Properties

Label 1620.3.t.c.269.4
Level $1620$
Weight $3$
Character 1620.269
Analytic conductor $44.142$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.1485512441856.6
Defining polynomial: \( x^{8} - 24x^{6} + 455x^{4} - 2904x^{2} + 14641 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 269.4
Root \(3.54921 + 2.04914i\) of defining polynomial
Character \(\chi\) \(=\) 1620.269
Dual form 1620.3.t.c.1349.4

$q$-expansion

\(f(q)\) \(=\) \(q+(3.62266 + 3.44621i) q^{5} +(5.87367 + 3.39116i) q^{7} +O(q^{10})\) \(q+(3.62266 + 3.44621i) q^{5} +(5.87367 + 3.39116i) q^{7} +(8.57321 + 4.94975i) q^{11} +(-17.6210 + 10.1735i) q^{13} -19.1833 q^{17} +12.0000 q^{19} +(4.79583 + 8.30662i) q^{23} +(1.24734 + 24.9689i) q^{25} +(-7.34847 - 4.24264i) q^{29} +(19.0000 + 32.9090i) q^{31} +(9.59166 + 32.5269i) q^{35} -6.78233i q^{37} +(-60.0125 + 34.6482i) q^{41} +(-58.7367 - 33.9116i) q^{43} +(-38.3667 + 66.4530i) q^{47} +(-1.50000 - 2.59808i) q^{49} +(14.0000 + 47.4763i) q^{55} +(72.2599 - 41.7193i) q^{59} +(35.0000 - 60.6218i) q^{61} +(-98.8949 - 23.8705i) q^{65} +(93.9787 - 54.2586i) q^{67} -118.794i q^{71} -13.5647i q^{73} +(33.5708 + 58.1464i) q^{77} +(-15.0000 + 25.9808i) q^{79} +(-67.1416 + 116.293i) q^{83} +(-69.4947 - 66.1097i) q^{85} -32.5269i q^{89} -138.000 q^{91} +(43.4719 + 41.3545i) q^{95} +(82.2314 + 47.4763i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 96 q^{19} - 84 q^{25} + 152 q^{31} - 12 q^{49} + 112 q^{55} + 280 q^{61} - 120 q^{79} - 368 q^{85} - 1104 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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
\(5\) 3.62266 + 3.44621i 0.724532 + 0.689241i
\(6\) 0 0
\(7\) 5.87367 + 3.39116i 0.839096 + 0.484452i 0.856957 0.515388i \(-0.172352\pi\)
−0.0178610 + 0.999840i \(0.505686\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 8.57321 + 4.94975i 0.779383 + 0.449977i 0.836212 0.548407i \(-0.184765\pi\)
−0.0568285 + 0.998384i \(0.518099\pi\)
\(12\) 0 0
\(13\) −17.6210 + 10.1735i −1.35546 + 0.782577i −0.989008 0.147860i \(-0.952762\pi\)
−0.366454 + 0.930436i \(0.619428\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −19.1833 −1.12843 −0.564215 0.825628i \(-0.690821\pi\)
−0.564215 + 0.825628i \(0.690821\pi\)
\(18\) 0 0
\(19\) 12.0000 0.631579 0.315789 0.948829i \(-0.397731\pi\)
0.315789 + 0.948829i \(0.397731\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.79583 + 8.30662i 0.208514 + 0.361158i 0.951247 0.308431i \(-0.0998038\pi\)
−0.742732 + 0.669588i \(0.766471\pi\)
\(24\) 0 0
\(25\) 1.24734 + 24.9689i 0.0498936 + 0.998755i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.34847 4.24264i −0.253395 0.146298i 0.367923 0.929856i \(-0.380069\pi\)
−0.621318 + 0.783558i \(0.713402\pi\)
\(30\) 0 0
\(31\) 19.0000 + 32.9090i 0.612903 + 1.06158i 0.990748 + 0.135711i \(0.0433318\pi\)
−0.377845 + 0.925869i \(0.623335\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 9.59166 + 32.5269i 0.274048 + 0.929340i
\(36\) 0 0
\(37\) 6.78233i 0.183306i −0.995791 0.0916531i \(-0.970785\pi\)
0.995791 0.0916531i \(-0.0292151\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −60.0125 + 34.6482i −1.46372 + 0.845079i −0.999181 0.0404739i \(-0.987113\pi\)
−0.464539 + 0.885553i \(0.653780\pi\)
\(42\) 0 0
\(43\) −58.7367 33.9116i −1.36597 0.788643i −0.375559 0.926798i \(-0.622549\pi\)
−0.990411 + 0.138155i \(0.955883\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −38.3667 + 66.4530i −0.816312 + 1.41389i 0.0920704 + 0.995752i \(0.470652\pi\)
−0.908382 + 0.418141i \(0.862682\pi\)
\(48\) 0 0
\(49\) −1.50000 2.59808i −0.0306122 0.0530220i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 14.0000 + 47.4763i 0.254545 + 0.863206i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 72.2599 41.7193i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(60\) 0 0
