Properties

Label 800.3.b.i.351.6
Level $800$
Weight $3$
Character 800.351
Analytic conductor $21.798$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 800.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.7984211488\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
Defining polynomial: \( x^{6} + 9x^{4} + 14x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 351.6
Root \(2.65109i\) of defining polynomial
Character \(\chi\) \(=\) 800.351
Dual form 800.3.b.i.351.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.30219i q^{3} +0.206625i q^{7} -19.1132 q^{9} +O(q^{10})\) \(q+5.30219i q^{3} +0.206625i q^{7} -19.1132 q^{9} -15.0176i q^{11} -11.6999 q^{13} +18.1911 q^{17} -19.3999i q^{19} -1.09556 q^{21} -27.2242i q^{23} -53.6220i q^{27} -44.4175 q^{29} -20.3822i q^{31} +79.6262 q^{33} +18.1089 q^{37} -62.0352i q^{39} -32.3043 q^{41} -4.06244i q^{43} -5.37588i q^{47} +48.9573 q^{49} +96.4527i q^{51} -79.1703 q^{53} +102.862 q^{57} +83.3999i q^{59} -36.7486 q^{61} -3.94925i q^{63} +4.51518i q^{67} +144.348 q^{69} -41.6530i q^{71} +41.5910 q^{73} +3.10301 q^{77} -15.5473i q^{79} +112.295 q^{81} -50.9862i q^{83} -235.510i q^{87} -10.8885 q^{89} -2.41749i q^{91} +108.070 q^{93} +12.1559 q^{97} +287.035i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 18 q^{9} + 80 q^{17} + 8 q^{21} - 44 q^{29} + 144 q^{33} + 208 q^{37} - 68 q^{41} + 62 q^{49} + 64 q^{53} + 400 q^{57} - 100 q^{61} + 184 q^{69} + 80 q^{73} + 400 q^{77} + 238 q^{81} - 76 q^{89} + 320 q^{93} + 208 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 5.30219i 1.76740i 0.468058 + 0.883698i \(0.344954\pi\)
−0.468058 + 0.883698i \(0.655046\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.206625i 0.0295178i 0.999891 + 0.0147589i \(0.00469807\pi\)
−0.999891 + 0.0147589i \(0.995302\pi\)
\(8\) 0 0
\(9\) −19.1132 −2.12369
\(10\) 0 0
\(11\) − 15.0176i − 1.36524i −0.730774 0.682619i \(-0.760841\pi\)
0.730774 0.682619i \(-0.239159\pi\)
\(12\) 0 0
\(13\) −11.6999 −0.899995 −0.449998 0.893030i \(-0.648575\pi\)
−0.449998 + 0.893030i \(0.648575\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 18.1911 1.07007 0.535033 0.844831i \(-0.320299\pi\)
0.535033 + 0.844831i \(0.320299\pi\)
\(18\) 0 0
\(19\) − 19.3999i − 1.02105i −0.859864 0.510523i \(-0.829452\pi\)
0.859864 0.510523i \(-0.170548\pi\)
\(20\) 0 0
\(21\) −1.09556 −0.0521696
\(22\) 0 0
\(23\) − 27.2242i − 1.18366i −0.806062 0.591831i \(-0.798405\pi\)
0.806062 0.591831i \(-0.201595\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 53.6220i − 1.98600i
\(28\) 0 0
\(29\) −44.4175 −1.53164 −0.765819 0.643056i \(-0.777666\pi\)
−0.765819 + 0.643056i \(0.777666\pi\)
\(30\) 0 0
\(31\) − 20.3822i − 0.657492i −0.944418 0.328746i \(-0.893374\pi\)
0.944418 0.328746i \(-0.106626\pi\)
\(32\) 0 0
\(33\) 79.6262 2.41292
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 18.1089 0.489431 0.244715 0.969595i \(-0.421305\pi\)
0.244715 + 0.969595i \(0.421305\pi\)
\(38\) 0 0
\(39\) − 62.0352i − 1.59065i
\(40\) 0 0
\(41\) −32.3043 −0.787910 −0.393955 0.919130i \(-0.628893\pi\)
−0.393955 + 0.919130i \(0.628893\pi\)
\(42\) 0 0
\(43\) − 4.06244i − 0.0944753i −0.998884 0.0472377i \(-0.984958\pi\)
0.998884 0.0472377i \(-0.0150418\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 5.37588i − 0.114380i −0.998363 0.0571902i \(-0.981786\pi\)
0.998363 0.0571902i \(-0.0182142\pi\)
\(48\) 0 0
\(49\) 48.9573 0.999129
\(50\) 0 0
\(51\) 96.4527i 1.89123i
\(52\) 0 0
\(53\) −79.1703 −1.49378 −0.746890 0.664948i \(-0.768454\pi\)
−0.746890 + 0.664948i \(0.768454\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 102.862 1.80459
\(58\) 0 0
\(59\) 83.3999i 1.41356i 0.707435 + 0.706779i \(0.249852\pi\)
−0.707435 + 0.706779i \(0.750148\pi\)
\(60\) 0 0
\(61\) −36.7486 −0.602435 −0.301218 0.953555i \(-0.597393\pi\)
−0.301218 + 0.953555i \(0.597393\pi\)
\(62\) 0 0
\(63\) − 3.94925i − 0.0626866i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.51518i 0.0673907i 0.999432 + 0.0336954i \(0.0107276\pi\)
−0.999432 + 0.0336954i \(0.989272\pi\)
\(68\) 0 0
\(69\) 144.348 2.09200
\(70\) 0 0
\(71\) − 41.6530i − 0.586662i −0.956011 0.293331i \(-0.905236\pi\)
0.956011 0.293331i \(-0.0947638\pi\)
\(72\) 0 0
\(73\) 41.5910 0.569740 0.284870 0.958566i \(-0.408050\pi\)
0.284870 + 0.958566i \(0.408050\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.10301 0.0402988
