Properties

Label 1840.2.e.b.369.1
Level $1840$
Weight $2$
Character 1840.369
Analytic conductor $14.692$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 369.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1840.369
Dual form 1840.2.e.b.369.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000i q^{3} +(2.00000 - 1.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{3} +(2.00000 - 1.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +2.00000i q^{13} +(-2.00000 - 4.00000i) q^{15} +5.00000i q^{17} +8.00000 q^{19} +2.00000 q^{21} +1.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -4.00000i q^{27} +5.00000 q^{29} +5.00000 q^{31} +(1.00000 + 2.00000i) q^{35} -7.00000i q^{37} +4.00000 q^{39} -7.00000 q^{41} -4.00000i q^{43} +(-2.00000 + 1.00000i) q^{45} -2.00000i q^{47} +6.00000 q^{49} +10.0000 q^{51} -1.00000i q^{53} -16.0000i q^{57} +3.00000 q^{59} -6.00000 q^{61} -1.00000i q^{63} +(2.00000 + 4.00000i) q^{65} +13.0000i q^{67} +2.00000 q^{69} -13.0000 q^{71} +8.00000i q^{73} +(-8.00000 - 6.00000i) q^{75} -14.0000 q^{79} -11.0000 q^{81} +3.00000i q^{83} +(5.00000 + 10.0000i) q^{85} -10.0000i q^{87} +14.0000 q^{89} -2.00000 q^{91} -10.0000i q^{93} +(16.0000 - 8.00000i) q^{95} -14.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 2q^{9} + O(q^{10}) \) \( 2q + 4q^{5} - 2q^{9} - 4q^{15} + 16q^{19} + 4q^{21} + 6q^{25} + 10q^{29} + 10q^{31} + 2q^{35} + 8q^{39} - 14q^{41} - 4q^{45} + 12q^{49} + 20q^{51} + 6q^{59} - 12q^{61} + 4q^{65} + 4q^{69} - 26q^{71} - 16q^{75} - 28q^{79} - 22q^{81} + 10q^{85} + 28q^{89} - 4q^{91} + 32q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 2.00000i 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) 0 0
\(5\) 2.00000 1.00000i 0.894427 0.447214i
\(6\) 0 0
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) −2.00000 4.00000i −0.516398 1.03280i
\(16\) 0 0
\(17\) 5.00000i 1.21268i 0.795206 + 0.606339i \(0.207363\pi\)
−0.795206 + 0.606339i \(0.792637\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 3.00000 4.00000i 0.600000 0.800000i
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 + 2.00000i 0.169031 + 0.338062i
\(36\) 0 0
\(37\) 7.00000i 1.15079i −0.817875 0.575396i \(-0.804848\pi\)
0.817875 0.575396i \(-0.195152\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) −2.00000 + 1.00000i −0.298142 + 0.149071i
\(46\) 0 0
\(47\) 2.00000i 0.291730i −0.989305 0.145865i \(-0.953403\pi\)
0.989305 0.145865i \(-0.0465965\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 10.0000 1.40028
\(52\) 0 0
\(53\) 1.00000i 0.137361i −0.997639 0.0686803i \(-0.978121\pi\)
0.997639 0.0686803i \(-0.0218788\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 16.0000i 2.11925i
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 2.00000 + 4.00000i 0.248069 + 0.496139i
\(66\) 0 0
\(67\) 13.0000i 1.58820i 0.607785 + 0.794101i \(0.292058\pi\)
−0.607785 + 0.794101i \(0.707942\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) −13.0000 −1.54282 −0.771408 0.636341i \(-0.780447\pi\)
−0.771408 + 0.636341i \(0.780447\pi\)
\(72\) 0 0
\(73\) 8.00000i 0.936329i 0.883641 + 0.468165i \(0.155085\pi\)
−0.883641 + 0.468165i \(0.844915\pi\)
