Properties

Label 5520.2.be.c.1471.12
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 1471.12
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.c.1471.11

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{5} -2.13359 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{5} -2.13359 q^{7} -1.00000 q^{9} +4.16538 q^{11} +2.81823 q^{13} -1.00000 q^{15} -1.82147i q^{17} -1.32589 q^{19} -2.13359i q^{21} +(4.63191 - 1.24316i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -4.92560 q^{29} -10.5363i q^{31} +4.16538i q^{33} -2.13359i q^{35} -4.95490i q^{37} +2.81823i q^{39} +0.656543 q^{41} -9.49430 q^{43} -1.00000i q^{45} -7.28458i q^{47} -2.44777 q^{49} +1.82147 q^{51} +7.24831i q^{53} +4.16538i q^{55} -1.32589i q^{57} -8.42939i q^{59} -3.54481i q^{61} +2.13359 q^{63} +2.81823i q^{65} -10.6962 q^{67} +(1.24316 + 4.63191i) q^{69} +0.897920i q^{71} +4.48027 q^{73} -1.00000i q^{75} -8.88724 q^{77} +7.73462 q^{79} +1.00000 q^{81} -3.14065 q^{83} +1.82147 q^{85} -4.92560i q^{87} -5.21158i q^{89} -6.01297 q^{91} +10.5363 q^{93} -1.32589i q^{95} -0.00964558i q^{97} -4.16538 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 8q^{7} - 32q^{9} + O(q^{10}) \) \( 32q - 8q^{7} - 32q^{9} + 8q^{11} - 8q^{13} - 32q^{15} - 32q^{25} + 4q^{29} + 20q^{41} + 52q^{49} - 4q^{51} + 8q^{63} + 32q^{67} - 40q^{73} - 24q^{77} + 32q^{79} + 32q^{81} - 4q^{85} - 48q^{91} - 8q^{99} + O(q^{100}) \)

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
\(7\) −2.13359 −0.806423 −0.403211 0.915107i \(-0.632106\pi\)
−0.403211 + 0.915107i \(0.632106\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.16538 1.25591 0.627955 0.778249i \(-0.283892\pi\)
0.627955 + 0.778249i \(0.283892\pi\)
\(12\) 0 0
\(13\) 2.81823 0.781638 0.390819 0.920468i \(-0.372192\pi\)
0.390819 + 0.920468i \(0.372192\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 1.82147i 0.441772i −0.975300 0.220886i \(-0.929105\pi\)
0.975300 0.220886i \(-0.0708948\pi\)
\(18\) 0 0
\(19\) −1.32589 −0.304180 −0.152090 0.988367i \(-0.548600\pi\)
−0.152090 + 0.988367i \(0.548600\pi\)
\(20\) 0 0
\(21\) 2.13359i 0.465589i
\(22\) 0 0
\(23\) 4.63191 1.24316i 0.965819 0.259217i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −4.92560 −0.914662 −0.457331 0.889297i \(-0.651194\pi\)
−0.457331 + 0.889297i \(0.651194\pi\)
\(30\) 0 0
\(31\) 10.5363i 1.89237i −0.323628 0.946184i \(-0.604903\pi\)
0.323628 0.946184i \(-0.395097\pi\)
\(32\) 0 0
\(33\) 4.16538i 0.725100i
\(34\) 0 0
\(35\) 2.13359i 0.360643i
\(36\) 0 0
\(37\) 4.95490i 0.814580i −0.913299 0.407290i \(-0.866474\pi\)
0.913299 0.407290i \(-0.133526\pi\)
\(38\) 0 0
\(39\) 2.81823i 0.451279i
\(40\) 0 0
\(41\) 0.656543 0.102535 0.0512674 0.998685i \(-0.483674\pi\)
0.0512674 + 0.998685i \(0.483674\pi\)
\(42\) 0 0
\(43\) −9.49430 −1.44787 −0.723934 0.689870i \(-0.757668\pi\)
−0.723934 + 0.689870i \(0.757668\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 7.28458i 1.06257i −0.847195 0.531283i \(-0.821710\pi\)
0.847195 0.531283i \(-0.178290\pi\)
\(48\) 0 0
\(49\) −2.44777 −0.349682
\(50\) 0 0
\(51\) 1.82147 0.255057
\(52\) 0 0
\(53\) 7.24831i 0.995631i 0.867283 + 0.497816i \(0.165864\pi\)
−0.867283 + 0.497816i \(0.834136\pi\)
\(54\) 0 0
\(55\) 4.16538i 0.561660i
\(56\) 0 0
\(57\) 1.32589i 0.175619i
\(58\) 0 0
\(59\) 8.42939i 1.09741i −0.836015 0.548707i \(-0.815120\pi\)
0.836015 0.548707i \(-0.184880\pi\)
\(60\) 0 0
