Properties

Label 3150.2.b.f.251.1
Level 3150
Weight 2
Character 3150.251
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.b (of order \(2\), degree \(1\), 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: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.1
Root \(-1.91681i\)
Character \(\chi\) = 3150.251
Dual form 3150.2.b.f.251.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +(-2.27220 + 1.35539i) q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +(-2.27220 + 1.35539i) q^{7} +1.00000i q^{8} -2.71078i q^{11} +6.54441i q^{13} +(1.35539 + 2.27220i) q^{14} +1.00000 q^{16} +1.53186 q^{17} -2.30177i q^{19} -2.71078 q^{22} -3.83363i q^{23} +6.54441 q^{26} +(2.27220 - 1.35539i) q^{28} +3.83363i q^{29} -3.25519i q^{31} -1.00000i q^{32} -1.53186i q^{34} -3.01255 q^{37} -2.30177 q^{38} +6.54441 q^{41} +0.468142 q^{43} +2.71078i q^{44} -3.83363 q^{46} -9.11980 q^{47} +(3.32583 - 6.15945i) q^{49} -6.54441i q^{52} -9.25519i q^{53} +(-1.35539 - 2.27220i) q^{56} +3.83363 q^{58} -11.1961 q^{59} +4.78705i q^{61} -3.25519 q^{62} -1.00000 q^{64} -13.5570 q^{67} -1.53186 q^{68} -2.30177i q^{71} -11.4216i q^{73} +3.01255i q^{74} +2.30177i q^{76} +(3.67417 + 6.15945i) q^{77} -12.6768 q^{79} -6.54441i q^{82} +13.0888 q^{83} -0.468142i q^{86} +2.71078 q^{88} +9.60812 q^{89} +(-8.87024 - 14.8702i) q^{91} +3.83363i q^{92} +9.11980i q^{94} -16.9224i q^{97} +(-6.15945 - 3.32583i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + 4q^{7} + O(q^{10}) \) \( 8q - 8q^{4} + 4q^{7} + 8q^{16} + 8q^{26} - 4q^{28} + 8q^{37} - 8q^{38} + 8q^{41} + 16q^{43} - 8q^{46} - 40q^{47} + 4q^{49} + 8q^{58} + 40q^{62} - 8q^{64} - 32q^{67} + 52q^{77} + 8q^{79} + 16q^{83} + 8q^{89} - 4q^{91} - 4q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −2.27220 + 1.35539i −0.858813 + 0.512290i
\(8\) 1.00000i 0.353553i
\(9\) 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
\(13\) 6.54441i 1.81509i 0.419952 + 0.907546i \(0.362047\pi\)
−0.419952 + 0.907546i \(0.637953\pi\)
\(14\) 1.35539 + 2.27220i 0.362244 + 0.607272i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.53186 0.371530 0.185765 0.982594i \(-0.440524\pi\)
0.185765 + 0.982594i \(0.440524\pi\)
\(18\) 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
\(22\) −2.71078 −0.577941
\(23\) 3.83363i 0.799366i −0.916653 0.399683i \(-0.869120\pi\)
0.916653 0.399683i \(-0.130880\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6.54441 1.28346
\(27\) 0 0
\(28\) 2.27220 1.35539i 0.429406 0.256145i
\(29\) 3.83363i 0.711887i 0.934508 + 0.355943i \(0.115840\pi\)
−0.934508 + 0.355943i \(0.884160\pi\)
\(30\) 0 0
\(31\) 3.25519i 0.584650i −0.956319 0.292325i \(-0.905571\pi\)
0.956319 0.292325i \(-0.0944289\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 1.53186i 0.262711i
\(35\) 0 0
\(36\) 0 0
\(37\) −3.01255 −0.495260 −0.247630 0.968855i \(-0.579652\pi\)
−0.247630 + 0.968855i \(0.579652\pi\)
\(38\) −2.30177 −0.373396
\(39\) 0 0
\(40\) 0 0
\(41\) 6.54441 1.02207 0.511033 0.859561i \(-0.329263\pi\)
0.511033 + 0.859561i \(0.329263\pi\)
\(42\) 0 0
\(43\) 0.468142 0.0713910 0.0356955 0.999363i \(-0.488635\pi\)
0.0356955 + 0.999363i \(0.488635\pi\)
\(44\) 2.71078i 0.408666i
\(45\) 0 0
\(46\) −3.83363 −0.565237
\(47\) −9.11980 −1.33026 −0.665130 0.746728i \(-0.731624\pi\)
−0.665130 + 0.746728i \(0.731624\pi\)
\(48\) 0 0
\(49\) 3.32583 6.15945i 0.475118 0.879922i
