Properties

Label 1200.3.bg.h.193.1
Level $1200$
Weight $3$
Character 1200.193
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 193.1
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1200.193
Dual form 1200.3.bg.h.1057.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.22474 - 1.22474i) q^{3} +(2.44949 - 2.44949i) q^{7} +3.00000i q^{9} +O(q^{10})\) \(q+(-1.22474 - 1.22474i) q^{3} +(2.44949 - 2.44949i) q^{7} +3.00000i q^{9} -6.00000 q^{11} +(12.2474 + 12.2474i) q^{13} +(-14.6969 + 14.6969i) q^{17} -10.0000i q^{19} -6.00000 q^{21} +(-29.3939 - 29.3939i) q^{23} +(3.67423 - 3.67423i) q^{27} -48.0000i q^{29} +26.0000 q^{31} +(7.34847 + 7.34847i) q^{33} +(-31.8434 + 31.8434i) q^{37} -30.0000i q^{39} +30.0000 q^{41} +(-29.3939 - 29.3939i) q^{43} +(-14.6969 + 14.6969i) q^{47} +37.0000i q^{49} +36.0000 q^{51} +(-14.6969 - 14.6969i) q^{53} +(-12.2474 + 12.2474i) q^{57} +78.0000i q^{59} +2.00000 q^{61} +(7.34847 + 7.34847i) q^{63} +(-63.6867 + 63.6867i) q^{67} +72.0000i q^{69} -120.000 q^{71} +(-83.2827 - 83.2827i) q^{73} +(-14.6969 + 14.6969i) q^{77} -74.0000i q^{79} -9.00000 q^{81} +(44.0908 + 44.0908i) q^{83} +(-58.7878 + 58.7878i) q^{87} -150.000i q^{89} +60.0000 q^{91} +(-31.8434 - 31.8434i) q^{93} +(-4.89898 + 4.89898i) q^{97} -18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 24q^{11} - 24q^{21} + 104q^{31} + 120q^{41} + 144q^{51} + 8q^{61} - 480q^{71} - 36q^{81} + 240q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\)

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
\(3\) −1.22474 1.22474i −0.408248 0.408248i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.44949 2.44949i 0.349927 0.349927i −0.510155 0.860082i \(-0.670412\pi\)
0.860082 + 0.510155i \(0.170412\pi\)
\(8\) 0 0
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) −6.00000 −0.545455 −0.272727 0.962091i \(-0.587926\pi\)
−0.272727 + 0.962091i \(0.587926\pi\)
\(12\) 0 0
\(13\) 12.2474 + 12.2474i 0.942111 + 0.942111i 0.998414 0.0563023i \(-0.0179311\pi\)
−0.0563023 + 0.998414i \(0.517931\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −14.6969 + 14.6969i −0.864526 + 0.864526i −0.991860 0.127334i \(-0.959358\pi\)
0.127334 + 0.991860i \(0.459358\pi\)
\(18\) 0 0
\(19\) 10.0000i 0.526316i −0.964753 0.263158i \(-0.915236\pi\)
0.964753 0.263158i \(-0.0847640\pi\)
\(20\) 0 0
\(21\) −6.00000 −0.285714
\(22\) 0 0
\(23\) −29.3939 29.3939i −1.27799 1.27799i −0.941788 0.336206i \(-0.890856\pi\)
−0.336206 0.941788i \(-0.609144\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.67423 3.67423i 0.136083 0.136083i
\(28\) 0 0
\(29\) 48.0000i 1.65517i −0.561339 0.827586i \(-0.689713\pi\)
0.561339 0.827586i \(-0.310287\pi\)
\(30\) 0 0
\(31\) 26.0000 0.838710 0.419355 0.907822i \(-0.362256\pi\)
0.419355 + 0.907822i \(0.362256\pi\)
\(32\) 0 0
\(33\) 7.34847 + 7.34847i 0.222681 + 0.222681i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −31.8434 + 31.8434i −0.860632 + 0.860632i −0.991411 0.130780i \(-0.958252\pi\)
0.130780 + 0.991411i \(0.458252\pi\)
\(38\) 0 0
\(39\) 30.0000i 0.769231i
\(40\) 0 0
\(41\) 30.0000 0.731707 0.365854 0.930672i \(-0.380777\pi\)
0.365854 + 0.930672i \(0.380777\pi\)
\(42\) 0 0
\(43\) −29.3939 29.3939i −0.683579 0.683579i 0.277226 0.960805i \(-0.410585\pi\)
−0.960805 + 0.277226i \(0.910585\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −14.6969 + 14.6969i −0.312701 + 0.312701i −0.845955 0.533254i \(-0.820969\pi\)
0.533254 + 0.845955i \(0.320969\pi\)
\(48\) 0 0
\(49\) 37.0000i 0.755102i
\(50\) 0 0
\(51\) 36.0000 0.705882
\(52\) 0 0
\(53\) −14.6969 14.6969i −0.277301 0.277301i 0.554730 0.832031i \(-0.312822\pi\)
−0.832031 + 0.554730i \(0.812822\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −12.2474 + 12.2474i −0.214868 + 0.214868i
