Properties

Label 5520.2.be.c.1471.4
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 1471.4
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.c.1471.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{5} -1.52302 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{5} -1.52302 q^{7} -1.00000 q^{9} -6.43261 q^{11} -4.50229 q^{13} -1.00000 q^{15} +1.65230i q^{17} +5.57593 q^{19} -1.52302i q^{21} +(2.99135 - 3.74857i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -2.30375 q^{29} -1.77291i q^{31} -6.43261i q^{33} -1.52302i q^{35} +5.43051i q^{37} -4.50229i q^{39} -2.07053 q^{41} -4.75591 q^{43} -1.00000i q^{45} -2.99382i q^{47} -4.68040 q^{49} -1.65230 q^{51} -2.72374i q^{53} -6.43261i q^{55} +5.57593i q^{57} -3.99055i q^{59} +9.74609i q^{61} +1.52302 q^{63} -4.50229i q^{65} +3.71642 q^{67} +(3.74857 + 2.99135i) q^{69} -2.39857i q^{71} -9.23774 q^{73} -1.00000i q^{75} +9.79702 q^{77} +11.4088 q^{79} +1.00000 q^{81} +9.35972 q^{83} -1.65230 q^{85} -2.30375i q^{87} -2.66224i q^{89} +6.85710 q^{91} +1.77291 q^{93} +5.57593i q^{95} -8.66759i q^{97} +6.43261 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 8q^{7} - 32q^{9} + O(q^{10}) \) \( 32q - 8q^{7} - 32q^{9} + 8q^{11} - 8q^{13} - 32q^{15} - 32q^{25} + 4q^{29} + 20q^{41} + 52q^{49} - 4q^{51} + 8q^{63} + 32q^{67} - 40q^{73} - 24q^{77} + 32q^{79} + 32q^{81} - 4q^{85} - 48q^{91} - 8q^{99} + O(q^{100}) \)

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
\(7\) −1.52302 −0.575649 −0.287824 0.957683i \(-0.592932\pi\)
−0.287824 + 0.957683i \(0.592932\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −6.43261 −1.93950 −0.969752 0.244092i \(-0.921510\pi\)
−0.969752 + 0.244092i \(0.921510\pi\)
\(12\) 0 0
\(13\) −4.50229 −1.24871 −0.624356 0.781140i \(-0.714639\pi\)
−0.624356 + 0.781140i \(0.714639\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 1.65230i 0.400743i 0.979720 + 0.200371i \(0.0642148\pi\)
−0.979720 + 0.200371i \(0.935785\pi\)
\(18\) 0 0
\(19\) 5.57593 1.27921 0.639603 0.768705i \(-0.279099\pi\)
0.639603 + 0.768705i \(0.279099\pi\)
\(20\) 0 0
\(21\) 1.52302i 0.332351i
\(22\) 0 0
\(23\) 2.99135 3.74857i 0.623740 0.781632i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −2.30375 −0.427796 −0.213898 0.976856i \(-0.568616\pi\)
−0.213898 + 0.976856i \(0.568616\pi\)
\(30\) 0 0
\(31\) 1.77291i 0.318424i −0.987244 0.159212i \(-0.949105\pi\)
0.987244 0.159212i \(-0.0508953\pi\)
\(32\) 0 0
\(33\) 6.43261i 1.11977i
\(34\) 0 0
\(35\) 1.52302i 0.257438i
\(36\) 0 0
\(37\) 5.43051i 0.892771i 0.894841 + 0.446385i \(0.147289\pi\)
−0.894841 + 0.446385i \(0.852711\pi\)
\(38\) 0 0
\(39\) 4.50229i 0.720944i
\(40\) 0 0
\(41\) −2.07053 −0.323362 −0.161681 0.986843i \(-0.551692\pi\)
−0.161681 + 0.986843i \(0.551692\pi\)
\(42\) 0 0
\(43\) −4.75591 −0.725270 −0.362635 0.931931i \(-0.618123\pi\)
−0.362635 + 0.931931i \(0.618123\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 2.99382i 0.436693i −0.975871 0.218347i \(-0.929934\pi\)
0.975871 0.218347i \(-0.0700663\pi\)
\(48\) 0 0
\(49\) −4.68040 −0.668628
\(50\) 0 0
\(51\) −1.65230 −0.231369
\(52\) 0 0
\(53\) 2.72374i 0.374134i −0.982347 0.187067i \(-0.940102\pi\)
0.982347 0.187067i \(-0.0598982\pi\)
\(54\) 0 0
\(55\) 6.43261i 0.867373i
\(56\) 0 0
\(57\) 5.57593i 0.738550i
\(58\) 0 0
\(59\) 3.99055i 0.519525i −0.965673 0.259763i \(-0.916356\pi\)
0.965673 0.259763i \(-0.0836443\pi\)
\(60\) 0 0
