Properties

Label 4004.2.m.c.2157.16
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 2157.16
Character \(\chi\) = 4004.2157
Dual form 4004.2.m.c.2157.15

$q$-expansion

\(f(q)\) \(=\) \(q\)\(-0.819032 q^{3}\) \(-1.69723i q^{5}\) \(-1.00000i q^{7}\) \(-2.32919 q^{9}\) \(+O(q^{10})\) \(q\)\(-0.819032 q^{3}\) \(-1.69723i q^{5}\) \(-1.00000i q^{7}\) \(-2.32919 q^{9}\) \(+1.00000i q^{11}\) \(+(-1.93282 + 3.04372i) q^{13}\) \(+1.39008i q^{15}\) \(-2.32085 q^{17}\) \(-2.77344i q^{19}\) \(+0.819032i q^{21}\) \(-0.366187 q^{23}\) \(+2.11942 q^{25}\) \(+4.36478 q^{27}\) \(+2.95197 q^{29}\) \(-9.83529i q^{31}\) \(-0.819032i q^{33}\) \(-1.69723 q^{35}\) \(+7.39804i q^{37}\) \(+(1.58304 - 2.49290i) q^{39}\) \(-9.67457i q^{41}\) \(-10.1237 q^{43}\) \(+3.95316i q^{45}\) \(+10.4378i q^{47}\) \(-1.00000 q^{49}\) \(+1.90085 q^{51}\) \(-7.02888 q^{53}\) \(+1.69723 q^{55}\) \(+2.27154i q^{57}\) \(+11.9025i q^{59}\) \(-8.21477 q^{61}\) \(+2.32919i q^{63}\) \(+(5.16588 + 3.28043i) q^{65}\) \(+10.3012i q^{67}\) \(+0.299919 q^{69}\) \(-0.127032i q^{71}\) \(+11.1281i q^{73}\) \(-1.73587 q^{75}\) \(+1.00000 q^{77}\) \(+14.2135 q^{79}\) \(+3.41266 q^{81}\) \(-4.44302i q^{83}\) \(+3.93902i q^{85}\) \(-2.41776 q^{87}\) \(+5.15887i q^{89}\) \(+(3.04372 + 1.93282i) q^{91}\) \(+8.05542i q^{93}\) \(-4.70716 q^{95}\) \(-4.55893i q^{97}\) \(-2.32919i q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 80q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut -\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 80q^{69} \) \(\mathstrut -\mathstrut 36q^{75} \) \(\mathstrut +\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 132q^{81} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

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

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.819032 −0.472869 −0.236434 0.971647i \(-0.575979\pi\)
−0.236434 + 0.971647i \(0.575979\pi\)
\(4\) 0 0
\(5\) 1.69723i 0.759023i −0.925187 0.379511i \(-0.876092\pi\)
0.925187 0.379511i \(-0.123908\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −2.32919 −0.776395
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) −1.93282 + 3.04372i −0.536067 + 0.844175i
\(14\) 0 0
\(15\) 1.39008i 0.358918i
\(16\) 0 0
\(17\) −2.32085 −0.562890 −0.281445 0.959577i \(-0.590814\pi\)
−0.281445 + 0.959577i \(0.590814\pi\)
\(18\) 0 0
\(19\) 2.77344i 0.636271i −0.948045 0.318136i \(-0.896943\pi\)
0.948045 0.318136i \(-0.103057\pi\)
\(20\) 0 0
\(21\) 0.819032i 0.178728i
\(22\) 0 0
\(23\) −0.366187 −0.0763553 −0.0381776 0.999271i \(-0.512155\pi\)
−0.0381776 + 0.999271i \(0.512155\pi\)
\(24\) 0 0
\(25\) 2.11942 0.423884
\(26\) 0 0
\(27\) 4.36478 0.840002
\(28\) 0 0
\(29\) 2.95197 0.548168 0.274084 0.961706i \(-0.411625\pi\)
0.274084 + 0.961706i \(0.411625\pi\)
\(30\) 0 0
\(31\) 9.83529i 1.76647i −0.468931 0.883235i \(-0.655361\pi\)
0.468931 0.883235i \(-0.344639\pi\)
\(32\) 0 0
\(33\) 0.819032i 0.142575i
\(34\) 0 0
\(35\) −1.69723 −0.286884
\(36\) 0 0
\(37\) 7.39804i 1.21623i 0.793849 + 0.608115i \(0.208074\pi\)
−0.793849 + 0.608115i \(0.791926\pi\)
\(38\) 0 0
\(39\) 1.58304 2.49290i 0.253489 0.399184i
\(40\) 0 0
\(41\) 9.67457i 1.51091i −0.655199 0.755457i \(-0.727415\pi\)
0.655199 0.755457i \(-0.272585\pi\)
\(42\) 0 0
\(43\) −10.1237 −1.54385 −0.771924 0.635715i \(-0.780705\pi\)
−0.771924 + 0.635715i \(0.780705\pi\)
\(44\) 0 0
\(45\) 3.95316i 0.589302i
\(46\) 0 0
\(47\) 10.4378i 1.52250i 0.648457 + 0.761251i \(0.275415\pi\)
