Properties

Label 1148.2.d.a.1065.5
Level $1148$
Weight $2$
Character 1148.1065
Analytic conductor $9.167$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 39 x^{18} + 531 x^{16} + 3488 x^{14} + 12661 x^{12} + 27027 x^{10} + 34540 x^{8} + 25909 x^{6} + 10677 x^{4} + 2092 x^{2} + 144\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1065.5
Root \(1.20144i\) of defining polynomial
Character \(\chi\) \(=\) 1148.1065
Dual form 1148.2.d.a.1065.16

$q$-expansion

\(f(q)\) \(=\) \(q-2.23543i q^{3} +0.468853 q^{5} +1.00000i q^{7} -1.99714 q^{9} +O(q^{10})\) \(q-2.23543i q^{3} +0.468853 q^{5} +1.00000i q^{7} -1.99714 q^{9} +4.98364i q^{11} +3.78611i q^{13} -1.04809i q^{15} +1.22201i q^{17} +7.89875i q^{19} +2.23543 q^{21} -0.249482 q^{23} -4.78018 q^{25} -2.24183i q^{27} +8.31531i q^{29} +3.17006 q^{31} +11.1406 q^{33} +0.468853i q^{35} -7.74950 q^{37} +8.46357 q^{39} +(-3.82365 - 5.13612i) q^{41} +4.90962 q^{43} -0.936364 q^{45} +1.80770i q^{47} -1.00000 q^{49} +2.73172 q^{51} -1.74173i q^{53} +2.33660i q^{55} +17.6571 q^{57} +2.25456 q^{59} -1.09904 q^{61} -1.99714i q^{63} +1.77513i q^{65} -11.8286i q^{67} +0.557699i q^{69} -12.3547i q^{71} -2.91035 q^{73} +10.6857i q^{75} -4.98364 q^{77} -3.79118i q^{79} -11.0029 q^{81} +8.75742 q^{83} +0.572943i q^{85} +18.5883 q^{87} +15.9289i q^{89} -3.78611 q^{91} -7.08645i q^{93} +3.70335i q^{95} +4.29519i q^{97} -9.95302i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{5} - 20q^{9} + O(q^{10}) \) \( 20q + 4q^{5} - 20q^{9} + 4q^{21} + 8q^{31} + 20q^{37} + 4q^{39} - 16q^{41} + 20q^{43} - 4q^{45} - 20q^{49} + 52q^{51} - 36q^{57} + 20q^{59} - 4q^{61} - 12q^{73} + 8q^{77} + 20q^{81} - 48q^{83} + 44q^{87} - 4q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.23543i 1.29062i −0.763919 0.645312i \(-0.776727\pi\)
0.763919 0.645312i \(-0.223273\pi\)
\(4\) 0 0
\(5\) 0.468853 0.209677 0.104839 0.994489i \(-0.466567\pi\)
0.104839 + 0.994489i \(0.466567\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.99714 −0.665713
\(10\) 0 0
\(11\) 4.98364i 1.50263i 0.659946 + 0.751313i \(0.270579\pi\)
−0.659946 + 0.751313i \(0.729421\pi\)
\(12\) 0 0
\(13\) 3.78611i 1.05008i 0.851078 + 0.525039i \(0.175949\pi\)
−0.851078 + 0.525039i \(0.824051\pi\)
\(14\) 0 0
\(15\) 1.04809i 0.270615i
\(16\) 0 0
\(17\) 1.22201i 0.296381i 0.988959 + 0.148191i \(0.0473449\pi\)
−0.988959 + 0.148191i \(0.952655\pi\)
\(18\) 0 0
\(19\) 7.89875i 1.81210i 0.423173 + 0.906049i \(0.360916\pi\)
−0.423173 + 0.906049i \(0.639084\pi\)
\(20\) 0 0
\(21\) 2.23543 0.487810
\(22\) 0 0
\(23\) −0.249482 −0.0520206 −0.0260103 0.999662i \(-0.508280\pi\)
−0.0260103 + 0.999662i \(0.508280\pi\)
\(24\) 0 0
\(25\) −4.78018 −0.956035
\(26\) 0 0
\(27\) 2.24183i 0.431440i
\(28\) 0 0
\(29\) 8.31531i 1.54411i 0.635553 + 0.772057i \(0.280772\pi\)
−0.635553 + 0.772057i \(0.719228\pi\)
\(30\) 0 0
\(31\) 3.17006 0.569360 0.284680 0.958623i \(-0.408113\pi\)
0.284680 + 0.958623i \(0.408113\pi\)
\(32\) 0 0
\(33\) 11.1406 1.93933
\(34\) 0 0
\(35\) 0.468853i 0.0792506i
\(36\) 0 0
\(37\) −7.74950 −1.27401 −0.637005 0.770860i \(-0.719827\pi\)
−0.637005 + 0.770860i \(0.719827\pi\)
\(38\) 0 0
\(39\) 8.46357 1.35526
\(40\) 0 0
\(41\) −3.82365 5.13612i −0.597153 0.802127i
\(42\) 0 0
\(43\) 4.90962 0.748711 0.374355 0.927285i \(-0.377864\pi\)
0.374355 + 0.927285i \(0.377864\pi\)
\(44\) 0 0
\(45\) −0.936364 −0.139585
\(46\) 0 0
\(47\) 1.80770i 0.263680i 0.991271 + 0.131840i \(0.0420885\pi\)
