Properties

Label 8550.2.a.ci.1.2
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.993.1
Defining polynomial: \(x^{3} - x^{2} - 6 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 950)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.25342\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.0778929 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.0778929 q^{7} -1.00000 q^{8} +4.50684 q^{11} -5.33131 q^{13} +0.0778929 q^{14} +1.00000 q^{16} -7.33131 q^{17} +1.00000 q^{19} -4.50684 q^{22} -3.40920 q^{23} +5.33131 q^{26} -0.0778929 q^{28} +1.33131 q^{29} -2.50684 q^{31} -1.00000 q^{32} +7.33131 q^{34} -5.50684 q^{37} -1.00000 q^{38} +0.506836 q^{43} +4.50684 q^{44} +3.40920 q^{46} +5.66262 q^{47} -6.99393 q^{49} -5.33131 q^{52} +12.9358 q^{53} +0.0778929 q^{56} -1.33131 q^{58} -7.56499 q^{59} -2.15579 q^{61} +2.50684 q^{62} +1.00000 q^{64} -4.58473 q^{67} -7.33131 q^{68} +10.8579 q^{71} -5.09763 q^{73} +5.50684 q^{74} +1.00000 q^{76} -0.351050 q^{77} +17.0137 q^{79} +13.1695 q^{83} -0.506836 q^{86} -4.50684 q^{88} -15.0137 q^{89} +0.415271 q^{91} -3.40920 q^{92} -5.66262 q^{94} +7.67629 q^{97} +6.99393 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} + 2q^{7} - 3q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} + 2q^{7} - 3q^{8} - 2q^{11} - 6q^{13} - 2q^{14} + 3q^{16} - 12q^{17} + 3q^{19} + 2q^{22} + 2q^{23} + 6q^{26} + 2q^{28} - 6q^{29} + 8q^{31} - 3q^{32} + 12q^{34} - q^{37} - 3q^{38} - 14q^{43} - 2q^{44} - 2q^{46} - 3q^{47} + 9q^{49} - 6q^{52} + 10q^{53} - 2q^{56} + 6q^{58} - 6q^{59} - 2q^{61} - 8q^{62} + 3q^{64} + 4q^{67} - 12q^{68} + 6q^{71} - 12q^{73} + q^{74} + 3q^{76} + 10q^{77} + 20q^{79} + 4q^{83} + 14q^{86} + 2q^{88} - 14q^{89} + 19q^{91} + 2q^{92} + 3q^{94} - 28q^{97} - 9q^{98} + O(q^{100}) \)

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
\(7\) −0.0778929 −0.0294407 −0.0147204 0.999892i \(-0.504686\pi\)
−0.0147204 + 0.999892i \(0.504686\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 4.50684 1.35886 0.679431 0.733739i \(-0.262227\pi\)
0.679431 + 0.733739i \(0.262227\pi\)
\(12\) 0 0
\(13\) −5.33131 −1.47864 −0.739320 0.673354i \(-0.764853\pi\)
−0.739320 + 0.673354i \(0.764853\pi\)
\(14\) 0.0778929 0.0208177
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.33131 −1.77810 −0.889052 0.457806i \(-0.848635\pi\)
−0.889052 + 0.457806i \(0.848635\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −4.50684 −0.960861
\(23\) −3.40920 −0.710868 −0.355434 0.934701i \(-0.615667\pi\)
−0.355434 + 0.934701i \(0.615667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 5.33131 1.04556
\(27\) 0 0
\(28\) −0.0778929 −0.0147204
\(29\) 1.33131 0.247218 0.123609 0.992331i \(-0.460553\pi\)
0.123609 + 0.992331i \(0.460553\pi\)
\(30\) 0 0
\(31\) −2.50684 −0.450241 −0.225121 0.974331i \(-0.572278\pi\)
−0.225121 + 0.974331i \(0.572278\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 7.33131 1.25731
\(35\) 0 0
\(36\) 0 0
\(37\) −5.50684 −0.905318 −0.452659 0.891684i \(-0.649525\pi\)
−0.452659 + 0.891684i \(0.649525\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0.506836 0.0772918 0.0386459 0.999253i \(-0.487696\pi\)
0.0386459 + 0.999253i \(0.487696\pi\)
\(44\) 4.50684 0.679431
\(45\) 0 0
\(46\) 3.40920 0.502660
\(47\) 5.66262 0.825978 0.412989 0.910736i \(-0.364485\pi\)
0.412989 + 0.910736i \(0.364485\pi\)
\(48\) 0 0
\(49\) −6.99393 −0.999133
\(50\) 0 0
\(51\) 0 0
\(52\) −5.33131 −0.739320
\(53\) 12.9358 1.77687 0.888433 0.459006i \(-0.151794\pi\)
0.888433 + 0.459006i \(0.151794\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.0778929 0.0104089
\(57\) 0 0
\(58\) −1.33131 −0.174810
\(59\) −7.56499 −0.984878 −0.492439 0.870347i \(-0.663894\pi\)
−0.492439 + 0.870347i \(0.663894\pi\)
\(60\) 0 0
\(61\) −2.15579 −0.276020 −0.138010 0.990431i \(-0.544071\pi\)
−0.138010 + 0.990431i \(0.544071\pi\)
\(62\) 2.50684 0.318369
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.58473 −0.560114 −0.280057 0.959983i \(-0.590353\pi\)
−0.280057 + 0.959983i \(0.590353\pi\)
\(68\) −7.33131 −0.889052
\(69\) 0 0
\(70\) 0 0
\(71\) 10.8579 1.28859 0.644297 0.764775i \(-0.277150\pi\)
0.644297 + 0.764775i \(0.277150\pi\)
\(72\) 0 0
\(73\) −5.09763 −0.596633 −0.298316 0.954467i \(-0.596425\pi\)
−0.298316 + 0.954467i \(0.596425\pi\)
