Properties

Label 1344.4.p.c.223.20
Level $1344$
Weight $4$
Character 1344.223
Analytic conductor $79.299$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 223.20
Character \(\chi\) \(=\) 1344.223
Dual form 1344.4.p.c.223.19

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000i q^{3} -3.15485 q^{5} +(6.11026 + 17.4833i) q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} -3.15485 q^{5} +(6.11026 + 17.4833i) q^{7} -9.00000 q^{9} +60.9479 q^{11} -59.6460 q^{13} -9.46455i q^{15} -21.4250i q^{17} -95.5205i q^{19} +(-52.4498 + 18.3308i) q^{21} -46.3236i q^{23} -115.047 q^{25} -27.0000i q^{27} +107.590i q^{29} -94.2549 q^{31} +182.844i q^{33} +(-19.2770 - 55.1571i) q^{35} -131.650i q^{37} -178.938i q^{39} -283.782i q^{41} -373.255 q^{43} +28.3937 q^{45} -136.224 q^{47} +(-268.330 + 213.655i) q^{49} +64.2751 q^{51} +298.133i q^{53} -192.281 q^{55} +286.561 q^{57} +468.435i q^{59} +563.070 q^{61} +(-54.9923 - 157.349i) q^{63} +188.174 q^{65} +160.990 q^{67} +138.971 q^{69} -409.730i q^{71} -930.545i q^{73} -345.141i q^{75} +(372.407 + 1065.57i) q^{77} +442.826i q^{79} +81.0000 q^{81} -190.709i q^{83} +67.5928i q^{85} -322.771 q^{87} -829.404i q^{89} +(-364.453 - 1042.81i) q^{91} -282.765i q^{93} +301.353i q^{95} -1030.40i q^{97} -548.531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 288q^{9} + O(q^{10}) \) \( 32q - 288q^{9} - 224q^{13} - 72q^{21} + 1120q^{25} - 752q^{49} - 672q^{57} - 544q^{61} + 1536q^{65} - 144q^{69} - 1632q^{77} + 2592q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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 0
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) −3.15485 −0.282178 −0.141089 0.989997i \(-0.545060\pi\)
−0.141089 + 0.989997i \(0.545060\pi\)
\(6\) 0 0
\(7\) 6.11026 + 17.4833i 0.329923 + 0.944008i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 60.9479 1.67059 0.835294 0.549803i \(-0.185297\pi\)
0.835294 + 0.549803i \(0.185297\pi\)
\(12\) 0 0
\(13\) −59.6460 −1.27253 −0.636263 0.771472i \(-0.719521\pi\)
−0.636263 + 0.771472i \(0.719521\pi\)
\(14\) 0 0
\(15\) 9.46455i 0.162916i
\(16\) 0 0
\(17\) 21.4250i 0.305667i −0.988252 0.152833i \(-0.951160\pi\)
0.988252 0.152833i \(-0.0488398\pi\)
\(18\) 0 0
\(19\) 95.5205i 1.15336i −0.816969 0.576682i \(-0.804347\pi\)
0.816969 0.576682i \(-0.195653\pi\)
\(20\) 0 0
\(21\) −52.4498 + 18.3308i −0.545023 + 0.190481i
\(22\) 0 0
\(23\) 46.3236i 0.419963i −0.977705 0.209981i \(-0.932660\pi\)
0.977705 0.209981i \(-0.0673403\pi\)
\(24\) 0 0
\(25\) −115.047 −0.920375
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 107.590i 0.688932i 0.938799 + 0.344466i \(0.111940\pi\)
−0.938799 + 0.344466i \(0.888060\pi\)
\(30\) 0 0
\(31\) −94.2549 −0.546086 −0.273043 0.962002i \(-0.588030\pi\)
−0.273043 + 0.962002i \(0.588030\pi\)
\(32\) 0 0
\(33\) 182.844i 0.964515i
\(34\) 0 0
\(35\) −19.2770 55.1571i −0.0930971 0.266379i
\(36\) 0 0
\(37\) 131.650i 0.584949i −0.956273 0.292475i \(-0.905521\pi\)
0.956273 0.292475i \(-0.0944787\pi\)
\(38\) 0 0
\(39\) 178.938i 0.734693i
\(40\) 0 0
\(41\) 283.782i 1.08096i −0.841358 0.540479i \(-0.818243\pi\)
0.841358 0.540479i \(-0.181757\pi\)
\(42\) 0 0
\(43\) −373.255 −1.32374 −0.661871 0.749618i \(-0.730237\pi\)
−0.661871 + 0.749618i \(0.730237\pi\)
\(44\) 0 0
\(45\) 28.3937 0.0940595
\(46\) 0 0
\(47\) −136.224 −0.422773 −0.211387 0.977403i \(-0.567798\pi\)
−0.211387 + 0.977403i \(0.567798\pi\)
\(48\) 0 0
\(49\) −268.330 + 213.655i −0.782302 + 0.622900i
\(50\) 0 0
\(51\) 64.2751 0.176477
\(52\) 0 0
\(53\) 298.133i 0.772674i 0.922358 + 0.386337i \(0.126260\pi\)
