Properties

Label 136.4.b.b.33.4
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: \( x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 33.4
Root \(9.30031i\) of defining polynomial
Character \(\chi\) \(=\) 136.33
Dual form 136.4.b.b.33.5

$q$-expansion

\(f(q)\) \(=\) \(q-2.62097i q^{3} +11.5318i q^{5} -23.0485i q^{7} +20.1305 q^{9} +O(q^{10})\) \(q-2.62097i q^{3} +11.5318i q^{5} -23.0485i q^{7} +20.1305 q^{9} -1.19452i q^{11} +58.3731 q^{13} +30.2245 q^{15} +(-54.8640 - 43.6227i) q^{17} +138.971 q^{19} -60.4096 q^{21} -180.911i q^{23} -7.98158 q^{25} -123.528i q^{27} -115.027i q^{29} +210.726i q^{31} -3.13080 q^{33} +265.790 q^{35} +210.661i q^{37} -152.994i q^{39} +297.500i q^{41} +174.746 q^{43} +232.140i q^{45} -199.717 q^{47} -188.234 q^{49} +(-114.334 + 143.797i) q^{51} -706.217 q^{53} +13.7749 q^{55} -364.238i q^{57} -182.746 q^{59} -489.499i q^{61} -463.978i q^{63} +673.144i q^{65} -167.043 q^{67} -474.163 q^{69} +818.157i q^{71} +1090.15i q^{73} +20.9195i q^{75} -27.5319 q^{77} +110.196i q^{79} +219.760 q^{81} +118.672 q^{83} +(503.047 - 632.679i) q^{85} -301.483 q^{87} +400.332 q^{89} -1345.41i q^{91} +552.306 q^{93} +1602.58i q^{95} -924.543i q^{97} -24.0463i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 132 q^{9}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

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

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
\(3\) 2.62097i 0.504407i −0.967674 0.252203i \(-0.918845\pi\)
0.967674 0.252203i \(-0.0811552\pi\)
\(4\) 0 0
\(5\) 11.5318i 1.03143i 0.856759 + 0.515716i \(0.172474\pi\)
−0.856759 + 0.515716i \(0.827526\pi\)
\(6\) 0 0
\(7\) 23.0485i 1.24450i −0.782817 0.622251i \(-0.786218\pi\)
0.782817 0.622251i \(-0.213782\pi\)
\(8\) 0 0
\(9\) 20.1305 0.745574
\(10\) 0 0
\(11\) 1.19452i 0.0327419i −0.999866 0.0163710i \(-0.994789\pi\)
0.999866 0.0163710i \(-0.00521127\pi\)
\(12\) 0 0
\(13\) 58.3731 1.24537 0.622684 0.782474i \(-0.286042\pi\)
0.622684 + 0.782474i \(0.286042\pi\)
\(14\) 0 0
\(15\) 30.2245 0.520262
\(16\) 0 0
\(17\) −54.8640 43.6227i −0.782734 0.622357i
\(18\) 0 0
\(19\) 138.971 1.67800 0.839001 0.544130i \(-0.183140\pi\)
0.839001 + 0.544130i \(0.183140\pi\)
\(20\) 0 0
\(21\) −60.4096 −0.627736
\(22\) 0 0
\(23\) 180.911i 1.64011i −0.572284 0.820055i \(-0.693943\pi\)
0.572284 0.820055i \(-0.306057\pi\)
\(24\) 0 0
\(25\) −7.98158 −0.0638527
\(26\) 0 0
\(27\) 123.528i 0.880479i
\(28\) 0 0
\(29\) 115.027i 0.736551i −0.929717 0.368275i \(-0.879948\pi\)
0.929717 0.368275i \(-0.120052\pi\)
\(30\) 0 0
\(31\) 210.726i 1.22088i 0.792061 + 0.610442i \(0.209008\pi\)
−0.792061 + 0.610442i \(0.790992\pi\)
\(32\) 0 0
\(33\) −3.13080 −0.0165152
\(34\) 0 0
\(35\) 265.790 1.28362
\(36\) 0 0
\(37\) 210.661i 0.936010i 0.883726 + 0.468005i \(0.155027\pi\)
−0.883726 + 0.468005i \(0.844973\pi\)
\(38\) 0 0
\(39\) 152.994i 0.628172i
\(40\) 0 0
\(41\) 297.500i 1.13321i 0.823989 + 0.566605i \(0.191743\pi\)
−0.823989 + 0.566605i \(0.808257\pi\)
\(42\) 0 0
\(43\) 174.746 0.619734 0.309867 0.950780i \(-0.399715\pi\)
0.309867 + 0.950780i \(0.399715\pi\)
\(44\) 0 0
\(45\) 232.140i 0.769009i
\(46\) 0 0
\(47\) −199.717 −0.619823 −0.309911 0.950765i \(-0.600299\pi\)
−0.309911 + 0.950765i \(0.600299\pi\)
\(48\) 0 0
\(49\) −188.234 −0.548787
\(50\) 0 0
\(51\) −114.334 + 143.797i −0.313921 + 0.394816i
\(52\) 0 0
\(53\) −706.217 −1.83031 −0.915154 0.403105i \(-0.867931\pi\)
