Properties

Label 3150.3.c.b.449.7
Level $3150$
Weight $3$
Character 3150.449
Analytic conductor $85.831$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3150.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(85.8312832735\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
Defining polynomial: \(x^{8} + x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.7
Root \(-1.28897 - 0.581861i\) of defining polynomial
Character \(\chi\) \(=\) 3150.449
Dual form 3150.3.c.b.449.6

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} +2.82843 q^{8} +O(q^{10})\) \(q+1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} +2.82843 q^{8} +12.1382i q^{11} -18.5830i q^{13} +3.74166i q^{14} +4.00000 q^{16} +10.9015 q^{17} -20.0000 q^{19} +17.1660i q^{22} -12.1382 q^{23} -26.2803i q^{26} +5.29150i q^{28} +41.8367i q^{29} +25.1660 q^{31} +5.65685 q^{32} +15.4170 q^{34} -38.0000i q^{37} -28.2843 q^{38} +60.6337i q^{41} +83.4980i q^{43} +24.2764i q^{44} -17.1660 q^{46} +16.9706 q^{47} -7.00000 q^{49} -37.1660i q^{52} +94.0424 q^{53} +7.48331i q^{56} +59.1660i q^{58} -58.2175i q^{59} +15.6680 q^{61} +35.5901 q^{62} +8.00000 q^{64} +132.664i q^{67} +21.8029 q^{68} +12.1382i q^{71} -76.9150i q^{73} -53.7401i q^{74} -40.0000 q^{76} -32.1147 q^{77} -33.6680 q^{79} +85.7490i q^{82} -60.5764 q^{83} +118.084i q^{86} +34.3320i q^{88} -4.77506i q^{89} +49.1660 q^{91} -24.2764 q^{92} +24.0000 q^{94} +188.413i q^{97} -9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4} + O(q^{10}) \) \( 8 q + 16 q^{4} + 32 q^{16} - 160 q^{19} + 32 q^{31} + 208 q^{34} + 32 q^{46} - 56 q^{49} + 464 q^{61} + 64 q^{64} - 320 q^{76} - 608 q^{79} + 224 q^{91} + 192 q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 0.707107
\(3\) 0 0
\(4\) 2.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.64575i 0.377964i
\(8\) 2.82843 0.353553
\(9\) 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
\(13\) − 18.5830i − 1.42946i −0.699399 0.714731i \(-0.746549\pi\)
0.699399 0.714731i \(-0.253451\pi\)
\(14\) 3.74166i 0.267261i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 10.9015 0.641262 0.320631 0.947204i \(-0.396105\pi\)
0.320631 + 0.947204i \(0.396105\pi\)
\(18\) 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
\(22\) 17.1660i 0.780273i
\(23\) −12.1382 −0.527748 −0.263874 0.964557i \(-0.585000\pi\)
−0.263874 + 0.964557i \(0.585000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) − 26.2803i − 1.01078i
\(27\) 0 0
\(28\) 5.29150i 0.188982i
\(29\) 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 \(-0.366957\pi\)
0.405903 + 0.913916i \(0.366957\pi\)
\(32\) 5.65685 0.176777
\(33\) 0 0
\(34\) 15.4170 0.453441
\(35\) 0 0
\(36\) 0 0
\(37\) − 38.0000i − 1.02703i −0.858082 0.513514i \(-0.828344\pi\)
0.858082 0.513514i \(-0.171656\pi\)
\(38\) −28.2843 −0.744323
\(39\) 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
\(43\) 83.4980i 1.94181i 0.239455 + 0.970907i \(0.423031\pi\)
−0.239455 + 0.970907i \(0.576969\pi\)
\(44\) 24.2764i 0.551736i
\(45\) 0 0
\(46\) −17.1660 −0.373174
\(47\) 16.9706 0.361076 0.180538 0.983568i \(-0.442216\pi\)
0.180538 + 0.983568i \(0.442216\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) − 37.1660i − 0.714731i
\(53\) 94.0424 1.77439 0.887193 0.461399i \(-0.152652\pi\)
0.887193 + 0.461399i \(0.152652\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 7.48331i 0.133631i
\(57\) 0 0
\(58\) 59.1660i 1.02010i
\(59\) − 58.2175i − 0.986738i −0.869820 0.493369i \(-0.835765\pi\)
0.869820 0.493369i \(-0.164235\pi\)
\(60\) 0 0
\(61\) 15.6680 0.256852 0.128426 0.991719i \(-0.459008\pi\)
0.128426 + 0.991719i \(0.459008\pi\)
\(62\) 35.5901 0.574034
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 132.664i 1.98006i 0.140856 + 0.990030i \(0.455015\pi\)
−0.140856 + 0.990030i \(0.544985\pi\)
\(68\) 21.8029 0.320631
\(69\) 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
\(73\) − 76.9150i − 1.05363i −0.849980 0.526815i \(-0.823386\pi\)
0.849980 0.526815i \(-0.176614\pi\)
\(74\) − 53.7401i − 0.726218i
\(75\) 0 0
\(76\) −40.0000 −0.526316
\(77\) −32.1147 −0.417074
