Properties

Label 5520.2.be.c.1471.1
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 1471.1
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.c.1471.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} -1.00000i q^{5} +3.34851 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -1.00000i q^{5} +3.34851 q^{7} -1.00000 q^{9} -3.80844 q^{11} -4.01046 q^{13} -1.00000 q^{15} -3.43500i q^{17} -2.65201 q^{19} -3.34851i q^{21} +(-4.71918 - 0.853990i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +2.73385 q^{29} -5.29319i q^{31} +3.80844i q^{33} -3.34851i q^{35} +1.26784i q^{37} +4.01046i q^{39} -0.928703 q^{41} +6.49805 q^{43} +1.00000i q^{45} +1.22656i q^{47} +4.21251 q^{49} -3.43500 q^{51} +10.0810i q^{53} +3.80844i q^{55} +2.65201i q^{57} +15.1536i q^{59} +5.10695i q^{61} -3.34851 q^{63} +4.01046i q^{65} +3.01447 q^{67} +(-0.853990 + 4.71918i) q^{69} +4.10114i q^{71} -12.9930 q^{73} +1.00000i q^{75} -12.7526 q^{77} +11.9924 q^{79} +1.00000 q^{81} -12.4426 q^{83} -3.43500 q^{85} -2.73385i q^{87} -0.329105i q^{89} -13.4291 q^{91} -5.29319 q^{93} +2.65201i q^{95} +4.88722i q^{97} +3.80844 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 8q^{7} - 32q^{9} + O(q^{10}) \) \( 32q - 8q^{7} - 32q^{9} + 8q^{11} - 8q^{13} - 32q^{15} - 32q^{25} + 4q^{29} + 20q^{41} + 52q^{49} - 4q^{51} + 8q^{63} + 32q^{67} - 40q^{73} - 24q^{77} + 32q^{79} + 32q^{81} - 4q^{85} - 48q^{91} - 8q^{99} + O(q^{100}) \)

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
\(7\) 3.34851 1.26562 0.632809 0.774308i \(-0.281902\pi\)
0.632809 + 0.774308i \(0.281902\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.80844 −1.14829 −0.574143 0.818755i \(-0.694665\pi\)
−0.574143 + 0.818755i \(0.694665\pi\)
\(12\) 0 0
\(13\) −4.01046 −1.11230 −0.556151 0.831081i \(-0.687722\pi\)
−0.556151 + 0.831081i \(0.687722\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 3.43500i 0.833109i −0.909111 0.416554i \(-0.863238\pi\)
0.909111 0.416554i \(-0.136762\pi\)
\(18\) 0 0
\(19\) −2.65201 −0.608413 −0.304207 0.952606i \(-0.598391\pi\)
−0.304207 + 0.952606i \(0.598391\pi\)
\(20\) 0 0
\(21\) 3.34851i 0.730705i
\(22\) 0 0
\(23\) −4.71918 0.853990i −0.984018 0.178069i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 2.73385 0.507664 0.253832 0.967248i \(-0.418309\pi\)
0.253832 + 0.967248i \(0.418309\pi\)
\(30\) 0 0
\(31\) 5.29319i 0.950684i −0.879801 0.475342i \(-0.842324\pi\)
0.879801 0.475342i \(-0.157676\pi\)
\(32\) 0 0
\(33\) 3.80844i 0.662964i
\(34\) 0 0
\(35\) 3.34851i 0.566001i
\(36\) 0 0
\(37\) 1.26784i 0.208431i 0.994555 + 0.104216i \(0.0332332\pi\)
−0.994555 + 0.104216i \(0.966767\pi\)
\(38\) 0 0
\(39\) 4.01046i 0.642188i
\(40\) 0 0
\(41\) −0.928703 −0.145039 −0.0725195 0.997367i \(-0.523104\pi\)
−0.0725195 + 0.997367i \(0.523104\pi\)
\(42\) 0 0
\(43\) 6.49805 0.990944 0.495472 0.868624i \(-0.334995\pi\)
0.495472 + 0.868624i \(0.334995\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 1.22656i 0.178912i 0.995991 + 0.0894562i \(0.0285129\pi\)
−0.995991 + 0.0894562i \(0.971487\pi\)
\(48\) 0 0
\(49\) 4.21251 0.601788
\(50\) 0 0
\(51\) −3.43500 −0.480996
\(52\) 0 0
\(53\) 10.0810i 1.38473i 0.721548 + 0.692365i \(0.243431\pi\)
−0.721548 + 0.692365i \(0.756569\pi\)
\(54\) 0 0
\(55\) 3.80844i 0.513530i
\(56\) 0 0
\(57\) 2.65201i 0.351267i
\(58\) 0 0
\(59\) 15.1536i 1.97283i 0.164279 + 0.986414i \(0.447470\pi\)
−0.164279 + 0.986414i \(0.552530\pi\)
