Properties

Label 1008.2.cs.q.271.2
Level 1008
Weight 2
Character 1008.271
Analytic conductor 8.049
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cs (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 271.2
Root \(-1.32288 + 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 1008.271
Dual form 1008.2.cs.q.703.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.50000 + 0.866025i) q^{5} +2.64575 q^{7} +O(q^{10})\) \(q+(1.50000 + 0.866025i) q^{5} +2.64575 q^{7} +(3.96863 - 2.29129i) q^{11} -3.46410i q^{13} +(-4.50000 + 2.59808i) q^{17} +(1.32288 - 2.29129i) q^{19} +(3.96863 + 2.29129i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(-1.32288 - 2.29129i) q^{31} +(3.96863 + 2.29129i) q^{35} +(-3.50000 + 6.06218i) q^{37} -3.46410i q^{41} +9.16515i q^{43} +(3.96863 - 6.87386i) q^{47} +7.00000 q^{49} +(1.50000 + 2.59808i) q^{53} +7.93725 q^{55} +(3.96863 + 6.87386i) q^{59} +(1.50000 + 0.866025i) q^{61} +(3.00000 - 5.19615i) q^{65} +(3.96863 - 2.29129i) q^{67} -9.16515i q^{71} +(-4.50000 + 2.59808i) q^{73} +(10.5000 - 6.06218i) q^{77} +(-3.96863 - 2.29129i) q^{79} -9.00000 q^{85} +(-1.50000 - 0.866025i) q^{89} -9.16515i q^{91} +(3.96863 - 2.29129i) q^{95} +3.46410i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{5} + O(q^{10}) \) \( 4q + 6q^{5} - 18q^{17} - 4q^{25} - 14q^{37} + 28q^{49} + 6q^{53} + 6q^{61} + 12q^{65} - 18q^{73} + 42q^{77} - 36q^{85} - 6q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) 1.50000 + 0.866025i 0.670820 + 0.387298i 0.796387 0.604787i \(-0.206742\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) 0 0
\(7\) 2.64575 1.00000
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.96863 2.29129i 1.19659 0.690849i 0.236794 0.971560i \(-0.423903\pi\)
0.959792 + 0.280711i \(0.0905701\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i −0.877058 0.480384i \(-0.840497\pi\)
0.877058 0.480384i \(-0.159503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.50000 + 2.59808i −1.09141 + 0.630126i −0.933952 0.357400i \(-0.883663\pi\)
−0.157459 + 0.987526i \(0.550330\pi\)
\(18\) 0 0
\(19\) 1.32288 2.29129i 0.303488 0.525657i −0.673435 0.739246i \(-0.735182\pi\)
0.976924 + 0.213589i \(0.0685153\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.96863 + 2.29129i 0.827516 + 0.477767i 0.853001 0.521909i \(-0.174780\pi\)
−0.0254855 + 0.999675i \(0.508113\pi\)
\(24\) 0 0
\(25\) −1.00000 1.73205i −0.200000 0.346410i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −1.32288 2.29129i −0.237595 0.411527i 0.722428 0.691446i \(-0.243026\pi\)
−0.960024 + 0.279918i \(0.909693\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.96863 + 2.29129i 0.670820 + 0.387298i
\(36\) 0 0
\(37\) −3.50000 + 6.06218i −0.575396 + 0.996616i 0.420602 + 0.907245i \(0.361819\pi\)
−0.995998 + 0.0893706i \(0.971514\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.46410i 0.541002i −0.962720 0.270501i \(-0.912811\pi\)
0.962720 0.270501i \(-0.0871893\pi\)
\(42\) 0 0
\(43\) 9.16515i 1.39767i 0.715282 + 0.698836i \(0.246298\pi\)
−0.715282 + 0.698836i \(0.753702\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.96863 6.87386i 0.578884 1.00266i −0.416724 0.909033i \(-0.636822\pi\)
0.995608 0.0936230i \(-0.0298448\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.50000 + 2.59808i 0.206041 + 0.356873i 0.950464 0.310835i \(-0.100609\pi\)
−0.744423 + 0.667708i \(0.767275\pi\)
\(54\) 0 0
\(55\) 7.93725 1.07026
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.96863 + 6.87386i 0.516671 + 0.894901i 0.999813 + 0.0193585i \(0.00616237\pi\)
