Properties

Label 4730.2.a.be.1.11
Level 4730
Weight 2
Character 4730.1
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(2.72061\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.72061 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.72061 q^{6} +1.91272 q^{7} +1.00000 q^{8} +4.40172 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.72061 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.72061 q^{6} +1.91272 q^{7} +1.00000 q^{8} +4.40172 q^{9} +1.00000 q^{10} -1.00000 q^{11} +2.72061 q^{12} +1.20875 q^{13} +1.91272 q^{14} +2.72061 q^{15} +1.00000 q^{16} -3.77479 q^{17} +4.40172 q^{18} +5.93546 q^{19} +1.00000 q^{20} +5.20376 q^{21} -1.00000 q^{22} +0.791254 q^{23} +2.72061 q^{24} +1.00000 q^{25} +1.20875 q^{26} +3.81352 q^{27} +1.91272 q^{28} +0.597002 q^{29} +2.72061 q^{30} +7.15695 q^{31} +1.00000 q^{32} -2.72061 q^{33} -3.77479 q^{34} +1.91272 q^{35} +4.40172 q^{36} +1.12735 q^{37} +5.93546 q^{38} +3.28853 q^{39} +1.00000 q^{40} -10.3674 q^{41} +5.20376 q^{42} +1.00000 q^{43} -1.00000 q^{44} +4.40172 q^{45} +0.791254 q^{46} -12.0994 q^{47} +2.72061 q^{48} -3.34151 q^{49} +1.00000 q^{50} -10.2697 q^{51} +1.20875 q^{52} -11.3466 q^{53} +3.81352 q^{54} -1.00000 q^{55} +1.91272 q^{56} +16.1481 q^{57} +0.597002 q^{58} -4.42095 q^{59} +2.72061 q^{60} -2.65531 q^{61} +7.15695 q^{62} +8.41924 q^{63} +1.00000 q^{64} +1.20875 q^{65} -2.72061 q^{66} +15.5544 q^{67} -3.77479 q^{68} +2.15269 q^{69} +1.91272 q^{70} -4.84257 q^{71} +4.40172 q^{72} -2.10242 q^{73} +1.12735 q^{74} +2.72061 q^{75} +5.93546 q^{76} -1.91272 q^{77} +3.28853 q^{78} -0.235281 q^{79} +1.00000 q^{80} -2.83004 q^{81} -10.3674 q^{82} +10.7230 q^{83} +5.20376 q^{84} -3.77479 q^{85} +1.00000 q^{86} +1.62421 q^{87} -1.00000 q^{88} -1.84824 q^{89} +4.40172 q^{90} +2.31199 q^{91} +0.791254 q^{92} +19.4713 q^{93} -12.0994 q^{94} +5.93546 q^{95} +2.72061 q^{96} -0.701016 q^{97} -3.34151 q^{98} -4.40172 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{2} + 3q^{3} + 12q^{4} + 12q^{5} + 3q^{6} + 8q^{7} + 12q^{8} + 25q^{9} + O(q^{10}) \) \( 12q + 12q^{2} + 3q^{3} + 12q^{4} + 12q^{5} + 3q^{6} + 8q^{7} + 12q^{8} + 25q^{9} + 12q^{10} - 12q^{11} + 3q^{12} + 16q^{13} + 8q^{14} + 3q^{15} + 12q^{16} + 18q^{17} + 25q^{18} - 4q^{19} + 12q^{20} + 4q^{21} - 12q^{22} + 8q^{23} + 3q^{24} + 12q^{25} + 16q^{26} + 6q^{27} + 8q^{28} + 20q^{29} + 3q^{30} + 5q^{31} + 12q^{32} - 3q^{33} + 18q^{34} + 8q^{35} + 25q^{36} + 19q^{37} - 4q^{38} + 6q^{39} + 12q^{40} + 16q^{41} + 4q^{42} + 12q^{43} - 12q^{44} + 25q^{45} + 8q^{46} - q^{47} + 3q^{48} + 52q^{49} + 12q^{50} + q^{51} + 16q^{52} + 11q^{53} + 6q^{54} - 12q^{55} + 8q^{56} + 9q^{57} + 20q^{58} - 11q^{59} + 3q^{60} + 18q^{61} + 5q^{62} + 15q^{63} + 12q^{64} + 16q^{65} - 3q^{66} - 10q^{67} + 18q^{68} + 8q^{70} - 2q^{71} + 25q^{72} + 29q^{73} + 19q^{74} + 3q^{75} - 4q^{76} - 8q^{77} + 6q^{78} + 2q^{79} + 12q^{80} - 8q^{81} + 16q^{82} + 26q^{83} + 4q^{84} + 18q^{85} + 12q^{86} - 4q^{87} - 12q^{88} + 41q^{89} + 25q^{90} - 4q^{91} + 8q^{92} + 5q^{93} - q^{94} - 4q^{95} + 3q^{96} - 7q^{97} + 52q^{98} - 25q^{99} + O(q^{100}) \)

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
\(3\) 2.72061 1.57074 0.785372 0.619024i \(-0.212471\pi\)
0.785372 + 0.619024i \(0.212471\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.72061 1.11068
\(7\) 1.91272 0.722939 0.361469 0.932384i \(-0.382275\pi\)
0.361469 + 0.932384i \(0.382275\pi\)
\(8\) 1.00000 0.353553
\(9\) 4.40172 1.46724
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) 2.72061 0.785372
\(13\) 1.20875 0.335246 0.167623 0.985851i \(-0.446391\pi\)
0.167623 + 0.985851i \(0.446391\pi\)
\(14\) 1.91272 0.511195
\(15\) 2.72061 0.702458
\(16\) 1.00000 0.250000
\(17\) −3.77479 −0.915521 −0.457760 0.889076i \(-0.651348\pi\)
−0.457760 + 0.889076i \(0.651348\pi\)
\(18\) 4.40172 1.03749
\(19\) 5.93546 1.36169 0.680844 0.732428i \(-0.261613\pi\)
0.680844 + 0.732428i \(0.261613\pi\)
\(20\) 1.00000 0.223607
\(21\) 5.20376 1.13555
\(22\) −1.00000 −0.213201
\(23\) 0.791254 0.164988 0.0824939 0.996592i \(-0.473711\pi\)
0.0824939 + 0.996592i \(0.473711\pi\)
\(24\) 2.72061 0.555342
\(25\) 1.00000 0.200000
\(26\) 1.20875 0.237055
\(27\) 3.81352 0.733913
\(28\) 1.91272 0.361469
\(29\) 0.597002 0.110860 0.0554302 0.998463i \(-0.482347\pi\)
0.0554302 + 0.998463i \(0.482347\pi\)
\(30\) 2.72061 0.496713
\(31\) 7.15695 1.28543 0.642713 0.766107i \(-0.277809\pi\)
0.642713 + 0.766107i \(0.277809\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.72061 −0.473597
\(34\) −3.77479 −0.647371
\(35\) 1.91272 0.323308
\(36\) 4.40172 0.733619
\(37\) 1.12735 0.185335 0.0926676 0.995697i \(-0.470461\pi\)
0.0926676 + 0.995697i \(0.470461\pi\)
\(38\) 5.93546 0.962859
\(39\) 3.28853 0.526586
\(40\) 1.00000 0.158114
\(41\) −10.3674 −1.61912 −0.809558 0.587039i \(-0.800293\pi\)
−0.809558 + 0.587039i \(0.800293\pi\)
\(42\) 5.20376 0.802957
\(43\) 1.00000 0.152499
\(44\) −1.00000 −0.150756
\(45\) 4.40172 0.656169
\(46\) 0.791254 0.116664
\(47\) −12.0994 −1.76488 −0.882439 0.470426i \(-0.844100\pi\)
−0.882439 + 0.470426i \(0.844100\pi\)
\(48\) 2.72061 0.392686
\(49\) −3.34151 −0.477359
\(50\) 1.00000 0.141421
\(51\) −10.2697 −1.43805
\(52\) 1.20875 0.167623