\(58\) 0 0
\(59\) 78.0000i 1.32203i 0.750371 + 0.661017i \(0.229875\pi\)
−0.750371 + 0.661017i \(0.770125\pi\)
\(60\) 0 0
\(61\) 2.00000 0.0327869 0.0163934 0.999866i \(-0.494782\pi\)
0.0163934 + 0.999866i \(0.494782\pi\)
\(62\) 0 0
\(63\) 7.34847 + 7.34847i 0.116642 + 0.116642i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −63.6867 + 63.6867i −0.950548 + 0.950548i −0.998834 0.0482853i \(-0.984624\pi\)
0.0482853 + 0.998834i \(0.484624\pi\)
\(68\) 0 0
\(69\) 72.0000i 1.04348i
\(70\) 0 0
\(71\) −120.000 −1.69014 −0.845070 0.534655i \(-0.820442\pi\)
−0.845070 + 0.534655i \(0.820442\pi\)
\(72\) 0 0
\(73\) −83.2827 83.2827i −1.14086 1.14086i −0.988293 0.152565i \(-0.951247\pi\)
−0.152565 0.988293i \(-0.548753\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −14.6969 + 14.6969i −0.190869 + 0.190869i
\(78\) 0 0
\(79\) 74.0000i 0.936709i −0.883541 0.468354i \(-0.844847\pi\)
0.883541 0.468354i \(-0.155153\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) 0 0
\(83\) 44.0908 + 44.0908i 0.531215 + 0.531215i 0.920934 0.389719i \(-0.127428\pi\)
−0.389719 + 0.920934i \(0.627428\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −58.7878 + 58.7878i −0.675721 + 0.675721i
\(88\) 0 0
\(89\) 150.000i 1.68539i −0.538389 0.842697i \(-0.680967\pi\)
0.538389 0.842697i \(-0.319033\pi\)
\(90\) 0 0
\(91\) 60.0000 0.659341
\(92\) 0 0
\(93\) −31.8434 31.8434i −0.342402 0.342402i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.89898 + 4.89898i −0.0505049 + 0.0505049i −0.731908 0.681403i \(-0.761370\pi\)
0.681403 + 0.731908i \(0.261370\pi\)
\(98\) 0 0
\(99\) 18.0000i 0.181818i
\(100\) 0 0
\(101\) −12.0000 −0.118812 −0.0594059 0.998234i \(-0.518921\pi\)
−0.0594059 + 0.998234i \(0.518921\pi\)
\(102\) 0 0
\(103\) 41.6413 + 41.6413i 0.404285 + 0.404285i 0.879740 0.475455i \(-0.157717\pi\)
−0.475455 + 0.879740i \(0.657717\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −102.879 + 102.879i −0.961482 + 0.961482i −0.999285 0.0378032i \(-0.987964\pi\)
0.0378032 + 0.999285i \(0.487964\pi\)
\(108\) 0 0
\(109\) 74.0000i 0.678899i 0.940624 + 0.339450i \(0.110241\pi\)
−0.940624 + 0.339450i \(0.889759\pi\)
\(110\) 0 0
\(111\) 78.0000 0.702703
\(112\) 0 0
\(113\) −132.272 132.272i −1.17055 1.17055i −0.982078 0.188475i \(-0.939646\pi\)
−0.188475 0.982078i \(-0.560354\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −36.7423 + 36.7423i −0.314037 + 0.314037i
\(118\) 0 0
\(119\) 72.0000i 0.605042i
\(120\) 0 0
\(121\) −85.0000 −0.702479
\(122\) 0 0
\(123\) −36.7423 36.7423i −0.298718 0.298718i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −95.5301 + 95.5301i −0.752206 + 0.752206i −0.974890 0.222685i \(-0.928518\pi\)
0.222685 + 0.974890i \(0.428518\pi\)
\(128\) 0 0
\(129\) 72.0000i 0.558140i
\(130\) 0 0
\(131\) −102.000 −0.778626 −0.389313 0.921106i \(-0.627288\pi\)
−0.389313 + 0.921106i \(0.627288\pi\)
\(132\) 0 0
\(133\) −24.4949 24.4949i −0.184172 0.184172i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −44.0908 + 44.0908i −0.321831 + 0.321831i −0.849469 0.527638i \(-0.823078\pi\)
0.527638 + 0.849469i \(0.323078\pi\)
\(138\) 0 0
\(139\) 122.000i 0.877698i 0.898561 + 0.438849i \(0.144614\pi\)
−0.898561 + 0.438849i \(0.855386\pi\)
\(140\) 0 0
\(141\) 36.0000 0.255319
\(142\) 0 0
\(143\) −73.4847 73.4847i −0.513879 0.513879i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 45.3156 45.3156i 0.308269 0.308269i
\(148\) 0 0
\(149\) 36.0000i 0.241611i −0.992676 0.120805i \(-0.961452\pi\)
0.992676 0.120805i \(-0.0385477\pi\)
\(150\) 0 0
\(151\) −70.0000 −0.463576 −0.231788 0.972766i \(-0.574458\pi\)
−0.231788 + 0.972766i \(0.574458\pi\)
\(152\) 0 0
