Properties

Label 684.3.y.g.145.2
Level $684$
Weight $3$
Character 684.145
Analytic conductor $18.638$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.y (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 34 x^{6} + 921 x^{4} - 7990 x^{2} + 55225\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.2
Root \(2.69047 + 1.55335i\) of defining polynomial
Character \(\chi\) \(=\) 684.145
Dual form 684.3.y.g.217.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.19676 - 3.80490i) q^{5} +1.44949 q^{7} +O(q^{10})\) \(q+(-2.19676 - 3.80490i) q^{5} +1.44949 q^{7} +15.1554 q^{11} +(-14.8485 - 8.57277i) q^{13} +(9.77447 + 16.9299i) q^{17} +(-3.34847 - 18.7026i) q^{19} +(-2.19676 + 3.80490i) q^{23} +(2.84847 - 4.93369i) q^{25} +(-29.3234 - 16.9299i) q^{29} -5.62475i q^{31} +(-3.18418 - 5.51517i) q^{35} -23.7238i q^{37} +(-32.2857 + 18.6401i) q^{41} +(-15.9722 - 27.6647i) q^{43} +(39.0979 - 67.7195i) q^{47} -46.8990 q^{49} +(-9.55255 - 5.51517i) q^{53} +(-33.2929 - 57.6649i) q^{55} +(-78.4177 + 45.2745i) q^{59} +(57.1413 - 98.9717i) q^{61} +75.3293i q^{65} +(-98.3105 - 56.7596i) q^{67} +(87.9702 - 50.7896i) q^{71} +(10.0505 + 17.4080i) q^{73} +21.9676 q^{77} +(-45.8258 + 26.4575i) q^{79} -29.2237 q^{83} +(42.9444 - 74.3819i) q^{85} +(126.846 + 73.2347i) q^{89} +(-21.5227 - 12.4261i) q^{91} +(-63.8059 + 53.8258i) q^{95} +(27.0000 - 15.5885i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7} + O(q^{10}) \) \( 8 q - 8 q^{7} - 60 q^{13} + 32 q^{19} - 36 q^{25} - 20 q^{43} - 336 q^{49} + 8 q^{55} + 124 q^{61} - 228 q^{67} + 100 q^{73} - 396 q^{79} + 128 q^{85} - 84 q^{91} + 216 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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
\(5\) −2.19676 3.80490i −0.439352 0.760981i 0.558287 0.829648i \(-0.311459\pi\)
−0.997640 + 0.0686670i \(0.978125\pi\)
\(6\) 0 0
\(7\) 1.44949 0.207070 0.103535 0.994626i \(-0.466985\pi\)
0.103535 + 0.994626i \(0.466985\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 15.1554 1.37777 0.688883 0.724873i \(-0.258102\pi\)
0.688883 + 0.724873i \(0.258102\pi\)
\(12\) 0 0
\(13\) −14.8485 8.57277i −1.14219 0.659444i −0.195218 0.980760i \(-0.562541\pi\)
−0.946972 + 0.321316i \(0.895875\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 9.77447 + 16.9299i 0.574969 + 0.995875i 0.996045 + 0.0888501i \(0.0283192\pi\)
−0.421076 + 0.907025i \(0.638347\pi\)
\(18\) 0 0
\(19\) −3.34847 18.7026i −0.176235 0.984348i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.19676 + 3.80490i −0.0955114 + 0.165431i −0.909822 0.414999i \(-0.863782\pi\)
0.814311 + 0.580429i \(0.197115\pi\)
\(24\) 0 0
\(25\) 2.84847 4.93369i 0.113939 0.197348i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −29.3234 16.9299i −1.01115 0.583789i −0.0996235 0.995025i \(-0.531764\pi\)
−0.911529 + 0.411236i \(0.865097\pi\)
\(30\) 0 0
\(31\) 5.62475i 0.181443i −0.995876 0.0907217i \(-0.971083\pi\)
0.995876 0.0907217i \(-0.0289174\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.18418 5.51517i −0.0909767 0.157576i
\(36\) 0 0
\(37\) 23.7238i 0.641184i −0.947217 0.320592i \(-0.896118\pi\)
0.947217 0.320592i \(-0.103882\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −32.2857 + 18.6401i −0.787456 + 0.454638i −0.839066 0.544030i \(-0.816898\pi\)
0.0516104 + 0.998667i \(0.483565\pi\)
\(42\) 0 0
\(43\) −15.9722 27.6647i −0.371446 0.643364i 0.618342 0.785909i \(-0.287805\pi\)
−0.989788 + 0.142545i \(0.954471\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 39.0979 67.7195i 0.831870 1.44084i −0.0646836 0.997906i \(-0.520604\pi\)
0.896554 0.442935i \(-0.146063\pi\)
\(48\) 0 0
\(49\) −46.8990 −0.957122
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.55255 5.51517i −0.180237 0.104060i 0.407167 0.913354i \(-0.366517\pi\)
