Properties

Label 8550.2.a.cf.1.3
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +3.35026 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +3.35026 q^{7} -1.00000 q^{8} +1.61213 q^{11} -1.35026 q^{13} -3.35026 q^{14} +1.00000 q^{16} -6.96239 q^{17} -1.00000 q^{19} -1.61213 q^{22} +1.35026 q^{23} +1.35026 q^{26} +3.35026 q^{28} -3.61213 q^{29} -2.31265 q^{31} -1.00000 q^{32} +6.96239 q^{34} +11.2750 q^{37} +1.00000 q^{38} +3.35026 q^{41} -10.3127 q^{43} +1.61213 q^{44} -1.35026 q^{46} +4.57452 q^{47} +4.22425 q^{49} -1.35026 q^{52} -11.9248 q^{53} -3.35026 q^{56} +3.61213 q^{58} -1.03761 q^{59} +2.00000 q^{61} +2.31265 q^{62} +1.00000 q^{64} -9.92478 q^{67} -6.96239 q^{68} +0.775746 q^{71} -3.22425 q^{73} -11.2750 q^{74} -1.00000 q^{76} +5.40105 q^{77} +14.3127 q^{79} -3.35026 q^{82} -10.8872 q^{83} +10.3127 q^{86} -1.61213 q^{88} +2.57452 q^{89} -4.52373 q^{91} +1.35026 q^{92} -4.57452 q^{94} -1.16362 q^{97} -4.22425 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} - 3q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} - 3q^{8} + 4q^{11} + 6q^{13} + 3q^{16} - 10q^{17} - 3q^{19} - 4q^{22} - 6q^{23} - 6q^{26} - 10q^{29} + 14q^{31} - 3q^{32} + 10q^{34} + 2q^{37} + 3q^{38} - 10q^{43} + 4q^{44} + 6q^{46} + 2q^{47} + 11q^{49} + 6q^{52} - 14q^{53} + 10q^{58} - 14q^{59} + 6q^{61} - 14q^{62} + 3q^{64} - 8q^{67} - 10q^{68} + 4q^{71} - 8q^{73} - 2q^{74} - 3q^{76} - 24q^{77} + 22q^{79} + 10q^{86} - 4q^{88} - 4q^{89} - 32q^{91} - 6q^{92} - 2q^{94} - 6q^{97} - 11q^{98} + O(q^{100}) \)

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
\(7\) 3.35026 1.26628 0.633140 0.774037i \(-0.281766\pi\)
0.633140 + 0.774037i \(0.281766\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 1.61213 0.486075 0.243037 0.970017i \(-0.421856\pi\)
0.243037 + 0.970017i \(0.421856\pi\)
\(12\) 0 0
\(13\) −1.35026 −0.374495 −0.187248 0.982313i \(-0.559957\pi\)
−0.187248 + 0.982313i \(0.559957\pi\)
\(14\) −3.35026 −0.895395
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.96239 −1.68863 −0.844314 0.535849i \(-0.819992\pi\)
−0.844314 + 0.535849i \(0.819992\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −1.61213 −0.343707
\(23\) 1.35026 0.281549 0.140775 0.990042i \(-0.455041\pi\)
0.140775 + 0.990042i \(0.455041\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.35026 0.264808
\(27\) 0 0
\(28\) 3.35026 0.633140
\(29\) −3.61213 −0.670755 −0.335378 0.942084i \(-0.608864\pi\)
−0.335378 + 0.942084i \(0.608864\pi\)
\(30\) 0 0
\(31\) −2.31265 −0.415364 −0.207682 0.978196i \(-0.566592\pi\)
−0.207682 + 0.978196i \(0.566592\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.96239 1.19404
\(35\) 0 0
\(36\) 0 0
\(37\) 11.2750 1.85360 0.926802 0.375549i \(-0.122546\pi\)
0.926802 + 0.375549i \(0.122546\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 3.35026 0.523223 0.261611 0.965173i \(-0.415746\pi\)
0.261611 + 0.965173i \(0.415746\pi\)
\(42\) 0 0
\(43\) −10.3127 −1.57266 −0.786332 0.617804i \(-0.788023\pi\)
−0.786332 + 0.617804i \(0.788023\pi\)
\(44\) 1.61213 0.243037
\(45\) 0 0
\(46\) −1.35026 −0.199085
\(47\) 4.57452 0.667262 0.333631 0.942704i \(-0.391726\pi\)
0.333631 + 0.942704i \(0.391726\pi\)
\(48\) 0 0
\(49\) 4.22425 0.603465
\(50\) 0 0
\(51\) 0 0
\(52\) −1.35026 −0.187248
\(53\) −11.9248 −1.63799 −0.818997 0.573798i \(-0.805470\pi\)
−0.818997 + 0.573798i \(0.805470\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.35026 −0.447698
\(57\) 0 0
\(58\) 3.61213 0.474295
\(59\) −1.03761 −0.135085 −0.0675427 0.997716i \(-0.521516\pi\)
−0.0675427 + 0.997716i \(0.521516\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 2.31265 0.293707
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −9.92478 −1.21250 −0.606252 0.795272i \(-0.707328\pi\)
−0.606252 + 0.795272i \(0.707328\pi\)
\(68\) −6.96239 −0.844314
\(69\) 0 0
\(70\) 0 0
\(71\) 0.775746 0.0920641 0.0460321 0.998940i \(-0.485342\pi\)
0.0460321 + 0.998940i \(0.485342\pi\)
\(72\) 0 0
\(73\) −3.22425 −0.377370 −0.188685 0.982038i \(-0.560423\pi\)
−0.188685 + 0.982038i \(0.560423\pi\)
\(74\) −11.2750 −1.31070
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 5.40105 0.615506
\(78\) 0 0
\(79\) 14.3127 1.61030 0.805149 0.593072i \(-0.202085\pi\)
0.805149 + 0.593072i \(0.202085\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −3.35026 −0.369975