−0.483141 + 0.875542i \(0.660504\pi\)
\(60\) 0 0
\(61\) 1.50000 + 0.866025i 0.192055 + 0.110883i 0.592944 0.805243i \(-0.297965\pi\)
−0.400889 + 0.916127i \(0.631299\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 5.19615i 0.372104 0.644503i
\(66\) 0 0
\(67\) 3.96863 2.29129i 0.484845 0.279925i −0.237588 0.971366i \(-0.576357\pi\)
0.722433 + 0.691441i \(0.243024\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.16515i 1.08770i −0.839181 0.543852i \(-0.816965\pi\)
0.839181 0.543852i \(-0.183035\pi\)
\(72\) 0 0
\(73\) −4.50000 + 2.59808i −0.526685 + 0.304082i −0.739666 0.672975i \(-0.765016\pi\)
0.212980 + 0.977056i \(0.431683\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.5000 6.06218i 1.19659 0.690849i
\(78\) 0 0
\(79\) −3.96863 2.29129i −0.446505 0.257790i 0.259848 0.965650i \(-0.416328\pi\)
−0.706353 + 0.707860i \(0.749661\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.50000 0.866025i −0.159000 0.0917985i 0.418389 0.908268i \(-0.362595\pi\)
−0.577389 + 0.816469i \(0.695928\pi\)
\(90\) 0 0
\(91\) 9.16515i 0.960769i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.96863 2.29129i 0.407173 0.235081i
\(96\) 0 0
\(97\) 3.46410i 0.351726i 0.984415 + 0.175863i \(0.0562716\pi\)
−0.984415 + 0.175863i \(0.943728\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.50000 + 2.59808i −0.447767 + 0.258518i −0.706887 0.707327i \(-0.749901\pi\)
0.259120 + 0.965845i \(0.416568\pi\)
\(102\) 0 0
\(103\) −1.32288 + 2.29129i −0.130347 + 0.225767i −0.923810 0.382851i \(-0.874942\pi\)
0.793463 + 0.608618i \(0.208276\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.9059 + 6.87386i 1.15098 + 0.664521i 0.949127 0.314893i \(-0.101969\pi\)
0.201858 + 0.979415i \(0.435302\pi\)
\(108\) 0 0
\(109\) 3.50000 + 6.06218i 0.335239 + 0.580651i 0.983531 0.180741i \(-0.0578495\pi\)
−0.648292 + 0.761392i \(0.724516\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 3.96863 + 6.87386i 0.370076 + 0.640991i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −11.9059 + 6.87386i −1.09141 + 0.630126i
\(120\) 0 0
\(121\) 5.00000 8.66025i 0.454545 0.787296i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) 9.16515i 0.813276i −0.913589 0.406638i \(-0.866701\pi\)
0.913589 0.406638i \(-0.133299\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.96863 + 6.87386i −0.346741 + 0.600572i −0.985668 0.168694i \(-0.946045\pi\)
0.638928 + 0.769267i \(0.279378\pi\)
\(132\) 0 0
\(133\) 3.50000 6.06218i 0.303488 0.525657i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.5000 + 18.1865i 0.897076 + 1.55378i 0.831215 + 0.555952i \(0.187646\pi\)
0.0658609 + 0.997829i \(0.479021\pi\)
\(138\) 0 0
\(139\) 10.5830 0.897639 0.448819 0.893622i \(-0.351845\pi\)
0.448819 + 0.893622i \(0.351845\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −7.93725 13.7477i −0.663747 1.14964i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.50000 + 12.9904i −0.614424 + 1.06421i 0.376061 + 0.926595i \(0.377278\pi\)
−0.990485 + 0.137619i \(0.956055\pi\)
\(150\) 0 0
\(151\) −19.8431 + 11.4564i −1.61481 + 0.932312i −0.626578 + 0.779359i \(0.715545\pi\)
−0.988233 + 0.152953i \(0.951122\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.58258i 0.368081i
\(156\) 0 0
\(157\) −4.50000 + 2.59808i −0.359139 + 0.207349i −0.668703 0.743530i \(-0.733150\pi\)
0.309564 + 0.950879i \(0.399817\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 10.5000 + 6.06218i 0.827516 + 0.477767i
\(162\) 0 0
