Properties

Label 800.2.n.j.607.1
Level $800$
Weight $2$
Character 800.607
Analytic conductor $6.388$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 607.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 800.607
Dual form 800.2.n.j.543.1

$q$-expansion

\(f(q)\) \(=\) \(q+(2.00000 + 2.00000i) q^{3} +(2.00000 - 2.00000i) q^{7} +5.00000i q^{9} +O(q^{10})\) \(q+(2.00000 + 2.00000i) q^{3} +(2.00000 - 2.00000i) q^{7} +5.00000i q^{9} +(1.00000 - 1.00000i) q^{13} +(5.00000 + 5.00000i) q^{17} -4.00000 q^{19} +8.00000 q^{21} +(2.00000 + 2.00000i) q^{23} +(-4.00000 + 4.00000i) q^{27} -4.00000i q^{29} +4.00000i q^{31} +(-1.00000 - 1.00000i) q^{37} +4.00000 q^{39} +(-6.00000 - 6.00000i) q^{43} +(-2.00000 + 2.00000i) q^{47} -1.00000i q^{49} +20.0000i q^{51} +(7.00000 - 7.00000i) q^{53} +(-8.00000 - 8.00000i) q^{57} -4.00000 q^{59} -4.00000 q^{61} +(10.0000 + 10.0000i) q^{63} +(-10.0000 + 10.0000i) q^{67} +8.00000i q^{69} -12.0000i q^{71} +(3.00000 - 3.00000i) q^{73} -16.0000 q^{79} -1.00000 q^{81} +(-2.00000 - 2.00000i) q^{83} +(8.00000 - 8.00000i) q^{87} -4.00000i q^{91} +(-8.00000 + 8.00000i) q^{93} +(3.00000 + 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} + 4q^{7} + O(q^{10}) \) \( 2q + 4q^{3} + 4q^{7} + 2q^{13} + 10q^{17} - 8q^{19} + 16q^{21} + 4q^{23} - 8q^{27} - 2q^{37} + 8q^{39} - 12q^{43} - 4q^{47} + 14q^{53} - 16q^{57} - 8q^{59} - 8q^{61} + 20q^{63} - 20q^{67} + 6q^{73} - 32q^{79} - 2q^{81} - 4q^{83} + 16q^{87} - 16q^{93} + 6q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) 2.00000 + 2.00000i 1.15470 + 1.15470i 0.985599 + 0.169102i \(0.0540867\pi\)
0.169102 + 0.985599i \(0.445913\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000 2.00000i 0.755929 0.755929i −0.219650 0.975579i \(-0.570491\pi\)
0.975579 + 0.219650i \(0.0704915\pi\)
\(8\) 0 0
\(9\) 5.00000i 1.66667i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 1.00000 1.00000i 0.277350 0.277350i −0.554700 0.832050i \(-0.687167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 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
\(21\) 8.00000 1.74574
\(22\) 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
\(25\) 0 0
\(26\) 0 0
\(27\) −4.00000 + 4.00000i −0.769800 + 0.769800i
\(28\) 0 0
\(29\) 4.00000i 0.742781i −0.928477 0.371391i \(-0.878881\pi\)
0.928477 0.371391i \(-0.121119\pi\)
\(30\) 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
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 1.00000i −0.164399 0.164399i 0.620113 0.784512i \(-0.287087\pi\)
−0.784512 + 0.620113i \(0.787087\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 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 0.0816682 0.996660i \(-0.473975\pi\)
−0.996660 + 0.0816682i \(0.973975\pi\)
\(44\) 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
\(51\) 20.0000i 2.80056i
\(52\) 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
\(57\) −8.00000 8.00000i −1.05963 1.05963i
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 10.0000 + 10.0000i 1.25988 + 1.25988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −10.0000 + 10.0000i −1.22169 + 1.22169i −0.254665 + 0.967029i \(0.581965\pi\)
−0.967029 + 0.254665i \(0.918035\pi\)
\(68\) 0 0
\(69\) 8.00000i 0.963087i
\(70\) 0 0
\(71\) 12.0000i 1.42414i −0.702109 0.712069i \(-0.747758\pi\)
0.702109 0.712069i \(-0.252242\pi\)
\(72\) 0 0
\(73\) 3.00000 3.00000i 0.351123 0.351123i −0.509404 0.860527i \(-0.670134\pi\)
0.860527 + 0.509404i \(0.170134\pi\)
\(74\) 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
\(81\) −1.00000 −0.111111
\(82\) 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
\(85\) 0 0
\(86\) 0 0
\(87\) 8.00000 8.00000i 0.857690 0.857690i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 4.00000i 0.419314i
\(92\) 0 0
