Properties

Label 1344.4.p.d.223.11
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 223.11
Character \(\chi\) \(=\) 1344.223
Dual form 1344.4.p.d.223.12

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} +15.4018 q^{5} +(-6.29285 - 17.4184i) q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +15.4018 q^{5} +(-6.29285 - 17.4184i) q^{7} -9.00000 q^{9} +16.8703 q^{11} +9.10290 q^{13} -46.2054i q^{15} +73.9573i q^{17} +151.262i q^{19} +(-52.2551 + 18.8786i) q^{21} +0.264643i q^{23} +112.215 q^{25} +27.0000i q^{27} +279.925i q^{29} -147.800 q^{31} -50.6110i q^{33} +(-96.9212 - 268.274i) q^{35} +418.498i q^{37} -27.3087i q^{39} +164.377i q^{41} +266.668 q^{43} -138.616 q^{45} +277.719 q^{47} +(-263.800 + 219.223i) q^{49} +221.872 q^{51} -276.852i q^{53} +259.833 q^{55} +453.785 q^{57} +212.008i q^{59} +174.541 q^{61} +(56.6357 + 156.765i) q^{63} +140.201 q^{65} -317.501 q^{67} +0.793928 q^{69} -106.093i q^{71} +755.118i q^{73} -336.645i q^{75} +(-106.163 - 293.854i) q^{77} -194.202i q^{79} +81.0000 q^{81} -997.764i q^{83} +1139.08i q^{85} +839.776 q^{87} -988.320i q^{89} +(-57.2832 - 158.558i) q^{91} +443.400i q^{93} +2329.70i q^{95} +1021.78i q^{97} -151.833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 288q^{9} + O(q^{10}) \) \( 32q - 288q^{9} + 224q^{13} + 72q^{21} + 1120q^{25} - 752q^{49} - 672q^{57} + 544q^{61} + 1536q^{65} + 144q^{69} + 1632q^{77} + 2592q^{81} + O(q^{100}) \)

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
\(5\) 15.4018 1.37758 0.688789 0.724962i \(-0.258143\pi\)
0.688789 + 0.724962i \(0.258143\pi\)
\(6\) 0 0
\(7\) −6.29285 17.4184i −0.339782 0.940504i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 16.8703 0.462418 0.231209 0.972904i \(-0.425732\pi\)
0.231209 + 0.972904i \(0.425732\pi\)
\(12\) 0 0
\(13\) 9.10290 0.194207 0.0971034 0.995274i \(-0.469042\pi\)
0.0971034 + 0.995274i \(0.469042\pi\)
\(14\) 0 0
\(15\) 46.2054i 0.795345i
\(16\) 0 0
\(17\) 73.9573i 1.05513i 0.849513 + 0.527567i \(0.176896\pi\)
−0.849513 + 0.527567i \(0.823104\pi\)
\(18\) 0 0
\(19\) 151.262i 1.82641i 0.407499 + 0.913206i \(0.366401\pi\)
−0.407499 + 0.913206i \(0.633599\pi\)
\(20\) 0 0
\(21\) −52.2551 + 18.8786i −0.543000 + 0.196173i
\(22\) 0 0
\(23\) 0.264643i 0.00239921i 0.999999 + 0.00119960i \(0.000381846\pi\)
−0.999999 + 0.00119960i \(0.999618\pi\)
\(24\) 0 0
\(25\) 112.215 0.897721
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 279.925i 1.79244i 0.443607 + 0.896221i \(0.353699\pi\)
−0.443607 + 0.896221i \(0.646301\pi\)
\(30\) 0 0
\(31\) −147.800 −0.856311 −0.428156 0.903705i \(-0.640836\pi\)
−0.428156 + 0.903705i \(0.640836\pi\)
\(32\) 0 0
\(33\) 50.6110i 0.266977i
\(34\) 0 0
\(35\) −96.9212 268.274i −0.468076 1.29562i
\(36\) 0 0
\(37\) 418.498i 1.85948i 0.368221 + 0.929738i \(0.379967\pi\)
−0.368221 + 0.929738i \(0.620033\pi\)
\(38\) 0 0
\(39\) 27.3087i 0.112125i
\(40\) 0 0
\(41\) 164.377i 0.626131i 0.949731 + 0.313066i \(0.101356\pi\)
−0.949731 + 0.313066i \(0.898644\pi\)
\(42\) 0 0
\(43\) 266.668 0.945731 0.472865 0.881135i \(-0.343220\pi\)
0.472865 + 0.881135i \(0.343220\pi\)
\(44\) 0 0
\(45\) −138.616 −0.459193
\(46\) 0 0
\(47\) 277.719 0.861903 0.430952 0.902375i \(-0.358178\pi\)
0.430952 + 0.902375i \(0.358178\pi\)
\(48\) 0 0
\(49\) −263.800 + 219.223i −0.769096 + 0.639133i
\(50\) 0 0
\(51\) 221.872 0.609182
\(52\) 0 0
\(53\) 276.852i 0.717519i −0.933430 0.358760i \(-0.883200\pi\)
0.933430 0.358760i \(-0.116800\pi\)
\(54\) 0 0
\(55\) 259.833 0.637017
\(56\) 0 0
\(57\) 453.785 1.05448
