Properties

Label 4004.2.m.c.2157.4
Level 4004
Weight 2
Character 4004.2157
Analytic conductor 31.972
Analytic rank 0
Dimension 36
CM No

Related objects

Downloads

Learn more about

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 2157.4
Character \(\chi\) = 4004.2157
Dual form 4004.2.m.c.2157.3

$q$-expansion

\(f(q)\) \(=\) \(q\)\(-3.10315 q^{3}\) \(-1.71558i q^{5}\) \(-1.00000i q^{7}\) \(+6.62954 q^{9}\) \(+O(q^{10})\) \(q\)\(-3.10315 q^{3}\) \(-1.71558i q^{5}\) \(-1.00000i q^{7}\) \(+6.62954 q^{9}\) \(+1.00000i q^{11}\) \(+(1.66202 + 3.19964i) q^{13}\) \(+5.32371i q^{15}\) \(+5.08192 q^{17}\) \(+7.32222i q^{19}\) \(+3.10315i q^{21}\) \(-3.22025 q^{23}\) \(+2.05677 q^{25}\) \(-11.2630 q^{27}\) \(-9.04384 q^{29}\) \(+4.98361i q^{31}\) \(-3.10315i q^{33}\) \(-1.71558 q^{35}\) \(-1.36062i q^{37}\) \(+(-5.15750 - 9.92896i) q^{39}\) \(-12.1147i q^{41}\) \(+8.57894 q^{43}\) \(-11.3735i q^{45}\) \(-0.265550i q^{47}\) \(-1.00000 q^{49}\) \(-15.7700 q^{51}\) \(-9.73208 q^{53}\) \(+1.71558 q^{55}\) \(-22.7219i q^{57}\) \(-0.610665i q^{59}\) \(-1.47181 q^{61}\) \(-6.62954i q^{63}\) \(+(5.48925 - 2.85134i) q^{65}\) \(+12.4920i q^{67}\) \(+9.99293 q^{69}\) \(-1.07609i q^{71}\) \(+7.37610i q^{73}\) \(-6.38248 q^{75}\) \(+1.00000 q^{77}\) \(-10.6681 q^{79}\) \(+15.0622 q^{81}\) \(-1.00686i q^{83}\) \(-8.71846i q^{85}\) \(+28.0644 q^{87}\) \(-2.76495i q^{89}\) \(+(3.19964 - 1.66202i) q^{91}\) \(-15.4649i q^{93}\) \(+12.5619 q^{95}\) \(-1.77103i q^{97}\) \(+6.62954i q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 80q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut -\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 80q^{69} \) \(\mathstrut -\mathstrut 36q^{75} \) \(\mathstrut +\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 132q^{81} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
\(3\) −3.10315 −1.79160 −0.895802 0.444453i \(-0.853398\pi\)
−0.895802 + 0.444453i \(0.853398\pi\)
\(4\) 0 0
\(5\) 1.71558i 0.767232i −0.923493 0.383616i \(-0.874679\pi\)
0.923493 0.383616i \(-0.125321\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 6.62954 2.20985
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) 1.66202 + 3.19964i 0.460962 + 0.887420i
\(14\) 0 0
\(15\) 5.32371i 1.37458i
\(16\) 0 0
\(17\) 5.08192 1.23255 0.616273 0.787532i \(-0.288642\pi\)
0.616273 + 0.787532i \(0.288642\pi\)
\(18\) 0 0
\(19\) 7.32222i 1.67983i 0.542717 + 0.839916i \(0.317396\pi\)
−0.542717 + 0.839916i \(0.682604\pi\)
\(20\) 0 0
\(21\) 3.10315i 0.677163i
\(22\) 0 0
\(23\) −3.22025 −0.671469 −0.335735 0.941957i \(-0.608985\pi\)
−0.335735 + 0.941957i \(0.608985\pi\)
\(24\) 0 0
\(25\) 2.05677 0.411355
\(26\) 0 0
\(27\) −11.2630 −2.16757
\(28\) 0 0
\(29\) −9.04384 −1.67940 −0.839700 0.543051i \(-0.817269\pi\)
−0.839700 + 0.543051i \(0.817269\pi\)
\(30\) 0 0
\(31\) 4.98361i 0.895082i 0.894263 + 0.447541i \(0.147700\pi\)
−0.894263 + 0.447541i \(0.852300\pi\)
\(32\) 0 0
\(33\) 3.10315i 0.540189i
\(34\) 0 0
\(35\) −1.71558 −0.289986
\(36\) 0 0
\(37\) 1.36062i 0.223685i −0.993726 0.111843i \(-0.964325\pi\)
0.993726 0.111843i \(-0.0356753\pi\)
\(38\) 0 0
\(39\) −5.15750 9.92896i −0.825861 1.58991i
\(40\) 0 0
\(41\) 12.1147i 1.89199i −0.324174 0.945997i \(-0.605086\pi\)
0.324174 0.945997i \(-0.394914\pi\)
\(42\) 0 0
\(43\) 8.57894 1.30828 0.654138 0.756375i \(-0.273032\pi\)
0.654138 + 0.756375i \(0.273032\pi\)
\(44\) 0 0
\(45\) 11.3735i 1.69547i
\(46\) 0 0