\(78\) 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
\(82\) 85.7490i 1.04572i
\(83\) −60.5764 −0.729836 −0.364918 0.931040i \(-0.618903\pi\)
−0.364918 + 0.931040i \(0.618903\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 118.084i 1.37307i
\(87\) 0 0
\(88\) 34.3320i 0.390137i
\(89\) − 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
\(92\) −24.2764 −0.263874
\(93\) 0 0
\(94\) 24.0000 0.255319
\(95\) 0 0
\(96\) 0 0
\(97\) 188.413i 1.94240i 0.238260 + 0.971201i \(0.423423\pi\)
−0.238260 + 0.971201i \(0.576577\pi\)
\(98\) −9.89949 −0.101015
\(99\) 0 0
\(100\) 0 0
\(101\) − 106.713i − 1.05656i −0.849069 0.528282i \(-0.822836\pi\)
0.849069 0.528282i \(-0.177164\pi\)
\(102\) 0 0
\(103\) 131.498i 1.27668i 0.769755 + 0.638340i \(0.220379\pi\)
−0.769755 + 0.638340i \(0.779621\pi\)
\(104\) − 52.5607i − 0.505391i
\(105\) 0 0
\(106\) 132.996 1.25468
\(107\) −82.3793 −0.769900 −0.384950 0.922937i \(-0.625781\pi\)
−0.384950 + 0.922937i \(0.625781\pi\)
\(108\) 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
\(112\) 10.5830i 0.0944911i
\(113\) −28.5190 −0.252381 −0.126190 0.992006i \(-0.540275\pi\)
−0.126190 + 0.992006i \(0.540275\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 83.6734i 0.721322i
\(117\) 0 0
\(118\) − 82.3320i − 0.697729i
\(119\) 28.8426i 0.242374i
\(120\) 0 0
\(121\) −26.3360 −0.217653
\(122\) 22.1579 0.181622
\(123\) 0 0
\(124\) 50.3320 0.405903
\(125\) 0 0
\(126\) 0 0
\(127\) − 129.668i − 1.02101i −0.859875 0.510504i \(-0.829459\pi\)
0.859875 0.510504i \(-0.170541\pi\)
\(128\) 11.3137 0.0883883
\(129\) 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
\(133\) − 52.9150i − 0.397857i
\(134\) 187.615i 1.40011i
\(135\) 0 0
\(136\) 30.8340 0.226721
\(137\) 76.9573 0.561732 0.280866 0.959747i \(-0.409378\pi\)
0.280866 + 0.959747i \(0.409378\pi\)
\(138\) 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
\(142\) 17.1660i 0.120887i
\(143\) 225.564 1.57737
\(144\) 0 0
\(145\) 0 0
\(146\) − 108.774i − 0.745029i
\(147\) 0 0
\(148\) − 76.0000i − 0.513514i
\(149\) 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 \(-0.599826\pi\)
−0.308497 + 0.951225i \(0.599826\pi\)
\(152\) −56.5685 −0.372161
\(153\) 0 0
\(154\) −45.4170 −0.294916
\(155\) 0 0
\(156\) 0 0
\(157\) 184.996i 1.17832i 0.808017 + 0.589159i \(0.200541\pi\)
−0.808017 + 0.589159i \(0.799459\pi\)
\(158\) −47.6137 −0.301353
\(159\) 0 0
\(160\) 0 0
\(161\) − 32.1147i − 0.199470i
\(162\) 0 0
\(163\) 86.9961i 0.533718i 0.963736 + 0.266859i \(0.0859858\pi\)
−0.963736 + 0.266859i \(0.914014\pi\)
\(164\) 121.267i 0.739435i
\(165\) 0 0
\(166\) −85.6680 −0.516072
\(167\) −60.5764 −0.362733 −0.181366 0.983416i \(-0.558052\pi\)
−0.181366 + 0.983416i \(0.558052\pi\)
\(168\) 0 0
\(169\) −176.328 −1.04336
\(170\) 0 0
\(171\) 0 0
\(172\) 166.996i 0.970907i
\(173\) 162.572 0.939721 0.469860 0.882741i \(-0.344304\pi\)
0.469860 + 0.882741i \(0.344304\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 48.5528i 0.275868i
\(177\) 0 0
\(178\) − 6.75295i − 0.0379379i
\(179\) 223.091i 1.24632i 0.782095 + 0.623159i \(0.214151\pi\)
−0.782095 + 0.623159i \(0.785849\pi\)
\(180\) 0 0
\(181\) 188.915 1.04373 0.521865 0.853028i \(-0.325237\pi\)
0.521865 + 0.853028i \(0.325237\pi\)
\(182\) 69.5312 0.382040
\(183\) 0 0
\(184\) −34.3320 −0.186587
\(185\) 0 0
\(186\) 0 0
\(187\) 132.324i 0.707616i
\(188\) 33.9411 0.180538
\(189\) 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
\(193\) 134.000i 0.694301i 0.937810 + 0.347150i \(0.112851\pi\)
−0.937810 + 0.347150i \(0.887149\pi\)
\(194\) 266.456i 1.37349i
\(195\) 0 0
\(196\) −14.0000 −0.0714286
\(197\) 188.560 0.957157 0.478579 0.878045i \(-0.341152\pi\)
0.478579 + 0.878045i \(0.341152\pi\)
\(198\) 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
\(202\) − 150.915i − 0.747104i
\(203\) −110.689 −0.545268
\(204\) 0 0
\(205\) 0 0
\(206\) 185.966i 0.902749i
\(207\) 0 0
\(208\) − 74.3320i − 0.357365i
\(209\) − 242.764i − 1.16155i
\(210\) 0 0
\(211\) −84.5020 −0.400483 −0.200242 0.979747i \(-0.564173\pi\)
−0.200242 + 0.979747i \(0.564173\pi\)
\(212\) 188.085 0.887193
