Properties

Label 8003.2.a.b.1.20
Level 8003
Weight 2
Character 8003.1
Self dual yes
Analytic conductor 63.904
Analytic rank 1
Dimension 153
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8003 = 53 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9042767376\)
Analytic rank: \(1\)
Dimension: \(153\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 8003.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.32654 q^{2} -0.105675 q^{3} +3.41278 q^{4} -1.05522 q^{5} +0.245856 q^{6} +0.945293 q^{7} -3.28688 q^{8} -2.98883 q^{9} +O(q^{10})\) \(q-2.32654 q^{2} -0.105675 q^{3} +3.41278 q^{4} -1.05522 q^{5} +0.245856 q^{6} +0.945293 q^{7} -3.28688 q^{8} -2.98883 q^{9} +2.45502 q^{10} +2.00424 q^{11} -0.360644 q^{12} +4.81051 q^{13} -2.19926 q^{14} +0.111511 q^{15} +0.821491 q^{16} +5.66382 q^{17} +6.95363 q^{18} +0.270587 q^{19} -3.60125 q^{20} -0.0998935 q^{21} -4.66293 q^{22} +2.04703 q^{23} +0.347340 q^{24} -3.88650 q^{25} -11.1918 q^{26} +0.632868 q^{27} +3.22608 q^{28} -5.94578 q^{29} -0.259433 q^{30} +3.29742 q^{31} +4.66253 q^{32} -0.211797 q^{33} -13.1771 q^{34} -0.997497 q^{35} -10.2002 q^{36} -5.04927 q^{37} -0.629531 q^{38} -0.508349 q^{39} +3.46840 q^{40} -5.06296 q^{41} +0.232406 q^{42} +1.81524 q^{43} +6.84001 q^{44} +3.15389 q^{45} -4.76249 q^{46} +12.2205 q^{47} -0.0868108 q^{48} -6.10642 q^{49} +9.04209 q^{50} -0.598522 q^{51} +16.4172 q^{52} -1.00000 q^{53} -1.47239 q^{54} -2.11492 q^{55} -3.10706 q^{56} -0.0285942 q^{57} +13.8331 q^{58} -11.0180 q^{59} +0.380560 q^{60} -7.84960 q^{61} -7.67157 q^{62} -2.82532 q^{63} -12.4905 q^{64} -5.07617 q^{65} +0.492754 q^{66} -11.3588 q^{67} +19.3293 q^{68} -0.216319 q^{69} +2.32071 q^{70} -6.29987 q^{71} +9.82393 q^{72} -1.26449 q^{73} +11.7473 q^{74} +0.410705 q^{75} +0.923454 q^{76} +1.89459 q^{77} +1.18269 q^{78} -9.25154 q^{79} -0.866858 q^{80} +8.89962 q^{81} +11.7792 q^{82} -15.9049 q^{83} -0.340914 q^{84} -5.97660 q^{85} -4.22323 q^{86} +0.628318 q^{87} -6.58768 q^{88} -9.97305 q^{89} -7.33765 q^{90} +4.54735 q^{91} +6.98606 q^{92} -0.348453 q^{93} -28.4314 q^{94} -0.285530 q^{95} -0.492711 q^{96} +13.8158 q^{97} +14.2068 q^{98} -5.99033 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 153q - 9q^{2} - 17q^{3} + 137q^{4} - 31q^{5} - 10q^{6} - 17q^{7} - 30q^{8} + 136q^{9} + O(q^{10}) \) \( 153q - 9q^{2} - 17q^{3} + 137q^{4} - 31q^{5} - 10q^{6} - 17q^{7} - 30q^{8} + 136q^{9} - 34q^{10} - q^{11} - 60q^{12} - 101q^{13} - 16q^{14} - 14q^{15} + 97q^{16} - 12q^{17} - 45q^{18} - 45q^{19} - 52q^{20} - 76q^{21} - 46q^{22} - 28q^{23} - 30q^{24} + 84q^{25} - 22q^{26} - 68q^{27} - 64q^{28} - 14q^{29} - q^{30} - 70q^{31} - 54q^{32} - 85q^{33} - 59q^{34} - 16q^{35} + 87q^{36} - 167q^{37} - 48q^{38} - 28q^{39} - 68q^{40} - 38q^{41} + 2q^{42} - 71q^{43} - 10q^{44} - 151q^{45} - 37q^{46} - 37q^{47} - 166q^{48} + 74q^{49} - 3q^{50} - 11q^{51} - 183q^{52} - 153q^{53} - 40q^{54} - 88q^{55} - 69q^{56} - 26q^{57} - 43q^{58} - 34q^{59} - 12q^{60} - 90q^{61} - 37q^{62} - 36q^{63} + 58q^{64} - 19q^{65} + 52q^{66} - 86q^{67} - 22q^{68} - 81q^{69} - 144q^{70} - 50q^{71} - 190q^{72} - 171q^{73} - 14q^{74} - 69q^{75} - 88q^{76} - 72q^{77} - 61q^{78} - 13q^{79} - 84q^{80} + 117q^{81} - 124q^{82} - 72q^{83} - 106q^{84} - 193q^{85} - 44q^{86} - 65q^{87} - 89q^{88} - 10q^{89} - 152q^{90} - 67q^{91} - 29q^{92} - 129q^{93} - 43q^{94} - 29q^{95} - 106q^{96} - 177q^{97} - 69q^{98} - 11q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.32654 −1.64511 −0.822555 0.568685i \(-0.807452\pi\)
−0.822555 + 0.568685i \(0.807452\pi\)
\(3\) −0.105675 −0.0610113 −0.0305056 0.999535i \(-0.509712\pi\)
−0.0305056 + 0.999535i \(0.509712\pi\)
\(4\) 3.41278 1.70639
\(5\) −1.05522 −0.471911 −0.235955 0.971764i \(-0.575822\pi\)
−0.235955 + 0.971764i \(0.575822\pi\)
\(6\) 0.245856 0.100370
\(7\) 0.945293 0.357287 0.178644 0.983914i \(-0.442829\pi\)
0.178644 + 0.983914i \(0.442829\pi\)
\(8\) −3.28688 −1.16209
\(9\) −2.98883 −0.996278
\(10\) 2.45502 0.776346
\(11\) 2.00424 0.604300 0.302150 0.953260i \(-0.402296\pi\)
0.302150 + 0.953260i \(0.402296\pi\)
\(12\) −0.360644 −0.104109
\(13\) 4.81051 1.33420 0.667098 0.744970i \(-0.267536\pi\)
0.667098 + 0.744970i \(0.267536\pi\)
\(14\) −2.19926 −0.587777
\(15\) 0.111511 0.0287919
\(16\) 0.821491 0.205373
\(17\) 5.66382 1.37368 0.686839 0.726810i \(-0.258998\pi\)
0.686839 + 0.726810i \(0.258998\pi\)
\(18\) 6.95363 1.63899
\(19\) 0.270587 0.0620770 0.0310385 0.999518i \(-0.490119\pi\)
0.0310385 + 0.999518i \(0.490119\pi\)
\(20\) −3.60125 −0.805263
\(21\) −0.0998935 −0.0217986
\(22\) −4.66293 −0.994140
\(23\) 2.04703 0.426835 0.213418 0.976961i \(-0.431540\pi\)
0.213418 + 0.976961i \(0.431540\pi\)
\(24\) 0.347340 0.0709004
\(25\) −3.88650 −0.777300
\(26\) −11.1918 −2.19490
\(27\) 0.632868 0.121795
\(28\) 3.22608 0.609671
\(29\) −5.94578 −1.10410 −0.552052 0.833810i \(-0.686155\pi\)
−0.552052 + 0.833810i \(0.686155\pi\)
\(30\) −0.259433 −0.0473658
\(31\) 3.29742 0.592234 0.296117 0.955152i \(-0.404308\pi\)
0.296117 + 0.955152i \(0.404308\pi\)
\(32\) 4.66253 0.824226
\(33\) −0.211797 −0.0368691
\(34\) −13.1771 −2.25985
\(35\) −0.997497 −0.168608
\(36\) −10.2002 −1.70004
\(37\) −5.04927 −0.830095 −0.415047 0.909800i \(-0.636235\pi\)
−0.415047 + 0.909800i \(0.636235\pi\)
\(38\) −0.629531 −0.102123
