Properties

Label 4033.2.a.f.1.3
Level 4033
Weight 2
Character 4033.1
Self dual yes
Analytic conductor 32.204
Analytic rank 0
Dimension 85
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4033 = 37 \cdot 109 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4033.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.2036671352\)
Analytic rank: \(0\)
Dimension: \(85\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Character \(\chi\) = 4033.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.66672 q^{2} -1.32046 q^{3} +5.11137 q^{4} +0.282486 q^{5} +3.52128 q^{6} -2.11170 q^{7} -8.29715 q^{8} -1.25639 q^{9} +O(q^{10})\) \(q-2.66672 q^{2} -1.32046 q^{3} +5.11137 q^{4} +0.282486 q^{5} +3.52128 q^{6} -2.11170 q^{7} -8.29715 q^{8} -1.25639 q^{9} -0.753309 q^{10} -3.12811 q^{11} -6.74935 q^{12} +0.585254 q^{13} +5.63131 q^{14} -0.373010 q^{15} +11.9034 q^{16} +2.84963 q^{17} +3.35044 q^{18} +2.44831 q^{19} +1.44389 q^{20} +2.78841 q^{21} +8.34178 q^{22} +7.78873 q^{23} +10.9560 q^{24} -4.92020 q^{25} -1.56071 q^{26} +5.62038 q^{27} -10.7937 q^{28} +5.73527 q^{29} +0.994712 q^{30} -6.47507 q^{31} -15.1487 q^{32} +4.13054 q^{33} -7.59914 q^{34} -0.596526 q^{35} -6.42189 q^{36} +1.00000 q^{37} -6.52895 q^{38} -0.772803 q^{39} -2.34382 q^{40} -10.9864 q^{41} -7.43591 q^{42} -3.71535 q^{43} -15.9889 q^{44} -0.354913 q^{45} -20.7703 q^{46} -2.88172 q^{47} -15.7179 q^{48} -2.54071 q^{49} +13.1208 q^{50} -3.76281 q^{51} +2.99145 q^{52} -2.19641 q^{53} -14.9880 q^{54} -0.883646 q^{55} +17.5211 q^{56} -3.23289 q^{57} -15.2943 q^{58} +5.22603 q^{59} -1.90659 q^{60} +5.20577 q^{61} +17.2672 q^{62} +2.65313 q^{63} +16.5904 q^{64} +0.165326 q^{65} -11.0150 q^{66} -10.7881 q^{67} +14.5655 q^{68} -10.2847 q^{69} +1.59077 q^{70} +6.53403 q^{71} +10.4245 q^{72} -3.02260 q^{73} -2.66672 q^{74} +6.49692 q^{75} +12.5142 q^{76} +6.60564 q^{77} +2.06085 q^{78} -7.81238 q^{79} +3.36253 q^{80} -3.65230 q^{81} +29.2975 q^{82} +0.0341406 q^{83} +14.2526 q^{84} +0.804978 q^{85} +9.90779 q^{86} -7.57318 q^{87} +25.9544 q^{88} -8.92098 q^{89} +0.946452 q^{90} -1.23588 q^{91} +39.8111 q^{92} +8.55005 q^{93} +7.68473 q^{94} +0.691612 q^{95} +20.0031 q^{96} +0.398466 q^{97} +6.77534 q^{98} +3.93013 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 85q + 11q^{2} + 21q^{3} + 93q^{4} + 12q^{5} + 4q^{6} + 17q^{7} + 30q^{8} + 98q^{9} + O(q^{10}) \) \( 85q + 11q^{2} + 21q^{3} + 93q^{4} + 12q^{5} + 4q^{6} + 17q^{7} + 30q^{8} + 98q^{9} + 9q^{10} + 37q^{11} + 44q^{12} + 14q^{13} + 26q^{14} + 27q^{15} + 85q^{16} + 34q^{17} + 3q^{18} + 15q^{19} + 15q^{20} + 17q^{21} + q^{22} + 72q^{23} + 15q^{24} + 85q^{25} + 33q^{26} + 69q^{27} + 7q^{28} + 19q^{29} - 9q^{30} + 23q^{31} + 51q^{32} + 32q^{33} + 49q^{34} + 40q^{35} + 121q^{36} + 85q^{37} + 84q^{38} + 39q^{39} + 22q^{40} + 55q^{41} - 28q^{42} + 78q^{44} + 28q^{45} + 17q^{46} + 184q^{47} + 97q^{48} + 88q^{49} + 26q^{50} + 27q^{51} + 73q^{52} + 64q^{53} + 31q^{54} + 39q^{55} + 68q^{56} - 33q^{57} + 28q^{58} + 60q^{59} - 22q^{60} + 7q^{61} + 70q^{62} + 28q^{63} + 102q^{64} + 17q^{65} - 15q^{66} + 82q^{67} + 92q^{68} + 22q^{69} - 41q^{70} + 113q^{71} - 19q^{73} + 11q^{74} + 45q^{75} + 34q^{76} + 64q^{77} + 29q^{78} + 23q^{79} + 54q^{80} + 149q^{81} + 4q^{82} + 100q^{83} - 49q^{84} - 5q^{85} - 24q^{86} + 65q^{87} + 14q^{88} + 84q^{89} - 21q^{90} + 32q^{91} + 95q^{92} + 19q^{93} - 47q^{94} + 102q^{95} + 29q^{96} + 7q^{97} + 26q^{98} + 107q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.66672 −1.88565 −0.942826 0.333284i \(-0.891843\pi\)
−0.942826 + 0.333284i \(0.891843\pi\)
\(3\) −1.32046 −0.762366 −0.381183 0.924500i \(-0.624483\pi\)
−0.381183 + 0.924500i \(0.624483\pi\)
\(4\) 5.11137 2.55569
\(5\) 0.282486 0.126331 0.0631657 0.998003i \(-0.479880\pi\)
0.0631657 + 0.998003i \(0.479880\pi\)
\(6\) 3.52128 1.43756
\(7\) −2.11170 −0.798149 −0.399075 0.916918i \(-0.630669\pi\)
−0.399075 + 0.916918i \(0.630669\pi\)
\(8\) −8.29715 −2.93348
\(9\) −1.25639 −0.418798
\(10\) −0.753309 −0.238217
\(11\) −3.12811 −0.943161 −0.471580 0.881823i \(-0.656316\pi\)
−0.471580 + 0.881823i \(0.656316\pi\)
\(12\) −6.74935 −1.94837
\(13\) 0.585254 0.162320 0.0811602 0.996701i \(-0.474137\pi\)
0.0811602 + 0.996701i \(0.474137\pi\)
\(14\) 5.63131 1.50503
\(15\) −0.373010 −0.0963108
\(16\) 11.9034 2.97585
\(17\) 2.84963 0.691136 0.345568 0.938394i \(-0.387686\pi\)
0.345568 + 0.938394i \(0.387686\pi\)
\(18\) 3.35044 0.789707
\(19\) 2.44831 0.561681 0.280840 0.959754i \(-0.409387\pi\)
0.280840 + 0.959754i \(0.409387\pi\)
\(20\) 1.44389 0.322863
\(21\) 2.78841 0.608482
\(22\) 8.34178 1.77847
\(23\) 7.78873 1.62406 0.812032 0.583613i \(-0.198362\pi\)
0.812032 + 0.583613i \(0.198362\pi\)
\(24\) 10.9560 2.23639
\(25\) −4.92020 −0.984040
\(26\) −1.56071 −0.306080
\(27\) 5.62038 1.08164
\(28\) −10.7937 −2.03982
\(29\) 5.73527 1.06501 0.532506 0.846426i \(-0.321250\pi\)
0.532506 + 0.846426i \(0.321250\pi\)
\(30\) 0.994712 0.181609
\(31\) −6.47507 −1.16296 −0.581478 0.813562i \(-0.697525\pi\)
−0.581478 + 0.813562i \(0.697525\pi\)
\(32\) −15.1487 −2.67793
\(33\) 4.13054 0.719034
\(34\) −7.59914 −1.30324
\(35\) −0.596526 −0.100831
\(36\) −6.42189 −1.07032
\(37\) 1.00000 0.164399
\(38\) −6.52895 −1.05914
\(39\) −0.772803 −0.123748
