Properties

Label 4004.2.m.c.2157.7
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 2157.7
Character \(\chi\) = 4004.2157
Dual form 4004.2.m.c.2157.8

$q$-expansion

\(f(q)\) \(=\) \(q\)\(-2.10949 q^{3}\) \(-3.39046i q^{5}\) \(+1.00000i q^{7}\) \(+1.44995 q^{9}\) \(+O(q^{10})\) \(q\)\(-2.10949 q^{3}\) \(-3.39046i q^{5}\) \(+1.00000i q^{7}\) \(+1.44995 q^{9}\) \(-1.00000i q^{11}\) \(+(-3.53014 + 0.733543i) q^{13}\) \(+7.15215i q^{15}\) \(+6.05719 q^{17}\) \(-1.08356i q^{19}\) \(-2.10949i q^{21}\) \(-6.38895 q^{23}\) \(-6.49523 q^{25}\) \(+3.26981 q^{27}\) \(-1.88226 q^{29}\) \(+5.69756i q^{31}\) \(+2.10949i q^{33}\) \(+3.39046 q^{35}\) \(+5.35035i q^{37}\) \(+(7.44681 - 1.54740i) q^{39}\) \(-1.69745i q^{41}\) \(+8.15280 q^{43}\) \(-4.91601i q^{45}\) \(+2.26952i q^{47}\) \(-1.00000 q^{49}\) \(-12.7776 q^{51}\) \(-12.6554 q^{53}\) \(-3.39046 q^{55}\) \(+2.28577i q^{57}\) \(+9.70115i q^{59}\) \(+1.04222 q^{61}\) \(+1.44995i q^{63}\) \(+(2.48705 + 11.9688i) q^{65}\) \(-6.67103i q^{67}\) \(+13.4774 q^{69}\) \(+13.9085i q^{71}\) \(-8.29126i q^{73}\) \(+13.7016 q^{75}\) \(+1.00000 q^{77}\) \(-1.23170 q^{79}\) \(-11.2475 q^{81}\) \(+10.9980i q^{83}\) \(-20.5367i q^{85}\) \(+3.97062 q^{87}\) \(+2.96113i q^{89}\) \(+(-0.733543 - 3.53014i) q^{91}\) \(-12.0189i q^{93}\) \(-3.67379 q^{95}\) \(-0.535647i q^{97}\) \(-1.44995i q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 80q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut -\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 80q^{69} \) \(\mathstrut -\mathstrut 36q^{75} \) \(\mathstrut +\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 132q^{81} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
\(3\) −2.10949 −1.21792 −0.608958 0.793203i \(-0.708412\pi\)
−0.608958 + 0.793203i \(0.708412\pi\)
\(4\) 0 0
\(5\) 3.39046i 1.51626i −0.652103 0.758130i \(-0.726113\pi\)
0.652103 0.758130i \(-0.273887\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 1.44995 0.483317
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) −3.53014 + 0.733543i −0.979086 + 0.203448i
\(14\) 0 0
\(15\) 7.15215i 1.84668i
\(16\) 0 0
\(17\) 6.05719 1.46908 0.734542 0.678563i \(-0.237397\pi\)
0.734542 + 0.678563i \(0.237397\pi\)
\(18\) 0 0
\(19\) 1.08356i 0.248587i −0.992246 0.124293i \(-0.960334\pi\)
0.992246 0.124293i \(-0.0396664\pi\)
\(20\) 0 0
\(21\) 2.10949i 0.460329i
\(22\) 0 0
\(23\) −6.38895 −1.33219 −0.666094 0.745867i \(-0.732035\pi\)
−0.666094 + 0.745867i \(0.732035\pi\)
\(24\) 0 0
\(25\) −6.49523 −1.29905
\(26\) 0 0
\(27\) 3.26981 0.629276
\(28\) 0 0
\(29\) −1.88226 −0.349528 −0.174764 0.984610i \(-0.555916\pi\)
−0.174764 + 0.984610i \(0.555916\pi\)
\(30\) 0 0
\(31\) 5.69756i 1.02331i 0.859191 + 0.511656i \(0.170968\pi\)
−0.859191 + 0.511656i \(0.829032\pi\)
\(32\) 0 0
\(33\) 2.10949i 0.367215i
\(34\) 0 0
\(35\) 3.39046 0.573093
\(36\) 0 0
\(37\) 5.35035i 0.879592i 0.898098 + 0.439796i \(0.144949\pi\)
−0.898098 + 0.439796i \(0.855051\pi\)
\(38\) 0 0
\(39\) 7.44681 1.54740i 1.19244 0.247783i
\(40\) 0 0
\(41\) 1.69745i 0.265098i −0.991176 0.132549i \(-0.957684\pi\)
0.991176 0.132549i \(-0.0423162\pi\)
\(42\) 0 0
\(43\) 8.15280 1.24329 0.621645 0.783299i \(-0.286465\pi\)
0.621645 + 0.783299i \(0.286465\pi\)
\(44\) 0 0
\(45\) 4.91601i 0.732835i
\(46\) 0 0
\(47\) 2.26952i 0.331044i 0.986206 + 0.165522i \(0.0529308\pi\)
−0.986206 + 0.165522i \(0.947069\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −12.7776 −1.78922
