Properties

Label 3150.3.e.e.701.4
Level $3150$
Weight $3$
Character 3150.701
Analytic conductor $85.831$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(85.8312832735\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 701.4
Root \(1.16372i\) of defining polynomial
Character \(\chi\) \(=\) 3150.701
Dual form 3150.3.e.e.701.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} +2.64575 q^{7} -2.82843i q^{8} +O(q^{10})\) \(q+1.41421i q^{2} -2.00000 q^{4} +2.64575 q^{7} -2.82843i q^{8} -12.1382i q^{11} +18.5830 q^{13} +3.74166i q^{14} +4.00000 q^{16} +10.9015i q^{17} +20.0000 q^{19} +17.1660 q^{22} +12.1382i q^{23} +26.2803i q^{26} -5.29150 q^{28} +41.8367i q^{29} +25.1660 q^{31} +5.65685i q^{32} -15.4170 q^{34} -38.0000 q^{37} +28.2843i q^{38} -60.6337i q^{41} -83.4980 q^{43} +24.2764i q^{44} -17.1660 q^{46} +16.9706i q^{47} +7.00000 q^{49} -37.1660 q^{52} -94.0424i q^{53} -7.48331i q^{56} -59.1660 q^{58} -58.2175i q^{59} +15.6680 q^{61} +35.5901i q^{62} -8.00000 q^{64} +132.664 q^{67} -21.8029i q^{68} -12.1382i q^{71} +76.9150 q^{73} -53.7401i q^{74} -40.0000 q^{76} -32.1147i q^{77} +33.6680 q^{79} +85.7490 q^{82} +60.5764i q^{83} -118.084i q^{86} -34.3320 q^{88} -4.77506i q^{89} +49.1660 q^{91} -24.2764i q^{92} -24.0000 q^{94} +188.413 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + O(q^{10}) \) \( 4 q - 8 q^{4} + 32 q^{13} + 16 q^{16} + 80 q^{19} - 16 q^{22} + 16 q^{31} - 104 q^{34} - 152 q^{37} - 80 q^{43} + 16 q^{46} + 28 q^{49} - 64 q^{52} - 152 q^{58} + 232 q^{61} - 32 q^{64} + 192 q^{67} + 96 q^{73} - 160 q^{76} + 304 q^{79} + 216 q^{82} + 32 q^{88} + 112 q^{91} - 96 q^{94} + 288 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 0.707107i
\(3\) 0 0
\(4\) −2.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.64575 0.377964
\(8\) − 2.82843i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) − 12.1382i − 1.10347i −0.834019 0.551736i \(-0.813965\pi\)
0.834019 0.551736i \(-0.186035\pi\)
\(12\) 0 0
\(13\) 18.5830 1.42946 0.714731 0.699399i \(-0.246549\pi\)
0.714731 + 0.699399i \(0.246549\pi\)
\(14\) 3.74166i 0.267261i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 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
\(22\) 17.1660 0.780273
\(23\) 12.1382i 0.527748i 0.964557 + 0.263874i \(0.0850003\pi\)
−0.964557 + 0.263874i \(0.915000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 26.2803i 1.01078i
\(27\) 0 0
\(28\) −5.29150 −0.188982
\(29\) 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 \(0.366957\pi\)
\(32\) 5.65685i 0.176777i
\(33\) 0 0
\(34\) −15.4170 −0.453441
\(35\) 0 0
\(36\) 0 0
\(37\) −38.0000 −1.02703 −0.513514 0.858082i \(-0.671656\pi\)
−0.513514 + 0.858082i \(0.671656\pi\)
\(38\) 28.2843i 0.744323i
\(39\) 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 \(0.923031\pi\)
\(44\) 24.2764i 0.551736i
\(45\) 0 0
\(46\) −17.1660 −0.373174
\(47\) 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
\(52\) −37.1660 −0.714731
\(53\) − 94.0424i − 1.77439i −0.461399 0.887193i \(-0.652652\pi\)
0.461399 0.887193i \(-0.347348\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 7.48331i − 0.133631i
\(57\) 0 0
\(58\) −59.1660 −1.02010
\(59\) − 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 \(0.459008\pi\)
\(62\) 35.5901i 0.574034i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 132.664 1.98006 0.990030 0.140856i \(-0.0449853\pi\)
0.990030 + 0.140856i \(0.0449853\pi\)
\(68\) − 21.8029i − 0.320631i
\(69\) 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 \(-0.323386\pi\)
0.526815 + 0.849980i \(0.323386\pi\)
