Properties

Label 8550.2.a.bu.1.2
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: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
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 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +4.44949 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +4.44949 q^{7} -1.00000 q^{8} -3.44949 q^{11} -2.44949 q^{13} -4.44949 q^{14} +1.00000 q^{16} -4.44949 q^{17} +1.00000 q^{19} +3.44949 q^{22} +1.00000 q^{23} +2.44949 q^{26} +4.44949 q^{28} +4.34847 q^{29} -3.00000 q^{31} -1.00000 q^{32} +4.44949 q^{34} -7.79796 q^{37} -1.00000 q^{38} +0.898979 q^{41} -2.44949 q^{43} -3.44949 q^{44} -1.00000 q^{46} +7.79796 q^{47} +12.7980 q^{49} -2.44949 q^{52} +7.44949 q^{53} -4.44949 q^{56} -4.34847 q^{58} -6.44949 q^{59} -9.44949 q^{61} +3.00000 q^{62} +1.00000 q^{64} +15.2474 q^{67} -4.44949 q^{68} +1.55051 q^{71} -1.00000 q^{73} +7.79796 q^{74} +1.00000 q^{76} -15.3485 q^{77} -5.00000 q^{79} -0.898979 q^{82} -8.34847 q^{83} +2.44949 q^{86} +3.44949 q^{88} -2.10102 q^{89} -10.8990 q^{91} +1.00000 q^{92} -7.79796 q^{94} +1.55051 q^{97} -12.7980 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 4q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 4q^{7} - 2q^{8} - 2q^{11} - 4q^{14} + 2q^{16} - 4q^{17} + 2q^{19} + 2q^{22} + 2q^{23} + 4q^{28} - 6q^{29} - 6q^{31} - 2q^{32} + 4q^{34} + 4q^{37} - 2q^{38} - 8q^{41} - 2q^{44} - 2q^{46} - 4q^{47} + 6q^{49} + 10q^{53} - 4q^{56} + 6q^{58} - 8q^{59} - 14q^{61} + 6q^{62} + 2q^{64} + 6q^{67} - 4q^{68} + 8q^{71} - 2q^{73} - 4q^{74} + 2q^{76} - 16q^{77} - 10q^{79} + 8q^{82} - 2q^{83} + 2q^{88} - 14q^{89} - 12q^{91} + 2q^{92} + 4q^{94} + 8q^{97} - 6q^{98} + O(q^{100}) \)

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
\(7\) 4.44949 1.68175 0.840875 0.541230i \(-0.182041\pi\)
0.840875 + 0.541230i \(0.182041\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −3.44949 −1.04006 −0.520030 0.854148i \(-0.674079\pi\)
−0.520030 + 0.854148i \(0.674079\pi\)
\(12\) 0 0
\(13\) −2.44949 −0.679366 −0.339683 0.940540i \(-0.610320\pi\)
−0.339683 + 0.940540i \(0.610320\pi\)
\(14\) −4.44949 −1.18918
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.44949 −1.07916 −0.539580 0.841934i \(-0.681417\pi\)
−0.539580 + 0.841934i \(0.681417\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 3.44949 0.735434
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.44949 0.480384
\(27\) 0 0
\(28\) 4.44949 0.840875
\(29\) 4.34847 0.807490 0.403745 0.914871i \(-0.367708\pi\)
0.403745 + 0.914871i \(0.367708\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 4.44949 0.763081
\(35\) 0 0
\(36\) 0 0
\(37\) −7.79796 −1.28198 −0.640988 0.767551i \(-0.721475\pi\)
−0.640988 + 0.767551i \(0.721475\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 0.898979 0.140397 0.0701985 0.997533i \(-0.477637\pi\)
0.0701985 + 0.997533i \(0.477637\pi\)
\(42\) 0 0
\(43\) −2.44949 −0.373544 −0.186772 0.982403i \(-0.559803\pi\)
−0.186772 + 0.982403i \(0.559803\pi\)
\(44\) −3.44949 −0.520030
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 7.79796 1.13745 0.568725 0.822528i \(-0.307437\pi\)
0.568725 + 0.822528i \(0.307437\pi\)
\(48\) 0 0
\(49\) 12.7980 1.82828
\(50\) 0 0
\(51\) 0 0
\(52\) −2.44949 −0.339683
\(53\) 7.44949 1.02327 0.511633 0.859204i \(-0.329041\pi\)
0.511633 + 0.859204i \(0.329041\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.44949 −0.594588
\(57\) 0 0
\(58\) −4.34847 −0.570982
\(59\) −6.44949 −0.839652 −0.419826 0.907605i \(-0.637909\pi\)
−0.419826 + 0.907605i \(0.637909\pi\)
\(60\) 0 0
\(61\) −9.44949 −1.20988 −0.604942 0.796270i \(-0.706804\pi\)
−0.604942 + 0.796270i \(0.706804\pi\)
\(62\) 3.00000 0.381000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 15.2474 1.86277 0.931386 0.364033i \(-0.118600\pi\)
0.931386 + 0.364033i \(0.118600\pi\)
\(68\) −4.44949 −0.539580
\(69\) 0 0
\(70\) 0 0
\(71\) 1.55051 0.184012 0.0920059 0.995758i \(-0.470672\pi\)
0.0920059 + 0.995758i \(0.470672\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 7.79796 0.906494
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −15.3485 −1.74912
\(78\) 0 0
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.898979 −0.0992757