\(52\) 0 0
\(53\) −12.6554 −1.73835 −0.869176 0.494503i \(-0.835350\pi\)
−0.869176 + 0.494503i \(0.835350\pi\)
\(54\) 0 0
\(55\) −3.39046 −0.457170
\(56\) 0 0
\(57\) 2.28577i 0.302758i
\(58\) 0 0
\(59\) 9.70115i 1.26298i 0.775383 + 0.631491i \(0.217557\pi\)
−0.775383 + 0.631491i \(0.782443\pi\)
\(60\) 0 0
\(61\) 1.04222 0.133442 0.0667211 0.997772i \(-0.478746\pi\)
0.0667211 + 0.997772i \(0.478746\pi\)
\(62\) 0 0
\(63\) 1.44995i 0.182677i
\(64\) 0 0
\(65\) 2.48705 + 11.9688i 0.308480 + 1.48455i
\(66\) 0 0
\(67\) 6.67103i 0.814996i −0.913206 0.407498i \(-0.866401\pi\)
0.913206 0.407498i \(-0.133599\pi\)
\(68\) 0 0
\(69\) 13.4774 1.62249
\(70\) 0 0
\(71\) 13.9085i 1.65063i 0.564671 + 0.825316i \(0.309003\pi\)
−0.564671 + 0.825316i \(0.690997\pi\)
\(72\) 0 0
\(73\) 8.29126i 0.970419i −0.874398 0.485209i \(-0.838743\pi\)
0.874398 0.485209i \(-0.161257\pi\)
\(74\) 0 0
\(75\) 13.7016 1.58213
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −1.23170 −0.138577 −0.0692883 0.997597i \(-0.522073\pi\)
−0.0692883 + 0.997597i \(0.522073\pi\)
\(80\) 0 0
\(81\) −11.2475 −1.24972
\(82\) 0 0
\(83\) 10.9980i 1.20719i 0.797291 + 0.603595i \(0.206266\pi\)
−0.797291 + 0.603595i \(0.793734\pi\)
\(84\) 0 0
\(85\) 20.5367i 2.22751i
\(86\) 0 0
\(87\) 3.97062 0.425695
\(88\) 0 0
\(89\) 2.96113i 0.313879i 0.987608 + 0.156940i \(0.0501628\pi\)
−0.987608 + 0.156940i \(0.949837\pi\)
\(90\) 0 0
\(91\) −0.733543 3.53014i −0.0768962 0.370060i
\(92\) 0 0
\(93\) 12.0189i 1.24631i
\(94\) 0 0
\(95\) −3.67379 −0.376922
\(96\) 0 0
\(97\) 0.535647i 0.0543867i −0.999630 0.0271933i \(-0.991343\pi\)
0.999630 0.0271933i \(-0.00865698\pi\)
\(98\) 0 0
\(99\) 1.44995i 0.145726i
\(100\) 0 0
\(101\) −9.46249 −0.941553 −0.470776 0.882253i \(-0.656026\pi\)
−0.470776 + 0.882253i \(0.656026\pi\)
\(102\) 0 0
\(103\) −16.0615 −1.58259 −0.791294 0.611436i \(-0.790592\pi\)
−0.791294 + 0.611436i \(0.790592\pi\)
\(104\) 0 0
\(105\) −7.15215 −0.697978
\(106\) 0 0
\(107\) 18.7042 1.80820 0.904100 0.427321i \(-0.140543\pi\)
0.904100 + 0.427321i \(0.140543\pi\)
\(108\) 0 0
\(109\) 12.9856i 1.24380i −0.783097 0.621899i \(-0.786361\pi\)
0.783097 0.621899i \(-0.213639\pi\)
\(110\) 0 0
\(111\) 11.2865i 1.07127i
\(112\) 0 0
\(113\) 1.94235 0.182721 0.0913605 0.995818i \(-0.470878\pi\)
0.0913605 + 0.995818i \(0.470878\pi\)
\(114\) 0 0
\(115\) 21.6615i 2.01995i
\(116\) 0 0
\(117\) −5.11854 + 1.06360i −0.473209 + 0.0983300i
\(118\) 0 0
\(119\) 6.05719i 0.555262i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 3.58077i 0.322867i
\(124\) 0 0
\(125\) 5.06952i 0.453432i
\(126\) 0 0
\(127\) 20.2352 1.79558 0.897791 0.440422i \(-0.145171\pi\)
0.897791 + 0.440422i \(0.145171\pi\)
\(128\) 0 0
\(129\) −17.1983 −1.51422
\(130\) 0 0
\(131\) 15.1914 1.32728 0.663640 0.748052i \(-0.269011\pi\)
0.663640 + 0.748052i \(0.269011\pi\)
\(132\) 0 0
\(133\) 1.08356 0.0939570
\(134\) 0 0
\(135\) 11.0862i 0.954146i
\(136\) 0 0
\(137\) 6.51302i 0.556445i 0.960517 + 0.278222i \(0.0897452\pi\)
−0.960517 + 0.278222i \(0.910255\pi\)
\(138\) 0 0
\(139\) 9.47873 0.803975 0.401988 0.915645i \(-0.368319\pi\)
0.401988 + 0.915645i \(0.368319\pi\)
\(140\) 0 0
\(141\) 4.78753i 0.403183i
\(142\) 0 0
\(143\) 0.733543 + 3.53014i 0.0613419 + 0.295205i
\(144\) 0 0
\(145\) 6.38175i 0.529975i
\(146\) 0 0
\(147\) 2.10949 0.173988
\(148\) 0 0
\(149\) 8.14389i 0.667173i −0.942719 0.333587i \(-0.891741\pi\)
0.942719 0.333587i \(-0.108259\pi\)
