Properties

Label 8550.2.a.by.1.1
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.2720937282\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2850)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.73205 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.73205 q^{7} +1.00000 q^{8} -0.267949 q^{11} +0.732051 q^{13} -2.73205 q^{14} +1.00000 q^{16} -4.19615 q^{17} -1.00000 q^{19} -0.267949 q^{22} +7.92820 q^{23} +0.732051 q^{26} -2.73205 q^{28} -1.73205 q^{29} +4.46410 q^{31} +1.00000 q^{32} -4.19615 q^{34} -2.00000 q^{37} -1.00000 q^{38} -10.9282 q^{41} +2.19615 q^{43} -0.267949 q^{44} +7.92820 q^{46} +3.46410 q^{47} +0.464102 q^{49} +0.732051 q^{52} -1.73205 q^{53} -2.73205 q^{56} -1.73205 q^{58} -2.19615 q^{59} -6.66025 q^{61} +4.46410 q^{62} +1.00000 q^{64} -3.73205 q^{67} -4.19615 q^{68} -1.80385 q^{71} +4.46410 q^{73} -2.00000 q^{74} -1.00000 q^{76} +0.732051 q^{77} -12.4641 q^{79} -10.9282 q^{82} -0.267949 q^{83} +2.19615 q^{86} -0.267949 q^{88} +16.8564 q^{89} -2.00000 q^{91} +7.92820 q^{92} +3.46410 q^{94} +9.12436 q^{97} +0.464102 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} - 4 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} - 4 q^{22} + 2 q^{23} - 2 q^{26} - 2 q^{28} + 2 q^{31} + 2 q^{32} + 2 q^{34} - 4 q^{37} - 2 q^{38} - 8 q^{41} - 6 q^{43} - 4 q^{44} + 2 q^{46} - 6 q^{49} - 2 q^{52} - 2 q^{56} + 6 q^{59} + 4 q^{61} + 2 q^{62} + 2 q^{64} - 4 q^{67} + 2 q^{68} - 14 q^{71} + 2 q^{73} - 4 q^{74} - 2 q^{76} - 2 q^{77} - 18 q^{79} - 8 q^{82} - 4 q^{83} - 6 q^{86} - 4 q^{88} + 6 q^{89} - 4 q^{91} + 2 q^{92} - 6 q^{97} - 6 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −2.73205 −1.03262 −0.516309 0.856402i \(-0.672694\pi\)
−0.516309 + 0.856402i \(0.672694\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −0.267949 −0.0807897 −0.0403949 0.999184i \(-0.512862\pi\)
−0.0403949 + 0.999184i \(0.512862\pi\)
\(12\) 0 0
\(13\) 0.732051 0.203034 0.101517 0.994834i \(-0.467630\pi\)
0.101517 + 0.994834i \(0.467630\pi\)
\(14\) −2.73205 −0.730171
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.19615 −1.01772 −0.508858 0.860850i \(-0.669932\pi\)
−0.508858 + 0.860850i \(0.669932\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −0.267949 −0.0571270
\(23\) 7.92820 1.65314 0.826572 0.562831i \(-0.190288\pi\)
0.826572 + 0.562831i \(0.190288\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.732051 0.143567
\(27\) 0 0
\(28\) −2.73205 −0.516309
\(29\) −1.73205 −0.321634 −0.160817 0.986984i \(-0.551413\pi\)
−0.160817 + 0.986984i \(0.551413\pi\)
\(30\) 0 0
\(31\) 4.46410 0.801776 0.400888 0.916127i \(-0.368702\pi\)
0.400888 + 0.916127i \(0.368702\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.19615 −0.719634
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −10.9282 −1.70670 −0.853349 0.521340i \(-0.825432\pi\)
−0.853349 + 0.521340i \(0.825432\pi\)
\(42\) 0 0
\(43\) 2.19615 0.334910 0.167455 0.985880i \(-0.446445\pi\)
0.167455 + 0.985880i \(0.446445\pi\)
\(44\) −0.267949 −0.0403949
\(45\) 0 0
\(46\) 7.92820 1.16895
\(47\) 3.46410 0.505291 0.252646 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) 0 0
\(49\) 0.464102 0.0663002
\(50\) 0 0
\(51\) 0 0
\(52\) 0.732051 0.101517
\(53\) −1.73205 −0.237915 −0.118958 0.992899i \(-0.537955\pi\)
−0.118958 + 0.992899i \(0.537955\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.73205 −0.365086
\(57\) 0 0
\(58\) −1.73205 −0.227429
\(59\) −2.19615 −0.285915 −0.142957 0.989729i \(-0.545661\pi\)
−0.142957 + 0.989729i \(0.545661\pi\)
\(60\) 0 0
\(61\) −6.66025 −0.852758 −0.426379 0.904545i \(-0.640211\pi\)
−0.426379 + 0.904545i \(0.640211\pi\)
\(62\) 4.46410 0.566941
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −3.73205 −0.455943 −0.227971 0.973668i \(-0.573209\pi\)
−0.227971 + 0.973668i \(0.573209\pi\)
\(68\) −4.19615 −0.508858
\(69\) 0 0
\(70\) 0 0
\(71\) −1.80385 −0.214077 −0.107039 0.994255i \(-0.534137\pi\)
−0.107039 + 0.994255i \(0.534137\pi\)
\(72\) 0 0
\(73\) 4.46410 0.522484 0.261242 0.965273i \(-0.415868\pi\)
0.261242 + 0.965273i \(0.415868\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 0.732051 0.0834249
\(78\) 0 0
\(79\) −12.4641 −1.40232 −0.701160 0.713003i \(-0.747334\pi\)