\(39\) −0.508349 −0.0814010
\(40\) 3.46840 0.548401
\(41\) −5.06296 −0.790701 −0.395350 0.918530i \(-0.629377\pi\)
−0.395350 + 0.918530i \(0.629377\pi\)
\(42\) 0.232406 0.0358610
\(43\) 1.81524 0.276822 0.138411 0.990375i \(-0.455801\pi\)
0.138411 + 0.990375i \(0.455801\pi\)
\(44\) 6.84001 1.03117
\(45\) 3.15389 0.470154
\(46\) −4.76249 −0.702191
\(47\) 12.2205 1.78254 0.891269 0.453474i \(-0.149816\pi\)
0.891269 + 0.453474i \(0.149816\pi\)
\(48\) −0.0868108 −0.0125301
\(49\) −6.10642 −0.872346
\(50\) 9.04209 1.27874
\(51\) −0.598522 −0.0838098
\(52\) 16.4172 2.27666
\(53\) −1.00000 −0.137361
\(54\) −1.47239 −0.200367
\(55\) −2.11492 −0.285176
\(56\) −3.10706 −0.415199
\(57\) −0.0285942 −0.00378739
\(58\) 13.8331 1.81637
\(59\) −11.0180 −1.43442 −0.717211 0.696856i \(-0.754581\pi\)
−0.717211 + 0.696856i \(0.754581\pi\)
\(60\) 0.380560 0.0491301
\(61\) −7.84960 −1.00504 −0.502519 0.864566i \(-0.667593\pi\)
−0.502519 + 0.864566i \(0.667593\pi\)
\(62\) −7.67157 −0.974290
\(63\) −2.82532 −0.355957
\(64\) −12.4905 −1.56132
\(65\) −5.07617 −0.629622
\(66\) 0.492754 0.0606538
\(67\) −11.3588 −1.38769 −0.693847 0.720122i \(-0.744086\pi\)
−0.693847 + 0.720122i \(0.744086\pi\)
\(68\) 19.3293 2.34403
\(69\) −0.216319 −0.0260418
\(70\) 2.32071 0.277378
\(71\) −6.29987 −0.747657 −0.373829 0.927498i \(-0.621955\pi\)
−0.373829 + 0.927498i \(0.621955\pi\)
\(72\) 9.82393 1.15776
\(73\) −1.26449 −0.147997 −0.0739985 0.997258i \(-0.523576\pi\)
−0.0739985 + 0.997258i \(0.523576\pi\)
\(74\) 11.7473 1.36560
\(75\) 0.410705 0.0474241
\(76\) 0.923454 0.105927
\(77\) 1.89459 0.215909
\(78\) 1.18269 0.133914
\(79\) −9.25154 −1.04088 −0.520440 0.853899i \(-0.674232\pi\)
−0.520440 + 0.853899i \(0.674232\pi\)
\(80\) −0.866858 −0.0969177
\(81\) 8.89962 0.988847
\(82\) 11.7792 1.30079
\(83\) −15.9049 −1.74579 −0.872893 0.487912i \(-0.837759\pi\)
−0.872893 + 0.487912i \(0.837759\pi\)
\(84\) −0.340914 −0.0371968
\(85\) −5.97660 −0.648253
\(86\) −4.22323 −0.455402
\(87\) 0.628318 0.0673628
\(88\) −6.58768 −0.702249
\(89\) −9.97305 −1.05714 −0.528570 0.848889i \(-0.677272\pi\)
−0.528570 + 0.848889i \(0.677272\pi\)
\(90\) −7.33765 −0.773456
\(91\) 4.54735 0.476691
\(92\) 6.98606 0.728347
\(93\) −0.348453 −0.0361329
\(94\) −28.4314 −2.93247
\(95\) −0.285530 −0.0292948
\(96\) −0.492711 −0.0502871
\(97\) 13.8158 1.40278 0.701389 0.712779i \(-0.252564\pi\)
0.701389 + 0.712779i \(0.252564\pi\)
\(98\) 14.2068 1.43511
\(99\) −5.99033 −0.602051
\(100\) −13.2638 −1.32638
\(101\) 11.6367 1.15790 0.578949 0.815364i \(-0.303463\pi\)
0.578949 + 0.815364i \(0.303463\pi\)
\(102\) 1.39248 0.137876
\(103\) −1.85596 −0.182873 −0.0914365 0.995811i \(-0.529146\pi\)
−0.0914365 + 0.995811i \(0.529146\pi\)
\(104\) −15.8116 −1.55045
\(105\) 0.105410 0.0102870
\(106\) 2.32654 0.225973
\(107\) 14.4919 1.40099 0.700493 0.713660i \(-0.252964\pi\)
0.700493 + 0.713660i \(0.252964\pi\)
\(108\) 2.15984 0.207830
\(109\) 11.5306 1.10443 0.552215 0.833702i \(-0.313783\pi\)
0.552215 + 0.833702i \(0.313783\pi\)
\(110\) 4.92044 0.469146
\(111\) 0.533580 0.0506451
\(112\) 0.776550 0.0733771
\(113\) 10.6900 1.00563 0.502814 0.864395i \(-0.332298\pi\)
0.502814 + 0.864395i \(0.332298\pi\)
\(114\) 0.0665255 0.00623068
\(115\) −2.16008 −0.201428
\(116\) −20.2916 −1.88403
\(117\) −14.3778 −1.32923
\(118\) 25.6338 2.35978
\(119\) 5.35397 0.490798
\(120\) −0.366521 −0.0334587
\(121\) −6.98304 −0.634821
\(122\) 18.2624 1.65340
\(123\) 0.535026 0.0482417
\(124\) 11.2534 1.01058
\(125\) 9.37726 0.838727
\(126\) 6.57322 0.585589
\(127\) 8.78273 0.779341 0.389671 0.920954i \(-0.372589\pi\)
0.389671 + 0.920954i \(0.372589\pi\)
\(128\) 19.7346 1.74431
\(129\) −0.191825 −0.0168892
\(130\) 11.8099 1.03580
\(131\) −0.515780 −0.0450639 −0.0225320 0.999746i \(-0.507173\pi\)
−0.0225320 + 0.999746i \(0.507173\pi\)
\(132\) −0.722816 −0.0629130
\(133\) 0.255784 0.0221793
\(134\) 26.4266 2.28291
\(135\) −0.667818 −0.0574766
\(136\) −18.6163 −1.59633
\(137\) 6.81055 0.581865 0.290932 0.956744i \(-0.406035\pi\)
0.290932 + 0.956744i \(0.406035\pi\)
\(138\) 0.503275 0.0428416
\(139\) −17.8245 −1.51186 −0.755928 0.654655i \(-0.772814\pi\)
−0.755928 + 0.654655i \(0.772814\pi\)
\(140\) −3.40424 −0.287710
\(141\) −1.29139 −0.108755
\(142\) 14.6569 1.22998
\(143\) 9.64140 0.806255
\(144\) −2.45530 −0.204608
\(145\) 6.27413 0.521039
\(146\) 2.94188 0.243471
\(147\) 0.645294 0.0532229
\(148\) −17.2320 −1.41646
\(149\) 24.2110 1.98345 0.991723 0.128394i \(-0.0409823\pi\)
0.991723 + 0.128394i \(0.0409823\pi\)
\(150\) −0.955520 −0.0780178
\(151\) −1.00000 −0.0813788
\(152\) −0.889387 −0.0721388
\(153\) −16.9282 −1.36856
\(154\) −4.40784 −0.355194
\(155\) −3.47952 −0.279482
\(156\) −1.73488 −0.138902
\(157\) −13.5873 −1.08439 −0.542194 0.840254i \(-0.682406\pi\)
−0.542194 + 0.840254i \(0.682406\pi\)
\(158\) 21.5240 1.71236
\(159\) 0.105675 0.00838054
\(160\) −4.92001 −0.388961
\(161\) 1.93504 0.152503
\(162\) −20.7053 −1.62676
\(163\) −11.0521 −0.865671 −0.432835 0.901473i \(-0.642487\pi\)
−0.432835 + 0.901473i \(0.642487\pi\)
\(164\) −17.2787 −1.34924
\(165\) 0.223493 0.0173989
\(166\) 37.0033 2.87201
\(167\) 13.3755 1.03503 0.517515 0.855674i \(-0.326857\pi\)
0.517515 + 0.855674i \(0.326857\pi\)
\(168\) 0.328338 0.0253318
\(169\) 10.1410 0.780079
