Properties

Label 4004.2.m.c.2157.2
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 2157.2
Character \(\chi\) = 4004.2157
Dual form 4004.2.m.c.2157.1

$q$-expansion

\(f(q)\) \(=\) \(q\)\(-3.27485 q^{3}\) \(+2.17871i q^{5}\) \(-1.00000i q^{7}\) \(+7.72466 q^{9}\) \(+O(q^{10})\) \(q\)\(-3.27485 q^{3}\) \(+2.17871i q^{5}\) \(-1.00000i q^{7}\) \(+7.72466 q^{9}\) \(+1.00000i q^{11}\) \(+(3.04907 - 1.92437i) q^{13}\) \(-7.13497i q^{15}\) \(-4.89205 q^{17}\) \(-2.89530i q^{19}\) \(+3.27485i q^{21}\) \(-3.83267 q^{23}\) \(+0.253205 q^{25}\) \(-15.4725 q^{27}\) \(+3.96938 q^{29}\) \(+0.338018i q^{31}\) \(-3.27485i q^{33}\) \(+2.17871 q^{35}\) \(-9.41513i q^{37}\) \(+(-9.98525 + 6.30202i) q^{39}\) \(+1.25457i q^{41}\) \(-7.90954 q^{43}\) \(+16.8298i q^{45}\) \(+11.2549i q^{47}\) \(-1.00000 q^{49}\) \(+16.0207 q^{51}\) \(-1.19463 q^{53}\) \(-2.17871 q^{55}\) \(+9.48167i q^{57}\) \(+9.32058i q^{59}\) \(+12.2280 q^{61}\) \(-7.72466i q^{63}\) \(+(4.19264 + 6.64305i) q^{65}\) \(+1.95290i q^{67}\) \(+12.5514 q^{69}\) \(-5.38691i q^{71}\) \(+0.975589i q^{73}\) \(-0.829209 q^{75}\) \(+1.00000 q^{77}\) \(+5.42795 q^{79}\) \(+27.4963 q^{81}\) \(-4.61107i q^{83}\) \(-10.6584i q^{85}\) \(-12.9991 q^{87}\) \(+2.06453i q^{89}\) \(+(-1.92437 - 3.04907i) q^{91}\) \(-1.10696i q^{93}\) \(+6.30802 q^{95}\) \(+0.930612i q^{97}\) \(+7.72466i q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 80q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut -\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 80q^{69} \) \(\mathstrut -\mathstrut 36q^{75} \) \(\mathstrut +\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 132q^{81} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
\(3\) −3.27485 −1.89074 −0.945368 0.326004i \(-0.894298\pi\)
−0.945368 + 0.326004i \(0.894298\pi\)
\(4\) 0 0
\(5\) 2.17871i 0.974351i 0.873304 + 0.487175i \(0.161973\pi\)
−0.873304 + 0.487175i \(0.838027\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 7.72466 2.57489
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) 3.04907 1.92437i 0.845659 0.533723i
\(14\) 0 0
\(15\) 7.13497i 1.84224i
\(16\) 0 0
\(17\) −4.89205 −1.18650 −0.593248 0.805020i \(-0.702155\pi\)
−0.593248 + 0.805020i \(0.702155\pi\)
\(18\) 0 0
\(19\) 2.89530i 0.664227i −0.943239 0.332113i \(-0.892238\pi\)
0.943239 0.332113i \(-0.107762\pi\)
\(20\) 0 0
\(21\) 3.27485i 0.714631i
\(22\) 0 0
\(23\) −3.83267 −0.799167 −0.399583 0.916697i \(-0.630845\pi\)
−0.399583 + 0.916697i \(0.630845\pi\)
\(24\) 0 0
\(25\) 0.253205 0.0506410
\(26\) 0 0
\(27\) −15.4725 −2.97769
\(28\) 0 0
\(29\) 3.96938 0.737095 0.368548 0.929609i \(-0.379855\pi\)
0.368548 + 0.929609i \(0.379855\pi\)
\(30\) 0 0
\(31\) 0.338018i 0.0607098i 0.999539 + 0.0303549i \(0.00966374\pi\)
−0.999539 + 0.0303549i \(0.990336\pi\)
\(32\) 0 0
\(33\) 3.27485i 0.570079i
\(34\) 0 0
\(35\) 2.17871 0.368270
\(36\) 0 0
\(37\) 9.41513i 1.54784i −0.633284 0.773919i \(-0.718294\pi\)
0.633284 0.773919i \(-0.281706\pi\)
\(38\) 0 0
\(39\) −9.98525 + 6.30202i −1.59892 + 1.00913i
\(40\) 0 0
\(41\) 1.25457i 0.195931i 0.995190 + 0.0979656i \(0.0312335\pi\)
−0.995190 + 0.0979656i \(0.968766\pi\)
\(42\) 0 0
\(43\) −7.90954 −1.20619 −0.603097 0.797668i \(-0.706067\pi\)
−0.603097 + 0.797668i \(0.706067\pi\)
\(44\) 0 0
\(45\) 16.8298i 2.50884i
\(46\) 0 0
\(47\) 11.2549i 1.64169i 0.571150 + 0.820845i \(0.306497\pi\)
−0.571150 + 0.820845i \(0.693503\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 16.0207 2.24335
