Properties

Label 8550.2.a.cq.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 \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +4.42864 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +4.42864 q^{7} +1.00000 q^{8} -2.62222 q^{11} -5.80642 q^{13} +4.42864 q^{14} +1.00000 q^{16} -3.80642 q^{17} +1.00000 q^{19} -2.62222 q^{22} -2.62222 q^{23} -5.80642 q^{26} +4.42864 q^{28} -3.37778 q^{29} -4.42864 q^{31} +1.00000 q^{32} -3.80642 q^{34} -5.80642 q^{37} +1.00000 q^{38} -5.67307 q^{41} +10.9906 q^{43} -2.62222 q^{44} -2.62222 q^{46} +2.62222 q^{47} +12.6128 q^{49} -5.80642 q^{52} -6.00000 q^{53} +4.42864 q^{56} -3.37778 q^{58} +1.05086 q^{59} +4.75557 q^{61} -4.42864 q^{62} +1.00000 q^{64} -15.6128 q^{67} -3.80642 q^{68} -15.6128 q^{71} -11.6128 q^{73} -5.80642 q^{74} +1.00000 q^{76} -11.6128 q^{77} +4.42864 q^{79} -5.67307 q^{82} +11.9081 q^{83} +10.9906 q^{86} -2.62222 q^{88} -12.4286 q^{89} -25.7146 q^{91} -2.62222 q^{92} +2.62222 q^{94} -7.37778 q^{97} +12.6128 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} + 3q^{8} - 8q^{11} - 4q^{13} + 3q^{16} + 2q^{17} + 3q^{19} - 8q^{22} - 8q^{23} - 4q^{26} - 10q^{29} + 3q^{32} + 2q^{34} - 4q^{37} + 3q^{38} - 4q^{41} + 6q^{43} - 8q^{44} - 8q^{46} + 8q^{47} + 11q^{49} - 4q^{52} - 18q^{53} - 10q^{58} - 10q^{59} + 14q^{61} + 3q^{64} - 20q^{67} + 2q^{68} - 20q^{71} - 8q^{73} - 4q^{74} + 3q^{76} - 8q^{77} - 4q^{82} - 4q^{83} + 6q^{86} - 8q^{88} - 24q^{89} - 24q^{91} - 8q^{92} + 8q^{94} - 22q^{97} + 11q^{98} + O(q^{100}) \)

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
\(7\) 4.42864 1.67387 0.836934 0.547304i \(-0.184346\pi\)
0.836934 + 0.547304i \(0.184346\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −2.62222 −0.790628 −0.395314 0.918546i \(-0.629364\pi\)
−0.395314 + 0.918546i \(0.629364\pi\)
\(12\) 0 0
\(13\) −5.80642 −1.61041 −0.805206 0.592995i \(-0.797945\pi\)
−0.805206 + 0.592995i \(0.797945\pi\)
\(14\) 4.42864 1.18360
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.80642 −0.923193 −0.461597 0.887090i \(-0.652723\pi\)
−0.461597 + 0.887090i \(0.652723\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −2.62222 −0.559058
\(23\) −2.62222 −0.546770 −0.273385 0.961905i \(-0.588143\pi\)
−0.273385 + 0.961905i \(0.588143\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −5.80642 −1.13873
\(27\) 0 0
\(28\) 4.42864 0.836934
\(29\) −3.37778 −0.627239 −0.313619 0.949549i \(-0.601542\pi\)
−0.313619 + 0.949549i \(0.601542\pi\)
\(30\) 0 0
\(31\) −4.42864 −0.795407 −0.397704 0.917514i \(-0.630193\pi\)
−0.397704 + 0.917514i \(0.630193\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.80642 −0.652796
\(35\) 0 0
\(36\) 0 0
\(37\) −5.80642 −0.954570 −0.477285 0.878749i \(-0.658379\pi\)
−0.477285 + 0.878749i \(0.658379\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −5.67307 −0.885985 −0.442992 0.896525i \(-0.646083\pi\)
−0.442992 + 0.896525i \(0.646083\pi\)
\(42\) 0 0
\(43\) 10.9906 1.67606 0.838028 0.545627i \(-0.183709\pi\)
0.838028 + 0.545627i \(0.183709\pi\)
\(44\) −2.62222 −0.395314
\(45\) 0 0
\(46\) −2.62222 −0.386625
\(47\) 2.62222 0.382489 0.191245 0.981542i \(-0.438748\pi\)
0.191245 + 0.981542i \(0.438748\pi\)
\(48\) 0 0
\(49\) 12.6128 1.80184
\(50\) 0 0
\(51\) 0 0
\(52\) −5.80642 −0.805206
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.42864 0.591802
\(57\) 0 0
\(58\) −3.37778 −0.443525
\(59\) 1.05086 0.136810 0.0684048 0.997658i \(-0.478209\pi\)
0.0684048 + 0.997658i \(0.478209\pi\)
\(60\) 0 0
\(61\) 4.75557 0.608888 0.304444 0.952530i \(-0.401529\pi\)
0.304444 + 0.952530i \(0.401529\pi\)
\(62\) −4.42864 −0.562438
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −15.6128 −1.90741 −0.953706 0.300739i \(-0.902767\pi\)
−0.953706 + 0.300739i \(0.902767\pi\)
\(68\) −3.80642 −0.461597
\(69\) 0 0
\(70\) 0 0
\(71\) −15.6128 −1.85290 −0.926452 0.376413i \(-0.877157\pi\)
−0.926452 + 0.376413i \(0.877157\pi\)
\(72\) 0 0
\(73\) −11.6128 −1.35918 −0.679591 0.733592i \(-0.737843\pi\)
−0.679591 + 0.733592i \(0.737843\pi\)
\(74\) −5.80642 −0.674983
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −11.6128 −1.32341
\(78\) 0 0
\(79\) 4.42864 0.498261 0.249130 0.968470i \(-0.419855\pi\)
0.249130 + 0.968470i \(0.419855\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −5.67307 −0.626486
