Properties

Label 4004.2.m.c.2157.9
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 2157.9
Character \(\chi\) = 4004.2157
Dual form 4004.2.m.c.2157.10

$q$-expansion

\(f(q)\) \(=\) \(q\)\(-1.64159 q^{3}\) \(-4.02832i q^{5}\) \(-1.00000i q^{7}\) \(-0.305195 q^{9}\) \(+O(q^{10})\) \(q\)\(-1.64159 q^{3}\) \(-4.02832i q^{5}\) \(-1.00000i q^{7}\) \(-0.305195 q^{9}\) \(+1.00000i q^{11}\) \(+(3.58935 + 0.341402i) q^{13}\) \(+6.61284i q^{15}\) \(+2.93822 q^{17}\) \(-5.26423i q^{19}\) \(+1.64159i q^{21}\) \(-6.16583 q^{23}\) \(-11.2274 q^{25}\) \(+5.42576 q^{27}\) \(+2.51264 q^{29}\) \(-9.67831i q^{31}\) \(-1.64159i q^{33}\) \(-4.02832 q^{35}\) \(-0.752499i q^{37}\) \(+(-5.89223 - 0.560440i) q^{39}\) \(-2.05391i q^{41}\) \(+3.54942 q^{43}\) \(+1.22942i q^{45}\) \(-10.3141i q^{47}\) \(-1.00000 q^{49}\) \(-4.82335 q^{51}\) \(+7.99742 q^{53}\) \(+4.02832 q^{55}\) \(+8.64168i q^{57}\) \(-3.00461i q^{59}\) \(+6.61169 q^{61}\) \(+0.305195i q^{63}\) \(+(1.37528 - 14.4591i) q^{65}\) \(-8.37934i q^{67}\) \(+10.1217 q^{69}\) \(+7.73832i q^{71}\) \(-6.96682i q^{73}\) \(+18.4307 q^{75}\) \(+1.00000 q^{77}\) \(-11.5285 q^{79}\) \(-7.99127 q^{81}\) \(-13.4835i q^{83}\) \(-11.8361i q^{85}\) \(-4.12471 q^{87}\) \(+12.5207i q^{89}\) \(+(0.341402 - 3.58935i) q^{91}\) \(+15.8878i q^{93}\) \(-21.2060 q^{95}\) \(+12.2072i q^{97}\) \(-0.305195i q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 80q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut -\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 80q^{69} \) \(\mathstrut -\mathstrut 36q^{75} \) \(\mathstrut +\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 132q^{81} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
\(3\) −1.64159 −0.947770 −0.473885 0.880587i \(-0.657149\pi\)
−0.473885 + 0.880587i \(0.657149\pi\)
\(4\) 0 0
\(5\) 4.02832i 1.80152i −0.434317 0.900760i \(-0.643010\pi\)
0.434317 0.900760i \(-0.356990\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −0.305195 −0.101732
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) 3.58935 + 0.341402i 0.995507 + 0.0946878i
\(14\) 0 0
\(15\) 6.61284i 1.70743i
\(16\) 0 0
\(17\) 2.93822 0.712624 0.356312 0.934367i \(-0.384034\pi\)
0.356312 + 0.934367i \(0.384034\pi\)
\(18\) 0 0
\(19\) 5.26423i 1.20770i −0.797099 0.603848i \(-0.793633\pi\)
0.797099 0.603848i \(-0.206367\pi\)
\(20\) 0 0
\(21\) 1.64159i 0.358223i
\(22\) 0 0
\(23\) −6.16583 −1.28566 −0.642832 0.766007i \(-0.722241\pi\)
−0.642832 + 0.766007i \(0.722241\pi\)
\(24\) 0 0
\(25\) −11.2274 −2.24548
\(26\) 0 0
\(27\) 5.42576 1.04419
\(28\) 0 0
\(29\) 2.51264 0.466585 0.233292 0.972407i \(-0.425050\pi\)
0.233292 + 0.972407i \(0.425050\pi\)
\(30\) 0 0
\(31\) 9.67831i 1.73827i −0.494571 0.869137i \(-0.664675\pi\)
0.494571 0.869137i \(-0.335325\pi\)
\(32\) 0 0
\(33\) 1.64159i 0.285763i
\(34\) 0 0
\(35\) −4.02832 −0.680911
\(36\) 0 0
\(37\) 0.752499i 0.123710i −0.998085 0.0618550i \(-0.980298\pi\)
0.998085 0.0618550i \(-0.0197016\pi\)
\(38\) 0 0
\(39\) −5.89223 0.560440i −0.943512 0.0897422i
\(40\) 0 0
\(41\) 2.05391i 0.320766i −0.987055 0.160383i \(-0.948727\pi\)
0.987055 0.160383i \(-0.0512730\pi\)
\(42\) 0 0
\(43\) 3.54942 0.541281 0.270641 0.962680i \(-0.412764\pi\)
0.270641 + 0.962680i \(0.412764\pi\)
\(44\) 0 0
\(45\) 1.22942i 0.183272i
\(46\) 0 0
\(47\) 10.3141i 1.50447i −0.658894 0.752236i \(-0.728976\pi\)
0.658894 0.752236i \(-0.271024\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −4.82335 −0.675404
