Properties

Label 1440.2.x.i.127.1
Level $1440$
Weight $2$
Character 1440.127
Analytic conductor $11.498$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
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 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 127.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1440.127
Dual form 1440.2.x.i.703.1

$q$-expansion

\(f(q)\) \(=\) \(q+(2.00000 - 1.00000i) q^{5} +(-2.00000 + 2.00000i) q^{7} +O(q^{10})\) \(q+(2.00000 - 1.00000i) q^{5} +(-2.00000 + 2.00000i) q^{7} +(-1.00000 + 1.00000i) q^{13} +(5.00000 + 5.00000i) q^{17} -4.00000 q^{19} +(2.00000 + 2.00000i) q^{23} +(3.00000 - 4.00000i) q^{25} +4.00000i q^{29} +4.00000i q^{31} +(-2.00000 + 6.00000i) q^{35} +(1.00000 + 1.00000i) q^{37} +(6.00000 + 6.00000i) q^{43} +(-2.00000 + 2.00000i) q^{47} -1.00000i q^{49} +(7.00000 - 7.00000i) q^{53} +4.00000 q^{59} -4.00000 q^{61} +(-1.00000 + 3.00000i) q^{65} +(10.0000 - 10.0000i) q^{67} +12.0000i q^{71} +(-3.00000 + 3.00000i) q^{73} -16.0000 q^{79} +(-2.00000 - 2.00000i) q^{83} +(15.0000 + 5.00000i) q^{85} -4.00000i q^{91} +(-8.00000 + 4.00000i) q^{95} +(-3.00000 - 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 4 q^{7} + O(q^{10}) \) \( 2 q + 4 q^{5} - 4 q^{7} - 2 q^{13} + 10 q^{17} - 8 q^{19} + 4 q^{23} + 6 q^{25} - 4 q^{35} + 2 q^{37} + 12 q^{43} - 4 q^{47} + 14 q^{53} + 8 q^{59} - 8 q^{61} - 2 q^{65} + 20 q^{67} - 6 q^{73} - 32 q^{79} - 4 q^{83} + 30 q^{85} - 16 q^{95} - 6 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\) \(-1\)

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 0
\(4\) 0 0
\(5\) 2.00000 1.00000i 0.894427 0.447214i
\(6\) 0 0
\(7\) −2.00000 + 2.00000i −0.755929 + 0.755929i −0.975579 0.219650i \(-0.929509\pi\)
0.219650 + 0.975579i \(0.429509\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −1.00000 + 1.00000i −0.277350 + 0.277350i −0.832050 0.554700i \(-0.812833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.00000 + 5.00000i 1.21268 + 1.21268i 0.970143 + 0.242536i \(0.0779791\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 + 2.00000i 0.417029 + 0.417029i 0.884178 0.467150i \(-0.154719\pi\)
−0.467150 + 0.884178i \(0.654719\pi\)
\(24\) 0 0
\(25\) 3.00000 4.00000i 0.600000 0.800000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000i 0.742781i 0.928477 + 0.371391i \(0.121119\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 + 6.00000i −0.338062 + 1.01419i
\(36\) 0 0
\(37\) 1.00000 + 1.00000i 0.164399 + 0.164399i 0.784512 0.620113i \(-0.212913\pi\)
−0.620113 + 0.784512i \(0.712913\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 6.00000 + 6.00000i 0.914991 + 0.914991i 0.996660 0.0816682i \(-0.0260248\pi\)
−0.0816682 + 0.996660i \(0.526025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.00000 + 2.00000i −0.291730 + 0.291730i −0.837763 0.546033i \(-0.816137\pi\)
0.546033 + 0.837763i \(0.316137\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.00000 7.00000i 0.961524 0.961524i −0.0377628 0.999287i \(-0.512023\pi\)
0.999287 + 0.0377628i \(0.0120231\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.00000 + 3.00000i −0.124035 + 0.372104i
\(66\) 0 0
\(67\) 10.0000 10.0000i 1.22169 1.22169i 0.254665 0.967029i \(-0.418035\pi\)
0.967029 0.254665i \(-0.0819652\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000i 1.42414i 0.702109 + 0.712069i \(0.252242\pi\)
−0.702109 + 0.712069i \(0.747758\pi\)
\(72\) 0 0
\(73\) −3.00000 + 3.00000i −0.351123 + 0.351123i −0.860527 0.509404i \(-0.829866\pi\)
0.509404 + 0.860527i \(0.329866\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.00000 2.00000i −0.219529 0.219529i 0.588771 0.808300i \(-0.299612\pi\)
−0.808300 + 0.588771i \(0.799612\pi\)
\(84\) 0 0
\(85\) 15.0000 + 5.00000i 1.62698 + 0.542326i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 4.00000i 0.419314i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 + 4.00000i −0.820783 + 0.410391i
