Properties

Label 1728.3.g.j.703.8
Level $1728$
Weight $3$
Character 1728.703
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.56070144.2
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 703.8
Root \(0.500000 - 1.19293i\) of defining polynomial
Character \(\chi\) \(=\) 1728.703
Dual form 1728.3.g.j.703.7

$q$-expansion

\(f(q)\) \(=\) \(q+6.59655 q^{5} +4.56106i q^{7} +O(q^{10})\) \(q+6.59655 q^{5} +4.56106i q^{7} +1.16066i q^{11} -13.5580 q^{13} +5.32636 q^{17} -25.1900i q^{19} -10.4309i q^{23} +18.5144 q^{25} +47.0288 q^{29} -22.3862i q^{31} +30.0872i q^{35} +60.7288 q^{37} +8.81696 q^{41} +29.1160i q^{43} -78.2321i q^{47} +28.1968 q^{49} +62.7623 q^{53} +7.65637i q^{55} +109.116i q^{59} +66.4997 q^{61} -89.4360 q^{65} +81.9685i q^{67} +40.7276i q^{71} -4.29030 q^{73} -5.29386 q^{77} -5.20371i q^{79} -115.114i q^{83} +35.1356 q^{85} -141.504 q^{89} -61.8388i q^{91} -166.167i q^{95} +136.769 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} - 8 q^{13} + 24 q^{17} + 24 q^{25} + 128 q^{29} - 24 q^{37} + 160 q^{41} - 144 q^{49} + 48 q^{53} + 136 q^{61} - 280 q^{65} + 72 q^{73} - 520 q^{77} - 96 q^{85} - 168 q^{89} + 104 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 6.59655 1.31931 0.659655 0.751569i \(-0.270703\pi\)
0.659655 + 0.751569i \(0.270703\pi\)
\(6\) 0 0
\(7\) 4.56106i 0.651580i 0.945442 + 0.325790i \(0.105630\pi\)
−0.945442 + 0.325790i \(0.894370\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.16066i 0.105515i 0.998607 + 0.0527575i \(0.0168010\pi\)
−0.998607 + 0.0527575i \(0.983199\pi\)
\(12\) 0 0
\(13\) −13.5580 −1.04292 −0.521461 0.853275i \(-0.674613\pi\)
−0.521461 + 0.853275i \(0.674613\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.32636 0.313315 0.156658 0.987653i \(-0.449928\pi\)
0.156658 + 0.987653i \(0.449928\pi\)
\(18\) 0 0
\(19\) − 25.1900i − 1.32579i −0.748712 0.662896i \(-0.769327\pi\)
0.748712 0.662896i \(-0.230673\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 10.4309i − 0.453515i −0.973951 0.226758i \(-0.927187\pi\)
0.973951 0.226758i \(-0.0728125\pi\)
\(24\) 0 0
\(25\) 18.5144 0.740577
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 47.0288 1.62168 0.810842 0.585265i \(-0.199010\pi\)
0.810842 + 0.585265i \(0.199010\pi\)
\(30\) 0 0
\(31\) − 22.3862i − 0.722135i −0.932540 0.361067i \(-0.882412\pi\)
0.932540 0.361067i \(-0.117588\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 30.0872i 0.859635i
\(36\) 0 0
\(37\) 60.7288 1.64132 0.820659 0.571418i \(-0.193606\pi\)
0.820659 + 0.571418i \(0.193606\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.81696 0.215048 0.107524 0.994202i \(-0.465708\pi\)
0.107524 + 0.994202i \(0.465708\pi\)
\(42\) 0 0
\(43\) 29.1160i 0.677116i 0.940945 + 0.338558i \(0.109939\pi\)
−0.940945 + 0.338558i \(0.890061\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 78.2321i − 1.66451i −0.554392 0.832256i \(-0.687049\pi\)
0.554392 0.832256i \(-0.312951\pi\)
\(48\) 0 0
\(49\) 28.1968 0.575444
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 62.7623 1.18419 0.592097 0.805866i \(-0.298300\pi\)
0.592097 + 0.805866i \(0.298300\pi\)
\(54\) 0 0
\(55\) 7.65637i 0.139207i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 109.116i 1.84942i 0.380666 + 0.924712i \(0.375695\pi\)
−0.380666 + 0.924712i \(0.624305\pi\)
\(60\) 0 0
\(61\) 66.4997 1.09016 0.545079 0.838384i \(-0.316500\pi\)
0.545079 + 0.838384i \(0.316500\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −89.4360 −1.37594
\(66\) 0 0
\(67\) 81.9685i 1.22341i 0.791086 + 0.611705i \(0.209516\pi\)
−0.791086 + 0.611705i \(0.790484\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 40.7276i 0.573629i 0.957986 + 0.286814i \(0.0925963\pi\)
−0.957986 + 0.286814i \(0.907404\pi\)
\(72\) 0 0
\(73\) −4.29030 −0.0587713 −0.0293856 0.999568i \(-0.509355\pi\)
−0.0293856 + 0.999568i \(0.509355\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.29386 −0.0687514
\(78\) 0 0
\(79\) − 5.20371i − 0.0658698i −0.999457 0.0329349i \(-0.989515\pi\)