\(83\) −10.8872 −1.19502 −0.597511 0.801861i \(-0.703844\pi\)
−0.597511 + 0.801861i \(0.703844\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.3127 1.11204
\(87\) 0 0
\(88\) −1.61213 −0.171853
\(89\) 2.57452 0.272898 0.136449 0.990647i \(-0.456431\pi\)
0.136449 + 0.990647i \(0.456431\pi\)
\(90\) 0 0
\(91\) −4.52373 −0.474216
\(92\) 1.35026 0.140775
\(93\) 0 0
\(94\) −4.57452 −0.471825
\(95\) 0 0
\(96\) 0 0
\(97\) −1.16362 −0.118148 −0.0590738 0.998254i \(-0.518815\pi\)
−0.0590738 + 0.998254i \(0.518815\pi\)
\(98\) −4.22425 −0.426714
\(99\) 0 0
\(100\) 0 0
\(101\) −8.88717 −0.884306 −0.442153 0.896940i \(-0.645785\pi\)
−0.442153 + 0.896940i \(0.645785\pi\)
\(102\) 0 0
\(103\) −7.03761 −0.693436 −0.346718 0.937969i \(-0.612704\pi\)
−0.346718 + 0.937969i \(0.612704\pi\)
\(104\) 1.35026 0.132404
\(105\) 0 0
\(106\) 11.9248 1.15824
\(107\) −0.775746 −0.0749942 −0.0374971 0.999297i \(-0.511938\pi\)
−0.0374971 + 0.999297i \(0.511938\pi\)
\(108\) 0 0
\(109\) −20.1622 −1.93119 −0.965594 0.260052i \(-0.916260\pi\)
−0.965594 + 0.260052i \(0.916260\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.35026 0.316570
\(113\) 11.1490 1.04881 0.524406 0.851468i \(-0.324287\pi\)
0.524406 + 0.851468i \(0.324287\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.61213 −0.335378
\(117\) 0 0
\(118\) 1.03761 0.0955199
\(119\) −23.3258 −2.13827
\(120\) 0 0
\(121\) −8.40105 −0.763732
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) −2.31265 −0.207682
\(125\) 0 0
\(126\) 0 0
\(127\) −13.7381 −1.21906 −0.609531 0.792762i \(-0.708642\pi\)
−0.609531 + 0.792762i \(0.708642\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −5.61213 −0.490334 −0.245167 0.969481i \(-0.578843\pi\)
−0.245167 + 0.969481i \(0.578843\pi\)
\(132\) 0 0
\(133\) −3.35026 −0.290505
\(134\) 9.92478 0.857370
\(135\) 0 0
\(136\) 6.96239 0.597020
\(137\) −17.6629 −1.50904 −0.754522 0.656275i \(-0.772131\pi\)
−0.754522 + 0.656275i \(0.772131\pi\)
\(138\) 0 0
\(139\) 10.7005 0.907607 0.453803 0.891102i \(-0.350067\pi\)
0.453803 + 0.891102i \(0.350067\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.775746 −0.0650992
\(143\) −2.17679 −0.182033
\(144\) 0 0
\(145\) 0 0
\(146\) 3.22425 0.266841
\(147\) 0 0
\(148\) 11.2750 0.926802
\(149\) −1.03761 −0.0850044 −0.0425022 0.999096i \(-0.513533\pi\)
−0.0425022 + 0.999096i \(0.513533\pi\)
\(150\) 0 0
\(151\) 1.16362 0.0946940 0.0473470 0.998879i \(-0.484923\pi\)
0.0473470 + 0.998879i \(0.484923\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) −5.40105 −0.435229
\(155\) 0 0
\(156\) 0 0
\(157\) −10.9624 −0.874894 −0.437447 0.899244i \(-0.644117\pi\)
−0.437447 + 0.899244i \(0.644117\pi\)
\(158\) −14.3127 −1.13865
\(159\) 0 0
\(160\) 0 0
\(161\) 4.52373 0.356520
\(162\) 0 0
\(163\) −21.0132 −1.64588 −0.822939 0.568129i \(-0.807667\pi\)
−0.822939 + 0.568129i \(0.807667\pi\)
\(164\) 3.35026 0.261611
\(165\) 0 0
\(166\) 10.8872 0.845008
\(167\) 9.92478 0.768002 0.384001 0.923333i \(-0.374546\pi\)
0.384001 + 0.923333i \(0.374546\pi\)
\(168\) 0 0
\(169\) −11.1768 −0.859753
\(170\) 0 0
\(171\) 0 0
\(172\) −10.3127 −0.786332
\(173\) 14.6253 1.11194 0.555971 0.831202i \(-0.312347\pi\)
0.555971 + 0.831202i \(0.312347\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.61213 0.121519
\(177\) 0 0
\(178\) −2.57452 −0.192968
\(179\) −11.7381 −0.877349 −0.438675 0.898646i \(-0.644552\pi\)
−0.438675 + 0.898646i \(0.644552\pi\)
\(180\) 0 0
\(181\) 21.4617 1.59523 0.797617 0.603164i \(-0.206094\pi\)
0.797617 + 0.603164i \(0.206094\pi\)
\(182\) 4.52373 0.335321
\(183\) 0 0
\(184\) −1.35026 −0.0995426
\(185\) 0 0
\(186\) 0 0
\(187\) −11.2243 −0.820799
\(188\) 4.57452 0.333631
\(189\) 0 0
\(190\) 0 0
\(191\) 21.2750 1.53941 0.769704 0.638401i \(-0.220404\pi\)
0.769704 + 0.638401i \(0.220404\pi\)
\(192\) 0 0
\(193\) 14.1622 1.01942 0.509709 0.860347i \(-0.329753\pi\)
0.509709 + 0.860347i \(0.329753\pi\)
\(194\) 1.16362 0.0835430
\(195\) 0 0
\(196\) 4.22425 0.301732
\(197\) 3.87399 0.276011 0.138005 0.990431i \(-0.455931\pi\)
0.138005 + 0.990431i \(0.455931\pi\)
\(198\) 0 0
\(199\) 3.47627 0.246426 0.123213 0.992380i \(-0.460680\pi\)
0.123213 + 0.992380i \(0.460680\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 8.88717 0.625299
\(203\) −12.1016 −0.849364