\(96\) 0 0
\(97\) −3.00000 3.00000i −0.304604 0.304604i 0.538208 0.842812i \(-0.319101\pi\)
−0.842812 + 0.538208i \(0.819101\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 6.00000 + 6.00000i 0.591198 + 0.591198i 0.937955 0.346757i \(-0.112717\pi\)
−0.346757 + 0.937955i \(0.612717\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000 6.00000i 0.580042 0.580042i −0.354873 0.934915i \(-0.615476\pi\)
0.934915 + 0.354873i \(0.115476\pi\)
\(108\) 0 0
\(109\) 10.0000i 0.957826i 0.877862 + 0.478913i \(0.158969\pi\)
−0.877862 + 0.478913i \(0.841031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.00000 9.00000i 0.846649 0.846649i −0.143065 0.989713i \(-0.545696\pi\)
0.989713 + 0.143065i \(0.0456957\pi\)
\(114\) 0 0
\(115\) 6.00000 + 2.00000i 0.559503 + 0.186501i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −20.0000 −1.83340
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.00000 11.0000i 0.178885 0.983870i
\(126\) 0 0
\(127\) −10.0000 + 10.0000i −0.887357 + 0.887357i −0.994268 0.106912i \(-0.965904\pi\)
0.106912 + 0.994268i \(0.465904\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000i 0.698963i 0.936943 + 0.349482i \(0.113642\pi\)
−0.936943 + 0.349482i \(0.886358\pi\)
\(132\) 0 0
\(133\) 8.00000 8.00000i 0.693688 0.693688i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.00000 1.00000i −0.0854358 0.0854358i 0.663097 0.748533i \(-0.269242\pi\)
−0.748533 + 0.663097i \(0.769242\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 4.00000 + 8.00000i 0.332182 + 0.664364i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 18.0000i 1.47462i 0.675556 + 0.737309i \(0.263904\pi\)
−0.675556 + 0.737309i \(0.736096\pi\)
\(150\) 0 0
\(151\) 12.0000i 0.976546i −0.872691 0.488273i \(-0.837627\pi\)
0.872691 0.488273i \(-0.162373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 + 8.00000i 0.321288 + 0.642575i
\(156\) 0 0
\(157\) −9.00000 9.00000i −0.718278 0.718278i 0.249974 0.968252i \(-0.419578\pi\)
−0.968252 + 0.249974i \(0.919578\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) −2.00000 2.00000i −0.156652 0.156652i 0.624429 0.781081i \(-0.285332\pi\)
−0.781081 + 0.624429i \(0.785332\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.00000 + 2.00000i −0.154765 + 0.154765i −0.780242 0.625478i \(-0.784904\pi\)
0.625478 + 0.780242i \(0.284904\pi\)
\(168\) 0 0
\(169\) 11.0000i 0.846154i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −13.0000 + 13.0000i −0.988372 + 0.988372i −0.999933 0.0115615i \(-0.996320\pi\)
0.0115615 + 0.999933i \(0.496320\pi\)
\(174\) 0 0
\(175\) 2.00000 + 14.0000i 0.151186 + 1.05830i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.00000 + 1.00000i 0.220564 + 0.0735215i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.0000i 1.44715i −0.690246 0.723575i \(-0.742498\pi\)
0.690246 0.723575i \(-0.257502\pi\)
\(192\) 0 0
\(193\) −5.00000 + 5.00000i −0.359908 + 0.359908i −0.863779 0.503871i \(-0.831909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.00000 5.00000i −0.356235 0.356235i 0.506188 0.862423i \(-0.331054\pi\)
−0.862423 + 0.506188i \(0.831054\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.00000 8.00000i −0.561490 0.561490i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000i 1.10149i −0.834675 0.550743i \(-0.814345\pi\)
0.834675 0.550743i \(-0.185655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 18.0000 + 6.00000i 1.22759 + 0.409197i
\(216\) 0 0
\(217\) −8.00000 8.00000i −0.543075 0.543075i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −10.0000 −0.672673
\(222\) 0 0
\(223\) 10.0000 + 10.0000i 0.669650 + 0.669650i 0.957635 0.287985i \(-0.0929854\pi\)
−0.287985 + 0.957635i \(0.592985\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.0000 10.0000i 0.663723 0.663723i −0.292532 0.956256i \(-0.594498\pi\)
