Properties

Label 1148.2.d.a.1065.17
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 1065.17
Root \(1.40239i\) of defining polynomial
Character \(\chi\) \(=\) 1148.1065
Dual form 1148.2.d.a.1065.4

$q$-expansion

\(f(q)\) \(=\) \(q+2.39349i q^{3} +3.99395 q^{5} -1.00000i q^{7} -2.72882 q^{9} +O(q^{10})\) \(q+2.39349i q^{3} +3.99395 q^{5} -1.00000i q^{7} -2.72882 q^{9} +0.655586i q^{11} -1.32178i q^{13} +9.55950i q^{15} +1.55031i q^{17} +3.21822i q^{19} +2.39349 q^{21} -1.09317 q^{23} +10.9517 q^{25} +0.649079i q^{27} +4.14410i q^{29} -3.62085 q^{31} -1.56914 q^{33} -3.99395i q^{35} +6.76585 q^{37} +3.16367 q^{39} +(6.37033 + 0.647203i) q^{41} +6.56003 q^{43} -10.8988 q^{45} +2.13764i q^{47} -1.00000 q^{49} -3.71066 q^{51} -11.6378i q^{53} +2.61838i q^{55} -7.70279 q^{57} -8.90983 q^{59} -4.81119 q^{61} +2.72882i q^{63} -5.27912i q^{65} +0.603316i q^{67} -2.61650i q^{69} -5.43199i q^{71} -10.0698 q^{73} +26.2127i q^{75} +0.655586 q^{77} +4.77184i q^{79} -9.74001 q^{81} +4.09012 q^{83} +6.19186i q^{85} -9.91888 q^{87} +2.88671i q^{89} -1.32178 q^{91} -8.66648i q^{93} +12.8534i q^{95} +15.8789i q^{97} -1.78897i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{5} - 20q^{9} + O(q^{10}) \) \( 20q + 4q^{5} - 20q^{9} + 4q^{21} + 8q^{31} + 20q^{37} + 4q^{39} - 16q^{41} + 20q^{43} - 4q^{45} - 20q^{49} + 52q^{51} - 36q^{57} + 20q^{59} - 4q^{61} - 12q^{73} + 8q^{77} + 20q^{81} - 48q^{83} + 44q^{87} - 4q^{91} + O(q^{100}) \)

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
\(3\) 2.39349i 1.38188i 0.722910 + 0.690942i \(0.242804\pi\)
−0.722910 + 0.690942i \(0.757196\pi\)
\(4\) 0 0
\(5\) 3.99395 1.78615 0.893075 0.449908i \(-0.148543\pi\)
0.893075 + 0.449908i \(0.148543\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −2.72882 −0.909605
\(10\) 0 0
\(11\) 0.655586i 0.197667i 0.995104 + 0.0988333i \(0.0315111\pi\)
−0.995104 + 0.0988333i \(0.968489\pi\)
\(12\) 0 0
\(13\) 1.32178i 0.366596i −0.983057 0.183298i \(-0.941323\pi\)
0.983057 0.183298i \(-0.0586773\pi\)
\(14\) 0 0
\(15\) 9.55950i 2.46825i
\(16\) 0 0
\(17\) 1.55031i 0.376005i 0.982169 + 0.188003i \(0.0602013\pi\)
−0.982169 + 0.188003i \(0.939799\pi\)
\(18\) 0 0
\(19\) 3.21822i 0.738311i 0.929368 + 0.369155i \(0.120353\pi\)
−0.929368 + 0.369155i \(0.879647\pi\)
\(20\) 0 0
\(21\) 2.39349 0.522303
\(22\) 0 0
\(23\) −1.09317 −0.227942 −0.113971 0.993484i \(-0.536357\pi\)
−0.113971 + 0.993484i \(0.536357\pi\)
\(24\) 0 0
\(25\) 10.9517 2.19033
\(26\) 0 0
\(27\) 0.649079i 0.124915i
\(28\) 0 0
\(29\) 4.14410i 0.769540i 0.923012 + 0.384770i \(0.125719\pi\)
−0.923012 + 0.384770i \(0.874281\pi\)
\(30\) 0 0
\(31\) −3.62085 −0.650323 −0.325162 0.945658i \(-0.605419\pi\)
−0.325162 + 0.945658i \(0.605419\pi\)
\(32\) 0 0
\(33\) −1.56914 −0.273153
\(34\) 0 0
\(35\) 3.99395i 0.675101i
\(36\) 0 0
\(37\) 6.76585 1.11230 0.556149 0.831082i \(-0.312278\pi\)
0.556149 + 0.831082i \(0.312278\pi\)
\(38\) 0 0
\(39\) 3.16367 0.506593
\(40\) 0 0
\(41\) 6.37033 + 0.647203i 0.994879 + 0.101076i
\(42\) 0 0
\(43\) 6.56003 1.00040 0.500198 0.865911i \(-0.333261\pi\)
0.500198 + 0.865911i \(0.333261\pi\)
\(44\) 0 0
\(45\) −10.8988 −1.62469
\(46\) 0 0
\(47\) 2.13764i 0.311806i 0.987772 + 0.155903i \(0.0498288\pi\)
−0.987772 + 0.155903i \(0.950171\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −3.71066 −0.519596
\(52\) 0 0
\(53\) 11.6378i 1.59857i −0.600950 0.799287i \(-0.705211\pi\)
0.600950 0.799287i \(-0.294789\pi\)