\(83\) −8.34847 −0.916364 −0.458182 0.888859i \(-0.651499\pi\)
−0.458182 + 0.888859i \(0.651499\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.44949 0.264135
\(87\) 0 0
\(88\) 3.44949 0.367717
\(89\) −2.10102 −0.222708 −0.111354 0.993781i \(-0.535519\pi\)
−0.111354 + 0.993781i \(0.535519\pi\)
\(90\) 0 0
\(91\) −10.8990 −1.14252
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −7.79796 −0.804298
\(95\) 0 0
\(96\) 0 0
\(97\) 1.55051 0.157430 0.0787152 0.996897i \(-0.474918\pi\)
0.0787152 + 0.996897i \(0.474918\pi\)
\(98\) −12.7980 −1.29279
\(99\) 0 0
\(100\) 0 0
\(101\) 1.55051 0.154282 0.0771408 0.997020i \(-0.475421\pi\)
0.0771408 + 0.997020i \(0.475421\pi\)
\(102\) 0 0
\(103\) 1.89898 0.187112 0.0935560 0.995614i \(-0.470177\pi\)
0.0935560 + 0.995614i \(0.470177\pi\)
\(104\) 2.44949 0.240192
\(105\) 0 0
\(106\) −7.44949 −0.723558
\(107\) −17.3485 −1.67714 −0.838570 0.544794i \(-0.816608\pi\)
−0.838570 + 0.544794i \(0.816608\pi\)
\(108\) 0 0
\(109\) −2.89898 −0.277672 −0.138836 0.990315i \(-0.544336\pi\)
−0.138836 + 0.990315i \(0.544336\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.44949 0.420437
\(113\) 16.7980 1.58022 0.790110 0.612966i \(-0.210024\pi\)
0.790110 + 0.612966i \(0.210024\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 4.34847 0.403745
\(117\) 0 0
\(118\) 6.44949 0.593724
\(119\) −19.7980 −1.81488
\(120\) 0 0
\(121\) 0.898979 0.0817254
\(122\) 9.44949 0.855517
\(123\) 0 0
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) 0 0
\(127\) 0.101021 0.00896412 0.00448206 0.999990i \(-0.498573\pi\)
0.00448206 + 0.999990i \(0.498573\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −9.24745 −0.807953 −0.403977 0.914769i \(-0.632372\pi\)
−0.403977 + 0.914769i \(0.632372\pi\)
\(132\) 0 0
\(133\) 4.44949 0.385820
\(134\) −15.2474 −1.31718
\(135\) 0 0
\(136\) 4.44949 0.381541
\(137\) 4.89898 0.418548 0.209274 0.977857i \(-0.432890\pi\)
0.209274 + 0.977857i \(0.432890\pi\)
\(138\) 0 0
\(139\) −22.2474 −1.88700 −0.943502 0.331367i \(-0.892490\pi\)
−0.943502 + 0.331367i \(0.892490\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.55051 −0.130116
\(143\) 8.44949 0.706582
\(144\) 0 0
\(145\) 0 0
\(146\) 1.00000 0.0827606
\(147\) 0 0
\(148\) −7.79796 −0.640988
\(149\) −5.79796 −0.474987 −0.237494 0.971389i \(-0.576326\pi\)
−0.237494 + 0.971389i \(0.576326\pi\)
\(150\) 0 0
\(151\) −23.7980 −1.93665 −0.968325 0.249692i \(-0.919671\pi\)
−0.968325 + 0.249692i \(0.919671\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 15.3485 1.23681
\(155\) 0 0
\(156\) 0 0
\(157\) −4.89898 −0.390981 −0.195491 0.980706i \(-0.562630\pi\)
−0.195491 + 0.980706i \(0.562630\pi\)
\(158\) 5.00000 0.397779
\(159\) 0 0
\(160\) 0 0
\(161\) 4.44949 0.350669
\(162\) 0 0
\(163\) 19.7980 1.55070 0.775348 0.631534i \(-0.217575\pi\)
0.775348 + 0.631534i \(0.217575\pi\)
\(164\) 0.898979 0.0701985
\(165\) 0 0
\(166\) 8.34847 0.647967
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) −2.44949 −0.186772
\(173\) 1.65153 0.125564 0.0627818 0.998027i \(-0.480003\pi\)
0.0627818 + 0.998027i \(0.480003\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.44949 −0.260015
\(177\) 0 0
\(178\) 2.10102 0.157478
\(179\) −25.7980 −1.92823 −0.964115 0.265485i \(-0.914468\pi\)
−0.964115 + 0.265485i \(0.914468\pi\)
\(180\) 0 0
\(181\) −14.4495 −1.07402 −0.537011 0.843575i \(-0.680447\pi\)
−0.537011 + 0.843575i \(0.680447\pi\)
\(182\) 10.8990 0.807886
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 15.3485 1.12239
\(188\) 7.79796 0.568725
\(189\) 0 0
\(190\) 0 0
\(191\) 0.101021 0.00730959 0.00365479 0.999993i \(-0.498837\pi\)
0.00365479 + 0.999993i \(0.498837\pi\)
\(192\) 0 0
\(193\) −15.3485 −1.10481 −0.552403 0.833577i \(-0.686289\pi\)
−0.552403 + 0.833577i \(0.686289\pi\)
\(194\) −1.55051 −0.111320
\(195\) 0 0
\(196\) 12.7980 0.914140
\(197\) −27.3485 −1.94850 −0.974249 0.225475i \(-0.927606\pi\)
−0.974249 + 0.225475i \(0.927606\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −1.55051 −0.109094
\(203\) 19.3485 1.35800
\(204\) 0 0
\(205\) 0 0
\(206\) −1.89898 −0.132308
\(207\) 0 0
\(208\) −2.44949 −0.169842
\(209\) −3.44949 −0.238606
\(210\) 0 0