0.956256 + 0.292532i \(0.0944979\pi\)
\(228\) 0 0
\(229\) 20.0000i 1.32164i −0.750546 0.660819i \(-0.770209\pi\)
0.750546 0.660819i \(-0.229791\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.00000 5.00000i 0.327561 0.327561i −0.524097 0.851658i \(-0.675597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) −2.00000 + 6.00000i −0.130466 + 0.391397i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −16.0000 −1.03065 −0.515325 0.856995i \(-0.672329\pi\)
−0.515325 + 0.856995i \(0.672329\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.00000 2.00000i −0.0638877 0.127775i
\(246\) 0 0
\(247\) 4.00000 4.00000i 0.254514 0.254514i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.0000i 1.51487i −0.652913 0.757433i \(-0.726453\pi\)
0.652913 0.757433i \(-0.273547\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.00000 7.00000i −0.436648 0.436648i 0.454234 0.890882i \(-0.349913\pi\)
−0.890882 + 0.454234i \(0.849913\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.00000 + 6.00000i 0.369976 + 0.369976i 0.867468 0.497492i \(-0.165746\pi\)
−0.497492 + 0.867468i \(0.665746\pi\)
\(264\) 0 0
\(265\) 7.00000 21.0000i 0.430007 1.29002i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.0000i 0.609711i 0.952399 + 0.304855i \(0.0986081\pi\)
−0.952399 + 0.304855i \(0.901392\pi\)
\(270\) 0 0
\(271\) 20.0000i 1.21491i −0.794353 0.607457i \(-0.792190\pi\)
0.794353 0.607457i \(-0.207810\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −9.00000 9.00000i −0.540758 0.540758i 0.382993 0.923751i \(-0.374893\pi\)
−0.923751 + 0.382993i \(0.874893\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) −6.00000 6.00000i −0.356663 0.356663i 0.505918 0.862581i \(-0.331154\pi\)
−0.862581 + 0.505918i \(0.831154\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 33.0000i 1.94118i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.00000 5.00000i 0.292103 0.292103i −0.545807 0.837911i \(-0.683777\pi\)
0.837911 + 0.545807i \(0.183777\pi\)
\(294\) 0 0
\(295\) 8.00000 4.00000i 0.465778 0.232889i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8.00000 + 4.00000i −0.458079 + 0.229039i
\(306\) 0 0
\(307\) 10.0000 10.0000i 0.570730 0.570730i −0.361602 0.932332i \(-0.617770\pi\)
0.932332 + 0.361602i \(0.117770\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.0000i 1.58773i −0.608091 0.793867i \(-0.708065\pi\)
0.608091 0.793867i \(-0.291935\pi\)
\(312\) 0 0
\(313\) 15.0000 15.0000i 0.847850 0.847850i −0.142014 0.989865i \(-0.545358\pi\)
0.989865 + 0.142014i \(0.0453579\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −11.0000 11.0000i −0.617822 0.617822i 0.327151 0.944972i \(-0.393912\pi\)
−0.944972 + 0.327151i \(0.893912\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −20.0000 20.0000i −1.11283 1.11283i
\(324\) 0 0
\(325\) 1.00000 + 7.00000i 0.0554700 + 0.388290i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 8.00000i 0.441054i
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 10.0000 30.0000i 0.546358 1.63908i
\(336\) 0 0
\(337\) −23.0000 23.0000i −1.25289 1.25289i −0.954419 0.298471i \(-0.903523\pi\)
−0.298471 0.954419i \(-0.596477\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −12.0000 12.0000i −0.647939 0.647939i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.0000 18.0000i 0.966291 0.966291i −0.0331594 0.999450i \(-0.510557\pi\)
0.999450 + 0.0331594i \(0.0105569\pi\)
\(348\) 0 0
\(349\) 20.0000i 1.07058i 0.844670 + 0.535288i \(0.179797\pi\)
−0.844670 + 0.535288i \(0.820203\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.00000 + 9.00000i −0.479022 + 0.479022i −0.904819 0.425797i \(-0.859994\pi\)
0.425797 + 0.904819i \(0.359994\pi\)
\(354\) 0 0
\(355\) 12.0000 + 24.0000i 0.636894 + 1.27379i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.00000 + 9.00000i −0.157027 + 0.471082i