\(53\) −11.3466 −1.55857 −0.779287 0.626667i \(-0.784419\pi\)
−0.779287 + 0.626667i \(0.784419\pi\)
\(54\) 3.81352 0.518955
\(55\) −1.00000 −0.134840
\(56\) 1.91272 0.255598
\(57\) 16.1481 2.13887
\(58\) 0.597002 0.0783902
\(59\) −4.42095 −0.575558 −0.287779 0.957697i \(-0.592917\pi\)
−0.287779 + 0.957697i \(0.592917\pi\)
\(60\) 2.72061 0.351229
\(61\) −2.65531 −0.339977 −0.169989 0.985446i \(-0.554373\pi\)
−0.169989 + 0.985446i \(0.554373\pi\)
\(62\) 7.15695 0.908933
\(63\) 8.41924 1.06072
\(64\) 1.00000 0.125000
\(65\) 1.20875 0.149926
\(66\) −2.72061 −0.334884
\(67\) 15.5544 1.90027 0.950134 0.311841i \(-0.100946\pi\)
0.950134 + 0.311841i \(0.100946\pi\)
\(68\) −3.77479 −0.457760
\(69\) 2.15269 0.259154
\(70\) 1.91272 0.228613
\(71\) −4.84257 −0.574708 −0.287354 0.957825i \(-0.592776\pi\)
−0.287354 + 0.957825i \(0.592776\pi\)
\(72\) 4.40172 0.518747
\(73\) −2.10242 −0.246069 −0.123035 0.992402i \(-0.539263\pi\)
−0.123035 + 0.992402i \(0.539263\pi\)
\(74\) 1.12735 0.131052
\(75\) 2.72061 0.314149
\(76\) 5.93546 0.680844
\(77\) −1.91272 −0.217974
\(78\) 3.28853 0.372352
\(79\) −0.235281 −0.0264711 −0.0132356 0.999912i \(-0.504213\pi\)
−0.0132356 + 0.999912i \(0.504213\pi\)
\(80\) 1.00000 0.111803
\(81\) −2.83004 −0.314449
\(82\) −10.3674 −1.14489
\(83\) 10.7230 1.17700 0.588499 0.808498i \(-0.299719\pi\)
0.588499 + 0.808498i \(0.299719\pi\)
\(84\) 5.20376 0.567776
\(85\) −3.77479 −0.409433
\(86\) 1.00000 0.107833
\(87\) 1.62421 0.174133
\(88\) −1.00000 −0.106600
\(89\) −1.84824 −0.195913 −0.0979566 0.995191i \(-0.531231\pi\)
−0.0979566 + 0.995191i \(0.531231\pi\)
\(90\) 4.40172 0.463982
\(91\) 2.31199 0.242362
\(92\) 0.791254 0.0824939
\(93\) 19.4713 2.01908
\(94\) −12.0994 −1.24796
\(95\) 5.93546 0.608966
\(96\) 2.72061 0.277671
\(97\) −0.701016 −0.0711774 −0.0355887 0.999367i \(-0.511331\pi\)
−0.0355887 + 0.999367i \(0.511331\pi\)
\(98\) −3.34151 −0.337544
\(99\) −4.40172 −0.442389
\(100\) 1.00000 0.100000
\(101\) −3.90878 −0.388938 −0.194469 0.980909i \(-0.562298\pi\)
−0.194469 + 0.980909i \(0.562298\pi\)
\(102\) −10.2697 −1.01685
\(103\) 6.85994 0.675930 0.337965 0.941159i \(-0.390261\pi\)
0.337965 + 0.941159i \(0.390261\pi\)
\(104\) 1.20875 0.118527
\(105\) 5.20376 0.507835
\(106\) −11.3466 −1.10208
\(107\) 11.6856 1.12969 0.564844 0.825198i \(-0.308936\pi\)
0.564844 + 0.825198i \(0.308936\pi\)
\(108\) 3.81352 0.366957
\(109\) 15.9177 1.52464 0.762321 0.647200i \(-0.224060\pi\)
0.762321 + 0.647200i \(0.224060\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 3.06708 0.291114
\(112\) 1.91272 0.180735
\(113\) 2.03490 0.191427 0.0957137 0.995409i \(-0.469487\pi\)
0.0957137 + 0.995409i \(0.469487\pi\)
\(114\) 16.1481 1.51241
\(115\) 0.791254 0.0737848
\(116\) 0.597002 0.0554302
\(117\) 5.32056 0.491886
\(118\) −4.42095 −0.406981
\(119\) −7.22010 −0.661866
\(120\) 2.72061 0.248357
\(121\) 1.00000 0.0909091
\(122\) −2.65531 −0.240400
\(123\) −28.2057 −2.54322
\(124\) 7.15695 0.642713
\(125\) 1.00000 0.0894427
\(126\) 8.41924 0.750045
\(127\) 8.31619 0.737942 0.368971 0.929441i \(-0.379710\pi\)
0.368971 + 0.929441i \(0.379710\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.72061 0.239536
\(130\) 1.20875 0.106014
\(131\) −8.39059 −0.733090 −0.366545 0.930400i \(-0.619459\pi\)
−0.366545 + 0.930400i \(0.619459\pi\)
\(132\) −2.72061 −0.236799
\(133\) 11.3529 0.984418
\(134\) 15.5544 1.34369
\(135\) 3.81352 0.328216
\(136\) −3.77479 −0.323686
\(137\) −15.8708 −1.35593 −0.677967 0.735092i \(-0.737139\pi\)
−0.677967 + 0.735092i \(0.737139\pi\)
\(138\) 2.15269 0.183249
\(139\) 9.81204 0.832246 0.416123 0.909308i \(-0.363389\pi\)
0.416123 + 0.909308i \(0.363389\pi\)
\(140\) 1.91272 0.161654
\(141\) −32.9177 −2.77217
\(142\) −4.84257 −0.406380
\(143\) −1.20875 −0.101080
\(144\) 4.40172 0.366810
\(145\) 0.597002 0.0495783
\(146\) −2.10242 −0.173997
\(147\) −9.09096 −0.749809
\(148\) 1.12735 0.0926676
\(149\) −3.06332 −0.250957 −0.125478 0.992096i \(-0.540047\pi\)
−0.125478 + 0.992096i \(0.540047\pi\)
\(150\) 2.72061 0.222137
\(151\) −3.50114 −0.284919 −0.142460 0.989801i \(-0.545501\pi\)
−0.142460 + 0.989801i \(0.545501\pi\)
\(152\) 5.93546 0.481430
\(153\) −16.6156 −1.34329
\(154\) −1.91272 −0.154131
\(155\) 7.15695 0.574860
\(156\) 3.28853 0.263293
\(157\) −6.52120 −0.520448 −0.260224 0.965548i \(-0.583796\pi\)
−0.260224 + 0.965548i \(0.583796\pi\)
\(158\) −0.235281 −0.0187179
\(159\) −30.8696 −2.44812
\(160\) 1.00000 0.0790569
\(161\) 1.51345 0.119276
\(162\) −2.83004 −0.222349
\(163\) −0.0708121 −0.00554643 −0.00277321 0.999996i \(-0.500883\pi\)
−0.00277321 + 0.999996i \(0.500883\pi\)
\(164\) −10.3674 −0.809558
\(165\) −2.72061 −0.211799
\(166\) 10.7230 0.832263
\(167\) −9.52120 −0.736773 −0.368386 0.929673i \(-0.620090\pi\)
−0.368386 + 0.929673i \(0.620090\pi\)
\(168\) 5.20376 0.401478
\(169\) −11.5389 −0.887610
\(170\) −3.77479 −0.289513
\(171\) 26.1262 1.99792
\(172\) 1.00000 0.0762493
\(173\) −10.1851 −0.774359 −0.387180 0.922004i \(-0.626551\pi\)
−0.387180 + 0.922004i \(0.626551\pi\)
\(174\) 1.62421 0.123131
\(175\) 1.91272 0.144588
\(176\) −1.00000 −0.0753778
\(177\) −12.0277 −0.904055
\(178\) −1.84824 −0.138532
\(179\) 16.5986 1.24063 0.620317 0.784351i \(-0.287004\pi\)