0.999457 0.0329349i \(-0.0104854\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 115.114i − 1.38691i −0.720498 0.693457i \(-0.756087\pi\)
0.720498 0.693457i \(-0.243913\pi\)
\(84\) 0 0
\(85\) 35.1356 0.413359
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −141.504 −1.58993 −0.794965 0.606656i \(-0.792511\pi\)
−0.794965 + 0.606656i \(0.792511\pi\)
\(90\) 0 0
\(91\) − 61.8388i − 0.679547i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 166.167i − 1.74913i
\(96\) 0 0
\(97\) 136.769 1.40999 0.704994 0.709213i \(-0.250950\pi\)
0.704994 + 0.709213i \(0.250950\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 45.2350 0.447871 0.223935 0.974604i \(-0.428110\pi\)
0.223935 + 0.974604i \(0.428110\pi\)
\(102\) 0 0
\(103\) 114.604i 1.11266i 0.830963 + 0.556328i \(0.187790\pi\)
−0.830963 + 0.556328i \(0.812210\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 194.930i 1.82178i 0.412652 + 0.910889i \(0.364603\pi\)
−0.412652 + 0.910889i \(0.635397\pi\)
\(108\) 0 0
\(109\) −6.35150 −0.0582706 −0.0291353 0.999575i \(-0.509275\pi\)
−0.0291353 + 0.999575i \(0.509275\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −48.7068 −0.431034 −0.215517 0.976500i \(-0.569144\pi\)
−0.215517 + 0.976500i \(0.569144\pi\)
\(114\) 0 0
\(115\) − 68.8076i − 0.598327i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 24.2938i 0.204150i
\(120\) 0 0
\(121\) 119.653 0.988867
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −42.7824 −0.342259
\(126\) 0 0
\(127\) − 86.3509i − 0.679929i −0.940438 0.339964i \(-0.889585\pi\)
0.940438 0.339964i \(-0.110415\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 90.6946i 0.692325i 0.938175 + 0.346162i \(0.112515\pi\)
−0.938175 + 0.346162i \(0.887485\pi\)
\(132\) 0 0
\(133\) 114.893 0.863859
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −148.355 −1.08288 −0.541442 0.840738i \(-0.682122\pi\)
−0.541442 + 0.840738i \(0.682122\pi\)
\(138\) 0 0
\(139\) − 101.426i − 0.729684i −0.931069 0.364842i \(-0.881123\pi\)
0.931069 0.364842i \(-0.118877\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 15.7363i − 0.110044i
\(144\) 0 0
\(145\) 310.228 2.13950
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −219.221 −1.47128 −0.735639 0.677373i \(-0.763118\pi\)
−0.735639 + 0.677373i \(0.763118\pi\)
\(150\) 0 0
\(151\) 26.1638i 0.173270i 0.996240 + 0.0866350i \(0.0276114\pi\)
−0.996240 + 0.0866350i \(0.972389\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 147.671i − 0.952719i
\(156\) 0 0
\(157\) 257.715 1.64150 0.820748 0.571290i \(-0.193557\pi\)
0.820748 + 0.571290i \(0.193557\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 47.5757 0.295501
\(162\) 0 0
\(163\) 140.310i 0.860796i 0.902639 + 0.430398i \(0.141627\pi\)
−0.902639 + 0.430398i \(0.858373\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 28.2435i − 0.169123i −0.996418 0.0845613i \(-0.973051\pi\)
0.996418 0.0845613i \(-0.0269489\pi\)
\(168\) 0 0
\(169\) 14.8193 0.0876881
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 127.532 0.737177 0.368589 0.929593i \(-0.379841\pi\)
0.368589 + 0.929593i \(0.379841\pi\)
\(174\) 0 0
\(175\) 84.4453i 0.482545i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 167.254i − 0.934378i −0.884157 0.467189i \(-0.845267\pi\)
0.884157 0.467189i \(-0.154733\pi\)
\(180\) 0 0
\(181\) 166.841 0.921775 0.460887 0.887459i \(-0.347531\pi\)
0.460887 + 0.887459i \(0.347531\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 400.600 2.16541
\(186\) 0 0
\(187\) 6.18211i 0.0330594i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 163.153i 0.854203i 0.904204 + 0.427102i \(0.140465\pi\)
−0.904204 + 0.427102i \(0.859535\pi\)
\(192\) 0 0
\(193\) −137.611 −0.713010 −0.356505 0.934293i \(-0.616032\pi\)
−0.356505 + 0.934293i \(0.616032\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 132.978 0.675017 0.337509 0.941322i \(-0.390416\pi\)
0.337509 + 0.941322i \(0.390416\pi\)
\(198\) 0 0
\(199\) − 371.097i − 1.86481i −0.361418 0.932404i \(-0.617707\pi\)
0.361418 0.932404i \(-0.382293\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 214.501i 1.05666i
\(204\) 0 0
\(205\) 58.1615 0.283715
