Properties

Label 1200.3.l.k.401.1
Level $1200$
Weight $3$
Character 1200.401
Analytic conductor $32.698$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-35}) \)
Defining polynomial: \(x^{2} - x + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 401.1
Root \(0.500000 + 2.95804i\) of defining polynomial
Character \(\chi\) \(=\) 1200.401
Dual form 1200.3.l.k.401.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.500000 - 2.95804i) q^{3} -8.00000 q^{7} +(-8.50000 + 2.95804i) q^{9} +O(q^{10})\) \(q+(-0.500000 - 2.95804i) q^{3} -8.00000 q^{7} +(-8.50000 + 2.95804i) q^{9} -17.7482i q^{11} +2.00000 q^{13} -17.7482i q^{17} -11.0000 q^{19} +(4.00000 + 23.6643i) q^{21} -35.4965i q^{23} +(13.0000 + 23.6643i) q^{27} +35.4965i q^{29} +46.0000 q^{31} +(-52.5000 + 8.87412i) q^{33} -16.0000 q^{37} +(-1.00000 - 5.91608i) q^{39} +53.2447i q^{41} -62.0000 q^{43} +35.4965i q^{47} +15.0000 q^{49} +(-52.5000 + 8.87412i) q^{51} -35.4965i q^{53} +(5.50000 + 32.5384i) q^{57} +70.9930i q^{59} -16.0000 q^{61} +(68.0000 - 23.6643i) q^{63} -113.000 q^{67} +(-105.000 + 17.7482i) q^{69} +106.489i q^{71} +101.000 q^{73} +141.986i q^{77} -68.0000 q^{79} +(63.5000 - 50.2867i) q^{81} +17.7482i q^{83} +(105.000 - 17.7482i) q^{87} +53.2447i q^{89} -16.0000 q^{91} +(-23.0000 - 136.070i) q^{93} -22.0000 q^{97} +(52.5000 + 150.860i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - 16q^{7} - 17q^{9} + O(q^{10}) \) \( 2q - q^{3} - 16q^{7} - 17q^{9} + 4q^{13} - 22q^{19} + 8q^{21} + 26q^{27} + 92q^{31} - 105q^{33} - 32q^{37} - 2q^{39} - 124q^{43} + 30q^{49} - 105q^{51} + 11q^{57} - 32q^{61} + 136q^{63} - 226q^{67} - 210q^{69} + 202q^{73} - 136q^{79} + 127q^{81} + 210q^{87} - 32q^{91} - 46q^{93} - 44q^{97} + 105q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) −0.500000 2.95804i −0.166667 0.986013i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −8.00000 −1.14286 −0.571429 0.820652i \(-0.693611\pi\)
−0.571429 + 0.820652i \(0.693611\pi\)
\(8\) 0 0
\(9\) −8.50000 + 2.95804i −0.944444 + 0.328671i
\(10\) 0 0
\(11\) 17.7482i 1.61348i −0.590909 0.806738i \(-0.701231\pi\)
0.590909 0.806738i \(-0.298769\pi\)
\(12\) 0 0
\(13\) 2.00000 0.153846 0.0769231 0.997037i \(-0.475490\pi\)
0.0769231 + 0.997037i \(0.475490\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 17.7482i 1.04401i −0.852941 0.522007i \(-0.825183\pi\)
0.852941 0.522007i \(-0.174817\pi\)
\(18\) 0 0
\(19\) −11.0000 −0.578947 −0.289474 0.957186i \(-0.593480\pi\)
−0.289474 + 0.957186i \(0.593480\pi\)
\(20\) 0 0
\(21\) 4.00000 + 23.6643i 0.190476 + 1.12687i
\(22\) 0 0
\(23\) 35.4965i 1.54333i −0.636032 0.771663i \(-0.719425\pi\)
0.636032 0.771663i \(-0.280575\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 13.0000 + 23.6643i 0.481481 + 0.876456i
\(28\) 0 0
\(29\) 35.4965i 1.22402i 0.790851 + 0.612008i \(0.209638\pi\)
−0.790851 + 0.612008i \(0.790362\pi\)
\(30\) 0 0
\(31\) 46.0000 1.48387 0.741935 0.670471i \(-0.233908\pi\)
0.741935 + 0.670471i \(0.233908\pi\)
\(32\) 0 0
\(33\) −52.5000 + 8.87412i −1.59091 + 0.268913i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −16.0000 −0.432432 −0.216216 0.976346i \(-0.569372\pi\)
−0.216216 + 0.976346i \(0.569372\pi\)
\(38\) 0 0
\(39\) −1.00000 5.91608i −0.0256410 0.151694i
\(40\) 0 0
\(41\) 53.2447i 1.29865i 0.760510 + 0.649326i \(0.224949\pi\)
−0.760510 + 0.649326i \(0.775051\pi\)
\(42\) 0 0
\(43\) −62.0000 −1.44186 −0.720930 0.693008i \(-0.756285\pi\)
−0.720930 + 0.693008i \(0.756285\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 35.4965i 0.755244i 0.925960 + 0.377622i \(0.123258\pi\)
