Properties

Label 4006.2.a.g.1.20
Level 4006
Weight 2
Character 4006.1
Self dual yes
Analytic conductor 31.988
Analytic rank 1
Dimension 40
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4006.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9880710497\)
Analytic rank: \(1\)
Dimension: \(40\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) = 4006.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.0866100 q^{3} +1.00000 q^{4} +2.47499 q^{5} +0.0866100 q^{6} +0.719708 q^{7} -1.00000 q^{8} -2.99250 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.0866100 q^{3} +1.00000 q^{4} +2.47499 q^{5} +0.0866100 q^{6} +0.719708 q^{7} -1.00000 q^{8} -2.99250 q^{9} -2.47499 q^{10} -4.02218 q^{11} -0.0866100 q^{12} -1.17365 q^{13} -0.719708 q^{14} -0.214358 q^{15} +1.00000 q^{16} +0.882057 q^{17} +2.99250 q^{18} +0.0145269 q^{19} +2.47499 q^{20} -0.0623338 q^{21} +4.02218 q^{22} -5.06055 q^{23} +0.0866100 q^{24} +1.12556 q^{25} +1.17365 q^{26} +0.519010 q^{27} +0.719708 q^{28} +5.12559 q^{29} +0.214358 q^{30} +7.34428 q^{31} -1.00000 q^{32} +0.348361 q^{33} -0.882057 q^{34} +1.78127 q^{35} -2.99250 q^{36} +5.66296 q^{37} -0.0145269 q^{38} +0.101650 q^{39} -2.47499 q^{40} +7.20437 q^{41} +0.0623338 q^{42} +2.79702 q^{43} -4.02218 q^{44} -7.40639 q^{45} +5.06055 q^{46} +6.06598 q^{47} -0.0866100 q^{48} -6.48202 q^{49} -1.12556 q^{50} -0.0763950 q^{51} -1.17365 q^{52} -13.1686 q^{53} -0.519010 q^{54} -9.95484 q^{55} -0.719708 q^{56} -0.00125817 q^{57} -5.12559 q^{58} -15.0453 q^{59} -0.214358 q^{60} -11.4828 q^{61} -7.34428 q^{62} -2.15372 q^{63} +1.00000 q^{64} -2.90477 q^{65} -0.348361 q^{66} +9.75512 q^{67} +0.882057 q^{68} +0.438294 q^{69} -1.78127 q^{70} +14.9650 q^{71} +2.99250 q^{72} -11.9589 q^{73} -5.66296 q^{74} -0.0974845 q^{75} +0.0145269 q^{76} -2.89479 q^{77} -0.101650 q^{78} -8.14549 q^{79} +2.47499 q^{80} +8.93254 q^{81} -7.20437 q^{82} -12.2879 q^{83} -0.0623338 q^{84} +2.18308 q^{85} -2.79702 q^{86} -0.443927 q^{87} +4.02218 q^{88} -17.7820 q^{89} +7.40639 q^{90} -0.844686 q^{91} -5.06055 q^{92} -0.636088 q^{93} -6.06598 q^{94} +0.0359538 q^{95} +0.0866100 q^{96} +0.575278 q^{97} +6.48202 q^{98} +12.0364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + O(q^{10}) \) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + 23q^{10} - 28q^{11} - q^{12} - 12q^{13} - 12q^{14} - 14q^{15} + 40q^{16} - 10q^{17} - 29q^{18} + 3q^{19} - 23q^{20} - 40q^{21} + 28q^{22} - 10q^{23} + q^{24} + 43q^{25} + 12q^{26} - 7q^{27} + 12q^{28} - 37q^{29} + 14q^{30} - 16q^{31} - 40q^{32} - 11q^{33} + 10q^{34} - 24q^{35} + 29q^{36} - 17q^{37} - 3q^{38} - 10q^{39} + 23q^{40} - 58q^{41} + 40q^{42} + 37q^{43} - 28q^{44} - 66q^{45} + 10q^{46} - 34q^{47} - q^{48} + 28q^{49} - 43q^{50} - 7q^{51} - 12q^{52} - 43q^{53} + 7q^{54} + 28q^{55} - 12q^{56} - 25q^{57} + 37q^{58} - 92q^{59} - 14q^{60} - 37q^{61} + 16q^{62} + 14q^{63} + 40q^{64} - 54q^{65} + 11q^{66} - 10q^{67} - 10q^{68} - 49q^{69} + 24q^{70} - 87q^{71} - 29q^{72} + 12q^{73} + 17q^{74} - 31q^{75} + 3q^{76} - 53q^{77} + 10q^{78} + 14q^{79} - 23q^{80} - 4q^{81} + 58q^{82} - 42q^{83} - 40q^{84} - 24q^{85} - 37q^{86} + 24q^{87} + 28q^{88} - 118q^{89} + 66q^{90} - 35q^{91} - 10q^{92} - 22q^{93} + 34q^{94} - 18q^{95} + q^{96} + 2q^{97} - 28q^{98} - 48q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.0866100 −0.0500043 −0.0250021 0.999687i \(-0.507959\pi\)
−0.0250021 + 0.999687i \(0.507959\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.47499 1.10685 0.553424 0.832900i \(-0.313321\pi\)
0.553424 + 0.832900i \(0.313321\pi\)
\(6\) 0.0866100 0.0353584
\(7\) 0.719708 0.272024 0.136012 0.990707i \(-0.456571\pi\)
0.136012 + 0.990707i \(0.456571\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.99250 −0.997500
\(10\) −2.47499 −0.782659
\(11\) −4.02218 −1.21273 −0.606366 0.795185i \(-0.707373\pi\)
−0.606366 + 0.795185i \(0.707373\pi\)
\(12\) −0.0866100 −0.0250021
\(13\) −1.17365 −0.325512 −0.162756 0.986666i \(-0.552038\pi\)
−0.162756 + 0.986666i \(0.552038\pi\)
\(14\) −0.719708 −0.192350
\(15\) −0.214358 −0.0553471
\(16\) 1.00000 0.250000
\(17\) 0.882057 0.213930 0.106965 0.994263i \(-0.465887\pi\)
0.106965 + 0.994263i \(0.465887\pi\)
\(18\) 2.99250 0.705339
\(19\) 0.0145269 0.00333269 0.00166635 0.999999i \(-0.499470\pi\)
0.00166635 + 0.999999i \(0.499470\pi\)
\(20\) 2.47499 0.553424
\(21\) −0.0623338 −0.0136024
\(22\) 4.02218 0.857532
\(23\) −5.06055 −1.05520 −0.527599 0.849493i \(-0.676908\pi\)
−0.527599 + 0.849493i \(0.676908\pi\)
\(24\) 0.0866100 0.0176792
\(25\) 1.12556 0.225111
\(26\) 1.17365 0.230172
\(27\) 0.519010 0.0998835
\(28\) 0.719708 0.136012
\(29\) 5.12559 0.951799 0.475899 0.879500i \(-0.342123\pi\)
0.475899 + 0.879500i \(0.342123\pi\)
\(30\) 0.214358 0.0391363
\(31\) 7.34428 1.31907 0.659536 0.751673i \(-0.270753\pi\)
0.659536 + 0.751673i \(0.270753\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.348361 0.0606418
\(34\) −0.882057 −0.151272
\(35\) 1.78127 0.301089
\(36\) −2.99250 −0.498750
\(37\) 5.66296 0.930985 0.465492 0.885052i \(-0.345877\pi\)
0.465492 + 0.885052i \(0.345877\pi\)
\(38\) −0.0145269 −0.00235657
\(39\) 0.101650 0.0162770
\(40\) −2.47499 −0.391330
\(41\) 7.20437 1.12513 0.562567 0.826752i \(-0.309814\pi\)
0.562567 + 0.826752i \(0.309814\pi\)
\(42\) 0.0623338 0.00961832
\(43\) 2.79702 0.426541 0.213271 0.976993i \(-0.431588\pi\)