−0.648457 + 0.761251i \(0.724585\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 1.90085 0.266173
\(52\) 0 0
\(53\) −7.02888 −0.965490 −0.482745 0.875761i \(-0.660360\pi\)
−0.482745 + 0.875761i \(0.660360\pi\)
\(54\) 0 0
\(55\) 1.69723 0.228854
\(56\) 0 0
\(57\) 2.27154i 0.300873i
\(58\) 0 0
\(59\) 11.9025i 1.54958i 0.632219 + 0.774790i \(0.282144\pi\)
−0.632219 + 0.774790i \(0.717856\pi\)
\(60\) 0 0
\(61\) −8.21477 −1.05179 −0.525897 0.850548i \(-0.676270\pi\)
−0.525897 + 0.850548i \(0.676270\pi\)
\(62\) 0 0
\(63\) 2.32919i 0.293450i
\(64\) 0 0
\(65\) 5.16588 + 3.28043i 0.640748 + 0.406887i
\(66\) 0 0
\(67\) 10.3012i 1.25849i 0.777208 + 0.629244i \(0.216635\pi\)
−0.777208 + 0.629244i \(0.783365\pi\)
\(68\) 0 0
\(69\) 0.299919 0.0361060
\(70\) 0 0
\(71\) 0.127032i 0.0150759i −0.999972 0.00753794i \(-0.997601\pi\)
0.999972 0.00753794i \(-0.00239942\pi\)
\(72\) 0 0
\(73\) 11.1281i 1.30244i 0.758888 + 0.651221i \(0.225743\pi\)
−0.758888 + 0.651221i \(0.774257\pi\)
\(74\) 0 0
\(75\) −1.73587 −0.200441
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 14.2135 1.59915 0.799574 0.600567i \(-0.205058\pi\)
0.799574 + 0.600567i \(0.205058\pi\)
\(80\) 0 0
\(81\) 3.41266 0.379185
\(82\) 0 0
\(83\) 4.44302i 0.487684i −0.969815 0.243842i \(-0.921592\pi\)
0.969815 0.243842i \(-0.0784079\pi\)
\(84\) 0 0
\(85\) 3.93902i 0.427246i
\(86\) 0 0
\(87\) −2.41776 −0.259211
\(88\) 0 0
\(89\) 5.15887i 0.546839i 0.961895 + 0.273419i \(0.0881547\pi\)
−0.961895 + 0.273419i \(0.911845\pi\)
\(90\) 0 0
\(91\) 3.04372 + 1.93282i 0.319068 + 0.202614i
\(92\) 0 0
\(93\) 8.05542i 0.835308i
\(94\) 0 0
\(95\) −4.70716 −0.482945
\(96\) 0 0
\(97\) 4.55893i 0.462890i −0.972848 0.231445i \(-0.925655\pi\)
0.972848 0.231445i \(-0.0743453\pi\)
\(98\) 0 0
\(99\) 2.32919i 0.234092i
\(100\) 0 0
\(101\) 16.4229 1.63414 0.817068 0.576541i \(-0.195598\pi\)
0.817068 + 0.576541i \(0.195598\pi\)
\(102\) 0 0
\(103\) −5.61320 −0.553085 −0.276543 0.961002i \(-0.589189\pi\)
−0.276543 + 0.961002i \(0.589189\pi\)
\(104\) 0 0
\(105\) 1.39008 0.135658
\(106\) 0 0
\(107\) 6.32562 0.611520 0.305760 0.952109i \(-0.401089\pi\)
0.305760 + 0.952109i \(0.401089\pi\)
\(108\) 0 0
\(109\) 0.445333i 0.0426552i −0.999773 0.0213276i \(-0.993211\pi\)
0.999773 0.0213276i \(-0.00678930\pi\)
\(110\) 0 0
\(111\) 6.05924i 0.575117i
\(112\) 0 0
\(113\) 1.69007 0.158988 0.0794940 0.996835i \(-0.474670\pi\)
0.0794940 + 0.996835i \(0.474670\pi\)
\(114\) 0 0
\(115\) 0.621502i 0.0579554i
\(116\) 0 0
\(117\) 4.50189 7.08938i 0.416200 0.655414i
\(118\) 0 0
\(119\) 2.32085i 0.212752i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 7.92378i 0.714463i
\(124\) 0 0
\(125\) 12.0833i 1.08076i
\(126\) 0 0
\(127\) 13.3288 1.18274 0.591369 0.806401i \(-0.298588\pi\)
0.591369 + 0.806401i \(0.298588\pi\)
\(128\) 0 0
\(129\) 8.29163 0.730037
\(130\) 0 0
\(131\) −15.1942 −1.32752 −0.663761 0.747944i \(-0.731041\pi\)
−0.663761 + 0.747944i \(0.731041\pi\)
\(132\) 0 0
\(133\) −2.77344 −0.240488
\(134\) 0 0
\(135\) 7.40802i 0.637581i
\(136\) 0 0
\(137\) 3.05099i 0.260663i −0.991470 0.130332i \(-0.958396\pi\)
0.991470 0.130332i \(-0.0416042\pi\)
\(138\) 0 0
\(139\) 15.1151 1.28205 0.641023 0.767522i \(-0.278510\pi\)
0.641023 + 0.767522i \(0.278510\pi\)
\(140\) 0 0
\(141\) 8.54886i 0.719944i
\(142\) 0 0
\(143\) −3.04372 1.93282i −0.254528 0.161630i
\(144\) 0 0
\(145\) 5.01017i 0.416072i
\(146\) 0 0
\(147\) 0.819032 0.0675527
\(148\) 0 0
\(149\) 4.07402i 0.333757i 0.985977 + 0.166878i \(0.0533688\pi\)
