Properties

Label 8004.2.a.e.1.1
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 13 x^{7} + 32 x^{6} + 40 x^{5} - 79 x^{4} - 39 x^{3} + 58 x^{2} + 9 x - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.86352\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.70748 q^{5} -1.59469 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.70748 q^{5} -1.59469 q^{7} +1.00000 q^{9} -2.15216 q^{11} -3.62850 q^{13} +3.70748 q^{15} +6.51792 q^{17} -5.86961 q^{19} +1.59469 q^{21} -1.00000 q^{23} +8.74544 q^{25} -1.00000 q^{27} +1.00000 q^{29} +10.0687 q^{31} +2.15216 q^{33} +5.91228 q^{35} +1.99282 q^{37} +3.62850 q^{39} -2.83754 q^{41} -1.53121 q^{43} -3.70748 q^{45} -1.23502 q^{47} -4.45697 q^{49} -6.51792 q^{51} -7.71284 q^{53} +7.97910 q^{55} +5.86961 q^{57} +7.21271 q^{59} +1.77725 q^{61} -1.59469 q^{63} +13.4526 q^{65} +6.75121 q^{67} +1.00000 q^{69} +6.62781 q^{71} +9.42253 q^{73} -8.74544 q^{75} +3.43203 q^{77} +14.8112 q^{79} +1.00000 q^{81} +11.5780 q^{83} -24.1651 q^{85} -1.00000 q^{87} -9.06238 q^{89} +5.78633 q^{91} -10.0687 q^{93} +21.7615 q^{95} +0.456077 q^{97} -2.15216 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} - 2q^{11} - 7q^{13} + 3q^{15} - q^{19} - 7q^{21} - 9q^{23} + 2q^{25} - 9q^{27} + 9q^{29} + 8q^{31} + 2q^{33} - 5q^{35} - 8q^{37} + 7q^{39} - 19q^{41} - 3q^{43} - 3q^{45} - 3q^{47} - 18q^{49} - 17q^{53} + 9q^{55} + q^{57} - 10q^{59} + q^{61} + 7q^{63} - 16q^{65} + 12q^{67} + 9q^{69} - 7q^{71} + 13q^{73} - 2q^{75} - 15q^{77} - 10q^{79} + 9q^{81} + 9q^{83} - 6q^{85} - 9q^{87} - 5q^{89} - 18q^{91} - 8q^{93} + 31q^{95} - 7q^{97} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −3.70748 −1.65804 −0.829019 0.559221i \(-0.811100\pi\)
−0.829019 + 0.559221i \(0.811100\pi\)
\(6\) 0 0
\(7\) −1.59469 −0.602736 −0.301368 0.953508i \(-0.597443\pi\)
−0.301368 + 0.953508i \(0.597443\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.15216 −0.648901 −0.324450 0.945903i \(-0.605179\pi\)
−0.324450 + 0.945903i \(0.605179\pi\)
\(12\) 0 0
\(13\) −3.62850 −1.00637 −0.503183 0.864180i \(-0.667838\pi\)
−0.503183 + 0.864180i \(0.667838\pi\)
\(14\) 0 0
\(15\) 3.70748 0.957268
\(16\) 0 0
\(17\) 6.51792 1.58083 0.790414 0.612573i \(-0.209865\pi\)
0.790414 + 0.612573i \(0.209865\pi\)
\(18\) 0 0
\(19\) −5.86961 −1.34658 −0.673291 0.739378i \(-0.735120\pi\)
−0.673291 + 0.739378i \(0.735120\pi\)
\(20\) 0 0
\(21\) 1.59469 0.347990
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 8.74544 1.74909
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 10.0687 1.80839 0.904197 0.427115i \(-0.140470\pi\)
0.904197 + 0.427115i \(0.140470\pi\)
\(32\) 0 0
\(33\) 2.15216 0.374643
\(34\) 0 0
\(35\) 5.91228 0.999358
\(36\) 0 0
\(37\) 1.99282 0.327618 0.163809 0.986492i \(-0.447622\pi\)
0.163809 + 0.986492i \(0.447622\pi\)
\(38\) 0 0
\(39\) 3.62850 0.581026
\(40\) 0 0
\(41\) −2.83754 −0.443149 −0.221574 0.975143i \(-0.571120\pi\)
−0.221574 + 0.975143i \(0.571120\pi\)
\(42\) 0 0
\(43\) −1.53121 −0.233508 −0.116754 0.993161i \(-0.537249\pi\)
−0.116754 + 0.993161i \(0.537249\pi\)
\(44\) 0 0
\(45\) −3.70748 −0.552679
\(46\) 0 0
\(47\) −1.23502 −0.180147 −0.0900734 0.995935i \(-0.528710\pi\)
−0.0900734 + 0.995935i \(0.528710\pi\)
\(48\) 0 0
\(49\) −4.45697 −0.636710
\(50\) 0 0
\(51\) −6.51792 −0.912691
\(52\) 0 0
\(53\) −7.71284 −1.05944 −0.529720 0.848173i \(-0.677703\pi\)
−0.529720 + 0.848173i \(0.677703\pi\)
\(54\) 0 0
\(55\) 7.97910 1.07590
\(56\) 0 0
\(57\) 5.86961 0.777449
\(58\) 0 0
\(59\) 7.21271 0.939015 0.469507 0.882929i \(-0.344432\pi\)
0.469507 + 0.882929i \(0.344432\pi\)
\(60\) 0 0
\(61\) 1.77725 0.227554 0.113777 0.993506i \(-0.463705\pi\)
0.113777 + 0.993506i \(0.463705\pi\)
\(62\) 0 0
\(63\) −1.59469 −0.200912
\(64\) 0 0
\(65\) 13.4526 1.66859
\(66\) 0 0
\(67\) 6.75121 0.824792 0.412396 0.911005i \(-0.364692\pi\)
0.412396 + 0.911005i \(0.364692\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 6.62781 0.786576 0.393288 0.919415i \(-0.371338\pi\)
0.393288 + 0.919415i \(0.371338\pi\)
\(72\) 0 0
\(73\) 9.42253 1.10282 0.551412 0.834233i \(-0.314089\pi\)
0.551412 + 0.834233i \(0.314089\pi\)