\(78\) 0 0
\(79\) − 15.5473i − 0.196801i −0.995147 0.0984004i \(-0.968627\pi\)
0.995147 0.0984004i \(-0.0313726\pi\)
\(80\) 0 0
\(81\) 112.295 1.38636
\(82\) 0 0
\(83\) − 50.9862i − 0.614291i −0.951663 0.307146i \(-0.900626\pi\)
0.951663 0.307146i \(-0.0993739\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 235.510i − 2.70701i
\(88\) 0 0
\(89\) −10.8885 −0.122343 −0.0611713 0.998127i \(-0.519484\pi\)
−0.0611713 + 0.998127i \(0.519484\pi\)
\(90\) 0 0
\(91\) − 2.41749i − 0.0265659i
\(92\) 0 0
\(93\) 108.070 1.16205
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 12.1559 0.125318 0.0626592 0.998035i \(-0.480042\pi\)
0.0626592 + 0.998035i \(0.480042\pi\)
\(98\) 0 0
\(99\) 287.035i 2.89934i
\(100\) 0 0
\(101\) 127.723 1.26459 0.632294 0.774728i \(-0.282113\pi\)
0.632294 + 0.774728i \(0.282113\pi\)
\(102\) 0 0
\(103\) 4.77575i 0.0463665i 0.999731 + 0.0231833i \(0.00738012\pi\)
−0.999731 + 0.0231833i \(0.992620\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 107.213i − 1.00199i −0.865449 0.500997i \(-0.832967\pi\)
0.865449 0.500997i \(-0.167033\pi\)
\(108\) 0 0
\(109\) −53.6689 −0.492376 −0.246188 0.969222i \(-0.579178\pi\)
−0.246188 + 0.969222i \(0.579178\pi\)
\(110\) 0 0
\(111\) 96.0170i 0.865018i
\(112\) 0 0
\(113\) 20.5063 0.181471 0.0907356 0.995875i \(-0.471078\pi\)
0.0907356 + 0.995875i \(0.471078\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 223.623 1.91131
\(118\) 0 0
\(119\) 3.75873i 0.0315860i
\(120\) 0 0
\(121\) −104.529 −0.863876
\(122\) 0 0
\(123\) − 171.283i − 1.39255i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 138.477i 1.09037i 0.838316 + 0.545184i \(0.183540\pi\)
−0.838316 + 0.545184i \(0.816460\pi\)
\(128\) 0 0
\(129\) 21.5398 0.166975
\(130\) 0 0
\(131\) − 219.105i − 1.67256i −0.548304 0.836279i \(-0.684726\pi\)
0.548304 0.836279i \(-0.315274\pi\)
\(132\) 0 0
\(133\) 4.00849 0.0301390
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 59.7821 0.436366 0.218183 0.975908i \(-0.429987\pi\)
0.218183 + 0.975908i \(0.429987\pi\)
\(138\) 0 0
\(139\) − 26.6524i − 0.191744i −0.995394 0.0958718i \(-0.969436\pi\)
0.995394 0.0958718i \(-0.0305639\pi\)
\(140\) 0 0
\(141\) 28.5039 0.202155
\(142\) 0 0
\(143\) 175.705i 1.22871i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 259.581i 1.76586i
\(148\) 0 0
\(149\) −143.463 −0.962838 −0.481419 0.876490i \(-0.659879\pi\)
−0.481419 + 0.876490i \(0.659879\pi\)
\(150\) 0 0
\(151\) − 83.4937i − 0.552939i −0.961023 0.276469i \(-0.910836\pi\)
0.961023 0.276469i \(-0.0891644\pi\)
\(152\) 0 0
\(153\) −347.690 −2.27249
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −169.673 −1.08072 −0.540360 0.841434i \(-0.681712\pi\)
−0.540360 + 0.841434i \(0.681712\pi\)
\(158\) 0 0
\(159\) − 419.776i − 2.64010i
\(160\) 0 0
\(161\) 5.62520 0.0349391
\(162\) 0 0
\(163\) 275.478i 1.69005i 0.534726 + 0.845026i \(0.320415\pi\)
−0.534726 + 0.845026i \(0.679585\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 132.481i − 0.793299i −0.917970 0.396650i \(-0.870173\pi\)
0.917970 0.396650i \(-0.129827\pi\)
\(168\) 0 0
\(169\) −32.1115 −0.190009
\(170\) 0 0
\(171\) 370.793i 2.16838i
\(172\) 0 0
\(173\) 272.614 1.57580 0.787901 0.615801i \(-0.211168\pi\)
0.787901 + 0.615801i \(0.211168\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −442.202 −2.49831
\(178\) 0 0
\(179\) − 157.523i − 0.880014i −0.897994 0.440007i \(-0.854976\pi\)
0.897994 0.440007i \(-0.145024\pi\)
\(180\) 0 0
\(181\) −335.063 −1.85118 −0.925590 0.378529i \(-0.876430\pi\)
−0.925590 + 0.378529i \(0.876430\pi\)
\(182\) 0 0
\(183\) − 194.848i − 1.06474i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 273.187i − 1.46090i
\(188\) 0 0
\(189\) 11.0796 0.0586223
\(190\) 0 0
\(191\) − 298.575i − 1.56322i −0.623766 0.781611i \(-0.714398\pi\)
0.623766 0.781611i \(-0.285602\pi\)
\(192\) 0 0
\(193\) −191.915 −0.994376 −0.497188 0.867643i \(-0.665634\pi\)
−0.497188 + 0.867643i \(0.665634\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −59.2472 −0.300747 −0.150374 0.988629i \(-0.548048\pi\)
−0.150374 + 0.988629i \(0.548048\pi\)
\(198\) 0 0
\(199\) − 309.100i − 1.55327i −0.629953 0.776633i \(-0.716926\pi\)
0.629953 0.776633i \(-0.283074\pi\)
\(200\) 0 0
\(201\) −23.9403 −0.119106
\(202\) 0 0
\(203\) − 9.17775i − 0.0452106i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 520.342i 2.51373i
