Properties

Label 3864.2.a.t.1.1
Level $3864$
Weight $2$
Character 3864.1
Self dual yes
Analytic conductor $30.854$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.8541953410\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.39605.1
Defining polynomial: \(x^{4} - 2 x^{3} - 8 x^{2} + 9 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.57685\) of defining polynomial
Character \(\chi\) \(=\) 3864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.57685 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.57685 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.81840 q^{11} -3.91214 q^{13} -3.57685 q^{15} -7.21699 q^{17} -0.335294 q^{19} -1.00000 q^{21} -1.00000 q^{23} +7.79384 q^{25} +1.00000 q^{27} +7.11144 q^{29} -1.21699 q^{31} -4.81840 q^{33} +3.57685 q^{35} +1.55915 q^{37} -3.91214 q^{39} -0.0422591 q^{41} +8.12325 q^{43} -3.57685 q^{45} +4.10555 q^{47} +1.00000 q^{49} -7.21699 q^{51} +6.35299 q^{53} +17.2347 q^{55} -0.335294 q^{57} -2.08099 q^{59} +10.9603 q^{61} -1.00000 q^{63} +13.9931 q^{65} +1.19929 q^{67} -1.00000 q^{69} -6.89939 q^{71} -2.73388 q^{73} +7.79384 q^{75} +4.81840 q^{77} +16.5582 q^{79} +1.00000 q^{81} -13.0413 q^{83} +25.8141 q^{85} +7.11144 q^{87} -9.17139 q^{89} +3.91214 q^{91} -1.21699 q^{93} +1.19929 q^{95} -13.7820 q^{97} -4.81840 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 2q^{5} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 2q^{5} - 4q^{7} + 4q^{9} + 2q^{13} - 2q^{15} - 10q^{17} + 4q^{19} - 4q^{21} - 4q^{23} + 4q^{27} + 11q^{29} + 14q^{31} + 2q^{35} + 13q^{37} + 2q^{39} + 7q^{41} + 12q^{43} - 2q^{45} + 15q^{47} + 4q^{49} - 10q^{51} + q^{53} + 31q^{55} + 4q^{57} + 5q^{59} + 3q^{61} - 4q^{63} + 25q^{65} + 5q^{67} - 4q^{69} + 5q^{71} - 6q^{73} + 26q^{79} + 4q^{81} + 2q^{83} + 5q^{85} + 11q^{87} + 7q^{89} - 2q^{91} + 14q^{93} + 5q^{95} - 27q^{97} + O(q^{100}) \)

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
\(5\) −3.57685 −1.59961 −0.799807 0.600257i \(-0.795065\pi\)
−0.799807 + 0.600257i \(0.795065\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.81840 −1.45280 −0.726401 0.687271i \(-0.758809\pi\)
−0.726401 + 0.687271i \(0.758809\pi\)
\(12\) 0 0
\(13\) −3.91214 −1.08503 −0.542516 0.840045i \(-0.682528\pi\)
−0.542516 + 0.840045i \(0.682528\pi\)
\(14\) 0 0
\(15\) −3.57685 −0.923538
\(16\) 0 0
\(17\) −7.21699 −1.75038 −0.875189 0.483782i \(-0.839263\pi\)
−0.875189 + 0.483782i \(0.839263\pi\)
\(18\) 0 0
\(19\) −0.335294 −0.0769217 −0.0384608 0.999260i \(-0.512245\pi\)
−0.0384608 + 0.999260i \(0.512245\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 7.79384 1.55877
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 7.11144 1.32056 0.660280 0.751019i \(-0.270438\pi\)
0.660280 + 0.751019i \(0.270438\pi\)
\(30\) 0 0
\(31\) −1.21699 −0.218578 −0.109289 0.994010i \(-0.534857\pi\)
−0.109289 + 0.994010i \(0.534857\pi\)
\(32\) 0 0
\(33\) −4.81840 −0.838776
\(34\) 0 0
\(35\) 3.57685 0.604598
\(36\) 0 0
\(37\) 1.55915 0.256323 0.128161 0.991753i \(-0.459092\pi\)
0.128161 + 0.991753i \(0.459092\pi\)
\(38\) 0 0
\(39\) −3.91214 −0.626444
\(40\) 0 0
\(41\) −0.0422591 −0.00659976 −0.00329988 0.999995i \(-0.501050\pi\)
−0.00329988 + 0.999995i \(0.501050\pi\)
\(42\) 0 0
\(43\) 8.12325 1.23878 0.619392 0.785082i \(-0.287379\pi\)
0.619392 + 0.785082i \(0.287379\pi\)
\(44\) 0 0
\(45\) −3.57685 −0.533205
\(46\) 0 0
\(47\) 4.10555 0.598857 0.299428 0.954119i \(-0.403204\pi\)
0.299428 + 0.954119i \(0.403204\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −7.21699 −1.01058
\(52\) 0 0
\(53\) 6.35299 0.872650 0.436325 0.899789i \(-0.356280\pi\)
0.436325 + 0.899789i \(0.356280\pi\)
\(54\) 0 0
\(55\) 17.2347 2.32392
\(56\) 0 0
\(57\) −0.335294 −0.0444107
\(58\) 0 0
\(59\) −2.08099 −0.270922 −0.135461 0.990783i \(-0.543252\pi\)
−0.135461 + 0.990783i \(0.543252\pi\)
\(60\) 0 0
\(61\) 10.9603 1.40332 0.701660 0.712512i \(-0.252443\pi\)
0.701660 + 0.712512i \(0.252443\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 13.9931 1.73563
\(66\) 0 0
\(67\) 1.19929 0.146517 0.0732586 0.997313i \(-0.476660\pi\)
0.0732586 + 0.997313i \(0.476660\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −6.89939 −0.818807 −0.409404 0.912353i \(-0.634263\pi\)