\(50\) 0 0
\(51\) 0 0
\(52\) 6.54441i 0.907546i
\(53\) 9.25519i 1.27130i −0.771978 0.635649i \(-0.780732\pi\)
0.771978 0.635649i \(-0.219268\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.35539 2.27220i −0.181122 0.303636i
\(57\) 0 0
\(58\) 3.83363 0.503380
\(59\) −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 \(0.0991444\pi\)
−0.951884 + 0.306459i \(0.900856\pi\)
\(62\) −3.25519 −0.413410
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −13.5570 −1.65625 −0.828123 0.560546i \(-0.810591\pi\)
−0.828123 + 0.560546i \(0.810591\pi\)
\(68\) −1.53186 −0.185765
\(69\) 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
\(73\) 11.4216i 1.33679i −0.743805 0.668397i \(-0.766981\pi\)
0.743805 0.668397i \(-0.233019\pi\)
\(74\) 3.01255i 0.350202i
\(75\) 0 0
\(76\) 2.30177i 0.264031i
\(77\) 3.67417 + 6.15945i 0.418711 + 0.701935i
\(78\) 0 0
\(79\) −12.6768 −1.42625 −0.713123 0.701039i \(-0.752720\pi\)
−0.713123 + 0.701039i \(0.752720\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.54441i 0.722709i
\(83\) 13.0888 1.43668 0.718342 0.695690i \(-0.244901\pi\)
0.718342 + 0.695690i \(0.244901\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.468142i 0.0504811i
\(87\) 0 0
\(88\) 2.71078 0.288970
\(89\) 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
\(92\) 3.83363i 0.399683i
\(93\) 0 0
\(94\) 9.11980i 0.940635i
\(95\) 0 0
\(96\) 0 0
\(97\) 16.9224i 1.71821i −0.511796 0.859107i \(-0.671020\pi\)
0.511796 0.859107i \(-0.328980\pi\)
\(98\) −6.15945 3.32583i −0.622199 0.335959i
\(99\) 0 0
\(100\) 0 0
\(101\) −17.7405 −1.76524 −0.882622 0.470084i \(-0.844224\pi\)
−0.882622 + 0.470084i \(0.844224\pi\)
\(102\) 0 0
\(103\) 4.05913i 0.399958i 0.979800 + 0.199979i \(0.0640874\pi\)
−0.979800 + 0.199979i \(0.935913\pi\)
\(104\) −6.54441 −0.641732
\(105\) 0 0
\(106\) −9.25519 −0.898944
\(107\) 6.31891i 0.610872i 0.952213 + 0.305436i \(0.0988022\pi\)
−0.952213 + 0.305436i \(0.901198\pi\)
\(108\) 0 0
\(109\) −12.0251 −1.15180 −0.575898 0.817522i \(-0.695347\pi\)
−0.575898 + 0.817522i \(0.695347\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.27220 + 1.35539i −0.214703 + 0.128072i
\(113\) 9.06372i 0.852643i 0.904572 + 0.426321i \(0.140191\pi\)
−0.904572 + 0.426321i \(0.859809\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.83363i 0.355943i
\(117\) 0 0
\(118\) 11.1961i 1.03068i
\(119\) −3.48069 + 2.07627i −0.319075 + 0.190331i
\(120\) 0 0
\(121\) 3.65166 0.331969
\(122\) 4.78705 0.433399
\(123\) 0 0
\(124\) 3.25519i 0.292325i
\(125\) 0 0
\(126\) 0 0
\(127\) −21.6332 −1.91964 −0.959819 0.280619i \(-0.909460\pi\)
−0.959819 + 0.280619i \(0.909460\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 2.51931 0.220113 0.110056 0.993925i \(-0.464897\pi\)
0.110056 + 0.993925i \(0.464897\pi\)
\(132\) 0 0
\(133\) 3.11980 + 5.23009i 0.270521 + 0.453506i
\(134\) 13.5570i 1.17114i
\(135\) 0 0
\(136\) 1.53186i 0.131356i
\(137\) 1.39646i 0.119308i −0.998219 0.0596539i \(-0.981000\pi\)
0.998219 0.0596539i \(-0.0189997\pi\)
\(138\) 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
\(142\) −2.30177 −0.193160
\(143\) 17.7405 1.48353
\(144\) 0 0
\(145\) 0 0
\(146\) −11.4216 −0.945256
\(147\) 0 0
\(148\) 3.01255 0.247630
\(149\) 4.65166i 0.381078i 0.981680 + 0.190539i \(0.0610236\pi\)
−0.981680 + 0.190539i \(0.938976\pi\)
\(150\) 0 0
\(151\) −3.58794 −0.291982 −0.145991 0.989286i \(-0.546637\pi\)