0.620317 + 0.784351i \(0.287004\pi\)
\(180\) 4.40172 0.328085
\(181\) 12.9709 0.964119 0.482060 0.876138i \(-0.339889\pi\)
0.482060 + 0.876138i \(0.339889\pi\)
\(182\) 2.31199 0.171376
\(183\) −7.22405 −0.534017
\(184\) 0.791254 0.0583320
\(185\) 1.12735 0.0828844
\(186\) 19.4713 1.42770
\(187\) 3.77479 0.276040
\(188\) −12.0994 −0.882439
\(189\) 7.29419 0.530574
\(190\) 5.93546 0.430604
\(191\) −4.33138 −0.313408 −0.156704 0.987646i \(-0.550087\pi\)
−0.156704 + 0.987646i \(0.550087\pi\)
\(192\) 2.72061 0.196343
\(193\) 1.94220 0.139802 0.0699012 0.997554i \(-0.477732\pi\)
0.0699012 + 0.997554i \(0.477732\pi\)
\(194\) −0.701016 −0.0503300
\(195\) 3.28853 0.235496
\(196\) −3.34151 −0.238680
\(197\) 7.27531 0.518344 0.259172 0.965831i \(-0.416550\pi\)
0.259172 + 0.965831i \(0.416550\pi\)
\(198\) −4.40172 −0.312816
\(199\) −12.3540 −0.875749 −0.437875 0.899036i \(-0.644269\pi\)
−0.437875 + 0.899036i \(0.644269\pi\)
\(200\) 1.00000 0.0707107
\(201\) 42.3174 2.98484
\(202\) −3.90878 −0.275021
\(203\) 1.14190 0.0801453
\(204\) −10.2697 −0.719025
\(205\) −10.3674 −0.724091
\(206\) 6.85994 0.477955
\(207\) 3.48288 0.242077
\(208\) 1.20875 0.0838114
\(209\) −5.93546 −0.410565
\(210\) 5.20376 0.359093
\(211\) −5.49748 −0.378462 −0.189231 0.981933i \(-0.560599\pi\)
−0.189231 + 0.981933i \(0.560599\pi\)
\(212\) −11.3466 −0.779287
\(213\) −13.1747 −0.902719
\(214\) 11.6856 0.798810
\(215\) 1.00000 0.0681994
\(216\) 3.81352 0.259477
\(217\) 13.6892 0.929285
\(218\) 15.9177 1.07808
\(219\) −5.71986 −0.386512
\(220\) −1.00000 −0.0674200
\(221\) −4.56276 −0.306925
\(222\) 3.06708 0.205849
\(223\) −19.9298 −1.33460 −0.667298 0.744791i \(-0.732549\pi\)
−0.667298 + 0.744791i \(0.732549\pi\)
\(224\) 1.91272 0.127799
\(225\) 4.40172 0.293448
\(226\) 2.03490 0.135360
\(227\) 7.77896 0.516307 0.258154 0.966104i \(-0.416886\pi\)
0.258154 + 0.966104i \(0.416886\pi\)
\(228\) 16.1481 1.06943
\(229\) −8.96590 −0.592483 −0.296242 0.955113i \(-0.595733\pi\)
−0.296242 + 0.955113i \(0.595733\pi\)
\(230\) 0.791254 0.0521738
\(231\) −5.20376 −0.342382
\(232\) 0.597002 0.0391951
\(233\) 16.7903 1.09997 0.549984 0.835175i \(-0.314634\pi\)
0.549984 + 0.835175i \(0.314634\pi\)
\(234\) 5.32056 0.347816
\(235\) −12.0994 −0.789278
\(236\) −4.42095 −0.287779
\(237\) −0.640107 −0.0415794
\(238\) −7.22010 −0.468010
\(239\) −8.02935 −0.519375 −0.259688 0.965693i \(-0.583620\pi\)
−0.259688 + 0.965693i \(0.583620\pi\)
\(240\) 2.72061 0.175615
\(241\) −1.12866 −0.0727034 −0.0363517 0.999339i \(-0.511574\pi\)
−0.0363517 + 0.999339i \(0.511574\pi\)
\(242\) 1.00000 0.0642824
\(243\) −19.1400 −1.22783
\(244\) −2.65531 −0.169989
\(245\) −3.34151 −0.213482
\(246\) −28.2057 −1.79833
\(247\) 7.17447 0.456500
\(248\) 7.15695 0.454467
\(249\) 29.1730 1.84876
\(250\) 1.00000 0.0632456
\(251\) −7.67613 −0.484513 −0.242256 0.970212i \(-0.577888\pi\)
−0.242256 + 0.970212i \(0.577888\pi\)
\(252\) 8.41924 0.530362
\(253\) −0.791254 −0.0497457
\(254\) 8.31619 0.521804
\(255\) −10.2697 −0.643115
\(256\) 1.00000 0.0625000
\(257\) −9.37594 −0.584855 −0.292428 0.956288i \(-0.594463\pi\)
−0.292428 + 0.956288i \(0.594463\pi\)
\(258\) 2.72061 0.169378
\(259\) 2.15630 0.133986
\(260\) 1.20875 0.0749632
\(261\) 2.62783 0.162659
\(262\) −8.39059 −0.518373
\(263\) 19.6649 1.21259 0.606296 0.795239i \(-0.292655\pi\)
0.606296 + 0.795239i \(0.292655\pi\)
\(264\) −2.72061 −0.167442
\(265\) −11.3466 −0.697016
\(266\) 11.3529 0.696089
\(267\) −5.02834 −0.307730
\(268\) 15.5544 0.950134
\(269\) 2.38473 0.145400 0.0726999 0.997354i \(-0.476838\pi\)
0.0726999 + 0.997354i \(0.476838\pi\)
\(270\) 3.81352 0.232084
\(271\) −4.10433 −0.249320 −0.124660 0.992199i \(-0.539784\pi\)
−0.124660 + 0.992199i \(0.539784\pi\)
\(272\) −3.77479 −0.228880
\(273\) 6.29002 0.380689
\(274\) −15.8708 −0.958791
\(275\) −1.00000 −0.0603023
\(276\) 2.15269 0.129577
\(277\) −27.0052 −1.62258 −0.811291 0.584642i \(-0.801235\pi\)
−0.811291 + 0.584642i \(0.801235\pi\)
\(278\) 9.81204 0.588487
\(279\) 31.5029 1.88603
\(280\) 1.91272 0.114307
\(281\) −1.83659 −0.109562 −0.0547808 0.998498i \(-0.517446\pi\)
−0.0547808 + 0.998498i \(0.517446\pi\)
\(282\) −32.9177 −1.96022
\(283\) −2.44280 −0.145209 −0.0726046 0.997361i \(-0.523131\pi\)
−0.0726046 + 0.997361i \(0.523131\pi\)
\(284\) −4.84257 −0.287354
\(285\) 16.1481 0.956530
\(286\) −1.20875 −0.0714746
\(287\) −19.8299 −1.17052
\(288\) 4.40172 0.259374
\(289\) −2.75097 −0.161822
\(290\) 0.597002 0.0350571
\(291\) −1.90719 −0.111801
\(292\) −2.10242 −0.123035
\(293\) −7.62521 −0.445469 −0.222735 0.974879i \(-0.571498\pi\)
−0.222735 + 0.974879i \(0.571498\pi\)
\(294\) −9.09096 −0.530195
\(295\) −4.42095 −0.257397
\(296\) 1.12735 0.0655259
\(297\) −3.81352 −0.221283
\(298\) −3.06332 −0.177453
\(299\) 0.956425 0.0553115
\(300\) 2.72061 0.157074
\(301\) 1.91272 0.110247
\(302\) −3.50114 −0.201468
\(303\) −10.6343 −0.610923
\(304\) 5.93546 0.340422
\(305\) −2.65531 −0.152042
\(306\) −16.6156 −0.949848
\(307\) −10.9760 −0.626434 −0.313217 0.949682i \(-0.601407\pi\)
−0.313217 + 0.949682i \(0.601407\pi\)
\(308\) −1.91272 −0.108987
\(309\) 18.6632 1.06171
\(310\) 7.15695 0.406487
\(311\) 8.53162 0.483784 0.241892 0.970303i \(-0.422232\pi\)
