Properties

Label 136.4.b.b.33.6
Level $136$
Weight $4$
Character 136.33
Analytic conductor $8.024$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 95 x^{6} + 756 x^{4} + 1780 x^{2} + 1152\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 33.6
Root \(1.60125i\) of defining polynomial
Character \(\chi\) \(=\) 136.33
Dual form 136.4.b.b.33.3

$q$-expansion

\(f(q)\) \(=\) \(q+2.95309i q^{3} +4.89575i q^{5} -4.46235i q^{7} +18.2793 q^{9} +O(q^{10})\) \(q+2.95309i q^{3} +4.89575i q^{5} -4.46235i q^{7} +18.2793 q^{9} +60.3865i q^{11} -56.1938 q^{13} -14.4576 q^{15} +(25.6861 + 65.2168i) q^{17} -134.845 q^{19} +13.1777 q^{21} +39.1633i q^{23} +101.032 q^{25} +133.714i q^{27} -113.700i q^{29} +306.516i q^{31} -178.326 q^{33} +21.8465 q^{35} -61.9621i q^{37} -165.945i q^{39} -317.272i q^{41} +122.660 q^{43} +89.4907i q^{45} +303.233 q^{47} +323.087 q^{49} +(-192.591 + 75.8532i) q^{51} +133.743 q^{53} -295.637 q^{55} -398.210i q^{57} -130.660 q^{59} -772.618i q^{61} -81.5685i q^{63} -275.111i q^{65} +378.907 q^{67} -115.653 q^{69} -465.116i q^{71} +664.801i q^{73} +298.355i q^{75} +269.466 q^{77} -925.763i q^{79} +98.6724 q^{81} -723.561 q^{83} +(-319.285 + 125.752i) q^{85} +335.766 q^{87} +889.439 q^{89} +250.757i q^{91} -905.170 q^{93} -660.168i q^{95} +1506.43i q^{97} +1103.82i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 132 q^{9} + O(q^{10}) \) \( 8 q - 132 q^{9} + 44 q^{13} + 24 q^{15} + 28 q^{17} + 48 q^{19} + 308 q^{21} - 520 q^{25} + 812 q^{33} - 1064 q^{35} + 8 q^{43} + 312 q^{47} - 1124 q^{49} + 408 q^{51} + 472 q^{53} + 1416 q^{55} - 72 q^{59} - 624 q^{67} - 180 q^{69} + 1660 q^{77} + 3156 q^{81} + 2472 q^{83} - 2160 q^{85} - 6664 q^{87} + 68 q^{89} - 4036 q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 2.95309i 0.568322i 0.958777 + 0.284161i \(0.0917150\pi\)
−0.958777 + 0.284161i \(0.908285\pi\)
\(4\) 0 0
\(5\) 4.89575i 0.437889i 0.975737 + 0.218944i \(0.0702614\pi\)
−0.975737 + 0.218944i \(0.929739\pi\)
\(6\) 0 0
\(7\) 4.46235i 0.240944i −0.992717 0.120472i \(-0.961559\pi\)
0.992717 0.120472i \(-0.0384408\pi\)
\(8\) 0 0
\(9\) 18.2793 0.677010
\(10\) 0 0
\(11\) 60.3865i 1.65520i 0.561318 + 0.827600i \(0.310294\pi\)
−0.561318 + 0.827600i \(0.689706\pi\)
\(12\) 0 0
\(13\) −56.1938 −1.19887 −0.599437 0.800422i \(-0.704609\pi\)
−0.599437 + 0.800422i \(0.704609\pi\)
\(14\) 0 0
\(15\) −14.4576 −0.248862
\(16\) 0 0
\(17\) 25.6861 + 65.2168i 0.366458 + 0.930435i
\(18\) 0 0
\(19\) −134.845 −1.62819 −0.814095 0.580731i \(-0.802767\pi\)
−0.814095 + 0.580731i \(0.802767\pi\)
\(20\) 0 0
\(21\) 13.1777 0.136934
\(22\) 0 0
\(23\) 39.1633i 0.355049i 0.984116 + 0.177524i \(0.0568088\pi\)
−0.984116 + 0.177524i \(0.943191\pi\)
\(24\) 0 0
\(25\) 101.032 0.808253
\(26\) 0 0
\(27\) 133.714i 0.953082i
\(28\) 0 0
\(29\) 113.700i 0.728053i −0.931389 0.364027i \(-0.881402\pi\)
0.931389 0.364027i \(-0.118598\pi\)
\(30\) 0 0
\(31\) 306.516i 1.77587i 0.459969 + 0.887935i \(0.347861\pi\)
−0.459969 + 0.887935i \(0.652139\pi\)
\(32\) 0 0
\(33\) −178.326 −0.940686
\(34\) 0 0
\(35\) 21.8465 0.105507
\(36\) 0 0
\(37\) 61.9621i 0.275311i −0.990480 0.137656i \(-0.956043\pi\)
0.990480 0.137656i \(-0.0439567\pi\)
\(38\) 0 0
\(39\) 165.945i 0.681346i
\(40\) 0 0
\(41\) 317.272i 1.20853i −0.796785 0.604263i \(-0.793468\pi\)
0.796785 0.604263i \(-0.206532\pi\)
\(42\) 0 0
\(43\) 122.660 0.435009 0.217505 0.976059i \(-0.430208\pi\)
0.217505 + 0.976059i \(0.430208\pi\)
\(44\) 0 0
\(45\) 89.4907i 0.296455i
\(46\) 0 0