0.213271 + 0.976993i \(0.431588\pi\)
\(44\) −4.02218 −0.606366
\(45\) −7.40639 −1.10408
\(46\) 5.06055 0.746138
\(47\) 6.06598 0.884814 0.442407 0.896814i \(-0.354125\pi\)
0.442407 + 0.896814i \(0.354125\pi\)
\(48\) −0.0866100 −0.0125011
\(49\) −6.48202 −0.926003
\(50\) −1.12556 −0.159178
\(51\) −0.0763950 −0.0106974
\(52\) −1.17365 −0.162756
\(53\) −13.1686 −1.80885 −0.904424 0.426635i \(-0.859699\pi\)
−0.904424 + 0.426635i \(0.859699\pi\)
\(54\) −0.519010 −0.0706283
\(55\) −9.95484 −1.34231
\(56\) −0.719708 −0.0961750
\(57\) −0.00125817 −0.000166649 0
\(58\) −5.12559 −0.673023
\(59\) −15.0453 −1.95873 −0.979363 0.202107i \(-0.935221\pi\)
−0.979363 + 0.202107i \(0.935221\pi\)
\(60\) −0.214358 −0.0276736
\(61\) −11.4828 −1.47022 −0.735110 0.677948i \(-0.762869\pi\)
−0.735110 + 0.677948i \(0.762869\pi\)
\(62\) −7.34428 −0.932725
\(63\) −2.15372 −0.271344
\(64\) 1.00000 0.125000
\(65\) −2.90477 −0.360293
\(66\) −0.348361 −0.0428803
\(67\) 9.75512 1.19178 0.595889 0.803067i \(-0.296800\pi\)
0.595889 + 0.803067i \(0.296800\pi\)
\(68\) 0.882057 0.106965
\(69\) 0.438294 0.0527644
\(70\) −1.78127 −0.212902
\(71\) 14.9650 1.77601 0.888007 0.459831i \(-0.152090\pi\)
0.888007 + 0.459831i \(0.152090\pi\)
\(72\) 2.99250 0.352669
\(73\) −11.9589 −1.39969 −0.699845 0.714295i \(-0.746747\pi\)
−0.699845 + 0.714295i \(0.746747\pi\)
\(74\) −5.66296 −0.658306
\(75\) −0.0974845 −0.0112565
\(76\) 0.0145269 0.00166635
\(77\) −2.89479 −0.329892
\(78\) −0.101650 −0.0115096
\(79\) −8.14549 −0.916439 −0.458220 0.888839i \(-0.651513\pi\)
−0.458220 + 0.888839i \(0.651513\pi\)
\(80\) 2.47499 0.276712
\(81\) 8.93254 0.992505
\(82\) −7.20437 −0.795589
\(83\) −12.2879 −1.34877 −0.674386 0.738379i \(-0.735591\pi\)
−0.674386 + 0.738379i \(0.735591\pi\)
\(84\) −0.0623338 −0.00680118
\(85\) 2.18308 0.236788
\(86\) −2.79702 −0.301610
\(87\) −0.443927 −0.0475940
\(88\) 4.02218 0.428766
\(89\) −17.7820 −1.88489 −0.942445 0.334360i \(-0.891480\pi\)
−0.942445 + 0.334360i \(0.891480\pi\)
\(90\) 7.40639 0.780702
\(91\) −0.844686 −0.0885471
\(92\) −5.06055 −0.527599
\(93\) −0.636088 −0.0659593
\(94\) −6.06598 −0.625658
\(95\) 0.0359538 0.00368878
\(96\) 0.0866100 0.00883959
\(97\) 0.575278 0.0584106 0.0292053 0.999573i \(-0.490702\pi\)
0.0292053 + 0.999573i \(0.490702\pi\)
\(98\) 6.48202 0.654783
\(99\) 12.0364 1.20970
\(100\) 1.12556 0.112556
\(101\) −11.5157 −1.14585 −0.572926 0.819607i \(-0.694192\pi\)
−0.572926 + 0.819607i \(0.694192\pi\)
\(102\) 0.0763950 0.00756423
\(103\) 0.875520 0.0862675 0.0431338 0.999069i \(-0.486266\pi\)
0.0431338 + 0.999069i \(0.486266\pi\)
\(104\) 1.17365 0.115086
\(105\) −0.154275 −0.0150557
\(106\) 13.1686 1.27905
\(107\) −2.63094 −0.254343 −0.127171 0.991881i \(-0.540590\pi\)
−0.127171 + 0.991881i \(0.540590\pi\)
\(108\) 0.519010 0.0499418
\(109\) −10.4638 −1.00225 −0.501123 0.865376i \(-0.667080\pi\)
−0.501123 + 0.865376i \(0.667080\pi\)
\(110\) 9.95484 0.949157
\(111\) −0.490469 −0.0465532
\(112\) 0.719708 0.0680060
\(113\) −7.93666 −0.746618 −0.373309 0.927707i \(-0.621777\pi\)
−0.373309 + 0.927707i \(0.621777\pi\)
\(114\) 0.00125817 0.000117839 0
\(115\) −12.5248 −1.16794
\(116\) 5.12559 0.475899
\(117\) 3.51215 0.324698
\(118\) 15.0453 1.38503
\(119\) 0.634823 0.0581942
\(120\) 0.214358 0.0195682
\(121\) 5.17793 0.470721
\(122\) 11.4828 1.03960
\(123\) −0.623970 −0.0562615
\(124\) 7.34428 0.659536
\(125\) −9.58919 −0.857683
\(126\) 2.15372 0.191869
\(127\) 10.7592 0.954727 0.477363 0.878706i \(-0.341593\pi\)
0.477363 + 0.878706i \(0.341593\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.242250 −0.0213289
\(130\) 2.90477 0.254765
\(131\) −5.75133 −0.502496 −0.251248 0.967923i \(-0.580841\pi\)
−0.251248 + 0.967923i \(0.580841\pi\)
\(132\) 0.348361 0.0303209
\(133\) 0.0104551 0.000906572 0
\(134\) −9.75512 −0.842714
\(135\) 1.28454 0.110556
\(136\) −0.882057 −0.0756358
\(137\) −7.35468 −0.628353 −0.314176 0.949365i \(-0.601728\pi\)
−0.314176 + 0.949365i \(0.601728\pi\)
\(138\) −0.438294 −0.0373101
\(139\) −8.55859 −0.725930 −0.362965 0.931803i \(-0.618236\pi\)
−0.362965 + 0.931803i \(0.618236\pi\)
\(140\) 1.78127 0.150544
\(141\) −0.525374 −0.0442445
\(142\) −14.9650 −1.25583
\(143\) 4.72064 0.394760
\(144\) −2.99250 −0.249375
\(145\) 12.6858 1.05350
\(146\) 11.9589 0.989730
\(147\) 0.561408 0.0463041
\(148\) 5.66296 0.465492
\(149\) 3.19817 0.262004 0.131002 0.991382i \(-0.458181\pi\)
0.131002 + 0.991382i \(0.458181\pi\)
\(150\) 0.0974845 0.00795957
\(151\) −4.40206 −0.358234 −0.179117 0.983828i \(-0.557324\pi\)
−0.179117 + 0.983828i \(0.557324\pi\)
\(152\) −0.0145269 −0.00117828
\(153\) −2.63956 −0.213395
\(154\) 2.89479 0.233269
\(155\) 18.1770 1.46001
\(156\) 0.101650 0.00813851
\(157\) −22.5756 −1.80173 −0.900866 0.434097i \(-0.857067\pi\)
−0.900866 + 0.434097i \(0.857067\pi\)
\(158\) 8.14549 0.648020
\(159\) 1.14053 0.0904501
\(160\) −2.47499 −0.195665
\(161\) −3.64212 −0.287039
\(162\) −8.93254 −0.701807
\(163\) −7.40401 −0.579927 −0.289964 0.957038i \(-0.593643\pi\)
−0.289964 + 0.957038i \(0.593643\pi\)
\(164\) 7.20437 0.562567
\(165\) 0.862188 0.0671213
\(166\) 12.2879 0.953725
\(167\) −21.9499 −1.69853 −0.849267 0.527963i \(-0.822956\pi\)
−0.849267 + 0.527963i \(0.822956\pi\)
\(168\) 0.0623338 0.00480916
\(169\) −11.6225 −0.894042
\(170\) −2.18308 −0.167435
\(171\) −0.0434716 −0.00332436