\(213\) 0 0
\(214\) −116.502 −0.544402
\(215\) 0 0
\(216\) 0 0
\(217\) 66.5830i 0.306834i
\(218\) −47.8429 −0.219463
\(219\) 0 0
\(220\) 0 0
\(221\) − 202.582i − 0.916660i
\(222\) 0 0
\(223\) − 158.494i − 0.710736i −0.934727 0.355368i \(-0.884356\pi\)
0.934727 0.355368i \(-0.115644\pi\)
\(224\) 14.9666i 0.0668153i
\(225\) 0 0
\(226\) −40.3320 −0.178460
\(227\) −101.823 −0.448561 −0.224281 0.974525i \(-0.572003\pi\)
−0.224281 + 0.974525i \(0.572003\pi\)
\(228\) 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
\(232\) 118.332i 0.510052i
\(233\) −26.2748 −0.112767 −0.0563836 0.998409i \(-0.517957\pi\)
−0.0563836 + 0.998409i \(0.517957\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) − 116.435i − 0.493369i
\(237\) 0 0
\(238\) 40.7895i 0.171385i
\(239\) 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 \(0.247730\pi\)
\(242\) −37.2447 −0.153904
\(243\) 0 0
\(244\) 31.3360 0.128426
\(245\) 0 0
\(246\) 0 0
\(247\) 371.660i 1.50470i
\(248\) 71.1802 0.287017
\(249\) 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
\(253\) − 147.336i − 0.582356i
\(254\) − 183.378i − 0.721961i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 254.730 0.991169 0.495584 0.868560i \(-0.334954\pi\)
0.495584 + 0.868560i \(0.334954\pi\)
\(258\) 0 0
\(259\) 100.539 0.388180
\(260\) 0 0
\(261\) 0 0
\(262\) 209.328i 0.798962i
\(263\) 261.979 0.996117 0.498059 0.867143i \(-0.334046\pi\)
0.498059 + 0.867143i \(0.334046\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) − 74.8331i − 0.281328i
\(267\) 0 0
\(268\) 265.328i 0.990030i
\(269\) 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 \(-0.499315\pi\)
0.00215131 + 0.999998i \(0.499315\pi\)
\(272\) 43.6058 0.160316
\(273\) 0 0
\(274\) 108.834 0.397204
\(275\) 0 0
\(276\) 0 0
\(277\) − 32.0000i − 0.115523i −0.998330 0.0577617i \(-0.981604\pi\)
0.998330 0.0577617i \(-0.0183964\pi\)
\(278\) −307.348 −1.10557
\(279\) 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
\(283\) 16.3399i 0.0577381i 0.999583 + 0.0288691i \(0.00919059\pi\)
−0.999583 + 0.0288691i \(0.990809\pi\)
\(284\) 24.2764i 0.0854803i
\(285\) 0 0
\(286\) 318.996 1.11537
\(287\) −160.422 −0.558961
\(288\) 0 0
\(289\) −170.158 −0.588782
\(290\) 0 0
\(291\) 0 0
\(292\) − 153.830i − 0.526815i
\(293\) 368.921 1.25912 0.629558 0.776953i \(-0.283236\pi\)
0.629558 + 0.776953i \(0.283236\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) − 107.480i − 0.363109i
\(297\) 0 0
\(298\) 228.996i 0.768443i
\(299\) 225.564i 0.754396i
\(300\) 0 0
\(301\) −220.915 −0.733937
\(302\) −131.757 −0.436280
\(303\) 0 0
\(304\) −80.0000 −0.263158
\(305\) 0 0
\(306\) 0 0
\(307\) 192.664i 0.627570i 0.949494 + 0.313785i \(0.101597\pi\)
−0.949494 + 0.313785i \(0.898403\pi\)
\(308\) −64.2293 −0.208537
\(309\) 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
\(313\) 43.3281i 0.138428i 0.997602 + 0.0692142i \(0.0220492\pi\)
−0.997602 + 0.0692142i \(0.977951\pi\)
\(314\) 261.624i 0.833197i
\(315\) 0 0
\(316\) −67.3360 −0.213088
\(317\) 251.724 0.794083 0.397042 0.917801i \(-0.370037\pi\)
0.397042 + 0.917801i \(0.370037\pi\)
\(318\) 0 0
\(319\) −507.822 −1.59192
\(320\) 0 0
\(321\) 0 0
\(322\) − 45.4170i − 0.141047i
\(323\) −218.029 −0.675013
\(324\) 0 0
\(325\) 0 0
\(326\) 123.031i 0.377396i
\(327\) 0 0
\(328\) 171.498i 0.522860i
\(329\) 44.8999i 0.136474i
\(330\) 0 0
\(331\) 361.490 1.09212 0.546058 0.837748i \(-0.316128\pi\)
0.546058 + 0.837748i \(0.316128\pi\)
\(332\) −121.153 −0.364918
\(333\) 0 0
\(334\) −85.6680 −0.256491
\(335\) 0 0
\(336\) 0 0
\(337\) 298.834i 0.886748i 0.896337 + 0.443374i \(0.146219\pi\)
−0.896337 + 0.443374i \(0.853781\pi\)
\(338\) −249.366 −0.737768
\(339\) 0 0
\(340\) 0 0
\(341\) 305.470i 0.895807i
\(342\) 0 0
\(343\) − 18.5203i − 0.0539949i
\(344\) 236.168i 0.686535i
\(345\) 0 0
\(346\) 229.911 0.664483
\(347\) −206.120 −0.594006 −0.297003 0.954876i \(-0.595987\pi\)
−0.297003 + 0.954876i \(0.595987\pi\)
\(348\) 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
\(352\) 68.6640i 0.195068i
\(353\) −185.439 −0.525324 −0.262662 0.964888i \(-0.584600\pi\)
