Properties

Label 272.4.b.f.33.8
Level $272$
Weight $4$
Character 272.33
Analytic conductor $16.049$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 272.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.0485195216\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 136)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 33.8
Root \(2.20783i\) of defining polynomial
Character \(\chi\) \(=\) 272.33
Dual form 272.4.b.f.33.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.26065i q^{3} +16.4090i q^{5} -34.5010i q^{7} -58.7597 q^{9} +O(q^{10})\) \(q+9.26065i q^{3} +16.4090i q^{5} -34.5010i q^{7} -58.7597 q^{9} -7.42843i q^{11} -42.0242 q^{13} -151.958 q^{15} +(-26.0870 + 65.0574i) q^{17} -59.9095 q^{19} +319.502 q^{21} -49.4728i q^{23} -144.255 q^{25} -294.116i q^{27} -259.369i q^{29} +92.2215i q^{31} +68.7922 q^{33} +566.126 q^{35} +207.564i q^{37} -389.172i q^{39} +176.800i q^{41} +19.0809 q^{43} -964.188i q^{45} -80.1389 q^{47} -847.318 q^{49} +(-602.474 - 241.583i) q^{51} +319.869 q^{53} +121.893 q^{55} -554.801i q^{57} -11.0809 q^{59} +712.648i q^{61} +2027.27i q^{63} -689.575i q^{65} -484.980 q^{67} +458.150 q^{69} +443.968i q^{71} +337.468i q^{73} -1335.89i q^{75} -256.288 q^{77} -840.905i q^{79} +1137.19 q^{81} -456.004 q^{83} +(-1067.53 - 428.062i) q^{85} +2401.93 q^{87} -1205.43 q^{89} +1449.88i q^{91} -854.032 q^{93} -983.055i q^{95} -638.484i q^{97} +436.493i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 132 q^{9} + 44 q^{13} - 24 q^{15} + 28 q^{17} - 48 q^{19} + 308 q^{21} - 520 q^{25} + 812 q^{33} + 1064 q^{35} - 8 q^{43} - 312 q^{47} - 1124 q^{49} - 408 q^{51} + 472 q^{53} - 1416 q^{55} + 72 q^{59} + 624 q^{67} - 180 q^{69} + 1660 q^{77} + 3156 q^{81} - 2472 q^{83} - 2160 q^{85} + 6664 q^{87} + 68 q^{89} - 4036 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/272\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(239\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 9.26065i 1.78221i 0.453794 + 0.891107i \(0.350070\pi\)
−0.453794 + 0.891107i \(0.649930\pi\)
\(4\) 0 0
\(5\) 16.4090i 1.46766i 0.679331 + 0.733832i \(0.262270\pi\)
−0.679331 + 0.733832i \(0.737730\pi\)
\(6\) 0 0
\(7\) 34.5010i 1.86288i −0.363898 0.931439i \(-0.618554\pi\)
0.363898 0.931439i \(-0.381446\pi\)
\(8\) 0 0
\(9\) −58.7597 −2.17629
\(10\) 0 0
\(11\) 7.42843i 0.203614i −0.994804 0.101807i \(-0.967538\pi\)
0.994804 0.101807i \(-0.0324625\pi\)
\(12\) 0 0
\(13\) −42.0242 −0.896571 −0.448285 0.893890i \(-0.647965\pi\)
−0.448285 + 0.893890i \(0.647965\pi\)
\(14\) 0 0
\(15\) −151.958 −2.61569
\(16\) 0 0
\(17\) −26.0870 + 65.0574i −0.372178 + 0.928161i
\(18\) 0 0
\(19\) −59.9095 −0.723378 −0.361689 0.932299i \(-0.617800\pi\)
−0.361689 + 0.932299i \(0.617800\pi\)
\(20\) 0 0
\(21\) 319.502 3.32005
\(22\) 0 0
\(23\) 49.4728i 0.448513i −0.974530 0.224256i \(-0.928005\pi\)
0.974530 0.224256i \(-0.0719953\pi\)
\(24\) 0 0
\(25\) −144.255 −1.15404
\(26\) 0 0
\(27\) 294.116i 2.09639i
\(28\) 0 0
\(29\) 259.369i 1.66081i −0.557157 0.830407i \(-0.688108\pi\)
0.557157 0.830407i \(-0.311892\pi\)
\(30\) 0 0
\(31\) 92.2215i 0.534306i 0.963654 + 0.267153i \(0.0860829\pi\)
−0.963654 + 0.267153i \(0.913917\pi\)
\(32\) 0 0
\(33\) 68.7922 0.362884
\(34\) 0 0
\(35\) 566.126 2.73408
\(36\) 0 0
\(37\) 207.564i 0.922252i 0.887335 + 0.461126i \(0.152554\pi\)
−0.887335 + 0.461126i \(0.847446\pi\)
\(38\) 0 0
\(39\) 389.172i 1.59788i
\(40\) 0 0
\(41\) 176.800i 0.673454i 0.941602 + 0.336727i \(0.109320\pi\)
