Properties

Label 8281.2.a.cp.1.8
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1241179138\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 16 x^{10} + 88 x^{8} - 197 x^{6} + 172 x^{4} - 36 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.499987\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.499987 q^{2} +0.849601 q^{3} -1.75001 q^{4} -1.04248 q^{5} +0.424789 q^{6} -1.87496 q^{8} -2.27818 q^{9} +O(q^{10})\) \(q+0.499987 q^{2} +0.849601 q^{3} -1.75001 q^{4} -1.04248 q^{5} +0.424789 q^{6} -1.87496 q^{8} -2.27818 q^{9} -0.521224 q^{10} +3.96730 q^{11} -1.48681 q^{12} -0.885688 q^{15} +2.56257 q^{16} -0.142035 q^{17} -1.13906 q^{18} +5.50977 q^{19} +1.82435 q^{20} +1.98360 q^{22} +4.39098 q^{23} -1.59297 q^{24} -3.91325 q^{25} -4.48435 q^{27} -8.39759 q^{29} -0.442832 q^{30} -2.84652 q^{31} +5.03117 q^{32} +3.37063 q^{33} -0.0710158 q^{34} +3.98684 q^{36} +0.843187 q^{37} +2.75481 q^{38} +1.95460 q^{40} -12.0974 q^{41} +4.82323 q^{43} -6.94284 q^{44} +2.37494 q^{45} +2.19543 q^{46} +4.55648 q^{47} +2.17717 q^{48} -1.95657 q^{50} -0.120673 q^{51} -0.279600 q^{53} -2.24211 q^{54} -4.13582 q^{55} +4.68111 q^{57} -4.19868 q^{58} +10.7815 q^{59} +1.54997 q^{60} -5.86354 q^{61} -1.42322 q^{62} -2.60963 q^{64} +1.68527 q^{66} -5.14447 q^{67} +0.248564 q^{68} +3.73058 q^{69} -3.69880 q^{71} +4.27148 q^{72} +6.61281 q^{73} +0.421582 q^{74} -3.32470 q^{75} -9.64216 q^{76} +11.9227 q^{79} -2.67142 q^{80} +3.02462 q^{81} -6.04853 q^{82} -2.87321 q^{83} +0.148068 q^{85} +2.41155 q^{86} -7.13461 q^{87} -7.43852 q^{88} -1.74765 q^{89} +1.18744 q^{90} -7.68427 q^{92} -2.41841 q^{93} +2.27818 q^{94} -5.74379 q^{95} +4.27449 q^{96} +2.70291 q^{97} -9.03822 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 6q^{3} + 8q^{4} + 2q^{9} + O(q^{10}) \) \( 12q + 6q^{3} + 8q^{4} + 2q^{9} + 24q^{10} - 2q^{12} + 16q^{16} + 34q^{17} + 30q^{22} + 6q^{23} - 10q^{25} + 12q^{27} + 2q^{29} + 22q^{30} - 26q^{36} + 38q^{38} + 2q^{40} + 22q^{43} - 38q^{48} + 8q^{51} + 16q^{53} + 30q^{55} - 10q^{61} + 82q^{62} - 2q^{64} + 68q^{66} + 22q^{68} + 14q^{69} + 66q^{74} + 2q^{75} + 70q^{79} - 28q^{81} + 10q^{82} - 20q^{87} - 28q^{88} - 66q^{92} - 2q^{94} + 4q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.499987 0.353544 0.176772 0.984252i \(-0.443434\pi\)
0.176772 + 0.984252i \(0.443434\pi\)
\(3\) 0.849601 0.490518 0.245259 0.969458i \(-0.421127\pi\)
0.245259 + 0.969458i \(0.421127\pi\)
\(4\) −1.75001 −0.875007
\(5\) −1.04248 −0.466209 −0.233105 0.972452i \(-0.574888\pi\)
−0.233105 + 0.972452i \(0.574888\pi\)
\(6\) 0.424789 0.173420
\(7\) 0 0
\(8\) −1.87496 −0.662897
\(9\) −2.27818 −0.759392
\(10\) −0.521224 −0.164825
\(11\) 3.96730 1.19619 0.598094 0.801426i \(-0.295925\pi\)
0.598094 + 0.801426i \(0.295925\pi\)
\(12\) −1.48681 −0.429206
\(13\) 0 0
\(14\) 0 0
\(15\) −0.885688 −0.228684
\(16\) 2.56257 0.640643
\(17\) −0.142035 −0.0344486 −0.0172243 0.999852i \(-0.505483\pi\)
−0.0172243 + 0.999852i \(0.505483\pi\)
\(18\) −1.13906 −0.268479
\(19\) 5.50977 1.26403 0.632014 0.774957i \(-0.282229\pi\)
0.632014 + 0.774957i \(0.282229\pi\)
\(20\) 1.82435 0.407936
\(21\) 0 0
\(22\) 1.98360 0.422905
\(23\) 4.39098 0.915582 0.457791 0.889060i \(-0.348641\pi\)
0.457791 + 0.889060i \(0.348641\pi\)
\(24\) −1.59297 −0.325163
\(25\) −3.91325 −0.782649
\(26\) 0 0
\(27\) −4.48435 −0.863013
\(28\) 0 0
\(29\) −8.39759 −1.55939 −0.779697 0.626157i \(-0.784627\pi\)
−0.779697 + 0.626157i \(0.784627\pi\)
\(30\) −0.442832 −0.0808498
\(31\) −2.84652 −0.511251 −0.255625 0.966776i \(-0.582281\pi\)
−0.255625 + 0.966776i \(0.582281\pi\)
\(32\) 5.03117 0.889393
\(33\) 3.37063 0.586751
\(34\) −0.0710158 −0.0121791
\(35\) 0 0
\(36\) 3.98684 0.664473
\(37\) 0.843187 0.138619 0.0693095 0.997595i \(-0.477920\pi\)
0.0693095 + 0.997595i \(0.477920\pi\)
\(38\) 2.75481 0.446889
\(39\) 0 0
\(40\) 1.95460 0.309049
\(41\) −12.0974 −1.88929 −0.944647 0.328089i \(-0.893595\pi\)
−0.944647 + 0.328089i \(0.893595\pi\)
\(42\) 0 0
\(43\) 4.82323 0.735536 0.367768 0.929918i \(-0.380122\pi\)
0.367768 + 0.929918i \(0.380122\pi\)
\(44\) −6.94284 −1.04667
\(45\) 2.37494 0.354036
\(46\) 2.19543 0.323699
\(47\) 4.55648 0.664630 0.332315 0.943168i \(-0.392170\pi\)
0.332315 + 0.943168i \(0.392170\pi\)
\(48\) 2.17717 0.314247
\(49\) 0 0
\(50\) −1.95657 −0.276701
\(51\) −0.120673 −0.0168977
\(52\) 0 0
\(53\) −0.279600 −0.0384060 −0.0192030 0.999816i \(-0.506113\pi\)
−0.0192030 + 0.999816i \(0.506113\pi\)
\(54\) −2.24211 −0.305113
\(55\) −4.13582 −0.557673
\(56\) 0 0
\(57\) 4.68111 0.620028
\(58\) −4.19868 −0.551314
\(59\) 10.7815 1.40363 0.701815 0.712359i \(-0.252373\pi\)
0.701815 + 0.712359i \(0.252373\pi\)
\(60\) 1.54997 0.200100
\(61\) −5.86354 −0.750749 −0.375374 0.926873i \(-0.622486\pi\)
−0.375374 + 0.926873i \(0.622486\pi\)
\(62\) −1.42322 −0.180750
\(63\) 0 0
\(64\) −2.60963 −0.326204
\(65\) 0 0
\(66\) 1.68527 0.207442
\(67\) −5.14447 −0.628497 −0.314248 0.949341i \(-0.601753\pi\)