\(74\) 0 0
\(75\) −8.00000 6.00000i −0.923760 0.692820i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 3.00000i 0.329293i 0.986353 + 0.164646i \(0.0526483\pi\)
−0.986353 + 0.164646i \(0.947352\pi\)
\(84\) 0 0
\(85\) 5.00000 + 10.0000i 0.542326 + 1.08465i
\(86\) 0 0
\(87\) 10.0000i 1.07211i
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 10.0000i 1.03695i
\(94\) 0 0
\(95\) 16.0000 8.00000i 1.64157 0.820783i
\(96\) 0 0
\(97\) 14.0000i 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.0000 1.49256 0.746278 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 4.00000 2.00000i 0.390360 0.195180i
\(106\) 0 0
\(107\) 9.00000i 0.870063i 0.900415 + 0.435031i \(0.143263\pi\)
−0.900415 + 0.435031i \(0.856737\pi\)
\(108\) 0 0
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) −14.0000 −1.32882
\(112\) 0 0
\(113\) 1.00000i 0.0940721i −0.998893 0.0470360i \(-0.985022\pi\)
0.998893 0.0470360i \(-0.0149776\pi\)
\(114\) 0 0
\(115\) 1.00000 + 2.00000i 0.0932505 + 0.186501i
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) −5.00000 −0.458349
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 14.0000i 1.26234i
\(124\) 0 0
\(125\) 2.00000 11.0000i 0.178885 0.983870i
\(126\) 0 0
\(127\) 20.0000i 1.77471i −0.461084 0.887357i \(-0.652539\pi\)
0.461084 0.887357i \(-0.347461\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) 0 0
\(135\) −4.00000 8.00000i −0.344265 0.688530i
\(136\) 0 0
\(137\) 6.00000i 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 0 0
\(139\) 9.00000 0.763370 0.381685 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 10.0000 5.00000i 0.830455 0.415227i
\(146\) 0 0
\(147\) 12.0000i 0.989743i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 5.00000i 0.404226i
\(154\) 0 0
\(155\) 10.0000 5.00000i 0.803219 0.401610i
\(156\) 0 0
\(157\) 3.00000i 0.239426i 0.992809 + 0.119713i \(0.0381975\pi\)
−0.992809 + 0.119713i \(0.961803\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 24.0000i 1.87983i −0.341415 0.939913i \(-0.610906\pi\)
0.341415 0.939913i \(-0.389094\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.0000i 1.23812i 0.785345 + 0.619059i \(0.212486\pi\)
−0.785345 + 0.619059i \(0.787514\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) 0 0
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 4.00000 + 3.00000i 0.302372 + 0.226779i
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 12.0000i 0.887066i
\(184\) 0 0
\(185\) −7.00000 14.0000i −0.514650 1.02930i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 12.0000i 0.863779i −0.901927 0.431889i \(-0.857847\pi\)
0.901927 0.431889i \(-0.142153\pi\)
\(194\) 0 0
\(195\) 8.00000 4.00000i 0.572892 0.286446i
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) 26.0000 1.83390
\(202\) 0 0
\(203\) 5.00000i 0.350931i
\(204\) 0 0
\(205\) −14.0000 + 7.00000i −0.977802 + 0.488901i
\(206\) 0 0
\(207\) 1.00000i 0.0695048i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 9.00000 0.619586 0.309793 0.950804i \(-0.399740\pi\)
0.309793 + 0.950804i \(0.399740\pi\)
\(212\) 0 0
\(213\) 26.0000i 1.78149i
\(214\) 0 0
\(215\) −4.00000 8.00000i −0.272798 0.545595i
\(216\) 0 0