\(208\) 0 0
\(209\) −291.340 −1.39397
\(210\) 0 0
\(211\) 205.693i 0.974850i 0.873165 + 0.487425i \(0.162064\pi\)
−0.873165 + 0.487425i \(0.837936\pi\)
\(212\) 0 0
\(213\) 220.852 1.03686
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.21147 0.0194077
\(218\) 0 0
\(219\) 220.523i 1.00696i
\(220\) 0 0
\(221\) −212.835 −0.963054
\(222\) 0 0
\(223\) − 228.723i − 1.02567i −0.858488 0.512833i \(-0.828596\pi\)
0.858488 0.512833i \(-0.171404\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 282.403i 1.24407i 0.782991 + 0.622033i \(0.213693\pi\)
−0.782991 + 0.622033i \(0.786307\pi\)
\(228\) 0 0
\(229\) −49.2525 −0.215076 −0.107538 0.994201i \(-0.534297\pi\)
−0.107538 + 0.994201i \(0.534297\pi\)
\(230\) 0 0
\(231\) 16.4527i 0.0712240i
\(232\) 0 0
\(233\) 124.273 0.533363 0.266681 0.963785i \(-0.414073\pi\)
0.266681 + 0.963785i \(0.414073\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 82.4345 0.347825
\(238\) 0 0
\(239\) 80.4527i 0.336622i 0.985734 + 0.168311i \(0.0538313\pi\)
−0.985734 + 0.168311i \(0.946169\pi\)
\(240\) 0 0
\(241\) −1.20979 −0.00501988 −0.00250994 0.999997i \(-0.500799\pi\)
−0.00250994 + 0.999997i \(0.500799\pi\)
\(242\) 0 0
\(243\) 112.812i 0.464247i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 226.977i 0.918936i
\(248\) 0 0
\(249\) 270.338 1.08570
\(250\) 0 0
\(251\) − 211.853i − 0.844034i −0.906588 0.422017i \(-0.861322\pi\)
0.906588 0.422017i \(-0.138678\pi\)
\(252\) 0 0
\(253\) −408.843 −1.61598
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 182.646 0.710685 0.355342 0.934736i \(-0.384364\pi\)
0.355342 + 0.934736i \(0.384364\pi\)
\(258\) 0 0
\(259\) 3.74175i 0.0144469i
\(260\) 0 0
\(261\) 848.960 3.25272
\(262\) 0 0
\(263\) 74.6636i 0.283892i 0.989874 + 0.141946i \(0.0453359\pi\)
−0.989874 + 0.141946i \(0.954664\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 57.7329i − 0.216228i
\(268\) 0 0
\(269\) 184.089 0.684344 0.342172 0.939637i \(-0.388837\pi\)
0.342172 + 0.939637i \(0.388837\pi\)
\(270\) 0 0
\(271\) − 234.746i − 0.866222i −0.901341 0.433111i \(-0.857416\pi\)
0.901341 0.433111i \(-0.142584\pi\)
\(272\) 0 0
\(273\) 12.8180 0.0469524
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −452.208 −1.63252 −0.816260 0.577684i \(-0.803957\pi\)
−0.816260 + 0.577684i \(0.803957\pi\)
\(278\) 0 0
\(279\) 389.570i 1.39631i
\(280\) 0 0
\(281\) −196.110 −0.697900 −0.348950 0.937141i \(-0.613462\pi\)
−0.348950 + 0.937141i \(0.613462\pi\)
\(282\) 0 0
\(283\) 418.449i 1.47862i 0.673366 + 0.739309i \(0.264848\pi\)
−0.673366 + 0.739309i \(0.735152\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 6.67486i − 0.0232574i
\(288\) 0 0
\(289\) 41.9170 0.145042
\(290\) 0 0
\(291\) 64.4527i 0.221487i
\(292\) 0 0
\(293\) −286.666 −0.978383 −0.489191 0.872176i \(-0.662708\pi\)
−0.489191 + 0.872176i \(0.662708\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −805.275 −2.71136
\(298\) 0 0
\(299\) 318.522i 1.06529i
\(300\) 0 0
\(301\) 0.839400 0.00278870
\(302\) 0 0
\(303\) 677.214i 2.23503i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 261.715i 0.852493i 0.904607 + 0.426247i \(0.140164\pi\)
−0.904607 + 0.426247i \(0.859836\pi\)
\(308\) 0 0
\(309\) −25.3219 −0.0819480
\(310\) 0 0
\(311\) 578.904i 1.86143i 0.365747 + 0.930714i \(0.380813\pi\)
−0.365747 + 0.930714i \(0.619187\pi\)
\(312\) 0 0
\(313\) −99.3124 −0.317292 −0.158646 0.987336i \(-0.550713\pi\)
−0.158646 + 0.987336i \(0.550713\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −191.623 −0.604489 −0.302245 0.953230i \(-0.597736\pi\)
−0.302245 + 0.953230i \(0.597736\pi\)
\(318\) 0 0
\(319\) 667.045i 2.09105i
\(320\) 0 0
\(321\) 568.466 1.77092
\(322\) 0 0
\(323\) − 352.905i − 1.09259i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 284.563i − 0.870222i
\(328\) 0 0
\(329\) 1.11079 0.00337626
\(330\) 0 0
\(331\) − 530.187i − 1.60177i −0.598816 0.800886i \(-0.704362\pi\)
0.598816 0.800886i \(-0.295638\pi\)
\(332\) 0 0
\(333\) −346.120 −1.03940
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 487.427 1.44637 0.723186 0.690653i \(-0.242677\pi\)
0.723186 + 0.690653i \(0.242677\pi\)
\(338\) 0 0
\(339\) 108.728i 0.320731i
\(340\) 0 0
\(341\) −306.093 −0.897633
\(342\) 0 0
\(343\) 20.2404i 0.0590099i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 310.497i 0.894804i 0.894333 + 0.447402i \(0.147651\pi\)