\(172\) 2.79702 0.213271
\(173\) −4.99367 −0.379661 −0.189831 0.981817i \(-0.560794\pi\)
−0.189831 + 0.981817i \(0.560794\pi\)
\(174\) 0.443927 0.0336541
\(175\) 0.810072 0.0612357
\(176\) −4.02218 −0.303183
\(177\) 1.30307 0.0979447
\(178\) 17.7820 1.33282
\(179\) 5.25556 0.392819 0.196410 0.980522i \(-0.437072\pi\)
0.196410 + 0.980522i \(0.437072\pi\)
\(180\) −7.40639 −0.552040
\(181\) 17.6280 1.31028 0.655139 0.755508i \(-0.272610\pi\)
0.655139 + 0.755508i \(0.272610\pi\)
\(182\) 0.844686 0.0626123
\(183\) 0.994523 0.0735173
\(184\) 5.06055 0.373069
\(185\) 14.0157 1.03046
\(186\) 0.636088 0.0466403
\(187\) −3.54779 −0.259440
\(188\) 6.06598 0.442407
\(189\) 0.373535 0.0271707
\(190\) −0.0359538 −0.00260836
\(191\) −0.288421 −0.0208694 −0.0104347 0.999946i \(-0.503322\pi\)
−0.0104347 + 0.999946i \(0.503322\pi\)
\(192\) −0.0866100 −0.00625054
\(193\) −2.30616 −0.166001 −0.0830006 0.996549i \(-0.526450\pi\)
−0.0830006 + 0.996549i \(0.526450\pi\)
\(194\) −0.575278 −0.0413025
\(195\) 0.251582 0.0180162
\(196\) −6.48202 −0.463002
\(197\) 16.6893 1.18906 0.594530 0.804073i \(-0.297338\pi\)
0.594530 + 0.804073i \(0.297338\pi\)
\(198\) −12.0364 −0.855387
\(199\) 13.6845 0.970067 0.485034 0.874496i \(-0.338807\pi\)
0.485034 + 0.874496i \(0.338807\pi\)
\(200\) −1.12556 −0.0795889
\(201\) −0.844890 −0.0595940
\(202\) 11.5157 0.810240
\(203\) 3.68893 0.258912
\(204\) −0.0763950 −0.00534872
\(205\) 17.8307 1.24535
\(206\) −0.875520 −0.0610003
\(207\) 15.1437 1.05256
\(208\) −1.17365 −0.0813781
\(209\) −0.0584297 −0.00404166
\(210\) 0.154275 0.0106460
\(211\) −0.344938 −0.0237465 −0.0118732 0.999930i \(-0.503779\pi\)
−0.0118732 + 0.999930i \(0.503779\pi\)
\(212\) −13.1686 −0.904424
\(213\) −1.29611 −0.0888083
\(214\) 2.63094 0.179847
\(215\) 6.92258 0.472116
\(216\) −0.519010 −0.0353142
\(217\) 5.28574 0.358819
\(218\) 10.4638 0.708695
\(219\) 1.03576 0.0699904
\(220\) −9.95484 −0.671155
\(221\) −1.03523 −0.0696370
\(222\) 0.490469 0.0329181
\(223\) 8.24741 0.552288 0.276144 0.961116i \(-0.410943\pi\)
0.276144 + 0.961116i \(0.410943\pi\)
\(224\) −0.719708 −0.0480875
\(225\) −3.36823 −0.224549
\(226\) 7.93666 0.527939
\(227\) 15.9491 1.05858 0.529291 0.848441i \(-0.322458\pi\)
0.529291 + 0.848441i \(0.322458\pi\)
\(228\) −0.00125817 −8.33244e−5 0
\(229\) 17.6910 1.16906 0.584528 0.811374i \(-0.301280\pi\)
0.584528 + 0.811374i \(0.301280\pi\)
\(230\) 12.5248 0.825861
\(231\) 0.250718 0.0164960
\(232\) −5.12559 −0.336512
\(233\) 4.52282 0.296300 0.148150 0.988965i \(-0.452668\pi\)
0.148150 + 0.988965i \(0.452668\pi\)
\(234\) −3.51215 −0.229596
\(235\) 15.0132 0.979354
\(236\) −15.0453 −0.979363
\(237\) 0.705481 0.0458259
\(238\) −0.634823 −0.0411495
\(239\) 10.2411 0.662440 0.331220 0.943554i \(-0.392540\pi\)
0.331220 + 0.943554i \(0.392540\pi\)
\(240\) −0.214358 −0.0138368
\(241\) 13.4817 0.868430 0.434215 0.900809i \(-0.357026\pi\)
0.434215 + 0.900809i \(0.357026\pi\)
\(242\) −5.17793 −0.332850
\(243\) −2.33068 −0.149513
\(244\) −11.4828 −0.735110
\(245\) −16.0429 −1.02494
\(246\) 0.623970 0.0397829
\(247\) −0.0170495 −0.00108483
\(248\) −7.34428 −0.466363
\(249\) 1.06425 0.0674443
\(250\) 9.58919 0.606474
\(251\) −13.3357 −0.841743 −0.420871 0.907120i \(-0.638276\pi\)
−0.420871 + 0.907120i \(0.638276\pi\)
\(252\) −2.15372 −0.135672
\(253\) 20.3545 1.27967
\(254\) −10.7592 −0.675094
\(255\) −0.189076 −0.0118404
\(256\) 1.00000 0.0625000
\(257\) −23.0432 −1.43739 −0.718697 0.695324i \(-0.755261\pi\)
−0.718697 + 0.695324i \(0.755261\pi\)
\(258\) 0.242250 0.0150818
\(259\) 4.07567 0.253250
\(260\) −2.90477 −0.180146
\(261\) −15.3383 −0.949419
\(262\) 5.75133 0.355318
\(263\) −17.2883 −1.06604 −0.533021 0.846102i \(-0.678943\pi\)
−0.533021 + 0.846102i \(0.678943\pi\)
\(264\) −0.348361 −0.0214401
\(265\) −32.5921 −2.00212
\(266\) −0.0104551 −0.000641043 0
\(267\) 1.54010 0.0942526
\(268\) 9.75512 0.595889
\(269\) −10.7977 −0.658346 −0.329173 0.944270i \(-0.606770\pi\)
−0.329173 + 0.944270i \(0.606770\pi\)
\(270\) −1.28454 −0.0781748
\(271\) −8.91559 −0.541584 −0.270792 0.962638i \(-0.587285\pi\)
−0.270792 + 0.962638i \(0.587285\pi\)
\(272\) 0.882057 0.0534826
\(273\) 0.0731582 0.00442774
\(274\) 7.35468 0.444312
\(275\) −4.52719 −0.273000
\(276\) 0.438294 0.0263822
\(277\) 4.51833 0.271480 0.135740 0.990745i \(-0.456659\pi\)
0.135740 + 0.990745i \(0.456659\pi\)
\(278\) 8.55859 0.513310
\(279\) −21.9778 −1.31577
\(280\) −1.78127 −0.106451
\(281\) 14.9366 0.891040 0.445520 0.895272i \(-0.353019\pi\)
0.445520 + 0.895272i \(0.353019\pi\)
\(282\) 0.525374 0.0312856
\(283\) −15.6586 −0.930805 −0.465402 0.885099i \(-0.654090\pi\)
−0.465402 + 0.885099i \(0.654090\pi\)
\(284\) 14.9650 0.888007
\(285\) −0.00311396 −0.000184455 0
\(286\) −4.72064 −0.279137
\(287\) 5.18504 0.306063
\(288\) 2.99250 0.176335
\(289\) −16.2220 −0.954234
\(290\) −12.6858 −0.744934
\(291\) −0.0498248 −0.00292078
\(292\) −11.9589 −0.699845
\(293\) 23.7676 1.38852 0.694258 0.719726i \(-0.255733\pi\)
0.694258 + 0.719726i \(0.255733\pi\)
\(294\) −0.561408 −0.0327420
\(295\) −37.2368 −2.16801
\(296\) −5.66296 −0.329153
\(297\) −2.08755 −0.121132
\(298\) −3.19817 −0.185265
\(299\) 5.93933 0.343480
\(300\) −0.0974845 −0.00562827
\(301\) 2.01304 0.116029
\(302\) 4.40206 0.253310
\(303\) 0.997372 0.0572975
\(304\) 0.0145269 0.000833173 0
\(305\) −28.4197 −1.62731