\(170\) 13.9048 1.06645
\(171\) −0.808740 −0.0618459
\(172\) 6.19501 0.472365
\(173\) −11.2195 −0.853004 −0.426502 0.904487i \(-0.640254\pi\)
−0.426502 + 0.904487i \(0.640254\pi\)
\(174\) −1.46181 −0.110819
\(175\) −3.67388 −0.277719
\(176\) 1.64646 0.124107
\(177\) 1.16432 0.0875159
\(178\) 23.2027 1.73911
\(179\) −18.7923 −1.40460 −0.702302 0.711879i \(-0.747844\pi\)
−0.702302 + 0.711879i \(0.747844\pi\)
\(180\) 10.7635 0.802266
\(181\) 9.04545 0.672343 0.336172 0.941801i \(-0.390868\pi\)
0.336172 + 0.941801i \(0.390868\pi\)
\(182\) −10.5796 −0.784210
\(183\) 0.829503 0.0613187
\(184\) −6.72834 −0.496020
\(185\) 5.32811 0.391731
\(186\) 0.810690 0.0594427
\(187\) 11.3516 0.830113
\(188\) 41.7057 3.04170
\(189\) 0.598246 0.0435160
\(190\) 0.664297 0.0481932
\(191\) −27.2592 −1.97240 −0.986202 0.165547i \(-0.947061\pi\)
−0.986202 + 0.165547i \(0.947061\pi\)
\(192\) 1.31993 0.0952579
\(193\) −25.5753 −1.84095 −0.920476 0.390798i \(-0.872199\pi\)
−0.920476 + 0.390798i \(0.872199\pi\)
\(194\) −32.1429 −2.30772
\(195\) 0.536423 0.0384140
\(196\) −20.8398 −1.48856
\(197\) −25.1873 −1.79452 −0.897259 0.441504i \(-0.854445\pi\)
−0.897259 + 0.441504i \(0.854445\pi\)
\(198\) 13.9367 0.990440
\(199\) −12.9728 −0.919620 −0.459810 0.888017i \(-0.652082\pi\)
−0.459810 + 0.888017i \(0.652082\pi\)
\(200\) 12.7745 0.903290
\(201\) 1.20033 0.0846650
\(202\) −27.0733 −1.90487
\(203\) −5.62051 −0.394482
\(204\) −2.04262 −0.143012
\(205\) 5.34256 0.373140
\(206\) 4.31795 0.300846
\(207\) −6.11823 −0.425246
\(208\) 3.95179 0.274008
\(209\) 0.542321 0.0375131
\(210\) −0.245241 −0.0169232
\(211\) −19.2950 −1.32832 −0.664161 0.747589i \(-0.731211\pi\)
−0.664161 + 0.747589i \(0.731211\pi\)
\(212\) −3.41278 −0.234390
\(213\) 0.665737 0.0456155
\(214\) −33.7160 −2.30478
\(215\) −1.91549 −0.130635
\(216\) −2.08016 −0.141537
\(217\) 3.11703 0.211598
\(218\) −26.8264 −1.81691
\(219\) 0.133624 0.00902949
\(220\) −7.21775 −0.486621
\(221\) 27.2459 1.83275
\(222\) −1.24139 −0.0833168
\(223\) 4.64432 0.311007 0.155503 0.987835i \(-0.450300\pi\)
0.155503 + 0.987835i \(0.450300\pi\)
\(224\) 4.40745 0.294485
\(225\) 11.6161 0.774407
\(226\) −24.8706 −1.65437
\(227\) 18.5004 1.22791 0.613957 0.789340i \(-0.289577\pi\)
0.613957 + 0.789340i \(0.289577\pi\)
\(228\) −0.0975856 −0.00646277
\(229\) 12.9241 0.854047 0.427023 0.904241i \(-0.359562\pi\)
0.427023 + 0.904241i \(0.359562\pi\)
\(230\) 5.02550 0.331372
\(231\) −0.200210 −0.0131729
\(232\) 19.5431 1.28306
\(233\) 4.88210 0.319837 0.159919 0.987130i \(-0.448877\pi\)
0.159919 + 0.987130i \(0.448877\pi\)
\(234\) 33.4505 2.18673
\(235\) −12.8953 −0.841199
\(236\) −37.6020 −2.44768
\(237\) 0.977653 0.0635054
\(238\) −12.4562 −0.807416
\(239\) 13.1707 0.851943 0.425971 0.904737i \(-0.359932\pi\)
0.425971 + 0.904737i \(0.359932\pi\)
\(240\) 0.0916049 0.00591307
\(241\) 2.09034 0.134651 0.0673253 0.997731i \(-0.478553\pi\)
0.0673253 + 0.997731i \(0.478553\pi\)
\(242\) 16.2463 1.04435
\(243\) −2.83907 −0.182126
\(244\) −26.7889 −1.71499
\(245\) 6.44365 0.411669
\(246\) −1.24476 −0.0793629
\(247\) 1.30166 0.0828228
\(248\) −10.8382 −0.688227
\(249\) 1.68074 0.106513
\(250\) −21.8165 −1.37980
\(251\) −6.82059 −0.430512 −0.215256 0.976558i \(-0.569059\pi\)
−0.215256 + 0.976558i \(0.569059\pi\)
\(252\) −9.64220 −0.607402
\(253\) 4.10273 0.257937
\(254\) −20.4333 −1.28210
\(255\) 0.631575 0.0395508
\(256\) −20.9323 −1.30827
\(257\) 9.96701 0.621725 0.310862 0.950455i \(-0.399382\pi\)
0.310862 + 0.950455i \(0.399382\pi\)
\(258\) 0.446288 0.0277847
\(259\) −4.77304 −0.296582
\(260\) −17.3238 −1.07438
\(261\) 17.7709 1.09999
\(262\) 1.19998 0.0741351
\(263\) 17.4636 1.07685 0.538426 0.842673i \(-0.319019\pi\)
0.538426 + 0.842673i \(0.319019\pi\)
\(264\) 0.696151 0.0428451
\(265\) 1.05522 0.0648219
\(266\) −0.595092 −0.0364874
\(267\) 1.05390 0.0644975
\(268\) −38.7649 −2.36794
\(269\) −3.40003 −0.207303 −0.103652 0.994614i \(-0.533053\pi\)
−0.103652 + 0.994614i \(0.533053\pi\)
\(270\) 1.55370 0.0945554
\(271\) 9.85897 0.598890 0.299445 0.954114i \(-0.403199\pi\)
0.299445 + 0.954114i \(0.403199\pi\)
\(272\) 4.65278 0.282116
\(273\) −0.480539 −0.0290835
\(274\) −15.8450 −0.957232
\(275\) −7.78947 −0.469723
\(276\) −0.738249 −0.0444374
\(277\) 27.4781 1.65100 0.825500 0.564401i \(-0.190893\pi\)
0.825500 + 0.564401i \(0.190893\pi\)
\(278\) 41.4694 2.48717
\(279\) −9.85543 −0.590029
\(280\) 3.27865 0.195937
\(281\) −22.4075 −1.33672 −0.668360 0.743838i \(-0.733004\pi\)
−0.668360 + 0.743838i \(0.733004\pi\)
\(282\) 3.00448 0.178914
\(283\) 4.32478 0.257081 0.128541 0.991704i \(-0.458971\pi\)
0.128541 + 0.991704i \(0.458971\pi\)
\(284\) −21.5001 −1.27579
\(285\) 0.0301733 0.00178731
\(286\) −22.4311 −1.32638
\(287\) −4.78598 −0.282507
\(288\) −13.9355 −0.821158
\(289\) 15.0788 0.886990
\(290\) −14.5970 −0.857166
\(291\) −1.45998 −0.0855853
\(292\) −4.31541 −0.252540
\(293\) −10.7815 −0.629862 −0.314931 0.949115i \(-0.601981\pi\)
−0.314931 + 0.949115i \(0.601981\pi\)
\(294\) −1.50130 −0.0875576
\(295\) 11.6265 0.676919
\(296\) 16.5963 0.964642
\(297\) 1.26842 0.0736010
\(298\) −56.3279 −3.26299
\(299\) 9.84726 0.569482
\(300\) 1.40164 0.0809239
\(301\) 1.71594 0.0989049
\(302\) 2.32654 0.133877
\(303\) −1.22971 −0.0706448
\(304\) 0.222285 0.0127489
\(305\) 8.28309 0.474288