−0.985977 + 0.166878i \(0.946631\pi\)
\(150\) 0 0
\(151\) 17.4602i 1.42089i 0.703750 + 0.710447i \(0.251507\pi\)
−0.703750 + 0.710447i \(0.748493\pi\)
\(152\) 0 0
\(153\) 5.40570 0.437025
\(154\) 0 0
\(155\) −16.6927 −1.34079
\(156\) 0 0
\(157\) 14.0108 1.11818 0.559091 0.829106i \(-0.311150\pi\)
0.559091 + 0.829106i \(0.311150\pi\)
\(158\) 0 0
\(159\) 5.75688 0.456550
\(160\) 0 0
\(161\) 0.366187i 0.0288596i
\(162\) 0 0
\(163\) 12.1130i 0.948766i 0.880319 + 0.474383i \(0.157329\pi\)
−0.880319 + 0.474383i \(0.842671\pi\)
\(164\) 0 0
\(165\) −1.39008 −0.108218
\(166\) 0 0
\(167\) 11.6051i 0.898033i 0.893523 + 0.449016i \(0.148226\pi\)
−0.893523 + 0.449016i \(0.851774\pi\)
\(168\) 0 0
\(169\) −5.52843 11.7659i −0.425264 0.905069i
\(170\) 0 0
\(171\) 6.45986i 0.493998i
\(172\) 0 0
\(173\) 14.4572 1.09916 0.549582 0.835440i \(-0.314787\pi\)
0.549582 + 0.835440i \(0.314787\pi\)
\(174\) 0 0
\(175\) 2.11942i 0.160213i
\(176\) 0 0
\(177\) 9.74857i 0.732747i
\(178\) 0 0
\(179\) −2.06658 −0.154464 −0.0772318 0.997013i \(-0.524608\pi\)
−0.0772318 + 0.997013i \(0.524608\pi\)
\(180\) 0 0
\(181\) 9.45225 0.702580 0.351290 0.936267i \(-0.385743\pi\)
0.351290 + 0.936267i \(0.385743\pi\)
\(182\) 0 0
\(183\) 6.72817 0.497360
\(184\) 0 0
\(185\) 12.5562 0.923147
\(186\) 0 0
\(187\) 2.32085i 0.169718i
\(188\) 0 0
\(189\) 4.36478i 0.317491i
\(190\) 0 0
\(191\) −0.193160 −0.0139765 −0.00698827 0.999976i \(-0.502224\pi\)
−0.00698827 + 0.999976i \(0.502224\pi\)
\(192\) 0 0
\(193\) 21.0349i 1.51413i 0.653340 + 0.757064i \(0.273367\pi\)
−0.653340 + 0.757064i \(0.726633\pi\)
\(194\) 0 0
\(195\) −4.23102 2.68678i −0.302990 0.192404i
\(196\) 0 0
\(197\) 6.65046i 0.473826i 0.971531 + 0.236913i \(0.0761356\pi\)
−0.971531 + 0.236913i \(0.923864\pi\)
\(198\) 0 0
\(199\) 19.4598 1.37947 0.689735 0.724062i \(-0.257727\pi\)
0.689735 + 0.724062i \(0.257727\pi\)
\(200\) 0 0
\(201\) 8.43699i 0.595099i
\(202\) 0 0
\(203\) 2.95197i 0.207188i
\(204\) 0 0
\(205\) −16.4199 −1.14682
\(206\) 0 0
\(207\) 0.852917 0.0592819
\(208\) 0 0
\(209\) 2.77344 0.191843
\(210\) 0 0
\(211\) 10.1153 0.696367 0.348183 0.937426i \(-0.386799\pi\)
0.348183 + 0.937426i \(0.386799\pi\)
\(212\) 0 0
\(213\) 0.104043i 0.00712891i
\(214\) 0 0
\(215\) 17.1822i 1.17182i
\(216\) 0 0
\(217\) −9.83529 −0.667663
\(218\) 0 0
\(219\) 9.11425i 0.615884i
\(220\) 0 0
\(221\) 4.48579 7.06402i 0.301747 0.475178i
\(222\) 0 0
\(223\) 20.6441i 1.38243i 0.722647 + 0.691217i \(0.242925\pi\)
−0.722647 + 0.691217i \(0.757075\pi\)
\(224\) 0 0
\(225\) −4.93652 −0.329102
\(226\) 0 0
\(227\) 29.0731i 1.92965i 0.262896 + 0.964824i \(0.415322\pi\)
−0.262896 + 0.964824i \(0.584678\pi\)
\(228\) 0 0
\(229\) 1.72489i 0.113984i 0.998375 + 0.0569921i \(0.0181510\pi\)
−0.998375 + 0.0569921i \(0.981849\pi\)
\(230\) 0 0
\(231\) −0.819032 −0.0538884
\(232\) 0 0
\(233\) 9.70662 0.635902 0.317951 0.948107i \(-0.397005\pi\)
0.317951 + 0.948107i \(0.397005\pi\)
\(234\) 0 0
\(235\) 17.7152 1.15561
\(236\) 0 0
\(237\) −11.6414 −0.756187
\(238\) 0 0
\(239\) 9.18724i 0.594273i 0.954835 + 0.297137i \(0.0960317\pi\)
−0.954835 + 0.297137i \(0.903968\pi\)
\(240\) 0 0
\(241\) 14.7453i 0.949826i −0.880033 0.474913i \(-0.842480\pi\)
0.880033 0.474913i \(-0.157520\pi\)
\(242\) 0 0
\(243\) −15.8894 −1.01931
\(244\) 0 0
\(245\) 1.69723i 0.108432i
\(246\) 0 0
\(247\) 8.44158 + 5.36056i 0.537125 + 0.341084i
\(248\) 0 0
\(249\) 3.63897i 0.230611i
\(250\) 0 0
\(251\) 29.9720 1.89182 0.945909 0.324432i \(-0.105173\pi\)
