Properties

Label 3762.2.a.y.1.2
Level $3762$
Weight $2$
Character 3762.1
Self dual yes
Analytic conductor $30.040$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.0397212404\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 3762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +3.56155 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +3.56155 q^{7} +1.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} -3.56155 q^{13} +3.56155 q^{14} +1.00000 q^{16} -3.56155 q^{17} -1.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} -5.56155 q^{23} -1.00000 q^{25} -3.56155 q^{26} +3.56155 q^{28} -6.68466 q^{29} +2.00000 q^{31} +1.00000 q^{32} -3.56155 q^{34} -7.12311 q^{35} +3.12311 q^{37} -1.00000 q^{38} -2.00000 q^{40} -2.00000 q^{41} -1.00000 q^{44} -5.56155 q^{46} -8.00000 q^{47} +5.68466 q^{49} -1.00000 q^{50} -3.56155 q^{52} +7.80776 q^{53} +2.00000 q^{55} +3.56155 q^{56} -6.68466 q^{58} -4.68466 q^{59} -10.2462 q^{61} +2.00000 q^{62} +1.00000 q^{64} +7.12311 q^{65} +4.68466 q^{67} -3.56155 q^{68} -7.12311 q^{70} -6.00000 q^{71} +2.68466 q^{73} +3.12311 q^{74} -1.00000 q^{76} -3.56155 q^{77} -4.00000 q^{79} -2.00000 q^{80} -2.00000 q^{82} +2.24621 q^{83} +7.12311 q^{85} -1.00000 q^{88} +9.12311 q^{89} -12.6847 q^{91} -5.56155 q^{92} -8.00000 q^{94} +2.00000 q^{95} +1.12311 q^{97} +5.68466 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8} - 4 q^{10} - 2 q^{11} - 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} - 2 q^{19} - 4 q^{20} - 2 q^{22} - 7 q^{23} - 2 q^{25} - 3 q^{26} + 3 q^{28} - q^{29} + 4 q^{31} + 2 q^{32} - 3 q^{34} - 6 q^{35} - 2 q^{37} - 2 q^{38} - 4 q^{40} - 4 q^{41} - 2 q^{44} - 7 q^{46} - 16 q^{47} - q^{49} - 2 q^{50} - 3 q^{52} - 5 q^{53} + 4 q^{55} + 3 q^{56} - q^{58} + 3 q^{59} - 4 q^{61} + 4 q^{62} + 2 q^{64} + 6 q^{65} - 3 q^{67} - 3 q^{68} - 6 q^{70} - 12 q^{71} - 7 q^{73} - 2 q^{74} - 2 q^{76} - 3 q^{77} - 8 q^{79} - 4 q^{80} - 4 q^{82} - 12 q^{83} + 6 q^{85} - 2 q^{88} + 10 q^{89} - 13 q^{91} - 7 q^{92} - 16 q^{94} + 4 q^{95} - 6 q^{97} - q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 3.56155 1.34614 0.673070 0.739579i \(-0.264975\pi\)
0.673070 + 0.739579i \(0.264975\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.56155 −0.987797 −0.493899 0.869520i \(-0.664429\pi\)
−0.493899 + 0.869520i \(0.664429\pi\)
\(14\) 3.56155 0.951865
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.56155 −0.863803 −0.431902 0.901921i \(-0.642157\pi\)
−0.431902 + 0.901921i \(0.642157\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −5.56155 −1.15966 −0.579832 0.814736i \(-0.696882\pi\)
−0.579832 + 0.814736i \(0.696882\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −3.56155 −0.698478
\(27\) 0 0
\(28\) 3.56155 0.673070
\(29\) −6.68466 −1.24131 −0.620655 0.784084i \(-0.713133\pi\)
−0.620655 + 0.784084i \(0.713133\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.56155 −0.610801
\(35\) −7.12311 −1.20402
\(36\) 0 0
\(37\) 3.12311 0.513435 0.256718 0.966486i \(-0.417359\pi\)
0.256718 + 0.966486i \(0.417359\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −5.56155 −0.820006
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 5.68466 0.812094
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −3.56155 −0.493899
\(53\) 7.80776 1.07248 0.536239 0.844066i \(-0.319844\pi\)
0.536239 + 0.844066i \(0.319844\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 3.56155 0.475933
\(57\) 0 0
\(58\) −6.68466 −0.877739
\(59\) −4.68466 −0.609891 −0.304945 0.952370i \(-0.598638\pi\)
−0.304945 + 0.952370i \(0.598638\pi\)
\(60\) 0 0
\(61\) −10.2462 −1.31189 −0.655946 0.754807i \(-0.727730\pi\)
−0.655946 + 0.754807i \(0.727730\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 7.12311 0.883513
\(66\) 0 0
\(67\) 4.68466 0.572322 0.286161 0.958182i \(-0.407621\pi\)
0.286161 + 0.958182i \(0.407621\pi\)
\(68\) −3.56155 −0.431902
\(69\) 0 0
\(70\) −7.12311 −0.851374
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 2.68466 0.314216 0.157108 0.987581i \(-0.449783\pi\)
0.157108 + 0.987581i \(0.449783\pi\)
\(74\) 3.12311 0.363054
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −3.56155 −0.405877
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) 2.24621 0.246554 0.123277 0.992372i \(-0.460660\pi\)
0.123277 + 0.992372i \(0.460660\pi\)
\(84\) 0 0
\(85\) 7.12311 0.772609
\(86\) 0 0
