Properties

Label 1008.3.d.a.449.2
Level $1008$
Weight $3$
Character 1008.449
Analytic conductor $27.466$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.16372i\) of defining polynomial
Character \(\chi\) \(=\) 1008.449
Dual form 1008.3.d.a.449.3

$q$-expansion

\(f(q)\) \(=\) \(q-6.06910i q^{5} +2.64575 q^{7} +O(q^{10})\) \(q-6.06910i q^{5} +2.64575 q^{7} +12.1382i q^{11} -18.5830 q^{13} -10.9015i q^{17} -20.0000 q^{19} +12.1382i q^{23} -11.8340 q^{25} +41.8367i q^{29} -25.1660 q^{31} -16.0573i q^{35} +38.0000 q^{37} -60.6337i q^{41} -83.4980 q^{43} +16.9706i q^{47} +7.00000 q^{49} +94.0424i q^{53} +73.6680 q^{55} +58.2175i q^{59} +15.6680 q^{61} +112.782i q^{65} +132.664 q^{67} +12.1382i q^{71} -76.9150 q^{73} +32.1147i q^{77} -33.6680 q^{79} +60.5764i q^{83} -66.1621 q^{85} -4.77506i q^{89} -49.1660 q^{91} +121.382i q^{95} -188.413 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q - 32 q^{13} - 80 q^{19} - 132 q^{25} - 16 q^{31} + 152 q^{37} - 80 q^{43} + 28 q^{49} + 464 q^{55} + 232 q^{61} + 192 q^{67} - 96 q^{73} - 304 q^{79} + 328 q^{85} - 112 q^{91} - 288 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) − 6.06910i − 1.21382i −0.794770 0.606910i \(-0.792409\pi\)
0.794770 0.606910i \(-0.207591\pi\)
\(6\) 0 0
\(7\) 2.64575 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 12.1382i 1.10347i 0.834019 + 0.551736i \(0.186035\pi\)
−0.834019 + 0.551736i \(0.813965\pi\)
\(12\) 0 0
\(13\) −18.5830 −1.42946 −0.714731 0.699399i \(-0.753451\pi\)
−0.714731 + 0.699399i \(0.753451\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 10.9015i − 0.641262i −0.947204 0.320631i \(-0.896105\pi\)
0.947204 0.320631i \(-0.103895\pi\)
\(18\) 0 0
\(19\) −20.0000 −1.05263 −0.526316 0.850289i \(-0.676427\pi\)
−0.526316 + 0.850289i \(0.676427\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 12.1382i 0.527748i 0.964557 + 0.263874i \(0.0850003\pi\)
−0.964557 + 0.263874i \(0.915000\pi\)
\(24\) 0 0
\(25\) −11.8340 −0.473360
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 41.8367i 1.44264i 0.692600 + 0.721322i \(0.256465\pi\)
−0.692600 + 0.721322i \(0.743535\pi\)
\(30\) 0 0
\(31\) −25.1660 −0.811807 −0.405903 0.913916i \(-0.633043\pi\)
−0.405903 + 0.913916i \(0.633043\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 16.0573i − 0.458781i
\(36\) 0 0
\(37\) 38.0000 1.02703 0.513514 0.858082i \(-0.328344\pi\)
0.513514 + 0.858082i \(0.328344\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 60.6337i − 1.47887i −0.673227 0.739435i \(-0.735092\pi\)
0.673227 0.739435i \(-0.264908\pi\)
\(42\) 0 0
\(43\) −83.4980 −1.94181 −0.970907 0.239455i \(-0.923031\pi\)
−0.970907 + 0.239455i \(0.923031\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 16.9706i 0.361076i 0.983568 + 0.180538i \(0.0577838\pi\)
−0.983568 + 0.180538i \(0.942216\pi\)
\(48\) 0 0
\(49\) 7.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 94.0424i 1.77439i 0.461399 + 0.887193i \(0.347348\pi\)
−0.461399 + 0.887193i \(0.652652\pi\)
\(54\) 0 0
\(55\) 73.6680 1.33942
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 58.2175i 0.986738i 0.869820 + 0.493369i \(0.164235\pi\)
−0.869820 + 0.493369i \(0.835765\pi\)
\(60\) 0 0
\(61\) 15.6680 0.256852 0.128426 0.991719i \(-0.459008\pi\)
0.128426 + 0.991719i \(0.459008\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 112.782i 1.73511i
\(66\) 0 0
\(67\) 132.664 1.98006 0.990030 0.140856i \(-0.0449853\pi\)
0.990030 + 0.140856i \(0.0449853\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.1382i 0.170961i 0.996340 + 0.0854803i \(0.0272425\pi\)
−0.996340 + 0.0854803i \(0.972758\pi\)
\(72\) 0 0
\(73\) −76.9150 −1.05363 −0.526815 0.849980i \(-0.676614\pi\)
−0.526815 + 0.849980i \(0.676614\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 32.1147i 0.417074i
\(78\) 0 0
\(79\) −33.6680 −0.426177 −0.213088 0.977033i \(-0.568352\pi\)
−0.213088 + 0.977033i \(0.568352\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 60.5764i 0.729836i 0.931040 + 0.364918i \(0.118903\pi\)
−0.931040 + 0.364918i \(0.881097\pi\)
\(84\) 0 0
\(85\) −66.1621 −0.778377
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 4.77506i − 0.0536523i −0.999640 0.0268262i \(-0.991460\pi\)