\(74\) − 53.7401i − 0.726218i
\(75\) 0 0
\(76\) −40.0000 −0.526316
\(77\) − 32.1147i − 0.417074i
\(78\) 0 0
\(79\) 33.6680 0.426177 0.213088 0.977033i \(-0.431648\pi\)
0.213088 + 0.977033i \(0.431648\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 85.7490 1.04572
\(83\) 60.5764i 0.729836i 0.931040 + 0.364918i \(0.118903\pi\)
−0.931040 + 0.364918i \(0.881097\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) − 118.084i − 1.37307i
\(87\) 0 0
\(88\) −34.3320 −0.390137
\(89\) − 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
\(92\) − 24.2764i − 0.263874i
\(93\) 0 0
\(94\) −24.0000 −0.255319
\(95\) 0 0
\(96\) 0 0
\(97\) 188.413 1.94240 0.971201 0.238260i \(-0.0765771\pi\)
0.971201 + 0.238260i \(0.0765771\pi\)
\(98\) 9.89949i 0.101015i
\(99\) 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 \(0.720379\pi\)
\(104\) − 52.5607i − 0.505391i
\(105\) 0 0
\(106\) 132.996 1.25468
\(107\) − 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
\(112\) 10.5830 0.0944911
\(113\) 28.5190i 0.252381i 0.992006 + 0.126190i \(0.0402750\pi\)
−0.992006 + 0.126190i \(0.959725\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) − 83.6734i − 0.721322i
\(117\) 0 0
\(118\) 82.3320 0.697729
\(119\) 28.8426i 0.242374i
\(120\) 0 0
\(121\) −26.3360 −0.217653
\(122\) 22.1579i 0.181622i
\(123\) 0 0
\(124\) −50.3320 −0.405903
\(125\) 0 0
\(126\) 0 0
\(127\) −129.668 −1.02101 −0.510504 0.859875i \(-0.670541\pi\)
−0.510504 + 0.859875i \(0.670541\pi\)
\(128\) − 11.3137i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) − 148.017i − 1.12990i −0.825124 0.564952i \(-0.808895\pi\)
0.825124 0.564952i \(-0.191105\pi\)
\(132\) 0 0
\(133\) 52.9150 0.397857
\(134\) 187.615i 1.40011i
\(135\) 0 0
\(136\) 30.8340 0.226721
\(137\) 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
\(142\) 17.1660 0.120887
\(143\) − 225.564i − 1.57737i
\(144\) 0 0
\(145\) 0 0
\(146\) 108.774i 0.745029i
\(147\) 0 0
\(148\) 76.0000 0.513514
\(149\) 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 \(0.599826\pi\)
\(152\) − 56.5685i − 0.372161i
\(153\) 0 0
\(154\) 45.4170 0.294916
\(155\) 0 0
\(156\) 0 0
\(157\) 184.996 1.17832 0.589159 0.808017i \(-0.299459\pi\)
0.589159 + 0.808017i \(0.299459\pi\)
\(158\) 47.6137i 0.301353i
\(159\) 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 \(0.585986\pi\)
\(164\) 121.267i 0.739435i
\(165\) 0 0
\(166\) −85.6680 −0.516072
\(167\) − 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
\(172\) 166.996 0.970907
\(173\) − 162.572i − 0.939721i −0.882741 0.469860i \(-0.844304\pi\)
0.882741 0.469860i \(-0.155696\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 48.5528i − 0.275868i
\(177\) 0 0
\(178\) 6.75295 0.0379379
\(179\) 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 \(0.325237\pi\)
\(182\) 69.5312i 0.382040i
\(183\) 0 0
\(184\) 34.3320 0.186587
\(185\) 0 0
\(186\) 0 0
\(187\) 132.324 0.707616
\(188\) − 33.9411i − 0.180538i
\(189\) 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 \(-0.612851\pi\)
−0.347150 + 0.937810i \(0.612851\pi\)
\(194\) 266.456i 1.37349i
\(195\) 0 0
\(196\) −14.0000 −0.0714286
\(197\) 188.560i 0.957157i 0.878045 + 0.478579i \(0.158848\pi\)
−0.878045 + 0.478579i \(0.841152\pi\)
\(198\) 0 0
\(199\) 102.494 0.515046 0.257523 0.966272i \(-0.417094\pi\)
0.257523 + 0.966272i \(0.417094\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −150.915 −0.747104
\(203\) 110.689i 0.545268i
\(204\) 0 0
\(205\) 0 0
\(206\) − 185.966i − 0.902749i
\(207\) 0 0
\(208\) 74.3320 0.357365
\(209\) − 242.764i − 1.16155i
\(210\) 0 0