\(163\) −3.96863 2.29129i −0.310847 0.179468i 0.336459 0.941698i \(-0.390771\pi\)
−0.647305 + 0.762231i \(0.724104\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.8745 −1.22841 −0.614203 0.789148i \(-0.710522\pi\)
−0.614203 + 0.789148i \(0.710522\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −19.5000 11.2583i −1.48256 0.855955i −0.482754 0.875756i \(-0.660363\pi\)
−0.999804 + 0.0198012i \(0.993697\pi\)
\(174\) 0 0
\(175\) −2.64575 4.58258i −0.200000 0.346410i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −11.9059 + 6.87386i −0.889887 + 0.513777i −0.873906 0.486096i \(-0.838421\pi\)
−0.0159817 + 0.999872i \(0.505087\pi\)
\(180\) 0 0
\(181\) 20.7846i 1.54491i −0.635071 0.772454i \(-0.719029\pi\)
0.635071 0.772454i \(-0.280971\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −10.5000 + 6.06218i −0.771975 + 0.445700i
\(186\) 0 0
\(187\) −11.9059 + 20.6216i −0.870644 + 1.50800i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.8431 11.4564i −1.43580 0.828959i −0.438245 0.898856i \(-0.644400\pi\)
−0.997554 + 0.0698969i \(0.977733\pi\)
\(192\) 0 0
\(193\) 5.50000 + 9.52628i 0.395899 + 0.685717i 0.993215 0.116289i \(-0.0370998\pi\)
−0.597317 + 0.802005i \(0.703766\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −6.61438 11.4564i −0.468881 0.812125i 0.530486 0.847693i \(-0.322009\pi\)
−0.999367 + 0.0355680i \(0.988676\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3.00000 5.19615i 0.209529 0.362915i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.1244i 0.838659i
\(210\) 0 0
\(211\) 9.16515i 0.630955i 0.948933 + 0.315478i \(0.102165\pi\)
−0.948933 + 0.315478i \(0.897835\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −7.93725 + 13.7477i −0.541316 + 0.937587i
\(216\) 0 0
\(217\) −3.50000 6.06218i −0.237595 0.411527i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 9.00000 + 15.5885i 0.605406 + 1.04859i
\(222\) 0 0
\(223\) −10.5830 −0.708690 −0.354345 0.935115i \(-0.615296\pi\)
−0.354345 + 0.935115i \(0.615296\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.9059 + 20.6216i 0.790221 + 1.36870i 0.925830 + 0.377941i \(0.123368\pi\)
−0.135609 + 0.990762i \(0.543299\pi\)
\(228\) 0 0
\(229\) −1.50000 0.866025i −0.0991228 0.0572286i 0.449619 0.893220i \(-0.351560\pi\)
−0.548742 + 0.835992i \(0.684893\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.5000 18.1865i 0.687878 1.19144i −0.284645 0.958633i \(-0.591876\pi\)
0.972523 0.232806i \(-0.0747909\pi\)
\(234\) 0 0
\(235\) 11.9059 6.87386i 0.776654 0.448401i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −16.5000 + 9.52628i −1.06286 + 0.613642i −0.926222 0.376980i \(-0.876963\pi\)
−0.136637 + 0.990621i \(0.543629\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 10.5000 + 6.06218i 0.670820 + 0.387298i
\(246\) 0 0
\(247\) −7.93725 4.58258i −0.505035 0.291582i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −15.8745 −1.00199 −0.500995 0.865450i \(-0.667033\pi\)
−0.500995 + 0.865450i \(0.667033\pi\)
\(252\) 0 0
\(253\) 21.0000 1.32026
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.50000 + 0.866025i 0.0935674 + 0.0540212i 0.546054 0.837750i \(-0.316129\pi\)
−0.452486 + 0.891771i \(0.649463\pi\)
\(258\) 0 0
\(259\) −9.26013 + 16.0390i −0.575396 + 0.996616i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.96863 + 2.29129i −0.244716 + 0.141287i −0.617342 0.786695i \(-0.711791\pi\)
0.372626 + 0.927981i \(0.378457\pi\)
\(264\) 0 0
\(265\) 5.19615i 0.319197i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −25.5000 + 14.7224i −1.55476 + 0.897643i −0.557019 + 0.830500i \(0.688055\pi\)
