Properties

Label 7623.2.a.dc.1.9
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 693)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(0.400970\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.400970 q^{2} -1.83922 q^{4} -2.30562 q^{5} +1.00000 q^{7} -1.53941 q^{8} +O(q^{10})\) \(q+0.400970 q^{2} -1.83922 q^{4} -2.30562 q^{5} +1.00000 q^{7} -1.53941 q^{8} -0.924485 q^{10} +4.49155 q^{13} +0.400970 q^{14} +3.06119 q^{16} -7.46771 q^{17} -1.34729 q^{19} +4.24055 q^{20} +4.27146 q^{23} +0.315883 q^{25} +1.80098 q^{26} -1.83922 q^{28} -1.12589 q^{29} -8.11240 q^{31} +4.30627 q^{32} -2.99433 q^{34} -2.30562 q^{35} +11.0657 q^{37} -0.540224 q^{38} +3.54930 q^{40} +1.16172 q^{41} +9.95451 q^{43} +1.71273 q^{46} -12.8441 q^{47} +1.00000 q^{49} +0.126660 q^{50} -8.26096 q^{52} -11.7297 q^{53} -1.53941 q^{56} -0.451447 q^{58} +1.04893 q^{59} -6.69637 q^{61} -3.25283 q^{62} -4.39569 q^{64} -10.3558 q^{65} -2.46445 q^{67} +13.7348 q^{68} -0.924485 q^{70} -8.42283 q^{71} -10.7770 q^{73} +4.43703 q^{74} +2.47797 q^{76} +2.71147 q^{79} -7.05793 q^{80} +0.465816 q^{82} +9.12912 q^{83} +17.2177 q^{85} +3.99146 q^{86} -5.28542 q^{89} +4.49155 q^{91} -7.85617 q^{92} -5.15009 q^{94} +3.10634 q^{95} +5.74601 q^{97} +0.400970 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 20q^{4} + 16q^{7} + O(q^{10}) \) \( 16q + 20q^{4} + 16q^{7} - 6q^{10} + 32q^{16} + 10q^{19} + 44q^{25} + 20q^{28} + 30q^{31} + 12q^{34} + 38q^{37} - 68q^{40} + 16q^{43} + 80q^{46} + 16q^{49} + 2q^{52} + 18q^{58} - 28q^{61} + 34q^{64} + 52q^{67} - 6q^{70} - 14q^{73} - 14q^{76} + 54q^{79} + 64q^{82} + 30q^{85} - 60q^{94} + 108q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.400970 0.283529 0.141764 0.989900i \(-0.454722\pi\)
0.141764 + 0.989900i \(0.454722\pi\)
\(3\) 0 0
\(4\) −1.83922 −0.919611
\(5\) −2.30562 −1.03110 −0.515552 0.856858i \(-0.672413\pi\)
−0.515552 + 0.856858i \(0.672413\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.53941 −0.544265
\(9\) 0 0
\(10\) −0.924485 −0.292348
\(11\) 0 0
\(12\) 0 0
\(13\) 4.49155 1.24573 0.622866 0.782329i \(-0.285968\pi\)
0.622866 + 0.782329i \(0.285968\pi\)
\(14\) 0.400970 0.107164
\(15\) 0 0
\(16\) 3.06119 0.765297
\(17\) −7.46771 −1.81119 −0.905593 0.424148i \(-0.860574\pi\)
−0.905593 + 0.424148i \(0.860574\pi\)
\(18\) 0 0
\(19\) −1.34729 −0.309090 −0.154545 0.987986i \(-0.549391\pi\)
−0.154545 + 0.987986i \(0.549391\pi\)
\(20\) 4.24055 0.948216
\(21\) 0 0
\(22\) 0 0
\(23\) 4.27146 0.890661 0.445330 0.895366i \(-0.353086\pi\)
0.445330 + 0.895366i \(0.353086\pi\)
\(24\) 0 0
\(25\) 0.315883 0.0631766
\(26\) 1.80098 0.353201
\(27\) 0 0
\(28\) −1.83922 −0.347580
\(29\) −1.12589 −0.209072 −0.104536 0.994521i \(-0.533336\pi\)
−0.104536 + 0.994521i \(0.533336\pi\)
\(30\) 0 0
\(31\) −8.11240 −1.45703 −0.728515 0.685030i \(-0.759789\pi\)
−0.728515 + 0.685030i \(0.759789\pi\)
\(32\) 4.30627 0.761249
\(33\) 0 0
\(34\) −2.99433 −0.513523
\(35\) −2.30562 −0.389721
\(36\) 0 0
\(37\) 11.0657 1.81920 0.909598 0.415490i \(-0.136390\pi\)
0.909598 + 0.415490i \(0.136390\pi\)
\(38\) −0.540224 −0.0876359
\(39\) 0 0
\(40\) 3.54930 0.561194
\(41\) 1.16172 0.181430 0.0907152 0.995877i \(-0.471085\pi\)
0.0907152 + 0.995877i \(0.471085\pi\)
\(42\) 0 0
\(43\) 9.95451 1.51805 0.759024 0.651063i \(-0.225676\pi\)
0.759024 + 0.651063i \(0.225676\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.71273 0.252528
\(47\) −12.8441 −1.87350 −0.936751 0.349998i \(-0.886182\pi\)
−0.936751 + 0.349998i \(0.886182\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0.126660 0.0179124
\(51\) 0 0
\(52\) −8.26096 −1.14559
\(53\) −11.7297 −1.61120 −0.805602 0.592457i \(-0.798158\pi\)
−0.805602 + 0.592457i \(0.798158\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.53941 −0.205713
\(57\) 0 0
\(58\) −0.451447 −0.0592779
\(59\) 1.04893 0.136559 0.0682793 0.997666i \(-0.478249\pi\)
0.0682793 + 0.997666i \(0.478249\pi\)
\(60\) 0 0
\(61\) −6.69637 −0.857382 −0.428691 0.903451i \(-0.641025\pi\)
−0.428691 + 0.903451i \(0.641025\pi\)
\(62\) −3.25283 −0.413110
\(63\) 0 0
\(64\) −4.39569 −0.549461
\(65\) −10.3558 −1.28448
\(66\) 0 0
\(67\) −2.46445 −0.301080 −0.150540 0.988604i \(-0.548101\pi\)
−0.150540 + 0.988604i \(0.548101\pi\)
\(68\) 13.7348 1.66559
\(69\) 0 0
\(70\) −0.924485 −0.110497
\(71\) −8.42283 −0.999606 −0.499803 0.866139i \(-0.666594\pi\)
−0.499803 + 0.866139i \(0.666594\pi\)
\(72\) 0 0
\(73\) −10.7770 −1.26135 −0.630673 0.776048i \(-0.717221\pi\)