\(204\) 0 0
\(205\) 0 0
\(206\) 7.03761 0.490334
\(207\) 0 0
\(208\) −1.35026 −0.0936238
\(209\) −1.61213 −0.111513
\(210\) 0 0
\(211\) 9.92478 0.683250 0.341625 0.939836i \(-0.389023\pi\)
0.341625 + 0.939836i \(0.389023\pi\)
\(212\) −11.9248 −0.818997
\(213\) 0 0
\(214\) 0.775746 0.0530289
\(215\) 0 0
\(216\) 0 0
\(217\) −7.74798 −0.525967
\(218\) 20.1622 1.36556
\(219\) 0 0
\(220\) 0 0
\(221\) 9.40105 0.632383
\(222\) 0 0
\(223\) −7.03761 −0.471273 −0.235637 0.971841i \(-0.575718\pi\)
−0.235637 + 0.971841i \(0.575718\pi\)
\(224\) −3.35026 −0.223849
\(225\) 0 0
\(226\) −11.1490 −0.741623
\(227\) −14.5501 −0.965723 −0.482861 0.875697i \(-0.660402\pi\)
−0.482861 + 0.875697i \(0.660402\pi\)
\(228\) 0 0
\(229\) 11.4010 0.753402 0.376701 0.926335i \(-0.377058\pi\)
0.376701 + 0.926335i \(0.377058\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.61213 0.237148
\(233\) 21.9149 1.43569 0.717847 0.696201i \(-0.245128\pi\)
0.717847 + 0.696201i \(0.245128\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.03761 −0.0675427
\(237\) 0 0
\(238\) 23.3258 1.51199
\(239\) 13.2750 0.858691 0.429345 0.903140i \(-0.358744\pi\)
0.429345 + 0.903140i \(0.358744\pi\)
\(240\) 0 0
\(241\) 21.3258 1.37372 0.686859 0.726791i \(-0.258989\pi\)
0.686859 + 0.726791i \(0.258989\pi\)
\(242\) 8.40105 0.540040
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) 1.35026 0.0859151
\(248\) 2.31265 0.146853
\(249\) 0 0
\(250\) 0 0
\(251\) −16.3127 −1.02965 −0.514823 0.857297i \(-0.672142\pi\)
−0.514823 + 0.857297i \(0.672142\pi\)
\(252\) 0 0
\(253\) 2.17679 0.136854
\(254\) 13.7381 0.862007
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −24.5501 −1.53139 −0.765696 0.643203i \(-0.777605\pi\)
−0.765696 + 0.643203i \(0.777605\pi\)
\(258\) 0 0
\(259\) 37.7743 2.34718
\(260\) 0 0
\(261\) 0 0
\(262\) 5.61213 0.346718
\(263\) −30.3488 −1.87139 −0.935695 0.352810i \(-0.885226\pi\)
−0.935695 + 0.352810i \(0.885226\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 3.35026 0.205418
\(267\) 0 0
\(268\) −9.92478 −0.606252
\(269\) −14.3127 −0.872658 −0.436329 0.899787i \(-0.643722\pi\)
−0.436329 + 0.899787i \(0.643722\pi\)
\(270\) 0 0
\(271\) −7.32582 −0.445012 −0.222506 0.974931i \(-0.571424\pi\)
−0.222506 + 0.974931i \(0.571424\pi\)
\(272\) −6.96239 −0.422157
\(273\) 0 0
\(274\) 17.6629 1.06706
\(275\) 0 0
\(276\) 0 0
\(277\) −14.4387 −0.867535 −0.433767 0.901025i \(-0.642816\pi\)
−0.433767 + 0.901025i \(0.642816\pi\)
\(278\) −10.7005 −0.641775
\(279\) 0 0
\(280\) 0 0
\(281\) 11.9756 0.714402 0.357201 0.934028i \(-0.383731\pi\)
0.357201 + 0.934028i \(0.383731\pi\)
\(282\) 0 0
\(283\) −24.4894 −1.45575 −0.727873 0.685712i \(-0.759491\pi\)
−0.727873 + 0.685712i \(0.759491\pi\)
\(284\) 0.775746 0.0460321
\(285\) 0 0
\(286\) 2.17679 0.128716
\(287\) 11.2243 0.662547
\(288\) 0 0
\(289\) 31.4749 1.85146
\(290\) 0 0
\(291\) 0 0
\(292\) −3.22425 −0.188685
\(293\) 18.1016 1.05751 0.528753 0.848776i \(-0.322660\pi\)
0.528753 + 0.848776i \(0.322660\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −11.2750 −0.655348
\(297\) 0 0
\(298\) 1.03761 0.0601072
\(299\) −1.82321 −0.105439
\(300\) 0 0
\(301\) −34.5501 −1.99143
\(302\) −1.16362 −0.0669588
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −29.9248 −1.70790 −0.853949 0.520357i \(-0.825799\pi\)
−0.853949 + 0.520357i \(0.825799\pi\)
\(308\) 5.40105 0.307753
\(309\) 0 0
\(310\) 0 0
\(311\) 21.2750 1.20640 0.603198 0.797591i \(-0.293893\pi\)
0.603198 + 0.797591i \(0.293893\pi\)
\(312\) 0 0
\(313\) −17.4010 −0.983565 −0.491783 0.870718i \(-0.663655\pi\)
−0.491783 + 0.870718i \(0.663655\pi\)
\(314\) 10.9624 0.618643
\(315\) 0 0
\(316\) 14.3127 0.805149
\(317\) 14.1016 0.792023 0.396012 0.918246i \(-0.370394\pi\)
0.396012 + 0.918246i \(0.370394\pi\)
\(318\) 0 0
\(319\) −5.82321 −0.326037
\(320\) 0 0
\(321\) 0 0
\(322\) −4.52373 −0.252098
\(323\) 6.96239 0.387398
\(324\) 0 0
\(325\) 0 0
\(326\) 21.0132 1.16381
\(327\) 0 0
\(328\) −3.35026 −0.184987
\(329\) 15.3258 0.844940
\(330\) 0 0
\(331\) 6.85097 0.376563 0.188282 0.982115i \(-0.439708\pi\)
0.188282 + 0.982115i \(0.439708\pi\)
\(332\) −10.8872 −0.597511
\(333\) 0 0
\(334\) −9.92478 −0.543060
\(335\) 0 0
\(336\) 0 0
\(337\) −24.2374 −1.32030 −0.660148 0.751135i \(-0.729507\pi\)