\(54\) 0 0
\(55\) 2.61838i 0.353062i
\(56\) 0 0
\(57\) −7.70279 −1.02026
\(58\) 0 0
\(59\) −8.90983 −1.15996 −0.579980 0.814630i \(-0.696940\pi\)
−0.579980 + 0.814630i \(0.696940\pi\)
\(60\) 0 0
\(61\) −4.81119 −0.616010 −0.308005 0.951385i \(-0.599661\pi\)
−0.308005 + 0.951385i \(0.599661\pi\)
\(62\) 0 0
\(63\) 2.72882i 0.343798i
\(64\) 0 0
\(65\) 5.27912i 0.654795i
\(66\) 0 0
\(67\) 0.603316i 0.0737068i 0.999321 + 0.0368534i \(0.0117335\pi\)
−0.999321 + 0.0368534i \(0.988267\pi\)
\(68\) 0 0
\(69\) 2.61650i 0.314990i
\(70\) 0 0
\(71\) 5.43199i 0.644658i −0.946628 0.322329i \(-0.895534\pi\)
0.946628 0.322329i \(-0.104466\pi\)
\(72\) 0 0
\(73\) −10.0698 −1.17858 −0.589292 0.807920i \(-0.700593\pi\)
−0.589292 + 0.807920i \(0.700593\pi\)
\(74\) 0 0
\(75\) 26.2127i 3.02678i
\(76\) 0 0
\(77\) 0.655586 0.0747110
\(78\) 0 0
\(79\) 4.77184i 0.536874i 0.963297 + 0.268437i \(0.0865071\pi\)
−0.963297 + 0.268437i \(0.913493\pi\)
\(80\) 0 0
\(81\) −9.74001 −1.08222
\(82\) 0 0
\(83\) 4.09012 0.448949 0.224474 0.974480i \(-0.427934\pi\)
0.224474 + 0.974480i \(0.427934\pi\)
\(84\) 0 0
\(85\) 6.19186i 0.671601i
\(86\) 0 0
\(87\) −9.91888 −1.06342
\(88\) 0 0
\(89\) 2.88671i 0.305990i 0.988227 + 0.152995i \(0.0488919\pi\)
−0.988227 + 0.152995i \(0.951108\pi\)
\(90\) 0 0
\(91\) −1.32178 −0.138560
\(92\) 0 0
\(93\) 8.66648i 0.898672i
\(94\) 0 0
\(95\) 12.8534i 1.31873i
\(96\) 0 0
\(97\) 15.8789i 1.61226i 0.591739 + 0.806130i \(0.298442\pi\)
−0.591739 + 0.806130i \(0.701558\pi\)
\(98\) 0 0
\(99\) 1.78897i 0.179799i
\(100\) 0 0
\(101\) 17.8206i 1.77321i −0.462525 0.886606i \(-0.653056\pi\)
0.462525 0.886606i \(-0.346944\pi\)
\(102\) 0 0
\(103\) 13.7897 1.35874 0.679368 0.733797i \(-0.262254\pi\)
0.679368 + 0.733797i \(0.262254\pi\)
\(104\) 0 0
\(105\) 9.55950 0.932912
\(106\) 0 0
\(107\) −12.8832 −1.24546 −0.622731 0.782436i \(-0.713977\pi\)
−0.622731 + 0.782436i \(0.713977\pi\)
\(108\) 0 0
\(109\) 16.1047i 1.54255i −0.636502 0.771275i \(-0.719619\pi\)
0.636502 0.771275i \(-0.280381\pi\)
\(110\) 0 0
\(111\) 16.1940i 1.53707i
\(112\) 0 0
\(113\) 0.712810 0.0670556 0.0335278 0.999438i \(-0.489326\pi\)
0.0335278 + 0.999438i \(0.489326\pi\)
\(114\) 0 0
\(115\) −4.36608 −0.407139
\(116\) 0 0
\(117\) 3.60689i 0.333457i
\(118\) 0 0
\(119\) 1.55031 0.142117
\(120\) 0 0
\(121\) 10.5702 0.960928
\(122\) 0 0
\(123\) −1.54908 + 15.2474i −0.139676 + 1.37481i
\(124\) 0 0
\(125\) 23.7706 2.12611
\(126\) 0 0
\(127\) −7.30114 −0.647871 −0.323936 0.946079i \(-0.605006\pi\)
−0.323936 + 0.946079i \(0.605006\pi\)
\(128\) 0 0
\(129\) 15.7014i 1.38243i
\(130\) 0 0
\(131\) −5.57734 −0.487294 −0.243647 0.969864i \(-0.578344\pi\)
−0.243647 + 0.969864i \(0.578344\pi\)
\(132\) 0 0
\(133\) 3.21822 0.279055
\(134\) 0 0
\(135\) 2.59239i 0.223117i
\(136\) 0 0
\(137\) 3.00928i 0.257100i −0.991703 0.128550i \(-0.958968\pi\)
0.991703 0.128550i \(-0.0410323\pi\)
\(138\) 0 0
\(139\) −4.75504 −0.403317 −0.201659 0.979456i \(-0.564633\pi\)
−0.201659 + 0.979456i \(0.564633\pi\)
\(140\) 0 0
\(141\) −5.11642 −0.430880
\(142\) 0 0
\(143\) 0.866540 0.0724637
\(144\) 0 0
\(145\) 16.5513i 1.37451i
\(146\) 0 0
\(147\) 2.39349i 0.197412i
\(148\) 0 0
\(149\) 15.3223i 1.25525i 0.778517 + 0.627624i \(0.215972\pi\)
−0.778517 + 0.627624i \(0.784028\pi\)
\(150\) 0 0
\(151\) 17.9639i 1.46188i −0.682442 0.730940i \(-0.739082\pi\)
0.682442 0.730940i \(-0.260918\pi\)
\(152\) 0 0
\(153\) 4.23051i 0.342016i