−0.991271 + 0.131840i \(0.957911\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 2.73172 0.382517
\(52\) 0 0
\(53\) 1.74173i 0.239244i −0.992819 0.119622i \(-0.961832\pi\)
0.992819 0.119622i \(-0.0381683\pi\)
\(54\) 0 0
\(55\) 2.33660i 0.315066i
\(56\) 0 0
\(57\) 17.6571 2.33874
\(58\) 0 0
\(59\) 2.25456 0.293519 0.146760 0.989172i \(-0.453116\pi\)
0.146760 + 0.989172i \(0.453116\pi\)
\(60\) 0 0
\(61\) −1.09904 −0.140718 −0.0703589 0.997522i \(-0.522414\pi\)
−0.0703589 + 0.997522i \(0.522414\pi\)
\(62\) 0 0
\(63\) 1.99714i 0.251616i
\(64\) 0 0
\(65\) 1.77513i 0.220177i
\(66\) 0 0
\(67\) 11.8286i 1.44509i −0.691323 0.722545i \(-0.742972\pi\)
0.691323 0.722545i \(-0.257028\pi\)
\(68\) 0 0
\(69\) 0.557699i 0.0671391i
\(70\) 0 0
\(71\) 12.3547i 1.46624i −0.680100 0.733119i \(-0.738064\pi\)
0.680100 0.733119i \(-0.261936\pi\)
\(72\) 0 0
\(73\) −2.91035 −0.340630 −0.170315 0.985390i \(-0.554479\pi\)
−0.170315 + 0.985390i \(0.554479\pi\)
\(74\) 0 0
\(75\) 10.6857i 1.23388i
\(76\) 0 0
\(77\) −4.98364 −0.567939
\(78\) 0 0
\(79\) 3.79118i 0.426542i −0.976993 0.213271i \(-0.931588\pi\)
0.976993 0.213271i \(-0.0684117\pi\)
\(80\) 0 0
\(81\) −11.0029 −1.22254
\(82\) 0 0
\(83\) 8.75742 0.961252 0.480626 0.876926i \(-0.340410\pi\)
0.480626 + 0.876926i \(0.340410\pi\)
\(84\) 0 0
\(85\) 0.572943i 0.0621444i
\(86\) 0 0
\(87\) 18.5883 1.99287
\(88\) 0 0
\(89\) 15.9289i 1.68846i 0.535982 + 0.844229i \(0.319941\pi\)
−0.535982 + 0.844229i \(0.680059\pi\)
\(90\) 0 0
\(91\) −3.78611 −0.396892
\(92\) 0 0
\(93\) 7.08645i 0.734831i
\(94\) 0 0
\(95\) 3.70335i 0.379956i
\(96\) 0 0
\(97\) 4.29519i 0.436111i 0.975936 + 0.218055i \(0.0699713\pi\)
−0.975936 + 0.218055i \(0.930029\pi\)
\(98\) 0 0
\(99\) 9.95302i 1.00032i
\(100\) 0 0
\(101\) 4.42421i 0.440225i −0.975474 0.220113i \(-0.929358\pi\)
0.975474 0.220113i \(-0.0706425\pi\)
\(102\) 0 0
\(103\) 14.9664 1.47469 0.737343 0.675519i \(-0.236080\pi\)
0.737343 + 0.675519i \(0.236080\pi\)
\(104\) 0 0
\(105\) 1.04809 0.102283
\(106\) 0 0
\(107\) 17.4446 1.68644 0.843218 0.537572i \(-0.180658\pi\)
0.843218 + 0.537572i \(0.180658\pi\)
\(108\) 0 0
\(109\) 5.31886i 0.509455i −0.967013 0.254727i \(-0.918014\pi\)
0.967013 0.254727i \(-0.0819857\pi\)
\(110\) 0 0
\(111\) 17.3234i 1.64427i
\(112\) 0 0
\(113\) 0.843987 0.0793957 0.0396978 0.999212i \(-0.487360\pi\)
0.0396978 + 0.999212i \(0.487360\pi\)
\(114\) 0 0
\(115\) −0.116970 −0.0109075
\(116\) 0 0
\(117\) 7.56138i 0.699050i
\(118\) 0 0
\(119\) −1.22201 −0.112022
\(120\) 0 0
\(121\) −13.8367 −1.25788
\(122\) 0 0
\(123\) −11.4814 + 8.54748i −1.03525 + 0.770701i
\(124\) 0 0
\(125\) −4.58546 −0.410136
\(126\) 0 0
\(127\) −8.18810 −0.726576 −0.363288 0.931677i \(-0.618346\pi\)
−0.363288 + 0.931677i \(0.618346\pi\)
\(128\) 0 0
\(129\) 10.9751i 0.966304i
\(130\) 0 0
\(131\) 5.65452 0.494038 0.247019 0.969011i \(-0.420549\pi\)
0.247019 + 0.969011i \(0.420549\pi\)
\(132\) 0 0
\(133\) −7.89875 −0.684909
\(134\) 0 0
\(135\) 1.05109i 0.0904631i
\(136\) 0 0
\(137\) 3.57141i 0.305126i −0.988294 0.152563i \(-0.951247\pi\)
0.988294 0.152563i \(-0.0487528\pi\)
\(138\) 0 0
\(139\) 20.9095 1.77352 0.886759 0.462232i \(-0.152951\pi\)
0.886759 + 0.462232i \(0.152951\pi\)
\(140\) 0 0
\(141\) 4.04098 0.340312
\(142\) 0 0
\(143\) −18.8686 −1.57787
\(144\) 0 0
\(145\) 3.89866i 0.323766i
\(146\) 0 0
\(147\) 2.23543i 0.184375i
\(148\) 0 0
\(149\) 14.6546i 1.20055i 0.799792 + 0.600277i \(0.204943\pi\)