\(60\) 0 0
\(61\) 5.10695i 0.653878i 0.945045 + 0.326939i \(0.106017\pi\)
−0.945045 + 0.326939i \(0.893983\pi\)
\(62\) 0 0
\(63\) −3.34851 −0.421873
\(64\) 0 0
\(65\) 4.01046i 0.497437i
\(66\) 0 0
\(67\) 3.01447 0.368276 0.184138 0.982900i \(-0.441051\pi\)
0.184138 + 0.982900i \(0.441051\pi\)
\(68\) 0 0
\(69\) −0.853990 + 4.71918i −0.102808 + 0.568123i
\(70\) 0 0
\(71\) 4.10114i 0.486716i 0.969937 + 0.243358i \(0.0782490\pi\)
−0.969937 + 0.243358i \(0.921751\pi\)
\(72\) 0 0
\(73\) −12.9930 −1.52072 −0.760360 0.649502i \(-0.774977\pi\)
−0.760360 + 0.649502i \(0.774977\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) −12.7526 −1.45329
\(78\) 0 0
\(79\) 11.9924 1.34925 0.674624 0.738161i \(-0.264306\pi\)
0.674624 + 0.738161i \(0.264306\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.4426 −1.36576 −0.682878 0.730532i \(-0.739272\pi\)
−0.682878 + 0.730532i \(0.739272\pi\)
\(84\) 0 0
\(85\) −3.43500 −0.372578
\(86\) 0 0
\(87\) 2.73385i 0.293100i
\(88\) 0 0
\(89\) 0.329105i 0.0348850i −0.999848 0.0174425i \(-0.994448\pi\)
0.999848 0.0174425i \(-0.00555241\pi\)
\(90\) 0 0
\(91\) −13.4291 −1.40775
\(92\) 0 0
\(93\) −5.29319 −0.548878
\(94\) 0 0
\(95\) 2.65201i 0.272091i
\(96\) 0 0
\(97\) 4.88722i 0.496222i 0.968732 + 0.248111i \(0.0798098\pi\)
−0.968732 + 0.248111i \(0.920190\pi\)
\(98\) 0 0
\(99\) 3.80844 0.382762
\(100\) 0 0
\(101\) −10.6619 −1.06089 −0.530447 0.847718i \(-0.677976\pi\)
−0.530447 + 0.847718i \(0.677976\pi\)
\(102\) 0 0
\(103\) 17.5524 1.72949 0.864747 0.502208i \(-0.167479\pi\)
0.864747 + 0.502208i \(0.167479\pi\)
\(104\) 0 0
\(105\) −3.34851 −0.326781
\(106\) 0 0
\(107\) −9.06743 −0.876581 −0.438291 0.898833i \(-0.644416\pi\)
−0.438291 + 0.898833i \(0.644416\pi\)
\(108\) 0 0
\(109\) 1.91151i 0.183090i −0.995801 0.0915449i \(-0.970819\pi\)
0.995801 0.0915449i \(-0.0291805\pi\)
\(110\) 0 0
\(111\) 1.26784 0.120338
\(112\) 0 0
\(113\) 9.47968i 0.891774i −0.895089 0.445887i \(-0.852888\pi\)
0.895089 0.445887i \(-0.147112\pi\)
\(114\) 0 0
\(115\) −0.853990 + 4.71918i −0.0796350 + 0.440066i
\(116\) 0 0
\(117\) 4.01046 0.370767
\(118\) 0 0
\(119\) 11.5021i 1.05440i
\(120\) 0 0
\(121\) 3.50419 0.318563
\(122\) 0 0
\(123\) 0.928703i 0.0837384i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 14.5043i 1.28704i 0.765427 + 0.643522i \(0.222528\pi\)
−0.765427 + 0.643522i \(0.777472\pi\)
\(128\) 0 0
\(129\) 6.49805i 0.572122i
\(130\) 0 0
\(131\) 16.8636i 1.47338i 0.676231 + 0.736690i \(0.263612\pi\)
−0.676231 + 0.736690i \(0.736388\pi\)
\(132\) 0 0
\(133\) −8.88028 −0.770018
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 5.29294i 0.452206i −0.974103 0.226103i \(-0.927401\pi\)
0.974103 0.226103i \(-0.0725987\pi\)
\(138\) 0 0
\(139\) 19.7415i 1.67445i 0.546858 + 0.837225i \(0.315824\pi\)
−0.546858 + 0.837225i \(0.684176\pi\)
\(140\) 0 0
\(141\) 1.22656 0.103295
\(142\) 0 0
\(143\) 15.2736 1.27724
\(144\) 0 0
\(145\) 2.73385i 0.227034i
\(146\) 0 0
\(147\) 4.21251i 0.347442i
\(148\) 0 0
\(149\) 11.2478i 0.921454i −0.887542 0.460727i \(-0.847589\pi\)
0.887542 0.460727i \(-0.152411\pi\)
\(150\) 0 0
\(151\) 11.2361i 0.914382i 0.889369 + 0.457191i \(0.151144\pi\)
−0.889369 + 0.457191i \(0.848856\pi\)
\(152\) 0 0
\(153\) 3.43500i 0.277703i
\(154\) 0 0
\(155\) −5.29319 −0.425159
\(156\) 0 0
\(157\) 2.61969i 0.209074i −0.994521 0.104537i \(-0.966664\pi\)
0.994521 0.104537i \(-0.0333361\pi\)
\(158\) 0 0