−0.922358 + 0.386337i \(0.873740\pi\)
\(54\) 0 0
\(55\) −192.281 −0.471404
\(56\) 0 0
\(57\) 286.561 0.665895
\(58\) 0 0
\(59\) 468.435i 1.03365i 0.856092 + 0.516823i \(0.172885\pi\)
−0.856092 + 0.516823i \(0.827115\pi\)
\(60\) 0 0
\(61\) 563.070 1.18186 0.590932 0.806721i \(-0.298760\pi\)
0.590932 + 0.806721i \(0.298760\pi\)
\(62\) 0 0
\(63\) −54.9923 157.349i −0.109974 0.314669i
\(64\) 0 0
\(65\) 188.174 0.359079
\(66\) 0 0
\(67\) 160.990 0.293553 0.146777 0.989170i \(-0.453110\pi\)
0.146777 + 0.989170i \(0.453110\pi\)
\(68\) 0 0
\(69\) 138.971 0.242466
\(70\) 0 0
\(71\) 409.730i 0.684873i −0.939541 0.342437i \(-0.888748\pi\)
0.939541 0.342437i \(-0.111252\pi\)
\(72\) 0 0
\(73\) 930.545i 1.49195i −0.665976 0.745973i \(-0.731985\pi\)
0.665976 0.745973i \(-0.268015\pi\)
\(74\) 0 0
\(75\) 345.141i 0.531379i
\(76\) 0 0
\(77\) 372.407 + 1065.57i 0.551165 + 1.57705i
\(78\) 0 0
\(79\) 442.826i 0.630656i 0.948983 + 0.315328i \(0.102115\pi\)
−0.948983 + 0.315328i \(0.897885\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 190.709i 0.252205i −0.992017 0.126102i \(-0.959753\pi\)
0.992017 0.126102i \(-0.0402468\pi\)
\(84\) 0 0
\(85\) 67.5928i 0.0862526i
\(86\) 0 0
\(87\) −322.771 −0.397755
\(88\) 0 0
\(89\) 829.404i 0.987828i −0.869511 0.493914i \(-0.835566\pi\)
0.869511 0.493914i \(-0.164434\pi\)
\(90\) 0 0
\(91\) −364.453 1042.81i −0.419835 1.20127i
\(92\) 0 0
\(93\) 282.765i 0.315283i
\(94\) 0 0
\(95\) 301.353i 0.325454i
\(96\) 0 0
\(97\) 1030.40i 1.07857i −0.842123 0.539285i \(-0.818694\pi\)
0.842123 0.539285i \(-0.181306\pi\)
\(98\) 0 0
\(99\) −548.531 −0.556863
\(100\) 0 0
\(101\) −1784.29 −1.75785 −0.878926 0.476958i \(-0.841739\pi\)
−0.878926 + 0.476958i \(0.841739\pi\)
\(102\) 0 0
\(103\) −288.271 −0.275769 −0.137885 0.990448i \(-0.544030\pi\)
−0.137885 + 0.990448i \(0.544030\pi\)
\(104\) 0 0
\(105\) 165.471 57.8309i 0.153794 0.0537497i
\(106\) 0 0
\(107\) 292.479 0.264252 0.132126 0.991233i \(-0.457820\pi\)
0.132126 + 0.991233i \(0.457820\pi\)
\(108\) 0 0
\(109\) 518.765i 0.455860i 0.973678 + 0.227930i \(0.0731957\pi\)
−0.973678 + 0.227930i \(0.926804\pi\)
\(110\) 0 0
\(111\) 394.950 0.337721
\(112\) 0 0
\(113\) −1101.69 −0.917156 −0.458578 0.888654i \(-0.651641\pi\)
−0.458578 + 0.888654i \(0.651641\pi\)
\(114\) 0 0
\(115\) 146.144i 0.118504i
\(116\) 0 0
\(117\) 536.814 0.424175
\(118\) 0 0
\(119\) 374.580 130.913i 0.288552 0.100847i
\(120\) 0 0
\(121\) 2383.64 1.79087
\(122\) 0 0
\(123\) 851.345 0.624091
\(124\) 0 0
\(125\) 757.312 0.541889
\(126\) 0 0
\(127\) 1739.50i 1.21540i −0.794168 0.607698i \(-0.792093\pi\)
0.794168 0.607698i \(-0.207907\pi\)
\(128\) 0 0
\(129\) 1119.77i 0.764263i
\(130\) 0 0
\(131\) 2546.59i 1.69845i −0.528033 0.849224i \(-0.677070\pi\)
0.528033 0.849224i \(-0.322930\pi\)
\(132\) 0 0
\(133\) 1670.01 583.655i 1.08878 0.380521i
\(134\) 0 0
\(135\) 85.1810i 0.0543053i
\(136\) 0 0
\(137\) −197.078 −0.122902 −0.0614509 0.998110i \(-0.519573\pi\)
−0.0614509 + 0.998110i \(0.519573\pi\)
\(138\) 0 0
\(139\) 810.877i 0.494803i 0.968913 + 0.247402i \(0.0795767\pi\)
−0.968913 + 0.247402i \(0.920423\pi\)
\(140\) 0 0
\(141\) 408.673i 0.244088i
\(142\) 0 0
\(143\) −3635.30 −2.12587
\(144\) 0 0
\(145\) 339.432i 0.194402i
\(146\) 0 0
\(147\) −640.964 804.989i −0.359631 0.451662i
\(148\) 0 0
\(149\) 1838.58i 1.01089i −0.862860 0.505443i \(-0.831329\pi\)
0.862860 0.505443i \(-0.168671\pi\)
\(150\) 0 0
\(151\) 43.6496i 0.0235242i −0.999931 0.0117621i \(-0.996256\pi\)
0.999931 0.0117621i \(-0.00374408\pi\)
