Properties

Label 3840.2.d.ba.2689.2
Level $3840$
Weight $2$
Character 3840.2689
Analytic conductor $30.663$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.6625543762\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2689.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3840.2689
Dual form 3840.2.d.ba.2689.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +(1.00000 + 2.00000i) q^{5} -2.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +(1.00000 + 2.00000i) q^{5} -2.00000i q^{7} +1.00000 q^{9} +2.00000i q^{11} +2.00000 q^{13} +(1.00000 + 2.00000i) q^{15} -6.00000i q^{17} +8.00000i q^{19} -2.00000i q^{21} +4.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +1.00000 q^{27} +8.00000i q^{29} +2.00000i q^{33} +(4.00000 - 2.00000i) q^{35} -10.0000 q^{37} +2.00000 q^{39} -2.00000 q^{41} +12.0000 q^{43} +(1.00000 + 2.00000i) q^{45} +3.00000 q^{49} -6.00000i q^{51} -10.0000 q^{53} +(-4.00000 + 2.00000i) q^{55} +8.00000i q^{57} +6.00000i q^{59} -2.00000i q^{61} -2.00000i q^{63} +(2.00000 + 4.00000i) q^{65} +8.00000 q^{67} +4.00000i q^{69} +4.00000 q^{71} -4.00000i q^{73} +(-3.00000 + 4.00000i) q^{75} +4.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +4.00000 q^{83} +(12.0000 - 6.00000i) q^{85} +8.00000i q^{87} +6.00000 q^{89} -4.00000i q^{91} +(-16.0000 + 8.00000i) q^{95} -8.00000i q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{5} + 2q^{9} + 4q^{13} + 2q^{15} - 6q^{25} + 2q^{27} + 8q^{35} - 20q^{37} + 4q^{39} - 4q^{41} + 24q^{43} + 2q^{45} + 6q^{49} - 20q^{53} - 8q^{55} + 4q^{65} + 16q^{67} + 8q^{71} - 6q^{75} + 8q^{77} + 16q^{79} + 2q^{81} + 8q^{83} + 24q^{85} + 12q^{89} - 32q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 + 2.00000i 0.447214 + 0.894427i
\(6\) 0 0
\(7\) 2.00000i 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 1.00000 + 2.00000i 0.258199 + 0.516398i
\(16\) 0 0
\(17\) 6.00000i 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) 8.00000i 1.83533i 0.397360 + 0.917663i \(0.369927\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 8.00000i 1.48556i 0.669534 + 0.742781i \(0.266494\pi\)
−0.669534 + 0.742781i \(0.733506\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 0 0
\(35\) 4.00000 2.00000i 0.676123 0.338062i
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 0 0
\(45\) 1.00000 + 2.00000i 0.149071 + 0.298142i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 6.00000i 0.840168i
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) −4.00000 + 2.00000i −0.539360 + 0.269680i
\(56\) 0 0
\(57\) 8.00000i 1.05963i
\(58\) 0 0
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 2.00000i 0.256074i −0.991769 0.128037i \(-0.959132\pi\)
0.991769 0.128037i \(-0.0408676\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) 2.00000 + 4.00000i 0.248069 + 0.496139i
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 4.00000i 0.481543i
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) 0 0
\(75\) −3.00000 + 4.00000i −0.346410 + 0.461880i
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 12.0000 6.00000i 1.30158 0.650791i
\(86\) 0 0
\(87\) 8.00000i 0.857690i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 4.00000i 0.419314i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −16.0000 + 8.00000i −1.64157 + 0.820783i
\(96\) 0 0
\(97\) 8.00000i 0.812277i −0.913812 0.406138i \(-0.866875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) 0 0