−0.925960 + 0.377622i \(0.876742\pi\)
\(48\) 0 0
\(49\) 15.0000 0.306122
\(50\) 0 0
\(51\) −52.5000 + 8.87412i −1.02941 + 0.174002i
\(52\) 0 0
\(53\) 35.4965i 0.669745i −0.942263 0.334872i \(-0.891307\pi\)
0.942263 0.334872i \(-0.108693\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.50000 + 32.5384i 0.0964912 + 0.570850i
\(58\) 0 0
\(59\) 70.9930i 1.20327i 0.798771 + 0.601635i \(0.205484\pi\)
−0.798771 + 0.601635i \(0.794516\pi\)
\(60\) 0 0
\(61\) −16.0000 −0.262295 −0.131148 0.991363i \(-0.541866\pi\)
−0.131148 + 0.991363i \(0.541866\pi\)
\(62\) 0 0
\(63\) 68.0000 23.6643i 1.07937 0.375624i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −113.000 −1.68657 −0.843284 0.537469i \(-0.819381\pi\)
−0.843284 + 0.537469i \(0.819381\pi\)
\(68\) 0 0
\(69\) −105.000 + 17.7482i −1.52174 + 0.257221i
\(70\) 0 0
\(71\) 106.489i 1.49985i 0.661522 + 0.749926i \(0.269911\pi\)
−0.661522 + 0.749926i \(0.730089\pi\)
\(72\) 0 0
\(73\) 101.000 1.38356 0.691781 0.722108i \(-0.256826\pi\)
0.691781 + 0.722108i \(0.256826\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 141.986i 1.84397i
\(78\) 0 0
\(79\) −68.0000 −0.860759 −0.430380 0.902648i \(-0.641620\pi\)
−0.430380 + 0.902648i \(0.641620\pi\)
\(80\) 0 0
\(81\) 63.5000 50.2867i 0.783951 0.620823i
\(82\) 0 0
\(83\) 17.7482i 0.213834i 0.994268 + 0.106917i \(0.0340979\pi\)
−0.994268 + 0.106917i \(0.965902\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 105.000 17.7482i 1.20690 0.204003i
\(88\) 0 0
\(89\) 53.2447i 0.598255i 0.954213 + 0.299128i \(0.0966956\pi\)
−0.954213 + 0.299128i \(0.903304\pi\)
\(90\) 0 0
\(91\) −16.0000 −0.175824
\(92\) 0 0
\(93\) −23.0000 136.070i −0.247312 1.46312i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −22.0000 −0.226804 −0.113402 0.993549i \(-0.536175\pi\)
−0.113402 + 0.993549i \(0.536175\pi\)
\(98\) 0 0
\(99\) 52.5000 + 150.860i 0.530303 + 1.52384i
\(100\) 0 0
\(101\) 141.986i 1.40580i −0.711288 0.702901i \(-0.751888\pi\)
0.711288 0.702901i \(-0.248112\pi\)
\(102\) 0 0
\(103\) −26.0000 −0.252427 −0.126214 0.992003i \(-0.540282\pi\)
−0.126214 + 0.992003i \(0.540282\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 17.7482i 0.165871i 0.996555 + 0.0829357i \(0.0264296\pi\)
−0.996555 + 0.0829357i \(0.973570\pi\)
\(108\) 0 0
\(109\) 176.000 1.61468 0.807339 0.590087i \(-0.200907\pi\)
0.807339 + 0.590087i \(0.200907\pi\)
\(110\) 0 0
\(111\) 8.00000 + 47.3286i 0.0720721 + 0.426384i
\(112\) 0 0
\(113\) 124.238i 1.09945i −0.835346 0.549724i \(-0.814733\pi\)
0.835346 0.549724i \(-0.185267\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −17.0000 + 5.91608i −0.145299 + 0.0505648i
\(118\) 0 0
\(119\) 141.986i 1.19316i
\(120\) 0 0
\(121\) −194.000 −1.60331
\(122\) 0 0
\(123\) 157.500 26.6224i 1.28049 0.216442i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 106.000 0.834646 0.417323 0.908758i \(-0.362968\pi\)
0.417323 + 0.908758i \(0.362968\pi\)
\(128\) 0 0
\(129\) 31.0000 + 183.398i 0.240310 + 1.42169i
\(130\) 0 0
\(131\) 70.9930i 0.541931i 0.962589 + 0.270965i \(0.0873429\pi\)
−0.962589 + 0.270965i \(0.912657\pi\)
\(132\) 0 0
\(133\) 88.0000 0.661654
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 53.2447i 0.388648i 0.980937 + 0.194324i \(0.0622512\pi\)
−0.980937 + 0.194324i \(0.937749\pi\)
\(138\) 0 0
\(139\) 127.000 0.913669 0.456835 0.889552i \(-0.348983\pi\)
0.456835 + 0.889552i \(0.348983\pi\)
\(140\) 0 0
\(141\) 105.000 17.7482i 0.744681 0.125874i
\(142\) 0 0
\(143\) 35.4965i 0.248227i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −7.50000 44.3706i −0.0510204 0.301841i
\(148\) 0 0