\(150\) 0 0
\(151\) 10.6160i 0.863918i −0.901893 0.431959i \(-0.857822\pi\)
0.901893 0.431959i \(-0.142178\pi\)
\(152\) 0 0
\(153\) 8.78263 0.710034
\(154\) 0 0
\(155\) 19.3173 1.55161
\(156\) 0 0
\(157\) 14.6183 1.16667 0.583335 0.812232i \(-0.301748\pi\)
0.583335 + 0.812232i \(0.301748\pi\)
\(158\) 0 0
\(159\) 26.6964 2.11717
\(160\) 0 0
\(161\) 6.38895i 0.503520i
\(162\) 0 0
\(163\) 19.3167i 1.51300i −0.653994 0.756500i \(-0.726908\pi\)
0.653994 0.756500i \(-0.273092\pi\)
\(164\) 0 0
\(165\) 7.15215 0.556794
\(166\) 0 0
\(167\) 15.4575i 1.19614i 0.801444 + 0.598070i \(0.204065\pi\)
−0.801444 + 0.598070i \(0.795935\pi\)
\(168\) 0 0
\(169\) 11.9238 5.17902i 0.917218 0.398386i
\(170\) 0 0
\(171\) 1.57112i 0.120146i
\(172\) 0 0
\(173\) 12.7061 0.966026 0.483013 0.875613i \(-0.339542\pi\)
0.483013 + 0.875613i \(0.339542\pi\)
\(174\) 0 0
\(175\) 6.49523i 0.490993i
\(176\) 0 0
\(177\) 20.4645i 1.53821i
\(178\) 0 0
\(179\) 10.8917 0.814086 0.407043 0.913409i \(-0.366560\pi\)
0.407043 + 0.913409i \(0.366560\pi\)
\(180\) 0 0
\(181\) 12.5373 0.931887 0.465944 0.884814i \(-0.345715\pi\)
0.465944 + 0.884814i \(0.345715\pi\)
\(182\) 0 0
\(183\) −2.19855 −0.162521
\(184\) 0 0
\(185\) 18.1402 1.33369
\(186\) 0 0
\(187\) 6.05719i 0.442946i
\(188\) 0 0
\(189\) 3.26981i 0.237844i
\(190\) 0 0
\(191\) −15.4529 −1.11813 −0.559065 0.829124i \(-0.688840\pi\)
−0.559065 + 0.829124i \(0.688840\pi\)
\(192\) 0 0
\(193\) 20.6057i 1.48323i 0.670825 + 0.741616i \(0.265940\pi\)
−0.670825 + 0.741616i \(0.734060\pi\)
\(194\) 0 0
\(195\) −5.24641 25.2481i −0.375703 1.80805i
\(196\) 0 0
\(197\) 1.80816i 0.128826i −0.997923 0.0644131i \(-0.979482\pi\)
0.997923 0.0644131i \(-0.0205175\pi\)
\(198\) 0 0
\(199\) −14.9470 −1.05956 −0.529782 0.848134i \(-0.677726\pi\)
−0.529782 + 0.848134i \(0.677726\pi\)
\(200\) 0 0
\(201\) 14.0725i 0.992596i
\(202\) 0 0
\(203\) 1.88226i 0.132109i
\(204\) 0 0
\(205\) −5.75516 −0.401958
\(206\) 0 0
\(207\) −9.26368 −0.643870
\(208\) 0 0
\(209\) −1.08356 −0.0749518
\(210\) 0 0
\(211\) −6.27173 −0.431764 −0.215882 0.976420i \(-0.569263\pi\)
−0.215882 + 0.976420i \(0.569263\pi\)
\(212\) 0 0
\(213\) 29.3398i 2.01033i
\(214\) 0 0
\(215\) 27.6417i 1.88515i
\(216\) 0 0
\(217\) −5.69756 −0.386775
\(218\) 0 0
\(219\) 17.4903i 1.18189i
\(220\) 0 0
\(221\) −21.3827 + 4.44321i −1.43836 + 0.298882i
\(222\) 0 0
\(223\) 23.7143i 1.58802i 0.607902 + 0.794012i \(0.292011\pi\)
−0.607902 + 0.794012i \(0.707989\pi\)
\(224\) 0 0
\(225\) −9.41778 −0.627852
\(226\) 0 0
\(227\) 27.3569i 1.81574i 0.419254 + 0.907869i \(0.362292\pi\)
−0.419254 + 0.907869i \(0.637708\pi\)
\(228\) 0 0
\(229\) 2.97069i 0.196309i −0.995171 0.0981543i \(-0.968706\pi\)
0.995171 0.0981543i \(-0.0312939\pi\)
\(230\) 0 0
\(231\) −2.10949 −0.138794
\(232\) 0 0
\(233\) −1.47412 −0.0965731 −0.0482865 0.998834i \(-0.515376\pi\)
−0.0482865 + 0.998834i \(0.515376\pi\)
\(234\) 0 0
\(235\) 7.69473 0.501949
\(236\) 0 0
\(237\) 2.59825 0.168774
\(238\) 0 0
\(239\) 11.7293i 0.758706i −0.925252 0.379353i \(-0.876147\pi\)
0.925252 0.379353i \(-0.123853\pi\)
\(240\) 0 0
\(241\) 1.11251i 0.0716628i 0.999358 + 0.0358314i \(0.0114079\pi\)
−0.999358 + 0.0358314i \(0.988592\pi\)
\(242\) 0 0
\(243\) 13.9171 0.892779
\(244\) 0 0
\(245\) 3.39046i 0.216609i
\(246\) 0 0
\(247\) 0.794841 + 3.82514i 0.0505745 + 0.243388i
\(248\) 0 0
\(249\) 23.2002i 1.47025i
\(250\) 0 0
\(251\) 29.3845 1.85473 0.927366 0.374156i \(-0.122068\pi\)