\(153\) −44.0908 44.0908i −0.288175 0.288175i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −139.621 + 139.621i −0.889305 + 0.889305i −0.994456 0.105151i \(-0.966467\pi\)
0.105151 + 0.994456i \(0.466467\pi\)
\(158\) 0 0
\(159\) 36.0000i 0.226415i
\(160\) 0 0
\(161\) −144.000 −0.894410
\(162\) 0 0
\(163\) −97.9796 97.9796i −0.601102 0.601102i 0.339503 0.940605i \(-0.389741\pi\)
−0.940605 + 0.339503i \(0.889741\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −29.3939 + 29.3939i −0.176011 + 0.176011i −0.789614 0.613603i \(-0.789719\pi\)
0.613603 + 0.789614i \(0.289719\pi\)
\(168\) 0 0
\(169\) 131.000i 0.775148i
\(170\) 0 0
\(171\) 30.0000 0.175439
\(172\) 0 0
\(173\) 191.060 + 191.060i 1.10439 + 1.10439i 0.993874 + 0.110520i \(0.0352517\pi\)
0.110520 + 0.993874i \(0.464748\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 95.5301 95.5301i 0.539718 0.539718i
\(178\) 0 0
\(179\) 222.000i 1.24022i −0.784514 0.620112i \(-0.787087\pi\)
0.784514 0.620112i \(-0.212913\pi\)
\(180\) 0 0
\(181\) 190.000 1.04972 0.524862 0.851187i \(-0.324117\pi\)
0.524862 + 0.851187i \(0.324117\pi\)
\(182\) 0 0
\(183\) −2.44949 2.44949i −0.0133852 0.0133852i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 88.1816 88.1816i 0.471560 0.471560i
\(188\) 0 0
\(189\) 18.0000i 0.0952381i
\(190\) 0 0
\(191\) −204.000 −1.06806 −0.534031 0.845465i \(-0.679324\pi\)
−0.534031 + 0.845465i \(0.679324\pi\)
\(192\) 0 0
\(193\) 68.5857 + 68.5857i 0.355366 + 0.355366i 0.862102 0.506735i \(-0.169148\pi\)
−0.506735 + 0.862102i \(0.669148\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −205.757 + 205.757i −1.04445 + 1.04445i −0.0454876 + 0.998965i \(0.514484\pi\)
−0.998965 + 0.0454876i \(0.985516\pi\)
\(198\) 0 0
\(199\) 46.0000i 0.231156i 0.993298 + 0.115578i \(0.0368720\pi\)
−0.993298 + 0.115578i \(0.963128\pi\)
\(200\) 0 0
\(201\) 156.000 0.776119
\(202\) 0 0
\(203\) −117.576 117.576i −0.579190 0.579190i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 88.1816 88.1816i 0.425998 0.425998i
\(208\) 0 0
\(209\) 60.0000i 0.287081i
\(210\) 0 0
\(211\) 310.000 1.46919 0.734597 0.678504i \(-0.237371\pi\)
0.734597 + 0.678504i \(0.237371\pi\)
\(212\) 0 0
\(213\) 146.969 + 146.969i 0.689997 + 0.689997i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 63.6867 63.6867i 0.293487 0.293487i
\(218\) 0 0
\(219\) 204.000i 0.931507i
\(220\) 0 0
\(221\) −360.000 −1.62896
\(222\) 0 0
\(223\) −183.712 183.712i −0.823819 0.823819i 0.162834 0.986653i \(-0.447936\pi\)
−0.986653 + 0.162834i \(0.947936\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 73.4847 73.4847i 0.323721 0.323721i −0.526472 0.850193i \(-0.676485\pi\)
0.850193 + 0.526472i \(0.176485\pi\)
\(228\) 0 0
\(229\) 242.000i 1.05677i −0.849005 0.528384i \(-0.822798\pi\)
0.849005 0.528384i \(-0.177202\pi\)
\(230\) 0 0
\(231\) 36.0000 0.155844
\(232\) 0 0
\(233\) −73.4847 73.4847i −0.315385 0.315385i 0.531607 0.846991i \(-0.321589\pi\)
−0.846991 + 0.531607i \(0.821589\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −90.6311 + 90.6311i −0.382410 + 0.382410i
\(238\) 0 0
\(239\) 324.000i 1.35565i −0.735224 0.677824i \(-0.762923\pi\)
0.735224 0.677824i \(-0.237077\pi\)
\(240\) 0 0
\(241\) 398.000 1.65145 0.825726 0.564071i \(-0.190766\pi\)
0.825726 + 0.564071i \(0.190766\pi\)
\(242\) 0 0
\(243\) 11.0227 + 11.0227i 0.0453609 + 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 122.474 122.474i 0.495848 0.495848i
\(248\) 0 0
\(249\) 108.000i 0.433735i
\(250\) 0 0
\(251\) 162.000 0.645418 0.322709 0.946498i \(-0.395406\pi\)
0.322709 + 0.946498i \(0.395406\pi\)
\(252\) 0 0
\(253\) 176.363 + 176.363i 0.697088 + 0.697088i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 102.879 102.879i 0.400306 0.400306i −0.478035 0.878341i \(-0.658651\pi\)
0.878341 + 0.478035i \(0.158651\pi\)