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 29.2372 0.139891
\(210\) 0 0
\(211\) 239.571i 1.13541i 0.823233 + 0.567704i \(0.192168\pi\)
−0.823233 + 0.567704i \(0.807832\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 192.065i 0.893326i
\(216\) 0 0
\(217\) 102.105 0.470528
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −72.2147 −0.326763
\(222\) 0 0
\(223\) − 121.555i − 0.545089i −0.962143 0.272545i \(-0.912135\pi\)
0.962143 0.272545i \(-0.0878652\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 223.023i − 0.982481i −0.871024 0.491241i \(-0.836544\pi\)
0.871024 0.491241i \(-0.163456\pi\)
\(228\) 0 0
\(229\) 139.599 0.609605 0.304802 0.952416i \(-0.401410\pi\)
0.304802 + 0.952416i \(0.401410\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 103.682 0.444986 0.222493 0.974934i \(-0.428581\pi\)
0.222493 + 0.974934i \(0.428581\pi\)
\(234\) 0 0
\(235\) − 516.061i − 2.19601i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 110.844i − 0.463783i −0.972742 0.231892i \(-0.925508\pi\)
0.972742 0.231892i \(-0.0744915\pi\)
\(240\) 0 0
\(241\) −80.6426 −0.334616 −0.167308 0.985905i \(-0.553508\pi\)
−0.167308 + 0.985905i \(0.553508\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 186.001 0.759188
\(246\) 0 0
\(247\) 341.526i 1.38270i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 471.180i − 1.87721i −0.344990 0.938606i \(-0.612118\pi\)
0.344990 0.938606i \(-0.387882\pi\)
\(252\) 0 0
\(253\) 12.1067 0.0478526
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 330.588 1.28633 0.643167 0.765726i \(-0.277620\pi\)
0.643167 + 0.765726i \(0.277620\pi\)
\(258\) 0 0
\(259\) 276.988i 1.06945i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 494.228i 1.87919i 0.342283 + 0.939597i \(0.388800\pi\)
−0.342283 + 0.939597i \(0.611200\pi\)
\(264\) 0 0
\(265\) 414.014 1.56232
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −214.548 −0.797576 −0.398788 0.917043i \(-0.630569\pi\)
−0.398788 + 0.917043i \(0.630569\pi\)
\(270\) 0 0
\(271\) − 283.519i − 1.04620i −0.852272 0.523099i \(-0.824776\pi\)
0.852272 0.523099i \(-0.175224\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 21.4890i 0.0781419i
\(276\) 0 0
\(277\) −107.432 −0.387842 −0.193921 0.981017i \(-0.562121\pi\)
−0.193921 + 0.981017i \(0.562121\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 27.7120 0.0986193 0.0493096 0.998784i \(-0.484298\pi\)
0.0493096 + 0.998784i \(0.484298\pi\)
\(282\) 0 0
\(283\) 311.956i 1.10232i 0.834400 + 0.551159i \(0.185814\pi\)
−0.834400 + 0.551159i \(0.814186\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 40.2147i 0.140121i
\(288\) 0 0
\(289\) −260.630 −0.901834
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −536.810 −1.83212 −0.916058 0.401045i \(-0.868647\pi\)
−0.916058 + 0.401045i \(0.868647\pi\)
\(294\) 0 0
\(295\) 719.789i 2.43996i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 141.421i 0.472982i
\(300\) 0 0
\(301\) −132.800 −0.441195
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 438.668 1.43826
\(306\) 0 0
\(307\) − 296.075i − 0.964415i −0.876057 0.482208i \(-0.839835\pi\)
0.876057 0.482208i \(-0.160165\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 274.720i − 0.883345i −0.897176 0.441673i \(-0.854385\pi\)
0.897176 0.441673i \(-0.145615\pi\)
\(312\) 0 0
\(313\) 204.294 0.652697 0.326348 0.945250i \(-0.394182\pi\)
0.326348 + 0.945250i \(0.394182\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −27.6889 −0.0873466 −0.0436733 0.999046i \(-0.513906\pi\)
−0.0436733 + 0.999046i \(0.513906\pi\)
\(318\) 0 0
\(319\) 54.5847i 0.171112i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 134.171i − 0.415390i
\(324\) 0 0
\(325\) −251.018 −0.772364
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 356.821 1.08456
\(330\) 0 0
\(331\) − 77.6640i − 0.234634i −0.993094 0.117317i \(-0.962571\pi\)
0.993094 0.117317i \(-0.0374294\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 540.709i 1.61406i
\(336\) 0 0
\(337\) 314.619 0.933588 0.466794 0.884366i \(-0.345409\pi\)