\(74\) 5.50684 0.640157
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −0.351050 −0.0400059
\(78\) 0 0
\(79\) 17.0137 1.91419 0.957094 0.289778i \(-0.0935815\pi\)
0.957094 + 0.289778i \(0.0935815\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.1695 1.44554 0.722768 0.691091i \(-0.242870\pi\)
0.722768 + 0.691091i \(0.242870\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.506836 −0.0546535
\(87\) 0 0
\(88\) −4.50684 −0.480430
\(89\) −15.0137 −1.59145 −0.795723 0.605661i \(-0.792909\pi\)
−0.795723 + 0.605661i \(0.792909\pi\)
\(90\) 0 0
\(91\) 0.415271 0.0435322
\(92\) −3.40920 −0.355434
\(93\) 0 0
\(94\) −5.66262 −0.584055
\(95\) 0 0
\(96\) 0 0
\(97\) 7.67629 0.779410 0.389705 0.920940i \(-0.372577\pi\)
0.389705 + 0.920940i \(0.372577\pi\)
\(98\) 6.99393 0.706494
\(99\) 0 0
\(100\) 0 0
\(101\) 4.15579 0.413516 0.206758 0.978392i \(-0.433709\pi\)
0.206758 + 0.978392i \(0.433709\pi\)
\(102\) 0 0
\(103\) 2.35105 0.231656 0.115828 0.993269i \(-0.463048\pi\)
0.115828 + 0.993269i \(0.463048\pi\)
\(104\) 5.33131 0.522778
\(105\) 0 0
\(106\) −12.9358 −1.25643
\(107\) 14.0334 1.35666 0.678331 0.734757i \(-0.262704\pi\)
0.678331 + 0.734757i \(0.262704\pi\)
\(108\) 0 0
\(109\) −0.0778929 −0.00746078 −0.00373039 0.999993i \(-0.501187\pi\)
−0.00373039 + 0.999993i \(0.501187\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.0778929 −0.00736018
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.33131 0.123609
\(117\) 0 0
\(118\) 7.56499 0.696414
\(119\) 0.571057 0.0523487
\(120\) 0 0
\(121\) 9.31157 0.846506
\(122\) 2.15579 0.195176
\(123\) 0 0
\(124\) −2.50684 −0.225121
\(125\) 0 0
\(126\) 0 0
\(127\) 17.8321 1.58234 0.791171 0.611596i \(-0.209472\pi\)
0.791171 + 0.611596i \(0.209472\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 1.49316 0.130458 0.0652292 0.997870i \(-0.479222\pi\)
0.0652292 + 0.997870i \(0.479222\pi\)
\(132\) 0 0
\(133\) −0.0778929 −0.00675417
\(134\) 4.58473 0.396060
\(135\) 0 0
\(136\) 7.33131 0.628655
\(137\) 8.42894 0.720133 0.360067 0.932927i \(-0.382754\pi\)
0.360067 + 0.932927i \(0.382754\pi\)
\(138\) 0 0
\(139\) 8.81841 0.747968 0.373984 0.927435i \(-0.377992\pi\)
0.373984 + 0.927435i \(0.377992\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −10.8579 −0.911174
\(143\) −24.0273 −2.00927
\(144\) 0 0
\(145\) 0 0
\(146\) 5.09763 0.421883
\(147\) 0 0
\(148\) −5.50684 −0.452659
\(149\) −17.6763 −1.44810 −0.724049 0.689748i \(-0.757721\pi\)
−0.724049 + 0.689748i \(0.757721\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 0.351050 0.0282884
\(155\) 0 0
\(156\) 0 0
\(157\) 0.506836 0.0404499 0.0202250 0.999795i \(-0.493562\pi\)
0.0202250 + 0.999795i \(0.493562\pi\)
\(158\) −17.0137 −1.35354
\(159\) 0 0
\(160\) 0 0
\(161\) 0.265553 0.0209285
\(162\) 0 0
\(163\) 0.830542 0.0650531 0.0325265 0.999471i \(-0.489645\pi\)
0.0325265 + 0.999471i \(0.489645\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −13.1695 −1.02215
\(167\) −16.5068 −1.27734 −0.638669 0.769482i \(-0.720515\pi\)
−0.638669 + 0.769482i \(0.720515\pi\)
\(168\) 0 0
\(169\) 15.4229 1.18638
\(170\) 0 0
\(171\) 0 0
\(172\) 0.506836 0.0386459
\(173\) −18.8321 −1.43178 −0.715888 0.698215i \(-0.753978\pi\)
−0.715888 + 0.698215i \(0.753978\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.50684 0.339716
\(177\) 0 0
\(178\) 15.0137 1.12532
\(179\) 7.15579 0.534849 0.267424 0.963579i \(-0.413827\pi\)
0.267424 + 0.963579i \(0.413827\pi\)
\(180\) 0 0
\(181\) 12.5205 0.930642 0.465321 0.885142i \(-0.345939\pi\)
0.465321 + 0.885142i \(0.345939\pi\)
\(182\) −0.415271 −0.0307819
\(183\) 0 0
\(184\) 3.40920 0.251330
\(185\) 0 0
\(186\) 0 0
\(187\) −33.0410 −2.41620
\(188\) 5.66262 0.412989
\(189\) 0 0
\(190\) 0 0
\(191\) 4.90237 0.354723 0.177361 0.984146i \(-0.443244\pi\)
0.177361 + 0.984146i \(0.443244\pi\)
\(192\) 0 0
\(193\) 18.1558 1.30688 0.653441 0.756977i \(-0.273325\pi\)
0.653441 + 0.756977i \(0.273325\pi\)
\(194\) −7.67629 −0.551126
\(195\) 0 0
\(196\) −6.99393 −0.499567
\(197\) 2.98633 0.212767 0.106384 0.994325i \(-0.466073\pi\)
0.106384 + 0.994325i \(0.466073\pi\)
\(198\) 0 0
\(199\) −3.06422 −0.217217 −0.108608 0.994085i \(-0.534639\pi\)
−0.108608 + 0.994085i \(0.534639\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −4.15579 −0.292400
\(203\) −0.103700 −0.00727829