\(149\) 106.489i 0.714694i 0.933972 + 0.357347i \(0.116319\pi\)
−0.933972 + 0.357347i \(0.883681\pi\)
\(150\) 0 0
\(151\) −164.000 −1.08609 −0.543046 0.839703i \(-0.682729\pi\)
−0.543046 + 0.839703i \(0.682729\pi\)
\(152\) 0 0
\(153\) 52.5000 + 150.860i 0.343137 + 0.986013i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 38.0000 0.242038 0.121019 0.992650i \(-0.461384\pi\)
0.121019 + 0.992650i \(0.461384\pi\)
\(158\) 0 0
\(159\) −105.000 + 17.7482i −0.660377 + 0.111624i
\(160\) 0 0
\(161\) 283.972i 1.76380i
\(162\) 0 0
\(163\) −47.0000 −0.288344 −0.144172 0.989553i \(-0.546052\pi\)
−0.144172 + 0.989553i \(0.546052\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 177.482i 1.06277i 0.847131 + 0.531384i \(0.178328\pi\)
−0.847131 + 0.531384i \(0.821672\pi\)
\(168\) 0 0
\(169\) −165.000 −0.976331
\(170\) 0 0
\(171\) 93.5000 32.5384i 0.546784 0.190283i
\(172\) 0 0
\(173\) 70.9930i 0.410364i 0.978724 + 0.205182i \(0.0657786\pi\)
−0.978724 + 0.205182i \(0.934221\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 210.000 35.4965i 1.18644 0.200545i
\(178\) 0 0
\(179\) 17.7482i 0.0991522i 0.998770 + 0.0495761i \(0.0157870\pi\)
−0.998770 + 0.0495761i \(0.984213\pi\)
\(180\) 0 0
\(181\) −106.000 −0.585635 −0.292818 0.956168i \(-0.594593\pi\)
−0.292818 + 0.956168i \(0.594593\pi\)
\(182\) 0 0
\(183\) 8.00000 + 47.3286i 0.0437158 + 0.258626i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −315.000 −1.68449
\(188\) 0 0
\(189\) −104.000 189.315i −0.550265 1.00166i
\(190\) 0 0
\(191\) 248.475i 1.30092i −0.759541 0.650459i \(-0.774577\pi\)
0.759541 0.650459i \(-0.225423\pi\)
\(192\) 0 0
\(193\) −193.000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 35.4965i 0.180185i −0.995933 0.0900926i \(-0.971284\pi\)
0.995933 0.0900926i \(-0.0287163\pi\)
\(198\) 0 0
\(199\) 4.00000 0.0201005 0.0100503 0.999949i \(-0.496801\pi\)
0.0100503 + 0.999949i \(0.496801\pi\)
\(200\) 0 0
\(201\) 56.5000 + 334.259i 0.281095 + 1.66298i
\(202\) 0 0
\(203\) 283.972i 1.39888i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 105.000 + 301.720i 0.507246 + 1.45758i
\(208\) 0 0
\(209\) 195.231i 0.934118i
\(210\) 0 0
\(211\) −209.000 −0.990521 −0.495261 0.868744i \(-0.664927\pi\)
−0.495261 + 0.868744i \(0.664927\pi\)
\(212\) 0 0
\(213\) 315.000 53.2447i 1.47887 0.249975i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −368.000 −1.69585
\(218\) 0 0
\(219\) −50.5000 298.762i −0.230594 1.36421i
\(220\) 0 0
\(221\) 35.4965i 0.160618i
\(222\) 0 0
\(223\) 148.000 0.663677 0.331839 0.943336i \(-0.392331\pi\)
0.331839 + 0.943336i \(0.392331\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 116.000 0.506550 0.253275 0.967394i \(-0.418492\pi\)
0.253275 + 0.967394i \(0.418492\pi\)
\(230\) 0 0
\(231\) 420.000 70.9930i 1.81818 0.307329i
\(232\) 0 0
\(233\) 212.979i 0.914072i 0.889448 + 0.457036i \(0.151089\pi\)
−0.889448 + 0.457036i \(0.848911\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 34.0000 + 201.147i 0.143460 + 0.848720i
\(238\) 0 0
\(239\) 177.482i 0.742604i −0.928512 0.371302i \(-0.878911\pi\)
0.928512 0.371302i \(-0.121089\pi\)
\(240\) 0 0
\(241\) 59.0000 0.244813 0.122407 0.992480i \(-0.460939\pi\)
0.122407 + 0.992480i \(0.460939\pi\)
\(242\) 0 0
\(243\) −180.500 162.692i −0.742798 0.669515i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −22.0000 −0.0890688
\(248\) 0 0
\(249\) 52.5000 8.87412i 0.210843 0.0356390i
\(250\) 0 0
\(251\) 124.238i 0.494971i −0.968892 0.247485i \(-0.920396\pi\)
0.968892 0.247485i \(-0.0796042\pi\)
\(252\) 0 0
\(253\) −630.000 −2.49012
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 496.951i 1.93366i −0.255420 0.966830i \(-0.582214\pi\)