\(61\) 9.74609i 1.24786i 0.781480 + 0.623930i \(0.214465\pi\)
−0.781480 + 0.623930i \(0.785535\pi\)
\(62\) 0 0
\(63\) 1.52302 0.191883
\(64\) 0 0
\(65\) 4.50229i 0.558441i
\(66\) 0 0
\(67\) 3.71642 0.454033 0.227017 0.973891i \(-0.427103\pi\)
0.227017 + 0.973891i \(0.427103\pi\)
\(68\) 0 0
\(69\) 3.74857 + 2.99135i 0.451275 + 0.360117i
\(70\) 0 0
\(71\) 2.39857i 0.284658i −0.989819 0.142329i \(-0.954541\pi\)
0.989819 0.142329i \(-0.0454591\pi\)
\(72\) 0 0
\(73\) −9.23774 −1.08120 −0.540598 0.841281i \(-0.681802\pi\)
−0.540598 + 0.841281i \(0.681802\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) 9.79702 1.11647
\(78\) 0 0
\(79\) 11.4088 1.28360 0.641798 0.766874i \(-0.278189\pi\)
0.641798 + 0.766874i \(0.278189\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.35972 1.02736 0.513681 0.857981i \(-0.328281\pi\)
0.513681 + 0.857981i \(0.328281\pi\)
\(84\) 0 0
\(85\) −1.65230 −0.179218
\(86\) 0 0
\(87\) 2.30375i 0.246988i
\(88\) 0 0
\(89\) 2.66224i 0.282197i −0.989996 0.141099i \(-0.954937\pi\)
0.989996 0.141099i \(-0.0450634\pi\)
\(90\) 0 0
\(91\) 6.85710 0.718820
\(92\) 0 0
\(93\) 1.77291 0.183842
\(94\) 0 0
\(95\) 5.57593i 0.572078i
\(96\) 0 0
\(97\) 8.66759i 0.880060i −0.897983 0.440030i \(-0.854968\pi\)
0.897983 0.440030i \(-0.145032\pi\)
\(98\) 0 0
\(99\) 6.43261 0.646501
\(100\) 0 0
\(101\) 18.6952 1.86024 0.930121 0.367254i \(-0.119702\pi\)
0.930121 + 0.367254i \(0.119702\pi\)
\(102\) 0 0
\(103\) 7.63952 0.752744 0.376372 0.926469i \(-0.377171\pi\)
0.376372 + 0.926469i \(0.377171\pi\)
\(104\) 0 0
\(105\) 1.52302 0.148632
\(106\) 0 0
\(107\) 6.71244 0.648916 0.324458 0.945900i \(-0.394818\pi\)
0.324458 + 0.945900i \(0.394818\pi\)
\(108\) 0 0
\(109\) 2.70608i 0.259196i −0.991567 0.129598i \(-0.958631\pi\)
0.991567 0.129598i \(-0.0413687\pi\)
\(110\) 0 0
\(111\) −5.43051 −0.515441
\(112\) 0 0
\(113\) 0.212216i 0.0199636i 0.999950 + 0.00998181i \(0.00317736\pi\)
−0.999950 + 0.00998181i \(0.996823\pi\)
\(114\) 0 0
\(115\) 3.74857 + 2.99135i 0.349556 + 0.278945i
\(116\) 0 0
\(117\) 4.50229 0.416237
\(118\) 0 0
\(119\) 2.51650i 0.230687i
\(120\) 0 0
\(121\) 30.3784 2.76168
\(122\) 0 0
\(123\) 2.07053i 0.186693i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 16.8377i 1.49410i 0.664767 + 0.747051i \(0.268531\pi\)
−0.664767 + 0.747051i \(0.731469\pi\)
\(128\) 0 0
\(129\) 4.75591i 0.418735i
\(130\) 0 0
\(131\) 11.1242i 0.971929i −0.873979 0.485964i \(-0.838468\pi\)
0.873979 0.485964i \(-0.161532\pi\)
\(132\) 0 0
\(133\) −8.49227 −0.736374
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 5.56165i 0.475164i 0.971368 + 0.237582i \(0.0763548\pi\)
−0.971368 + 0.237582i \(0.923645\pi\)
\(138\) 0 0
\(139\) 5.63364i 0.477839i −0.971039 0.238920i \(-0.923207\pi\)
0.971039 0.238920i \(-0.0767932\pi\)
\(140\) 0 0
\(141\) 2.99382 0.252125
\(142\) 0 0
\(143\) 28.9615 2.42188
\(144\) 0 0
\(145\) 2.30375i 0.191316i
\(146\) 0 0
\(147\) 4.68040i 0.386033i
\(148\) 0 0
\(149\) 10.5221i 0.862001i −0.902352 0.431000i \(-0.858161\pi\)
0.902352 0.431000i \(-0.141839\pi\)
\(150\) 0 0
\(151\) 17.5034i 1.42441i −0.701972 0.712205i \(-0.747697\pi\)
0.701972 0.712205i \(-0.252303\pi\)
\(152\) 0 0
\(153\) 1.65230i 0.133581i
\(154\) 0 0
\(155\) 1.77291 0.142404
\(156\) 0 0
\(157\) 7.31873i 0.584098i 0.956403 + 0.292049i \(0.0943371\pi\)
−0.956403 + 0.292049i \(0.905663\pi\)
\(158\) 0 0
\(159\) 2.72374 0.216007
\(160\) 0 0