−0.941602 + 0.336727i \(0.890680\pi\)
\(42\) 0 0
\(43\) 19.0809 0.0676700 0.0338350 0.999427i \(-0.489228\pi\)
0.0338350 + 0.999427i \(0.489228\pi\)
\(44\) 0 0
\(45\) 964.188i 3.19406i
\(46\) 0 0
\(47\) −80.1389 −0.248712 −0.124356 0.992238i \(-0.539686\pi\)
−0.124356 + 0.992238i \(0.539686\pi\)
\(48\) 0 0
\(49\) −847.318 −2.47031
\(50\) 0 0
\(51\) −602.474 241.583i −1.65418 0.663301i
\(52\) 0 0
\(53\) 319.869 0.829006 0.414503 0.910048i \(-0.363955\pi\)
0.414503 + 0.910048i \(0.363955\pi\)
\(54\) 0 0
\(55\) 121.893 0.298837
\(56\) 0 0
\(57\) 554.801i 1.28921i
\(58\) 0 0
\(59\) −11.0809 −0.0244510 −0.0122255 0.999925i \(-0.503892\pi\)
−0.0122255 + 0.999925i \(0.503892\pi\)
\(60\) 0 0
\(61\) 712.648i 1.49582i 0.663798 + 0.747912i \(0.268943\pi\)
−0.663798 + 0.747912i \(0.731057\pi\)
\(62\) 0 0
\(63\) 2027.27i 4.05415i
\(64\) 0 0
\(65\) 689.575i 1.31587i
\(66\) 0 0
\(67\) −484.980 −0.884325 −0.442163 0.896935i \(-0.645789\pi\)
−0.442163 + 0.896935i \(0.645789\pi\)
\(68\) 0 0
\(69\) 458.150 0.799345
\(70\) 0 0
\(71\) 443.968i 0.742103i 0.928612 + 0.371051i \(0.121003\pi\)
−0.928612 + 0.371051i \(0.878997\pi\)
\(72\) 0 0
\(73\) 337.468i 0.541063i 0.962711 + 0.270531i \(0.0871994\pi\)
−0.962711 + 0.270531i \(0.912801\pi\)
\(74\) 0 0
\(75\) 1335.89i 2.05674i
\(76\) 0 0
\(77\) −256.288 −0.379309
\(78\) 0 0
\(79\) 840.905i 1.19759i −0.800904 0.598793i \(-0.795647\pi\)
0.800904 0.598793i \(-0.204353\pi\)
\(80\) 0 0
\(81\) 1137.19 1.55993
\(82\) 0 0
\(83\) −456.004 −0.603047 −0.301524 0.953459i \(-0.597495\pi\)
−0.301524 + 0.953459i \(0.597495\pi\)
\(84\) 0 0
\(85\) −1067.53 428.062i −1.36223 0.546233i
\(86\) 0 0
\(87\) 2401.93 2.95993
\(88\) 0 0
\(89\) −1205.43 −1.43568 −0.717841 0.696207i \(-0.754870\pi\)
−0.717841 + 0.696207i \(0.754870\pi\)
\(90\) 0 0
\(91\) 1449.88i 1.67020i
\(92\) 0 0
\(93\) −854.032 −0.952247
\(94\) 0 0
\(95\) 983.055i 1.06168i
\(96\) 0 0
\(97\) 638.484i 0.668332i −0.942514 0.334166i \(-0.891545\pi\)
0.942514 0.334166i \(-0.108455\pi\)
\(98\) 0 0
\(99\) 436.493i 0.443123i
\(100\) 0 0
\(101\) 739.395 0.728441 0.364220 0.931313i \(-0.381335\pi\)
0.364220 + 0.931313i \(0.381335\pi\)
\(102\) 0 0
\(103\) −1588.54 −1.51965 −0.759824 0.650129i \(-0.774715\pi\)
−0.759824 + 0.650129i \(0.774715\pi\)
\(104\) 0 0
\(105\) 5242.70i 4.87271i
\(106\) 0 0
\(107\) 202.298i 0.182775i −0.995815 0.0913875i \(-0.970870\pi\)
0.995815 0.0913875i \(-0.0291302\pi\)
\(108\) 0 0
\(109\) 1244.71i 1.09378i 0.837204 + 0.546890i \(0.184188\pi\)
−0.837204 + 0.546890i \(0.815812\pi\)
\(110\) 0 0
\(111\) −1922.18 −1.64365
\(112\) 0 0
\(113\) 1492.28i 1.24232i −0.783685 0.621159i \(-0.786662\pi\)
0.783685 0.621159i \(-0.213338\pi\)
\(114\) 0 0
\(115\) 811.798 0.658266
\(116\) 0 0
\(117\) 2469.33 1.95119
\(118\) 0 0
\(119\) 2244.54 + 900.028i 1.72905 + 0.693323i
\(120\) 0 0
\(121\) 1275.82 0.958541
\(122\) 0 0
\(123\) −1637.29 −1.20024
\(124\) 0 0
\(125\) 315.954i 0.226078i
\(126\) 0 0
\(127\) −2054.49 −1.43548 −0.717741 0.696311i \(-0.754824\pi\)
−0.717741 + 0.696311i \(0.754824\pi\)
\(128\) 0 0
\(129\) 176.702i 0.120602i
\(130\) 0 0
\(131\) 1892.72i 1.26235i 0.775640 + 0.631175i \(0.217427\pi\)
−0.775640 + 0.631175i \(0.782573\pi\)
\(132\) 0 0
\(133\) 2066.94i 1.34757i
\(134\) 0 0
\(135\) 4826.14 3.07680
\(136\) 0 0
\(137\) −848.618 −0.529214 −0.264607 0.964356i \(-0.585242\pi\)
−0.264607 + 0.964356i \(0.585242\pi\)
\(138\) 0 0
\(139\) 20.6414i 0.0125955i 0.999980 + 0.00629777i \(0.00200466\pi\)
−0.999980 + 0.00629777i \(0.997995\pi\)