−0.997743 + 0.0671428i \(0.978612\pi\)
\(270\) 0 0
\(271\) 14.5516 25.2042i 0.883949 1.53104i 0.0370348 0.999314i \(-0.488209\pi\)
0.846914 0.531730i \(-0.178458\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.93725 4.58258i −0.478634 0.276340i
\(276\) 0 0
\(277\) 8.50000 + 14.7224i 0.510716 + 0.884585i 0.999923 + 0.0124177i \(0.00395278\pi\)
−0.489207 + 0.872167i \(0.662714\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −6.61438 11.4564i −0.393184 0.681015i 0.599684 0.800237i \(-0.295293\pi\)
−0.992868 + 0.119223i \(0.961960\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 9.16515i 0.541002i
\(288\) 0 0
\(289\) 5.00000 8.66025i 0.294118 0.509427i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 20.7846i 1.21425i −0.794606 0.607125i \(-0.792323\pi\)
0.794606 0.607125i \(-0.207677\pi\)
\(294\) 0 0
\(295\) 13.7477i 0.800424i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.93725 13.7477i 0.459023 0.795052i
\(300\) 0 0
\(301\) 24.2487i 1.39767i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.50000 + 2.59808i 0.0858898 + 0.148765i
\(306\) 0 0
\(307\) −10.5830 −0.604004 −0.302002 0.953307i \(-0.597655\pi\)
−0.302002 + 0.953307i \(0.597655\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −11.9059 20.6216i −0.675121 1.16934i −0.976434 0.215818i \(-0.930758\pi\)
0.301313 0.953525i \(1.59742\pi\)
\(312\) 0 0
\(313\) −22.5000 12.9904i −1.27178 0.734260i −0.296453 0.955047i \(-0.595804\pi\)
−0.975322 + 0.220788i \(0.929137\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.50000 12.9904i 0.421242 0.729612i −0.574819 0.818280i \(-0.694928\pi\)
0.996061 + 0.0886679i \(0.0282610\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 13.7477i 0.764944i
\(324\) 0 0
\(325\) −6.00000 + 3.46410i −0.332820 + 0.192154i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 10.5000 18.1865i 0.578884 1.00266i
\(330\) 0 0
\(331\) 11.9059 + 6.87386i 0.654406 + 0.377822i 0.790142 0.612923i \(-0.210007\pi\)
−0.135736 + 0.990745i \(0.543340\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.93725 0.433659
\(336\) 0 0
\(337\) −28.0000 −1.52526 −0.762629 0.646837i \(-0.776092\pi\)
−0.762629 + 0.646837i \(0.776092\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −10.5000 6.06218i −0.568607 0.328285i
\(342\) 0 0
\(343\) 18.5203 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.96863 + 2.29129i −0.213047 + 0.123003i −0.602727 0.797948i \(-0.705919\pi\)
0.389680 + 0.920950i \(0.372586\pi\)
\(348\) 0 0
\(349\) 3.46410i 0.185429i 0.995693 + 0.0927146i \(0.0295544\pi\)
−0.995693 + 0.0927146i \(0.970446\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −16.5000 + 9.52628i −0.878206 + 0.507033i −0.870067 0.492934i \(-0.835924\pi\)
−0.00813978 + 0.999967i \(0.502591\pi\)
\(354\) 0 0
\(355\) 7.93725 13.7477i 0.421266 0.729654i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.9059 + 6.87386i 0.628368 + 0.362789i 0.780120 0.625630i \(-0.215158\pi\)
−0.151752 + 0.988419i \(0.548491\pi\)
\(360\) 0 0
\(361\) 6.00000 + 10.3923i 0.315789 + 0.546963i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −9.00000 −0.471082
\(366\) 0 0
\(367\) 17.1974 + 29.7867i 0.897696 + 1.55486i 0.830432 + 0.557120i \(0.188094\pi\)
0.0672642 + 0.997735i \(0.478573\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.96863 + 6.87386i 0.206041 + 0.356873i
\(372\) 0 0
\(373\) 0.500000 0.866025i 0.0258890 0.0448411i −0.852791 0.522253i \(-0.825092\pi\)