−0.409404 + 0.912353i \(0.634263\pi\)
\(72\) 0 0
\(73\) −2.73388 −0.319977 −0.159988 0.987119i \(-0.551146\pi\)
−0.159988 + 0.987119i \(0.551146\pi\)
\(74\) 0 0
\(75\) 7.79384 0.899955
\(76\) 0 0
\(77\) 4.81840 0.549108
\(78\) 0 0
\(79\) 16.5582 1.86294 0.931470 0.363819i \(-0.118527\pi\)
0.931470 + 0.363819i \(0.118527\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.0413 −1.43147 −0.715733 0.698374i \(-0.753907\pi\)
−0.715733 + 0.698374i \(0.753907\pi\)
\(84\) 0 0
\(85\) 25.8141 2.79993
\(86\) 0 0
\(87\) 7.11144 0.762426
\(88\) 0 0
\(89\) −9.17139 −0.972165 −0.486083 0.873913i \(-0.661575\pi\)
−0.486083 + 0.873913i \(0.661575\pi\)
\(90\) 0 0
\(91\) 3.91214 0.410104
\(92\) 0 0
\(93\) −1.21699 −0.126196
\(94\) 0 0
\(95\) 1.19929 0.123045
\(96\) 0 0
\(97\) −13.7820 −1.39935 −0.699676 0.714460i \(-0.746672\pi\)
−0.699676 + 0.714460i \(0.746672\pi\)
\(98\) 0 0
\(99\) −4.81840 −0.484268
\(100\) 0 0
\(101\) 15.7178 1.56398 0.781989 0.623292i \(-0.214205\pi\)
0.781989 + 0.623292i \(0.214205\pi\)
\(102\) 0 0
\(103\) 3.70697 0.365258 0.182629 0.983182i \(-0.441539\pi\)
0.182629 + 0.983182i \(0.441539\pi\)
\(104\) 0 0
\(105\) 3.57685 0.349065
\(106\) 0 0
\(107\) −2.65289 −0.256465 −0.128232 0.991744i \(-0.540930\pi\)
−0.128232 + 0.991744i \(0.540930\pi\)
\(108\) 0 0
\(109\) −6.85713 −0.656794 −0.328397 0.944540i \(-0.606508\pi\)
−0.328397 + 0.944540i \(0.606508\pi\)
\(110\) 0 0
\(111\) 1.55915 0.147988
\(112\) 0 0
\(113\) 8.54306 0.803664 0.401832 0.915713i \(-0.368374\pi\)
0.401832 + 0.915713i \(0.368374\pi\)
\(114\) 0 0
\(115\) 3.57685 0.333543
\(116\) 0 0
\(117\) −3.91214 −0.361678
\(118\) 0 0
\(119\) 7.21699 0.661580
\(120\) 0 0
\(121\) 12.2170 1.11064
\(122\) 0 0
\(123\) −0.0422591 −0.00381037
\(124\) 0 0
\(125\) −9.99313 −0.893813
\(126\) 0 0
\(127\) −15.7718 −1.39952 −0.699761 0.714377i \(-0.746710\pi\)
−0.699761 + 0.714377i \(0.746710\pi\)
\(128\) 0 0
\(129\) 8.12325 0.715212
\(130\) 0 0
\(131\) −12.0987 −1.05707 −0.528534 0.848912i \(-0.677258\pi\)
−0.528534 + 0.848912i \(0.677258\pi\)
\(132\) 0 0
\(133\) 0.335294 0.0290737
\(134\) 0 0
\(135\) −3.57685 −0.307846
\(136\) 0 0
\(137\) −21.7145 −1.85519 −0.927595 0.373586i \(-0.878128\pi\)
−0.927595 + 0.373586i \(0.878128\pi\)
\(138\) 0 0
\(139\) 7.30485 0.619589 0.309795 0.950804i \(-0.399740\pi\)
0.309795 + 0.950804i \(0.399740\pi\)
\(140\) 0 0
\(141\) 4.10555 0.345750
\(142\) 0 0
\(143\) 18.8503 1.57634
\(144\) 0 0
\(145\) −25.4365 −2.11239
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 3.39030 0.277744 0.138872 0.990310i \(-0.455652\pi\)
0.138872 + 0.990310i \(0.455652\pi\)
\(150\) 0 0
\(151\) −18.8469 −1.53374 −0.766871 0.641802i \(-0.778187\pi\)
−0.766871 + 0.641802i \(0.778187\pi\)
\(152\) 0 0
\(153\) −7.21699 −0.583459
\(154\) 0 0
\(155\) 4.35299 0.349641
\(156\) 0 0
\(157\) 13.8243 1.10330 0.551649 0.834076i \(-0.313999\pi\)
0.551649 + 0.834076i \(0.313999\pi\)
\(158\) 0 0
\(159\) 6.35299 0.503825
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 13.7433 1.07646 0.538229 0.842799i \(-0.319094\pi\)
0.538229 + 0.842799i \(0.319094\pi\)
\(164\) 0 0
\(165\) 17.2347 1.34172
\(166\) 0 0
\(167\) −0.335294 −0.0259458 −0.0129729 0.999916i \(-0.504130\pi\)
−0.0129729 + 0.999916i \(0.504130\pi\)
\(168\) 0 0
\(169\) 2.30485 0.177296
\(170\) 0 0
\(171\) −0.335294 −0.0256406
\(172\) 0 0
\(173\) 0.323531 0.0245976 0.0122988 0.999924i \(-0.496085\pi\)
0.0122988 + 0.999924i \(0.496085\pi\)
\(174\) 0 0
\(175\) −7.79384 −0.589159
\(176\) 0 0
\(177\) −2.08099 −0.156417
\(178\) 0 0
\(179\) 5.06824 0.378818 0.189409 0.981898i \(-0.439343\pi\)
0.189409 + 0.981898i \(0.439343\pi\)
\(180\) 0 0
\(181\) −12.5818 −0.935197 −0.467599 0.883941i \(-0.654881\pi\)
−0.467599 + 0.883941i \(0.654881\pi\)
\(182\) 0 0
\(183\) 10.9603 0.810207
\(184\) 0 0
\(185\) −5.57685 −0.410018
\(186\) 0 0
\(187\) 34.7744 2.54295
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 7.06918 0.511508 0.255754 0.966742i \(-0.417676\pi\)
0.255754 + 0.966742i \(0.417676\pi\)
\(192\) 0 0
\(193\) 20.4247 1.47020 0.735101 0.677957i \(-0.237135\pi\)