\(47\) 0.265550i 0.0387344i −0.999812 0.0193672i \(-0.993835\pi\)
0.999812 0.0193672i \(-0.00616516\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −15.7700 −2.20824
\(52\) 0 0
\(53\) −9.73208 −1.33680 −0.668402 0.743800i \(-0.733022\pi\)
−0.668402 + 0.743800i \(0.733022\pi\)
\(54\) 0 0
\(55\) 1.71558 0.231329
\(56\) 0 0
\(57\) 22.7219i 3.00959i
\(58\) 0 0
\(59\) 0.610665i 0.0795018i −0.999210 0.0397509i \(-0.987344\pi\)
0.999210 0.0397509i \(-0.0126564\pi\)
\(60\) 0 0
\(61\) −1.47181 −0.188446 −0.0942229 0.995551i \(-0.530037\pi\)
−0.0942229 + 0.995551i \(0.530037\pi\)
\(62\) 0 0
\(63\) 6.62954i 0.835243i
\(64\) 0 0
\(65\) 5.48925 2.85134i 0.680857 0.353665i
\(66\) 0 0
\(67\) 12.4920i 1.52614i 0.646313 + 0.763072i \(0.276310\pi\)
−0.646313 + 0.763072i \(0.723690\pi\)
\(68\) 0 0
\(69\) 9.99293 1.20301
\(70\) 0 0
\(71\) 1.07609i 0.127708i −0.997959 0.0638541i \(-0.979661\pi\)
0.997959 0.0638541i \(-0.0203392\pi\)
\(72\) 0 0
\(73\) 7.37610i 0.863307i 0.902039 + 0.431654i \(0.142070\pi\)
−0.902039 + 0.431654i \(0.857930\pi\)
\(74\) 0 0
\(75\) −6.38248 −0.736985
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −10.6681 −1.20026 −0.600130 0.799902i \(-0.704885\pi\)
−0.600130 + 0.799902i \(0.704885\pi\)
\(80\) 0 0
\(81\) 15.0622 1.67357
\(82\) 0 0
\(83\) 1.00686i 0.110517i −0.998472 0.0552584i \(-0.982402\pi\)
0.998472 0.0552584i \(-0.0175983\pi\)
\(84\) 0 0
\(85\) 8.71846i 0.945649i
\(86\) 0 0
\(87\) 28.0644 3.00882
\(88\) 0 0
\(89\) 2.76495i 0.293084i −0.989204 0.146542i \(-0.953186\pi\)
0.989204 0.146542i \(-0.0468144\pi\)
\(90\) 0 0
\(91\) 3.19964 1.66202i 0.335413 0.174227i
\(92\) 0 0
\(93\) 15.4649i 1.60363i
\(94\) 0 0
\(95\) 12.5619 1.28882
\(96\) 0 0
\(97\) 1.77103i 0.179821i −0.995950 0.0899107i \(-0.971342\pi\)
0.995950 0.0899107i \(-0.0286582\pi\)
\(98\) 0 0
\(99\) 6.62954i 0.666294i
\(100\) 0 0
\(101\) 16.5672 1.64850 0.824250 0.566226i \(-0.191597\pi\)
0.824250 + 0.566226i \(0.191597\pi\)
\(102\) 0 0
\(103\) 12.8945 1.27053 0.635266 0.772294i \(-0.280891\pi\)
0.635266 + 0.772294i \(0.280891\pi\)
\(104\) 0 0
\(105\) 5.32371 0.519541
\(106\) 0 0
\(107\) −15.5121 −1.49961 −0.749806 0.661658i \(-0.769853\pi\)
−0.749806 + 0.661658i \(0.769853\pi\)
\(108\) 0 0
\(109\) 2.76650i 0.264982i −0.991184 0.132491i \(-0.957702\pi\)
0.991184 0.132491i \(-0.0422976\pi\)
\(110\) 0 0
\(111\) 4.22222i 0.400756i
\(112\) 0 0
\(113\) −7.63951 −0.718665 −0.359332 0.933210i \(-0.616996\pi\)
−0.359332 + 0.933210i \(0.616996\pi\)
\(114\) 0 0
\(115\) 5.52461i 0.515173i
\(116\) 0 0
\(117\) 11.0184 + 21.2121i 1.01865 + 1.96106i
\(118\) 0 0
\(119\) 5.08192i 0.465859i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 37.5937i 3.38971i
\(124\) 0 0
\(125\) 12.1065i 1.08284i
\(126\) 0 0
\(127\) −20.4565 −1.81522 −0.907612 0.419809i \(-0.862097\pi\)
−0.907612 + 0.419809i \(0.862097\pi\)
\(128\) 0 0
\(129\) −26.6217 −2.34391
\(130\) 0 0
\(131\) −15.1564 −1.32422 −0.662109 0.749408i \(-0.730338\pi\)
−0.662109 + 0.749408i \(0.730338\pi\)
\(132\) 0 0
\(133\) 7.32222 0.634917
\(134\) 0 0
\(135\) 19.3226i 1.66303i
\(136\) 0 0
\(137\) 7.89611i 0.674610i 0.941395 + 0.337305i \(0.109515\pi\)
−0.941395 + 0.337305i \(0.890485\pi\)
\(138\) 0 0
\(139\) −9.11731 −0.773320 −0.386660 0.922222i \(-0.626371\pi\)
−0.386660 + 0.922222i \(0.626371\pi\)
\(140\) 0 0
\(141\) 0.824040i 0.0693967i
\(142\) 0 0
\(143\) −3.19964 + 1.66202i −0.267567 + 0.138985i
\(144\) 0 0
\(145\) 15.5155i 1.28849i
\(146\) 0 0
\(147\) 3.10315 0.255943