0.878680 + 0.477412i \(0.158425\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 18.3303i 0.941564i −0.882249 0.470782i \(-0.843972\pi\)
0.882249 0.470782i \(-0.156028\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.96863 6.87386i 0.202787 0.351238i −0.746638 0.665230i \(-0.768333\pi\)
0.949425 + 0.313992i \(0.101667\pi\)
\(384\) 0 0
\(385\) 21.0000 1.07026
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 10.5000 + 18.1865i 0.532371 + 0.922094i 0.999286 + 0.0377914i \(0.0120322\pi\)
−0.466915 + 0.884302i \(0.654634\pi\)
\(390\) 0 0
\(391\) −23.8118 −1.20421
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.96863 6.87386i −0.199683 0.345862i
\(396\) 0 0
\(397\) −19.5000 11.2583i −0.978677 0.565039i −0.0768065 0.997046i \(-0.524472\pi\)
−0.901870 + 0.432007i \(0.857806\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.5000 18.1865i 0.524345 0.908192i −0.475253 0.879849i \(-0.657644\pi\)
0.999598 0.0283431i \(-0.00902310\pi\)
\(402\) 0 0
\(403\) −7.93725 + 4.58258i −0.395383 + 0.228274i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 32.0780i 1.59005i
\(408\) 0 0
\(409\) 16.5000 9.52628i 0.815872 0.471044i −0.0331186 0.999451i \(-0.510544\pi\)
0.848991 + 0.528407i \(0.177211\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.5000 + 18.1865i 0.516671 + 0.894901i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 31.7490 1.55104 0.775520 0.631322i \(-0.217488\pi\)
0.775520 + 0.631322i \(0.217488\pi\)
\(420\) 0 0
\(421\) 16.0000 0.779792 0.389896 0.920859i \(-0.372511\pi\)
0.389896 + 0.920859i \(0.372511\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 9.00000 + 5.19615i 0.436564 + 0.252050i
\(426\) 0 0
\(427\) 3.96863 + 2.29129i 0.192055 + 0.110883i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 19.8431 11.4564i 0.955810 0.551837i 0.0609292 0.998142i \(-0.480594\pi\)
0.894881 + 0.446305i \(0.147260\pi\)
\(432\) 0 0
\(433\) 3.46410i 0.166474i −0.996530 0.0832370i \(-0.973474\pi\)
0.996530 0.0832370i \(-0.0265259\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 10.5000 6.06218i 0.502283 0.289993i
\(438\) 0 0
\(439\) −14.5516 + 25.2042i −0.694512 + 1.20293i 0.275834 + 0.961205i \(0.411046\pi\)
−0.970345 + 0.241724i \(0.922287\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.96863 2.29129i −0.188555 0.108862i 0.402751 0.915310i \(-0.368054\pi\)
−0.591306 + 0.806447i \(0.701387\pi\)
\(444\) 0 0
\(445\) −1.50000 2.59808i −0.0711068 0.123161i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) −7.93725 13.7477i −0.373751 0.647355i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 7.93725 13.7477i 0.372104 0.644503i
\(456\) 0 0
\(457\) 0.500000 0.866025i 0.0233890 0.0405110i −0.854094 0.520119i \(-0.825888\pi\)
0.877483 + 0.479608i \(0.159221\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.46410i 0.161339i −0.996741 0.0806696i \(-0.974294\pi\)
0.996741 0.0806696i \(-0.0257059\pi\)
\(462\) 0 0
\(463\) 27.4955i 1.27782i −0.769281 0.638911i \(-0.779385\pi\)
0.769281 0.638911i \(-0.220615\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.96863 + 6.87386i −0.183646 + 0.318084i −0.943119 0.332454i \(-0.892123\pi\)
0.759473 + 0.650538i \(0.225457\pi\)
\(468\) 0 0
\(469\) 10.5000 6.06218i 0.484845 0.279925i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 21.0000 + 36.3731i 0.965581 + 1.67244i
\(474\) 0 0
\(475\) −5.29150 −0.242791
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.96863 + 6.87386i 0.181331 + 0.314075i 0.942334 0.334674i \(-0.108626\pi\)