\(83\) 11.9081 1.30709 0.653544 0.756889i \(-0.273282\pi\)
0.653544 + 0.756889i \(0.273282\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.9906 1.18515
\(87\) 0 0
\(88\) −2.62222 −0.279529
\(89\) −12.4286 −1.31743 −0.658717 0.752391i \(-0.728900\pi\)
−0.658717 + 0.752391i \(0.728900\pi\)
\(90\) 0 0
\(91\) −25.7146 −2.69562
\(92\) −2.62222 −0.273385
\(93\) 0 0
\(94\) 2.62222 0.270461
\(95\) 0 0
\(96\) 0 0
\(97\) −7.37778 −0.749101 −0.374550 0.927207i \(-0.622203\pi\)
−0.374550 + 0.927207i \(0.622203\pi\)
\(98\) 12.6128 1.27409
\(99\) 0 0
\(100\) 0 0
\(101\) −17.6731 −1.75854 −0.879268 0.476327i \(-0.841968\pi\)
−0.879268 + 0.476327i \(0.841968\pi\)
\(102\) 0 0
\(103\) 1.18421 0.116684 0.0583418 0.998297i \(-0.481419\pi\)
0.0583418 + 0.998297i \(0.481419\pi\)
\(104\) −5.80642 −0.569367
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 4.85728 0.469571 0.234785 0.972047i \(-0.424561\pi\)
0.234785 + 0.972047i \(0.424561\pi\)
\(108\) 0 0
\(109\) −6.04149 −0.578670 −0.289335 0.957228i \(-0.593434\pi\)
−0.289335 + 0.957228i \(0.593434\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.42864 0.418467
\(113\) 9.34614 0.879211 0.439606 0.898191i \(-0.355118\pi\)
0.439606 + 0.898191i \(0.355118\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.37778 −0.313619
\(117\) 0 0
\(118\) 1.05086 0.0967391
\(119\) −16.8573 −1.54530
\(120\) 0 0
\(121\) −4.12399 −0.374908
\(122\) 4.75557 0.430549
\(123\) 0 0
\(124\) −4.42864 −0.397704
\(125\) 0 0
\(126\) 0 0
\(127\) −6.42864 −0.570450 −0.285225 0.958461i \(-0.592068\pi\)
−0.285225 + 0.958461i \(0.592068\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 15.0923 1.31862 0.659312 0.751869i \(-0.270848\pi\)
0.659312 + 0.751869i \(0.270848\pi\)
\(132\) 0 0
\(133\) 4.42864 0.384012
\(134\) −15.6128 −1.34874
\(135\) 0 0
\(136\) −3.80642 −0.326398
\(137\) −7.53972 −0.644162 −0.322081 0.946712i \(-0.604382\pi\)
−0.322081 + 0.946712i \(0.604382\pi\)
\(138\) 0 0
\(139\) 0.387152 0.0328378 0.0164189 0.999865i \(-0.494773\pi\)
0.0164189 + 0.999865i \(0.494773\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −15.6128 −1.31020
\(143\) 15.2257 1.27324
\(144\) 0 0
\(145\) 0 0
\(146\) −11.6128 −0.961086
\(147\) 0 0
\(148\) −5.80642 −0.477285
\(149\) 14.5303 1.19037 0.595186 0.803588i \(-0.297078\pi\)
0.595186 + 0.803588i \(0.297078\pi\)
\(150\) 0 0
\(151\) 4.69535 0.382102 0.191051 0.981580i \(-0.438810\pi\)
0.191051 + 0.981580i \(0.438810\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) −11.6128 −0.935790
\(155\) 0 0
\(156\) 0 0
\(157\) 21.5210 1.71756 0.858781 0.512343i \(-0.171222\pi\)
0.858781 + 0.512343i \(0.171222\pi\)
\(158\) 4.42864 0.352324
\(159\) 0 0
\(160\) 0 0
\(161\) −11.6128 −0.915221
\(162\) 0 0
\(163\) 8.23506 0.645020 0.322510 0.946566i \(-0.395473\pi\)
0.322510 + 0.946566i \(0.395473\pi\)
\(164\) −5.67307 −0.442992
\(165\) 0 0
\(166\) 11.9081 0.924250
\(167\) 8.47013 0.655438 0.327719 0.944775i \(-0.393720\pi\)
0.327719 + 0.944775i \(0.393720\pi\)
\(168\) 0 0
\(169\) 20.7146 1.59343
\(170\) 0 0
\(171\) 0 0
\(172\) 10.9906 0.838028
\(173\) −22.4701 −1.70837 −0.854186 0.519967i \(-0.825944\pi\)
−0.854186 + 0.519967i \(0.825944\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.62222 −0.197657
\(177\) 0 0
\(178\) −12.4286 −0.931566
\(179\) 2.94914 0.220429 0.110215 0.993908i \(-0.464846\pi\)
0.110215 + 0.993908i \(0.464846\pi\)
\(180\) 0 0
\(181\) 22.8988 1.70205 0.851026 0.525124i \(-0.175981\pi\)
0.851026 + 0.525124i \(0.175981\pi\)
\(182\) −25.7146 −1.90609
\(183\) 0 0
\(184\) −2.62222 −0.193312
\(185\) 0 0
\(186\) 0 0
\(187\) 9.98126 0.729902
\(188\) 2.62222 0.191245
\(189\) 0 0
\(190\) 0 0
\(191\) 9.05086 0.654897 0.327448 0.944869i \(-0.393811\pi\)
0.327448 + 0.944869i \(0.393811\pi\)
\(192\) 0 0
\(193\) −18.7239 −1.34778 −0.673889 0.738833i \(-0.735377\pi\)
−0.673889 + 0.738833i \(0.735377\pi\)
\(194\) −7.37778 −0.529694
\(195\) 0 0
\(196\) 12.6128 0.900918
\(197\) 0.888922 0.0633331 0.0316665 0.999498i \(-0.489919\pi\)
0.0316665 + 0.999498i \(0.489919\pi\)
\(198\) 0 0
\(199\) 21.7146 1.53930 0.769652 0.638464i \(-0.220430\pi\)
0.769652 + 0.638464i \(0.220430\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −17.6731 −1.24347
\(203\) −14.9590 −1.04992
\(204\) 0 0
\(205\) 0 0