\(52\) 0 0
\(53\) 7.99742 1.09853 0.549265 0.835648i \(-0.314908\pi\)
0.549265 + 0.835648i \(0.314908\pi\)
\(54\) 0 0
\(55\) 4.02832 0.543179
\(56\) 0 0
\(57\) 8.64168i 1.14462i
\(58\) 0 0
\(59\) 3.00461i 0.391167i −0.980687 0.195583i \(-0.937340\pi\)
0.980687 0.195583i \(-0.0626600\pi\)
\(60\) 0 0
\(61\) 6.61169 0.846540 0.423270 0.906004i \(-0.360882\pi\)
0.423270 + 0.906004i \(0.360882\pi\)
\(62\) 0 0
\(63\) 0.305195i 0.0384510i
\(64\) 0 0
\(65\) 1.37528 14.4591i 0.170582 1.79343i
\(66\) 0 0
\(67\) 8.37934i 1.02370i −0.859075 0.511850i \(-0.828960\pi\)
0.859075 0.511850i \(-0.171040\pi\)
\(68\) 0 0
\(69\) 10.1217 1.21851
\(70\) 0 0
\(71\) 7.73832i 0.918370i 0.888341 + 0.459185i \(0.151858\pi\)
−0.888341 + 0.459185i \(0.848142\pi\)
\(72\) 0 0
\(73\) 6.96682i 0.815404i −0.913115 0.407702i \(-0.866330\pi\)
0.913115 0.407702i \(-0.133670\pi\)
\(74\) 0 0
\(75\) 18.4307 2.12819
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −11.5285 −1.29706 −0.648529 0.761190i \(-0.724615\pi\)
−0.648529 + 0.761190i \(0.724615\pi\)
\(80\) 0 0
\(81\) −7.99127 −0.887919
\(82\) 0 0
\(83\) 13.4835i 1.48000i −0.672606 0.740001i \(-0.734825\pi\)
0.672606 0.740001i \(-0.265175\pi\)
\(84\) 0 0
\(85\) 11.8361i 1.28381i
\(86\) 0 0
\(87\) −4.12471 −0.442215
\(88\) 0 0
\(89\) 12.5207i 1.32719i 0.748093 + 0.663594i \(0.230970\pi\)
−0.748093 + 0.663594i \(0.769030\pi\)
\(90\) 0 0
\(91\) 0.341402 3.58935i 0.0357886 0.376266i
\(92\) 0 0
\(93\) 15.8878i 1.64749i
\(94\) 0 0
\(95\) −21.2060 −2.17569
\(96\) 0 0
\(97\) 12.2072i 1.23945i 0.784817 + 0.619727i \(0.212757\pi\)
−0.784817 + 0.619727i \(0.787243\pi\)
\(98\) 0 0
\(99\) 0.305195i 0.0306733i
\(100\) 0 0
\(101\) −7.87840 −0.783930 −0.391965 0.919980i \(-0.628205\pi\)
−0.391965 + 0.919980i \(0.628205\pi\)
\(102\) 0 0
\(103\) −10.6895 −1.05327 −0.526634 0.850092i \(-0.676546\pi\)
−0.526634 + 0.850092i \(0.676546\pi\)
\(104\) 0 0
\(105\) 6.61284 0.645347
\(106\) 0 0
\(107\) −5.68314 −0.549410 −0.274705 0.961529i \(-0.588580\pi\)
−0.274705 + 0.961529i \(0.588580\pi\)
\(108\) 0 0
\(109\) 6.60605i 0.632745i 0.948635 + 0.316372i \(0.102465\pi\)
−0.948635 + 0.316372i \(0.897535\pi\)
\(110\) 0 0
\(111\) 1.23529i 0.117249i
\(112\) 0 0
\(113\) 17.9010 1.68399 0.841993 0.539489i \(-0.181383\pi\)
0.841993 + 0.539489i \(0.181383\pi\)
\(114\) 0 0
\(115\) 24.8379i 2.31615i
\(116\) 0 0
\(117\) −1.09545 0.104194i −0.101275 0.00963275i
\(118\) 0 0
\(119\) 2.93822i 0.269347i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 3.37166i 0.304013i
\(124\) 0 0
\(125\) 25.0859i 2.24375i
\(126\) 0 0
\(127\) 1.56725 0.139071 0.0695354 0.997579i \(-0.477848\pi\)
0.0695354 + 0.997579i \(0.477848\pi\)
\(128\) 0 0
\(129\) −5.82668 −0.513010
\(130\) 0 0
\(131\) 19.0267 1.66237 0.831187 0.555994i \(-0.187662\pi\)
0.831187 + 0.555994i \(0.187662\pi\)
\(132\) 0 0
\(133\) −5.26423 −0.456466
\(134\) 0 0
\(135\) 21.8567i 1.88113i
\(136\) 0 0
\(137\) 9.38227i 0.801582i 0.916170 + 0.400791i \(0.131265\pi\)
−0.916170 + 0.400791i \(0.868735\pi\)
\(138\) 0 0
\(139\) −19.9484 −1.69200 −0.846002 0.533180i \(-0.820997\pi\)
−0.846002 + 0.533180i \(0.820997\pi\)
\(140\) 0 0
\(141\) 16.9315i 1.42589i
\(142\) 0 0
\(143\) −0.341402 + 3.58935i −0.0285494 + 0.300157i
\(144\) 0 0
\(145\) 10.1217i 0.840562i
\(146\) 0 0
\(147\) 1.64159 0.135396
\(148\) 0 0
\(149\) 3.70508i 0.303532i 0.988416 + 0.151766i \(0.0484960\pi\)
−0.988416 + 0.151766i \(0.951504\pi\)
\(150\) 0 0