−0.701160 + 0.713003i \(0.747334\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −10.9282 −1.20682
\(83\) −0.267949 −0.0294112 −0.0147056 0.999892i \(-0.504681\pi\)
−0.0147056 + 0.999892i \(0.504681\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.19615 0.236817
\(87\) 0 0
\(88\) −0.267949 −0.0285635
\(89\) 16.8564 1.78678 0.893388 0.449286i \(-0.148322\pi\)
0.893388 + 0.449286i \(0.148322\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 7.92820 0.826572
\(93\) 0 0
\(94\) 3.46410 0.357295
\(95\) 0 0
\(96\) 0 0
\(97\) 9.12436 0.926438 0.463219 0.886244i \(-0.346694\pi\)
0.463219 + 0.886244i \(0.346694\pi\)
\(98\) 0.464102 0.0468813
\(99\) 0 0
\(100\) 0 0
\(101\) −12.7321 −1.26689 −0.633443 0.773789i \(-0.718359\pi\)
−0.633443 + 0.773789i \(0.718359\pi\)
\(102\) 0 0
\(103\) −10.4641 −1.03106 −0.515529 0.856872i \(-0.672405\pi\)
−0.515529 + 0.856872i \(0.672405\pi\)
\(104\) 0.732051 0.0717835
\(105\) 0 0
\(106\) −1.73205 −0.168232
\(107\) 8.19615 0.792352 0.396176 0.918175i \(-0.370337\pi\)
0.396176 + 0.918175i \(0.370337\pi\)
\(108\) 0 0
\(109\) −18.3923 −1.76166 −0.880832 0.473430i \(-0.843016\pi\)
−0.880832 + 0.473430i \(0.843016\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.73205 −0.258155
\(113\) −7.39230 −0.695410 −0.347705 0.937604i \(-0.613039\pi\)
−0.347705 + 0.937604i \(0.613039\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.73205 −0.160817
\(117\) 0 0
\(118\) −2.19615 −0.202172
\(119\) 11.4641 1.05091
\(120\) 0 0
\(121\) −10.9282 −0.993473
\(122\) −6.66025 −0.602991
\(123\) 0 0
\(124\) 4.46410 0.400888
\(125\) 0 0
\(126\) 0 0
\(127\) −5.00000 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −13.7321 −1.19977 −0.599887 0.800084i \(-0.704788\pi\)
−0.599887 + 0.800084i \(0.704788\pi\)
\(132\) 0 0
\(133\) 2.73205 0.236899
\(134\) −3.73205 −0.322400
\(135\) 0 0
\(136\) −4.19615 −0.359817
\(137\) −8.39230 −0.717003 −0.358501 0.933529i \(-0.616712\pi\)
−0.358501 + 0.933529i \(0.616712\pi\)
\(138\) 0 0
\(139\) 3.12436 0.265004 0.132502 0.991183i \(-0.457699\pi\)
0.132502 + 0.991183i \(0.457699\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.80385 −0.151376
\(143\) −0.196152 −0.0164031
\(144\) 0 0
\(145\) 0 0
\(146\) 4.46410 0.369452
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −5.85641 −0.479776 −0.239888 0.970801i \(-0.577111\pi\)
−0.239888 + 0.970801i \(0.577111\pi\)
\(150\) 0 0
\(151\) −4.53590 −0.369126 −0.184563 0.982821i \(-0.559087\pi\)
−0.184563 + 0.982821i \(0.559087\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 0.732051 0.0589903
\(155\) 0 0
\(156\) 0 0
\(157\) −13.8564 −1.10586 −0.552931 0.833227i \(-0.686491\pi\)
−0.552931 + 0.833227i \(0.686491\pi\)
\(158\) −12.4641 −0.991591
\(159\) 0 0
\(160\) 0 0
\(161\) −21.6603 −1.70707
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) −10.9282 −0.853349
\(165\) 0 0
\(166\) −0.267949 −0.0207969
\(167\) −4.92820 −0.381356 −0.190678 0.981653i \(-0.561069\pi\)
−0.190678 + 0.981653i \(0.561069\pi\)
\(168\) 0 0
\(169\) −12.4641 −0.958777
\(170\) 0 0
\(171\) 0 0
\(172\) 2.19615 0.167455
\(173\) 17.5885 1.33723 0.668613 0.743611i \(-0.266888\pi\)
0.668613 + 0.743611i \(0.266888\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.267949 −0.0201974
\(177\) 0 0
\(178\) 16.8564 1.26344
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 14.5885 1.08435 0.542176 0.840265i \(-0.317601\pi\)
0.542176 + 0.840265i \(0.317601\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) 7.92820 0.584475
\(185\) 0 0
\(186\) 0 0
\(187\) 1.12436 0.0822210
\(188\) 3.46410 0.252646
\(189\) 0 0
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) −16.0526 −1.15549 −0.577744 0.816218i \(-0.696067\pi\)
−0.577744 + 0.816218i \(0.696067\pi\)
\(194\) 9.12436 0.655091
\(195\) 0 0
\(196\) 0.464102 0.0331501
\(197\) 6.58846 0.469408 0.234704 0.972067i \(-0.424588\pi\)
0.234704 + 0.972067i \(0.424588\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −12.7321 −0.895824
\(203\) 4.73205 0.332125
\(204\) 0 0
\(205\) 0 0
\(206\) −10.4641 −0.729069
\(207\) 0 0
\(208\) 0.732051 0.0507586
\(209\) 0.267949 0.0185344
\(210\) 0 0
\(211\) −13.7321 −0.945353 −0.472677 0.881236i \(-0.656712\pi\)
−0.472677 + 0.881236i \(0.656712\pi\)