\(154\) 0 0
\(155\) −14.4615 −1.16157
\(156\) 0 0
\(157\) 11.6104i 0.926608i −0.886199 0.463304i \(-0.846664\pi\)
0.886199 0.463304i \(-0.153336\pi\)
\(158\) 0 0
\(159\) 27.8550 2.20904
\(160\) 0 0
\(161\) 1.09317i 0.0861541i
\(162\) 0 0
\(163\) −15.1493 −1.18659 −0.593293 0.804987i \(-0.702172\pi\)
−0.593293 + 0.804987i \(0.702172\pi\)
\(164\) 0 0
\(165\) −6.26708 −0.487891
\(166\) 0 0
\(167\) 4.26888i 0.330336i −0.986265 0.165168i \(-0.947183\pi\)
0.986265 0.165168i \(-0.0528166\pi\)
\(168\) 0 0
\(169\) 11.2529 0.865608
\(170\) 0 0
\(171\) 8.78193i 0.671571i
\(172\) 0 0
\(173\) −11.8219 −0.898800 −0.449400 0.893331i \(-0.648362\pi\)
−0.449400 + 0.893331i \(0.648362\pi\)
\(174\) 0 0
\(175\) 10.9517i 0.827867i
\(176\) 0 0
\(177\) 21.3256i 1.60293i
\(178\) 0 0
\(179\) 9.10793i 0.680758i −0.940288 0.340379i \(-0.889445\pi\)
0.940288 0.340379i \(-0.110555\pi\)
\(180\) 0 0
\(181\) 17.0621i 1.26821i −0.773245 0.634107i \(-0.781368\pi\)
0.773245 0.634107i \(-0.218632\pi\)
\(182\) 0 0
\(183\) 11.5156i 0.851255i
\(184\) 0 0
\(185\) 27.0225 1.98673
\(186\) 0 0
\(187\) −1.01636 −0.0743237
\(188\) 0 0
\(189\) 0.649079 0.0472136
\(190\) 0 0
\(191\) 18.8163i 1.36150i −0.732516 0.680750i \(-0.761654\pi\)
0.732516 0.680750i \(-0.238346\pi\)
\(192\) 0 0
\(193\) 19.1003i 1.37487i −0.726245 0.687436i \(-0.758736\pi\)
0.726245 0.687436i \(-0.241264\pi\)
\(194\) 0 0
\(195\) 12.6356 0.904851
\(196\) 0 0
\(197\) −7.18645 −0.512014 −0.256007 0.966675i \(-0.582407\pi\)
−0.256007 + 0.966675i \(0.582407\pi\)
\(198\) 0 0
\(199\) 6.49015i 0.460074i −0.973182 0.230037i \(-0.926115\pi\)
0.973182 0.230037i \(-0.0738848\pi\)
\(200\) 0 0
\(201\) −1.44403 −0.101854
\(202\) 0 0
\(203\) 4.14410 0.290859
\(204\) 0 0
\(205\) 25.4428 + 2.58490i 1.77700 + 0.180537i
\(206\) 0 0
\(207\) 2.98307 0.207338
\(208\) 0 0
\(209\) −2.10982 −0.145939
\(210\) 0 0
\(211\) 2.84127i 0.195601i 0.995206 + 0.0978004i \(0.0311807\pi\)
−0.995206 + 0.0978004i \(0.968819\pi\)
\(212\) 0 0
\(213\) 13.0014 0.890844
\(214\) 0 0
\(215\) 26.2004 1.78686
\(216\) 0 0
\(217\) 3.62085i 0.245799i
\(218\) 0 0
\(219\) 24.1021i 1.62867i
\(220\) 0 0
\(221\) 2.04917 0.137842
\(222\) 0 0
\(223\) 2.15843 0.144539 0.0722695 0.997385i \(-0.476976\pi\)
0.0722695 + 0.997385i \(0.476976\pi\)
\(224\) 0 0
\(225\) −29.8850 −1.99234
\(226\) 0 0
\(227\) 4.85412i 0.322179i 0.986940 + 0.161090i \(0.0515008\pi\)
−0.986940 + 0.161090i \(0.948499\pi\)
\(228\) 0 0
\(229\) 17.9050i 1.18319i −0.806233 0.591597i \(-0.798497\pi\)
0.806233 0.591597i \(-0.201503\pi\)
\(230\) 0 0
\(231\) 1.56914i 0.103242i
\(232\) 0 0
\(233\) 1.77544i 0.116313i −0.998307 0.0581563i \(-0.981478\pi\)
0.998307 0.0581563i \(-0.0185222\pi\)
\(234\) 0 0
\(235\) 8.53761i 0.556932i
\(236\) 0 0
\(237\) −11.4214 −0.741898
\(238\) 0 0
\(239\) 12.9001i 0.834438i 0.908806 + 0.417219i \(0.136995\pi\)
−0.908806 + 0.417219i \(0.863005\pi\)
\(240\) 0 0
\(241\) −23.1109 −1.48870 −0.744352 0.667787i \(-0.767242\pi\)
−0.744352 + 0.667787i \(0.767242\pi\)
\(242\) 0 0
\(243\) 21.3654i 1.37059i
\(244\) 0 0
\(245\) −3.99395 −0.255164
\(246\) 0 0
\(247\) 4.25378 0.270661
\(248\) 0 0
\(249\) 9.78967i 0.620395i
\(250\) 0 0
\(251\) 21.5199 1.35832 0.679162 0.733988i \(-0.262343\pi\)
0.679162 + 0.733988i \(0.262343\pi\)
\(252\) 0 0
\(253\) 0.716669i 0.0450566i
\(254\) 0 0
\(255\) −14.8202 −0.928076
\(256\) 0 0
\(257\) 3.90586i 0.243641i 0.992552 + 0.121821i \(0.0388733\pi\)