0.466794 + 0.884366i \(0.345409\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 25.9828 0.0761960
\(342\) 0 0
\(343\) 352.099i 1.02653i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 485.244i − 1.39840i −0.714928 0.699198i \(-0.753540\pi\)
0.714928 0.699198i \(-0.246460\pi\)
\(348\) 0 0
\(349\) 61.2165 0.175405 0.0877027 0.996147i \(-0.472047\pi\)
0.0877027 + 0.996147i \(0.472047\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −274.644 −0.778028 −0.389014 0.921232i \(-0.627184\pi\)
−0.389014 + 0.921232i \(0.627184\pi\)
\(354\) 0 0
\(355\) 268.662i 0.756793i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 406.229i − 1.13156i −0.824557 0.565779i \(-0.808576\pi\)
0.824557 0.565779i \(-0.191424\pi\)
\(360\) 0 0
\(361\) −273.538 −0.757722
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −28.3012 −0.0775375
\(366\) 0 0
\(367\) − 678.650i − 1.84918i −0.380959 0.924592i \(-0.624406\pi\)
0.380959 0.924592i \(-0.375594\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 286.263i 0.771597i
\(372\) 0 0
\(373\) −17.1447 −0.0459645 −0.0229822 0.999736i \(-0.507316\pi\)
−0.0229822 + 0.999736i \(0.507316\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −637.617 −1.69129
\(378\) 0 0
\(379\) − 669.773i − 1.76721i −0.468231 0.883606i \(-0.655109\pi\)
0.468231 0.883606i \(-0.344891\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 282.840i 0.738486i 0.929333 + 0.369243i \(0.120383\pi\)
−0.929333 + 0.369243i \(0.879617\pi\)
\(384\) 0 0
\(385\) −34.9212 −0.0907043
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −197.725 −0.508290 −0.254145 0.967166i \(-0.581794\pi\)
−0.254145 + 0.967166i \(0.581794\pi\)
\(390\) 0 0
\(391\) − 55.5584i − 0.142093i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 34.3265i − 0.0869026i
\(396\) 0 0
\(397\) 546.375 1.37626 0.688129 0.725588i \(-0.258432\pi\)
0.688129 + 0.725588i \(0.258432\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −586.152 −1.46173 −0.730863 0.682525i \(-0.760882\pi\)
−0.730863 + 0.682525i \(0.760882\pi\)
\(402\) 0 0
\(403\) 303.512i 0.753131i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 70.4857i 0.173184i
\(408\) 0 0
\(409\) −614.056 −1.50136 −0.750679 0.660667i \(-0.770274\pi\)
−0.750679 + 0.660667i \(0.770274\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −497.685 −1.20505
\(414\) 0 0
\(415\) − 759.354i − 1.82977i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 509.128i 1.21510i 0.794281 + 0.607551i \(0.207848\pi\)
−0.794281 + 0.607551i \(0.792152\pi\)
\(420\) 0 0
\(421\) −478.025 −1.13545 −0.567725 0.823218i \(-0.692176\pi\)
−0.567725 + 0.823218i \(0.692176\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 98.6144 0.232034
\(426\) 0 0
\(427\) 303.309i 0.710325i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 158.810i 0.368468i 0.982882 + 0.184234i \(0.0589804\pi\)
−0.982882 + 0.184234i \(0.941020\pi\)
\(432\) 0 0
\(433\) −367.803 −0.849429 −0.424715 0.905327i \(-0.639626\pi\)
−0.424715 + 0.905327i \(0.639626\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −262.754 −0.601267
\(438\) 0 0
\(439\) 145.096i 0.330514i 0.986251 + 0.165257i \(0.0528453\pi\)
−0.986251 + 0.165257i \(0.947155\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 771.496i − 1.74153i −0.491703 0.870763i \(-0.663626\pi\)
0.491703 0.870763i \(-0.336374\pi\)
\(444\) 0 0
\(445\) −933.436 −2.09761
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 56.2630 0.125307 0.0626536 0.998035i \(-0.480044\pi\)
0.0626536 + 0.998035i \(0.480044\pi\)
\(450\) 0 0
\(451\) 10.2335i 0.0226908i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 407.923i − 0.896533i
\(456\) 0 0
\(457\) −804.889 −1.76125 −0.880623 0.473817i \(-0.842876\pi\)
−0.880623 + 0.473817i \(0.842876\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 46.1301 0.100065 0.0500327 0.998748i \(-0.484067\pi\)
0.0500327 + 0.998748i \(0.484067\pi\)
\(462\) 0 0
\(463\) 273.418i 0.590536i 0.955414 + 0.295268i \(0.0954090\pi\)
−0.955414 + 0.295268i \(0.904591\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 109.131i 0.233684i 0.993150 + 0.116842i \(0.0372772\pi\)