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 9.12311 0.967047 0.483524 0.875331i \(-0.339357\pi\)
0.483524 + 0.875331i \(0.339357\pi\)
\(90\) 0 0
\(91\) −12.6847 −1.32971
\(92\) −5.56155 −0.579832
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) 1.12311 0.114034 0.0570170 0.998373i \(-0.481841\pi\)
0.0570170 + 0.998373i \(0.481841\pi\)
\(98\) 5.68466 0.574237
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 15.1231 1.50481 0.752403 0.658703i \(-0.228895\pi\)
0.752403 + 0.658703i \(0.228895\pi\)
\(102\) 0 0
\(103\) 11.3693 1.12025 0.560126 0.828407i \(-0.310753\pi\)
0.560126 + 0.828407i \(0.310753\pi\)
\(104\) −3.56155 −0.349239
\(105\) 0 0
\(106\) 7.80776 0.758357
\(107\) −18.9309 −1.83012 −0.915058 0.403322i \(-0.867855\pi\)
−0.915058 + 0.403322i \(0.867855\pi\)
\(108\) 0 0
\(109\) −18.6847 −1.78967 −0.894833 0.446401i \(-0.852705\pi\)
−0.894833 + 0.446401i \(0.852705\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) 3.56155 0.336535
\(113\) −5.12311 −0.481941 −0.240971 0.970532i \(-0.577466\pi\)
−0.240971 + 0.970532i \(0.577466\pi\)
\(114\) 0 0
\(115\) 11.1231 1.03723
\(116\) −6.68466 −0.620655
\(117\) 0 0
\(118\) −4.68466 −0.431258
\(119\) −12.6847 −1.16280
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −10.2462 −0.927648
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −14.2462 −1.26415 −0.632073 0.774909i \(-0.717796\pi\)
−0.632073 + 0.774909i \(0.717796\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 7.12311 0.624738
\(131\) 7.12311 0.622349 0.311174 0.950353i \(-0.399278\pi\)
0.311174 + 0.950353i \(0.399278\pi\)
\(132\) 0 0
\(133\) −3.56155 −0.308826
\(134\) 4.68466 0.404693
\(135\) 0 0
\(136\) −3.56155 −0.305401
\(137\) −8.43845 −0.720945 −0.360473 0.932770i \(-0.617385\pi\)
−0.360473 + 0.932770i \(0.617385\pi\)
\(138\) 0 0
\(139\) −17.3693 −1.47325 −0.736623 0.676303i \(-0.763581\pi\)
−0.736623 + 0.676303i \(0.763581\pi\)
\(140\) −7.12311 −0.602012
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 3.56155 0.297832
\(144\) 0 0
\(145\) 13.3693 1.11026
\(146\) 2.68466 0.222184
\(147\) 0 0
\(148\) 3.12311 0.256718
\(149\) 15.1231 1.23893 0.619467 0.785023i \(-0.287349\pi\)
0.619467 + 0.785023i \(0.287349\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) −3.56155 −0.286998
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 18.4924 1.47586 0.737928 0.674879i \(-0.235804\pi\)
0.737928 + 0.674879i \(0.235804\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) −2.00000 −0.158114
\(161\) −19.8078 −1.56107
\(162\) 0 0
\(163\) −13.3693 −1.04717 −0.523583 0.851975i \(-0.675405\pi\)
−0.523583 + 0.851975i \(0.675405\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 2.24621 0.174340
\(167\) 25.3693 1.96314 0.981568 0.191111i \(-0.0612092\pi\)
0.981568 + 0.191111i \(0.0612092\pi\)
\(168\) 0 0
\(169\) −0.315342 −0.0242570
\(170\) 7.12311 0.546317
\(171\) 0 0
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) −3.56155 −0.269228
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 9.12311 0.683806
\(179\) −22.2462 −1.66276 −0.831380 0.555704i \(-0.812449\pi\)
−0.831380 + 0.555704i \(0.812449\pi\)
\(180\) 0 0
\(181\) −19.1231 −1.42141 −0.710705 0.703491i \(-0.751624\pi\)
−0.710705 + 0.703491i \(0.751624\pi\)
\(182\) −12.6847 −0.940249
\(183\) 0 0
\(184\) −5.56155 −0.410003
\(185\) −6.24621 −0.459231
\(186\) 0 0
\(187\) 3.56155 0.260447
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) 20.6847 1.49669 0.748345 0.663310i \(-0.230849\pi\)
0.748345 + 0.663310i \(0.230849\pi\)
\(192\) 0 0
\(193\) −1.12311 −0.0808429 −0.0404215 0.999183i \(-0.512870\pi\)
−0.0404215 + 0.999183i \(0.512870\pi\)
\(194\) 1.12311 0.0806343
\(195\) 0 0
\(196\) 5.68466 0.406047
\(197\) −1.75379 −0.124952 −0.0624761 0.998046i \(-0.519900\pi\)
−0.0624761 + 0.998046i \(0.519900\pi\)
\(198\) 0 0
\(199\) −0.684658 −0.0485341 −0.0242671 0.999706i \(-0.507725\pi\)
−0.0242671 + 0.999706i \(0.507725\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 15.1231 1.06406
\(203\) −23.8078 −1.67098
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 11.3693 0.792138
\(207\) 0 0
\(208\) −3.56155 −0.246949
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −12.6847 −0.873248 −0.436624 0.899644i \(-0.643826\pi\)
−0.436624 + 0.899644i \(0.643826\pi\)
\(212\) 7.80776 0.536239
\(213\) 0 0
\(214\) −18.9309 −1.29409
