Properties

Label 3969.2.a.t.1.2
Level $3969$
Weight $2$
Character 3969.1
Self dual yes
Analytic conductor $31.693$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.6926245622\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.14013.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 6x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 567)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.53652\) of defining polynomial
Character \(\chi\) \(=\) 3969.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.53652 q^{2} +0.360904 q^{4} +3.15761 q^{5} +2.51851 q^{8} +O(q^{10})\) \(q-1.53652 q^{2} +0.360904 q^{4} +3.15761 q^{5} +2.51851 q^{8} -4.85173 q^{10} +5.74916 q^{11} +0.360904 q^{13} -4.59156 q^{16} +2.77684 q^{17} +7.23065 q^{19} +1.13959 q^{20} -8.83372 q^{22} +0.824381 q^{23} +4.97047 q^{25} -0.554537 q^{26} +4.27819 q^{29} +4.98199 q^{31} +2.01801 q^{32} -4.26668 q^{34} +7.49083 q^{37} -11.1101 q^{38} +7.95246 q^{40} -3.33138 q^{41} -7.86975 q^{43} +2.07489 q^{44} -1.26668 q^{46} -3.48149 q^{47} -7.63725 q^{50} +0.130252 q^{52} +2.91544 q^{53} +18.1536 q^{55} -6.57354 q^{58} +2.39878 q^{59} -3.20113 q^{61} -7.65494 q^{62} +6.08239 q^{64} +1.13959 q^{65} +1.89927 q^{67} +1.00217 q^{68} -1.60957 q^{71} +15.4138 q^{73} -11.5098 q^{74} +2.60957 q^{76} +5.47282 q^{79} -14.4983 q^{80} +5.11874 q^{82} -13.0348 q^{83} +8.76817 q^{85} +12.0921 q^{86} +14.4793 q^{88} -14.2677 q^{89} +0.297522 q^{92} +5.34939 q^{94} +22.8315 q^{95} -16.0053 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{8} + 7 q^{10} - 5 q^{11} + 5 q^{13} - q^{16} + 6 q^{17} + 8 q^{19} - 8 q^{20} - 7 q^{22} + 12 q^{23} + 8 q^{25} + q^{26} + 10 q^{29} + 18 q^{31} + 10 q^{32} - 20 q^{38} + 18 q^{40} - 5 q^{41} - 7 q^{43} - 13 q^{44} + 12 q^{46} - 21 q^{47} - 38 q^{50} + 25 q^{52} + 12 q^{53} + 26 q^{55} - 7 q^{58} + 6 q^{59} + 20 q^{61} + 18 q^{62} - 23 q^{64} - 8 q^{65} - 5 q^{67} + 51 q^{68} + 9 q^{71} + 6 q^{73} - 5 q^{76} - 10 q^{79} - 2 q^{80} + 35 q^{82} + 9 q^{83} - 9 q^{85} + 22 q^{86} + 18 q^{88} - 22 q^{89} + 36 q^{92} + 15 q^{94} + 16 q^{95} + 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.53652 −1.08649 −0.543243 0.839576i \(-0.682804\pi\)
−0.543243 + 0.839576i \(0.682804\pi\)
\(3\) 0 0
\(4\) 0.360904 0.180452
\(5\) 3.15761 1.41212 0.706062 0.708150i \(-0.250470\pi\)
0.706062 + 0.708150i \(0.250470\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.51851 0.890428
\(9\) 0 0
\(10\) −4.85173 −1.53425
\(11\) 5.74916 1.73344 0.866719 0.498797i \(-0.166225\pi\)
0.866719 + 0.498797i \(0.166225\pi\)
\(12\) 0 0
\(13\) 0.360904 0.100097 0.0500484 0.998747i \(-0.484062\pi\)
0.0500484 + 0.998747i \(0.484062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.59156 −1.14789
\(17\) 2.77684 0.673483 0.336741 0.941597i \(-0.390675\pi\)
0.336741 + 0.941597i \(0.390675\pi\)
\(18\) 0 0
\(19\) 7.23065 1.65883 0.829413 0.558636i \(-0.188675\pi\)
0.829413 + 0.558636i \(0.188675\pi\)
\(20\) 1.13959 0.254820
\(21\) 0 0
\(22\) −8.83372 −1.88336
\(23\) 0.824381 0.171895 0.0859476 0.996300i \(-0.472608\pi\)
0.0859476 + 0.996300i \(0.472608\pi\)
\(24\) 0 0
\(25\) 4.97047 0.994095
\(26\) −0.554537 −0.108754
\(27\) 0 0
\(28\) 0 0
\(29\) 4.27819 0.794440 0.397220 0.917723i \(-0.369975\pi\)
0.397220 + 0.917723i \(0.369975\pi\)
\(30\) 0 0
\(31\) 4.98199 0.894791 0.447396 0.894336i \(-0.352352\pi\)
0.447396 + 0.894336i \(0.352352\pi\)
\(32\) 2.01801 0.356738
\(33\) 0 0
\(34\) −4.26668 −0.731730
\(35\) 0 0
\(36\) 0 0
\(37\) 7.49083 1.23149 0.615743 0.787947i \(-0.288856\pi\)
0.615743 + 0.787947i \(0.288856\pi\)
\(38\) −11.1101 −1.80229
\(39\) 0 0
\(40\) 7.95246 1.25739
\(41\) −3.33138 −0.520274 −0.260137 0.965572i \(-0.583768\pi\)
−0.260137 + 0.965572i \(0.583768\pi\)
\(42\) 0 0
\(43\) −7.86975 −1.20013 −0.600063 0.799953i \(-0.704858\pi\)
−0.600063 + 0.799953i \(0.704858\pi\)
\(44\) 2.07489 0.312802
\(45\) 0 0
\(46\) −1.26668 −0.186762
\(47\) −3.48149 −0.507828 −0.253914 0.967227i \(-0.581718\pi\)
−0.253914 + 0.967227i \(0.581718\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −7.63725 −1.08007
\(51\) 0 0
\(52\) 0.130252 0.0180626
\(53\) 2.91544 0.400467 0.200233 0.979748i \(-0.435830\pi\)
0.200233 + 0.979748i \(0.435830\pi\)
\(54\) 0 0
\(55\) 18.1536 2.44783
\(56\) 0 0
\(57\) 0 0
\(58\) −6.57354 −0.863148
\(59\) 2.39878 0.312294 0.156147 0.987734i \(-0.450093\pi\)
0.156147 + 0.987734i \(0.450093\pi\)
\(60\) 0 0
\(61\) −3.20113 −0.409862 −0.204931 0.978776i \(-0.565697\pi\)
−0.204931 + 0.978776i \(0.565697\pi\)