\(211\) −84.5020 −0.400483 −0.200242 0.979747i \(-0.564173\pi\)
−0.200242 + 0.979747i \(0.564173\pi\)
\(212\) 188.085i 0.887193i
\(213\) 0 0
\(214\) 116.502 0.544402
\(215\) 0 0
\(216\) 0 0
\(217\) 66.5830 0.306834
\(218\) 47.8429i 0.219463i
\(219\) 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 \(0.384356\pi\)
\(224\) 14.9666i 0.0668153i
\(225\) 0 0
\(226\) −40.3320 −0.178460
\(227\) − 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
\(232\) 118.332 0.510052
\(233\) 26.2748i 0.112767i 0.998409 + 0.0563836i \(0.0179570\pi\)
−0.998409 + 0.0563836i \(0.982043\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 116.435i 0.493369i
\(237\) 0 0
\(238\) −40.7895 −0.171385
\(239\) 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 \(0.247730\pi\)
\(242\) − 37.2447i − 0.153904i
\(243\) 0 0
\(244\) −31.3360 −0.128426
\(245\) 0 0
\(246\) 0 0
\(247\) 371.660 1.50470
\(248\) − 71.1802i − 0.287017i
\(249\) 0 0
\(250\) 0 0
\(251\) 356.382i 1.41985i 0.704278 + 0.709924i \(0.251271\pi\)
−0.704278 + 0.709924i \(0.748729\pi\)
\(252\) 0 0
\(253\) 147.336 0.582356
\(254\) − 183.378i − 0.721961i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 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
\(262\) 209.328 0.798962
\(263\) − 261.979i − 0.996117i −0.867143 0.498059i \(-0.834046\pi\)
0.867143 0.498059i \(-0.165954\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 74.8331i 0.281328i
\(267\) 0 0
\(268\) −265.328 −0.990030
\(269\) 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 \(0.499315\pi\)
\(272\) 43.6058i 0.160316i
\(273\) 0 0
\(274\) −108.834 −0.397204
\(275\) 0 0
\(276\) 0 0
\(277\) −32.0000 −0.115523 −0.0577617 0.998330i \(-0.518396\pi\)
−0.0577617 + 0.998330i \(0.518396\pi\)
\(278\) 307.348i 1.10557i
\(279\) 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 \(0.509191\pi\)
\(284\) 24.2764i 0.0854803i
\(285\) 0 0
\(286\) 318.996 1.11537
\(287\) − 160.422i − 0.558961i
\(288\) 0 0
\(289\) 170.158 0.588782
\(290\) 0 0
\(291\) 0 0
\(292\) −153.830 −0.526815
\(293\) − 368.921i − 1.25912i −0.776953 0.629558i \(-0.783236\pi\)
0.776953 0.629558i \(-0.216764\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 107.480i 0.363109i
\(297\) 0 0
\(298\) −228.996 −0.768443
\(299\) 225.564i 0.754396i
\(300\) 0 0
\(301\) −220.915 −0.733937
\(302\) − 131.757i − 0.436280i
\(303\) 0 0
\(304\) 80.0000 0.263158
\(305\) 0 0
\(306\) 0 0
\(307\) 192.664 0.627570 0.313785 0.949494i \(-0.398403\pi\)
0.313785 + 0.949494i \(0.398403\pi\)
\(308\) 64.2293i 0.208537i
\(309\) 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 \(-0.522049\pi\)
−0.0692142 + 0.997602i \(0.522049\pi\)
\(314\) 261.624i 0.833197i
\(315\) 0 0
\(316\) −67.3360 −0.213088
\(317\) 251.724i 0.794083i 0.917801 + 0.397042i \(0.129963\pi\)
−0.917801 + 0.397042i \(0.870037\pi\)
\(318\) 0 0
\(319\) 507.822 1.59192
\(320\) 0 0
\(321\) 0 0
\(322\) −45.4170 −0.141047
\(323\) 218.029i 0.675013i
\(324\) 0 0
\(325\) 0 0
\(326\) − 123.031i − 0.377396i
\(327\) 0 0
\(328\) −171.498 −0.522860
\(329\) 44.8999i 0.136474i
\(330\) 0 0
\(331\) 361.490 1.09212 0.546058 0.837748i \(-0.316128\pi\)
0.546058 + 0.837748i \(0.316128\pi\)
\(332\) − 121.153i − 0.364918i
\(333\) 0 0
\(334\) 85.6680 0.256491
\(335\) 0 0
\(336\) 0 0
\(337\) 298.834 0.886748 0.443374 0.896337i \(-0.353781\pi\)
0.443374 + 0.896337i \(0.353781\pi\)
\(338\) 249.366i 0.737768i
\(339\) 0 0
\(340\) 0 0
\(341\) − 305.470i − 0.895807i
\(342\) 0 0
\(343\) 18.5203 0.0539949
\(344\) 236.168i 0.686535i
\(345\) 0 0
\(346\) 229.911 0.664483
\(347\) − 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