\(93\) −8.00000 + 8.00000i −0.829561 + 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.00000 + 3.00000i 0.304604 + 0.304604i 0.842812 0.538208i \(-0.180899\pi\)
−0.538208 + 0.842812i \(0.680899\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −6.00000 6.00000i −0.591198 0.591198i 0.346757 0.937955i \(-0.387283\pi\)
−0.937955 + 0.346757i \(0.887283\pi\)
\(104\) 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
\(111\) 4.00000i 0.379663i
\(112\) 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
\(115\) 0 0
\(116\) 0 0
\(117\) 5.00000 + 5.00000i 0.462250 + 0.462250i
\(118\) 0 0
\(119\) 20.0000 1.83340
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.0000 10.0000i 0.887357 0.887357i −0.106912 0.994268i \(-0.534096\pi\)
0.994268 + 0.106912i \(0.0340963\pi\)
\(128\) 0 0
\(129\) 24.0000i 2.11308i
\(130\) 0 0
\(131\) 8.00000i 0.698963i −0.936943 0.349482i \(-0.886358\pi\)
0.936943 0.349482i \(-0.113642\pi\)
\(132\) 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
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.00000 2.00000i 0.164957 0.164957i
\(148\) 0 0
\(149\) 18.0000i 1.47462i −0.675556 0.737309i \(-0.736096\pi\)
0.675556 0.737309i \(-0.263904\pi\)
\(150\) 0 0
\(151\) 12.0000i 0.976546i −0.872691 0.488273i \(-0.837627\pi\)
0.872691 0.488273i \(-0.162373\pi\)
\(152\) 0 0
\(153\) −25.0000 + 25.0000i −2.02113 + 2.02113i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 9.00000 + 9.00000i 0.718278 + 0.718278i 0.968252 0.249974i \(-0.0804222\pi\)
−0.249974 + 0.968252i \(0.580422\pi\)
\(158\) 0 0
\(159\) 28.0000 2.22054
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 2.00000 + 2.00000i 0.156652 + 0.156652i 0.781081 0.624429i \(-0.214668\pi\)
−0.624429 + 0.781081i \(0.714668\pi\)
\(164\) 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
\(171\) 20.0000i 1.52944i
\(172\) 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
\(175\) 0 0
\(176\) 0 0
\(177\) −8.00000 8.00000i −0.601317 0.601317i
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −8.00000 8.00000i −0.591377 0.591377i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 16.0000i 1.16383i
\(190\) 0 0
\(191\) 20.0000i 1.44715i 0.690246 + 0.723575i \(0.257502\pi\)
−0.690246 + 0.723575i \(0.742498\pi\)
\(192\) 0 0
\(193\) 5.00000 5.00000i 0.359908 0.359908i −0.503871 0.863779i \(-0.668091\pi\)
0.863779 + 0.503871i \(0.168091\pi\)
\(194\) 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
\(201\) −40.0000 −2.82138
\(202\) 0 0
\(203\) −8.00000 8.00000i −0.561490 0.561490i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −10.0000 + 10.0000i −0.695048 + 0.695048i
\(208\) 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
\(213\) 24.0000 24.0000i 1.64445 1.64445i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 8.00000 + 8.00000i 0.543075 + 0.543075i
\(218\) 0 0
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 10.0000 0.672673
\(222\) 0 0
\(223\) −10.0000 10.0000i −0.669650 0.669650i 0.287985 0.957635i \(-0.407015\pi\)
−0.957635 + 0.287985i \(0.907015\pi\)
\(224\) 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
\(235\) 0 0
\(236\) 0 0
\(237\) −32.0000 32.0000i −2.07862 2.07862i
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −16.0000 −1.03065 −0.515325 0.856995i \(-0.672329\pi\)
−0.515325 + 0.856995i \(0.672329\pi\)
\(242\) 0 0
\(243\) 10.0000 + 10.0000i 0.641500 + 0.641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 + 4.00000i −0.254514 + 0.254514i
\(248\) 0 0
\(249\) 8.00000i 0.506979i
\(250\) 0 0
\(251\) 24.0000i 1.51487i 0.652913 + 0.757433i \(0.273547\pi\)
−0.652913 + 0.757433i \(0.726453\pi\)
\(252\) 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
\(261\) 20.0000 1.23797
\(262\) 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
