Properties

Label 4032.3.d.j.449.2
Level $4032$
Weight $3$
Character 4032.449
Analytic conductor $109.864$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 4032.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(109.864042590\)
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 \(\chi\) \(=\) 4032.449
Dual form 4032.3.d.j.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 \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(-0.246549\pi\)
0.714731 + 0.699399i \(0.246549\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 10.9015i 0.641262i 0.947204 + 0.320631i \(0.103895\pi\)
−0.947204 + 0.320631i \(0.896105\pi\)
\(18\) 0 0
\(19\) 20.0000 1.05263 0.526316 0.850289i \(-0.323573\pi\)
0.526316 + 0.850289i \(0.323573\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 12.1382i − 0.527748i −0.964557 0.263874i \(-0.915000\pi\)
0.964557 0.263874i \(-0.0850003\pi\)
\(24\) 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 \(-0.671656\pi\)
−0.513514 + 0.858082i \(0.671656\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 60.6337i 1.47887i 0.673227 + 0.739435i \(0.264908\pi\)
−0.673227 + 0.739435i \(0.735092\pi\)
\(42\) 0 0
\(43\) 83.4980 1.94181 0.970907 0.239455i \(-0.0769689\pi\)
0.970907 + 0.239455i \(0.0769689\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 16.9706i − 0.361076i −0.983568 0.180538i \(-0.942216\pi\)
0.983568 0.180538i \(-0.0577838\pi\)
\(48\) 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 \(-0.540992\pi\)
−0.128426 + 0.991719i \(0.540992\pi\)
\(62\) 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 \(-0.955015\pi\)
−0.990030 + 0.140856i \(0.955015\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 12.1382i − 0.170961i −0.996340 0.0854803i \(-0.972758\pi\)
0.996340 0.0854803i \(-0.0272425\pi\)
\(72\) 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 \(0.00854006\pi\)
−0.999640 + 0.0268262i \(0.991460\pi\)
\(90\) 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 \(-0.549597\pi\)
−0.155184 + 0.987886i \(0.549597\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 28.5190i 0.252381i 0.992006 + 0.126190i \(0.0402750\pi\)
−0.992006 + 0.126190i \(0.959725\pi\)
\(114\) 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 \(0.0906216\pi\)
−0.959747 + 0.280866i \(0.909378\pi\)
\(138\) 0 0
\(139\) 217.328 1.56351 0.781756 0.623585i \(-0.214324\pi\)
0.781756 + 0.623585i \(0.214324\pi\)
\(140\) 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 \(-0.299459\pi\)
0.589159 + 0.808017i \(0.299459\pi\)
\(158\) 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 \(-0.414014\pi\)
0.266859 + 0.963736i \(0.414014\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 60.5764i 0.362733i 0.983416 + 0.181366i \(0.0580520\pi\)
−0.983416 + 0.181366i \(0.941948\pi\)
\(168\) 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 \(-0.674763\pi\)
−0.521865 + 0.853028i \(0.674763\pi\)
\(182\) 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 \(-0.796376\pi\)
0.802273 0.596957i \(-0.203624\pi\)
\(192\) 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 \(-0.564173\pi\)
−0.200242 + 0.979747i \(0.564173\pi\)
\(212\) 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 \(-0.300250\pi\)
0.587151 + 0.809478i \(0.300250\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 26.2748i 0.112767i 0.998409 + 0.0563836i \(0.0179570\pi\)
−0.998409 + 0.0563836i \(0.982043\pi\)
\(234\) 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 \(0.0618348\pi\)
−0.981191 + 0.193040i \(0.938165\pi\)
\(240\) 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 \(0.165046\pi\)
−0.868560 + 0.495584i \(0.834954\pi\)
\(258\) 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 \(0.165954\pi\)
−0.867143 + 0.498059i \(0.834046\pi\)
\(264\) 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 \(-0.518396\pi\)
−0.0577617 + 0.998330i \(0.518396\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 166.757i − 0.593441i −0.954964 0.296721i \(-0.904107\pi\)
0.954964 0.296721i \(-0.0958930\pi\)
\(282\) 0 0
\(283\) 16.3399 0.0577381 0.0288691 0.999583i \(-0.490809\pi\)
0.0288691 + 0.999583i \(0.490809\pi\)
\(284\) 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 \(-0.601597\pi\)
−0.313785 + 0.949494i \(0.601597\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 131.276i 0.422109i 0.977474 + 0.211055i \(0.0676898\pi\)
−0.977474 + 0.211055i \(0.932310\pi\)
\(312\) 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 \(-0.316128\pi\)
