Properties

Label 1638.2.c.b.883.1
Level $1638$
Weight $2$
Character 1638.883
Analytic conductor $13.079$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 883.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1638.883
Dual form 1638.2.c.b.883.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{5} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{5} +1.00000i q^{7} +1.00000i q^{8} -2.00000 q^{10} +(2.00000 + 3.00000i) q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +4.00000i q^{19} +2.00000i q^{20} +6.00000 q^{23} +1.00000 q^{25} +(3.00000 - 2.00000i) q^{26} -1.00000i q^{28} -1.00000i q^{32} -2.00000i q^{34} +2.00000 q^{35} -2.00000i q^{37} +4.00000 q^{38} +2.00000 q^{40} +4.00000 q^{43} -6.00000i q^{46} -8.00000i q^{47} -1.00000 q^{49} -1.00000i q^{50} +(-2.00000 - 3.00000i) q^{52} -4.00000 q^{53} -1.00000 q^{56} +6.00000i q^{59} +12.0000 q^{61} -1.00000 q^{64} +(6.00000 - 4.00000i) q^{65} -2.00000i q^{67} -2.00000 q^{68} -2.00000i q^{70} -14.0000i q^{73} -2.00000 q^{74} -4.00000i q^{76} -2.00000i q^{80} +14.0000i q^{83} -4.00000i q^{85} -4.00000i q^{86} -4.00000i q^{89} +(-3.00000 + 2.00000i) q^{91} -6.00000 q^{92} -8.00000 q^{94} +8.00000 q^{95} -2.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 4q^{10} + 4q^{13} + 2q^{14} + 2q^{16} + 4q^{17} + 12q^{23} + 2q^{25} + 6q^{26} + 4q^{35} + 8q^{38} + 4q^{40} + 8q^{43} - 2q^{49} - 4q^{52} - 8q^{53} - 2q^{56} + 24q^{61} - 2q^{64} + 12q^{65} - 4q^{68} - 4q^{74} - 6q^{91} - 12q^{92} - 16q^{94} + 16q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 2.00000i 0.894427i −0.894427 0.447214i \(-0.852416\pi\)
0.894427 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 2.00000 + 3.00000i 0.554700 + 0.832050i
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 2.00000i 0.447214i
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 3.00000 2.00000i 0.588348 0.392232i
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 2.00000i 0.342997i
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 2.00000i 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 6.00000i 0.884652i
\(47\) 8.00000i 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 1.00000i 0.141421i
\(51\) 0 0
\(52\) −2.00000 3.00000i −0.277350 0.416025i
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 6.00000 4.00000i 0.744208 0.496139i
\(66\) 0 0
\(67\) 2.00000i 0.244339i −0.992509 0.122169i \(-0.961015\pi\)
0.992509 0.122169i \(-0.0389851\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 2.00000i 0.239046i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 14.0000i 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 4.00000i 0.458831i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.00000i 0.223607i
\(81\) 0 0
\(82\) 0 0
\(83\) 14.0000i 1.53670i 0.640030 + 0.768350i \(0.278922\pi\)
−0.640030 + 0.768350i \(0.721078\pi\)
\(84\) 0 0
\(85\) 4.00000i 0.433861i
\(86\) 4.00000i 0.431331i
\(87\) 0 0
\(88\) 0 0
\(89\) 4.00000i 0.423999i −0.977270 0.212000i \(-0.932002\pi\)
0.977270 0.212000i \(-0.0679975\pi\)
\(90\) 0 0
\(91\) −3.00000 + 2.00000i −0.314485 + 0.209657i
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 2.00000i 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −3.00000 + 2.00000i −0.294174 + 0.196116i
\(105\) 0 0
\(106\) 4.00000i 0.388514i
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 6.00000i 0.574696i −0.957826 0.287348i \(-0.907226\pi\)
0.957826 0.287348i \(-0.0927736\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000i 0.0944911i
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 12.0000i 1.11901i
\(116\) 0 0
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) 2.00000i 0.183340i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 12.0000i 1.08643i
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000i 1.07331i
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) −4.00000 6.00000i −0.350823 0.526235i
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) 2.00000i 0.171499i
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) 0 0
