Properties

Label 7623.2.a.cr.1.5
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 6
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: \(6\)
Coefficient field: 6.6.3829849.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.300853\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.36768 q^{2} -0.129461 q^{4} +0.636509 q^{5} +1.00000 q^{7} -2.91241 q^{8} +O(q^{10})\) \(q+1.36768 q^{2} -0.129461 q^{4} +0.636509 q^{5} +1.00000 q^{7} -2.91241 q^{8} +0.870539 q^{10} -3.59486 q^{13} +1.36768 q^{14} -3.72432 q^{16} +1.46233 q^{17} -5.85378 q^{19} -0.0824033 q^{20} +8.56018 q^{23} -4.59486 q^{25} -4.91660 q^{26} -0.129461 q^{28} +7.73436 q^{29} -9.44863 q^{31} +0.731167 q^{32} +2.00000 q^{34} +0.636509 q^{35} +5.59486 q^{37} -8.00607 q^{38} -1.85378 q^{40} +8.37086 q^{41} +1.74108 q^{43} +11.7076 q^{46} -4.99900 q^{47} +1.00000 q^{49} -6.28428 q^{50} +0.465395 q^{52} +11.1062 q^{53} -2.91241 q^{56} +10.5781 q^{58} -13.3944 q^{59} +11.1897 q^{61} -12.9227 q^{62} +8.44863 q^{64} -2.28816 q^{65} -1.85378 q^{67} -0.189316 q^{68} +0.870539 q^{70} +9.66839 q^{71} +11.5949 q^{73} +7.65195 q^{74} +0.757838 q^{76} +4.51785 q^{79} -2.37056 q^{80} +11.4486 q^{82} +3.08948 q^{83} +0.930789 q^{85} +2.38123 q^{86} +4.19769 q^{89} -3.59486 q^{91} -1.10821 q^{92} -6.83702 q^{94} -3.72598 q^{95} +7.44863 q^{97} +1.36768 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4q^{4} + 6q^{7} + O(q^{10}) \) \( 6q + 4q^{4} + 6q^{7} + 10q^{10} + 4q^{13} + 8q^{16} - 2q^{25} + 4q^{28} + 4q^{31} + 12q^{34} + 8q^{37} + 24q^{40} + 20q^{43} + 6q^{49} - 18q^{52} - 2q^{58} + 16q^{61} - 10q^{64} + 24q^{67} + 10q^{70} + 44q^{73} + 54q^{76} + 8q^{79} + 8q^{82} - 36q^{85} + 4q^{91} + 34q^{94} - 16q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36768 0.967093 0.483547 0.875319i \(-0.339348\pi\)
0.483547 + 0.875319i \(0.339348\pi\)
\(3\) 0 0
\(4\) −0.129461 −0.0647306
\(5\) 0.636509 0.284656 0.142328 0.989820i \(-0.454541\pi\)
0.142328 + 0.989820i \(0.454541\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −2.91241 −1.02969
\(9\) 0 0
\(10\) 0.870539 0.275289
\(11\) 0 0
\(12\) 0 0
\(13\) −3.59486 −0.997034 −0.498517 0.866880i \(-0.666122\pi\)
−0.498517 + 0.866880i \(0.666122\pi\)
\(14\) 1.36768 0.365527
\(15\) 0 0
\(16\) −3.72432 −0.931079
\(17\) 1.46233 0.354668 0.177334 0.984151i \(-0.443253\pi\)
0.177334 + 0.984151i \(0.443253\pi\)
\(18\) 0 0
\(19\) −5.85378 −1.34295 −0.671474 0.741028i \(-0.734339\pi\)
−0.671474 + 0.741028i \(0.734339\pi\)
\(20\) −0.0824033 −0.0184259
\(21\) 0 0
\(22\) 0 0
\(23\) 8.56018 1.78492 0.892461 0.451125i \(-0.148977\pi\)
0.892461 + 0.451125i \(0.148977\pi\)
\(24\) 0 0
\(25\) −4.59486 −0.918971
\(26\) −4.91660 −0.964225
\(27\) 0 0
\(28\) −0.129461 −0.0244659
\(29\) 7.73436 1.43623 0.718117 0.695922i \(-0.245004\pi\)
0.718117 + 0.695922i \(0.245004\pi\)
\(30\) 0 0
\(31\) −9.44863 −1.69702 −0.848512 0.529175i \(-0.822501\pi\)
−0.848512 + 0.529175i \(0.822501\pi\)
\(32\) 0.731167 0.129253
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0.636509 0.107590
\(36\) 0 0
\(37\) 5.59486 0.919789 0.459894 0.887974i \(-0.347887\pi\)
0.459894 + 0.887974i \(0.347887\pi\)
\(38\) −8.00607 −1.29876
\(39\) 0 0
\(40\) −1.85378 −0.293108
\(41\) 8.37086 1.30731 0.653655 0.756793i \(-0.273235\pi\)
0.653655 + 0.756793i \(0.273235\pi\)
\(42\) 0 0
\(43\) 1.74108 0.265512 0.132756 0.991149i \(-0.457617\pi\)
0.132756 + 0.991149i \(0.457617\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 11.7076 1.72619
\(47\) −4.99900 −0.729180 −0.364590 0.931168i \(-0.618791\pi\)
−0.364590 + 0.931168i \(0.618791\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −6.28428 −0.888731
\(51\) 0 0
\(52\) 0.465395 0.0645386
\(53\) 11.1062 1.52556 0.762778 0.646660i \(-0.223835\pi\)
0.762778 + 0.646660i \(0.223835\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.91241 −0.389188
\(57\) 0 0
\(58\) 10.5781 1.38897
\(59\) −13.3944 −1.74380 −0.871900 0.489685i \(-0.837112\pi\)
−0.871900 + 0.489685i \(0.837112\pi\)
\(60\) 0 0
\(61\) 11.1897 1.43270 0.716348 0.697743i \(-0.245812\pi\)
0.716348 + 0.697743i \(0.245812\pi\)
\(62\) −12.9227 −1.64118
\(63\) 0 0
\(64\) 8.44863 1.05608
\(65\) −2.28816 −0.283811
\(66\) 0 0
\(67\) −1.85378 −0.226475 −0.113238 0.993568i \(-0.536122\pi\)
−0.113238 + 0.993568i \(0.536122\pi\)
\(68\) −0.189316 −0.0229579
\(69\) 0 0
\(70\) 0.870539 0.104049
\(71\) 9.66839 1.14743 0.573714 0.819056i \(-0.305502\pi\)