\(217\) 5.00000i 0.339422i
\(218\) 0 0
\(219\) 16.0000 1.08118
\(220\) 0 0
\(221\) −10.0000 −0.672673
\(222\) 0 0
\(223\) 14.0000i 0.937509i 0.883328 + 0.468755i \(0.155297\pi\)
−0.883328 + 0.468755i \(0.844703\pi\)
\(224\) 0 0
\(225\) −3.00000 + 4.00000i −0.200000 + 0.266667i
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 0 0
\(235\) −2.00000 4.00000i −0.130466 0.260931i
\(236\) 0 0
\(237\) 28.0000i 1.81880i
\(238\) 0 0
\(239\) −1.00000 −0.0646846 −0.0323423 0.999477i \(-0.510297\pi\)
−0.0323423 + 0.999477i \(0.510297\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0 0
\(243\) 10.0000i 0.641500i
\(244\) 0 0
\(245\) 12.0000 6.00000i 0.766652 0.383326i
\(246\) 0 0
\(247\) 16.0000i 1.01806i
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 20.0000 10.0000i 1.25245 0.626224i
\(256\) 0 0
\(257\) 22.0000i 1.37232i −0.727450 0.686161i \(-0.759294\pi\)
0.727450 0.686161i \(-0.240706\pi\)
\(258\) 0 0
\(259\) 7.00000 0.434959
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) 0 0
\(263\) 13.0000i 0.801614i 0.916162 + 0.400807i \(0.131270\pi\)
−0.916162 + 0.400807i \(0.868730\pi\)
\(264\) 0 0
\(265\) −1.00000 2.00000i −0.0614295 0.122859i
\(266\) 0 0
\(267\) 28.0000i 1.71357i
\(268\) 0 0
\(269\) 15.0000 0.914566 0.457283 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(270\) 0 0
\(271\) 15.0000 0.911185 0.455593 0.890188i \(-0.349427\pi\)
0.455593 + 0.890188i \(0.349427\pi\)
\(272\) 0 0
\(273\) 4.00000i 0.242091i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 26.0000i 1.56219i 0.624413 + 0.781094i \(0.285338\pi\)
−0.624413 + 0.781094i \(0.714662\pi\)
\(278\) 0 0
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) 11.0000i 0.653882i −0.945045 0.326941i \(-0.893982\pi\)
0.945045 0.326941i \(-0.106018\pi\)
\(284\) 0 0
\(285\) −16.0000 32.0000i −0.947758 1.89552i
\(286\) 0 0
\(287\) 7.00000i 0.413197i
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −28.0000 −1.64139
\(292\) 0 0
\(293\) 29.0000i 1.69420i 0.531435 + 0.847099i \(0.321653\pi\)
−0.531435 + 0.847099i \(0.678347\pi\)
\(294\) 0 0
\(295\) 6.00000 3.00000i 0.349334 0.174667i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) 30.0000i 1.72345i
\(304\) 0 0
\(305\) −12.0000 + 6.00000i −0.687118 + 0.343559i
\(306\) 0 0
\(307\) 14.0000i 0.799022i 0.916728 + 0.399511i \(0.130820\pi\)
−0.916728 + 0.399511i \(0.869180\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) 21.0000i 1.18699i 0.804838 + 0.593495i \(0.202252\pi\)
−0.804838 + 0.593495i \(0.797748\pi\)
\(314\) 0 0
\(315\) −1.00000 2.00000i −0.0563436 0.112687i
\(316\) 0 0
\(317\) 24.0000i 1.34797i 0.738743 + 0.673987i \(0.235420\pi\)
−0.738743 + 0.673987i \(0.764580\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) 40.0000i 2.22566i
\(324\) 0 0
\(325\) 8.00000 + 6.00000i 0.443760 + 0.332820i
\(326\) 0 0
\(327\) 36.0000i 1.99080i
\(328\) 0 0
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) 5.00000 0.274825 0.137412 0.990514i \(-0.456121\pi\)
0.137412 + 0.990514i \(0.456121\pi\)
\(332\) 0 0
\(333\) 7.00000i 0.383598i
\(334\) 0 0
\(335\) 13.0000 + 26.0000i 0.710266 + 1.42053i
\(336\) 0 0
\(337\) 26.0000i 1.41631i −0.706057 0.708155i \(-0.749528\pi\)
