Properties

Label 2842.2.a.p.1.2
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 1.2
Root \(2.13578\) of defining polynomial
Character \(\chi\) \(=\) 2842.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.19935 q^{3} +1.00000 q^{4} -2.13578 q^{5} +1.19935 q^{6} -1.00000 q^{8} -1.56155 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.19935 q^{3} +1.00000 q^{4} -2.13578 q^{5} +1.19935 q^{6} -1.00000 q^{8} -1.56155 q^{9} +2.13578 q^{10} -4.56155 q^{11} -1.19935 q^{12} +4.53448 q^{13} +2.56155 q^{15} +1.00000 q^{16} -5.20798 q^{17} +1.56155 q^{18} +6.14441 q^{19} -2.13578 q^{20} +4.56155 q^{22} +8.24621 q^{23} +1.19935 q^{24} -0.438447 q^{25} -4.53448 q^{26} +5.47091 q^{27} +1.00000 q^{29} -2.56155 q^{30} +4.53448 q^{31} -1.00000 q^{32} +5.47091 q^{33} +5.20798 q^{34} -1.56155 q^{36} -1.12311 q^{37} -6.14441 q^{38} -5.43845 q^{39} +2.13578 q^{40} -5.20798 q^{41} +7.68466 q^{43} -4.56155 q^{44} +3.33513 q^{45} -8.24621 q^{46} -2.13578 q^{47} -1.19935 q^{48} +0.438447 q^{50} +6.24621 q^{51} +4.53448 q^{52} -7.43845 q^{53} -5.47091 q^{54} +9.74247 q^{55} -7.36932 q^{57} -1.00000 q^{58} +5.20798 q^{59} +2.56155 q^{60} +4.27156 q^{61} -4.53448 q^{62} +1.00000 q^{64} -9.68466 q^{65} -5.47091 q^{66} -13.1231 q^{67} -5.20798 q^{68} -9.89012 q^{69} -13.3693 q^{71} +1.56155 q^{72} +16.6757 q^{73} +1.12311 q^{74} +0.525853 q^{75} +6.14441 q^{76} +5.43845 q^{78} -12.8078 q^{79} -2.13578 q^{80} -1.87689 q^{81} +5.20798 q^{82} +10.0054 q^{83} +11.1231 q^{85} -7.68466 q^{86} -1.19935 q^{87} +4.56155 q^{88} -2.80928 q^{89} -3.33513 q^{90} +8.24621 q^{92} -5.43845 q^{93} +2.13578 q^{94} -13.1231 q^{95} +1.19935 q^{96} -13.7511 q^{97} +7.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 2 q^{9}+O(q^{10}) \) 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
\(3\) −1.19935 −0.692447 −0.346223 0.938152i \(-0.612536\pi\)
−0.346223 + 0.938152i \(0.612536\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.13578 −0.955149 −0.477575 0.878591i \(-0.658484\pi\)
−0.477575 + 0.878591i \(0.658484\pi\)
\(6\) 1.19935 0.489634
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) −1.56155 −0.520518
\(10\) 2.13578 0.675393
\(11\) −4.56155 −1.37536 −0.687680 0.726014i \(-0.741371\pi\)
−0.687680 + 0.726014i \(0.741371\pi\)
\(12\) −1.19935 −0.346223
\(13\) 4.53448 1.25764 0.628820 0.777551i \(-0.283538\pi\)
0.628820 + 0.777551i \(0.283538\pi\)
\(14\) 0 0
\(15\) 2.56155 0.661390
\(16\) 1.00000 0.250000
\(17\) −5.20798 −1.26312 −0.631561 0.775326i \(-0.717585\pi\)
−0.631561 + 0.775326i \(0.717585\pi\)
\(18\) 1.56155 0.368062
\(19\) 6.14441 1.40962 0.704812 0.709394i \(-0.251031\pi\)
0.704812 + 0.709394i \(0.251031\pi\)
\(20\) −2.13578 −0.477575
\(21\) 0 0
\(22\) 4.56155 0.972526
\(23\) 8.24621 1.71945 0.859727 0.510754i \(-0.170634\pi\)
0.859727 + 0.510754i \(0.170634\pi\)
\(24\) 1.19935 0.244817
\(25\) −0.438447 −0.0876894
\(26\) −4.53448 −0.889286
\(27\) 5.47091 1.05288
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) −2.56155 −0.467673
\(31\) 4.53448 0.814418 0.407209 0.913335i \(-0.366502\pi\)
0.407209 + 0.913335i \(0.366502\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.47091 0.952363
\(34\) 5.20798 0.893162
\(35\) 0 0
\(36\) −1.56155 −0.260259
\(37\) −1.12311 −0.184637 −0.0923187 0.995730i \(-0.529428\pi\)
−0.0923187 + 0.995730i \(0.529428\pi\)
\(38\) −6.14441 −0.996755
\(39\) −5.43845 −0.870849
\(40\) 2.13578 0.337696
\(41\) −5.20798 −0.813351 −0.406675 0.913573i \(-0.633312\pi\)
−0.406675 + 0.913573i \(0.633312\pi\)
\(42\) 0 0
\(43\) 7.68466 1.17190 0.585950 0.810347i \(-0.300722\pi\)
0.585950 + 0.810347i \(0.300722\pi\)
\(44\) −4.56155 −0.687680
\(45\) 3.33513 0.497172
\(46\) −8.24621 −1.21584
\(47\) −2.13578 −0.311535 −0.155768 0.987794i \(-0.549785\pi\)
−0.155768 + 0.987794i \(0.549785\pi\)
\(48\) −1.19935 −0.173112
\(49\) 0 0
\(50\) 0.438447 0.0620058
\(51\) 6.24621 0.874645
\(52\) 4.53448 0.628820
\(53\) −7.43845 −1.02175 −0.510875 0.859655i \(-0.670678\pi\)
−0.510875 + 0.859655i \(0.670678\pi\)
\(54\) −5.47091 −0.744497
\(55\) 9.74247 1.31367
\(56\) 0 0
\(57\) −7.36932 −0.976090
\(58\) −1.00000 −0.131306
\(59\) 5.20798 0.678022 0.339011 0.940782i \(-0.389908\pi\)
0.339011 + 0.940782i \(0.389908\pi\)
\(60\) 2.56155 0.330695
\(61\) 4.27156 0.546917 0.273459 0.961884i \(-0.411832\pi\)
0.273459 + 0.961884i \(0.411832\pi\)
\(62\) −4.53448 −0.575880
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −9.68466 −1.20123
\(66\) −5.47091 −0.673423
\(67\) −13.1231 −1.60324 −0.801621 0.597832i \(-0.796029\pi\)
−0.801621 + 0.597832i \(0.796029\pi\)
\(68\) −5.20798 −0.631561
\(69\) −9.89012 −1.19063
\(70\) 0 0
\(71\) −13.3693 −1.58665 −0.793323 0.608801i \(-0.791651\pi\)
−0.793323 + 0.608801i \(0.791651\pi\)
\(72\) 1.56155 0.184031