\(159\) 10.0810 0.799474
\(160\) 0 0
\(161\) −15.8022 2.85959i −1.24539 0.225367i
\(162\) 0 0
\(163\) 16.4052i 1.28495i −0.766305 0.642477i \(-0.777907\pi\)
0.766305 0.642477i \(-0.222093\pi\)
\(164\) 0 0
\(165\) 3.80844 0.296486
\(166\) 0 0
\(167\) 6.17814i 0.478079i −0.971010 0.239039i \(-0.923167\pi\)
0.971010 0.239039i \(-0.0768325\pi\)
\(168\) 0 0
\(169\) 3.08380 0.237216
\(170\) 0 0
\(171\) 2.65201 0.202804
\(172\) 0 0
\(173\) −24.5549 −1.86687 −0.933436 0.358744i \(-0.883205\pi\)
−0.933436 + 0.358744i \(0.883205\pi\)
\(174\) 0 0
\(175\) −3.34851 −0.253124
\(176\) 0 0
\(177\) 15.1536 1.13901
\(178\) 0 0
\(179\) 11.9824i 0.895606i 0.894132 + 0.447803i \(0.147793\pi\)
−0.894132 + 0.447803i \(0.852207\pi\)
\(180\) 0 0
\(181\) 13.9533i 1.03714i −0.855036 0.518569i \(-0.826465\pi\)
0.855036 0.518569i \(-0.173535\pi\)
\(182\) 0 0
\(183\) 5.10695 0.377517
\(184\) 0 0
\(185\) 1.26784 0.0932133
\(186\) 0 0
\(187\) 13.0820i 0.956648i
\(188\) 0 0
\(189\) 3.34851i 0.243568i
\(190\) 0 0
\(191\) −17.2707 −1.24966 −0.624832 0.780759i \(-0.714833\pi\)
−0.624832 + 0.780759i \(0.714833\pi\)
\(192\) 0 0
\(193\) 3.94520 0.283982 0.141991 0.989868i \(-0.454650\pi\)
0.141991 + 0.989868i \(0.454650\pi\)
\(194\) 0 0
\(195\) 4.01046 0.287195
\(196\) 0 0
\(197\) −5.70577 −0.406519 −0.203260 0.979125i \(-0.565154\pi\)
−0.203260 + 0.979125i \(0.565154\pi\)
\(198\) 0 0
\(199\) −3.55709 −0.252156 −0.126078 0.992020i \(-0.540239\pi\)
−0.126078 + 0.992020i \(0.540239\pi\)
\(200\) 0 0
\(201\) 3.01447i 0.212624i
\(202\) 0 0
\(203\) 9.15433 0.642508
\(204\) 0 0
\(205\) 0.928703i 0.0648634i
\(206\) 0 0
\(207\) 4.71918 + 0.853990i 0.328006 + 0.0593564i
\(208\) 0 0
\(209\) 10.1000 0.698633
\(210\) 0 0
\(211\) 12.3538i 0.850468i −0.905083 0.425234i \(-0.860192\pi\)
0.905083 0.425234i \(-0.139808\pi\)
\(212\) 0 0
\(213\) 4.10114 0.281006
\(214\) 0 0
\(215\) 6.49805i 0.443164i
\(216\) 0 0
\(217\) 17.7243i 1.20320i
\(218\) 0 0
\(219\) 12.9930i 0.877988i
\(220\) 0 0
\(221\) 13.7759i 0.926669i
\(222\) 0 0
\(223\) 3.60984i 0.241732i −0.992669 0.120866i \(-0.961433\pi\)
0.992669 0.120866i \(-0.0385672\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 10.4598 0.694241 0.347120 0.937821i \(-0.387160\pi\)
0.347120 + 0.937821i \(0.387160\pi\)
\(228\) 0 0
\(229\) 6.63659i 0.438558i 0.975662 + 0.219279i \(0.0703705\pi\)
−0.975662 + 0.219279i \(0.929629\pi\)
\(230\) 0 0
\(231\) 12.7526i 0.839059i
\(232\) 0 0
\(233\) −28.4233 −1.86207 −0.931036 0.364927i \(-0.881094\pi\)
−0.931036 + 0.364927i \(0.881094\pi\)
\(234\) 0 0
\(235\) 1.22656 0.0800120
\(236\) 0 0
\(237\) 11.9924i 0.778989i
\(238\) 0 0
\(239\) 21.4035i 1.38448i −0.721667 0.692240i \(-0.756624\pi\)
0.721667 0.692240i \(-0.243376\pi\)
\(240\) 0 0
\(241\) 17.7463i 1.14314i 0.820554 + 0.571570i \(0.193665\pi\)
−0.820554 + 0.571570i \(0.806335\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 4.21251i 0.269128i
\(246\) 0 0
\(247\) 10.6358 0.676739
\(248\) 0 0
\(249\) 12.4426i 0.788520i
\(250\) 0 0
\(251\) −3.15845 −0.199360 −0.0996798 0.995020i \(-0.531782\pi\)
−0.0996798 + 0.995020i \(0.531782\pi\)
\(252\) 0 0
\(253\) 17.9727 + 3.25237i 1.12993 + 0.204475i
\(254\) 0 0
\(255\) 3.43500i 0.215108i
\(256\) 0 0
\(257\) −5.33676 −0.332898 −0.166449 0.986050i \(-0.553230\pi\)
−0.166449 + 0.986050i \(0.553230\pi\)
\(258\) 0 0
\(259\) 4.24537i 0.263794i
\(260\) 0 0
\(261\) −2.73385 −0.169221
\(262\) 0 0