0.735101 + 0.677957i \(0.237135\pi\)
\(194\) 0 0
\(195\) 13.9931 1.00207
\(196\) 0 0
\(197\) −6.87836 −0.490063 −0.245031 0.969515i \(-0.578798\pi\)
−0.245031 + 0.969515i \(0.578798\pi\)
\(198\) 0 0
\(199\) −18.3182 −1.29854 −0.649272 0.760556i \(-0.724926\pi\)
−0.649272 + 0.760556i \(0.724926\pi\)
\(200\) 0 0
\(201\) 1.19929 0.0845917
\(202\) 0 0
\(203\) −7.11144 −0.499125
\(204\) 0 0
\(205\) 0.151154 0.0105571
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 1.61558 0.111752
\(210\) 0 0
\(211\) 18.1596 1.25016 0.625078 0.780562i \(-0.285067\pi\)
0.625078 + 0.780562i \(0.285067\pi\)
\(212\) 0 0
\(213\) −6.89939 −0.472739
\(214\) 0 0
\(215\) −29.0556 −1.98158
\(216\) 0 0
\(217\) 1.21699 0.0826147
\(218\) 0 0
\(219\) −2.73388 −0.184739
\(220\) 0 0
\(221\) 28.2339 1.89922
\(222\) 0 0
\(223\) 25.5421 1.71042 0.855212 0.518278i \(-0.173427\pi\)
0.855212 + 0.518278i \(0.173427\pi\)
\(224\) 0 0
\(225\) 7.79384 0.519589
\(226\) 0 0
\(227\) 16.7423 1.11123 0.555613 0.831441i \(-0.312484\pi\)
0.555613 + 0.831441i \(0.312484\pi\)
\(228\) 0 0
\(229\) 26.5504 1.75450 0.877249 0.480036i \(-0.159376\pi\)
0.877249 + 0.480036i \(0.159376\pi\)
\(230\) 0 0
\(231\) 4.81840 0.317028
\(232\) 0 0
\(233\) 5.76692 0.377804 0.188902 0.981996i \(-0.439507\pi\)
0.188902 + 0.981996i \(0.439507\pi\)
\(234\) 0 0
\(235\) −14.6849 −0.957940
\(236\) 0 0
\(237\) 16.5582 1.07557
\(238\) 0 0
\(239\) 23.8023 1.53964 0.769822 0.638259i \(-0.220345\pi\)
0.769822 + 0.638259i \(0.220345\pi\)
\(240\) 0 0
\(241\) −12.1031 −0.779632 −0.389816 0.920893i \(-0.627461\pi\)
−0.389816 + 0.920893i \(0.627461\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −3.57685 −0.228516
\(246\) 0 0
\(247\) 1.31172 0.0834625
\(248\) 0 0
\(249\) −13.0413 −0.826457
\(250\) 0 0
\(251\) −7.80993 −0.492958 −0.246479 0.969148i \(-0.579274\pi\)
−0.246479 + 0.969148i \(0.579274\pi\)
\(252\) 0 0
\(253\) 4.81840 0.302930
\(254\) 0 0
\(255\) 25.8141 1.61654
\(256\) 0 0
\(257\) −2.26024 −0.140990 −0.0704948 0.997512i \(-0.522458\pi\)
−0.0704948 + 0.997512i \(0.522458\pi\)
\(258\) 0 0
\(259\) −1.55915 −0.0968810
\(260\) 0 0
\(261\) 7.11144 0.440187
\(262\) 0 0
\(263\) −20.0168 −1.23429 −0.617143 0.786851i \(-0.711710\pi\)
−0.617143 + 0.786851i \(0.711710\pi\)
\(264\) 0 0
\(265\) −22.7237 −1.39590
\(266\) 0 0
\(267\) −9.17139 −0.561280
\(268\) 0 0
\(269\) 30.1730 1.83968 0.919840 0.392294i \(-0.128318\pi\)
0.919840 + 0.392294i \(0.128318\pi\)
\(270\) 0 0
\(271\) 19.5735 1.18901 0.594503 0.804093i \(-0.297349\pi\)
0.594503 + 0.804093i \(0.297349\pi\)
\(272\) 0 0
\(273\) 3.91214 0.236774
\(274\) 0 0
\(275\) −37.5538 −2.26458
\(276\) 0 0
\(277\) −22.1140 −1.32870 −0.664350 0.747422i \(-0.731292\pi\)
−0.664350 + 0.747422i \(0.731292\pi\)
\(278\) 0 0
\(279\) −1.21699 −0.0728593
\(280\) 0 0
\(281\) −21.7424 −1.29704 −0.648520 0.761197i \(-0.724612\pi\)
−0.648520 + 0.761197i \(0.724612\pi\)
\(282\) 0 0
\(283\) 5.59788 0.332760 0.166380 0.986062i \(-0.446792\pi\)
0.166380 + 0.986062i \(0.446792\pi\)
\(284\) 0 0
\(285\) 1.19929 0.0710401
\(286\) 0 0
\(287\) 0.0422591 0.00249447
\(288\) 0 0
\(289\) 35.0850 2.06382
\(290\) 0 0
\(291\) −13.7820 −0.807917
\(292\) 0 0
\(293\) −14.6706 −0.857065 −0.428532 0.903526i \(-0.640969\pi\)
−0.428532 + 0.903526i \(0.640969\pi\)
\(294\) 0 0
\(295\) 7.44339 0.433371
\(296\) 0 0
\(297\) −4.81840 −0.279592
\(298\) 0 0
\(299\) 3.91214 0.226245
\(300\) 0 0
\(301\) −8.12325 −0.468216
\(302\) 0 0
\(303\) 15.7178 0.902964
\(304\) 0 0
\(305\) −39.2033 −2.24477
\(306\) 0 0
\(307\) 9.07604 0.517997 0.258999 0.965878i \(-0.416607\pi\)
0.258999 + 0.965878i \(0.416607\pi\)
\(308\) 0 0
\(309\) 3.70697 0.210882
\(310\) 0 0
\(311\) −4.10221 −0.232615 −0.116308 0.993213i \(-0.537106\pi\)
−0.116308 + 0.993213i \(0.537106\pi\)
\(312\) 0 0
\(313\) 2.09628 0.118489 0.0592444 0.998244i \(-0.481131\pi\)
0.0592444 + 0.998244i \(0.481131\pi\)
\(314\) 0 0
\(315\) 3.57685 0.201533
\(316\) 0 0
\(317\) 10.9063 0.612557 0.306278 0.951942i \(-0.400916\pi\)
0.306278 + 0.951942i \(0.400916\pi\)
\(318\) 0 0
\(319\) −34.2658 −1.91851
\(320\) 0 0
\(321\) −2.65289 −0.148070
\(322\) 0 0
\(323\) 2.41981 0.134642
\(324\) 0 0
