Properties

Label 3150.2.d.d.3149.5
Level 3150
Weight 2
Character 3150.3149
Analytic conductor 25.153
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.7442857984.4
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3149.5
Root \(-1.91681i\)
Character \(\chi\) = 3150.3149
Dual form 3150.2.d.d.3149.6

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +(1.35539 - 2.27220i) q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +(1.35539 - 2.27220i) q^{7} +1.00000 q^{8} -2.71078i q^{11} -6.54441 q^{13} +(1.35539 - 2.27220i) q^{14} +1.00000 q^{16} -1.53186i q^{17} -2.30177i q^{19} -2.71078i q^{22} -3.83363 q^{23} -6.54441 q^{26} +(1.35539 - 2.27220i) q^{28} -3.83363i q^{29} +3.25519i q^{31} +1.00000 q^{32} -1.53186i q^{34} -3.01255i q^{37} -2.30177i q^{38} -6.54441 q^{41} -0.468142i q^{43} -2.71078i q^{44} -3.83363 q^{46} +9.11980i q^{47} +(-3.32583 - 6.15945i) q^{49} -6.54441 q^{52} -9.25519 q^{53} +(1.35539 - 2.27220i) q^{56} -3.83363i q^{58} -11.1961 q^{59} -4.78705i q^{61} +3.25519i q^{62} +1.00000 q^{64} -13.5570i q^{67} -1.53186i q^{68} -2.30177i q^{71} +11.4216 q^{73} -3.01255i q^{74} -2.30177i q^{76} +(-6.15945 - 3.67417i) q^{77} +12.6768 q^{79} -6.54441 q^{82} +13.0888i q^{83} -0.468142i q^{86} -2.71078i q^{88} +9.60812 q^{89} +(-8.87024 + 14.8702i) q^{91} -3.83363 q^{92} +9.11980i q^{94} -16.9224 q^{97} +(-3.32583 - 6.15945i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{4} + 8q^{8} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{4} + 8q^{8} - 8q^{13} + 8q^{16} - 8q^{23} - 8q^{26} + 8q^{32} - 8q^{41} - 8q^{46} - 4q^{49} - 8q^{52} - 8q^{53} + 8q^{64} + 48q^{73} - 4q^{77} - 8q^{79} - 8q^{82} + 8q^{89} - 4q^{91} - 8q^{92} - 24q^{97} - 4q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.35539 2.27220i 0.512290 0.858813i
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 2.71078i 0.817332i −0.912684 0.408666i \(-0.865994\pi\)
0.912684 0.408666i \(-0.134006\pi\)
\(12\) 0 0
\(13\) −6.54441 −1.81509 −0.907546 0.419952i \(-0.862047\pi\)
−0.907546 + 0.419952i \(0.862047\pi\)
\(14\) 1.35539 2.27220i 0.362244 0.607272i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.53186i 0.371530i −0.982594 0.185765i \(-0.940524\pi\)
0.982594 0.185765i \(-0.0594763\pi\)
\(18\) 0 0
\(19\) 2.30177i 0.528062i −0.964514 0.264031i \(-0.914948\pi\)
0.964514 0.264031i \(-0.0850521\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.71078i 0.577941i
\(23\) −3.83363 −0.799366 −0.399683 0.916653i \(-0.630880\pi\)
−0.399683 + 0.916653i \(0.630880\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −6.54441 −1.28346
\(27\) 0 0
\(28\) 1.35539 2.27220i 0.256145 0.429406i
\(29\) 3.83363i 0.711887i −0.934508 0.355943i \(-0.884160\pi\)
0.934508 0.355943i \(-0.115840\pi\)
\(30\) 0 0
\(31\) 3.25519i 0.584650i 0.956319 + 0.292325i \(0.0944289\pi\)
−0.956319 + 0.292325i \(0.905571\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.53186i 0.262711i
\(35\) 0 0
\(36\) 0 0
\(37\) 3.01255i 0.495260i −0.968855 0.247630i \(-0.920348\pi\)
0.968855 0.247630i \(-0.0796518\pi\)
\(38\) 2.30177i 0.373396i
\(39\) 0 0
\(40\) 0 0
\(41\) −6.54441 −1.02207 −0.511033 0.859561i \(-0.670737\pi\)
−0.511033 + 0.859561i \(0.670737\pi\)
\(42\) 0 0
\(43\) 0.468142i 0.0713910i −0.999363 0.0356955i \(-0.988635\pi\)
0.999363 0.0356955i \(-0.0113647\pi\)
\(44\) 2.71078i 0.408666i
\(45\) 0 0
\(46\) −3.83363 −0.565237
\(47\) 9.11980i 1.33026i 0.746728 + 0.665130i \(0.231624\pi\)
−0.746728 + 0.665130i \(0.768376\pi\)
\(48\) 0 0
\(49\) −3.32583 6.15945i −0.475118 0.879922i
\(50\) 0 0
\(51\) 0 0
\(52\) −6.54441 −0.907546
\(53\) −9.25519 −1.27130 −0.635649 0.771978i \(-0.719268\pi\)
