Properties

Label 588.4.a.f.1.2
Level $588$
Weight $4$
Character 588.1
Self dual yes
Analytic conductor $34.693$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.6931230834\)
Analytic rank: \(0\)
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 1.2
Root \(-6.44622\) of defining polynomial
Character \(\chi\) \(=\) 588.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} +1.44622 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +1.44622 q^{5} +9.00000 q^{9} +46.1236 q^{11} +32.2311 q^{13} -4.33867 q^{15} -77.7849 q^{17} -12.6613 q^{19} -100.924 q^{23} -122.908 q^{25} -27.0000 q^{27} +213.908 q^{29} -42.0756 q^{31} -138.371 q^{33} +310.080 q^{37} -96.6933 q^{39} -44.0320 q^{41} +381.339 q^{43} +13.0160 q^{45} +358.064 q^{47} +233.355 q^{51} +184.984 q^{53} +66.7049 q^{55} +37.9840 q^{57} -454.650 q^{59} -11.8489 q^{61} +46.6133 q^{65} +590.359 q^{67} +302.773 q^{69} +494.366 q^{71} -975.650 q^{73} +368.725 q^{75} +299.334 q^{79} +81.0000 q^{81} +1406.07 q^{83} -112.494 q^{85} -641.725 q^{87} +695.259 q^{89} +126.227 q^{93} -18.3111 q^{95} -481.940 q^{97} +415.112 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 11 q^{5} + 18 q^{9} + O(q^{10}) \) \( 2 q - 6 q^{3} - 11 q^{5} + 18 q^{9} - 5 q^{11} - 5 q^{13} + 33 q^{15} - 100 q^{17} - 67 q^{19} + 76 q^{23} - 93 q^{25} - 54 q^{27} + 275 q^{29} - 362 q^{31} + 15 q^{33} - 5 q^{37} + 15 q^{39} + 162 q^{41} + 721 q^{43} - 99 q^{45} + 216 q^{47} + 300 q^{51} + 495 q^{53} + 703 q^{55} + 201 q^{57} - 173 q^{59} + 532 q^{61} + 510 q^{65} + 111 q^{67} - 228 q^{69} + 1600 q^{71} - 1215 q^{73} + 279 q^{75} + 1460 q^{79} + 162 q^{81} + 1409 q^{83} + 164 q^{85} - 825 q^{87} + 1974 q^{89} + 1086 q^{93} + 658 q^{95} - 561 q^{97} - 45 q^{99} + O(q^{100}) \)

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
\(5\) 1.44622 0.129354 0.0646770 0.997906i \(-0.479398\pi\)
0.0646770 + 0.997906i \(0.479398\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 46.1236 1.26425 0.632126 0.774866i \(-0.282182\pi\)
0.632126 + 0.774866i \(0.282182\pi\)
\(12\) 0 0
\(13\) 32.2311 0.687639 0.343819 0.939036i \(-0.388279\pi\)
0.343819 + 0.939036i \(0.388279\pi\)
\(14\) 0 0
\(15\) −4.33867 −0.0746826
\(16\) 0 0
\(17\) −77.7849 −1.10974 −0.554871 0.831937i \(-0.687232\pi\)
−0.554871 + 0.831937i \(0.687232\pi\)
\(18\) 0 0
\(19\) −12.6613 −0.152879 −0.0764397 0.997074i \(-0.524355\pi\)
−0.0764397 + 0.997074i \(0.524355\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −100.924 −0.914965 −0.457483 0.889219i \(-0.651249\pi\)
−0.457483 + 0.889219i \(0.651249\pi\)
\(24\) 0 0
\(25\) −122.908 −0.983268
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 213.908 1.36972 0.684859 0.728676i \(-0.259864\pi\)
0.684859 + 0.728676i \(0.259864\pi\)
\(30\) 0 0
\(31\) −42.0756 −0.243774 −0.121887 0.992544i \(-0.538895\pi\)
−0.121887 + 0.992544i \(0.538895\pi\)
\(32\) 0 0
\(33\) −138.371 −0.729916
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 310.080 1.37775 0.688876 0.724879i \(-0.258104\pi\)
0.688876 + 0.724879i \(0.258104\pi\)
\(38\) 0 0
\(39\) −96.6933 −0.397008
\(40\) 0 0
\(41\) −44.0320 −0.167723 −0.0838615 0.996477i \(-0.526725\pi\)
−0.0838615 + 0.996477i \(0.526725\pi\)
\(42\) 0 0
\(43\) 381.339 1.35241 0.676205 0.736714i \(-0.263624\pi\)
0.676205 + 0.736714i \(0.263624\pi\)
\(44\) 0 0
\(45\) 13.0160 0.0431180
\(46\) 0 0
\(47\) 358.064 1.11126 0.555628 0.831431i \(-0.312478\pi\)
0.555628 + 0.831431i \(0.312478\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 233.355 0.640710
\(52\) 0 0
\(53\) 184.984 0.479425 0.239712 0.970844i \(-0.422947\pi\)
0.239712 + 0.970844i \(0.422947\pi\)
\(54\) 0 0
\(55\) 66.7049 0.163536
\(56\) 0 0
\(57\) 37.9840 0.0882650
\(58\) 0 0
\(59\) −454.650 −1.00323 −0.501613 0.865092i \(-0.667260\pi\)
−0.501613 + 0.865092i \(0.667260\pi\)
\(60\) 0 0
\(61\) −11.8489 −0.0248704 −0.0124352 0.999923i \(-0.503958\pi\)
−0.0124352 + 0.999923i \(0.503958\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 46.6133 0.0889488
\(66\) 0 0
\(67\) 590.359 1.07648 0.538238 0.842793i \(-0.319090\pi\)
0.538238 + 0.842793i \(0.319090\pi\)
\(68\) 0 0
\(69\) 302.773 0.528255
\(70\) 0 0
\(71\) 494.366 0.826345 0.413172 0.910653i \(-0.364421\pi\)