\(74\) 0 0
\(75\) −8.74544 −1.00984
\(76\) 0 0
\(77\) 3.43203 0.391116
\(78\) 0 0
\(79\) 14.8112 1.66639 0.833194 0.552981i \(-0.186510\pi\)
0.833194 + 0.552981i \(0.186510\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.5780 1.27085 0.635426 0.772161i \(-0.280824\pi\)
0.635426 + 0.772161i \(0.280824\pi\)
\(84\) 0 0
\(85\) −24.1651 −2.62107
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −9.06238 −0.960610 −0.480305 0.877101i \(-0.659474\pi\)
−0.480305 + 0.877101i \(0.659474\pi\)
\(90\) 0 0
\(91\) 5.78633 0.606572
\(92\) 0 0
\(93\) −10.0687 −1.04408
\(94\) 0 0
\(95\) 21.7615 2.23268
\(96\) 0 0
\(97\) 0.456077 0.0463076 0.0231538 0.999732i \(-0.492629\pi\)
0.0231538 + 0.999732i \(0.492629\pi\)
\(98\) 0 0
\(99\) −2.15216 −0.216300
\(100\) 0 0
\(101\) 2.55110 0.253844 0.126922 0.991913i \(-0.459490\pi\)
0.126922 + 0.991913i \(0.459490\pi\)
\(102\) 0 0
\(103\) 6.99010 0.688755 0.344377 0.938831i \(-0.388090\pi\)
0.344377 + 0.938831i \(0.388090\pi\)
\(104\) 0 0
\(105\) −5.91228 −0.576980
\(106\) 0 0
\(107\) −7.77034 −0.751188 −0.375594 0.926784i \(-0.622561\pi\)
−0.375594 + 0.926784i \(0.622561\pi\)
\(108\) 0 0
\(109\) 9.83289 0.941820 0.470910 0.882181i \(-0.343926\pi\)
0.470910 + 0.882181i \(0.343926\pi\)
\(110\) 0 0
\(111\) −1.99282 −0.189150
\(112\) 0 0
\(113\) −5.13103 −0.482687 −0.241344 0.970440i \(-0.577588\pi\)
−0.241344 + 0.970440i \(0.577588\pi\)
\(114\) 0 0
\(115\) 3.70748 0.345725
\(116\) 0 0
\(117\) −3.62850 −0.335455
\(118\) 0 0
\(119\) −10.3941 −0.952821
\(120\) 0 0
\(121\) −6.36820 −0.578928
\(122\) 0 0
\(123\) 2.83754 0.255852
\(124\) 0 0
\(125\) −13.8861 −1.24201
\(126\) 0 0
\(127\) 10.5582 0.936889 0.468444 0.883493i \(-0.344815\pi\)
0.468444 + 0.883493i \(0.344815\pi\)
\(128\) 0 0
\(129\) 1.53121 0.134816
\(130\) 0 0
\(131\) 1.14566 0.100097 0.0500483 0.998747i \(-0.484062\pi\)
0.0500483 + 0.998747i \(0.484062\pi\)
\(132\) 0 0
\(133\) 9.36020 0.811632
\(134\) 0 0
\(135\) 3.70748 0.319089
\(136\) 0 0
\(137\) −0.658713 −0.0562777 −0.0281388 0.999604i \(-0.508958\pi\)
−0.0281388 + 0.999604i \(0.508958\pi\)
\(138\) 0 0
\(139\) −13.1747 −1.11746 −0.558730 0.829350i \(-0.688711\pi\)
−0.558730 + 0.829350i \(0.688711\pi\)
\(140\) 0 0
\(141\) 1.23502 0.104008
\(142\) 0 0
\(143\) 7.80912 0.653032
\(144\) 0 0
\(145\) −3.70748 −0.307890
\(146\) 0 0
\(147\) 4.45697 0.367605
\(148\) 0 0
\(149\) 1.22464 0.100327 0.0501633 0.998741i \(-0.484026\pi\)
0.0501633 + 0.998741i \(0.484026\pi\)
\(150\) 0 0
\(151\) −13.8166 −1.12438 −0.562189 0.827009i \(-0.690041\pi\)
−0.562189 + 0.827009i \(0.690041\pi\)
\(152\) 0 0
\(153\) 6.51792 0.526943
\(154\) 0 0
\(155\) −37.3296 −2.99839
\(156\) 0 0
\(157\) −4.33993 −0.346364 −0.173182 0.984890i \(-0.555405\pi\)
−0.173182 + 0.984890i \(0.555405\pi\)
\(158\) 0 0
\(159\) 7.71284 0.611668
\(160\) 0 0
\(161\) 1.59469 0.125679
\(162\) 0 0
\(163\) −19.1345 −1.49873 −0.749363 0.662159i \(-0.769640\pi\)
−0.749363 + 0.662159i \(0.769640\pi\)
\(164\) 0 0
\(165\) −7.97910 −0.621172
\(166\) 0 0
\(167\) −20.1088 −1.55607 −0.778034 0.628223i \(-0.783783\pi\)
−0.778034 + 0.628223i \(0.783783\pi\)
\(168\) 0 0
\(169\) 0.166035 0.0127719
\(170\) 0 0
\(171\) −5.86961 −0.448860
\(172\) 0 0
\(173\) 9.92281 0.754417 0.377209 0.926128i \(-0.376884\pi\)
0.377209 + 0.926128i \(0.376884\pi\)
\(174\) 0 0
\(175\) −13.9462 −1.05424
\(176\) 0 0
\(177\) −7.21271 −0.542141
\(178\) 0 0
\(179\) −0.900750 −0.0673252 −0.0336626 0.999433i \(-0.510717\pi\)
−0.0336626 + 0.999433i \(0.510717\pi\)
\(180\) 0 0
\(181\) −21.3878 −1.58974 −0.794871 0.606778i \(-0.792462\pi\)
−0.794871 + 0.606778i \(0.792462\pi\)
\(182\) 0 0
\(183\) −1.77725 −0.131378
\(184\) 0 0
\(185\) −7.38836 −0.543203
\(186\) 0 0
\(187\) −14.0276 −1.02580
\(188\) 0 0
\(189\) 1.59469 0.115997
\(190\) 0 0
\(191\) 4.06677 0.294261 0.147131 0.989117i \(-0.452996\pi\)
0.147131 + 0.989117i \(0.452996\pi\)
\(192\) 0 0
\(193\) 5.55739 0.400030 0.200015 0.979793i \(-0.435901\pi\)
0.200015 + 0.979793i \(0.435901\pi\)
\(194\) 0 0
\(195\) −13.4526 −0.963362
\(196\) 0 0
\(197\) 12.7694 0.909784 0.454892 0.890547i \(-0.349678\pi\)