0.999640 0.0268262i \(-0.00854006\pi\)
\(90\) 0 0
\(91\) −49.1660 −0.540286
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 121.382i 1.27771i
\(96\) 0 0
\(97\) −188.413 −1.94240 −0.971201 0.238260i \(-0.923423\pi\)
−0.971201 + 0.238260i \(0.923423\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 106.713i 1.05656i 0.849069 + 0.528282i \(0.177164\pi\)
−0.849069 + 0.528282i \(0.822836\pi\)
\(102\) 0 0
\(103\) −131.498 −1.27668 −0.638340 0.769755i \(-0.720379\pi\)
−0.638340 + 0.769755i \(0.720379\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 82.3793i − 0.769900i −0.922937 0.384950i \(-0.874219\pi\)
0.922937 0.384950i \(-0.125781\pi\)
\(108\) 0 0
\(109\) 33.8301 0.310367 0.155184 0.987886i \(-0.450403\pi\)
0.155184 + 0.987886i \(0.450403\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 28.5190i − 0.252381i −0.992006 0.126190i \(-0.959725\pi\)
0.992006 0.126190i \(-0.0402750\pi\)
\(114\) 0 0
\(115\) 73.6680 0.640591
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 28.8426i − 0.242374i
\(120\) 0 0
\(121\) −26.3360 −0.217653
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 79.9059i − 0.639247i
\(126\) 0 0
\(127\) −129.668 −1.02101 −0.510504 0.859875i \(-0.670541\pi\)
−0.510504 + 0.859875i \(0.670541\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 148.017i 1.12990i 0.825124 + 0.564952i \(0.191105\pi\)
−0.825124 + 0.564952i \(0.808895\pi\)
\(132\) 0 0
\(133\) −52.9150 −0.397857
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 76.9573i − 0.561732i −0.959747 0.280866i \(-0.909378\pi\)
0.959747 0.280866i \(-0.0906216\pi\)
\(138\) 0 0
\(139\) −217.328 −1.56351 −0.781756 0.623585i \(-0.785676\pi\)
−0.781756 + 0.623585i \(0.785676\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 225.564i − 1.57737i
\(144\) 0 0
\(145\) 253.911 1.75111
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 161.925i 1.08674i 0.839492 + 0.543371i \(0.182852\pi\)
−0.839492 + 0.543371i \(0.817148\pi\)
\(150\) 0 0
\(151\) 93.1660 0.616993 0.308497 0.951225i \(-0.400174\pi\)
0.308497 + 0.951225i \(0.400174\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 152.735i 0.985388i
\(156\) 0 0
\(157\) −184.996 −1.17832 −0.589159 0.808017i \(-0.700541\pi\)
−0.589159 + 0.808017i \(0.700541\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 32.1147i 0.199470i
\(162\) 0 0
\(163\) −86.9961 −0.533718 −0.266859 0.963736i \(-0.585986\pi\)
−0.266859 + 0.963736i \(0.585986\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 60.5764i − 0.362733i −0.983416 0.181366i \(-0.941948\pi\)
0.983416 0.181366i \(-0.0580520\pi\)
\(168\) 0 0
\(169\) 176.328 1.04336
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 162.572i 0.939721i 0.882741 + 0.469860i \(0.155696\pi\)
−0.882741 + 0.469860i \(0.844304\pi\)
\(174\) 0 0
\(175\) −31.3098 −0.178913
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 223.091i − 1.24632i −0.782095 0.623159i \(-0.785849\pi\)
0.782095 0.623159i \(-0.214151\pi\)
\(180\) 0 0
\(181\) 188.915 1.04373 0.521865 0.853028i \(-0.325237\pi\)
0.521865 + 0.853028i \(0.325237\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 230.626i − 1.24663i
\(186\) 0 0
\(187\) 132.324 0.707616
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 228.038i 1.19391i 0.802273 + 0.596957i \(0.203624\pi\)
−0.802273 + 0.596957i \(0.796376\pi\)
\(192\) 0 0
\(193\) 134.000 0.694301 0.347150 0.937810i \(-0.387149\pi\)
0.347150 + 0.937810i \(0.387149\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 188.560i − 0.957157i −0.878045 0.478579i \(-0.841152\pi\)
0.878045 0.478579i \(-0.158848\pi\)
\(198\) 0 0
\(199\) −102.494 −0.515046 −0.257523 0.966272i \(-0.582906\pi\)
−0.257523 + 0.966272i \(0.582906\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 110.689i 0.545268i
\(204\) 0 0
\(205\) −367.992 −1.79508
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 242.764i − 1.16155i
\(210\) 0 0
\(211\) 84.5020 0.400483 0.200242 0.979747i \(-0.435827\pi\)
0.200242 + 0.979747i \(0.435827\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 506.758i 2.35701i
\(216\) 0 0
\(217\) −66.5830 −0.306834
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 202.582i 0.916660i
\(222\) 0 0
