Properties

Label 240.2.f.b.49.1
Level $240$
Weight $2$
Character 240.49
Analytic conductor $1.916$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 240.49
Dual form 240.2.f.b.49.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +(1.00000 + 2.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(1.00000 + 2.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} +4.00000 q^{11} +(2.00000 - 1.00000i) q^{15} -4.00000i q^{17} +4.00000 q^{21} -4.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +1.00000i q^{27} +6.00000 q^{29} -4.00000 q^{31} -4.00000i q^{33} +(-8.00000 + 4.00000i) q^{35} +8.00000i q^{37} -10.0000 q^{41} -4.00000i q^{43} +(-1.00000 - 2.00000i) q^{45} -4.00000i q^{47} -9.00000 q^{49} -4.00000 q^{51} -12.0000i q^{53} +(4.00000 + 8.00000i) q^{55} +4.00000 q^{59} +2.00000 q^{61} -4.00000i q^{63} -4.00000i q^{67} -4.00000 q^{69} -8.00000i q^{73} +(4.00000 + 3.00000i) q^{75} +16.0000i q^{77} -12.0000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +(8.00000 - 4.00000i) q^{85} -6.00000i q^{87} +10.0000 q^{89} +4.00000i q^{93} -8.00000i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{5} - 2q^{9} + 8q^{11} + 4q^{15} + 8q^{21} - 6q^{25} + 12q^{29} - 8q^{31} - 16q^{35} - 20q^{41} - 2q^{45} - 18q^{49} - 8q^{51} + 8q^{55} + 8q^{59} + 4q^{61} - 8q^{69} + 8q^{75} - 24q^{79} + 2q^{81} + 16q^{85} + 20q^{89} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.00000 + 2.00000i 0.447214 + 0.894427i
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 2.00000 1.00000i 0.516398 0.258199i
\(16\) 0 0
\(17\) 4.00000i 0.970143i −0.874475 0.485071i \(-0.838794\pi\)
0.874475 0.485071i \(-0.161206\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 4.00000i 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 4.00000i 0.696311i
\(34\) 0 0
\(35\) −8.00000 + 4.00000i −1.35225 + 0.676123i
\(36\) 0 0
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) −1.00000 2.00000i −0.149071 0.298142i
\(46\) 0 0
\(47\) 4.00000i 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 12.0000i 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) 0 0
\(55\) 4.00000 + 8.00000i 0.539360 + 1.07872i
\(56\) 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
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 4.00000i 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 8.00000i 0.936329i −0.883641 0.468165i \(-0.844915\pi\)
0.883641 0.468165i \(-0.155085\pi\)
\(74\) 0 0
\(75\) 4.00000 + 3.00000i 0.461880 + 0.346410i
\(76\) 0 0
\(77\) 16.0000i 1.82337i
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000i 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 8.00000 4.00000i 0.867722 0.433861i
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000i 0.812277i −0.913812 0.406138i \(-0.866875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 4.00000 + 8.00000i 0.390360 + 0.780720i
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) 12.0000i 1.12887i 0.825479 + 0.564433i \(0.190905\pi\)
−0.825479 + 0.564433i \(0.809095\pi\)
\(114\) 0 0
\(115\) 8.00000 4.00000i 0.746004 0.373002i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 10.0000i 0.901670i
\(124\) 0 0
\(125\) −11.0000 2.00000i −0.983870 0.178885i
\(126\) 0 0
\(127\) 4.00000i 0.354943i 0.984126 + 0.177471i \(0.0567917\pi\)
−0.984126 + 0.177471i \(0.943208\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.00000 + 1.00000i −0.172133 + 0.0860663i
\(136\) 0 0
\(137\) 12.0000i 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.00000 + 12.0000i 0.498273 + 0.996546i
\(146\) 0 0
\(147\) 9.00000i 0.742307i
\(148\) 0 0
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 0 0