\(215\) 0 0
\(216\) 0 0
\(217\) 7.12311 0.483548
\(218\) −18.6847 −1.26548
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 12.6847 0.853262
\(222\) 0 0
\(223\) −24.7386 −1.65662 −0.828311 0.560269i \(-0.810698\pi\)
−0.828311 + 0.560269i \(0.810698\pi\)
\(224\) 3.56155 0.237966
\(225\) 0 0
\(226\) −5.12311 −0.340784
\(227\) 17.1771 1.14008 0.570041 0.821616i \(-0.306927\pi\)
0.570041 + 0.821616i \(0.306927\pi\)
\(228\) 0 0
\(229\) 21.1231 1.39585 0.697927 0.716169i \(-0.254106\pi\)
0.697927 + 0.716169i \(0.254106\pi\)
\(230\) 11.1231 0.733436
\(231\) 0 0
\(232\) −6.68466 −0.438869
\(233\) −20.7386 −1.35863 −0.679317 0.733845i \(-0.737724\pi\)
−0.679317 + 0.733845i \(0.737724\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) −4.68466 −0.304945
\(237\) 0 0
\(238\) −12.6847 −0.822224
\(239\) 4.93087 0.318951 0.159476 0.987202i \(-0.449020\pi\)
0.159476 + 0.987202i \(0.449020\pi\)
\(240\) 0 0
\(241\) 21.6155 1.39238 0.696189 0.717858i \(-0.254877\pi\)
0.696189 + 0.717858i \(0.254877\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −10.2462 −0.655946
\(245\) −11.3693 −0.726359
\(246\) 0 0
\(247\) 3.56155 0.226616
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) −8.87689 −0.560305 −0.280152 0.959956i \(-0.590385\pi\)
−0.280152 + 0.959956i \(0.590385\pi\)
\(252\) 0 0
\(253\) 5.56155 0.349652
\(254\) −14.2462 −0.893887
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) 11.1231 0.691156
\(260\) 7.12311 0.441756
\(261\) 0 0
\(262\) 7.12311 0.440067
\(263\) −17.1231 −1.05586 −0.527928 0.849289i \(-0.677031\pi\)
−0.527928 + 0.849289i \(0.677031\pi\)
\(264\) 0 0
\(265\) −15.6155 −0.959254
\(266\) −3.56155 −0.218373
\(267\) 0 0
\(268\) 4.68466 0.286161
\(269\) −11.1231 −0.678188 −0.339094 0.940753i \(-0.610120\pi\)
−0.339094 + 0.940753i \(0.610120\pi\)
\(270\) 0 0
\(271\) 13.3153 0.808849 0.404425 0.914571i \(-0.367472\pi\)
0.404425 + 0.914571i \(0.367472\pi\)
\(272\) −3.56155 −0.215951
\(273\) 0 0
\(274\) −8.43845 −0.509785
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −24.4924 −1.47161 −0.735804 0.677195i \(-0.763195\pi\)
−0.735804 + 0.677195i \(0.763195\pi\)
\(278\) −17.3693 −1.04174
\(279\) 0 0
\(280\) −7.12311 −0.425687
\(281\) 0.630683 0.0376234 0.0188117 0.999823i \(-0.494012\pi\)
0.0188117 + 0.999823i \(0.494012\pi\)
\(282\) 0 0
\(283\) −13.3693 −0.794723 −0.397362 0.917662i \(-0.630074\pi\)
−0.397362 + 0.917662i \(0.630074\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 3.56155 0.210599
\(287\) −7.12311 −0.420464
\(288\) 0 0
\(289\) −4.31534 −0.253844
\(290\) 13.3693 0.785073
\(291\) 0 0
\(292\) 2.68466 0.157108
\(293\) 24.9309 1.45648 0.728238 0.685324i \(-0.240339\pi\)
0.728238 + 0.685324i \(0.240339\pi\)
\(294\) 0 0
\(295\) 9.36932 0.545503
\(296\) 3.12311 0.181527
\(297\) 0 0
\(298\) 15.1231 0.876058
\(299\) 19.8078 1.14551
\(300\) 0 0
\(301\) 0 0
\(302\) 4.00000 0.230174
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 20.4924 1.17339
\(306\) 0 0
\(307\) −5.75379 −0.328386 −0.164193 0.986428i \(-0.552502\pi\)
−0.164193 + 0.986428i \(0.552502\pi\)
\(308\) −3.56155 −0.202938
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) −15.8078 −0.896376 −0.448188 0.893939i \(-0.647930\pi\)
−0.448188 + 0.893939i \(0.647930\pi\)
\(312\) 0 0
\(313\) 3.56155 0.201311 0.100655 0.994921i \(-0.467906\pi\)
0.100655 + 0.994921i \(0.467906\pi\)
\(314\) 18.4924 1.04359
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −30.0540 −1.68800 −0.844000 0.536344i \(-0.819805\pi\)
−0.844000 + 0.536344i \(0.819805\pi\)
\(318\) 0 0
\(319\) 6.68466 0.374269
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) −19.8078 −1.10384
\(323\) 3.56155 0.198170
\(324\) 0 0
\(325\) 3.56155 0.197559
\(326\) −13.3693 −0.740458
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) −28.4924 −1.57084
\(330\) 0 0
\(331\) −10.4384 −0.573749 −0.286874 0.957968i \(-0.592616\pi\)
−0.286874 + 0.957968i \(0.592616\pi\)
\(332\) 2.24621 0.123277
\(333\) 0 0
\(334\) 25.3693 1.38815
\(335\) −9.36932 −0.511900
\(336\) 0 0
\(337\) 2.87689 0.156714 0.0783572 0.996925i \(-0.475033\pi\)
0.0783572 + 0.996925i \(0.475033\pi\)
\(338\) −0.315342 −0.0171523
\(339\) 0 0
\(340\) 7.12311 0.386305
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) −4.68466 −0.252948
\(344\) 0 0
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 31.6155 1.69721 0.848605 0.529027i \(-0.177443\pi\)
