Properties

Label 4004.2.m.c.2157.18
Level 4004
Weight 2
Character 4004.2157
Analytic conductor 31.972
Analytic rank 0
Dimension 36
CM No

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Newspace parameters

Level: \( N \) = \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4004.m (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(36\)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2157.18
Character \(\chi\) = 4004.2157
Dual form 4004.2.m.c.2157.17

$q$-expansion

\(f(q)\) \(=\) \(q\)\(-0.722868 q^{3}\) \(-0.383791i q^{5}\) \(+1.00000i q^{7}\) \(-2.47746 q^{9}\) \(+O(q^{10})\) \(q\)\(-0.722868 q^{3}\) \(-0.383791i q^{5}\) \(+1.00000i q^{7}\) \(-2.47746 q^{9}\) \(-1.00000i q^{11}\) \(+(3.37304 - 1.27380i) q^{13}\) \(+0.277430i q^{15}\) \(-5.43236 q^{17}\) \(-5.21859i q^{19}\) \(-0.722868i q^{21}\) \(+2.81438 q^{23}\) \(+4.85270 q^{25}\) \(+3.95948 q^{27}\) \(+1.99936 q^{29}\) \(+2.19118i q^{31}\) \(+0.722868i q^{33}\) \(+0.383791 q^{35}\) \(+8.80912i q^{37}\) \(+(-2.43827 + 0.920790i) q^{39}\) \(-5.89210i q^{41}\) \(-9.25357 q^{43}\) \(+0.950828i q^{45}\) \(+8.28744i q^{47}\) \(-1.00000 q^{49}\) \(+3.92688 q^{51}\) \(-0.632174 q^{53}\) \(-0.383791 q^{55}\) \(+3.77235i q^{57}\) \(+13.2980i q^{59}\) \(-13.3069 q^{61}\) \(-2.47746i q^{63}\) \(+(-0.488874 - 1.29454i) q^{65}\) \(-7.84695i q^{67}\) \(-2.03442 q^{69}\) \(-8.46182i q^{71}\) \(-13.6678i q^{73}\) \(-3.50786 q^{75}\) \(+1.00000 q^{77}\) \(-13.9841 q^{79}\) \(+4.57021 q^{81}\) \(-13.5504i q^{83}\) \(+2.08489i q^{85}\) \(-1.44528 q^{87}\) \(+3.98493i q^{89}\) \(+(1.27380 + 3.37304i) q^{91}\) \(-1.58393i q^{93}\) \(-2.00285 q^{95}\) \(-4.98484i q^{97}\) \(+2.47746i q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 80q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut -\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 80q^{69} \) \(\mathstrut -\mathstrut 36q^{75} \) \(\mathstrut +\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 132q^{81} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times\).

\(n\) \(365\) \(925\) \(2003\) \(3433\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.722868 −0.417348 −0.208674 0.977985i \(-0.566915\pi\)
−0.208674 + 0.977985i \(0.566915\pi\)
\(4\) 0 0
\(5\) 0.383791i 0.171637i −0.996311 0.0858183i \(-0.972650\pi\)
0.996311 0.0858183i \(-0.0273505\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −2.47746 −0.825821
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) 3.37304 1.27380i 0.935514 0.353289i
\(14\) 0 0
\(15\) 0.277430i 0.0716322i
\(16\) 0 0
\(17\) −5.43236 −1.31754 −0.658771 0.752344i \(-0.728923\pi\)
−0.658771 + 0.752344i \(0.728923\pi\)
\(18\) 0 0
\(19\) 5.21859i 1.19723i −0.801038 0.598614i \(-0.795718\pi\)
0.801038 0.598614i \(-0.204282\pi\)
\(20\) 0 0
\(21\) 0.722868i 0.157743i
\(22\) 0 0
\(23\) 2.81438 0.586838 0.293419 0.955984i \(-0.405207\pi\)
0.293419 + 0.955984i \(0.405207\pi\)
\(24\) 0 0
\(25\) 4.85270 0.970541
\(26\) 0 0
\(27\) 3.95948 0.762002
\(28\) 0 0
\(29\) 1.99936 0.371272 0.185636 0.982619i \(-0.440565\pi\)
0.185636 + 0.982619i \(0.440565\pi\)
\(30\) 0 0
\(31\) 2.19118i 0.393548i 0.980449 + 0.196774i \(0.0630465\pi\)
−0.980449 + 0.196774i \(0.936954\pi\)
\(32\) 0 0
\(33\) 0.722868i 0.125835i
\(34\) 0 0
\(35\) 0.383791 0.0648726
\(36\) 0 0
\(37\) 8.80912i 1.44821i 0.689690 + 0.724105i \(0.257747\pi\)
−0.689690 + 0.724105i \(0.742253\pi\)
\(38\) 0 0
\(39\) −2.43827 + 0.920790i −0.390435 + 0.147444i
\(40\) 0 0
\(41\) 5.89210i 0.920191i −0.887869 0.460096i \(-0.847815\pi\)