\(99\) 2.00000i 0.201008i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 2.00000i 0.197066i 0.995134 + 0.0985329i \(0.0314150\pi\)
−0.995134 + 0.0985329i \(0.968585\pi\)
\(104\) 0 0
\(105\) 4.00000 2.00000i 0.390360 0.195180i
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 6.00000i 0.574696i −0.957826 0.287348i \(-0.907226\pi\)
0.957826 0.287348i \(-0.0927736\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) 0 0
\(115\) −8.00000 + 4.00000i −0.746004 + 0.373002i
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) −11.0000 2.00000i −0.983870 0.178885i
\(126\) 0 0
\(127\) 18.0000i 1.59724i 0.601834 + 0.798621i \(0.294437\pi\)
−0.601834 + 0.798621i \(0.705563\pi\)
\(128\) 0 0
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 18.0000i 1.57267i 0.617802 + 0.786334i \(0.288023\pi\)
−0.617802 + 0.786334i \(0.711977\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) 0 0
\(135\) 1.00000 + 2.00000i 0.0860663 + 0.172133i
\(136\) 0 0
\(137\) 10.0000i 0.854358i 0.904167 + 0.427179i \(0.140493\pi\)
−0.904167 + 0.427179i \(0.859507\pi\)
\(138\) 0 0
\(139\) 4.00000i 0.339276i −0.985506 0.169638i \(-0.945740\pi\)
0.985506 0.169638i \(-0.0542598\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000i 0.334497i
\(144\) 0 0
\(145\) −16.0000 + 8.00000i −1.32873 + 0.664364i
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) 12.0000i 0.983078i 0.870855 + 0.491539i \(0.163566\pi\)
−0.870855 + 0.491539i \(0.836434\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 0 0
\(165\) −4.00000 + 2.00000i −0.311400 + 0.155700i
\(166\) 0 0
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 8.00000i 0.611775i
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) 8.00000 + 6.00000i 0.604743 + 0.453557i
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 0 0
\(179\) 22.0000i 1.64436i −0.569230 0.822179i \(-0.692758\pi\)
0.569230 0.822179i \(-0.307242\pi\)
\(180\) 0 0
\(181\) 14.0000i 1.04061i −0.853980 0.520306i \(-0.825818\pi\)
0.853980 0.520306i \(-0.174182\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) 0 0
\(185\) −10.0000 20.0000i −0.735215 1.47043i
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 0 0
\(189\) 2.00000i 0.145479i
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 4.00000i 0.287926i 0.989583 + 0.143963i \(0.0459847\pi\)
−0.989583 + 0.143963i \(0.954015\pi\)
\(194\) 0 0
\(195\) 2.00000 + 4.00000i 0.143223 + 0.286446i
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) 16.0000 1.12298
\(204\) 0 0
\(205\) −2.00000 4.00000i −0.139686 0.279372i
\(206\) 0 0
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 4.00000i 0.275371i −0.990476 0.137686i \(-0.956034\pi\)
0.990476 0.137686i \(-0.0439664\pi\)
\(212\) 0 0
\(213\) 4.00000 0.274075
\(214\) 0 0
\(215\) 12.0000 + 24.0000i 0.818393 + 1.63679i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 4.00000i 0.270295i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) 6.00000i 0.401790i 0.979613 + 0.200895i \(0.0643850\pi\)
−0.979613 + 0.200895i \(0.935615\pi\)
\(224\) 0 0
\(225\) −3.00000 + 4.00000i −0.200000 + 0.266667i
\(226\) 0 0
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) 6.00000i 0.396491i −0.980152 0.198246i \(-0.936476\pi\)
0.980152 0.198246i \(-0.0635244\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) 6.00000i 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.00000 + 6.00000i 0.191663 + 0.383326i