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.0000i 0.609711i −0.952399 0.304855i \(-0.901392\pi\)
0.952399 0.304855i \(-0.0986081\pi\)
\(270\) 0 0
\(271\) 20.0000i 1.21491i −0.794353 0.607457i \(-0.792190\pi\)
0.794353 0.607457i \(-0.207810\pi\)
\(272\) 0 0
\(273\) 8.00000 8.00000i 0.484182 0.484182i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 9.00000 + 9.00000i 0.540758 + 0.540758i 0.923751 0.382993i \(-0.125107\pi\)
−0.382993 + 0.923751i \(0.625107\pi\)
\(278\) 0 0
\(279\) −20.0000 −1.19737
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) 6.00000 + 6.00000i 0.356663 + 0.356663i 0.862581 0.505918i \(-0.168846\pi\)
−0.505918 + 0.862581i \(0.668846\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 33.0000i 1.94118i
\(290\) 0 0
\(291\) 12.0000i 0.703452i
\(292\) 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
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 0 0
\(303\) −12.0000 12.0000i −0.689382 0.689382i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −10.0000 + 10.0000i −0.570730 + 0.570730i −0.932332 0.361602i \(-0.882230\pi\)
0.361602 + 0.932332i \(0.382230\pi\)
\(308\) 0 0
\(309\) 24.0000i 1.36531i
\(310\) 0 0
\(311\) 28.0000i 1.58773i 0.608091 + 0.793867i \(0.291935\pi\)
−0.608091 + 0.793867i \(0.708065\pi\)
\(312\) 0 0
\(313\) −15.0000 + 15.0000i −0.847850 + 0.847850i −0.989865 0.142014i \(-0.954642\pi\)
0.142014 + 0.989865i \(0.454642\pi\)
\(314\) 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
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) −20.0000 20.0000i −1.11283 1.11283i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −20.0000 + 20.0000i −1.10600 + 1.10600i
\(328\) 0 0
\(329\) 8.00000i 0.441054i
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 5.00000 5.00000i 0.273998 0.273998i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 23.0000 + 23.0000i 1.25289 + 1.25289i 0.954419 + 0.298471i \(0.0964767\pi\)
0.298471 + 0.954419i \(0.403523\pi\)
\(338\) 0 0
\(339\) 36.0000 1.95525
\(340\) 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
\(351\) 8.00000i 0.427008i
\(352\) 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
\(355\) 0 0
\(356\) 0 0
\(357\) 40.0000 + 40.0000i 2.11702 + 2.11702i
\(358\) 0 0
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 22.0000 + 22.0000i 1.15470 + 1.15470i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −22.0000 + 22.0000i −1.14839 + 1.14839i −0.161521 + 0.986869i \(0.551640\pi\)
−0.986869 + 0.161521i \(0.948360\pi\)
\(368\) 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 \(0.529178\pi\)
−0.995802 + 0.0915371i \(0.970822\pi\)
\(374\) 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
\(381\) 40.0000 2.04926
\(382\) 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
\(387\) 30.0000 30.0000i 1.52499 1.52499i
\(388\) 0 0
\(389\) 18.0000i 0.912636i 0.889817 + 0.456318i \(0.150832\pi\)
−0.889817 + 0.456318i \(0.849168\pi\)
\(390\) 0 0
\(391\) 20.0000i 1.01144i
\(392\) 0 0
\(393\) 16.0000 16.0000i 0.807093 0.807093i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −13.0000 13.0000i −0.652451 0.652451i 0.301131 0.953583i \(-0.402636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) 0 0
\(399\) −32.0000 −1.60200
\(400\) 0 0
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 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
\(411\) 4.00000i 0.197305i
\(412\) 0 0
\(413\) −8.00000 + 8.00000i −0.393654 + 0.393654i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 24.0000 + 24.0000i 1.17529 + 1.17529i
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) −10.0000 10.0000i −0.486217 0.486217i
\(424\) 0 0
\(425\) 0 0
\(426\) 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 \(-0.969287\pi\)
0.995349 0.0963366i \(-0.0307125\pi\)
\(432\) 0 0