\(73\) 16.6757 1.95174 0.975869 0.218356i \(-0.0700694\pi\)
0.975869 + 0.218356i \(0.0700694\pi\)
\(74\) 1.12311 0.130558
\(75\) 0.525853 0.0607203
\(76\) 6.14441 0.704812
\(77\) 0 0
\(78\) 5.43845 0.615783
\(79\) −12.8078 −1.44099 −0.720493 0.693462i \(-0.756084\pi\)
−0.720493 + 0.693462i \(0.756084\pi\)
\(80\) −2.13578 −0.238787
\(81\) −1.87689 −0.208544
\(82\) 5.20798 0.575126
\(83\) 10.0054 1.09823 0.549117 0.835745i \(-0.314964\pi\)
0.549117 + 0.835745i \(0.314964\pi\)
\(84\) 0 0
\(85\) 11.1231 1.20647
\(86\) −7.68466 −0.828658
\(87\) −1.19935 −0.128584
\(88\) 4.56155 0.486263
\(89\) −2.80928 −0.297783 −0.148891 0.988854i \(-0.547571\pi\)
−0.148891 + 0.988854i \(0.547571\pi\)
\(90\) −3.33513 −0.351554
\(91\) 0 0
\(92\) 8.24621 0.859727
\(93\) −5.43845 −0.563941
\(94\) 2.13578 0.220289
\(95\) −13.1231 −1.34640
\(96\) 1.19935 0.122408
\(97\) −13.7511 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(98\) 0 0
\(99\) 7.12311 0.715899
\(100\) −0.438447 −0.0438447
\(101\) 18.9591 1.88650 0.943250 0.332084i \(-0.107752\pi\)
0.943250 + 0.332084i \(0.107752\pi\)
\(102\) −6.24621 −0.618467
\(103\) −11.4677 −1.12994 −0.564972 0.825110i \(-0.691113\pi\)
−0.564972 + 0.825110i \(0.691113\pi\)
\(104\) −4.53448 −0.444643
\(105\) 0 0
\(106\) 7.43845 0.722486
\(107\) −9.12311 −0.881964 −0.440982 0.897516i \(-0.645370\pi\)
−0.440982 + 0.897516i \(0.645370\pi\)
\(108\) 5.47091 0.526439
\(109\) −1.68466 −0.161361 −0.0806805 0.996740i \(-0.525709\pi\)
−0.0806805 + 0.996740i \(0.525709\pi\)
\(110\) −9.74247 −0.928908
\(111\) 1.34700 0.127852
\(112\) 0 0
\(113\) 5.36932 0.505103 0.252551 0.967583i \(-0.418730\pi\)
0.252551 + 0.967583i \(0.418730\pi\)
\(114\) 7.36932 0.690200
\(115\) −17.6121 −1.64234
\(116\) 1.00000 0.0928477
\(117\) −7.08084 −0.654624
\(118\) −5.20798 −0.479434
\(119\) 0 0
\(120\) −2.56155 −0.233837
\(121\) 9.80776 0.891615
\(122\) −4.27156 −0.386729
\(123\) 6.24621 0.563202
\(124\) 4.53448 0.407209
\(125\) 11.6153 1.03891
\(126\) 0 0
\(127\) 14.2462 1.26415 0.632073 0.774909i \(-0.282204\pi\)
0.632073 + 0.774909i \(0.282204\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −9.21662 −0.811478
\(130\) 9.68466 0.849401
\(131\) 10.9418 0.955991 0.477996 0.878362i \(-0.341363\pi\)
0.477996 + 0.878362i \(0.341363\pi\)
\(132\) 5.47091 0.476182
\(133\) 0 0
\(134\) 13.1231 1.13366
\(135\) −11.6847 −1.00566
\(136\) 5.20798 0.446581
\(137\) 8.24621 0.704521 0.352261 0.935902i \(-0.385413\pi\)
0.352261 + 0.935902i \(0.385413\pi\)
\(138\) 9.89012 0.841903
\(139\) −18.0227 −1.52866 −0.764331 0.644824i \(-0.776931\pi\)
−0.764331 + 0.644824i \(0.776931\pi\)
\(140\) 0 0
\(141\) 2.56155 0.215722
\(142\) 13.3693 1.12193
\(143\) −20.6843 −1.72971
\(144\) −1.56155 −0.130129
\(145\) −2.13578 −0.177367
\(146\) −16.6757 −1.38009
\(147\) 0 0
\(148\) −1.12311 −0.0923187
\(149\) −9.68466 −0.793398 −0.396699 0.917949i \(-0.629844\pi\)
−0.396699 + 0.917949i \(0.629844\pi\)
\(150\) −0.525853 −0.0429357
\(151\) −12.2462 −0.996583 −0.498291 0.867010i \(-0.666039\pi\)
−0.498291 + 0.867010i \(0.666039\pi\)
\(152\) −6.14441 −0.498378
\(153\) 8.13254 0.657477
\(154\) 0 0
\(155\) −9.68466 −0.777890
\(156\) −5.43845 −0.435424
\(157\) −20.8319 −1.66257 −0.831285 0.555847i \(-0.812394\pi\)
−0.831285 + 0.555847i \(0.812394\pi\)
\(158\) 12.8078 1.01893
\(159\) 8.92132 0.707507
\(160\) 2.13578 0.168848
\(161\) 0 0
\(162\) 1.87689 0.147463
\(163\) −8.56155 −0.670593 −0.335296 0.942113i \(-0.608836\pi\)
−0.335296 + 0.942113i \(0.608836\pi\)
\(164\) −5.20798 −0.406675
\(165\) −11.6847 −0.909649
\(166\) −10.0054 −0.776569
\(167\) −4.79741 −0.371235 −0.185617 0.982622i \(-0.559429\pi\)
−0.185617 + 0.982622i \(0.559429\pi\)
\(168\) 0 0
\(169\) 7.56155 0.581658
\(170\) −11.1231 −0.853103
\(171\) −9.59482 −0.733734
\(172\) 7.68466 0.585950
\(173\) −8.13254 −0.618306 −0.309153 0.951012i \(-0.600045\pi\)
−0.309153 + 0.951012i \(0.600045\pi\)
\(174\) 1.19935 0.0909227
\(175\) 0 0
\(176\) −4.56155 −0.343840
\(177\) −6.24621 −0.469494
\(178\) 2.80928 0.210564
\(179\) −18.2462 −1.36379 −0.681893 0.731452i \(-0.738843\pi\)
−0.681893 + 0.731452i \(0.738843\pi\)
\(180\) 3.33513 0.248586
\(181\) 1.08407 0.0805785 0.0402893 0.999188i \(-0.487172\pi\)
0.0402893 + 0.999188i \(0.487172\pi\)
\(182\) 0 0
\(183\) −5.12311 −0.378711
\(184\) −8.24621 −0.607919
\(185\) 2.39871 0.176356
\(186\) 5.43845 0.398766
\(187\) 23.7565 1.73725
\(188\) −2.13578 −0.155768
\(189\) 0 0
\(190\) 13.1231 0.952050
\(191\) 6.24621 0.451960 0.225980 0.974132i \(-0.427442\pi\)
0.225980 + 0.974132i \(0.427442\pi\)
\(192\) −1.19935 −0.0865558
\(193\) 21.3693 1.53820 0.769099 0.639130i \(-0.220706\pi\)
0.769099 + 0.639130i \(0.220706\pi\)
\(194\) 13.7511 0.987272
\(195\) 11.6153 0.831791
\(196\) 0 0