\(223\) 158.494 0.710736 0.355368 0.934727i \(-0.384356\pi\)
0.355368 + 0.934727i \(0.384356\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 101.823i − 0.448561i −0.974525 0.224281i \(-0.927997\pi\)
0.974525 0.224281i \(-0.0720032\pi\)
\(228\) 0 0
\(229\) −268.915 −1.17430 −0.587151 0.809478i \(-0.699750\pi\)
−0.587151 + 0.809478i \(0.699750\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 26.2748i − 0.112767i −0.998409 0.0563836i \(-0.982043\pi\)
0.998409 0.0563836i \(-0.0179570\pi\)
\(234\) 0 0
\(235\) 102.996 0.438281
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 92.2733i − 0.386081i −0.981191 0.193040i \(-0.938165\pi\)
0.981191 0.193040i \(-0.0618348\pi\)
\(240\) 0 0
\(241\) 343.247 1.42426 0.712131 0.702047i \(-0.247730\pi\)
0.712131 + 0.702047i \(0.247730\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 42.4837i − 0.173403i
\(246\) 0 0
\(247\) 371.660 1.50470
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 356.382i − 1.41985i −0.704278 0.709924i \(-0.748729\pi\)
0.704278 0.709924i \(-0.251271\pi\)
\(252\) 0 0
\(253\) −147.336 −0.582356
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 254.730i − 0.991169i −0.868560 0.495584i \(-0.834954\pi\)
0.868560 0.495584i \(-0.165046\pi\)
\(258\) 0 0
\(259\) 100.539 0.388180
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 261.979i − 0.996117i −0.867143 0.498059i \(-0.834046\pi\)
0.867143 0.498059i \(-0.165954\pi\)
\(264\) 0 0
\(265\) 570.753 2.15378
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 93.6246i 0.348047i 0.984742 + 0.174023i \(0.0556768\pi\)
−0.984742 + 0.174023i \(0.944323\pi\)
\(270\) 0 0
\(271\) −1.16601 −0.00430262 −0.00215131 0.999998i \(-0.500685\pi\)
−0.00215131 + 0.999998i \(0.500685\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 143.643i − 0.522339i
\(276\) 0 0
\(277\) 32.0000 0.115523 0.0577617 0.998330i \(-0.481604\pi\)
0.0577617 + 0.998330i \(0.481604\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 166.757i 0.593441i 0.954964 + 0.296721i \(0.0958930\pi\)
−0.954964 + 0.296721i \(0.904107\pi\)
\(282\) 0 0
\(283\) −16.3399 −0.0577381 −0.0288691 0.999583i \(-0.509191\pi\)
−0.0288691 + 0.999583i \(0.509191\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 160.422i − 0.558961i
\(288\) 0 0
\(289\) 170.158 0.588782
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 368.921i 1.25912i 0.776953 + 0.629558i \(0.216764\pi\)
−0.776953 + 0.629558i \(0.783236\pi\)
\(294\) 0 0
\(295\) 353.328 1.19772
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 225.564i − 0.754396i
\(300\) 0 0
\(301\) −220.915 −0.733937
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 95.0906i − 0.311772i
\(306\) 0 0
\(307\) 192.664 0.627570 0.313785 0.949494i \(-0.398403\pi\)
0.313785 + 0.949494i \(0.398403\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 131.276i − 0.422109i −0.977474 0.211055i \(-0.932310\pi\)
0.977474 0.211055i \(-0.0676898\pi\)
\(312\) 0 0
\(313\) 43.3281 0.138428 0.0692142 0.997602i \(-0.477951\pi\)
0.0692142 + 0.997602i \(0.477951\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 251.724i − 0.794083i −0.917801 0.397042i \(-0.870037\pi\)
0.917801 0.397042i \(-0.129963\pi\)
\(318\) 0 0
\(319\) −507.822 −1.59192
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 218.029i 0.675013i
\(324\) 0 0
\(325\) 219.911 0.676650
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 44.8999i 0.136474i
\(330\) 0 0
\(331\) −361.490 −1.09212 −0.546058 0.837748i \(-0.683872\pi\)
−0.546058 + 0.837748i \(0.683872\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 805.151i − 2.40344i
\(336\) 0 0
\(337\) −298.834 −0.886748 −0.443374 0.896337i \(-0.646219\pi\)
−0.443374 + 0.896337i \(0.646219\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 305.470i − 0.895807i
\(342\) 0 0
\(343\) 18.5203 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 206.120i − 0.594006i −0.954876 0.297003i \(-0.904013\pi\)
0.954876 0.297003i \(-0.0959872\pi\)
\(348\) 0 0
\(349\) 434.324 1.24448 0.622241 0.782826i \(-0.286222\pi\)
0.622241 + 0.782826i \(0.286222\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 185.439i − 0.525324i −0.964888 0.262662i \(-0.915400\pi\)
0.964888 0.262662i \(-0.0846005\pi\)