\(140\) 0 0
\(141\) 742.139i 0.443258i
\(142\) 0 0
\(143\) 312.174i 0.182555i
\(144\) 0 0
\(145\) 4255.98 2.43752
\(146\) 0 0
\(147\) 7846.71i 4.40263i
\(148\) 0 0
\(149\) −1202.59 −0.661206 −0.330603 0.943770i \(-0.607252\pi\)
−0.330603 + 0.943770i \(0.607252\pi\)
\(150\) 0 0
\(151\) 2080.77 1.12140 0.560698 0.828020i \(-0.310533\pi\)
0.560698 + 0.828020i \(0.310533\pi\)
\(152\) 0 0
\(153\) 1532.87 3822.76i 0.809966 2.01994i
\(154\) 0 0
\(155\) −1513.26 −0.784182
\(156\) 0 0
\(157\) 1153.74 0.586486 0.293243 0.956038i \(-0.405266\pi\)
0.293243 + 0.956038i \(0.405266\pi\)
\(158\) 0 0
\(159\) 2962.19i 1.47747i
\(160\) 0 0
\(161\) −1706.86 −0.835524
\(162\) 0 0
\(163\) 1461.35i 0.702221i −0.936334 0.351110i \(-0.885804\pi\)
0.936334 0.351110i \(-0.114196\pi\)
\(164\) 0 0
\(165\) 1128.81i 0.532592i
\(166\) 0 0
\(167\) 4157.98i 1.92667i 0.268296 + 0.963337i \(0.413539\pi\)
−0.268296 + 0.963337i \(0.586461\pi\)
\(168\) 0 0
\(169\) −430.965 −0.196160
\(170\) 0 0
\(171\) 3520.27 1.57428
\(172\) 0 0
\(173\) 2504.08i 1.10047i 0.835010 + 0.550235i \(0.185462\pi\)
−0.835010 + 0.550235i \(0.814538\pi\)
\(174\) 0 0
\(175\) 4976.94i 2.14983i
\(176\) 0 0
\(177\) 102.616i 0.0435769i
\(178\) 0 0
\(179\) −839.685 −0.350620 −0.175310 0.984513i \(-0.556093\pi\)
−0.175310 + 0.984513i \(0.556093\pi\)
\(180\) 0 0
\(181\) 500.491i 0.205532i −0.994706 0.102766i \(-0.967231\pi\)
0.994706 0.102766i \(-0.0327692\pi\)
\(182\) 0 0
\(183\) −6599.59 −2.66588
\(184\) 0 0
\(185\) −3405.92 −1.35356
\(186\) 0 0
\(187\) 483.275 + 193.786i 0.188987 + 0.0757808i
\(188\) 0 0
\(189\) −10147.3 −3.90532
\(190\) 0 0
\(191\) −199.937 −0.0757430 −0.0378715 0.999283i \(-0.512058\pi\)
−0.0378715 + 0.999283i \(0.512058\pi\)
\(192\) 0 0
\(193\) 2758.52i 1.02882i 0.857544 + 0.514411i \(0.171989\pi\)
−0.857544 + 0.514411i \(0.828011\pi\)
\(194\) 0 0
\(195\) 6385.92 2.34515
\(196\) 0 0
\(197\) 3379.87i 1.22236i −0.791490 0.611182i \(-0.790694\pi\)
0.791490 0.611182i \(-0.209306\pi\)
\(198\) 0 0
\(199\) 2729.46i 0.972293i −0.873877 0.486147i \(-0.838402\pi\)
0.873877 0.486147i \(-0.161598\pi\)
\(200\) 0 0
\(201\) 4491.24i 1.57606i
\(202\) 0 0
\(203\) −8948.48 −3.09389
\(204\) 0 0
\(205\) −2901.12 −0.988404
\(206\) 0 0
\(207\) 2907.01i 0.976092i
\(208\) 0 0
\(209\) 445.034i 0.147290i
\(210\) 0 0
\(211\) 2149.01i 0.701157i −0.936533 0.350579i \(-0.885985\pi\)
0.936533 0.350579i \(-0.114015\pi\)
\(212\) 0 0
\(213\) −4111.43 −1.32259
\(214\) 0 0
\(215\) 313.098i 0.0993168i
\(216\) 0 0
\(217\) 3181.73 0.995346
\(218\) 0 0
\(219\) −3125.17 −0.964290
\(220\) 0 0
\(221\) 1096.29 2733.99i 0.333684 0.832162i
\(222\) 0 0
\(223\) 115.665 0.0347332 0.0173666 0.999849i \(-0.494472\pi\)
0.0173666 + 0.999849i \(0.494472\pi\)
\(224\) 0 0
\(225\) 8476.38 2.51152
\(226\) 0 0
\(227\) 150.739i 0.0440743i 0.999757 + 0.0220372i \(0.00701521\pi\)
−0.999757 + 0.0220372i \(0.992985\pi\)
\(228\) 0 0
\(229\) −252.262 −0.0727946 −0.0363973 0.999337i \(-0.511588\pi\)
−0.0363973 + 0.999337i \(0.511588\pi\)
\(230\) 0 0
\(231\) 2373.40i 0.676009i
\(232\) 0 0
\(233\) 1025.45i 0.288323i −0.989554 0.144161i \(-0.953952\pi\)
0.989554 0.144161i \(-0.0460484\pi\)
\(234\) 0 0
\(235\) 1315.00i 0.365026i
\(236\) 0 0
\(237\) 7787.33 2.13435
\(238\) 0 0
\(239\) −3803.45 −1.02939 −0.514696 0.857373i \(-0.672095\pi\)
−0.514696 + 0.857373i \(0.672095\pi\)
\(240\) 0 0
\(241\) 2073.99i 0.554346i −0.960820 0.277173i \(-0.910603\pi\)
0.960820 0.277173i \(-0.0893975\pi\)
\(242\) 0 0