\(204\) 0 0
\(205\) 0 0
\(206\) −2.35105 −0.163805
\(207\) 0 0
\(208\) −5.33131 −0.369660
\(209\) 4.50684 0.311744
\(210\) 0 0
\(211\) 19.2534 1.32546 0.662730 0.748858i \(-0.269398\pi\)
0.662730 + 0.748858i \(0.269398\pi\)
\(212\) 12.9358 0.888433
\(213\) 0 0
\(214\) −14.0334 −0.959304
\(215\) 0 0
\(216\) 0 0
\(217\) 0.195265 0.0132554
\(218\) 0.0778929 0.00527557
\(219\) 0 0
\(220\) 0 0
\(221\) 39.0855 2.62918
\(222\) 0 0
\(223\) −14.5068 −0.971450 −0.485725 0.874112i \(-0.661444\pi\)
−0.485725 + 0.874112i \(0.661444\pi\)
\(224\) 0.0778929 0.00520444
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 21.5984 1.43354 0.716768 0.697312i \(-0.245621\pi\)
0.716768 + 0.697312i \(0.245621\pi\)
\(228\) 0 0
\(229\) −19.0137 −1.25646 −0.628229 0.778028i \(-0.716220\pi\)
−0.628229 + 0.778028i \(0.716220\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.33131 −0.0874048
\(233\) 6.01367 0.393969 0.196984 0.980407i \(-0.436885\pi\)
0.196984 + 0.980407i \(0.436885\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −7.56499 −0.492439
\(237\) 0 0
\(238\) −0.571057 −0.0370161
\(239\) −15.4092 −0.996739 −0.498369 0.866965i \(-0.666068\pi\)
−0.498369 + 0.866965i \(0.666068\pi\)
\(240\) 0 0
\(241\) −4.81841 −0.310381 −0.155190 0.987885i \(-0.549599\pi\)
−0.155190 + 0.987885i \(0.549599\pi\)
\(242\) −9.31157 −0.598570
\(243\) 0 0
\(244\) −2.15579 −0.138010
\(245\) 0 0
\(246\) 0 0
\(247\) −5.33131 −0.339223
\(248\) 2.50684 0.159184
\(249\) 0 0
\(250\) 0 0
\(251\) −1.52051 −0.0959736 −0.0479868 0.998848i \(-0.515281\pi\)
−0.0479868 + 0.998848i \(0.515281\pi\)
\(252\) 0 0
\(253\) −15.3647 −0.965972
\(254\) −17.8321 −1.11888
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −16.5068 −1.02967 −0.514834 0.857290i \(-0.672146\pi\)
−0.514834 + 0.857290i \(0.672146\pi\)
\(258\) 0 0
\(259\) 0.428943 0.0266532
\(260\) 0 0
\(261\) 0 0
\(262\) −1.49316 −0.0922480
\(263\) 18.0273 1.11161 0.555807 0.831311i \(-0.312409\pi\)
0.555807 + 0.831311i \(0.312409\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0.0778929 0.00477592
\(267\) 0 0
\(268\) −4.58473 −0.280057
\(269\) −20.2089 −1.23216 −0.616080 0.787683i \(-0.711280\pi\)
−0.616080 + 0.787683i \(0.711280\pi\)
\(270\) 0 0
\(271\) 6.08396 0.369574 0.184787 0.982779i \(-0.440840\pi\)
0.184787 + 0.982779i \(0.440840\pi\)
\(272\) −7.33131 −0.444526
\(273\) 0 0
\(274\) −8.42894 −0.509211
\(275\) 0 0
\(276\) 0 0
\(277\) −31.3647 −1.88452 −0.942262 0.334877i \(-0.891305\pi\)
−0.942262 + 0.334877i \(0.891305\pi\)
\(278\) −8.81841 −0.528893
\(279\) 0 0
\(280\) 0 0
\(281\) −11.3252 −0.675607 −0.337804 0.941217i \(-0.609684\pi\)
−0.337804 + 0.941217i \(0.609684\pi\)
\(282\) 0 0
\(283\) 26.1437 1.55408 0.777039 0.629452i \(-0.216721\pi\)
0.777039 + 0.629452i \(0.216721\pi\)
\(284\) 10.8579 0.644297
\(285\) 0 0
\(286\) 24.0273 1.42077
\(287\) 0 0
\(288\) 0 0
\(289\) 36.7481 2.16165
\(290\) 0 0
\(291\) 0 0
\(292\) −5.09763 −0.298316
\(293\) −1.33131 −0.0777760 −0.0388880 0.999244i \(-0.512382\pi\)
−0.0388880 + 0.999244i \(0.512382\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 5.50684 0.320078
\(297\) 0 0
\(298\) 17.6763 1.02396
\(299\) 18.1755 1.05112
\(300\) 0 0
\(301\) −0.0394789 −0.00227553
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −2.84421 −0.162328 −0.0811639 0.996701i \(-0.525864\pi\)
−0.0811639 + 0.996701i \(0.525864\pi\)
\(308\) −0.351050 −0.0200030
\(309\) 0 0
\(310\) 0 0
\(311\) 16.3895 0.929361 0.464681 0.885478i \(-0.346169\pi\)
0.464681 + 0.885478i \(0.346169\pi\)
\(312\) 0 0
\(313\) −21.0471 −1.18965 −0.594826 0.803855i \(-0.702779\pi\)
−0.594826 + 0.803855i \(0.702779\pi\)
\(314\) −0.506836 −0.0286024
\(315\) 0 0
\(316\) 17.0137 0.957094
\(317\) 30.8902 1.73497 0.867484 0.497465i \(-0.165736\pi\)
0.867484 + 0.497465i \(0.165736\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 0 0
\(322\) −0.265553 −0.0147987
\(323\) −7.33131 −0.407925
\(324\) 0 0
\(325\) 0 0
\(326\) −0.830542 −0.0459995
\(327\) 0 0
\(328\) 0 0
\(329\) −0.441078 −0.0243174
\(330\) 0 0
\(331\) 28.3845 1.56015 0.780076 0.625685i \(-0.215181\pi\)
0.780076 + 0.625685i \(0.215181\pi\)
\(332\) 13.1695 0.722768
\(333\) 0 0
\(334\) 16.5068 0.903214
\(335\) 0 0
\(336\) 0 0
\(337\) −17.8716 −0.973526 −0.486763 0.873534i \(-0.661822\pi\)