\(62\) −7.65494 −0.972178
\(63\) 0 0
\(64\) 6.08239 0.760298
\(65\) 1.13959 0.141349
\(66\) 0 0
\(67\) 1.89927 0.232033 0.116017 0.993247i \(-0.462987\pi\)
0.116017 + 0.993247i \(0.462987\pi\)
\(68\) 1.00217 0.121531
\(69\) 0 0
\(70\) 0 0
\(71\) −1.60957 −0.191021 −0.0955104 0.995428i \(-0.530448\pi\)
−0.0955104 + 0.995428i \(0.530448\pi\)
\(72\) 0 0
\(73\) 15.4138 1.80404 0.902022 0.431689i \(-0.142082\pi\)
0.902022 + 0.431689i \(0.142082\pi\)
\(74\) −11.5098 −1.33799
\(75\) 0 0
\(76\) 2.60957 0.299338
\(77\) 0 0
\(78\) 0 0
\(79\) 5.47282 0.615740 0.307870 0.951428i \(-0.400384\pi\)
0.307870 + 0.951428i \(0.400384\pi\)
\(80\) −14.4983 −1.62096
\(81\) 0 0
\(82\) 5.11874 0.565270
\(83\) −13.0348 −1.43076 −0.715380 0.698735i \(-0.753746\pi\)
−0.715380 + 0.698735i \(0.753746\pi\)
\(84\) 0 0
\(85\) 8.76817 0.951041
\(86\) 12.0921 1.30392
\(87\) 0 0
\(88\) 14.4793 1.54350
\(89\) −14.2677 −1.51237 −0.756185 0.654358i \(-0.772939\pi\)
−0.756185 + 0.654358i \(0.772939\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.297522 0.0310188
\(93\) 0 0
\(94\) 5.34939 0.551748
\(95\) 22.8315 2.34247
\(96\) 0 0
\(97\) −16.0053 −1.62509 −0.812547 0.582896i \(-0.801920\pi\)
−0.812547 + 0.582896i \(0.801920\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.79386 0.179386
\(101\) 3.00749 0.299257 0.149628 0.988742i \(-0.452192\pi\)
0.149628 + 0.988742i \(0.452192\pi\)
\(102\) 0 0
\(103\) −9.72063 −0.957802 −0.478901 0.877869i \(-0.658965\pi\)
−0.478901 + 0.877869i \(0.658965\pi\)
\(104\) 0.908940 0.0891289
\(105\) 0 0
\(106\) −4.47964 −0.435101
\(107\) −10.4243 −1.00775 −0.503877 0.863775i \(-0.668093\pi\)
−0.503877 + 0.863775i \(0.668093\pi\)
\(108\) 0 0
\(109\) −4.67427 −0.447714 −0.223857 0.974622i \(-0.571865\pi\)
−0.223857 + 0.974622i \(0.571865\pi\)
\(110\) −27.8934 −2.65953
\(111\) 0 0
\(112\) 0 0
\(113\) −4.68664 −0.440882 −0.220441 0.975400i \(-0.570750\pi\)
−0.220441 + 0.975400i \(0.570750\pi\)
\(114\) 0 0
\(115\) 2.60307 0.242737
\(116\) 1.54402 0.143358
\(117\) 0 0
\(118\) −3.68578 −0.339304
\(119\) 0 0
\(120\) 0 0
\(121\) 22.0529 2.00481
\(122\) 4.91860 0.445310
\(123\) 0 0
\(124\) 1.79802 0.161467
\(125\) −0.0932326 −0.00833898
\(126\) 0 0
\(127\) −9.15945 −0.812770 −0.406385 0.913702i \(-0.633211\pi\)
−0.406385 + 0.913702i \(0.633211\pi\)
\(128\) −13.3818 −1.18279
\(129\) 0 0
\(130\) −1.75101 −0.153574
\(131\) 9.21165 0.804825 0.402413 0.915458i \(-0.368172\pi\)
0.402413 + 0.915458i \(0.368172\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −2.91828 −0.252101
\(135\) 0 0
\(136\) 6.99350 0.599688
\(137\) −6.82438 −0.583046 −0.291523 0.956564i \(-0.594162\pi\)
−0.291523 + 0.956564i \(0.594162\pi\)
\(138\) 0 0
\(139\) 5.68664 0.482334 0.241167 0.970484i \(-0.422470\pi\)
0.241167 + 0.970484i \(0.422470\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.47314 0.207541
\(143\) 2.07489 0.173511
\(144\) 0 0
\(145\) 13.5088 1.12185
\(146\) −23.6836 −1.96007
\(147\) 0 0
\(148\) 2.70347 0.222224
\(149\) −21.5885 −1.76860 −0.884301 0.466918i \(-0.845364\pi\)
−0.884301 + 0.466918i \(0.845364\pi\)
\(150\) 0 0
\(151\) −5.55553 −0.452102 −0.226051 0.974115i \(-0.572582\pi\)
−0.226051 + 0.974115i \(0.572582\pi\)
\(152\) 18.2105 1.47706
\(153\) 0 0
\(154\) 0 0
\(155\) 15.7311 1.26356
\(156\) 0 0
\(157\) −6.07120 −0.484534 −0.242267 0.970210i \(-0.577891\pi\)
−0.242267 + 0.970210i \(0.577891\pi\)
\(158\) −8.40911 −0.668993
\(159\) 0 0
\(160\) 6.37209 0.503758
\(161\) 0 0
\(162\) 0 0
\(163\) −3.85259 −0.301758 −0.150879 0.988552i \(-0.548210\pi\)
−0.150879 + 0.988552i \(0.548210\pi\)
\(164\) −1.20231 −0.0938844
\(165\) 0 0
\(166\) 20.0283 1.55450
\(167\) −3.53837 −0.273807 −0.136904 0.990584i \(-0.543715\pi\)
−0.136904 + 0.990584i \(0.543715\pi\)
\(168\) 0 0
\(169\) −12.8697 −0.989981
\(170\) −13.4725 −1.03329
\(171\) 0 0
\(172\) −2.84022 −0.216565
\(173\) −9.85358 −0.749154 −0.374577 0.927196i \(-0.622212\pi\)
−0.374577 + 0.927196i \(0.622212\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −26.3976 −1.98979
\(177\) 0 0
\(178\) 21.9226 1.64317
\(179\) 19.8971 1.48718 0.743590 0.668636i \(-0.233122\pi\)
0.743590 + 0.668636i \(0.233122\pi\)
\(180\) 0 0
\(181\) −12.0930 −0.898869 −0.449434 0.893313i \(-0.648374\pi\)
−0.449434 + 0.893313i \(0.648374\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2.07621 0.153060
\(185\) 23.6531 1.73901
\(186\) 0 0
\(187\) 15.9645 1.16744
\(188\) −1.25648 −0.0916385
\(189\) 0 0
\(190\) −35.0812 −2.54506
\(191\) −9.71964 −0.703288 −0.351644 0.936134i \(-0.614377\pi\)