−0.799792 + 0.600277i \(0.795057\pi\)
\(150\) 0 0
\(151\) 1.70827i 0.139017i −0.997581 0.0695087i \(-0.977857\pi\)
0.997581 0.0695087i \(-0.0221431\pi\)
\(152\) 0 0
\(153\) 2.44052i 0.197305i
\(154\) 0 0
\(155\) 1.48629 0.119382
\(156\) 0 0
\(157\) 14.1603i 1.13011i −0.825052 0.565057i \(-0.808854\pi\)
0.825052 0.565057i \(-0.191146\pi\)
\(158\) 0 0
\(159\) −3.89350 −0.308775
\(160\) 0 0
\(161\) 0.249482i 0.0196619i
\(162\) 0 0
\(163\) −20.0614 −1.57133 −0.785664 0.618654i \(-0.787678\pi\)
−0.785664 + 0.618654i \(0.787678\pi\)
\(164\) 0 0
\(165\) 5.22329 0.406633
\(166\) 0 0
\(167\) 12.5991i 0.974946i 0.873138 + 0.487473i \(0.162081\pi\)
−0.873138 + 0.487473i \(0.837919\pi\)
\(168\) 0 0
\(169\) −1.33461 −0.102662
\(170\) 0 0
\(171\) 15.7749i 1.20634i
\(172\) 0 0
\(173\) 0.134256 0.0102073 0.00510364 0.999987i \(-0.498375\pi\)
0.00510364 + 0.999987i \(0.498375\pi\)
\(174\) 0 0
\(175\) 4.78018i 0.361347i
\(176\) 0 0
\(177\) 5.03992i 0.378823i
\(178\) 0 0
\(179\) 14.9327i 1.11613i −0.829799 0.558063i \(-0.811545\pi\)
0.829799 0.558063i \(-0.188455\pi\)
\(180\) 0 0
\(181\) 22.6490i 1.68349i 0.539878 + 0.841744i \(0.318471\pi\)
−0.539878 + 0.841744i \(0.681529\pi\)
\(182\) 0 0
\(183\) 2.45683i 0.181614i
\(184\) 0 0
\(185\) −3.63337 −0.267131
\(186\) 0 0
\(187\) −6.09007 −0.445350
\(188\) 0 0
\(189\) 2.24183 0.163069
\(190\) 0 0
\(191\) 19.7885i 1.43185i 0.698178 + 0.715924i \(0.253994\pi\)
−0.698178 + 0.715924i \(0.746006\pi\)
\(192\) 0 0
\(193\) 23.4346i 1.68686i −0.537237 0.843431i \(-0.680532\pi\)
0.537237 0.843431i \(-0.319468\pi\)
\(194\) 0 0
\(195\) 3.96817 0.284166
\(196\) 0 0
\(197\) 15.6808 1.11721 0.558606 0.829433i \(-0.311336\pi\)
0.558606 + 0.829433i \(0.311336\pi\)
\(198\) 0 0
\(199\) 18.6879i 1.32475i 0.749171 + 0.662376i \(0.230452\pi\)
−0.749171 + 0.662376i \(0.769548\pi\)
\(200\) 0 0
\(201\) −26.4419 −1.86507
\(202\) 0 0
\(203\) −8.31531 −0.583621
\(204\) 0 0
\(205\) −1.79273 2.40808i −0.125209 0.168188i
\(206\) 0 0
\(207\) 0.498250 0.0346308
\(208\) 0 0
\(209\) −39.3646 −2.72290
\(210\) 0 0
\(211\) 17.8016i 1.22551i −0.790272 0.612756i \(-0.790061\pi\)
0.790272 0.612756i \(-0.209939\pi\)
\(212\) 0 0
\(213\) −27.6181 −1.89236
\(214\) 0 0
\(215\) 2.30189 0.156988
\(216\) 0 0
\(217\) 3.17006i 0.215198i
\(218\) 0 0
\(219\) 6.50587i 0.439626i
\(220\) 0 0
\(221\) −4.62666 −0.311223
\(222\) 0 0
\(223\) −18.4729 −1.23704 −0.618518 0.785771i \(-0.712267\pi\)
−0.618518 + 0.785771i \(0.712267\pi\)
\(224\) 0 0
\(225\) 9.54667 0.636445
\(226\) 0 0
\(227\) 11.1615i 0.740816i −0.928869 0.370408i \(-0.879218\pi\)
0.928869 0.370408i \(-0.120782\pi\)
\(228\) 0 0
\(229\) 12.2692i 0.810774i 0.914145 + 0.405387i \(0.132863\pi\)
−0.914145 + 0.405387i \(0.867137\pi\)
\(230\) 0 0
\(231\) 11.1406i 0.732996i
\(232\) 0 0
\(233\) 5.38249i 0.352619i 0.984335 + 0.176309i \(0.0564159\pi\)
−0.984335 + 0.176309i \(0.943584\pi\)
\(234\) 0 0
\(235\) 0.847545i 0.0552878i
\(236\) 0 0
\(237\) −8.47492 −0.550505
\(238\) 0 0
\(239\) 1.79783i 0.116292i 0.998308 + 0.0581461i \(0.0185189\pi\)
−0.998308 + 0.0581461i \(0.981481\pi\)
\(240\) 0 0
\(241\) −16.4358 −1.05873 −0.529363 0.848396i \(-0.677569\pi\)
−0.529363 + 0.848396i \(0.677569\pi\)
\(242\) 0 0
\(243\) 17.8706i 1.14640i
\(244\) 0 0
\(245\) −0.468853 −0.0299539
\(246\) 0 0
\(247\) −29.9055 −1.90284
\(248\) 0 0
\(249\) 19.5766i 1.24062i
\(250\) 0 0
\(251\) 20.8489 1.31597 0.657986 0.753031i \(-0.271409\pi\)
0.657986 + 0.753031i \(0.271409\pi\)