0.241892 + 0.970303i \(0.422232\pi\)
\(312\) 3.28853 0.186176
\(313\) −10.9279 −0.617682 −0.308841 0.951114i \(-0.599941\pi\)
−0.308841 + 0.951114i \(0.599941\pi\)
\(314\) −6.52120 −0.368013
\(315\) 8.41924 0.474370
\(316\) −0.235281 −0.0132356
\(317\) 5.05291 0.283800 0.141900 0.989881i \(-0.454679\pi\)
0.141900 + 0.989881i \(0.454679\pi\)
\(318\) −30.8696 −1.73108
\(319\) −0.597002 −0.0334257
\(320\) 1.00000 0.0559017
\(321\) 31.7919 1.77445
\(322\) 1.51345 0.0843410
\(323\) −22.4051 −1.24665
\(324\) −2.83004 −0.157224
\(325\) 1.20875 0.0670492
\(326\) −0.0708121 −0.00392192
\(327\) 43.3059 2.39482
\(328\) −10.3674 −0.572444
\(329\) −23.1427 −1.27590
\(330\) −2.72061 −0.149765
\(331\) 31.7736 1.74644 0.873218 0.487329i \(-0.162029\pi\)
0.873218 + 0.487329i \(0.162029\pi\)
\(332\) 10.7230 0.588499
\(333\) 4.96227 0.271931
\(334\) −9.52120 −0.520977
\(335\) 15.5544 0.849826
\(336\) 5.20376 0.283888
\(337\) 16.5299 0.900441 0.450221 0.892917i \(-0.351345\pi\)
0.450221 + 0.892917i \(0.351345\pi\)
\(338\) −11.5389 −0.627635
\(339\) 5.53617 0.300683
\(340\) −3.77479 −0.204717
\(341\) −7.15695 −0.387571
\(342\) 26.1262 1.41274
\(343\) −19.7804 −1.06804
\(344\) 1.00000 0.0539164
\(345\) 2.15269 0.115897
\(346\) −10.1851 −0.547555
\(347\) 2.73720 0.146941 0.0734703 0.997297i \(-0.476593\pi\)
0.0734703 + 0.997297i \(0.476593\pi\)
\(348\) 1.62421 0.0870667
\(349\) 14.0391 0.751498 0.375749 0.926722i \(-0.377386\pi\)
0.375749 + 0.926722i \(0.377386\pi\)
\(350\) 1.91272 0.102239
\(351\) 4.60958 0.246041
\(352\) −1.00000 −0.0533002
\(353\) −31.8160 −1.69340 −0.846698 0.532074i \(-0.821413\pi\)
−0.846698 + 0.532074i \(0.821413\pi\)
\(354\) −12.0277 −0.639263
\(355\) −4.84257 −0.257017
\(356\) −1.84824 −0.0979566
\(357\) −19.6431 −1.03962
\(358\) 16.5986 0.877261
\(359\) −3.43246 −0.181158 −0.0905792 0.995889i \(-0.528872\pi\)
−0.0905792 + 0.995889i \(0.528872\pi\)
\(360\) 4.40172 0.231991
\(361\) 16.2297 0.854197
\(362\) 12.9709 0.681735
\(363\) 2.72061 0.142795
\(364\) 2.31199 0.121181
\(365\) −2.10242 −0.110046
\(366\) −7.22405 −0.377607
\(367\) 30.7599 1.60566 0.802828 0.596211i \(-0.203328\pi\)
0.802828 + 0.596211i \(0.203328\pi\)
\(368\) 0.791254 0.0412470
\(369\) −45.6344 −2.37563
\(370\) 1.12735 0.0586081
\(371\) −21.7028 −1.12675
\(372\) 19.4713 1.00954
\(373\) 26.0052 1.34650 0.673250 0.739415i \(-0.264898\pi\)
0.673250 + 0.739415i \(0.264898\pi\)
\(374\) 3.77479 0.195190
\(375\) 2.72061 0.140492
\(376\) −12.0994 −0.623979
\(377\) 0.721623 0.0371655
\(378\) 7.29419 0.375173
\(379\) −18.6814 −0.959601 −0.479800 0.877378i \(-0.659291\pi\)
−0.479800 + 0.877378i \(0.659291\pi\)
\(380\) 5.93546 0.304483
\(381\) 22.6251 1.15912
\(382\) −4.33138 −0.221613
\(383\) 17.2652 0.882212 0.441106 0.897455i \(-0.354586\pi\)
0.441106 + 0.897455i \(0.354586\pi\)
\(384\) 2.72061 0.138836
\(385\) −1.91272 −0.0974811
\(386\) 1.94220 0.0988552
\(387\) 4.40172 0.223752
\(388\) −0.701016 −0.0355887
\(389\) 11.3509 0.575513 0.287757 0.957704i \(-0.407091\pi\)
0.287757 + 0.957704i \(0.407091\pi\)
\(390\) 3.28853 0.166521
\(391\) −2.98682 −0.151050
\(392\) −3.34151 −0.168772
\(393\) −22.8275 −1.15150
\(394\) 7.27531 0.366525
\(395\) −0.235281 −0.0118383
\(396\) −4.40172 −0.221195
\(397\) 33.4940 1.68101 0.840507 0.541801i \(-0.182257\pi\)
0.840507 + 0.541801i \(0.182257\pi\)
\(398\) −12.3540 −0.619248
\(399\) 30.8867 1.54627
\(400\) 1.00000 0.0500000
\(401\) −10.6671 −0.532691 −0.266345 0.963878i \(-0.585816\pi\)
−0.266345 + 0.963878i \(0.585816\pi\)
\(402\) 42.3174 2.11060
\(403\) 8.65093 0.430934
\(404\) −3.90878 −0.194469
\(405\) −2.83004 −0.140626
\(406\) 1.14190 0.0566713
\(407\) −1.12735 −0.0558806
\(408\) −10.2697 −0.508427
\(409\) −5.64176 −0.278967 −0.139483 0.990224i \(-0.544544\pi\)
−0.139483 + 0.990224i \(0.544544\pi\)
\(410\) −10.3674 −0.512010
\(411\) −43.1783 −2.12983
\(412\) 6.85994 0.337965
\(413\) −8.45602 −0.416093
\(414\) 3.48288 0.171174
\(415\) 10.7230 0.526369
\(416\) 1.20875 0.0592636
\(417\) 26.6947 1.30725
\(418\) −5.93546 −0.290313
\(419\) −12.2933 −0.600566 −0.300283 0.953850i \(-0.597081\pi\)
−0.300283 + 0.953850i \(0.597081\pi\)
\(420\) 5.20376 0.253917
\(421\) 9.99247 0.487003 0.243502 0.969900i \(-0.421704\pi\)
0.243502 + 0.969900i \(0.421704\pi\)
\(422\) −5.49748 −0.267613
\(423\) −53.2581 −2.58950
\(424\) −11.3466 −0.551039
\(425\) −3.77479 −0.183104
\(426\) −13.1747 −0.638319
\(427\) −5.07885 −0.245783
\(428\) 11.6856 0.564844
\(429\) −3.28853 −0.158772
\(430\) 1.00000 0.0482243
\(431\) −20.3287 −0.979200 −0.489600 0.871947i \(-0.662857\pi\)
−0.489600 + 0.871947i \(0.662857\pi\)
\(432\) 3.81352 0.183478
\(433\) −33.7115 −1.62007 −0.810035 0.586381i \(-0.800552\pi\)
−0.810035 + 0.586381i \(0.800552\pi\)
\(434\) 13.6892 0.657103
\(435\) 1.62421 0.0778748
\(436\) 15.9177 0.762321
\(437\) 4.69646 0.224662
\(438\) −5.71986 −0.273305
\(439\) −8.79633 −0.419826 −0.209913 0.977720i \(-0.567318\pi\)
−0.209913 + 0.977720i \(0.567318\pi\)
\(440\) −1.00000 −0.0476731
\(441\) −14.7084 −0.700400
\(442\) −4.56276 −0.217028
\(443\) −20.3959 −0.969040 −0.484520 0.874780i \(-0.661006\pi\)
−0.484520 + 0.874780i \(0.661006\pi\)
\(444\) 3.06708 0.145557
\(445\) −1.84824 −0.0876150
\(446\) −19.9298 −0.943702
