Properties

Label 2352.4.a.bx.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
Defining polynomial: \(x^{2} - x - 48\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(7.44622\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -12.4462 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -12.4462 q^{5} +9.00000 q^{9} +51.1236 q^{11} -37.2311 q^{13} -37.3387 q^{15} -22.2151 q^{17} +54.3387 q^{19} -176.924 q^{23} +29.9084 q^{25} +27.0000 q^{27} +61.0916 q^{29} +319.924 q^{31} +153.371 q^{33} -315.080 q^{37} -111.693 q^{39} +206.032 q^{41} -339.661 q^{43} -112.016 q^{45} +142.064 q^{47} -66.6453 q^{51} +310.016 q^{53} -636.295 q^{55} +163.016 q^{57} -281.650 q^{59} +543.849 q^{61} +463.387 q^{65} +479.359 q^{67} -530.773 q^{69} -1105.63 q^{71} -239.350 q^{73} +89.7253 q^{75} -1160.67 q^{79} +81.0000 q^{81} -2.93158 q^{83} +276.494 q^{85} +183.275 q^{87} +1278.74 q^{89} +959.773 q^{93} -676.311 q^{95} -79.0596 q^{97} +460.112 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} - 11q^{5} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} - 11q^{5} + 18q^{9} + 5q^{11} - 5q^{13} - 33q^{15} - 100q^{17} + 67q^{19} - 76q^{23} - 93q^{25} + 54q^{27} + 275q^{29} + 362q^{31} + 15q^{33} - 5q^{37} - 15q^{39} + 162q^{41} - 721q^{43} - 99q^{45} - 216q^{47} - 300q^{51} + 495q^{53} - 703q^{55} + 201q^{57} + 173q^{59} + 532q^{61} + 510q^{65} - 111q^{67} - 228q^{69} - 1600q^{71} - 1215q^{73} - 279q^{75} - 1460q^{79} + 162q^{81} - 1409q^{83} + 164q^{85} + 825q^{87} + 1974q^{89} + 1086q^{93} - 658q^{95} - 561q^{97} + 45q^{99} + O(q^{100}) \)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −12.4462 −1.11322 −0.556612 0.830773i \(-0.687899\pi\)
−0.556612 + 0.830773i \(0.687899\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 51.1236 1.40130 0.700651 0.713504i \(-0.252893\pi\)
0.700651 + 0.713504i \(0.252893\pi\)
\(12\) 0 0
\(13\) −37.2311 −0.794312 −0.397156 0.917751i \(-0.630003\pi\)
−0.397156 + 0.917751i \(0.630003\pi\)
\(14\) 0 0
\(15\) −37.3387 −0.642720
\(16\) 0 0
\(17\) −22.2151 −0.316939 −0.158469 0.987364i \(-0.550656\pi\)
−0.158469 + 0.987364i \(0.550656\pi\)
\(18\) 0 0
\(19\) 54.3387 0.656113 0.328056 0.944658i \(-0.393606\pi\)
0.328056 + 0.944658i \(0.393606\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −176.924 −1.60397 −0.801985 0.597345i \(-0.796222\pi\)
−0.801985 + 0.597345i \(0.796222\pi\)
\(24\) 0 0
\(25\) 29.9084 0.239268
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 61.0916 0.391187 0.195593 0.980685i \(-0.437337\pi\)
0.195593 + 0.980685i \(0.437337\pi\)
\(30\) 0 0
\(31\) 319.924 1.85355 0.926776 0.375614i \(-0.122568\pi\)
0.926776 + 0.375614i \(0.122568\pi\)
\(32\) 0 0
\(33\) 153.371 0.809043
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −315.080 −1.39997 −0.699984 0.714158i \(-0.746810\pi\)
−0.699984 + 0.714158i \(0.746810\pi\)
\(38\) 0 0
\(39\) −111.693 −0.458596
\(40\) 0 0
\(41\) 206.032 0.784800 0.392400 0.919795i \(-0.371645\pi\)
0.392400 + 0.919795i \(0.371645\pi\)
\(42\) 0 0
\(43\) −339.661 −1.20460 −0.602301 0.798269i \(-0.705749\pi\)
−0.602301 + 0.798269i \(0.705749\pi\)
\(44\) 0 0
\(45\) −112.016 −0.371075
\(46\) 0 0
\(47\) 142.064 0.440897 0.220449 0.975399i \(-0.429248\pi\)
0.220449 + 0.975399i \(0.429248\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −66.6453 −0.182985
\(52\) 0 0
\(53\) 310.016 0.803471 0.401736 0.915756i \(-0.368407\pi\)
0.401736 + 0.915756i \(0.368407\pi\)
\(54\) 0 0
\(55\) −636.295 −1.55996
\(56\) 0 0
\(57\) 163.016 0.378807
\(58\) 0 0
\(59\) −281.650 −0.621486 −0.310743 0.950494i \(-0.600578\pi\)
−0.310743 + 0.950494i \(0.600578\pi\)
\(60\) 0 0
\(61\) 543.849 1.14152 0.570760 0.821117i \(-0.306649\pi\)
0.570760 + 0.821117i \(0.306649\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 463.387 0.884247
\(66\) 0 0
\(67\) 479.359 0.874075 0.437038 0.899443i \(-0.356028\pi\)
0.437038 + 0.899443i \(0.356028\pi\)
\(68\) 0 0
\(69\) −530.773 −0.926052
\(70\) 0 0
\(71\) −1105.63 −1.84809 −0.924046 0.382280i \(-0.875139\pi\)
−0.924046 + 0.382280i \(0.875139\pi\)
\(72\) 0 0
\(73\) −239.350 −0.383751 −0.191876 0.981419i \(-0.561457\pi\)
−0.191876 + 0.981419i \(0.561457\pi\)
\(74\) 0 0
