Properties

Label 637.2.a.e.1.1
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.61803 q^{2} +2.23607 q^{3} +4.85410 q^{4} -2.23607 q^{5} -5.85410 q^{6} -7.47214 q^{8} +2.00000 q^{9} +O(q^{10})\) \(q-2.61803 q^{2} +2.23607 q^{3} +4.85410 q^{4} -2.23607 q^{5} -5.85410 q^{6} -7.47214 q^{8} +2.00000 q^{9} +5.85410 q^{10} -3.00000 q^{11} +10.8541 q^{12} +1.00000 q^{13} -5.00000 q^{15} +9.85410 q^{16} -1.47214 q^{17} -5.23607 q^{18} -3.00000 q^{19} -10.8541 q^{20} +7.85410 q^{22} -8.23607 q^{23} -16.7082 q^{24} -2.61803 q^{26} -2.23607 q^{27} +4.47214 q^{29} +13.0902 q^{30} -5.00000 q^{31} -10.8541 q^{32} -6.70820 q^{33} +3.85410 q^{34} +9.70820 q^{36} +4.70820 q^{37} +7.85410 q^{38} +2.23607 q^{39} +16.7082 q^{40} +4.47214 q^{41} -8.00000 q^{43} -14.5623 q^{44} -4.47214 q^{45} +21.5623 q^{46} +7.47214 q^{47} +22.0344 q^{48} -3.29180 q^{51} +4.85410 q^{52} -7.47214 q^{53} +5.85410 q^{54} +6.70820 q^{55} -6.70820 q^{57} -11.7082 q^{58} +1.47214 q^{59} -24.2705 q^{60} -3.00000 q^{61} +13.0902 q^{62} +8.70820 q^{64} -2.23607 q^{65} +17.5623 q^{66} -3.00000 q^{67} -7.14590 q^{68} -18.4164 q^{69} -8.94427 q^{71} -14.9443 q^{72} +2.70820 q^{73} -12.3262 q^{74} -14.5623 q^{76} -5.85410 q^{78} -2.70820 q^{79} -22.0344 q^{80} -11.0000 q^{81} -11.7082 q^{82} +3.29180 q^{85} +20.9443 q^{86} +10.0000 q^{87} +22.4164 q^{88} -2.23607 q^{89} +11.7082 q^{90} -39.9787 q^{92} -11.1803 q^{93} -19.5623 q^{94} +6.70820 q^{95} -24.2705 q^{96} -9.41641 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} + 3q^{4} - 5q^{6} - 6q^{8} + 4q^{9} + O(q^{10}) \) \( 2q - 3q^{2} + 3q^{4} - 5q^{6} - 6q^{8} + 4q^{9} + 5q^{10} - 6q^{11} + 15q^{12} + 2q^{13} - 10q^{15} + 13q^{16} + 6q^{17} - 6q^{18} - 6q^{19} - 15q^{20} + 9q^{22} - 12q^{23} - 20q^{24} - 3q^{26} + 15q^{30} - 10q^{31} - 15q^{32} + q^{34} + 6q^{36} - 4q^{37} + 9q^{38} + 20q^{40} - 16q^{43} - 9q^{44} + 23q^{46} + 6q^{47} + 15q^{48} - 20q^{51} + 3q^{52} - 6q^{53} + 5q^{54} - 10q^{58} - 6q^{59} - 15q^{60} - 6q^{61} + 15q^{62} + 4q^{64} + 15q^{66} - 6q^{67} - 21q^{68} - 10q^{69} - 12q^{72} - 8q^{73} - 9q^{74} - 9q^{76} - 5q^{78} + 8q^{79} - 15q^{80} - 22q^{81} - 10q^{82} + 20q^{85} + 24q^{86} + 20q^{87} + 18q^{88} + 10q^{90} - 33q^{92} - 19q^{94} - 15q^{96} + 8q^{97} - 12q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.61803 −1.85123 −0.925615 0.378467i \(-0.876451\pi\)
−0.925615 + 0.378467i \(0.876451\pi\)
\(3\) 2.23607 1.29099 0.645497 0.763763i \(-0.276650\pi\)
0.645497 + 0.763763i \(0.276650\pi\)
\(4\) 4.85410 2.42705
\(5\) −2.23607 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) −5.85410 −2.38993
\(7\) 0 0
\(8\) −7.47214 −2.64180
\(9\) 2.00000 0.666667
\(10\) 5.85410 1.85123
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 10.8541 3.13331
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −5.00000 −1.29099
\(16\) 9.85410 2.46353
\(17\) −1.47214 −0.357045 −0.178523 0.983936i \(-0.557132\pi\)
−0.178523 + 0.983936i \(0.557132\pi\)
\(18\) −5.23607 −1.23415
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) −10.8541 −2.42705
\(21\) 0 0
\(22\) 7.85410 1.67450
\(23\) −8.23607 −1.71734 −0.858669 0.512530i \(-0.828708\pi\)
−0.858669 + 0.512530i \(0.828708\pi\)
\(24\) −16.7082 −3.41055
\(25\) 0 0
\(26\) −2.61803 −0.513439
\(27\) −2.23607 −0.430331
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 13.0902 2.38993
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) −10.8541 −1.91875
\(33\) −6.70820 −1.16775
\(34\) 3.85410 0.660973
\(35\) 0 0
\(36\) 9.70820 1.61803
\(37\) 4.70820 0.774024 0.387012 0.922075i \(-0.373507\pi\)
0.387012 + 0.922075i \(0.373507\pi\)
\(38\) 7.85410 1.27410
\(39\) 2.23607 0.358057
\(40\) 16.7082 2.64180
\(41\) 4.47214 0.698430 0.349215 0.937043i \(-0.386448\pi\)
0.349215 + 0.937043i \(0.386448\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −14.5623 −2.19535
\(45\) −4.47214 −0.666667
\(46\) 21.5623 3.17919
\(47\) 7.47214 1.08992 0.544962 0.838461i \(-0.316544\pi\)
0.544962 + 0.838461i \(0.316544\pi\)
\(48\) 22.0344 3.18040
\(49\) 0 0
\(50\) 0 0
\(51\) −3.29180 −0.460944
\(52\) 4.85410 0.673143
\(53\) −7.47214 −1.02638 −0.513188 0.858276i \(-0.671536\pi\)
−0.513188 + 0.858276i \(0.671536\pi\)
\(54\) 5.85410 0.796642
\(55\) 6.70820 0.904534
\(56\) 0 0
\(57\) −6.70820 −0.888523
\(58\) −11.7082 −1.53736
\(59\) 1.47214 0.191656 0.0958279 0.995398i \(-0.469450\pi\)
0.0958279 + 0.995398i \(0.469450\pi\)
\(60\) −24.2705 −3.13331
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) 13.0902 1.66245
\(63\) 0 0
\(64\) 8.70820 1.08853
\(65\) −2.23607 −0.277350
\(66\) 17.5623 2.16177
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) −7.14590 −0.866567
\(69\) −18.4164 −2.21707
\(70\) 0 0
\(71\) −8.94427 −1.06149 −0.530745 0.847532i \(-0.678088\pi\)