\(433\) 19.0000 19.0000i 0.913082 0.913082i −0.0834318 0.996513i \(-0.526588\pi\)
0.996513 + 0.0834318i \(0.0265881\pi\)
\(434\) 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
\(441\) 5.00000 0.238095
\(442\) 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
\(447\) 36.0000 36.0000i 1.70274 1.70274i
\(448\) 0 0
\(449\) 26.0000i 1.22702i −0.789689 0.613508i \(-0.789758\pi\)
0.789689 0.613508i \(-0.210242\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 24.0000 24.0000i 1.12762 1.12762i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −15.0000 15.0000i −0.701670 0.701670i 0.263099 0.964769i \(-0.415256\pi\)
−0.964769 + 0.263099i \(0.915256\pi\)
\(458\) 0 0
\(459\) −40.0000 −1.86704
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) −22.0000 22.0000i −1.02243 1.02243i −0.999743 0.0226840i \(-0.992779\pi\)
−0.0226840 0.999743i \(-0.507221\pi\)
\(464\) 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
\(471\) 36.0000i 1.65879i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 35.0000 + 35.0000i 1.60254 + 1.60254i
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 16.0000 + 16.0000i 0.728025 + 0.728025i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −6.00000 + 6.00000i −0.271886 + 0.271886i −0.829859 0.557973i \(-0.811579\pi\)
0.557973 + 0.829859i \(0.311579\pi\)
\(488\) 0 0
\(489\) 8.00000i 0.361773i
\(490\) 0 0
\(491\) 16.0000i 0.722070i 0.932552 + 0.361035i \(0.117576\pi\)
−0.932552 + 0.361035i \(0.882424\pi\)
\(492\) 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
\(501\) −8.00000 −0.357414
\(502\) 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
\(505\) 0 0
\(506\) 0 0
\(507\) −22.0000 + 22.0000i −0.977054 + 0.977054i
\(508\) 0 0
\(509\) 36.0000i 1.59567i 0.602875 + 0.797836i \(0.294022\pi\)
−0.602875 + 0.797836i \(0.705978\pi\)
\(510\) 0 0
\(511\) 12.0000i 0.530849i
\(512\) 0 0
\(513\) 16.0000 16.0000i 0.706417 0.706417i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −52.0000 −2.28255
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 14.0000 + 14.0000i 0.612177 + 0.612177i 0.943513 0.331336i \(-0.107499\pi\)
−0.331336 + 0.943513i \(0.607499\pi\)
\(524\) 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
\(531\) 20.0000i 0.867926i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −24.0000 24.0000i −1.03568 1.03568i
\(538\) 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
\(543\) −20.0000 20.0000i −0.858282 0.858282i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −6.00000 + 6.00000i −0.256541 + 0.256541i −0.823646 0.567104i \(-0.808064\pi\)
0.567104 + 0.823646i \(0.308064\pi\)
\(548\) 0 0
\(549\) 20.0000i 0.853579i
\(550\) 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
\(565\) 0 0
\(566\) 0 0
\(567\) −2.00000 + 2.00000i −0.0839921 + 0.0839921i
\(568\) 0 0
\(569\) 2.00000i 0.0838444i −0.999121 0.0419222i \(-0.986652\pi\)
0.999121 0.0419222i \(-0.0133482\pi\)
\(570\) 0 0
\(571\) 16.0000i 0.669579i 0.942293 + 0.334790i \(0.108665\pi\)
−0.942293 + 0.334790i \(0.891335\pi\)
\(572\) 0 0
\(573\) −40.0000 + 40.0000i −1.67102 + 1.67102i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −15.0000 15.0000i −0.624458 0.624458i 0.322210 0.946668i \(-0.395574\pi\)
−0.946668 + 0.322210i \(0.895574\pi\)
\(578\) 0 0
\(579\) 20.0000 0.831172
\(580\) 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
\(591\) 20.0000i 0.822690i
\(592\) 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
\(595\) 0 0
\(596\) 0 0
\(597\) −48.0000 48.0000i −1.96451 1.96451i
\(598\) 0 0
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 0 0
\(603\) −50.0000 50.0000i −2.03616 2.03616i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −18.0000 + 18.0000i −0.730597 + 0.730597i −0.970738 0.240141i \(-0.922806\pi\)
0.240141 + 0.970738i \(0.422806\pi\)
\(608\) 0 0
\(609\) 32.0000i 1.29671i