\(61\) 35.0000 60.6218i 0.573770 0.993800i −0.422404 0.906408i \(-0.638813\pi\)
0.996174 0.0873918i \(-0.0278532\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −98.8949 23.8705i −1.52146 0.367238i
\(66\) 0 0
\(67\) 93.9787 54.2586i 1.40267 0.809830i 0.408002 0.912981i \(-0.366226\pi\)
0.994666 + 0.103151i \(0.0328924\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 118.794i 1.67315i −0.547849 0.836577i \(-0.684553\pi\)
0.547849 0.836577i \(-0.315447\pi\)
\(72\) 0 0
\(73\) 13.5647i 0.185817i −0.995675 0.0929086i \(-0.970384\pi\)
0.995675 0.0929086i \(-0.0296164\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 33.5708 + 58.1464i 0.435985 + 0.755148i
\(78\) 0 0
\(79\) −15.0000 + 25.9808i −0.189873 + 0.328870i −0.945208 0.326469i \(-0.894141\pi\)
0.755334 + 0.655339i \(0.227474\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −67.1416 + 116.293i −0.808935 + 1.40112i 0.104666 + 0.994507i \(0.466623\pi\)
−0.913602 + 0.406610i \(0.866711\pi\)
\(84\) 0 0
\(85\) −69.4947 66.1097i −0.817584 0.777761i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 32.5269i 0.365471i −0.983162 0.182735i \(-0.941505\pi\)
0.983162 0.182735i \(-0.0584952\pi\)
\(90\) 0 0
\(91\) −138.000 −1.51648
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 43.4719 + 41.3545i 0.457599 + 0.435310i
\(96\) 0 0
\(97\) 82.2314 + 47.4763i 0.847746 + 0.489446i 0.859890 0.510480i \(-0.170532\pi\)
−0.0121436 + 0.999926i \(0.503866\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 68.5857 + 39.5980i 0.679066 + 0.392059i 0.799503 0.600662i \(-0.205096\pi\)
−0.120437 + 0.992721i \(0.538430\pi\)
\(102\) 0 0
\(103\) −76.3577 + 44.0851i −0.741337 + 0.428011i −0.822555 0.568685i \(-0.807452\pi\)
0.0812182 + 0.996696i \(0.474119\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −57.5500 −0.537850 −0.268925 0.963161i \(-0.586668\pi\)
−0.268925 + 0.963161i \(0.586668\pi\)
\(108\) 0 0
\(109\) 74.0000 0.678899 0.339450 0.940624i \(-0.389759\pi\)
0.339450 + 0.940624i \(0.389759\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 86.3250 + 149.519i 0.763938 + 1.32318i 0.940806 + 0.338945i \(0.110070\pi\)
−0.176869 + 0.984234i \(0.556597\pi\)
\(114\) 0 0
\(115\) −11.2527 + 46.6195i −0.0978492 + 0.405387i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −112.677 65.0538i −0.946862 0.546671i
\(120\) 0 0
\(121\) −11.5000 19.9186i −0.0950413 0.164616i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −81.5291 + 94.7523i −0.652233 + 0.758018i
\(126\) 0 0
\(127\) 169.558i 1.33510i −0.744563 0.667552i \(-0.767342\pi\)
0.744563 0.667552i \(-0.232658\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −143.295 + 82.7315i −1.09386 + 0.631538i −0.934600 0.355699i \(-0.884243\pi\)
−0.159256 + 0.987237i \(0.550909\pi\)
\(132\) 0 0
\(133\) 70.4840 + 40.6940i 0.529955 + 0.305970i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −57.5500 + 99.6795i −0.420073 + 0.727587i −0.995946 0.0899515i \(-0.971329\pi\)
0.575873 + 0.817539i \(0.304662\pi\)
\(138\) 0 0
\(139\) −31.0000 53.6936i −0.223022 0.386285i 0.732702 0.680549i \(-0.238259\pi\)
−0.955724 + 0.294264i \(0.904925\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −201.425 −1.40857
\(144\) 0 0
\(145\) −12.0000 40.6940i −0.0827586 0.280648i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −137.171 + 79.1960i −0.920614 + 0.531517i −0.883831 0.467807i \(-0.845044\pi\)
−0.0367828 + 0.999323i \(0.511711\pi\)
\(150\) 0 0
\(151\) −35.0000 + 60.6218i −0.231788 + 0.401469i −0.958334 0.285649i \(-0.907791\pi\)