\(306\) 39.3841 2.25144
\(307\) −16.9354 −0.966556 −0.483278 0.875467i \(-0.660554\pi\)
−0.483278 + 0.875467i \(0.660554\pi\)
\(308\) 6.46582 0.368424
\(309\) 0.196128 0.0111573
\(310\) 8.09523 0.459778
\(311\) 17.4501 0.989505 0.494753 0.869034i \(-0.335259\pi\)
0.494753 + 0.869034i \(0.335259\pi\)
\(312\) 1.67088 0.0945950
\(313\) 4.11461 0.232571 0.116286 0.993216i \(-0.462901\pi\)
0.116286 + 0.993216i \(0.462901\pi\)
\(314\) 31.6114 1.78394
\(315\) 2.98135 0.167980
\(316\) −31.5734 −1.77614
\(317\) −6.94553 −0.390100 −0.195050 0.980793i \(-0.562487\pi\)
−0.195050 + 0.980793i \(0.562487\pi\)
\(318\) −0.245856 −0.0137869
\(319\) −11.9168 −0.667210
\(320\) 13.1803 0.736802
\(321\) −1.53143 −0.0854759
\(322\) −4.50195 −0.250884
\(323\) 1.53256 0.0852737
\(324\) 30.3724 1.68736
\(325\) −18.6961 −1.03707
\(326\) 25.7132 1.42412
\(327\) −1.21849 −0.0673827
\(328\) 16.6413 0.918863
\(329\) 11.5519 0.636878
\(330\) −0.519966 −0.0286232
\(331\) −2.27204 −0.124883 −0.0624414 0.998049i \(-0.519889\pi\)
−0.0624414 + 0.998049i \(0.519889\pi\)
\(332\) −54.2798 −2.97899
\(333\) 15.0914 0.827005
\(334\) −31.1187 −1.70274
\(335\) 11.9861 0.654868
\(336\) −0.0820617 −0.00447683
\(337\) −8.13127 −0.442939 −0.221469 0.975167i \(-0.571085\pi\)
−0.221469 + 0.975167i \(0.571085\pi\)
\(338\) −23.5935 −1.28332
\(339\) −1.12966 −0.0613547
\(340\) −20.3968 −1.10617
\(341\) 6.60881 0.357887
\(342\) 1.88156 0.101743
\(343\) −12.3894 −0.668965
\(344\) −5.96648 −0.321691
\(345\) 0.228265 0.0122894
\(346\) 26.1026 1.40329
\(347\) 1.24494 0.0668321 0.0334161 0.999442i \(-0.489361\pi\)
0.0334161 + 0.999442i \(0.489361\pi\)
\(348\) 2.14431 0.114947
\(349\) −31.3306 −1.67709 −0.838544 0.544833i \(-0.816593\pi\)
−0.838544 + 0.544833i \(0.816593\pi\)
\(350\) 8.54743 0.456879
\(351\) 3.04442 0.162499
\(352\) 9.34480 0.498080
\(353\) −7.46509 −0.397327 −0.198663 0.980068i \(-0.563660\pi\)
−0.198663 + 0.980068i \(0.563660\pi\)
\(354\) −2.70884 −0.143973
\(355\) 6.64778 0.352828
\(356\) −34.0358 −1.80389
\(357\) −0.565779 −0.0299442
\(358\) 43.7210 2.31073
\(359\) 18.2435 0.962854 0.481427 0.876486i \(-0.340119\pi\)
0.481427 + 0.876486i \(0.340119\pi\)
\(360\) −10.3665 −0.546360
\(361\) −18.9268 −0.996146
\(362\) −21.0446 −1.10608
\(363\) 0.737930 0.0387313
\(364\) 15.5191 0.813421
\(365\) 1.33432 0.0698414
\(366\) −1.92987 −0.100876
\(367\) 33.6861 1.75840 0.879199 0.476455i \(-0.158078\pi\)
0.879199 + 0.476455i \(0.158078\pi\)
\(368\) 1.68162 0.0876604
\(369\) 15.1323 0.787758
\(370\) −12.3961 −0.644440
\(371\) −0.945293 −0.0490772
\(372\) −1.18919 −0.0616568
\(373\) −31.5618 −1.63421 −0.817104 0.576490i \(-0.804422\pi\)
−0.817104 + 0.576490i \(0.804422\pi\)
\(374\) −26.4100 −1.36563
\(375\) −0.990938 −0.0511718
\(376\) −40.1672 −2.07146
\(377\) −28.6022 −1.47309
\(378\) −1.39184 −0.0715886
\(379\) 10.3087 0.529523 0.264761 0.964314i \(-0.414707\pi\)
0.264761 + 0.964314i \(0.414707\pi\)
\(380\) −0.974451 −0.0499883
\(381\) −0.928112 −0.0475486
\(382\) 63.4195 3.24482
\(383\) 1.01110 0.0516647 0.0258323 0.999666i \(-0.491776\pi\)
0.0258323 + 0.999666i \(0.491776\pi\)
\(384\) −2.08545 −0.106423
\(385\) −1.99922 −0.101890
\(386\) 59.5020 3.02857
\(387\) −5.42545 −0.275791
\(388\) 47.1501 2.39368
\(389\) 22.5036 1.14098 0.570488 0.821306i \(-0.306754\pi\)
0.570488 + 0.821306i \(0.306754\pi\)
\(390\) −1.24801 −0.0631953
\(391\) 11.5940 0.586334
\(392\) 20.0711 1.01374
\(393\) 0.0545049 0.00274941
\(394\) 58.5991 2.95218
\(395\) 9.76245 0.491202
\(396\) −20.4437 −1.02733
\(397\) 23.3877 1.17379 0.586896 0.809662i \(-0.300350\pi\)
0.586896 + 0.809662i \(0.300350\pi\)
\(398\) 30.1818 1.51288
\(399\) −0.0270299 −0.00135319
\(400\) −3.19273 −0.159636
\(401\) 36.5891 1.82717 0.913587 0.406644i \(-0.133301\pi\)
0.913587 + 0.406644i \(0.133301\pi\)
\(402\) −2.79262 −0.139283
\(403\) 15.8623 0.790156
\(404\) 39.7135 1.97582
\(405\) −9.39110 −0.466648
\(406\) 13.0763 0.648967
\(407\) −10.1199 −0.501626
\(408\) 1.96727 0.0973943
\(409\) 18.8697 0.933047 0.466524 0.884509i \(-0.345506\pi\)
0.466524 + 0.884509i \(0.345506\pi\)
\(410\) −12.4297 −0.613857
\(411\) −0.719703 −0.0355003
\(412\) −6.33397 −0.312052
\(413\) −10.4152 −0.512501
\(414\) 14.2343 0.699577
\(415\) 16.7832 0.823855
\(416\) 22.4291 1.09968
\(417\) 1.88360 0.0922403
\(418\) −1.26173 −0.0617132
\(419\) 13.1710 0.643447 0.321724 0.946834i \(-0.395738\pi\)
0.321724 + 0.946834i \(0.395738\pi\)
\(420\) 0.359741 0.0175536
\(421\) −33.5068 −1.63302 −0.816510 0.577331i \(-0.804094\pi\)
−0.816510 + 0.577331i \(0.804094\pi\)
\(422\) 44.8905 2.18524
\(423\) −36.5249 −1.77590
\(424\) 3.28688 0.159625
\(425\) −22.0124 −1.06776
\(426\) −1.54886 −0.0750426
\(427\) −7.42017 −0.359087
\(428\) 49.4576 2.39063
\(429\) −1.01885 −0.0491906
\(430\) 4.45645 0.214909
\(431\) −12.4916 −0.601700 −0.300850 0.953671i \(-0.597270\pi\)
−0.300850 + 0.953671i \(0.597270\pi\)
\(432\) 0.519895 0.0250135
\(433\) −29.8047 −1.43232 −0.716162 0.697935i \(-0.754103\pi\)
−0.716162 + 0.697935i \(0.754103\pi\)
\(434\) −7.25188 −0.348101
\(435\) −0.663017 −0.0317892
\(436\) 39.3513 1.88459
\(437\) 0.553900 0.0264966
\(438\) −0.310882 −0.0148545
\(439\) −25.7071 −1.22693 −0.613467 0.789721i \(-0.710225\pi\)
−0.613467 + 0.789721i \(0.710225\pi\)
\(440\) 6.95149 0.331399