0.546058 + 0.837748i \(0.316128\pi\)
\(332\) 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 \(-0.713778\pi\)
−0.622241 + 0.782826i \(0.713778\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 185.439i 0.525324i 0.964888 + 0.262662i \(0.0846005\pi\)
−0.964888 + 0.262662i \(0.915400\pi\)
\(354\) 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 \(-0.744265\pi\)
0.694254 0.719731i \(-0.255735\pi\)
\(360\) 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 \(-0.318513\pi\)
0.539764 + 0.841816i \(0.318513\pi\)
\(374\) 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 \(-0.323629\pi\)
0.526166 + 0.850382i \(0.323629\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 744.804i 1.94466i 0.233614 + 0.972329i \(0.424945\pi\)
−0.233614 + 0.972329i \(0.575055\pi\)
\(384\) 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 \(-0.462096\pi\)
0.118796 + 0.992919i \(0.462096\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 103.593i 0.258335i 0.991623 + 0.129168i \(0.0412306\pi\)
−0.991623 + 0.129168i \(0.958769\pi\)
\(402\) 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 \(-0.247889\pi\)
0.711782 + 0.702401i \(0.247889\pi\)
\(422\) 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 \(0.308712\pi\)
−0.565424 + 0.824800i \(0.691288\pi\)
\(432\) 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 \(0.146006\pi\)
−0.896633 + 0.442775i \(0.853994\pi\)
\(450\) 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 \(-0.737575\pi\)
0.678973 0.734163i \(-0.262425\pi\)
\(480\) 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 \(-0.227967\pi\)
0.754319 + 0.656508i \(0.227967\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 662.540i − 1.31718i −0.752504 0.658588i \(-0.771154\pi\)
0.752504 0.658588i \(-0.228846\pi\)
\(504\) 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 \(-0.759561\pi\)
0.728025 0.685551i \(-0.240439\pi\)
\(522\) 0 0
\(523\) −232.000 −0.443595 −0.221797 0.975093i \(-0.571192\pi\)
−0.221797 + 0.975093i \(0.571192\pi\)
\(524\) 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 \(-0.548930\pi\)
−0.153113 + 0.988209i \(0.548930\pi\)
\(542\) 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 \(-0.412887\pi\)
0.270270 + 0.962784i \(0.412887\pi\)
\(548\) 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 \(0.172359\pi\)
−0.856946 + 0.515406i \(0.827641\pi\)
\(570\) 0 0
\(571\) 951.644 1.66663 0.833314 0.552800i \(-0.186441\pi\)
0.833314 + 0.552800i \(0.186441\pi\)
\(572\) 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 \(-0.941361\pi\)
0.983079 0.183180i \(-0.0586390\pi\)
\(594\) 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 \(-0.954093\pi\)
0.989618 0.143722i \(-0.0459072\pi\)
\(600\) 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 \(-0.426835\pi\)
0.227837 + 0.973699i \(0.426835\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 358.380i − 0.580843i −0.956899 0.290422i \(-0.906204\pi\)
0.956899 0.290422i \(-0.0937955\pi\)
\(618\) 0 0
\(619\) −983.644 −1.58909 −0.794543 0.607208i \(-0.792290\pi\)
−0.794543 + 0.607208i \(0.792290\pi\)
\(620\) 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 \(0.0782418\pi\)
−0.969942 + 0.243336i \(0.921758\pi\)
\(642\) 0 0
\(643\) −604.000 −0.939347 −0.469673 0.882840i \(-0.655628\pi\)
−0.469673 + 0.882840i \(0.655628\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 179.600i 0.277588i 0.990321 + 0.138794i \(0.0443226\pi\)
−0.990321 + 0.138794i \(0.955677\pi\)
\(648\) 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 \(-0.370888\pi\)
0.394587 + 0.918858i \(0.370888\pi\)
\(662\) 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 \(-0.511745\pi\)
−0.0368888 + 0.999319i \(0.511745\pi\)
\(692\) 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 \(-0.487539\pi\)
0.0391382 + 0.999234i \(0.487539\pi\)
\(710\) 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 \(-0.752430\pi\)
0.712484 0.701688i \(-0.247570\pi\)
\(720\) 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 \(-0.422576\pi\)
0.240844 + 0.970564i \(0.422576\pi\)
\(734\) 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 \(-0.428395\pi\)
0.223061 + 0.974805i \(0.428395\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 112.061i − 0.150822i −0.997153 0.0754112i \(-0.975973\pi\)
0.997153 0.0754112i \(-0.0240270\pi\)
\(744\) 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 \(-0.516442\pi\)