−0.314248 + 0.949341i \(0.601753\pi\)
\(68\) 0.248564 0.0301428
\(69\) 3.73058 0.449109
\(70\) 0 0
\(71\) −3.69880 −0.438967 −0.219484 0.975616i \(-0.570437\pi\)
−0.219484 + 0.975616i \(0.570437\pi\)
\(72\) 4.27148 0.503399
\(73\) 6.61281 0.773970 0.386985 0.922086i \(-0.373516\pi\)
0.386985 + 0.922086i \(0.373516\pi\)
\(74\) 0.421582 0.0490079
\(75\) −3.32470 −0.383903
\(76\) −9.64216 −1.10603
\(77\) 0 0
\(78\) 0 0
\(79\) 11.9227 1.34141 0.670705 0.741725i \(-0.265992\pi\)
0.670705 + 0.741725i \(0.265992\pi\)
\(80\) −2.67142 −0.298674
\(81\) 3.02462 0.336069
\(82\) −6.04853 −0.667948
\(83\) −2.87321 −0.315376 −0.157688 0.987489i \(-0.550404\pi\)
−0.157688 + 0.987489i \(0.550404\pi\)
\(84\) 0 0
\(85\) 0.148068 0.0160603
\(86\) 2.41155 0.260044
\(87\) −7.13461 −0.764910
\(88\) −7.43852 −0.792949
\(89\) −1.74765 −0.185250 −0.0926252 0.995701i \(-0.529526\pi\)
−0.0926252 + 0.995701i \(0.529526\pi\)
\(90\) 1.18744 0.125167
\(91\) 0 0
\(92\) −7.68427 −0.801141
\(93\) −2.41841 −0.250777
\(94\) 2.27818 0.234976
\(95\) −5.74379 −0.589301
\(96\) 4.27449 0.436263
\(97\) 2.70291 0.274438 0.137219 0.990541i \(-0.456184\pi\)
0.137219 + 0.990541i \(0.456184\pi\)
\(98\) 0 0
\(99\) −9.03822 −0.908376
\(100\) 6.84823 0.684823
\(101\) 11.4722 1.14153 0.570765 0.821114i \(-0.306647\pi\)
0.570765 + 0.821114i \(0.306647\pi\)
\(102\) −0.0603351 −0.00597407
\(103\) 4.16950 0.410834 0.205417 0.978675i \(-0.434145\pi\)
0.205417 + 0.978675i \(0.434145\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −0.139796 −0.0135782
\(107\) 8.48742 0.820510 0.410255 0.911971i \(-0.365440\pi\)
0.410255 + 0.911971i \(0.365440\pi\)
\(108\) 7.84767 0.755142
\(109\) −6.43036 −0.615917 −0.307958 0.951400i \(-0.599646\pi\)
−0.307958 + 0.951400i \(0.599646\pi\)
\(110\) −2.06785 −0.197162
\(111\) 0.716373 0.0679951
\(112\) 0 0
\(113\) 10.9633 1.03134 0.515670 0.856788i \(-0.327543\pi\)
0.515670 + 0.856788i \(0.327543\pi\)
\(114\) 2.34049 0.219207
\(115\) −4.57749 −0.426853
\(116\) 14.6959 1.36448
\(117\) 0 0
\(118\) 5.39060 0.496245
\(119\) 0 0
\(120\) 1.66063 0.151594
\(121\) 4.73951 0.430864
\(122\) −2.93169 −0.265423
\(123\) −10.2780 −0.926732
\(124\) 4.98145 0.447348
\(125\) 9.29184 0.831087
\(126\) 0 0
\(127\) −2.00787 −0.178170 −0.0890849 0.996024i \(-0.528394\pi\)
−0.0890849 + 0.996024i \(0.528394\pi\)
\(128\) −11.3671 −1.00472
\(129\) 4.09782 0.360793
\(130\) 0 0
\(131\) −12.4502 −1.08778 −0.543890 0.839156i \(-0.683049\pi\)
−0.543890 + 0.839156i \(0.683049\pi\)
\(132\) −5.89864 −0.513411
\(133\) 0 0
\(134\) −2.57217 −0.222201
\(135\) 4.67482 0.402344
\(136\) 0.266310 0.0228359
\(137\) 5.24518 0.448126 0.224063 0.974575i \(-0.428068\pi\)
0.224063 + 0.974575i \(0.428068\pi\)
\(138\) 1.86524 0.158780
\(139\) −20.7385 −1.75902 −0.879510 0.475881i \(-0.842129\pi\)
−0.879510 + 0.475881i \(0.842129\pi\)
\(140\) 0 0
\(141\) 3.87119 0.326013
\(142\) −1.84935 −0.155194
\(143\) 0 0
\(144\) −5.83800 −0.486500
\(145\) 8.75428 0.727004
\(146\) 3.30631 0.273633
\(147\) 0 0
\(148\) −1.47559 −0.121293
\(149\) −0.0113760 −0.000931956 0 −0.000465978 1.00000i \(-0.500148\pi\)
−0.000465978 1.00000i \(0.500148\pi\)
\(150\) −1.66231 −0.135727
\(151\) −18.9054 −1.53850 −0.769251 0.638947i \(-0.779370\pi\)
−0.769251 + 0.638947i \(0.779370\pi\)
\(152\) −10.3306 −0.837920
\(153\) 0.323582 0.0261600
\(154\) 0 0
\(155\) 2.96743 0.238350
\(156\) 0 0
\(157\) 19.7937 1.57971 0.789856 0.613292i \(-0.210155\pi\)
0.789856 + 0.613292i \(0.210155\pi\)
\(158\) 5.96119 0.474247
\(159\) −0.237549 −0.0188388
\(160\) −5.24486 −0.414643
\(161\) 0 0
\(162\) 1.51227 0.118815
\(163\) 8.93255 0.699651 0.349825 0.936815i \(-0.386241\pi\)
0.349825 + 0.936815i \(0.386241\pi\)
\(164\) 21.1706 1.65314
\(165\) −3.51380 −0.273549
\(166\) −1.43657 −0.111499
\(167\) 6.13469 0.474716 0.237358 0.971422i \(-0.423719\pi\)
0.237358 + 0.971422i \(0.423719\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0.0740322 0.00567801
\(171\) −12.5522 −0.959893
\(172\) −8.44072 −0.643599
\(173\) 24.2628 1.84466 0.922332 0.386399i \(-0.126281\pi\)
0.922332 + 0.386399i \(0.126281\pi\)
\(174\) −3.56721 −0.270429
\(175\) 0 0
\(176\) 10.1665 0.766330
\(177\) 9.15997 0.688506
\(178\) −0.873801 −0.0654942
\(179\) −4.13675 −0.309195 −0.154598 0.987978i \(-0.549408\pi\)
−0.154598 + 0.987978i \(0.549408\pi\)
\(180\) −4.15618 −0.309784
\(181\) 7.86568 0.584651 0.292326 0.956319i \(-0.405571\pi\)
0.292326 + 0.956319i \(0.405571\pi\)
\(182\) 0 0
\(183\) −4.98167 −0.368256
\(184\) −8.23289 −0.606937
\(185\) −0.879001 −0.0646254
\(186\) −1.20917 −0.0886608
\(187\) −0.563498 −0.0412070
\(188\) −7.97389 −0.581556
\(189\) 0 0
\(190\) −2.87182 −0.208344
\(191\) −6.47866 −0.468780 −0.234390 0.972143i \(-0.575309\pi\)
−0.234390 + 0.972143i \(0.575309\pi\)
\(192\) −2.21715 −0.160009
\(193\) −4.82928 −0.347619 −0.173810 0.984779i \(-0.555608\pi\)
−0.173810 + 0.984779i \(0.555608\pi\)
\(194\) 1.35142 0.0970261
\(195\) 0 0
\(196\) 0 0
\(197\) 25.8362 1.84075 0.920377 0.391032i \(-0.127882\pi\)