−0.630673 + 0.776048i \(0.717221\pi\)
\(74\) 4.43703 0.515794
\(75\) 0 0
\(76\) 2.47797 0.284243
\(77\) 0 0
\(78\) 0 0
\(79\) 2.71147 0.305065 0.152532 0.988298i \(-0.451257\pi\)
0.152532 + 0.988298i \(0.451257\pi\)
\(80\) −7.05793 −0.789101
\(81\) 0 0
\(82\) 0.465816 0.0514407
\(83\) 9.12912 1.00205 0.501025 0.865433i \(-0.332956\pi\)
0.501025 + 0.865433i \(0.332956\pi\)
\(84\) 0 0
\(85\) 17.2177 1.86752
\(86\) 3.99146 0.430410
\(87\) 0 0
\(88\) 0 0
\(89\) −5.28542 −0.560253 −0.280126 0.959963i \(-0.590376\pi\)
−0.280126 + 0.959963i \(0.590376\pi\)
\(90\) 0 0
\(91\) 4.49155 0.470842
\(92\) −7.85617 −0.819062
\(93\) 0 0
\(94\) −5.15009 −0.531191
\(95\) 3.10634 0.318704
\(96\) 0 0
\(97\) 5.74601 0.583419 0.291709 0.956507i \(-0.405776\pi\)
0.291709 + 0.956507i \(0.405776\pi\)
\(98\) 0.400970 0.0405041
\(99\) 0 0
\(100\) −0.580979 −0.0580979
\(101\) −9.56357 −0.951611 −0.475805 0.879551i \(-0.657843\pi\)
−0.475805 + 0.879551i \(0.657843\pi\)
\(102\) 0 0
\(103\) −0.807595 −0.0795747 −0.0397873 0.999208i \(-0.512668\pi\)
−0.0397873 + 0.999208i \(0.512668\pi\)
\(104\) −6.91435 −0.678008
\(105\) 0 0
\(106\) −4.70328 −0.456823
\(107\) 16.8478 1.62874 0.814371 0.580345i \(-0.197082\pi\)
0.814371 + 0.580345i \(0.197082\pi\)
\(108\) 0 0
\(109\) 9.41870 0.902147 0.451074 0.892487i \(-0.351041\pi\)
0.451074 + 0.892487i \(0.351041\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.06119 0.289255
\(113\) 4.35139 0.409344 0.204672 0.978831i \(-0.434387\pi\)
0.204672 + 0.978831i \(0.434387\pi\)
\(114\) 0 0
\(115\) −9.84836 −0.918365
\(116\) 2.07076 0.192265
\(117\) 0 0
\(118\) 0.420588 0.0387183
\(119\) −7.46771 −0.684564
\(120\) 0 0
\(121\) 0 0
\(122\) −2.68504 −0.243093
\(123\) 0 0
\(124\) 14.9205 1.33990
\(125\) 10.7998 0.965963
\(126\) 0 0
\(127\) −6.34739 −0.563240 −0.281620 0.959526i \(-0.590872\pi\)
−0.281620 + 0.959526i \(0.590872\pi\)
\(128\) −10.3751 −0.917036
\(129\) 0 0
\(130\) −4.15237 −0.364187
\(131\) 15.2292 1.33058 0.665290 0.746585i \(-0.268308\pi\)
0.665290 + 0.746585i \(0.268308\pi\)
\(132\) 0 0
\(133\) −1.34729 −0.116825
\(134\) −0.988170 −0.0853649
\(135\) 0 0
\(136\) 11.4959 0.985765
\(137\) 19.0993 1.63176 0.815880 0.578221i \(-0.196253\pi\)
0.815880 + 0.578221i \(0.196253\pi\)
\(138\) 0 0
\(139\) −3.93905 −0.334106 −0.167053 0.985948i \(-0.553425\pi\)
−0.167053 + 0.985948i \(0.553425\pi\)
\(140\) 4.24055 0.358392
\(141\) 0 0
\(142\) −3.37730 −0.283417
\(143\) 0 0
\(144\) 0 0
\(145\) 2.59587 0.215575
\(146\) −4.32124 −0.357628
\(147\) 0 0
\(148\) −20.3523 −1.67295
\(149\) 20.3815 1.66972 0.834859 0.550464i \(-0.185549\pi\)
0.834859 + 0.550464i \(0.185549\pi\)
\(150\) 0 0
\(151\) −4.91718 −0.400154 −0.200077 0.979780i \(-0.564119\pi\)
−0.200077 + 0.979780i \(0.564119\pi\)
\(152\) 2.07404 0.168227
\(153\) 0 0
\(154\) 0 0
\(155\) 18.7041 1.50235
\(156\) 0 0
\(157\) 2.24200 0.178931 0.0894653 0.995990i \(-0.471484\pi\)
0.0894653 + 0.995990i \(0.471484\pi\)
\(158\) 1.08722 0.0864946
\(159\) 0 0
\(160\) −9.92863 −0.784927
\(161\) 4.27146 0.336638
\(162\) 0 0
\(163\) −9.83745 −0.770529 −0.385264 0.922806i \(-0.625890\pi\)
−0.385264 + 0.922806i \(0.625890\pi\)
\(164\) −2.13666 −0.166845
\(165\) 0 0
\(166\) 3.66050 0.284110
\(167\) −8.64481 −0.668955 −0.334478 0.942404i \(-0.608560\pi\)
−0.334478 + 0.942404i \(0.608560\pi\)
\(168\) 0 0
\(169\) 7.17400 0.551847
\(170\) 6.90378 0.529496
\(171\) 0 0
\(172\) −18.3086 −1.39601
\(173\) −15.9672 −1.21397 −0.606984 0.794714i \(-0.707621\pi\)
−0.606984 + 0.794714i \(0.707621\pi\)
\(174\) 0 0
\(175\) 0.315883 0.0238785
\(176\) 0 0
\(177\) 0 0
\(178\) −2.11929 −0.158848
\(179\) 17.5759 1.31368 0.656841 0.754029i \(-0.271892\pi\)
0.656841 + 0.754029i \(0.271892\pi\)
\(180\) 0 0
\(181\) 20.7645 1.54341 0.771707 0.635979i \(-0.219403\pi\)
0.771707 + 0.635979i \(0.219403\pi\)
\(182\) 1.80098 0.133497
\(183\) 0 0
\(184\) −6.57554 −0.484756
\(185\) −25.5134 −1.87578
\(186\) 0 0
\(187\) 0 0
\(188\) 23.6231 1.72289
\(189\) 0 0
\(190\) 1.24555 0.0903618
\(191\) −10.5001 −0.759762 −0.379881 0.925035i \(-0.624035\pi\)
−0.379881 + 0.925035i \(0.624035\pi\)
\(192\) 0 0
\(193\) −12.9500 −0.932164 −0.466082 0.884742i \(-0.654335\pi\)
−0.466082 + 0.884742i \(0.654335\pi\)
\(194\) 2.30398 0.165416
\(195\) 0 0
\(196\) −1.83922 −0.131373
\(197\) −0.664214 −0.0473233 −0.0236617 0.999720i \(-0.507532\pi\)
−0.0236617 + 0.999720i \(0.507532\pi\)
\(198\) 0 0