\(151\) 1.48027i 0.120462i −0.998184 0.0602312i \(-0.980816\pi\)
0.998184 0.0602312i \(-0.0191838\pi\)
\(152\) 0 0
\(153\) −0.896732 −0.0724965
\(154\) 0 0
\(155\) −38.9873 −3.13154
\(156\) 0 0
\(157\) −12.3875 −0.988626 −0.494313 0.869284i \(-0.664580\pi\)
−0.494313 + 0.869284i \(0.664580\pi\)
\(158\) 0 0
\(159\) −13.1285 −1.04115
\(160\) 0 0
\(161\) 6.16583i 0.485935i
\(162\) 0 0
\(163\) 12.4673i 0.976512i 0.872700 + 0.488256i \(0.162367\pi\)
−0.872700 + 0.488256i \(0.837633\pi\)
\(164\) 0 0
\(165\) −6.61284 −0.514809
\(166\) 0 0
\(167\) 3.38570i 0.261993i 0.991383 + 0.130997i \(0.0418177\pi\)
−0.991383 + 0.130997i \(0.958182\pi\)
\(168\) 0 0
\(169\) 12.7669 + 2.45082i 0.982068 + 0.188525i
\(170\) 0 0
\(171\) 1.60662i 0.122861i
\(172\) 0 0
\(173\) 0.526945 0.0400629 0.0200314 0.999799i \(-0.493623\pi\)
0.0200314 + 0.999799i \(0.493623\pi\)
\(174\) 0 0
\(175\) 11.2274i 0.848710i
\(176\) 0 0
\(177\) 4.93232i 0.370736i
\(178\) 0 0
\(179\) −8.84103 −0.660809 −0.330405 0.943839i \(-0.607185\pi\)
−0.330405 + 0.943839i \(0.607185\pi\)
\(180\) 0 0
\(181\) −12.6176 −0.937859 −0.468929 0.883236i \(-0.655360\pi\)
−0.468929 + 0.883236i \(0.655360\pi\)
\(182\) 0 0
\(183\) −10.8537 −0.802325
\(184\) 0 0
\(185\) −3.03131 −0.222866
\(186\) 0 0
\(187\) 2.93822i 0.214864i
\(188\) 0 0
\(189\) 5.42576i 0.394666i
\(190\) 0 0
\(191\) −9.28352 −0.671732 −0.335866 0.941910i \(-0.609029\pi\)
−0.335866 + 0.941910i \(0.609029\pi\)
\(192\) 0 0
\(193\) 1.60103i 0.115245i 0.998338 + 0.0576225i \(0.0183520\pi\)
−0.998338 + 0.0576225i \(0.981648\pi\)
\(194\) 0 0
\(195\) −2.25763 + 23.7358i −0.161672 + 1.69976i
\(196\) 0 0
\(197\) 17.9711i 1.28039i −0.768213 0.640194i \(-0.778854\pi\)
0.768213 0.640194i \(-0.221146\pi\)
\(198\) 0 0
\(199\) 8.13459 0.576645 0.288323 0.957533i \(-0.406902\pi\)
0.288323 + 0.957533i \(0.406902\pi\)
\(200\) 0 0
\(201\) 13.7554i 0.970232i
\(202\) 0 0
\(203\) 2.51264i 0.176352i
\(204\) 0 0
\(205\) −8.27379 −0.577867
\(206\) 0 0
\(207\) 1.88178 0.130793
\(208\) 0 0
\(209\) 5.26423 0.364134
\(210\) 0 0
\(211\) 21.6911 1.49328 0.746638 0.665231i \(-0.231667\pi\)
0.746638 + 0.665231i \(0.231667\pi\)
\(212\) 0 0
\(213\) 12.7031i 0.870403i
\(214\) 0 0
\(215\) 14.2982i 0.975129i
\(216\) 0 0
\(217\) −9.67831 −0.657006
\(218\) 0 0
\(219\) 11.4366i 0.772816i
\(220\) 0 0
\(221\) 10.5463 + 1.00311i 0.709422 + 0.0674768i
\(222\) 0 0
\(223\) 23.0277i 1.54205i 0.636806 + 0.771024i \(0.280255\pi\)
−0.636806 + 0.771024i \(0.719745\pi\)
\(224\) 0 0
\(225\) 3.42654 0.228436
\(226\) 0 0
\(227\) 1.16344i 0.0772201i −0.999254 0.0386100i \(-0.987707\pi\)
0.999254 0.0386100i \(-0.0122930\pi\)
\(228\) 0 0
\(229\) 0.913327i 0.0603543i 0.999545 + 0.0301772i \(0.00960715\pi\)
−0.999545 + 0.0301772i \(0.990393\pi\)
\(230\) 0 0
\(231\) −1.64159 −0.108008
\(232\) 0 0
\(233\) −6.51944 −0.427103 −0.213551 0.976932i \(-0.568503\pi\)
−0.213551 + 0.976932i \(0.568503\pi\)
\(234\) 0 0
\(235\) −41.5487 −2.71034
\(236\) 0 0
\(237\) 18.9250 1.22931
\(238\) 0 0
\(239\) 28.2270i 1.82585i −0.408124 0.912927i \(-0.633817\pi\)
0.408124 0.912927i \(-0.366183\pi\)
\(240\) 0 0
\(241\) 6.11291i 0.393767i 0.980427 + 0.196884i \(0.0630821\pi\)
−0.980427 + 0.196884i \(0.936918\pi\)
\(242\) 0 0
\(243\) −3.15893 −0.202645
\(244\) 0 0
\(245\) 4.02832i 0.257360i
\(246\) 0 0
\(247\) 1.79722 18.8952i 0.114354 1.20227i
\(248\) 0 0
\(249\) 22.1343i 1.40270i
\(250\) 0 0
\(251\) −1.60455 −0.101278 −0.0506392 0.998717i \(-0.516126\pi\)