\(243\) 2590.02i 0.683743i
\(244\) 0 0
\(245\) 13903.6i 3.62559i
\(246\) 0 0
\(247\) 2517.65 0.648560
\(248\) 0 0
\(249\) 4222.89i 1.07476i
\(250\) 0 0
\(251\) 1242.05 0.312341 0.156170 0.987730i \(-0.450085\pi\)
0.156170 + 0.987730i \(0.450085\pi\)
\(252\) 0 0
\(253\) −367.505 −0.0913236
\(254\) 0 0
\(255\) 3964.13 9885.99i 0.973504 2.42778i
\(256\) 0 0
\(257\) 1469.82 0.356751 0.178375 0.983963i \(-0.442916\pi\)
0.178375 + 0.983963i \(0.442916\pi\)
\(258\) 0 0
\(259\) 7161.16 1.71804
\(260\) 0 0
\(261\) 15240.4i 3.61441i
\(262\) 0 0
\(263\) 4308.70 1.01021 0.505106 0.863057i \(-0.331453\pi\)
0.505106 + 0.863057i \(0.331453\pi\)
\(264\) 0 0
\(265\) 5248.72i 1.21670i
\(266\) 0 0
\(267\) 11163.1i 2.55869i
\(268\) 0 0
\(269\) 3812.72i 0.864184i 0.901830 + 0.432092i \(0.142224\pi\)
−0.901830 + 0.432092i \(0.857776\pi\)
\(270\) 0 0
\(271\) −2859.05 −0.640868 −0.320434 0.947271i \(-0.603829\pi\)
−0.320434 + 0.947271i \(0.603829\pi\)
\(272\) 0 0
\(273\) −13426.8 −2.97666
\(274\) 0 0
\(275\) 1071.59i 0.234979i
\(276\) 0 0
\(277\) 5377.92i 1.16653i 0.812283 + 0.583263i \(0.198224\pi\)
−0.812283 + 0.583263i \(0.801776\pi\)
\(278\) 0 0
\(279\) 5418.91i 1.16280i
\(280\) 0 0
\(281\) 1867.29 0.396418 0.198209 0.980160i \(-0.436488\pi\)
0.198209 + 0.980160i \(0.436488\pi\)
\(282\) 0 0
\(283\) 4296.76i 0.902529i 0.892390 + 0.451265i \(0.149027\pi\)
−0.892390 + 0.451265i \(0.850973\pi\)
\(284\) 0 0
\(285\) 9103.73 1.89213
\(286\) 0 0
\(287\) 6099.79 1.25456
\(288\) 0 0
\(289\) −3551.94 3394.31i −0.722967 0.690883i
\(290\) 0 0
\(291\) 5912.78 1.19111
\(292\) 0 0
\(293\) 8816.10 1.75782 0.878911 0.476985i \(-0.158270\pi\)
0.878911 + 0.476985i \(0.158270\pi\)
\(294\) 0 0
\(295\) 181.826i 0.0358859i
\(296\) 0 0
\(297\) −2184.82 −0.426855
\(298\) 0 0
\(299\) 2079.06i 0.402123i
\(300\) 0 0
\(301\) 658.309i 0.126061i
\(302\) 0 0
\(303\) 6847.28i 1.29824i
\(304\) 0 0
\(305\) −11693.8 −2.19537
\(306\) 0 0
\(307\) −6143.84 −1.14217 −0.571087 0.820890i \(-0.693478\pi\)
−0.571087 + 0.820890i \(0.693478\pi\)
\(308\) 0 0
\(309\) 14710.9i 2.70834i
\(310\) 0 0
\(311\) 7409.80i 1.35103i −0.737345 0.675516i \(-0.763921\pi\)
0.737345 0.675516i \(-0.236079\pi\)
\(312\) 0 0
\(313\) 9455.98i 1.70761i 0.520589 + 0.853807i \(0.325712\pi\)
−0.520589 + 0.853807i \(0.674288\pi\)
\(314\) 0 0
\(315\) −33265.4 −5.95014
\(316\) 0 0
\(317\) 1045.23i 0.185191i 0.995704 + 0.0925957i \(0.0295164\pi\)
−0.995704 + 0.0925957i \(0.970484\pi\)
\(318\) 0 0
\(319\) −1926.71 −0.338165
\(320\) 0 0
\(321\) 1873.42 0.325744
\(322\) 0 0
\(323\) 1562.86 3897.56i 0.269226 0.671412i
\(324\) 0 0
\(325\) 6062.20 1.03468
\(326\) 0 0
\(327\) −11526.9 −1.94935
\(328\) 0 0
\(329\) 2764.87i 0.463320i
\(330\) 0 0
\(331\) 6370.04 1.05779 0.528896 0.848687i \(-0.322606\pi\)
0.528896 + 0.848687i \(0.322606\pi\)
\(332\) 0 0
\(333\) 12196.4i 2.00708i
\(334\) 0 0
\(335\) 7958.04i 1.29789i
\(336\) 0 0
\(337\) 9216.37i 1.48976i 0.667201 + 0.744878i \(0.267492\pi\)
−0.667201 + 0.744878i \(0.732508\pi\)
\(338\) 0 0
\(339\) 13819.5 2.21408
\(340\) 0 0
\(341\) 685.062 0.108792
\(342\) 0 0
\(343\) 17399.4i 2.73901i
\(344\) 0 0
\(345\) 7517.79i 1.17317i
\(346\) 0 0
\(347\) 7810.63i 1.20835i 0.796852 + 0.604174i \(0.206497\pi\)
−0.796852 + 0.604174i \(0.793503\pi\)
\(348\) 0 0
\(349\) 8377.96 1.28499 0.642495 0.766290i \(-0.277899\pi\)
0.642495 + 0.766290i \(0.277899\pi\)
\(350\) 0 0
\(351\) 12360.0i 1.87956i
\(352\) 0 0
\(353\) 500.551 0.0754721 0.0377360 0.999288i \(-0.487985\pi\)