\(447\) −8.33409 −0.394189
\(448\) 1.91272 0.0903674
\(449\) 23.4319 1.10582 0.552910 0.833241i \(-0.313517\pi\)
0.552910 + 0.833241i \(0.313517\pi\)
\(450\) 4.40172 0.207499
\(451\) 10.3674 0.488182
\(452\) 2.03490 0.0957137
\(453\) −9.52525 −0.447535
\(454\) 7.77896 0.365084
\(455\) 2.31199 0.108388
\(456\) 16.1481 0.756203
\(457\) −19.0784 −0.892450 −0.446225 0.894921i \(-0.647232\pi\)
−0.446225 + 0.894921i \(0.647232\pi\)
\(458\) −8.96590 −0.418949
\(459\) −14.3953 −0.671913
\(460\) 0.791254 0.0368924
\(461\) 34.5032 1.60698 0.803488 0.595321i \(-0.202975\pi\)
0.803488 + 0.595321i \(0.202975\pi\)
\(462\) −5.20376 −0.242101
\(463\) −32.9370 −1.53071 −0.765356 0.643607i \(-0.777437\pi\)
−0.765356 + 0.643607i \(0.777437\pi\)
\(464\) 0.597002 0.0277151
\(465\) 19.4713 0.902958
\(466\) 16.7903 0.777794
\(467\) 36.3551 1.68231 0.841156 0.540792i \(-0.181875\pi\)
0.841156 + 0.540792i \(0.181875\pi\)
\(468\) 5.32056 0.245943
\(469\) 29.7511 1.37378
\(470\) −12.0994 −0.558104
\(471\) −17.7416 −0.817492
\(472\) −4.42095 −0.203490
\(473\) −1.00000 −0.0459800
\(474\) −0.640107 −0.0294011
\(475\) 5.93546 0.272338
\(476\) −7.22010 −0.330933
\(477\) −49.9445 −2.28680
\(478\) −8.02935 −0.367254
\(479\) 8.98706 0.410629 0.205315 0.978696i \(-0.434178\pi\)
0.205315 + 0.978696i \(0.434178\pi\)
\(480\) 2.72061 0.124178
\(481\) 1.36268 0.0621328
\(482\) −1.12866 −0.0514090
\(483\) 4.11749 0.187352
\(484\) 1.00000 0.0454545
\(485\) −0.701016 −0.0318315
\(486\) −19.1400 −0.868208
\(487\) −6.66801 −0.302156 −0.151078 0.988522i \(-0.548275\pi\)
−0.151078 + 0.988522i \(0.548275\pi\)
\(488\) −2.65531 −0.120200
\(489\) −0.192652 −0.00871202
\(490\) −3.34151 −0.150954
\(491\) 27.9655 1.26206 0.631032 0.775757i \(-0.282632\pi\)
0.631032 + 0.775757i \(0.282632\pi\)
\(492\) −28.2057 −1.27161
\(493\) −2.25356 −0.101495
\(494\) 7.17447 0.322795
\(495\) −4.40172 −0.197842
\(496\) 7.15695 0.321357
\(497\) −9.26247 −0.415479
\(498\) 29.1730 1.30727
\(499\) −27.9426 −1.25088 −0.625442 0.780271i \(-0.715081\pi\)
−0.625442 + 0.780271i \(0.715081\pi\)
\(500\) 1.00000 0.0447214
\(501\) −25.9035 −1.15728
\(502\) −7.67613 −0.342602
\(503\) −9.30955 −0.415092 −0.207546 0.978225i \(-0.566548\pi\)
−0.207546 + 0.978225i \(0.566548\pi\)
\(504\) 8.41924 0.375023
\(505\) −3.90878 −0.173938
\(506\) −0.791254 −0.0351755
\(507\) −31.3929 −1.39421
\(508\) 8.31619 0.368971
\(509\) −2.82087 −0.125033 −0.0625165 0.998044i \(-0.519913\pi\)
−0.0625165 + 0.998044i \(0.519913\pi\)
\(510\) −10.2697 −0.454751
\(511\) −4.02133 −0.177893
\(512\) 1.00000 0.0441942
\(513\) 22.6350 0.999361
\(514\) −9.37594 −0.413555
\(515\) 6.85994 0.302285
\(516\) 2.72061 0.119768
\(517\) 12.0994 0.532131
\(518\) 2.15630 0.0947424
\(519\) −27.7097 −1.21632
\(520\) 1.20875 0.0530070
\(521\) −17.0892 −0.748692 −0.374346 0.927289i \(-0.622133\pi\)
−0.374346 + 0.927289i \(0.622133\pi\)
\(522\) 2.62783 0.115017
\(523\) −0.934960 −0.0408830 −0.0204415 0.999791i \(-0.506507\pi\)
−0.0204415 + 0.999791i \(0.506507\pi\)
\(524\) −8.39059 −0.366545
\(525\) 5.20376 0.227111
\(526\) 19.6649 0.857431
\(527\) −27.0160 −1.17683
\(528\) −2.72061 −0.118399
\(529\) −22.3739 −0.972779
\(530\) −11.3466 −0.492864
\(531\) −19.4597 −0.844481
\(532\) 11.3529 0.492209
\(533\) −12.5316 −0.542802
\(534\) −5.02834 −0.217598
\(535\) 11.6856 0.505212
\(536\) 15.5544 0.671846
\(537\) 45.1582 1.94872
\(538\) 2.38473 0.102813
\(539\) 3.34151 0.143929
\(540\) 3.81352 0.164108
\(541\) −23.5381 −1.01198 −0.505991 0.862539i \(-0.668873\pi\)
−0.505991 + 0.862539i \(0.668873\pi\)
\(542\) −4.10433 −0.176296
\(543\) 35.2887 1.51438
\(544\) −3.77479 −0.161843
\(545\) 15.9177 0.681840
\(546\) 6.29002 0.269188
\(547\) 21.2179 0.907211 0.453606 0.891203i \(-0.350138\pi\)
0.453606 + 0.891203i \(0.350138\pi\)
\(548\) −15.8708 −0.677967
\(549\) −11.6879 −0.498828
\(550\) −1.00000 −0.0426401
\(551\) 3.54348 0.150957
\(552\) 2.15269 0.0916247
\(553\) −0.450025 −0.0191370
\(554\) −27.0052 −1.14734
\(555\) 3.06708 0.130190
\(556\) 9.81204 0.416123
\(557\) 22.9932 0.974254 0.487127 0.873331i \(-0.338045\pi\)
0.487127 + 0.873331i \(0.338045\pi\)
\(558\) 31.5029 1.33362
\(559\) 1.20875 0.0511245
\(560\) 1.91272 0.0808270
\(561\) 10.2697 0.433588
\(562\) −1.83659 −0.0774718
\(563\) 3.75876 0.158413 0.0792065 0.996858i \(-0.474761\pi\)
0.0792065 + 0.996858i \(0.474761\pi\)
\(564\) −32.9177 −1.38609
\(565\) 2.03490 0.0856089
\(566\) −2.44280 −0.102678
\(567\) −5.41306 −0.227327
\(568\) −4.84257 −0.203190
\(569\) −32.0681 −1.34436 −0.672182 0.740386i \(-0.734643\pi\)
−0.672182 + 0.740386i \(0.734643\pi\)
\(570\) 16.1481 0.676369
\(571\) 29.5503 1.23664 0.618320 0.785927i \(-0.287814\pi\)
0.618320 + 0.785927i \(0.287814\pi\)
\(572\) −1.20875 −0.0505402
\(573\) −11.7840 −0.492284
\(574\) −19.8299 −0.827684
\(575\) 0.791254 0.0329976
\(576\) 4.40172 0.183405
\(577\) −30.9087 −1.28675 −0.643373 0.765553i \(-0.722466\pi\)
−0.643373 + 0.765553i \(0.722466\pi\)
\(578\) −2.75097 −0.114425
\(579\) 5.28396 0.219594
\(580\) 0.597002 0.0247891
\(581\) 20.5100 0.850897
\(582\) −1.90719 −0.0790556
\(583\) 11.3466 0.469928
\(584\) −2.10242 −0.0869987
\(585\) 5.32056 0.219978
\(586\) −7.62521 −0.314994