−0.530745 + 0.847532i \(0.678088\pi\)
\(72\) −14.9443 −1.76120
\(73\) 2.70820 0.316971 0.158486 0.987361i \(-0.449339\pi\)
0.158486 + 0.987361i \(0.449339\pi\)
\(74\) −12.3262 −1.43290
\(75\) 0 0
\(76\) −14.5623 −1.67041
\(77\) 0 0
\(78\) −5.85410 −0.662847
\(79\) −2.70820 −0.304697 −0.152348 0.988327i \(-0.548684\pi\)
−0.152348 + 0.988327i \(0.548684\pi\)
\(80\) −22.0344 −2.46353
\(81\) −11.0000 −1.22222
\(82\) −11.7082 −1.29295
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 3.29180 0.357045
\(86\) 20.9443 2.25848
\(87\) 10.0000 1.07211
\(88\) 22.4164 2.38960
\(89\) −2.23607 −0.237023 −0.118511 0.992953i \(-0.537812\pi\)
−0.118511 + 0.992953i \(0.537812\pi\)
\(90\) 11.7082 1.23415
\(91\) 0 0
\(92\) −39.9787 −4.16807
\(93\) −11.1803 −1.15935
\(94\) −19.5623 −2.01770
\(95\) 6.70820 0.688247
\(96\) −24.2705 −2.47710
\(97\) −9.41641 −0.956091 −0.478046 0.878335i \(-0.658655\pi\)
−0.478046 + 0.878335i \(0.658655\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) 8.61803 0.853313
\(103\) 2.70820 0.266847 0.133424 0.991059i \(-0.457403\pi\)
0.133424 + 0.991059i \(0.457403\pi\)
\(104\) −7.47214 −0.732703
\(105\) 0 0
\(106\) 19.5623 1.90006
\(107\) −9.76393 −0.943915 −0.471957 0.881621i \(-0.656452\pi\)
−0.471957 + 0.881621i \(0.656452\pi\)
\(108\) −10.8541 −1.04444
\(109\) −2.70820 −0.259399 −0.129699 0.991553i \(-0.541401\pi\)
−0.129699 + 0.991553i \(0.541401\pi\)
\(110\) −17.5623 −1.67450
\(111\) 10.5279 0.999261
\(112\) 0 0
\(113\) 2.94427 0.276974 0.138487 0.990364i \(-0.455776\pi\)
0.138487 + 0.990364i \(0.455776\pi\)
\(114\) 17.5623 1.64486
\(115\) 18.4164 1.71734
\(116\) 21.7082 2.01556
\(117\) 2.00000 0.184900
\(118\) −3.85410 −0.354799
\(119\) 0 0
\(120\) 37.3607 3.41055
\(121\) −2.00000 −0.181818
\(122\) 7.85410 0.711077
\(123\) 10.0000 0.901670
\(124\) −24.2705 −2.17956
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) −11.4164 −1.01304 −0.506521 0.862228i \(-0.669069\pi\)
−0.506521 + 0.862228i \(0.669069\pi\)
\(128\) −1.09017 −0.0963583
\(129\) −17.8885 −1.57500
\(130\) 5.85410 0.513439
\(131\) 8.23607 0.719589 0.359794 0.933032i \(-0.382847\pi\)
0.359794 + 0.933032i \(0.382847\pi\)
\(132\) −32.5623 −2.83418
\(133\) 0 0
\(134\) 7.85410 0.678491
\(135\) 5.00000 0.430331
\(136\) 11.0000 0.943242
\(137\) −8.23607 −0.703655 −0.351827 0.936065i \(-0.614440\pi\)
−0.351827 + 0.936065i \(0.614440\pi\)
\(138\) 48.2148 4.10431
\(139\) 23.4164 1.98615 0.993077 0.117466i \(-0.0374771\pi\)
0.993077 + 0.117466i \(0.0374771\pi\)
\(140\) 0 0
\(141\) 16.7082 1.40708
\(142\) 23.4164 1.96506
\(143\) −3.00000 −0.250873
\(144\) 19.7082 1.64235
\(145\) −10.0000 −0.830455
\(146\) −7.09017 −0.586787
\(147\) 0 0
\(148\) 22.8541 1.87860
\(149\) 0.708204 0.0580183 0.0290092 0.999579i \(-0.490765\pi\)
0.0290092 + 0.999579i \(0.490765\pi\)
\(150\) 0 0
\(151\) 20.4164 1.66146 0.830732 0.556673i \(-0.187922\pi\)
0.830732 + 0.556673i \(0.187922\pi\)
\(152\) 22.4164 1.81821
\(153\) −2.94427 −0.238030
\(154\) 0 0
\(155\) 11.1803 0.898027
\(156\) 10.8541 0.869024
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) 7.09017 0.564064
\(159\) −16.7082 −1.32505
\(160\) 24.2705 1.91875
\(161\) 0 0
\(162\) 28.7984 2.26261
\(163\) −16.4164 −1.28583 −0.642916 0.765937i \(-0.722276\pi\)
−0.642916 + 0.765937i \(0.722276\pi\)
\(164\) 21.7082 1.69513
\(165\) 15.0000 1.16775
\(166\) 0 0
\(167\) −22.4721 −1.73895 −0.869473 0.493980i \(-0.835541\pi\)
−0.869473 + 0.493980i \(0.835541\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −8.61803 −0.660973
\(171\) −6.00000 −0.458831
\(172\) −38.8328 −2.96097
\(173\) 16.4164 1.24812 0.624058 0.781378i \(-0.285483\pi\)
0.624058 + 0.781378i \(0.285483\pi\)
\(174\) −26.1803 −1.98473
\(175\) 0 0
\(176\) −29.5623 −2.22834
\(177\) 3.29180 0.247427
\(178\) 5.85410 0.438783
\(179\) −20.1246 −1.50418 −0.752092 0.659058i \(-0.770955\pi\)
−0.752092 + 0.659058i \(0.770955\pi\)
\(180\) −21.7082 −1.61803
\(181\) 25.4164 1.88919 0.944593 0.328243i \(-0.106456\pi\)
0.944593 + 0.328243i \(0.106456\pi\)
\(182\) 0 0
\(183\) −6.70820 −0.495885
\(184\) 61.5410 4.53686
\(185\) −10.5279 −0.774024
\(186\) 29.2705 2.14622
\(187\) 4.41641 0.322960
\(188\) 36.2705 2.64530
\(189\) 0 0
\(190\) −17.5623 −1.27410
\(191\) 11.1803 0.808981 0.404491 0.914542i \(-0.367449\pi\)
0.404491 + 0.914542i \(0.367449\pi\)
\(192\) 19.4721 1.40528
\(193\) 0.708204 0.0509776 0.0254888 0.999675i \(-0.491886\pi\)
0.0254888 + 0.999675i \(0.491886\pi\)
\(194\) 24.6525 1.76994
\(195\) −5.00000 −0.358057
\(196\) 0 0
\(197\) −9.05573 −0.645194 −0.322597 0.946536i \(-0.604556\pi\)
−0.322597 + 0.946536i \(0.604556\pi\)
\(198\) 15.7082 1.11633
\(199\) 20.7082 1.46797 0.733983 0.679168i \(-0.237659\pi\)
0.733983 + 0.679168i \(0.237659\pi\)
