Properties

Label 1344.4.p.c.223.4
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 223.4
Character \(\chi\) \(=\) 1344.223
Dual form 1344.4.p.c.223.3

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000i q^{3} -12.8225 q^{5} +(6.32908 + 17.4053i) q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} -12.8225 q^{5} +(6.32908 + 17.4053i) q^{7} -9.00000 q^{9} +25.0429 q^{11} +64.1604 q^{13} -38.4675i q^{15} -77.4251i q^{17} +49.9995i q^{19} +(-52.2158 + 18.9872i) q^{21} +80.5687i q^{23} +39.4169 q^{25} -27.0000i q^{27} -159.100i q^{29} +96.5596 q^{31} +75.1287i q^{33} +(-81.1547 - 223.179i) q^{35} -274.101i q^{37} +192.481i q^{39} -299.796i q^{41} -385.824 q^{43} +115.403 q^{45} +418.362 q^{47} +(-262.886 + 220.318i) q^{49} +232.275 q^{51} -665.385i q^{53} -321.113 q^{55} -149.999 q^{57} -445.655i q^{59} +599.783 q^{61} +(-56.9617 - 156.647i) q^{63} -822.697 q^{65} +675.504 q^{67} -241.706 q^{69} +877.093i q^{71} +696.060i q^{73} +118.251i q^{75} +(158.498 + 435.878i) q^{77} -1248.56i q^{79} +81.0000 q^{81} +238.303i q^{83} +992.784i q^{85} +477.299 q^{87} +743.503i q^{89} +(406.076 + 1116.73i) q^{91} +289.679i q^{93} -641.120i q^{95} +1552.16i q^{97} -225.386 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 288q^{9} + O(q^{10}) \) \( 32q - 288q^{9} - 224q^{13} - 72q^{21} + 1120q^{25} - 752q^{49} - 672q^{57} - 544q^{61} + 1536q^{65} - 144q^{69} - 1632q^{77} + 2592q^{81} + O(q^{100}) \)

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
\(5\) −12.8225 −1.14688 −0.573440 0.819247i \(-0.694392\pi\)
−0.573440 + 0.819247i \(0.694392\pi\)
\(6\) 0 0
\(7\) 6.32908 + 17.4053i 0.341738 + 0.939795i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 25.0429 0.686429 0.343214 0.939257i \(-0.388484\pi\)
0.343214 + 0.939257i \(0.388484\pi\)
\(12\) 0 0
\(13\) 64.1604 1.36884 0.684419 0.729089i \(-0.260056\pi\)
0.684419 + 0.729089i \(0.260056\pi\)
\(14\) 0 0
\(15\) 38.4675i 0.662152i
\(16\) 0 0
\(17\) 77.4251i 1.10461i −0.833643 0.552304i \(-0.813749\pi\)
0.833643 0.552304i \(-0.186251\pi\)
\(18\) 0 0
\(19\) 49.9995i 0.603720i 0.953352 + 0.301860i \(0.0976075\pi\)
−0.953352 + 0.301860i \(0.902393\pi\)
\(20\) 0 0
\(21\) −52.2158 + 18.9872i −0.542591 + 0.197303i
\(22\) 0 0
\(23\) 80.5687i 0.730423i 0.930925 + 0.365212i \(0.119003\pi\)
−0.930925 + 0.365212i \(0.880997\pi\)
\(24\) 0 0
\(25\) 39.4169 0.315335
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 159.100i 1.01876i −0.860541 0.509381i \(-0.829874\pi\)
0.860541 0.509381i \(-0.170126\pi\)
\(30\) 0 0
\(31\) 96.5596 0.559440 0.279720 0.960082i \(-0.409758\pi\)
0.279720 + 0.960082i \(0.409758\pi\)
\(32\) 0 0
\(33\) 75.1287i 0.396310i
\(34\) 0 0
\(35\) −81.1547 223.179i −0.391933 1.07783i
\(36\) 0 0
\(37\) 274.101i 1.21789i −0.793212 0.608946i \(-0.791593\pi\)
0.793212 0.608946i \(-0.208407\pi\)
\(38\) 0 0
\(39\) 192.481i 0.790299i
\(40\) 0 0
\(41\) 299.796i 1.14196i −0.820964 0.570980i \(-0.806564\pi\)
0.820964 0.570980i \(-0.193436\pi\)
\(42\) 0 0
\(43\) −385.824 −1.36832 −0.684158 0.729334i \(-0.739830\pi\)
−0.684158 + 0.729334i \(0.739830\pi\)
\(44\) 0 0
\(45\) 115.403 0.382294
\(46\) 0 0
\(47\) 418.362 1.29839 0.649196 0.760622i \(-0.275106\pi\)
0.649196 + 0.760622i \(0.275106\pi\)
\(48\) 0 0
\(49\) −262.886 + 220.318i −0.766430 + 0.642327i
\(50\) 0 0
\(51\) 232.275 0.637746
\(52\) 0 0
\(53\) 665.385i 1.72448i −0.506497 0.862242i \(-0.669060\pi\)
0.506497 0.862242i \(-0.330940\pi\)
\(54\) 0 0
\(55\) −321.113 −0.787252
\(56\) 0 0
\(57\) −149.999 −0.348558