0.454892 + 0.890547i \(0.349678\pi\)
\(198\) 0 0
\(199\) 21.2840 1.50878 0.754391 0.656425i \(-0.227932\pi\)
0.754391 + 0.656425i \(0.227932\pi\)
\(200\) 0 0
\(201\) −6.75121 −0.476194
\(202\) 0 0
\(203\) −1.59469 −0.111925
\(204\) 0 0
\(205\) 10.5201 0.734757
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 12.6323 0.873798
\(210\) 0 0
\(211\) −9.34034 −0.643015 −0.321508 0.946907i \(-0.604190\pi\)
−0.321508 + 0.946907i \(0.604190\pi\)
\(212\) 0 0
\(213\) −6.62781 −0.454130
\(214\) 0 0
\(215\) 5.67695 0.387165
\(216\) 0 0
\(217\) −16.0565 −1.08998
\(218\) 0 0
\(219\) −9.42253 −0.636716
\(220\) 0 0
\(221\) −23.6503 −1.59089
\(222\) 0 0
\(223\) 21.2923 1.42584 0.712918 0.701247i \(-0.247373\pi\)
0.712918 + 0.701247i \(0.247373\pi\)
\(224\) 0 0
\(225\) 8.74544 0.583029
\(226\) 0 0
\(227\) −6.71940 −0.445982 −0.222991 0.974820i \(-0.571582\pi\)
−0.222991 + 0.974820i \(0.571582\pi\)
\(228\) 0 0
\(229\) 15.5515 1.02767 0.513835 0.857889i \(-0.328224\pi\)
0.513835 + 0.857889i \(0.328224\pi\)
\(230\) 0 0
\(231\) −3.43203 −0.225811
\(232\) 0 0
\(233\) 3.60530 0.236191 0.118096 0.993002i \(-0.462321\pi\)
0.118096 + 0.993002i \(0.462321\pi\)
\(234\) 0 0
\(235\) 4.57883 0.298690
\(236\) 0 0
\(237\) −14.8112 −0.962089
\(238\) 0 0
\(239\) 9.39572 0.607759 0.303879 0.952710i \(-0.401718\pi\)
0.303879 + 0.952710i \(0.401718\pi\)
\(240\) 0 0
\(241\) 12.5234 0.806704 0.403352 0.915045i \(-0.367845\pi\)
0.403352 + 0.915045i \(0.367845\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 16.5241 1.05569
\(246\) 0 0
\(247\) 21.2979 1.35515
\(248\) 0 0
\(249\) −11.5780 −0.733727
\(250\) 0 0
\(251\) −13.9199 −0.878614 −0.439307 0.898337i \(-0.644776\pi\)
−0.439307 + 0.898337i \(0.644776\pi\)
\(252\) 0 0
\(253\) 2.15216 0.135305
\(254\) 0 0
\(255\) 24.1651 1.51328
\(256\) 0 0
\(257\) −16.4930 −1.02881 −0.514403 0.857548i \(-0.671987\pi\)
−0.514403 + 0.857548i \(0.671987\pi\)
\(258\) 0 0
\(259\) −3.17793 −0.197467
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −30.7133 −1.89387 −0.946933 0.321431i \(-0.895836\pi\)
−0.946933 + 0.321431i \(0.895836\pi\)
\(264\) 0 0
\(265\) 28.5952 1.75659
\(266\) 0 0
\(267\) 9.06238 0.554609
\(268\) 0 0
\(269\) 7.55230 0.460472 0.230236 0.973135i \(-0.426050\pi\)
0.230236 + 0.973135i \(0.426050\pi\)
\(270\) 0 0
\(271\) −2.21164 −0.134348 −0.0671739 0.997741i \(-0.521398\pi\)
−0.0671739 + 0.997741i \(0.521398\pi\)
\(272\) 0 0
\(273\) −5.78633 −0.350205
\(274\) 0 0
\(275\) −18.8216 −1.13498
\(276\) 0 0
\(277\) 7.31297 0.439394 0.219697 0.975568i \(-0.429493\pi\)
0.219697 + 0.975568i \(0.429493\pi\)
\(278\) 0 0
\(279\) 10.0687 0.602798
\(280\) 0 0
\(281\) −32.8318 −1.95858 −0.979290 0.202461i \(-0.935106\pi\)
−0.979290 + 0.202461i \(0.935106\pi\)
\(282\) 0 0
\(283\) 12.8415 0.763347 0.381673 0.924297i \(-0.375348\pi\)
0.381673 + 0.924297i \(0.375348\pi\)
\(284\) 0 0
\(285\) −21.7615 −1.28904
\(286\) 0 0
\(287\) 4.52498 0.267101
\(288\) 0 0
\(289\) 25.4833 1.49902
\(290\) 0 0
\(291\) −0.456077 −0.0267357
\(292\) 0 0
\(293\) −14.4325 −0.843157 −0.421578 0.906792i \(-0.638524\pi\)
−0.421578 + 0.906792i \(0.638524\pi\)
\(294\) 0 0
\(295\) −26.7410 −1.55692
\(296\) 0 0
\(297\) 2.15216 0.124881
\(298\) 0 0
\(299\) 3.62850 0.209842
\(300\) 0 0
\(301\) 2.44181 0.140744
\(302\) 0 0
\(303\) −2.55110 −0.146557
\(304\) 0 0
\(305\) −6.58914 −0.377293
\(306\) 0 0
\(307\) 23.1124 1.31910 0.659548 0.751663i \(-0.270748\pi\)
0.659548 + 0.751663i \(0.270748\pi\)
\(308\) 0 0
\(309\) −6.99010 −0.397653
\(310\) 0 0
\(311\) −28.4686 −1.61430 −0.807152 0.590344i \(-0.798992\pi\)
−0.807152 + 0.590344i \(0.798992\pi\)
\(312\) 0 0
\(313\) −13.2820 −0.750742 −0.375371 0.926875i \(-0.622485\pi\)
−0.375371 + 0.926875i \(0.622485\pi\)
\(314\) 0 0
\(315\) 5.91228 0.333119
\(316\) 0 0
\(317\) −19.8968 −1.11752 −0.558758 0.829331i \(-0.688722\pi\)
−0.558758 + 0.829331i \(0.688722\pi\)
\(318\) 0 0
\(319\) −2.15216 −0.120498
\(320\) 0 0
\(321\) 7.77034 0.433698
\(322\) 0 0
\(323\) −38.2577 −2.12871
\(324\) 0 0
\(325\) −31.7328 −1.76022
\(326\) 0 0
\(327\) −9.83289 −0.543760
\(328\) 0 0
\(329\) 1.96948 0.108581
\(330\) 0 0