\(200\) 0 0
\(201\) −6.70820 −0.473160
\(202\) −23.5623 −1.65784
\(203\) 0 0
\(204\) −15.9787 −1.11873
\(205\) −10.0000 −0.698430
\(206\) −7.09017 −0.493996
\(207\) −16.4721 −1.14489
\(208\) 9.85410 0.683259
\(209\) 9.00000 0.622543
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −36.2705 −2.49107
\(213\) −20.0000 −1.37038
\(214\) 25.5623 1.74740
\(215\) 17.8885 1.21999
\(216\) 16.7082 1.13685
\(217\) 0 0
\(218\) 7.09017 0.480207
\(219\) 6.05573 0.409208
\(220\) 32.5623 2.19535
\(221\) −1.47214 −0.0990266
\(222\) −27.5623 −1.84986
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −7.70820 −0.512742
\(227\) −5.94427 −0.394535 −0.197268 0.980350i \(-0.563207\pi\)
−0.197268 + 0.980350i \(0.563207\pi\)
\(228\) −32.5623 −2.15649
\(229\) −24.1246 −1.59420 −0.797100 0.603848i \(-0.793633\pi\)
−0.797100 + 0.603848i \(0.793633\pi\)
\(230\) −48.2148 −3.17919
\(231\) 0 0
\(232\) −33.4164 −2.19389
\(233\) 11.9443 0.782495 0.391248 0.920285i \(-0.372044\pi\)
0.391248 + 0.920285i \(0.372044\pi\)
\(234\) −5.23607 −0.342292
\(235\) −16.7082 −1.08992
\(236\) 7.14590 0.465158
\(237\) −6.05573 −0.393362
\(238\) 0 0
\(239\) 19.4164 1.25594 0.627972 0.778236i \(-0.283885\pi\)
0.627972 + 0.778236i \(0.283885\pi\)
\(240\) −49.2705 −3.18040
\(241\) −4.70820 −0.303282 −0.151641 0.988436i \(-0.548456\pi\)
−0.151641 + 0.988436i \(0.548456\pi\)
\(242\) 5.23607 0.336587
\(243\) −17.8885 −1.14755
\(244\) −14.5623 −0.932256
\(245\) 0 0
\(246\) −26.1803 −1.66920
\(247\) −3.00000 −0.190885
\(248\) 37.3607 2.37241
\(249\) 0 0
\(250\) −29.2705 −1.85123
\(251\) −1.52786 −0.0964379 −0.0482190 0.998837i \(-0.515355\pi\)
−0.0482190 + 0.998837i \(0.515355\pi\)
\(252\) 0 0
\(253\) 24.7082 1.55339
\(254\) 29.8885 1.87537
\(255\) 7.36068 0.460944
\(256\) −14.5623 −0.910144
\(257\) 0.0557281 0.00347622 0.00173811 0.999998i \(-0.499447\pi\)
0.00173811 + 0.999998i \(0.499447\pi\)
\(258\) 46.8328 2.91568
\(259\) 0 0
\(260\) −10.8541 −0.673143
\(261\) 8.94427 0.553637
\(262\) −21.5623 −1.33212
\(263\) 26.1246 1.61091 0.805456 0.592655i \(-0.201920\pi\)
0.805456 + 0.592655i \(0.201920\pi\)
\(264\) 50.1246 3.08496
\(265\) 16.7082 1.02638
\(266\) 0 0
\(267\) −5.00000 −0.305995
\(268\) −14.5623 −0.889534
\(269\) −13.4721 −0.821411 −0.410705 0.911768i \(-0.634717\pi\)
−0.410705 + 0.911768i \(0.634717\pi\)
\(270\) −13.0902 −0.796642
\(271\) 20.4164 1.24021 0.620104 0.784519i \(-0.287090\pi\)
0.620104 + 0.784519i \(0.287090\pi\)
\(272\) −14.5066 −0.879590
\(273\) 0 0
\(274\) 21.5623 1.30263
\(275\) 0 0
\(276\) −89.3951 −5.38095
\(277\) −0.416408 −0.0250195 −0.0125098 0.999922i \(-0.503982\pi\)
−0.0125098 + 0.999922i \(0.503982\pi\)
\(278\) −61.3050 −3.67683
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 26.9443 1.60736 0.803680 0.595061i \(-0.202872\pi\)
0.803680 + 0.595061i \(0.202872\pi\)
\(282\) −43.7426 −2.60484
\(283\) −26.1246 −1.55295 −0.776473 0.630150i \(-0.782993\pi\)
−0.776473 + 0.630150i \(0.782993\pi\)
\(284\) −43.4164 −2.57629
\(285\) 15.0000 0.888523
\(286\) 7.85410 0.464423
\(287\) 0 0
\(288\) −21.7082 −1.27917
\(289\) −14.8328 −0.872519
\(290\) 26.1803 1.53736
\(291\) −21.0557 −1.23431
\(292\) 13.1459 0.769305
\(293\) 14.9443 0.873054 0.436527 0.899691i \(-0.356208\pi\)
0.436527 + 0.899691i \(0.356208\pi\)
\(294\) 0 0
\(295\) −3.29180 −0.191656
\(296\) −35.1803 −2.04482
\(297\) 6.70820 0.389249
\(298\) −1.85410 −0.107405
\(299\) −8.23607 −0.476304
\(300\) 0 0
\(301\) 0 0
\(302\) −53.4508 −3.07575
\(303\) 20.1246 1.15613
\(304\) −29.5623 −1.69551
\(305\) 6.70820 0.384111
\(306\) 7.70820 0.440649
\(307\) −19.4164 −1.10815 −0.554076 0.832466i \(-0.686928\pi\)
−0.554076 + 0.832466i \(0.686928\pi\)
\(308\) 0 0
\(309\) 6.05573 0.344498
\(310\) −29.2705 −1.66245
\(311\) 27.7639 1.57435 0.787174 0.616731i \(-0.211543\pi\)
0.787174 + 0.616731i \(0.211543\pi\)
\(312\) −16.7082 −0.945916
\(313\) 5.58359 0.315603 0.157802 0.987471i \(-0.449559\pi\)
0.157802 + 0.987471i \(0.449559\pi\)
\(314\) 18.3262 1.03421
\(315\) 0 0
\(316\) −13.1459 −0.739515
\(317\) −8.23607 −0.462584 −0.231292 0.972884i \(-0.574295\pi\)
−0.231292 + 0.972884i \(0.574295\pi\)
\(318\) 43.7426 2.45297
\(319\) −13.4164 −0.751175
\(320\) −19.4721 −1.08853
\(321\) −21.8328 −1.21859
\(322\) 0 0
\(323\) 4.41641 0.245736
\(324\) −53.3951 −2.96640
\(325\) 0 0
\(326\) 42.9787 2.38037
\(327\) −6.05573 −0.334883
\(328\) −33.4164 −1.84511
\(329\) 0 0
\(330\) −39.2705 −2.16177
\(331\) −1.58359 −0.0870421 −0.0435210 0.999053i \(-0.513858\pi\)
−0.0435210 + 0.999053i \(0.513858\pi\)
\(332\) 0 0
\(333\) 9.41641 0.516016
\(334\) 58.8328 3.21919
\(335\) 6.70820 0.366508
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) −2.61803 −0.142402
\(339\) 6.58359 0.357572