0.0377360 + 0.999288i \(0.487985\pi\)
\(354\) 0 0
\(355\) −7285.06 −1.08916
\(356\) 0 0
\(357\) −8334.84 + 20786.0i −1.23565 + 3.08154i
\(358\) 0 0
\(359\) 12643.0 1.85870 0.929351 0.369198i \(-0.120368\pi\)
0.929351 + 0.369198i \(0.120368\pi\)
\(360\) 0 0
\(361\) −3269.85 −0.476724
\(362\) 0 0
\(363\) 11814.9i 1.70833i
\(364\) 0 0
\(365\) −5537.50 −0.794099
\(366\) 0 0
\(367\) 640.446i 0.0910927i 0.998962 + 0.0455464i \(0.0145029\pi\)
−0.998962 + 0.0455464i \(0.985497\pi\)
\(368\) 0 0
\(369\) 10388.7i 1.46563i
\(370\) 0 0
\(371\) 11035.8i 1.54434i
\(372\) 0 0
\(373\) −1138.01 −0.157972 −0.0789861 0.996876i \(-0.525168\pi\)
−0.0789861 + 0.996876i \(0.525168\pi\)
\(374\) 0 0
\(375\) 2925.94 0.402920
\(376\) 0 0
\(377\) 10899.8i 1.48904i
\(378\) 0 0
\(379\) 12320.8i 1.66987i 0.550352 + 0.834933i \(0.314493\pi\)
−0.550352 + 0.834933i \(0.685507\pi\)
\(380\) 0 0
\(381\) 19025.9i 2.55833i
\(382\) 0 0
\(383\) 9934.77 1.32544 0.662719 0.748868i \(-0.269402\pi\)
0.662719 + 0.748868i \(0.269402\pi\)
\(384\) 0 0
\(385\) 4205.43i 0.556698i
\(386\) 0 0
\(387\) −1121.19 −0.147269
\(388\) 0 0
\(389\) 3323.46 0.433178 0.216589 0.976263i \(-0.430507\pi\)
0.216589 + 0.976263i \(0.430507\pi\)
\(390\) 0 0
\(391\) 3218.57 + 1290.60i 0.416292 + 0.166927i
\(392\) 0 0
\(393\) −17527.8 −2.24978
\(394\) 0 0
\(395\) 13798.4 1.75765
\(396\) 0 0
\(397\) 3094.45i 0.391199i −0.980684 0.195599i \(-0.937335\pi\)
0.980684 0.195599i \(-0.0626652\pi\)
\(398\) 0 0
\(399\) −19141.2 −2.40165
\(400\) 0 0
\(401\) 6261.50i 0.779761i −0.920865 0.389881i \(-0.872516\pi\)
0.920865 0.389881i \(-0.127484\pi\)
\(402\) 0 0
\(403\) 3875.54i 0.479043i
\(404\) 0 0
\(405\) 18660.2i 2.28946i
\(406\) 0 0
\(407\) 1541.88 0.187784
\(408\) 0 0
\(409\) 13966.7 1.68853 0.844264 0.535927i \(-0.180038\pi\)
0.844264 + 0.535927i \(0.180038\pi\)
\(410\) 0 0
\(411\) 7858.76i 0.943172i
\(412\) 0 0
\(413\) 382.302i 0.0455492i
\(414\) 0 0
\(415\) 7482.56i 0.885071i
\(416\) 0 0
\(417\) −191.153 −0.0224479
\(418\) 0 0
\(419\) 6308.88i 0.735582i 0.929908 + 0.367791i \(0.119886\pi\)
−0.929908 + 0.367791i \(0.880114\pi\)
\(420\) 0 0
\(421\) −3142.63 −0.363806 −0.181903 0.983317i \(-0.558226\pi\)
−0.181903 + 0.983317i \(0.558226\pi\)
\(422\) 0 0
\(423\) 4708.94 0.541268
\(424\) 0 0
\(425\) 3763.18 9384.85i 0.429508 1.07113i
\(426\) 0 0
\(427\) 24587.1 2.78654
\(428\) 0 0
\(429\) −2890.94 −0.325351
\(430\) 0 0
\(431\) 6900.34i 0.771178i −0.922671 0.385589i \(-0.873998\pi\)
0.922671 0.385589i \(-0.126002\pi\)
\(432\) 0 0
\(433\) −15236.0 −1.69098 −0.845490 0.533991i \(-0.820692\pi\)
−0.845490 + 0.533991i \(0.820692\pi\)
\(434\) 0 0
\(435\) 39413.2i 4.34418i
\(436\) 0 0
\(437\) 2963.89i 0.324444i
\(438\) 0 0
\(439\) 9594.64i 1.04311i −0.853216 0.521557i \(-0.825351\pi\)
0.853216 0.521557i \(-0.174649\pi\)
\(440\) 0 0
\(441\) 49788.1 5.37611
\(442\) 0 0
\(443\) 3375.02 0.361968 0.180984 0.983486i \(-0.442072\pi\)
0.180984 + 0.983486i \(0.442072\pi\)
\(444\) 0 0
\(445\) 19780.0i 2.10710i
\(446\) 0 0
\(447\) 11136.7i 1.17841i
\(448\) 0 0
\(449\) 2918.05i 0.306707i −0.988171 0.153354i \(-0.950993\pi\)
0.988171 0.153354i \(-0.0490073\pi\)
\(450\) 0 0
\(451\) 1313.35 0.137125
\(452\) 0 0
\(453\) 19269.3i 1.99857i
\(454\) 0 0
\(455\) −23791.0 −2.45130
\(456\) 0 0
\(457\) −3829.50 −0.391983 −0.195992 0.980606i \(-0.562793\pi\)
−0.195992 + 0.980606i \(0.562793\pi\)
\(458\) 0 0
\(459\) 19134.4 + 7672.60i 1.94579 + 0.780232i
\(460\) 0 0
\(461\) 6988.84 0.706080 0.353040 0.935608i \(-0.385148\pi\)