\(58\) 0 0
\(59\) 212.008i 0.467815i 0.972259 + 0.233908i \(0.0751513\pi\)
−0.972259 + 0.233908i \(0.924849\pi\)
\(60\) 0 0
\(61\) 174.541 0.366355 0.183177 0.983080i \(-0.441362\pi\)
0.183177 + 0.983080i \(0.441362\pi\)
\(62\) 0 0
\(63\) 56.6357 + 156.765i 0.113261 + 0.313501i
\(64\) 0 0
\(65\) 140.201 0.267535
\(66\) 0 0
\(67\) −317.501 −0.578940 −0.289470 0.957187i \(-0.593479\pi\)
−0.289470 + 0.957187i \(0.593479\pi\)
\(68\) 0 0
\(69\) 0.793928 0.00138518
\(70\) 0 0
\(71\) 106.093i 0.177338i −0.996061 0.0886688i \(-0.971739\pi\)
0.996061 0.0886688i \(-0.0282613\pi\)
\(72\) 0 0
\(73\) 755.118i 1.21068i 0.795966 + 0.605342i \(0.206964\pi\)
−0.795966 + 0.605342i \(0.793036\pi\)
\(74\) 0 0
\(75\) 336.645i 0.518299i
\(76\) 0 0
\(77\) −106.163 293.854i −0.157121 0.434906i
\(78\) 0 0
\(79\) 194.202i 0.276575i −0.990392 0.138288i \(-0.955840\pi\)
0.990392 0.138288i \(-0.0441598\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 997.764i 1.31950i −0.751484 0.659752i \(-0.770661\pi\)
0.751484 0.659752i \(-0.229339\pi\)
\(84\) 0 0
\(85\) 1139.08i 1.45353i
\(86\) 0 0
\(87\) 839.776 1.03487
\(88\) 0 0
\(89\) 988.320i 1.17710i −0.808462 0.588549i \(-0.799700\pi\)
0.808462 0.588549i \(-0.200300\pi\)
\(90\) 0 0
\(91\) −57.2832 158.558i −0.0659880 0.182652i
\(92\) 0 0
\(93\) 443.400i 0.494391i
\(94\) 0 0
\(95\) 2329.70i 2.51602i
\(96\) 0 0
\(97\) 1021.78i 1.06954i 0.844996 + 0.534772i \(0.179602\pi\)
−0.844996 + 0.534772i \(0.820398\pi\)
\(98\) 0 0
\(99\) −151.833 −0.154139
\(100\) 0 0
\(101\) 914.669 0.901118 0.450559 0.892747i \(-0.351225\pi\)
0.450559 + 0.892747i \(0.351225\pi\)
\(102\) 0 0
\(103\) −469.098 −0.448753 −0.224377 0.974503i \(-0.572035\pi\)
−0.224377 + 0.974503i \(0.572035\pi\)
\(104\) 0 0
\(105\) −804.823 + 290.763i −0.748025 + 0.270244i
\(106\) 0 0
\(107\) −1553.58 −1.40364 −0.701822 0.712352i \(-0.747630\pi\)
−0.701822 + 0.712352i \(0.747630\pi\)
\(108\) 0 0
\(109\) 1424.25i 1.25155i −0.780005 0.625773i \(-0.784784\pi\)
0.780005 0.625773i \(-0.215216\pi\)
\(110\) 0 0
\(111\) 1255.49 1.07357
\(112\) 0 0
\(113\) 784.755 0.653305 0.326653 0.945144i \(-0.394079\pi\)
0.326653 + 0.945144i \(0.394079\pi\)
\(114\) 0 0
\(115\) 4.07597i 0.00330510i
\(116\) 0 0
\(117\) −81.9261 −0.0647356
\(118\) 0 0
\(119\) 1288.22 465.402i 0.992358 0.358516i
\(120\) 0 0
\(121\) −1046.39 −0.786170
\(122\) 0 0
\(123\) 493.131 0.361497
\(124\) 0 0
\(125\) −196.910 −0.140897
\(126\) 0 0
\(127\) 2509.27i 1.75324i 0.481182 + 0.876620i \(0.340207\pi\)
−0.481182 + 0.876620i \(0.659793\pi\)
\(128\) 0 0
\(129\) 800.003i 0.546018i
\(130\) 0 0
\(131\) 2050.10i 1.36731i −0.729805 0.683655i \(-0.760389\pi\)
0.729805 0.683655i \(-0.239611\pi\)
\(132\) 0 0
\(133\) 2634.73 951.868i 1.71775 0.620582i
\(134\) 0 0
\(135\) 415.848i 0.265115i
\(136\) 0 0
\(137\) −2516.47 −1.56932 −0.784658 0.619929i \(-0.787161\pi\)
−0.784658 + 0.619929i \(0.787161\pi\)
\(138\) 0 0
\(139\) 270.678i 0.165170i −0.996584 0.0825849i \(-0.973682\pi\)
0.996584 0.0825849i \(-0.0263176\pi\)
\(140\) 0 0
\(141\) 833.157i 0.497620i
\(142\) 0 0
\(143\) 153.569 0.0898048
\(144\) 0 0
\(145\) 4311.35i 2.46923i
\(146\) 0 0
\(147\) 657.668 + 791.400i 0.369004 + 0.444038i
\(148\) 0 0
\(149\) 238.893i 0.131348i −0.997841 0.0656742i \(-0.979080\pi\)
0.997841 0.0656742i \(-0.0209198\pi\)
\(150\) 0 0
\(151\) 2859.49i 1.54107i −0.637397 0.770535i \(-0.719989\pi\)
0.637397 0.770535i \(-0.280011\pi\)
\(152\) 0 0
\(153\) 665.616i 0.351711i
\(154\) 0 0
\(155\) −2276.38 −1.17964
\(156\) 0 0