−0.262662 + 0.964888i \(0.584600\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) − 9.55012i − 0.0268262i
\(357\) 0 0
\(358\) 315.498i 0.881279i
\(359\) − 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
\(362\) 267.166 0.738028
\(363\) 0 0
\(364\) 98.3320 0.270143
\(365\) 0 0
\(366\) 0 0
\(367\) 117.490i 0.320137i 0.987106 + 0.160068i \(0.0511715\pi\)
−0.987106 + 0.160068i \(0.948829\pi\)
\(368\) −48.5528 −0.131937
\(369\) 0 0
\(370\) 0 0
\(371\) 248.813i 0.670655i
\(372\) 0 0
\(373\) − 402.664i − 1.07953i −0.841816 0.539764i \(-0.818513\pi\)
0.841816 0.539764i \(-0.181487\pi\)
\(374\) 187.135i 0.500360i
\(375\) 0 0
\(376\) 48.0000 0.127660
\(377\) 777.451 2.06221
\(378\) 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
\(382\) 322.494i 0.844225i
\(383\) 744.804 1.94466 0.972329 0.233614i \(-0.0750553\pi\)
0.972329 + 0.233614i \(0.0750553\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 189.505i 0.490945i
\(387\) 0 0
\(388\) 376.826i 0.971201i
\(389\) − 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
\(392\) −19.7990 −0.0505076
\(393\) 0 0
\(394\) 266.664 0.676812
\(395\) 0 0
\(396\) 0 0
\(397\) 94.3241i 0.237592i 0.992919 + 0.118796i \(0.0379035\pi\)
−0.992919 + 0.118796i \(0.962096\pi\)
\(398\) −144.949 −0.364192
\(399\) 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
\(403\) − 467.660i − 1.16045i
\(404\) − 213.426i − 0.528282i
\(405\) 0 0
\(406\) −156.539 −0.385563
\(407\) 461.252 1.13330
\(408\) 0 0
\(409\) 9.75689 0.0238555 0.0119277 0.999929i \(-0.496203\pi\)
0.0119277 + 0.999929i \(0.496203\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 262.996i 0.638340i
\(413\) 154.029 0.372952
\(414\) 0 0
\(415\) 0 0
\(416\) − 105.121i − 0.252696i
\(417\) 0 0
\(418\) − 343.320i − 0.821340i
\(419\) 339.411i 0.810051i 0.914305 + 0.405025i \(0.132737\pi\)
−0.914305 + 0.405025i \(0.867263\pi\)
\(420\) 0 0
\(421\) −599.320 −1.42356 −0.711782 0.702401i \(-0.752111\pi\)
−0.711782 + 0.702401i \(0.752111\pi\)
\(422\) −119.504 −0.283184
\(423\) 0 0
\(424\) 265.992 0.627340
\(425\) 0 0
\(426\) 0 0
\(427\) 41.4536i 0.0970810i
\(428\) −164.759 −0.384950
\(429\) 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
\(433\) 377.984i 0.872943i 0.899718 + 0.436471i \(0.143772\pi\)
−0.899718 + 0.436471i \(0.856228\pi\)
\(434\) 94.1626i 0.216964i
\(435\) 0 0
\(436\) −67.6601 −0.155184
\(437\) 242.764 0.555524
\(438\) 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
\(442\) − 286.494i − 0.648177i
\(443\) 36.6438 0.0827174 0.0413587 0.999144i \(-0.486831\pi\)
0.0413587 + 0.999144i \(0.486831\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) − 224.144i − 0.502566i
\(447\) 0 0
\(448\) 21.1660i 0.0472456i
\(449\) − 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
\(452\) −57.0381 −0.126190
\(453\) 0 0
\(454\) −144.000 −0.317181
\(455\) 0 0
\(456\) 0 0
\(457\) − 344.324i − 0.753445i −0.926326 0.376722i \(-0.877051\pi\)
0.926326 0.376722i \(-0.122949\pi\)
\(458\) 380.303 0.830357
\(459\) 0 0
\(460\) 0 0
\(461\) 370.936i 0.804634i 0.915500 + 0.402317i \(0.131795\pi\)
−0.915500 + 0.402317i \(0.868205\pi\)
\(462\) 0 0
\(463\) 78.3320i 0.169184i 0.996416 + 0.0845918i \(0.0269586\pi\)
−0.996416 + 0.0845918i \(0.973041\pi\)
\(464\) 167.347i 0.360661i
\(465\) 0 0
\(466\) −37.1581 −0.0797385
\(467\) 399.758 0.856014 0.428007 0.903775i \(-0.359216\pi\)
0.428007 + 0.903775i \(0.359216\pi\)
\(468\) 0 0
\(469\) −350.996 −0.748392
\(470\) 0 0
\(471\) 0 0
\(472\) − 164.664i − 0.348864i
\(473\) −1013.52 −2.14274
\(474\) 0 0
\(475\) 0 0
\(476\) 57.6851i 0.121187i
\(477\) 0 0
\(478\) 130.494i 0.273000i
\(479\) − 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
\(482\) 485.425 1.00711
\(483\) 0 0
\(484\) −52.6719 −0.108826
\(485\) 0 0
\(486\) 0 0
\(487\) 82.5098i 0.169425i 0.996405 + 0.0847124i \(0.0269971\pi\)
−0.996405 + 0.0847124i \(0.973003\pi\)
\(488\) 44.3157 0.0908109
\(489\) 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
\(493\) 456.081i 0.925114i
\(494\) 525.607i 1.06398i
\(495\) 0 0
\(496\) 100.664 0.202952
\(497\) −32.1147 −0.0646170