0.945909 + 0.324432i \(0.105173\pi\)
\(252\) 0 0
\(253\) 0.366187i 0.0230220i
\(254\) 0 0
\(255\) 3.22618i 0.202031i
\(256\) 0 0
\(257\) −14.6901 −0.916345 −0.458173 0.888863i \(-0.651496\pi\)
−0.458173 + 0.888863i \(0.651496\pi\)
\(258\) 0 0
\(259\) 7.39804 0.459692
\(260\) 0 0
\(261\) −6.87570 −0.425595
\(262\) 0 0
\(263\) 8.47278 0.522454 0.261227 0.965277i \(-0.415873\pi\)
0.261227 + 0.965277i \(0.415873\pi\)
\(264\) 0 0
\(265\) 11.9296i 0.732829i
\(266\) 0 0
\(267\) 4.22528i 0.258583i
\(268\) 0 0
\(269\) −16.9317 −1.03234 −0.516171 0.856486i \(-0.672643\pi\)
−0.516171 + 0.856486i \(0.672643\pi\)
\(270\) 0 0
\(271\) 1.06282i 0.0645616i −0.999479 0.0322808i \(-0.989723\pi\)
0.999479 0.0322808i \(-0.0102771\pi\)
\(272\) 0 0
\(273\) −2.49290 1.58304i −0.150877 0.0958100i
\(274\) 0 0
\(275\) 2.11942i 0.127806i
\(276\) 0 0
\(277\) −16.9601 −1.01904 −0.509518 0.860460i \(-0.670176\pi\)
−0.509518 + 0.860460i \(0.670176\pi\)
\(278\) 0 0
\(279\) 22.9082i 1.37148i
\(280\) 0 0
\(281\) 1.53421i 0.0915236i −0.998952 0.0457618i \(-0.985428\pi\)
0.998952 0.0457618i \(-0.0145715\pi\)
\(282\) 0 0
\(283\) 24.4847 1.45546 0.727732 0.685861i \(-0.240574\pi\)
0.727732 + 0.685861i \(0.240574\pi\)
\(284\) 0 0
\(285\) 3.85532 0.228369
\(286\) 0 0
\(287\) −9.67457 −0.571072
\(288\) 0 0
\(289\) −11.6136 −0.683155
\(290\) 0 0
\(291\) 3.73392i 0.218886i
\(292\) 0 0
\(293\) 11.4585i 0.669414i −0.942322 0.334707i \(-0.891363\pi\)
0.942322 0.334707i \(-0.108637\pi\)
\(294\) 0 0
\(295\) 20.2013 1.17617
\(296\) 0 0
\(297\) 4.36478i 0.253270i
\(298\) 0 0
\(299\) 0.707773 1.11457i 0.0409315 0.0644572i
\(300\) 0 0
\(301\) 10.1237i 0.583519i
\(302\) 0 0
\(303\) −13.4509 −0.772732
\(304\) 0 0
\(305\) 13.9423i 0.798336i
\(306\) 0 0
\(307\) 16.2231i 0.925902i −0.886384 0.462951i \(-0.846791\pi\)
0.886384 0.462951i \(-0.153209\pi\)
\(308\) 0 0
\(309\) 4.59740 0.261537
\(310\) 0 0
\(311\) −32.9393 −1.86782 −0.933908 0.357514i \(-0.883625\pi\)
−0.933908 + 0.357514i \(0.883625\pi\)
\(312\) 0 0
\(313\) −6.11898 −0.345865 −0.172933 0.984934i \(-0.555324\pi\)
−0.172933 + 0.984934i \(0.555324\pi\)
\(314\) 0 0
\(315\) 3.95316 0.222735
\(316\) 0 0
\(317\) 5.56290i 0.312444i −0.987722 0.156222i \(-0.950069\pi\)
0.987722 0.156222i \(-0.0499315\pi\)
\(318\) 0 0
\(319\) 2.95197i 0.165279i
\(320\) 0 0
\(321\) −5.18089 −0.289169
\(322\) 0 0
\(323\) 6.43675i 0.358151i
\(324\) 0 0
\(325\) −4.09645 + 6.45092i −0.227230 + 0.357832i
\(326\) 0 0
\(327\) 0.364743i 0.0201703i
\(328\) 0 0
\(329\) 10.4378 0.575452
\(330\) 0 0
\(331\) 6.30178i 0.346377i 0.984889 + 0.173188i \(0.0554070\pi\)
−0.984889 + 0.173188i \(0.944593\pi\)
\(332\) 0 0
\(333\) 17.2314i 0.944276i
\(334\) 0 0
\(335\) 17.4834 0.955221
\(336\) 0 0
\(337\) −13.2305 −0.720709 −0.360355 0.932815i \(-0.617344\pi\)
−0.360355 + 0.932815i \(0.617344\pi\)
\(338\) 0 0
\(339\) −1.38422 −0.0751805
\(340\) 0 0
\(341\) 9.83529 0.532611
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0.509031i 0.0274053i
\(346\) 0 0
\(347\) −33.1097 −1.77742 −0.888710 0.458470i \(-0.848398\pi\)
−0.888710 + 0.458470i \(0.848398\pi\)
\(348\) 0 0
\(349\) 3.39507i 0.181734i 0.995863 + 0.0908670i \(0.0289638\pi\)
−0.995863 + 0.0908670i \(0.971036\pi\)
\(350\) 0 0
\(351\) −8.43632 + 13.2851i −0.450297 + 0.709109i
\(352\) 0 0
\(353\) 26.9734i 1.43565i −0.696224 0.717824i \(-0.745138\pi\)
0.696224 0.717824i \(-0.254862\pi\)
\(354\) 0 0
\(355\) −0.215601 −0.0114429
\(356\) 0 0
\(357\) 1.90085i 0.100604i