\(40\) −2.34382 −0.370591
\(41\) −10.9864 −1.71578 −0.857890 0.513833i \(-0.828225\pi\)
−0.857890 + 0.513833i \(0.828225\pi\)
\(42\) −7.43591 −1.14739
\(43\) −3.71535 −0.566586 −0.283293 0.959033i \(-0.591427\pi\)
−0.283293 + 0.959033i \(0.591427\pi\)
\(44\) −15.9889 −2.41042
\(45\) −0.354913 −0.0529073
\(46\) −20.7703 −3.06242
\(47\) −2.88172 −0.420342 −0.210171 0.977665i \(-0.567402\pi\)
−0.210171 + 0.977665i \(0.567402\pi\)
\(48\) −15.7179 −2.26868
\(49\) −2.54071 −0.362958
\(50\) 13.1208 1.85556
\(51\) −3.76281 −0.526899
\(52\) 2.99145 0.414840
\(53\) −2.19641 −0.301700 −0.150850 0.988557i \(-0.548201\pi\)
−0.150850 + 0.988557i \(0.548201\pi\)
\(54\) −14.9880 −2.03960
\(55\) −0.883646 −0.119151
\(56\) 17.5211 2.34136
\(57\) −3.23289 −0.428207
\(58\) −15.2943 −2.00824
\(59\) 5.22603 0.680372 0.340186 0.940358i \(-0.389510\pi\)
0.340186 + 0.940358i \(0.389510\pi\)
\(60\) −1.90659 −0.246140
\(61\) 5.20577 0.666531 0.333265 0.942833i \(-0.391850\pi\)
0.333265 + 0.942833i \(0.391850\pi\)
\(62\) 17.2672 2.19293
\(63\) 2.65313 0.334263
\(64\) 16.5904 2.07380
\(65\) 0.165326 0.0205062
\(66\) −11.0150 −1.35585
\(67\) −10.7881 −1.31797 −0.658987 0.752154i \(-0.729015\pi\)
−0.658987 + 0.752154i \(0.729015\pi\)
\(68\) 14.5655 1.76633
\(69\) −10.2847 −1.23813
\(70\) 1.59077 0.190133
\(71\) 6.53403 0.775446 0.387723 0.921776i \(-0.373262\pi\)
0.387723 + 0.921776i \(0.373262\pi\)
\(72\) 10.4245 1.22854
\(73\) −3.02260 −0.353769 −0.176884 0.984232i \(-0.556602\pi\)
−0.176884 + 0.984232i \(0.556602\pi\)
\(74\) −2.66672 −0.309999
\(75\) 6.49692 0.750199
\(76\) 12.5142 1.43548
\(77\) 6.60564 0.752783
\(78\) 2.06085 0.233345
\(79\) −7.81238 −0.878962 −0.439481 0.898252i \(-0.644838\pi\)
−0.439481 + 0.898252i \(0.644838\pi\)
\(80\) 3.36253 0.375943
\(81\) −3.65230 −0.405811
\(82\) 29.2975 3.23537
\(83\) 0.0341406 0.00374742 0.00187371 0.999998i \(-0.499404\pi\)
0.00187371 + 0.999998i \(0.499404\pi\)
\(84\) 14.2526 1.55509
\(85\) 0.804978 0.0873121
\(86\) 9.90779 1.06838
\(87\) −7.57318 −0.811930
\(88\) 25.9544 2.76675
\(89\) −8.92098 −0.945622 −0.472811 0.881164i \(-0.656761\pi\)
−0.472811 + 0.881164i \(0.656761\pi\)
\(90\) 0.946452 0.0997648
\(91\) −1.23588 −0.129556
\(92\) 39.8111 4.15060
\(93\) 8.55005 0.886599
\(94\) 7.68473 0.792620
\(95\) 0.691612 0.0709579
\(96\) 20.0031 2.04156
\(97\) 0.398466 0.0404580 0.0202290 0.999795i \(-0.493560\pi\)
0.0202290 + 0.999795i \(0.493560\pi\)
\(98\) 6.77534 0.684413
\(99\) 3.93013 0.394993
\(100\) −25.1490 −2.51490
\(101\) −10.9169 −1.08627 −0.543137 0.839644i \(-0.682764\pi\)
−0.543137 + 0.839644i \(0.682764\pi\)
\(102\) 10.0343 0.993548
\(103\) −6.64691 −0.654940 −0.327470 0.944862i \(-0.606196\pi\)
−0.327470 + 0.944862i \(0.606196\pi\)
\(104\) −4.85594 −0.476164
\(105\) 0.787687 0.0768704
\(106\) 5.85721 0.568902
\(107\) 14.4923 1.40103 0.700513 0.713640i \(-0.252955\pi\)
0.700513 + 0.713640i \(0.252955\pi\)
\(108\) 28.7279 2.76434
\(109\) −1.00000 −0.0957826
\(110\) 2.35643 0.224677
\(111\) −1.32046 −0.125332
\(112\) −25.1364 −2.37517
\(113\) −3.81063 −0.358474 −0.179237 0.983806i \(-0.557363\pi\)
−0.179237 + 0.983806i \(0.557363\pi\)
\(114\) 8.62120 0.807449
\(115\) 2.20021 0.205170
\(116\) 29.3151 2.72184
\(117\) −0.735310 −0.0679794
\(118\) −13.9363 −1.28294
\(119\) −6.01757 −0.551629
\(120\) 3.09492 0.282526
\(121\) −1.21493 −0.110448
\(122\) −13.8823 −1.25685
\(123\) 14.5070 1.30805
\(124\) −33.0965 −2.97215
\(125\) −2.80231 −0.250647
\(126\) −7.07514 −0.630304
\(127\) 1.33653 0.118598 0.0592988 0.998240i \(-0.481114\pi\)
0.0592988 + 0.998240i \(0.481114\pi\)
\(128\) −13.9445 −1.23253
\(129\) 4.90597 0.431946
\(130\) −0.440877 −0.0386675
\(131\) 19.6894 1.72027 0.860136 0.510064i \(-0.170378\pi\)
0.860136 + 0.510064i \(0.170378\pi\)
\(132\) 21.1127 1.83762
\(133\) −5.17011 −0.448305
\(134\) 28.7687 2.48524
\(135\) 1.58768 0.136646
\(136\) −23.6438 −2.02744
\(137\) 6.73024 0.575004 0.287502 0.957780i \(-0.407175\pi\)
0.287502 + 0.957780i \(0.407175\pi\)
\(138\) 27.4263 2.33469
\(139\) −13.6558 −1.15827 −0.579137 0.815231i \(-0.696610\pi\)
−0.579137 + 0.815231i \(0.696610\pi\)
\(140\) −3.04907 −0.257693
\(141\) 3.80519 0.320455
\(142\) −17.4244 −1.46222
\(143\) −1.83074 −0.153094
\(144\) −14.9553 −1.24628
\(145\) 1.62013 0.134545
\(146\) 8.06041 0.667085
\(147\) 3.35489 0.276707
\(148\) 5.11137 0.420152
\(149\) −21.4240 −1.75512 −0.877562 0.479464i \(-0.840831\pi\)
−0.877562 + 0.479464i \(0.840831\pi\)
\(150\) −17.3254 −1.41462
\(151\) −18.8277 −1.53218 −0.766089 0.642734i \(-0.777800\pi\)
−0.766089 + 0.642734i \(0.777800\pi\)
\(152\) −20.3140 −1.64768
\(153\) −3.58025 −0.289446
\(154\) −17.6154 −1.41949
\(155\) −1.82911 −0.146918
\(156\) −3.95009 −0.316260
\(157\) 4.51224 0.360116 0.180058 0.983656i \(-0.442371\pi\)
0.180058 + 0.983656i \(0.442371\pi\)
\(158\) 20.8334 1.65742
\(159\) 2.90027 0.230006
\(160\) −4.27928 −0.338306
\(161\) −16.4475 −1.29624
\(162\) 9.73964 0.765218
\(163\) 12.5702 0.984577 0.492288 0.870432i \(-0.336161\pi\)
0.492288 + 0.870432i \(0.336161\pi\)
\(164\) −56.1553 −4.38500
\(165\) 1.16682 0.0908366
\(166\) −0.0910432 −0.00706632
\(167\) 17.8598 1.38203 0.691016 0.722840i \(-0.257164\pi\)
0.691016 + 0.722840i \(0.257164\pi\)
\(168\) −23.1359 −1.78497