−0.660148 + 0.751135i \(0.729507\pi\)
\(338\) 11.1768 0.607937
\(339\) 0 0
\(340\) 0 0
\(341\) −3.72829 −0.201898
\(342\) 0 0
\(343\) −9.29948 −0.502125
\(344\) 10.3127 0.556021
\(345\) 0 0
\(346\) −14.6253 −0.786261
\(347\) 0.962389 0.0516637 0.0258319 0.999666i \(-0.491777\pi\)
0.0258319 + 0.999666i \(0.491777\pi\)
\(348\) 0 0
\(349\) −1.37470 −0.0735860 −0.0367930 0.999323i \(-0.511714\pi\)
−0.0367930 + 0.999323i \(0.511714\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.61213 −0.0859267
\(353\) 28.7367 1.52950 0.764751 0.644326i \(-0.222862\pi\)
0.764751 + 0.644326i \(0.222862\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.57452 0.136449
\(357\) 0 0
\(358\) 11.7381 0.620380
\(359\) −31.1998 −1.64666 −0.823332 0.567561i \(-0.807887\pi\)
−0.823332 + 0.567561i \(0.807887\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −21.4617 −1.12800
\(363\) 0 0
\(364\) −4.52373 −0.237108
\(365\) 0 0
\(366\) 0 0
\(367\) −20.1260 −1.05057 −0.525285 0.850927i \(-0.676041\pi\)
−0.525285 + 0.850927i \(0.676041\pi\)
\(368\) 1.35026 0.0703873
\(369\) 0 0
\(370\) 0 0
\(371\) −39.9511 −2.07416
\(372\) 0 0
\(373\) 17.9756 0.930739 0.465370 0.885116i \(-0.345921\pi\)
0.465370 + 0.885116i \(0.345921\pi\)
\(374\) 11.2243 0.580392
\(375\) 0 0
\(376\) −4.57452 −0.235913
\(377\) 4.87732 0.251195
\(378\) 0 0
\(379\) 1.67276 0.0859240 0.0429620 0.999077i \(-0.486321\pi\)
0.0429620 + 0.999077i \(0.486321\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −21.2750 −1.08853
\(383\) 31.8496 1.62744 0.813718 0.581260i \(-0.197440\pi\)
0.813718 + 0.581260i \(0.197440\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14.1622 −0.720837
\(387\) 0 0
\(388\) −1.16362 −0.0590738
\(389\) 10.3371 0.524111 0.262056 0.965053i \(-0.415600\pi\)
0.262056 + 0.965053i \(0.415600\pi\)
\(390\) 0 0
\(391\) −9.40105 −0.475431
\(392\) −4.22425 −0.213357
\(393\) 0 0
\(394\) −3.87399 −0.195169
\(395\) 0 0
\(396\) 0 0
\(397\) 31.4372 1.57779 0.788895 0.614528i \(-0.210654\pi\)
0.788895 + 0.614528i \(0.210654\pi\)
\(398\) −3.47627 −0.174250
\(399\) 0 0
\(400\) 0 0
\(401\) −5.94921 −0.297090 −0.148545 0.988906i \(-0.547459\pi\)
−0.148545 + 0.988906i \(0.547459\pi\)
\(402\) 0 0
\(403\) 3.12268 0.155552
\(404\) −8.88717 −0.442153
\(405\) 0 0
\(406\) 12.1016 0.600591
\(407\) 18.1768 0.900990
\(408\) 0 0
\(409\) −30.9986 −1.53278 −0.766391 0.642375i \(-0.777949\pi\)
−0.766391 + 0.642375i \(0.777949\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −7.03761 −0.346718
\(413\) −3.47627 −0.171056
\(414\) 0 0
\(415\) 0 0
\(416\) 1.35026 0.0662020
\(417\) 0 0
\(418\) 1.61213 0.0788517
\(419\) 34.3390 1.67757 0.838785 0.544463i \(-0.183266\pi\)
0.838785 + 0.544463i \(0.183266\pi\)
\(420\) 0 0
\(421\) 22.8627 1.11426 0.557131 0.830425i \(-0.311902\pi\)
0.557131 + 0.830425i \(0.311902\pi\)
\(422\) −9.92478 −0.483131
\(423\) 0 0
\(424\) 11.9248 0.579118
\(425\) 0 0
\(426\) 0 0
\(427\) 6.70052 0.324261
\(428\) −0.775746 −0.0374971
\(429\) 0 0
\(430\) 0 0
\(431\) −11.5975 −0.558634 −0.279317 0.960199i \(-0.590108\pi\)
−0.279317 + 0.960199i \(0.590108\pi\)
\(432\) 0 0
\(433\) −27.8350 −1.33766 −0.668832 0.743414i \(-0.733205\pi\)
−0.668832 + 0.743414i \(0.733205\pi\)
\(434\) 7.74798 0.371915
\(435\) 0 0
\(436\) −20.1622 −0.965594
\(437\) −1.35026 −0.0645918
\(438\) 0 0
\(439\) −38.7875 −1.85123 −0.925613 0.378471i \(-0.876450\pi\)
−0.925613 + 0.378471i \(0.876450\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −9.40105 −0.447162
\(443\) −29.2144 −1.38802 −0.694009 0.719966i \(-0.744157\pi\)
−0.694009 + 0.719966i \(0.744157\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 7.03761 0.333241
\(447\) 0 0
\(448\) 3.35026 0.158285
\(449\) 20.4993 0.967421 0.483711 0.875228i \(-0.339289\pi\)
0.483711 + 0.875228i \(0.339289\pi\)
\(450\) 0 0
\(451\) 5.40105 0.254325
\(452\) 11.1490 0.524406
\(453\) 0 0
\(454\) 14.5501 0.682869
\(455\) 0 0
\(456\) 0 0
\(457\) −6.44851 −0.301648 −0.150824 0.988561i \(-0.548193\pi\)
−0.150824 + 0.988561i \(0.548193\pi\)
\(458\) −11.4010 −0.532736
\(459\) 0 0
\(460\) 0 0
\(461\) 13.5125 0.629338 0.314669 0.949201i \(-0.398106\pi\)
0.314669 + 0.949201i \(0.398106\pi\)
\(462\) 0 0
\(463\) −6.20123 −0.288196 −0.144098 0.989563i \(-0.546028\pi\)
−0.144098 + 0.989563i \(0.546028\pi\)
\(464\) −3.61213 −0.167689