\(258\) 0 0
\(259\) 156.000i 0.602317i
\(260\) 0 0
\(261\) 144.000 0.551724
\(262\) 0 0
\(263\) 117.576 + 117.576i 0.447055 + 0.447055i 0.894374 0.447319i \(-0.147621\pi\)
−0.447319 + 0.894374i \(0.647621\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −183.712 + 183.712i −0.688059 + 0.688059i
\(268\) 0 0
\(269\) 48.0000i 0.178439i 0.996012 + 0.0892193i \(0.0284372\pi\)
−0.996012 + 0.0892193i \(0.971563\pi\)
\(270\) 0 0
\(271\) 46.0000 0.169742 0.0848708 0.996392i \(-0.472952\pi\)
0.0848708 + 0.996392i \(0.472952\pi\)
\(272\) 0 0
\(273\) −73.4847 73.4847i −0.269175 0.269175i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 80.8332 80.8332i 0.291816 0.291816i −0.545981 0.837798i \(-0.683843\pi\)
0.837798 + 0.545981i \(0.183843\pi\)
\(278\) 0 0
\(279\) 78.0000i 0.279570i
\(280\) 0 0
\(281\) −414.000 −1.47331 −0.736655 0.676269i \(-0.763596\pi\)
−0.736655 + 0.676269i \(0.763596\pi\)
\(282\) 0 0
\(283\) 279.242 + 279.242i 0.986720 + 0.986720i 0.999913 0.0131927i \(-0.00419950\pi\)
−0.0131927 + 0.999913i \(0.504199\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 73.4847 73.4847i 0.256044 0.256044i
\(288\) 0 0
\(289\) 143.000i 0.494810i
\(290\) 0 0
\(291\) 12.0000 0.0412371
\(292\) 0 0
\(293\) 235.151 + 235.151i 0.802563 + 0.802563i 0.983496 0.180932i \(-0.0579115\pi\)
−0.180932 + 0.983496i \(0.557912\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −22.0454 + 22.0454i −0.0742270 + 0.0742270i
\(298\) 0 0
\(299\) 720.000i 2.40803i
\(300\) 0 0
\(301\) −144.000 −0.478405
\(302\) 0 0
\(303\) 14.6969 + 14.6969i 0.0485047 + 0.0485047i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 44.0908 44.0908i 0.143618 0.143618i −0.631642 0.775260i \(-0.717619\pi\)
0.775260 + 0.631642i \(0.217619\pi\)
\(308\) 0 0
\(309\) 102.000i 0.330097i
\(310\) 0 0
\(311\) −204.000 −0.655949 −0.327974 0.944687i \(-0.606366\pi\)
−0.327974 + 0.944687i \(0.606366\pi\)
\(312\) 0 0
\(313\) −372.322 372.322i −1.18953 1.18953i −0.977198 0.212331i \(-0.931895\pi\)
−0.212331 0.977198i \(-0.568105\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 352.727 352.727i 1.11270 1.11270i 0.119918 0.992784i \(-0.461737\pi\)
0.992784 0.119918i \(-0.0382632\pi\)
\(318\) 0 0
\(319\) 288.000i 0.902821i
\(320\) 0 0
\(321\) 252.000 0.785047
\(322\) 0 0
\(323\) 146.969 + 146.969i 0.455014 + 0.455014i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 90.6311 90.6311i 0.277159 0.277159i
\(328\) 0 0
\(329\) 72.0000i 0.218845i
\(330\) 0 0
\(331\) −542.000 −1.63746 −0.818731 0.574177i \(-0.805322\pi\)
−0.818731 + 0.574177i \(0.805322\pi\)
\(332\) 0 0
\(333\) −95.5301 95.5301i −0.286877 0.286877i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 347.828 347.828i 1.03213 1.03213i 0.0326628 0.999466i \(-0.489601\pi\)
0.999466 0.0326628i \(-0.0103987\pi\)
\(338\) 0 0
\(339\) 324.000i 0.955752i
\(340\) 0 0
\(341\) −156.000 −0.457478
\(342\) 0 0
\(343\) 210.656 + 210.656i 0.614158 + 0.614158i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 191.060 191.060i 0.550606 0.550606i −0.376010 0.926616i \(-0.622704\pi\)
0.926616 + 0.376010i \(0.122704\pi\)
\(348\) 0 0
\(349\) 358.000i 1.02579i 0.858452 + 0.512894i \(0.171427\pi\)
−0.858452 + 0.512894i \(0.828573\pi\)
\(350\) 0 0
\(351\) 90.0000 0.256410
\(352\) 0 0
\(353\) −426.211 426.211i −1.20740 1.20740i −0.971867 0.235530i \(-0.924317\pi\)
−0.235530 0.971867i \(-0.575683\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 88.1816 88.1816i 0.247007 0.247007i
\(358\) 0 0
\(359\) 132.000i 0.367688i −0.982955 0.183844i \(-0.941146\pi\)
0.982955 0.183844i \(-0.0588541\pi\)
\(360\) 0 0
\(361\) 261.000 0.722992
\(362\) 0 0