\(157\) −566.367 −0.287904 −0.143952 0.989585i \(-0.545981\pi\)
−0.143952 + 0.989585i \(0.545981\pi\)
\(158\) 0 0
\(159\) −830.555 −0.414260
\(160\) 0 0
\(161\) 4.60965 1.66536i 0.00225647 0.000815208i
\(162\) 0 0
\(163\) 1330.25 0.639222 0.319611 0.947549i \(-0.396448\pi\)
0.319611 + 0.947549i \(0.396448\pi\)
\(164\) 0 0
\(165\) 779.500i 0.367782i
\(166\) 0 0
\(167\) 2770.08 1.28356 0.641782 0.766887i \(-0.278195\pi\)
0.641782 + 0.766887i \(0.278195\pi\)
\(168\) 0 0
\(169\) −2114.14 −0.962284
\(170\) 0 0
\(171\) 1361.36i 0.608804i
\(172\) 0 0
\(173\) 1389.16 0.610498 0.305249 0.952273i \(-0.401260\pi\)
0.305249 + 0.952273i \(0.401260\pi\)
\(174\) 0 0
\(175\) −706.153 1954.61i −0.305029 0.844310i
\(176\) 0 0
\(177\) 636.024 0.270093
\(178\) 0 0
\(179\) 2337.71 0.976136 0.488068 0.872806i \(-0.337702\pi\)
0.488068 + 0.872806i \(0.337702\pi\)
\(180\) 0 0
\(181\) 3318.66 1.36284 0.681420 0.731893i \(-0.261363\pi\)
0.681420 + 0.731893i \(0.261363\pi\)
\(182\) 0 0
\(183\) 523.622i 0.211515i
\(184\) 0 0
\(185\) 6445.62i 2.56157i
\(186\) 0 0
\(187\) 1247.69i 0.487913i
\(188\) 0 0
\(189\) 470.296 169.907i 0.181000 0.0653911i
\(190\) 0 0
\(191\) 1614.06i 0.611463i 0.952118 + 0.305731i \(0.0989010\pi\)
−0.952118 + 0.305731i \(0.901099\pi\)
\(192\) 0 0
\(193\) −1248.70 −0.465719 −0.232859 0.972510i \(-0.574808\pi\)
−0.232859 + 0.972510i \(0.574808\pi\)
\(194\) 0 0
\(195\) 420.603i 0.154461i
\(196\) 0 0
\(197\) 4151.12i 1.50129i 0.660703 + 0.750647i \(0.270258\pi\)
−0.660703 + 0.750647i \(0.729742\pi\)
\(198\) 0 0
\(199\) 2928.99 1.04337 0.521685 0.853138i \(-0.325304\pi\)
0.521685 + 0.853138i \(0.325304\pi\)
\(200\) 0 0
\(201\) 952.504i 0.334251i
\(202\) 0 0
\(203\) 4875.85 1761.53i 1.68580 0.609040i
\(204\) 0 0
\(205\) 2531.70i 0.862545i
\(206\) 0 0
\(207\) 2.38178i 0.000799737i
\(208\) 0 0
\(209\) 2551.84i 0.844566i
\(210\) 0 0
\(211\) −3360.79 −1.09652 −0.548261 0.836307i \(-0.684710\pi\)
−0.548261 + 0.836307i \(0.684710\pi\)
\(212\) 0 0
\(213\) −318.280 −0.102386
\(214\) 0 0
\(215\) 4107.16 1.30282
\(216\) 0 0
\(217\) 930.083 + 2574.43i 0.290959 + 0.805364i
\(218\) 0 0
\(219\) 2265.35 0.698988
\(220\) 0 0
\(221\) 673.226i 0.204914i
\(222\) 0 0
\(223\) 2329.45 0.699515 0.349757 0.936840i \(-0.386264\pi\)
0.349757 + 0.936840i \(0.386264\pi\)
\(224\) 0 0
\(225\) −1009.94 −0.299240
\(226\) 0 0
\(227\) 832.888i 0.243528i −0.992559 0.121764i \(-0.961145\pi\)
0.992559 0.121764i \(-0.0388550\pi\)
\(228\) 0 0
\(229\) 1434.36 0.413908 0.206954 0.978351i \(-0.433645\pi\)
0.206954 + 0.978351i \(0.433645\pi\)
\(230\) 0 0
\(231\) −881.562 + 318.488i −0.251093 + 0.0907140i
\(232\) 0 0
\(233\) 2388.42 0.671547 0.335773 0.941943i \(-0.391002\pi\)
0.335773 + 0.941943i \(0.391002\pi\)
\(234\) 0 0
\(235\) 4277.37 1.18734
\(236\) 0 0
\(237\) −582.606 −0.159681
\(238\) 0 0
\(239\) 4808.99i 1.30154i 0.759276 + 0.650769i \(0.225553\pi\)
−0.759276 + 0.650769i \(0.774447\pi\)
\(240\) 0 0
\(241\) 2269.59i 0.606627i −0.952891 0.303313i \(-0.901907\pi\)
0.952891 0.303313i \(-0.0980929\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) −4062.99 + 3376.42i −1.05949 + 0.880455i
\(246\) 0 0
\(247\) 1376.92i 0.354702i
\(248\) 0 0
\(249\) −2993.29 −0.761816
\(250\) 0 0
\(251\) 2242.07i 0.563818i 0.959441 + 0.281909i \(0.0909676\pi\)
−0.959441 + 0.281909i \(0.909032\pi\)
\(252\) 0 0
\(253\) 4.46461i 0.00110944i
\(254\) 0 0
\(255\) 3417.23 0.839196
\(256\) 0 0
\(257\) 787.861i 0.191227i 0.995419 + 0.0956136i \(0.0304813\pi\)
−0.995419 + 0.0956136i \(0.969519\pi\)
\(258\) 0 0