\(58\) 0 0
\(59\) 445.655i 0.983380i −0.870770 0.491690i \(-0.836379\pi\)
0.870770 0.491690i \(-0.163621\pi\)
\(60\) 0 0
\(61\) 599.783 1.25892 0.629461 0.777032i \(-0.283276\pi\)
0.629461 + 0.777032i \(0.283276\pi\)
\(62\) 0 0
\(63\) −56.9617 156.647i −0.113913 0.313265i
\(64\) 0 0
\(65\) −822.697 −1.56989
\(66\) 0 0
\(67\) 675.504 1.23173 0.615866 0.787851i \(-0.288806\pi\)
0.615866 + 0.787851i \(0.288806\pi\)
\(68\) 0 0
\(69\) −241.706 −0.421710
\(70\) 0 0
\(71\) 877.093i 1.46608i 0.680185 + 0.733041i \(0.261900\pi\)
−0.680185 + 0.733041i \(0.738100\pi\)
\(72\) 0 0
\(73\) 696.060i 1.11600i 0.829842 + 0.557998i \(0.188430\pi\)
−0.829842 + 0.557998i \(0.811570\pi\)
\(74\) 0 0
\(75\) 118.251i 0.182059i
\(76\) 0 0
\(77\) 158.498 + 435.878i 0.234579 + 0.645102i
\(78\) 0 0
\(79\) 1248.56i 1.77815i −0.457760 0.889076i \(-0.651348\pi\)
0.457760 0.889076i \(-0.348652\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 238.303i 0.315146i 0.987507 + 0.157573i \(0.0503670\pi\)
−0.987507 + 0.157573i \(0.949633\pi\)
\(84\) 0 0
\(85\) 992.784i 1.26685i
\(86\) 0 0
\(87\) 477.299 0.588182
\(88\) 0 0
\(89\) 743.503i 0.885519i 0.896640 + 0.442759i \(0.146000\pi\)
−0.896640 + 0.442759i \(0.854000\pi\)
\(90\) 0 0
\(91\) 406.076 + 1116.73i 0.467784 + 1.28643i
\(92\) 0 0
\(93\) 289.679i 0.322993i
\(94\) 0 0
\(95\) 641.120i 0.692395i
\(96\) 0 0
\(97\) 1552.16i 1.62472i 0.583153 + 0.812362i \(0.301819\pi\)
−0.583153 + 0.812362i \(0.698181\pi\)
\(98\) 0 0
\(99\) −225.386 −0.228810
\(100\) 0 0
\(101\) 1202.46 1.18464 0.592322 0.805701i \(-0.298211\pi\)
0.592322 + 0.805701i \(0.298211\pi\)
\(102\) 0 0
\(103\) 404.562 0.387016 0.193508 0.981099i \(-0.438013\pi\)
0.193508 + 0.981099i \(0.438013\pi\)
\(104\) 0 0
\(105\) 669.537 243.464i 0.622287 0.226282i
\(106\) 0 0
\(107\) 1065.72 0.962871 0.481436 0.876481i \(-0.340116\pi\)
0.481436 + 0.876481i \(0.340116\pi\)
\(108\) 0 0
\(109\) 359.515i 0.315920i 0.987445 + 0.157960i \(0.0504918\pi\)
−0.987445 + 0.157960i \(0.949508\pi\)
\(110\) 0 0
\(111\) 822.304 0.703150
\(112\) 0 0
\(113\) 828.461 0.689691 0.344845 0.938660i \(-0.387931\pi\)
0.344845 + 0.938660i \(0.387931\pi\)
\(114\) 0 0
\(115\) 1033.09i 0.837708i
\(116\) 0 0
\(117\) −577.443 −0.456279
\(118\) 0 0
\(119\) 1347.60 490.029i 1.03811 0.377486i
\(120\) 0 0
\(121\) −703.854 −0.528816
\(122\) 0 0
\(123\) 899.389 0.659310
\(124\) 0 0
\(125\) 1097.39 0.785229
\(126\) 0 0
\(127\) 876.572i 0.612466i 0.951957 + 0.306233i \(0.0990687\pi\)
−0.951957 + 0.306233i \(0.900931\pi\)
\(128\) 0 0
\(129\) 1157.47i 0.789998i
\(130\) 0 0
\(131\) 1083.04i 0.722335i −0.932501 0.361167i \(-0.882378\pi\)
0.932501 0.361167i \(-0.117622\pi\)
\(132\) 0 0
\(133\) −870.254 + 316.451i −0.567373 + 0.206314i
\(134\) 0 0
\(135\) 346.208i 0.220717i
\(136\) 0 0
\(137\) −751.836 −0.468859 −0.234430 0.972133i \(-0.575322\pi\)
−0.234430 + 0.972133i \(0.575322\pi\)
\(138\) 0 0
\(139\) 543.555i 0.331681i −0.986153 0.165841i \(-0.946966\pi\)
0.986153 0.165841i \(-0.0530337\pi\)
\(140\) 0 0
\(141\) 1255.09i 0.749626i
\(142\) 0 0
\(143\) 1606.76 0.939609
\(144\) 0 0
\(145\) 2040.06i 1.16840i
\(146\) 0 0
\(147\) −660.955 788.657i −0.370848 0.442499i
\(148\) 0 0
\(149\) 1056.44i 0.580855i −0.956897 0.290427i \(-0.906203\pi\)
0.956897 0.290427i \(-0.0937974\pi\)
\(150\) 0 0
\(151\) 2092.13i 1.12752i 0.825939 + 0.563759i \(0.190645\pi\)
−0.825939 + 0.563759i \(0.809355\pi\)
\(152\) 0 0
\(153\) 696.826i 0.368203i
\(154\) 0 0
\(155\) −1238.14 −0.641610
\(156\) 0 0