−0.635649 + 0.771978i \(0.719268\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.35539 2.27220i 0.181122 0.303636i
\(57\) 0 0
\(58\) 3.83363i 0.503380i
\(59\) −11.1961 −1.45760 −0.728802 0.684725i \(-0.759922\pi\)
−0.728802 + 0.684725i \(0.759922\pi\)
\(60\) 0 0
\(61\) 4.78705i 0.612919i −0.951884 0.306459i \(-0.900856\pi\)
0.951884 0.306459i \(-0.0991444\pi\)
\(62\) 3.25519i 0.413410i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 13.5570i 1.65625i −0.560546 0.828123i \(-0.689409\pi\)
0.560546 0.828123i \(-0.310591\pi\)
\(68\) 1.53186i 0.185765i
\(69\) 0 0
\(70\) 0 0
\(71\) 2.30177i 0.273170i −0.990628 0.136585i \(-0.956387\pi\)
0.990628 0.136585i \(-0.0436126\pi\)
\(72\) 0 0
\(73\) 11.4216 1.33679 0.668397 0.743805i \(-0.266981\pi\)
0.668397 + 0.743805i \(0.266981\pi\)
\(74\) 3.01255i 0.350202i
\(75\) 0 0
\(76\) 2.30177i 0.264031i
\(77\) −6.15945 3.67417i −0.701935 0.418711i
\(78\) 0 0
\(79\) 12.6768 1.42625 0.713123 0.701039i \(-0.247280\pi\)
0.713123 + 0.701039i \(0.247280\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.54441 −0.722709
\(83\) 13.0888i 1.43668i 0.695690 + 0.718342i \(0.255099\pi\)
−0.695690 + 0.718342i \(0.744901\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.468142i 0.0504811i
\(87\) 0 0
\(88\) 2.71078i 0.288970i
\(89\) 9.60812 1.01846 0.509230 0.860631i \(-0.329930\pi\)
0.509230 + 0.860631i \(0.329930\pi\)
\(90\) 0 0
\(91\) −8.87024 + 14.8702i −0.929853 + 1.55882i
\(92\) −3.83363 −0.399683
\(93\) 0 0
\(94\) 9.11980i 0.940635i
\(95\) 0 0
\(96\) 0 0
\(97\) −16.9224 −1.71821 −0.859107 0.511796i \(-0.828980\pi\)
−0.859107 + 0.511796i \(0.828980\pi\)
\(98\) −3.32583 6.15945i −0.335959 0.622199i
\(99\) 0 0
\(100\) 0 0
\(101\) 17.7405 1.76524 0.882622 0.470084i \(-0.155776\pi\)
0.882622 + 0.470084i \(0.155776\pi\)
\(102\) 0 0
\(103\) −4.05913 −0.399958 −0.199979 0.979800i \(-0.564087\pi\)
−0.199979 + 0.979800i \(0.564087\pi\)
\(104\) −6.54441 −0.641732
\(105\) 0 0
\(106\) −9.25519 −0.898944
\(107\) −6.31891 −0.610872 −0.305436 0.952213i \(-0.598802\pi\)
−0.305436 + 0.952213i \(0.598802\pi\)
\(108\) 0 0
\(109\) 12.0251 1.15180 0.575898 0.817522i \(-0.304653\pi\)
0.575898 + 0.817522i \(0.304653\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.35539 2.27220i 0.128072 0.214703i
\(113\) 9.06372 0.852643 0.426321 0.904572i \(-0.359809\pi\)
0.426321 + 0.904572i \(0.359809\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.83363i 0.355943i
\(117\) 0 0
\(118\) −11.1961 −1.03068
\(119\) −3.48069 2.07627i −0.319075 0.190331i
\(120\) 0 0
\(121\) 3.65166 0.331969
\(122\) 4.78705i 0.433399i
\(123\) 0 0
\(124\) 3.25519i 0.292325i
\(125\) 0 0
\(126\) 0 0
\(127\) 21.6332i 1.91964i −0.280619 0.959819i \(-0.590540\pi\)
0.280619 0.959819i \(-0.409460\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −2.51931 −0.220113 −0.110056 0.993925i \(-0.535103\pi\)
−0.110056 + 0.993925i \(0.535103\pi\)
\(132\) 0 0
\(133\) −5.23009 3.11980i −0.453506 0.270521i
\(134\) 13.5570i 1.17114i
\(135\) 0 0
\(136\) 1.53186i 0.131356i
\(137\) 1.39646 0.119308 0.0596539 0.998219i \(-0.481000\pi\)
0.0596539 + 0.998219i \(0.481000\pi\)
\(138\) 0 0
\(139\) 6.13539i 0.520397i −0.965555 0.260199i \(-0.916212\pi\)
0.965555 0.260199i \(-0.0837881\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.30177i 0.193160i
\(143\) 17.7405i 1.48353i
\(144\) 0 0
\(145\) 0 0
\(146\) 11.4216 0.945256
\(147\) 0 0
\(148\) 3.01255i 0.247630i
\(149\) 4.65166i 0.381078i −0.981680 0.190539i \(-0.938976\pi\)
0.981680 0.190539i \(-0.0610236\pi\)
\(150\) 0 0
\(151\) −3.58794 −0.291982 −0.145991 0.989286i \(-0.546637\pi\)
−0.145991 + 0.989286i \(0.546637\pi\)
\(152\) 2.30177i 0.186698i
\(153\) 0 0
\(154\) −6.15945 3.67417i −0.496343 0.296073i