−0.0506392 + 0.998717i \(0.516126\pi\)
\(252\) 0 0
\(253\) 6.16583i 0.387642i
\(254\) 0 0
\(255\) 19.4300i 1.21675i
\(256\) 0 0
\(257\) 7.52738 0.469545 0.234773 0.972050i \(-0.424565\pi\)
0.234773 + 0.972050i \(0.424565\pi\)
\(258\) 0 0
\(259\) −0.752499 −0.0467580
\(260\) 0 0
\(261\) −0.766844 −0.0474665
\(262\) 0 0
\(263\) 20.9839 1.29392 0.646961 0.762523i \(-0.276039\pi\)
0.646961 + 0.762523i \(0.276039\pi\)
\(264\) 0 0
\(265\) 32.2162i 1.97903i
\(266\) 0 0
\(267\) 20.5538i 1.25787i
\(268\) 0 0
\(269\) −12.7451 −0.777085 −0.388542 0.921431i \(-0.627021\pi\)
−0.388542 + 0.921431i \(0.627021\pi\)
\(270\) 0 0
\(271\) 16.6891i 1.01379i 0.862007 + 0.506896i \(0.169207\pi\)
−0.862007 + 0.506896i \(0.830793\pi\)
\(272\) 0 0
\(273\) −0.560440 + 5.89223i −0.0339194 + 0.356614i
\(274\) 0 0
\(275\) 11.2274i 0.677036i
\(276\) 0 0
\(277\) −31.0363 −1.86479 −0.932396 0.361439i \(-0.882285\pi\)
−0.932396 + 0.361439i \(0.882285\pi\)
\(278\) 0 0
\(279\) 2.95377i 0.176838i
\(280\) 0 0
\(281\) 29.9221i 1.78500i −0.451047 0.892500i \(-0.648949\pi\)
0.451047 0.892500i \(-0.351051\pi\)
\(282\) 0 0
\(283\) −2.97429 −0.176803 −0.0884015 0.996085i \(-0.528176\pi\)
−0.0884015 + 0.996085i \(0.528176\pi\)
\(284\) 0 0
\(285\) 34.8115 2.06205
\(286\) 0 0
\(287\) −2.05391 −0.121238
\(288\) 0 0
\(289\) −8.36684 −0.492167
\(290\) 0 0
\(291\) 20.0392i 1.17472i
\(292\) 0 0
\(293\) 20.8555i 1.21839i 0.793020 + 0.609195i \(0.208507\pi\)
−0.793020 + 0.609195i \(0.791493\pi\)
\(294\) 0 0
\(295\) −12.1035 −0.704695
\(296\) 0 0
\(297\) 5.42576i 0.314835i
\(298\) 0 0
\(299\) −22.1313 2.10502i −1.27989 0.121737i
\(300\) 0 0
\(301\) 3.54942i 0.204585i
\(302\) 0 0
\(303\) 12.9331 0.742986
\(304\) 0 0
\(305\) 26.6340i 1.52506i
\(306\) 0 0
\(307\) 19.1950i 1.09552i 0.836636 + 0.547759i \(0.184519\pi\)
−0.836636 + 0.547759i \(0.815481\pi\)
\(308\) 0 0
\(309\) 17.5478 0.998257
\(310\) 0 0
\(311\) 27.2009 1.54242 0.771211 0.636580i \(-0.219652\pi\)
0.771211 + 0.636580i \(0.219652\pi\)
\(312\) 0 0
\(313\) 0.807068 0.0456182 0.0228091 0.999740i \(-0.492739\pi\)
0.0228091 + 0.999740i \(0.492739\pi\)
\(314\) 0 0
\(315\) 1.22942 0.0692702
\(316\) 0 0
\(317\) 11.3870i 0.639559i −0.947492 0.319779i \(-0.896391\pi\)
0.947492 0.319779i \(-0.103609\pi\)
\(318\) 0 0
\(319\) 2.51264i 0.140681i
\(320\) 0 0
\(321\) 9.32936 0.520714
\(322\) 0 0
\(323\) 15.4675i 0.860634i
\(324\) 0 0
\(325\) −40.2990 3.83304i −2.23539 0.212619i
\(326\) 0 0
\(327\) 10.8444i 0.599697i
\(328\) 0 0
\(329\) −10.3141 −0.568637
\(330\) 0 0
\(331\) 34.2151i 1.88063i 0.340305 + 0.940315i \(0.389470\pi\)
−0.340305 + 0.940315i \(0.610530\pi\)
\(332\) 0 0
\(333\) 0.229659i 0.0125852i
\(334\) 0 0
\(335\) −33.7547 −1.84422
\(336\) 0 0
\(337\) −2.44148 −0.132996 −0.0664980 0.997787i \(-0.521183\pi\)
−0.0664980 + 0.997787i \(0.521183\pi\)
\(338\) 0 0
\(339\) −29.3860 −1.59603
\(340\) 0 0
\(341\) 9.67831 0.524110
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 40.7736i 2.19518i
\(346\) 0 0
\(347\) −20.2875 −1.08909 −0.544546 0.838731i \(-0.683298\pi\)
−0.544546 + 0.838731i \(0.683298\pi\)
\(348\) 0 0
\(349\) 22.7273i 1.21656i 0.793721 + 0.608282i \(0.208141\pi\)
−0.793721 + 0.608282i \(0.791859\pi\)
\(350\) 0 0
\(351\) 19.4750 + 1.85236i 1.03950 + 0.0988719i
\(352\) 0 0
\(353\) 16.1117i 0.857537i 0.903414 + 0.428768i \(0.141052\pi\)
−0.903414 + 0.428768i \(0.858948\pi\)
\(354\) 0 0
\(355\) 31.1724 1.65446
\(356\) 0 0
\(357\) 4.82335i 0.255279i