\(152\) 0 0
\(153\) 192.825i 0.101889i
\(154\) 0 0
\(155\) 297.360 0.154094
\(156\) 0 0
\(157\) 193.518 0.0983721 0.0491860 0.998790i \(-0.484337\pi\)
0.0491860 + 0.998790i \(0.484337\pi\)
\(158\) 0 0
\(159\) −894.399 −0.446103
\(160\) 0 0
\(161\) 809.888 283.049i 0.396448 0.138555i
\(162\) 0 0
\(163\) 1656.29 0.795893 0.397946 0.917409i \(-0.369723\pi\)
0.397946 + 0.917409i \(0.369723\pi\)
\(164\) 0 0
\(165\) 576.844i 0.272165i
\(166\) 0 0
\(167\) −4172.57 −1.93343 −0.966716 0.255851i \(-0.917644\pi\)
−0.966716 + 0.255851i \(0.917644\pi\)
\(168\) 0 0
\(169\) 1360.65 0.619322
\(170\) 0 0
\(171\) 859.684i 0.384454i
\(172\) 0 0
\(173\) 3172.52 1.39423 0.697116 0.716958i \(-0.254466\pi\)
0.697116 + 0.716958i \(0.254466\pi\)
\(174\) 0 0
\(175\) −702.966 2011.40i −0.303653 0.868842i
\(176\) 0 0
\(177\) −1405.31 −0.596776
\(178\) 0 0
\(179\) −137.757 −0.0575221 −0.0287611 0.999586i \(-0.509156\pi\)
−0.0287611 + 0.999586i \(0.509156\pi\)
\(180\) 0 0
\(181\) 978.118 0.401674 0.200837 0.979625i \(-0.435634\pi\)
0.200837 + 0.979625i \(0.435634\pi\)
\(182\) 0 0
\(183\) 1689.21i 0.682350i
\(184\) 0 0
\(185\) 415.336i 0.165060i
\(186\) 0 0
\(187\) 1305.81i 0.510644i
\(188\) 0 0
\(189\) 472.048 164.977i 0.181674 0.0634937i
\(190\) 0 0
\(191\) 4051.60i 1.53489i −0.641116 0.767444i \(-0.721528\pi\)
0.641116 0.767444i \(-0.278472\pi\)
\(192\) 0 0
\(193\) 1175.58 0.438444 0.219222 0.975675i \(-0.429648\pi\)
0.219222 + 0.975675i \(0.429648\pi\)
\(194\) 0 0
\(195\) 564.523i 0.207315i
\(196\) 0 0
\(197\) 3303.51i 1.19475i 0.801963 + 0.597374i \(0.203789\pi\)
−0.801963 + 0.597374i \(0.796211\pi\)
\(198\) 0 0
\(199\) −4452.76 −1.58617 −0.793086 0.609110i \(-0.791527\pi\)
−0.793086 + 0.609110i \(0.791527\pi\)
\(200\) 0 0
\(201\) 482.970i 0.169483i
\(202\) 0 0
\(203\) −1881.03 + 657.405i −0.650358 + 0.227295i
\(204\) 0 0
\(205\) 895.289i 0.305023i
\(206\) 0 0
\(207\) 416.912i 0.139988i
\(208\) 0 0
\(209\) 5821.77i 1.92680i
\(210\) 0 0
\(211\) 5875.61 1.91703 0.958516 0.285038i \(-0.0920061\pi\)
0.958516 + 0.285038i \(0.0920061\pi\)
\(212\) 0 0
\(213\) 1229.19 0.395412
\(214\) 0 0
\(215\) 1177.57 0.373532
\(216\) 0 0
\(217\) −575.921 1647.88i −0.180166 0.515510i
\(218\) 0 0
\(219\) 2791.63 0.861375
\(220\) 0 0
\(221\) 1277.92i 0.388969i
\(222\) 0 0
\(223\) 1909.89 0.573524 0.286762 0.958002i \(-0.407421\pi\)
0.286762 + 0.958002i \(0.407421\pi\)
\(224\) 0 0
\(225\) 1035.42 0.306792
\(226\) 0 0
\(227\) 4211.95i 1.23153i −0.787931 0.615764i \(-0.788848\pi\)
0.787931 0.615764i \(-0.211152\pi\)
\(228\) 0 0
\(229\) −135.681 −0.0391532 −0.0195766 0.999808i \(-0.506232\pi\)
−0.0195766 + 0.999808i \(0.506232\pi\)
\(230\) 0 0
\(231\) −3196.70 + 1117.22i −0.910509 + 0.318215i
\(232\) 0 0
\(233\) −613.936 −0.172619 −0.0863096 0.996268i \(-0.527507\pi\)
−0.0863096 + 0.996268i \(0.527507\pi\)
\(234\) 0 0
\(235\) 429.767 0.119298
\(236\) 0 0
\(237\) −1328.48 −0.364109
\(238\) 0 0
\(239\) 4611.45i 1.24807i 0.781395 + 0.624037i \(0.214508\pi\)
−0.781395 + 0.624037i \(0.785492\pi\)
\(240\) 0 0
\(241\) 4934.31i 1.31887i −0.751763 0.659433i \(-0.770796\pi\)
0.751763 0.659433i \(-0.229204\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) 846.540 674.048i 0.220749 0.175769i
\(246\) 0 0
\(247\) 5697.42i 1.46768i
\(248\) 0 0
\(249\) 572.126 0.145611
\(250\) 0 0
\(251\) 3655.89i 0.919353i −0.888086 0.459676i \(-0.847965\pi\)
0.888086 0.459676i \(-0.152035\pi\)
\(252\) 0 0
\(253\) 2823.32i 0.701585i
\(254\) 0 0
\(255\) −202.779 −0.0497980
\(256\) 0 0
\(257\) 4419.08i 1.07259i −0.844032 0.536294i \(-0.819824\pi\)