−0.915154 + 0.403105i \(0.867931\pi\)
\(54\) 0 0
\(55\) 13.7749 0.0337711
\(56\) 0 0
\(57\) 364.238i 0.846396i
\(58\) 0 0
\(59\) −182.746 −0.403247 −0.201623 0.979463i \(-0.564622\pi\)
−0.201623 + 0.979463i \(0.564622\pi\)
\(60\) 0 0
\(61\) 489.499i 1.02744i −0.857958 0.513720i \(-0.828267\pi\)
0.857958 0.513720i \(-0.171733\pi\)
\(62\) 0 0
\(63\) 463.978i 0.927869i
\(64\) 0 0
\(65\) 673.144i 1.28451i
\(66\) 0 0
\(67\) −167.043 −0.304590 −0.152295 0.988335i \(-0.548666\pi\)
−0.152295 + 0.988335i \(0.548666\pi\)
\(68\) 0 0
\(69\) −474.163 −0.827283
\(70\) 0 0
\(71\) 818.157i 1.36757i 0.729684 + 0.683784i \(0.239667\pi\)
−0.729684 + 0.683784i \(0.760333\pi\)
\(72\) 0 0
\(73\) 1090.15i 1.74784i 0.486071 + 0.873919i \(0.338430\pi\)
−0.486071 + 0.873919i \(0.661570\pi\)
\(74\) 0 0
\(75\) 20.9195i 0.0322077i
\(76\) 0 0
\(77\) −27.5319 −0.0407474
\(78\) 0 0
\(79\) 110.196i 0.156937i 0.996917 + 0.0784685i \(0.0250030\pi\)
−0.996917 + 0.0784685i \(0.974997\pi\)
\(80\) 0 0
\(81\) 219.760 0.301454
\(82\) 0 0
\(83\) 118.672 0.156939 0.0784696 0.996917i \(-0.474997\pi\)
0.0784696 + 0.996917i \(0.474997\pi\)
\(84\) 0 0
\(85\) 503.047 632.679i 0.641919 0.807337i
\(86\) 0 0
\(87\) −301.483 −0.371521
\(88\) 0 0
\(89\) 400.332 0.476799 0.238399 0.971167i \(-0.423377\pi\)
0.238399 + 0.971167i \(0.423377\pi\)
\(90\) 0 0
\(91\) 1345.41i 1.54986i
\(92\) 0 0
\(93\) 552.306 0.615823
\(94\) 0 0
\(95\) 1602.58i 1.73075i
\(96\) 0 0
\(97\) 924.543i 0.967764i −0.875133 0.483882i \(-0.839226\pi\)
0.875133 0.483882i \(-0.160774\pi\)
\(98\) 0 0
\(99\) 24.0463i 0.0244115i
\(100\) 0 0
\(101\) 223.816 0.220500 0.110250 0.993904i \(-0.464835\pi\)
0.110250 + 0.993904i \(0.464835\pi\)
\(102\) 0 0
\(103\) −1280.06 −1.22454 −0.612271 0.790648i \(-0.709744\pi\)
−0.612271 + 0.790648i \(0.709744\pi\)
\(104\) 0 0
\(105\) 696.629i 0.647467i
\(106\) 0 0
\(107\) 1099.03i 0.992969i −0.868046 0.496485i \(-0.834624\pi\)
0.868046 0.496485i \(-0.165376\pi\)
\(108\) 0 0
\(109\) 733.360i 0.644433i 0.946666 + 0.322216i \(0.104428\pi\)
−0.946666 + 0.322216i \(0.895572\pi\)
\(110\) 0 0
\(111\) 552.136 0.472130
\(112\) 0 0
\(113\) 1637.39i 1.36312i −0.731761 0.681561i \(-0.761301\pi\)
0.731761 0.681561i \(-0.238699\pi\)
\(114\) 0 0
\(115\) 2086.22 1.69166
\(116\) 0 0
\(117\) 1175.08 0.928513
\(118\) 0 0
\(119\) −1005.44 + 1264.53i −0.774525 + 0.974115i
\(120\) 0 0
\(121\) 1329.57 0.998928
\(122\) 0 0
\(123\) 779.739 0.571599
\(124\) 0 0
\(125\) 1349.43i 0.965573i
\(126\) 0 0
\(127\) −1502.71 −1.04995 −0.524977 0.851117i \(-0.675926\pi\)
−0.524977 + 0.851117i \(0.675926\pi\)
\(128\) 0 0
\(129\) 458.006i 0.312598i
\(130\) 0 0
\(131\) 702.381i 0.468453i −0.972182 0.234226i \(-0.924744\pi\)
0.972182 0.234226i \(-0.0752557\pi\)
\(132\) 0 0
\(133\) 3203.07i 2.08828i
\(134\) 0 0
\(135\) 1424.49 0.908155
\(136\) 0 0
\(137\) 18.9889 0.0118418 0.00592092 0.999982i \(-0.498115\pi\)
0.00592092 + 0.999982i \(0.498115\pi\)
\(138\) 0 0
\(139\) 276.969i 0.169009i −0.996423 0.0845043i \(-0.973069\pi\)
0.996423 0.0845043i \(-0.0269307\pi\)
\(140\) 0 0
\(141\) 523.452i 0.312643i
\(142\) 0 0
\(143\) 69.7277i 0.0407757i
\(144\) 0 0
\(145\) 1326.46 0.759702
\(146\) 0 0
\(147\) 493.357i 0.276812i
\(148\) 0 0
\(149\) 128.491 0.0706468 0.0353234 0.999376i \(-0.488754\pi\)
0.0353234 + 0.999376i \(0.488754\pi\)
\(150\) 0 0
\(151\) −2097.44 −1.13038 −0.565190 0.824961i \(-0.691197\pi\)
−0.565190 + 0.824961i \(0.691197\pi\)
\(152\) 0 0