0.887869 0.460096i \(-0.152185\pi\)
\(42\) 0 0
\(43\) −9.25357 −1.41116 −0.705578 0.708632i \(-0.749313\pi\)
−0.705578 + 0.708632i \(0.749313\pi\)
\(44\) 0 0
\(45\) 0.950828i 0.141741i
\(46\) 0 0
\(47\) 8.28744i 1.20885i 0.796663 + 0.604424i \(0.206597\pi\)
−0.796663 + 0.604424i \(0.793403\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 3.92688 0.549873
\(52\) 0 0
\(53\) −0.632174 −0.0868358 −0.0434179 0.999057i \(-0.513825\pi\)
−0.0434179 + 0.999057i \(0.513825\pi\)
\(54\) 0 0
\(55\) −0.383791 −0.0517504
\(56\) 0 0
\(57\) 3.77235i 0.499660i
\(58\) 0 0
\(59\) 13.2980i 1.73125i 0.500694 + 0.865625i \(0.333078\pi\)
−0.500694 + 0.865625i \(0.666922\pi\)
\(60\) 0 0
\(61\) −13.3069 −1.70377 −0.851884 0.523731i \(-0.824540\pi\)
−0.851884 + 0.523731i \(0.824540\pi\)
\(62\) 0 0
\(63\) 2.47746i 0.312131i
\(64\) 0 0
\(65\) −0.488874 1.29454i −0.0606373 0.160569i
\(66\) 0 0
\(67\) 7.84695i 0.958657i −0.877636 0.479329i \(-0.840880\pi\)
0.877636 0.479329i \(-0.159120\pi\)
\(68\) 0 0
\(69\) −2.03442 −0.244916
\(70\) 0 0
\(71\) 8.46182i 1.00423i −0.864800 0.502116i \(-0.832555\pi\)
0.864800 0.502116i \(-0.167445\pi\)
\(72\) 0 0
\(73\) 13.6678i 1.59970i −0.600203 0.799848i \(-0.704914\pi\)
0.600203 0.799848i \(-0.295086\pi\)
\(74\) 0 0
\(75\) −3.50786 −0.405053
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −13.9841 −1.57333 −0.786666 0.617379i \(-0.788195\pi\)
−0.786666 + 0.617379i \(0.788195\pi\)
\(80\) 0 0
\(81\) 4.57021 0.507801
\(82\) 0 0
\(83\) 13.5504i 1.48735i −0.668540 0.743677i \(-0.733080\pi\)
0.668540 0.743677i \(-0.266920\pi\)
\(84\) 0 0
\(85\) 2.08489i 0.226138i
\(86\) 0 0
\(87\) −1.44528 −0.154950
\(88\) 0 0
\(89\) 3.98493i 0.422402i 0.977443 + 0.211201i \(0.0677374\pi\)
−0.977443 + 0.211201i \(0.932263\pi\)
\(90\) 0 0
\(91\) 1.27380 + 3.37304i 0.133531 + 0.353591i
\(92\) 0 0
\(93\) 1.58393i 0.164246i
\(94\) 0 0
\(95\) −2.00285 −0.205488
\(96\) 0 0
\(97\) 4.98484i 0.506134i −0.967449 0.253067i \(-0.918561\pi\)
0.967449 0.253067i \(-0.0814393\pi\)
\(98\) 0 0
\(99\) 2.47746i 0.248994i
\(100\) 0 0
\(101\) −0.966532 −0.0961735 −0.0480867 0.998843i \(-0.515312\pi\)
−0.0480867 + 0.998843i \(0.515312\pi\)
\(102\) 0 0
\(103\) −11.4249 −1.12573 −0.562864 0.826550i \(-0.690300\pi\)
−0.562864 + 0.826550i \(0.690300\pi\)
\(104\) 0 0
\(105\) −0.277430 −0.0270744
\(106\) 0 0
\(107\) 12.2340 1.18271 0.591353 0.806413i \(-0.298594\pi\)
0.591353 + 0.806413i \(0.298594\pi\)
\(108\) 0 0
\(109\) 1.65127i 0.158163i 0.996868 + 0.0790814i \(0.0251987\pi\)
−0.996868 + 0.0790814i \(0.974801\pi\)
\(110\) 0 0
\(111\) 6.36783i 0.604407i
\(112\) 0 0
\(113\) −7.25772 −0.682749 −0.341375 0.939927i \(-0.610892\pi\)
−0.341375 + 0.939927i \(0.610892\pi\)
\(114\) 0 0
\(115\) 1.08013i 0.100723i
\(116\) 0 0
\(117\) −8.35659 + 3.15579i −0.772567 + 0.291753i
\(118\) 0 0
\(119\) 5.43236i 0.497984i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 4.25921i 0.384040i
\(124\) 0 0
\(125\) 3.78138i 0.338217i
\(126\) 0 0
\(127\) −8.03568 −0.713052 −0.356526 0.934285i \(-0.616039\pi\)
−0.356526 + 0.934285i \(0.616039\pi\)
\(128\) 0 0
\(129\) 6.68911 0.588943
\(130\) 0 0
\(131\) 14.5041 1.26723 0.633616 0.773648i \(-0.281570\pi\)
0.633616 + 0.773648i \(0.281570\pi\)
\(132\) 0 0
\(133\) 5.21859 0.452509
\(134\) 0 0
\(135\) 1.51961i 0.130788i
\(136\) 0 0
\(137\) 4.28927i 0.366457i 0.983070 + 0.183228i \(0.0586548\pi\)
−0.983070 + 0.183228i \(0.941345\pi\)
\(138\) 0 0
\(139\) −16.6819 −1.41494 −0.707469 0.706745i \(-0.750163\pi\)
−0.707469 + 0.706745i \(0.750163\pi\)