\(157\) 1185.70 0.602735 0.301367 0.953508i \(-0.402557\pi\)
0.301367 + 0.953508i \(0.402557\pi\)
\(158\) 0 0
\(159\) 1996.15 0.995631
\(160\) 0 0
\(161\) −1402.32 + 509.925i −0.686448 + 0.249613i
\(162\) 0 0
\(163\) 903.635 0.434222 0.217111 0.976147i \(-0.430337\pi\)
0.217111 + 0.976147i \(0.430337\pi\)
\(164\) 0 0
\(165\) 963.338i 0.454520i
\(166\) 0 0
\(167\) −474.632 −0.219929 −0.109964 0.993936i \(-0.535074\pi\)
−0.109964 + 0.993936i \(0.535074\pi\)
\(168\) 0 0
\(169\) 1919.55 0.873715
\(170\) 0 0
\(171\) 449.996i 0.201240i
\(172\) 0 0
\(173\) −1641.33 −0.721317 −0.360658 0.932698i \(-0.617448\pi\)
−0.360658 + 0.932698i \(0.617448\pi\)
\(174\) 0 0
\(175\) 249.473 + 686.061i 0.107762 + 0.296351i
\(176\) 0 0
\(177\) 1336.97 0.567755
\(178\) 0 0
\(179\) 1802.11 0.752491 0.376246 0.926520i \(-0.377215\pi\)
0.376246 + 0.926520i \(0.377215\pi\)
\(180\) 0 0
\(181\) 2068.45 0.849429 0.424715 0.905327i \(-0.360374\pi\)
0.424715 + 0.905327i \(0.360374\pi\)
\(182\) 0 0
\(183\) 1799.35i 0.726840i
\(184\) 0 0
\(185\) 3514.67i 1.39678i
\(186\) 0 0
\(187\) 1938.95i 0.758234i
\(188\) 0 0
\(189\) 469.942 170.885i 0.180864 0.0657675i
\(190\) 0 0
\(191\) 2179.61i 0.825711i 0.910797 + 0.412855i \(0.135469\pi\)
−0.910797 + 0.412855i \(0.864531\pi\)
\(192\) 0 0
\(193\) −4489.81 −1.67453 −0.837263 0.546801i \(-0.815845\pi\)
−0.837263 + 0.546801i \(0.815845\pi\)
\(194\) 0 0
\(195\) 2468.09i 0.906378i
\(196\) 0 0
\(197\) 4519.08i 1.63437i 0.576376 + 0.817185i \(0.304467\pi\)
−0.576376 + 0.817185i \(0.695533\pi\)
\(198\) 0 0
\(199\) 826.598 0.294452 0.147226 0.989103i \(-0.452966\pi\)
0.147226 + 0.989103i \(0.452966\pi\)
\(200\) 0 0
\(201\) 2026.51i 0.711140i
\(202\) 0 0
\(203\) 2769.17 1006.95i 0.957428 0.348150i
\(204\) 0 0
\(205\) 3844.14i 1.30969i
\(206\) 0 0
\(207\) 725.118i 0.243474i
\(208\) 0 0
\(209\) 1252.13i 0.414411i
\(210\) 0 0
\(211\) −4709.17 −1.53646 −0.768229 0.640175i \(-0.778862\pi\)
−0.768229 + 0.640175i \(0.778862\pi\)
\(212\) 0 0
\(213\) −2631.28 −0.846443
\(214\) 0 0
\(215\) 4947.23 1.56930
\(216\) 0 0
\(217\) 611.133 + 1680.65i 0.191182 + 0.525759i
\(218\) 0 0
\(219\) −2088.18 −0.644321
\(220\) 0 0
\(221\) 4967.62i 1.51203i
\(222\) 0 0
\(223\) 5775.09 1.73421 0.867104 0.498127i \(-0.165979\pi\)
0.867104 + 0.498127i \(0.165979\pi\)
\(224\) 0 0
\(225\) −354.752 −0.105112
\(226\) 0 0
\(227\) 2280.11i 0.666678i −0.942807 0.333339i \(-0.891825\pi\)
0.942807 0.333339i \(-0.108175\pi\)
\(228\) 0 0
\(229\) 3692.99 1.06567 0.532837 0.846218i \(-0.321126\pi\)
0.532837 + 0.846218i \(0.321126\pi\)
\(230\) 0 0
\(231\) −1307.63 + 475.495i −0.372450 + 0.135434i
\(232\) 0 0
\(233\) 3140.07 0.882887 0.441443 0.897289i \(-0.354467\pi\)
0.441443 + 0.897289i \(0.354467\pi\)
\(234\) 0 0
\(235\) −5364.45 −1.48910
\(236\) 0 0
\(237\) 3745.68 1.02662
\(238\) 0 0
\(239\) 310.784i 0.0841127i −0.999115 0.0420563i \(-0.986609\pi\)
0.999115 0.0420563i \(-0.0133909\pi\)
\(240\) 0 0
\(241\) 457.833i 0.122372i −0.998126 0.0611859i \(-0.980512\pi\)
0.998126 0.0611859i \(-0.0194883\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) 3370.85 2825.04i 0.879004 0.736673i
\(246\) 0 0
\(247\) 3207.99i 0.826394i
\(248\) 0 0
\(249\) −714.909 −0.181950
\(250\) 0 0
\(251\) 807.916i 0.203168i 0.994827 + 0.101584i \(0.0323911\pi\)
−0.994827 + 0.101584i \(0.967609\pi\)
\(252\) 0 0
\(253\) 2017.67i 0.501383i
\(254\) 0 0
\(255\) −2978.35 −0.731418
\(256\) 0 0
\(257\) 1749.36i 0.424599i −0.977205 0.212299i \(-0.931905\pi\)
0.977205 0.212299i \(-0.0680952\pi\)
\(258\) 0 0