\(354\) 0 0
\(355\) 73.6680 0.207515
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 516.767i 1.43946i 0.694254 + 0.719731i \(0.255735\pi\)
−0.694254 + 0.719731i \(0.744265\pi\)
\(360\) 0 0
\(361\) 39.0000 0.108033
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 466.805i 1.27892i
\(366\) 0 0
\(367\) 117.490 0.320137 0.160068 0.987106i \(-0.448829\pi\)
0.160068 + 0.987106i \(0.448829\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 248.813i 0.670655i
\(372\) 0 0
\(373\) −402.664 −1.07953 −0.539764 0.841816i \(-0.681487\pi\)
−0.539764 + 0.841816i \(0.681487\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 777.451i − 2.06221i
\(378\) 0 0
\(379\) −398.834 −1.05233 −0.526166 0.850382i \(-0.676371\pi\)
−0.526166 + 0.850382i \(0.676371\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 744.804i − 1.94466i −0.233614 0.972329i \(-0.575055\pi\)
0.233614 0.972329i \(-0.424945\pi\)
\(384\) 0 0
\(385\) 194.907 0.506252
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 535.162i − 1.37574i −0.725834 0.687869i \(-0.758546\pi\)
0.725834 0.687869i \(-0.241454\pi\)
\(390\) 0 0
\(391\) 132.324 0.338425
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 204.334i 0.517302i
\(396\) 0 0
\(397\) −94.3241 −0.237592 −0.118796 0.992919i \(-0.537904\pi\)
−0.118796 + 0.992919i \(0.537904\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 103.593i − 0.258335i −0.991623 0.129168i \(-0.958769\pi\)
0.991623 0.129168i \(-0.0412306\pi\)
\(402\) 0 0
\(403\) 467.660 1.16045
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 461.252i 1.13330i
\(408\) 0 0
\(409\) −9.75689 −0.0238555 −0.0119277 0.999929i \(-0.503797\pi\)
−0.0119277 + 0.999929i \(0.503797\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 154.029i 0.372952i
\(414\) 0 0
\(415\) 367.644 0.885890
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 339.411i − 0.810051i −0.914305 0.405025i \(-0.867263\pi\)
0.914305 0.405025i \(-0.132737\pi\)
\(420\) 0 0
\(421\) −599.320 −1.42356 −0.711782 0.702401i \(-0.752111\pi\)
−0.711782 + 0.702401i \(0.752111\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 129.008i 0.303548i
\(426\) 0 0
\(427\) 41.4536 0.0970810
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 710.978i − 1.64960i −0.565424 0.824800i \(-0.691288\pi\)
0.565424 0.824800i \(-0.308712\pi\)
\(432\) 0 0
\(433\) 377.984 0.872943 0.436471 0.899718i \(-0.356228\pi\)
0.436471 + 0.899718i \(0.356228\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 242.764i − 0.555524i
\(438\) 0 0
\(439\) −528.146 −1.20307 −0.601533 0.798848i \(-0.705443\pi\)
−0.601533 + 0.798848i \(0.705443\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 36.6438i − 0.0827174i −0.999144 0.0413587i \(-0.986831\pi\)
0.999144 0.0413587i \(-0.0131686\pi\)
\(444\) 0 0
\(445\) −28.9803 −0.0651243
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 397.612i − 0.885550i −0.896633 0.442775i \(-0.853994\pi\)
0.896633 0.442775i \(-0.146006\pi\)
\(450\) 0 0
\(451\) 735.984 1.63189
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 298.393i 0.655810i
\(456\) 0 0
\(457\) 344.324 0.753445 0.376722 0.926326i \(-0.377051\pi\)
0.376722 + 0.926326i \(0.377051\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 370.936i − 0.804634i −0.915500 0.402317i \(-0.868205\pi\)
0.915500 0.402317i \(-0.131795\pi\)
\(462\) 0 0
\(463\) −78.3320 −0.169184 −0.0845918 0.996416i \(-0.526959\pi\)
−0.0845918 + 0.996416i \(0.526959\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 399.758i 0.856014i 0.903775 + 0.428007i \(0.140784\pi\)
−0.903775 + 0.428007i \(0.859216\pi\)
\(468\) 0 0
\(469\) 350.996 0.748392
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 1013.52i − 2.14274i
\(474\) 0 0
\(475\) 236.680 0.498273
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 703.328i 1.46833i 0.678973 + 0.734163i \(0.262425\pi\)
−0.678973 + 0.734163i \(0.737575\pi\)
\(480\) 0 0
\(481\) −706.154 −1.46810
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1143.50i 2.35773i
\(486\) 0 0
\(487\) 82.5098 0.169425 0.0847124 0.996405i \(-0.473003\pi\)
0.0847124 + 0.996405i \(0.473003\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 184.203i 0.375158i 0.982249 + 0.187579i \(0.0600641\pi\)