\(325\) −30.4906 −1.69131
\(326\) 0 0
\(327\) −6.85713 −0.379200
\(328\) 0 0
\(329\) −4.10555 −0.226347
\(330\) 0 0
\(331\) 29.6368 1.62899 0.814493 0.580173i \(-0.197015\pi\)
0.814493 + 0.580173i \(0.197015\pi\)
\(332\) 0 0
\(333\) 1.55915 0.0854410
\(334\) 0 0
\(335\) −4.28969 −0.234371
\(336\) 0 0
\(337\) −31.0731 −1.69266 −0.846331 0.532658i \(-0.821193\pi\)
−0.846331 + 0.532658i \(0.821193\pi\)
\(338\) 0 0
\(339\) 8.54306 0.463995
\(340\) 0 0
\(341\) 5.86395 0.317551
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 3.57685 0.192571
\(346\) 0 0
\(347\) 13.9019 0.746295 0.373147 0.927772i \(-0.378279\pi\)
0.373147 + 0.927772i \(0.378279\pi\)
\(348\) 0 0
\(349\) −15.8868 −0.850403 −0.425201 0.905099i \(-0.639797\pi\)
−0.425201 + 0.905099i \(0.639797\pi\)
\(350\) 0 0
\(351\) −3.91214 −0.208815
\(352\) 0 0
\(353\) −26.3917 −1.40469 −0.702345 0.711837i \(-0.747864\pi\)
−0.702345 + 0.711837i \(0.747864\pi\)
\(354\) 0 0
\(355\) 24.6781 1.30978
\(356\) 0 0
\(357\) 7.21699 0.381964
\(358\) 0 0
\(359\) 22.1208 1.16749 0.583747 0.811936i \(-0.301586\pi\)
0.583747 + 0.811936i \(0.301586\pi\)
\(360\) 0 0
\(361\) −18.8876 −0.994083
\(362\) 0 0
\(363\) 12.2170 0.641226
\(364\) 0 0
\(365\) 9.77868 0.511840
\(366\) 0 0
\(367\) 2.69756 0.140811 0.0704057 0.997518i \(-0.477571\pi\)
0.0704057 + 0.997518i \(0.477571\pi\)
\(368\) 0 0
\(369\) −0.0422591 −0.00219992
\(370\) 0 0
\(371\) −6.35299 −0.329831
\(372\) 0 0
\(373\) −12.2862 −0.636154 −0.318077 0.948065i \(-0.603037\pi\)
−0.318077 + 0.948065i \(0.603037\pi\)
\(374\) 0 0
\(375\) −9.99313 −0.516043
\(376\) 0 0
\(377\) −27.8209 −1.43285
\(378\) 0 0
\(379\) 19.8198 1.01808 0.509038 0.860744i \(-0.330001\pi\)
0.509038 + 0.860744i \(0.330001\pi\)
\(380\) 0 0
\(381\) −15.7718 −0.808015
\(382\) 0 0
\(383\) 7.92297 0.404845 0.202422 0.979298i \(-0.435119\pi\)
0.202422 + 0.979298i \(0.435119\pi\)
\(384\) 0 0
\(385\) −17.2347 −0.878361
\(386\) 0 0
\(387\) 8.12325 0.412928
\(388\) 0 0
\(389\) −12.2508 −0.621139 −0.310569 0.950551i \(-0.600520\pi\)
−0.310569 + 0.950551i \(0.600520\pi\)
\(390\) 0 0
\(391\) 7.21699 0.364979
\(392\) 0 0
\(393\) −12.0987 −0.610298
\(394\) 0 0
\(395\) −59.2260 −2.97999
\(396\) 0 0
\(397\) −8.19924 −0.411508 −0.205754 0.978604i \(-0.565965\pi\)
−0.205754 + 0.978604i \(0.565965\pi\)
\(398\) 0 0
\(399\) 0.335294 0.0167857
\(400\) 0 0
\(401\) −5.88758 −0.294012 −0.147006 0.989136i \(-0.546964\pi\)
−0.147006 + 0.989136i \(0.546964\pi\)
\(402\) 0 0
\(403\) 4.76104 0.237164
\(404\) 0 0
\(405\) −3.57685 −0.177735
\(406\) 0 0
\(407\) −7.51262 −0.372387
\(408\) 0 0
\(409\) 4.83684 0.239167 0.119583 0.992824i \(-0.461844\pi\)
0.119583 + 0.992824i \(0.461844\pi\)
\(410\) 0 0
\(411\) −21.7145 −1.07109
\(412\) 0 0
\(413\) 2.08099 0.102399
\(414\) 0 0
\(415\) 46.6466 2.28979
\(416\) 0 0
\(417\) 7.30485 0.357720
\(418\) 0 0
\(419\) −3.31920 −0.162154 −0.0810769 0.996708i \(-0.525836\pi\)
−0.0810769 + 0.996708i \(0.525836\pi\)
\(420\) 0 0
\(421\) −29.8209 −1.45338 −0.726692 0.686964i \(-0.758943\pi\)
−0.726692 + 0.686964i \(0.758943\pi\)
\(422\) 0 0
\(423\) 4.10555 0.199619
\(424\) 0 0
\(425\) −56.2481 −2.72843
\(426\) 0 0
\(427\) −10.9603 −0.530405
\(428\) 0 0
\(429\) 18.8503 0.910099
\(430\) 0 0
\(431\) 8.23611 0.396719 0.198360 0.980129i \(-0.436439\pi\)
0.198360 + 0.980129i \(0.436439\pi\)
\(432\) 0 0
\(433\) −17.2101 −0.827066 −0.413533 0.910489i \(-0.635705\pi\)
−0.413533 + 0.910489i \(0.635705\pi\)
\(434\) 0 0
\(435\) −25.4365 −1.21959
\(436\) 0 0
\(437\) 0.335294 0.0160393
\(438\) 0 0
\(439\) 23.8200 1.13687 0.568433 0.822729i \(-0.307550\pi\)
0.568433 + 0.822729i \(0.307550\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 37.7050 1.79142 0.895709 0.444640i \(-0.146669\pi\)
0.895709 + 0.444640i \(0.146669\pi\)
\(444\) 0 0
\(445\) 32.8047 1.55509
\(446\) 0 0
\(447\) 3.39030 0.160356
\(448\) 0 0
\(449\) 33.1552 1.56469 0.782346 0.622843i \(-0.214023\pi\)
0.782346 + 0.622843i \(0.214023\pi\)
\(450\) 0 0
\(451\) 0.203621 0.00958815
\(452\) 0 0
\(453\) −18.8469 −0.885506
\(454\) 0 0
\(455\) −13.9931 −0.656008
\(456\) 0 0
\(457\) −8.92062 −0.417289 −0.208644 0.977992i \(-0.566905\pi\)