\(161\) −4.55590 + 5.70917i −0.359055 + 0.449946i
\(162\) 0 0
\(163\) 9.06133i 0.709738i 0.934916 + 0.354869i \(0.115475\pi\)
−0.934916 + 0.354869i \(0.884525\pi\)
\(164\) 0 0
\(165\) 6.43261 0.500778
\(166\) 0 0
\(167\) 8.02942i 0.621335i −0.950519 0.310667i \(-0.899447\pi\)
0.950519 0.310667i \(-0.100553\pi\)
\(168\) 0 0
\(169\) 7.27066 0.559281
\(170\) 0 0
\(171\) −5.57593 −0.426402
\(172\) 0 0
\(173\) −0.927933 −0.0705495 −0.0352747 0.999378i \(-0.511231\pi\)
−0.0352747 + 0.999378i \(0.511231\pi\)
\(174\) 0 0
\(175\) 1.52302 0.115130
\(176\) 0 0
\(177\) 3.99055 0.299948
\(178\) 0 0
\(179\) 25.1922i 1.88295i 0.337082 + 0.941475i \(0.390560\pi\)
−0.337082 + 0.941475i \(0.609440\pi\)
\(180\) 0 0
\(181\) 1.69810i 0.126219i 0.998007 + 0.0631095i \(0.0201017\pi\)
−0.998007 + 0.0631095i \(0.979898\pi\)
\(182\) 0 0
\(183\) −9.74609 −0.720452
\(184\) 0 0
\(185\) −5.43051 −0.399259
\(186\) 0 0
\(187\) 10.6286i 0.777242i
\(188\) 0 0
\(189\) 1.52302i 0.110784i
\(190\) 0 0
\(191\) 13.0724 0.945886 0.472943 0.881093i \(-0.343192\pi\)
0.472943 + 0.881093i \(0.343192\pi\)
\(192\) 0 0
\(193\) 16.0209 1.15321 0.576606 0.817023i \(-0.304377\pi\)
0.576606 + 0.817023i \(0.304377\pi\)
\(194\) 0 0
\(195\) 4.50229 0.322416
\(196\) 0 0
\(197\) −11.9113 −0.848647 −0.424323 0.905511i \(-0.639488\pi\)
−0.424323 + 0.905511i \(0.639488\pi\)
\(198\) 0 0
\(199\) −7.54875 −0.535117 −0.267558 0.963542i \(-0.586217\pi\)
−0.267558 + 0.963542i \(0.586217\pi\)
\(200\) 0 0
\(201\) 3.71642i 0.262136i
\(202\) 0 0
\(203\) 3.50867 0.246261
\(204\) 0 0
\(205\) 2.07053i 0.144612i
\(206\) 0 0
\(207\) −2.99135 + 3.74857i −0.207913 + 0.260544i
\(208\) 0 0
\(209\) −35.8678 −2.48103
\(210\) 0 0
\(211\) 4.69539i 0.323244i 0.986853 + 0.161622i \(0.0516726\pi\)
−0.986853 + 0.161622i \(0.948327\pi\)
\(212\) 0 0
\(213\) 2.39857 0.164347
\(214\) 0 0
\(215\) 4.75591i 0.324351i
\(216\) 0 0
\(217\) 2.70018i 0.183300i
\(218\) 0 0
\(219\) 9.23774i 0.624229i
\(220\) 0 0
\(221\) 7.43916i 0.500412i
\(222\) 0 0
\(223\) 24.1553i 1.61756i −0.588110 0.808781i \(-0.700128\pi\)
0.588110 0.808781i \(-0.299872\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 12.4903 0.829010 0.414505 0.910047i \(-0.363955\pi\)
0.414505 + 0.910047i \(0.363955\pi\)
\(228\) 0 0
\(229\) 29.8509i 1.97260i 0.164951 + 0.986302i \(0.447253\pi\)
−0.164951 + 0.986302i \(0.552747\pi\)
\(230\) 0 0
\(231\) 9.79702i 0.644596i
\(232\) 0 0
\(233\) −0.897082 −0.0587698 −0.0293849 0.999568i \(-0.509355\pi\)
−0.0293849 + 0.999568i \(0.509355\pi\)
\(234\) 0 0
\(235\) 2.99382 0.195295
\(236\) 0 0
\(237\) 11.4088i 0.741084i
\(238\) 0 0
\(239\) 25.9763i 1.68027i −0.542381 0.840133i \(-0.682477\pi\)
0.542381 0.840133i \(-0.317523\pi\)
\(240\) 0 0
\(241\) 8.15296i 0.525179i −0.964908 0.262589i \(-0.915424\pi\)
0.964908 0.262589i \(-0.0845764\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 4.68040i 0.299020i
\(246\) 0 0
\(247\) −25.1045 −1.59736
\(248\) 0 0
\(249\) 9.35972i 0.593148i
\(250\) 0 0
\(251\) −6.31194 −0.398406 −0.199203 0.979958i \(-0.563835\pi\)
−0.199203 + 0.979958i \(0.563835\pi\)
\(252\) 0 0
\(253\) −19.2422 + 24.1131i −1.20975 + 1.51598i
\(254\) 0 0
\(255\) 1.65230i 0.103471i
\(256\) 0 0
\(257\) −10.8718 −0.678164 −0.339082 0.940757i \(-0.610116\pi\)
−0.339082 + 0.940757i \(0.610116\pi\)
\(258\) 0 0
\(259\) 8.27080i 0.513923i
\(260\) 0 0
\(261\) 2.30375 0.142599
\(262\) 0 0