\(441\) 18.2511 0.869099
\(442\) −63.3885 −3.01508
\(443\) −1.37364 −0.0652636 −0.0326318 0.999467i \(-0.510389\pi\)
−0.0326318 + 0.999467i \(0.510389\pi\)
\(444\) 1.82099 0.0864203
\(445\) 10.5238 0.498876
\(446\) −10.8052 −0.511641
\(447\) −2.55849 −0.121013
\(448\) −11.8072 −0.557838
\(449\) −3.08402 −0.145544 −0.0727719 0.997349i \(-0.523185\pi\)
−0.0727719 + 0.997349i \(0.523185\pi\)
\(450\) −27.0253 −1.27398
\(451\) −10.1474 −0.477821
\(452\) 36.4825 1.71599
\(453\) 0.105675 0.00496503
\(454\) −43.0418 −2.02005
\(455\) −4.79847 −0.224956
\(456\) 0.0939857 0.00440128
\(457\) 27.5201 1.28734 0.643669 0.765304i \(-0.277411\pi\)
0.643669 + 0.765304i \(0.277411\pi\)
\(458\) −30.0683 −1.40500
\(459\) 3.58445 0.167308
\(460\) −7.37186 −0.343715
\(461\) 21.4450 0.998792 0.499396 0.866374i \(-0.333555\pi\)
0.499396 + 0.866374i \(0.333555\pi\)
\(462\) 0.465797 0.0216708
\(463\) 5.50370 0.255779 0.127889 0.991788i \(-0.459180\pi\)
0.127889 + 0.991788i \(0.459180\pi\)
\(464\) −4.88441 −0.226753
\(465\) 0.367697 0.0170515
\(466\) −11.3584 −0.526168
\(467\) −13.3843 −0.619351 −0.309676 0.950842i \(-0.600220\pi\)
−0.309676 + 0.950842i \(0.600220\pi\)
\(468\) −49.0683 −2.26818
\(469\) −10.7374 −0.495806
\(470\) 30.0015 1.38387
\(471\) 1.43584 0.0661598
\(472\) 36.2148 1.66692
\(473\) 3.63817 0.167283
\(474\) −2.27455 −0.104473
\(475\) −1.05164 −0.0482524
\(476\) 18.2719 0.837491
\(477\) 2.98883 0.136849
\(478\) −30.6422 −1.40154
\(479\) −25.0429 −1.14424 −0.572119 0.820171i \(-0.693878\pi\)
−0.572119 + 0.820171i \(0.693878\pi\)
\(480\) 0.519921 0.0237310
\(481\) −24.2896 −1.10751
\(482\) −4.86326 −0.221515
\(483\) −0.204485 −0.00930439
\(484\) −23.8315 −1.08325
\(485\) −14.5787 −0.661986
\(486\) 6.60520 0.299618
\(487\) −14.1657 −0.641911 −0.320956 0.947094i \(-0.604004\pi\)
−0.320956 + 0.947094i \(0.604004\pi\)
\(488\) 25.8007 1.16794
\(489\) 1.16793 0.0528157
\(490\) −14.9914 −0.677242
\(491\) −20.4328 −0.922121 −0.461061 0.887369i \(-0.652531\pi\)
−0.461061 + 0.887369i \(0.652531\pi\)
\(492\) 1.82592 0.0823190
\(493\) −33.6758 −1.51668
\(494\) −3.02837 −0.136253
\(495\) 6.32114 0.284114
\(496\) 2.70880 0.121629
\(497\) −5.95523 −0.267129
\(498\) −3.91031 −0.175225
\(499\) −24.5326 −1.09823 −0.549114 0.835747i \(-0.685035\pi\)
−0.549114 + 0.835747i \(0.685035\pi\)
\(500\) 32.0025 1.43119
\(501\) −1.41346 −0.0631485
\(502\) 15.8684 0.708239
\(503\) −27.2903 −1.21681 −0.608406 0.793626i \(-0.708191\pi\)
−0.608406 + 0.793626i \(0.708191\pi\)
\(504\) 9.28650 0.413653
\(505\) −12.2794 −0.546424
\(506\) −9.54516 −0.424334
\(507\) −1.07165 −0.0475936
\(508\) 29.9735 1.32986
\(509\) 37.3185 1.65411 0.827057 0.562117i \(-0.190013\pi\)
0.827057 + 0.562117i \(0.190013\pi\)
\(510\) −1.46938 −0.0650654
\(511\) −1.19531 −0.0528775
\(512\) 9.23051 0.407935
\(513\) 0.171246 0.00756069
\(514\) −23.1886 −1.02281
\(515\) 1.95845 0.0862997
\(516\) −0.654656 −0.0288196
\(517\) 24.4927 1.07719
\(518\) 11.1047 0.487911
\(519\) 1.18562 0.0520429
\(520\) 16.6848 0.731675
\(521\) 29.2842 1.28296 0.641482 0.767138i \(-0.278320\pi\)
0.641482 + 0.767138i \(0.278320\pi\)
\(522\) −41.3448 −1.80961
\(523\) 31.4200 1.37390 0.686950 0.726705i \(-0.258949\pi\)
0.686950 + 0.726705i \(0.258949\pi\)
\(524\) −1.76024 −0.0768965
\(525\) 0.388236 0.0169440
\(526\) −40.6297 −1.77154
\(527\) 18.6760 0.813538
\(528\) −0.173989 −0.00757192
\(529\) −18.8097 −0.817812
\(530\) −2.45502 −0.106639
\(531\) 32.9310 1.42908
\(532\) 0.872935 0.0378465
\(533\) −24.3554 −1.05495
\(534\) −2.45193 −0.106106
\(535\) −15.2922 −0.661140
\(536\) 37.3349 1.61262
\(537\) 1.98587 0.0856966
\(538\) 7.91029 0.341037
\(539\) −12.2387 −0.527159
\(540\) −2.27911 −0.0980774
\(541\) −18.8032 −0.808415 −0.404207 0.914667i \(-0.632453\pi\)
−0.404207 + 0.914667i \(0.632453\pi\)
\(542\) −22.9373 −0.985240
\(543\) −0.955875 −0.0410205
\(544\) 26.4077 1.13222
\(545\) −12.1674 −0.521193
\(546\) 1.11799 0.0478456
\(547\) −44.2854 −1.89351 −0.946754 0.321958i \(-0.895659\pi\)
−0.946754 + 0.321958i \(0.895659\pi\)
\(548\) 23.2429 0.992887
\(549\) 23.4611 1.00130
\(550\) 18.1225 0.772745
\(551\) −1.60885 −0.0685394
\(552\) 0.711015 0.0302628
\(553\) −8.74542 −0.371893
\(554\) −63.9289 −2.71608
\(555\) −0.563046 −0.0239000
\(556\) −60.8311 −2.57981
\(557\) 24.4583 1.03633 0.518166 0.855280i \(-0.326615\pi\)
0.518166 + 0.855280i \(0.326615\pi\)
\(558\) 22.9290 0.970663
\(559\) 8.73224 0.369334
\(560\) −0.819435 −0.0346275
\(561\) −1.19958 −0.0506463
\(562\) 52.1319 2.19905
\(563\) −23.3970 −0.986067 −0.493033 0.870010i \(-0.664112\pi\)
−0.493033 + 0.870010i \(0.664112\pi\)
\(564\) −4.40724 −0.185578
\(565\) −11.2803 −0.474567
\(566\) −10.0618 −0.422927
\(567\) 8.41275 0.353302
\(568\) 20.7069 0.868843
\(569\) 32.9512 1.38138 0.690692 0.723149i \(-0.257306\pi\)
0.690692 + 0.723149i \(0.257306\pi\)
\(570\) −0.0701993 −0.00294033
\(571\) −35.6656 −1.49256 −0.746280 0.665632i \(-0.768162\pi\)
−0.746280 + 0.665632i \(0.768162\pi\)
\(572\) 32.9040 1.37578
\(573\) 2.88060 0.120339
\(574\) 11.1348 0.464756
\(575\) −7.95578 −0.331779
\(576\) 37.3321 1.55550
\(577\) 12.4895 0.519944 0.259972 0.965616i \(-0.416287\pi\)
0.259972 + 0.965616i \(0.416287\pi\)
\(578\) −35.0814 −1.45920
\(579\) 2.70266 0.112319
\(580\) 21.4122 0.889094
\(581\) −15.0348 −0.623747