\(246\) 0 0
\(247\) 16.0000i 1.01806i
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 2.00000i 0.126239i −0.998006 0.0631194i \(-0.979895\pi\)
0.998006 0.0631194i \(-0.0201049\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) 12.0000 6.00000i 0.751469 0.375735i
\(256\) 0 0
\(257\) 6.00000i 0.374270i 0.982334 + 0.187135i \(0.0599201\pi\)
−0.982334 + 0.187135i \(0.940080\pi\)
\(258\) 0 0
\(259\) 20.0000i 1.24274i
\(260\) 0 0
\(261\) 8.00000i 0.495188i
\(262\) 0 0
\(263\) 12.0000i 0.739952i −0.929041 0.369976i \(-0.879366\pi\)
0.929041 0.369976i \(-0.120634\pi\)
\(264\) 0 0
\(265\) −10.0000 20.0000i −0.614295 1.22859i
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) 24.0000i 1.46331i 0.681677 + 0.731653i \(0.261251\pi\)
−0.681677 + 0.731653i \(0.738749\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 4.00000i 0.242091i
\(274\) 0 0
\(275\) −8.00000 6.00000i −0.482418 0.361814i
\(276\) 0 0
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) −16.0000 + 8.00000i −0.947758 + 0.473879i
\(286\) 0 0
\(287\) 4.00000i 0.236113i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 8.00000i 0.468968i
\(292\) 0 0
\(293\) 22.0000 1.28525 0.642627 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 0 0
\(295\) −12.0000 + 6.00000i −0.698667 + 0.349334i
\(296\) 0 0
\(297\) 2.00000i 0.116052i
\(298\) 0 0
\(299\) 8.00000i 0.462652i
\(300\) 0 0
\(301\) 24.0000i 1.38334i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.00000 2.00000i 0.229039 0.114520i
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) 2.00000i 0.113776i
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 4.00000i 0.226093i −0.993590 0.113047i \(-0.963939\pi\)
0.993590 0.113047i \(-0.0360610\pi\)
\(314\) 0 0
\(315\) 4.00000 2.00000i 0.225374 0.112687i
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −16.0000 −0.895828
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 48.0000 2.67079
\(324\) 0 0
\(325\) −6.00000 + 8.00000i −0.332820 + 0.443760i
\(326\) 0 0
\(327\) 6.00000i 0.331801i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 16.0000i 0.879440i 0.898135 + 0.439720i \(0.144922\pi\)
−0.898135 + 0.439720i \(0.855078\pi\)
\(332\) 0 0
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) 8.00000 + 16.0000i 0.437087 + 0.874173i
\(336\) 0 0
\(337\) 28.0000i 1.52526i −0.646837 0.762629i \(-0.723908\pi\)
0.646837 0.762629i \(-0.276092\pi\)
\(338\) 0 0
\(339\) 2.00000i 0.108625i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) −8.00000 + 4.00000i −0.430706 + 0.215353i
\(346\) 0 0
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 0 0
\(349\) 22.0000i 1.17763i −0.808267 0.588817i \(-0.799594\pi\)
0.808267 0.588817i \(-0.200406\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 0 0
\(355\) 4.00000 + 8.00000i 0.212298 + 0.424596i
\(356\) 0 0
\(357\) −12.0000 −0.635107
\(358\) 0 0
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −45.0000 −2.36842
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 8.00000 4.00000i 0.418739 0.209370i
\(366\) 0 0
\(367\) 14.0000i 0.730794i −0.930852 0.365397i \(-0.880933\pi\)
0.930852 0.365397i \(-0.119067\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 20.0000i 1.03835i
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) −11.0000 2.00000i −0.568038 0.103280i
\(376\) 0 0