0.573714 + 0.819056i \(0.305502\pi\)
\(72\) 0 0
\(73\) 11.5949 1.35708 0.678538 0.734566i \(-0.262614\pi\)
0.678538 + 0.734566i \(0.262614\pi\)
\(74\) 7.65195 0.889521
\(75\) 0 0
\(76\) 0.757838 0.0869299
\(77\) 0 0
\(78\) 0 0
\(79\) 4.51785 0.508297 0.254149 0.967165i \(-0.418205\pi\)
0.254149 + 0.967165i \(0.418205\pi\)
\(80\) −2.37056 −0.265037
\(81\) 0 0
\(82\) 11.4486 1.26429
\(83\) 3.08948 0.339114 0.169557 0.985520i \(-0.445766\pi\)
0.169557 + 0.985520i \(0.445766\pi\)
\(84\) 0 0
\(85\) 0.930789 0.100958
\(86\) 2.38123 0.256775
\(87\) 0 0
\(88\) 0 0
\(89\) 4.19769 0.444954 0.222477 0.974938i \(-0.428586\pi\)
0.222477 + 0.974938i \(0.428586\pi\)
\(90\) 0 0
\(91\) −3.59486 −0.376843
\(92\) −1.10821 −0.115539
\(93\) 0 0
\(94\) −6.83702 −0.705185
\(95\) −3.72598 −0.382278
\(96\) 0 0
\(97\) 7.44863 0.756294 0.378147 0.925746i \(-0.376561\pi\)
0.378147 + 0.925746i \(0.376561\pi\)
\(98\) 1.36768 0.138156
\(99\) 0 0
\(100\) 0.594856 0.0594856
\(101\) −6.93304 −0.689863 −0.344932 0.938628i \(-0.612098\pi\)
−0.344932 + 0.938628i \(0.612098\pi\)
\(102\) 0 0
\(103\) −6.67187 −0.657399 −0.328699 0.944435i \(-0.606610\pi\)
−0.328699 + 0.944435i \(0.606610\pi\)
\(104\) 10.4697 1.02664
\(105\) 0 0
\(106\) 15.1897 1.47536
\(107\) −12.1214 −1.17182 −0.585908 0.810378i \(-0.699262\pi\)
−0.585908 + 0.810378i \(0.699262\pi\)
\(108\) 0 0
\(109\) 13.4486 1.28815 0.644073 0.764964i \(-0.277243\pi\)
0.644073 + 0.764964i \(0.277243\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.72432 −0.351915
\(113\) 9.99801 0.940533 0.470267 0.882524i \(-0.344158\pi\)
0.470267 + 0.882524i \(0.344158\pi\)
\(114\) 0 0
\(115\) 5.44863 0.508088
\(116\) −1.00130 −0.0929683
\(117\) 0 0
\(118\) −18.3192 −1.68642
\(119\) 1.46233 0.134052
\(120\) 0 0
\(121\) 0 0
\(122\) 15.3039 1.38555
\(123\) 0 0
\(124\) 1.22323 0.109850
\(125\) −6.10721 −0.546246
\(126\) 0 0
\(127\) −12.6383 −1.12147 −0.560736 0.827995i \(-0.689482\pi\)
−0.560736 + 0.827995i \(0.689482\pi\)
\(128\) 10.0927 0.892074
\(129\) 0 0
\(130\) −3.12946 −0.274472
\(131\) 4.19769 0.366754 0.183377 0.983043i \(-0.441297\pi\)
0.183377 + 0.983043i \(0.441297\pi\)
\(132\) 0 0
\(133\) −5.85378 −0.507587
\(134\) −2.53537 −0.219023
\(135\) 0 0
\(136\) −4.25892 −0.365200
\(137\) 16.9556 1.44861 0.724305 0.689479i \(-0.242161\pi\)
0.724305 + 0.689479i \(0.242161\pi\)
\(138\) 0 0
\(139\) 12.5178 1.06175 0.530875 0.847450i \(-0.321863\pi\)
0.530875 + 0.847450i \(0.321863\pi\)
\(140\) −0.0824033 −0.00696435
\(141\) 0 0
\(142\) 13.2232 1.10967
\(143\) 0 0
\(144\) 0 0
\(145\) 4.92299 0.408832
\(146\) 15.8580 1.31242
\(147\) 0 0
\(148\) −0.724317 −0.0595385
\(149\) −3.91530 −0.320754 −0.160377 0.987056i \(-0.551271\pi\)
−0.160377 + 0.987056i \(0.551271\pi\)
\(150\) 0 0
\(151\) 0.930789 0.0757466 0.0378733 0.999283i \(-0.487942\pi\)
0.0378733 + 0.999283i \(0.487942\pi\)
\(152\) 17.0486 1.38283
\(153\) 0 0
\(154\) 0 0
\(155\) −6.01414 −0.483068
\(156\) 0 0
\(157\) 5.70756 0.455513 0.227756 0.973718i \(-0.426861\pi\)
0.227756 + 0.973718i \(0.426861\pi\)
\(158\) 6.17895 0.491571
\(159\) 0 0
\(160\) 0.465395 0.0367927
\(161\) 8.56018 0.674637
\(162\) 0 0
\(163\) −1.85378 −0.145199 −0.0725996 0.997361i \(-0.523130\pi\)
−0.0725996 + 0.997361i \(0.523130\pi\)
\(164\) −1.08370 −0.0846230
\(165\) 0 0
\(166\) 4.22540 0.327955
\(167\) 23.1345 1.79020 0.895101 0.445864i \(-0.147103\pi\)
0.895101 + 0.445864i \(0.147103\pi\)
\(168\) 0 0
\(169\) −0.0770108 −0.00592391
\(170\) 1.27302 0.0976361
\(171\) 0 0
\(172\) −0.225402 −0.0171868
\(173\) −7.09785 −0.539639 −0.269820 0.962911i \(-0.586964\pi\)
−0.269820 + 0.962911i \(0.586964\pi\)
\(174\) 0 0
\(175\) −4.59486 −0.347338
\(176\) 0 0
\(177\) 0 0
\(178\) 5.74108 0.430312
\(179\) −5.09207 −0.380599 −0.190300 0.981726i \(-0.560946\pi\)
−0.190300 + 0.981726i \(0.560946\pi\)
\(180\) 0 0
\(181\) −19.4486 −1.44561 −0.722803 0.691054i \(-0.757147\pi\)
−0.722803 + 0.691054i \(0.757147\pi\)
\(182\) −4.91660 −0.364443
\(183\) 0 0
\(184\) −24.9308 −1.83792
\(185\) 3.56118 0.261823
\(186\) 0 0
\(187\) 0 0
\(188\) 0.647177 0.0472003
\(189\) 0 0
\(190\) −5.09594 −0.369698
\(191\) −20.7746 −1.50320 −0.751599 0.659620i \(-0.770717\pi\)
−0.751599 + 0.659620i \(0.770717\pi\)
\(192\) 0 0
\(193\) 21.4486 1.54391 0.771953 0.635679i \(-0.219280\pi\)
0.771953 + 0.635679i \(0.219280\pi\)
\(194\) 10.1873 0.731407
\(195\) 0 0
\(196\) −0.129461 −0.00924723