0.927366 + 0.374156i \(0.122068\pi\)
\(252\) 0 0
\(253\) 6.38895i 0.401670i
\(254\) 0 0
\(255\) 43.3219i 2.71292i
\(256\) 0 0
\(257\) −3.21605 −0.200612 −0.100306 0.994957i \(-0.531982\pi\)
−0.100306 + 0.994957i \(0.531982\pi\)
\(258\) 0 0
\(259\) −5.35035 −0.332455
\(260\) 0 0
\(261\) −2.72919 −0.168933
\(262\) 0 0
\(263\) 17.9637 1.10769 0.553844 0.832621i \(-0.313160\pi\)
0.553844 + 0.832621i \(0.313160\pi\)
\(264\) 0 0
\(265\) 42.9076i 2.63579i
\(266\) 0 0
\(267\) 6.24648i 0.382278i
\(268\) 0 0
\(269\) −9.53047 −0.581083 −0.290541 0.956862i \(-0.593835\pi\)
−0.290541 + 0.956862i \(0.593835\pi\)
\(270\) 0 0
\(271\) 20.9006i 1.26962i −0.772669 0.634810i \(-0.781079\pi\)
0.772669 0.634810i \(-0.218921\pi\)
\(272\) 0 0
\(273\) 1.54740 + 7.44681i 0.0936530 + 0.450701i
\(274\) 0 0
\(275\) 6.49523i 0.391677i
\(276\) 0 0
\(277\) 13.3475 0.801975 0.400987 0.916084i \(-0.368667\pi\)
0.400987 + 0.916084i \(0.368667\pi\)
\(278\) 0 0
\(279\) 8.26118i 0.494584i
\(280\) 0 0
\(281\) 0.183295i 0.0109345i 0.999985 + 0.00546724i \(0.00174028\pi\)
−0.999985 + 0.00546724i \(0.998260\pi\)
\(282\) 0 0
\(283\) 3.80597 0.226241 0.113121 0.993581i \(-0.463915\pi\)
0.113121 + 0.993581i \(0.463915\pi\)
\(284\) 0 0
\(285\) 7.74982 0.459060
\(286\) 0 0
\(287\) 1.69745 0.100198
\(288\) 0 0
\(289\) 19.6895 1.15821
\(290\) 0 0
\(291\) 1.12994i 0.0662384i
\(292\) 0 0
\(293\) 22.4190i 1.30973i 0.755744 + 0.654867i \(0.227275\pi\)
−0.755744 + 0.654867i \(0.772725\pi\)
\(294\) 0 0
\(295\) 32.8914 1.91501
\(296\) 0 0
\(297\) 3.26981i 0.189734i
\(298\) 0 0
\(299\) 22.5539 4.68657i 1.30433 0.271031i
\(300\) 0 0
\(301\) 8.15280i 0.469919i
\(302\) 0 0
\(303\) 19.9610 1.14673
\(304\) 0 0
\(305\) 3.53360i 0.202333i
\(306\) 0 0
\(307\) 29.4548i 1.68107i 0.541755 + 0.840537i \(0.317760\pi\)
−0.541755 + 0.840537i \(0.682240\pi\)
\(308\) 0 0
\(309\) 33.8816 1.92746
\(310\) 0 0
\(311\) 1.68743 0.0956851 0.0478426 0.998855i \(-0.484765\pi\)
0.0478426 + 0.998855i \(0.484765\pi\)
\(312\) 0 0
\(313\) 14.3653 0.811974 0.405987 0.913879i \(-0.366928\pi\)
0.405987 + 0.913879i \(0.366928\pi\)
\(314\) 0 0
\(315\) 4.91601 0.276986
\(316\) 0 0
\(317\) 24.0176i 1.34896i −0.738293 0.674480i \(-0.764368\pi\)
0.738293 0.674480i \(-0.235632\pi\)
\(318\) 0 0
\(319\) 1.88226i 0.105387i
\(320\) 0 0
\(321\) −39.4563 −2.20223
\(322\) 0 0
\(323\) 6.56336i 0.365195i
\(324\) 0 0
\(325\) 22.9291 4.76453i 1.27188 0.264289i
\(326\) 0 0
\(327\) 27.3931i 1.51484i
\(328\) 0 0
\(329\) −2.26952 −0.125123
\(330\) 0 0
\(331\) 26.4829i 1.45563i −0.685772 0.727816i \(-0.740535\pi\)
0.685772 0.727816i \(-0.259465\pi\)
\(332\) 0 0
\(333\) 7.75775i 0.425122i
\(334\) 0 0
\(335\) −22.6179 −1.23575
\(336\) 0 0
\(337\) 10.7375 0.584907 0.292453 0.956280i \(-0.405528\pi\)
0.292453 + 0.956280i \(0.405528\pi\)
\(338\) 0 0
\(339\) −4.09737 −0.222539
\(340\) 0 0
\(341\) 5.69756 0.308540
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 45.6947i 2.46012i
\(346\) 0 0
\(347\) 5.38449 0.289055 0.144527 0.989501i \(-0.453834\pi\)
0.144527 + 0.989501i \(0.453834\pi\)
\(348\) 0 0
\(349\) 24.2633i 1.29879i 0.760453 + 0.649393i \(0.224977\pi\)
−0.760453 + 0.649393i \(0.775023\pi\)
\(350\) 0 0
\(351\) −11.5429 + 2.39855i −0.616115 + 0.128025i
\(352\) 0 0
\(353\) 9.61034i 0.511507i 0.966742 + 0.255753i \(0.0823235\pi\)
−0.966742 + 0.255753i \(0.917676\pi\)
\(354\) 0 0
\(355\) 47.1562 2.50279
\(356\) 0 0
\(357\) 12.7776i 0.676262i
\(358\) 0 0