\(52\) 0 0
\(53\) −1.19463 −0.164094 −0.0820472 0.996628i \(-0.526146\pi\)
−0.0820472 + 0.996628i \(0.526146\pi\)
\(54\) 0 0
\(55\) −2.17871 −0.293778
\(56\) 0 0
\(57\) 9.48167i 1.25588i
\(58\) 0 0
\(59\) 9.32058i 1.21344i 0.794917 + 0.606718i \(0.207514\pi\)
−0.794917 + 0.606718i \(0.792486\pi\)
\(60\) 0 0
\(61\) 12.2280 1.56564 0.782818 0.622251i \(-0.213781\pi\)
0.782818 + 0.622251i \(0.213781\pi\)
\(62\) 0 0
\(63\) 7.72466i 0.973215i
\(64\) 0 0
\(65\) 4.19264 + 6.64305i 0.520034 + 0.823968i
\(66\) 0 0
\(67\) 1.95290i 0.238585i 0.992859 + 0.119293i \(0.0380627\pi\)
−0.992859 + 0.119293i \(0.961937\pi\)
\(68\) 0 0
\(69\) 12.5514 1.51101
\(70\) 0 0
\(71\) 5.38691i 0.639309i −0.947534 0.319654i \(-0.896433\pi\)
0.947534 0.319654i \(-0.103567\pi\)
\(72\) 0 0
\(73\) 0.975589i 0.114184i 0.998369 + 0.0570920i \(0.0181828\pi\)
−0.998369 + 0.0570920i \(0.981817\pi\)
\(74\) 0 0
\(75\) −0.829209 −0.0957488
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 5.42795 0.610692 0.305346 0.952241i \(-0.401228\pi\)
0.305346 + 0.952241i \(0.401228\pi\)
\(80\) 0 0
\(81\) 27.4963 3.05515
\(82\) 0 0
\(83\) 4.61107i 0.506131i −0.967449 0.253065i \(-0.918561\pi\)
0.967449 0.253065i \(-0.0814388\pi\)
\(84\) 0 0
\(85\) 10.6584i 1.15606i
\(86\) 0 0
\(87\) −12.9991 −1.39365
\(88\) 0 0
\(89\) 2.06453i 0.218840i 0.993996 + 0.109420i \(0.0348993\pi\)
−0.993996 + 0.109420i \(0.965101\pi\)
\(90\) 0 0
\(91\) −1.92437 3.04907i −0.201728 0.319629i
\(92\) 0 0
\(93\) 1.10696i 0.114786i
\(94\) 0 0
\(95\) 6.30802 0.647190
\(96\) 0 0
\(97\) 0.930612i 0.0944894i 0.998883 + 0.0472447i \(0.0150441\pi\)
−0.998883 + 0.0472447i \(0.984956\pi\)
\(98\) 0 0
\(99\) 7.72466i 0.776357i
\(100\) 0 0
\(101\) −13.0014 −1.29369 −0.646844 0.762622i \(-0.723912\pi\)
−0.646844 + 0.762622i \(0.723912\pi\)
\(102\) 0 0
\(103\) −15.9032 −1.56699 −0.783495 0.621398i \(-0.786565\pi\)
−0.783495 + 0.621398i \(0.786565\pi\)
\(104\) 0 0
\(105\) −7.13497 −0.696301
\(106\) 0 0
\(107\) 3.07538 0.297308 0.148654 0.988889i \(-0.452506\pi\)
0.148654 + 0.988889i \(0.452506\pi\)
\(108\) 0 0
\(109\) 16.6995i 1.59952i 0.600318 + 0.799762i \(0.295041\pi\)
−0.600318 + 0.799762i \(0.704959\pi\)
\(110\) 0 0
\(111\) 30.8332i 2.92656i
\(112\) 0 0
\(113\) 8.43772 0.793754 0.396877 0.917872i \(-0.370094\pi\)
0.396877 + 0.917872i \(0.370094\pi\)
\(114\) 0 0
\(115\) 8.35029i 0.778668i
\(116\) 0 0
\(117\) 23.5530 14.8651i 2.17748 1.37428i
\(118\) 0 0
\(119\) 4.89205i 0.448454i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 4.10854i 0.370454i
\(124\) 0 0
\(125\) 11.4452i 1.02369i
\(126\) 0 0
\(127\) −2.18598 −0.193975 −0.0969874 0.995286i \(-0.530921\pi\)
−0.0969874 + 0.995286i \(0.530921\pi\)
\(128\) 0 0
\(129\) 25.9026 2.28059
\(130\) 0 0
\(131\) −18.1555 −1.58626 −0.793129 0.609054i \(-0.791549\pi\)
−0.793129 + 0.609054i \(0.791549\pi\)
\(132\) 0 0
\(133\) −2.89530 −0.251054
\(134\) 0 0
\(135\) 33.7103i 2.90132i
\(136\) 0 0
\(137\) 13.5418i 1.15696i 0.815698 + 0.578478i \(0.196353\pi\)
−0.815698 + 0.578478i \(0.803647\pi\)
\(138\) 0 0
\(139\) 2.84495 0.241306 0.120653 0.992695i \(-0.461501\pi\)
0.120653 + 0.992695i \(0.461501\pi\)
\(140\) 0 0
\(141\) 36.8580i 3.10401i
\(142\) 0 0
\(143\) 1.92437 + 3.04907i 0.160924 + 0.254976i
\(144\) 0 0
\(145\) 8.64814i 0.718189i
\(146\) 0 0
\(147\) 3.27485 0.270105
\(148\) 0 0
\(149\) 21.4754i 1.75933i −0.475594 0.879665i \(-0.657767\pi\)
0.475594 0.879665i \(-0.342233\pi\)
