Properties

Label 160.3.h.b.159.3
Level $160$
Weight $3$
Character 160.159
Analytic conductor $4.360$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 160.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35968422976\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
Defining polynomial: \(x^{6} + 9 x^{4} + 14 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 159.3
Root \(-1.37720i\) of defining polynomial
Character \(\chi\) \(=\) 160.159
Dual form 160.3.h.b.159.4

$q$-expansion

\(f(q)\) \(=\) \(q-0.547781 q^{3} +(-3.75441 - 3.30219i) q^{5} +10.0566 q^{7} -8.69994 q^{9} +O(q^{10})\) \(q-0.547781 q^{3} +(-3.75441 - 3.30219i) q^{5} +10.0566 q^{7} -8.69994 q^{9} -17.2087i q^{11} -4.41325i q^{13} +(2.05659 + 1.80888i) q^{15} -27.0176i q^{17} -4.82650i q^{19} -5.50881 q^{21} +15.2653 q^{23} +(3.19112 + 24.7955i) q^{25} +9.69569 q^{27} -2.38225 q^{29} +38.0352i q^{31} +9.42663i q^{33} +(-37.7565 - 33.2087i) q^{35} -16.5691i q^{37} +2.41749i q^{39} -13.3177 q^{41} -59.7918 q^{43} +(32.6631 + 28.7288i) q^{45} +62.4388 q^{47} +52.1351 q^{49} +14.7997i q^{51} +71.5952i q^{53} +(-56.8265 + 64.6086i) q^{55} +2.64386i q^{57} +68.8265i q^{59} +40.9439 q^{61} -87.4917 q^{63} +(-14.5734 + 16.5691i) q^{65} +51.0080 q^{67} -8.36206 q^{69} -40.4527i q^{71} +35.8441i q^{73} +(-1.74804 - 13.5825i) q^{75} -173.061i q^{77} -126.800i q^{79} +72.9883 q^{81} +75.1490 q^{83} +(-89.2172 + 101.435i) q^{85} +1.30495 q^{87} -106.523 q^{89} -44.3822i q^{91} -20.8350i q^{93} +(-15.9380 + 18.1206i) q^{95} -85.4351i q^{97} +149.715i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4q^{3} - 2q^{5} + 12q^{7} + 18q^{9} + O(q^{10}) \) \( 6q + 4q^{3} - 2q^{5} + 12q^{7} + 18q^{9} - 36q^{15} + 8q^{21} - 68q^{23} - 10q^{25} + 184q^{27} + 44q^{29} - 108q^{35} - 68q^{41} - 76q^{43} - 6q^{45} + 268q^{47} - 62q^{49} - 288q^{55} - 100q^{61} - 172q^{63} + 308q^{67} - 184q^{69} - 284q^{75} + 238q^{81} - 204q^{83} - 32q^{85} + 584q^{87} + 76q^{89} - 32q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −0.547781 −0.182594 −0.0912969 0.995824i \(-0.529101\pi\)
−0.0912969 + 0.995824i \(0.529101\pi\)
\(4\) 0 0
\(5\) −3.75441 3.30219i −0.750881 0.660437i
\(6\) 0 0
\(7\) 10.0566 1.43666 0.718328 0.695705i \(-0.244908\pi\)
0.718328 + 0.695705i \(0.244908\pi\)
\(8\) 0 0
\(9\) −8.69994 −0.966660
\(10\) 0 0
\(11\) 17.2087i 1.56443i −0.623008 0.782216i \(-0.714089\pi\)
0.623008 0.782216i \(-0.285911\pi\)
\(12\) 0 0
\(13\) 4.41325i 0.339481i −0.985489 0.169740i \(-0.945707\pi\)
0.985489 0.169740i \(-0.0542929\pi\)
\(14\) 0 0
\(15\) 2.05659 + 1.80888i 0.137106 + 0.120592i
\(16\) 0 0
\(17\) 27.0176i 1.58927i −0.607086 0.794636i \(-0.707662\pi\)
0.607086 0.794636i \(-0.292338\pi\)
\(18\) 0 0
\(19\) 4.82650i 0.254026i −0.991901 0.127013i \(-0.959461\pi\)
0.991901 0.127013i \(-0.0405390\pi\)
\(20\) 0 0
\(21\) −5.50881 −0.262324
\(22\) 0 0
\(23\) 15.2653 0.663710 0.331855 0.943330i \(-0.392325\pi\)
0.331855 + 0.943330i \(0.392325\pi\)
\(24\) 0 0
\(25\) 3.19112 + 24.7955i 0.127645 + 0.991820i
\(26\) 0 0
\(27\) 9.69569 0.359100
\(28\) 0 0
\(29\) −2.38225 −0.0821465 −0.0410733 0.999156i \(-0.513078\pi\)
−0.0410733 + 0.999156i \(0.513078\pi\)
\(30\) 0 0
\(31\) 38.0352i 1.22694i 0.789717 + 0.613472i \(0.210228\pi\)
−0.789717 + 0.613472i \(0.789772\pi\)
\(32\) 0 0
\(33\) 9.42663i 0.285655i
\(34\) 0 0
\(35\) −37.7565 33.2087i −1.07876 0.948821i
\(36\) 0 0
\(37\) 16.5691i 0.447814i −0.974610 0.223907i \(-0.928119\pi\)
0.974610 0.223907i \(-0.0718812\pi\)
\(38\) 0 0
\(39\) 2.41749i 0.0619870i
\(40\) 0 0
\(41\) −13.3177 −0.324822 −0.162411 0.986723i \(-0.551927\pi\)
−0.162411 + 0.986723i \(0.551927\pi\)
\(42\) 0 0