−0.351644 + 0.936134i \(0.614377\pi\)
\(192\) 0 0
\(193\) −2.30185 −0.165691 −0.0828454 0.996562i \(-0.526401\pi\)
−0.0828454 + 0.996562i \(0.526401\pi\)
\(194\) 24.5925 1.76564
\(195\) 0 0
\(196\) 0 0
\(197\) −5.06470 −0.360845 −0.180422 0.983589i \(-0.557746\pi\)
−0.180422 + 0.983589i \(0.557746\pi\)
\(198\) 0 0
\(199\) 9.95332 0.705572 0.352786 0.935704i \(-0.385234\pi\)
0.352786 + 0.935704i \(0.385234\pi\)
\(200\) 12.5182 0.885169
\(201\) 0 0
\(202\) −4.62108 −0.325138
\(203\) 0 0
\(204\) 0 0
\(205\) −10.5192 −0.734691
\(206\) 14.9360 1.04064
\(207\) 0 0
\(208\) −1.65711 −0.114900
\(209\) 41.5702 2.87547
\(210\) 0 0
\(211\) −23.3007 −1.60408 −0.802042 0.597267i \(-0.796253\pi\)
−0.802042 + 0.597267i \(0.796253\pi\)
\(212\) 1.05219 0.0722650
\(213\) 0 0
\(214\) 16.0172 1.09491
\(215\) −24.8496 −1.69473
\(216\) 0 0
\(217\) 0 0
\(218\) 7.18212 0.486435
\(219\) 0 0
\(220\) 6.55170 0.441715
\(221\) 1.00217 0.0674134
\(222\) 0 0
\(223\) 11.8754 0.795235 0.397618 0.917551i \(-0.369837\pi\)
0.397618 + 0.917551i \(0.369837\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 7.20113 0.479012
\(227\) 5.96081 0.395633 0.197816 0.980239i \(-0.436615\pi\)
0.197816 + 0.980239i \(0.436615\pi\)
\(228\) 0 0
\(229\) 23.7206 1.56750 0.783752 0.621074i \(-0.213304\pi\)
0.783752 + 0.621074i \(0.213304\pi\)
\(230\) −3.99968 −0.263731
\(231\) 0 0
\(232\) 10.7747 0.707392
\(233\) 13.8592 0.907948 0.453974 0.891015i \(-0.350006\pi\)
0.453974 + 0.891015i \(0.350006\pi\)
\(234\) 0 0
\(235\) −10.9932 −0.717116
\(236\) 0.865728 0.0563541
\(237\) 0 0
\(238\) 0 0
\(239\) −22.3426 −1.44522 −0.722610 0.691256i \(-0.757058\pi\)
−0.722610 + 0.691256i \(0.757058\pi\)
\(240\) 0 0
\(241\) −9.72063 −0.626161 −0.313080 0.949727i \(-0.601361\pi\)
−0.313080 + 0.949727i \(0.601361\pi\)
\(242\) −33.8847 −2.17819
\(243\) 0 0
\(244\) −1.15530 −0.0739604
\(245\) 0 0
\(246\) 0 0
\(247\) 2.60957 0.166043
\(248\) 12.5472 0.796747
\(249\) 0 0
\(250\) 0.143254 0.00906018
\(251\) −6.51950 −0.411507 −0.205754 0.978604i \(-0.565965\pi\)
−0.205754 + 0.978604i \(0.565965\pi\)
\(252\) 0 0
\(253\) 4.73950 0.297970
\(254\) 14.0737 0.883063
\(255\) 0 0
\(256\) 8.39661 0.524788
\(257\) 5.53002 0.344953 0.172477 0.985014i \(-0.444823\pi\)
0.172477 + 0.985014i \(0.444823\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0.411283 0.0255067
\(261\) 0 0
\(262\) −14.1539 −0.874431
\(263\) −8.69499 −0.536156 −0.268078 0.963397i \(-0.586388\pi\)
−0.268078 + 0.963397i \(0.586388\pi\)
\(264\) 0 0
\(265\) 9.20581 0.565509
\(266\) 0 0
\(267\) 0 0
\(268\) 0.685455 0.0418709
\(269\) −3.77901 −0.230410 −0.115205 0.993342i \(-0.536753\pi\)
−0.115205 + 0.993342i \(0.536753\pi\)
\(270\) 0 0
\(271\) −6.61990 −0.402130 −0.201065 0.979578i \(-0.564440\pi\)
−0.201065 + 0.979578i \(0.564440\pi\)
\(272\) −12.7500 −0.773083
\(273\) 0 0
\(274\) 10.4858 0.633472
\(275\) 28.5761 1.72320
\(276\) 0 0
\(277\) −25.2658 −1.51808 −0.759038 0.651046i \(-0.774330\pi\)
−0.759038 + 0.651046i \(0.774330\pi\)
\(278\) −8.73765 −0.524049
\(279\) 0 0
\(280\) 0 0
\(281\) −7.42442 −0.442904 −0.221452 0.975171i \(-0.571080\pi\)
−0.221452 + 0.975171i \(0.571080\pi\)
\(282\) 0 0
\(283\) −15.4203 −0.916640 −0.458320 0.888787i \(-0.651549\pi\)
−0.458320 + 0.888787i \(0.651549\pi\)
\(284\) −0.580900 −0.0344701
\(285\) 0 0
\(286\) −3.18812 −0.188518
\(287\) 0 0
\(288\) 0 0
\(289\) −9.28916 −0.546421
\(290\) −20.7567 −1.21887
\(291\) 0 0
\(292\) 5.56289 0.325543
\(293\) 30.5797 1.78649 0.893243 0.449574i \(-0.148424\pi\)
0.893243 + 0.449574i \(0.148424\pi\)
\(294\) 0 0
\(295\) 7.57440 0.440999
\(296\) 18.8657 1.09655
\(297\) 0 0
\(298\) 33.1713 1.92156
\(299\) 0.297522 0.0172061
\(300\) 0 0
\(301\) 0 0
\(302\) 8.53620 0.491203
\(303\) 0 0
\(304\) −33.1999 −1.90415
\(305\) −10.1079 −0.578776
\(306\) 0 0
\(307\) 28.7794 1.64252 0.821262 0.570551i \(-0.193270\pi\)
0.821262 + 0.570551i \(0.193270\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −24.1713 −1.37284
\(311\) −2.57255 −0.145876 −0.0729380 0.997336i \(-0.523238\pi\)
−0.0729380 + 0.997336i \(0.523238\pi\)
\(312\) 0 0
\(313\) 18.5145 1.04650 0.523250 0.852179i \(-0.324719\pi\)
0.523250 + 0.852179i \(0.324719\pi\)
\(314\) 9.32854 0.526440
\(315\) 0 0
\(316\) 1.97516 0.111111
\(317\) −15.5182 −0.871588 −0.435794 0.900046i \(-0.643532\pi\)
−0.435794 + 0.900046i \(0.643532\pi\)
\(318\) 0 0
\(319\) 24.5960 1.37711
\(320\) 19.2058 1.07364
\(321\) 0 0
\(322\) 0 0
\(323\) 20.0784 1.11719
\(324\) 0 0
\(325\) 1.79386 0.0995056