\(150\) 0 0
\(151\) 19.1883i 1.56152i −0.624832 0.780759i \(-0.714833\pi\)
0.624832 0.780759i \(-0.285167\pi\)
\(152\) 0 0
\(153\) −37.7894 −3.05509
\(154\) 0 0
\(155\) −0.736444 −0.0591526
\(156\) 0 0
\(157\) −16.3380 −1.30391 −0.651956 0.758256i \(-0.726051\pi\)
−0.651956 + 0.758256i \(0.726051\pi\)
\(158\) 0 0
\(159\) 3.91222 0.310259
\(160\) 0 0
\(161\) 3.83267i 0.302057i
\(162\) 0 0
\(163\) 17.5545i 1.37497i −0.726197 0.687486i \(-0.758714\pi\)
0.726197 0.687486i \(-0.241286\pi\)
\(164\) 0 0
\(165\) 7.13497 0.555456
\(166\) 0 0
\(167\) 18.6797i 1.44548i 0.691122 + 0.722738i \(0.257117\pi\)
−0.691122 + 0.722738i \(0.742883\pi\)
\(168\) 0 0
\(169\) 5.59363 11.7350i 0.430279 0.902696i
\(170\) 0 0
\(171\) 22.3652i 1.71031i
\(172\) 0 0
\(173\) −6.72541 −0.511324 −0.255662 0.966766i \(-0.582293\pi\)
−0.255662 + 0.966766i \(0.582293\pi\)
\(174\) 0 0
\(175\) 0.253205i 0.0191405i
\(176\) 0 0
\(177\) 30.5235i 2.29429i
\(178\) 0 0
\(179\) −8.51974 −0.636795 −0.318398 0.947957i \(-0.603145\pi\)
−0.318398 + 0.947957i \(0.603145\pi\)
\(180\) 0 0
\(181\) −24.7965 −1.84311 −0.921554 0.388250i \(-0.873080\pi\)
−0.921554 + 0.388250i \(0.873080\pi\)
\(182\) 0 0
\(183\) −40.0449 −2.96021
\(184\) 0 0
\(185\) 20.5129 1.50814
\(186\) 0 0
\(187\) 4.89205i 0.357742i
\(188\) 0 0
\(189\) 15.4725i 1.12546i
\(190\) 0 0
\(191\) 16.2007 1.17224 0.586120 0.810224i \(-0.300655\pi\)
0.586120 + 0.810224i \(0.300655\pi\)
\(192\) 0 0
\(193\) 17.4161i 1.25364i 0.779166 + 0.626818i \(0.215643\pi\)
−0.779166 + 0.626818i \(0.784357\pi\)
\(194\) 0 0
\(195\) −13.7303 21.7550i −0.983247 1.55791i
\(196\) 0 0
\(197\) 12.7956i 0.911647i −0.890070 0.455824i \(-0.849345\pi\)
0.890070 0.455824i \(-0.150655\pi\)
\(198\) 0 0
\(199\) 3.08910 0.218980 0.109490 0.993988i \(-0.465078\pi\)
0.109490 + 0.993988i \(0.465078\pi\)
\(200\) 0 0
\(201\) 6.39547i 0.451102i
\(202\) 0 0
\(203\) 3.96938i 0.278596i
\(204\) 0 0
\(205\) −2.73335 −0.190906
\(206\) 0 0
\(207\) −29.6060 −2.05776
\(208\) 0 0
\(209\) 2.89530 0.200272
\(210\) 0 0
\(211\) −3.04110 −0.209358 −0.104679 0.994506i \(-0.533382\pi\)
−0.104679 + 0.994506i \(0.533382\pi\)
\(212\) 0 0
\(213\) 17.6413i 1.20876i
\(214\) 0 0
\(215\) 17.2326i 1.17525i
\(216\) 0 0
\(217\) 0.338018 0.0229461
\(218\) 0 0
\(219\) 3.19491i 0.215892i
\(220\) 0 0
\(221\) −14.9162 + 9.41410i −1.00337 + 0.633261i
\(222\) 0 0
\(223\) 16.8016i 1.12512i −0.826756 0.562561i \(-0.809816\pi\)
0.826756 0.562561i \(-0.190184\pi\)
\(224\) 0 0
\(225\) 1.95592 0.130395
\(226\) 0 0
\(227\) 21.4933i 1.42656i −0.700880 0.713280i \(-0.747209\pi\)
0.700880 0.713280i \(-0.252791\pi\)
\(228\) 0 0
\(229\) 11.4208i 0.754711i −0.926069 0.377355i \(-0.876834\pi\)
0.926069 0.377355i \(-0.123166\pi\)
\(230\) 0 0
\(231\) −3.27485 −0.215469
\(232\) 0 0
\(233\) −11.9996 −0.786118 −0.393059 0.919513i \(-0.628583\pi\)
−0.393059 + 0.919513i \(0.628583\pi\)
\(234\) 0 0
\(235\) −24.5211 −1.59958
\(236\) 0 0
\(237\) −17.7757 −1.15466
\(238\) 0 0
\(239\) 7.58559i 0.490671i −0.969438 0.245335i \(-0.921102\pi\)
0.969438 0.245335i \(-0.0788981\pi\)
\(240\) 0 0
\(241\) 21.0927i 1.35870i 0.733814 + 0.679351i \(0.237739\pi\)
−0.733814 + 0.679351i \(0.762261\pi\)
\(242\) 0 0
\(243\) −43.6288 −2.79879
\(244\) 0 0
\(245\) 2.17871i 0.139193i
\(246\) 0 0
\(247\) −5.57161 8.82796i −0.354513 0.561709i
\(248\) 0 0
\(249\) 15.1006i 0.956960i
\(250\) 0 0
\(251\) −2.10029 −0.132569 −0.0662847 0.997801i \(-0.521115\pi\)