\(199\) 13.3550 0.946710 0.473355 0.880872i \(-0.343043\pi\)
0.473355 + 0.880872i \(0.343043\pi\)
\(200\) −0.486274 −0.0343848
\(201\) 0 0
\(202\) −3.83470 −0.269809
\(203\) −1.12589 −0.0790218
\(204\) 0 0
\(205\) −2.67849 −0.187074
\(206\) −0.323821 −0.0225617
\(207\) 0 0
\(208\) 13.7495 0.953354
\(209\) 0 0
\(210\) 0 0
\(211\) 13.0795 0.900428 0.450214 0.892921i \(-0.351348\pi\)
0.450214 + 0.892921i \(0.351348\pi\)
\(212\) 21.5736 1.48168
\(213\) 0 0
\(214\) 6.75548 0.461795
\(215\) −22.9513 −1.56527
\(216\) 0 0
\(217\) −8.11240 −0.550706
\(218\) 3.77662 0.255785
\(219\) 0 0
\(220\) 0 0
\(221\) −33.5416 −2.25625
\(222\) 0 0
\(223\) 10.2293 0.685004 0.342502 0.939517i \(-0.388726\pi\)
0.342502 + 0.939517i \(0.388726\pi\)
\(224\) 4.30627 0.287725
\(225\) 0 0
\(226\) 1.74478 0.116061
\(227\) 2.98867 0.198365 0.0991824 0.995069i \(-0.468377\pi\)
0.0991824 + 0.995069i \(0.468377\pi\)
\(228\) 0 0
\(229\) 20.1017 1.32836 0.664179 0.747574i \(-0.268781\pi\)
0.664179 + 0.747574i \(0.268781\pi\)
\(230\) −3.94890 −0.260383
\(231\) 0 0
\(232\) 1.73321 0.113791
\(233\) 4.67052 0.305976 0.152988 0.988228i \(-0.451110\pi\)
0.152988 + 0.988228i \(0.451110\pi\)
\(234\) 0 0
\(235\) 29.6136 1.93178
\(236\) −1.92921 −0.125581
\(237\) 0 0
\(238\) −2.99433 −0.194094
\(239\) −22.7673 −1.47269 −0.736347 0.676604i \(-0.763451\pi\)
−0.736347 + 0.676604i \(0.763451\pi\)
\(240\) 0 0
\(241\) −4.83714 −0.311587 −0.155794 0.987790i \(-0.549793\pi\)
−0.155794 + 0.987790i \(0.549793\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 12.3161 0.788459
\(245\) −2.30562 −0.147301
\(246\) 0 0
\(247\) −6.05143 −0.385043
\(248\) 12.4883 0.793011
\(249\) 0 0
\(250\) 4.33039 0.273878
\(251\) 16.0556 1.01342 0.506710 0.862117i \(-0.330862\pi\)
0.506710 + 0.862117i \(0.330862\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −2.54512 −0.159695
\(255\) 0 0
\(256\) 4.63128 0.289455
\(257\) −12.1743 −0.759411 −0.379706 0.925107i \(-0.623975\pi\)
−0.379706 + 0.925107i \(0.623975\pi\)
\(258\) 0 0
\(259\) 11.0657 0.687591
\(260\) 19.0466 1.18122
\(261\) 0 0
\(262\) 6.10645 0.377258
\(263\) 18.8786 1.16411 0.582053 0.813151i \(-0.302250\pi\)
0.582053 + 0.813151i \(0.302250\pi\)
\(264\) 0 0
\(265\) 27.0443 1.66132
\(266\) −0.540224 −0.0331233
\(267\) 0 0
\(268\) 4.53267 0.276877
\(269\) 0.711037 0.0433527 0.0216763 0.999765i \(-0.493100\pi\)
0.0216763 + 0.999765i \(0.493100\pi\)
\(270\) 0 0
\(271\) 14.7238 0.894404 0.447202 0.894433i \(-0.352420\pi\)
0.447202 + 0.894433i \(0.352420\pi\)
\(272\) −22.8601 −1.38609
\(273\) 0 0
\(274\) 7.65823 0.462651
\(275\) 0 0
\(276\) 0 0
\(277\) 17.5829 1.05645 0.528227 0.849103i \(-0.322857\pi\)
0.528227 + 0.849103i \(0.322857\pi\)
\(278\) −1.57944 −0.0947286
\(279\) 0 0
\(280\) 3.54930 0.212111
\(281\) −14.9728 −0.893200 −0.446600 0.894734i \(-0.647365\pi\)
−0.446600 + 0.894734i \(0.647365\pi\)
\(282\) 0 0
\(283\) 17.4418 1.03681 0.518404 0.855136i \(-0.326526\pi\)
0.518404 + 0.855136i \(0.326526\pi\)
\(284\) 15.4915 0.919249
\(285\) 0 0
\(286\) 0 0
\(287\) 1.16172 0.0685742
\(288\) 0 0
\(289\) 38.7667 2.28039
\(290\) 1.04087 0.0611217
\(291\) 0 0
\(292\) 19.8212 1.15995
\(293\) 26.0123 1.51966 0.759828 0.650124i \(-0.225283\pi\)
0.759828 + 0.650124i \(0.225283\pi\)
\(294\) 0 0
\(295\) −2.41843 −0.140806
\(296\) −17.0347 −0.990124
\(297\) 0 0
\(298\) 8.17237 0.473413
\(299\) 19.1855 1.10952
\(300\) 0 0
\(301\) 9.95451 0.573768
\(302\) −1.97164 −0.113455
\(303\) 0 0
\(304\) −4.12431 −0.236546
\(305\) 15.4393 0.884051
\(306\) 0 0
\(307\) −23.5921 −1.34647 −0.673236 0.739428i \(-0.735096\pi\)
−0.673236 + 0.739428i \(0.735096\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 7.49979 0.425960
\(311\) 4.55191 0.258115 0.129058 0.991637i \(-0.458805\pi\)
0.129058 + 0.991637i \(0.458805\pi\)
\(312\) 0 0
\(313\) −13.1273 −0.742000 −0.371000 0.928633i \(-0.620985\pi\)
−0.371000 + 0.928633i \(0.620985\pi\)
\(314\) 0.898973 0.0507320
\(315\) 0 0
\(316\) −4.98700 −0.280541
\(317\) −8.94352 −0.502318 −0.251159 0.967946i \(-0.580812\pi\)
−0.251159 + 0.967946i \(0.580812\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 10.1348 0.566552
\(321\) 0 0
\(322\) 1.71273 0.0954466
\(323\) 10.0612 0.559820
\(324\) 0 0
\(325\) 1.41880 0.0787010
\(326\) −3.94453 −0.218467
\(327\) 0 0
\(328\) −1.78837 −0.0987462
\(329\) −12.8441 −0.708117
\(330\) 0 0
\(331\) 25.9302 1.42525 0.712627 0.701543i \(-0.247505\pi\)
0.712627 + 0.701543i \(0.247505\pi\)