\(340\) 15.9787 0.866567
\(341\) 15.0000 0.812296
\(342\) 15.7082 0.849402
\(343\) 0 0
\(344\) 59.7771 3.22296
\(345\) 41.1803 2.21707
\(346\) −42.9787 −2.31055
\(347\) 23.0689 1.23840 0.619201 0.785232i \(-0.287456\pi\)
0.619201 + 0.785232i \(0.287456\pi\)
\(348\) 48.5410 2.60207
\(349\) 29.4164 1.57462 0.787312 0.616555i \(-0.211472\pi\)
0.787312 + 0.616555i \(0.211472\pi\)
\(350\) 0 0
\(351\) −2.23607 −0.119352
\(352\) 32.5623 1.73558
\(353\) −17.2918 −0.920349 −0.460175 0.887828i \(-0.652213\pi\)
−0.460175 + 0.887828i \(0.652213\pi\)
\(354\) −8.61803 −0.458043
\(355\) 20.0000 1.06149
\(356\) −10.8541 −0.575266
\(357\) 0 0
\(358\) 52.6869 2.78459
\(359\) 11.9443 0.630395 0.315197 0.949026i \(-0.397929\pi\)
0.315197 + 0.949026i \(0.397929\pi\)
\(360\) 33.4164 1.76120
\(361\) −10.0000 −0.526316
\(362\) −66.5410 −3.49732
\(363\) −4.47214 −0.234726
\(364\) 0 0
\(365\) −6.05573 −0.316971
\(366\) 17.5623 0.917996
\(367\) 12.7082 0.663363 0.331681 0.943391i \(-0.392384\pi\)
0.331681 + 0.943391i \(0.392384\pi\)
\(368\) −81.1591 −4.23071
\(369\) 8.94427 0.465620
\(370\) 27.5623 1.43290
\(371\) 0 0
\(372\) −54.2705 −2.81379
\(373\) −1.58359 −0.0819953 −0.0409976 0.999159i \(-0.513054\pi\)
−0.0409976 + 0.999159i \(0.513054\pi\)
\(374\) −11.5623 −0.597873
\(375\) 25.0000 1.29099
\(376\) −55.8328 −2.87936
\(377\) 4.47214 0.230327
\(378\) 0 0
\(379\) −15.4164 −0.791888 −0.395944 0.918275i \(-0.629583\pi\)
−0.395944 + 0.918275i \(0.629583\pi\)
\(380\) 32.5623 1.67041
\(381\) −25.5279 −1.30783
\(382\) −29.2705 −1.49761
\(383\) −15.0000 −0.766464 −0.383232 0.923652i \(-0.625189\pi\)
−0.383232 + 0.923652i \(0.625189\pi\)
\(384\) −2.43769 −0.124398
\(385\) 0 0
\(386\) −1.85410 −0.0943713
\(387\) −16.0000 −0.813326
\(388\) −45.7082 −2.32048
\(389\) 1.47214 0.0746403 0.0373201 0.999303i \(-0.488118\pi\)
0.0373201 + 0.999303i \(0.488118\pi\)
\(390\) 13.0902 0.662847
\(391\) 12.1246 0.613168
\(392\) 0 0
\(393\) 18.4164 0.928985
\(394\) 23.7082 1.19440
\(395\) 6.05573 0.304697
\(396\) −29.1246 −1.46357
\(397\) 26.1246 1.31116 0.655578 0.755127i \(-0.272425\pi\)
0.655578 + 0.755127i \(0.272425\pi\)
\(398\) −54.2148 −2.71754
\(399\) 0 0
\(400\) 0 0
\(401\) 14.2361 0.710915 0.355458 0.934692i \(-0.384325\pi\)
0.355458 + 0.934692i \(0.384325\pi\)
\(402\) 17.5623 0.875928
\(403\) −5.00000 −0.249068
\(404\) 43.6869 2.17351
\(405\) 24.5967 1.22222
\(406\) 0 0
\(407\) −14.1246 −0.700131
\(408\) 24.5967 1.21772
\(409\) −8.70820 −0.430593 −0.215296 0.976549i \(-0.569072\pi\)
−0.215296 + 0.976549i \(0.569072\pi\)
\(410\) 26.1803 1.29295
\(411\) −18.4164 −0.908414
\(412\) 13.1459 0.647652
\(413\) 0 0
\(414\) 43.1246 2.11946
\(415\) 0 0
\(416\) −10.8541 −0.532166
\(417\) 52.3607 2.56411
\(418\) −23.5623 −1.15247
\(419\) −32.9443 −1.60943 −0.804717 0.593659i \(-0.797683\pi\)
−0.804717 + 0.593659i \(0.797683\pi\)
\(420\) 0 0
\(421\) 13.4164 0.653876 0.326938 0.945046i \(-0.393983\pi\)
0.326938 + 0.945046i \(0.393983\pi\)
\(422\) −10.4721 −0.509776
\(423\) 14.9443 0.726615
\(424\) 55.8328 2.71148
\(425\) 0 0
\(426\) 52.3607 2.53688
\(427\) 0 0
\(428\) −47.3951 −2.29093
\(429\) −6.70820 −0.323875
\(430\) −46.8328 −2.25848
\(431\) −31.3607 −1.51059 −0.755295 0.655385i \(-0.772507\pi\)
−0.755295 + 0.655385i \(0.772507\pi\)
\(432\) −22.0344 −1.06013
\(433\) −29.4164 −1.41366 −0.706831 0.707382i \(-0.749876\pi\)
−0.706831 + 0.707382i \(0.749876\pi\)
\(434\) 0 0
\(435\) −22.3607 −1.07211
\(436\) −13.1459 −0.629574
\(437\) 24.7082 1.18195
\(438\) −15.8541 −0.757538
\(439\) −24.1246 −1.15140 −0.575702 0.817659i \(-0.695271\pi\)
−0.575702 + 0.817659i \(0.695271\pi\)
\(440\) −50.1246 −2.38960
\(441\) 0 0
\(442\) 3.85410 0.183321
\(443\) 2.23607 0.106239 0.0531194 0.998588i \(-0.483084\pi\)
0.0531194 + 0.998588i \(0.483084\pi\)
\(444\) 51.1033 2.42526
\(445\) 5.00000 0.237023
\(446\) −10.4721 −0.495870
\(447\) 1.58359 0.0749013
\(448\) 0 0
\(449\) −34.3607 −1.62158 −0.810790 0.585337i \(-0.800962\pi\)
−0.810790 + 0.585337i \(0.800962\pi\)
\(450\) 0 0
\(451\) −13.4164 −0.631754
\(452\) 14.2918 0.672230
\(453\) 45.6525 2.14494
\(454\) 15.5623 0.730375
\(455\) 0 0
\(456\) 50.1246 2.34730
\(457\) −6.12461 −0.286497 −0.143249 0.989687i \(-0.545755\pi\)
−0.143249 + 0.989687i \(0.545755\pi\)
\(458\) 63.1591 2.95123
\(459\) 3.29180 0.153648
\(460\) 89.3951 4.16807
\(461\) −34.3607 −1.60034 −0.800168 0.599776i \(-0.795256\pi\)
−0.800168 + 0.599776i \(0.795256\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 44.0689 2.04585
\(465\) 25.0000 1.15935
\(466\) −31.2705 −1.44858
\(467\) −9.65248 −0.446663 −0.223332 0.974743i \(-0.571693\pi\)
−0.223332 + 0.974743i \(0.571693\pi\)
\(468\) 9.70820 0.448762
\(469\) 0 0
\(470\) 43.7426 2.01770
\(471\) −15.6525 −0.721228
