Properties

Label 5520.2.be.c.1471.14
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.be (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.14
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.c.1471.13

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{5} -3.74981 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{5} -3.74981 q^{7} -1.00000 q^{9} +0.951741 q^{11} -0.958125 q^{13} -1.00000 q^{15} +7.12519i q^{17} -8.14480 q^{19} -3.74981i q^{21} +(3.34339 + 3.43827i) q^{23} -1.00000 q^{25} -1.00000i q^{27} +3.97878 q^{29} +10.2840i q^{31} +0.951741i q^{33} -3.74981i q^{35} +7.97275i q^{37} -0.958125i q^{39} +2.05735 q^{41} +4.47788 q^{43} -1.00000i q^{45} -5.73241i q^{47} +7.06110 q^{49} -7.12519 q^{51} -5.84488i q^{53} +0.951741i q^{55} -8.14480i q^{57} -5.24677i q^{59} -1.55681i q^{61} +3.74981 q^{63} -0.958125i q^{65} -8.31703 q^{67} +(-3.43827 + 3.34339i) q^{69} -14.9335i q^{71} +2.66136 q^{73} -1.00000i q^{75} -3.56885 q^{77} -15.4270 q^{79} +1.00000 q^{81} +3.86616 q^{83} -7.12519 q^{85} +3.97878i q^{87} +3.08766i q^{89} +3.59279 q^{91} -10.2840 q^{93} -8.14480i q^{95} +5.79266i q^{97} -0.951741 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 8q^{7} - 32q^{9} + O(q^{10}) \) \( 32q - 8q^{7} - 32q^{9} + 8q^{11} - 8q^{13} - 32q^{15} - 32q^{25} + 4q^{29} + 20q^{41} + 52q^{49} - 4q^{51} + 8q^{63} + 32q^{67} - 40q^{73} - 24q^{77} + 32q^{79} + 32q^{81} - 4q^{85} - 48q^{91} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1841\) \(4417\) \(4831\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) −3.74981 −1.41730 −0.708648 0.705562i \(-0.750695\pi\)
−0.708648 + 0.705562i \(0.750695\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0.951741 0.286961 0.143480 0.989653i \(-0.454171\pi\)
0.143480 + 0.989653i \(0.454171\pi\)
\(12\) 0 0
\(13\) −0.958125 −0.265736 −0.132868 0.991134i \(-0.542419\pi\)
−0.132868 + 0.991134i \(0.542419\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 7.12519i 1.72811i 0.503396 + 0.864056i \(0.332084\pi\)
−0.503396 + 0.864056i \(0.667916\pi\)
\(18\) 0 0
\(19\) −8.14480 −1.86855 −0.934273 0.356559i \(-0.883950\pi\)
−0.934273 + 0.356559i \(0.883950\pi\)
\(20\) 0 0
\(21\) 3.74981i 0.818276i
\(22\) 0 0
\(23\) 3.34339 + 3.43827i 0.697145 + 0.716930i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 3.97878 0.738842 0.369421 0.929262i \(-0.379556\pi\)
0.369421 + 0.929262i \(0.379556\pi\)
\(30\) 0 0
\(31\) 10.2840i 1.84707i 0.383518 + 0.923534i \(0.374713\pi\)
−0.383518 + 0.923534i \(0.625287\pi\)
\(32\) 0 0
\(33\) 0.951741i 0.165677i
\(34\) 0 0
\(35\) 3.74981i 0.633834i
\(36\) 0 0
\(37\) 7.97275i 1.31071i 0.755320 + 0.655356i \(0.227481\pi\)
−0.755320 + 0.655356i \(0.772519\pi\)
\(38\) 0 0
\(39\) 0.958125i 0.153423i
\(40\) 0 0
\(41\) 2.05735 0.321305 0.160652 0.987011i \(-0.448640\pi\)
0.160652 + 0.987011i \(0.448640\pi\)
\(42\) 0 0
\(43\) 4.47788 0.682870 0.341435 0.939905i \(-0.389087\pi\)
0.341435 + 0.939905i \(0.389087\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 5.73241i 0.836158i −0.908411 0.418079i \(-0.862704\pi\)
0.908411 0.418079i \(-0.137296\pi\)
\(48\) 0 0
\(49\) 7.06110 1.00873
\(50\) 0 0
\(51\) −7.12519 −0.997726
\(52\) 0 0
\(53\) 5.84488i 0.802856i −0.915891 0.401428i \(-0.868514\pi\)
0.915891 0.401428i \(-0.131486\pi\)
\(54\) 0 0
\(55\) 0.951741i 0.128333i
\(56\) 0 0
\(57\) 8.14480i 1.07881i
\(58\) 0 0
\(59\) 5.24677i 0.683071i −0.939869 0.341536i \(-0.889053\pi\)