\(259\) 4770.80 1734.81i 1.14457 0.416200i
\(260\) 0 0
\(261\) 1431.90i 0.339587i
\(262\) 0 0
\(263\) 1360.35i 0.318945i −0.987202 0.159473i \(-0.949021\pi\)
0.987202 0.159473i \(-0.0509793\pi\)
\(264\) 0 0
\(265\) 8531.91i 1.97778i
\(266\) 0 0
\(267\) −2230.51 −0.511255
\(268\) 0 0
\(269\) 2224.06 0.504102 0.252051 0.967714i \(-0.418895\pi\)
0.252051 + 0.967714i \(0.418895\pi\)
\(270\) 0 0
\(271\) −2918.59 −0.654212 −0.327106 0.944988i \(-0.606073\pi\)
−0.327106 + 0.944988i \(0.606073\pi\)
\(272\) 0 0
\(273\) −3350.18 + 1218.23i −0.742719 + 0.270075i
\(274\) 0 0
\(275\) 987.113 0.216455
\(276\) 0 0
\(277\) 1632.84i 0.354180i −0.984195 0.177090i \(-0.943332\pi\)
0.984195 0.177090i \(-0.0566684\pi\)
\(278\) 0 0
\(279\) −869.037 −0.186480
\(280\) 0 0
\(281\) −7615.42 −1.61672 −0.808360 0.588689i \(-0.799644\pi\)
−0.808360 + 0.588689i \(0.799644\pi\)
\(282\) 0 0
\(283\) 2867.07i 0.602226i −0.953589 0.301113i \(-0.902642\pi\)
0.953589 0.301113i \(-0.0973581\pi\)
\(284\) 0 0
\(285\) 1923.36 0.399754
\(286\) 0 0
\(287\) 5218.03 1897.43i 1.07321 0.390251i
\(288\) 0 0
\(289\) −1081.64 −0.220159
\(290\) 0 0
\(291\) −4656.49 −0.938035
\(292\) 0 0
\(293\) −1133.07 −0.225920 −0.112960 0.993600i \(-0.536033\pi\)
−0.112960 + 0.993600i \(0.536033\pi\)
\(294\) 0 0
\(295\) 5714.42i 1.12782i
\(296\) 0 0
\(297\) 676.158i 0.132103i
\(298\) 0 0
\(299\) 5169.32i 0.999831i
\(300\) 0 0
\(301\) −2441.91 6715.36i −0.467606 1.28594i
\(302\) 0 0
\(303\) 3607.37i 0.683954i
\(304\) 0 0
\(305\) −7690.72 −1.44383
\(306\) 0 0
\(307\) 206.149i 0.0383242i 0.999816 + 0.0191621i \(0.00609985\pi\)
−0.999816 + 0.0191621i \(0.993900\pi\)
\(308\) 0 0
\(309\) 1213.69i 0.223444i
\(310\) 0 0
\(311\) 6296.00 1.14795 0.573977 0.818871i \(-0.305400\pi\)
0.573977 + 0.818871i \(0.305400\pi\)
\(312\) 0 0
\(313\) 7997.75i 1.44428i −0.691747 0.722140i \(-0.743158\pi\)
0.691747 0.722140i \(-0.256842\pi\)
\(314\) 0 0
\(315\) 730.392 + 2008.61i 0.130644 + 0.359278i
\(316\) 0 0
\(317\) 5830.42i 1.03303i 0.856279 + 0.516513i \(0.172770\pi\)
−0.856279 + 0.516513i \(0.827230\pi\)
\(318\) 0 0
\(319\) 3984.32i 0.699307i
\(320\) 0 0
\(321\) 3197.17i 0.555914i
\(322\) 0 0
\(323\) 3871.22 0.666874
\(324\) 0 0
\(325\) 2529.00 0.431643
\(326\) 0 0
\(327\) −1078.55 −0.182397
\(328\) 0 0
\(329\) 2647.85 + 7281.70i 0.443710 + 1.22022i
\(330\) 0 0
\(331\) −5022.44 −0.834013 −0.417007 0.908903i \(-0.636921\pi\)
−0.417007 + 0.908903i \(0.636921\pi\)
\(332\) 0 0
\(333\) 2466.91i 0.405964i
\(334\) 0 0
\(335\) −8661.66 −1.41265
\(336\) 0 0
\(337\) 8480.49 1.37081 0.685403 0.728164i \(-0.259626\pi\)
0.685403 + 0.728164i \(0.259626\pi\)
\(338\) 0 0
\(339\) 2485.38i 0.398193i
\(340\) 0 0
\(341\) 2418.13 0.384015
\(342\) 0 0
\(343\) −5498.52 3181.18i −0.865575 0.500780i
\(344\) 0 0
\(345\) 3099.28 0.483651
\(346\) 0 0
\(347\) 9890.27 1.53008 0.765040 0.643983i \(-0.222719\pi\)
0.765040 + 0.643983i \(0.222719\pi\)
\(348\) 0 0
\(349\) −6913.69 −1.06041 −0.530203 0.847871i \(-0.677884\pi\)
−0.530203 + 0.847871i \(0.677884\pi\)
\(350\) 0 0
\(351\) 1732.33i 0.263433i
\(352\) 0 0
\(353\) 5062.99i 0.763388i 0.924289 + 0.381694i \(0.124659\pi\)
−0.924289 + 0.381694i \(0.875341\pi\)
\(354\) 0 0
\(355\) 11246.5i 1.68142i
\(356\) 0 0
\(357\) 1470.09 + 4042.81i 0.217942 + 0.599350i
\(358\) 0 0
\(359\) 7551.38i 1.11016i −0.831798 0.555079i \(-0.812688\pi\)
0.831798 0.555079i \(-0.187312\pi\)
\(360\) 0 0
\(361\) 4359.05 0.635522
\(362\) 0 0
\(363\) 2111.56i 0.305312i
\(364\) 0 0
\(365\) 8925.25i 1.27991i
\(366\) 0 0