\(326\) 5.91960 0.327856
\(327\) 0 0
\(328\) −8.39011 −0.463266
\(329\) 0 0
\(330\) 0 0
\(331\) 31.5904 1.73636 0.868182 0.496246i \(-0.165289\pi\)
0.868182 + 0.496246i \(0.165289\pi\)
\(332\) −4.70433 −0.258183
\(333\) 0 0
\(334\) 5.43679 0.297488
\(335\) 5.99716 0.327660
\(336\) 0 0
\(337\) −11.4081 −0.621440 −0.310720 0.950502i \(-0.600570\pi\)
−0.310720 + 0.950502i \(0.600570\pi\)
\(338\) 19.7747 1.07560
\(339\) 0 0
\(340\) 3.16446 0.171617
\(341\) 28.6422 1.55106
\(342\) 0 0
\(343\) 0 0
\(344\) −19.8200 −1.06862
\(345\) 0 0
\(346\) 15.1403 0.813945
\(347\) 4.08140 0.219101 0.109550 0.993981i \(-0.465059\pi\)
0.109550 + 0.993981i \(0.465059\pi\)
\(348\) 0 0
\(349\) 24.2779 1.29956 0.649782 0.760120i \(-0.274860\pi\)
0.649782 + 0.760120i \(0.274860\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 11.6019 0.618383
\(353\) −34.0790 −1.81384 −0.906922 0.421299i \(-0.861574\pi\)
−0.906922 + 0.421299i \(0.861574\pi\)
\(354\) 0 0
\(355\) −5.08239 −0.269745
\(356\) −5.14926 −0.272910
\(357\) 0 0
\(358\) −30.5724 −1.61580
\(359\) −5.48931 −0.289715 −0.144857 0.989453i \(-0.546272\pi\)
−0.144857 + 0.989453i \(0.546272\pi\)
\(360\) 0 0
\(361\) 33.2823 1.75170
\(362\) 18.5812 0.976608
\(363\) 0 0
\(364\) 0 0
\(365\) 48.6706 2.54754
\(366\) 0 0
\(367\) 13.6391 0.711955 0.355978 0.934495i \(-0.384148\pi\)
0.355978 + 0.934495i \(0.384148\pi\)
\(368\) −3.78519 −0.197317
\(369\) 0 0
\(370\) −36.3435 −1.88941
\(371\) 0 0
\(372\) 0 0
\(373\) 8.92379 0.462056 0.231028 0.972947i \(-0.425791\pi\)
0.231028 + 0.972947i \(0.425791\pi\)
\(374\) −24.5298 −1.26841
\(375\) 0 0
\(376\) −8.76817 −0.452184
\(377\) 1.54402 0.0795209
\(378\) 0 0
\(379\) 29.7035 1.52576 0.762882 0.646537i \(-0.223784\pi\)
0.762882 + 0.646537i \(0.223784\pi\)
\(380\) 8.23999 0.422703
\(381\) 0 0
\(382\) 14.9344 0.764113
\(383\) −17.7101 −0.904944 −0.452472 0.891779i \(-0.649458\pi\)
−0.452472 + 0.891779i \(0.649458\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.53685 0.180021
\(387\) 0 0
\(388\) −5.77638 −0.293251
\(389\) −21.3255 −1.08125 −0.540624 0.841264i \(-0.681812\pi\)
−0.540624 + 0.841264i \(0.681812\pi\)
\(390\) 0 0
\(391\) 2.28917 0.115768
\(392\) 0 0
\(393\) 0 0
\(394\) 7.78203 0.392053
\(395\) 17.2810 0.869501
\(396\) 0 0
\(397\) 2.23566 0.112205 0.0561024 0.998425i \(-0.482133\pi\)
0.0561024 + 0.998425i \(0.482133\pi\)
\(398\) −15.2935 −0.766594
\(399\) 0 0
\(400\) −22.8222 −1.14111
\(401\) −8.73702 −0.436306 −0.218153 0.975915i \(-0.570003\pi\)
−0.218153 + 0.975915i \(0.570003\pi\)
\(402\) 0 0
\(403\) 1.79802 0.0895656
\(404\) 1.08542 0.0540014
\(405\) 0 0
\(406\) 0 0
\(407\) 43.0660 2.13470
\(408\) 0 0
\(409\) 28.6920 1.41873 0.709363 0.704843i \(-0.248983\pi\)
0.709363 + 0.704843i \(0.248983\pi\)
\(410\) 16.1630 0.798232
\(411\) 0 0
\(412\) −3.50821 −0.172837
\(413\) 0 0
\(414\) 0 0
\(415\) −41.1589 −2.02041
\(416\) 0.728309 0.0357083
\(417\) 0 0
\(418\) −63.8736 −3.12416
\(419\) 8.64442 0.422307 0.211154 0.977453i \(-0.432278\pi\)
0.211154 + 0.977453i \(0.432278\pi\)
\(420\) 0 0
\(421\) 18.4669 0.900024 0.450012 0.893022i \(-0.351420\pi\)
0.450012 + 0.893022i \(0.351420\pi\)
\(422\) 35.8020 1.74282
\(423\) 0 0
\(424\) 7.34257 0.356587
\(425\) 13.8022 0.669506
\(426\) 0 0
\(427\) 0 0
\(428\) −3.76216 −0.181851
\(429\) 0 0
\(430\) 38.1819 1.84130
\(431\) 5.80736 0.279731 0.139865 0.990171i \(-0.455333\pi\)
0.139865 + 0.990171i \(0.455333\pi\)
\(432\) 0 0
\(433\) −3.63877 −0.174868 −0.0874341 0.996170i \(-0.527867\pi\)
−0.0874341 + 0.996170i \(0.527867\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.68696 −0.0807908
\(437\) 5.96081 0.285144
\(438\) 0 0
\(439\) 18.6619 0.890684 0.445342 0.895361i \(-0.353082\pi\)
0.445342 + 0.895361i \(0.353082\pi\)
\(440\) 45.7200 2.17961
\(441\) 0 0
\(442\) −1.53986 −0.0732437
\(443\) 26.0911 1.23962 0.619812 0.784750i \(-0.287209\pi\)
0.619812 + 0.784750i \(0.287209\pi\)
\(444\) 0 0
\(445\) −45.0517 −2.13565
\(446\) −18.2468 −0.864012
\(447\) 0 0
\(448\) 0 0
\(449\) 5.63824 0.266085 0.133042 0.991110i \(-0.457525\pi\)
0.133042 + 0.991110i \(0.457525\pi\)
\(450\) 0 0
\(451\) −19.1526 −0.901862
\(452\) −1.69142 −0.0795579
\(453\) 0 0
\(454\) −9.15892 −0.429849
\(455\) 0 0
\(456\) 0 0
\(457\) −19.9923 −0.935201 −0.467601 0.883940i \(-0.654881\pi\)
−0.467601 + 0.883940i \(0.654881\pi\)
\(458\) −36.4473 −1.70307
\(459\) 0 0
\(460\) 0.939457 0.0438024
\(461\) 37.4495 1.74420 0.872098 0.489332i \(-0.162759\pi\)
0.872098 + 0.489332i \(0.162759\pi\)
\(462\) 0 0