0.255420 0.966830i \(-0.417786\pi\)
\(258\) 0 0
\(259\) 128.000 0.494208
\(260\) 0 0
\(261\) −105.000 301.720i −0.402299 1.15602i
\(262\) 0 0
\(263\) 248.475i 0.944773i 0.881391 + 0.472387i \(0.156607\pi\)
−0.881391 + 0.472387i \(0.843393\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 157.500 26.6224i 0.589888 0.0997092i
\(268\) 0 0
\(269\) 354.965i 1.31957i −0.751454 0.659786i \(-0.770647\pi\)
0.751454 0.659786i \(-0.229353\pi\)
\(270\) 0 0
\(271\) 178.000 0.656827 0.328413 0.944534i \(-0.393486\pi\)
0.328413 + 0.944534i \(0.393486\pi\)
\(272\) 0 0
\(273\) 8.00000 + 47.3286i 0.0293040 + 0.173365i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −202.000 −0.729242 −0.364621 0.931156i \(-0.618801\pi\)
−0.364621 + 0.931156i \(0.618801\pi\)
\(278\) 0 0
\(279\) −391.000 + 136.070i −1.40143 + 0.487706i
\(280\) 0 0
\(281\) 70.9930i 0.252644i 0.991989 + 0.126322i \(0.0403173\pi\)
−0.991989 + 0.126322i \(0.959683\pi\)
\(282\) 0 0
\(283\) −197.000 −0.696113 −0.348057 0.937474i \(-0.613158\pi\)
−0.348057 + 0.937474i \(0.613158\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 425.958i 1.48417i
\(288\) 0 0
\(289\) −26.0000 −0.0899654
\(290\) 0 0
\(291\) 11.0000 + 65.0769i 0.0378007 + 0.223632i
\(292\) 0 0
\(293\) 425.958i 1.45378i −0.686754 0.726890i \(-0.740965\pi\)
0.686754 0.726890i \(-0.259035\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 420.000 230.727i 1.41414 0.776859i
\(298\) 0 0
\(299\) 70.9930i 0.237435i
\(300\) 0 0
\(301\) 496.000 1.64784
\(302\) 0 0
\(303\) −420.000 + 70.9930i −1.38614 + 0.234300i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 127.000 0.413681 0.206840 0.978375i \(-0.433682\pi\)
0.206840 + 0.978375i \(0.433682\pi\)
\(308\) 0 0
\(309\) 13.0000 + 76.9090i 0.0420712 + 0.248897i
\(310\) 0 0
\(311\) 283.972i 0.913093i 0.889700 + 0.456546i \(0.150914\pi\)
−0.889700 + 0.456546i \(0.849086\pi\)
\(312\) 0 0
\(313\) −58.0000 −0.185304 −0.0926518 0.995699i \(-0.529534\pi\)
−0.0926518 + 0.995699i \(0.529534\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 283.972i 0.895810i 0.894081 + 0.447905i \(0.147830\pi\)
−0.894081 + 0.447905i \(0.852170\pi\)
\(318\) 0 0
\(319\) 630.000 1.97492
\(320\) 0 0
\(321\) 52.5000 8.87412i 0.163551 0.0276452i
\(322\) 0 0
\(323\) 195.231i 0.604429i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −88.0000 520.615i −0.269113 1.59209i
\(328\) 0 0
\(329\) 283.972i 0.863136i
\(330\) 0 0
\(331\) −257.000 −0.776435 −0.388218 0.921568i \(-0.626909\pi\)
−0.388218 + 0.921568i \(0.626909\pi\)
\(332\) 0 0
\(333\) 136.000 47.3286i 0.408408 0.142128i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −127.000 −0.376855 −0.188427 0.982087i \(-0.560339\pi\)
−0.188427 + 0.982087i \(0.560339\pi\)
\(338\) 0 0
\(339\) −367.500 + 62.1188i −1.08407 + 0.183241i
\(340\) 0 0
\(341\) 816.419i 2.39419i
\(342\) 0 0
\(343\) 272.000 0.793003
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 372.713i 1.07410i −0.843550 0.537050i \(-0.819538\pi\)
0.843550 0.537050i \(-0.180462\pi\)
\(348\) 0 0
\(349\) −292.000 −0.836676 −0.418338 0.908291i \(-0.637387\pi\)
−0.418338 + 0.908291i \(0.637387\pi\)
\(350\) 0 0
\(351\) 26.0000 + 47.3286i 0.0740741 + 0.134839i
\(352\) 0 0
\(353\) 638.937i 1.81002i 0.425392 + 0.905009i \(0.360136\pi\)
−0.425392 + 0.905009i \(0.639864\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 420.000 70.9930i 1.17647 0.198860i
\(358\) 0 0
\(359\) 283.972i 0.791008i −0.918464 0.395504i \(-0.870570\pi\)
0.918464 0.395504i \(-0.129430\pi\)
\(360\) 0 0
\(361\) −240.000 −0.664820