\(148\) 2.00000i 0.164399i
\(149\) 6.00000i 0.491539i 0.969328 + 0.245770i \(0.0790407\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.00000 0.638470 0.319235 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −2.00000 −0.158114
\(161\) 6.00000i 0.472866i
\(162\) 0 0
\(163\) 14.0000i 1.09656i −0.836293 0.548282i \(-0.815282\pi\)
0.836293 0.548282i \(-0.184718\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 14.0000 1.08661
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) −5.00000 + 12.0000i −0.384615 + 0.923077i
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 1.00000i 0.0755929i
\(176\) 0 0
\(177\) 0 0
\(178\) −4.00000 −0.299813
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 2.00000 + 3.00000i 0.148250 + 0.222375i
\(183\) 0 0
\(184\) 6.00000i 0.442326i
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) 8.00000i 0.583460i
\(189\) 0 0
\(190\) 8.00000i 0.580381i
\(191\) −2.00000 −0.144715 −0.0723575 0.997379i \(-0.523052\pi\)
−0.0723575 + 0.997379i \(0.523052\pi\)
\(192\) 0 0
\(193\) 4.00000i 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.0000i 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 0 0
\(202\) 2.00000i 0.140720i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 14.0000i 0.975426i
\(207\) 0 0
\(208\) 2.00000 + 3.00000i 0.138675 + 0.208013i
\(209\) 0 0
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) 4.00000 0.274721
\(213\) 0 0
\(214\) 12.0000i 0.820303i
\(215\) 8.00000i 0.545595i
\(216\) 0 0
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 0 0
\(220\) 0 0
\(221\) 4.00000 + 6.00000i 0.269069 + 0.403604i
\(222\) 0 0
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 14.0000i 0.931266i
\(227\) 22.0000i 1.46019i 0.683345 + 0.730096i \(0.260525\pi\)
−0.683345 + 0.730096i \(0.739475\pi\)
\(228\) 0 0
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) −12.0000 −0.791257
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −16.0000 −1.04372
\(236\) 6.00000i 0.390567i
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) 16.0000i 1.03495i 0.855697 + 0.517477i \(0.173129\pi\)
−0.855697 + 0.517477i \(0.826871\pi\)
\(240\) 0 0
\(241\) 10.0000i 0.644157i 0.946713 + 0.322078i \(0.104381\pi\)
−0.946713 + 0.322078i \(0.895619\pi\)
\(242\) 11.0000i 0.707107i
\(243\) 0 0
\(244\) −12.0000 −0.768221
\(245\) 2.00000i 0.127775i
\(246\) 0 0
\(247\) −12.0000 + 8.00000i −0.763542 + 0.509028i
\(248\) 0 0
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 12.0000i 0.752947i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) −6.00000 + 4.00000i −0.372104 + 0.248069i
\(261\) 0 0
\(262\) 12.0000i 0.741362i
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 8.00000i 0.491436i
\(266\) 4.00000i 0.245256i
\(267\) 0 0
\(268\) 2.00000i 0.122169i
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 20.0000i 1.19952i
\(279\) 0 0
\(280\) 2.00000i 0.119523i
\(281\) 30.0000i 1.78965i −0.446417 0.894825i \(-0.647300\pi\)
0.446417 0.894825i \(-0.352700\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 14.0000i 0.819288i
\(293\) 6.00000i 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 12.0000 + 18.0000i 0.693978 + 1.04097i
\(300\) 0 0
\(301\) 4.00000i 0.230556i
\(302\) 0 0
\(303\) 0 0
\(304\) 4.00000i 0.229416i
\(305\) 24.0000i 1.37424i
\(306\) 0 0
\(307\) 28.0000i 1.59804i 0.601302 + 0.799022i \(0.294649\pi\)
−0.601302 + 0.799022i \(0.705351\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 8.00000i 0.451466i
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 2.00000i 0.111803i
\(321\) 0 0
\(322\) 6.00000 0.334367
\(323\) 8.00000i 0.445132i
\(324\) 0 0
\(325\) 2.00000 + 3.00000i 0.110940 + 0.166410i
\(326\) −14.0000 −0.775388
\(327\) 0 0
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 10.0000i 0.549650i −0.961494 0.274825i \(-0.911380\pi\)
0.961494 0.274825i \(-0.0886199\pi\)
\(332\) 14.0000i 0.768350i
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 12.0000 + 5.00000i 0.652714 + 0.271964i
\(339\) 0 0