\(206\) 1.18421 0.0825077
\(207\) 0 0
\(208\) −5.80642 −0.402603
\(209\) −2.62222 −0.181382
\(210\) 0 0
\(211\) −3.61285 −0.248719 −0.124359 0.992237i \(-0.539688\pi\)
−0.124359 + 0.992237i \(0.539688\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 4.85728 0.332037
\(215\) 0 0
\(216\) 0 0
\(217\) −19.6128 −1.33141
\(218\) −6.04149 −0.409181
\(219\) 0 0
\(220\) 0 0
\(221\) 22.1017 1.48672
\(222\) 0 0
\(223\) 21.3876 1.43222 0.716111 0.697987i \(-0.245921\pi\)
0.716111 + 0.697987i \(0.245921\pi\)
\(224\) 4.42864 0.295901
\(225\) 0 0
\(226\) 9.34614 0.621696
\(227\) 8.47013 0.562182 0.281091 0.959681i \(-0.409304\pi\)
0.281091 + 0.959681i \(0.409304\pi\)
\(228\) 0 0
\(229\) −16.9590 −1.12068 −0.560341 0.828262i \(-0.689330\pi\)
−0.560341 + 0.828262i \(0.689330\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.37778 −0.221762
\(233\) −17.9081 −1.17320 −0.586600 0.809876i \(-0.699534\pi\)
−0.586600 + 0.809876i \(0.699534\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.05086 0.0684048
\(237\) 0 0
\(238\) −16.8573 −1.09270
\(239\) 28.2766 1.82906 0.914529 0.404520i \(-0.132562\pi\)
0.914529 + 0.404520i \(0.132562\pi\)
\(240\) 0 0
\(241\) −11.5111 −0.741498 −0.370749 0.928733i \(-0.620899\pi\)
−0.370749 + 0.928733i \(0.620899\pi\)
\(242\) −4.12399 −0.265100
\(243\) 0 0
\(244\) 4.75557 0.304444
\(245\) 0 0
\(246\) 0 0
\(247\) −5.80642 −0.369454
\(248\) −4.42864 −0.281219
\(249\) 0 0
\(250\) 0 0
\(251\) 7.74620 0.488936 0.244468 0.969657i \(-0.421387\pi\)
0.244468 + 0.969657i \(0.421387\pi\)
\(252\) 0 0
\(253\) 6.87601 0.432291
\(254\) −6.42864 −0.403369
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −13.8796 −0.865783 −0.432891 0.901446i \(-0.642507\pi\)
−0.432891 + 0.901446i \(0.642507\pi\)
\(258\) 0 0
\(259\) −25.7146 −1.59782
\(260\) 0 0
\(261\) 0 0
\(262\) 15.0923 0.932408
\(263\) −18.6222 −1.14830 −0.574148 0.818752i \(-0.694666\pi\)
−0.574148 + 0.818752i \(0.694666\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.42864 0.271537
\(267\) 0 0
\(268\) −15.6128 −0.953706
\(269\) −15.8479 −0.966264 −0.483132 0.875547i \(-0.660501\pi\)
−0.483132 + 0.875547i \(0.660501\pi\)
\(270\) 0 0
\(271\) 1.51114 0.0917951 0.0458975 0.998946i \(-0.485385\pi\)
0.0458975 + 0.998946i \(0.485385\pi\)
\(272\) −3.80642 −0.230798
\(273\) 0 0
\(274\) −7.53972 −0.455491
\(275\) 0 0
\(276\) 0 0
\(277\) −1.05086 −0.0631398 −0.0315699 0.999502i \(-0.510051\pi\)
−0.0315699 + 0.999502i \(0.510051\pi\)
\(278\) 0.387152 0.0232199
\(279\) 0 0
\(280\) 0 0
\(281\) −3.45091 −0.205864 −0.102932 0.994688i \(-0.532822\pi\)
−0.102932 + 0.994688i \(0.532822\pi\)
\(282\) 0 0
\(283\) −15.5812 −0.926206 −0.463103 0.886304i \(-0.653264\pi\)
−0.463103 + 0.886304i \(0.653264\pi\)
\(284\) −15.6128 −0.926452
\(285\) 0 0
\(286\) 15.2257 0.900314
\(287\) −25.1240 −1.48302
\(288\) 0 0
\(289\) −2.51114 −0.147714
\(290\) 0 0
\(291\) 0 0
\(292\) −11.6128 −0.679591
\(293\) 7.12399 0.416188 0.208094 0.978109i \(-0.433274\pi\)
0.208094 + 0.978109i \(0.433274\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −5.80642 −0.337492
\(297\) 0 0
\(298\) 14.5303 0.841721
\(299\) 15.2257 0.880525
\(300\) 0 0
\(301\) 48.6735 2.80550
\(302\) 4.69535 0.270187
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −1.12399 −0.0641492 −0.0320746 0.999485i \(-0.510211\pi\)
−0.0320746 + 0.999485i \(0.510211\pi\)
\(308\) −11.6128 −0.661703
\(309\) 0 0
\(310\) 0 0
\(311\) −14.9491 −0.847688 −0.423844 0.905735i \(-0.639320\pi\)
−0.423844 + 0.905735i \(0.639320\pi\)
\(312\) 0 0
\(313\) −3.14272 −0.177637 −0.0888185 0.996048i \(-0.528309\pi\)
−0.0888185 + 0.996048i \(0.528309\pi\)
\(314\) 21.5210 1.21450
\(315\) 0 0
\(316\) 4.42864 0.249130
\(317\) 10.5906 0.594826 0.297413 0.954749i \(-0.403876\pi\)
0.297413 + 0.954749i \(0.403876\pi\)
\(318\) 0 0
\(319\) 8.85728 0.495912
\(320\) 0 0
\(321\) 0 0
\(322\) −11.6128 −0.647159
\(323\) −3.80642 −0.211795
\(324\) 0 0
\(325\) 0 0
\(326\) 8.23506 0.456098
\(327\) 0 0
\(328\) −5.67307 −0.313243
\(329\) 11.6128 0.640237
\(330\) 0 0
\(331\) −15.1427 −0.832319 −0.416160 0.909292i \(-0.636624\pi\)
−0.416160 + 0.909292i \(0.636624\pi\)
\(332\) 11.9081 0.653544
\(333\) 0 0
\(334\) 8.47013 0.463465
\(335\) 0 0
\(336\) 0 0
\(337\) −11.1111 −0.605259 −0.302629 0.953108i \(-0.597864\pi\)