\(153\) −1104.44 878.147i −0.583586 0.464013i
\(154\) 0 0
\(155\) −2430.04 −1.25926
\(156\) 0 0
\(157\) 2911.59 1.48007 0.740033 0.672571i \(-0.234810\pi\)
0.740033 + 0.672571i \(0.234810\pi\)
\(158\) 0 0
\(159\) 1850.98i 0.923220i
\(160\) 0 0
\(161\) −4169.73 −2.04112
\(162\) 0 0
\(163\) 1574.26i 0.756478i 0.925708 + 0.378239i \(0.123470\pi\)
−0.925708 + 0.378239i \(0.876530\pi\)
\(164\) 0 0
\(165\) 36.1037i 0.0170344i
\(166\) 0 0
\(167\) 1290.10i 0.597788i −0.954286 0.298894i \(-0.903382\pi\)
0.954286 0.298894i \(-0.0966177\pi\)
\(168\) 0 0
\(169\) 1210.41 0.550939
\(170\) 0 0
\(171\) 2797.55 1.25107
\(172\) 0 0
\(173\) 2370.19i 1.04163i 0.853669 + 0.520816i \(0.174372\pi\)
−0.853669 + 0.520816i \(0.825628\pi\)
\(174\) 0 0
\(175\) 183.964i 0.0794648i
\(176\) 0 0
\(177\) 478.974i 0.203400i
\(178\) 0 0
\(179\) 1719.87 0.718153 0.359077 0.933308i \(-0.383092\pi\)
0.359077 + 0.933308i \(0.383092\pi\)
\(180\) 0 0
\(181\) 61.7101i 0.0253418i 0.999920 + 0.0126709i \(0.00403339\pi\)
−0.999920 + 0.0126709i \(0.995967\pi\)
\(182\) 0 0
\(183\) −1282.96 −0.518248
\(184\) 0 0
\(185\) −2429.29 −0.965431
\(186\) 0 0
\(187\) −52.1082 + 65.5361i −0.0203771 + 0.0256282i
\(188\) 0 0
\(189\) −2847.13 −1.09576
\(190\) 0 0
\(191\) −2565.34 −0.971839 −0.485919 0.874004i \(-0.661515\pi\)
−0.485919 + 0.874004i \(0.661515\pi\)
\(192\) 0 0
\(193\) 1991.65i 0.742810i 0.928471 + 0.371405i \(0.121124\pi\)
−0.928471 + 0.371405i \(0.878876\pi\)
\(194\) 0 0
\(195\) 1764.29 0.647917
\(196\) 0 0
\(197\) 3831.76i 1.38579i 0.721036 + 0.692897i \(0.243666\pi\)
−0.721036 + 0.692897i \(0.756334\pi\)
\(198\) 0 0
\(199\) 2171.38i 0.773492i 0.922186 + 0.386746i \(0.126401\pi\)
−0.922186 + 0.386746i \(0.873599\pi\)
\(200\) 0 0
\(201\) 437.815i 0.153637i
\(202\) 0 0
\(203\) −2651.20 −0.916639
\(204\) 0 0
\(205\) −3430.69 −1.16883
\(206\) 0 0
\(207\) 3641.83i 1.22282i
\(208\) 0 0
\(209\) 166.003i 0.0549410i
\(210\) 0 0
\(211\) 2069.97i 0.675369i −0.941259 0.337685i \(-0.890356\pi\)
0.941259 0.337685i \(-0.109644\pi\)
\(212\) 0 0
\(213\) 2144.37 0.689811
\(214\) 0 0
\(215\) 2015.13i 0.639214i
\(216\) 0 0
\(217\) 4856.91 1.51939
\(218\) 0 0
\(219\) 2857.25 0.881622
\(220\) 0 0
\(221\) −3202.58 2546.39i −0.974791 0.775062i
\(222\) 0 0
\(223\) −4297.20 −1.29041 −0.645206 0.764008i \(-0.723229\pi\)
−0.645206 + 0.764008i \(0.723229\pi\)
\(224\) 0 0
\(225\) −160.673 −0.0476069
\(226\) 0 0
\(227\) 336.979i 0.0985291i −0.998786 0.0492646i \(-0.984312\pi\)
0.998786 0.0492646i \(-0.0156877\pi\)
\(228\) 0 0
\(229\) −3076.57 −0.887798 −0.443899 0.896077i \(-0.646405\pi\)
−0.443899 + 0.896077i \(0.646405\pi\)
\(230\) 0 0
\(231\) 72.1604i 0.0205533i
\(232\) 0 0
\(233\) 6537.86i 1.83824i 0.393980 + 0.919119i \(0.371098\pi\)
−0.393980 + 0.919119i \(0.628902\pi\)
\(234\) 0 0
\(235\) 2303.09i 0.639305i
\(236\) 0 0
\(237\) 288.821 0.0791601
\(238\) 0 0
\(239\) −3432.31 −0.928943 −0.464472 0.885588i \(-0.653756\pi\)
−0.464472 + 0.885588i \(0.653756\pi\)
\(240\) 0 0
\(241\) 1312.48i 0.350807i 0.984497 + 0.175404i \(0.0561230\pi\)
−0.984497 + 0.175404i \(0.943877\pi\)
\(242\) 0 0
\(243\) 3911.24i 1.03253i
\(244\) 0 0
\(245\) 2170.67i 0.566037i
\(246\) 0 0
\(247\) 8112.14 2.08973
\(248\) 0 0
\(249\) 311.037i 0.0791612i
\(250\) 0 0
\(251\) 6498.55 1.63420 0.817101 0.576494i \(-0.195580\pi\)
0.817101 + 0.576494i \(0.195580\pi\)
\(252\) 0 0
\(253\) −216.102 −0.0537004
\(254\) 0 0
\(255\) −1658.24 1318.47i −0.407226 0.323788i