\(587\) 25.2434 1.04191 0.520953 0.853586i \(-0.325577\pi\)
0.520953 + 0.853586i \(0.325577\pi\)
\(588\) −9.09096 −0.374905
\(589\) 42.4798 1.75035
\(590\) −4.42095 −0.182007
\(591\) 19.7933 0.814186
\(592\) 1.12735 0.0463338
\(593\) −3.53476 −0.145155 −0.0725776 0.997363i \(-0.523122\pi\)
−0.0725776 + 0.997363i \(0.523122\pi\)
\(594\) −3.81352 −0.156471
\(595\) −7.22010 −0.295995
\(596\) −3.06332 −0.125478
\(597\) −33.6103 −1.37558
\(598\) 0.956425 0.0391111
\(599\) 12.1990 0.498436 0.249218 0.968447i \(-0.419826\pi\)
0.249218 + 0.968447i \(0.419826\pi\)
\(600\) 2.72061 0.111068
\(601\) 26.4493 1.07889 0.539445 0.842021i \(-0.318634\pi\)
0.539445 + 0.842021i \(0.318634\pi\)
\(602\) 1.91272 0.0779565
\(603\) 68.4659 2.78815
\(604\) −3.50114 −0.142460
\(605\) 1.00000 0.0406558
\(606\) −10.6343 −0.431987
\(607\) −9.77084 −0.396586 −0.198293 0.980143i \(-0.563540\pi\)
−0.198293 + 0.980143i \(0.563540\pi\)
\(608\) 5.93546 0.240715
\(609\) 3.10665 0.125888
\(610\) −2.65531 −0.107510
\(611\) −14.6251 −0.591668
\(612\) −16.6156 −0.671644
\(613\) −29.1266 −1.17641 −0.588205 0.808712i \(-0.700165\pi\)
−0.588205 + 0.808712i \(0.700165\pi\)
\(614\) −10.9760 −0.442956
\(615\) −28.2057 −1.13736
\(616\) −1.91272 −0.0770656
\(617\) 30.5532 1.23003 0.615013 0.788517i \(-0.289151\pi\)
0.615013 + 0.788517i \(0.289151\pi\)
\(618\) 18.6632 0.750745
\(619\) 18.4567 0.741836 0.370918 0.928666i \(-0.379043\pi\)
0.370918 + 0.928666i \(0.379043\pi\)
\(620\) 7.15695 0.287430
\(621\) 3.01747 0.121087
\(622\) 8.53162 0.342087
\(623\) −3.53516 −0.141633
\(624\) 3.28853 0.131646
\(625\) 1.00000 0.0400000
\(626\) −10.9279 −0.436767
\(627\) −16.1481 −0.644892
\(628\) −6.52120 −0.260224
\(629\) −4.25551 −0.169678
\(630\) 8.41924 0.335430
\(631\) −27.1736 −1.08176 −0.540882 0.841098i \(-0.681910\pi\)
−0.540882 + 0.841098i \(0.681910\pi\)
\(632\) −0.235281 −0.00935896
\(633\) −14.9565 −0.594467
\(634\) 5.05291 0.200677
\(635\) 8.31619 0.330018
\(636\) −30.8696 −1.22406
\(637\) −4.03904 −0.160033
\(638\) −0.597002 −0.0236355
\(639\) −21.3156 −0.843233
\(640\) 1.00000 0.0395285
\(641\) −18.7156 −0.739222 −0.369611 0.929187i \(-0.620509\pi\)
−0.369611 + 0.929187i \(0.620509\pi\)
\(642\) 31.7919 1.25473
\(643\) −22.4437 −0.885094 −0.442547 0.896745i \(-0.645925\pi\)
−0.442547 + 0.896745i \(0.645925\pi\)
\(644\) 1.51345 0.0596381
\(645\) 2.72061 0.107124
\(646\) −22.4051 −0.881518
\(647\) 6.70090 0.263440 0.131720 0.991287i \(-0.457950\pi\)
0.131720 + 0.991287i \(0.457950\pi\)
\(648\) −2.83004 −0.111174
\(649\) 4.42095 0.173537
\(650\) 1.20875 0.0474109
\(651\) 37.2430 1.45967
\(652\) −0.0708121 −0.00277321
\(653\) −23.8883 −0.934822 −0.467411 0.884040i \(-0.654813\pi\)
−0.467411 + 0.884040i \(0.654813\pi\)
\(654\) 43.3059 1.69339
\(655\) −8.39059 −0.327848
\(656\) −10.3674 −0.404779
\(657\) −9.25425 −0.361043
\(658\) −23.1427 −0.902197
\(659\) 6.89215 0.268480 0.134240 0.990949i \(-0.457141\pi\)
0.134240 + 0.990949i \(0.457141\pi\)
\(660\) −2.72061 −0.105900
\(661\) −34.7551 −1.35182 −0.675908 0.736986i \(-0.736248\pi\)
−0.675908 + 0.736986i \(0.736248\pi\)
\(662\) 31.7736 1.23492
\(663\) −12.4135 −0.482100
\(664\) 10.7230 0.416131
\(665\) 11.3529 0.440245
\(666\) 4.96227 0.192284
\(667\) 0.472380 0.0182906
\(668\) −9.52120 −0.368386
\(669\) −54.2211 −2.09631
\(670\) 15.5544 0.600918
\(671\) 2.65531 0.102507
\(672\) 5.20376 0.200739
\(673\) −26.8790 −1.03611 −0.518055 0.855347i \(-0.673344\pi\)
−0.518055 + 0.855347i \(0.673344\pi\)
\(674\) 16.5299 0.636708
\(675\) 3.81352 0.146783
\(676\) −11.5389 −0.443805
\(677\) −13.6546 −0.524787 −0.262394 0.964961i \(-0.584512\pi\)
−0.262394 + 0.964961i \(0.584512\pi\)
\(678\) 5.53617 0.212615
\(679\) −1.34084 −0.0514569
\(680\) −3.77479 −0.144757
\(681\) 21.1635 0.810987
\(682\) −7.15695 −0.274054
\(683\) 19.7840 0.757014 0.378507 0.925598i \(-0.376438\pi\)
0.378507 + 0.925598i \(0.376438\pi\)
\(684\) 26.1262 0.998961
\(685\) −15.8708 −0.606393
\(686\) −19.7804 −0.755219
\(687\) −24.3927 −0.930640
\(688\) 1.00000 0.0381246
\(689\) −13.7151 −0.522505
\(690\) 2.15269 0.0819516
\(691\) −24.6686 −0.938437 −0.469218 0.883082i \(-0.655464\pi\)
−0.469218 + 0.883082i \(0.655464\pi\)
\(692\) −10.1851 −0.387180
\(693\) −8.41924 −0.319820
\(694\) 2.73720 0.103903
\(695\) 9.81204 0.372192
\(696\) 1.62421 0.0615655
\(697\) 39.1348 1.48234
\(698\) 14.0391 0.531389
\(699\) 45.6798 1.72777
\(700\) 1.91272 0.0722939
\(701\) −0.950137 −0.0358862 −0.0179431 0.999839i \(-0.505712\pi\)
−0.0179431 + 0.999839i \(0.505712\pi\)
\(702\) 4.60958 0.173977
\(703\) 6.69134 0.252369
\(704\) −1.00000 −0.0376889
\(705\) −32.9177 −1.23975
\(706\) −31.8160 −1.19741
\(707\) −7.47639 −0.281179
\(708\) −12.0277 −0.452027
\(709\) −9.66206 −0.362866 −0.181433 0.983403i \(-0.558074\pi\)
−0.181433 + 0.983403i \(0.558074\pi\)
\(710\) −4.84257 −0.181739
\(711\) −1.03564 −0.0388395
\(712\) −1.84824 −0.0692658
\(713\) 5.66297 0.212080
\(714\) −19.6431 −0.735124
\(715\) −1.20875 −0.0452045
\(716\) 16.5986 0.620317
\(717\) −21.8447 −0.815806
\(718\) −3.43246 −0.128098
\(719\) 30.0475 1.12058 0.560290 0.828296i \(-0.310690\pi\)
0.560290 + 0.828296i \(0.310690\pi\)
\(720\) 4.40172 0.164042
\(721\) 13.1211 0.488656
\(722\) 16.2297 0.604008