\(472\) −11.0000 −0.506316
\(473\) 24.0000 1.10352
\(474\) 15.8541 0.728203
\(475\) 0 0
\(476\) 0 0
\(477\) −14.9443 −0.684251
\(478\) −50.8328 −2.32504
\(479\) −23.8328 −1.08895 −0.544475 0.838777i \(-0.683271\pi\)
−0.544475 + 0.838777i \(0.683271\pi\)
\(480\) 54.2705 2.47710
\(481\) 4.70820 0.214676
\(482\) 12.3262 0.561445
\(483\) 0 0
\(484\) −9.70820 −0.441282
\(485\) 21.0557 0.956091
\(486\) 46.8328 2.12438
\(487\) −21.8328 −0.989339 −0.494670 0.869081i \(-0.664711\pi\)
−0.494670 + 0.869081i \(0.664711\pi\)
\(488\) 22.4164 1.01474
\(489\) −36.7082 −1.66000
\(490\) 0 0
\(491\) −25.5279 −1.15206 −0.576028 0.817430i \(-0.695398\pi\)
−0.576028 + 0.817430i \(0.695398\pi\)
\(492\) 48.5410 2.18840
\(493\) −6.58359 −0.296510
\(494\) 7.85410 0.353373
\(495\) 13.4164 0.603023
\(496\) −49.2705 −2.21231
\(497\) 0 0
\(498\) 0 0
\(499\) 26.4164 1.18256 0.591280 0.806466i \(-0.298623\pi\)
0.591280 + 0.806466i \(0.298623\pi\)
\(500\) 54.2705 2.42705
\(501\) −50.2492 −2.24497
\(502\) 4.00000 0.178529
\(503\) −20.9443 −0.933859 −0.466929 0.884295i \(-0.654640\pi\)
−0.466929 + 0.884295i \(0.654640\pi\)
\(504\) 0 0
\(505\) −20.1246 −0.895533
\(506\) −64.6869 −2.87568
\(507\) 2.23607 0.0993073
\(508\) −55.4164 −2.45871
\(509\) 20.2361 0.896948 0.448474 0.893796i \(-0.351968\pi\)
0.448474 + 0.893796i \(0.351968\pi\)
\(510\) −19.2705 −0.853313
\(511\) 0 0
\(512\) 40.3050 1.78124
\(513\) 6.70820 0.296174
\(514\) −0.145898 −0.00643529
\(515\) −6.05573 −0.266847
\(516\) −86.8328 −3.82260
\(517\) −22.4164 −0.985872
\(518\) 0 0
\(519\) 36.7082 1.61131
\(520\) 16.7082 0.732703
\(521\) 17.9443 0.786153 0.393076 0.919506i \(-0.371411\pi\)
0.393076 + 0.919506i \(0.371411\pi\)
\(522\) −23.4164 −1.02491
\(523\) −32.7082 −1.43023 −0.715115 0.699007i \(-0.753626\pi\)
−0.715115 + 0.699007i \(0.753626\pi\)
\(524\) 39.9787 1.74648
\(525\) 0 0
\(526\) −68.3951 −2.98217
\(527\) 7.36068 0.320636
\(528\) −66.1033 −2.87678
\(529\) 44.8328 1.94925
\(530\) −43.7426 −1.90006
\(531\) 2.94427 0.127771
\(532\) 0 0
\(533\) 4.47214 0.193710
\(534\) 13.0902 0.566467
\(535\) 21.8328 0.943915
\(536\) 22.4164 0.968241
\(537\) −45.0000 −1.94189
\(538\) 35.2705 1.52062
\(539\) 0 0
\(540\) 24.2705 1.04444
\(541\) 1.29180 0.0555387 0.0277693 0.999614i \(-0.491160\pi\)
0.0277693 + 0.999614i \(0.491160\pi\)
\(542\) −53.4508 −2.29591
\(543\) 56.8328 2.43893
\(544\) 15.9787 0.685082
\(545\) 6.05573 0.259399
\(546\) 0 0
\(547\) −4.58359 −0.195980 −0.0979901 0.995187i \(-0.531241\pi\)
−0.0979901 + 0.995187i \(0.531241\pi\)
\(548\) −39.9787 −1.70781
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −13.4164 −0.571558
\(552\) 137.610 5.85707
\(553\) 0 0
\(554\) 1.09017 0.0463169
\(555\) −23.5410 −0.999261
\(556\) 113.666 4.82050
\(557\) −18.7082 −0.792692 −0.396346 0.918101i \(-0.629722\pi\)
−0.396346 + 0.918101i \(0.629722\pi\)
\(558\) 26.1803 1.10830
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 9.87539 0.416939
\(562\) −70.5410 −2.97559
\(563\) −12.5967 −0.530890 −0.265445 0.964126i \(-0.585519\pi\)
−0.265445 + 0.964126i \(0.585519\pi\)
\(564\) 81.1033 3.41507
\(565\) −6.58359 −0.276974
\(566\) 68.3951 2.87486
\(567\) 0 0
\(568\) 66.8328 2.80424
\(569\) 25.4721 1.06785 0.533924 0.845533i \(-0.320717\pi\)
0.533924 + 0.845533i \(0.320717\pi\)
\(570\) −39.2705 −1.64486
\(571\) −36.1246 −1.51177 −0.755884 0.654706i \(-0.772793\pi\)
−0.755884 + 0.654706i \(0.772793\pi\)
\(572\) −14.5623 −0.608881
\(573\) 25.0000 1.04439
\(574\) 0 0
\(575\) 0 0
\(576\) 17.4164 0.725684
\(577\) 19.2918 0.803128 0.401564 0.915831i \(-0.368467\pi\)
0.401564 + 0.915831i \(0.368467\pi\)
\(578\) 38.8328 1.61523
\(579\) 1.58359 0.0658118
\(580\) −48.5410 −2.01556
\(581\) 0 0
\(582\) 55.1246 2.28499
\(583\) 22.4164 0.928393
\(584\) −20.2361 −0.837374
\(585\) −4.47214 −0.184900
\(586\) −39.1246 −1.61622
\(587\) 6.11146 0.252247 0.126123 0.992015i \(-0.459746\pi\)
0.126123 + 0.992015i \(0.459746\pi\)
\(588\) 0 0
\(589\) 15.0000 0.618064
\(590\) 8.61803 0.354799
\(591\) −20.2492 −0.832942
\(592\) 46.3951 1.90683
\(593\) −27.7639 −1.14013 −0.570064 0.821600i \(-0.693082\pi\)
−0.570064 + 0.821600i \(0.693082\pi\)
\(594\) −17.5623 −0.720590
\(595\) 0 0
\(596\) 3.43769 0.140813
\(597\) 46.3050 1.89514
\(598\) 21.5623 0.881748
\(599\) −17.0689 −0.697416 −0.348708 0.937231i \(-0.613379\pi\)
−0.348708 + 0.937231i \(0.613379\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) −6.00000 −0.244339
\(604\) 99.1033 4.03246
\(605\) 4.47214 0.181818
\(606\) −52.6869 −2.14026
\(607\) −24.1246 −0.979188 −0.489594 0.871951i \(-0.662855\pi\)
−0.489594 + 0.871951i \(0.662855\pi\)
\(608\) 32.5623 1.32058
\(609\) 0 0
\(610\) −17.5623 −0.711077
\(611\) 7.47214 0.302290
\(612\) −14.2918 −0.577712