\(153\) 4.00000i 0.323381i
\(154\) 0 0
\(155\) −4.00000 8.00000i −0.321288 0.642575i
\(156\) 0 0
\(157\) 8.00000i 0.638470i −0.947676 0.319235i \(-0.896574\pi\)
0.947676 0.319235i \(-0.103426\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 0 0
\(163\) 20.0000i 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) 0 0
\(165\) 8.00000 4.00000i 0.622799 0.311400i
\(166\) 0 0
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.00000i 0.304114i −0.988372 0.152057i \(-0.951410\pi\)
0.988372 0.152057i \(-0.0485898\pi\)
\(174\) 0 0
\(175\) −16.0000 12.0000i −1.20949 0.907115i
\(176\) 0 0
\(177\) 4.00000i 0.300658i
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) 0 0
\(185\) −16.0000 + 8.00000i −1.17634 + 0.588172i
\(186\) 0 0
\(187\) 16.0000i 1.17004i
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) 16.0000i 1.15171i 0.817554 + 0.575853i \(0.195330\pi\)
−0.817554 + 0.575853i \(0.804670\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.00000i 0.284988i 0.989796 + 0.142494i \(0.0455122\pi\)
−0.989796 + 0.142494i \(0.954488\pi\)
\(198\) 0 0
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 24.0000i 1.68447i
\(204\) 0 0
\(205\) −10.0000 20.0000i −0.698430 1.39686i
\(206\) 0 0
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.00000 4.00000i 0.545595 0.272798i
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 20.0000i 1.33930i 0.742677 + 0.669650i \(0.233556\pi\)
−0.742677 + 0.669650i \(0.766444\pi\)
\(224\) 0 0
\(225\) 3.00000 4.00000i 0.200000 0.266667i
\(226\) 0 0
\(227\) 20.0000i 1.32745i −0.747978 0.663723i \(-0.768975\pi\)
0.747978 0.663723i \(-0.231025\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) 0 0
\(233\) 20.0000i 1.31024i 0.755523 + 0.655122i \(0.227383\pi\)
−0.755523 + 0.655122i \(0.772617\pi\)
\(234\) 0 0
\(235\) 8.00000 4.00000i 0.521862 0.260931i
\(236\) 0 0
\(237\) 12.0000i 0.779484i
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) −9.00000 18.0000i −0.574989 1.14998i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 16.0000i 1.00591i
\(254\) 0 0
\(255\) −4.00000 8.00000i −0.250490 0.500979i
\(256\) 0 0
\(257\) 12.0000i 0.748539i −0.927320 0.374270i \(-0.877893\pi\)
0.927320 0.374270i \(-0.122107\pi\)
\(258\) 0 0
\(259\) −32.0000 −1.98838
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 4.00000i 0.246651i −0.992366 0.123325i \(-0.960644\pi\)
0.992366 0.123325i \(-0.0393559\pi\)
\(264\) 0 0
\(265\) 24.0000 12.0000i 1.47431 0.737154i
\(266\) 0 0
\(267\) 10.0000i 0.611990i
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −12.0000 + 16.0000i −0.723627 + 0.964836i
\(276\) 0 0
\(277\) 32.0000i 1.92269i 0.275340 + 0.961347i \(0.411209\pi\)
−0.275340 + 0.961347i \(0.588791\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 28.0000i 1.66443i 0.554455 + 0.832214i \(0.312927\pi\)
−0.554455 + 0.832214i \(0.687073\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 40.0000i 2.36113i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −8.00000 −0.468968
\(292\) 0 0
\(293\) 12.0000i 0.701047i 0.936554 + 0.350524i \(0.113996\pi\)
−0.936554 + 0.350524i \(0.886004\pi\)
\(294\) 0 0
\(295\) 4.00000 + 8.00000i 0.232889 + 0.465778i
\(296\) 0 0
\(297\) 4.00000i 0.232104i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 0 0
\(303\) 2.00000i 0.114897i
\(304\) 0 0
\(305\) 2.00000 + 4.00000i 0.114520 + 0.229039i
\(306\) 0 0
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 16.0000i 0.904373i 0.891923 + 0.452187i \(0.149356\pi\)
−0.891923 + 0.452187i \(0.850644\pi\)
\(314\) 0 0
\(315\) 8.00000 4.00000i 0.450749 0.225374i
\(316\) 0 0
\(317\) 4.00000i 0.224662i −0.993671 0.112331i \(-0.964168\pi\)
