Properties

Label 4598.2.a.bb.1.1
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -0.618034 q^{5} -2.00000 q^{6} +1.85410 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -0.618034 q^{5} -2.00000 q^{6} +1.85410 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.618034 q^{10} -2.00000 q^{12} -6.47214 q^{13} +1.85410 q^{14} +1.23607 q^{15} +1.00000 q^{16} +5.61803 q^{17} +1.00000 q^{18} +1.00000 q^{19} -0.618034 q^{20} -3.70820 q^{21} -1.14590 q^{23} -2.00000 q^{24} -4.61803 q^{25} -6.47214 q^{26} +4.00000 q^{27} +1.85410 q^{28} +1.52786 q^{29} +1.23607 q^{30} +2.00000 q^{31} +1.00000 q^{32} +5.61803 q^{34} -1.14590 q^{35} +1.00000 q^{36} +4.00000 q^{37} +1.00000 q^{38} +12.9443 q^{39} -0.618034 q^{40} -9.23607 q^{41} -3.70820 q^{42} +2.09017 q^{43} -0.618034 q^{45} -1.14590 q^{46} +3.61803 q^{47} -2.00000 q^{48} -3.56231 q^{49} -4.61803 q^{50} -11.2361 q^{51} -6.47214 q^{52} -9.23607 q^{53} +4.00000 q^{54} +1.85410 q^{56} -2.00000 q^{57} +1.52786 q^{58} -12.4721 q^{59} +1.23607 q^{60} +10.5623 q^{61} +2.00000 q^{62} +1.85410 q^{63} +1.00000 q^{64} +4.00000 q^{65} +7.23607 q^{67} +5.61803 q^{68} +2.29180 q^{69} -1.14590 q^{70} -12.9443 q^{71} +1.00000 q^{72} -14.0000 q^{73} +4.00000 q^{74} +9.23607 q^{75} +1.00000 q^{76} +12.9443 q^{78} -9.70820 q^{79} -0.618034 q^{80} -11.0000 q^{81} -9.23607 q^{82} +10.7984 q^{83} -3.70820 q^{84} -3.47214 q^{85} +2.09017 q^{86} -3.05573 q^{87} +6.76393 q^{89} -0.618034 q^{90} -12.0000 q^{91} -1.14590 q^{92} -4.00000 q^{93} +3.61803 q^{94} -0.618034 q^{95} -2.00000 q^{96} -4.76393 q^{97} -3.56231 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} + q^{5} - 4 q^{6} - 3 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} + q^{5} - 4 q^{6} - 3 q^{7} + 2 q^{8} + 2 q^{9} + q^{10} - 4 q^{12} - 4 q^{13} - 3 q^{14} - 2 q^{15} + 2 q^{16} + 9 q^{17} + 2 q^{18} + 2 q^{19} + q^{20} + 6 q^{21} - 9 q^{23} - 4 q^{24} - 7 q^{25} - 4 q^{26} + 8 q^{27} - 3 q^{28} + 12 q^{29} - 2 q^{30} + 4 q^{31} + 2 q^{32} + 9 q^{34} - 9 q^{35} + 2 q^{36} + 8 q^{37} + 2 q^{38} + 8 q^{39} + q^{40} - 14 q^{41} + 6 q^{42} - 7 q^{43} + q^{45} - 9 q^{46} + 5 q^{47} - 4 q^{48} + 13 q^{49} - 7 q^{50} - 18 q^{51} - 4 q^{52} - 14 q^{53} + 8 q^{54} - 3 q^{56} - 4 q^{57} + 12 q^{58} - 16 q^{59} - 2 q^{60} + q^{61} + 4 q^{62} - 3 q^{63} + 2 q^{64} + 8 q^{65} + 10 q^{67} + 9 q^{68} + 18 q^{69} - 9 q^{70} - 8 q^{71} + 2 q^{72} - 28 q^{73} + 8 q^{74} + 14 q^{75} + 2 q^{76} + 8 q^{78} - 6 q^{79} + q^{80} - 22 q^{81} - 14 q^{82} - 3 q^{83} + 6 q^{84} + 2 q^{85} - 7 q^{86} - 24 q^{87} + 18 q^{89} + q^{90} - 24 q^{91} - 9 q^{92} - 8 q^{93} + 5 q^{94} + q^{95} - 4 q^{96} - 14 q^{97} + 13 q^{98}+O(q^{100}) \) 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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.618034 −0.276393 −0.138197 0.990405i \(-0.544131\pi\)
−0.138197 + 0.990405i \(0.544131\pi\)
\(6\) −2.00000 −0.816497
\(7\) 1.85410 0.700785 0.350392 0.936603i \(-0.386048\pi\)
0.350392 + 0.936603i \(0.386048\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −0.618034 −0.195440
\(11\) 0 0
\(12\) −2.00000 −0.577350
\(13\) −6.47214 −1.79505 −0.897524 0.440966i \(-0.854636\pi\)
−0.897524 + 0.440966i \(0.854636\pi\)
\(14\) 1.85410 0.495530
\(15\) 1.23607 0.319151
\(16\) 1.00000 0.250000
\(17\) 5.61803 1.36257 0.681287 0.732017i \(-0.261421\pi\)
0.681287 + 0.732017i \(0.261421\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) −0.618034 −0.138197
\(21\) −3.70820 −0.809196
\(22\) 0 0
\(23\) −1.14590 −0.238936 −0.119468 0.992838i \(-0.538119\pi\)
−0.119468 + 0.992838i \(0.538119\pi\)
\(24\) −2.00000 −0.408248
\(25\) −4.61803 −0.923607
\(26\) −6.47214 −1.26929
\(27\) 4.00000 0.769800
\(28\) 1.85410 0.350392
\(29\) 1.52786 0.283717 0.141859 0.989887i \(-0.454692\pi\)
0.141859 + 0.989887i \(0.454692\pi\)
\(30\) 1.23607 0.225674
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 5.61803 0.963485
\(35\) −1.14590 −0.193692
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 1.00000 0.162221
\(39\) 12.9443 2.07274
\(40\) −0.618034 −0.0977198
\(41\) −9.23607 −1.44243 −0.721216 0.692711i \(-0.756416\pi\)
−0.721216 + 0.692711i \(0.756416\pi\)
\(42\) −3.70820 −0.572188
\(43\) 2.09017 0.318748 0.159374 0.987218i \(-0.449052\pi\)
0.159374 + 0.987218i \(0.449052\pi\)
\(44\) 0 0
\(45\) −0.618034 −0.0921311
\(46\) −1.14590 −0.168953
\(47\) 3.61803 0.527744 0.263872 0.964558i \(-0.415000\pi\)
0.263872 + 0.964558i \(0.415000\pi\)
\(48\) −2.00000 −0.288675
\(49\) −3.56231 −0.508901
\(50\) −4.61803 −0.653089
\(51\) −11.2361 −1.57336
\(52\) −6.47214 −0.897524
\(53\) −9.23607 −1.26867 −0.634336 0.773058i \(-0.718726\pi\)
−0.634336 + 0.773058i \(0.718726\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 1.85410 0.247765
\(57\) −2.00000 −0.264906
\(58\) 1.52786 0.200618
\(59\) −12.4721 −1.62373 −0.811867 0.583843i \(-0.801549\pi\)
−0.811867 + 0.583843i \(0.801549\pi\)
\(60\) 1.23607 0.159576
