Properties

Label 4012.2.b.b.237.45
Level 4012
Weight 2
Character 4012.237
Analytic conductor 32.036
Analytic rank 0
Dimension 46
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(46\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 237.45
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.2

$q$-expansion

\(f(q)\) \(=\) \(q+3.20235i q^{3} +1.49915i q^{5} -2.89750i q^{7} -7.25507 q^{9} +O(q^{10})\) \(q+3.20235i q^{3} +1.49915i q^{5} -2.89750i q^{7} -7.25507 q^{9} -0.590335i q^{11} +6.18813 q^{13} -4.80082 q^{15} +(3.80163 + 1.59611i) q^{17} -2.41715 q^{19} +9.27883 q^{21} +6.41697i q^{23} +2.75254 q^{25} -13.6262i q^{27} +6.45965i q^{29} -1.17429i q^{31} +1.89046 q^{33} +4.34380 q^{35} -0.792669i q^{37} +19.8166i q^{39} +3.09509i q^{41} -0.984651 q^{43} -10.8765i q^{45} +11.2473 q^{47} -1.39553 q^{49} +(-5.11131 + 12.1742i) q^{51} -4.09299 q^{53} +0.885003 q^{55} -7.74056i q^{57} +1.00000 q^{59} -8.04025i q^{61} +21.0216i q^{63} +9.27696i q^{65} -13.0346 q^{67} -20.5494 q^{69} +7.20065i q^{71} -3.29579i q^{73} +8.81460i q^{75} -1.71050 q^{77} +10.9932i q^{79} +21.8708 q^{81} +1.87501 q^{83} +(-2.39282 + 5.69924i) q^{85} -20.6861 q^{87} -2.40337 q^{89} -17.9301i q^{91} +3.76049 q^{93} -3.62367i q^{95} +11.0905i q^{97} +4.28292i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times\).

\(n\) \(2007\) \(3129\) \(3777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 3.20235i 1.84888i 0.381328 + 0.924440i \(0.375467\pi\)
−0.381328 + 0.924440i \(0.624533\pi\)
\(4\) 0 0
\(5\) 1.49915i 0.670442i 0.942140 + 0.335221i \(0.108811\pi\)
−0.942140 + 0.335221i \(0.891189\pi\)
\(6\) 0 0
\(7\) 2.89750i 1.09515i −0.836755 0.547577i \(-0.815550\pi\)
0.836755 0.547577i \(-0.184450\pi\)
\(8\) 0 0
\(9\) −7.25507 −2.41836
\(10\) 0 0
\(11\) 0.590335i 0.177993i −0.996032 0.0889963i \(-0.971634\pi\)
0.996032 0.0889963i \(-0.0283659\pi\)
\(12\) 0 0
\(13\) 6.18813 1.71628 0.858139 0.513417i \(-0.171621\pi\)
0.858139 + 0.513417i \(0.171621\pi\)
\(14\) 0 0
\(15\) −4.80082 −1.23957
\(16\) 0 0
\(17\) 3.80163 + 1.59611i 0.922032 + 0.387114i
\(18\) 0 0
\(19\) −2.41715 −0.554531 −0.277266 0.960793i \(-0.589428\pi\)
−0.277266 + 0.960793i \(0.589428\pi\)
\(20\) 0 0
\(21\) 9.27883 2.02481
\(22\) 0 0
\(23\) 6.41697i 1.33803i 0.743249 + 0.669015i \(0.233284\pi\)
−0.743249 + 0.669015i \(0.766716\pi\)
\(24\) 0 0
\(25\) 2.75254 0.550508
\(26\) 0 0
\(27\) 13.6262i 2.62237i
\(28\) 0 0
\(29\) 6.45965i 1.19953i 0.800177 + 0.599763i \(0.204739\pi\)
−0.800177 + 0.599763i \(0.795261\pi\)
\(30\) 0 0
\(31\) 1.17429i 0.210909i −0.994424 0.105454i \(-0.966370\pi\)
0.994424 0.105454i \(-0.0336297\pi\)
\(32\) 0 0
\(33\) 1.89046 0.329087
\(34\) 0 0
\(35\) 4.34380 0.734237
\(36\) 0 0
\(37\) 0.792669i 0.130314i −0.997875 0.0651570i \(-0.979245\pi\)
0.997875 0.0651570i \(-0.0207548\pi\)
\(38\) 0 0
\(39\) 19.8166i 3.17319i
\(40\) 0 0
\(41\) 3.09509i 0.483372i 0.970355 + 0.241686i \(0.0777003\pi\)
−0.970355 + 0.241686i \(0.922300\pi\)
\(42\) 0 0
\(43\) −0.984651 −0.150158 −0.0750789 0.997178i \(-0.523921\pi\)
−0.0750789 + 0.997178i \(0.523921\pi\)
\(44\) 0 0
\(45\) 10.8765i 1.62137i
\(46\) 0 0
\(47\) 11.2473 1.64059 0.820295 0.571940i \(-0.193809\pi\)
0.820295 + 0.571940i \(0.193809\pi\)
\(48\) 0 0
\(49\) −1.39553 −0.199362
\(50\) 0 0
\(51\) −5.11131 + 12.1742i −0.715727 + 1.70473i
\(52\) 0 0
\(53\) −4.09299 −0.562216 −0.281108 0.959676i \(-0.590702\pi\)
