Properties

Label 3840.2.d.bc.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 480)
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.bc.2689.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +(1.00000 + 2.00000i) q^{5} -2.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +(1.00000 + 2.00000i) q^{5} -2.00000i q^{7} +1.00000 q^{9} -6.00000i q^{11} +2.00000 q^{13} +(1.00000 + 2.00000i) q^{15} +6.00000i q^{17} +4.00000i q^{19} -2.00000i q^{21} -8.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +1.00000 q^{27} +8.00000 q^{31} -6.00000i q^{33} +(4.00000 - 2.00000i) q^{35} -2.00000 q^{37} +2.00000 q^{39} +6.00000 q^{41} -4.00000 q^{43} +(1.00000 + 2.00000i) q^{45} -4.00000i q^{47} +3.00000 q^{49} +6.00000i q^{51} +6.00000 q^{53} +(12.0000 - 6.00000i) q^{55} +4.00000i q^{57} +6.00000i q^{59} -6.00000i q^{61} -2.00000i q^{63} +(2.00000 + 4.00000i) q^{65} -8.00000i q^{69} -4.00000 q^{71} -12.0000i q^{73} +(-3.00000 + 4.00000i) q^{75} -12.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} +(-12.0000 + 6.00000i) q^{85} +14.0000 q^{89} -4.00000i q^{91} +8.00000 q^{93} +(-8.00000 + 4.00000i) q^{95} -8.00000i q^{97} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{5} + 2q^{9} + 4q^{13} + 2q^{15} - 6q^{25} + 2q^{27} + 16q^{31} + 8q^{35} - 4q^{37} + 4q^{39} + 12q^{41} - 8q^{43} + 2q^{45} + 6q^{49} + 12q^{53} + 24q^{55} + 4q^{65} - 8q^{71} - 6q^{75} - 24q^{77} - 16q^{79} + 2q^{81} + 24q^{83} - 24q^{85} + 28q^{89} + 16q^{93} - 16q^{95} + O(q^{100}) \)

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
\(11\) 6.00000i 1.80907i −0.426401 0.904534i \(-0.640219\pi\)
0.426401 0.904534i \(-0.359781\pi\)
\(12\) 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 \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) 8.00000i 1.66812i −0.551677 0.834058i \(-0.686012\pi\)
0.551677 0.834058i \(-0.313988\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 6.00000i 1.04447i
\(34\) 0 0
\(35\) 4.00000 2.00000i 0.676123 0.338062i
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 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
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 6.00000i 0.840168i
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 12.0000 6.00000i 1.61808 0.809040i
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 6.00000i 0.768221i −0.923287 0.384111i \(-0.874508\pi\)
0.923287 0.384111i \(-0.125492\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) 2.00000 + 4.00000i 0.248069 + 0.496139i
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 8.00000i 0.963087i
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 12.0000i 1.40449i −0.711934 0.702247i \(-0.752180\pi\)
0.711934 0.702247i \(-0.247820\pi\)
\(74\) 0 0
\(75\) −3.00000 + 4.00000i −0.346410 + 0.461880i
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −12.0000 + 6.00000i −1.30158 + 0.650791i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 4.00000i 0.419314i
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) −8.00000 + 4.00000i −0.820783 + 0.410391i
\(96\) 0 0
\(97\) 8.00000i 0.812277i −0.913812 0.406138i \(-0.866875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) 0 0
\(99\) 6.00000i 0.603023i
\(100\) 0 0
\(101\) 8.00000i 0.796030i −0.917379 0.398015i \(-0.869699\pi\)
0.917379 0.398015i \(-0.130301\pi\)
\(102\) 0 0
\(103\) 6.00000i 0.591198i −0.955312 0.295599i \(-0.904481\pi\)
0.955312 0.295599i \(-0.0955191\pi\)
\(104\) 0 0
\(105\) 4.00000 2.00000i 0.390360 0.195180i