\(61\) 10.5623 1.35236 0.676182 0.736734i \(-0.263633\pi\)
0.676182 + 0.736734i \(0.263633\pi\)
\(62\) 2.00000 0.254000
\(63\) 1.85410 0.233595
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 7.23607 0.884026 0.442013 0.897009i \(-0.354264\pi\)
0.442013 + 0.897009i \(0.354264\pi\)
\(68\) 5.61803 0.681287
\(69\) 2.29180 0.275900
\(70\) −1.14590 −0.136961
\(71\) −12.9443 −1.53620 −0.768101 0.640328i \(-0.778798\pi\)
−0.768101 + 0.640328i \(0.778798\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 4.00000 0.464991
\(75\) 9.23607 1.06649
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 12.9443 1.46565
\(79\) −9.70820 −1.09226 −0.546129 0.837701i \(-0.683899\pi\)
−0.546129 + 0.837701i \(0.683899\pi\)
\(80\) −0.618034 −0.0690983
\(81\) −11.0000 −1.22222
\(82\) −9.23607 −1.01995
\(83\) 10.7984 1.18528 0.592638 0.805469i \(-0.298087\pi\)
0.592638 + 0.805469i \(0.298087\pi\)
\(84\) −3.70820 −0.404598
\(85\) −3.47214 −0.376606
\(86\) 2.09017 0.225389
\(87\) −3.05573 −0.327608
\(88\) 0 0
\(89\) 6.76393 0.716975 0.358488 0.933534i \(-0.383293\pi\)
0.358488 + 0.933534i \(0.383293\pi\)
\(90\) −0.618034 −0.0651465
\(91\) −12.0000 −1.25794
\(92\) −1.14590 −0.119468
\(93\) −4.00000 −0.414781
\(94\) 3.61803 0.373172
\(95\) −0.618034 −0.0634089
\(96\) −2.00000 −0.204124
\(97\) −4.76393 −0.483704 −0.241852 0.970313i \(-0.577755\pi\)
−0.241852 + 0.970313i \(0.577755\pi\)
\(98\) −3.56231 −0.359847
\(99\) 0 0
\(100\) −4.61803 −0.461803
\(101\) 5.56231 0.553470 0.276735 0.960946i \(-0.410748\pi\)
0.276735 + 0.960946i \(0.410748\pi\)
\(102\) −11.2361 −1.11254
\(103\) −9.23607 −0.910057 −0.455028 0.890477i \(-0.650371\pi\)
−0.455028 + 0.890477i \(0.650371\pi\)
\(104\) −6.47214 −0.634645
\(105\) 2.29180 0.223656
\(106\) −9.23607 −0.897086
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 4.00000 0.384900
\(109\) −1.23607 −0.118394 −0.0591969 0.998246i \(-0.518854\pi\)
−0.0591969 + 0.998246i \(0.518854\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 1.85410 0.175196
\(113\) 14.4721 1.36142 0.680712 0.732551i \(-0.261671\pi\)
0.680712 + 0.732551i \(0.261671\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0.708204 0.0660404
\(116\) 1.52786 0.141859
\(117\) −6.47214 −0.598349
\(118\) −12.4721 −1.14815
\(119\) 10.4164 0.954871
\(120\) 1.23607 0.112837
\(121\) 0 0
\(122\) 10.5623 0.956266
\(123\) 18.4721 1.66558
\(124\) 2.00000 0.179605
\(125\) 5.94427 0.531672
\(126\) 1.85410 0.165177
\(127\) 7.41641 0.658100 0.329050 0.944313i \(-0.393272\pi\)
0.329050 + 0.944313i \(0.393272\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.18034 −0.368058
\(130\) 4.00000 0.350823
\(131\) −0.909830 −0.0794922 −0.0397461 0.999210i \(-0.512655\pi\)
−0.0397461 + 0.999210i \(0.512655\pi\)
\(132\) 0 0
\(133\) 1.85410 0.160771
\(134\) 7.23607 0.625101
\(135\) −2.47214 −0.212768
\(136\) 5.61803 0.481742
\(137\) −15.0902 −1.28924 −0.644620 0.764503i \(-0.722984\pi\)
−0.644620 + 0.764503i \(0.722984\pi\)
\(138\) 2.29180 0.195091
\(139\) 5.32624 0.451766 0.225883 0.974154i \(-0.427473\pi\)
0.225883 + 0.974154i \(0.427473\pi\)
\(140\) −1.14590 −0.0968461
\(141\) −7.23607 −0.609387
\(142\) −12.9443 −1.08626
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −0.944272 −0.0784175
\(146\) −14.0000 −1.15865
\(147\) 7.12461 0.587628
\(148\) 4.00000 0.328798
\(149\) −20.4721 −1.67714 −0.838571 0.544792i \(-0.816609\pi\)
−0.838571 + 0.544792i \(0.816609\pi\)
\(150\) 9.23607 0.754122
\(151\) −0.291796 −0.0237460 −0.0118730 0.999930i \(-0.503779\pi\)
−0.0118730 + 0.999930i \(0.503779\pi\)
\(152\) 1.00000 0.0811107
\(153\) 5.61803 0.454191
\(154\) 0 0
\(155\) −1.23607 −0.0992834
\(156\) 12.9443 1.03637
\(157\) 7.67376 0.612433 0.306216 0.951962i \(-0.400937\pi\)
0.306216 + 0.951962i \(0.400937\pi\)
\(158\) −9.70820 −0.772343
\(159\) 18.4721 1.46494
\(160\) −0.618034 −0.0488599
\(161\) −2.12461 −0.167443
\(162\) −11.0000 −0.864242
\(163\) −16.3262 −1.27877 −0.639385 0.768887i \(-0.720811\pi\)
−0.639385 + 0.768887i \(0.720811\pi\)
\(164\) −9.23607 −0.721216
\(165\) 0 0
\(166\) 10.7984 0.838116
\(167\) −25.4164 −1.96678 −0.983390 0.181503i \(-0.941904\pi\)
−0.983390 + 0.181503i \(0.941904\pi\)
\(168\) −3.70820 −0.286094
\(169\) 28.8885 2.22220
\(170\) −3.47214 −0.266301
\(171\) 1.00000 0.0764719
\(172\) 2.09017 0.159374
\(173\) −3.05573 −0.232323 −0.116161 0.993230i \(-0.537059\pi\)
−0.116161 + 0.993230i \(0.537059\pi\)
\(174\) −3.05573 −0.231654
\(175\) −8.56231 −0.647249
\(176\) 0 0
\(177\) 24.9443 1.87493
\(178\) 6.76393 0.506978
\(179\) −20.1803 −1.50835 −0.754175 0.656674i \(-0.771963\pi\)
−0.754175 + 0.656674i \(0.771963\pi\)
\(180\) −0.618034 −0.0460655
\(181\) −13.7082 −1.01892 −0.509461 0.860494i \(-0.670155\pi\)
−0.509461 + 0.860494i \(0.670155\pi\)
\(182\) −12.0000 −0.889499
\(183\) −21.1246 −1.56158
\(184\) −1.14590 −0.0844767
\(185\) −2.47214 −0.181755
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) 3.61803 0.263872
\(189\) 7.41641 0.539464