\(252\) 0 0
\(253\) 1.24333i 0.0781675i
\(254\) 0 0
\(255\) 1.28077 0.0802051
\(256\) 0 0
\(257\) 11.7446i 0.732606i −0.930496 0.366303i \(-0.880623\pi\)
0.930496 0.366303i \(-0.119377\pi\)
\(258\) 0 0
\(259\) 7.74950i 0.481530i
\(260\) 0 0
\(261\) 16.6068i 1.02794i
\(262\) 0 0
\(263\) 21.4258i 1.32117i 0.750750 + 0.660586i \(0.229692\pi\)
−0.750750 + 0.660586i \(0.770308\pi\)
\(264\) 0 0
\(265\) 0.816613i 0.0501641i
\(266\) 0 0
\(267\) 35.6079 2.17917
\(268\) 0 0
\(269\) 15.9515 0.972580 0.486290 0.873797i \(-0.338350\pi\)
0.486290 + 0.873797i \(0.338350\pi\)
\(270\) 0 0
\(271\) −24.5899 −1.49373 −0.746865 0.664976i \(-0.768442\pi\)
−0.746865 + 0.664976i \(0.768442\pi\)
\(272\) 0 0
\(273\) 8.46357i 0.512239i
\(274\) 0 0
\(275\) 23.8227i 1.43656i
\(276\) 0 0
\(277\) 31.9459 1.91944 0.959721 0.280956i \(-0.0906514\pi\)
0.959721 + 0.280956i \(0.0906514\pi\)
\(278\) 0 0
\(279\) −6.33106 −0.379030
\(280\) 0 0
\(281\) 14.0011i 0.835235i −0.908623 0.417618i \(-0.862865\pi\)
0.908623 0.417618i \(-0.137135\pi\)
\(282\) 0 0
\(283\) 1.43345 0.0852097 0.0426049 0.999092i \(-0.486434\pi\)
0.0426049 + 0.999092i \(0.486434\pi\)
\(284\) 0 0
\(285\) 8.27858 0.490381
\(286\) 0 0
\(287\) 5.13612 3.82365i 0.303176 0.225703i
\(288\) 0 0
\(289\) 15.5067 0.912158
\(290\) 0 0
\(291\) 9.60159 0.562855
\(292\) 0 0
\(293\) 14.0309i 0.819695i −0.912154 0.409847i \(-0.865582\pi\)
0.912154 0.409847i \(-0.134418\pi\)
\(294\) 0 0
\(295\) 1.05706 0.0615444
\(296\) 0 0
\(297\) 11.1725 0.648292
\(298\) 0 0
\(299\) 0.944566i 0.0546257i
\(300\) 0 0
\(301\) 4.90962i 0.282986i
\(302\) 0 0
\(303\) −9.89000 −0.568166
\(304\) 0 0
\(305\) −0.515289 −0.0295053
\(306\) 0 0
\(307\) −6.11435 −0.348964 −0.174482 0.984660i \(-0.555825\pi\)
−0.174482 + 0.984660i \(0.555825\pi\)
\(308\) 0 0
\(309\) 33.4564i 1.90327i
\(310\) 0 0
\(311\) 2.22194i 0.125995i 0.998014 + 0.0629973i \(0.0200660\pi\)
−0.998014 + 0.0629973i \(0.979934\pi\)
\(312\) 0 0
\(313\) 4.57533i 0.258613i −0.991605 0.129307i \(-0.958725\pi\)
0.991605 0.129307i \(-0.0412751\pi\)
\(314\) 0 0
\(315\) 0.936364i 0.0527581i
\(316\) 0 0
\(317\) 7.39003i 0.415066i 0.978228 + 0.207533i \(0.0665434\pi\)
−0.978228 + 0.207533i \(0.933457\pi\)
\(318\) 0 0
\(319\) −41.4406 −2.32023
\(320\) 0 0
\(321\) 38.9962i 2.17656i
\(322\) 0 0
\(323\) −9.65236 −0.537072
\(324\) 0 0
\(325\) 18.0983i 1.00391i
\(326\) 0 0
\(327\) −11.8899 −0.657515
\(328\) 0 0
\(329\) −1.80770 −0.0996617
\(330\) 0 0
\(331\) 4.35261i 0.239241i −0.992820 0.119620i \(-0.961832\pi\)
0.992820 0.119620i \(-0.0381678\pi\)
\(332\) 0 0
\(333\) 15.4768 0.848124
\(334\) 0 0
\(335\) 5.54586i 0.303003i
\(336\) 0 0
\(337\) 0.788204 0.0429362 0.0214681 0.999770i \(-0.493166\pi\)
0.0214681 + 0.999770i \(0.493166\pi\)
\(338\) 0 0
\(339\) 1.88667i 0.102470i
\(340\) 0 0
\(341\) 15.7985i 0.855535i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0.261479i 0.0140775i
\(346\) 0 0
\(347\) 4.37888i 0.235070i 0.993069 + 0.117535i \(0.0374993\pi\)
−0.993069 + 0.117535i \(0.962501\pi\)
\(348\) 0 0
\(349\) −17.5805 −0.941061 −0.470530 0.882384i \(-0.655937\pi\)
−0.470530 + 0.882384i \(0.655937\pi\)
\(350\) 0 0
\(351\) 8.48779 0.453045
\(352\) 0 0
\(353\) 32.6489 1.73773 0.868864 0.495052i \(-0.164851\pi\)
0.868864 + 0.495052i \(0.164851\pi\)
\(354\) 0 0
\(355\) 5.79255i 0.307437i
\(356\) 0 0
\(357\) 2.73172i 0.144578i
\(358\) 0 0
\(359\) 28.0591 1.48090 0.740450 0.672111i \(-0.234612\pi\)
0.740450 + 0.672111i \(0.234612\pi\)