−0.145991 + 0.989286i \(0.546637\pi\)
\(152\) 2.30177 0.186698
\(153\) 0 0
\(154\) 6.15945 3.67417i 0.496343 0.296073i
\(155\) 0 0
\(156\) 0 0
\(157\) 13.1228i 1.04732i −0.851928 0.523658i \(-0.824567\pi\)
0.851928 0.523658i \(-0.175433\pi\)
\(158\) 12.6768i 1.00851i
\(159\) 0 0
\(160\) 0 0
\(161\) 5.19606 + 8.71078i 0.409507 + 0.686506i
\(162\) 0 0
\(163\) −16.6207 −1.30183 −0.650916 0.759150i \(-0.725615\pi\)
−0.650916 + 0.759150i \(0.725615\pi\)
\(164\) −6.54441 −0.511033
\(165\) 0 0
\(166\) 13.0888i 1.01589i
\(167\) 8.62068 0.667088 0.333544 0.942735i \(-0.391755\pi\)
0.333544 + 0.942735i \(0.391755\pi\)
\(168\) 0 0
\(169\) −29.8293 −2.29456
\(170\) 0 0
\(171\) 0 0
\(172\) −0.468142 −0.0356955
\(173\) −10.6517 −0.809830 −0.404915 0.914354i \(-0.632699\pi\)
−0.404915 + 0.914354i \(0.632699\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.71078i 0.204333i
\(177\) 0 0
\(178\) 9.60812i 0.720159i
\(179\) 23.1961i 1.73376i −0.498521 0.866878i \(-0.666123\pi\)
0.498521 0.866878i \(-0.333877\pi\)
\(180\) 0 0
\(181\) 9.11980i 0.677869i 0.940810 + 0.338935i \(0.110067\pi\)
−0.940810 + 0.338935i \(0.889933\pi\)
\(182\) −14.8702 + 8.87024i −1.10226 + 0.657506i
\(183\) 0 0
\(184\) 3.83363 0.282619
\(185\) 0 0
\(186\) 0 0
\(187\) 4.15253i 0.303663i
\(188\) 9.11980 0.665130
\(189\) 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
\(193\) 18.1525 1.30665 0.653324 0.757078i \(-0.273374\pi\)
0.653324 + 0.757078i \(0.273374\pi\)
\(194\) −16.9224 −1.21496
\(195\) 0 0
\(196\) −3.32583 + 6.15945i −0.237559 + 0.439961i
\(197\) 4.60354i 0.327988i 0.986461 + 0.163994i \(0.0524378\pi\)
−0.986461 + 0.163994i \(0.947562\pi\)
\(198\) 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
\(202\) 17.7405i 1.24822i
\(203\) −5.19606 8.71078i −0.364692 0.611377i
\(204\) 0 0
\(205\) 0 0
\(206\) 4.05913 0.282813
\(207\) 0 0
\(208\) 6.54441i 0.453773i
\(209\) −6.23960 −0.431602
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 9.25519i 0.635649i
\(213\) 0 0
\(214\) 6.31891 0.431952
\(215\) 0 0
\(216\) 0 0
\(217\) 4.41206 + 7.39646i 0.299510 + 0.502105i
\(218\) 12.0251i 0.814443i
\(219\) 0 0
\(220\) 0 0
\(221\) 10.0251i 0.674361i
\(222\) 0 0
\(223\) 26.4513i 1.77131i −0.464347 0.885654i \(-0.653711\pi\)
0.464347 0.885654i \(-0.346289\pi\)
\(224\) 1.35539 + 2.27220i 0.0905609 + 0.151818i
\(225\) 0 0
\(226\) 9.06372 0.602909
\(227\) 15.0637 0.999814 0.499907 0.866079i \(-0.333368\pi\)
0.499907 + 0.866079i \(0.333368\pi\)
\(228\) 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
\(232\) −3.83363 −0.251690
\(233\) 20.9957i 1.37547i −0.725961 0.687736i \(-0.758605\pi\)
0.725961 0.687736i \(-0.241395\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 11.1961 0.728802
\(237\) 0 0
\(238\) 2.07627 + 3.48069i 0.134584 + 0.225620i
\(239\) 9.69823i 0.627326i 0.949534 + 0.313663i \(0.101556\pi\)
−0.949534 + 0.313663i \(0.898444\pi\)
\(240\) 0 0
\(241\) 24.7019i 1.59119i 0.605831 + 0.795593i \(0.292841\pi\)
−0.605831 + 0.795593i \(0.707159\pi\)
\(242\) 3.65166i 0.234737i
\(243\) 0 0
\(244\) 4.78705i 0.306459i
\(245\) 0 0
\(246\) 0 0
\(247\) 15.0637 0.958481
\(248\) 3.25519 0.206705
\(249\) 0 0
\(250\) 0 0
\(251\) −3.60812 −0.227743 −0.113871 0.993495i \(-0.536325\pi\)
−0.113871 + 0.993495i \(0.536325\pi\)
\(252\) 0 0
\(253\) −10.3921 −0.653348