\(463\) 2.45513 0.114099 0.0570497 0.998371i \(-0.481831\pi\)
0.0570497 + 0.998371i \(0.481831\pi\)
\(464\) −19.6436 −0.911929
\(465\) 0 0
\(466\) −21.2950 −0.986473
\(467\) 33.4107 1.54606 0.773032 0.634367i \(-0.218739\pi\)
0.773032 + 0.634367i \(0.218739\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 16.8913 0.779136
\(471\) 0 0
\(472\) 6.04135 0.278076
\(473\) −45.2445 −2.08034
\(474\) 0 0
\(475\) 35.9398 1.64903
\(476\) 0 0
\(477\) 0 0
\(478\) 34.3299 1.57021
\(479\) 34.6936 1.58519 0.792595 0.609748i \(-0.208729\pi\)
0.792595 + 0.609748i \(0.208729\pi\)
\(480\) 0 0
\(481\) 2.70347 0.123268
\(482\) 14.9360 0.680315
\(483\) 0 0
\(484\) 7.95896 0.361771
\(485\) −50.5385 −2.29483
\(486\) 0 0
\(487\) 0.959817 0.0434935 0.0217467 0.999764i \(-0.493077\pi\)
0.0217467 + 0.999764i \(0.493077\pi\)
\(488\) −8.06207 −0.364953
\(489\) 0 0
\(490\) 0 0
\(491\) −15.7738 −0.711862 −0.355931 0.934512i \(-0.615836\pi\)
−0.355931 + 0.934512i \(0.615836\pi\)
\(492\) 0 0
\(493\) 11.8799 0.535042
\(494\) −4.00966 −0.180403
\(495\) 0 0
\(496\) −22.8751 −1.02712
\(497\) 0 0
\(498\) 0 0
\(499\) −19.1287 −0.856320 −0.428160 0.903703i \(-0.640838\pi\)
−0.428160 + 0.903703i \(0.640838\pi\)
\(500\) −0.0336480 −0.00150478
\(501\) 0 0
\(502\) 10.0174 0.447097
\(503\) −33.3898 −1.48878 −0.744388 0.667747i \(-0.767259\pi\)
−0.744388 + 0.667747i \(0.767259\pi\)
\(504\) 0 0
\(505\) 9.49648 0.422588
\(506\) −7.28235 −0.323740
\(507\) 0 0
\(508\) −3.30568 −0.146666
\(509\) 33.5176 1.48564 0.742822 0.669489i \(-0.233487\pi\)
0.742822 + 0.669489i \(0.233487\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 13.8619 0.612617
\(513\) 0 0
\(514\) −8.49701 −0.374787
\(515\) −30.6939 −1.35254
\(516\) 0 0
\(517\) −20.0157 −0.880287
\(518\) 0 0
\(519\) 0 0
\(520\) 2.87007 0.125861
\(521\) −26.7243 −1.17081 −0.585407 0.810740i \(-0.699065\pi\)
−0.585407 + 0.810740i \(0.699065\pi\)
\(522\) 0 0
\(523\) −17.0644 −0.746173 −0.373086 0.927797i \(-0.621701\pi\)
−0.373086 + 0.927797i \(0.621701\pi\)
\(524\) 3.32452 0.145232
\(525\) 0 0
\(526\) 13.3600 0.582526
\(527\) 13.8342 0.602626
\(528\) 0 0
\(529\) −22.3204 −0.970452
\(530\) −14.1449 −0.614417
\(531\) 0 0
\(532\) 0 0
\(533\) −1.20231 −0.0520777
\(534\) 0 0
\(535\) −32.9158 −1.42307
\(536\) 4.78334 0.206609
\(537\) 0 0
\(538\) 5.80654 0.250338
\(539\) 0 0
\(540\) 0 0
\(541\) 12.3426 0.530648 0.265324 0.964159i \(-0.414521\pi\)
0.265324 + 0.964159i \(0.414521\pi\)
\(542\) 10.1716 0.436909
\(543\) 0 0
\(544\) 5.60370 0.240257
\(545\) −14.7595 −0.632227
\(546\) 0 0
\(547\) 23.4424 1.00233 0.501163 0.865353i \(-0.332906\pi\)
0.501163 + 0.865353i \(0.332906\pi\)
\(548\) −2.46294 −0.105212
\(549\) 0 0
\(550\) −43.9078 −1.87223
\(551\) 30.9341 1.31784
\(552\) 0 0
\(553\) 0 0
\(554\) 38.8215 1.64937
\(555\) 0 0
\(556\) 2.05233 0.0870381
\(557\) 24.5115 1.03858 0.519292 0.854597i \(-0.326196\pi\)
0.519292 + 0.854597i \(0.326196\pi\)
\(558\) 0 0
\(559\) −2.84022 −0.120129
\(560\) 0 0
\(561\) 0 0
\(562\) 11.4078 0.481209
\(563\) −13.3714 −0.563538 −0.281769 0.959482i \(-0.590921\pi\)
−0.281769 + 0.959482i \(0.590921\pi\)
\(564\) 0 0
\(565\) −14.7985 −0.622580
\(566\) 23.6936 0.995916
\(567\) 0 0
\(568\) −4.05372 −0.170090
\(569\) 14.2488 0.597341 0.298670 0.954356i \(-0.403457\pi\)
0.298670 + 0.954356i \(0.403457\pi\)
\(570\) 0 0
\(571\) −29.3237 −1.22716 −0.613579 0.789633i \(-0.710271\pi\)
−0.613579 + 0.789633i \(0.710271\pi\)
\(572\) 0.748837 0.0313105
\(573\) 0 0
\(574\) 0 0
\(575\) 4.09756 0.170880
\(576\) 0 0
\(577\) 18.1001 0.753516 0.376758 0.926312i \(-0.377039\pi\)
0.376758 + 0.926312i \(0.377039\pi\)
\(578\) 14.2730 0.593679
\(579\) 0 0
\(580\) 4.87539 0.202440
\(581\) 0 0
\(582\) 0 0
\(583\) 16.7613 0.694184
\(584\) 38.8197 1.60637
\(585\) 0 0
\(586\) −46.9865 −1.94099
\(587\) 6.52804 0.269441 0.134721 0.990884i \(-0.456986\pi\)
0.134721 + 0.990884i \(0.456986\pi\)
\(588\) 0 0
\(589\) 36.0230 1.48430
\(590\) −11.6382 −0.479139
\(591\) 0 0
\(592\) −34.3946 −1.41361
\(593\) −25.4405 −1.04471 −0.522357 0.852727i \(-0.674947\pi\)
−0.522357 + 0.852727i \(0.674947\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −7.79138 −0.319147
\(597\) 0 0
\(598\) −0.457150 −0.0186942
\(599\) −6.17931 −0.252480 −0.126240 0.992000i \(-0.540291\pi\)
−0.126240 + 0.992000i \(0.540291\pi\)
\(600\) 0 0
\(601\) 12.9344 0.527607 0.263804 0.964576i \(-0.415023\pi\)
0.263804 + 0.964576i \(0.415023\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −2.00501 −0.0815828
\(605\) 69.6342 2.83103
\(606\) 0 0