\(360\) 0 0
\(361\) −43.3903 −2.28370
\(362\) 0 0
\(363\) 30.9310i 1.62345i
\(364\) 0 0
\(365\) −1.36452 −0.0714224
\(366\) 0 0
\(367\) 19.8918 1.03834 0.519172 0.854670i \(-0.326240\pi\)
0.519172 + 0.854670i \(0.326240\pi\)
\(368\) 0 0
\(369\) 7.63635 + 10.2575i 0.397532 + 0.533986i
\(370\) 0 0
\(371\) 1.74173 0.0904259
\(372\) 0 0
\(373\) 2.48402 0.128617 0.0643087 0.997930i \(-0.479516\pi\)
0.0643087 + 0.997930i \(0.479516\pi\)
\(374\) 0 0
\(375\) 10.2505i 0.529332i
\(376\) 0 0
\(377\) −31.4827 −1.62144
\(378\) 0 0
\(379\) −11.5135 −0.591409 −0.295705 0.955279i \(-0.595554\pi\)
−0.295705 + 0.955279i \(0.595554\pi\)
\(380\) 0 0
\(381\) 18.3039i 0.937738i
\(382\) 0 0
\(383\) 36.6688i 1.87369i 0.349745 + 0.936845i \(0.386268\pi\)
−0.349745 + 0.936845i \(0.613732\pi\)
\(384\) 0 0
\(385\) −2.33660 −0.119084
\(386\) 0 0
\(387\) −9.80519 −0.498426
\(388\) 0 0
\(389\) 23.4388 1.18839 0.594197 0.804319i \(-0.297470\pi\)
0.594197 + 0.804319i \(0.297470\pi\)
\(390\) 0 0
\(391\) 0.304870i 0.0154179i
\(392\) 0 0
\(393\) 12.6403i 0.637618i
\(394\) 0 0
\(395\) 1.77751i 0.0894361i
\(396\) 0 0
\(397\) 32.4793i 1.63009i −0.579397 0.815046i \(-0.696712\pi\)
0.579397 0.815046i \(-0.303288\pi\)
\(398\) 0 0
\(399\) 17.6571i 0.883960i
\(400\) 0 0
\(401\) −4.98971 −0.249174 −0.124587 0.992209i \(-0.539761\pi\)
−0.124587 + 0.992209i \(0.539761\pi\)
\(402\) 0 0
\(403\) 12.0022i 0.597873i
\(404\) 0 0
\(405\) −5.15872 −0.256339
\(406\) 0 0
\(407\) 38.6207i 1.91436i
\(408\) 0 0
\(409\) 21.2544 1.05096 0.525481 0.850805i \(-0.323885\pi\)
0.525481 + 0.850805i \(0.323885\pi\)
\(410\) 0 0
\(411\) −7.98364 −0.393804
\(412\) 0 0
\(413\) 2.25456i 0.110940i
\(414\) 0 0
\(415\) 4.10594 0.201553
\(416\) 0 0
\(417\) 46.7416i 2.28895i
\(418\) 0 0
\(419\) 0.415097 0.0202788 0.0101394 0.999949i \(-0.496772\pi\)
0.0101394 + 0.999949i \(0.496772\pi\)
\(420\) 0 0
\(421\) 24.4243i 1.19037i −0.803590 0.595183i \(-0.797079\pi\)
0.803590 0.595183i \(-0.202921\pi\)
\(422\) 0 0
\(423\) 3.61023i 0.175535i
\(424\) 0 0
\(425\) 5.84143i 0.283351i
\(426\) 0 0
\(427\) 1.09904i 0.0531863i
\(428\) 0 0
\(429\) 42.1794i 2.03644i
\(430\) 0 0
\(431\) 33.4445 1.61096 0.805482 0.592620i \(-0.201907\pi\)
0.805482 + 0.592620i \(0.201907\pi\)
\(432\) 0 0
\(433\) 5.74246 0.275965 0.137983 0.990435i \(-0.455938\pi\)
0.137983 + 0.990435i \(0.455938\pi\)
\(434\) 0 0
\(435\) 8.71517 0.417860
\(436\) 0 0
\(437\) 1.97060i 0.0942664i
\(438\) 0 0
\(439\) 17.9479i 0.856606i 0.903635 + 0.428303i \(0.140888\pi\)
−0.903635 + 0.428303i \(0.859112\pi\)
\(440\) 0 0
\(441\) 1.99714 0.0951018
\(442\) 0 0
\(443\) −12.7389 −0.605245 −0.302622 0.953111i \(-0.597862\pi\)
−0.302622 + 0.953111i \(0.597862\pi\)
\(444\) 0 0
\(445\) 7.46830i 0.354032i
\(446\) 0 0
\(447\) 32.7593 1.54946
\(448\) 0 0
\(449\) −33.6046 −1.58590 −0.792949 0.609289i \(-0.791455\pi\)
−0.792949 + 0.609289i \(0.791455\pi\)
\(450\) 0 0
\(451\) 25.5966 19.0557i 1.20530 0.897297i
\(452\) 0 0
\(453\) −3.81872 −0.179419
\(454\) 0 0
\(455\) −1.77513 −0.0832193
\(456\) 0 0
\(457\) 30.3766i 1.42096i 0.703718 + 0.710480i \(0.251522\pi\)
−0.703718 + 0.710480i \(0.748478\pi\)
\(458\) 0 0
\(459\) 2.73954 0.127871
\(460\) 0 0
\(461\) −10.4830 −0.488244 −0.244122 0.969745i \(-0.578500\pi\)
−0.244122 + 0.969745i \(0.578500\pi\)
\(462\) 0 0
\(463\) 16.9928i 0.789723i −0.918741 0.394862i \(-0.870792\pi\)
0.918741 0.394862i \(-0.129208\pi\)
\(464\) 0 0
\(465\) 3.32250i 0.154077i