−0.0662847 + 0.997801i \(0.521115\pi\)
\(252\) 0 0
\(253\) 3.83267i 0.240958i
\(254\) 0 0
\(255\) 34.9046i 2.18581i
\(256\) 0 0
\(257\) −26.6802 −1.66427 −0.832133 0.554576i \(-0.812881\pi\)
−0.832133 + 0.554576i \(0.812881\pi\)
\(258\) 0 0
\(259\) −9.41513 −0.585028
\(260\) 0 0
\(261\) 30.6621 1.89794
\(262\) 0 0
\(263\) 14.8812 0.917615 0.458808 0.888536i \(-0.348277\pi\)
0.458808 + 0.888536i \(0.348277\pi\)
\(264\) 0 0
\(265\) 2.60275i 0.159886i
\(266\) 0 0
\(267\) 6.76102i 0.413768i
\(268\) 0 0
\(269\) −21.7648 −1.32702 −0.663512 0.748166i \(-0.730935\pi\)
−0.663512 + 0.748166i \(0.730935\pi\)
\(270\) 0 0
\(271\) 15.6386i 0.949976i −0.879992 0.474988i \(-0.842452\pi\)
0.879992 0.474988i \(-0.157548\pi\)
\(272\) 0 0
\(273\) 6.30202 + 9.98525i 0.381415 + 0.604335i
\(274\) 0 0
\(275\) 0.253205i 0.0152688i
\(276\) 0 0
\(277\) −12.6493 −0.760026 −0.380013 0.924981i \(-0.624080\pi\)
−0.380013 + 0.924981i \(0.624080\pi\)
\(278\) 0 0
\(279\) 2.61107i 0.156321i
\(280\) 0 0
\(281\) 14.2607i 0.850725i 0.905023 + 0.425362i \(0.139853\pi\)
−0.905023 + 0.425362i \(0.860147\pi\)
\(282\) 0 0
\(283\) −15.3957 −0.915182 −0.457591 0.889163i \(-0.651288\pi\)
−0.457591 + 0.889163i \(0.651288\pi\)
\(284\) 0 0
\(285\) −20.6578 −1.22367
\(286\) 0 0
\(287\) 1.25457 0.0740550
\(288\) 0 0
\(289\) 6.93216 0.407774
\(290\) 0 0
\(291\) 3.04762i 0.178655i
\(292\) 0 0
\(293\) 17.0707i 0.997283i 0.866808 + 0.498642i \(0.166168\pi\)
−0.866808 + 0.498642i \(0.833832\pi\)
\(294\) 0 0
\(295\) −20.3069 −1.18231
\(296\) 0 0
\(297\) 15.4725i 0.897808i
\(298\) 0 0
\(299\) −11.6861 + 7.37546i −0.675823 + 0.426534i
\(300\) 0 0
\(301\) 7.90954i 0.455898i
\(302\) 0 0
\(303\) 42.5777 2.44602
\(304\) 0 0
\(305\) 26.6413i 1.52548i
\(306\) 0 0
\(307\) 10.8541i 0.619477i −0.950822 0.309739i \(-0.899758\pi\)
0.950822 0.309739i \(-0.100242\pi\)
\(308\) 0 0
\(309\) 52.0807 2.96277
\(310\) 0 0
\(311\) −26.3877 −1.49631 −0.748155 0.663524i \(-0.769060\pi\)
−0.748155 + 0.663524i \(0.769060\pi\)
\(312\) 0 0
\(313\) 22.6577 1.28069 0.640344 0.768088i \(-0.278792\pi\)
0.640344 + 0.768088i \(0.278792\pi\)
\(314\) 0 0
\(315\) 16.8298 0.948253
\(316\) 0 0
\(317\) 24.8773i 1.39725i −0.715489 0.698624i \(-0.753796\pi\)
0.715489 0.698624i \(-0.246204\pi\)
\(318\) 0 0
\(319\) 3.96938i 0.222243i
\(320\) 0 0
\(321\) −10.0714 −0.562131
\(322\) 0 0
\(323\) 14.1639i 0.788103i
\(324\) 0 0
\(325\) 0.772039 0.487259i 0.0428250 0.0270283i
\(326\) 0 0
\(327\) 54.6884i 3.02428i
\(328\) 0 0
\(329\) 11.2549 0.620501
\(330\) 0 0
\(331\) 12.1567i 0.668191i −0.942539 0.334096i \(-0.891569\pi\)
0.942539 0.334096i \(-0.108431\pi\)
\(332\) 0 0
\(333\) 72.7287i 3.98551i
\(334\) 0 0
\(335\) −4.25482 −0.232466
\(336\) 0 0
\(337\) −32.7652 −1.78483 −0.892416 0.451213i \(-0.850991\pi\)
−0.892416 + 0.451213i \(0.850991\pi\)
\(338\) 0 0
\(339\) −27.6323 −1.50078
\(340\) 0 0
\(341\) −0.338018 −0.0183047
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 27.3460i 1.47226i
\(346\) 0 0
\(347\) −6.46324 −0.346965 −0.173482 0.984837i \(-0.555502\pi\)
−0.173482 + 0.984837i \(0.555502\pi\)
\(348\) 0 0
\(349\) 11.6995i 0.626261i −0.949710 0.313131i \(-0.898622\pi\)
0.949710 0.313131i \(-0.101378\pi\)
\(350\) 0 0
\(351\) −47.1768 + 29.7749i −2.51811 + 1.58926i
\(352\) 0 0
\(353\) 11.2069i 0.596481i −0.954491 0.298241i \(-0.903600\pi\)
0.954491 0.298241i \(-0.0963998\pi\)
\(354\) 0 0
\(355\) 11.7365 0.622911
\(356\) 0 0
\(357\) 16.0207i 0.847908i
\(358\) 0 0