\(190\) −0.618034 −0.0448369
\(191\) −24.0344 −1.73907 −0.869536 0.493870i \(-0.835582\pi\)
−0.869536 + 0.493870i \(0.835582\pi\)
\(192\) −2.00000 −0.144338
\(193\) −21.1246 −1.52058 −0.760291 0.649582i \(-0.774944\pi\)
−0.760291 + 0.649582i \(0.774944\pi\)
\(194\) −4.76393 −0.342030
\(195\) −8.00000 −0.572892
\(196\) −3.56231 −0.254450
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −5.56231 −0.394301 −0.197151 0.980373i \(-0.563169\pi\)
−0.197151 + 0.980373i \(0.563169\pi\)
\(200\) −4.61803 −0.326544
\(201\) −14.4721 −1.02079
\(202\) 5.56231 0.391362
\(203\) 2.83282 0.198825
\(204\) −11.2361 −0.786682
\(205\) 5.70820 0.398678
\(206\) −9.23607 −0.643507
\(207\) −1.14590 −0.0796454
\(208\) −6.47214 −0.448762
\(209\) 0 0
\(210\) 2.29180 0.158149
\(211\) −11.4164 −0.785938 −0.392969 0.919552i \(-0.628552\pi\)
−0.392969 + 0.919552i \(0.628552\pi\)
\(212\) −9.23607 −0.634336
\(213\) 25.8885 1.77385
\(214\) 6.00000 0.410152
\(215\) −1.29180 −0.0880998
\(216\) 4.00000 0.272166
\(217\) 3.70820 0.251729
\(218\) −1.23607 −0.0837171
\(219\) 28.0000 1.89206
\(220\) 0 0
\(221\) −36.3607 −2.44588
\(222\) −8.00000 −0.536925
\(223\) 3.70820 0.248320 0.124160 0.992262i \(-0.460376\pi\)
0.124160 + 0.992262i \(0.460376\pi\)
\(224\) 1.85410 0.123882
\(225\) −4.61803 −0.307869
\(226\) 14.4721 0.962672
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) −2.00000 −0.132453
\(229\) 24.2705 1.60384 0.801920 0.597431i \(-0.203812\pi\)
0.801920 + 0.597431i \(0.203812\pi\)
\(230\) 0.708204 0.0466976
\(231\) 0 0
\(232\) 1.52786 0.100309
\(233\) −18.7984 −1.23152 −0.615761 0.787933i \(-0.711151\pi\)
−0.615761 + 0.787933i \(0.711151\pi\)
\(234\) −6.47214 −0.423097
\(235\) −2.23607 −0.145865
\(236\) −12.4721 −0.811867
\(237\) 19.4164 1.26123
\(238\) 10.4164 0.675195
\(239\) 2.67376 0.172951 0.0864756 0.996254i \(-0.472440\pi\)
0.0864756 + 0.996254i \(0.472440\pi\)
\(240\) 1.23607 0.0797878
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 10.5623 0.676182
\(245\) 2.20163 0.140657
\(246\) 18.4721 1.17774
\(247\) −6.47214 −0.411812
\(248\) 2.00000 0.127000
\(249\) −21.5967 −1.36864
\(250\) 5.94427 0.375949
\(251\) 10.8541 0.685105 0.342552 0.939499i \(-0.388709\pi\)
0.342552 + 0.939499i \(0.388709\pi\)
\(252\) 1.85410 0.116797
\(253\) 0 0
\(254\) 7.41641 0.465347
\(255\) 6.94427 0.434867
\(256\) 1.00000 0.0625000
\(257\) −10.9443 −0.682685 −0.341342 0.939939i \(-0.610882\pi\)
−0.341342 + 0.939939i \(0.610882\pi\)
\(258\) −4.18034 −0.260257
\(259\) 7.41641 0.460833
\(260\) 4.00000 0.248069
\(261\) 1.52786 0.0945724
\(262\) −0.909830 −0.0562095
\(263\) 6.47214 0.399089 0.199544 0.979889i \(-0.436054\pi\)
0.199544 + 0.979889i \(0.436054\pi\)
\(264\) 0 0
\(265\) 5.70820 0.350652
\(266\) 1.85410 0.113682
\(267\) −13.5279 −0.827892
\(268\) 7.23607 0.442013
\(269\) 18.6525 1.13726 0.568631 0.822593i \(-0.307473\pi\)
0.568631 + 0.822593i \(0.307473\pi\)
\(270\) −2.47214 −0.150449
\(271\) −31.2148 −1.89616 −0.948081 0.318028i \(-0.896980\pi\)
−0.948081 + 0.318028i \(0.896980\pi\)
\(272\) 5.61803 0.340643
\(273\) 24.0000 1.45255
\(274\) −15.0902 −0.911631
\(275\) 0 0
\(276\) 2.29180 0.137950
\(277\) −3.52786 −0.211969 −0.105984 0.994368i \(-0.533799\pi\)
−0.105984 + 0.994368i \(0.533799\pi\)
\(278\) 5.32624 0.319447
\(279\) 2.00000 0.119737
\(280\) −1.14590 −0.0684805
\(281\) 4.65248 0.277543 0.138772 0.990324i \(-0.455685\pi\)
0.138772 + 0.990324i \(0.455685\pi\)
\(282\) −7.23607 −0.430902
\(283\) −9.56231 −0.568420 −0.284210 0.958762i \(-0.591731\pi\)
−0.284210 + 0.958762i \(0.591731\pi\)
\(284\) −12.9443 −0.768101
\(285\) 1.23607 0.0732183
\(286\) 0 0
\(287\) −17.1246 −1.01083
\(288\) 1.00000 0.0589256
\(289\) 14.5623 0.856606
\(290\) −0.944272 −0.0554496
\(291\) 9.52786 0.558533
\(292\) −14.0000 −0.819288
\(293\) 0.652476 0.0381180 0.0190590 0.999818i \(-0.493933\pi\)
0.0190590 + 0.999818i \(0.493933\pi\)
\(294\) 7.12461 0.415516
\(295\) 7.70820 0.448789
\(296\) 4.00000 0.232495
\(297\) 0 0
\(298\) −20.4721 −1.18592
\(299\) 7.41641 0.428902
\(300\) 9.23607 0.533245
\(301\) 3.87539 0.223374
\(302\) −0.291796 −0.0167910
\(303\) −11.1246 −0.639092
\(304\) 1.00000 0.0573539
\(305\) −6.52786 −0.373784
\(306\) 5.61803 0.321162
\(307\) −27.4164 −1.56474 −0.782369 0.622816i \(-0.785989\pi\)
−0.782369 + 0.622816i \(0.785989\pi\)
\(308\) 0 0
\(309\) 18.4721 1.05084
\(310\) −1.23607 −0.0702039
\(311\) −4.20163 −0.238252 −0.119126 0.992879i \(-0.538009\pi\)
−0.119126 + 0.992879i \(0.538009\pi\)
\(312\) 12.9443 0.732825
\(313\) −0.0901699 −0.00509671 −0.00254835 0.999997i \(-0.500811\pi\)
−0.00254835 + 0.999997i \(0.500811\pi\)
\(314\) 7.67376 0.433055
\(315\) −1.14590 −0.0645640
\(316\) −9.70820 −0.546129
\(317\) 2.18034 0.122460 0.0612300 0.998124i \(-0.480498\pi\)
0.0612300 + 0.998124i \(0.480498\pi\)
\(318\) 18.4721 1.03587
\(319\) 0 0
\(320\) −0.618034 −0.0345492
\(321\) −12.0000 −0.669775
\(322\) −2.12461 −0.118400
\(323\) 5.61803 0.312596