\(43\) −59.7918 −1.39051 −0.695253 0.718765i \(-0.744708\pi\)
−0.695253 + 0.718765i \(0.744708\pi\)
\(44\) 0 0
\(45\) 32.6631 + 28.7288i 0.725846 + 0.638418i
\(46\) 0 0
\(47\) 62.4388 1.32849 0.664243 0.747517i \(-0.268754\pi\)
0.664243 + 0.747517i \(0.268754\pi\)
\(48\) 0 0
\(49\) 52.1351 1.06398
\(50\) 0 0
\(51\) 14.7997i 0.290191i
\(52\) 0 0
\(53\) 71.5952i 1.35085i 0.737427 + 0.675427i \(0.236041\pi\)
−0.737427 + 0.675427i \(0.763959\pi\)
\(54\) 0 0
\(55\) −56.8265 + 64.6086i −1.03321 + 1.17470i
\(56\) 0 0
\(57\) 2.64386i 0.0463836i
\(58\) 0 0
\(59\) 68.8265i 1.16655i 0.812274 + 0.583275i \(0.198229\pi\)
−0.812274 + 0.583275i \(0.801771\pi\)
\(60\) 0 0
\(61\) 40.9439 0.671212 0.335606 0.942002i \(-0.391059\pi\)
0.335606 + 0.942002i \(0.391059\pi\)
\(62\) 0 0
\(63\) −87.4917 −1.38876
\(64\) 0 0
\(65\) −14.5734 + 16.5691i −0.224206 + 0.254910i
\(66\) 0 0
\(67\) 51.0080 0.761313 0.380656 0.924717i \(-0.375698\pi\)
0.380656 + 0.924717i \(0.375698\pi\)
\(68\) 0 0
\(69\) −8.36206 −0.121189
\(70\) 0 0
\(71\) 40.4527i 0.569757i −0.958564 0.284878i \(-0.908047\pi\)
0.958564 0.284878i \(-0.0919532\pi\)
\(72\) 0 0
\(73\) 35.8441i 0.491015i 0.969395 + 0.245508i \(0.0789547\pi\)
−0.969395 + 0.245508i \(0.921045\pi\)
\(74\) 0 0
\(75\) −1.74804 13.5825i −0.0233072 0.181100i
\(76\) 0 0
\(77\) 173.061i 2.24755i
\(78\) 0 0
\(79\) 126.800i 1.60506i −0.596612 0.802530i \(-0.703487\pi\)
0.596612 0.802530i \(-0.296513\pi\)
\(80\) 0 0
\(81\) 72.9883 0.901090
\(82\) 0 0
\(83\) 75.1490 0.905409 0.452705 0.891661i \(-0.350459\pi\)
0.452705 + 0.891661i \(0.350459\pi\)
\(84\) 0 0
\(85\) −89.2172 + 101.435i −1.04961 + 1.19335i
\(86\) 0 0
\(87\) 1.30495 0.0149994
\(88\) 0 0
\(89\) −106.523 −1.19689 −0.598445 0.801164i \(-0.704215\pi\)
−0.598445 + 0.801164i \(0.704215\pi\)
\(90\) 0 0
\(91\) 44.3822i 0.487717i
\(92\) 0 0
\(93\) 20.8350i 0.224032i
\(94\) 0 0
\(95\) −15.9380 + 18.1206i −0.167768 + 0.190744i
\(96\) 0 0
\(97\) 85.4351i 0.880774i −0.897808 0.440387i \(-0.854841\pi\)
0.897808 0.440387i \(-0.145159\pi\)
\(98\) 0 0
\(99\) 149.715i 1.51227i
\(100\) 0 0
\(101\) −83.2877 −0.824631 −0.412315 0.911041i \(-0.635280\pi\)
−0.412315 + 0.911041i \(0.635280\pi\)
\(102\) 0 0
\(103\) 47.2653 0.458887 0.229443 0.973322i \(-0.426309\pi\)
0.229443 + 0.973322i \(0.426309\pi\)
\(104\) 0 0
\(105\) 20.6823 + 18.1911i 0.196974 + 0.173249i
\(106\) 0 0
\(107\) 189.628 1.77223 0.886114 0.463467i \(-0.153395\pi\)
0.886114 + 0.463467i \(0.153395\pi\)
\(108\) 0 0
\(109\) 84.5617 0.775795 0.387898 0.921702i \(-0.373201\pi\)
0.387898 + 0.921702i \(0.373201\pi\)
\(110\) 0 0
\(111\) 9.07626i 0.0817681i
\(112\) 0 0
\(113\) 114.558i 1.01379i −0.862007 0.506896i \(-0.830793\pi\)
0.862007 0.506896i \(-0.169207\pi\)
\(114\) 0 0
\(115\) −57.3123 50.4090i −0.498368 0.438339i
\(116\) 0 0
\(117\) 38.3950i 0.328162i
\(118\) 0 0
\(119\) 271.705i 2.28324i
\(120\) 0 0
\(121\) −175.141 −1.44745
\(122\) 0 0
\(123\) 7.29518 0.0593104
\(124\) 0 0
\(125\) 69.8986 103.630i 0.559189 0.829040i
\(126\) 0 0
\(127\) 44.4121 0.349701 0.174851 0.984595i \(-0.444056\pi\)
0.174851 + 0.984595i \(0.444056\pi\)
\(128\) 0 0
\(129\) 32.7528 0.253898
\(130\) 0 0
\(131\) 47.1200i 0.359695i −0.983695 0.179847i \(-0.942440\pi\)
0.983695 0.179847i \(-0.0575604\pi\)
\(132\) 0 0
\(133\) 48.5381i 0.364948i
\(134\) 0 0
\(135\) −36.4016 32.0170i −0.269641 0.237163i
\(136\) 0 0
\(137\) 62.8617i 0.458845i −0.973327 0.229422i \(-0.926316\pi\)
0.973327 0.229422i \(-0.0736837\pi\)
\(138\) 0 0
\(139\) 128.320i 0.923167i 0.887097 + 0.461584i \(0.152719\pi\)
−0.887097 + 0.461584i \(0.847281\pi\)
\(140\) 0 0
\(141\) −34.2028 −0.242573
\(142\) 0 0
\(143\) −75.9465 −0.531094