−0.894333 + 0.447402i \(0.852349\pi\)
\(348\) 0 0
\(349\) 253.004 0.724941 0.362471 0.931995i \(-0.381933\pi\)
0.362471 + 0.931995i \(0.381933\pi\)
\(350\) 0 0
\(351\) 627.374i 1.78739i
\(352\) 0 0
\(353\) 322.639 0.913992 0.456996 0.889469i \(-0.348925\pi\)
0.456996 + 0.889469i \(0.348925\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −19.9295 −0.0558250
\(358\) 0 0
\(359\) − 254.975i − 0.710236i −0.934822 0.355118i \(-0.884441\pi\)
0.934822 0.355118i \(-0.115559\pi\)
\(360\) 0 0
\(361\) −15.3550 −0.0425347
\(362\) 0 0
\(363\) − 554.232i − 1.52681i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 207.935i 0.566581i 0.959034 + 0.283291i \(0.0914261\pi\)
−0.959034 + 0.283291i \(0.908574\pi\)
\(368\) 0 0
\(369\) 617.438 1.67327
\(370\) 0 0
\(371\) − 16.3585i − 0.0440931i
\(372\) 0 0
\(373\) −203.826 −0.546450 −0.273225 0.961950i \(-0.588090\pi\)
−0.273225 + 0.961950i \(0.588090\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 519.682 1.37847
\(378\) 0 0
\(379\) − 454.099i − 1.19815i −0.800692 0.599076i \(-0.795535\pi\)
0.800692 0.599076i \(-0.204465\pi\)
\(380\) 0 0
\(381\) −734.229 −1.92711
\(382\) 0 0
\(383\) − 541.569i − 1.41402i −0.707205 0.707009i \(-0.750044\pi\)
0.707205 0.707009i \(-0.249956\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 77.6462i 0.200636i
\(388\) 0 0
\(389\) 423.431 1.08851 0.544256 0.838919i \(-0.316812\pi\)
0.544256 + 0.838919i \(0.316812\pi\)
\(390\) 0 0
\(391\) − 495.240i − 1.26660i
\(392\) 0 0
\(393\) 1161.74 2.95607
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.7772 0.0296654 0.0148327 0.999890i \(-0.495278\pi\)
0.0148327 + 0.999890i \(0.495278\pi\)
\(398\) 0 0
\(399\) 21.2538i 0.0532676i
\(400\) 0 0
\(401\) 127.442 0.317809 0.158905 0.987294i \(-0.449204\pi\)
0.158905 + 0.987294i \(0.449204\pi\)
\(402\) 0 0
\(403\) 238.471i 0.591739i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 271.953i − 0.668190i
\(408\) 0 0
\(409\) −608.012 −1.48658 −0.743290 0.668969i \(-0.766736\pi\)
−0.743290 + 0.668969i \(0.766736\pi\)
\(410\) 0 0
\(411\) 316.976i 0.771231i
\(412\) 0 0
\(413\) −17.2325 −0.0417251
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 141.316 0.338887
\(418\) 0 0
\(419\) − 565.630i − 1.34995i −0.737840 0.674976i \(-0.764154\pi\)
0.737840 0.674976i \(-0.235846\pi\)
\(420\) 0 0
\(421\) 711.356 1.68968 0.844841 0.535018i \(-0.179695\pi\)
0.844841 + 0.535018i \(0.179695\pi\)
\(422\) 0 0
\(423\) 102.750i 0.242908i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 7.59316i − 0.0177826i
\(428\) 0 0
\(429\) −931.622 −2.17161
\(430\) 0 0
\(431\) 309.254i 0.717526i 0.933429 + 0.358763i \(0.116801\pi\)
−0.933429 + 0.358763i \(0.883199\pi\)
\(432\) 0 0
\(433\) −187.374 −0.432735 −0.216368 0.976312i \(-0.569421\pi\)
−0.216368 + 0.976312i \(0.569421\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −528.147 −1.20857
\(438\) 0 0
\(439\) − 289.657i − 0.659811i −0.944014 0.329906i \(-0.892983\pi\)
0.944014 0.329906i \(-0.107017\pi\)
\(440\) 0 0
\(441\) −935.730 −2.12184
\(442\) 0 0
\(443\) − 295.516i − 0.667079i −0.942736 0.333539i \(-0.891757\pi\)
0.942736 0.333539i \(-0.108243\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 760.667i − 1.70172i
\(448\) 0 0
\(449\) 604.409 1.34612 0.673061 0.739587i \(-0.264979\pi\)
0.673061 + 0.739587i \(0.264979\pi\)
\(450\) 0 0
\(451\) 485.134i 1.07569i
\(452\) 0 0
\(453\) 442.699 0.977262
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −392.507 −0.858877 −0.429438 0.903096i \(-0.641288\pi\)
−0.429438 + 0.903096i \(0.641288\pi\)
\(458\) 0 0
\(459\) − 975.444i − 2.12515i
\(460\) 0 0
\(461\) −400.277 −0.868279 −0.434139 0.900846i \(-0.642947\pi\)
−0.434139 + 0.900846i \(0.642947\pi\)
\(462\) 0 0
\(463\) 732.679i 1.58246i 0.611518 + 0.791230i \(0.290559\pi\)
−0.611518 + 0.791230i \(0.709441\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 592.126i − 1.26794i −0.773360 0.633968i \(-0.781425\pi\)
0.773360 0.633968i \(-0.218575\pi\)
\(468\) 0 0
\(469\) −0.932947 −0.00198923
\(470\) 0 0
\(471\) − 899.639i − 1.91006i
\(472\) 0 0
\(473\) −61.0082 −0.128981
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1513.20 3.17232
\(478\) 0 0
\(479\) − 309.151i − 0.645409i −0.946500 0.322705i \(-0.895408\pi\)
0.946500 0.322705i \(-0.104592\pi\)
\(480\) 0 0
\(481\) −211.873 −0.440485
\(482\) 0 0