\(211\) −3.65153 −0.251382 −0.125691 0.992069i \(-0.540115\pi\)
−0.125691 + 0.992069i \(0.540115\pi\)
\(212\) 7.44949 0.511633
\(213\) 0 0
\(214\) 17.3485 1.18592
\(215\) 0 0
\(216\) 0 0
\(217\) −13.3485 −0.906153
\(218\) 2.89898 0.196344
\(219\) 0 0
\(220\) 0 0
\(221\) 10.8990 0.733145
\(222\) 0 0
\(223\) 16.1010 1.07820 0.539102 0.842240i \(-0.318764\pi\)
0.539102 + 0.842240i \(0.318764\pi\)
\(224\) −4.44949 −0.297294
\(225\) 0 0
\(226\) −16.7980 −1.11738
\(227\) −29.5959 −1.96435 −0.982175 0.187969i \(-0.939810\pi\)
−0.982175 + 0.187969i \(0.939810\pi\)
\(228\) 0 0
\(229\) 7.24745 0.478925 0.239462 0.970906i \(-0.423029\pi\)
0.239462 + 0.970906i \(0.423029\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.34847 −0.285491
\(233\) −6.24745 −0.409284 −0.204642 0.978837i \(-0.565603\pi\)
−0.204642 + 0.978837i \(0.565603\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.44949 −0.419826
\(237\) 0 0
\(238\) 19.7980 1.28331
\(239\) 8.69694 0.562558 0.281279 0.959626i \(-0.409241\pi\)
0.281279 + 0.959626i \(0.409241\pi\)
\(240\) 0 0
\(241\) −4.44949 −0.286617 −0.143308 0.989678i \(-0.545774\pi\)
−0.143308 + 0.989678i \(0.545774\pi\)
\(242\) −0.898979 −0.0577886
\(243\) 0 0
\(244\) −9.44949 −0.604942
\(245\) 0 0
\(246\) 0 0
\(247\) −2.44949 −0.155857
\(248\) 3.00000 0.190500
\(249\) 0 0
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) −3.44949 −0.216868
\(254\) −0.101021 −0.00633859
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 21.4949 1.34081 0.670407 0.741993i \(-0.266119\pi\)
0.670407 + 0.741993i \(0.266119\pi\)
\(258\) 0 0
\(259\) −34.6969 −2.15596
\(260\) 0 0
\(261\) 0 0
\(262\) 9.24745 0.571309
\(263\) 6.79796 0.419180 0.209590 0.977789i \(-0.432787\pi\)
0.209590 + 0.977789i \(0.432787\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.44949 −0.272816
\(267\) 0 0
\(268\) 15.2474 0.931386
\(269\) 2.89898 0.176754 0.0883769 0.996087i \(-0.471832\pi\)
0.0883769 + 0.996087i \(0.471832\pi\)
\(270\) 0 0
\(271\) 22.9444 1.39377 0.696886 0.717182i \(-0.254568\pi\)
0.696886 + 0.717182i \(0.254568\pi\)
\(272\) −4.44949 −0.269790
\(273\) 0 0
\(274\) −4.89898 −0.295958
\(275\) 0 0
\(276\) 0 0
\(277\) 21.0454 1.26450 0.632248 0.774766i \(-0.282132\pi\)
0.632248 + 0.774766i \(0.282132\pi\)
\(278\) 22.2474 1.33431
\(279\) 0 0
\(280\) 0 0
\(281\) −7.00000 −0.417585 −0.208792 0.977960i \(-0.566953\pi\)
−0.208792 + 0.977960i \(0.566953\pi\)
\(282\) 0 0
\(283\) 29.7980 1.77130 0.885652 0.464349i \(-0.153712\pi\)
0.885652 + 0.464349i \(0.153712\pi\)
\(284\) 1.55051 0.0920059
\(285\) 0 0
\(286\) −8.44949 −0.499629
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) 2.79796 0.164586
\(290\) 0 0
\(291\) 0 0
\(292\) −1.00000 −0.0585206
\(293\) 23.2474 1.35813 0.679065 0.734078i \(-0.262385\pi\)
0.679065 + 0.734078i \(0.262385\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 7.79796 0.453247
\(297\) 0 0
\(298\) 5.79796 0.335867
\(299\) −2.44949 −0.141658
\(300\) 0 0
\(301\) −10.8990 −0.628207
\(302\) 23.7980 1.36942
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −16.3485 −0.933056 −0.466528 0.884506i \(-0.654495\pi\)
−0.466528 + 0.884506i \(0.654495\pi\)
\(308\) −15.3485 −0.874560
\(309\) 0 0
\(310\) 0 0
\(311\) 23.7980 1.34946 0.674729 0.738065i \(-0.264260\pi\)
0.674729 + 0.738065i \(0.264260\pi\)
\(312\) 0 0
\(313\) 31.8990 1.80304 0.901518 0.432741i \(-0.142453\pi\)
0.901518 + 0.432741i \(0.142453\pi\)
\(314\) 4.89898 0.276465
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) −15.2474 −0.856382 −0.428191 0.903688i \(-0.640849\pi\)
−0.428191 + 0.903688i \(0.640849\pi\)
\(318\) 0 0
\(319\) −15.0000 −0.839839
\(320\) 0 0
\(321\) 0 0
\(322\) −4.44949 −0.247960
\(323\) −4.44949 −0.247576
\(324\) 0 0
\(325\) 0 0
\(326\) −19.7980 −1.09651
\(327\) 0 0
\(328\) −0.898979 −0.0496378
\(329\) 34.6969 1.91290
\(330\) 0 0
\(331\) −22.3485 −1.22838 −0.614191 0.789157i \(-0.710518\pi\)
−0.614191 + 0.789157i \(0.710518\pi\)
\(332\) −8.34847 −0.458182
\(333\) 0 0
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) −24.8990 −1.35633 −0.678167 0.734908i \(-0.737225\pi\)
−0.678167 + 0.734908i \(0.737225\pi\)
\(338\) 7.00000 0.380750
\(339\) 0 0
\(340\) 0 0