\(263\) 23.5415 1.45163 0.725814 0.687891i \(-0.241463\pi\)
0.725814 + 0.687891i \(0.241463\pi\)
\(264\) 0 0
\(265\) 2.72374 0.167318
\(266\) 0 0
\(267\) 2.66224 0.162927
\(268\) 0 0
\(269\) 13.5049 0.823408 0.411704 0.911318i \(-0.364934\pi\)
0.411704 + 0.911318i \(0.364934\pi\)
\(270\) 0 0
\(271\) 18.2034i 1.10578i 0.833255 + 0.552889i \(0.186475\pi\)
−0.833255 + 0.552889i \(0.813525\pi\)
\(272\) 0 0
\(273\) 6.85710i 0.415011i
\(274\) 0 0
\(275\) 6.43261 0.387901
\(276\) 0 0
\(277\) −9.79569 −0.588566 −0.294283 0.955718i \(-0.595081\pi\)
−0.294283 + 0.955718i \(0.595081\pi\)
\(278\) 0 0
\(279\) 1.77291i 0.106141i
\(280\) 0 0
\(281\) 10.6935i 0.637923i −0.947768 0.318962i \(-0.896666\pi\)
0.947768 0.318962i \(-0.103334\pi\)
\(282\) 0 0
\(283\) −27.2186 −1.61797 −0.808987 0.587826i \(-0.799984\pi\)
−0.808987 + 0.587826i \(0.799984\pi\)
\(284\) 0 0
\(285\) −5.57593 −0.330290
\(286\) 0 0
\(287\) 3.15346 0.186143
\(288\) 0 0
\(289\) 14.2699 0.839405
\(290\) 0 0
\(291\) 8.66759 0.508103
\(292\) 0 0
\(293\) 23.1769i 1.35401i 0.735979 + 0.677004i \(0.236722\pi\)
−0.735979 + 0.677004i \(0.763278\pi\)
\(294\) 0 0
\(295\) 3.99055 0.232339
\(296\) 0 0
\(297\) 6.43261i 0.373258i
\(298\) 0 0
\(299\) −13.4680 + 16.8772i −0.778872 + 0.976033i
\(300\) 0 0
\(301\) 7.24337 0.417501
\(302\) 0 0
\(303\) 18.6952i 1.07401i
\(304\) 0 0
\(305\) −9.74609 −0.558060
\(306\) 0 0
\(307\) 23.1278i 1.31997i −0.751278 0.659985i \(-0.770562\pi\)
0.751278 0.659985i \(-0.229438\pi\)
\(308\) 0 0
\(309\) 7.63952i 0.434597i
\(310\) 0 0
\(311\) 15.5382i 0.881091i −0.897730 0.440545i \(-0.854785\pi\)
0.897730 0.440545i \(-0.145215\pi\)
\(312\) 0 0
\(313\) 1.93700i 0.109486i −0.998500 0.0547428i \(-0.982566\pi\)
0.998500 0.0547428i \(-0.0174339\pi\)
\(314\) 0 0
\(315\) 1.52302i 0.0858127i
\(316\) 0 0
\(317\) −20.0281 −1.12489 −0.562446 0.826834i \(-0.690140\pi\)
−0.562446 + 0.826834i \(0.690140\pi\)
\(318\) 0 0
\(319\) 14.8191 0.829713
\(320\) 0 0
\(321\) 6.71244i 0.374652i
\(322\) 0 0
\(323\) 9.21313i 0.512632i
\(324\) 0 0
\(325\) 4.50229 0.249742
\(326\) 0 0
\(327\) 2.70608 0.149647
\(328\) 0 0
\(329\) 4.55966i 0.251382i
\(330\) 0 0
\(331\) 19.6804i 1.08173i −0.841109 0.540866i \(-0.818097\pi\)
0.841109 0.540866i \(-0.181903\pi\)
\(332\) 0 0
\(333\) 5.43051i 0.297590i
\(334\) 0 0
\(335\) 3.71642i 0.203050i
\(336\) 0 0
\(337\) 3.73958i 0.203708i 0.994799 + 0.101854i \(0.0324774\pi\)
−0.994799 + 0.101854i \(0.967523\pi\)
\(338\) 0 0
\(339\) −0.212216 −0.0115260
\(340\) 0 0
\(341\) 11.4044i 0.617585i
\(342\) 0 0
\(343\) 17.7895 0.960544
\(344\) 0 0
\(345\) −2.99135 + 3.74857i −0.161049 + 0.201816i
\(346\) 0 0
\(347\) 32.8358i 1.76272i −0.472449 0.881358i \(-0.656630\pi\)
0.472449 0.881358i \(-0.343370\pi\)
\(348\) 0 0
\(349\) 17.3391 0.928139 0.464069 0.885799i \(-0.346389\pi\)
0.464069 + 0.885799i \(0.346389\pi\)
\(350\) 0 0
\(351\) 4.50229i 0.240315i
\(352\) 0 0
\(353\) 7.90388 0.420681 0.210340 0.977628i \(-0.432543\pi\)
0.210340 + 0.977628i \(0.432543\pi\)
\(354\) 0 0
\(355\) 2.39857 0.127303
\(356\) 0 0
\(357\) 2.51650 0.133187
\(358\) 0 0
\(359\) 6.85014 0.361537 0.180768 0.983526i \(-0.442142\pi\)
0.180768 + 0.983526i \(0.442142\pi\)
\(360\) 0 0
\(361\) 12.0910 0.636368
\(362\) 0 0
\(363\) 30.3784i 1.59446i
\(364\) 0 0
\(365\) 9.23774i 0.483526i
\(366\) 0 0
\(367\) −22.4163 −1.17012 −0.585060 0.810990i \(-0.698929\pi\)
−0.585060 + 0.810990i \(0.698929\pi\)
\(368\) 0 0