−0.208644 + 0.977992i \(0.566905\pi\)
\(458\) 0 0
\(459\) −7.21699 −0.336860
\(460\) 0 0
\(461\) −13.2484 −0.617040 −0.308520 0.951218i \(-0.599834\pi\)
−0.308520 + 0.951218i \(0.599834\pi\)
\(462\) 0 0
\(463\) 12.9357 0.601174 0.300587 0.953754i \(-0.402817\pi\)
0.300587 + 0.953754i \(0.402817\pi\)
\(464\) 0 0
\(465\) 4.35299 0.201865
\(466\) 0 0
\(467\) 0.959348 0.0443933 0.0221967 0.999754i \(-0.492934\pi\)
0.0221967 + 0.999754i \(0.492934\pi\)
\(468\) 0 0
\(469\) −1.19929 −0.0553783
\(470\) 0 0
\(471\) 13.8243 0.636989
\(472\) 0 0
\(473\) −39.1411 −1.79971
\(474\) 0 0
\(475\) −2.61323 −0.119903
\(476\) 0 0
\(477\) 6.35299 0.290883
\(478\) 0 0
\(479\) 29.8621 1.36443 0.682217 0.731150i \(-0.261016\pi\)
0.682217 + 0.731150i \(0.261016\pi\)
\(480\) 0 0
\(481\) −6.09962 −0.278119
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 49.2962 2.23842
\(486\) 0 0
\(487\) 37.3021 1.69032 0.845160 0.534513i \(-0.179505\pi\)
0.845160 + 0.534513i \(0.179505\pi\)
\(488\) 0 0
\(489\) 13.7433 0.621493
\(490\) 0 0
\(491\) 25.2794 1.14084 0.570422 0.821351i \(-0.306780\pi\)
0.570422 + 0.821351i \(0.306780\pi\)
\(492\) 0 0
\(493\) −51.3232 −2.31148
\(494\) 0 0
\(495\) 17.2347 0.774642
\(496\) 0 0
\(497\) 6.89939 0.309480
\(498\) 0 0
\(499\) −25.0944 −1.12338 −0.561689 0.827348i \(-0.689848\pi\)
−0.561689 + 0.827348i \(0.689848\pi\)
\(500\) 0 0
\(501\) −0.335294 −0.0149798
\(502\) 0 0
\(503\) −26.2736 −1.17148 −0.585740 0.810499i \(-0.699196\pi\)
−0.585740 + 0.810499i \(0.699196\pi\)
\(504\) 0 0
\(505\) −56.2202 −2.50176
\(506\) 0 0
\(507\) 2.30485 0.102362
\(508\) 0 0
\(509\) −3.86636 −0.171373 −0.0856866 0.996322i \(-0.527308\pi\)
−0.0856866 + 0.996322i \(0.527308\pi\)
\(510\) 0 0
\(511\) 2.73388 0.120940
\(512\) 0 0
\(513\) −0.335294 −0.0148036
\(514\) 0 0
\(515\) −13.2592 −0.584272
\(516\) 0 0
\(517\) −19.7822 −0.870021
\(518\) 0 0
\(519\) 0.323531 0.0142014
\(520\) 0 0
\(521\) 27.2599 1.19428 0.597138 0.802138i \(-0.296304\pi\)
0.597138 + 0.802138i \(0.296304\pi\)
\(522\) 0 0
\(523\) −11.7634 −0.514377 −0.257189 0.966361i \(-0.582796\pi\)
−0.257189 + 0.966361i \(0.582796\pi\)
\(524\) 0 0
\(525\) −7.79384 −0.340151
\(526\) 0 0
\(527\) 8.78301 0.382594
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −2.08099 −0.0903073
\(532\) 0 0
\(533\) 0.165323 0.00716096
\(534\) 0 0
\(535\) 9.48899 0.410245
\(536\) 0 0
\(537\) 5.06824 0.218711
\(538\) 0 0
\(539\) −4.81840 −0.207543
\(540\) 0 0
\(541\) 39.8833 1.71472 0.857359 0.514720i \(-0.172104\pi\)
0.857359 + 0.514720i \(0.172104\pi\)
\(542\) 0 0
\(543\) −12.5818 −0.539936
\(544\) 0 0
\(545\) 24.5269 1.05062
\(546\) 0 0
\(547\) −12.5504 −0.536615 −0.268307 0.963333i \(-0.586464\pi\)
−0.268307 + 0.963333i \(0.586464\pi\)
\(548\) 0 0
\(549\) 10.9603 0.467773
\(550\) 0 0
\(551\) −2.38442 −0.101580
\(552\) 0 0
\(553\) −16.5582 −0.704125
\(554\) 0 0
\(555\) −5.57685 −0.236724
\(556\) 0 0
\(557\) −37.8951 −1.60567 −0.802833 0.596204i \(-0.796675\pi\)
−0.802833 + 0.596204i \(0.796675\pi\)
\(558\) 0 0
\(559\) −31.7793 −1.34412
\(560\) 0 0
\(561\) 34.7744 1.46817
\(562\) 0 0
\(563\) −18.2498 −0.769139 −0.384570 0.923096i \(-0.625650\pi\)
−0.384570 + 0.923096i \(0.625650\pi\)
\(564\) 0 0
\(565\) −30.5572 −1.28555
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 15.5076 0.650113 0.325057 0.945695i \(-0.394617\pi\)
0.325057 + 0.945695i \(0.394617\pi\)
\(570\) 0 0
\(571\) 32.8773 1.37587 0.687936 0.725771i \(-0.258517\pi\)
0.687936 + 0.725771i \(0.258517\pi\)
\(572\) 0 0
\(573\) 7.06918 0.295319
\(574\) 0 0
\(575\) −7.79384 −0.325026
\(576\) 0 0
\(577\) −24.1384 −1.00489 −0.502446 0.864608i \(-0.667567\pi\)
−0.502446 + 0.864608i \(0.667567\pi\)
\(578\) 0 0
\(579\) 20.4247 0.848822
\(580\) 0 0
\(581\) 13.0413 0.541043
\(582\) 0 0
\(583\) −30.6113 −1.26779
\(584\) 0 0
\(585\) 13.9931 0.578545
\(586\) 0 0
\(587\) 8.53806 0.352404 0.176202 0.984354i \(-0.443619\pi\)
0.176202 + 0.984354i \(0.443619\pi\)
\(588\) 0 0
\(589\) 0.408049 0.0168134
\(590\) 0 0
\(591\) −6.87836 −0.282938
\(592\) 0 0
\(593\) 16.8469 0.691820 0.345910 0.938268i \(-0.387570\pi\)
0.345910 + 0.938268i \(0.387570\pi\)
\(594\) 0 0