\(331\) −10.0353 −0.551592 −0.275796 0.961216i \(-0.588941\pi\)
−0.275796 + 0.961216i \(0.588941\pi\)
\(332\) 0 0
\(333\) 1.99282 0.109206
\(334\) 0 0
\(335\) −25.0300 −1.36754
\(336\) 0 0
\(337\) −14.7464 −0.803290 −0.401645 0.915796i \(-0.631561\pi\)
−0.401645 + 0.915796i \(0.631561\pi\)
\(338\) 0 0
\(339\) 5.13103 0.278680
\(340\) 0 0
\(341\) −21.6695 −1.17347
\(342\) 0 0
\(343\) 18.2703 0.986503
\(344\) 0 0
\(345\) −3.70748 −0.199604
\(346\) 0 0
\(347\) 22.1533 1.18925 0.594625 0.804003i \(-0.297300\pi\)
0.594625 + 0.804003i \(0.297300\pi\)
\(348\) 0 0
\(349\) −11.5717 −0.619417 −0.309708 0.950832i \(-0.600231\pi\)
−0.309708 + 0.950832i \(0.600231\pi\)
\(350\) 0 0
\(351\) 3.62850 0.193675
\(352\) 0 0
\(353\) 32.8443 1.74813 0.874063 0.485813i \(-0.161476\pi\)
0.874063 + 0.485813i \(0.161476\pi\)
\(354\) 0 0
\(355\) −24.5725 −1.30417
\(356\) 0 0
\(357\) 10.3941 0.550112
\(358\) 0 0
\(359\) −8.11658 −0.428377 −0.214188 0.976792i \(-0.568711\pi\)
−0.214188 + 0.976792i \(0.568711\pi\)
\(360\) 0 0
\(361\) 15.4523 0.813281
\(362\) 0 0
\(363\) 6.36820 0.334244
\(364\) 0 0
\(365\) −34.9339 −1.82852
\(366\) 0 0
\(367\) 21.6880 1.13210 0.566051 0.824370i \(-0.308470\pi\)
0.566051 + 0.824370i \(0.308470\pi\)
\(368\) 0 0
\(369\) −2.83754 −0.147716
\(370\) 0 0
\(371\) 12.2996 0.638562
\(372\) 0 0
\(373\) −24.7291 −1.28042 −0.640211 0.768199i \(-0.721153\pi\)
−0.640211 + 0.768199i \(0.721153\pi\)
\(374\) 0 0
\(375\) 13.8861 0.717078
\(376\) 0 0
\(377\) −3.62850 −0.186877
\(378\) 0 0
\(379\) −9.48141 −0.487027 −0.243514 0.969897i \(-0.578300\pi\)
−0.243514 + 0.969897i \(0.578300\pi\)
\(380\) 0 0
\(381\) −10.5582 −0.540913
\(382\) 0 0
\(383\) −18.4914 −0.944868 −0.472434 0.881366i \(-0.656625\pi\)
−0.472434 + 0.881366i \(0.656625\pi\)
\(384\) 0 0
\(385\) −12.7242 −0.648484
\(386\) 0 0
\(387\) −1.53121 −0.0778360
\(388\) 0 0
\(389\) −12.7820 −0.648073 −0.324036 0.946045i \(-0.605040\pi\)
−0.324036 + 0.946045i \(0.605040\pi\)
\(390\) 0 0
\(391\) −6.51792 −0.329625
\(392\) 0 0
\(393\) −1.14566 −0.0577908
\(394\) 0 0
\(395\) −54.9122 −2.76293
\(396\) 0 0
\(397\) 8.93842 0.448607 0.224303 0.974519i \(-0.427989\pi\)
0.224303 + 0.974519i \(0.427989\pi\)
\(398\) 0 0
\(399\) −9.36020 −0.468596
\(400\) 0 0
\(401\) −24.4730 −1.22212 −0.611062 0.791583i \(-0.709257\pi\)
−0.611062 + 0.791583i \(0.709257\pi\)
\(402\) 0 0
\(403\) −36.5344 −1.81991
\(404\) 0 0
\(405\) −3.70748 −0.184226
\(406\) 0 0
\(407\) −4.28888 −0.212592
\(408\) 0 0
\(409\) 8.34547 0.412657 0.206329 0.978483i \(-0.433848\pi\)
0.206329 + 0.978483i \(0.433848\pi\)
\(410\) 0 0
\(411\) 0.658713 0.0324919
\(412\) 0 0
\(413\) −11.5020 −0.565978
\(414\) 0 0
\(415\) −42.9253 −2.10712
\(416\) 0 0
\(417\) 13.1747 0.645166
\(418\) 0 0
\(419\) −14.2208 −0.694733 −0.347367 0.937729i \(-0.612924\pi\)
−0.347367 + 0.937729i \(0.612924\pi\)
\(420\) 0 0
\(421\) 28.2369 1.37618 0.688092 0.725624i \(-0.258449\pi\)
0.688092 + 0.725624i \(0.258449\pi\)
\(422\) 0 0
\(423\) −1.23502 −0.0600489
\(424\) 0 0
\(425\) 57.0021 2.76501
\(426\) 0 0
\(427\) −2.83416 −0.137155
\(428\) 0 0
\(429\) −7.80912 −0.377028
\(430\) 0 0
\(431\) −37.0946 −1.78678 −0.893391 0.449280i \(-0.851681\pi\)
−0.893391 + 0.449280i \(0.851681\pi\)
\(432\) 0 0
\(433\) −6.57756 −0.316098 −0.158049 0.987431i \(-0.550520\pi\)
−0.158049 + 0.987431i \(0.550520\pi\)
\(434\) 0 0
\(435\) 3.70748 0.177760
\(436\) 0 0
\(437\) 5.86961 0.280782
\(438\) 0 0
\(439\) 22.0232 1.05111 0.525554 0.850760i \(-0.323858\pi\)
0.525554 + 0.850760i \(0.323858\pi\)
\(440\) 0 0
\(441\) −4.45697 −0.212237
\(442\) 0 0
\(443\) 39.3018 1.86729 0.933643 0.358206i \(-0.116611\pi\)
0.933643 + 0.358206i \(0.116611\pi\)
\(444\) 0 0
\(445\) 33.5986 1.59273
\(446\) 0 0
\(447\) −1.22464 −0.0579236
\(448\) 0 0
\(449\) −25.5687 −1.20666 −0.603330 0.797492i \(-0.706160\pi\)
−0.603330 + 0.797492i \(0.706160\pi\)
\(450\) 0 0
\(451\) 6.10683 0.287560
\(452\) 0 0
\(453\) 13.8166 0.649160
\(454\) 0 0
\(455\) −21.4527 −1.00572
\(456\) 0 0
\(457\) −22.2440 −1.04053 −0.520265 0.854005i \(-0.674167\pi\)
−0.520265 + 0.854005i \(0.674167\pi\)
\(458\) 0 0
\(459\) −6.51792 −0.304230