−0.281108 + 0.959676i \(0.590702\pi\)
\(54\) 0 0
\(55\) 0.885003 0.119334
\(56\) 0 0
\(57\) 7.74056i 1.02526i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 8.04025i 1.02945i −0.857356 0.514725i \(-0.827894\pi\)
0.857356 0.514725i \(-0.172106\pi\)
\(62\) 0 0
\(63\) 21.0216i 2.64847i
\(64\) 0 0
\(65\) 9.27696i 1.15066i
\(66\) 0 0
\(67\) −13.0346 −1.59243 −0.796217 0.605011i \(-0.793169\pi\)
−0.796217 + 0.605011i \(0.793169\pi\)
\(68\) 0 0
\(69\) −20.5494 −2.47386
\(70\) 0 0
\(71\) 7.20065i 0.854560i 0.904119 + 0.427280i \(0.140528\pi\)
−0.904119 + 0.427280i \(0.859472\pi\)
\(72\) 0 0
\(73\) 3.29579i 0.385743i −0.981224 0.192871i \(-0.938220\pi\)
0.981224 0.192871i \(-0.0617800\pi\)
\(74\) 0 0
\(75\) 8.81460i 1.01782i
\(76\) 0 0
\(77\) −1.71050 −0.194929
\(78\) 0 0
\(79\) 10.9932i 1.23683i 0.785852 + 0.618414i \(0.212225\pi\)
−0.785852 + 0.618414i \(0.787775\pi\)
\(80\) 0 0
\(81\) 21.8708 2.43009
\(82\) 0 0
\(83\) 1.87501 0.205809 0.102904 0.994691i \(-0.467186\pi\)
0.102904 + 0.994691i \(0.467186\pi\)
\(84\) 0 0
\(85\) −2.39282 + 5.69924i −0.259537 + 0.618169i
\(86\) 0 0
\(87\) −20.6861 −2.21778
\(88\) 0 0
\(89\) −2.40337 −0.254756 −0.127378 0.991854i \(-0.540656\pi\)
−0.127378 + 0.991854i \(0.540656\pi\)
\(90\) 0 0
\(91\) 17.9301i 1.87959i
\(92\) 0 0
\(93\) 3.76049 0.389945
\(94\) 0 0
\(95\) 3.62367i 0.371781i
\(96\) 0 0
\(97\) 11.0905i 1.12607i 0.826432 + 0.563037i \(0.190367\pi\)
−0.826432 + 0.563037i \(0.809633\pi\)
\(98\) 0 0
\(99\) 4.28292i 0.430450i
\(100\) 0 0
\(101\) −0.857720 −0.0853463 −0.0426732 0.999089i \(-0.513587\pi\)
−0.0426732 + 0.999089i \(0.513587\pi\)
\(102\) 0 0
\(103\) 4.74585 0.467623 0.233811 0.972282i \(-0.424880\pi\)
0.233811 + 0.972282i \(0.424880\pi\)
\(104\) 0 0
\(105\) 13.9104i 1.35752i
\(106\) 0 0
\(107\) 5.06741i 0.489885i 0.969538 + 0.244942i \(0.0787691\pi\)
−0.969538 + 0.244942i \(0.921231\pi\)
\(108\) 0 0
\(109\) 8.53677i 0.817674i 0.912607 + 0.408837i \(0.134066\pi\)
−0.912607 + 0.408837i \(0.865934\pi\)
\(110\) 0 0
\(111\) 2.53841 0.240935
\(112\) 0 0
\(113\) 6.83003i 0.642515i 0.946992 + 0.321258i \(0.104106\pi\)
−0.946992 + 0.321258i \(0.895894\pi\)
\(114\) 0 0
\(115\) −9.62002 −0.897072
\(116\) 0 0
\(117\) −44.8953 −4.15057
\(118\) 0 0
\(119\) 4.62474 11.0153i 0.423949 1.00977i
\(120\) 0 0
\(121\) 10.6515 0.968319
\(122\) 0 0
\(123\) −9.91157 −0.893696
\(124\) 0 0
\(125\) 11.6222i 1.03953i
\(126\) 0 0
\(127\) −20.7281 −1.83932 −0.919660 0.392716i \(-0.871536\pi\)
−0.919660 + 0.392716i \(0.871536\pi\)
\(128\) 0 0
\(129\) 3.15320i 0.277624i
\(130\) 0 0
\(131\) 0.975048i 0.0851904i −0.999092 0.0425952i \(-0.986437\pi\)
0.999092 0.0425952i \(-0.0135626\pi\)
\(132\) 0 0
\(133\) 7.00369i 0.607297i
\(134\) 0 0
\(135\) 20.4278 1.75815
\(136\) 0 0
\(137\) −6.17245 −0.527348 −0.263674 0.964612i \(-0.584934\pi\)
−0.263674 + 0.964612i \(0.584934\pi\)
\(138\) 0 0
\(139\) 14.3370i 1.21605i −0.793918 0.608025i \(-0.791962\pi\)
0.793918 0.608025i \(-0.208038\pi\)
\(140\) 0 0
\(141\) 36.0179i 3.03325i
\(142\) 0 0
\(143\) 3.65307i 0.305485i
\(144\) 0 0
\(145\) −9.68401 −0.804213
\(146\) 0 0
\(147\) 4.46899i 0.368596i
\(148\) 0 0
\(149\) 10.5727 0.866146 0.433073 0.901359i \(-0.357429\pi\)
0.433073 + 0.901359i \(0.357429\pi\)
\(150\) 0 0
\(151\) −8.52537 −0.693785 −0.346893 0.937905i \(-0.612763\pi\)
−0.346893 + 0.937905i \(0.612763\pi\)
\(152\) 0 0