\(169\) −12.6575 −0.973652
\(170\) −2.14665 −0.164640
\(171\) −3.07604 −0.235231
\(172\) −18.9906 −1.44802
\(173\) 9.45773 0.719058 0.359529 0.933134i \(-0.382937\pi\)
0.359529 + 0.933134i \(0.382937\pi\)
\(174\) 20.1955 1.53102
\(175\) 10.3900 0.785411
\(176\) −37.2351 −2.80670
\(177\) −6.90075 −0.518692
\(178\) 23.7897 1.78311
\(179\) −0.235481 −0.0176006 −0.00880032 0.999961i \(-0.502801\pi\)
−0.00880032 + 0.999961i \(0.502801\pi\)
\(180\) −1.81409 −0.135214
\(181\) 7.88157 0.585832 0.292916 0.956138i \(-0.405374\pi\)
0.292916 + 0.956138i \(0.405374\pi\)
\(182\) 3.29575 0.244297
\(183\) −6.87400 −0.508140
\(184\) −64.6243 −4.76416
\(185\) 0.282486 0.0207688
\(186\) −22.8005 −1.67182
\(187\) −8.91394 −0.651852
\(188\) −14.7296 −1.07426
\(189\) −11.8686 −0.863313
\(190\) −1.84433 −0.133802
\(191\) 22.0817 1.59778 0.798888 0.601479i \(-0.205422\pi\)
0.798888 + 0.601479i \(0.205422\pi\)
\(192\) −21.9069 −1.58099
\(193\) 13.3583 0.961549 0.480774 0.876844i \(-0.340356\pi\)
0.480774 + 0.876844i \(0.340356\pi\)
\(194\) −1.06259 −0.0762898
\(195\) −0.218306 −0.0156332
\(196\) −12.9865 −0.927607
\(197\) −5.84967 −0.416772 −0.208386 0.978047i \(-0.566821\pi\)
−0.208386 + 0.978047i \(0.566821\pi\)
\(198\) −10.4806 −0.744820
\(199\) 16.9420 1.20099 0.600493 0.799630i \(-0.294971\pi\)
0.600493 + 0.799630i \(0.294971\pi\)
\(200\) 40.8236 2.88667
\(201\) 14.2452 1.00478
\(202\) 29.1123 2.04834
\(203\) −12.1112 −0.850039
\(204\) −19.2331 −1.34659
\(205\) −3.10349 −0.216757
\(206\) 17.7254 1.23499
\(207\) −9.78571 −0.680154
\(208\) 6.96651 0.483040
\(209\) −7.65858 −0.529755
\(210\) −2.10054 −0.144951
\(211\) 8.97733 0.618025 0.309012 0.951058i \(-0.400002\pi\)
0.309012 + 0.951058i \(0.400002\pi\)
\(212\) −11.2267 −0.771052
\(213\) −8.62790 −0.591174
\(214\) −38.6469 −2.64185
\(215\) −1.04953 −0.0715776
\(216\) −46.6332 −3.17298
\(217\) 13.6734 0.928213
\(218\) 2.66672 0.180613
\(219\) 3.99121 0.269701
\(220\) −4.51664 −0.304512
\(221\) 1.66776 0.112185
\(222\) 3.52128 0.236333
\(223\) −4.98364 −0.333729 −0.166864 0.985980i \(-0.553364\pi\)
−0.166864 + 0.985980i \(0.553364\pi\)
\(224\) 31.9895 2.13739
\(225\) 6.18171 0.412114
\(226\) 10.1619 0.675957
\(227\) −4.94560 −0.328251 −0.164125 0.986439i \(-0.552480\pi\)
−0.164125 + 0.986439i \(0.552480\pi\)
\(228\) −16.5245 −1.09436
\(229\) 14.5668 0.962602 0.481301 0.876555i \(-0.340164\pi\)
0.481301 + 0.876555i \(0.340164\pi\)
\(230\) −5.86732 −0.386880
\(231\) −8.72247 −0.573896
\(232\) −47.5864 −3.12420
\(233\) −0.487090 −0.0319103 −0.0159552 0.999873i \(-0.505079\pi\)
−0.0159552 + 0.999873i \(0.505079\pi\)
\(234\) 1.96086 0.128186
\(235\) −0.814045 −0.0531024
\(236\) 26.7122 1.73882
\(237\) 10.3159 0.670091
\(238\) 16.0471 1.04018
\(239\) −18.6924 −1.20911 −0.604556 0.796563i \(-0.706650\pi\)
−0.604556 + 0.796563i \(0.706650\pi\)
\(240\) −4.44008 −0.286606
\(241\) 14.6817 0.945734 0.472867 0.881134i \(-0.343219\pi\)
0.472867 + 0.881134i \(0.343219\pi\)
\(242\) 3.23987 0.208267
\(243\) −12.0384 −0.772267
\(244\) 26.6086 1.70344
\(245\) −0.717713 −0.0458530
\(246\) −38.6861 −2.46653
\(247\) 1.43288 0.0911723
\(248\) 53.7246 3.41151
\(249\) −0.0450812 −0.00285690
\(250\) 7.47298 0.472632
\(251\) 14.6712 0.926037 0.463019 0.886349i \(-0.346766\pi\)
0.463019 + 0.886349i \(0.346766\pi\)
\(252\) 13.5611 0.854271
\(253\) −24.3640 −1.53175
\(254\) −3.56414 −0.223634
\(255\) −1.06294 −0.0665638
\(256\) 4.00530 0.250331
\(257\) 7.71964 0.481538 0.240769 0.970582i \(-0.422600\pi\)
0.240769 + 0.970582i \(0.422600\pi\)
\(258\) −13.0828 −0.814501
\(259\) −2.11170 −0.131215
\(260\) 0.845043 0.0524073
\(261\) −7.20575 −0.446025
\(262\) −52.5061 −3.24384
\(263\) −6.70759 −0.413608 −0.206804 0.978382i \(-0.566306\pi\)
−0.206804 + 0.978382i \(0.566306\pi\)
\(264\) −34.2717 −2.10927
\(265\) −0.620455 −0.0381142
\(266\) 13.7872 0.845348
\(267\) 11.7798 0.720910
\(268\) −55.1419 −3.36833
\(269\) −12.4120 −0.756774 −0.378387 0.925647i \(-0.623521\pi\)
−0.378387 + 0.925647i \(0.623521\pi\)
\(270\) −4.23388 −0.257666
\(271\) −25.6953 −1.56088 −0.780439 0.625232i \(-0.785004\pi\)
−0.780439 + 0.625232i \(0.785004\pi\)
\(272\) 33.9202 2.05671
\(273\) 1.63193 0.0987690
\(274\) −17.9476 −1.08426
\(275\) 15.3909 0.928108
\(276\) −52.5689 −3.16428
\(277\) 13.2777 0.797782 0.398891 0.916998i \(-0.369395\pi\)
0.398891 + 0.916998i \(0.369395\pi\)
\(278\) 36.4162 2.18410
\(279\) 8.13523 0.487043
\(280\) 4.94946 0.295787
\(281\) −17.2384 −1.02836 −0.514178 0.857683i \(-0.671903\pi\)
−0.514178 + 0.857683i \(0.671903\pi\)
\(282\) −10.1474 −0.604267
\(283\) 2.17497 0.129289 0.0646444 0.997908i \(-0.479409\pi\)
0.0646444 + 0.997908i \(0.479409\pi\)
\(284\) 33.3978 1.98180
\(285\) −0.913245 −0.0540959
\(286\) 4.88206 0.288682
\(287\) 23.1999 1.36945
\(288\) 19.0327 1.12151
\(289\) −8.87963 −0.522331
\(290\) −4.32043 −0.253704
\(291\) −0.526157 −0.0308439
\(292\) −15.4496 −0.904121
\(293\) 20.2592 1.18355 0.591777 0.806102i \(-0.298427\pi\)
0.591777 + 0.806102i \(0.298427\pi\)
\(294\) −8.94655 −0.521773
\(295\) 1.47628 0.0859523
\(296\) −8.29715 −0.482262
\(297\) −17.5812 −1.02016
\(298\) 57.1317 3.30955
\(299\) 4.55839 0.263619
\(300\) 33.2082 1.91727
\(301\) 7.84573 0.452220
\(302\) 50.2082 2.88916