\(723\) −3.07064 −0.114198
\(724\) 12.9709 0.482060
\(725\) 0.597002 0.0221721
\(726\) 2.72061 0.100971
\(727\) 36.8592 1.36703 0.683515 0.729936i \(-0.260450\pi\)
0.683515 + 0.729936i \(0.260450\pi\)
\(728\) 2.31199 0.0856880
\(729\) −43.5824 −1.61416
\(730\) −2.10242 −0.0778140
\(731\) −3.77479 −0.139616
\(732\) −7.22405 −0.267009
\(733\) −25.6801 −0.948517 −0.474259 0.880386i \(-0.657284\pi\)
−0.474259 + 0.880386i \(0.657284\pi\)
\(734\) 30.7599 1.13537
\(735\) −9.09096 −0.335325
\(736\) 0.791254 0.0291660
\(737\) −15.5544 −0.572953
\(738\) −45.6344 −1.67982
\(739\) −40.8543 −1.50285 −0.751424 0.659820i \(-0.770633\pi\)
−0.751424 + 0.659820i \(0.770633\pi\)
\(740\) 1.12735 0.0414422
\(741\) 19.5189 0.717046
\(742\) −21.7028 −0.796735
\(743\) 48.6927 1.78636 0.893180 0.449699i \(-0.148469\pi\)
0.893180 + 0.449699i \(0.148469\pi\)
\(744\) 19.4713 0.713851
\(745\) −3.06332 −0.112231
\(746\) 26.0052 0.952119
\(747\) 47.1994 1.72694
\(748\) 3.77479 0.138020
\(749\) 22.3512 0.816695
\(750\) 2.72061 0.0993426
\(751\) 28.7921 1.05064 0.525319 0.850906i \(-0.323946\pi\)
0.525319 + 0.850906i \(0.323946\pi\)
\(752\) −12.0994 −0.441220
\(753\) −20.8837 −0.761046
\(754\) 0.721623 0.0262800
\(755\) −3.50114 −0.127420
\(756\) 7.29419 0.265287
\(757\) 2.21937 0.0806642 0.0403321 0.999186i \(-0.487158\pi\)
0.0403321 + 0.999186i \(0.487158\pi\)
\(758\) −18.6814 −0.678540
\(759\) −2.15269 −0.0781378
\(760\) 5.93546 0.215302
\(761\) 2.96812 0.107594 0.0537972 0.998552i \(-0.482868\pi\)
0.0537972 + 0.998552i \(0.482868\pi\)
\(762\) 22.6251 0.819621
\(763\) 30.4461 1.10222
\(764\) −4.33138 −0.156704
\(765\) −16.6156 −0.600737
\(766\) 17.2652 0.623818
\(767\) −5.34380 −0.192953
\(768\) 2.72061 0.0981715
\(769\) −19.5103 −0.703561 −0.351781 0.936082i \(-0.614424\pi\)
−0.351781 + 0.936082i \(0.614424\pi\)
\(770\) −1.91272 −0.0689295
\(771\) −25.5083 −0.918658
\(772\) 1.94220 0.0699012
\(773\) −34.3646 −1.23601 −0.618005 0.786174i \(-0.712059\pi\)
−0.618005 + 0.786174i \(0.712059\pi\)
\(774\) 4.40172 0.158216
\(775\) 7.15695 0.257085
\(776\) −0.701016 −0.0251650
\(777\) 5.86645 0.210458
\(778\) 11.3509 0.406949
\(779\) −61.5354 −2.20473
\(780\) 3.28853 0.117748
\(781\) 4.84257 0.173281
\(782\) −2.98682 −0.106808
\(783\) 2.27668 0.0813619
\(784\) −3.34151 −0.119340
\(785\) −6.52120 −0.232752
\(786\) −22.8275 −0.814231
\(787\) 19.7549 0.704186 0.352093 0.935965i \(-0.385470\pi\)
0.352093 + 0.935965i \(0.385470\pi\)
\(788\) 7.27531 0.259172
\(789\) 53.5006 1.90467
\(790\) −0.235281 −0.00837091
\(791\) 3.89219 0.138390
\(792\) −4.40172 −0.156408
\(793\) −3.20959 −0.113976
\(794\) 33.4940 1.18866
\(795\) −30.8696 −1.09483
\(796\) −12.3540 −0.437875
\(797\) −30.6839 −1.08688 −0.543440 0.839448i \(-0.682878\pi\)
−0.543440 + 0.839448i \(0.682878\pi\)
\(798\) 30.8867 1.09338
\(799\) 45.6727 1.61578
\(800\) 1.00000 0.0353553
\(801\) −8.13543 −0.287451
\(802\) −10.6671 −0.376669
\(803\) 2.10242 0.0741927
\(804\) 42.3174 1.49242
\(805\) 1.51345 0.0533419
\(806\) 8.65093 0.304716
\(807\) 6.48793 0.228386
\(808\) −3.90878 −0.137510
\(809\) 31.2009 1.09697 0.548483 0.836162i \(-0.315205\pi\)
0.548483 + 0.836162i \(0.315205\pi\)
\(810\) −2.83004 −0.0994374
\(811\) 30.9511 1.08684 0.543420 0.839461i \(-0.317129\pi\)
0.543420 + 0.839461i \(0.317129\pi\)
\(812\) 1.14190 0.0400727
\(813\) −11.1663 −0.391619
\(814\) −1.12735 −0.0395136
\(815\) −0.0708121 −0.00248044
\(816\) −10.2697 −0.359512
\(817\) 5.93546 0.207656
\(818\) −5.64176 −0.197259
\(819\) 10.1767 0.355603
\(820\) −10.3674 −0.362045
\(821\) 49.2147 1.71761 0.858803 0.512306i \(-0.171209\pi\)
0.858803 + 0.512306i \(0.171209\pi\)
\(822\) −43.1783 −1.50602
\(823\) 6.63026 0.231116 0.115558 0.993301i \(-0.463134\pi\)
0.115558 + 0.993301i \(0.463134\pi\)
\(824\) 6.85994 0.238977
\(825\) −2.72061 −0.0947195
\(826\) −8.45602 −0.294222
\(827\) −11.4359 −0.397667 −0.198833 0.980033i \(-0.563715\pi\)
−0.198833 + 0.980033i \(0.563715\pi\)
\(828\) 3.48288 0.121038
\(829\) 32.8379 1.14051 0.570254 0.821468i \(-0.306845\pi\)
0.570254 + 0.821468i \(0.306845\pi\)
\(830\) 10.7230 0.372199
\(831\) −73.4705 −2.54866
\(832\) 1.20875 0.0419057
\(833\) 12.6135 0.437032
\(834\) 26.6947 0.924363
\(835\) −9.52120 −0.329495
\(836\) −5.93546 −0.205282
\(837\) 27.2932 0.943391
\(838\) −12.2933 −0.424664
\(839\) 17.6411 0.609039 0.304520 0.952506i \(-0.401504\pi\)
0.304520 + 0.952506i \(0.401504\pi\)
\(840\) 5.20376 0.179547
\(841\) −28.6436 −0.987710
\(842\) 9.99247 0.344363
\(843\) −4.99664 −0.172093
\(844\) −5.49748 −0.189231
\(845\) −11.5389 −0.396951
\(846\) −53.2581 −1.83105
\(847\) 1.91272 0.0657217
\(848\) −11.3466 −0.389644
\(849\) −6.64590 −0.228087
\(850\) −3.77479 −0.129474
\(851\) 0.892020 0.0305781
\(852\) −13.1747 −0.451359
\(853\) 30.5864 1.04726 0.523629 0.851946i \(-0.324578\pi\)
0.523629 + 0.851946i \(0.324578\pi\)
\(854\) −5.07885 −0.173795
\(855\) 26.1262 0.893498
\(856\) 11.6856 0.399405
\(857\) 48.9983 1.67375 0.836875 0.547394i \(-0.184380\pi\)
0.836875 + 0.547394i \(0.184380\pi\)
\(858\) −3.28853 −0.112268
\(859\) −36.0523 −1.23009 −0.615044 0.788493i \(-0.710862\pi\)
−0.615044 + 0.788493i \(0.710862\pi\)
\(860\) 1.00000 0.0340997
\(861\) −53.9494 −1.83859