\(256\) 0 0
\(257\) −4874.66 −1.18316 −0.591582 0.806245i \(-0.701496\pi\)
−0.591582 + 0.806245i \(0.701496\pi\)
\(258\) 0 0
\(259\) 4855.41 1.16487
\(260\) 0 0
\(261\) 2315.55i 0.549153i
\(262\) 0 0
\(263\) 1482.50 0.347584 0.173792 0.984782i \(-0.444398\pi\)
0.173792 + 0.984782i \(0.444398\pi\)
\(264\) 0 0
\(265\) 8143.92i 1.88784i
\(266\) 0 0
\(267\) 1049.26i 0.240501i
\(268\) 0 0
\(269\) 5058.75i 1.14661i 0.819343 + 0.573304i \(0.194338\pi\)
−0.819343 + 0.573304i \(0.805662\pi\)
\(270\) 0 0
\(271\) 1133.70 0.254122 0.127061 0.991895i \(-0.459446\pi\)
0.127061 + 0.991895i \(0.459446\pi\)
\(272\) 0 0
\(273\) −3526.29 −0.781761
\(274\) 0 0
\(275\) 9.53415i 0.00209066i
\(276\) 0 0
\(277\) 3698.34i 0.802208i −0.916033 0.401104i \(-0.868627\pi\)
0.916033 0.401104i \(-0.131373\pi\)
\(278\) 0 0
\(279\) 4242.01i 0.910260i
\(280\) 0 0
\(281\) 2190.08 0.464945 0.232472 0.972603i \(-0.425318\pi\)
0.232472 + 0.972603i \(0.425318\pi\)
\(282\) 0 0
\(283\) 8159.66i 1.71393i 0.515378 + 0.856963i \(0.327652\pi\)
−0.515378 + 0.856963i \(0.672348\pi\)
\(284\) 0 0
\(285\) 4200.31 0.873000
\(286\) 0 0
\(287\) 6856.92 1.41028
\(288\) 0 0
\(289\) 1107.12 + 4786.63i 0.225345 + 0.974279i
\(290\) 0 0
\(291\) −2423.20 −0.488147
\(292\) 0 0
\(293\) 4777.49 0.952574 0.476287 0.879290i \(-0.341982\pi\)
0.476287 + 0.879290i \(0.341982\pi\)
\(294\) 0 0
\(295\) 2107.39i 0.415922i
\(296\) 0 0
\(297\) −147.556 −0.0288286
\(298\) 0 0
\(299\) 10560.3i 2.04254i
\(300\) 0 0
\(301\) 4027.65i 0.771261i
\(302\) 0 0
\(303\) 586.615i 0.111222i
\(304\) 0 0
\(305\) 5644.78 1.05974
\(306\) 0 0
\(307\) 531.103 0.0987351 0.0493675 0.998781i \(-0.484279\pi\)
0.0493675 + 0.998781i \(0.484279\pi\)
\(308\) 0 0
\(309\) 3355.00i 0.617667i
\(310\) 0 0
\(311\) 4767.26i 0.869217i 0.900619 + 0.434609i \(0.143113\pi\)
−0.900619 + 0.434609i \(0.856887\pi\)
\(312\) 0 0
\(313\) 8613.47i 1.55547i −0.628593 0.777735i \(-0.716369\pi\)
0.628593 0.777735i \(-0.283631\pi\)
\(314\) 0 0
\(315\) 5350.48 0.957034
\(316\) 0 0
\(317\) 2453.20i 0.434654i −0.976099 0.217327i \(-0.930266\pi\)
0.976099 0.217327i \(-0.0697338\pi\)
\(318\) 0 0
\(319\) −137.402 −0.0241161
\(320\) 0 0
\(321\) −2880.54 −0.500860
\(322\) 0 0
\(323\) −7624.48 6062.27i −1.31343 1.04432i
\(324\) 0 0
\(325\) −465.909 −0.0795200
\(326\) 0 0
\(327\) 1922.12 0.325056
\(328\) 0 0
\(329\) 4603.17i 0.771371i
\(330\) 0 0
\(331\) −11003.5 −1.82721 −0.913605 0.406604i \(-0.866713\pi\)
−0.913605 + 0.406604i \(0.866713\pi\)
\(332\) 0 0
\(333\) 4240.70i 0.697865i
\(334\) 0 0
\(335\) 1926.30i 0.314164i
\(336\) 0 0
\(337\) 3664.37i 0.592318i 0.955139 + 0.296159i \(0.0957058\pi\)
−0.955139 + 0.296159i \(0.904294\pi\)
\(338\) 0 0
\(339\) −4291.56 −0.687568
\(340\) 0 0
\(341\) 251.716 0.0399741
\(342\) 0 0
\(343\) 3567.12i 0.561535i
\(344\) 0 0
\(345\) 5467.94i 0.853286i
\(346\) 0 0
\(347\) 3173.37i 0.490938i 0.969404 + 0.245469i \(0.0789419\pi\)
−0.969404 + 0.245469i \(0.921058\pi\)
\(348\) 0 0
\(349\) −2185.47 −0.335202 −0.167601 0.985855i \(-0.553602\pi\)
−0.167601 + 0.985855i \(0.553602\pi\)
\(350\) 0 0
\(351\) 7210.70i 1.09652i
\(352\) 0 0
\(353\) 1216.36 0.183400 0.0917002 0.995787i \(-0.470770\pi\)
0.0917002 + 0.995787i \(0.470770\pi\)
\(354\) 0 0
\(355\) −9434.80 −1.41055
\(356\) 0 0
\(357\) 3314.31 + 2635.23i 0.491350 + 0.390675i
\(358\) 0 0
\(359\) −7448.93 −1.09510 −0.547548 0.836774i \(-0.684439\pi\)
−0.547548 + 0.836774i \(0.684439\pi\)
\(360\) 0 0