\(460\) 0 0
\(461\) 25.4443 1.18506 0.592530 0.805548i \(-0.298129\pi\)
0.592530 + 0.805548i \(0.298129\pi\)
\(462\) 0 0
\(463\) 38.2248 1.77646 0.888228 0.459403i \(-0.151937\pi\)
0.888228 + 0.459403i \(0.151937\pi\)
\(464\) 0 0
\(465\) 37.3296 1.73112
\(466\) 0 0
\(467\) −8.60328 −0.398112 −0.199056 0.979988i \(-0.563788\pi\)
−0.199056 + 0.979988i \(0.563788\pi\)
\(468\) 0 0
\(469\) −10.7661 −0.497131
\(470\) 0 0
\(471\) 4.33993 0.199973
\(472\) 0 0
\(473\) 3.29542 0.151524
\(474\) 0 0
\(475\) −51.3323 −2.35529
\(476\) 0 0
\(477\) −7.71284 −0.353147
\(478\) 0 0
\(479\) 27.0877 1.23767 0.618834 0.785522i \(-0.287605\pi\)
0.618834 + 0.785522i \(0.287605\pi\)
\(480\) 0 0
\(481\) −7.23097 −0.329704
\(482\) 0 0
\(483\) −1.59469 −0.0725608
\(484\) 0 0
\(485\) −1.69090 −0.0767798
\(486\) 0 0
\(487\) 33.0312 1.49679 0.748394 0.663254i \(-0.230825\pi\)
0.748394 + 0.663254i \(0.230825\pi\)
\(488\) 0 0
\(489\) 19.1345 0.865290
\(490\) 0 0
\(491\) −10.1438 −0.457783 −0.228892 0.973452i \(-0.573510\pi\)
−0.228892 + 0.973452i \(0.573510\pi\)
\(492\) 0 0
\(493\) 6.51792 0.293552
\(494\) 0 0
\(495\) 7.97910 0.358634
\(496\) 0 0
\(497\) −10.5693 −0.474097
\(498\) 0 0
\(499\) 30.0106 1.34346 0.671730 0.740796i \(-0.265552\pi\)
0.671730 + 0.740796i \(0.265552\pi\)
\(500\) 0 0
\(501\) 20.1088 0.898396
\(502\) 0 0
\(503\) −39.1935 −1.74755 −0.873775 0.486331i \(-0.838335\pi\)
−0.873775 + 0.486331i \(0.838335\pi\)
\(504\) 0 0
\(505\) −9.45816 −0.420882
\(506\) 0 0
\(507\) −0.166035 −0.00737388
\(508\) 0 0
\(509\) −25.1744 −1.11584 −0.557918 0.829896i \(-0.688400\pi\)
−0.557918 + 0.829896i \(0.688400\pi\)
\(510\) 0 0
\(511\) −15.0260 −0.664711
\(512\) 0 0
\(513\) 5.86961 0.259150
\(514\) 0 0
\(515\) −25.9157 −1.14198
\(516\) 0 0
\(517\) 2.65797 0.116897
\(518\) 0 0
\(519\) −9.92281 −0.435563
\(520\) 0 0
\(521\) 32.4677 1.42244 0.711218 0.702971i \(-0.248144\pi\)
0.711218 + 0.702971i \(0.248144\pi\)
\(522\) 0 0
\(523\) 15.7931 0.690583 0.345292 0.938495i \(-0.387780\pi\)
0.345292 + 0.938495i \(0.387780\pi\)
\(524\) 0 0
\(525\) 13.9462 0.608664
\(526\) 0 0
\(527\) 65.6271 2.85876
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 7.21271 0.313005
\(532\) 0 0
\(533\) 10.2960 0.445970
\(534\) 0 0
\(535\) 28.8084 1.24550
\(536\) 0 0
\(537\) 0.900750 0.0388702
\(538\) 0 0
\(539\) 9.59212 0.413162
\(540\) 0 0
\(541\) 17.7211 0.761891 0.380946 0.924597i \(-0.375599\pi\)
0.380946 + 0.924597i \(0.375599\pi\)
\(542\) 0 0
\(543\) 21.3878 0.917838
\(544\) 0 0
\(545\) −36.4553 −1.56157
\(546\) 0 0
\(547\) −36.0363 −1.54080 −0.770400 0.637560i \(-0.779944\pi\)
−0.770400 + 0.637560i \(0.779944\pi\)
\(548\) 0 0
\(549\) 1.77725 0.0758513
\(550\) 0 0
\(551\) −5.86961 −0.250054
\(552\) 0 0
\(553\) −23.6192 −1.00439
\(554\) 0 0
\(555\) 7.38836 0.313619
\(556\) 0 0
\(557\) −17.0363 −0.721850 −0.360925 0.932595i \(-0.617539\pi\)
−0.360925 + 0.932595i \(0.617539\pi\)
\(558\) 0 0
\(559\) 5.55601 0.234994
\(560\) 0 0
\(561\) 14.0276 0.592246
\(562\) 0 0
\(563\) −29.6423 −1.24927 −0.624637 0.780915i \(-0.714753\pi\)
−0.624637 + 0.780915i \(0.714753\pi\)
\(564\) 0 0
\(565\) 19.0232 0.800313
\(566\) 0 0
\(567\) −1.59469 −0.0669706
\(568\) 0 0
\(569\) −2.88610 −0.120991 −0.0604957 0.998168i \(-0.519268\pi\)
−0.0604957 + 0.998168i \(0.519268\pi\)
\(570\) 0 0
\(571\) −32.5461 −1.36201 −0.681007 0.732277i \(-0.738458\pi\)
−0.681007 + 0.732277i \(0.738458\pi\)
\(572\) 0 0
\(573\) −4.06677 −0.169892
\(574\) 0 0
\(575\) −8.74544 −0.364710
\(576\) 0 0
\(577\) −8.33309 −0.346911 −0.173456 0.984842i \(-0.555493\pi\)
−0.173456 + 0.984842i \(0.555493\pi\)
\(578\) 0 0
\(579\) −5.55739 −0.230957
\(580\) 0 0
\(581\) −18.4633 −0.765988
\(582\) 0 0
\(583\) 16.5993 0.687471
\(584\) 0 0
\(585\) 13.4526 0.556197
\(586\) 0 0
\(587\) 9.11774 0.376329 0.188165 0.982137i \(-0.439746\pi\)
0.188165 + 0.982137i \(0.439746\pi\)
\(588\) 0 0
\(589\) −59.0994 −2.43515
\(590\) 0 0
\(591\) −12.7694 −0.525264
\(592\) 0 0
\(593\) −43.9819 −1.80612 −0.903061 0.429512i \(-0.858685\pi\)
−0.903061 + 0.429512i \(0.858685\pi\)
\(594\) 0 0
\(595\) 38.5358 1.57981
\(596\) 0 0