−0.302629 + 0.953108i \(0.597864\pi\)
\(338\) 20.7146 1.12672
\(339\) 0 0
\(340\) 0 0
\(341\) 11.6128 0.628871
\(342\) 0 0
\(343\) 24.8573 1.34217
\(344\) 10.9906 0.592575
\(345\) 0 0
\(346\) −22.4701 −1.20800
\(347\) −14.1936 −0.761951 −0.380976 0.924585i \(-0.624412\pi\)
−0.380976 + 0.924585i \(0.624412\pi\)
\(348\) 0 0
\(349\) 32.3684 1.73264 0.866321 0.499488i \(-0.166479\pi\)
0.866321 + 0.499488i \(0.166479\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.62222 −0.139765
\(353\) −11.8064 −0.628393 −0.314196 0.949358i \(-0.601735\pi\)
−0.314196 + 0.949358i \(0.601735\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −12.4286 −0.658717
\(357\) 0 0
\(358\) 2.94914 0.155867
\(359\) 10.8287 0.571517 0.285758 0.958302i \(-0.407755\pi\)
0.285758 + 0.958302i \(0.407755\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 22.8988 1.20353
\(363\) 0 0
\(364\) −25.7146 −1.34781
\(365\) 0 0
\(366\) 0 0
\(367\) 1.46965 0.0767151 0.0383576 0.999264i \(-0.487787\pi\)
0.0383576 + 0.999264i \(0.487787\pi\)
\(368\) −2.62222 −0.136692
\(369\) 0 0
\(370\) 0 0
\(371\) −26.5718 −1.37954
\(372\) 0 0
\(373\) −24.3783 −1.26226 −0.631129 0.775678i \(-0.717408\pi\)
−0.631129 + 0.775678i \(0.717408\pi\)
\(374\) 9.98126 0.516119
\(375\) 0 0
\(376\) 2.62222 0.135230
\(377\) 19.6128 1.01011
\(378\) 0 0
\(379\) 18.9590 0.973858 0.486929 0.873442i \(-0.338117\pi\)
0.486929 + 0.873442i \(0.338117\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 9.05086 0.463082
\(383\) −26.1017 −1.33374 −0.666868 0.745176i \(-0.732365\pi\)
−0.666868 + 0.745176i \(0.732365\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −18.7239 −0.953023
\(387\) 0 0
\(388\) −7.37778 −0.374550
\(389\) 3.57136 0.181075 0.0905376 0.995893i \(-0.471141\pi\)
0.0905376 + 0.995893i \(0.471141\pi\)
\(390\) 0 0
\(391\) 9.98126 0.504774
\(392\) 12.6128 0.637045
\(393\) 0 0
\(394\) 0.888922 0.0447832
\(395\) 0 0
\(396\) 0 0
\(397\) −31.4193 −1.57689 −0.788444 0.615107i \(-0.789113\pi\)
−0.788444 + 0.615107i \(0.789113\pi\)
\(398\) 21.7146 1.08845
\(399\) 0 0
\(400\) 0 0
\(401\) −12.0415 −0.601323 −0.300662 0.953731i \(-0.597207\pi\)
−0.300662 + 0.953731i \(0.597207\pi\)
\(402\) 0 0
\(403\) 25.7146 1.28093
\(404\) −17.6731 −0.879268
\(405\) 0 0
\(406\) −14.9590 −0.742402
\(407\) 15.2257 0.754710
\(408\) 0 0
\(409\) 26.4701 1.30886 0.654432 0.756121i \(-0.272908\pi\)
0.654432 + 0.756121i \(0.272908\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.18421 0.0583418
\(413\) 4.65386 0.229001
\(414\) 0 0
\(415\) 0 0
\(416\) −5.80642 −0.284683
\(417\) 0 0
\(418\) −2.62222 −0.128257
\(419\) 25.9684 1.26864 0.634319 0.773072i \(-0.281281\pi\)
0.634319 + 0.773072i \(0.281281\pi\)
\(420\) 0 0
\(421\) −29.2672 −1.42640 −0.713198 0.700963i \(-0.752754\pi\)
−0.713198 + 0.700963i \(0.752754\pi\)
\(422\) −3.61285 −0.175871
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 21.0607 1.01920
\(428\) 4.85728 0.234785
\(429\) 0 0
\(430\) 0 0
\(431\) −14.8385 −0.714747 −0.357374 0.933961i \(-0.616328\pi\)
−0.357374 + 0.933961i \(0.616328\pi\)
\(432\) 0 0
\(433\) 5.27607 0.253552 0.126776 0.991931i \(-0.459537\pi\)
0.126776 + 0.991931i \(0.459537\pi\)
\(434\) −19.6128 −0.941447
\(435\) 0 0
\(436\) −6.04149 −0.289335
\(437\) −2.62222 −0.125438
\(438\) 0 0
\(439\) 31.8578 1.52049 0.760244 0.649638i \(-0.225079\pi\)
0.760244 + 0.649638i \(0.225079\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 22.1017 1.05127
\(443\) −4.94914 −0.235141 −0.117570 0.993065i \(-0.537511\pi\)
−0.117570 + 0.993065i \(0.537511\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 21.3876 1.01273
\(447\) 0 0
\(448\) 4.42864 0.209234
\(449\) −9.75605 −0.460416 −0.230208 0.973141i \(-0.573941\pi\)
−0.230208 + 0.973141i \(0.573941\pi\)
\(450\) 0 0
\(451\) 14.8760 0.700484
\(452\) 9.34614 0.439606
\(453\) 0 0
\(454\) 8.47013 0.397523
\(455\) 0 0
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −16.9590 −0.792442
\(459\) 0 0
\(460\) 0 0
\(461\) 36.1245 1.68248 0.841242 0.540659i \(-0.181825\pi\)
0.841242 + 0.540659i \(0.181825\pi\)
\(462\) 0 0
\(463\) −35.1209 −1.63221 −0.816104 0.577905i \(-0.803870\pi\)
−0.816104 + 0.577905i \(0.803870\pi\)
\(464\) −3.37778 −0.156810
\(465\) 0 0
\(466\) −17.9081 −0.829578
\(467\) −4.56199 −0.211104 −0.105552 0.994414i \(-0.533661\pi\)