0.726546 + 0.687118i \(0.241124\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −44.5805 + 184.696i −0.287616 + 1.19159i
\(156\) 0 0
\(157\) 41.1157 23.7382i 0.261883 0.151198i −0.363310 0.931668i \(-0.618353\pi\)
0.625193 + 0.780470i \(0.285020\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 65.0538i 0.404061i
\(162\) 0 0
\(163\) 94.9526i 0.582531i 0.956642 + 0.291266i \(0.0940764\pi\)
−0.956642 + 0.291266i \(0.905924\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 100.712 + 174.439i 0.603069 + 1.04455i 0.992354 + 0.123428i \(0.0393888\pi\)
−0.389285 + 0.921117i \(0.627278\pi\)
\(168\) 0 0
\(169\) 122.500 212.176i 0.724852 1.25548i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −134.283 + 232.585i −0.776204 + 1.34442i 0.157911 + 0.987453i \(0.449524\pi\)
−0.934115 + 0.356971i \(0.883809\pi\)
\(174\) 0 0
\(175\) −77.3471 + 150.889i −0.441983 + 0.862222i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 134.350i 0.750560i 0.926911 + 0.375280i \(0.122453\pi\)
−0.926911 + 0.375280i \(0.877547\pi\)
\(180\) 0 0
\(181\) −22.0000 −0.121547 −0.0607735 0.998152i \(-0.519357\pi\)
−0.0607735 + 0.998152i \(0.519357\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 23.3733 24.5701i 0.126342 0.132811i
\(186\) 0 0
\(187\) −164.463 94.9526i −0.879480 0.507768i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −169.015 97.5807i −0.884894 0.510894i −0.0126252 0.999920i \(-0.504019\pi\)
−0.872269 + 0.489026i \(0.837352\pi\)
\(192\) 0 0
\(193\) 234.947 135.647i 1.21734 0.702832i 0.252993 0.967468i \(-0.418585\pi\)
0.964348 + 0.264636i \(0.0852518\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 163.058 0.827707 0.413853 0.910344i \(-0.364183\pi\)
0.413853 + 0.910344i \(0.364183\pi\)
\(198\) 0 0
\(199\) 294.000 1.47739 0.738693 0.674042i \(-0.235443\pi\)
0.738693 + 0.674042i \(0.235443\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −28.7750 49.8397i −0.141749 0.245516i
\(204\) 0 0
\(205\) −336.810 81.2966i −1.64297 0.396569i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 102.879 + 59.3970i 0.492242 + 0.284196i
\(210\) 0 0
\(211\) 21.0000 + 36.3731i 0.0995261 + 0.172384i 0.911489 0.411325i \(-0.134934\pi\)
−0.811963 + 0.583710i \(0.801601\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −95.9166 325.269i −0.446124 1.51288i
\(216\) 0 0
\(217\) 257.729i 1.18769i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 338.030 195.161i 1.52955 0.883084i
\(222\) 0 0
\(223\) 170.336 + 98.3438i 0.763841 + 0.441004i 0.830673 0.556761i \(-0.187956\pi\)
−0.0668324 + 0.997764i \(0.521289\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −134.283 + 232.585i −0.591556 + 1.02461i 0.402467 + 0.915435i \(0.368153\pi\)
−0.994023 + 0.109171i \(0.965180\pi\)
\(228\) 0 0
\(229\) 211.000 + 365.463i 0.921397 + 1.59591i 0.797255 + 0.603643i \(0.206285\pi\)
0.124142 + 0.992264i \(0.460382\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 211.017 0.905651 0.452825 0.891599i \(-0.350416\pi\)
0.452825 + 0.891599i \(0.350416\pi\)
\(234\) 0 0
\(235\) −368.000 + 108.517i −1.56596 + 0.461776i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −61.2372 + 35.3553i −0.256223 + 0.147930i −0.622610 0.782532i \(-0.713928\pi\)
0.366388 + 0.930462i \(0.380594\pi\)
\(240\) 0 0
\(241\) −140.000 + 242.487i −0.580913 + 1.00617i 0.414459 + 0.910068i \(0.363971\pi\)
−0.995371 + 0.0961024i \(0.969362\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.51951 14.5813i 0.0143654 0.0595153i
\(246\) 0 0
\(247\) −211.452 + 122.082i −0.856081 + 0.494259i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 80.6102i 0.321156i 0.987023 + 0.160578i \(0.0513358\pi\)
−0.987023 + 0.160578i \(0.948664\pi\)
\(252\) 0 0