\(466\) 0 0
\(467\) 25.3701 1.17399 0.586994 0.809591i \(-0.300311\pi\)
0.586994 + 0.809591i \(0.300311\pi\)
\(468\) 0 0
\(469\) 11.8286 0.546193
\(470\) 0 0
\(471\) −31.6543 −1.45855
\(472\) 0 0
\(473\) 24.4678i 1.12503i
\(474\) 0 0
\(475\) 37.7574i 1.73243i
\(476\) 0 0
\(477\) 3.47847i 0.159268i
\(478\) 0 0
\(479\) 5.92313i 0.270635i 0.990802 + 0.135317i \(0.0432054\pi\)
−0.990802 + 0.135317i \(0.956795\pi\)
\(480\) 0 0
\(481\) 29.3404i 1.33781i
\(482\) 0 0
\(483\) −0.557699 −0.0253762
\(484\) 0 0
\(485\) 2.01381i 0.0914425i
\(486\) 0 0
\(487\) 5.12441 0.232209 0.116105 0.993237i \(-0.462959\pi\)
0.116105 + 0.993237i \(0.462959\pi\)
\(488\) 0 0
\(489\) 44.8457i 2.02799i
\(490\) 0 0
\(491\) 18.8111 0.848933 0.424467 0.905444i \(-0.360462\pi\)
0.424467 + 0.905444i \(0.360462\pi\)
\(492\) 0 0
\(493\) −10.1614 −0.457646
\(494\) 0 0
\(495\) 4.66650i 0.209744i
\(496\) 0 0
\(497\) 12.3547 0.554186
\(498\) 0 0
\(499\) 2.07851i 0.0930470i 0.998917 + 0.0465235i \(0.0148142\pi\)
−0.998917 + 0.0465235i \(0.985186\pi\)
\(500\) 0 0
\(501\) 28.1643 1.25829
\(502\) 0 0
\(503\) 26.1912i 1.16781i 0.811822 + 0.583905i \(0.198476\pi\)
−0.811822 + 0.583905i \(0.801524\pi\)
\(504\) 0 0
\(505\) 2.07430i 0.0923053i
\(506\) 0 0
\(507\) 2.98343i 0.132499i
\(508\) 0 0
\(509\) 31.6951i 1.40486i 0.711752 + 0.702431i \(0.247902\pi\)
−0.711752 + 0.702431i \(0.752098\pi\)
\(510\) 0 0
\(511\) 2.91035i 0.128746i
\(512\) 0 0
\(513\) 17.7076 0.781811
\(514\) 0 0
\(515\) 7.01705 0.309208
\(516\) 0 0
\(517\) −9.00893 −0.396212
\(518\) 0 0
\(519\) 0.300119i 0.0131738i
\(520\) 0 0
\(521\) 37.0562i 1.62346i 0.584033 + 0.811730i \(0.301474\pi\)
−0.584033 + 0.811730i \(0.698526\pi\)
\(522\) 0 0
\(523\) 3.75364 0.164135 0.0820676 0.996627i \(-0.473848\pi\)
0.0820676 + 0.996627i \(0.473848\pi\)
\(524\) 0 0
\(525\) −10.6857 −0.466364
\(526\) 0 0
\(527\) 3.87385i 0.168748i
\(528\) 0 0
\(529\) −22.9378 −0.997294
\(530\) 0 0
\(531\) −4.50268 −0.195400
\(532\) 0 0
\(533\) 19.4459 14.4767i 0.842296 0.627057i
\(534\) 0 0
\(535\) 8.17896 0.353607
\(536\) 0 0
\(537\) −33.3811 −1.44050
\(538\) 0 0
\(539\) 4.98364i 0.214661i
\(540\) 0 0
\(541\) 12.0339 0.517380 0.258690 0.965960i \(-0.416709\pi\)
0.258690 + 0.965960i \(0.416709\pi\)
\(542\) 0 0
\(543\) 50.6302 2.17275
\(544\) 0 0
\(545\) 2.49376i 0.106821i
\(546\) 0 0
\(547\) 5.14839i 0.220129i 0.993924 + 0.110065i \(0.0351058\pi\)
−0.993924 + 0.110065i \(0.964894\pi\)
\(548\) 0 0
\(549\) 2.19494 0.0936776
\(550\) 0 0
\(551\) −65.6806 −2.79809
\(552\) 0 0
\(553\) 3.79118 0.161218
\(554\) 0 0
\(555\) 8.12214i 0.344766i
\(556\) 0 0
\(557\) 46.7199i 1.97959i 0.142508 + 0.989794i \(0.454483\pi\)
−0.142508 + 0.989794i \(0.545517\pi\)
\(558\) 0 0
\(559\) 18.5884i 0.786204i
\(560\) 0 0
\(561\) 13.6139i 0.574779i
\(562\) 0 0
\(563\) 19.4210i 0.818496i 0.912423 + 0.409248i \(0.134209\pi\)
−0.912423 + 0.409248i \(0.865791\pi\)
\(564\) 0 0
\(565\) 0.395706 0.0166475
\(566\) 0 0
\(567\) 11.0029i 0.462076i
\(568\) 0 0
\(569\) −11.9853 −0.502451 −0.251225 0.967929i \(-0.580833\pi\)
−0.251225 + 0.967929i \(0.580833\pi\)
\(570\) 0 0
\(571\) 28.2952i 1.18412i −0.805895 0.592059i \(-0.798315\pi\)
0.805895 0.592059i \(-0.201685\pi\)
\(572\) 0 0
\(573\) 44.2359 1.84798
\(574\) 0 0
\(575\) 1.19257 0.0497335
\(576\) 0 0
\(577\) 10.4014i 0.433016i 0.976281 + 0.216508i \(0.0694667\pi\)
−0.976281 + 0.216508i \(0.930533\pi\)
\(578\) 0 0
\(579\) −52.3864 −2.17711
\(580\) 0 0