0.353040 + 0.935608i \(0.385148\pi\)
\(462\) 0 0
\(463\) −8669.67 −0.870224 −0.435112 0.900376i \(-0.643291\pi\)
−0.435112 + 0.900376i \(0.643291\pi\)
\(464\) 0 0
\(465\) 14013.8i 1.39758i
\(466\) 0 0
\(467\) −12728.0 −1.26120 −0.630600 0.776108i \(-0.717191\pi\)
−0.630600 + 0.776108i \(0.717191\pi\)
\(468\) 0 0
\(469\) 16732.3i 1.64739i
\(470\) 0 0
\(471\) 10684.4i 1.04524i
\(472\) 0 0
\(473\) 141.741i 0.0137786i
\(474\) 0 0
\(475\) 8642.24 0.834807
\(476\) 0 0
\(477\) −18795.4 −1.80415
\(478\) 0 0
\(479\) 13282.8i 1.26703i −0.773730 0.633516i \(-0.781611\pi\)
0.773730 0.633516i \(-0.218389\pi\)
\(480\) 0 0
\(481\) 8722.72i 0.826864i
\(482\) 0 0
\(483\) 15806.6i 1.48908i
\(484\) 0 0
\(485\) 10476.9 0.980888
\(486\) 0 0
\(487\) 5014.37i 0.466577i −0.972408 0.233288i \(-0.925051\pi\)
0.972408 0.233288i \(-0.0749486\pi\)
\(488\) 0 0
\(489\) 13533.1 1.25151
\(490\) 0 0
\(491\) −14518.4 −1.33443 −0.667217 0.744863i \(-0.732515\pi\)
−0.667217 + 0.744863i \(0.732515\pi\)
\(492\) 0 0
\(493\) 16873.9 + 6766.16i 1.54150 + 0.618119i
\(494\) 0 0
\(495\) −7162.40 −0.650356
\(496\) 0 0
\(497\) 15317.3 1.38245
\(498\) 0 0
\(499\) 18158.2i 1.62901i 0.580158 + 0.814504i \(0.302991\pi\)
−0.580158 + 0.814504i \(0.697009\pi\)
\(500\) 0 0
\(501\) −38505.6 −3.43374
\(502\) 0 0
\(503\) 12331.2i 1.09309i −0.837431 0.546543i \(-0.815943\pi\)
0.837431 0.546543i \(-0.184057\pi\)
\(504\) 0 0
\(505\) 12132.7i 1.06911i
\(506\) 0 0
\(507\) 3991.01i 0.349600i
\(508\) 0 0
\(509\) −4051.78 −0.352833 −0.176417 0.984316i \(-0.556451\pi\)
−0.176417 + 0.984316i \(0.556451\pi\)
\(510\) 0 0
\(511\) 11643.0 1.00793
\(512\) 0 0
\(513\) 17620.3i 1.51648i
\(514\) 0 0
\(515\) 26066.4i 2.23033i
\(516\) 0 0
\(517\) 595.307i 0.0506413i
\(518\) 0 0
\(519\) −23189.4 −1.96127
\(520\) 0 0
\(521\) 6101.23i 0.513052i 0.966537 + 0.256526i \(0.0825779\pi\)
−0.966537 + 0.256526i \(0.917422\pi\)
\(522\) 0 0
\(523\) 1233.70 0.103147 0.0515734 0.998669i \(-0.483576\pi\)
0.0515734 + 0.998669i \(0.483576\pi\)
\(524\) 0 0
\(525\) −46089.7 −3.83146
\(526\) 0 0
\(527\) −5999.69 2405.78i −0.495922 0.198857i
\(528\) 0 0
\(529\) 9719.44 0.798836
\(530\) 0 0
\(531\) 651.110 0.0532124
\(532\) 0 0
\(533\) 7429.90i 0.603799i
\(534\) 0 0
\(535\) 3319.51 0.268252
\(536\) 0 0
\(537\) 7776.03i 0.624880i
\(538\) 0 0
\(539\) 6294.24i 0.502991i
\(540\) 0 0
\(541\) 8968.57i 0.712734i 0.934346 + 0.356367i \(0.115985\pi\)
−0.934346 + 0.356367i \(0.884015\pi\)
\(542\) 0 0
\(543\) 4634.87 0.366301
\(544\) 0 0
\(545\) −20424.5 −1.60530
\(546\) 0 0
\(547\) 7187.71i 0.561836i 0.959732 + 0.280918i \(0.0906389\pi\)
−0.959732 + 0.280918i \(0.909361\pi\)
\(548\) 0 0
\(549\) 41875.0i 3.25534i
\(550\) 0 0
\(551\) 15538.7i 1.20140i
\(552\) 0 0
\(553\) −29012.0 −2.23095
\(554\) 0 0
\(555\) 31541.0i 2.41233i
\(556\) 0 0
\(557\) 12502.2 0.951047 0.475524 0.879703i \(-0.342259\pi\)
0.475524 + 0.879703i \(0.342259\pi\)
\(558\) 0 0
\(559\) −801.860 −0.0606709
\(560\) 0 0
\(561\) −1794.58 + 4475.44i −0.135058 + 0.336815i
\(562\) 0 0
\(563\) −20391.1 −1.52643 −0.763216 0.646143i \(-0.776381\pi\)
−0.763216 + 0.646143i \(0.776381\pi\)
\(564\) 0 0
\(565\) 24486.8 1.82331
\(566\) 0 0
\(567\) 39234.2i 2.90597i
\(568\) 0 0
\(569\) 3407.53 0.251057 0.125528 0.992090i \(-0.459937\pi\)
0.125528 + 0.992090i \(0.459937\pi\)
\(570\) 0 0
\(571\) 9682.41i 0.709626i −0.934937 0.354813i \(-0.884545\pi\)
0.934937 0.354813i \(-0.115455\pi\)
\(572\) 0 0
\(573\) 1851.54i 0.134990i
\(574\) 0 0