\(358\) 0 0
\(359\) 22.2390i 1.17373i 0.809684 + 0.586866i \(0.199638\pi\)
−0.809684 + 0.586866i \(0.800362\pi\)
\(360\) 0 0
\(361\) 11.3080 0.595159
\(362\) 0 0
\(363\) 0.819032 0.0429881
\(364\) 0 0
\(365\) 18.8869 0.988584
\(366\) 0 0
\(367\) −9.31898 −0.486447 −0.243223 0.969970i \(-0.578205\pi\)
−0.243223 + 0.969970i \(0.578205\pi\)
\(368\) 0 0
\(369\) 22.5339i 1.17307i
\(370\) 0 0
\(371\) 7.02888i 0.364921i
\(372\) 0 0
\(373\) −5.75842 −0.298160 −0.149080 0.988825i \(-0.547631\pi\)
−0.149080 + 0.988825i \(0.547631\pi\)
\(374\) 0 0
\(375\) 9.89659i 0.511058i
\(376\) 0 0
\(377\) −5.70563 + 8.98498i −0.293855 + 0.462750i
\(378\) 0 0
\(379\) 11.5494i 0.593251i 0.954994 + 0.296625i \(0.0958613\pi\)
−0.954994 + 0.296625i \(0.904139\pi\)
\(380\) 0 0
\(381\) −10.9167 −0.559280
\(382\) 0 0
\(383\) 1.63226i 0.0834043i −0.999130 0.0417022i \(-0.986722\pi\)
0.999130 0.0417022i \(-0.0132781\pi\)
\(384\) 0 0
\(385\) 1.69723i 0.0864987i
\(386\) 0 0
\(387\) 23.5799 1.19864
\(388\) 0 0
\(389\) −34.1493 −1.73144 −0.865720 0.500528i \(-0.833139\pi\)
−0.865720 + 0.500528i \(0.833139\pi\)
\(390\) 0 0
\(391\) 0.849866 0.0429796
\(392\) 0 0
\(393\) 12.4445 0.627744
\(394\) 0 0
\(395\) 24.1236i 1.21379i
\(396\) 0 0
\(397\) 22.1368i 1.11102i 0.831511 + 0.555508i \(0.187476\pi\)
−0.831511 + 0.555508i \(0.812524\pi\)
\(398\) 0 0
\(399\) 2.27154 0.113719
\(400\) 0 0
\(401\) 17.3380i 0.865819i 0.901437 + 0.432910i \(0.142513\pi\)
−0.901437 + 0.432910i \(0.857487\pi\)
\(402\) 0 0
\(403\) 29.9358 + 19.0098i 1.49121 + 0.946946i
\(404\) 0 0
\(405\) 5.79206i 0.287810i
\(406\) 0 0
\(407\) −7.39804 −0.366707
\(408\) 0 0
\(409\) 5.96973i 0.295184i −0.989048 0.147592i \(-0.952848\pi\)
0.989048 0.147592i \(-0.0471523\pi\)
\(410\) 0 0
\(411\) 2.49886i 0.123260i
\(412\) 0 0
\(413\) 11.9025 0.585686
\(414\) 0 0
\(415\) −7.54081 −0.370164
\(416\) 0 0
\(417\) −12.3798 −0.606239
\(418\) 0 0
\(419\) −17.1740 −0.839006 −0.419503 0.907754i \(-0.637796\pi\)
−0.419503 + 0.907754i \(0.637796\pi\)
\(420\) 0 0
\(421\) 14.8026i 0.721436i −0.932675 0.360718i \(-0.882532\pi\)
0.932675 0.360718i \(-0.117468\pi\)
\(422\) 0 0
\(423\) 24.3115i 1.18206i
\(424\) 0 0
\(425\) −4.91887 −0.238600
\(426\) 0 0
\(427\) 8.21477i 0.397541i
\(428\) 0 0
\(429\) 2.49290 + 1.58304i 0.120359 + 0.0764299i
\(430\) 0 0
\(431\) 1.17641i 0.0566655i 0.999599 + 0.0283327i \(0.00901980\pi\)
−0.999599 + 0.0283327i \(0.990980\pi\)
\(432\) 0 0
\(433\) 33.7302 1.62097 0.810485 0.585760i \(-0.199204\pi\)
0.810485 + 0.585760i \(0.199204\pi\)
\(434\) 0 0
\(435\) 4.10349i 0.196747i
\(436\) 0 0
\(437\) 1.01560i 0.0485827i
\(438\) 0 0
\(439\) −33.0484 −1.57731 −0.788656 0.614835i \(-0.789223\pi\)
−0.788656 + 0.614835i \(0.789223\pi\)
\(440\) 0 0
\(441\) 2.32919 0.110914
\(442\) 0 0
\(443\) 28.5853 1.35813 0.679064 0.734079i \(-0.262386\pi\)
0.679064 + 0.734079i \(0.262386\pi\)
\(444\) 0 0
\(445\) 8.75577 0.415063
\(446\) 0 0
\(447\) 3.33676i 0.157823i
\(448\) 0 0
\(449\) 0.750950i 0.0354395i 0.999843 + 0.0177198i \(0.00564067\pi\)
−0.999843 + 0.0177198i \(0.994359\pi\)
\(450\) 0 0
\(451\) 9.67457 0.455557
\(452\) 0 0
\(453\) 14.3005i 0.671897i
\(454\) 0 0
\(455\) 3.28043 5.16588i 0.153789 0.242180i
\(456\) 0 0
\(457\) 22.9719i 1.07458i −0.843398 0.537290i \(-0.819448\pi\)
0.843398 0.537290i \(-0.180552\pi\)
\(458\) 0 0
\(459\) −10.1300 −0.472828
\(460\) 0 0
\(461\) 13.4739i 0.627541i −0.949499 0.313770i \(-0.898408\pi\)
0.949499 0.313770i \(-0.101592\pi\)
\(462\) 0 0