−0.587404 + 0.809294i \(0.699850\pi\)
\(54\) 0 0
\(55\) −33.2929 57.6649i −0.605325 1.04845i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −78.4177 + 45.2745i −1.32911 + 0.767364i −0.985163 0.171624i \(-0.945099\pi\)
−0.343951 + 0.938988i \(0.611765\pi\)
\(60\) 0 0
\(61\) 57.1413 98.9717i 0.936743 1.62249i 0.165247 0.986252i \(-0.447158\pi\)
0.771496 0.636234i \(-0.219509\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 75.3293i 1.15891i
\(66\) 0 0
\(67\) −98.3105 56.7596i −1.46732 0.847158i −0.467989 0.883734i \(-0.655021\pi\)
−0.999331 + 0.0365763i \(0.988355\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 87.9702 50.7896i 1.23902 0.715347i 0.270124 0.962825i \(-0.412935\pi\)
0.968893 + 0.247478i \(0.0796019\pi\)
\(72\) 0 0
\(73\) 10.0505 + 17.4080i 0.137678 + 0.238466i 0.926617 0.376006i \(-0.122703\pi\)
−0.788939 + 0.614471i \(0.789369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 21.9676 0.285294
\(78\) 0 0
\(79\) −45.8258 + 26.4575i −0.580073 + 0.334905i −0.761162 0.648561i \(-0.775371\pi\)
0.181089 + 0.983467i \(0.442038\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −29.2237 −0.352092 −0.176046 0.984382i \(-0.556331\pi\)
−0.176046 + 0.984382i \(0.556331\pi\)
\(84\) 0 0
\(85\) 42.9444 74.3819i 0.505228 0.875081i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 126.846 + 73.2347i 1.42524 + 0.822862i 0.996740 0.0806807i \(-0.0257094\pi\)
0.428498 + 0.903543i \(0.359043\pi\)
\(90\) 0 0
\(91\) −21.5227 12.4261i −0.236513 0.136551i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −63.8059 + 53.8258i −0.671641 + 0.566587i
\(96\) 0 0
\(97\) 27.0000 15.5885i 0.278351 0.160706i −0.354326 0.935122i \(-0.615290\pi\)
0.632676 + 0.774416i \(0.281956\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 66.0026 114.320i 0.653491 1.13188i −0.328778 0.944407i \(-0.606637\pi\)
0.982270 0.187473i \(-0.0600297\pi\)
\(102\) 0 0
\(103\) 41.3156i 0.401122i 0.979681 + 0.200561i \(0.0642765\pi\)
−0.979681 + 0.200561i \(0.935723\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 33.8598i 0.316446i −0.987403 0.158223i \(-0.949423\pi\)
0.987403 0.158223i \(-0.0505766\pi\)
\(108\) 0 0
\(109\) −30.3031 + 17.4955i −0.278010 + 0.160509i −0.632522 0.774542i \(-0.717980\pi\)
0.354512 + 0.935051i \(0.384647\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 163.227i 1.44448i −0.691641 0.722241i \(-0.743112\pi\)
0.691641 0.722241i \(-0.256888\pi\)
\(114\) 0 0
\(115\) 19.3031 0.167853
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 14.1680 + 24.5397i 0.119059 + 0.206216i
\(120\) 0 0
\(121\) 108.687 0.898237
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −134.868 −1.07894
\(126\) 0 0
\(127\) −135.576 78.2746i −1.06752 0.616335i −0.140020 0.990149i \(-0.544717\pi\)
−0.927504 + 0.373814i \(0.878050\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 107.963 + 186.997i 0.824145 + 1.42746i 0.902571 + 0.430541i \(0.141677\pi\)
−0.0784255 + 0.996920i \(0.524989\pi\)
\(132\) 0 0
\(133\) −4.85357 27.1092i −0.0364930 0.203829i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −29.7673 + 51.5584i −0.217279 + 0.376339i −0.953975 0.299885i \(-0.903052\pi\)
0.736696 + 0.676224i \(0.236385\pi\)
\(138\) 0 0
\(139\) −1.82577 + 3.16232i −0.0131350 + 0.0227505i −0.872518 0.488582i \(-0.837514\pi\)
0.859383 + 0.511332i \(0.170848\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −225.035 129.924i −1.57367 0.908559i
\(144\) 0 0
\(145\) 148.764i 1.02596i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 29.0017 + 50.2325i 0.194643 + 0.337131i 0.946783 0.321872i \(-0.104312\pi\)
−0.752141 + 0.659003i \(0.770979\pi\)
\(150\) 0 0
\(151\) 8.64258i 0.0572356i 0.999590 + 0.0286178i \(0.00911058\pi\)