\(332\) −16.7905 −0.921497
\(333\) 0 0
\(334\) −3.46631 −0.189668
\(335\) 5.68208 0.310445
\(336\) 0 0
\(337\) −15.7560 −0.858285 −0.429142 0.903237i \(-0.641184\pi\)
−0.429142 + 0.903237i \(0.641184\pi\)
\(338\) 2.87656 0.156464
\(339\) 0 0
\(340\) −31.6672 −1.71739
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −15.3241 −0.826220
\(345\) 0 0
\(346\) −6.40239 −0.344195
\(347\) −2.22968 −0.119696 −0.0598478 0.998208i \(-0.519062\pi\)
−0.0598478 + 0.998208i \(0.519062\pi\)
\(348\) 0 0
\(349\) 9.61826 0.514854 0.257427 0.966298i \(-0.417125\pi\)
0.257427 + 0.966298i \(0.417125\pi\)
\(350\) 0.126660 0.00677024
\(351\) 0 0
\(352\) 0 0
\(353\) 9.98576 0.531488 0.265744 0.964044i \(-0.414382\pi\)
0.265744 + 0.964044i \(0.414382\pi\)
\(354\) 0 0
\(355\) 19.4198 1.03070
\(356\) 9.72106 0.515215
\(357\) 0 0
\(358\) 7.04740 0.372466
\(359\) 10.2469 0.540808 0.270404 0.962747i \(-0.412843\pi\)
0.270404 + 0.962747i \(0.412843\pi\)
\(360\) 0 0
\(361\) −17.1848 −0.904463
\(362\) 8.32595 0.437602
\(363\) 0 0
\(364\) −8.26096 −0.432992
\(365\) 24.8475 1.30058
\(366\) 0 0
\(367\) 3.69263 0.192754 0.0963769 0.995345i \(-0.469275\pi\)
0.0963769 + 0.995345i \(0.469275\pi\)
\(368\) 13.0757 0.681620
\(369\) 0 0
\(370\) −10.2301 −0.531838
\(371\) −11.7297 −0.608978
\(372\) 0 0
\(373\) 17.2657 0.893986 0.446993 0.894538i \(-0.352495\pi\)
0.446993 + 0.894538i \(0.352495\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 19.7723 1.01968
\(377\) −5.05697 −0.260447
\(378\) 0 0
\(379\) −5.18952 −0.266568 −0.133284 0.991078i \(-0.542552\pi\)
−0.133284 + 0.991078i \(0.542552\pi\)
\(380\) −5.71326 −0.293084
\(381\) 0 0
\(382\) −4.21023 −0.215414
\(383\) 17.6212 0.900400 0.450200 0.892928i \(-0.351353\pi\)
0.450200 + 0.892928i \(0.351353\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5.19258 −0.264295
\(387\) 0 0
\(388\) −10.5682 −0.536519
\(389\) −10.6188 −0.538396 −0.269198 0.963085i \(-0.586759\pi\)
−0.269198 + 0.963085i \(0.586759\pi\)
\(390\) 0 0
\(391\) −31.8980 −1.61315
\(392\) −1.53941 −0.0777521
\(393\) 0 0
\(394\) −0.266330 −0.0134175
\(395\) −6.25163 −0.314554
\(396\) 0 0
\(397\) 22.6262 1.13557 0.567787 0.823175i \(-0.307800\pi\)
0.567787 + 0.823175i \(0.307800\pi\)
\(398\) 5.35495 0.268419
\(399\) 0 0
\(400\) 0.966976 0.0483488
\(401\) −4.60950 −0.230187 −0.115094 0.993355i \(-0.536717\pi\)
−0.115094 + 0.993355i \(0.536717\pi\)
\(402\) 0 0
\(403\) −36.4372 −1.81507
\(404\) 17.5895 0.875112
\(405\) 0 0
\(406\) −0.451447 −0.0224049
\(407\) 0 0
\(408\) 0 0
\(409\) 21.0486 1.04079 0.520394 0.853926i \(-0.325785\pi\)
0.520394 + 0.853926i \(0.325785\pi\)
\(410\) −1.07399 −0.0530408
\(411\) 0 0
\(412\) 1.48535 0.0731778
\(413\) 1.04893 0.0516143
\(414\) 0 0
\(415\) −21.0483 −1.03322
\(416\) 19.3418 0.948311
\(417\) 0 0
\(418\) 0 0
\(419\) −14.0827 −0.687984 −0.343992 0.938973i \(-0.611779\pi\)
−0.343992 + 0.938973i \(0.611779\pi\)
\(420\) 0 0
\(421\) 27.5676 1.34356 0.671781 0.740750i \(-0.265530\pi\)
0.671781 + 0.740750i \(0.265530\pi\)
\(422\) 5.24448 0.255297
\(423\) 0 0
\(424\) 18.0569 0.876922
\(425\) −2.35892 −0.114424
\(426\) 0 0
\(427\) −6.69637 −0.324060
\(428\) −30.9869 −1.49781
\(429\) 0 0
\(430\) −9.20279 −0.443798
\(431\) 34.4396 1.65890 0.829448 0.558584i \(-0.188655\pi\)
0.829448 + 0.558584i \(0.188655\pi\)
\(432\) 0 0
\(433\) 16.1385 0.775564 0.387782 0.921751i \(-0.373241\pi\)
0.387782 + 0.921751i \(0.373241\pi\)
\(434\) −3.25283 −0.156141
\(435\) 0 0
\(436\) −17.3231 −0.829625
\(437\) −5.75491 −0.275294
\(438\) 0 0
\(439\) 19.3036 0.921309 0.460655 0.887580i \(-0.347615\pi\)
0.460655 + 0.887580i \(0.347615\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −13.4492 −0.639712
\(443\) −13.1312 −0.623880 −0.311940 0.950102i \(-0.600979\pi\)
−0.311940 + 0.950102i \(0.600979\pi\)
\(444\) 0 0
\(445\) 12.1862 0.577679
\(446\) 4.10164 0.194218
\(447\) 0 0
\(448\) −4.39569 −0.207677
\(449\) 12.7511 0.601763 0.300882 0.953662i \(-0.402719\pi\)
0.300882 + 0.953662i \(0.402719\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −8.00317 −0.376437
\(453\) 0 0
\(454\) 1.19837 0.0562421
\(455\) −10.3558 −0.485488
\(456\) 0 0
\(457\) −9.19436 −0.430094 −0.215047 0.976604i \(-0.568991\pi\)
−0.215047 + 0.976604i \(0.568991\pi\)
\(458\) 8.06018 0.376628
\(459\) 0 0
\(460\) 18.1133 0.844539
\(461\) −16.0714 −0.748521 −0.374260 0.927324i \(-0.622103\pi\)
−0.374260 + 0.927324i \(0.622103\pi\)
\(462\) 0 0