\(303\) 14.4153 0.828139
\(304\) 29.1432 1.67148
\(305\) 1.47055 0.0842037
\(306\) 9.54751 0.545795
\(307\) 7.81015 0.445749 0.222875 0.974847i \(-0.428456\pi\)
0.222875 + 0.974847i \(0.428456\pi\)
\(308\) 33.7639 1.92388
\(309\) 8.77697 0.499304
\(310\) 4.87772 0.277036
\(311\) −7.67064 −0.434962 −0.217481 0.976065i \(-0.569784\pi\)
−0.217481 + 0.976065i \(0.569784\pi\)
\(312\) 6.41206 0.363012
\(313\) −17.4415 −0.985851 −0.492926 0.870071i \(-0.664073\pi\)
−0.492926 + 0.870071i \(0.664073\pi\)
\(314\) −12.0329 −0.679054
\(315\) 0.749471 0.0422279
\(316\) −39.9320 −2.24635
\(317\) 7.93855 0.445873 0.222937 0.974833i \(-0.428436\pi\)
0.222937 + 0.974833i \(0.428436\pi\)
\(318\) −7.73419 −0.433712
\(319\) −17.9406 −1.00448
\(320\) 4.68654 0.261986
\(321\) −19.1365 −1.06809
\(322\) 43.8608 2.44427
\(323\) 6.97677 0.388198
\(324\) −18.6683 −1.03713
\(325\) −2.87957 −0.159730
\(326\) −33.5212 −1.85657
\(327\) 1.32046 0.0730214
\(328\) 91.1554 5.03321
\(329\) 6.08535 0.335496
\(330\) −3.11157 −0.171286
\(331\) 24.9422 1.37095 0.685475 0.728096i \(-0.259595\pi\)
0.685475 + 0.728096i \(0.259595\pi\)
\(332\) 0.174505 0.00957722
\(333\) −1.25639 −0.0688499
\(334\) −47.6270 −2.60603
\(335\) −3.04748 −0.166501
\(336\) 33.1916 1.81075
\(337\) 27.5242 1.49934 0.749669 0.661813i \(-0.230213\pi\)
0.749669 + 0.661813i \(0.230213\pi\)
\(338\) 33.7539 1.83597
\(339\) 5.03177 0.273288
\(340\) 4.11454 0.223142
\(341\) 20.2547 1.09685
\(342\) 8.20292 0.443563
\(343\) 20.1471 1.08784
\(344\) 30.8268 1.66207
\(345\) −2.90528 −0.156415
\(346\) −25.2211 −1.35589
\(347\) 27.3134 1.46626 0.733129 0.680089i \(-0.238059\pi\)
0.733129 + 0.680089i \(0.238059\pi\)
\(348\) −38.7093 −2.07504
\(349\) −15.6535 −0.837914 −0.418957 0.908006i \(-0.637604\pi\)
−0.418957 + 0.908006i \(0.637604\pi\)
\(350\) −27.7072 −1.48101
\(351\) 3.28936 0.175573
\(352\) 47.3866 2.52572
\(353\) 29.4434 1.56712 0.783558 0.621319i \(-0.213403\pi\)
0.783558 + 0.621319i \(0.213403\pi\)
\(354\) 18.4023 0.978074
\(355\) 1.84577 0.0979632
\(356\) −45.5985 −2.41671
\(357\) 7.94594 0.420544
\(358\) 0.627960 0.0331887
\(359\) 10.1877 0.537687 0.268843 0.963184i \(-0.413359\pi\)
0.268843 + 0.963184i \(0.413359\pi\)
\(360\) 2.94476 0.155203
\(361\) −13.0058 −0.684515
\(362\) −21.0179 −1.10468
\(363\) 1.60426 0.0842019
\(364\) −6.31707 −0.331104
\(365\) −0.853841 −0.0446921
\(366\) 18.3310 0.958176
\(367\) −5.63779 −0.294290 −0.147145 0.989115i \(-0.547008\pi\)
−0.147145 + 0.989115i \(0.547008\pi\)
\(368\) 92.7123 4.83296
\(369\) 13.8032 0.718565
\(370\) −0.753309 −0.0391627
\(371\) 4.63817 0.240802
\(372\) 43.7025 2.26587
\(373\) −10.9166 −0.565238 −0.282619 0.959232i \(-0.591203\pi\)
−0.282619 + 0.959232i \(0.591203\pi\)
\(374\) 23.7709 1.22917
\(375\) 3.70034 0.191085
\(376\) 23.9101 1.23307
\(377\) 3.35659 0.172873
\(378\) 31.6502 1.62791
\(379\) 30.1883 1.55067 0.775334 0.631551i \(-0.217582\pi\)
0.775334 + 0.631551i \(0.217582\pi\)
\(380\) 3.53509 0.181346
\(381\) −1.76483 −0.0904148
\(382\) −58.8857 −3.01285
\(383\) −13.5349 −0.691599 −0.345799 0.938308i \(-0.612392\pi\)
−0.345799 + 0.938308i \(0.612392\pi\)
\(384\) 18.4131 0.939641
\(385\) 1.86600 0.0951001
\(386\) −35.6227 −1.81315
\(387\) 4.66794 0.237285
\(388\) 2.03671 0.103398
\(389\) 8.36783 0.424266 0.212133 0.977241i \(-0.431959\pi\)
0.212133 + 0.977241i \(0.431959\pi\)
\(390\) 0.582160 0.0294788
\(391\) 22.1950 1.12245
\(392\) 21.0806 1.06473
\(393\) −25.9990 −1.31148
\(394\) 15.5994 0.785887
\(395\) −2.20689 −0.111040
\(396\) 20.0884 1.00948
\(397\) 4.24958 0.213281 0.106640 0.994298i \(-0.465991\pi\)
0.106640 + 0.994298i \(0.465991\pi\)
\(398\) −45.1795 −2.26464
\(399\) 6.82690 0.341773
\(400\) −58.5671 −2.92835
\(401\) 3.72419 0.185977 0.0929885 0.995667i \(-0.470358\pi\)
0.0929885 + 0.995667i \(0.470358\pi\)
\(402\) −37.9879 −1.89466
\(403\) −3.78956 −0.188772
\(404\) −55.8005 −2.77618
\(405\) −1.03172 −0.0512667
\(406\) 32.2971 1.60288
\(407\) −3.12811 −0.155055
\(408\) 31.2206 1.54565
\(409\) 12.9227 0.638984 0.319492 0.947589i \(-0.396488\pi\)
0.319492 + 0.947589i \(0.396488\pi\)
\(410\) 8.27612 0.408728
\(411\) −8.88700 −0.438363
\(412\) −33.9749 −1.67382
\(413\) −11.0358 −0.543038
\(414\) 26.0957 1.28253
\(415\) 0.00964422 0.000473416 0
\(416\) −8.86582 −0.434682
\(417\) 18.0320 0.883028
\(418\) 20.4233 0.998934
\(419\) 23.5784 1.15188 0.575940 0.817492i \(-0.304636\pi\)
0.575940 + 0.817492i \(0.304636\pi\)
\(420\) 4.02616 0.196457
\(421\) −13.2193 −0.644268 −0.322134 0.946694i \(-0.604400\pi\)
−0.322134 + 0.946694i \(0.604400\pi\)
\(422\) −23.9400 −1.16538
\(423\) 3.62058 0.176038
\(424\) 18.2240 0.885034
\(425\) −14.0207 −0.680105
\(426\) 23.0082 1.11475
\(427\) −10.9930 −0.531991
\(428\) 74.0757 3.58058
\(429\) 2.41741 0.116714
\(430\) 2.79881 0.134971
\(431\) 30.0175 1.44589 0.722947 0.690904i \(-0.242787\pi\)
0.722947 + 0.690904i \(0.242787\pi\)
\(432\) 66.9016 3.21880
\(433\) 31.0488 1.49211 0.746056 0.665883i \(-0.231945\pi\)
0.746056 + 0.665883i \(0.231945\pi\)
\(434\) −36.4631 −1.75029
\(435\) −2.13931 −0.102572
\(436\) −5.11137 −0.244790
\(437\) 19.0692 0.912205
\(438\) −10.6434 −0.508563
\(439\) −16.5762 −0.791137 −0.395569 0.918436i \(-0.629452\pi\)
−0.395569 + 0.918436i \(0.629452\pi\)