\(359\) 4.81775i 0.254271i 0.991885 + 0.127136i \(0.0405783\pi\)
−0.991885 + 0.127136i \(0.959422\pi\)
\(360\) 0 0
\(361\) 10.6173 0.558803
\(362\) 0 0
\(363\) 3.27485 0.171885
\(364\) 0 0
\(365\) −2.12553 −0.111255
\(366\) 0 0
\(367\) 19.8322 1.03523 0.517615 0.855614i \(-0.326820\pi\)
0.517615 + 0.855614i \(0.326820\pi\)
\(368\) 0 0
\(369\) 9.69114i 0.504500i
\(370\) 0 0
\(371\) 1.19463i 0.0620219i
\(372\) 0 0
\(373\) 22.0693 1.14271 0.571354 0.820704i \(-0.306419\pi\)
0.571354 + 0.820704i \(0.306419\pi\)
\(374\) 0 0
\(375\) 37.4814i 1.93553i
\(376\) 0 0
\(377\) 12.1029 7.63854i 0.623331 0.393405i
\(378\) 0 0
\(379\) 15.1259i 0.776966i 0.921456 + 0.388483i \(0.127001\pi\)
−0.921456 + 0.388483i \(0.872999\pi\)
\(380\) 0 0
\(381\) 7.15877 0.366755
\(382\) 0 0
\(383\) 12.8829i 0.658285i 0.944280 + 0.329143i \(0.106760\pi\)
−0.944280 + 0.329143i \(0.893240\pi\)
\(384\) 0 0
\(385\) 2.17871i 0.111038i
\(386\) 0 0
\(387\) −61.0985 −3.10581
\(388\) 0 0
\(389\) −24.6105 −1.24780 −0.623901 0.781503i \(-0.714453\pi\)
−0.623901 + 0.781503i \(0.714453\pi\)
\(390\) 0 0
\(391\) 18.7496 0.948208
\(392\) 0 0
\(393\) 59.4567 2.99919
\(394\) 0 0
\(395\) 11.8260i 0.595028i
\(396\) 0 0
\(397\) 27.5294i 1.38166i −0.723016 0.690831i \(-0.757245\pi\)
0.723016 0.690831i \(-0.242755\pi\)
\(398\) 0 0
\(399\) 9.48167 0.474677
\(400\) 0 0
\(401\) 3.88250i 0.193883i −0.995290 0.0969414i \(-0.969094\pi\)
0.995290 0.0969414i \(-0.0309059\pi\)
\(402\) 0 0
\(403\) 0.650470 + 1.03064i 0.0324022 + 0.0513398i
\(404\) 0 0
\(405\) 59.9067i 2.97679i
\(406\) 0 0
\(407\) 9.41513 0.466691
\(408\) 0 0
\(409\) 11.8113i 0.584033i −0.956413 0.292016i \(-0.905674\pi\)
0.956413 0.292016i \(-0.0943262\pi\)
\(410\) 0 0
\(411\) 44.3475i 2.18750i
\(412\) 0 0
\(413\) 9.32058 0.458636
\(414\) 0 0
\(415\) 10.0462 0.493149
\(416\) 0 0
\(417\) −9.31680 −0.456246
\(418\) 0 0
\(419\) 4.72086 0.230629 0.115315 0.993329i \(-0.463212\pi\)
0.115315 + 0.993329i \(0.463212\pi\)
\(420\) 0 0
\(421\) 17.6560i 0.860501i −0.902710 0.430251i \(-0.858425\pi\)
0.902710 0.430251i \(-0.141575\pi\)
\(422\) 0 0
\(423\) 86.9400i 4.22717i
\(424\) 0 0
\(425\) −1.23869 −0.0600854
\(426\) 0 0
\(427\) 12.2280i 0.591755i
\(428\) 0 0
\(429\) −6.30202 9.98525i −0.304264 0.482092i
\(430\) 0 0
\(431\) 28.4949i 1.37255i 0.727341 + 0.686277i \(0.240756\pi\)
−0.727341 + 0.686277i \(0.759244\pi\)
\(432\) 0 0
\(433\) −2.16389 −0.103990 −0.0519949 0.998647i \(-0.516558\pi\)
−0.0519949 + 0.998647i \(0.516558\pi\)
\(434\) 0 0
\(435\) 28.3214i 1.35791i
\(436\) 0 0
\(437\) 11.0967i 0.530828i
\(438\) 0 0
\(439\) 12.4914 0.596181 0.298091 0.954538i \(-0.403650\pi\)
0.298091 + 0.954538i \(0.403650\pi\)
\(440\) 0 0
\(441\) −7.72466 −0.367841
\(442\) 0 0
\(443\) −30.1958 −1.43464 −0.717322 0.696742i \(-0.754632\pi\)
−0.717322 + 0.696742i \(0.754632\pi\)
\(444\) 0 0
\(445\) −4.49802 −0.213226
\(446\) 0 0
\(447\) 70.3286i 3.32643i
\(448\) 0 0
\(449\) 25.7787i 1.21657i −0.793719 0.608285i \(-0.791858\pi\)
0.793719 0.608285i \(-0.208142\pi\)
\(450\) 0 0
\(451\) −1.25457 −0.0590755
\(452\) 0 0
\(453\) 62.8387i 2.95242i
\(454\) 0 0
\(455\) 6.64305 4.19264i 0.311431 0.196554i
\(456\) 0 0
\(457\) 25.2388i 1.18062i 0.807176 + 0.590310i \(0.200995\pi\)
−0.807176 + 0.590310i \(0.799005\pi\)
\(458\) 0 0
\(459\) 75.6925 3.53302
\(460\) 0 0
\(461\) 0.737940i 0.0343693i −0.999852 0.0171847i \(-0.994530\pi\)
0.999852 0.0171847i \(-0.00547031\pi\)
\(462\) 0 0