\(361\) 12453.8 1.81569
\(362\) 0 0
\(363\) 3484.78i 0.503866i
\(364\) 0 0
\(365\) −12571.3 −1.80278
\(366\) 0 0
\(367\) 2374.59i 0.337746i 0.985638 + 0.168873i \(0.0540128\pi\)
−0.985638 + 0.168873i \(0.945987\pi\)
\(368\) 0 0
\(369\) 5988.81i 0.844892i
\(370\) 0 0
\(371\) 16277.2i 2.27782i
\(372\) 0 0
\(373\) 12231.7 1.69795 0.848974 0.528435i \(-0.177221\pi\)
0.848974 + 0.528435i \(0.177221\pi\)
\(374\) 0 0
\(375\) 3536.82 0.487041
\(376\) 0 0
\(377\) 6714.48i 0.917276i
\(378\) 0 0
\(379\) 14129.9i 1.91505i 0.288358 + 0.957523i \(0.406891\pi\)
−0.288358 + 0.957523i \(0.593109\pi\)
\(380\) 0 0
\(381\) 3938.57i 0.529604i
\(382\) 0 0
\(383\) 13306.4 1.77526 0.887631 0.460556i \(-0.152350\pi\)
0.887631 + 0.460556i \(0.152350\pi\)
\(384\) 0 0
\(385\) 317.491i 0.0420282i
\(386\) 0 0
\(387\) 3517.73 0.462058
\(388\) 0 0
\(389\) −8007.84 −1.04374 −0.521868 0.853026i \(-0.674765\pi\)
−0.521868 + 0.853026i \(0.674765\pi\)
\(390\) 0 0
\(391\) −7891.83 + 9925.50i −1.02073 + 1.28377i
\(392\) 0 0
\(393\) −1840.92 −0.236291
\(394\) 0 0
\(395\) −1270.76 −0.161870
\(396\) 0 0
\(397\) 13185.2i 1.66687i −0.552619 0.833434i \(-0.686372\pi\)
0.552619 0.833434i \(-0.313628\pi\)
\(398\) 0 0
\(399\) −8395.15 −1.05334
\(400\) 0 0
\(401\) 3205.81i 0.399228i 0.979875 + 0.199614i \(0.0639688\pi\)
−0.979875 + 0.199614i \(0.936031\pi\)
\(402\) 0 0
\(403\) 12300.7i 1.52045i
\(404\) 0 0
\(405\) 2534.22i 0.310929i
\(406\) 0 0
\(407\) 251.638 0.0306468
\(408\) 0 0
\(409\) −15693.3 −1.89727 −0.948636 0.316369i \(-0.897536\pi\)
−0.948636 + 0.316369i \(0.897536\pi\)
\(410\) 0 0
\(411\) 49.7695i 0.00597311i
\(412\) 0 0
\(413\) 4212.03i 0.501842i
\(414\) 0 0
\(415\) 1368.50i 0.161872i
\(416\) 0 0
\(417\) −725.929 −0.0852491
\(418\) 0 0
\(419\) 6910.65i 0.805745i −0.915256 0.402873i \(-0.868012\pi\)
0.915256 0.402873i \(-0.131988\pi\)
\(420\) 0 0
\(421\) 9225.05 1.06794 0.533968 0.845505i \(-0.320700\pi\)
0.533968 + 0.845505i \(0.320700\pi\)
\(422\) 0 0
\(423\) −4020.40 −0.462124
\(424\) 0 0
\(425\) 437.902 + 348.178i 0.0499796 + 0.0397391i
\(426\) 0 0
\(427\) −11282.2 −1.27865
\(428\) 0 0
\(429\) −182.755 −0.0205675
\(430\) 0 0
\(431\) 8822.74i 0.986024i 0.870022 + 0.493012i \(0.164104\pi\)
−0.870022 + 0.493012i \(0.835896\pi\)
\(432\) 0 0
\(433\) −3190.95 −0.354151 −0.177075 0.984197i \(-0.556664\pi\)
−0.177075 + 0.984197i \(0.556664\pi\)
\(434\) 0 0
\(435\) 3476.63i 0.383199i
\(436\) 0 0
\(437\) 25141.3i 2.75211i
\(438\) 0 0
\(439\) 15069.5i 1.63834i −0.573553 0.819168i \(-0.694435\pi\)
0.573553 0.819168i \(-0.305565\pi\)
\(440\) 0 0
\(441\) −3789.24 −0.409161
\(442\) 0 0
\(443\) −3026.24 −0.324562 −0.162281 0.986745i \(-0.551885\pi\)
−0.162281 + 0.986745i \(0.551885\pi\)
\(444\) 0 0
\(445\) 4616.53i 0.491786i
\(446\) 0 0
\(447\) 336.771i 0.0356347i
\(448\) 0 0
\(449\) 4641.02i 0.487802i −0.969800 0.243901i \(-0.921573\pi\)
0.969800 0.243901i \(-0.0784272\pi\)
\(450\) 0 0
\(451\) 355.369 0.0371035
\(452\) 0 0
\(453\) 5497.34i 0.570171i
\(454\) 0 0
\(455\) 15515.0 1.59858
\(456\) 0 0
\(457\) 17393.2 1.78035 0.890174 0.455620i \(-0.150582\pi\)
0.890174 + 0.455620i \(0.150582\pi\)
\(458\) 0 0
\(459\) −5388.62 + 6777.23i −0.547972 + 0.689181i
\(460\) 0 0
\(461\) 2451.93 0.247718 0.123859 0.992300i \(-0.460473\pi\)
0.123859 + 0.992300i \(0.460473\pi\)
\(462\) 0 0
\(463\) −14586.3 −1.46411 −0.732054 0.681246i \(-0.761438\pi\)
−0.732054 + 0.681246i \(0.761438\pi\)
\(464\) 0 0