−0.982249 + 0.187579i \(0.939936\pi\)
\(492\) 0 0
\(493\) 456.081 0.925114
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 32.1147i 0.0646170i
\(498\) 0 0
\(499\) −752.810 −1.50864 −0.754319 0.656508i \(-0.772033\pi\)
−0.754319 + 0.656508i \(0.772033\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 662.540i 1.31718i 0.752504 + 0.658588i \(0.228846\pi\)
−0.752504 + 0.658588i \(0.771154\pi\)
\(504\) 0 0
\(505\) 647.652 1.28248
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 949.115i 1.86467i 0.361601 + 0.932333i \(0.382230\pi\)
−0.361601 + 0.932333i \(0.617770\pi\)
\(510\) 0 0
\(511\) −203.498 −0.398235
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 798.075i 1.54966i
\(516\) 0 0
\(517\) −205.992 −0.398437
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 714.344i 1.37110i 0.728025 + 0.685551i \(0.240439\pi\)
−0.728025 + 0.685551i \(0.759561\pi\)
\(522\) 0 0
\(523\) 232.000 0.443595 0.221797 0.975093i \(-0.428808\pi\)
0.221797 + 0.975093i \(0.428808\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 274.346i 0.520581i
\(528\) 0 0
\(529\) 381.664 0.721482
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1126.76i 2.11399i
\(534\) 0 0
\(535\) −499.969 −0.934521
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 84.9674i 0.157639i
\(540\) 0 0
\(541\) 165.668 0.306225 0.153113 0.988209i \(-0.451070\pi\)
0.153113 + 0.988209i \(0.451070\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 205.318i − 0.376730i
\(546\) 0 0
\(547\) −295.676 −0.540541 −0.270270 0.962784i \(-0.587113\pi\)
−0.270270 + 0.962784i \(0.587113\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 836.734i − 1.51857i
\(552\) 0 0
\(553\) −89.0771 −0.161080
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 76.8426i 0.137958i 0.997618 + 0.0689790i \(0.0219742\pi\)
−0.997618 + 0.0689790i \(0.978026\pi\)
\(558\) 0 0
\(559\) 1551.64 2.77575
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 1016.33i − 1.80521i −0.430470 0.902605i \(-0.641652\pi\)
0.430470 0.902605i \(-0.358348\pi\)
\(564\) 0 0
\(565\) −173.085 −0.306345
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 586.533i − 1.03081i −0.856946 0.515406i \(-0.827641\pi\)
0.856946 0.515406i \(-0.172359\pi\)
\(570\) 0 0
\(571\) −951.644 −1.66663 −0.833314 0.552800i \(-0.813559\pi\)
−0.833314 + 0.552800i \(0.813559\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 143.643i − 0.249815i
\(576\) 0 0
\(577\) −148.672 −0.257664 −0.128832 0.991666i \(-0.541123\pi\)
−0.128832 + 0.991666i \(0.541123\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 160.270i 0.275852i
\(582\) 0 0
\(583\) −1141.51 −1.95799
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 332.564i − 0.566548i −0.959039 0.283274i \(-0.908579\pi\)
0.959039 0.283274i \(-0.0914206\pi\)
\(588\) 0 0
\(589\) 503.320 0.854533
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 217.251i 0.366359i 0.983079 + 0.183180i \(0.0586390\pi\)
−0.983079 + 0.183180i \(0.941361\pi\)
\(594\) 0 0
\(595\) −175.048 −0.294199
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 172.179i 0.287444i 0.989618 + 0.143722i \(0.0459072\pi\)
−0.989618 + 0.143722i \(0.954093\pi\)
\(600\) 0 0
\(601\) −418.000 −0.695507 −0.347754 0.937586i \(-0.613055\pi\)
−0.347754 + 0.937586i \(0.613055\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 159.836i 0.264191i
\(606\) 0 0
\(607\) −627.158 −1.03321 −0.516605 0.856224i \(-0.672804\pi\)
−0.516605 + 0.856224i \(0.672804\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 315.364i − 0.516144i
\(612\) 0 0
\(613\) −279.328 −0.455674 −0.227837 0.973699i \(-0.573165\pi\)
−0.227837 + 0.973699i \(0.573165\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 358.380i 0.580843i 0.956899 + 0.290422i \(0.0937955\pi\)
−0.956899 + 0.290422i \(0.906204\pi\)
\(618\) 0 0
\(619\) 983.644 1.58909 0.794543 0.607208i \(-0.207710\pi\)
0.794543 + 0.607208i \(0.207710\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 12.6336i − 0.0202787i
\(624\) 0 0
\(625\) −780.806 −1.24929
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 414.256i − 0.658594i
\(630\) 0 0
\(631\) 298.996 0.473845 0.236922 0.971529i \(-0.423861\pi\)