\(369\) 2.07053 0.107787
\(370\) 0 0
\(371\) 4.14832i 0.215370i
\(372\) 0 0
\(373\) 31.4598i 1.62893i 0.580214 + 0.814464i \(0.302969\pi\)
−0.580214 + 0.814464i \(0.697031\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 10.3722 0.534195
\(378\) 0 0
\(379\) 9.23870 0.474560 0.237280 0.971441i \(-0.423744\pi\)
0.237280 + 0.971441i \(0.423744\pi\)
\(380\) 0 0
\(381\) −16.8377 −0.862620
\(382\) 0 0
\(383\) 20.2393 1.03418 0.517089 0.855932i \(-0.327016\pi\)
0.517089 + 0.855932i \(0.327016\pi\)
\(384\) 0 0
\(385\) 9.79702i 0.499302i
\(386\) 0 0
\(387\) 4.75591 0.241757
\(388\) 0 0
\(389\) 15.0549i 0.763316i −0.924304 0.381658i \(-0.875353\pi\)
0.924304 0.381658i \(-0.124647\pi\)
\(390\) 0 0
\(391\) 6.19378 + 4.94262i 0.313233 + 0.249959i
\(392\) 0 0
\(393\) 11.1242 0.561143
\(394\) 0 0
\(395\) 11.4088i 0.574041i
\(396\) 0 0
\(397\) 30.0861 1.50998 0.754989 0.655737i \(-0.227642\pi\)
0.754989 + 0.655737i \(0.227642\pi\)
\(398\) 0 0
\(399\) 8.49227i 0.425145i
\(400\) 0 0
\(401\) 34.8412i 1.73989i −0.493151 0.869944i \(-0.664155\pi\)
0.493151 0.869944i \(-0.335845\pi\)
\(402\) 0 0
\(403\) 7.98216i 0.397620i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 34.9324i 1.73153i
\(408\) 0 0
\(409\) 25.1224 1.24223 0.621113 0.783721i \(-0.286681\pi\)
0.621113 + 0.783721i \(0.286681\pi\)
\(410\) 0 0
\(411\) −5.56165 −0.274336
\(412\) 0 0
\(413\) 6.07770i 0.299064i
\(414\) 0 0
\(415\) 9.35972i 0.459451i
\(416\) 0 0
\(417\) 5.63364 0.275881
\(418\) 0 0
\(419\) −22.1667 −1.08291 −0.541456 0.840729i \(-0.682127\pi\)
−0.541456 + 0.840729i \(0.682127\pi\)
\(420\) 0 0
\(421\) 7.99216i 0.389514i −0.980852 0.194757i \(-0.937608\pi\)
0.980852 0.194757i \(-0.0623918\pi\)
\(422\) 0 0
\(423\) 2.99382i 0.145564i
\(424\) 0 0
\(425\) 1.65230i 0.0801485i
\(426\) 0 0
\(427\) 14.8435i 0.718329i
\(428\) 0 0
\(429\) 28.9615i 1.39827i
\(430\) 0 0
\(431\) 38.2692 1.84336 0.921682 0.387946i \(-0.126815\pi\)
0.921682 + 0.387946i \(0.126815\pi\)
\(432\) 0 0
\(433\) 22.5291i 1.08268i 0.840805 + 0.541339i \(0.182082\pi\)
−0.840805 + 0.541339i \(0.817918\pi\)
\(434\) 0 0
\(435\) 2.30375 0.110457
\(436\) 0 0
\(437\) 16.6796 20.9018i 0.797892 0.999868i
\(438\) 0 0
\(439\) 17.6828i 0.843954i −0.906606 0.421977i \(-0.861336\pi\)
0.906606 0.421977i \(-0.138664\pi\)
\(440\) 0 0
\(441\) 4.68040 0.222876
\(442\) 0 0
\(443\) 14.2484i 0.676962i −0.940973 0.338481i \(-0.890087\pi\)
0.940973 0.338481i \(-0.109913\pi\)
\(444\) 0 0
\(445\) 2.66224 0.126202
\(446\) 0 0
\(447\) 10.5221 0.497676
\(448\) 0 0
\(449\) −23.2577 −1.09760 −0.548800 0.835954i \(-0.684915\pi\)
−0.548800 + 0.835954i \(0.684915\pi\)
\(450\) 0 0
\(451\) 13.3189 0.627162
\(452\) 0 0
\(453\) 17.5034 0.822383
\(454\) 0 0
\(455\) 6.85710i 0.321466i
\(456\) 0 0
\(457\) 19.8933i 0.930569i −0.885161 0.465285i \(-0.845952\pi\)
0.885161 0.465285i \(-0.154048\pi\)
\(458\) 0 0
\(459\) 1.65230 0.0771230
\(460\) 0 0
\(461\) 5.03647 0.234572 0.117286 0.993098i \(-0.462581\pi\)
0.117286 + 0.993098i \(0.462581\pi\)
\(462\) 0 0
\(463\) 14.1754i 0.658788i 0.944193 + 0.329394i \(0.106844\pi\)
−0.944193 + 0.329394i \(0.893156\pi\)
\(464\) 0 0
\(465\) 1.77291i 0.0822167i
\(466\) 0 0
\(467\) 11.8246 0.547176 0.273588 0.961847i \(-0.411790\pi\)
0.273588 + 0.961847i \(0.411790\pi\)
\(468\) 0 0
\(469\) −5.66020 −0.261364
\(470\) 0 0
\(471\) −7.31873 −0.337229
\(472\) 0 0
\(473\) 30.5929 1.40666
\(474\) 0 0