\(498\) 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
\(502\) − 504.000i − 1.00398i
\(503\) −662.540 −1.31718 −0.658588 0.752504i \(-0.728846\pi\)
−0.658588 + 0.752504i \(0.728846\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) − 208.365i − 0.411788i
\(507\) 0 0
\(508\) − 259.336i − 0.510504i
\(509\) 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
\(512\) 22.6274 0.0441942
\(513\) 0 0
\(514\) 360.243 0.700862
\(515\) 0 0
\(516\) 0 0
\(517\) 205.992i 0.398437i
\(518\) 142.183 0.274485
\(519\) 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
\(523\) − 232.000i − 0.443595i −0.975093 0.221797i \(-0.928808\pi\)
0.975093 0.221797i \(-0.0711923\pi\)
\(524\) 296.035i 0.564952i
\(525\) 0 0
\(526\) 370.494 0.704361
\(527\) 274.346 0.520581
\(528\) 0 0
\(529\) −381.664 −0.721482
\(530\) 0 0
\(531\) 0 0
\(532\) − 105.830i − 0.198929i
\(533\) 1126.76 2.11399
\(534\) 0 0
\(535\) 0 0
\(536\) 375.231i 0.700057i
\(537\) 0 0
\(538\) 132.405i 0.246106i
\(539\) − 84.9674i − 0.157639i
\(540\) 0 0
\(541\) 165.668 0.306225 0.153113 0.988209i \(-0.451070\pi\)
0.153113 + 0.988209i \(0.451070\pi\)
\(542\) 1.64899 0.00304241
\(543\) 0 0
\(544\) 61.6680 0.113360
\(545\) 0 0
\(546\) 0 0
\(547\) − 295.676i − 0.540541i −0.962784 0.270270i \(-0.912887\pi\)
0.962784 0.270270i \(-0.0871131\pi\)
\(548\) 153.915 0.280866
\(549\) 0 0
\(550\) 0 0
\(551\) − 836.734i − 1.51857i
\(552\) 0 0
\(553\) − 89.0771i − 0.161080i
\(554\) − 45.2548i − 0.0816874i
\(555\) 0 0
\(556\) −434.656 −0.781756
\(557\) −76.8426 −0.137958 −0.0689790 0.997618i \(-0.521974\pi\)
−0.0689790 + 0.997618i \(0.521974\pi\)
\(558\) 0 0
\(559\) 1551.64 2.77575
\(560\) 0 0
\(561\) 0 0
\(562\) − 235.830i − 0.419626i
\(563\) 1016.33 1.80521 0.902605 0.430470i \(-0.141652\pi\)
0.902605 + 0.430470i \(0.141652\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 23.1081i 0.0408270i
\(567\) 0 0
\(568\) 34.3320i 0.0604437i
\(569\) − 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 \(-0.186441\pi\)
0.833314 + 0.552800i \(0.186441\pi\)
\(572\) 451.129 0.788686
\(573\) 0 0
\(574\) −226.871 −0.395245
\(575\) 0 0
\(576\) 0 0
\(577\) 148.672i 0.257664i 0.991666 + 0.128832i \(0.0411227\pi\)
−0.991666 + 0.128832i \(0.958877\pi\)
\(578\) −240.640 −0.416332
\(579\) 0 0
\(580\) 0 0
\(581\) − 160.270i − 0.275852i
\(582\) 0 0
\(583\) 1141.51i 1.95799i
\(584\) − 217.549i − 0.372515i
\(585\) 0 0
\(586\) 521.733 0.890330
\(587\) −332.564 −0.566548 −0.283274 0.959039i \(-0.591421\pi\)
−0.283274 + 0.959039i \(0.591421\pi\)
\(588\) 0 0
\(589\) −503.320 −0.854533
\(590\) 0 0
\(591\) 0 0
\(592\) − 152.000i − 0.256757i
\(593\) 217.251 0.366359 0.183180 0.983079i \(-0.441361\pi\)
0.183180 + 0.983079i \(0.441361\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 323.849i 0.543371i
\(597\) 0 0
\(598\) 318.996i 0.533438i
\(599\) − 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 \(0.613055\pi\)
\(602\) −312.421 −0.518972
\(603\) 0 0
\(604\) −186.332 −0.308497
\(605\) 0 0
\(606\) 0 0
\(607\) − 627.158i − 1.03321i −0.856224 0.516605i \(-0.827196\pi\)
0.856224 0.516605i \(-0.172804\pi\)
\(608\) −113.137 −0.186081
\(609\) 0 0
\(610\) 0 0
\(611\) − 315.364i − 0.516144i
\(612\) 0 0
\(613\) − 279.328i − 0.455674i −0.973699 0.227837i \(-0.926835\pi\)
0.973699 0.227837i \(-0.0731653\pi\)
\(614\) 272.468i 0.443759i
\(615\) 0 0
\(616\) −90.8340 −0.147458
\(617\) −358.380 −0.580843 −0.290422 0.956899i \(-0.593796\pi\)
−0.290422 + 0.956899i \(0.593796\pi\)
\(618\) 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
\(622\) − 185.652i − 0.298476i
\(623\) 12.6336 0.0202787
\(624\) 0 0
\(625\) 0 0
\(626\) 61.2752i 0.0978836i
\(627\) 0 0
\(628\) 369.992i 0.589159i
\(629\) − 414.256i − 0.658594i
\(630\) 0 0
\(631\) −298.996 −0.473845 −0.236922 0.971529i \(-0.576139\pi\)
−0.236922 + 0.971529i \(0.576139\pi\)
\(632\) −95.2274 −0.150676
\(633\) 0 0
\(634\) 355.992 0.561502
\(635\) 0 0
\(636\) 0 0
\(637\) 130.081i 0.204209i
\(638\) −718.169 −1.12566
\(639\) 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
\(643\) − 604.000i − 0.939347i −0.882840 0.469673i \(-0.844372\pi\)