\(352\) 68.6640 0.195068
\(353\) 185.439i 0.525324i 0.964888 + 0.262662i \(0.0846005\pi\)
−0.964888 + 0.262662i \(0.915400\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 9.55012i 0.0268262i
\(357\) 0 0
\(358\) −315.498 −0.881279
\(359\) − 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
\(362\) 267.166i 0.738028i
\(363\) 0 0
\(364\) −98.3320 −0.270143
\(365\) 0 0
\(366\) 0 0
\(367\) 117.490 0.320137 0.160068 0.987106i \(-0.448829\pi\)
0.160068 + 0.987106i \(0.448829\pi\)
\(368\) 48.5528i 0.131937i
\(369\) 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 \(0.318513\pi\)
\(374\) 187.135i 0.500360i
\(375\) 0 0
\(376\) 48.0000 0.127660
\(377\) 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
\(382\) 322.494 0.844225
\(383\) − 744.804i − 1.94466i −0.233614 0.972329i \(-0.575055\pi\)
0.233614 0.972329i \(-0.424945\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 189.505i − 0.490945i
\(387\) 0 0
\(388\) −376.826 −0.971201
\(389\) − 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
\(392\) − 19.7990i − 0.0505076i
\(393\) 0 0
\(394\) −266.664 −0.676812
\(395\) 0 0
\(396\) 0 0
\(397\) 94.3241 0.237592 0.118796 0.992919i \(-0.462096\pi\)
0.118796 + 0.992919i \(0.462096\pi\)
\(398\) 144.949i 0.364192i
\(399\) 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
\(404\) − 213.426i − 0.528282i
\(405\) 0 0
\(406\) −156.539 −0.385563
\(407\) 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
\(412\) 262.996 0.638340
\(413\) − 154.029i − 0.372952i
\(414\) 0 0
\(415\) 0 0
\(416\) 105.121i 0.252696i
\(417\) 0 0
\(418\) 343.320 0.821340
\(419\) 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 \(0.752111\pi\)
\(422\) − 119.504i − 0.283184i
\(423\) 0 0
\(424\) −265.992 −0.627340
\(425\) 0 0
\(426\) 0 0
\(427\) 41.4536 0.0970810
\(428\) 164.759i 0.384950i
\(429\) 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 \(-0.643772\pi\)
−0.436471 + 0.899718i \(0.643772\pi\)
\(434\) 94.1626i 0.216964i
\(435\) 0 0
\(436\) −67.6601 −0.155184
\(437\) 242.764i 0.555524i
\(438\) 0 0
\(439\) 528.146 1.20307 0.601533 0.798848i \(-0.294557\pi\)
0.601533 + 0.798848i \(0.294557\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −286.494 −0.648177
\(443\) − 36.6438i − 0.0827174i −0.999144 0.0413587i \(-0.986831\pi\)
0.999144 0.0413587i \(-0.0131686\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 224.144i 0.502566i
\(447\) 0 0
\(448\) −21.1660 −0.0472456
\(449\) − 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
\(452\) − 57.0381i − 0.126190i
\(453\) 0 0
\(454\) 144.000 0.317181
\(455\) 0 0
\(456\) 0 0
\(457\) −344.324 −0.753445 −0.376722 0.926326i \(-0.622949\pi\)
−0.376722 + 0.926326i \(0.622949\pi\)
\(458\) − 380.303i − 0.830357i
\(459\) 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 \(0.526959\pi\)
\(464\) 167.347i 0.360661i
\(465\) 0 0
\(466\) −37.1581 −0.0797385
\(467\) 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
\(472\) −164.664 −0.348864
\(473\) 1013.52i 2.14274i
\(474\) 0 0
\(475\) 0 0
\(476\) − 57.6851i − 0.121187i
\(477\) 0 0
\(478\) −130.494 −0.273000
\(479\) − 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
\(482\) 485.425i 1.00711i
\(483\) 0 0
\(484\) 52.6719 0.108826
\(485\) 0 0
\(486\) 0 0
\(487\) 82.5098 0.169425 0.0847124 0.996405i \(-0.473003\pi\)
0.0847124 + 0.996405i \(0.473003\pi\)
\(488\) − 44.3157i − 0.0908109i
\(489\) 0 0
\(490\) 0 0
\(491\) − 184.203i − 0.375158i −0.982249 0.187579i \(-0.939936\pi\)
0.982249 0.187579i \(-0.0600641\pi\)
\(492\) 0 0
\(493\) −456.081 −0.925114
