Properties

Label 1728.3.g.j.703.7
Level $1728$
Weight $3$
Character 1728.703
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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

$q$-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 \(-0.894370\pi\)
0.945442 0.325790i \(-0.105630\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.16066i − 0.105515i −0.998607 0.0527575i \(-0.983199\pi\)
0.998607 0.0527575i \(-0.0168010\pi\)
\(12\) 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 \(0.230673\pi\)
−0.748712 + 0.662896i \(0.769327\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 10.4309i 0.453515i 0.973951 + 0.226758i \(0.0728125\pi\)
−0.973951 + 0.226758i \(0.927187\pi\)
\(24\) 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 \(0.117588\pi\)
−0.932540 + 0.361067i \(0.882412\pi\)
\(32\) 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 \(-0.890061\pi\)
0.940945 0.338558i \(-0.109939\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 78.2321i 1.66451i 0.554392 + 0.832256i \(0.312951\pi\)
−0.554392 + 0.832256i \(0.687049\pi\)
\(48\) 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 \(-0.624305\pi\)
0.380666 0.924712i \(-0.375695\pi\)
\(60\) 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 \(-0.790484\pi\)
0.791086 0.611705i \(-0.209516\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 40.7276i − 0.573629i −0.957986 0.286814i \(-0.907404\pi\)
0.957986 0.286814i \(-0.0925963\pi\)
\(72\) 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 \(0.0104854\pi\)
−0.999457 + 0.0329349i \(0.989515\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 115.114i 1.38691i 0.720498 + 0.693457i \(0.243913\pi\)
−0.720498 + 0.693457i \(0.756087\pi\)
\(84\) 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 \(-0.812210\pi\)
0.830963 0.556328i \(-0.187790\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 194.930i − 1.82178i −0.412652 0.910889i \(-0.635397\pi\)
0.412652 0.910889i \(-0.364603\pi\)
\(108\) 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 \(0.110415\pi\)
−0.940438 + 0.339964i \(0.889585\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 90.6946i − 0.692325i −0.938175 0.346162i \(-0.887485\pi\)
0.938175 0.346162i \(-0.112515\pi\)
\(132\) 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 \(0.118877\pi\)
−0.931069 + 0.364842i \(0.881123\pi\)
\(140\) 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 \(-0.972389\pi\)
0.996240 0.0866350i \(-0.0276114\pi\)
\(152\) 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 \(-0.858373\pi\)
0.902639 0.430398i \(-0.141627\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 28.2435i 0.169123i 0.996418 + 0.0845613i \(0.0269489\pi\)
−0.996418 + 0.0845613i \(0.973051\pi\)
\(168\) 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 \(0.154733\pi\)
−0.884157 + 0.467189i \(0.845267\pi\)
\(180\) 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 \(-0.859535\pi\)
0.904204 0.427102i \(-0.140465\pi\)
\(192\) 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 \(0.382293\pi\)
−0.361418 + 0.932404i \(0.617707\pi\)
\(200\) 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 \(-0.807832\pi\)
0.823233 0.567704i \(-0.192168\pi\)
\(212\) 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 \(0.0878652\pi\)
−0.962143 + 0.272545i \(0.912135\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 223.023i 0.982481i 0.871024 + 0.491241i \(0.163456\pi\)
−0.871024 + 0.491241i \(0.836544\pi\)
\(228\) 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 \(0.0744915\pi\)
−0.972742 + 0.231892i \(0.925508\pi\)
\(240\) 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 \(0.387882\pi\)
−0.344990 + 0.938606i \(0.612118\pi\)
\(252\) 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 \(-0.611200\pi\)
0.342283 0.939597i \(-0.388800\pi\)
\(264\) 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 \(0.175224\pi\)
−0.852272 + 0.523099i \(0.824776\pi\)
\(272\) 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 \(-0.814186\pi\)
0.834400 0.551159i \(-0.185814\pi\)
\(284\) 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 \(0.160165\pi\)
−0.876057 + 0.482208i \(0.839835\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 274.720i 0.883345i 0.897176 + 0.441673i \(0.145615\pi\)
−0.897176 + 0.441673i \(0.854385\pi\)
\(312\) 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 \(0.0374294\pi\)
−0.993094 + 0.117317i \(0.962571\pi\)
\(332\) 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 \(0.246460\pi\)
−0.714928 + 0.699198i \(0.753540\pi\)
\(348\) 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 \(0.191424\pi\)
−0.824557 + 0.565779i \(0.808576\pi\)
\(360\) 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 \(0.375594\pi\)
−0.380959 + 0.924592i \(0.624406\pi\)
\(368\) 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 \(0.344891\pi\)
−0.468231 + 0.883606i \(0.655109\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 282.840i − 0.738486i −0.929333 0.369243i \(-0.879617\pi\)
0.929333 0.369243i \(-0.120383\pi\)
\(384\) 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 \(-0.792152\pi\)
0.794281 0.607551i \(-0.207848\pi\)
\(420\) 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 \(-0.941020\pi\)
0.982882 0.184234i \(-0.0589804\pi\)
\(432\) 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 \(-0.947155\pi\)
0.986251 0.165257i \(-0.0528453\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 771.496i 1.74153i 0.491703 + 0.870763i \(0.336374\pi\)
−0.491703 + 0.870763i \(0.663626\pi\)
\(444\) 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 \(-0.904591\pi\)
0.955414 0.295268i \(-0.0954090\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 109.131i − 0.233684i −0.993150 0.116842i \(-0.962723\pi\)