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 2.00000i 0.191565i −0.995402 0.0957826i \(-0.969465\pi\)
0.995402 0.0957826i \(-0.0305354\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 2.00000i 0.188144i −0.995565 0.0940721i \(-0.970012\pi\)
0.995565 0.0940721i \(-0.0299884\pi\)
\(114\) 0 0
\(115\) 16.0000 8.00000i 1.49201 0.746004i
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −25.0000 −2.27273
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) −11.0000 2.00000i −0.983870 0.178885i
\(126\) 0 0
\(127\) 22.0000i 1.95218i −0.217357 0.976092i \(-0.569744\pi\)
0.217357 0.976092i \(-0.430256\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 10.0000i 0.873704i 0.899533 + 0.436852i \(0.143907\pi\)
−0.899533 + 0.436852i \(0.856093\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) 0 0
\(135\) 1.00000 + 2.00000i 0.0860663 + 0.172133i
\(136\) 0 0
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) 16.0000i 1.35710i 0.734553 + 0.678551i \(0.237392\pi\)
−0.734553 + 0.678551i \(0.762608\pi\)
\(140\) 0 0
\(141\) 4.00000i 0.336861i
\(142\) 0 0
\(143\) 12.0000i 1.00349i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) 4.00000i 0.327693i −0.986486 0.163846i \(-0.947610\pi\)
0.986486 0.163846i \(-0.0523901\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 8.00000 + 16.0000i 0.642575 + 1.28515i
\(156\) 0 0
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 0 0
\(165\) 12.0000 6.00000i 0.934199 0.467099i
\(166\) 0 0
\(167\) 16.0000i 1.23812i 0.785345 + 0.619059i \(0.212486\pi\)
−0.785345 + 0.619059i \(0.787514\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 4.00000i 0.305888i
\(172\) 0 0
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 8.00000 + 6.00000i 0.604743 + 0.453557i
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 0 0
\(179\) 2.00000i 0.149487i 0.997203 + 0.0747435i \(0.0238138\pi\)
−0.997203 + 0.0747435i \(0.976186\pi\)
\(180\) 0 0
\(181\) 22.0000i 1.63525i 0.575753 + 0.817624i \(0.304709\pi\)
−0.575753 + 0.817624i \(0.695291\pi\)
\(182\) 0 0
\(183\) 6.00000i 0.443533i
\(184\) 0 0
\(185\) −2.00000 4.00000i −0.147043 0.294086i
\(186\) 0 0
\(187\) 36.0000 2.63258
\(188\) 0 0
\(189\) 2.00000i 0.145479i
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) 12.0000i 0.863779i 0.901927 + 0.431889i \(0.142153\pi\)
−0.901927 + 0.431889i \(0.857847\pi\)
\(194\) 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
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 6.00000 + 12.0000i 0.419058 + 0.838116i
\(206\) 0 0
\(207\) 8.00000i 0.556038i
\(208\) 0 0
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) 16.0000i 1.10149i 0.834675 + 0.550743i \(0.185655\pi\)
−0.834675 + 0.550743i \(0.814345\pi\)
\(212\) 0 0
\(213\) −4.00000 −0.274075
\(214\) 0 0
\(215\) −4.00000 8.00000i −0.272798 0.545595i
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) 12.0000i 0.810885i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) 10.0000i 0.669650i −0.942280 0.334825i \(-0.891323\pi\)
0.942280 0.334825i \(-0.108677\pi\)
\(224\) 0 0
\(225\) −3.00000 + 4.00000i −0.200000 + 0.266667i
\(226\) 0 0
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 0 0
\(233\) 22.0000i 1.44127i 0.693316 + 0.720634i \(0.256149\pi\)
−0.693316 + 0.720634i \(0.743851\pi\)
\(234\) 0 0
\(235\) 8.00000 4.00000i 0.521862 0.260931i
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 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
\(247\) 8.00000i 0.509028i