\(465\) 0 0
\(466\) −21.9149 −1.01519
\(467\) −16.5599 −0.766302 −0.383151 0.923686i \(-0.625161\pi\)
−0.383151 + 0.923686i \(0.625161\pi\)
\(468\) 0 0
\(469\) −33.2506 −1.53537
\(470\) 0 0
\(471\) 0 0
\(472\) 1.03761 0.0477599
\(473\) −16.6253 −0.764432
\(474\) 0 0
\(475\) 0 0
\(476\) −23.3258 −1.06914
\(477\) 0 0
\(478\) −13.2750 −0.607186
\(479\) 30.5256 1.39475 0.697376 0.716705i \(-0.254351\pi\)
0.697376 + 0.716705i \(0.254351\pi\)
\(480\) 0 0
\(481\) −15.2243 −0.694166
\(482\) −21.3258 −0.971365
\(483\) 0 0
\(484\) −8.40105 −0.381866
\(485\) 0 0
\(486\) 0 0
\(487\) 2.51388 0.113915 0.0569574 0.998377i \(-0.481860\pi\)
0.0569574 + 0.998377i \(0.481860\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) 0 0
\(491\) −1.46168 −0.0659648 −0.0329824 0.999456i \(-0.510501\pi\)
−0.0329824 + 0.999456i \(0.510501\pi\)
\(492\) 0 0
\(493\) 25.1490 1.13266
\(494\) −1.35026 −0.0607511
\(495\) 0 0
\(496\) −2.31265 −0.103841
\(497\) 2.59895 0.116579
\(498\) 0 0
\(499\) −5.55149 −0.248519 −0.124259 0.992250i \(-0.539656\pi\)
−0.124259 + 0.992250i \(0.539656\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 16.3127 0.728069
\(503\) 6.90175 0.307734 0.153867 0.988092i \(-0.450827\pi\)
0.153867 + 0.988092i \(0.450827\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −2.17679 −0.0967703
\(507\) 0 0
\(508\) −13.7381 −0.609531
\(509\) −8.76116 −0.388331 −0.194166 0.980969i \(-0.562200\pi\)
−0.194166 + 0.980969i \(0.562200\pi\)
\(510\) 0 0
\(511\) −10.8021 −0.477857
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 24.5501 1.08286
\(515\) 0 0
\(516\) 0 0
\(517\) 7.37470 0.324339
\(518\) −37.7743 −1.65971
\(519\) 0 0
\(520\) 0 0
\(521\) −39.4518 −1.72842 −0.864208 0.503135i \(-0.832180\pi\)
−0.864208 + 0.503135i \(0.832180\pi\)
\(522\) 0 0
\(523\) 37.5026 1.63987 0.819937 0.572453i \(-0.194008\pi\)
0.819937 + 0.572453i \(0.194008\pi\)
\(524\) −5.61213 −0.245167
\(525\) 0 0
\(526\) 30.3488 1.32327
\(527\) 16.1016 0.701395
\(528\) 0 0
\(529\) −21.1768 −0.920730
\(530\) 0 0
\(531\) 0 0
\(532\) −3.35026 −0.145252
\(533\) −4.52373 −0.195945
\(534\) 0 0
\(535\) 0 0
\(536\) 9.92478 0.428685
\(537\) 0 0
\(538\) 14.3127 0.617062
\(539\) 6.81003 0.293329
\(540\) 0 0
\(541\) −30.7757 −1.32315 −0.661576 0.749878i \(-0.730112\pi\)
−0.661576 + 0.749878i \(0.730112\pi\)
\(542\) 7.32582 0.314671
\(543\) 0 0
\(544\) 6.96239 0.298510
\(545\) 0 0
\(546\) 0 0
\(547\) 1.77433 0.0758649 0.0379325 0.999280i \(-0.487923\pi\)
0.0379325 + 0.999280i \(0.487923\pi\)
\(548\) −17.6629 −0.754522
\(549\) 0 0
\(550\) 0 0
\(551\) 3.61213 0.153882
\(552\) 0 0
\(553\) 47.9511 2.03909
\(554\) 14.4387 0.613440
\(555\) 0 0
\(556\) 10.7005 0.453803
\(557\) −8.90175 −0.377179 −0.188590 0.982056i \(-0.560392\pi\)
−0.188590 + 0.982056i \(0.560392\pi\)
\(558\) 0 0
\(559\) 13.9248 0.588955
\(560\) 0 0
\(561\) 0 0
\(562\) −11.9756 −0.505159
\(563\) −10.9525 −0.461595 −0.230797 0.973002i \(-0.574133\pi\)
−0.230797 + 0.973002i \(0.574133\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 24.4894 1.02937
\(567\) 0 0
\(568\) −0.775746 −0.0325496
\(569\) 10.2012 0.427658 0.213829 0.976871i \(-0.431406\pi\)
0.213829 + 0.976871i \(0.431406\pi\)
\(570\) 0 0
\(571\) −25.6531 −1.07355 −0.536774 0.843726i \(-0.680357\pi\)
−0.536774 + 0.843726i \(0.680357\pi\)
\(572\) −2.17679 −0.0910163
\(573\) 0 0
\(574\) −11.2243 −0.468491
\(575\) 0 0
\(576\) 0 0
\(577\) −6.44851 −0.268455 −0.134227 0.990951i \(-0.542855\pi\)
−0.134227 + 0.990951i \(0.542855\pi\)
\(578\) −31.4749 −1.30918
\(579\) 0 0
\(580\) 0 0
\(581\) −36.4749 −1.51323
\(582\) 0 0
\(583\) −19.2243 −0.796187
\(584\) 3.22425 0.133421
\(585\) 0 0
\(586\) −18.1016 −0.747769
\(587\) −41.8397 −1.72691 −0.863455 0.504426i \(-0.831704\pi\)
−0.863455 + 0.504426i \(0.831704\pi\)
\(588\) 0 0
\(589\) 2.31265 0.0952911
\(590\) 0 0
\(591\) 0 0
\(592\) 11.2750 0.463401
\(593\) 8.73672 0.358774 0.179387 0.983779i \(-0.442589\pi\)
0.179387 + 0.983779i \(0.442589\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.03761 −0.0425022
\(597\) 0 0
\(598\) 1.82321 0.0745565
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 30.6253 1.24923 0.624616 0.780932i \(-0.285255\pi\)
0.624616 + 0.780932i \(0.285255\pi\)
\(602\) 34.5501 1.40816
\(603\) 0 0
\(604\) 1.16362 0.0473470
\(605\) 0 0
\(606\) 0 0