−0.486763 + 0.873534i \(0.661822\pi\)
\(338\) −15.4229 −0.838894
\(339\) 0 0
\(340\) 0 0
\(341\) −11.2979 −0.611816
\(342\) 0 0
\(343\) 1.09003 0.0588560
\(344\) −0.506836 −0.0273268
\(345\) 0 0
\(346\) 18.8321 1.01242
\(347\) −33.0410 −1.77373 −0.886867 0.462024i \(-0.847123\pi\)
−0.886867 + 0.462024i \(0.847123\pi\)
\(348\) 0 0
\(349\) −19.7158 −1.05536 −0.527681 0.849443i \(-0.676938\pi\)
−0.527681 + 0.849443i \(0.676938\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.50684 −0.240215
\(353\) 5.90997 0.314556 0.157278 0.987554i \(-0.449728\pi\)
0.157278 + 0.987554i \(0.449728\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −15.0137 −0.795723
\(357\) 0 0
\(358\) −7.15579 −0.378195
\(359\) 7.00760 0.369847 0.184924 0.982753i \(-0.440796\pi\)
0.184924 + 0.982753i \(0.440796\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −12.5205 −0.658063
\(363\) 0 0
\(364\) 0.415271 0.0217661
\(365\) 0 0
\(366\) 0 0
\(367\) 17.7158 0.924756 0.462378 0.886683i \(-0.346996\pi\)
0.462378 + 0.886683i \(0.346996\pi\)
\(368\) −3.40920 −0.177717
\(369\) 0 0
\(370\) 0 0
\(371\) −1.00760 −0.0523122
\(372\) 0 0
\(373\) 15.4487 0.799902 0.399951 0.916536i \(-0.369027\pi\)
0.399951 + 0.916536i \(0.369027\pi\)
\(374\) 33.0410 1.70851
\(375\) 0 0
\(376\) −5.66262 −0.292027
\(377\) −7.09763 −0.365547
\(378\) 0 0
\(379\) 34.4563 1.76990 0.884950 0.465685i \(-0.154192\pi\)
0.884950 + 0.465685i \(0.154192\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4.90237 −0.250827
\(383\) 12.0273 0.614569 0.307284 0.951618i \(-0.400580\pi\)
0.307284 + 0.951618i \(0.400580\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −18.1558 −0.924105
\(387\) 0 0
\(388\) 7.67629 0.389705
\(389\) 15.3647 0.779022 0.389511 0.921022i \(-0.372644\pi\)
0.389511 + 0.921022i \(0.372644\pi\)
\(390\) 0 0
\(391\) 24.9939 1.26400
\(392\) 6.99393 0.353247
\(393\) 0 0
\(394\) −2.98633 −0.150449
\(395\) 0 0
\(396\) 0 0
\(397\) 1.32524 0.0665121 0.0332560 0.999447i \(-0.489412\pi\)
0.0332560 + 0.999447i \(0.489412\pi\)
\(398\) 3.06422 0.153596
\(399\) 0 0
\(400\) 0 0
\(401\) 6.46736 0.322964 0.161482 0.986876i \(-0.448373\pi\)
0.161482 + 0.986876i \(0.448373\pi\)
\(402\) 0 0
\(403\) 13.3647 0.665744
\(404\) 4.15579 0.206758
\(405\) 0 0
\(406\) 0.103700 0.00514653
\(407\) −24.8184 −1.23020
\(408\) 0 0
\(409\) −1.36472 −0.0674812 −0.0337406 0.999431i \(-0.510742\pi\)
−0.0337406 + 0.999431i \(0.510742\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.35105 0.115828
\(413\) 0.589259 0.0289955
\(414\) 0 0
\(415\) 0 0
\(416\) 5.33131 0.261389
\(417\) 0 0
\(418\) −4.50684 −0.220437
\(419\) −16.6231 −0.812094 −0.406047 0.913852i \(-0.633093\pi\)
−0.406047 + 0.913852i \(0.633093\pi\)
\(420\) 0 0
\(421\) −31.1128 −1.51635 −0.758174 0.652053i \(-0.773908\pi\)
−0.758174 + 0.652053i \(0.773908\pi\)
\(422\) −19.2534 −0.937242
\(423\) 0 0
\(424\) −12.9358 −0.628217
\(425\) 0 0
\(426\) 0 0
\(427\) 0.167920 0.00812623
\(428\) 14.0334 0.678331
\(429\) 0 0
\(430\) 0 0
\(431\) 10.1831 0.490504 0.245252 0.969459i \(-0.421129\pi\)
0.245252 + 0.969459i \(0.421129\pi\)
\(432\) 0 0
\(433\) 22.1953 1.06664 0.533318 0.845915i \(-0.320945\pi\)
0.533318 + 0.845915i \(0.320945\pi\)
\(434\) −0.195265 −0.00937300
\(435\) 0 0
\(436\) −0.0778929 −0.00373039
\(437\) −3.40920 −0.163084
\(438\) 0 0
\(439\) −9.32524 −0.445070 −0.222535 0.974925i \(-0.571433\pi\)
−0.222535 + 0.974925i \(0.571433\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −39.0855 −1.85911
\(443\) 13.9879 0.664584 0.332292 0.943177i \(-0.392178\pi\)
0.332292 + 0.943177i \(0.392178\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 14.5068 0.686919
\(447\) 0 0
\(448\) −0.0778929 −0.00368009
\(449\) −13.4932 −0.636782 −0.318391 0.947960i \(-0.603142\pi\)
−0.318391 + 0.947960i \(0.603142\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −21.5984 −1.01366
\(455\) 0 0
\(456\) 0 0
\(457\) 9.68236 0.452922 0.226461 0.974020i \(-0.427284\pi\)
0.226461 + 0.974020i \(0.427284\pi\)
\(458\) 19.0137 0.888451
\(459\) 0 0
\(460\) 0 0
\(461\) −8.66262 −0.403459 −0.201729 0.979441i \(-0.564656\pi\)
−0.201729 + 0.979441i \(0.564656\pi\)
\(462\) 0 0
\(463\) 28.0015 1.30134 0.650671 0.759360i \(-0.274488\pi\)
0.650671 + 0.759360i \(0.274488\pi\)
\(464\) 1.33131 0.0618046
\(465\) 0 0
\(466\) −6.01367 −0.278578