\(595\) −25.8141 −1.05827
\(596\) 0 0
\(597\) −18.3182 −0.749715
\(598\) 0 0
\(599\) −28.7000 −1.17265 −0.586326 0.810075i \(-0.699426\pi\)
−0.586326 + 0.810075i \(0.699426\pi\)
\(600\) 0 0
\(601\) 10.6840 0.435810 0.217905 0.975970i \(-0.430078\pi\)
0.217905 + 0.975970i \(0.430078\pi\)
\(602\) 0 0
\(603\) 1.19929 0.0488391
\(604\) 0 0
\(605\) −43.6983 −1.77659
\(606\) 0 0
\(607\) 0.751144 0.0304880 0.0152440 0.999884i \(-0.495147\pi\)
0.0152440 + 0.999884i \(0.495147\pi\)
\(608\) 0 0
\(609\) −7.11144 −0.288170
\(610\) 0 0
\(611\) −16.0615 −0.649779
\(612\) 0 0
\(613\) 47.1618 1.90485 0.952424 0.304777i \(-0.0985820\pi\)
0.952424 + 0.304777i \(0.0985820\pi\)
\(614\) 0 0
\(615\) 0.151154 0.00609513
\(616\) 0 0
\(617\) 16.8503 0.678366 0.339183 0.940720i \(-0.389849\pi\)
0.339183 + 0.940720i \(0.389849\pi\)
\(618\) 0 0
\(619\) −43.0665 −1.73099 −0.865494 0.500920i \(-0.832995\pi\)
−0.865494 + 0.500920i \(0.832995\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 9.17139 0.367444
\(624\) 0 0
\(625\) −3.22528 −0.129011
\(626\) 0 0
\(627\) 1.61558 0.0645200
\(628\) 0 0
\(629\) −11.2524 −0.448662
\(630\) 0 0
\(631\) 41.5881 1.65560 0.827798 0.561026i \(-0.189593\pi\)
0.827798 + 0.561026i \(0.189593\pi\)
\(632\) 0 0
\(633\) 18.1596 0.721778
\(634\) 0 0
\(635\) 56.4134 2.23870
\(636\) 0 0
\(637\) −3.91214 −0.155005
\(638\) 0 0
\(639\) −6.89939 −0.272936
\(640\) 0 0
\(641\) −2.08786 −0.0824655 −0.0412327 0.999150i \(-0.513129\pi\)
−0.0412327 + 0.999150i \(0.513129\pi\)
\(642\) 0 0
\(643\) 20.3997 0.804486 0.402243 0.915533i \(-0.368231\pi\)
0.402243 + 0.915533i \(0.368231\pi\)
\(644\) 0 0
\(645\) −29.0556 −1.14406
\(646\) 0 0
\(647\) 21.8750 0.859994 0.429997 0.902830i \(-0.358515\pi\)
0.429997 + 0.902830i \(0.358515\pi\)
\(648\) 0 0
\(649\) 10.0271 0.393596
\(650\) 0 0
\(651\) 1.21699 0.0476976
\(652\) 0 0
\(653\) −18.8788 −0.738784 −0.369392 0.929274i \(-0.620434\pi\)
−0.369392 + 0.929274i \(0.620434\pi\)
\(654\) 0 0
\(655\) 43.2752 1.69090
\(656\) 0 0
\(657\) −2.73388 −0.106659
\(658\) 0 0
\(659\) 10.4552 0.407277 0.203638 0.979046i \(-0.434723\pi\)
0.203638 + 0.979046i \(0.434723\pi\)
\(660\) 0 0
\(661\) 2.21860 0.0862934 0.0431467 0.999069i \(-0.486262\pi\)
0.0431467 + 0.999069i \(0.486262\pi\)
\(662\) 0 0
\(663\) 28.2339 1.09651
\(664\) 0 0
\(665\) −1.19929 −0.0465067
\(666\) 0 0
\(667\) −7.11144 −0.275356
\(668\) 0 0
\(669\) 25.5421 0.987514
\(670\) 0 0
\(671\) −52.8110 −2.03875
\(672\) 0 0
\(673\) 31.8037 1.22594 0.612971 0.790105i \(-0.289974\pi\)
0.612971 + 0.790105i \(0.289974\pi\)
\(674\) 0 0
\(675\) 7.79384 0.299985
\(676\) 0 0
\(677\) 39.2001 1.50658 0.753291 0.657687i \(-0.228465\pi\)
0.753291 + 0.657687i \(0.228465\pi\)
\(678\) 0 0
\(679\) 13.7820 0.528906
\(680\) 0 0
\(681\) 16.7423 0.641567
\(682\) 0 0
\(683\) −22.1242 −0.846558 −0.423279 0.905999i \(-0.639121\pi\)
−0.423279 + 0.905999i \(0.639121\pi\)
\(684\) 0 0
\(685\) 77.6693 2.96759
\(686\) 0 0
\(687\) 26.5504 1.01296
\(688\) 0 0
\(689\) −24.8538 −0.946854
\(690\) 0 0
\(691\) −22.1115 −0.841161 −0.420580 0.907255i \(-0.638174\pi\)
−0.420580 + 0.907255i \(0.638174\pi\)
\(692\) 0 0
\(693\) 4.81840 0.183036
\(694\) 0 0
\(695\) −26.1283 −0.991104
\(696\) 0 0
\(697\) 0.304983 0.0115521
\(698\) 0 0
\(699\) 5.76692 0.218125
\(700\) 0 0
\(701\) −33.0353 −1.24773 −0.623864 0.781533i \(-0.714438\pi\)
−0.623864 + 0.781533i \(0.714438\pi\)
\(702\) 0 0
\(703\) −0.522774 −0.0197168
\(704\) 0 0
\(705\) −14.6849 −0.553067
\(706\) 0 0
\(707\) −15.7178 −0.591128
\(708\) 0 0
\(709\) −11.9265 −0.447909 −0.223954 0.974600i \(-0.571897\pi\)
−0.223954 + 0.974600i \(0.571897\pi\)
\(710\) 0 0
\(711\) 16.5582 0.620980
\(712\) 0 0
\(713\) 1.21699 0.0455767
\(714\) 0 0
\(715\) −67.4245 −2.52153
\(716\) 0 0
\(717\) 23.8023 0.888914
\(718\) 0 0
\(719\) −39.0840 −1.45759 −0.728793 0.684734i \(-0.759918\pi\)
−0.728793 + 0.684734i \(0.759918\pi\)
\(720\) 0 0
\(721\) −3.70697 −0.138055
\(722\) 0 0
\(723\) −12.1031 −0.450121
\(724\) 0 0
\(725\) 55.4254 2.05845
\(726\) 0 0
\(727\) 27.2170 1.00942 0.504711 0.863288i \(-0.331599\pi\)
0.504711 + 0.863288i \(0.331599\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −58.6254 −2.16834