−0.999590 + 0.0286178i \(0.990889\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −21.4016 + 12.3562i −0.138075 + 0.0797176i
\(156\) 0 0
\(157\) 60.0301 + 103.975i 0.382357 + 0.662262i 0.991399 0.130876i \(-0.0417790\pi\)
−0.609041 + 0.793138i \(0.708446\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.18418 + 5.51517i −0.0197775 + 0.0342557i
\(162\) 0 0
\(163\) −51.8332 −0.317995 −0.158997 0.987279i \(-0.550826\pi\)
−0.158997 + 0.987279i \(0.550826\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 125.215 + 72.2932i 0.749793 + 0.432893i 0.825619 0.564228i \(-0.190826\pi\)
−0.0758260 + 0.997121i \(0.524159\pi\)
\(168\) 0 0
\(169\) 62.4847 + 108.227i 0.369732 + 0.640394i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −57.0161 + 32.9182i −0.329573 + 0.190279i −0.655651 0.755064i \(-0.727606\pi\)
0.326079 + 0.945343i \(0.394273\pi\)
\(174\) 0 0
\(175\) 4.12883 7.15134i 0.0235933 0.0408648i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 60.1097i 0.335808i −0.985803 0.167904i \(-0.946300\pi\)
0.985803 0.167904i \(-0.0536999\pi\)
\(180\) 0 0
\(181\) 218.530 + 126.168i 1.20735 + 0.697063i 0.962179 0.272417i \(-0.0878231\pi\)
0.245169 + 0.969480i \(0.421156\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −90.2668 + 52.1155i −0.487928 + 0.281706i
\(186\) 0 0
\(187\) 148.136 + 256.579i 0.792172 + 1.37208i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 211.533 1.10750 0.553750 0.832683i \(-0.313196\pi\)
0.553750 + 0.832683i \(0.313196\pi\)
\(192\) 0 0
\(193\) 48.4546 27.9753i 0.251060 0.144950i −0.369189 0.929354i \(-0.620365\pi\)
0.620250 + 0.784405i \(0.287031\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −95.1265 −0.482876 −0.241438 0.970416i \(-0.577619\pi\)
−0.241438 + 0.970416i \(0.577619\pi\)
\(198\) 0 0
\(199\) −40.4115 + 69.9947i −0.203073 + 0.351732i −0.949517 0.313716i \(-0.898426\pi\)
0.746444 + 0.665448i \(0.231759\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −42.5040 24.5397i −0.209379 0.120885i
\(204\) 0 0
\(205\) 141.848 + 81.8960i 0.691941 + 0.399492i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −50.7475 283.446i −0.242811 1.35620i
\(210\) 0 0
\(211\) −46.0982 + 26.6148i −0.218475 + 0.126137i −0.605244 0.796040i \(-0.706924\pi\)
0.386769 + 0.922177i \(0.373591\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −70.1742 + 121.545i −0.326392 + 0.565327i
\(216\) 0 0
\(217\) 8.15301i 0.0375715i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 335.177i 1.51664i
\(222\) 0 0
\(223\) −171.356 + 98.9324i −0.768412 + 0.443643i −0.832308 0.554314i \(-0.812981\pi\)
0.0638959 + 0.997957i \(0.479647\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 125.946i 0.554829i −0.960750 0.277415i \(-0.910522\pi\)
0.960750 0.277415i \(-0.0894775\pi\)
\(228\) 0 0
\(229\) 193.586 0.845352 0.422676 0.906281i \(-0.361091\pi\)
0.422676 + 0.906281i \(0.361091\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 148.592 + 257.369i 0.637734 + 1.10459i 0.985929 + 0.167165i \(0.0534613\pi\)
−0.348195 + 0.937422i \(0.613205\pi\)
\(234\) 0 0
\(235\) −343.555 −1.46194
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −104.801 −0.438499 −0.219250 0.975669i \(-0.570361\pi\)
−0.219250 + 0.975669i \(0.570361\pi\)
\(240\) 0 0
\(241\) 356.787 + 205.991i 1.48044 + 0.854735i 0.999755 0.0221551i \(-0.00705275\pi\)
0.480690 + 0.876890i \(0.340386\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 103.026 + 178.446i 0.420514 + 0.728352i
\(246\) 0 0
\(247\) −110.614 + 306.411i −0.447828 + 1.24053i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −87.8705 + 152.196i −0.350082 + 0.606359i −0.986263 0.165180i \(-0.947180\pi\)
0.636182 + 0.771539i \(0.280513\pi\)
\(252\) 0 0