\(465\) 6369.07i 0.635179i
\(466\) 0 0
\(467\) 15195.6 1.50572 0.752859 0.658182i \(-0.228674\pi\)
0.752859 + 0.658182i \(0.228674\pi\)
\(468\) 0 0
\(469\) 3850.09i 0.379063i
\(470\) 0 0
\(471\) 7631.21i 0.746555i
\(472\) 0 0
\(473\) 208.738i 0.0202913i
\(474\) 0 0
\(475\) −1109.21 −0.107145
\(476\) 0 0
\(477\) −14216.5 −1.36463
\(478\) 0 0
\(479\) 4422.10i 0.421819i −0.977506 0.210909i \(-0.932358\pi\)
0.977506 0.210909i \(-0.0676425\pi\)
\(480\) 0 0
\(481\) 12296.9i 1.16568i
\(482\) 0 0
\(483\) 10928.8i 1.02956i
\(484\) 0 0
\(485\) 10661.6 0.998183
\(486\) 0 0
\(487\) 17772.7i 1.65371i −0.562414 0.826856i \(-0.690127\pi\)
0.562414 0.826856i \(-0.309873\pi\)
\(488\) 0 0
\(489\) 4126.11 0.381573
\(490\) 0 0
\(491\) 4845.34 0.445350 0.222675 0.974893i \(-0.428521\pi\)
0.222675 + 0.974893i \(0.428521\pi\)
\(492\) 0 0
\(493\) −5017.79 + 6310.84i −0.458397 + 0.576523i
\(494\) 0 0
\(495\) 277.296 0.0251788
\(496\) 0 0
\(497\) 18857.3 1.70194
\(498\) 0 0
\(499\) 1060.23i 0.0951153i −0.998868 0.0475576i \(-0.984856\pi\)
0.998868 0.0475576i \(-0.0151438\pi\)
\(500\) 0 0
\(501\) −3381.31 −0.301528
\(502\) 0 0
\(503\) 2803.19i 0.248485i 0.992252 + 0.124243i \(0.0396501\pi\)
−0.992252 + 0.124243i \(0.960350\pi\)
\(504\) 0 0
\(505\) 2580.99i 0.227431i
\(506\) 0 0
\(507\) 3172.46i 0.277898i
\(508\) 0 0
\(509\) 1658.97 0.144465 0.0722323 0.997388i \(-0.476988\pi\)
0.0722323 + 0.997388i \(0.476988\pi\)
\(510\) 0 0
\(511\) 25126.3 2.17519
\(512\) 0 0
\(513\) 17166.7i 1.47745i
\(514\) 0 0
\(515\) 14761.3i 1.26303i
\(516\) 0 0
\(517\) 238.565i 0.0202942i
\(518\) 0 0
\(519\) 6212.21 0.525406
\(520\) 0 0
\(521\) 12553.6i 1.05563i −0.849359 0.527815i \(-0.823011\pi\)
0.849359 0.527815i \(-0.176989\pi\)
\(522\) 0 0
\(523\) −12031.6 −1.00593 −0.502967 0.864306i \(-0.667758\pi\)
−0.502967 + 0.864306i \(0.667758\pi\)
\(524\) 0 0
\(525\) 482.164 0.0400826
\(526\) 0 0
\(527\) 9192.42 11561.2i 0.759826 0.955628i
\(528\) 0 0
\(529\) −20561.8 −1.68996
\(530\) 0 0
\(531\) −3678.78 −0.300650
\(532\) 0 0
\(533\) 17366.0i 1.41126i
\(534\) 0 0
\(535\) 12673.8 1.02418
\(536\) 0 0
\(537\) 4507.75i 0.362241i
\(538\) 0 0
\(539\) 224.849i 0.0179683i
\(540\) 0 0
\(541\) 10632.7i 0.844984i −0.906367 0.422492i \(-0.861155\pi\)
0.906367 0.422492i \(-0.138845\pi\)
\(542\) 0 0
\(543\) 161.741 0.0127826
\(544\) 0 0
\(545\) −8456.94 −0.664689
\(546\) 0 0
\(547\) 17370.0i 1.35774i 0.734257 + 0.678872i \(0.237531\pi\)
−0.734257 + 0.678872i \(0.762469\pi\)
\(548\) 0 0
\(549\) 9853.85i 0.766033i
\(550\) 0 0
\(551\) 15985.4i 1.23593i
\(552\) 0 0
\(553\) 2539.86 0.195309
\(554\) 0 0
\(555\) 6367.10i 0.486970i
\(556\) 0 0
\(557\) 1084.77 0.0825190 0.0412595 0.999148i \(-0.486863\pi\)
0.0412595 + 0.999148i \(0.486863\pi\)
\(558\) 0 0
\(559\) 10200.5 0.771797
\(560\) 0 0
\(561\) 171.768 + 136.574i 0.0129270 + 0.0102784i
\(562\) 0 0
\(563\) −5494.27 −0.411289 −0.205645 0.978627i \(-0.565929\pi\)
−0.205645 + 0.978627i \(0.565929\pi\)
\(564\) 0 0
\(565\) 18882.0 1.40597
\(566\) 0 0
\(567\) 5065.14i 0.375160i
\(568\) 0 0
\(569\) −11612.4 −0.855569 −0.427784 0.903881i \(-0.640706\pi\)
−0.427784 + 0.903881i \(0.640706\pi\)
\(570\) 0 0
\(571\) 3413.97i 0.250210i 0.992143 + 0.125105i \(0.0399268\pi\)
−0.992143 + 0.125105i \(0.960073\pi\)
\(572\) 0 0
\(573\) 6723.68i 0.490202i
\(574\) 0 0
\(575\) 1443.96i 0.104725i
\(576\) 0 0
\(577\) 1355.70 0.0978141 0.0489070 0.998803i \(-0.484426\pi\)
0.0489070 + 0.998803i \(0.484426\pi\)