\(212\) −1.73205 −0.118958
\(213\) 0 0
\(214\) 8.19615 0.560277
\(215\) 0 0
\(216\) 0 0
\(217\) −12.1962 −0.827929
\(218\) −18.3923 −1.24568
\(219\) 0 0
\(220\) 0 0
\(221\) −3.07180 −0.206631
\(222\) 0 0
\(223\) −10.4641 −0.700728 −0.350364 0.936614i \(-0.613942\pi\)
−0.350364 + 0.936614i \(0.613942\pi\)
\(224\) −2.73205 −0.182543
\(225\) 0 0
\(226\) −7.39230 −0.491729
\(227\) 6.00000 0.398234 0.199117 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) 0 0
\(229\) −9.73205 −0.643112 −0.321556 0.946891i \(-0.604206\pi\)
−0.321556 + 0.946891i \(0.604206\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.73205 −0.113715
\(233\) −3.12436 −0.204683 −0.102342 0.994749i \(-0.532634\pi\)
−0.102342 + 0.994749i \(0.532634\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2.19615 −0.142957
\(237\) 0 0
\(238\) 11.4641 0.743107
\(239\) 14.0000 0.905585 0.452792 0.891616i \(-0.350428\pi\)
0.452792 + 0.891616i \(0.350428\pi\)
\(240\) 0 0
\(241\) −19.8038 −1.27568 −0.637839 0.770170i \(-0.720171\pi\)
−0.637839 + 0.770170i \(0.720171\pi\)
\(242\) −10.9282 −0.702492
\(243\) 0 0
\(244\) −6.66025 −0.426379
\(245\) 0 0
\(246\) 0 0
\(247\) −0.732051 −0.0465793
\(248\) 4.46410 0.283471
\(249\) 0 0
\(250\) 0 0
\(251\) −4.53590 −0.286303 −0.143152 0.989701i \(-0.545724\pi\)
−0.143152 + 0.989701i \(0.545724\pi\)
\(252\) 0 0
\(253\) −2.12436 −0.133557
\(254\) −5.00000 −0.313728
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.00000 −0.561405 −0.280702 0.959795i \(-0.590567\pi\)
−0.280702 + 0.959795i \(0.590567\pi\)
\(258\) 0 0
\(259\) 5.46410 0.339523
\(260\) 0 0
\(261\) 0 0
\(262\) −13.7321 −0.848369
\(263\) −15.0000 −0.924940 −0.462470 0.886635i \(-0.653037\pi\)
−0.462470 + 0.886635i \(0.653037\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.73205 0.167513
\(267\) 0 0
\(268\) −3.73205 −0.227971
\(269\) 27.4641 1.67452 0.837258 0.546808i \(-0.184157\pi\)
0.837258 + 0.546808i \(0.184157\pi\)
\(270\) 0 0
\(271\) −26.1962 −1.59130 −0.795651 0.605755i \(-0.792871\pi\)
−0.795651 + 0.605755i \(0.792871\pi\)
\(272\) −4.19615 −0.254429
\(273\) 0 0
\(274\) −8.39230 −0.506998
\(275\) 0 0
\(276\) 0 0
\(277\) 0.803848 0.0482985 0.0241493 0.999708i \(-0.492312\pi\)
0.0241493 + 0.999708i \(0.492312\pi\)
\(278\) 3.12436 0.187386
\(279\) 0 0
\(280\) 0 0
\(281\) 23.7846 1.41887 0.709435 0.704770i \(-0.248950\pi\)
0.709435 + 0.704770i \(0.248950\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −1.80385 −0.107039
\(285\) 0 0
\(286\) −0.196152 −0.0115987
\(287\) 29.8564 1.76237
\(288\) 0 0
\(289\) 0.607695 0.0357468
\(290\) 0 0
\(291\) 0 0
\(292\) 4.46410 0.261242
\(293\) 24.2679 1.41775 0.708874 0.705335i \(-0.249203\pi\)
0.708874 + 0.705335i \(0.249203\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −5.85641 −0.339253
\(299\) 5.80385 0.335645
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) −4.53590 −0.261012
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −23.7321 −1.35446 −0.677230 0.735772i \(-0.736820\pi\)
−0.677230 + 0.735772i \(0.736820\pi\)
\(308\) 0.732051 0.0417125
\(309\) 0 0
\(310\) 0 0
\(311\) −5.32051 −0.301698 −0.150849 0.988557i \(-0.548201\pi\)
−0.150849 + 0.988557i \(0.548201\pi\)
\(312\) 0 0
\(313\) 9.92820 0.561175 0.280588 0.959828i \(-0.409471\pi\)
0.280588 + 0.959828i \(0.409471\pi\)
\(314\) −13.8564 −0.781962
\(315\) 0 0
\(316\) −12.4641 −0.701160
\(317\) −1.73205 −0.0972817 −0.0486408 0.998816i \(-0.515489\pi\)
−0.0486408 + 0.998816i \(0.515489\pi\)
\(318\) 0 0
\(319\) 0.464102 0.0259847
\(320\) 0 0
\(321\) 0 0
\(322\) −21.6603 −1.20708
\(323\) 4.19615 0.233480
\(324\) 0 0
\(325\) 0 0
\(326\) −2.00000 −0.110770
\(327\) 0 0
\(328\) −10.9282 −0.603409
\(329\) −9.46410 −0.521773
\(330\) 0 0
\(331\) −25.7321 −1.41436 −0.707181 0.707033i \(-0.750033\pi\)
−0.707181 + 0.707033i \(0.750033\pi\)
\(332\) −0.267949 −0.0147056
\(333\) 0 0
\(334\) −4.92820 −0.269659
\(335\) 0 0
\(336\) 0 0
\(337\) 17.4641 0.951330 0.475665 0.879626i \(-0.342207\pi\)
0.475665 + 0.879626i \(0.342207\pi\)
\(338\) −12.4641 −0.677958
\(339\) 0 0
\(340\) 0 0
\(341\) −1.19615 −0.0647753
\(342\) 0 0
\(343\) 17.8564 0.964155
\(344\) 2.19615 0.118409
\(345\) 0 0
\(346\) 17.5885 0.945561