\(467\) −16.9742 −0.785472 −0.392736 0.919651i \(-0.628471\pi\)
−0.392736 + 0.919651i \(0.628471\pi\)
\(468\) 0 0
\(469\) 0.357118 0.0164902
\(470\) 0 0
\(471\) 0 0
\(472\) 7.56499 0.348207
\(473\) 2.28423 0.105029
\(474\) 0 0
\(475\) 0 0
\(476\) 0.571057 0.0261743
\(477\) 0 0
\(478\) 15.4092 0.704801
\(479\) −10.0532 −0.459340 −0.229670 0.973269i \(-0.573765\pi\)
−0.229670 + 0.973269i \(0.573765\pi\)
\(480\) 0 0
\(481\) 29.3587 1.33864
\(482\) 4.81841 0.219472
\(483\) 0 0
\(484\) 9.31157 0.423253
\(485\) 0 0
\(486\) 0 0
\(487\) 19.2089 0.870440 0.435220 0.900324i \(-0.356671\pi\)
0.435220 + 0.900324i \(0.356671\pi\)
\(488\) 2.15579 0.0975878
\(489\) 0 0
\(490\) 0 0
\(491\) −5.32524 −0.240325 −0.120162 0.992754i \(-0.538342\pi\)
−0.120162 + 0.992754i \(0.538342\pi\)
\(492\) 0 0
\(493\) −9.76025 −0.439580
\(494\) 5.33131 0.239867
\(495\) 0 0
\(496\) −2.50684 −0.112560
\(497\) −0.845752 −0.0379372
\(498\) 0 0
\(499\) −11.9605 −0.535426 −0.267713 0.963499i \(-0.586268\pi\)
−0.267713 + 0.963499i \(0.586268\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1.52051 0.0678636
\(503\) 19.3313 0.861941 0.430970 0.902366i \(-0.358171\pi\)
0.430970 + 0.902366i \(0.358171\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 15.3647 0.683045
\(507\) 0 0
\(508\) 17.8321 0.791171
\(509\) 36.1573 1.60265 0.801323 0.598232i \(-0.204130\pi\)
0.801323 + 0.598232i \(0.204130\pi\)
\(510\) 0 0
\(511\) 0.397069 0.0175653
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 16.5068 0.728085
\(515\) 0 0
\(516\) 0 0
\(517\) 25.5205 1.12239
\(518\) −0.428943 −0.0188467
\(519\) 0 0
\(520\) 0 0
\(521\) 31.5205 1.38094 0.690469 0.723362i \(-0.257404\pi\)
0.690469 + 0.723362i \(0.257404\pi\)
\(522\) 0 0
\(523\) 5.59840 0.244801 0.122400 0.992481i \(-0.460941\pi\)
0.122400 + 0.992481i \(0.460941\pi\)
\(524\) 1.49316 0.0652292
\(525\) 0 0
\(526\) −18.0273 −0.786030
\(527\) 18.3784 0.800575
\(528\) 0 0
\(529\) −11.3773 −0.494667
\(530\) 0 0
\(531\) 0 0
\(532\) −0.0778929 −0.00337708
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 4.58473 0.198030
\(537\) 0 0
\(538\) 20.2089 0.871269
\(539\) −31.5205 −1.35768
\(540\) 0 0
\(541\) 15.1968 0.653362 0.326681 0.945135i \(-0.394070\pi\)
0.326681 + 0.945135i \(0.394070\pi\)
\(542\) −6.08396 −0.261328
\(543\) 0 0
\(544\) 7.33131 0.314327
\(545\) 0 0
\(546\) 0 0
\(547\) −26.6505 −1.13949 −0.569746 0.821821i \(-0.692959\pi\)
−0.569746 + 0.821821i \(0.692959\pi\)
\(548\) 8.42894 0.360067
\(549\) 0 0
\(550\) 0 0
\(551\) 1.33131 0.0567158
\(552\) 0 0
\(553\) −1.32524 −0.0563551
\(554\) 31.3647 1.33256
\(555\) 0 0
\(556\) 8.81841 0.373984
\(557\) −12.8458 −0.544292 −0.272146 0.962256i \(-0.587733\pi\)
−0.272146 + 0.962256i \(0.587733\pi\)
\(558\) 0 0
\(559\) −2.70210 −0.114287
\(560\) 0 0
\(561\) 0 0
\(562\) 11.3252 0.477727
\(563\) −10.1968 −0.429744 −0.214872 0.976642i \(-0.568933\pi\)
−0.214872 + 0.976642i \(0.568933\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −26.1437 −1.09890
\(567\) 0 0
\(568\) −10.8579 −0.455587
\(569\) 24.3784 1.02200 0.510998 0.859582i \(-0.329276\pi\)
0.510998 + 0.859582i \(0.329276\pi\)
\(570\) 0 0
\(571\) 8.32371 0.348336 0.174168 0.984716i \(-0.444276\pi\)
0.174168 + 0.984716i \(0.444276\pi\)
\(572\) −24.0273 −1.00463
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −7.29290 −0.303607 −0.151804 0.988411i \(-0.548508\pi\)
−0.151804 + 0.988411i \(0.548508\pi\)
\(578\) −36.7481 −1.52852
\(579\) 0 0
\(580\) 0 0
\(581\) −1.02581 −0.0425576
\(582\) 0 0
\(583\) 58.2994 2.41452
\(584\) 5.09763 0.210942
\(585\) 0 0
\(586\) 1.33131 0.0549959
\(587\) −5.53264 −0.228357 −0.114178 0.993460i \(-0.536424\pi\)
−0.114178 + 0.993460i \(0.536424\pi\)
\(588\) 0 0
\(589\) −2.50684 −0.103292
\(590\) 0 0
\(591\) 0 0
\(592\) −5.50684 −0.226330
\(593\) 26.3252 1.08105 0.540524 0.841329i \(-0.318226\pi\)
0.540524 + 0.841329i \(0.318226\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −17.6763 −0.724049
\(597\) 0 0
\(598\) −18.1755 −0.743252
\(599\) −22.2994 −0.911130 −0.455565 0.890202i \(-0.650563\pi\)
−0.455565 + 0.890202i \(0.650563\pi\)
\(600\) 0 0
\(601\) 32.4947 1.32549 0.662743 0.748847i \(-0.269392\pi\)
0.662743 + 0.748847i \(0.269392\pi\)
\(602\) 0.0394789 0.00160904
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) 0 0