\(197\) −20.2462 −1.44248 −0.721241 0.692684i \(-0.756428\pi\)
−0.721241 + 0.692684i \(0.756428\pi\)
\(198\) −7.12311 −0.506217
\(199\) 14.1617 1.00389 0.501947 0.864898i \(-0.332617\pi\)
0.501947 + 0.864898i \(0.332617\pi\)
\(200\) 0.438447 0.0310029
\(201\) 15.7392 1.11016
\(202\) −18.9591 −1.33396
\(203\) 0 0
\(204\) 6.24621 0.437322
\(205\) 11.1231 0.776871
\(206\) 11.4677 0.798991
\(207\) −12.8769 −0.895006
\(208\) 4.53448 0.314410
\(209\) −28.0281 −1.93874
\(210\) 0 0
\(211\) −7.68466 −0.529034 −0.264517 0.964381i \(-0.585212\pi\)
−0.264517 + 0.964381i \(0.585212\pi\)
\(212\) −7.43845 −0.510875
\(213\) 16.0345 1.09867
\(214\) 9.12311 0.623643
\(215\) −16.4127 −1.11934
\(216\) −5.47091 −0.372248
\(217\) 0 0
\(218\) 1.68466 0.114099
\(219\) −20.0000 −1.35147
\(220\) 9.74247 0.656837
\(221\) −23.6155 −1.58855
\(222\) −1.34700 −0.0904047
\(223\) −6.67026 −0.446674 −0.223337 0.974741i \(-0.571695\pi\)
−0.223337 + 0.974741i \(0.571695\pi\)
\(224\) 0 0
\(225\) 0.684658 0.0456439
\(226\) −5.36932 −0.357162
\(227\) −3.86098 −0.256263 −0.128131 0.991757i \(-0.540898\pi\)
−0.128131 + 0.991757i \(0.540898\pi\)
\(228\) −7.36932 −0.488045
\(229\) −11.9935 −0.792555 −0.396277 0.918131i \(-0.629698\pi\)
−0.396277 + 0.918131i \(0.629698\pi\)
\(230\) 17.6121 1.16131
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) 15.9309 1.04367 0.521833 0.853048i \(-0.325248\pi\)
0.521833 + 0.853048i \(0.325248\pi\)
\(234\) 7.08084 0.462889
\(235\) 4.56155 0.297563
\(236\) 5.20798 0.339011
\(237\) 15.3610 0.997806
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 2.56155 0.165348
\(241\) 17.4644 1.12498 0.562492 0.826803i \(-0.309843\pi\)
0.562492 + 0.826803i \(0.309843\pi\)
\(242\) −9.80776 −0.630467
\(243\) −14.1617 −0.908472
\(244\) 4.27156 0.273459
\(245\) 0 0
\(246\) −6.24621 −0.398244
\(247\) 27.8617 1.77280
\(248\) −4.53448 −0.287940
\(249\) −12.0000 −0.760469
\(250\) −11.6153 −0.734617
\(251\) −6.52262 −0.411704 −0.205852 0.978583i \(-0.565997\pi\)
−0.205852 + 0.978583i \(0.565997\pi\)
\(252\) 0 0
\(253\) −37.6155 −2.36487
\(254\) −14.2462 −0.893887
\(255\) −13.3405 −0.835416
\(256\) 1.00000 0.0625000
\(257\) −16.9386 −1.05660 −0.528300 0.849058i \(-0.677170\pi\)
−0.528300 + 0.849058i \(0.677170\pi\)
\(258\) 9.21662 0.573801
\(259\) 0 0
\(260\) −9.68466 −0.600617
\(261\) −1.56155 −0.0966577
\(262\) −10.9418 −0.675988
\(263\) 4.80776 0.296459 0.148230 0.988953i \(-0.452643\pi\)
0.148230 + 0.988953i \(0.452643\pi\)
\(264\) −5.47091 −0.336711
\(265\) 15.8869 0.975923
\(266\) 0 0
\(267\) 3.36932 0.206199
\(268\) −13.1231 −0.801621
\(269\) −7.72197 −0.470817 −0.235408 0.971897i \(-0.575643\pi\)
−0.235408 + 0.971897i \(0.575643\pi\)
\(270\) 11.6847 0.711106
\(271\) −21.0949 −1.28142 −0.640711 0.767782i \(-0.721360\pi\)
−0.640711 + 0.767782i \(0.721360\pi\)
\(272\) −5.20798 −0.315780
\(273\) 0 0
\(274\) −8.24621 −0.498172
\(275\) 2.00000 0.120605
\(276\) −9.89012 −0.595315
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 18.0227 1.08093
\(279\) −7.08084 −0.423919
\(280\) 0 0
\(281\) 16.1771 0.965044 0.482522 0.875884i \(-0.339721\pi\)
0.482522 + 0.875884i \(0.339721\pi\)
\(282\) −2.56155 −0.152538
\(283\) −27.6175 −1.64169 −0.820844 0.571152i \(-0.806497\pi\)
−0.820844 + 0.571152i \(0.806497\pi\)
\(284\) −13.3693 −0.793323
\(285\) 15.7392 0.932312
\(286\) 20.6843 1.22309
\(287\) 0 0
\(288\) 1.56155 0.0920154
\(289\) 10.1231 0.595477
\(290\) 2.13578 0.125417
\(291\) 16.4924 0.966803
\(292\) 16.6757 0.975869
\(293\) 15.7392 0.919496 0.459748 0.888049i \(-0.347940\pi\)
0.459748 + 0.888049i \(0.347940\pi\)
\(294\) 0 0
\(295\) −11.1231 −0.647612
\(296\) 1.12311 0.0652792
\(297\) −24.9559 −1.44809
\(298\) 9.68466 0.561017
\(299\) 37.3923 2.16245
\(300\) 0.525853 0.0303601
\(301\) 0 0
\(302\) 12.2462 0.704690
\(303\) −22.7386 −1.30630
\(304\) 6.14441 0.352406
\(305\) −9.12311 −0.522388
\(306\) −8.13254 −0.464907
\(307\) −25.4817 −1.45432 −0.727159 0.686469i \(-0.759160\pi\)
−0.727159 + 0.686469i \(0.759160\pi\)
\(308\) 0 0
\(309\) 13.7538 0.782426
\(310\) 9.68466 0.550052
\(311\) 12.4041 0.703372 0.351686 0.936118i \(-0.385608\pi\)
0.351686 + 0.936118i \(0.385608\pi\)
\(312\) 5.43845 0.307891
\(313\) −8.92132 −0.504263 −0.252131 0.967693i \(-0.581132\pi\)
−0.252131 + 0.967693i \(0.581132\pi\)
\(314\) 20.8319 1.17561
\(315\) 0 0
\(316\) −12.8078 −0.720493
\(317\) −18.4924 −1.03864 −0.519319 0.854580i \(-0.673814\pi\)
−0.519319 + 0.854580i \(0.673814\pi\)
\(318\) −8.92132 −0.500283
\(319\) −4.56155 −0.255398
\(320\) −2.13578 −0.119394
\(321\) 10.9418 0.610713
\(322\) 0 0
\(323\) −32.0000 −1.78053
\(324\) −1.87689 −0.104272
\(325\) −1.98813 −0.110282
\(326\) 8.56155 0.474181
\(327\) 2.02050 0.111734
\(328\) 5.20798 0.287563
\(329\) 0 0
\(330\) 11.6847 0.643219