\(483\) 29.8259i 0.0617513i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 570.769i 1.17201i 0.810307 + 0.586005i \(0.199300\pi\)
−0.810307 + 0.586005i \(0.800700\pi\)
\(488\) 0 0
\(489\) −1460.64 −2.98699
\(490\) 0 0
\(491\) − 301.659i − 0.614378i −0.951649 0.307189i \(-0.900612\pi\)
0.951649 0.307189i \(-0.0993883\pi\)
\(492\) 0 0
\(493\) −808.004 −1.63895
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.60653 0.0173170
\(498\) 0 0
\(499\) 517.758i 1.03759i 0.854898 + 0.518795i \(0.173619\pi\)
−0.854898 + 0.518795i \(0.826381\pi\)
\(500\) 0 0
\(501\) 702.439 1.40207
\(502\) 0 0
\(503\) − 406.671i − 0.808491i −0.914650 0.404246i \(-0.867534\pi\)
0.914650 0.404246i \(-0.132466\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 170.261i − 0.335821i
\(508\) 0 0
\(509\) −627.097 −1.23202 −0.616009 0.787739i \(-0.711252\pi\)
−0.616009 + 0.787739i \(0.711252\pi\)
\(510\) 0 0
\(511\) 8.59372i 0.0168175i
\(512\) 0 0
\(513\) −1040.26 −2.02780
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −80.7330 −0.156157
\(518\) 0 0
\(519\) 1445.45i 2.78507i
\(520\) 0 0
\(521\) −111.743 −0.214478 −0.107239 0.994233i \(-0.534201\pi\)
−0.107239 + 0.994233i \(0.534201\pi\)
\(522\) 0 0
\(523\) 769.813i 1.47192i 0.677027 + 0.735959i \(0.263268\pi\)
−0.677027 + 0.735959i \(0.736732\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 370.776i − 0.703560i
\(528\) 0 0
\(529\) −212.160 −0.401058
\(530\) 0 0
\(531\) − 1594.04i − 3.00195i
\(532\) 0 0
\(533\) 377.958 0.709115
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 835.214 1.55533
\(538\) 0 0
\(539\) − 735.222i − 1.36405i
\(540\) 0 0
\(541\) −225.558 −0.416927 −0.208463 0.978030i \(-0.566846\pi\)
−0.208463 + 0.978030i \(0.566846\pi\)
\(542\) 0 0
\(543\) − 1776.57i − 3.27177i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 882.346i − 1.61306i −0.591190 0.806532i \(-0.701342\pi\)
0.591190 0.806532i \(-0.298658\pi\)
\(548\) 0 0
\(549\) 702.382 1.27938
\(550\) 0 0
\(551\) 861.694i 1.56387i
\(552\) 0 0
\(553\) 3.21245 0.00580913
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −303.119 −0.544199 −0.272100 0.962269i \(-0.587718\pi\)
−0.272100 + 0.962269i \(0.587718\pi\)
\(558\) 0 0
\(559\) 47.5303i 0.0850273i
\(560\) 0 0
\(561\) 1448.49 2.58198
\(562\) 0 0
\(563\) − 344.003i − 0.611017i −0.952189 0.305508i \(-0.901174\pi\)
0.952189 0.305508i \(-0.0988264\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 23.2029i 0.0409223i
\(568\) 0 0
\(569\) 228.925 0.402329 0.201165 0.979557i \(-0.435527\pi\)
0.201165 + 0.979557i \(0.435527\pi\)
\(570\) 0 0
\(571\) 371.169i 0.650033i 0.945708 + 0.325017i \(0.105370\pi\)
−0.945708 + 0.325017i \(0.894630\pi\)
\(572\) 0 0
\(573\) 1583.10 2.76283
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 580.289 1.00570 0.502850 0.864374i \(-0.332285\pi\)
0.502850 + 0.864374i \(0.332285\pi\)
\(578\) 0 0
\(579\) − 1017.57i − 1.75746i
\(580\) 0 0
\(581\) 10.5350 0.0181325
\(582\) 0 0
\(583\) 1188.95i 2.03936i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 65.0801i − 0.110869i −0.998462 0.0554345i \(-0.982346\pi\)
0.998462 0.0554345i \(-0.0176544\pi\)
\(588\) 0 0
\(589\) −395.413 −0.671329
\(590\) 0 0
\(591\) − 314.140i − 0.531539i
\(592\) 0 0
\(593\) 1002.90 1.69124 0.845619 0.533787i \(-0.179232\pi\)
0.845619 + 0.533787i \(0.179232\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1638.91 2.74524
\(598\) 0 0
\(599\) 888.567i 1.48342i 0.670722 + 0.741709i \(0.265984\pi\)
−0.670722 + 0.741709i \(0.734016\pi\)
\(600\) 0 0
\(601\) 132.065 0.219742 0.109871 0.993946i \(-0.464956\pi\)
0.109871 + 0.993946i \(0.464956\pi\)
\(602\) 0 0
\(603\) − 86.2995i − 0.143117i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 700.090i − 1.15336i −0.816970 0.576680i \(-0.804348\pi\)
0.816970 0.576680i \(-0.195652\pi\)
\(608\) 0 0
\(609\) 48.6621 0.0799050
\(610\) 0 0
\(611\) 62.8975i 0.102942i
\(612\) 0 0
\(613\) 727.420 1.18666 0.593328 0.804961i \(-0.297814\pi\)
0.593328 + 0.804961i \(0.297814\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −99.7137 −0.161611 −0.0808053 0.996730i \(-0.525749\pi\)
−0.0808053 + 0.996730i \(0.525749\pi\)
\(618\) 0 0
\(619\) 721.349i 1.16535i 0.812707 + 0.582673i \(0.197993\pi\)
−0.812707 + 0.582673i \(0.802007\pi\)
\(620\) 0 0
\(621\) −1459.82 −2.35075
\(622\) 0 0
\(623\) − 2.24983i − 0.00361129i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 1544.74i − 2.46370i