\(581\) 8.75742i 0.363319i
\(582\) 0 0
\(583\) 8.68014 0.359495
\(584\) 0 0
\(585\) 3.54517i 0.146575i
\(586\) 0 0
\(587\) 21.5780i 0.890619i 0.895377 + 0.445310i \(0.146906\pi\)
−0.895377 + 0.445310i \(0.853094\pi\)
\(588\) 0 0
\(589\) 25.0396i 1.03174i
\(590\) 0 0
\(591\) 35.0534i 1.44190i
\(592\) 0 0
\(593\) 14.5336i 0.596823i −0.954437 0.298412i \(-0.903543\pi\)
0.954437 0.298412i \(-0.0964568\pi\)
\(594\) 0 0
\(595\) −0.572943 −0.0234884
\(596\) 0 0
\(597\) 41.7755 1.70976
\(598\) 0 0
\(599\) 5.12899 0.209565 0.104782 0.994495i \(-0.466585\pi\)
0.104782 + 0.994495i \(0.466585\pi\)
\(600\) 0 0
\(601\) 4.80002i 0.195797i 0.995196 + 0.0978984i \(0.0312120\pi\)
−0.995196 + 0.0978984i \(0.968788\pi\)
\(602\) 0 0
\(603\) 23.6233i 0.962015i
\(604\) 0 0
\(605\) −6.48738 −0.263750
\(606\) 0 0
\(607\) −17.8802 −0.725735 −0.362868 0.931841i \(-0.618202\pi\)
−0.362868 + 0.931841i \(0.618202\pi\)
\(608\) 0 0
\(609\) 18.5883i 0.753235i
\(610\) 0 0
\(611\) −6.84415 −0.276885
\(612\) 0 0
\(613\) −9.92497 −0.400866 −0.200433 0.979707i \(-0.564235\pi\)
−0.200433 + 0.979707i \(0.564235\pi\)
\(614\) 0 0
\(615\) −5.38310 + 4.00751i −0.217068 + 0.161598i
\(616\) 0 0
\(617\) 27.0895 1.09058 0.545291 0.838247i \(-0.316419\pi\)
0.545291 + 0.838247i \(0.316419\pi\)
\(618\) 0 0
\(619\) −18.1664 −0.730170 −0.365085 0.930974i \(-0.618960\pi\)
−0.365085 + 0.930974i \(0.618960\pi\)
\(620\) 0 0
\(621\) 0.559295i 0.0224437i
\(622\) 0 0
\(623\) −15.9289 −0.638177
\(624\) 0 0
\(625\) 21.7510 0.870039
\(626\) 0 0
\(627\) 87.9966i 3.51425i
\(628\) 0 0
\(629\) 9.46997i 0.377592i
\(630\) 0 0
\(631\) 11.1736 0.444812 0.222406 0.974954i \(-0.428609\pi\)
0.222406 + 0.974954i \(0.428609\pi\)
\(632\) 0 0
\(633\) −39.7942 −1.58168
\(634\) 0 0
\(635\) −3.83901 −0.152347
\(636\) 0 0
\(637\) 3.78611i 0.150011i
\(638\) 0 0
\(639\) 24.6741i 0.976093i
\(640\) 0 0
\(641\) 23.0981i 0.912320i −0.889898 0.456160i \(-0.849224\pi\)
0.889898 0.456160i \(-0.150776\pi\)
\(642\) 0 0
\(643\) 8.28421i 0.326697i 0.986568 + 0.163349i \(0.0522295\pi\)
−0.986568 + 0.163349i \(0.947770\pi\)
\(644\) 0 0
\(645\) 5.14571i 0.202612i
\(646\) 0 0
\(647\) 49.0736 1.92928 0.964642 0.263565i \(-0.0848985\pi\)
0.964642 + 0.263565i \(0.0848985\pi\)
\(648\) 0 0
\(649\) 11.2359i 0.441050i
\(650\) 0 0
\(651\) 7.08645 0.277740
\(652\) 0 0
\(653\) 33.5708i 1.31373i −0.754010 0.656863i \(-0.771883\pi\)
0.754010 0.656863i \(-0.228117\pi\)
\(654\) 0 0
\(655\) 2.65114 0.103589
\(656\) 0 0
\(657\) 5.81236 0.226762
\(658\) 0 0
\(659\) 28.9895i 1.12927i −0.825341 0.564635i \(-0.809017\pi\)
0.825341 0.564635i \(-0.190983\pi\)
\(660\) 0 0
\(661\) −1.64974 −0.0641673 −0.0320836 0.999485i \(-0.510214\pi\)
−0.0320836 + 0.999485i \(0.510214\pi\)
\(662\) 0 0
\(663\) 10.3426i 0.401672i
\(664\) 0 0
\(665\) −3.70335 −0.143610
\(666\) 0 0
\(667\) 2.07452i 0.0803258i
\(668\) 0 0
\(669\) 41.2948i 1.59655i
\(670\) 0 0
\(671\) 5.47723i 0.211446i
\(672\) 0 0
\(673\) 38.9087i 1.49982i −0.661540 0.749910i \(-0.730097\pi\)
0.661540 0.749910i \(-0.269903\pi\)
\(674\) 0 0
\(675\) 10.7163i 0.412472i
\(676\) 0 0
\(677\) 16.5432 0.635805 0.317903 0.948123i \(-0.397021\pi\)
0.317903 + 0.948123i \(0.397021\pi\)
\(678\) 0 0
\(679\) −4.29519 −0.164834
\(680\) 0 0
\(681\) −24.9508 −0.956116
\(682\) 0 0
\(683\) 21.1382i 0.808830i 0.914576 + 0.404415i \(0.132525\pi\)
−0.914576 + 0.404415i \(0.867475\pi\)
\(684\) 0 0
\(685\) 1.67447i 0.0639781i
\(686\) 0 0
\(687\) 27.4270 1.04641