\(440\) 7.33174 0.349527
\(441\) 3.19212 0.152006
\(442\) −4.44743 −0.211543
\(443\) 5.52690 0.262591 0.131295 0.991343i \(-0.458086\pi\)
0.131295 + 0.991343i \(0.458086\pi\)
\(444\) −6.74935 −0.320310
\(445\) −2.52005 −0.119462
\(446\) 13.2899 0.629297
\(447\) 28.2895 1.33805
\(448\) −35.0340 −1.65520
\(449\) −38.5165 −1.81770 −0.908852 0.417119i \(-0.863040\pi\)
−0.908852 + 0.417119i \(0.863040\pi\)
\(450\) −16.4849 −0.777103
\(451\) 34.3665 1.61826
\(452\) −19.4776 −0.916147
\(453\) 24.8612 1.16808
\(454\) 13.1885 0.618967
\(455\) −0.349120 −0.0163670
\(456\) 26.8238 1.25614
\(457\) 14.1776 0.663201 0.331601 0.943420i \(-0.392411\pi\)
0.331601 + 0.943420i \(0.392411\pi\)
\(458\) −38.8456 −1.81513
\(459\) 16.0160 0.747562
\(460\) 11.2461 0.524351
\(461\) −37.5338 −1.74812 −0.874061 0.485816i \(-0.838523\pi\)
−0.874061 + 0.485816i \(0.838523\pi\)
\(462\) 23.2603 1.08217
\(463\) −14.8066 −0.688123 −0.344061 0.938947i \(-0.611803\pi\)
−0.344061 + 0.938947i \(0.611803\pi\)
\(464\) 68.2691 3.16931
\(465\) 2.41527 0.112005
\(466\) 1.29893 0.0601718
\(467\) 11.5446 0.534218 0.267109 0.963666i \(-0.413932\pi\)
0.267109 + 0.963666i \(0.413932\pi\)
\(468\) −3.75844 −0.173734
\(469\) 22.7812 1.05194
\(470\) 2.17083 0.100133
\(471\) −5.95822 −0.274540
\(472\) −43.3612 −1.99586
\(473\) 11.6220 0.534382
\(474\) −27.5096 −1.26356
\(475\) −12.0462 −0.552717
\(476\) −30.7580 −1.40979
\(477\) 2.75956 0.126351
\(478\) 49.8474 2.27997
\(479\) 33.7500 1.54207 0.771037 0.636790i \(-0.219738\pi\)
0.771037 + 0.636790i \(0.219738\pi\)
\(480\) 5.65060 0.257913
\(481\) 0.585254 0.0266853
\(482\) −39.1520 −1.78333
\(483\) 21.7182 0.988213
\(484\) −6.20995 −0.282271
\(485\) 0.112561 0.00511112
\(486\) 32.1031 1.45623
\(487\) 33.5691 1.52116 0.760581 0.649243i \(-0.224914\pi\)
0.760581 + 0.649243i \(0.224914\pi\)
\(488\) −43.1930 −1.95526
\(489\) −16.5985 −0.750608
\(490\) 1.91394 0.0864628
\(491\) 9.22414 0.416279 0.208140 0.978099i \(-0.433259\pi\)
0.208140 + 0.978099i \(0.433259\pi\)
\(492\) 74.1507 3.34297
\(493\) 16.3434 0.736068
\(494\) −3.82110 −0.171919
\(495\) 1.11021 0.0499001
\(496\) −77.0752 −3.46078
\(497\) −13.7979 −0.618922
\(498\) 0.120219 0.00538713
\(499\) 26.6585 1.19340 0.596698 0.802466i \(-0.296479\pi\)
0.596698 + 0.802466i \(0.296479\pi\)
\(500\) −14.3237 −0.640574
\(501\) −23.5831 −1.05361
\(502\) −39.1239 −1.74618
\(503\) −17.0258 −0.759144 −0.379572 0.925162i \(-0.623929\pi\)
−0.379572 + 0.925162i \(0.623929\pi\)
\(504\) −22.0134 −0.980555
\(505\) −3.08387 −0.137231
\(506\) 64.9719 2.88835
\(507\) 16.7137 0.742280
\(508\) 6.83149 0.303098
\(509\) −6.59566 −0.292347 −0.146174 0.989259i \(-0.546696\pi\)
−0.146174 + 0.989259i \(0.546696\pi\)
\(510\) 2.83456 0.125516
\(511\) 6.38284 0.282360
\(512\) 17.2080 0.760495
\(513\) 13.7604 0.607538
\(514\) −20.5861 −0.908013
\(515\) −1.87766 −0.0827395
\(516\) 25.0762 1.10392
\(517\) 9.01434 0.396450
\(518\) 5.63131 0.247426
\(519\) −12.4885 −0.548185
\(520\) −1.37173 −0.0601545
\(521\) −17.9292 −0.785495 −0.392747 0.919646i \(-0.628475\pi\)
−0.392747 + 0.919646i \(0.628475\pi\)
\(522\) 19.2157 0.841048
\(523\) −29.4305 −1.28691 −0.643454 0.765485i \(-0.722499\pi\)
−0.643454 + 0.765485i \(0.722499\pi\)
\(524\) 100.640 4.39648
\(525\) −13.7196 −0.598771
\(526\) 17.8872 0.779921
\(527\) −18.4515 −0.803761
\(528\) 49.1673 2.13973
\(529\) 37.6644 1.63758
\(530\) 1.65458 0.0718702
\(531\) −6.56595 −0.284938
\(532\) −26.4263 −1.14573
\(533\) −6.42981 −0.278506
\(534\) −31.4133 −1.35939
\(535\) 4.09387 0.176994
\(536\) 89.5103 3.86626
\(537\) 0.310942 0.0134181
\(538\) 33.0993 1.42701
\(539\) 7.94761 0.342328
\(540\) 8.11521 0.349223
\(541\) −5.63290 −0.242177 −0.121089 0.992642i \(-0.538639\pi\)
−0.121089 + 0.992642i \(0.538639\pi\)
\(542\) 68.5221 2.94327
\(543\) −10.4073 −0.446619
\(544\) −43.1680 −1.85081
\(545\) −0.282486 −0.0121004
\(546\) −4.35190 −0.186244
\(547\) −12.2042 −0.521812 −0.260906 0.965364i \(-0.584021\pi\)
−0.260906 + 0.965364i \(0.584021\pi\)
\(548\) 34.4008 1.46953
\(549\) −6.54049 −0.279141
\(550\) −41.0432 −1.75009
\(551\) 14.0417 0.598197
\(552\) 85.3336 3.63204
\(553\) 16.4974 0.701542
\(554\) −35.4079 −1.50434
\(555\) −0.373010 −0.0158334
\(556\) −69.8001 −2.96018
\(557\) −39.0490 −1.65456 −0.827279 0.561791i \(-0.810112\pi\)
−0.827279 + 0.561791i \(0.810112\pi\)
\(558\) −21.6943 −0.918395
\(559\) −2.17443 −0.0919685
\(560\) −7.10068 −0.300058
\(561\) 11.7705 0.496950
\(562\) 45.9699 1.93912
\(563\) 5.69860 0.240167 0.120084 0.992764i \(-0.461684\pi\)
0.120084 + 0.992764i \(0.461684\pi\)
\(564\) 19.4497 0.818982
\(565\) −1.07645 −0.0452865
\(566\) −5.80004 −0.243794
\(567\) 7.71257 0.323898
\(568\) −54.2138 −2.27476
\(569\) 19.5106 0.817925 0.408962 0.912551i \(-0.365891\pi\)
0.408962 + 0.912551i \(0.365891\pi\)
\(570\) 2.43536 0.102006
\(571\) 26.1481 1.09427 0.547133 0.837046i \(-0.315719\pi\)
0.547133 + 0.837046i \(0.315719\pi\)
\(572\) −9.35760 −0.391261
\(573\) −29.1580 −1.21809
\(574\) −61.8676 −2.58230
\(575\) −38.3221 −1.59814
\(576\) −20.8440 −0.868501
\(577\) −35.5889 −1.48158 −0.740792 0.671734i \(-0.765550\pi\)
−0.740792 + 0.671734i \(0.765550\pi\)
\(578\) 23.6795 0.984936
\(579\) −17.6390 −0.733052
\(580\) 8.28109 0.343854
\(581\) −0.0720948 −0.00299100