\(155\) 0 0
\(156\) 0 0
\(157\) −13.1228 −1.04732 −0.523658 0.851928i \(-0.675433\pi\)
−0.523658 + 0.851928i \(0.675433\pi\)
\(158\) 12.6768 1.00851
\(159\) 0 0
\(160\) 0 0
\(161\) −5.19606 + 8.71078i −0.409507 + 0.686506i
\(162\) 0 0
\(163\) 16.6207i 1.30183i 0.759150 + 0.650916i \(0.225615\pi\)
−0.759150 + 0.650916i \(0.774385\pi\)
\(164\) −6.54441 −0.511033
\(165\) 0 0
\(166\) 13.0888i 1.01589i
\(167\) 8.62068i 0.667088i −0.942735 0.333544i \(-0.891755\pi\)
0.942735 0.333544i \(-0.108245\pi\)
\(168\) 0 0
\(169\) 29.8293 2.29456
\(170\) 0 0
\(171\) 0 0
\(172\) 0.468142i 0.0356955i
\(173\) 10.6517i 0.809830i −0.914354 0.404915i \(-0.867301\pi\)
0.914354 0.404915i \(-0.132699\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.71078i 0.204333i
\(177\) 0 0
\(178\) 9.60812 0.720159
\(179\) 23.1961i 1.73376i 0.498521 + 0.866878i \(0.333877\pi\)
−0.498521 + 0.866878i \(0.666123\pi\)
\(180\) 0 0
\(181\) 9.11980i 0.677869i −0.940810 0.338935i \(-0.889933\pi\)
0.940810 0.338935i \(-0.110067\pi\)
\(182\) −8.87024 + 14.8702i −0.657506 + 1.10226i
\(183\) 0 0
\(184\) −3.83363 −0.282619
\(185\) 0 0
\(186\) 0 0
\(187\) −4.15253 −0.303663
\(188\) 9.11980i 0.665130i
\(189\) 0 0
\(190\) 0 0
\(191\) 9.69823i 0.701739i −0.936424 0.350870i \(-0.885886\pi\)
0.936424 0.350870i \(-0.114114\pi\)
\(192\) 0 0
\(193\) 18.1525i 1.30665i −0.757078 0.653324i \(-0.773374\pi\)
0.757078 0.653324i \(-0.226626\pi\)
\(194\) −16.9224 −1.21496
\(195\) 0 0
\(196\) −3.32583 6.15945i −0.237559 0.439961i
\(197\) −4.60354 −0.327988 −0.163994 0.986461i \(-0.552438\pi\)
−0.163994 + 0.986461i \(0.552438\pi\)
\(198\) 0 0
\(199\) 11.7405i 0.832260i −0.909305 0.416130i \(-0.863386\pi\)
0.909305 0.416130i \(-0.136614\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 17.7405 1.24822
\(203\) −8.71078 5.19606i −0.611377 0.364692i
\(204\) 0 0
\(205\) 0 0
\(206\) −4.05913 −0.282813
\(207\) 0 0
\(208\) −6.54441 −0.453773
\(209\) −6.23960 −0.431602
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −9.25519 −0.635649
\(213\) 0 0
\(214\) −6.31891 −0.431952
\(215\) 0 0
\(216\) 0 0
\(217\) 7.39646 + 4.41206i 0.502105 + 0.299510i
\(218\) 12.0251 0.814443
\(219\) 0 0
\(220\) 0 0
\(221\) 10.0251i 0.674361i
\(222\) 0 0
\(223\) 26.4513 1.77131 0.885654 0.464347i \(-0.153711\pi\)
0.885654 + 0.464347i \(0.153711\pi\)
\(224\) 1.35539 2.27220i 0.0905609 0.151818i
\(225\) 0 0
\(226\) 9.06372 0.602909
\(227\) 15.0637i 0.999814i −0.866079 0.499907i \(-0.833368\pi\)
0.866079 0.499907i \(-0.166632\pi\)
\(228\) 0 0
\(229\) 11.3655i 0.751052i −0.926812 0.375526i \(-0.877462\pi\)
0.926812 0.375526i \(-0.122538\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.83363i 0.251690i
\(233\) −20.9957 −1.37547 −0.687736 0.725961i \(-0.741395\pi\)
−0.687736 + 0.725961i \(0.741395\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −11.1961 −0.728802
\(237\) 0 0
\(238\) −3.48069 2.07627i −0.225620 0.134584i
\(239\) 9.69823i 0.627326i −0.949534 0.313663i \(-0.898444\pi\)
0.949534 0.313663i \(-0.101556\pi\)
\(240\) 0 0
\(241\) 24.7019i 1.59119i −0.605831 0.795593i \(-0.707159\pi\)
0.605831 0.795593i \(-0.292841\pi\)
\(242\) 3.65166 0.234737
\(243\) 0 0
\(244\) 4.78705i 0.306459i
\(245\) 0 0
\(246\) 0 0
\(247\) 15.0637i 0.958481i
\(248\) 3.25519i 0.206705i
\(249\) 0 0
\(250\) 0 0
\(251\) 3.60812 0.227743 0.113871 0.993495i \(-0.463675\pi\)
0.113871 + 0.993495i \(0.463675\pi\)
\(252\) 0 0
\(253\) 10.3921i 0.653348i
\(254\) 21.6332i 1.35739i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 28.7983i 1.79639i 0.439598 + 0.898195i \(0.355121\pi\)
−0.439598 + 0.898195i \(0.644879\pi\)
\(258\) 0 0
\(259\) −6.84513 4.08319i −0.425336 0.253717i