\(377\) 16.0000i 0.824042i
\(378\) 0 0
\(379\) 4.00000i 0.205466i 0.994709 + 0.102733i \(0.0327588\pi\)
−0.994709 + 0.102733i \(0.967241\pi\)
\(380\) 0 0
\(381\) 18.0000i 0.922168i
\(382\) 0 0
\(383\) 24.0000i 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) 4.00000 + 8.00000i 0.203859 + 0.407718i
\(386\) 0 0
\(387\) 12.0000 0.609994
\(388\) 0 0
\(389\) 12.0000i 0.608424i −0.952604 0.304212i \(-0.901607\pi\)
0.952604 0.304212i \(-0.0983931\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) 18.0000i 0.907980i
\(394\) 0 0
\(395\) 8.00000 + 16.0000i 0.402524 + 0.805047i
\(396\) 0 0
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000 + 2.00000i 0.0496904 + 0.0993808i
\(406\) 0 0
\(407\) 20.0000i 0.991363i
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 10.0000i 0.493264i
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 4.00000 + 8.00000i 0.196352 + 0.392705i
\(416\) 0 0
\(417\) 4.00000i 0.195881i
\(418\) 0 0
\(419\) 6.00000i 0.293119i 0.989202 + 0.146560i \(0.0468200\pi\)
−0.989202 + 0.146560i \(0.953180\pi\)
\(420\) 0 0
\(421\) 26.0000i 1.26716i −0.773676 0.633581i \(-0.781584\pi\)
0.773676 0.633581i \(-0.218416\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 24.0000 + 18.0000i 1.16417 + 0.873128i
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 0 0
\(429\) 4.00000i 0.193122i
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 12.0000i 0.576683i −0.957528 0.288342i \(-0.906896\pi\)
0.957528 0.288342i \(-0.0931039\pi\)
\(434\) 0 0
\(435\) −16.0000 + 8.00000i −0.767141 + 0.383571i
\(436\) 0 0
\(437\) −32.0000 −1.53077
\(438\) 0 0
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 6.00000 + 12.0000i 0.284427 + 0.568855i
\(446\) 0 0
\(447\) 12.0000i 0.567581i
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 4.00000i 0.188353i
\(452\) 0 0
\(453\) −16.0000 −0.751746
\(454\) 0 0
\(455\) 8.00000 4.00000i 0.375046 0.187523i
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 6.00000i 0.280056i
\(460\) 0 0
\(461\) 12.0000i 0.558896i 0.960161 + 0.279448i \(0.0901514\pi\)
−0.960161 + 0.279448i \(0.909849\pi\)
\(462\) 0 0
\(463\) 22.0000i 1.02243i −0.859454 0.511213i \(-0.829196\pi\)
0.859454 0.511213i \(-0.170804\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) 16.0000i 0.738811i
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) 24.0000i 1.10352i
\(474\) 0 0
\(475\) −32.0000 24.0000i −1.46826 1.10120i
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) 0 0
\(483\) 8.00000 0.364013
\(484\) 0 0
\(485\) 16.0000 8.00000i 0.726523 0.363261i
\(486\) 0 0
\(487\) 34.0000i 1.54069i 0.637629 + 0.770344i \(0.279915\pi\)
−0.637629 + 0.770344i \(0.720085\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 34.0000i 1.53440i −0.641409 0.767199i \(-0.721650\pi\)
0.641409 0.767199i \(-0.278350\pi\)
\(492\) 0 0
\(493\) 48.0000 2.16181
\(494\) 0 0
\(495\) −4.00000 + 2.00000i −0.179787 + 0.0898933i
\(496\) 0 0
\(497\) 8.00000i 0.358849i
\(498\) 0 0
\(499\) 40.0000i 1.79065i −0.445418 0.895323i \(-0.646945\pi\)
0.445418 0.895323i \(-0.353055\pi\)
\(500\) 0 0
\(501\) 12.0000i 0.536120i
\(502\) 0 0
\(503\) 16.0000i 0.713405i −0.934218 0.356702i \(-0.883901\pi\)
0.934218 0.356702i \(-0.116099\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) 8.00000i 0.354594i −0.984157 0.177297i \(-0.943265\pi\)
0.984157 0.177297i \(-0.0567353\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 0 0