\(607\) 2.51388 0.102035 0.0510176 0.998698i \(-0.483754\pi\)
0.0510176 + 0.998698i \(0.483754\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −6.17679 −0.249886
\(612\) 0 0
\(613\) 2.96239 0.119650 0.0598249 0.998209i \(-0.480946\pi\)
0.0598249 + 0.998209i \(0.480946\pi\)
\(614\) 29.9248 1.20767
\(615\) 0 0
\(616\) −5.40105 −0.217614
\(617\) −18.3371 −0.738223 −0.369112 0.929385i \(-0.620338\pi\)
−0.369112 + 0.929385i \(0.620338\pi\)
\(618\) 0 0
\(619\) −40.7269 −1.63695 −0.818476 0.574541i \(-0.805180\pi\)
−0.818476 + 0.574541i \(0.805180\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −21.2750 −0.853051
\(623\) 8.62530 0.345565
\(624\) 0 0
\(625\) 0 0
\(626\) 17.4010 0.695486
\(627\) 0 0
\(628\) −10.9624 −0.437447
\(629\) −78.5012 −3.13005
\(630\) 0 0
\(631\) −5.92478 −0.235862 −0.117931 0.993022i \(-0.537626\pi\)
−0.117931 + 0.993022i \(0.537626\pi\)
\(632\) −14.3127 −0.569327
\(633\) 0 0
\(634\) −14.1016 −0.560045
\(635\) 0 0
\(636\) 0 0
\(637\) −5.70385 −0.225995
\(638\) 5.82321 0.230543
\(639\) 0 0
\(640\) 0 0
\(641\) −19.0494 −0.752405 −0.376202 0.926537i \(-0.622770\pi\)
−0.376202 + 0.926537i \(0.622770\pi\)
\(642\) 0 0
\(643\) 1.06205 0.0418831 0.0209416 0.999781i \(-0.493334\pi\)
0.0209416 + 0.999781i \(0.493334\pi\)
\(644\) 4.52373 0.178260
\(645\) 0 0
\(646\) −6.96239 −0.273932
\(647\) 26.6497 1.04771 0.523855 0.851808i \(-0.324493\pi\)
0.523855 + 0.851808i \(0.324493\pi\)
\(648\) 0 0
\(649\) −1.67276 −0.0656616
\(650\) 0 0
\(651\) 0 0
\(652\) −21.0132 −0.822939
\(653\) −9.64832 −0.377568 −0.188784 0.982019i \(-0.560455\pi\)
−0.188784 + 0.982019i \(0.560455\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.35026 0.130806
\(657\) 0 0
\(658\) −15.3258 −0.597463
\(659\) −10.0654 −0.392091 −0.196046 0.980595i \(-0.562810\pi\)
−0.196046 + 0.980595i \(0.562810\pi\)
\(660\) 0 0
\(661\) 13.5633 0.527549 0.263775 0.964584i \(-0.415032\pi\)
0.263775 + 0.964584i \(0.415032\pi\)
\(662\) −6.85097 −0.266270
\(663\) 0 0
\(664\) 10.8872 0.422504
\(665\) 0 0
\(666\) 0 0
\(667\) −4.87732 −0.188850
\(668\) 9.92478 0.384001
\(669\) 0 0
\(670\) 0 0
\(671\) 3.22425 0.124471
\(672\) 0 0
\(673\) 5.93937 0.228946 0.114473 0.993426i \(-0.463482\pi\)
0.114473 + 0.993426i \(0.463482\pi\)
\(674\) 24.2374 0.933591
\(675\) 0 0
\(676\) −11.1768 −0.429877
\(677\) −24.9525 −0.959004 −0.479502 0.877541i \(-0.659183\pi\)
−0.479502 + 0.877541i \(0.659183\pi\)
\(678\) 0 0
\(679\) −3.89843 −0.149608
\(680\) 0 0
\(681\) 0 0
\(682\) 3.72829 0.142763
\(683\) 45.6239 1.74575 0.872875 0.487944i \(-0.162253\pi\)
0.872875 + 0.487944i \(0.162253\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 9.29948 0.355056
\(687\) 0 0
\(688\) −10.3127 −0.393166
\(689\) 16.1016 0.613421
\(690\) 0 0
\(691\) −11.7480 −0.446914 −0.223457 0.974714i \(-0.571734\pi\)
−0.223457 + 0.974714i \(0.571734\pi\)
\(692\) 14.6253 0.555971
\(693\) 0 0
\(694\) −0.962389 −0.0365318
\(695\) 0 0
\(696\) 0 0
\(697\) −23.3258 −0.883529
\(698\) 1.37470 0.0520331
\(699\) 0 0
\(700\) 0 0
\(701\) 43.8105 1.65470 0.827350 0.561686i \(-0.189847\pi\)
0.827350 + 0.561686i \(0.189847\pi\)
\(702\) 0 0
\(703\) −11.2750 −0.425246
\(704\) 1.61213 0.0607593
\(705\) 0 0
\(706\) −28.7367 −1.08152
\(707\) −29.7743 −1.11978
\(708\) 0 0
\(709\) −29.5975 −1.11156 −0.555779 0.831330i \(-0.687580\pi\)
−0.555779 + 0.831330i \(0.687580\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −2.57452 −0.0964840
\(713\) −3.12268 −0.116945
\(714\) 0 0
\(715\) 0 0
\(716\) −11.7381 −0.438675
\(717\) 0 0
\(718\) 31.1998 1.16437
\(719\) −7.45183 −0.277906 −0.138953 0.990299i \(-0.544374\pi\)
−0.138953 + 0.990299i \(0.544374\pi\)
\(720\) 0 0
\(721\) −23.5778 −0.878085
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) 21.4617 0.797617
\(725\) 0 0
\(726\) 0 0
\(727\) 0.600863 0.0222848 0.0111424 0.999938i \(-0.496453\pi\)
0.0111424 + 0.999938i \(0.496453\pi\)
\(728\) 4.52373 0.167661
\(729\) 0 0
\(730\) 0 0
\(731\) 71.8007 2.65564
\(732\) 0 0
\(733\) 36.0625 1.33200 0.666000 0.745952i \(-0.268005\pi\)
0.666000 + 0.745952i \(0.268005\pi\)
\(734\) 20.1260 0.742865
\(735\) 0 0
\(736\) −1.35026 −0.0497713
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) −14.1768 −0.521502 −0.260751 0.965406i \(-0.583970\pi\)