\(47\) 303.233 0.941086 0.470543 0.882377i \(-0.344058\pi\)
0.470543 + 0.882377i \(0.344058\pi\)
\(48\) 0 0
\(49\) 323.087 0.941946
\(50\) 0 0
\(51\) −192.591 + 75.8532i −0.528786 + 0.208266i
\(52\) 0 0
\(53\) 133.743 0.346623 0.173311 0.984867i \(-0.444553\pi\)
0.173311 + 0.984867i \(0.444553\pi\)
\(54\) 0 0
\(55\) −295.637 −0.724794
\(56\) 0 0
\(57\) 398.210i 0.925336i
\(58\) 0 0
\(59\) −130.660 −0.288312 −0.144156 0.989555i \(-0.546047\pi\)
−0.144156 + 0.989555i \(0.546047\pi\)
\(60\) 0 0
\(61\) 772.618i 1.62170i −0.585256 0.810849i \(-0.699006\pi\)
0.585256 0.810849i \(-0.300994\pi\)
\(62\) 0 0
\(63\) 81.5685i 0.163122i
\(64\) 0 0
\(65\) 275.111i 0.524974i
\(66\) 0 0
\(67\) 378.907 0.690909 0.345455 0.938435i \(-0.387725\pi\)
0.345455 + 0.938435i \(0.387725\pi\)
\(68\) 0 0
\(69\) −115.653 −0.201782
\(70\) 0 0
\(71\) 465.116i 0.777452i −0.921353 0.388726i \(-0.872915\pi\)
0.921353 0.388726i \(-0.127085\pi\)
\(72\) 0 0
\(73\) 664.801i 1.06588i 0.846154 + 0.532939i \(0.178913\pi\)
−0.846154 + 0.532939i \(0.821087\pi\)
\(74\) 0 0
\(75\) 298.355i 0.459348i
\(76\) 0 0
\(77\) 269.466 0.398811
\(78\) 0 0
\(79\) 925.763i 1.31844i −0.751952 0.659218i \(-0.770887\pi\)
0.751952 0.659218i \(-0.229113\pi\)
\(80\) 0 0
\(81\) 98.6724 0.135353
\(82\) 0 0
\(83\) −723.561 −0.956882 −0.478441 0.878120i \(-0.658798\pi\)
−0.478441 + 0.878120i \(0.658798\pi\)
\(84\) 0 0
\(85\) −319.285 + 125.752i −0.407427 + 0.160468i
\(86\) 0 0
\(87\) 335.766 0.413769
\(88\) 0 0
\(89\) 889.439 1.05933 0.529665 0.848207i \(-0.322318\pi\)
0.529665 + 0.848207i \(0.322318\pi\)
\(90\) 0 0
\(91\) 250.757i 0.288862i
\(92\) 0 0
\(93\) −905.170 −1.00927
\(94\) 0 0
\(95\) 660.168i 0.712967i
\(96\) 0 0
\(97\) 1506.43i 1.57686i 0.615127 + 0.788428i \(0.289105\pi\)
−0.615127 + 0.788428i \(0.710895\pi\)
\(98\) 0 0
\(99\) 1103.82i 1.12059i
\(100\) 0 0
\(101\) 606.815 0.597825 0.298912 0.954281i \(-0.403376\pi\)
0.298912 + 0.954281i \(0.403376\pi\)
\(102\) 0 0
\(103\) 582.633 0.557364 0.278682 0.960383i \(-0.410102\pi\)
0.278682 + 0.960383i \(0.410102\pi\)
\(104\) 0 0
\(105\) 64.5147i 0.0599619i
\(106\) 0 0
\(107\) 14.1057i 0.0127444i 0.999980 + 0.00637221i \(0.00202835\pi\)
−0.999980 + 0.00637221i \(0.997972\pi\)
\(108\) 0 0
\(109\) 1599.48i 1.40553i 0.711422 + 0.702765i \(0.248051\pi\)
−0.711422 + 0.702765i \(0.751949\pi\)
\(110\) 0 0
\(111\) 182.980 0.156465
\(112\) 0 0
\(113\) 620.655i 0.516693i −0.966052 0.258347i \(-0.916822\pi\)
0.966052 0.258347i \(-0.0831777\pi\)
\(114\) 0 0
\(115\) −191.734 −0.155472
\(116\) 0 0
\(117\) −1027.18 −0.811650
\(118\) 0 0
\(119\) 291.020 114.620i 0.224183 0.0882960i
\(120\) 0 0
\(121\) −2315.52 −1.73969
\(122\) 0 0
\(123\) 936.932 0.686832
\(124\) 0 0
\(125\) 1106.59i 0.791814i
\(126\) 0 0
\(127\) −970.396 −0.678021 −0.339011 0.940783i \(-0.610092\pi\)
−0.339011 + 0.940783i \(0.610092\pi\)
\(128\) 0 0
\(129\) 362.224i 0.247225i
\(130\) 0 0
\(131\) 705.717i 0.470678i 0.971913 + 0.235339i \(0.0756200\pi\)
−0.971913 + 0.235339i \(0.924380\pi\)
\(132\) 0 0
\(133\) 601.727i 0.392303i
\(134\) 0 0
\(135\) −654.628 −0.417344
\(136\) 0 0
\(137\) −2470.01 −1.54035 −0.770173 0.637835i \(-0.779830\pi\)
−0.770173 + 0.637835i \(0.779830\pi\)
\(138\) 0 0
\(139\) 435.484i 0.265736i 0.991134 + 0.132868i \(0.0424186\pi\)
−0.991134 + 0.132868i \(0.957581\pi\)
\(140\) 0 0
\(141\) 895.473i 0.534840i
\(142\) 0 0
\(143\) 3393.35i 1.98438i
\(144\) 0 0
\(145\) 556.646 0.318806
\(146\) 0 0
\(147\) 954.105i 0.535328i
\(148\) 0 0