\(363\) 104.103 + 104.103i 0.286786 + 0.286786i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −120.025 + 120.025i −0.327044 + 0.327044i −0.851461 0.524418i \(-0.824283\pi\)
0.524418 + 0.851461i \(0.324283\pi\)
\(368\) 0 0
\(369\) 90.0000i 0.243902i
\(370\) 0 0
\(371\) −72.0000 −0.194070
\(372\) 0 0
\(373\) 409.065 + 409.065i 1.09669 + 1.09669i 0.994795 + 0.101893i \(0.0324900\pi\)
0.101893 + 0.994795i \(0.467510\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 587.878 587.878i 1.55936 1.55936i
\(378\) 0 0
\(379\) 26.0000i 0.0686016i 0.999412 + 0.0343008i \(0.0109204\pi\)
−0.999412 + 0.0343008i \(0.989080\pi\)
\(380\) 0 0
\(381\) 234.000 0.614173
\(382\) 0 0
\(383\) 426.211 + 426.211i 1.11282 + 1.11282i 0.992767 + 0.120056i \(0.0383074\pi\)
0.120056 + 0.992767i \(0.461693\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 88.1816 88.1816i 0.227860 0.227860i
\(388\) 0 0
\(389\) 72.0000i 0.185090i −0.995709 0.0925450i \(-0.970500\pi\)
0.995709 0.0925450i \(-0.0295002\pi\)
\(390\) 0 0
\(391\) 864.000 2.20972
\(392\) 0 0
\(393\) 124.924 + 124.924i 0.317873 + 0.317873i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −227.803 + 227.803i −0.573810 + 0.573810i −0.933191 0.359381i \(-0.882988\pi\)
0.359381 + 0.933191i \(0.382988\pi\)
\(398\) 0 0
\(399\) 60.0000i 0.150376i
\(400\) 0 0
\(401\) −414.000 −1.03242 −0.516209 0.856462i \(-0.672657\pi\)
−0.516209 + 0.856462i \(0.672657\pi\)
\(402\) 0 0
\(403\) 318.434 + 318.434i 0.790158 + 0.790158i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 191.060 191.060i 0.469435 0.469435i
\(408\) 0 0
\(409\) 482.000i 1.17848i −0.807957 0.589242i \(-0.799426\pi\)
0.807957 0.589242i \(-0.200574\pi\)
\(410\) 0 0
\(411\) 108.000 0.262774
\(412\) 0 0
\(413\) 191.060 + 191.060i 0.462615 + 0.462615i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 149.419 149.419i 0.358319 0.358319i
\(418\) 0 0
\(419\) 126.000i 0.300716i −0.988632 0.150358i \(-0.951957\pi\)
0.988632 0.150358i \(-0.0480426\pi\)
\(420\) 0 0
\(421\) 430.000 1.02138 0.510689 0.859766i \(-0.329390\pi\)
0.510689 + 0.859766i \(0.329390\pi\)
\(422\) 0 0
\(423\) −44.0908 44.0908i −0.104234 0.104234i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4.89898 4.89898i 0.0114730 0.0114730i
\(428\) 0 0
\(429\) 180.000i 0.419580i
\(430\) 0 0
\(431\) −228.000 −0.529002 −0.264501 0.964385i \(-0.585207\pi\)
−0.264501 + 0.964385i \(0.585207\pi\)
\(432\) 0 0
\(433\) −249.848 249.848i −0.577016 0.577016i 0.357064 0.934080i \(-0.383778\pi\)
−0.934080 + 0.357064i \(0.883778\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −293.939 + 293.939i −0.672629 + 0.672629i
\(438\) 0 0
\(439\) 194.000i 0.441913i −0.975284 0.220957i \(-0.929082\pi\)
0.975284 0.220957i \(-0.0709180\pi\)
\(440\) 0 0
\(441\) −111.000 −0.251701
\(442\) 0 0
\(443\) −338.030 338.030i −0.763046 0.763046i 0.213825 0.976872i \(-0.431408\pi\)
−0.976872 + 0.213825i \(0.931408\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −44.0908 + 44.0908i −0.0986372 + 0.0986372i
\(448\) 0 0
\(449\) 6.00000i 0.0133630i −0.999978 0.00668151i \(-0.997873\pi\)
0.999978 0.00668151i \(-0.00212681\pi\)
\(450\) 0 0
\(451\) −180.000 −0.399113
\(452\) 0 0
\(453\) 85.7321 + 85.7321i 0.189254 + 0.189254i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 367.423 367.423i 0.803990 0.803990i −0.179727 0.983717i \(-0.557521\pi\)
0.983717 + 0.179727i \(0.0575213\pi\)
\(458\) 0 0
\(459\) 108.000i 0.235294i
\(460\) 0 0
\(461\) −204.000 −0.442516 −0.221258 0.975215i \(-0.571016\pi\)
−0.221258 + 0.975215i \(0.571016\pi\)
\(462\) 0 0
\(463\) −100.429 100.429i −0.216909 0.216909i 0.590285 0.807195i \(-0.299015\pi\)
−0.807195 + 0.590285i \(0.799015\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 191.060 191.060i 0.409122 0.409122i −0.472310 0.881432i \(-0.656580\pi\)