\(153\) −27.5811 11.5799i −2.22980 0.936179i
\(154\) 0 0
\(155\) 1.76044 0.141402
\(156\) 0 0
\(157\) −3.33072 −0.265820 −0.132910 0.991128i \(-0.542432\pi\)
−0.132910 + 0.991128i \(0.542432\pi\)
\(158\) 0 0
\(159\) 13.1072i 1.03947i
\(160\) 0 0
\(161\) 18.5932 1.46535
\(162\) 0 0
\(163\) 18.0669i 1.41511i 0.706660 + 0.707553i \(0.250201\pi\)
−0.706660 + 0.707553i \(0.749799\pi\)
\(164\) 0 0
\(165\) 2.83409i 0.220634i
\(166\) 0 0
\(167\) 20.7391i 1.60484i −0.596760 0.802420i \(-0.703545\pi\)
0.596760 0.802420i \(-0.296455\pi\)
\(168\) 0 0
\(169\) 25.2929 1.94561
\(170\) 0 0
\(171\) 17.5366 1.34105
\(172\) 0 0
\(173\) 23.2415i 1.76702i 0.468413 + 0.883509i \(0.344826\pi\)
−0.468413 + 0.883509i \(0.655174\pi\)
\(174\) 0 0
\(175\) 7.97549i 0.602891i
\(176\) 0 0
\(177\) 3.20235i 0.240704i
\(178\) 0 0
\(179\) −4.61019 −0.344582 −0.172291 0.985046i \(-0.555117\pi\)
−0.172291 + 0.985046i \(0.555117\pi\)
\(180\) 0 0
\(181\) 13.3108i 0.989385i 0.869068 + 0.494693i \(0.164719\pi\)
−0.869068 + 0.494693i \(0.835281\pi\)
\(182\) 0 0
\(183\) 25.7477 1.90333
\(184\) 0 0
\(185\) 1.18833 0.0873680
\(186\) 0 0
\(187\) 0.942240 2.24424i 0.0689034 0.164115i
\(188\) 0 0
\(189\) −39.4821 −2.87190
\(190\) 0 0
\(191\) −4.20553 −0.304301 −0.152151 0.988357i \(-0.548620\pi\)
−0.152151 + 0.988357i \(0.548620\pi\)
\(192\) 0 0
\(193\) 9.59831i 0.690902i −0.938437 0.345451i \(-0.887726\pi\)
0.938437 0.345451i \(-0.112274\pi\)
\(194\) 0 0
\(195\) −29.7081 −2.12744
\(196\) 0 0
\(197\) 7.91120i 0.563649i −0.959466 0.281825i \(-0.909060\pi\)
0.959466 0.281825i \(-0.0909397\pi\)
\(198\) 0 0
\(199\) 11.5702i 0.820192i −0.912042 0.410096i \(-0.865495\pi\)
0.912042 0.410096i \(-0.134505\pi\)
\(200\) 0 0
\(201\) 41.7415i 2.94422i
\(202\) 0 0
\(203\) 18.7169 1.31367
\(204\) 0 0
\(205\) −4.64001 −0.324073
\(206\) 0 0
\(207\) 46.5556i 3.23583i
\(208\) 0 0
\(209\) 1.42693i 0.0987025i
\(210\) 0 0
\(211\) 7.58369i 0.522083i −0.965328 0.261041i \(-0.915934\pi\)
0.965328 0.261041i \(-0.0840659\pi\)
\(212\) 0 0
\(213\) −23.0590 −1.57998
\(214\) 0 0
\(215\) 1.47614i 0.100672i
\(216\) 0 0
\(217\) −3.40251 −0.230978
\(218\) 0 0
\(219\) 10.5543 0.713192
\(220\) 0 0
\(221\) 23.5250 + 9.87694i 1.58246 + 0.664395i
\(222\) 0 0
\(223\) −3.38200 −0.226475 −0.113238 0.993568i \(-0.536122\pi\)
−0.113238 + 0.993568i \(0.536122\pi\)
\(224\) 0 0
\(225\) −19.9699 −1.33132
\(226\) 0 0
\(227\) 27.0399i 1.79470i −0.441316 0.897352i \(-0.645488\pi\)
0.441316 0.897352i \(-0.354512\pi\)
\(228\) 0 0
\(229\) 8.10125 0.535346 0.267673 0.963510i \(-0.413745\pi\)
0.267673 + 0.963510i \(0.413745\pi\)
\(230\) 0 0
\(231\) 5.47762i 0.360401i
\(232\) 0 0
\(233\) 28.2230i 1.84895i −0.381240 0.924476i \(-0.624503\pi\)
0.381240 0.924476i \(-0.375497\pi\)
\(234\) 0 0
\(235\) 16.8615i 1.09992i
\(236\) 0 0
\(237\) −35.2040 −2.28675
\(238\) 0 0
\(239\) 6.85304 0.443287 0.221643 0.975128i \(-0.428858\pi\)
0.221643 + 0.975128i \(0.428858\pi\)
\(240\) 0 0
\(241\) 15.2678i 0.983485i −0.870741 0.491743i \(-0.836360\pi\)
0.870741 0.491743i \(-0.163640\pi\)
\(242\) 0 0
\(243\) 29.1594i 1.87057i
\(244\) 0 0
\(245\) 2.09212i 0.133660i
\(246\) 0 0
\(247\) −14.9576 −0.951730
\(248\) 0 0
\(249\) 6.00444i 0.380516i
\(250\) 0 0
\(251\) 17.2257 1.08728 0.543639 0.839319i \(-0.317046\pi\)
0.543639 + 0.839319i \(0.317046\pi\)
\(252\) 0 0
\(253\) 3.78816 0.238160
\(254\) 0 0
\(255\) −18.2510 7.66264i −1.14292 0.479853i
\(256\) 0 0