\(254\) 21.6332i 1.35739i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −28.7983 −1.79639 −0.898195 0.439598i \(-0.855121\pi\)
−0.898195 + 0.439598i \(0.855121\pi\)
\(258\) 0 0
\(259\) 6.84513 4.08319i 0.425336 0.253717i
\(260\) 0 0
\(261\) 0 0
\(262\) 2.51931i 0.155643i
\(263\) 12.7699i 0.787426i −0.919233 0.393713i \(-0.871190\pi\)
0.919233 0.393713i \(-0.128810\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 5.23009 3.11980i 0.320677 0.191287i
\(267\) 0 0
\(268\) 13.5570 0.828123
\(269\) 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 \(0.0209596\pi\)
−0.997833 + 0.0657989i \(0.979040\pi\)
\(272\) 1.53186 0.0928825
\(273\) 0 0
\(274\) −1.39646 −0.0843634
\(275\) 0 0
\(276\) 0 0
\(277\) −5.92373 −0.355923 −0.177961 0.984037i \(-0.556950\pi\)
−0.177961 + 0.984037i \(0.556950\pi\)
\(278\) −6.13539 −0.367977
\(279\) 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
\(283\) 18.0594i 1.07352i 0.843735 + 0.536759i \(0.180352\pi\)
−0.843735 + 0.536759i \(0.819648\pi\)
\(284\) 2.30177i 0.136585i
\(285\) 0 0
\(286\) 17.7405i 1.04902i
\(287\) −14.8702 + 8.87024i −0.877762 + 0.523594i
\(288\) 0 0
\(289\) −14.6534 −0.861965
\(290\) 0 0
\(291\) 0 0
\(292\) 11.4216i 0.668397i
\(293\) −29.1139 −1.70085 −0.850427 0.526094i \(-0.823656\pi\)
−0.850427 + 0.526094i \(0.823656\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.01255i 0.175101i
\(297\) 0 0
\(298\) 4.65166 0.269463
\(299\) 25.0888 1.45092
\(300\) 0 0
\(301\) −1.06372 + 0.634516i −0.0613115 + 0.0365729i
\(302\) 3.58794i 0.206463i
\(303\) 0 0
\(304\) 2.30177i 0.132015i
\(305\) 0 0
\(306\) 0 0
\(307\) 7.39646i 0.422138i −0.977471 0.211069i \(-0.932305\pi\)
0.977471 0.211069i \(-0.0676945\pi\)
\(308\) −3.67417 6.15945i −0.209355 0.350967i
\(309\) 0 0
\(310\) 0 0
\(311\) −1.97490 −0.111986 −0.0559931 0.998431i \(-0.517832\pi\)
−0.0559931 + 0.998431i \(0.517832\pi\)
\(312\) 0 0
\(313\) 0.961388i 0.0543409i −0.999631 0.0271704i \(-0.991350\pi\)
0.999631 0.0271704i \(-0.00864968\pi\)
\(314\) −13.1228 −0.740565
\(315\) 0 0
\(316\) 12.6768 0.713123
\(317\) 17.1620i 0.963916i 0.876194 + 0.481958i \(0.160074\pi\)
−0.876194 + 0.481958i \(0.839926\pi\)
\(318\) 0 0
\(319\) 10.3921 0.581848
\(320\) 0 0
\(321\) 0 0
\(322\) 8.71078 5.19606i 0.485433 0.289565i
\(323\) 3.52598i 0.196191i
\(324\) 0 0
\(325\) 0 0
\(326\) 16.6207i 0.920534i
\(327\) 0 0
\(328\) 6.54441i 0.361355i
\(329\) 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 \(0.413739\pi\)
\(332\) −13.0888 −0.718342
\(333\) 0 0
\(334\) 8.62068i 0.471702i
\(335\) 0 0
\(336\) 0 0
\(337\) −3.15078 −0.171634 −0.0858169 0.996311i \(-0.527350\pi\)
−0.0858169 + 0.996311i \(0.527350\pi\)
\(338\) 29.8293i 1.62250i
\(339\) 0 0
\(340\) 0 0
\(341\) −8.82412 −0.477853
\(342\) 0 0
\(343\) 0.791511 + 18.5033i 0.0427376 + 0.999086i
\(344\) 0.468142i 0.0252405i
\(345\) 0 0
\(346\) 10.6517i 0.572637i
\(347\) 30.0594i 1.61367i −0.590775 0.806836i \(-0.701178\pi\)
0.590775 0.806836i \(-0.298822\pi\)
\(348\) 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
\(352\) −2.71078 −0.144485
\(353\) 16.5956 0.883293 0.441647 0.897189i \(-0.354395\pi\)
0.441647 + 0.897189i \(0.354395\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −9.60812 −0.509230
\(357\) 0 0
\(358\) −23.1961 −1.22595
\(359\) 15.0076i 0.792073i 0.918235 + 0.396036i \(0.129615\pi\)
−0.918235 + 0.396036i \(0.870385\pi\)
\(360\) 0 0
\(361\) 13.7019 0.721151
