Properties

Label 8281.2.a.cp.1.5
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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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 1.5
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 \(-0.556566\pi\)
−0.176772 + 0.984252i \(0.556566\pi\)
\(3\) 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 \(-0.425112\pi\)
0.233105 + 0.972452i \(0.425112\pi\)
\(6\) −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 \(-0.704075\pi\)
−0.598094 + 0.801426i \(0.704075\pi\)
\(12\) −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 \(-0.717771\pi\)
−0.632014 + 0.774957i \(0.717771\pi\)
\(20\) −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 \(-0.417719\pi\)
0.255625 + 0.966776i \(0.417719\pi\)
\(32\) −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 \(-0.522080\pi\)
−0.0693095 + 0.997595i \(0.522080\pi\)
\(38\) 2.75481 0.446889
\(39\) 0 0
\(40\) 1.95460 0.309049
\(41\) 12.0974 1.88929 0.944647 0.328089i \(-0.106405\pi\)
0.944647 + 0.328089i \(0.106405\pi\)
\(42\) 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 \(-0.607830\pi\)
−0.332315 + 0.943168i \(0.607830\pi\)
\(48\) 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 \(-0.747627\pi\)
−0.701815 + 0.712359i \(0.747627\pi\)
\(60\) −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 \(-0.398247\pi\)
0.314248 + 0.949341i \(0.398247\pi\)
\(68\) 0.248564 0.0301428
\(69\) 3.73058 0.449109
\(70\) 0 0
\(71\) 3.69880 0.438967 0.219484 0.975616i \(-0.429563\pi\)
0.219484 + 0.975616i \(0.429563\pi\)
\(72\) −4.27148 −0.503399
\(73\) −6.61281 −0.773970 −0.386985 0.922086i \(-0.626484\pi\)
−0.386985 + 0.922086i \(0.626484\pi\)
\(74\) 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 \(-0.449596\pi\)
0.157688 + 0.987489i \(0.449596\pi\)
\(84\) 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 \(-0.470474\pi\)
0.0926252 + 0.995701i \(0.470474\pi\)
\(90\) 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 \(-0.543816\pi\)
−0.137219 + 0.990541i \(0.543816\pi\)
\(98\) 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 \(-0.400354\pi\)
0.307958 + 0.951400i \(0.400354\pi\)
\(110\) 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 \(-0.571932\pi\)
−0.224063 + 0.974575i \(0.571932\pi\)
\(138\) −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 \(-0.499852\pi\)
0.000465978 1.00000i \(0.499852\pi\)
\(150\) 1.66231 0.135727
\(151\) 18.9054 1.53850 0.769251 0.638947i \(-0.220630\pi\)
0.769251 + 0.638947i \(0.220630\pi\)
\(152\) −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 \(-0.613759\pi\)
−0.349825 + 0.936815i \(0.613759\pi\)
\(164\) −21.1706 −1.65314
\(165\) −3.51380 −0.273549
\(166\) −1.43657 −0.111499
\(167\) −6.13469 −0.474716 −0.237358 0.971422i \(-0.576281\pi\)
−0.237358 + 0.971422i \(0.576281\pi\)
\(168\) 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 \(-0.444392\pi\)
0.173810 + 0.984779i \(0.444392\pi\)
\(194\) 1.35142 0.0970261
\(195\) 0 0
\(196\) 0 0
\(197\) −25.8362 −1.84075 −0.920377 0.391032i \(-0.872118\pi\)
−0.920377 + 0.391032i \(0.872118\pi\)
\(198\) −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 \(-0.625916\pi\)
−0.385340 + 0.922775i \(0.625916\pi\)
\(224\) 0 0
\(225\) 8.91507 0.594338
\(226\) −5.48150 −0.364624
\(227\) 17.9045 1.18836 0.594181 0.804332i \(-0.297476\pi\)
0.594181 + 0.804332i \(0.297476\pi\)
\(228\) 8.19200 0.542528
\(229\) −3.86350 −0.255307 −0.127654 0.991819i \(-0.540745\pi\)
−0.127654 + 0.991819i \(0.540745\pi\)
\(230\) −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 \(-0.418774\pi\)
0.252419 + 0.967618i \(0.418774\pi\)
\(240\) 2.26964 0.146505
\(241\) −21.7653 −1.40202 −0.701012 0.713150i \(-0.747268\pi\)
−0.701012 + 0.713150i \(0.747268\pi\)
\(242\) −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 \(-0.475189\pi\)
0.0778665 + 0.996964i \(0.475189\pi\)
\(272\) −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 \(-0.561651\pi\)
−0.192473 + 0.981302i \(0.561651\pi\)
\(282\) 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 \(-0.749936\pi\)
−0.706964 + 0.707249i \(0.749936\pi\)
\(294\) 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 \(-0.743809\pi\)
−0.693220 + 0.720726i \(0.743809\pi\)