\(148\) 0 0
\(149\) 8.08924i 0.662697i 0.943509 + 0.331348i \(0.107504\pi\)
−0.943509 + 0.331348i \(0.892496\pi\)
\(150\) 0 0
\(151\) 20.1840i 1.64255i −0.570535 0.821273i \(-0.693264\pi\)
0.570535 0.821273i \(-0.306736\pi\)
\(152\) 0 0
\(153\) 33.6908 2.72374
\(154\) 0 0
\(155\) 8.54979 0.686736
\(156\) 0 0
\(157\) −14.6475 −1.16900 −0.584498 0.811395i \(-0.698709\pi\)
−0.584498 + 0.811395i \(0.698709\pi\)
\(158\) 0 0
\(159\) 30.2001 2.39502
\(160\) 0 0
\(161\) 3.22025i 0.253791i
\(162\) 0 0
\(163\) 12.8860i 1.00931i −0.863322 0.504653i \(-0.831620\pi\)
0.863322 0.504653i \(-0.168380\pi\)
\(164\) 0 0
\(165\) −5.32371 −0.414450
\(166\) 0 0
\(167\) 17.2959i 1.33840i 0.743083 + 0.669200i \(0.233363\pi\)
−0.743083 + 0.669200i \(0.766637\pi\)
\(168\) 0 0
\(169\) −7.47537 + 10.6357i −0.575029 + 0.818133i
\(170\) 0 0
\(171\) 48.5429i 3.71217i
\(172\) 0 0
\(173\) −12.4223 −0.944450 −0.472225 0.881478i \(-0.656549\pi\)
−0.472225 + 0.881478i \(0.656549\pi\)
\(174\) 0 0
\(175\) 2.05677i 0.155478i
\(176\) 0 0
\(177\) 1.89498i 0.142436i
\(178\) 0 0
\(179\) −11.1096 −0.830370 −0.415185 0.909737i \(-0.636283\pi\)
−0.415185 + 0.909737i \(0.636283\pi\)
\(180\) 0 0
\(181\) 18.5362 1.37779 0.688894 0.724862i \(-0.258097\pi\)
0.688894 + 0.724862i \(0.258097\pi\)
\(182\) 0 0
\(183\) 4.56724 0.337620
\(184\) 0 0
\(185\) −2.33426 −0.171619
\(186\) 0 0
\(187\) 5.08192i 0.371627i
\(188\) 0 0
\(189\) 11.2630i 0.819263i
\(190\) 0 0
\(191\) −18.5932 −1.34535 −0.672677 0.739936i \(-0.734856\pi\)
−0.672677 + 0.739936i \(0.734856\pi\)
\(192\) 0 0
\(193\) 3.28669i 0.236581i 0.992979 + 0.118291i \(0.0377415\pi\)
−0.992979 + 0.118291i \(0.962259\pi\)
\(194\) 0 0
\(195\) −17.0340 + 8.84812i −1.21983 + 0.633627i
\(196\) 0 0
\(197\) 17.4262i 1.24156i 0.783984 + 0.620781i \(0.213185\pi\)
−0.783984 + 0.620781i \(0.786815\pi\)
\(198\) 0 0
\(199\) 6.64221 0.470854 0.235427 0.971892i \(-0.424351\pi\)
0.235427 + 0.971892i \(0.424351\pi\)
\(200\) 0 0
\(201\) 38.7646i 2.73425i
\(202\) 0 0
\(203\) 9.04384i 0.634753i
\(204\) 0 0
\(205\) −20.7837 −1.45160
\(206\) 0 0
\(207\) −21.3488 −1.48384
\(208\) 0 0
\(209\) −7.32222 −0.506488
\(210\) 0 0
\(211\) 6.13924 0.422642 0.211321 0.977417i \(-0.432223\pi\)
0.211321 + 0.977417i \(0.432223\pi\)
\(212\) 0 0
\(213\) 3.33926i 0.228803i
\(214\) 0 0
\(215\) 14.7179i 1.00375i
\(216\) 0 0
\(217\) 4.98361 0.338309
\(218\) 0 0
\(219\) 22.8891i 1.54670i
\(220\) 0 0
\(221\) 8.44626 + 16.2603i 0.568157 + 1.09379i
\(222\) 0 0
\(223\) 8.90531i 0.596344i 0.954512 + 0.298172i \(0.0963769\pi\)
−0.954512 + 0.298172i \(0.903623\pi\)
\(224\) 0 0
\(225\) 13.6355 0.909031
\(226\) 0 0
\(227\) 14.1194i 0.937138i 0.883427 + 0.468569i \(0.155230\pi\)
−0.883427 + 0.468569i \(0.844770\pi\)
\(228\) 0 0
\(229\) 2.92634i 0.193378i −0.995315 0.0966890i \(-0.969175\pi\)
0.995315 0.0966890i \(-0.0308252\pi\)
\(230\) 0 0
\(231\) −3.10315 −0.204172
\(232\) 0 0
\(233\) −9.49861 −0.622275 −0.311137 0.950365i \(-0.600710\pi\)
−0.311137 + 0.950365i \(0.600710\pi\)
\(234\) 0 0
\(235\) −0.455573 −0.0297183
\(236\) 0 0
\(237\) 33.1049 2.15039
\(238\) 0 0
\(239\) 16.9333i 1.09533i −0.836699 0.547663i \(-0.815518\pi\)
0.836699 0.547663i \(-0.184482\pi\)
\(240\) 0 0
\(241\) 5.70871i 0.367730i −0.982951 0.183865i \(-0.941139\pi\)
0.982951 0.183865i \(-0.0588609\pi\)
\(242\) 0 0
\(243\) −12.9512 −0.830818
\(244\) 0 0
\(245\) 1.71558i 0.109605i
\(246\) 0 0
\(247\) −23.4284 + 12.1697i −1.49072 + 0.774338i
\(248\) 0 0