\(367\) 12597.1 1.79173 0.895864 0.444329i \(-0.146558\pi\)
0.895864 + 0.444329i \(0.146558\pi\)
\(368\) 0 0
\(369\) 2698.17i 0.380653i
\(370\) 0 0
\(371\) 11581.2 4211.27i 1.62066 0.589322i
\(372\) 0 0
\(373\) 9549.28i 1.32558i 0.748803 + 0.662792i \(0.230629\pi\)
−0.748803 + 0.662792i \(0.769371\pi\)
\(374\) 0 0
\(375\) 3292.17i 0.453352i
\(376\) 0 0
\(377\) 10207.9i 1.39452i
\(378\) 0 0
\(379\) 1603.13 0.217276 0.108638 0.994081i \(-0.465351\pi\)
0.108638 + 0.994081i \(0.465351\pi\)
\(380\) 0 0
\(381\) −2629.72 −0.353608
\(382\) 0 0
\(383\) 402.889 0.0537511 0.0268756 0.999639i \(-0.491444\pi\)
0.0268756 + 0.999639i \(0.491444\pi\)
\(384\) 0 0
\(385\) −2032.35 5589.05i −0.269034 0.739855i
\(386\) 0 0
\(387\) 3472.42 0.456105
\(388\) 0 0
\(389\) 8540.21i 1.11313i −0.830806 0.556563i \(-0.812120\pi\)
0.830806 0.556563i \(-0.187880\pi\)
\(390\) 0 0
\(391\) 6238.04 0.806831
\(392\) 0 0
\(393\) 3249.13 0.417040
\(394\) 0 0
\(395\) 16009.7i 2.03933i
\(396\) 0 0
\(397\) −11977.7 −1.51421 −0.757106 0.653292i \(-0.773387\pi\)
−0.757106 + 0.653292i \(0.773387\pi\)
\(398\) 0 0
\(399\) −949.352 2610.76i −0.119115 0.327573i
\(400\) 0 0
\(401\) 58.8089 0.00732364 0.00366182 0.999993i \(-0.498834\pi\)
0.00366182 + 0.999993i \(0.498834\pi\)
\(402\) 0 0
\(403\) 6195.30 0.765782
\(404\) 0 0
\(405\) −1038.62 −0.127431
\(406\) 0 0
\(407\) 6864.29i 0.835996i
\(408\) 0 0
\(409\) 805.417i 0.0973723i 0.998814 + 0.0486862i \(0.0155034\pi\)
−0.998814 + 0.0486862i \(0.984497\pi\)
\(410\) 0 0
\(411\) 2255.51i 0.270696i
\(412\) 0 0
\(413\) 7756.75 2820.59i 0.924176 0.336058i
\(414\) 0 0
\(415\) 3055.64i 0.361435i
\(416\) 0 0
\(417\) 1630.66 0.191496
\(418\) 0 0
\(419\) 7784.37i 0.907616i 0.891099 + 0.453808i \(0.149935\pi\)
−0.891099 + 0.453808i \(0.850065\pi\)
\(420\) 0 0
\(421\) 4998.41i 0.578640i 0.957232 + 0.289320i \(0.0934292\pi\)
−0.957232 + 0.289320i \(0.906571\pi\)
\(422\) 0 0
\(423\) −3765.26 −0.432797
\(424\) 0 0
\(425\) 3051.86i 0.348322i
\(426\) 0 0
\(427\) 3796.07 + 10439.4i 0.430222 + 1.18313i
\(428\) 0 0
\(429\) 4820.28i 0.542484i
\(430\) 0 0
\(431\) 5983.17i 0.668675i 0.942453 + 0.334338i \(0.108513\pi\)
−0.942453 + 0.334338i \(0.891487\pi\)
\(432\) 0 0
\(433\) 3739.75i 0.415060i −0.978229 0.207530i \(-0.933458\pi\)
0.978229 0.207530i \(-0.0665425\pi\)
\(434\) 0 0
\(435\) −6120.18 −0.674575
\(436\) 0 0
\(437\) −4028.40 −0.440971
\(438\) 0 0
\(439\) 8032.79 0.873313 0.436656 0.899628i \(-0.356163\pi\)
0.436656 + 0.899628i \(0.356163\pi\)
\(440\) 0 0
\(441\) 2365.97 1982.86i 0.255477 0.214109i
\(442\) 0 0
\(443\) 10025.1 1.07519 0.537595 0.843203i \(-0.319333\pi\)
0.537595 + 0.843203i \(0.319333\pi\)
\(444\) 0 0
\(445\) 9533.58i 1.01558i
\(446\) 0 0
\(447\) 3169.33 0.335357
\(448\) 0 0
\(449\) 4911.85 0.516268 0.258134 0.966109i \(-0.416892\pi\)
0.258134 + 0.966109i \(0.416892\pi\)
\(450\) 0 0
\(451\) 7507.77i 0.783873i
\(452\) 0 0
\(453\) −6276.40 −0.650973
\(454\) 0 0
\(455\) −5206.91 14319.3i −0.536492 1.47538i
\(456\) 0 0
\(457\) 10828.7 1.10842 0.554208 0.832378i \(-0.313021\pi\)
0.554208 + 0.832378i \(0.313021\pi\)
\(458\) 0 0
\(459\) −2090.48 −0.212582
\(460\) 0 0
\(461\) 7105.06 0.717821 0.358911 0.933372i \(-0.383148\pi\)
0.358911 + 0.933372i \(0.383148\pi\)
\(462\) 0 0
\(463\) 14241.0i 1.42945i 0.699403 + 0.714727i \(0.253449\pi\)
−0.699403 + 0.714727i \(0.746551\pi\)
\(464\) 0 0
\(465\) 3714.41i 0.370434i
\(466\) 0 0
\(467\) 5568.41i 0.551767i −0.961191 0.275884i \(-0.911030\pi\)
0.961191 0.275884i \(-0.0889705\pi\)
\(468\) 0 0