\(610\) 0 0
\(611\) 4.00000i 0.161823i
\(612\) 0 0
\(613\) 9.00000 9.00000i 0.363507 0.363507i −0.501596 0.865102i \(-0.667253\pi\)
0.865102 + 0.501596i \(0.167253\pi\)
\(614\) 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
\(621\) −16.0000 −0.642058
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 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
\(633\) 32.0000 32.0000i 1.27189 1.27189i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.00000 1.00000i −0.0396214 0.0396214i
\(638\) 0 0
\(639\) 60.0000 2.37356
\(640\) 0 0
\(641\) −48.0000 −1.89589 −0.947943 0.318440i \(-0.896841\pi\)
−0.947943 + 0.318440i \(0.896841\pi\)
\(642\) 0 0
\(643\) −10.0000 10.0000i −0.394362 0.394362i 0.481877 0.876239i \(-0.339955\pi\)
−0.876239 + 0.481877i \(0.839955\pi\)
\(644\) 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
\(651\) 32.0000i 1.25418i
\(652\) 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
\(655\) 0 0
\(656\) 0 0
\(657\) 15.0000 + 15.0000i 0.585206 + 0.585206i
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 12.0000 0.466746 0.233373 0.972387i \(-0.425024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) 0 0
\(663\) 20.0000 + 20.0000i 0.776736 + 0.776736i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8.00000 8.00000i 0.309761 0.309761i
\(668\) 0 0
\(669\) 40.0000i 1.54649i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −5.00000 + 5.00000i −0.192736 + 0.192736i −0.796877 0.604141i \(-0.793516\pi\)
0.604141 + 0.796877i \(0.293516\pi\)
\(674\) 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
\(681\) 40.0000 1.53280
\(682\) 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
\(685\) 0 0
\(686\) 0 0
\(687\) 40.0000 40.0000i 1.52610 1.52610i
\(688\) 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
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 20.0000 0.756469
\(700\) 0 0
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 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
\(711\) 80.0000i 3.00023i
\(712\) 0 0
\(713\) −8.00000 + 8.00000i −0.299602 + 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −16.0000 16.0000i −0.597531 0.597531i
\(718\) 0 0
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) 0 0
\(723\) −32.0000 32.0000i −1.19009 1.19009i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 18.0000 18.0000i 0.667583 0.667583i −0.289573 0.957156i \(-0.593513\pi\)
0.957156 + 0.289573i \(0.0935133\pi\)
\(728\) 0 0
\(729\) 43.0000i 1.59259i
\(730\) 0 0
\(731\) 60.0000i 2.21918i
\(732\) 0 0
\(733\) −21.0000 + 21.0000i −0.775653 + 0.775653i −0.979088 0.203436i \(-0.934789\pi\)
0.203436 + 0.979088i \(0.434789\pi\)
\(734\) 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
\(741\) −16.0000 −0.587775
\(742\) 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
\(745\) 0 0
\(746\) 0 0
\(747\) 10.0000 10.0000i 0.365881 0.365881i
\(748\) 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
\(753\) −48.0000 + 48.0000i −1.74922 + 1.74922i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.00000 + 1.00000i 0.0363456 + 0.0363456i 0.725046 0.688700i \(-0.241818\pi\)
−0.688700 + 0.725046i \(0.741818\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 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
\(771\) 28.0000i 1.00840i
\(772\) 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
\(775\) 0 0
\(776\) 0 0
\(777\) −8.00000 8.00000i −0.286998 0.286998i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 16.0000 + 16.0000i 0.571793 + 0.571793i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 30.0000 30.0000i 1.06938 1.06938i 0.0719783 0.997406i \(-0.477069\pi\)
0.997406 0.0719783i \(-0.0229312\pi\)
\(788\) 0 0
\(789\) 24.0000i 0.854423i
\(790\) 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