−0.992552 + 0.121821i \(0.961127\pi\)
\(258\) 0 0
\(259\) 6.76585i 0.420409i
\(260\) 0 0
\(261\) 11.3085i 0.699978i
\(262\) 0 0
\(263\) 8.92846i 0.550553i −0.961365 0.275276i \(-0.911231\pi\)
0.961365 0.275276i \(-0.0887693\pi\)
\(264\) 0 0
\(265\) 46.4808i 2.85529i
\(266\) 0 0
\(267\) −6.90931 −0.422843
\(268\) 0 0
\(269\) 4.60411 0.280718 0.140359 0.990101i \(-0.455174\pi\)
0.140359 + 0.990101i \(0.455174\pi\)
\(270\) 0 0
\(271\) 21.6257 1.31367 0.656834 0.754035i \(-0.271895\pi\)
0.656834 + 0.754035i \(0.271895\pi\)
\(272\) 0 0
\(273\) 3.16367i 0.191474i
\(274\) 0 0
\(275\) 7.17975i 0.432955i
\(276\) 0 0
\(277\) −24.6943 −1.48374 −0.741868 0.670546i \(-0.766060\pi\)
−0.741868 + 0.670546i \(0.766060\pi\)
\(278\) 0 0
\(279\) 9.88062 0.591537
\(280\) 0 0
\(281\) 5.86304i 0.349760i 0.984590 + 0.174880i \(0.0559537\pi\)
−0.984590 + 0.174880i \(0.944046\pi\)
\(282\) 0 0
\(283\) 20.0268 1.19047 0.595235 0.803552i \(-0.297059\pi\)
0.595235 + 0.803552i \(0.297059\pi\)
\(284\) 0 0
\(285\) −30.7646 −1.82234
\(286\) 0 0
\(287\) 0.647203 6.37033i 0.0382032 0.376029i
\(288\) 0 0
\(289\) 14.5965 0.858620
\(290\) 0 0
\(291\) −38.0061 −2.22796
\(292\) 0 0
\(293\) 25.5700i 1.49382i 0.664927 + 0.746908i \(0.268463\pi\)
−0.664927 + 0.746908i \(0.731537\pi\)
\(294\) 0 0
\(295\) −35.5854 −2.07186
\(296\) 0 0
\(297\) −0.425527 −0.0246916
\(298\) 0 0
\(299\) 1.44493i 0.0835627i
\(300\) 0 0
\(301\) 6.56003i 0.378114i
\(302\) 0 0
\(303\) 42.6534 2.45038
\(304\) 0 0
\(305\) −19.2157 −1.10029
\(306\) 0 0
\(307\) −21.7097 −1.23904 −0.619519 0.784982i \(-0.712672\pi\)
−0.619519 + 0.784982i \(0.712672\pi\)
\(308\) 0 0
\(309\) 33.0055i 1.87762i
\(310\) 0 0
\(311\) 23.8936i 1.35488i 0.735577 + 0.677441i \(0.236911\pi\)
−0.735577 + 0.677441i \(0.763089\pi\)
\(312\) 0 0
\(313\) 9.27380i 0.524186i 0.965043 + 0.262093i \(0.0844127\pi\)
−0.965043 + 0.262093i \(0.915587\pi\)
\(314\) 0 0
\(315\) 10.8988i 0.614075i
\(316\) 0 0
\(317\) 32.8603i 1.84562i 0.385255 + 0.922810i \(0.374114\pi\)
−0.385255 + 0.922810i \(0.625886\pi\)
\(318\) 0 0
\(319\) −2.71682 −0.152112
\(320\) 0 0
\(321\) 30.8358i 1.72108i
\(322\) 0 0
\(323\) −4.98924 −0.277609
\(324\) 0 0
\(325\) 14.4757i 0.802966i
\(326\) 0 0
\(327\) 38.5465 2.13163
\(328\) 0 0
\(329\) 2.13764 0.117852
\(330\) 0 0
\(331\) 5.23593i 0.287793i 0.989593 + 0.143897i \(0.0459632\pi\)
−0.989593 + 0.143897i \(0.954037\pi\)
\(332\) 0 0
\(333\) −18.4627 −1.01175
\(334\) 0 0
\(335\) 2.40962i 0.131651i
\(336\) 0 0
\(337\) −15.6868 −0.854513 −0.427257 0.904130i \(-0.640520\pi\)
−0.427257 + 0.904130i \(0.640520\pi\)
\(338\) 0 0
\(339\) 1.70611i 0.0926631i
\(340\) 0 0
\(341\) 2.37378i 0.128547i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 10.4502i 0.562619i
\(346\) 0 0
\(347\) 11.2286i 0.602782i −0.953501 0.301391i \(-0.902549\pi\)
0.953501 0.301391i \(-0.0974510\pi\)
\(348\) 0 0
\(349\) 15.3491 0.821617 0.410809 0.911722i \(-0.365246\pi\)
0.410809 + 0.911722i \(0.365246\pi\)
\(350\) 0 0
\(351\) 0.857940 0.0457934
\(352\) 0 0
\(353\) 2.16013 0.114972 0.0574861 0.998346i \(-0.481692\pi\)
0.0574861 + 0.998346i \(0.481692\pi\)
\(354\) 0 0
\(355\) 21.6951i 1.15146i
\(356\) 0 0
\(357\) 3.71066i 0.196389i
\(358\) 0 0
\(359\) 21.8268 1.15198 0.575988 0.817458i \(-0.304617\pi\)
0.575988 + 0.817458i \(0.304617\pi\)
\(360\) 0 0
\(361\) 8.64305 0.454898
\(362\) 0 0
\(363\) 25.2997i 1.32789i
\(364\) 0 0
\(365\) −40.2184 −2.10513