0.882840 0.469673i \(-0.155628\pi\)
\(644\) − 64.2293i − 0.0997350i
\(645\) 0 0
\(646\) −308.340 −0.477306
\(647\) −179.600 −0.277588 −0.138794 0.990321i \(-0.544323\pi\)
−0.138794 + 0.990321i \(0.544323\pi\)
\(648\) 0 0
\(649\) 706.656 1.08884
\(650\) 0 0
\(651\) 0 0
\(652\) 173.992i 0.266859i
\(653\) 481.892 0.737966 0.368983 0.929436i \(-0.379706\pi\)
0.368983 + 0.929436i \(0.379706\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 242.535i 0.369718i
\(657\) 0 0
\(658\) 63.4980i 0.0965016i
\(659\) 877.408i 1.33142i 0.746209 + 0.665711i \(0.231872\pi\)
−0.746209 + 0.665711i \(0.768128\pi\)
\(660\) 0 0
\(661\) −521.644 −0.789175 −0.394587 0.918858i \(-0.629112\pi\)
−0.394587 + 0.918858i \(0.629112\pi\)
\(662\) 511.224 0.772242
\(663\) 0 0
\(664\) −171.336 −0.258036
\(665\) 0 0
\(666\) 0 0
\(667\) − 507.822i − 0.761353i
\(668\) −121.153 −0.181366
\(669\) 0 0
\(670\) 0 0
\(671\) 190.181i 0.283429i
\(672\) 0 0
\(673\) − 659.992i − 0.980672i −0.871534 0.490336i \(-0.836874\pi\)
0.871534 0.490336i \(-0.163126\pi\)
\(674\) 422.615i 0.627025i
\(675\) 0 0
\(676\) −352.656 −0.521681
\(677\) −1016.28 −1.50115 −0.750573 0.660787i \(-0.770222\pi\)
−0.750573 + 0.660787i \(0.770222\pi\)
\(678\) 0 0
\(679\) −498.494 −0.734159
\(680\) 0 0
\(681\) 0 0
\(682\) 432.000i 0.633431i
\(683\) −235.114 −0.344238 −0.172119 0.985076i \(-0.555061\pi\)
−0.172119 + 0.985076i \(0.555061\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) − 26.1916i − 0.0381802i
\(687\) 0 0
\(688\) 333.992i 0.485454i
\(689\) − 1747.59i − 2.53642i
\(690\) 0 0
\(691\) −50.9803 −0.0737776 −0.0368888 0.999319i \(-0.511745\pi\)
−0.0368888 + 0.999319i \(0.511745\pi\)
\(692\) 325.143 0.469860
\(693\) 0 0
\(694\) −291.498 −0.420026
\(695\) 0 0
\(696\) 0 0
\(697\) 660.996i 0.948344i
\(698\) −614.227 −0.879982
\(699\) 0 0
\(700\) 0 0
\(701\) − 141.530i − 0.201898i −0.994892 0.100949i \(-0.967812\pi\)
0.994892 0.100949i \(-0.0321879\pi\)
\(702\) 0 0
\(703\) 760.000i 1.08108i
\(704\) 97.1056i 0.137934i
\(705\) 0 0
\(706\) −262.251 −0.371460
\(707\) 282.336 0.399344
\(708\) 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
\(712\) − 13.5059i − 0.0189690i
\(713\) −305.470 −0.428429
\(714\) 0 0
\(715\) 0 0
\(716\) 446.182i 0.623159i
\(717\) 0 0
\(718\) − 730.818i − 1.01785i
\(719\) − 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
\(722\) 55.1543 0.0763910
\(723\) 0 0
\(724\) 377.830 0.521865
\(725\) 0 0
\(726\) 0 0
\(727\) 365.182i 0.502313i 0.967946 + 0.251157i \(0.0808109\pi\)
−0.967946 + 0.251157i \(0.919189\pi\)
\(728\) 139.062 0.191020
\(729\) 0 0
\(730\) 0 0
\(731\) 910.251i 1.24521i
\(732\) 0 0
\(733\) − 353.077i − 0.481688i −0.970564 0.240844i \(-0.922576\pi\)
0.970564 0.240844i \(-0.0774242\pi\)
\(734\) 166.156i 0.226371i
\(735\) 0 0
\(736\) −68.6640 −0.0932935
\(737\) −1610.30 −2.18494
\(738\) 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
\(742\) 351.875i 0.474224i
\(743\) −112.061 −0.150822 −0.0754112 0.997153i \(-0.524027\pi\)
−0.0754112 + 0.997153i \(0.524027\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) − 569.453i − 0.763342i
\(747\) 0 0
\(748\) 264.648i 0.353808i
\(749\) − 217.955i − 0.290995i
\(750\) 0 0
\(751\) −144.826 −0.192844 −0.0964222 0.995341i \(-0.530740\pi\)
−0.0964222 + 0.995341i \(0.530740\pi\)
\(752\) 67.8823 0.0902690
\(753\) 0 0
\(754\) 1099.48 1.45820
\(755\) 0 0
\(756\) 0 0
\(757\) − 78.1699i − 0.103263i −0.998666 0.0516314i \(-0.983558\pi\)
0.998666 0.0516314i \(-0.0164421\pi\)
\(758\) −564.036 −0.744111
\(759\) 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
\(763\) − 89.5059i − 0.117308i
\(764\) 456.076i 0.596957i
\(765\) 0 0
\(766\) 1053.31 1.37508
\(767\) −1081.86 −1.41050
\(768\) 0 0
\(769\) −729.320 −0.948401 −0.474200 0.880417i \(-0.657263\pi\)
−0.474200 + 0.880417i \(0.657263\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 268.000i 0.347150i
\(773\) −434.559 −0.562172 −0.281086 0.959683i \(-0.590695\pi\)
−0.281086 + 0.959683i \(0.590695\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 532.913i 0.686743i