\(306\) 2.63956 0.150893
\(307\) −8.32429 −0.475092 −0.237546 0.971376i \(-0.576343\pi\)
−0.237546 + 0.971376i \(0.576343\pi\)
\(308\) −2.89479 −0.164946
\(309\) −0.0758287 −0.00431374
\(310\) −18.1770 −1.03238
\(311\) −11.8177 −0.670119 −0.335059 0.942197i \(-0.608756\pi\)
−0.335059 + 0.942197i \(0.608756\pi\)
\(312\) −0.101650 −0.00575479
\(313\) −12.3763 −0.699551 −0.349775 0.936834i \(-0.613742\pi\)
−0.349775 + 0.936834i \(0.613742\pi\)
\(314\) 22.5756 1.27402
\(315\) −5.33044 −0.300336
\(316\) −8.14549 −0.458220
\(317\) 0.620603 0.0348566 0.0174283 0.999848i \(-0.494452\pi\)
0.0174283 + 0.999848i \(0.494452\pi\)
\(318\) −1.14053 −0.0639579
\(319\) −20.6161 −1.15428
\(320\) 2.47499 0.138356
\(321\) 0.227866 0.0127182
\(322\) 3.64212 0.202967
\(323\) 0.0128135 0.000712964 0
\(324\) 8.93254 0.496252
\(325\) −1.32101 −0.0732766
\(326\) 7.40401 0.410070
\(327\) 0.906266 0.0501166
\(328\) −7.20437 −0.397795
\(329\) 4.36573 0.240691
\(330\) −0.862188 −0.0474619
\(331\) −9.37988 −0.515565 −0.257783 0.966203i \(-0.582992\pi\)
−0.257783 + 0.966203i \(0.582992\pi\)
\(332\) −12.2879 −0.674386
\(333\) −16.9464 −0.928657
\(334\) 21.9499 1.20105
\(335\) 24.1438 1.31912
\(336\) −0.0623338 −0.00340059
\(337\) 2.50685 0.136557 0.0682784 0.997666i \(-0.478249\pi\)
0.0682784 + 0.997666i \(0.478249\pi\)
\(338\) 11.6225 0.632183
\(339\) 0.687394 0.0373341
\(340\) 2.18308 0.118394
\(341\) −29.5400 −1.59968
\(342\) 0.0434716 0.00235068
\(343\) −9.70311 −0.523919
\(344\) −2.79702 −0.150805
\(345\) 1.08477 0.0584022
\(346\) 4.99367 0.268461
\(347\) −0.0754958 −0.00405283 −0.00202641 0.999998i \(-0.500645\pi\)
−0.00202641 + 0.999998i \(0.500645\pi\)
\(348\) −0.443927 −0.0237970
\(349\) 24.9751 1.33689 0.668444 0.743763i \(-0.266961\pi\)
0.668444 + 0.743763i \(0.266961\pi\)
\(350\) −0.810072 −0.0433002
\(351\) −0.609137 −0.0325133
\(352\) 4.02218 0.214383
\(353\) 8.13318 0.432886 0.216443 0.976295i \(-0.430555\pi\)
0.216443 + 0.976295i \(0.430555\pi\)
\(354\) −1.30307 −0.0692574
\(355\) 37.0381 1.96578
\(356\) −17.7820 −0.942445
\(357\) −0.0549820 −0.00290996
\(358\) −5.25556 −0.277765
\(359\) 28.6150 1.51024 0.755122 0.655584i \(-0.227578\pi\)
0.755122 + 0.655584i \(0.227578\pi\)
\(360\) 7.40639 0.390351
\(361\) −18.9998 −0.999989
\(362\) −17.6280 −0.926507
\(363\) −0.448460 −0.0235380
\(364\) −0.844686 −0.0442736
\(365\) −29.5982 −1.54924
\(366\) −0.994523 −0.0519846
\(367\) −9.09315 −0.474658 −0.237329 0.971429i \(-0.576272\pi\)
−0.237329 + 0.971429i \(0.576272\pi\)
\(368\) −5.06055 −0.263800
\(369\) −21.5591 −1.12232
\(370\) −14.0157 −0.728644
\(371\) −9.47755 −0.492050
\(372\) −0.636088 −0.0329796
\(373\) −36.6472 −1.89752 −0.948760 0.315998i \(-0.897661\pi\)
−0.948760 + 0.315998i \(0.897661\pi\)
\(374\) 3.54779 0.183452
\(375\) 0.830520 0.0428878
\(376\) −6.06598 −0.312829
\(377\) −6.01566 −0.309822
\(378\) −0.373535 −0.0192126
\(379\) −18.0899 −0.929218 −0.464609 0.885516i \(-0.653805\pi\)
−0.464609 + 0.885516i \(0.653805\pi\)
\(380\) 0.0359538 0.00184439
\(381\) −0.931856 −0.0477404
\(382\) 0.288421 0.0147569
\(383\) 27.9297 1.42714 0.713570 0.700584i \(-0.247077\pi\)
0.713570 + 0.700584i \(0.247077\pi\)
\(384\) 0.0866100 0.00441980
\(385\) −7.16457 −0.365140
\(386\) 2.30616 0.117381
\(387\) −8.37007 −0.425475
\(388\) 0.575278 0.0292053
\(389\) 4.12087 0.208937 0.104468 0.994528i \(-0.466686\pi\)
0.104468 + 0.994528i \(0.466686\pi\)
\(390\) −0.251582 −0.0127394
\(391\) −4.46370 −0.225739
\(392\) 6.48202 0.327392
\(393\) 0.498123 0.0251270
\(394\) −16.6893 −0.840793
\(395\) −20.1600 −1.01436
\(396\) 12.0364 0.604850
\(397\) 16.3734 0.821756 0.410878 0.911690i \(-0.365222\pi\)
0.410878 + 0.911690i \(0.365222\pi\)
\(398\) −13.6845 −0.685941
\(399\) −0.000905515 0 −4.53325e−5 0
\(400\) 1.12556 0.0562779
\(401\) −22.5009 −1.12364 −0.561820 0.827260i \(-0.689899\pi\)
−0.561820 + 0.827260i \(0.689899\pi\)
\(402\) 0.844890 0.0421393
\(403\) −8.61963 −0.429374
\(404\) −11.5157 −0.572926
\(405\) 22.1079 1.09855
\(406\) −3.68893 −0.183078
\(407\) −22.7774 −1.12904
\(408\) 0.0763950 0.00378211
\(409\) 14.9244 0.737966 0.368983 0.929436i \(-0.379706\pi\)
0.368983 + 0.929436i \(0.379706\pi\)
\(410\) −17.8307 −0.880596
\(411\) 0.636988 0.0314203
\(412\) 0.875520 0.0431338
\(413\) −10.8282 −0.532820
\(414\) −15.1437 −0.744272
\(415\) −30.4124 −1.49288
\(416\) 1.17365 0.0575430
\(417\) 0.741259 0.0362996
\(418\) 0.0584297 0.00285789
\(419\) −28.6308 −1.39871 −0.699353 0.714776i \(-0.746528\pi\)
−0.699353 + 0.714776i \(0.746528\pi\)
\(420\) −0.154275 −0.00752787
\(421\) 15.6243 0.761480 0.380740 0.924682i \(-0.375669\pi\)
0.380740 + 0.924682i \(0.375669\pi\)
\(422\) 0.344938 0.0167913
\(423\) −18.1524 −0.882602
\(424\) 13.1686 0.639524
\(425\) 0.992806 0.0481582
\(426\) 1.29611 0.0627969
\(427\) −8.26425 −0.399935
\(428\) −2.63094 −0.127171
\(429\) −0.408854 −0.0197397
\(430\) −6.92258 −0.333837
\(431\) 22.3042 1.07436 0.537178 0.843469i \(-0.319490\pi\)
0.537178 + 0.843469i \(0.319490\pi\)
\(432\) 0.519010 0.0249709
\(433\) 35.6307 1.71230 0.856152 0.516725i \(-0.172849\pi\)
0.856152 + 0.516725i \(0.172849\pi\)
\(434\) −5.28574 −0.253724
\(435\) −1.09871 −0.0526793
\(436\) −10.4638 −0.501123
\(437\) −0.0735140 −0.00351665
\(438\) −1.03576 −0.0494907
\(439\) 37.7141 1.80000 0.899999 0.435893i \(-0.143567\pi\)
0.899999 + 0.435893i \(0.143567\pi\)