\(249\) 3.12443i 0.198002i
\(250\) 0 0
\(251\) 17.0766 1.07787 0.538934 0.842348i \(-0.318827\pi\)
0.538934 + 0.842348i \(0.318827\pi\)
\(252\) 0 0
\(253\) 3.22025i 0.202456i
\(254\) 0 0
\(255\) 27.0547i 1.69423i
\(256\) 0 0
\(257\) −17.6369 −1.10016 −0.550080 0.835112i \(-0.685403\pi\)
−0.550080 + 0.835112i \(0.685403\pi\)
\(258\) 0 0
\(259\) −1.36062 −0.0845451
\(260\) 0 0
\(261\) −59.9565 −3.71121
\(262\) 0 0
\(263\) 22.4613 1.38502 0.692510 0.721408i \(-0.256505\pi\)
0.692510 + 0.721408i \(0.256505\pi\)
\(264\) 0 0
\(265\) 16.6962i 1.02564i
\(266\) 0 0
\(267\) 8.58006i 0.525091i
\(268\) 0 0
\(269\) 6.52344 0.397741 0.198871 0.980026i \(-0.436273\pi\)
0.198871 + 0.980026i \(0.436273\pi\)
\(270\) 0 0
\(271\) 2.68382i 0.163030i −0.996672 0.0815151i \(-0.974024\pi\)
0.996672 0.0815151i \(-0.0259759\pi\)
\(272\) 0 0
\(273\) −9.92896 + 5.15750i −0.600928 + 0.312146i
\(274\) 0 0
\(275\) 2.05677i 0.124028i
\(276\) 0 0
\(277\) 17.1046 1.02771 0.513856 0.857876i \(-0.328216\pi\)
0.513856 + 0.857876i \(0.328216\pi\)
\(278\) 0 0
\(279\) 33.0390i 1.97799i
\(280\) 0 0
\(281\) 32.5978i 1.94462i 0.233696 + 0.972310i \(0.424918\pi\)
−0.233696 + 0.972310i \(0.575082\pi\)
\(282\) 0 0
\(283\) −1.68236 −0.100006 −0.0500030 0.998749i \(-0.515923\pi\)
−0.0500030 + 0.998749i \(0.515923\pi\)
\(284\) 0 0
\(285\) −38.9814 −2.30906
\(286\) 0 0
\(287\) −12.1147 −0.715107
\(288\) 0 0
\(289\) 8.82590 0.519171
\(290\) 0 0
\(291\) 5.49579i 0.322169i
\(292\) 0 0
\(293\) 10.9268i 0.638353i 0.947695 + 0.319176i \(0.103406\pi\)
−0.947695 + 0.319176i \(0.896594\pi\)
\(294\) 0 0
\(295\) −1.04765 −0.0609963
\(296\) 0 0
\(297\) 11.2630i 0.653546i
\(298\) 0 0
\(299\) −5.35213 10.3036i −0.309522 0.595875i
\(300\) 0 0
\(301\) 8.57894i 0.494482i
\(302\) 0 0
\(303\) −51.4106 −2.95346
\(304\) 0 0
\(305\) 2.52501i 0.144582i
\(306\) 0 0
\(307\) 28.4746i 1.62513i 0.582872 + 0.812564i \(0.301929\pi\)
−0.582872 + 0.812564i \(0.698071\pi\)
\(308\) 0 0
\(309\) −40.0135 −2.27629
\(310\) 0 0
\(311\) −4.49257 −0.254750 −0.127375 0.991855i \(-0.540655\pi\)
−0.127375 + 0.991855i \(0.540655\pi\)
\(312\) 0 0
\(313\) −0.176055 −0.00995119 −0.00497560 0.999988i \(-0.501584\pi\)
−0.00497560 + 0.999988i \(0.501584\pi\)
\(314\) 0 0
\(315\) −11.3735 −0.640826
\(316\) 0 0
\(317\) 5.56199i 0.312392i 0.987726 + 0.156196i \(0.0499232\pi\)
−0.987726 + 0.156196i \(0.950077\pi\)
\(318\) 0 0
\(319\) 9.04384i 0.506358i
\(320\) 0 0
\(321\) 48.1364 2.68671
\(322\) 0 0
\(323\) 37.2109i 2.07047i
\(324\) 0 0
\(325\) 3.41840 + 6.58093i 0.189619 + 0.365045i
\(326\) 0 0
\(327\) 8.58486i 0.474744i
\(328\) 0 0
\(329\) −0.265550 −0.0146402
\(330\) 0 0
\(331\) 20.3149i 1.11661i 0.829636 + 0.558305i \(0.188548\pi\)
−0.829636 + 0.558305i \(0.811452\pi\)
\(332\) 0 0
\(333\) 9.02031i 0.494310i
\(334\) 0 0
\(335\) 21.4311 1.17091
\(336\) 0 0
\(337\) −29.0083 −1.58019 −0.790093 0.612987i \(-0.789968\pi\)
−0.790093 + 0.612987i \(0.789968\pi\)
\(338\) 0 0
\(339\) 23.7065 1.28756
\(340\) 0 0
\(341\) −4.98361 −0.269877
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 17.1437i 0.922986i
\(346\) 0 0
\(347\) −23.8427 −1.27995 −0.639973 0.768397i \(-0.721054\pi\)
−0.639973 + 0.768397i \(0.721054\pi\)
\(348\) 0 0
\(349\) 29.8451i 1.59757i 0.601615 + 0.798786i \(0.294524\pi\)
−0.601615 + 0.798786i \(0.705476\pi\)
\(350\) 0 0
\(351\) −18.7194 36.0375i −0.999165 1.92354i
\(352\) 0 0
\(353\) 15.4542i 0.822544i −0.911513 0.411272i \(-0.865085\pi\)
0.911513 0.411272i \(-0.134915\pi\)