\(61\) 3.54481i 0.453866i −0.973910 0.226933i \(-0.927130\pi\)
0.973910 0.226933i \(-0.0728698\pi\)
\(62\) 0 0
\(63\) 2.13359 0.268808
\(64\) 0 0
\(65\) 2.81823i 0.349559i
\(66\) 0 0
\(67\) −10.6962 −1.30674 −0.653372 0.757037i \(-0.726646\pi\)
−0.653372 + 0.757037i \(0.726646\pi\)
\(68\) 0 0
\(69\) 1.24316 + 4.63191i 0.149659 + 0.557616i
\(70\) 0 0
\(71\) 0.897920i 0.106564i 0.998580 + 0.0532818i \(0.0169681\pi\)
−0.998580 + 0.0532818i \(0.983032\pi\)
\(72\) 0 0
\(73\) 4.48027 0.524375 0.262188 0.965017i \(-0.415556\pi\)
0.262188 + 0.965017i \(0.415556\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) −8.88724 −1.01280
\(78\) 0 0
\(79\) 7.73462 0.870213 0.435106 0.900379i \(-0.356711\pi\)
0.435106 + 0.900379i \(0.356711\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.14065 −0.344731 −0.172365 0.985033i \(-0.555141\pi\)
−0.172365 + 0.985033i \(0.555141\pi\)
\(84\) 0 0
\(85\) 1.82147 0.197566
\(86\) 0 0
\(87\) 4.92560i 0.528080i
\(88\) 0 0
\(89\) 5.21158i 0.552427i −0.961096 0.276213i \(-0.910920\pi\)
0.961096 0.276213i \(-0.0890796\pi\)
\(90\) 0 0
\(91\) −6.01297 −0.630331
\(92\) 0 0
\(93\) 10.5363 1.09256
\(94\) 0 0
\(95\) 1.32589i 0.136034i
\(96\) 0 0
\(97\) 0.00964558i 0.000979360i −1.00000 0.000489680i \(-0.999844\pi\)
1.00000 0.000489680i \(-0.000155870\pi\)
\(98\) 0 0
\(99\) −4.16538 −0.418637
\(100\) 0 0
\(101\) −6.02561 −0.599571 −0.299785 0.954007i \(-0.596915\pi\)
−0.299785 + 0.954007i \(0.596915\pi\)
\(102\) 0 0
\(103\) 6.10648 0.601690 0.300845 0.953673i \(-0.402731\pi\)
0.300845 + 0.953673i \(0.402731\pi\)
\(104\) 0 0
\(105\) 2.13359 0.208218
\(106\) 0 0
\(107\) 3.50759 0.339091 0.169546 0.985522i \(-0.445770\pi\)
0.169546 + 0.985522i \(0.445770\pi\)
\(108\) 0 0
\(109\) 9.73697i 0.932632i 0.884618 + 0.466316i \(0.154419\pi\)
−0.884618 + 0.466316i \(0.845581\pi\)
\(110\) 0 0
\(111\) 4.95490 0.470298
\(112\) 0 0
\(113\) 1.94423i 0.182898i 0.995810 + 0.0914490i \(0.0291499\pi\)
−0.995810 + 0.0914490i \(0.970850\pi\)
\(114\) 0 0
\(115\) 1.24316 + 4.63191i 0.115925 + 0.431927i
\(116\) 0 0
\(117\) −2.81823 −0.260546
\(118\) 0 0
\(119\) 3.88628i 0.356255i
\(120\) 0 0
\(121\) 6.35043 0.577312
\(122\) 0 0
\(123\) 0.656543i 0.0591985i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 15.3785i 1.36462i −0.731061 0.682312i \(-0.760974\pi\)
0.731061 0.682312i \(-0.239026\pi\)
\(128\) 0 0
\(129\) 9.49430i 0.835927i
\(130\) 0 0
\(131\) 1.02037i 0.0891498i 0.999006 + 0.0445749i \(0.0141933\pi\)
−0.999006 + 0.0445749i \(0.985807\pi\)
\(132\) 0 0
\(133\) 2.82891 0.245298
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 2.73571i 0.233727i −0.993148 0.116864i \(-0.962716\pi\)
0.993148 0.116864i \(-0.0372840\pi\)
\(138\) 0 0
\(139\) 20.8691i 1.77009i −0.465504 0.885046i \(-0.654127\pi\)
0.465504 0.885046i \(-0.345873\pi\)
\(140\) 0 0
\(141\) 7.28458 0.613472
\(142\) 0 0
\(143\) 11.7390 0.981667
\(144\) 0 0
\(145\) 4.92560i 0.409049i
\(146\) 0 0
\(147\) 2.44777i 0.201889i
\(148\) 0 0
\(149\) 5.40737i 0.442989i −0.975162 0.221494i \(-0.928907\pi\)
0.975162 0.221494i \(-0.0710934\pi\)
\(150\) 0 0
\(151\) 2.34996i 0.191237i −0.995418 0.0956185i \(-0.969517\pi\)
0.995418 0.0956185i \(-0.0304829\pi\)
\(152\) 0 0
\(153\) 1.82147i 0.147257i
\(154\) 0 0
\(155\) 10.5363 0.846293
\(156\) 0 0
\(157\) 7.86091i 0.627369i 0.949527 + 0.313685i \(0.101563\pi\)
−0.949527 + 0.313685i \(0.898437\pi\)
\(158\) 0 0
\(159\) −7.24831 −0.574828
\(160\) 0 0