\(463\) 19.0845i 0.886931i 0.896291 + 0.443466i \(0.146251\pi\)
−0.896291 + 0.443466i \(0.853749\pi\)
\(464\) 0 0
\(465\) 2.41174 0.111842
\(466\) 0 0
\(467\) 14.9860 0.693470 0.346735 0.937963i \(-0.387290\pi\)
0.346735 + 0.937963i \(0.387290\pi\)
\(468\) 0 0
\(469\) 1.95290 0.0901768
\(470\) 0 0
\(471\) 53.5045 2.46536
\(472\) 0 0
\(473\) 7.90954i 0.363681i
\(474\) 0 0
\(475\) 0.733104i 0.0336371i
\(476\) 0 0
\(477\) −9.22807 −0.422524
\(478\) 0 0
\(479\) 38.5988i 1.76363i −0.471600 0.881813i \(-0.656323\pi\)
0.471600 0.881813i \(-0.343677\pi\)
\(480\) 0 0
\(481\) −18.1182 28.7074i −0.826117 1.30894i
\(482\) 0 0
\(483\) 12.5514i 0.571109i
\(484\) 0 0
\(485\) −2.02754 −0.0920658
\(486\) 0 0
\(487\) 17.5038i 0.793173i 0.917997 + 0.396587i \(0.129805\pi\)
−0.917997 + 0.396587i \(0.870195\pi\)
\(488\) 0 0
\(489\) 57.4883i 2.59971i
\(490\) 0 0
\(491\) 26.6205 1.20137 0.600683 0.799487i \(-0.294895\pi\)
0.600683 + 0.799487i \(0.294895\pi\)
\(492\) 0 0
\(493\) −19.4184 −0.874561
\(494\) 0 0
\(495\) −16.8298 −0.756444
\(496\) 0 0
\(497\) −5.38691 −0.241636
\(498\) 0 0
\(499\) 30.0002i 1.34299i 0.741007 + 0.671497i \(0.234348\pi\)
−0.741007 + 0.671497i \(0.765652\pi\)
\(500\) 0 0
\(501\) 61.1732i 2.73302i
\(502\) 0 0
\(503\) −12.4859 −0.556719 −0.278360 0.960477i \(-0.589791\pi\)
−0.278360 + 0.960477i \(0.589791\pi\)
\(504\) 0 0
\(505\) 28.3264i 1.26051i
\(506\) 0 0
\(507\) −18.3183 + 38.4305i −0.813544 + 1.70676i
\(508\) 0 0
\(509\) 1.54412i 0.0684420i 0.999414 + 0.0342210i \(0.0108950\pi\)
−0.999414 + 0.0342210i \(0.989105\pi\)
\(510\) 0 0
\(511\) 0.975589 0.0431575
\(512\) 0 0
\(513\) 44.7976i 1.97786i
\(514\) 0 0
\(515\) 34.6486i 1.52680i
\(516\) 0 0
\(517\) −11.2549 −0.494988
\(518\) 0 0
\(519\) 22.0247 0.966779
\(520\) 0 0
\(521\) −26.3243 −1.15329 −0.576644 0.816996i \(-0.695638\pi\)
−0.576644 + 0.816996i \(0.695638\pi\)
\(522\) 0 0
\(523\) 5.84825 0.255726 0.127863 0.991792i \(-0.459188\pi\)
0.127863 + 0.991792i \(0.459188\pi\)
\(524\) 0 0
\(525\) 0.829209i 0.0361896i
\(526\) 0 0
\(527\) 1.65360i 0.0720319i
\(528\) 0 0
\(529\) −8.31065 −0.361333
\(530\) 0 0
\(531\) 71.9983i 3.12446i
\(532\) 0 0
\(533\) 2.41426 + 3.82527i 0.104573 + 0.165691i
\(534\) 0 0
\(535\) 6.70037i 0.289682i
\(536\) 0 0
\(537\) 27.9009 1.20401
\(538\) 0 0
\(539\) 1.00000i 0.0430730i
\(540\) 0 0
\(541\) 8.30458i 0.357042i 0.983936 + 0.178521i \(0.0571312\pi\)
−0.983936 + 0.178521i \(0.942869\pi\)
\(542\) 0 0
\(543\) 81.2048 3.48483
\(544\) 0 0
\(545\) −36.3835 −1.55850
\(546\) 0 0
\(547\) 15.0828 0.644894 0.322447 0.946588i \(-0.395495\pi\)
0.322447 + 0.946588i \(0.395495\pi\)
\(548\) 0 0
\(549\) 94.4572 4.03133
\(550\) 0 0
\(551\) 11.4925i 0.489598i
\(552\) 0 0
\(553\) 5.42795i 0.230820i
\(554\) 0 0
\(555\) −67.1767 −2.85149
\(556\) 0 0
\(557\) 46.4880i 1.96976i −0.173240 0.984880i \(-0.555424\pi\)
0.173240 0.984880i \(-0.444576\pi\)
\(558\) 0 0
\(559\) −24.1167 + 15.2208i −1.02003 + 0.643773i
\(560\) 0 0
\(561\) 16.0207i 0.676396i
\(562\) 0 0
\(563\) −19.0508 −0.802895 −0.401447 0.915882i \(-0.631493\pi\)
−0.401447 + 0.915882i \(0.631493\pi\)
\(564\) 0 0
\(565\) 18.3834i 0.773394i
\(566\) 0 0
\(567\) 27.4963i 1.15474i
\(568\) 0 0
\(569\) −20.9533 −0.878407 −0.439203 0.898388i \(-0.644739\pi\)
−0.439203 + 0.898388i \(0.644739\pi\)
\(570\) 0 0
\(571\) 10.3037 0.431196 0.215598 0.976482i \(-0.430830\pi\)
0.215598 + 0.976482i \(0.430830\pi\)
\(572\) 0 0
\(573\) −53.0548 −2.21640
\(574\) 0 0
\(575\) −0.970451 −0.0404706