\(513\) 8.00000i 0.353209i
\(514\) 0 0
\(515\) −4.00000 + 2.00000i −0.176261 + 0.0881305i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 8.00000 + 6.00000i 0.349149 + 0.261861i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 6.00000i 0.260378i
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) 4.00000 + 8.00000i 0.172935 + 0.345870i
\(536\) 0 0
\(537\) 22.0000i 0.949370i
\(538\) 0 0
\(539\) 6.00000i 0.258438i
\(540\) 0 0
\(541\) 10.0000i 0.429934i −0.976621 0.214967i \(-0.931036\pi\)
0.976621 0.214967i \(-0.0689643\pi\)
\(542\) 0 0
\(543\) 14.0000i 0.600798i
\(544\) 0 0
\(545\) 12.0000 6.00000i 0.514024 0.257012i
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 0 0
\(549\) 2.00000i 0.0853579i
\(550\) 0 0
\(551\) −64.0000 −2.72649
\(552\) 0 0
\(553\) 16.0000i 0.680389i
\(554\) 0 0
\(555\) −10.0000 20.0000i −0.424476 0.848953i
\(556\) 0 0
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) −4.00000 + 2.00000i −0.168281 + 0.0841406i
\(566\) 0 0
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) 16.0000i 0.669579i 0.942293 + 0.334790i \(0.108665\pi\)
−0.942293 + 0.334790i \(0.891335\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −16.0000 12.0000i −0.667246 0.500435i
\(576\) 0 0
\(577\) 16.0000i 0.666089i −0.942911 0.333044i \(-0.891924\pi\)
0.942911 0.333044i \(-0.108076\pi\)
\(578\) 0 0
\(579\) 4.00000i 0.166234i
\(580\) 0 0
\(581\) 8.00000i 0.331896i
\(582\) 0 0
\(583\) 20.0000i 0.828315i
\(584\) 0 0
\(585\) 2.00000 + 4.00000i 0.0826898 + 0.165380i
\(586\) 0 0
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 34.0000i 1.39621i 0.715994 + 0.698106i \(0.245974\pi\)
−0.715994 + 0.698106i \(0.754026\pi\)
\(594\) 0 0
\(595\) −12.0000 24.0000i −0.491952 0.983904i
\(596\) 0 0
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) 0 0
\(605\) 7.00000 + 14.0000i 0.284590 + 0.569181i
\(606\) 0 0
\(607\) 22.0000i 0.892952i 0.894795 + 0.446476i \(0.147321\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(608\) 0 0
\(609\) 16.0000 0.648353
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) 0 0
\(615\) −2.00000 4.00000i −0.0806478 0.161296i
\(616\) 0 0
\(617\) 10.0000i 0.402585i −0.979531 0.201292i \(-0.935486\pi\)
0.979531 0.201292i \(-0.0645141\pi\)
\(618\) 0 0
\(619\) 20.0000i 0.803868i 0.915669 + 0.401934i \(0.131662\pi\)
−0.915669 + 0.401934i \(0.868338\pi\)
\(620\) 0 0
\(621\) 4.00000i 0.160514i
\(622\) 0 0
\(623\) 12.0000i 0.480770i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) −16.0000 −0.638978
\(628\) 0 0
\(629\) 60.0000i 2.39236i
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) 4.00000i 0.158986i
\(634\) 0 0
\(635\) −36.0000 + 18.0000i −1.42862 + 0.714308i
\(636\) 0 0
\(637\) 6.00000 0.237729
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 24.0000 0.946468 0.473234 0.880937i \(-0.343087\pi\)
0.473234 + 0.880937i \(0.343087\pi\)
\(644\) 0 0
\(645\) 12.0000 + 24.0000i 0.472500 + 0.944999i
\(646\) 0 0
\(647\) 8.00000i 0.314512i −0.987558 0.157256i \(-0.949735\pi\)
0.987558 0.157256i \(-0.0502649\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) −36.0000 + 18.0000i −1.40664 + 0.703318i
\(656\) 0 0
\(657\) 4.00000i 0.156055i
\(658\) 0 0
\(659\) 30.0000i 1.16863i 0.811525 + 0.584317i \(0.198638\pi\)
−0.811525 + 0.584317i \(0.801362\pi\)
\(660\) 0 0
\(661\) 2.00000i 0.0777910i 0.999243 + 0.0388955i \(0.0123839\pi\)