\(197\) −3.27879 −0.233604 −0.116802 0.993155i \(-0.537264\pi\)
−0.116802 + 0.993155i \(0.537264\pi\)
\(198\) 0 0
\(199\) 16.6383 1.17946 0.589731 0.807600i \(-0.299234\pi\)
0.589731 + 0.807600i \(0.299234\pi\)
\(200\) 13.3821 0.946259
\(201\) 0 0
\(202\) −9.48215 −0.667162
\(203\) 7.73436 0.542845
\(204\) 0 0
\(205\) 5.32813 0.372133
\(206\) −9.12495 −0.635766
\(207\) 0 0
\(208\) 13.3884 0.928317
\(209\) 0 0
\(210\) 0 0
\(211\) 22.3794 1.54066 0.770332 0.637644i \(-0.220091\pi\)
0.770332 + 0.637644i \(0.220091\pi\)
\(212\) −1.43783 −0.0987502
\(213\) 0 0
\(214\) −16.5781 −1.13326
\(215\) 1.10821 0.0755794
\(216\) 0 0
\(217\) −9.44863 −0.641415
\(218\) 18.3934 1.24576
\(219\) 0 0
\(220\) 0 0
\(221\) −5.25688 −0.353616
\(222\) 0 0
\(223\) −8.51785 −0.570397 −0.285199 0.958468i \(-0.592060\pi\)
−0.285199 + 0.958468i \(0.592060\pi\)
\(224\) 0.731167 0.0488532
\(225\) 0 0
\(226\) 13.6740 0.909583
\(227\) −23.8641 −1.58391 −0.791957 0.610576i \(-0.790938\pi\)
−0.791957 + 0.610576i \(0.790938\pi\)
\(228\) 0 0
\(229\) 4.77677 0.315658 0.157829 0.987466i \(-0.449551\pi\)
0.157829 + 0.987466i \(0.449551\pi\)
\(230\) 7.45197 0.491368
\(231\) 0 0
\(232\) −22.5256 −1.47888
\(233\) −22.5666 −1.47838 −0.739192 0.673495i \(-0.764792\pi\)
−0.739192 + 0.673495i \(0.764792\pi\)
\(234\) 0 0
\(235\) −3.18191 −0.207565
\(236\) 1.73405 0.112877
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) 20.1381 1.30263 0.651313 0.758809i \(-0.274219\pi\)
0.651313 + 0.758809i \(0.274219\pi\)
\(240\) 0 0
\(241\) 0.922989 0.0594550 0.0297275 0.999558i \(-0.490536\pi\)
0.0297275 + 0.999558i \(0.490536\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −1.44863 −0.0927393
\(245\) 0.636509 0.0406651
\(246\) 0 0
\(247\) 21.0435 1.33897
\(248\) 27.5183 1.74742
\(249\) 0 0
\(250\) −8.35269 −0.528271
\(251\) 10.8483 0.684741 0.342371 0.939565i \(-0.388770\pi\)
0.342371 + 0.939565i \(0.388770\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −17.2852 −1.08457
\(255\) 0 0
\(256\) −3.09377 −0.193361
\(257\) 6.81546 0.425137 0.212568 0.977146i \(-0.431817\pi\)
0.212568 + 0.977146i \(0.431817\pi\)
\(258\) 0 0
\(259\) 5.59486 0.347647
\(260\) 0.296228 0.0183713
\(261\) 0 0
\(262\) 5.74108 0.354685
\(263\) 7.92367 0.488595 0.244297 0.969700i \(-0.421443\pi\)
0.244297 + 0.969700i \(0.421443\pi\)
\(264\) 0 0
\(265\) 7.06921 0.434258
\(266\) −8.00607 −0.490884
\(267\) 0 0
\(268\) 0.239993 0.0146599
\(269\) 26.4101 1.61025 0.805127 0.593103i \(-0.202097\pi\)
0.805127 + 0.593103i \(0.202097\pi\)
\(270\) 0 0
\(271\) 18.1462 1.10230 0.551152 0.834405i \(-0.314188\pi\)
0.551152 + 0.834405i \(0.314188\pi\)
\(272\) −5.44620 −0.330224
\(273\) 0 0
\(274\) 23.1897 1.40094
\(275\) 0 0
\(276\) 0 0
\(277\) 4.29244 0.257908 0.128954 0.991651i \(-0.458838\pi\)
0.128954 + 0.991651i \(0.458838\pi\)
\(278\) 17.1204 1.02681
\(279\) 0 0
\(280\) −1.85378 −0.110784
\(281\) −12.3107 −0.734393 −0.367197 0.930143i \(-0.619682\pi\)
−0.367197 + 0.930143i \(0.619682\pi\)
\(282\) 0 0
\(283\) −18.3716 −1.09208 −0.546040 0.837759i \(-0.683865\pi\)
−0.546040 + 0.837759i \(0.683865\pi\)
\(284\) −1.25168 −0.0742737
\(285\) 0 0
\(286\) 0 0
\(287\) 8.37086 0.494117
\(288\) 0 0
\(289\) −14.8616 −0.874211
\(290\) 6.73306 0.395379
\(291\) 0 0
\(292\) −1.50108 −0.0878443
\(293\) 3.46493 0.202424 0.101212 0.994865i \(-0.467728\pi\)
0.101212 + 0.994865i \(0.467728\pi\)
\(294\) 0 0
\(295\) −8.52565 −0.496382
\(296\) −16.2945 −0.947101
\(297\) 0 0
\(298\) −5.35486 −0.310199
\(299\) −30.7726 −1.77963
\(300\) 0 0
\(301\) 1.74108 0.100354
\(302\) 1.27302 0.0732540
\(303\) 0 0
\(304\) 21.8013 1.25039
\(305\) 7.12236 0.407825
\(306\) 0 0
\(307\) 9.03569 0.515694 0.257847 0.966186i \(-0.416987\pi\)
0.257847 + 0.966186i \(0.416987\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −8.22540 −0.467171
\(311\) 8.72499 0.494749 0.247374 0.968920i \(-0.420432\pi\)
0.247374 + 0.968920i \(0.420432\pi\)
\(312\) 0 0
\(313\) −14.1205 −0.798138 −0.399069 0.916921i \(-0.630667\pi\)
−0.399069 + 0.916921i \(0.630667\pi\)
\(314\) 7.80609 0.440523
\(315\) 0 0
\(316\) −0.584886 −0.0329024
\(317\) 16.0121 0.899332 0.449666 0.893197i \(-0.351543\pi\)
0.449666 + 0.893197i \(0.351543\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 5.37763 0.300619
\(321\) 0 0
\(322\) 11.7076 0.652437
\(323\) −8.56018 −0.476301
\(324\) 0 0
\(325\) 16.5178 0.916245
\(326\) −2.53537 −0.140421
\(327\) 0 0
\(328\) −24.3794 −1.34613