\(259\) 7289.56 2633.54i 1.74885 0.631817i
\(260\) 0 0
\(261\) 2519.33i 0.597481i
\(262\) 0 0
\(263\) 6696.97i 1.57016i −0.619392 0.785082i \(-0.712621\pi\)
0.619392 0.785082i \(-0.287379\pi\)
\(264\) 0 0
\(265\) 4264.01i 0.988439i
\(266\) 0 0
\(267\) −2964.96 −0.679597
\(268\) 0 0
\(269\) 4374.92 0.991612 0.495806 0.868433i \(-0.334873\pi\)
0.495806 + 0.868433i \(0.334873\pi\)
\(270\) 0 0
\(271\) 8058.24 1.80629 0.903143 0.429340i \(-0.141254\pi\)
0.903143 + 0.429340i \(0.141254\pi\)
\(272\) 0 0
\(273\) −475.673 + 171.850i −0.105454 + 0.0380982i
\(274\) 0 0
\(275\) 1893.11 0.415122
\(276\) 0 0
\(277\) 4224.44i 0.916324i −0.888869 0.458162i \(-0.848508\pi\)
0.888869 0.458162i \(-0.151492\pi\)
\(278\) 0 0
\(279\) 1330.20 0.285437
\(280\) 0 0
\(281\) 3763.77 0.799031 0.399516 0.916726i \(-0.369178\pi\)
0.399516 + 0.916726i \(0.369178\pi\)
\(282\) 0 0
\(283\) 1604.16i 0.336952i 0.985706 + 0.168476i \(0.0538846\pi\)
−0.985706 + 0.168476i \(0.946115\pi\)
\(284\) 0 0
\(285\) 6989.10 1.45263
\(286\) 0 0
\(287\) 2863.18 1034.40i 0.588879 0.212748i
\(288\) 0 0
\(289\) −556.685 −0.113309
\(290\) 0 0
\(291\) 3065.33 0.617502
\(292\) 0 0
\(293\) −3124.58 −0.623004 −0.311502 0.950245i \(-0.600832\pi\)
−0.311502 + 0.950245i \(0.600832\pi\)
\(294\) 0 0
\(295\) 3265.30i 0.644452i
\(296\) 0 0
\(297\) 455.499i 0.0889924i
\(298\) 0 0
\(299\) 2.40902i 0.000465943i
\(300\) 0 0
\(301\) −1678.10 4644.92i −0.321342 0.889464i
\(302\) 0 0
\(303\) 2744.01i 0.520261i
\(304\) 0 0
\(305\) 2688.24 0.504683
\(306\) 0 0
\(307\) 9789.14i 1.81986i −0.414767 0.909928i \(-0.636137\pi\)
0.414767 0.909928i \(-0.363863\pi\)
\(308\) 0 0
\(309\) 1407.29i 0.259088i
\(310\) 0 0
\(311\) −3916.61 −0.714117 −0.357059 0.934082i \(-0.616220\pi\)
−0.357059 + 0.934082i \(0.616220\pi\)
\(312\) 0 0
\(313\) 5571.72i 1.00617i 0.864236 + 0.503087i \(0.167802\pi\)
−0.864236 + 0.503087i \(0.832198\pi\)
\(314\) 0 0
\(315\) 872.290 + 2414.47i 0.156025 + 0.431873i
\(316\) 0 0
\(317\) 10110.5i 1.79136i 0.444700 + 0.895680i \(0.353310\pi\)
−0.444700 + 0.895680i \(0.646690\pi\)
\(318\) 0 0
\(319\) 4722.44i 0.828858i
\(320\) 0 0
\(321\) 4660.73i 0.810395i
\(322\) 0 0
\(323\) −11186.9 −1.92711
\(324\) 0 0
\(325\) 1021.48 0.174344
\(326\) 0 0
\(327\) −4272.75 −0.722580
\(328\) 0 0
\(329\) −1747.64 4837.41i −0.292859 0.810624i
\(330\) 0 0
\(331\) 396.980 0.0659214 0.0329607 0.999457i \(-0.489506\pi\)
0.0329607 + 0.999457i \(0.489506\pi\)
\(332\) 0 0
\(333\) 3766.48i 0.619825i
\(334\) 0 0
\(335\) −4890.09 −0.797535
\(336\) 0 0
\(337\) 7848.71 1.26868 0.634342 0.773053i \(-0.281271\pi\)
0.634342 + 0.773053i \(0.281271\pi\)
\(338\) 0 0
\(339\) 2354.26i 0.377186i
\(340\) 0 0
\(341\) −2493.43 −0.395974
\(342\) 0 0
\(343\) 5478.56 + 3215.44i 0.862432 + 0.506173i
\(344\) 0 0
\(345\) 12.2279 0.00190820
\(346\) 0 0
\(347\) −12522.0 −1.93722 −0.968609 0.248591i \(-0.920033\pi\)
−0.968609 + 0.248591i \(0.920033\pi\)
\(348\) 0 0
\(349\) −5091.86 −0.780978 −0.390489 0.920608i \(-0.627694\pi\)
−0.390489 + 0.920608i \(0.627694\pi\)
\(350\) 0 0
\(351\) 245.778i 0.0373751i
\(352\) 0 0
\(353\) 9690.22i 1.46107i 0.682874 + 0.730536i \(0.260730\pi\)
−0.682874 + 0.730536i \(0.739270\pi\)
\(354\) 0 0
\(355\) 1634.03i 0.244296i
\(356\) 0 0
\(357\) −1396.21 3864.65i −0.206989 0.572938i
\(358\) 0 0
\(359\) 9222.74i 1.35587i 0.735122 + 0.677935i \(0.237125\pi\)
−0.735122 + 0.677935i \(0.762875\pi\)
\(360\) 0 0
\(361\) −16021.1 −2.33578
\(362\) 0 0
\(363\) 3139.18i 0.453895i
\(364\) 0 0
\(365\) 11630.2i 1.66781i
\(366\) 0 0