0.920377 + 0.391032i \(0.127882\pi\)
\(198\) −4.51899 −0.321151
\(199\) 17.1146 1.21322 0.606612 0.794998i \(-0.292528\pi\)
0.606612 + 0.794998i \(0.292528\pi\)
\(200\) 7.33717 0.518816
\(201\) −4.37075 −0.308289
\(202\) 5.73596 0.403581
\(203\) 0 0
\(204\) 0.211180 0.0147856
\(205\) 12.6112 0.880806
\(206\) 2.08470 0.145248
\(207\) −10.0034 −0.695286
\(208\) 0 0
\(209\) 21.8589 1.51201
\(210\) 0 0
\(211\) 18.2911 1.25921 0.629607 0.776914i \(-0.283216\pi\)
0.629607 + 0.776914i \(0.283216\pi\)
\(212\) 0.489304 0.0336055
\(213\) −3.14251 −0.215321
\(214\) 4.24360 0.290086
\(215\) −5.02810 −0.342913
\(216\) 8.40796 0.572089
\(217\) 0 0
\(218\) −3.21509 −0.217754
\(219\) 5.61825 0.379646
\(220\) 7.23773 0.487968
\(221\) 0 0
\(222\) 0.358177 0.0240392
\(223\) 11.5087 0.770679 0.385340 0.922775i \(-0.374084\pi\)
0.385340 + 0.922775i \(0.374084\pi\)
\(224\) 0 0
\(225\) 8.91507 0.594338
\(226\) 5.48150 0.364624
\(227\) −17.9045 −1.18836 −0.594181 0.804332i \(-0.702524\pi\)
−0.594181 + 0.804332i \(0.702524\pi\)
\(228\) −8.19200 −0.542528
\(229\) 3.86350 0.255307 0.127654 0.991819i \(-0.459255\pi\)
0.127654 + 0.991819i \(0.459255\pi\)
\(230\) −2.28868 −0.150911
\(231\) 0 0
\(232\) 15.7451 1.03372
\(233\) 25.0642 1.64201 0.821004 0.570922i \(-0.193414\pi\)
0.821004 + 0.570922i \(0.193414\pi\)
\(234\) 0 0
\(235\) −4.75001 −0.309857
\(236\) −18.8678 −1.22819
\(237\) 10.1295 0.657985
\(238\) 0 0
\(239\) −7.80462 −0.504839 −0.252419 0.967618i \(-0.581226\pi\)
−0.252419 + 0.967618i \(0.581226\pi\)
\(240\) −2.26964 −0.146505
\(241\) 21.7653 1.40202 0.701012 0.713150i \(-0.252732\pi\)
0.701012 + 0.713150i \(0.252732\pi\)
\(242\) 2.36969 0.152329
\(243\) 16.0228 1.02786
\(244\) 10.2613 0.656910
\(245\) 0 0
\(246\) −5.13884 −0.327640
\(247\) 0 0
\(248\) 5.33711 0.338907
\(249\) −2.44109 −0.154698
\(250\) 4.64579 0.293826
\(251\) 7.67980 0.484745 0.242372 0.970183i \(-0.422074\pi\)
0.242372 + 0.970183i \(0.422074\pi\)
\(252\) 0 0
\(253\) 17.4203 1.09521
\(254\) −1.00391 −0.0629909
\(255\) 0.125799 0.00787784
\(256\) −0.464141 −0.0290088
\(257\) 13.6237 0.849826 0.424913 0.905234i \(-0.360305\pi\)
0.424913 + 0.905234i \(0.360305\pi\)
\(258\) 2.04886 0.127556
\(259\) 0 0
\(260\) 0 0
\(261\) 19.1312 1.18419
\(262\) −6.22494 −0.384578
\(263\) −11.7232 −0.722880 −0.361440 0.932395i \(-0.617715\pi\)
−0.361440 + 0.932395i \(0.617715\pi\)
\(264\) −6.31978 −0.388956
\(265\) 0.291476 0.0179052
\(266\) 0 0
\(267\) −1.48481 −0.0908686
\(268\) 9.00289 0.549939
\(269\) 9.19876 0.560858 0.280429 0.959875i \(-0.409523\pi\)
0.280429 + 0.959875i \(0.409523\pi\)
\(270\) 2.33735 0.142246
\(271\) −2.56369 −0.155733 −0.0778665 0.996964i \(-0.524811\pi\)
−0.0778665 + 0.996964i \(0.524811\pi\)
\(272\) −0.363976 −0.0220693
\(273\) 0 0
\(274\) 2.62252 0.158432
\(275\) −15.5250 −0.936195
\(276\) −6.52857 −0.392974
\(277\) −0.933882 −0.0561115 −0.0280558 0.999606i \(-0.508932\pi\)
−0.0280558 + 0.999606i \(0.508932\pi\)
\(278\) −10.3690 −0.621891
\(279\) 6.48488 0.388240
\(280\) 0 0
\(281\) 6.45288 0.384947 0.192473 0.981302i \(-0.438349\pi\)
0.192473 + 0.981302i \(0.438349\pi\)
\(282\) 1.93554 0.115260
\(283\) 22.1746 1.31814 0.659071 0.752081i \(-0.270950\pi\)
0.659071 + 0.752081i \(0.270950\pi\)
\(284\) 6.47296 0.384099
\(285\) −4.87994 −0.289062
\(286\) 0 0
\(287\) 0 0
\(288\) −11.4619 −0.675398
\(289\) −16.9798 −0.998813
\(290\) 4.37702 0.257028
\(291\) 2.29639 0.134617
\(292\) −11.5725 −0.677229
\(293\) 24.2026 1.41393 0.706964 0.707249i \(-0.250064\pi\)
0.706964 + 0.707249i \(0.250064\pi\)
\(294\) 0 0
\(295\) −11.2394 −0.654385
\(296\) −1.58094 −0.0918902
\(297\) −17.7908 −1.03233
\(298\) −0.00568784 −0.000329487 0
\(299\) 0 0
\(300\) 5.81827 0.335918
\(301\) 0 0
\(302\) −9.45246 −0.543928
\(303\) 9.74683 0.559940
\(304\) 14.1192 0.809791
\(305\) 6.11259 0.350006
\(306\) 0.161787 0.00924872
\(307\) 24.2924 1.38644 0.693220 0.720726i \(-0.256191\pi\)
0.693220 + 0.720726i \(0.256191\pi\)
\(308\) 0 0
\(309\) 3.54242 0.201521
\(310\) 1.48367 0.0842671
\(311\) 3.98711 0.226088 0.113044 0.993590i \(-0.463940\pi\)
0.113044 + 0.993590i \(0.463940\pi\)
\(312\) 0 0
\(313\) 28.4754 1.60953 0.804763 0.593597i \(-0.202293\pi\)
0.804763 + 0.593597i \(0.202293\pi\)
\(314\) 9.89661 0.558498
\(315\) 0 0
\(316\) −20.8649 −1.17374
\(317\) −16.8161 −0.944487 −0.472244 0.881468i \(-0.656556\pi\)
−0.472244 + 0.881468i \(0.656556\pi\)
\(318\) −0.118771 −0.00666036
\(319\) −33.3158 −1.86533
\(320\) 2.72048 0.152079
\(321\) 7.21093 0.402475
\(322\) 0 0
\(323\) −0.782582 −0.0435440
\(324\) −5.29313 −0.294063
\(325\) 0 0
\(326\) 4.46615 0.247357
\(327\) −5.46324 −0.302118
\(328\) 22.6821 1.25241
\(329\) 0 0
\(330\) −1.75685 −0.0967115
\(331\) −6.20917 −0.341287 −0.170644 0.985333i \(-0.554585\pi\)
−0.170644 + 0.985333i \(0.554585\pi\)
\(332\) 5.02816 0.275956
\(333\) −1.92093 −0.105266
\(334\) 3.06726 0.167833
\(335\) 5.36298 0.293011
\(336\) 0 0