\(340\) 4.00000i 0.216930i
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 4.00000i 0.215666i
\(345\) 0 0
\(346\) 6.00000i 0.322562i
\(347\) 32.0000 1.71785 0.858925 0.512101i \(-0.171133\pi\)
0.858925 + 0.512101i \(0.171133\pi\)
\(348\) 0 0
\(349\) 14.0000i 0.749403i 0.927146 + 0.374701i \(0.122255\pi\)
−0.927146 + 0.374701i \(0.877745\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 0 0
\(353\) 24.0000i 1.27739i 0.769460 + 0.638696i \(0.220526\pi\)
−0.769460 + 0.638696i \(0.779474\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 4.00000i 0.212000i
\(357\) 0 0
\(358\) 20.0000i 1.05703i
\(359\) 24.0000i 1.26667i −0.773877 0.633336i \(-0.781685\pi\)
0.773877 0.633336i \(-0.218315\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 12.0000i 0.630706i
\(363\) 0 0
\(364\) 3.00000 2.00000i 0.157243 0.104828i
\(365\) −28.0000 −1.46559
\(366\) 0 0
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) 4.00000i 0.207950i
\(371\) 4.00000i 0.207670i
\(372\) 0 0
\(373\) 34.0000 1.76045 0.880227 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 0 0
\(378\) 0 0
\(379\) 26.0000i 1.33553i −0.744372 0.667765i \(-0.767251\pi\)
0.744372 0.667765i \(-0.232749\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) 2.00000i 0.102329i
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) 0 0
\(388\) 2.00000i 0.101535i
\(389\) −20.0000 −1.01404 −0.507020 0.861934i \(-0.669253\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 1.00000i 0.0505076i
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) 18.0000i 0.903394i 0.892171 + 0.451697i \(0.149181\pi\)
−0.892171 + 0.451697i \(0.850819\pi\)
\(398\) 10.0000i 0.501255i
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 30.0000i 1.49813i 0.662497 + 0.749064i \(0.269497\pi\)
−0.662497 + 0.749064i \(0.730503\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 14.0000i 0.692255i 0.938187 + 0.346128i \(0.112504\pi\)
−0.938187 + 0.346128i \(0.887496\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −14.0000 −0.689730
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 28.0000 1.37447
\(416\) 3.00000 2.00000i 0.147087 0.0980581i
\(417\) 0 0
\(418\) 0 0
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 30.0000i 1.46211i −0.682318 0.731055i \(-0.739028\pi\)
0.682318 0.731055i \(-0.260972\pi\)
\(422\) 28.0000i 1.36302i
\(423\) 0 0
\(424\) 4.00000i 0.194257i
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 12.0000i 0.580721i
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.00000i 0.287348i
\(437\) 24.0000i 1.14808i
\(438\) 0 0
\(439\) 30.0000 1.43182 0.715911 0.698192i \(-0.246012\pi\)
0.715911 + 0.698192i \(0.246012\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6.00000 4.00000i 0.285391 0.190261i
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) −8.00000 −0.379236
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) 14.0000i 0.660701i −0.943858 0.330350i \(-0.892833\pi\)
0.943858 0.330350i \(-0.107167\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) 22.0000 1.03251
\(455\) 4.00000 + 6.00000i 0.187523 + 0.281284i
\(456\) 0 0
\(457\) 8.00000i 0.374224i 0.982339 + 0.187112i \(0.0599128\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) 14.0000 0.654177
\(459\) 0 0
\(460\) 12.0000i 0.559503i
\(461\) 10.0000i 0.465746i −0.972507 0.232873i \(-0.925187\pi\)
0.972507 0.232873i \(-0.0748127\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 6.00000i 0.277945i
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 2.00000 0.0923514
\(470\) 16.0000i 0.738025i
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000i 0.183533i
\(476\) 2.00000i 0.0916698i
\(477\) 0 0
\(478\) 16.0000 0.731823
\(479\) 16.0000i 0.731059i 0.930800 + 0.365529i \(0.119112\pi\)
−0.930800 + 0.365529i \(0.880888\pi\)
\(480\) 0 0
\(481\) 6.00000 4.00000i 0.273576 0.182384i
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) 28.0000i 1.26880i 0.773004 + 0.634401i \(0.218753\pi\)
−0.773004 + 0.634401i \(0.781247\pi\)
\(488\) 12.0000i 0.543214i