0.848605 + 0.529027i \(0.177443\pi\)
\(348\) 0 0
\(349\) −19.6155 −1.05000 −0.524998 0.851104i \(-0.675934\pi\)
−0.524998 + 0.851104i \(0.675934\pi\)
\(350\) −3.56155 −0.190373
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 23.1771 1.23359 0.616796 0.787123i \(-0.288430\pi\)
0.616796 + 0.787123i \(0.288430\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 9.12311 0.483524
\(357\) 0 0
\(358\) −22.2462 −1.17575
\(359\) −12.9309 −0.682465 −0.341233 0.939979i \(-0.610844\pi\)
−0.341233 + 0.939979i \(0.610844\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −19.1231 −1.00509
\(363\) 0 0
\(364\) −12.6847 −0.664857
\(365\) −5.36932 −0.281043
\(366\) 0 0
\(367\) 13.7538 0.717942 0.358971 0.933349i \(-0.383128\pi\)
0.358971 + 0.933349i \(0.383128\pi\)
\(368\) −5.56155 −0.289916
\(369\) 0 0
\(370\) −6.24621 −0.324725
\(371\) 27.8078 1.44371
\(372\) 0 0
\(373\) 7.56155 0.391522 0.195761 0.980652i \(-0.437282\pi\)
0.195761 + 0.980652i \(0.437282\pi\)
\(374\) 3.56155 0.184164
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 23.8078 1.22616
\(378\) 0 0
\(379\) 2.43845 0.125255 0.0626273 0.998037i \(-0.480052\pi\)
0.0626273 + 0.998037i \(0.480052\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) 20.6847 1.05832
\(383\) 2.00000 0.102195 0.0510976 0.998694i \(-0.483728\pi\)
0.0510976 + 0.998694i \(0.483728\pi\)
\(384\) 0 0
\(385\) 7.12311 0.363027
\(386\) −1.12311 −0.0571646
\(387\) 0 0
\(388\) 1.12311 0.0570170
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 19.8078 1.00172
\(392\) 5.68466 0.287119
\(393\) 0 0
\(394\) −1.75379 −0.0883546
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −26.9848 −1.35433 −0.677165 0.735831i \(-0.736792\pi\)
−0.677165 + 0.735831i \(0.736792\pi\)
\(398\) −0.684658 −0.0343188
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 24.2462 1.21080 0.605399 0.795922i \(-0.293014\pi\)
0.605399 + 0.795922i \(0.293014\pi\)
\(402\) 0 0
\(403\) −7.12311 −0.354827
\(404\) 15.1231 0.752403
\(405\) 0 0
\(406\) −23.8078 −1.18156
\(407\) −3.12311 −0.154807
\(408\) 0 0
\(409\) −0.246211 −0.0121744 −0.00608718 0.999981i \(-0.501938\pi\)
−0.00608718 + 0.999981i \(0.501938\pi\)
\(410\) 4.00000 0.197546
\(411\) 0 0
\(412\) 11.3693 0.560126
\(413\) −16.6847 −0.820998
\(414\) 0 0
\(415\) −4.49242 −0.220524
\(416\) −3.56155 −0.174619
\(417\) 0 0
\(418\) 1.00000 0.0489116
\(419\) 23.1231 1.12964 0.564819 0.825215i \(-0.308946\pi\)
0.564819 + 0.825215i \(0.308946\pi\)
\(420\) 0 0
\(421\) −1.06913 −0.0521062 −0.0260531 0.999661i \(-0.508294\pi\)
−0.0260531 + 0.999661i \(0.508294\pi\)
\(422\) −12.6847 −0.617480
\(423\) 0 0
\(424\) 7.80776 0.379179
\(425\) 3.56155 0.172761
\(426\) 0 0
\(427\) −36.4924 −1.76599
\(428\) −18.9309 −0.915058
\(429\) 0 0
\(430\) 0 0
\(431\) 25.8617 1.24572 0.622858 0.782335i \(-0.285971\pi\)
0.622858 + 0.782335i \(0.285971\pi\)
\(432\) 0 0
\(433\) 31.3693 1.50751 0.753757 0.657154i \(-0.228240\pi\)
0.753757 + 0.657154i \(0.228240\pi\)
\(434\) 7.12311 0.341920
\(435\) 0 0
\(436\) −18.6847 −0.894833
\(437\) 5.56155 0.266045
\(438\) 0 0
\(439\) 31.6155 1.50893 0.754463 0.656342i \(-0.227897\pi\)
0.754463 + 0.656342i \(0.227897\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) 12.6847 0.603348
\(443\) 17.8617 0.848637 0.424318 0.905513i \(-0.360514\pi\)
0.424318 + 0.905513i \(0.360514\pi\)
\(444\) 0 0
\(445\) −18.2462 −0.864953
\(446\) −24.7386 −1.17141
\(447\) 0 0
\(448\) 3.56155 0.168268
\(449\) 17.6155 0.831328 0.415664 0.909518i \(-0.363549\pi\)
0.415664 + 0.909518i \(0.363549\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) −5.12311 −0.240971
\(453\) 0 0
\(454\) 17.1771 0.806160
\(455\) 25.3693 1.18933
\(456\) 0 0
\(457\) 32.4384 1.51741 0.758703 0.651436i \(-0.225833\pi\)
0.758703 + 0.651436i \(0.225833\pi\)
\(458\) 21.1231 0.987018
\(459\) 0 0
\(460\) 11.1231 0.518617
\(461\) 22.7386 1.05904 0.529522 0.848296i \(-0.322371\pi\)
0.529522 + 0.848296i \(0.322371\pi\)
\(462\) 0 0
\(463\) 36.4924 1.69595 0.847973 0.530039i \(-0.177823\pi\)
0.847973 + 0.530039i \(0.177823\pi\)
\(464\) −6.68466 −0.310327
\(465\) 0 0
\(466\) −20.7386 −0.960699
\(467\) −26.2462 −1.21453 −0.607265 0.794499i \(-0.707733\pi\)
−0.607265 + 0.794499i \(0.707733\pi\)
\(468\) 0 0
\(469\) 16.6847 0.770426
\(470\) 16.0000 0.738025
\(471\) 0 0