\(149\) 2428.83 1.33542 0.667709 0.744422i \(-0.267275\pi\)
0.667709 + 0.744422i \(0.267275\pi\)
\(150\) 0 0
\(151\) 2608.09 1.40558 0.702792 0.711396i \(-0.251937\pi\)
0.702792 + 0.711396i \(0.251937\pi\)
\(152\) 0 0
\(153\) 469.523 + 1192.12i 0.248096 + 0.629914i
\(154\) 0 0
\(155\) −1500.63 −0.777634
\(156\) 0 0
\(157\) 164.006 0.0833703 0.0416851 0.999131i \(-0.486727\pi\)
0.0416851 + 0.999131i \(0.486727\pi\)
\(158\) 0 0
\(159\) 394.955i 0.196993i
\(160\) 0 0
\(161\) 174.760 0.0855469
\(162\) 0 0
\(163\) 289.416i 0.139072i 0.997579 + 0.0695362i \(0.0221519\pi\)
−0.997579 + 0.0695362i \(0.977848\pi\)
\(164\) 0 0
\(165\) 873.041i 0.411916i
\(166\) 0 0
\(167\) 2147.52i 0.995090i −0.867438 0.497545i \(-0.834235\pi\)
0.867438 0.497545i \(-0.165765\pi\)
\(168\) 0 0
\(169\) 960.747 0.437299
\(170\) 0 0
\(171\) −2464.87 −1.10230
\(172\) 0 0
\(173\) 4383.99i 1.92664i −0.268357 0.963320i \(-0.586481\pi\)
0.268357 0.963320i \(-0.413519\pi\)
\(174\) 0 0
\(175\) 450.839i 0.194744i
\(176\) 0 0
\(177\) 385.849i 0.163854i
\(178\) 0 0
\(179\) 3793.10 1.58385 0.791926 0.610617i \(-0.209079\pi\)
0.791926 + 0.610617i \(0.209079\pi\)
\(180\) 0 0
\(181\) 1726.75i 0.709108i 0.935036 + 0.354554i \(0.115367\pi\)
−0.935036 + 0.354554i \(0.884633\pi\)
\(182\) 0 0
\(183\) 2281.61 0.921646
\(184\) 0 0
\(185\) 303.351 0.120556
\(186\) 0 0
\(187\) −3938.21 + 1551.09i −1.54006 + 0.606561i
\(188\) 0 0
\(189\) 596.677 0.229640
\(190\) 0 0
\(191\) 1286.01 0.487184 0.243592 0.969878i \(-0.421674\pi\)
0.243592 + 0.969878i \(0.421674\pi\)
\(192\) 0 0
\(193\) 1682.07i 0.627349i −0.949531 0.313674i \(-0.898440\pi\)
0.949531 0.313674i \(-0.101560\pi\)
\(194\) 0 0
\(195\) 812.426 0.298354
\(196\) 0 0
\(197\) 415.578i 0.150298i 0.997172 + 0.0751489i \(0.0239432\pi\)
−0.997172 + 0.0751489i \(0.976057\pi\)
\(198\) 0 0
\(199\) 2761.31i 0.983637i −0.870698 0.491819i \(-0.836332\pi\)
0.870698 0.491819i \(-0.163668\pi\)
\(200\) 0 0
\(201\) 1118.95i 0.392659i
\(202\) 0 0
\(203\) −507.369 −0.175420
\(204\) 0 0
\(205\) 1553.28 0.529200
\(206\) 0 0
\(207\) 715.877i 0.240372i
\(208\) 0 0
\(209\) 8142.83i 2.69498i
\(210\) 0 0
\(211\) 2713.80i 0.885431i −0.896662 0.442715i \(-0.854015\pi\)
0.896662 0.442715i \(-0.145985\pi\)
\(212\) 0 0
\(213\) 1373.53 0.441843
\(214\) 0 0
\(215\) 600.510i 0.190486i
\(216\) 0 0
\(217\) 1367.78 0.427886
\(218\) 0 0
\(219\) −1963.22 −0.605762
\(220\) 0 0
\(221\) −1443.40 3664.78i −0.439337 1.11547i
\(222\) 0 0
\(223\) −6395.76 −1.92059 −0.960296 0.278984i \(-0.910002\pi\)
−0.960296 + 0.278984i \(0.910002\pi\)
\(224\) 0 0
\(225\) 1846.79 0.547196
\(226\) 0 0
\(227\) 3844.30i 1.12403i 0.827127 + 0.562016i \(0.189974\pi\)
−0.827127 + 0.562016i \(0.810026\pi\)
\(228\) 0 0
\(229\) −4849.35 −1.39936 −0.699682 0.714455i \(-0.746675\pi\)
−0.699682 + 0.714455i \(0.746675\pi\)
\(230\) 0 0
\(231\) 795.755i 0.226653i
\(232\) 0 0
\(233\) 2719.86i 0.764738i 0.924010 + 0.382369i \(0.124892\pi\)
−0.924010 + 0.382369i \(0.875108\pi\)
\(234\) 0 0
\(235\) 1484.55i 0.412091i
\(236\) 0 0
\(237\) 2733.86 0.749296
\(238\) 0 0
\(239\) 2524.92 0.683362 0.341681 0.939816i \(-0.389004\pi\)
0.341681 + 0.939816i \(0.389004\pi\)
\(240\) 0 0
\(241\) 1943.29i 0.519413i 0.965688 + 0.259706i \(0.0836258\pi\)
−0.965688 + 0.259706i \(0.916374\pi\)
\(242\) 0 0
\(243\) 3901.66i 1.03001i
\(244\) 0 0
\(245\) 1581.75i 0.412468i
\(246\) 0 0
\(247\) 7577.47 1.95200
\(248\) 0 0
\(249\) 2136.74i 0.543817i
\(250\) 0 0
\(251\) 2332.58 0.586578 0.293289 0.956024i \(-0.405250\pi\)