\(777\) 0 0
\(778\) − 756.834i − 0.972794i
\(779\) − 1212.67i − 1.55671i
\(780\) 0 0
\(781\) −147.336 −0.188650
\(782\) −187.135 −0.239303
\(783\) 0 0
\(784\) −28.0000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 15.3517i 0.0195066i 0.999952 + 0.00975331i \(0.00310462\pi\)
−0.999952 + 0.00975331i \(0.996895\pi\)
\(788\) 377.120 0.478579
\(789\) 0 0
\(790\) 0 0
\(791\) − 75.4543i − 0.0953910i
\(792\) 0 0
\(793\) − 291.158i − 0.367160i
\(794\) 133.394i 0.168003i
\(795\) 0 0
\(796\) −204.988 −0.257523
\(797\) 1043.48 1.30927 0.654633 0.755947i \(-0.272823\pi\)
0.654633 + 0.755947i \(0.272823\pi\)
\(798\) 0 0
\(799\) 185.004 0.231544
\(800\) 0 0
\(801\) 0 0
\(802\) 146.502i 0.182671i
\(803\) 933.610 1.16265
\(804\) 0 0
\(805\) 0 0
\(806\) − 661.371i − 0.820560i
\(807\) 0 0
\(808\) − 301.830i − 0.373552i
\(809\) 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 \(-0.399775\pi\)
0.309689 + 0.950838i \(0.399775\pi\)
\(812\) −221.379 −0.272634
\(813\) 0 0
\(814\) 652.308 0.801362
\(815\) 0 0
\(816\) 0 0
\(817\) − 1669.96i − 2.04402i
\(818\) 13.7983 0.0168684
\(819\) 0 0
\(820\) 0 0
\(821\) 23.1137i 0.0281531i 0.999901 + 0.0140765i \(0.00448085\pi\)
−0.999901 + 0.0140765i \(0.995519\pi\)
\(822\) 0 0
\(823\) − 600.664i − 0.729847i −0.931038 0.364923i \(-0.881095\pi\)
0.931038 0.364923i \(-0.118905\pi\)
\(824\) 371.933i 0.451375i
\(825\) 0 0
\(826\) 217.830 0.263717
\(827\) 1309.21 1.58308 0.791540 0.611118i \(-0.209280\pi\)
0.791540 + 0.611118i \(0.209280\pi\)
\(828\) 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
\(832\) − 148.664i − 0.178683i
\(833\) −76.3102 −0.0916089
\(834\) 0 0
\(835\) 0 0
\(836\) − 485.528i − 0.580775i
\(837\) 0 0
\(838\) 480.000i 0.572792i
\(839\) 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
\(842\) −847.567 −1.00661
\(843\) 0 0
\(844\) −169.004 −0.200242
\(845\) 0 0
\(846\) 0 0
\(847\) − 69.6784i − 0.0822649i
\(848\) 376.170 0.443596
\(849\) 0 0
\(850\) 0 0
\(851\) 461.252i 0.542011i
\(852\) 0 0
\(853\) 137.012i 0.160623i 0.996770 + 0.0803117i \(0.0255916\pi\)
−0.996770 + 0.0803117i \(0.974408\pi\)
\(854\) 58.6242i 0.0686466i
\(855\) 0 0
\(856\) −233.004 −0.272201
\(857\) −466.141 −0.543922 −0.271961 0.962308i \(-0.587672\pi\)
−0.271961 + 0.962308i \(0.587672\pi\)
\(858\) 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
\(862\) − 1005.47i − 1.16644i
\(863\) 0.114603 0.000132796 0 6.63982e−5 1.00000i \(-0.499979\pi\)
6.63982e−5 1.00000i \(0.499979\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 534.550i 0.617264i
\(867\) 0 0
\(868\) 133.166i 0.153417i
\(869\) − 408.669i − 0.470275i
\(870\) 0 0
\(871\) 2465.30 2.83042
\(872\) −95.6858 −0.109731
\(873\) 0 0
\(874\) 343.320 0.392815
\(875\) 0 0
\(876\) 0 0
\(877\) − 997.304i − 1.13718i −0.822622 0.568589i \(-0.807490\pi\)
0.822622 0.568589i \(-0.192510\pi\)
\(878\) −746.912 −0.850697
\(879\) 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
\(883\) − 1549.47i − 1.75478i −0.479774 0.877392i \(-0.659281\pi\)
0.479774 0.877392i \(-0.340719\pi\)
\(884\) − 405.164i − 0.458330i
\(885\) 0 0
\(886\) 51.8222 0.0584900
\(887\) 894.493 1.00845 0.504224 0.863573i \(-0.331779\pi\)
0.504224 + 0.863573i \(0.331779\pi\)
\(888\) 0 0
\(889\) 343.069 0.385905
\(890\) 0 0
\(891\) 0 0
\(892\) − 316.988i − 0.355368i
\(893\) −339.411 −0.380080
\(894\) 0 0
\(895\) 0 0
\(896\) 29.9333i 0.0334077i
\(897\) 0 0
\(898\) − 562.308i − 0.626179i
\(899\) 1052.86i 1.17115i
\(900\) 0 0
\(901\) 1025.20 1.13785
\(902\) −1040.84 −1.15392
\(903\) 0 0
\(904\) −80.6640 −0.0892301
\(905\) 0 0
\(906\) 0 0
\(907\) 135.838i 0.149766i 0.997192 + 0.0748831i \(0.0238584\pi\)
−0.997192 + 0.0748831i \(0.976142\pi\)
\(908\) −203.647 −0.224281
\(909\) 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
\(913\) − 735.289i − 0.805355i
\(914\) − 486.948i − 0.532766i
\(915\) 0 0
\(916\) 537.830 0.587151
\(917\) −391.617 −0.427063
\(918\) 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
\(922\) 524.583i 0.568962i
\(923\) 225.564 0.244382
\(924\) 0 0
\(925\) 0 0
\(926\) 110.778i 0.119631i
\(927\) 0 0