\(576\) 0 0
\(577\) 21.7471i 0.905343i −0.891677 0.452672i \(-0.850471\pi\)
0.891677 0.452672i \(-0.149529\pi\)
\(578\) 0 0
\(579\) 57.0350i 2.37030i
\(580\) 0 0
\(581\) −4.61107 −0.191299
\(582\) 0 0
\(583\) 1.19463i 0.0494763i
\(584\) 0 0
\(585\) 32.3867 + 51.3152i 1.33903 + 2.12162i
\(586\) 0 0
\(587\) 42.1122i 1.73816i −0.494675 0.869078i \(-0.664713\pi\)
0.494675 0.869078i \(-0.335287\pi\)
\(588\) 0 0
\(589\) 0.978661 0.0403250
\(590\) 0 0
\(591\) 41.9036i 1.72368i
\(592\) 0 0
\(593\) 1.68629i 0.0692476i 0.999400 + 0.0346238i \(0.0110233\pi\)
−0.999400 + 0.0346238i \(0.988977\pi\)
\(594\) 0 0
\(595\) −10.6584 −0.436951
\(596\) 0 0
\(597\) −10.1163 −0.414034
\(598\) 0 0
\(599\) −45.7342 −1.86865 −0.934324 0.356424i \(-0.883996\pi\)
−0.934324 + 0.356424i \(0.883996\pi\)
\(600\) 0 0
\(601\) 32.6835 1.33319 0.666594 0.745421i \(-0.267752\pi\)
0.666594 + 0.745421i \(0.267752\pi\)
\(602\) 0 0
\(603\) 15.0855i 0.614330i
\(604\) 0 0
\(605\) 2.17871i 0.0885773i
\(606\) 0 0
\(607\) −20.9722 −0.851234 −0.425617 0.904903i \(-0.639943\pi\)
−0.425617 + 0.904903i \(0.639943\pi\)
\(608\) 0 0
\(609\) 12.9991i 0.526751i
\(610\) 0 0
\(611\) 21.6585 + 34.3168i 0.876209 + 1.38831i
\(612\) 0 0
\(613\) 32.3811i 1.30786i −0.756555 0.653930i \(-0.773119\pi\)
0.756555 0.653930i \(-0.226881\pi\)
\(614\) 0 0
\(615\) 8.95133 0.360952
\(616\) 0 0
\(617\) 8.51325i 0.342731i −0.985208 0.171365i \(-0.945182\pi\)
0.985208 0.171365i \(-0.0548178\pi\)
\(618\) 0 0
\(619\) 0.843990i 0.0339228i −0.999856 0.0169614i \(-0.994601\pi\)
0.999856 0.0169614i \(-0.00539925\pi\)
\(620\) 0 0
\(621\) 59.3012 2.37967
\(622\) 0 0
\(623\) 2.06453 0.0827136
\(624\) 0 0
\(625\) −23.6699 −0.946794
\(626\) 0 0
\(627\) −9.48167 −0.378661
\(628\) 0 0
\(629\) 46.0593i 1.83650i
\(630\) 0 0
\(631\) 46.8526i 1.86517i −0.360948 0.932586i \(-0.617547\pi\)
0.360948 0.932586i \(-0.382453\pi\)
\(632\) 0 0
\(633\) 9.95916 0.395841
\(634\) 0 0
\(635\) 4.76263i 0.188999i
\(636\) 0 0
\(637\) −3.04907 + 1.92437i −0.120808 + 0.0762462i
\(638\) 0 0
\(639\) 41.6120i 1.64615i
\(640\) 0 0
\(641\) 18.5344 0.732064 0.366032 0.930602i \(-0.380716\pi\)
0.366032 + 0.930602i \(0.380716\pi\)
\(642\) 0 0
\(643\) 13.4921i 0.532075i 0.963963 + 0.266037i \(0.0857145\pi\)
−0.963963 + 0.266037i \(0.914286\pi\)
\(644\) 0 0
\(645\) 56.4343i 2.22210i
\(646\) 0 0
\(647\) −13.2199 −0.519729 −0.259865 0.965645i \(-0.583678\pi\)
−0.259865 + 0.965645i \(0.583678\pi\)
\(648\) 0 0
\(649\) −9.32058 −0.365865
\(650\) 0 0
\(651\) −1.10696 −0.0433851
\(652\) 0 0
\(653\) −8.18417 −0.320271 −0.160136 0.987095i \(-0.551193\pi\)
−0.160136 + 0.987095i \(0.551193\pi\)
\(654\) 0 0
\(655\) 39.5557i 1.54557i
\(656\) 0 0
\(657\) 7.53609i 0.294011i
\(658\) 0 0
\(659\) −26.9898 −1.05138 −0.525688 0.850678i \(-0.676192\pi\)
−0.525688 + 0.850678i \(0.676192\pi\)
\(660\) 0 0
\(661\) 1.73606i 0.0675249i 0.999430 + 0.0337625i \(0.0107490\pi\)
−0.999430 + 0.0337625i \(0.989251\pi\)
\(662\) 0 0
\(663\) 48.8483 30.8298i 1.89711 1.19733i
\(664\) 0 0
\(665\) 6.30802i 0.244615i
\(666\) 0 0
\(667\) −15.2133 −0.589062
\(668\) 0 0
\(669\) 55.0229i 2.12731i
\(670\) 0 0
\(671\) 12.2280i 0.472057i
\(672\) 0 0
\(673\) −1.56865 −0.0604672 −0.0302336 0.999543i \(-0.509625\pi\)
−0.0302336 + 0.999543i \(0.509625\pi\)
\(674\) 0 0
\(675\) −3.91773 −0.150793
\(676\) 0 0
\(677\) 36.5238 1.40372 0.701861 0.712314i \(-0.252353\pi\)
0.701861 + 0.712314i \(0.252353\pi\)
\(678\) 0 0