\(732\) 0 0
\(733\) 47.6585 1.76031 0.880153 0.474691i \(-0.157440\pi\)
0.880153 + 0.474691i \(0.157440\pi\)
\(734\) 0 0
\(735\) −3.57685 −0.131934
\(736\) 0 0
\(737\) −5.77868 −0.212861
\(738\) 0 0
\(739\) 16.1341 0.593502 0.296751 0.954955i \(-0.404097\pi\)
0.296751 + 0.954955i \(0.404097\pi\)
\(740\) 0 0
\(741\) 1.31172 0.0481871
\(742\) 0 0
\(743\) −25.2347 −0.925771 −0.462885 0.886418i \(-0.653186\pi\)
−0.462885 + 0.886418i \(0.653186\pi\)
\(744\) 0 0
\(745\) −12.1266 −0.444284
\(746\) 0 0
\(747\) −13.0413 −0.477155
\(748\) 0 0
\(749\) 2.65289 0.0969346
\(750\) 0 0
\(751\) 2.57586 0.0939945 0.0469973 0.998895i \(-0.485035\pi\)
0.0469973 + 0.998895i \(0.485035\pi\)
\(752\) 0 0
\(753\) −7.80993 −0.284610
\(754\) 0 0
\(755\) 67.4126 2.45339
\(756\) 0 0
\(757\) −36.1086 −1.31239 −0.656194 0.754592i \(-0.727835\pi\)
−0.656194 + 0.754592i \(0.727835\pi\)
\(758\) 0 0
\(759\) 4.81840 0.174897
\(760\) 0 0
\(761\) 0.687163 0.0249097 0.0124548 0.999922i \(-0.496035\pi\)
0.0124548 + 0.999922i \(0.496035\pi\)
\(762\) 0 0
\(763\) 6.85713 0.248245
\(764\) 0 0
\(765\) 25.8141 0.933310
\(766\) 0 0
\(767\) 8.14113 0.293959
\(768\) 0 0
\(769\) 31.1426 1.12303 0.561515 0.827467i \(-0.310219\pi\)
0.561515 + 0.827467i \(0.310219\pi\)
\(770\) 0 0
\(771\) −2.26024 −0.0814004
\(772\) 0 0
\(773\) 44.8425 1.61287 0.806437 0.591320i \(-0.201393\pi\)
0.806437 + 0.591320i \(0.201393\pi\)
\(774\) 0 0
\(775\) −9.48503 −0.340712
\(776\) 0 0
\(777\) −1.55915 −0.0559343
\(778\) 0 0
\(779\) 0.0141692 0.000507665 0
\(780\) 0 0
\(781\) 33.2440 1.18957
\(782\) 0 0
\(783\) 7.11144 0.254142
\(784\) 0 0
\(785\) −49.4474 −1.76485
\(786\) 0 0
\(787\) 12.3412 0.439917 0.219959 0.975509i \(-0.429408\pi\)
0.219959 + 0.975509i \(0.429408\pi\)
\(788\) 0 0
\(789\) −20.0168 −0.712616
\(790\) 0 0
\(791\) −8.54306 −0.303756
\(792\) 0 0
\(793\) −42.8782 −1.52265
\(794\) 0 0
\(795\) −22.7237 −0.805926
\(796\) 0 0
\(797\) 52.4818 1.85900 0.929500 0.368823i \(-0.120239\pi\)
0.929500 + 0.368823i \(0.120239\pi\)
\(798\) 0 0
\(799\) −29.6297 −1.04823
\(800\) 0 0
\(801\) −9.17139 −0.324055
\(802\) 0 0
\(803\) 13.1729 0.464863
\(804\) 0 0
\(805\) −3.57685 −0.126067
\(806\) 0 0
\(807\) 30.1730 1.06214
\(808\) 0 0
\(809\) −33.6408 −1.18275 −0.591373 0.806398i \(-0.701414\pi\)
−0.591373 + 0.806398i \(0.701414\pi\)
\(810\) 0 0
\(811\) −10.6500 −0.373972 −0.186986 0.982363i \(-0.559872\pi\)
−0.186986 + 0.982363i \(0.559872\pi\)
\(812\) 0 0
\(813\) 19.5735 0.686473
\(814\) 0 0
\(815\) −49.1577 −1.72192
\(816\) 0 0
\(817\) −2.72368 −0.0952893
\(818\) 0 0
\(819\) 3.91214 0.136701
\(820\) 0 0
\(821\) 50.3310 1.75656 0.878282 0.478142i \(-0.158690\pi\)
0.878282 + 0.478142i \(0.158690\pi\)
\(822\) 0 0
\(823\) −18.1916 −0.634120 −0.317060 0.948405i \(-0.602696\pi\)
−0.317060 + 0.948405i \(0.602696\pi\)
\(824\) 0 0
\(825\) −37.5538 −1.30746
\(826\) 0 0
\(827\) 33.1298 1.15203 0.576017 0.817438i \(-0.304606\pi\)
0.576017 + 0.817438i \(0.304606\pi\)
\(828\) 0 0
\(829\) −9.34555 −0.324584 −0.162292 0.986743i \(-0.551889\pi\)
−0.162292 + 0.986743i \(0.551889\pi\)
\(830\) 0 0
\(831\) −22.1140 −0.767125
\(832\) 0 0
\(833\) −7.21699 −0.250054
\(834\) 0 0
\(835\) 1.19929 0.0415033
\(836\) 0 0
\(837\) −1.21699 −0.0420653
\(838\) 0 0
\(839\) 11.4482 0.395236 0.197618 0.980279i \(-0.436679\pi\)
0.197618 + 0.980279i \(0.436679\pi\)
\(840\) 0 0
\(841\) 21.5725 0.743880
\(842\) 0 0
\(843\) −21.7424 −0.748847
\(844\) 0 0
\(845\) −8.24409 −0.283605
\(846\) 0 0
\(847\) −12.2170 −0.419781
\(848\) 0 0
\(849\) 5.59788 0.192119
\(850\) 0 0
\(851\) −1.55915 −0.0534470
\(852\) 0 0
\(853\) −53.2336 −1.82269 −0.911343 0.411648i \(-0.864953\pi\)
−0.911343 + 0.411648i \(0.864953\pi\)
\(854\) 0 0
\(855\) 1.19929 0.0410150
\(856\) 0 0
\(857\) −7.95025 −0.271575 −0.135788 0.990738i \(-0.543357\pi\)
−0.135788 + 0.990738i \(0.543357\pi\)
\(858\) 0 0
\(859\) −16.1664 −0.551592 −0.275796 0.961216i \(-0.588941\pi\)
−0.275796 + 0.961216i \(0.588941\pi\)
\(860\) 0 0
\(861\) 0.0422591 0.00144019
\(862\) 0 0
\(863\) −25.2500 −0.859519 −0.429760 0.902943i \(-0.641402\pi\)
−0.429760 + 0.902943i \(0.641402\pi\)
\(864\) 0 0