\(260\) 0 0
\(261\) 0 0
\(262\) −2.51931 −0.155643
\(263\) −12.7699 −0.787426 −0.393713 0.919233i \(-0.628810\pi\)
−0.393713 + 0.919233i \(0.628810\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −5.23009 3.11980i −0.320677 0.191287i
\(267\) 0 0
\(268\) 13.5570i 0.828123i
\(269\) 8.43716 0.514423 0.257211 0.966355i \(-0.417196\pi\)
0.257211 + 0.966355i \(0.417196\pi\)
\(270\) 0 0
\(271\) 2.16637i 0.131598i −0.997833 0.0657989i \(-0.979040\pi\)
0.997833 0.0657989i \(-0.0209596\pi\)
\(272\) 1.53186i 0.0928825i
\(273\) 0 0
\(274\) 1.39646 0.0843634
\(275\) 0 0
\(276\) 0 0
\(277\) 5.92373i 0.355923i −0.984037 0.177961i \(-0.943050\pi\)
0.984037 0.177961i \(-0.0569502\pi\)
\(278\) 6.13539i 0.367977i
\(279\) 0 0
\(280\) 0 0
\(281\) 27.3566i 1.63196i 0.578083 + 0.815978i \(0.303801\pi\)
−0.578083 + 0.815978i \(0.696199\pi\)
\(282\) 0 0
\(283\) −18.0594 −1.07352 −0.536759 0.843735i \(-0.680352\pi\)
−0.536759 + 0.843735i \(0.680352\pi\)
\(284\) 2.30177i 0.136585i
\(285\) 0 0
\(286\) 17.7405i 1.04902i
\(287\) −8.87024 + 14.8702i −0.523594 + 0.877762i
\(288\) 0 0
\(289\) 14.6534 0.861965
\(290\) 0 0
\(291\) 0 0
\(292\) 11.4216 0.668397
\(293\) 29.1139i 1.70085i −0.526094 0.850427i \(-0.676344\pi\)
0.526094 0.850427i \(-0.323656\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.01255i 0.175101i
\(297\) 0 0
\(298\) 4.65166i 0.269463i
\(299\) 25.0888 1.45092
\(300\) 0 0
\(301\) −1.06372 0.634516i −0.0613115 0.0365729i
\(302\) −3.58794 −0.206463
\(303\) 0 0
\(304\) 2.30177i 0.132015i
\(305\) 0 0
\(306\) 0 0
\(307\) −7.39646 −0.422138 −0.211069 0.977471i \(-0.567695\pi\)
−0.211069 + 0.977471i \(0.567695\pi\)
\(308\) −6.15945 3.67417i −0.350967 0.209355i
\(309\) 0 0
\(310\) 0 0
\(311\) 1.97490 0.111986 0.0559931 0.998431i \(-0.482168\pi\)
0.0559931 + 0.998431i \(0.482168\pi\)
\(312\) 0 0
\(313\) 0.961388 0.0543409 0.0271704 0.999631i \(-0.491350\pi\)
0.0271704 + 0.999631i \(0.491350\pi\)
\(314\) −13.1228 −0.740565
\(315\) 0 0
\(316\) 12.6768 0.713123
\(317\) −17.1620 −0.963916 −0.481958 0.876194i \(-0.660074\pi\)
−0.481958 + 0.876194i \(0.660074\pi\)
\(318\) 0 0
\(319\) −10.3921 −0.581848
\(320\) 0 0
\(321\) 0 0
\(322\) −5.19606 + 8.71078i −0.289565 + 0.485433i
\(323\) −3.52598 −0.196191
\(324\) 0 0
\(325\) 0 0
\(326\) 16.6207i 0.920534i
\(327\) 0 0
\(328\) −6.54441 −0.361355
\(329\) 20.7220 + 12.3609i 1.14244 + 0.681478i
\(330\) 0 0
\(331\) 9.74047 0.535385 0.267692 0.963504i \(-0.413739\pi\)
0.267692 + 0.963504i \(0.413739\pi\)
\(332\) 13.0888i 0.718342i
\(333\) 0 0
\(334\) 8.62068i 0.471702i
\(335\) 0 0
\(336\) 0 0
\(337\) 3.15078i 0.171634i −0.996311 0.0858169i \(-0.972650\pi\)
0.996311 0.0858169i \(-0.0273500\pi\)
\(338\) 29.8293 1.62250
\(339\) 0 0
\(340\) 0 0
\(341\) 8.82412 0.477853
\(342\) 0 0
\(343\) −18.5033 0.791511i −0.999086 0.0427376i
\(344\) 0.468142i 0.0252405i
\(345\) 0 0
\(346\) 10.6517i 0.572637i
\(347\) 30.0594 1.61367 0.806836 0.590775i \(-0.201178\pi\)
0.806836 + 0.590775i \(0.201178\pi\)
\(348\) 0 0
\(349\) 27.5180i 1.47301i 0.676434 + 0.736503i \(0.263524\pi\)
−0.676434 + 0.736503i \(0.736476\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.71078i 0.144485i
\(353\) 16.5956i 0.883293i 0.897189 + 0.441647i \(0.145605\pi\)
−0.897189 + 0.441647i \(0.854395\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 9.60812 0.509230
\(357\) 0 0
\(358\) 23.1961i 1.22595i
\(359\) 15.0076i 0.792073i −0.918235 0.396036i \(-0.870385\pi\)
0.918235 0.396036i \(-0.129615\pi\)
\(360\) 0 0
\(361\) 13.7019 0.721151
\(362\) 9.11980i 0.479326i
\(363\) 0 0
\(364\) −8.87024 + 14.8702i −0.464927 + 0.779412i
\(365\) 0 0
\(366\) 0 0
\(367\) 14.2598 0.744354 0.372177 0.928162i \(-0.378611\pi\)