\(606\) 0 0
\(607\) 8.35105 0.338959 0.169479 0.985534i \(-0.445791\pi\)
0.169479 + 0.985534i \(0.445791\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −30.1892 −1.22132
\(612\) 0 0
\(613\) −38.2994 −1.54690 −0.773450 0.633858i \(-0.781471\pi\)
−0.773450 + 0.633858i \(0.781471\pi\)
\(614\) 2.84421 0.114783
\(615\) 0 0
\(616\) 0.351050 0.0141442
\(617\) 14.3526 0.577813 0.288907 0.957357i \(-0.406708\pi\)
0.288907 + 0.957357i \(0.406708\pi\)
\(618\) 0 0
\(619\) −8.62314 −0.346593 −0.173297 0.984870i \(-0.555442\pi\)
−0.173297 + 0.984870i \(0.555442\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −16.3895 −0.657158
\(623\) 1.16946 0.0468533
\(624\) 0 0
\(625\) 0 0
\(626\) 21.0471 0.841211
\(627\) 0 0
\(628\) 0.506836 0.0202250
\(629\) 40.3723 1.60975
\(630\) 0 0
\(631\) −10.3374 −0.411525 −0.205762 0.978602i \(-0.565967\pi\)
−0.205762 + 0.978602i \(0.565967\pi\)
\(632\) −17.0137 −0.676768
\(633\) 0 0
\(634\) −30.8902 −1.22681
\(635\) 0 0
\(636\) 0 0
\(637\) 37.2868 1.47736
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) 0 0
\(641\) −5.88369 −0.232392 −0.116196 0.993226i \(-0.537070\pi\)
−0.116196 + 0.993226i \(0.537070\pi\)
\(642\) 0 0
\(643\) −25.5084 −1.00595 −0.502976 0.864300i \(-0.667762\pi\)
−0.502976 + 0.864300i \(0.667762\pi\)
\(644\) 0.265553 0.0104642
\(645\) 0 0
\(646\) 7.33131 0.288447
\(647\) 5.09157 0.200170 0.100085 0.994979i \(-0.468089\pi\)
0.100085 + 0.994979i \(0.468089\pi\)
\(648\) 0 0
\(649\) −34.0942 −1.33831
\(650\) 0 0
\(651\) 0 0
\(652\) 0.830542 0.0325265
\(653\) 11.1816 0.437570 0.218785 0.975773i \(-0.429791\pi\)
0.218785 + 0.975773i \(0.429791\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0.441078 0.0171950
\(659\) 13.7542 0.535787 0.267894 0.963449i \(-0.413672\pi\)
0.267894 + 0.963449i \(0.413672\pi\)
\(660\) 0 0
\(661\) −12.6171 −0.490747 −0.245374 0.969429i \(-0.578911\pi\)
−0.245374 + 0.969429i \(0.578911\pi\)
\(662\) −28.3845 −1.10319
\(663\) 0 0
\(664\) −13.1695 −0.511074
\(665\) 0 0
\(666\) 0 0
\(667\) −4.53871 −0.175740
\(668\) −16.5068 −0.638669
\(669\) 0 0
\(670\) 0 0
\(671\) −9.71577 −0.375073
\(672\) 0 0
\(673\) 2.35105 0.0906263 0.0453132 0.998973i \(-0.485571\pi\)
0.0453132 + 0.998973i \(0.485571\pi\)
\(674\) 17.8716 0.688387
\(675\) 0 0
\(676\) 15.4229 0.593188
\(677\) 23.7663 0.913414 0.456707 0.889617i \(-0.349029\pi\)
0.456707 + 0.889617i \(0.349029\pi\)
\(678\) 0 0
\(679\) −0.597928 −0.0229464
\(680\) 0 0
\(681\) 0 0
\(682\) 11.2979 0.432619
\(683\) −28.1695 −1.07787 −0.538937 0.842346i \(-0.681174\pi\)
−0.538937 + 0.842346i \(0.681174\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.09003 −0.0416174
\(687\) 0 0
\(688\) 0.506836 0.0193229
\(689\) −68.9647 −2.62734
\(690\) 0 0
\(691\) 2.32371 0.0883979 0.0441990 0.999023i \(-0.485926\pi\)
0.0441990 + 0.999023i \(0.485926\pi\)
\(692\) −18.8321 −0.715888
\(693\) 0 0
\(694\) 33.0410 1.25422
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 19.7158 0.746253
\(699\) 0 0
\(700\) 0 0
\(701\) −23.7036 −0.895274 −0.447637 0.894215i \(-0.647734\pi\)
−0.447637 + 0.894215i \(0.647734\pi\)
\(702\) 0 0
\(703\) −5.50684 −0.207694
\(704\) 4.50684 0.169858
\(705\) 0 0
\(706\) −5.90997 −0.222425
\(707\) −0.323706 −0.0121742
\(708\) 0 0
\(709\) 9.05315 0.339998 0.169999 0.985444i \(-0.445624\pi\)
0.169999 + 0.985444i \(0.445624\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 15.0137 0.562661
\(713\) 8.54631 0.320062
\(714\) 0 0
\(715\) 0 0
\(716\) 7.15579 0.267424
\(717\) 0 0
\(718\) −7.00760 −0.261521
\(719\) −35.0734 −1.30802 −0.654008 0.756488i \(-0.726914\pi\)
−0.654008 + 0.756488i \(0.726914\pi\)
\(720\) 0 0
\(721\) −0.183130 −0.00682012
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) 12.5205 0.465321
\(725\) 0 0
\(726\) 0 0
\(727\) 30.1386 1.11778 0.558890 0.829242i \(-0.311227\pi\)
0.558890 + 0.829242i \(0.311227\pi\)
\(728\) −0.415271 −0.0153910
\(729\) 0 0
\(730\) 0 0
\(731\) −3.71577 −0.137433
\(732\) 0 0
\(733\) 47.8321 1.76672 0.883359 0.468697i \(-0.155276\pi\)
0.883359 + 0.468697i \(0.155276\pi\)
\(734\) −17.7158 −0.653901
\(735\) 0 0
\(736\) 3.40920 0.125665
\(737\) −20.6626 −0.761117
\(738\) 0 0
\(739\) −22.4674 −0.826475 −0.413238 0.910623i \(-0.635602\pi\)
−0.413238 + 0.910623i \(0.635602\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.00760 0.0369903