\(582\) 3.39669 0.140797
\(583\) −2.00424 −0.0830070
\(584\) 4.15621 0.171985
\(585\) 15.1718 0.627278
\(586\) 25.0835 1.03619
\(587\) 11.3770 0.469581 0.234790 0.972046i \(-0.424560\pi\)
0.234790 + 0.972046i \(0.424560\pi\)
\(588\) 2.20224 0.0908190
\(589\) 0.892239 0.0367641
\(590\) −27.0494 −1.11361
\(591\) 2.66166 0.109486
\(592\) −4.14793 −0.170479
\(593\) −5.62730 −0.231086 −0.115543 0.993302i \(-0.536861\pi\)
−0.115543 + 0.993302i \(0.536861\pi\)
\(594\) −2.95102 −0.121082
\(595\) −5.64964 −0.231613
\(596\) 82.6269 3.38453
\(597\) 1.37090 0.0561072
\(598\) −22.9100 −0.936861
\(599\) −22.0472 −0.900823 −0.450412 0.892821i \(-0.648723\pi\)
−0.450412 + 0.892821i \(0.648723\pi\)
\(600\) −1.34994 −0.0551109
\(601\) 5.80486 0.236785 0.118393 0.992967i \(-0.462226\pi\)
0.118393 + 0.992967i \(0.462226\pi\)
\(602\) −3.99219 −0.162709
\(603\) 33.9495 1.38253
\(604\) −3.41278 −0.138864
\(605\) 7.36867 0.299579
\(606\) 2.86096 0.116219
\(607\) 17.0397 0.691621 0.345810 0.938304i \(-0.387604\pi\)
0.345810 + 0.938304i \(0.387604\pi\)
\(608\) 1.26162 0.0511654
\(609\) 0.593945 0.0240679
\(610\) −19.2709 −0.780257
\(611\) 58.7867 2.37826
\(612\) −57.7722 −2.33530
\(613\) −25.1182 −1.01451 −0.507256 0.861795i \(-0.669340\pi\)
−0.507256 + 0.861795i \(0.669340\pi\)
\(614\) 39.4009 1.59009
\(615\) −0.564573 −0.0227658
\(616\) −6.22729 −0.250905
\(617\) −35.5411 −1.43083 −0.715414 0.698700i \(-0.753762\pi\)
−0.715414 + 0.698700i \(0.753762\pi\)
\(618\) −0.456298 −0.0183550
\(619\) 9.59120 0.385503 0.192751 0.981248i \(-0.438259\pi\)
0.192751 + 0.981248i \(0.438259\pi\)
\(620\) −11.8748 −0.476904
\(621\) 1.29550 0.0519866
\(622\) −40.5983 −1.62785
\(623\) −9.42746 −0.377703
\(624\) −0.417604 −0.0167176
\(625\) 9.53739 0.381496
\(626\) −9.57279 −0.382606
\(627\) −0.0573096 −0.00228872
\(628\) −46.3705 −1.85039
\(629\) −28.5981 −1.14028
\(630\) −6.93623 −0.276346
\(631\) −17.1033 −0.680871 −0.340436 0.940268i \(-0.610575\pi\)
−0.340436 + 0.940268i \(0.610575\pi\)
\(632\) 30.4087 1.20959
\(633\) 2.03899 0.0810426
\(634\) 16.1590 0.641757
\(635\) −9.26775 −0.367780
\(636\) 0.360644 0.0143005
\(637\) −29.3750 −1.16388
\(638\) 27.7248 1.09763
\(639\) 18.8293 0.744874
\(640\) −20.8245 −0.823159
\(641\) 8.39218 0.331471 0.165736 0.986170i \(-0.447000\pi\)
0.165736 + 0.986170i \(0.447000\pi\)
\(642\) 3.56292 0.140617
\(643\) 4.57793 0.180536 0.0902680 0.995918i \(-0.471228\pi\)
0.0902680 + 0.995918i \(0.471228\pi\)
\(644\) 6.60387 0.260229
\(645\) 0.202418 0.00797022
\(646\) −3.56555 −0.140285
\(647\) 37.0464 1.45644 0.728221 0.685342i \(-0.240347\pi\)
0.728221 + 0.685342i \(0.240347\pi\)
\(648\) −29.2520 −1.14913
\(649\) −22.0827 −0.866821
\(650\) 43.4971 1.70610
\(651\) −0.329391 −0.0129098
\(652\) −37.7185 −1.47717
\(653\) 11.6546 0.456079 0.228039 0.973652i \(-0.426768\pi\)
0.228039 + 0.973652i \(0.426768\pi\)
\(654\) 2.83487 0.110852
\(655\) 0.544264 0.0212661
\(656\) −4.15917 −0.162388
\(657\) 3.77934 0.147446
\(658\) −26.8760 −1.04774
\(659\) 0.398028 0.0155050 0.00775248 0.999970i \(-0.497532\pi\)
0.00775248 + 0.999970i \(0.497532\pi\)
\(660\) 0.762733 0.0296894
\(661\) 1.51832 0.0590559 0.0295279 0.999564i \(-0.490600\pi\)
0.0295279 + 0.999564i \(0.490600\pi\)
\(662\) 5.28600 0.205446
\(663\) −2.87920 −0.111819
\(664\) 52.2774 2.02875
\(665\) −0.269910 −0.0104667
\(666\) −35.1107 −1.36051
\(667\) −12.1712 −0.471270
\(668\) 45.6477 1.76616
\(669\) −0.490787 −0.0189749
\(670\) −27.8860 −1.07733
\(671\) −15.7325 −0.607345
\(672\) −0.465756 −0.0179669
\(673\) 24.7578 0.954343 0.477171 0.878810i \(-0.341662\pi\)
0.477171 + 0.878810i \(0.341662\pi\)
\(674\) 18.9177 0.728683
\(675\) −2.45964 −0.0946716
\(676\) 34.6091 1.33112
\(677\) −9.46609 −0.363812 −0.181906 0.983316i \(-0.558227\pi\)
−0.181906 + 0.983316i \(0.558227\pi\)
\(678\) 2.62819 0.100935
\(679\) 13.0599 0.501195
\(680\) 19.6444 0.753327
\(681\) −1.95502 −0.0749166
\(682\) −15.3756 −0.588764
\(683\) −29.9564 −1.14625 −0.573124 0.819469i \(-0.694269\pi\)
−0.573124 + 0.819469i \(0.694269\pi\)
\(684\) −2.76005 −0.105533
\(685\) −7.18666 −0.274588
\(686\) 28.8244 1.10052
\(687\) −1.36575 −0.0521065
\(688\) 1.49121 0.0568517
\(689\) −4.81051 −0.183266
\(690\) −0.531068 −0.0202174
\(691\) 13.8166 0.525609 0.262804 0.964849i \(-0.415353\pi\)
0.262804 + 0.964849i \(0.415353\pi\)
\(692\) −38.2897 −1.45556
\(693\) −5.66262 −0.215105
\(694\) −2.89641 −0.109946
\(695\) 18.8089 0.713461
\(696\) −2.06521 −0.0782814
\(697\) −28.6757 −1.08617
\(698\) 72.8918 2.75900
\(699\) −0.515915 −0.0195137
\(700\) −12.5381 −0.473897
\(701\) −25.0831 −0.947377 −0.473688 0.880692i \(-0.657078\pi\)
−0.473688 + 0.880692i \(0.657078\pi\)
\(702\) −7.08295 −0.267329
\(703\) −1.36627 −0.0515297
\(704\) −25.0340 −0.943503
\(705\) 1.36271 0.0513227
\(706\) 17.3678 0.653646
\(707\) 11.0001 0.413702
\(708\) 3.97357 0.149336
\(709\) −39.9093 −1.49882 −0.749412 0.662104i \(-0.769664\pi\)
−0.749412 + 0.662104i \(0.769664\pi\)
\(710\) −15.4663 −0.580441
\(711\) 27.6513 1.03700
\(712\) 32.7802 1.22849
\(713\) 6.74992 0.252786
\(714\) 1.31631 0.0492615
\(715\) −10.1739 −0.380480
\(716\) −64.1340 −2.39680
\(717\) −1.39181 −0.0519781
\(718\) −42.4441 −1.58400
\(719\) 11.7807 0.439348 0.219674 0.975573i \(-0.429501\pi\)
0.219674 + 0.975573i \(0.429501\pi\)