\(475\) −5.57593 −0.255841
\(476\) 0 0
\(477\) 2.72374i 0.124711i
\(478\) 0 0
\(479\) −25.6625 −1.17255 −0.586276 0.810112i \(-0.699406\pi\)
−0.586276 + 0.810112i \(0.699406\pi\)
\(480\) 0 0
\(481\) 24.4498i 1.11481i
\(482\) 0 0
\(483\) −5.70917 4.55590i −0.259776 0.207301i
\(484\) 0 0
\(485\) 8.66759 0.393575
\(486\) 0 0
\(487\) 17.8398i 0.808397i 0.914671 + 0.404199i \(0.132450\pi\)
−0.914671 + 0.404199i \(0.867550\pi\)
\(488\) 0 0
\(489\) −9.06133 −0.409768
\(490\) 0 0
\(491\) 20.0228i 0.903616i −0.892115 0.451808i \(-0.850779\pi\)
0.892115 0.451808i \(-0.149221\pi\)
\(492\) 0 0
\(493\) 3.80650i 0.171436i
\(494\) 0 0
\(495\) 6.43261i 0.289124i
\(496\) 0 0
\(497\) 3.65308i 0.163863i
\(498\) 0 0
\(499\) 24.8428i 1.11212i −0.831144 0.556058i \(-0.812313\pi\)
0.831144 0.556058i \(-0.187687\pi\)
\(500\) 0 0
\(501\) 8.02942 0.358728
\(502\) 0 0
\(503\) −5.67105 −0.252860 −0.126430 0.991976i \(-0.540352\pi\)
−0.126430 + 0.991976i \(0.540352\pi\)
\(504\) 0 0
\(505\) 18.6952i 0.831925i
\(506\) 0 0
\(507\) 7.27066i 0.322901i
\(508\) 0 0
\(509\) 12.8447 0.569331 0.284666 0.958627i \(-0.408117\pi\)
0.284666 + 0.958627i \(0.408117\pi\)
\(510\) 0 0
\(511\) 14.0693 0.622389
\(512\) 0 0
\(513\) 5.57593i 0.246183i
\(514\) 0 0
\(515\) 7.63952i 0.336637i
\(516\) 0 0
\(517\) 19.2581i 0.846968i
\(518\) 0 0
\(519\) 0.927933i 0.0407317i
\(520\) 0 0
\(521\) 12.9412i 0.566963i 0.958978 + 0.283481i \(0.0914894\pi\)
−0.958978 + 0.283481i \(0.908511\pi\)
\(522\) 0 0
\(523\) −2.93781 −0.128462 −0.0642308 0.997935i \(-0.520459\pi\)
−0.0642308 + 0.997935i \(0.520459\pi\)
\(524\) 0 0
\(525\) 1.52302i 0.0664702i
\(526\) 0 0
\(527\) 2.92939 0.127606
\(528\) 0 0
\(529\) −5.10362 22.4266i −0.221897 0.975070i
\(530\) 0 0
\(531\) 3.99055i 0.173175i
\(532\) 0 0
\(533\) 9.32212 0.403786
\(534\) 0 0
\(535\) 6.71244i 0.290204i
\(536\) 0 0
\(537\) −25.1922 −1.08712
\(538\) 0 0
\(539\) 30.1072 1.29681
\(540\) 0 0
\(541\) 29.0680 1.24973 0.624865 0.780733i \(-0.285154\pi\)
0.624865 + 0.780733i \(0.285154\pi\)
\(542\) 0 0
\(543\) −1.69810 −0.0728726
\(544\) 0 0
\(545\) 2.70608 0.115916
\(546\) 0 0
\(547\) 35.0271i 1.49765i 0.662768 + 0.748825i \(0.269382\pi\)
−0.662768 + 0.748825i \(0.730618\pi\)
\(548\) 0 0
\(549\) 9.74609i 0.415953i
\(550\) 0 0
\(551\) −12.8456 −0.547240
\(552\) 0 0
\(553\) −17.3759 −0.738900
\(554\) 0 0
\(555\) 5.43051i 0.230512i
\(556\) 0 0
\(557\) 17.5781i 0.744809i −0.928071 0.372404i \(-0.878533\pi\)
0.928071 0.372404i \(-0.121467\pi\)
\(558\) 0 0
\(559\) 21.4125 0.905653
\(560\) 0 0
\(561\) 10.6286 0.448741
\(562\) 0 0
\(563\) −19.7102 −0.830686 −0.415343 0.909665i \(-0.636339\pi\)
−0.415343 + 0.909665i \(0.636339\pi\)
\(564\) 0 0
\(565\) −0.212216 −0.00892800
\(566\) 0 0
\(567\) −1.52302 −0.0639610
\(568\) 0 0
\(569\) 25.5038i 1.06918i −0.845113 0.534588i \(-0.820467\pi\)
0.845113 0.534588i \(-0.179533\pi\)
\(570\) 0 0
\(571\) −44.2420 −1.85147 −0.925735 0.378173i \(-0.876552\pi\)
−0.925735 + 0.378173i \(0.876552\pi\)
\(572\) 0 0
\(573\) 13.0724i 0.546107i
\(574\) 0 0
\(575\) −2.99135 + 3.74857i −0.124748 + 0.156326i
\(576\) 0 0
\(577\) −30.3335 −1.26280 −0.631400 0.775457i \(-0.717519\pi\)
−0.631400 + 0.775457i \(0.717519\pi\)
\(578\) 0 0
\(579\) 16.0209i 0.665807i
\(580\) 0 0
\(581\) −14.2551 −0.591400
\(582\) 0 0
\(583\) 17.5208i 0.725635i
\(584\) 0 0
\(585\) 4.50229i 0.186147i
\(586\) 0 0
\(587\) 42.0137i 1.73409i −0.498230 0.867045i \(-0.666016\pi\)