\(161\) −9.88261 + 2.65240i −0.778859 + 0.209039i
\(162\) 0 0
\(163\) 0.585998i 0.0458989i −0.999737 0.0229494i \(-0.992694\pi\)
0.999737 0.0229494i \(-0.00730568\pi\)
\(164\) 0 0
\(165\) −4.16538 −0.324275
\(166\) 0 0
\(167\) 3.15561i 0.244189i −0.992518 0.122094i \(-0.961039\pi\)
0.992518 0.122094i \(-0.0389611\pi\)
\(168\) 0 0
\(169\) −5.05755 −0.389043
\(170\) 0 0
\(171\) 1.32589 0.101393
\(172\) 0 0
\(173\) 1.97459 0.150126 0.0750628 0.997179i \(-0.476084\pi\)
0.0750628 + 0.997179i \(0.476084\pi\)
\(174\) 0 0
\(175\) 2.13359 0.161285
\(176\) 0 0
\(177\) 8.42939 0.633592
\(178\) 0 0
\(179\) 25.9565i 1.94008i −0.242943 0.970041i \(-0.578113\pi\)
0.242943 0.970041i \(-0.421887\pi\)
\(180\) 0 0
\(181\) 15.5579i 1.15641i 0.815892 + 0.578204i \(0.196246\pi\)
−0.815892 + 0.578204i \(0.803754\pi\)
\(182\) 0 0
\(183\) 3.54481 0.262040
\(184\) 0 0
\(185\) 4.95490 0.364291
\(186\) 0 0
\(187\) 7.58713i 0.554826i
\(188\) 0 0
\(189\) 2.13359i 0.155196i
\(190\) 0 0
\(191\) 15.2728 1.10510 0.552552 0.833479i \(-0.313654\pi\)
0.552552 + 0.833479i \(0.313654\pi\)
\(192\) 0 0
\(193\) 7.77339 0.559541 0.279771 0.960067i \(-0.409742\pi\)
0.279771 + 0.960067i \(0.409742\pi\)
\(194\) 0 0
\(195\) −2.81823 −0.201818
\(196\) 0 0
\(197\) 14.3268 1.02074 0.510369 0.859955i \(-0.329509\pi\)
0.510369 + 0.859955i \(0.329509\pi\)
\(198\) 0 0
\(199\) 16.9685 1.20287 0.601433 0.798923i \(-0.294597\pi\)
0.601433 + 0.798923i \(0.294597\pi\)
\(200\) 0 0
\(201\) 10.6962i 0.754450i
\(202\) 0 0
\(203\) 10.5092 0.737604
\(204\) 0 0
\(205\) 0.656543i 0.0458550i
\(206\) 0 0
\(207\) −4.63191 + 1.24316i −0.321940 + 0.0864058i
\(208\) 0 0
\(209\) −5.52285 −0.382023
\(210\) 0 0
\(211\) 10.6786i 0.735142i −0.929995 0.367571i \(-0.880189\pi\)
0.929995 0.367571i \(-0.119811\pi\)
\(212\) 0 0
\(213\) −0.897920 −0.0615245
\(214\) 0 0
\(215\) 9.49430i 0.647506i
\(216\) 0 0
\(217\) 22.4801i 1.52605i
\(218\) 0 0
\(219\) 4.48027i 0.302748i
\(220\) 0 0
\(221\) 5.13333i 0.345305i
\(222\) 0 0
\(223\) 5.49270i 0.367818i 0.982943 + 0.183909i \(0.0588752\pi\)
−0.982943 + 0.183909i \(0.941125\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −3.53360 −0.234534 −0.117267 0.993100i \(-0.537413\pi\)
−0.117267 + 0.993100i \(0.537413\pi\)
\(228\) 0 0
\(229\) 6.15959i 0.407037i −0.979071 0.203518i \(-0.934762\pi\)
0.979071 0.203518i \(-0.0652377\pi\)
\(230\) 0 0
\(231\) 8.88724i 0.584738i
\(232\) 0 0
\(233\) 17.9823 1.17806 0.589030 0.808111i \(-0.299510\pi\)
0.589030 + 0.808111i \(0.299510\pi\)
\(234\) 0 0
\(235\) 7.28458 0.475194
\(236\) 0 0
\(237\) 7.73462i 0.502418i
\(238\) 0 0
\(239\) 16.4389i 1.06335i −0.846950 0.531673i \(-0.821564\pi\)
0.846950 0.531673i \(-0.178436\pi\)
\(240\) 0 0
\(241\) 15.9706i 1.02875i −0.857564 0.514377i \(-0.828023\pi\)
0.857564 0.514377i \(-0.171977\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 2.44777i 0.156383i
\(246\) 0 0
\(247\) −3.73667 −0.237759
\(248\) 0 0
\(249\) 3.14065i 0.199030i
\(250\) 0 0
\(251\) −25.9346 −1.63698 −0.818488 0.574523i \(-0.805187\pi\)
−0.818488 + 0.574523i \(0.805187\pi\)
\(252\) 0 0
\(253\) 19.2937 5.17825i 1.21298 0.325554i
\(254\) 0 0
\(255\) 1.82147i 0.114065i
\(256\) 0 0
\(257\) −9.54194 −0.595210 −0.297605 0.954689i \(-0.596188\pi\)
−0.297605 + 0.954689i \(0.596188\pi\)
\(258\) 0 0
\(259\) 10.5717i 0.656896i
\(260\) 0 0
\(261\) 4.92560 0.304887
\(262\) 0 0
\(263\) 10.6982 0.659681 0.329840 0.944037i \(-0.393005\pi\)