\(440\) 9.95484 0.474578
\(441\) 19.3974 0.923688
\(442\) 1.03523 0.0492408
\(443\) 18.8613 0.896128 0.448064 0.894001i \(-0.352114\pi\)
0.448064 + 0.894001i \(0.352114\pi\)
\(444\) −0.490469 −0.0232766
\(445\) −44.0103 −2.08629
\(446\) −8.24741 −0.390526
\(447\) −0.276993 −0.0131013
\(448\) 0.719708 0.0340030
\(449\) −33.1802 −1.56587 −0.782935 0.622103i \(-0.786279\pi\)
−0.782935 + 0.622103i \(0.786279\pi\)
\(450\) 3.36823 0.158780
\(451\) −28.9773 −1.36449
\(452\) −7.93666 −0.373309
\(453\) 0.381262 0.0179133
\(454\) −15.9491 −0.748530
\(455\) −2.09059 −0.0980082
\(456\) 0.00125817 5.89193e−5 0
\(457\) 2.62314 0.122705 0.0613526 0.998116i \(-0.480459\pi\)
0.0613526 + 0.998116i \(0.480459\pi\)
\(458\) −17.6910 −0.826647
\(459\) 0.457797 0.0213681
\(460\) −12.5248 −0.583972
\(461\) −12.1494 −0.565852 −0.282926 0.959142i \(-0.591305\pi\)
−0.282926 + 0.959142i \(0.591305\pi\)
\(462\) −0.250718 −0.0116645
\(463\) 9.59647 0.445986 0.222993 0.974820i \(-0.428417\pi\)
0.222993 + 0.974820i \(0.428417\pi\)
\(464\) 5.12559 0.237950
\(465\) −1.57431 −0.0730069
\(466\) −4.52282 −0.209516
\(467\) −29.1248 −1.34773 −0.673867 0.738853i \(-0.735368\pi\)
−0.673867 + 0.738853i \(0.735368\pi\)
\(468\) 3.51215 0.162349
\(469\) 7.02083 0.324192
\(470\) −15.0132 −0.692508
\(471\) 1.95528 0.0900943
\(472\) 15.0453 0.692514
\(473\) −11.2501 −0.517281
\(474\) −0.705481 −0.0324038
\(475\) 0.0163508 0.000750227 0
\(476\) 0.634823 0.0290971
\(477\) 39.4070 1.80432
\(478\) −10.2411 −0.468416
\(479\) −9.68670 −0.442597 −0.221298 0.975206i \(-0.571029\pi\)
−0.221298 + 0.975206i \(0.571029\pi\)
\(480\) 0.214358 0.00978408
\(481\) −6.64634 −0.303047
\(482\) −13.4817 −0.614073
\(483\) 0.315444 0.0143532
\(484\) 5.17793 0.235360
\(485\) 1.42380 0.0646516
\(486\) 2.33068 0.105722
\(487\) 36.6179 1.65931 0.829657 0.558274i \(-0.188536\pi\)
0.829657 + 0.558274i \(0.188536\pi\)
\(488\) 11.4828 0.519801
\(489\) 0.641261 0.0289988
\(490\) 16.0429 0.724745
\(491\) 19.3351 0.872581 0.436291 0.899806i \(-0.356292\pi\)
0.436291 + 0.899806i \(0.356292\pi\)
\(492\) −0.623970 −0.0281307
\(493\) 4.52107 0.203619
\(494\) 0.0170495 0.000767092 0
\(495\) 29.7898 1.33895
\(496\) 7.34428 0.329768
\(497\) 10.7704 0.483118
\(498\) −1.06425 −0.0476903
\(499\) 36.7305 1.64428 0.822141 0.569284i \(-0.192779\pi\)
0.822141 + 0.569284i \(0.192779\pi\)
\(500\) −9.58919 −0.428842
\(501\) 1.90108 0.0849340
\(502\) 13.3357 0.595202
\(503\) 11.9816 0.534235 0.267117 0.963664i \(-0.413929\pi\)
0.267117 + 0.963664i \(0.413929\pi\)
\(504\) 2.15372 0.0959345
\(505\) −28.5011 −1.26828
\(506\) −20.3545 −0.904866
\(507\) 1.00663 0.0447059
\(508\) 10.7592 0.477363
\(509\) 5.65869 0.250817 0.125409 0.992105i \(-0.459976\pi\)
0.125409 + 0.992105i \(0.459976\pi\)
\(510\) 0.189076 0.00837245
\(511\) −8.60695 −0.380749
\(512\) −1.00000 −0.0441942
\(513\) 0.00753959 0.000332881 0
\(514\) 23.0432 1.01639
\(515\) 2.16690 0.0954850
\(516\) −0.242250 −0.0106644
\(517\) −24.3985 −1.07304
\(518\) −4.07567 −0.179075
\(519\) 0.432501 0.0189847
\(520\) 2.90477 0.127383
\(521\) 29.3943 1.28779 0.643894 0.765114i \(-0.277318\pi\)
0.643894 + 0.765114i \(0.277318\pi\)
\(522\) 15.3383 0.671341
\(523\) 33.7248 1.47468 0.737342 0.675520i \(-0.236081\pi\)
0.737342 + 0.675520i \(0.236081\pi\)
\(524\) −5.75133 −0.251248
\(525\) −0.0701603 −0.00306205
\(526\) 17.2883 0.753805
\(527\) 6.47808 0.282190
\(528\) 0.348361 0.0151605
\(529\) 2.60920 0.113443
\(530\) 32.5921 1.41571
\(531\) 45.0229 1.95383
\(532\) 0.0104551 0.000453286 0
\(533\) −8.45542 −0.366245
\(534\) −1.54010 −0.0666467
\(535\) −6.51155 −0.281519
\(536\) −9.75512 −0.421357
\(537\) −0.455184 −0.0196426
\(538\) 10.7977 0.465521
\(539\) 26.0719 1.12299
\(540\) 1.28454 0.0552779
\(541\) 12.0531 0.518201 0.259101 0.965850i \(-0.416574\pi\)
0.259101 + 0.965850i \(0.416574\pi\)
\(542\) 8.91559 0.382957
\(543\) −1.52676 −0.0655195
\(544\) −0.882057 −0.0378179
\(545\) −25.8977 −1.10933
\(546\) −0.0731582 −0.00313088
\(547\) −8.05868 −0.344564 −0.172282 0.985048i \(-0.555114\pi\)
−0.172282 + 0.985048i \(0.555114\pi\)
\(548\) −7.35468 −0.314176
\(549\) 34.3622 1.46654
\(550\) 4.52719 0.193040
\(551\) 0.0744588 0.00317205
\(552\) −0.438294 −0.0186550
\(553\) −5.86237 −0.249293
\(554\) −4.51833 −0.191965
\(555\) −1.21390 −0.0515273
\(556\) −8.55859 −0.362965
\(557\) −33.2645 −1.40946 −0.704731 0.709475i \(-0.748932\pi\)
−0.704731 + 0.709475i \(0.748932\pi\)
\(558\) 21.9778 0.930393
\(559\) −3.28272 −0.138844
\(560\) 1.78127 0.0752722
\(561\) 0.307274 0.0129731
\(562\) −14.9366 −0.630060
\(563\) −19.7919 −0.834131 −0.417065 0.908876i \(-0.636941\pi\)
−0.417065 + 0.908876i \(0.636941\pi\)
\(564\) −0.525374 −0.0221222
\(565\) −19.6431 −0.826392
\(566\) 15.6586 0.658178
\(567\) 6.42882 0.269985
\(568\) −14.9650 −0.627916
\(569\) −12.6053 −0.528443 −0.264222 0.964462i \(-0.585115\pi\)
−0.264222 + 0.964462i \(0.585115\pi\)
\(570\) 0.00311396 0.000130429 0
\(571\) −14.3011 −0.598482 −0.299241 0.954178i \(-0.596733\pi\)
−0.299241 + 0.954178i \(0.596733\pi\)
\(572\) 4.72064 0.197380
\(573\) 0.0249801 0.00104356
\(574\) −5.18504 −0.216419
\(575\) −5.69594 −0.237537
\(576\) −2.99250 −0.124687
\(577\) −4.21658 −0.175539 −0.0877693 0.996141i \(-0.527974\pi\)
−0.0877693 + 0.996141i \(0.527974\pi\)
\(578\) 16.2220 0.674745
\(579\) 0.199737 0.00830077
\(580\) 12.6858 0.526748