\(329\) −4.99900 −0.275604
\(330\) 0 0
\(331\) −13.0357 −0.716506 −0.358253 0.933624i \(-0.616628\pi\)
−0.358253 + 0.933624i \(0.616628\pi\)
\(332\) −0.399967 −0.0219511
\(333\) 0 0
\(334\) 31.6405 1.73129
\(335\) −1.17995 −0.0644674
\(336\) 0 0
\(337\) 22.4843 1.22480 0.612400 0.790548i \(-0.290204\pi\)
0.612400 + 0.790548i \(0.290204\pi\)
\(338\) −0.105326 −0.00572897
\(339\) 0 0
\(340\) −0.120501 −0.00653509
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −5.07074 −0.273396
\(345\) 0 0
\(346\) −9.70756 −0.521881
\(347\) 2.87565 0.154373 0.0771865 0.997017i \(-0.475406\pi\)
0.0771865 + 0.997017i \(0.475406\pi\)
\(348\) 0 0
\(349\) −7.07701 −0.378824 −0.189412 0.981898i \(-0.560658\pi\)
−0.189412 + 0.981898i \(0.560658\pi\)
\(350\) −6.28428 −0.335909
\(351\) 0 0
\(352\) 0 0
\(353\) −3.93981 −0.209695 −0.104847 0.994488i \(-0.533435\pi\)
−0.104847 + 0.994488i \(0.533435\pi\)
\(354\) 0 0
\(355\) 6.15402 0.326622
\(356\) −0.543438 −0.0288022
\(357\) 0 0
\(358\) −6.96431 −0.368075
\(359\) −10.5628 −0.557482 −0.278741 0.960366i \(-0.589917\pi\)
−0.278741 + 0.960366i \(0.589917\pi\)
\(360\) 0 0
\(361\) 15.2667 0.803512
\(362\) −26.5994 −1.39804
\(363\) 0 0
\(364\) 0.465395 0.0243933
\(365\) 7.38023 0.386299
\(366\) 0 0
\(367\) 21.8616 1.14117 0.570583 0.821240i \(-0.306717\pi\)
0.570583 + 0.821240i \(0.306717\pi\)
\(368\) −31.8808 −1.66190
\(369\) 0 0
\(370\) 4.87054 0.253207
\(371\) 11.1062 0.576606
\(372\) 0 0
\(373\) 6.25892 0.324075 0.162037 0.986785i \(-0.448193\pi\)
0.162037 + 0.986785i \(0.448193\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 14.5592 0.750832
\(377\) −27.8039 −1.43197
\(378\) 0 0
\(379\) 22.3716 1.14915 0.574577 0.818451i \(-0.305167\pi\)
0.574577 + 0.818451i \(0.305167\pi\)
\(380\) 0.482371 0.0247451
\(381\) 0 0
\(382\) −28.4129 −1.45373
\(383\) −22.9697 −1.17370 −0.586848 0.809697i \(-0.699632\pi\)
−0.586848 + 0.809697i \(0.699632\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 29.3348 1.49310
\(387\) 0 0
\(388\) −0.964310 −0.0489554
\(389\) −38.0598 −1.92971 −0.964854 0.262788i \(-0.915358\pi\)
−0.964854 + 0.262788i \(0.915358\pi\)
\(390\) 0 0
\(391\) 12.5178 0.633055
\(392\) −2.91241 −0.147099
\(393\) 0 0
\(394\) −4.48432 −0.225917
\(395\) 2.87565 0.144690
\(396\) 0 0
\(397\) −17.6027 −0.883452 −0.441726 0.897150i \(-0.645634\pi\)
−0.441726 + 0.897150i \(0.645634\pi\)
\(398\) 22.7559 1.14065
\(399\) 0 0
\(400\) 17.1127 0.855635
\(401\) 6.95755 0.347443 0.173722 0.984795i \(-0.444421\pi\)
0.173722 + 0.984795i \(0.444421\pi\)
\(402\) 0 0
\(403\) 33.9665 1.69199
\(404\) 0.897560 0.0446553
\(405\) 0 0
\(406\) 10.5781 0.524982
\(407\) 0 0
\(408\) 0 0
\(409\) 14.6719 0.725477 0.362739 0.931891i \(-0.381842\pi\)
0.362739 + 0.931891i \(0.381842\pi\)
\(410\) 7.28716 0.359887
\(411\) 0 0
\(412\) 0.863748 0.0425538
\(413\) −13.3944 −0.659094
\(414\) 0 0
\(415\) 1.96648 0.0965307
\(416\) −2.62844 −0.128870
\(417\) 0 0
\(418\) 0 0
\(419\) 16.2700 0.794843 0.397421 0.917636i \(-0.369905\pi\)
0.397421 + 0.917636i \(0.369905\pi\)
\(420\) 0 0
\(421\) 20.7846 1.01298 0.506489 0.862246i \(-0.330943\pi\)
0.506489 + 0.862246i \(0.330943\pi\)
\(422\) 30.6078 1.48996
\(423\) 0 0
\(424\) −32.3459 −1.57086
\(425\) −6.71922 −0.325930
\(426\) 0 0
\(427\) 11.1897 0.541508
\(428\) 1.56925 0.0758524
\(429\) 0 0
\(430\) 1.51568 0.0730924
\(431\) 33.4394 1.61072 0.805360 0.592786i \(-0.201972\pi\)
0.805360 + 0.592786i \(0.201972\pi\)
\(432\) 0 0
\(433\) −29.7076 −1.42765 −0.713827 0.700322i \(-0.753040\pi\)
−0.713827 + 0.700322i \(0.753040\pi\)
\(434\) −12.9227 −0.620308
\(435\) 0 0
\(436\) −1.74108 −0.0833825
\(437\) −50.1094 −2.39706
\(438\) 0 0
\(439\) −4.81809 −0.229955 −0.114977 0.993368i \(-0.536680\pi\)
−0.114977 + 0.993368i \(0.536680\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −7.18971 −0.341980
\(443\) −7.12236 −0.338393 −0.169197 0.985582i \(-0.554117\pi\)
−0.169197 + 0.985582i \(0.554117\pi\)
\(444\) 0 0
\(445\) 2.67187 0.126659
\(446\) −11.6497 −0.551627
\(447\) 0 0
\(448\) 8.44863 0.399160
\(449\) −32.2104 −1.52010 −0.760052 0.649862i \(-0.774827\pi\)
−0.760052 + 0.649862i \(0.774827\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −1.29435 −0.0608813
\(453\) 0 0
\(454\) −32.6383 −1.53179
\(455\) −2.28816 −0.107271
\(456\) 0 0
\(457\) −19.8281 −0.927517 −0.463759 0.885962i \(-0.653500\pi\)
−0.463759 + 0.885962i \(0.653500\pi\)
\(458\) 6.53307 0.305270