0.293289 + 0.956024i \(0.405250\pi\)
\(252\) 0 0
\(253\) −2364.93 −0.587676
\(254\) 0 0
\(255\) −371.358 942.876i −0.0911974 0.231550i
\(256\) 0 0
\(257\) −807.810 −0.196069 −0.0980346 0.995183i \(-0.531256\pi\)
−0.0980346 + 0.995183i \(0.531256\pi\)
\(258\) 0 0
\(259\) −276.497 −0.0663347
\(260\) 0 0
\(261\) 2078.35i 0.492900i
\(262\) 0 0
\(263\) −6420.82 −1.50542 −0.752708 0.658354i \(-0.771253\pi\)
−0.752708 + 0.658354i \(0.771253\pi\)
\(264\) 0 0
\(265\) 654.772i 0.151782i
\(266\) 0 0
\(267\) 2626.59i 0.602040i
\(268\) 0 0
\(269\) 6335.06i 1.43589i −0.696097 0.717947i \(-0.745082\pi\)
0.696097 0.717947i \(-0.254918\pi\)
\(270\) 0 0
\(271\) −1429.35 −0.320394 −0.160197 0.987085i \(-0.551213\pi\)
−0.160197 + 0.987085i \(0.551213\pi\)
\(272\) 0 0
\(273\) −740.506 −0.164167
\(274\) 0 0
\(275\) 6100.94i 1.33782i
\(276\) 0 0
\(277\) 7878.89i 1.70901i 0.519441 + 0.854506i \(0.326140\pi\)
−0.519441 + 0.854506i \(0.673860\pi\)
\(278\) 0 0
\(279\) 5602.90i 1.20228i
\(280\) 0 0
\(281\) 2967.99 0.630090 0.315045 0.949077i \(-0.397980\pi\)
0.315045 + 0.949077i \(0.397980\pi\)
\(282\) 0 0
\(283\) 401.204i 0.0842725i 0.999112 + 0.0421363i \(0.0134164\pi\)
−0.999112 + 0.0421363i \(0.986584\pi\)
\(284\) 0 0
\(285\) 1949.53 0.405195
\(286\) 0 0
\(287\) −1415.78 −0.291188
\(288\) 0 0
\(289\) −3593.45 + 3350.32i −0.731417 + 0.681930i
\(290\) 0 0
\(291\) −4448.63 −0.896162
\(292\) 0 0
\(293\) 7390.62 1.47360 0.736800 0.676111i \(-0.236336\pi\)
0.736800 + 0.676111i \(0.236336\pi\)
\(294\) 0 0
\(295\) 639.676i 0.126249i
\(296\) 0 0
\(297\) −8074.49 −1.57754
\(298\) 0 0
\(299\) 2200.74i 0.425659i
\(300\) 0 0
\(301\) 547.350i 0.104813i
\(302\) 0 0
\(303\) 1791.98i 0.339757i
\(304\) 0 0
\(305\) 3782.54 0.710123
\(306\) 0 0
\(307\) −50.2688 −0.00934524 −0.00467262 0.999989i \(-0.501487\pi\)
−0.00467262 + 0.999989i \(0.501487\pi\)
\(308\) 0 0
\(309\) 1720.57i 0.316762i
\(310\) 0 0
\(311\) 215.028i 0.0392062i 0.999808 + 0.0196031i \(0.00624026\pi\)
−0.999808 + 0.0196031i \(0.993760\pi\)
\(312\) 0 0
\(313\) 4639.52i 0.837831i 0.908025 + 0.418915i \(0.137590\pi\)
−0.908025 + 0.418915i \(0.862410\pi\)
\(314\) 0 0
\(315\) 399.339 0.0714292
\(316\) 0 0
\(317\) 3965.63i 0.702624i 0.936259 + 0.351312i \(0.114264\pi\)
−0.936259 + 0.351312i \(0.885736\pi\)
\(318\) 0 0
\(319\) 6865.94 1.20507
\(320\) 0 0
\(321\) −41.6554 −0.00724293
\(322\) 0 0
\(323\) −3463.64 8794.17i −0.596663 1.51492i
\(324\) 0 0
\(325\) −5677.36 −0.968994
\(326\) 0 0
\(327\) −4723.41 −0.798793
\(328\) 0 0
\(329\) 1353.13i 0.226749i
\(330\) 0 0
\(331\) 7497.94 1.24509 0.622544 0.782585i \(-0.286099\pi\)
0.622544 + 0.782585i \(0.286099\pi\)
\(332\) 0 0
\(333\) 1132.62i 0.186388i
\(334\) 0 0
\(335\) 1855.03i 0.302541i
\(336\) 0 0
\(337\) 798.500i 0.129071i −0.997915 0.0645357i \(-0.979443\pi\)
0.997915 0.0645357i \(-0.0205567\pi\)
\(338\) 0 0
\(339\) 1832.85 0.293648
\(340\) 0 0
\(341\) −18509.4 −2.93942
\(342\) 0 0
\(343\) 2972.32i 0.467901i
\(344\) 0 0
\(345\) 566.206i 0.0883581i
\(346\) 0 0
\(347\) 4014.43i 0.621054i 0.950564 + 0.310527i \(0.100506\pi\)
−0.950564 + 0.310527i \(0.899494\pi\)
\(348\) 0 0
\(349\) 4747.53 0.728165 0.364083 0.931367i \(-0.381383\pi\)
0.364083 + 0.931367i \(0.381383\pi\)
\(350\) 0 0
\(351\) 7513.88i 1.14262i
\(352\) 0 0
\(353\) 9653.02 1.45546 0.727731 0.685862i \(-0.240575\pi\)
0.727731 + 0.685862i \(0.240575\pi\)
\(354\) 0 0
\(355\) 2277.09 0.340438
\(356\) 0 0
\(357\) 338.484 + 859.408i 0.0501805 + 0.127408i