\(253\) −33.2929 + 57.6649i −0.131592 + 0.227925i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 373.283 + 215.515i 1.45246 + 0.838579i 0.998621 0.0525037i \(-0.0167201\pi\)
0.453841 + 0.891083i \(0.350053\pi\)
\(258\) 0 0
\(259\) 34.3874i 0.132770i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 60.0781 + 104.058i 0.228434 + 0.395659i 0.957344 0.288950i \(-0.0933062\pi\)
−0.728910 + 0.684609i \(0.759973\pi\)
\(264\) 0 0
\(265\) 48.4621i 0.182876i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 415.487 239.882i 1.54456 0.891754i 0.546021 0.837772i \(-0.316142\pi\)
0.998542 0.0539820i \(-0.0171914\pi\)
\(270\) 0 0
\(271\) −63.0908 109.276i −0.232807 0.403234i 0.725826 0.687879i \(-0.241458\pi\)
−0.958633 + 0.284644i \(0.908125\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 43.1697 74.7722i 0.156981 0.271899i
\(276\) 0 0
\(277\) 459.918 1.66036 0.830178 0.557499i \(-0.188239\pi\)
0.830178 + 0.557499i \(0.188239\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 65.2371 + 37.6647i 0.232161 + 0.134038i 0.611568 0.791192i \(-0.290539\pi\)
−0.379408 + 0.925230i \(0.623872\pi\)
\(282\) 0 0
\(283\) 178.753 + 309.609i 0.631634 + 1.09402i 0.987218 + 0.159378i \(0.0509490\pi\)
−0.355583 + 0.934645i \(0.615718\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −46.7978 + 27.0187i −0.163058 + 0.0941418i
\(288\) 0 0
\(289\) −46.5806 + 80.6800i −0.161179 + 0.279169i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 228.294i 0.779161i 0.920992 + 0.389581i \(0.127380\pi\)
−0.920992 + 0.389581i \(0.872620\pi\)
\(294\) 0 0
\(295\) 344.530 + 198.915i 1.16790 + 0.674287i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 65.2371 37.6647i 0.218184 0.125969i
\(300\) 0 0
\(301\) −23.1515 40.0996i −0.0769154 0.133221i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −502.104 −1.64624
\(306\) 0 0
\(307\) −83.6969 + 48.3224i −0.272628 + 0.157402i −0.630082 0.776529i \(-0.716979\pi\)
0.357453 + 0.933931i \(0.383645\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −414.722 −1.33351 −0.666755 0.745277i \(-0.732317\pi\)
−0.666755 + 0.745277i \(0.732317\pi\)
\(312\) 0 0
\(313\) 164.000 284.056i 0.523962 0.907528i −0.475649 0.879635i \(-0.657787\pi\)
0.999611 0.0278932i \(-0.00887983\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −461.253 266.304i −1.45506 0.840077i −0.456295 0.889829i \(-0.650824\pi\)
−0.998762 + 0.0497515i \(0.984157\pi\)
\(318\) 0 0
\(319\) −444.409 256.579i −1.39313 0.804324i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 283.904 239.497i 0.878958 0.741478i
\(324\) 0 0
\(325\) −84.5908 + 48.8385i −0.260279 + 0.150272i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 56.6720 98.1588i 0.172255 0.298355i
\(330\) 0 0
\(331\) 138.310i 0.417856i −0.977931 0.208928i \(-0.933003\pi\)
0.977931 0.208928i \(-0.0669975\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 498.749i 1.48880i
\(336\) 0 0
\(337\) −200.409 + 115.706i −0.594686 + 0.343342i −0.766948 0.641709i \(-0.778226\pi\)
0.172262 + 0.985051i \(0.444892\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 85.2454i 0.249986i
\(342\) 0 0
\(343\) −139.005 −0.405261
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −291.137 504.264i −0.839012 1.45321i −0.890722 0.454549i \(-0.849800\pi\)
0.0517097 0.998662i \(-0.483533\pi\)
\(348\) 0 0
\(349\) 160.444 0.459725 0.229862 0.973223i \(-0.426172\pi\)
0.229862 + 0.973223i \(0.426172\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −162.959 −0.461641 −0.230821 0.972996i \(-0.574141\pi\)
−0.230821 + 0.972996i \(0.574141\pi\)
\(354\) 0 0
\(355\) −386.499 223.146i −1.08873 0.628579i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −130.674 226.334i −0.363994 0.630456i 0.624620 0.780928i \(-0.285254\pi\)
−0.988614 + 0.150473i \(0.951920\pi\)
\(360\) 0 0