\(472\) −4.68466 −0.215629
\(473\) 0 0
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) −12.6847 −0.581400
\(477\) 0 0
\(478\) 4.93087 0.225533
\(479\) −11.3693 −0.519477 −0.259739 0.965679i \(-0.583636\pi\)
−0.259739 + 0.965679i \(0.583636\pi\)
\(480\) 0 0
\(481\) −11.1231 −0.507170
\(482\) 21.6155 0.984560
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −2.24621 −0.101995
\(486\) 0 0
\(487\) 22.8769 1.03665 0.518326 0.855183i \(-0.326556\pi\)
0.518326 + 0.855183i \(0.326556\pi\)
\(488\) −10.2462 −0.463824
\(489\) 0 0
\(490\) −11.3693 −0.513613
\(491\) 14.6307 0.660273 0.330137 0.943933i \(-0.392905\pi\)
0.330137 + 0.943933i \(0.392905\pi\)
\(492\) 0 0
\(493\) 23.8078 1.07225
\(494\) 3.56155 0.160242
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −21.3693 −0.958545
\(498\) 0 0
\(499\) 26.2462 1.17494 0.587471 0.809245i \(-0.300124\pi\)
0.587471 + 0.809245i \(0.300124\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) −8.87689 −0.396195
\(503\) −9.31534 −0.415351 −0.207675 0.978198i \(-0.566590\pi\)
−0.207675 + 0.978198i \(0.566590\pi\)
\(504\) 0 0
\(505\) −30.2462 −1.34594
\(506\) 5.56155 0.247241
\(507\) 0 0
\(508\) −14.2462 −0.632073
\(509\) 6.63068 0.293900 0.146950 0.989144i \(-0.453054\pi\)
0.146950 + 0.989144i \(0.453054\pi\)
\(510\) 0 0
\(511\) 9.56155 0.422978
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) −22.7386 −1.00198
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 11.1231 0.488721
\(519\) 0 0
\(520\) 7.12311 0.312369
\(521\) −1.12311 −0.0492042 −0.0246021 0.999697i \(-0.507832\pi\)
−0.0246021 + 0.999697i \(0.507832\pi\)
\(522\) 0 0
\(523\) −6.05398 −0.264722 −0.132361 0.991202i \(-0.542256\pi\)
−0.132361 + 0.991202i \(0.542256\pi\)
\(524\) 7.12311 0.311174
\(525\) 0 0
\(526\) −17.1231 −0.746603
\(527\) −7.12311 −0.310287
\(528\) 0 0
\(529\) 7.93087 0.344820
\(530\) −15.6155 −0.678295
\(531\) 0 0
\(532\) −3.56155 −0.154413
\(533\) 7.12311 0.308536
\(534\) 0 0
\(535\) 37.8617 1.63691
\(536\) 4.68466 0.202346
\(537\) 0 0
\(538\) −11.1231 −0.479551
\(539\) −5.68466 −0.244856
\(540\) 0 0
\(541\) 43.2311 1.85865 0.929324 0.369265i \(-0.120391\pi\)
0.929324 + 0.369265i \(0.120391\pi\)
\(542\) 13.3153 0.571943
\(543\) 0 0
\(544\) −3.56155 −0.152700
\(545\) 37.3693 1.60073
\(546\) 0 0
\(547\) 6.73863 0.288123 0.144062 0.989569i \(-0.453984\pi\)
0.144062 + 0.989569i \(0.453984\pi\)
\(548\) −8.43845 −0.360473
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 6.68466 0.284776
\(552\) 0 0
\(553\) −14.2462 −0.605811
\(554\) −24.4924 −1.04058
\(555\) 0 0
\(556\) −17.3693 −0.736623
\(557\) −8.87689 −0.376126 −0.188063 0.982157i \(-0.560221\pi\)
−0.188063 + 0.982157i \(0.560221\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −7.12311 −0.301006
\(561\) 0 0
\(562\) 0.630683 0.0266038
\(563\) −6.73863 −0.284000 −0.142000 0.989867i \(-0.545353\pi\)
−0.142000 + 0.989867i \(0.545353\pi\)
\(564\) 0 0
\(565\) 10.2462 0.431061
\(566\) −13.3693 −0.561954
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −3.36932 −0.141249 −0.0706246 0.997503i \(-0.522499\pi\)
−0.0706246 + 0.997503i \(0.522499\pi\)
\(570\) 0 0
\(571\) 35.6155 1.49046 0.745232 0.666806i \(-0.232339\pi\)
0.745232 + 0.666806i \(0.232339\pi\)
\(572\) 3.56155 0.148916
\(573\) 0 0
\(574\) −7.12311 −0.297313
\(575\) 5.56155 0.231933
\(576\) 0 0
\(577\) 3.94602 0.164275 0.0821376 0.996621i \(-0.473825\pi\)
0.0821376 + 0.996621i \(0.473825\pi\)
\(578\) −4.31534 −0.179495
\(579\) 0 0
\(580\) 13.3693 0.555131
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) −7.80776 −0.323365
\(584\) 2.68466 0.111092
\(585\) 0 0
\(586\) 24.9309 1.02988
\(587\) 7.50758 0.309871 0.154935 0.987925i \(-0.450483\pi\)
0.154935 + 0.987925i \(0.450483\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 9.36932 0.385729
\(591\) 0 0
\(592\) 3.12311 0.128359
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) 25.3693 1.04004
\(596\) 15.1231 0.619467
\(597\) 0 0
\(598\) 19.8078 0.810000
\(599\) −39.8617 −1.62871 −0.814353 0.580370i \(-0.802908\pi\)
−0.814353 + 0.580370i \(0.802908\pi\)
\(600\) 0 0
\(601\) −3.36932 −0.137437 −0.0687187 0.997636i \(-0.521891\pi\)
−0.0687187 + 0.997636i \(0.521891\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 4.00000 0.162758
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) 27.6155 1.12088 0.560440 0.828195i \(-0.310632\pi\)