0.881432 + 0.472310i \(0.156580\pi\)
\(468\) 0 0
\(469\) 312.000i 0.665245i
\(470\) 0 0
\(471\) 342.000 0.726115
\(472\) 0 0
\(473\) 176.363 + 176.363i 0.372861 + 0.372861i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 44.0908 44.0908i 0.0924336 0.0924336i
\(478\) 0 0
\(479\) 888.000i 1.85386i 0.375232 + 0.926931i \(0.377563\pi\)
−0.375232 + 0.926931i \(0.622437\pi\)
\(480\) 0 0
\(481\) −780.000 −1.62162
\(482\) 0 0
\(483\) 176.363 + 176.363i 0.365141 + 0.365141i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 531.539 531.539i 1.09146 1.09146i 0.0960831 0.995373i \(-0.469369\pi\)
0.995373 0.0960831i \(-0.0306315\pi\)
\(488\) 0 0
\(489\) 240.000i 0.490798i
\(490\) 0 0
\(491\) −534.000 −1.08758 −0.543788 0.839223i \(-0.683011\pi\)
−0.543788 + 0.839223i \(0.683011\pi\)
\(492\) 0 0
\(493\) 705.453 + 705.453i 1.43094 + 1.43094i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −293.939 + 293.939i −0.591426 + 0.591426i
\(498\) 0 0
\(499\) 658.000i 1.31864i 0.751864 + 0.659319i \(0.229155\pi\)
−0.751864 + 0.659319i \(0.770845\pi\)
\(500\) 0 0
\(501\) 72.0000 0.143713
\(502\) 0 0
\(503\) −191.060 191.060i −0.379841 0.379841i 0.491203 0.871045i \(-0.336557\pi\)
−0.871045 + 0.491203i \(0.836557\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 160.442 160.442i 0.316453 0.316453i
\(508\) 0 0
\(509\) 324.000i 0.636542i −0.948000 0.318271i \(-0.896898\pi\)
0.948000 0.318271i \(-0.103102\pi\)
\(510\) 0 0
\(511\) −408.000 −0.798434
\(512\) 0 0
\(513\) −36.7423 36.7423i −0.0716225 0.0716225i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 88.1816 88.1816i 0.170564 0.170564i
\(518\) 0 0
\(519\) 468.000i 0.901734i
\(520\) 0 0
\(521\) 342.000 0.656430 0.328215 0.944603i \(-0.393553\pi\)
0.328215 + 0.944603i \(0.393553\pi\)
\(522\) 0 0
\(523\) −578.080 578.080i −1.10531 1.10531i −0.993758 0.111557i \(-0.964416\pi\)
−0.111557 0.993758i \(-0.535584\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −382.120 + 382.120i −0.725086 + 0.725086i
\(528\) 0 0
\(529\) 1199.00i 2.26654i
\(530\) 0 0
\(531\) −234.000 −0.440678
\(532\) 0 0
\(533\) 367.423 + 367.423i 0.689350 + 0.689350i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −271.893 + 271.893i −0.506319 + 0.506319i
\(538\) 0 0
\(539\) 222.000i 0.411874i
\(540\) 0 0
\(541\) 98.0000 0.181146 0.0905730 0.995890i \(-0.471130\pi\)
0.0905730 + 0.995890i \(0.471130\pi\)
\(542\) 0 0
\(543\) −232.702 232.702i −0.428548 0.428548i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 519.292 519.292i 0.949345 0.949345i −0.0494323 0.998777i \(-0.515741\pi\)
0.998777 + 0.0494323i \(0.0157412\pi\)
\(548\) 0 0
\(549\) 6.00000i 0.0109290i
\(550\) 0 0
\(551\) −480.000 −0.871143
\(552\) 0 0
\(553\) −181.262 181.262i −0.327780 0.327780i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −396.817 + 396.817i −0.712419 + 0.712419i −0.967041 0.254622i \(-0.918049\pi\)
0.254622 + 0.967041i \(0.418049\pi\)
\(558\) 0 0
\(559\) 720.000i 1.28801i
\(560\) 0 0
\(561\) −216.000 −0.385027
\(562\) 0 0
\(563\) 73.4847 + 73.4847i 0.130523 + 0.130523i 0.769350 0.638827i \(-0.220580\pi\)
−0.638827 + 0.769350i \(0.720580\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −22.0454 + 22.0454i −0.0388808 + 0.0388808i
\(568\) 0 0
\(569\) 762.000i 1.33919i −0.742726 0.669596i \(-0.766467\pi\)
0.742726 0.669596i \(-0.233533\pi\)
\(570\) 0 0
\(571\) 850.000 1.48862 0.744308 0.667836i \(-0.232779\pi\)
0.744308 + 0.667836i \(0.232779\pi\)
\(572\) 0 0
\(573\) 249.848 + 249.848i 0.436035 + 0.436035i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −4.89898 + 4.89898i −0.00849043 + 0.00849043i −0.711339 0.702849i \(-0.751911\pi\)
0.702849 + 0.711339i \(0.251911\pi\)
\(578\) 0 0
\(579\) 168.000i 0.290155i
\(580\) 0 0