\(361\) −338.576 + 125.250i −0.937882 + 0.346954i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 44.1572 76.4825i 0.120979 0.209541i
\(366\) 0 0
\(367\) 229.750 397.938i 0.626021 1.08430i −0.362321 0.932053i \(-0.618016\pi\)
0.988342 0.152247i \(-0.0486510\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −13.8463 7.99418i −0.0373216 0.0215477i
\(372\) 0 0
\(373\) 48.6194i 0.130347i −0.997874 0.0651734i \(-0.979240\pi\)
0.997874 0.0651734i \(-0.0207601\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 290.272 + 502.766i 0.769952 + 1.33360i
\(378\) 0 0
\(379\) 523.962i 1.38249i −0.722622 0.691243i \(-0.757063\pi\)
0.722622 0.691243i \(-0.242937\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 339.665 196.106i 0.886855 0.512026i 0.0139424 0.999903i \(-0.495562\pi\)
0.872912 + 0.487877i \(0.162229\pi\)
\(384\) 0 0
\(385\) −48.2577 83.5847i −0.125345 0.217103i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −266.108 + 460.912i −0.684081 + 1.18486i 0.289644 + 0.957135i \(0.406463\pi\)
−0.973725 + 0.227729i \(0.926870\pi\)
\(390\) 0 0
\(391\) −85.8888 −0.219664
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 201.337 + 116.242i 0.509713 + 0.294283i
\(396\) 0 0
\(397\) −62.3332 107.964i −0.157010 0.271950i 0.776779 0.629774i \(-0.216852\pi\)
−0.933789 + 0.357823i \(0.883519\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −174.975 + 101.022i −0.436348 + 0.251926i −0.702047 0.712130i \(-0.747730\pi\)
0.265699 + 0.964056i \(0.414397\pi\)
\(402\) 0 0
\(403\) −48.2196 + 83.5189i −0.119652 + 0.207243i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 359.544i 0.883401i
\(408\) 0 0
\(409\) 473.091 + 273.139i 1.15670 + 0.667822i 0.950511 0.310690i \(-0.100560\pi\)
0.206190 + 0.978512i \(0.433893\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −113.666 + 65.6249i −0.275219 + 0.158898i
\(414\) 0 0
\(415\) 64.1975 + 111.193i 0.154693 + 0.267936i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 529.552 1.26385 0.631924 0.775031i \(-0.282266\pi\)
0.631924 + 0.775031i \(0.282266\pi\)
\(420\) 0 0
\(421\) 225.681 130.297i 0.536060 0.309494i −0.207421 0.978252i \(-0.566507\pi\)
0.743480 + 0.668758i \(0.233174\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 111.369 0.262045
\(426\) 0 0
\(427\) 82.8258 143.458i 0.193971 0.335968i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 208.226 + 120.219i 0.483123 + 0.278931i 0.721717 0.692188i \(-0.243353\pi\)
−0.238594 + 0.971119i \(0.576686\pi\)
\(432\) 0 0
\(433\) −715.847 413.295i −1.65323 0.954491i −0.975733 0.218963i \(-0.929733\pi\)
−0.677494 0.735528i \(-0.736934\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 78.5174 + 28.3446i 0.179674 + 0.0648618i
\(438\) 0 0
\(439\) 516.901 298.433i 1.17745 0.679801i 0.222027 0.975040i \(-0.428733\pi\)
0.955423 + 0.295239i \(0.0953993\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −259.517 + 449.497i −0.585818 + 1.01467i 0.408955 + 0.912554i \(0.365893\pi\)
−0.994773 + 0.102112i \(0.967440\pi\)
\(444\) 0 0
\(445\) 643.517i 1.44611i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 50.6169i 0.112732i −0.998410 0.0563662i \(-0.982049\pi\)
0.998410 0.0563662i \(-0.0179514\pi\)
\(450\) 0 0
\(451\) −489.303 + 282.499i −1.08493 + 0.626384i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 109.189i 0.239976i
\(456\) 0 0
\(457\) 219.788 0.480936 0.240468 0.970657i \(-0.422699\pi\)
0.240468 + 0.970657i \(0.422699\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −23.1770 40.1437i −0.0502754 0.0870796i 0.839793 0.542907i \(-0.182677\pi\)
−0.890068 + 0.455828i \(0.849343\pi\)
\(462\) 0 0
\(463\) −678.308 −1.46503 −0.732514 0.680752i \(-0.761653\pi\)
−0.732514 + 0.680752i \(0.761653\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 687.475 1.47211 0.736054 0.676923i \(-0.236687\pi\)