0.413172 + 0.910653i \(0.364421\pi\)
\(72\) 0 0
\(73\) −975.650 −1.56426 −0.782131 0.623114i \(-0.785867\pi\)
−0.782131 + 0.623114i \(0.785867\pi\)
\(74\) 0 0
\(75\) 368.725 0.567690
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 299.334 0.426300 0.213150 0.977019i \(-0.431628\pi\)
0.213150 + 0.977019i \(0.431628\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1406.07 1.85947 0.929735 0.368229i \(-0.120036\pi\)
0.929735 + 0.368229i \(0.120036\pi\)
\(84\) 0 0
\(85\) −112.494 −0.143550
\(86\) 0 0
\(87\) −641.725 −0.790807
\(88\) 0 0
\(89\) 695.259 0.828059 0.414030 0.910263i \(-0.364121\pi\)
0.414030 + 0.910263i \(0.364121\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 126.227 0.140743
\(94\) 0 0
\(95\) −18.3111 −0.0197756
\(96\) 0 0
\(97\) −481.940 −0.504470 −0.252235 0.967666i \(-0.581166\pi\)
−0.252235 + 0.967666i \(0.581166\pi\)
\(98\) 0 0
\(99\) 415.112 0.421417
\(100\) 0 0
\(101\) 1184.10 1.16656 0.583281 0.812270i \(-0.301769\pi\)
0.583281 + 0.812270i \(0.301769\pi\)
\(102\) 0 0
\(103\) 1283.53 1.22786 0.613932 0.789359i \(-0.289587\pi\)
0.613932 + 0.789359i \(0.289587\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1613.37 1.45767 0.728833 0.684692i \(-0.240063\pi\)
0.728833 + 0.684692i \(0.240063\pi\)
\(108\) 0 0
\(109\) −153.833 −0.135179 −0.0675895 0.997713i \(-0.521531\pi\)
−0.0675895 + 0.997713i \(0.521531\pi\)
\(110\) 0 0
\(111\) −930.240 −0.795446
\(112\) 0 0
\(113\) 1581.08 1.31625 0.658123 0.752910i \(-0.271350\pi\)
0.658123 + 0.752910i \(0.271350\pi\)
\(114\) 0 0
\(115\) −145.959 −0.118354
\(116\) 0 0
\(117\) 290.080 0.229213
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 796.382 0.598334
\(122\) 0 0
\(123\) 132.096 0.0968349
\(124\) 0 0
\(125\) −358.531 −0.256544
\(126\) 0 0
\(127\) 1916.30 1.33893 0.669465 0.742844i \(-0.266523\pi\)
0.669465 + 0.742844i \(0.266523\pi\)
\(128\) 0 0
\(129\) −1144.02 −0.780814
\(130\) 0 0
\(131\) −2500.48 −1.66770 −0.833849 0.551993i \(-0.813867\pi\)
−0.833849 + 0.551993i \(0.813867\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −39.0480 −0.0248942
\(136\) 0 0
\(137\) 290.892 0.181406 0.0907030 0.995878i \(-0.471089\pi\)
0.0907030 + 0.995878i \(0.471089\pi\)
\(138\) 0 0
\(139\) 1348.77 0.823028 0.411514 0.911403i \(-0.365000\pi\)
0.411514 + 0.911403i \(0.365000\pi\)
\(140\) 0 0
\(141\) −1074.19 −0.641584
\(142\) 0 0
\(143\) 1486.61 0.869349
\(144\) 0 0
\(145\) 309.359 0.177178
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2848.81 −1.56633 −0.783165 0.621814i \(-0.786396\pi\)
−0.783165 + 0.621814i \(0.786396\pi\)
\(150\) 0 0
\(151\) −1489.31 −0.802639 −0.401320 0.915938i \(-0.631448\pi\)
−0.401320 + 0.915938i \(0.631448\pi\)
\(152\) 0 0
\(153\) −700.064 −0.369914
\(154\) 0 0
\(155\) −60.8506 −0.0315331
\(156\) 0 0
\(157\) −3643.38 −1.85206 −0.926030 0.377449i \(-0.876801\pi\)
−0.926030 + 0.377449i \(0.876801\pi\)
\(158\) 0 0
\(159\) −554.952 −0.276796
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 769.163 0.369604 0.184802 0.982776i \(-0.440836\pi\)
0.184802 + 0.982776i \(0.440836\pi\)
\(164\) 0 0
\(165\) −200.115 −0.0944176
\(166\) 0 0
\(167\) 2399.78 1.11198 0.555991 0.831188i \(-0.312339\pi\)
0.555991 + 0.831188i \(0.312339\pi\)
\(168\) 0 0
\(169\) −1158.16 −0.527153
\(170\) 0 0
\(171\) −113.952 −0.0509598
\(172\) 0 0
\(173\) 3336.65 1.46636 0.733181 0.680033i \(-0.238035\pi\)
0.733181 + 0.680033i \(0.238035\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1363.95 0.579213
\(178\) 0 0
\(179\) 2461.77 1.02794 0.513970 0.857808i \(-0.328174\pi\)
0.513970 + 0.857808i \(0.328174\pi\)
\(180\) 0 0
\(181\) −1316.74 −0.540732 −0.270366 0.962758i \(-0.587145\pi\)
−0.270366 + 0.962758i \(0.587145\pi\)
\(182\) 0 0
\(183\) 35.5466 0.0143589
\(184\) 0 0
\(185\) 448.445 0.178218
\(186\) 0 0
\(187\) −3587.72 −1.40299
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3267.97 −1.23802 −0.619010 0.785383i \(-0.712466\pi\)
−0.619010 + 0.785383i \(0.712466\pi\)
\(192\) 0 0
\(193\) 233.672 0.0871506 0.0435753 0.999050i \(-0.486125\pi\)
0.0435753 + 0.999050i \(0.486125\pi\)
\(194\) 0 0
\(195\) −139.840 −0.0513546
\(196\) 0 0
\(197\) −31.8632 −0.0115236 −0.00576182 0.999983i \(-0.501834\pi\)