\(324\) −11.0000 −0.611111
\(325\) 29.8885 1.65792
\(326\) −16.3262 −0.904227
\(327\) 2.47214 0.136709
\(328\) −9.23607 −0.509977
\(329\) 6.70820 0.369835
\(330\) 0 0
\(331\) −9.70820 −0.533611 −0.266806 0.963750i \(-0.585968\pi\)
−0.266806 + 0.963750i \(0.585968\pi\)
\(332\) 10.7984 0.592638
\(333\) 4.00000 0.219199
\(334\) −25.4164 −1.39072
\(335\) −4.47214 −0.244339
\(336\) −3.70820 −0.202299
\(337\) −17.0557 −0.929085 −0.464542 0.885551i \(-0.653781\pi\)
−0.464542 + 0.885551i \(0.653781\pi\)
\(338\) 28.8885 1.57133
\(339\) −28.9443 −1.57204
\(340\) −3.47214 −0.188303
\(341\) 0 0
\(342\) 1.00000 0.0540738
\(343\) −19.5836 −1.05741
\(344\) 2.09017 0.112694
\(345\) −1.41641 −0.0762568
\(346\) −3.05573 −0.164277
\(347\) 8.56231 0.459649 0.229824 0.973232i \(-0.426185\pi\)
0.229824 + 0.973232i \(0.426185\pi\)
\(348\) −3.05573 −0.163804
\(349\) 20.9787 1.12296 0.561482 0.827489i \(-0.310231\pi\)
0.561482 + 0.827489i \(0.310231\pi\)
\(350\) −8.56231 −0.457675
\(351\) −25.8885 −1.38183
\(352\) 0 0
\(353\) 15.3820 0.818699 0.409350 0.912378i \(-0.365756\pi\)
0.409350 + 0.912378i \(0.365756\pi\)
\(354\) 24.9443 1.32577
\(355\) 8.00000 0.424596
\(356\) 6.76393 0.358488
\(357\) −20.8328 −1.10259
\(358\) −20.1803 −1.06656
\(359\) −6.09017 −0.321427 −0.160713 0.987001i \(-0.551379\pi\)
−0.160713 + 0.987001i \(0.551379\pi\)
\(360\) −0.618034 −0.0325733
\(361\) 1.00000 0.0526316
\(362\) −13.7082 −0.720487
\(363\) 0 0
\(364\) −12.0000 −0.628971
\(365\) 8.65248 0.452891
\(366\) −21.1246 −1.10420
\(367\) −21.5623 −1.12554 −0.562772 0.826612i \(-0.690265\pi\)
−0.562772 + 0.826612i \(0.690265\pi\)
\(368\) −1.14590 −0.0597341
\(369\) −9.23607 −0.480810
\(370\) −2.47214 −0.128520
\(371\) −17.1246 −0.889066
\(372\) −4.00000 −0.207390
\(373\) −30.3607 −1.57202 −0.786008 0.618216i \(-0.787856\pi\)
−0.786008 + 0.618216i \(0.787856\pi\)
\(374\) 0 0
\(375\) −11.8885 −0.613922
\(376\) 3.61803 0.186586
\(377\) −9.88854 −0.509286
\(378\) 7.41641 0.381459
\(379\) 1.23607 0.0634925 0.0317463 0.999496i \(-0.489893\pi\)
0.0317463 + 0.999496i \(0.489893\pi\)
\(380\) −0.618034 −0.0317045
\(381\) −14.8328 −0.759908
\(382\) −24.0344 −1.22971
\(383\) 14.9443 0.763617 0.381808 0.924242i \(-0.375301\pi\)
0.381808 + 0.924242i \(0.375301\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) −21.1246 −1.07521
\(387\) 2.09017 0.106249
\(388\) −4.76393 −0.241852
\(389\) 13.0344 0.660872 0.330436 0.943828i \(-0.392804\pi\)
0.330436 + 0.943828i \(0.392804\pi\)
\(390\) −8.00000 −0.405096
\(391\) −6.43769 −0.325568
\(392\) −3.56231 −0.179924
\(393\) 1.81966 0.0917897
\(394\) −18.0000 −0.906827
\(395\) 6.00000 0.301893
\(396\) 0 0
\(397\) 20.8541 1.04664 0.523319 0.852137i \(-0.324694\pi\)
0.523319 + 0.852137i \(0.324694\pi\)
\(398\) −5.56231 −0.278813
\(399\) −3.70820 −0.185642
\(400\) −4.61803 −0.230902
\(401\) 12.4721 0.622829 0.311414 0.950274i \(-0.399197\pi\)
0.311414 + 0.950274i \(0.399197\pi\)
\(402\) −14.4721 −0.721805
\(403\) −12.9443 −0.644800
\(404\) 5.56231 0.276735
\(405\) 6.79837 0.337814
\(406\) 2.83282 0.140590
\(407\) 0 0
\(408\) −11.2361 −0.556268
\(409\) −22.9443 −1.13452 −0.567261 0.823538i \(-0.691997\pi\)
−0.567261 + 0.823538i \(0.691997\pi\)
\(410\) 5.70820 0.281908
\(411\) 30.1803 1.48869
\(412\) −9.23607 −0.455028
\(413\) −23.1246 −1.13789
\(414\) −1.14590 −0.0563178
\(415\) −6.67376 −0.327602
\(416\) −6.47214 −0.317323
\(417\) −10.6525 −0.521654
\(418\) 0 0
\(419\) 9.32624 0.455617 0.227808 0.973706i \(-0.426844\pi\)
0.227808 + 0.973706i \(0.426844\pi\)
\(420\) 2.29180 0.111828
\(421\) 28.0000 1.36464 0.682318 0.731055i \(-0.260972\pi\)
0.682318 + 0.731055i \(0.260972\pi\)
\(422\) −11.4164 −0.555742
\(423\) 3.61803 0.175915
\(424\) −9.23607 −0.448543
\(425\) −25.9443 −1.25848
\(426\) 25.8885 1.25430
\(427\) 19.5836 0.947716
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) −1.29180 −0.0622959
\(431\) 35.8885 1.72869 0.864345 0.502899i \(-0.167733\pi\)
0.864345 + 0.502899i \(0.167733\pi\)
\(432\) 4.00000 0.192450
\(433\) 5.52786 0.265652 0.132826 0.991139i \(-0.457595\pi\)
0.132826 + 0.991139i \(0.457595\pi\)
\(434\) 3.70820 0.177999
\(435\) 1.88854 0.0905487
\(436\) −1.23607 −0.0591969
\(437\) −1.14590 −0.0548157
\(438\) 28.0000 1.33789
\(439\) 23.1246 1.10368 0.551839 0.833951i \(-0.313926\pi\)
0.551839 + 0.833951i \(0.313926\pi\)
\(440\) 0 0
\(441\) −3.56231 −0.169634
\(442\) −36.3607 −1.72950
\(443\) 14.2148 0.675365 0.337682 0.941260i \(-0.390357\pi\)
0.337682 + 0.941260i \(0.390357\pi\)
\(444\) −8.00000 −0.379663
\(445\) −4.18034 −0.198167
\(446\) 3.70820 0.175589
\(447\) 40.9443 1.93660
\(448\) 1.85410 0.0875981
\(449\) −8.18034 −0.386054 −0.193027 0.981193i \(-0.561831\pi\)
−0.193027 + 0.981193i \(0.561831\pi\)
\(450\) −4.61803 −0.217696
\(451\) 0 0
\(452\) 14.4721 0.680712
\(453\) 0.583592 0.0274196
\(454\) 0 0
\(455\) 7.41641 0.347687
\(456\) −2.00000 −0.0936586
\(457\) −23.3820 −1.09376 −0.546881 0.837210i \(-0.684185\pi\)