−0.0516314 + 0.998666i \(0.516442\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 1465.50i − 1.92576i −0.269928 0.962880i \(-0.587000\pi\)
0.269928 0.962880i \(-0.413000\pi\)
\(762\) 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 \(-0.503105\pi\)
−0.00975331 + 0.999952i \(0.503105\pi\)
\(788\) 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 \(-0.777448\pi\)
0.765378 0.643581i \(-0.222552\pi\)
\(810\) 0 0
\(811\) 502.316 0.619379 0.309689 0.950838i \(-0.399775\pi\)
0.309689 + 0.950838i \(0.399775\pi\)
\(812\) 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 \(-0.377608\pi\)
0.375102 + 0.926984i \(0.377608\pi\)
\(830\) 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 \(0.251014\pi\)
−0.704851 + 0.709355i \(0.748986\pi\)
\(840\) 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 \(-0.525592\pi\)
−0.0803117 + 0.996770i \(0.525592\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 466.141i − 0.543922i −0.962308 0.271961i \(-0.912328\pi\)
0.962308 0.271961i \(-0.0876722\pi\)
\(858\) 0 0
\(859\) 23.9843 0.0279211 0.0139606 0.999903i \(-0.495556\pi\)
0.0139606 + 0.999903i \(0.495556\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0.114603i 0 0.000132796i 1.00000 6.63982e-5i \(2.11352e-5\pi\)
−1.00000 6.63982e-5i \(0.999979\pi\)
\(864\) 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 \(-0.692510\pi\)
−0.568589 + 0.822622i \(0.692510\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 935.649i − 1.06203i −0.847362 0.531015i \(-0.821811\pi\)
0.847362 0.531015i \(-0.178189\pi\)
\(882\) 0 0
\(883\) −1549.47 −1.75478 −0.877392 0.479774i \(-0.840719\pi\)
−0.877392 + 0.479774i \(0.840719\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 894.493i − 1.00845i −0.863573 0.504224i \(-0.831779\pi\)
0.863573 0.504224i \(-0.168221\pi\)
\(888\) 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 \(-0.523858\pi\)
−0.0748831 + 0.997192i \(0.523858\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 1242.01i − 1.36335i −0.731655 0.681675i \(-0.761252\pi\)
0.731655 0.681675i \(-0.238748\pi\)
\(912\) 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 \(0.108602\pi\)
−0.942359 + 0.334604i \(0.891398\pi\)
\(930\) 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 \(0.0272724\pi\)
−0.996332 + 0.0855740i \(0.972728\pi\)
\(954\) 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 \(-0.944487\pi\)
0.984831 0.173516i \(-0.0555129\pi\)
\(978\) 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 \(-0.920173\pi\)
0.968718 0.248162i \(-0.0798266\pi\)
\(984\) 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 \(-0.574265\pi\)
−0.231200 + 0.972906i \(0.574265\pi\)
\(998\) 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 4032.3.d.j.449.2 4
3.2 odd 2 inner 4032.3.d.j.449.3 4
4.3 odd 2 4032.3.d.i.449.2 4
8.3 odd 2 126.3.b.a.71.4 yes 4
8.5 even 2 1008.3.d.a.449.3 4
12.11 even 2 4032.3.d.i.449.3 4
24.5 odd 2 1008.3.d.a.449.2 4
24.11 even 2 126.3.b.a.71.1 4
40.3 even 4 3150.3.c.b.449.7 8
40.19 odd 2 3150.3.e.e.701.2 4
40.27 even 4 3150.3.c.b.449.1 8
56.3 even 6 882.3.s.i.863.2 8
56.11 odd 6 882.3.s.e.863.1 8
56.19 even 6 882.3.s.i.557.3 8
56.27 even 2 882.3.b.f.197.3 4
56.51 odd 6 882.3.s.e.557.4 8
72.11 even 6 1134.3.q.c.701.2 8
72.43 odd 6 1134.3.q.c.701.3 8
72.59 even 6 1134.3.q.c.1079.3 8
72.67 odd 6 1134.3.q.c.1079.2 8
120.59 even 2 3150.3.e.e.701.4 4
120.83 odd 4 3150.3.c.b.449.4 8
120.107 odd 4 3150.3.c.b.449.6 8
168.11 even 6 882.3.s.e.863.4 8
168.59 odd 6 882.3.s.i.863.3 8
168.83 odd 2 882.3.b.f.197.2 4
168.107 even 6 882.3.s.e.557.1 8
168.131 odd 6 882.3.s.i.557.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.b.a.71.1 4 24.11 even 2
126.3.b.a.71.4 yes 4 8.3 odd 2
882.3.b.f.197.2 4 168.83 odd 2
882.3.b.f.197.3 4 56.27 even 2
882.3.s.e.557.1 8 168.107 even 6
882.3.s.e.557.4 8 56.51 odd 6
882.3.s.e.863.1 8 56.11 odd 6
882.3.s.e.863.4 8 168.11 even 6
882.3.s.i.557.2 8 168.131 odd 6
882.3.s.i.557.3 8 56.19 even 6
882.3.s.i.863.2 8 56.3 even 6
882.3.s.i.863.3 8 168.59 odd 6
1008.3.d.a.449.2 4 24.5 odd 2
1008.3.d.a.449.3 4 8.5 even 2
1134.3.q.c.701.2 8 72.11 even 6
1134.3.q.c.701.3 8 72.43 odd 6
1134.3.q.c.1079.2 8 72.67 odd 6
1134.3.q.c.1079.3 8 72.59 even 6
3150.3.c.b.449.1 8 40.27 even 4
3150.3.c.b.449.4 8 120.83 odd 4
3150.3.c.b.449.6 8 120.107 odd 4
3150.3.c.b.449.7 8 40.3 even 4
3150.3.e.e.701.2 4 40.19 odd 2
3150.3.e.e.701.4 4 120.59 even 2
4032.3.d.i.449.2 4 4.3 odd 2
4032.3.d.i.449.3 4 12.11 even 2
4032.3.d.j.449.2 4 1.1 even 1 trivial
4032.3.d.j.449.3 4 3.2 odd 2 inner