0.993671 0.112331i \(-0.0358318\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.00000i 0.110600i
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) 8.00000i 0.438397i
\(334\) 0 0
\(335\) 8.00000 4.00000i 0.437087 0.218543i
\(336\) 0 0
\(337\) 8.00000i 0.435788i 0.975972 + 0.217894i \(0.0699187\pi\)
−0.975972 + 0.217894i \(0.930081\pi\)
\(338\) 0 0
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 0 0
\(345\) −4.00000 8.00000i −0.215353 0.430706i
\(346\) 0 0
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.0000i 1.49029i −0.666903 0.745145i \(-0.732380\pi\)
0.666903 0.745145i \(-0.267620\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 16.0000i 0.846810i
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) 16.0000 8.00000i 0.837478 0.418739i
\(366\) 0 0
\(367\) 4.00000i 0.208798i 0.994535 + 0.104399i \(0.0332919\pi\)
−0.994535 + 0.104399i \(0.966708\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 48.0000 2.49204
\(372\) 0 0
\(373\) 24.0000i 1.24267i 0.783544 + 0.621336i \(0.213410\pi\)
−0.783544 + 0.621336i \(0.786590\pi\)
\(374\) 0 0
\(375\) −2.00000 + 11.0000i −0.103280 + 0.568038i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) 28.0000i 1.43073i 0.698749 + 0.715367i \(0.253740\pi\)
−0.698749 + 0.715367i \(0.746260\pi\)
\(384\) 0 0
\(385\) −32.0000 + 16.0000i −1.63087 + 0.815436i
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) 34.0000 1.72387 0.861934 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 12.0000i 0.605320i
\(394\) 0 0
\(395\) −12.0000 24.0000i −0.603786 1.20757i
\(396\) 0 0
\(397\) 8.00000i 0.401508i 0.979642 + 0.200754i \(0.0643393\pi\)
−0.979642 + 0.200754i \(0.935661\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000 + 2.00000i 0.0496904 + 0.0993808i
\(406\) 0 0
\(407\) 32.0000i 1.58618i
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 0 0
\(413\) 16.0000i 0.787309i
\(414\) 0 0
\(415\) 8.00000 4.00000i 0.392705 0.196352i
\(416\) 0 0
\(417\) 16.0000i 0.783523i
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 4.00000i 0.194487i
\(424\) 0 0
\(425\) 16.0000 + 12.0000i 0.776114 + 0.582086i
\(426\) 0 0
\(427\) 8.00000i 0.387147i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) 16.0000i 0.768911i 0.923144 + 0.384455i \(0.125611\pi\)
−0.923144 + 0.384455i \(0.874389\pi\)
\(434\) 0 0
\(435\) 12.0000 6.00000i 0.575356 0.287678i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −12.0000 −0.572729 −0.286364 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 36.0000i 1.71041i −0.518289 0.855206i \(-0.673431\pi\)
0.518289 0.855206i \(-0.326569\pi\)
\(444\) 0 0
\(445\) 10.0000 + 20.0000i 0.474045 + 0.948091i
\(446\) 0 0
\(447\) 2.00000i 0.0945968i
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 0 0
\(453\) 20.0000i 0.939682i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 40.0000i 1.87112i −0.353166 0.935561i \(-0.614895\pi\)
0.353166 0.935561i \(-0.385105\pi\)
\(458\) 0 0
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 12.0000i 0.557687i −0.960337 0.278844i \(-0.910049\pi\)
0.960337 0.278844i \(-0.0899511\pi\)
\(464\) 0 0
\(465\) −8.00000 + 4.00000i −0.370991 + 0.185496i
\(466\) 0 0
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) −8.00000 −0.368621
\(472\) 0 0
\(473\) 16.0000i 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000i 0.549442i
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 16.0000i 0.728025i
\(484\) 0 0
\(485\) 16.0000 8.00000i 0.726523 0.363261i
\(486\) 0 0
\(487\) 12.0000i 0.543772i −0.962329 0.271886i \(-0.912353\pi\)
0.962329 0.271886i \(-0.0876473\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) 24.0000i 1.08091i