0.993150 0.116842i \(-0.0372772\pi\)
\(468\) 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 \(-0.985842\pi\)
0.999011 0.0444653i \(-0.0141584\pi\)
\(480\) 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 \(0.151394\pi\)
−0.889009 + 0.457889i \(0.848606\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 565.533i − 1.15180i −0.817521 0.575899i \(-0.804652\pi\)
0.817521 0.575899i \(-0.195348\pi\)
\(492\) 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 \(-0.623971\pi\)
0.379696 0.925111i \(-0.376029\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 406.646i 0.808441i 0.914662 + 0.404220i \(0.132457\pi\)
−0.914662 + 0.404220i \(0.867543\pi\)
\(504\) 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 \(0.118683\pi\)
−0.931292 + 0.364274i \(0.881317\pi\)
\(524\) 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 \(-0.994493\pi\)
0.999850 0.0173012i \(-0.00550740\pi\)
\(548\) 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 \(0.142813\pi\)
−0.901029 + 0.433760i \(0.857187\pi\)
\(564\) 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 \(0.280189\pi\)
−0.636966 + 0.770892i \(0.719811\pi\)
\(572\) 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 \(-0.692604\pi\)
0.568831 0.822455i \(-0.307396\pi\)
\(588\) 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 \(-0.877931\pi\)
0.927364 0.374161i \(-0.122069\pi\)
\(600\) 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 \(0.0224160\pi\)
−0.997521 + 0.0703636i \(0.977584\pi\)
\(608\) 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 \(0.0152291\pi\)
−0.998856 + 0.0478253i \(0.984771\pi\)
\(620\) 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 \(-0.982325\pi\)
0.998459 0.0554984i \(-0.0176748\pi\)
\(632\) 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 \(-0.913058\pi\)
0.962930 0.269753i \(-0.0869419\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 746.221i − 1.15335i −0.816972 0.576677i \(-0.804349\pi\)
0.816972 0.576677i \(-0.195651\pi\)
\(648\) 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 \(0.212995\pi\)
−0.784354 + 0.620314i \(0.787005\pi\)
\(660\) 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 \(-0.884097\pi\)
0.934438 0.356126i \(-0.115903\pi\)
\(684\) 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 \(-0.685503\pi\)
0.550344 0.834938i \(-0.314497\pi\)
\(692\) 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 \(0.0221247\pi\)
−0.997585 + 0.0694508i \(0.977875\pi\)
\(720\) 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 \(-0.863250\pi\)
0.909127 0.416519i \(-0.136750\pi\)
\(728\) 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 \(-0.947761\pi\)
0.986564 0.163377i \(-0.0522386\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 660.096i − 0.888420i −0.895923 0.444210i \(-0.853484\pi\)
0.895923 0.444210i \(-0.146516\pi\)
\(744\) 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 \(0.136454\pi\)
−0.909514 + 0.415674i \(0.863546\pi\)
\(752\) 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 \(-0.753947\pi\)
0.715820 0.698285i \(-0.246053\pi\)
\(788\) 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 \(0.250620\pi\)
−0.705728 + 0.708483i \(0.749380\pi\)
\(812\) 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 \(-0.908499\pi\)
0.958967 0.283517i \(-0.0915011\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 340.554i − 0.411794i −0.978574 0.205897i \(-0.933989\pi\)
0.978574 0.205897i \(-0.0660112\pi\)
\(828\) 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 \(-0.943255\pi\)
0.984152 0.177327i \(-0.0567450\pi\)
\(840\) 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 \(-0.992508\pi\)
0.999723 0.0235346i \(-0.00749200\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 1233.37i − 1.42917i −0.699548 0.714585i \(-0.746615\pi\)
0.699548 0.714585i \(-0.253385\pi\)
\(864\) 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 \(0.161028\pi\)
−0.874747 + 0.484580i \(0.838972\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 767.876i − 0.865700i −0.901466 0.432850i \(-0.857508\pi\)
0.901466 0.432850i \(-0.142492\pi\)
\(888\) 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 \(0.177011\pi\)
−0.849323 + 0.527874i \(0.822989\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 413.981i 0.454425i 0.973845 + 0.227212i \(0.0729611\pi\)
−0.973845 + 0.227212i \(0.927039\pi\)
\(912\) 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 \(-0.622556\pi\)
0.375579 0.926790i \(-0.377444\pi\)
\(920\) 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 \(-0.995841\pi\)
0.999915 0.0130653i \(-0.00415893\pi\)
\(948\) 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 \(-0.978750\pi\)
0.997772 0.0667090i \(-0.0212499\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 595.591i − 0.613379i −0.951810 0.306689i \(-0.900779\pi\)
0.951810 0.306689i \(-0.0992212\pi\)
\(972\) 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 \(-0.762364\pi\)
0.734031 0.679115i \(-0.237636\pi\)
\(984\) 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 \(-0.899104\pi\)
0.950183 0.311692i \(-0.100896\pi\)
\(992\) 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.703.7 8
3.2 odd 2 1728.3.g.m.703.1 8
4.3 odd 2 inner 1728.3.g.j.703.8 8
8.3 odd 2 864.3.g.d.703.2 yes 8
8.5 even 2 864.3.g.d.703.1 yes 8
12.11 even 2 1728.3.g.m.703.2 8
24.5 odd 2 864.3.g.b.703.7 8
24.11 even 2 864.3.g.b.703.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.g.b.703.7 8 24.5 odd 2
864.3.g.b.703.8 yes 8 24.11 even 2
864.3.g.d.703.1 yes 8 8.5 even 2
864.3.g.d.703.2 yes 8 8.3 odd 2
1728.3.g.j.703.7 8 1.1 even 1 trivial
1728.3.g.j.703.8 8 4.3 odd 2 inner
1728.3.g.m.703.1 8 3.2 odd 2
1728.3.g.m.703.2 8 12.11 even 2