\(359\) 12.6139i 0.665736i 0.942973 + 0.332868i \(0.108016\pi\)
−0.942973 + 0.332868i \(0.891984\pi\)
\(360\) 0 0
\(361\) 17.8259 0.938205
\(362\) 0 0
\(363\) 2.10949 0.110720
\(364\) 0 0
\(365\) −28.1112 −1.47141
\(366\) 0 0
\(367\) 30.7005 1.60255 0.801276 0.598294i \(-0.204155\pi\)
0.801276 + 0.598294i \(0.204155\pi\)
\(368\) 0 0
\(369\) 2.46123i 0.128126i
\(370\) 0 0
\(371\) 12.6554i 0.657035i
\(372\) 0 0
\(373\) −12.9676 −0.671435 −0.335717 0.941963i \(-0.608979\pi\)
−0.335717 + 0.941963i \(0.608979\pi\)
\(374\) 0 0
\(375\) 10.6941i 0.552242i
\(376\) 0 0
\(377\) 6.64466 1.38072i 0.342218 0.0711108i
\(378\) 0 0
\(379\) 21.9438i 1.12718i −0.826056 0.563588i \(-0.809421\pi\)
0.826056 0.563588i \(-0.190579\pi\)
\(380\) 0 0
\(381\) −42.6859 −2.18687
\(382\) 0 0
\(383\) 22.3287i 1.14094i 0.821317 + 0.570472i \(0.193240\pi\)
−0.821317 + 0.570472i \(0.806760\pi\)
\(384\) 0 0
\(385\) 3.39046i 0.172794i
\(386\) 0 0
\(387\) 11.8212 0.600904
\(388\) 0 0
\(389\) 10.7973 0.547445 0.273723 0.961809i \(-0.411745\pi\)
0.273723 + 0.961809i \(0.411745\pi\)
\(390\) 0 0
\(391\) −38.6991 −1.95710
\(392\) 0 0
\(393\) −32.0462 −1.61652
\(394\) 0 0
\(395\) 4.17602i 0.210118i
\(396\) 0 0
\(397\) 9.71532i 0.487598i −0.969826 0.243799i \(-0.921606\pi\)
0.969826 0.243799i \(-0.0783937\pi\)
\(398\) 0 0
\(399\) −2.28577 −0.114432
\(400\) 0 0
\(401\) 17.3367i 0.865754i 0.901453 + 0.432877i \(0.142501\pi\)
−0.901453 + 0.432877i \(0.857499\pi\)
\(402\) 0 0
\(403\) −4.17940 20.1132i −0.208191 1.00191i
\(404\) 0 0
\(405\) 38.1342i 1.89490i
\(406\) 0 0
\(407\) 5.35035 0.265207
\(408\) 0 0
\(409\) 4.64721i 0.229790i −0.993378 0.114895i \(-0.963347\pi\)
0.993378 0.114895i \(-0.0366532\pi\)
\(410\) 0 0
\(411\) 13.7392i 0.677703i
\(412\) 0 0
\(413\) −9.70115 −0.477362
\(414\) 0 0
\(415\) 37.2884 1.83041
\(416\) 0 0
\(417\) −19.9953 −0.979174
\(418\) 0 0
\(419\) 14.8602 0.725967 0.362983 0.931796i \(-0.381758\pi\)
0.362983 + 0.931796i \(0.381758\pi\)
\(420\) 0 0
\(421\) 13.4579i 0.655896i −0.944696 0.327948i \(-0.893643\pi\)
0.944696 0.327948i \(-0.106357\pi\)
\(422\) 0 0
\(423\) 3.29070i 0.159999i
\(424\) 0 0
\(425\) −39.3428 −1.90841
\(426\) 0 0
\(427\) 1.04222i 0.0504364i
\(428\) 0 0
\(429\) −1.54740 7.44681i −0.0747093 0.359535i
\(430\) 0 0
\(431\) 30.9138i 1.48906i 0.667587 + 0.744532i \(0.267327\pi\)
−0.667587 + 0.744532i \(0.732673\pi\)
\(432\) 0 0
\(433\) 3.08516 0.148263 0.0741316 0.997248i \(-0.476382\pi\)
0.0741316 + 0.997248i \(0.476382\pi\)
\(434\) 0 0
\(435\) 13.4622i 0.645465i
\(436\) 0 0
\(437\) 6.92285i 0.331165i
\(438\) 0 0
\(439\) 26.5908 1.26911 0.634555 0.772878i \(-0.281183\pi\)
0.634555 + 0.772878i \(0.281183\pi\)
\(440\) 0 0
\(441\) −1.44995 −0.0690453
\(442\) 0 0
\(443\) 13.8072 0.655999 0.328000 0.944678i \(-0.393625\pi\)
0.328000 + 0.944678i \(0.393625\pi\)
\(444\) 0 0
\(445\) 10.0396 0.475923
\(446\) 0 0
\(447\) 17.1795i 0.812560i
\(448\) 0 0
\(449\) 33.1115i 1.56263i 0.624138 + 0.781314i \(0.285450\pi\)
−0.624138 + 0.781314i \(0.714550\pi\)
\(450\) 0 0
\(451\) −1.69745 −0.0799300
\(452\) 0 0
\(453\) 22.3944i 1.05218i
\(454\) 0 0
\(455\) −11.9688 + 2.48705i −0.561107 + 0.116595i
\(456\) 0 0
\(457\) 29.7724i 1.39269i −0.717705 0.696347i \(-0.754807\pi\)
0.717705 0.696347i \(-0.245193\pi\)
\(458\) 0 0
\(459\) 19.8059 0.924459
\(460\) 0 0
\(461\) 19.8433i 0.924194i −0.886829 0.462097i \(-0.847097\pi\)
0.886829 0.462097i \(-0.152903\pi\)
\(462\) 0 0