0.939869 0.341536i \(-0.110947\pi\)
\(60\) 0 0
\(61\) 1.55681i 0.199329i −0.995021 0.0996644i \(-0.968223\pi\)
0.995021 0.0996644i \(-0.0317769\pi\)
\(62\) 0 0
\(63\) 3.74981 0.472432
\(64\) 0 0
\(65\) 0.958125i 0.118841i
\(66\) 0 0
\(67\) −8.31703 −1.01609 −0.508044 0.861331i \(-0.669631\pi\)
−0.508044 + 0.861331i \(0.669631\pi\)
\(68\) 0 0
\(69\) −3.43827 + 3.34339i −0.413920 + 0.402497i
\(70\) 0 0
\(71\) 14.9335i 1.77228i −0.463414 0.886142i \(-0.653376\pi\)
0.463414 0.886142i \(-0.346624\pi\)
\(72\) 0 0
\(73\) 2.66136 0.311488 0.155744 0.987797i \(-0.450222\pi\)
0.155744 + 0.987797i \(0.450222\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) −3.56885 −0.406708
\(78\) 0 0
\(79\) −15.4270 −1.73567 −0.867834 0.496854i \(-0.834488\pi\)
−0.867834 + 0.496854i \(0.834488\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.86616 0.424366 0.212183 0.977230i \(-0.431943\pi\)
0.212183 + 0.977230i \(0.431943\pi\)
\(84\) 0 0
\(85\) −7.12519 −0.772835
\(86\) 0 0
\(87\) 3.97878i 0.426570i
\(88\) 0 0
\(89\) 3.08766i 0.327291i 0.986519 + 0.163646i \(0.0523253\pi\)
−0.986519 + 0.163646i \(0.947675\pi\)
\(90\) 0 0
\(91\) 3.59279 0.376627
\(92\) 0 0
\(93\) −10.2840 −1.06640
\(94\) 0 0
\(95\) 8.14480i 0.835639i
\(96\) 0 0
\(97\) 5.79266i 0.588156i 0.955781 + 0.294078i \(0.0950125\pi\)
−0.955781 + 0.294078i \(0.904987\pi\)
\(98\) 0 0
\(99\) −0.951741 −0.0956536
\(100\) 0 0
\(101\) −1.61688 −0.160886 −0.0804428 0.996759i \(-0.525633\pi\)
−0.0804428 + 0.996759i \(0.525633\pi\)
\(102\) 0 0
\(103\) 1.88732 0.185963 0.0929817 0.995668i \(-0.470360\pi\)
0.0929817 + 0.995668i \(0.470360\pi\)
\(104\) 0 0
\(105\) 3.74981 0.365944
\(106\) 0 0
\(107\) −10.4072 −1.00611 −0.503053 0.864255i \(-0.667790\pi\)
−0.503053 + 0.864255i \(0.667790\pi\)
\(108\) 0 0
\(109\) 14.8316i 1.42061i −0.703895 0.710304i \(-0.748558\pi\)
0.703895 0.710304i \(-0.251442\pi\)
\(110\) 0 0
\(111\) −7.97275 −0.756740
\(112\) 0 0
\(113\) 2.06992i 0.194721i −0.995249 0.0973607i \(-0.968960\pi\)
0.995249 0.0973607i \(-0.0310400\pi\)
\(114\) 0 0
\(115\) −3.43827 + 3.34339i −0.320621 + 0.311773i
\(116\) 0 0
\(117\) 0.958125 0.0885787
\(118\) 0 0
\(119\) 26.7181i 2.44925i
\(120\) 0 0
\(121\) −10.0942 −0.917654
\(122\) 0 0
\(123\) 2.05735i 0.185505i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 1.77521i 0.157525i −0.996893 0.0787624i \(-0.974903\pi\)
0.996893 0.0787624i \(-0.0250968\pi\)
\(128\) 0 0
\(129\) 4.47788i 0.394255i
\(130\) 0 0
\(131\) 8.26875i 0.722444i 0.932480 + 0.361222i \(0.117640\pi\)
−0.932480 + 0.361222i \(0.882360\pi\)
\(132\) 0 0
\(133\) 30.5415 2.64828
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 11.9628i 1.02205i 0.859564 + 0.511027i \(0.170735\pi\)
−0.859564 + 0.511027i \(0.829265\pi\)
\(138\) 0 0
\(139\) 11.6365i 0.986993i −0.869748 0.493497i \(-0.835718\pi\)
0.869748 0.493497i \(-0.164282\pi\)
\(140\) 0 0
\(141\) 5.73241 0.482756
\(142\) 0 0
\(143\) −0.911887 −0.0762558
\(144\) 0 0
\(145\) 3.97878i 0.330420i
\(146\) 0 0
\(147\) 7.06110i 0.582390i
\(148\) 0 0
\(149\) 9.88485i 0.809798i 0.914361 + 0.404899i \(0.132693\pi\)
−0.914361 + 0.404899i \(0.867307\pi\)
\(150\) 0 0
\(151\) 15.5120i 1.26235i −0.775640 0.631175i \(-0.782573\pi\)
0.775640 0.631175i \(-0.217427\pi\)
\(152\) 0 0
\(153\) 7.12519i 0.576037i
\(154\) 0 0
\(155\) −10.2840 −0.826034
\(156\) 0 0
\(157\) 14.2536i 1.13756i −0.822488 0.568782i \(-0.807415\pi\)