\(494\) 0 0
\(495\) −4.00000 8.00000i −0.179787 0.359573i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 36.0000i 1.60516i −0.596544 0.802580i \(-0.703460\pi\)
0.596544 0.802580i \(-0.296540\pi\)
\(504\) 0 0
\(505\) −2.00000 4.00000i −0.0889988 0.177998i
\(506\) 0 0
\(507\) 13.0000i 0.577350i
\(508\) 0 0
\(509\) −42.0000 −1.86162 −0.930809 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) 0 0
\(511\) 32.0000 1.41560
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.00000 + 4.00000i −0.352522 + 0.176261i
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) −4.00000 −0.175581
\(520\) 0 0
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) 0 0
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) 0 0
\(525\) −12.0000 + 16.0000i −0.523723 + 0.698297i
\(526\) 0 0
\(527\) 16.0000i 0.696971i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −24.0000 + 12.0000i −1.03761 + 0.518805i
\(536\) 0 0
\(537\) 4.00000i 0.172613i
\(538\) 0 0
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 10.0000i 0.429141i
\(544\) 0 0
\(545\) 2.00000 + 4.00000i 0.0856706 + 0.171341i
\(546\) 0 0
\(547\) 28.0000i 1.19719i 0.801050 + 0.598597i \(0.204275\pi\)
−0.801050 + 0.598597i \(0.795725\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 48.0000i 2.04117i
\(554\) 0 0
\(555\) 8.00000 + 16.0000i 0.339581 + 0.679162i
\(556\) 0 0
\(557\) 12.0000i 0.508456i 0.967144 + 0.254228i \(0.0818214\pi\)
−0.967144 + 0.254228i \(0.918179\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) 0 0
\(563\) 20.0000i 0.842900i −0.906852 0.421450i \(-0.861521\pi\)
0.906852 0.421450i \(-0.138479\pi\)
\(564\) 0 0
\(565\) −24.0000 + 12.0000i −1.00969 + 0.504844i
\(566\) 0 0
\(567\) 4.00000i 0.167984i
\(568\) 0 0
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 24.0000i 1.00261i
\(574\) 0 0
\(575\) 16.0000 + 12.0000i 0.667246 + 0.500435i
\(576\) 0 0
\(577\) 32.0000i 1.33218i 0.745873 + 0.666089i \(0.232033\pi\)
−0.745873 + 0.666089i \(0.767967\pi\)
\(578\) 0 0
\(579\) 16.0000 0.664937
\(580\) 0 0
\(581\) 16.0000 0.663792
\(582\) 0 0
\(583\) 48.0000i 1.98796i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.0000i 1.48588i −0.669359 0.742940i \(-0.733431\pi\)
0.669359 0.742940i \(-0.266569\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 4.00000 0.164538
\(592\) 0 0
\(593\) 36.0000i 1.47834i −0.673517 0.739171i \(-0.735217\pi\)
0.673517 0.739171i \(-0.264783\pi\)
\(594\) 0 0
\(595\) 16.0000 + 32.0000i 0.655936 + 1.31187i
\(596\) 0 0
\(597\) 12.0000i 0.491127i
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) 5.00000 + 10.0000i 0.203279 + 0.406558i
\(606\) 0 0
\(607\) 28.0000i 1.13648i −0.822861 0.568242i \(-0.807624\pi\)
0.822861 0.568242i \(-0.192376\pi\)
\(608\) 0 0
\(609\) 24.0000 0.972529
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 8.00000i 0.323117i −0.986863 0.161558i \(-0.948348\pi\)
0.986863 0.161558i \(-0.0516520\pi\)
\(614\) 0 0
\(615\) −20.0000 + 10.0000i −0.806478 + 0.403239i
\(616\) 0 0
\(617\) 44.0000i 1.77137i 0.464283 + 0.885687i \(0.346312\pi\)
−0.464283 + 0.885687i \(0.653688\pi\)
\(618\) 0 0
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) 40.0000i 1.60257i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 32.0000 1.27592
\(630\) 0 0
\(631\) 28.0000 1.11466 0.557331 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.00000 + 4.00000i −0.317470 + 0.158735i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 14.0000 0.552967 0.276483 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(642\) 0 0
\(643\) 36.0000i 1.41970i −0.704352 0.709851i \(-0.748762\pi\)
0.704352 0.709851i \(-0.251238\pi\)