\(358\) 0 0
\(359\) 25.8362i 1.36358i −0.731546 0.681792i \(-0.761201\pi\)
0.731546 0.681792i \(-0.238799\pi\)
\(360\) 0 0
\(361\) −8.71209 −0.458531
\(362\) 0 0
\(363\) 1.64159 0.0861609
\(364\) 0 0
\(365\) −28.0646 −1.46897
\(366\) 0 0
\(367\) −26.5198 −1.38432 −0.692160 0.721744i \(-0.743341\pi\)
−0.692160 + 0.721744i \(0.743341\pi\)
\(368\) 0 0
\(369\) 0.626842i 0.0326321i
\(370\) 0 0
\(371\) 7.99742i 0.415206i
\(372\) 0 0
\(373\) −11.0852 −0.573969 −0.286985 0.957935i \(-0.592653\pi\)
−0.286985 + 0.957935i \(0.592653\pi\)
\(374\) 0 0
\(375\) 41.1806i 2.12656i
\(376\) 0 0
\(377\) 9.01873 + 0.857818i 0.464488 + 0.0441799i
\(378\) 0 0
\(379\) 28.8736i 1.48314i 0.670876 + 0.741570i \(0.265918\pi\)
−0.670876 + 0.741570i \(0.734082\pi\)
\(380\) 0 0
\(381\) −2.57277 −0.131807
\(382\) 0 0
\(383\) 14.5209i 0.741981i −0.928637 0.370990i \(-0.879018\pi\)
0.928637 0.370990i \(-0.120982\pi\)
\(384\) 0 0
\(385\) 4.02832i 0.205302i
\(386\) 0 0
\(387\) −1.08327 −0.0550655
\(388\) 0 0
\(389\) 19.0845 0.967624 0.483812 0.875172i \(-0.339252\pi\)
0.483812 + 0.875172i \(0.339252\pi\)
\(390\) 0 0
\(391\) −18.1166 −0.916195
\(392\) 0 0
\(393\) −31.2340 −1.57555
\(394\) 0 0
\(395\) 46.4405i 2.33668i
\(396\) 0 0
\(397\) 26.3360i 1.32177i −0.750488 0.660884i \(-0.770181\pi\)
0.750488 0.660884i \(-0.229819\pi\)
\(398\) 0 0
\(399\) 8.64168 0.432625
\(400\) 0 0
\(401\) 24.7685i 1.23688i −0.785831 0.618441i \(-0.787765\pi\)
0.785831 0.618441i \(-0.212235\pi\)
\(402\) 0 0
\(403\) 3.30419 34.7388i 0.164593 1.73046i
\(404\) 0 0
\(405\) 32.1914i 1.59960i
\(406\) 0 0
\(407\) 0.752499 0.0373000
\(408\) 0 0
\(409\) 4.99360i 0.246917i −0.992350 0.123459i \(-0.960601\pi\)
0.992350 0.123459i \(-0.0393987\pi\)
\(410\) 0 0
\(411\) 15.4018i 0.759715i
\(412\) 0 0
\(413\) −3.00461 −0.147847
\(414\) 0 0
\(415\) −54.3157 −2.66625
\(416\) 0 0
\(417\) 32.7471 1.60363
\(418\) 0 0
\(419\) −11.8262 −0.577750 −0.288875 0.957367i \(-0.593281\pi\)
−0.288875 + 0.957367i \(0.593281\pi\)
\(420\) 0 0
\(421\) 13.6380i 0.664675i 0.943160 + 0.332338i \(0.107837\pi\)
−0.943160 + 0.332338i \(0.892163\pi\)
\(422\) 0 0
\(423\) 3.14782i 0.153052i
\(424\) 0 0
\(425\) −32.9885 −1.60018
\(426\) 0 0
\(427\) 6.61169i 0.319962i
\(428\) 0 0
\(429\) 0.560440 5.89223i 0.0270583 0.284480i
\(430\) 0 0
\(431\) 0.439608i 0.0211752i −0.999944 0.0105876i \(-0.996630\pi\)
0.999944 0.0105876i \(-0.00337020\pi\)
\(432\) 0 0
\(433\) −2.37775 −0.114268 −0.0571338 0.998367i \(-0.518196\pi\)
−0.0571338 + 0.998367i \(0.518196\pi\)
\(434\) 0 0
\(435\) 16.6157i 0.796660i
\(436\) 0 0
\(437\) 32.4583i 1.55269i
\(438\) 0 0
\(439\) 12.5617 0.599538 0.299769 0.954012i \(-0.403090\pi\)
0.299769 + 0.954012i \(0.403090\pi\)
\(440\) 0 0
\(441\) 0.305195 0.0145331
\(442\) 0 0
\(443\) −38.6964 −1.83852 −0.919261 0.393648i \(-0.871213\pi\)
−0.919261 + 0.393648i \(0.871213\pi\)
\(444\) 0 0
\(445\) 50.4373 2.39096
\(446\) 0 0
\(447\) 6.08221i 0.287678i
\(448\) 0 0
\(449\) 6.18606i 0.291938i −0.989289 0.145969i \(-0.953370\pi\)
0.989289 0.145969i \(-0.0466300\pi\)
\(450\) 0 0
\(451\) 2.05391 0.0967146
\(452\) 0 0
\(453\) 2.42999i 0.114171i
\(454\) 0 0
\(455\) −14.4591 1.37528i −0.677851 0.0644739i
\(456\) 0 0
\(457\) 10.2781i 0.480787i −0.970676 0.240393i \(-0.922724\pi\)
0.970676 0.240393i \(-0.0772765\pi\)
\(458\) 0 0
\(459\) 15.9421 0.744114
\(460\) 0 0
\(461\) 34.8204i 1.62175i −0.585222 0.810873i \(-0.698993\pi\)
0.585222 0.810873i \(-0.301007\pi\)
\(462\) 0 0