\(257\) −20.3501 −1.26941 −0.634704 0.772756i \(-0.718878\pi\)
−0.634704 + 0.772756i \(0.718878\pi\)
\(258\) 0 0
\(259\) −2.29676 −0.142714
\(260\) 0 0
\(261\) 46.8652i 2.90088i
\(262\) 0 0
\(263\) −25.7298 −1.58657 −0.793285 0.608850i \(-0.791631\pi\)
−0.793285 + 0.608850i \(0.791631\pi\)
\(264\) 0 0
\(265\) 6.13602i 0.376933i
\(266\) 0 0
\(267\) 7.69643i 0.471014i
\(268\) 0 0
\(269\) 18.8539i 1.14954i 0.818314 + 0.574772i \(0.194909\pi\)
−0.818314 + 0.574772i \(0.805091\pi\)
\(270\) 0 0
\(271\) −5.01751 −0.304792 −0.152396 0.988320i \(-0.548699\pi\)
−0.152396 + 0.988320i \(0.548699\pi\)
\(272\) 0 0
\(273\) 57.4186 3.47513
\(274\) 0 0
\(275\) 1.62492i 0.0979863i
\(276\) 0 0
\(277\) 18.7009i 1.12363i −0.827263 0.561814i \(-0.810104\pi\)
0.827263 0.561814i \(-0.189896\pi\)
\(278\) 0 0
\(279\) 8.51956i 0.510053i
\(280\) 0 0
\(281\) 26.0941 1.55664 0.778321 0.627867i \(-0.216072\pi\)
0.778321 + 0.627867i \(0.216072\pi\)
\(282\) 0 0
\(283\) 2.86388i 0.170240i −0.996371 0.0851199i \(-0.972873\pi\)
0.996371 0.0851199i \(-0.0271273\pi\)
\(284\) 0 0
\(285\) 11.6043 0.687379
\(286\) 0 0
\(287\) 8.96803 0.529366
\(288\) 0 0
\(289\) 11.9049 + 12.1357i 0.700286 + 0.713863i
\(290\) 0 0
\(291\) −35.5158 −2.08197
\(292\) 0 0
\(293\) −0.0831991 −0.00486054 −0.00243027 0.999997i \(-0.500774\pi\)
−0.00243027 + 0.999997i \(0.500774\pi\)
\(294\) 0 0
\(295\) 1.49915i 0.0872841i
\(296\) 0 0
\(297\) −8.04404 −0.466763
\(298\) 0 0
\(299\) 39.7090i 2.29643i
\(300\) 0 0
\(301\) 2.85303i 0.164446i
\(302\) 0 0
\(303\) 2.74672i 0.157795i
\(304\) 0 0
\(305\) 12.0536 0.690186
\(306\) 0 0
\(307\) 5.09004 0.290504 0.145252 0.989395i \(-0.453601\pi\)
0.145252 + 0.989395i \(0.453601\pi\)
\(308\) 0 0
\(309\) 15.1979i 0.864578i
\(310\) 0 0
\(311\) 11.7670i 0.667246i 0.942707 + 0.333623i \(0.108271\pi\)
−0.942707 + 0.333623i \(0.891729\pi\)
\(312\) 0 0
\(313\) 3.63405i 0.205409i 0.994712 + 0.102704i \(0.0327495\pi\)
−0.994712 + 0.102704i \(0.967250\pi\)
\(314\) 0 0
\(315\) −31.5146 −1.77565
\(316\) 0 0
\(317\) 3.13271i 0.175950i −0.996123 0.0879751i \(-0.971960\pi\)
0.996123 0.0879751i \(-0.0280396\pi\)
\(318\) 0 0
\(319\) 3.81336 0.213507
\(320\) 0 0
\(321\) −16.2276 −0.905738
\(322\) 0 0
\(323\) −9.18911 3.85804i −0.511296 0.214667i
\(324\) 0 0
\(325\) 17.0331 0.944824
\(326\) 0 0
\(327\) −27.3378 −1.51178
\(328\) 0 0
\(329\) 32.5892i 1.79670i
\(330\) 0 0
\(331\) 17.4901 0.961342 0.480671 0.876901i \(-0.340393\pi\)
0.480671 + 0.876901i \(0.340393\pi\)
\(332\) 0 0
\(333\) 5.75087i 0.315146i
\(334\) 0 0
\(335\) 19.5409i 1.06763i
\(336\) 0 0
\(337\) 31.8398i 1.73442i 0.497940 + 0.867211i \(0.334090\pi\)
−0.497940 + 0.867211i \(0.665910\pi\)
\(338\) 0 0
\(339\) −21.8722 −1.18793
\(340\) 0 0
\(341\) −0.693225 −0.0375402
\(342\) 0 0
\(343\) 16.2390i 0.876822i
\(344\) 0 0
\(345\) 30.8067i 1.65858i
\(346\) 0 0
\(347\) 19.6264i 1.05360i 0.849989 + 0.526800i \(0.176608\pi\)
−0.849989 + 0.526800i \(0.823392\pi\)
\(348\) 0 0
\(349\) −26.5184 −1.41950 −0.709748 0.704455i \(-0.751191\pi\)
−0.709748 + 0.704455i \(0.751191\pi\)
\(350\) 0 0
\(351\) 84.3209i 4.50072i
\(352\) 0 0
\(353\) −0.813971 −0.0433233 −0.0216617 0.999765i \(-0.506896\pi\)
−0.0216617 + 0.999765i \(0.506896\pi\)
\(354\) 0 0
\(355\) −10.7949 −0.572933
\(356\) 0 0
\(357\) 35.2747 + 14.8101i 1.86694 + 0.783831i
\(358\) 0 0
\(359\) 30.7778 1.62439 0.812196 0.583384i \(-0.198272\pi\)