\(341\) 10.3485 0.560401
\(342\) 0 0
\(343\) 25.7980 1.39296
\(344\) 2.44949 0.132068
\(345\) 0 0
\(346\) −1.65153 −0.0887868
\(347\) 23.5959 1.26670 0.633348 0.773867i \(-0.281680\pi\)
0.633348 + 0.773867i \(0.281680\pi\)
\(348\) 0 0
\(349\) −25.9444 −1.38877 −0.694386 0.719603i \(-0.744324\pi\)
−0.694386 + 0.719603i \(0.744324\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.44949 0.183858
\(353\) 18.2474 0.971214 0.485607 0.874177i \(-0.338599\pi\)
0.485607 + 0.874177i \(0.338599\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.10102 −0.111354
\(357\) 0 0
\(358\) 25.7980 1.36346
\(359\) 22.8990 1.20856 0.604281 0.796771i \(-0.293460\pi\)
0.604281 + 0.796771i \(0.293460\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 14.4495 0.759448
\(363\) 0 0
\(364\) −10.8990 −0.571262
\(365\) 0 0
\(366\) 0 0
\(367\) −5.55051 −0.289734 −0.144867 0.989451i \(-0.546275\pi\)
−0.144867 + 0.989451i \(0.546275\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 0 0
\(371\) 33.1464 1.72088
\(372\) 0 0
\(373\) −22.4495 −1.16239 −0.581195 0.813764i \(-0.697415\pi\)
−0.581195 + 0.813764i \(0.697415\pi\)
\(374\) −15.3485 −0.793650
\(375\) 0 0
\(376\) −7.79796 −0.402149
\(377\) −10.6515 −0.548582
\(378\) 0 0
\(379\) −1.30306 −0.0669338 −0.0334669 0.999440i \(-0.510655\pi\)
−0.0334669 + 0.999440i \(0.510655\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −0.101021 −0.00516866
\(383\) −16.2474 −0.830206 −0.415103 0.909774i \(-0.636254\pi\)
−0.415103 + 0.909774i \(0.636254\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 15.3485 0.781217
\(387\) 0 0
\(388\) 1.55051 0.0787152
\(389\) −21.5959 −1.09496 −0.547478 0.836820i \(-0.684412\pi\)
−0.547478 + 0.836820i \(0.684412\pi\)
\(390\) 0 0
\(391\) −4.44949 −0.225020
\(392\) −12.7980 −0.646395
\(393\) 0 0
\(394\) 27.3485 1.37780
\(395\) 0 0
\(396\) 0 0
\(397\) 23.9444 1.20173 0.600867 0.799349i \(-0.294822\pi\)
0.600867 + 0.799349i \(0.294822\pi\)
\(398\) 10.0000 0.501255
\(399\) 0 0
\(400\) 0 0
\(401\) −11.2020 −0.559403 −0.279702 0.960087i \(-0.590236\pi\)
−0.279702 + 0.960087i \(0.590236\pi\)
\(402\) 0 0
\(403\) 7.34847 0.366053
\(404\) 1.55051 0.0771408
\(405\) 0 0
\(406\) −19.3485 −0.960248
\(407\) 26.8990 1.33333
\(408\) 0 0
\(409\) −22.8990 −1.13228 −0.566141 0.824309i \(-0.691564\pi\)
−0.566141 + 0.824309i \(0.691564\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.89898 0.0935560
\(413\) −28.6969 −1.41208
\(414\) 0 0
\(415\) 0 0
\(416\) 2.44949 0.120096
\(417\) 0 0
\(418\) 3.44949 0.168720
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 3.65153 0.177754
\(423\) 0 0
\(424\) −7.44949 −0.361779
\(425\) 0 0
\(426\) 0 0
\(427\) −42.0454 −2.03472
\(428\) −17.3485 −0.838570
\(429\) 0 0
\(430\) 0 0
\(431\) 5.10102 0.245708 0.122854 0.992425i \(-0.460795\pi\)
0.122854 + 0.992425i \(0.460795\pi\)
\(432\) 0 0
\(433\) 4.65153 0.223538 0.111769 0.993734i \(-0.464348\pi\)
0.111769 + 0.993734i \(0.464348\pi\)
\(434\) 13.3485 0.640747
\(435\) 0 0
\(436\) −2.89898 −0.138836
\(437\) 1.00000 0.0478365
\(438\) 0 0
\(439\) −12.1010 −0.577550 −0.288775 0.957397i \(-0.593248\pi\)
−0.288775 + 0.957397i \(0.593248\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −10.8990 −0.518412
\(443\) −21.2474 −1.00950 −0.504748 0.863267i \(-0.668415\pi\)
−0.504748 + 0.863267i \(0.668415\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −16.1010 −0.762405
\(447\) 0 0
\(448\) 4.44949 0.210219
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) −3.10102 −0.146021
\(452\) 16.7980 0.790110
\(453\) 0 0
\(454\) 29.5959 1.38901
\(455\) 0 0
\(456\) 0 0
\(457\) −24.8990 −1.16473 −0.582363 0.812929i \(-0.697872\pi\)
−0.582363 + 0.812929i \(0.697872\pi\)
\(458\) −7.24745 −0.338651
\(459\) 0 0
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) −14.6969 −0.683025 −0.341512 0.939877i \(-0.610939\pi\)
−0.341512 + 0.939877i \(0.610939\pi\)
\(464\) 4.34847 0.201873
\(465\) 0 0
\(466\) 6.24745 0.289407
\(467\) 6.34847 0.293772 0.146886 0.989153i \(-0.453075\pi\)
0.146886 + 0.989153i \(0.453075\pi\)
\(468\) 0 0
\(469\) 67.8434 3.13272
\(470\) 0 0
\(471\) 0 0
\(472\) 6.44949 0.296862
\(473\) 8.44949 0.388508
\(474\) 0 0
\(475\) 0 0