\(308\) 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 \(-0.343444\pi\)
0.472244 + 0.881468i \(0.343444\pi\)
\(318\) 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 \(-0.445415\pi\)
0.170644 + 0.985333i \(0.445415\pi\)
\(332\) −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 \(-0.357645\pi\)
0.432462 + 0.901652i \(0.357645\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 19.9602 1.06388
\(353\) 11.8424 0.630306 0.315153 0.949041i \(-0.397944\pi\)
0.315153 + 0.949041i \(0.397944\pi\)
\(354\) 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 \(-0.810350\pi\)
−0.827698 + 0.561174i \(0.810350\pi\)
\(360\) −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 \(-0.567019\pi\)
−0.208995 + 0.977917i \(0.567019\pi\)
\(380\) 10.0517 0.515642
\(381\) −1.70589 −0.0873955
\(382\) 3.23925 0.165734
\(383\) −22.3711 −1.14311 −0.571555 0.820564i \(-0.693660\pi\)
−0.571555 + 0.820564i \(0.693660\pi\)
\(384\) 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 \(-0.509561\pi\)
−0.0300311 + 0.999549i \(0.509561\pi\)
\(398\) −8.55708 −0.428928
\(399\) 0 0
\(400\) −10.0280 −0.501399
\(401\) 36.2749 1.81148 0.905741 0.423831i \(-0.139315\pi\)
0.905741 + 0.423831i \(0.139315\pi\)
\(402\) −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 \(-0.618162\pi\)
−0.362750 + 0.931887i \(0.618162\pi\)
\(410\) −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 \(-0.520428\pi\)
−0.0641317 + 0.997941i \(0.520428\pi\)
\(422\) −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 \(-0.649892\pi\)
−0.453689 + 0.891160i \(0.649892\pi\)
\(432\) −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 \(-0.361501\pi\)
0.421509 + 0.906824i \(0.361501\pi\)
\(450\) −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 \(-0.450922\pi\)
0.153571 + 0.988138i \(0.450922\pi\)
\(458\) 1.93170 0.0902624
\(459\) 0.636936 0.0297296
\(460\) −8.01066 −0.373499
\(461\) 5.11364 0.238166 0.119083 0.992884i \(-0.462005\pi\)
0.119083 + 0.992884i \(0.462005\pi\)
\(462\) 0 0
\(463\) 33.3239 1.54869 0.774347 0.632761i \(-0.218079\pi\)
0.774347 + 0.632761i \(0.218079\pi\)
\(464\) −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 \(-0.288306\pi\)
0.617104 + 0.786882i \(0.288306\pi\)
\(480\) −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 \(-0.240947\pi\)
0.726928 + 0.686713i \(0.240947\pi\)
\(488\) −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 \(-0.512815\pi\)
−0.0402484 + 0.999190i \(0.512815\pi\)
\(500\) 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 \(-0.671825\pi\)
−0.513969 + 0.857809i \(0.671825\pi\)
\(510\) 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 \(-0.662009\pi\)
−0.487275 + 0.873248i \(0.662009\pi\)
\(542\) −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 \(-0.422855\pi\)
0.239993 + 0.970775i \(0.422855\pi\)
\(558\) 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 \(-0.329095\pi\)
0.511487 + 0.859291i \(0.329095\pi\)
\(578\) 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 \(-0.360752\pi\)
0.423639 + 0.905831i \(0.360752\pi\)
\(588\) 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 \(-0.212119\pi\)
0.786057 + 0.618154i \(0.212119\pi\)
\(594\) −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 \(-0.734986\pi\)
−0.672980 + 0.739660i \(0.734986\pi\)
\(614\) 12.1459 0.490168
\(615\) 10.7145 0.432051
\(616\) 0 0
\(617\) −6.76038 −0.272162 −0.136081 0.990698i \(-0.543451\pi\)
−0.136081 + 0.990698i \(0.543451\pi\)
\(618\) −1.77116 −0.0712466
\(619\) −17.6186 −0.708152 −0.354076 0.935217i \(-0.615205\pi\)
−0.354076 + 0.935217i \(0.615205\pi\)
\(620\) −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 \(-0.398486\pi\)
0.313537 + 0.949576i \(0.398486\pi\)
\(632\) 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 \(-0.378229\pi\)
0.373293 + 0.927713i \(0.378229\pi\)
\(644\) 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 \(-0.528470\pi\)
−0.0893231 + 0.996003i \(0.528470\pi\)
\(662\) −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 \(-0.656577\pi\)
−0.472302 + 0.881437i \(0.656577\pi\)
\(684\) −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 \(-0.431314\pi\)
0.214113 + 0.976809i \(0.431314\pi\)