0.812196 + 0.583384i \(0.198272\pi\)
\(360\) 0 0
\(361\) −13.1574 −0.692495
\(362\) 0 0
\(363\) 34.1099i 1.79030i
\(364\) 0 0
\(365\) 4.94089 0.258618
\(366\) 0 0
\(367\) 1.65170i 0.0862179i 0.999070 + 0.0431089i \(0.0137263\pi\)
−0.999070 + 0.0431089i \(0.986274\pi\)
\(368\) 0 0
\(369\) 22.4551i 1.16896i
\(370\) 0 0
\(371\) 11.8595i 0.615713i
\(372\) 0 0
\(373\) 26.2501 1.35918 0.679589 0.733593i \(-0.262158\pi\)
0.679589 + 0.733593i \(0.262158\pi\)
\(374\) 0 0
\(375\) −37.2185 −1.92196
\(376\) 0 0
\(377\) 39.9731i 2.05872i
\(378\) 0 0
\(379\) 15.5132i 0.796860i 0.917199 + 0.398430i \(0.130445\pi\)
−0.917199 + 0.398430i \(0.869555\pi\)
\(380\) 0 0
\(381\) 66.3787i 3.40068i
\(382\) 0 0
\(383\) 2.65909 0.135873 0.0679367 0.997690i \(-0.478358\pi\)
0.0679367 + 0.997690i \(0.478358\pi\)
\(384\) 0 0
\(385\) 2.56430i 0.130689i
\(386\) 0 0
\(387\) 7.14371 0.363135
\(388\) 0 0
\(389\) −29.5116 −1.49630 −0.748148 0.663532i \(-0.769057\pi\)
−0.748148 + 0.663532i \(0.769057\pi\)
\(390\) 0 0
\(391\) −10.2422 + 24.3950i −0.517970 + 1.23371i
\(392\) 0 0
\(393\) 3.12245 0.157507
\(394\) 0 0
\(395\) −16.4805 −0.829222
\(396\) 0 0
\(397\) 1.50052i 0.0753089i 0.999291 + 0.0376544i \(0.0119886\pi\)
−0.999291 + 0.0376544i \(0.988011\pi\)
\(398\) 0 0
\(399\) −22.4283 −1.12282
\(400\) 0 0
\(401\) 33.6682i 1.68131i 0.541573 + 0.840654i \(0.317829\pi\)
−0.541573 + 0.840654i \(0.682171\pi\)
\(402\) 0 0
\(403\) 7.26666i 0.361978i
\(404\) 0 0
\(405\) 32.7877i 1.62923i
\(406\) 0 0
\(407\) −0.467940 −0.0231949
\(408\) 0 0
\(409\) −23.2070 −1.14751 −0.573755 0.819027i \(-0.694514\pi\)
−0.573755 + 0.819027i \(0.694514\pi\)
\(410\) 0 0
\(411\) 19.7664i 0.975003i
\(412\) 0 0
\(413\) 2.89750i 0.142577i
\(414\) 0 0
\(415\) 2.81093i 0.137983i
\(416\) 0 0
\(417\) 45.9122 2.24833
\(418\) 0 0
\(419\) 16.2702i 0.794850i 0.917635 + 0.397425i \(0.130096\pi\)
−0.917635 + 0.397425i \(0.869904\pi\)
\(420\) 0 0
\(421\) 7.79891 0.380096 0.190048 0.981775i \(-0.439136\pi\)
0.190048 + 0.981775i \(0.439136\pi\)
\(422\) 0 0
\(423\) −81.6001 −3.96753
\(424\) 0 0
\(425\) 10.4641 + 4.39336i 0.507586 + 0.213109i
\(426\) 0 0
\(427\) −23.2967 −1.12741
\(428\) 0 0
\(429\) 11.6984 0.564805
\(430\) 0 0
\(431\) 7.39257i 0.356088i −0.984023 0.178044i \(-0.943023\pi\)
0.984023 0.178044i \(-0.0569769\pi\)
\(432\) 0 0
\(433\) 29.8433 1.43418 0.717089 0.696981i \(-0.245474\pi\)
0.717089 + 0.696981i \(0.245474\pi\)
\(434\) 0 0
\(435\) 31.0116i 1.48689i
\(436\) 0 0
\(437\) 15.5108i 0.741980i
\(438\) 0 0
\(439\) 21.3004i 1.01661i −0.861176 0.508307i \(-0.830271\pi\)
0.861176 0.508307i \(-0.169729\pi\)
\(440\) 0 0
\(441\) 10.1247 0.482128
\(442\) 0 0
\(443\) −30.2734 −1.43833 −0.719166 0.694838i \(-0.755476\pi\)
−0.719166 + 0.694838i \(0.755476\pi\)
\(444\) 0 0
\(445\) 3.60302i 0.170799i
\(446\) 0 0
\(447\) 33.8574i 1.60140i
\(448\) 0 0
\(449\) 4.03317i 0.190337i 0.995461 + 0.0951685i \(0.0303390\pi\)
−0.995461 + 0.0951685i \(0.969661\pi\)
\(450\) 0 0
\(451\) 1.82714 0.0860366
\(452\) 0 0
\(453\) 27.3013i 1.28273i
\(454\) 0 0
\(455\) 26.8800 1.26015
\(456\) 0 0
\(457\) 12.7730 0.597495 0.298748 0.954332i \(-0.403431\pi\)
0.298748 + 0.954332i \(0.403431\pi\)
\(458\) 0 0
\(459\) 21.7490 51.8020i 1.01516 2.41791i
\(460\) 0 0
\(461\) 10.1186 0.471269 0.235634 0.971842i \(-0.424283\pi\)
0.235634 + 0.971842i \(0.424283\pi\)
\(462\) 0 0
\(463\) 19.2242 0.893422 0.446711 0.894678i \(-0.352595\pi\)
