970\) −32.2561 −1.03568
\(971\) −28.3291 −0.909123 −0.454561 0.890715i \(-0.650204\pi\)
−0.454561 + 0.890715i \(0.650204\pi\)
\(972\) 15.1328 0.485385
\(973\) 3.62222 0.116123
\(974\) −22.2019 −0.711394
\(975\) 4.42498 0.141713
\(976\) 4.80597 0.153835
\(977\) −15.7603 −0.504216 −0.252108 0.967699i \(-0.581124\pi\)
−0.252108 + 0.967699i \(0.581124\pi\)
\(978\) 0.516056 0.0165016
\(979\) 3.85309 0.123145
\(980\) 2.91631 0.0931581
\(981\) −49.4625 −1.57922
\(982\) −31.0334 −0.990316
\(983\) −20.4920 −0.653594 −0.326797 0.945094i \(-0.605969\pi\)
−0.326797 + 0.945094i \(0.605969\pi\)
\(984\) 4.74397 0.151232
\(985\) 59.7405 1.90349
\(986\) −1.10131 −0.0350729
\(987\) 0.886703 0.0282241
\(988\) 0 0
\(989\) −42.0842 −1.33820
\(990\) −18.5787 −0.590469
\(991\) 6.85697 0.217819 0.108909 0.994052i \(-0.465264\pi\)
0.108909 + 0.994052i \(0.465264\pi\)
\(992\) 4.99865 0.158707
\(993\) 3.07328 0.0975277
\(994\) −8.70323 −0.276050
\(995\) 39.2328 1.24376
\(996\) −9.79567 −0.310388
\(997\) 11.7171 0.371083 0.185541 0.982636i \(-0.440596\pi\)
0.185541 + 0.982636i \(0.440596\pi\)
\(998\) −24.0668 −0.761822
\(999\) −31.1174 −0.984510
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 5054.2.a.bi.1.5 yes 8
19.18 odd 2 5054.2.a.bf.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.bf.1.4 8 19.18 odd 2
5054.2.a.bi.1.5 yes 8 1.1 even 1 trivial