0.822488 0.568782i \(-0.192585\pi\)
\(158\) 0 0
\(159\) 5.84488 0.463529
\(160\) 0 0
\(161\) −12.5371 12.8929i −0.988061 1.01610i
\(162\) 0 0
\(163\) 5.48701i 0.429776i −0.976639 0.214888i \(-0.931061\pi\)
0.976639 0.214888i \(-0.0689386\pi\)
\(164\) 0 0
\(165\) −0.951741 −0.0740929
\(166\) 0 0
\(167\) 14.3069i 1.10710i −0.832817 0.553549i \(-0.813273\pi\)
0.832817 0.553549i \(-0.186727\pi\)
\(168\) 0 0
\(169\) −12.0820 −0.929384
\(170\) 0 0
\(171\) 8.14480 0.622849
\(172\) 0 0
\(173\) 21.9452 1.66846 0.834231 0.551415i \(-0.185912\pi\)
0.834231 + 0.551415i \(0.185912\pi\)
\(174\) 0 0
\(175\) 3.74981 0.283459
\(176\) 0 0
\(177\) 5.24677 0.394371
\(178\) 0 0
\(179\) 15.2193i 1.13754i 0.822496 + 0.568771i \(0.192581\pi\)
−0.822496 + 0.568771i \(0.807419\pi\)
\(180\) 0 0
\(181\) 13.5401i 1.00642i −0.864163 0.503212i \(-0.832151\pi\)
0.864163 0.503212i \(-0.167849\pi\)
\(182\) 0 0
\(183\) 1.55681 0.115082
\(184\) 0 0
\(185\) −7.97275 −0.586168
\(186\) 0 0
\(187\) 6.78133i 0.495900i
\(188\) 0 0
\(189\) 3.74981i 0.272759i
\(190\) 0 0
\(191\) −26.2962 −1.90273 −0.951363 0.308072i \(-0.900316\pi\)
−0.951363 + 0.308072i \(0.900316\pi\)
\(192\) 0 0
\(193\) 8.11360 0.584030 0.292015 0.956414i \(-0.405674\pi\)
0.292015 + 0.956414i \(0.405674\pi\)
\(194\) 0 0
\(195\) 0.958125 0.0686128
\(196\) 0 0
\(197\) −15.2265 −1.08484 −0.542420 0.840107i \(-0.682492\pi\)
−0.542420 + 0.840107i \(0.682492\pi\)
\(198\) 0 0
\(199\) −2.38626 −0.169157 −0.0845786 0.996417i \(-0.526954\pi\)
−0.0845786 + 0.996417i \(0.526954\pi\)
\(200\) 0 0
\(201\) 8.31703i 0.586638i
\(202\) 0 0
\(203\) −14.9197 −1.04716
\(204\) 0 0
\(205\) 2.05735i 0.143692i
\(206\) 0 0
\(207\) −3.34339 3.43827i −0.232382 0.238977i
\(208\) 0 0
\(209\) −7.75174 −0.536199
\(210\) 0 0
\(211\) 10.5539i 0.726558i 0.931680 + 0.363279i \(0.118343\pi\)
−0.931680 + 0.363279i \(0.881657\pi\)
\(212\) 0 0
\(213\) 14.9335 1.02323
\(214\) 0 0
\(215\) 4.47788i 0.305389i
\(216\) 0 0
\(217\) 38.5632i 2.61784i
\(218\) 0 0
\(219\) 2.66136i 0.179838i
\(220\) 0 0
\(221\) 6.82682i 0.459222i
\(222\) 0 0
\(223\) 27.0582i 1.81195i −0.423333 0.905974i \(-0.639140\pi\)
0.423333 0.905974i \(-0.360860\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −12.0307 −0.798507 −0.399253 0.916841i \(-0.630731\pi\)
−0.399253 + 0.916841i \(0.630731\pi\)
\(228\) 0 0
\(229\) 13.4864i 0.891207i 0.895230 + 0.445604i \(0.147011\pi\)
−0.895230 + 0.445604i \(0.852989\pi\)
\(230\) 0 0
\(231\) 3.56885i 0.234813i
\(232\) 0 0
\(233\) −9.71906 −0.636717 −0.318358 0.947970i \(-0.603132\pi\)
−0.318358 + 0.947970i \(0.603132\pi\)
\(234\) 0 0
\(235\) 5.73241 0.373941
\(236\) 0 0
\(237\) 15.4270i 1.00209i
\(238\) 0 0
\(239\) 21.9024i 1.41675i 0.705836 + 0.708375i \(0.250572\pi\)
−0.705836 + 0.708375i \(0.749428\pi\)
\(240\) 0 0
\(241\) 1.37300i 0.0884430i −0.999022 0.0442215i \(-0.985919\pi\)
0.999022 0.0442215i \(-0.0140807\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 7.06110i 0.451117i
\(246\) 0 0
\(247\) 7.80374 0.496540
\(248\) 0 0
\(249\) 3.86616i 0.245008i
\(250\) 0 0
\(251\) 26.3289 1.66186 0.830932 0.556374i \(-0.187808\pi\)
0.830932 + 0.556374i \(0.187808\pi\)
\(252\) 0 0
\(253\) 3.18204 + 3.27235i 0.200053 + 0.205731i
\(254\) 0 0
\(255\) 7.12519i 0.446196i
\(256\) 0 0
\(257\) 6.42940 0.401055 0.200527 0.979688i \(-0.435734\pi\)
0.200527 + 0.979688i \(0.435734\pi\)
\(258\) 0 0
\(259\) 29.8963i 1.85767i
\(260\) 0 0
\(261\) −3.97878 −0.246281