\(144\) 0 0
\(145\) 8.94393 + 7.86663i 0.0616823 + 0.0542526i
\(146\) 0 0
\(147\) −28.5586 −0.194276
\(148\) 0 0
\(149\) −165.561 −1.11115 −0.555574 0.831467i \(-0.687501\pi\)
−0.555574 + 0.831467i \(0.687501\pi\)
\(150\) 0 0
\(151\) 218.558i 1.44741i 0.690111 + 0.723704i \(0.257562\pi\)
−0.690111 + 0.723704i \(0.742438\pi\)
\(152\) 0 0
\(153\) 235.052i 1.53628i
\(154\) 0 0
\(155\) 125.599 142.800i 0.810319 0.921289i
\(156\) 0 0
\(157\) 174.293i 1.11014i 0.831802 + 0.555072i \(0.187309\pi\)
−0.831802 + 0.555072i \(0.812691\pi\)
\(158\) 0 0
\(159\) 39.2185i 0.246657i
\(160\) 0 0
\(161\) 153.517 0.953524
\(162\) 0 0
\(163\) −52.6353 −0.322916 −0.161458 0.986880i \(-0.551620\pi\)
−0.161458 + 0.986880i \(0.551620\pi\)
\(164\) 0 0
\(165\) 31.1285 35.3914i 0.188657 0.214493i
\(166\) 0 0
\(167\) −76.6812 −0.459169 −0.229584 0.973289i \(-0.573737\pi\)
−0.229584 + 0.973289i \(0.573737\pi\)
\(168\) 0 0
\(169\) 149.523 0.884753
\(170\) 0 0
\(171\) 41.9902i 0.245557i
\(172\) 0 0
\(173\) 9.72437i 0.0562102i −0.999605 0.0281051i \(-0.991053\pi\)
0.999605 0.0281051i \(-0.00894731\pi\)
\(174\) 0 0
\(175\) 32.0918 + 249.358i 0.183382 + 1.42490i
\(176\) 0 0
\(177\) 37.7019i 0.213005i
\(178\) 0 0
\(179\) 155.502i 0.868728i 0.900738 + 0.434364i \(0.143027\pi\)
−0.900738 + 0.434364i \(0.856973\pi\)
\(180\) 0 0
\(181\) 250.346 1.38313 0.691563 0.722316i \(-0.256923\pi\)
0.691563 + 0.722316i \(0.256923\pi\)
\(182\) 0 0
\(183\) −22.4283 −0.122559
\(184\) 0 0
\(185\) −54.7144 + 62.2072i −0.295753 + 0.336255i
\(186\) 0 0
\(187\) −464.939 −2.48631
\(188\) 0 0
\(189\) 97.5056 0.515903
\(190\) 0 0
\(191\) 111.128i 0.581825i −0.956750 0.290912i \(-0.906041\pi\)
0.956750 0.290912i \(-0.0939588\pi\)
\(192\) 0 0
\(193\) 10.2701i 0.0532130i 0.999646 + 0.0266065i \(0.00847011\pi\)
−0.999646 + 0.0266065i \(0.991530\pi\)
\(194\) 0 0
\(195\) 7.98302 9.07626i 0.0409386 0.0465449i
\(196\) 0 0
\(197\) 163.213i 0.828492i 0.910165 + 0.414246i \(0.135955\pi\)
−0.910165 + 0.414246i \(0.864045\pi\)
\(198\) 0 0
\(199\) 195.028i 0.980041i 0.871711 + 0.490021i \(0.163011\pi\)
−0.871711 + 0.490021i \(0.836989\pi\)
\(200\) 0 0
\(201\) −27.9412 −0.139011
\(202\) 0 0
\(203\) −23.9573 −0.118016
\(204\) 0 0
\(205\) 50.0000 + 43.9775i 0.243902 + 0.214524i
\(206\) 0 0
\(207\) −132.807 −0.641582
\(208\) 0 0
\(209\) −83.0580 −0.397407
\(210\) 0 0
\(211\) 297.038i 1.40776i −0.710317 0.703881i \(-0.751449\pi\)
0.710317 0.703881i \(-0.248551\pi\)
\(212\) 0 0
\(213\) 22.1592i 0.104034i
\(214\) 0 0
\(215\) 224.483 + 197.444i 1.04411 + 0.918342i
\(216\) 0 0
\(217\) 382.505i 1.76270i
\(218\) 0 0
\(219\) 19.6347i 0.0896563i
\(220\) 0 0
\(221\) −119.236 −0.539527
\(222\) 0 0
\(223\) 405.335 1.81764 0.908822 0.417184i \(-0.136983\pi\)
0.908822 + 0.417184i \(0.136983\pi\)
\(224\) 0 0
\(225\) −27.7626 215.719i −0.123389 0.958752i
\(226\) 0 0
\(227\) −36.6013 −0.161239 −0.0806196 0.996745i \(-0.525690\pi\)
−0.0806196 + 0.996745i \(0.525690\pi\)
\(228\) 0 0
\(229\) −91.1467 −0.398021 −0.199010 0.979997i \(-0.563773\pi\)
−0.199010 + 0.979997i \(0.563773\pi\)
\(230\) 0 0
\(231\) 94.7997i 0.410388i
\(232\) 0 0
\(233\) 338.802i 1.45409i −0.686592 0.727043i \(-0.740894\pi\)
0.686592 0.727043i \(-0.259106\pi\)
\(234\) 0 0
\(235\) −234.421 206.185i −0.997535 0.877382i
\(236\) 0 0
\(237\) 69.4585i 0.293074i
\(238\) 0 0
\(239\) 30.7997i 0.128869i −0.997922 0.0644346i \(-0.979476\pi\)
0.997922 0.0644346i \(-0.0205244\pi\)
\(240\) 0 0
\(241\) 240.282 0.997020 0.498510 0.866884i \(-0.333881\pi\)
0.498510 + 0.866884i \(0.333881\pi\)
\(242\) 0 0
\(243\) −127.243 −0.523633
\(244\) 0 0
\(245\) −195.736 172.160i −0.798923 0.702693i
\(246\) 0 0
\(247\) −21.3005 −0.0862370