\(366\) 0 0
\(367\) 5.62271 0.293503 0.146751 0.989173i \(-0.453118\pi\)
0.146751 + 0.989173i \(0.453118\pi\)
\(368\) 0 0
\(369\) −17.3835 1.76610i −0.904947 0.0919394i
\(370\) 0 0
\(371\) −11.6378 −0.604204
\(372\) 0 0
\(373\) −19.9786 −1.03445 −0.517227 0.855848i \(-0.673036\pi\)
−0.517227 + 0.855848i \(0.673036\pi\)
\(374\) 0 0
\(375\) 56.8948i 2.93804i
\(376\) 0 0
\(377\) 5.47759 0.282110
\(378\) 0 0
\(379\) 5.35101 0.274863 0.137432 0.990511i \(-0.456115\pi\)
0.137432 + 0.990511i \(0.456115\pi\)
\(380\) 0 0
\(381\) 17.4752i 0.895283i
\(382\) 0 0
\(383\) 35.8618i 1.83245i −0.400663 0.916226i \(-0.631220\pi\)
0.400663 0.916226i \(-0.368780\pi\)
\(384\) 0 0
\(385\) 2.61838 0.133445
\(386\) 0 0
\(387\) −17.9011 −0.909964
\(388\) 0 0
\(389\) 34.7920 1.76402 0.882011 0.471228i \(-0.156189\pi\)
0.882011 + 0.471228i \(0.156189\pi\)
\(390\) 0 0
\(391\) 1.69476i 0.0857075i
\(392\) 0 0
\(393\) 13.3493i 0.673384i
\(394\) 0 0
\(395\) 19.0585i 0.958937i
\(396\) 0 0
\(397\) 7.53806i 0.378324i −0.981946 0.189162i \(-0.939423\pi\)
0.981946 0.189162i \(-0.0605772\pi\)
\(398\) 0 0
\(399\) 7.70279i 0.385622i
\(400\) 0 0
\(401\) 6.84450 0.341798 0.170899 0.985289i \(-0.445333\pi\)
0.170899 + 0.985289i \(0.445333\pi\)
\(402\) 0 0
\(403\) 4.78596i 0.238406i
\(404\) 0 0
\(405\) −38.9011 −1.93301
\(406\) 0 0
\(407\) 4.43560i 0.219864i
\(408\) 0 0
\(409\) 17.2857 0.854725 0.427363 0.904080i \(-0.359443\pi\)
0.427363 + 0.904080i \(0.359443\pi\)
\(410\) 0 0
\(411\) 7.20269 0.355282
\(412\) 0 0
\(413\) 8.90983i 0.438424i
\(414\) 0 0
\(415\) 16.3357 0.801889
\(416\) 0 0
\(417\) 11.3812i 0.557338i
\(418\) 0 0
\(419\) 27.5236 1.34462 0.672308 0.740271i \(-0.265303\pi\)
0.672308 + 0.740271i \(0.265303\pi\)
\(420\) 0 0
\(421\) 20.0785i 0.978564i −0.872126 0.489282i \(-0.837259\pi\)
0.872126 0.489282i \(-0.162741\pi\)
\(422\) 0 0
\(423\) 5.83321i 0.283620i
\(424\) 0 0
\(425\) 16.9784i 0.823575i
\(426\) 0 0
\(427\) 4.81119i 0.232830i
\(428\) 0 0
\(429\) 2.07406i 0.100137i
\(430\) 0 0
\(431\) −4.46647 −0.215142 −0.107571 0.994197i \(-0.534307\pi\)
−0.107571 + 0.994197i \(0.534307\pi\)
\(432\) 0 0
\(433\) −0.143515 −0.00689691 −0.00344845 0.999994i \(-0.501098\pi\)
−0.00344845 + 0.999994i \(0.501098\pi\)
\(434\) 0 0
\(435\) −39.6155 −1.89942
\(436\) 0 0
\(437\) 3.51807i 0.168292i
\(438\) 0 0
\(439\) 14.6831i 0.700787i −0.936603 0.350393i \(-0.886048\pi\)
0.936603 0.350393i \(-0.113952\pi\)
\(440\) 0 0
\(441\) 2.72882 0.129944
\(442\) 0 0
\(443\) −29.0704 −1.38117 −0.690587 0.723249i \(-0.742648\pi\)
−0.690587 + 0.723249i \(0.742648\pi\)
\(444\) 0 0
\(445\) 11.5294i 0.546544i
\(446\) 0 0
\(447\) −36.6737 −1.73461
\(448\) 0 0
\(449\) −27.1979 −1.28355 −0.641774 0.766894i \(-0.721801\pi\)
−0.641774 + 0.766894i \(0.721801\pi\)
\(450\) 0 0
\(451\) −0.424298 + 4.17630i −0.0199794 + 0.196654i
\(452\) 0 0
\(453\) 42.9965 2.02015
\(454\) 0 0
\(455\) −5.27912 −0.247489
\(456\) 0 0
\(457\) 1.02827i 0.0481005i −0.999711 0.0240503i \(-0.992344\pi\)
0.999711 0.0240503i \(-0.00765617\pi\)
\(458\) 0 0
\(459\) −1.00627 −0.0469688
\(460\) 0 0
\(461\) −0.925752 −0.0431166 −0.0215583 0.999768i \(-0.506863\pi\)
−0.0215583 + 0.999768i \(0.506863\pi\)
\(462\) 0 0
\(463\) 1.17805i 0.0547485i −0.999625 0.0273743i \(-0.991285\pi\)
0.999625 0.0273743i \(-0.00871458\pi\)
\(464\) 0 0
\(465\) 34.6135i 1.60516i
\(466\) 0 0
\(467\) 0.0671690 0.00310821 0.00155410 0.999999i \(-0.499505\pi\)