\(331\) −17.3002 −0.950904 −0.475452 0.879742i \(-0.657716\pi\)
−0.475452 + 0.879742i \(0.657716\pi\)
\(332\) 10.0054 0.549117
\(333\) 1.75379 0.0961070
\(334\) 4.79741 0.262503
\(335\) 28.0281 1.53134
\(336\) 0 0
\(337\) −7.75379 −0.422376 −0.211188 0.977445i \(-0.567733\pi\)
−0.211188 + 0.977445i \(0.567733\pi\)
\(338\) −7.56155 −0.411294
\(339\) −6.43971 −0.349757
\(340\) 11.1231 0.603235
\(341\) −20.6843 −1.12012
\(342\) 9.59482 0.518829
\(343\) 0 0
\(344\) −7.68466 −0.414329
\(345\) 21.1231 1.13723
\(346\) 8.13254 0.437208
\(347\) −22.8769 −1.22810 −0.614048 0.789269i \(-0.710460\pi\)
−0.614048 + 0.789269i \(0.710460\pi\)
\(348\) −1.19935 −0.0642921
\(349\) 14.1293 0.756324 0.378162 0.925739i \(-0.376556\pi\)
0.378162 + 0.925739i \(0.376556\pi\)
\(350\) 0 0
\(351\) 24.8078 1.32414
\(352\) 4.56155 0.243132
\(353\) 4.79741 0.255340 0.127670 0.991817i \(-0.459250\pi\)
0.127670 + 0.991817i \(0.459250\pi\)
\(354\) 6.24621 0.331982
\(355\) 28.5539 1.51548
\(356\) −2.80928 −0.148891
\(357\) 0 0
\(358\) 18.2462 0.964342
\(359\) −16.3153 −0.861091 −0.430545 0.902569i \(-0.641679\pi\)
−0.430545 + 0.902569i \(0.641679\pi\)
\(360\) −3.33513 −0.175777
\(361\) 18.7538 0.987042
\(362\) −1.08407 −0.0569776
\(363\) −11.7630 −0.617396
\(364\) 0 0
\(365\) −35.6155 −1.86420
\(366\) 5.12311 0.267789
\(367\) 7.60669 0.397066 0.198533 0.980094i \(-0.436382\pi\)
0.198533 + 0.980094i \(0.436382\pi\)
\(368\) 8.24621 0.429863
\(369\) 8.13254 0.423363
\(370\) −2.39871 −0.124703
\(371\) 0 0
\(372\) −5.43845 −0.281970
\(373\) −1.19224 −0.0617316 −0.0308658 0.999524i \(-0.509826\pi\)
−0.0308658 + 0.999524i \(0.509826\pi\)
\(374\) −23.7565 −1.22842
\(375\) −13.9309 −0.719387
\(376\) 2.13578 0.110144
\(377\) 4.53448 0.233538
\(378\) 0 0
\(379\) −0.492423 −0.0252940 −0.0126470 0.999920i \(-0.504026\pi\)
−0.0126470 + 0.999920i \(0.504026\pi\)
\(380\) −13.1231 −0.673201
\(381\) −17.0862 −0.875354
\(382\) −6.24621 −0.319584
\(383\) −6.14441 −0.313965 −0.156982 0.987601i \(-0.550177\pi\)
−0.156982 + 0.987601i \(0.550177\pi\)
\(384\) 1.19935 0.0612042
\(385\) 0 0
\(386\) −21.3693 −1.08767
\(387\) −12.0000 −0.609994
\(388\) −13.7511 −0.698106
\(389\) 32.9848 1.67240 0.836199 0.548426i \(-0.184773\pi\)
0.836199 + 0.548426i \(0.184773\pi\)
\(390\) −11.6153 −0.588165
\(391\) −42.9461 −2.17188
\(392\) 0 0
\(393\) −13.1231 −0.661973
\(394\) 20.2462 1.01999
\(395\) 27.3546 1.37636
\(396\) 7.12311 0.357950
\(397\) −37.3600 −1.87504 −0.937521 0.347928i \(-0.886885\pi\)
−0.937521 + 0.347928i \(0.886885\pi\)
\(398\) −14.1617 −0.709861
\(399\) 0 0
\(400\) −0.438447 −0.0219224
\(401\) 34.8078 1.73822 0.869108 0.494622i \(-0.164693\pi\)
0.869108 + 0.494622i \(0.164693\pi\)
\(402\) −15.7392 −0.785002
\(403\) 20.5616 1.02424
\(404\) 18.9591 0.943250
\(405\) 4.00863 0.199191
\(406\) 0 0
\(407\) 5.12311 0.253943
\(408\) −6.24621 −0.309234
\(409\) 17.7274 0.876562 0.438281 0.898838i \(-0.355588\pi\)
0.438281 + 0.898838i \(0.355588\pi\)
\(410\) −11.1231 −0.549331
\(411\) −9.89012 −0.487843
\(412\) −11.4677 −0.564972
\(413\) 0 0
\(414\) 12.8769 0.632865
\(415\) −21.3693 −1.04898
\(416\) −4.53448 −0.222321
\(417\) 21.6155 1.05852
\(418\) 28.0281 1.37090
\(419\) 25.2188 1.23202 0.616009 0.787739i \(-0.288749\pi\)
0.616009 + 0.787739i \(0.288749\pi\)
\(420\) 0 0
\(421\) −22.8769 −1.11495 −0.557476 0.830193i \(-0.688230\pi\)
−0.557476 + 0.830193i \(0.688230\pi\)
\(422\) 7.68466 0.374083
\(423\) 3.33513 0.162160
\(424\) 7.43845 0.361243
\(425\) 2.28343 0.110762
\(426\) −16.0345 −0.776875
\(427\) 0 0
\(428\) −9.12311 −0.440982
\(429\) 24.8078 1.19773
\(430\) 16.4127 0.791492
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 5.47091 0.263219
\(433\) −25.7446 −1.23721 −0.618604 0.785703i \(-0.712301\pi\)
−0.618604 + 0.785703i \(0.712301\pi\)
\(434\) 0 0
\(435\) 2.56155 0.122817
\(436\) −1.68466 −0.0806805
\(437\) 50.6681 2.42378
\(438\) 20.0000 0.955637
\(439\) 8.24782 0.393647 0.196824 0.980439i \(-0.436937\pi\)
0.196824 + 0.980439i \(0.436937\pi\)
\(440\) −9.74247 −0.464454
\(441\) 0 0
\(442\) 23.6155 1.12328
\(443\) 24.8769 1.18194 0.590968 0.806695i \(-0.298746\pi\)
0.590968 + 0.806695i \(0.298746\pi\)
\(444\) 1.34700 0.0639258
\(445\) 6.00000 0.284427
\(446\) 6.67026 0.315846
\(447\) 11.6153 0.549386
\(448\) 0 0
\(449\) 5.36932 0.253394 0.126697 0.991941i \(-0.459562\pi\)
0.126697 + 0.991941i \(0.459562\pi\)
\(450\) −0.684658 −0.0322751
\(451\) 23.7565 1.11865
\(452\) 5.36932 0.252551
\(453\) 14.6875 0.690080
\(454\) 3.86098 0.181205
\(455\) 0 0
\(456\) 7.36932 0.345100
\(457\) 4.63068 0.216614 0.108307 0.994117i \(-0.465457\pi\)
0.108307 + 0.994117i \(0.465457\pi\)
\(458\) 11.9935 0.560421
\(459\) −28.4924 −1.32991
\(460\) −17.6121 −0.821168
\(461\) −22.7048 −1.05747 −0.528734 0.848788i \(-0.677333\pi\)