−0.00576182 + 0.999983i \(0.501834\pi\)
\(198\) 0 0
\(199\) 1478.90 0.526817 0.263408 0.964684i \(-0.415153\pi\)
0.263408 + 0.964684i \(0.415153\pi\)
\(200\) 0 0
\(201\) −1771.08 −0.621503
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −63.6800 −0.0216956
\(206\) 0 0
\(207\) −908.320 −0.304988
\(208\) 0 0
\(209\) −583.986 −0.193278
\(210\) 0 0
\(211\) −4498.67 −1.46778 −0.733889 0.679269i \(-0.762297\pi\)
−0.733889 + 0.679269i \(0.762297\pi\)
\(212\) 0 0
\(213\) −1483.10 −0.477090
\(214\) 0 0
\(215\) 551.500 0.174940
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2926.95 0.903127
\(220\) 0 0
\(221\) −2507.09 −0.763101
\(222\) 0 0
\(223\) −5382.75 −1.61639 −0.808196 0.588913i \(-0.799556\pi\)
−0.808196 + 0.588913i \(0.799556\pi\)
\(224\) 0 0
\(225\) −1106.18 −0.327756
\(226\) 0 0
\(227\) 5425.57 1.58638 0.793188 0.608977i \(-0.208420\pi\)
0.793188 + 0.608977i \(0.208420\pi\)
\(228\) 0 0
\(229\) −1989.75 −0.574175 −0.287088 0.957904i \(-0.592687\pi\)
−0.287088 + 0.957904i \(0.592687\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6297.66 −1.77070 −0.885351 0.464924i \(-0.846082\pi\)
−0.885351 + 0.464924i \(0.846082\pi\)
\(234\) 0 0
\(235\) 517.840 0.143745
\(236\) 0 0
\(237\) −898.003 −0.246125
\(238\) 0 0
\(239\) 3395.77 0.919054 0.459527 0.888164i \(-0.348019\pi\)
0.459527 + 0.888164i \(0.348019\pi\)
\(240\) 0 0
\(241\) 6373.94 1.70366 0.851829 0.523820i \(-0.175494\pi\)
0.851829 + 0.523820i \(0.175494\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −408.089 −0.105126
\(248\) 0 0
\(249\) −4218.21 −1.07357
\(250\) 0 0
\(251\) 2650.91 0.666630 0.333315 0.942815i \(-0.391833\pi\)
0.333315 + 0.942815i \(0.391833\pi\)
\(252\) 0 0
\(253\) −4654.99 −1.15675
\(254\) 0 0
\(255\) 337.483 0.0828784
\(256\) 0 0
\(257\) 1872.07 0.454383 0.227192 0.973850i \(-0.427046\pi\)
0.227192 + 0.973850i \(0.427046\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1925.18 0.456572
\(262\) 0 0
\(263\) 6126.10 1.43632 0.718158 0.695880i \(-0.244985\pi\)
0.718158 + 0.695880i \(0.244985\pi\)
\(264\) 0 0
\(265\) 267.528 0.0620155
\(266\) 0 0
\(267\) −2085.78 −0.478080
\(268\) 0 0
\(269\) 8504.12 1.92753 0.963764 0.266755i \(-0.0859515\pi\)
0.963764 + 0.266755i \(0.0859515\pi\)
\(270\) 0 0
\(271\) −2121.54 −0.475552 −0.237776 0.971320i \(-0.576418\pi\)
−0.237776 + 0.971320i \(0.576418\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5668.97 −1.24310
\(276\) 0 0
\(277\) −8021.37 −1.73992 −0.869959 0.493125i \(-0.835855\pi\)
−0.869959 + 0.493125i \(0.835855\pi\)
\(278\) 0 0
\(279\) −378.680 −0.0812580
\(280\) 0 0
\(281\) −8244.17 −1.75020 −0.875100 0.483943i \(-0.839204\pi\)
−0.875100 + 0.483943i \(0.839204\pi\)
\(282\) 0 0
\(283\) −6101.28 −1.28157 −0.640784 0.767722i \(-0.721390\pi\)
−0.640784 + 0.767722i \(0.721390\pi\)
\(284\) 0 0
\(285\) 54.9333 0.0114174
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1137.49 0.231526
\(290\) 0 0
\(291\) 1445.82 0.291256
\(292\) 0 0
\(293\) 1965.98 0.391993 0.195996 0.980605i \(-0.437206\pi\)
0.195996 + 0.980605i \(0.437206\pi\)
\(294\) 0 0
\(295\) −657.524 −0.129771
\(296\) 0 0
\(297\) −1245.34 −0.243305
\(298\) 0 0
\(299\) −3252.91 −0.629165
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −3552.31 −0.673515
\(304\) 0 0
\(305\) −17.1361 −0.00321708
\(306\) 0 0
\(307\) −997.810 −0.185498 −0.0927492 0.995690i \(-0.529565\pi\)
−0.0927492 + 0.995690i \(0.529565\pi\)
\(308\) 0 0
\(309\) −3850.59 −0.708908
\(310\) 0 0
\(311\) 6901.43 1.25834 0.629171 0.777267i \(-0.283395\pi\)
0.629171 + 0.777267i \(0.283395\pi\)
\(312\) 0 0
\(313\) 6341.71 1.14522 0.572612 0.819827i \(-0.305930\pi\)
0.572612 + 0.819827i \(0.305930\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1224.97 0.217039 0.108519 0.994094i \(-0.465389\pi\)
0.108519 + 0.994094i \(0.465389\pi\)
\(318\) 0 0
\(319\) 9866.22 1.73167
\(320\) 0 0
\(321\) −4840.10 −0.841583
\(322\) 0 0
\(323\) 984.860 0.169657
\(324\) 0 0
\(325\) −3961.48 −0.676133
\(326\) 0 0
\(327\) 461.499 0.0780457
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3736.30 0.620440 0.310220 0.950665i \(-0.399597\pi\)
0.310220 + 0.950665i \(0.399597\pi\)
\(332\) 0 0
\(333\) 2790.72 0.459251
\(334\) 0 0
\(335\) 853.790 0.139246