0.706057 0.708155i \(-0.250472\pi\)
\(338\) 0 0
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.0000i 0.701934i
\(344\) 0 0
\(345\) 4.00000 2.00000i 0.215353 0.107676i
\(346\) 0 0
\(347\) 8.00000i 0.429463i 0.976673 + 0.214731i \(0.0688876\pi\)
−0.976673 + 0.214731i \(0.931112\pi\)
\(348\) 0 0
\(349\) −23.0000 −1.23116 −0.615581 0.788074i \(-0.711079\pi\)
−0.615581 + 0.788074i \(0.711079\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) −26.0000 + 13.0000i −1.37994 + 0.689968i
\(356\) 0 0
\(357\) 10.0000i 0.529256i
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 22.0000i 1.15470i
\(364\) 0 0
\(365\) 8.00000 + 16.0000i 0.418739 + 0.837478i
\(366\) 0 0
\(367\) 13.0000i 0.678594i −0.940679 0.339297i \(-0.889811\pi\)
0.940679 0.339297i \(-0.110189\pi\)
\(368\) 0 0
\(369\) 7.00000 0.364405
\(370\) 0 0
\(371\) 1.00000 0.0519174
\(372\) 0 0
\(373\) 6.00000i 0.310668i −0.987862 0.155334i \(-0.950355\pi\)
0.987862 0.155334i \(-0.0496454\pi\)
\(374\) 0 0
\(375\) −22.0000 4.00000i −1.13608 0.206559i
\(376\) 0 0
\(377\) 10.0000i 0.515026i
\(378\) 0 0
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) 0 0
\(381\) −40.0000 −2.04926
\(382\) 0 0
\(383\) 3.00000i 0.153293i −0.997058 0.0766464i \(-0.975579\pi\)
0.997058 0.0766464i \(-0.0244213\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) −22.0000 −1.11544 −0.557722 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 0 0
\(391\) −5.00000 −0.252861
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −28.0000 + 14.0000i −1.40883 + 0.704416i
\(396\) 0 0
\(397\) 16.0000i 0.803017i −0.915855 0.401508i \(-0.868486\pi\)
0.915855 0.401508i \(-0.131514\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 10.0000i 0.498135i
\(404\) 0 0
\(405\) −22.0000 + 11.0000i −1.09319 + 0.546594i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 19.0000 0.939490 0.469745 0.882802i \(-0.344346\pi\)
0.469745 + 0.882802i \(0.344346\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 0 0
\(413\) 3.00000i 0.147620i
\(414\) 0 0
\(415\) 3.00000 + 6.00000i 0.147264 + 0.294528i
\(416\) 0 0
\(417\) 18.0000i 0.881464i
\(418\) 0 0
\(419\) −2.00000 −0.0977064 −0.0488532 0.998806i \(-0.515557\pi\)
−0.0488532 + 0.998806i \(0.515557\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 2.00000i 0.0972433i
\(424\) 0 0
\(425\) 20.0000 + 15.0000i 0.970143 + 0.727607i
\(426\) 0 0
\(427\) 6.00000i 0.290360i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 19.0000i 0.913082i 0.889702 + 0.456541i \(0.150912\pi\)
−0.889702 + 0.456541i \(0.849088\pi\)
\(434\) 0 0
\(435\) −10.0000 20.0000i −0.479463 0.958927i
\(436\) 0 0
\(437\) 8.00000i 0.382692i
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 24.0000i 1.14027i 0.821549 + 0.570137i \(0.193110\pi\)
−0.821549 + 0.570137i \(0.806890\pi\)
\(444\) 0 0
\(445\) 28.0000 14.0000i 1.32733 0.663664i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 16.0000i 0.751746i
\(454\) 0 0
\(455\) −4.00000 + 2.00000i −0.187523 + 0.0937614i
\(456\) 0 0
\(457\) 1.00000i 0.0467780i −0.999726 0.0233890i \(-0.992554\pi\)
0.999726 0.0233890i \(-0.00744563\pi\)
\(458\) 0 0