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 14.0000i 0.883672i 0.897096 + 0.441836i \(0.145673\pi\)
−0.897096 + 0.441836i \(0.854327\pi\)
\(252\) 0 0
\(253\) −48.0000 −3.01773
\(254\) 0 0
\(255\) −12.0000 + 6.00000i −0.751469 + 0.375735i
\(256\) 0 0
\(257\) 6.00000i 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) 0 0
\(259\) 4.00000i 0.248548i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 24.0000i 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) 0 0
\(265\) 6.00000 + 12.0000i 0.368577 + 0.737154i
\(266\) 0 0
\(267\) 14.0000 0.856786
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 4.00000i 0.242091i
\(274\) 0 0
\(275\) 24.0000 + 18.0000i 1.44725 + 1.08544i
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) −8.00000 −0.475551 −0.237775 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) 0 0
\(285\) −8.00000 + 4.00000i −0.473879 + 0.236940i
\(286\) 0 0
\(287\) 12.0000i 0.708338i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 8.00000i 0.468968i
\(292\) 0 0
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 0 0
\(295\) −12.0000 + 6.00000i −0.698667 + 0.349334i
\(296\) 0 0
\(297\) 6.00000i 0.348155i
\(298\) 0 0
\(299\) 16.0000i 0.925304i
\(300\) 0 0
\(301\) 8.00000i 0.461112i
\(302\) 0 0
\(303\) 8.00000i 0.459588i
\(304\) 0 0
\(305\) 12.0000 6.00000i 0.687118 0.343559i
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 6.00000i 0.341328i
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 28.0000i 1.58265i −0.611393 0.791327i \(-0.709391\pi\)
0.611393 0.791327i \(-0.290609\pi\)
\(314\) 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
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) −6.00000 + 8.00000i −0.332820 + 0.443760i
\(326\) 0 0
\(327\) 2.00000i 0.110600i
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 4.00000i 0.219860i −0.993939 0.109930i \(-0.964937\pi\)
0.993939 0.109930i \(-0.0350627\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 20.0000i 1.08947i −0.838608 0.544735i \(-0.816630\pi\)
0.838608 0.544735i \(-0.183370\pi\)
\(338\) 0 0
\(339\) 2.00000i 0.108625i
\(340\) 0 0
\(341\) 48.0000i 2.59935i
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 16.0000 8.00000i 0.861411 0.430706i
\(346\) 0 0
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 0 0
\(349\) 14.0000i 0.749403i 0.927146 + 0.374701i \(0.122255\pi\)
−0.927146 + 0.374701i \(0.877745\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 6.00000i 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) −4.00000 8.00000i −0.212298 0.424596i
\(356\) 0 0
\(357\) 12.0000 0.635107
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) 24.0000 12.0000i 1.25622 0.628109i
\(366\) 0 0
\(367\) 10.0000i 0.521996i 0.965339 + 0.260998i \(0.0840516\pi\)
−0.965339 + 0.260998i \(0.915948\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 12.0000i 0.623009i
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) −11.0000 2.00000i −0.568038 0.103280i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.00000i 0.410932i 0.978664 + 0.205466i \(0.0658711\pi\)
−0.978664 + 0.205466i \(0.934129\pi\)
\(380\) 0 0
\(381\) 22.0000i 1.12709i
\(382\) 0 0
\(383\) 12.0000i 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) 0 0
\(385\) −12.0000 24.0000i −0.611577 1.22315i
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 36.0000i 1.82527i 0.408773 + 0.912636i \(0.365957\pi\)
−0.408773 + 0.912636i \(0.634043\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) 0 0