0.498230 0.867045i \(-0.333984\pi\)
\(588\) 0 0
\(589\) 9.88562i 0.407330i
\(590\) 0 0
\(591\) 11.9113i 0.489967i
\(592\) 0 0
\(593\) −45.9637 −1.88750 −0.943752 0.330654i \(-0.892731\pi\)
−0.943752 + 0.330654i \(0.892731\pi\)
\(594\) 0 0
\(595\) 2.51650 0.103166
\(596\) 0 0
\(597\) 7.54875i 0.308950i
\(598\) 0 0
\(599\) 4.06616i 0.166139i 0.996544 + 0.0830694i \(0.0264723\pi\)
−0.996544 + 0.0830694i \(0.973528\pi\)
\(600\) 0 0
\(601\) −27.3871 −1.11714 −0.558571 0.829456i \(-0.688650\pi\)
−0.558571 + 0.829456i \(0.688650\pi\)
\(602\) 0 0
\(603\) −3.71642 −0.151344
\(604\) 0 0
\(605\) 30.3784i 1.23506i
\(606\) 0 0
\(607\) 24.1569i 0.980496i −0.871583 0.490248i \(-0.836906\pi\)
0.871583 0.490248i \(-0.163094\pi\)
\(608\) 0 0
\(609\) 3.50867i 0.142179i
\(610\) 0 0
\(611\) 13.4791i 0.545304i
\(612\) 0 0
\(613\) 21.9152i 0.885147i 0.896732 + 0.442574i \(0.145934\pi\)
−0.896732 + 0.442574i \(0.854066\pi\)
\(614\) 0 0
\(615\) 2.07053 0.0834917
\(616\) 0 0
\(617\) 6.74213i 0.271428i −0.990748 0.135714i \(-0.956667\pi\)
0.990748 0.135714i \(-0.0433328\pi\)
\(618\) 0 0
\(619\) −19.3523 −0.777836 −0.388918 0.921272i \(-0.627151\pi\)
−0.388918 + 0.921272i \(0.627151\pi\)
\(620\) 0 0
\(621\) −3.74857 2.99135i −0.150425 0.120039i
\(622\) 0 0
\(623\) 4.05466i 0.162446i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 35.8678i 1.43242i
\(628\) 0 0
\(629\) −8.97286 −0.357771
\(630\) 0 0
\(631\) −16.1921 −0.644598 −0.322299 0.946638i \(-0.604456\pi\)
−0.322299 + 0.946638i \(0.604456\pi\)
\(632\) 0 0
\(633\) −4.69539 −0.186625
\(634\) 0 0
\(635\) −16.8377 −0.668183
\(636\) 0 0
\(637\) 21.0725 0.834924
\(638\) 0 0
\(639\) 2.39857i 0.0948859i
\(640\) 0 0
\(641\) 25.9194i 1.02376i −0.859058 0.511878i \(-0.828950\pi\)
0.859058 0.511878i \(-0.171050\pi\)
\(642\) 0 0
\(643\) 23.0513 0.909055 0.454527 0.890733i \(-0.349808\pi\)
0.454527 + 0.890733i \(0.349808\pi\)
\(644\) 0 0
\(645\) 4.75591 0.187264
\(646\) 0 0
\(647\) 38.7579i 1.52373i 0.647735 + 0.761866i \(0.275716\pi\)
−0.647735 + 0.761866i \(0.724284\pi\)
\(648\) 0 0
\(649\) 25.6696i 1.00762i
\(650\) 0 0
\(651\) −2.70018 −0.105829
\(652\) 0 0
\(653\) −2.54972 −0.0997781 −0.0498891 0.998755i \(-0.515887\pi\)
−0.0498891 + 0.998755i \(0.515887\pi\)
\(654\) 0 0
\(655\) 11.1242 0.434660
\(656\) 0 0
\(657\) 9.23774 0.360399
\(658\) 0 0
\(659\) −43.3121 −1.68720 −0.843600 0.536972i \(-0.819568\pi\)
−0.843600 + 0.536972i \(0.819568\pi\)
\(660\) 0 0
\(661\) 9.39709i 0.365504i −0.983159 0.182752i \(-0.941499\pi\)
0.983159 0.182752i \(-0.0585006\pi\)
\(662\) 0 0
\(663\) 7.43916 0.288913
\(664\) 0 0
\(665\) 8.49227i 0.329316i
\(666\) 0 0
\(667\) −6.89134 + 8.63579i −0.266834 + 0.334379i
\(668\) 0 0
\(669\) 24.1553 0.933900
\(670\) 0 0
\(671\) 62.6928i 2.42023i
\(672\) 0 0
\(673\) −2.72747 −0.105136 −0.0525681 0.998617i \(-0.516741\pi\)
−0.0525681 + 0.998617i \(0.516741\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 12.1061i 0.465276i −0.972563 0.232638i \(-0.925264\pi\)
0.972563 0.232638i \(-0.0747358\pi\)
\(678\) 0 0
\(679\) 13.2009i 0.506606i
\(680\) 0 0
\(681\) 12.4903i 0.478629i
\(682\) 0 0
\(683\) 20.7694i 0.794720i 0.917663 + 0.397360i \(0.130074\pi\)
−0.917663 + 0.397360i \(0.869926\pi\)
\(684\) 0 0
\(685\) −5.56165 −0.212500
\(686\) 0 0
\(687\) −29.8509 −1.13888
\(688\) 0 0
\(689\) 12.2631i 0.467186i
\(690\) 0 0
\(691\) 6.08978i 0.231666i −0.993269 0.115833i \(-0.963046\pi\)