\(582\) 1.40311 0.0581608
\(583\) 6.87062 0.284552
\(584\) 25.0790 1.03777
\(585\) −0.207714 −0.00858793
\(586\) −54.0255 −2.23177
\(587\) 8.67563 0.358082 0.179041 0.983842i \(-0.442701\pi\)
0.179041 + 0.983842i \(0.442701\pi\)
\(588\) 17.1481 0.707176
\(589\) −15.8530 −0.653210
\(590\) −3.93682 −0.162076
\(591\) 7.72424 0.317733
\(592\) 11.9034 0.489226
\(593\) 45.8131 1.88132 0.940660 0.339352i \(-0.110208\pi\)
0.940660 + 0.339352i \(0.110208\pi\)
\(594\) 46.8840 1.92367
\(595\) −1.69988 −0.0696881
\(596\) −109.506 −4.48554
\(597\) −22.3712 −0.915592
\(598\) −12.1559 −0.497093
\(599\) 15.4604 0.631695 0.315847 0.948810i \(-0.397711\pi\)
0.315847 + 0.948810i \(0.397711\pi\)
\(600\) −53.9059 −2.20070
\(601\) −10.3946 −0.424003 −0.212001 0.977269i \(-0.567998\pi\)
−0.212001 + 0.977269i \(0.567998\pi\)
\(602\) −20.9223 −0.852730
\(603\) 13.5541 0.551964
\(604\) −96.2355 −3.91577
\(605\) −0.343200 −0.0139531
\(606\) −38.4416 −1.56158
\(607\) 7.48731 0.303900 0.151950 0.988388i \(-0.451445\pi\)
0.151950 + 0.988388i \(0.451445\pi\)
\(608\) −37.0886 −1.50414
\(609\) 15.9923 0.648041
\(610\) −3.92155 −0.158779
\(611\) −1.68654 −0.0682301
\(612\) −18.3000 −0.739733
\(613\) 3.85021 0.155509 0.0777543 0.996973i \(-0.475225\pi\)
0.0777543 + 0.996973i \(0.475225\pi\)
\(614\) −20.8275 −0.840528
\(615\) 4.09802 0.165248
\(616\) −54.8080 −2.20828
\(617\) 7.29444 0.293663 0.146831 0.989162i \(-0.453092\pi\)
0.146831 + 0.989162i \(0.453092\pi\)
\(618\) −23.4057 −0.941514
\(619\) −34.2151 −1.37522 −0.687610 0.726081i \(-0.741340\pi\)
−0.687610 + 0.726081i \(0.741340\pi\)
\(620\) −9.34928 −0.375476
\(621\) 43.7757 1.75666
\(622\) 20.4554 0.820187
\(623\) 18.8385 0.754747
\(624\) −9.19898 −0.368254
\(625\) 23.8094 0.952376
\(626\) 46.5115 1.85897
\(627\) 10.1128 0.403868
\(628\) 23.0637 0.920343
\(629\) 2.84963 0.113622
\(630\) −1.99863 −0.0796272
\(631\) −22.3144 −0.888324 −0.444162 0.895947i \(-0.646498\pi\)
−0.444162 + 0.895947i \(0.646498\pi\)
\(632\) 64.8205 2.57842
\(633\) −11.8542 −0.471161
\(634\) −21.1698 −0.840762
\(635\) 0.377550 0.0149826
\(636\) 14.8244 0.587824
\(637\) −1.48696 −0.0589155
\(638\) 47.8423 1.89410
\(639\) −8.20930 −0.324755
\(640\) −3.93912 −0.155708
\(641\) 27.0793 1.06957 0.534783 0.844989i \(-0.320393\pi\)
0.534783 + 0.844989i \(0.320393\pi\)
\(642\) 51.0316 2.01406
\(643\) 12.4365 0.490448 0.245224 0.969466i \(-0.421139\pi\)
0.245224 + 0.969466i \(0.421139\pi\)
\(644\) −84.0693 −3.31280
\(645\) 1.38586 0.0545684
\(646\) −18.6051 −0.732006
\(647\) 37.0323 1.45589 0.727944 0.685636i \(-0.240476\pi\)
0.727944 + 0.685636i \(0.240476\pi\)
\(648\) 30.3037 1.19044
\(649\) −16.3476 −0.641700
\(650\) 7.67899 0.301195
\(651\) −18.0552 −0.707638
\(652\) 64.2512 2.51627
\(653\) 17.8963 0.700335 0.350167 0.936687i \(-0.386125\pi\)
0.350167 + 0.936687i \(0.386125\pi\)
\(654\) −3.52128 −0.137693
\(655\) 5.56198 0.217324
\(656\) −130.775 −5.10590
\(657\) 3.79757 0.148157
\(658\) −16.2279 −0.632629
\(659\) 0.450954 0.0175667 0.00878333 0.999961i \(-0.497204\pi\)
0.00878333 + 0.999961i \(0.497204\pi\)
\(660\) 5.96403 0.232150
\(661\) 37.5598 1.46091 0.730454 0.682962i \(-0.239309\pi\)
0.730454 + 0.682962i \(0.239309\pi\)
\(662\) −66.5139 −2.58514
\(663\) −2.20220 −0.0855264
\(664\) −0.283269 −0.0109930
\(665\) −1.46048 −0.0566350
\(666\) 3.35044 0.129827
\(667\) 44.6705 1.72965
\(668\) 91.2880 3.53204
\(669\) 6.58068 0.254424
\(670\) 8.12676 0.313964
\(671\) −16.2842 −0.628645
\(672\) −42.2407 −1.62947
\(673\) 14.0539 0.541738 0.270869 0.962616i \(-0.412689\pi\)
0.270869 + 0.962616i \(0.412689\pi\)
\(674\) −73.3992 −2.82723
\(675\) −27.6534 −1.06438
\(676\) −64.6971 −2.48835
\(677\) 3.39839 0.130611 0.0653053 0.997865i \(-0.479198\pi\)
0.0653053 + 0.997865i \(0.479198\pi\)
\(678\) −13.4183 −0.515327
\(679\) −0.841441 −0.0322916
\(680\) −6.67902 −0.256129
\(681\) 6.53045 0.250247
\(682\) −54.0136 −2.06829
\(683\) −32.1149 −1.22884 −0.614421 0.788978i \(-0.710611\pi\)
−0.614421 + 0.788978i \(0.710611\pi\)
\(684\) −15.7228 −0.601176
\(685\) 1.90120 0.0726410
\(686\) −53.7267 −2.05130
\(687\) −19.2349 −0.733855
\(688\) −44.2253 −1.68607
\(689\) −1.28546 −0.0489721
\(690\) 7.74755 0.294944
\(691\) −6.53753 −0.248699 −0.124350 0.992238i \(-0.539684\pi\)
−0.124350 + 0.992238i \(0.539684\pi\)
\(692\) 48.3420 1.83769
\(693\) −8.29928 −0.315264
\(694\) −72.8370 −2.76485
\(695\) −3.85758 −0.146326
\(696\) 62.8358 2.38178
\(697\) −31.3070 −1.18584
\(698\) 41.7435 1.58001
\(699\) 0.643181 0.0243274
\(700\) 53.1072 2.00726
\(701\) −44.2499 −1.67130 −0.835648 0.549266i \(-0.814907\pi\)
−0.835648 + 0.549266i \(0.814907\pi\)
\(702\) −8.77177 −0.331069
\(703\) 2.44831 0.0923398
\(704\) −51.8965 −1.95592
\(705\) 1.07491 0.0404835
\(706\) −78.5173 −2.95504
\(707\) 23.0533 0.867009
\(708\) −35.2723 −1.32562
\(709\) −0.275455 −0.0103449 −0.00517247 0.999987i \(-0.501646\pi\)
−0.00517247 + 0.999987i \(0.501646\pi\)
\(710\) −4.92214 −0.184725
\(711\) 9.81542 0.368107
\(712\) 74.0187 2.77397
\(713\) −50.4326 −1.88871
\(714\) −21.1896 −0.792999
\(715\) −0.517158 −0.0193406
\(716\) −1.20363 −0.0449817
\(717\) 24.6825 0.921787
\(718\) −27.1677 −1.01389
\(719\) 34.6886 1.29366 0.646832 0.762632i \(-0.276093\pi\)