−0.761003 + 0.648748i \(0.775293\pi\)
\(480\) 0 0
\(481\) 21.0000 + 12.1244i 0.957518 + 0.552823i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.00000 + 5.19615i −0.136223 + 0.235945i
\(486\) 0 0
\(487\) 35.7176 20.6216i 1.61852 0.934453i 0.631218 0.775606i \(-0.282555\pi\)
0.987303 0.158848i \(-0.0507779\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 18.3303i 0.827235i −0.910451 0.413617i \(-0.864265\pi\)
0.910451 0.413617i \(-0.135735\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.2487i 1.08770i
\(498\) 0 0
\(499\) 27.7804 + 16.0390i 1.24362 + 0.718005i 0.969830 0.243784i \(-0.0783888\pi\)
0.273791 + 0.961789i \(0.411722\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 31.7490 1.41562 0.707809 0.706404i \(-0.249684\pi\)
0.707809 + 0.706404i \(0.249684\pi\)
\(504\) 0 0
\(505\) −9.00000 −0.400495
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −22.5000 12.9904i −0.997295 0.575789i −0.0898481 0.995955i \(-0.528638\pi\)
−0.907447 + 0.420167i \(0.861972\pi\)
\(510\) 0 0
\(511\) −11.9059 + 6.87386i −0.526685 + 0.304082i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.96863 + 2.29129i −0.174879 + 0.100966i
\(516\) 0 0
\(517\) 36.3731i 1.59969i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −37.5000 + 21.6506i −1.64290 + 0.948532i −0.663111 + 0.748521i \(0.730764\pi\)
−0.979794 + 0.200011i \(0.935902\pi\)
\(522\) 0 0
\(523\) 6.61438 11.4564i 0.289227 0.500955i −0.684399 0.729108i \(-0.739935\pi\)
0.973625 + 0.228153i \(0.0732686\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.9059 + 6.87386i 0.518628 + 0.299430i
\(528\) 0 0
\(529\) −1.00000 1.73205i −0.0434783 0.0753066i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 11.9059 + 20.6216i 0.514736 + 0.891549i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 27.7804 16.0390i 1.19659 0.690849i
\(540\) 0 0
\(541\) −6.50000 + 11.2583i −0.279457 + 0.484033i −0.971250 0.238062i \(-0.923488\pi\)
0.691793 + 0.722096i \(0.256821\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 12.1244i 0.519350i
\(546\) 0 0
\(547\) 18.3303i 0.783747i 0.920019 + 0.391874i \(0.128173\pi\)
−0.920019 + 0.391874i \(0.871827\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −10.5000 6.06218i −0.446505 0.257790i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.50000 2.59808i −0.0635570 0.110084i 0.832496 0.554031i \(-0.186911\pi\)
−0.896053 + 0.443947i \(0.853578\pi\)
\(558\) 0 0
\(559\) 31.7490 1.34284
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.96863 + 6.87386i 0.167258 + 0.289699i 0.937455 0.348107i \(-0.113175\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(564\) 0 0
\(565\) −18.0000 10.3923i −0.757266 0.437208i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10.5000 + 18.1865i −0.440183 + 0.762419i −0.997703 0.0677445i \(-0.978420\pi\)
0.557520 + 0.830164i \(0.311753\pi\)
\(570\) 0 0
\(571\) 11.9059 6.87386i 0.498246 0.287662i −0.229743 0.973251i \(-0.573789\pi\)
0.727989 + 0.685589i \(0.240455\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 9.16515i 0.382213i
\(576\) 0 0
\(577\) −16.5000 + 9.52628i −0.686904 + 0.396584i −0.802451 0.596718i \(-0.796471\pi\)
0.115547 + 0.993302i \(0.463138\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 11.9059 + 6.87386i 0.493091 + 0.284686i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 31.7490 1.31042 0.655211 0.755446i \(-0.272580\pi\)
0.655211 + 0.755446i \(0.272580\pi\)
\(588\) 0 0
\(589\) −7.00000 −0.288430