\(928\) 236.664i 0.255026i
\(929\) − 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
\(932\) −52.5495 −0.0563836
\(933\) 0 0
\(934\) 565.344 0.605293
\(935\) 0 0
\(936\) 0 0
\(937\) − 1262.00i − 1.34685i −0.739255 0.673426i \(-0.764822\pi\)
0.739255 0.673426i \(-0.235178\pi\)
\(938\) −496.383 −0.529193
\(939\) 0 0
\(940\) 0 0
\(941\) − 672.410i − 0.714569i −0.933996 0.357285i \(-0.883703\pi\)
0.933996 0.357285i \(-0.116297\pi\)
\(942\) 0 0
\(943\) − 735.984i − 0.780471i
\(944\) − 232.870i − 0.246684i
\(945\) 0 0
\(946\) −1433.33 −1.51515
\(947\) −1159.75 −1.22465 −0.612327 0.790605i \(-0.709766\pi\)
−0.612327 + 0.790605i \(0.709766\pi\)
\(948\) 0 0
\(949\) −1429.31 −1.50612
\(950\) 0 0
\(951\) 0 0
\(952\) 81.5791i 0.0856923i
\(953\) −163.104 −0.171148 −0.0855740 0.996332i \(-0.527272\pi\)
−0.0855740 + 0.996332i \(0.527272\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 184.547i 0.193040i
\(957\) 0 0
\(958\) − 994.656i − 1.03826i
\(959\) 203.610i 0.212315i
\(960\) 0 0
\(961\) −327.672 −0.340970
\(962\) −998.653 −1.03810
\(963\) 0 0
\(964\) 686.494 0.712131
\(965\) 0 0
\(966\) 0 0
\(967\) − 887.012i − 0.917282i −0.888622 0.458641i \(-0.848336\pi\)
0.888622 0.458641i \(-0.151664\pi\)
\(968\) −74.4893 −0.0769518
\(969\) 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
\(973\) − 574.996i − 0.590952i
\(974\) 116.687i 0.119801i
\(975\) 0 0
\(976\) 62.6719 0.0642130
\(977\) −339.051 −0.347032 −0.173516 0.984831i \(-0.555513\pi\)
−0.173516 + 0.984831i \(0.555513\pi\)
\(978\) 0 0
\(979\) 57.9606 0.0592039
\(980\) 0 0
\(981\) 0 0
\(982\) 260.502i 0.265277i
\(983\) −487.887 −0.496324 −0.248162 0.968718i \(-0.579827\pi\)
−0.248162 + 0.968718i \(0.579827\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 644.996i 0.654154i
\(987\) 0 0
\(988\) 743.320i 0.752348i
\(989\) − 1013.52i − 1.02479i
\(990\) 0 0
\(991\) −937.474 −0.945988 −0.472994 0.881066i \(-0.656827\pi\)
−0.472994 + 0.881066i \(0.656827\pi\)
\(992\) 142.360 0.143509
\(993\) 0 0
\(994\) −45.4170 −0.0456911
\(995\) 0 0
\(996\) 0 0
\(997\) − 461.012i − 0.462399i −0.972906 0.231200i \(-0.925735\pi\)
0.972906 0.231200i \(-0.0742650\pi\)
\(998\) −1064.63 −1.06677
\(999\) 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 3150.3.c.b.449.7 8
3.2 odd 2 inner 3150.3.c.b.449.4 8
5.2 odd 4 126.3.b.a.71.4 yes 4
5.3 odd 4 3150.3.e.e.701.2 4
5.4 even 2 inner 3150.3.c.b.449.1 8
15.2 even 4 126.3.b.a.71.1 4
15.8 even 4 3150.3.e.e.701.4 4
15.14 odd 2 inner 3150.3.c.b.449.6 8
20.7 even 4 1008.3.d.a.449.3 4
35.2 odd 12 882.3.s.e.557.4 8
35.12 even 12 882.3.s.i.557.3 8
35.17 even 12 882.3.s.i.863.2 8
35.27 even 4 882.3.b.f.197.3 4
35.32 odd 12 882.3.s.e.863.1 8
40.27 even 4 4032.3.d.j.449.2 4
40.37 odd 4 4032.3.d.i.449.2 4
45.2 even 12 1134.3.q.c.701.2 8
45.7 odd 12 1134.3.q.c.701.3 8
45.22 odd 12 1134.3.q.c.1079.2 8
45.32 even 12 1134.3.q.c.1079.3 8
60.47 odd 4 1008.3.d.a.449.2 4
105.2 even 12 882.3.s.e.557.1 8
105.17 odd 12 882.3.s.i.863.3 8
105.32 even 12 882.3.s.e.863.4 8
105.47 odd 12 882.3.s.i.557.2 8
105.62 odd 4 882.3.b.f.197.2 4
120.77 even 4 4032.3.d.i.449.3 4
120.107 odd 4 4032.3.d.j.449.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.b.a.71.1 4 15.2 even 4
126.3.b.a.71.4 yes 4 5.2 odd 4
882.3.b.f.197.2 4 105.62 odd 4
882.3.b.f.197.3 4 35.27 even 4
882.3.s.e.557.1 8 105.2 even 12
882.3.s.e.557.4 8 35.2 odd 12
882.3.s.e.863.1 8 35.32 odd 12
882.3.s.e.863.4 8 105.32 even 12
882.3.s.i.557.2 8 105.47 odd 12
882.3.s.i.557.3 8 35.12 even 12
882.3.s.i.863.2 8 35.17 even 12
882.3.s.i.863.3 8 105.17 odd 12
1008.3.d.a.449.2 4 60.47 odd 4
1008.3.d.a.449.3 4 20.7 even 4
1134.3.q.c.701.2 8 45.2 even 12
1134.3.q.c.701.3 8 45.7 odd 12
1134.3.q.c.1079.2 8 45.22 odd 12
1134.3.q.c.1079.3 8 45.32 even 12
3150.3.c.b.449.1 8 5.4 even 2 inner
3150.3.c.b.449.4 8 3.2 odd 2 inner
3150.3.c.b.449.6 8 15.14 odd 2 inner
3150.3.c.b.449.7 8 1.1 even 1 trivial
3150.3.e.e.701.2 4 5.3 odd 4
3150.3.e.e.701.4 4 15.8 even 4
4032.3.d.i.449.2 4 40.37 odd 4
4032.3.d.i.449.3 4 120.77 even 4
4032.3.d.j.449.2 4 40.27 even 4
4032.3.d.j.449.3 4 120.107 odd 4