Properties

Label 5733.2.a.v.1.2
Level $5733$
Weight $2$
Character 5733.1
Self dual yes
Analytic conductor $45.778$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5733.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.7782354788\)
Analytic rank: \(0\)
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 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 5733.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.61803 q^{2} +4.85410 q^{4} -2.23607 q^{5} +7.47214 q^{8} +O(q^{10})\) \(q+2.61803 q^{2} +4.85410 q^{4} -2.23607 q^{5} +7.47214 q^{8} -5.85410 q^{10} +3.00000 q^{11} -1.00000 q^{13} +9.85410 q^{16} -1.47214 q^{17} +3.00000 q^{19} -10.8541 q^{20} +7.85410 q^{22} +8.23607 q^{23} -2.61803 q^{26} -4.47214 q^{29} +5.00000 q^{31} +10.8541 q^{32} -3.85410 q^{34} +4.70820 q^{37} +7.85410 q^{38} -16.7082 q^{40} +4.47214 q^{41} -8.00000 q^{43} +14.5623 q^{44} +21.5623 q^{46} +7.47214 q^{47} -4.85410 q^{52} +7.47214 q^{53} -6.70820 q^{55} -11.7082 q^{58} +1.47214 q^{59} +3.00000 q^{61} +13.0902 q^{62} +8.70820 q^{64} +2.23607 q^{65} -3.00000 q^{67} -7.14590 q^{68} +8.94427 q^{71} -2.70820 q^{73} +12.3262 q^{74} +14.5623 q^{76} -2.70820 q^{79} -22.0344 q^{80} +11.7082 q^{82} +3.29180 q^{85} -20.9443 q^{86} +22.4164 q^{88} -2.23607 q^{89} +39.9787 q^{92} +19.5623 q^{94} -6.70820 q^{95} +9.41641 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} + O(q^{10}) \) \( 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} - 5 q^{10} + 6 q^{11} - 2 q^{13} + 13 q^{16} + 6 q^{17} + 6 q^{19} - 15 q^{20} + 9 q^{22} + 12 q^{23} - 3 q^{26} + 10 q^{31} + 15 q^{32} - q^{34} - 4 q^{37} + 9 q^{38} - 20 q^{40} - 16 q^{43} + 9 q^{44} + 23 q^{46} + 6 q^{47} - 3 q^{52} + 6 q^{53} - 10 q^{58} - 6 q^{59} + 6 q^{61} + 15 q^{62} + 4 q^{64} - 6 q^{67} - 21 q^{68} + 8 q^{73} + 9 q^{74} + 9 q^{76} + 8 q^{79} - 15 q^{80} + 10 q^{82} + 20 q^{85} - 24 q^{86} + 18 q^{88} + 33 q^{92} + 19 q^{94} - 8 q^{97} + O(q^{100}) \)

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 \(-0.123549\pi\)
0.925615 + 0.378467i \(0.123549\pi\)
\(3\) 0 0
\(4\) 4.85410 2.42705
\(5\) −2.23607 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 7.47214 2.64180
\(9\) 0 0
\(10\) −5.85410 −1.85123
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 9.85410 2.46353
\(17\) −1.47214 −0.357045 −0.178523 0.983936i \(-0.557132\pi\)
−0.178523 + 0.983936i \(0.557132\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) −10.8541 −2.42705
\(21\) 0 0
\(22\) 7.85410 1.67450
\(23\) 8.23607 1.71734 0.858669 0.512530i \(-0.171292\pi\)
0.858669 + 0.512530i \(0.171292\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.61803 −0.513439
\(27\) 0 0
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 10.8541 1.91875
\(33\) 0 0
\(34\) −3.85410 −0.660973
\(35\) 0 0
\(36\) 0 0
\(37\) 4.70820 0.774024 0.387012 0.922075i \(-0.373507\pi\)
0.387012 + 0.922075i \(0.373507\pi\)
\(38\) 7.85410 1.27410
\(39\) 0 0
\(40\) −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
\(45\) 0 0
\(46\) 21.5623 3.17919
\(47\) 7.47214 1.08992 0.544962 0.838461i \(-0.316544\pi\)
0.544962 + 0.838461i \(0.316544\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) −4.85410 −0.673143
\(53\) 7.47214 1.02638 0.513188 0.858276i \(-0.328464\pi\)
0.513188 + 0.858276i \(0.328464\pi\)
\(54\) 0 0
\(55\) −6.70820 −0.904534
\(56\) 0 0
\(57\) 0 0
\(58\) −11.7082 −1.53736
\(59\) 1.47214 0.191656 0.0958279 0.995398i \(-0.469450\pi\)
0.0958279 + 0.995398i \(0.469450\pi\)
\(60\) 0 0
\(61\) 3.00000 0.384111 0.192055 0.981384i \(-0.438485\pi\)
0.192055 + 0.981384i \(0.438485\pi\)
\(62\) 13.0902 1.66245
\(63\) 0 0
\(64\) 8.70820 1.08853
\(65\) 2.23607 0.277350
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) −7.14590 −0.866567
\(69\) 0 0
\(70\) 0 0