\(459\) 0 0
\(460\) −0.705387 −0.0328888
\(461\) 33.8866 1.57826 0.789128 0.614229i \(-0.210533\pi\)
0.789128 + 0.614229i \(0.210533\pi\)
\(462\) 0 0
\(463\) 1.56134 0.0725614 0.0362807 0.999342i \(-0.488449\pi\)
0.0362807 + 0.999342i \(0.488449\pi\)
\(464\) −28.8052 −1.33725
\(465\) 0 0
\(466\) −30.8637 −1.42974
\(467\) 5.94241 0.274982 0.137491 0.990503i \(-0.456096\pi\)
0.137491 + 0.990503i \(0.456096\pi\)
\(468\) 0 0
\(469\) −1.85378 −0.0855995
\(470\) −4.35183 −0.200735
\(471\) 0 0
\(472\) 39.0100 1.79558
\(473\) 0 0
\(474\) 0 0
\(475\) 26.8973 1.23413
\(476\) −0.189316 −0.00867727
\(477\) 0 0
\(478\) 27.5424 1.25976
\(479\) 29.1700 1.33281 0.666405 0.745590i \(-0.267832\pi\)
0.666405 + 0.745590i \(0.267832\pi\)
\(480\) 0 0
\(481\) −20.1127 −0.917060
\(482\) 1.26235 0.0574985
\(483\) 0 0
\(484\) 0 0
\(485\) 4.74112 0.215283
\(486\) 0 0
\(487\) 15.4151 0.698525 0.349263 0.937025i \(-0.386432\pi\)
0.349263 + 0.937025i \(0.386432\pi\)
\(488\) −32.5891 −1.47524
\(489\) 0 0
\(490\) 0.870539 0.0393269
\(491\) 24.6654 1.11313 0.556567 0.830803i \(-0.312118\pi\)
0.556567 + 0.830803i \(0.312118\pi\)
\(492\) 0 0
\(493\) 11.3102 0.509386
\(494\) 28.7807 1.29490
\(495\) 0 0
\(496\) 35.1897 1.58006
\(497\) 9.66839 0.433687
\(498\) 0 0
\(499\) 2.43866 0.109170 0.0545848 0.998509i \(-0.482616\pi\)
0.0545848 + 0.998509i \(0.482616\pi\)
\(500\) 0.790648 0.0353588
\(501\) 0 0
\(502\) 14.8370 0.662209
\(503\) 18.2286 0.812772 0.406386 0.913702i \(-0.366789\pi\)
0.406386 + 0.913702i \(0.366789\pi\)
\(504\) 0 0
\(505\) −4.41294 −0.196373
\(506\) 0 0
\(507\) 0 0
\(508\) 1.63618 0.0725936
\(509\) −20.2312 −0.896731 −0.448365 0.893850i \(-0.647994\pi\)
−0.448365 + 0.893850i \(0.647994\pi\)
\(510\) 0 0
\(511\) 11.5949 0.512926
\(512\) −24.4166 −1.07907
\(513\) 0 0
\(514\) 9.32134 0.411147
\(515\) −4.24671 −0.187132
\(516\) 0 0
\(517\) 0 0
\(518\) 7.65195 0.336207
\(519\) 0 0
\(520\) 6.66407 0.292239
\(521\) 33.7904 1.48038 0.740191 0.672396i \(-0.234735\pi\)
0.740191 + 0.672396i \(0.234735\pi\)
\(522\) 0 0
\(523\) 14.8895 0.651071 0.325536 0.945530i \(-0.394455\pi\)
0.325536 + 0.945530i \(0.394455\pi\)
\(524\) −0.543438 −0.0237402
\(525\) 0 0
\(526\) 10.8370 0.472516
\(527\) −13.8171 −0.601881
\(528\) 0 0
\(529\) 50.2767 2.18594
\(530\) 9.66839 0.419968
\(531\) 0 0
\(532\) 0.757838 0.0328564
\(533\) −30.0921 −1.30343
\(534\) 0 0
\(535\) −7.71536 −0.333564
\(536\) 5.39897 0.233200
\(537\) 0 0
\(538\) 36.1205 1.55727
\(539\) 0 0
\(540\) 0 0
\(541\) 24.2254 1.04153 0.520766 0.853700i \(-0.325647\pi\)
0.520766 + 0.853700i \(0.325647\pi\)
\(542\) 24.8182 1.06603
\(543\) 0 0
\(544\) 1.06921 0.0458420
\(545\) 8.56018 0.366678
\(546\) 0 0
\(547\) −18.0870 −0.773343 −0.386672 0.922217i \(-0.626375\pi\)
−0.386672 + 0.922217i \(0.626375\pi\)
\(548\) −2.19509 −0.0937695
\(549\) 0 0
\(550\) 0 0
\(551\) −45.2752 −1.92879
\(552\) 0 0
\(553\) 4.51785 0.192118
\(554\) 5.87067 0.249421
\(555\) 0 0
\(556\) −1.62058 −0.0687277
\(557\) 24.4761 1.03709 0.518543 0.855052i \(-0.326475\pi\)
0.518543 + 0.855052i \(0.326475\pi\)
\(558\) 0 0
\(559\) −6.25892 −0.264724
\(560\) −2.37056 −0.100175
\(561\) 0 0
\(562\) −16.8370 −0.710227
\(563\) −37.5163 −1.58113 −0.790563 0.612381i \(-0.790212\pi\)
−0.790563 + 0.612381i \(0.790212\pi\)
\(564\) 0 0
\(565\) 6.36382 0.267728
\(566\) −25.1264 −1.05614
\(567\) 0 0
\(568\) −28.1584 −1.18150
\(569\) −19.3123 −0.809613 −0.404806 0.914402i \(-0.632661\pi\)
−0.404806 + 0.914402i \(0.632661\pi\)
\(570\) 0 0
\(571\) −8.81029 −0.368699 −0.184350 0.982861i \(-0.559018\pi\)
−0.184350 + 0.982861i \(0.559018\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 11.4486 0.477857
\(575\) −39.3328 −1.64029
\(576\) 0 0
\(577\) −31.6740 −1.31861 −0.659304 0.751877i \(-0.729149\pi\)
−0.659304 + 0.751877i \(0.729149\pi\)
\(578\) −20.3258 −0.845443
\(579\) 0 0
\(580\) −0.637336 −0.0264640
\(581\) 3.08948 0.128173
\(582\) 0 0
\(583\) 0 0
\(584\) −33.7690 −1.39737
\(585\) 0 0
\(586\) 4.73891 0.195762
\(587\) 13.7240 0.566450 0.283225 0.959054i \(-0.408596\pi\)
0.283225 + 0.959054i \(0.408596\pi\)
\(588\) 0 0
\(589\) 55.3102 2.27902
\(590\) −11.6603 −0.480048
\(591\) 0 0
\(592\) −20.8370 −0.856396
\(593\) 4.73795 0.194564 0.0972822 0.995257i \(-0.468985\pi\)
0.0972822 + 0.995257i \(0.468985\pi\)
\(594\) 0 0
\(595\) 0.930789 0.0381586
\(596\) 0.506880 0.0207626
\(597\) 0 0
\(598\) −42.0870 −1.72106