\(581\) −8.84369 −0.366898
\(582\) 0.0498248 0.00206530
\(583\) 52.9665 2.19365
\(584\) 11.9589 0.494865
\(585\) 8.69253 0.359392
\(586\) −23.7676 −0.981829
\(587\) 2.92899 0.120892 0.0604462 0.998171i \(-0.480748\pi\)
0.0604462 + 0.998171i \(0.480748\pi\)
\(588\) 0.561408 0.0231521
\(589\) 0.106689 0.00439606
\(590\) 37.2368 1.53302
\(591\) −1.44546 −0.0594581
\(592\) 5.66296 0.232746
\(593\) −33.9955 −1.39603 −0.698013 0.716085i \(-0.745932\pi\)
−0.698013 + 0.716085i \(0.745932\pi\)
\(594\) 2.08755 0.0856533
\(595\) 1.57118 0.0644121
\(596\) 3.19817 0.131002
\(597\) −1.18521 −0.0485075
\(598\) −5.93933 −0.242877
\(599\) −20.2876 −0.828928 −0.414464 0.910066i \(-0.636031\pi\)
−0.414464 + 0.910066i \(0.636031\pi\)
\(600\) 0.0974845 0.00397979
\(601\) −23.9482 −0.976868 −0.488434 0.872601i \(-0.662432\pi\)
−0.488434 + 0.872601i \(0.662432\pi\)
\(602\) −2.01304 −0.0820452
\(603\) −29.1922 −1.18880
\(604\) −4.40206 −0.179117
\(605\) 12.8153 0.521016
\(606\) −0.997372 −0.0405155
\(607\) 11.6635 0.473409 0.236704 0.971582i \(-0.423933\pi\)
0.236704 + 0.971582i \(0.423933\pi\)
\(608\) −0.0145269 −0.000589142 0
\(609\) −0.319498 −0.0129467
\(610\) 28.4197 1.15068
\(611\) −7.11935 −0.288018
\(612\) −2.63956 −0.106698
\(613\) −10.2199 −0.412779 −0.206389 0.978470i \(-0.566171\pi\)
−0.206389 + 0.978470i \(0.566171\pi\)
\(614\) 8.32429 0.335941
\(615\) −1.54432 −0.0622729
\(616\) 2.89479 0.116635
\(617\) −45.4631 −1.83028 −0.915138 0.403140i \(-0.867919\pi\)
−0.915138 + 0.403140i \(0.867919\pi\)
\(618\) 0.0758287 0.00305028
\(619\) −13.1784 −0.529683 −0.264842 0.964292i \(-0.585320\pi\)
−0.264842 + 0.964292i \(0.585320\pi\)
\(620\) 18.1770 0.730006
\(621\) −2.62648 −0.105397
\(622\) 11.8177 0.473845
\(623\) −12.7979 −0.512735
\(624\) 0.101650 0.00406925
\(625\) −29.3609 −1.17444
\(626\) 12.3763 0.494657
\(627\) 0.00506059 0.000202101 0
\(628\) −22.5756 −0.900866
\(629\) 4.99506 0.199166
\(630\) 5.33044 0.212370
\(631\) −6.51397 −0.259317 −0.129658 0.991559i \(-0.541388\pi\)
−0.129658 + 0.991559i \(0.541388\pi\)
\(632\) 8.14549 0.324010
\(633\) 0.0298750 0.00118743
\(634\) −0.620603 −0.0246473
\(635\) 26.6289 1.05674
\(636\) 1.14053 0.0452251
\(637\) 7.60763 0.301425
\(638\) 20.6161 0.816197
\(639\) −44.7826 −1.77157
\(640\) −2.47499 −0.0978324
\(641\) 18.2644 0.721402 0.360701 0.932682i \(-0.382538\pi\)
0.360701 + 0.932682i \(0.382538\pi\)
\(642\) −0.227866 −0.00899314
\(643\) 27.6039 1.08859 0.544296 0.838893i \(-0.316797\pi\)
0.544296 + 0.838893i \(0.316797\pi\)
\(644\) −3.64212 −0.143520
\(645\) −0.599565 −0.0236078
\(646\) −0.0128135 −0.000504142 0
\(647\) 31.8749 1.25313 0.626567 0.779368i \(-0.284460\pi\)
0.626567 + 0.779368i \(0.284460\pi\)
\(648\) −8.93254 −0.350903
\(649\) 60.5148 2.37541
\(650\) 1.32101 0.0518144
\(651\) −0.457797 −0.0179425
\(652\) −7.40401 −0.289964
\(653\) −17.2519 −0.675119 −0.337560 0.941304i \(-0.609601\pi\)
−0.337560 + 0.941304i \(0.609601\pi\)
\(654\) −0.906266 −0.0354378
\(655\) −14.2345 −0.556187
\(656\) 7.20437 0.281283
\(657\) 35.7871 1.39619
\(658\) −4.36573 −0.170194
\(659\) −1.95689 −0.0762295 −0.0381148 0.999273i \(-0.512135\pi\)
−0.0381148 + 0.999273i \(0.512135\pi\)
\(660\) 0.862188 0.0335606
\(661\) 40.5283 1.57637 0.788185 0.615439i \(-0.211021\pi\)
0.788185 + 0.615439i \(0.211021\pi\)
\(662\) 9.37988 0.364560
\(663\) 0.0896611 0.00348215
\(664\) 12.2879 0.476863
\(665\) 0.0258762 0.00100344
\(666\) 16.9464 0.656660
\(667\) −25.9383 −1.00434
\(668\) −21.9499 −0.849267
\(669\) −0.714308 −0.0276167
\(670\) −24.1438 −0.932756
\(671\) 46.1858 1.78298
\(672\) 0.0623338 0.00240458
\(673\) 42.4721 1.63718 0.818590 0.574378i \(-0.194756\pi\)
0.818590 + 0.574378i \(0.194756\pi\)
\(674\) −2.50685 −0.0965602
\(675\) 0.584176 0.0224849
\(676\) −11.6225 −0.447021
\(677\) −5.57011 −0.214077 −0.107038 0.994255i \(-0.534137\pi\)
−0.107038 + 0.994255i \(0.534137\pi\)
\(678\) −0.687394 −0.0263992
\(679\) 0.414032 0.0158891
\(680\) −2.18308 −0.0837173
\(681\) −1.38135 −0.0529336
\(682\) 29.5400 1.13115
\(683\) −5.21261 −0.199455 −0.0997276 0.995015i \(-0.531797\pi\)
−0.0997276 + 0.995015i \(0.531797\pi\)
\(684\) −0.0434716 −0.00166218
\(685\) −18.2027 −0.695490
\(686\) 9.70311 0.370467
\(687\) −1.53222 −0.0584578
\(688\) 2.79702 0.106635
\(689\) 15.4554 0.588802
\(690\) −1.08477 −0.0412966
\(691\) 45.8844 1.74552 0.872762 0.488147i \(-0.162327\pi\)
0.872762 + 0.488147i \(0.162327\pi\)
\(692\) −4.99367 −0.189831
\(693\) 8.66266 0.329067
\(694\) 0.0754958 0.00286578
\(695\) −21.1824 −0.803494
\(696\) 0.443927 0.0168270
\(697\) 6.35467 0.240700
\(698\) −24.9751 −0.945322
\(699\) −0.391721 −0.0148163
\(700\) 0.810072 0.0306178
\(701\) −17.4338 −0.658465 −0.329232 0.944249i \(-0.606790\pi\)
−0.329232 + 0.944249i \(0.606790\pi\)
\(702\) 0.609137 0.0229904
\(703\) 0.0822651 0.00310269
\(704\) −4.02218 −0.151592
\(705\) −1.30029 −0.0489719
\(706\) −8.13318 −0.306096
\(707\) −8.28792 −0.311699
\(708\) 1.30307 0.0489724
\(709\) 2.47271 0.0928646 0.0464323 0.998921i \(-0.485215\pi\)
0.0464323 + 0.998921i \(0.485215\pi\)
\(710\) −37.0381 −1.39001
\(711\) 24.3754 0.914148
\(712\) 17.7820 0.666409
\(713\) −37.1661 −1.39188
\(714\) 0.0549820 0.00205765
\(715\) 11.6835 0.436939
\(716\) 5.25556 0.196410
\(717\) −0.886979 −0.0331248
\(718\) −28.6150 −1.06790
\(719\) 9.37847 0.349758 0.174879 0.984590i \(-0.444047\pi\)