\(263\) 12.1518 0.749312 0.374656 0.927164i \(-0.377761\pi\)
0.374656 + 0.927164i \(0.377761\pi\)
\(264\) 0 0
\(265\) 10.0810 0.619270
\(266\) 0 0
\(267\) −0.329105 −0.0201409
\(268\) 0 0
\(269\) −1.21138 −0.0738589 −0.0369295 0.999318i \(-0.511758\pi\)
−0.0369295 + 0.999318i \(0.511758\pi\)
\(270\) 0 0
\(271\) 12.8041i 0.777794i 0.921281 + 0.388897i \(0.127144\pi\)
−0.921281 + 0.388897i \(0.872856\pi\)
\(272\) 0 0
\(273\) 13.4291i 0.812764i
\(274\) 0 0
\(275\) 3.80844 0.229657
\(276\) 0 0
\(277\) −25.9384 −1.55849 −0.779243 0.626721i \(-0.784396\pi\)
−0.779243 + 0.626721i \(0.784396\pi\)
\(278\) 0 0
\(279\) 5.29319i 0.316895i
\(280\) 0 0
\(281\) 12.0016i 0.715954i −0.933730 0.357977i \(-0.883467\pi\)
0.933730 0.357977i \(-0.116533\pi\)
\(282\) 0 0
\(283\) 17.9652 1.06792 0.533959 0.845510i \(-0.320704\pi\)
0.533959 + 0.845510i \(0.320704\pi\)
\(284\) 0 0
\(285\) 2.65201 0.157092
\(286\) 0 0
\(287\) −3.10977 −0.183564
\(288\) 0 0
\(289\) 5.20080 0.305930
\(290\) 0 0
\(291\) 4.88722 0.286494
\(292\) 0 0
\(293\) 24.4657i 1.42930i 0.699481 + 0.714651i \(0.253415\pi\)
−0.699481 + 0.714651i \(0.746585\pi\)
\(294\) 0 0
\(295\) 15.1536 0.882275
\(296\) 0 0
\(297\) 3.80844i 0.220988i
\(298\) 0 0
\(299\) 18.9261 + 3.42489i 1.09453 + 0.198067i
\(300\) 0 0
\(301\) 21.7588 1.25416
\(302\) 0 0
\(303\) 10.6619i 0.612507i
\(304\) 0 0
\(305\) 5.10695 0.292423
\(306\) 0 0
\(307\) 33.9420i 1.93717i 0.248677 + 0.968587i \(0.420004\pi\)
−0.248677 + 0.968587i \(0.579996\pi\)
\(308\) 0 0
\(309\) 17.5524i 0.998523i
\(310\) 0 0
\(311\) 3.90724i 0.221559i 0.993845 + 0.110780i \(0.0353348\pi\)
−0.993845 + 0.110780i \(0.964665\pi\)
\(312\) 0 0
\(313\) 34.7059i 1.96169i −0.194782 0.980847i \(-0.562400\pi\)
0.194782 0.980847i \(-0.437600\pi\)
\(314\) 0 0
\(315\) 3.34851i 0.188667i
\(316\) 0 0
\(317\) −16.0862 −0.903492 −0.451746 0.892147i \(-0.649199\pi\)
−0.451746 + 0.892147i \(0.649199\pi\)
\(318\) 0 0
\(319\) −10.4117 −0.582944
\(320\) 0 0
\(321\) 9.06743i 0.506094i
\(322\) 0 0
\(323\) 9.10965i 0.506874i
\(324\) 0 0
\(325\) 4.01046 0.222460
\(326\) 0 0
\(327\) −1.91151 −0.105707
\(328\) 0 0
\(329\) 4.10715i 0.226435i
\(330\) 0 0
\(331\) 3.21976i 0.176974i −0.996077 0.0884871i \(-0.971797\pi\)
0.996077 0.0884871i \(-0.0282032\pi\)
\(332\) 0 0
\(333\) 1.26784i 0.0694771i
\(334\) 0 0
\(335\) 3.01447i 0.164698i
\(336\) 0 0
\(337\) 3.42112i 0.186360i 0.995649 + 0.0931801i \(0.0297032\pi\)
−0.995649 + 0.0931801i \(0.970297\pi\)
\(338\) 0 0
\(339\) −9.47968 −0.514866
\(340\) 0 0
\(341\) 20.1588i 1.09166i
\(342\) 0 0
\(343\) −9.33392 −0.503984
\(344\) 0 0
\(345\) 4.71918 + 0.853990i 0.254072 + 0.0459773i
\(346\) 0 0
\(347\) 14.3395i 0.769784i −0.922962 0.384892i \(-0.874239\pi\)
0.922962 0.384892i \(-0.125761\pi\)
\(348\) 0 0
\(349\) −17.8040 −0.953025 −0.476512 0.879168i \(-0.658099\pi\)
−0.476512 + 0.879168i \(0.658099\pi\)
\(350\) 0 0
\(351\) 4.01046i 0.214063i
\(352\) 0 0
\(353\) 7.05096 0.375285 0.187642 0.982237i \(-0.439915\pi\)
0.187642 + 0.982237i \(0.439915\pi\)
\(354\) 0 0
\(355\) 4.10114 0.217666
\(356\) 0 0
\(357\) −11.5021 −0.608757
\(358\) 0 0
\(359\) 11.2467 0.593579 0.296790 0.954943i \(-0.404084\pi\)
0.296790 + 0.954943i \(0.404084\pi\)
\(360\) 0 0
\(361\) −11.9668 −0.629834
\(362\) 0 0
\(363\) 3.50419i 0.183922i
\(364\) 0 0
\(365\) 12.9930i 0.680086i
\(366\) 0 0
\(367\) −35.8800 −1.87292 −0.936461 0.350772i \(-0.885919\pi\)
−0.936461 + 0.350772i \(0.885919\pi\)