\(463\) 21.4380i 0.996309i 0.867088 + 0.498154i \(0.165989\pi\)
−0.867088 + 0.498154i \(0.834011\pi\)
\(464\) 0 0
\(465\) 64.0011 2.96798
\(466\) 0 0
\(467\) 26.6020 1.23099 0.615496 0.788140i \(-0.288956\pi\)
0.615496 + 0.788140i \(0.288956\pi\)
\(468\) 0 0
\(469\) −8.37934 −0.386922
\(470\) 0 0
\(471\) 20.3351 0.936991
\(472\) 0 0
\(473\) 3.54942i 0.163202i
\(474\) 0 0
\(475\) 59.1035i 2.71185i
\(476\) 0 0
\(477\) −2.44077 −0.111755
\(478\) 0 0
\(479\) 22.7042i 1.03738i −0.854962 0.518690i \(-0.826420\pi\)
0.854962 0.518690i \(-0.173580\pi\)
\(480\) 0 0
\(481\) 0.256904 2.70098i 0.0117138 0.123154i
\(482\) 0 0
\(483\) 10.1217i 0.460555i
\(484\) 0 0
\(485\) 49.1746 2.23290
\(486\) 0 0
\(487\) 17.9735i 0.814455i −0.913327 0.407228i \(-0.866496\pi\)
0.913327 0.407228i \(-0.133504\pi\)
\(488\) 0 0
\(489\) 20.4661i 0.925509i
\(490\) 0 0
\(491\) −21.9664 −0.991328 −0.495664 0.868514i \(-0.665075\pi\)
−0.495664 + 0.868514i \(0.665075\pi\)
\(492\) 0 0
\(493\) 7.38269 0.332500
\(494\) 0 0
\(495\) −1.22942 −0.0552585
\(496\) 0 0
\(497\) 7.73832 0.347111
\(498\) 0 0
\(499\) 30.3035i 1.35657i 0.734798 + 0.678286i \(0.237277\pi\)
−0.734798 + 0.678286i \(0.762723\pi\)
\(500\) 0 0
\(501\) 5.55792i 0.248309i
\(502\) 0 0
\(503\) −22.4016 −0.998839 −0.499419 0.866360i \(-0.666453\pi\)
−0.499419 + 0.866360i \(0.666453\pi\)
\(504\) 0 0
\(505\) 31.7367i 1.41227i
\(506\) 0 0
\(507\) −20.9579 4.02323i −0.930775 0.178678i
\(508\) 0 0
\(509\) 20.1063i 0.891198i −0.895233 0.445599i \(-0.852991\pi\)
0.895233 0.445599i \(-0.147009\pi\)
\(510\) 0 0
\(511\) −6.96682 −0.308194
\(512\) 0 0
\(513\) 28.5624i 1.26106i
\(514\) 0 0
\(515\) 43.0608i 1.89749i
\(516\) 0 0
\(517\) 10.3141 0.453615
\(518\) 0 0
\(519\) −0.865025 −0.0379704
\(520\) 0 0
\(521\) 41.7507 1.82913 0.914565 0.404439i \(-0.132533\pi\)
0.914565 + 0.404439i \(0.132533\pi\)
\(522\) 0 0
\(523\) −2.53254 −0.110740 −0.0553700 0.998466i \(-0.517634\pi\)
−0.0553700 + 0.998466i \(0.517634\pi\)
\(524\) 0 0
\(525\) 18.4307i 0.804382i
\(526\) 0 0
\(527\) 28.4370i 1.23874i
\(528\) 0 0
\(529\) 15.0174 0.652932
\(530\) 0 0
\(531\) 0.916992i 0.0397940i
\(532\) 0 0
\(533\) 0.701207 7.37219i 0.0303726 0.319325i
\(534\) 0 0
\(535\) 22.8935i 0.989773i
\(536\) 0 0
\(537\) 14.5133 0.626295
\(538\) 0 0
\(539\) 1.00000i 0.0430730i
\(540\) 0 0
\(541\) 37.6168i 1.61727i 0.588308 + 0.808637i \(0.299794\pi\)
−0.588308 + 0.808637i \(0.700206\pi\)
\(542\) 0 0
\(543\) 20.7129 0.888875
\(544\) 0 0
\(545\) 26.6113 1.13990
\(546\) 0 0
\(547\) 17.2805 0.738860 0.369430 0.929259i \(-0.379553\pi\)
0.369430 + 0.929259i \(0.379553\pi\)
\(548\) 0 0
\(549\) −2.01785 −0.0861199
\(550\) 0 0
\(551\) 13.2271i 0.563493i
\(552\) 0 0
\(553\) 11.5285i 0.490242i
\(554\) 0 0
\(555\) 4.97615 0.211226
\(556\) 0 0
\(557\) 29.4108i 1.24618i 0.782152 + 0.623088i \(0.214122\pi\)
−0.782152 + 0.623088i \(0.785878\pi\)
\(558\) 0 0
\(559\) 12.7401 + 1.21178i 0.538849 + 0.0512527i
\(560\) 0 0
\(561\) 4.82335i 0.203642i
\(562\) 0 0
\(563\) 26.1535 1.10224 0.551119 0.834426i \(-0.314201\pi\)
0.551119 + 0.834426i \(0.314201\pi\)
\(564\) 0 0
\(565\) 72.1110i 3.03373i
\(566\) 0 0
\(567\) 7.99127i 0.335602i
\(568\) 0 0
\(569\) 23.8334 0.999149 0.499574 0.866271i \(-0.333490\pi\)
0.499574 + 0.866271i \(0.333490\pi\)
\(570\) 0 0
\(571\) −27.8518 −1.16556 −0.582781 0.812629i \(-0.698036\pi\)
−0.582781 + 0.812629i \(0.698036\pi\)
\(572\) 0 0
\(573\) 15.2397 0.636647
\(574\) 0 0
\(575\) 69.2261 2.88693