\(599\) −19.6664 −0.803547 −0.401774 0.915739i \(-0.631606\pi\)
−0.401774 + 0.915739i \(0.631606\pi\)
\(600\) 0 0
\(601\) 0.697587 0.0284552 0.0142276 0.999899i \(-0.495471\pi\)
0.0142276 + 0.999899i \(0.495471\pi\)
\(602\) 2.38123 0.0970517
\(603\) 0 0
\(604\) −0.120501 −0.00490312
\(605\) 0 0
\(606\) 0 0
\(607\) −37.2689 −1.51270 −0.756349 0.654169i \(-0.773019\pi\)
−0.756349 + 0.654169i \(0.773019\pi\)
\(608\) −4.28009 −0.173581
\(609\) 0 0
\(610\) 9.74108 0.394405
\(611\) 17.9707 0.727017
\(612\) 0 0
\(613\) 5.22323 0.210964 0.105482 0.994421i \(-0.466361\pi\)
0.105482 + 0.994421i \(0.466361\pi\)
\(614\) 12.3579 0.498724
\(615\) 0 0
\(616\) 0 0
\(617\) 6.17895 0.248755 0.124378 0.992235i \(-0.460307\pi\)
0.124378 + 0.992235i \(0.460307\pi\)
\(618\) 0 0
\(619\) −17.1562 −0.689566 −0.344783 0.938683i \(-0.612047\pi\)
−0.344783 + 0.938683i \(0.612047\pi\)
\(620\) 0.778599 0.0312693
\(621\) 0 0
\(622\) 11.9330 0.478468
\(623\) 4.19769 0.168177
\(624\) 0 0
\(625\) 19.0870 0.763479
\(626\) −19.3123 −0.771874
\(627\) 0 0
\(628\) −0.738908 −0.0294856
\(629\) 8.18155 0.326220
\(630\) 0 0
\(631\) 9.03569 0.359705 0.179853 0.983694i \(-0.442438\pi\)
0.179853 + 0.983694i \(0.442438\pi\)
\(632\) −13.1578 −0.523391
\(633\) 0 0
\(634\) 21.8994 0.869738
\(635\) −8.04442 −0.319233
\(636\) 0 0
\(637\) −3.59486 −0.142433
\(638\) 0 0
\(639\) 0 0
\(640\) 6.42407 0.253934
\(641\) −5.25688 −0.207634 −0.103817 0.994596i \(-0.533106\pi\)
−0.103817 + 0.994596i \(0.533106\pi\)
\(642\) 0 0
\(643\) −45.6740 −1.80121 −0.900604 0.434640i \(-0.856875\pi\)
−0.900604 + 0.434640i \(0.856875\pi\)
\(644\) −1.10821 −0.0436697
\(645\) 0 0
\(646\) −11.7076 −0.460628
\(647\) −2.63911 −0.103754 −0.0518770 0.998653i \(-0.516520\pi\)
−0.0518770 + 0.998653i \(0.516520\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 22.5911 0.886095
\(651\) 0 0
\(652\) 0.239993 0.00939883
\(653\) 3.30330 0.129268 0.0646341 0.997909i \(-0.479412\pi\)
0.0646341 + 0.997909i \(0.479412\pi\)
\(654\) 0 0
\(655\) 2.67187 0.104398
\(656\) −31.1758 −1.21721
\(657\) 0 0
\(658\) −6.83702 −0.266535
\(659\) −3.72598 −0.145144 −0.0725719 0.997363i \(-0.523121\pi\)
−0.0725719 + 0.997363i \(0.523121\pi\)
\(660\) 0 0
\(661\) −29.5356 −1.14880 −0.574401 0.818574i \(-0.694765\pi\)
−0.574401 + 0.818574i \(0.694765\pi\)
\(662\) −17.8286 −0.692928
\(663\) 0 0
\(664\) −8.99783 −0.349184
\(665\) −3.72598 −0.144487
\(666\) 0 0
\(667\) 66.2075 2.56356
\(668\) −2.99502 −0.115881
\(669\) 0 0
\(670\) −1.61379 −0.0623460
\(671\) 0 0
\(672\) 0 0
\(673\) 6.84381 0.263809 0.131905 0.991262i \(-0.457891\pi\)
0.131905 + 0.991262i \(0.457891\pi\)
\(674\) 30.7513 1.18450
\(675\) 0 0
\(676\) 0.00996992 0.000383458 0
\(677\) 33.5080 1.28782 0.643908 0.765103i \(-0.277312\pi\)
0.643908 + 0.765103i \(0.277312\pi\)
\(678\) 0 0
\(679\) 7.44863 0.285852
\(680\) −2.71084 −0.103956
\(681\) 0 0
\(682\) 0 0
\(683\) −3.98386 −0.152438 −0.0762191 0.997091i \(-0.524285\pi\)
−0.0762191 + 0.997091i \(0.524285\pi\)
\(684\) 0 0
\(685\) 10.7924 0.412355
\(686\) 1.36768 0.0522181
\(687\) 0 0
\(688\) −6.48432 −0.247213
\(689\) −39.9253 −1.52103
\(690\) 0 0
\(691\) −17.8616 −0.679486 −0.339743 0.940518i \(-0.610340\pi\)
−0.339743 + 0.940518i \(0.610340\pi\)
\(692\) 0.918896 0.0349312
\(693\) 0 0
\(694\) 3.93296 0.149293
\(695\) 7.96772 0.302233
\(696\) 0 0
\(697\) 12.2410 0.463661
\(698\) −9.67906 −0.366358
\(699\) 0 0
\(700\) 0.594856 0.0224834
\(701\) 50.6773 1.91406 0.957029 0.289994i \(-0.0936533\pi\)
0.957029 + 0.289994i \(0.0936533\pi\)
\(702\) 0 0
\(703\) −32.7510 −1.23523
\(704\) 0 0
\(705\) 0 0
\(706\) −5.38838 −0.202795
\(707\) −6.93304 −0.260744
\(708\) 0 0
\(709\) −3.97428 −0.149257 −0.0746286 0.997211i \(-0.523777\pi\)
−0.0746286 + 0.997211i \(0.523777\pi\)
\(710\) 8.41671 0.315874
\(711\) 0 0
\(712\) −12.2254 −0.458166
\(713\) −80.8820 −3.02906
\(714\) 0 0
\(715\) 0 0
\(716\) 0.659226 0.0246364
\(717\) 0 0
\(718\) −14.4465 −0.539137
\(719\) 44.0022 1.64100 0.820502 0.571643i \(-0.193694\pi\)
0.820502 + 0.571643i \(0.193694\pi\)
\(720\) 0 0
\(721\) −6.67187 −0.248473
\(722\) 20.8799 0.777071
\(723\) 0 0
\(724\) 2.51785 0.0935750
\(725\) −35.5383 −1.31986
\(726\) 0 0
\(727\) 39.0022 1.44651 0.723255 0.690581i \(-0.242645\pi\)
0.723255 + 0.690581i \(0.242645\pi\)
\(728\) 10.4697 0.388033
\(729\) 0 0
\(730\) 10.0938 0.373587
\(731\) 2.54604 0.0941686
\(732\) 0 0
\(733\) 3.18971 0.117815 0.0589073 0.998263i \(-0.481238\pi\)