\(463\) 5.00306i 0.232512i −0.993219 0.116256i \(-0.962911\pi\)
0.993219 0.116256i \(-0.0370893\pi\)
\(464\) 0 0
\(465\) −40.7498 −1.88973
\(466\) 0 0
\(467\) 0.849550 0.0393125 0.0196563 0.999807i \(-0.493743\pi\)
0.0196563 + 0.999807i \(0.493743\pi\)
\(468\) 0 0
\(469\) 6.67103 0.308040
\(470\) 0 0
\(471\) −30.8372 −1.42090
\(472\) 0 0
\(473\) 8.15280i 0.374866i
\(474\) 0 0
\(475\) 7.03801i 0.322926i
\(476\) 0 0
\(477\) −18.3497 −0.840176
\(478\) 0 0
\(479\) 4.54583i 0.207704i −0.994593 0.103852i \(-0.966883\pi\)
0.994593 0.103852i \(-0.0331168\pi\)
\(480\) 0 0
\(481\) −3.92471 18.8875i −0.178951 0.861196i
\(482\) 0 0
\(483\) 13.4774i 0.613245i
\(484\) 0 0
\(485\) −1.81609 −0.0824644
\(486\) 0 0
\(487\) 4.20647i 0.190613i −0.995448 0.0953066i \(-0.969617\pi\)
0.995448 0.0953066i \(-0.0303832\pi\)
\(488\) 0 0
\(489\) 40.7484i 1.84271i
\(490\) 0 0
\(491\) 2.33800 0.105513 0.0527563 0.998607i \(-0.483199\pi\)
0.0527563 + 0.998607i \(0.483199\pi\)
\(492\) 0 0
\(493\) −11.4012 −0.513486
\(494\) 0 0
\(495\) −4.91601 −0.220958
\(496\) 0 0
\(497\) −13.9085 −0.623880
\(498\) 0 0
\(499\) 2.93176i 0.131244i −0.997845 0.0656218i \(-0.979097\pi\)
0.997845 0.0656218i \(-0.0209031\pi\)
\(500\) 0 0
\(501\) 32.6075i 1.45680i
\(502\) 0 0
\(503\) −27.9960 −1.24828 −0.624140 0.781312i \(-0.714551\pi\)
−0.624140 + 0.781312i \(0.714551\pi\)
\(504\) 0 0
\(505\) 32.0822i 1.42764i
\(506\) 0 0
\(507\) −25.1532 + 10.9251i −1.11709 + 0.485201i
\(508\) 0 0
\(509\) 41.9840i 1.86091i −0.366411 0.930453i \(-0.619414\pi\)
0.366411 0.930453i \(-0.380586\pi\)
\(510\) 0 0
\(511\) 8.29126 0.366784
\(512\) 0 0
\(513\) 3.54305i 0.156430i
\(514\) 0 0
\(515\) 54.4560i 2.39962i
\(516\) 0 0
\(517\) 2.26952 0.0998134
\(518\) 0 0
\(519\) −26.8034 −1.17654
\(520\) 0 0
\(521\) −23.6736 −1.03716 −0.518581 0.855029i \(-0.673539\pi\)
−0.518581 + 0.855029i \(0.673539\pi\)
\(522\) 0 0
\(523\) 16.8305 0.735945 0.367973 0.929837i \(-0.380052\pi\)
0.367973 + 0.929837i \(0.380052\pi\)
\(524\) 0 0
\(525\) 13.7016i 0.597988i
\(526\) 0 0
\(527\) 34.5112i 1.50333i
\(528\) 0 0
\(529\) 17.8187 0.774727
\(530\) 0 0
\(531\) 14.0662i 0.610421i
\(532\) 0 0
\(533\) 1.24516 + 5.99226i 0.0539337 + 0.259554i
\(534\) 0 0
\(535\) 63.4158i 2.74170i
\(536\) 0 0
\(537\) −22.9760 −0.991488
\(538\) 0 0
\(539\) 1.00000i 0.0430730i
\(540\) 0 0
\(541\) 10.8116i 0.464826i −0.972617 0.232413i \(-0.925338\pi\)
0.972617 0.232413i \(-0.0746621\pi\)
\(542\) 0 0
\(543\) −26.4472 −1.13496
\(544\) 0 0
\(545\) −44.0273 −1.88592
\(546\) 0 0
\(547\) −17.0580 −0.729349 −0.364675 0.931135i \(-0.618820\pi\)
−0.364675 + 0.931135i \(0.618820\pi\)
\(548\) 0 0
\(549\) 1.51116 0.0644949
\(550\) 0 0
\(551\) 2.03956i 0.0868880i
\(552\) 0 0
\(553\) 1.23170i 0.0523770i
\(554\) 0 0
\(555\) −38.2665 −1.62432
\(556\) 0 0
\(557\) 42.5401i 1.80248i 0.433319 + 0.901241i \(0.357342\pi\)
−0.433319 + 0.901241i \(0.642658\pi\)
\(558\) 0 0
\(559\) −28.7805 + 5.98043i −1.21729 + 0.252945i
\(560\) 0 0
\(561\) 12.7776i 0.539470i
\(562\) 0 0
\(563\) 8.44992 0.356122 0.178061 0.984019i \(-0.443018\pi\)
0.178061 + 0.984019i \(0.443018\pi\)
\(564\) 0 0
\(565\) 6.58546i 0.277053i
\(566\) 0 0
\(567\) 11.2475i 0.472350i
\(568\) 0 0
\(569\) −33.5192 −1.40520 −0.702599 0.711586i \(-0.747977\pi\)
−0.702599 + 0.711586i \(0.747977\pi\)
\(570\) 0 0
\(571\) −20.8664 −0.873233 −0.436617 0.899648i \(-0.643823\pi\)
−0.436617 + 0.899648i \(0.643823\pi\)
\(572\) 0 0
\(573\) 32.5977 1.36179
\(574\) 0 0