\(576\) 0 0
\(577\) 4.16564i 0.173418i 0.996234 + 0.0867089i \(0.0276350\pi\)
−0.996234 + 0.0867089i \(0.972365\pi\)
\(578\) 0 0
\(579\) 2.62823i 0.109226i
\(580\) 0 0
\(581\) −13.4835 −0.559388
\(582\) 0 0
\(583\) 7.99742i 0.331219i
\(584\) 0 0
\(585\) −0.419727 + 4.41284i −0.0173536 + 0.182448i
\(586\) 0 0
\(587\) 7.64129i 0.315390i −0.987488 0.157695i \(-0.949594\pi\)
0.987488 0.157695i \(-0.0504063\pi\)
\(588\) 0 0
\(589\) −50.9488 −2.09931
\(590\) 0 0
\(591\) 29.5011i 1.21351i
\(592\) 0 0
\(593\) 4.71886i 0.193780i 0.995295 + 0.0968902i \(0.0308896\pi\)
−0.995295 + 0.0968902i \(0.969110\pi\)
\(594\) 0 0
\(595\) −11.8361 −0.485233
\(596\) 0 0
\(597\) −13.3536 −0.546527
\(598\) 0 0
\(599\) −12.8076 −0.523303 −0.261651 0.965162i \(-0.584267\pi\)
−0.261651 + 0.965162i \(0.584267\pi\)
\(600\) 0 0
\(601\) 41.9380 1.71069 0.855343 0.518063i \(-0.173347\pi\)
0.855343 + 0.518063i \(0.173347\pi\)
\(602\) 0 0
\(603\) 2.55733i 0.104143i
\(604\) 0 0
\(605\) 4.02832i 0.163775i
\(606\) 0 0
\(607\) −19.2450 −0.781129 −0.390564 0.920576i \(-0.627720\pi\)
−0.390564 + 0.920576i \(0.627720\pi\)
\(608\) 0 0
\(609\) 4.12471i 0.167142i
\(610\) 0 0
\(611\) 3.52126 37.0211i 0.142455 1.49771i
\(612\) 0 0
\(613\) 4.20708i 0.169922i 0.996384 + 0.0849612i \(0.0270766\pi\)
−0.996384 + 0.0849612i \(0.972923\pi\)
\(614\) 0 0
\(615\) 13.5821 0.547685
\(616\) 0 0
\(617\) 2.71815i 0.109428i −0.998502 0.0547142i \(-0.982575\pi\)
0.998502 0.0547142i \(-0.0174248\pi\)
\(618\) 0 0
\(619\) 21.8252i 0.877231i −0.898675 0.438615i \(-0.855469\pi\)
0.898675 0.438615i \(-0.144531\pi\)
\(620\) 0 0
\(621\) −33.4543 −1.34248
\(622\) 0 0
\(623\) 12.5207 0.501630
\(624\) 0 0
\(625\) 44.9171 1.79668
\(626\) 0 0
\(627\) −8.64168 −0.345116
\(628\) 0 0
\(629\) 2.21101i 0.0881588i
\(630\) 0 0
\(631\) 38.0053i 1.51297i −0.654013 0.756484i \(-0.726916\pi\)
0.654013 0.756484i \(-0.273084\pi\)
\(632\) 0 0
\(633\) −35.6078 −1.41528
\(634\) 0 0
\(635\) 6.31338i 0.250539i
\(636\) 0 0
\(637\) −3.58935 0.341402i −0.142215 0.0135268i
\(638\) 0 0
\(639\) 2.36170i 0.0934273i
\(640\) 0 0
\(641\) −1.15512 −0.0456246 −0.0228123 0.999740i \(-0.507262\pi\)
−0.0228123 + 0.999740i \(0.507262\pi\)
\(642\) 0 0
\(643\) 15.6633i 0.617699i 0.951111 + 0.308849i \(0.0999439\pi\)
−0.951111 + 0.308849i \(0.900056\pi\)
\(644\) 0 0
\(645\) 23.4717i 0.924199i
\(646\) 0 0
\(647\) 12.0538 0.473884 0.236942 0.971524i \(-0.423855\pi\)
0.236942 + 0.971524i \(0.423855\pi\)
\(648\) 0 0
\(649\) 3.00461 0.117941
\(650\) 0 0
\(651\) 15.8878 0.622691
\(652\) 0 0
\(653\) −15.7742 −0.617291 −0.308646 0.951177i \(-0.599876\pi\)
−0.308646 + 0.951177i \(0.599876\pi\)
\(654\) 0 0
\(655\) 76.6458i 2.99480i
\(656\) 0 0
\(657\) 2.12624i 0.0829525i
\(658\) 0 0
\(659\) 14.1915 0.552821 0.276411 0.961040i \(-0.410855\pi\)
0.276411 + 0.961040i \(0.410855\pi\)
\(660\) 0 0
\(661\) 31.8285i 1.23798i −0.785397 0.618992i \(-0.787541\pi\)
0.785397 0.618992i \(-0.212459\pi\)
\(662\) 0 0
\(663\) −17.3127 1.64670i −0.672369 0.0639525i
\(664\) 0 0
\(665\) 21.2060i 0.822333i
\(666\) 0 0
\(667\) −15.4925 −0.599871
\(668\) 0 0
\(669\) 37.8019i 1.46151i
\(670\) 0 0
\(671\) 6.61169i 0.255241i
\(672\) 0 0
\(673\) 40.2144 1.55015 0.775076 0.631868i \(-0.217712\pi\)
0.775076 + 0.631868i \(0.217712\pi\)
\(674\) 0 0
\(675\) −60.9171 −2.34470
\(676\) 0 0
\(677\) −30.5715 −1.17496 −0.587480 0.809239i \(-0.699880\pi\)
−0.587480 + 0.809239i \(0.699880\pi\)
\(678\) 0 0