\(613\) −18.1246 −0.732046 −0.366023 0.930606i \(-0.619281\pi\)
−0.366023 + 0.930606i \(0.619281\pi\)
\(614\) 50.8328 2.05145
\(615\) −22.3607 −0.901670
\(616\) 0 0
\(617\) −4.47214 −0.180041 −0.0900207 0.995940i \(-0.528693\pi\)
−0.0900207 + 0.995940i \(0.528693\pi\)
\(618\) −15.8541 −0.637746
\(619\) −17.0000 −0.683288 −0.341644 0.939829i \(-0.610984\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) 54.2705 2.17956
\(621\) 18.4164 0.739025
\(622\) −72.6869 −2.91448
\(623\) 0 0
\(624\) 22.0344 0.882084
\(625\) −25.0000 −1.00000
\(626\) −14.6180 −0.584254
\(627\) 20.1246 0.803700
\(628\) −33.9787 −1.35590
\(629\) −6.93112 −0.276362
\(630\) 0 0
\(631\) 22.8328 0.908960 0.454480 0.890757i \(-0.349825\pi\)
0.454480 + 0.890757i \(0.349825\pi\)
\(632\) 20.2361 0.804948
\(633\) 8.94427 0.355503
\(634\) 21.5623 0.856349
\(635\) 25.5279 1.01304
\(636\) −81.1033 −3.21596
\(637\) 0 0
\(638\) 35.1246 1.39060
\(639\) −17.8885 −0.707660
\(640\) 2.43769 0.0963583
\(641\) 11.9443 0.471770 0.235885 0.971781i \(-0.424201\pi\)
0.235885 + 0.971781i \(0.424201\pi\)
\(642\) 57.1591 2.25589
\(643\) −34.8328 −1.37367 −0.686836 0.726812i \(-0.741001\pi\)
−0.686836 + 0.726812i \(0.741001\pi\)
\(644\) 0 0
\(645\) 40.0000 1.57500
\(646\) −11.5623 −0.454913
\(647\) −20.2361 −0.795562 −0.397781 0.917480i \(-0.630220\pi\)
−0.397781 + 0.917480i \(0.630220\pi\)
\(648\) 82.1935 3.22887
\(649\) −4.41641 −0.173359
\(650\) 0 0
\(651\) 0 0
\(652\) −79.6869 −3.12078
\(653\) 4.52786 0.177189 0.0885945 0.996068i \(-0.471762\pi\)
0.0885945 + 0.996068i \(0.471762\pi\)
\(654\) 15.8541 0.619944
\(655\) −18.4164 −0.719589
\(656\) 44.0689 1.72060
\(657\) 5.41641 0.211314
\(658\) 0 0
\(659\) −8.94427 −0.348419 −0.174210 0.984709i \(-0.555737\pi\)
−0.174210 + 0.984709i \(0.555737\pi\)
\(660\) 72.8115 2.83418
\(661\) 6.70820 0.260919 0.130459 0.991454i \(-0.458355\pi\)
0.130459 + 0.991454i \(0.458355\pi\)
\(662\) 4.14590 0.161135
\(663\) −3.29180 −0.127843
\(664\) 0 0
\(665\) 0 0
\(666\) −24.6525 −0.955264
\(667\) −36.8328 −1.42617
\(668\) −109.082 −4.22051
\(669\) 8.94427 0.345806
\(670\) −17.5623 −0.678491
\(671\) 9.00000 0.347441
\(672\) 0 0
\(673\) −9.41641 −0.362976 −0.181488 0.983393i \(-0.558091\pi\)
−0.181488 + 0.983393i \(0.558091\pi\)
\(674\) −47.1246 −1.81517
\(675\) 0 0
\(676\) 4.85410 0.186696
\(677\) 2.88854 0.111016 0.0555079 0.998458i \(-0.482322\pi\)
0.0555079 + 0.998458i \(0.482322\pi\)
\(678\) −17.2361 −0.661947
\(679\) 0 0
\(680\) −24.5967 −0.943242
\(681\) −13.2918 −0.509343
\(682\) −39.2705 −1.50375
\(683\) −13.4721 −0.515497 −0.257748 0.966212i \(-0.582981\pi\)
−0.257748 + 0.966212i \(0.582981\pi\)
\(684\) −29.1246 −1.11361
\(685\) 18.4164 0.703655
\(686\) 0 0
\(687\) −53.9443 −2.05810
\(688\) −78.8328 −3.00547
\(689\) −7.47214 −0.284666
\(690\) −107.812 −4.10431
\(691\) 51.8328 1.97181 0.985907 0.167297i \(-0.0535037\pi\)
0.985907 + 0.167297i \(0.0535037\pi\)
\(692\) 79.6869 3.02924
\(693\) 0 0
\(694\) −60.3951 −2.29257
\(695\) −52.3607 −1.98615
\(696\) −74.7214 −2.83231
\(697\) −6.58359 −0.249371
\(698\) −77.0132 −2.91499
\(699\) 26.7082 1.01020
\(700\) 0 0
\(701\) −22.3607 −0.844551 −0.422276 0.906467i \(-0.638769\pi\)
−0.422276 + 0.906467i \(0.638769\pi\)
\(702\) 5.85410 0.220949
\(703\) −14.1246 −0.532720
\(704\) −26.1246 −0.984608
\(705\) −37.3607 −1.40708
\(706\) 45.2705 1.70378
\(707\) 0 0
\(708\) 15.9787 0.600517
\(709\) −50.1246 −1.88247 −0.941235 0.337753i \(-0.890333\pi\)
−0.941235 + 0.337753i \(0.890333\pi\)
\(710\) −52.3607 −1.96506
\(711\) −5.41641 −0.203131
\(712\) 16.7082 0.626166
\(713\) 41.1803 1.54222
\(714\) 0 0
\(715\) 6.70820 0.250873
\(716\) −97.6869 −3.65073
\(717\) 43.4164 1.62142
\(718\) −31.2705 −1.16701
\(719\) −24.7082 −0.921461 −0.460730 0.887540i \(-0.652412\pi\)
−0.460730 + 0.887540i \(0.652412\pi\)
\(720\) −44.0689 −1.64235
\(721\) 0 0
\(722\) 26.1803 0.974331
\(723\) −10.5279 −0.391535
\(724\) 123.374 4.58515
\(725\) 0 0
\(726\) 11.7082 0.434532
\(727\) 38.8328 1.44023 0.720115 0.693855i \(-0.244089\pi\)
0.720115 + 0.693855i \(0.244089\pi\)
\(728\) 0 0
\(729\) −7.00000 −0.259259
\(730\) 15.8541 0.586787
\(731\) 11.7771 0.435591
\(732\) −32.5623 −1.20354
\(733\) −28.7082 −1.06036 −0.530181 0.847885i \(-0.677876\pi\)
−0.530181 + 0.847885i \(0.677876\pi\)
\(734\) −33.2705 −1.22804
\(735\) 0 0
\(736\) 89.3951 3.29515
\(737\) 9.00000 0.331519
\(738\) −23.4164 −0.861970
\(739\) −17.8328 −0.655991 −0.327995 0.944679i \(-0.606373\pi\)
−0.327995 + 0.944679i \(0.606373\pi\)
\(740\) −51.1033 −1.87860
\(741\) −6.70820 −0.246432
\(742\) 0 0
\(743\) 32.9443 1.20861 0.604304 0.796754i \(-0.293451\pi\)
0.604304 + 0.796754i \(0.293451\pi\)
\(744\) 83.5410 3.06276
\(745\) −1.58359 −0.0580183
\(746\) 4.14590 0.151792