0.329840 + 0.944037i \(0.393005\pi\)
\(264\) 0 0
\(265\) −7.24831 −0.445260
\(266\) 0 0
\(267\) 5.21158 0.318944
\(268\) 0 0
\(269\) 5.41224 0.329990 0.164995 0.986294i \(-0.447239\pi\)
0.164995 + 0.986294i \(0.447239\pi\)
\(270\) 0 0
\(271\) 1.28927i 0.0783179i −0.999233 0.0391589i \(-0.987532\pi\)
0.999233 0.0391589i \(-0.0124679\pi\)
\(272\) 0 0
\(273\) 6.01297i 0.363922i
\(274\) 0 0
\(275\) −4.16538 −0.251182
\(276\) 0 0
\(277\) 11.7340 0.705027 0.352513 0.935807i \(-0.385327\pi\)
0.352513 + 0.935807i \(0.385327\pi\)
\(278\) 0 0
\(279\) 10.5363i 0.630790i
\(280\) 0 0
\(281\) 13.4017i 0.799480i −0.916628 0.399740i \(-0.869100\pi\)
0.916628 0.399740i \(-0.130900\pi\)
\(282\) 0 0
\(283\) 15.1691 0.901707 0.450853 0.892598i \(-0.351120\pi\)
0.450853 + 0.892598i \(0.351120\pi\)
\(284\) 0 0
\(285\) 1.32589 0.0785390
\(286\) 0 0
\(287\) −1.40080 −0.0826865
\(288\) 0 0
\(289\) 13.6822 0.804838
\(290\) 0 0
\(291\) 0.00964558 0.000565434
\(292\) 0 0
\(293\) 0.465132i 0.0271733i −0.999908 0.0135867i \(-0.995675\pi\)
0.999908 0.0135867i \(-0.00432490\pi\)
\(294\) 0 0
\(295\) 8.42939 0.490778
\(296\) 0 0
\(297\) 4.16538i 0.241700i
\(298\) 0 0
\(299\) 13.0538 3.50352i 0.754921 0.202614i
\(300\) 0 0
\(301\) 20.2570 1.16759
\(302\) 0 0
\(303\) 6.02561i 0.346162i
\(304\) 0 0
\(305\) 3.54481 0.202975
\(306\) 0 0
\(307\) 22.1194i 1.26242i 0.775613 + 0.631209i \(0.217441\pi\)
−0.775613 + 0.631209i \(0.782559\pi\)
\(308\) 0 0
\(309\) 6.10648i 0.347386i
\(310\) 0 0
\(311\) 23.0765i 1.30855i −0.756257 0.654275i \(-0.772974\pi\)
0.756257 0.654275i \(-0.227026\pi\)
\(312\) 0 0
\(313\) 13.5444i 0.765576i 0.923836 + 0.382788i \(0.125036\pi\)
−0.923836 + 0.382788i \(0.874964\pi\)
\(314\) 0 0
\(315\) 2.13359i 0.120214i
\(316\) 0 0
\(317\) 24.2326 1.36104 0.680520 0.732730i \(-0.261754\pi\)
0.680520 + 0.732730i \(0.261754\pi\)
\(318\) 0 0
\(319\) −20.5170 −1.14873
\(320\) 0 0
\(321\) 3.50759i 0.195775i
\(322\) 0 0
\(323\) 2.41507i 0.134378i
\(324\) 0 0
\(325\) −2.81823 −0.156328
\(326\) 0 0
\(327\) −9.73697 −0.538455
\(328\) 0 0
\(329\) 15.5423i 0.856877i
\(330\) 0 0
\(331\) 24.5464i 1.34919i 0.738186 + 0.674597i \(0.235683\pi\)
−0.738186 + 0.674597i \(0.764317\pi\)
\(332\) 0 0
\(333\) 4.95490i 0.271527i
\(334\) 0 0
\(335\) 10.6962i 0.584394i
\(336\) 0 0
\(337\) 3.21466i 0.175113i −0.996160 0.0875567i \(-0.972094\pi\)
0.996160 0.0875567i \(-0.0279059\pi\)
\(338\) 0 0
\(339\) −1.94423 −0.105596
\(340\) 0 0
\(341\) 43.8876i 2.37665i
\(342\) 0 0
\(343\) 20.1577 1.08841
\(344\) 0 0
\(345\) −4.63191 + 1.24316i −0.249373 + 0.0669296i
\(346\) 0 0
\(347\) 2.76560i 0.148465i 0.997241 + 0.0742327i \(0.0236508\pi\)
−0.997241 + 0.0742327i \(0.976349\pi\)
\(348\) 0 0
\(349\) 3.14049 0.168107 0.0840533 0.996461i \(-0.473213\pi\)
0.0840533 + 0.996461i \(0.473213\pi\)
\(350\) 0 0
\(351\) 2.81823i 0.150426i
\(352\) 0 0
\(353\) −31.9728 −1.70174 −0.850871 0.525375i \(-0.823925\pi\)
−0.850871 + 0.525375i \(0.823925\pi\)
\(354\) 0 0
\(355\) −0.897920 −0.0476567
\(356\) 0 0
\(357\) −3.88628 −0.205684
\(358\) 0 0
\(359\) −8.48175 −0.447650 −0.223825 0.974629i \(-0.571854\pi\)
−0.223825 + 0.974629i \(0.571854\pi\)
\(360\) 0 0
\(361\) −17.2420 −0.907474
\(362\) 0 0
\(363\) 6.35043i 0.333311i
\(364\) 0 0
\(365\) 4.48027i 0.234508i
\(366\) 0 0
\(367\) −5.00433 −0.261224 −0.130612 0.991434i \(-0.541694\pi\)
−0.130612 + 0.991434i \(0.541694\pi\)
\(368\) 0 0