\(463\) 32.9203 1.52994 0.764968 0.644068i \(-0.222755\pi\)
0.764968 + 0.644068i \(0.222755\pi\)
\(464\) −3.44655 −0.160002
\(465\) 0 0
\(466\) 1.87274 0.0867530
\(467\) 8.50986 0.393790 0.196895 0.980425i \(-0.436914\pi\)
0.196895 + 0.980425i \(0.436914\pi\)
\(468\) 0 0
\(469\) −2.46445 −0.113798
\(470\) 11.8742 0.547714
\(471\) 0 0
\(472\) −1.61473 −0.0743240
\(473\) 0 0
\(474\) 0 0
\(475\) −0.425586 −0.0195272
\(476\) 13.7348 0.629533
\(477\) 0 0
\(478\) −9.12901 −0.417551
\(479\) 9.52240 0.435089 0.217545 0.976050i \(-0.430195\pi\)
0.217545 + 0.976050i \(0.430195\pi\)
\(480\) 0 0
\(481\) 49.7023 2.26623
\(482\) −1.93955 −0.0883439
\(483\) 0 0
\(484\) 0 0
\(485\) −13.2481 −0.601566
\(486\) 0 0
\(487\) −1.09438 −0.0495910 −0.0247955 0.999693i \(-0.507893\pi\)
−0.0247955 + 0.999693i \(0.507893\pi\)
\(488\) 10.3085 0.466643
\(489\) 0 0
\(490\) −0.924485 −0.0417640
\(491\) −20.6684 −0.932751 −0.466375 0.884587i \(-0.654440\pi\)
−0.466375 + 0.884587i \(0.654440\pi\)
\(492\) 0 0
\(493\) 8.40780 0.378668
\(494\) −2.42644 −0.109171
\(495\) 0 0
\(496\) −24.8336 −1.11506
\(497\) −8.42283 −0.377816
\(498\) 0 0
\(499\) 23.8350 1.06700 0.533500 0.845800i \(-0.320876\pi\)
0.533500 + 0.845800i \(0.320876\pi\)
\(500\) −19.8632 −0.888311
\(501\) 0 0
\(502\) 6.43781 0.287333
\(503\) −15.1096 −0.673706 −0.336853 0.941557i \(-0.609362\pi\)
−0.336853 + 0.941557i \(0.609362\pi\)
\(504\) 0 0
\(505\) 22.0500 0.981210
\(506\) 0 0
\(507\) 0 0
\(508\) 11.6743 0.517962
\(509\) −1.67629 −0.0743004 −0.0371502 0.999310i \(-0.511828\pi\)
−0.0371502 + 0.999310i \(0.511828\pi\)
\(510\) 0 0
\(511\) −10.7770 −0.476744
\(512\) 22.6072 0.999105
\(513\) 0 0
\(514\) −4.88152 −0.215315
\(515\) 1.86201 0.0820498
\(516\) 0 0
\(517\) 0 0
\(518\) 4.43703 0.194952
\(519\) 0 0
\(520\) 15.9419 0.699097
\(521\) 35.9717 1.57595 0.787974 0.615709i \(-0.211130\pi\)
0.787974 + 0.615709i \(0.211130\pi\)
\(522\) 0 0
\(523\) −15.9612 −0.697933 −0.348967 0.937135i \(-0.613467\pi\)
−0.348967 + 0.937135i \(0.613467\pi\)
\(524\) −28.0099 −1.22362
\(525\) 0 0
\(526\) 7.56976 0.330057
\(527\) 60.5811 2.63895
\(528\) 0 0
\(529\) −4.75463 −0.206723
\(530\) 10.8440 0.471032
\(531\) 0 0
\(532\) 2.47797 0.107434
\(533\) 5.21793 0.226014
\(534\) 0 0
\(535\) −38.8447 −1.67940
\(536\) 3.79380 0.163867
\(537\) 0 0
\(538\) 0.285104 0.0122917
\(539\) 0 0
\(540\) 0 0
\(541\) −7.68655 −0.330471 −0.165235 0.986254i \(-0.552838\pi\)
−0.165235 + 0.986254i \(0.552838\pi\)
\(542\) 5.90379 0.253589
\(543\) 0 0
\(544\) −32.1580 −1.37876
\(545\) −21.7159 −0.930208
\(546\) 0 0
\(547\) 15.2442 0.651794 0.325897 0.945405i \(-0.394334\pi\)
0.325897 + 0.945405i \(0.394334\pi\)
\(548\) −35.1278 −1.50058
\(549\) 0 0
\(550\) 0 0
\(551\) 1.51690 0.0646221
\(552\) 0 0
\(553\) 2.71147 0.115304
\(554\) 7.05022 0.299535
\(555\) 0 0
\(556\) 7.24479 0.307248
\(557\) 2.09493 0.0887649 0.0443825 0.999015i \(-0.485868\pi\)
0.0443825 + 0.999015i \(0.485868\pi\)
\(558\) 0 0
\(559\) 44.7112 1.89108
\(560\) −7.05793 −0.298252
\(561\) 0 0
\(562\) −6.00363 −0.253248
\(563\) −9.86821 −0.415895 −0.207948 0.978140i \(-0.566678\pi\)
−0.207948 + 0.978140i \(0.566678\pi\)
\(564\) 0 0
\(565\) −10.0326 −0.422077
\(566\) 6.99365 0.293965
\(567\) 0 0
\(568\) 12.9662 0.544051
\(569\) −15.2708 −0.640186 −0.320093 0.947386i \(-0.603714\pi\)
−0.320093 + 0.947386i \(0.603714\pi\)
\(570\) 0 0
\(571\) −20.5400 −0.859573 −0.429787 0.902930i \(-0.641411\pi\)
−0.429787 + 0.902930i \(0.641411\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0.465816 0.0194428
\(575\) 1.34928 0.0562689
\(576\) 0 0
\(577\) 44.2582 1.84249 0.921246 0.388980i \(-0.127172\pi\)
0.921246 + 0.388980i \(0.127172\pi\)
\(578\) 15.5443 0.646557
\(579\) 0 0
\(580\) −4.77438 −0.198245
\(581\) 9.12912 0.378740
\(582\) 0 0
\(583\) 0 0
\(584\) 16.5902 0.686507
\(585\) 0 0
\(586\) 10.4302 0.430866
\(587\) −3.56942 −0.147326 −0.0736628 0.997283i \(-0.523469\pi\)
−0.0736628 + 0.997283i \(0.523469\pi\)
\(588\) 0 0
\(589\) 10.9298 0.450354
\(590\) −0.969716 −0.0399226
\(591\) 0 0
\(592\) 33.8743 1.39222
\(593\) −11.2563 −0.462240 −0.231120 0.972925i \(-0.574239\pi\)
−0.231120 + 0.972925i \(0.574239\pi\)
\(594\) 0 0
\(595\) 17.2177 0.705857
\(596\) −37.4861 −1.53549
\(597\) 0 0
\(598\) 7.69280 0.314582
\(599\) 43.6263 1.78252 0.891262 0.453488i \(-0.149821\pi\)
0.891262 + 0.453488i \(0.149821\pi\)
\(600\) 0 0
\(601\) −17.7286 −0.723166 −0.361583 0.932340i \(-0.617764\pi\)