\(358\) 0 0
\(359\) −6641.04 −0.976325 −0.488162 0.872753i \(-0.662333\pi\)
−0.488162 + 0.872753i \(0.662333\pi\)
\(360\) 0 0
\(361\) 11324.2 1.65100
\(362\) 0 0
\(363\) 6837.94i 0.988703i
\(364\) 0 0
\(365\) −3254.70 −0.466736
\(366\) 0 0
\(367\) 3305.96i 0.470217i 0.971969 + 0.235108i \(0.0755445\pi\)
−0.971969 + 0.235108i \(0.924455\pi\)
\(368\) 0 0
\(369\) 5799.50i 0.818185i
\(370\) 0 0
\(371\) 596.808i 0.0835168i
\(372\) 0 0
\(373\) 238.635 0.0331261 0.0165631 0.999863i \(-0.494728\pi\)
0.0165631 + 0.999863i \(0.494728\pi\)
\(374\) 0 0
\(375\) −3267.87 −0.450005
\(376\) 0 0
\(377\) 6389.23i 0.872844i
\(378\) 0 0
\(379\) 11081.6i 1.50191i −0.660353 0.750955i \(-0.729593\pi\)
0.660353 0.750955i \(-0.270407\pi\)
\(380\) 0 0
\(381\) 2865.66i 0.385334i
\(382\) 0 0
\(383\) 7956.45 1.06150 0.530752 0.847527i \(-0.321910\pi\)
0.530752 + 0.847527i \(0.321910\pi\)
\(384\) 0 0
\(385\) 1319.24i 0.174635i
\(386\) 0 0
\(387\) 2242.13 0.294506
\(388\) 0 0
\(389\) 5128.06 0.668388 0.334194 0.942504i \(-0.391536\pi\)
0.334194 + 0.942504i \(0.391536\pi\)
\(390\) 0 0
\(391\) −2554.11 + 1005.95i −0.330349 + 0.130110i
\(392\) 0 0
\(393\) −2084.04 −0.267496
\(394\) 0 0
\(395\) 4532.30 0.577329
\(396\) 0 0
\(397\) 2933.96i 0.370910i −0.982653 0.185455i \(-0.940624\pi\)
0.982653 0.185455i \(-0.0593759\pi\)
\(398\) 0 0
\(399\) −1776.95 −0.222955
\(400\) 0 0
\(401\) 5278.07i 0.657293i 0.944453 + 0.328646i \(0.106592\pi\)
−0.944453 + 0.328646i \(0.893408\pi\)
\(402\) 0 0
\(403\) 17224.3i 2.12904i
\(404\) 0 0
\(405\) 483.075i 0.0592696i
\(406\) 0 0
\(407\) 3741.67 0.455695
\(408\) 0 0
\(409\) −6880.00 −0.831770 −0.415885 0.909417i \(-0.636528\pi\)
−0.415885 + 0.909417i \(0.636528\pi\)
\(410\) 0 0
\(411\) 7294.16i 0.875412i
\(412\) 0 0
\(413\) 583.049i 0.0694672i
\(414\) 0 0
\(415\) 3542.37i 0.419008i
\(416\) 0 0
\(417\) −1286.02 −0.151024
\(418\) 0 0
\(419\) 3048.43i 0.355431i −0.984082 0.177715i \(-0.943129\pi\)
0.984082 0.177715i \(-0.0568707\pi\)
\(420\) 0 0
\(421\) 3607.83 0.417660 0.208830 0.977952i \(-0.433034\pi\)
0.208830 + 0.977952i \(0.433034\pi\)
\(422\) 0 0
\(423\) 5542.88 0.637125
\(424\) 0 0
\(425\) 2595.11 + 6588.96i 0.296191 + 0.752027i
\(426\) 0 0
\(427\) −3447.69 −0.390739
\(428\) 0 0
\(429\) 10020.8 1.12776
\(430\) 0 0
\(431\) 8866.93i 0.990962i 0.868619 + 0.495481i \(0.165008\pi\)
−0.868619 + 0.495481i \(0.834992\pi\)
\(432\) 0 0
\(433\) −11353.2 −1.26005 −0.630024 0.776575i \(-0.716955\pi\)
−0.630024 + 0.776575i \(0.716955\pi\)
\(434\) 0 0
\(435\) 1643.82i 0.181185i
\(436\) 0 0
\(437\) 5280.99i 0.578087i
\(438\) 0 0
\(439\) 2786.53i 0.302947i −0.988461 0.151474i \(-0.951598\pi\)
0.988461 0.151474i \(-0.0484019\pi\)
\(440\) 0 0
\(441\) 5905.80 0.637707
\(442\) 0 0
\(443\) 15244.4 1.63495 0.817474 0.575965i \(-0.195374\pi\)
0.817474 + 0.575965i \(0.195374\pi\)
\(444\) 0 0
\(445\) 4354.47i 0.463869i
\(446\) 0 0
\(447\) 7172.54i 0.758947i
\(448\) 0 0
\(449\) 17473.1i 1.83654i −0.395958 0.918269i \(-0.629587\pi\)
0.395958 0.918269i \(-0.370413\pi\)
\(450\) 0 0
\(451\) 19158.9 2.00035
\(452\) 0 0
\(453\) 7701.91i 0.798824i
\(454\) 0 0
\(455\) −1227.64 −0.126489
\(456\) 0 0
\(457\) −16422.8 −1.68102 −0.840508 0.541800i \(-0.817743\pi\)
−0.840508 + 0.541800i \(0.817743\pi\)
\(458\) 0 0
\(459\) −8720.37 + 3434.58i −0.886780 + 0.349264i
\(460\) 0 0
\(461\) −12802.9 −1.29347 −0.646737 0.762713i \(-0.723867\pi\)
−0.646737 + 0.762713i \(0.723867\pi\)
\(462\) 0 0
\(463\) −2291.00 −0.229961 −0.114980 0.993368i \(-0.536680\pi\)