\(862\) −20.3287 −0.692399
\(863\) 40.1399 1.36638 0.683190 0.730241i \(-0.260592\pi\)
0.683190 + 0.730241i \(0.260592\pi\)
\(864\) 3.81352 0.129739
\(865\) −10.1851 −0.346304
\(866\) −33.7115 −1.14556
\(867\) −7.48430 −0.254180
\(868\) 13.6892 0.464642
\(869\) 0.235281 0.00798135
\(870\) 1.62421 0.0550658
\(871\) 18.8013 0.637057
\(872\) 15.9177 0.539042
\(873\) −3.08567 −0.104434
\(874\) 4.69646 0.158860
\(875\) 1.91272 0.0646616
\(876\) −5.71986 −0.193256
\(877\) 58.5488 1.97705 0.988526 0.151049i \(-0.0482652\pi\)
0.988526 + 0.151049i \(0.0482652\pi\)
\(878\) −8.79633 −0.296862
\(879\) −20.7452 −0.699719
\(880\) −1.00000 −0.0337100
\(881\) 49.4346 1.66550 0.832748 0.553653i \(-0.186766\pi\)
0.832748 + 0.553653i \(0.186766\pi\)
\(882\) −14.7084 −0.495258
\(883\) −46.3441 −1.55960 −0.779801 0.626028i \(-0.784680\pi\)
−0.779801 + 0.626028i \(0.784680\pi\)
\(884\) −4.56276 −0.153462
\(885\) −12.0277 −0.404306
\(886\) −20.3959 −0.685215
\(887\) 25.9287 0.870602 0.435301 0.900285i \(-0.356642\pi\)
0.435301 + 0.900285i \(0.356642\pi\)
\(888\) 3.06708 0.102924
\(889\) 15.9065 0.533487
\(890\) −1.84824 −0.0619532
\(891\) 2.83004 0.0948099
\(892\) −19.9298 −0.667298
\(893\) −71.8156 −2.40322
\(894\) −8.33409 −0.278734
\(895\) 16.5986 0.554829
\(896\) 1.91272 0.0638994
\(897\) 2.60206 0.0868802
\(898\) 23.4319 0.781933
\(899\) 4.27271 0.142503
\(900\) 4.40172 0.146724
\(901\) 42.8310 1.42691
\(902\) 10.3674 0.345197
\(903\) 5.20376 0.173170
\(904\) 2.03490 0.0676798
\(905\) 12.9709 0.431167
\(906\) −9.52525 −0.316455
\(907\) −50.9527 −1.69186 −0.845929 0.533296i \(-0.820953\pi\)
−0.845929 + 0.533296i \(0.820953\pi\)
\(908\) 7.77896 0.258154
\(909\) −17.2053 −0.570665
\(910\) 2.31199 0.0766417
\(911\) −25.6757 −0.850674 −0.425337 0.905035i \(-0.639844\pi\)
−0.425337 + 0.905035i \(0.639844\pi\)
\(912\) 16.1481 0.534716
\(913\) −10.7230 −0.354878
\(914\) −19.0784 −0.631058
\(915\) −7.22405 −0.238820
\(916\) −8.96590 −0.296242
\(917\) −16.0488 −0.529979
\(918\) −14.3953 −0.475114
\(919\) 43.9593 1.45008 0.725042 0.688704i \(-0.241820\pi\)
0.725042 + 0.688704i \(0.241820\pi\)
\(920\) 0.791254 0.0260869
\(921\) −29.8615 −0.983969
\(922\) 34.5032 1.13630
\(923\) −5.85344 −0.192668
\(924\) −5.20376 −0.171191
\(925\) 1.12735 0.0370670
\(926\) −32.9370 −1.08238
\(927\) 30.1955 0.991751
\(928\) 0.597002 0.0195975
\(929\) −6.10050 −0.200151 −0.100075 0.994980i \(-0.531908\pi\)
−0.100075 + 0.994980i \(0.531908\pi\)
\(930\) 19.4713 0.638488
\(931\) −19.8334 −0.650015
\(932\) 16.7903 0.549984
\(933\) 23.2112 0.759901
\(934\) 36.3551 1.18957
\(935\) 3.77479 0.123449
\(936\) 5.32056 0.173908
\(937\) 8.15730 0.266487 0.133244 0.991083i \(-0.457461\pi\)
0.133244 + 0.991083i \(0.457461\pi\)
\(938\) 29.7511 0.971408
\(939\) −29.7306 −0.970220
\(940\) −12.0994 −0.394639
\(941\) −0.279419 −0.00910880 −0.00455440 0.999990i \(-0.501450\pi\)
−0.00455440 + 0.999990i \(0.501450\pi\)
\(942\) −17.7416 −0.578054
\(943\) −8.20325 −0.267135
\(944\) −4.42095 −0.143890
\(945\) 7.29419 0.237280
\(946\) −1.00000 −0.0325128
\(947\) 43.0750 1.39975 0.699875 0.714265i \(-0.253239\pi\)
0.699875 + 0.714265i \(0.253239\pi\)
\(948\) −0.640107 −0.0207897
\(949\) −2.54129 −0.0824937
\(950\) 5.93546 0.192572
\(951\) 13.7470 0.445777
\(952\) −7.22010 −0.234005
\(953\) 35.2696 1.14249 0.571247 0.820778i \(-0.306460\pi\)
0.571247 + 0.820778i \(0.306460\pi\)
\(954\) −49.9445 −1.61701
\(955\) −4.33138 −0.140160
\(956\) −8.02935 −0.259688
\(957\) −1.62421 −0.0525032
\(958\) 8.98706 0.290359
\(959\) −30.3564 −0.980258
\(960\) 2.72061 0.0878073
\(961\) 20.2219 0.652320
\(962\) 1.36268 0.0439345
\(963\) 51.4366 1.65752
\(964\) −1.12866 −0.0363517
\(965\) 1.94220 0.0625215
\(966\) 4.11749 0.132478
\(967\) −50.2807 −1.61692 −0.808459 0.588553i \(-0.799698\pi\)
−0.808459 + 0.588553i \(0.799698\pi\)
\(968\) 1.00000 0.0321412
\(969\) −60.9556 −1.95818
\(970\) −0.701016 −0.0225083
\(971\) 38.1828 1.22535 0.612673 0.790337i \(-0.290094\pi\)
0.612673 + 0.790337i \(0.290094\pi\)
\(972\) −19.1400 −0.613916
\(973\) 18.7676 0.601663
\(974\) −6.66801 −0.213657
\(975\) 3.28853 0.105317
\(976\) −2.65531 −0.0849943
\(977\) −14.4882 −0.463520 −0.231760 0.972773i \(-0.574448\pi\)
−0.231760 + 0.972773i \(0.574448\pi\)
\(978\) −0.192652 −0.00616033
\(979\) 1.84824 0.0590700
\(980\) −3.34151 −0.106741
\(981\) 70.0653 2.23701
\(982\) 27.9655 0.892413
\(983\) −42.9493 −1.36987 −0.684935 0.728604i \(-0.740170\pi\)
−0.684935 + 0.728604i \(0.740170\pi\)
\(984\) −28.2057 −0.899164
\(985\) 7.27531 0.231811
\(986\) −2.25356 −0.0717678
\(987\) −62.9623 −2.00411
\(988\) 7.17447 0.228250
\(989\) 0.791254 0.0251604
\(990\) −4.40172 −0.139896
\(991\) 23.1905 0.736669 0.368334 0.929693i \(-0.379928\pi\)
0.368334 + 0.929693i \(0.379928\pi\)
\(992\) 7.15695 0.227233
\(993\) 86.4436 2.74321
\(994\) −9.26247 −0.293788
\(995\) −12.3540 −0.391647
\(996\) 29.1730 0.924381
\(997\) −33.9549 −1.07536 −0.537681 0.843149i \(-0.680699\pi\)
−0.537681 + 0.843149i \(0.680699\pi\)
\(998\) −27.9426 −0.884508
\(999\) 4.29918 0.136020
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 4730.2.a.be.1.11 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.be.1.11 12 1.1 even 1 trivial