\(575\) 7136.69i 0.517601i
\(576\) 0 0
\(577\) −10357.7 −0.747309 −0.373654 0.927568i \(-0.621895\pi\)
−0.373654 + 0.927568i \(0.621895\pi\)
\(578\) 0 0
\(579\) −25545.7 −1.83358
\(580\) 0 0
\(581\) 15732.6i 1.12340i
\(582\) 0 0
\(583\) 2376.12i 0.168798i
\(584\) 0 0
\(585\) 40519.2i 2.86370i
\(586\) 0 0
\(587\) 760.786 0.0534940 0.0267470 0.999642i \(-0.491485\pi\)
0.0267470 + 0.999642i \(0.491485\pi\)
\(588\) 0 0
\(589\) 5524.95i 0.386505i
\(590\) 0 0
\(591\) 31299.8 2.17851
\(592\) 0 0
\(593\) 23861.5 1.65241 0.826203 0.563373i \(-0.190497\pi\)
0.826203 + 0.563373i \(0.190497\pi\)
\(594\) 0 0
\(595\) −14768.5 + 36830.7i −1.01757 + 2.53767i
\(596\) 0 0
\(597\) 25276.6 1.73283
\(598\) 0 0
\(599\) −40.2985 −0.00274884 −0.00137442 0.999999i \(-0.500437\pi\)
−0.00137442 + 0.999999i \(0.500437\pi\)
\(600\) 0 0
\(601\) 28351.4i 1.92426i −0.272594 0.962129i \(-0.587882\pi\)
0.272594 0.962129i \(-0.412118\pi\)
\(602\) 0 0
\(603\) 28497.3 1.92454
\(604\) 0 0
\(605\) 20934.9i 1.40682i
\(606\) 0 0
\(607\) 380.879i 0.0254686i −0.999919 0.0127343i \(-0.995946\pi\)
0.999919 0.0127343i \(-0.00405356\pi\)
\(608\) 0 0
\(609\) 82868.8i 5.51398i
\(610\) 0 0
\(611\) 3367.78 0.222988
\(612\) 0 0
\(613\) −18449.5 −1.21561 −0.607804 0.794087i \(-0.707949\pi\)
−0.607804 + 0.794087i \(0.707949\pi\)
\(614\) 0 0
\(615\) 26866.2i 1.76155i
\(616\) 0 0
\(617\) 13692.2i 0.893397i 0.894685 + 0.446698i \(0.147400\pi\)
−0.894685 + 0.446698i \(0.852600\pi\)
\(618\) 0 0
\(619\) 15000.2i 0.974004i 0.873401 + 0.487002i \(0.161910\pi\)
−0.873401 + 0.487002i \(0.838090\pi\)
\(620\) 0 0
\(621\) −14550.7 −0.940259
\(622\) 0 0
\(623\) 41588.7i 2.67450i
\(624\) 0 0
\(625\) −12847.4 −0.822232
\(626\) 0 0
\(627\) −4121.31 −0.262503
\(628\) 0 0
\(629\) −13503.6 5414.73i −0.855998 0.343242i
\(630\) 0 0
\(631\) 25659.3 1.61883 0.809413 0.587240i \(-0.199785\pi\)
0.809413 + 0.587240i \(0.199785\pi\)
\(632\) 0 0
\(633\) 19901.3 1.24961
\(634\) 0 0
\(635\) 33712.0i 2.10680i
\(636\) 0 0
\(637\) 35607.9 2.21481
\(638\) 0 0
\(639\) 26087.4i 1.61503i
\(640\) 0 0
\(641\) 6709.60i 0.413437i 0.978400 + 0.206719i \(0.0662785\pi\)
−0.978400 + 0.206719i \(0.933722\pi\)
\(642\) 0 0
\(643\) 11976.4i 0.734528i 0.930117 + 0.367264i \(0.119705\pi\)
−0.930117 + 0.367264i \(0.880295\pi\)
\(644\) 0 0
\(645\) −2899.49 −0.177004
\(646\) 0 0
\(647\) −21265.3 −1.29215 −0.646077 0.763272i \(-0.723592\pi\)
−0.646077 + 0.763272i \(0.723592\pi\)
\(648\) 0 0
\(649\) 82.3137i 0.00497857i
\(650\) 0 0
\(651\) 29464.9i 1.77392i
\(652\) 0 0
\(653\) 19671.9i 1.17890i −0.807806 0.589449i \(-0.799345\pi\)
0.807806 0.589449i \(-0.200655\pi\)
\(654\) 0 0
\(655\) −31057.6 −1.85271
\(656\) 0 0
\(657\) 19829.5i 1.17751i
\(658\) 0 0
\(659\) 14782.6 0.873824 0.436912 0.899504i \(-0.356072\pi\)
0.436912 + 0.899504i \(0.356072\pi\)
\(660\) 0 0
\(661\) 27770.8 1.63413 0.817063 0.576548i \(-0.195601\pi\)
0.817063 + 0.576548i \(0.195601\pi\)
\(662\) 0 0
\(663\) 25318.5 + 10152.3i 1.48309 + 0.594697i
\(664\) 0 0
\(665\) −33916.4 −1.97777
\(666\) 0 0
\(667\) −12831.7 −0.744896
\(668\) 0 0
\(669\) 1071.14i 0.0619021i
\(670\) 0 0
\(671\) 5293.86 0.304571
\(672\) 0 0
\(673\) 21891.3i 1.25386i 0.779077 + 0.626929i \(0.215688\pi\)
−0.779077 + 0.626929i \(0.784312\pi\)
\(674\) 0 0
\(675\) 42427.6i 2.41932i
\(676\) 0 0
\(677\) 1946.42i 0.110498i −0.998473 0.0552488i \(-0.982405\pi\)
0.998473 0.0552488i \(-0.0175952\pi\)
\(678\) 0 0
\(679\) −22028.3 −1.24502
\(680\) 0 0
\(681\) −1395.94 −0.0785498
\(682\) 0 0