\(581\) 216.000 0.371773
\(582\) 0 0
\(583\) 88.1816 + 88.1816i 0.151255 + 0.151255i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −132.272 + 132.272i −0.225336 + 0.225336i −0.810741 0.585405i \(-0.800936\pi\)
0.585405 + 0.810741i \(0.300936\pi\)
\(588\) 0 0
\(589\) 260.000i 0.441426i
\(590\) 0 0
\(591\) 504.000 0.852792
\(592\) 0 0
\(593\) 102.879 + 102.879i 0.173488 + 0.173488i 0.788510 0.615022i \(-0.210853\pi\)
−0.615022 + 0.788510i \(0.710853\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 56.3383 56.3383i 0.0943690 0.0943690i
\(598\) 0 0
\(599\) 732.000i 1.22204i 0.791616 + 0.611018i \(0.209240\pi\)
−0.791616 + 0.611018i \(0.790760\pi\)
\(600\) 0 0
\(601\) 778.000 1.29451 0.647255 0.762274i \(-0.275917\pi\)
0.647255 + 0.762274i \(0.275917\pi\)
\(602\) 0 0
\(603\) −191.060 191.060i −0.316849 0.316849i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 105.328 105.328i 0.173522 0.173522i −0.615003 0.788525i \(-0.710845\pi\)
0.788525 + 0.615003i \(0.210845\pi\)
\(608\) 0 0
\(609\) 288.000i 0.472906i
\(610\) 0 0
\(611\) −360.000 −0.589198
\(612\) 0 0
\(613\) −71.0352 71.0352i −0.115881 0.115881i 0.646788 0.762670i \(-0.276112\pi\)
−0.762670 + 0.646788i \(0.776112\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 455.605 455.605i 0.738420 0.738420i −0.233852 0.972272i \(-0.575133\pi\)
0.972272 + 0.233852i \(0.0751332\pi\)
\(618\) 0 0
\(619\) 362.000i 0.584814i −0.956294 0.292407i \(-0.905544\pi\)
0.956294 0.292407i \(-0.0944562\pi\)
\(620\) 0 0
\(621\) −216.000 −0.347826
\(622\) 0 0
\(623\) −367.423 367.423i −0.589765 0.589765i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 73.4847 73.4847i 0.117200 0.117200i
\(628\) 0 0
\(629\) 936.000i 1.48808i
\(630\) 0 0
\(631\) 478.000 0.757528 0.378764 0.925493i \(-0.376349\pi\)
0.378764 + 0.925493i \(0.376349\pi\)
\(632\) 0 0
\(633\) −379.671 379.671i −0.599796 0.599796i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −453.156 + 453.156i −0.711390 + 0.711390i
\(638\) 0 0
\(639\) 360.000i 0.563380i
\(640\) 0 0
\(641\) 354.000 0.552262 0.276131 0.961120i \(-0.410948\pi\)
0.276131 + 0.961120i \(0.410948\pi\)
\(642\) 0 0
\(643\) −264.545 264.545i −0.411423 0.411423i 0.470811 0.882234i \(-0.343961\pi\)
−0.882234 + 0.470811i \(0.843961\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 484.999 484.999i 0.749612 0.749612i −0.224794 0.974406i \(-0.572171\pi\)
0.974406 + 0.224794i \(0.0721710\pi\)
\(648\) 0 0
\(649\) 468.000i 0.721109i
\(650\) 0 0
\(651\) −156.000 −0.239631
\(652\) 0 0
\(653\) 29.3939 + 29.3939i 0.0450136 + 0.0450136i 0.729255 0.684242i \(-0.239867\pi\)
−0.684242 + 0.729255i \(0.739867\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 249.848 249.848i 0.380286 0.380286i
\(658\) 0 0
\(659\) 414.000i 0.628225i −0.949386 0.314112i \(-0.898293\pi\)
0.949386 0.314112i \(-0.101707\pi\)
\(660\) 0 0
\(661\) −1202.00 −1.81846 −0.909228 0.416298i \(-0.863327\pi\)
−0.909228 + 0.416298i \(0.863327\pi\)
\(662\) 0 0
\(663\) 440.908 + 440.908i 0.665020 + 0.665020i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1410.91 + 1410.91i −2.11530 + 2.11530i
\(668\) 0 0
\(669\) 450.000i 0.672646i
\(670\) 0 0
\(671\) −12.0000 −0.0178838
\(672\) 0 0
\(673\) −171.464 171.464i −0.254776 0.254776i 0.568149 0.822925i \(-0.307660\pi\)
−0.822925 + 0.568149i \(0.807660\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −235.151 + 235.151i −0.347343 + 0.347343i −0.859119 0.511776i \(-0.828988\pi\)
0.511776 + 0.859119i \(0.328988\pi\)
\(678\) 0 0
\(679\) 24.0000i 0.0353461i
\(680\) 0 0
\(681\) −180.000 −0.264317
\(682\) 0 0
\(683\) 690.756 + 690.756i 1.01136 + 1.01136i 0.999935 + 0.0114212i \(0.00363555\pi\)