\(679\) 0.930612 0.0357136
\(680\) 0 0
\(681\) 70.3873i 2.69725i
\(682\) 0 0
\(683\) 31.0436i 1.18785i −0.804521 0.593924i \(-0.797578\pi\)
0.804521 0.593924i \(-0.202422\pi\)
\(684\) 0 0
\(685\) −29.5038 −1.12728
\(686\) 0 0
\(687\) 37.4016i 1.42696i
\(688\) 0 0
\(689\) −3.64249 + 2.29890i −0.138768 + 0.0875810i
\(690\) 0 0
\(691\) 30.8777i 1.17464i 0.809353 + 0.587322i \(0.199818\pi\)
−0.809353 + 0.587322i \(0.800182\pi\)
\(692\) 0 0
\(693\) 7.72466 0.293435
\(694\) 0 0
\(695\) 6.19834i 0.235117i
\(696\) 0 0
\(697\) 6.13743i 0.232472i
\(698\) 0 0
\(699\) 39.2968 1.48634
\(700\) 0 0
\(701\) 23.3763 0.882911 0.441455 0.897283i \(-0.354462\pi\)
0.441455 + 0.897283i \(0.354462\pi\)
\(702\) 0 0
\(703\) −27.2596 −1.02812
\(704\) 0 0
\(705\) 80.3031 3.02439
\(706\) 0 0
\(707\) 13.0014i 0.488968i
\(708\) 0 0
\(709\) 41.4087i 1.55514i 0.628798 + 0.777568i \(0.283547\pi\)
−0.628798 + 0.777568i \(0.716453\pi\)
\(710\) 0 0
\(711\) 41.9291 1.57246
\(712\) 0 0
\(713\) 1.29551i 0.0485172i
\(714\) 0 0
\(715\) −6.64305 + 4.19264i −0.248436 + 0.156796i
\(716\) 0 0
\(717\) 24.8417i 0.927730i
\(718\) 0 0
\(719\) 36.0533 1.34456 0.672281 0.740296i \(-0.265315\pi\)
0.672281 + 0.740296i \(0.265315\pi\)
\(720\) 0 0
\(721\) 15.9032i 0.592267i
\(722\) 0 0
\(723\) 69.0756i 2.56895i
\(724\) 0 0
\(725\) 1.00507 0.0373273
\(726\) 0 0
\(727\) 16.5955 0.615492 0.307746 0.951469i \(-0.400425\pi\)
0.307746 + 0.951469i \(0.400425\pi\)
\(728\) 0 0
\(729\) 60.3889 2.23662
\(730\) 0 0
\(731\) 38.6939 1.43114
\(732\) 0 0
\(733\) 18.3864i 0.679117i −0.940585 0.339559i \(-0.889722\pi\)
0.940585 0.339559i \(-0.110278\pi\)
\(734\) 0 0
\(735\) 7.13497i 0.263177i
\(736\) 0 0
\(737\) −1.95290 −0.0719362
\(738\) 0 0
\(739\) 16.0008i 0.588600i −0.955713 0.294300i \(-0.904914\pi\)
0.955713 0.294300i \(-0.0950865\pi\)
\(740\) 0 0
\(741\) 18.2462 + 28.9103i 0.670291 + 1.06204i
\(742\) 0 0
\(743\) 38.0070i 1.39434i 0.716905 + 0.697171i \(0.245558\pi\)
−0.716905 + 0.697171i \(0.754442\pi\)
\(744\) 0 0
\(745\) 46.7887 1.71420
\(746\) 0 0
\(747\) 35.6189i 1.30323i
\(748\) 0 0
\(749\) 3.07538i 0.112372i
\(750\) 0 0
\(751\) −29.0475 −1.05996 −0.529980 0.848010i \(-0.677800\pi\)
−0.529980 + 0.848010i \(0.677800\pi\)
\(752\) 0 0
\(753\) 6.87815 0.250654
\(754\) 0 0
\(755\) 41.8057 1.52147
\(756\) 0 0
\(757\) −46.7817 −1.70031 −0.850154 0.526534i \(-0.823491\pi\)
−0.850154 + 0.526534i \(0.823491\pi\)
\(758\) 0 0
\(759\) 12.5514i 0.455588i
\(760\) 0 0
\(761\) 22.9367i 0.831454i −0.909489 0.415727i \(-0.863527\pi\)
0.909489 0.415727i \(-0.136473\pi\)
\(762\) 0 0
\(763\) 16.6995 0.604563
\(764\) 0 0
\(765\) 82.3323i 2.97673i
\(766\) 0 0
\(767\) 17.9362 + 28.4191i 0.647639 + 1.02615i
\(768\) 0 0
\(769\) 28.9721i 1.04476i 0.852713 + 0.522380i \(0.174956\pi\)
−0.852713 + 0.522380i \(0.825044\pi\)
\(770\) 0 0
\(771\) 87.3738 3.14669
\(772\) 0 0
\(773\) 30.7776i 1.10699i −0.832852 0.553496i \(-0.813293\pi\)
0.832852 0.553496i \(-0.186707\pi\)
\(774\) 0 0
\(775\) 0.0855878i 0.00307440i
\(776\) 0 0
\(777\) 30.8332 1.10613
\(778\) 0 0
\(779\) 3.63236 0.130143
\(780\) 0 0
\(781\) 5.38691 0.192759
\(782\) 0 0
\(783\) −61.4164 −2.19484
\(784\) 0 0
\(785\) 35.5958i 1.27047i
\(786\) 0 0
\(787\) 12.9562i 0.461838i −0.972973 0.230919i \(-0.925827\pi\)
0.972973 0.230919i \(-0.0741732\pi\)
\(788\) 0 0
\(789\) −48.7338 −1.73497
\(790\) 0 0
\(791\) 8.43772i 0.300011i
\(792\) 0 0
\(793\) 37.2840 23.5312i 1.32399 0.835617i
\(794\) 0 </