\(248\) 0 0
\(249\) −41.1652 −0.165322
\(250\) 0 0
\(251\) 86.0268i 0.342736i 0.985207 + 0.171368i \(0.0548187\pi\)
−0.985207 + 0.171368i \(0.945181\pi\)
\(252\) 0 0
\(253\) 262.697i 1.03833i
\(254\) 0 0
\(255\) 48.8715 55.5642i 0.191653 0.217899i
\(256\) 0 0
\(257\) 355.963i 1.38507i 0.721383 + 0.692536i \(0.243507\pi\)
−0.721383 + 0.692536i \(0.756493\pi\)
\(258\) 0 0
\(259\) 166.629i 0.643355i
\(260\) 0 0
\(261\) 20.7254 0.0794077
\(262\) 0 0
\(263\) −73.1254 −0.278043 −0.139022 0.990289i \(-0.544396\pi\)
−0.139022 + 0.990289i \(0.544396\pi\)
\(264\) 0 0
\(265\) 236.421 268.798i 0.892154 1.01433i
\(266\) 0 0
\(267\) 58.3514 0.218545
\(268\) 0 0
\(269\) 268.002 0.996290 0.498145 0.867094i \(-0.334015\pi\)
0.498145 + 0.867094i \(0.334015\pi\)
\(270\) 0 0
\(271\) 229.412i 0.846538i 0.906004 + 0.423269i \(0.139117\pi\)
−0.906004 + 0.423269i \(0.860883\pi\)
\(272\) 0 0
\(273\) 24.3118i 0.0890541i
\(274\) 0 0
\(275\) 426.699 54.9153i 1.55163 0.199692i
\(276\) 0 0
\(277\) 126.327i 0.456053i −0.973655 0.228026i \(-0.926773\pi\)
0.973655 0.228026i \(-0.0732272\pi\)
\(278\) 0 0
\(279\) 330.904i 1.18604i
\(280\) 0 0
\(281\) −458.746 −1.63255 −0.816275 0.577664i \(-0.803964\pi\)
−0.816275 + 0.577664i \(0.803964\pi\)
\(282\) 0 0
\(283\) 465.558 1.64508 0.822541 0.568706i \(-0.192556\pi\)
0.822541 + 0.568706i \(0.192556\pi\)
\(284\) 0 0
\(285\) 8.73053 9.92614i 0.0306335 0.0348286i
\(286\) 0 0
\(287\) −133.931 −0.466657
\(288\) 0 0
\(289\) −440.952 −1.52579
\(290\) 0 0
\(291\) 46.7997i 0.160824i
\(292\) 0 0
\(293\) 165.218i 0.563884i 0.959431 + 0.281942i \(0.0909786\pi\)
−0.959431 + 0.281942i \(0.909021\pi\)
\(294\) 0 0
\(295\) 227.278 258.403i 0.770434 0.875941i
\(296\) 0 0
\(297\) 166.851i 0.561787i
\(298\) 0 0
\(299\) 67.3697i 0.225317i
\(300\) 0 0
\(301\) −601.302 −1.99768
\(302\) 0 0
\(303\) 45.6234 0.150572
\(304\) 0 0
\(305\) −153.720 135.205i −0.504000 0.443293i
\(306\) 0 0
\(307\) −235.339 −0.766577 −0.383288 0.923629i \(-0.625208\pi\)
−0.383288 + 0.923629i \(0.625208\pi\)
\(308\) 0 0
\(309\) −25.8911 −0.0837898
\(310\) 0 0
\(311\) 210.665i 0.677381i −0.940898 0.338691i \(-0.890016\pi\)
0.940898 0.338691i \(-0.109984\pi\)
\(312\) 0 0
\(313\) 318.738i 1.01833i −0.860668 0.509166i \(-0.829954\pi\)
0.860668 0.509166i \(-0.170046\pi\)
\(314\) 0 0
\(315\) 328.479 + 288.914i 1.04279 + 0.917187i
\(316\) 0 0
\(317\) 70.3950i 0.222066i −0.993817 0.111033i \(-0.964584\pi\)
0.993817 0.111033i \(-0.0354160\pi\)
\(318\) 0 0
\(319\) 40.9955i 0.128513i
\(320\) 0 0
\(321\) −103.875 −0.323598
\(322\) 0 0
\(323\) −130.401 −0.403717
\(324\) 0 0
\(325\) 109.429 14.0832i 0.336704 0.0433330i
\(326\) 0 0
\(327\) −46.3213 −0.141655
\(328\) 0 0
\(329\) 627.922 1.90858
\(330\) 0 0
\(331\) 280.807i 0.848359i −0.905578 0.424180i \(-0.860562\pi\)
0.905578 0.424180i \(-0.139438\pi\)
\(332\) 0 0
\(333\) 144.150i 0.432884i
\(334\) 0 0
\(335\) −191.505 168.438i −0.571656 0.502800i
\(336\) 0 0
\(337\) 120.969i 0.358959i 0.983762 + 0.179480i \(0.0574414\pi\)
−0.983762 + 0.179480i \(0.942559\pi\)
\(338\) 0 0
\(339\) 62.7530i 0.185112i
\(340\) 0 0
\(341\) 654.539 1.91947
\(342\) 0 0
\(343\) 31.5280 0.0919182
\(344\) 0 0
\(345\) 31.3946 + 27.6131i 0.0909988 + 0.0800380i
\(346\) 0 0
\(347\) −214.333 −0.617675 −0.308838 0.951115i \(-0.599940\pi\)
−0.308838 + 0.951115i \(0.599940\pi\)
\(348\) 0 0
\(349\) −418.041 −1.19782 −0.598912 0.800815i \(-0.704400\pi\)
−0.598912 + 0.800815i \(0.704400\pi\)
\(350\) 0 0
\(351\) 42.7895i 0.121907i
\(352\) 0 0
\(353\) 364.929i 1.03379i 0.856048 + 0.516897i \(0.172913\pi\)
−0.856048 + 0.516897i \(0.827087\pi\)
\(354\) 0 0
\(355\) −133.583 + 151.876i −0.376289 + 0.427820i