\(253\) 94.9526i 0.375307i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) 23.0000 39.8372i 0.0888031 0.153811i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 33.5708 58.1464i 0.127646 0.221089i −0.795118 0.606454i \(-0.792591\pi\)
0.922764 + 0.385365i \(0.125925\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 263.044i 0.977858i −0.872324 0.488929i \(-0.837388\pi\)
0.872324 0.488929i \(-0.162612\pi\)
\(270\) 0 0
\(271\) 322.000 1.18819 0.594096 0.804394i \(-0.297510\pi\)
0.594096 + 0.804394i \(0.297510\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −112.896 + 220.237i −0.410530 + 0.800863i
\(276\) 0 0
\(277\) −29.3684 16.9558i −0.106023 0.0612124i 0.446051 0.895008i \(-0.352830\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −282.916 163.342i −1.00682 0.581287i −0.0965601 0.995327i \(-0.530784\pi\)
−0.910259 + 0.414040i \(0.864117\pi\)
\(282\) 0 0
\(283\) −93.9787 + 54.2586i −0.332080 + 0.191727i −0.656764 0.754096i \(-0.728075\pi\)
0.324684 + 0.945823i \(0.394742\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −469.991 −1.63760
\(288\) 0 0
\(289\) 79.0000 0.273356
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 100.712 + 174.439i 0.343729 + 0.595355i 0.985122 0.171857i \(-0.0549767\pi\)
−0.641393 + 0.767212i \(0.721643\pi\)
\(294\) 0 0
\(295\) 405.547 + 97.8877i 1.37473 + 0.331823i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −169.015 97.5807i −0.565267 0.326357i
\(300\) 0 0
\(301\) −230.000 398.372i −0.764120 1.32349i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 335.708 98.9949i 1.10068 0.324574i
\(306\) 0 0
\(307\) 162.776i 0.530215i −0.964219 0.265107i \(-0.914593\pi\)
0.964219 0.265107i \(-0.0854074\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 154.318 89.0955i 0.496199 0.286481i −0.230944 0.972967i \(-0.574181\pi\)
0.727142 + 0.686487i \(0.240848\pi\)
\(312\) 0 0
\(313\) 46.9894 + 27.1293i 0.150126 + 0.0866751i 0.573181 0.819429i \(-0.305709\pi\)
−0.423055 + 0.906104i \(0.639042\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 139.079 240.892i 0.438735 0.759912i −0.558857 0.829264i \(-0.688760\pi\)
0.997592 + 0.0693522i \(0.0220932\pi\)
\(318\) 0 0
\(319\) −42.0000 72.7461i −0.131661 0.228044i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −230.200 −0.712693
\(324\) 0 0
\(325\) −276.000 427.287i −0.849231 1.31473i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −450.706 + 260.215i −1.36993 + 0.790928i
\(330\) 0 0
\(331\) −20.0000 + 34.6410i −0.0604230 + 0.104656i −0.894655 0.446759i \(-0.852578\pi\)
0.834232 + 0.551414i \(0.185912\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 527.439 + 127.309i 1.57445 + 0.380028i
\(336\) 0 0
\(337\) −270.189 + 155.994i −0.801747 + 0.462889i −0.844082 0.536215i \(-0.819854\pi\)
0.0423345 + 0.999103i \(0.486520\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 376.181i 1.10317i
\(342\) 0 0
\(343\) 352.681i 1.02822i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −201.425 348.878i −0.580475 1.00541i −0.995423 0.0955674i \(-0.969533\pi\)
0.414948 0.909845i \(-0.363800\pi\)
\(348\) 0 0
\(349\) 203.000 351.606i 0.581662 1.00747i −0.413621 0.910449i \(-0.635736\pi\)
0.995283 0.0970186i \(-0.0309306\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 268.567 465.171i 0.760812 1.31776i −0.181621 0.983369i \(-0.558134\pi\)
0.942433 0.334396i \(-0.108532\pi\)
\(354\) 0 0
\(355\) 409.388 430.350i 1.15321 1.21225i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 161.220i 0.449082i 0.974465 + 0.224541i \(0.0720882\pi\)
−0.974465 + 0.224541i \(0.927912\pi\)
\(360\) 0 0
\(361\) −217.000 −0.601108
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 46.7466 49.1402i 0.128073 0.134631i
\(366\) 0 0