\(494\) 525.607i 1.06398i
\(495\) 0 0
\(496\) 100.664 0.202952
\(497\) − 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
\(502\) −504.000 −1.00398
\(503\) 662.540i 1.31718i 0.752504 + 0.658588i \(0.228846\pi\)
−0.752504 + 0.658588i \(0.771154\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 208.365i 0.411788i
\(507\) 0 0
\(508\) 259.336 0.510504
\(509\) 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
\(512\) 22.6274i 0.0441942i
\(513\) 0 0
\(514\) −360.243 −0.700862
\(515\) 0 0
\(516\) 0 0
\(517\) 205.992 0.398437
\(518\) − 142.183i − 0.274485i
\(519\) 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 \(0.428808\pi\)
\(524\) 296.035i 0.564952i
\(525\) 0 0
\(526\) 370.494 0.704361
\(527\) 274.346i 0.520581i
\(528\) 0 0
\(529\) 381.664 0.721482
\(530\) 0 0
\(531\) 0 0
\(532\) −105.830 −0.198929
\(533\) − 1126.76i − 2.11399i
\(534\) 0 0
\(535\) 0 0
\(536\) − 375.231i − 0.700057i
\(537\) 0 0
\(538\) −132.405 −0.246106
\(539\) − 84.9674i − 0.157639i
\(540\) 0 0
\(541\) 165.668 0.306225 0.153113 0.988209i \(-0.451070\pi\)
0.153113 + 0.988209i \(0.451070\pi\)
\(542\) 1.64899i 0.00304241i
\(543\) 0 0
\(544\) −61.6680 −0.113360
\(545\) 0 0
\(546\) 0 0
\(547\) −295.676 −0.540541 −0.270270 0.962784i \(-0.587113\pi\)
−0.270270 + 0.962784i \(0.587113\pi\)
\(548\) − 153.915i − 0.280866i
\(549\) 0 0
\(550\) 0 0
\(551\) 836.734i 1.51857i
\(552\) 0 0
\(553\) 89.0771 0.161080
\(554\) − 45.2548i − 0.0816874i
\(555\) 0 0
\(556\) −434.656 −0.781756
\(557\) − 76.8426i − 0.137958i −0.997618 0.0689790i \(-0.978026\pi\)
0.997618 0.0689790i \(-0.0219742\pi\)
\(558\) 0 0
\(559\) −1551.64 −2.77575
\(560\) 0 0
\(561\) 0 0
\(562\) −235.830 −0.419626
\(563\) − 1016.33i − 1.80521i −0.430470 0.902605i \(-0.641652\pi\)
0.430470 0.902605i \(-0.358348\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) − 23.1081i − 0.0408270i
\(567\) 0 0
\(568\) −34.3320 −0.0604437
\(569\) − 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 \(0.186441\pi\)
\(572\) 451.129i 0.788686i
\(573\) 0 0
\(574\) 226.871 0.395245
\(575\) 0 0
\(576\) 0 0
\(577\) 148.672 0.257664 0.128832 0.991666i \(-0.458877\pi\)
0.128832 + 0.991666i \(0.458877\pi\)
\(578\) 240.640i 0.416332i
\(579\) 0 0
\(580\) 0 0
\(581\) 160.270i 0.275852i
\(582\) 0 0
\(583\) −1141.51 −1.95799
\(584\) − 217.549i − 0.372515i
\(585\) 0 0
\(586\) 521.733 0.890330
\(587\) − 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
\(592\) −152.000 −0.256757
\(593\) − 217.251i − 0.366359i −0.983079 0.183180i \(-0.941361\pi\)
0.983079 0.183180i \(-0.0586390\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 323.849i − 0.543371i
\(597\) 0 0
\(598\) −318.996 −0.533438
\(599\) − 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 \(0.613055\pi\)
\(602\) − 312.421i − 0.518972i
\(603\) 0 0
\(604\) 186.332 0.308497
\(605\) 0 0
\(606\) 0 0
\(607\) −627.158 −1.03321 −0.516605 0.856224i \(-0.672804\pi\)
−0.516605 + 0.856224i \(0.672804\pi\)
\(608\) 113.137i 0.186081i
\(609\) 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 \(0.426835\pi\)
\(614\) 272.468i 0.443759i
\(615\) 0 0
\(616\) −90.8340 −0.147458
\(617\) − 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
\(622\) −185.652 −0.298476
\(623\) − 12.6336i − 0.0202787i
\(624\) 0 0
\(625\) 0 0
\(626\) − 61.2752i − 0.0978836i
\(627\) 0 0
\(628\) −369.992 −0.589159
\(629\) − 414.256i − 0.658594i
\(630\) 0 0
\(631\) −298.996 −0.473845 −0.236922 0.971529i \(-0.576139\pi\)
−0.236922 + 0.971529i \(0.576139\pi\)
\(632\) − 95.2274i − 0.150676i
\(633\) 0 0
\(634\) −355.992 −0.561502
\(635\) 0 0