\(356\) 0 0
\(357\) 148.835i 0.416905i
\(358\) 0 0
\(359\) 242.169i 0.674567i 0.941403 + 0.337283i \(0.109508\pi\)
−0.941403 + 0.337283i \(0.890492\pi\)
\(360\) 0 0
\(361\) 337.705 0.935471
\(362\) 0 0
\(363\) 95.9389 0.264295
\(364\) 0 0
\(365\) 118.364 134.573i 0.324285 0.368694i
\(366\) 0 0
\(367\) −224.564 −0.611891 −0.305945 0.952049i \(-0.598972\pi\)
−0.305945 + 0.952049i \(0.598972\pi\)
\(368\) 0 0
\(369\) 115.863 0.313992
\(370\) 0 0
\(371\) 720.004i 1.94071i
\(372\) 0 0
\(373\) 572.174i 1.53398i −0.641660 0.766989i \(-0.721754\pi\)
0.641660 0.766989i \(-0.278246\pi\)
\(374\) 0 0
\(375\) −38.2891 + 56.7666i −0.102104 + 0.151378i
\(376\) 0 0
\(377\) 10.5135i 0.0278872i
\(378\) 0 0
\(379\) 122.896i 0.324263i 0.986769 + 0.162132i \(0.0518369\pi\)
−0.986769 + 0.162132i \(0.948163\pi\)
\(380\) 0 0
\(381\) −24.3281 −0.0638533
\(382\) 0 0
\(383\) −289.717 −0.756442 −0.378221 0.925715i \(-0.623464\pi\)
−0.378221 + 0.925715i \(0.623464\pi\)
\(384\) 0 0
\(385\) −571.481 + 649.743i −1.48437 + 1.68764i
\(386\) 0 0
\(387\) 520.185 1.34415
\(388\) 0 0
\(389\) 111.590 0.286863 0.143432 0.989660i \(-0.454186\pi\)
0.143432 + 0.989660i \(0.454186\pi\)
\(390\) 0 0
\(391\) 412.433i 1.05482i
\(392\) 0 0
\(393\) 25.8114i 0.0656780i
\(394\) 0 0
\(395\) −418.716 + 476.058i −1.06004 + 1.20521i
\(396\) 0 0
\(397\) 705.314i 1.77661i 0.459253 + 0.888305i \(0.348117\pi\)
−0.459253 + 0.888305i \(0.651883\pi\)
\(398\) 0 0
\(399\) 26.5883i 0.0666373i
\(400\) 0 0
\(401\) 432.052 1.07744 0.538718 0.842486i \(-0.318909\pi\)
0.538718 + 0.842486i \(0.318909\pi\)
\(402\) 0 0
\(403\) 167.859 0.416524
\(404\) 0 0
\(405\) −274.028 241.021i −0.676612 0.595114i
\(406\) 0 0
\(407\) −285.134 −0.700575
\(408\) 0 0
\(409\) −98.8102 −0.241590 −0.120795 0.992677i \(-0.538544\pi\)
−0.120795 + 0.992677i \(0.538544\pi\)
\(410\) 0 0
\(411\) 34.4345i 0.0837822i
\(412\) 0 0
\(413\) 692.160i 1.67593i
\(414\) 0 0
\(415\) −282.140 248.156i −0.679855 0.597966i
\(416\) 0 0
\(417\) 70.2914i 0.168565i
\(418\) 0 0
\(419\) 204.980i 0.489213i −0.969622 0.244607i \(-0.921341\pi\)
0.969622 0.244607i \(-0.0786588\pi\)
\(420\) 0 0
\(421\) 449.956 1.06878 0.534390 0.845238i \(-0.320541\pi\)
0.534390 + 0.845238i \(0.320541\pi\)
\(422\) 0 0
\(423\) −543.214 −1.28419
\(424\) 0 0
\(425\) 669.915 86.2166i 1.57627 0.202863i
\(426\) 0 0
\(427\) 411.756 0.964301
\(428\) 0 0
\(429\) 41.6021 0.0969745
\(430\) 0 0
\(431\) 314.588i 0.729903i −0.931026 0.364952i \(-0.881086\pi\)
0.931026 0.364952i \(-0.118914\pi\)
\(432\) 0 0
\(433\) 330.441i 0.763143i −0.924339 0.381571i \(-0.875383\pi\)
0.924339 0.381571i \(-0.124617\pi\)
\(434\) 0 0
\(435\) −4.89932 4.30919i −0.0112628 0.00990619i
\(436\) 0 0
\(437\) 73.6781i 0.168600i
\(438\) 0 0
\(439\) 664.815i 1.51439i 0.653191 + 0.757193i \(0.273430\pi\)
−0.653191 + 0.757193i \(0.726570\pi\)
\(440\) 0 0
\(441\) −453.572 −1.02851
\(442\) 0 0
\(443\) −312.860 −0.706229 −0.353115 0.935580i \(-0.614877\pi\)
−0.353115 + 0.935580i \(0.614877\pi\)
\(444\) 0 0
\(445\) 399.931 + 351.760i 0.898722 + 0.790471i
\(446\) 0 0
\(447\) 90.6912 0.202889
\(448\) 0 0
\(449\) −246.330 −0.548620 −0.274310 0.961641i \(-0.588449\pi\)
−0.274310 + 0.961641i \(0.588449\pi\)
\(450\) 0 0
\(451\) 229.181i 0.508161i
\(452\) 0 0
\(453\) 119.722i 0.264287i
\(454\) 0 0
\(455\) −146.558 + 166.629i −0.322107 + 0.366218i
\(456\) 0 0
\(457\) 707.094i 1.54725i −0.633642 0.773626i \(-0.718441\pi\)
0.633642 0.773626i \(-0.281559\pi\)
\(458\) 0 0
\(459\) 261.955i 0.570707i
\(460\) 0 0
\(461\) −611.288 −1.32600 −0.663002 0.748618i \(-0.730718\pi\)
−0.663002 + 0.748618i \(0.730718\pi\)
\(462\) 0 0
\(463\) 334.063 0.721519 0.360760 0.932659i \(-0.382518\pi\)