\(575\) 41.4977 1.73058
\(576\) 0 0
\(577\) 3.23241i 0.134567i −0.997734 0.0672836i \(-0.978567\pi\)
0.997734 0.0672836i \(-0.0214332\pi\)
\(578\) 0 0
\(579\) 43.4676i 1.80645i
\(580\) 0 0
\(581\) −10.9980 −0.456275
\(582\) 0 0
\(583\) 12.6554i 0.524133i
\(584\) 0 0
\(585\) 3.60610 + 17.3542i 0.149094 + 0.717508i
\(586\) 0 0
\(587\) 31.4601i 1.29850i −0.760575 0.649250i \(-0.775083\pi\)
0.760575 0.649250i \(-0.224917\pi\)
\(588\) 0 0
\(589\) 6.17367 0.254382
\(590\) 0 0
\(591\) 3.81430i 0.156899i
\(592\) 0 0
\(593\) 5.56692i 0.228606i −0.993446 0.114303i \(-0.963537\pi\)
0.993446 0.114303i \(-0.0364635\pi\)
\(594\) 0 0
\(595\) 20.5367 0.841921
\(596\) 0 0
\(597\) 31.5305 1.29046
\(598\) 0 0
\(599\) −29.8832 −1.22099 −0.610497 0.792018i \(-0.709030\pi\)
−0.610497 + 0.792018i \(0.709030\pi\)
\(600\) 0 0
\(601\) −10.2407 −0.417727 −0.208864 0.977945i \(-0.566977\pi\)
−0.208864 + 0.977945i \(0.566977\pi\)
\(602\) 0 0
\(603\) 9.67267i 0.393902i
\(604\) 0 0
\(605\) 3.39046i 0.137842i
\(606\) 0 0
\(607\) −4.65268 −0.188846 −0.0944232 0.995532i \(-0.530101\pi\)
−0.0944232 + 0.995532i \(0.530101\pi\)
\(608\) 0 0
\(609\) 3.97062i 0.160898i
\(610\) 0 0
\(611\) −1.66479 8.01174i −0.0673502 0.324120i
\(612\) 0 0
\(613\) 5.25496i 0.212246i 0.994353 + 0.106123i \(0.0338437\pi\)
−0.994353 + 0.106123i \(0.966156\pi\)
\(614\) 0 0
\(615\) 12.1404 0.489550
\(616\) 0 0
\(617\) 19.6452i 0.790884i −0.918491 0.395442i \(-0.870591\pi\)
0.918491 0.395442i \(-0.129409\pi\)
\(618\) 0 0
\(619\) 15.5124i 0.623497i 0.950165 + 0.311748i \(0.100915\pi\)
−0.950165 + 0.311748i \(0.899085\pi\)
\(620\) 0 0
\(621\) −20.8907 −0.838314
\(622\) 0 0
\(623\) −2.96113 −0.118635
\(624\) 0 0
\(625\) −15.2881 −0.611525
\(626\) 0 0
\(627\) 2.28577 0.0912849
\(628\) 0 0
\(629\) 32.4081i 1.29219i
\(630\) 0 0
\(631\) 37.7412i 1.50246i 0.660043 + 0.751228i \(0.270538\pi\)
−0.660043 + 0.751228i \(0.729462\pi\)
\(632\) 0 0
\(633\) 13.2302 0.525851
\(634\) 0 0
\(635\) 68.6066i 2.72257i
\(636\) 0 0
\(637\) 3.53014 0.733543i 0.139869 0.0290640i
\(638\) 0 0
\(639\) 20.1666i 0.797779i
\(640\) 0 0
\(641\) −12.2744 −0.484811 −0.242406 0.970175i \(-0.577936\pi\)
−0.242406 + 0.970175i \(0.577936\pi\)
\(642\) 0 0
\(643\) 39.5059i 1.55796i 0.627048 + 0.778980i \(0.284263\pi\)
−0.627048 + 0.778980i \(0.715737\pi\)
\(644\) 0 0
\(645\) 58.3100i 2.29595i
\(646\) 0 0
\(647\) 11.5680 0.454786 0.227393 0.973803i \(-0.426980\pi\)
0.227393 + 0.973803i \(0.426980\pi\)
\(648\) 0 0
\(649\) 9.70115 0.380803
\(650\) 0 0
\(651\) 12.0189 0.471060
\(652\) 0 0
\(653\) 20.4016 0.798375 0.399188 0.916869i \(-0.369292\pi\)
0.399188 + 0.916869i \(0.369292\pi\)
\(654\) 0 0
\(655\) 51.5059i 2.01250i
\(656\) 0 0
\(657\) 12.0219i 0.469020i
\(658\) 0 0
\(659\) −36.1249 −1.40723 −0.703614 0.710582i \(-0.748432\pi\)
−0.703614 + 0.710582i \(0.748432\pi\)
\(660\) 0 0
\(661\) 31.8050i 1.23707i 0.785757 + 0.618535i \(0.212274\pi\)
−0.785757 + 0.618535i \(0.787726\pi\)
\(662\) 0 0
\(663\) 45.1067 9.37290i 1.75180 0.364013i
\(664\) 0 0
\(665\) 3.67379i 0.142463i
\(666\) 0 0
\(667\) 12.0257 0.465637
\(668\) 0 0
\(669\) 50.0250i 1.93408i
\(670\) 0 0
\(671\) 1.04222i 0.0402343i
\(672\) 0 0
\(673\) −38.5632 −1.48650 −0.743251 0.669013i \(-0.766717\pi\)
−0.743251 + 0.669013i \(0.766717\pi\)
\(674\) 0 0
\(675\) −21.2382 −0.817458
\(676\) 0 0
\(677\) 31.8742 1.22503 0.612513 0.790460i \(-0.290159\pi\)
0.612513 + 0.790460i \(0.290159\pi\)
\(678\) 0 0
\(679\) 0.535647 0.0205562