\(463\) 23.5879i 1.09622i 0.836405 + 0.548112i \(0.184653\pi\)
−0.836405 + 0.548112i \(0.815347\pi\)
\(464\) 0 0
\(465\) 13.6719 0.634018
\(466\) 0 0
\(467\) 27.0362 1.25109 0.625544 0.780189i \(-0.284877\pi\)
0.625544 + 0.780189i \(0.284877\pi\)
\(468\) 0 0
\(469\) 10.3012 0.475664
\(470\) 0 0
\(471\) −11.4753 −0.528753
\(472\) 0 0
\(473\) 10.1237i 0.465487i
\(474\) 0 0
\(475\) 5.87809i 0.269705i
\(476\) 0 0
\(477\) 16.3716 0.749602
\(478\) 0 0
\(479\) 7.28707i 0.332955i 0.986045 + 0.166477i \(0.0532393\pi\)
−0.986045 + 0.166477i \(0.946761\pi\)
\(480\) 0 0
\(481\) −22.5175 14.2991i −1.02671 0.651981i
\(482\) 0 0
\(483\) 0.299919i 0.0136468i
\(484\) 0 0
\(485\) −7.73755 −0.351344
\(486\) 0 0
\(487\) 19.3351i 0.876155i −0.898937 0.438077i \(-0.855660\pi\)
0.898937 0.438077i \(-0.144340\pi\)
\(488\) 0 0
\(489\) 9.92096i 0.448642i
\(490\) 0 0
\(491\) 2.36812 0.106872 0.0534358 0.998571i \(-0.482983\pi\)
0.0534358 + 0.998571i \(0.482983\pi\)
\(492\) 0 0
\(493\) −6.85110 −0.308558
\(494\) 0 0
\(495\) −3.95316 −0.177681
\(496\) 0 0
\(497\) −0.127032 −0.00569814
\(498\) 0 0
\(499\) 39.6681i 1.77579i 0.460048 + 0.887894i \(0.347832\pi\)
−0.460048 + 0.887894i \(0.652168\pi\)
\(500\) 0 0
\(501\) 9.50499i 0.424652i
\(502\) 0 0
\(503\) 4.75868 0.212179 0.106089 0.994357i \(-0.466167\pi\)
0.106089 + 0.994357i \(0.466167\pi\)
\(504\) 0 0
\(505\) 27.8733i 1.24035i
\(506\) 0 0
\(507\) 4.52797 + 9.63666i 0.201094 + 0.427979i
\(508\) 0 0
\(509\) 4.79819i 0.212676i 0.994330 + 0.106338i \(0.0339126\pi\)
−0.994330 + 0.106338i \(0.966087\pi\)
\(510\) 0 0
\(511\) 11.1281 0.492277
\(512\) 0 0
\(513\) 12.1055i 0.534469i
\(514\) 0 0
\(515\) 9.52688i 0.419804i
\(516\) 0 0
\(517\) −10.4378 −0.459052
\(518\) 0 0
\(519\) −11.8410 −0.519760
\(520\) 0 0
\(521\) −32.9554 −1.44380 −0.721902 0.691996i \(-0.756732\pi\)
−0.721902 + 0.691996i \(0.756732\pi\)
\(522\) 0 0
\(523\) −0.616555 −0.0269601 −0.0134800 0.999909i \(-0.504291\pi\)
−0.0134800 + 0.999909i \(0.504291\pi\)
\(524\) 0 0
\(525\) 1.73587i 0.0757598i
\(526\) 0 0
\(527\) 22.8263i 0.994328i
\(528\) 0 0
\(529\) −22.8659 −0.994170
\(530\) 0 0
\(531\) 27.7232i 1.20309i
\(532\) 0 0
\(533\) 29.4466 + 18.6992i 1.27548 + 0.809951i
\(534\) 0 0
\(535\) 10.7360i 0.464158i
\(536\) 0 0
\(537\) 1.69260 0.0730410
\(538\) 0 0
\(539\) 1.00000i 0.0430730i
\(540\) 0 0
\(541\) 35.2915i 1.51730i 0.651499 + 0.758650i \(0.274141\pi\)
−0.651499 + 0.758650i \(0.725859\pi\)
\(542\) 0 0
\(543\) −7.74170 −0.332228
\(544\) 0 0
\(545\) −0.755832 −0.0323763
\(546\) 0 0
\(547\) −27.8694 −1.19161 −0.595804 0.803130i \(-0.703166\pi\)
−0.595804 + 0.803130i \(0.703166\pi\)
\(548\) 0 0
\(549\) 19.1337 0.816608
\(550\) 0 0
\(551\) 8.18713i 0.348784i
\(552\) 0 0
\(553\) 14.2135i 0.604421i
\(554\) 0 0
\(555\) −10.2839 −0.436527
\(556\) 0 0
\(557\) 20.7880i 0.880815i −0.897798 0.440408i \(-0.854834\pi\)
0.897798 0.440408i \(-0.145166\pi\)
\(558\) 0 0
\(559\) 19.5672 30.8136i 0.827606 1.30328i
\(560\) 0 0
\(561\) 1.90085i 0.0802542i
\(562\) 0 0
\(563\) 36.3083 1.53021 0.765106 0.643905i \(-0.222687\pi\)
0.765106 + 0.643905i \(0.222687\pi\)
\(564\) 0 0
\(565\) 2.86843i 0.120676i
\(566\) 0 0
\(567\) 3.41266i 0.143318i
\(568\) 0 0
\(569\) −30.4117 −1.27493 −0.637463 0.770481i \(-0.720016\pi\)
−0.637463 + 0.770481i \(0.720016\pi\)
\(570\) 0 0
\(571\) 27.8569 1.16578 0.582888 0.812553i \(-0.301923\pi\)
0.582888 + 0.812553i \(0.301923\pi\)
\(572\) 0 0
\(573\) 0.158204 0.00660906
\(574\) 0 0
\(575\) −0.776104 −0.0323658