0.560440 + 0.828195i \(0.310632\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 20.4924 0.829714
\(611\) 28.4924 1.15268
\(612\) 0 0
\(613\) 1.75379 0.0708349 0.0354174 0.999373i \(-0.488724\pi\)
0.0354174 + 0.999373i \(0.488724\pi\)
\(614\) −5.75379 −0.232204
\(615\) 0 0
\(616\) −3.56155 −0.143499
\(617\) 4.73863 0.190770 0.0953851 0.995440i \(-0.469592\pi\)
0.0953851 + 0.995440i \(0.469592\pi\)
\(618\) 0 0
\(619\) −26.2462 −1.05492 −0.527462 0.849579i \(-0.676856\pi\)
−0.527462 + 0.849579i \(0.676856\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) −15.8078 −0.633834
\(623\) 32.4924 1.30178
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 3.56155 0.142348
\(627\) 0 0
\(628\) 18.4924 0.737928
\(629\) −11.1231 −0.443507
\(630\) 0 0
\(631\) 36.4924 1.45274 0.726370 0.687304i \(-0.241206\pi\)
0.726370 + 0.687304i \(0.241206\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) −30.0540 −1.19360
\(635\) 28.4924 1.13069
\(636\) 0 0
\(637\) −20.2462 −0.802184
\(638\) 6.68466 0.264648
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) 9.61553 0.379791 0.189895 0.981804i \(-0.439185\pi\)
0.189895 + 0.981804i \(0.439185\pi\)
\(642\) 0 0
\(643\) −37.3693 −1.47370 −0.736851 0.676055i \(-0.763688\pi\)
−0.736851 + 0.676055i \(0.763688\pi\)
\(644\) −19.8078 −0.780535
\(645\) 0 0
\(646\) 3.56155 0.140127
\(647\) −22.5464 −0.886390 −0.443195 0.896425i \(-0.646155\pi\)
−0.443195 + 0.896425i \(0.646155\pi\)
\(648\) 0 0
\(649\) 4.68466 0.183889
\(650\) 3.56155 0.139696
\(651\) 0 0
\(652\) −13.3693 −0.523583
\(653\) −25.6155 −1.00241 −0.501207 0.865328i \(-0.667110\pi\)
−0.501207 + 0.865328i \(0.667110\pi\)
\(654\) 0 0
\(655\) −14.2462 −0.556646
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) −28.4924 −1.11075
\(659\) −47.4233 −1.84735 −0.923675 0.383178i \(-0.874830\pi\)
−0.923675 + 0.383178i \(0.874830\pi\)
\(660\) 0 0
\(661\) −22.4384 −0.872754 −0.436377 0.899764i \(-0.643739\pi\)
−0.436377 + 0.899764i \(0.643739\pi\)
\(662\) −10.4384 −0.405702
\(663\) 0 0
\(664\) 2.24621 0.0871699
\(665\) 7.12311 0.276222
\(666\) 0 0
\(667\) 37.1771 1.43950
\(668\) 25.3693 0.981568
\(669\) 0 0
\(670\) −9.36932 −0.361968
\(671\) 10.2462 0.395551
\(672\) 0 0
\(673\) −18.8769 −0.727651 −0.363825 0.931467i \(-0.618530\pi\)
−0.363825 + 0.931467i \(0.618530\pi\)
\(674\) 2.87689 0.110814
\(675\) 0 0
\(676\) −0.315342 −0.0121285
\(677\) 39.1771 1.50570 0.752849 0.658194i \(-0.228679\pi\)
0.752849 + 0.658194i \(0.228679\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) 7.12311 0.273159
\(681\) 0 0
\(682\) −2.00000 −0.0765840
\(683\) 4.49242 0.171898 0.0859489 0.996300i \(-0.472608\pi\)
0.0859489 + 0.996300i \(0.472608\pi\)
\(684\) 0 0
\(685\) 16.8769 0.644833
\(686\) −4.68466 −0.178861
\(687\) 0 0
\(688\) 0 0
\(689\) −27.8078 −1.05939
\(690\) 0 0
\(691\) −43.6155 −1.65921 −0.829606 0.558349i \(-0.811435\pi\)
−0.829606 + 0.558349i \(0.811435\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 31.6155 1.20011
\(695\) 34.7386 1.31771
\(696\) 0 0
\(697\) 7.12311 0.269807
\(698\) −19.6155 −0.742459
\(699\) 0 0
\(700\) −3.56155 −0.134614
\(701\) −24.9848 −0.943665 −0.471832 0.881688i \(-0.656407\pi\)
−0.471832 + 0.881688i \(0.656407\pi\)
\(702\) 0 0
\(703\) −3.12311 −0.117790
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 23.1771 0.872281
\(707\) 53.8617 2.02568
\(708\) 0 0
\(709\) −9.50758 −0.357065 −0.178532 0.983934i \(-0.557135\pi\)
−0.178532 + 0.983934i \(0.557135\pi\)
\(710\) 12.0000 0.450352
\(711\) 0 0
\(712\) 9.12311 0.341903
\(713\) −11.1231 −0.416564
\(714\) 0 0
\(715\) −7.12311 −0.266389
\(716\) −22.2462 −0.831380
\(717\) 0 0
\(718\) −12.9309 −0.482576
\(719\) 6.43845 0.240114 0.120057 0.992767i \(-0.461692\pi\)
0.120057 + 0.992767i \(0.461692\pi\)
\(720\) 0 0
\(721\) 40.4924 1.50802
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) −19.1231 −0.710705
\(725\) 6.68466 0.248262
\(726\) 0 0
\(727\) −39.4233 −1.46213 −0.731064 0.682308i \(-0.760976\pi\)
−0.731064 + 0.682308i \(0.760976\pi\)
\(728\) −12.6847 −0.470125
\(729\) 0 0
\(730\) −5.36932 −0.198727
\(731\) 0 0
\(732\) 0 0
\(733\) −48.1080 −1.77691 −0.888454 0.458966i \(-0.848220\pi\)
−0.888454 + 0.458966i \(0.848220\pi\)
\(734\) 13.7538 0.507662
\(735\) 0 0
\(736\) −5.56155 −0.205002
\(737\) −4.68466 −0.172562
\(738\) 0 0