\(607\) 23.0756 0.936608 0.468304 0.883567i \(-0.344865\pi\)
0.468304 + 0.883567i \(0.344865\pi\)
\(608\) 14.5916 0.591766
\(609\) 0 0
\(610\) 15.5310 0.628832
\(611\) −1.25648 −0.0508319
\(612\) 0 0
\(613\) −17.6273 −0.711958 −0.355979 0.934494i \(-0.615853\pi\)
−0.355979 + 0.934494i \(0.615853\pi\)
\(614\) −44.2201 −1.78458
\(615\) 0 0
\(616\) 0 0
\(617\) −21.6441 −0.871358 −0.435679 0.900102i \(-0.643492\pi\)
−0.435679 + 0.900102i \(0.643492\pi\)
\(618\) 0 0
\(619\) 9.57941 0.385029 0.192514 0.981294i \(-0.438336\pi\)
0.192514 + 0.981294i \(0.438336\pi\)
\(620\) 5.67743 0.228011
\(621\) 0 0
\(622\) 3.95278 0.158492
\(623\) 0 0
\(624\) 0 0
\(625\) −25.1468 −1.00587
\(626\) −28.4479 −1.13701
\(627\) 0 0
\(628\) −2.19112 −0.0874352
\(629\) 20.8008 0.829384
\(630\) 0 0
\(631\) −31.1742 −1.24103 −0.620514 0.784196i \(-0.713076\pi\)
−0.620514 + 0.784196i \(0.713076\pi\)
\(632\) 13.7833 0.548272
\(633\) 0 0
\(634\) 23.8441 0.946968
\(635\) −28.9219 −1.14773
\(636\) 0 0
\(637\) 0 0
\(638\) −37.7924 −1.49621
\(639\) 0 0
\(640\) −42.2543 −1.67025
\(641\) 8.25214 0.325940 0.162970 0.986631i \(-0.447893\pi\)
0.162970 + 0.986631i \(0.447893\pi\)
\(642\) 0 0
\(643\) −24.4318 −0.963495 −0.481748 0.876310i \(-0.659998\pi\)
−0.481748 + 0.876310i \(0.659998\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −30.8509 −1.21381
\(647\) 38.6864 1.52092 0.760461 0.649384i \(-0.224973\pi\)
0.760461 + 0.649384i \(0.224973\pi\)
\(648\) 0 0
\(649\) 13.7910 0.541343
\(650\) −2.75631 −0.108111
\(651\) 0 0
\(652\) −1.39041 −0.0544528
\(653\) −7.65430 −0.299536 −0.149768 0.988721i \(-0.547853\pi\)
−0.149768 + 0.988721i \(0.547853\pi\)
\(654\) 0 0
\(655\) 29.0867 1.13651
\(656\) 15.2962 0.597217
\(657\) 0 0
\(658\) 0 0
\(659\) −38.5145 −1.50031 −0.750156 0.661261i \(-0.770022\pi\)
−0.750156 + 0.661261i \(0.770022\pi\)
\(660\) 0 0
\(661\) −32.2132 −1.25295 −0.626474 0.779443i \(-0.715502\pi\)
−0.626474 + 0.779443i \(0.715502\pi\)
\(662\) −48.5393 −1.88654
\(663\) 0 0
\(664\) −32.8284 −1.27399
\(665\) 0 0
\(666\) 0 0
\(667\) 3.52686 0.136561
\(668\) −1.27701 −0.0494091
\(669\) 0 0
\(670\) −9.21478 −0.355998
\(671\) −18.4038 −0.710470
\(672\) 0 0
\(673\) 1.26136 0.0486218 0.0243109 0.999704i \(-0.492261\pi\)
0.0243109 + 0.999704i \(0.492261\pi\)
\(674\) 17.5288 0.675186
\(675\) 0 0
\(676\) −4.64474 −0.178644
\(677\) 14.8318 0.570031 0.285016 0.958523i \(-0.408001\pi\)
0.285016 + 0.958523i \(0.408001\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 22.0827 0.846833
\(681\) 0 0
\(682\) −44.0095 −1.68521
\(683\) 5.53120 0.211646 0.105823 0.994385i \(-0.466252\pi\)
0.105823 + 0.994385i \(0.466252\pi\)
\(684\) 0 0
\(685\) −21.5487 −0.823334
\(686\) 0 0
\(687\) 0 0
\(688\) 36.1344 1.37761
\(689\) 1.05219 0.0400854
\(690\) 0 0
\(691\) −8.63792 −0.328602 −0.164301 0.986410i \(-0.552537\pi\)
−0.164301 + 0.986410i \(0.552537\pi\)
\(692\) −3.55620 −0.135186
\(693\) 0 0
\(694\) −6.27116 −0.238050
\(695\) 17.9562 0.681116
\(696\) 0 0
\(697\) −9.25070 −0.350395
\(698\) −37.3035 −1.41196
\(699\) 0 0
\(700\) 0 0
\(701\) −26.5897 −1.00428 −0.502140 0.864786i \(-0.667454\pi\)
−0.502140 + 0.864786i \(0.667454\pi\)
\(702\) 0 0
\(703\) 54.1636 2.04282
\(704\) 34.9686 1.31793
\(705\) 0 0
\(706\) 52.3632 1.97072
\(707\) 0 0
\(708\) 0 0
\(709\) 10.1261 0.380294 0.190147 0.981756i \(-0.439104\pi\)
0.190147 + 0.981756i \(0.439104\pi\)
\(710\) 7.80921 0.293074
\(711\) 0 0
\(712\) −35.9333 −1.34666
\(713\) 4.10705 0.153810
\(714\) 0 0
\(715\) 6.55170 0.245020
\(716\) 7.18094 0.268364
\(717\) 0 0
\(718\) 8.43445 0.314771
\(719\) 32.3876 1.20785 0.603927 0.797040i \(-0.293602\pi\)
0.603927 + 0.797040i \(0.293602\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −51.1391 −1.90320
\(723\) 0 0
\(724\) −4.36443 −0.162203
\(725\) 21.2646 0.789749
\(726\) 0 0
\(727\) −25.2348 −0.935907 −0.467953 0.883753i \(-0.655008\pi\)
−0.467953 + 0.883753i \(0.655008\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −74.7835 −2.76786
\(731\) −21.8530 −0.808264
\(732\) 0 0
\(733\) −2.84672 −0.105146 −0.0525731 0.998617i \(-0.516742\pi\)
−0.0525731 + 0.998617i \(0.516742\pi\)
\(734\) −20.9568 −0.773529
\(735\) 0 0
\(736\) 1.66361 0.0613215
\(737\) 10.9192 0.402215
\(738\) 0 0
\(739\) 24.5417 0.902779 0.451390 0.892327i \(-0.350928\pi\)
0.451390 + 0.892327i \(0.350928\pi\)
\(740\) 8.53649 0.313808
\(741\) 0 0
\(742\) 0 0
\(743\) −4.59070 −0.168416 −0.0842082 0.996448i \(-0.526836\pi\)
−0.0842082 + 0.996448i \(0.526836\pi\)
\(744\) 0 0
\(745\) −68.1681 −2.49748