\(140\) 0 0
\(141\) 5.99073i 0.504510i
\(142\) 0 0
\(143\) −1.27380 3.37304i −0.106521 0.282068i
\(144\) 0 0
\(145\) 0.767338i 0.0637240i
\(146\) 0 0
\(147\) 0.722868 0.0596211
\(148\) 0 0
\(149\) 19.6040i 1.60602i 0.595967 + 0.803009i \(0.296769\pi\)
−0.595967 + 0.803009i \(0.703231\pi\)
\(150\) 0 0
\(151\) 2.47295i 0.201246i −0.994925 0.100623i \(-0.967916\pi\)
0.994925 0.100623i \(-0.0320835\pi\)
\(152\) 0 0
\(153\) 13.4585 1.08805
\(154\) 0 0
\(155\) 0.840956 0.0675472
\(156\) 0 0
\(157\) −10.1146 −0.807234 −0.403617 0.914928i \(-0.632247\pi\)
−0.403617 + 0.914928i \(0.632247\pi\)
\(158\) 0 0
\(159\) 0.456978 0.0362407
\(160\) 0 0
\(161\) 2.81438i 0.221804i
\(162\) 0 0
\(163\) 22.9361i 1.79650i −0.439488 0.898248i \(-0.644840\pi\)
0.439488 0.898248i \(-0.355160\pi\)
\(164\) 0 0
\(165\) 0.277430 0.0215979
\(166\) 0 0
\(167\) 20.6241i 1.59594i −0.602699 0.797969i \(-0.705908\pi\)
0.602699 0.797969i \(-0.294092\pi\)
\(168\) 0 0
\(169\) 9.75486 8.59318i 0.750374 0.661014i
\(170\) 0 0
\(171\) 12.9289i 0.988695i
\(172\) 0 0
\(173\) −5.66630 −0.430801 −0.215400 0.976526i \(-0.569106\pi\)
−0.215400 + 0.976526i \(0.569106\pi\)
\(174\) 0 0
\(175\) 4.85270i 0.366830i
\(176\) 0 0
\(177\) 9.61268i 0.722533i
\(178\) 0 0
\(179\) −12.5552 −0.938418 −0.469209 0.883087i \(-0.655461\pi\)
−0.469209 + 0.883087i \(0.655461\pi\)
\(180\) 0 0
\(181\) −14.9079 −1.10810 −0.554049 0.832484i \(-0.686918\pi\)
−0.554049 + 0.832484i \(0.686918\pi\)
\(182\) 0 0
\(183\) 9.61909 0.711064
\(184\) 0 0
\(185\) 3.38086 0.248566
\(186\) 0 0
\(187\) 5.43236i 0.397254i
\(188\) 0 0
\(189\) 3.95948i 0.288010i
\(190\) 0 0
\(191\) −17.7347 −1.28324 −0.641618 0.767025i \(-0.721736\pi\)
−0.641618 + 0.767025i \(0.721736\pi\)
\(192\) 0 0
\(193\) 15.4927i 1.11519i 0.830115 + 0.557593i \(0.188275\pi\)
−0.830115 + 0.557593i \(0.811725\pi\)
\(194\) 0 0
\(195\) 0.353391 + 0.935785i 0.0253069 + 0.0670129i
\(196\) 0 0
\(197\) 5.02539i 0.358044i −0.983845 0.179022i \(-0.942707\pi\)
0.983845 0.179022i \(-0.0572934\pi\)
\(198\) 0 0
\(199\) −16.5290 −1.17171 −0.585855 0.810416i \(-0.699241\pi\)
−0.585855 + 0.810416i \(0.699241\pi\)
\(200\) 0 0
\(201\) 5.67231i 0.400094i
\(202\) 0 0
\(203\) 1.99936i 0.140328i
\(204\) 0 0
\(205\) −2.26134 −0.157939
\(206\) 0 0
\(207\) −6.97251 −0.484623
\(208\) 0 0
\(209\) −5.21859 −0.360978
\(210\) 0 0
\(211\) −23.0461 −1.58656 −0.793279 0.608859i \(-0.791628\pi\)
−0.793279 + 0.608859i \(0.791628\pi\)
\(212\) 0 0
\(213\) 6.11677i 0.419114i
\(214\) 0 0
\(215\) 3.55144i 0.242206i
\(216\) 0 0
\(217\) −2.19118 −0.148747
\(218\) 0 0
\(219\) 9.88001i 0.667630i
\(220\) 0 0
\(221\) −18.3236 + 6.91975i −1.23258 + 0.465473i
\(222\) 0 0
\(223\) 3.12458i 0.209237i −0.994512 0.104619i \(-0.966638\pi\)
0.994512 0.104619i \(-0.0333622\pi\)
\(224\) 0 0
\(225\) −12.0224 −0.801493
\(226\) 0 0
\(227\) 11.5176i 0.764449i −0.924069 0.382225i \(-0.875158\pi\)
0.924069 0.382225i \(-0.124842\pi\)
\(228\) 0 0
\(229\) 3.82300i 0.252631i 0.991990 + 0.126315i \(0.0403151\pi\)
−0.991990 + 0.126315i \(0.959685\pi\)
\(230\) 0 0
\(231\) −0.722868 −0.0475612
\(232\) 0 0
\(233\) −4.54901 −0.298016 −0.149008 0.988836i \(-0.547608\pi\)
−0.149008 + 0.988836i \(0.547608\pi\)
\(234\) 0 0
\(235\) 3.18065 0.207483
\(236\) 0 0
\(237\) 10.1086 0.656627
\(238\) 0 0
\(239\) 12.3313i 0.797644i −0.917028 0.398822i \(-0.869419\pi\)
0.917028 0.398822i \(-0.130581\pi\)
\(240\) 0 0
\(241\) 21.7442i 1.40067i −0.713815 0.700335i \(-0.753034\pi\)
0.713815 0.700335i \(-0.246966\pi\)