\(362\) 9.11980 0.479326
\(363\) 0 0
\(364\) 8.87024 + 14.8702i 0.464927 + 0.779412i
\(365\) 0 0
\(366\) 0 0
\(367\) 14.2598i 0.744354i 0.928162 + 0.372177i \(0.121389\pi\)
−0.928162 + 0.372177i \(0.878611\pi\)
\(368\) 3.83363i 0.199842i
\(369\) 0 0
\(370\) 0 0
\(371\) 12.5444 + 21.0297i 0.651273 + 1.09181i
\(372\) 0 0
\(373\) −8.98745 −0.465352 −0.232676 0.972554i \(-0.574748\pi\)
−0.232676 + 0.972554i \(0.574748\pi\)
\(374\) −4.15253 −0.214722
\(375\) 0 0
\(376\) 9.11980i 0.470318i
\(377\) −25.0888 −1.29214
\(378\) 0 0
\(379\) −34.5447 −1.77444 −0.887220 0.461346i \(-0.847367\pi\)
−0.887220 + 0.461346i \(0.847367\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −9.69823 −0.496205
\(383\) 2.88020 0.147171 0.0735857 0.997289i \(-0.476556\pi\)
0.0735857 + 0.997289i \(0.476556\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 18.1525i 0.923940i
\(387\) 0 0
\(388\) 16.9224i 0.859107i
\(389\) 35.1620i 1.78279i −0.453231 0.891393i \(-0.649729\pi\)
0.453231 0.891393i \(-0.350271\pi\)
\(390\) 0 0
\(391\) 5.87257i 0.296989i
\(392\) 6.15945 + 3.32583i 0.311099 + 0.167980i
\(393\) 0 0
\(394\) 4.60354 0.231923
\(395\) 0 0
\(396\) 0 0
\(397\) 16.1185i 0.808965i 0.914546 + 0.404482i \(0.132548\pi\)
−0.914546 + 0.404482i \(0.867452\pi\)
\(398\) −11.7405 −0.588497
\(399\) 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
\(403\) 21.3033 1.06119
\(404\) 17.7405 0.882622
\(405\) 0 0
\(406\) −8.71078 + 5.19606i −0.432309 + 0.257876i
\(407\) 8.16637i 0.404792i
\(408\) 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
\(412\) 4.05913i 0.199979i
\(413\) 25.4397 15.1751i 1.25181 0.746715i
\(414\) 0 0
\(415\) 0 0
\(416\) 6.54441 0.320866
\(417\) 0 0
\(418\) 6.23960i 0.305189i
\(419\) 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 \(0.344657\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 0 0
\(424\) 9.25519 0.449472
\(425\) 0 0
\(426\) 0 0
\(427\) −6.48833 10.8772i −0.313992 0.526383i
\(428\) 6.31891i 0.305436i
\(429\) 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
\(433\) 31.6473i 1.52087i −0.649412 0.760437i \(-0.724985\pi\)
0.649412 0.760437i \(-0.275015\pi\)
\(434\) 7.39646 4.41206i 0.355042 0.211786i
\(435\) 0 0
\(436\) 12.0251 0.575898
\(437\) −8.82412 −0.422115
\(438\) 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
\(442\) 10.0251 0.476846
\(443\) 3.78551i 0.179855i 0.995948 + 0.0899275i \(0.0286635\pi\)
−0.995948 + 0.0899275i \(0.971336\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −26.4513 −1.25250
\(447\) 0 0
\(448\) 2.27220 1.35539i 0.107352 0.0640362i
\(449\) 24.9307i 1.17655i −0.808661 0.588275i \(-0.799807\pi\)
0.808661 0.588275i \(-0.200193\pi\)
\(450\) 0 0
\(451\) 17.7405i 0.835366i
\(452\) 9.06372i 0.426321i
\(453\) 0 0
\(454\) 15.0637i 0.706975i
\(455\) 0 0
\(456\) 0 0
\(457\) 31.3535 1.46666 0.733328 0.679875i \(-0.237966\pi\)
0.733328 + 0.679875i \(0.237966\pi\)
\(458\) −11.3655 −0.531074
\(459\) 0 0
\(460\) 0 0
\(461\) −9.52598 −0.443669 −0.221835 0.975084i \(-0.571205\pi\)
−0.221835 + 0.975084i \(0.571205\pi\)
\(462\) 0 0
\(463\) −34.0003 −1.58013 −0.790063 0.613026i \(-0.789952\pi\)
−0.790063 + 0.613026i \(0.789952\pi\)
\(464\) 3.83363i 0.177972i
\(465\) 0 0
\(466\) −20.9957 −0.972605
\(467\) 39.2665 1.81703 0.908517 0.417847i \(-0.137215\pi\)
0.908517 + 0.417847i \(0.137215\pi\)
\(468\) 0 0
\(469\) 30.8042 18.3750i 1.42241 0.848478i