−0.260751 + 0.965406i \(0.583970\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 39.9511 1.46665
\(743\) −24.9986 −0.917109 −0.458555 0.888666i \(-0.651633\pi\)
−0.458555 + 0.888666i \(0.651633\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −17.9756 −0.658132
\(747\) 0 0
\(748\) −11.2243 −0.410399
\(749\) −2.59895 −0.0949637
\(750\) 0 0
\(751\) −10.2111 −0.372608 −0.186304 0.982492i \(-0.559651\pi\)
−0.186304 + 0.982492i \(0.559651\pi\)
\(752\) 4.57452 0.166815
\(753\) 0 0
\(754\) −4.87732 −0.177621
\(755\) 0 0
\(756\) 0 0
\(757\) 44.4847 1.61682 0.808412 0.588617i \(-0.200327\pi\)
0.808412 + 0.588617i \(0.200327\pi\)
\(758\) −1.67276 −0.0607574
\(759\) 0 0
\(760\) 0 0
\(761\) −27.1490 −0.984152 −0.492076 0.870552i \(-0.663762\pi\)
−0.492076 + 0.870552i \(0.663762\pi\)
\(762\) 0 0
\(763\) −67.5487 −2.44543
\(764\) 21.2750 0.769704
\(765\) 0 0
\(766\) −31.8496 −1.15077
\(767\) 1.40105 0.0505889
\(768\) 0 0
\(769\) −31.4010 −1.13235 −0.566175 0.824285i \(-0.691578\pi\)
−0.566175 + 0.824285i \(0.691578\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14.1622 0.509709
\(773\) −4.02635 −0.144818 −0.0724088 0.997375i \(-0.523069\pi\)
−0.0724088 + 0.997375i \(0.523069\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.16362 0.0417715
\(777\) 0 0
\(778\) −10.3371 −0.370603
\(779\) −3.35026 −0.120036
\(780\) 0 0
\(781\) 1.25060 0.0447500
\(782\) 9.40105 0.336181
\(783\) 0 0
\(784\) 4.22425 0.150866
\(785\) 0 0
\(786\) 0 0
\(787\) 18.2981 0.652255 0.326128 0.945326i \(-0.394256\pi\)
0.326128 + 0.945326i \(0.394256\pi\)
\(788\) 3.87399 0.138005
\(789\) 0 0
\(790\) 0 0
\(791\) 37.3522 1.32809
\(792\) 0 0
\(793\) −2.70052 −0.0958984
\(794\) −31.4372 −1.11567
\(795\) 0 0
\(796\) 3.47627 0.123213
\(797\) 8.55008 0.302859 0.151430 0.988468i \(-0.451612\pi\)
0.151430 + 0.988468i \(0.451612\pi\)
\(798\) 0 0
\(799\) −31.8496 −1.12676
\(800\) 0 0
\(801\) 0 0
\(802\) 5.94921 0.210074
\(803\) −5.19791 −0.183430
\(804\) 0 0
\(805\) 0 0
\(806\) −3.12268 −0.109992
\(807\) 0 0
\(808\) 8.88717 0.312649
\(809\) −50.7269 −1.78346 −0.891731 0.452566i \(-0.850509\pi\)
−0.891731 + 0.452566i \(0.850509\pi\)
\(810\) 0 0
\(811\) 10.3272 0.362638 0.181319 0.983424i \(-0.441963\pi\)
0.181319 + 0.983424i \(0.441963\pi\)
\(812\) −12.1016 −0.424682
\(813\) 0 0
\(814\) −18.1768 −0.637096
\(815\) 0 0
\(816\) 0 0
\(817\) 10.3127 0.360794
\(818\) 30.9986 1.08384
\(819\) 0 0
\(820\) 0 0
\(821\) −36.5139 −1.27434 −0.637172 0.770722i \(-0.719896\pi\)
−0.637172 + 0.770722i \(0.719896\pi\)
\(822\) 0 0
\(823\) 23.0982 0.805154 0.402577 0.915386i \(-0.368115\pi\)
0.402577 + 0.915386i \(0.368115\pi\)
\(824\) 7.03761 0.245167
\(825\) 0 0
\(826\) 3.47627 0.120955
\(827\) 3.37470 0.117350 0.0586749 0.998277i \(-0.481312\pi\)
0.0586749 + 0.998277i \(0.481312\pi\)
\(828\) 0 0
\(829\) 22.6399 0.786316 0.393158 0.919471i \(-0.371383\pi\)
0.393158 + 0.919471i \(0.371383\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.35026 −0.0468119
\(833\) −29.4109 −1.01903
\(834\) 0 0
\(835\) 0 0
\(836\) −1.61213 −0.0557566
\(837\) 0 0
\(838\) −34.3390 −1.18622
\(839\) 9.02776 0.311673 0.155836 0.987783i \(-0.450193\pi\)
0.155836 + 0.987783i \(0.450193\pi\)
\(840\) 0 0
\(841\) −15.9525 −0.550088
\(842\) −22.8627 −0.787902
\(843\) 0 0
\(844\) 9.92478 0.341625
\(845\) 0 0
\(846\) 0 0
\(847\) −28.1457 −0.967098
\(848\) −11.9248 −0.409499
\(849\) 0 0
\(850\) 0 0
\(851\) 15.2243 0.521881
\(852\) 0 0
\(853\) −43.1852 −1.47863 −0.739317 0.673358i \(-0.764851\pi\)
−0.739317 + 0.673358i \(0.764851\pi\)
\(854\) −6.70052 −0.229287
\(855\) 0 0
\(856\) 0.775746 0.0265145
\(857\) 33.0249 1.12811 0.564055 0.825737i \(-0.309241\pi\)
0.564055 + 0.825737i \(0.309241\pi\)
\(858\) 0 0
\(859\) 15.1754 0.517777 0.258889 0.965907i \(-0.416644\pi\)
0.258889 + 0.965907i \(0.416644\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 11.5975 0.395014
\(863\) 39.4763 1.34379 0.671894 0.740647i \(-0.265481\pi\)
0.671894 + 0.740647i \(0.265481\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 27.8350 0.945871
\(867\) 0 0
\(868\) −7.74798 −0.262984
\(869\) 23.0738 0.782725
\(870\) 0 0
\(871\) 13.4010 0.454077
\(872\) 20.1622 0.682778
\(873\) 0 0
\(874\) 1.35026 0.0456733
\(875\) 0 0
\(876\) 0 0
\(877\) −52.5256 −1.77366 −0.886832 0.462091i \(-0.847099\pi\)
−0.886832 + 0.462091i \(0.847099\pi\)