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 19.5000 + 11.2583i 0.800769 + 0.462324i 0.843740 0.536752i \(-0.180349\pi\)
−0.0429710 + 0.999076i \(0.513682\pi\)
\(594\) 0 0
\(595\) −23.8118 −0.976187
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.96863 + 2.29129i −0.162154 + 0.0936195i −0.578881 0.815412i \(-0.696510\pi\)
0.416727 + 0.909032i \(0.363177\pi\)
\(600\) 0 0
\(601\) 3.46410i 0.141304i 0.997501 + 0.0706518i \(0.0225079\pi\)
−0.997501 + 0.0706518i \(0.977492\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 15.0000 8.66025i 0.609837 0.352089i
\(606\) 0 0
\(607\) 1.32288 2.29129i 0.0536939 0.0930005i −0.837929 0.545779i \(-0.816234\pi\)
0.891623 + 0.452778i \(0.149567\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −23.8118 13.7477i −0.963321 0.556174i
\(612\) 0 0
\(613\) 3.50000 + 6.06218i 0.141364 + 0.244849i 0.928010 0.372554i \(-0.121518\pi\)
−0.786647 + 0.617403i \(0.788185\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −9.26013 16.0390i −0.372196 0.644662i 0.617707 0.786408i \(-0.288062\pi\)
−0.989903 + 0.141746i \(0.954728\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.96863 2.29129i −0.159000 0.0917985i
\(624\) 0 0
\(625\) 5.50000 9.52628i 0.220000 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 36.3731i 1.45029i
\(630\) 0 0
\(631\) 45.8258i 1.82429i −0.409863 0.912147i \(-0.634423\pi\)
0.409863 0.912147i \(-0.365577\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 7.93725 13.7477i 0.314980 0.545562i
\(636\) 0 0
\(637\) 24.2487i 0.960769i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10.5000 18.1865i −0.414725 0.718325i 0.580674 0.814136i \(-0.302789\pi\)
−0.995400 + 0.0958109i \(0.969456\pi\)
\(642\) 0 0
\(643\) 21.1660 0.834706 0.417353 0.908744i \(-0.362958\pi\)
0.417353 + 0.908744i \(0.362958\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.8431 + 34.3693i 0.780114 + 1.35120i 0.931875 + 0.362780i \(0.118172\pi\)
−0.151761 + 0.988417i \(0.548494\pi\)
\(648\) 0 0
\(649\) 31.5000 + 18.1865i 1.23648 + 0.713884i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −10.5000 + 18.1865i −0.410897 + 0.711694i −0.994988 0.0999939i \(-0.968118\pi\)
0.584091 + 0.811688i \(0.301451\pi\)
\(654\) 0 0
\(655\) −11.9059 + 6.87386i −0.465201 + 0.268584i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9.16515i 0.357024i −0.983938 0.178512i \(-0.942872\pi\)
0.983938 0.178512i \(-0.0571283\pi\)
\(660\) 0 0
\(661\) −4.50000 + 2.59808i −0.175030 + 0.101053i −0.584955 0.811065i \(-0.698888\pi\)
0.409926 + 0.912119i \(0.365555\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 10.5000 6.06218i 0.407173 0.235081i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 7.93725 0.306414
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.50000 + 0.866025i 0.0576497 + 0.0332841i 0.528548 0.848904i \(-0.322737\pi\)
−0.470898 + 0.882188i \(0.656070\pi\)
\(678\) 0 0
\(679\) 9.16515i 0.351726i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 3.96863 2.29129i 0.151855 0.0876737i −0.422147 0.906527i \(-0.638723\pi\)
0.574002 + 0.818854i \(0.305390\pi\)
\(684\) 0 0
\(685\) 36.3731i 1.38974i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.00000 5.19615i 0.342873 0.197958i
\(690\) 0 0
\(691\) −1.32288 + 2.29129i −0.0503246 + 0.0871647i −0.890090 0.455784i \(-0.849359\pi\)
0.839766 + 0.542949i \(0.182692\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 15.8745 + 9.16515i 0.602154 + 0.347654i
\(696\) 0 0