−0.361583 + 0.932340i \(0.617764\pi\)
\(602\) 3.99146 0.162680
\(603\) 0 0
\(604\) 9.04378 0.367986
\(605\) 0 0
\(606\) 0 0
\(607\) −7.16734 −0.290913 −0.145457 0.989365i \(-0.546465\pi\)
−0.145457 + 0.989365i \(0.546465\pi\)
\(608\) −5.80181 −0.235294
\(609\) 0 0
\(610\) 6.19069 0.250654
\(611\) −57.6898 −2.33388
\(612\) 0 0
\(613\) −12.3784 −0.499957 −0.249979 0.968251i \(-0.580424\pi\)
−0.249979 + 0.968251i \(0.580424\pi\)
\(614\) −9.45972 −0.381763
\(615\) 0 0
\(616\) 0 0
\(617\) −6.11777 −0.246292 −0.123146 0.992389i \(-0.539298\pi\)
−0.123146 + 0.992389i \(0.539298\pi\)
\(618\) 0 0
\(619\) 38.8234 1.56045 0.780223 0.625501i \(-0.215106\pi\)
0.780223 + 0.625501i \(0.215106\pi\)
\(620\) −34.4010 −1.38158
\(621\) 0 0
\(622\) 1.82518 0.0731831
\(623\) −5.28542 −0.211756
\(624\) 0 0
\(625\) −26.4796 −1.05919
\(626\) −5.26366 −0.210378
\(627\) 0 0
\(628\) −4.12353 −0.164547
\(629\) −82.6357 −3.29490
\(630\) 0 0
\(631\) −9.70799 −0.386469 −0.193234 0.981153i \(-0.561898\pi\)
−0.193234 + 0.981153i \(0.561898\pi\)
\(632\) −4.17408 −0.166036
\(633\) 0 0
\(634\) −3.58609 −0.142422
\(635\) 14.6347 0.580759
\(636\) 0 0
\(637\) 4.49155 0.177962
\(638\) 0 0
\(639\) 0 0
\(640\) 23.9210 0.945560
\(641\) 14.8483 0.586473 0.293236 0.956040i \(-0.405268\pi\)
0.293236 + 0.956040i \(0.405268\pi\)
\(642\) 0 0
\(643\) −1.05956 −0.0417851 −0.0208926 0.999782i \(-0.506651\pi\)
−0.0208926 + 0.999782i \(0.506651\pi\)
\(644\) −7.85617 −0.309576
\(645\) 0 0
\(646\) 4.03424 0.158725
\(647\) 12.5622 0.493872 0.246936 0.969032i \(-0.420576\pi\)
0.246936 + 0.969032i \(0.420576\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0.568898 0.0223140
\(651\) 0 0
\(652\) 18.0933 0.708587
\(653\) −29.6143 −1.15890 −0.579449 0.815009i \(-0.696732\pi\)
−0.579449 + 0.815009i \(0.696732\pi\)
\(654\) 0 0
\(655\) −35.1127 −1.37197
\(656\) 3.55625 0.138848
\(657\) 0 0
\(658\) −5.15009 −0.200771
\(659\) 9.45208 0.368201 0.184100 0.982907i \(-0.441063\pi\)
0.184100 + 0.982907i \(0.441063\pi\)
\(660\) 0 0
\(661\) −8.94284 −0.347836 −0.173918 0.984760i \(-0.555643\pi\)
−0.173918 + 0.984760i \(0.555643\pi\)
\(662\) 10.3972 0.404100
\(663\) 0 0
\(664\) −14.0535 −0.545381
\(665\) 3.10634 0.120459
\(666\) 0 0
\(667\) −4.80918 −0.186212
\(668\) 15.8997 0.615179
\(669\) 0 0
\(670\) 2.27834 0.0880201
\(671\) 0 0
\(672\) 0 0
\(673\) 7.68207 0.296122 0.148061 0.988978i \(-0.452697\pi\)
0.148061 + 0.988978i \(0.452697\pi\)
\(674\) −6.31769 −0.243348
\(675\) 0 0
\(676\) −13.1946 −0.507484
\(677\) 35.0474 1.34698 0.673490 0.739197i \(-0.264795\pi\)
0.673490 + 0.739197i \(0.264795\pi\)
\(678\) 0 0
\(679\) 5.74601 0.220512
\(680\) −26.5052 −1.01643
\(681\) 0 0
\(682\) 0 0
\(683\) 23.5098 0.899577 0.449789 0.893135i \(-0.351499\pi\)
0.449789 + 0.893135i \(0.351499\pi\)
\(684\) 0 0
\(685\) −44.0356 −1.68251
\(686\) 0.400970 0.0153091
\(687\) 0 0
\(688\) 30.4726 1.16176
\(689\) −52.6847 −2.00713
\(690\) 0 0
\(691\) −16.2445 −0.617968 −0.308984 0.951067i \(-0.599989\pi\)
−0.308984 + 0.951067i \(0.599989\pi\)
\(692\) 29.3673 1.11638
\(693\) 0 0
\(694\) −0.894036 −0.0339371
\(695\) 9.08195 0.344498
\(696\) 0 0
\(697\) −8.67540 −0.328604
\(698\) 3.85664 0.145976
\(699\) 0 0
\(700\) −0.580979 −0.0219589
\(701\) 22.5424 0.851413 0.425706 0.904861i \(-0.360026\pi\)
0.425706 + 0.904861i \(0.360026\pi\)
\(702\) 0 0
\(703\) −14.9088 −0.562295
\(704\) 0 0
\(705\) 0 0
\(706\) 4.00399 0.150692
\(707\) −9.56357 −0.359675
\(708\) 0 0
\(709\) 24.1185 0.905791 0.452896 0.891564i \(-0.350391\pi\)
0.452896 + 0.891564i \(0.350391\pi\)
\(710\) 7.78678 0.292233
\(711\) 0 0
\(712\) 8.13644 0.304926
\(713\) −34.6518 −1.29772
\(714\) 0 0
\(715\) 0 0
\(716\) −32.3259 −1.20808
\(717\) 0 0
\(718\) 4.10868 0.153335
\(719\) −32.2087 −1.20118 −0.600591 0.799557i \(-0.705068\pi\)
−0.600591 + 0.799557i \(0.705068\pi\)
\(720\) 0 0
\(721\) −0.807595 −0.0300764
\(722\) −6.89059 −0.256441
\(723\) 0 0
\(724\) −38.1906 −1.41934
\(725\) −0.355648 −0.0132084
\(726\) 0 0
\(727\) 7.29214 0.270450 0.135225 0.990815i \(-0.456824\pi\)
0.135225 + 0.990815i \(0.456824\pi\)
\(728\) −6.91435 −0.256263
\(729\) 0 0
\(730\) 9.96313 0.368752
\(731\) −74.3374 −2.74947
\(732\) 0 0
\(733\) 5.81868 0.214918 0.107459 0.994210i \(-0.465729\pi\)
0.107459 + 0.994210i \(0.465729\pi\)
\(734\) 1.48063 0.0546512
\(735\) 0 0
\(736\) 18.3941 0.678014
\(737\) 0 0
\(738\) 0 0
\(739\) −14.1113 −0.519093 −0.259546 0.965731i \(-0.583573\pi\)