0.446711 + 0.894678i \(0.352595\pi\)
\(464\) 0 0
\(465\) 5.63756i 0.261435i
\(466\) 0 0
\(467\) −7.98823 −0.369651 −0.184826 0.982771i \(-0.559172\pi\)
−0.184826 + 0.982771i \(0.559172\pi\)
\(468\) 0 0
\(469\) 37.7679i 1.74396i
\(470\) 0 0
\(471\) 10.6661i 0.491470i
\(472\) 0 0
\(473\) 0.581274i 0.0267270i
\(474\) 0 0
\(475\) −6.65329 −0.305274
\(476\) 0 0
\(477\) 29.6949 1.35964
\(478\) 0 0
\(479\) 31.4216i 1.43569i −0.696202 0.717846i \(-0.745128\pi\)
0.696202 0.717846i \(-0.254872\pi\)
\(480\) 0 0
\(481\) 4.90514i 0.223655i
\(482\) 0 0
\(483\) 59.5420i 2.70925i
\(484\) 0 0
\(485\) −16.6264 −0.754967
\(486\) 0 0
\(487\) 21.3236i 0.966266i −0.875547 0.483133i \(-0.839499\pi\)
0.875547 0.483133i \(-0.160501\pi\)
\(488\) 0 0
\(489\) −57.8565 −2.61636
\(490\) 0 0
\(491\) −7.32703 −0.330664 −0.165332 0.986238i \(-0.552870\pi\)
−0.165332 + 0.986238i \(0.552870\pi\)
\(492\) 0 0
\(493\) −10.3103 + 24.5572i −0.464353 + 1.10600i
\(494\) 0 0
\(495\) −6.42076 −0.288592
\(496\) 0 0
\(497\) 20.8639 0.935874
\(498\) 0 0
\(499\) 37.5609i 1.68146i 0.541457 + 0.840729i \(0.317873\pi\)
−0.541457 + 0.840729i \(0.682127\pi\)
\(500\) 0 0
\(501\) 66.4139 2.96716
\(502\) 0 0
\(503\) 24.6177i 1.09765i 0.835937 + 0.548825i \(0.184925\pi\)
−0.835937 + 0.548825i \(0.815075\pi\)
\(504\) 0 0
\(505\) 1.28585i 0.0572198i
\(506\) 0 0
\(507\) 80.9970i 3.59720i
\(508\) 0 0
\(509\) 11.0162 0.488285 0.244143 0.969739i \(-0.421493\pi\)
0.244143 + 0.969739i \(0.421493\pi\)
\(510\) 0 0
\(511\) −9.54956 −0.422448
\(512\) 0 0
\(513\) 32.9366i 1.45419i
\(514\) 0 0
\(515\) 7.11476i 0.313514i
\(516\) 0 0
\(517\) 6.63969i 0.292013i
\(518\) 0 0
\(519\) −74.4275 −3.26701
\(520\) 0 0
\(521\) 31.0672i 1.36108i 0.732711 + 0.680539i \(0.238255\pi\)
−0.732711 + 0.680539i \(0.761745\pi\)
\(522\) 0 0
\(523\) −1.65603 −0.0724131 −0.0362066 0.999344i \(-0.511527\pi\)
−0.0362066 + 0.999344i \(0.511527\pi\)
\(524\) 0 0
\(525\) 25.5403 1.11467
\(526\) 0 0
\(527\) 1.87430 4.46422i 0.0816457 0.194465i
\(528\) 0 0
\(529\) −18.1775 −0.790326
\(530\) 0 0
\(531\) −7.25507 −0.314843
\(532\) 0 0
\(533\) 19.1528i 0.829600i
\(534\) 0 0
\(535\) −7.59682 −0.328439
\(536\) 0 0
\(537\) 14.7635i 0.637091i
\(538\) 0 0
\(539\) 0.823831i 0.0354849i
\(540\) 0 0
\(541\) 14.3785i 0.618179i 0.951033 + 0.309090i \(0.100024\pi\)
−0.951033 + 0.309090i \(0.899976\pi\)
\(542\) 0 0
\(543\) −42.6260 −1.82925
\(544\) 0 0
\(545\) −12.7979 −0.548203
\(546\) 0 0
\(547\) 31.8208i 1.36056i 0.732953 + 0.680279i \(0.238142\pi\)
−0.732953 + 0.680279i \(0.761858\pi\)
\(548\) 0 0
\(549\) 58.3326i 2.48957i
\(550\) 0 0
\(551\) 15.6139i 0.665175i
\(552\) 0 0
\(553\) 31.8528 1.35452
\(554\) 0 0
\(555\) 3.80546i 0.161533i
\(556\) 0 0
\(557\) 24.4982 1.03802 0.519011 0.854767i \(-0.326300\pi\)
0.519011 + 0.854767i \(0.326300\pi\)
\(558\) 0 0
\(559\) −6.09315 −0.257713
\(560\) 0 0
\(561\) 7.18684 + 3.01739i 0.303429 + 0.127394i
\(562\) 0 0
\(563\) 28.6707 1.20833 0.604163 0.796861i \(-0.293508\pi\)
0.604163 + 0.796861i \(0.293508\pi\)
\(564\) 0 0
\(565\) −10.2393 −0.430769
\(566\) 0 0
\(567\) 63.3708i 2.66132i
\(568\) 0 0
\(569\) 33.9326 1.42253 0.711265 0.702924i \(-0.248123\pi\)
0.711265 + 0.702924i \(0.248123\pi\)
\(570\) 0 0
\(571\) 38.1381i 1.59603i 0.602637 + 0.798016i \(0.294117\pi\)
−0.602637 + 0.798016i \(0.705883\pi\)
\(572\) 0 0
\(573\) 13.4676i 0.562617i
\(574\) 0 0
\(575\) 17.6630i 0.736596i