−0.528734 + 0.848788i \(0.677333\pi\)
\(462\) 0 0
\(463\) 1.36932 0.0636376 0.0318188 0.999494i \(-0.489870\pi\)
0.0318188 + 0.999494i \(0.489870\pi\)
\(464\) 1.00000 0.0464238
\(465\) 11.6153 0.538648
\(466\) −15.9309 −0.737983
\(467\) −11.6153 −0.537493 −0.268747 0.963211i \(-0.586609\pi\)
−0.268747 + 0.963211i \(0.586609\pi\)
\(468\) −7.08084 −0.327312
\(469\) 0 0
\(470\) −4.56155 −0.210409
\(471\) 24.9848 1.15124
\(472\) −5.20798 −0.239717
\(473\) −35.0540 −1.61178
\(474\) −15.3610 −0.705555
\(475\) −2.69400 −0.123609
\(476\) 0 0
\(477\) 11.6155 0.531838
\(478\) 0 0
\(479\) −6.70263 −0.306251 −0.153126 0.988207i \(-0.548934\pi\)
−0.153126 + 0.988207i \(0.548934\pi\)
\(480\) −2.56155 −0.116918
\(481\) −5.09271 −0.232207
\(482\) −17.4644 −0.795483
\(483\) 0 0
\(484\) 9.80776 0.445807
\(485\) 29.3693 1.33359
\(486\) 14.1617 0.642387
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −4.27156 −0.193364
\(489\) 10.2683 0.464350
\(490\) 0 0
\(491\) 20.1771 0.910579 0.455289 0.890343i \(-0.349536\pi\)
0.455289 + 0.890343i \(0.349536\pi\)
\(492\) 6.24621 0.281601
\(493\) −5.20798 −0.234556
\(494\) −27.8617 −1.25356
\(495\) −15.2134 −0.683791
\(496\) 4.53448 0.203604
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) −5.12311 −0.229342 −0.114671 0.993404i \(-0.536581\pi\)
−0.114671 + 0.993404i \(0.536581\pi\)
\(500\) 11.6153 0.519453
\(501\) 5.75379 0.257060
\(502\) 6.52262 0.291119
\(503\) −14.6552 −0.653441 −0.326721 0.945121i \(-0.605944\pi\)
−0.326721 + 0.945121i \(0.605944\pi\)
\(504\) 0 0
\(505\) −40.4924 −1.80189
\(506\) 37.6155 1.67221
\(507\) −9.06897 −0.402767
\(508\) 14.2462 0.632073
\(509\) −29.8685 −1.32390 −0.661950 0.749548i \(-0.730271\pi\)
−0.661950 + 0.749548i \(0.730271\pi\)
\(510\) 13.3405 0.590729
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 33.6155 1.48416
\(514\) 16.9386 0.747129
\(515\) 24.4924 1.07927
\(516\) −9.21662 −0.405739
\(517\) 9.74247 0.428473
\(518\) 0 0
\(519\) 9.75379 0.428144
\(520\) 9.68466 0.424700
\(521\) −17.2339 −0.755030 −0.377515 0.926003i \(-0.623221\pi\)
−0.377515 + 0.926003i \(0.623221\pi\)
\(522\) 1.56155 0.0683473
\(523\) 44.4732 1.94468 0.972338 0.233580i \(-0.0750439\pi\)
0.972338 + 0.233580i \(0.0750439\pi\)
\(524\) 10.9418 0.477996
\(525\) 0 0
\(526\) −4.80776 −0.209628
\(527\) −23.6155 −1.02871
\(528\) 5.47091 0.238091
\(529\) 45.0000 1.95652
\(530\) −15.8869 −0.690082
\(531\) −8.13254 −0.352922
\(532\) 0 0
\(533\) −23.6155 −1.02290
\(534\) −3.36932 −0.145805
\(535\) 19.4849 0.842407
\(536\) 13.1231 0.566832
\(537\) 21.8836 0.944349
\(538\) 7.72197 0.332918
\(539\) 0 0
\(540\) −11.6847 −0.502828
\(541\) 0.876894 0.0377006 0.0188503 0.999822i \(-0.493999\pi\)
0.0188503 + 0.999822i \(0.493999\pi\)
\(542\) 21.0949 0.906102
\(543\) −1.30019 −0.0557963
\(544\) 5.20798 0.223291
\(545\) 3.59806 0.154124
\(546\) 0 0
\(547\) −33.6155 −1.43730 −0.718648 0.695374i \(-0.755239\pi\)
−0.718648 + 0.695374i \(0.755239\pi\)
\(548\) 8.24621 0.352261
\(549\) −6.67026 −0.284680
\(550\) −2.00000 −0.0852803
\(551\) 6.14441 0.261761
\(552\) 9.89012 0.420951
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) −2.87689 −0.122117
\(556\) −18.0227 −0.764331
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 7.08084 0.299756
\(559\) 34.8460 1.47383
\(560\) 0 0
\(561\) −28.4924 −1.20295
\(562\) −16.1771 −0.682389
\(563\) 15.3610 0.647390 0.323695 0.946161i \(-0.395075\pi\)
0.323695 + 0.946161i \(0.395075\pi\)
\(564\) 2.56155 0.107861
\(565\) −11.4677 −0.482449
\(566\) 27.6175 1.16085
\(567\) 0 0
\(568\) 13.3693 0.560964
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) −15.7392 −0.659244
\(571\) 25.1231 1.05137 0.525685 0.850680i \(-0.323809\pi\)
0.525685 + 0.850680i \(0.323809\pi\)
\(572\) −20.6843 −0.864854
\(573\) −7.49141 −0.312958
\(574\) 0 0
\(575\) −3.61553 −0.150778
\(576\) −1.56155 −0.0650647
\(577\) 39.9063 1.66132 0.830661 0.556779i \(-0.187963\pi\)
0.830661 + 0.556779i \(0.187963\pi\)
\(578\) −10.1231 −0.421066
\(579\) −25.6294 −1.06512
\(580\) −2.13578 −0.0886834
\(581\) 0 0
\(582\) −16.4924 −0.683633
\(583\) 33.9309 1.40527
\(584\) −16.6757 −0.690044
\(585\) 15.1231 0.625263
\(586\) −15.7392 −0.650182
\(587\) 18.8438 0.777767 0.388884 0.921287i \(-0.372861\pi\)
0.388884 + 0.921287i \(0.372861\pi\)
\(588\) 0 0
\(589\) 27.8617 1.14802
\(590\) 11.1231 0.457931
\(591\) 24.2824 0.998842
\(592\) −1.12311 −0.0461594
\(593\) −28.7016 −1.17863 −0.589316 0.807903i \(-0.700603\pi\)
−0.589316 + 0.807903i \(0.700603\pi\)
\(594\) 24.9559 1.02395
\(595\) 0 0
\(596\) −9.68466 −0.396699
\(597\) −16.9848 −0.695143
\(598\) −37.3923 −1.52909
\(599\) −12.3153 −0.503191 −0.251596 0.967832i \(-0.580955\pi\)
−0.251596 + 0.967832i \(0.580955\pi\)
\(600\) −0.525853 −0.0214679