\(393\) 10.0000i 0.504433i
\(394\) 0 0
\(395\) −8.00000 16.0000i −0.402524 0.805047i
\(396\) 0 0
\(397\) −38.0000 −1.90717 −0.953583 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) 0 0
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 0 0
\(405\) 1.00000 + 2.00000i 0.0496904 + 0.0993808i
\(406\) 0 0
\(407\) 12.0000i 0.594818i
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 6.00000i 0.295958i
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 12.0000 + 24.0000i 0.589057 + 1.17811i
\(416\) 0 0
\(417\) 16.0000i 0.783523i
\(418\) 0 0
\(419\) 26.0000i 1.27018i −0.772437 0.635092i \(-0.780962\pi\)
0.772437 0.635092i \(-0.219038\pi\)
\(420\) 0 0
\(421\) 2.00000i 0.0974740i 0.998812 + 0.0487370i \(0.0155196\pi\)
−0.998812 + 0.0487370i \(0.984480\pi\)
\(422\) 0 0
\(423\) 4.00000i 0.194487i
\(424\) 0 0
\(425\) −24.0000 18.0000i −1.16417 0.873128i
\(426\) 0 0
\(427\) −12.0000 −0.580721
\(428\) 0 0
\(429\) 12.0000i 0.579365i
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) 28.0000i 1.34559i 0.739827 + 0.672797i \(0.234907\pi\)
−0.739827 + 0.672797i \(0.765093\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 32.0000 1.53077
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 0 0
\(445\) 14.0000 + 28.0000i 0.663664 + 1.32733i
\(446\) 0 0
\(447\) 4.00000i 0.189194i
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 36.0000i 1.69517i
\(452\) 0 0
\(453\) −8.00000 −0.375873
\(454\) 0 0
\(455\) 8.00000 4.00000i 0.375046 0.187523i
\(456\) 0 0
\(457\) 16.0000i 0.748448i 0.927338 + 0.374224i \(0.122091\pi\)
−0.927338 + 0.374224i \(0.877909\pi\)
\(458\) 0 0
\(459\) 6.00000i 0.280056i
\(460\) 0 0
\(461\) 20.0000i 0.931493i −0.884918 0.465746i \(-0.845786\pi\)
0.884918 0.465746i \(-0.154214\pi\)
\(462\) 0 0
\(463\) 14.0000i 0.650635i −0.945605 0.325318i \(-0.894529\pi\)
0.945605 0.325318i \(-0.105471\pi\)
\(464\) 0 0
\(465\) 8.00000 + 16.0000i 0.370991 + 0.741982i
\(466\) 0 0
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) 0 0
\(473\) 24.0000i 1.10352i
\(474\) 0 0
\(475\) −16.0000 12.0000i −0.734130 0.550598i
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) −28.0000 −1.27935 −0.639676 0.768644i \(-0.720932\pi\)
−0.639676 + 0.768644i \(0.720932\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) −16.0000 −0.728025
\(484\) 0 0
\(485\) 16.0000 8.00000i 0.726523 0.363261i
\(486\) 0 0
\(487\) 10.0000i 0.453143i 0.973995 + 0.226572i \(0.0727517\pi\)
−0.973995 + 0.226572i \(0.927248\pi\)
\(488\) 0 0
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) 14.0000i 0.631811i 0.948791 + 0.315906i \(0.102308\pi\)
−0.948791 + 0.315906i \(0.897692\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 12.0000 6.00000i 0.539360 0.269680i
\(496\) 0 0
\(497\) 8.00000i 0.358849i
\(498\) 0 0
\(499\) 36.0000i 1.61158i 0.592200 + 0.805791i \(0.298259\pi\)
−0.592200 + 0.805791i \(0.701741\pi\)
\(500\) 0 0
\(501\) 16.0000i 0.714827i
\(502\) 0 0
\(503\) 4.00000i 0.178351i −0.996016 0.0891756i \(-0.971577\pi\)
0.996016 0.0891756i \(-0.0284232\pi\)
\(504\) 0 0
\(505\) 16.0000 8.00000i 0.711991 0.355995i
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −24.0000 −1.06170
\(512\) 0 0
\(513\) 4.00000i 0.176604i
\(514\) 0 0
\(515\) 12.0000 6.00000i 0.528783 0.264392i
\(516\) 0 0
\(517\) −24.0000 −1.05552
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) 0 0
\(523\) −24.0000 −1.04945 −0.524723 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(524\) 0 0
\(525\) 8.00000 + 6.00000i 0.349149 + 0.261861i