\(347\) −20.7846 −1.11578 −0.557888 0.829916i \(-0.688388\pi\)
−0.557888 + 0.829916i \(0.688388\pi\)
\(348\) 0 0
\(349\) 21.0526 1.12692 0.563459 0.826144i \(-0.309470\pi\)
0.563459 + 0.826144i \(0.309470\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.267949 −0.0142817
\(353\) 22.0526 1.17374 0.586870 0.809681i \(-0.300360\pi\)
0.586870 + 0.809681i \(0.300360\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 16.8564 0.893388
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 3.07180 0.162123 0.0810616 0.996709i \(-0.474169\pi\)
0.0810616 + 0.996709i \(0.474169\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 14.5885 0.766752
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) −16.7321 −0.873406 −0.436703 0.899606i \(-0.643854\pi\)
−0.436703 + 0.899606i \(0.643854\pi\)
\(368\) 7.92820 0.413286
\(369\) 0 0
\(370\) 0 0
\(371\) 4.73205 0.245676
\(372\) 0 0
\(373\) 13.5167 0.699866 0.349933 0.936775i \(-0.386204\pi\)
0.349933 + 0.936775i \(0.386204\pi\)
\(374\) 1.12436 0.0581390
\(375\) 0 0
\(376\) 3.46410 0.178647
\(377\) −1.26795 −0.0653027
\(378\) 0 0
\(379\) 20.3923 1.04748 0.523741 0.851877i \(-0.324536\pi\)
0.523741 + 0.851877i \(0.324536\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −3.00000 −0.153493
\(383\) −25.5167 −1.30384 −0.651920 0.758288i \(-0.726036\pi\)
−0.651920 + 0.758288i \(0.726036\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −16.0526 −0.817054
\(387\) 0 0
\(388\) 9.12436 0.463219
\(389\) 33.7128 1.70931 0.854654 0.519198i \(-0.173769\pi\)
0.854654 + 0.519198i \(0.173769\pi\)
\(390\) 0 0
\(391\) −33.2679 −1.68243
\(392\) 0.464102 0.0234407
\(393\) 0 0
\(394\) 6.58846 0.331922
\(395\) 0 0
\(396\) 0 0
\(397\) 8.26795 0.414956 0.207478 0.978240i \(-0.433474\pi\)
0.207478 + 0.978240i \(0.433474\pi\)
\(398\) −14.0000 −0.701757
\(399\) 0 0
\(400\) 0 0
\(401\) 10.8564 0.542143 0.271072 0.962559i \(-0.412622\pi\)
0.271072 + 0.962559i \(0.412622\pi\)
\(402\) 0 0
\(403\) 3.26795 0.162788
\(404\) −12.7321 −0.633443
\(405\) 0 0
\(406\) 4.73205 0.234848
\(407\) 0.535898 0.0265635
\(408\) 0 0
\(409\) 22.3923 1.10723 0.553614 0.832773i \(-0.313248\pi\)
0.553614 + 0.832773i \(0.313248\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −10.4641 −0.515529
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) 0 0
\(416\) 0.732051 0.0358917
\(417\) 0 0
\(418\) 0.267949 0.0131058
\(419\) 13.8564 0.676930 0.338465 0.940979i \(-0.390092\pi\)
0.338465 + 0.940979i \(0.390092\pi\)
\(420\) 0 0
\(421\) −34.7846 −1.69530 −0.847649 0.530557i \(-0.821983\pi\)
−0.847649 + 0.530557i \(0.821983\pi\)
\(422\) −13.7321 −0.668466
\(423\) 0 0
\(424\) −1.73205 −0.0841158
\(425\) 0 0
\(426\) 0 0
\(427\) 18.1962 0.880574
\(428\) 8.19615 0.396176
\(429\) 0 0
\(430\) 0 0
\(431\) −5.32051 −0.256280 −0.128140 0.991756i \(-0.540901\pi\)
−0.128140 + 0.991756i \(0.540901\pi\)
\(432\) 0 0
\(433\) 13.8038 0.663371 0.331685 0.943390i \(-0.392383\pi\)
0.331685 + 0.943390i \(0.392383\pi\)
\(434\) −12.1962 −0.585434
\(435\) 0 0
\(436\) −18.3923 −0.880832
\(437\) −7.92820 −0.379257
\(438\) 0 0
\(439\) −15.7846 −0.753358 −0.376679 0.926344i \(-0.622934\pi\)
−0.376679 + 0.926344i \(0.622934\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.07180 −0.146110
\(443\) −20.6603 −0.981598 −0.490799 0.871273i \(-0.663295\pi\)
−0.490799 + 0.871273i \(0.663295\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −10.4641 −0.495490
\(447\) 0 0
\(448\) −2.73205 −0.129077
\(449\) −17.5359 −0.827570 −0.413785 0.910375i \(-0.635794\pi\)
−0.413785 + 0.910375i \(0.635794\pi\)
\(450\) 0 0
\(451\) 2.92820 0.137884
\(452\) −7.39230 −0.347705
\(453\) 0 0
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) 18.9282 0.885424 0.442712 0.896664i \(-0.354016\pi\)
0.442712 + 0.896664i \(0.354016\pi\)
\(458\) −9.73205 −0.454749
\(459\) 0 0
\(460\) 0 0
\(461\) −9.85641 −0.459059 −0.229529 0.973302i \(-0.573719\pi\)
−0.229529 + 0.973302i \(0.573719\pi\)
\(462\) 0 0
\(463\) 0.679492 0.0315787 0.0157893 0.999875i \(-0.494974\pi\)
0.0157893 + 0.999875i \(0.494974\pi\)
\(464\) −1.73205 −0.0804084
\(465\) 0 0
\(466\) −3.12436 −0.144733
\(467\) −3.19615 −0.147900 −0.0739501 0.997262i \(-0.523561\pi\)
−0.0739501 + 0.997262i \(0.523561\pi\)
\(468\) 0 0