0.0589073 + 0.998263i \(0.481238\pi\)
\(734\) 29.8996 1.10361
\(735\) 0 0
\(736\) 6.25892 0.230707
\(737\) 0 0
\(738\) 0 0
\(739\) −49.1718 −1.80881 −0.904407 0.426671i \(-0.859686\pi\)
−0.904407 + 0.426671i \(0.859686\pi\)
\(740\) −0.461035 −0.0169480
\(741\) 0 0
\(742\) 15.1897 0.557632
\(743\) −42.3505 −1.55369 −0.776845 0.629692i \(-0.783181\pi\)
−0.776845 + 0.629692i \(0.783181\pi\)
\(744\) 0 0
\(745\) −2.49212 −0.0913044
\(746\) 8.56018 0.313410
\(747\) 0 0
\(748\) 0 0
\(749\) −12.1214 −0.442905
\(750\) 0 0
\(751\) 38.0792 1.38953 0.694765 0.719237i \(-0.255509\pi\)
0.694765 + 0.719237i \(0.255509\pi\)
\(752\) 18.6179 0.678924
\(753\) 0 0
\(754\) −38.0267 −1.38485
\(755\) 0.592456 0.0215617
\(756\) 0 0
\(757\) −19.6818 −0.715349 −0.357674 0.933846i \(-0.616430\pi\)
−0.357674 + 0.933846i \(0.616430\pi\)
\(758\) 30.5971 1.11134
\(759\) 0 0
\(760\) 10.8516 0.393629
\(761\) 21.6232 0.783839 0.391919 0.920000i \(-0.371811\pi\)
0.391919 + 0.920000i \(0.371811\pi\)
\(762\) 0 0
\(763\) 13.4486 0.486873
\(764\) 2.68951 0.0973030
\(765\) 0 0
\(766\) −31.4151 −1.13507
\(767\) 48.1509 1.73863
\(768\) 0 0
\(769\) −35.0100 −1.26249 −0.631246 0.775583i \(-0.717456\pi\)
−0.631246 + 0.775583i \(0.717456\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.77677 −0.0999381
\(773\) −40.9127 −1.47153 −0.735764 0.677238i \(-0.763177\pi\)
−0.735764 + 0.677238i \(0.763177\pi\)
\(774\) 0 0
\(775\) 43.4151 1.55952
\(776\) −21.6935 −0.778752
\(777\) 0 0
\(778\) −52.0535 −1.86621
\(779\) −49.0012 −1.75565
\(780\) 0 0
\(781\) 0 0
\(782\) 17.1204 0.612223
\(783\) 0 0
\(784\) −3.72432 −0.133011
\(785\) 3.63291 0.129664
\(786\) 0 0
\(787\) 25.3359 0.903128 0.451564 0.892239i \(-0.350866\pi\)
0.451564 + 0.892239i \(0.350866\pi\)
\(788\) 0.424476 0.0151213
\(789\) 0 0
\(790\) 3.93296 0.139928
\(791\) 9.99801 0.355488
\(792\) 0 0
\(793\) −40.2254 −1.42845
\(794\) −24.0747 −0.854380
\(795\) 0 0
\(796\) −2.15402 −0.0763473
\(797\) 16.4839 0.583888 0.291944 0.956435i \(-0.405698\pi\)
0.291944 + 0.956435i \(0.405698\pi\)
\(798\) 0 0
\(799\) −7.31021 −0.258617
\(800\) −3.35961 −0.118780
\(801\) 0 0
\(802\) 9.51568 0.336010
\(803\) 0 0
\(804\) 0 0
\(805\) 5.44863 0.192039
\(806\) 46.4552 1.63631
\(807\) 0 0
\(808\) 20.1919 0.710348
\(809\) −41.7826 −1.46900 −0.734499 0.678610i \(-0.762583\pi\)
−0.734499 + 0.678610i \(0.762583\pi\)
\(810\) 0 0
\(811\) 9.33593 0.327829 0.163914 0.986475i \(-0.447588\pi\)
0.163914 + 0.986475i \(0.447588\pi\)
\(812\) −1.00130 −0.0351387
\(813\) 0 0
\(814\) 0 0
\(815\) −1.17995 −0.0413317
\(816\) 0 0
\(817\) −10.1919 −0.356569
\(818\) 20.0664 0.701604
\(819\) 0 0
\(820\) −0.689787 −0.0240884
\(821\) 11.5534 0.403217 0.201608 0.979466i \(-0.435383\pi\)
0.201608 + 0.979466i \(0.435383\pi\)
\(822\) 0 0
\(823\) −21.2689 −0.741387 −0.370693 0.928755i \(-0.620880\pi\)
−0.370693 + 0.928755i \(0.620880\pi\)
\(824\) 19.4312 0.676919
\(825\) 0 0
\(826\) −18.3192 −0.637406
\(827\) 7.54504 0.262367 0.131183 0.991358i \(-0.458122\pi\)
0.131183 + 0.991358i \(0.458122\pi\)
\(828\) 0 0
\(829\) −32.6048 −1.13241 −0.566206 0.824264i \(-0.691589\pi\)
−0.566206 + 0.824264i \(0.691589\pi\)
\(830\) 2.68951 0.0933542
\(831\) 0 0
\(832\) −30.3716 −1.05295
\(833\) 1.46233 0.0506669
\(834\) 0 0
\(835\) 14.7253 0.509591
\(836\) 0 0
\(837\) 0 0
\(838\) 22.2521 0.768687
\(839\) 51.7838 1.78777 0.893887 0.448292i \(-0.147968\pi\)
0.893887 + 0.448292i \(0.147968\pi\)
\(840\) 0 0
\(841\) 30.8203 1.06277
\(842\) 28.4266 0.979644
\(843\) 0 0
\(844\) −2.89727 −0.0997281
\(845\) −0.0490181 −0.00168627
\(846\) 0 0
\(847\) 0 0
\(848\) −41.3631 −1.42041
\(849\) 0 0
\(850\) −9.18971 −0.315205
\(851\) 47.8930 1.64175
\(852\) 0 0
\(853\) −32.9842 −1.12936 −0.564680 0.825310i \(-0.691000\pi\)
−0.564680 + 0.825310i \(0.691000\pi\)
\(854\) 15.3039 0.523689
\(855\) 0 0
\(856\) 35.3024 1.20661
\(857\) −35.1596 −1.20103 −0.600515 0.799614i \(-0.705038\pi\)
−0.600515 + 0.799614i \(0.705038\pi\)
\(858\) 0 0
\(859\) −46.3794 −1.58245 −0.791223 0.611528i \(-0.790555\pi\)
−0.791223 + 0.611528i \(0.790555\pi\)
\(860\) −0.143471 −0.00489230
\(861\) 0 0
\(862\) 45.7343 1.55772
\(863\) 16.0121 0.545060 0.272530 0.962147i \(-0.412140\pi\)
0.272530 + 0.962147i \(0.412140\pi\)
\(864\) 0 0
\(865\) −4.51785 −0.153611
\(866\) −40.6303 −1.38067
\(867\) 0 0
\(868\) 1.22323 0.0415192
\(869\) 0 0
\(870\) 0 0
\(871\) 6.66407 0.225803