\(476\) −19.7980 −0.907438
\(477\) 0 0
\(478\) −8.69694 −0.397789
\(479\) −16.5959 −0.758287 −0.379143 0.925338i \(-0.623781\pi\)
−0.379143 + 0.925338i \(0.623781\pi\)
\(480\) 0 0
\(481\) 19.1010 0.870932
\(482\) 4.44949 0.202669
\(483\) 0 0
\(484\) 0.898979 0.0408627
\(485\) 0 0
\(486\) 0 0
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 9.44949 0.427758
\(489\) 0 0
\(490\) 0 0
\(491\) −43.5959 −1.96746 −0.983728 0.179664i \(-0.942499\pi\)
−0.983728 + 0.179664i \(0.942499\pi\)
\(492\) 0 0
\(493\) −19.3485 −0.871411
\(494\) 2.44949 0.110208
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) 6.89898 0.309462
\(498\) 0 0
\(499\) −23.8434 −1.06738 −0.533688 0.845682i \(-0.679194\pi\)
−0.533688 + 0.845682i \(0.679194\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −18.0000 −0.803379
\(503\) −29.7980 −1.32863 −0.664313 0.747455i \(-0.731276\pi\)
−0.664313 + 0.747455i \(0.731276\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 3.44949 0.153349
\(507\) 0 0
\(508\) 0.101021 0.00448206
\(509\) 30.1464 1.33622 0.668108 0.744064i \(-0.267104\pi\)
0.668108 + 0.744064i \(0.267104\pi\)
\(510\) 0 0
\(511\) −4.44949 −0.196834
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −21.4949 −0.948099
\(515\) 0 0
\(516\) 0 0
\(517\) −26.8990 −1.18302
\(518\) 34.6969 1.52450
\(519\) 0 0
\(520\) 0 0
\(521\) −15.6969 −0.687695 −0.343848 0.939025i \(-0.611730\pi\)
−0.343848 + 0.939025i \(0.611730\pi\)
\(522\) 0 0
\(523\) −33.3939 −1.46021 −0.730106 0.683334i \(-0.760529\pi\)
−0.730106 + 0.683334i \(0.760529\pi\)
\(524\) −9.24745 −0.403977
\(525\) 0 0
\(526\) −6.79796 −0.296405
\(527\) 13.3485 0.581468
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) 4.44949 0.192910
\(533\) −2.20204 −0.0953810
\(534\) 0 0
\(535\) 0 0
\(536\) −15.2474 −0.658589
\(537\) 0 0
\(538\) −2.89898 −0.124984
\(539\) −44.1464 −1.90152
\(540\) 0 0
\(541\) 22.1464 0.952149 0.476075 0.879405i \(-0.342059\pi\)
0.476075 + 0.879405i \(0.342059\pi\)
\(542\) −22.9444 −0.985546
\(543\) 0 0
\(544\) 4.44949 0.190770
\(545\) 0 0
\(546\) 0 0
\(547\) −46.6413 −1.99424 −0.997120 0.0758461i \(-0.975834\pi\)
−0.997120 + 0.0758461i \(0.975834\pi\)
\(548\) 4.89898 0.209274
\(549\) 0 0
\(550\) 0 0
\(551\) 4.34847 0.185251
\(552\) 0 0
\(553\) −22.2474 −0.946058
\(554\) −21.0454 −0.894134
\(555\) 0 0
\(556\) −22.2474 −0.943502
\(557\) −27.3485 −1.15879 −0.579396 0.815046i \(-0.696711\pi\)
−0.579396 + 0.815046i \(0.696711\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) 0 0
\(562\) 7.00000 0.295277
\(563\) −6.24745 −0.263299 −0.131649 0.991296i \(-0.542027\pi\)
−0.131649 + 0.991296i \(0.542027\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −29.7980 −1.25250
\(567\) 0 0
\(568\) −1.55051 −0.0650580
\(569\) 33.1918 1.39147 0.695737 0.718297i \(-0.255078\pi\)
0.695737 + 0.718297i \(0.255078\pi\)
\(570\) 0 0
\(571\) −10.2474 −0.428842 −0.214421 0.976741i \(-0.568787\pi\)
−0.214421 + 0.976741i \(0.568787\pi\)
\(572\) 8.44949 0.353291
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) 0 0
\(577\) 0.101021 0.00420554 0.00210277 0.999998i \(-0.499331\pi\)
0.00210277 + 0.999998i \(0.499331\pi\)
\(578\) −2.79796 −0.116380
\(579\) 0 0
\(580\) 0 0
\(581\) −37.1464 −1.54109
\(582\) 0 0
\(583\) −25.6969 −1.06426
\(584\) 1.00000 0.0413803
\(585\) 0 0
\(586\) −23.2474 −0.960343
\(587\) 33.4495 1.38061 0.690304 0.723519i \(-0.257477\pi\)
0.690304 + 0.723519i \(0.257477\pi\)
\(588\) 0 0
\(589\) −3.00000 −0.123613
\(590\) 0 0
\(591\) 0 0
\(592\) −7.79796 −0.320494
\(593\) −8.49490 −0.348844 −0.174422 0.984671i \(-0.555806\pi\)
−0.174422 + 0.984671i \(0.555806\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −5.79796 −0.237494
\(597\) 0 0
\(598\) 2.44949 0.100167
\(599\) −16.4495 −0.672108 −0.336054 0.941843i \(-0.609092\pi\)
−0.336054 + 0.941843i \(0.609092\pi\)
\(600\) 0 0
\(601\) 17.1464 0.699417 0.349709 0.936858i \(-0.386281\pi\)
0.349709 + 0.936858i \(0.386281\pi\)
\(602\) 10.8990 0.444209
\(603\) 0 0
\(604\) −23.7980 −0.968325
\(605\) 0 0
\(606\) 0 0
\(607\) 46.1918 1.87487 0.937434 0.348162i \(-0.113194\pi\)
0.937434 + 0.348162i \(0.113194\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −19.1010 −0.772745