−0.546881 + 0.837210i \(0.684185\pi\)
\(458\) 24.2705 1.13409
\(459\) 22.4721 1.04891
\(460\) 0.708204 0.0330202
\(461\) 5.20163 0.242264 0.121132 0.992636i \(-0.461348\pi\)
0.121132 + 0.992636i \(0.461348\pi\)
\(462\) 0 0
\(463\) 2.79837 0.130051 0.0650257 0.997884i \(-0.479287\pi\)
0.0650257 + 0.997884i \(0.479287\pi\)
\(464\) 1.52786 0.0709293
\(465\) 2.47214 0.114643
\(466\) −18.7984 −0.870818
\(467\) 31.0902 1.43868 0.719341 0.694657i \(-0.244444\pi\)
0.719341 + 0.694657i \(0.244444\pi\)
\(468\) −6.47214 −0.299175
\(469\) 13.4164 0.619512
\(470\) −2.23607 −0.103142
\(471\) −15.3475 −0.707177
\(472\) −12.4721 −0.574077
\(473\) 0 0
\(474\) 19.4164 0.891825
\(475\) −4.61803 −0.211890
\(476\) 10.4164 0.477435
\(477\) −9.23607 −0.422891
\(478\) 2.67376 0.122295
\(479\) −10.0902 −0.461032 −0.230516 0.973069i \(-0.574041\pi\)
−0.230516 + 0.973069i \(0.574041\pi\)
\(480\) 1.23607 0.0564185
\(481\) −25.8885 −1.18042
\(482\) 10.0000 0.455488
\(483\) 4.24922 0.193346
\(484\) 0 0
\(485\) 2.94427 0.133693
\(486\) 10.0000 0.453609
\(487\) 10.0000 0.453143 0.226572 0.973995i \(-0.427248\pi\)
0.226572 + 0.973995i \(0.427248\pi\)
\(488\) 10.5623 0.478133
\(489\) 32.6525 1.47660
\(490\) 2.20163 0.0994593
\(491\) −10.1459 −0.457878 −0.228939 0.973441i \(-0.573526\pi\)
−0.228939 + 0.973441i \(0.573526\pi\)
\(492\) 18.4721 0.832788
\(493\) 8.58359 0.386586
\(494\) −6.47214 −0.291195
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −24.0000 −1.07655
\(498\) −21.5967 −0.967773
\(499\) 17.5623 0.786197 0.393098 0.919496i \(-0.371403\pi\)
0.393098 + 0.919496i \(0.371403\pi\)
\(500\) 5.94427 0.265836
\(501\) 50.8328 2.27104
\(502\) 10.8541 0.484442
\(503\) −29.3050 −1.30664 −0.653322 0.757080i \(-0.726625\pi\)
−0.653322 + 0.757080i \(0.726625\pi\)
\(504\) 1.85410 0.0825883
\(505\) −3.43769 −0.152975
\(506\) 0 0
\(507\) −57.7771 −2.56597
\(508\) 7.41641 0.329050
\(509\) 20.2918 0.899418 0.449709 0.893175i \(-0.351528\pi\)
0.449709 + 0.893175i \(0.351528\pi\)
\(510\) 6.94427 0.307498
\(511\) −25.9574 −1.14829
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) −10.9443 −0.482731
\(515\) 5.70820 0.251534
\(516\) −4.18034 −0.184029
\(517\) 0 0
\(518\) 7.41641 0.325858
\(519\) 6.11146 0.268263
\(520\) 4.00000 0.175412
\(521\) 2.18034 0.0955224 0.0477612 0.998859i \(-0.484791\pi\)
0.0477612 + 0.998859i \(0.484791\pi\)
\(522\) 1.52786 0.0668728
\(523\) −29.1246 −1.27353 −0.636765 0.771058i \(-0.719728\pi\)
−0.636765 + 0.771058i \(0.719728\pi\)
\(524\) −0.909830 −0.0397461
\(525\) 17.1246 0.747379
\(526\) 6.47214 0.282199
\(527\) 11.2361 0.489451
\(528\) 0 0
\(529\) −21.6869 −0.942909
\(530\) 5.70820 0.247949
\(531\) −12.4721 −0.541245
\(532\) 1.85410 0.0803855
\(533\) 59.7771 2.58923
\(534\) −13.5279 −0.585408
\(535\) −3.70820 −0.160320
\(536\) 7.23607 0.312551
\(537\) 40.3607 1.74169
\(538\) 18.6525 0.804165
\(539\) 0 0
\(540\) −2.47214 −0.106384
\(541\) 27.1459 1.16709 0.583547 0.812080i \(-0.301665\pi\)
0.583547 + 0.812080i \(0.301665\pi\)
\(542\) −31.2148 −1.34079
\(543\) 27.4164 1.17655
\(544\) 5.61803 0.240871
\(545\) 0.763932 0.0327233
\(546\) 24.0000 1.02711
\(547\) 6.29180 0.269018 0.134509 0.990912i \(-0.457054\pi\)
0.134509 + 0.990912i \(0.457054\pi\)
\(548\) −15.0902 −0.644620
\(549\) 10.5623 0.450788
\(550\) 0 0
\(551\) 1.52786 0.0650892
\(552\) 2.29180 0.0975453
\(553\) −18.0000 −0.765438
\(554\) −3.52786 −0.149885
\(555\) 4.94427 0.209873
\(556\) 5.32624 0.225883
\(557\) 35.4508 1.50210 0.751050 0.660245i \(-0.229548\pi\)
0.751050 + 0.660245i \(0.229548\pi\)
\(558\) 2.00000 0.0846668
\(559\) −13.5279 −0.572168
\(560\) −1.14590 −0.0484230
\(561\) 0 0
\(562\) 4.65248 0.196253
\(563\) −9.70820 −0.409152 −0.204576 0.978851i \(-0.565582\pi\)
−0.204576 + 0.978851i \(0.565582\pi\)
\(564\) −7.23607 −0.304693
\(565\) −8.94427 −0.376288
\(566\) −9.56231 −0.401934
\(567\) −20.3951 −0.856515
\(568\) −12.9443 −0.543130
\(569\) −16.6525 −0.698108 −0.349054 0.937103i \(-0.613497\pi\)
−0.349054 + 0.937103i \(0.613497\pi\)
\(570\) 1.23607 0.0517732
\(571\) 6.14590 0.257198 0.128599 0.991697i \(-0.458952\pi\)
0.128599 + 0.991697i \(0.458952\pi\)
\(572\) 0 0
\(573\) 48.0689 2.00811
\(574\) −17.1246 −0.714767
\(575\) 5.29180 0.220683
\(576\) 1.00000 0.0416667
\(577\) 12.4721 0.519222 0.259611 0.965713i \(-0.416406\pi\)
0.259611 + 0.965713i \(0.416406\pi\)
\(578\) 14.5623 0.605712
\(579\) 42.2492 1.75582
\(580\) −0.944272 −0.0392088
\(581\) 20.0213 0.830623
\(582\) 9.52786 0.394943
\(583\) 0 0
\(584\) −14.0000 −0.579324
\(585\) 4.00000 0.165380
\(586\) 0.652476 0.0269535
\(587\) 28.9443 1.19466 0.597329 0.801996i \(-0.296229\pi\)
0.597329 + 0.801996i \(0.296229\pi\)
\(588\) 7.12461 0.293814
\(589\) 2.00000 0.0824086
\(590\) 7.70820 0.317342
\(591\) 36.0000 1.48084
\(592\) 4.00000 0.164399
\(593\) −44.4508 −1.82538 −0.912689 0.408655i \(-0.865998\pi\)
−0.912689 + 0.408655i \(0.865998\pi\)
\(594\) 0 0
\(595\) −6.43769 −0.263920