\(720\) 2.59089 0.0965569
\(721\) −1.75442 −0.0653382
\(722\) 44.0339 1.63877
\(723\) −0.220896 −0.00821521
\(724\) 30.8701 1.14728
\(725\) 23.1083 0.858220
\(726\) −1.71682 −0.0637172
\(727\) 48.5814 1.80178 0.900892 0.434043i \(-0.142914\pi\)
0.900892 + 0.434043i \(0.142914\pi\)
\(728\) −14.9466 −0.553957
\(729\) −26.3988 −0.977735
\(730\) −3.10434 −0.114897
\(731\) 10.2812 0.380264
\(732\) 2.83091 0.104633
\(733\) −45.4763 −1.67970 −0.839852 0.542815i \(-0.817358\pi\)
−0.839852 + 0.542815i \(0.817358\pi\)
\(734\) −78.3719 −2.89276
\(735\) −0.680930 −0.0251165
\(736\) 9.54433 0.351809
\(737\) −22.7657 −0.838584
\(738\) −35.2059 −1.29595
\(739\) 0.141290 0.00519742 0.00259871 0.999997i \(-0.499173\pi\)
0.00259871 + 0.999997i \(0.499173\pi\)
\(740\) 18.1837 0.668445
\(741\) −0.137553 −0.00505313
\(742\) 2.19926 0.0807374
\(743\) −18.1142 −0.664545 −0.332272 0.943183i \(-0.607815\pi\)
−0.332272 + 0.943183i \(0.607815\pi\)
\(744\) 1.14532 0.0419896
\(745\) −25.5481 −0.936010
\(746\) 73.4297 2.68845
\(747\) 47.5370 1.73929
\(748\) 38.7406 1.41650
\(749\) 13.6991 0.500554
\(750\) 2.30545 0.0841833
\(751\) −44.4699 −1.62273 −0.811364 0.584541i \(-0.801275\pi\)
−0.811364 + 0.584541i \(0.801275\pi\)
\(752\) 10.0390 0.366085
\(753\) 0.720763 0.0262661
\(754\) 66.5442 2.42340
\(755\) 1.05522 0.0384036
\(756\) 2.04168 0.0742552
\(757\) 16.5346 0.600959 0.300480 0.953788i \(-0.402853\pi\)
0.300480 + 0.953788i \(0.402853\pi\)
\(758\) −23.9836 −0.871123
\(759\) −0.433555 −0.0157370
\(760\) 0.938503 0.0340431
\(761\) −47.3243 −1.71551 −0.857753 0.514063i \(-0.828140\pi\)
−0.857753 + 0.514063i \(0.828140\pi\)
\(762\) 2.15929 0.0782227
\(763\) 10.8998 0.394599
\(764\) −93.0294 −3.36569
\(765\) 17.8631 0.645840
\(766\) −2.35236 −0.0849941
\(767\) −53.0022 −1.91380
\(768\) 2.21201 0.0798191
\(769\) 19.6585 0.708903 0.354452 0.935074i \(-0.384668\pi\)
0.354452 + 0.935074i \(0.384668\pi\)
\(770\) 4.65126 0.167620
\(771\) −1.05326 −0.0379322
\(772\) −87.2829 −3.14138
\(773\) −41.1572 −1.48032 −0.740160 0.672431i \(-0.765250\pi\)
−0.740160 + 0.672431i \(0.765250\pi\)
\(774\) 12.6225 0.453707
\(775\) −12.8154 −0.460343
\(776\) −45.4107 −1.63015
\(777\) 0.504389 0.0180949
\(778\) −52.3554 −1.87703
\(779\) −1.36997 −0.0490843
\(780\) 1.83069 0.0655492
\(781\) −12.6264 −0.451809
\(782\) −26.9739 −0.964584
\(783\) −3.76289 −0.134475
\(784\) −5.01637 −0.179156
\(785\) 14.3377 0.511734
\(786\) −0.126808 −0.00452308
\(787\) −9.52652 −0.339584 −0.169792 0.985480i \(-0.554310\pi\)
−0.169792 + 0.985480i \(0.554310\pi\)
\(788\) −85.9585 −3.06215
\(789\) −1.84546 −0.0657001
\(790\) −22.7127 −0.808082
\(791\) 10.1052 0.359298
\(792\) 19.6895 0.699635
\(793\) −37.7606 −1.34092
\(794\) −54.4123 −1.93102
\(795\) −0.111511 −0.00395487
\(796\) −44.2734 −1.56923
\(797\) 6.28050 0.222467 0.111233 0.993794i \(-0.464520\pi\)
0.111233 + 0.993794i \(0.464520\pi\)
\(798\) 0.0628861 0.00222614
\(799\) 69.2145 2.44863
\(800\) −18.1209 −0.640671
\(801\) 29.8078 1.05321
\(802\) −85.1260 −3.00590
\(803\) −2.53433 −0.0894346
\(804\) 4.09647 0.144471
\(805\) −2.04191 −0.0719678
\(806\) −36.9042 −1.29989
\(807\) 0.359297 0.0126478
\(808\) −38.2485 −1.34558
\(809\) −40.6225 −1.42821 −0.714105 0.700038i \(-0.753166\pi\)
−0.714105 + 0.700038i \(0.753166\pi\)
\(810\) 21.8487 0.767687
\(811\) 3.23697 0.113665 0.0568327 0.998384i \(-0.481900\pi\)
0.0568327 + 0.998384i \(0.481900\pi\)
\(812\) −19.1815 −0.673140
\(813\) −1.04184 −0.0365390
\(814\) 23.5444 0.825231
\(815\) 11.6625 0.408520
\(816\) −0.491680 −0.0172123
\(817\) 0.491181 0.0171843
\(818\) −43.9011 −1.53497
\(819\) −13.5913 −0.474917
\(820\) 18.2330 0.636722
\(821\) −4.72802 −0.165009 −0.0825046 0.996591i \(-0.526292\pi\)
−0.0825046 + 0.996591i \(0.526292\pi\)
\(822\) 1.67442 0.0584019
\(823\) −33.8363 −1.17946 −0.589729 0.807601i \(-0.700765\pi\)
−0.589729 + 0.807601i \(0.700765\pi\)
\(824\) 6.10031 0.212514
\(825\) 0.823149 0.0286584
\(826\) 24.2315 0.843120
\(827\) −51.8419 −1.80272 −0.901360 0.433070i \(-0.857430\pi\)
−0.901360 + 0.433070i \(0.857430\pi\)
\(828\) −20.8802 −0.725636
\(829\) 6.74425 0.234238 0.117119 0.993118i \(-0.462634\pi\)
0.117119 + 0.993118i \(0.462634\pi\)
\(830\) −39.0468 −1.35533
\(831\) −2.90374 −0.100730
\(832\) −60.0858 −2.08310
\(833\) −34.5856 −1.19832
\(834\) −4.38226 −0.151745
\(835\) −14.1142 −0.488442
\(836\) 1.85082 0.0640119
\(837\) 2.08683 0.0721314
\(838\) −30.6429 −1.05854
\(839\) 6.13693 0.211870 0.105935 0.994373i \(-0.466216\pi\)
0.105935 + 0.994373i \(0.466216\pi\)
\(840\) −0.346470 −0.0119544
\(841\) 6.35230 0.219045
\(842\) 77.9548 2.68650
\(843\) 2.36791 0.0815551
\(844\) −65.8495 −2.26663
\(845\) −10.7011 −0.368128
\(846\) 84.9766 2.92156
\(847\) −6.60102 −0.226814
\(848\) −0.821491 −0.0282101
\(849\) −0.457019 −0.0156849
\(850\) 51.2127 1.75658
\(851\) −10.3360 −0.354314
\(852\) 2.27201 0.0778378
\(853\) −30.3243 −1.03828 −0.519142 0.854688i \(-0.673748\pi\)
−0.519142 + 0.854688i \(0.673748\pi\)
\(854\) 17.2633 0.590738
\(855\) 0.853402 0.0291857
\(856\) −47.6331 −1.62807
\(857\) −49.2665 −1.68291 −0.841456 0.540325i \(-0.818301\pi\)
−0.841456 + 0.540325i \(0.818301\pi\)
\(858\) 2.37040 0.0809240
\(859\) 52.8682 1.80384 0.901920 0.431904i \(-0.142158\pi\)
0.901920 + 0.431904i \(0.142158\pi\)