\(628\) 0 0
\(629\) 329.422 0.523723
\(630\) 0 0
\(631\) 796.856i 1.26285i 0.775438 + 0.631423i \(0.217529\pi\)
−0.775438 + 0.631423i \(0.782471\pi\)
\(632\) 0 0
\(633\) −1090.62 −1.72295
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −572.797 −0.899211
\(638\) 0 0
\(639\) 796.121i 1.24589i
\(640\) 0 0
\(641\) 205.013 0.319833 0.159917 0.987131i \(-0.448877\pi\)
0.159917 + 0.987131i \(0.448877\pi\)
\(642\) 0 0
\(643\) − 495.044i − 0.769897i −0.922938 0.384949i \(-0.874219\pi\)
0.922938 0.384949i \(-0.125781\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1121.84i 1.73391i 0.498389 + 0.866953i \(0.333925\pi\)
−0.498389 + 0.866953i \(0.666075\pi\)
\(648\) 0 0
\(649\) 1252.47 1.92984
\(650\) 0 0
\(651\) 22.3300i 0.0343011i
\(652\) 0 0
\(653\) 622.987 0.954038 0.477019 0.878893i \(-0.341717\pi\)
0.477019 + 0.878893i \(0.341717\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −794.936 −1.20995
\(658\) 0 0
\(659\) − 264.030i − 0.400653i −0.979729 0.200326i \(-0.935800\pi\)
0.979729 0.200326i \(-0.0642002\pi\)
\(660\) 0 0
\(661\) 1285.15 1.94425 0.972124 0.234469i \(-0.0753352\pi\)
0.972124 + 0.234469i \(0.0753352\pi\)
\(662\) 0 0
\(663\) − 1128.49i − 1.70210i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1209.23i 1.81294i
\(668\) 0 0
\(669\) 1212.73 1.81276
\(670\) 0 0
\(671\) 551.876i 0.822468i
\(672\) 0 0
\(673\) −1244.16 −1.84868 −0.924342 0.381565i \(-0.875385\pi\)
−0.924342 + 0.381565i \(0.875385\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 963.335 1.42295 0.711473 0.702713i \(-0.248028\pi\)
0.711473 + 0.702713i \(0.248028\pi\)
\(678\) 0 0
\(679\) 2.51170i 0.00369912i
\(680\) 0 0
\(681\) −1497.35 −2.19876
\(682\) 0 0
\(683\) 770.819i 1.12858i 0.825577 + 0.564289i \(0.190850\pi\)
−0.825577 + 0.564289i \(0.809150\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 261.146i − 0.380125i
\(688\) 0 0
\(689\) 926.287 1.34439
\(690\) 0 0
\(691\) 408.765i 0.591555i 0.955257 + 0.295778i \(0.0955788\pi\)
−0.955257 + 0.295778i \(0.904421\pi\)
\(692\) 0 0
\(693\) −59.3084 −0.0855821
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −587.652 −0.843116
\(698\) 0 0
\(699\) 658.921i 0.942663i
\(700\) 0 0
\(701\) 1335.62 1.90530 0.952650 0.304068i \(-0.0983450\pi\)
0.952650 + 0.304068i \(0.0983450\pi\)
\(702\) 0 0
\(703\) − 351.311i − 0.499731i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 26.3908i 0.0373279i
\(708\) 0 0
\(709\) −362.956 −0.511927 −0.255964 0.966686i \(-0.582393\pi\)
−0.255964 + 0.966686i \(0.582393\pi\)
\(710\) 0 0
\(711\) 297.158i 0.417943i
\(712\) 0 0
\(713\) −554.891 −0.778249
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −426.575 −0.594945
\(718\) 0 0
\(719\) 648.098i 0.901388i 0.892679 + 0.450694i \(0.148823\pi\)
−0.892679 + 0.450694i \(0.851177\pi\)
\(720\) 0 0
\(721\) −0.986788 −0.00136864
\(722\) 0 0
\(723\) − 6.41453i − 0.00887211i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 431.123i − 0.593017i −0.955030 0.296508i \(-0.904178\pi\)
0.955030 0.296508i \(-0.0958222\pi\)
\(728\) 0 0
\(729\) 412.506 0.565852
\(730\) 0 0
\(731\) − 73.9003i − 0.101095i
\(732\) 0 0
\(733\) −1464.87 −1.99846 −0.999231 0.0392126i \(-0.987515\pi\)
−0.999231 + 0.0392126i \(0.987515\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 67.8073 0.0920044
\(738\) 0 0
\(739\) − 99.1951i − 0.134229i −0.997745 0.0671144i \(-0.978621\pi\)
0.997745 0.0671144i \(-0.0213793\pi\)
\(740\) 0 0
\(741\) −1203.48 −1.62412
\(742\) 0 0
\(743\) 602.719i 0.811196i 0.914052 + 0.405598i \(0.132937\pi\)
−0.914052 + 0.405598i \(0.867063\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 974.508i 1.30456i
\(748\) 0 0
\(749\) 22.1529 0.0295767
\(750\) 0 0
\(751\) − 541.472i − 0.721001i −0.932759 0.360501i \(-0.882606\pi\)
0.932759 0.360501i \(-0.117394\pi\)
\(752\) 0 0
\(753\) 1123.28 1.49174
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1092.79 −1.44358 −0.721789 0.692113i \(-0.756680\pi\)
−0.721789 + 0.692113i \(0.756680\pi\)
\(758\) 0 0
\(759\) − 2167.76i − 2.85608i
\(760\) 0 0
\(761\) −18.7706 −0.0246657 −0.0123328 0.999924i \(-0.503926\pi\)
−0.0123328 + 0.999924i \(0.503926\pi\)
\(762\) 0 0
\(763\) − 11.0893i − 0.0145338i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 975.773i − 1.27219i
\(768\) 0 0
\(769\) −39.0830 −0.0508231 −0.0254116 0.999677i \(-0.508090\pi\)