\(596\) −20.4721 −0.838571
\(597\) 11.1246 0.455300
\(598\) 7.41641 0.303279
\(599\) 15.5967 0.637266 0.318633 0.947878i \(-0.396776\pi\)
0.318633 + 0.947878i \(0.396776\pi\)
\(600\) 9.23607 0.377061
\(601\) 31.7082 1.29340 0.646702 0.762743i \(-0.276148\pi\)
0.646702 + 0.762743i \(0.276148\pi\)
\(602\) 3.87539 0.157949
\(603\) 7.23607 0.294675
\(604\) −0.291796 −0.0118730
\(605\) 0 0
\(606\) −11.1246 −0.451906
\(607\) 24.9443 1.01246 0.506228 0.862399i \(-0.331039\pi\)
0.506228 + 0.862399i \(0.331039\pi\)
\(608\) 1.00000 0.0405554
\(609\) −5.66563 −0.229583
\(610\) −6.52786 −0.264305
\(611\) −23.4164 −0.947326
\(612\) 5.61803 0.227096
\(613\) −7.72949 −0.312191 −0.156096 0.987742i \(-0.549891\pi\)
−0.156096 + 0.987742i \(0.549891\pi\)
\(614\) −27.4164 −1.10644
\(615\) −11.4164 −0.460354
\(616\) 0 0
\(617\) 42.7214 1.71990 0.859949 0.510381i \(-0.170495\pi\)
0.859949 + 0.510381i \(0.170495\pi\)
\(618\) 18.4721 0.743058
\(619\) 12.5623 0.504922 0.252461 0.967607i \(-0.418760\pi\)
0.252461 + 0.967607i \(0.418760\pi\)
\(620\) −1.23607 −0.0496417
\(621\) −4.58359 −0.183933
\(622\) −4.20163 −0.168470
\(623\) 12.5410 0.502445
\(624\) 12.9443 0.518186
\(625\) 19.4164 0.776656
\(626\) −0.0901699 −0.00360392
\(627\) 0 0
\(628\) 7.67376 0.306216
\(629\) 22.4721 0.896023
\(630\) −1.14590 −0.0456537
\(631\) −24.3607 −0.969783 −0.484892 0.874574i \(-0.661141\pi\)
−0.484892 + 0.874574i \(0.661141\pi\)
\(632\) −9.70820 −0.386172
\(633\) 22.8328 0.907523
\(634\) 2.18034 0.0865924
\(635\) −4.58359 −0.181894
\(636\) 18.4721 0.732468
\(637\) 23.0557 0.913501
\(638\) 0 0
\(639\) −12.9443 −0.512067
\(640\) −0.618034 −0.0244299
\(641\) 21.3050 0.841495 0.420747 0.907178i \(-0.361768\pi\)
0.420747 + 0.907178i \(0.361768\pi\)
\(642\) −12.0000 −0.473602
\(643\) 20.9230 0.825122 0.412561 0.910930i \(-0.364634\pi\)
0.412561 + 0.910930i \(0.364634\pi\)
\(644\) −2.12461 −0.0837214
\(645\) 2.58359 0.101729
\(646\) 5.61803 0.221039
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) 29.8885 1.17233
\(651\) −7.41641 −0.290672
\(652\) −16.3262 −0.639385
\(653\) −1.61803 −0.0633186 −0.0316593 0.999499i \(-0.510079\pi\)
−0.0316593 + 0.999499i \(0.510079\pi\)
\(654\) 2.47214 0.0966682
\(655\) 0.562306 0.0219711
\(656\) −9.23607 −0.360608
\(657\) −14.0000 −0.546192
\(658\) 6.70820 0.261513
\(659\) 20.2918 0.790456 0.395228 0.918583i \(-0.370666\pi\)
0.395228 + 0.918583i \(0.370666\pi\)
\(660\) 0 0
\(661\) 19.5967 0.762225 0.381113 0.924529i \(-0.375541\pi\)
0.381113 + 0.924529i \(0.375541\pi\)
\(662\) −9.70820 −0.377320
\(663\) 72.7214 2.82426
\(664\) 10.7984 0.419058
\(665\) −1.14590 −0.0444360
\(666\) 4.00000 0.154997
\(667\) −1.75078 −0.0677903
\(668\) −25.4164 −0.983390
\(669\) −7.41641 −0.286735
\(670\) −4.47214 −0.172774
\(671\) 0 0
\(672\) −3.70820 −0.143047
\(673\) −45.0132 −1.73513 −0.867565 0.497324i \(-0.834316\pi\)
−0.867565 + 0.497324i \(0.834316\pi\)
\(674\) −17.0557 −0.656962
\(675\) −18.4721 −0.710993
\(676\) 28.8885 1.11110
\(677\) 33.7082 1.29551 0.647756 0.761848i \(-0.275708\pi\)
0.647756 + 0.761848i \(0.275708\pi\)
\(678\) −28.9443 −1.11160
\(679\) −8.83282 −0.338972
\(680\) −3.47214 −0.133150
\(681\) 0 0
\(682\) 0 0
\(683\) −20.0689 −0.767914 −0.383957 0.923351i \(-0.625439\pi\)
−0.383957 + 0.923351i \(0.625439\pi\)
\(684\) 1.00000 0.0382360
\(685\) 9.32624 0.356337
\(686\) −19.5836 −0.747705
\(687\) −48.5410 −1.85196
\(688\) 2.09017 0.0796870
\(689\) 59.7771 2.27733
\(690\) −1.41641 −0.0539217
\(691\) −20.3262 −0.773247 −0.386623 0.922238i \(-0.626359\pi\)
−0.386623 + 0.922238i \(0.626359\pi\)
\(692\) −3.05573 −0.116161
\(693\) 0 0
\(694\) 8.56231 0.325021
\(695\) −3.29180 −0.124865
\(696\) −3.05573 −0.115827
\(697\) −51.8885 −1.96542
\(698\) 20.9787 0.794056
\(699\) 37.5967 1.42204
\(700\) −8.56231 −0.323625
\(701\) 14.7426 0.556822 0.278411 0.960462i \(-0.410192\pi\)
0.278411 + 0.960462i \(0.410192\pi\)
\(702\) −25.8885 −0.977100
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) 4.47214 0.168430
\(706\) 15.3820 0.578908
\(707\) 10.3131 0.387863
\(708\) 24.9443 0.937463
\(709\) 5.20163 0.195351 0.0976756 0.995218i \(-0.468859\pi\)
0.0976756 + 0.995218i \(0.468859\pi\)
\(710\) 8.00000 0.300235
\(711\) −9.70820 −0.364086
\(712\) 6.76393 0.253489
\(713\) −2.29180 −0.0858284
\(714\) −20.8328 −0.779649
\(715\) 0 0
\(716\) −20.1803 −0.754175
\(717\) −5.34752 −0.199707
\(718\) −6.09017 −0.227283
\(719\) −20.3262 −0.758041 −0.379020 0.925388i \(-0.623739\pi\)
−0.379020 + 0.925388i \(0.623739\pi\)
\(720\) −0.618034 −0.0230328
\(721\) −17.1246 −0.637754
\(722\) 1.00000 0.0372161
\(723\) −20.0000 −0.743808
\(724\) −13.7082 −0.509461
\(725\) −7.05573 −0.262043
\(726\) 0 0
\(727\) 18.8541 0.699260 0.349630 0.936888i \(-0.386307\pi\)
0.349630 + 0.936888i \(0.386307\pi\)
\(728\) −12.0000 −0.444750
\(729\) 13.0000 0.481481
\(730\) 8.65248 0.320242
\(731\) 11.7426 0.434317
\(732\) −21.1246 −0.780788
\(733\) 44.3262 1.63723 0.818613 0.574345i \(-0.194743\pi\)