0.174879 + 0.984590i \(0.444047\pi\)
\(720\) −7.40639 −0.276020
\(721\) 0.630118 0.0234668
\(722\) 18.9998 0.707099
\(723\) −1.16765 −0.0434252
\(724\) 17.6280 0.655139
\(725\) 5.76915 0.214261
\(726\) 0.448460 0.0166439
\(727\) −28.2806 −1.04887 −0.524435 0.851450i \(-0.675723\pi\)
−0.524435 + 0.851450i \(0.675723\pi\)
\(728\) 0.844686 0.0313061
\(729\) −26.5958 −0.985029
\(730\) 29.5982 1.09548
\(731\) 2.46713 0.0912501
\(732\) 0.994523 0.0367586
\(733\) 53.3296 1.96977 0.984887 0.173197i \(-0.0554096\pi\)
0.984887 + 0.173197i \(0.0554096\pi\)
\(734\) 9.09315 0.335634
\(735\) 1.38948 0.0512516
\(736\) 5.06055 0.186534
\(737\) −39.2368 −1.44531
\(738\) 21.5591 0.793600
\(739\) −42.0566 −1.54708 −0.773539 0.633748i \(-0.781515\pi\)
−0.773539 + 0.633748i \(0.781515\pi\)
\(740\) 14.0157 0.515229
\(741\) 0.00147665 5.42463e−5 0
\(742\) 9.47755 0.347932
\(743\) 13.0948 0.480400 0.240200 0.970723i \(-0.422787\pi\)
0.240200 + 0.970723i \(0.422787\pi\)
\(744\) 0.636088 0.0233201
\(745\) 7.91542 0.289998
\(746\) 36.6472 1.34175
\(747\) 36.7715 1.34540
\(748\) −3.54779 −0.129720
\(749\) −1.89351 −0.0691873
\(750\) −0.830520 −0.0303263
\(751\) 9.10727 0.332329 0.166165 0.986098i \(-0.446862\pi\)
0.166165 + 0.986098i \(0.446862\pi\)
\(752\) 6.06598 0.221204
\(753\) 1.15501 0.0420908
\(754\) 6.01566 0.219077
\(755\) −10.8950 −0.396511
\(756\) 0.373535 0.0135854
\(757\) 52.9668 1.92511 0.962556 0.271083i \(-0.0873819\pi\)
0.962556 + 0.271083i \(0.0873819\pi\)
\(758\) 18.0899 0.657057
\(759\) −1.76290 −0.0639892
\(760\) −0.0359538 −0.00130418
\(761\) −28.3104 −1.02625 −0.513126 0.858313i \(-0.671513\pi\)
−0.513126 + 0.858313i \(0.671513\pi\)
\(762\) 0.931856 0.0337576
\(763\) −7.53085 −0.272635
\(764\) −0.288421 −0.0104347
\(765\) −6.53286 −0.236196
\(766\) −27.9297 −1.00914
\(767\) 17.6579 0.637590
\(768\) −0.0866100 −0.00312527
\(769\) −51.9878 −1.87473 −0.937365 0.348350i \(-0.886742\pi\)
−0.937365 + 0.348350i \(0.886742\pi\)
\(770\) 7.16457 0.258193
\(771\) 1.99577 0.0718758
\(772\) −2.30616 −0.0830006
\(773\) 13.3517 0.480229 0.240115 0.970745i \(-0.422815\pi\)
0.240115 + 0.970745i \(0.422815\pi\)
\(774\) 8.37007 0.300856
\(775\) 8.26641 0.296938
\(776\) −0.575278 −0.0206513
\(777\) −0.352994 −0.0126636
\(778\) −4.12087 −0.147741
\(779\) 0.104657 0.00374972
\(780\) 0.251582 0.00900809
\(781\) −60.1917 −2.15383
\(782\) 4.46370 0.159622
\(783\) 2.66023 0.0950690
\(784\) −6.48202 −0.231501
\(785\) −55.8744 −1.99424
\(786\) −0.498123 −0.0177674
\(787\) 47.2718 1.68506 0.842529 0.538651i \(-0.181066\pi\)
0.842529 + 0.538651i \(0.181066\pi\)
\(788\) 16.6893 0.594530
\(789\) 1.49734 0.0533066
\(790\) 20.1600 0.717260
\(791\) −5.71207 −0.203098
\(792\) −12.0364 −0.427694
\(793\) 13.4768 0.478575
\(794\) −16.3734 −0.581069
\(795\) 2.82280 0.100115
\(796\) 13.6845 0.485034
\(797\) 18.6392 0.660233 0.330117 0.943940i \(-0.392912\pi\)
0.330117 + 0.943940i \(0.392912\pi\)
\(798\) 0.000905515 0 3.20549e−5 0
\(799\) 5.35054 0.189289
\(800\) −1.12556 −0.0397945
\(801\) 53.2127 1.88018
\(802\) 22.5009 0.794533
\(803\) 48.1010 1.69745
\(804\) −0.844890 −0.0297970
\(805\) −9.01419 −0.317709
\(806\) 8.61963 0.303614
\(807\) 0.935186 0.0329201
\(808\) 11.5157 0.405120
\(809\) 6.00277 0.211046 0.105523 0.994417i \(-0.466348\pi\)
0.105523 + 0.994417i \(0.466348\pi\)
\(810\) −22.1079 −0.776793
\(811\) 33.4766 1.17552 0.587762 0.809034i \(-0.300009\pi\)
0.587762 + 0.809034i \(0.300009\pi\)
\(812\) 3.68893 0.129456
\(813\) 0.772179 0.0270815
\(814\) 22.7774 0.798349
\(815\) −18.3248 −0.641891
\(816\) −0.0763950 −0.00267436
\(817\) 0.0406319 0.00142153
\(818\) −14.9244 −0.521821
\(819\) 2.52772 0.0883257
\(820\) 17.8307 0.622676
\(821\) −3.13609 −0.109450 −0.0547251 0.998501i \(-0.517428\pi\)
−0.0547251 + 0.998501i \(0.517428\pi\)
\(822\) −0.636988 −0.0222175
\(823\) 53.9959 1.88218 0.941090 0.338156i \(-0.109803\pi\)
0.941090 + 0.338156i \(0.109803\pi\)
\(824\) −0.875520 −0.0305002
\(825\) 0.392100 0.0136512
\(826\) 10.8282 0.376761
\(827\) 14.1570 0.492287 0.246143 0.969233i \(-0.420837\pi\)
0.246143 + 0.969233i \(0.420837\pi\)
\(828\) 15.1437 0.526280
\(829\) 11.2896 0.392105 0.196052 0.980593i \(-0.437188\pi\)
0.196052 + 0.980593i \(0.437188\pi\)
\(830\) 30.4124 1.05563
\(831\) −0.391332 −0.0135752
\(832\) −1.17365 −0.0406890
\(833\) −5.71751 −0.198100
\(834\) −0.741259 −0.0256677
\(835\) −54.3257 −1.88002
\(836\) −0.0584297 −0.00202083
\(837\) 3.81176 0.131754
\(838\) 28.6308 0.989035
\(839\) 25.2091 0.870316 0.435158 0.900354i \(-0.356693\pi\)
0.435158 + 0.900354i \(0.356693\pi\)
\(840\) 0.154275 0.00532301
\(841\) −2.72829 −0.0940791
\(842\) −15.6243 −0.538448
\(843\) −1.29365 −0.0445558
\(844\) −0.344938 −0.0118732
\(845\) −28.7656 −0.989568
\(846\) 18.1524 0.624094
\(847\) 3.72659 0.128047
\(848\) −13.1686 −0.452212
\(849\) 1.35619 0.0465442
\(850\) −0.992806 −0.0340530
\(851\) −28.6577 −0.982374
\(852\) −1.29611 −0.0444041
\(853\) 37.2095 1.27403 0.637014 0.770852i \(-0.280169\pi\)
0.637014 + 0.770852i \(0.280169\pi\)
\(854\) 8.26425 0.282797
\(855\) −0.107592 −0.00367956
\(856\) 2.63094 0.0899237
\(857\) −36.3804 −1.24273 −0.621365 0.783521i \(-0.713422\pi\)
−0.621365 + 0.783521i \(0.713422\pi\)
\(858\) 0.408854 0.0139581
\(859\) −30.8665 −1.05315 −0.526575 0.850128i \(-0.676524\pi\)
−0.526575 + 0.850128i \(0.676524\pi\)