\(337\) −7.69650 −0.419255 −0.209628 0.977781i \(-0.567225\pi\)
−0.209628 + 0.977781i \(0.567225\pi\)
\(338\) 0 0
\(339\) 9.31442 0.505890
\(340\) −0.259122 −0.0140528
\(341\) −11.2930 −0.611552
\(342\) −6.27594 −0.339364
\(343\) 0 0
\(344\) −9.04335 −0.487585
\(345\) −3.88904 −0.209379
\(346\) 12.1311 0.652170
\(347\) 30.4094 1.63246 0.816231 0.577725i \(-0.196059\pi\)
0.816231 + 0.577725i \(0.196059\pi\)
\(348\) 12.4857 0.669302
\(349\) −16.1581 −0.864924 −0.432462 0.901652i \(-0.642355\pi\)
−0.432462 + 0.901652i \(0.642355\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 19.9602 1.06388
\(353\) −11.8424 −0.630306 −0.315153 0.949041i \(-0.602056\pi\)
−0.315153 + 0.949041i \(0.602056\pi\)
\(354\) 4.57986 0.243417
\(355\) 3.85591 0.204651
\(356\) 3.05841 0.162095
\(357\) 0 0
\(358\) −2.06832 −0.109314
\(359\) 31.3653 1.65540 0.827698 0.561174i \(-0.189650\pi\)
0.827698 + 0.561174i \(0.189650\pi\)
\(360\) −4.45292 −0.234689
\(361\) 11.3575 0.597765
\(362\) 3.93273 0.206700
\(363\) 4.02669 0.211346
\(364\) 0 0
\(365\) −6.89369 −0.360832
\(366\) −2.49077 −0.130195
\(367\) 24.0774 1.25683 0.628415 0.777878i \(-0.283704\pi\)
0.628415 + 0.777878i \(0.283704\pi\)
\(368\) 11.2522 0.586562
\(369\) 27.5600 1.43472
\(370\) −0.439489 −0.0228479
\(371\) 0 0
\(372\) 4.23225 0.219432
\(373\) −18.3922 −0.952314 −0.476157 0.879360i \(-0.657971\pi\)
−0.476157 + 0.879360i \(0.657971\pi\)
\(374\) −0.281741 −0.0145685
\(375\) 7.89436 0.407663
\(376\) −8.54320 −0.440582
\(377\) 0 0
\(378\) 0 0
\(379\) 8.13740 0.417990 0.208995 0.977917i \(-0.432981\pi\)
0.208995 + 0.977917i \(0.432981\pi\)
\(380\) 10.0517 0.515642
\(381\) −1.70589 −0.0873955
\(382\) −3.23925 −0.165734
\(383\) 22.3711 1.14311 0.571555 0.820564i \(-0.306340\pi\)
0.571555 + 0.820564i \(0.306340\pi\)
\(384\) −9.65752 −0.492833
\(385\) 0 0
\(386\) −2.41458 −0.122899
\(387\) −10.9882 −0.558560
\(388\) −4.73012 −0.240136
\(389\) −21.3946 −1.08475 −0.542374 0.840137i \(-0.682475\pi\)
−0.542374 + 0.840137i \(0.682475\pi\)
\(390\) 0 0
\(391\) −0.623674 −0.0315406
\(392\) 0 0
\(393\) −10.5777 −0.533576
\(394\) 12.9178 0.650788
\(395\) −12.4291 −0.625377
\(396\) 15.8170 0.794835
\(397\) 1.19673 0.0600622 0.0300311 0.999549i \(-0.490439\pi\)
0.0300311 + 0.999549i \(0.490439\pi\)
\(398\) 8.55708 0.428928
\(399\) 0 0
\(400\) −10.0280 −0.501399
\(401\) −36.2749 −1.81148 −0.905741 0.423831i \(-0.860685\pi\)
−0.905741 + 0.423831i \(0.860685\pi\)
\(402\) −2.18532 −0.108994
\(403\) 0 0
\(404\) −20.0766 −0.998846
\(405\) −3.15309 −0.156679
\(406\) 0 0
\(407\) 3.34518 0.165814
\(408\) 0.226257 0.0112014
\(409\) 14.6723 0.725500 0.362750 0.931887i \(-0.381838\pi\)
0.362750 + 0.931887i \(0.381838\pi\)
\(410\) 6.30544 0.311404
\(411\) 4.45631 0.219814
\(412\) −7.29669 −0.359482
\(413\) 0 0
\(414\) −5.00158 −0.245814
\(415\) 2.99525 0.147031
\(416\) 0 0
\(417\) −17.6195 −0.862830
\(418\) 10.9292 0.534563
\(419\) −5.93348 −0.289870 −0.144935 0.989441i \(-0.546297\pi\)
−0.144935 + 0.989441i \(0.546297\pi\)
\(420\) 0 0
\(421\) 2.63174 0.128263 0.0641317 0.997941i \(-0.479572\pi\)
0.0641317 + 0.997941i \(0.479572\pi\)
\(422\) 9.14532 0.445187
\(423\) −10.3805 −0.504715
\(424\) 0.524238 0.0254593
\(425\) 0.555819 0.0269612
\(426\) −1.57121 −0.0761255
\(427\) 0 0
\(428\) −14.8531 −0.717952
\(429\) 0 0
\(430\) −2.51398 −0.121235
\(431\) 18.8377 0.907378 0.453689 0.891160i \(-0.350108\pi\)
0.453689 + 0.891160i \(0.350108\pi\)
\(432\) −11.4915 −0.552884
\(433\) 19.1355 0.919591 0.459796 0.888025i \(-0.347923\pi\)
0.459796 + 0.888025i \(0.347923\pi\)
\(434\) 0 0
\(435\) 7.43765 0.356608
\(436\) 11.2532 0.538931
\(437\) 24.1933 1.15732
\(438\) 2.80905 0.134222
\(439\) −1.26511 −0.0603803 −0.0301901 0.999544i \(-0.509611\pi\)
−0.0301901 + 0.999544i \(0.509611\pi\)
\(440\) 7.75448 0.369680
\(441\) 0 0
\(442\) 0 0
\(443\) −20.9392 −0.994853 −0.497426 0.867506i \(-0.665722\pi\)
−0.497426 + 0.867506i \(0.665722\pi\)
\(444\) −1.25366 −0.0594961
\(445\) 1.82188 0.0863654
\(446\) 5.75419 0.272469
\(447\) −0.00966505 −0.000457141 0
\(448\) 0 0
\(449\) −17.8632 −0.843018 −0.421509 0.906824i \(-0.638499\pi\)
−0.421509 + 0.906824i \(0.638499\pi\)
\(450\) 4.45741 0.210125
\(451\) −47.9940 −2.25995
\(452\) −19.1859 −0.902429
\(453\) −16.0621 −0.754662
\(454\) −8.95199 −0.420138
\(455\) 0 0
\(456\) −8.77687 −0.411015
\(457\) −6.56597 −0.307143 −0.153571 0.988138i \(-0.549078\pi\)
−0.153571 + 0.988138i \(0.549078\pi\)
\(458\) 1.93170 0.0902624
\(459\) 0.636936 0.0297296
\(460\) 8.01066 0.373499
\(461\) −5.11364 −0.238166 −0.119083 0.992884i \(-0.537995\pi\)
−0.119083 + 0.992884i \(0.537995\pi\)
\(462\) 0 0
\(463\) −33.3239 −1.54869 −0.774347 0.632761i \(-0.781921\pi\)
−0.774347 + 0.632761i \(0.781921\pi\)
\(464\) −21.5194 −0.999015
\(465\) 2.52113 0.116915
\(466\) 12.5317 0.580522
\(467\) −12.9494 −0.599229 −0.299614 0.954060i \(-0.596858\pi\)
−0.299614 + 0.954060i \(0.596858\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −2.37494 −0.109548