\(368\) 0 0
\(369\) 0.928703 0.0483464
\(370\) 0 0
\(371\) 33.7563i 1.75254i
\(372\) 0 0
\(373\) 9.99825i 0.517690i 0.965919 + 0.258845i \(0.0833418\pi\)
−0.965919 + 0.258845i \(0.916658\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −10.9640 −0.564675
\(378\) 0 0
\(379\) −24.3987 −1.25328 −0.626638 0.779311i \(-0.715569\pi\)
−0.626638 + 0.779311i \(0.715569\pi\)
\(380\) 0 0
\(381\) 14.5043 0.743075
\(382\) 0 0
\(383\) −8.66954 −0.442993 −0.221496 0.975161i \(-0.571094\pi\)
−0.221496 + 0.975161i \(0.571094\pi\)
\(384\) 0 0
\(385\) 12.7526i 0.649932i
\(386\) 0 0
\(387\) −6.49805 −0.330315
\(388\) 0 0
\(389\) 6.29587i 0.319213i −0.987181 0.159607i \(-0.948977\pi\)
0.987181 0.159607i \(-0.0510226\pi\)
\(390\) 0 0
\(391\) −2.93345 + 16.2104i −0.148351 + 0.819794i
\(392\) 0 0
\(393\) 16.8636 0.850656
\(394\) 0 0
\(395\) 11.9924i 0.603402i
\(396\) 0 0
\(397\) 6.23934 0.313143 0.156572 0.987667i \(-0.449956\pi\)
0.156572 + 0.987667i \(0.449956\pi\)
\(398\) 0 0
\(399\) 8.88028i 0.444570i
\(400\) 0 0
\(401\) 25.7877i 1.28777i −0.765120 0.643887i \(-0.777321\pi\)
0.765120 0.643887i \(-0.222679\pi\)
\(402\) 0 0
\(403\) 21.2281i 1.05745i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 4.82848i 0.239339i
\(408\) 0 0
\(409\) 19.7280 0.975485 0.487743 0.872987i \(-0.337820\pi\)
0.487743 + 0.872987i \(0.337820\pi\)
\(410\) 0 0
\(411\) −5.29294 −0.261082
\(412\) 0 0
\(413\) 50.7419i 2.49685i
\(414\) 0 0
\(415\) 12.4426i 0.610785i
\(416\) 0 0
\(417\) 19.7415 0.966744
\(418\) 0 0
\(419\) 0.455871 0.0222707 0.0111354 0.999938i \(-0.496455\pi\)
0.0111354 + 0.999938i \(0.496455\pi\)
\(420\) 0 0
\(421\) 10.9952i 0.535873i 0.963437 + 0.267936i \(0.0863417\pi\)
−0.963437 + 0.267936i \(0.913658\pi\)
\(422\) 0 0
\(423\) 1.22656i 0.0596375i
\(424\) 0 0
\(425\) 3.43500i 0.166622i
\(426\) 0 0
\(427\) 17.1007i 0.827559i
\(428\) 0 0
\(429\) 15.2736i 0.737416i
\(430\) 0 0
\(431\) 4.29748 0.207003 0.103501 0.994629i \(-0.466995\pi\)
0.103501 + 0.994629i \(0.466995\pi\)
\(432\) 0 0
\(433\) 0.952676i 0.0457827i −0.999738 0.0228914i \(-0.992713\pi\)
0.999738 0.0228914i \(-0.00728718\pi\)
\(434\) 0 0
\(435\) −2.73385 −0.131078
\(436\) 0 0
\(437\) 12.5153 + 2.26479i 0.598689 + 0.108340i
\(438\) 0 0
\(439\) 25.4726i 1.21574i −0.794037 0.607870i \(-0.792024\pi\)
0.794037 0.607870i \(-0.207976\pi\)
\(440\) 0 0
\(441\) −4.21251 −0.200596
\(442\) 0 0
\(443\) 15.7371i 0.747694i 0.927490 + 0.373847i \(0.121961\pi\)
−0.927490 + 0.373847i \(0.878039\pi\)
\(444\) 0 0
\(445\) −0.329105 −0.0156011
\(446\) 0 0
\(447\) −11.2478 −0.532002
\(448\) 0 0
\(449\) −22.3906 −1.05668 −0.528338 0.849034i \(-0.677185\pi\)
−0.528338 + 0.849034i \(0.677185\pi\)
\(450\) 0 0
\(451\) 3.53691 0.166546
\(452\) 0 0
\(453\) 11.2361 0.527918
\(454\) 0 0
\(455\) 13.4291i 0.629564i
\(456\) 0 0
\(457\) 9.16148i 0.428556i −0.976773 0.214278i \(-0.931260\pi\)
0.976773 0.214278i \(-0.0687398\pi\)
\(458\) 0 0
\(459\) 3.43500 0.160332
\(460\) 0 0
\(461\) −16.8603 −0.785264 −0.392632 0.919696i \(-0.628435\pi\)
−0.392632 + 0.919696i \(0.628435\pi\)
\(462\) 0 0
\(463\) 20.5902i 0.956906i 0.878113 + 0.478453i \(0.158802\pi\)
−0.878113 + 0.478453i \(0.841198\pi\)
\(464\) 0 0
\(465\) 5.29319i 0.245466i
\(466\) 0 0
\(467\) 22.7394 1.05225 0.526126 0.850406i \(-0.323644\pi\)
0.526126 + 0.850406i \(0.323644\pi\)
\(468\) 0 0
\(469\) 10.0940 0.466097
\(470\) 0 0
\(471\) −2.61969 −0.120709