\(747\) 0 0
\(748\) 21.4377 0.783840
\(749\) 0 0
\(750\) −65.4508 −2.38993
\(751\) −10.1246 −0.369452 −0.184726 0.982790i \(-0.559140\pi\)
−0.184726 + 0.982790i \(0.559140\pi\)
\(752\) 73.6312 2.68505
\(753\) −3.41641 −0.124501
\(754\) −11.7082 −0.426388
\(755\) −45.6525 −1.66146
\(756\) 0 0
\(757\) −52.8328 −1.92024 −0.960121 0.279586i \(-0.909803\pi\)
−0.960121 + 0.279586i \(0.909803\pi\)
\(758\) 40.3607 1.46597
\(759\) 55.2492 2.00542
\(760\) −50.1246 −1.81821
\(761\) 33.5410 1.21586 0.607931 0.793990i \(-0.292000\pi\)
0.607931 + 0.793990i \(0.292000\pi\)
\(762\) 66.8328 2.42110
\(763\) 0 0
\(764\) 54.2705 1.96344
\(765\) 6.58359 0.238030
\(766\) 39.2705 1.41890
\(767\) 1.47214 0.0531557
\(768\) −32.5623 −1.17499
\(769\) −46.0000 −1.65880 −0.829401 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) 0.124612 0.00448778
\(772\) 3.43769 0.123725
\(773\) 47.0689 1.69295 0.846475 0.532428i \(-0.178720\pi\)
0.846475 + 0.532428i \(0.178720\pi\)
\(774\) 41.8885 1.50565
\(775\) 0 0
\(776\) 70.3607 2.52580
\(777\) 0 0
\(778\) −3.85410 −0.138176
\(779\) −13.4164 −0.480693
\(780\) −24.2705 −0.869024
\(781\) 26.8328 0.960154
\(782\) −31.7426 −1.13511
\(783\) −10.0000 −0.357371
\(784\) 0 0
\(785\) 15.6525 0.558661
\(786\) −48.2148 −1.71976
\(787\) −20.4164 −0.727766 −0.363883 0.931445i \(-0.618549\pi\)
−0.363883 + 0.931445i \(0.618549\pi\)
\(788\) −43.9574 −1.56592
\(789\) 58.4164 2.07968
\(790\) −15.8541 −0.564064
\(791\) 0 0
\(792\) 44.8328 1.59306
\(793\) −3.00000 −0.106533
\(794\) −68.3951 −2.42725
\(795\) 37.3607 1.32505
\(796\) 100.520 3.56283
\(797\) −9.05573 −0.320770 −0.160385 0.987055i \(-0.551274\pi\)
−0.160385 + 0.987055i \(0.551274\pi\)
\(798\) 0 0
\(799\) −11.0000 −0.389152
\(800\) 0 0
\(801\) −4.47214 −0.158015
\(802\) −37.2705 −1.31607
\(803\) −8.12461 −0.286711
\(804\) −32.5623 −1.14838
\(805\) 0 0
\(806\) 13.0902 0.461082
\(807\) −30.1246 −1.06044
\(808\) −67.2492 −2.36582
\(809\) 22.4164 0.788119 0.394059 0.919085i \(-0.371070\pi\)
0.394059 + 0.919085i \(0.371070\pi\)
\(810\) −64.3951 −2.26261
\(811\) 14.8328 0.520851 0.260425 0.965494i \(-0.416137\pi\)
0.260425 + 0.965494i \(0.416137\pi\)
\(812\) 0 0
\(813\) 45.6525 1.60110
\(814\) 36.9787 1.29610
\(815\) 36.7082 1.28583
\(816\) −32.4377 −1.13555
\(817\) 24.0000 0.839654
\(818\) 22.7984 0.797126
\(819\) 0 0
\(820\) −48.5410 −1.69513
\(821\) 38.2361 1.33445 0.667224 0.744857i \(-0.267482\pi\)
0.667224 + 0.744857i \(0.267482\pi\)
\(822\) 48.2148 1.68168
\(823\) 34.1246 1.18951 0.594755 0.803907i \(-0.297249\pi\)
0.594755 + 0.803907i \(0.297249\pi\)
\(824\) −20.2361 −0.704957
\(825\) 0 0
\(826\) 0 0
\(827\) 26.8328 0.933068 0.466534 0.884503i \(-0.345502\pi\)
0.466534 + 0.884503i \(0.345502\pi\)
\(828\) −79.9574 −2.77871
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 0 0
\(831\) −0.931116 −0.0323001
\(832\) 8.70820 0.301903
\(833\) 0 0
\(834\) −137.082 −4.74676
\(835\) 50.2492 1.73895
\(836\) 43.6869 1.51094
\(837\) 11.1803 0.386449
\(838\) 86.2492 2.97943
\(839\) 5.88854 0.203295 0.101648 0.994820i \(-0.467589\pi\)
0.101648 + 0.994820i \(0.467589\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) −35.1246 −1.21047
\(843\) 60.2492 2.07509
\(844\) 19.4164 0.668340
\(845\) −2.23607 −0.0769231
\(846\) −39.1246 −1.34513
\(847\) 0 0
\(848\) −73.6312 −2.52851
\(849\) −58.4164 −2.00485
\(850\) 0 0
\(851\) −38.7771 −1.32926
\(852\) −97.0820 −3.32598
\(853\) 52.2492 1.78898 0.894490 0.447089i \(-0.147539\pi\)
0.894490 + 0.447089i \(0.147539\pi\)
\(854\) 0 0
\(855\) 13.4164 0.458831
\(856\) 72.9574 2.49363
\(857\) 1.36068 0.0464799 0.0232400 0.999730i \(-0.492602\pi\)
0.0232400 + 0.999730i \(0.492602\pi\)
\(858\) 17.5623 0.599567
\(859\) −20.7082 −0.706555 −0.353277 0.935519i \(-0.614933\pi\)
−0.353277 + 0.935519i \(0.614933\pi\)
\(860\) 86.8328 2.96097
\(861\) 0 0
\(862\) 82.1033 2.79645
\(863\) 23.9443 0.815072 0.407536 0.913189i \(-0.366388\pi\)
0.407536 + 0.913189i \(0.366388\pi\)
\(864\) 24.2705 0.825700
\(865\) −36.7082 −1.24812
\(866\) 77.0132 2.61701
\(867\) −33.1672 −1.12642
\(868\) 0 0
\(869\) 8.12461 0.275609
\(870\) 58.5410 1.98473
\(871\) −3.00000 −0.101651
\(872\) 20.2361 0.685280
\(873\) −18.8328 −0.637394
\(874\) −64.6869 −2.18807
\(875\) 0 0
\(876\) 29.3951 0.993169
\(877\) 32.1246 1.08477 0.542386 0.840130i \(-0.317521\pi\)
0.542386 + 0.840130i \(0.317521\pi\)
\(878\) 63.1591 2.13151
\(879\) 33.4164 1.12711
\(880\) 66.1033 2.22834
\(881\) −43.3050 −1.45898 −0.729490 0.683991i \(-0.760243\pi\)
−0.729490 + 0.683991i \(0.760243\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −7.14590 −0.240343
\(885\) −7.36068 −0.247427
\(886\) −5.85410 −0.196672
\(887\) 20.2361 0.679461 0.339730 0.940523i \(-0.389664\pi\)