0.818613 + 0.574345i \(0.194743\pi\)
\(734\) −21.5623 −0.795879
\(735\) −4.40325 −0.162416
\(736\) −1.14590 −0.0422384
\(737\) 0 0
\(738\) −9.23607 −0.339984
\(739\) −43.4508 −1.59837 −0.799183 0.601088i \(-0.794734\pi\)
−0.799183 + 0.601088i \(0.794734\pi\)
\(740\) −2.47214 −0.0908775
\(741\) 12.9443 0.475520
\(742\) −17.1246 −0.628664
\(743\) 28.6525 1.05116 0.525579 0.850745i \(-0.323849\pi\)
0.525579 + 0.850745i \(0.323849\pi\)
\(744\) −4.00000 −0.146647
\(745\) 12.6525 0.463551
\(746\) −30.3607 −1.11158
\(747\) 10.7984 0.395092
\(748\) 0 0
\(749\) 11.1246 0.406484
\(750\) −11.8885 −0.434108
\(751\) −37.4164 −1.36534 −0.682672 0.730725i \(-0.739182\pi\)
−0.682672 + 0.730725i \(0.739182\pi\)
\(752\) 3.61803 0.131936
\(753\) −21.7082 −0.791091
\(754\) −9.88854 −0.360120
\(755\) 0.180340 0.00656324
\(756\) 7.41641 0.269732
\(757\) 1.41641 0.0514802 0.0257401 0.999669i \(-0.491806\pi\)
0.0257401 + 0.999669i \(0.491806\pi\)
\(758\) 1.23607 0.0448960
\(759\) 0 0
\(760\) −0.618034 −0.0224184
\(761\) 51.3050 1.85980 0.929902 0.367809i \(-0.119892\pi\)
0.929902 + 0.367809i \(0.119892\pi\)
\(762\) −14.8328 −0.537336
\(763\) −2.29180 −0.0829686
\(764\) −24.0344 −0.869536
\(765\) −3.47214 −0.125535
\(766\) 14.9443 0.539958
\(767\) 80.7214 2.91468
\(768\) −2.00000 −0.0721688
\(769\) 34.5066 1.24434 0.622170 0.782883i \(-0.286251\pi\)
0.622170 + 0.782883i \(0.286251\pi\)
\(770\) 0 0
\(771\) 21.8885 0.788297
\(772\) −21.1246 −0.760291
\(773\) −33.5967 −1.20839 −0.604196 0.796836i \(-0.706505\pi\)
−0.604196 + 0.796836i \(0.706505\pi\)
\(774\) 2.09017 0.0751296
\(775\) −9.23607 −0.331769
\(776\) −4.76393 −0.171015
\(777\) −14.8328 −0.532124
\(778\) 13.0344 0.467307
\(779\) −9.23607 −0.330916
\(780\) −8.00000 −0.286446
\(781\) 0 0
\(782\) −6.43769 −0.230211
\(783\) 6.11146 0.218406
\(784\) −3.56231 −0.127225
\(785\) −4.74265 −0.169272
\(786\) 1.81966 0.0649051
\(787\) −42.9443 −1.53080 −0.765399 0.643556i \(-0.777458\pi\)
−0.765399 + 0.643556i \(0.777458\pi\)
\(788\) −18.0000 −0.641223
\(789\) −12.9443 −0.460828
\(790\) 6.00000 0.213470
\(791\) 26.8328 0.954065
\(792\) 0 0
\(793\) −68.3607 −2.42756
\(794\) 20.8541 0.740084
\(795\) −11.4164 −0.404898
\(796\) −5.56231 −0.197151
\(797\) −53.7771 −1.90488 −0.952441 0.304723i \(-0.901436\pi\)
−0.952441 + 0.304723i \(0.901436\pi\)
\(798\) −3.70820 −0.131269
\(799\) 20.3262 0.719091
\(800\) −4.61803 −0.163272
\(801\) 6.76393 0.238992
\(802\) 12.4721 0.440406
\(803\) 0 0
\(804\) −14.4721 −0.510393
\(805\) 1.31308 0.0462801
\(806\) −12.9443 −0.455943
\(807\) −37.3050 −1.31320
\(808\) 5.56231 0.195681
\(809\) 5.56231 0.195560 0.0977801 0.995208i \(-0.468826\pi\)
0.0977801 + 0.995208i \(0.468826\pi\)
\(810\) 6.79837 0.238871
\(811\) −40.2492 −1.41334 −0.706671 0.707543i \(-0.749804\pi\)
−0.706671 + 0.707543i \(0.749804\pi\)
\(812\) 2.83282 0.0994123
\(813\) 62.4296 2.18950
\(814\) 0 0
\(815\) 10.0902 0.353443
\(816\) −11.2361 −0.393341
\(817\) 2.09017 0.0731258
\(818\) −22.9443 −0.802228
\(819\) −12.0000 −0.419314
\(820\) 5.70820 0.199339
\(821\) 3.90983 0.136454 0.0682270 0.997670i \(-0.478266\pi\)
0.0682270 + 0.997670i \(0.478266\pi\)
\(822\) 30.1803 1.05266
\(823\) 13.8541 0.482924 0.241462 0.970410i \(-0.422373\pi\)
0.241462 + 0.970410i \(0.422373\pi\)
\(824\) −9.23607 −0.321754
\(825\) 0 0
\(826\) −23.1246 −0.804608
\(827\) 28.2492 0.982322 0.491161 0.871069i \(-0.336573\pi\)
0.491161 + 0.871069i \(0.336573\pi\)
\(828\) −1.14590 −0.0398227
\(829\) −18.8328 −0.654091 −0.327045 0.945009i \(-0.606053\pi\)
−0.327045 + 0.945009i \(0.606053\pi\)
\(830\) −6.67376 −0.231650
\(831\) 7.05573 0.244760
\(832\) −6.47214 −0.224381
\(833\) −20.0132 −0.693415
\(834\) −10.6525 −0.368865
\(835\) 15.7082 0.543605
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 9.32624 0.322170
\(839\) −24.1803 −0.834798 −0.417399 0.908723i \(-0.637058\pi\)
−0.417399 + 0.908723i \(0.637058\pi\)
\(840\) 2.29180 0.0790745
\(841\) −26.6656 −0.919505
\(842\) 28.0000 0.964944
\(843\) −9.30495 −0.320480
\(844\) −11.4164 −0.392969
\(845\) −17.8541 −0.614200
\(846\) 3.61803 0.124391
\(847\) 0 0
\(848\) −9.23607 −0.317168
\(849\) 19.1246 0.656355
\(850\) −25.9443 −0.889881
\(851\) −4.58359 −0.157124
\(852\) 25.8885 0.886927
\(853\) −4.27051 −0.146219 −0.0731097 0.997324i \(-0.523292\pi\)
−0.0731097 + 0.997324i \(0.523292\pi\)
\(854\) 19.5836 0.670137
\(855\) −0.618034 −0.0211363
\(856\) 6.00000 0.205076
\(857\) −19.0132 −0.649477 −0.324739 0.945804i \(-0.605276\pi\)
−0.324739 + 0.945804i \(0.605276\pi\)
\(858\) 0 0
\(859\) 33.6869 1.14938 0.574691 0.818370i \(-0.305122\pi\)
0.574691 + 0.818370i \(0.305122\pi\)
\(860\) −1.29180 −0.0440499
\(861\) 34.2492 1.16721
\(862\) 35.8885 1.22237
\(863\) 48.1803 1.64008 0.820039 0.572308i \(-0.193952\pi\)
0.820039 + 0.572308i \(0.193952\pi\)
\(864\) 4.00000 0.136083
\(865\) 1.88854 0.0642124
\(866\) 5.52786 0.187844
\(867\) −29.1246 −0.989124
\(868\) 3.70820 0.125865