0.736054 + 0.676923i \(0.236687\pi\)
\(468\) 0 0
\(469\) −142.500 82.2724i −0.303838 0.175421i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −242.065 419.269i −0.511766 0.886405i
\(474\) 0 0
\(475\) −101.811 36.7535i −0.214339 0.0773758i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −194.946 + 337.656i −0.406985 + 0.704919i −0.994550 0.104258i \(-0.966753\pi\)
0.587565 + 0.809177i \(0.300087\pi\)
\(480\) 0 0
\(481\) −203.379 + 352.262i −0.422824 + 0.732353i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −118.625 68.4883i −0.244588 0.141213i
\(486\) 0 0
\(487\) 179.520i 0.368624i −0.982868 0.184312i \(-0.940994\pi\)
0.982868 0.184312i \(-0.0590057\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −253.593 439.235i −0.516482 0.894573i −0.999817 0.0191376i \(-0.993908\pi\)
0.483335 0.875436i \(-0.339425\pi\)
\(492\) 0 0
\(493\) 661.923i 1.34264i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 127.512 73.6191i 0.256563 0.148127i
\(498\) 0 0
\(499\) −359.729 623.069i −0.720900 1.24864i −0.960639 0.277799i \(-0.910395\pi\)
0.239739 0.970837i \(-0.422938\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 315.102 545.773i 0.626445 1.08504i −0.361814 0.932250i \(-0.617842\pi\)
0.988259 0.152785i \(-0.0488242\pi\)
\(504\) 0 0
\(505\) −579.968 −1.14845
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −83.0780 47.9651i −0.163218 0.0942340i 0.416166 0.909289i \(-0.363374\pi\)
−0.579384 + 0.815055i \(0.696707\pi\)
\(510\) 0 0
\(511\) 14.5681 + 25.2327i 0.0285090 + 0.0493791i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 157.202 90.7606i 0.305246 0.176234i
\(516\) 0 0
\(517\) 592.545 1026.32i 1.14612 1.98514i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 253.352i 0.486281i −0.969991 0.243140i \(-0.921822\pi\)
0.969991 0.243140i \(-0.0781776\pi\)
\(522\) 0 0
\(523\) −406.309 234.583i −0.776882 0.448533i 0.0584420 0.998291i \(-0.481387\pi\)
−0.835324 + 0.549758i \(0.814720\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 95.2263 54.9789i 0.180695 0.104324i
\(528\) 0 0
\(529\) 254.848 + 441.410i 0.481755 + 0.834424i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 639.191 1.19923
\(534\) 0 0
\(535\) −128.833 + 74.3819i −0.240810 + 0.139032i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −710.774 −1.31869
\(540\) 0 0
\(541\) 461.257 798.921i 0.852601 1.47675i −0.0262519 0.999655i \(-0.508357\pi\)
0.878853 0.477093i \(-0.158309\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 133.137 + 76.8668i 0.244289 + 0.141040i
\(546\) 0 0
\(547\) 159.901 + 92.3187i 0.292323 + 0.168773i 0.638989 0.769216i \(-0.279353\pi\)
−0.346666 + 0.937989i \(0.612686\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −218.444 + 605.114i −0.396451 + 1.09821i
\(552\) 0 0
\(553\) −66.4240 + 38.3499i −0.120116 + 0.0693488i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −392.410 + 679.674i −0.704507 + 1.22024i 0.262363 + 0.964969i \(0.415498\pi\)
−0.966869 + 0.255272i \(0.917835\pi\)
\(558\) 0 0
\(559\) 547.704i 0.979792i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 700.448i 1.24413i 0.782964 + 0.622067i \(0.213707\pi\)
−0.782964 + 0.622067i \(0.786293\pi\)
\(564\) 0 0
\(565\) −621.061 + 358.570i −1.09922 + 0.634637i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 251.046i 0.441206i −0.975364 0.220603i \(-0.929198\pi\)
0.975364 0.220603i \(-0.0708025\pi\)
\(570\) 0 0
\(571\) 728.378 1.27562 0.637809 0.770194i \(-0.279841\pi\)
0.637809 + 0.770194i \(0.279841\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 12.5148 + 21.6763i 0.0217649 + 0.0376979i
\(576\) 0 0
\(577\) −200.504 −0.347494 −0.173747 0.984790i \(-0.555588\pi\)
−0.173747 + 0.984790i \(0.555588\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −42.3594 −0.0729078