\(262\) 0 0
\(263\) 9.71951 0.599330 0.299665 0.954044i \(-0.403125\pi\)
0.299665 + 0.954044i \(0.403125\pi\)
\(264\) 0 0
\(265\) 5.84488 0.359048
\(266\) 0 0
\(267\) −3.08766 −0.188962
\(268\) 0 0
\(269\) −2.98681 −0.182109 −0.0910544 0.995846i \(-0.529024\pi\)
−0.0910544 + 0.995846i \(0.529024\pi\)
\(270\) 0 0
\(271\) 16.5815i 1.00726i −0.863920 0.503629i \(-0.831998\pi\)
0.863920 0.503629i \(-0.168002\pi\)
\(272\) 0 0
\(273\) 3.59279i 0.217446i
\(274\) 0 0
\(275\) −0.951741 −0.0573922
\(276\) 0 0
\(277\) 25.1673 1.51216 0.756078 0.654481i \(-0.227113\pi\)
0.756078 + 0.654481i \(0.227113\pi\)
\(278\) 0 0
\(279\) 10.2840i 0.615689i
\(280\) 0 0
\(281\) 1.83384i 0.109398i −0.998503 0.0546989i \(-0.982580\pi\)
0.998503 0.0546989i \(-0.0174199\pi\)
\(282\) 0 0
\(283\) −13.6656 −0.812337 −0.406169 0.913798i \(-0.633135\pi\)
−0.406169 + 0.913798i \(0.633135\pi\)
\(284\) 0 0
\(285\) 8.14480 0.482456
\(286\) 0 0
\(287\) −7.71469 −0.455384
\(288\) 0 0
\(289\) −33.7683 −1.98637
\(290\) 0 0
\(291\) −5.79266 −0.339572
\(292\) 0 0
\(293\) 2.30871i 0.134876i −0.997723 0.0674381i \(-0.978517\pi\)
0.997723 0.0674381i \(-0.0214825\pi\)
\(294\) 0 0
\(295\) 5.24677 0.305479
\(296\) 0 0
\(297\) 0.951741i 0.0552256i
\(298\) 0 0
\(299\) −3.20339 3.29430i −0.185257 0.190514i
\(300\) 0 0
\(301\) −16.7912 −0.967829
\(302\) 0 0
\(303\) 1.61688i 0.0928873i
\(304\) 0 0
\(305\) 1.55681 0.0891425
\(306\) 0 0
\(307\) 11.6583i 0.665375i 0.943037 + 0.332687i \(0.107955\pi\)
−0.943037 + 0.332687i \(0.892045\pi\)
\(308\) 0 0
\(309\) 1.88732i 0.107366i
\(310\) 0 0
\(311\) 26.1374i 1.48212i −0.671440 0.741059i \(-0.734324\pi\)
0.671440 0.741059i \(-0.265676\pi\)
\(312\) 0 0
\(313\) 9.60260i 0.542771i −0.962471 0.271386i \(-0.912518\pi\)
0.962471 0.271386i \(-0.0874819\pi\)
\(314\) 0 0
\(315\) 3.74981i 0.211278i
\(316\) 0 0
\(317\) 21.8211 1.22560 0.612799 0.790239i \(-0.290044\pi\)
0.612799 + 0.790239i \(0.290044\pi\)
\(318\) 0 0
\(319\) 3.78677 0.212019
\(320\) 0 0
\(321\) 10.4072i 0.580876i
\(322\) 0 0
\(323\) 58.0332i 3.22906i
\(324\) 0 0
\(325\) 0.958125 0.0531472
\(326\) 0 0
\(327\) 14.8316 0.820188
\(328\) 0 0
\(329\) 21.4955i 1.18508i
\(330\) 0 0
\(331\) 23.3643i 1.28422i 0.766614 + 0.642108i \(0.221940\pi\)
−0.766614 + 0.642108i \(0.778060\pi\)
\(332\) 0 0
\(333\) 7.97275i 0.436904i
\(334\) 0 0
\(335\) 8.31703i 0.454408i
\(336\) 0 0
\(337\) 18.8389i 1.02622i 0.858323 + 0.513109i \(0.171506\pi\)
−0.858323 + 0.513109i \(0.828494\pi\)
\(338\) 0 0
\(339\) 2.06992 0.112422
\(340\) 0 0
\(341\) 9.78774i 0.530036i
\(342\) 0 0
\(343\) −0.229106 −0.0123706
\(344\) 0 0
\(345\) −3.34339 3.43827i −0.180002 0.185110i
\(346\) 0 0
\(347\) 13.8095i 0.741331i 0.928766 + 0.370666i \(0.120870\pi\)
−0.928766 + 0.370666i \(0.879130\pi\)
\(348\) 0 0
\(349\) 30.0976 1.61109 0.805544 0.592537i \(-0.201873\pi\)
0.805544 + 0.592537i \(0.201873\pi\)
\(350\) 0 0
\(351\) 0.958125i 0.0511409i
\(352\) 0 0
\(353\) 0.774223 0.0412077 0.0206039 0.999788i \(-0.493441\pi\)
0.0206039 + 0.999788i \(0.493441\pi\)
\(354\) 0 0
\(355\) 14.9335 0.792589
\(356\) 0 0
\(357\) 26.7181 1.41407
\(358\) 0 0
\(359\) −5.96220 −0.314673 −0.157336 0.987545i \(-0.550291\pi\)
−0.157336 + 0.987545i \(0.550291\pi\)
\(360\) 0 0
\(361\) 47.3378 2.49146
\(362\) 0 0
\(363\) 10.0942i 0.529808i
\(364\) 0 0
\(365\) 2.66136i 0.139302i
\(366\) 0 0
\(367\) 11.9835 0.625535 0.312768 0.949830i \(-0.398744\pi\)