\(692\) −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 \(-0.647304\pi\)
−0.446427 + 0.894820i \(0.647304\pi\)
\(710\) −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 \(-0.530949\pi\)
−0.0970761 + 0.995277i \(0.530949\pi\)
\(734\) −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 \(-0.458018\pi\)
0.131509 + 0.991315i \(0.458018\pi\)
\(740\) 1.53826 0.0565477
\(741\) 0 0
\(742\) 0 0
\(743\) −0.713641 −0.0261810 −0.0130905 0.999914i \(-0.504167\pi\)
−0.0130905 + 0.999914i \(0.504167\pi\)
\(744\) 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 \(-0.548185\pi\)
−0.150800 + 0.988564i \(0.548185\pi\)
\(762\) 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 \(-0.347549\pi\)
0.460838 + 0.887485i \(0.347549\pi\)
\(770\) 0 0
\(771\) 11.5748 0.416855
\(772\) −8.45131 −0.304169
\(773\) 8.40077 0.302155 0.151077 0.988522i \(-0.451726\pi\)
0.151077 + 0.988522i \(0.451726\pi\)
\(774\) 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 \(-0.285117\pi\)
0.624956 + 0.780660i \(0.285117\pi\)
\(788\) 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 \(-0.441758\pi\)
0.181953 + 0.983307i \(0.441758\pi\)
\(812\) 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 \(-0.522548\pi\)
−0.0707762 + 0.997492i \(0.522548\pi\)
\(822\) 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 \(-0.365544\pi\)
0.409957 + 0.912105i \(0.365544\pi\)
\(828\) 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 \(-0.563933\pi\)
−0.199503 + 0.979897i \(0.563933\pi\)
\(840\) 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 \(-0.491725\pi\)
0.0259953 + 0.999662i \(0.491725\pi\)
\(854\) 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 \(-0.753895\pi\)
−0.715706 + 0.698401i \(0.753895\pi\)
\(864\) 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 \(-0.325528\pi\)
0.521084 + 0.853505i \(0.325528\pi\)
\(878\) 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 \(-0.552577\pi\)
−0.164424 + 0.986390i \(0.552577\pi\)
\(930\) −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 \(-0.589855\pi\)
−0.278553 + 0.960421i \(0.589855\pi\)
\(942\) −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 \(-0.384999\pi\)
0.353479 + 0.935443i \(0.384999\pi\)
\(948\) −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 \(-0.307372\pi\)
0.568893 + 0.822412i \(0.307372\pi\)
\(968\) 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 \(-0.0633477\pi\)
0.980262 + 0.197702i \(0.0633477\pi\)
\(978\) 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 \(-0.464445\pi\)
0.111467 + 0.993768i \(0.464445\pi\)
\(984\) 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.1.5 12
7.2 even 3 1183.2.e.j.508.8 24
7.4 even 3 1183.2.e.j.170.8 24
7.6 odd 2 8281.2.a.co.1.5 12
13.2 odd 12 637.2.q.g.589.3 12
13.7 odd 12 637.2.q.g.491.3 12
13.12 even 2 inner 8281.2.a.cp.1.8 12
91.2 odd 12 91.2.k.b.4.4 12
91.20 even 12 637.2.q.i.491.3 12
91.25 even 6 1183.2.e.j.170.5 24
91.33 even 12 637.2.u.g.361.4 12
91.41 even 12 637.2.q.i.589.3 12
91.46 odd 12 91.2.k.b.23.3 yes 12
91.51 even 6 1183.2.e.j.508.5 24
91.54 even 12 637.2.k.i.459.4 12
91.59 even 12 637.2.k.i.569.3 12
91.67 odd 12 91.2.u.b.30.4 yes 12
91.72 odd 12 91.2.u.b.88.4 yes 12
91.80 even 12 637.2.u.g.30.4 12
91.90 odd 2 8281.2.a.co.1.8 12
273.2 even 12 819.2.bm.f.550.3 12
273.137 even 12 819.2.bm.f.478.4 12
273.158 even 12 819.2.do.e.667.3 12
273.254 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 91.2 odd 12
91.2.k.b.23.3 yes 12 91.46 odd 12
91.2.u.b.30.4 yes 12 91.67 odd 12
91.2.u.b.88.4 yes 12 91.72 odd 12
637.2.k.i.459.4 12 91.54 even 12
637.2.k.i.569.3 12 91.59 even 12
637.2.q.g.491.3 12 13.7 odd 12
637.2.q.g.589.3 12 13.2 odd 12
637.2.q.i.491.3 12 91.20 even 12
637.2.q.i.589.3 12 91.41 even 12
637.2.u.g.30.4 12 91.80 even 12
637.2.u.g.361.4 12 91.33 even 12
819.2.bm.f.478.4 12 273.137 even 12
819.2.bm.f.550.3 12 273.2 even 12
819.2.do.e.361.3 12 273.254 even 12
819.2.do.e.667.3 12 273.158 even 12
1183.2.e.j.170.5 24 91.25 even 6
1183.2.e.j.170.8 24 7.4 even 3
1183.2.e.j.508.5 24 91.51 even 6
1183.2.e.j.508.8 24 7.2 even 3
8281.2.a.co.1.5 12 7.6 odd 2
8281.2.a.co.1.8 12 91.90 odd 2
8281.2.a.cp.1.5 12 1.1 even 1 trivial
8281.2.a.cp.1.8 12 13.12 even 2 inner