\(578\) 0 0
\(579\) 5220.07 0.374678
\(580\) 0 0
\(581\) 2735.22i 0.195311i
\(582\) 0 0
\(583\) 843.589i 0.0599278i
\(584\) 0 0
\(585\) 13550.7i 0.957698i
\(586\) 0 0
\(587\) 25232.2 1.77418 0.887092 0.461593i \(-0.152722\pi\)
0.887092 + 0.461593i \(0.152722\pi\)
\(588\) 0 0
\(589\) 29284.7i 2.04865i
\(590\) 0 0
\(591\) 10042.9 0.699004
\(592\) 0 0
\(593\) −14327.4 −0.992171 −0.496086 0.868274i \(-0.665230\pi\)
−0.496086 + 0.868274i \(0.665230\pi\)
\(594\) 0 0
\(595\) −14582.3 11594.5i −1.00473 0.798870i
\(596\) 0 0
\(597\) 5691.13 0.390155
\(598\) 0 0
\(599\) −2523.52 −0.172134 −0.0860671 0.996289i \(-0.527430\pi\)
−0.0860671 + 0.996289i \(0.527430\pi\)
\(600\) 0 0
\(601\) 17286.6i 1.17327i −0.809853 0.586634i \(-0.800453\pi\)
0.809853 0.586634i \(-0.199547\pi\)
\(602\) 0 0
\(603\) −3362.66 −0.227095
\(604\) 0 0
\(605\) 15332.3i 1.03033i
\(606\) 0 0
\(607\) 6413.65i 0.428867i 0.976739 + 0.214433i \(0.0687905\pi\)
−0.976739 + 0.214433i \(0.931210\pi\)
\(608\) 0 0
\(609\) 6948.73i 0.462359i
\(610\) 0 0
\(611\) −11658.1 −0.771907
\(612\) 0 0
\(613\) 485.788 0.0320078 0.0160039 0.999872i \(-0.494906\pi\)
0.0160039 + 0.999872i \(0.494906\pi\)
\(614\) 0 0
\(615\) 8991.76i 0.589566i
\(616\) 0 0
\(617\) 6830.32i 0.445670i 0.974856 + 0.222835i \(0.0715311\pi\)
−0.974856 + 0.222835i \(0.928469\pi\)
\(618\) 0 0
\(619\) 10667.8i 0.692689i −0.938107 0.346344i \(-0.887423\pi\)
0.938107 0.346344i \(-0.112577\pi\)
\(620\) 0 0
\(621\) −22347.5 −1.44408
\(622\) 0 0
\(623\) 9227.05i 0.593377i
\(624\) 0 0
\(625\) −16559.0 −1.05978
\(626\) 0 0
\(627\) −435.090 −0.0277126
\(628\) 0 0
\(629\) 9189.58 11557.7i 0.582532 0.732647i
\(630\) 0 0
\(631\) −15251.2 −0.962188 −0.481094 0.876669i \(-0.659760\pi\)
−0.481094 + 0.876669i \(0.659760\pi\)
\(632\) 0 0
\(633\) −5425.35 −0.340661
\(634\) 0 0
\(635\) 17328.9i 1.08296i
\(636\) 0 0
\(637\) −10987.8 −0.683442
\(638\) 0 0
\(639\) 16469.9i 1.01962i
\(640\) 0 0
\(641\) 12000.9i 0.739480i 0.929135 + 0.369740i \(0.120553\pi\)
−0.929135 + 0.369740i \(0.879447\pi\)
\(642\) 0 0
\(643\) 5642.30i 0.346051i −0.984917 0.173025i \(-0.944646\pi\)
0.984917 0.173025i \(-0.0553543\pi\)
\(644\) 0 0
\(645\) 5281.62 0.322424
\(646\) 0 0
\(647\) 7790.69 0.473391 0.236695 0.971584i \(-0.423936\pi\)
0.236695 + 0.971584i \(0.423936\pi\)
\(648\) 0 0
\(649\) 218.294i 0.0132031i
\(650\) 0 0
\(651\) 12729.8i 0.766393i
\(652\) 0 0
\(653\) 6755.02i 0.404815i −0.979301 0.202408i \(-0.935123\pi\)
0.979301 0.202408i \(-0.0648766\pi\)
\(654\) 0 0
\(655\) 8099.69 0.483177
\(656\) 0 0
\(657\) 21945.2i 1.30314i
\(658\) 0 0
\(659\) −14303.3 −0.845492 −0.422746 0.906248i \(-0.638934\pi\)
−0.422746 + 0.906248i \(0.638934\pi\)
\(660\) 0 0
\(661\) −26136.0 −1.53793 −0.768966 0.639289i \(-0.779229\pi\)
−0.768966 + 0.639289i \(0.779229\pi\)
\(662\) 0 0
\(663\) −6674.03 + 8393.88i −0.390947 + 0.491691i
\(664\) 0 0
\(665\) 36937.0 2.15392
\(666\) 0 0
\(667\) −20809.6 −1.20802
\(668\) 0 0
\(669\) 11262.9i 0.650893i
\(670\) 0 0
\(671\) −584.715 −0.0336404
\(672\) 0 0
\(673\) 19026.5i 1.08977i −0.838509 0.544887i \(-0.816572\pi\)
0.838509 0.544887i \(-0.183428\pi\)
\(674\) 0 0
\(675\) 985.948i 0.0562210i
\(676\) 0 0
\(677\) 25683.6i 1.45805i −0.684486 0.729026i \(-0.739974\pi\)
0.684486 0.729026i \(-0.260026\pi\)
\(678\) 0 0
\(679\) −21309.3 −1.20439
\(680\) 0 0
\(681\) −883.214 −0.0496988
\(682\) 0 0
\(683\) 28196.4i 1.57966i 0.613327 + 0.789829i \(0.289831\pi\)