\(472\) 0 0
\(473\) −24.7474 −1.13789
\(474\) 0 0
\(475\) 2.65201 0.121683
\(476\) 0 0
\(477\) 10.0810i 0.461576i
\(478\) 0 0
\(479\) −17.6618 −0.806987 −0.403494 0.914982i \(-0.632204\pi\)
−0.403494 + 0.914982i \(0.632204\pi\)
\(480\) 0 0
\(481\) 5.08461i 0.231838i
\(482\) 0 0
\(483\) −2.85959 + 15.8022i −0.130116 + 0.719026i
\(484\) 0 0
\(485\) 4.88722 0.221917
\(486\) 0 0
\(487\) 30.1379i 1.36568i 0.730568 + 0.682840i \(0.239255\pi\)
−0.730568 + 0.682840i \(0.760745\pi\)
\(488\) 0 0
\(489\) −16.4052 −0.741868
\(490\) 0 0
\(491\) 35.7563i 1.61366i −0.590785 0.806829i \(-0.701182\pi\)
0.590785 0.806829i \(-0.298818\pi\)
\(492\) 0 0
\(493\) 9.39077i 0.422939i
\(494\) 0 0
\(495\) 3.80844i 0.171177i
\(496\) 0 0
\(497\) 13.7327i 0.615997i
\(498\) 0 0
\(499\) 1.38923i 0.0621903i 0.999516 + 0.0310951i \(0.00989948\pi\)
−0.999516 + 0.0310951i \(0.990101\pi\)
\(500\) 0 0
\(501\) −6.17814 −0.276019
\(502\) 0 0
\(503\) 19.9672 0.890295 0.445147 0.895457i \(-0.353151\pi\)
0.445147 + 0.895457i \(0.353151\pi\)
\(504\) 0 0
\(505\) 10.6619i 0.474446i
\(506\) 0 0
\(507\) 3.08380i 0.136957i
\(508\) 0 0
\(509\) −21.3968 −0.948395 −0.474198 0.880418i \(-0.657262\pi\)
−0.474198 + 0.880418i \(0.657262\pi\)
\(510\) 0 0
\(511\) −43.5073 −1.92465
\(512\) 0 0
\(513\) 2.65201i 0.117089i
\(514\) 0 0
\(515\) 17.5524i 0.773453i
\(516\) 0 0
\(517\) 4.67128i 0.205443i
\(518\) 0 0
\(519\) 24.5549i 1.07784i
\(520\) 0 0
\(521\) 40.4491i 1.77211i 0.463581 + 0.886054i \(0.346564\pi\)
−0.463581 + 0.886054i \(0.653436\pi\)
\(522\) 0 0
\(523\) 33.8959 1.48216 0.741082 0.671414i \(-0.234313\pi\)
0.741082 + 0.671414i \(0.234313\pi\)
\(524\) 0 0
\(525\) 3.34851i 0.146141i
\(526\) 0 0
\(527\) −18.1821 −0.792024
\(528\) 0 0
\(529\) 21.5414 + 8.06027i 0.936583 + 0.350447i
\(530\) 0 0
\(531\) 15.1536i 0.657609i
\(532\) 0 0
\(533\) 3.72453 0.161327
\(534\) 0 0
\(535\) 9.06743i 0.392019i
\(536\) 0 0
\(537\) 11.9824 0.517078
\(538\) 0 0
\(539\) −16.0431 −0.691025
\(540\) 0 0
\(541\) −13.7424 −0.590834 −0.295417 0.955368i \(-0.595459\pi\)
−0.295417 + 0.955368i \(0.595459\pi\)
\(542\) 0 0
\(543\) −13.9533 −0.598792
\(544\) 0 0
\(545\) −1.91151 −0.0818803
\(546\) 0 0
\(547\) 22.3514i 0.955676i −0.878448 0.477838i \(-0.841421\pi\)
0.878448 0.477838i \(-0.158579\pi\)
\(548\) 0 0
\(549\) 5.10695i 0.217959i
\(550\) 0 0
\(551\) −7.25021 −0.308869
\(552\) 0 0
\(553\) 40.1566 1.70763
\(554\) 0 0
\(555\) 1.26784i 0.0538167i
\(556\) 0 0
\(557\) 24.9665i 1.05787i −0.848664 0.528933i \(-0.822592\pi\)
0.848664 0.528933i \(-0.177408\pi\)
\(558\) 0 0
\(559\) −26.0602 −1.10223
\(560\) 0 0
\(561\) 13.0820 0.552321
\(562\) 0 0
\(563\) 33.3847 1.40700 0.703499 0.710697i \(-0.251620\pi\)
0.703499 + 0.710697i \(0.251620\pi\)
\(564\) 0 0
\(565\) −9.47968 −0.398813
\(566\) 0 0
\(567\) 3.34851 0.140624
\(568\) 0 0
\(569\) 9.74274i 0.408437i 0.978925 + 0.204219i \(0.0654653\pi\)
−0.978925 + 0.204219i \(0.934535\pi\)
\(570\) 0 0
\(571\) −16.9759 −0.710420 −0.355210 0.934786i \(-0.615591\pi\)
−0.355210 + 0.934786i \(0.615591\pi\)
\(572\) 0 0
\(573\) 17.2707i 0.721494i
\(574\) 0 0
\(575\) 4.71918 + 0.853990i 0.196804 + 0.0356138i
\(576\) 0 0
\(577\) 19.1108 0.795592 0.397796 0.917474i \(-0.369775\pi\)
0.397796 + 0.917474i \(0.369775\pi\)
\(578\) 0 0
\(579\) 3.94520i 0.163957i
\(580\) 0 0
\(581\) −41.6643 −1.72853
\(582\) 0 0
\(583\) 38.3928i 1.59007i
\(584\) 0 0
\(585\) 4.01046i 0.165812i
\(586\) 0 0