0.339730 + 0.940523i \(0.389664\pi\)
\(888\) −78.6656 −2.63985
\(889\) 0 0
\(890\) −13.0902 −0.438783
\(891\) 33.0000 1.10554
\(892\) 19.4164 0.650109
\(893\) −22.4164 −0.750136
\(894\) −4.14590 −0.138660
\(895\) 45.0000 1.50418
\(896\) 0 0
\(897\) −18.4164 −0.614906
\(898\) 89.9574 3.00192
\(899\) −22.3607 −0.745770
\(900\) 0 0
\(901\) 11.0000 0.366463
\(902\) 35.1246 1.16952
\(903\) 0 0
\(904\) −22.0000 −0.731709
\(905\) −56.8328 −1.88919
\(906\) −119.520 −3.97078
\(907\) 18.7082 0.621196 0.310598 0.950541i \(-0.399471\pi\)
0.310598 + 0.950541i \(0.399471\pi\)
\(908\) −28.8541 −0.957557
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) −34.2492 −1.13473 −0.567364 0.823467i \(-0.692037\pi\)
−0.567364 + 0.823467i \(0.692037\pi\)
\(912\) −66.1033 −2.18890
\(913\) 0 0
\(914\) 16.0344 0.530372
\(915\) 15.0000 0.495885
\(916\) −117.103 −3.86920
\(917\) 0 0
\(918\) −8.61803 −0.284438
\(919\) 20.1246 0.663850 0.331925 0.943306i \(-0.392302\pi\)
0.331925 + 0.943306i \(0.392302\pi\)
\(920\) −137.610 −4.53686
\(921\) −43.4164 −1.43062
\(922\) 89.9574 2.96259
\(923\) −8.94427 −0.294404
\(924\) 0 0
\(925\) 0 0
\(926\) 62.8328 2.06481
\(927\) 5.41641 0.177898
\(928\) −48.5410 −1.59344
\(929\) 24.8197 0.814307 0.407153 0.913360i \(-0.366521\pi\)
0.407153 + 0.913360i \(0.366521\pi\)
\(930\) −65.4508 −2.14622
\(931\) 0 0
\(932\) 57.9787 1.89916
\(933\) 62.0820 2.03247
\(934\) 25.2705 0.826876
\(935\) −9.87539 −0.322960
\(936\) −14.9443 −0.488469
\(937\) −25.4164 −0.830318 −0.415159 0.909749i \(-0.636274\pi\)
−0.415159 + 0.909749i \(0.636274\pi\)
\(938\) 0 0
\(939\) 12.4853 0.407442
\(940\) −81.1033 −2.64530
\(941\) −50.2361 −1.63765 −0.818825 0.574044i \(-0.805374\pi\)
−0.818825 + 0.574044i \(0.805374\pi\)
\(942\) 40.9787 1.33516
\(943\) −36.8328 −1.19944
\(944\) 14.5066 0.472149
\(945\) 0 0
\(946\) −62.8328 −2.04287
\(947\) −22.5279 −0.732057 −0.366029 0.930604i \(-0.619283\pi\)
−0.366029 + 0.930604i \(0.619283\pi\)
\(948\) −29.3951 −0.954709
\(949\) 2.70820 0.0879120
\(950\) 0 0
\(951\) −18.4164 −0.597193
\(952\) 0 0
\(953\) −41.7771 −1.35329 −0.676646 0.736308i \(-0.736567\pi\)
−0.676646 + 0.736308i \(0.736567\pi\)
\(954\) 39.1246 1.26671
\(955\) −25.0000 −0.808981
\(956\) 94.2492 3.04824
\(957\) −30.0000 −0.969762
\(958\) 62.3951 2.01589
\(959\) 0 0
\(960\) −43.5410 −1.40528
\(961\) −6.00000 −0.193548
\(962\) −12.3262 −0.397414
\(963\) −19.5279 −0.629277
\(964\) −22.8541 −0.736081
\(965\) −1.58359 −0.0509776
\(966\) 0 0
\(967\) 16.5836 0.533292 0.266646 0.963794i \(-0.414084\pi\)
0.266646 + 0.963794i \(0.414084\pi\)
\(968\) 14.9443 0.480327
\(969\) 9.87539 0.317243
\(970\) −55.1246 −1.76994
\(971\) 11.2918 0.362371 0.181185 0.983449i \(-0.442007\pi\)
0.181185 + 0.983449i \(0.442007\pi\)
\(972\) −86.8328 −2.78516
\(973\) 0 0
\(974\) 57.1591 1.83149
\(975\) 0 0
\(976\) −29.5623 −0.946266
\(977\) −2.34752 −0.0751040 −0.0375520 0.999295i \(-0.511956\pi\)
−0.0375520 + 0.999295i \(0.511956\pi\)
\(978\) 96.1033 3.07305
\(979\) 6.70820 0.214395
\(980\) 0 0
\(981\) −5.41641 −0.172933
\(982\) 66.8328 2.13272
\(983\) −7.47214 −0.238324 −0.119162 0.992875i \(-0.538021\pi\)
−0.119162 + 0.992875i \(0.538021\pi\)
\(984\) −74.7214 −2.38203
\(985\) 20.2492 0.645194
\(986\) 17.2361 0.548908
\(987\) 0 0
\(988\) −14.5623 −0.463289
\(989\) 65.8885 2.09513
\(990\) −35.1246 −1.11633
\(991\) 30.7082 0.975478 0.487739 0.872989i \(-0.337822\pi\)
0.487739 + 0.872989i \(0.337822\pi\)
\(992\) 54.2705 1.72309
\(993\) −3.54102 −0.112371
\(994\) 0 0
\(995\) −46.3050 −1.46797
\(996\) 0 0
\(997\) −26.4164 −0.836616 −0.418308 0.908305i \(-0.637377\pi\)
−0.418308 + 0.908305i \(0.637377\pi\)
\(998\) −69.1591 −2.18919
\(999\) −10.5279 −0.333087
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 637.2.a.e.1.1 2
3.2 odd 2 5733.2.a.w.1.2 2
7.2 even 3 637.2.e.h.508.2 4
7.3 odd 6 91.2.e.b.79.2 yes 4
7.4 even 3 637.2.e.h.79.2 4
7.5 odd 6 91.2.e.b.53.2 4
7.6 odd 2 637.2.a.f.1.1 2
13.12 even 2 8281.2.a.ba.1.2 2
21.5 even 6 819.2.j.c.235.1 4
21.17 even 6 819.2.j.c.352.1 4
21.20 even 2 5733.2.a.v.1.2 2
28.3 even 6 1456.2.r.j.625.1 4
28.19 even 6 1456.2.r.j.417.1 4
91.12 odd 6 1183.2.e.d.508.1 4
91.38 odd 6 1183.2.e.d.170.1 4
91.90 odd 2 8281.2.a.z.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.b.53.2 4 7.5 odd 6
91.2.e.b.79.2 yes 4 7.3 odd 6
637.2.a.e.1.1 2 1.1 even 1 trivial
637.2.a.f.1.1 2 7.6 odd 2
637.2.e.h.79.2 4 7.4 even 3
637.2.e.h.508.2 4 7.2 even 3
819.2.j.c.235.1 4 21.5 even 6
819.2.j.c.352.1 4 21.17 even 6
1183.2.e.d.170.1 4 91.38 odd 6
1183.2.e.d.508.1 4 91.12 odd 6
1456.2.r.j.417.1 4 28.19 even 6
1456.2.r.j.625.1 4 28.3 even 6
5733.2.a.v.1.2 2 21.20 even 2
5733.2.a.w.1.2 2 3.2 odd 2
8281.2.a.z.1.2 2 91.90 odd 2
8281.2.a.ba.1.2 2 13.12 even 2