0.993269 0.115833i \(-0.0369538\pi\)
\(692\) 0 0
\(693\) −9.79702 −0.372158
\(694\) 0 0
\(695\) 5.63364 0.213696
\(696\) 0 0
\(697\) 3.42114i 0.129585i
\(698\) 0 0
\(699\) 0.897082i 0.0339308i
\(700\) 0 0
\(701\) 10.6195i 0.401094i −0.979684 0.200547i \(-0.935728\pi\)
0.979684 0.200547i \(-0.0642720\pi\)
\(702\) 0 0
\(703\) 30.2802i 1.14204i
\(704\) 0 0
\(705\) 2.99382i 0.112754i
\(706\) 0 0
\(707\) −28.4732 −1.07085
\(708\) 0 0
\(709\) 24.7363i 0.928991i −0.885575 0.464496i \(-0.846236\pi\)
0.885575 0.464496i \(-0.153764\pi\)
\(710\) 0 0
\(711\) −11.4088 −0.427865
\(712\) 0 0
\(713\) −6.64588 5.30340i −0.248890 0.198614i
\(714\) 0 0
\(715\) 28.9615i 1.08310i
\(716\) 0 0
\(717\) 25.9763 0.970101
\(718\) 0 0
\(719\) 51.3791i 1.91612i −0.286571 0.958059i \(-0.592515\pi\)
0.286571 0.958059i \(-0.407485\pi\)
\(720\) 0 0
\(721\) −11.6352 −0.433316
\(722\) 0 0
\(723\) 8.15296 0.303212
\(724\) 0 0
\(725\) 2.30375 0.0855593
\(726\) 0 0
\(727\) −5.94441 −0.220466 −0.110233 0.993906i \(-0.535160\pi\)
−0.110233 + 0.993906i \(0.535160\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 7.85822i 0.290647i
\(732\) 0 0
\(733\) 47.7046i 1.76201i 0.473106 + 0.881005i \(0.343133\pi\)
−0.473106 + 0.881005i \(0.656867\pi\)
\(734\) 0 0
\(735\) 4.68040 0.172639
\(736\) 0 0
\(737\) −23.9063 −0.880599
\(738\) 0 0
\(739\) 30.6154i 1.12621i 0.826387 + 0.563103i \(0.190393\pi\)
−0.826387 + 0.563103i \(0.809607\pi\)
\(740\) 0 0
\(741\) 25.1045i 0.922236i
\(742\) 0 0
\(743\) −4.15462 −0.152418 −0.0762092 0.997092i \(-0.524282\pi\)
−0.0762092 + 0.997092i \(0.524282\pi\)
\(744\) 0 0
\(745\) 10.5221 0.385498
\(746\) 0 0
\(747\) −9.35972 −0.342454
\(748\) 0 0
\(749\) −10.2232 −0.373548
\(750\) 0 0
\(751\) 4.49223 0.163924 0.0819619 0.996635i \(-0.473881\pi\)
0.0819619 + 0.996635i \(0.473881\pi\)
\(752\) 0 0
\(753\) 6.31194i 0.230020i
\(754\) 0 0
\(755\) 17.5034 0.637015
\(756\) 0 0
\(757\) 7.70763i 0.280138i −0.990142 0.140069i \(-0.955267\pi\)
0.990142 0.140069i \(-0.0447325\pi\)
\(758\) 0 0
\(759\) −24.1131 19.2422i −0.875250 0.698448i
\(760\) 0 0
\(761\) −2.85989 −0.103671 −0.0518355 0.998656i \(-0.516507\pi\)
−0.0518355 + 0.998656i \(0.516507\pi\)
\(762\) 0 0
\(763\) 4.12143i 0.149206i
\(764\) 0 0
\(765\) 1.65230 0.0597392
\(766\) 0 0
\(767\) 17.9666i 0.648737i
\(768\) 0 0
\(769\) 0.422431i 0.0152333i −0.999971 0.00761663i \(-0.997576\pi\)
0.999971 0.00761663i \(-0.00242447\pi\)
\(770\) 0 0
\(771\) 10.8718i 0.391538i
\(772\) 0 0
\(773\) 13.0769i 0.470342i −0.971954 0.235171i \(-0.924435\pi\)
0.971954 0.235171i \(-0.0755650\pi\)
\(774\) 0 0
\(775\) 1.77291i 0.0636848i
\(776\) 0 0
\(777\) 8.27080 0.296713
\(778\) 0 0
\(779\) −11.5451 −0.413647
\(780\) 0 0
\(781\) 15.4291i 0.552095i
\(782\) 0 0
\(783\) 2.30375i 0.0823295i
\(784\) 0 0
\(785\) −7.31873 −0.261217
\(786\) 0 0
\(787\) 44.8315 1.59807 0.799035 0.601285i \(-0.205344\pi\)
0.799035 + 0.601285i \(0.205344\pi\)
\(788\) 0 0
\(789\) 23.5415i 0.838098i
\(790\) 0 0
\(791\) 0.323210i 0.0114920i
\(792\) 0 0
\(793\) 43.8798i 1.55822i
\(794\) 0 0
\(795\) 2.72374i 0.0966011i
\(796\) 0 0
\(797\) 27.9408i 0.989715i 0.868974 + 0.494858i \(0.164780\pi\)
−0.868974 + 0.494858i \(0.835220\pi\)
\(798\) 0 0
\(799\) 4.94670 0.175002
\(800\) 0 0
\(801\) 2.66224i 0.0940657i
\(802\) 0 0
\(803\) 59.4228 2.09698
\(804\) 0 0
\(805\) −5.70917 4.55590i −0.201222 0.160574i
\(806\) 0 0