0.844032 0.536294i \(-0.180176\pi\)
\(258\) 0 0
\(259\) 2301.67 804.415i 0.552197 0.192988i
\(260\) 0 0
\(261\) 968.314i 0.229644i
\(262\) 0 0
\(263\) 5949.45i 1.39490i −0.716633 0.697450i \(-0.754318\pi\)
0.716633 0.697450i \(-0.245682\pi\)
\(264\) 0 0
\(265\) 940.565i 0.218032i
\(266\) 0 0
\(267\) 2488.21 0.570323
\(268\) 0 0
\(269\) 735.471 0.166701 0.0833503 0.996520i \(-0.473438\pi\)
0.0833503 + 0.996520i \(0.473438\pi\)
\(270\) 0 0
\(271\) −8337.62 −1.86891 −0.934454 0.356084i \(-0.884112\pi\)
−0.934454 + 0.356084i \(0.884112\pi\)
\(272\) 0 0
\(273\) 3128.42 1093.36i 0.693556 0.242392i
\(274\) 0 0
\(275\) −7011.86 −1.53757
\(276\) 0 0
\(277\) 3922.04i 0.850731i 0.905022 + 0.425366i \(0.139855\pi\)
−0.905022 + 0.425366i \(0.860145\pi\)
\(278\) 0 0
\(279\) 848.294 0.182029
\(280\) 0 0
\(281\) 6465.12 1.37251 0.686257 0.727359i \(-0.259253\pi\)
0.686257 + 0.727359i \(0.259253\pi\)
\(282\) 0 0
\(283\) 868.108i 0.182345i 0.995835 + 0.0911726i \(0.0290615\pi\)
−0.995835 + 0.0911726i \(0.970939\pi\)
\(284\) 0 0
\(285\) −904.059 −0.187901
\(286\) 0 0
\(287\) 4961.43 1733.98i 1.02043 0.356633i
\(288\) 0 0
\(289\) 4453.97 0.906568
\(290\) 0 0
\(291\) 3091.20 0.622713
\(292\) 0 0
\(293\) 4595.00 0.916186 0.458093 0.888904i \(-0.348533\pi\)
0.458093 + 0.888904i \(0.348533\pi\)
\(294\) 0 0
\(295\) 1477.84i 0.291673i
\(296\) 0 0
\(297\) 1645.59i 0.321505i
\(298\) 0 0
\(299\) 2763.02i 0.534413i
\(300\) 0 0
\(301\) −2280.69 6525.73i −0.436733 1.24962i
\(302\) 0 0
\(303\) 5352.86i 1.01490i
\(304\) 0 0
\(305\) −1776.40 −0.333497
\(306\) 0 0
\(307\) 3217.92i 0.598230i −0.954217 0.299115i \(-0.903309\pi\)
0.954217 0.299115i \(-0.0966914\pi\)
\(308\) 0 0
\(309\) 864.814i 0.159215i
\(310\) 0 0
\(311\) 7883.65 1.43743 0.718715 0.695305i \(-0.244731\pi\)
0.718715 + 0.695305i \(0.244731\pi\)
\(312\) 0 0
\(313\) 2670.55i 0.482263i 0.970492 + 0.241131i \(0.0775185\pi\)
−0.970492 + 0.241131i \(0.922482\pi\)
\(314\) 0 0
\(315\) 173.493 + 496.414i 0.0310324 + 0.0887929i
\(316\) 0 0
\(317\) 5005.09i 0.886794i 0.896325 + 0.443397i \(0.146227\pi\)
−0.896325 + 0.443397i \(0.853773\pi\)
\(318\) 0 0
\(319\) 6557.41i 1.15092i
\(320\) 0 0
\(321\) 877.436i 0.152566i
\(322\) 0 0
\(323\) −2046.53 −0.352545
\(324\) 0 0
\(325\) 6862.09 1.17120
\(326\) 0 0
\(327\) −1556.30 −0.263191
\(328\) 0 0
\(329\) −832.365 2381.65i −0.139483 0.399101i
\(330\) 0 0
\(331\) −10962.6 −1.82042 −0.910210 0.414148i \(-0.864080\pi\)
−0.910210 + 0.414148i \(0.864080\pi\)
\(332\) 0 0
\(333\) 1184.85i 0.194983i
\(334\) 0 0
\(335\) −507.900 −0.0828344
\(336\) 0 0
\(337\) −10584.5 −1.71090 −0.855448 0.517888i \(-0.826718\pi\)
−0.855448 + 0.517888i \(0.826718\pi\)
\(338\) 0 0
\(339\) 3305.08i 0.529520i
\(340\) 0 0
\(341\) −5744.63 −0.912285
\(342\) 0 0
\(343\) −5374.94 3385.79i −0.846121 0.532990i
\(344\) 0 0
\(345\) −438.432 −0.0684185
\(346\) 0 0
\(347\) −3267.07 −0.505434 −0.252717 0.967540i \(-0.581324\pi\)
−0.252717 + 0.967540i \(0.581324\pi\)
\(348\) 0 0
\(349\) 5781.37 0.886732 0.443366 0.896341i \(-0.353784\pi\)
0.443366 + 0.896341i \(0.353784\pi\)
\(350\) 0 0
\(351\) 1610.44i 0.244898i
\(352\) 0 0
\(353\) 10650.3i 1.60583i 0.596093 + 0.802915i \(0.296719\pi\)
−0.596093 + 0.802915i \(0.703281\pi\)
\(354\) 0 0
\(355\) 1292.64i 0.193256i
\(356\) 0 0
\(357\) 392.738 + 1123.74i 0.0582238 + 0.166596i
\(358\) 0 0
\(359\) 9179.33i 1.34949i −0.738051 0.674745i \(-0.764254\pi\)
0.738051 0.674745i \(-0.235746\pi\)
\(360\) 0 0
\(361\) −2265.16 −0.330247
\(362\) 0 0
\(363\) 7150.93i 1.03396i
\(364\) 0 0