0.312768 + 0.949830i \(0.398744\pi\)
\(368\) 0 0
\(369\) −2.05735 −0.107102
\(370\) 0 0
\(371\) 21.9172i 1.13788i
\(372\) 0 0
\(373\) 13.8905i 0.719221i −0.933102 0.359611i \(-0.882910\pi\)
0.933102 0.359611i \(-0.117090\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −3.81217 −0.196337
\(378\) 0 0
\(379\) −24.1266 −1.23930 −0.619650 0.784878i \(-0.712725\pi\)
−0.619650 + 0.784878i \(0.712725\pi\)
\(380\) 0 0
\(381\) 1.77521 0.0909469
\(382\) 0 0
\(383\) 16.2488 0.830276 0.415138 0.909759i \(-0.363733\pi\)
0.415138 + 0.909759i \(0.363733\pi\)
\(384\) 0 0
\(385\) 3.56885i 0.181886i
\(386\) 0 0
\(387\) −4.47788 −0.227623
\(388\) 0 0
\(389\) 10.2099i 0.517662i −0.965923 0.258831i \(-0.916663\pi\)
0.965923 0.258831i \(-0.0833372\pi\)
\(390\) 0 0
\(391\) −24.4983 + 23.8223i −1.23893 + 1.20474i
\(392\) 0 0
\(393\) −8.26875 −0.417103
\(394\) 0 0
\(395\) 15.4270i 0.776214i
\(396\) 0 0
\(397\) 20.9003 1.04896 0.524479 0.851423i \(-0.324260\pi\)
0.524479 + 0.851423i \(0.324260\pi\)
\(398\) 0 0
\(399\) 30.5415i 1.52899i
\(400\) 0 0
\(401\) 4.00427i 0.199964i 0.994989 + 0.0999818i \(0.0318785\pi\)
−0.994989 + 0.0999818i \(0.968122\pi\)
\(402\) 0 0
\(403\) 9.85339i 0.490832i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 7.58799i 0.376123i
\(408\) 0 0
\(409\) −7.27963 −0.359955 −0.179977 0.983671i \(-0.557602\pi\)
−0.179977 + 0.983671i \(0.557602\pi\)
\(410\) 0 0
\(411\) −11.9628 −0.590084
\(412\) 0 0
\(413\) 19.6744i 0.968114i
\(414\) 0 0
\(415\) 3.86616i 0.189782i
\(416\) 0 0
\(417\) 11.6365 0.569841
\(418\) 0 0
\(419\) −20.0067 −0.977393 −0.488697 0.872454i \(-0.662527\pi\)
−0.488697 + 0.872454i \(0.662527\pi\)
\(420\) 0 0
\(421\) 4.66630i 0.227422i −0.993514 0.113711i \(-0.963726\pi\)
0.993514 0.113711i \(-0.0362738\pi\)
\(422\) 0 0
\(423\) 5.73241i 0.278719i
\(424\) 0 0
\(425\) 7.12519i 0.345622i
\(426\) 0 0
\(427\) 5.83774i 0.282508i
\(428\) 0 0
\(429\) 0.911887i 0.0440263i
\(430\) 0 0
\(431\) −32.6123 −1.57088 −0.785440 0.618937i \(-0.787563\pi\)
−0.785440 + 0.618937i \(0.787563\pi\)
\(432\) 0 0
\(433\) 0.0237330i 0.00114053i −1.00000 0.000570267i \(-0.999818\pi\)
1.00000 0.000570267i \(-0.000181522\pi\)
\(434\) 0 0
\(435\) −3.97878 −0.190768
\(436\) 0 0
\(437\) −27.2313 28.0041i −1.30265 1.33962i
\(438\) 0 0
\(439\) 2.75508i 0.131493i −0.997836 0.0657464i \(-0.979057\pi\)
0.997836 0.0657464i \(-0.0209428\pi\)
\(440\) 0 0
\(441\) −7.06110 −0.336243
\(442\) 0 0
\(443\) 10.7028i 0.508506i 0.967138 + 0.254253i \(0.0818296\pi\)
−0.967138 + 0.254253i \(0.918170\pi\)
\(444\) 0 0
\(445\) −3.08766 −0.146369
\(446\) 0 0
\(447\) −9.88485 −0.467537
\(448\) 0 0
\(449\) 18.3727 0.867060 0.433530 0.901139i \(-0.357268\pi\)
0.433530 + 0.901139i \(0.357268\pi\)
\(450\) 0 0
\(451\) 1.95807 0.0922018
\(452\) 0 0
\(453\) 15.5120 0.728818
\(454\) 0 0
\(455\) 3.59279i 0.168433i
\(456\) 0 0
\(457\) 8.10281i 0.379033i 0.981878 + 0.189517i \(0.0606921\pi\)
−0.981878 + 0.189517i \(0.939308\pi\)
\(458\) 0 0
\(459\) 7.12519 0.332575
\(460\) 0 0
\(461\) −16.0210 −0.746174 −0.373087 0.927796i \(-0.621701\pi\)
−0.373087 + 0.927796i \(0.621701\pi\)
\(462\) 0 0
\(463\) 22.1086i 1.02747i 0.857948 + 0.513737i \(0.171739\pi\)
−0.857948 + 0.513737i \(0.828261\pi\)
\(464\) 0 0
\(465\) 10.2840i 0.476911i
\(466\) 0 0
\(467\) 1.95676 0.0905479 0.0452739 0.998975i \(-0.485584\pi\)
0.0452739 + 0.998975i \(0.485584\pi\)
\(468\) 0 0
\(469\) 31.1873 1.44010
\(470\) 0 0