0.236922 + 0.971529i \(0.423861\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 786.968i 1.23932i
\(636\) 0 0
\(637\) −130.081 −0.204209
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 311.957i − 0.486672i −0.969942 0.243336i \(-0.921758\pi\)
0.969942 0.243336i \(-0.0782418\pi\)
\(642\) 0 0
\(643\) 604.000 0.939347 0.469673 0.882840i \(-0.344372\pi\)
0.469673 + 0.882840i \(0.344372\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 179.600i − 0.277588i −0.990321 0.138794i \(-0.955677\pi\)
0.990321 0.138794i \(-0.0443226\pi\)
\(648\) 0 0
\(649\) −706.656 −1.08884
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 481.892i 0.737966i 0.929436 + 0.368983i \(0.120294\pi\)
−0.929436 + 0.368983i \(0.879706\pi\)
\(654\) 0 0
\(655\) 898.332 1.37150
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 877.408i − 1.33142i −0.746209 0.665711i \(-0.768128\pi\)
0.746209 0.665711i \(-0.231872\pi\)
\(660\) 0 0
\(661\) −521.644 −0.789175 −0.394587 0.918858i \(-0.629112\pi\)
−0.394587 + 0.918858i \(0.629112\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 321.147i 0.482927i
\(666\) 0 0
\(667\) −507.822 −0.761353
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 190.181i 0.283429i
\(672\) 0 0
\(673\) −659.992 −0.980672 −0.490336 0.871534i \(-0.663126\pi\)
−0.490336 + 0.871534i \(0.663126\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1016.28i 1.50115i 0.660787 + 0.750573i \(0.270222\pi\)
−0.660787 + 0.750573i \(0.729778\pi\)
\(678\) 0 0
\(679\) −498.494 −0.734159
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 235.114i 0.344238i 0.985076 + 0.172119i \(0.0550613\pi\)
−0.985076 + 0.172119i \(0.944939\pi\)
\(684\) 0 0
\(685\) −467.061 −0.681841
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 1747.59i − 2.53642i
\(690\) 0 0
\(691\) 50.9803 0.0737776 0.0368888 0.999319i \(-0.488255\pi\)
0.0368888 + 0.999319i \(0.488255\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1318.99i 1.89782i
\(696\) 0 0
\(697\) −660.996 −0.948344
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 141.530i 0.201898i 0.994892 + 0.100949i \(0.0321879\pi\)
−0.994892 + 0.100949i \(0.967812\pi\)
\(702\) 0 0
\(703\) −760.000 −1.08108
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 282.336i 0.399344i
\(708\) 0 0
\(709\) −55.4980 −0.0782765 −0.0391382 0.999234i \(-0.512461\pi\)
−0.0391382 + 0.999234i \(0.512461\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 305.470i − 0.428429i
\(714\) 0 0
\(715\) −1368.97 −1.91465
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1009.03i 1.40338i 0.712484 + 0.701688i \(0.247570\pi\)
−0.712484 + 0.701688i \(0.752430\pi\)
\(720\) 0 0
\(721\) −347.911 −0.482540
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 495.095i − 0.682890i
\(726\) 0 0
\(727\) 365.182 0.502313 0.251157 0.967946i \(-0.419189\pi\)
0.251157 + 0.967946i \(0.419189\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 910.251i 1.24521i
\(732\) 0 0
\(733\) −353.077 −0.481688 −0.240844 0.970564i \(-0.577424\pi\)
−0.240844 + 0.970564i \(0.577424\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1610.30i 2.18494i
\(738\) 0 0
\(739\) −329.684 −0.446121 −0.223061 0.974805i \(-0.571605\pi\)
−0.223061 + 0.974805i \(0.571605\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 112.061i 0.150822i 0.997153 + 0.0754112i \(0.0240270\pi\)
−0.997153 + 0.0754112i \(0.975973\pi\)
\(744\) 0 0
\(745\) 982.737 1.31911
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 217.955i − 0.290995i
\(750\) 0 0
\(751\) 144.826 0.192844 0.0964222 0.995341i \(-0.469260\pi\)
0.0964222 + 0.995341i \(0.469260\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 565.434i − 0.748919i
\(756\) 0 0
\(757\) 78.1699 0.103263 0.0516314 0.998666i \(-0.483558\pi\)
0.0516314 + 0.998666i \(0.483558\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1465.50i 1.92576i 0.269928 + 0.962880i \(0.413000\pi\)
−0.269928 + 0.962880i \(0.587000\pi\)
\(762\) 0 0
\(763\) 89.5059 0.117308
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1081.86i − 1.41050i
\(768\) 0 0
\(769\) 729.320 0.948401 0.474200 0.880417i \(-0.342737\pi\)
0.474200 + 0.880417i \(0.342737\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 434.559i − 0.562172i −0.959683 0.281086i \(-0.909305\pi\)