\(75\) 89.7253 0.138141
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1160.67 −1.65298 −0.826488 0.562954i \(-0.809665\pi\)
−0.826488 + 0.562954i \(0.809665\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −2.93158 −0.00387690 −0.00193845 0.999998i \(-0.500617\pi\)
−0.00193845 + 0.999998i \(0.500617\pi\)
\(84\) 0 0
\(85\) 276.494 0.352824
\(86\) 0 0
\(87\) 183.275 0.225852
\(88\) 0 0
\(89\) 1278.74 1.52299 0.761496 0.648169i \(-0.224465\pi\)
0.761496 + 0.648169i \(0.224465\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 959.773 1.07015
\(94\) 0 0
\(95\) −676.311 −0.730401
\(96\) 0 0
\(97\) −79.0596 −0.0827555 −0.0413777 0.999144i \(-0.513175\pi\)
−0.0413777 + 0.999144i \(0.513175\pi\)
\(98\) 0 0
\(99\) 460.112 0.467101
\(100\) 0 0
\(101\) −1372.10 −1.35178 −0.675889 0.737004i \(-0.736240\pi\)
−0.675889 + 0.737004i \(0.736240\pi\)
\(102\) 0 0
\(103\) 258.531 0.247318 0.123659 0.992325i \(-0.460537\pi\)
0.123659 + 0.992325i \(0.460537\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1262.37 1.14054 0.570270 0.821458i \(-0.306839\pi\)
0.570270 + 0.821458i \(0.306839\pi\)
\(108\) 0 0
\(109\) 276.833 0.243264 0.121632 0.992575i \(-0.461187\pi\)
0.121632 + 0.992575i \(0.461187\pi\)
\(110\) 0 0
\(111\) −945.240 −0.808272
\(112\) 0 0
\(113\) 52.9156 0.0440520 0.0220260 0.999757i \(-0.492988\pi\)
0.0220260 + 0.999757i \(0.492988\pi\)
\(114\) 0 0
\(115\) 2202.04 1.78558
\(116\) 0 0
\(117\) −335.080 −0.264771
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1282.62 0.963650
\(122\) 0 0
\(123\) 618.096 0.453104
\(124\) 0 0
\(125\) 1183.53 0.846866
\(126\) 0 0
\(127\) −443.700 −0.310016 −0.155008 0.987913i \(-0.549540\pi\)
−0.155008 + 0.987913i \(0.549540\pi\)
\(128\) 0 0
\(129\) −1018.98 −0.695477
\(130\) 0 0
\(131\) −2153.48 −1.43627 −0.718133 0.695906i \(-0.755003\pi\)
−0.718133 + 0.695906i \(0.755003\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −336.048 −0.214240
\(136\) 0 0
\(137\) 263.108 0.164079 0.0820394 0.996629i \(-0.473857\pi\)
0.0820394 + 0.996629i \(0.473857\pi\)
\(138\) 0 0
\(139\) 1165.77 0.711360 0.355680 0.934608i \(-0.384249\pi\)
0.355680 + 0.934608i \(0.384249\pi\)
\(140\) 0 0
\(141\) 426.192 0.254552
\(142\) 0 0
\(143\) −1903.39 −1.11307
\(144\) 0 0
\(145\) −760.359 −0.435479
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1765.19 −0.970539 −0.485270 0.874364i \(-0.661279\pi\)
−0.485270 + 0.874364i \(0.661279\pi\)
\(150\) 0 0
\(151\) −2692.31 −1.45098 −0.725488 0.688235i \(-0.758386\pi\)
−0.725488 + 0.688235i \(0.758386\pi\)
\(152\) 0 0
\(153\) −199.936 −0.105646
\(154\) 0 0
\(155\) −3981.85 −2.06342
\(156\) 0 0
\(157\) 1941.38 0.986873 0.493437 0.869782i \(-0.335740\pi\)
0.493437 + 0.869782i \(0.335740\pi\)
\(158\) 0 0
\(159\) 930.048 0.463884
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2102.84 −1.01047 −0.505236 0.862981i \(-0.668595\pi\)
−0.505236 + 0.862981i \(0.668595\pi\)
\(164\) 0 0
\(165\) −1908.89 −0.900646
\(166\) 0 0
\(167\) −2344.22 −1.08623 −0.543116 0.839658i \(-0.682756\pi\)
−0.543116 + 0.839658i \(0.682756\pi\)
\(168\) 0 0
\(169\) −810.844 −0.369069
\(170\) 0 0
\(171\) 489.048 0.218704
\(172\) 0 0
\(173\) −3470.65 −1.52525 −0.762626 0.646840i \(-0.776090\pi\)
−0.762626 + 0.646840i \(0.776090\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −844.949 −0.358815
\(178\) 0 0
\(179\) 955.771 0.399093 0.199547 0.979888i \(-0.436053\pi\)
0.199547 + 0.979888i \(0.436053\pi\)
\(180\) 0 0
\(181\) −4220.26 −1.73309 −0.866546 0.499098i \(-0.833665\pi\)
−0.866546 + 0.499098i \(0.833665\pi\)
\(182\) 0 0
\(183\) 1631.55 0.659057
\(184\) 0 0
\(185\) 3921.56 1.55848
\(186\) 0 0
\(187\) −1135.72 −0.444127
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3518.03 1.33275 0.666377 0.745615i \(-0.267844\pi\)
0.666377 + 0.745615i \(0.267844\pi\)
\(192\) 0 0
\(193\) −5017.67 −1.87140 −0.935699 0.352799i \(-0.885230\pi\)
−0.935699 + 0.352799i \(0.885230\pi\)
\(194\) 0 0
\(195\) 1390.16 0.510520
\(196\) 0 0
\(197\) −2838.14 −1.02644 −0.513221 0.858257i \(-0.671548\pi\)
−0.513221 + 0.858257i \(0.671548\pi\)
\(198\) 0 0
\(199\) 354.901 0.126424 0.0632118 0.998000i \(-0.479866\pi\)
0.0632118 + 0.998000i \(0.479866\pi\)