\(587\) 20.4134i 0.842550i 0.906933 + 0.421275i \(0.138417\pi\)
−0.906933 + 0.421275i \(0.861583\pi\)
\(588\) 0 0
\(589\) 14.0376i 0.578409i
\(590\) 0 0
\(591\) 5.70577i 0.234704i
\(592\) 0 0
\(593\) 40.1285 1.64788 0.823940 0.566677i \(-0.191771\pi\)
0.823940 + 0.566677i \(0.191771\pi\)
\(594\) 0 0
\(595\) −11.5021 −0.471541
\(596\) 0 0
\(597\) 3.55709i 0.145582i
\(598\) 0 0
\(599\) 4.23805i 0.173162i 0.996245 + 0.0865809i \(0.0275941\pi\)
−0.996245 + 0.0865809i \(0.972406\pi\)
\(600\) 0 0
\(601\) −37.2617 −1.51994 −0.759968 0.649960i \(-0.774786\pi\)
−0.759968 + 0.649960i \(0.774786\pi\)
\(602\) 0 0
\(603\) −3.01447 −0.122759
\(604\) 0 0
\(605\) 3.50419i 0.142466i
\(606\) 0 0
\(607\) 4.56947i 0.185469i −0.995691 0.0927346i \(-0.970439\pi\)
0.995691 0.0927346i \(-0.0295608\pi\)
\(608\) 0 0
\(609\) 9.15433i 0.370952i
\(610\) 0 0
\(611\) 4.91908i 0.199005i
\(612\) 0 0
\(613\) 6.85443i 0.276848i −0.990373 0.138424i \(-0.955796\pi\)
0.990373 0.138424i \(-0.0442036\pi\)
\(614\) 0 0
\(615\) 0.928703 0.0374489
\(616\) 0 0
\(617\) 1.07697i 0.0433574i 0.999765 + 0.0216787i \(0.00690108\pi\)
−0.999765 + 0.0216787i \(0.993099\pi\)
\(618\) 0 0
\(619\) 39.6508 1.59370 0.796850 0.604177i \(-0.206498\pi\)
0.796850 + 0.604177i \(0.206498\pi\)
\(620\) 0 0
\(621\) 0.853990 4.71918i 0.0342694 0.189374i
\(622\) 0 0
\(623\) 1.10201i 0.0441511i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 10.1000i 0.403356i
\(628\) 0 0
\(629\) 4.35502 0.173646
\(630\) 0 0
\(631\) −12.7755 −0.508583 −0.254291 0.967128i \(-0.581842\pi\)
−0.254291 + 0.967128i \(0.581842\pi\)
\(632\) 0 0
\(633\) −12.3538 −0.491018
\(634\) 0 0
\(635\) 14.5043 0.575584
\(636\) 0 0
\(637\) −16.8941 −0.669370
\(638\) 0 0
\(639\) 4.10114i 0.162239i
\(640\) 0 0
\(641\) 6.23062i 0.246095i −0.992401 0.123047i \(-0.960733\pi\)
0.992401 0.123047i \(-0.0392667\pi\)
\(642\) 0 0
\(643\) 23.6102 0.931095 0.465547 0.885023i \(-0.345857\pi\)
0.465547 + 0.885023i \(0.345857\pi\)
\(644\) 0 0
\(645\) −6.49805 −0.255861
\(646\) 0 0
\(647\) 17.3269i 0.681189i −0.940210 0.340595i \(-0.889372\pi\)
0.940210 0.340595i \(-0.110628\pi\)
\(648\) 0 0
\(649\) 57.7114i 2.26537i
\(650\) 0 0
\(651\) −17.7243 −0.694669
\(652\) 0 0
\(653\) 7.82842 0.306350 0.153175 0.988199i \(-0.451050\pi\)
0.153175 + 0.988199i \(0.451050\pi\)
\(654\) 0 0
\(655\) 16.8636 0.658915
\(656\) 0 0
\(657\) 12.9930 0.506906
\(658\) 0 0
\(659\) −11.1695 −0.435103 −0.217551 0.976049i \(-0.569807\pi\)
−0.217551 + 0.976049i \(0.569807\pi\)
\(660\) 0 0
\(661\) 15.4691i 0.601680i −0.953675 0.300840i \(-0.902733\pi\)
0.953675 0.300840i \(-0.0972670\pi\)
\(662\) 0 0
\(663\) 13.7759 0.535012
\(664\) 0 0
\(665\) 8.88028i 0.344363i
\(666\) 0 0
\(667\) −12.9016 2.33468i −0.499550 0.0903993i
\(668\) 0 0
\(669\) −3.60984 −0.139564
\(670\) 0 0
\(671\) 19.4495i 0.750839i
\(672\) 0 0
\(673\) 47.5831 1.83420 0.917098 0.398663i \(-0.130526\pi\)
0.917098 + 0.398663i \(0.130526\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 49.1013i 1.88712i −0.331206 0.943558i \(-0.607456\pi\)
0.331206 0.943558i \(-0.392544\pi\)
\(678\) 0 0
\(679\) 16.3649i 0.628028i
\(680\) 0 0
\(681\) 10.4598i 0.400820i
\(682\) 0 0
\(683\) 36.1616i 1.38368i 0.722049 + 0.691842i \(0.243200\pi\)
−0.722049 + 0.691842i \(0.756800\pi\)
\(684\) 0 0
\(685\) −5.29294 −0.202233
\(686\) 0 0
\(687\) 6.63659 0.253202
\(688\) 0 0
\(689\) 40.4294i 1.54024i
\(690\) 0 0