\(369\) −0.656543 −0.0341783
\(370\) 0 0
\(371\) 15.4649i 0.802900i
\(372\) 0 0
\(373\) 13.6544i 0.706998i 0.935435 + 0.353499i \(0.115008\pi\)
−0.935435 + 0.353499i \(0.884992\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −13.8815 −0.714934
\(378\) 0 0
\(379\) 0.658572 0.0338285 0.0169143 0.999857i \(-0.494616\pi\)
0.0169143 + 0.999857i \(0.494616\pi\)
\(380\) 0 0
\(381\) 15.3785 0.787866
\(382\) 0 0
\(383\) −13.6020 −0.695028 −0.347514 0.937675i \(-0.612974\pi\)
−0.347514 + 0.937675i \(0.612974\pi\)
\(384\) 0 0
\(385\) 8.88724i 0.452936i
\(386\) 0 0
\(387\) 9.49430 0.482622
\(388\) 0 0
\(389\) 32.7804i 1.66203i 0.556247 + 0.831017i \(0.312241\pi\)
−0.556247 + 0.831017i \(0.687759\pi\)
\(390\) 0 0
\(391\) −2.26438 8.43688i −0.114515 0.426672i
\(392\) 0 0
\(393\) −1.02037 −0.0514707
\(394\) 0 0
\(395\) 7.73462i 0.389171i
\(396\) 0 0
\(397\) 7.70037 0.386470 0.193235 0.981152i \(-0.438102\pi\)
0.193235 + 0.981152i \(0.438102\pi\)
\(398\) 0 0
\(399\) 2.82891i 0.141623i
\(400\) 0 0
\(401\) 0.572120i 0.0285703i −0.999898 0.0142851i \(-0.995453\pi\)
0.999898 0.0142851i \(-0.00454726\pi\)
\(402\) 0 0
\(403\) 29.6937i 1.47915i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 20.6390i 1.02304i
\(408\) 0 0
\(409\) 2.81116 0.139003 0.0695014 0.997582i \(-0.477859\pi\)
0.0695014 + 0.997582i \(0.477859\pi\)
\(410\) 0 0
\(411\) 2.73571 0.134942
\(412\) 0 0
\(413\) 17.9849i 0.884980i
\(414\) 0 0
\(415\) 3.14065i 0.154168i
\(416\) 0 0
\(417\) 20.8691 1.02196
\(418\) 0 0
\(419\) 19.4496 0.950174 0.475087 0.879939i \(-0.342417\pi\)
0.475087 + 0.879939i \(0.342417\pi\)
\(420\) 0 0
\(421\) 10.1604i 0.495186i 0.968864 + 0.247593i \(0.0796395\pi\)
−0.968864 + 0.247593i \(0.920360\pi\)
\(422\) 0 0
\(423\) 7.28458i 0.354188i
\(424\) 0 0
\(425\) 1.82147i 0.0883544i
\(426\) 0 0
\(427\) 7.56318i 0.366008i
\(428\) 0 0
\(429\) 11.7390i 0.566766i
\(430\) 0 0
\(431\) −25.8128 −1.24336 −0.621679 0.783272i \(-0.713549\pi\)
−0.621679 + 0.783272i \(0.713549\pi\)
\(432\) 0 0
\(433\) 37.3789i 1.79631i 0.439676 + 0.898157i \(0.355093\pi\)
−0.439676 + 0.898157i \(0.644907\pi\)
\(434\) 0 0
\(435\) 4.92560 0.236165
\(436\) 0 0
\(437\) −6.14140 + 1.64830i −0.293783 + 0.0788488i
\(438\) 0 0
\(439\) 5.86390i 0.279869i 0.990161 + 0.139934i \(0.0446892\pi\)
−0.990161 + 0.139934i \(0.955311\pi\)
\(440\) 0 0
\(441\) 2.44777 0.116561
\(442\) 0 0
\(443\) 12.8409i 0.610091i 0.952338 + 0.305046i \(0.0986718\pi\)
−0.952338 + 0.305046i \(0.901328\pi\)
\(444\) 0 0
\(445\) 5.21158 0.247053
\(446\) 0 0
\(447\) 5.40737 0.255760
\(448\) 0 0
\(449\) −22.9627 −1.08368 −0.541838 0.840483i \(-0.682271\pi\)
−0.541838 + 0.840483i \(0.682271\pi\)
\(450\) 0 0
\(451\) 2.73476 0.128775
\(452\) 0 0
\(453\) 2.34996 0.110411
\(454\) 0 0
\(455\) 6.01297i 0.281892i
\(456\) 0 0
\(457\) 26.7913i 1.25324i 0.779323 + 0.626622i \(0.215563\pi\)
−0.779323 + 0.626622i \(0.784437\pi\)
\(458\) 0 0
\(459\) −1.82147 −0.0850190
\(460\) 0 0
\(461\) 21.6671 1.00914 0.504569 0.863371i \(-0.331651\pi\)
0.504569 + 0.863371i \(0.331651\pi\)
\(462\) 0 0
\(463\) 35.5825i 1.65366i −0.562454 0.826829i \(-0.690143\pi\)
0.562454 0.826829i \(-0.309857\pi\)
\(464\) 0 0
\(465\) 10.5363i 0.488608i
\(466\) 0 0
\(467\) −18.6086 −0.861104 −0.430552 0.902566i \(-0.641681\pi\)
−0.430552 + 0.902566i \(0.641681\pi\)
\(468\) 0 0
\(469\) 22.8213 1.05379
\(470\) 0 0
\(471\) −7.86091 −0.362212
\(472\) 0 0
\(473\) −39.5474 −1.81839