\(576\) 0 0
\(577\) 43.2004 1.79845 0.899227 0.437481i \(-0.144129\pi\)
0.899227 + 0.437481i \(0.144129\pi\)
\(578\) 0 0
\(579\) 30.7372 1.27739
\(580\) 0 0
\(581\) 5.43284i 0.225392i
\(582\) 0 0
\(583\) 2.41624i 0.100070i
\(584\) 0 0
\(585\) 67.3050i 2.78272i
\(586\) 0 0
\(587\) −41.9962 −1.73337 −0.866684 0.498857i \(-0.833753\pi\)
−0.866684 + 0.498857i \(0.833753\pi\)
\(588\) 0 0
\(589\) 2.83843i 0.116956i
\(590\) 0 0
\(591\) 25.3345 1.04212
\(592\) 0 0
\(593\) 15.6415 0.642319 0.321160 0.947025i \(-0.395927\pi\)
0.321160 + 0.947025i \(0.395927\pi\)
\(594\) 0 0
\(595\) 16.5136 + 6.93320i 0.676990 + 0.284233i
\(596\) 0 0
\(597\) 37.0520 1.51644
\(598\) 0 0
\(599\) −45.3050 −1.85111 −0.925556 0.378611i \(-0.876402\pi\)
−0.925556 + 0.378611i \(0.876402\pi\)
\(600\) 0 0
\(601\) 21.2217i 0.865650i −0.901478 0.432825i \(-0.857517\pi\)
0.901478 0.432825i \(-0.142483\pi\)
\(602\) 0 0
\(603\) 94.5672 3.85107
\(604\) 0 0
\(605\) 15.9682i 0.649201i
\(606\) 0 0
\(607\) 25.1740i 1.02178i −0.859646 0.510890i \(-0.829316\pi\)
0.859646 0.510890i \(-0.170684\pi\)
\(608\) 0 0
\(609\) 59.9380i 2.42881i
\(610\) 0 0
\(611\) 69.5999 2.81571
\(612\) 0 0
\(613\) −46.0852 −1.86136 −0.930681 0.365831i \(-0.880785\pi\)
−0.930681 + 0.365831i \(0.880785\pi\)
\(614\) 0 0
\(615\) 14.8590i 0.599171i
\(616\) 0 0
\(617\) 9.76642i 0.393181i 0.980486 + 0.196591i \(0.0629870\pi\)
−0.980486 + 0.196591i \(0.937013\pi\)
\(618\) 0 0
\(619\) 23.7517i 0.954661i −0.878724 0.477330i \(-0.841605\pi\)
0.878724 0.477330i \(-0.158395\pi\)
\(620\) 0 0
\(621\) 87.4391 3.50881
\(622\) 0 0
\(623\) 6.96377i 0.278997i
\(624\) 0 0
\(625\) −3.66084 −0.146434
\(626\) 0 0
\(627\) −4.56952 −0.182489
\(628\) 0 0
\(629\) 1.26519 3.01344i 0.0504464 0.120154i
\(630\) 0 0
\(631\) −36.4965 −1.45290 −0.726452 0.687217i \(-0.758832\pi\)
−0.726452 + 0.687217i \(0.758832\pi\)
\(632\) 0 0
\(633\) 24.2856 0.965268
\(634\) 0 0
\(635\) 31.0746i 1.23316i
\(636\) 0 0
\(637\) −8.63573 −0.342160
\(638\) 0 0
\(639\) 52.2412i 2.06663i
\(640\) 0 0
\(641\) 8.05177i 0.318026i −0.987277 0.159013i \(-0.949169\pi\)
0.987277 0.159013i \(-0.0508311\pi\)
\(642\) 0 0
\(643\) 36.0512i 1.42172i 0.703333 + 0.710860i \(0.251694\pi\)
−0.703333 + 0.710860i \(0.748306\pi\)
\(644\) 0 0
\(645\) 4.72713 0.186131
\(646\) 0 0
\(647\) 46.2667 1.81893 0.909466 0.415779i \(-0.136491\pi\)
0.909466 + 0.415779i \(0.136491\pi\)
\(648\) 0 0
\(649\) 0.590335i 0.0231727i
\(650\) 0 0
\(651\) 10.8960i 0.427050i
\(652\) 0 0
\(653\) 2.21066i 0.0865098i 0.999064 + 0.0432549i \(0.0137728\pi\)
−0.999064 + 0.0432549i \(0.986227\pi\)
\(654\) 0 0
\(655\) 1.46175 0.0571152
\(656\) 0 0
\(657\) 23.9112i 0.932863i
\(658\) 0 0
\(659\) −27.0082 −1.05209 −0.526045 0.850456i \(-0.676326\pi\)
−0.526045 + 0.850456i \(0.676326\pi\)
\(660\) 0 0
\(661\) −19.1698 −0.745619 −0.372809 0.927908i \(-0.621605\pi\)
−0.372809 + 0.927908i \(0.621605\pi\)
\(662\) 0 0
\(663\) −31.6295 + 75.3354i −1.22839 + 2.92578i
\(664\) 0 0
\(665\) −10.4996 −0.407157
\(666\) 0 0
\(667\) −41.4514 −1.60500
\(668\) 0 0
\(669\) 10.8304i 0.418726i
\(670\) 0 0
\(671\) −4.74644 −0.183234
\(672\) 0 0
\(673\) 41.4412i 1.59744i −0.601701 0.798721i \(-0.705510\pi\)
0.601701 0.798721i \(-0.294490\pi\)
\(674\) 0 0
\(675\) 37.5067i 1.44363i
\(676\) 0 0
\(677\) 10.0301i 0.385487i −0.981249 0.192744i \(-0.938261\pi\)
0.981249 0.192744i \(-0.0617386\pi\)
\(678\) 0 0