\(469\) 10.1962 0.470815
\(470\) 0 0
\(471\) 0 0
\(472\) −2.19615 −0.101086
\(473\) −0.588457 −0.0270573
\(474\) 0 0
\(475\) 0 0
\(476\) 11.4641 0.525456
\(477\) 0 0
\(478\) 14.0000 0.640345
\(479\) −30.8564 −1.40987 −0.704933 0.709274i \(-0.749023\pi\)
−0.704933 + 0.709274i \(0.749023\pi\)
\(480\) 0 0
\(481\) −1.46410 −0.0667573
\(482\) −19.8038 −0.902041
\(483\) 0 0
\(484\) −10.9282 −0.496737
\(485\) 0 0
\(486\) 0 0
\(487\) −29.8564 −1.35292 −0.676461 0.736478i \(-0.736487\pi\)
−0.676461 + 0.736478i \(0.736487\pi\)
\(488\) −6.66025 −0.301496
\(489\) 0 0
\(490\) 0 0
\(491\) −10.9282 −0.493183 −0.246591 0.969120i \(-0.579311\pi\)
−0.246591 + 0.969120i \(0.579311\pi\)
\(492\) 0 0
\(493\) 7.26795 0.327332
\(494\) −0.732051 −0.0329365
\(495\) 0 0
\(496\) 4.46410 0.200444
\(497\) 4.92820 0.221060
\(498\) 0 0
\(499\) −31.3731 −1.40445 −0.702226 0.711954i \(-0.747810\pi\)
−0.702226 + 0.711954i \(0.747810\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −4.53590 −0.202447
\(503\) 9.85641 0.439475 0.219738 0.975559i \(-0.429480\pi\)
0.219738 + 0.975559i \(0.429480\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −2.12436 −0.0944391
\(507\) 0 0
\(508\) −5.00000 −0.221839
\(509\) −38.1244 −1.68983 −0.844916 0.534899i \(-0.820350\pi\)
−0.844916 + 0.534899i \(0.820350\pi\)
\(510\) 0 0
\(511\) −12.1962 −0.539526
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −9.00000 −0.396973
\(515\) 0 0
\(516\) 0 0
\(517\) −0.928203 −0.0408223
\(518\) 5.46410 0.240079
\(519\) 0 0
\(520\) 0 0
\(521\) 38.1769 1.67256 0.836280 0.548302i \(-0.184726\pi\)
0.836280 + 0.548302i \(0.184726\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −13.7321 −0.599887
\(525\) 0 0
\(526\) −15.0000 −0.654031
\(527\) −18.7321 −0.815981
\(528\) 0 0
\(529\) 39.8564 1.73289
\(530\) 0 0
\(531\) 0 0
\(532\) 2.73205 0.118449
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) 0 0
\(536\) −3.73205 −0.161200
\(537\) 0 0
\(538\) 27.4641 1.18406
\(539\) −0.124356 −0.00535638
\(540\) 0 0
\(541\) 6.80385 0.292520 0.146260 0.989246i \(-0.453276\pi\)
0.146260 + 0.989246i \(0.453276\pi\)
\(542\) −26.1962 −1.12522
\(543\) 0 0
\(544\) −4.19615 −0.179909
\(545\) 0 0
\(546\) 0 0
\(547\) −15.4449 −0.660375 −0.330187 0.943915i \(-0.607112\pi\)
−0.330187 + 0.943915i \(0.607112\pi\)
\(548\) −8.39230 −0.358501
\(549\) 0 0
\(550\) 0 0
\(551\) 1.73205 0.0737878
\(552\) 0 0
\(553\) 34.0526 1.44806
\(554\) 0.803848 0.0341522
\(555\) 0 0
\(556\) 3.12436 0.132502
\(557\) −14.1962 −0.601510 −0.300755 0.953701i \(-0.597239\pi\)
−0.300755 + 0.953701i \(0.597239\pi\)
\(558\) 0 0
\(559\) 1.60770 0.0679983
\(560\) 0 0
\(561\) 0 0
\(562\) 23.7846 1.00329
\(563\) 9.66025 0.407131 0.203566 0.979061i \(-0.434747\pi\)
0.203566 + 0.979061i \(0.434747\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) −1.80385 −0.0756878
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 4.19615 0.175604 0.0878018 0.996138i \(-0.472016\pi\)
0.0878018 + 0.996138i \(0.472016\pi\)
\(572\) −0.196152 −0.00820154
\(573\) 0 0
\(574\) 29.8564 1.24618
\(575\) 0 0
\(576\) 0 0
\(577\) 31.2487 1.30090 0.650450 0.759549i \(-0.274580\pi\)
0.650450 + 0.759549i \(0.274580\pi\)
\(578\) 0.607695 0.0252768
\(579\) 0 0
\(580\) 0 0
\(581\) 0.732051 0.0303706
\(582\) 0 0
\(583\) 0.464102 0.0192211
\(584\) 4.46410 0.184726
\(585\) 0 0
\(586\) 24.2679 1.00250
\(587\) 15.7321 0.649331 0.324666 0.945829i \(-0.394748\pi\)
0.324666 + 0.945829i \(0.394748\pi\)
\(588\) 0 0
\(589\) −4.46410 −0.183940
\(590\) 0 0
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) 48.3923 1.98723 0.993617 0.112807i \(-0.0359843\pi\)
0.993617 + 0.112807i \(0.0359843\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −5.85641 −0.239888
\(597\) 0 0
\(598\) 5.80385 0.237337
\(599\) −13.6603 −0.558143 −0.279071 0.960270i \(-0.590027\pi\)
−0.279071 + 0.960270i \(0.590027\pi\)
\(600\) 0 0
\(601\) 28.0526 1.14429 0.572144 0.820153i \(-0.306112\pi\)
0.572144 + 0.820153i \(0.306112\pi\)
\(602\) −6.00000 −0.244542
\(603\) 0 0
\(604\) −4.53590 −0.184563
\(605\) 0 0
\(606\) 0 0
\(607\) 29.7846 1.20892 0.604460 0.796635i \(-0.293389\pi\)
0.604460 + 0.796635i \(0.293389\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 2.53590 0.102591