\(71\) 8.94427 1.06149 0.530745 0.847532i \(-0.321912\pi\)
0.530745 + 0.847532i \(0.321912\pi\)
\(72\) 0 0
\(73\) −2.70820 −0.316971 −0.158486 0.987361i \(-0.550661\pi\)
−0.158486 + 0.987361i \(0.550661\pi\)
\(74\) 12.3262 1.43290
\(75\) 0 0
\(76\) 14.5623 1.67041
\(77\) 0 0
\(78\) 0 0
\(79\) −2.70820 −0.304697 −0.152348 0.988327i \(-0.548684\pi\)
−0.152348 + 0.988327i \(0.548684\pi\)
\(80\) −22.0344 −2.46353
\(81\) 0 0
\(82\) 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
\(87\) 0 0
\(88\) 22.4164 2.38960
\(89\) −2.23607 −0.237023 −0.118511 0.992953i \(-0.537812\pi\)
−0.118511 + 0.992953i \(0.537812\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 39.9787 4.16807
\(93\) 0 0
\(94\) 19.5623 2.01770
\(95\) −6.70820 −0.688247
\(96\) 0 0
\(97\) 9.41641 0.956091 0.478046 0.878335i \(-0.341345\pi\)
0.478046 + 0.878335i \(0.341345\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) 0 0
\(103\) −2.70820 −0.266847 −0.133424 0.991059i \(-0.542597\pi\)
−0.133424 + 0.991059i \(0.542597\pi\)
\(104\) −7.47214 −0.732703
\(105\) 0 0
\(106\) 19.5623 1.90006
\(107\) 9.76393 0.943915 0.471957 0.881621i \(-0.343548\pi\)
0.471957 + 0.881621i \(0.343548\pi\)
\(108\) 0 0
\(109\) −2.70820 −0.259399 −0.129699 0.991553i \(-0.541401\pi\)
−0.129699 + 0.991553i \(0.541401\pi\)
\(110\) −17.5623 −1.67450
\(111\) 0 0
\(112\) 0 0
\(113\) −2.94427 −0.276974 −0.138487 0.990364i \(-0.544224\pi\)
−0.138487 + 0.990364i \(0.544224\pi\)
\(114\) 0 0
\(115\) −18.4164 −1.71734
\(116\) −21.7082 −2.01556
\(117\) 0 0
\(118\) 3.85410 0.354799
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 7.85410 0.711077
\(123\) 0 0
\(124\) 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
\(129\) 0 0
\(130\) 5.85410 0.513439
\(131\) 8.23607 0.719589 0.359794 0.933032i \(-0.382847\pi\)
0.359794 + 0.933032i \(0.382847\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −7.85410 −0.678491
\(135\) 0 0
\(136\) −11.0000 −0.943242
\(137\) 8.23607 0.703655 0.351827 0.936065i \(-0.385560\pi\)
0.351827 + 0.936065i \(0.385560\pi\)
\(138\) 0 0
\(139\) −23.4164 −1.98615 −0.993077 0.117466i \(-0.962523\pi\)
−0.993077 + 0.117466i \(0.962523\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 23.4164 1.96506
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) 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 \(-0.509235\pi\)
−0.0290092 + 0.999579i \(0.509235\pi\)
\(150\) 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
\(153\) 0 0
\(154\) 0 0
\(155\) −11.1803 −0.898027
\(156\) 0 0
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) −7.09017 −0.564064
\(159\) 0 0
\(160\) −24.2705 −1.91875
\(161\) 0 0
\(162\) 0 0
\(163\) −16.4164 −1.28583 −0.642916 0.765937i \(-0.722276\pi\)
−0.642916 + 0.765937i \(0.722276\pi\)
\(164\) 21.7082 1.69513
\(165\) 0 0
\(166\) 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
\(171\) 0 0
\(172\) −38.8328 −2.96097
\(173\) 16.4164 1.24812 0.624058 0.781378i \(-0.285483\pi\)
0.624058 + 0.781378i \(0.285483\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 29.5623 2.22834
\(177\) 0 0
\(178\) −5.85410 −0.438783
\(179\) 20.1246 1.50418 0.752092 0.659058i \(-0.229045\pi\)
0.752092 + 0.659058i \(0.229045\pi\)
\(180\) 0 0
\(181\) −25.4164 −1.88919 −0.944593 0.328243i \(-0.893544\pi\)
−0.944593 + 0.328243i \(0.893544\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 61.5410 4.53686
\(185\) −10.5279 −0.774024
\(186\) 0 0
\(187\) −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 \(-0.632551\pi\)
−0.404491 + 0.914542i \(0.632551\pi\)
\(192\) 0 0
\(193\) 0.708204 0.0509776 0.0254888 0.999675i \(-0.491886\pi\)
0.0254888 + 0.999675i \(0.491886\pi\)
\(194\) 24.6525 1.76994
\(195\) 0 0
\(196\) 0 0
\(197\) 9.05573 0.645194 0.322597 0.946536i \(-0.395444\pi\)
0.322597 + 0.946536i \(0.395444\pi\)
\(198\) 0 0