\(242\) 0 0
\(243\) −15.1821 −0.973932
\(244\) 0 0
\(245\) 0.383791i 0.0245195i
\(246\) 0 0
\(247\) −6.64745 17.6025i −0.422967 1.12002i
\(248\) 0 0
\(249\) 9.79517i 0.620744i
\(250\) 0 0
\(251\) 15.3695 0.970115 0.485057 0.874482i \(-0.338799\pi\)
0.485057 + 0.874482i \(0.338799\pi\)
\(252\) 0 0
\(253\) 2.81438i 0.176938i
\(254\) 0 0
\(255\) 1.50710i 0.0943783i
\(256\) 0 0
\(257\) 12.7666 0.796358 0.398179 0.917308i \(-0.369642\pi\)
0.398179 + 0.917308i \(0.369642\pi\)
\(258\) 0 0
\(259\) −8.80912 −0.547372
\(260\) 0 0
\(261\) −4.95335 −0.306605
\(262\) 0 0
\(263\) −27.6129 −1.70268 −0.851342 0.524610i \(-0.824211\pi\)
−0.851342 + 0.524610i \(0.824211\pi\)
\(264\) 0 0
\(265\) 0.242623i 0.0149042i
\(266\) 0 0
\(267\) 2.88058i 0.176289i
\(268\) 0 0
\(269\) 28.1801 1.71817 0.859084 0.511835i \(-0.171034\pi\)
0.859084 + 0.511835i \(0.171034\pi\)
\(270\) 0 0
\(271\) 3.96706i 0.240981i 0.992714 + 0.120491i \(0.0384468\pi\)
−0.992714 + 0.120491i \(0.961553\pi\)
\(272\) 0 0
\(273\) −0.920790 2.43827i −0.0557287 0.147571i
\(274\) 0 0
\(275\) 4.85270i 0.292629i
\(276\) 0 0
\(277\) 25.0315 1.50400 0.751998 0.659166i \(-0.229090\pi\)
0.751998 + 0.659166i \(0.229090\pi\)
\(278\) 0 0
\(279\) 5.42857i 0.325000i
\(280\) 0 0
\(281\) 3.59437i 0.214422i −0.994236 0.107211i \(-0.965808\pi\)
0.994236 0.107211i \(-0.0341920\pi\)
\(282\) 0 0
\(283\) −11.2143 −0.666618 −0.333309 0.942818i \(-0.608165\pi\)
−0.333309 + 0.942818i \(0.608165\pi\)
\(284\) 0 0
\(285\) 1.44780 0.0857600
\(286\) 0 0
\(287\) 5.89210 0.347800
\(288\) 0 0
\(289\) 12.5105 0.735914
\(290\) 0 0
\(291\) 3.60338i 0.211234i
\(292\) 0 0
\(293\) 10.2580i 0.599279i 0.954052 + 0.299640i \(0.0968665\pi\)
−0.954052 + 0.299640i \(0.903134\pi\)
\(294\) 0 0
\(295\) 5.10365 0.297146
\(296\) 0 0
\(297\) 3.95948i 0.229752i
\(298\) 0 0
\(299\) 9.49302 3.58496i 0.548995 0.207323i
\(300\) 0 0
\(301\) 9.25357i 0.533367i
\(302\) 0 0
\(303\) 0.698674 0.0401378
\(304\) 0 0
\(305\) 5.10705i 0.292429i
\(306\) 0 0
\(307\) 18.1345i 1.03499i 0.855687 + 0.517494i \(0.173135\pi\)
−0.855687 + 0.517494i \(0.826865\pi\)
\(308\) 0 0
\(309\) 8.25868 0.469820
\(310\) 0 0
\(311\) −21.3181 −1.20884 −0.604420 0.796666i \(-0.706595\pi\)
−0.604420 + 0.796666i \(0.706595\pi\)
\(312\) 0 0
\(313\) −0.529307 −0.0299182 −0.0149591 0.999888i \(-0.504762\pi\)
−0.0149591 + 0.999888i \(0.504762\pi\)
\(314\) 0 0
\(315\) −0.950828 −0.0535731
\(316\) 0 0
\(317\) 3.03911i 0.170693i −0.996351 0.0853467i \(-0.972800\pi\)
0.996351 0.0853467i \(-0.0271998\pi\)
\(318\) 0 0
\(319\) 1.99936i 0.111943i
\(320\) 0 0
\(321\) −8.84356 −0.493600
\(322\) 0 0
\(323\) 28.3493i 1.57740i
\(324\) 0 0
\(325\) 16.3684 6.18138i 0.907955 0.342881i
\(326\) 0 0
\(327\) 1.19365i 0.0660089i
\(328\) 0 0
\(329\) −8.28744 −0.456902
\(330\) 0 0
\(331\) 15.8844i 0.873087i 0.899683 + 0.436543i \(0.143797\pi\)
−0.899683 + 0.436543i \(0.856203\pi\)
\(332\) 0 0
\(333\) 21.8243i 1.19596i
\(334\) 0 0
\(335\) −3.01159 −0.164541
\(336\) 0 0
\(337\) 16.1118 0.877667 0.438833 0.898568i \(-0.355392\pi\)
0.438833 + 0.898568i \(0.355392\pi\)
\(338\) 0 0
\(339\) 5.24637 0.284944
\(340\) 0 0
\(341\) 2.19118 0.118659
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0.780793i 0.0420365i
\(346\) 0 0
\(347\) −2.13479 −0.114602 −0.0573008 0.998357i \(-0.518249\pi\)
−0.0573008 + 0.998357i \(0.518249\pi\)
\(348\) 0 0
\(349\) 30.3659i 1.62545i 0.582647 + 0.812725i \(0.302017\pi\)
−0.582647 + 0.812725i \(0.697983\pi\)
\(350\) 0 0
\(351\) 13.3555 5.04359i 0.712864 0.269207i