\(597\) −21.2840 −0.871096
\(598\) 0 0
\(599\) 4.50050 0.183885 0.0919427 0.995764i \(-0.470692\pi\)
0.0919427 + 0.995764i \(0.470692\pi\)
\(600\) 0 0
\(601\) −39.4696 −1.61000 −0.804999 0.593277i \(-0.797834\pi\)
−0.804999 + 0.593277i \(0.797834\pi\)
\(602\) 0 0
\(603\) 6.75121 0.274931
\(604\) 0 0
\(605\) 23.6100 0.959884
\(606\) 0 0
\(607\) 15.2929 0.620718 0.310359 0.950619i \(-0.399551\pi\)
0.310359 + 0.950619i \(0.399551\pi\)
\(608\) 0 0
\(609\) 1.59469 0.0646200
\(610\) 0 0
\(611\) 4.48129 0.181293
\(612\) 0 0
\(613\) −34.1919 −1.38100 −0.690499 0.723334i \(-0.742609\pi\)
−0.690499 + 0.723334i \(0.742609\pi\)
\(614\) 0 0
\(615\) −10.5201 −0.424212
\(616\) 0 0
\(617\) −44.8672 −1.80629 −0.903143 0.429340i \(-0.858746\pi\)
−0.903143 + 0.429340i \(0.858746\pi\)
\(618\) 0 0
\(619\) −32.4722 −1.30517 −0.652585 0.757716i \(-0.726315\pi\)
−0.652585 + 0.757716i \(0.726315\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 14.4517 0.578994
\(624\) 0 0
\(625\) 7.75549 0.310220
\(626\) 0 0
\(627\) −12.6323 −0.504487
\(628\) 0 0
\(629\) 12.9891 0.517908
\(630\) 0 0
\(631\) 8.91524 0.354910 0.177455 0.984129i \(-0.443214\pi\)
0.177455 + 0.984129i \(0.443214\pi\)
\(632\) 0 0
\(633\) 9.34034 0.371245
\(634\) 0 0
\(635\) −39.1444 −1.55340
\(636\) 0 0
\(637\) 16.1721 0.640763
\(638\) 0 0
\(639\) 6.62781 0.262192
\(640\) 0 0
\(641\) −2.82470 −0.111569 −0.0557845 0.998443i \(-0.517766\pi\)
−0.0557845 + 0.998443i \(0.517766\pi\)
\(642\) 0 0
\(643\) 50.0196 1.97258 0.986290 0.165023i \(-0.0527697\pi\)
0.986290 + 0.165023i \(0.0527697\pi\)
\(644\) 0 0
\(645\) −5.67695 −0.223530
\(646\) 0 0
\(647\) −17.4253 −0.685058 −0.342529 0.939507i \(-0.611283\pi\)
−0.342529 + 0.939507i \(0.611283\pi\)
\(648\) 0 0
\(649\) −15.5229 −0.609328
\(650\) 0 0
\(651\) 16.0565 0.629302
\(652\) 0 0
\(653\) 2.53400 0.0991633 0.0495816 0.998770i \(-0.484211\pi\)
0.0495816 + 0.998770i \(0.484211\pi\)
\(654\) 0 0
\(655\) −4.24751 −0.165964
\(656\) 0 0
\(657\) 9.42253 0.367608
\(658\) 0 0
\(659\) 46.7865 1.82254 0.911272 0.411805i \(-0.135101\pi\)
0.911272 + 0.411805i \(0.135101\pi\)
\(660\) 0 0
\(661\) 38.2474 1.48765 0.743827 0.668373i \(-0.233009\pi\)
0.743827 + 0.668373i \(0.233009\pi\)
\(662\) 0 0
\(663\) 23.6503 0.918501
\(664\) 0 0
\(665\) −34.7028 −1.34572
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −21.2923 −0.823207
\(670\) 0 0
\(671\) −3.82493 −0.147660
\(672\) 0 0
\(673\) 9.98712 0.384975 0.192488 0.981299i \(-0.438345\pi\)
0.192488 + 0.981299i \(0.438345\pi\)
\(674\) 0 0
\(675\) −8.74544 −0.336612
\(676\) 0 0
\(677\) 16.4733 0.633122 0.316561 0.948572i \(-0.397472\pi\)
0.316561 + 0.948572i \(0.397472\pi\)
\(678\) 0 0
\(679\) −0.727301 −0.0279112
\(680\) 0 0
\(681\) 6.71940 0.257488
\(682\) 0 0
\(683\) 13.4867 0.516053 0.258026 0.966138i \(-0.416928\pi\)
0.258026 + 0.966138i \(0.416928\pi\)
\(684\) 0 0
\(685\) 2.44217 0.0933104
\(686\) 0 0
\(687\) −15.5515 −0.593325
\(688\) 0 0
\(689\) 27.9860 1.06618
\(690\) 0 0
\(691\) −18.9990 −0.722758 −0.361379 0.932419i \(-0.617694\pi\)
−0.361379 + 0.932419i \(0.617694\pi\)
\(692\) 0 0
\(693\) 3.43203 0.130372
\(694\) 0 0
\(695\) 48.8448 1.85279
\(696\) 0 0
\(697\) −18.4948 −0.700542
\(698\) 0 0
\(699\) −3.60530 −0.136365
\(700\) 0 0
\(701\) −15.3046 −0.578045 −0.289023 0.957322i \(-0.593330\pi\)
−0.289023 + 0.957322i \(0.593330\pi\)
\(702\) 0 0
\(703\) −11.6971 −0.441165
\(704\) 0 0
\(705\) −4.57883 −0.172449
\(706\) 0 0
\(707\) −4.06821 −0.153001
\(708\) 0 0
\(709\) −23.1417 −0.869104 −0.434552 0.900647i \(-0.643093\pi\)
−0.434552 + 0.900647i \(0.643093\pi\)
\(710\) 0 0
\(711\) 14.8112 0.555462
\(712\) 0 0
\(713\) −10.0687 −0.377076
\(714\) 0 0
\(715\) −28.9522 −1.08275
\(716\) 0 0
\(717\) −9.39572 −0.350890
\(718\) 0 0
\(719\) 4.55539 0.169887 0.0849436 0.996386i \(-0.472929\pi\)
0.0849436 + 0.996386i \(0.472929\pi\)
\(720\) 0 0
\(721\) −11.1470 −0.415137
\(722\) 0 0
\(723\) −12.5234 −0.465751
\(724\) 0 0
\(725\) 8.74544 0.324797
\(726\) 0 0
\(727\) 22.0600 0.818159 0.409079 0.912499i \(-0.365850\pi\)
0.409079 + 0.912499i \(0.365850\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −9.98033 −0.369136