0.00155410 + 0.999999i \(0.499505\pi\)
\(468\) 0 0
\(469\) 0.603316 0.0278586
\(470\) 0 0
\(471\) 27.7893 1.28047
\(472\) 0 0
\(473\) 4.30066i 0.197745i
\(474\) 0 0
\(475\) 35.2448i 1.61714i
\(476\) 0 0
\(477\) 31.7574i 1.45407i
\(478\) 0 0
\(479\) 13.0763i 0.597470i 0.954336 + 0.298735i \(0.0965646\pi\)
−0.954336 + 0.298735i \(0.903435\pi\)
\(480\) 0 0
\(481\) 8.94296i 0.407764i
\(482\) 0 0
\(483\) −2.61650 −0.119055
\(484\) 0 0
\(485\) 63.4196i 2.87974i
\(486\) 0 0
\(487\) −26.5480 −1.20301 −0.601503 0.798870i \(-0.705431\pi\)
−0.601503 + 0.798870i \(0.705431\pi\)
\(488\) 0 0
\(489\) 36.2598i 1.63973i
\(490\) 0 0
\(491\) −12.5104 −0.564586 −0.282293 0.959328i \(-0.591095\pi\)
−0.282293 + 0.959328i \(0.591095\pi\)
\(492\) 0 0
\(493\) −6.42464 −0.289351
\(494\) 0 0
\(495\) 7.14507i 0.321147i
\(496\) 0 0
\(497\) −5.43199 −0.243658
\(498\) 0 0
\(499\) 25.9760i 1.16284i −0.813602 0.581422i \(-0.802496\pi\)
0.813602 0.581422i \(-0.197504\pi\)
\(500\) 0 0
\(501\) 10.2175 0.456486
\(502\) 0 0
\(503\) 42.3299i 1.88740i 0.330806 + 0.943699i \(0.392679\pi\)
−0.330806 + 0.943699i \(0.607321\pi\)
\(504\) 0 0
\(505\) 71.1745i 3.16722i
\(506\) 0 0
\(507\) 26.9337i 1.19617i
\(508\) 0 0
\(509\) 18.1242i 0.803342i 0.915784 + 0.401671i \(0.131571\pi\)
−0.915784 + 0.401671i \(0.868429\pi\)
\(510\) 0 0
\(511\) 10.0698i 0.445463i
\(512\) 0 0
\(513\) −2.08888 −0.0922263
\(514\) 0 0
\(515\) 55.0753 2.42691
\(516\) 0 0
\(517\) −1.40140 −0.0616337
\(518\) 0 0
\(519\) 28.2956i 1.24204i
\(520\) 0 0
\(521\) 17.2055i 0.753786i 0.926257 + 0.376893i \(0.123008\pi\)
−0.926257 + 0.376893i \(0.876992\pi\)
\(522\) 0 0
\(523\) 41.5379 1.81633 0.908163 0.418616i \(-0.137485\pi\)
0.908163 + 0.418616i \(0.137485\pi\)
\(524\) 0 0
\(525\) 26.2127 1.14402
\(526\) 0 0
\(527\) 5.61343i 0.244525i
\(528\) 0 0
\(529\) −21.8050 −0.948042
\(530\) 0 0
\(531\) 24.3133 1.05511
\(532\) 0 0
\(533\) 0.855460 8.42017i 0.0370541 0.364718i
\(534\) 0 0
\(535\) −51.4547 −2.22458
\(536\) 0 0
\(537\) 21.7998 0.940730
\(538\) 0 0
\(539\) 0.655586i 0.0282381i
\(540\) 0 0
\(541\) 34.2684 1.47332 0.736658 0.676266i \(-0.236403\pi\)
0.736658 + 0.676266i \(0.236403\pi\)
\(542\) 0 0
\(543\) 40.8380 1.75252
\(544\) 0 0
\(545\) 64.3214i 2.75523i
\(546\) 0 0
\(547\) 28.0006i 1.19722i −0.801041 0.598610i \(-0.795720\pi\)
0.801041 0.598610i \(-0.204280\pi\)
\(548\) 0 0
\(549\) 13.1289 0.560326
\(550\) 0 0
\(551\) −13.3366 −0.568160
\(552\) 0 0
\(553\) 4.77184 0.202919
\(554\) 0 0
\(555\) 64.6781i 2.74543i
\(556\) 0 0
\(557\) 37.4618i 1.58731i 0.608370 + 0.793654i \(0.291824\pi\)
−0.608370 + 0.793654i \(0.708176\pi\)
\(558\) 0 0
\(559\) 8.67091i 0.366741i
\(560\) 0 0
\(561\) 2.43265i 0.102707i
\(562\) 0 0
\(563\) 1.87581i 0.0790559i 0.999218 + 0.0395279i \(0.0125854\pi\)
−0.999218 + 0.0395279i \(0.987415\pi\)
\(564\) 0 0
\(565\) 2.84693 0.119771
\(566\) 0 0
\(567\) 9.74001i 0.409042i
\(568\) 0 0
\(569\) −40.1225 −1.68202 −0.841012 0.541017i \(-0.818040\pi\)
−0.841012 + 0.541017i \(0.818040\pi\)
\(570\) 0 0
\(571\) 20.3308i 0.850817i −0.905001 0.425409i \(-0.860130\pi\)
0.905001 0.425409i \(-0.139870\pi\)
\(572\) 0 0
\(573\) 45.0367 1.88144
\(574\) 0 0
\(575\) −11.9720 −0.499269
\(576\) 0 0
\(577\) 34.0647i 1.41813i 0.705143 + 0.709065i \(0.250883\pi\)
−0.705143 + 0.709065i \(0.749117\pi\)
\(578\) 0 0
\(579\) 45.7165 1.89991
\(580\) 0 0
\(581\) 4.09012i 0.169687i
\(582\) 0 0
\(583\) 7.62957 0.315985
\(584\) 0 0