−0.993150 + 0.116842i \(0.962723\pi\)
\(468\) 0 0
\(469\) −373.863 −0.797150
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −33.7939 −0.0714459
\(474\) 0 0
\(475\) − 466.379i − 0.981850i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 42.5978i 0.0889306i 0.999011 + 0.0444653i \(0.0141584\pi\)
−0.999011 + 0.0444653i \(0.985842\pi\)
\(480\) 0 0
\(481\) −823.361 −1.71177
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 902.202 1.86021
\(486\) 0 0
\(487\) − 445.984i − 0.915779i −0.889009 0.457889i \(-0.848606\pi\)
0.889009 0.457889i \(-0.151394\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 565.533i 1.15180i 0.817521 + 0.575899i \(0.195348\pi\)
−0.817521 + 0.575899i \(0.804652\pi\)
\(492\) 0 0
\(493\) 250.492 0.508098
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −185.761 −0.373765
\(498\) 0 0
\(499\) 923.261i 1.85022i 0.379696 + 0.925111i \(0.376029\pi\)
−0.379696 + 0.925111i \(0.623971\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 406.646i − 0.808441i −0.914662 0.404220i \(-0.867543\pi\)
0.914662 0.404220i \(-0.132457\pi\)
\(504\) 0 0
\(505\) 298.394 0.590880
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 674.031 1.32423 0.662113 0.749404i \(-0.269660\pi\)
0.662113 + 0.749404i \(0.269660\pi\)
\(510\) 0 0
\(511\) − 19.5683i − 0.0382942i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 755.988i 1.46794i
\(516\) 0 0
\(517\) 90.8011 0.175631
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −513.618 −0.985831 −0.492916 0.870077i \(-0.664069\pi\)
−0.492916 + 0.870077i \(0.664069\pi\)
\(522\) 0 0
\(523\) − 381.030i − 0.728547i −0.931292 0.364274i \(-0.881317\pi\)
0.931292 0.364274i \(-0.118683\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 119.237i − 0.226256i
\(528\) 0 0
\(529\) 420.197 0.794324
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −119.540 −0.224278
\(534\) 0 0
\(535\) 1285.87i 2.40349i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 32.7270i 0.0607179i
\(540\) 0 0
\(541\) −626.913 −1.15880 −0.579402 0.815042i \(-0.696714\pi\)
−0.579402 + 0.815042i \(0.696714\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −41.8980 −0.0768770
\(546\) 0 0
\(547\) 18.9275i 0.0346023i 0.999850 + 0.0173012i \(0.00550740\pi\)
−0.999850 + 0.0173012i \(0.994493\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 1184.66i − 2.15001i
\(552\) 0 0
\(553\) 23.7344 0.0429194
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −26.9003 −0.0482949 −0.0241475 0.999708i \(-0.507687\pi\)
−0.0241475 + 0.999708i \(0.507687\pi\)
\(558\) 0 0
\(559\) − 394.755i − 0.706180i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 488.414i − 0.867520i −0.901029 0.433760i \(-0.857187\pi\)
0.901029 0.433760i \(-0.142813\pi\)
\(564\) 0 0
\(565\) −321.297 −0.568667
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 266.712 0.468739 0.234369 0.972148i \(-0.424698\pi\)
0.234369 + 0.972148i \(0.424698\pi\)
\(570\) 0 0
\(571\) − 880.359i − 1.54178i −0.636966 0.770892i \(-0.719811\pi\)
0.636966 0.770892i \(-0.280189\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 193.121i − 0.335863i
\(576\) 0 0
\(577\) 82.9805 0.143814 0.0719069 0.997411i \(-0.477092\pi\)
0.0719069 + 0.997411i \(0.477092\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 525.041 0.903685
\(582\) 0 0
\(583\) 72.8460i 0.124950i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 965.562i 1.64491i 0.568831 + 0.822455i \(0.307396\pi\)
−0.568831 + 0.822455i \(0.692604\pi\)
\(588\) 0 0
\(589\) −563.909 −0.957400
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 289.460 0.488128 0.244064 0.969759i \(-0.421519\pi\)
0.244064 + 0.969759i \(0.421519\pi\)
\(594\) 0 0
\(595\) 160.255i 0.269337i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 448.245i 0.748322i 0.927364 + 0.374161i \(0.122069\pi\)
−0.927364 + 0.374161i \(0.877931\pi\)
\(600\) 0 0
\(601\) −229.348 −0.381610 −0.190805 0.981628i \(-0.561110\pi\)
−0.190805 + 0.981628i \(0.561110\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 789.296 1.30462
\(606\) 0 0