\(471\) 16.8168 0.774877
\(472\) −20.2148 −0.930463
\(473\) 19.1352 0.879838
\(474\) 5.06464 0.232627
\(475\) −21.5611 −0.989290
\(476\) 0 0
\(477\) 0.636979 0.0291652
\(478\) −3.90221 −0.178483
\(479\) −27.0119 −1.23421 −0.617104 0.786882i \(-0.711694\pi\)
−0.617104 + 0.786882i \(0.711694\pi\)
\(480\) −4.45605 −0.203390
\(481\) 0 0
\(482\) 10.8823 0.495677
\(483\) 0 0
\(484\) −8.29420 −0.377009
\(485\) −2.81771 −0.127946
\(486\) 8.01117 0.363394
\(487\) −32.0838 −1.45386 −0.726928 0.686713i \(-0.759053\pi\)
−0.726928 + 0.686713i \(0.759053\pi\)
\(488\) 10.9939 0.497669
\(489\) 7.58910 0.343191
\(490\) 0 0
\(491\) −28.6040 −1.29088 −0.645440 0.763811i \(-0.723326\pi\)
−0.645440 + 0.763811i \(0.723326\pi\)
\(492\) 17.9866 0.810897
\(493\) 1.19276 0.0537190
\(494\) 0 0
\(495\) 9.42212 0.423493
\(496\) −7.29442 −0.327529
\(497\) 0 0
\(498\) −1.22051 −0.0546924
\(499\) 1.79816 0.0804969 0.0402484 0.999190i \(-0.487185\pi\)
0.0402484 + 0.999190i \(0.487185\pi\)
\(500\) −16.2608 −0.727207
\(501\) 5.21204 0.232857
\(502\) 3.83980 0.171379
\(503\) 29.0772 1.29649 0.648245 0.761432i \(-0.275503\pi\)
0.648245 + 0.761432i \(0.275503\pi\)
\(504\) 0 0
\(505\) −11.9595 −0.532191
\(506\) 8.70994 0.387204
\(507\) 0 0
\(508\) 3.51380 0.155900
\(509\) 23.1913 1.02794 0.513969 0.857809i \(-0.328175\pi\)
0.513969 + 0.857809i \(0.328175\pi\)
\(510\) 0.0628979 0.00278516
\(511\) 0 0
\(512\) 22.5022 0.994464
\(513\) −24.7077 −1.09087
\(514\) 6.81169 0.300451
\(515\) −4.34660 −0.191534
\(516\) −7.17125 −0.315696
\(517\) 18.0769 0.795022
\(518\) 0 0
\(519\) 20.6137 0.904840
\(520\) 0 0
\(521\) 33.2510 1.45675 0.728376 0.685178i \(-0.240275\pi\)
0.728376 + 0.685178i \(0.240275\pi\)
\(522\) 9.56535 0.418664
\(523\) 38.7121 1.69276 0.846380 0.532579i \(-0.178777\pi\)
0.846380 + 0.532579i \(0.178777\pi\)
\(524\) 21.7881 0.951816
\(525\) 0 0
\(526\) −5.86142 −0.255570
\(527\) 0.404307 0.0176119
\(528\) 8.63748 0.375898
\(529\) −3.71931 −0.161709
\(530\) 0.145734 0.00633029
\(531\) −24.5622 −1.06591
\(532\) 0 0
\(533\) 0 0
\(534\) −0.742383 −0.0321260
\(535\) −8.84793 −0.382529
\(536\) 9.64566 0.416629
\(537\) −3.51459 −0.151666
\(538\) 4.59926 0.198288
\(539\) 0 0
\(540\) −8.18100 −0.352054
\(541\) 22.6675 0.974551 0.487275 0.873248i \(-0.337991\pi\)
0.487275 + 0.873248i \(0.337991\pi\)
\(542\) −1.28181 −0.0550585
\(543\) 6.68269 0.286782
\(544\) −0.714603 −0.0306384
\(545\) 6.70349 0.287146
\(546\) 0 0
\(547\) −9.21134 −0.393848 −0.196924 0.980419i \(-0.563095\pi\)
−0.196924 + 0.980419i \(0.563095\pi\)
\(548\) −9.17913 −0.392113
\(549\) 13.3582 0.570113
\(550\) −7.76231 −0.330986
\(551\) −46.2688 −1.97112
\(552\) −6.99468 −0.297713
\(553\) 0 0
\(554\) −0.466928 −0.0198379
\(555\) −0.746801 −0.0316999
\(556\) 36.2927 1.53915
\(557\) −11.3281 −0.479986 −0.239993 0.970775i \(-0.577145\pi\)
−0.239993 + 0.970775i \(0.577145\pi\)
\(558\) 3.24236 0.137260
\(559\) 0 0
\(560\) 0 0
\(561\) −0.478748 −0.0202128
\(562\) 3.22636 0.136096
\(563\) 32.6386 1.37555 0.687777 0.725922i \(-0.258586\pi\)
0.687777 + 0.725922i \(0.258586\pi\)
\(564\) −6.77463 −0.285264
\(565\) −11.4290 −0.480820
\(566\) 11.0870 0.466021
\(567\) 0 0
\(568\) 6.93510 0.290990
\(569\) 35.0091 1.46766 0.733829 0.679335i \(-0.237731\pi\)
0.733829 + 0.679335i \(0.237731\pi\)
\(570\) −2.43990 −0.102196
\(571\) 26.2546 1.09872 0.549360 0.835586i \(-0.314872\pi\)
0.549360 + 0.835586i \(0.314872\pi\)
\(572\) 0 0
\(573\) −5.50428 −0.229945
\(574\) 0 0
\(575\) −17.1830 −0.716580
\(576\) 5.94520 0.247717
\(577\) −24.5727 −1.02297 −0.511487 0.859291i \(-0.670905\pi\)
−0.511487 + 0.859291i \(0.670905\pi\)
\(578\) −8.48969 −0.353124
\(579\) −4.10296 −0.170513
\(580\) −15.3201 −0.636133
\(581\) 0 0
\(582\) 1.14817 0.0475930
\(583\) −1.10926 −0.0459408
\(584\) −12.3987 −0.513063
\(585\) 0 0
\(586\) 12.1010 0.499886
\(587\) −20.5279 −0.847279 −0.423639 0.905831i \(-0.639248\pi\)
−0.423639 + 0.905831i \(0.639248\pi\)
\(588\) 0 0
\(589\) −15.6837 −0.646235
\(590\) −5.61957 −0.231354
\(591\) 21.9505 0.902923
\(592\) 2.16073 0.0888054
\(593\) −38.2835 −1.57211 −0.786057 0.618154i \(-0.787881\pi\)
−0.786057 + 0.618154i \(0.787881\pi\)
\(594\) −8.89515 −0.364972
\(595\) 0 0
\(596\) 0.0199081 0.000815468 0
\(597\) 14.5406 0.595107
\(598\) 0 0
\(599\) 14.0713 0.574939 0.287470 0.957790i \(-0.407186\pi\)
0.287470 + 0.957790i \(0.407186\pi\)
\(600\) 6.23367 0.254488
\(601\) −20.2342 −0.825369 −0.412685 0.910874i \(-0.635409\pi\)
−0.412685 + 0.910874i \(0.635409\pi\)
\(602\) 0 0
\(603\) 11.7200 0.477276
\(604\) 33.0847 1.34620
\(605\) −4.94082 −0.200873
\(606\) 4.87328 0.197964
\(607\) 6.55127 0.265908 0.132954 0.991122i \(-0.457554\pi\)
0.132954 + 0.991122i \(0.457554\pi\)
\(608\) 27.7205 1.12422
\(609\) 0 0
\(610\) 3.05621 0.123742
\(611\) 0 0
\(612\) −0.566272 −0.0228902
\(613\) 33.3244 1.34596 0.672980 0.739660i \(-0.265014\pi\)
0.672980 + 0.739660i \(0.265014\pi\)
\(614\) 12.1459 0.490168
\(615\) 10.7145 0.432051
\(616\) 0 0