\(585\) 14.4058i 0.595605i
\(586\) 0 0
\(587\) 44.2327i 1.82568i 0.408317 + 0.912840i \(0.366116\pi\)
−0.408317 + 0.912840i \(0.633884\pi\)
\(588\) 0 0
\(589\) 11.6527i 0.480141i
\(590\) 0 0
\(591\) 17.2007i 0.707544i
\(592\) 0 0
\(593\) 28.1565i 1.15625i −0.815949 0.578124i \(-0.803785\pi\)
0.815949 0.578124i \(-0.196215\pi\)
\(594\) 0 0
\(595\) 6.19186 0.253841
\(596\) 0 0
\(597\) 15.5341 0.635770
\(598\) 0 0
\(599\) 21.4765 0.877505 0.438753 0.898608i \(-0.355420\pi\)
0.438753 + 0.898608i \(0.355420\pi\)
\(600\) 0 0
\(601\) 21.6559i 0.883363i 0.897172 + 0.441681i \(0.145618\pi\)
−0.897172 + 0.441681i \(0.854382\pi\)
\(602\) 0 0
\(603\) 1.64634i 0.0670441i
\(604\) 0 0
\(605\) 42.2169 1.71636
\(606\) 0 0
\(607\) −42.1733 −1.71176 −0.855880 0.517174i \(-0.826984\pi\)
−0.855880 + 0.517174i \(0.826984\pi\)
\(608\) 0 0
\(609\) 9.91888i 0.401933i
\(610\) 0 0
\(611\) 2.82548 0.114307
\(612\) 0 0
\(613\) −35.3343 −1.42714 −0.713570 0.700584i \(-0.752923\pi\)
−0.713570 + 0.700584i \(0.752923\pi\)
\(614\) 0 0
\(615\) −6.18694 + 60.8972i −0.249482 + 2.45561i
\(616\) 0 0
\(617\) 2.82448 0.113709 0.0568546 0.998382i \(-0.481893\pi\)
0.0568546 + 0.998382i \(0.481893\pi\)
\(618\) 0 0
\(619\) 12.2616 0.492834 0.246417 0.969164i \(-0.420747\pi\)
0.246417 + 0.969164i \(0.420747\pi\)
\(620\) 0 0
\(621\) 0.709556i 0.0284735i
\(622\) 0 0
\(623\) 2.88671 0.115653
\(624\) 0 0
\(625\) 40.1804 1.60722
\(626\) 0 0
\(627\) 5.04984i 0.201671i
\(628\) 0 0
\(629\) 10.4892i 0.418230i
\(630\) 0 0
\(631\) 14.9954 0.596956 0.298478 0.954416i \(-0.403521\pi\)
0.298478 + 0.954416i \(0.403521\pi\)
\(632\) 0 0
\(633\) −6.80055 −0.270298
\(634\) 0 0
\(635\) −29.1604 −1.15719
\(636\) 0 0
\(637\) 1.32178i 0.0523708i
\(638\) 0 0
\(639\) 14.8229i 0.586385i
\(640\) 0 0
\(641\) 0.00935095i 0.000369340i 1.00000 0.000184670i \(5.87824e-5\pi\)
−1.00000 0.000184670i \(0.999941\pi\)
\(642\) 0 0
\(643\) 38.9540i 1.53619i −0.640333 0.768097i \(-0.721204\pi\)
0.640333 0.768097i \(-0.278796\pi\)
\(644\) 0 0
\(645\) 62.7106i 2.46923i
\(646\) 0 0
\(647\) −37.9934 −1.49368 −0.746838 0.665006i \(-0.768429\pi\)
−0.746838 + 0.665006i \(0.768429\pi\)
\(648\) 0 0
\(649\) 5.84116i 0.229286i
\(650\) 0 0
\(651\) −8.66648 −0.339666
\(652\) 0 0
\(653\) 28.5842i 1.11859i 0.828970 + 0.559293i \(0.188927\pi\)
−0.828970 + 0.559293i \(0.811073\pi\)
\(654\) 0 0
\(655\) −22.2756 −0.870380
\(656\) 0 0
\(657\) 27.4787 1.07205
\(658\) 0 0
\(659\) 2.10799i 0.0821157i 0.999157 + 0.0410579i \(0.0130728\pi\)
−0.999157 + 0.0410579i \(0.986927\pi\)
\(660\) 0 0
\(661\) −17.6021 −0.684642 −0.342321 0.939583i \(-0.611213\pi\)
−0.342321 + 0.939583i \(0.611213\pi\)
\(662\) 0 0
\(663\) 4.90467i 0.190482i
\(664\) 0 0
\(665\) 12.8534 0.498434
\(666\) 0 0
\(667\) 4.53022i 0.175411i
\(668\) 0 0
\(669\) 5.16619i 0.199736i
\(670\) 0 0
\(671\) 3.15415i 0.121765i
\(672\) 0 0
\(673\) 26.9746i 1.03979i 0.854229 + 0.519897i \(0.174030\pi\)
−0.854229 + 0.519897i \(0.825970\pi\)
\(674\) 0 0
\(675\) 7.10849i 0.273606i
\(676\) 0 0
\(677\) −15.8186 −0.607958 −0.303979 0.952679i \(-0.598315\pi\)
−0.303979 + 0.952679i \(0.598315\pi\)
\(678\) 0 0
\(679\) 15.8789 0.609377
\(680\) 0 0
\(681\) −11.6183 −0.445214
\(682\) 0 0
\(683\) 26.0082i 0.995177i 0.867413 + 0.497588i \(0.165781\pi\)
−0.867413 + 0.497588i \(0.834219\pi\)
\(684\) 0 0
\(685\) 12.0189i 0.459219i
\(686\) 0 0
\(687\) 42.8555 1.63504
\(688\) 0 0
\(689\) −15.3826 −0.586030