\(612\) 0 0
\(613\) 37.1918 1.50216 0.751082 0.660209i \(-0.229532\pi\)
0.751082 + 0.660209i \(0.229532\pi\)
\(614\) 16.3485 0.659771
\(615\) 0 0
\(616\) 15.3485 0.618407
\(617\) −28.2929 −1.13903 −0.569514 0.821982i \(-0.692868\pi\)
−0.569514 + 0.821982i \(0.692868\pi\)
\(618\) 0 0
\(619\) 45.1464 1.81459 0.907294 0.420497i \(-0.138144\pi\)
0.907294 + 0.420497i \(0.138144\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −23.7980 −0.954211
\(623\) −9.34847 −0.374539
\(624\) 0 0
\(625\) 0 0
\(626\) −31.8990 −1.27494
\(627\) 0 0
\(628\) −4.89898 −0.195491
\(629\) 34.6969 1.38346
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 5.00000 0.198889
\(633\) 0 0
\(634\) 15.2474 0.605554
\(635\) 0 0
\(636\) 0 0
\(637\) −31.3485 −1.24207
\(638\) 15.0000 0.593856
\(639\) 0 0
\(640\) 0 0
\(641\) −37.7980 −1.49293 −0.746465 0.665425i \(-0.768250\pi\)
−0.746465 + 0.665425i \(0.768250\pi\)
\(642\) 0 0
\(643\) −0.202041 −0.00796772 −0.00398386 0.999992i \(-0.501268\pi\)
−0.00398386 + 0.999992i \(0.501268\pi\)
\(644\) 4.44949 0.175334
\(645\) 0 0
\(646\) 4.44949 0.175063
\(647\) −25.8990 −1.01819 −0.509097 0.860709i \(-0.670021\pi\)
−0.509097 + 0.860709i \(0.670021\pi\)
\(648\) 0 0
\(649\) 22.2474 0.873289
\(650\) 0 0
\(651\) 0 0
\(652\) 19.7980 0.775348
\(653\) 14.6969 0.575136 0.287568 0.957760i \(-0.407153\pi\)
0.287568 + 0.957760i \(0.407153\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.898979 0.0350993
\(657\) 0 0
\(658\) −34.6969 −1.35263
\(659\) 18.0454 0.702949 0.351475 0.936197i \(-0.385680\pi\)
0.351475 + 0.936197i \(0.385680\pi\)
\(660\) 0 0
\(661\) 23.5959 0.917775 0.458887 0.888494i \(-0.348248\pi\)
0.458887 + 0.888494i \(0.348248\pi\)
\(662\) 22.3485 0.868598
\(663\) 0 0
\(664\) 8.34847 0.323983
\(665\) 0 0
\(666\) 0 0
\(667\) 4.34847 0.168373
\(668\) −18.0000 −0.696441
\(669\) 0 0
\(670\) 0 0
\(671\) 32.5959 1.25835
\(672\) 0 0
\(673\) −45.3485 −1.74806 −0.874028 0.485876i \(-0.838501\pi\)
−0.874028 + 0.485876i \(0.838501\pi\)
\(674\) 24.8990 0.959073
\(675\) 0 0
\(676\) −7.00000 −0.269231
\(677\) −22.3485 −0.858921 −0.429461 0.903086i \(-0.641296\pi\)
−0.429461 + 0.903086i \(0.641296\pi\)
\(678\) 0 0
\(679\) 6.89898 0.264759
\(680\) 0 0
\(681\) 0 0
\(682\) −10.3485 −0.396263
\(683\) 35.6413 1.36378 0.681889 0.731456i \(-0.261159\pi\)
0.681889 + 0.731456i \(0.261159\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −25.7980 −0.984971
\(687\) 0 0
\(688\) −2.44949 −0.0933859
\(689\) −18.2474 −0.695172
\(690\) 0 0
\(691\) 9.75255 0.371005 0.185502 0.982644i \(-0.440609\pi\)
0.185502 + 0.982644i \(0.440609\pi\)
\(692\) 1.65153 0.0627818
\(693\) 0 0
\(694\) −23.5959 −0.895689
\(695\) 0 0
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 25.9444 0.982010
\(699\) 0 0
\(700\) 0 0
\(701\) 32.4949 1.22732 0.613658 0.789572i \(-0.289698\pi\)
0.613658 + 0.789572i \(0.289698\pi\)
\(702\) 0 0
\(703\) −7.79796 −0.294106
\(704\) −3.44949 −0.130008
\(705\) 0 0
\(706\) −18.2474 −0.686752
\(707\) 6.89898 0.259463
\(708\) 0 0
\(709\) −12.7526 −0.478932 −0.239466 0.970905i \(-0.576972\pi\)
−0.239466 + 0.970905i \(0.576972\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.10102 0.0787391
\(713\) −3.00000 −0.112351
\(714\) 0 0
\(715\) 0 0
\(716\) −25.7980 −0.964115
\(717\) 0 0
\(718\) −22.8990 −0.854582
\(719\) 19.2020 0.716115 0.358058 0.933699i \(-0.383439\pi\)
0.358058 + 0.933699i \(0.383439\pi\)
\(720\) 0 0
\(721\) 8.44949 0.314675
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) −14.4495 −0.537011
\(725\) 0 0
\(726\) 0 0
\(727\) 22.4949 0.834290 0.417145 0.908840i \(-0.363031\pi\)
0.417145 + 0.908840i \(0.363031\pi\)
\(728\) 10.8990 0.403943
\(729\) 0 0
\(730\) 0 0
\(731\) 10.8990 0.403113
\(732\) 0 0
\(733\) 4.14643 0.153152 0.0765759 0.997064i \(-0.475601\pi\)
0.0765759 + 0.997064i \(0.475601\pi\)
\(734\) 5.55051 0.204873
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −52.5959 −1.93740
\(738\) 0 0
\(739\) 32.8990 1.21021 0.605104 0.796146i \(-0.293131\pi\)
0.605104 + 0.796146i \(0.293131\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −33.1464 −1.21684
\(743\) 53.3939 1.95883 0.979416 0.201854i \(-0.0646966\pi\)