\(697\) 9.00000 + 15.5885i 0.340899 + 0.590455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) 9.26013 + 16.0390i 0.349252 + 0.604923i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −11.9059 + 6.87386i −0.447767 + 0.258518i
\(708\) 0 0
\(709\) −17.5000 + 30.3109i −0.657226 + 1.13835i 0.324104 + 0.946021i \(0.394937\pi\)
−0.981331 + 0.192328i \(0.938396\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.1244i 0.454061i
\(714\) 0 0
\(715\) 27.4955i 1.02827i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −3.96863 + 6.87386i −0.148005 + 0.256352i −0.930490 0.366317i \(-0.880618\pi\)
0.782485 + 0.622669i \(0.213952\pi\)
\(720\) 0 0
\(721\) −3.50000 + 6.06218i −0.130347 + 0.225767i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −21.1660 −0.785004 −0.392502 0.919751i \(-0.628390\pi\)
−0.392502 + 0.919751i \(0.628390\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −23.8118 41.2432i −0.880710 1.52543i
\(732\) 0 0
\(733\) 19.5000 + 11.2583i 0.720249 + 0.415836i 0.814844 0.579680i \(-0.196822\pi\)
−0.0945954 + 0.995516i \(0.530156\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.5000 18.1865i 0.386772 0.669910i
\(738\) 0 0
\(739\) −19.8431 + 11.4564i −0.729942 + 0.421432i −0.818401 0.574648i \(-0.805139\pi\)
0.0884593 + 0.996080i \(0.471806\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 45.8258i 1.68118i 0.541669 + 0.840592i \(0.317793\pi\)
−0.541669 + 0.840592i \(0.682207\pi\)
\(744\) 0 0
\(745\) −22.5000 + 12.9904i −0.824336 + 0.475931i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 31.5000 + 18.1865i 1.15098 + 0.664521i
\(750\) 0 0
\(751\) −11.9059 6.87386i −0.434452 0.250831i 0.266790 0.963755i \(-0.414037\pi\)
−0.701241 + 0.712924i \(0.747370\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −39.6863 −1.44433
\(756\) 0 0
\(757\) −28.0000 −1.01768 −0.508839 0.860862i \(-0.669925\pi\)
−0.508839 + 0.860862i \(0.669925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −22.5000 12.9904i −0.815624 0.470901i 0.0332809 0.999446i \(-0.489404\pi\)
−0.848905 + 0.528545i \(0.822738\pi\)
\(762\) 0 0
\(763\) 9.26013 + 16.0390i 0.335239 + 0.580651i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 23.8118 13.7477i 0.859793 0.496402i
\(768\) 0 0
\(769\) 51.9615i 1.87378i −0.349624 0.936890i \(-0.613691\pi\)
0.349624 0.936890i \(-0.386309\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25.5000 14.7224i 0.917171 0.529529i 0.0344397 0.999407i \(-0.489035\pi\)
0.882732 + 0.469878i \(0.155702\pi\)
\(774\) 0 0
\(775\) −2.64575 + 4.58258i −0.0950382 + 0.164611i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −7.93725 4.58258i −0.284382 0.164188i
\(780\) 0 0
\(781\) −21.0000 36.3731i −0.751439 1.30153i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9.00000 −0.321224
\(786\) 0 0
\(787\) 6.61438 + 11.4564i 0.235777 + 0.408378i 0.959498 0.281715i \(-0.0909031\pi\)
−0.723721 + 0.690093i \(0.757570\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −31.7490 −1.12887
\(792\) 0 0
\(793\) 3.00000 5.19615i 0.106533 0.184521i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.46410i 0.122705i 0.998116 + 0.0613524i \(0.0195413\pi\)
−0.998116 + 0.0613524i \(0.980459\pi\)
\(798\) 0 0
\(799\) 41.2432i 1.45908i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.9059 + 20.6216i −0.420149 + 0.727720i
\(804\) 0 0
\(805\) 10.5000 + 18.1865i 0.370076 + 0.640991i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −22.5000 38.9711i −0.791058 1.37015i −0.925312 0.379206i \(-0.876197\pi\)
0.134255 0.990947i \(-0.457136\pi\)
\(810\) 0 0
\(811\) −21.1660 −0.743239 −0.371620