\(367\) −2968.62 −0.422236 −0.211118 0.977461i \(-0.567711\pi\)
−0.211118 + 0.977461i \(0.567711\pi\)
\(368\) 0 0
\(369\) 1479.39i 0.208710i
\(370\) 0 0
\(371\) −4822.31 + 1742.19i −0.674830 + 0.243800i
\(372\) 0 0
\(373\) 2883.02i 0.400207i −0.979775 0.200103i \(-0.935872\pi\)
0.979775 0.200103i \(-0.0641278\pi\)
\(374\) 0 0
\(375\) 590.730i 0.0813471i
\(376\) 0 0
\(377\) 2548.13i 0.348105i
\(378\) 0 0
\(379\) −2727.26 −0.369630 −0.184815 0.982773i \(-0.559169\pi\)
−0.184815 + 0.982773i \(0.559169\pi\)
\(380\) 0 0
\(381\) 7527.81 1.01223
\(382\) 0 0
\(383\) −10418.3 −1.38995 −0.694976 0.719033i \(-0.744585\pi\)
−0.694976 + 0.719033i \(0.744585\pi\)
\(384\) 0 0
\(385\) −1635.09 4525.88i −0.216447 0.599117i
\(386\) 0 0
\(387\) −2400.01 −0.315244
\(388\) 0 0
\(389\) 9095.84i 1.18555i −0.805369 0.592773i \(-0.798033\pi\)
0.805369 0.592773i \(-0.201967\pi\)
\(390\) 0 0
\(391\) −19.5723 −0.00253149
\(392\) 0 0
\(393\) −6150.29 −0.789417
\(394\) 0 0
\(395\) 2991.06i 0.381004i
\(396\) 0 0
\(397\) 1349.74 0.170633 0.0853167 0.996354i \(-0.472810\pi\)
0.0853167 + 0.996354i \(0.472810\pi\)
\(398\) 0 0
\(399\) −2855.60 7904.20i −0.358293 0.991742i
\(400\) 0 0
\(401\) 1830.81 0.227996 0.113998 0.993481i \(-0.463634\pi\)
0.113998 + 0.993481i \(0.463634\pi\)
\(402\) 0 0
\(403\) −1345.41 −0.166301
\(404\) 0 0
\(405\) 1247.54 0.153064
\(406\) 0 0
\(407\) 7060.20i 0.859855i
\(408\) 0 0
\(409\) 14709.9i 1.77838i −0.457542 0.889188i \(-0.651270\pi\)
0.457542 0.889188i \(-0.348730\pi\)
\(410\) 0 0
\(411\) 7549.40i 0.906045i
\(412\) 0 0
\(413\) 3692.84 1334.14i 0.439982 0.158955i
\(414\) 0 0
\(415\) 15367.3i 1.81772i
\(416\) 0 0
\(417\) −812.034 −0.0953608
\(418\) 0 0
\(419\) 6721.49i 0.783690i −0.920031 0.391845i \(-0.871837\pi\)
0.920031 0.391845i \(-0.128163\pi\)
\(420\) 0 0
\(421\) 4541.66i 0.525765i 0.964828 + 0.262882i \(0.0846731\pi\)
−0.964828 + 0.262882i \(0.915327\pi\)
\(422\) 0 0
\(423\) −2499.47 −0.287301
\(424\) 0 0
\(425\) 8299.13i 0.947216i
\(426\) 0 0
\(427\) −1098.36 3040.22i −0.124481 0.344558i
\(428\) 0 0
\(429\) 460.707i 0.0518488i
\(430\) 0 0
\(431\) 10747.7i 1.20116i 0.799564 + 0.600580i \(0.205064\pi\)
−0.799564 + 0.600580i \(0.794936\pi\)
\(432\) 0 0
\(433\) 10058.7i 1.11638i −0.829714 0.558188i \(-0.811497\pi\)
0.829714 0.558188i \(-0.188503\pi\)
\(434\) 0 0
\(435\) 12934.1 1.42561
\(436\) 0 0
\(437\) −40.0303 −0.00438194
\(438\) 0 0
\(439\) −2944.00 −0.320067 −0.160034 0.987112i \(-0.551160\pi\)
−0.160034 + 0.987112i \(0.551160\pi\)
\(440\) 0 0
\(441\) 2374.20 1973.00i 0.256365 0.213044i
\(442\) 0 0
\(443\) −8667.71 −0.929606 −0.464803 0.885414i \(-0.653875\pi\)
−0.464803 + 0.885414i \(0.653875\pi\)
\(444\) 0 0
\(445\) 15221.9i 1.62154i
\(446\) 0 0
\(447\) −716.680 −0.0758340
\(448\) 0 0
\(449\) 2980.91 0.313314 0.156657 0.987653i \(-0.449928\pi\)
0.156657 + 0.987653i \(0.449928\pi\)
\(450\) 0 0
\(451\) 2773.10i 0.289534i
\(452\) 0 0
\(453\) −8578.46 −0.889738
\(454\) 0 0
\(455\) −882.263 2442.07i −0.0909036 0.251618i
\(456\) 0 0
\(457\) 5915.60 0.605515 0.302757 0.953068i \(-0.402093\pi\)
0.302757 + 0.953068i \(0.402093\pi\)
\(458\) 0 0
\(459\) −1996.85 −0.203061
\(460\) 0 0
\(461\) 5755.48 0.581474 0.290737 0.956803i \(-0.406099\pi\)
0.290737 + 0.956803i \(0.406099\pi\)
\(462\) 0 0
\(463\) 6502.53i 0.652696i −0.945250 0.326348i \(-0.894182\pi\)
0.945250 0.326348i \(-0.105818\pi\)
\(464\) 0 0
\(465\) 6829.15i 0.681063i
\(466\) 0 0
\(467\) 4507.81i 0.446674i −0.974741 0.223337i \(-0.928305\pi\)
0.974741 0.223337i \(-0.0716950\pi\)
\(468\) 0 0