\(336\) 0 0
\(337\) −3928.18 −0.634960 −0.317480 0.948265i \(-0.602837\pi\)
−0.317480 + 0.948265i \(0.602837\pi\)
\(338\) 0 0
\(339\) −4743.25 −0.759936
\(340\) 0 0
\(341\) −1940.67 −0.308192
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 437.877 0.0683320
\(346\) 0 0
\(347\) 542.005 0.0838512 0.0419256 0.999121i \(-0.486651\pi\)
0.0419256 + 0.999121i \(0.486651\pi\)
\(348\) 0 0
\(349\) −2331.24 −0.357561 −0.178780 0.983889i \(-0.557215\pi\)
−0.178780 + 0.983889i \(0.557215\pi\)
\(350\) 0 0
\(351\) −870.240 −0.132336
\(352\) 0 0
\(353\) 2560.30 0.386037 0.193018 0.981195i \(-0.438172\pi\)
0.193018 + 0.981195i \(0.438172\pi\)
\(354\) 0 0
\(355\) 714.963 0.106891
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4568.53 −0.671638 −0.335819 0.941927i \(-0.609013\pi\)
−0.335819 + 0.941927i \(0.609013\pi\)
\(360\) 0 0
\(361\) −6698.69 −0.976628
\(362\) 0 0
\(363\) −2389.15 −0.345448
\(364\) 0 0
\(365\) −1411.01 −0.202344
\(366\) 0 0
\(367\) 6381.83 0.907707 0.453854 0.891076i \(-0.350049\pi\)
0.453854 + 0.891076i \(0.350049\pi\)
\(368\) 0 0
\(369\) −396.288 −0.0559077
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6372.01 −0.884532 −0.442266 0.896884i \(-0.645825\pi\)
−0.442266 + 0.896884i \(0.645825\pi\)
\(374\) 0 0
\(375\) 1075.59 0.148116
\(376\) 0 0
\(377\) 6894.51 0.941870
\(378\) 0 0
\(379\) 1494.59 0.202564 0.101282 0.994858i \(-0.467706\pi\)
0.101282 + 0.994858i \(0.467706\pi\)
\(380\) 0 0
\(381\) −5748.90 −0.773031
\(382\) 0 0
\(383\) −8820.26 −1.17675 −0.588374 0.808589i \(-0.700232\pi\)
−0.588374 + 0.808589i \(0.700232\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3432.05 0.450803
\(388\) 0 0
\(389\) −9878.47 −1.28755 −0.643777 0.765213i \(-0.722634\pi\)
−0.643777 + 0.765213i \(0.722634\pi\)
\(390\) 0 0
\(391\) 7850.40 1.01537
\(392\) 0 0
\(393\) 7501.45 0.962845
\(394\) 0 0
\(395\) 432.904 0.0551437
\(396\) 0 0
\(397\) −1941.80 −0.245481 −0.122740 0.992439i \(-0.539168\pi\)
−0.122740 + 0.992439i \(0.539168\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7584.96 0.944576 0.472288 0.881444i \(-0.343428\pi\)
0.472288 + 0.881444i \(0.343428\pi\)
\(402\) 0 0
\(403\) −1356.14 −0.167628
\(404\) 0 0
\(405\) 117.144 0.0143727
\(406\) 0 0
\(407\) 14302.0 1.74183
\(408\) 0 0
\(409\) −8707.80 −1.05275 −0.526373 0.850254i \(-0.676448\pi\)
−0.526373 + 0.850254i \(0.676448\pi\)
\(410\) 0 0
\(411\) −872.677 −0.104735
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 2033.49 0.240530
\(416\) 0 0
\(417\) −4046.30 −0.475175
\(418\) 0 0
\(419\) −6647.96 −0.775117 −0.387558 0.921845i \(-0.626681\pi\)
−0.387558 + 0.921845i \(0.626681\pi\)
\(420\) 0 0
\(421\) 11670.6 1.35105 0.675524 0.737338i \(-0.263917\pi\)
0.675524 + 0.737338i \(0.263917\pi\)
\(422\) 0 0
\(423\) 3222.58 0.370418
\(424\) 0 0
\(425\) 9560.42 1.09117
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −4459.84 −0.501919
\(430\) 0 0
\(431\) −1565.64 −0.174975 −0.0874876 0.996166i \(-0.527884\pi\)
−0.0874876 + 0.996166i \(0.527884\pi\)
\(432\) 0 0
\(433\) −15446.4 −1.71434 −0.857168 0.515037i \(-0.827778\pi\)
−0.857168 + 0.515037i \(0.827778\pi\)
\(434\) 0 0
\(435\) −928.077 −0.102294
\(436\) 0 0
\(437\) 1277.84 0.139879
\(438\) 0 0
\(439\) −10697.4 −1.16300 −0.581502 0.813545i \(-0.697535\pi\)
−0.581502 + 0.813545i \(0.697535\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1053.96 −0.113037 −0.0565183 0.998402i \(-0.518000\pi\)
−0.0565183 + 0.998402i \(0.518000\pi\)
\(444\) 0 0
\(445\) 1005.50 0.107113
\(446\) 0 0
\(447\) 8546.42 0.904321
\(448\) 0 0
\(449\) 1139.33 0.119752 0.0598759 0.998206i \(-0.480930\pi\)
0.0598759 + 0.998206i \(0.480930\pi\)
\(450\) 0 0
\(451\) −2030.91 −0.212044
\(452\) 0 0
\(453\) 4467.94 0.463404
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5704.16 0.583872 0.291936 0.956438i \(-0.405701\pi\)
0.291936 + 0.956438i \(0.405701\pi\)
\(458\) 0 0
\(459\) 2100.19 0.213570
\(460\) 0 0
\(461\) 6476.39 0.654307 0.327154 0.944971i \(-0.393911\pi\)
0.327154 + 0.944971i \(0.393911\pi\)
\(462\) 0 0
\(463\) −232.366 −0.0233239 −0.0116619 0.999932i \(-0.503712\pi\)
−0.0116619 + 0.999932i \(0.503712\pi\)
\(464\) 0 0
\(465\) 182.552 0.0182057
\(466\) 0 0