\(200\) 0 0
\(201\) 1438.08 0.504648
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2564.32 −0.873658
\(206\) 0 0
\(207\) −1592.32 −0.534656
\(208\) 0 0
\(209\) 2777.99 0.919413
\(210\) 0 0
\(211\) −752.672 −0.245574 −0.122787 0.992433i \(-0.539183\pi\)
−0.122787 + 0.992433i \(0.539183\pi\)
\(212\) 0 0
\(213\) −3316.90 −1.06700
\(214\) 0 0
\(215\) 4227.50 1.34099
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −718.051 −0.221559
\(220\) 0 0
\(221\) 827.093 0.251748
\(222\) 0 0
\(223\) −3077.75 −0.924221 −0.462111 0.886822i \(-0.652908\pi\)
−0.462111 + 0.886822i \(0.652908\pi\)
\(224\) 0 0
\(225\) 269.176 0.0797558
\(226\) 0 0
\(227\) −6217.43 −1.81791 −0.908955 0.416894i \(-0.863119\pi\)
−0.908955 + 0.416894i \(0.863119\pi\)
\(228\) 0 0
\(229\) −503.254 −0.145223 −0.0726113 0.997360i \(-0.523133\pi\)
−0.0726113 + 0.997360i \(0.523133\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −268.340 −0.0754486 −0.0377243 0.999288i \(-0.512011\pi\)
−0.0377243 + 0.999288i \(0.512011\pi\)
\(234\) 0 0
\(235\) −1768.16 −0.490817
\(236\) 0 0
\(237\) −3482.00 −0.954346
\(238\) 0 0
\(239\) 5189.77 1.40459 0.702297 0.711884i \(-0.252158\pi\)
0.702297 + 0.711884i \(0.252158\pi\)
\(240\) 0 0
\(241\) −6170.94 −1.64940 −0.824699 0.565571i \(-0.808656\pi\)
−0.824699 + 0.565571i \(0.808656\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2023.09 −0.521158
\(248\) 0 0
\(249\) −8.79474 −0.00223833
\(250\) 0 0
\(251\) 1891.91 0.475763 0.237882 0.971294i \(-0.423547\pi\)
0.237882 + 0.971294i \(0.423547\pi\)
\(252\) 0 0
\(253\) −9045.01 −2.24765
\(254\) 0 0
\(255\) 829.483 0.203703
\(256\) 0 0
\(257\) 6539.93 1.58735 0.793676 0.608340i \(-0.208164\pi\)
0.793676 + 0.608340i \(0.208164\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 549.824 0.130396
\(262\) 0 0
\(263\) −5375.90 −1.26043 −0.630214 0.776422i \(-0.717033\pi\)
−0.630214 + 0.776422i \(0.717033\pi\)
\(264\) 0 0
\(265\) −3858.53 −0.894443
\(266\) 0 0
\(267\) 3836.22 0.879300
\(268\) 0 0
\(269\) 3238.88 0.734119 0.367060 0.930197i \(-0.380364\pi\)
0.367060 + 0.930197i \(0.380364\pi\)
\(270\) 0 0
\(271\) 1357.46 0.304279 0.152140 0.988359i \(-0.451384\pi\)
0.152140 + 0.988359i \(0.451384\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1529.03 0.335286
\(276\) 0 0
\(277\) −2561.63 −0.555645 −0.277823 0.960632i \(-0.589613\pi\)
−0.277823 + 0.960632i \(0.589613\pi\)
\(278\) 0 0
\(279\) 2879.32 0.617851
\(280\) 0 0
\(281\) 1786.17 0.379196 0.189598 0.981862i \(-0.439282\pi\)
0.189598 + 0.981862i \(0.439282\pi\)
\(282\) 0 0
\(283\) −7388.28 −1.55190 −0.775950 0.630794i \(-0.782729\pi\)
−0.775950 + 0.630794i \(0.782729\pi\)
\(284\) 0 0
\(285\) −2028.93 −0.421697
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4419.49 −0.899550
\(290\) 0 0
\(291\) −237.179 −0.0477789
\(292\) 0 0
\(293\) −492.981 −0.0982945 −0.0491472 0.998792i \(-0.515650\pi\)
−0.0491472 + 0.998792i \(0.515650\pi\)
\(294\) 0 0
\(295\) 3505.48 0.691853
\(296\) 0 0
\(297\) 1380.34 0.269681
\(298\) 0 0
\(299\) 6587.09 1.27405
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −4116.31 −0.780449
\(304\) 0 0
\(305\) −6768.86 −1.27077
\(306\) 0 0
\(307\) −988.810 −0.183825 −0.0919126 0.995767i \(-0.529298\pi\)
−0.0919126 + 0.995767i \(0.529298\pi\)
\(308\) 0 0
\(309\) 775.592 0.142789
\(310\) 0 0
\(311\) −9596.57 −1.74975 −0.874874 0.484351i \(-0.839056\pi\)
−0.874874 + 0.484351i \(0.839056\pi\)
\(312\) 0 0
\(313\) −965.713 −0.174394 −0.0871970 0.996191i \(-0.527791\pi\)
−0.0871970 + 0.996191i \(0.527791\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −8985.97 −1.59212 −0.796061 0.605217i \(-0.793086\pi\)
−0.796061 + 0.605217i \(0.793086\pi\)
\(318\) 0 0
\(319\) 3123.22 0.548171
\(320\) 0 0
\(321\) 3787.10 0.658491
\(322\) 0 0
\(323\) −1207.14 −0.207947
\(324\) 0 0
\(325\) −1113.52 −0.190053
\(326\) 0 0
\(327\) 830.499 0.140449
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3807.30 0.632230 0.316115 0.948721i \(-0.397621\pi\)
0.316115 + 0.948721i \(0.397621\pi\)
\(332\) 0 0
\(333\) −2835.72 −0.466656
\(334\) 0 0
\(335\) −5966.21 −0.973041
\(336\) 0 0
\(337\) −1649.82 −0.266681 −0.133340 0.991070i \(-0.542570\pi\)
−0.133340 + 0.991070i \(0.542570\pi\)