\(354\) 0 0
\(355\) −1.84612 −0.0979819
\(356\) 0 0
\(357\) 15.7700i 0.834635i
\(358\) 0 0
\(359\) 8.37938i 0.442247i −0.975246 0.221123i \(-0.929028\pi\)
0.975246 0.221123i \(-0.0709723\pi\)
\(360\) 0 0
\(361\) −34.6149 −1.82183
\(362\) 0 0
\(363\) 3.10315 0.162873
\(364\) 0 0
\(365\) 12.6543 0.662357
\(366\) 0 0
\(367\) 5.78081 0.301756 0.150878 0.988552i \(-0.451790\pi\)
0.150878 + 0.988552i \(0.451790\pi\)
\(368\) 0 0
\(369\) 80.3147i 4.18102i
\(370\) 0 0
\(371\) 9.73208i 0.505265i
\(372\) 0 0
\(373\) 4.26380 0.220771 0.110386 0.993889i \(-0.464791\pi\)
0.110386 + 0.993889i \(0.464791\pi\)
\(374\) 0 0
\(375\) 37.5682i 1.94002i
\(376\) 0 0
\(377\) −15.0311 28.9370i −0.774139 1.49033i
\(378\) 0 0
\(379\) 17.0712i 0.876890i 0.898758 + 0.438445i \(0.144471\pi\)
−0.898758 + 0.438445i \(0.855529\pi\)
\(380\) 0 0
\(381\) 63.4797 3.25216
\(382\) 0 0
\(383\) 1.28626i 0.0657250i 0.999460 + 0.0328625i \(0.0104623\pi\)
−0.999460 + 0.0328625i \(0.989538\pi\)
\(384\) 0 0
\(385\) 1.71558i 0.0874342i
\(386\) 0 0
\(387\) 56.8744 2.89109
\(388\) 0 0
\(389\) 22.8254 1.15729 0.578646 0.815579i \(-0.303581\pi\)
0.578646 + 0.815579i \(0.303581\pi\)
\(390\) 0 0
\(391\) −16.3651 −0.827617
\(392\) 0 0
\(393\) 47.0325 2.37247
\(394\) 0 0
\(395\) 18.3021i 0.920878i
\(396\) 0 0
\(397\) 9.90885i 0.497311i 0.968592 + 0.248656i \(0.0799887\pi\)
−0.968592 + 0.248656i \(0.920011\pi\)
\(398\) 0 0
\(399\) −22.7219 −1.13752
\(400\) 0 0
\(401\) 8.17088i 0.408035i 0.978967 + 0.204017i \(0.0653999\pi\)
−0.978967 + 0.204017i \(0.934600\pi\)
\(402\) 0 0
\(403\) −15.9457 + 8.28286i −0.794314 + 0.412599i
\(404\) 0 0
\(405\) 25.8404i 1.28402i
\(406\) 0 0
\(407\) 1.36062 0.0674436
\(408\) 0 0
\(409\) 14.9797i 0.740700i 0.928892 + 0.370350i \(0.120762\pi\)
−0.928892 + 0.370350i \(0.879238\pi\)
\(410\) 0 0
\(411\) 24.5028i 1.20863i
\(412\) 0 0
\(413\) −0.610665 −0.0300489
\(414\) 0 0
\(415\) −1.72735 −0.0847920
\(416\) 0 0
\(417\) 28.2924 1.38548
\(418\) 0 0
\(419\) 17.2674 0.843566 0.421783 0.906697i \(-0.361404\pi\)
0.421783 + 0.906697i \(0.361404\pi\)
\(420\) 0 0
\(421\) 30.4363i 1.48338i −0.670746 0.741688i \(-0.734026\pi\)
0.670746 0.741688i \(-0.265974\pi\)
\(422\) 0 0
\(423\) 1.76047i 0.0855971i
\(424\) 0 0
\(425\) 10.4524 0.507014
\(426\) 0 0
\(427\) 1.47181i 0.0712258i
\(428\) 0 0
\(429\) 9.92896 5.15750i 0.479375 0.249007i
\(430\) 0 0
\(431\) 22.2055i 1.06960i −0.844978 0.534801i \(-0.820387\pi\)
0.844978 0.534801i \(-0.179613\pi\)
\(432\) 0 0
\(433\) 21.2497 1.02119 0.510597 0.859820i \(-0.329425\pi\)
0.510597 + 0.859820i \(0.329425\pi\)
\(434\) 0 0
\(435\) 48.1468i 2.30846i
\(436\) 0 0
\(437\) 23.5794i 1.12796i
\(438\) 0 0
\(439\) −4.51477 −0.215478 −0.107739 0.994179i \(-0.534361\pi\)
−0.107739 + 0.994179i \(0.534361\pi\)
\(440\) 0 0
\(441\) −6.62954 −0.315692
\(442\) 0 0
\(443\) −19.5619 −0.929416 −0.464708 0.885464i \(-0.653841\pi\)
−0.464708 + 0.885464i \(0.653841\pi\)
\(444\) 0 0
\(445\) −4.74351 −0.224864
\(446\) 0 0
\(447\) 25.1021i 1.18729i
\(448\) 0 0
\(449\) 13.2952i 0.627439i −0.949516 0.313720i \(-0.898425\pi\)
0.949516 0.313720i \(-0.101575\pi\)
\(450\) 0 0
\(451\) 12.1147 0.570458
\(452\) 0 0
\(453\) 62.6338i 2.94279i
\(454\) 0 0
\(455\) −2.85134 5.48925i −0.133673 0.257340i
\(456\) 0 0
\(457\) 34.9067i 1.63287i 0.577439 + 0.816434i \(0.304052\pi\)
−0.577439 + 0.816434i \(0.695948\pi\)
\(458\) 0 0
\(459\) −57.2377 −2.67163
\(460\) 0 0
\(461\) 34.2750i 1.59635i 0.602428 + 0.798173i \(0.294200\pi\)