0.372177 + 0.928162i \(0.378611\pi\)
\(368\) −3.83363 −0.199842
\(369\) 0 0
\(370\) 0 0
\(371\) −12.5444 + 21.0297i −0.651273 + 1.09181i
\(372\) 0 0
\(373\) 8.98745i 0.465352i 0.972554 + 0.232676i \(0.0747482\pi\)
−0.972554 + 0.232676i \(0.925252\pi\)
\(374\) −4.15253 −0.214722
\(375\) 0 0
\(376\) 9.11980i 0.470318i
\(377\) 25.0888i 1.29214i
\(378\) 0 0
\(379\) 34.5447 1.77444 0.887220 0.461346i \(-0.152633\pi\)
0.887220 + 0.461346i \(0.152633\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 9.69823i 0.496205i
\(383\) 2.88020i 0.147171i 0.997289 + 0.0735857i \(0.0234443\pi\)
−0.997289 + 0.0735857i \(0.976556\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 18.1525i 0.923940i
\(387\) 0 0
\(388\) −16.9224 −0.859107
\(389\) 35.1620i 1.78279i 0.453231 + 0.891393i \(0.350271\pi\)
−0.453231 + 0.891393i \(0.649729\pi\)
\(390\) 0 0
\(391\) 5.87257i 0.296989i
\(392\) −3.32583 6.15945i −0.167980 0.311099i
\(393\) 0 0
\(394\) −4.60354 −0.231923
\(395\) 0 0
\(396\) 0 0
\(397\) 16.1185 0.808965 0.404482 0.914546i \(-0.367452\pi\)
0.404482 + 0.914546i \(0.367452\pi\)
\(398\) 11.7405i 0.588497i
\(399\) 0 0
\(400\) 0 0
\(401\) 16.6256i 0.830243i 0.909766 + 0.415121i \(0.136261\pi\)
−0.909766 + 0.415121i \(0.863739\pi\)
\(402\) 0 0
\(403\) 21.3033i 1.06119i
\(404\) 17.7405 0.882622
\(405\) 0 0
\(406\) −8.71078 5.19606i −0.432309 0.257876i
\(407\) −8.16637 −0.404792
\(408\) 0 0
\(409\) 12.0000i 0.593362i 0.954977 + 0.296681i \(0.0958798\pi\)
−0.954977 + 0.296681i \(0.904120\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.05913 −0.199979
\(413\) −15.1751 + 25.4397i −0.746715 + 1.25181i
\(414\) 0 0
\(415\) 0 0
\(416\) −6.54441 −0.320866
\(417\) 0 0
\(418\) −6.23960 −0.305189
\(419\) 36.2849 1.77263 0.886316 0.463080i \(-0.153256\pi\)
0.886316 + 0.463080i \(0.153256\pi\)
\(420\) 0 0
\(421\) 19.2414 0.937766 0.468883 0.883260i \(-0.344657\pi\)
0.468883 + 0.883260i \(0.344657\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −9.25519 −0.449472
\(425\) 0 0
\(426\) 0 0
\(427\) −10.8772 6.48833i −0.526383 0.313992i
\(428\) −6.31891 −0.305436
\(429\) 0 0
\(430\) 0 0
\(431\) 17.2974i 0.833188i −0.909093 0.416594i \(-0.863224\pi\)
0.909093 0.416594i \(-0.136776\pi\)
\(432\) 0 0
\(433\) 31.6473 1.52087 0.760437 0.649412i \(-0.224985\pi\)
0.760437 + 0.649412i \(0.224985\pi\)
\(434\) 7.39646 + 4.41206i 0.355042 + 0.211786i
\(435\) 0 0
\(436\) 12.0251 0.575898
\(437\) 8.82412i 0.422115i
\(438\) 0 0
\(439\) 13.0095i 0.620910i −0.950588 0.310455i \(-0.899519\pi\)
0.950588 0.310455i \(-0.100481\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 10.0251i 0.476846i
\(443\) 3.78551 0.179855 0.0899275 0.995948i \(-0.471336\pi\)
0.0899275 + 0.995948i \(0.471336\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 26.4513 1.25250
\(447\) 0 0
\(448\) 1.35539 2.27220i 0.0640362 0.107352i
\(449\) 24.9307i 1.17655i 0.808661 + 0.588275i \(0.200193\pi\)
−0.808661 + 0.588275i \(0.799807\pi\)
\(450\) 0 0
\(451\) 17.7405i 0.835366i
\(452\) 9.06372 0.426321
\(453\) 0 0
\(454\) 15.0637i 0.706975i
\(455\) 0 0
\(456\) 0 0
\(457\) 31.3535i 1.46666i 0.679875 + 0.733328i \(0.262034\pi\)
−0.679875 + 0.733328i \(0.737966\pi\)
\(458\) 11.3655i 0.531074i
\(459\) 0 0
\(460\) 0 0
\(461\) 9.52598 0.443669 0.221835 0.975084i \(-0.428795\pi\)
0.221835 + 0.975084i \(0.428795\pi\)
\(462\) 0 0
\(463\) 34.0003i 1.58013i 0.613026 + 0.790063i \(0.289952\pi\)
−0.613026 + 0.790063i \(0.710048\pi\)
\(464\) 3.83363i 0.177972i
\(465\) 0 0
\(466\) −20.9957 −0.972605
\(467\) 39.2665i 1.81703i −0.417847 0.908517i \(-0.637215\pi\)
0.417847 0.908517i \(-0.362785\pi\)
\(468\) 0 0
\(469\) −30.8042 18.3750i −1.42241 0.848478i