−0.114980 + 0.993368i \(0.536680\pi\)
\(464\) 0 0
\(465\) 4431.48i 0.441946i
\(466\) 0 0
\(467\) −14464.0 −1.43322 −0.716610 0.697474i \(-0.754307\pi\)
−0.716610 + 0.697474i \(0.754307\pi\)
\(468\) 0 0
\(469\) 1690.82i 0.166471i
\(470\) 0 0
\(471\) 484.325i 0.0473812i
\(472\) 0 0
\(473\) 7406.97i 0.720028i
\(474\) 0 0
\(475\) −13623.6 −1.31599
\(476\) 0 0
\(477\) 2444.72 0.234667
\(478\) 0 0
\(479\) 6988.15i 0.666590i 0.942823 + 0.333295i \(0.108161\pi\)
−0.942823 + 0.333295i \(0.891839\pi\)
\(480\) 0 0
\(481\) 3481.89i 0.330063i
\(482\) 0 0
\(483\) 516.083i 0.0486182i
\(484\) 0 0
\(485\) −7375.11 −0.690488
\(486\) 0 0
\(487\) 8494.76i 0.790420i 0.918591 + 0.395210i \(0.129328\pi\)
−0.918591 + 0.395210i \(0.870672\pi\)
\(488\) 0 0
\(489\) −854.670 −0.0790379
\(490\) 0 0
\(491\) −5195.82 −0.477565 −0.238782 0.971073i \(-0.576748\pi\)
−0.238782 + 0.971073i \(0.576748\pi\)
\(492\) 0 0
\(493\) 7415.14 2920.50i 0.677406 0.266801i
\(494\) 0 0
\(495\) −5404.03 −0.490693
\(496\) 0 0
\(497\) −2075.51 −0.187323
\(498\) 0 0
\(499\) 8790.68i 0.788627i −0.918976 0.394313i \(-0.870982\pi\)
0.918976 0.394313i \(-0.129018\pi\)
\(500\) 0 0
\(501\) 6341.81 0.565531
\(502\) 0 0
\(503\) 7764.74i 0.688295i 0.938916 + 0.344148i \(0.111832\pi\)
−0.938916 + 0.344148i \(0.888168\pi\)
\(504\) 0 0
\(505\) 2970.81i 0.261781i
\(506\) 0 0
\(507\) 2837.17i 0.248527i
\(508\) 0 0
\(509\) 9919.29 0.863782 0.431891 0.901926i \(-0.357847\pi\)
0.431891 + 0.901926i \(0.357847\pi\)
\(510\) 0 0
\(511\) 2966.58 0.256817
\(512\) 0 0
\(513\) 18030.6i 1.55180i
\(514\) 0 0
\(515\) 2852.42i 0.244064i
\(516\) 0 0
\(517\) 18311.2i 1.55769i
\(518\) 0 0
\(519\) 12946.3 1.09495
\(520\) 0 0
\(521\) 4230.44i 0.355737i 0.984054 + 0.177868i \(0.0569201\pi\)
−0.984054 + 0.177868i \(0.943080\pi\)
\(522\) 0 0
\(523\) 10566.8 0.883465 0.441732 0.897147i \(-0.354364\pi\)
0.441732 + 0.897147i \(0.354364\pi\)
\(524\) 0 0
\(525\) 1331.37 0.110677
\(526\) 0 0
\(527\) −19990.0 + 7873.20i −1.65233 + 0.650782i
\(528\) 0 0
\(529\) 10633.2 0.873941
\(530\) 0 0
\(531\) −2388.36 −0.195190
\(532\) 0 0
\(533\) 17828.7i 1.44887i
\(534\) 0 0
\(535\) −69.0581 −0.00558064
\(536\) 0 0
\(537\) 11201.3i 0.900138i
\(538\) 0 0
\(539\) 19510.1i 1.55911i
\(540\) 0 0
\(541\) 1980.86i 0.157419i −0.996898 0.0787094i \(-0.974920\pi\)
0.996898 0.0787094i \(-0.0250799\pi\)
\(542\) 0 0
\(543\) −5099.25 −0.403001
\(544\) 0 0
\(545\) −7830.67 −0.615466
\(546\) 0 0
\(547\) 3940.22i 0.307992i 0.988071 + 0.153996i \(0.0492143\pi\)
−0.988071 + 0.153996i \(0.950786\pi\)
\(548\) 0 0
\(549\) 14122.9i 1.09791i
\(550\) 0 0
\(551\) 15331.9i 1.18541i
\(552\) 0 0
\(553\) −4131.08 −0.317670
\(554\) 0 0
\(555\) 895.822i 0.0685145i
\(556\) 0 0
\(557\) 3709.12 0.282155 0.141077 0.989999i \(-0.454943\pi\)
0.141077 + 0.989999i \(0.454943\pi\)
\(558\) 0 0
\(559\) −6892.71 −0.521522
\(560\) 0 0
\(561\) −4580.50 11629.9i −0.344722 0.875247i
\(562\) 0 0
\(563\) −3393.34 −0.254018 −0.127009 0.991902i \(-0.540538\pi\)
−0.127009 + 0.991902i \(0.540538\pi\)
\(564\) 0 0
\(565\) 3038.57 0.226254
\(566\) 0 0
\(567\) 440.311i 0.0326125i
\(568\) 0 0
\(569\) −2927.65 −0.215700 −0.107850 0.994167i \(-0.534397\pi\)
−0.107850 + 0.994167i \(0.534397\pi\)
\(570\) 0 0
\(571\) 19781.8i 1.44981i −0.688846 0.724907i \(-0.741883\pi\)
0.688846 0.724907i \(-0.258117\pi\)
\(572\) 0 0
\(573\) 3797.69i 0.276877i
\(574\) 0 0
\(575\) 3956.74i 0.286969i
\(576\) 0 0
\(577\) 17520.1 1.26407 0.632037 0.774938i \(-0.282219\pi\)