\(459\) 20.0000 0.933520
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) 0 0
\(465\) −10.0000 20.0000i −0.463739 0.927478i
\(466\) 0 0
\(467\) 27.0000i 1.24941i −0.780860 0.624705i \(-0.785219\pi\)
0.780860 0.624705i \(-0.214781\pi\)
\(468\) 0 0
\(469\) −13.0000 −0.600284
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 24.0000 32.0000i 1.10120 1.46826i
\(476\) 0 0
\(477\) 1.00000i 0.0457869i
\(478\) 0 0
\(479\) −28.0000 −1.27935 −0.639676 0.768644i \(-0.720932\pi\)
−0.639676 + 0.768644i \(0.720932\pi\)
\(480\) 0 0
\(481\) 14.0000 0.638345
\(482\) 0 0
\(483\) 2.00000i 0.0910032i
\(484\) 0 0
\(485\) −14.0000 28.0000i −0.635707 1.27141i
\(486\) 0 0
\(487\) 32.0000i 1.45006i −0.688718 0.725029i \(-0.741826\pi\)
0.688718 0.725029i \(-0.258174\pi\)
\(488\) 0 0
\(489\) −48.0000 −2.17064
\(490\) 0 0
\(491\) 31.0000 1.39901 0.699505 0.714628i \(-0.253404\pi\)
0.699505 + 0.714628i \(0.253404\pi\)
\(492\) 0 0
\(493\) 25.0000i 1.12594i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13.0000i 0.583130i
\(498\) 0 0
\(499\) −11.0000 −0.492428 −0.246214 0.969216i \(-0.579187\pi\)
−0.246214 + 0.969216i \(0.579187\pi\)
\(500\) 0 0
\(501\) 32.0000 1.42965
\(502\) 0 0
\(503\) 9.00000i 0.401290i 0.979664 + 0.200645i \(0.0643038\pi\)
−0.979664 + 0.200645i \(0.935696\pi\)
\(504\) 0 0
\(505\) 30.0000 15.0000i 1.33498 0.667491i
\(506\) 0 0
\(507\) 18.0000i 0.799408i
\(508\) 0 0
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 0 0
\(513\) 32.0000i 1.41283i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) −20.0000 −0.876216 −0.438108 0.898922i \(-0.644351\pi\)
−0.438108 + 0.898922i \(0.644351\pi\)
\(522\) 0 0
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) 0 0
\(525\) 6.00000 8.00000i 0.261861 0.349149i
\(526\) 0 0
\(527\) 25.0000i 1.08902i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −3.00000 −0.130189
\(532\) 0 0
\(533\) 14.0000i 0.606407i
\(534\) 0 0
\(535\) 9.00000 + 18.0000i 0.389104 + 0.778208i
\(536\) 0 0
\(537\) 8.00000i 0.345225i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) 28.0000i 1.20160i
\(544\) 0 0
\(545\) −36.0000 + 18.0000i −1.54207 + 0.771035i
\(546\) 0 0
\(547\) 4.00000i 0.171028i −0.996337 0.0855138i \(-0.972747\pi\)
0.996337 0.0855138i \(-0.0272532\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 40.0000 1.70406
\(552\) 0 0
\(553\) 14.0000i 0.595341i
\(554\) 0 0
\(555\) −28.0000 + 14.0000i −1.18853 + 0.594267i
\(556\) 0 0
\(557\) 33.0000i 1.39825i 0.714997 + 0.699127i \(0.246428\pi\)
−0.714997 + 0.699127i \(0.753572\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 25.0000i 1.05362i 0.849982 + 0.526812i \(0.176613\pi\)
−0.849982 + 0.526812i \(0.823387\pi\)
\(564\) 0 0
\(565\) −1.00000 2.00000i −0.0420703 0.0841406i
\(566\) 0 0
\(567\) 11.0000i 0.461957i
\(568\) 0 0
\(569\) 12.0000 0.503066 0.251533 0.967849i \(-0.419065\pi\)
0.251533 + 0.967849i \(0.419065\pi\)
\(570\) 0 0
\(571\) −18.0000 −0.753277 −0.376638 0.926360i \(-0.622920\pi\)
−0.376638 + 0.926360i \(0.622920\pi\)
\(572\) 0 0
\(573\) 16.0000i 0.668410i
\(574\) 0 0
\(575\) 4.00000 + 3.00000i 0.166812 + 0.125109i
\(576\) 0 0
\(577\) 22.0000i 0.915872i −0.888985 0.457936i \(-0.848589\pi\)