\(617\) 6.76038 0.272162 0.136081 0.990698i \(-0.456549\pi\)
0.136081 + 0.990698i \(0.456549\pi\)
\(618\) 1.77116 0.0712466
\(619\) 17.6186 0.708152 0.354076 0.935217i \(-0.384795\pi\)
0.354076 + 0.935217i \(0.384795\pi\)
\(620\) −5.19304 −0.208558
\(621\) −19.6907 −0.790159
\(622\) 1.99350 0.0799321
\(623\) 0 0
\(624\) 0 0
\(625\) 9.87972 0.395189
\(626\) 14.2373 0.569038
\(627\) 18.5714 0.741669
\(628\) −34.6393 −1.38226
\(629\) −0.119762 −0.00477524
\(630\) 0 0
\(631\) −15.7519 −0.627074 −0.313537 0.949576i \(-0.601514\pi\)
−0.313537 + 0.949576i \(0.601514\pi\)
\(632\) −22.3546 −0.889216
\(633\) 15.5402 0.617667
\(634\) −8.40783 −0.333918
\(635\) 2.09316 0.0830644
\(636\) 0.415713 0.0164841
\(637\) 0 0
\(638\) −16.6575 −0.659475
\(639\) 8.42653 0.333348
\(640\) 11.8499 0.468410
\(641\) −20.9405 −0.827099 −0.413550 0.910482i \(-0.635711\pi\)
−0.413550 + 0.910482i \(0.635711\pi\)
\(642\) 3.60537 0.142293
\(643\) −18.9315 −0.746586 −0.373293 0.927713i \(-0.621771\pi\)
−0.373293 + 0.927713i \(0.621771\pi\)
\(644\) 0 0
\(645\) −4.27188 −0.168205
\(646\) −0.391280 −0.0153947
\(647\) −37.6768 −1.48123 −0.740614 0.671930i \(-0.765465\pi\)
−0.740614 + 0.671930i \(0.765465\pi\)
\(648\) −5.67104 −0.222779
\(649\) 42.7735 1.67901
\(650\) 0 0
\(651\) 0 0
\(652\) −15.6321 −0.612199
\(653\) 29.0326 1.13613 0.568066 0.822983i \(-0.307692\pi\)
0.568066 + 0.822983i \(0.307692\pi\)
\(654\) −2.73155 −0.106812
\(655\) 12.9790 0.507133
\(656\) −31.0004 −1.21036
\(657\) −15.0651 −0.587747
\(658\) 0 0
\(659\) −1.41830 −0.0552493 −0.0276247 0.999618i \(-0.508794\pi\)
−0.0276247 + 0.999618i \(0.508794\pi\)
\(660\) 6.14919 0.239357
\(661\) 4.59298 0.178646 0.0893231 0.996003i \(-0.471530\pi\)
0.0893231 + 0.996003i \(0.471530\pi\)
\(662\) −3.10450 −0.120660
\(663\) 0 0
\(664\) 5.38715 0.209062
\(665\) 0 0
\(666\) −0.960439 −0.0372162
\(667\) −36.8736 −1.42775
\(668\) −10.7358 −0.415380
\(669\) 9.77780 0.378032
\(670\) 2.68142 0.103592
\(671\) −23.2624 −0.898036
\(672\) 0 0
\(673\) −4.20223 −0.161984 −0.0809920 0.996715i \(-0.525809\pi\)
−0.0809920 + 0.996715i \(0.525809\pi\)
\(674\) −3.84815 −0.148225
\(675\) 17.5484 0.675436
\(676\) 0 0
\(677\) 8.08708 0.310812 0.155406 0.987851i \(-0.450331\pi\)
0.155406 + 0.987851i \(0.450331\pi\)
\(678\) 4.65709 0.178854
\(679\) 0 0
\(680\) −0.277622 −0.0106463
\(681\) −15.2117 −0.582912
\(682\) −5.64636 −0.216210
\(683\) 24.6865 0.944604 0.472302 0.881437i \(-0.343423\pi\)
0.472302 + 0.881437i \(0.343423\pi\)
\(684\) 21.9666 0.839912
\(685\) −5.46797 −0.208920
\(686\) 0 0
\(687\) 3.28244 0.125233
\(688\) 12.3599 0.471216
\(689\) 0 0
\(690\) −1.94447 −0.0740246
\(691\) −11.2567 −0.428225 −0.214113 0.976809i \(-0.568686\pi\)
−0.214113 + 0.976809i \(0.568686\pi\)
\(692\) −42.4602 −1.61409
\(693\) 0 0
\(694\) 15.2043 0.577147
\(695\) 21.6194 0.820071
\(696\) 13.3771 0.507057
\(697\) 1.71826 0.0650836
\(698\) −8.07884 −0.305789
\(699\) 21.2946 0.805434
\(700\) 0 0
\(701\) −22.2305 −0.839635 −0.419818 0.907608i \(-0.637906\pi\)
−0.419818 + 0.907608i \(0.637906\pi\)
\(702\) 0 0
\(703\) 4.64576 0.175218
\(704\) −10.3532 −0.390201
\(705\) −4.03562 −0.151990
\(706\) −5.92103 −0.222841
\(707\) 0 0
\(708\) −16.0301 −0.602447
\(709\) 23.7741 0.892854 0.446427 0.894820i \(-0.352696\pi\)
0.446427 + 0.894820i \(0.352696\pi\)
\(710\) 1.92790 0.0723530
\(711\) −27.1620 −1.01866
\(712\) 3.27677 0.122802
\(713\) −12.4990 −0.468092
\(714\) 0 0
\(715\) 0 0
\(716\) 7.23937 0.270548
\(717\) −6.63082 −0.247632
\(718\) 15.6822 0.585255
\(719\) 20.7808 0.774992 0.387496 0.921871i \(-0.373340\pi\)
0.387496 + 0.921871i \(0.373340\pi\)
\(720\) 6.08597 0.226811
\(721\) 0 0
\(722\) 5.67861 0.211336
\(723\) 18.4918 0.687717
\(724\) −13.7650 −0.511574
\(725\) 32.8618 1.22046
\(726\) 2.01329 0.0747203
\(727\) 26.7719 0.992915 0.496457 0.868061i \(-0.334634\pi\)
0.496457 + 0.868061i \(0.334634\pi\)
\(728\) 0 0
\(729\) 4.53910 0.168115
\(730\) −3.44675 −0.127570
\(731\) −0.685069 −0.0253382
\(732\) 8.71799 0.322226
\(733\) 5.25647 0.194152 0.0970761 0.995277i \(-0.469051\pi\)
0.0970761 + 0.995277i \(0.469051\pi\)
\(734\) 12.0384 0.444345
\(735\) 0 0
\(736\) 22.0917 0.814312
\(737\) −20.4097 −0.751800
\(738\) 13.7796 0.507235
\(739\) −7.15001 −0.263017 −0.131509 0.991315i \(-0.541982\pi\)
−0.131509 + 0.991315i \(0.541982\pi\)
\(740\) 1.53826 0.0565477
\(741\) 0 0
\(742\) 0 0
\(743\) 0.713641 0.0261810 0.0130905 0.999914i \(-0.495833\pi\)
0.0130905 + 0.999914i \(0.495833\pi\)
\(744\) 4.53441 0.166240
\(745\) 0.0118592 0.000434486 0
\(746\) −9.19587 −0.336685
\(747\) 6.54569 0.239494
\(748\) 0.986128 0.0360564
\(749\) 0 0
\(750\) 3.94707 0.144127
\(751\) −25.7013 −0.937854 −0.468927 0.883237i \(-0.655359\pi\)
−0.468927 + 0.883237i \(0.655359\pi\)
\(752\) 11.6763 0.425791
\(753\) 6.52477 0.237776
\(754\) 0 0
\(755\) 19.7084 0.717263
\(756\) 0 0
\(757\) 16.3885 0.595650 0.297825 0.954621i \(-0.403739\pi\)
0.297825 + 0.954621i \(0.403739\pi\)
\(758\) 4.06859 0.147778