−0.602428 + 0.798173i \(0.705800\pi\)
\(462\) 0 0
\(463\) 33.9899i 1.57964i −0.613337 0.789822i \(-0.710173\pi\)
0.613337 0.789822i \(-0.289827\pi\)
\(464\) 0 0
\(465\) −26.5313 −1.23036
\(466\) 0 0
\(467\) −34.5307 −1.59789 −0.798946 0.601403i \(-0.794609\pi\)
−0.798946 + 0.601403i \(0.794609\pi\)
\(468\) 0 0
\(469\) 12.4920 0.576828
\(470\) 0 0
\(471\) 45.4533 2.09438
\(472\) 0 0
\(473\) 8.57894i 0.394460i
\(474\) 0 0
\(475\) 15.0601i 0.691007i
\(476\) 0 0
\(477\) −64.5192 −2.95413
\(478\) 0 0
\(479\) 15.1076i 0.690284i −0.938551 0.345142i \(-0.887831\pi\)
0.938551 0.345142i \(-0.112169\pi\)
\(480\) 0 0
\(481\) 4.35351 2.26139i 0.198503 0.103110i
\(482\) 0 0
\(483\) 9.99293i 0.454694i
\(484\) 0 0
\(485\) −3.03836 −0.137965
\(486\) 0 0
\(487\) 38.6163i 1.74987i 0.484239 + 0.874936i \(0.339097\pi\)
−0.484239 + 0.874936i \(0.660903\pi\)
\(488\) 0 0
\(489\) 39.9871i 1.80828i
\(490\) 0 0
\(491\) −11.7053 −0.528254 −0.264127 0.964488i \(-0.585084\pi\)
−0.264127 + 0.964488i \(0.585084\pi\)
\(492\) 0 0
\(493\) −45.9601 −2.06994
\(494\) 0 0
\(495\) 11.3735 0.511202
\(496\) 0 0
\(497\) −1.07609 −0.0482692
\(498\) 0 0
\(499\) 1.85956i 0.0832455i −0.999133 0.0416227i \(-0.986747\pi\)
0.999133 0.0416227i \(-0.0132528\pi\)
\(500\) 0 0
\(501\) 53.6719i 2.39788i
\(502\) 0 0
\(503\) 3.45118 0.153881 0.0769403 0.997036i \(-0.475485\pi\)
0.0769403 + 0.997036i \(0.475485\pi\)
\(504\) 0 0
\(505\) 28.4224i 1.26478i
\(506\) 0 0
\(507\) 23.1972 33.0043i 1.03022 1.46577i
\(508\) 0 0
\(509\) 30.5030i 1.35202i −0.736891 0.676011i \(-0.763707\pi\)
0.736891 0.676011i \(-0.236293\pi\)
\(510\) 0 0
\(511\) 7.37610 0.326299
\(512\) 0 0
\(513\) 82.4702i 3.64115i
\(514\) 0 0
\(515\) 22.1216i 0.974792i
\(516\) 0 0
\(517\) 0.265550 0.0116789
\(518\) 0 0
\(519\) 38.5483 1.69208
\(520\) 0 0
\(521\) 22.3987 0.981307 0.490653 0.871355i \(-0.336758\pi\)
0.490653 + 0.871355i \(0.336758\pi\)
\(522\) 0 0
\(523\) 4.09703 0.179151 0.0895753 0.995980i \(-0.471449\pi\)
0.0895753 + 0.995980i \(0.471449\pi\)
\(524\) 0 0
\(525\) 6.38248i 0.278554i
\(526\) 0 0
\(527\) 25.3263i 1.10323i
\(528\) 0 0
\(529\) −12.6300 −0.549129
\(530\) 0 0
\(531\) 4.04843i 0.175687i
\(532\) 0 0
\(533\) 38.7626 20.1349i 1.67899 0.872137i
\(534\) 0 0
\(535\) 26.6123i 1.15055i
\(536\) 0 0
\(537\) 34.4747 1.48770
\(538\) 0 0
\(539\) 1.00000i 0.0430730i
\(540\) 0 0
\(541\) 4.60972i 0.198187i −0.995078 0.0990936i \(-0.968406\pi\)
0.995078 0.0990936i \(-0.0315943\pi\)
\(542\) 0 0
\(543\) −57.5207 −2.46845
\(544\) 0 0
\(545\) −4.74616 −0.203303
\(546\) 0 0
\(547\) −28.7361 −1.22867 −0.614333 0.789047i \(-0.710575\pi\)
−0.614333 + 0.789047i \(0.710575\pi\)
\(548\) 0 0
\(549\) −9.75742 −0.416436
\(550\) 0 0
\(551\) 66.2210i 2.82111i
\(552\) 0 0
\(553\) 10.6681i 0.453656i
\(554\) 0 0
\(555\) 7.24357 0.307472
\(556\) 0 0
\(557\) 10.8600i 0.460152i 0.973173 + 0.230076i \(0.0738976\pi\)
−0.973173 + 0.230076i \(0.926102\pi\)
\(558\) 0 0
\(559\) 14.2584 + 27.4495i 0.603065 + 1.16099i
\(560\) 0 0
\(561\) 15.7700i 0.665808i
\(562\) 0 0
\(563\) 23.4139 0.986777 0.493388 0.869809i \(-0.335758\pi\)
0.493388 + 0.869809i \(0.335758\pi\)
\(564\) 0 0
\(565\) 13.1062i 0.551383i
\(566\) 0 0
\(567\) 15.0622i 0.632552i
\(568\) 0 0
\(569\) 26.5389 1.11257 0.556285 0.830991i \(-0.312226\pi\)
0.556285 + 0.830991i \(0.312226\pi\)
\(570\) 0 0
\(571\) 31.0785 1.30060 0.650298 0.759679i \(-0.274644\pi\)
0.650298 + 0.759679i \(0.274644\pi\)