0.632037 + 0.774938i \(0.282219\pi\)
\(578\) 0 0
\(579\) 4967.31 0.356536
\(580\) 0 0
\(581\) 3228.79i 0.230555i
\(582\) 0 0
\(583\) 8076.26i 0.573730i
\(584\) 0 0
\(585\) 5028.83i 0.355413i
\(586\) 0 0
\(587\) 7846.90 0.551748 0.275874 0.961194i \(-0.411033\pi\)
0.275874 + 0.961194i \(0.411033\pi\)
\(588\) 0 0
\(589\) 41332.3i 2.89145i
\(590\) 0 0
\(591\) −1227.24 −0.0854175
\(592\) 0 0
\(593\) 10572.9 0.732173 0.366086 0.930581i \(-0.380697\pi\)
0.366086 + 0.930581i \(0.380697\pi\)
\(594\) 0 0
\(595\) 561.152 + 1424.76i 0.0386638 + 0.0981672i
\(596\) 0 0
\(597\) 8154.38 0.559023
\(598\) 0 0
\(599\) −27156.5 −1.85240 −0.926198 0.377038i \(-0.876943\pi\)
−0.926198 + 0.377038i \(0.876943\pi\)
\(600\) 0 0
\(601\) 11180.6i 0.758849i 0.925223 + 0.379424i \(0.123878\pi\)
−0.925223 + 0.379424i \(0.876122\pi\)
\(602\) 0 0
\(603\) 6926.15 0.467752
\(604\) 0 0
\(605\) 11336.2i 0.761790i
\(606\) 0 0
\(607\) 19444.2i 1.30019i 0.759851 + 0.650097i \(0.225272\pi\)
−0.759851 + 0.650097i \(0.774728\pi\)
\(608\) 0 0
\(609\) 1498.30i 0.0996952i
\(610\) 0 0
\(611\) −17039.8 −1.12824
\(612\) 0 0
\(613\) −16630.9 −1.09579 −0.547893 0.836548i \(-0.684570\pi\)
−0.547893 + 0.836548i \(0.684570\pi\)
\(614\) 0 0
\(615\) 4586.98i 0.300756i
\(616\) 0 0
\(617\) 8035.03i 0.524276i 0.965030 + 0.262138i \(0.0844275\pi\)
−0.965030 + 0.262138i \(0.915572\pi\)
\(618\) 0 0
\(619\) 16161.6i 1.04942i −0.851282 0.524709i \(-0.824174\pi\)
0.851282 0.524709i \(-0.175826\pi\)
\(620\) 0 0
\(621\) −5236.67 −0.338390
\(622\) 0 0
\(623\) 3968.99i 0.255239i
\(624\) 0 0
\(625\) 7211.35 0.461527
\(626\) 0 0
\(627\) 24046.5 1.53162
\(628\) 0 0
\(629\) 4040.97 1591.56i 0.256159 0.100890i
\(630\) 0 0
\(631\) −10043.9 −0.633665 −0.316833 0.948481i \(-0.602619\pi\)
−0.316833 + 0.948481i \(0.602619\pi\)
\(632\) 0 0
\(633\) 8014.09 0.503210
\(634\) 0 0
\(635\) 4750.81i 0.296898i
\(636\) 0 0
\(637\) −18155.5 −1.12927
\(638\) 0 0
\(639\) 8501.98i 0.526343i
\(640\) 0 0
\(641\) 15316.1i 0.943757i −0.881664 0.471879i \(-0.843576\pi\)
0.881664 0.471879i \(-0.156424\pi\)
\(642\) 0 0
\(643\) 24684.6i 1.51394i −0.653449 0.756970i \(-0.726679\pi\)
0.653449 0.756970i \(-0.273321\pi\)
\(644\) 0 0
\(645\) −1773.36 −0.108257
\(646\) 0 0
\(647\) 7774.11 0.472383 0.236192 0.971707i \(-0.424101\pi\)
0.236192 + 0.971707i \(0.424101\pi\)
\(648\) 0 0
\(649\) 7890.07i 0.477214i
\(650\) 0 0
\(651\) 4039.18i 0.243177i
\(652\) 0 0
\(653\) 752.623i 0.0451032i −0.999746 0.0225516i \(-0.992821\pi\)
0.999746 0.0225516i \(-0.00717901\pi\)
\(654\) 0 0
\(655\) −3455.01 −0.206105
\(656\) 0 0
\(657\) 12152.1i 0.721611i
\(658\) 0 0
\(659\) −4887.68 −0.288918 −0.144459 0.989511i \(-0.546144\pi\)
−0.144459 + 0.989511i \(0.546144\pi\)
\(660\) 0 0
\(661\) −7122.91 −0.419136 −0.209568 0.977794i \(-0.567206\pi\)
−0.209568 + 0.977794i \(0.567206\pi\)
\(662\) 0 0
\(663\) 10822.4 4262.48i 0.633948 0.249685i
\(664\) 0 0
\(665\) −2945.90 −0.171785
\(666\) 0 0
\(667\) 4452.87 0.258494
\(668\) 0 0
\(669\) 18887.2i 1.09151i
\(670\) 0 0
\(671\) 46655.6 2.68423
\(672\) 0 0
\(673\) 32249.6i 1.84715i −0.383421 0.923574i \(-0.625254\pi\)
0.383421 0.923574i \(-0.374746\pi\)
\(674\) 0 0
\(675\) 13509.3i 0.770331i
\(676\) 0 0
\(677\) 17447.9i 0.990511i 0.868748 + 0.495255i \(0.164925\pi\)
−0.868748 + 0.495255i \(0.835075\pi\)
\(678\) 0 0
\(679\) 6722.23 0.379934
\(680\) 0 0
\(681\) −11352.6 −0.638812
\(682\) 0 0
\(683\) 13935.3i 0.780701i −0.920666 0.390351i \(-0.872354\pi\)