\(582\) 0 0
\(583\) −144.773 83.5847i −0.248324 0.143370i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −257.221 445.519i −0.438195 0.758977i 0.559355 0.828928i \(-0.311049\pi\)
−0.997550 + 0.0699515i \(0.977716\pi\)
\(588\) 0 0
\(589\) −105.197 + 18.8343i −0.178603 + 0.0319767i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 399.444 691.858i 0.673599 1.16671i −0.303277 0.952902i \(-0.598081\pi\)
0.976876 0.213805i \(-0.0685859\pi\)
\(594\) 0 0
\(595\) 62.2474 107.816i 0.104618 0.181203i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −147.881 85.3793i −0.246880 0.142536i 0.371455 0.928451i \(-0.378859\pi\)
−0.618335 + 0.785915i \(0.712192\pi\)
\(600\) 0 0
\(601\) 836.789i 1.39233i 0.717883 + 0.696164i \(0.245111\pi\)
−0.717883 + 0.696164i \(0.754889\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −238.759 413.543i −0.394643 0.683542i
\(606\) 0 0
\(607\) 285.238i 0.469914i −0.972006 0.234957i \(-0.924505\pi\)
0.972006 0.234957i \(-0.0754949\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1161.09 + 670.354i −1.90031 + 1.09714i
\(612\) 0 0
\(613\) −330.388 572.249i −0.538969 0.933522i −0.998960 0.0455986i \(-0.985480\pi\)
0.459990 0.887924i \(-0.347853\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −372.340 + 644.912i −0.603468 + 1.04524i 0.388823 + 0.921312i \(0.372882\pi\)
−0.992292 + 0.123926i \(0.960452\pi\)
\(618\) 0 0
\(619\) 94.1964 0.152175 0.0760876 0.997101i \(-0.475757\pi\)
0.0760876 + 0.997101i \(0.475757\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 183.862 + 106.153i 0.295124 + 0.170390i
\(624\) 0 0
\(625\) 225.061 + 389.817i 0.360097 + 0.623707i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 401.641 231.888i 0.638539 0.368661i
\(630\) 0 0
\(631\) 284.195 492.240i 0.450388 0.780094i −0.548022 0.836464i \(-0.684619\pi\)
0.998410 + 0.0563693i \(0.0179524\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 687.802i 1.08315i
\(636\) 0 0
\(637\) 696.378 + 402.054i 1.09322 + 0.631168i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 233.256 134.670i 0.363894 0.210094i −0.306894 0.951744i \(-0.599290\pi\)
0.670787 + 0.741650i \(0.265956\pi\)
\(642\) 0 0
\(643\) 577.725 + 1000.65i 0.898483 + 1.55622i 0.829434 + 0.558605i \(0.188663\pi\)
0.0690498 + 0.997613i \(0.478003\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1039.16 −1.60611 −0.803057 0.595902i \(-0.796795\pi\)
−0.803057 + 0.595902i \(0.796795\pi\)
\(648\) 0 0
\(649\) −1188.45 + 686.154i −1.83121 + 1.05725i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1115.67 1.70852 0.854262 0.519843i \(-0.174009\pi\)
0.854262 + 0.519843i \(0.174009\pi\)
\(654\) 0 0
\(655\) 474.338 821.578i 0.724181 1.25432i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −86.7060 50.0597i −0.131572 0.0759632i 0.432769 0.901505i \(-0.357536\pi\)
−0.564341 + 0.825542i \(0.690870\pi\)
\(660\) 0 0
\(661\) −759.303 438.384i −1.14872 0.663213i −0.200144 0.979766i \(-0.564141\pi\)
−0.948575 + 0.316553i \(0.897474\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −92.4859 + 78.0200i −0.139077 + 0.117323i
\(666\) 0 0
\(667\) 128.833 74.3819i 0.193153 0.111517i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 866.001 1499.96i 1.29061 2.23541i
\(672\) 0 0
\(673\) 838.032i 1.24522i 0.782533 + 0.622609i \(0.213927\pi\)
−0.782533 + 0.622609i \(0.786073\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 572.195i 0.845193i −0.906318 0.422596i \(-0.861119\pi\)
0.906318 0.422596i \(-0.138881\pi\)
\(678\) 0 0
\(679\) 39.1362 22.5953i 0.0576380 0.0332773i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 191.437i 0.280289i 0.990131 + 0.140144i \(0.0447567\pi\)
−0.990131 + 0.140144i \(0.955243\pi\)
\(684\) 0 0
\(685\) 261.566 0.381849