\(199\) −20.7082 −1.46797 −0.733983 0.679168i \(-0.762341\pi\)
−0.733983 + 0.679168i \(0.762341\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 23.5623 1.65784
\(203\) 0 0
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) −7.09017 −0.493996
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) 25.5623 1.74740
\(215\) 17.8885 1.21999
\(216\) 0 0
\(217\) 0 0
\(218\) −7.09017 −0.480207
\(219\) 0 0
\(220\) −32.5623 −2.19535
\(221\) 1.47214 0.0990266
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 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 \(0.563207\pi\)
\(228\) 0 0
\(229\) 24.1246 1.59420 0.797100 0.603848i \(-0.206367\pi\)
0.797100 + 0.603848i \(0.206367\pi\)
\(230\) −48.2148 −3.17919
\(231\) 0 0
\(232\) −33.4164 −2.19389
\(233\) −11.9443 −0.782495 −0.391248 0.920285i \(-0.627956\pi\)
−0.391248 + 0.920285i \(0.627956\pi\)
\(234\) 0 0
\(235\) −16.7082 −1.08992
\(236\) 7.14590 0.465158
\(237\) 0 0
\(238\) 0 0
\(239\) −19.4164 −1.25594 −0.627972 0.778236i \(-0.716115\pi\)
−0.627972 + 0.778236i \(0.716115\pi\)
\(240\) 0 0
\(241\) 4.70820 0.303282 0.151641 0.988436i \(-0.451544\pi\)
0.151641 + 0.988436i \(0.451544\pi\)
\(242\) −5.23607 −0.336587
\(243\) 0 0
\(244\) 14.5623 0.932256
\(245\) 0 0
\(246\) 0 0
\(247\) −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
\(255\) 0 0
\(256\) −14.5623 −0.910144
\(257\) 0.0557281 0.00347622 0.00173811 0.999998i \(-0.499447\pi\)
0.00173811 + 0.999998i \(0.499447\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 10.8541 0.673143
\(261\) 0 0
\(262\) 21.5623 1.33212
\(263\) −26.1246 −1.61091 −0.805456 0.592655i \(-0.798080\pi\)
−0.805456 + 0.592655i \(0.798080\pi\)
\(264\) 0 0
\(265\) −16.7082 −1.02638
\(266\) 0 0
\(267\) 0 0
\(268\) −14.5623 −0.889534
\(269\) −13.4721 −0.821411 −0.410705 0.911768i \(-0.634717\pi\)
−0.410705 + 0.911768i \(0.634717\pi\)
\(270\) 0 0
\(271\) −20.4164 −1.24021 −0.620104 0.784519i \(-0.712910\pi\)
−0.620104 + 0.784519i \(0.712910\pi\)
\(272\) −14.5066 −0.879590
\(273\) 0 0
\(274\) 21.5623 1.30263
\(275\) 0 0
\(276\) 0 0
\(277\) −0.416408 −0.0250195 −0.0125098 0.999922i \(-0.503982\pi\)
−0.0125098 + 0.999922i \(0.503982\pi\)
\(278\) −61.3050 −3.67683
\(279\) 0 0
\(280\) 0 0
\(281\) −26.9443 −1.60736 −0.803680 0.595061i \(-0.797128\pi\)
−0.803680 + 0.595061i \(0.797128\pi\)
\(282\) 0 0
\(283\) 26.1246 1.55295 0.776473 0.630150i \(-0.217007\pi\)
0.776473 + 0.630150i \(0.217007\pi\)
\(284\) 43.4164 2.57629
\(285\) 0 0
\(286\) −7.85410 −0.464423
\(287\) 0 0
\(288\) 0 0
\(289\) −14.8328 −0.872519
\(290\) 26.1803 1.53736
\(291\) 0 0
\(292\) −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
\(297\) 0 0
\(298\) −1.85410 −0.107405
\(299\) −8.23607 −0.476304
\(300\) 0 0
\(301\) 0 0
\(302\) 53.4508 3.07575
\(303\) 0 0
\(304\) 29.5623 1.69551
\(305\) −6.70820 −0.384111
\(306\) 0 0
\(307\) 19.4164 1.10815 0.554076 0.832466i \(-0.313072\pi\)
0.554076 + 0.832466i \(0.313072\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −29.2705 −1.66245
\(311\) 27.7639 1.57435 0.787174 0.616731i \(-0.211543\pi\)
0.787174 + 0.616731i \(0.211543\pi\)
\(312\) 0 0
\(313\) −5.58359 −0.315603 −0.157802 0.987471i \(-0.550441\pi\)
−0.157802 + 0.987471i \(0.550441\pi\)
\(314\) 18.3262 1.03421
\(315\) 0 0
\(316\) −13.1459 −0.739515
\(317\) 8.23607 0.462584 0.231292 0.972884i \(-0.425705\pi\)
0.231292 + 0.972884i \(0.425705\pi\)
\(318\) 0 0
\(319\) −13.4164 −0.751175
\(320\) −19.4721 −1.08853
\(321\) 0 0
\(322\) 0 0
\(323\) −4.41641 −0.245736
\(324\) 0 0
\(325\) 0 0
\(326\) −42.9787 −2.38037
\(327\) 0 0
\(328\) 33.4164 1.84511
\(329\) 0 0
\(330\) 0 0
\(331\) −1.58359 −0.0870421 −0.0435210 0.999053i \(-0.513858\pi\)
−0.0435210 + 0.999053i \(0.513858\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −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