\(746\) −13.7116 −0.502018
\(747\) 0 0
\(748\) 5.76165 0.210667
\(749\) 0 0
\(750\) 0 0
\(751\) 31.4356 1.14710 0.573551 0.819170i \(-0.305566\pi\)
0.573551 + 0.819170i \(0.305566\pi\)
\(752\) 15.9855 0.582930
\(753\) 0 0
\(754\) −2.37242 −0.0863983
\(755\) −17.5422 −0.638425
\(756\) 0 0
\(757\) 29.5432 1.07376 0.536882 0.843657i \(-0.319602\pi\)
0.536882 + 0.843657i \(0.319602\pi\)
\(758\) −45.6401 −1.65772
\(759\) 0 0
\(760\) 57.5015 2.08580
\(761\) −46.4873 −1.68516 −0.842582 0.538567i \(-0.818966\pi\)
−0.842582 + 0.538567i \(0.818966\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −3.50785 −0.126910
\(765\) 0 0
\(766\) 27.2120 0.983209
\(767\) 0.865728 0.0312596
\(768\) 0 0
\(769\) −14.1706 −0.511006 −0.255503 0.966808i \(-0.582241\pi\)
−0.255503 + 0.966808i \(0.582241\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.830747 −0.0298992
\(773\) −34.0085 −1.22320 −0.611600 0.791167i \(-0.709474\pi\)
−0.611600 + 0.791167i \(0.709474\pi\)
\(774\) 0 0
\(775\) 24.7628 0.889507
\(776\) −40.3096 −1.44703
\(777\) 0 0
\(778\) 32.7672 1.17476
\(779\) −24.0880 −0.863043
\(780\) 0 0
\(781\) −9.25368 −0.331123
\(782\) −3.51737 −0.125781
\(783\) 0 0
\(784\) 0 0
\(785\) −19.1705 −0.684223
\(786\) 0 0
\(787\) 29.1059 1.03751 0.518757 0.854921i \(-0.326395\pi\)
0.518757 + 0.854921i \(0.326395\pi\)
\(788\) −1.82787 −0.0651151
\(789\) 0 0
\(790\) −26.5527 −0.944701
\(791\) 0 0
\(792\) 0 0
\(793\) −1.15530 −0.0410259
\(794\) −3.43515 −0.121909
\(795\) 0 0
\(796\) 3.59219 0.127322
\(797\) −18.5038 −0.655439 −0.327720 0.944775i \(-0.606280\pi\)
−0.327720 + 0.944775i \(0.606280\pi\)
\(798\) 0 0
\(799\) −9.66754 −0.342013
\(800\) 10.0305 0.354631
\(801\) 0 0
\(802\) 13.4246 0.474040
\(803\) 88.6162 3.12720
\(804\) 0 0
\(805\) 0 0
\(806\) −2.76270 −0.0973118
\(807\) 0 0
\(808\) 7.57440 0.266466
\(809\) −41.0813 −1.44434 −0.722172 0.691714i \(-0.756856\pi\)
−0.722172 + 0.691714i \(0.756856\pi\)
\(810\) 0 0
\(811\) 43.1361 1.51471 0.757357 0.653001i \(-0.226490\pi\)
0.757357 + 0.653001i \(0.226490\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −66.1719 −2.31932
\(815\) −12.1650 −0.426120
\(816\) 0 0
\(817\) −56.9034 −1.99080
\(818\) −44.0859 −1.54143
\(819\) 0 0
\(820\) −3.79641 −0.132576
\(821\) −13.1748 −0.459802 −0.229901 0.973214i \(-0.573840\pi\)
−0.229901 + 0.973214i \(0.573840\pi\)
\(822\) 0 0
\(823\) −11.9817 −0.417654 −0.208827 0.977953i \(-0.566965\pi\)
−0.208827 + 0.977953i \(0.566965\pi\)
\(824\) −24.4815 −0.852853
\(825\) 0 0
\(826\) 0 0
\(827\) −29.9879 −1.04278 −0.521391 0.853318i \(-0.674587\pi\)
−0.521391 + 0.853318i \(0.674587\pi\)
\(828\) 0 0
\(829\) −26.9238 −0.935102 −0.467551 0.883966i \(-0.654864\pi\)
−0.467551 + 0.883966i \(0.654864\pi\)
\(830\) 63.2416 2.19515
\(831\) 0 0
\(832\) 2.19516 0.0761034
\(833\) 0 0
\(834\) 0 0
\(835\) −11.1728 −0.386650
\(836\) 15.0028 0.518884
\(837\) 0 0
\(838\) −13.2823 −0.458831
\(839\) −11.2238 −0.387490 −0.193745 0.981052i \(-0.562063\pi\)
−0.193745 + 0.981052i \(0.562063\pi\)
\(840\) 0 0
\(841\) −10.6971 −0.368864
\(842\) −28.3749 −0.977864
\(843\) 0 0
\(844\) −8.40930 −0.289460
\(845\) −40.6376 −1.39798
\(846\) 0 0
\(847\) 0 0
\(848\) −13.3864 −0.459691
\(849\) 0 0
\(850\) −21.2074 −0.727408
\(851\) 6.17529 0.211686
\(852\) 0 0
\(853\) 12.1934 0.417496 0.208748 0.977970i \(-0.433061\pi\)
0.208748 + 0.977970i \(0.433061\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −26.2537 −0.897332
\(857\) −37.4382 −1.27887 −0.639433 0.768847i \(-0.720831\pi\)
−0.639433 + 0.768847i \(0.720831\pi\)
\(858\) 0 0
\(859\) −0.757734 −0.0258535 −0.0129268 0.999916i \(-0.504115\pi\)
−0.0129268 + 0.999916i \(0.504115\pi\)
\(860\) −8.96830 −0.305817
\(861\) 0 0
\(862\) −8.92314 −0.303923
\(863\) −23.5592 −0.801965 −0.400983 0.916086i \(-0.631331\pi\)
−0.400983 + 0.916086i \(0.631331\pi\)
\(864\) 0 0
\(865\) −31.1137 −1.05790
\(866\) 5.59106 0.189992
\(867\) 0 0
\(868\) 0 0
\(869\) 31.4641 1.06735
\(870\) 0 0
\(871\) 0.685455 0.0232258
\(872\) −11.7722 −0.398657
\(873\) 0 0
\(874\) −9.15892 −0.309805
\(875\) 0 0
\(876\) 0 0
\(877\) 36.6739 1.23839 0.619196 0.785237i \(-0.287459\pi\)
0.619196 + 0.785237i \(0.287459\pi\)
\(878\) −28.6744 −0.967716
\(879\) 0 0
\(880\) −83.3532 −2.80984
\(881\) −39.4357 −1.32862 −0.664311 0.747456i \(-0.731275\pi\)
−0.664311 + 0.747456i \(0.731275\pi\)
\(882\) 0 0
\(883\) −8.76912 −0.295105 −0.147552 0.989054i \(-0.547139\pi\)
−0.147552 + 0.989054i \(0.547139\pi\)
\(884\) 0.361688 0.0121649
\(885\) 0 0
\(886\) −40.0895 −1.34683