\(470\) 0 0
\(471\) 0 0
\(472\) 11.1961i 0.515341i
\(473\) 1.26903i 0.0583502i
\(474\) 0 0
\(475\) 0 0
\(476\) 3.48069 2.07627i 0.159537 0.0951655i
\(477\) 0 0
\(478\) 9.69823 0.443587
\(479\) −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
\(482\) 24.7019 1.12514
\(483\) 0 0
\(484\) −3.65166 −0.165984
\(485\) 0 0
\(486\) 0 0
\(487\) 12.8091 0.580436 0.290218 0.956961i \(-0.406272\pi\)
0.290218 + 0.956961i \(0.406272\pi\)
\(488\) −4.78705 −0.216700
\(489\) 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
\(493\) 5.87257i 0.264487i
\(494\) 15.0637i 0.677749i
\(495\) 0 0
\(496\) 3.25519i 0.146162i
\(497\) 3.11980 + 5.23009i 0.139942 + 0.234602i
\(498\) 0 0
\(499\) 35.5511 1.59149 0.795743 0.605635i \(-0.207081\pi\)
0.795743 + 0.605635i \(0.207081\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 3.60812i 0.161038i
\(503\) 8.23372 0.367123 0.183562 0.983008i \(-0.441237\pi\)
0.183562 + 0.983008i \(0.441237\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 10.3921i 0.461986i
\(507\) 0 0
\(508\) 21.6332 0.959819
\(509\) −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
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 28.7983i 1.27024i
\(515\) 0 0
\(516\) 0 0
\(517\) 24.7218i 1.08726i
\(518\) −4.08319 6.84513i −0.179405 0.300758i
\(519\) 0 0
\(520\) 0 0
\(521\) −1.39363 −0.0610561 −0.0305281 0.999534i \(-0.509719\pi\)
−0.0305281 + 0.999534i \(0.509719\pi\)
\(522\) 0 0
\(523\) 20.4853i 0.895759i −0.894094 0.447879i \(-0.852179\pi\)
0.894094 0.447879i \(-0.147821\pi\)
\(524\) −2.51931 −0.110056
\(525\) 0 0
\(526\) −12.7699 −0.556795
\(527\) 4.98649i 0.217215i
\(528\) 0 0
\(529\) 8.30331 0.361013
\(530\) 0 0
\(531\) 0 0
\(532\) −3.11980 5.23009i −0.135260 0.226753i
\(533\) 42.8293i 1.85514i
\(534\) 0 0
\(535\) 0 0
\(536\) 13.5570i 0.585572i
\(537\) 0 0
\(538\) 8.43716i 0.363752i
\(539\) −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 \(0.500596\pi\)
\(542\) 2.16637 0.0930537
\(543\) 0 0
\(544\) 1.53186i 0.0656779i
\(545\) 0 0
\(546\) 0 0
\(547\) 5.40443 0.231077 0.115538 0.993303i \(-0.463141\pi\)
0.115538 + 0.993303i \(0.463141\pi\)
\(548\) 1.39646i 0.0596539i
\(549\) 0 0
\(550\) 0 0
\(551\) 8.82412 0.375920
\(552\) 0 0
\(553\) 28.8042 17.1820i 1.22488 0.730652i
\(554\) 5.92373i 0.251675i
\(555\) 0 0
\(556\) 6.13539i 0.260199i
\(557\) 0.127431i 0.00539940i −0.999996 0.00269970i \(-0.999141\pi\)
0.999996 0.00269970i \(-0.000859343\pi\)
\(558\) 0 0
\(559\) 3.06372i 0.129581i
\(560\) 0 0
\(561\) 0 0
\(562\) 27.3566 1.15397
\(563\) −6.23960 −0.262968 −0.131484 0.991318i \(-0.541974\pi\)
−0.131484 + 0.991318i \(0.541974\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 18.0594 0.759092
\(567\) 0 0
\(568\) 2.30177 0.0965801
\(569\) 11.6391i 0.487937i 0.969783 + 0.243968i \(0.0784493\pi\)
−0.969783 + 0.243968i \(0.921551\pi\)
\(570\) 0 0
\(571\) −35.9181 −1.50313 −0.751563 0.659661i \(-0.770700\pi\)
−0.751563 + 0.659661i \(0.770700\pi\)
\(572\) −17.7405 −0.741766
\(573\) 0 0
\(574\) 8.87024 + 14.8702i 0.370237 + 0.620672i
\(575\) 0 0
\(576\) 0 0
\(577\) 7.15687i 0.297944i 0.988841 + 0.148972i \(0.0475965\pi\)
−0.988841 + 0.148972i \(0.952404\pi\)
\(578\) 14.6534i 0.609502i
\(579\) 0 0
\(580\) 0 0
\(581\) −29.7405 + 17.7405i −1.23384 + 0.735999i
\(582\) 0 0
\(583\) −25.0888 −1.03907
\(584\) 11.4216 0.472628
\(585\) 0 0
\(586\) 29.1139i 1.20269i