\(365\) 2935.73i 0.420995i
\(366\) 0 0
\(367\) 2199.66 0.312865 0.156433 0.987689i \(-0.450001\pi\)
0.156433 + 0.987689i \(0.450001\pi\)
\(368\) 0 0
\(369\) 2554.03i 0.360319i
\(370\) 0 0
\(371\) −5212.34 + 1821.67i −0.729410 + 0.254923i
\(372\) 0 0
\(373\) 1961.66i 0.272309i 0.990688 + 0.136154i \(0.0434743\pi\)
−0.990688 + 0.136154i \(0.956526\pi\)
\(374\) 0 0
\(375\) 2271.94i 0.312859i
\(376\) 0 0
\(377\) 6417.34i 0.876684i
\(378\) 0 0
\(379\) 4839.03 0.655842 0.327921 0.944705i \(-0.393652\pi\)
0.327921 + 0.944705i \(0.393652\pi\)
\(380\) 0 0
\(381\) 5218.49 0.701709
\(382\) 0 0
\(383\) −14701.3 −1.96136 −0.980678 0.195628i \(-0.937326\pi\)
−0.980678 + 0.195628i \(0.937326\pi\)
\(384\) 0 0
\(385\) −1174.89 3361.71i −0.155527 0.445009i
\(386\) 0 0
\(387\) 3359.30 0.441248
\(388\) 0 0
\(389\) 15110.1i 1.96945i 0.174126 + 0.984723i \(0.444290\pi\)
−0.174126 + 0.984723i \(0.555710\pi\)
\(390\) 0 0
\(391\) −992.486 −0.128369
\(392\) 0 0
\(393\) 7639.77 0.980599
\(394\) 0 0
\(395\) 1397.05i 0.177958i
\(396\) 0 0
\(397\) −11309.2 −1.42971 −0.714854 0.699274i \(-0.753507\pi\)
−0.714854 + 0.699274i \(0.753507\pi\)
\(398\) 0 0
\(399\) 1750.96 + 5010.03i 0.219694 + 0.628610i
\(400\) 0 0
\(401\) −12823.1 −1.59690 −0.798450 0.602061i \(-0.794346\pi\)
−0.798450 + 0.602061i \(0.794346\pi\)
\(402\) 0 0
\(403\) 5621.93 0.694909
\(404\) 0 0
\(405\) −255.543 −0.0313532
\(406\) 0 0
\(407\) 8023.79i 0.977210i
\(408\) 0 0
\(409\) 3805.30i 0.460049i 0.973185 + 0.230025i \(0.0738807\pi\)
−0.973185 + 0.230025i \(0.926119\pi\)
\(410\) 0 0
\(411\) 591.235i 0.0709574i
\(412\) 0 0
\(413\) −8189.78 + 2862.26i −0.975770 + 0.341023i
\(414\) 0 0
\(415\) 601.658i 0.0711668i
\(416\) 0 0
\(417\) −2432.63 −0.285675
\(418\) 0 0
\(419\) 8027.30i 0.935941i −0.883744 0.467970i \(-0.844985\pi\)
0.883744 0.467970i \(-0.155015\pi\)
\(420\) 0 0
\(421\) 16646.2i 1.92705i −0.267622 0.963524i \(-0.586238\pi\)
0.267622 0.963524i \(-0.413762\pi\)
\(422\) 0 0
\(423\) 1226.02 0.140924
\(424\) 0 0
\(425\) 2464.89i 0.281328i
\(426\) 0 0
\(427\) 3440.50 + 9844.30i 0.389924 + 1.11569i
\(428\) 0 0
\(429\) 10905.9i 1.22737i
\(430\) 0 0
\(431\) 602.051i 0.0672849i 0.999434 + 0.0336425i \(0.0107107\pi\)
−0.999434 + 0.0336425i \(0.989289\pi\)
\(432\) 0 0
\(433\) 9231.81i 1.02460i 0.858806 + 0.512301i \(0.171207\pi\)
−0.858806 + 0.512301i \(0.828793\pi\)
\(434\) 0 0
\(435\) 1018.30 0.112238
\(436\) 0 0
\(437\) −4424.85 −0.484369
\(438\) 0 0
\(439\) 11079.3 1.20452 0.602262 0.798298i \(-0.294266\pi\)
0.602262 + 0.798298i \(0.294266\pi\)
\(440\) 0 0
\(441\) 2414.97 1922.89i 0.260767 0.207633i
\(442\) 0 0
\(443\) −3757.07 −0.402943 −0.201471 0.979494i \(-0.564572\pi\)
−0.201471 + 0.979494i \(0.564572\pi\)
\(444\) 0 0
\(445\) 2616.65i 0.278744i
\(446\) 0 0
\(447\) 5515.73 0.583635
\(448\) 0 0
\(449\) −5309.74 −0.558090 −0.279045 0.960278i \(-0.590018\pi\)
−0.279045 + 0.960278i \(0.590018\pi\)
\(450\) 0 0
\(451\) 17295.9i 1.80583i
\(452\) 0 0
\(453\) 130.949 0.0135817
\(454\) 0 0
\(455\) 1149.79 + 3289.90i 0.118469 + 0.338974i
\(456\) 0 0
\(457\) 17826.5 1.82470 0.912350 0.409412i \(-0.134266\pi\)
0.912350 + 0.409412i \(0.134266\pi\)
\(458\) 0 0
\(459\) −578.476 −0.0588256
\(460\) 0 0
\(461\) −7613.64 −0.769203 −0.384601 0.923083i \(-0.625661\pi\)
−0.384601 + 0.923083i \(0.625661\pi\)
\(462\) 0 0
\(463\) 831.086i 0.0834208i −0.999130 0.0417104i \(-0.986719\pi\)
0.999130 0.0417104i \(-0.0132807\pi\)
\(464\) 0 0
\(465\) 892.080i 0.0889661i
\(466\) 0 0
\(467\) 11189.4i 1.10875i −0.832267 0.554374i \(-0.812958\pi\)