\(467\) 1037.17 0.102772 0.0513858 0.998679i \(-0.483636\pi\)
0.0513858 + 0.998679i \(0.483636\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 10930.1 1.06929
\(472\) 0 0
\(473\) 17588.7 1.70979
\(474\) 0 0
\(475\) 1556.18 0.150321
\(476\) 0 0
\(477\) 1664.86 0.159808
\(478\) 0 0
\(479\) −10284.6 −0.981039 −0.490519 0.871430i \(-0.663193\pi\)
−0.490519 + 0.871430i \(0.663193\pi\)
\(480\) 0 0
\(481\) 9994.22 0.947396
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −696.993 −0.0652553
\(486\) 0 0
\(487\) 10845.8 1.00918 0.504591 0.863359i \(-0.331643\pi\)
0.504591 + 0.863359i \(0.331643\pi\)
\(488\) 0 0
\(489\) −2307.49 −0.213391
\(490\) 0 0
\(491\) 8442.11 0.775941 0.387971 0.921672i \(-0.373176\pi\)
0.387971 + 0.921672i \(0.373176\pi\)
\(492\) 0 0
\(493\) −16638.8 −1.52003
\(494\) 0 0
\(495\) 600.344 0.0545120
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 9810.85 0.880148 0.440074 0.897961i \(-0.354952\pi\)
0.440074 + 0.897961i \(0.354952\pi\)
\(500\) 0 0
\(501\) −7199.35 −0.642003
\(502\) 0 0
\(503\) 6433.96 0.570330 0.285165 0.958478i \(-0.407952\pi\)
0.285165 + 0.958478i \(0.407952\pi\)
\(504\) 0 0
\(505\) 1712.48 0.150900
\(506\) 0 0
\(507\) 3474.47 0.304352
\(508\) 0 0
\(509\) 20560.6 1.79044 0.895220 0.445624i \(-0.147018\pi\)
0.895220 + 0.445624i \(0.147018\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 341.856 0.0294217
\(514\) 0 0
\(515\) 1856.27 0.158829
\(516\) 0 0
\(517\) 16515.2 1.40491
\(518\) 0 0
\(519\) −10009.9 −0.846605
\(520\) 0 0
\(521\) −19126.1 −1.60831 −0.804156 0.594418i \(-0.797383\pi\)
−0.804156 + 0.594418i \(0.797383\pi\)
\(522\) 0 0
\(523\) 3044.66 0.254558 0.127279 0.991867i \(-0.459376\pi\)
0.127279 + 0.991867i \(0.459376\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3272.84 0.270526
\(528\) 0 0
\(529\) −1981.26 −0.162839
\(530\) 0 0
\(531\) −4091.85 −0.334409
\(532\) 0 0
\(533\) −1419.20 −0.115333
\(534\) 0 0
\(535\) 2333.29 0.188555
\(536\) 0 0
\(537\) −7385.31 −0.593482
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5895.09 −0.468484 −0.234242 0.972178i \(-0.575261\pi\)
−0.234242 + 0.972178i \(0.575261\pi\)
\(542\) 0 0
\(543\) 3950.22 0.312192
\(544\) 0 0
\(545\) −222.476 −0.0174860
\(546\) 0 0
\(547\) 11151.1 0.871641 0.435821 0.900034i \(-0.356458\pi\)
0.435821 + 0.900034i \(0.356458\pi\)
\(548\) 0 0
\(549\) −106.640 −0.00829013
\(550\) 0 0
\(551\) −2708.37 −0.209402
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1345.33 −0.102894
\(556\) 0 0
\(557\) 1744.35 0.132694 0.0663468 0.997797i \(-0.478866\pi\)
0.0663468 + 0.997797i \(0.478866\pi\)
\(558\) 0 0
\(559\) 12291.0 0.929969
\(560\) 0 0
\(561\) 10763.1 0.810019
\(562\) 0 0
\(563\) −11524.4 −0.862691 −0.431345 0.902187i \(-0.641961\pi\)
−0.431345 + 0.902187i \(0.641961\pi\)
\(564\) 0 0
\(565\) 2286.60 0.170262
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2822.76 0.207973 0.103986 0.994579i \(-0.466840\pi\)
0.103986 + 0.994579i \(0.466840\pi\)
\(570\) 0 0
\(571\) −5167.64 −0.378737 −0.189369 0.981906i \(-0.560644\pi\)
−0.189369 + 0.981906i \(0.560644\pi\)
\(572\) 0 0
\(573\) 9803.90 0.714771
\(574\) 0 0
\(575\) 12404.5 0.899656
\(576\) 0 0
\(577\) 14715.2 1.06170 0.530851 0.847465i \(-0.321872\pi\)
0.530851 + 0.847465i \(0.321872\pi\)
\(578\) 0 0
\(579\) −701.016 −0.0503164
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 8532.12 0.606114
\(584\) 0 0
\(585\) 419.520 0.0296496
\(586\) 0 0
\(587\) −9981.64 −0.701851 −0.350925 0.936403i \(-0.614133\pi\)
−0.350925 + 0.936403i \(0.614133\pi\)
\(588\) 0 0
\(589\) 532.733 0.0372680
\(590\) 0 0
\(591\) 95.5895 0.00665317
\(592\) 0 0
\(593\) 1675.21 0.116008 0.0580039 0.998316i \(-0.481526\pi\)
0.0580039 + 0.998316i \(0.481526\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −4436.70 −0.304158
\(598\) 0 0
\(599\) 1039.91 0.0709341 0.0354671 0.999371i \(-0.488708\pi\)
0.0354671 + 0.999371i \(0.488708\pi\)
\(600\) 0 0
\(601\) −4472.61 −0.303563 −0.151782 0.988414i \(-0.548501\pi\)
−0.151782 + 0.988414i \(0.548501\pi\)
\(602\) 0 0
\(603\) 5313.23 0.358825
\(604\) 0 0
\(605\) 1151.75 0.0773969
\(606\) 0 0
\(607\) 17790.8 1.18963 0.594814 0.803863i \(-0.297226\pi\)