\(338\) 0 0
\(339\) 158.747 0.0254334
\(340\) 0 0
\(341\) 16355.7 2.59739
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 6606.12 1.03090
\(346\) 0 0
\(347\) −5709.99 −0.883368 −0.441684 0.897171i \(-0.645619\pi\)
−0.441684 + 0.897171i \(0.645619\pi\)
\(348\) 0 0
\(349\) 447.244 0.0685973 0.0342986 0.999412i \(-0.489080\pi\)
0.0342986 + 0.999412i \(0.489080\pi\)
\(350\) 0 0
\(351\) −1005.24 −0.152865
\(352\) 0 0
\(353\) 10645.7 1.60514 0.802569 0.596560i \(-0.203466\pi\)
0.802569 + 0.596560i \(0.203466\pi\)
\(354\) 0 0
\(355\) 13761.0 2.05734
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9097.47 1.33745 0.668727 0.743508i \(-0.266839\pi\)
0.668727 + 0.743508i \(0.266839\pi\)
\(360\) 0 0
\(361\) −3906.31 −0.569516
\(362\) 0 0
\(363\) 3847.85 0.556363
\(364\) 0 0
\(365\) 2979.01 0.427201
\(366\) 0 0
\(367\) 5287.83 0.752104 0.376052 0.926598i \(-0.377281\pi\)
0.376052 + 0.926598i \(0.377281\pi\)
\(368\) 0 0
\(369\) 1854.29 0.261600
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 5895.01 0.818317 0.409159 0.912463i \(-0.365822\pi\)
0.409159 + 0.912463i \(0.365822\pi\)
\(374\) 0 0
\(375\) 3550.59 0.488938
\(376\) 0 0
\(377\) −2274.51 −0.310724
\(378\) 0 0
\(379\) −3842.41 −0.520769 −0.260384 0.965505i \(-0.583849\pi\)
−0.260384 + 0.965505i \(0.583849\pi\)
\(380\) 0 0
\(381\) −1331.10 −0.178988
\(382\) 0 0
\(383\) 5013.74 0.668903 0.334452 0.942413i \(-0.391449\pi\)
0.334452 + 0.942413i \(0.391449\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3056.95 −0.401534
\(388\) 0 0
\(389\) 11182.5 1.45752 0.728758 0.684771i \(-0.240098\pi\)
0.728758 + 0.684771i \(0.240098\pi\)
\(390\) 0 0
\(391\) 3930.40 0.508360
\(392\) 0 0
\(393\) −6460.45 −0.829228
\(394\) 0 0
\(395\) 14445.9 1.84013
\(396\) 0 0
\(397\) 3406.80 0.430686 0.215343 0.976539i \(-0.430913\pi\)
0.215343 + 0.976539i \(0.430913\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 83.0401 0.0103412 0.00517061 0.999987i \(-0.498354\pi\)
0.00517061 + 0.999987i \(0.498354\pi\)
\(402\) 0 0
\(403\) −11911.1 −1.47230
\(404\) 0 0
\(405\) −1008.14 −0.123692
\(406\) 0 0
\(407\) −16108.0 −1.96178
\(408\) 0 0
\(409\) −2456.20 −0.296947 −0.148473 0.988916i \(-0.547436\pi\)
−0.148473 + 0.988916i \(0.547436\pi\)
\(410\) 0 0
\(411\) 789.323 0.0947309
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 36.4871 0.00431586
\(416\) 0 0
\(417\) 3497.30 0.410704
\(418\) 0 0
\(419\) −3437.96 −0.400848 −0.200424 0.979709i \(-0.564232\pi\)
−0.200424 + 0.979709i \(0.564232\pi\)
\(420\) 0 0
\(421\) −5347.62 −0.619067 −0.309533 0.950889i \(-0.600173\pi\)
−0.309533 + 0.950889i \(0.600173\pi\)
\(422\) 0 0
\(423\) 1278.58 0.146966
\(424\) 0 0
\(425\) −664.419 −0.0758331
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −5710.16 −0.642632
\(430\) 0 0
\(431\) −851.643 −0.0951791 −0.0475895 0.998867i \(-0.515154\pi\)
−0.0475895 + 0.998867i \(0.515154\pi\)
\(432\) 0 0
\(433\) 3433.42 0.381061 0.190531 0.981681i \(-0.438979\pi\)
0.190531 + 0.981681i \(0.438979\pi\)
\(434\) 0 0
\(435\) −2281.08 −0.251424
\(436\) 0 0
\(437\) −9613.84 −1.05238
\(438\) 0 0
\(439\) −9738.39 −1.05874 −0.529371 0.848390i \(-0.677572\pi\)
−0.529371 + 0.848390i \(0.677572\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −9934.96 −1.06552 −0.532759 0.846267i \(-0.678845\pi\)
−0.532759 + 0.846267i \(0.678845\pi\)
\(444\) 0 0
\(445\) −15915.5 −1.69543
\(446\) 0 0
\(447\) −5295.58 −0.560341
\(448\) 0 0
\(449\) −7557.33 −0.794327 −0.397163 0.917748i \(-0.630005\pi\)
−0.397163 + 0.917748i \(0.630005\pi\)
\(450\) 0 0
\(451\) 10533.1 1.09974
\(452\) 0 0
\(453\) −8076.94 −0.837721
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 14011.8 1.43424 0.717118 0.696951i \(-0.245461\pi\)
0.717118 + 0.696951i \(0.245461\pi\)
\(458\) 0 0
\(459\) −599.808 −0.0609949
\(460\) 0 0
\(461\) 1669.61 0.168680 0.0843399 0.996437i \(-0.473122\pi\)
0.0843399 + 0.996437i \(0.473122\pi\)
\(462\) 0 0
\(463\) −14785.4 −1.48409 −0.742046 0.670349i \(-0.766144\pi\)
−0.742046 + 0.670349i \(0.766144\pi\)
\(464\) 0 0
\(465\) −11945.6 −1.19132
\(466\) 0 0
\(467\) 4603.17 0.456122 0.228061 0.973647i \(-0.426761\pi\)