\(743\) 11.7926 0.432629 0.216314 0.976324i \(-0.430596\pi\)
0.216314 + 0.976324i \(0.430596\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −15.4487 −0.565616
\(747\) 0 0
\(748\) −33.0410 −1.20810
\(749\) −1.09310 −0.0399411
\(750\) 0 0
\(751\) −2.03948 −0.0744216 −0.0372108 0.999307i \(-0.511847\pi\)
−0.0372108 + 0.999307i \(0.511847\pi\)
\(752\) 5.66262 0.206495
\(753\) 0 0
\(754\) 7.09763 0.258481
\(755\) 0 0
\(756\) 0 0
\(757\) 43.3526 1.57568 0.787838 0.615882i \(-0.211200\pi\)
0.787838 + 0.615882i \(0.211200\pi\)
\(758\) −34.4563 −1.25151
\(759\) 0 0
\(760\) 0 0
\(761\) −6.62921 −0.240309 −0.120154 0.992755i \(-0.538339\pi\)
−0.120154 + 0.992755i \(0.538339\pi\)
\(762\) 0 0
\(763\) 0.00606730 0.000219651 0
\(764\) 4.90237 0.177361
\(765\) 0 0
\(766\) −12.0273 −0.434566
\(767\) 40.3313 1.45628
\(768\) 0 0
\(769\) −19.6429 −0.708340 −0.354170 0.935181i \(-0.615237\pi\)
−0.354170 + 0.935181i \(0.615237\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 18.1558 0.653441
\(773\) −14.4107 −0.518318 −0.259159 0.965835i \(-0.583445\pi\)
−0.259159 + 0.965835i \(0.583445\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −7.67629 −0.275563
\(777\) 0 0
\(778\) −15.3647 −0.550852
\(779\) 0 0
\(780\) 0 0
\(781\) 48.9347 1.75102
\(782\) −24.9939 −0.893781
\(783\) 0 0
\(784\) −6.99393 −0.249783
\(785\) 0 0
\(786\) 0 0
\(787\) 24.3176 0.866830 0.433415 0.901194i \(-0.357308\pi\)
0.433415 + 0.901194i \(0.357308\pi\)
\(788\) 2.98633 0.106384
\(789\) 0 0
\(790\) 0 0
\(791\) −0.467357 −0.0166173
\(792\) 0 0
\(793\) 11.4932 0.408134
\(794\) −1.32524 −0.0470311
\(795\) 0 0
\(796\) −3.06422 −0.108608
\(797\) 7.30397 0.258720 0.129360 0.991598i \(-0.458708\pi\)
0.129360 + 0.991598i \(0.458708\pi\)
\(798\) 0 0
\(799\) −41.5144 −1.46868
\(800\) 0 0
\(801\) 0 0
\(802\) −6.46736 −0.228370
\(803\) −22.9742 −0.810742
\(804\) 0 0
\(805\) 0 0
\(806\) −13.3647 −0.470752
\(807\) 0 0
\(808\) −4.15579 −0.146200
\(809\) 0.584729 0.0205580 0.0102790 0.999947i \(-0.496728\pi\)
0.0102790 + 0.999947i \(0.496728\pi\)
\(810\) 0 0
\(811\) 1.90997 0.0670682 0.0335341 0.999438i \(-0.489324\pi\)
0.0335341 + 0.999438i \(0.489324\pi\)
\(812\) −0.103700 −0.00363914
\(813\) 0 0
\(814\) 24.8184 0.869885
\(815\) 0 0
\(816\) 0 0
\(817\) 0.506836 0.0177319
\(818\) 1.36472 0.0477164
\(819\) 0 0
\(820\) 0 0
\(821\) 38.2226 1.33398 0.666989 0.745067i \(-0.267583\pi\)
0.666989 + 0.745067i \(0.267583\pi\)
\(822\) 0 0
\(823\) −45.7481 −1.59468 −0.797340 0.603531i \(-0.793760\pi\)
−0.797340 + 0.603531i \(0.793760\pi\)
\(824\) −2.35105 −0.0819027
\(825\) 0 0
\(826\) −0.589259 −0.0205029
\(827\) 22.6687 0.788268 0.394134 0.919053i \(-0.371045\pi\)
0.394134 + 0.919053i \(0.371045\pi\)
\(828\) 0 0
\(829\) −4.63028 −0.160816 −0.0804081 0.996762i \(-0.525622\pi\)
−0.0804081 + 0.996762i \(0.525622\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −5.33131 −0.184830
\(833\) 51.2747 1.77656
\(834\) 0 0
\(835\) 0 0
\(836\) 4.50684 0.155872
\(837\) 0 0
\(838\) 16.6231 0.574237
\(839\) 50.4826 1.74285 0.871426 0.490527i \(-0.163196\pi\)
0.871426 + 0.490527i \(0.163196\pi\)
\(840\) 0 0
\(841\) −27.2276 −0.938883
\(842\) 31.1128 1.07222
\(843\) 0 0
\(844\) 19.2534 0.662730
\(845\) 0 0
\(846\) 0 0
\(847\) −0.725305 −0.0249218
\(848\) 12.9358 0.444216
\(849\) 0 0
\(850\) 0 0
\(851\) 18.7739 0.643562
\(852\) 0 0
\(853\) 36.2105 1.23982 0.619912 0.784672i \(-0.287168\pi\)
0.619912 + 0.784672i \(0.287168\pi\)
\(854\) −0.167920 −0.00574611
\(855\) 0 0
\(856\) −14.0334 −0.479652
\(857\) −25.1968 −0.860706 −0.430353 0.902661i \(-0.641611\pi\)
−0.430353 + 0.902661i \(0.641611\pi\)
\(858\) 0 0
\(859\) 18.8579 0.643423 0.321711 0.946838i \(-0.395742\pi\)
0.321711 + 0.946838i \(0.395742\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −10.1831 −0.346839
\(863\) −15.0137 −0.511071 −0.255536 0.966800i \(-0.582252\pi\)
−0.255536 + 0.966800i \(0.582252\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −22.1953 −0.754226
\(867\) 0 0
\(868\) 0.195265 0.00662771
\(869\) 76.6778 2.60112
\(870\) 0 0
\(871\) 24.4426 0.828206
\(872\) 0.0778929 0.00263779
\(873\) 0 0
\(874\) 3.40920 0.115318
\(875\) 0 0
\(876\) 0 0
\(877\) 35.1128 1.18568 0.592838 0.805322i \(-0.298007\pi\)
0.592838 + 0.805322i \(0.298007\pi\)