−0.999243 + 0.0388955i \(0.987616\pi\)
\(662\) 0 0
\(663\) 12.0000i 0.466041i
\(664\) 0 0
\(665\) 16.0000 + 32.0000i 0.620453 + 1.24091i
\(666\) 0 0
\(667\) −32.0000 −1.23904
\(668\) 0 0
\(669\) 6.00000i 0.231973i
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 20.0000i 0.770943i −0.922720 0.385472i \(-0.874039\pi\)
0.922720 0.385472i \(-0.125961\pi\)
\(674\) 0 0
\(675\) −3.00000 + 4.00000i −0.115470 + 0.153960i
\(676\) 0 0
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 0 0
\(679\) −16.0000 −0.614024
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) −20.0000 + 10.0000i −0.764161 + 0.382080i
\(686\) 0 0
\(687\) 6.00000i 0.228914i
\(688\) 0 0
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) 48.0000i 1.82601i −0.407953 0.913003i \(-0.633757\pi\)
0.407953 0.913003i \(-0.366243\pi\)
\(692\) 0 0
\(693\) 4.00000 0.151947
\(694\) 0 0
\(695\) 8.00000 4.00000i 0.303457 0.151729i
\(696\) 0 0
\(697\) 12.0000i 0.454532i
\(698\) 0 0
\(699\) 6.00000i 0.226941i
\(700\) 0 0
\(701\) 8.00000i 0.302156i 0.988522 + 0.151078i \(0.0482744\pi\)
−0.988522 + 0.151078i \(0.951726\pi\)
\(702\) 0 0
\(703\) 80.0000i 3.01726i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 46.0000i 1.72757i −0.503864 0.863783i \(-0.668089\pi\)
0.503864 0.863783i \(-0.331911\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −8.00000 + 4.00000i −0.299183 + 0.149592i
\(716\) 0 0
\(717\) −20.0000 −0.746914
\(718\) 0 0
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) 0 0
\(725\) −32.0000 24.0000i −1.18845 0.891338i
\(726\) 0 0
\(727\) 30.0000i 1.11264i −0.830969 0.556319i \(-0.812213\pi\)
0.830969 0.556319i \(-0.187787\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 72.0000i 2.66302i
\(732\) 0 0
\(733\) −42.0000 −1.55131 −0.775653 0.631160i \(-0.782579\pi\)
−0.775653 + 0.631160i \(0.782579\pi\)
\(734\) 0 0
\(735\) 3.00000 + 6.00000i 0.110657 + 0.221313i
\(736\) 0 0
\(737\) 16.0000i 0.589368i
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 16.0000i 0.587775i
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −24.0000 + 12.0000i −0.879292 + 0.439646i
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 8.00000i 0.292314i
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 2.00000i 0.0728841i
\(754\) 0 0
\(755\) −16.0000 32.0000i −0.582300 1.16460i
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 2.00000 0.0724999 0.0362500 0.999343i \(-0.488459\pi\)
0.0362500 + 0.999343i \(0.488459\pi\)
\(762\) 0 0
\(763\) −12.0000 −0.434429
\(764\) 0 0
\(765\) 12.0000 6.00000i 0.433861 0.216930i
\(766\) 0 0
\(767\) 12.0000i 0.433295i
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 6.00000i 0.216085i
\(772\) 0 0
\(773\) −38.0000 −1.36677 −0.683383 0.730061i \(-0.739492\pi\)
−0.683383 + 0.730061i \(0.739492\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 20.0000i 0.717496i
\(778\) 0 0
\(779\) 16.0000i 0.573259i
\(780\) 0 0
\(781\) 8.00000i 0.286263i
\(782\) 0 0
\(783\) 8.00000i 0.285897i
\(784\) 0 0
\(785\) 14.0000 + 28.0000i 0.499681 + 0.999363i
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 0 0
\(789\) 12.0000i 0.427211i
\(790\) 0 0
\(791\) 4.00000 0.142224
\(792\) 0 0
\(793\) 4.00000i 0.142044i
\(794\) 0 0
\(795\) −10.0000 20.0000i −0.354663 0.709327i
\(796\) 0 0
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 8.00000 0.282314
\(804\) 0 0