\(526\) 0 0
\(527\) 48.0000i 2.09091i
\(528\) 0 0
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 6.00000i 0.260378i
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 12.0000 + 24.0000i 0.518805 + 1.03761i
\(536\) 0 0
\(537\) 2.00000i 0.0863064i
\(538\) 0 0
\(539\) 18.0000i 0.775315i
\(540\) 0 0
\(541\) 2.00000i 0.0859867i 0.999075 + 0.0429934i \(0.0136894\pi\)
−0.999075 + 0.0429934i \(0.986311\pi\)
\(542\) 0 0
\(543\) 22.0000i 0.944110i
\(544\) 0 0
\(545\) 4.00000 2.00000i 0.171341 0.0856706i
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) 6.00000i 0.256074i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 16.0000i 0.680389i
\(554\) 0 0
\(555\) −2.00000 4.00000i −0.0848953 0.169791i
\(556\) 0 0
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 36.0000 1.51992
\(562\) 0 0
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 4.00000 2.00000i 0.168281 0.0841406i
\(566\) 0 0
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) −2.00000 −0.0838444 −0.0419222 0.999121i \(-0.513348\pi\)
−0.0419222 + 0.999121i \(0.513348\pi\)
\(570\) 0 0
\(571\) 28.0000i 1.17176i 0.810397 + 0.585882i \(0.199252\pi\)
−0.810397 + 0.585882i \(0.800748\pi\)
\(572\) 0 0
\(573\) 20.0000 0.835512
\(574\) 0 0
\(575\) 32.0000 + 24.0000i 1.33449 + 1.00087i
\(576\) 0 0
\(577\) 16.0000i 0.666089i 0.942911 + 0.333044i \(0.108076\pi\)
−0.942911 + 0.333044i \(0.891924\pi\)
\(578\) 0 0
\(579\) 12.0000i 0.498703i
\(580\) 0 0
\(581\) 24.0000i 0.995688i
\(582\) 0 0
\(583\) 36.0000i 1.49097i
\(584\) 0 0
\(585\) 2.00000 + 4.00000i 0.0826898 + 0.165380i
\(586\) 0 0
\(587\) −20.0000 −0.825488 −0.412744 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) 0 0
\(589\) 32.0000i 1.31854i
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 2.00000i 0.0821302i −0.999156 0.0410651i \(-0.986925\pi\)
0.999156 0.0410651i \(-0.0130751\pi\)
\(594\) 0 0
\(595\) 12.0000 + 24.0000i 0.491952 + 0.983904i
\(596\) 0 0
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −25.0000 50.0000i −1.01639 2.03279i
\(606\) 0 0
\(607\) 22.0000i 0.892952i 0.894795 + 0.446476i \(0.147321\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000i 0.323645i
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 0 0
\(615\) 6.00000 + 12.0000i 0.241943 + 0.483887i
\(616\) 0 0
\(617\) 38.0000i 1.52982i −0.644136 0.764911i \(-0.722783\pi\)
0.644136 0.764911i \(-0.277217\pi\)
\(618\) 0 0
\(619\) 24.0000i 0.964641i −0.875995 0.482321i \(-0.839794\pi\)
0.875995 0.482321i \(-0.160206\pi\)
\(620\) 0 0
\(621\) 8.00000i 0.321029i
\(622\) 0 0
\(623\) 28.0000i 1.12180i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 24.0000 0.958468
\(628\) 0 0
\(629\) 12.0000i 0.478471i
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 16.0000i 0.635943i
\(634\) 0 0
\(635\) 44.0000 22.0000i 1.74609 0.873043i
\(636\) 0 0
\(637\) 6.00000 0.237729
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) −48.0000 −1.89294 −0.946468 0.322799i \(-0.895376\pi\)
−0.946468 + 0.322799i \(0.895376\pi\)
\(644\) 0 0
\(645\) −4.00000 8.00000i −0.157500 0.315000i
\(646\) 0 0
\(647\) 12.0000i 0.471769i −0.971781 0.235884i \(-0.924201\pi\)
0.971781 0.235884i \(-0.0757987\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 16.0000i 0.627089i
\(652\) 0 0
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 0 0
\(655\) −20.0000 + 10.0000i −0.781465 + 0.390732i
\(656\) 0 0
\(657\) 12.0000i 0.468165i
\(658\) 0 0
\(659\) 30.0000i 1.16863i 0.811525 + 0.584317i \(0.198638\pi\)