0.360760 + 0.932659i \(0.382518\pi\)
\(464\) 0 0
\(465\) −68.8010 + 78.2230i −0.147959 + 0.168222i
\(466\) 0 0
\(467\) −128.864 −0.275939 −0.137970 0.990436i \(-0.544058\pi\)
−0.137970 + 0.990436i \(0.544058\pi\)
\(468\) 0 0
\(469\) 512.966 1.09374
\(470\) 0 0
\(471\) 95.4742i 0.202705i
\(472\) 0 0
\(473\) 1028.94i 2.17535i
\(474\) 0 0
\(475\) 119.675 15.4020i 0.251948 0.0324252i
\(476\) 0 0
\(477\) 622.874i 1.30582i
\(478\) 0 0
\(479\) 510.257i 1.06525i 0.846350 + 0.532627i \(0.178795\pi\)
−0.846350 + 0.532627i \(0.821205\pi\)
\(480\) 0 0
\(481\) −73.1237 −0.152024
\(482\) 0 0
\(483\) −84.0939 −0.174107
\(484\) 0 0
\(485\) −282.123 + 320.758i −0.581696 + 0.661357i
\(486\) 0 0
\(487\) 616.472 1.26586 0.632928 0.774211i \(-0.281853\pi\)
0.632928 + 0.774211i \(0.281853\pi\)
\(488\) 0 0
\(489\) 28.8326 0.0589624
\(490\) 0 0
\(491\) 603.190i 1.22849i 0.789114 + 0.614247i \(0.210540\pi\)
−0.789114 + 0.614247i \(0.789460\pi\)
\(492\) 0 0
\(493\) 64.3627i 0.130553i
\(494\) 0 0
\(495\) 494.387 562.091i 0.998761 1.13554i
\(496\) 0 0
\(497\) 406.817i 0.818545i
\(498\) 0 0
\(499\) 867.976i 1.73943i −0.493553 0.869716i \(-0.664302\pi\)
0.493553 0.869716i \(-0.335698\pi\)
\(500\) 0 0
\(501\) 42.0045 0.0838413
\(502\) 0 0
\(503\) 57.8408 0.114992 0.0574958 0.998346i \(-0.481688\pi\)
0.0574958 + 0.998346i \(0.481688\pi\)
\(504\) 0 0
\(505\) 312.696 + 275.032i 0.619200 + 0.544617i
\(506\) 0 0
\(507\) −81.9060 −0.161550
\(508\) 0 0
\(509\) −168.498 −0.331038 −0.165519 0.986207i \(-0.552930\pi\)
−0.165519 + 0.986207i \(0.552930\pi\)
\(510\) 0 0
\(511\) 360.470i 0.705420i
\(512\) 0 0
\(513\) 46.7962i 0.0912207i
\(514\) 0 0
\(515\) −177.453 156.079i −0.344569 0.303066i
\(516\) 0 0
\(517\) 1074.49i 2.07833i
\(518\) 0 0
\(519\) 5.32682i 0.0102636i
\(520\) 0 0
\(521\) −87.1060 −0.167190 −0.0835951 0.996500i \(-0.526640\pi\)
−0.0835951 + 0.996500i \(0.526640\pi\)
\(522\) 0 0
\(523\) −243.469 −0.465524 −0.232762 0.972534i \(-0.574776\pi\)
−0.232762 + 0.972534i \(0.574776\pi\)
\(524\) 0 0
\(525\) −17.5793 136.594i −0.0334844 0.260179i
\(526\) 0 0
\(527\) 1027.62 1.94995
\(528\) 0 0
\(529\) −295.969 −0.559488
\(530\) 0 0
\(531\) 598.786i 1.12766i
\(532\) 0 0
\(533\) 58.7743i 0.110271i
\(534\) 0 0
\(535\) −711.942 626.189i −1.33073 1.17045i
\(536\) 0 0
\(537\) 85.1812i 0.158624i
\(538\) 0 0
\(539\) 897.179i 1.66452i
\(540\) 0 0
\(541\) −812.616 −1.50206 −0.751032 0.660266i \(-0.770443\pi\)
−0.751032 + 0.660266i \(0.770443\pi\)
\(542\) 0 0
\(543\) −137.135 −0.252550
\(544\) 0 0
\(545\) −317.479 279.238i −0.582530 0.512364i
\(546\) 0 0
\(547\) 104.713 0.191431 0.0957155 0.995409i \(-0.469486\pi\)
0.0957155 + 0.995409i \(0.469486\pi\)
\(548\) 0 0
\(549\) −356.210 −0.648833
\(550\) 0 0
\(551\) 11.4979i 0.0208674i
\(552\) 0 0
\(553\) 1275.17i 2.30592i
\(554\) 0 0
\(555\) 29.9715 34.0759i 0.0540027 0.0613981i
\(556\) 0 0
\(557\) 260.018i 0.466818i −0.972379 0.233409i \(-0.925012\pi\)
0.972379 0.233409i \(-0.0749882\pi\)
\(558\) 0 0
\(559\) 263.876i 0.472050i
\(560\) 0 0
\(561\) 254.685 0.453984
\(562\) 0 0
\(563\) −39.9073 −0.0708833 −0.0354416 0.999372i \(-0.511284\pi\)
−0.0354416 + 0.999372i \(0.511284\pi\)
\(564\) 0 0
\(565\) −378.293 + 430.099i −0.669546 + 0.761237i
\(566\) 0 0
\(567\) 734.014 1.29456
\(568\) 0 0
\(569\) −211.588 −0.371860 −0.185930 0.982563i \(-0.559530\pi\)
−0.185930 + 0.982563i \(0.559530\pi\)
\(570\) 0 0
\(571\) 556.938i 0.975372i −0.873019 0.487686i \(-0.837841\pi\)
0.873019 0.487686i \(-0.162159\pi\)
\(572\) 0 0
\(573\) 60.8741i 0.106237i
\(574\) 0 0
\(575\) 48.7136 + 378.512i 0.0847193 + 0.658281i
\(576\) 0 0
\(577\) 516.233i 0.894684i 0.894363 + 0.447342i \(0.147629\pi\)