\(362\) 0 0
\(363\) 97.0000 + 573.860i 0.267218 + 1.58088i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −668.000 −1.82016 −0.910082 0.414429i \(-0.863981\pi\)
−0.910082 + 0.414429i \(0.863981\pi\)
\(368\) 0 0
\(369\) −157.500 452.580i −0.426829 1.22650i
\(370\) 0 0
\(371\) 283.972i 0.765423i
\(372\) 0 0
\(373\) 242.000 0.648794 0.324397 0.945921i \(-0.394839\pi\)
0.324397 + 0.945921i \(0.394839\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 70.9930i 0.188310i
\(378\) 0 0
\(379\) −191.000 −0.503958 −0.251979 0.967733i \(-0.581081\pi\)
−0.251979 + 0.967733i \(0.581081\pi\)
\(380\) 0 0
\(381\) −53.0000 313.552i −0.139108 0.822972i
\(382\) 0 0
\(383\) 283.972i 0.741441i 0.928745 + 0.370720i \(0.120889\pi\)
−0.928745 + 0.370720i \(0.879111\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 527.000 183.398i 1.36176 0.473898i
\(388\) 0 0
\(389\) 603.440i 1.55126i −0.631188 0.775630i \(-0.717432\pi\)
0.631188 0.775630i \(-0.282568\pi\)
\(390\) 0 0
\(391\) −630.000 −1.61125
\(392\) 0 0
\(393\) 210.000 35.4965i 0.534351 0.0903218i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −772.000 −1.94458 −0.972292 0.233769i \(-0.924894\pi\)
−0.972292 + 0.233769i \(0.924894\pi\)
\(398\) 0 0
\(399\) −44.0000 260.308i −0.110276 0.652400i
\(400\) 0 0
\(401\) 266.224i 0.663899i 0.943297 + 0.331950i \(0.107706\pi\)
−0.943297 + 0.331950i \(0.892294\pi\)
\(402\) 0 0
\(403\) 92.0000 0.228288
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 283.972i 0.697719i
\(408\) 0 0
\(409\) 701.000 1.71394 0.856968 0.515369i \(-0.172345\pi\)
0.856968 + 0.515369i \(0.172345\pi\)
\(410\) 0 0
\(411\) 157.500 26.6224i 0.383212 0.0647746i
\(412\) 0 0
\(413\) 567.944i 1.37517i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −63.5000 375.671i −0.152278 0.900890i
\(418\) 0 0
\(419\) 301.720i 0.720096i −0.932934 0.360048i \(-0.882760\pi\)
0.932934 0.360048i \(-0.117240\pi\)
\(420\) 0 0
\(421\) −148.000 −0.351544 −0.175772 0.984431i \(-0.556242\pi\)
−0.175772 + 0.984431i \(0.556242\pi\)
\(422\) 0 0
\(423\) −105.000 301.720i −0.248227 0.713286i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 128.000 0.299766
\(428\) 0 0
\(429\) −105.000 + 17.7482i −0.244755 + 0.0413712i
\(430\) 0 0
\(431\) 212.979i 0.494151i −0.968996 0.247075i \(-0.920531\pi\)
0.968996 0.247075i \(-0.0794695\pi\)
\(432\) 0 0
\(433\) −463.000 −1.06928 −0.534642 0.845079i \(-0.679554\pi\)
−0.534642 + 0.845079i \(0.679554\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 390.461i 0.893504i
\(438\) 0 0
\(439\) 394.000 0.897494 0.448747 0.893659i \(-0.351870\pi\)
0.448747 + 0.893659i \(0.351870\pi\)
\(440\) 0 0
\(441\) −127.500 + 44.3706i −0.289116 + 0.100614i
\(442\) 0 0
\(443\) 656.685i 1.48236i 0.671307 + 0.741179i \(0.265733\pi\)
−0.671307 + 0.741179i \(0.734267\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 315.000 53.2447i 0.704698 0.119116i
\(448\) 0 0
\(449\) 124.238i 0.276699i 0.990383 + 0.138349i \(0.0441797\pi\)
−0.990383 + 0.138349i \(0.955820\pi\)
\(450\) 0 0
\(451\) 945.000 2.09534
\(452\) 0 0
\(453\) 82.0000 + 485.119i 0.181015 + 1.07090i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 509.000 1.11379 0.556893 0.830584i \(-0.311993\pi\)
0.556893 + 0.830584i \(0.311993\pi\)
\(458\) 0 0
\(459\) 420.000 230.727i 0.915033 0.502673i
\(460\) 0 0
\(461\) 212.979i 0.461993i −0.972955 0.230997i \(-0.925801\pi\)
0.972955 0.230997i \(-0.0741986\pi\)
\(462\) 0 0
\(463\) 64.0000 0.138229 0.0691145 0.997609i \(-0.477983\pi\)
0.0691145 + 0.997609i \(0.477983\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 638.937i 1.36817i 0.729401 + 0.684086i \(0.239799\pi\)