\(860\) 6.92258 0.236058
\(861\) −0.449076 −0.0153045
\(862\) −22.3042 −0.759684
\(863\) 5.45571 0.185714 0.0928572 0.995679i \(-0.470400\pi\)
0.0928572 + 0.995679i \(0.470400\pi\)
\(864\) −0.519010 −0.0176571
\(865\) −12.3593 −0.420227
\(866\) −35.6307 −1.21078
\(867\) 1.40498 0.0477158
\(868\) 5.28574 0.179410
\(869\) 32.7626 1.11140
\(870\) 1.09871 0.0372499
\(871\) −11.4491 −0.387938
\(872\) 10.4638 0.354348
\(873\) −1.72152 −0.0582645
\(874\) 0.0735140 0.00248665
\(875\) −6.90141 −0.233310
\(876\) 1.03576 0.0349952
\(877\) −43.4652 −1.46772 −0.733858 0.679303i \(-0.762282\pi\)
−0.733858 + 0.679303i \(0.762282\pi\)
\(878\) −37.7141 −1.27279
\(879\) −2.05851 −0.0694318
\(880\) −9.95484 −0.335578
\(881\) −50.4024 −1.69810 −0.849050 0.528312i \(-0.822825\pi\)
−0.849050 + 0.528312i \(0.822825\pi\)
\(882\) −19.3974 −0.653146
\(883\) −50.0310 −1.68368 −0.841838 0.539730i \(-0.818526\pi\)
−0.841838 + 0.539730i \(0.818526\pi\)
\(884\) −1.03523 −0.0348185
\(885\) 3.22508 0.108410
\(886\) −18.8613 −0.633658
\(887\) 55.8349 1.87475 0.937376 0.348318i \(-0.113247\pi\)
0.937376 + 0.348318i \(0.113247\pi\)
\(888\) 0.490469 0.0164591
\(889\) 7.74349 0.259708
\(890\) 44.0103 1.47523
\(891\) −35.9283 −1.20364
\(892\) 8.24741 0.276144
\(893\) 0.0881197 0.00294881
\(894\) 0.276993 0.00926403
\(895\) 13.0074 0.434791
\(896\) −0.719708 −0.0240437
\(897\) −0.514405 −0.0171755
\(898\) 33.1802 1.10724
\(899\) 37.6438 1.25549
\(900\) −3.36823 −0.112274
\(901\) −11.6155 −0.386967
\(902\) 28.9773 0.964837
\(903\) −0.174349 −0.00580197
\(904\) 7.93666 0.263969
\(905\) 43.6290 1.45028
\(906\) −0.381262 −0.0126666
\(907\) 20.9035 0.694089 0.347044 0.937849i \(-0.387185\pi\)
0.347044 + 0.937849i \(0.387185\pi\)
\(908\) 15.9491 0.529291
\(909\) 34.4606 1.14299
\(910\) 2.09059 0.0693023
\(911\) 45.1890 1.49718 0.748589 0.663034i \(-0.230732\pi\)
0.748589 + 0.663034i \(0.230732\pi\)
\(912\) −0.00125817 −4.16622e−5 0
\(913\) 49.4241 1.63570
\(914\) −2.62314 −0.0867657
\(915\) 2.46143 0.0813724
\(916\) 17.6910 0.584528
\(917\) −4.13928 −0.136691
\(918\) −0.457797 −0.0151095
\(919\) −28.3254 −0.934369 −0.467185 0.884160i \(-0.654732\pi\)
−0.467185 + 0.884160i \(0.654732\pi\)
\(920\) 12.5248 0.412930
\(921\) 0.720966 0.0237566
\(922\) 12.1494 0.400118
\(923\) −17.5636 −0.578114
\(924\) 0.250718 0.00824801
\(925\) 6.37399 0.209575
\(926\) −9.59647 −0.315359
\(927\) −2.61999 −0.0860518
\(928\) −5.12559 −0.168256
\(929\) 6.47936 0.212581 0.106290 0.994335i \(-0.466103\pi\)
0.106290 + 0.994335i \(0.466103\pi\)
\(930\) 1.57431 0.0516236
\(931\) −0.0941635 −0.00308608
\(932\) 4.52282 0.148150
\(933\) 1.02353 0.0335088
\(934\) 29.1248 0.952992
\(935\) −8.78074 −0.287161
\(936\) −3.51215 −0.114798
\(937\) 4.60177 0.150333 0.0751666 0.997171i \(-0.476051\pi\)
0.0751666 + 0.997171i \(0.476051\pi\)
\(938\) −7.02083 −0.229238
\(939\) 1.07191 0.0349805
\(940\) 15.0132 0.489677
\(941\) −6.91735 −0.225499 −0.112750 0.993623i \(-0.535966\pi\)
−0.112750 + 0.993623i \(0.535966\pi\)
\(942\) −1.95528 −0.0637063
\(943\) −36.4581 −1.18724
\(944\) −15.0453 −0.489682
\(945\) 0.924495 0.0300738
\(946\) 11.2501 0.365773
\(947\) −18.6923 −0.607418 −0.303709 0.952765i \(-0.598225\pi\)
−0.303709 + 0.952765i \(0.598225\pi\)
\(948\) 0.705481 0.0229129
\(949\) 14.0356 0.455616
\(950\) −0.0163508 −0.000530491 0
\(951\) −0.0537504 −0.00174298
\(952\) −0.634823 −0.0205747
\(953\) 14.4918 0.469436 0.234718 0.972064i \(-0.424583\pi\)
0.234718 + 0.972064i \(0.424583\pi\)
\(954\) −39.4070 −1.27585
\(955\) −0.713837 −0.0230992
\(956\) 10.2411 0.331220
\(957\) 1.78556 0.0577188
\(958\) 9.68670 0.312963
\(959\) −5.29322 −0.170927
\(960\) −0.214358 −0.00691839
\(961\) 22.9385 0.739952
\(962\) 6.64634 0.214287
\(963\) 7.87309 0.253707
\(964\) 13.4817 0.434215
\(965\) −5.70772 −0.183738
\(966\) −0.315444 −0.0101492
\(967\) 11.9258 0.383508 0.191754 0.981443i \(-0.438582\pi\)
0.191754 + 0.981443i \(0.438582\pi\)
\(968\) −5.17793 −0.166425
\(969\) −0.00110978 −3.56513e−5 0
\(970\) −1.42380 −0.0457156
\(971\) −10.6936 −0.343175 −0.171587 0.985169i \(-0.554890\pi\)
−0.171587 + 0.985169i \(0.554890\pi\)
\(972\) −2.33068 −0.0747565
\(973\) −6.15968 −0.197470
\(974\) −36.6179 −1.17331
\(975\) 0.114413 0.00366414
\(976\) −11.4828 −0.367555
\(977\) 56.5515 1.80924 0.904622 0.426215i \(-0.140153\pi\)
0.904622 + 0.426215i \(0.140153\pi\)
\(978\) −0.641261 −0.0205053
\(979\) 71.5225 2.28587
\(980\) −16.0429 −0.512472
\(981\) 31.3128 0.999741
\(982\) −19.3351 −0.617008
\(983\) −3.58442 −0.114325 −0.0571626 0.998365i \(-0.518205\pi\)
−0.0571626 + 0.998365i \(0.518205\pi\)
\(984\) 0.623970 0.0198914
\(985\) 41.3057 1.31611
\(986\) −4.52107 −0.143980
\(987\) −0.378116 −0.0120356
\(988\) −0.0170495 −0.000542416 0
\(989\) −14.1545 −0.450086
\(990\) −29.7898 −0.946783
\(991\) 12.6564 0.402043 0.201022 0.979587i \(-0.435574\pi\)
0.201022 + 0.979587i \(0.435574\pi\)
\(992\) −7.34428 −0.233181
\(993\) 0.812391 0.0257805
\(994\) −10.7704 −0.341616
\(995\) 33.8689 1.07372
\(996\) 1.06425 0.0337222
\(997\) 53.6616 1.69948 0.849739 0.527203i \(-0.176759\pi\)
0.849739 + 0.527203i \(0.176759\pi\)
\(998\) −36.7305 −1.16268
\(999\) 2.93913 0.0929901
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 4006.2.a.g.1.20 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4006.2.a.g.1.20 40 1.1 even 1 trivial