\(367\) 217.326 + 125.473i 0.592168 + 0.341889i 0.765954 0.642895i \(-0.222267\pi\)
−0.173786 + 0.984783i \(0.555600\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −88.1051 + 50.8675i −0.236207 + 0.136374i −0.613432 0.789748i \(-0.710212\pi\)
0.377225 + 0.926121i \(0.376878\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 172.650 0.457957
\(378\) 0 0
\(379\) 538.000 1.41953 0.709763 0.704441i \(-0.248802\pi\)
0.709763 + 0.704441i \(0.248802\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −134.283 232.585i −0.350609 0.607273i 0.635747 0.771897i \(-0.280692\pi\)
−0.986356 + 0.164625i \(0.947359\pi\)
\(384\) 0 0
\(385\) −78.7686 + 326.336i −0.204594 + 0.847627i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 428.661 + 247.487i 1.10196 + 0.636214i 0.936734 0.350042i \(-0.113833\pi\)
0.165222 + 0.986256i \(0.447166\pi\)
\(390\) 0 0
\(391\) −92.0000 159.349i −0.235294 0.407541i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −143.875 + 42.4264i −0.364240 + 0.107409i
\(396\) 0 0
\(397\) 278.076i 0.700442i 0.936667 + 0.350221i \(0.113894\pi\)
−0.936667 + 0.350221i \(0.886106\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 214.330 123.744i 0.534490 0.308588i −0.208353 0.978054i \(-0.566810\pi\)
0.742843 + 0.669466i \(0.233477\pi\)
\(402\) 0 0
\(403\) −669.598 386.593i −1.66153 0.959287i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 33.5708 58.1464i 0.0824836 0.142866i
\(408\) 0 0
\(409\) 121.000 + 209.578i 0.295844 + 0.512416i 0.975181 0.221410i \(-0.0710660\pi\)
−0.679337 + 0.733826i \(0.737733\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 565.908 1.37024
\(414\) 0 0
\(415\) −644.000 + 189.905i −1.55181 + 0.457603i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 684.632 395.273i 1.63397 0.943372i 0.651115 0.758979i \(-0.274302\pi\)
0.982853 0.184392i \(-0.0590317\pi\)
\(420\) 0 0
\(421\) −179.000 + 310.037i −0.425178 + 0.736430i −0.996437 0.0843398i \(-0.973122\pi\)
0.571259 + 0.820770i \(0.306455\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −23.9281 478.986i −0.0563015 1.12703i
\(426\) 0 0
\(427\) 411.157 237.382i 0.962897 0.555929i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.1421i 0.0328124i −0.999865 0.0164062i \(-0.994778\pi\)
0.999865 0.0164062i \(-0.00522249\pi\)
\(432\) 0 0
\(433\) 257.729i 0.595216i −0.954688 0.297608i \(-0.903811\pi\)
0.954688 0.297608i \(-0.0961888\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 57.5500 + 99.6795i 0.131693 + 0.228100i
\(438\) 0 0
\(439\) 345.000 597.558i 0.785877 1.36118i −0.142597 0.989781i \(-0.545545\pi\)
0.928474 0.371398i \(-0.121121\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −201.425 + 348.878i −0.454684 + 0.787535i −0.998670 0.0515587i \(-0.983581\pi\)
0.543986 + 0.839094i \(0.316914\pi\)
\(444\) 0 0
\(445\) 112.094 117.834i 0.251898 0.264795i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 80.6102i 0.179533i −0.995963 0.0897663i \(-0.971388\pi\)
0.995963 0.0897663i \(-0.0286120\pi\)
\(450\) 0 0
\(451\) −686.000 −1.52106
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −499.927 475.576i −1.09874 1.04522i
\(456\) 0 0
\(457\) −293.684 169.558i −0.642633 0.371025i 0.142995 0.989723i \(-0.454327\pi\)
−0.785628 + 0.618699i \(0.787660\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 558.484 + 322.441i 1.21146 + 0.699438i 0.963078 0.269224i \(-0.0867673\pi\)
0.248384 + 0.968662i \(0.420101\pi\)
\(462\) 0 0
\(463\) 229.073 132.255i 0.494758 0.285649i −0.231788 0.972766i \(-0.574458\pi\)
0.726546 + 0.687117i \(0.241124\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 460.400 0.985867 0.492933 0.870067i \(-0.335925\pi\)
0.492933 + 0.870067i \(0.335925\pi\)
\(468\) 0 0
\(469\) 736.000 1.56930