\(607\) − 85.4214i − 0.140727i −0.997521 0.0703636i \(-0.977584\pi\)
0.997521 0.0703636i \(-0.0224160\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1060.67i 1.73596i
\(612\) 0 0
\(613\) −534.037 −0.871187 −0.435593 0.900144i \(-0.643461\pi\)
−0.435593 + 0.900144i \(0.643461\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −151.146 −0.244970 −0.122485 0.992470i \(-0.539086\pi\)
−0.122485 + 0.992470i \(0.539086\pi\)
\(618\) 0 0
\(619\) − 59.2078i − 0.0956507i −0.998856 0.0478253i \(-0.984771\pi\)
0.998856 0.0478253i \(-0.0152291\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 645.407i − 1.03597i
\(624\) 0 0
\(625\) −745.077 −1.19212
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 323.463 0.514250
\(630\) 0 0
\(631\) 70.0390i 0.110997i 0.998459 + 0.0554984i \(0.0176748\pi\)
−0.998459 + 0.0554984i \(0.982325\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 569.618i − 0.897036i
\(636\) 0 0
\(637\) −382.292 −0.600144
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −195.338 −0.304739 −0.152370 0.988324i \(-0.548690\pi\)
−0.152370 + 0.988324i \(0.548690\pi\)
\(642\) 0 0
\(643\) 346.902i 0.539505i 0.962930 + 0.269753i \(0.0869419\pi\)
−0.962930 + 0.269753i \(0.913058\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 746.221i 1.15335i 0.816972 + 0.576677i \(0.195651\pi\)
−0.816972 + 0.576677i \(0.804349\pi\)
\(648\) 0 0
\(649\) −126.647 −0.195142
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 850.046 1.30176 0.650878 0.759183i \(-0.274401\pi\)
0.650878 + 0.759183i \(0.274401\pi\)
\(654\) 0 0
\(655\) 598.271i 0.913391i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 817.574i − 1.24063i −0.784354 0.620314i \(-0.787005\pi\)
0.784354 0.620314i \(-0.212995\pi\)
\(660\) 0 0
\(661\) 670.001 1.01362 0.506809 0.862058i \(-0.330825\pi\)
0.506809 + 0.862058i \(0.330825\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 757.898 1.13970
\(666\) 0 0
\(667\) − 490.551i − 0.735459i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 77.1838i 0.115028i
\(672\) 0 0
\(673\) −41.6868 −0.0619418 −0.0309709 0.999520i \(-0.509860\pi\)
−0.0309709 + 0.999520i \(0.509860\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 71.8534 0.106135 0.0530675 0.998591i \(-0.483100\pi\)
0.0530675 + 0.998591i \(0.483100\pi\)
\(678\) 0 0
\(679\) 623.811i 0.918720i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 486.468i 0.712252i 0.934438 + 0.356126i \(0.115903\pi\)
−0.934438 + 0.356126i \(0.884097\pi\)
\(684\) 0 0
\(685\) −978.632 −1.42866
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −850.931 −1.23502
\(690\) 0 0
\(691\) 1153.88i 1.66988i 0.550344 + 0.834938i \(0.314497\pi\)
−0.550344 + 0.834938i \(0.685503\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 669.062i − 0.962679i
\(696\) 0 0
\(697\) 46.9623 0.0673778
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 278.673 0.397537 0.198768 0.980047i \(-0.436306\pi\)
0.198768 + 0.980047i \(0.436306\pi\)
\(702\) 0 0
\(703\) − 1529.76i − 2.17605i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 206.319i 0.291824i
\(708\) 0 0
\(709\) 128.606 0.181390 0.0906950 0.995879i \(-0.471091\pi\)
0.0906950 + 0.995879i \(0.471091\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −233.507 −0.327499
\(714\) 0 0
\(715\) − 103.805i − 0.145182i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 99.8703i − 0.138902i −0.997585 0.0694508i \(-0.977875\pi\)
0.997585 0.0694508i \(-0.0221247\pi\)
\(720\) 0 0
\(721\) −522.714 −0.724984
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 870.712 1.20098
\(726\) 0 0
\(727\) 605.619i 0.833039i 0.909127 + 0.416519i \(0.136750\pi\)
−0.909127 + 0.416519i \(0.863250\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 155.082i 0.212151i
\(732\) 0 0
\(733\) −11.5244 −0.0157223 −0.00786113 0.999969i \(-0.502502\pi\)
−0.00786113 + 0.999969i \(0.502502\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −95.1379 −0.129088
\(738\) 0 0
\(739\) 241.471i 0.326753i 0.986564 + 0.163377i \(0.0522386\pi\)
−0.986564 + 0.163377i \(0.947761\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 660.096i 0.888420i 0.895923 + 0.444210i \(0.146516\pi\)