−0.811525 + 0.584317i \(0.801362\pi\)
\(660\) 0 0
\(661\) 6.00000i 0.233373i 0.993169 + 0.116686i \(0.0372273\pi\)
−0.993169 + 0.116686i \(0.962773\pi\)
\(662\) 0 0
\(663\) 12.0000i 0.466041i
\(664\) 0 0
\(665\) 8.00000 + 16.0000i 0.310227 + 0.620453i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 10.0000i 0.386622i
\(670\) 0 0
\(671\) −36.0000 −1.38976
\(672\) 0 0
\(673\) 28.0000i 1.07932i −0.841883 0.539660i \(-0.818553\pi\)
0.841883 0.539660i \(-0.181447\pi\)
\(674\) 0 0
\(675\) −3.00000 + 4.00000i −0.115470 + 0.153960i
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −16.0000 −0.614024
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) −12.0000 + 6.00000i −0.458496 + 0.229248i
\(686\) 0 0
\(687\) 14.0000i 0.534133i
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 20.0000i 0.760836i −0.924815 0.380418i \(-0.875780\pi\)
0.924815 0.380418i \(-0.124220\pi\)
\(692\) 0 0
\(693\) −12.0000 −0.455842
\(694\) 0 0
\(695\) −32.0000 + 16.0000i −1.21383 + 0.606915i
\(696\) 0 0
\(697\) 36.0000i 1.36360i
\(698\) 0 0
\(699\) 22.0000i 0.832116i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 8.00000i 0.301726i
\(704\) 0 0
\(705\) 8.00000 4.00000i 0.301297 0.150649i
\(706\) 0 0
\(707\) −16.0000 −0.601742
\(708\) 0 0
\(709\) 10.0000i 0.375558i −0.982211 0.187779i \(-0.939871\pi\)
0.982211 0.187779i \(-0.0601289\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 64.0000i 2.39682i
\(714\) 0 0
\(715\) 24.0000 12.0000i 0.897549 0.448775i
\(716\) 0 0
\(717\) 20.0000 0.746914
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 26.0000i 0.964287i 0.876092 + 0.482143i \(0.160142\pi\)
−0.876092 + 0.482143i \(0.839858\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000i 0.887672i
\(732\) 0 0
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 0 0
\(735\) 3.00000 + 6.00000i 0.110657 + 0.221313i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 20.0000i 0.735712i −0.929883 0.367856i \(-0.880092\pi\)
0.929883 0.367856i \(-0.119908\pi\)
\(740\) 0 0
\(741\) 8.00000i 0.293887i
\(742\) 0 0
\(743\) 12.0000i 0.440237i 0.975473 + 0.220119i \(0.0706445\pi\)
−0.975473 + 0.220119i \(0.929356\pi\)
\(744\) 0 0
\(745\) 8.00000 4.00000i 0.293097 0.146549i
\(746\) 0 0
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 24.0000i 0.876941i
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) 14.0000i 0.510188i
\(754\) 0 0
\(755\) −8.00000 16.0000i −0.291150 0.582300i
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 0 0
\(759\) −48.0000 −1.74229
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) 0 0
\(765\) −12.0000 + 6.00000i −0.433861 + 0.216930i
\(766\) 0 0
\(767\) 12.0000i 0.433295i
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 6.00000i 0.216085i
\(772\) 0 0
\(773\) 26.0000 0.935155 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) 0 0
\(775\) −24.0000 + 32.0000i −0.862105 + 1.14947i
\(776\) 0 0
\(777\) 4.00000i 0.143499i
\(778\) 0 0
\(779\) 24.0000i 0.859889i
\(780\) 0 0
\(781\) 24.0000i 0.858788i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 22.0000 + 44.0000i 0.785214 + 1.57043i
\(786\) 0 0
\(787\) 24.0000 0.855508 0.427754 0.903895i \(-0.359305\pi\)
0.427754 + 0.903895i \(0.359305\pi\)
\(788\) 0 0
\(789\) 24.0000i 0.854423i
\(790\) 0 0
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) 0 0
\(795\) 6.00000 + 12.0000i 0.212798 + 0.425596i
\(796\) 0 0
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) 0 0
\(803\) −72.0000 −2.54082