−0.105552 + 0.994414i \(0.533661\pi\)
\(468\) 0 0
\(469\) −69.1437 −3.19276
\(470\) 0 0
\(471\) 0 0
\(472\) 1.05086 0.0483695
\(473\) −28.8198 −1.32514
\(474\) 0 0
\(475\) 0 0
\(476\) −16.8573 −0.772652
\(477\) 0 0
\(478\) 28.2766 1.29334
\(479\) −37.4005 −1.70887 −0.854437 0.519555i \(-0.826098\pi\)
−0.854437 + 0.519555i \(0.826098\pi\)
\(480\) 0 0
\(481\) 33.7146 1.53725
\(482\) −11.5111 −0.524318
\(483\) 0 0
\(484\) −4.12399 −0.187454
\(485\) 0 0
\(486\) 0 0
\(487\) 20.9175 0.947862 0.473931 0.880562i \(-0.342835\pi\)
0.473931 + 0.880562i \(0.342835\pi\)
\(488\) 4.75557 0.215274
\(489\) 0 0
\(490\) 0 0
\(491\) 15.8666 0.716052 0.358026 0.933712i \(-0.383450\pi\)
0.358026 + 0.933712i \(0.383450\pi\)
\(492\) 0 0
\(493\) 12.8573 0.579063
\(494\) −5.80642 −0.261243
\(495\) 0 0
\(496\) −4.42864 −0.198852
\(497\) −69.1437 −3.10152
\(498\) 0 0
\(499\) 16.7368 0.749244 0.374622 0.927178i \(-0.377773\pi\)
0.374622 + 0.927178i \(0.377773\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 7.74620 0.345730
\(503\) 21.9684 0.979521 0.489760 0.871857i \(-0.337084\pi\)
0.489760 + 0.871857i \(0.337084\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6.87601 0.305676
\(507\) 0 0
\(508\) −6.42864 −0.285225
\(509\) 15.2573 0.676270 0.338135 0.941098i \(-0.390204\pi\)
0.338135 + 0.941098i \(0.390204\pi\)
\(510\) 0 0
\(511\) −51.4291 −2.27509
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −13.8796 −0.612201
\(515\) 0 0
\(516\) 0 0
\(517\) −6.87601 −0.302407
\(518\) −25.7146 −1.12983
\(519\) 0 0
\(520\) 0 0
\(521\) 18.3269 0.802917 0.401459 0.915877i \(-0.368503\pi\)
0.401459 + 0.915877i \(0.368503\pi\)
\(522\) 0 0
\(523\) −24.8573 −1.08693 −0.543466 0.839431i \(-0.682888\pi\)
−0.543466 + 0.839431i \(0.682888\pi\)
\(524\) 15.0923 0.659312
\(525\) 0 0
\(526\) −18.6222 −0.811967
\(527\) 16.8573 0.734315
\(528\) 0 0
\(529\) −16.1240 −0.701043
\(530\) 0 0
\(531\) 0 0
\(532\) 4.42864 0.192006
\(533\) 32.9403 1.42680
\(534\) 0 0
\(535\) 0 0
\(536\) −15.6128 −0.674372
\(537\) 0 0
\(538\) −15.8479 −0.683252
\(539\) −33.0736 −1.42458
\(540\) 0 0
\(541\) −1.34614 −0.0578751 −0.0289376 0.999581i \(-0.509212\pi\)
−0.0289376 + 0.999581i \(0.509212\pi\)
\(542\) 1.51114 0.0649089
\(543\) 0 0
\(544\) −3.80642 −0.163199
\(545\) 0 0
\(546\) 0 0
\(547\) 8.59057 0.367306 0.183653 0.982991i \(-0.441208\pi\)
0.183653 + 0.982991i \(0.441208\pi\)
\(548\) −7.53972 −0.322081
\(549\) 0 0
\(550\) 0 0
\(551\) −3.37778 −0.143898
\(552\) 0 0
\(553\) 19.6128 0.834023
\(554\) −1.05086 −0.0446466
\(555\) 0 0
\(556\) 0.387152 0.0164189
\(557\) −5.86665 −0.248578 −0.124289 0.992246i \(-0.539665\pi\)
−0.124289 + 0.992246i \(0.539665\pi\)
\(558\) 0 0
\(559\) −63.8163 −2.69914
\(560\) 0 0
\(561\) 0 0
\(562\) −3.45091 −0.145568
\(563\) 31.4291 1.32458 0.662290 0.749248i \(-0.269585\pi\)
0.662290 + 0.749248i \(0.269585\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −15.5812 −0.654927
\(567\) 0 0
\(568\) −15.6128 −0.655101
\(569\) 7.95851 0.333638 0.166819 0.985988i \(-0.446650\pi\)
0.166819 + 0.985988i \(0.446650\pi\)
\(570\) 0 0
\(571\) −30.8385 −1.29055 −0.645276 0.763949i \(-0.723258\pi\)
−0.645276 + 0.763949i \(0.723258\pi\)
\(572\) 15.2257 0.636618
\(573\) 0 0
\(574\) −25.1240 −1.04865
\(575\) 0 0
\(576\) 0 0
\(577\) −32.0000 −1.33218 −0.666089 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(578\) −2.51114 −0.104450
\(579\) 0 0
\(580\) 0 0
\(581\) 52.7368 2.18789
\(582\) 0 0
\(583\) 15.7333 0.651606
\(584\) −11.6128 −0.480543
\(585\) 0 0
\(586\) 7.12399 0.294289
\(587\) 17.8064 0.734950 0.367475 0.930033i \(-0.380222\pi\)
0.367475 + 0.930033i \(0.380222\pi\)
\(588\) 0 0
\(589\) −4.42864 −0.182479
\(590\) 0 0
\(591\) 0 0
\(592\) −5.80642 −0.238643
\(593\) −13.3176 −0.546887 −0.273443 0.961888i \(-0.588163\pi\)
−0.273443 + 0.961888i \(0.588163\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 14.5303 0.595186
\(597\) 0 0
\(598\) 15.2257 0.622625
\(599\) 43.8163 1.79028 0.895142 0.445781i \(-0.147074\pi\)
0.895142 + 0.445781i \(0.147074\pi\)
\(600\) 0 0
\(601\) −9.73329 −0.397029 −0.198515 0.980098i \(-0.563612\pi\)
−0.198515 + 0.980098i \(0.563612\pi\)
\(602\) 48.6735 1.98379
\(603\) 0 0
\(604\) 4.69535 0.191051
\(605\) 0 0
\(606\) 0 0