0.888985 0.457936i \(-0.151411\pi\)
\(578\) 0 0
\(579\) −24.0000 −0.997406
\(580\) 0 0
\(581\) −3.00000 −0.124461
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −2.00000 4.00000i −0.0826898 0.165380i
\(586\) 0 0
\(587\) 12.0000i 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 0 0
\(589\) 40.0000 1.64817
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) 0 0
\(595\) −10.0000 + 5.00000i −0.409960 + 0.204980i
\(596\) 0 0
\(597\) 4.00000i 0.163709i
\(598\) 0 0
\(599\) −48.0000 −1.96123 −0.980613 0.195952i \(-0.937220\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) 0 0
\(601\) −25.0000 −1.01977 −0.509886 0.860242i \(-0.670312\pi\)
−0.509886 + 0.860242i \(0.670312\pi\)
\(602\) 0 0
\(603\) 13.0000i 0.529401i
\(604\) 0 0
\(605\) −22.0000 + 11.0000i −0.894427 + 0.447214i
\(606\) 0 0
\(607\) 38.0000i 1.54237i −0.636610 0.771186i \(-0.719664\pi\)
0.636610 0.771186i \(-0.280336\pi\)
\(608\) 0 0
\(609\) 10.0000 0.405220
\(610\) 0 0
\(611\) 4.00000 0.161823
\(612\) 0 0
\(613\) 6.00000i 0.242338i 0.992632 + 0.121169i \(0.0386643\pi\)
−0.992632 + 0.121169i \(0.961336\pi\)
\(614\) 0 0
\(615\) 14.0000 + 28.0000i 0.564534 + 1.12907i
\(616\) 0 0
\(617\) 47.0000i 1.89215i 0.323949 + 0.946074i \(0.394989\pi\)
−0.323949 + 0.946074i \(0.605011\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) 14.0000i 0.560898i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 35.0000 1.39554
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 0 0
\(633\) 18.0000i 0.715436i
\(634\) 0 0
\(635\) −20.0000 40.0000i −0.793676 1.58735i
\(636\) 0 0
\(637\) 12.0000i 0.475457i
\(638\) 0 0
\(639\) 13.0000 0.514272
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) 37.0000i 1.45914i −0.683907 0.729569i \(-0.739721\pi\)
0.683907 0.729569i \(-0.260279\pi\)
\(644\) 0 0
\(645\) −16.0000 + 8.00000i −0.629999 + 0.315000i
\(646\) 0 0
\(647\) 18.0000i 0.707653i −0.935311 0.353827i \(-0.884880\pi\)
0.935311 0.353827i \(-0.115120\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) 0 0
\(653\) 2.00000i 0.0782660i 0.999234 + 0.0391330i \(0.0124596\pi\)
−0.999234 + 0.0391330i \(0.987540\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.00000i 0.312110i
\(658\) 0 0
\(659\) 22.0000 0.856998 0.428499 0.903542i \(-0.359042\pi\)
0.428499 + 0.903542i \(0.359042\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) 0 0
\(663\) 20.0000i 0.776736i
\(664\) 0 0
\(665\) 8.00000 + 16.0000i 0.310227 + 0.620453i
\(666\) 0 0
\(667\) 5.00000i 0.193601i
\(668\) 0 0
\(669\) 28.0000 1.08254
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 24.0000i 0.925132i 0.886585 + 0.462566i \(0.153071\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 0 0
\(675\) −16.0000 12.0000i −0.615840 0.461880i
\(676\) 0 0
\(677\) 3.00000i 0.115299i 0.998337 + 0.0576497i \(0.0183606\pi\)
−0.998337 + 0.0576497i \(0.981639\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 46.0000i 1.76014i −0.474843 0.880071i \(-0.657495\pi\)
0.474843 0.880071i \(-0.342505\pi\)
\(684\) 0 0
\(685\) −6.00000 12.0000i −0.229248 0.458496i
\(686\) 0 0
\(687\) 8.00000i 0.305219i
\(688\) 0 0