\(352\) 0 0
\(353\) 2.02651i 0.107860i 0.998545 + 0.0539301i \(0.0171748\pi\)
−0.998545 + 0.0539301i \(0.982825\pi\)
\(354\) 0 0
\(355\) −3.24757 −0.172363
\(356\) 0 0
\(357\) 3.92688i 0.207832i
\(358\) 0 0
\(359\) 11.8666i 0.626295i 0.949705 + 0.313147i \(0.101383\pi\)
−0.949705 + 0.313147i \(0.898617\pi\)
\(360\) 0 0
\(361\) −8.23372 −0.433354
\(362\) 0 0
\(363\) 0.722868 0.0379407
\(364\) 0 0
\(365\) −5.24558 −0.274566
\(366\) 0 0
\(367\) −21.2420 −1.10882 −0.554412 0.832242i \(-0.687057\pi\)
−0.554412 + 0.832242i \(0.687057\pi\)
\(368\) 0 0
\(369\) 14.5975i 0.759913i
\(370\) 0 0
\(371\) 0.632174i 0.0328209i
\(372\) 0 0
\(373\) −17.6162 −0.912131 −0.456066 0.889946i \(-0.650742\pi\)
−0.456066 + 0.889946i \(0.650742\pi\)
\(374\) 0 0
\(375\) 2.73344i 0.141154i
\(376\) 0 0
\(377\) 6.74394 2.54679i 0.347331 0.131166i
\(378\) 0 0
\(379\) 16.5983i 0.852595i −0.904583 0.426298i \(-0.859818\pi\)
0.904583 0.426298i \(-0.140182\pi\)
\(380\) 0 0
\(381\) 5.80874 0.297591
\(382\) 0 0
\(383\) 18.4942i 0.945008i −0.881329 0.472504i \(-0.843350\pi\)
0.881329 0.472504i \(-0.156650\pi\)
\(384\) 0 0
\(385\) 0.383791i 0.0195598i
\(386\) 0 0
\(387\) 22.9254 1.16536
\(388\) 0 0
\(389\) −17.1720 −0.870653 −0.435327 0.900273i \(-0.643367\pi\)
−0.435327 + 0.900273i \(0.643367\pi\)
\(390\) 0 0
\(391\) −15.2887 −0.773183
\(392\) 0 0
\(393\) −10.4846 −0.528877
\(394\) 0 0
\(395\) 5.36697i 0.270041i
\(396\) 0 0
\(397\) 1.69929i 0.0852847i 0.999090 + 0.0426424i \(0.0135776\pi\)
−0.999090 + 0.0426424i \(0.986422\pi\)
\(398\) 0 0
\(399\) −3.77235 −0.188854
\(400\) 0 0
\(401\) 16.0941i 0.803703i −0.915705 0.401851i \(-0.868367\pi\)
0.915705 0.401851i \(-0.131633\pi\)
\(402\) 0 0
\(403\) 2.79113 + 7.39095i 0.139036 + 0.368169i
\(404\) 0 0
\(405\) 1.75400i 0.0871572i
\(406\) 0 0
\(407\) 8.80912 0.436652
\(408\) 0 0
\(409\) 23.5297i 1.16347i −0.813379 0.581734i \(-0.802374\pi\)
0.813379 0.581734i \(-0.197626\pi\)
\(410\) 0 0
\(411\) 3.10057i 0.152940i
\(412\) 0 0
\(413\) −13.2980 −0.654351
\(414\) 0 0
\(415\) −5.20054 −0.255284
\(416\) 0 0
\(417\) 12.0588 0.590521
\(418\) 0 0
\(419\) 6.71087 0.327847 0.163924 0.986473i \(-0.447585\pi\)
0.163924 + 0.986473i \(0.447585\pi\)
\(420\) 0 0
\(421\) 12.7391i 0.620864i 0.950596 + 0.310432i \(0.100474\pi\)
−0.950596 + 0.310432i \(0.899526\pi\)
\(422\) 0 0
\(423\) 20.5318i 0.998292i
\(424\) 0 0
\(425\) −26.3616 −1.27873
\(426\) 0 0
\(427\) 13.3069i 0.643964i
\(428\) 0 0
\(429\) 0.920790 + 2.43827i 0.0444562 + 0.117721i
\(430\) 0 0
\(431\) 8.34857i 0.402137i 0.979577 + 0.201068i \(0.0644413\pi\)
−0.979577 + 0.201068i \(0.935559\pi\)
\(432\) 0 0
\(433\) −3.00531 −0.144426 −0.0722130 0.997389i \(-0.523006\pi\)
−0.0722130 + 0.997389i \(0.523006\pi\)
\(434\) 0 0
\(435\) 0.554684i 0.0265951i
\(436\) 0 0
\(437\) 14.6871i 0.702579i
\(438\) 0 0
\(439\) 15.5404 0.741704 0.370852 0.928692i \(-0.379066\pi\)
0.370852 + 0.928692i \(0.379066\pi\)
\(440\) 0 0
\(441\) 2.47746 0.117974
\(442\) 0 0
\(443\) −25.6406 −1.21822 −0.609111 0.793085i \(-0.708473\pi\)
−0.609111 + 0.793085i \(0.708473\pi\)
\(444\) 0 0
\(445\) 1.52938 0.0724996
\(446\) 0 0
\(447\) 14.1711i 0.670268i
\(448\) 0 0
\(449\) 9.03275i 0.426282i −0.977021 0.213141i \(-0.931631\pi\)
0.977021 0.213141i \(-0.0683694\pi\)
\(450\) 0 0
\(451\) −5.89210 −0.277448
\(452\) 0 0
\(453\) 1.78761i 0.0839894i
\(454\) 0 0
\(455\) 1.29454 0.488874i 0.0606892 0.0229188i
\(456\) 0 0
\(457\) 10.1393i 0.474295i 0.971474 + 0.237148i \(0.0762125\pi\)
−0.971474 + 0.237148i \(0.923787\pi\)
\(458\) 0 0