\(759\) 14.8004 0.537219
\(760\) 10.7694 0.390646
\(761\) 8.31998 0.301599 0.150800 0.988564i \(-0.451815\pi\)
0.150800 + 0.988564i \(0.451815\pi\)
\(762\) −0.852923 −0.0308981
\(763\) 0 0
\(764\) 11.3378 0.410185
\(765\) −0.337326 −0.0121960
\(766\) 11.1853 0.404140
\(767\) 0 0
\(768\) −0.394335 −0.0142293
\(769\) −25.5588 −0.921675 −0.460838 0.887485i \(-0.652451\pi\)
−0.460838 + 0.887485i \(0.652451\pi\)
\(770\) 0 0
\(771\) 11.5748 0.416855
\(772\) 8.45131 0.304169
\(773\) −8.40077 −0.302155 −0.151077 0.988522i \(-0.548274\pi\)
−0.151077 + 0.988522i \(0.548274\pi\)
\(774\) −5.49394 −0.197476
\(775\) 11.1391 0.400130
\(776\) −5.06783 −0.181925
\(777\) 0 0
\(778\) −10.6970 −0.383506
\(779\) −66.6537 −2.38812
\(780\) 0 0
\(781\) −14.6743 −0.525087
\(782\) −0.311829 −0.0111510
\(783\) 37.6577 1.34578
\(784\) 0 0
\(785\) −20.6345 −0.736476
\(786\) −5.28872 −0.188642
\(787\) −35.0644 −1.24991 −0.624956 0.780660i \(-0.714883\pi\)
−0.624956 + 0.780660i \(0.714883\pi\)
\(788\) −45.2137 −1.61067
\(789\) −9.96001 −0.354586
\(790\) −6.21440 −0.221098
\(791\) 0 0
\(792\) 16.9463 0.602160
\(793\) 0 0
\(794\) 0.598349 0.0212346
\(795\) 0.247639 0.00878284
\(796\) −29.9508 −1.06158
\(797\) −24.5752 −0.870500 −0.435250 0.900310i \(-0.643340\pi\)
−0.435250 + 0.900310i \(0.643340\pi\)
\(798\) 0 0
\(799\) −0.647181 −0.0228956
\(800\) −19.6882 −0.696083
\(801\) 3.98145 0.140678
\(802\) −18.1370 −0.640439
\(803\) 26.2350 0.925814
\(804\) 7.64887 0.269755
\(805\) 0 0
\(806\) 0 0
\(807\) 7.81528 0.275111
\(808\) −21.5099 −0.756717
\(809\) −31.8012 −1.11807 −0.559035 0.829144i \(-0.688828\pi\)
−0.559035 + 0.829144i \(0.688828\pi\)
\(810\) −1.57651 −0.0553927
\(811\) −10.3633 −0.363905 −0.181953 0.983307i \(-0.558242\pi\)
−0.181953 + 0.983307i \(0.558242\pi\)
\(812\) 0 0
\(813\) −2.17811 −0.0763898
\(814\) 1.67254 0.0586226
\(815\) −9.31196 −0.326184
\(816\) −0.309235 −0.0108254
\(817\) 26.5749 0.929737
\(818\) 7.33597 0.256496
\(819\) 0 0
\(820\) −22.0698 −0.770711
\(821\) 4.05592 0.141552 0.0707762 0.997492i \(-0.477452\pi\)
0.0707762 + 0.997492i \(0.477452\pi\)
\(822\) 2.22810 0.0777138
\(823\) 8.71697 0.303854 0.151927 0.988392i \(-0.451452\pi\)
0.151927 + 0.988392i \(0.451452\pi\)
\(824\) −7.81764 −0.272340
\(825\) −13.1901 −0.459220
\(826\) 0 0
\(827\) −23.5788 −0.819915 −0.409957 0.912105i \(-0.634456\pi\)
−0.409957 + 0.912105i \(0.634456\pi\)
\(828\) 17.5061 0.608380
\(829\) −29.0406 −1.00862 −0.504311 0.863522i \(-0.668253\pi\)
−0.504311 + 0.863522i \(0.668253\pi\)
\(830\) 1.49759 0.0519820
\(831\) −0.793427 −0.0275237
\(832\) 0 0
\(833\) 0 0
\(834\) −8.80951 −0.305048
\(835\) −6.39526 −0.221317
\(836\) −38.2534 −1.32302
\(837\) 12.7648 0.441216
\(838\) −2.96666 −0.102482
\(839\) 11.5574 0.399007 0.199503 0.979897i \(-0.436067\pi\)
0.199503 + 0.979897i \(0.436067\pi\)
\(840\) 0 0
\(841\) 41.5196 1.43171
\(842\) 1.31584 0.0453467
\(843\) 5.48238 0.188823
\(844\) −32.0097 −1.10182
\(845\) 0 0
\(846\) −5.19009 −0.178439
\(847\) 0 0
\(848\) −0.716496 −0.0246046
\(849\) 18.8395 0.646572
\(850\) 0.277902 0.00953197
\(851\) 3.70241 0.126917
\(852\) 5.49943 0.188408
\(853\) −1.51845 −0.0519906 −0.0259953 0.999662i \(-0.508275\pi\)
−0.0259953 + 0.999662i \(0.508275\pi\)
\(854\) 0 0
\(855\) 13.0854 0.447511
\(856\) −15.9136 −0.543914
\(857\) −6.00119 −0.204997 −0.102498 0.994733i \(-0.532684\pi\)
−0.102498 + 0.994733i \(0.532684\pi\)
\(858\) 0 0
\(859\) 1.67604 0.0571858 0.0285929 0.999591i \(-0.490897\pi\)
0.0285929 + 0.999591i \(0.490897\pi\)
\(860\) 8.79924 0.300051
\(861\) 0 0
\(862\) 9.41858 0.320798
\(863\) 42.0504 1.43141 0.715706 0.698401i \(-0.246105\pi\)
0.715706 + 0.698401i \(0.246105\pi\)
\(864\) −22.5615 −0.767558
\(865\) −25.2933 −0.859999
\(866\) 9.56747 0.325116
\(867\) −14.4261 −0.489936
\(868\) 0 0
\(869\) 47.3010 1.60458
\(870\) 3.71873 0.126077
\(871\) 0 0
\(872\) 12.0566 0.408290
\(873\) −6.15770 −0.208407
\(874\) 12.0963 0.409164
\(875\) 0 0
\(876\) −9.83201 −0.332193
\(877\) −30.8630 −1.04217 −0.521084 0.853505i \(-0.674472\pi\)
−0.521084 + 0.853505i \(0.674472\pi\)
\(878\) −0.632537 −0.0213471
\(879\) 20.5625 0.693557
\(880\) −10.5983 −0.357270
\(881\) −55.7338 −1.87772 −0.938860 0.344298i \(-0.888117\pi\)
−0.938860 + 0.344298i \(0.888117\pi\)
\(882\) 0 0
\(883\) 2.92007 0.0982681 0.0491341 0.998792i \(-0.484354\pi\)
0.0491341 + 0.998792i \(0.484354\pi\)
\(884\) 0 0
\(885\) −9.54904 −0.320988
\(886\) −10.4693 −0.351724
\(887\) 1.87909 0.0630937 0.0315468 0.999502i \(-0.489957\pi\)
0.0315468 + 0.999502i \(0.489957\pi\)
\(888\) −1.34317 −0.0450737
\(889\) 0 0
\(890\) 0.910916 0.0305340
\(891\) 11.9996 0.402002
\(892\) −20.1404 −0.674349
\(893\) 25.1051 0.840111
\(894\) −0.00483239 −0.000161619 0
\(895\) 4.31246 0.144150
\(896\) 0 0
\(897\) 0 0
\(898\) −8.93138 −0.298044
\(899\) 23.9039 0.797241
\(900\) −15.6015 −0.520050
\(901\) 0.0397131 0.00132304
\(902\) −23.9964 −0.798991
\(903\) 0 0
\(904\) −20.5557 −0.683672