\(739\) 20.9848 0.771940 0.385970 0.922511i \(-0.373867\pi\)
0.385970 + 0.922511i \(0.373867\pi\)
\(740\) −6.24621 −0.229615
\(741\) 0 0
\(742\) 27.8078 1.02086
\(743\) 25.7538 0.944815 0.472407 0.881380i \(-0.343385\pi\)
0.472407 + 0.881380i \(0.343385\pi\)
\(744\) 0 0
\(745\) −30.2462 −1.10814
\(746\) 7.56155 0.276848
\(747\) 0 0
\(748\) 3.56155 0.130223
\(749\) −67.4233 −2.46359
\(750\) 0 0
\(751\) −41.2311 −1.50454 −0.752271 0.658853i \(-0.771042\pi\)
−0.752271 + 0.658853i \(0.771042\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 23.8078 0.867028
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −9.50758 −0.345559 −0.172779 0.984961i \(-0.555275\pi\)
−0.172779 + 0.984961i \(0.555275\pi\)
\(758\) 2.43845 0.0885684
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) 22.1922 0.804468 0.402234 0.915537i \(-0.368234\pi\)
0.402234 + 0.915537i \(0.368234\pi\)
\(762\) 0 0
\(763\) −66.5464 −2.40914
\(764\) 20.6847 0.748345
\(765\) 0 0
\(766\) 2.00000 0.0722629
\(767\) 16.6847 0.602448
\(768\) 0 0
\(769\) −1.31534 −0.0474324 −0.0237162 0.999719i \(-0.507550\pi\)
−0.0237162 + 0.999719i \(0.507550\pi\)
\(770\) 7.12311 0.256699
\(771\) 0 0
\(772\) −1.12311 −0.0404215
\(773\) −9.06913 −0.326194 −0.163097 0.986610i \(-0.552148\pi\)
−0.163097 + 0.986610i \(0.552148\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 1.12311 0.0403171
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) 19.8078 0.708324
\(783\) 0 0
\(784\) 5.68466 0.203024
\(785\) −36.9848 −1.32005
\(786\) 0 0
\(787\) 3.31534 0.118179 0.0590896 0.998253i \(-0.481180\pi\)
0.0590896 + 0.998253i \(0.481180\pi\)
\(788\) −1.75379 −0.0624761
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) −18.2462 −0.648761
\(792\) 0 0
\(793\) 36.4924 1.29588
\(794\) −26.9848 −0.957656
\(795\) 0 0
\(796\) −0.684658 −0.0242671
\(797\) 35.8078 1.26838 0.634188 0.773179i \(-0.281335\pi\)
0.634188 + 0.773179i \(0.281335\pi\)
\(798\) 0 0
\(799\) 28.4924 1.00799
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 24.2462 0.856163
\(803\) −2.68466 −0.0947395
\(804\) 0 0
\(805\) 39.6155 1.39626
\(806\) −7.12311 −0.250901
\(807\) 0 0
\(808\) 15.1231 0.532029
\(809\) −36.9309 −1.29842 −0.649210 0.760609i \(-0.724900\pi\)
−0.649210 + 0.760609i \(0.724900\pi\)
\(810\) 0 0
\(811\) 20.6847 0.726337 0.363168 0.931724i \(-0.381695\pi\)
0.363168 + 0.931724i \(0.381695\pi\)
\(812\) −23.8078 −0.835489
\(813\) 0 0
\(814\) −3.12311 −0.109465
\(815\) 26.7386 0.936613
\(816\) 0 0
\(817\) 0 0
\(818\) −0.246211 −0.00860857
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) −44.9848 −1.56998 −0.784991 0.619507i \(-0.787332\pi\)
−0.784991 + 0.619507i \(0.787332\pi\)
\(822\) 0 0
\(823\) 14.9309 0.520457 0.260229 0.965547i \(-0.416202\pi\)
0.260229 + 0.965547i \(0.416202\pi\)
\(824\) 11.3693 0.396069
\(825\) 0 0
\(826\) −16.6847 −0.580534
\(827\) −21.0691 −0.732645 −0.366323 0.930488i \(-0.619383\pi\)
−0.366323 + 0.930488i \(0.619383\pi\)
\(828\) 0 0
\(829\) −33.1771 −1.15229 −0.576144 0.817348i \(-0.695443\pi\)
−0.576144 + 0.817348i \(0.695443\pi\)
\(830\) −4.49242 −0.155934
\(831\) 0 0
\(832\) −3.56155 −0.123475
\(833\) −20.2462 −0.701490
\(834\) 0 0
\(835\) −50.7386 −1.75588
\(836\) 1.00000 0.0345857
\(837\) 0 0
\(838\) 23.1231 0.798774
\(839\) −1.12311 −0.0387739 −0.0193870 0.999812i \(-0.506171\pi\)
−0.0193870 + 0.999812i \(0.506171\pi\)
\(840\) 0 0
\(841\) 15.6847 0.540850
\(842\) −1.06913 −0.0368447
\(843\) 0 0
\(844\) −12.6847 −0.436624
\(845\) 0.630683 0.0216962
\(846\) 0 0
\(847\) 3.56155 0.122376
\(848\) 7.80776 0.268120
\(849\) 0 0
\(850\) 3.56155 0.122160
\(851\) −17.3693 −0.595413
\(852\) 0 0
\(853\) 38.3542 1.31322 0.656611 0.754230i \(-0.271989\pi\)
0.656611 + 0.754230i \(0.271989\pi\)
\(854\) −36.4924 −1.24874
\(855\) 0 0
\(856\) −18.9309 −0.647044
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 0 0
\(859\) −41.8617 −1.42830 −0.714152 0.699991i \(-0.753188\pi\)
−0.714152 + 0.699991i \(0.753188\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 25.8617 0.880854
\(863\) 16.2462 0.553027 0.276514 0.961010i \(-0.410821\pi\)
0.276514 + 0.961010i \(0.410821\pi\)
\(864\) 0 0
\(865\) 36.0000 1.22404
\(866\) 31.3693 1.06597
\(867\) 0 0
\(868\) 7.12311 0.241774
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) −16.6847 −0.565338
\(872\) −18.6847 −0.632742
\(873\) 0 0
\(874\) 5.56155 0.188122