\(688\) 0 0
\(689\) 6.59436 0.251225
\(690\) 0 0
\(691\) 39.5315i 1.50385i −0.659250 0.751924i \(-0.729126\pi\)
0.659250 0.751924i \(-0.270874\pi\)
\(692\) 0 0
\(693\) 9.95302 0.378084
\(694\) 0 0
\(695\) 9.80346 0.371867
\(696\) 0 0
\(697\) 6.27639 4.67253i 0.237735 0.176985i
\(698\) 0 0
\(699\) 12.0322 0.455099
\(700\) 0 0
\(701\) −25.9383 −0.979675 −0.489838 0.871814i \(-0.662944\pi\)
−0.489838 + 0.871814i \(0.662944\pi\)
\(702\) 0 0
\(703\) 61.2113i 2.30863i
\(704\) 0 0
\(705\) 1.89463 0.0713557
\(706\) 0 0
\(707\) 4.42421 0.166389
\(708\) 0 0
\(709\) 29.6216i 1.11246i 0.831028 + 0.556231i \(0.187753\pi\)
−0.831028 + 0.556231i \(0.812247\pi\)
\(710\) 0 0
\(711\) 7.57152i 0.283954i
\(712\) 0 0
\(713\) −0.790874 −0.0296185
\(714\) 0 0
\(715\) −8.84660 −0.330844
\(716\) 0 0
\(717\) 4.01893 0.150090
\(718\) 0 0
\(719\) 27.4212i 1.02264i −0.859391 0.511318i \(-0.829157\pi\)
0.859391 0.511318i \(-0.170843\pi\)
\(720\) 0 0
\(721\) 14.9664i 0.557379i
\(722\) 0 0
\(723\) 36.7411i 1.36642i
\(724\) 0 0
\(725\) 39.7487i 1.47623i
\(726\) 0 0
\(727\) 45.9715i 1.70499i −0.522738 0.852493i \(-0.675089\pi\)
0.522738 0.852493i \(-0.324911\pi\)
\(728\) 0 0
\(729\) 6.93989 0.257033
\(730\) 0 0
\(731\) 5.99961i 0.221904i
\(732\) 0 0
\(733\) −29.9746 −1.10714 −0.553569 0.832803i \(-0.686734\pi\)
−0.553569 + 0.832803i \(0.686734\pi\)
\(734\) 0 0
\(735\) 1.04809i 0.0386593i
\(736\) 0 0
\(737\) 58.9494 2.17143
\(738\) 0 0
\(739\) −13.1355 −0.483196 −0.241598 0.970376i \(-0.577672\pi\)
−0.241598 + 0.970376i \(0.577672\pi\)
\(740\) 0 0
\(741\) 66.8516i 2.45586i
\(742\) 0 0
\(743\) 4.72772 0.173443 0.0867216 0.996233i \(-0.472361\pi\)
0.0867216 + 0.996233i \(0.472361\pi\)
\(744\) 0 0
\(745\) 6.87086i 0.251729i
\(746\) 0 0
\(747\) −17.4898 −0.639917
\(748\) 0 0
\(749\) 17.4446i 0.637413i
\(750\) 0 0
\(751\) 35.5122i 1.29586i 0.761701 + 0.647929i \(0.224364\pi\)
−0.761701 + 0.647929i \(0.775636\pi\)
\(752\) 0 0
\(753\) 46.6062i 1.69843i
\(754\) 0 0
\(755\) 0.800929i 0.0291488i
\(756\) 0 0
\(757\) 45.9485i 1.67003i 0.550230 + 0.835013i \(0.314540\pi\)
−0.550230 + 0.835013i \(0.685460\pi\)
\(758\) 0 0
\(759\) −2.77937 −0.100885
\(760\) 0 0
\(761\) 32.9598 1.19479 0.597396 0.801947i \(-0.296202\pi\)
0.597396 + 0.801947i \(0.296202\pi\)
\(762\) 0 0
\(763\) 5.31886 0.192556
\(764\) 0 0
\(765\) 1.14425i 0.0413703i
\(766\) 0 0
\(767\) 8.53603i 0.308218i
\(768\) 0 0
\(769\) −20.4944 −0.739046 −0.369523 0.929222i \(-0.620479\pi\)
−0.369523 + 0.929222i \(0.620479\pi\)
\(770\) 0 0
\(771\) −26.2541 −0.945520
\(772\) 0 0
\(773\) 6.62253i 0.238196i 0.992883 + 0.119098i \(0.0380002\pi\)
−0.992883 + 0.119098i \(0.962000\pi\)
\(774\) 0 0
\(775\) −15.1535 −0.544329
\(776\) 0 0
\(777\) −17.3234 −0.621475
\(778\) 0 0
\(779\) 40.5689 30.2020i 1.45353 1.08210i
\(780\) 0 0
\(781\) 61.5716 2.20321
\(782\) 0 0
\(783\) 18.6415 0.666192
\(784\) 0 0
\(785\) 6.63909i 0.236959i
\(786\) 0 0
\(787\) 30.6191 1.09145 0.545726 0.837964i \(-0.316254\pi\)
0.545726 + 0.837964i \(0.316254\pi\)
\(788\) 0 0
\(789\) 47.8959 1.70514
\(790\) 0 0
\(791\) 0.843987i 0.0300087i
\(792\) 0 0
\(793\) 4.16109i 0.147765i
\(794\) 0 0
\(795\) −1.82548 −0.0647431
\(796\) 0 0
\(797\) 28.0187 0.992475 0.496237 0.868187i \(-0.334715\pi\)
0.496237 + 0.868187i \(0.334715\pi\)
\(798\) 0 0
\(799\) −2.20903 −0.0781498
\(800\) 0 0
\(801\) 31.8122i 1.12403i
\(802\) 0 0
\(803\) 14.5041i 0.511840i
\(804\) 0 0
\(805\) 0.116970i 0.00412266i
\(806\) 0