\(469\) 4275.32 + 11757.3i 0.420929 + 1.15757i
\(470\) 0 0
\(471\) 3557.11i 0.347989i
\(472\) 0 0
\(473\) −9662.15 −0.939252
\(474\) 0 0
\(475\) 1970.83i 0.190374i
\(476\) 0 0
\(477\) 5988.46i 0.574828i
\(478\) 0 0
\(479\) −14755.8 −1.40754 −0.703768 0.710430i \(-0.748501\pi\)
−0.703768 + 0.710430i \(0.748501\pi\)
\(480\) 0 0
\(481\) 17586.4i 1.66710i
\(482\) 0 0
\(483\) −1529.78 4206.96i −0.144114 0.396321i
\(484\) 0 0
\(485\) 19902.6i 1.86336i
\(486\) 0 0
\(487\) 2870.19i 0.267065i 0.991044 + 0.133533i \(0.0426321\pi\)
−0.991044 + 0.133533i \(0.957368\pi\)
\(488\) 0 0
\(489\) 2710.91i 0.250698i
\(490\) 0 0
\(491\) −13050.2 −1.19949 −0.599745 0.800191i \(-0.704731\pi\)
−0.599745 + 0.800191i \(0.704731\pi\)
\(492\) 0 0
\(493\) −12318.3 −1.12533
\(494\) 0 0
\(495\) 2890.02 0.262417
\(496\) 0 0
\(497\) −15266.0 + 5551.19i −1.37782 + 0.501016i
\(498\) 0 0
\(499\) −10021.7 −0.899063 −0.449531 0.893265i \(-0.648409\pi\)
−0.449531 + 0.893265i \(0.648409\pi\)
\(500\) 0 0
\(501\) 1423.90i 0.126976i
\(502\) 0 0
\(503\) −13625.2 −1.20779 −0.603894 0.797065i \(-0.706385\pi\)
−0.603894 + 0.797065i \(0.706385\pi\)
\(504\) 0 0
\(505\) −15418.5 −1.35865
\(506\) 0 0
\(507\) 5758.66i 0.504440i
\(508\) 0 0
\(509\) 6931.67 0.603617 0.301808 0.953369i \(-0.402410\pi\)
0.301808 + 0.953369i \(0.402410\pi\)
\(510\) 0 0
\(511\) −12115.1 + 4405.42i −1.04881 + 0.381378i
\(512\) 0 0
\(513\) 1349.99 0.116186
\(514\) 0 0
\(515\) −5187.50 −0.443861
\(516\) 0 0
\(517\) 10477.0 0.891253
\(518\) 0 0
\(519\) 4923.98i 0.416452i
\(520\) 0 0
\(521\) 20497.7i 1.72365i −0.507207 0.861824i \(-0.669322\pi\)
0.507207 0.861824i \(-0.330678\pi\)
\(522\) 0 0
\(523\) 17220.5i 1.43977i 0.694094 + 0.719885i \(0.255805\pi\)
−0.694094 + 0.719885i \(0.744195\pi\)
\(524\) 0 0
\(525\) −2058.18 + 748.418i −0.171098 + 0.0622165i
\(526\) 0 0
\(527\) 7476.14i 0.617961i
\(528\) 0 0
\(529\) 5675.68 0.466482
\(530\) 0 0
\(531\) 4010.90i 0.327793i
\(532\) 0 0
\(533\) 19235.0i 1.56316i
\(534\) 0 0
\(535\) −13665.2 −1.10430
\(536\) 0 0
\(537\) 5406.33i 0.434451i
\(538\) 0 0
\(539\) −6583.41 + 5517.41i −0.526100 + 0.440912i
\(540\) 0 0
\(541\) 6118.24i 0.486218i 0.969999 + 0.243109i \(0.0781672\pi\)
−0.969999 + 0.243109i \(0.921833\pi\)
\(542\) 0 0
\(543\) 6205.35i 0.490418i
\(544\) 0 0
\(545\) 4609.89i 0.362323i
\(546\) 0 0
\(547\) −391.181 −0.0305772 −0.0152886 0.999883i \(-0.504867\pi\)
−0.0152886 + 0.999883i \(0.504867\pi\)
\(548\) 0 0
\(549\) −5398.04 −0.419641
\(550\) 0 0
\(551\) 7954.91 0.615047
\(552\) 0 0
\(553\) 21731.5 7902.23i 1.67110 0.607662i
\(554\) 0 0
\(555\) −10544.0 −0.806429
\(556\) 0 0
\(557\) 14903.1i 1.13369i −0.823824 0.566846i \(-0.808163\pi\)
0.823824 0.566846i \(-0.191837\pi\)
\(558\) 0 0
\(559\) −24754.6 −1.87300
\(560\) 0 0
\(561\) 5816.84 0.437767
\(562\) 0 0
\(563\) 18651.1i 1.39618i −0.716010 0.698090i \(-0.754034\pi\)
0.716010 0.698090i \(-0.245966\pi\)
\(564\) 0 0
\(565\) −10623.0 −0.790993
\(566\) 0 0
\(567\) 512.655 + 1409.83i 0.0379709 + 0.104422i
\(568\) 0 0
\(569\) −14380.9 −1.05954 −0.529770 0.848141i \(-0.677722\pi\)
−0.529770 + 0.848141i \(0.677722\pi\)
\(570\) 0 0
\(571\) −26145.9 −1.91623 −0.958117 0.286376i \(-0.907549\pi\)
−0.958117 + 0.286376i \(0.907549\pi\)
\(572\) 0 0
\(573\) −6538.82 −0.476724
\(574\) 0 0
\(575\) 3175.77i 0.230328i
\(576\) 0 0
\(577\) 18408.8i 1.32820i 0.747645 + 0.664099i \(0.231185\pi\)
−0.747645 + 0.664099i \(0.768815\pi\)
\(578\) 0 0
\(579\) 13469.4i 0.966788i
\(580\) 0 0
\(581\) −4147.72 + 1508.24i −0.296173 + 0.107697i