−0.0254116 + 0.999677i \(0.508090\pi\)
\(770\) 0 0
\(771\) 968.423i 1.25606i
\(772\) 0 0
\(773\) −31.2171 −0.0403843 −0.0201922 0.999796i \(-0.506428\pi\)
−0.0201922 + 0.999796i \(0.506428\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −19.8395 −0.0255334
\(778\) 0 0
\(779\) 626.699i 0.804492i
\(780\) 0 0
\(781\) −625.529 −0.800933
\(782\) 0 0
\(783\) 2381.75i 3.04183i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 988.563i 1.25612i 0.778167 + 0.628058i \(0.216150\pi\)
−0.778167 + 0.628058i \(0.783850\pi\)
\(788\) 0 0
\(789\) −395.880 −0.501750
\(790\) 0 0
\(791\) 4.23710i 0.00535663i
\(792\) 0 0
\(793\) 429.956 0.542189
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 118.920 0.149210 0.0746048 0.997213i \(-0.476230\pi\)
0.0746048 + 0.997213i \(0.476230\pi\)
\(798\) 0 0
\(799\) − 97.7933i − 0.122395i
\(800\) 0 0
\(801\) 208.114 0.259818
\(802\) 0 0
\(803\) − 624.598i − 0.777831i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 976.072i 1.20951i
\(808\) 0 0
\(809\) −1214.04 −1.50067 −0.750337 0.661056i \(-0.770109\pi\)
−0.750337 + 0.661056i \(0.770109\pi\)
\(810\) 0 0
\(811\) 706.666i 0.871351i 0.900104 + 0.435676i \(0.143491\pi\)
−0.900104 + 0.435676i \(0.856509\pi\)
\(812\) 0 0
\(813\) 1244.67 1.53096
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −78.8108 −0.0964636
\(818\) 0 0
\(819\) 46.2060i 0.0564176i
\(820\) 0 0
\(821\) 991.775 1.20801 0.604004 0.796981i \(-0.293571\pi\)
0.604004 + 0.796981i \(0.293571\pi\)
\(822\) 0 0
\(823\) 523.998i 0.636693i 0.947974 + 0.318347i \(0.103128\pi\)
−0.947974 + 0.318347i \(0.896872\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 318.794i − 0.385483i −0.981250 0.192741i \(-0.938262\pi\)
0.981250 0.192741i \(-0.0617379\pi\)
\(828\) 0 0
\(829\) 1034.11 1.24742 0.623711 0.781655i \(-0.285624\pi\)
0.623711 + 0.781655i \(0.285624\pi\)
\(830\) 0 0
\(831\) − 2397.69i − 2.88531i
\(832\) 0 0
\(833\) 890.588 1.06913
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1092.94 −1.30578
\(838\) 0 0
\(839\) 445.284i 0.530732i 0.964148 + 0.265366i \(0.0854928\pi\)
−0.964148 + 0.265366i \(0.914507\pi\)
\(840\) 0 0
\(841\) 1131.91 1.34591
\(842\) 0 0
\(843\) − 1039.81i − 1.23346i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 21.5983i − 0.0254997i
\(848\) 0 0
\(849\) −2218.69 −2.61330
\(850\) 0 0
\(851\) − 493.002i − 0.579321i
\(852\) 0 0
\(853\) −1200.49 −1.40737 −0.703684 0.710513i \(-0.748463\pi\)
−0.703684 + 0.710513i \(0.748463\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1223.15 1.42724 0.713622 0.700531i \(-0.247054\pi\)
0.713622 + 0.700531i \(0.247054\pi\)
\(858\) 0 0
\(859\) − 210.033i − 0.244509i −0.992499 0.122254i \(-0.960988\pi\)
0.992499 0.122254i \(-0.0390124\pi\)
\(860\) 0 0
\(861\) 35.3914 0.0411050
\(862\) 0 0
\(863\) − 1560.47i − 1.80819i −0.427333 0.904095i \(-0.640547\pi\)
0.427333 0.904095i \(-0.359453\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 222.252i 0.256346i
\(868\) 0 0
\(869\) −233.483 −0.268680
\(870\) 0 0
\(871\) − 52.8273i − 0.0606513i
\(872\) 0 0
\(873\) −232.338 −0.266137
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1011.66 1.15354 0.576771 0.816906i \(-0.304313\pi\)
0.576771 + 0.816906i \(0.304313\pi\)
\(878\) 0 0
\(879\) − 1519.96i − 1.72919i
\(880\) 0 0
\(881\) 266.455 0.302446 0.151223 0.988500i \(-0.451679\pi\)
0.151223 + 0.988500i \(0.451679\pi\)
\(882\) 0 0
\(883\) − 469.871i − 0.532130i −0.963955 0.266065i \(-0.914276\pi\)
0.963955 0.266065i \(-0.0857237\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 764.559i 0.861961i 0.902361 + 0.430980i \(0.141832\pi\)
−0.902361 + 0.430980i \(0.858168\pi\)
\(888\) 0 0
\(889\) −28.6127 −0.0321853
\(890\) 0 0
\(891\) − 1686.41i − 1.89271i
\(892\) 0 0
\(893\) −104.291 −0.116788
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −1688.86 −1.88279
\(898\) 0 0
\(899\) 905.328i 1.00704i
\(900\) 0 0
\(901\) −1440.20 −1.59844
\(902\) 0 0
\(903\) 4.45065i 0.00492874i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1333.20i 1.46990i 0.678123 + 0.734948i \(0.262794\pi\)
−0.678123 + 0.734948i \(0.737206\pi\)
\(908\) 0 0
\(909\) −2441.20 −2.68559
\(910\) 0 0
\(911\) 1496.11i 1.64227i 0.570736 + 0.821134i \(0.306658\pi\)
−0.570736 + 0.821134i \(0.693342\pi\)
\(912\) 0 0
\(913\) −765.691 −0.838654
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 45.2725 0.0493702