−0.894363 + 0.447342i \(0.852371\pi\)
\(578\) 0 0
\(579\) 5.62577i 0.00971636i
\(580\) 0 0
\(581\) 755.742 1.30076
\(582\) 0 0
\(583\) 1232.06 2.11332
\(584\) 0 0
\(585\) 126.787 144.150i 0.216731 0.246411i
\(586\) 0 0
\(587\) 934.677 1.59229 0.796147 0.605103i \(-0.206868\pi\)
0.796147 + 0.605103i \(0.206868\pi\)
\(588\) 0 0
\(589\) 183.577 0.311676
\(590\) 0 0
\(591\) 89.4050i 0.151277i
\(592\) 0 0
\(593\) 634.665i 1.07026i 0.844769 + 0.535131i \(0.179738\pi\)
−0.844769 + 0.535131i \(0.820262\pi\)
\(594\) 0 0
\(595\) −897.221 + 1020.09i −1.50794 + 1.71444i
\(596\) 0 0
\(597\) 106.833i 0.178949i
\(598\) 0 0
\(599\) 914.029i 1.52593i 0.646443 + 0.762963i \(0.276256\pi\)
−0.646443 + 0.762963i \(0.723744\pi\)
\(600\) 0 0
\(601\) −598.569 −0.995955 −0.497977 0.867190i \(-0.665924\pi\)
−0.497977 + 0.867190i \(0.665924\pi\)
\(602\) 0 0
\(603\) −443.766 −0.735930
\(604\) 0 0
\(605\) 657.550 + 578.348i 1.08686 + 0.955948i
\(606\) 0 0
\(607\) −201.830 −0.332504 −0.166252 0.986083i \(-0.553166\pi\)
−0.166252 + 0.986083i \(0.553166\pi\)
\(608\) 0 0
\(609\) 13.1234 0.0215490
\(610\) 0 0
\(611\) 275.558i 0.450995i
\(612\) 0 0
\(613\) 71.9205i 0.117325i 0.998278 + 0.0586627i \(0.0186836\pi\)
−0.998278 + 0.0586627i \(0.981316\pi\)
\(614\) 0 0
\(615\) −27.3891 24.0900i −0.0445350 0.0391708i
\(616\) 0 0
\(617\) 204.485i 0.331418i −0.986175 0.165709i \(-0.947009\pi\)
0.986175 0.165709i \(-0.0529912\pi\)
\(618\) 0 0
\(619\) 287.512i 0.464479i −0.972659 0.232239i \(-0.925395\pi\)
0.972659 0.232239i \(-0.0746053\pi\)
\(620\) 0 0
\(621\) 148.008 0.238338
\(622\) 0 0
\(623\) −1071.26 −1.71952
\(624\) 0 0
\(625\) −604.633 + 158.251i −0.967414 + 0.253202i
\(626\) 0 0
\(627\) 45.4976 0.0725640
\(628\) 0 0
\(629\) −447.658 −0.711699
\(630\) 0 0
\(631\) 274.203i 0.434554i 0.976110 + 0.217277i \(0.0697175\pi\)
−0.976110 + 0.217277i \(0.930283\pi\)
\(632\) 0 0
\(633\) 162.712i 0.257049i
\(634\) 0 0
\(635\) −166.741 146.657i −0.262584 0.230956i
\(636\) 0 0
\(637\) 230.085i 0.361201i
\(638\) 0 0
\(639\) 351.936i 0.550761i
\(640\) 0 0
\(641\) 681.328 1.06291 0.531457 0.847085i \(-0.321645\pi\)
0.531457 + 0.847085i \(0.321645\pi\)
\(642\) 0 0
\(643\) 527.270 0.820016 0.410008 0.912082i \(-0.365526\pi\)
0.410008 + 0.912082i \(0.365526\pi\)
\(644\) 0 0
\(645\) −122.967 108.156i −0.190647 0.167684i
\(646\) 0 0
\(647\) −551.627 −0.852592 −0.426296 0.904584i \(-0.640182\pi\)
−0.426296 + 0.904584i \(0.640182\pi\)
\(648\) 0 0
\(649\) 1184.42 1.82499
\(650\) 0 0
\(651\) 209.529i 0.321857i
\(652\) 0 0
\(653\) 699.422i 1.07109i −0.844507 0.535545i \(-0.820106\pi\)
0.844507 0.535545i \(-0.179894\pi\)
\(654\) 0 0
\(655\) −155.599 + 176.908i −0.237556 + 0.270088i
\(656\) 0 0
\(657\) 311.842i 0.474645i
\(658\) 0 0
\(659\) 38.3257i 0.0581574i 0.999577 + 0.0290787i \(0.00925734\pi\)
−0.999577 + 0.0290787i \(0.990743\pi\)
\(660\) 0 0
\(661\) 392.364 0.593591 0.296796 0.954941i \(-0.404082\pi\)
0.296796 + 0.954941i \(0.404082\pi\)
\(662\) 0 0
\(663\) 65.3150 0.0985143
\(664\) 0 0
\(665\) −160.282 + 182.232i −0.241026 + 0.274033i
\(666\) 0 0
\(667\) −36.3658 −0.0545215
\(668\) 0 0
\(669\) −222.035 −0.331890
\(670\) 0 0
\(671\) 704.594i 1.05007i
\(672\) 0 0
\(673\) 427.290i 0.634904i 0.948274 + 0.317452i \(0.102827\pi\)
−0.948274 + 0.317452i \(0.897173\pi\)
\(674\) 0 0
\(675\) 30.9402 + 240.409i 0.0458373 + 0.356162i
\(676\) 0 0
\(677\) 131.234i 0.193847i 0.995292 + 0.0969233i \(0.0309002\pi\)
−0.995292 + 0.0969233i \(0.969100\pi\)
\(678\) 0 0
\(679\) 859.186i 1.26537i
\(680\) 0 0
\(681\) 20.0495 0.0294413
\(682\) 0 0
\(683\) −896.229 −1.31219 −0.656097 0.754676i \(-0.727794\pi\)
−0.656097 + 0.754676i \(0.727794\pi\)