\(865\) −1.15722 −0.0393467
\(866\) 0 0
\(867\) 35.0850 1.19155
\(868\) 0 0
\(869\) −79.7839 −2.70648
\(870\) 0 0
\(871\) −4.69181 −0.158976
\(872\) 0 0
\(873\) −13.7820 −0.466451
\(874\) 0 0
\(875\) 9.99313 0.337830
\(876\) 0 0
\(877\) −2.08452 −0.0703892 −0.0351946 0.999380i \(-0.511205\pi\)
−0.0351946 + 0.999380i \(0.511205\pi\)
\(878\) 0 0
\(879\) −14.6706 −0.494827
\(880\) 0 0
\(881\) 7.69769 0.259342 0.129671 0.991557i \(-0.458608\pi\)
0.129671 + 0.991557i \(0.458608\pi\)
\(882\) 0 0
\(883\) 57.9600 1.95051 0.975255 0.221083i \(-0.0709592\pi\)
0.975255 + 0.221083i \(0.0709592\pi\)
\(884\) 0 0
\(885\) 7.44339 0.250207
\(886\) 0 0
\(887\) 39.4481 1.32454 0.662269 0.749266i \(-0.269594\pi\)
0.662269 + 0.749266i \(0.269594\pi\)
\(888\) 0 0
\(889\) 15.7718 0.528970
\(890\) 0 0
\(891\) −4.81840 −0.161423
\(892\) 0 0
\(893\) −1.37657 −0.0460651
\(894\) 0 0
\(895\) −18.1283 −0.605963
\(896\) 0 0
\(897\) 3.91214 0.130623
\(898\) 0 0
\(899\) −8.65455 −0.288645
\(900\) 0 0
\(901\) −45.8495 −1.52747
\(902\) 0 0
\(903\) −8.12325 −0.270325
\(904\) 0 0
\(905\) 45.0032 1.49596
\(906\) 0 0
\(907\) −30.4448 −1.01090 −0.505452 0.862855i \(-0.668674\pi\)
−0.505452 + 0.862855i \(0.668674\pi\)
\(908\) 0 0
\(909\) 15.7178 0.521326
\(910\) 0 0
\(911\) 45.3234 1.50163 0.750815 0.660513i \(-0.229661\pi\)
0.750815 + 0.660513i \(0.229661\pi\)
\(912\) 0 0
\(913\) 62.8381 2.07964
\(914\) 0 0
\(915\) −39.2033 −1.29602
\(916\) 0 0
\(917\) 12.0987 0.399534
\(918\) 0 0
\(919\) 47.2201 1.55765 0.778824 0.627243i \(-0.215817\pi\)
0.778824 + 0.627243i \(0.215817\pi\)
\(920\) 0 0
\(921\) 9.07604 0.299066
\(922\) 0 0
\(923\) 26.9914 0.888433
\(924\) 0 0
\(925\) 12.1518 0.399548
\(926\) 0 0
\(927\) 3.70697 0.121753
\(928\) 0 0
\(929\) −42.4145 −1.39157 −0.695787 0.718248i \(-0.744944\pi\)
−0.695787 + 0.718248i \(0.744944\pi\)
\(930\) 0 0
\(931\) −0.335294 −0.0109888
\(932\) 0 0
\(933\) −4.10221 −0.134300
\(934\) 0 0
\(935\) −124.383 −4.06774
\(936\) 0 0
\(937\) −2.68451 −0.0876991 −0.0438495 0.999038i \(-0.513962\pi\)
−0.0438495 + 0.999038i \(0.513962\pi\)
\(938\) 0 0
\(939\) 2.09628 0.0684095
\(940\) 0 0
\(941\) −42.9923 −1.40151 −0.700755 0.713402i \(-0.747153\pi\)
−0.700755 + 0.713402i \(0.747153\pi\)
\(942\) 0 0
\(943\) 0.0422591 0.00137615
\(944\) 0 0
\(945\) 3.57685 0.116355
\(946\) 0 0
\(947\) −27.2997 −0.887122 −0.443561 0.896244i \(-0.646285\pi\)
−0.443561 + 0.896244i \(0.646285\pi\)
\(948\) 0 0
\(949\) 10.6953 0.347185
\(950\) 0 0
\(951\) 10.9063 0.353660
\(952\) 0 0
\(953\) −36.8015 −1.19212 −0.596059 0.802941i \(-0.703268\pi\)
−0.596059 + 0.802941i \(0.703268\pi\)
\(954\) 0 0
\(955\) −25.2854 −0.818215
\(956\) 0 0
\(957\) −34.2658 −1.10765
\(958\) 0 0
\(959\) 21.7145 0.701196
\(960\) 0 0
\(961\) −29.5189 −0.952224
\(962\) 0 0
\(963\) −2.65289 −0.0854882
\(964\) 0 0
\(965\) −73.0561 −2.35176
\(966\) 0 0
\(967\) −38.9270 −1.25181 −0.625904 0.779900i \(-0.715270\pi\)
−0.625904 + 0.779900i \(0.715270\pi\)
\(968\) 0 0
\(969\) 2.41981 0.0777356
\(970\) 0 0
\(971\) −56.2523 −1.80522 −0.902612 0.430456i \(-0.858353\pi\)
−0.902612 + 0.430456i \(0.858353\pi\)
\(972\) 0 0
\(973\) −7.30485 −0.234183
\(974\) 0 0
\(975\) −30.4906 −0.976481
\(976\) 0 0
\(977\) 43.0147 1.37616 0.688082 0.725633i \(-0.258453\pi\)
0.688082 + 0.725633i \(0.258453\pi\)
\(978\) 0 0
\(979\) 44.1914 1.41236
\(980\) 0 0
\(981\) −6.85713 −0.218931
\(982\) 0 0
\(983\) −32.3824 −1.03284 −0.516420 0.856336i \(-0.672736\pi\)
−0.516420 + 0.856336i \(0.672736\pi\)
\(984\) 0 0
\(985\) 24.6028 0.783911
\(986\) 0 0
\(987\) −4.10555 −0.130681
\(988\) 0 0
\(989\) −8.12325 −0.258304
\(990\) 0 0
\(991\) −16.9957 −0.539885 −0.269943 0.962876i \(-0.587005\pi\)
−0.269943 + 0.962876i \(0.587005\pi\)
\(992\) 0 0
\(993\) 29.6368 0.940495
\(994\) 0 0
\(995\) 65.5215 2.07717
\(996\) 0 0
\(997\) 17.6882 0.560192 0.280096 0.959972i \(-0.409634\pi\)
0.280096 + 0.959972i \(0.409634\pi\)
\(998\) 0 0
\(999\) 1.55915 0.0493294
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 3864.2.a.t.1.1 4
4.3 odd 2 7728.2.a.bz.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.t.1.1 4 1.1 even 1 trivial
7728.2.a.bz.1.1 4 4.3 odd 2