0.832267 0.554374i \(-0.187042\pi\)
\(468\) 0 0
\(469\) 983.691 + 2814.63i 0.0968500 + 0.277117i
\(470\) 0 0
\(471\) 580.554i 0.0567951i
\(472\) 0 0
\(473\) −22749.1 −2.21143
\(474\) 0 0
\(475\) 10989.3i 1.06153i
\(476\) 0 0
\(477\) 2683.20i 0.257558i
\(478\) 0 0
\(479\) 8503.67 0.811154 0.405577 0.914061i \(-0.367071\pi\)
0.405577 + 0.914061i \(0.367071\pi\)
\(480\) 0 0
\(481\) 7852.40i 0.744363i
\(482\) 0 0
\(483\) 849.147 + 2429.66i 0.0799949 + 0.228889i
\(484\) 0 0
\(485\) 3250.76i 0.304349i
\(486\) 0 0
\(487\) 1829.08i 0.170192i 0.996373 + 0.0850962i \(0.0271198\pi\)
−0.996373 + 0.0850962i \(0.972880\pi\)
\(488\) 0 0
\(489\) 4968.87i 0.459509i
\(490\) 0 0
\(491\) −15970.1 −1.46786 −0.733931 0.679224i \(-0.762316\pi\)
−0.733931 + 0.679224i \(0.762316\pi\)
\(492\) 0 0
\(493\) 2305.13 0.210584
\(494\) 0 0
\(495\) 1730.53 0.157135
\(496\) 0 0
\(497\) 7163.42 2503.55i 0.646526 0.225955i
\(498\) 0 0
\(499\) −1903.84 −0.170796 −0.0853982 0.996347i \(-0.527216\pi\)
−0.0853982 + 0.996347i \(0.527216\pi\)
\(500\) 0 0
\(501\) 12517.7i 1.11627i
\(502\) 0 0
\(503\) 5674.20 0.502982 0.251491 0.967860i \(-0.419079\pi\)
0.251491 + 0.967860i \(0.419079\pi\)
\(504\) 0 0
\(505\) 5629.16 0.496028
\(506\) 0 0
\(507\) 4081.95i 0.357566i
\(508\) 0 0
\(509\) −10624.8 −0.925219 −0.462609 0.886562i \(-0.653087\pi\)
−0.462609 + 0.886562i \(0.653087\pi\)
\(510\) 0 0
\(511\) 16269.0 5685.87i 1.40841 0.492227i
\(512\) 0 0
\(513\) −2579.05 −0.221965
\(514\) 0 0
\(515\) 909.453 0.0778161
\(516\) 0 0
\(517\) −8302.58 −0.706280
\(518\) 0 0
\(519\) 9517.56i 0.804961i
\(520\) 0 0
\(521\) 695.563i 0.0584898i 0.999572 + 0.0292449i \(0.00931026\pi\)
−0.999572 + 0.0292449i \(0.990690\pi\)
\(522\) 0 0
\(523\) 11831.5i 0.989211i 0.869118 + 0.494605i \(0.164687\pi\)
−0.869118 + 0.494605i \(0.835313\pi\)
\(524\) 0 0
\(525\) 6034.19 2108.90i 0.501626 0.175314i
\(526\) 0 0
\(527\) 2019.42i 0.166921i
\(528\) 0 0
\(529\) 10021.1 0.823631
\(530\) 0 0
\(531\) 4215.92i 0.344549i
\(532\) 0 0
\(533\) 16926.5i 1.37555i
\(534\) 0 0
\(535\) −922.726 −0.0745662
\(536\) 0 0
\(537\) 413.272i 0.0332104i
\(538\) 0 0
\(539\) −16354.1 + 13021.8i −1.30690 + 1.04061i
\(540\) 0 0
\(541\) 10553.5i 0.838687i −0.907828 0.419343i \(-0.862260\pi\)
0.907828 0.419343i \(-0.137740\pi\)
\(542\) 0 0
\(543\) 2934.36i 0.231907i
\(544\) 0 0
\(545\) 1636.63i 0.128634i
\(546\) 0 0
\(547\) 1001.89 0.0783136 0.0391568 0.999233i \(-0.487533\pi\)
0.0391568 + 0.999233i \(0.487533\pi\)
\(548\) 0 0
\(549\) −5067.63 −0.393955
\(550\) 0 0
\(551\) 10277.1 0.794589
\(552\) 0 0
\(553\) −7742.05 + 2705.78i −0.595344 + 0.208068i
\(554\) 0 0
\(555\) −1246.01 −0.0952975
\(556\) 0 0
\(557\) 12853.2i 0.977752i 0.872353 + 0.488876i \(0.162593\pi\)
−0.872353 + 0.488876i \(0.837407\pi\)
\(558\) 0 0
\(559\) 22263.2 1.68450
\(560\) 0 0
\(561\) 3917.43 0.294820
\(562\) 0 0
\(563\) 10159.5i 0.760518i 0.924880 + 0.380259i \(0.124165\pi\)
−0.924880 + 0.380259i \(0.875835\pi\)
\(564\) 0 0
\(565\) 3475.68 0.258802
\(566\) 0 0
\(567\) 494.931 + 1416.14i 0.0366581 + 0.104890i
\(568\) 0 0
\(569\) −9104.92 −0.670823 −0.335411 0.942072i \(-0.608875\pi\)
−0.335411 + 0.942072i \(0.608875\pi\)
\(570\) 0 0
\(571\) 21714.0 1.59143 0.795713 0.605674i \(-0.207097\pi\)
0.795713 + 0.605674i \(0.207097\pi\)
\(572\) 0 0
\(573\) 12154.8 0.886168
\(574\) 0 0
\(575\) 5329.39i 0.386523i
\(576\) 0 0
\(577\) 12878.6i 0.929191i 0.885523 + 0.464595i \(0.153800\pi\)
−0.885523 + 0.464595i \(0.846200\pi\)
\(578\) 0 0
\(579\) 3526.73i 0.253136i
\(580\) 0 0
\(581\) 3334.21 1165.28i 0.238083 0.0832081i