\(612\) 0 0
\(613\) −2.78461 −0.112469 −0.0562347 0.998418i \(-0.517909\pi\)
−0.0562347 + 0.998418i \(0.517909\pi\)
\(614\) −23.7321 −0.957748
\(615\) 0 0
\(616\) 0.732051 0.0294952
\(617\) −25.6077 −1.03093 −0.515463 0.856912i \(-0.672380\pi\)
−0.515463 + 0.856912i \(0.672380\pi\)
\(618\) 0 0
\(619\) 10.1962 0.409818 0.204909 0.978781i \(-0.434310\pi\)
0.204909 + 0.978781i \(0.434310\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −5.32051 −0.213333
\(623\) −46.0526 −1.84506
\(624\) 0 0
\(625\) 0 0
\(626\) 9.92820 0.396811
\(627\) 0 0
\(628\) −13.8564 −0.552931
\(629\) 8.39230 0.334623
\(630\) 0 0
\(631\) −10.9282 −0.435045 −0.217522 0.976055i \(-0.569798\pi\)
−0.217522 + 0.976055i \(0.569798\pi\)
\(632\) −12.4641 −0.495795
\(633\) 0 0
\(634\) −1.73205 −0.0687885
\(635\) 0 0
\(636\) 0 0
\(637\) 0.339746 0.0134612
\(638\) 0.464102 0.0183740
\(639\) 0 0
\(640\) 0 0
\(641\) −17.4641 −0.689791 −0.344895 0.938641i \(-0.612086\pi\)
−0.344895 + 0.938641i \(0.612086\pi\)
\(642\) 0 0
\(643\) 40.9282 1.61405 0.807025 0.590517i \(-0.201076\pi\)
0.807025 + 0.590517i \(0.201076\pi\)
\(644\) −21.6603 −0.853534
\(645\) 0 0
\(646\) 4.19615 0.165095
\(647\) 11.3923 0.447878 0.223939 0.974603i \(-0.428108\pi\)
0.223939 + 0.974603i \(0.428108\pi\)
\(648\) 0 0
\(649\) 0.588457 0.0230990
\(650\) 0 0
\(651\) 0 0
\(652\) −2.00000 −0.0783260
\(653\) 10.5359 0.412302 0.206151 0.978520i \(-0.433906\pi\)
0.206151 + 0.978520i \(0.433906\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −10.9282 −0.426675
\(657\) 0 0
\(658\) −9.46410 −0.368949
\(659\) −38.5885 −1.50319 −0.751596 0.659623i \(-0.770716\pi\)
−0.751596 + 0.659623i \(0.770716\pi\)
\(660\) 0 0
\(661\) 46.6410 1.81413 0.907063 0.420996i \(-0.138319\pi\)
0.907063 + 0.420996i \(0.138319\pi\)
\(662\) −25.7321 −1.00010
\(663\) 0 0
\(664\) −0.267949 −0.0103984
\(665\) 0 0
\(666\) 0 0
\(667\) −13.7321 −0.531707
\(668\) −4.92820 −0.190678
\(669\) 0 0
\(670\) 0 0
\(671\) 1.78461 0.0688941
\(672\) 0 0
\(673\) −44.9808 −1.73388 −0.866940 0.498412i \(-0.833917\pi\)
−0.866940 + 0.498412i \(0.833917\pi\)
\(674\) 17.4641 0.672692
\(675\) 0 0
\(676\) −12.4641 −0.479389
\(677\) −8.66025 −0.332841 −0.166420 0.986055i \(-0.553221\pi\)
−0.166420 + 0.986055i \(0.553221\pi\)
\(678\) 0 0
\(679\) −24.9282 −0.956657
\(680\) 0 0
\(681\) 0 0
\(682\) −1.19615 −0.0458030
\(683\) 4.98076 0.190584 0.0952918 0.995449i \(-0.469622\pi\)
0.0952918 + 0.995449i \(0.469622\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 17.8564 0.681761
\(687\) 0 0
\(688\) 2.19615 0.0837275
\(689\) −1.26795 −0.0483050
\(690\) 0 0
\(691\) −11.8038 −0.449040 −0.224520 0.974470i \(-0.572081\pi\)
−0.224520 + 0.974470i \(0.572081\pi\)
\(692\) 17.5885 0.668613
\(693\) 0 0
\(694\) −20.7846 −0.788973
\(695\) 0 0
\(696\) 0 0
\(697\) 45.8564 1.73694
\(698\) 21.0526 0.796851
\(699\) 0 0
\(700\) 0 0
\(701\) −24.3923 −0.921285 −0.460642 0.887586i \(-0.652381\pi\)
−0.460642 + 0.887586i \(0.652381\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) −0.267949 −0.0100987
\(705\) 0 0
\(706\) 22.0526 0.829959
\(707\) 34.7846 1.30821
\(708\) 0 0
\(709\) −39.3013 −1.47599 −0.737995 0.674806i \(-0.764227\pi\)
−0.737995 + 0.674806i \(0.764227\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 16.8564 0.631721
\(713\) 35.3923 1.32545
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 3.07180 0.114638
\(719\) 50.1769 1.87128 0.935642 0.352952i \(-0.114822\pi\)
0.935642 + 0.352952i \(0.114822\pi\)
\(720\) 0 0
\(721\) 28.5885 1.06469
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 14.5885 0.542176
\(725\) 0 0
\(726\) 0 0
\(727\) 38.1051 1.41324 0.706620 0.707593i \(-0.250219\pi\)
0.706620 + 0.707593i \(0.250219\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 0 0
\(730\) 0 0
\(731\) −9.21539 −0.340844
\(732\) 0 0
\(733\) 24.5167 0.905544 0.452772 0.891626i \(-0.350435\pi\)
0.452772 + 0.891626i \(0.350435\pi\)
\(734\) −16.7321 −0.617591
\(735\) 0 0
\(736\) 7.92820 0.292237
\(737\) 1.00000 0.0368355
\(738\) 0 0
\(739\) 9.17691 0.337578 0.168789 0.985652i \(-0.446014\pi\)
0.168789 + 0.985652i \(0.446014\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.73205 0.173719
\(743\) 18.9282 0.694408 0.347204 0.937790i \(-0.387131\pi\)