0.594814 + 0.803863i \(0.297226\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 11540.8 0.764142
\(612\) 0 0
\(613\) −4326.74 −0.285082 −0.142541 0.989789i \(-0.545527\pi\)
−0.142541 + 0.989789i \(0.545527\pi\)
\(614\) 0 0
\(615\) 191.040 0.0125260
\(616\) 0 0
\(617\) −18866.2 −1.23100 −0.615498 0.788139i \(-0.711045\pi\)
−0.615498 + 0.788139i \(0.711045\pi\)
\(618\) 0 0
\(619\) 10179.4 0.660975 0.330488 0.943810i \(-0.392787\pi\)
0.330488 + 0.943810i \(0.392787\pi\)
\(620\) 0 0
\(621\) 2724.96 0.176085
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 14845.0 0.950083
\(626\) 0 0
\(627\) 1751.96 0.111589
\(628\) 0 0
\(629\) −24119.5 −1.52895
\(630\) 0 0
\(631\) −1661.72 −0.104837 −0.0524184 0.998625i \(-0.516693\pi\)
−0.0524184 + 0.998625i \(0.516693\pi\)
\(632\) 0 0
\(633\) 13496.0 0.847423
\(634\) 0 0
\(635\) 2771.39 0.173196
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 4449.30 0.275448
\(640\) 0 0
\(641\) −2277.64 −0.140346 −0.0701728 0.997535i \(-0.522355\pi\)
−0.0701728 + 0.997535i \(0.522355\pi\)
\(642\) 0 0
\(643\) −1217.38 −0.0746638 −0.0373319 0.999303i \(-0.511886\pi\)
−0.0373319 + 0.999303i \(0.511886\pi\)
\(644\) 0 0
\(645\) −1654.50 −0.101001
\(646\) 0 0
\(647\) −10731.8 −0.652103 −0.326052 0.945352i \(-0.605718\pi\)
−0.326052 + 0.945352i \(0.605718\pi\)
\(648\) 0 0
\(649\) −20970.1 −1.26833
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 15228.1 0.912588 0.456294 0.889829i \(-0.349177\pi\)
0.456294 + 0.889829i \(0.349177\pi\)
\(654\) 0 0
\(655\) −3616.26 −0.215723
\(656\) 0 0
\(657\) −8780.85 −0.521421
\(658\) 0 0
\(659\) 10590.8 0.626038 0.313019 0.949747i \(-0.398660\pi\)
0.313019 + 0.949747i \(0.398660\pi\)
\(660\) 0 0
\(661\) 3868.30 0.227624 0.113812 0.993502i \(-0.463694\pi\)
0.113812 + 0.993502i \(0.463694\pi\)
\(662\) 0 0
\(663\) 7521.28 0.440577
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −21588.6 −1.25324
\(668\) 0 0
\(669\) 16148.2 0.933225
\(670\) 0 0
\(671\) −546.512 −0.0314424
\(672\) 0 0
\(673\) 11028.7 0.631687 0.315843 0.948811i \(-0.397713\pi\)
0.315843 + 0.948811i \(0.397713\pi\)
\(674\) 0 0
\(675\) 3318.53 0.189230
\(676\) 0 0
\(677\) −18792.5 −1.06684 −0.533422 0.845849i \(-0.679094\pi\)
−0.533422 + 0.845849i \(0.679094\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −16276.7 −0.915895
\(682\) 0 0
\(683\) −20243.8 −1.13412 −0.567062 0.823675i \(-0.691920\pi\)
−0.567062 + 0.823675i \(0.691920\pi\)
\(684\) 0 0
\(685\) 420.695 0.0234656
\(686\) 0 0
\(687\) 5969.24 0.331500
\(688\) 0 0
\(689\) 5962.24 0.329671
\(690\) 0 0
\(691\) 19202.9 1.05718 0.528591 0.848877i \(-0.322721\pi\)
0.528591 + 0.848877i \(0.322721\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1950.62 0.106462
\(696\) 0 0
\(697\) 3425.02 0.186129
\(698\) 0 0
\(699\) 18893.0 1.02231
\(700\) 0 0
\(701\) −22156.9 −1.19380 −0.596900 0.802316i \(-0.703601\pi\)
−0.596900 + 0.802316i \(0.703601\pi\)
\(702\) 0 0
\(703\) −3926.03 −0.210630
\(704\) 0 0
\(705\) −1553.52 −0.0829914
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −27209.1 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(710\) 0 0
\(711\) 2694.01 0.142100
\(712\) 0 0
\(713\) 4246.45 0.223045
\(714\) 0 0
\(715\) 2149.97 0.112454
\(716\) 0 0
\(717\) −10187.3 −0.530616
\(718\) 0 0
\(719\) −21552.3 −1.11789 −0.558947 0.829203i \(-0.688794\pi\)
−0.558947 + 0.829203i \(0.688794\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −19121.8 −0.983607
\(724\) 0 0
\(725\) −26291.2 −1.34680
\(726\) 0 0
\(727\) 20599.1 1.05086 0.525431 0.850836i \(-0.323904\pi\)
0.525431 + 0.850836i \(0.323904\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −29662.4 −1.50082
\(732\) 0 0
\(733\) −19761.2 −0.995764 −0.497882 0.867245i \(-0.665889\pi\)
−0.497882 + 0.867245i \(0.665889\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 27229.5 1.36094
\(738\) 0 0
\(739\) 33408.9 1.66301 0.831506 0.555516i \(-0.187479\pi\)
0.831506 + 0.555516i \(0.187479\pi\)
\(740\) 0 0
\(741\) 1224.27 0.0606944
\(742\) 0 0
\(743\) −36225.4 −1.78867 −0.894335 0.447398i \(-0.852351\pi\)
−0.894335 + 0.447398i \(0.852351\pi\)
\(744\) 0 0
\(745\) −4120.00 −0.202611
\(746\) 0 0
\(747\) 12654.6 0.619824
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −27320.4 −1.32748 −0.663740 0.747964i \(-0.731032\pi\)