−0.259546 + 0.965731i \(0.583573\pi\)
\(740\) 46.9248 1.72499
\(741\) 0 0
\(742\) −4.70328 −0.172663
\(743\) −2.28107 −0.0836843 −0.0418422 0.999124i \(-0.513323\pi\)
−0.0418422 + 0.999124i \(0.513323\pi\)
\(744\) 0 0
\(745\) −46.9920 −1.72165
\(746\) 6.92304 0.253471
\(747\) 0 0
\(748\) 0 0
\(749\) 16.8478 0.615607
\(750\) 0 0
\(751\) −25.4870 −0.930033 −0.465016 0.885302i \(-0.653952\pi\)
−0.465016 + 0.885302i \(0.653952\pi\)
\(752\) −39.3181 −1.43378
\(753\) 0 0
\(754\) −2.02770 −0.0738443
\(755\) 11.3371 0.412601
\(756\) 0 0
\(757\) 25.1462 0.913953 0.456976 0.889479i \(-0.348932\pi\)
0.456976 + 0.889479i \(0.348932\pi\)
\(758\) −2.08084 −0.0755796
\(759\) 0 0
\(760\) −4.78195 −0.173460
\(761\) −28.4523 −1.03139 −0.515697 0.856771i \(-0.672467\pi\)
−0.515697 + 0.856771i \(0.672467\pi\)
\(762\) 0 0
\(763\) 9.41870 0.340980
\(764\) 19.3120 0.698685
\(765\) 0 0
\(766\) 7.06556 0.255289
\(767\) 4.71130 0.170115
\(768\) 0 0
\(769\) −45.0119 −1.62317 −0.811586 0.584233i \(-0.801395\pi\)
−0.811586 + 0.584233i \(0.801395\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 23.8180 0.857229
\(773\) −30.8653 −1.11015 −0.555074 0.831801i \(-0.687310\pi\)
−0.555074 + 0.831801i \(0.687310\pi\)
\(774\) 0 0
\(775\) −2.56257 −0.0920502
\(776\) −8.84548 −0.317534
\(777\) 0 0
\(778\) −4.25783 −0.152651
\(779\) −1.56518 −0.0560783
\(780\) 0 0
\(781\) 0 0
\(782\) −12.7902 −0.457375
\(783\) 0 0
\(784\) 3.06119 0.109328
\(785\) −5.16919 −0.184496
\(786\) 0 0
\(787\) −35.6535 −1.27091 −0.635456 0.772137i \(-0.719188\pi\)
−0.635456 + 0.772137i \(0.719188\pi\)
\(788\) 1.22164 0.0435191
\(789\) 0 0
\(790\) −2.50672 −0.0891850
\(791\) 4.35139 0.154718
\(792\) 0 0
\(793\) −30.0771 −1.06807
\(794\) 9.07241 0.321968
\(795\) 0 0
\(796\) −24.5628 −0.870605
\(797\) −44.3444 −1.57076 −0.785379 0.619015i \(-0.787532\pi\)
−0.785379 + 0.619015i \(0.787532\pi\)
\(798\) 0 0
\(799\) 95.9158 3.39326
\(800\) 1.36028 0.0480931
\(801\) 0 0
\(802\) −1.84827 −0.0652647
\(803\) 0 0
\(804\) 0 0
\(805\) −9.84836 −0.347109
\(806\) −14.6102 −0.514624
\(807\) 0 0
\(808\) 14.7223 0.517928
\(809\) −6.41321 −0.225476 −0.112738 0.993625i \(-0.535962\pi\)
−0.112738 + 0.993625i \(0.535962\pi\)
\(810\) 0 0
\(811\) 34.5913 1.21466 0.607332 0.794448i \(-0.292240\pi\)
0.607332 + 0.794448i \(0.292240\pi\)
\(812\) 2.07076 0.0726693
\(813\) 0 0
\(814\) 0 0
\(815\) 22.6814 0.794496
\(816\) 0 0
\(817\) −13.4116 −0.469214
\(818\) 8.43987 0.295093
\(819\) 0 0
\(820\) 4.92634 0.172035
\(821\) 13.8043 0.481772 0.240886 0.970553i \(-0.422562\pi\)
0.240886 + 0.970553i \(0.422562\pi\)
\(822\) 0 0
\(823\) −56.7973 −1.97983 −0.989915 0.141666i \(-0.954754\pi\)
−0.989915 + 0.141666i \(0.954754\pi\)
\(824\) 1.24322 0.0433097
\(825\) 0 0
\(826\) 0.420588 0.0146341
\(827\) −5.29390 −0.184087 −0.0920434 0.995755i \(-0.529340\pi\)
−0.0920434 + 0.995755i \(0.529340\pi\)
\(828\) 0 0
\(829\) 30.6618 1.06493 0.532463 0.846453i \(-0.321266\pi\)
0.532463 + 0.846453i \(0.321266\pi\)
\(830\) −8.43973 −0.292947
\(831\) 0 0
\(832\) −19.7434 −0.684481
\(833\) −7.46771 −0.258741
\(834\) 0 0
\(835\) 19.9316 0.689763
\(836\) 0 0
\(837\) 0 0
\(838\) −5.64673 −0.195063
\(839\) 15.4714 0.534133 0.267067 0.963678i \(-0.413946\pi\)
0.267067 + 0.963678i \(0.413946\pi\)
\(840\) 0 0
\(841\) −27.7324 −0.956289
\(842\) 11.0538 0.380938
\(843\) 0 0
\(844\) −24.0561 −0.828044
\(845\) −16.5405 −0.569011
\(846\) 0 0
\(847\) 0 0
\(848\) −35.9069 −1.23305
\(849\) 0 0
\(850\) −0.945857 −0.0324426
\(851\) 47.2668 1.62029
\(852\) 0 0
\(853\) −36.3631 −1.24505 −0.622524 0.782600i \(-0.713893\pi\)
−0.622524 + 0.782600i \(0.713893\pi\)
\(854\) −2.68504 −0.0918803
\(855\) 0 0
\(856\) −25.9358 −0.886467
\(857\) −17.0833 −0.583554 −0.291777 0.956486i \(-0.594246\pi\)
−0.291777 + 0.956486i \(0.594246\pi\)
\(858\) 0 0
\(859\) 21.3315 0.727822 0.363911 0.931434i \(-0.381441\pi\)
0.363911 + 0.931434i \(0.381441\pi\)
\(860\) 42.2126 1.43944
\(861\) 0 0
\(862\) 13.8092 0.470345
\(863\) −14.9495 −0.508887 −0.254443 0.967088i \(-0.581892\pi\)
−0.254443 + 0.967088i \(0.581892\pi\)
\(864\) 0 0
\(865\) 36.8144 1.25173
\(866\) 6.47104 0.219895
\(867\) 0 0
\(868\) 14.9205 0.506435
\(869\) 0 0
\(870\) 0 0
\(871\) −11.0692 −0.375065
\(872\) −14.4993 −0.491007
\(873\) 0 0
\(874\) −2.30755 −0.0780539
\(875\) 10.7998 0.365100
\(876\) 0 0
\(877\) 17.2499 0.582488 0.291244 0.956649i \(-0.405931\pi\)
0.291244 + 0.956649i \(0.405931\pi\)
\(878\) 7.74015 0.261218