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −335.708 581.464i −0.709743 1.22931i
\(474\) 0 0
\(475\) 14.9681 + 299.626i 0.0315118 + 0.630792i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 68.5857 + 39.5980i 0.143185 + 0.0826680i 0.569881 0.821727i \(-0.306989\pi\)
−0.426696 + 0.904395i \(0.640323\pi\)
\(480\) 0 0
\(481\) 69.0000 + 119.512i 0.143451 + 0.248465i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 134.283 + 455.377i 0.276873 + 0.938921i
\(486\) 0 0
\(487\) 793.533i 1.62943i 0.579861 + 0.814715i \(0.303107\pi\)
−0.579861 + 0.814715i \(0.696893\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −488.673 + 282.136i −0.995261 + 0.574614i −0.906843 0.421469i \(-0.861515\pi\)
−0.0884184 + 0.996083i \(0.528181\pi\)
\(492\) 0 0
\(493\) 140.968 + 81.3880i 0.285939 + 0.165087i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 402.850 697.756i 0.810563 1.40394i
\(498\) 0 0
\(499\) 36.0000 + 62.3538i 0.0721443 + 0.124958i 0.899841 0.436218i \(-0.143682\pi\)
−0.827697 + 0.561176i \(0.810349\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −230.200 −0.457654 −0.228827 0.973467i \(-0.573489\pi\)
−0.228827 + 0.973467i \(0.573489\pi\)
\(504\) 0 0
\(505\) 112.000 + 379.810i 0.221782 + 0.752100i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 426.211 246.073i 0.837350 0.483444i −0.0190125 0.999819i \(-0.506052\pi\)
0.856363 + 0.516375i \(0.172719\pi\)
\(510\) 0 0
\(511\) 46.0000 79.6743i 0.0900196 0.155918i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −428.545 103.439i −0.832125 0.200852i
\(516\) 0 0
\(517\) −657.851 + 379.810i −1.27244 + 0.734643i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 69.2965i 0.133007i 0.997786 + 0.0665033i \(0.0211843\pi\)
−0.997786 + 0.0665033i \(0.978816\pi\)
\(522\) 0 0
\(523\) 339.116i 0.648406i 0.945987 + 0.324203i \(0.105096\pi\)
−0.945987 + 0.324203i \(0.894904\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −364.483 631.303i −0.691619 1.19792i
\(528\) 0 0
\(529\) 218.500 378.453i 0.413043 0.715412i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 704.987 1221.07i 1.32268 2.29095i
\(534\) 0 0
\(535\) −208.484 198.329i −0.389690 0.370708i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 29.6985i 0.0550992i
\(540\) 0 0
\(541\) −590.000 −1.09057 −0.545287 0.838250i \(-0.683579\pi\)
−0.545287 + 0.838250i \(0.683579\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 268.077 + 255.019i 0.491884 + 0.467925i
\(546\) 0 0
\(547\) 199.705 + 115.300i 0.365091 + 0.210785i 0.671312 0.741175i \(-0.265731\pi\)
−0.306221 + 0.951961i \(0.599065\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −88.1816 50.9117i −0.160039 0.0923987i
\(552\) 0 0
\(553\) −176.210 + 101.735i −0.318644 + 0.183969i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 690.600 1.23986 0.619928 0.784659i \(-0.287162\pi\)
0.619928 + 0.784659i \(0.287162\pi\)
\(558\) 0 0
\(559\) 1380.00 2.46869
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.59166 + 16.6132i 0.0170367 + 0.0295084i 0.874418 0.485173i \(-0.161243\pi\)
−0.857381 + 0.514682i \(0.827910\pi\)
\(564\) 0 0
\(565\) −202.548 + 839.151i −0.358492 + 1.48522i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 94.3054 + 54.4472i 0.165739 + 0.0956893i 0.580575 0.814207i \(-0.302828\pi\)
−0.414836 + 0.909896i \(0.636161\pi\)
\(570\) 0 0
\(571\) 352.000 + 609.682i 0.616462 + 1.06774i 0.990126 + 0.140180i \(0.0447681\pi\)
−0.373664 + 0.927564i \(0.621899\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −201.425 + 130.108i −0.350304 + 0.226274i
\(576\) 0 0
\(577\) 1125.87i 1.95124i −0.219462 0.975621i \(-0.570430\pi\)