\(459\) −21.5093 −1.00397
\(460\) 0 0
\(461\) 37.1766i 1.73149i −0.500486 0.865745i \(-0.666845\pi\)
0.500486 0.865745i \(-0.333155\pi\)
\(462\) 0 0
\(463\) 18.4199i 0.856046i 0.903768 + 0.428023i \(0.140790\pi\)
−0.903768 + 0.428023i \(0.859210\pi\)
\(464\) 0 0
\(465\) −0.607900 −0.0281907
\(466\) 0 0
\(467\) −18.7425 −0.867298 −0.433649 0.901082i \(-0.642774\pi\)
−0.433649 + 0.901082i \(0.642774\pi\)
\(468\) 0 0
\(469\) 7.84695 0.362338
\(470\) 0 0
\(471\) 7.31153 0.336897
\(472\) 0 0
\(473\) 9.25357i 0.425480i
\(474\) 0 0
\(475\) 25.3243i 1.16196i
\(476\) 0 0
\(477\) 1.56619 0.0717108
\(478\) 0 0
\(479\) 10.2727i 0.469373i −0.972071 0.234686i \(-0.924594\pi\)
0.972071 0.234686i \(-0.0754063\pi\)
\(480\) 0 0
\(481\) 11.2211 + 29.7135i 0.511636 + 1.35482i
\(482\) 0 0
\(483\) 2.03442i 0.0925694i
\(484\) 0 0
\(485\) −1.91314 −0.0868711
\(486\) 0 0
\(487\) 39.0239i 1.76834i 0.467162 + 0.884172i \(0.345276\pi\)
−0.467162 + 0.884172i \(0.654724\pi\)
\(488\) 0 0
\(489\) 16.5798i 0.749764i
\(490\) 0 0
\(491\) 4.02434 0.181616 0.0908079 0.995868i \(-0.471055\pi\)
0.0908079 + 0.995868i \(0.471055\pi\)
\(492\) 0 0
\(493\) −10.8613 −0.489167
\(494\) 0 0
\(495\) 0.950828 0.0427365
\(496\) 0 0
\(497\) 8.46182 0.379564
\(498\) 0 0
\(499\) 11.7324i 0.525212i 0.964903 + 0.262606i \(0.0845820\pi\)
−0.964903 + 0.262606i \(0.915418\pi\)
\(500\) 0 0
\(501\) 14.9085i 0.666061i
\(502\) 0 0
\(503\) 9.52353 0.424633 0.212317 0.977201i \(-0.431899\pi\)
0.212317 + 0.977201i \(0.431899\pi\)
\(504\) 0 0
\(505\) 0.370946i 0.0165069i
\(506\) 0 0
\(507\) −7.05147 + 6.21173i −0.313167 + 0.275873i
\(508\) 0 0
\(509\) 4.56408i 0.202299i −0.994871 0.101150i \(-0.967748\pi\)
0.994871 0.101150i \(-0.0322521\pi\)
\(510\) 0 0
\(511\) 13.6678 0.604628
\(512\) 0 0
\(513\) 20.6629i 0.912290i
\(514\) 0 0
\(515\) 4.38477i 0.193216i
\(516\) 0 0
\(517\) 8.28744 0.364481
\(518\) 0 0
\(519\) 4.09599 0.179794
\(520\) 0 0
\(521\) 27.7904 1.21752 0.608759 0.793355i \(-0.291668\pi\)
0.608759 + 0.793355i \(0.291668\pi\)
\(522\) 0 0
\(523\) −30.8738 −1.35002 −0.675009 0.737809i \(-0.735860\pi\)
−0.675009 + 0.737809i \(0.735860\pi\)
\(524\) 0 0
\(525\) 3.50786i 0.153096i
\(526\) 0 0
\(527\) 11.9033i 0.518515i
\(528\) 0 0
\(529\) −15.0793 −0.655621
\(530\) 0 0
\(531\) 32.9452i 1.42970i
\(532\) 0 0
\(533\) −7.50536 19.8743i −0.325093 0.860852i
\(534\) 0 0
\(535\) 4.69530i 0.202996i
\(536\) 0 0
\(537\) 9.07573 0.391647
\(538\) 0 0
\(539\) 1.00000i 0.0430730i
\(540\) 0 0
\(541\) 25.6566i 1.10306i 0.834154 + 0.551532i \(0.185957\pi\)
−0.834154 + 0.551532i \(0.814043\pi\)
\(542\) 0 0
\(543\) 10.7765 0.462462
\(544\) 0 0
\(545\) 0.633742 0.0271465
\(546\) 0 0
\(547\) −17.7340 −0.758251 −0.379126 0.925345i \(-0.623775\pi\)
−0.379126 + 0.925345i \(0.623775\pi\)
\(548\) 0 0
\(549\) 32.9672 1.40701
\(550\) 0 0
\(551\) 10.4339i 0.444498i
\(552\) 0 0
\(553\) 13.9841i 0.594664i
\(554\) 0 0
\(555\) −2.44392 −0.103738
\(556\) 0 0
\(557\) 4.60249i 0.195014i −0.995235 0.0975069i \(-0.968913\pi\)
0.995235 0.0975069i \(-0.0310868\pi\)
\(558\) 0 0
\(559\) −31.2127 + 11.7872i −1.32016 + 0.498546i
\(560\) 0 0
\(561\) 3.92688i 0.165793i
\(562\) 0 0
\(563\) 15.6375 0.659041 0.329521 0.944148i \(-0.393113\pi\)
0.329521 + 0.944148i \(0.393113\pi\)
\(564\) 0 0
\(565\) 2.78545i 0.117185i
\(566\) 0 0
\(567\) 4.57021i 0.191931i
\(568\) 0 0
\(569\) 27.8266 1.16655 0.583276 0.812274i \(-0.301771\pi\)
0.583276 + 0.812274i \(0.301771\pi\)
\(570\) 0 0
\(571\) −32.8460 −1.37456 −0.687280 0.726392i \(-0.741196\pi\)