−0.729401 + 0.684086i \(0.760201\pi\)
\(468\) 0 0
\(469\) 904.000 1.92751
\(470\) 0 0
\(471\) −19.0000 112.406i −0.0403397 0.238653i
\(472\) 0 0
\(473\) 1100.39i 2.32641i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 105.000 + 301.720i 0.220126 + 0.632537i
\(478\) 0 0
\(479\) 248.475i 0.518738i −0.965778 0.259369i \(-0.916485\pi\)
0.965778 0.259369i \(-0.0835145\pi\)
\(480\) 0 0
\(481\) −32.0000 −0.0665281
\(482\) 0 0
\(483\) 840.000 141.986i 1.73913 0.293967i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 382.000 0.784394 0.392197 0.919881i \(-0.371715\pi\)
0.392197 + 0.919881i \(0.371715\pi\)
\(488\) 0 0
\(489\) 23.5000 + 139.028i 0.0480573 + 0.284311i
\(490\) 0 0
\(491\) 709.930i 1.44589i −0.690908 0.722943i \(-0.742789\pi\)
0.690908 0.722943i \(-0.257211\pi\)
\(492\) 0 0
\(493\) 630.000 1.27789
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 851.915i 1.71412i
\(498\) 0 0
\(499\) 94.0000 0.188377 0.0941884 0.995554i \(-0.469974\pi\)
0.0941884 + 0.995554i \(0.469974\pi\)
\(500\) 0 0
\(501\) 525.000 88.7412i 1.04790 0.177128i
\(502\) 0 0
\(503\) 780.923i 1.55253i −0.630407 0.776265i \(-0.717112\pi\)
0.630407 0.776265i \(-0.282888\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 82.5000 + 488.077i 0.162722 + 0.962676i
\(508\) 0 0
\(509\) 319.468i 0.627639i −0.949483 0.313820i \(-0.898391\pi\)
0.949483 0.313820i \(-0.101609\pi\)
\(510\) 0 0
\(511\) −808.000 −1.58121
\(512\) 0 0
\(513\) −143.000 260.308i −0.278752 0.507422i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 630.000 1.21857
\(518\) 0 0
\(519\) 210.000 35.4965i 0.404624 0.0683940i
\(520\) 0 0
\(521\) 798.671i 1.53296i 0.642270 + 0.766479i \(0.277993\pi\)
−0.642270 + 0.766479i \(0.722007\pi\)
\(522\) 0 0
\(523\) −467.000 −0.892925 −0.446463 0.894802i \(-0.647316\pi\)
−0.446463 + 0.894802i \(0.647316\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 816.419i 1.54918i
\(528\) 0 0
\(529\) −731.000 −1.38185
\(530\) 0 0
\(531\) −210.000 603.440i −0.395480 1.13642i
\(532\) 0 0
\(533\) 106.489i 0.199793i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 52.5000 8.87412i 0.0977654 0.0165254i
\(538\) 0 0
\(539\) 266.224i 0.493921i
\(540\) 0 0
\(541\) −976.000 −1.80407 −0.902033 0.431667i \(-0.857926\pi\)
−0.902033 + 0.431667i \(0.857926\pi\)
\(542\) 0 0
\(543\) 53.0000 + 313.552i 0.0976059 + 0.577444i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −779.000 −1.42413 −0.712066 0.702113i \(-0.752240\pi\)
−0.712066 + 0.702113i \(0.752240\pi\)
\(548\) 0 0
\(549\) 136.000 47.3286i 0.247723 0.0862088i
\(550\) 0 0
\(551\) 390.461i 0.708641i
\(552\) 0 0
\(553\) 544.000 0.983725
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 745.426i 1.33829i −0.743133 0.669144i \(-0.766661\pi\)
0.743133 0.669144i \(-0.233339\pi\)
\(558\) 0 0
\(559\) −124.000 −0.221825
\(560\) 0 0
\(561\) 157.500 + 931.783i 0.280749 + 1.66093i
\(562\) 0 0
\(563\) 70.9930i 0.126098i −0.998010 0.0630488i \(-0.979918\pi\)
0.998010 0.0630488i \(-0.0200824\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −508.000 + 402.293i −0.895944 + 0.709512i
\(568\) 0 0
\(569\) 976.153i 1.71556i 0.514018 + 0.857780i \(0.328157\pi\)
−0.514018 + 0.857780i \(0.671843\pi\)
\(570\) 0 0
\(571\) 286.000 0.500876 0.250438 0.968133i \(-0.419425\pi\)
0.250438 + 0.968133i \(0.419425\pi\)
\(572\) 0 0
\(573\) −735.000 + 124.238i −1.28272 + 0.216820i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −487.000 −0.844021 −0.422010 0.906591i \(-0.638675\pi\)
−0.422010 + 0.906591i \(0.638675\pi\)
\(578\) 0 0
\(579\) 96.5000 + 570.902i 0.166667 + 0.986013i