\(572\) 0 0
\(573\) 57.6974 2.41034
\(574\) 0 0
\(575\) −6.62333 −0.276212
\(576\) 0 0
\(577\) 44.7936i 1.86478i 0.361451 + 0.932391i \(0.382281\pi\)
−0.361451 + 0.932391i \(0.617719\pi\)
\(578\) 0 0
\(579\) 10.1991i 0.423860i
\(580\) 0 0
\(581\) −1.00686 −0.0417714
\(582\) 0 0
\(583\) 9.73208i 0.403062i
\(584\) 0 0
\(585\) 36.3912 18.9030i 1.50459 0.781545i
\(586\) 0 0
\(587\) 31.7357i 1.30987i 0.755685 + 0.654935i \(0.227304\pi\)
−0.755685 + 0.654935i \(0.772696\pi\)
\(588\) 0 0
\(589\) −36.4910 −1.50359
\(590\) 0 0
\(591\) 54.0760i 2.22439i
\(592\) 0 0
\(593\) 28.3865i 1.16569i 0.812582 + 0.582847i \(0.198061\pi\)
−0.812582 + 0.582847i \(0.801939\pi\)
\(594\) 0 0
\(595\) −8.71846 −0.357422
\(596\) 0 0
\(597\) −20.6118 −0.843583
\(598\) 0 0
\(599\) 21.3055 0.870518 0.435259 0.900305i \(-0.356657\pi\)
0.435259 + 0.900305i \(0.356657\pi\)
\(600\) 0 0
\(601\) −16.0139 −0.653219 −0.326610 0.945159i \(-0.605906\pi\)
−0.326610 + 0.945159i \(0.605906\pi\)
\(602\) 0 0
\(603\) 82.8164i 3.37255i
\(604\) 0 0
\(605\) 1.71558i 0.0697484i
\(606\) 0 0
\(607\) −17.8074 −0.722782 −0.361391 0.932414i \(-0.617698\pi\)
−0.361391 + 0.932414i \(0.617698\pi\)
\(608\) 0 0
\(609\) 28.0644i 1.13723i
\(610\) 0 0
\(611\) 0.849663 0.441349i 0.0343737 0.0178551i
\(612\) 0 0
\(613\) 32.7247i 1.32174i −0.750500 0.660870i \(-0.770187\pi\)
0.750500 0.660870i \(-0.229813\pi\)
\(614\) 0 0
\(615\) 64.4950 2.60069
\(616\) 0 0
\(617\) 10.4929i 0.422430i −0.977440 0.211215i \(-0.932258\pi\)
0.977440 0.211215i \(-0.0677420\pi\)
\(618\) 0 0
\(619\) 40.5207i 1.62867i 0.580398 + 0.814333i \(0.302897\pi\)
−0.580398 + 0.814333i \(0.697103\pi\)
\(620\) 0 0
\(621\) 36.2697 1.45545
\(622\) 0 0
\(623\) −2.76495 −0.110775
\(624\) 0 0
\(625\) −10.4858 −0.419432
\(626\) 0 0
\(627\) 22.7219 0.907427
\(628\) 0 0
\(629\) 6.91458i 0.275702i
\(630\) 0 0
\(631\) 8.34947i 0.332387i 0.986093 + 0.166193i \(0.0531476\pi\)
−0.986093 + 0.166193i \(0.946852\pi\)
\(632\) 0 0
\(633\) −19.0510 −0.757208
\(634\) 0 0
\(635\) 35.0949i 1.39270i
\(636\) 0 0
\(637\) −1.66202 3.19964i −0.0658517 0.126774i
\(638\) 0 0
\(639\) 7.13397i 0.282216i
\(640\) 0 0
\(641\) 32.6733 1.29052 0.645258 0.763964i \(-0.276750\pi\)
0.645258 + 0.763964i \(0.276750\pi\)
\(642\) 0 0
\(643\) 32.3767i 1.27681i −0.769699 0.638407i \(-0.779594\pi\)
0.769699 0.638407i \(-0.220406\pi\)
\(644\) 0 0
\(645\) 45.6718i 1.79833i
\(646\) 0 0
\(647\) 22.3589 0.879021 0.439510 0.898238i \(-0.355152\pi\)
0.439510 + 0.898238i \(0.355152\pi\)
\(648\) 0 0
\(649\) 0.610665 0.0239707
\(650\) 0 0
\(651\) −15.4649 −0.606116
\(652\) 0 0
\(653\) 21.3370 0.834982 0.417491 0.908681i \(-0.362910\pi\)
0.417491 + 0.908681i \(0.362910\pi\)
\(654\) 0 0
\(655\) 26.0020i 1.01598i
\(656\) 0 0
\(657\) 48.9001i 1.90778i
\(658\) 0 0
\(659\) −38.7806 −1.51068 −0.755339 0.655334i \(-0.772528\pi\)
−0.755339 + 0.655334i \(0.772528\pi\)
\(660\) 0 0
\(661\) 28.3898i 1.10423i −0.833767 0.552117i \(-0.813820\pi\)
0.833767 0.552117i \(-0.186180\pi\)
\(662\) 0 0
\(663\) −26.2100 50.4582i −1.01791 1.95963i
\(664\) 0 0
\(665\) 12.5619i 0.487128i
\(666\) 0 0
\(667\) 29.1235 1.12766
\(668\) 0 0
\(669\) 27.6345i 1.06841i
\(670\) 0 0
\(671\) 1.47181i 0.0568186i
\(672\) 0 0
\(673\) 26.7052 1.02941 0.514705 0.857367i \(-0.327901\pi\)
0.514705 + 0.857367i \(0.327901\pi\)
\(674\) 0 0
\(675\) −23.1655 −0.891639
\(676\) 0 0
\(677\) −40.2714 −1.54775 −0.773877 0.633336i \(-0.781685\pi\)
−0.773877 + 0.633336i \(0.781685\pi\)