−0.687280 + 0.726392i \(0.741196\pi\)
\(572\) 0 0
\(573\) 12.8198 0.535555
\(574\) 0 0
\(575\) 13.6573 0.569550
\(576\) 0 0
\(577\) 13.7664i 0.573102i 0.958065 + 0.286551i \(0.0925088\pi\)
−0.958065 + 0.286551i \(0.907491\pi\)
\(578\) 0 0
\(579\) 11.1991i 0.465420i
\(580\) 0 0
\(581\) 13.5504 0.562167
\(582\) 0 0
\(583\) 0.632174i 0.0261820i
\(584\) 0 0
\(585\) 1.21117 + 3.20719i 0.0500756 + 0.132601i
\(586\) 0 0
\(587\) 19.0750i 0.787310i −0.919258 0.393655i \(-0.871211\pi\)
0.919258 0.393655i \(-0.128789\pi\)
\(588\) 0 0
\(589\) 11.4349 0.471166
\(590\) 0 0
\(591\) 3.63269i 0.149429i
\(592\) 0 0
\(593\) 21.8115i 0.895690i 0.894111 + 0.447845i \(0.147808\pi\)
−0.894111 + 0.447845i \(0.852192\pi\)
\(594\) 0 0
\(595\) −2.08489 −0.0854722
\(596\) 0 0
\(597\) 11.9483 0.489011
\(598\) 0 0
\(599\) 1.30419 0.0532880 0.0266440 0.999645i \(-0.491518\pi\)
0.0266440 + 0.999645i \(0.491518\pi\)
\(600\) 0 0
\(601\) 44.8904 1.83112 0.915559 0.402183i \(-0.131748\pi\)
0.915559 + 0.402183i \(0.131748\pi\)
\(602\) 0 0
\(603\) 19.4405i 0.791679i
\(604\) 0 0
\(605\) 0.383791i 0.0156033i
\(606\) 0 0
\(607\) −4.20928 −0.170849 −0.0854247 0.996345i \(-0.527225\pi\)
−0.0854247 + 0.996345i \(0.527225\pi\)
\(608\) 0 0
\(609\) 1.44528i 0.0585655i
\(610\) 0 0
\(611\) 10.5566 + 27.9539i 0.427073 + 1.13089i
\(612\) 0 0
\(613\) 18.2079i 0.735411i 0.929942 + 0.367706i \(0.119857\pi\)
−0.929942 + 0.367706i \(0.880143\pi\)
\(614\) 0 0
\(615\) 1.63465 0.0659153
\(616\) 0 0
\(617\) 14.2443i 0.573455i 0.958012 + 0.286727i \(0.0925673\pi\)
−0.958012 + 0.286727i \(0.907433\pi\)
\(618\) 0 0
\(619\) 30.1836i 1.21318i −0.795014 0.606591i \(-0.792537\pi\)
0.795014 0.606591i \(-0.207463\pi\)
\(620\) 0 0
\(621\) 11.1435 0.447172
\(622\) 0 0
\(623\) −3.98493 −0.159653
\(624\) 0 0
\(625\) 22.8123 0.912490
\(626\) 0 0
\(627\) 3.77235 0.150653
\(628\) 0 0
\(629\) 47.8543i 1.90808i
\(630\) 0 0
\(631\) 1.93512i 0.0770358i 0.999258 + 0.0385179i \(0.0122637\pi\)
−0.999258 + 0.0385179i \(0.987736\pi\)
\(632\) 0 0
\(633\) 16.6593 0.662146
\(634\) 0 0
\(635\) 3.08402i 0.122386i
\(636\) 0 0
\(637\) −3.37304 + 1.27380i −0.133645 + 0.0504698i
\(638\) 0 0
\(639\) 20.9638i 0.829316i
\(640\) 0 0
\(641\) 45.5002 1.79715 0.898574 0.438821i \(-0.144604\pi\)
0.898574 + 0.438821i \(0.144604\pi\)
\(642\) 0 0
\(643\) 18.6451i 0.735292i −0.929966 0.367646i \(-0.880164\pi\)
0.929966 0.367646i \(-0.119836\pi\)
\(644\) 0 0
\(645\) 2.56722i 0.101084i
\(646\) 0 0
\(647\) 27.6333 1.08638 0.543189 0.839611i \(-0.317217\pi\)
0.543189 + 0.839611i \(0.317217\pi\)
\(648\) 0 0
\(649\) 13.2980 0.521991
\(650\) 0 0
\(651\) 1.58393 0.0620793
\(652\) 0 0
\(653\) 10.6935 0.418468 0.209234 0.977866i \(-0.432903\pi\)
0.209234 + 0.977866i \(0.432903\pi\)
\(654\) 0 0
\(655\) 5.56656i 0.217503i
\(656\) 0 0
\(657\) 33.8615i 1.32106i
\(658\) 0 0
\(659\) −11.1509 −0.434376 −0.217188 0.976130i \(-0.569688\pi\)
−0.217188 + 0.976130i \(0.569688\pi\)
\(660\) 0 0
\(661\) 6.18156i 0.240435i −0.992748 0.120217i \(-0.961641\pi\)
0.992748 0.120217i \(-0.0383591\pi\)
\(662\) 0 0
\(663\) 13.2455 5.00206i 0.514414 0.194264i
\(664\) 0 0
\(665\) 2.00285i 0.0776672i
\(666\) 0 0
\(667\) 5.62696 0.217877
\(668\) 0 0
\(669\) 2.25866i 0.0873247i
\(670\) 0 0
\(671\) 13.3069i 0.513705i
\(672\) 0 0
\(673\) 24.1526 0.931015 0.465507 0.885044i \(-0.345872\pi\)
0.465507 + 0.885044i \(0.345872\pi\)
\(674\) 0 0
\(675\) 19.2142 0.739554
\(676\) 0 0
\(677\) 25.1643 0.967143 0.483571 0.875305i \(-0.339339\pi\)
0.483571 + 0.875305i \(0.339339\pi\)