\(366\) 0 0
\(367\) 22.0000 22.0000i 1.14839 1.14839i 0.161521 0.986869i \(-0.448360\pi\)
0.986869 0.161521i \(-0.0516401\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 28.0000i 1.45369i
\(372\) 0 0
\(373\) 21.0000 21.0000i 1.08734 1.08734i 0.0915371 0.995802i \(-0.470822\pi\)
0.995802 0.0915371i \(-0.0291780\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 4.00000i −0.206010 0.206010i
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 22.0000 + 22.0000i 1.12415 + 1.12415i 0.991111 + 0.133036i \(0.0424727\pi\)
0.133036 + 0.991111i \(0.457527\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 18.0000i 0.912636i −0.889817 0.456318i \(-0.849168\pi\)
0.889817 0.456318i \(-0.150832\pi\)
\(390\) 0 0
\(391\) 20.0000i 1.01144i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −32.0000 + 16.0000i −1.61009 + 0.805047i
\(396\) 0 0
\(397\) 13.0000 + 13.0000i 0.652451 + 0.652451i 0.953583 0.301131i \(-0.0973643\pi\)
−0.301131 + 0.953583i \(0.597364\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −34.0000 −1.69788 −0.848939 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) 0 0
\(403\) −4.00000 4.00000i −0.199254 0.199254i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.00000i 0.0988936i 0.998777 + 0.0494468i \(0.0157458\pi\)
−0.998777 + 0.0494468i \(0.984254\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.00000 + 8.00000i −0.393654 + 0.393654i
\(414\) 0 0
\(415\) −6.00000 2.00000i −0.294528 0.0981761i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 35.0000 5.00000i 1.69775 0.242536i
\(426\) 0 0
\(427\) 8.00000 8.00000i 0.387147 0.387147i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.00000i 0.192673i 0.995349 + 0.0963366i \(0.0307125\pi\)
−0.995349 + 0.0963366i \(0.969287\pi\)
\(432\) 0 0
\(433\) −19.0000 + 19.0000i −0.913082 + 0.913082i −0.996513 0.0834318i \(-0.973412\pi\)
0.0834318 + 0.996513i \(0.473412\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8.00000 8.00000i −0.382692 0.382692i
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 22.0000 + 22.0000i 1.04525 + 1.04525i 0.998926 + 0.0463251i \(0.0147510\pi\)
0.0463251 + 0.998926i \(0.485249\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 26.0000i 1.22702i 0.789689 + 0.613508i \(0.210242\pi\)
−0.789689 + 0.613508i \(0.789758\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4.00000 8.00000i −0.187523 0.375046i
\(456\) 0 0
\(457\) 15.0000 + 15.0000i 0.701670 + 0.701670i 0.964769 0.263099i \(-0.0847444\pi\)
−0.263099 + 0.964769i \(0.584744\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) 22.0000 + 22.0000i 1.02243 + 1.02243i 0.999743 + 0.0226840i \(0.00722117\pi\)
0.0226840 + 0.999743i \(0.492779\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.00000 2.00000i 0.0925490 0.0925490i −0.659317 0.751865i \(-0.729154\pi\)
0.751865 + 0.659317i \(0.229154\pi\)
\(468\) 0 0
\(469\) 40.0000i 1.84703i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −12.0000 + 16.0000i −0.550598 + 0.734130i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −9.00000 3.00000i −0.408669 0.136223i
\(486\) 0 0
\(487\) 6.00000 6.00000i 0.271886 0.271886i −0.557973 0.829859i \(-0.688421\pi\)
0.829859 + 0.557973i \(0.188421\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 16.0000i 0.722070i −0.932552 0.361035i \(-0.882424\pi\)
0.932552 0.361035i \(-0.117576\pi\)
\(492\) 0 0
\(493\) −20.0000 + 20.0000i −0.900755 + 0.900755i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.0000 24.0000i −1.07655 1.07655i
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 10.0000 + 10.0000i 0.445878 + 0.445878i 0.893982 0.448104i \(-0.147900\pi\)
−0.448104 + 0.893982i \(0.647900\pi\)
\(504\) 0 0
\(505\) 12.0000 6.00000i 0.533993 0.266996i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 36.0000i 1.59567i −0.602875 0.797836i \(-0.705978\pi\)