0.979416 + 0.201854i \(0.0646966\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 22.4495 0.821934
\(747\) 0 0
\(748\) 15.3485 0.561196
\(749\) −77.1918 −2.82053
\(750\) 0 0
\(751\) −23.7980 −0.868400 −0.434200 0.900817i \(-0.642969\pi\)
−0.434200 + 0.900817i \(0.642969\pi\)
\(752\) 7.79796 0.284362
\(753\) 0 0
\(754\) 10.6515 0.387906
\(755\) 0 0
\(756\) 0 0
\(757\) 48.1464 1.74991 0.874956 0.484203i \(-0.160890\pi\)
0.874956 + 0.484203i \(0.160890\pi\)
\(758\) 1.30306 0.0473293
\(759\) 0 0
\(760\) 0 0
\(761\) −51.3485 −1.86138 −0.930690 0.365808i \(-0.880793\pi\)
−0.930690 + 0.365808i \(0.880793\pi\)
\(762\) 0 0
\(763\) −12.8990 −0.466974
\(764\) 0.101021 0.00365479
\(765\) 0 0
\(766\) 16.2474 0.587044
\(767\) 15.7980 0.570431
\(768\) 0 0
\(769\) −7.89898 −0.284844 −0.142422 0.989806i \(-0.545489\pi\)
−0.142422 + 0.989806i \(0.545489\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −15.3485 −0.552403
\(773\) −18.4949 −0.665215 −0.332608 0.943065i \(-0.607928\pi\)
−0.332608 + 0.943065i \(0.607928\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.55051 −0.0556601
\(777\) 0 0
\(778\) 21.5959 0.774251
\(779\) 0.898979 0.0322093
\(780\) 0 0
\(781\) −5.34847 −0.191383
\(782\) 4.44949 0.159113
\(783\) 0 0
\(784\) 12.7980 0.457070
\(785\) 0 0
\(786\) 0 0
\(787\) −33.4495 −1.19235 −0.596173 0.802856i \(-0.703313\pi\)
−0.596173 + 0.802856i \(0.703313\pi\)
\(788\) −27.3485 −0.974249
\(789\) 0 0
\(790\) 0 0
\(791\) 74.7423 2.65753
\(792\) 0 0
\(793\) 23.1464 0.821954
\(794\) −23.9444 −0.849755
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) −3.50510 −0.124157 −0.0620786 0.998071i \(-0.519773\pi\)
−0.0620786 + 0.998071i \(0.519773\pi\)
\(798\) 0 0
\(799\) −34.6969 −1.22749
\(800\) 0 0
\(801\) 0 0
\(802\) 11.2020 0.395558
\(803\) 3.44949 0.121730
\(804\) 0 0
\(805\) 0 0
\(806\) −7.34847 −0.258839
\(807\) 0 0
\(808\) −1.55051 −0.0545468
\(809\) 3.55051 0.124829 0.0624146 0.998050i \(-0.480120\pi\)
0.0624146 + 0.998050i \(0.480120\pi\)
\(810\) 0 0
\(811\) −19.4495 −0.682964 −0.341482 0.939888i \(-0.610929\pi\)
−0.341482 + 0.939888i \(0.610929\pi\)
\(812\) 19.3485 0.678998
\(813\) 0 0
\(814\) −26.8990 −0.942809
\(815\) 0 0
\(816\) 0 0
\(817\) −2.44949 −0.0856968
\(818\) 22.8990 0.800644
\(819\) 0 0
\(820\) 0 0
\(821\) 23.1464 0.807816 0.403908 0.914800i \(-0.367652\pi\)
0.403908 + 0.914800i \(0.367652\pi\)
\(822\) 0 0
\(823\) −31.7980 −1.10841 −0.554204 0.832381i \(-0.686977\pi\)
−0.554204 + 0.832381i \(0.686977\pi\)
\(824\) −1.89898 −0.0661541
\(825\) 0 0
\(826\) 28.6969 0.998494
\(827\) −0.247449 −0.00860463 −0.00430232 0.999991i \(-0.501369\pi\)
−0.00430232 + 0.999991i \(0.501369\pi\)
\(828\) 0 0
\(829\) −39.6413 −1.37680 −0.688400 0.725331i \(-0.741687\pi\)
−0.688400 + 0.725331i \(0.741687\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.44949 −0.0849208
\(833\) −56.9444 −1.97301
\(834\) 0 0
\(835\) 0 0
\(836\) −3.44949 −0.119303
\(837\) 0 0
\(838\) 0 0
\(839\) −10.6515 −0.367732 −0.183866 0.982951i \(-0.558861\pi\)
−0.183866 + 0.982951i \(0.558861\pi\)
\(840\) 0 0
\(841\) −10.0908 −0.347959
\(842\) −22.0000 −0.758170
\(843\) 0 0
\(844\) −3.65153 −0.125691
\(845\) 0 0
\(846\) 0 0
\(847\) 4.00000 0.137442
\(848\) 7.44949 0.255817
\(849\) 0 0
\(850\) 0 0
\(851\) −7.79796 −0.267311
\(852\) 0 0
\(853\) 1.10102 0.0376982 0.0188491 0.999822i \(-0.494000\pi\)
0.0188491 + 0.999822i \(0.494000\pi\)
\(854\) 42.0454 1.43876
\(855\) 0 0
\(856\) 17.3485 0.592958
\(857\) 36.4949 1.24664 0.623321 0.781966i \(-0.285783\pi\)
0.623321 + 0.781966i \(0.285783\pi\)
\(858\) 0 0
\(859\) −4.20204 −0.143372 −0.0716859 0.997427i \(-0.522838\pi\)
−0.0716859 + 0.997427i \(0.522838\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −5.10102 −0.173741
\(863\) 17.3031 0.589003 0.294502 0.955651i \(-0.404846\pi\)
0.294502 + 0.955651i \(0.404846\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −4.65153 −0.158065
\(867\) 0 0
\(868\) −13.3485 −0.453077
\(869\) 17.2474 0.585080
\(870\) 0 0
\(871\) −37.3485 −1.26550
\(872\) 2.89898 0.0981718
\(873\) 0 0
\(874\) −1.00000 −0.0338255
\(875\) 0 0
\(876\) 0 0
\(877\) −9.10102 −0.307320 −0.153660 0.988124i \(-0.549106\pi\)
−0.153660 + 0.988124i \(0.549106\pi\)