0.228061 + 0.973647i \(0.426761\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 5824.14 0.569772
\(472\) 0 0
\(473\) −17364.7 −1.68801
\(474\) 0 0
\(475\) 1625.18 0.156987
\(476\) 0 0
\(477\) 2790.14 0.267824
\(478\) 0 0
\(479\) 3477.35 0.331700 0.165850 0.986151i \(-0.446963\pi\)
0.165850 + 0.986151i \(0.446963\pi\)
\(480\) 0 0
\(481\) 11730.8 1.11201
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 983.993 0.0921254
\(486\) 0 0
\(487\) −4344.17 −0.404216 −0.202108 0.979363i \(-0.564779\pi\)
−0.202108 + 0.979363i \(0.564779\pi\)
\(488\) 0 0
\(489\) −6308.51 −0.583396
\(490\) 0 0
\(491\) −4982.89 −0.457993 −0.228997 0.973427i \(-0.573544\pi\)
−0.228997 + 0.973427i \(0.573544\pi\)
\(492\) 0 0
\(493\) −1357.16 −0.123982
\(494\) 0 0
\(495\) −5726.66 −0.519988
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −15326.2 −1.37494 −0.687468 0.726215i \(-0.741278\pi\)
−0.687468 + 0.726215i \(0.741278\pi\)
\(500\) 0 0
\(501\) −7032.65 −0.627137
\(502\) 0 0
\(503\) −1516.04 −0.134387 −0.0671936 0.997740i \(-0.521405\pi\)
−0.0671936 + 0.997740i \(0.521405\pi\)
\(504\) 0 0
\(505\) 17077.5 1.50483
\(506\) 0 0
\(507\) −2432.53 −0.213082
\(508\) 0 0
\(509\) −2653.64 −0.231081 −0.115541 0.993303i \(-0.536860\pi\)
−0.115541 + 0.993303i \(0.536860\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1467.14 0.126269
\(514\) 0 0
\(515\) −3217.73 −0.275321
\(516\) 0 0
\(517\) 7262.82 0.617830
\(518\) 0 0
\(519\) −10411.9 −0.880604
\(520\) 0 0
\(521\) 13132.1 1.10428 0.552139 0.833752i \(-0.313812\pi\)
0.552139 + 0.833752i \(0.313812\pi\)
\(522\) 0 0
\(523\) −3086.34 −0.258042 −0.129021 0.991642i \(-0.541184\pi\)
−0.129021 + 0.991642i \(0.541184\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7107.16 −0.587462
\(528\) 0 0
\(529\) 19135.3 1.57272
\(530\) 0 0
\(531\) −2534.85 −0.207162
\(532\) 0 0
\(533\) −7670.80 −0.623376
\(534\) 0 0
\(535\) −15711.7 −1.26968
\(536\) 0 0
\(537\) 2867.31 0.230416
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 926.095 0.0735969 0.0367985 0.999323i \(-0.488284\pi\)
0.0367985 + 0.999323i \(0.488284\pi\)
\(542\) 0 0
\(543\) −12660.8 −1.00060
\(544\) 0 0
\(545\) −3445.52 −0.270807
\(546\) 0 0
\(547\) −592.871 −0.0463425 −0.0231712 0.999732i \(-0.507376\pi\)
−0.0231712 + 0.999732i \(0.507376\pi\)
\(548\) 0 0
\(549\) 4894.64 0.380507
\(550\) 0 0
\(551\) 3319.63 0.256663
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 11764.7 0.899788
\(556\) 0 0
\(557\) −12245.3 −0.931512 −0.465756 0.884913i \(-0.654217\pi\)
−0.465756 + 0.884913i \(0.654217\pi\)
\(558\) 0 0
\(559\) 12646.0 0.956829
\(560\) 0 0
\(561\) −3407.15 −0.256417
\(562\) 0 0
\(563\) 14594.6 1.09252 0.546261 0.837615i \(-0.316051\pi\)
0.546261 + 0.837615i \(0.316051\pi\)
\(564\) 0 0
\(565\) −658.599 −0.0490398
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 22911.2 1.68803 0.844015 0.536320i \(-0.180186\pi\)
0.844015 + 0.536320i \(0.180186\pi\)
\(570\) 0 0
\(571\) −5904.64 −0.432752 −0.216376 0.976310i \(-0.569424\pi\)
−0.216376 + 0.976310i \(0.569424\pi\)
\(572\) 0 0
\(573\) 10554.1 0.769465
\(574\) 0 0
\(575\) −5291.53 −0.383778
\(576\) 0 0
\(577\) −9513.21 −0.686378 −0.343189 0.939266i \(-0.611507\pi\)
−0.343189 + 0.939266i \(0.611507\pi\)
\(578\) 0 0
\(579\) −15053.0 −1.08045
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 15849.1 1.12591
\(584\) 0 0
\(585\) 4170.48 0.294749
\(586\) 0 0
\(587\) −22790.6 −1.60250 −0.801252 0.598327i \(-0.795833\pi\)
−0.801252 + 0.598327i \(0.795833\pi\)
\(588\) 0 0
\(589\) 17384.3 1.21614
\(590\) 0 0
\(591\) −8514.41 −0.592616
\(592\) 0 0
\(593\) 18262.8 1.26469 0.632346 0.774686i \(-0.282092\pi\)
0.632346 + 0.774686i \(0.282092\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1064.70 0.0729907
\(598\) 0 0
\(599\) −6958.09 −0.474624 −0.237312 0.971433i \(-0.576266\pi\)
−0.237312 + 0.971433i \(0.576266\pi\)
\(600\) 0 0
\(601\) −2305.39 −0.156471 −0.0782353 0.996935i \(-0.524929\pi\)
−0.0782353 + 0.996935i \(0.524929\pi\)
\(602\) 0 0
\(603\) 4314.23 0.291358
\(604\) 0 0
\(605\) −15963.7 −1.07276
\(606\) 0 0
\(607\) −16179.2 −1.08187 −0.540935 0.841064i \(-0.681929\pi\)
−0.540935 + 0.841064i \(0.681929\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5289.20 −0.350210