\(471\) 14.2536 0.656773
\(472\) 0 0
\(473\) 4.26178 0.195957
\(474\) 0 0
\(475\) 8.14480 0.373709
\(476\) 0 0
\(477\) 5.84488i 0.267619i
\(478\) 0 0
\(479\) −8.00074 −0.365563 −0.182782 0.983154i \(-0.558510\pi\)
−0.182782 + 0.983154i \(0.558510\pi\)
\(480\) 0 0
\(481\) 7.63889i 0.348303i
\(482\) 0 0
\(483\) 12.8929 12.5371i 0.586647 0.570457i
\(484\) 0 0
\(485\) −5.79266 −0.263031
\(486\) 0 0
\(487\) 33.1196i 1.50079i 0.660990 + 0.750395i \(0.270137\pi\)
−0.660990 + 0.750395i \(0.729863\pi\)
\(488\) 0 0
\(489\) 5.48701 0.248131
\(490\) 0 0
\(491\) 0.232141i 0.0104764i 0.999986 + 0.00523818i \(0.00166737\pi\)
−0.999986 + 0.00523818i \(0.998333\pi\)
\(492\) 0 0
\(493\) 28.3496i 1.27680i
\(494\) 0 0
\(495\) 0.951741i 0.0427776i
\(496\) 0 0
\(497\) 55.9979i 2.51185i
\(498\) 0 0
\(499\) 9.30458i 0.416530i 0.978072 + 0.208265i \(0.0667817\pi\)
−0.978072 + 0.208265i \(0.933218\pi\)
\(500\) 0 0
\(501\) 14.3069 0.639183
\(502\) 0 0
\(503\) −31.1845 −1.39045 −0.695225 0.718792i \(-0.744695\pi\)
−0.695225 + 0.718792i \(0.744695\pi\)
\(504\) 0 0
\(505\) 1.61688i 0.0719502i
\(506\) 0 0
\(507\) 12.0820i 0.536580i
\(508\) 0 0
\(509\) 2.38407 0.105672 0.0528361 0.998603i \(-0.483174\pi\)
0.0528361 + 0.998603i \(0.483174\pi\)
\(510\) 0 0
\(511\) −9.97959 −0.441471
\(512\) 0 0
\(513\) 8.14480i 0.359602i
\(514\) 0 0
\(515\) 1.88732i 0.0831654i
\(516\) 0 0
\(517\) 5.45577i 0.239945i
\(518\) 0 0
\(519\) 21.9452i 0.963287i
\(520\) 0 0
\(521\) 27.9158i 1.22302i −0.791238 0.611508i \(-0.790563\pi\)
0.791238 0.611508i \(-0.209437\pi\)
\(522\) 0 0
\(523\) −1.88398 −0.0823809 −0.0411904 0.999151i \(-0.513115\pi\)
−0.0411904 + 0.999151i \(0.513115\pi\)
\(524\) 0 0
\(525\) 3.74981i 0.163655i
\(526\) 0 0
\(527\) −73.2757 −3.19194
\(528\) 0 0
\(529\) −0.643468 + 22.9910i −0.0279769 + 0.999609i
\(530\) 0 0
\(531\) 5.24677i 0.227690i
\(532\) 0 0
\(533\) −1.97120 −0.0853822
\(534\) 0 0
\(535\) 10.4072i 0.449944i
\(536\) 0 0
\(537\) −15.2193 −0.656760
\(538\) 0 0
\(539\) 6.72034 0.289465
\(540\) 0 0
\(541\) −13.5904 −0.584299 −0.292149 0.956373i \(-0.594370\pi\)
−0.292149 + 0.956373i \(0.594370\pi\)
\(542\) 0 0
\(543\) 13.5401 0.581060
\(544\) 0 0
\(545\) 14.8316 0.635315
\(546\) 0 0
\(547\) 18.5944i 0.795040i −0.917594 0.397520i \(-0.869871\pi\)
0.917594 0.397520i \(-0.130129\pi\)
\(548\) 0 0
\(549\) 1.55681i 0.0664429i
\(550\) 0 0
\(551\) −32.4064 −1.38056
\(552\) 0 0
\(553\) 57.8482 2.45996
\(554\) 0 0
\(555\) 7.97275i 0.338424i
\(556\) 0 0
\(557\) 28.2050i 1.19508i −0.801837 0.597542i \(-0.796144\pi\)
0.801837 0.597542i \(-0.203856\pi\)
\(558\) 0 0
\(559\) −4.29037 −0.181463
\(560\) 0 0
\(561\) −6.78133 −0.286308
\(562\) 0 0
\(563\) 33.3843 1.40698 0.703491 0.710705i \(-0.251624\pi\)
0.703491 + 0.710705i \(0.251624\pi\)
\(564\) 0 0
\(565\) 2.06992 0.0870820
\(566\) 0 0
\(567\) −3.74981 −0.157477
\(568\) 0 0
\(569\) 5.36287i 0.224823i 0.993662 + 0.112412i \(0.0358575\pi\)
−0.993662 + 0.112412i \(0.964142\pi\)
\(570\) 0 0
\(571\) −18.9916 −0.794775 −0.397388 0.917651i \(-0.630083\pi\)
−0.397388 + 0.917651i \(0.630083\pi\)
\(572\) 0 0
\(573\) 26.2962i 1.09854i
\(574\) 0 0
\(575\) −3.34339 3.43827i −0.139429 0.143386i
\(576\) 0 0
\(577\) −10.4590 −0.435414 −0.217707 0.976014i \(-0.569858\pi\)
−0.217707 + 0.976014i \(0.569858\pi\)
\(578\) 0 0
\(579\) 8.11360i 0.337190i
\(580\) 0 0
\(581\) −14.4974 −0.601453
\(582\) 0 0
\(583\) 5.56281i 0.230388i
\(584\) 0 0