\(636\) 0 0
\(637\) 130.081 0.204209
\(638\) 718.169i 1.12566i
\(639\) 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 \(0.344372\pi\)
\(644\) − 64.2293i − 0.0997350i
\(645\) 0 0
\(646\) −308.340 −0.477306
\(647\) − 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
\(652\) 173.992 0.266859
\(653\) − 481.892i − 0.737966i −0.929436 0.368983i \(-0.879706\pi\)
0.929436 0.368983i \(-0.120294\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) − 242.535i − 0.369718i
\(657\) 0 0
\(658\) −63.4980 −0.0965016
\(659\) 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 \(0.629112\pi\)
\(662\) 511.224i 0.772242i
\(663\) 0 0
\(664\) 171.336 0.258036
\(665\) 0 0
\(666\) 0 0
\(667\) −507.822 −0.761353
\(668\) 121.153i 0.181366i
\(669\) 0 0
\(670\) 0 0
\(671\) − 190.181i − 0.283429i
\(672\) 0 0
\(673\) 659.992 0.980672 0.490336 0.871534i \(-0.336874\pi\)
0.490336 + 0.871534i \(0.336874\pi\)
\(674\) 422.615i 0.627025i
\(675\) 0 0
\(676\) −352.656 −0.521681
\(677\) − 1016.28i − 1.50115i −0.660787 0.750573i \(-0.729778\pi\)
0.660787 0.750573i \(-0.270222\pi\)
\(678\) 0 0
\(679\) 498.494 0.734159
\(680\) 0 0
\(681\) 0 0
\(682\) 432.000 0.633431
\(683\) 235.114i 0.344238i 0.985076 + 0.172119i \(0.0550613\pi\)
−0.985076 + 0.172119i \(0.944939\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 26.1916i 0.0381802i
\(687\) 0 0
\(688\) −333.992 −0.485454
\(689\) − 1747.59i − 2.53642i
\(690\) 0 0
\(691\) −50.9803 −0.0737776 −0.0368888 0.999319i \(-0.511745\pi\)
−0.0368888 + 0.999319i \(0.511745\pi\)
\(692\) 325.143i 0.469860i
\(693\) 0 0
\(694\) 291.498 0.420026
\(695\) 0 0
\(696\) 0 0
\(697\) 660.996 0.948344
\(698\) 614.227i 0.879982i
\(699\) 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
\(704\) 97.1056i 0.137934i
\(705\) 0 0
\(706\) −262.251 −0.371460
\(707\) 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
\(712\) −13.5059 −0.0189690
\(713\) 305.470i 0.428429i
\(714\) 0 0
\(715\) 0 0
\(716\) − 446.182i − 0.623159i
\(717\) 0 0
\(718\) 730.818 1.01785
\(719\) − 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
\(722\) 55.1543i 0.0763910i
\(723\) 0 0
\(724\) −377.830 −0.521865
\(725\) 0 0
\(726\) 0 0
\(727\) 365.182 0.502313 0.251157 0.967946i \(-0.419189\pi\)
0.251157 + 0.967946i \(0.419189\pi\)
\(728\) − 139.062i − 0.191020i
\(729\) 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 \(0.422576\pi\)
\(734\) 166.156i 0.226371i
\(735\) 0 0
\(736\) −68.6640 −0.0932935
\(737\) − 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
\(742\) 351.875 0.474224
\(743\) 112.061i 0.150822i 0.997153 + 0.0754112i \(0.0240270\pi\)
−0.997153 + 0.0754112i \(0.975973\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 569.453i 0.763342i
\(747\) 0 0
\(748\) −264.648 −0.353808
\(749\) − 217.955i − 0.290995i
\(750\) 0 0
\(751\) −144.826 −0.192844 −0.0964222 0.995341i \(-0.530740\pi\)
−0.0964222 + 0.995341i \(0.530740\pi\)
\(752\) 67.8823i 0.0902690i
\(753\) 0 0
\(754\) −1099.48 −1.45820
\(755\) 0 0
\(756\) 0 0
\(757\) −78.1699 −0.103263 −0.0516314 0.998666i \(-0.516442\pi\)
−0.0516314 + 0.998666i \(0.516442\pi\)
\(758\) 564.036i 0.744111i
\(759\) 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
\(764\) 456.076i 0.596957i
\(765\) 0 0
\(766\) 1053.31 1.37508
\(767\) − 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
\(772\) 268.000 0.347150
\(773\) 434.559i 0.562172i 0.959683 + 0.281086i \(0.0906947\pi\)
−0.959683 + 0.281086i \(0.909305\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) − 532.913i − 0.686743i
\(777\) 0 0
\(778\) 756.834 0.972794
\(779\) − 1212.67i − 1.55671i