\(918\) 0 0
\(919\) − 564.228i − 0.613959i −0.951716 0.306980i \(-0.900682\pi\)
0.951716 0.306980i \(-0.0993183\pi\)
\(920\) 0 0
\(921\) −1387.66 −1.50669
\(922\) 0 0
\(923\) 487.337i 0.527993i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 91.2798i − 0.0984680i
\(928\) 0 0
\(929\) −1449.16 −1.55991 −0.779957 0.625833i \(-0.784759\pi\)
−0.779957 + 0.625833i \(0.784759\pi\)
\(930\) 0 0
\(931\) − 949.765i − 1.02016i
\(932\) 0 0
\(933\) −3069.46 −3.28988
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 258.795 0.276196 0.138098 0.990419i \(-0.455901\pi\)
0.138098 + 0.990419i \(0.455901\pi\)
\(938\) 0 0
\(939\) − 526.573i − 0.560781i
\(940\) 0 0
\(941\) 20.3444 0.0216200 0.0108100 0.999942i \(-0.496559\pi\)
0.0108100 + 0.999942i \(0.496559\pi\)
\(942\) 0 0
\(943\) 879.461i 0.932620i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 613.539i − 0.647876i −0.946078 0.323938i \(-0.894993\pi\)
0.946078 0.323938i \(-0.105007\pi\)
\(948\) 0 0
\(949\) −486.612 −0.512763
\(950\) 0 0
\(951\) − 1016.02i − 1.06837i
\(952\) 0 0
\(953\) −81.6126 −0.0856376 −0.0428188 0.999083i \(-0.513634\pi\)
−0.0428188 + 0.999083i \(0.513634\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −3536.80 −3.69571
\(958\) 0 0
\(959\) 12.3525i 0.0128806i
\(960\) 0 0
\(961\) 545.564 0.567704
\(962\) 0 0
\(963\) 2049.19i 2.12792i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 127.482i 0.131833i 0.997825 + 0.0659165i \(0.0209971\pi\)
−0.997825 + 0.0659165i \(0.979003\pi\)
\(968\) 0 0
\(969\) 1871.17 1.93103
\(970\) 0 0
\(971\) − 1122.62i − 1.15615i −0.815983 0.578076i \(-0.803804\pi\)
0.815983 0.578076i \(-0.196196\pi\)
\(972\) 0 0
\(973\) 5.50703 0.00565985
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −424.837 −0.434838 −0.217419 0.976078i \(-0.569764\pi\)
−0.217419 + 0.976078i \(0.569764\pi\)
\(978\) 0 0
\(979\) 163.519i 0.167027i
\(980\) 0 0
\(981\) 1025.78 1.04565
\(982\) 0 0
\(983\) 663.324i 0.674795i 0.941362 + 0.337398i \(0.109547\pi\)
−0.941362 + 0.337398i \(0.890453\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 5.88961i 0.00596719i
\(988\) 0 0
\(989\) −110.597 −0.111827
\(990\) 0 0
\(991\) 1771.36i 1.78745i 0.448616 + 0.893724i \(0.351917\pi\)
−0.448616 + 0.893724i \(0.648083\pi\)
\(992\) 0 0
\(993\) 2811.15 2.83097
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −447.066 −0.448411 −0.224205 0.974542i \(-0.571979\pi\)
−0.224205 + 0.974542i \(0.571979\pi\)
\(998\) 0 0
\(999\) − 971.038i − 0.972010i
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 800.3.b.i.351.6 6
4.3 odd 2 inner 800.3.b.i.351.1 6
5.2 odd 4 160.3.h.b.159.5 yes 6
5.3 odd 4 160.3.h.a.159.2 yes 6
5.4 even 2 800.3.b.h.351.1 6
8.3 odd 2 1600.3.b.w.1151.6 6
8.5 even 2 1600.3.b.w.1151.1 6
15.2 even 4 1440.3.j.b.1279.4 6
15.8 even 4 1440.3.j.a.1279.3 6
20.3 even 4 160.3.h.b.159.6 yes 6
20.7 even 4 160.3.h.a.159.1 6
20.19 odd 2 800.3.b.h.351.6 6
40.3 even 4 320.3.h.f.319.1 6
40.13 odd 4 320.3.h.g.319.5 6
40.19 odd 2 1600.3.b.v.1151.1 6
40.27 even 4 320.3.h.g.319.6 6
40.29 even 2 1600.3.b.v.1151.6 6
40.37 odd 4 320.3.h.f.319.2 6
60.23 odd 4 1440.3.j.b.1279.3 6
60.47 odd 4 1440.3.j.a.1279.4 6
80.3 even 4 1280.3.e.h.639.6 6
80.13 odd 4 1280.3.e.i.639.1 6
80.27 even 4 1280.3.e.i.639.6 6
80.37 odd 4 1280.3.e.h.639.1 6
80.43 even 4 1280.3.e.f.639.1 6
80.53 odd 4 1280.3.e.g.639.6 6
80.67 even 4 1280.3.e.g.639.1 6
80.77 odd 4 1280.3.e.f.639.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.3.h.a.159.1 6 20.7 even 4
160.3.h.a.159.2 yes 6 5.3 odd 4
160.3.h.b.159.5 yes 6 5.2 odd 4
160.3.h.b.159.6 yes 6 20.3 even 4
320.3.h.f.319.1 6 40.3 even 4
320.3.h.f.319.2 6 40.37 odd 4
320.3.h.g.319.5 6 40.13 odd 4
320.3.h.g.319.6 6 40.27 even 4
800.3.b.h.351.1 6 5.4 even 2
800.3.b.h.351.6 6 20.19 odd 2
800.3.b.i.351.1 6 4.3 odd 2 inner
800.3.b.i.351.6 6 1.1 even 1 trivial
1280.3.e.f.639.1 6 80.43 even 4
1280.3.e.f.639.6 6 80.77 odd 4
1280.3.e.g.639.1 6 80.67 even 4
1280.3.e.g.639.6 6 80.53 odd 4
1280.3.e.h.639.1 6 80.37 odd 4
1280.3.e.h.639.6 6 80.3 even 4
1280.3.e.i.639.1 6 80.13 odd 4
1280.3.e.i.639.6 6 80.27 even 4
1440.3.j.a.1279.3 6 15.8 even 4
1440.3.j.a.1279.4 6 60.47 odd 4
1440.3.j.b.1279.3 6 60.23 odd 4
1440.3.j.b.1279.4 6 15.2 even 4
1600.3.b.v.1151.1 6 40.19 odd 2
1600.3.b.v.1151.6 6 40.29 even 2
1600.3.b.w.1151.1 6 8.5 even 2
1600.3.b.w.1151.6 6 8.3 odd 2