\(470\) 0 0
\(471\) 0 0
\(472\) −11.1961 −0.515341
\(473\) −1.26903 −0.0583502
\(474\) 0 0
\(475\) 0 0
\(476\) −3.48069 2.07627i −0.159537 0.0951655i
\(477\) 0 0
\(478\) 9.69823i 0.443587i
\(479\) −34.0251 −1.55465 −0.777323 0.629101i \(-0.783423\pi\)
−0.777323 + 0.629101i \(0.783423\pi\)
\(480\) 0 0
\(481\) 19.7154i 0.898944i
\(482\) 24.7019i 1.12514i
\(483\) 0 0
\(484\) 3.65166 0.165984
\(485\) 0 0
\(486\) 0 0
\(487\) 12.8091i 0.580436i 0.956961 + 0.290218i \(0.0937278\pi\)
−0.956961 + 0.290218i \(0.906272\pi\)
\(488\) 4.78705i 0.216700i
\(489\) 0 0
\(490\) 0 0
\(491\) 11.6471i 0.525625i −0.964847 0.262812i \(-0.915350\pi\)
0.964847 0.262812i \(-0.0846500\pi\)
\(492\) 0 0
\(493\) −5.87257 −0.264487
\(494\) 15.0637i 0.677749i
\(495\) 0 0
\(496\) 3.25519i 0.146162i
\(497\) −5.23009 3.11980i −0.234602 0.139942i
\(498\) 0 0
\(499\) −35.5511 −1.59149 −0.795743 0.605635i \(-0.792919\pi\)
−0.795743 + 0.605635i \(0.792919\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 3.60812 0.161038
\(503\) 8.23372i 0.367123i 0.983008 + 0.183562i \(0.0587627\pi\)
−0.983008 + 0.183562i \(0.941237\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 10.3921i 0.461986i
\(507\) 0 0
\(508\) 21.6332i 0.959819i
\(509\) −31.0888 −1.37799 −0.688994 0.724767i \(-0.741947\pi\)
−0.688994 + 0.724767i \(0.741947\pi\)
\(510\) 0 0
\(511\) 15.4807 25.9521i 0.684826 1.14805i
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 28.7983i 1.27024i
\(515\) 0 0
\(516\) 0 0
\(517\) 24.7218 1.08726
\(518\) −6.84513 4.08319i −0.300758 0.179405i
\(519\) 0 0
\(520\) 0 0
\(521\) 1.39363 0.0610561 0.0305281 0.999534i \(-0.490281\pi\)
0.0305281 + 0.999534i \(0.490281\pi\)
\(522\) 0 0
\(523\) 20.4853 0.895759 0.447879 0.894094i \(-0.352179\pi\)
0.447879 + 0.894094i \(0.352179\pi\)
\(524\) −2.51931 −0.110056
\(525\) 0 0
\(526\) −12.7699 −0.556795
\(527\) 4.98649 0.217215
\(528\) 0 0
\(529\) −8.30331 −0.361013
\(530\) 0 0
\(531\) 0 0
\(532\) −5.23009 3.11980i −0.226753 0.135260i
\(533\) 42.8293 1.85514
\(534\) 0 0
\(535\) 0 0
\(536\) 13.5570i 0.585572i
\(537\) 0 0
\(538\) 8.43716 0.363752
\(539\) −16.6969 + 9.01560i −0.719188 + 0.388329i
\(540\) 0 0
\(541\) −0.0870615 −0.00374306 −0.00187153 0.999998i \(-0.500596\pi\)
−0.00187153 + 0.999998i \(0.500596\pi\)
\(542\) 2.16637i 0.0930537i
\(543\) 0 0
\(544\) 1.53186i 0.0656779i
\(545\) 0 0
\(546\) 0 0
\(547\) 5.40443i 0.231077i 0.993303 + 0.115538i \(0.0368593\pi\)
−0.993303 + 0.115538i \(0.963141\pi\)
\(548\) 1.39646 0.0596539
\(549\) 0 0
\(550\) 0 0
\(551\) −8.82412 −0.375920
\(552\) 0 0
\(553\) 17.1820 28.8042i 0.730652 1.22488i
\(554\) 5.92373i 0.251675i
\(555\) 0 0
\(556\) 6.13539i 0.260199i
\(557\) 0.127431 0.00539940 0.00269970 0.999996i \(-0.499141\pi\)
0.00269970 + 0.999996i \(0.499141\pi\)
\(558\) 0 0
\(559\) 3.06372i 0.129581i
\(560\) 0 0
\(561\) 0 0
\(562\) 27.3566i 1.15397i
\(563\) 6.23960i 0.262968i −0.991318 0.131484i \(-0.958026\pi\)
0.991318 0.131484i \(-0.0419741\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −18.0594 −0.759092
\(567\) 0 0
\(568\) 2.30177i 0.0965801i
\(569\) 11.6391i 0.487937i −0.969783 0.243968i \(-0.921551\pi\)
0.969783 0.243968i \(-0.0784493\pi\)
\(570\) 0 0
\(571\) −35.9181 −1.50313 −0.751563 0.659661i \(-0.770700\pi\)
−0.751563 + 0.659661i \(0.770700\pi\)
\(572\) 17.7405i 0.741766i
\(573\) 0 0
\(574\) −8.87024 + 14.8702i −0.370237 + 0.620672i
\(575\) 0 0
\(576\) 0 0
\(577\) 7.15687 0.297944 0.148972 0.988841i \(-0.452404\pi\)
0.148972 + 0.988841i \(0.452404\pi\)
\(578\) 14.6534 0.609502
\(579\) 0 0
\(580\) 0 0
\(581\) 29.7405 + 17.7405i 1.23384 + 0.735999i
\(582\) 0 0
\(583\) 25.0888i 1.03907i
\(584\) 11.4216 0.472628