−0.895923 + 0.444210i \(0.853484\pi\)
\(744\) 0 0
\(745\) −1446.10 −1.94107
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −889.088 −1.18703
\(750\) 0 0
\(751\) − 624.342i − 0.831347i −0.909514 0.415674i \(-0.863546\pi\)
0.909514 0.415674i \(-0.136454\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 172.590i 0.228597i
\(756\) 0 0
\(757\) 291.603 0.385209 0.192604 0.981276i \(-0.438307\pi\)
0.192604 + 0.981276i \(0.438307\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1333.83 −1.75274 −0.876369 0.481640i \(-0.840041\pi\)
−0.876369 + 0.481640i \(0.840041\pi\)
\(762\) 0 0
\(763\) − 28.9696i − 0.0379680i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1479.40i − 1.92881i
\(768\) 0 0
\(769\) −75.3889 −0.0980349 −0.0490175 0.998798i \(-0.515609\pi\)
−0.0490175 + 0.998798i \(0.515609\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1331.18 −1.72210 −0.861049 0.508521i \(-0.830192\pi\)
−0.861049 + 0.508521i \(0.830192\pi\)
\(774\) 0 0
\(775\) − 414.467i − 0.534796i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 222.100i − 0.285109i
\(780\) 0 0
\(781\) −47.2711 −0.0605264
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1700.03 2.16564
\(786\) 0 0
\(787\) 1099.10i 1.39657i 0.715820 + 0.698285i \(0.246053\pi\)
−0.715820 + 0.698285i \(0.753947\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 222.155i − 0.280853i
\(792\) 0 0
\(793\) −901.603 −1.13695
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −640.713 −0.803906 −0.401953 0.915660i \(-0.631668\pi\)
−0.401953 + 0.915660i \(0.631668\pi\)
\(798\) 0 0
\(799\) − 416.692i − 0.521517i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 4.97960i − 0.00620125i
\(804\) 0 0
\(805\) 313.835 0.389858
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −733.593 −0.906790 −0.453395 0.891310i \(-0.649787\pi\)
−0.453395 + 0.891310i \(0.649787\pi\)
\(810\) 0 0
\(811\) − 1149.16i − 1.41697i −0.705728 0.708483i \(-0.749380\pi\)
0.705728 0.708483i \(-0.250620\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 925.560i 1.13566i
\(816\) 0 0
\(817\) 733.433 0.897715
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −293.112 −0.357019 −0.178509 0.983938i \(-0.557127\pi\)
−0.178509 + 0.983938i \(0.557127\pi\)
\(822\) 0 0
\(823\) 466.668i 0.567033i 0.958967 + 0.283517i \(0.0915011\pi\)
−0.958967 + 0.283517i \(0.908499\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 340.554i 0.411794i 0.978574 + 0.205897i \(0.0660112\pi\)
−0.978574 + 0.205897i \(0.933989\pi\)
\(828\) 0 0
\(829\) 1368.33 1.65058 0.825289 0.564710i \(-0.191012\pi\)
0.825289 + 0.564710i \(0.191012\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 150.186 0.180295
\(834\) 0 0
\(835\) − 186.309i − 0.223125i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 297.555i 0.354654i 0.984152 + 0.177327i \(0.0567450\pi\)
−0.984152 + 0.177327i \(0.943255\pi\)
\(840\) 0 0
\(841\) 1370.71 1.62986
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 97.7562 0.115688
\(846\) 0 0
\(847\) 545.744i 0.644325i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 633.453i − 0.744363i
\(852\) 0 0
\(853\) −941.528 −1.10378 −0.551892 0.833916i \(-0.686094\pi\)
−0.551892 + 0.833916i \(0.686094\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 458.626 0.535153 0.267576 0.963537i \(-0.413777\pi\)
0.267576 + 0.963537i \(0.413777\pi\)
\(858\) 0 0
\(859\) 40.4325i 0.0470693i 0.999723 + 0.0235346i \(0.00749200\pi\)
−0.999723 + 0.0235346i \(0.992508\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1233.37i 1.42917i 0.699548 + 0.714585i \(0.253385\pi\)
−0.699548 + 0.714585i \(0.746615\pi\)
\(864\) 0 0
\(865\) 841.268 0.972565
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6.03976 0.00695025
\(870\) 0 0
\(871\) − 1111.33i − 1.27592i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 195.133i − 0.223009i
\(876\) 0 0
\(877\) −584.678 −0.666680 −0.333340 0.942807i \(-0.608176\pi\)
−0.333340 + 0.942807i \(0.608176\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −733.997 −0.833140 −0.416570 0.909104i \(-0.636768\pi\)