\(576\) 0 0
\(577\) 21.5422i 0.896812i −0.893830 0.448406i \(-0.851992\pi\)
0.893830 0.448406i \(-0.148008\pi\)
\(578\) 0 0
\(579\) 17.2283i 0.715984i
\(580\) 0 0
\(581\) −4.44302 −0.184327
\(582\) 0 0
\(583\) 7.02888i 0.291106i
\(584\) 0 0
\(585\) −12.0323 7.64073i −0.497474 0.315905i
\(586\) 0 0
\(587\) 8.25568i 0.340748i −0.985379 0.170374i \(-0.945502\pi\)
0.985379 0.170374i \(-0.0544977\pi\)
\(588\) 0 0
\(589\) −27.2776 −1.12395
\(590\) 0 0
\(591\) 5.44694i 0.224057i
\(592\) 0 0
\(593\) 0.692417i 0.0284342i −0.999899 0.0142171i \(-0.995474\pi\)
0.999899 0.0142171i \(-0.00452559\pi\)
\(594\) 0 0
\(595\) 3.93902 0.161484
\(596\) 0 0
\(597\) −15.9382 −0.652308
\(598\) 0 0
\(599\) −13.4964 −0.551450 −0.275725 0.961237i \(-0.588918\pi\)
−0.275725 + 0.961237i \(0.588918\pi\)
\(600\) 0 0
\(601\) 40.9950 1.67222 0.836111 0.548560i \(-0.184824\pi\)
0.836111 + 0.548560i \(0.184824\pi\)
\(602\) 0 0
\(603\) 23.9933i 0.977084i
\(604\) 0 0
\(605\) 1.69723i 0.0690021i
\(606\) 0 0
\(607\) −9.68506 −0.393105 −0.196552 0.980493i \(-0.562975\pi\)
−0.196552 + 0.980493i \(0.562975\pi\)
\(608\) 0 0
\(609\) 2.41776i 0.0979727i
\(610\) 0 0
\(611\) −31.7696 20.1743i −1.28526 0.816164i
\(612\) 0 0
\(613\) 47.2799i 1.90962i 0.297223 + 0.954808i \(0.403940\pi\)
−0.297223 + 0.954808i \(0.596060\pi\)
\(614\) 0 0
\(615\) 13.4485 0.542294
\(616\) 0 0
\(617\) 12.4585i 0.501562i −0.968044 0.250781i \(-0.919313\pi\)
0.968044 0.250781i \(-0.0806873\pi\)
\(618\) 0 0
\(619\) 34.8410i 1.40038i 0.713958 + 0.700188i \(0.246901\pi\)
−0.713958 + 0.700188i \(0.753099\pi\)
\(620\) 0 0
\(621\) −1.59832 −0.0641385
\(622\) 0 0
\(623\) 5.15887 0.206686
\(624\) 0 0
\(625\) −9.91095 −0.396438
\(626\) 0 0
\(627\) −2.27154 −0.0907165
\(628\) 0 0
\(629\) 17.1698i 0.684604i
\(630\) 0 0
\(631\) 30.9578i 1.23241i −0.787586 0.616205i \(-0.788669\pi\)
0.787586 0.616205i \(-0.211331\pi\)
\(632\) 0 0
\(633\) −8.28477 −0.329290
\(634\) 0 0
\(635\) 22.6220i 0.897726i
\(636\) 0 0
\(637\) 1.93282 3.04372i 0.0765810 0.120596i
\(638\) 0 0
\(639\) 0.295880i 0.0117048i
\(640\) 0 0
\(641\) −37.1952 −1.46912 −0.734560 0.678543i \(-0.762612\pi\)
−0.734560 + 0.678543i \(0.762612\pi\)
\(642\) 0 0
\(643\) 0.449713i 0.0177349i 0.999961 + 0.00886747i \(0.00282264\pi\)
−0.999961 + 0.00886747i \(0.997177\pi\)
\(644\) 0 0
\(645\) 14.0728i 0.554115i
\(646\) 0 0
\(647\) −8.79899 −0.345924 −0.172962 0.984928i \(-0.555334\pi\)
−0.172962 + 0.984928i \(0.555334\pi\)
\(648\) 0 0
\(649\) −11.9025 −0.467216
\(650\) 0 0
\(651\) 8.05542 0.315717
\(652\) 0 0
\(653\) −36.0064 −1.40904 −0.704519 0.709685i \(-0.748837\pi\)
−0.704519 + 0.709685i \(0.748837\pi\)
\(654\) 0 0
\(655\) 25.7880i 1.00762i
\(656\) 0 0
\(657\) 25.9194i 1.01121i
\(658\) 0 0
\(659\) 0.969131 0.0377520 0.0188760 0.999822i \(-0.493991\pi\)
0.0188760 + 0.999822i \(0.493991\pi\)
\(660\) 0 0
\(661\) 2.92011i 0.113579i 0.998386 + 0.0567896i \(0.0180864\pi\)
−0.998386 + 0.0567896i \(0.981914\pi\)
\(662\) 0 0
\(663\) −3.67401 + 5.78567i −0.142687 + 0.224697i
\(664\) 0 0
\(665\) 4.70716i 0.182536i
\(666\) 0 0
\(667\) −1.08097 −0.0418555
\(668\) 0 0
\(669\) 16.9082i 0.653710i
\(670\) 0 0
\(671\) 8.21477i 0.317128i
\(672\) 0 0
\(673\) 18.8429 0.726342 0.363171 0.931723i \(-0.381694\pi\)
0.363171 + 0.931723i \(0.381694\pi\)
\(674\) 0 0
\(675\) 9.25080 0.356063
\(676\) 0 0
\(677\) −19.6292 −0.754410 −0.377205 0.926130i \(-0.623115\pi\)
−0.377205 + 0.926130i \(0.623115\pi\)
\(678\) 0 0
\(679\) −4.55893 −0.174956