\(585\) 0 0
\(586\) 29.1139i 1.20269i
\(587\) 4.15253i 0.171393i 0.996321 + 0.0856967i \(0.0273116\pi\)
−0.996321 + 0.0856967i \(0.972688\pi\)
\(588\) 0 0
\(589\) 7.49270 0.308731
\(590\) 0 0
\(591\) 0 0
\(592\) 3.01255i 0.123815i
\(593\) 29.7966i 1.22360i 0.791013 + 0.611799i \(0.209554\pi\)
−0.791013 + 0.611799i \(0.790446\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.65166i 0.190539i
\(597\) 0 0
\(598\) 25.0888 1.02596
\(599\) 35.4249i 1.44742i 0.690104 + 0.723710i \(0.257565\pi\)
−0.690104 + 0.723710i \(0.742435\pi\)
\(600\) 0 0
\(601\) 27.4948i 1.12154i −0.827973 0.560768i \(-0.810506\pi\)
0.827973 0.560768i \(-0.189494\pi\)
\(602\) −1.06372 0.634516i −0.0433538 0.0258609i
\(603\) 0 0
\(604\) −3.58794 −0.145991
\(605\) 0 0
\(606\) 0 0
\(607\) 4.76499 0.193405 0.0967025 0.995313i \(-0.469170\pi\)
0.0967025 + 0.995313i \(0.469170\pi\)
\(608\) 2.30177i 0.0933490i
\(609\) 0 0
\(610\) 0 0
\(611\) 59.6837i 2.41454i
\(612\) 0 0
\(613\) 10.7981i 0.436130i 0.975934 + 0.218065i \(0.0699744\pi\)
−0.975934 + 0.218065i \(0.930026\pi\)
\(614\) −7.39646 −0.298497
\(615\) 0 0
\(616\) −6.15945 3.67417i −0.248171 0.148037i
\(617\) 34.9025 1.40512 0.702561 0.711623i \(-0.252040\pi\)
0.702561 + 0.711623i \(0.252040\pi\)
\(618\) 0 0
\(619\) 1.26107i 0.0506866i −0.999679 0.0253433i \(-0.991932\pi\)
0.999679 0.0253433i \(-0.00806789\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 1.97490 0.0791861
\(623\) 13.0228 21.8316i 0.521746 0.874666i
\(624\) 0 0
\(625\) 0 0
\(626\) 0.961388 0.0384248
\(627\) 0 0
\(628\) −13.1228 −0.523658
\(629\) −4.61480 −0.184004
\(630\) 0 0
\(631\) 16.2145 0.645489 0.322744 0.946486i \(-0.395395\pi\)
0.322744 + 0.946486i \(0.395395\pi\)
\(632\) 12.6768 0.504254
\(633\) 0 0
\(634\) −17.1620 −0.681592
\(635\) 0 0
\(636\) 0 0
\(637\) 21.7656 + 40.3100i 0.862384 + 1.59714i
\(638\) −10.3921 −0.411428
\(639\) 0 0
\(640\) 0 0
\(641\) 19.6893i 0.777681i 0.921305 + 0.388840i \(0.127124\pi\)
−0.921305 + 0.388840i \(0.872876\pi\)
\(642\) 0 0
\(643\) −17.1508 −0.676361 −0.338180 0.941081i \(-0.609811\pi\)
−0.338180 + 0.941081i \(0.609811\pi\)
\(644\) −5.19606 + 8.71078i −0.204754 + 0.343253i
\(645\) 0 0
\(646\) −3.52598 −0.138728
\(647\) 29.0578i 1.14238i −0.820817 0.571191i \(-0.806482\pi\)
0.820817 0.571191i \(-0.193518\pi\)
\(648\) 0 0
\(649\) 30.3501i 1.19135i
\(650\) 0 0
\(651\) 0 0
\(652\) 16.6207i 0.650916i
\(653\) −9.11183 −0.356574 −0.178287 0.983979i \(-0.557055\pi\)
−0.178287 + 0.983979i \(0.557055\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −6.54441 −0.255516
\(657\) 0 0
\(658\) 20.7220 + 12.3609i 0.807829 + 0.481878i
\(659\) 37.5539i 1.46289i −0.681899 0.731446i \(-0.738846\pi\)
0.681899 0.731446i \(-0.261154\pi\)
\(660\) 0 0
\(661\) 9.50275i 0.369614i 0.982775 + 0.184807i \(0.0591660\pi\)
−0.982775 + 0.184807i \(0.940834\pi\)
\(662\) 9.74047 0.378574
\(663\) 0 0
\(664\) 13.0888i 0.507945i
\(665\) 0 0
\(666\) 0 0
\(667\) 14.6967i 0.569058i
\(668\) 8.62068i 0.333544i
\(669\) 0 0
\(670\) 0 0
\(671\) −12.9767 −0.500958
\(672\) 0 0
\(673\) 49.6335i 1.91323i 0.291355 + 0.956615i \(0.405894\pi\)
−0.291355 + 0.956615i \(0.594106\pi\)
\(674\) 3.15078i 0.121363i
\(675\) 0 0
\(676\) 29.8293 1.14728
\(677\) 37.5312i 1.44244i −0.692706 0.721220i \(-0.743582\pi\)
0.692706 0.721220i \(-0.256418\pi\)
\(678\) 0 0
\(679\) −22.9365 + 38.4513i −0.880224 + 1.47562i
\(680\) 0 0
\(681\) 0 0
\(682\) 8.82412 0.337893
\(683\) −9.01560 −0.344972 −0.172486 0.985012i \(-0.555180\pi\)
−0.172486 + 0.985012i \(0.555180\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −18.5033 0.791511i −0.706461 0.0302200i
\(687\) 0 0
\(688\) 0.468142i 0.0178478i