\(879\) 0 0
\(880\) 0 0
\(881\) 2.58251 0.0870069 0.0435035 0.999053i \(-0.486148\pi\)
0.0435035 + 0.999053i \(0.486148\pi\)
\(882\) 0 0
\(883\) 36.2072 1.21847 0.609234 0.792990i \(-0.291477\pi\)
0.609234 + 0.792990i \(0.291477\pi\)
\(884\) 61.6904 2.07487
\(885\) 0 0
\(886\) −5.26520 −0.176888
\(887\) −53.1757 −1.78546 −0.892732 0.450587i \(-0.851215\pi\)
−0.892732 + 0.450587i \(0.851215\pi\)
\(888\) 0 0
\(889\) −6.34739 −0.212885
\(890\) 4.88629 0.163789
\(891\) 0 0
\(892\) −18.8140 −0.629938
\(893\) 17.3047 0.579081
\(894\) 0 0
\(895\) −40.5233 −1.35454
\(896\) −10.3751 −0.346607
\(897\) 0 0
\(898\) 5.11282 0.170617
\(899\) 9.13365 0.304624
\(900\) 0 0
\(901\) 87.5943 2.91819
\(902\) 0 0
\(903\) 0 0
\(904\) −6.69858 −0.222792
\(905\) −47.8751 −1.59142
\(906\) 0 0
\(907\) 48.9616 1.62574 0.812872 0.582443i \(-0.197903\pi\)
0.812872 + 0.582443i \(0.197903\pi\)
\(908\) −5.49682 −0.182418
\(909\) 0 0
\(910\) −4.15237 −0.137650
\(911\) −0.998096 −0.0330684 −0.0165342 0.999863i \(-0.505263\pi\)
−0.0165342 + 0.999863i \(0.505263\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −3.68667 −0.121944
\(915\) 0 0
\(916\) −36.9715 −1.22157
\(917\) 15.2292 0.502912
\(918\) 0 0
\(919\) 53.3533 1.75996 0.879982 0.475008i \(-0.157555\pi\)
0.879982 + 0.475008i \(0.157555\pi\)
\(920\) 15.1607 0.499834
\(921\) 0 0
\(922\) −6.44416 −0.212227
\(923\) −37.8315 −1.24524
\(924\) 0 0
\(925\) 3.49547 0.114930
\(926\) 13.2001 0.433781
\(927\) 0 0
\(928\) −4.84837 −0.159156
\(929\) −2.72724 −0.0894778 −0.0447389 0.998999i \(-0.514246\pi\)
−0.0447389 + 0.998999i \(0.514246\pi\)
\(930\) 0 0
\(931\) −1.34729 −0.0441557
\(932\) −8.59013 −0.281379
\(933\) 0 0
\(934\) 3.41220 0.111651
\(935\) 0 0
\(936\) 0 0
\(937\) 12.6735 0.414024 0.207012 0.978338i \(-0.433626\pi\)
0.207012 + 0.978338i \(0.433626\pi\)
\(938\) −0.988170 −0.0322649
\(939\) 0 0
\(940\) −54.4659 −1.77648
\(941\) 6.00255 0.195678 0.0978388 0.995202i \(-0.468807\pi\)
0.0978388 + 0.995202i \(0.468807\pi\)
\(942\) 0 0
\(943\) 4.96225 0.161593
\(944\) 3.21096 0.104508
\(945\) 0 0
\(946\) 0 0
\(947\) 20.2935 0.659450 0.329725 0.944077i \(-0.393044\pi\)
0.329725 + 0.944077i \(0.393044\pi\)
\(948\) 0 0
\(949\) −48.4052 −1.57130
\(950\) −0.170647 −0.00553653
\(951\) 0 0
\(952\) 11.4959 0.372584
\(953\) 25.6882 0.832122 0.416061 0.909337i \(-0.363410\pi\)
0.416061 + 0.909337i \(0.363410\pi\)
\(954\) 0 0
\(955\) 24.2093 0.783394
\(956\) 41.8741 1.35431
\(957\) 0 0
\(958\) 3.81820 0.123360
\(959\) 19.0993 0.616747
\(960\) 0 0
\(961\) 34.8111 1.12294
\(962\) 19.9291 0.642541
\(963\) 0 0
\(964\) 8.89657 0.286539
\(965\) 29.8579 0.961158
\(966\) 0 0
\(967\) −3.86788 −0.124383 −0.0621914 0.998064i \(-0.519809\pi\)
−0.0621914 + 0.998064i \(0.519809\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −5.31210 −0.170561
\(971\) 25.8216 0.828656 0.414328 0.910128i \(-0.364017\pi\)
0.414328 + 0.910128i \(0.364017\pi\)
\(972\) 0 0
\(973\) −3.93905 −0.126280
\(974\) −0.438813 −0.0140605
\(975\) 0 0
\(976\) −20.4988 −0.656152
\(977\) 52.2227 1.67075 0.835376 0.549679i \(-0.185250\pi\)
0.835376 + 0.549679i \(0.185250\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 4.24055 0.135459
\(981\) 0 0
\(982\) −8.28740 −0.264462
\(983\) 32.4955 1.03645 0.518223 0.855246i \(-0.326594\pi\)
0.518223 + 0.855246i \(0.326594\pi\)
\(984\) 0 0
\(985\) 1.53143 0.0487953
\(986\) 3.37128 0.107363
\(987\) 0 0
\(988\) 11.1299 0.354090
\(989\) 42.5203 1.35207
\(990\) 0 0
\(991\) 1.37062 0.0435391 0.0217695 0.999763i \(-0.493070\pi\)
0.0217695 + 0.999763i \(0.493070\pi\)
\(992\) −34.9342 −1.10916
\(993\) 0 0
\(994\) −3.37730 −0.107122
\(995\) −30.7915 −0.976157
\(996\) 0 0
\(997\) 8.64360 0.273746 0.136873 0.990589i \(-0.456295\pi\)
0.136873 + 0.990589i \(0.456295\pi\)
\(998\) 9.55712 0.302525
\(999\) 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 7623.2.a.dc.1.9 16
3.2 odd 2 inner 7623.2.a.dc.1.8 16
11.3 even 5 693.2.m.k.64.5 yes 32
11.4 even 5 693.2.m.k.379.5 yes 32
11.10 odd 2 7623.2.a.db.1.8 16
33.14 odd 10 693.2.m.k.64.4 32
33.26 odd 10 693.2.m.k.379.4 yes 32
33.32 even 2 7623.2.a.db.1.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.m.k.64.4 32 33.14 odd 10
693.2.m.k.64.5 yes 32 11.3 even 5
693.2.m.k.379.4 yes 32 33.26 odd 10
693.2.m.k.379.5 yes 32 11.4 even 5
7623.2.a.db.1.8 16 11.10 odd 2
7623.2.a.db.1.9 16 33.32 even 2
7623.2.a.dc.1.8 16 3.2 odd 2 inner
7623.2.a.dc.1.9 16 1.1 even 1 trivial