\(860\) −6.53713 −0.222914
\(861\) 0.505757 0.0172361
\(862\) 29.0622 0.989863
\(863\) 52.0226 1.77087 0.885434 0.464765i \(-0.153861\pi\)
0.885434 + 0.464765i \(0.153861\pi\)
\(864\) 2.95076 0.100387
\(865\) 11.8391 0.402542
\(866\) 69.3418 2.35633
\(867\) −1.59345 −0.0541164
\(868\) 10.6377 0.361068
\(869\) −18.5423 −0.629003
\(870\) 1.54253 0.0522968
\(871\) −54.6415 −1.85146
\(872\) −37.8997 −1.28344
\(873\) −41.2930 −1.39756
\(874\) −1.28867 −0.0435899
\(875\) 8.86426 0.299667
\(876\) 0.456030 0.0154078
\(877\) 9.80581 0.331119 0.165559 0.986200i \(-0.447057\pi\)
0.165559 + 0.986200i \(0.447057\pi\)
\(878\) 59.8085 2.01844
\(879\) 1.13933 0.0384287
\(880\) −1.73739 −0.0585674
\(881\) −43.3366 −1.46005 −0.730024 0.683421i \(-0.760491\pi\)
−0.730024 + 0.683421i \(0.760491\pi\)
\(882\) −42.4618 −1.42976
\(883\) 2.69451 0.0906774 0.0453387 0.998972i \(-0.485563\pi\)
0.0453387 + 0.998972i \(0.485563\pi\)
\(884\) 92.9840 3.12739
\(885\) −1.22862 −0.0412997
\(886\) 3.19582 0.107366
\(887\) 12.1789 0.408927 0.204463 0.978874i \(-0.434455\pi\)
0.204463 + 0.978874i \(0.434455\pi\)
\(888\) −1.75381 −0.0588540
\(889\) 8.30226 0.278449
\(890\) −24.4840 −0.820707
\(891\) 17.8369 0.597560
\(892\) 15.8500 0.530699
\(893\) 3.30670 0.110655
\(894\) 5.95243 0.199079
\(895\) 19.8301 0.662848
\(896\) 18.6550 0.623220
\(897\) −1.04061 −0.0347448
\(898\) 7.17508 0.239436
\(899\) −19.6057 −0.653887
\(900\) 39.6432 1.32144
\(901\) −5.66382 −0.188689
\(902\) 23.6082 0.786068
\(903\) −0.181331 −0.00603431
\(904\) −35.1366 −1.16863
\(905\) −9.54499 −0.317286
\(906\) −0.245856 −0.00816802
\(907\) −21.4756 −0.713084 −0.356542 0.934279i \(-0.616044\pi\)
−0.356542 + 0.934279i \(0.616044\pi\)
\(908\) 63.1377 2.09530
\(909\) −34.7802 −1.15359
\(910\) 11.1638 0.370077
\(911\) −3.06655 −0.101599 −0.0507997 0.998709i \(-0.516177\pi\)
−0.0507997 + 0.998709i \(0.516177\pi\)
\(912\) −0.0234899 −0.000777828 0
\(913\) −31.8771 −1.05498
\(914\) −64.0266 −2.11781
\(915\) −0.875313 −0.0289369
\(916\) 44.1070 1.45734
\(917\) −0.487563 −0.0161008
\(918\) −8.33935 −0.275240
\(919\) −31.8045 −1.04913 −0.524567 0.851369i \(-0.675773\pi\)
−0.524567 + 0.851369i \(0.675773\pi\)
\(920\) 7.09991 0.234077
\(921\) 1.78965 0.0589708
\(922\) −49.8926 −1.64312
\(923\) −30.3056 −0.997522
\(924\) −0.683273 −0.0224780
\(925\) 19.6240 0.645233
\(926\) −12.8046 −0.420784
\(927\) 5.54715 0.182192
\(928\) −27.7224 −0.910031
\(929\) 35.0498 1.14995 0.574973 0.818172i \(-0.305012\pi\)
0.574973 + 0.818172i \(0.305012\pi\)
\(930\) −0.855460 −0.0280517
\(931\) −1.65232 −0.0541526
\(932\) 16.6615 0.545767
\(933\) −1.84403 −0.0603710
\(934\) 31.1391 1.01890
\(935\) −11.9785 −0.391740
\(936\) 47.2581 1.54468
\(937\) 16.7390 0.546839 0.273419 0.961895i \(-0.411845\pi\)
0.273419 + 0.961895i \(0.411845\pi\)
\(938\) 24.9809 0.815655
\(939\) −0.434810 −0.0141895
\(940\) −44.0089 −1.43541
\(941\) 21.0001 0.684583 0.342291 0.939594i \(-0.388797\pi\)
0.342291 + 0.939594i \(0.388797\pi\)
\(942\) −3.34053 −0.108840
\(943\) −10.3640 −0.337499
\(944\) −9.05119 −0.294591
\(945\) −0.631284 −0.0205357
\(946\) −8.46435 −0.275200
\(947\) 27.2102 0.884214 0.442107 0.896962i \(-0.354231\pi\)
0.442107 + 0.896962i \(0.354231\pi\)
\(948\) 3.33651 0.108365
\(949\) −6.08283 −0.197457
\(950\) 2.44667 0.0793806
\(951\) 0.733966 0.0238005
\(952\) −17.5978 −0.570349
\(953\) −58.4981 −1.89494 −0.947470 0.319846i \(-0.896369\pi\)
−0.947470 + 0.319846i \(0.896369\pi\)
\(954\) −6.95363 −0.225132
\(955\) 28.7645 0.930799
\(956\) 44.9487 1.45375
\(957\) 1.25930 0.0407073
\(958\) 58.2632 1.88240
\(959\) 6.43797 0.207893
\(960\) −1.39282 −0.0449532
\(961\) −20.1270 −0.649259
\(962\) 56.5106 1.82197
\(963\) −43.3139 −1.39577
\(964\) 7.13387 0.229766
\(965\) 26.9877 0.868766
\(966\) 0.475742 0.0153068
\(967\) −24.8688 −0.799726 −0.399863 0.916575i \(-0.630942\pi\)
−0.399863 + 0.916575i \(0.630942\pi\)
\(968\) 22.9524 0.737718
\(969\) −0.161952 −0.00520266
\(970\) 33.9180 1.08904
\(971\) −42.4789 −1.36321 −0.681606 0.731719i \(-0.738718\pi\)
−0.681606 + 0.731719i \(0.738718\pi\)
\(972\) −9.68910 −0.310778
\(973\) −16.8494 −0.540167
\(974\) 32.9571 1.05601
\(975\) 1.97570 0.0632730
\(976\) −6.44838 −0.206407
\(977\) −26.2243 −0.838990 −0.419495 0.907758i \(-0.637793\pi\)
−0.419495 + 0.907758i \(0.637793\pi\)
\(978\) −2.71724 −0.0868876
\(979\) −19.9883 −0.638830
\(980\) 21.9907 0.702468
\(981\) −34.4630 −1.10032
\(982\) 47.5378 1.51699
\(983\) 52.1294 1.66267 0.831335 0.555772i \(-0.187577\pi\)
0.831335 + 0.555772i \(0.187577\pi\)
\(984\) −1.75857 −0.0560610
\(985\) 26.5782 0.846853
\(986\) 78.3480 2.49511
\(987\) −1.22075 −0.0388568
\(988\) 4.44228 0.141328
\(989\) 3.71585 0.118157
\(990\) −14.7064 −0.467399
\(991\) −21.5566 −0.684768 −0.342384 0.939560i \(-0.611234\pi\)
−0.342384 + 0.939560i \(0.611234\pi\)
\(992\) 15.3743 0.488134
\(993\) 0.240097 0.00761926
\(994\) 13.8551 0.439456
\(995\) 13.6893 0.433978
\(996\) 5.73599 0.181752
\(997\) −9.97917 −0.316043 −0.158022 0.987436i \(-0.550512\pi\)
−0.158022 + 0.987436i \(0.550512\pi\)
\(998\) 57.0759 1.80671
\(999\) −3.19552 −0.101102
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 8003.2.a.b.1.20 153
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8003.2.a.b.1.20 153 1.1 even 1 trivial