0.602875 0.797836i \(-0.294022\pi\)
\(510\) 0 0
\(511\) 12.0000i 0.530849i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 18.0000 + 6.00000i 0.793175 + 0.264392i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −14.0000 14.0000i −0.612177 0.612177i 0.331336 0.943513i \(-0.392501\pi\)
−0.943513 + 0.331336i \(0.892501\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −20.0000 + 20.0000i −0.871214 + 0.871214i
\(528\) 0 0
\(529\) 15.0000i 0.652174i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 6.00000 18.0000i 0.259403 0.778208i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.0000 + 20.0000i 0.428353 + 0.856706i
\(546\) 0 0
\(547\) 6.00000 6.00000i 0.256541 0.256541i −0.567104 0.823646i \(-0.691936\pi\)
0.823646 + 0.567104i \(0.191936\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 16.0000i 0.681623i
\(552\) 0 0
\(553\) 32.0000 32.0000i 1.36078 1.36078i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.0000 + 15.0000i 0.635570 + 0.635570i 0.949460 0.313889i \(-0.101632\pi\)
−0.313889 + 0.949460i \(0.601632\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.00000 6.00000i −0.252870 0.252870i 0.569276 0.822146i \(-0.307223\pi\)
−0.822146 + 0.569276i \(0.807223\pi\)
\(564\) 0 0
\(565\) 9.00000 27.0000i 0.378633 1.13590i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.00000i 0.0838444i 0.999121 + 0.0419222i \(0.0133482\pi\)
−0.999121 + 0.0419222i \(0.986652\pi\)
\(570\) 0 0
\(571\) 16.0000i 0.669579i 0.942293 + 0.334790i \(0.108665\pi\)
−0.942293 + 0.334790i \(0.891335\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14.0000 2.00000i 0.583840 0.0834058i
\(576\) 0 0
\(577\) 15.0000 + 15.0000i 0.624458 + 0.624458i 0.946668 0.322210i \(-0.104426\pi\)
−0.322210 + 0.946668i \(0.604426\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −14.0000 + 14.0000i −0.577842 + 0.577842i −0.934308 0.356466i \(-0.883981\pi\)
0.356466 + 0.934308i \(0.383981\pi\)
\(588\) 0 0
\(589\) 16.0000i 0.659269i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.00000 + 1.00000i −0.0410651 + 0.0410651i −0.727341 0.686276i \(-0.759244\pi\)
0.686276 + 0.727341i \(0.259244\pi\)
\(594\) 0 0
\(595\) −40.0000 + 20.0000i −1.63984 + 0.819920i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 22.0000 11.0000i 0.894427 0.447214i
\(606\) 0 0
\(607\) 18.0000 18.0000i 0.730597 0.730597i −0.240141 0.970738i \(-0.577194\pi\)
0.970738 + 0.240141i \(0.0771936\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.00000i 0.161823i
\(612\) 0 0
\(613\) −9.00000 + 9.00000i −0.363507 + 0.363507i −0.865102 0.501596i \(-0.832747\pi\)
0.501596 + 0.865102i \(0.332747\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 29.0000 + 29.0000i 1.16750 + 1.16750i 0.982795 + 0.184701i \(0.0591318\pi\)
0.184701 + 0.982795i \(0.440868\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10.0000i 0.398726i
\(630\) 0 0
\(631\) 4.00000i 0.159237i −0.996825 0.0796187i \(-0.974630\pi\)
0.996825 0.0796187i \(-0.0253703\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −10.0000 + 30.0000i −0.396838 + 1.19051i
\(636\) 0 0
\(637\) 1.00000 + 1.00000i 0.0396214 + 0.0396214i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 48.0000 1.89589 0.947943 0.318440i \(-0.103159\pi\)
0.947943 + 0.318440i \(0.103159\pi\)
\(642\) 0 0
\(643\) 10.0000 + 10.0000i 0.394362 + 0.394362i 0.876239 0.481877i \(-0.160045\pi\)
−0.481877 + 0.876239i \(0.660045\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.0000 10.0000i 0.393141 0.393141i −0.482665 0.875805i \(-0.660331\pi\)
0.875805 + 0.482665i \(0.160331\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.00000 + 1.00000i −0.0391330 + 0.0391330i −0.726403 0.687270i \(-0.758809\pi\)
0.687270 + 0.726403i \(0.258809\pi\)
\(654\) 0 0