0.920666 0.390351i \(-0.127646\pi\)
\(684\) 0 0
\(685\) 12092.6i 0.674500i
\(686\) 0 0
\(687\) 14320.6i 0.795289i
\(688\) 0 0
\(689\) −7515.53 −0.415557
\(690\) 0 0
\(691\) 22358.6i 1.23092i 0.788170 + 0.615458i \(0.211029\pi\)
−0.788170 + 0.615458i \(0.788971\pi\)
\(692\) 0 0
\(693\) 4925.64 0.269999
\(694\) 0 0
\(695\) −2132.02 −0.116363
\(696\) 0 0
\(697\) 20691.5 8149.47i 1.12445 0.442874i
\(698\) 0 0
\(699\) −8031.99 −0.434617
\(700\) 0 0
\(701\) 30456.5 1.64098 0.820491 0.571660i \(-0.193700\pi\)
0.820491 + 0.571660i \(0.193700\pi\)
\(702\) 0 0
\(703\) 8355.30i 0.448259i
\(704\) 0 0
\(705\) −4384.01 −0.234201
\(706\) 0 0
\(707\) 2707.82i 0.144042i
\(708\) 0 0
\(709\) 36945.7i 1.95702i 0.206210 + 0.978508i \(0.433887\pi\)
−0.206210 + 0.978508i \(0.566113\pi\)
\(710\) 0 0
\(711\) 16922.3i 0.892595i
\(712\) 0 0
\(713\) −12004.2 −0.630520
\(714\) 0 0
\(715\) 16613.0 0.868937
\(716\) 0 0
\(717\) 7456.31i 0.388369i
\(718\) 0 0
\(719\) 12442.4i 0.645375i −0.946506 0.322687i \(-0.895414\pi\)
0.946506 0.322687i \(-0.104586\pi\)
\(720\) 0 0
\(721\) 2599.91i 0.134294i
\(722\) 0 0
\(723\) −5738.71 −0.295194
\(724\) 0 0
\(725\) 11487.3i 0.588451i
\(726\) 0 0
\(727\) −29943.3 −1.52756 −0.763780 0.645477i \(-0.776659\pi\)
−0.763780 + 0.645477i \(0.776659\pi\)
\(728\) 0 0
\(729\) −8857.78 −0.450022
\(730\) 0 0
\(731\) 3150.64 + 7999.46i 0.159413 + 0.404748i
\(732\) 0 0
\(733\) 19615.5 0.988423 0.494212 0.869342i \(-0.335457\pi\)
0.494212 + 0.869342i \(0.335457\pi\)
\(734\) 0 0
\(735\) −4671.06 −0.234414
\(736\) 0 0
\(737\) 22880.9i 1.14359i
\(738\) 0 0
\(739\) −3459.26 −0.172194 −0.0860968 0.996287i \(-0.527439\pi\)
−0.0860968 + 0.996287i \(0.527439\pi\)
\(740\) 0 0
\(741\) 22376.9i 1.10936i
\(742\) 0 0
\(743\) 37392.9i 1.84632i −0.384421 0.923158i \(-0.625599\pi\)
0.384421 0.923158i \(-0.374401\pi\)
\(744\) 0 0
\(745\) 11890.9i 0.584765i
\(746\) 0 0
\(747\) −13226.2 −0.647819
\(748\) 0 0
\(749\) 62.9447 0.00307069
\(750\) 0 0
\(751\) 34379.7i 1.67048i −0.549884 0.835241i \(-0.685328\pi\)
0.549884 0.835241i \(-0.314672\pi\)
\(752\) 0 0
\(753\) 6888.31i 0.333365i
\(754\) 0 0
\(755\) 12768.5i 0.615489i
\(756\) 0 0
\(757\) −20120.9 −0.966057 −0.483028 0.875605i \(-0.660463\pi\)
−0.483028 + 0.875605i \(0.660463\pi\)
\(758\) 0 0
\(759\) 6983.86i 0.333989i
\(760\) 0 0
\(761\) −21739.7 −1.03556 −0.517781 0.855513i \(-0.673242\pi\)
−0.517781 + 0.855513i \(0.673242\pi\)
\(762\) 0 0
\(763\) 7137.46 0.338654
\(764\) 0 0
\(765\) −5836.29 + 2298.66i −0.275832 + 0.108638i
\(766\) 0 0
\(767\) 7342.26 0.345650
\(768\) 0 0
\(769\) 26884.9 1.26072 0.630360 0.776303i \(-0.282907\pi\)
0.630360 + 0.776303i \(0.282907\pi\)
\(770\) 0 0
\(771\) 2385.53i 0.111430i
\(772\) 0 0
\(773\) −21927.0 −1.02026 −0.510128 0.860099i \(-0.670402\pi\)
−0.510128 + 0.860099i \(0.670402\pi\)
\(774\) 0 0
\(775\) 30967.9i 1.43535i
\(776\) 0 0
\(777\) 816.519i 0.0376994i
\(778\) 0 0
\(779\) 42782.6i 1.96771i
\(780\) 0 0
\(781\) 28086.7 1.28684
\(782\) 0 0
\(783\) 15203.2 0.693894
\(784\) 0 0
\(785\) 802.934i 0.0365069i
\(786\) 0 0
\(787\) 28401.4i 1.28641i −0.765695 0.643203i \(-0.777605\pi\)
0.765695 0.643203i \(-0.222395\pi\)
\(788\) 0 0
\(789\) 18961.2i 0.855561i
\(790\) 0 0
\(791\) −2769.58 −0.124494
\(792\) 0 0
\(793\) 43416.3i 1.94421i
\(794\) 0 0
\(795\) −1933.60 −0.0862612
\(796\) 0 0
\(797\) −19462.7 −0.864999 −0.432499 0.901634i \(-0.642368\pi\)
−0.432499 + 0.901634i \(0.642368\pi\)
\(798\) 0 0
\(799\) 7788.86 + 19775.9i 0.344869 + 0.875619i
\(800\) 0 0