\(644\) 0 0
\(645\) −4.00000 8.00000i −0.157500 0.315000i
\(646\) 0 0
\(647\) 12.0000i 0.471769i 0.971781 + 0.235884i \(0.0757987\pi\)
−0.971781 + 0.235884i \(0.924201\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) 0 0
\(653\) 4.00000i 0.156532i 0.996933 + 0.0782660i \(0.0249384\pi\)
−0.996933 + 0.0782660i \(0.975062\pi\)
\(654\) 0 0
\(655\) −12.0000 24.0000i −0.468879 0.937758i
\(656\) 0 0
\(657\) 8.00000i 0.312110i
\(658\) 0 0
\(659\) 28.0000 1.09073 0.545363 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 24.0000i 0.929284i
\(668\) 0 0
\(669\) 20.0000 0.773245
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −4.00000 3.00000i −0.153960 0.115470i
\(676\) 0 0
\(677\) 4.00000i 0.153732i −0.997041 0.0768662i \(-0.975509\pi\)
0.997041 0.0768662i \(-0.0244914\pi\)
\(678\) 0 0
\(679\) 32.0000 1.22805
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 44.0000i 1.68361i 0.539779 + 0.841807i \(0.318508\pi\)
−0.539779 + 0.841807i \(0.681492\pi\)
\(684\) 0 0
\(685\) 24.0000 12.0000i 0.916993 0.458496i
\(686\) 0 0
\(687\) 26.0000i 0.991962i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) 0 0
\(693\) 16.0000i 0.607790i
\(694\) 0 0
\(695\) 16.0000 + 32.0000i 0.606915 + 1.21383i
\(696\) 0 0
\(697\) 40.0000i 1.51511i
\(698\) 0 0
\(699\) 20.0000 0.756469
\(700\) 0 0
\(701\) 26.0000 0.982006 0.491003 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −4.00000 8.00000i −0.150649 0.301297i
\(706\) 0 0
\(707\) 8.00000i 0.300871i
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 0 0
\(713\) 16.0000i 0.599205i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 8.00000i 0.298765i
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) 2.00000i 0.0743808i
\(724\) 0 0
\(725\) −18.0000 + 24.0000i −0.668503 + 0.891338i
\(726\) 0 0
\(727\) 12.0000i 0.445055i −0.974926 0.222528i \(-0.928569\pi\)
0.974926 0.222528i \(-0.0714308\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) 16.0000i 0.590973i −0.955347 0.295487i \(-0.904518\pi\)
0.955347 0.295487i \(-0.0954818\pi\)
\(734\) 0 0
\(735\) −18.0000 + 9.00000i −0.663940 + 0.331970i
\(736\) 0 0
\(737\) 16.0000i 0.589368i
\(738\) 0 0
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36.0000i 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 0 0
\(745\) 2.00000 + 4.00000i 0.0732743 + 0.146549i
\(746\) 0 0
\(747\) 4.00000i 0.146352i
\(748\) 0 0
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) 0 0
\(753\) 12.0000i 0.437304i
\(754\) 0 0
\(755\) 20.0000 + 40.0000i 0.727875 + 1.45575i
\(756\) 0 0
\(757\) 16.0000i 0.581530i −0.956795 0.290765i \(-0.906090\pi\)
0.956795 0.290765i \(-0.0939098\pi\)
\(758\) 0 0
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 8.00000i 0.289619i
\(764\) 0 0
\(765\) −8.00000 + 4.00000i −0.289241 + 0.144620i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 0 0
\(773\) 12.0000i 0.431610i −0.976436 0.215805i \(-0.930762\pi\)
0.976436 0.215805i \(-0.0692376\pi\)
\(774\) 0 0
\(775\) 12.0000 16.0000i 0.431053 0.574737i
\(776\) 0 0
\(777\) 32.0000i 1.14799i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) 0 0
\(785\) 16.0000 8.00000i 0.571064 0.285532i
\(786\) 0 0
\(787\) 28.0000i 0.998092i 0.866575 + 0.499046i \(0.166316\pi\)
−0.866575 + 0.499046i \(0.833684\pi\)
\(788\) 0 0
\(789\) −4.00000 −0.142404
\(790\) 0 0
\(791\) −48.0000 −1.70668
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −12.0000 24.0000i −0.425596 0.851192i
\(796\) 0 0
\(797\) 28.0000i 0.991811i −0.868377 0.495905i \(-0.834836\pi\)
0.868377 0.495905i \(-0.165164\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) 0 0
\(803\) 32.0000i 1.12926i
\(804\) 0