\(878\) 9.32524 0.314712
\(879\) 0 0
\(880\) 0 0
\(881\) 41.3389 1.39274 0.696372 0.717681i \(-0.254797\pi\)
0.696372 + 0.717681i \(0.254797\pi\)
\(882\) 0 0
\(883\) −12.6353 −0.425211 −0.212605 0.977138i \(-0.568195\pi\)
−0.212605 + 0.977138i \(0.568195\pi\)
\(884\) 39.0855 1.31459
\(885\) 0 0
\(886\) −13.9879 −0.469932
\(887\) −4.15579 −0.139538 −0.0697688 0.997563i \(-0.522226\pi\)
−0.0697688 + 0.997563i \(0.522226\pi\)
\(888\) 0 0
\(889\) −1.38899 −0.0465853
\(890\) 0 0
\(891\) 0 0
\(892\) −14.5068 −0.485725
\(893\) 5.66262 0.189492
\(894\) 0 0
\(895\) 0 0
\(896\) 0.0778929 0.00260222
\(897\) 0 0
\(898\) 13.4932 0.450273
\(899\) −3.33738 −0.111308
\(900\) 0 0
\(901\) −94.8362 −3.15945
\(902\) 0 0
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) −41.7542 −1.38643 −0.693213 0.720733i \(-0.743805\pi\)
−0.693213 + 0.720733i \(0.743805\pi\)
\(908\) 21.5984 0.716768
\(909\) 0 0
\(910\) 0 0
\(911\) 9.12998 0.302490 0.151245 0.988496i \(-0.451672\pi\)
0.151245 + 0.988496i \(0.451672\pi\)
\(912\) 0 0
\(913\) 59.3526 1.96428
\(914\) −9.68236 −0.320264
\(915\) 0 0
\(916\) −19.0137 −0.628229
\(917\) −0.116307 −0.00384079
\(918\) 0 0
\(919\) 19.0197 0.627403 0.313702 0.949522i \(-0.398431\pi\)
0.313702 + 0.949522i \(0.398431\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 8.66262 0.285288
\(923\) −57.8868 −1.90537
\(924\) 0 0
\(925\) 0 0
\(926\) −28.0015 −0.920188
\(927\) 0 0
\(928\) −1.33131 −0.0437024
\(929\) 7.77239 0.255004 0.127502 0.991838i \(-0.459304\pi\)
0.127502 + 0.991838i \(0.459304\pi\)
\(930\) 0 0
\(931\) −6.99393 −0.229217
\(932\) 6.01367 0.196984
\(933\) 0 0
\(934\) 16.9742 0.555413
\(935\) 0 0
\(936\) 0 0
\(937\) −45.9818 −1.50216 −0.751080 0.660211i \(-0.770467\pi\)
−0.751080 + 0.660211i \(0.770467\pi\)
\(938\) −0.357118 −0.0116603
\(939\) 0 0
\(940\) 0 0
\(941\) 40.6242 1.32431 0.662156 0.749366i \(-0.269642\pi\)
0.662156 + 0.749366i \(0.269642\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −7.56499 −0.246219
\(945\) 0 0
\(946\) −2.28423 −0.0742666
\(947\) 2.28423 0.0742274 0.0371137 0.999311i \(-0.488184\pi\)
0.0371137 + 0.999311i \(0.488184\pi\)
\(948\) 0 0
\(949\) 27.1771 0.882205
\(950\) 0 0
\(951\) 0 0
\(952\) −0.571057 −0.0185081
\(953\) 57.5084 1.86288 0.931439 0.363896i \(-0.118554\pi\)
0.931439 + 0.363896i \(0.118554\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −15.4092 −0.498369
\(957\) 0 0
\(958\) 10.0532 0.324803
\(959\) −0.656555 −0.0212013
\(960\) 0 0
\(961\) −24.7158 −0.797283
\(962\) −29.3587 −0.946561
\(963\) 0 0
\(964\) −4.81841 −0.155190
\(965\) 0 0
\(966\) 0 0
\(967\) −4.70210 −0.151209 −0.0756047 0.997138i \(-0.524089\pi\)
−0.0756047 + 0.997138i \(0.524089\pi\)
\(968\) −9.31157 −0.299285
\(969\) 0 0
\(970\) 0 0
\(971\) −14.9863 −0.480934 −0.240467 0.970657i \(-0.577301\pi\)
−0.240467 + 0.970657i \(0.577301\pi\)
\(972\) 0 0
\(973\) −0.686891 −0.0220207
\(974\) −19.2089 −0.615494
\(975\) 0 0
\(976\) −2.15579 −0.0690050
\(977\) −8.89737 −0.284652 −0.142326 0.989820i \(-0.545458\pi\)
−0.142326 + 0.989820i \(0.545458\pi\)
\(978\) 0 0
\(979\) −67.6642 −2.16256
\(980\) 0 0
\(981\) 0 0
\(982\) 5.32524 0.169935
\(983\) 1.60947 0.0513341 0.0256671 0.999671i \(-0.491829\pi\)
0.0256671 + 0.999671i \(0.491829\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 9.76025 0.310830
\(987\) 0 0
\(988\) −5.33131 −0.169612
\(989\) −1.72791 −0.0549443
\(990\) 0 0
\(991\) 16.8974 0.536763 0.268381 0.963313i \(-0.413511\pi\)
0.268381 + 0.963313i \(0.413511\pi\)
\(992\) 2.50684 0.0795921
\(993\) 0 0
\(994\) 0.845752 0.0268256
\(995\) 0 0
\(996\) 0 0
\(997\) 24.1558 0.765021 0.382511 0.923951i \(-0.375060\pi\)
0.382511 + 0.923951i \(0.375060\pi\)
\(998\) 11.9605 0.378604
\(999\) 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.ci.1.2 3
3.2 odd 2 950.2.a.l.1.3 yes 3
5.4 even 2 8550.2.a.cp.1.2 3
12.11 even 2 7600.2.a.bk.1.1 3
15.2 even 4 950.2.b.h.799.4 6
15.8 even 4 950.2.b.h.799.3 6
15.14 odd 2 950.2.a.j.1.1 3
60.59 even 2 7600.2.a.bz.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.j.1.1 3 15.14 odd 2
950.2.a.l.1.3 yes 3 3.2 odd 2
950.2.b.h.799.3 6 15.8 even 4
950.2.b.h.799.4 6 15.2 even 4
7600.2.a.bk.1.1 3 12.11 even 2
7600.2.a.bz.1.3 3 60.59 even 2
8550.2.a.ci.1.2 3 1.1 even 1 trivial
8550.2.a.cp.1.2 3 5.4 even 2