\(680\) 0 0
\(681\) 57.7090i 2.21142i
\(682\) 0 0
\(683\) 15.9904i 0.611856i −0.952055 0.305928i \(-0.901033\pi\)
0.952055 0.305928i \(-0.0989666\pi\)
\(684\) 0 0
\(685\) 22.0821 0.843715
\(686\) 0 0
\(687\) 6.26664i 0.239087i
\(688\) 0 0
\(689\) 44.6754 9.28327i 1.70200 0.353664i
\(690\) 0 0
\(691\) 31.6269i 1.20314i 0.798819 + 0.601572i \(0.205459\pi\)
−0.798819 + 0.601572i \(0.794541\pi\)
\(692\) 0 0
\(693\) 1.44995 0.0550791
\(694\) 0 0
\(695\) 32.1373i 1.21904i
\(696\) 0 0
\(697\) 10.2818i 0.389451i
\(698\) 0 0
\(699\) 3.10965 0.117618
\(700\) 0 0
\(701\) −23.8564 −0.901044 −0.450522 0.892765i \(-0.648762\pi\)
−0.450522 + 0.892765i \(0.648762\pi\)
\(702\) 0 0
\(703\) 5.79745 0.218655
\(704\) 0 0
\(705\) −16.2320 −0.611331
\(706\) 0 0
\(707\) 9.46249i 0.355873i
\(708\) 0 0
\(709\) 40.2315i 1.51093i 0.655192 + 0.755463i \(0.272588\pi\)
−0.655192 + 0.755463i \(0.727412\pi\)
\(710\) 0 0
\(711\) −1.78590 −0.0669765
\(712\) 0 0
\(713\) 36.4014i 1.36324i
\(714\) 0 0
\(715\) 11.9688 2.48705i 0.447608 0.0930103i
\(716\) 0 0
\(717\) 24.7429i 0.924039i
\(718\) 0 0
\(719\) −37.9884 −1.41673 −0.708363 0.705848i \(-0.750566\pi\)
−0.708363 + 0.705848i \(0.750566\pi\)
\(720\) 0 0
\(721\) 16.0615i 0.598162i
\(722\) 0 0
\(723\) 2.34682i 0.0872792i
\(724\) 0 0
\(725\) 12.2257 0.454053
\(726\) 0 0
\(727\) 8.83158 0.327545 0.163773 0.986498i \(-0.447634\pi\)
0.163773 + 0.986498i \(0.447634\pi\)
\(728\) 0 0
\(729\) 4.38458 0.162392
\(730\) 0 0
\(731\) 49.3830 1.82650
\(732\) 0 0
\(733\) 41.0304i 1.51549i −0.652550 0.757746i \(-0.726301\pi\)
0.652550 0.757746i \(-0.273699\pi\)
\(734\) 0 0
\(735\) 7.15215i 0.263811i
\(736\) 0 0
\(737\) −6.67103 −0.245731
\(738\) 0 0
\(739\) 2.90919i 0.107016i 0.998567 + 0.0535081i \(0.0170403\pi\)
−0.998567 + 0.0535081i \(0.982960\pi\)
\(740\) 0 0
\(741\) −1.67671 8.06910i −0.0615955 0.296426i
\(742\) 0 0
\(743\) 31.5498i 1.15745i −0.815523 0.578724i \(-0.803551\pi\)
0.815523 0.578724i \(-0.196449\pi\)
\(744\) 0 0
\(745\) −27.6115 −1.01161
\(746\) 0 0
\(747\) 15.9466i 0.583456i
\(748\) 0 0
\(749\) 18.7042i 0.683435i
\(750\) 0 0
\(751\) −47.2106 −1.72274 −0.861369 0.507979i \(-0.830393\pi\)
−0.861369 + 0.507979i \(0.830393\pi\)
\(752\) 0 0
\(753\) −61.9863 −2.25891
\(754\) 0 0
\(755\) −35.9932 −1.30993
\(756\) 0 0
\(757\) 44.3429 1.61167 0.805835 0.592140i \(-0.201717\pi\)
0.805835 + 0.592140i \(0.201717\pi\)
\(758\) 0 0
\(759\) 13.4774i 0.489200i
\(760\) 0 0
\(761\) 38.2642i 1.38708i 0.720420 + 0.693538i \(0.243949\pi\)
−0.720420 + 0.693538i \(0.756051\pi\)
\(762\) 0 0
\(763\) 12.9856 0.470112
\(764\) 0 0
\(765\) 29.7772i 1.07660i
\(766\) 0 0
\(767\) −7.11621 34.2465i −0.256951 1.23657i
\(768\) 0 0
\(769\) 37.0112i 1.33466i 0.744763 + 0.667329i \(0.232563\pi\)
−0.744763 + 0.667329i \(0.767437\pi\)
\(770\) 0 0
\(771\) 6.78423 0.244328
\(772\) 0 0
\(773\) 26.1438i 0.940325i −0.882580 0.470163i \(-0.844195\pi\)
0.882580 0.470163i \(-0.155805\pi\)
\(774\) 0 0
\(775\) 37.0069i 1.32933i
\(776\) 0 0
\(777\) 11.2865 0.404902
\(778\) 0 0
\(779\) −1.83930 −0.0658999
\(780\) 0 0
\(781\) 13.9085 0.497684
\(782\) 0 0
\(783\) −6.15465 −0.219949
\(784\) 0 0
\(785\) 49.5629i 1.76897i
\(786\) 0 0
\(787\) 28.6164i 1.02006i −0.860155 0.510032i \(-0.829634\pi\)
0.860155 0.510032i \(-0.170366\pi\)
\(788\) 0 0
\(789\) −37.8942 −1.34907
\(790\) 0 0
\(791\) 1.94235i 0.0690620i
\(792\) 0 0
\(793\) −3.67918 + 0.764511i −0.130651 + 0.0271486i
\(794\) 0