0.347204 + 0.937790i \(0.387131\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 13.5167 0.494880
\(747\) 0 0
\(748\) 1.12436 0.0411105
\(749\) −22.3923 −0.818197
\(750\) 0 0
\(751\) 0.535898 0.0195552 0.00977760 0.999952i \(-0.496888\pi\)
0.00977760 + 0.999952i \(0.496888\pi\)
\(752\) 3.46410 0.126323
\(753\) 0 0
\(754\) −1.26795 −0.0461760
\(755\) 0 0
\(756\) 0 0
\(757\) 17.5885 0.639263 0.319632 0.947542i \(-0.396441\pi\)
0.319632 + 0.947542i \(0.396441\pi\)
\(758\) 20.3923 0.740682
\(759\) 0 0
\(760\) 0 0
\(761\) −4.73205 −0.171537 −0.0857684 0.996315i \(-0.527334\pi\)
−0.0857684 + 0.996315i \(0.527334\pi\)
\(762\) 0 0
\(763\) 50.2487 1.81913
\(764\) −3.00000 −0.108536
\(765\) 0 0
\(766\) −25.5167 −0.921954
\(767\) −1.60770 −0.0580505
\(768\) 0 0
\(769\) 20.7128 0.746923 0.373462 0.927646i \(-0.378171\pi\)
0.373462 + 0.927646i \(0.378171\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −16.0526 −0.577744
\(773\) 10.6795 0.384115 0.192057 0.981384i \(-0.438484\pi\)
0.192057 + 0.981384i \(0.438484\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 9.12436 0.327545
\(777\) 0 0
\(778\) 33.7128 1.20866
\(779\) 10.9282 0.391544
\(780\) 0 0
\(781\) 0.483340 0.0172952
\(782\) −33.2679 −1.18966
\(783\) 0 0
\(784\) 0.464102 0.0165751
\(785\) 0 0
\(786\) 0 0
\(787\) −22.1244 −0.788648 −0.394324 0.918971i \(-0.629021\pi\)
−0.394324 + 0.918971i \(0.629021\pi\)
\(788\) 6.58846 0.234704
\(789\) 0 0
\(790\) 0 0
\(791\) 20.1962 0.718093
\(792\) 0 0
\(793\) −4.87564 −0.173139
\(794\) 8.26795 0.293419
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) −20.0000 −0.708436 −0.354218 0.935163i \(-0.615253\pi\)
−0.354218 + 0.935163i \(0.615253\pi\)
\(798\) 0 0
\(799\) −14.5359 −0.514243
\(800\) 0 0
\(801\) 0 0
\(802\) 10.8564 0.383353
\(803\) −1.19615 −0.0422113
\(804\) 0 0
\(805\) 0 0
\(806\) 3.26795 0.115109
\(807\) 0 0
\(808\) −12.7321 −0.447912
\(809\) 7.51666 0.264272 0.132136 0.991232i \(-0.457816\pi\)
0.132136 + 0.991232i \(0.457816\pi\)
\(810\) 0 0
\(811\) −54.3731 −1.90930 −0.954648 0.297736i \(-0.903769\pi\)
−0.954648 + 0.297736i \(0.903769\pi\)
\(812\) 4.73205 0.166062
\(813\) 0 0
\(814\) 0.535898 0.0187832
\(815\) 0 0
\(816\) 0 0
\(817\) −2.19615 −0.0768336
\(818\) 22.3923 0.782929
\(819\) 0 0
\(820\) 0 0
\(821\) −47.5167 −1.65834 −0.829171 0.558994i \(-0.811187\pi\)
−0.829171 + 0.558994i \(0.811187\pi\)
\(822\) 0 0
\(823\) 34.7846 1.21252 0.606258 0.795268i \(-0.292670\pi\)
0.606258 + 0.795268i \(0.292670\pi\)
\(824\) −10.4641 −0.364534
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) −3.26795 −0.113638 −0.0568189 0.998385i \(-0.518096\pi\)
−0.0568189 + 0.998385i \(0.518096\pi\)
\(828\) 0 0
\(829\) −4.19615 −0.145738 −0.0728692 0.997342i \(-0.523216\pi\)
−0.0728692 + 0.997342i \(0.523216\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.732051 0.0253793
\(833\) −1.94744 −0.0674748
\(834\) 0 0
\(835\) 0 0
\(836\) 0.267949 0.00926722
\(837\) 0 0
\(838\) 13.8564 0.478662
\(839\) 15.4115 0.532066 0.266033 0.963964i \(-0.414287\pi\)
0.266033 + 0.963964i \(0.414287\pi\)
\(840\) 0 0
\(841\) −26.0000 −0.896552
\(842\) −34.7846 −1.19876
\(843\) 0 0
\(844\) −13.7321 −0.472677
\(845\) 0 0
\(846\) 0 0
\(847\) 29.8564 1.02588
\(848\) −1.73205 −0.0594789
\(849\) 0 0
\(850\) 0 0
\(851\) −15.8564 −0.543551
\(852\) 0 0
\(853\) −11.4641 −0.392523 −0.196262 0.980552i \(-0.562880\pi\)
−0.196262 + 0.980552i \(0.562880\pi\)
\(854\) 18.1962 0.622660
\(855\) 0 0
\(856\) 8.19615 0.280139
\(857\) 29.0718 0.993074 0.496537 0.868016i \(-0.334605\pi\)
0.496537 + 0.868016i \(0.334605\pi\)
\(858\) 0 0
\(859\) 43.0718 1.46959 0.734795 0.678289i \(-0.237278\pi\)
0.734795 + 0.678289i \(0.237278\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −5.32051 −0.181217
\(863\) 20.6795 0.703938 0.351969 0.936012i \(-0.385512\pi\)
0.351969 + 0.936012i \(0.385512\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 13.8038 0.469074
\(867\) 0 0
\(868\) −12.1962 −0.413964
\(869\) 3.33975 0.113293
\(870\) 0 0
\(871\) −2.73205 −0.0925720
\(872\) −18.3923 −0.622842
\(873\) 0 0
\(874\) −7.92820 −0.268175
\(875\) 0 0
\(876\) 0 0
\(877\) −31.1769 −1.05277 −0.526385 0.850246i \(-0.676453\pi\)