0.646832 + 0.762632i \(0.276093\pi\)
\(720\) −4.22466 −0.157444
\(721\) 14.0363 0.522740
\(722\) 34.6827 1.29076
\(723\) −19.3866 −0.720996
\(724\) 40.2856 1.49720
\(725\) −28.2187 −1.04802
\(726\) −4.27811 −0.158776
\(727\) 34.3683 1.27465 0.637325 0.770595i \(-0.280041\pi\)
0.637325 + 0.770595i \(0.280041\pi\)
\(728\) 10.2543 0.380050
\(729\) 26.8532 0.994561
\(730\) 2.27695 0.0842737
\(731\) −10.5874 −0.391588
\(732\) −35.1356 −1.29865
\(733\) 13.3248 0.492164 0.246082 0.969249i \(-0.420857\pi\)
0.246082 + 0.969249i \(0.420857\pi\)
\(734\) 15.0344 0.554929
\(735\) 0.947709 0.0349568
\(736\) −117.989 −4.34913
\(737\) 33.7463 1.24306
\(738\) −36.8091 −1.35496
\(739\) 39.6322 1.45789 0.728947 0.684570i \(-0.240010\pi\)
0.728947 + 0.684570i \(0.240010\pi\)
\(740\) 1.44389 0.0530784
\(741\) −1.89206 −0.0695067
\(742\) −12.3687 −0.454069
\(743\) 14.6256 0.536563 0.268281 0.963341i \(-0.413544\pi\)
0.268281 + 0.963341i \(0.413544\pi\)
\(744\) −70.9410 −2.60082
\(745\) −6.05197 −0.221727
\(746\) 29.1114 1.06584
\(747\) −0.0428940 −0.00156941
\(748\) −45.5625 −1.66593
\(749\) −30.6035 −1.11823
\(750\) −9.86774 −0.360319
\(751\) 23.8520 0.870373 0.435186 0.900340i \(-0.356682\pi\)
0.435186 + 0.900340i \(0.356682\pi\)
\(752\) −34.3023 −1.25087
\(753\) −19.3727 −0.705980
\(754\) −8.95108 −0.325979
\(755\) −5.31856 −0.193562
\(756\) −60.6648 −2.20636
\(757\) −33.3624 −1.21258 −0.606289 0.795245i \(-0.707342\pi\)
−0.606289 + 0.795245i \(0.707342\pi\)
\(758\) −80.5036 −2.92402
\(759\) 32.1716 1.16776
\(760\) −5.73841 −0.208154
\(761\) 10.3425 0.374914 0.187457 0.982273i \(-0.439976\pi\)
0.187457 + 0.982273i \(0.439976\pi\)
\(762\) 4.70629 0.170491
\(763\) 2.11170 0.0764488
\(764\) 112.868 4.08342
\(765\) −1.01137 −0.0365661
\(766\) 36.0936 1.30412
\(767\) 3.05856 0.110438
\(768\) −5.28882 −0.190844
\(769\) −37.6772 −1.35868 −0.679338 0.733826i \(-0.737733\pi\)
−0.679338 + 0.733826i \(0.737733\pi\)
\(770\) −4.97609 −0.179326
\(771\) −10.1935 −0.367108
\(772\) 68.2790 2.45742
\(773\) 10.0205 0.360414 0.180207 0.983629i \(-0.442323\pi\)
0.180207 + 0.983629i \(0.442323\pi\)
\(774\) −12.4481 −0.447437
\(775\) 31.8586 1.14440
\(776\) −3.30613 −0.118683
\(777\) 2.78841 0.100034
\(778\) −22.3146 −0.800018
\(779\) −26.8980 −0.963721
\(780\) −1.11584 −0.0399536
\(781\) −20.4392 −0.731370
\(782\) −59.1877 −2.11655
\(783\) 32.2344 1.15196
\(784\) −30.2430 −1.08011
\(785\) 1.27464 0.0454940
\(786\) 69.3321 2.47299
\(787\) −6.49592 −0.231555 −0.115777 0.993275i \(-0.536936\pi\)
−0.115777 + 0.993275i \(0.536936\pi\)
\(788\) −29.8999 −1.06514
\(789\) 8.85709 0.315321
\(790\) 5.88514 0.209384
\(791\) 8.04692 0.286116
\(792\) −32.6089 −1.15871
\(793\) 3.04670 0.108191
\(794\) −11.3324 −0.402173
\(795\) 0.819284 0.0290570
\(796\) 86.5969 3.06935
\(797\) 36.6040 1.29658 0.648290 0.761393i \(-0.275484\pi\)
0.648290 + 0.761393i \(0.275484\pi\)
\(798\) −18.2054 −0.644465
\(799\) −8.21183 −0.290514
\(800\) 74.5344 2.63519
\(801\) 11.2083 0.396024
\(802\) −9.93135 −0.350688
\(803\) 9.45502 0.333661
\(804\) 72.8125 2.56790
\(805\) −4.64618 −0.163756
\(806\) 10.1057 0.355958
\(807\) 16.3895 0.576939
\(808\) 90.5793 3.18657
\(809\) 12.9219 0.454311 0.227155 0.973859i \(-0.427058\pi\)
0.227155 + 0.973859i \(0.427058\pi\)
\(810\) 2.75131 0.0966711
\(811\) −1.96637 −0.0690487 −0.0345244 0.999404i \(-0.510992\pi\)
−0.0345244 + 0.999404i \(0.510992\pi\)
\(812\) −61.9048 −2.17243
\(813\) 33.9295 1.18996
\(814\) 8.34178 0.292379
\(815\) 3.55091 0.124383
\(816\) −44.7902 −1.56797
\(817\) −9.09634 −0.318241
\(818\) −34.4611 −1.20490
\(819\) 1.55276 0.0542577
\(820\) −15.8631 −0.553963
\(821\) 29.4988 1.02952 0.514758 0.857336i \(-0.327882\pi\)
0.514758 + 0.857336i \(0.327882\pi\)
\(822\) 23.6991 0.826601
\(823\) 11.8907 0.414484 0.207242 0.978290i \(-0.433551\pi\)
0.207242 + 0.978290i \(0.433551\pi\)
\(824\) 55.1504 1.92126
\(825\) −20.3231 −0.707558
\(826\) 29.4294 1.02398
\(827\) −11.2458 −0.391054 −0.195527 0.980698i \(-0.562642\pi\)
−0.195527 + 0.980698i \(0.562642\pi\)
\(828\) −50.0184 −1.73826
\(829\) −17.7578 −0.616753 −0.308376 0.951264i \(-0.599786\pi\)
−0.308376 + 0.951264i \(0.599786\pi\)
\(830\) −0.0257184 −0.000892699 0
\(831\) −17.5327 −0.608202
\(832\) 9.70959 0.336619
\(833\) −7.24006 −0.250853
\(834\) −48.0861 −1.66509
\(835\) 5.04513 0.174594
\(836\) −39.1459 −1.35389
\(837\) −36.3924 −1.25790
\(838\) −62.8769 −2.17205
\(839\) −5.36417 −0.185192 −0.0925958 0.995704i \(-0.529516\pi\)
−0.0925958 + 0.995704i \(0.529516\pi\)
\(840\) −6.53555 −0.225498
\(841\) 3.89331 0.134252
\(842\) 35.2521 1.21487
\(843\) 22.7626 0.783984
\(844\) 45.8865 1.57948
\(845\) −3.57555 −0.123003
\(846\) −9.65505 −0.331947
\(847\) 2.56557 0.0881541
\(848\) −26.1447 −0.897814
\(849\) −2.87196 −0.0985655
\(850\) 37.3893 1.28244
\(851\) 7.78873 0.266994
\(852\) −44.1004 −1.51086
\(853\) −10.0531 −0.344212 −0.172106 0.985078i \(-0.555057\pi\)
−0.172106 + 0.985078i \(0.555057\pi\)
\(854\) 29.3153 1.00315
\(855\) −0.868937 −0.0297170
\(856\) −120.245 −4.10989
\(857\) −48.6023 −1.66022 −0.830111 0.557598i \(-0.811723\pi\)
−0.830111 + 0.557598i \(0.811723\pi\)
\(858\) −6.44656 −0.220082
\(859\) 18.0666 0.616426 0.308213 0.951317i \(-0.400269\pi\)
0.308213 + 0.951317i \(0.400269\pi\)