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 94.5605 + 163.784i 0.137243 + 0.237712i
\(690\) 0 0
\(691\) 1090.79 1.57856 0.789281 0.614033i \(-0.210454\pi\)
0.789281 + 0.614033i \(0.210454\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.0431 0.0230836
\(696\) 0 0
\(697\) −631.151 364.395i −0.905525 0.522805i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 404.337 + 700.331i 0.576800 + 0.999046i 0.995844 + 0.0910803i \(0.0290320\pi\)
−0.419044 + 0.907966i \(0.637635\pi\)
\(702\) 0 0
\(703\) −443.697 + 79.4384i −0.631148 + 0.112999i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 95.6701 165.706i 0.135318 0.234378i
\(708\) 0 0
\(709\) 153.803 266.395i 0.216930 0.375733i −0.736938 0.675960i \(-0.763729\pi\)
0.953868 + 0.300227i \(0.0970625\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 21.4016 + 12.3562i 0.0300163 + 0.0173299i
\(714\) 0 0
\(715\) 1141.65i 1.59671i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 156.569 + 271.185i 0.217759 + 0.377169i 0.954122 0.299417i \(-0.0967920\pi\)
−0.736364 + 0.676586i \(0.763459\pi\)
\(720\) 0 0
\(721\) 59.8865i 0.0830604i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −167.054 + 96.4485i −0.230419 + 0.133032i
\(726\) 0 0
\(727\) 295.082 + 511.098i 0.405890 + 0.703023i 0.994425 0.105449i \(-0.0336281\pi\)
−0.588534 + 0.808472i \(0.700295\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 312.240 540.815i 0.427140 0.739829i
\(732\) 0 0
\(733\) −775.514 −1.05800 −0.529000 0.848622i \(-0.677433\pi\)
−0.529000 + 0.848622i \(0.677433\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1489.94 860.215i −2.02162 1.16718i
\(738\) 0 0
\(739\) 473.628 + 820.348i 0.640904 + 1.11008i 0.985231 + 0.171229i \(0.0547738\pi\)
−0.344327 + 0.938850i \(0.611893\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −487.913 + 281.697i −0.656680 + 0.379134i −0.791011 0.611802i \(-0.790445\pi\)
0.134331 + 0.990937i \(0.457112\pi\)
\(744\) 0 0
\(745\) 127.420 220.698i 0.171033 0.296239i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 49.0794i 0.0655265i
\(750\) 0 0
\(751\) 863.689 + 498.651i 1.15005 + 0.663982i 0.948900 0.315578i \(-0.102198\pi\)
0.201152 + 0.979560i \(0.435532\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 32.8842 18.9857i 0.0435552 0.0251466i
\(756\) 0 0
\(757\) 595.954 + 1032.22i 0.787258 + 1.36357i 0.927641 + 0.373473i \(0.121833\pi\)
−0.140383 + 0.990097i \(0.544833\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −516.905 −0.679244 −0.339622 0.940562i \(-0.610299\pi\)
−0.339622 + 0.940562i \(0.610299\pi\)
\(762\) 0 0
\(763\) −43.9240 + 25.3595i −0.0575675 + 0.0332366i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1552.51 2.02413
\(768\) 0 0
\(769\) −142.307 + 246.483i −0.185055 + 0.320524i −0.943595 0.331102i \(-0.892580\pi\)
0.758540 + 0.651626i \(0.225913\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 792.031 + 457.280i 1.02462 + 0.591565i 0.915439 0.402457i \(-0.131844\pi\)
0.109181 + 0.994022i \(0.465177\pi\)
\(774\) 0 0
\(775\) −27.7508 16.0219i −0.0358074 0.0206734i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 456.727 + 541.411i 0.586299 + 0.695007i
\(780\) 0 0
\(781\) 1333.23 769.738i 1.70708 0.985581i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 263.744 456.818i 0.335979 0.581933i
\(786\) 0 0
\(787\) 199.823i 0.253905i 0.991909 + 0.126952i \(0.0405195\pi\)
−0.991909 + 0.126952i \(0.959480\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 236.595i 0.299109i
\(792\) 0 0
\(793\) −1696.92 + 979.719i −2.13988 + 1.23546i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1490.18i 1.86973i 0.355003 + 0.934865i \(0.384480\pi\)
−0.355003 + 0.934865i \(0.615520\pi\)
\(798\) 0 0
\(799\) 1528.64 1.91320
\(800\) 0 0
\(801\) 0 0