\(601\) −25.2188 −1.02870 −0.514348 0.857582i \(-0.671966\pi\)
−0.514348 + 0.857582i \(0.671966\pi\)
\(602\) 0 0
\(603\) 20.4924 0.834516
\(604\) −12.2462 −0.498291
\(605\) −20.9472 −0.851626
\(606\) 22.7386 0.923694
\(607\) 0.558221 0.0226575 0.0113287 0.999936i \(-0.496394\pi\)
0.0113287 + 0.999936i \(0.496394\pi\)
\(608\) −6.14441 −0.249189
\(609\) 0 0
\(610\) 9.12311 0.369384
\(611\) −9.68466 −0.391799
\(612\) 8.13254 0.328739
\(613\) −17.0540 −0.688804 −0.344402 0.938822i \(-0.611918\pi\)
−0.344402 + 0.938822i \(0.611918\pi\)
\(614\) 25.4817 1.02836
\(615\) −13.3405 −0.537942
\(616\) 0 0
\(617\) −17.3693 −0.699262 −0.349631 0.936887i \(-0.613693\pi\)
−0.349631 + 0.936887i \(0.613693\pi\)
\(618\) −13.7538 −0.553259
\(619\) −30.8050 −1.23816 −0.619078 0.785329i \(-0.712494\pi\)
−0.619078 + 0.785329i \(0.712494\pi\)
\(620\) −9.68466 −0.388945
\(621\) 45.1143 1.81037
\(622\) −12.4041 −0.497359
\(623\) 0 0
\(624\) −5.43845 −0.217712
\(625\) −22.6155 −0.904621
\(626\) 8.92132 0.356568
\(627\) 33.6155 1.34247
\(628\) −20.8319 −0.831285
\(629\) 5.84912 0.233220
\(630\) 0 0
\(631\) 5.75379 0.229055 0.114527 0.993420i \(-0.463465\pi\)
0.114527 + 0.993420i \(0.463465\pi\)
\(632\) 12.8078 0.509465
\(633\) 9.21662 0.366328
\(634\) 18.4924 0.734428
\(635\) −30.4268 −1.20745
\(636\) 8.92132 0.353753
\(637\) 0 0
\(638\) 4.56155 0.180594
\(639\) 20.8769 0.825877
\(640\) 2.13578 0.0844241
\(641\) −17.3693 −0.686047 −0.343023 0.939327i \(-0.611451\pi\)
−0.343023 + 0.939327i \(0.611451\pi\)
\(642\) −10.9418 −0.431839
\(643\) −15.6240 −0.616149 −0.308074 0.951362i \(-0.599685\pi\)
−0.308074 + 0.951362i \(0.599685\pi\)
\(644\) 0 0
\(645\) 19.6847 0.775083
\(646\) 32.0000 1.25902
\(647\) −15.2134 −0.598100 −0.299050 0.954237i \(-0.596670\pi\)
−0.299050 + 0.954237i \(0.596670\pi\)
\(648\) 1.87689 0.0737314
\(649\) −23.7565 −0.932524
\(650\) 1.98813 0.0779810
\(651\) 0 0
\(652\) −8.56155 −0.335296
\(653\) −9.86174 −0.385920 −0.192960 0.981207i \(-0.561809\pi\)
−0.192960 + 0.981207i \(0.561809\pi\)
\(654\) −2.02050 −0.0790078
\(655\) −23.3693 −0.913115
\(656\) −5.20798 −0.203338
\(657\) −26.0399 −1.01591
\(658\) 0 0
\(659\) 19.3002 0.751829 0.375914 0.926654i \(-0.377329\pi\)
0.375914 + 0.926654i \(0.377329\pi\)
\(660\) −11.6847 −0.454825
\(661\) −4.15628 −0.161661 −0.0808303 0.996728i \(-0.525757\pi\)
−0.0808303 + 0.996728i \(0.525757\pi\)
\(662\) 17.3002 0.672391
\(663\) 28.3234 1.09999
\(664\) −10.0054 −0.388285
\(665\) 0 0
\(666\) −1.75379 −0.0679579
\(667\) 8.24621 0.319295
\(668\) −4.79741 −0.185617
\(669\) 8.00000 0.309298
\(670\) −28.0281 −1.08282
\(671\) −19.4849 −0.752208
\(672\) 0 0
\(673\) −36.6695 −1.41351 −0.706753 0.707461i \(-0.749841\pi\)
−0.706753 + 0.707461i \(0.749841\pi\)
\(674\) 7.75379 0.298665
\(675\) −2.39871 −0.0923262
\(676\) 7.56155 0.290829
\(677\) 38.9699 1.49773 0.748867 0.662720i \(-0.230598\pi\)
0.748867 + 0.662720i \(0.230598\pi\)
\(678\) 6.43971 0.247315
\(679\) 0 0
\(680\) −11.1231 −0.426552
\(681\) 4.63068 0.177448
\(682\) 20.6843 0.792042
\(683\) −5.75379 −0.220163 −0.110081 0.993923i \(-0.535111\pi\)
−0.110081 + 0.993923i \(0.535111\pi\)
\(684\) −9.59482 −0.366867
\(685\) −17.6121 −0.672923
\(686\) 0 0
\(687\) 14.3845 0.548802
\(688\) 7.68466 0.292975
\(689\) −33.7295 −1.28499
\(690\) −21.1231 −0.804143
\(691\) 17.7274 0.674381 0.337190 0.941436i \(-0.390523\pi\)
0.337190 + 0.941436i \(0.390523\pi\)
\(692\) −8.13254 −0.309153
\(693\) 0 0
\(694\) 22.8769 0.868395
\(695\) 38.4924 1.46010
\(696\) 1.19935 0.0454614
\(697\) 27.1231 1.02736
\(698\) −14.1293 −0.534802
\(699\) −19.1067 −0.722683
\(700\) 0 0
\(701\) 25.0540 0.946276 0.473138 0.880988i \(-0.343121\pi\)
0.473138 + 0.880988i \(0.343121\pi\)
\(702\) −24.8078 −0.936309
\(703\) −6.90082 −0.260269
\(704\) −4.56155 −0.171920
\(705\) −5.47091 −0.206046
\(706\) −4.79741 −0.180553
\(707\) 0 0
\(708\) −6.24621 −0.234747
\(709\) 31.9309 1.19919 0.599594 0.800304i \(-0.295329\pi\)
0.599594 + 0.800304i \(0.295329\pi\)
\(710\) −28.5539 −1.07161
\(711\) 20.0000 0.750059
\(712\) 2.80928 0.105282
\(713\) 37.3923 1.40035
\(714\) 0 0
\(715\) 44.1771 1.65213
\(716\) −18.2462 −0.681893
\(717\) 0 0
\(718\) 16.3153 0.608883
\(719\) 10.1207 0.377438 0.188719 0.982031i \(-0.439567\pi\)
0.188719 + 0.982031i \(0.439567\pi\)
\(720\) 3.33513 0.124293
\(721\) 0 0
\(722\) −18.7538 −0.697944
\(723\) −20.9460 −0.778991
\(724\) 1.08407 0.0402893
\(725\) −0.438447 −0.0162835
\(726\) 11.7630 0.436565
\(727\) 20.1261 0.746435 0.373217 0.927744i \(-0.378255\pi\)
0.373217 + 0.927744i \(0.378255\pi\)
\(728\) 0 0
\(729\) 22.6155 0.837612
\(730\) 35.6155 1.31819
\(731\) −40.0216 −1.48025
\(732\) −5.12311 −0.189355
\(733\) −27.2069 −1.00491 −0.502455 0.864603i \(-0.667570\pi\)