\(732\) 0 0
\(733\) −38.6690 −1.42827 −0.714137 0.700006i \(-0.753181\pi\)
−0.714137 + 0.700006i \(0.753181\pi\)
\(734\) 0 0
\(735\) −16.5241 −0.609502
\(736\) 0 0
\(737\) −14.5297 −0.535208
\(738\) 0 0
\(739\) 24.2883 0.893458 0.446729 0.894669i \(-0.352589\pi\)
0.446729 + 0.894669i \(0.352589\pi\)
\(740\) 0 0
\(741\) −21.2979 −0.782398
\(742\) 0 0
\(743\) 24.0568 0.882559 0.441279 0.897370i \(-0.354525\pi\)
0.441279 + 0.897370i \(0.354525\pi\)
\(744\) 0 0
\(745\) −4.54034 −0.166345
\(746\) 0 0
\(747\) 11.5780 0.423618
\(748\) 0 0
\(749\) 12.3913 0.452767
\(750\) 0 0
\(751\) −0.261425 −0.00953954 −0.00476977 0.999989i \(-0.501518\pi\)
−0.00476977 + 0.999989i \(0.501518\pi\)
\(752\) 0 0
\(753\) 13.9199 0.507268
\(754\) 0 0
\(755\) 51.2248 1.86426
\(756\) 0 0
\(757\) −40.2089 −1.46142 −0.730709 0.682689i \(-0.760810\pi\)
−0.730709 + 0.682689i \(0.760810\pi\)
\(758\) 0 0
\(759\) −2.15216 −0.0781185
\(760\) 0 0
\(761\) 14.8187 0.537176 0.268588 0.963255i \(-0.413443\pi\)
0.268588 + 0.963255i \(0.413443\pi\)
\(762\) 0 0
\(763\) −15.6804 −0.567668
\(764\) 0 0
\(765\) −24.1651 −0.873690
\(766\) 0 0
\(767\) −26.1713 −0.944992
\(768\) 0 0
\(769\) −15.1156 −0.545083 −0.272542 0.962144i \(-0.587864\pi\)
−0.272542 + 0.962144i \(0.587864\pi\)
\(770\) 0 0
\(771\) 16.4930 0.593982
\(772\) 0 0
\(773\) 23.2648 0.836777 0.418388 0.908268i \(-0.362595\pi\)
0.418388 + 0.908268i \(0.362595\pi\)
\(774\) 0 0
\(775\) 88.0553 3.16304
\(776\) 0 0
\(777\) 3.17793 0.114008
\(778\) 0 0
\(779\) 16.6552 0.596736
\(780\) 0 0
\(781\) −14.2641 −0.510410
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 16.0902 0.574284
\(786\) 0 0
\(787\) −19.9419 −0.710853 −0.355427 0.934704i \(-0.615664\pi\)
−0.355427 + 0.934704i \(0.615664\pi\)
\(788\) 0 0
\(789\) 30.7133 1.09342
\(790\) 0 0
\(791\) 8.18240 0.290933
\(792\) 0 0
\(793\) −6.44877 −0.229003
\(794\) 0 0
\(795\) −28.5952 −1.01417
\(796\) 0 0
\(797\) −17.5348 −0.621113 −0.310557 0.950555i \(-0.600515\pi\)
−0.310557 + 0.950555i \(0.600515\pi\)
\(798\) 0 0
\(799\) −8.04979 −0.284781
\(800\) 0 0
\(801\) −9.06238 −0.320203
\(802\) 0 0
\(803\) −20.2788 −0.715623
\(804\) 0 0
\(805\) −5.91228 −0.208381
\(806\) 0 0
\(807\) −7.55230 −0.265854
\(808\) 0 0
\(809\) 13.8664 0.487517 0.243759 0.969836i \(-0.421619\pi\)
0.243759 + 0.969836i \(0.421619\pi\)
\(810\) 0 0
\(811\) 6.06197 0.212865 0.106432 0.994320i \(-0.466057\pi\)
0.106432 + 0.994320i \(0.466057\pi\)
\(812\) 0 0
\(813\) 2.21164 0.0775657
\(814\) 0 0
\(815\) 70.9407 2.48494
\(816\) 0 0
\(817\) 8.98763 0.314437
\(818\) 0 0
\(819\) 5.78633 0.202191
\(820\) 0 0
\(821\) −22.6540 −0.790631 −0.395316 0.918545i \(-0.629365\pi\)
−0.395316 + 0.918545i \(0.629365\pi\)
\(822\) 0 0
\(823\) −20.5866 −0.717605 −0.358803 0.933413i \(-0.616815\pi\)
−0.358803 + 0.933413i \(0.616815\pi\)
\(824\) 0 0
\(825\) 18.8216 0.655284
\(826\) 0 0
\(827\) −2.94930 −0.102557 −0.0512786 0.998684i \(-0.516330\pi\)
−0.0512786 + 0.998684i \(0.516330\pi\)
\(828\) 0 0
\(829\) −51.3779 −1.78443 −0.892214 0.451612i \(-0.850849\pi\)
−0.892214 + 0.451612i \(0.850849\pi\)
\(830\) 0 0
\(831\) −7.31297 −0.253684
\(832\) 0 0
\(833\) −29.0502 −1.00653
\(834\) 0 0
\(835\) 74.5531 2.58002
\(836\) 0 0
\(837\) −10.0687 −0.348026
\(838\) 0 0
\(839\) 46.5085 1.60565 0.802826 0.596214i \(-0.203329\pi\)
0.802826 + 0.596214i \(0.203329\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 32.8318 1.13079
\(844\) 0 0
\(845\) −0.615572 −0.0211763
\(846\) 0 0
\(847\) 10.1553 0.348940
\(848\) 0 0
\(849\) −12.8415 −0.440718
\(850\) 0 0
\(851\) −1.99282 −0.0683131
\(852\) 0 0
\(853\) 33.7426 1.15532 0.577662 0.816276i \(-0.303965\pi\)
0.577662 + 0.816276i \(0.303965\pi\)
\(854\) 0 0
\(855\) 21.7615 0.744227
\(856\) 0 0
\(857\) −10.6128 −0.362527 −0.181263 0.983435i \(-0.558019\pi\)
−0.181263 + 0.983435i \(0.558019\pi\)
\(858\) 0 0
\(859\) −38.9227 −1.32802 −0.664012 0.747722i \(-0.731148\pi\)
−0.664012 + 0.747722i \(0.731148\pi\)
\(860\) 0 0
\(861\) −4.52498 −0.154211
\(862\) 0 0
\(863\) −13.2174 −0.449924 −0.224962 0.974368i \(-0.572226\pi\)
−0.224962 + 0.974368i \(0.572226\pi\)