−0.526385 + 0.850246i \(0.676453\pi\)
\(878\) −15.7846 −0.532705
\(879\) 0 0
\(880\) 0 0
\(881\) 33.4641 1.12743 0.563717 0.825968i \(-0.309371\pi\)
0.563717 + 0.825968i \(0.309371\pi\)
\(882\) 0 0
\(883\) 30.7321 1.03422 0.517108 0.855920i \(-0.327009\pi\)
0.517108 + 0.855920i \(0.327009\pi\)
\(884\) −3.07180 −0.103316
\(885\) 0 0
\(886\) −20.6603 −0.694095
\(887\) 28.3923 0.953320 0.476660 0.879088i \(-0.341847\pi\)
0.476660 + 0.879088i \(0.341847\pi\)
\(888\) 0 0
\(889\) 13.6603 0.458150
\(890\) 0 0
\(891\) 0 0
\(892\) −10.4641 −0.350364
\(893\) −3.46410 −0.115922
\(894\) 0 0
\(895\) 0 0
\(896\) −2.73205 −0.0912714
\(897\) 0 0
\(898\) −17.5359 −0.585181
\(899\) −7.73205 −0.257878
\(900\) 0 0
\(901\) 7.26795 0.242130
\(902\) 2.92820 0.0974985
\(903\) 0 0
\(904\) −7.39230 −0.245864
\(905\) 0 0
\(906\) 0 0
\(907\) 14.6795 0.487425 0.243712 0.969848i \(-0.421635\pi\)
0.243712 + 0.969848i \(0.421635\pi\)
\(908\) 6.00000 0.199117
\(909\) 0 0
\(910\) 0 0
\(911\) 42.1051 1.39500 0.697502 0.716582i \(-0.254295\pi\)
0.697502 + 0.716582i \(0.254295\pi\)
\(912\) 0 0
\(913\) 0.0717968 0.00237613
\(914\) 18.9282 0.626089
\(915\) 0 0
\(916\) −9.73205 −0.321556
\(917\) 37.5167 1.23891
\(918\) 0 0
\(919\) −49.5167 −1.63340 −0.816702 0.577060i \(-0.804200\pi\)
−0.816702 + 0.577060i \(0.804200\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −9.85641 −0.324603
\(923\) −1.32051 −0.0434651
\(924\) 0 0
\(925\) 0 0
\(926\) 0.679492 0.0223295
\(927\) 0 0
\(928\) −1.73205 −0.0568574
\(929\) −29.0718 −0.953815 −0.476907 0.878954i \(-0.658242\pi\)
−0.476907 + 0.878954i \(0.658242\pi\)
\(930\) 0 0
\(931\) −0.464102 −0.0152103
\(932\) −3.12436 −0.102342
\(933\) 0 0
\(934\) −3.19615 −0.104581
\(935\) 0 0
\(936\) 0 0
\(937\) −54.7846 −1.78974 −0.894868 0.446332i \(-0.852730\pi\)
−0.894868 + 0.446332i \(0.852730\pi\)
\(938\) 10.1962 0.332916
\(939\) 0 0
\(940\) 0 0
\(941\) 15.7321 0.512850 0.256425 0.966564i \(-0.417455\pi\)
0.256425 + 0.966564i \(0.417455\pi\)
\(942\) 0 0
\(943\) −86.6410 −2.82142
\(944\) −2.19615 −0.0714787
\(945\) 0 0
\(946\) −0.588457 −0.0191324
\(947\) 21.8564 0.710238 0.355119 0.934821i \(-0.384440\pi\)
0.355119 + 0.934821i \(0.384440\pi\)
\(948\) 0 0
\(949\) 3.26795 0.106082
\(950\) 0 0
\(951\) 0 0
\(952\) 11.4641 0.371554
\(953\) 43.2487 1.40096 0.700482 0.713670i \(-0.252969\pi\)
0.700482 + 0.713670i \(0.252969\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 14.0000 0.452792
\(957\) 0 0
\(958\) −30.8564 −0.996925
\(959\) 22.9282 0.740390
\(960\) 0 0
\(961\) −11.0718 −0.357155
\(962\) −1.46410 −0.0472045
\(963\) 0 0
\(964\) −19.8038 −0.637839
\(965\) 0 0
\(966\) 0 0
\(967\) 58.1051 1.86853 0.934267 0.356573i \(-0.116055\pi\)
0.934267 + 0.356573i \(0.116055\pi\)
\(968\) −10.9282 −0.351246
\(969\) 0 0
\(970\) 0 0
\(971\) −55.7654 −1.78960 −0.894798 0.446471i \(-0.852681\pi\)
−0.894798 + 0.446471i \(0.852681\pi\)
\(972\) 0 0
\(973\) −8.53590 −0.273648
\(974\) −29.8564 −0.956661
\(975\) 0 0
\(976\) −6.66025 −0.213190
\(977\) 5.46410 0.174812 0.0874060 0.996173i \(-0.472142\pi\)
0.0874060 + 0.996173i \(0.472142\pi\)
\(978\) 0 0
\(979\) −4.51666 −0.144353
\(980\) 0 0
\(981\) 0 0
\(982\) −10.9282 −0.348733
\(983\) −47.6603 −1.52013 −0.760063 0.649849i \(-0.774832\pi\)
−0.760063 + 0.649849i \(0.774832\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 7.26795 0.231459
\(987\) 0 0
\(988\) −0.732051 −0.0232896
\(989\) 17.4115 0.553655
\(990\) 0 0
\(991\) −5.67949 −0.180415 −0.0902075 0.995923i \(-0.528753\pi\)
−0.0902075 + 0.995923i \(0.528753\pi\)
\(992\) 4.46410 0.141735
\(993\) 0 0
\(994\) 4.92820 0.156313
\(995\) 0 0
\(996\) 0 0
\(997\) 24.9474 0.790093 0.395047 0.918661i \(-0.370728\pi\)
0.395047 + 0.918661i \(0.370728\pi\)
\(998\) −31.3731 −0.993097
\(999\) 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 8550.2.a.by.1.1 2
3.2 odd 2 2850.2.a.bd.1.1 2
5.4 even 2 8550.2.a.bs.1.2 2
15.2 even 4 2850.2.d.x.799.1 4
15.8 even 4 2850.2.d.x.799.4 4
15.14 odd 2 2850.2.a.bi.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.bd.1.1 2 3.2 odd 2
2850.2.a.bi.1.2 yes 2 15.14 odd 2
2850.2.d.x.799.1 4 15.2 even 4
2850.2.d.x.799.4 4 15.8 even 4
8550.2.a.bs.1.2 2 5.4 even 2
8550.2.a.by.1.1 2 1.1 even 1 trivial