\(339\) 0 0
\(340\) 15.9787 0.866567
\(341\) 15.0000 0.812296
\(342\) 0 0
\(343\) 0 0
\(344\) −59.7771 −3.22296
\(345\) 0 0
\(346\) 42.9787 2.31055
\(347\) −23.0689 −1.23840 −0.619201 0.785232i \(-0.712544\pi\)
−0.619201 + 0.785232i \(0.712544\pi\)
\(348\) 0 0
\(349\) −29.4164 −1.57462 −0.787312 0.616555i \(-0.788528\pi\)
−0.787312 + 0.616555i \(0.788528\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 32.5623 1.73558
\(353\) −17.2918 −0.920349 −0.460175 0.887828i \(-0.652213\pi\)
−0.460175 + 0.887828i \(0.652213\pi\)
\(354\) 0 0
\(355\) −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 \(-0.602071\pi\)
−0.315197 + 0.949026i \(0.602071\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) −66.5410 −3.49732
\(363\) 0 0
\(364\) 0 0
\(365\) 6.05573 0.316971
\(366\) 0 0
\(367\) −12.7082 −0.663363 −0.331681 0.943391i \(-0.607616\pi\)
−0.331681 + 0.943391i \(0.607616\pi\)
\(368\) 81.1591 4.23071
\(369\) 0 0
\(370\) −27.5623 −1.43290
\(371\) 0 0
\(372\) 0 0
\(373\) −1.58359 −0.0819953 −0.0409976 0.999159i \(-0.513054\pi\)
−0.0409976 + 0.999159i \(0.513054\pi\)
\(374\) −11.5623 −0.597873
\(375\) 0 0
\(376\) 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
\(381\) 0 0
\(382\) −29.2705 −1.49761
\(383\) −15.0000 −0.766464 −0.383232 0.923652i \(-0.625189\pi\)
−0.383232 + 0.923652i \(0.625189\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.85410 0.0943713
\(387\) 0 0
\(388\) 45.7082 2.32048
\(389\) −1.47214 −0.0746403 −0.0373201 0.999303i \(-0.511882\pi\)
−0.0373201 + 0.999303i \(0.511882\pi\)
\(390\) 0 0
\(391\) −12.1246 −0.613168
\(392\) 0 0
\(393\) 0 0
\(394\) 23.7082 1.19440
\(395\) 6.05573 0.304697
\(396\) 0 0
\(397\) −26.1246 −1.31116 −0.655578 0.755127i \(-0.727575\pi\)
−0.655578 + 0.755127i \(0.727575\pi\)
\(398\) −54.2148 −2.71754
\(399\) 0 0
\(400\) 0 0
\(401\) −14.2361 −0.710915 −0.355458 0.934692i \(-0.615675\pi\)
−0.355458 + 0.934692i \(0.615675\pi\)
\(402\) 0 0
\(403\) −5.00000 −0.249068
\(404\) 43.6869 2.17351
\(405\) 0 0
\(406\) 0 0
\(407\) 14.1246 0.700131
\(408\) 0 0
\(409\) 8.70820 0.430593 0.215296 0.976549i \(-0.430928\pi\)
0.215296 + 0.976549i \(0.430928\pi\)
\(410\) −26.1803 −1.29295
\(411\) 0 0
\(412\) −13.1459 −0.647652
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −10.8541 −0.532166
\(417\) 0 0
\(418\) 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
\(423\) 0 0
\(424\) 55.8328 2.71148
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 47.3951 2.29093
\(429\) 0 0
\(430\) 46.8328 2.25848
\(431\) 31.3607 1.51059 0.755295 0.655385i \(-0.227493\pi\)
0.755295 + 0.655385i \(0.227493\pi\)
\(432\) 0 0
\(433\) 29.4164 1.41366 0.706831 0.707382i \(-0.250124\pi\)
0.706831 + 0.707382i \(0.250124\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −13.1459 −0.629574
\(437\) 24.7082 1.18195
\(438\) 0 0
\(439\) 24.1246 1.15140 0.575702 0.817659i \(-0.304729\pi\)
0.575702 + 0.817659i \(0.304729\pi\)
\(440\) −50.1246 −2.38960
\(441\) 0 0
\(442\) 3.85410 0.183321
\(443\) −2.23607 −0.106239 −0.0531194 0.998588i \(-0.516916\pi\)
−0.0531194 + 0.998588i \(0.516916\pi\)
\(444\) 0 0
\(445\) 5.00000 0.237023
\(446\) −10.4721 −0.495870
\(447\) 0 0
\(448\) 0 0
\(449\) 34.3607 1.62158 0.810790 0.585337i \(-0.199038\pi\)
0.810790 + 0.585337i \(0.199038\pi\)
\(450\) 0 0
\(451\) 13.4164 0.631754
\(452\) −14.2918 −0.672230
\(453\) 0 0
\(454\) −15.5623 −0.730375
\(455\) 0 0
\(456\) 0 0
\(457\) −6.12461 −0.286497 −0.143249 0.989687i \(-0.545755\pi\)
−0.143249 + 0.989687i \(0.545755\pi\)
\(458\) 63.1591 2.95123
\(459\) 0 0
\(460\) −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
\(465\) 0 0
\(466\) −31.2705 −1.44858
\(467\) −9.65248 −0.446663 −0.223332 0.974743i \(-0.571693\pi\)
−0.223332 + 0.974743i \(0.571693\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −43.7426 −2.01770