\(612\) 0 0
\(613\) 20540.7 1.35340 0.676699 0.736260i \(-0.263410\pi\)
0.676699 + 0.736260i \(0.263410\pi\)
\(614\) 0 0
\(615\) −7692.96 −0.504407
\(616\) 0 0
\(617\) 6918.19 0.451403 0.225702 0.974196i \(-0.427533\pi\)
0.225702 + 0.974196i \(0.427533\pi\)
\(618\) 0 0
\(619\) −8081.62 −0.524762 −0.262381 0.964964i \(-0.584508\pi\)
−0.262381 + 0.964964i \(0.584508\pi\)
\(620\) 0 0
\(621\) −4776.96 −0.308684
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −18469.0 −1.18202
\(626\) 0 0
\(627\) 8333.96 0.530823
\(628\) 0 0
\(629\) 6999.54 0.443704
\(630\) 0 0
\(631\) 27293.3 1.72191 0.860957 0.508677i \(-0.169865\pi\)
0.860957 + 0.508677i \(0.169865\pi\)
\(632\) 0 0
\(633\) −2258.02 −0.141782
\(634\) 0 0
\(635\) 5522.39 0.345117
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −9950.70 −0.616031
\(640\) 0 0
\(641\) 19255.6 1.18651 0.593254 0.805015i \(-0.297843\pi\)
0.593254 + 0.805015i \(0.297843\pi\)
\(642\) 0 0
\(643\) −19996.4 −1.22641 −0.613204 0.789925i \(-0.710120\pi\)
−0.613204 + 0.789925i \(0.710120\pi\)
\(644\) 0 0
\(645\) 12682.5 0.774222
\(646\) 0 0
\(647\) 7064.20 0.429246 0.214623 0.976697i \(-0.431148\pi\)
0.214623 + 0.976697i \(0.431148\pi\)
\(648\) 0 0
\(649\) −14398.9 −0.870890
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −13821.1 −0.828269 −0.414134 0.910216i \(-0.635916\pi\)
−0.414134 + 0.910216i \(0.635916\pi\)
\(654\) 0 0
\(655\) 26802.7 1.59889
\(656\) 0 0
\(657\) −2154.15 −0.127917
\(658\) 0 0
\(659\) 7802.80 0.461235 0.230617 0.973044i \(-0.425925\pi\)
0.230617 + 0.973044i \(0.425925\pi\)
\(660\) 0 0
\(661\) −15817.3 −0.930744 −0.465372 0.885115i \(-0.654079\pi\)
−0.465372 + 0.885115i \(0.654079\pi\)
\(662\) 0 0
\(663\) 2481.28 0.145347
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −10808.6 −0.627452
\(668\) 0 0
\(669\) −9233.25 −0.533599
\(670\) 0 0
\(671\) 27803.5 1.59962
\(672\) 0 0
\(673\) 2943.30 0.168582 0.0842911 0.996441i \(-0.473137\pi\)
0.0842911 + 0.996441i \(0.473137\pi\)
\(674\) 0 0
\(675\) 807.528 0.0460471
\(676\) 0 0
\(677\) 3171.48 0.180044 0.0900220 0.995940i \(-0.471306\pi\)
0.0900220 + 0.995940i \(0.471306\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −18652.3 −1.04957
\(682\) 0 0
\(683\) 24453.2 1.36995 0.684975 0.728567i \(-0.259813\pi\)
0.684975 + 0.728567i \(0.259813\pi\)
\(684\) 0 0
\(685\) −3274.70 −0.182656
\(686\) 0 0
\(687\) −1509.76 −0.0838443
\(688\) 0 0
\(689\) −11542.2 −0.638207
\(690\) 0 0
\(691\) 8595.89 0.473232 0.236616 0.971603i \(-0.423962\pi\)
0.236616 + 0.971603i \(0.423962\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −14509.4 −0.791903
\(696\) 0 0
\(697\) −4577.02 −0.248733
\(698\) 0 0
\(699\) −805.019 −0.0435602
\(700\) 0 0
\(701\) −21476.1 −1.15712 −0.578561 0.815639i \(-0.696385\pi\)
−0.578561 + 0.815639i \(0.696385\pi\)
\(702\) 0 0
\(703\) −17121.0 −0.918537
\(704\) 0 0
\(705\) −5304.48 −0.283373
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −13538.9 −0.717158 −0.358579 0.933499i \(-0.616739\pi\)
−0.358579 + 0.933499i \(0.616739\pi\)
\(710\) 0 0
\(711\) −10446.0 −0.550992
\(712\) 0 0
\(713\) −56602.5 −2.97304
\(714\) 0 0
\(715\) 23690.0 1.23910
\(716\) 0 0
\(717\) 15569.3 0.810943
\(718\) 0 0
\(719\) 13883.7 0.720131 0.360065 0.932927i \(-0.382754\pi\)
0.360065 + 0.932927i \(0.382754\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −18512.8 −0.952281
\(724\) 0 0
\(725\) 1827.15 0.0935983
\(726\) 0 0
\(727\) −18292.9 −0.933215 −0.466607 0.884465i \(-0.654524\pi\)
−0.466607 + 0.884465i \(0.654524\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 7545.61 0.381785
\(732\) 0 0
\(733\) −14245.8 −0.717848 −0.358924 0.933367i \(-0.616856\pi\)
−0.358924 + 0.933367i \(0.616856\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 24506.5 1.22484
\(738\) 0 0
\(739\) 1363.89 0.0678913 0.0339456 0.999424i \(-0.489193\pi\)
0.0339456 + 0.999424i \(0.489193\pi\)
\(740\) 0 0
\(741\) −6069.27 −0.300891
\(742\) 0 0
\(743\) −21789.4 −1.07588 −0.537938 0.842984i \(-0.680797\pi\)
−0.537938 + 0.842984i \(0.680797\pi\)
\(744\) 0 0
\(745\) 21970.0 1.08043
\(746\) 0 0
\(747\) −26.3842 −0.00129230
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2119.55 0.102987 0.0514937 0.998673i \(-0.483602\pi\)