\(780\) 0 0
\(781\) −147.336 −0.188650
\(782\) − 187.135i − 0.239303i
\(783\) 0 0
\(784\) 28.0000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 15.3517 0.0195066 0.00975331 0.999952i \(-0.496895\pi\)
0.00975331 + 0.999952i \(0.496895\pi\)
\(788\) − 377.120i − 0.478579i
\(789\) 0 0
\(790\) 0 0
\(791\) 75.4543i 0.0953910i
\(792\) 0 0
\(793\) 291.158 0.367160
\(794\) 133.394i 0.168003i
\(795\) 0 0
\(796\) −204.988 −0.257523
\(797\) 1043.48i 1.30927i 0.755947 + 0.654633i \(0.227177\pi\)
−0.755947 + 0.654633i \(0.772823\pi\)
\(798\) 0 0
\(799\) −185.004 −0.231544
\(800\) 0 0
\(801\) 0 0
\(802\) 146.502 0.182671
\(803\) − 933.610i − 1.16265i
\(804\) 0 0
\(805\) 0 0
\(806\) 661.371i 0.820560i
\(807\) 0 0
\(808\) 301.830 0.373552
\(809\) 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 \(0.399775\pi\)
\(812\) − 221.379i − 0.272634i
\(813\) 0 0
\(814\) −652.308 −0.801362
\(815\) 0 0
\(816\) 0 0
\(817\) −1669.96 −2.04402
\(818\) − 13.7983i − 0.0168684i
\(819\) 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 \(0.381095\pi\)
\(824\) 371.933i 0.451375i
\(825\) 0 0
\(826\) 217.830 0.263717
\(827\) 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
\(832\) −148.664 −0.178683
\(833\) 76.3102i 0.0916089i
\(834\) 0 0
\(835\) 0 0
\(836\) 485.528i 0.580775i
\(837\) 0 0
\(838\) −480.000 −0.572792
\(839\) 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
\(842\) − 847.567i − 1.00661i
\(843\) 0 0
\(844\) 169.004 0.200242
\(845\) 0 0
\(846\) 0 0
\(847\) −69.6784 −0.0822649
\(848\) − 376.170i − 0.443596i
\(849\) 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 \(0.525592\pi\)
\(854\) 58.6242i 0.0686466i
\(855\) 0 0
\(856\) −233.004 −0.272201
\(857\) − 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
\(862\) −1005.47 −1.16644
\(863\) − 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
\(865\) 0 0
\(866\) − 534.550i − 0.617264i
\(867\) 0 0
\(868\) −133.166 −0.153417
\(869\) − 408.669i − 0.470275i
\(870\) 0 0
\(871\) 2465.30 2.83042
\(872\) − 95.6858i − 0.109731i
\(873\) 0 0
\(874\) −343.320 −0.392815
\(875\) 0 0
\(876\) 0 0
\(877\) −997.304 −1.13718 −0.568589 0.822622i \(-0.692510\pi\)
−0.568589 + 0.822622i \(0.692510\pi\)
\(878\) 746.912i 0.850697i
\(879\) 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 \(0.159281\pi\)
\(884\) − 405.164i − 0.458330i
\(885\) 0 0
\(886\) 51.8222 0.0584900
\(887\) 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
\(892\) −316.988 −0.355368
\(893\) 339.411i 0.380080i
\(894\) 0 0
\(895\) 0 0
\(896\) − 29.9333i − 0.0334077i
\(897\) 0 0
\(898\) 562.308 0.626179
\(899\) 1052.86i 1.17115i
\(900\) 0 0
\(901\) 1025.20 1.13785
\(902\) − 1040.84i − 1.15392i
\(903\) 0 0
\(904\) 80.6640 0.0892301
\(905\) 0 0
\(906\) 0 0
\(907\) 135.838 0.149766 0.0748831 0.997192i \(-0.476142\pi\)
0.0748831 + 0.997192i \(0.476142\pi\)
\(908\) 203.647i 0.224281i
\(909\) 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
\(914\) − 486.948i − 0.532766i
\(915\) 0 0
\(916\) 537.830 0.587151
\(917\) − 391.617i − 0.427063i
\(918\) 0 0
\(919\) −388.162 −0.422374 −0.211187 0.977446i \(-0.567733\pi\)
−0.211187 + 0.977446i \(0.567733\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 524.583 0.568962
\(923\) − 225.564i − 0.244382i
\(924\) 0 0
\(925\) 0 0
\(926\) − 110.778i − 0.119631i
\(927\) 0 0
\(928\) −236.664 −0.255026
\(929\) − 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
\(932\) − 52.5495i − 0.0563836i
\(933\) 0 0
\(934\) −565.344 −0.605293
\(935\) 0 0
\(936\) 0 0
\(937\) −1262.00 −1.34685 −0.673426 0.739255i \(-0.735178\pi\)