\(905\) −8.19978 −0.272570
\(906\) −8.03082 −0.266806
\(907\) −35.6164 −1.18262 −0.591311 0.806444i \(-0.701389\pi\)
−0.591311 + 0.806444i \(0.701389\pi\)
\(908\) 31.3331 1.03982
\(909\) −26.1358 −0.866869
\(910\) 0 0
\(911\) 44.2844 1.46721 0.733604 0.679577i \(-0.237837\pi\)
0.733604 + 0.679577i \(0.237837\pi\)
\(912\) 11.9957 0.397217
\(913\) −11.3989 −0.377249
\(914\) −3.28290 −0.108589
\(915\) 5.19327 0.171684
\(916\) −6.76118 −0.223396
\(917\) 0 0
\(918\) 0.318459 0.0105107
\(919\) 26.1128 0.861382 0.430691 0.902499i \(-0.358270\pi\)
0.430691 + 0.902499i \(0.358270\pi\)
\(920\) 8.58259 0.282959
\(921\) 20.6389 0.680074
\(922\) −2.55675 −0.0842021
\(923\) 0 0
\(924\) 0 0
\(925\) −3.29960 −0.108490
\(926\) −16.6615 −0.547531
\(927\) −9.49887 −0.311984
\(928\) −42.2497 −1.38691
\(929\) 10.0231 0.328849 0.164424 0.986390i \(-0.447423\pi\)
0.164424 + 0.986390i \(0.447423\pi\)
\(930\) 1.26053 0.0413345
\(931\) 0 0
\(932\) −43.8626 −1.43677
\(933\) 3.38745 0.110900
\(934\) −6.47455 −0.211854
\(935\) 0.587432 0.0192111
\(936\) 0 0
\(937\) −0.916838 −0.0299518 −0.0149759 0.999888i \(-0.504767\pi\)
−0.0149759 + 0.999888i \(0.504767\pi\)
\(938\) 0 0
\(939\) 24.1928 0.789501
\(940\) 8.31259 0.271127
\(941\) 17.0896 0.557107 0.278553 0.960421i \(-0.410145\pi\)
0.278553 + 0.960421i \(0.410145\pi\)
\(942\) 8.40817 0.273953
\(943\) −53.1193 −1.72980
\(944\) 27.6284 0.899227
\(945\) 0 0
\(946\) 9.56735 0.311062
\(947\) −21.7555 −0.706957 −0.353479 0.935443i \(-0.615001\pi\)
−0.353479 + 0.935443i \(0.615001\pi\)
\(948\) −17.7268 −0.575741
\(949\) 0 0
\(950\) −10.7802 −0.349757
\(951\) −14.2870 −0.463288
\(952\) 0 0
\(953\) −17.8081 −0.576862 −0.288431 0.957501i \(-0.593134\pi\)
−0.288431 + 0.957501i \(0.593134\pi\)
\(954\) 0.318481 0.0103112
\(955\) 6.75385 0.218549
\(956\) 13.6582 0.441737
\(957\) −28.3052 −0.914976
\(958\) −13.5056 −0.436347
\(959\) 0 0
\(960\) 2.31132 0.0745975
\(961\) −22.8973 −0.738623
\(962\) 0 0
\(963\) −19.3359 −0.623089
\(964\) −38.0895 −1.22678
\(965\) 5.03441 0.162063
\(966\) 0 0
\(967\) −35.3813 −1.13779 −0.568893 0.822412i \(-0.692628\pi\)
−0.568893 + 0.822412i \(0.692628\pi\)
\(968\) −8.88637 −0.285619
\(969\) −0.664883 −0.0213591
\(970\) −1.40882 −0.0452344
\(971\) 49.7067 1.59516 0.797582 0.603211i \(-0.206112\pi\)
0.797582 + 0.603211i \(0.206112\pi\)
\(972\) −28.0401 −0.899385
\(973\) 0 0
\(974\) −16.0415 −0.514002
\(975\) 0 0
\(976\) −15.0257 −0.480962
\(977\) −61.2801 −1.96052 −0.980262 0.197702i \(-0.936652\pi\)
−0.980262 + 0.197702i \(0.936652\pi\)
\(978\) 3.79445 0.121333
\(979\) −6.93346 −0.221594
\(980\) 0 0
\(981\) 14.6495 0.467723
\(982\) −14.3016 −0.456383
\(983\) −6.98962 −0.222934 −0.111467 0.993768i \(-0.535555\pi\)
−0.111467 + 0.993768i \(0.535555\pi\)
\(984\) 19.2707 0.614328
\(985\) −26.9336 −0.858176
\(986\) 0.596362 0.0189920
\(987\) 0 0
\(988\) 0 0
\(989\) 21.1787 0.673443
\(990\) 4.71094 0.149723
\(991\) −32.7341 −1.03983 −0.519917 0.854217i \(-0.674037\pi\)
−0.519917 + 0.854217i \(0.674037\pi\)
\(992\) −14.3213 −0.454703
\(993\) −5.27532 −0.167407
\(994\) 0 0
\(995\) −17.8416 −0.565616
\(996\) 4.27193 0.135361
\(997\) 57.7897 1.83022 0.915109 0.403207i \(-0.132104\pi\)
0.915109 + 0.403207i \(0.132104\pi\)
\(998\) 0.899058 0.0284592
\(999\) −3.78114 −0.119630
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 8281.2.a.cp.1.8 12
7.2 even 3 1183.2.e.j.508.5 24
7.4 even 3 1183.2.e.j.170.5 24
7.6 odd 2 8281.2.a.co.1.8 12
13.6 odd 12 637.2.q.g.491.3 12
13.11 odd 12 637.2.q.g.589.3 12
13.12 even 2 inner 8281.2.a.cp.1.5 12
91.6 even 12 637.2.q.i.491.3 12
91.11 odd 12 91.2.u.b.30.4 yes 12
91.19 even 12 637.2.u.g.361.4 12
91.24 even 12 637.2.u.g.30.4 12
91.25 even 6 1183.2.e.j.170.8 24
91.32 odd 12 91.2.k.b.23.3 yes 12
91.37 odd 12 91.2.k.b.4.4 12
91.45 even 12 637.2.k.i.569.3 12
91.51 even 6 1183.2.e.j.508.8 24
91.58 odd 12 91.2.u.b.88.4 yes 12
91.76 even 12 637.2.q.i.589.3 12
91.89 even 12 637.2.k.i.459.4 12
91.90 odd 2 8281.2.a.co.1.5 12
273.11 even 12 819.2.do.e.667.3 12
273.32 even 12 819.2.bm.f.478.4 12
273.128 even 12 819.2.bm.f.550.3 12
273.149 even 12 819.2.do.e.361.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.k.b.4.4 12 91.37 odd 12
91.2.k.b.23.3 yes 12 91.32 odd 12
91.2.u.b.30.4 yes 12 91.11 odd 12
91.2.u.b.88.4 yes 12 91.58 odd 12
637.2.k.i.459.4 12 91.89 even 12
637.2.k.i.569.3 12 91.45 even 12
637.2.q.g.491.3 12 13.6 odd 12
637.2.q.g.589.3 12 13.11 odd 12
637.2.q.i.491.3 12 91.6 even 12
637.2.q.i.589.3 12 91.76 even 12
637.2.u.g.30.4 12 91.24 even 12
637.2.u.g.361.4 12 91.19 even 12
819.2.bm.f.478.4 12 273.32 even 12
819.2.bm.f.550.3 12 273.128 even 12
819.2.do.e.361.3 12 273.149 even 12
819.2.do.e.667.3 12 273.11 even 12
1183.2.e.j.170.5 24 7.4 even 3
1183.2.e.j.170.8 24 91.25 even 6
1183.2.e.j.508.5 24 7.2 even 3
1183.2.e.j.508.8 24 91.51 even 6
8281.2.a.co.1.5 12 91.90 odd 2
8281.2.a.co.1.8 12 7.6 odd 2
8281.2.a.cp.1.5 12 13.12 even 2 inner
8281.2.a.cp.1.8 12 1.1 even 1 trivial