\(655\) 8.00000 + 16.0000i 0.312586 + 0.625172i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) 12.0000 0.466746 0.233373 0.972387i \(-0.425024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.00000 24.0000i 0.310227 0.930680i
\(666\) 0 0
\(667\) −8.00000 + 8.00000i −0.309761 + 0.309761i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 5.00000 5.00000i 0.192736 0.192736i −0.604141 0.796877i \(-0.706484\pi\)
0.796877 + 0.604141i \(0.206484\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.00000 + 3.00000i 0.115299 + 0.115299i 0.762402 0.647103i \(-0.224020\pi\)
−0.647103 + 0.762402i \(0.724020\pi\)
\(678\) 0 0
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −22.0000 22.0000i −0.841807 0.841807i 0.147287 0.989094i \(-0.452946\pi\)
−0.989094 + 0.147287i \(0.952946\pi\)
\(684\) 0 0
\(685\) −3.00000 1.00000i −0.114624 0.0382080i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 14.0000i 0.533358i
\(690\) 0 0
\(691\) 32.0000i 1.21734i 0.793424 + 0.608669i \(0.208296\pi\)
−0.793424 + 0.608669i \(0.791704\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 24.0000 12.0000i 0.910372 0.455186i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −20.0000 −0.755390 −0.377695 0.925930i \(-0.623283\pi\)
−0.377695 + 0.925930i \(0.623283\pi\)
\(702\) 0 0
\(703\) −4.00000 4.00000i −0.150863 0.150863i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.0000 + 12.0000i −0.451306 + 0.451306i
\(708\) 0 0
\(709\) 12.0000i 0.450669i −0.974281 0.225335i \(-0.927652\pi\)
0.974281 0.225335i \(-0.0723476\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −8.00000 + 8.00000i −0.299602 + 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 16.0000 + 12.0000i 0.594225 + 0.445669i
\(726\) 0 0
\(727\) −18.0000 + 18.0000i −0.667583 + 0.667583i −0.957156 0.289573i \(-0.906487\pi\)
0.289573 + 0.957156i \(0.406487\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 60.0000i 2.21918i
\(732\) 0 0
\(733\) 21.0000 21.0000i 0.775653 0.775653i −0.203436 0.979088i \(-0.565211\pi\)
0.979088 + 0.203436i \(0.0652108\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 30.0000 + 30.0000i 1.10059 + 1.10059i 0.994339 + 0.106254i \(0.0338857\pi\)
0.106254 + 0.994339i \(0.466114\pi\)
\(744\) 0 0
\(745\) 18.0000 + 36.0000i 0.659469 + 1.31894i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 24.0000i 0.876941i
\(750\) 0 0
\(751\) 44.0000i 1.60558i −0.596260 0.802791i \(-0.703347\pi\)
0.596260 0.802791i \(-0.296653\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −12.0000 24.0000i −0.436725 0.873449i
\(756\) 0 0
\(757\) −1.00000 1.00000i −0.0363456 0.0363456i 0.688700 0.725046i \(-0.258182\pi\)
−0.725046 + 0.688700i \(0.758182\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) −20.0000 20.0000i −0.724049 0.724049i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.00000 + 4.00000i −0.144432 + 0.144432i
\(768\) 0 0
\(769\) 40.0000i 1.44244i 0.692708 + 0.721218i \(0.256418\pi\)
−0.692708 + 0.721218i \(0.743582\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.00000 1.00000i 0.0359675 0.0359675i −0.688894 0.724862i \(-0.741904\pi\)
0.724862 + 0.688894i \(0.241904\pi\)
\(774\) 0 0
\(775\) 16.0000 + 12.0000i 0.574737 + 0.431053i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −27.0000 9.00000i −0.963671 0.321224i
\(786\) 0 0
\(787\) −30.0000 + 30.0000i −1.06938 + 1.06938i −0.0719783 + 0.997406i \(0.522931\pi\)
−0.997406 + 0.0719783i \(0.977069\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 36.0000i 1.28001i
\(792\) 0 0
\(793\) 4.00000 4.00000i 0.142044 0.142044i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 29.0000 + 29.0000i 1.02723 + 1.02723i 0.999619 + 0.0276140i \(0.00879094\pi\)
0.0276140 + 0.999619i \(0.491209\pi\)
\(798\) 0 0
\(799\) −20.0000 −0.707549
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0