\(860\) −5.36456 −0.182930
\(861\) −30.6345 −1.04402
\(862\) −80.0482 −2.72645
\(863\) 44.5630 1.51694 0.758471 0.651707i \(-0.225947\pi\)
0.758471 + 0.651707i \(0.225947\pi\)
\(864\) −85.1412 −2.89656
\(865\) 2.67167 0.0908395
\(866\) −82.7984 −2.81360
\(867\) 11.7252 0.398208
\(868\) 69.8900 2.37222
\(869\) 24.4380 0.829002
\(870\) 5.70494 0.193416
\(871\) −6.31377 −0.213934
\(872\) 8.29715 0.280977
\(873\) −0.500629 −0.0169437
\(874\) −50.8522 −1.72010
\(875\) 5.91766 0.200053
\(876\) 20.4006 0.689272
\(877\) 6.73318 0.227363 0.113682 0.993517i \(-0.463736\pi\)
0.113682 + 0.993517i \(0.463736\pi\)
\(878\) 44.2039 1.49181
\(879\) −26.7514 −0.902302
\(880\) −10.5184 −0.354574
\(881\) 27.2479 0.918004 0.459002 0.888435i \(-0.348207\pi\)
0.459002 + 0.888435i \(0.348207\pi\)
\(882\) −8.51249 −0.286630
\(883\) −8.70316 −0.292885 −0.146442 0.989219i \(-0.546782\pi\)
−0.146442 + 0.989219i \(0.546782\pi\)
\(884\) 8.52452 0.286711
\(885\) −1.94936 −0.0655271
\(886\) −14.7387 −0.495155
\(887\) −46.2493 −1.55290 −0.776450 0.630179i \(-0.782981\pi\)
−0.776450 + 0.630179i \(0.782981\pi\)
\(888\) 10.9560 0.367660
\(889\) −2.82235 −0.0946586
\(890\) 6.72025 0.225263
\(891\) 11.4248 0.382745
\(892\) −25.4732 −0.852907
\(893\) −7.05535 −0.236098
\(894\) −75.4400 −2.52309
\(895\) −0.0665199 −0.00222351
\(896\) 29.4467 0.983745
\(897\) −6.01916 −0.200974
\(898\) 102.712 3.42756
\(899\) −37.1363 −1.23856
\(900\) 31.5970 1.05323
\(901\) −6.25895 −0.208516
\(902\) −91.6457 −3.05147
\(903\) −10.3599 −0.344757
\(904\) 31.6174 1.05158
\(905\) 2.22643 0.0740090
\(906\) −66.2978 −2.20260
\(907\) 16.4131 0.544987 0.272494 0.962158i \(-0.412152\pi\)
0.272494 + 0.962158i \(0.412152\pi\)
\(908\) −25.2788 −0.838907
\(909\) 13.7160 0.454929
\(910\) 0.931002 0.0308624
\(911\) −4.28492 −0.141966 −0.0709829 0.997478i \(-0.522614\pi\)
−0.0709829 + 0.997478i \(0.522614\pi\)
\(912\) −38.4823 −1.27428
\(913\) −0.106795 −0.00353441
\(914\) −37.8077 −1.25057
\(915\) −1.94180 −0.0641941
\(916\) 74.4564 2.46011
\(917\) −41.5782 −1.37303
\(918\) −42.7101 −1.40964
\(919\) 25.2287 0.832218 0.416109 0.909315i \(-0.363394\pi\)
0.416109 + 0.909315i \(0.363394\pi\)
\(920\) −18.2554 −0.601864
\(921\) −10.3130 −0.339824
\(922\) 100.092 3.29635
\(923\) 3.82407 0.125871
\(924\) −44.5838 −1.46670
\(925\) −4.92020 −0.161775
\(926\) 39.4851 1.29756
\(927\) 8.35114 0.274287
\(928\) −86.8816 −2.85203
\(929\) 4.31329 0.141514 0.0707571 0.997494i \(-0.477458\pi\)
0.0707571 + 0.997494i \(0.477458\pi\)
\(930\) −6.44083 −0.211203
\(931\) −6.22044 −0.203867
\(932\) −2.48970 −0.0815528
\(933\) 10.1287 0.331600
\(934\) −30.7860 −1.00735
\(935\) −2.51806 −0.0823494
\(936\) 6.10097 0.199416
\(937\) 23.4464 0.765959 0.382980 0.923757i \(-0.374898\pi\)
0.382980 + 0.923757i \(0.374898\pi\)
\(938\) −60.7511 −1.98359
\(939\) 23.0307 0.751580
\(940\) −4.16089 −0.135713
\(941\) −26.1473 −0.852379 −0.426189 0.904634i \(-0.640144\pi\)
−0.426189 + 0.904634i \(0.640144\pi\)
\(942\) 15.8889 0.517688
\(943\) −85.5698 −2.78654
\(944\) 62.2075 2.02468
\(945\) −3.35271 −0.109064
\(946\) −30.9927 −1.00766
\(947\) 20.3697 0.661928 0.330964 0.943643i \(-0.392626\pi\)
0.330964 + 0.943643i \(0.392626\pi\)
\(948\) 52.7285 1.71254
\(949\) −1.76899 −0.0574238
\(950\) 32.1237 1.04223
\(951\) −10.4825 −0.339919
\(952\) 49.9286 1.61820
\(953\) −53.3495 −1.72816 −0.864080 0.503354i \(-0.832099\pi\)
−0.864080 + 0.503354i \(0.832099\pi\)
\(954\) −7.35895 −0.238255
\(955\) 6.23777 0.201849
\(956\) −95.5439 −3.09011
\(957\) 23.6897 0.765780
\(958\) −90.0015 −2.90782
\(959\) −14.2123 −0.458939
\(960\) −6.18838 −0.199729
\(961\) 10.9265 0.352468
\(962\) −1.56071 −0.0503192
\(963\) −18.2081 −0.586746
\(964\) 75.0438 2.41700
\(965\) 3.77352 0.121474
\(966\) −57.9163 −1.86343
\(967\) −15.5422 −0.499802 −0.249901 0.968271i \(-0.580398\pi\)
−0.249901 + 0.968271i \(0.580398\pi\)
\(968\) 10.0804 0.323998
\(969\) −9.21252 −0.295949
\(970\) −0.300168 −0.00963780
\(971\) −40.5433 −1.30110 −0.650548 0.759465i \(-0.725461\pi\)
−0.650548 + 0.759465i \(0.725461\pi\)
\(972\) −61.5330 −1.97367
\(973\) 28.8371 0.924475
\(974\) −89.5193 −2.86838
\(975\) 3.80235 0.121773
\(976\) 61.9663 1.98349
\(977\) 8.40900 0.269028 0.134514 0.990912i \(-0.457053\pi\)
0.134514 + 0.990912i \(0.457053\pi\)
\(978\) 44.2634 1.41539
\(979\) 27.9058 0.891873
\(980\) −3.66850 −0.117186
\(981\) 1.25639 0.0401135
\(982\) −24.5982 −0.784959
\(983\) 25.0169 0.797916 0.398958 0.916969i \(-0.369372\pi\)
0.398958 + 0.916969i \(0.369372\pi\)
\(984\) −120.367 −3.83715
\(985\) −1.65245 −0.0526514
\(986\) −43.5831 −1.38797
\(987\) −8.03544 −0.255771
\(988\) 7.32401 0.233008
\(989\) −28.9379 −0.920172
\(990\) −2.96061 −0.0940942
\(991\) 50.3166 1.59836 0.799180 0.601091i \(-0.205267\pi\)
0.799180 + 0.601091i \(0.205267\pi\)
\(992\) 98.0885 3.11431
\(993\) −32.9352 −1.04517
\(994\) 36.7952 1.16707
\(995\) 4.78587 0.151722
\(996\) −0.230427 −0.00730135
\(997\) −42.2446 −1.33790 −0.668949 0.743308i \(-0.733256\pi\)
−0.668949 + 0.743308i \(0.733256\pi\)
\(998\) −71.0905 −2.25033
\(999\) 5.62038 0.177821
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 4033.2.a.f.1.3 85
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4033.2.a.f.1.3 85 1.1 even 1 trivial