\(689\) 60.5698 2.30752
\(690\) 0 0
\(691\) 8.15841i 0.310361i −0.987886 0.155180i \(-0.950404\pi\)
0.987886 0.155180i \(-0.0495958\pi\)
\(692\) 10.6517i 0.404915i
\(693\) 0 0
\(694\) 30.0594 1.14104
\(695\) 0 0
\(696\) 0 0
\(697\) 10.0251i 0.379728i
\(698\) 27.5180i 1.04157i
\(699\) 0 0
\(700\) 0 0
\(701\) 34.3440i 1.29716i −0.761148 0.648578i \(-0.775364\pi\)
0.761148 0.648578i \(-0.224636\pi\)
\(702\) 0 0
\(703\) −6.93420 −0.261528
\(704\) 2.71078i 0.102166i
\(705\) 0 0
\(706\) 16.5956i 0.624583i
\(707\) 24.0453 40.3100i 0.904316 1.51601i
\(708\) 0 0
\(709\) 5.06372 0.190172 0.0950859 0.995469i \(-0.469687\pi\)
0.0950859 + 0.995469i \(0.469687\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 9.60812 0.360080
\(713\) 12.4792i 0.467349i
\(714\) 0 0
\(715\) 0 0
\(716\) 23.1961i 0.866878i
\(717\) 0 0
\(718\) 15.0076i 0.560080i
\(719\) −18.1274 −0.676039 −0.338020 0.941139i \(-0.609757\pi\)
−0.338020 + 0.941139i \(0.609757\pi\)
\(720\) 0 0
\(721\) −5.50171 + 9.22317i −0.204894 + 0.343489i
\(722\) 13.7019 0.509930
\(723\) 0 0
\(724\) 9.11980i 0.338935i
\(725\) 0 0
\(726\) 0 0
\(727\) −11.1168 −0.412298 −0.206149 0.978521i \(-0.566093\pi\)
−0.206149 + 0.978521i \(0.566093\pi\)
\(728\) −8.87024 + 14.8702i −0.328753 + 0.551128i
\(729\) 0 0
\(730\) 0 0
\(731\) −0.717127 −0.0265239
\(732\) 0 0
\(733\) −20.8342 −0.769529 −0.384765 0.923015i \(-0.625717\pi\)
−0.384765 + 0.923015i \(0.625717\pi\)
\(734\) 14.2598 0.526338
\(735\) 0 0
\(736\) −3.83363 −0.141309
\(737\) −36.7500 −1.35370
\(738\) 0 0
\(739\) −30.9818 −1.13968 −0.569842 0.821754i \(-0.692996\pi\)
−0.569842 + 0.821754i \(0.692996\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −12.5444 + 21.0297i −0.460520 + 0.772024i
\(743\) 45.8449 1.68189 0.840943 0.541124i \(-0.182001\pi\)
0.840943 + 0.541124i \(0.182001\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 8.98745i 0.329054i
\(747\) 0 0
\(748\) −4.15253 −0.151832
\(749\) −8.56459 + 14.3579i −0.312943 + 0.524624i
\(750\) 0 0
\(751\) −35.2665 −1.28689 −0.643446 0.765492i \(-0.722496\pi\)
−0.643446 + 0.765492i \(0.722496\pi\)
\(752\) 9.11980i 0.332565i
\(753\) 0 0
\(754\) 25.0888i 0.913681i
\(755\) 0 0
\(756\) 0 0
\(757\) 12.3661i 0.449452i 0.974422 + 0.224726i \(0.0721488\pi\)
−0.974422 + 0.224726i \(0.927851\pi\)
\(758\) 34.5447 1.25472
\(759\) 0 0
\(760\) 0 0
\(761\) −1.50580 −0.0545851 −0.0272926 0.999627i \(-0.508689\pi\)
−0.0272926 + 0.999627i \(0.508689\pi\)
\(762\) 0 0
\(763\) 16.2987 27.3235i 0.590053 0.989177i
\(764\) 9.69823i 0.350870i
\(765\) 0 0
\(766\) 2.88020i 0.104066i
\(767\) 73.2716 2.64569
\(768\) 0 0
\(769\) 3.76558i 0.135790i 0.997692 + 0.0678951i \(0.0216283\pi\)
−0.997692 + 0.0678951i \(0.978372\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 18.1525i 0.653324i
\(773\) 15.1911i 0.546388i −0.961959 0.273194i \(-0.911920\pi\)
0.961959 0.273194i \(-0.0880800\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −16.9224 −0.607480
\(777\) 0 0
\(778\) 35.1620i 1.26062i
\(779\) 15.0637i 0.539714i
\(780\) 0 0
\(781\) −6.23960 −0.223270
\(782\) 5.87257i 0.210003i
\(783\) 0 0
\(784\) −3.32583 6.15945i −0.118780 0.219980i
\(785\) 0 0
\(786\) 0 0
\(787\) 23.8198 0.849084 0.424542 0.905408i \(-0.360435\pi\)
0.424542 + 0.905408i \(0.360435\pi\)
\(788\) −4.60354 −0.163994
\(789\) 0 0
\(790\) 0 0
\(791\) 12.2849 20.5946i 0.436800 0.732260i
\(792\) 0 0
\(793\) 31.3284i 1.11250i
\(794\) 16.1185 0.572024
\(795\) 0 0
\(796\) 11.7405i 0.416130i
\(797\) 10.6517i 0.377301i −0.982044 0.188650i \(-0.939589\pi\)
0.982044 0.188650i \(-0.0604113\pi\)
\(798\) 0 0
\(799\) 13.9702 0.494231
\(800\) 0 0
\(801\) 0 0
\(802\) 16.6256i 0.587070i
\(803\) 30.9614i 1.09260i
\(804\) 0