\(607\) 31.9398 1.29640 0.648198 0.761472i \(-0.275523\pi\)
0.648198 + 0.761472i \(0.275523\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −15.2257 −0.615966
\(612\) 0 0
\(613\) −11.8064 −0.476857 −0.238428 0.971160i \(-0.576632\pi\)
−0.238428 + 0.971160i \(0.576632\pi\)
\(614\) −1.12399 −0.0453603
\(615\) 0 0
\(616\) −11.6128 −0.467895
\(617\) 26.5620 1.06935 0.534673 0.845059i \(-0.320435\pi\)
0.534673 + 0.845059i \(0.320435\pi\)
\(618\) 0 0
\(619\) 7.34614 0.295266 0.147633 0.989042i \(-0.452834\pi\)
0.147633 + 0.989042i \(0.452834\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −14.9491 −0.599406
\(623\) −55.0420 −2.20521
\(624\) 0 0
\(625\) 0 0
\(626\) −3.14272 −0.125608
\(627\) 0 0
\(628\) 21.5210 0.858781
\(629\) 22.1017 0.881253
\(630\) 0 0
\(631\) −30.7556 −1.22436 −0.612180 0.790718i \(-0.709707\pi\)
−0.612180 + 0.790718i \(0.709707\pi\)
\(632\) 4.42864 0.176162
\(633\) 0 0
\(634\) 10.5906 0.420605
\(635\) 0 0
\(636\) 0 0
\(637\) −73.2355 −2.90170
\(638\) 8.85728 0.350663
\(639\) 0 0
\(640\) 0 0
\(641\) 39.3876 1.55572 0.777859 0.628439i \(-0.216306\pi\)
0.777859 + 0.628439i \(0.216306\pi\)
\(642\) 0 0
\(643\) −24.1146 −0.950988 −0.475494 0.879719i \(-0.657731\pi\)
−0.475494 + 0.879719i \(0.657731\pi\)
\(644\) −11.6128 −0.457610
\(645\) 0 0
\(646\) −3.80642 −0.149762
\(647\) −47.6829 −1.87461 −0.937304 0.348512i \(-0.886687\pi\)
−0.937304 + 0.348512i \(0.886687\pi\)
\(648\) 0 0
\(649\) −2.75557 −0.108166
\(650\) 0 0
\(651\) 0 0
\(652\) 8.23506 0.322510
\(653\) 21.7462 0.850995 0.425497 0.904960i \(-0.360099\pi\)
0.425497 + 0.904960i \(0.360099\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −5.67307 −0.221496
\(657\) 0 0
\(658\) 11.6128 0.452716
\(659\) 16.1936 0.630812 0.315406 0.948957i \(-0.397859\pi\)
0.315406 + 0.948957i \(0.397859\pi\)
\(660\) 0 0
\(661\) −22.5116 −0.875600 −0.437800 0.899072i \(-0.644242\pi\)
−0.437800 + 0.899072i \(0.644242\pi\)
\(662\) −15.1427 −0.588539
\(663\) 0 0
\(664\) 11.9081 0.462125
\(665\) 0 0
\(666\) 0 0
\(667\) 8.85728 0.342955
\(668\) 8.47013 0.327719
\(669\) 0 0
\(670\) 0 0
\(671\) −12.4701 −0.481404
\(672\) 0 0
\(673\) −9.66323 −0.372490 −0.186245 0.982503i \(-0.559632\pi\)
−0.186245 + 0.982503i \(0.559632\pi\)
\(674\) −11.1111 −0.427983
\(675\) 0 0
\(676\) 20.7146 0.796714
\(677\) −6.85728 −0.263547 −0.131773 0.991280i \(-0.542067\pi\)
−0.131773 + 0.991280i \(0.542067\pi\)
\(678\) 0 0
\(679\) −32.6735 −1.25390
\(680\) 0 0
\(681\) 0 0
\(682\) 11.6128 0.444679
\(683\) 30.3051 1.15959 0.579797 0.814761i \(-0.303132\pi\)
0.579797 + 0.814761i \(0.303132\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 24.8573 0.949055
\(687\) 0 0
\(688\) 10.9906 0.419014
\(689\) 34.8385 1.32724
\(690\) 0 0
\(691\) 7.61285 0.289606 0.144803 0.989460i \(-0.453745\pi\)
0.144803 + 0.989460i \(0.453745\pi\)
\(692\) −22.4701 −0.854186
\(693\) 0 0
\(694\) −14.1936 −0.538781
\(695\) 0 0
\(696\) 0 0
\(697\) 21.5941 0.817935
\(698\) 32.3684 1.22516
\(699\) 0 0
\(700\) 0 0
\(701\) 32.3654 1.22242 0.611211 0.791467i \(-0.290683\pi\)
0.611211 + 0.791467i \(0.290683\pi\)
\(702\) 0 0
\(703\) −5.80642 −0.218993
\(704\) −2.62222 −0.0988285
\(705\) 0 0
\(706\) −11.8064 −0.444341
\(707\) −78.2677 −2.94356
\(708\) 0 0
\(709\) −49.8992 −1.87401 −0.937003 0.349322i \(-0.886412\pi\)
−0.937003 + 0.349322i \(0.886412\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −12.4286 −0.465783
\(713\) 11.6128 0.434905
\(714\) 0 0
\(715\) 0 0
\(716\) 2.94914 0.110215
\(717\) 0 0
\(718\) 10.8287 0.404123
\(719\) 26.4415 0.986103 0.493052 0.870000i \(-0.335881\pi\)
0.493052 + 0.870000i \(0.335881\pi\)
\(720\) 0 0
\(721\) 5.24443 0.195313
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 22.8988 0.851026
\(725\) 0 0
\(726\) 0 0
\(727\) −48.1245 −1.78484 −0.892419 0.451208i \(-0.850993\pi\)
−0.892419 + 0.451208i \(0.850993\pi\)
\(728\) −25.7146 −0.953045
\(729\) 0 0
\(730\) 0 0
\(731\) −41.8350 −1.54732
\(732\) 0 0
\(733\) 14.0286 0.518157 0.259079 0.965856i \(-0.416581\pi\)
0.259079 + 0.965856i \(0.416581\pi\)
\(734\) 1.46965 0.0542458
\(735\) 0 0
\(736\) −2.62222 −0.0966562
\(737\) 40.9403 1.50805
\(738\) 0 0
\(739\) 10.1847 0.374650 0.187325 0.982298i \(-0.440018\pi\)
0.187325 + 0.982298i \(0.440018\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −26.5718 −0.975483