\(582\) 0 0
\(583\) 18170.6i 1.29082i
\(584\) 0 0
\(585\) −1693.57 −0.119693
\(586\) 0 0
\(587\) 19024.7i 1.33771i 0.743394 + 0.668854i \(0.233215\pi\)
−0.743394 + 0.668854i \(0.766785\pi\)
\(588\) 0 0
\(589\) 9003.27i 0.629836i
\(590\) 0 0
\(591\) −9910.53 −0.689788
\(592\) 0 0
\(593\) 25477.9i 1.76434i 0.470934 + 0.882169i \(0.343917\pi\)
−0.470934 + 0.882169i \(0.656083\pi\)
\(594\) 0 0
\(595\) −1181.74 + 413.010i −0.0814232 + 0.0284567i
\(596\) 0 0
\(597\) 13358.3i 0.915776i
\(598\) 0 0
\(599\) 12475.5i 0.850976i 0.904964 + 0.425488i \(0.139898\pi\)
−0.904964 + 0.425488i \(0.860102\pi\)
\(600\) 0 0
\(601\) 947.642i 0.0643180i −0.999483 0.0321590i \(-0.989762\pi\)
0.999483 0.0321590i \(-0.0102383\pi\)
\(602\) 0 0
\(603\) −1448.91 −0.0978511
\(604\) 0 0
\(605\) −7520.04 −0.505344
\(606\) 0 0
\(607\) −17586.1 −1.17594 −0.587971 0.808882i \(-0.700073\pi\)
−0.587971 + 0.808882i \(0.700073\pi\)
\(608\) 0 0
\(609\) −1972.22 5643.10i −0.131229 0.375484i
\(610\) 0 0
\(611\) 8125.24 0.537990
\(612\) 0 0
\(613\) 9936.44i 0.654697i −0.944904 0.327348i \(-0.893845\pi\)
0.944904 0.327348i \(-0.106155\pi\)
\(614\) 0 0
\(615\) −2685.87 −0.176105
\(616\) 0 0
\(617\) −23950.9 −1.56277 −0.781383 0.624051i \(-0.785486\pi\)
−0.781383 + 0.624051i \(0.785486\pi\)
\(618\) 0 0
\(619\) 26145.1i 1.69767i 0.528654 + 0.848837i \(0.322697\pi\)
−0.528654 + 0.848837i \(0.677303\pi\)
\(620\) 0 0
\(621\) −1250.74 −0.0808218
\(622\) 0 0
\(623\) 14500.7 5067.87i 0.932517 0.325907i
\(624\) 0 0
\(625\) 11991.7 0.767466
\(626\) 0 0
\(627\) 17465.3 1.11244
\(628\) 0 0
\(629\) −2820.61 −0.178800
\(630\) 0 0
\(631\) 6978.35i 0.440260i 0.975471 + 0.220130i \(0.0706481\pi\)
−0.975471 + 0.220130i \(0.929352\pi\)
\(632\) 0 0
\(633\) 17626.8i 1.10680i
\(634\) 0 0
\(635\) 5487.85i 0.342958i
\(636\) 0 0
\(637\) 16004.8 12743.6i 0.995499 0.792656i
\(638\) 0 0
\(639\) 3687.57i 0.228291i
\(640\) 0 0
\(641\) 9353.09 0.576326 0.288163 0.957581i \(-0.406956\pi\)
0.288163 + 0.957581i \(0.406956\pi\)
\(642\) 0 0
\(643\) 11216.2i 0.687906i 0.938987 + 0.343953i \(0.111766\pi\)
−0.938987 + 0.343953i \(0.888234\pi\)
\(644\) 0 0
\(645\) 3532.70i 0.215659i
\(646\) 0 0
\(647\) 8875.46 0.539305 0.269653 0.962958i \(-0.413091\pi\)
0.269653 + 0.962958i \(0.413091\pi\)
\(648\) 0 0
\(649\) 28550.1i 1.72680i
\(650\) 0 0
\(651\) 4943.65 1727.76i 0.297630 0.104019i
\(652\) 0 0
\(653\) 12.5387i 0.000751419i −1.00000 0.000375710i \(-0.999880\pi\)
1.00000 0.000375710i \(-0.000119592\pi\)
\(654\) 0 0
\(655\) 8034.11i 0.479265i
\(656\) 0 0
\(657\) 8374.90i 0.497315i
\(658\) 0 0
\(659\) −127.456 −0.00753414 −0.00376707 0.999993i \(-0.501199\pi\)
−0.00376707 + 0.999993i \(0.501199\pi\)
\(660\) 0 0
\(661\) −18113.0 −1.06583 −0.532916 0.846168i \(-0.678904\pi\)
−0.532916 + 0.846168i \(0.678904\pi\)
\(662\) 0 0
\(663\) −3833.76 −0.224571
\(664\) 0 0
\(665\) −5268.63 + 1841.34i −0.307231 + 0.107375i
\(666\) 0 0
\(667\) 4983.98 0.289326
\(668\) 0 0
\(669\) 5729.67i 0.331124i
\(670\) 0 0
\(671\) 34317.9 1.97441
\(672\) 0 0
\(673\) −11689.6 −0.669540 −0.334770 0.942300i \(-0.608659\pi\)
−0.334770 + 0.942300i \(0.608659\pi\)
\(674\) 0 0
\(675\) 3106.27i 0.177126i
\(676\) 0 0
\(677\) −16101.4 −0.914070 −0.457035 0.889449i \(-0.651089\pi\)
−0.457035 + 0.889449i \(0.651089\pi\)
\(678\) 0 0
\(679\) 18014.8 6296.02i 1.01818 0.355845i
\(680\) 0 0
\(681\) 12635.8 0.711023
\(682\) 0 0
\(683\) −23914.0 −1.33974 −0.669870 0.742478i \(-0.733650\pi\)
−0.669870 + 0.742478i \(0.733650\pi\)
\(684\) 0 0
\(685\) 621.753 0.0346802
\(686\) 0 0