\(690\) 0 0
\(691\) 34.1744i 1.30006i −0.759910 0.650028i \(-0.774757\pi\)
0.759910 0.650028i \(-0.225243\pi\)
\(692\) 0 0
\(693\) −1.78897 −0.0679575
\(694\) 0 0
\(695\) −18.9914 −0.720385
\(696\) 0 0
\(697\) −1.00337 + 9.87598i −0.0380052 + 0.374079i
\(698\) 0 0
\(699\) 4.24950 0.160731
\(700\) 0 0
\(701\) 12.5140 0.472648 0.236324 0.971674i \(-0.424057\pi\)
0.236324 + 0.971674i \(0.424057\pi\)
\(702\) 0 0
\(703\) 21.7740i 0.821222i
\(704\) 0 0
\(705\) −20.4347 −0.769616
\(706\) 0 0
\(707\) −17.8206 −0.670211
\(708\) 0 0
\(709\) 27.6217i 1.03735i 0.854970 + 0.518677i \(0.173575\pi\)
−0.854970 + 0.518677i \(0.826425\pi\)
\(710\) 0 0
\(711\) 13.0215i 0.488343i
\(712\) 0 0
\(713\) 3.95821 0.148236
\(714\) 0 0
\(715\) 3.46092 0.129431
\(716\) 0 0
\(717\) −30.8763 −1.15310
\(718\) 0 0
\(719\) 34.2303i 1.27657i 0.769798 + 0.638287i \(0.220357\pi\)
−0.769798 + 0.638287i \(0.779643\pi\)
\(720\) 0 0
\(721\) 13.7897i 0.513554i
\(722\) 0 0
\(723\) 55.3158i 2.05722i
\(724\) 0 0
\(725\) 45.3848i 1.68555i
\(726\) 0 0
\(727\) 16.6439i 0.617288i −0.951178 0.308644i \(-0.900125\pi\)
0.951178 0.308644i \(-0.0998751\pi\)
\(728\) 0 0
\(729\) 21.9180 0.811778
\(730\) 0 0
\(731\) 10.1701i 0.376154i
\(732\) 0 0
\(733\) −27.4973 −1.01564 −0.507819 0.861464i \(-0.669548\pi\)
−0.507819 + 0.861464i \(0.669548\pi\)
\(734\) 0 0
\(735\) 9.55950i 0.352608i
\(736\) 0 0
\(737\) −0.395526 −0.0145694
\(738\) 0 0
\(739\) −34.7601 −1.27867 −0.639335 0.768929i \(-0.720790\pi\)
−0.639335 + 0.768929i \(0.720790\pi\)
\(740\) 0 0
\(741\) 10.1814i 0.374023i
\(742\) 0 0
\(743\) 3.27081 0.119994 0.0599971 0.998199i \(-0.480891\pi\)
0.0599971 + 0.998199i \(0.480891\pi\)
\(744\) 0 0
\(745\) 61.1964i 2.24206i
\(746\) 0 0
\(747\) −11.1612 −0.408366
\(748\) 0 0
\(749\) 12.8832i 0.470740i
\(750\) 0 0
\(751\) 41.0267i 1.49709i −0.663087 0.748543i \(-0.730754\pi\)
0.663087 0.748543i \(-0.269246\pi\)
\(752\) 0 0
\(753\) 51.5078i 1.87705i
\(754\) 0 0
\(755\) 71.7469i 2.61114i
\(756\) 0 0
\(757\) 24.3275i 0.884199i 0.896966 + 0.442100i \(0.145766\pi\)
−0.896966 + 0.442100i \(0.854234\pi\)
\(758\) 0 0
\(759\) 1.71534 0.0622630
\(760\) 0 0
\(761\) 2.77842 0.100718 0.0503588 0.998731i \(-0.483963\pi\)
0.0503588 + 0.998731i \(0.483963\pi\)
\(762\) 0 0
\(763\) −16.1047 −0.583029
\(764\) 0 0
\(765\) 16.8964i 0.610892i
\(766\) 0 0
\(767\) 11.7768i 0.425237i
\(768\) 0 0
\(769\) 26.0885 0.940775 0.470387 0.882460i \(-0.344114\pi\)
0.470387 + 0.882460i \(0.344114\pi\)
\(770\) 0 0
\(771\) −9.34867 −0.336684
\(772\) 0 0
\(773\) 47.8018i 1.71931i 0.510875 + 0.859655i \(0.329322\pi\)
−0.510875 + 0.859655i \(0.670678\pi\)
\(774\) 0 0
\(775\) −39.6543 −1.42442
\(776\) 0 0
\(777\) 16.1940 0.580957
\(778\) 0 0
\(779\) −2.08284 + 20.5011i −0.0746256 + 0.734529i
\(780\) 0 0
\(781\) 3.56114 0.127427
\(782\) 0 0
\(783\) −2.68985 −0.0961274
\(784\) 0 0
\(785\) 46.3712i 1.65506i
\(786\) 0 0
\(787\) −10.0915 −0.359725 −0.179862 0.983692i \(-0.557565\pi\)
−0.179862 + 0.983692i \(0.557565\pi\)
\(788\) 0 0
\(789\) 21.3702 0.760800
\(790\) 0 0
\(791\) 0.712810i 0.0253446i
\(792\) 0 0
\(793\) 6.35934i 0.225827i
\(794\) 0 0
\(795\) 111.251 3.94568
\(796\) 0 0
\(797\) 42.0491 1.48946 0.744728 0.667368i \(-0.232579\pi\)
0.744728 + 0.667368i \(0.232579\pi\)
\(798\) 0 0
\(799\) −3.31399 −0.117241
\(800\) 0 0
\(801\) 7.87729i 0.278330i
\(802\) 0 0
\(803\) 6.60164i 0.232967i
\(804\) 0 0
\(805\) 4.36608i 0.153884i
\(806\) 0