\(469\) 1997.99 + 5530.36i 0.196713 + 0.544495i
\(470\) 0 0
\(471\) 1699.10i 0.166222i
\(472\) 0 0
\(473\) 4498.77 0.437323
\(474\) 0 0
\(475\) 16973.9i 1.63961i
\(476\) 0 0
\(477\) 2491.67i 0.239173i
\(478\) 0 0
\(479\) 2320.76 0.221375 0.110687 0.993855i \(-0.464695\pi\)
0.110687 + 0.993855i \(0.464695\pi\)
\(480\) 0 0
\(481\) 3809.54i 0.361123i
\(482\) 0 0
\(483\) −4.99607 13.8289i −0.000470661 0.00130277i
\(484\) 0 0
\(485\) 15737.2i 1.47338i
\(486\) 0 0
\(487\) 8971.64i 0.834792i 0.908725 + 0.417396i \(0.137057\pi\)
−0.908725 + 0.417396i \(0.862943\pi\)
\(488\) 0 0
\(489\) 3990.75i 0.369055i
\(490\) 0 0
\(491\) −11942.9 −1.09771 −0.548857 0.835916i \(-0.684937\pi\)
−0.548857 + 0.835916i \(0.684937\pi\)
\(492\) 0 0
\(493\) −20702.5 −1.89127
\(494\) 0 0
\(495\) −2338.50 −0.212339
\(496\) 0 0
\(497\) −1847.98 + 667.630i −0.166787 + 0.0602561i
\(498\) 0 0
\(499\) 4865.01 0.436448 0.218224 0.975899i \(-0.429974\pi\)
0.218224 + 0.975899i \(0.429974\pi\)
\(500\) 0 0
\(501\) 8310.24i 0.741066i
\(502\) 0 0
\(503\) 5302.73 0.470054 0.235027 0.971989i \(-0.424482\pi\)
0.235027 + 0.971989i \(0.424482\pi\)
\(504\) 0 0
\(505\) 14087.5 1.24136
\(506\) 0 0
\(507\) 6342.41i 0.555575i
\(508\) 0 0
\(509\) −13255.3 −1.15428 −0.577142 0.816644i \(-0.695832\pi\)
−0.577142 + 0.816644i \(0.695832\pi\)
\(510\) 0 0
\(511\) 13152.9 4751.85i 1.13865 0.411368i
\(512\) 0 0
\(513\) −4084.07 −0.351493
\(514\) 0 0
\(515\) −7224.95 −0.618193
\(516\) 0 0
\(517\) 4685.21 0.398560
\(518\) 0 0
\(519\) 4167.49i 0.352471i
\(520\) 0 0
\(521\) 17961.3i 1.51036i −0.655517 0.755181i \(-0.727549\pi\)
0.655517 0.755181i \(-0.272451\pi\)
\(522\) 0 0
\(523\) 15183.8i 1.26948i 0.772724 + 0.634742i \(0.218894\pi\)
−0.772724 + 0.634742i \(0.781106\pi\)
\(524\) 0 0
\(525\) −5863.82 + 2118.46i −0.487463 + 0.176109i
\(526\) 0 0
\(527\) 10930.9i 0.903523i
\(528\) 0 0
\(529\) 12166.9 0.999994
\(530\) 0 0
\(531\) 1908.07i 0.155938i
\(532\) 0 0
\(533\) 1496.31i 0.121599i
\(534\) 0 0
\(535\) −23927.9 −1.93363
\(536\) 0 0
\(537\) 7013.12i 0.563572i
\(538\) 0 0
\(539\) −4450.40 + 3698.36i −0.355644 + 0.295547i
\(540\) 0 0
\(541\) 1840.34i 0.146252i 0.997323 + 0.0731262i \(0.0232976\pi\)
−0.997323 + 0.0731262i \(0.976702\pi\)
\(542\) 0 0
\(543\) 9955.98i 0.786836i
\(544\) 0 0
\(545\) 21936.0i 1.72410i
\(546\) 0 0
\(547\) 17887.3 1.39818 0.699091 0.715033i \(-0.253588\pi\)
0.699091 + 0.715033i \(0.253588\pi\)
\(548\) 0 0
\(549\) −1570.87 −0.122118
\(550\) 0 0
\(551\) −42342.0 −3.27374
\(552\) 0 0
\(553\) −3382.69 + 1222.09i −0.260120 + 0.0939753i
\(554\) 0 0
\(555\) 19336.8 1.47893
\(556\) 0 0
\(557\) 12246.4i 0.931590i 0.884893 + 0.465795i \(0.154232\pi\)
−0.884893 + 0.465795i \(0.845768\pi\)
\(558\) 0 0
\(559\) 2427.45 0.183667
\(560\) 0 0
\(561\) 3743.06 0.281697
\(562\) 0 0
\(563\) 18024.0i 1.34924i −0.738165 0.674621i \(-0.764307\pi\)
0.738165 0.674621i \(-0.235693\pi\)
\(564\) 0 0
\(565\) 12086.6 0.899979
\(566\) 0 0
\(567\) −509.721 1410.89i −0.0377536 0.104500i
\(568\) 0 0
\(569\) 21534.4 1.58659 0.793294 0.608839i \(-0.208364\pi\)
0.793294 + 0.608839i \(0.208364\pi\)
\(570\) 0 0
\(571\) 5542.36 0.406201 0.203100 0.979158i \(-0.434898\pi\)
0.203100 + 0.979158i \(0.434898\pi\)
\(572\) 0 0
\(573\) 4842.19 0.353028
\(574\) 0 0
\(575\) 29.6969i 0.00215382i
\(576\) 0 0
\(577\) 10857.0i 0.783334i −0.920107 0.391667i \(-0.871898\pi\)
0.920107 0.391667i \(-0.128102\pi\)
\(578\) 0 0
\(579\) 3746.11i 0.268883i
\(580\) 0 0
\(581\) −17379.4 + 6278.78i −1.24100 + 0.448344i
\(582\) 0 0