0.959683 0.281086i \(-0.0906947\pi\)
\(774\) 0 0
\(775\) 297.814 0.384277
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1212.67i 1.55671i
\(780\) 0 0
\(781\) −147.336 −0.188650
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1122.76i 1.43027i
\(786\) 0 0
\(787\) 15.3517 0.0195066 0.00975331 0.999952i \(-0.496895\pi\)
0.00975331 + 0.999952i \(0.496895\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 75.4543i − 0.0953910i
\(792\) 0 0
\(793\) −291.158 −0.367160
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1043.48i − 1.30927i −0.755947 0.654633i \(-0.772823\pi\)
0.755947 0.654633i \(-0.227177\pi\)
\(798\) 0 0
\(799\) 185.004 0.231544
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 933.610i − 1.16265i
\(804\) 0 0
\(805\) 194.907 0.242121
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1041.31i 1.28716i 0.765378 + 0.643581i \(0.222552\pi\)
−0.765378 + 0.643581i \(0.777448\pi\)
\(810\) 0 0
\(811\) −502.316 −0.619379 −0.309689 0.950838i \(-0.600225\pi\)
−0.309689 + 0.950838i \(0.600225\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 527.988i 0.647838i
\(816\) 0 0
\(817\) 1669.96 2.04402
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 23.1137i − 0.0281531i −0.999901 0.0140765i \(-0.995519\pi\)
0.999901 0.0140765i \(-0.00448085\pi\)
\(822\) 0 0
\(823\) 600.664 0.729847 0.364923 0.931038i \(-0.381095\pi\)
0.364923 + 0.931038i \(0.381095\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1309.21i 1.58308i 0.611118 + 0.791540i \(0.290720\pi\)
−0.611118 + 0.791540i \(0.709280\pi\)
\(828\) 0 0
\(829\) −621.919 −0.750204 −0.375102 0.926984i \(-0.622392\pi\)
−0.375102 + 0.926984i \(0.622392\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 76.3102i − 0.0916089i
\(834\) 0 0
\(835\) −367.644 −0.440293
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 1190.30i − 1.41871i −0.704851 0.709355i \(-0.748986\pi\)
0.704851 0.709355i \(-0.251014\pi\)
\(840\) 0 0
\(841\) −909.308 −1.08122
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 1070.15i − 1.26645i
\(846\) 0 0
\(847\) −69.6784 −0.0822649
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 461.252i 0.542011i
\(852\) 0 0
\(853\) 137.012 0.160623 0.0803117 0.996770i \(-0.474408\pi\)
0.0803117 + 0.996770i \(0.474408\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 466.141i 0.543922i 0.962308 + 0.271961i \(0.0876722\pi\)
−0.962308 + 0.271961i \(0.912328\pi\)
\(858\) 0 0
\(859\) −23.9843 −0.0279211 −0.0139606 0.999903i \(-0.504444\pi\)
−0.0139606 + 0.999903i \(0.504444\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 0.114603i 0 0.000132796i −1.00000 6.63982e-5i \(-0.999979\pi\)
1.00000 6.63982e-5i \(-2.11352e-5\pi\)
\(864\) 0 0
\(865\) 986.664 1.14065
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 408.669i − 0.470275i
\(870\) 0 0
\(871\) −2465.30 −2.83042
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 211.411i − 0.241613i
\(876\) 0 0
\(877\) 997.304 1.13718 0.568589 0.822622i \(-0.307490\pi\)
0.568589 + 0.822622i \(0.307490\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 935.649i 1.06203i 0.847362 + 0.531015i \(0.178189\pi\)
−0.847362 + 0.531015i \(0.821811\pi\)
\(882\) 0 0
\(883\) 1549.47 1.75478 0.877392 0.479774i \(-0.159281\pi\)
0.877392 + 0.479774i \(0.159281\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 894.493i 1.00845i 0.863573 + 0.504224i \(0.168221\pi\)
−0.863573 + 0.504224i \(0.831779\pi\)
\(888\) 0 0
\(889\) −343.069 −0.385905
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 339.411i − 0.380080i
\(894\) 0 0
\(895\) −1353.96 −1.51281
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 1052.86i − 1.17115i
\(900\) 0 0
\(901\) 1025.20 1.13785
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 1146.54i − 1.26690i
\(906\) 0 0
\(907\) 135.838 0.149766 0.0748831 0.997192i \(-0.476142\pi\)
0.0748831 + 0.997192i \(0.476142\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1242.01i 1.36335i 0.731655 + 0.681675i \(0.238748\pi\)
−0.731655 + 0.681675i \(0.761252\pi\)
\(912\) 0 0
\(913\) −735.289 −0.805355
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 391.617i 0.427063i
\(918\) 0 0