\(582\) 0 0
\(583\) 16663.2i 1.18374i
\(584\) 0 0
\(585\) 7404.28 0.523298
\(586\) 0 0
\(587\) 25942.5i 1.82412i 0.410053 + 0.912062i \(0.365510\pi\)
−0.410053 + 0.912062i \(0.634490\pi\)
\(588\) 0 0
\(589\) 4827.94i 0.337745i
\(590\) 0 0
\(591\) −13557.2 −0.943604
\(592\) 0 0
\(593\) 19221.9i 1.33111i −0.746347 0.665557i \(-0.768194\pi\)
0.746347 0.665557i \(-0.231806\pi\)
\(594\) 0 0
\(595\) −17279.7 + 6283.41i −1.19058 + 0.432932i
\(596\) 0 0
\(597\) 2479.79i 0.170002i
\(598\) 0 0
\(599\) 14280.0i 0.974068i −0.873383 0.487034i \(-0.838079\pi\)
0.873383 0.487034i \(-0.161921\pi\)
\(600\) 0 0
\(601\) 5677.17i 0.385319i 0.981266 + 0.192659i \(0.0617113\pi\)
−0.981266 + 0.192659i \(0.938289\pi\)
\(602\) 0 0
\(603\) −6079.54 −0.410577
\(604\) 0 0
\(605\) 9025.18 0.606489
\(606\) 0 0
\(607\) −16394.3 −1.09625 −0.548124 0.836397i \(-0.684658\pi\)
−0.548124 + 0.836397i \(0.684658\pi\)
\(608\) 0 0
\(609\) 3020.86 + 8307.52i 0.201004 + 0.552771i
\(610\) 0 0
\(611\) 26842.3 1.77729
\(612\) 0 0
\(613\) 2171.28i 0.143063i −0.997438 0.0715313i \(-0.977211\pi\)
0.997438 0.0715313i \(-0.0227886\pi\)
\(614\) 0 0
\(615\) −11532.4 −0.756150
\(616\) 0 0
\(617\) −6024.75 −0.393108 −0.196554 0.980493i \(-0.562975\pi\)
−0.196554 + 0.980493i \(0.562975\pi\)
\(618\) 0 0
\(619\) 13135.5i 0.852926i 0.904505 + 0.426463i \(0.140240\pi\)
−0.904505 + 0.426463i \(0.859760\pi\)
\(620\) 0 0
\(621\) 2175.36 0.140570
\(622\) 0 0
\(623\) −12940.9 + 4705.69i −0.832206 + 0.302615i
\(624\) 0 0
\(625\) −18998.4 −1.21590
\(626\) 0 0
\(627\) −3756.40 −0.239260
\(628\) 0 0
\(629\) −21222.3 −1.34529
\(630\) 0 0
\(631\) 13957.4i 0.880562i −0.897860 0.440281i \(-0.854879\pi\)
0.897860 0.440281i \(-0.145121\pi\)
\(632\) 0 0
\(633\) 14127.5i 0.887074i
\(634\) 0 0
\(635\) 11239.9i 0.702426i
\(636\) 0 0
\(637\) −16866.8 + 14135.7i −1.04912 + 0.879242i
\(638\) 0 0
\(639\) 7893.84i 0.488694i
\(640\) 0 0
\(641\) 24561.9 1.51348 0.756738 0.653718i \(-0.226792\pi\)
0.756738 + 0.653718i \(0.226792\pi\)
\(642\) 0 0
\(643\) 15778.3i 0.967708i −0.875149 0.483854i \(-0.839237\pi\)
0.875149 0.483854i \(-0.160763\pi\)
\(644\) 0 0
\(645\) 14841.7i 0.906033i
\(646\) 0 0
\(647\) 11167.7 0.678588 0.339294 0.940680i \(-0.389812\pi\)
0.339294 + 0.940680i \(0.389812\pi\)
\(648\) 0 0
\(649\) 11160.5i 0.675020i
\(650\) 0 0
\(651\) −5041.94 + 1833.40i −0.303547 + 0.110379i
\(652\) 0 0
\(653\) 15738.6i 0.943185i −0.881817 0.471592i \(-0.843679\pi\)
0.881817 0.471592i \(-0.156321\pi\)
\(654\) 0 0
\(655\) 13887.3i 0.828432i
\(656\) 0 0
\(657\) 6264.54i 0.371999i
\(658\) 0 0
\(659\) 21821.3 1.28989 0.644944 0.764230i \(-0.276881\pi\)
0.644944 + 0.764230i \(0.276881\pi\)
\(660\) 0 0
\(661\) 5703.63 0.335621 0.167810 0.985819i \(-0.446330\pi\)
0.167810 + 0.985819i \(0.446330\pi\)
\(662\) 0 0
\(663\) 14902.9 0.872970
\(664\) 0 0
\(665\) 11158.8 4057.69i 0.650709 0.236618i
\(666\) 0 0
\(667\) 12818.5 0.744127
\(668\) 0 0
\(669\) 17325.3i 1.00125i
\(670\) 0 0
\(671\) 15020.3 0.864161
\(672\) 0 0
\(673\) 30185.3 1.72891 0.864456 0.502708i \(-0.167663\pi\)
0.864456 + 0.502708i \(0.167663\pi\)
\(674\) 0 0
\(675\) 1064.26i 0.0606863i
\(676\) 0 0
\(677\) −12990.9 −0.737488 −0.368744 0.929531i \(-0.620212\pi\)
−0.368744 + 0.929531i \(0.620212\pi\)
\(678\) 0 0
\(679\) −27015.8 + 9823.76i −1.52691 + 0.555230i
\(680\) 0 0
\(681\) 6840.32 0.384907
\(682\) 0 0
\(683\) −3636.79 −0.203745 −0.101872 0.994797i \(-0.532483\pi\)
−0.101872 + 0.994797i \(0.532483\pi\)
\(684\) 0 0
\(685\) 9640.43 0.537725
\(686\) 0 0