\(678\) 0 0
\(679\) 4.98484 0.191301
\(680\) 0 0
\(681\) 8.32569i 0.319041i
\(682\) 0 0
\(683\) 38.7812i 1.48392i 0.670443 + 0.741961i \(0.266104\pi\)
−0.670443 + 0.741961i \(0.733896\pi\)
\(684\) 0 0
\(685\) 1.64618 0.0628974
\(686\) 0 0
\(687\) 2.76352i 0.105435i
\(688\) 0 0
\(689\) −2.13235 + 0.805265i −0.0812362 + 0.0306781i
\(690\) 0 0
\(691\) 3.27158i 0.124457i −0.998062 0.0622283i \(-0.980179\pi\)
0.998062 0.0622283i \(-0.0198207\pi\)
\(692\) 0 0
\(693\) −2.47746 −0.0941110
\(694\) 0 0
\(695\) 6.40235i 0.242855i
\(696\) 0 0
\(697\) 32.0080i 1.21239i
\(698\) 0 0
\(699\) 3.28833 0.124376
\(700\) 0 0
\(701\) −48.2566 −1.82263 −0.911313 0.411715i \(-0.864930\pi\)
−0.911313 + 0.411715i \(0.864930\pi\)
\(702\) 0 0
\(703\) 45.9712 1.73384
\(704\) 0 0
\(705\) −2.29919 −0.0865924
\(706\) 0 0
\(707\) 0.966532i 0.0363502i
\(708\) 0 0
\(709\) 47.1828i 1.77199i 0.463698 + 0.885993i \(0.346522\pi\)
−0.463698 + 0.885993i \(0.653478\pi\)
\(710\) 0 0
\(711\) 34.6450 1.29929
\(712\) 0 0
\(713\) 6.16681i 0.230949i
\(714\) 0 0
\(715\) −1.29454 + 0.488874i −0.0484132 + 0.0182828i
\(716\) 0 0
\(717\) 8.91388i 0.332895i
\(718\) 0 0
\(719\) −3.56804 −0.133066 −0.0665328 0.997784i \(-0.521194\pi\)
−0.0665328 + 0.997784i \(0.521194\pi\)
\(720\) 0 0
\(721\) 11.4249i 0.425485i
\(722\) 0 0
\(723\) 15.7182i 0.584566i
\(724\) 0 0
\(725\) 9.70232 0.360335
\(726\) 0 0
\(727\) 27.6748 1.02640 0.513202 0.858268i \(-0.328459\pi\)
0.513202 + 0.858268i \(0.328459\pi\)
\(728\) 0 0
\(729\) −2.73597 −0.101332
\(730\) 0 0
\(731\) 50.2687 1.85926
\(732\) 0 0
\(733\) 37.6774i 1.39165i −0.718212 0.695824i \(-0.755039\pi\)
0.718212 0.695824i \(-0.244961\pi\)
\(734\) 0 0
\(735\) 0.277430i 0.0102332i
\(736\) 0 0
\(737\) −7.84695 −0.289046
\(738\) 0 0
\(739\) 24.3642i 0.896252i −0.893970 0.448126i \(-0.852092\pi\)
0.893970 0.448126i \(-0.147908\pi\)
\(740\) 0 0
\(741\) 4.80523 + 12.7243i 0.176524 + 0.467439i
\(742\) 0 0
\(743\) 41.4463i 1.52052i 0.649622 + 0.760258i \(0.274927\pi\)
−0.649622 + 0.760258i \(0.725073\pi\)
\(744\) 0 0
\(745\) 7.52382 0.275652
\(746\) 0 0
\(747\) 33.5707i 1.22829i
\(748\) 0 0
\(749\) 12.2340i 0.447021i
\(750\) 0 0
\(751\) −36.2124 −1.32141 −0.660705 0.750645i \(-0.729743\pi\)
−0.660705 + 0.750645i \(0.729743\pi\)
\(752\) 0 0
\(753\) −11.1101 −0.404875
\(754\) 0 0
\(755\) −0.949095 −0.0345411
\(756\) 0 0
\(757\) 40.6886 1.47885 0.739426 0.673238i \(-0.235097\pi\)
0.739426 + 0.673238i \(0.235097\pi\)
\(758\) 0 0
\(759\) 2.03442i 0.0738448i
\(760\) 0 0
\(761\) 30.1176i 1.09176i −0.837863 0.545881i \(-0.816195\pi\)
0.837863 0.545881i \(-0.183805\pi\)
\(762\) 0 0
\(763\) −1.65127 −0.0597799
\(764\) 0 0
\(765\) 5.16524i 0.186750i
\(766\) 0 0
\(767\) 16.9390 + 44.8547i 0.611631 + 1.61961i
\(768\) 0 0
\(769\) 26.4775i 0.954803i −0.878685 0.477401i \(-0.841579\pi\)
0.878685 0.477401i \(-0.158421\pi\)
\(770\) 0 0
\(771\) −9.22856 −0.332358
\(772\) 0 0
\(773\) 24.1869i 0.869944i −0.900444 0.434972i \(-0.856758\pi\)
0.900444 0.434972i \(-0.143242\pi\)
\(774\) 0 0
\(775\) 10.6332i 0.381954i
\(776\) 0 0
\(777\) 6.36783 0.228444
\(778\) 0 0
\(779\) −30.7485 −1.10168
\(780\) 0 0
\(781\) −8.46182 −0.302788
\(782\) 0 0
\(783\) 7.91644 0.282911
\(784\) 0 0
\(785\) 3.88190i 0.138551i
\(786\) 0 0
\(787\) 2.31299i 0.0824493i 0.999150 + 0.0412247i \(0.0131259\pi\)
−0.999150 + 0.0412247i \(0.986874\pi\)
\(788\) 0 0
\(789\) 19.9605 0.710612
\(790\) 0 0
\(791\) 7.25772i 0.258055i
\(792\) 0 0
\(793\) −44.8846 + 16.9503i −1.59390 + 0.601922i
\(794\) 0