\(919\) 388.162 0.422374 0.211187 0.977446i \(-0.432267\pi\)
0.211187 + 0.977446i \(0.432267\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 225.564i − 0.244382i
\(924\) 0 0
\(925\) −449.692 −0.486153
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 621.694i − 0.669207i −0.942359 0.334604i \(-0.891398\pi\)
0.942359 0.334604i \(-0.108602\pi\)
\(930\) 0 0
\(931\) −140.000 −0.150376
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 803.089i − 0.858918i
\(936\) 0 0
\(937\) 1262.00 1.34685 0.673426 0.739255i \(-0.264822\pi\)
0.673426 + 0.739255i \(0.264822\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 672.410i 0.714569i 0.933996 + 0.357285i \(0.116297\pi\)
−0.933996 + 0.357285i \(0.883703\pi\)
\(942\) 0 0
\(943\) 735.984 0.780471
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1159.75i − 1.22465i −0.790605 0.612327i \(-0.790234\pi\)
0.790605 0.612327i \(-0.209766\pi\)
\(948\) 0 0
\(949\) 1429.31 1.50612
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 163.104i − 0.171148i −0.996332 0.0855740i \(-0.972728\pi\)
0.996332 0.0855740i \(-0.0272724\pi\)
\(954\) 0 0
\(955\) 1383.98 1.44920
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 203.610i − 0.212315i
\(960\) 0 0
\(961\) −327.672 −0.340970
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 813.260i − 0.842756i
\(966\) 0 0
\(967\) −887.012 −0.917282 −0.458641 0.888622i \(-0.651664\pi\)
−0.458641 + 0.888622i \(0.651664\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1416.32i 1.45862i 0.684183 + 0.729310i \(0.260159\pi\)
−0.684183 + 0.729310i \(0.739841\pi\)
\(972\) 0 0
\(973\) −574.996 −0.590952
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 339.051i 0.347032i 0.984831 + 0.173516i \(0.0555129\pi\)
−0.984831 + 0.173516i \(0.944487\pi\)
\(978\) 0 0
\(979\) 57.9606 0.0592039
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 487.887i 0.496324i 0.968718 + 0.248162i \(0.0798266\pi\)
−0.968718 + 0.248162i \(0.920173\pi\)
\(984\) 0 0
\(985\) −1144.39 −1.16182
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 1013.52i − 1.02479i
\(990\) 0 0
\(991\) 937.474 0.945988 0.472994 0.881066i \(-0.343173\pi\)
0.472994 + 0.881066i \(0.343173\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 622.047i 0.625173i
\(996\) 0 0
\(997\) 461.012 0.462399 0.231200 0.972906i \(-0.425735\pi\)
0.231200 + 0.972906i \(0.425735\pi\)
\(998\) 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 1008.3.d.a.449.2 4
3.2 odd 2 inner 1008.3.d.a.449.3 4
4.3 odd 2 126.3.b.a.71.1 4
8.3 odd 2 4032.3.d.i.449.3 4
8.5 even 2 4032.3.d.j.449.3 4
12.11 even 2 126.3.b.a.71.4 yes 4
20.3 even 4 3150.3.c.b.449.4 8
20.7 even 4 3150.3.c.b.449.6 8
20.19 odd 2 3150.3.e.e.701.4 4
24.5 odd 2 4032.3.d.j.449.2 4
24.11 even 2 4032.3.d.i.449.2 4
28.3 even 6 882.3.s.i.863.3 8
28.11 odd 6 882.3.s.e.863.4 8
28.19 even 6 882.3.s.i.557.2 8
28.23 odd 6 882.3.s.e.557.1 8
28.27 even 2 882.3.b.f.197.2 4
36.7 odd 6 1134.3.q.c.701.2 8
36.11 even 6 1134.3.q.c.701.3 8
36.23 even 6 1134.3.q.c.1079.2 8
36.31 odd 6 1134.3.q.c.1079.3 8
60.23 odd 4 3150.3.c.b.449.7 8
60.47 odd 4 3150.3.c.b.449.1 8
60.59 even 2 3150.3.e.e.701.2 4
84.11 even 6 882.3.s.e.863.1 8
84.23 even 6 882.3.s.e.557.4 8
84.47 odd 6 882.3.s.i.557.3 8
84.59 odd 6 882.3.s.i.863.2 8
84.83 odd 2 882.3.b.f.197.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.b.a.71.1 4 4.3 odd 2
126.3.b.a.71.4 yes 4 12.11 even 2
882.3.b.f.197.2 4 28.27 even 2
882.3.b.f.197.3 4 84.83 odd 2
882.3.s.e.557.1 8 28.23 odd 6
882.3.s.e.557.4 8 84.23 even 6
882.3.s.e.863.1 8 84.11 even 6
882.3.s.e.863.4 8 28.11 odd 6
882.3.s.i.557.2 8 28.19 even 6
882.3.s.i.557.3 8 84.47 odd 6
882.3.s.i.863.2 8 84.59 odd 6
882.3.s.i.863.3 8 28.3 even 6
1008.3.d.a.449.2 4 1.1 even 1 trivial
1008.3.d.a.449.3 4 3.2 odd 2 inner
1134.3.q.c.701.2 8 36.7 odd 6
1134.3.q.c.701.3 8 36.11 even 6
1134.3.q.c.1079.2 8 36.23 even 6
1134.3.q.c.1079.3 8 36.31 odd 6
3150.3.c.b.449.1 8 60.47 odd 4
3150.3.c.b.449.4 8 20.3 even 4
3150.3.c.b.449.6 8 20.7 even 4
3150.3.c.b.449.7 8 60.23 odd 4
3150.3.e.e.701.2 4 60.59 even 2
3150.3.e.e.701.4 4 20.19 odd 2
4032.3.d.i.449.2 4 24.11 even 2
4032.3.d.i.449.3 4 8.3 odd 2
4032.3.d.j.449.2 4 24.5 odd 2
4032.3.d.j.449.3 4 8.5 even 2