\(684\) 0 0
\(685\) −207.581 + 236.008i −0.303038 + 0.344538i
\(686\) 0 0
\(687\) 49.9285 0.0726761
\(688\) 0 0
\(689\) 315.968 0.458589
\(690\) 0 0
\(691\) 520.465i 0.753206i 0.926375 + 0.376603i \(0.122908\pi\)
−0.926375 + 0.376603i \(0.877092\pi\)
\(692\) 0 0
\(693\) 1505.62i 2.17262i
\(694\) 0 0
\(695\) 423.737 481.766i 0.609694 0.693189i
\(696\) 0 0
\(697\) 359.812i 0.516230i
\(698\) 0 0
\(699\) 185.589i 0.265507i
\(700\) 0 0
\(701\) 80.7527 0.115196 0.0575982 0.998340i \(-0.481656\pi\)
0.0575982 + 0.998340i \(0.481656\pi\)
\(702\) 0 0
\(703\) −79.9709 −0.113757
\(704\) 0 0
\(705\) 128.411 + 112.944i 0.182144 + 0.160204i
\(706\) 0 0
\(707\) −837.591 −1.18471
\(708\) 0 0
\(709\) −174.828 −0.246584 −0.123292 0.992370i \(-0.539345\pi\)
−0.123292 + 0.992370i \(0.539345\pi\)
\(710\) 0 0
\(711\) 1103.15i 1.55155i
\(712\) 0 0
\(713\) 580.621i 0.814335i
\(714\) 0 0
\(715\) 285.134 + 250.789i 0.398789 + 0.350755i
\(716\) 0 0
\(717\) 16.8715i 0.0235307i
\(718\) 0 0
\(719\) 889.905i 1.23770i 0.785510 + 0.618849i \(0.212401\pi\)
−0.785510 + 0.618849i \(0.787599\pi\)
\(720\) 0 0
\(721\) 475.328 0.659263
\(722\) 0 0
\(723\) −131.622 −0.182050
\(724\) 0 0
\(725\) −7.60205 59.0690i −0.0104856 0.0814745i
\(726\) 0 0
\(727\) −408.818 −0.562335 −0.281168 0.959659i \(-0.590722\pi\)
−0.281168 + 0.959659i \(0.590722\pi\)
\(728\) 0 0
\(729\) −587.194 −0.805478
\(730\) 0 0
\(731\) 1615.43i 2.20989i
\(732\) 0 0
\(733\) 388.369i 0.529835i −0.964271 0.264917i \(-0.914655\pi\)
0.964271 0.264917i \(-0.0853446\pi\)
\(734\) 0 0
\(735\) 107.221 + 94.3058i 0.145878 + 0.128307i
\(736\) 0 0
\(737\) 877.783i 1.19102i
\(738\) 0 0
\(739\) 109.561i 0.148256i 0.997249 + 0.0741280i \(0.0236173\pi\)
−0.997249 + 0.0741280i \(0.976383\pi\)
\(740\) 0 0
\(741\) 11.6680 0.0157463
\(742\) 0 0
\(743\) −732.717 −0.986160 −0.493080 0.869984i \(-0.664129\pi\)
−0.493080 + 0.869984i \(0.664129\pi\)
\(744\) 0 0
\(745\) 621.583 + 546.713i 0.834340 + 0.733844i
\(746\) 0 0
\(747\) −653.791 −0.875222
\(748\) 0 0
\(749\) 1907.02 2.54608
\(750\) 0 0
\(751\) 420.809i 0.560332i −0.959952 0.280166i \(-0.909610\pi\)
0.959952 0.280166i \(-0.0903895\pi\)
\(752\) 0 0
\(753\) 47.1238i 0.0625814i
\(754\) 0 0
\(755\) 721.721 820.557i 0.955922 1.08683i
\(756\) 0 0
\(757\) 1306.99i 1.72654i 0.504744 + 0.863269i \(0.331587\pi\)
−0.504744 + 0.863269i \(0.668413\pi\)
\(758\) 0 0
\(759\) 143.901i 0.189592i
\(760\) 0 0
\(761\) −1402.25 −1.84264 −0.921320 0.388804i \(-0.872888\pi\)
−0.921320 + 0.388804i \(0.872888\pi\)
\(762\) 0 0
\(763\) 850.402 1.11455
\(764\) 0 0
\(765\) 776.184 882.479i 1.01462 1.15357i
\(766\) 0 0
\(767\) 303.748 0.396021
\(768\) 0 0
\(769\) −359.952 −0.468078 −0.234039 0.972227i \(-0.575194\pi\)
−0.234039 + 0.972227i \(0.575194\pi\)
\(770\) 0 0
\(771\) 194.990i 0.252905i
\(772\) 0 0
\(773\) 437.539i 0.566027i −0.959116 0.283013i \(-0.908666\pi\)
0.959116 0.283013i \(-0.0913341\pi\)
\(774\) 0 0
\(775\) −943.103 + 121.375i −1.21691 + 0.156613i
\(776\) 0 0
\(777\) 91.2762i 0.117473i
\(778\) 0 0
\(779\) 64.2778i 0.0825132i
\(780\) 0 0
\(781\) −696.141 −0.891346
\(782\) 0 0
\(783\) −23.0975 −0.0294988
\(784\) 0 0
\(785\) 575.547 654.365i 0.733181 0.833586i
\(786\) 0 0
\(787\) 438.946 0.557746 0.278873 0.960328i \(-0.410039\pi\)
0.278873 + 0.960328i \(0.410039\pi\)
\(788\) 0 0
\(789\) 40.0567 0.0507690
\(790\) 0 0
\(791\) 1152.07i 1.45647i
\(792\) 0 0
\(793\) 180.696i 0.227864i
\(794\) 0 0
\(795\) −129.507 + 147.242i −0.162902 + 0.185210i
\(796\) 0 0
\(797\) 982.414i 1.23264i 0.787496 + 0.616320i \(0.211377\pi\)
−0.787496 + 0.616320i \(0.788623\pi\)
\(798\) 0 0
\(799\) 1686.95i 2.11133i
\(800\) 0 0
\(801\)