\(583\) 4670.58i 0.331794i
\(584\) 0 0
\(585\) −1261.81 −0.0891784
\(586\) 0 0
\(587\) 2426.15i 0.170593i 0.996356 + 0.0852963i \(0.0271837\pi\)
−0.996356 + 0.0852963i \(0.972816\pi\)
\(588\) 0 0
\(589\) 22356.5i 1.56398i
\(590\) 0 0
\(591\) 12453.4 0.866773
\(592\) 0 0
\(593\) 6562.03i 0.454418i −0.973846 0.227209i \(-0.927040\pi\)
0.973846 0.227209i \(-0.0729601\pi\)
\(594\) 0 0
\(595\) 19840.8 7168.03i 1.36705 0.493883i
\(596\) 0 0
\(597\) 8786.97i 0.602390i
\(598\) 0 0
\(599\) 14414.2i 0.983222i 0.870815 + 0.491611i \(0.163592\pi\)
−0.870815 + 0.491611i \(0.836408\pi\)
\(600\) 0 0
\(601\) 206.035i 0.0139839i 0.999976 + 0.00699196i \(0.00222563\pi\)
−0.999976 + 0.00699196i \(0.997774\pi\)
\(602\) 0 0
\(603\) 2857.51 0.192980
\(604\) 0 0
\(605\) −16116.3 −1.08301
\(606\) 0 0
\(607\) −9273.66 −0.620109 −0.310055 0.950719i \(-0.600347\pi\)
−0.310055 + 0.950719i \(0.600347\pi\)
\(608\) 0 0
\(609\) −5284.59 14627.5i −0.351629 0.973297i
\(610\) 0 0
\(611\) 2528.05 0.167388
\(612\) 0 0
\(613\) 1780.49i 0.117314i 0.998278 + 0.0586568i \(0.0186817\pi\)
−0.998278 + 0.0586568i \(0.981318\pi\)
\(614\) 0 0
\(615\) 7595.10 0.497990
\(616\) 0 0
\(617\) −10288.9 −0.671338 −0.335669 0.941980i \(-0.608962\pi\)
−0.335669 + 0.941980i \(0.608962\pi\)
\(618\) 0 0
\(619\) 28693.8i 1.86317i 0.363525 + 0.931585i \(0.381573\pi\)
−0.363525 + 0.931585i \(0.618427\pi\)
\(620\) 0 0
\(621\) −7.14535 −0.000461728
\(622\) 0 0
\(623\) −17214.9 + 6219.35i −1.10706 + 0.399956i
\(624\) 0 0
\(625\) −17059.7 −1.09182
\(626\) 0 0
\(627\) 7655.51 0.487610
\(628\) 0 0
\(629\) −30951.0 −1.96200
\(630\) 0 0
\(631\) 22169.2i 1.39864i −0.714808 0.699320i \(-0.753486\pi\)
0.714808 0.699320i \(-0.246514\pi\)
\(632\) 0 0
\(633\) 10082.4i 0.633077i
\(634\) 0 0
\(635\) 38647.2i 2.41523i
\(636\) 0 0
\(637\) −2401.34 + 1995.56i −0.149364 + 0.124124i
\(638\) 0 0
\(639\) 954.841i 0.0591125i
\(640\) 0 0
\(641\) 15310.8 0.943430 0.471715 0.881751i \(-0.343635\pi\)
0.471715 + 0.881751i \(0.343635\pi\)
\(642\) 0 0
\(643\) 9.01281i 0.000552769i −1.00000 0.000276385i \(-0.999912\pi\)
1.00000 0.000276385i \(-8.79760e-5\pi\)
\(644\) 0 0
\(645\) 12321.5i 0.752182i
\(646\) 0 0
\(647\) 20077.1 1.21996 0.609979 0.792417i \(-0.291178\pi\)
0.609979 + 0.792417i \(0.291178\pi\)
\(648\) 0 0
\(649\) 3576.65i 0.216326i
\(650\) 0 0
\(651\) 7723.30 2790.25i 0.464977 0.167985i
\(652\) 0 0
\(653\) 1796.00i 0.107631i −0.998551 0.0538154i \(-0.982862\pi\)
0.998551 0.0538154i \(-0.0171383\pi\)
\(654\) 0 0
\(655\) 31575.1i 1.88358i
\(656\) 0 0
\(657\) 6796.06i 0.403561i
\(658\) 0 0
\(659\) −12136.5 −0.717405 −0.358703 0.933452i \(-0.616781\pi\)
−0.358703 + 0.933452i \(0.616781\pi\)
\(660\) 0 0
\(661\) 18080.9 1.06394 0.531970 0.846763i \(-0.321452\pi\)
0.531970 + 0.846763i \(0.321452\pi\)
\(662\) 0 0
\(663\) 2019.68 0.118307
\(664\) 0 0
\(665\) 40579.6 14660.5i 2.36633 0.854900i
\(666\) 0 0
\(667\) −74.0802 −0.00430045
\(668\) 0 0
\(669\) 6988.36i 0.403865i
\(670\) 0 0
\(671\) 2944.56 0.169409
\(672\) 0 0
\(673\) −25019.4 −1.43303 −0.716513 0.697573i \(-0.754263\pi\)
−0.716513 + 0.697573i \(0.754263\pi\)
\(674\) 0 0
\(675\) 3029.81i 0.172766i
\(676\) 0 0
\(677\) 28258.5 1.60423 0.802114 0.597170i \(-0.203708\pi\)
0.802114 + 0.597170i \(0.203708\pi\)
\(678\) 0 0
\(679\) 17797.7 6429.89i 1.00591 0.363412i
\(680\) 0 0
\(681\) −2498.67 −0.140601
\(682\) 0 0
\(683\) −14966.8 −0.838491 −0.419245 0.907873i \(-0.637705\pi\)
−0.419245 + 0.907873i \(0.637705\pi\)
\(684\) 0 0
\(685\) −38758.1 −2.16185
\(686\) 0 0