0.0514937 + 0.998673i \(0.483602\pi\)
\(752\) 0 0
\(753\) 5675.74 0.274682
\(754\) 0 0
\(755\) 33509.1 1.61526
\(756\) 0 0
\(757\) 28202.4 1.35408 0.677038 0.735948i \(-0.263263\pi\)
0.677038 + 0.735948i \(0.263263\pi\)
\(758\) 0 0
\(759\) −27135.0 −1.29768
\(760\) 0 0
\(761\) −11147.0 −0.530985 −0.265493 0.964113i \(-0.585535\pi\)
−0.265493 + 0.964113i \(0.585535\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2488.45 0.117608
\(766\) 0 0
\(767\) 10486.1 0.493654
\(768\) 0 0
\(769\) 4109.29 0.192698 0.0963491 0.995348i \(-0.469283\pi\)
0.0963491 + 0.995348i \(0.469283\pi\)
\(770\) 0 0
\(771\) 19619.8 0.916459
\(772\) 0 0
\(773\) −13891.4 −0.646362 −0.323181 0.946337i \(-0.604752\pi\)
−0.323181 + 0.946337i \(0.604752\pi\)
\(774\) 0 0
\(775\) 9568.44 0.443495
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 11195.5 0.514917
\(780\) 0 0
\(781\) −56523.9 −2.58974
\(782\) 0 0
\(783\) 1649.47 0.0752839
\(784\) 0 0
\(785\) −24162.9 −1.09861
\(786\) 0 0
\(787\) −9342.35 −0.423150 −0.211575 0.977362i \(-0.567859\pi\)
−0.211575 + 0.977362i \(0.567859\pi\)
\(788\) 0 0
\(789\) −16127.7 −0.727708
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −20248.1 −0.906723
\(794\) 0 0
\(795\) −11575.6 −0.516407
\(796\) 0 0
\(797\) 24324.3 1.08107 0.540534 0.841322i \(-0.318222\pi\)
0.540534 + 0.841322i \(0.318222\pi\)
\(798\) 0 0
\(799\) −3155.97 −0.139737
\(800\) 0 0
\(801\) 11508.7 0.507664
\(802\) 0 0
\(803\) −12236.4 −0.537751
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 9716.65 0.423844
\(808\) 0 0
\(809\) 34454.9 1.49737 0.748684 0.662927i \(-0.230686\pi\)
0.748684 + 0.662927i \(0.230686\pi\)
\(810\) 0 0
\(811\) −8350.13 −0.361545 −0.180772 0.983525i \(-0.557860\pi\)
−0.180772 + 0.983525i \(0.557860\pi\)
\(812\) 0 0
\(813\) 4072.37 0.175676
\(814\) 0 0
\(815\) 26172.4 1.12488
\(816\) 0 0
\(817\) −18456.7 −0.790355
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −18834.8 −0.800655 −0.400327 0.916372i \(-0.631104\pi\)
−0.400327 + 0.916372i \(0.631104\pi\)
\(822\) 0 0
\(823\) −9656.76 −0.409008 −0.204504 0.978866i \(-0.565558\pi\)
−0.204504 + 0.978866i \(0.565558\pi\)
\(824\) 0 0
\(825\) 4587.08 0.193578
\(826\) 0 0
\(827\) 20759.6 0.872892 0.436446 0.899731i \(-0.356237\pi\)
0.436446 + 0.899731i \(0.356237\pi\)
\(828\) 0 0
\(829\) 15616.1 0.654245 0.327123 0.944982i \(-0.393921\pi\)
0.327123 + 0.944982i \(0.393921\pi\)
\(830\) 0 0
\(831\) −7684.90 −0.320802
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 29176.6 1.20922
\(836\) 0 0
\(837\) 8637.96 0.356716
\(838\) 0 0
\(839\) −417.027 −0.0171601 −0.00858007 0.999963i \(-0.502731\pi\)
−0.00858007 + 0.999963i \(0.502731\pi\)
\(840\) 0 0
\(841\) −20656.8 −0.846973
\(842\) 0 0
\(843\) 5358.52 0.218929
\(844\) 0 0
\(845\) 10092.0 0.410856
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −22164.8 −0.895990
\(850\) 0 0
\(851\) 55745.4 2.24551
\(852\) 0 0
\(853\) 24917.4 1.00018 0.500092 0.865972i \(-0.333300\pi\)
0.500092 + 0.865972i \(0.333300\pi\)
\(854\) 0 0
\(855\) −6086.80 −0.243467
\(856\) 0 0
\(857\) 44523.8 1.77468 0.887342 0.461112i \(-0.152549\pi\)
0.887342 + 0.461112i \(0.152549\pi\)
\(858\) 0 0
\(859\) −24146.4 −0.959097 −0.479548 0.877515i \(-0.659200\pi\)
−0.479548 + 0.877515i \(0.659200\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 26442.6 1.04301 0.521505 0.853248i \(-0.325371\pi\)
0.521505 + 0.853248i \(0.325371\pi\)
\(864\) 0 0
\(865\) 43196.5 1.69795
\(866\) 0 0
\(867\) −13258.5 −0.519355
\(868\) 0 0
\(869\) −59337.4 −2.31632
\(870\) 0 0
\(871\) −17847.1 −0.694288
\(872\) 0 0
\(873\) −711.536 −0.0275852
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −22516.1 −0.866950 −0.433475 0.901166i \(-0.642713\pi\)
−0.433475 + 0.901166i \(0.642713\pi\)
\(878\) 0 0
\(879\) −1478.94 −0.0567503
\(880\) 0 0
\(881\) −10120.6 −0.387027 −0.193514 0.981098i \(-0.561988\pi\)
−0.193514 + 0.981098i \(0.561988\pi\)
\(882\) 0 0
\(883\) 20748.5 0.790761 0.395380 0.918517i \(-0.370613\pi\)
0.395380 + 0.918517i \(0.370613\pi\)
\(884\) 0 0
\(885\) 10516.4 0.399442
\(886\) 0 0
\(887\) 25499.4 0.965259 0.482630 0.875825i \(-0.339682\pi\)