−0.613327 + 0.789829i \(0.710169\pi\)
\(684\) 0 0
\(685\) 218.976i 0.0122141i
\(686\) 0 0
\(687\) 8063.62i 0.447811i
\(688\) 0 0
\(689\) −41224.0 −2.27941
\(690\) 0 0
\(691\) 35037.4i 1.92892i 0.264221 + 0.964462i \(0.414885\pi\)
−0.264221 + 0.964462i \(0.585115\pi\)
\(692\) 0 0
\(693\) −554.230 −0.0303802
\(694\) 0 0
\(695\) 3193.94 0.174321
\(696\) 0 0
\(697\) 12977.7 16322.0i 0.705261 0.887002i
\(698\) 0 0
\(699\) 17135.6 0.927220
\(700\) 0 0
\(701\) −3539.51 −0.190707 −0.0953535 0.995443i \(-0.530398\pi\)
−0.0953535 + 0.995443i \(0.530398\pi\)
\(702\) 0 0
\(703\) 29275.6i 1.57063i
\(704\) 0 0
\(705\) −6036.33 −0.322470
\(706\) 0 0
\(707\) 5158.62i 0.274413i
\(708\) 0 0
\(709\) 7264.39i 0.384795i 0.981317 + 0.192398i \(0.0616263\pi\)
−0.981317 + 0.192398i \(0.938374\pi\)
\(710\) 0 0
\(711\) 2218.30i 0.117008i
\(712\) 0 0
\(713\) 38122.6 2.00239
\(714\) 0 0
\(715\) 804.084 0.0420574
\(716\) 0 0
\(717\) 8995.99i 0.468565i
\(718\) 0 0
\(719\) 9704.92i 0.503383i −0.967807 0.251692i \(-0.919013\pi\)
0.967807 0.251692i \(-0.0809869\pi\)
\(720\) 0 0
\(721\) 29503.4i 1.52395i
\(722\) 0 0
\(723\) 3439.99 0.176949
\(724\) 0 0
\(725\) 918.097i 0.0470307i
\(726\) 0 0
\(727\) 10937.3 0.557967 0.278983 0.960296i \(-0.410003\pi\)
0.278983 + 0.960296i \(0.410003\pi\)
\(728\) 0 0
\(729\) −4317.73 −0.219364
\(730\) 0 0
\(731\) −9587.29 7622.91i −0.485087 0.385696i
\(732\) 0 0
\(733\) 14196.3 0.715351 0.357675 0.933846i \(-0.383569\pi\)
0.357675 + 0.933846i \(0.383569\pi\)
\(734\) 0 0
\(735\) −5689.27 −0.285513
\(736\) 0 0
\(737\) 199.536i 0.00997287i
\(738\) 0 0
\(739\) 8477.05 0.421966 0.210983 0.977490i \(-0.432333\pi\)
0.210983 + 0.977490i \(0.432333\pi\)
\(740\) 0 0
\(741\) 21261.7i 1.05407i
\(742\) 0 0
\(743\) 5251.86i 0.259316i −0.991559 0.129658i \(-0.958612\pi\)
0.991559 0.129658i \(-0.0413880\pi\)
\(744\) 0 0
\(745\) 1481.72i 0.0728674i
\(746\) 0 0
\(747\) 2388.93 0.117010
\(748\) 0 0
\(749\) −25331.1 −1.23575
\(750\) 0 0
\(751\) 16018.0i 0.778302i −0.921174 0.389151i \(-0.872768\pi\)
0.921174 0.389151i \(-0.127232\pi\)
\(752\) 0 0
\(753\) 17032.5i 0.824303i
\(754\) 0 0
\(755\) 24187.2i 1.16591i
\(756\) 0 0
\(757\) −9344.23 −0.448642 −0.224321 0.974515i \(-0.572016\pi\)
−0.224321 + 0.974515i \(0.572016\pi\)
\(758\) 0 0
\(759\) 566.397i 0.0270868i
\(760\) 0 0
\(761\) 35043.1 1.66927 0.834633 0.550807i \(-0.185680\pi\)
0.834633 + 0.550807i \(0.185680\pi\)
\(762\) 0 0
\(763\) 16902.9 0.801998
\(764\) 0 0
\(765\) 10126.6 12736.1i 0.478598 0.601929i
\(766\) 0 0
\(767\) −10667.5 −0.502190
\(768\) 0 0
\(769\) −6719.33 −0.315091 −0.157546 0.987512i \(-0.550358\pi\)
−0.157546 + 0.987512i \(0.550358\pi\)
\(770\) 0 0
\(771\) 12776.4i 0.596796i
\(772\) 0 0
\(773\) 18716.9 0.870892 0.435446 0.900215i \(-0.356591\pi\)
0.435446 + 0.900215i \(0.356591\pi\)
\(774\) 0 0
\(775\) 1681.92i 0.0779568i
\(776\) 0 0
\(777\) 12725.9i 0.587567i
\(778\) 0 0
\(779\) 41343.7i 1.90153i
\(780\) 0 0
\(781\) 977.304 0.0447768
\(782\) 0 0
\(783\) −14209.0 −0.648518
\(784\) 0 0
\(785\) 33575.8i 1.52659i
\(786\) 0 0
\(787\) 15916.1i 0.720898i −0.932779 0.360449i \(-0.882623\pi\)
0.932779 0.360449i \(-0.117377\pi\)
\(788\) 0 0
\(789\) 3885.59i 0.175324i
\(790\) 0 0
\(791\) −37739.4 −1.69641
\(792\) 0 0
\(793\) 28573.5i 1.27954i
\(794\) 0 0
\(795\) −21345.0 −0.952239
\(796\) 0 0
\(797\) 39802.4 1.76898 0.884489 0.466562i \(-0.154508\pi\)
0.884489 + 0.466562i \(0.154508\pi\)
\(798\) 0 0