\(691\) 0.284481i 0.0108222i −0.999985 0.00541108i \(-0.998278\pi\)
0.999985 0.00541108i \(-0.00172241\pi\)
\(692\) 0 0
\(693\) 12.7526 0.484431
\(694\) 0 0
\(695\) 19.7415 0.748837
\(696\) 0 0
\(697\) 3.19009i 0.120833i
\(698\) 0 0
\(699\) 28.4233i 1.07507i
\(700\) 0 0
\(701\) 8.87200i 0.335091i −0.985864 0.167545i \(-0.946416\pi\)
0.985864 0.167545i \(-0.0535841\pi\)
\(702\) 0 0
\(703\) 3.36232i 0.126812i
\(704\) 0 0
\(705\) 1.22656i 0.0461950i
\(706\) 0 0
\(707\) −35.7013 −1.34269
\(708\) 0 0
\(709\) 40.1502i 1.50787i 0.656948 + 0.753936i \(0.271847\pi\)
−0.656948 + 0.753936i \(0.728153\pi\)
\(710\) 0 0
\(711\) −11.9924 −0.449750
\(712\) 0 0
\(713\) −4.52033 + 24.9795i −0.169288 + 0.935490i
\(714\) 0 0
\(715\) 15.2736i 0.571200i
\(716\) 0 0
\(717\) −21.4035 −0.799330
\(718\) 0 0
\(719\) 41.0118i 1.52948i −0.644337 0.764741i \(-0.722867\pi\)
0.644337 0.764741i \(-0.277133\pi\)
\(720\) 0 0
\(721\) 58.7745 2.18888
\(722\) 0 0
\(723\) 17.7463 0.659992
\(724\) 0 0
\(725\) −2.73385 −0.101533
\(726\) 0 0
\(727\) 32.7737 1.21551 0.607755 0.794124i \(-0.292070\pi\)
0.607755 + 0.794124i \(0.292070\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 22.3208i 0.825564i
\(732\) 0 0
\(733\) 22.7817i 0.841462i −0.907186 0.420731i \(-0.861774\pi\)
0.907186 0.420731i \(-0.138226\pi\)
\(734\) 0 0
\(735\) −4.21251 −0.155381
\(736\) 0 0
\(737\) −11.4804 −0.422887
\(738\) 0 0
\(739\) 42.0012i 1.54504i −0.634990 0.772520i \(-0.718996\pi\)
0.634990 0.772520i \(-0.281004\pi\)
\(740\) 0 0
\(741\) 10.6358i 0.390715i
\(742\) 0 0
\(743\) −7.19669 −0.264021 −0.132010 0.991248i \(-0.542143\pi\)
−0.132010 + 0.991248i \(0.542143\pi\)
\(744\) 0 0
\(745\) −11.2478 −0.412087
\(746\) 0 0
\(747\) 12.4426 0.455252
\(748\) 0 0
\(749\) −30.3624 −1.10942
\(750\) 0 0
\(751\) 3.66405 0.133703 0.0668515 0.997763i \(-0.478705\pi\)
0.0668515 + 0.997763i \(0.478705\pi\)
\(752\) 0 0
\(753\) 3.15845i 0.115100i
\(754\) 0 0
\(755\) 11.2361 0.408924
\(756\) 0 0
\(757\) 15.3288i 0.557134i −0.960417 0.278567i \(-0.910141\pi\)
0.960417 0.278567i \(-0.0898594\pi\)
\(758\) 0 0
\(759\) 3.25237 17.9727i 0.118053 0.652368i
\(760\) 0 0
\(761\) 52.0017 1.88506 0.942531 0.334120i \(-0.108439\pi\)
0.942531 + 0.334120i \(0.108439\pi\)
\(762\) 0 0
\(763\) 6.40072i 0.231722i
\(764\) 0 0
\(765\) 3.43500 0.124193
\(766\) 0 0
\(767\) 60.7728i 2.19438i
\(768\) 0 0
\(769\) 13.3406i 0.481074i 0.970640 + 0.240537i \(0.0773236\pi\)
−0.970640 + 0.240537i \(0.922676\pi\)
\(770\) 0 0
\(771\) 5.33676i 0.192199i
\(772\) 0 0
\(773\) 23.5703i 0.847766i −0.905717 0.423883i \(-0.860667\pi\)
0.905717 0.423883i \(-0.139333\pi\)
\(774\) 0 0
\(775\) 5.29319i 0.190137i
\(776\) 0 0
\(777\) 4.24537 0.152302
\(778\) 0 0
\(779\) 2.46293 0.0882437
\(780\) 0 0
\(781\) 15.6189i 0.558890i
\(782\) 0 0
\(783\) 2.73385i 0.0976999i
\(784\) 0 0
\(785\) −2.61969 −0.0935009
\(786\) 0 0
\(787\) 2.86992 0.102301 0.0511507 0.998691i \(-0.483711\pi\)
0.0511507 + 0.998691i \(0.483711\pi\)
\(788\) 0 0
\(789\) 12.1518i 0.432616i
\(790\) 0 0
\(791\) 31.7428i 1.12864i
\(792\) 0 0
\(793\) 20.4812i 0.727310i
\(794\) 0 0
\(795\) 10.0810i 0.357536i
\(796\) 0 0
\(797\) 7.88351i 0.279248i −0.990205 0.139624i \(-0.955411\pi\)
0.990205 0.139624i \(-0.0445894\pi\)
\(798\) 0 0
\(799\) 4.21323 0.149053
\(800\) 0 0
\(801\) 0.329105i 0.0116283i
\(802\) 0 0
\(803\) 49.4831 1.74622
\(804\) 0 0
\(805\) −2.85959 + 15.8022i −0.100787 + 0.556956i
\(806\) 0 0