\(587\) −4.15253 −0.171393 −0.0856967 0.996321i \(-0.527312\pi\)
−0.0856967 + 0.996321i \(0.527312\pi\)
\(588\) 0 0
\(589\) −7.49270 −0.308731
\(590\) 0 0
\(591\) 0 0
\(592\) −3.01255 −0.123815
\(593\) 29.7966 1.22360 0.611799 0.791013i \(-0.290446\pi\)
0.611799 + 0.791013i \(0.290446\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.65166i 0.190539i
\(597\) 0 0
\(598\) 25.0888i 1.02596i
\(599\) 35.4249i 1.44742i −0.690104 0.723710i \(-0.742435\pi\)
0.690104 0.723710i \(-0.257565\pi\)
\(600\) 0 0
\(601\) 27.4948i 1.12154i 0.827973 + 0.560768i \(0.189494\pi\)
−0.827973 + 0.560768i \(0.810506\pi\)
\(602\) 0.634516 + 1.06372i 0.0258609 + 0.0433538i
\(603\) 0 0
\(604\) 3.58794 0.145991
\(605\) 0 0
\(606\) 0 0
\(607\) 4.76499i 0.193405i 0.995313 + 0.0967025i \(0.0308296\pi\)
−0.995313 + 0.0967025i \(0.969170\pi\)
\(608\) −2.30177 −0.0933490
\(609\) 0 0
\(610\) 0 0
\(611\) 59.6837i 2.41454i
\(612\) 0 0
\(613\) −10.7981 −0.436130 −0.218065 0.975934i \(-0.569974\pi\)
−0.218065 + 0.975934i \(0.569974\pi\)
\(614\) −7.39646 −0.298497
\(615\) 0 0
\(616\) −6.15945 + 3.67417i −0.248171 + 0.148037i
\(617\) 34.9025i 1.40512i −0.711623 0.702561i \(-0.752040\pi\)
0.711623 0.702561i \(-0.247960\pi\)
\(618\) 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
\(622\) 1.97490i 0.0791861i
\(623\) −21.8316 + 13.0228i −0.874666 + 0.521746i
\(624\) 0 0
\(625\) 0 0
\(626\) −0.961388 −0.0384248
\(627\) 0 0
\(628\) 13.1228i 0.523658i
\(629\) −4.61480 −0.184004
\(630\) 0 0
\(631\) 16.2145 0.645489 0.322744 0.946486i \(-0.395395\pi\)
0.322744 + 0.946486i \(0.395395\pi\)
\(632\) 12.6768i 0.504254i
\(633\) 0 0
\(634\) 17.1620 0.681592
\(635\) 0 0
\(636\) 0 0
\(637\) 40.3100 + 21.7656i 1.59714 + 0.862384i
\(638\) 10.3921i 0.411428i
\(639\) 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
\(643\) 17.1508i 0.676361i 0.941081 + 0.338180i \(0.109811\pi\)
−0.941081 + 0.338180i \(0.890189\pi\)
\(644\) −5.19606 8.71078i −0.204754 0.343253i
\(645\) 0 0
\(646\) −3.52598 −0.138728
\(647\) 29.0578 1.14238 0.571191 0.820817i \(-0.306482\pi\)
0.571191 + 0.820817i \(0.306482\pi\)
\(648\) 0 0
\(649\) 30.3501i 1.19135i
\(650\) 0 0
\(651\) 0 0
\(652\) 16.6207 0.650916
\(653\) 9.11183i 0.356574i −0.983979 0.178287i \(-0.942945\pi\)
0.983979 0.178287i \(-0.0570555\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6.54441 0.255516
\(657\) 0 0
\(658\) −12.3609 20.7220i −0.481878 0.807829i
\(659\) 37.5539i 1.46289i 0.681899 + 0.731446i \(0.261154\pi\)
−0.681899 + 0.731446i \(0.738846\pi\)
\(660\) 0 0
\(661\) 9.50275i 0.369614i −0.982775 0.184807i \(-0.940834\pi\)
0.982775 0.184807i \(-0.0591660\pi\)
\(662\) 9.74047i 0.378574i
\(663\) 0 0
\(664\) 13.0888i 0.507945i
\(665\) 0 0
\(666\) 0 0
\(667\) 14.6967 0.569058
\(668\) −8.62068 −0.333544
\(669\) 0 0
\(670\) 0 0
\(671\) 12.9767 0.500958
\(672\) 0 0
\(673\) −49.6335 −1.91323 −0.956615 0.291355i \(-0.905894\pi\)
−0.956615 + 0.291355i \(0.905894\pi\)
\(674\) 3.15078i 0.121363i
\(675\) 0 0
\(676\) 29.8293 1.14728
\(677\) 37.5312 1.44244 0.721220 0.692706i \(-0.243582\pi\)
0.721220 + 0.692706i \(0.243582\pi\)
\(678\) 0 0
\(679\) 22.9365 + 38.4513i 0.880224 + 1.47562i
\(680\) 0 0
\(681\) 0 0
\(682\) 8.82412i 0.337893i
\(683\) 9.01560i 0.344972i −0.985012 0.172486i \(-0.944820\pi\)
0.985012 0.172486i \(-0.0551800\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 18.5033 0.791511i 0.706461 0.0302200i