\(580\) 0 0
\(581\) 141.986i 0.244382i
\(582\) 0 0
\(583\) −630.000 −1.08062
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 585.692i 0.997772i −0.866668 0.498886i \(-0.833743\pi\)
0.866668 0.498886i \(-0.166257\pi\)
\(588\) 0 0
\(589\) −506.000 −0.859083
\(590\) 0 0
\(591\) −105.000 + 17.7482i −0.177665 + 0.0300309i
\(592\) 0 0
\(593\) 88.7412i 0.149648i −0.997197 0.0748239i \(-0.976161\pi\)
0.997197 0.0748239i \(-0.0238395\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.00000 11.8322i −0.00335008 0.0198194i
\(598\) 0 0
\(599\) 638.937i 1.06667i 0.845903 + 0.533336i \(0.179062\pi\)
−0.845903 + 0.533336i \(0.820938\pi\)
\(600\) 0 0
\(601\) −271.000 −0.450915 −0.225458 0.974253i \(-0.572388\pi\)
−0.225458 + 0.974253i \(0.572388\pi\)
\(602\) 0 0
\(603\) 960.500 334.259i 1.59287 0.554326i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −404.000 −0.665568 −0.332784 0.943003i \(-0.607988\pi\)
−0.332784 + 0.943003i \(0.607988\pi\)
\(608\) 0 0
\(609\) −840.000 + 141.986i −1.37931 + 0.233146i
\(610\) 0 0
\(611\) 70.9930i 0.116191i
\(612\) 0 0
\(613\) −34.0000 −0.0554649 −0.0277325 0.999615i \(-0.508829\pi\)
−0.0277325 + 0.999615i \(0.508829\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 425.958i 0.690369i −0.938535 0.345185i \(-0.887816\pi\)
0.938535 0.345185i \(-0.112184\pi\)
\(618\) 0 0
\(619\) −1046.00 −1.68982 −0.844911 0.534907i \(-0.820347\pi\)
−0.844911 + 0.534907i \(0.820347\pi\)
\(620\) 0 0
\(621\) 840.000 461.454i 1.35266 0.743082i
\(622\) 0 0
\(623\) 425.958i 0.683720i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 577.500 97.6153i 0.921053 0.155686i
\(628\) 0 0
\(629\) 283.972i 0.451466i
\(630\) 0 0
\(631\) 106.000 0.167987 0.0839937 0.996466i \(-0.473232\pi\)
0.0839937 + 0.996466i \(0.473232\pi\)
\(632\) 0 0
\(633\) 104.500 + 618.230i 0.165087 + 0.976667i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 30.0000 0.0470958
\(638\) 0 0
\(639\) −315.000 905.160i −0.492958 1.41653i
\(640\) 0 0
\(641\) 212.979i 0.332260i 0.986104 + 0.166130i \(0.0531272\pi\)
−0.986104 + 0.166130i \(0.946873\pi\)
\(642\) 0 0
\(643\) −566.000 −0.880249 −0.440124 0.897937i \(-0.645066\pi\)
−0.440124 + 0.897937i \(0.645066\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 496.951i 0.768085i 0.923316 + 0.384042i \(0.125468\pi\)
−0.923316 + 0.384042i \(0.874532\pi\)
\(648\) 0 0
\(649\) 1260.00 1.94145
\(650\) 0 0
\(651\) 184.000 + 1088.56i 0.282642 + 1.67213i
\(652\) 0 0
\(653\) 496.951i 0.761027i −0.924775 0.380514i \(-0.875747\pi\)
0.924775 0.380514i \(-0.124253\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −858.500 + 298.762i −1.30670 + 0.454737i
\(658\) 0 0
\(659\) 550.195i 0.834894i 0.908701 + 0.417447i \(0.137075\pi\)
−0.908701 + 0.417447i \(0.862925\pi\)
\(660\) 0 0
\(661\) −298.000 −0.450832 −0.225416 0.974263i \(-0.572374\pi\)
−0.225416 + 0.974263i \(0.572374\pi\)
\(662\) 0 0
\(663\) −105.000 + 17.7482i −0.158371 + 0.0267696i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1260.00 1.88906
\(668\) 0 0
\(669\) −74.0000 437.790i −0.110613 0.654394i
\(670\) 0 0
\(671\) 283.972i 0.423207i
\(672\) 0 0
\(673\) −274.000 −0.407132 −0.203566 0.979061i \(-0.565253\pi\)
−0.203566 + 0.979061i \(0.565253\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 993.901i 1.46810i 0.679097 + 0.734048i \(0.262371\pi\)
−0.679097 + 0.734048i \(0.737629\pi\)
\(678\) 0 0
\(679\) 176.000 0.259205
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 124.238i 0.181900i 0.995855 + 0.0909500i \(0.0289903\pi\)
−0.995855 + 0.0909500i \(0.971010\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −58.0000 343.133i −0.0844250 0.499465i