\(474\) 0 0
\(475\) 1.32589 0.0608361
\(476\) 0 0
\(477\) 7.24831i 0.331877i
\(478\) 0 0
\(479\) −15.9689 −0.729637 −0.364818 0.931079i \(-0.618869\pi\)
−0.364818 + 0.931079i \(0.618869\pi\)
\(480\) 0 0
\(481\) 13.9641i 0.636706i
\(482\) 0 0
\(483\) −2.65240 9.88261i −0.120689 0.449674i
\(484\) 0 0
\(485\) 0.00964558 0.000437983
\(486\) 0 0
\(487\) 15.6968i 0.711288i 0.934621 + 0.355644i \(0.115738\pi\)
−0.934621 + 0.355644i \(0.884262\pi\)
\(488\) 0 0
\(489\) 0.585998 0.0264997
\(490\) 0 0
\(491\) 32.2500i 1.45542i 0.685884 + 0.727710i \(0.259416\pi\)
−0.685884 + 0.727710i \(0.740584\pi\)
\(492\) 0 0
\(493\) 8.97185i 0.404072i
\(494\) 0 0
\(495\) 4.16538i 0.187220i
\(496\) 0 0
\(497\) 1.91580i 0.0859353i
\(498\) 0 0
\(499\) 11.7721i 0.526993i −0.964660 0.263496i \(-0.915124\pi\)
0.964660 0.263496i \(-0.0848757\pi\)
\(500\) 0 0
\(501\) 3.15561 0.140982
\(502\) 0 0
\(503\) 7.65715 0.341415 0.170708 0.985322i \(-0.445395\pi\)
0.170708 + 0.985322i \(0.445395\pi\)
\(504\) 0 0
\(505\) 6.02561i 0.268136i
\(506\) 0 0
\(507\) 5.05755i 0.224614i
\(508\) 0 0
\(509\) 1.45059 0.0642961 0.0321481 0.999483i \(-0.489765\pi\)
0.0321481 + 0.999483i \(0.489765\pi\)
\(510\) 0 0
\(511\) −9.55907 −0.422868
\(512\) 0 0
\(513\) 1.32589i 0.0585395i
\(514\) 0 0
\(515\) 6.10648i 0.269084i
\(516\) 0 0
\(517\) 30.3431i 1.33449i
\(518\) 0 0
\(519\) 1.97459i 0.0866751i
\(520\) 0 0
\(521\) 6.69276i 0.293215i −0.989195 0.146607i \(-0.953165\pi\)
0.989195 0.146607i \(-0.0468354\pi\)
\(522\) 0 0
\(523\) 29.7009 1.29873 0.649365 0.760477i \(-0.275035\pi\)
0.649365 + 0.760477i \(0.275035\pi\)
\(524\) 0 0
\(525\) 2.13359i 0.0931177i
\(526\) 0 0
\(527\) −19.1915 −0.835995
\(528\) 0 0
\(529\) 19.9091 11.5164i 0.865613 0.500714i
\(530\) 0 0
\(531\) 8.42939i 0.365805i
\(532\) 0 0
\(533\) 1.85029 0.0801451
\(534\) 0 0
\(535\) 3.50759i 0.151646i
\(536\) 0 0
\(537\) 25.9565 1.12011
\(538\) 0 0
\(539\) −10.1959 −0.439169
\(540\) 0 0
\(541\) 43.6607 1.87712 0.938560 0.345117i \(-0.112161\pi\)
0.938560 + 0.345117i \(0.112161\pi\)
\(542\) 0 0
\(543\) −15.5579 −0.667652
\(544\) 0 0
\(545\) −9.73697 −0.417086
\(546\) 0 0
\(547\) 11.1381i 0.476231i −0.971237 0.238115i \(-0.923470\pi\)
0.971237 0.238115i \(-0.0765296\pi\)
\(548\) 0 0
\(549\) 3.54481i 0.151289i
\(550\) 0 0
\(551\) 6.53082 0.278222
\(552\) 0 0
\(553\) −16.5025 −0.701760
\(554\) 0 0
\(555\) 4.95490i 0.210324i
\(556\) 0 0
\(557\) 24.6097i 1.04275i 0.853328 + 0.521374i \(0.174580\pi\)
−0.853328 + 0.521374i \(0.825420\pi\)
\(558\) 0 0
\(559\) −26.7572 −1.13171
\(560\) 0 0
\(561\) 7.58713 0.320329
\(562\) 0 0
\(563\) −1.64103 −0.0691613 −0.0345807 0.999402i \(-0.511010\pi\)
−0.0345807 + 0.999402i \(0.511010\pi\)
\(564\) 0 0
\(565\) −1.94423 −0.0817945
\(566\) 0 0
\(567\) −2.13359 −0.0896026
\(568\) 0 0
\(569\) 39.1957i 1.64317i −0.570087 0.821585i \(-0.693090\pi\)
0.570087 0.821585i \(-0.306910\pi\)
\(570\) 0 0
\(571\) −43.5236 −1.82141 −0.910704 0.413060i \(-0.864460\pi\)
−0.910704 + 0.413060i \(0.864460\pi\)
\(572\) 0 0
\(573\) 15.2728i 0.638032i
\(574\) 0 0
\(575\) −4.63191 + 1.24316i −0.193164 + 0.0518435i
\(576\) 0 0
\(577\) −36.8449 −1.53387 −0.766936 0.641724i \(-0.778220\pi\)
−0.766936 + 0.641724i \(0.778220\pi\)
\(578\) 0 0
\(579\) 7.77339i 0.323051i
\(580\) 0 0
\(581\) 6.70086 0.277999
\(582\) 0 0
\(583\) 30.1920i 1.25042i
\(584\) 0 0
\(585\) 2.81823i 0.116520i
\(586\) 0 0