0.482630 + 0.875825i \(0.339682\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4141.01 0.155700
\(892\) 0 0
\(893\) 7719.57 0.289278
\(894\) 0 0
\(895\) −11895.7 −0.444280
\(896\) 0 0
\(897\) 19761.3 0.735574
\(898\) 0 0
\(899\) 19544.7 0.725085
\(900\) 0 0
\(901\) −6887.04 −0.254651
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 52526.3 1.92932
\(906\) 0 0
\(907\) −37807.0 −1.38408 −0.692040 0.721859i \(-0.743288\pi\)
−0.692040 + 0.721859i \(0.743288\pi\)
\(908\) 0 0
\(909\) −12348.9 −0.450593
\(910\) 0 0
\(911\) 3230.08 0.117472 0.0587362 0.998274i \(-0.481293\pi\)
0.0587362 + 0.998274i \(0.481293\pi\)
\(912\) 0 0
\(913\) −149.873 −0.00543271
\(914\) 0 0
\(915\) −20306.6 −0.733678
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 35343.1 1.26862 0.634310 0.773079i \(-0.281284\pi\)
0.634310 + 0.773079i \(0.281284\pi\)
\(920\) 0 0
\(921\) −2966.43 −0.106132
\(922\) 0 0
\(923\) 41164.0 1.46796
\(924\) 0 0
\(925\) −9423.55 −0.334967
\(926\) 0 0
\(927\) 2326.78 0.0824394
\(928\) 0 0
\(929\) −33030.2 −1.16651 −0.583254 0.812289i \(-0.698221\pi\)
−0.583254 + 0.812289i \(0.698221\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −28789.7 −1.01022
\(934\) 0 0
\(935\) 14135.4 0.494413
\(936\) 0 0
\(937\) −54695.9 −1.90698 −0.953488 0.301430i \(-0.902536\pi\)
−0.953488 + 0.301430i \(0.902536\pi\)
\(938\) 0 0
\(939\) −2897.14 −0.100686
\(940\) 0 0
\(941\) 18255.3 0.632418 0.316209 0.948690i \(-0.397590\pi\)
0.316209 + 0.948690i \(0.397590\pi\)
\(942\) 0 0
\(943\) −36452.1 −1.25879
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −6215.75 −0.213289 −0.106645 0.994297i \(-0.534011\pi\)
−0.106645 + 0.994297i \(0.534011\pi\)
\(948\) 0 0
\(949\) 8911.27 0.304818
\(950\) 0 0
\(951\) −26957.9 −0.919212
\(952\) 0 0
\(953\) 8594.53 0.292135 0.146067 0.989275i \(-0.453338\pi\)
0.146067 + 0.989275i \(0.453338\pi\)
\(954\) 0 0
\(955\) −43786.2 −1.48365
\(956\) 0 0
\(957\) 9369.65 0.316487
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 72560.6 2.43566
\(962\) 0 0
\(963\) 11361.3 0.380180
\(964\) 0 0
\(965\) 62451.1 2.08329
\(966\) 0 0
\(967\) 17168.1 0.570929 0.285464 0.958389i \(-0.407852\pi\)
0.285464 + 0.958389i \(0.407852\pi\)
\(968\) 0 0
\(969\) −3621.42 −0.120059
\(970\) 0 0
\(971\) 11926.2 0.394162 0.197081 0.980387i \(-0.436854\pi\)
0.197081 + 0.980387i \(0.436854\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −3340.57 −0.109727
\(976\) 0 0
\(977\) 14038.2 0.459696 0.229848 0.973227i \(-0.426177\pi\)
0.229848 + 0.973227i \(0.426177\pi\)
\(978\) 0 0
\(979\) 65373.8 2.13417
\(980\) 0 0
\(981\) 2491.50 0.0810880
\(982\) 0 0
\(983\) 20351.4 0.660334 0.330167 0.943923i \(-0.392895\pi\)
0.330167 + 0.943923i \(0.392895\pi\)
\(984\) 0 0
\(985\) 35324.1 1.14266
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 60094.4 1.93214
\(990\) 0 0
\(991\) 32505.7 1.04196 0.520978 0.853570i \(-0.325567\pi\)
0.520978 + 0.853570i \(0.325567\pi\)
\(992\) 0 0
\(993\) 11421.9 0.365018
\(994\) 0 0
\(995\) −4417.18 −0.140738
\(996\) 0 0
\(997\) 24847.1 0.789282 0.394641 0.918835i \(-0.370869\pi\)
0.394641 + 0.918835i \(0.370869\pi\)
\(998\) 0 0
\(999\) −8507.16 −0.269424
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 2352.4.a.bx.1.1 2
4.3 odd 2 588.4.a.f.1.1 2
7.3 odd 6 336.4.q.i.289.1 4
7.5 odd 6 336.4.q.i.193.1 4
7.6 odd 2 2352.4.a.bt.1.2 2
12.11 even 2 1764.4.a.y.1.2 2
28.3 even 6 84.4.i.a.37.1 yes 4
28.11 odd 6 588.4.i.j.373.2 4
28.19 even 6 84.4.i.a.25.1 4
28.23 odd 6 588.4.i.j.361.2 4
28.27 even 2 588.4.a.i.1.2 2
84.11 even 6 1764.4.k.q.1549.1 4
84.23 even 6 1764.4.k.q.361.1 4
84.47 odd 6 252.4.k.f.109.2 4
84.59 odd 6 252.4.k.f.37.2 4
84.83 odd 2 1764.4.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.i.a.25.1 4 28.19 even 6
84.4.i.a.37.1 yes 4 28.3 even 6
252.4.k.f.37.2 4 84.59 odd 6
252.4.k.f.109.2 4 84.47 odd 6
336.4.q.i.193.1 4 7.5 odd 6
336.4.q.i.289.1 4 7.3 odd 6
588.4.a.f.1.1 2 4.3 odd 2
588.4.a.i.1.2 2 28.27 even 2
588.4.i.j.361.2 4 28.23 odd 6
588.4.i.j.373.2 4 28.11 odd 6
1764.4.a.o.1.1 2 84.83 odd 2
1764.4.a.y.1.2 2 12.11 even 2
1764.4.k.q.361.1 4 84.23 even 6
1764.4.k.q.1549.1 4 84.11 even 6
2352.4.a.bt.1.2 2 7.6 odd 2
2352.4.a.bx.1.1 2 1.1 even 1 trivial