\(878\) 38.7875 1.30901
\(879\) 0 0
\(880\) 0 0
\(881\) 41.8007 1.40830 0.704150 0.710051i \(-0.251328\pi\)
0.704150 + 0.710051i \(0.251328\pi\)
\(882\) 0 0
\(883\) 36.9643 1.24395 0.621974 0.783038i \(-0.286331\pi\)
0.621974 + 0.783038i \(0.286331\pi\)
\(884\) 9.40105 0.316191
\(885\) 0 0
\(886\) 29.2144 0.981477
\(887\) 36.0724 1.21119 0.605596 0.795772i \(-0.292935\pi\)
0.605596 + 0.795772i \(0.292935\pi\)
\(888\) 0 0
\(889\) −46.0263 −1.54367
\(890\) 0 0
\(891\) 0 0
\(892\) −7.03761 −0.235637
\(893\) −4.57452 −0.153080
\(894\) 0 0
\(895\) 0 0
\(896\) −3.35026 −0.111924
\(897\) 0 0
\(898\) −20.4993 −0.684070
\(899\) 8.35359 0.278608
\(900\) 0 0
\(901\) 83.0249 2.76596
\(902\) −5.40105 −0.179835
\(903\) 0 0
\(904\) −11.1490 −0.370811
\(905\) 0 0
\(906\) 0 0
\(907\) 57.6239 1.91337 0.956685 0.291126i \(-0.0940297\pi\)
0.956685 + 0.291126i \(0.0940297\pi\)
\(908\) −14.5501 −0.482861
\(909\) 0 0
\(910\) 0 0
\(911\) −30.9234 −1.02454 −0.512268 0.858825i \(-0.671195\pi\)
−0.512268 + 0.858825i \(0.671195\pi\)
\(912\) 0 0
\(913\) −17.5515 −0.580870
\(914\) 6.44851 0.213298
\(915\) 0 0
\(916\) 11.4010 0.376701
\(917\) −18.8021 −0.620900
\(918\) 0 0
\(919\) 33.7743 1.11411 0.557056 0.830475i \(-0.311931\pi\)
0.557056 + 0.830475i \(0.311931\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −13.5125 −0.445009
\(923\) −1.04746 −0.0344776
\(924\) 0 0
\(925\) 0 0
\(926\) 6.20123 0.203785
\(927\) 0 0
\(928\) 3.61213 0.118574
\(929\) −48.6516 −1.59621 −0.798104 0.602519i \(-0.794164\pi\)
−0.798104 + 0.602519i \(0.794164\pi\)
\(930\) 0 0
\(931\) −4.22425 −0.138444
\(932\) 21.9149 0.717847
\(933\) 0 0
\(934\) 16.5599 0.541857
\(935\) 0 0
\(936\) 0 0
\(937\) 18.7005 0.610919 0.305460 0.952205i \(-0.401190\pi\)
0.305460 + 0.952205i \(0.401190\pi\)
\(938\) 33.2506 1.08567
\(939\) 0 0
\(940\) 0 0
\(941\) −38.4142 −1.25227 −0.626134 0.779716i \(-0.715364\pi\)
−0.626134 + 0.779716i \(0.715364\pi\)
\(942\) 0 0
\(943\) 4.52373 0.147313
\(944\) −1.03761 −0.0337714
\(945\) 0 0
\(946\) 16.6253 0.540535
\(947\) 44.0362 1.43098 0.715492 0.698621i \(-0.246203\pi\)
0.715492 + 0.698621i \(0.246203\pi\)
\(948\) 0 0
\(949\) 4.35359 0.141323
\(950\) 0 0
\(951\) 0 0
\(952\) 23.3258 0.755994
\(953\) −38.0724 −1.23329 −0.616643 0.787243i \(-0.711508\pi\)
−0.616643 + 0.787243i \(0.711508\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 13.2750 0.429345
\(957\) 0 0
\(958\) −30.5256 −0.986239
\(959\) −59.1754 −1.91087
\(960\) 0 0
\(961\) −25.6516 −0.827473
\(962\) 15.2243 0.490850
\(963\) 0 0
\(964\) 21.3258 0.686859
\(965\) 0 0
\(966\) 0 0
\(967\) −42.5256 −1.36753 −0.683766 0.729701i \(-0.739659\pi\)
−0.683766 + 0.729701i \(0.739659\pi\)
\(968\) 8.40105 0.270020
\(969\) 0 0
\(970\) 0 0
\(971\) 22.1378 0.710435 0.355217 0.934784i \(-0.384407\pi\)
0.355217 + 0.934784i \(0.384407\pi\)
\(972\) 0 0
\(973\) 35.8496 1.14928
\(974\) −2.51388 −0.0805499
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −21.7480 −0.695780 −0.347890 0.937535i \(-0.613102\pi\)
−0.347890 + 0.937535i \(0.613102\pi\)
\(978\) 0 0
\(979\) 4.15045 0.132649
\(980\) 0 0
\(981\) 0 0
\(982\) 1.46168 0.0466441
\(983\) 9.04746 0.288569 0.144285 0.989536i \(-0.453912\pi\)
0.144285 + 0.989536i \(0.453912\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −25.1490 −0.800908
\(987\) 0 0
\(988\) 1.35026 0.0429575
\(989\) −13.9248 −0.442782
\(990\) 0 0
\(991\) −26.0606 −0.827843 −0.413922 0.910313i \(-0.635841\pi\)
−0.413922 + 0.910313i \(0.635841\pi\)
\(992\) 2.31265 0.0734267
\(993\) 0 0
\(994\) −2.59895 −0.0824338
\(995\) 0 0
\(996\) 0 0
\(997\) −28.2130 −0.893514 −0.446757 0.894655i \(-0.647421\pi\)
−0.446757 + 0.894655i \(0.647421\pi\)
\(998\) 5.55149 0.175729
\(999\) 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.cf.1.3 3
3.2 odd 2 2850.2.a.bn.1.3 3
5.2 odd 4 1710.2.d.e.1369.3 6
5.3 odd 4 1710.2.d.e.1369.6 6
5.4 even 2 8550.2.a.cr.1.1 3
15.2 even 4 570.2.d.d.229.4 yes 6
15.8 even 4 570.2.d.d.229.1 6
15.14 odd 2 2850.2.a.bk.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.d.d.229.1 6 15.8 even 4
570.2.d.d.229.4 yes 6 15.2 even 4
1710.2.d.e.1369.3 6 5.2 odd 4
1710.2.d.e.1369.6 6 5.3 odd 4
2850.2.a.bk.1.1 3 15.14 odd 2
2850.2.a.bn.1.3 3 3.2 odd 2
8550.2.a.cf.1.3 3 1.1 even 1 trivial
8550.2.a.cr.1.1 3 5.4 even 2