0.219462 0.975621i \(-0.429570\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −788.736 + 455.377i −1.35755 + 0.783781i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 489.175 847.276i 0.833347 1.44340i −0.0620218 0.998075i \(-0.519755\pi\)
0.895369 0.445325i \(-0.146912\pi\)
\(588\) 0 0
\(589\) 228.000 + 394.908i 0.387097 + 0.670471i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) −184.000 623.974i −0.309244 1.04870i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 257.196 148.492i 0.429376 0.247901i −0.269705 0.962943i \(-0.586926\pi\)
0.699081 + 0.715043i \(0.253593\pi\)
\(600\) 0 0
\(601\) −10.0000 + 17.3205i −0.0166389 + 0.0288195i −0.874225 0.485521i \(-0.838630\pi\)
0.857586 + 0.514341i \(0.171963\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 26.9829 111.790i 0.0445999 0.184776i
\(606\) 0 0
\(607\) 5.87367 3.39116i 0.00967656 0.00558676i −0.495154 0.868805i \(-0.664888\pi\)
0.504830 + 0.863219i \(0.331555\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1561.29i 2.55531i
\(612\) 0 0
\(613\) 47.4763i 0.0774491i 0.999250 + 0.0387246i \(0.0123295\pi\)
−0.999250 + 0.0387246i \(0.987671\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −19.1833 33.2265i −0.0310913 0.0538517i 0.850061 0.526684i \(-0.176565\pi\)
−0.881152 + 0.472832i \(0.843232\pi\)
\(618\) 0 0
\(619\) −89.0000 + 154.153i −0.143780 + 0.249035i −0.928917 0.370287i \(-0.879259\pi\)
0.785137 + 0.619322i \(0.212593\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 110.304 191.052i 0.177053 0.306665i
\(624\) 0 0
\(625\) −621.888 + 62.2893i −0.995021 + 0.0996629i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 130.108i 0.206848i
\(630\) 0 0
\(631\) −154.000 −0.244057 −0.122029 0.992527i \(-0.538940\pi\)
−0.122029 + 0.992527i \(0.538940\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 584.333 614.252i 0.920209 0.967326i
\(636\) 0 0
\(637\) 52.8630 + 30.5205i 0.0829875 + 0.0479128i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 699.329 + 403.758i 1.09100 + 0.629888i 0.933842 0.357686i \(-0.116434\pi\)
0.157156 + 0.987574i \(0.449768\pi\)
\(642\) 0 0
\(643\) −881.051 + 508.675i −1.37022 + 0.791096i −0.990955 0.134193i \(-0.957156\pi\)
−0.379263 + 0.925289i \(0.623823\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −805.700 −1.24529 −0.622643 0.782506i \(-0.713941\pi\)
−0.622643 + 0.782506i \(0.713941\pi\)
\(648\) 0 0
\(649\) 826.000 1.27273
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 33.5708 + 58.1464i 0.0514101 + 0.0890450i 0.890585 0.454816i \(-0.150295\pi\)
−0.839175 + 0.543861i \(0.816962\pi\)
\(654\) 0 0
\(655\) −804.219 194.116i −1.22782 0.296361i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −454.380 262.337i −0.689500 0.398083i 0.113925 0.993489i \(-0.463658\pi\)
−0.803425 + 0.595406i \(0.796991\pi\)
\(660\) 0 0
\(661\) −37.0000 64.0859i −0.0559758 0.0969529i 0.836680 0.547693i \(-0.184494\pi\)
−0.892655 + 0.450740i \(0.851160\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 115.100 + 390.323i 0.173083 + 0.586952i
\(666\) 0 0
\(667\) 81.3880i 0.122021i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 600.125 346.482i 0.894374 0.516367i
\(672\) 0 0
\(673\) 599.114 + 345.899i 0.890214 + 0.513966i 0.874013 0.485903i \(-0.161509\pi\)
0.0162018 + 0.999869i \(0.494843\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 139.079 240.892i 0.205434 0.355823i −0.744837 0.667247i \(-0.767473\pi\)
0.950271 + 0.311424i \(0.100806\pi\)
\(678\) 0 0
\(679\) 322.000 + 557.720i 0.474227 + 0.821385i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 422.033 0.617911 0.308955 0.951077i \(-0.400021\pi\)
0.308955 + 0.951077i \(0.400021\pi\)
\(684\) 0 0