\(869\) 0 0
\(870\) 1.88854 0.0640276
\(871\) −46.8328 −1.58687
\(872\) −1.23607 −0.0418585
\(873\) −4.76393 −0.161235
\(874\) −1.14590 −0.0387606
\(875\) 11.0213 0.372587
\(876\) 28.0000 0.946032
\(877\) −14.5836 −0.492453 −0.246226 0.969212i \(-0.579191\pi\)
−0.246226 + 0.969212i \(0.579191\pi\)
\(878\) 23.1246 0.780418
\(879\) −1.30495 −0.0440149
\(880\) 0 0
\(881\) 25.4164 0.856301 0.428150 0.903708i \(-0.359165\pi\)
0.428150 + 0.903708i \(0.359165\pi\)
\(882\) −3.56231 −0.119949
\(883\) −24.3951 −0.820961 −0.410481 0.911869i \(-0.634639\pi\)
−0.410481 + 0.911869i \(0.634639\pi\)
\(884\) −36.3607 −1.22294
\(885\) −15.4164 −0.518217
\(886\) 14.2148 0.477555
\(887\) 12.1115 0.406663 0.203331 0.979110i \(-0.434823\pi\)
0.203331 + 0.979110i \(0.434823\pi\)
\(888\) −8.00000 −0.268462
\(889\) 13.7508 0.461186
\(890\) −4.18034 −0.140125
\(891\) 0 0
\(892\) 3.70820 0.124160
\(893\) 3.61803 0.121073
\(894\) 40.9443 1.36938
\(895\) 12.4721 0.416898
\(896\) 1.85410 0.0619412
\(897\) −14.8328 −0.495253
\(898\) −8.18034 −0.272981
\(899\) 3.05573 0.101914
\(900\) −4.61803 −0.153934
\(901\) −51.8885 −1.72866
\(902\) 0 0
\(903\) −7.75078 −0.257930
\(904\) 14.4721 0.481336
\(905\) 8.47214 0.281623
\(906\) 0.583592 0.0193886
\(907\) −30.8328 −1.02379 −0.511893 0.859049i \(-0.671056\pi\)
−0.511893 + 0.859049i \(0.671056\pi\)
\(908\) 0 0
\(909\) 5.56231 0.184490
\(910\) 7.41641 0.245852
\(911\) 13.4164 0.444505 0.222253 0.974989i \(-0.428659\pi\)
0.222253 + 0.974989i \(0.428659\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 0 0
\(914\) −23.3820 −0.773407
\(915\) 13.0557 0.431609
\(916\) 24.2705 0.801920
\(917\) −1.68692 −0.0557069
\(918\) 22.4721 0.741691
\(919\) −28.6869 −0.946294 −0.473147 0.880983i \(-0.656882\pi\)
−0.473147 + 0.880983i \(0.656882\pi\)
\(920\) 0.708204 0.0233488
\(921\) 54.8328 1.80680
\(922\) 5.20163 0.171306
\(923\) 83.7771 2.75756
\(924\) 0 0
\(925\) −18.4721 −0.607360
\(926\) 2.79837 0.0919603
\(927\) −9.23607 −0.303352
\(928\) 1.52786 0.0501546
\(929\) 48.2705 1.58370 0.791852 0.610713i \(-0.209117\pi\)
0.791852 + 0.610713i \(0.209117\pi\)
\(930\) 2.47214 0.0810645
\(931\) −3.56231 −0.116750
\(932\) −18.7984 −0.615761
\(933\) 8.40325 0.275110
\(934\) 31.0902 1.01730
\(935\) 0 0
\(936\) −6.47214 −0.211548
\(937\) 56.8115 1.85595 0.927976 0.372640i \(-0.121547\pi\)
0.927976 + 0.372640i \(0.121547\pi\)
\(938\) 13.4164 0.438061
\(939\) 0.180340 0.00588517
\(940\) −2.23607 −0.0729325
\(941\) 8.94427 0.291575 0.145787 0.989316i \(-0.453428\pi\)
0.145787 + 0.989316i \(0.453428\pi\)
\(942\) −15.3475 −0.500049
\(943\) 10.5836 0.344649
\(944\) −12.4721 −0.405933
\(945\) −4.58359 −0.149104
\(946\) 0 0
\(947\) −37.9230 −1.23233 −0.616166 0.787617i \(-0.711315\pi\)
−0.616166 + 0.787617i \(0.711315\pi\)
\(948\) 19.4164 0.630616
\(949\) 90.6099 2.94132
\(950\) −4.61803 −0.149829
\(951\) −4.36068 −0.141405
\(952\) 10.4164 0.337598
\(953\) 49.7771 1.61244 0.806219 0.591617i \(-0.201510\pi\)
0.806219 + 0.591617i \(0.201510\pi\)
\(954\) −9.23607 −0.299029
\(955\) 14.8541 0.480667
\(956\) 2.67376 0.0864756
\(957\) 0 0
\(958\) −10.0902 −0.325999
\(959\) −27.9787 −0.903480
\(960\) 1.23607 0.0398939
\(961\) −27.0000 −0.870968
\(962\) −25.8885 −0.834680
\(963\) 6.00000 0.193347
\(964\) 10.0000 0.322078
\(965\) 13.0557 0.420279
\(966\) 4.24922 0.136717
\(967\) −58.2705 −1.87385 −0.936927 0.349526i \(-0.886343\pi\)
−0.936927 + 0.349526i \(0.886343\pi\)
\(968\) 0 0
\(969\) −11.2361 −0.360955
\(970\) 2.94427 0.0945349
\(971\) −34.3607 −1.10269 −0.551343 0.834278i \(-0.685885\pi\)
−0.551343 + 0.834278i \(0.685885\pi\)
\(972\) 10.0000 0.320750
\(973\) 9.87539 0.316590
\(974\) 10.0000 0.320421
\(975\) −59.7771 −1.91440
\(976\) 10.5623 0.338091
\(977\) 8.83282 0.282587 0.141293 0.989968i \(-0.454874\pi\)
0.141293 + 0.989968i \(0.454874\pi\)
\(978\) 32.6525 1.04411
\(979\) 0 0
\(980\) 2.20163 0.0703284
\(981\) −1.23607 −0.0394646
\(982\) −10.1459 −0.323769
\(983\) −58.1378 −1.85431 −0.927153 0.374682i \(-0.877752\pi\)
−0.927153 + 0.374682i \(0.877752\pi\)
\(984\) 18.4721 0.588870
\(985\) 11.1246 0.354460
\(986\) 8.58359 0.273357
\(987\) −13.4164 −0.427049
\(988\) −6.47214 −0.205906
\(989\) −2.39512 −0.0761604
\(990\) 0 0
\(991\) −26.3607 −0.837375 −0.418687 0.908130i \(-0.637510\pi\)
−0.418687 + 0.908130i \(0.637510\pi\)
\(992\) 2.00000 0.0635001
\(993\) 19.4164 0.616161
\(994\) −24.0000 −0.761234
\(995\) 3.43769 0.108982
\(996\) −21.5967 −0.684319
\(997\) −21.2705 −0.673644 −0.336822 0.941568i \(-0.609352\pi\)
−0.336822 + 0.941568i \(0.609352\pi\)
\(998\) 17.5623 0.555925
\(999\) 16.0000 0.506218
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 4598.2.a.bb.1.1 2
11.5 even 5 418.2.f.d.267.1 yes 4
11.9 even 5 418.2.f.d.191.1 4
11.10 odd 2 4598.2.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.f.d.191.1 4 11.9 even 5
418.2.f.d.267.1 yes 4 11.5 even 5
4598.2.a.t.1.1 2 11.10 odd 2
4598.2.a.bb.1.1 2 1.1 even 1 trivial