−0.502455 + 0.864603i \(0.667570\pi\)
\(734\) −7.60669 −0.280768
\(735\) 0 0
\(736\) −8.24621 −0.303959
\(737\) 59.8617 2.20504
\(738\) −8.13254 −0.299363
\(739\) 15.0540 0.553769 0.276885 0.960903i \(-0.410698\pi\)
0.276885 + 0.960903i \(0.410698\pi\)
\(740\) 2.39871 0.0881782
\(741\) −33.4161 −1.22757
\(742\) 0 0
\(743\) −20.9848 −0.769859 −0.384930 0.922946i \(-0.625774\pi\)
−0.384930 + 0.922946i \(0.625774\pi\)
\(744\) 5.43845 0.199383
\(745\) 20.6843 0.757814
\(746\) 1.19224 0.0436509
\(747\) −15.6240 −0.571651
\(748\) 23.7565 0.868624
\(749\) 0 0
\(750\) 13.9309 0.508683
\(751\) −18.7386 −0.683782 −0.341891 0.939740i \(-0.611067\pi\)
−0.341891 + 0.939740i \(0.611067\pi\)
\(752\) −2.13578 −0.0778839
\(753\) 7.82292 0.285083
\(754\) −4.53448 −0.165136
\(755\) 26.1552 0.951885
\(756\) 0 0
\(757\) 32.4924 1.18096 0.590479 0.807053i \(-0.298939\pi\)
0.590479 + 0.807053i \(0.298939\pi\)
\(758\) 0.492423 0.0178856
\(759\) 45.1143 1.63754
\(760\) 13.1231 0.476025
\(761\) 53.3621 1.93438 0.967188 0.254064i \(-0.0817672\pi\)
0.967188 + 0.254064i \(0.0817672\pi\)
\(762\) 17.0862 0.618969
\(763\) 0 0
\(764\) 6.24621 0.225980
\(765\) −17.3693 −0.627989
\(766\) 6.14441 0.222007
\(767\) 23.6155 0.852707
\(768\) −1.19935 −0.0432779
\(769\) 34.8136 1.25541 0.627705 0.778451i \(-0.283994\pi\)
0.627705 + 0.778451i \(0.283994\pi\)
\(770\) 0 0
\(771\) 20.3153 0.731639
\(772\) 21.3693 0.769099
\(773\) −2.10341 −0.0756545 −0.0378272 0.999284i \(-0.512044\pi\)
−0.0378272 + 0.999284i \(0.512044\pi\)
\(774\) 12.0000 0.431331
\(775\) −1.98813 −0.0714158
\(776\) 13.7511 0.493636
\(777\) 0 0
\(778\) −32.9848 −1.18256
\(779\) −32.0000 −1.14652
\(780\) 11.6153 0.415895
\(781\) 60.9848 2.18221
\(782\) 42.9461 1.53575
\(783\) 5.47091 0.195514
\(784\) 0 0
\(785\) 44.4924 1.58800
\(786\) 13.1231 0.468086
\(787\) −17.7274 −0.631912 −0.315956 0.948774i \(-0.602325\pi\)
−0.315956 + 0.948774i \(0.602325\pi\)
\(788\) −20.2462 −0.721241
\(789\) −5.76621 −0.205282
\(790\) −27.3546 −0.973231
\(791\) 0 0
\(792\) −7.12311 −0.253109
\(793\) 19.3693 0.687825
\(794\) 37.3600 1.32586
\(795\) −19.0540 −0.675775
\(796\) 14.1617 0.501947
\(797\) −23.9871 −0.849665 −0.424833 0.905272i \(-0.639667\pi\)
−0.424833 + 0.905272i \(0.639667\pi\)
\(798\) 0 0
\(799\) 11.1231 0.393507
\(800\) 0.438447 0.0155014
\(801\) 4.38684 0.155001
\(802\) −34.8078 −1.22910
\(803\) −76.0669 −2.68434
\(804\) 15.7392 0.555080
\(805\) 0 0
\(806\) −20.5616 −0.724250
\(807\) 9.26137 0.326016
\(808\) −18.9591 −0.666978
\(809\) −22.9848 −0.808104 −0.404052 0.914736i \(-0.632399\pi\)
−0.404052 + 0.914736i \(0.632399\pi\)
\(810\) −4.00863 −0.140849
\(811\) 9.24898 0.324776 0.162388 0.986727i \(-0.448080\pi\)
0.162388 + 0.986727i \(0.448080\pi\)
\(812\) 0 0
\(813\) 25.3002 0.887316
\(814\) −5.12311 −0.179565
\(815\) 18.2856 0.640516
\(816\) 6.24621 0.218661
\(817\) 47.2177 1.65194
\(818\) −17.7274 −0.619823
\(819\) 0 0
\(820\) 11.1231 0.388436
\(821\) 11.9309 0.416390 0.208195 0.978087i \(-0.433241\pi\)
0.208195 + 0.978087i \(0.433241\pi\)
\(822\) 9.89012 0.344957
\(823\) 9.75379 0.339996 0.169998 0.985444i \(-0.445624\pi\)
0.169998 + 0.985444i \(0.445624\pi\)
\(824\) 11.4677 0.399495
\(825\) −2.39871 −0.0835122
\(826\) 0 0
\(827\) −37.6847 −1.31042 −0.655212 0.755445i \(-0.727421\pi\)
−0.655212 + 0.755445i \(0.727421\pi\)
\(828\) −12.8769 −0.447503
\(829\) −16.2651 −0.564910 −0.282455 0.959281i \(-0.591149\pi\)
−0.282455 + 0.959281i \(0.591149\pi\)
\(830\) 21.3693 0.741740
\(831\) 11.9935 0.416051
\(832\) 4.53448 0.157205
\(833\) 0 0
\(834\) −21.6155 −0.748485
\(835\) 10.2462 0.354585
\(836\) −28.0281 −0.969371
\(837\) 24.8078 0.857482
\(838\) −25.2188 −0.871168
\(839\) 16.2975 0.562651 0.281325 0.959612i \(-0.409226\pi\)
0.281325 + 0.959612i \(0.409226\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 22.8769 0.788390
\(843\) −19.4020 −0.668241
\(844\) −7.68466 −0.264517
\(845\) −16.1498 −0.555570
\(846\) −3.33513 −0.114664
\(847\) 0 0
\(848\) −7.43845 −0.255437
\(849\) 33.1231 1.13678
\(850\) −2.28343 −0.0783209
\(851\) −9.26137 −0.317476
\(852\) 16.0345 0.549334
\(853\) 45.9354 1.57280 0.786400 0.617718i \(-0.211943\pi\)
0.786400 + 0.617718i \(0.211943\pi\)
\(854\) 0 0
\(855\) 20.4924 0.700826
\(856\) 9.12311 0.311821
\(857\) −8.16491 −0.278908 −0.139454 0.990229i \(-0.544535\pi\)
−0.139454 + 0.990229i \(0.544535\pi\)
\(858\) −24.8078 −0.846923
\(859\) −12.4365 −0.424327 −0.212163 0.977234i \(-0.568051\pi\)
−0.212163 + 0.977234i \(0.568051\pi\)
\(860\) −16.4127 −0.559670
\(861\) 0 0
\(862\) −8.00000 −0.272481
\(863\) −14.0000 −0.476566 −0.238283 0.971196i \(-0.576585\pi\)
−0.238283 + 0.971196i \(0.576585\pi\)
\(864\) −5.47091 −0.186124
\(865\) 17.3693 0.590574
\(866\) 25.7446 0.874838
\(867\) −12.1412 −0.412336