\(743\) −42.9590 −1.57601 −0.788006 0.615667i \(-0.788887\pi\)
−0.788006 + 0.615667i \(0.788887\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −24.3783 −0.892552
\(747\) 0 0
\(748\) 9.98126 0.364951
\(749\) 21.5111 0.786000
\(750\) 0 0
\(751\) −7.18421 −0.262155 −0.131078 0.991372i \(-0.541844\pi\)
−0.131078 + 0.991372i \(0.541844\pi\)
\(752\) 2.62222 0.0956224
\(753\) 0 0
\(754\) 19.6128 0.714258
\(755\) 0 0
\(756\) 0 0
\(757\) −41.2543 −1.49941 −0.749706 0.661771i \(-0.769805\pi\)
−0.749706 + 0.661771i \(0.769805\pi\)
\(758\) 18.9590 0.688621
\(759\) 0 0
\(760\) 0 0
\(761\) −44.3051 −1.60606 −0.803030 0.595939i \(-0.796780\pi\)
−0.803030 + 0.595939i \(0.796780\pi\)
\(762\) 0 0
\(763\) −26.7556 −0.968617
\(764\) 9.05086 0.327448
\(765\) 0 0
\(766\) −26.1017 −0.943093
\(767\) −6.10171 −0.220320
\(768\) 0 0
\(769\) −6.59057 −0.237662 −0.118831 0.992914i \(-0.537915\pi\)
−0.118831 + 0.992914i \(0.537915\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −18.7239 −0.673889
\(773\) −8.83854 −0.317900 −0.158950 0.987287i \(-0.550811\pi\)
−0.158950 + 0.987287i \(0.550811\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −7.37778 −0.264847
\(777\) 0 0
\(778\) 3.57136 0.128039
\(779\) −5.67307 −0.203259
\(780\) 0 0
\(781\) 40.9403 1.46496
\(782\) 9.98126 0.356929
\(783\) 0 0
\(784\) 12.6128 0.450459
\(785\) 0 0
\(786\) 0 0
\(787\) 16.9403 0.603855 0.301927 0.953331i \(-0.402370\pi\)
0.301927 + 0.953331i \(0.402370\pi\)
\(788\) 0.888922 0.0316665
\(789\) 0 0
\(790\) 0 0
\(791\) 41.3907 1.47168
\(792\) 0 0
\(793\) −27.6128 −0.980561
\(794\) −31.4193 −1.11503
\(795\) 0 0
\(796\) 21.7146 0.769652
\(797\) −40.1847 −1.42341 −0.711707 0.702476i \(-0.752078\pi\)
−0.711707 + 0.702476i \(0.752078\pi\)
\(798\) 0 0
\(799\) −9.98126 −0.353112
\(800\) 0 0
\(801\) 0 0
\(802\) −12.0415 −0.425200
\(803\) 30.4514 1.07461
\(804\) 0 0
\(805\) 0 0
\(806\) 25.7146 0.905757
\(807\) 0 0
\(808\) −17.6731 −0.621736
\(809\) −29.1052 −1.02329 −0.511643 0.859198i \(-0.670963\pi\)
−0.511643 + 0.859198i \(0.670963\pi\)
\(810\) 0 0
\(811\) 38.7753 1.36158 0.680792 0.732477i \(-0.261636\pi\)
0.680792 + 0.732477i \(0.261636\pi\)
\(812\) −14.9590 −0.524958
\(813\) 0 0
\(814\) 15.2257 0.533660
\(815\) 0 0
\(816\) 0 0
\(817\) 10.9906 0.384514
\(818\) 26.4701 0.925506
\(819\) 0 0
\(820\) 0 0
\(821\) −2.53035 −0.0883098 −0.0441549 0.999025i \(-0.514060\pi\)
−0.0441549 + 0.999025i \(0.514060\pi\)
\(822\) 0 0
\(823\) 24.0415 0.838034 0.419017 0.907978i \(-0.362375\pi\)
0.419017 + 0.907978i \(0.362375\pi\)
\(824\) 1.18421 0.0412538
\(825\) 0 0
\(826\) 4.65386 0.161928
\(827\) −6.57184 −0.228525 −0.114263 0.993451i \(-0.536451\pi\)
−0.114263 + 0.993451i \(0.536451\pi\)
\(828\) 0 0
\(829\) −28.7338 −0.997965 −0.498983 0.866612i \(-0.666293\pi\)
−0.498983 + 0.866612i \(0.666293\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −5.80642 −0.201302
\(833\) −48.0098 −1.66344
\(834\) 0 0
\(835\) 0 0
\(836\) −2.62222 −0.0906912
\(837\) 0 0
\(838\) 25.9684 0.897062
\(839\) 50.5718 1.74593 0.872967 0.487780i \(-0.162193\pi\)
0.872967 + 0.487780i \(0.162193\pi\)
\(840\) 0 0
\(841\) −17.5906 −0.606571
\(842\) −29.2672 −1.00861
\(843\) 0 0
\(844\) −3.61285 −0.124359
\(845\) 0 0
\(846\) 0 0
\(847\) −18.2636 −0.627546
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 0 0
\(851\) 15.2257 0.521930
\(852\) 0 0
\(853\) 31.2355 1.06948 0.534742 0.845015i \(-0.320409\pi\)
0.534742 + 0.845015i \(0.320409\pi\)
\(854\) 21.0607 0.720682
\(855\) 0 0
\(856\) 4.85728 0.166018
\(857\) 27.3274 0.933486 0.466743 0.884393i \(-0.345427\pi\)
0.466743 + 0.884393i \(0.345427\pi\)
\(858\) 0 0
\(859\) −34.7753 −1.18652 −0.593258 0.805012i \(-0.702159\pi\)
−0.593258 + 0.805012i \(0.702159\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −14.8385 −0.505403
\(863\) 21.2444 0.723169 0.361584 0.932339i \(-0.382236\pi\)
0.361584 + 0.932339i \(0.382236\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 5.27607 0.179288
\(867\) 0 0
\(868\) −19.6128 −0.665703
\(869\) −11.6128 −0.393939
\(870\) 0 0
\(871\) 90.6548 3.07172
\(872\) −6.04149 −0.204591
\(873\) 0 0
\(874\) −2.62222 −0.0886978
\(875\) 0 0
\(876\) 0 0
\(877\) −22.3970 −0.756293 −0.378146 0.925746i \(-0.623438\pi\)
−0.378146 + 0.925746i \(0.623438\pi\)