\(471\) 0 0
\(472\) 11.0000 0.506316
\(473\) −24.0000 −1.10352
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −50.8328 −2.32504
\(479\) −23.8328 −1.08895 −0.544475 0.838777i \(-0.683271\pi\)
−0.544475 + 0.838777i \(0.683271\pi\)
\(480\) 0 0
\(481\) −4.70820 −0.214676
\(482\) 12.3262 0.561445
\(483\) 0 0
\(484\) −9.70820 −0.441282
\(485\) −21.0557 −0.956091
\(486\) 0 0
\(487\) −21.8328 −0.989339 −0.494670 0.869081i \(-0.664711\pi\)
−0.494670 + 0.869081i \(0.664711\pi\)
\(488\) 22.4164 1.01474
\(489\) 0 0
\(490\) 0 0
\(491\) 25.5279 1.15206 0.576028 0.817430i \(-0.304602\pi\)
0.576028 + 0.817430i \(0.304602\pi\)
\(492\) 0 0
\(493\) 6.58359 0.296510
\(494\) −7.85410 −0.353373
\(495\) 0 0
\(496\) 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
\(501\) 0 0
\(502\) −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
\(507\) 0 0
\(508\) −55.4164 −2.45871
\(509\) 20.2361 0.896948 0.448474 0.893796i \(-0.351968\pi\)
0.448474 + 0.893796i \(0.351968\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −40.3050 −1.78124
\(513\) 0 0
\(514\) 0.145898 0.00643529
\(515\) 6.05573 0.266847
\(516\) 0 0
\(517\) 22.4164 0.985872
\(518\) 0 0
\(519\) 0 0
\(520\) 16.7082 0.732703
\(521\) 17.9443 0.786153 0.393076 0.919506i \(-0.371411\pi\)
0.393076 + 0.919506i \(0.371411\pi\)
\(522\) 0 0
\(523\) 32.7082 1.43023 0.715115 0.699007i \(-0.246374\pi\)
0.715115 + 0.699007i \(0.246374\pi\)
\(524\) 39.9787 1.74648
\(525\) 0 0
\(526\) −68.3951 −2.98217
\(527\) −7.36068 −0.320636
\(528\) 0 0
\(529\) 44.8328 1.94925
\(530\) −43.7426 −1.90006
\(531\) 0 0
\(532\) 0 0
\(533\) −4.47214 −0.193710
\(534\) 0 0
\(535\) −21.8328 −0.943915
\(536\) −22.4164 −0.968241
\(537\) 0 0
\(538\) −35.2705 −1.52062
\(539\) 0 0
\(540\) 0 0
\(541\) 1.29180 0.0555387 0.0277693 0.999614i \(-0.491160\pi\)
0.0277693 + 0.999614i \(0.491160\pi\)
\(542\) −53.4508 −2.29591
\(543\) 0 0
\(544\) −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
\(549\) 0 0
\(550\) 0 0
\(551\) −13.4164 −0.571558
\(552\) 0 0
\(553\) 0 0
\(554\) −1.09017 −0.0463169
\(555\) 0 0
\(556\) −113.666 −4.82050
\(557\) 18.7082 0.792692 0.396346 0.918101i \(-0.370278\pi\)
0.396346 + 0.918101i \(0.370278\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) −70.5410 −2.97559
\(563\) −12.5967 −0.530890 −0.265445 0.964126i \(-0.585519\pi\)
−0.265445 + 0.964126i \(0.585519\pi\)
\(564\) 0 0
\(565\) 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 \(-0.679283\pi\)
−0.533924 + 0.845533i \(0.679283\pi\)
\(570\) 0 0
\(571\) −36.1246 −1.51177 −0.755884 0.654706i \(-0.772793\pi\)
−0.755884 + 0.654706i \(0.772793\pi\)
\(572\) −14.5623 −0.608881
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −19.2918 −0.803128 −0.401564 0.915831i \(-0.631533\pi\)
−0.401564 + 0.915831i \(0.631533\pi\)
\(578\) −38.8328 −1.61523
\(579\) 0 0
\(580\) 48.5410 2.01556
\(581\) 0 0
\(582\) 0 0
\(583\) 22.4164 0.928393
\(584\) −20.2361 −0.837374
\(585\) 0 0
\(586\) 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
\(591\) 0 0
\(592\) 46.3951 1.90683
\(593\) −27.7639 −1.14013 −0.570064 0.821600i \(-0.693082\pi\)
−0.570064 + 0.821600i \(0.693082\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.43769 −0.140813
\(597\) 0 0
\(598\) −21.5623 −0.881748
\(599\) 17.0689 0.697416 0.348708 0.937231i \(-0.386621\pi\)
0.348708 + 0.937231i \(0.386621\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 99.1033 4.03246
\(605\) 4.47214 0.181818
\(606\) 0 0
\(607\) 24.1246 0.979188 0.489594 0.871951i \(-0.337145\pi\)
0.489594 + 0.871951i \(0.337145\pi\)
\(608\) 32.5623 1.32058
\(609\) 0 0
\(610\) −17.5623 −0.711077
\(611\) −7.47214 −0.302290
\(612\) 0 0
\(613\) −18.1246 −0.732046 −0.366023 0.930606i \(-0.619281\pi\)
−0.366023 + 0.930606i \(0.619281\pi\)