−0.663740 + 0.747964i \(0.731032\pi\)
\(752\) 0 0
\(753\) −7952.74 −0.384879
\(754\) 0 0
\(755\) −2153.88 −0.103825
\(756\) 0 0
\(757\) −9918.43 −0.476211 −0.238105 0.971239i \(-0.576526\pi\)
−0.238105 + 0.971239i \(0.576526\pi\)
\(758\) 0 0
\(759\) 13965.0 0.667848
\(760\) 0 0
\(761\) 5635.04 0.268423 0.134211 0.990953i \(-0.457150\pi\)
0.134211 + 0.990953i \(0.457150\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1012.45 −0.0478498
\(766\) 0 0
\(767\) −14653.9 −0.689857
\(768\) 0 0
\(769\) −12089.3 −0.566907 −0.283453 0.958986i \(-0.591480\pi\)
−0.283453 + 0.958986i \(0.591480\pi\)
\(770\) 0 0
\(771\) −5616.21 −0.262338
\(772\) 0 0
\(773\) 14171.4 0.659390 0.329695 0.944087i \(-0.393054\pi\)
0.329695 + 0.944087i \(0.393054\pi\)
\(774\) 0 0
\(775\) 5171.44 0.239695
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 557.504 0.0256414
\(780\) 0 0
\(781\) 22801.9 1.04471
\(782\) 0 0
\(783\) −5775.53 −0.263602
\(784\) 0 0
\(785\) −5269.14 −0.239571
\(786\) 0 0
\(787\) 25707.6 1.16439 0.582197 0.813047i \(-0.302193\pi\)
0.582197 + 0.813047i \(0.302193\pi\)
\(788\) 0 0
\(789\) −18378.3 −0.829258
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −381.903 −0.0171018
\(794\) 0 0
\(795\) −802.584 −0.0358047
\(796\) 0 0
\(797\) 6194.68 0.275316 0.137658 0.990480i \(-0.456043\pi\)
0.137658 + 0.990480i \(0.456043\pi\)
\(798\) 0 0
\(799\) −27852.0 −1.23321
\(800\) 0 0
\(801\) 6257.33 0.276020
\(802\) 0 0
\(803\) −45000.4 −1.97762
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −25512.4 −1.11286
\(808\) 0 0
\(809\) 30815.1 1.33919 0.669593 0.742729i \(-0.266469\pi\)
0.669593 + 0.742729i \(0.266469\pi\)
\(810\) 0 0
\(811\) −43024.1 −1.86286 −0.931431 0.363917i \(-0.881439\pi\)
−0.931431 + 0.363917i \(0.881439\pi\)
\(812\) 0 0
\(813\) 6364.63 0.274560
\(814\) 0 0
\(815\) 1112.38 0.0478098
\(816\) 0 0
\(817\) −4828.26 −0.206756
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −11152.2 −0.474075 −0.237038 0.971500i \(-0.576176\pi\)
−0.237038 + 0.971500i \(0.576176\pi\)
\(822\) 0 0
\(823\) −44912.8 −1.90226 −0.951130 0.308790i \(-0.900076\pi\)
−0.951130 + 0.308790i \(0.900076\pi\)
\(824\) 0 0
\(825\) 17006.9 0.717703
\(826\) 0 0
\(827\) −3213.42 −0.135117 −0.0675584 0.997715i \(-0.521521\pi\)
−0.0675584 + 0.997715i \(0.521521\pi\)
\(828\) 0 0
\(829\) 8794.91 0.368468 0.184234 0.982882i \(-0.441020\pi\)
0.184234 + 0.982882i \(0.441020\pi\)
\(830\) 0 0
\(831\) 24064.1 1.00454
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 3470.62 0.143839
\(836\) 0 0
\(837\) 1136.04 0.0469143
\(838\) 0 0
\(839\) −45817.0 −1.88532 −0.942658 0.333761i \(-0.891682\pi\)
−0.942658 + 0.333761i \(0.891682\pi\)
\(840\) 0 0
\(841\) 21367.8 0.876125
\(842\) 0 0
\(843\) 24732.5 1.01048
\(844\) 0 0
\(845\) −1674.95 −0.0681894
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 18303.8 0.739913
\(850\) 0 0
\(851\) −31294.6 −1.26060
\(852\) 0 0
\(853\) 16373.6 0.657234 0.328617 0.944463i \(-0.393417\pi\)
0.328617 + 0.944463i \(0.393417\pi\)
\(854\) 0 0
\(855\) −164.800 −0.00659186
\(856\) 0 0
\(857\) 4430.20 0.176584 0.0882922 0.996095i \(-0.471859\pi\)
0.0882922 + 0.996095i \(0.471859\pi\)
\(858\) 0 0
\(859\) −4944.39 −0.196392 −0.0981958 0.995167i \(-0.531307\pi\)
−0.0981958 + 0.995167i \(0.531307\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 13178.6 0.519821 0.259911 0.965633i \(-0.416307\pi\)
0.259911 + 0.965633i \(0.416307\pi\)
\(864\) 0 0
\(865\) 4825.53 0.189680
\(866\) 0 0
\(867\) −3412.47 −0.133672
\(868\) 0 0
\(869\) 13806.4 0.538951
\(870\) 0 0
\(871\) 19027.9 0.740226
\(872\) 0 0
\(873\) −4337.46 −0.168157
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −12985.9 −0.500003 −0.250001 0.968245i \(-0.580431\pi\)
−0.250001 + 0.968245i \(0.580431\pi\)
\(878\) 0 0
\(879\) −5897.94 −0.226317
\(880\) 0 0
\(881\) −36877.4 −1.41025 −0.705126 0.709082i \(-0.749110\pi\)
−0.705126 + 0.709082i \(0.749110\pi\)
\(882\) 0 0
\(883\) −24874.5 −0.948012 −0.474006 0.880522i \(-0.657192\pi\)
−0.474006 + 0.880522i \(0.657192\pi\)
\(884\) 0 0
\(885\) 1972.57 0.0749235
\(886\) 0 0
\(887\) −4410.64 −0.166961 −0.0834806 0.996509i \(-0.526604\pi\)