\(875\) 42.7386 1.44483
\(876\) 0 0
\(877\) −40.4384 −1.36551 −0.682755 0.730648i \(-0.739218\pi\)
−0.682755 + 0.730648i \(0.739218\pi\)
\(878\) 31.6155 1.06697
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) 28.7386 0.968229 0.484115 0.875005i \(-0.339142\pi\)
0.484115 + 0.875005i \(0.339142\pi\)
\(882\) 0 0
\(883\) −54.7386 −1.84210 −0.921051 0.389442i \(-0.872668\pi\)
−0.921051 + 0.389442i \(0.872668\pi\)
\(884\) 12.6847 0.426631
\(885\) 0 0
\(886\) 17.8617 0.600077
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 0 0
\(889\) −50.7386 −1.70172
\(890\) −18.2462 −0.611614
\(891\) 0 0
\(892\) −24.7386 −0.828311
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) 44.4924 1.48722
\(896\) 3.56155 0.118983
\(897\) 0 0
\(898\) 17.6155 0.587838
\(899\) −13.3693 −0.445892
\(900\) 0 0
\(901\) −27.8078 −0.926411
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) −5.12311 −0.170392
\(905\) 38.2462 1.27135
\(906\) 0 0
\(907\) 56.7926 1.88577 0.942884 0.333122i \(-0.108102\pi\)
0.942884 + 0.333122i \(0.108102\pi\)
\(908\) 17.1771 0.570041
\(909\) 0 0
\(910\) 25.3693 0.840985
\(911\) 46.9848 1.55668 0.778339 0.627845i \(-0.216063\pi\)
0.778339 + 0.627845i \(0.216063\pi\)
\(912\) 0 0
\(913\) −2.24621 −0.0743387
\(914\) 32.4384 1.07297
\(915\) 0 0
\(916\) 21.1231 0.697927
\(917\) 25.3693 0.837769
\(918\) 0 0
\(919\) 28.4384 0.938098 0.469049 0.883172i \(-0.344597\pi\)
0.469049 + 0.883172i \(0.344597\pi\)
\(920\) 11.1231 0.366718
\(921\) 0 0
\(922\) 22.7386 0.748857
\(923\) 21.3693 0.703380
\(924\) 0 0
\(925\) −3.12311 −0.102687
\(926\) 36.4924 1.19922
\(927\) 0 0
\(928\) −6.68466 −0.219435
\(929\) 16.9309 0.555484 0.277742 0.960656i \(-0.410414\pi\)
0.277742 + 0.960656i \(0.410414\pi\)
\(930\) 0 0
\(931\) −5.68466 −0.186307
\(932\) −20.7386 −0.679317
\(933\) 0 0
\(934\) −26.2462 −0.858802
\(935\) −7.12311 −0.232950
\(936\) 0 0
\(937\) −54.3002 −1.77391 −0.886955 0.461856i \(-0.847184\pi\)
−0.886955 + 0.461856i \(0.847184\pi\)
\(938\) 16.6847 0.544773
\(939\) 0 0
\(940\) 16.0000 0.521862
\(941\) −20.4384 −0.666274 −0.333137 0.942878i \(-0.608107\pi\)
−0.333137 + 0.942878i \(0.608107\pi\)
\(942\) 0 0
\(943\) 11.1231 0.362218
\(944\) −4.68466 −0.152473
\(945\) 0 0
\(946\) 0 0
\(947\) −28.9848 −0.941881 −0.470940 0.882165i \(-0.656085\pi\)
−0.470940 + 0.882165i \(0.656085\pi\)
\(948\) 0 0
\(949\) −9.56155 −0.310381
\(950\) 1.00000 0.0324443
\(951\) 0 0
\(952\) −12.6847 −0.411112
\(953\) 9.12311 0.295526 0.147763 0.989023i \(-0.452793\pi\)
0.147763 + 0.989023i \(0.452793\pi\)
\(954\) 0 0
\(955\) −41.3693 −1.33868
\(956\) 4.93087 0.159476
\(957\) 0 0
\(958\) −11.3693 −0.367326
\(959\) −30.0540 −0.970493
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −11.1231 −0.358623
\(963\) 0 0
\(964\) 21.6155 0.696189
\(965\) 2.24621 0.0723081
\(966\) 0 0
\(967\) −3.86174 −0.124185 −0.0620926 0.998070i \(-0.519777\pi\)
−0.0620926 + 0.998070i \(0.519777\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −2.24621 −0.0721215
\(971\) 14.2462 0.457183 0.228591 0.973522i \(-0.426588\pi\)
0.228591 + 0.973522i \(0.426588\pi\)
\(972\) 0 0
\(973\) −61.8617 −1.98320
\(974\) 22.8769 0.733023
\(975\) 0 0
\(976\) −10.2462 −0.327973
\(977\) −55.3693 −1.77142 −0.885711 0.464238i \(-0.846328\pi\)
−0.885711 + 0.464238i \(0.846328\pi\)
\(978\) 0 0
\(979\) −9.12311 −0.291576
\(980\) −11.3693 −0.363180
\(981\) 0 0
\(982\) 14.6307 0.466884
\(983\) 48.2462 1.53882 0.769408 0.638758i \(-0.220552\pi\)
0.769408 + 0.638758i \(0.220552\pi\)
\(984\) 0 0
\(985\) 3.50758 0.111761
\(986\) 23.8078 0.758194
\(987\) 0 0
\(988\) 3.56155 0.113308
\(989\) 0 0
\(990\) 0 0
\(991\) 21.1231 0.670998 0.335499 0.942041i \(-0.391095\pi\)
0.335499 + 0.942041i \(0.391095\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) −21.3693 −0.677794
\(995\) 1.36932 0.0434103
\(996\) 0 0
\(997\) −53.3693 −1.69022 −0.845112 0.534590i \(-0.820466\pi\)
−0.845112 + 0.534590i \(0.820466\pi\)
\(998\) 26.2462 0.830809
\(999\) 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 3762.2.a.y.1.2 2
3.2 odd 2 418.2.a.e.1.1 2
12.11 even 2 3344.2.a.k.1.2 2
33.32 even 2 4598.2.a.bj.1.1 2
57.56 even 2 7942.2.a.x.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.e.1.1 2 3.2 odd 2
3344.2.a.k.1.2 2 12.11 even 2
3762.2.a.y.1.2 2 1.1 even 1 trivial
4598.2.a.bj.1.1 2 33.32 even 2
7942.2.a.x.1.2 2 57.56 even 2