\(585\) 0.958125i 0.0396136i
\(586\) 0 0
\(587\) 36.1502i 1.49208i −0.665902 0.746040i \(-0.731953\pi\)
0.665902 0.746040i \(-0.268047\pi\)
\(588\) 0 0
\(589\) 83.7614i 3.45133i
\(590\) 0 0
\(591\) 15.2265i 0.626333i
\(592\) 0 0
\(593\) −34.5082 −1.41708 −0.708541 0.705669i \(-0.750646\pi\)
−0.708541 + 0.705669i \(0.750646\pi\)
\(594\) 0 0
\(595\) 26.7181 1.09534
\(596\) 0 0
\(597\) 2.38626i 0.0976629i
\(598\) 0 0
\(599\) 21.2595i 0.868641i 0.900758 + 0.434320i \(0.143011\pi\)
−0.900758 + 0.434320i \(0.856989\pi\)
\(600\) 0 0
\(601\) −21.8456 −0.891102 −0.445551 0.895257i \(-0.646992\pi\)
−0.445551 + 0.895257i \(0.646992\pi\)
\(602\) 0 0
\(603\) 8.31703 0.338696
\(604\) 0 0
\(605\) 10.0942i 0.410387i
\(606\) 0 0
\(607\) 32.1601i 1.30534i 0.757642 + 0.652670i \(0.226351\pi\)
−0.757642 + 0.652670i \(0.773649\pi\)
\(608\) 0 0
\(609\) 14.9197i 0.604577i
\(610\) 0 0
\(611\) 5.49237i 0.222197i
\(612\) 0 0
\(613\) 22.4575i 0.907050i 0.891243 + 0.453525i \(0.149834\pi\)
−0.891243 + 0.453525i \(0.850166\pi\)
\(614\) 0 0
\(615\) −2.05735 −0.0829605
\(616\) 0 0
\(617\) 22.5204i 0.906638i 0.891348 + 0.453319i \(0.149760\pi\)
−0.891348 + 0.453319i \(0.850240\pi\)
\(618\) 0 0
\(619\) 22.4685 0.903087 0.451544 0.892249i \(-0.350874\pi\)
0.451544 + 0.892249i \(0.350874\pi\)
\(620\) 0 0
\(621\) 3.43827 3.34339i 0.137973 0.134166i
\(622\) 0 0
\(623\) 11.5781i 0.463868i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.75174i 0.309575i
\(628\) 0 0
\(629\) −56.8073 −2.26506
\(630\) 0 0
\(631\) 40.1336 1.59769 0.798846 0.601535i \(-0.205444\pi\)
0.798846 + 0.601535i \(0.205444\pi\)
\(632\) 0 0
\(633\) −10.5539 −0.419478
\(634\) 0 0
\(635\) 1.77521 0.0704472
\(636\) 0 0
\(637\) −6.76541 −0.268055
\(638\) 0 0
\(639\) 14.9335i 0.590761i
\(640\) 0 0
\(641\) 17.8547i 0.705217i 0.935771 + 0.352609i \(0.114705\pi\)
−0.935771 + 0.352609i \(0.885295\pi\)
\(642\) 0 0
\(643\) −2.51422 −0.0991513 −0.0495757 0.998770i \(-0.515787\pi\)
−0.0495757 + 0.998770i \(0.515787\pi\)
\(644\) 0 0
\(645\) −4.47788 −0.176316
\(646\) 0 0
\(647\) 7.22628i 0.284095i 0.989860 + 0.142047i \(0.0453685\pi\)
−0.989860 + 0.142047i \(0.954631\pi\)
\(648\) 0 0
\(649\) 4.99356i 0.196015i
\(650\) 0 0
\(651\) 38.5632 1.51141
\(652\) 0 0
\(653\) −4.33613 −0.169686 −0.0848429 0.996394i \(-0.527039\pi\)
−0.0848429 + 0.996394i \(0.527039\pi\)
\(654\) 0 0
\(655\) −8.26875 −0.323087
\(656\) 0 0
\(657\) −2.66136 −0.103829
\(658\) 0 0
\(659\) 31.9108 1.24307 0.621534 0.783387i \(-0.286510\pi\)
0.621534 + 0.783387i \(0.286510\pi\)
\(660\) 0 0
\(661\) 48.6433i 1.89200i −0.324162 0.946002i \(-0.605082\pi\)
0.324162 0.946002i \(-0.394918\pi\)
\(662\) 0 0
\(663\) 6.82682 0.265132
\(664\) 0 0
\(665\) 30.5415i 1.18435i
\(666\) 0 0
\(667\) 13.3026 + 13.6802i 0.515080 + 0.529698i
\(668\) 0 0
\(669\) 27.0582 1.04613
\(670\) 0 0
\(671\) 1.48168i 0.0571995i
\(672\) 0 0
\(673\) −20.2021 −0.778733 −0.389366 0.921083i \(-0.627306\pi\)
−0.389366 + 0.921083i \(0.627306\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 25.4292i 0.977323i −0.872473 0.488662i \(-0.837485\pi\)
0.872473 0.488662i \(-0.162515\pi\)
\(678\) 0 0
\(679\) 21.7214i 0.833591i
\(680\) 0 0
\(681\) 12.0307i 0.461018i
\(682\) 0 0
\(683\) 8.95155i 0.342522i −0.985226 0.171261i \(-0.945216\pi\)
0.985226 0.171261i \(-0.0547841\pi\)
\(684\) 0 0
\(685\) −11.9628 −0.457077
\(686\) 0 0
\(687\) −13.4864 −0.514539
\(688\) 0 0
\(689\) 5.60013i 0.213348i
\(690\) 0 0