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 8.00000 + 12.0000i 0.359937 + 0.539906i
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 14.0000i 0.626726i 0.949633 + 0.313363i \(0.101456\pi\)
−0.949633 + 0.313363i \(0.898544\pi\)
\(500\) 12.0000i 0.536656i
\(501\) 0 0
\(502\) 28.0000i 1.24970i
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 0 0
\(505\) 4.00000i 0.177998i
\(506\) 0 0
\(507\) 0 0
\(508\) 12.0000 0.532414
\(509\) 26.0000i 1.15243i 0.817298 + 0.576215i \(0.195471\pi\)
−0.817298 + 0.576215i \(0.804529\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 18.0000i 0.793946i
\(515\) 28.0000i 1.23383i
\(516\) 0 0
\(517\) 0 0
\(518\) 2.00000i 0.0878750i
\(519\) 0 0
\(520\) 4.00000 + 6.00000i 0.175412 + 0.263117i
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 6.00000i 0.261612i
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 8.00000 0.347498
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) 0 0
\(534\) 0 0
\(535\) 24.0000i 1.03761i
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) 10.0000i 0.431131i
\(539\) 0 0
\(540\) 0 0
\(541\) 30.0000i 1.28980i −0.764267 0.644900i \(-0.776899\pi\)
0.764267 0.644900i \(-0.223101\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 2.00000i 0.0857493i
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 2.00000i 0.0854358i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 22.0000i 0.934690i
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 18.0000i 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) 8.00000 + 12.0000i 0.338364 + 0.507546i
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) −30.0000 −1.26547
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 28.0000i 1.17797i
\(566\) 16.0000i 0.672530i
\(567\) 0 0
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) 22.0000i 0.915872i −0.888985 0.457936i \(-0.848589\pi\)
0.888985 0.457936i \(-0.151411\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 0 0
\(580\) 0 0
\(581\) −14.0000 −0.580818
\(582\) 0 0
\(583\) 0 0
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 18.0000i 0.742940i −0.928445 0.371470i \(-0.878854\pi\)
0.928445 0.371470i \(-0.121146\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 12.0000i 0.494032i
\(591\) 0 0
\(592\) 2.00000i 0.0821995i
\(593\) 24.0000i 0.985562i 0.870153 + 0.492781i \(0.164020\pi\)
−0.870153 + 0.492781i \(0.835980\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) 6.00000i 0.245770i
\(597\) 0 0
\(598\) 18.0000 12.0000i 0.736075 0.490716i
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) 4.00000 0.163028
\(603\) 0 0
\(604\) 0 0
\(605\) 22.0000i 0.894427i
\(606\) 0 0
\(607\) 18.0000 0.730597 0.365299 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −24.0000 −0.971732
\(611\) 24.0000 16.0000i 0.970936 0.647291i
\(612\) 0 0
\(613\) 34.0000i 1.37325i −0.727013 0.686624i \(-0.759092\pi\)
0.727013 0.686624i \(-0.240908\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000i 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 0 0
\(619\) 16.0000i 0.643094i −0.946894 0.321547i \(-0.895797\pi\)
0.946894 0.321547i \(-0.104203\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 12.0000i 0.481156i
\(623\) 4.00000 0.160257
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 6.00000i 0.239808i
\(627\) 0 0
\(628\) −8.00000 −0.319235
\(629\) 4.00000i 0.159490i
\(630\) 0 0
\(631\) 20.0000i 0.796187i −0.917345 0.398094i \(-0.869672\pi\)
0.917345 0.398094i \(-0.130328\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 2.00000 0.0794301
\(635\) 24.0000i 0.952411i
\(636\) 0 0
\(637\) −2.00000 3.00000i −0.0792429 0.118864i
\(638\) 0 0
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) 44.0000i 1.73519i −0.497271 0.867595i \(-0.665665\pi\)
0.497271 0.867595i \(-0.334335\pi\)
\(644\) 6.00000i 0.236433i
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 3.00000 2.00000i 0.117670 0.0784465i
\(651\) 0 0
\(652\) 14.0000i 0.548282i
\(653\) −4.00000 −0.156532 −0.0782660 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 0 0