\(687\) 0 0
\(688\) 0.468142 0.0178478
\(689\) 60.5698 2.30752
\(690\) 0 0
\(691\) 8.15841i 0.310361i 0.987886 + 0.155180i \(0.0495958\pi\)
−0.987886 + 0.155180i \(0.950404\pi\)
\(692\) 10.6517 0.404915
\(693\) 0 0
\(694\) −30.0594 −1.14104
\(695\) 0 0
\(696\) 0 0
\(697\) 10.0251 0.379728
\(698\) 27.5180 1.04157
\(699\) 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
\(703\) 6.93420i 0.261528i
\(704\) 2.71078i 0.102166i
\(705\) 0 0
\(706\) 16.5956i 0.624583i
\(707\) 40.3100 24.0453i 1.51601 0.904316i
\(708\) 0 0
\(709\) −5.06372 −0.190172 −0.0950859 0.995469i \(-0.530313\pi\)
−0.0950859 + 0.995469i \(0.530313\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 9.60812i 0.360080i
\(713\) −12.4792 −0.467349
\(714\) 0 0
\(715\) 0 0
\(716\) 23.1961i 0.866878i
\(717\) 0 0
\(718\) 15.0076 0.560080
\(719\) −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
\(722\) 13.7019i 0.509930i
\(723\) 0 0
\(724\) 9.11980i 0.338935i
\(725\) 0 0
\(726\) 0 0
\(727\) 11.1168i 0.412298i −0.978521 0.206149i \(-0.933907\pi\)
0.978521 0.206149i \(-0.0660931\pi\)
\(728\) 14.8702 8.87024i 0.551128 0.328753i
\(729\) 0 0
\(730\) 0 0
\(731\) 0.717127 0.0265239
\(732\) 0 0
\(733\) 20.8342i 0.769529i 0.923015 + 0.384765i \(0.125717\pi\)
−0.923015 + 0.384765i \(0.874283\pi\)
\(734\) 14.2598 0.526338
\(735\) 0 0
\(736\) −3.83363 −0.141309
\(737\) 36.7500i 1.35370i
\(738\) 0 0
\(739\) 30.9818 1.13968 0.569842 0.821754i \(-0.307004\pi\)
0.569842 + 0.821754i \(0.307004\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 21.0297 12.5444i 0.772024 0.460520i
\(743\) 45.8449i 1.68189i 0.541124 + 0.840943i \(0.317999\pi\)
−0.541124 + 0.840943i \(0.682001\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 8.98745i 0.329054i
\(747\) 0 0
\(748\) 4.15253i 0.151832i
\(749\) −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 \(0.722496\pi\)
\(752\) −9.11980 −0.332565
\(753\) 0 0
\(754\) 25.0888i 0.913681i
\(755\) 0 0
\(756\) 0 0
\(757\) 12.3661 0.449452 0.224726 0.974422i \(-0.427851\pi\)
0.224726 + 0.974422i \(0.427851\pi\)
\(758\) 34.5447i 1.25472i
\(759\) 0 0
\(760\) 0 0
\(761\) 1.50580 0.0545851 0.0272926 0.999627i \(-0.491311\pi\)
0.0272926 + 0.999627i \(0.491311\pi\)
\(762\) 0 0
\(763\) 27.3235 16.2987i 0.989177 0.590053i
\(764\) 9.69823i 0.350870i
\(765\) 0 0
\(766\) 2.88020i 0.104066i
\(767\) 73.2716i 2.64569i
\(768\) 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
\(772\) −18.1525 −0.653324
\(773\) −15.1911 −0.546388 −0.273194 0.961959i \(-0.588080\pi\)
−0.273194 + 0.961959i \(0.588080\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 16.9224 0.607480
\(777\) 0 0
\(778\) −35.1620 −1.26062
\(779\) 15.0637i 0.539714i
\(780\) 0 0
\(781\) −6.23960 −0.223270
\(782\) −5.87257 −0.210003
\(783\) 0 0
\(784\) 3.32583 6.15945i 0.118780 0.219980i
\(785\) 0 0
\(786\) 0 0
\(787\) 23.8198i 0.849084i 0.905408 + 0.424542i \(0.139565\pi\)
−0.905408 + 0.424542i \(0.860435\pi\)
\(788\) 4.60354i 0.163994i
\(789\) 0 0
\(790\) 0 0
\(791\) −12.2849 20.5946i −0.436800 0.732260i
\(792\) 0 0
\(793\) −31.3284 −1.11250
\(794\) 16.1185 0.572024
\(795\) 0 0
\(796\) 11.7405i 0.416130i
\(797\) 10.6517 0.377301 0.188650 0.982044i \(-0.439589\pi\)
0.188650 + 0.982044i \(0.439589\pi\)
\(798\) 0 0
\(799\) −13.9702 −0.494231
\(800\) 0 0
\(801\) 0 0
\(802\) 16.6256 0.587070
\(803\) −30.9614 −1.09260