\(691\) 44.4175i 1.68972i −0.534987 0.844860i \(-0.679684\pi\)
0.534987 0.844860i \(-0.320316\pi\)
\(692\) 0 0
\(693\) 3.56885 0.135569
\(694\) 0 0
\(695\) 11.6365 0.441397
\(696\) 0 0
\(697\) 14.6590i 0.555250i
\(698\) 0 0
\(699\) 9.71906i 0.367609i
\(700\) 0 0
\(701\) 27.5566i 1.04080i −0.853923 0.520399i \(-0.825783\pi\)
0.853923 0.520399i \(-0.174217\pi\)
\(702\) 0 0
\(703\) 64.9365i 2.44913i
\(704\) 0 0
\(705\) 5.73241i 0.215895i
\(706\) 0 0
\(707\) 6.06300 0.228022
\(708\) 0 0
\(709\) 16.7602i 0.629444i −0.949184 0.314722i \(-0.898089\pi\)
0.949184 0.314722i \(-0.101911\pi\)
\(710\) 0 0
\(711\) 15.4270 0.578556
\(712\) 0 0
\(713\) −35.3593 + 34.3836i −1.32422 + 1.28767i
\(714\) 0 0
\(715\) 0.911887i 0.0341026i
\(716\) 0 0
\(717\) −21.9024 −0.817961
\(718\) 0 0
\(719\) 30.6843i 1.14433i 0.820138 + 0.572166i \(0.193897\pi\)
−0.820138 + 0.572166i \(0.806103\pi\)
\(720\) 0 0
\(721\) −7.07711 −0.263565
\(722\) 0 0
\(723\) 1.37300 0.0510626
\(724\) 0 0
\(725\) −3.97878 −0.147768
\(726\) 0 0
\(727\) −4.17679 −0.154909 −0.0774543 0.996996i \(-0.524679\pi\)
−0.0774543 + 0.996996i \(0.524679\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 31.9057i 1.18008i
\(732\) 0 0
\(733\) 7.01127i 0.258967i 0.991582 + 0.129484i \(0.0413320\pi\)
−0.991582 + 0.129484i \(0.958668\pi\)
\(734\) 0 0
\(735\) −7.06110 −0.260453
\(736\) 0 0
\(737\) −7.91566 −0.291577
\(738\) 0 0
\(739\) 48.1792i 1.77230i −0.463397 0.886151i \(-0.653370\pi\)
0.463397 0.886151i \(-0.346630\pi\)
\(740\) 0 0
\(741\) 7.80374i 0.286677i
\(742\) 0 0
\(743\) 51.0801 1.87395 0.936973 0.349403i \(-0.113615\pi\)
0.936973 + 0.349403i \(0.113615\pi\)
\(744\) 0 0
\(745\) −9.88485 −0.362153
\(746\) 0 0
\(747\) −3.86616 −0.141455
\(748\) 0 0
\(749\) 39.0252 1.42595
\(750\) 0 0
\(751\) −26.2376 −0.957423 −0.478712 0.877972i \(-0.658896\pi\)
−0.478712 + 0.877972i \(0.658896\pi\)
\(752\) 0 0
\(753\) 26.3289i 0.959478i
\(754\) 0 0
\(755\) 15.5120 0.564540
\(756\) 0 0
\(757\) 40.6730i 1.47828i 0.673550 + 0.739142i \(0.264769\pi\)
−0.673550 + 0.739142i \(0.735231\pi\)
\(758\) 0 0
\(759\) −3.27235 + 3.18204i −0.118779 + 0.115501i
\(760\) 0 0
\(761\) −40.2592 −1.45940 −0.729698 0.683770i \(-0.760339\pi\)
−0.729698 + 0.683770i \(0.760339\pi\)
\(762\) 0 0
\(763\) 55.6156i 2.01342i
\(764\) 0 0
\(765\) 7.12519 0.257612
\(766\) 0 0
\(767\) 5.02706i 0.181517i
\(768\) 0 0
\(769\) 23.5282i 0.848448i −0.905557 0.424224i \(-0.860547\pi\)
0.905557 0.424224i \(-0.139453\pi\)
\(770\) 0 0
\(771\) 6.42940i 0.231549i
\(772\) 0 0
\(773\) 14.8496i 0.534102i −0.963682 0.267051i \(-0.913951\pi\)
0.963682 0.267051i \(-0.0860492\pi\)
\(774\) 0 0
\(775\) 10.2840i 0.369413i
\(776\) 0 0
\(777\) 29.8963 1.07252
\(778\) 0 0
\(779\) −16.7567 −0.600372
\(780\) 0 0
\(781\) 14.2129i 0.508576i
\(782\) 0 0
\(783\) 3.97878i 0.142190i
\(784\) 0 0
\(785\) 14.2536 0.508734
\(786\) 0 0
\(787\) 38.8210 1.38382 0.691910 0.721984i \(-0.256770\pi\)
0.691910 + 0.721984i \(0.256770\pi\)
\(788\) 0 0
\(789\) 9.71951i 0.346024i
\(790\) 0 0
\(791\) 7.76180i 0.275978i
\(792\) 0 0
\(793\) 1.49162i 0.0529688i
\(794\) 0 0
\(795\) 5.84488i 0.207297i
\(796\) 0 0
\(797\) 20.6413i 0.731153i 0.930781 + 0.365576i \(0.119128\pi\)
−0.930781 + 0.365576i \(0.880872\pi\)
\(798\) 0 0
\(799\) 40.8445 1.44497
\(800\) 0 0
\(801\) 3.08766i 0.109097i
\(802\) 0 0
\(803\) 2.53292 0.0893849
\(804\) 0 0
\(805\) 12.8929 12.5371i 0.454415 0.441874i
\(806\) 0