\(614\) 50.8328 2.05145
\(615\) 0 0
\(616\) 0 0
\(617\) 4.47214 0.180041 0.0900207 0.995940i \(-0.471307\pi\)
0.0900207 + 0.995940i \(0.471307\pi\)
\(618\) 0 0
\(619\) 17.0000 0.683288 0.341644 0.939829i \(-0.389016\pi\)
0.341644 + 0.939829i \(0.389016\pi\)
\(620\) −54.2705 −2.17956
\(621\) 0 0
\(622\) 72.6869 2.91448
\(623\) 0 0
\(624\) 0 0
\(625\) −25.0000 −1.00000
\(626\) −14.6180 −0.584254
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 21.5623 0.856349
\(635\) 25.5279 1.01304
\(636\) 0 0
\(637\) 0 0
\(638\) −35.1246 −1.39060
\(639\) 0 0
\(640\) −2.43769 −0.0963583
\(641\) −11.9443 −0.471770 −0.235885 0.971781i \(-0.575799\pi\)
−0.235885 + 0.971781i \(0.575799\pi\)
\(642\) 0 0
\(643\) 34.8328 1.37367 0.686836 0.726812i \(-0.258999\pi\)
0.686836 + 0.726812i \(0.258999\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −11.5623 −0.454913
\(647\) −20.2361 −0.795562 −0.397781 0.917480i \(-0.630220\pi\)
−0.397781 + 0.917480i \(0.630220\pi\)
\(648\) 0 0
\(649\) 4.41641 0.173359
\(650\) 0 0
\(651\) 0 0
\(652\) −79.6869 −3.12078
\(653\) −4.52786 −0.177189 −0.0885945 0.996068i \(-0.528238\pi\)
−0.0885945 + 0.996068i \(0.528238\pi\)
\(654\) 0 0
\(655\) −18.4164 −0.719589
\(656\) 44.0689 1.72060
\(657\) 0 0
\(658\) 0 0
\(659\) 8.94427 0.348419 0.174210 0.984709i \(-0.444263\pi\)
0.174210 + 0.984709i \(0.444263\pi\)
\(660\) 0 0
\(661\) −6.70820 −0.260919 −0.130459 0.991454i \(-0.541645\pi\)
−0.130459 + 0.991454i \(0.541645\pi\)
\(662\) −4.14590 −0.161135
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −36.8328 −1.42617
\(668\) −109.082 −4.22051
\(669\) 0 0
\(670\) 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 \(0.482322\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 24.5967 0.943242
\(681\) 0 0
\(682\) 39.2705 1.50375
\(683\) 13.4721 0.515497 0.257748 0.966212i \(-0.417019\pi\)
0.257748 + 0.966212i \(0.417019\pi\)
\(684\) 0 0
\(685\) −18.4164 −0.703655
\(686\) 0 0
\(687\) 0 0
\(688\) −78.8328 −3.00547
\(689\) −7.47214 −0.284666
\(690\) 0 0
\(691\) −51.8328 −1.97181 −0.985907 0.167297i \(-0.946496\pi\)
−0.985907 + 0.167297i \(0.946496\pi\)
\(692\) 79.6869 3.02924
\(693\) 0 0
\(694\) −60.3951 −2.29257
\(695\) 52.3607 1.98615
\(696\) 0 0
\(697\) −6.58359 −0.249371
\(698\) −77.0132 −2.91499
\(699\) 0 0
\(700\) 0 0
\(701\) 22.3607 0.844551 0.422276 0.906467i \(-0.361231\pi\)
0.422276 + 0.906467i \(0.361231\pi\)
\(702\) 0 0
\(703\) 14.1246 0.532720
\(704\) 26.1246 0.984608
\(705\) 0 0
\(706\) −45.2705 −1.70378
\(707\) 0 0
\(708\) 0 0
\(709\) −50.1246 −1.88247 −0.941235 0.337753i \(-0.890333\pi\)
−0.941235 + 0.337753i \(0.890333\pi\)
\(710\) −52.3607 −1.96506
\(711\) 0 0
\(712\) −16.7082 −0.626166
\(713\) 41.1803 1.54222
\(714\) 0 0
\(715\) 6.70820 0.250873
\(716\) 97.6869 3.65073
\(717\) 0 0
\(718\) −31.2705 −1.16701
\(719\) −24.7082 −0.921461 −0.460730 0.887540i \(-0.652412\pi\)
−0.460730 + 0.887540i \(0.652412\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −26.1803 −0.974331
\(723\) 0 0
\(724\) −123.374 −4.58515
\(725\) 0 0
\(726\) 0 0
\(727\) −38.8328 −1.44023 −0.720115 0.693855i \(-0.755911\pi\)
−0.720115 + 0.693855i \(0.755911\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 15.8541 0.586787
\(731\) 11.7771 0.435591
\(732\) 0 0
\(733\) 28.7082 1.06036 0.530181 0.847885i \(-0.322124\pi\)
0.530181 + 0.847885i \(0.322124\pi\)
\(734\) −33.2705 −1.22804
\(735\) 0 0
\(736\) 89.3951 3.29515
\(737\) −9.00000 −0.331519
\(738\) 0 0
\(739\) −17.8328 −0.655991 −0.327995 0.944679i \(-0.606373\pi\)
−0.327995 + 0.944679i \(0.606373\pi\)
\(740\) −51.1033 −1.87860
\(741\) 0 0
\(742\) 0 0
\(743\) −32.9443 −1.20861 −0.604304 0.796754i \(-0.706549\pi\)
−0.604304 + 0.796754i \(0.706549\pi\)
\(744\) 0 0
\(745\) 1.58359 0.0580183
\(746\) −4.14590 −0.151792
\(747\) 0 0