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18.0000 9.00000i 0.682779 0.341389i
\(696\) 0 0
\(697\) 35.0000i 1.32572i
\(698\) 0 0
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) −28.0000 −1.05755 −0.528773 0.848763i \(-0.677348\pi\)
−0.528773 + 0.848763i \(0.677348\pi\)
\(702\) 0 0
\(703\) 56.0000i 2.11208i
\(704\) 0 0
\(705\) −8.00000 + 4.00000i −0.301297 + 0.150649i
\(706\) 0 0
\(707\) 15.0000i 0.564133i
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) 0 0
\(713\) 5.00000i 0.187251i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 2.00000i 0.0746914i
\(718\) 0 0
\(719\) −45.0000 −1.67822 −0.839108 0.543964i \(-0.816923\pi\)
−0.839108 + 0.543964i \(0.816923\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 12.0000i 0.446285i
\(724\) 0 0
\(725\) 15.0000 20.0000i 0.557086 0.742781i
\(726\) 0 0
\(727\) 27.0000i 1.00137i 0.865628 + 0.500687i \(0.166919\pi\)
−0.865628 + 0.500687i \(0.833081\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 20.0000 0.739727
\(732\) 0 0
\(733\) 7.00000i 0.258551i 0.991609 + 0.129275i \(0.0412651\pi\)
−0.991609 + 0.129275i \(0.958735\pi\)
\(734\) 0 0
\(735\) −12.0000 24.0000i −0.442627 0.885253i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 7.00000 0.257499 0.128750 0.991677i \(-0.458904\pi\)
0.128750 + 0.991677i \(0.458904\pi\)
\(740\) 0 0
\(741\) 32.0000 1.17555
\(742\) 0 0
\(743\) 32.0000i 1.17397i 0.809599 + 0.586983i \(0.199684\pi\)
−0.809599 + 0.586983i \(0.800316\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3.00000i 0.109764i
\(748\) 0 0
\(749\) −9.00000 −0.328853
\(750\) 0 0
\(751\) −14.0000 −0.510867 −0.255434 0.966827i \(-0.582218\pi\)
−0.255434 + 0.966827i \(0.582218\pi\)
\(752\) 0 0
\(753\) 36.0000i 1.31191i
\(754\) 0 0
\(755\) −16.0000 + 8.00000i −0.582300 + 0.291150i
\(756\) 0 0
\(757\) 35.0000i 1.27210i 0.771649 + 0.636048i \(0.219432\pi\)
−0.771649 + 0.636048i \(0.780568\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 25.0000 0.906249 0.453125 0.891447i \(-0.350309\pi\)
0.453125 + 0.891447i \(0.350309\pi\)
\(762\) 0 0
\(763\) 18.0000i 0.651644i
\(764\) 0 0
\(765\) −5.00000 10.0000i −0.180775 0.361551i
\(766\) 0 0
\(767\) 6.00000i 0.216647i
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) −44.0000 −1.58462
\(772\) 0 0
\(773\) 42.0000i 1.51064i −0.655359 0.755318i \(-0.727483\pi\)
0.655359 0.755318i \(-0.272517\pi\)
\(774\) 0 0
\(775\) 15.0000 20.0000i 0.538816 0.718421i
\(776\) 0 0
\(777\) 14.0000i 0.502247i
\(778\) 0 0
\(779\) −56.0000 −2.00641
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 20.0000i 0.714742i
\(784\) 0 0
\(785\) 3.00000 + 6.00000i 0.107075 + 0.214149i
\(786\) 0 0
\(787\) 17.0000i 0.605985i 0.952993 + 0.302992i \(0.0979856\pi\)
−0.952993 + 0.302992i \(0.902014\pi\)
\(788\) 0 0
\(789\) 26.0000 0.925625
\(790\) 0 0
\(791\) 1.00000 0.0355559
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) 0 0
\(795\) −4.00000 + 2.00000i −0.141865 + 0.0709327i
\(796\) 0 0
\(797\) 15.0000i 0.531327i 0.964066 + 0.265664i \(0.0855911\pi\)
−0.964066 + 0.265664i \(0.914409\pi\)
\(798\) 0 0
\(799\) 10.0000 0.353775
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 0 0
\(803\) 0 0