\(688\) 0 0
\(689\) 70.9930i 0.103038i
\(690\) 0 0
\(691\) 373.000 0.539797 0.269899 0.962889i \(-0.413010\pi\)
0.269899 + 0.962889i \(0.413010\pi\)
\(692\) 0 0
\(693\) −420.000 1206.88i −0.606061 1.74153i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 945.000 1.35581
\(698\) 0 0
\(699\) 630.000 106.489i 0.901288 0.152345i
\(700\) 0 0
\(701\) 674.433i 0.962101i −0.876693 0.481051i \(-0.840255\pi\)
0.876693 0.481051i \(-0.159745\pi\)
\(702\) 0 0
\(703\) 176.000 0.250356
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1135.89i 1.60663i
\(708\) 0 0
\(709\) −184.000 −0.259520 −0.129760 0.991545i \(-0.541421\pi\)
−0.129760 + 0.991545i \(0.541421\pi\)
\(710\) 0 0
\(711\) 578.000 201.147i 0.812940 0.282907i
\(712\) 0 0
\(713\) 1632.84i 2.29010i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −525.000 + 88.7412i −0.732218 + 0.123767i
\(718\) 0 0
\(719\) 638.937i 0.888646i 0.895867 + 0.444323i \(0.146556\pi\)
−0.895867 + 0.444323i \(0.853444\pi\)
\(720\) 0 0
\(721\) 208.000 0.288488
\(722\) 0 0
\(723\) −29.5000 174.524i −0.0408022 0.241389i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −758.000 −1.04264 −0.521320 0.853361i \(-0.674560\pi\)
−0.521320 + 0.853361i \(0.674560\pi\)
\(728\) 0 0
\(729\) −391.000 + 615.272i −0.536351 + 0.843995i
\(730\) 0 0
\(731\) 1100.39i 1.50532i
\(732\) 0 0
\(733\) 26.0000 0.0354707 0.0177353 0.999843i \(-0.494354\pi\)
0.0177353 + 0.999843i \(0.494354\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2005.55i 2.72124i
\(738\) 0 0
\(739\) −1298.00 −1.75643 −0.878214 0.478268i \(-0.841265\pi\)
−0.878214 + 0.478268i \(0.841265\pi\)
\(740\) 0 0
\(741\) 11.0000 + 65.0769i 0.0148448 + 0.0878230i
\(742\) 0 0
\(743\) 532.447i 0.716618i 0.933603 + 0.358309i \(0.116647\pi\)
−0.933603 + 0.358309i \(0.883353\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −52.5000 150.860i −0.0702811 0.201955i
\(748\) 0 0
\(749\) 141.986i 0.189567i
\(750\) 0 0
\(751\) 478.000 0.636485 0.318242 0.948009i \(-0.396907\pi\)
0.318242 + 0.948009i \(0.396907\pi\)
\(752\) 0 0
\(753\) −367.500 + 62.1188i −0.488048 + 0.0824951i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1072.00 −1.41612 −0.708058 0.706154i \(-0.750429\pi\)
−0.708058 + 0.706154i \(0.750429\pi\)
\(758\) 0 0
\(759\) 315.000 + 1863.57i 0.415020 + 2.45529i
\(760\) 0 0
\(761\) 550.195i 0.722990i −0.932374 0.361495i \(-0.882266\pi\)
0.932374 0.361495i \(-0.117734\pi\)
\(762\) 0 0
\(763\) −1408.00 −1.84535
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 141.986i 0.185119i
\(768\) 0 0
\(769\) −169.000 −0.219766 −0.109883 0.993945i \(-0.535048\pi\)
−0.109883 + 0.993945i \(0.535048\pi\)
\(770\) 0 0
\(771\) −1470.00 + 248.475i −1.90661 + 0.322277i
\(772\) 0 0
\(773\) 461.454i 0.596965i 0.954415 + 0.298483i \(0.0964805\pi\)
−0.954415 + 0.298483i \(0.903519\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −64.0000 378.629i −0.0823681 0.487296i
\(778\) 0 0
\(779\) 585.692i 0.751851i
\(780\) 0 0
\(781\) 1890.00 2.41997
\(782\) 0 0
\(783\) −840.000 + 461.454i −1.07280 + 0.589341i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −878.000 −1.11563 −0.557814 0.829966i \(-0.688360\pi\)
−0.557814 + 0.829966i \(0.688360\pi\)
\(788\) 0 0
\(789\) 735.000 124.238i 0.931559 0.157462i
\(790\) 0 0
\(791\) 993.901i 1.25651i
\(792\) 0 0
\(793\) −32.0000 −0.0403531
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 816.419i 1.02437i 0.858877 + 0.512183i \(0.171163\pi\)
−0.858877 + 0.512183i \(0.828837\pi\)
\(798\) 0 0
\(799\) 630.000 0.788486
\(800\) 0 0
\(801\)