\(655\) 24.0000i 0.937758i
\(656\) 0 0
\(657\) 0 0
\(658\) 8.00000i 0.311872i
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 10.0000i 0.388955i 0.980907 + 0.194477i \(0.0623011\pi\)
−0.980907 + 0.194477i \(0.937699\pi\)
\(662\) −10.0000 −0.388661
\(663\) 0 0
\(664\) −14.0000 −0.543305
\(665\) 8.00000i 0.310227i
\(666\) 0 0
\(667\) 0 0
\(668\) 12.0000i 0.464294i
\(669\) 0 0
\(670\) 4.00000i 0.154533i
\(671\) 0 0
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 22.0000i 0.847408i
\(675\) 0 0
\(676\) 5.00000 12.0000i 0.192308 0.461538i
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 4.00000 0.153393
\(681\) 0 0
\(682\) 0 0
\(683\) 16.0000i 0.612223i −0.951996 0.306111i \(-0.900972\pi\)
0.951996 0.306111i \(-0.0990280\pi\)
\(684\) 0 0
\(685\) 4.00000 0.152832
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −8.00000 12.0000i −0.304776 0.457164i
\(690\) 0 0
\(691\) 40.0000i 1.52167i −0.648944 0.760836i \(-0.724789\pi\)
0.648944 0.760836i \(-0.275211\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 32.0000i 1.21470i
\(695\) 40.0000i 1.51729i
\(696\) 0 0
\(697\) 0 0
\(698\) 14.0000 0.529908
\(699\) 0 0
\(700\) 1.00000i 0.0377964i
\(701\) 48.0000 1.81293 0.906467 0.422276i \(-0.138769\pi\)
0.906467 + 0.422276i \(0.138769\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 2.00000i 0.0752177i
\(708\) 0 0
\(709\) 46.0000i 1.72757i −0.503864 0.863783i \(-0.668089\pi\)
0.503864 0.863783i \(-0.331911\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 4.00000 0.149906
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 0 0
\(721\) 14.0000i 0.521387i
\(722\) 3.00000i 0.111648i
\(723\) 0 0
\(724\) −12.0000 −0.445976
\(725\) 0 0
\(726\) 0 0
\(727\) 38.0000 1.40934 0.704671 0.709534i \(-0.251095\pi\)
0.704671 + 0.709534i \(0.251095\pi\)
\(728\) −2.00000 3.00000i −0.0741249 0.111187i
\(729\) 0 0
\(730\) 28.0000i 1.03633i
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 14.0000i 0.517102i −0.965998 0.258551i \(-0.916755\pi\)
0.965998 0.258551i \(-0.0832450\pi\)
\(734\) 22.0000i 0.812035i
\(735\) 0 0
\(736\) 6.00000i 0.221163i
\(737\) 0 0
\(738\) 0 0
\(739\) 34.0000i 1.25071i 0.780340 + 0.625355i \(0.215046\pi\)
−0.780340 + 0.625355i \(0.784954\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) −4.00000 −0.146845
\(743\) 16.0000i 0.586983i −0.955962 0.293492i \(-0.905183\pi\)
0.955962 0.293492i \(-0.0948173\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) 34.0000i 1.24483i
\(747\) 0 0
\(748\) 0 0
\(749\) 12.0000i 0.438470i
\(750\) 0 0
\(751\) 12.0000 0.437886 0.218943 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 8.00000i 0.291730i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) −26.0000 −0.944363
\(759\) 0 0
\(760\) 8.00000i 0.290191i
\(761\) 40.0000i 1.45000i −0.688749 0.724999i \(-0.741840\pi\)
0.688749 0.724999i \(-0.258160\pi\)
\(762\) 0 0
\(763\) 6.00000 0.217215
\(764\) 2.00000 0.0723575
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) −18.0000 + 12.0000i −0.649942 + 0.433295i
\(768\) 0 0
\(769\) 26.0000i 0.937584i −0.883309 0.468792i \(-0.844689\pi\)
0.883309 0.468792i \(-0.155311\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.00000i 0.143963i
\(773\) 26.0000i 0.935155i −0.883952 0.467578i \(-0.845127\pi\)
0.883952 0.467578i \(-0.154873\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 20.0000i 0.717035i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 12.0000i 0.429119i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 16.0000i 0.571064i
\(786\) 0 0
\(787\) 48.0000i 1.71102i 0.517790 + 0.855508i \(0.326755\pi\)
−0.517790 + 0.855508i \(0.673245\pi\)
\(788\) 18.0000i 0.641223i
\(789\) 0 0
\(790\) 0 0
\(791\) 14.0000i 0.497783i
\(792\) 0 0
\(793\) 24.0000 + 36.0000i 0.852265 + 1.27840i
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) 10.0000 0.354441
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) 16.0000i 0.566039i
\(800\) 1.00000i 0.0353553i
\(801\) 0 0
\(802\) 30.0000 1.05934
\(803\) 0 0