−0.416570 + 0.909104i \(0.636768\pi\)
\(882\) 0 0
\(883\) − 855.769i − 0.969161i −0.874747 0.484580i \(-0.838972\pi\)
0.874747 0.484580i \(-0.161028\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 767.876i 0.865700i 0.901466 + 0.432850i \(0.142492\pi\)
−0.901466 + 0.432850i \(0.857508\pi\)
\(888\) 0 0
\(889\) 393.852 0.443028
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1970.67 −2.20680
\(894\) 0 0
\(895\) − 1103.30i − 1.23273i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 1052.80i − 1.17107i
\(900\) 0 0
\(901\) 334.294 0.371026
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1100.58 1.21611
\(906\) 0 0
\(907\) − 957.563i − 1.05575i −0.849323 0.527874i \(-0.822989\pi\)
0.849323 0.527874i \(-0.177011\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 413.981i − 0.454425i −0.973845 0.227212i \(-0.927039\pi\)
0.973845 0.227212i \(-0.0729611\pi\)
\(912\) 0 0
\(913\) 133.608 0.146340
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −413.663 −0.451105
\(918\) 0 0
\(919\) 1703.44i 1.85358i 0.375579 + 0.926790i \(0.377444\pi\)
−0.375579 + 0.926790i \(0.622556\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 552.185i − 0.598250i
\(924\) 0 0
\(925\) 1124.36 1.21552
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −645.209 −0.694520 −0.347260 0.937769i \(-0.612888\pi\)
−0.347260 + 0.937769i \(0.612888\pi\)
\(930\) 0 0
\(931\) − 710.277i − 0.762918i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 40.7806i 0.0436156i
\(936\) 0 0
\(937\) 360.662 0.384911 0.192456 0.981306i \(-0.438355\pi\)
0.192456 + 0.981306i \(0.438355\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −111.816 −0.118827 −0.0594134 0.998233i \(-0.518923\pi\)
−0.0594134 + 0.998233i \(0.518923\pi\)
\(942\) 0 0
\(943\) − 91.9685i − 0.0975275i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24.7456i 0.0261306i 0.999915 + 0.0130653i \(0.00415893\pi\)
−0.999915 + 0.0130653i \(0.995841\pi\)
\(948\) 0 0
\(949\) 58.1679 0.0612939
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1569.10 1.64649 0.823245 0.567686i \(-0.192161\pi\)
0.823245 + 0.567686i \(0.192161\pi\)
\(954\) 0 0
\(955\) 1076.25i 1.12696i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 676.657i − 0.705586i
\(960\) 0 0
\(961\) 459.859 0.478521
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −907.757 −0.940681
\(966\) 0 0
\(967\) 129.015i 0.133418i 0.997772 + 0.0667090i \(0.0212499\pi\)
−0.997772 + 0.0667090i \(0.978750\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 595.591i 0.613379i 0.951810 + 0.306689i \(0.0992212\pi\)
−0.951810 + 0.306689i \(0.900779\pi\)
\(972\) 0 0
\(973\) 462.610 0.475447
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1232.40 −1.26141 −0.630706 0.776022i \(-0.717235\pi\)
−0.630706 + 0.776022i \(0.717235\pi\)
\(978\) 0 0
\(979\) − 164.238i − 0.167761i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1335.14i 1.35823i 0.734031 + 0.679115i \(0.237636\pi\)
−0.734031 + 0.679115i \(0.762364\pi\)
\(984\) 0 0
\(985\) 877.198 0.890556
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 303.705 0.307083
\(990\) 0 0
\(991\) 617.773i 0.623384i 0.950183 + 0.311692i \(0.100896\pi\)
−0.950183 + 0.311692i \(0.899104\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 2447.96i − 2.46026i
\(996\) 0 0
\(997\) −622.311 −0.624183 −0.312092 0.950052i \(-0.601030\pi\)
−0.312092 + 0.950052i \(0.601030\pi\)
\(998\) 0 0
\(999\) 0 0
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 1728.3.g.j.703.8 8
3.2 odd 2 1728.3.g.m.703.2 8
4.3 odd 2 inner 1728.3.g.j.703.7 8
8.3 odd 2 864.3.g.d.703.1 yes 8
8.5 even 2 864.3.g.d.703.2 yes 8
12.11 even 2 1728.3.g.m.703.1 8
24.5 odd 2 864.3.g.b.703.8 yes 8
24.11 even 2 864.3.g.b.703.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.g.b.703.7 8 24.11 even 2
864.3.g.b.703.8 yes 8 24.5 odd 2
864.3.g.d.703.1 yes 8 8.3 odd 2
864.3.g.d.703.2 yes 8 8.5 even 2
1728.3.g.j.703.7 8 4.3 odd 2 inner
1728.3.g.j.703.8 8 1.1 even 1 trivial
1728.3.g.m.703.1 8 12.11 even 2
1728.3.g.m.703.2 8 3.2 odd 2