\(887\) 7.82601 0.262772 0.131386 0.991331i \(-0.458057\pi\)
0.131386 + 0.991331i \(0.458057\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 69.2230 2.32036
\(891\) 0 0
\(892\) 4.28587 0.143502
\(893\) −25.1734 −0.842397
\(894\) 0 0
\(895\) 62.8272 2.10008
\(896\) 0 0
\(897\) 0 0
\(898\) −8.66329 −0.289098
\(899\) 21.3139 0.710858
\(900\) 0 0
\(901\) 8.09571 0.269707
\(902\) 29.4285 0.979860
\(903\) 0 0
\(904\) −11.8033 −0.392573
\(905\) −38.1851 −1.26931
\(906\) 0 0
\(907\) 51.9332 1.72442 0.862208 0.506555i \(-0.169081\pi\)
0.862208 + 0.506555i \(0.169081\pi\)
\(908\) 2.15128 0.0713927
\(909\) 0 0
\(910\) 0 0
\(911\) −36.1524 −1.19778 −0.598891 0.800830i \(-0.704392\pi\)
−0.598891 + 0.800830i \(0.704392\pi\)
\(912\) 0 0
\(913\) −74.9394 −2.48013
\(914\) 30.7187 1.01608
\(915\) 0 0
\(916\) 8.56086 0.282859
\(917\) 0 0
\(918\) 0 0
\(919\) −11.1768 −0.368690 −0.184345 0.982862i \(-0.559016\pi\)
−0.184345 + 0.982862i \(0.559016\pi\)
\(920\) 6.55585 0.216140
\(921\) 0 0
\(922\) −57.5420 −1.89504
\(923\) −0.580900 −0.0191206
\(924\) 0 0
\(925\) 37.2330 1.22421
\(926\) −3.77236 −0.123967
\(927\) 0 0
\(928\) 8.63345 0.283407
\(929\) 32.8384 1.07739 0.538696 0.842500i \(-0.318917\pi\)
0.538696 + 0.842500i \(0.318917\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 5.00185 0.163841
\(933\) 0 0
\(934\) −51.3364 −1.67978
\(935\) 50.4096 1.64857
\(936\) 0 0
\(937\) 25.9566 0.847965 0.423983 0.905670i \(-0.360632\pi\)
0.423983 + 0.905670i \(0.360632\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −3.96748 −0.129405
\(941\) 29.4679 0.960627 0.480314 0.877097i \(-0.340523\pi\)
0.480314 + 0.877097i \(0.340523\pi\)
\(942\) 0 0
\(943\) −2.74632 −0.0894326
\(944\) −11.0141 −0.358479
\(945\) 0 0
\(946\) 69.5192 2.26026
\(947\) 11.5227 0.374439 0.187219 0.982318i \(-0.440052\pi\)
0.187219 + 0.982318i \(0.440052\pi\)
\(948\) 0 0
\(949\) 5.56289 0.180579
\(950\) −55.2223 −1.79165
\(951\) 0 0
\(952\) 0 0
\(953\) −29.2912 −0.948835 −0.474417 0.880300i \(-0.657341\pi\)
−0.474417 + 0.880300i \(0.657341\pi\)
\(954\) 0 0
\(955\) −30.6908 −0.993130
\(956\) −8.06352 −0.260793
\(957\) 0 0
\(958\) −53.3075 −1.72229
\(959\) 0 0
\(960\) 0 0
\(961\) −6.17981 −0.199349
\(962\) −4.15394 −0.133929
\(963\) 0 0
\(964\) −3.50821 −0.112992
\(965\) −7.26834 −0.233976
\(966\) 0 0
\(967\) 42.0803 1.35321 0.676606 0.736345i \(-0.263450\pi\)
0.676606 + 0.736345i \(0.263450\pi\)
\(968\) 55.5403 1.78513
\(969\) 0 0
\(970\) 77.6536 2.49331
\(971\) −35.1549 −1.12817 −0.564087 0.825715i \(-0.690772\pi\)
−0.564087 + 0.825715i \(0.690772\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −1.47478 −0.0472551
\(975\) 0 0
\(976\) 14.6981 0.470476
\(977\) 4.43025 0.141736 0.0708682 0.997486i \(-0.477423\pi\)
0.0708682 + 0.997486i \(0.477423\pi\)
\(978\) 0 0
\(979\) −82.0271 −2.62160
\(980\) 0 0
\(981\) 0 0
\(982\) 24.2368 0.773428
\(983\) 22.1601 0.706798 0.353399 0.935473i \(-0.385026\pi\)
0.353399 + 0.935473i \(0.385026\pi\)
\(984\) 0 0
\(985\) −15.9923 −0.509558
\(986\) −18.2537 −0.581316
\(987\) 0 0
\(988\) 0.941804 0.0299628
\(989\) −6.48767 −0.206296
\(990\) 0 0
\(991\) −36.7203 −1.16646 −0.583229 0.812307i \(-0.698211\pi\)
−0.583229 + 0.812307i \(0.698211\pi\)
\(992\) 10.0537 0.319206
\(993\) 0 0
\(994\) 0 0
\(995\) 31.4286 0.996355
\(996\) 0 0
\(997\) 39.9031 1.26374 0.631872 0.775073i \(-0.282287\pi\)
0.631872 + 0.775073i \(0.282287\pi\)
\(998\) 29.3918 0.930380
\(999\) 0 0
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 3969.2.a.t.1.2 4
3.2 odd 2 3969.2.a.w.1.3 4
7.3 odd 6 567.2.e.d.163.3 yes 8
7.5 odd 6 567.2.e.d.487.3 yes 8
7.6 odd 2 3969.2.a.s.1.2 4
21.5 even 6 567.2.e.c.487.2 yes 8
21.17 even 6 567.2.e.c.163.2 8
21.20 even 2 3969.2.a.x.1.3 4
63.5 even 6 567.2.h.k.298.3 8
63.31 odd 6 567.2.g.k.541.3 8
63.38 even 6 567.2.h.k.352.3 8
63.40 odd 6 567.2.h.j.298.2 8
63.47 even 6 567.2.g.j.109.2 8
63.52 odd 6 567.2.h.j.352.2 8
63.59 even 6 567.2.g.j.541.2 8
63.61 odd 6 567.2.g.k.109.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.e.c.163.2 8 21.17 even 6
567.2.e.c.487.2 yes 8 21.5 even 6
567.2.e.d.163.3 yes 8 7.3 odd 6
567.2.e.d.487.3 yes 8 7.5 odd 6
567.2.g.j.109.2 8 63.47 even 6
567.2.g.j.541.2 8 63.59 even 6
567.2.g.k.109.3 8 63.61 odd 6
567.2.g.k.541.3 8 63.31 odd 6
567.2.h.j.298.2 8 63.40 odd 6
567.2.h.j.352.2 8 63.52 odd 6
567.2.h.k.298.3 8 63.5 even 6
567.2.h.k.352.3 8 63.38 even 6
3969.2.a.s.1.2 4 7.6 odd 2
3969.2.a.t.1.2 4 1.1 even 1 trivial
3969.2.a.w.1.3 4 3.2 odd 2
3969.2.a.x.1.3 4 21.20 even 2