\(687\) 407.044i 0.0226051i
\(688\) 0 0
\(689\) 17782.4i 0.983247i
\(690\) 0 0
\(691\) 27330.4i 1.50463i −0.658805 0.752314i \(-0.728938\pi\)
0.658805 0.752314i \(-0.271062\pi\)
\(692\) 0 0
\(693\) −3351.66 9590.11i −0.183722 0.525683i
\(694\) 0 0
\(695\) 2558.20i 0.139623i
\(696\) 0 0
\(697\) −6080.04 −0.330413
\(698\) 0 0
\(699\) 1841.81i 0.0996617i
\(700\) 0 0
\(701\) 13641.6i 0.735002i 0.930023 + 0.367501i \(0.119787\pi\)
−0.930023 + 0.367501i \(0.880213\pi\)
\(702\) 0 0
\(703\) −12575.3 −0.674659
\(704\) 0 0
\(705\) 1289.30i 0.0688765i
\(706\) 0 0
\(707\) −10902.4 31195.1i −0.579956 1.65943i
\(708\) 0 0
\(709\) 25345.6i 1.34256i −0.741205 0.671278i \(-0.765746\pi\)
0.741205 0.671278i \(-0.234254\pi\)
\(710\) 0 0
\(711\) 3985.43i 0.210219i
\(712\) 0 0
\(713\) 4366.23i 0.229336i
\(714\) 0 0
\(715\) 11468.8 0.599874
\(716\) 0 0
\(717\) −13834.3 −0.720576
\(718\) 0 0
\(719\) −13143.0 −0.681715 −0.340857 0.940115i \(-0.610717\pi\)
−0.340857 + 0.940115i \(0.610717\pi\)
\(720\) 0 0
\(721\) −1761.41 5039.93i −0.0909826 0.260328i
\(722\) 0 0
\(723\) 14802.9 0.761447
\(724\) 0 0
\(725\) 12377.9i 0.634076i
\(726\) 0 0
\(727\) −7313.19 −0.373083 −0.186541 0.982447i \(-0.559728\pi\)
−0.186541 + 0.982447i \(0.559728\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 7997.02i 0.404624i
\(732\) 0 0
\(733\) −2577.28 −0.129869 −0.0649344 0.997890i \(-0.520684\pi\)
−0.0649344 + 0.997890i \(0.520684\pi\)
\(734\) 0 0
\(735\) 2022.15 + 2539.62i 0.101480 + 0.127449i
\(736\) 0 0
\(737\) 9812.00 0.490407
\(738\) 0 0
\(739\) −5495.97 −0.273576 −0.136788 0.990600i \(-0.543678\pi\)
−0.136788 + 0.990600i \(0.543678\pi\)
\(740\) 0 0
\(741\) −17092.3 −0.847368
\(742\) 0 0
\(743\) 16818.0i 0.830409i −0.909728 0.415204i \(-0.863710\pi\)
0.909728 0.415204i \(-0.136290\pi\)
\(744\) 0 0
\(745\) 5800.43i 0.285250i
\(746\) 0 0
\(747\) 1716.38i 0.0840683i
\(748\) 0 0
\(749\) 1787.12 + 5113.48i 0.0871828 + 0.249456i
\(750\) 0 0
\(751\) 10217.3i 0.496450i 0.968702 + 0.248225i \(0.0798472\pi\)
−0.968702 + 0.248225i \(0.920153\pi\)
\(752\) 0 0
\(753\) 10967.7 0.530789
\(754\) 0 0
\(755\) 137.708i 0.00663802i
\(756\) 0 0
\(757\) 2257.96i 0.108411i −0.998530 0.0542055i \(-0.982737\pi\)
0.998530 0.0542055i \(-0.0172626\pi\)
\(758\) 0 0
\(759\) 8469.97 0.405060
\(760\) 0 0
\(761\) 27933.8i 1.33061i 0.746570 + 0.665307i \(0.231699\pi\)
−0.746570 + 0.665307i \(0.768301\pi\)
\(762\) 0 0
\(763\) −9069.71 + 3169.79i −0.430335 + 0.150399i
\(764\) 0 0
\(765\) 608.336i 0.0287509i
\(766\) 0 0
\(767\) 27940.3i 1.31534i
\(768\) 0 0
\(769\) 33790.5i 1.58455i −0.610166 0.792274i \(-0.708897\pi\)
0.610166 0.792274i \(-0.291103\pi\)
\(770\) 0 0
\(771\) 13257.3 0.619259
\(772\) 0 0
\(773\) −18955.5 −0.881996 −0.440998 0.897508i \(-0.645375\pi\)
−0.440998 + 0.897508i \(0.645375\pi\)
\(774\) 0 0
\(775\) 10843.7 0.502604
\(776\) 0 0
\(777\) 2413.25 + 6905.02i 0.111422 + 0.318811i
\(778\) 0 0
\(779\) −27107.0 −1.24674
\(780\) 0 0
\(781\) 24972.2i 1.14414i
\(782\) 0 0
\(783\) 2904.94 0.132585
\(784\) 0 0
\(785\) −610.520 −0.0277585
\(786\) 0 0
\(787\) 3001.54i 0.135951i −0.997687 0.0679754i \(-0.978346\pi\)
0.997687 0.0679754i \(-0.0216539\pi\)
\(788\) 0 0
\(789\) 17848.3 0.805346
\(790\) 0 0
\(791\) −6731.63 19261.2i −0.302591 0.865803i
\(792\) 0 0
\(793\) −33584.9 −1.50395
\(794\) 0 0
\(795\) 2821.69 0.125881
\(796\) 0 0
\(797\) 14068.6 0.625265 0.312633 0.949874i \(-0.398789\pi\)
0.312633 + 0.949874i \(0.398789\pi\)
\(798\) 0 0
\(799\) 2918.61i 0.129228i
\(800\) 0 0
\(801\) 7464.64i 0.329276i
\(802\) 0