−0.673426 + 0.739255i \(0.735178\pi\)
\(938\) 496.383i 0.529193i
\(939\) 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
\(944\) − 232.870i − 0.246684i
\(945\) 0 0
\(946\) −1433.33 −1.51515
\(947\) − 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
\(952\) 81.5791 0.0856923
\(953\) 163.104i 0.171148i 0.996332 + 0.0855740i \(0.0272724\pi\)
−0.996332 + 0.0855740i \(0.972728\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) − 184.547i − 0.193040i
\(957\) 0 0
\(958\) 994.656 1.03826
\(959\) 203.610i 0.212315i
\(960\) 0 0
\(961\) −327.672 −0.340970
\(962\) − 998.653i − 1.03810i
\(963\) 0 0
\(964\) −686.494 −0.712131
\(965\) 0 0
\(966\) 0 0
\(967\) −887.012 −0.917282 −0.458641 0.888622i \(-0.651664\pi\)
−0.458641 + 0.888622i \(0.651664\pi\)
\(968\) 74.4893i 0.0769518i
\(969\) 0 0
\(970\) 0 0
\(971\) − 1416.32i − 1.45862i −0.684183 0.729310i \(-0.739841\pi\)
0.684183 0.729310i \(-0.260159\pi\)
\(972\) 0 0
\(973\) 574.996 0.590952
\(974\) 116.687i 0.119801i
\(975\) 0 0
\(976\) 62.6719 0.0642130
\(977\) − 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
\(982\) 260.502 0.265277
\(983\) 487.887i 0.496324i 0.968718 + 0.248162i \(0.0798266\pi\)
−0.968718 + 0.248162i \(0.920173\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) − 644.996i − 0.654154i
\(987\) 0 0
\(988\) −743.320 −0.752348
\(989\) − 1013.52i − 1.02479i
\(990\) 0 0
\(991\) −937.474 −0.945988 −0.472994 0.881066i \(-0.656827\pi\)
−0.472994 + 0.881066i \(0.656827\pi\)
\(992\) 142.360i 0.143509i
\(993\) 0 0
\(994\) 45.4170 0.0456911
\(995\) 0 0
\(996\) 0 0
\(997\) −461.012 −0.462399 −0.231200 0.972906i \(-0.574265\pi\)
−0.231200 + 0.972906i \(0.574265\pi\)
\(998\) 1064.63i 1.06677i
\(999\) 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.e.e.701.4 4
3.2 odd 2 inner 3150.3.e.e.701.2 4
5.2 odd 4 3150.3.c.b.449.4 8
5.3 odd 4 3150.3.c.b.449.6 8
5.4 even 2 126.3.b.a.71.1 4
15.2 even 4 3150.3.c.b.449.7 8
15.8 even 4 3150.3.c.b.449.1 8
15.14 odd 2 126.3.b.a.71.4 yes 4
20.19 odd 2 1008.3.d.a.449.2 4
35.4 even 6 882.3.s.e.863.4 8
35.9 even 6 882.3.s.e.557.1 8
35.19 odd 6 882.3.s.i.557.2 8
35.24 odd 6 882.3.s.i.863.3 8
35.34 odd 2 882.3.b.f.197.2 4
40.19 odd 2 4032.3.d.j.449.3 4
40.29 even 2 4032.3.d.i.449.3 4
45.4 even 6 1134.3.q.c.1079.3 8
45.14 odd 6 1134.3.q.c.1079.2 8
45.29 odd 6 1134.3.q.c.701.3 8
45.34 even 6 1134.3.q.c.701.2 8
60.59 even 2 1008.3.d.a.449.3 4
105.44 odd 6 882.3.s.e.557.4 8
105.59 even 6 882.3.s.i.863.2 8
105.74 odd 6 882.3.s.e.863.1 8
105.89 even 6 882.3.s.i.557.3 8
105.104 even 2 882.3.b.f.197.3 4
120.29 odd 2 4032.3.d.i.449.2 4
120.59 even 2 4032.3.d.j.449.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.b.a.71.1 4 5.4 even 2
126.3.b.a.71.4 yes 4 15.14 odd 2
882.3.b.f.197.2 4 35.34 odd 2
882.3.b.f.197.3 4 105.104 even 2
882.3.s.e.557.1 8 35.9 even 6
882.3.s.e.557.4 8 105.44 odd 6
882.3.s.e.863.1 8 105.74 odd 6
882.3.s.e.863.4 8 35.4 even 6
882.3.s.i.557.2 8 35.19 odd 6
882.3.s.i.557.3 8 105.89 even 6
882.3.s.i.863.2 8 105.59 even 6
882.3.s.i.863.3 8 35.24 odd 6
1008.3.d.a.449.2 4 20.19 odd 2
1008.3.d.a.449.3 4 60.59 even 2
1134.3.q.c.701.2 8 45.34 even 6
1134.3.q.c.701.3 8 45.29 odd 6
1134.3.q.c.1079.2 8 45.14 odd 6
1134.3.q.c.1079.3 8 45.4 even 6
3150.3.c.b.449.1 8 15.8 even 4
3150.3.c.b.449.4 8 5.2 odd 4
3150.3.c.b.449.6 8 5.3 odd 4
3150.3.c.b.449.7 8 15.2 even 4
3150.3.e.e.701.2 4 3.2 odd 2 inner
3150.3.e.e.701.4 4 1.1 even 1 trivial
4032.3.d.i.449.2 4 120.29 odd 2
4032.3.d.i.449.3 4 40.29 even 2
4032.3.d.j.449.2 4 120.59 even 2
4032.3.d.j.449.3 4 40.19 odd 2