0.0114212 + 0.999935i \(0.496364\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −296.388 + 296.388i −0.431424 + 0.431424i
\(688\) 0 0
\(689\) 360.000i 0.522496i
\(690\) 0 0
\(691\) −778.000 −1.12590 −0.562952 0.826489i \(-0.690335\pi\)
−0.562952 + 0.826489i \(0.690335\pi\)
\(692\) 0 0
\(693\) −44.0908 44.0908i −0.0636231 0.0636231i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −440.908 + 440.908i −0.632580 + 0.632580i
\(698\) 0 0
\(699\) 180.000i 0.257511i
\(700\) 0 0
\(701\) −84.0000 −0.119829 −0.0599144 0.998204i \(-0.519083\pi\)
−0.0599144 + 0.998204i \(0.519083\pi\)
\(702\) 0 0
\(703\) 318.434 + 318.434i 0.452964 + 0.452964i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −29.3939 + 29.3939i −0.0415755 + 0.0415755i
\(708\) 0 0
\(709\) 502.000i 0.708039i 0.935238 + 0.354020i \(0.115185\pi\)
−0.935238 + 0.354020i \(0.884815\pi\)
\(710\) 0 0
\(711\) 222.000 0.312236
\(712\) 0 0
\(713\) −764.241 764.241i −1.07187 1.07187i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −396.817 + 396.817i −0.553441 + 0.553441i
\(718\) 0 0
\(719\) 1356.00i 1.88595i 0.332860 + 0.942976i \(0.391986\pi\)
−0.332860 + 0.942976i \(0.608014\pi\)
\(720\) 0 0
\(721\) 204.000 0.282940
\(722\) 0 0
\(723\) −487.448 487.448i −0.674203 0.674203i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −815.680 + 815.680i −1.12198 + 1.12198i −0.130537 + 0.991443i \(0.541670\pi\)
−0.991443 + 0.130537i \(0.958330\pi\)
\(728\) 0 0
\(729\) 27.0000i 0.0370370i
\(730\) 0 0
\(731\) 864.000 1.18194
\(732\) 0 0
\(733\) −962.649 962.649i −1.31330 1.31330i −0.918968 0.394333i \(-0.870976\pi\)
−0.394333 0.918968i \(-0.629024\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 382.120 382.120i 0.518481 0.518481i
\(738\) 0 0
\(739\) 610.000i 0.825440i −0.910858 0.412720i \(-0.864579\pi\)
0.910858 0.412720i \(-0.135421\pi\)
\(740\) 0 0
\(741\) −300.000 −0.404858
\(742\) 0 0
\(743\) −44.0908 44.0908i −0.0593416 0.0593416i 0.676813 0.736155i \(-0.263360\pi\)
−0.736155 + 0.676813i \(0.763360\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −132.272 + 132.272i −0.177072 + 0.177072i
\(748\) 0 0
\(749\) 504.000i 0.672897i
\(750\) 0 0
\(751\) 1058.00 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(752\) 0 0
\(753\) −198.409 198.409i −0.263491 0.263491i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 26.9444 26.9444i 0.0355936 0.0355936i −0.689086 0.724680i \(-0.741988\pi\)
0.724680 + 0.689086i \(0.241988\pi\)
\(758\) 0 0
\(759\) 432.000i 0.569170i
\(760\) 0 0
\(761\) 1122.00 1.47438 0.737188 0.675688i \(-0.236153\pi\)
0.737188 + 0.675688i \(0.236153\pi\)
\(762\) 0 0
\(763\) 181.262 + 181.262i 0.237565 + 0.237565i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −955.301 + 955.301i −1.24550 + 1.24550i
\(768\) 0 0
\(769\) 274.000i 0.356307i −0.984003 0.178153i \(-0.942988\pi\)
0.984003 0.178153i \(-0.0570123\pi\)
\(770\) 0 0
\(771\) −252.000 −0.326848
\(772\) 0 0
\(773\) −382.120 382.120i −0.494334 0.494334i 0.415334 0.909669i \(-0.363665\pi\)
−0.909669 + 0.415334i \(0.863665\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 191.060 191.060i 0.245895 0.245895i
\(778\) 0 0
\(779\) 300.000i 0.385109i
\(780\) 0 0
\(781\) 720.000 0.921895
\(782\) 0 0
\(783\) −176.363 176.363i −0.225240 0.225240i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 421.312 421.312i 0.535340 0.535340i −0.386817 0.922157i \(-0.626425\pi\)
0.922157 + 0.386817i \(0.126425\pi\)
\(788\) 0 0
\(789\) 288.000i 0.365019i
\(790\) 0 0
\(791\) −648.000 −0.819216
\(792\) 0 0
\(793\) 24.4949 + 24.4949i 0.0308889 + 0.0308889i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −396.817 + 396.817i −0.497889 + 0.497889i −0.910780 0.412891i \(-0.864519\pi\)
0.412891 + 0.910780i \(0.364519\pi\)
\(798\) 0 0
\(799\) 432.000i 0.540676i
\(800\) 0