\(679\) 12.2072 0.468470
\(680\) 0 0
\(681\) 1.90988i 0.0731869i
\(682\) 0 0
\(683\) 1.56824i 0.0600071i −0.999550 0.0300036i \(-0.990448\pi\)
0.999550 0.0300036i \(-0.00955186\pi\)
\(684\) 0 0
\(685\) 37.7948 1.44407
\(686\) 0 0
\(687\) 1.49930i 0.0572020i
\(688\) 0 0
\(689\) 28.7056 + 2.73033i 1.09359 + 0.104017i
\(690\) 0 0
\(691\) 25.0856i 0.954302i −0.878821 0.477151i \(-0.841669\pi\)
0.878821 0.477151i \(-0.158331\pi\)
\(692\) 0 0
\(693\) −0.305195 −0.0115934
\(694\) 0 0
\(695\) 80.3587i 3.04818i
\(696\) 0 0
\(697\) 6.03483i 0.228586i
\(698\) 0 0
\(699\) 10.7022 0.404795
\(700\) 0 0
\(701\) 38.1712 1.44171 0.720854 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(702\) 0 0
\(703\) −3.96133 −0.149404
\(704\) 0 0
\(705\) 68.2057 2.56878
\(706\) 0 0
\(707\) 7.87840i 0.296298i
\(708\) 0 0
\(709\) 4.27537i 0.160565i 0.996772 + 0.0802824i \(0.0255822\pi\)
−0.996772 + 0.0802824i \(0.974418\pi\)
\(710\) 0 0
\(711\) 3.51844 0.131952
\(712\) 0 0
\(713\) 59.6748i 2.23484i
\(714\) 0 0
\(715\) 14.4591 + 1.37528i 0.540738 + 0.0514324i
\(716\) 0 0
\(717\) 46.3371i 1.73049i
\(718\) 0 0
\(719\) 3.95838 0.147623 0.0738113 0.997272i \(-0.476484\pi\)
0.0738113 + 0.997272i \(0.476484\pi\)
\(720\) 0 0
\(721\) 10.6895i 0.398098i
\(722\) 0 0
\(723\) 10.0349i 0.373201i
\(724\) 0 0
\(725\) −28.2103 −1.04770
\(726\) 0 0
\(727\) −46.2417 −1.71501 −0.857505 0.514475i \(-0.827987\pi\)
−0.857505 + 0.514475i \(0.827987\pi\)
\(728\) 0 0
\(729\) 29.1595 1.07998
\(730\) 0 0
\(731\) 10.4290 0.385730
\(732\) 0 0
\(733\) 10.5451i 0.389491i −0.980854 0.194746i \(-0.937612\pi\)
0.980854 0.194746i \(-0.0623881\pi\)
\(734\) 0 0
\(735\) 6.61284i 0.243918i
\(736\) 0 0
\(737\) 8.37934 0.308657
\(738\) 0 0
\(739\) 9.83102i 0.361640i 0.983516 + 0.180820i \(0.0578751\pi\)
−0.983516 + 0.180820i \(0.942125\pi\)
\(740\) 0 0
\(741\) −2.95028 + 31.0180i −0.108381 + 1.13948i
\(742\) 0 0
\(743\) 32.1412i 1.17915i 0.807715 + 0.589573i \(0.200704\pi\)
−0.807715 + 0.589573i \(0.799296\pi\)
\(744\) 0 0
\(745\) 14.9252 0.546819
\(746\) 0 0
\(747\) 4.11509i 0.150563i
\(748\) 0 0
\(749\) 5.68314i 0.207657i
\(750\) 0 0
\(751\) 35.6313 1.30020 0.650102 0.759847i \(-0.274726\pi\)
0.650102 + 0.759847i \(0.274726\pi\)
\(752\) 0 0
\(753\) 2.63401 0.0959886
\(754\) 0 0
\(755\) −5.96299 −0.217016
\(756\) 0 0
\(757\) 15.1061 0.549042 0.274521 0.961581i \(-0.411481\pi\)
0.274521 + 0.961581i \(0.411481\pi\)
\(758\) 0 0
\(759\) 10.1217i 0.367396i
\(760\) 0 0
\(761\) 35.0444i 1.27036i 0.772365 + 0.635179i \(0.219074\pi\)
−0.772365 + 0.635179i \(0.780926\pi\)
\(762\) 0 0
\(763\) 6.60605 0.239155
\(764\) 0 0
\(765\) 3.61232i 0.130604i
\(766\) 0 0
\(767\) 1.02578 10.7846i 0.0370387 0.389409i
\(768\) 0 0
\(769\) 32.3235i 1.16562i 0.812610 + 0.582808i \(0.198046\pi\)
−0.812610 + 0.582808i \(0.801954\pi\)
\(770\) 0 0
\(771\) −12.3568 −0.445021
\(772\) 0 0
\(773\) 24.4241i 0.878475i −0.898371 0.439238i \(-0.855249\pi\)
0.898371 0.439238i \(-0.144751\pi\)
\(774\) 0 0
\(775\) 108.662i 3.90325i
\(776\) 0 0
\(777\) 1.23529 0.0443158
\(778\) 0 0
\(779\) −10.8122 −0.387388
\(780\) 0 0
\(781\) −7.73832 −0.276899
\(782\) 0 0
\(783\) 13.6330 0.487202
\(784\) 0 0
\(785\) 49.9006i 1.78103i
\(786\) 0 0
\(787\) 22.8365i 0.814032i 0.913421 + 0.407016i \(0.133431\pi\)
−0.913421 + 0.407016i \(0.866569\pi\)
\(788\) 0 0
\(789\) −34.4469 −1.22634
\(790\) 0 0
\(791\) 17.9010i 0.636487i
\(792\) 0 0
\(793\) 23.7317 + 2.25724i 0.842736 + 0.0801569i
\(794\) 0