\(878\) 12.1010 0.408390
\(879\) 0 0
\(880\) 0 0
\(881\) −50.6969 −1.70802 −0.854012 0.520254i \(-0.825837\pi\)
−0.854012 + 0.520254i \(0.825837\pi\)
\(882\) 0 0
\(883\) 34.6515 1.16612 0.583058 0.812430i \(-0.301856\pi\)
0.583058 + 0.812430i \(0.301856\pi\)
\(884\) 10.8990 0.366572
\(885\) 0 0
\(886\) 21.2474 0.713822
\(887\) 4.89898 0.164492 0.0822458 0.996612i \(-0.473791\pi\)
0.0822458 + 0.996612i \(0.473791\pi\)
\(888\) 0 0
\(889\) 0.449490 0.0150754
\(890\) 0 0
\(891\) 0 0
\(892\) 16.1010 0.539102
\(893\) 7.79796 0.260949
\(894\) 0 0
\(895\) 0 0
\(896\) −4.44949 −0.148647
\(897\) 0 0
\(898\) −15.0000 −0.500556
\(899\) −13.0454 −0.435089
\(900\) 0 0
\(901\) −33.1464 −1.10427
\(902\) 3.10102 0.103253
\(903\) 0 0
\(904\) −16.7980 −0.558692
\(905\) 0 0
\(906\) 0 0
\(907\) −22.0000 −0.730498 −0.365249 0.930910i \(-0.619016\pi\)
−0.365249 + 0.930910i \(0.619016\pi\)
\(908\) −29.5959 −0.982175
\(909\) 0 0
\(910\) 0 0
\(911\) −6.49490 −0.215186 −0.107593 0.994195i \(-0.534314\pi\)
−0.107593 + 0.994195i \(0.534314\pi\)
\(912\) 0 0
\(913\) 28.7980 0.953073
\(914\) 24.8990 0.823585
\(915\) 0 0
\(916\) 7.24745 0.239462
\(917\) −41.1464 −1.35877
\(918\) 0 0
\(919\) 23.8434 0.786520 0.393260 0.919427i \(-0.371347\pi\)
0.393260 + 0.919427i \(0.371347\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 12.0000 0.395199
\(923\) −3.79796 −0.125011
\(924\) 0 0
\(925\) 0 0
\(926\) 14.6969 0.482971
\(927\) 0 0
\(928\) −4.34847 −0.142745
\(929\) 31.5959 1.03663 0.518314 0.855190i \(-0.326560\pi\)
0.518314 + 0.855190i \(0.326560\pi\)
\(930\) 0 0
\(931\) 12.7980 0.419436
\(932\) −6.24745 −0.204642
\(933\) 0 0
\(934\) −6.34847 −0.207728
\(935\) 0 0
\(936\) 0 0
\(937\) −19.3939 −0.633570 −0.316785 0.948497i \(-0.602603\pi\)
−0.316785 + 0.948497i \(0.602603\pi\)
\(938\) −67.8434 −2.21516
\(939\) 0 0
\(940\) 0 0
\(941\) −36.6413 −1.19447 −0.597237 0.802065i \(-0.703735\pi\)
−0.597237 + 0.802065i \(0.703735\pi\)
\(942\) 0 0
\(943\) 0.898979 0.0292748
\(944\) −6.44949 −0.209913
\(945\) 0 0
\(946\) −8.44949 −0.274717
\(947\) −8.00000 −0.259965 −0.129983 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) 0 0
\(949\) 2.44949 0.0795138
\(950\) 0 0
\(951\) 0 0
\(952\) 19.7980 0.641656
\(953\) 15.4949 0.501929 0.250964 0.967996i \(-0.419252\pi\)
0.250964 + 0.967996i \(0.419252\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 8.69694 0.281279
\(957\) 0 0
\(958\) 16.5959 0.536190
\(959\) 21.7980 0.703893
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −19.1010 −0.615842
\(963\) 0 0
\(964\) −4.44949 −0.143308
\(965\) 0 0
\(966\) 0 0
\(967\) −34.8990 −1.12228 −0.561138 0.827722i \(-0.689636\pi\)
−0.561138 + 0.827722i \(0.689636\pi\)
\(968\) −0.898979 −0.0288943
\(969\) 0 0
\(970\) 0 0
\(971\) −41.6413 −1.33633 −0.668167 0.744011i \(-0.732921\pi\)
−0.668167 + 0.744011i \(0.732921\pi\)
\(972\) 0 0
\(973\) −98.9898 −3.17347
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) −9.44949 −0.302471
\(977\) −19.5959 −0.626929 −0.313464 0.949600i \(-0.601490\pi\)
−0.313464 + 0.949600i \(0.601490\pi\)
\(978\) 0 0
\(979\) 7.24745 0.231629
\(980\) 0 0
\(981\) 0 0
\(982\) 43.5959 1.39120
\(983\) −50.4495 −1.60909 −0.804544 0.593892i \(-0.797590\pi\)
−0.804544 + 0.593892i \(0.797590\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 19.3485 0.616181
\(987\) 0 0
\(988\) −2.44949 −0.0779287
\(989\) −2.44949 −0.0778892
\(990\) 0 0
\(991\) 44.1010 1.40092 0.700458 0.713694i \(-0.252979\pi\)
0.700458 + 0.713694i \(0.252979\pi\)
\(992\) 3.00000 0.0952501
\(993\) 0 0
\(994\) −6.89898 −0.218822
\(995\) 0 0
\(996\) 0 0
\(997\) 39.4495 1.24938 0.624689 0.780874i \(-0.285226\pi\)
0.624689 + 0.780874i \(0.285226\pi\)
\(998\) 23.8434 0.754749
\(999\) 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.bu.1.2 2
3.2 odd 2 2850.2.a.bj.1.2 yes 2
5.4 even 2 8550.2.a.bv.1.1 2
15.2 even 4 2850.2.d.w.799.4 4
15.8 even 4 2850.2.d.w.799.1 4
15.14 odd 2 2850.2.a.bc.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.bc.1.1 2 15.14 odd 2
2850.2.a.bj.1.2 yes 2 3.2 odd 2
2850.2.d.w.799.1 4 15.8 even 4
2850.2.d.w.799.4 4 15.2 even 4
8550.2.a.bu.1.2 2 1.1 even 1 trivial
8550.2.a.bv.1.1 2 5.4 even 2