\(587\) 22.7311i 0.938214i −0.883141 0.469107i \(-0.844576\pi\)
0.883141 0.469107i \(-0.155424\pi\)
\(588\) 0 0
\(589\) 13.9699i 0.575621i
\(590\) 0 0
\(591\) 14.3268i 0.589324i
\(592\) 0 0
\(593\) −20.3848 −0.837103 −0.418551 0.908193i \(-0.637462\pi\)
−0.418551 + 0.908193i \(0.637462\pi\)
\(594\) 0 0
\(595\) −3.88628 −0.159322
\(596\) 0 0
\(597\) 16.9685i 0.694475i
\(598\) 0 0
\(599\) 40.9408i 1.67279i −0.548124 0.836397i \(-0.684658\pi\)
0.548124 0.836397i \(-0.315342\pi\)
\(600\) 0 0
\(601\) 37.7717 1.54074 0.770369 0.637598i \(-0.220072\pi\)
0.770369 + 0.637598i \(0.220072\pi\)
\(602\) 0 0
\(603\) 10.6962 0.435582
\(604\) 0 0
\(605\) 6.35043i 0.258182i
\(606\) 0 0
\(607\) 25.2408i 1.02449i 0.858839 + 0.512246i \(0.171186\pi\)
−0.858839 + 0.512246i \(0.828814\pi\)
\(608\) 0 0
\(609\) 10.5092i 0.425856i
\(610\) 0 0
\(611\) 20.5297i 0.830541i
\(612\) 0 0
\(613\) 0.132012i 0.00533192i 0.999996 + 0.00266596i \(0.000848603\pi\)
−0.999996 + 0.00266596i \(0.999151\pi\)
\(614\) 0 0
\(615\) −0.656543 −0.0264744
\(616\) 0 0
\(617\) 8.92309i 0.359230i 0.983737 + 0.179615i \(0.0574852\pi\)
−0.983737 + 0.179615i \(0.942515\pi\)
\(618\) 0 0
\(619\) 30.3910 1.22152 0.610758 0.791817i \(-0.290865\pi\)
0.610758 + 0.791817i \(0.290865\pi\)
\(620\) 0 0
\(621\) −1.24316 4.63191i −0.0498864 0.185872i
\(622\) 0 0
\(623\) 11.1194i 0.445489i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 5.52285i 0.220561i
\(628\) 0 0
\(629\) −9.02520 −0.359858
\(630\) 0 0
\(631\) −3.75251 −0.149385 −0.0746924 0.997207i \(-0.523798\pi\)
−0.0746924 + 0.997207i \(0.523798\pi\)
\(632\) 0 0
\(633\) 10.6786 0.424435
\(634\) 0 0
\(635\) 15.3785 0.610279
\(636\) 0 0
\(637\) −6.89840 −0.273325
\(638\) 0 0
\(639\) 0.897920i 0.0355212i
\(640\) 0 0
\(641\) 4.00796i 0.158305i −0.996863 0.0791524i \(-0.974779\pi\)
0.996863 0.0791524i \(-0.0252214\pi\)
\(642\) 0 0
\(643\) 43.5751 1.71843 0.859217 0.511612i \(-0.170952\pi\)
0.859217 + 0.511612i \(0.170952\pi\)
\(644\) 0 0
\(645\) 9.49430 0.373838
\(646\) 0 0
\(647\) 39.2034i 1.54124i −0.637293 0.770621i \(-0.719946\pi\)
0.637293 0.770621i \(-0.280054\pi\)
\(648\) 0 0
\(649\) 35.1117i 1.37825i
\(650\) 0 0
\(651\) −22.4801 −0.881065
\(652\) 0 0
\(653\) −40.2961 −1.57691 −0.788455 0.615093i \(-0.789119\pi\)
−0.788455 + 0.615093i \(0.789119\pi\)
\(654\) 0 0
\(655\) −1.02037 −0.0398690
\(656\) 0 0
\(657\) −4.48027 −0.174792
\(658\) 0 0
\(659\) 10.1481 0.395315 0.197658 0.980271i \(-0.436667\pi\)
0.197658 + 0.980271i \(0.436667\pi\)
\(660\) 0 0
\(661\) 8.80651i 0.342534i −0.985225 0.171267i \(-0.945214\pi\)
0.985225 0.171267i \(-0.0547860\pi\)
\(662\) 0 0
\(663\) 5.13333 0.199362
\(664\) 0 0
\(665\) 2.82891i 0.109701i
\(666\) 0 0
\(667\) −22.8149 + 6.12333i −0.883398 + 0.237096i
\(668\) 0 0
\(669\) −5.49270 −0.212360
\(670\) 0 0
\(671\) 14.7655i 0.570015i
\(672\) 0 0
\(673\) 15.2734 0.588748 0.294374 0.955690i \(-0.404889\pi\)
0.294374 + 0.955690i \(0.404889\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 37.8079i 1.45307i −0.687127 0.726537i \(-0.741129\pi\)
0.687127 0.726537i \(-0.258871\pi\)
\(678\) 0 0
\(679\) 0.0205798i 0.000789779i
\(680\) 0 0
\(681\) 3.53360i 0.135408i
\(682\) 0 0
\(683\) 3.32230i 0.127124i −0.997978 0.0635622i \(-0.979754\pi\)
0.997978 0.0635622i \(-0.0202461\pi\)
\(684\) 0 0
\(685\) 2.73571 0.104526
\(686\) 0 0
\(687\) 6.15959 0.235003
\(688\) 0 0
\(689\) 20.4274i 0.778223i
\(690\) 0 0