\(685\) −552.000 + 162.776i −0.805839 + 0.237629i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 34.0000 58.8897i 0.0492041 0.0852239i −0.840374 0.542006i \(-0.817665\pi\)
0.889578 + 0.456782i \(0.150998\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 72.7366 301.346i 0.104657 0.433591i
\(696\) 0 0
\(697\) 1151.24 664.668i 1.65171 0.953613i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 130.108i 0.185603i 0.995685 + 0.0928015i \(0.0295822\pi\)
−0.995685 + 0.0928015i \(0.970418\pi\)
\(702\) 0 0
\(703\) 81.3880i 0.115772i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 268.567 + 465.171i 0.379868 + 0.657950i
\(708\) 0 0
\(709\) 105.000 181.865i 0.148096 0.256510i −0.782428 0.622741i \(-0.786019\pi\)
0.930524 + 0.366232i \(0.119352\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −182.242 + 315.652i −0.255598 + 0.442709i
\(714\) 0 0
\(715\) −729.694 694.152i −1.02055 0.970841i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 121.622i 0.169155i −0.996417 0.0845774i \(-0.973046\pi\)
0.996417 0.0845774i \(-0.0269540\pi\)
\(720\) 0 0
\(721\) −598.000 −0.829404
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 96.7679 188.775i 0.133473 0.260379i
\(726\) 0 0
\(727\) −17.6210 10.1735i −0.0242380 0.0139938i 0.487832 0.872938i \(-0.337788\pi\)
−0.512070 + 0.858944i \(0.671121\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1126.77 + 650.538i 1.54140 + 0.889929i
\(732\) 0 0
\(733\) 1227.60 708.753i 1.67476 0.966922i 0.709842 0.704361i \(-0.248767\pi\)
0.964916 0.262560i \(-0.0845668\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1074.27 1.45762
\(738\) 0 0
\(739\) 416.000 0.562923 0.281461 0.959573i \(-0.409181\pi\)
0.281461 + 0.959573i \(0.409181\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −402.850 697.756i −0.542194 0.939107i −0.998778 0.0494266i \(-0.984261\pi\)
0.456584 0.889680i \(-0.349073\pi\)
\(744\) 0 0
\(745\) −769.851 185.821i −1.03336 0.249424i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −338.030 195.161i −0.451308 0.260563i
\(750\) 0 0
\(751\) −43.0000 74.4782i −0.0572570 0.0991720i 0.835976 0.548766i \(-0.184902\pi\)
−0.893233 + 0.449594i \(0.851569\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −335.708 + 98.9949i −0.444647 + 0.131119i
\(756\) 0 0
\(757\) 47.4763i 0.0627164i −0.999508 0.0313582i \(-0.990017\pi\)
0.999508 0.0313582i \(-0.00998326\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −319.658 + 184.555i −0.420050 + 0.242516i −0.695099 0.718914i \(-0.744639\pi\)
0.275048 + 0.961430i \(0.411306\pi\)
\(762\) 0 0
\(763\) 434.652 + 250.946i 0.569661 + 0.328894i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −848.862 + 1470.27i −1.10673 + 1.91691i
\(768\) 0 0
\(769\) −192.000 332.554i −0.249675 0.432450i 0.713761 0.700390i \(-0.246990\pi\)
−0.963436 + 0.267940i \(0.913657\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 201.425 0.260576 0.130288 0.991476i \(-0.458410\pi\)
0.130288 + 0.991476i \(0.458410\pi\)
\(774\) 0 0
\(775\) −798.000 + 515.457i −1.02968 + 0.665106i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −720.150 + 415.779i −0.924454 + 0.533734i
\(780\) 0 0
\(781\) 588.000 1018.45i 0.752881 1.30403i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 230.755 + 55.6978i 0.293955 + 0.0709526i
\(786\) 0 0
\(787\) −411.157 + 237.382i −0.522436 + 0.301628i −0.737931 0.674877i \(-0.764197\pi\)
0.215495 + 0.976505i \(0.430864\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1170.97i 1.48037i
\(792\) 0 0
\(793\) 1424.29i 1.79608i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −249.383 431.944i −0.312902 0.541963i 0.666087 0.745874i \(-0.267968\pi\)
−0.978989 + 0.203911i \(0.934635\pi\)
\(798\) 0 0
\(799\) 736.000 1274.79i 0.921151 1.59548i
\(800\) 0 0
\(801\)