\(748\) −21.4377 −0.783840
\(749\) 0 0
\(750\) 0 0
\(751\) −10.1246 −0.369452 −0.184726 0.982790i \(-0.559140\pi\)
−0.184726 + 0.982790i \(0.559140\pi\)
\(752\) 73.6312 2.68505
\(753\) 0 0
\(754\) 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
\(759\) 0 0
\(760\) −50.1246 −1.81821
\(761\) 33.5410 1.21586 0.607931 0.793990i \(-0.292000\pi\)
0.607931 + 0.793990i \(0.292000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −54.2705 −1.96344
\(765\) 0 0
\(766\) −39.2705 −1.41890
\(767\) −1.47214 −0.0531557
\(768\) 0 0
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.43769 0.123725
\(773\) 47.0689 1.69295 0.846475 0.532428i \(-0.178720\pi\)
0.846475 + 0.532428i \(0.178720\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 70.3607 2.52580
\(777\) 0 0
\(778\) −3.85410 −0.138176
\(779\) 13.4164 0.480693
\(780\) 0 0
\(781\) 26.8328 0.960154
\(782\) −31.7426 −1.13511
\(783\) 0 0
\(784\) 0 0
\(785\) −15.6525 −0.558661
\(786\) 0 0
\(787\) 20.4164 0.727766 0.363883 0.931445i \(-0.381451\pi\)
0.363883 + 0.931445i \(0.381451\pi\)
\(788\) 43.9574 1.56592
\(789\) 0 0
\(790\) 15.8541 0.564064
\(791\) 0 0
\(792\) 0 0
\(793\) −3.00000 −0.106533
\(794\) −68.3951 −2.42725
\(795\) 0 0
\(796\) −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
\(801\) 0 0
\(802\) −37.2705 −1.31607
\(803\) −8.12461 −0.286711
\(804\) 0 0
\(805\) 0 0
\(806\) −13.0902 −0.461082
\(807\) 0 0
\(808\) 67.2492 2.36582
\(809\) −22.4164 −0.788119 −0.394059 0.919085i \(-0.628930\pi\)
−0.394059 + 0.919085i \(0.628930\pi\)
\(810\) 0 0
\(811\) −14.8328 −0.520851 −0.260425 0.965494i \(-0.583863\pi\)
−0.260425 + 0.965494i \(0.583863\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 36.9787 1.29610
\(815\) 36.7082 1.28583
\(816\) 0 0
\(817\) −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 \(-0.732518\pi\)
−0.667224 + 0.744857i \(0.732518\pi\)
\(822\) 0 0
\(823\) 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 \(-0.654498\pi\)
−0.466534 + 0.884503i \(0.654498\pi\)
\(828\) 0 0
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −8.70820 −0.301903
\(833\) 0 0
\(834\) 0 0
\(835\) 50.2492 1.73895
\(836\) 43.6869 1.51094
\(837\) 0 0
\(838\) −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
\(843\) 0 0
\(844\) 19.4164 0.668340
\(845\) −2.23607 −0.0769231
\(846\) 0 0
\(847\) 0 0
\(848\) 73.6312 2.52851
\(849\) 0 0
\(850\) 0 0
\(851\) 38.7771 1.32926
\(852\) 0 0
\(853\) −52.2492 −1.78898 −0.894490 0.447089i \(-0.852461\pi\)
−0.894490 + 0.447089i \(0.852461\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 72.9574 2.49363
\(857\) 1.36068 0.0464799 0.0232400 0.999730i \(-0.492602\pi\)
0.0232400 + 0.999730i \(0.492602\pi\)
\(858\) 0 0
\(859\) 20.7082 0.706555 0.353277 0.935519i \(-0.385067\pi\)
0.353277 + 0.935519i \(0.385067\pi\)
\(860\) 86.8328 2.96097
\(861\) 0 0
\(862\) 82.1033 2.79645
\(863\) −23.9443 −0.815072 −0.407536 0.913189i \(-0.633612\pi\)
−0.407536 + 0.913189i \(0.633612\pi\)
\(864\) 0 0
\(865\) −36.7082 −1.24812
\(866\) 77.0132 2.61701
\(867\) 0 0
\(868\) 0 0
\(869\) −8.12461 −0.275609
\(870\) 0 0
\(871\) 3.00000 0.101651
\(872\) −20.2361 −0.685280
\(873\) 0 0
\(874\) 64.6869 2.18807
\(875\) 0 0
\(876\) 0 0
\(877\) 32.1246 1.08477 0.542386 0.840130i \(-0.317521\pi\)
0.542386 + 0.840130i \(0.317521\pi\)
\(878\) 63.1591 2.13151
\(879\) 0 0
\(880\) −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
\(885\) 0 0
\(886\) −5.85410 −0.196672
\(887\) 20.2361 0.679461 0.339730 0.940523i \(-0.389664\pi\)
0.339730 + 0.940523i \(0.389664\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 13.0902 0.438783
\(891\) 0 0
\(892\) −19.4164 −0.650109
\(893\) 22.4164 0.750136
\(894\) 0 0
\(895\) −45.0000 −1.50418
\(896\) 0 0
\(897\) 0 0
\(898\) 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
\(906\) 0 0