\(868\) 0 0
\(869\) 58.4233 1.98187
\(870\) −2.56155 −0.0868448
\(871\) −59.5065 −2.01630
\(872\) 1.68466 0.0570497
\(873\) 21.4731 0.726753
\(874\) −50.6681 −1.71387
\(875\) 0 0
\(876\) −20.0000 −0.675737
\(877\) −18.3153 −0.618465 −0.309233 0.950986i \(-0.600072\pi\)
−0.309233 + 0.950986i \(0.600072\pi\)
\(878\) −8.24782 −0.278351
\(879\) −18.8769 −0.636702
\(880\) 9.74247 0.328419
\(881\) 25.5141 0.859591 0.429795 0.902926i \(-0.358586\pi\)
0.429795 + 0.902926i \(0.358586\pi\)
\(882\) 0 0
\(883\) −26.8769 −0.904480 −0.452240 0.891896i \(-0.649375\pi\)
−0.452240 + 0.891896i \(0.649375\pi\)
\(884\) −23.6155 −0.794276
\(885\) 13.3405 0.448437
\(886\) −24.8769 −0.835756
\(887\) 50.1746 1.68470 0.842350 0.538932i \(-0.181172\pi\)
0.842350 + 0.538932i \(0.181172\pi\)
\(888\) −1.34700 −0.0452024
\(889\) 0 0
\(890\) −6.00000 −0.201120
\(891\) 8.56155 0.286823
\(892\) −6.67026 −0.223337
\(893\) −13.1231 −0.439148
\(894\) −11.6153 −0.388474
\(895\) 38.9699 1.30262
\(896\) 0 0
\(897\) −44.8466 −1.49738
\(898\) −5.36932 −0.179176
\(899\) 4.53448 0.151234
\(900\) 0.684658 0.0228219
\(901\) 38.7393 1.29059
\(902\) −23.7565 −0.791005
\(903\) 0 0
\(904\) −5.36932 −0.178581
\(905\) −2.31534 −0.0769646
\(906\) −14.6875 −0.487960
\(907\) −5.36932 −0.178285 −0.0891426 0.996019i \(-0.528413\pi\)
−0.0891426 + 0.996019i \(0.528413\pi\)
\(908\) −3.86098 −0.128131
\(909\) −29.6056 −0.981956
\(910\) 0 0
\(911\) −53.3002 −1.76591 −0.882957 0.469454i \(-0.844451\pi\)
−0.882957 + 0.469454i \(0.844451\pi\)
\(912\) −7.36932 −0.244022
\(913\) −45.6401 −1.51047
\(914\) −4.63068 −0.153169
\(915\) 10.9418 0.361725
\(916\) −11.9935 −0.396277
\(917\) 0 0
\(918\) 28.4924 0.940390
\(919\) 18.0000 0.593765 0.296883 0.954914i \(-0.404053\pi\)
0.296883 + 0.954914i \(0.404053\pi\)
\(920\) 17.6121 0.580653
\(921\) 30.5616 1.00704
\(922\) 22.7048 0.747743
\(923\) −60.6230 −1.99543
\(924\) 0 0
\(925\) 0.492423 0.0161908
\(926\) −1.36932 −0.0449985
\(927\) 17.9074 0.588156
\(928\) −1.00000 −0.0328266
\(929\) −7.49141 −0.245785 −0.122893 0.992420i \(-0.539217\pi\)
−0.122893 + 0.992420i \(0.539217\pi\)
\(930\) −11.6153 −0.380881
\(931\) 0 0
\(932\) 15.9309 0.521833
\(933\) −14.8769 −0.487048
\(934\) 11.6153 0.380065
\(935\) −50.7386 −1.65933
\(936\) 7.08084 0.231444
\(937\) 57.1078 1.86563 0.932816 0.360354i \(-0.117344\pi\)
0.932816 + 0.360354i \(0.117344\pi\)
\(938\) 0 0
\(939\) 10.6998 0.349175
\(940\) 4.56155 0.148781
\(941\) −2.66163 −0.0867667 −0.0433834 0.999058i \(-0.513814\pi\)
−0.0433834 + 0.999058i \(0.513814\pi\)
\(942\) −24.9848 −0.814050
\(943\) −42.9461 −1.39852
\(944\) 5.20798 0.169505
\(945\) 0 0
\(946\) 35.0540 1.13970
\(947\) −40.3153 −1.31007 −0.655036 0.755597i \(-0.727347\pi\)
−0.655036 + 0.755597i \(0.727347\pi\)
\(948\) 15.3610 0.498903
\(949\) 75.6155 2.45458
\(950\) 2.69400 0.0874049
\(951\) 22.1789 0.719201
\(952\) 0 0
\(953\) −0.946025 −0.0306447 −0.0153224 0.999883i \(-0.504877\pi\)
−0.0153224 + 0.999883i \(0.504877\pi\)
\(954\) −11.6155 −0.376067
\(955\) −13.3405 −0.431689
\(956\) 0 0
\(957\) 5.47091 0.176849
\(958\) 6.70263 0.216552
\(959\) 0 0
\(960\) 2.56155 0.0826738
\(961\) −10.4384 −0.336724
\(962\) 5.09271 0.164195
\(963\) 14.2462 0.459078
\(964\) 17.4644 0.562492
\(965\) −45.6401 −1.46921
\(966\) 0 0
\(967\) 5.93087 0.190724 0.0953620 0.995443i \(-0.469599\pi\)
0.0953620 + 0.995443i \(0.469599\pi\)
\(968\) −9.80776 −0.315233
\(969\) 38.3793 1.23292
\(970\) −29.3693 −0.942992
\(971\) 15.7392 0.505096 0.252548 0.967584i \(-0.418731\pi\)
0.252548 + 0.967584i \(0.418731\pi\)
\(972\) −14.1617 −0.454236
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) 2.38447 0.0763642
\(976\) 4.27156 0.136729
\(977\) −60.6695 −1.94099 −0.970495 0.241122i \(-0.922484\pi\)
−0.970495 + 0.241122i \(0.922484\pi\)
\(978\) −10.2683 −0.328345
\(979\) 12.8147 0.409559
\(980\) 0 0
\(981\) 2.63068 0.0839912
\(982\) −20.1771 −0.643877
\(983\) −32.2672 −1.02917 −0.514583 0.857441i \(-0.672053\pi\)
−0.514583 + 0.857441i \(0.672053\pi\)
\(984\) −6.24621 −0.199122
\(985\) 43.2414 1.37779
\(986\) 5.20798 0.165856
\(987\) 0 0
\(988\) 27.8617 0.886400
\(989\) 63.3693 2.01503
\(990\) 15.2134 0.483513
\(991\) 1.61553 0.0513189 0.0256595 0.999671i \(-0.491831\pi\)
0.0256595 + 0.999671i \(0.491831\pi\)
\(992\) −4.53448 −0.143970
\(993\) 20.7490 0.658450
\(994\) 0 0
\(995\) −30.2462 −0.958869
\(996\) −12.0000 −0.380235
\(997\) −11.7630 −0.372537 −0.186268 0.982499i \(-0.559639\pi\)
−0.186268 + 0.982499i \(0.559639\pi\)
\(998\) 5.12311 0.162169
\(999\) −6.14441 −0.194401
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.1.2 4
7.6 odd 2 inner 2842.2.a.p.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2842.2.a.p.1.2 4 1.1 even 1 trivial
2842.2.a.p.1.3 yes 4 7.6 odd 2 inner