\(683\) 10516.5i 0.589169i −0.955625 0.294584i \(-0.904819\pi\)
0.955625 0.294584i \(-0.0951812\pi\)
\(684\) 0 0
\(685\) 13925.0i 0.776709i
\(686\) 0 0
\(687\) 2336.12i 0.129736i
\(688\) 0 0
\(689\) −13442.2 −0.743263
\(690\) 0 0
\(691\) 26695.9i 1.46970i −0.678232 0.734848i \(-0.737253\pi\)
0.678232 0.734848i \(-0.262747\pi\)
\(692\) 0 0
\(693\) 15059.4 0.825484
\(694\) 0 0
\(695\) −338.704 −0.0184860
\(696\) 0 0
\(697\) −11502.2 4612.20i −0.625074 0.250645i
\(698\) 0 0
\(699\) 9496.30 0.513853
\(700\) 0 0
\(701\) −22317.0 −1.20243 −0.601214 0.799088i \(-0.705316\pi\)
−0.601214 + 0.799088i \(0.705316\pi\)
\(702\) 0 0
\(703\) 12435.1i 0.667137i
\(704\) 0 0
\(705\) 12177.7 0.650554
\(706\) 0 0
\(707\) 25509.8i 1.35700i
\(708\) 0 0
\(709\) 28249.7i 1.49639i −0.663478 0.748196i \(-0.730920\pi\)
0.663478 0.748196i \(-0.269080\pi\)
\(710\) 0 0
\(711\) 49411.3i 2.60629i
\(712\) 0 0
\(713\) 4562.46 0.239643
\(714\) 0 0
\(715\) −5122.46 −0.267929
\(716\) 0 0
\(717\) 35222.4i 1.83460i
\(718\) 0 0
\(719\) 30991.2i 1.60748i −0.594982 0.803739i \(-0.702841\pi\)
0.594982 0.803739i \(-0.297159\pi\)
\(720\) 0 0
\(721\) 54806.3i 2.83092i
\(722\) 0 0
\(723\) 19206.5 0.987963
\(724\) 0 0
\(725\) 37415.2i 1.91664i
\(726\) 0 0
\(727\) −13328.0 −0.679929 −0.339965 0.940438i \(-0.610415\pi\)
−0.339965 + 0.940438i \(0.610415\pi\)
\(728\) 0 0
\(729\) 6718.94 0.341358
\(730\) 0 0
\(731\) −497.764 + 1241.35i −0.0251853 + 0.0628086i
\(732\) 0 0
\(733\) −27976.7 −1.40974 −0.704872 0.709334i \(-0.748996\pi\)
−0.704872 + 0.709334i \(0.748996\pi\)
\(734\) 0 0
\(735\) 128757. 6.46158
\(736\) 0 0
\(737\) 3602.64i 0.180061i
\(738\) 0 0
\(739\) −33971.2 −1.69100 −0.845501 0.533974i \(-0.820698\pi\)
−0.845501 + 0.533974i \(0.820698\pi\)
\(740\) 0 0
\(741\) 23315.1i 1.15587i
\(742\) 0 0
\(743\) 25922.4i 1.27995i 0.768397 + 0.639974i \(0.221055\pi\)
−0.768397 + 0.639974i \(0.778945\pi\)
\(744\) 0 0
\(745\) 19733.2i 0.970428i
\(746\) 0 0
\(747\) 26794.6 1.31240
\(748\) 0 0
\(749\) −6979.49 −0.340488
\(750\) 0 0
\(751\) 18950.5i 0.920792i 0.887714 + 0.460396i \(0.152293\pi\)
−0.887714 + 0.460396i \(0.847707\pi\)
\(752\) 0 0
\(753\) 11502.2i 0.556658i
\(754\) 0 0
\(755\) 34143.4i 1.64583i
\(756\) 0 0
\(757\) −8184.07 −0.392940 −0.196470 0.980510i \(-0.562948\pi\)
−0.196470 + 0.980510i \(0.562948\pi\)
\(758\) 0 0
\(759\) 3403.34i 0.162758i
\(760\) 0 0
\(761\) −25803.9 −1.22916 −0.614579 0.788855i \(-0.710674\pi\)
−0.614579 + 0.788855i \(0.710674\pi\)
\(762\) 0 0
\(763\) 42943.9 2.03758
\(764\) 0 0
\(765\) 62727.6 + 25152.8i 2.96460 + 1.18876i
\(766\) 0 0
\(767\) 465.666 0.0219221
\(768\) 0 0
\(769\) 5222.53 0.244901 0.122451 0.992475i \(-0.460925\pi\)
0.122451 + 0.992475i \(0.460925\pi\)
\(770\) 0 0
\(771\) 13611.5i 0.635806i
\(772\) 0 0
\(773\) 32636.6 1.51857 0.759286 0.650757i \(-0.225548\pi\)
0.759286 + 0.650757i \(0.225548\pi\)
\(774\) 0 0
\(775\) 13303.4i 0.616610i
\(776\) 0 0
\(777\) 66317.0i 3.06192i
\(778\) 0 0
\(779\) 10592.0i 0.487162i
\(780\) 0 0
\(781\) 3297.99 0.151103
\(782\) 0 0
\(783\) −76284.5 −3.48172
\(784\) 0 0
\(785\) 18931.7i 0.860764i
\(786\) 0 0
\(787\) 17014.8i 0.770664i 0.922778 + 0.385332i \(0.125913\pi\)
−0.922778 + 0.385332i \(0.874087\pi\)
\(788\) 0 0
\(789\) 39901.4i 1.80042i
\(790\) 0 0
\(791\) −51485.1 −2.31429
\(792\) 0 0
\(793\) 29948.5i 1.34111i
\(794\) 0 0
\(795\) −48606.6 −2.16842
\(796\) 0 0
\(797\) −20826.6 −0.925616 −0.462808 0.886458i \(-0.653158\pi\)
−0.462808 + 0.886458i \(0.653158\pi\)
\(798\)