\(907\) 18.7082 0.621196 0.310598 0.950541i \(-0.399471\pi\)
0.310598 + 0.950541i \(0.399471\pi\)
\(908\) −28.8541 −0.957557
\(909\) 0 0
\(910\) 0 0
\(911\) 34.2492 1.13473 0.567364 0.823467i \(-0.307963\pi\)
0.567364 + 0.823467i \(0.307963\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −16.0344 −0.530372
\(915\) 0 0
\(916\) 117.103 3.86920
\(917\) 0 0
\(918\) 0 0
\(919\) 20.1246 0.663850 0.331925 0.943306i \(-0.392302\pi\)
0.331925 + 0.943306i \(0.392302\pi\)
\(920\) −137.610 −4.53686
\(921\) 0 0
\(922\) −89.9574 −2.96259
\(923\) −8.94427 −0.294404
\(924\) 0 0
\(925\) 0 0
\(926\) −62.8328 −2.06481
\(927\) 0 0
\(928\) −48.5410 −1.59344
\(929\) 24.8197 0.814307 0.407153 0.913360i \(-0.366521\pi\)
0.407153 + 0.913360i \(0.366521\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −57.9787 −1.89916
\(933\) 0 0
\(934\) −25.2705 −0.826876
\(935\) 9.87539 0.322960
\(936\) 0 0
\(937\) 25.4164 0.830318 0.415159 0.909749i \(-0.363726\pi\)
0.415159 + 0.909749i \(0.363726\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −81.1033 −2.64530
\(941\) −50.2361 −1.63765 −0.818825 0.574044i \(-0.805374\pi\)
−0.818825 + 0.574044i \(0.805374\pi\)
\(942\) 0 0
\(943\) 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 \(-0.380717\pi\)
0.366029 + 0.930604i \(0.380717\pi\)
\(948\) 0 0
\(949\) 2.70820 0.0879120
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 41.7771 1.35329 0.676646 0.736308i \(-0.263433\pi\)
0.676646 + 0.736308i \(0.263433\pi\)
\(954\) 0 0
\(955\) 25.0000 0.808981
\(956\) −94.2492 −3.04824
\(957\) 0 0
\(958\) −62.3951 −2.01589
\(959\) 0 0
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) −12.3262 −0.397414
\(963\) 0 0
\(964\) 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
\(969\) 0 0
\(970\) −55.1246 −1.76994
\(971\) 11.2918 0.362371 0.181185 0.983449i \(-0.442007\pi\)
0.181185 + 0.983449i \(0.442007\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −57.1591 −1.83149
\(975\) 0 0
\(976\) 29.5623 0.946266
\(977\) 2.34752 0.0751040 0.0375520 0.999295i \(-0.488044\pi\)
0.0375520 + 0.999295i \(0.488044\pi\)
\(978\) 0 0
\(979\) −6.70820 −0.214395
\(980\) 0 0
\(981\) 0 0
\(982\) 66.8328 2.13272
\(983\) −7.47214 −0.238324 −0.119162 0.992875i \(-0.538021\pi\)
−0.119162 + 0.992875i \(0.538021\pi\)
\(984\) 0 0
\(985\) −20.2492 −0.645194
\(986\) 17.2361 0.548908
\(987\) 0 0
\(988\) −14.5623 −0.463289
\(989\) −65.8885 −2.09513
\(990\) 0 0
\(991\) 30.7082 0.975478 0.487739 0.872989i \(-0.337822\pi\)
0.487739 + 0.872989i \(0.337822\pi\)
\(992\) 54.2705 1.72309
\(993\) 0 0
\(994\) 0 0
\(995\) 46.3050 1.46797
\(996\) 0 0
\(997\) 26.4164 0.836616 0.418308 0.908305i \(-0.362623\pi\)
0.418308 + 0.908305i \(0.362623\pi\)
\(998\) 69.1591 2.18919
\(999\) 0 0
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 5733.2.a.v.1.2 2
3.2 odd 2 637.2.a.f.1.1 2
7.2 even 3 819.2.j.c.235.1 4
7.4 even 3 819.2.j.c.352.1 4
7.6 odd 2 5733.2.a.w.1.2 2
21.2 odd 6 91.2.e.b.53.2 4
21.5 even 6 637.2.e.h.508.2 4
21.11 odd 6 91.2.e.b.79.2 yes 4
21.17 even 6 637.2.e.h.79.2 4
21.20 even 2 637.2.a.e.1.1 2
39.38 odd 2 8281.2.a.z.1.2 2
84.11 even 6 1456.2.r.j.625.1 4
84.23 even 6 1456.2.r.j.417.1 4
273.116 odd 6 1183.2.e.d.170.1 4
273.233 odd 6 1183.2.e.d.508.1 4
273.272 even 2 8281.2.a.ba.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.b.53.2 4 21.2 odd 6
91.2.e.b.79.2 yes 4 21.11 odd 6
637.2.a.e.1.1 2 21.20 even 2
637.2.a.f.1.1 2 3.2 odd 2
637.2.e.h.79.2 4 21.17 even 6
637.2.e.h.508.2 4 21.5 even 6
819.2.j.c.235.1 4 7.2 even 3
819.2.j.c.352.1 4 7.4 even 3
1183.2.e.d.170.1 4 273.116 odd 6
1183.2.e.d.508.1 4 273.233 odd 6
1456.2.r.j.417.1 4 84.23 even 6
1456.2.r.j.625.1 4 84.11 even 6
5733.2.a.v.1.2 2 1.1 even 1 trivial
5733.2.a.w.1.2 2 7.6 odd 2
8281.2.a.z.1.2 2 39.38 odd 2
8281.2.a.ba.1.2 2 273.272 even 2