\(878\) 31.8578 1.07515
\(879\) 0 0
\(880\) 0 0
\(881\) 20.3684 0.686229 0.343115 0.939294i \(-0.388518\pi\)
0.343115 + 0.939294i \(0.388518\pi\)
\(882\) 0 0
\(883\) −21.6829 −0.729688 −0.364844 0.931069i \(-0.618878\pi\)
−0.364844 + 0.931069i \(0.618878\pi\)
\(884\) 22.1017 0.743361
\(885\) 0 0
\(886\) −4.94914 −0.166270
\(887\) −7.14272 −0.239829 −0.119915 0.992784i \(-0.538262\pi\)
−0.119915 + 0.992784i \(0.538262\pi\)
\(888\) 0 0
\(889\) −28.4701 −0.954857
\(890\) 0 0
\(891\) 0 0
\(892\) 21.3876 0.716111
\(893\) 2.62222 0.0877491
\(894\) 0 0
\(895\) 0 0
\(896\) 4.42864 0.147950
\(897\) 0 0
\(898\) −9.75605 −0.325563
\(899\) 14.9590 0.498910
\(900\) 0 0
\(901\) 22.8385 0.760862
\(902\) 14.8760 0.495317
\(903\) 0 0
\(904\) 9.34614 0.310848
\(905\) 0 0
\(906\) 0 0
\(907\) −15.3461 −0.509560 −0.254780 0.966999i \(-0.582003\pi\)
−0.254780 + 0.966999i \(0.582003\pi\)
\(908\) 8.47013 0.281091
\(909\) 0 0
\(910\) 0 0
\(911\) −12.1204 −0.401568 −0.200784 0.979636i \(-0.564349\pi\)
−0.200784 + 0.979636i \(0.564349\pi\)
\(912\) 0 0
\(913\) −31.2257 −1.03342
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) −16.9590 −0.560341
\(917\) 66.8385 2.20720
\(918\) 0 0
\(919\) −12.2667 −0.404641 −0.202321 0.979319i \(-0.564848\pi\)
−0.202321 + 0.979319i \(0.564848\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 36.1245 1.18970
\(923\) 90.6548 2.98394
\(924\) 0 0
\(925\) 0 0
\(926\) −35.1209 −1.15415
\(927\) 0 0
\(928\) −3.37778 −0.110881
\(929\) 40.1847 1.31842 0.659208 0.751960i \(-0.270892\pi\)
0.659208 + 0.751960i \(0.270892\pi\)
\(930\) 0 0
\(931\) 12.6128 0.413369
\(932\) −17.9081 −0.586600
\(933\) 0 0
\(934\) −4.56199 −0.149273
\(935\) 0 0
\(936\) 0 0
\(937\) 24.7368 0.808117 0.404059 0.914733i \(-0.367599\pi\)
0.404059 + 0.914733i \(0.367599\pi\)
\(938\) −69.1437 −2.25762
\(939\) 0 0
\(940\) 0 0
\(941\) 1.66323 0.0542196 0.0271098 0.999632i \(-0.491370\pi\)
0.0271098 + 0.999632i \(0.491370\pi\)
\(942\) 0 0
\(943\) 14.8760 0.484430
\(944\) 1.05086 0.0342024
\(945\) 0 0
\(946\) −28.8198 −0.937013
\(947\) 14.8671 0.483117 0.241558 0.970386i \(-0.422341\pi\)
0.241558 + 0.970386i \(0.422341\pi\)
\(948\) 0 0
\(949\) 67.4291 2.18884
\(950\) 0 0
\(951\) 0 0
\(952\) −16.8573 −0.546348
\(953\) −8.83854 −0.286308 −0.143154 0.989700i \(-0.545725\pi\)
−0.143154 + 0.989700i \(0.545725\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 28.2766 0.914529
\(957\) 0 0
\(958\) −37.4005 −1.20836
\(959\) −33.3907 −1.07824
\(960\) 0 0
\(961\) −11.3872 −0.367327
\(962\) 33.7146 1.08700
\(963\) 0 0
\(964\) −11.5111 −0.370749
\(965\) 0 0
\(966\) 0 0
\(967\) −1.55262 −0.0499290 −0.0249645 0.999688i \(-0.507947\pi\)
−0.0249645 + 0.999688i \(0.507947\pi\)
\(968\) −4.12399 −0.132550
\(969\) 0 0
\(970\) 0 0
\(971\) −44.5433 −1.42946 −0.714731 0.699400i \(-0.753451\pi\)
−0.714731 + 0.699400i \(0.753451\pi\)
\(972\) 0 0
\(973\) 1.71456 0.0549662
\(974\) 20.9175 0.670240
\(975\) 0 0
\(976\) 4.75557 0.152222
\(977\) 15.8350 0.506607 0.253303 0.967387i \(-0.418483\pi\)
0.253303 + 0.967387i \(0.418483\pi\)
\(978\) 0 0
\(979\) 32.5906 1.04160
\(980\) 0 0
\(981\) 0 0
\(982\) 15.8666 0.506325
\(983\) 6.87601 0.219311 0.109655 0.993970i \(-0.465025\pi\)
0.109655 + 0.993970i \(0.465025\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 12.8573 0.409459
\(987\) 0 0
\(988\) −5.80642 −0.184727
\(989\) −28.8198 −0.916417
\(990\) 0 0
\(991\) −29.8765 −0.949058 −0.474529 0.880240i \(-0.657382\pi\)
−0.474529 + 0.880240i \(0.657382\pi\)
\(992\) −4.42864 −0.140609
\(993\) 0 0
\(994\) −69.1437 −2.19310
\(995\) 0 0
\(996\) 0 0
\(997\) −5.90813 −0.187112 −0.0935562 0.995614i \(-0.529823\pi\)
−0.0935562 + 0.995614i \(0.529823\pi\)
\(998\) 16.7368 0.529795
\(999\) 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.cq.1.3 3
3.2 odd 2 2850.2.a.bl.1.3 3
5.2 odd 4 1710.2.d.f.1369.4 6
5.3 odd 4 1710.2.d.f.1369.1 6
5.4 even 2 8550.2.a.ce.1.1 3
15.2 even 4 570.2.d.c.229.3 6
15.8 even 4 570.2.d.c.229.6 yes 6
15.14 odd 2 2850.2.a.bm.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.d.c.229.3 6 15.2 even 4
570.2.d.c.229.6 yes 6 15.8 even 4
1710.2.d.f.1369.1 6 5.3 odd 4
1710.2.d.f.1369.4 6 5.2 odd 4
2850.2.a.bl.1.3 3 3.2 odd 2
2850.2.a.bm.1.1 3 15.14 odd 2
8550.2.a.ce.1.1 3 5.4 even 2
8550.2.a.cq.1.3 3 1.1 even 1 trivial