\(691\) 8.07215i 0.307079i −0.988143 0.153540i \(-0.950933\pi\)
0.988143 0.153540i \(-0.0490672\pi\)
\(692\) 0 0
\(693\) 8.88724 0.337598
\(694\) 0 0
\(695\) 20.8691 0.791609
\(696\) 0 0
\(697\) 1.19588i 0.0452970i
\(698\) 0 0
\(699\) 17.9823i 0.680153i
\(700\) 0 0
\(701\) 45.3917i 1.71442i −0.514965 0.857211i \(-0.672195\pi\)
0.514965 0.857211i \(-0.327805\pi\)
\(702\) 0 0
\(703\) 6.56965i 0.247779i
\(704\) 0 0
\(705\) 7.28458i 0.274353i
\(706\) 0 0
\(707\) 12.8562 0.483508
\(708\) 0 0
\(709\) 24.3181i 0.913285i −0.889650 0.456643i \(-0.849052\pi\)
0.889650 0.456643i \(-0.150948\pi\)
\(710\) 0 0
\(711\) −7.73462 −0.290071
\(712\) 0 0
\(713\) −13.0983 48.8030i −0.490535 1.82769i
\(714\) 0 0
\(715\) 11.7390i 0.439015i
\(716\) 0 0
\(717\) 16.4389 0.613923
\(718\) 0 0
\(719\) 13.0606i 0.487079i 0.969891 + 0.243539i \(0.0783085\pi\)
−0.969891 + 0.243539i \(0.921691\pi\)
\(720\) 0 0
\(721\) −13.0288 −0.485216
\(722\) 0 0
\(723\) 15.9706 0.593951
\(724\) 0 0
\(725\) 4.92560 0.182932
\(726\) 0 0
\(727\) 44.1977 1.63920 0.819600 0.572935i \(-0.194195\pi\)
0.819600 + 0.572935i \(0.194195\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 17.2936i 0.639627i
\(732\) 0 0
\(733\) 6.33078i 0.233833i 0.993142 + 0.116916i \(0.0373010\pi\)
−0.993142 + 0.116916i \(0.962699\pi\)
\(734\) 0 0
\(735\) 2.44777 0.0902875
\(736\) 0 0
\(737\) −44.5537 −1.64115
\(738\) 0 0
\(739\) 18.0828i 0.665186i 0.943070 + 0.332593i \(0.107924\pi\)
−0.943070 + 0.332593i \(0.892076\pi\)
\(740\) 0 0
\(741\) 3.73667i 0.137270i
\(742\) 0 0
\(743\) 23.5187 0.862818 0.431409 0.902156i \(-0.358017\pi\)
0.431409 + 0.902156i \(0.358017\pi\)
\(744\) 0 0
\(745\) 5.40737 0.198111
\(746\) 0 0
\(747\) 3.14065 0.114910
\(748\) 0 0
\(749\) −7.48377 −0.273451
\(750\) 0 0
\(751\) −2.98033 −0.108754 −0.0543769 0.998520i \(-0.517317\pi\)
−0.0543769 + 0.998520i \(0.517317\pi\)
\(752\) 0 0
\(753\) 25.9346i 0.945109i
\(754\) 0 0
\(755\) 2.34996 0.0855238
\(756\) 0 0
\(757\) 16.0118i 0.581958i −0.956729 0.290979i \(-0.906019\pi\)
0.956729 0.290979i \(-0.0939810\pi\)
\(758\) 0 0
\(759\) 5.17825 + 19.2937i 0.187959 + 0.700316i
\(760\) 0 0
\(761\) 1.15577 0.0418968 0.0209484 0.999781i \(-0.493331\pi\)
0.0209484 + 0.999781i \(0.493331\pi\)
\(762\) 0 0
\(763\) 20.7747i 0.752096i
\(764\) 0 0
\(765\) −1.82147 −0.0658554
\(766\) 0 0
\(767\) 23.7560i 0.857780i
\(768\) 0 0
\(769\) 34.8520i 1.25679i −0.777893 0.628397i \(-0.783711\pi\)
0.777893 0.628397i \(-0.216289\pi\)
\(770\) 0 0
\(771\) 9.54194i 0.343645i
\(772\) 0 0
\(773\) 23.6988i 0.852387i −0.904632 0.426194i \(-0.859854\pi\)
0.904632 0.426194i \(-0.140146\pi\)
\(774\) 0 0
\(775\) 10.5363i 0.378474i
\(776\) 0 0
\(777\) −10.5717 −0.379259
\(778\) 0 0
\(779\) −0.870505 −0.0311891
\(780\) 0 0
\(781\) 3.74018i 0.133834i
\(782\) 0 0
\(783\) 4.92560i 0.176027i
\(784\) 0 0
\(785\) −7.86091 −0.280568
\(786\) 0 0
\(787\) −33.5648 −1.19646 −0.598229 0.801326i \(-0.704128\pi\)
−0.598229 + 0.801326i \(0.704128\pi\)
\(788\) 0 0
\(789\) 10.6982i 0.380867i
\(790\) 0 0
\(791\) 4.14821i 0.147493i
\(792\) 0 0
\(793\) 9.99009i 0.354759i
\(794\) 0 0
\(795\) 7.24831i 0.257071i
\(796\) 0 0
\(797\) 6.33001i 0.224221i 0.993696 + 0.112110i \(0.0357610\pi\)
−0.993696 + 0.112110i \(0.964239\pi\)
\(798\) 0 0
\(799\) −13.2687 −0.469411
\(800\) 0 0
\(801\) 5.21158i 0.184142i
\(802\) 0 0
\(803\) 18.6620 0.658569
\(804\) 0 0
\(805\) −2.65240 9.88261i −0.0934850 0.348316i
\(806\) 0 0