−0.0834806 + 0.996509i \(0.526604\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 3736.01 0.140472
\(892\) 0 0
\(893\) −4533.57 −0.169888
\(894\) 0 0
\(895\) 3560.27 0.132968
\(896\) 0 0
\(897\) 9758.72 0.363249
\(898\) 0 0
\(899\) −9000.32 −0.333901
\(900\) 0 0
\(901\) −14389.0 −0.532037
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1904.30 −0.0699459
\(906\) 0 0
\(907\) −7273.99 −0.266294 −0.133147 0.991096i \(-0.542508\pi\)
−0.133147 + 0.991096i \(0.542508\pi\)
\(908\) 0 0
\(909\) 10656.9 0.388854
\(910\) 0 0
\(911\) −49491.9 −1.79993 −0.899967 0.435957i \(-0.856410\pi\)
−0.899967 + 0.435957i \(0.856410\pi\)
\(912\) 0 0
\(913\) 64852.9 2.35084
\(914\) 0 0
\(915\) 51.4083 0.00185738
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −5015.90 −0.180043 −0.0900213 0.995940i \(-0.528694\pi\)
−0.0900213 + 0.995940i \(0.528694\pi\)
\(920\) 0 0
\(921\) 2993.43 0.107098
\(922\) 0 0
\(923\) 15934.0 0.568227
\(924\) 0 0
\(925\) −38111.4 −1.35470
\(926\) 0 0
\(927\) 11551.8 0.409288
\(928\) 0 0
\(929\) 52158.2 1.84204 0.921021 0.389514i \(-0.127357\pi\)
0.921021 + 0.389514i \(0.127357\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −20704.3 −0.726504
\(934\) 0 0
\(935\) −5188.63 −0.181483
\(936\) 0 0
\(937\) 14821.9 0.516766 0.258383 0.966042i \(-0.416810\pi\)
0.258383 + 0.966042i \(0.416810\pi\)
\(938\) 0 0
\(939\) −19025.1 −0.661195
\(940\) 0 0
\(941\) −26492.3 −0.917772 −0.458886 0.888495i \(-0.651751\pi\)
−0.458886 + 0.888495i \(0.651751\pi\)
\(942\) 0 0
\(943\) 4443.90 0.153461
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6410.25 0.219963 0.109982 0.993934i \(-0.464921\pi\)
0.109982 + 0.993934i \(0.464921\pi\)
\(948\) 0 0
\(949\) −31446.3 −1.07565
\(950\) 0 0
\(951\) −3674.92 −0.125307
\(952\) 0 0
\(953\) −25108.5 −0.853458 −0.426729 0.904380i \(-0.640334\pi\)
−0.426729 + 0.904380i \(0.640334\pi\)
\(954\) 0 0
\(955\) −4726.21 −0.160143
\(956\) 0 0
\(957\) −29598.7 −0.999779
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −28020.6 −0.940574
\(962\) 0 0
\(963\) 14520.3 0.485888
\(964\) 0 0
\(965\) 337.941 0.0112733
\(966\) 0 0
\(967\) 32928.1 1.09503 0.547516 0.836795i \(-0.315574\pi\)
0.547516 + 0.836795i \(0.315574\pi\)
\(968\) 0 0
\(969\) −2954.58 −0.0979513
\(970\) 0 0
\(971\) 31543.2 1.04250 0.521251 0.853403i \(-0.325465\pi\)
0.521251 + 0.853403i \(0.325465\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 11884.4 0.390365
\(976\) 0 0
\(977\) 53881.8 1.76441 0.882206 0.470864i \(-0.156058\pi\)
0.882206 + 0.470864i \(0.156058\pi\)
\(978\) 0 0
\(979\) 32067.8 1.04688
\(980\) 0 0
\(981\) −1384.50 −0.0450597
\(982\) 0 0
\(983\) −35688.6 −1.15798 −0.578988 0.815336i \(-0.696552\pi\)
−0.578988 + 0.815336i \(0.696552\pi\)
\(984\) 0 0
\(985\) −46.0812 −0.00149063
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −38486.4 −1.23741
\(990\) 0 0
\(991\) 14839.7 0.475681 0.237840 0.971304i \(-0.423561\pi\)
0.237840 + 0.971304i \(0.423561\pi\)
\(992\) 0 0
\(993\) −11208.9 −0.358211
\(994\) 0 0
\(995\) 2138.82 0.0681459
\(996\) 0 0
\(997\) −17372.1 −0.551834 −0.275917 0.961181i \(-0.588982\pi\)
−0.275917 + 0.961181i \(0.588982\pi\)
\(998\) 0 0
\(999\) −8372.16 −0.265149
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 588.4.a.f.1.2 2
3.2 odd 2 1764.4.a.y.1.1 2
4.3 odd 2 2352.4.a.bx.1.2 2
7.2 even 3 588.4.i.j.361.1 4
7.3 odd 6 84.4.i.a.37.2 yes 4
7.4 even 3 588.4.i.j.373.1 4
7.5 odd 6 84.4.i.a.25.2 4
7.6 odd 2 588.4.a.i.1.1 2
21.2 odd 6 1764.4.k.q.361.2 4
21.5 even 6 252.4.k.f.109.1 4
21.11 odd 6 1764.4.k.q.1549.2 4
21.17 even 6 252.4.k.f.37.1 4
21.20 even 2 1764.4.a.o.1.2 2
28.3 even 6 336.4.q.i.289.2 4
28.19 even 6 336.4.q.i.193.2 4
28.27 even 2 2352.4.a.bt.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.i.a.25.2 4 7.5 odd 6
84.4.i.a.37.2 yes 4 7.3 odd 6
252.4.k.f.37.1 4 21.17 even 6
252.4.k.f.109.1 4 21.5 even 6
336.4.q.i.193.2 4 28.19 even 6
336.4.q.i.289.2 4 28.3 even 6
588.4.a.f.1.2 2 1.1 even 1 trivial
588.4.a.i.1.1 2 7.6 odd 2
588.4.i.j.361.1 4 7.2 even 3
588.4.i.j.373.1 4 7.4 even 3
1764.4.a.o.1.2 2 21.20 even 2
1764.4.a.y.1.1 2 3.2 odd 2
1764.4.k.q.361.2 4 21.2 odd 6
1764.4.k.q.1549.2 4 21.11 odd 6
2352.4.a.bt.1.1 2 28.27 even 2
2352.4.a.bx.1.2 2 4.3 odd 2