\(687\) 4303.07i 0.238970i
\(688\) 0 0
\(689\) 2520.15i 0.139347i
\(690\) 0 0
\(691\) 11414.7i 0.628415i −0.949354 0.314208i \(-0.898261\pi\)
0.949354 0.314208i \(-0.101739\pi\)
\(692\) 0 0
\(693\) 955.463 + 2644.69i 0.0523738 + 0.144969i
\(694\) 0 0
\(695\) 4168.92i 0.227534i
\(696\) 0 0
\(697\) −12156.9 −0.660653
\(698\) 0 0
\(699\) 7165.25i 0.387718i
\(700\) 0 0
\(701\) 26445.0i 1.42484i −0.701753 0.712420i \(-0.747599\pi\)
0.701753 0.712420i \(-0.252401\pi\)
\(702\) 0 0
\(703\) −63302.7 −3.39617
\(704\) 0 0
\(705\) 12832.1i 0.685511i
\(706\) 0 0
\(707\) −5755.88 15932.1i −0.306184 0.847506i
\(708\) 0 0
\(709\) 4251.03i 0.225177i −0.993642 0.112589i \(-0.964086\pi\)
0.993642 0.112589i \(-0.0359143\pi\)
\(710\) 0 0
\(711\) 1747.82i 0.0921918i
\(712\) 0 0
\(713\) 39.1142i 0.00205447i
\(714\) 0 0
\(715\) 2365.24 0.123713
\(716\) 0 0
\(717\) 14427.0 0.751443
\(718\) 0 0
\(719\) −14049.0 −0.728703 −0.364352 0.931261i \(-0.618709\pi\)
−0.364352 + 0.931261i \(0.618709\pi\)
\(720\) 0 0
\(721\) 2951.96 + 8170.92i 0.152478 + 0.422054i
\(722\) 0 0
\(723\) −6808.76 −0.350236
\(724\) 0 0
\(725\) 31411.9i 1.60911i
\(726\) 0 0
\(727\) −22710.8 −1.15859 −0.579295 0.815118i \(-0.696672\pi\)
−0.579295 + 0.815118i \(0.696672\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 19722.0i 0.997873i
\(732\) 0 0
\(733\) 34196.7 1.72317 0.861586 0.507611i \(-0.169471\pi\)
0.861586 + 0.507611i \(0.169471\pi\)
\(734\) 0 0
\(735\) 10129.3 + 12189.0i 0.508331 + 0.611697i
\(736\) 0 0
\(737\) −5356.35 −0.267712
\(738\) 0 0
\(739\) −33629.7 −1.67400 −0.837001 0.547201i \(-0.815693\pi\)
−0.837001 + 0.547201i \(0.815693\pi\)
\(740\) 0 0
\(741\) 4130.76 0.204787
\(742\) 0 0
\(743\) 12955.5i 0.639693i 0.947469 + 0.319846i \(0.103631\pi\)
−0.947469 + 0.319846i \(0.896369\pi\)
\(744\) 0 0
\(745\) 3679.39i 0.180943i
\(746\) 0 0
\(747\) 8979.87i 0.439835i
\(748\) 0 0
\(749\) 9776.43 + 27060.8i 0.476933 + 1.32013i
\(750\) 0 0
\(751\) 13497.2i 0.655820i 0.944709 + 0.327910i \(0.106344\pi\)
−0.944709 + 0.327910i \(0.893656\pi\)
\(752\) 0 0
\(753\) 6726.21 0.325520
\(754\) 0 0
\(755\) 44041.2i 2.12295i
\(756\) 0 0
\(757\) 35571.1i 1.70786i −0.520386 0.853931i \(-0.674212\pi\)
0.520386 0.853931i \(-0.325788\pi\)
\(758\) 0 0
\(759\) 13.3938 0.000640534
\(760\) 0 0
\(761\) 16927.8i 0.806348i 0.915123 + 0.403174i \(0.132093\pi\)
−0.915123 + 0.403174i \(0.867907\pi\)
\(762\) 0 0
\(763\) −24808.1 + 8962.60i −1.17708 + 0.425253i
\(764\) 0 0
\(765\) 10251.7i 0.484510i
\(766\) 0 0
\(767\) 1929.89i 0.0908529i
\(768\) 0 0
\(769\) 15208.9i 0.713197i −0.934258 0.356599i \(-0.883936\pi\)
0.934258 0.356599i \(-0.116064\pi\)
\(770\) 0 0
\(771\) 2363.58 0.110405
\(772\) 0 0
\(773\) −10234.0 −0.476187 −0.238093 0.971242i \(-0.576522\pi\)
−0.238093 + 0.971242i \(0.576522\pi\)
\(774\) 0 0
\(775\) −16585.4 −0.768728
\(776\) 0 0
\(777\) −7900.63 21868.7i −0.364780 1.00970i
\(778\) 0 0
\(779\) −24864.0 −1.14357
\(780\) 0 0
\(781\) 1789.83i 0.0820041i
\(782\) 0 0
\(783\) −7557.99 −0.344956
\(784\) 0 0
\(785\) −8723.06 −0.396611
\(786\) 0 0
\(787\) 26421.8i 1.19674i 0.801220 + 0.598370i \(0.204185\pi\)
−0.801220 + 0.598370i \(0.795815\pi\)
\(788\) 0 0
\(789\) −20090.9 −0.906534
\(790\) 0 0
\(791\) −4938.34 13669.2i −0.221981 0.614437i
\(792\) 0 0
\(793\) 1588.83 0.0711486
\(794\) 0 0
\(795\) −12792.0 −0.570675
\(796\) 0 0
\(797\) 4607.96 0.204796 0.102398 0.994744i \(-0.467348\pi\)
0.102398 + 0.994744i \(0.467348\pi\)
\(798\) 0 0
\(799\) 20539.3i 0.909424i
\(800\) 0 0
\(801\) 8894.88i 0.392366i
\(802\) 0