\(872\) −39.1680 −1.32640
\(873\) 0 0
\(874\) −68.5334 −2.31818
\(875\) −6.10721 −0.206462
\(876\) 0 0
\(877\) 40.2410 1.35884 0.679421 0.733749i \(-0.262231\pi\)
0.679421 + 0.733749i \(0.262231\pi\)
\(878\) −6.58959 −0.222388
\(879\) 0 0
\(880\) 0 0
\(881\) −41.9996 −1.41500 −0.707501 0.706712i \(-0.750178\pi\)
−0.707501 + 0.706712i \(0.750178\pi\)
\(882\) 0 0
\(883\) 13.3359 0.448790 0.224395 0.974498i \(-0.427959\pi\)
0.224395 + 0.974498i \(0.427959\pi\)
\(884\) 0.680563 0.0228898
\(885\) 0 0
\(886\) −9.74108 −0.327258
\(887\) 38.2246 1.28346 0.641728 0.766932i \(-0.278218\pi\)
0.641728 + 0.766932i \(0.278218\pi\)
\(888\) 0 0
\(889\) −12.6383 −0.423877
\(890\) 3.65425 0.122491
\(891\) 0 0
\(892\) 1.10273 0.0369222
\(893\) 29.2631 0.979251
\(894\) 0 0
\(895\) −3.24115 −0.108340
\(896\) 10.0927 0.337172
\(897\) 0 0
\(898\) −44.0535 −1.47008
\(899\) −73.0791 −2.43732
\(900\) 0 0
\(901\) 16.2410 0.541066
\(902\) 0 0
\(903\) 0 0
\(904\) −29.1183 −0.968461
\(905\) −12.3792 −0.411500
\(906\) 0 0
\(907\) −39.9330 −1.32595 −0.662976 0.748641i \(-0.730707\pi\)
−0.662976 + 0.748641i \(0.730707\pi\)
\(908\) 3.08948 0.102528
\(909\) 0 0
\(910\) −3.12946 −0.103741
\(911\) −9.33878 −0.309408 −0.154704 0.987961i \(-0.549442\pi\)
−0.154704 + 0.987961i \(0.549442\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −27.1184 −0.896996
\(915\) 0 0
\(916\) −0.618406 −0.0204327
\(917\) 4.19769 0.138620
\(918\) 0 0
\(919\) 34.7924 1.14769 0.573847 0.818962i \(-0.305450\pi\)
0.573847 + 0.818962i \(0.305450\pi\)
\(920\) −15.8687 −0.523175
\(921\) 0 0
\(922\) 46.3459 1.52632
\(923\) −34.7565 −1.14402
\(924\) 0 0
\(925\) −25.7076 −0.845259
\(926\) 2.13540 0.0701737
\(927\) 0 0
\(928\) 5.65511 0.185638
\(929\) −31.5739 −1.03591 −0.517954 0.855409i \(-0.673306\pi\)
−0.517954 + 0.855409i \(0.673306\pi\)
\(930\) 0 0
\(931\) −5.85378 −0.191850
\(932\) 2.92150 0.0956968
\(933\) 0 0
\(934\) 8.12729 0.265933
\(935\) 0 0
\(936\) 0 0
\(937\) 50.0199 1.63408 0.817040 0.576581i \(-0.195613\pi\)
0.817040 + 0.576581i \(0.195613\pi\)
\(938\) −2.53537 −0.0827827
\(939\) 0 0
\(940\) 0.411934 0.0134358
\(941\) 51.5717 1.68119 0.840595 0.541664i \(-0.182205\pi\)
0.840595 + 0.541664i \(0.182205\pi\)
\(942\) 0 0
\(943\) 71.6561 2.33344
\(944\) 49.8849 1.62362
\(945\) 0 0
\(946\) 0 0
\(947\) −46.9709 −1.52635 −0.763175 0.646192i \(-0.776360\pi\)
−0.763175 + 0.646192i \(0.776360\pi\)
\(948\) 0 0
\(949\) −41.6818 −1.35305
\(950\) 36.7868 1.19352
\(951\) 0 0
\(952\) −4.25892 −0.138032
\(953\) −4.62354 −0.149771 −0.0748856 0.997192i \(-0.523859\pi\)
−0.0748856 + 0.997192i \(0.523859\pi\)
\(954\) 0 0
\(955\) −13.2232 −0.427894
\(956\) −2.60710 −0.0843198
\(957\) 0 0
\(958\) 39.8951 1.28895
\(959\) 16.9556 0.547523
\(960\) 0 0
\(961\) 58.2767 1.87989
\(962\) −27.5077 −0.886883
\(963\) 0 0
\(964\) −0.119491 −0.00384856
\(965\) 13.6523 0.439482
\(966\) 0 0
\(967\) −28.7432 −0.924321 −0.462160 0.886796i \(-0.652926\pi\)
−0.462160 + 0.886796i \(0.652926\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 6.48432 0.208199
\(971\) −8.48845 −0.272407 −0.136204 0.990681i \(-0.543490\pi\)
−0.136204 + 0.990681i \(0.543490\pi\)
\(972\) 0 0
\(973\) 12.5178 0.401304
\(974\) 21.0829 0.675539
\(975\) 0 0
\(976\) −41.6740 −1.33395
\(977\) −19.5016 −0.623911 −0.311956 0.950097i \(-0.600984\pi\)
−0.311956 + 0.950097i \(0.600984\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.0824033 −0.00263228
\(981\) 0 0
\(982\) 33.7343 1.07650
\(983\) −30.6078 −0.976238 −0.488119 0.872777i \(-0.662317\pi\)
−0.488119 + 0.872777i \(0.662317\pi\)
\(984\) 0 0
\(985\) −2.08698 −0.0664967
\(986\) 15.4687 0.492624
\(987\) 0 0
\(988\) −2.72432 −0.0866721
\(989\) 14.9039 0.473918
\(990\) 0 0
\(991\) −12.2332 −0.388600 −0.194300 0.980942i \(-0.562244\pi\)
−0.194300 + 0.980942i \(0.562244\pi\)
\(992\) −6.90853 −0.219346
\(993\) 0 0
\(994\) 13.2232 0.419415
\(995\) 10.5905 0.335740
\(996\) 0 0
\(997\) −0.292443 −0.00926176 −0.00463088 0.999989i \(-0.501474\pi\)
−0.00463088 + 0.999989i \(0.501474\pi\)
\(998\) 3.33530 0.105577
\(999\) 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.cr.1.5 yes 6
3.2 odd 2 inner 7623.2.a.cr.1.2 yes 6
11.10 odd 2 7623.2.a.cq.1.2 6
33.32 even 2 7623.2.a.cq.1.5 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7623.2.a.cq.1.2 6 11.10 odd 2
7623.2.a.cq.1.5 yes 6 33.32 even 2
7623.2.a.cr.1.2 yes 6 3.2 odd 2 inner
7623.2.a.cr.1.5 yes 6 1.1 even 1 trivial