\(864\) 0 0
\(865\) −36.7887 −1.25085
\(866\) 0 0
\(867\) −25.4833 −0.865458
\(868\) 0 0
\(869\) −31.8760 −1.08132
\(870\) 0 0
\(871\) −24.4968 −0.830042
\(872\) 0 0
\(873\) 0.456077 0.0154359
\(874\) 0 0
\(875\) 22.1441 0.748606
\(876\) 0 0
\(877\) 48.3073 1.63122 0.815610 0.578602i \(-0.196401\pi\)
0.815610 + 0.578602i \(0.196401\pi\)
\(878\) 0 0
\(879\) 14.4325 0.486797
\(880\) 0 0
\(881\) −28.7688 −0.969244 −0.484622 0.874724i \(-0.661043\pi\)
−0.484622 + 0.874724i \(0.661043\pi\)
\(882\) 0 0
\(883\) 7.21826 0.242914 0.121457 0.992597i \(-0.461243\pi\)
0.121457 + 0.992597i \(0.461243\pi\)
\(884\) 0 0
\(885\) 26.7410 0.898889
\(886\) 0 0
\(887\) 31.1158 1.04477 0.522383 0.852711i \(-0.325043\pi\)
0.522383 + 0.852711i \(0.325043\pi\)
\(888\) 0 0
\(889\) −16.8370 −0.564696
\(890\) 0 0
\(891\) −2.15216 −0.0721001
\(892\) 0 0
\(893\) 7.24911 0.242582
\(894\) 0 0
\(895\) 3.33951 0.111628
\(896\) 0 0
\(897\) −3.62850 −0.121152
\(898\) 0 0
\(899\) 10.0687 0.335810
\(900\) 0 0
\(901\) −50.2717 −1.67479
\(902\) 0 0
\(903\) −2.44181 −0.0812583
\(904\) 0 0
\(905\) 79.2949 2.63585
\(906\) 0 0
\(907\) −11.6633 −0.387272 −0.193636 0.981073i \(-0.562028\pi\)
−0.193636 + 0.981073i \(0.562028\pi\)
\(908\) 0 0
\(909\) 2.55110 0.0846146
\(910\) 0 0
\(911\) −48.7355 −1.61468 −0.807340 0.590087i \(-0.799094\pi\)
−0.807340 + 0.590087i \(0.799094\pi\)
\(912\) 0 0
\(913\) −24.9178 −0.824658
\(914\) 0 0
\(915\) 6.58914 0.217830
\(916\) 0 0
\(917\) −1.82697 −0.0603318
\(918\) 0 0
\(919\) 37.0018 1.22058 0.610289 0.792179i \(-0.291053\pi\)
0.610289 + 0.792179i \(0.291053\pi\)
\(920\) 0 0
\(921\) −23.1124 −0.761580
\(922\) 0 0
\(923\) −24.0490 −0.791583
\(924\) 0 0
\(925\) 17.4281 0.573033
\(926\) 0 0
\(927\) 6.99010 0.229585
\(928\) 0 0
\(929\) 46.1915 1.51549 0.757747 0.652549i \(-0.226300\pi\)
0.757747 + 0.652549i \(0.226300\pi\)
\(930\) 0 0
\(931\) 26.1607 0.857382
\(932\) 0 0
\(933\) 28.4686 0.932019
\(934\) 0 0
\(935\) 52.0071 1.70082
\(936\) 0 0
\(937\) 24.0838 0.786785 0.393392 0.919371i \(-0.371301\pi\)
0.393392 + 0.919371i \(0.371301\pi\)
\(938\) 0 0
\(939\) 13.2820 0.433441
\(940\) 0 0
\(941\) 30.9552 1.00911 0.504555 0.863380i \(-0.331657\pi\)
0.504555 + 0.863380i \(0.331657\pi\)
\(942\) 0 0
\(943\) 2.83754 0.0924029
\(944\) 0 0
\(945\) −5.91228 −0.192327
\(946\) 0 0
\(947\) 10.8455 0.352433 0.176216 0.984351i \(-0.443614\pi\)
0.176216 + 0.984351i \(0.443614\pi\)
\(948\) 0 0
\(949\) −34.1897 −1.10984
\(950\) 0 0
\(951\) 19.8968 0.645198
\(952\) 0 0
\(953\) 16.2568 0.526609 0.263304 0.964713i \(-0.415188\pi\)
0.263304 + 0.964713i \(0.415188\pi\)
\(954\) 0 0
\(955\) −15.0775 −0.487896
\(956\) 0 0
\(957\) 2.15216 0.0695695
\(958\) 0 0
\(959\) 1.05044 0.0339205
\(960\) 0 0
\(961\) 70.3790 2.27029
\(962\) 0 0
\(963\) −7.77034 −0.250396
\(964\) 0 0
\(965\) −20.6039 −0.663264
\(966\) 0 0
\(967\) −10.7315 −0.345102 −0.172551 0.985001i \(-0.555201\pi\)
−0.172551 + 0.985001i \(0.555201\pi\)
\(968\) 0 0
\(969\) 38.2577 1.22901
\(970\) 0 0
\(971\) −0.672735 −0.0215891 −0.0107946 0.999942i \(-0.503436\pi\)
−0.0107946 + 0.999942i \(0.503436\pi\)
\(972\) 0 0
\(973\) 21.0095 0.673533
\(974\) 0 0
\(975\) 31.7328 1.01626
\(976\) 0 0
\(977\) −1.61692 −0.0517299 −0.0258650 0.999665i \(-0.508234\pi\)
−0.0258650 + 0.999665i \(0.508234\pi\)
\(978\) 0 0
\(979\) 19.5037 0.623341
\(980\) 0 0
\(981\) 9.83289 0.313940
\(982\) 0 0
\(983\) −52.1299 −1.66268 −0.831342 0.555761i \(-0.812427\pi\)
−0.831342 + 0.555761i \(0.812427\pi\)
\(984\) 0 0
\(985\) −47.3424 −1.50846
\(986\) 0 0
\(987\) −1.96948 −0.0626892
\(988\) 0 0
\(989\) 1.53121 0.0486898
\(990\) 0 0
\(991\) 57.5804 1.82910 0.914550 0.404473i \(-0.132545\pi\)
0.914550 + 0.404473i \(0.132545\pi\)
\(992\) 0 0
\(993\) 10.0353 0.318462
\(994\) 0 0
\(995\) −78.9101 −2.50162
\(996\) 0 0
\(997\) −5.70739 −0.180755 −0.0903775 0.995908i \(-0.528807\pi\)
−0.0903775 + 0.995908i \(0.528807\pi\)
\(998\) 0 0
\(999\) −1.99282 −0.0630502
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 8004.2.a.e.1.1 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.e.1.1 9 1.1 even 1 trivial