\(679\) 32.1349 1.23322
\(680\) 0 0
\(681\) 86.5914 3.31819
\(682\) 0 0
\(683\) 15.2890i 0.585017i 0.956263 + 0.292508i \(0.0944900\pi\)
−0.956263 + 0.292508i \(0.905510\pi\)
\(684\) 0 0
\(685\) 9.25345i 0.353556i
\(686\) 0 0
\(687\) 25.9431i 0.989790i
\(688\) 0 0
\(689\) −25.3280 −0.964919
\(690\) 0 0
\(691\) 5.95101i 0.226387i −0.993573 0.113194i \(-0.963892\pi\)
0.993573 0.113194i \(-0.0361080\pi\)
\(692\) 0 0
\(693\) 12.4098 0.471409
\(694\) 0 0
\(695\) 21.4934 0.815290
\(696\) 0 0
\(697\) −4.94011 + 11.7664i −0.187120 + 0.445684i
\(698\) 0 0
\(699\) 90.3801 3.41849
\(700\) 0 0
\(701\) 16.9093 0.638657 0.319328 0.947644i \(-0.396543\pi\)
0.319328 + 0.947644i \(0.396543\pi\)
\(702\) 0 0
\(703\) 1.91600i 0.0722632i
\(704\) 0 0
\(705\) −53.9964 −2.03362
\(706\) 0 0
\(707\) 2.48525i 0.0934673i
\(708\) 0 0
\(709\) 22.6584i 0.850955i −0.904969 0.425477i \(-0.860106\pi\)
0.904969 0.425477i \(-0.139894\pi\)
\(710\) 0 0
\(711\) 79.7562i 2.99109i
\(712\) 0 0
\(713\) 7.53539 0.282202
\(714\) 0 0
\(715\) 5.47651 0.204810
\(716\) 0 0
\(717\) 21.9459i 0.819584i
\(718\) 0 0
\(719\) 2.00790i 0.0748820i −0.999299 0.0374410i \(-0.988079\pi\)
0.999299 0.0374410i \(-0.0119206\pi\)
\(720\) 0 0
\(721\) 13.7511i 0.512119i
\(722\) 0 0
\(723\) 48.8929 1.81835
\(724\) 0 0
\(725\) 17.7804i 0.660349i
\(726\) 0 0
\(727\) 31.8763 1.18223 0.591114 0.806588i \(-0.298688\pi\)
0.591114 + 0.806588i \(0.298688\pi\)
\(728\) 0 0
\(729\) −27.7662 −1.02838
\(730\) 0 0
\(731\) −3.74328 1.57161i −0.138450 0.0581282i
\(732\) 0 0
\(733\) 23.5568 0.870090 0.435045 0.900409i \(-0.356733\pi\)
0.435045 + 0.900409i \(0.356733\pi\)
\(734\) 0 0
\(735\) 6.69970 0.247122
\(736\) 0 0
\(737\) 7.69480i 0.283442i
\(738\) 0 0
\(739\) 5.97202 0.219684 0.109842 0.993949i \(-0.464965\pi\)
0.109842 + 0.993949i \(0.464965\pi\)
\(740\) 0 0
\(741\) 47.8996i 1.75963i
\(742\) 0 0
\(743\) 14.0233i 0.514464i 0.966350 + 0.257232i \(0.0828104\pi\)
−0.966350 + 0.257232i \(0.917190\pi\)
\(744\) 0 0
\(745\) 15.8500i 0.580701i
\(746\) 0 0
\(747\) −13.6033 −0.497719
\(748\) 0 0
\(749\) 14.6828 0.536499
\(750\) 0 0
\(751\) 22.4155i 0.817952i 0.912545 + 0.408976i \(0.134114\pi\)
−0.912545 + 0.408976i \(0.865886\pi\)
\(752\) 0 0
\(753\) 55.1629i 2.01025i
\(754\) 0 0
\(755\) 12.7808i 0.465143i
\(756\) 0 0
\(757\) −5.72469 −0.208067 −0.104034 0.994574i \(-0.533175\pi\)
−0.104034 + 0.994574i \(0.533175\pi\)
\(758\) 0 0
\(759\) 12.1310i 0.440329i
\(760\) 0 0
\(761\) −39.4142 −1.42876 −0.714382 0.699756i \(-0.753292\pi\)
−0.714382 + 0.699756i \(0.753292\pi\)
\(762\) 0 0
\(763\) 24.7353 0.895479
\(764\) 0 0
\(765\) 17.3600 41.3483i 0.627654 1.49495i
\(766\) 0 0
\(767\) 6.18813 0.223440
\(768\) 0 0
\(769\) 18.7805 0.677243 0.338621 0.940923i \(-0.390039\pi\)
0.338621 + 0.940923i \(0.390039\pi\)
\(770\) 0 0
\(771\) 65.1684i 2.34698i
\(772\) 0 0
\(773\) −14.6069 −0.525373 −0.262686 0.964881i \(-0.584608\pi\)
−0.262686 + 0.964881i \(0.584608\pi\)
\(774\) 0 0
\(775\) 3.23228i 0.116107i
\(776\) 0 0
\(777\) 7.35505i 0.263861i
\(778\) 0 0
\(779\) 7.48128i 0.268045i
\(780\) 0 0
\(781\) 4.25079 0.152105
\(782\) 0 0
\(783\) 88.0207 3.14560
\(784\) 0 0
\(785\) 4.99326i 0.178217i
\(786\) 0 0
\(787\) 19.0872i 0.680385i 0.940356 + 0.340193i \(0.110492\pi\)
−0.940356 + 0.340193i \(0.889508\pi\)
\(788\) 0 0
\(789\) 82.3961i 2.93338i
\(790\) 0 0
\(791\) 19.7900 0.703653
\(792\) 0 0
\(793\) 49.7541i 1.76682i