\(687\) 11079.0i 0.615267i
\(688\) 0 0
\(689\) 42691.3i 2.36054i
\(690\) 0 0
\(691\) 869.636i 0.0478763i −0.999713 0.0239381i \(-0.992380\pi\)
0.999713 0.0239381i \(-0.00762048\pi\)
\(692\) 0 0
\(693\) −1426.49 3922.90i −0.0781929 0.215034i
\(694\) 0 0
\(695\) 6969.74i 0.380399i
\(696\) 0 0
\(697\) −23211.7 −1.26142
\(698\) 0 0
\(699\) 9420.20i 0.509735i
\(700\) 0 0
\(701\) 8128.07i 0.437936i −0.975732 0.218968i \(-0.929731\pi\)
0.975732 0.218968i \(-0.0702690\pi\)
\(702\) 0 0
\(703\) 13704.9 0.735265
\(704\) 0 0
\(705\) 16093.4i 0.859732i
\(706\) 0 0
\(707\) 7610.45 + 20929.1i 0.404838 + 1.11332i
\(708\) 0 0
\(709\) 6488.87i 0.343716i 0.985122 + 0.171858i \(0.0549770\pi\)
−0.985122 + 0.171858i \(0.945023\pi\)
\(710\) 0 0
\(711\) 11237.0i 0.592717i
\(712\) 0 0
\(713\) 7779.69i 0.408628i
\(714\) 0 0
\(715\) −20602.7 −1.07762
\(716\) 0 0
\(717\) 932.351 0.0485625
\(718\) 0 0
\(719\) 19419.9 1.00729 0.503645 0.863911i \(-0.331992\pi\)
0.503645 + 0.863911i \(0.331992\pi\)
\(720\) 0 0
\(721\) 2560.50 + 7041.50i 0.132258 + 0.363716i
\(722\) 0 0
\(723\) 1373.50 0.0706514
\(724\) 0 0
\(725\) 6271.22i 0.321252i
\(726\) 0 0
\(727\) −29343.3 −1.49695 −0.748475 0.663163i \(-0.769214\pi\)
−0.748475 + 0.663163i \(0.769214\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 29872.4i 1.51145i
\(732\) 0 0
\(733\) −27788.9 −1.40028 −0.700140 0.714006i \(-0.746879\pi\)
−0.700140 + 0.714006i \(0.746879\pi\)
\(734\) 0 0
\(735\) 8475.11 + 10112.6i 0.425318 + 0.507493i
\(736\) 0 0
\(737\) 16916.6 0.845495
\(738\) 0 0
\(739\) −12588.2 −0.626607 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(740\) 0 0
\(741\) −9623.96 −0.477119
\(742\) 0 0
\(743\) 8137.53i 0.401799i −0.979612 0.200900i \(-0.935613\pi\)
0.979612 0.200900i \(-0.0643865\pi\)
\(744\) 0 0
\(745\) 13546.3i 0.666171i
\(746\) 0 0
\(747\) 2144.73i 0.105049i
\(748\) 0 0
\(749\) 6745.03 + 18549.2i 0.329050 + 0.904902i
\(750\) 0 0
\(751\) 2723.45i 0.132331i 0.997809 + 0.0661653i \(0.0210765\pi\)
−0.997809 + 0.0661653i \(0.978924\pi\)
\(752\) 0 0
\(753\) −2423.75 −0.117299
\(754\) 0 0
\(755\) 26826.4i 1.29313i
\(756\) 0 0
\(757\) 8168.46i 0.392190i −0.980585 0.196095i \(-0.937174\pi\)
0.980585 0.196095i \(-0.0628261\pi\)
\(758\) 0 0
\(759\) −6053.02 −0.289474
\(760\) 0 0
\(761\) 24255.5i 1.15540i −0.816248 0.577702i \(-0.803950\pi\)
0.816248 0.577702i \(-0.196050\pi\)
\(762\) 0 0
\(763\) −6257.46 + 2275.40i −0.296901 + 0.107962i
\(764\) 0 0
\(765\) 8935.06i 0.422285i
\(766\) 0 0
\(767\) 28593.4i 1.34609i
\(768\) 0 0
\(769\) 13226.7i 0.620241i 0.950697 + 0.310121i \(0.100369\pi\)
−0.950697 + 0.310121i \(0.899631\pi\)
\(770\) 0 0
\(771\) 5248.07 0.245142
\(772\) 0 0
\(773\) −24347.2 −1.13287 −0.566434 0.824107i \(-0.691677\pi\)
−0.566434 + 0.824107i \(0.691677\pi\)
\(774\) 0 0
\(775\) 3806.08 0.176411
\(776\) 0 0
\(777\) 5204.42 + 14312.4i 0.240293 + 0.660817i
\(778\) 0 0
\(779\) 14989.7 0.689423
\(780\) 0 0
\(781\) 21964.9i 1.00636i
\(782\) 0 0
\(783\) −4295.69 −0.196061
\(784\) 0 0
\(785\) −15203.7 −0.691265
\(786\) 0 0
\(787\) 31493.4i 1.42645i −0.700933 0.713227i \(-0.747233\pi\)
0.700933 0.713227i \(-0.252767\pi\)
\(788\) 0 0
\(789\) 4081.04 0.184143
\(790\) 0 0
\(791\) 5243.39 + 14419.6i 0.235694 + 0.648168i
\(792\) 0 0
\(793\) 38482.3 1.72326
\(794\) 0 0
\(795\) −25595.7 −1.14187
\(796\) 0 0
\(797\) 27985.5 1.24378 0.621892 0.783103i \(-0.286364\pi\)
0.621892 + 0.783103i \(0.286364\pi\)
\(798\) 0 0
\(799\) 32391.7i 1.43421i
\(800\) 0 0
\(801\) 6691.53i 0.295173i
\(802\) 0 0