\(678\) 0 0
\(679\) −1.77103 −0.0679661
\(680\) 0 0
\(681\) 43.8146i 1.67898i
\(682\) 0 0
\(683\) 14.8735i 0.569118i 0.958658 + 0.284559i \(0.0918472\pi\)
−0.958658 + 0.284559i \(0.908153\pi\)
\(684\) 0 0
\(685\) 13.5464 0.517583
\(686\) 0 0
\(687\) 9.08087i 0.346457i
\(688\) 0 0
\(689\) −16.1749 31.1391i −0.616216 1.18631i
\(690\) 0 0
\(691\) 20.5711i 0.782562i −0.920271 0.391281i \(-0.872032\pi\)
0.920271 0.391281i \(-0.127968\pi\)
\(692\) 0 0
\(693\) 6.62954 0.251835
\(694\) 0 0
\(695\) 15.6415i 0.593316i
\(696\) 0 0
\(697\) 61.5658i 2.33197i
\(698\) 0 0
\(699\) 29.4756 1.11487
\(700\) 0 0
\(701\) −21.1759 −0.799804 −0.399902 0.916558i \(-0.630956\pi\)
−0.399902 + 0.916558i \(0.630956\pi\)
\(702\) 0 0
\(703\) 9.96279 0.375754
\(704\) 0 0
\(705\) 1.41371 0.0532434
\(706\) 0 0
\(707\) 16.5672i 0.623074i
\(708\) 0 0
\(709\) 32.4743i 1.21960i 0.792557 + 0.609798i \(0.208750\pi\)
−0.792557 + 0.609798i \(0.791250\pi\)
\(710\) 0 0
\(711\) −70.7249 −2.65239
\(712\) 0 0
\(713\) 16.0485i 0.601020i
\(714\) 0 0
\(715\) 2.85134 + 5.48925i 0.106634 + 0.205286i
\(716\) 0 0
\(717\) 52.5466i 1.96239i
\(718\) 0 0
\(719\) −21.7515 −0.811195 −0.405598 0.914052i \(-0.632937\pi\)
−0.405598 + 0.914052i \(0.632937\pi\)
\(720\) 0 0
\(721\) 12.8945i 0.480216i
\(722\) 0 0
\(723\) 17.7150i 0.658827i
\(724\) 0 0
\(725\) −18.6011 −0.690829
\(726\) 0 0
\(727\) −50.4648 −1.87163 −0.935817 0.352485i \(-0.885337\pi\)
−0.935817 + 0.352485i \(0.885337\pi\)
\(728\) 0 0
\(729\) −4.99710 −0.185078
\(730\) 0 0
\(731\) 43.5975 1.61251
\(732\) 0 0
\(733\) 12.3476i 0.456068i −0.973653 0.228034i \(-0.926770\pi\)
0.973653 0.228034i \(-0.0732298\pi\)
\(734\) 0 0
\(735\) 5.32371i 0.196368i
\(736\) 0 0
\(737\) −12.4920 −0.460150
\(738\) 0 0
\(739\) 31.3855i 1.15454i −0.816555 0.577268i \(-0.804119\pi\)
0.816555 0.577268i \(-0.195881\pi\)
\(740\) 0 0
\(741\) 72.7020 37.7643i 2.67077 1.38731i
\(742\) 0 0
\(743\) 38.4254i 1.40969i −0.709360 0.704846i \(-0.751016\pi\)
0.709360 0.704846i \(-0.248984\pi\)
\(744\) 0 0
\(745\) 13.8778 0.508442
\(746\) 0 0
\(747\) 6.67499i 0.244225i
\(748\) 0 0
\(749\) 15.5121i 0.566800i
\(750\) 0 0
\(751\) 40.3811 1.47353 0.736763 0.676151i \(-0.236353\pi\)
0.736763 + 0.676151i \(0.236353\pi\)
\(752\) 0 0
\(753\) −52.9913 −1.93111
\(754\) 0 0
\(755\) −34.6273 −1.26021
\(756\) 0 0
\(757\) 22.2675 0.809326 0.404663 0.914466i \(-0.367389\pi\)
0.404663 + 0.914466i \(0.367389\pi\)
\(758\) 0 0
\(759\) 9.99293i 0.362720i
\(760\) 0 0
\(761\) 19.9937i 0.724771i −0.932028 0.362386i \(-0.881962\pi\)
0.932028 0.362386i \(-0.118038\pi\)
\(762\) 0 0
\(763\) −2.76650 −0.100154
\(764\) 0 0
\(765\) 57.7993i 2.08974i
\(766\) 0 0
\(767\) 1.95391 1.01494i 0.0705515 0.0366473i
\(768\) 0 0
\(769\) 36.9427i 1.33219i −0.745868 0.666093i \(-0.767965\pi\)
0.745868 0.666093i \(-0.232035\pi\)
\(770\) 0 0
\(771\) 54.7299 1.97105
\(772\) 0 0
\(773\) 38.8776i 1.39833i −0.714960 0.699165i \(-0.753555\pi\)
0.714960 0.699165i \(-0.246445\pi\)
\(774\) 0 0
\(775\) 10.2502i 0.368196i
\(776\) 0 0
\(777\) 4.22222 0.151471
\(778\) 0 0
\(779\) 88.7063 3.17823
\(780\) 0 0
\(781\) 1.07609 0.0385055
\(782\) 0 0
\(783\) 101.861 3.64021
\(784\) 0 0
\(785\) 25.1290i 0.896892i
\(786\) 0 0
\(787\) 19.8181i 0.706437i 0.935541 + 0.353219i \(0.114913\pi\)
−0.935541 + 0.353219i \(0.885087\pi\)
\(788\) 0 0
\(789\) −69.7006 −2.48141
\(790\) 0 0
\(791\) 7.63951i 0.271630i
\(792\) 0 0
\(793\) −2.44618 4.70926i −0.0868663 0.167231i
\(794\) 0