\(680\) 0 0
\(681\) 23.8118i 0.912470i
\(682\) 0 0
\(683\) 42.0865i 1.61040i 0.593006 + 0.805198i \(0.297941\pi\)
−0.593006 + 0.805198i \(0.702059\pi\)
\(684\) 0 0
\(685\) −5.17822 −0.197849
\(686\) 0 0
\(687\) 1.41274i 0.0538995i
\(688\) 0 0
\(689\) 13.5855 21.3939i 0.517568 0.815043i
\(690\) 0 0
\(691\) 0.595349i 0.0226481i 0.999936 + 0.0113241i \(0.00360464\pi\)
−0.999936 + 0.0113241i \(0.996395\pi\)
\(692\) 0 0
\(693\) −2.32919 −0.0884785
\(694\) 0 0
\(695\) 25.6537i 0.973102i
\(696\) 0 0
\(697\) 22.4533i 0.850478i
\(698\) 0 0
\(699\) −7.95004 −0.300698
\(700\) 0 0
\(701\) −8.54504 −0.322742 −0.161371 0.986894i \(-0.551592\pi\)
−0.161371 + 0.986894i \(0.551592\pi\)
\(702\) 0 0
\(703\) 20.5180 0.773853
\(704\) 0 0
\(705\) −14.5094 −0.546454
\(706\) 0 0
\(707\) 16.4229i 0.617646i
\(708\) 0 0
\(709\) 2.69377i 0.101167i −0.998720 0.0505833i \(-0.983892\pi\)
0.998720 0.0505833i \(-0.0161080\pi\)
\(710\) 0 0
\(711\) −33.1060 −1.24157
\(712\) 0 0
\(713\) 3.60155i 0.134879i
\(714\) 0 0
\(715\) −3.28043 + 5.16588i −0.122681 + 0.193193i
\(716\) 0 0
\(717\) 7.52465i 0.281013i
\(718\) 0 0
\(719\) −7.17053 −0.267416 −0.133708 0.991021i \(-0.542688\pi\)
−0.133708 + 0.991021i \(0.542688\pi\)
\(720\) 0 0
\(721\) 5.61320i 0.209047i
\(722\) 0 0
\(723\) 12.0769i 0.449143i
\(724\) 0 0
\(725\) 6.25648 0.232360
\(726\) 0 0
\(727\) −11.7863 −0.437128 −0.218564 0.975823i \(-0.570137\pi\)
−0.218564 + 0.975823i \(0.570137\pi\)
\(728\) 0 0
\(729\) 2.77595 0.102813
\(730\) 0 0
\(731\) 23.4956 0.869016
\(732\) 0 0
\(733\) 16.2355i 0.599673i 0.953991 + 0.299837i \(0.0969321\pi\)
−0.953991 + 0.299837i \(0.903068\pi\)
\(734\) 0 0
\(735\) 1.39008i 0.0512740i
\(736\) 0 0
\(737\) −10.3012 −0.379448
\(738\) 0 0
\(739\) 4.53787i 0.166928i −0.996511 0.0834642i \(-0.973402\pi\)
0.996511 0.0834642i \(-0.0265984\pi\)
\(740\) 0 0
\(741\) −6.91392 4.39047i −0.253989 0.161288i
\(742\) 0 0
\(743\) 7.32658i 0.268786i −0.990928 0.134393i \(-0.957092\pi\)
0.990928 0.134393i \(-0.0429085\pi\)
\(744\) 0 0
\(745\) 6.91454 0.253329
\(746\) 0 0
\(747\) 10.3486i 0.378636i
\(748\) 0 0
\(749\) 6.32562i 0.231133i
\(750\) 0 0
\(751\) −7.03656 −0.256768 −0.128384 0.991725i \(-0.540979\pi\)
−0.128384 + 0.991725i \(0.540979\pi\)
\(752\) 0 0
\(753\) −24.5481 −0.894582
\(754\) 0 0
\(755\) 29.6340 1.07849
\(756\) 0 0
\(757\) 13.4980 0.490594 0.245297 0.969448i \(-0.421114\pi\)
0.245297 + 0.969448i \(0.421114\pi\)
\(758\) 0 0
\(759\) 0.299919i 0.0108864i
\(760\) 0 0
\(761\) 13.4622i 0.488005i −0.969775 0.244002i \(-0.921540\pi\)
0.969775 0.244002i \(-0.0784605\pi\)
\(762\) 0 0
\(763\) −0.445333 −0.0161222
\(764\) 0 0
\(765\) 9.17470i 0.331712i
\(766\) 0 0
\(767\) −36.2280 23.0054i −1.30812 0.830678i
\(768\) 0 0
\(769\) 11.3243i 0.408366i 0.978933 + 0.204183i \(0.0654538\pi\)
−0.978933 + 0.204183i \(0.934546\pi\)
\(770\) 0 0
\(771\) 12.0317 0.433311
\(772\) 0 0
\(773\) 39.8536i 1.43343i −0.697364 0.716717i \(-0.745644\pi\)
0.697364 0.716717i \(-0.254356\pi\)
\(774\) 0 0
\(775\) 20.8451i 0.748778i
\(776\) 0 0
\(777\) −6.05924 −0.217374
\(778\) 0 0
\(779\) −26.8318 −0.961351
\(780\) 0 0
\(781\) 0.127032 0.00454555
\(782\) 0 0
\(783\) 12.8847 0.460462
\(784\) 0 0
\(785\) 23.7795i 0.848726i
\(786\) 0 0
\(787\) 18.8270i 0.671109i 0.942021 + 0.335555i \(0.108924\pi\)
−0.942021 + 0.335555i \(0.891076\pi\)
\(788\) 0 0
\(789\) −6.93948 −0.247052
\(790\) 0 0
\(791\) 1.69007i 0.0600918i
\(792\) 0 0
\(793\) 15.8777 25.0035i 0.563832 0.887899i
\(794\)