Properties

Label 6762.2.a.bo.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.70156 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.70156 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.70156 q^{10} -4.00000 q^{11} -1.00000 q^{12} -5.70156 q^{13} +3.70156 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +5.40312 q^{19} -3.70156 q^{20} +4.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +8.70156 q^{25} +5.70156 q^{26} -1.00000 q^{27} +0.298438 q^{29} -3.70156 q^{30} +2.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} +4.29844 q^{37} -5.40312 q^{38} +5.70156 q^{39} +3.70156 q^{40} +0.298438 q^{41} -1.70156 q^{43} -4.00000 q^{44} -3.70156 q^{45} -1.00000 q^{46} +11.1047 q^{47} -1.00000 q^{48} -8.70156 q^{50} +4.00000 q^{51} -5.70156 q^{52} -9.40312 q^{53} +1.00000 q^{54} +14.8062 q^{55} -5.40312 q^{57} -0.298438 q^{58} +7.40312 q^{59} +3.70156 q^{60} +2.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +21.1047 q^{65} -4.00000 q^{66} +14.8062 q^{67} -4.00000 q^{68} -1.00000 q^{69} -7.40312 q^{71} -1.00000 q^{72} -1.40312 q^{73} -4.29844 q^{74} -8.70156 q^{75} +5.40312 q^{76} -5.70156 q^{78} +8.00000 q^{79} -3.70156 q^{80} +1.00000 q^{81} -0.298438 q^{82} -13.4031 q^{83} +14.8062 q^{85} +1.70156 q^{86} -0.298438 q^{87} +4.00000 q^{88} +11.4031 q^{89} +3.70156 q^{90} +1.00000 q^{92} -2.00000 q^{93} -11.1047 q^{94} -20.0000 q^{95} +1.00000 q^{96} -6.29844 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} + q^{10} - 8 q^{11} - 2 q^{12} - 5 q^{13} + q^{15} + 2 q^{16} - 8 q^{17} - 2 q^{18} - 2 q^{19} - q^{20} + 8 q^{22} + 2 q^{23} + 2 q^{24} + 11 q^{25} + 5 q^{26} - 2 q^{27} + 7 q^{29} - q^{30} + 4 q^{31} - 2 q^{32} + 8 q^{33} + 8 q^{34} + 2 q^{36} + 15 q^{37} + 2 q^{38} + 5 q^{39} + q^{40} + 7 q^{41} + 3 q^{43} - 8 q^{44} - q^{45} - 2 q^{46} + 3 q^{47} - 2 q^{48} - 11 q^{50} + 8 q^{51} - 5 q^{52} - 6 q^{53} + 2 q^{54} + 4 q^{55} + 2 q^{57} - 7 q^{58} + 2 q^{59} + q^{60} + 4 q^{61} - 4 q^{62} + 2 q^{64} + 23 q^{65} - 8 q^{66} + 4 q^{67} - 8 q^{68} - 2 q^{69} - 2 q^{71} - 2 q^{72} + 10 q^{73} - 15 q^{74} - 11 q^{75} - 2 q^{76} - 5 q^{78} + 16 q^{79} - q^{80} + 2 q^{81} - 7 q^{82} - 14 q^{83} + 4 q^{85} - 3 q^{86} - 7 q^{87} + 8 q^{88} + 10 q^{89} + q^{90} + 2 q^{92} - 4 q^{93} - 3 q^{94} - 40 q^{95} + 2 q^{96} - 19 q^{97} - 8 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −3.70156 −1.65539 −0.827694 0.561179i \(-0.810348\pi\)
−0.827694 + 0.561179i \(0.810348\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.70156 1.17054
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) −5.70156 −1.58133 −0.790664 0.612250i \(-0.790265\pi\)
−0.790664 + 0.612250i \(0.790265\pi\)
\(14\) 0 0
\(15\) 3.70156 0.955739
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.40312 1.23956 0.619781 0.784775i \(-0.287221\pi\)
0.619781 + 0.784775i \(0.287221\pi\)
\(20\) −3.70156 −0.827694
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 8.70156 1.74031
\(26\) 5.70156 1.11817
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0.298438 0.0554185 0.0277093 0.999616i \(-0.491179\pi\)
0.0277093 + 0.999616i \(0.491179\pi\)
\(30\) −3.70156 −0.675810
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.29844 0.706659 0.353329 0.935499i \(-0.385049\pi\)
0.353329 + 0.935499i \(0.385049\pi\)
\(38\) −5.40312 −0.876502
\(39\) 5.70156 0.912981
\(40\) 3.70156 0.585268
\(41\) 0.298438 0.0466082 0.0233041 0.999728i \(-0.492581\pi\)
0.0233041 + 0.999728i \(0.492581\pi\)
\(42\) 0 0
\(43\) −1.70156 −0.259486 −0.129743 0.991548i \(-0.541415\pi\)
−0.129743 + 0.991548i \(0.541415\pi\)
\(44\) −4.00000 −0.603023
\(45\) −3.70156 −0.551796
\(46\) −1.00000 −0.147442
\(47\) 11.1047 1.61978 0.809892 0.586578i \(-0.199525\pi\)
0.809892 + 0.586578i \(0.199525\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −8.70156 −1.23059
\(51\) 4.00000 0.560112
\(52\) −5.70156 −0.790664
\(53\) −9.40312 −1.29162 −0.645809 0.763499i \(-0.723480\pi\)
−0.645809 + 0.763499i \(0.723480\pi\)
\(54\) 1.00000 0.136083
\(55\) 14.8062 1.99647
\(56\) 0 0
\(57\) −5.40312 −0.715661
\(58\) −0.298438 −0.0391868
\(59\) 7.40312 0.963805 0.481902 0.876225i \(-0.339946\pi\)
0.481902 + 0.876225i \(0.339946\pi\)
\(60\) 3.70156 0.477870
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 21.1047 2.61771
\(66\) −4.00000 −0.492366
\(67\) 14.8062 1.80887 0.904436 0.426610i \(-0.140292\pi\)
0.904436 + 0.426610i \(0.140292\pi\)
\(68\) −4.00000 −0.485071
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −7.40312 −0.878589 −0.439295 0.898343i \(-0.644772\pi\)
−0.439295 + 0.898343i \(0.644772\pi\)
\(72\) −1.00000 −0.117851
\(73\) −1.40312 −0.164223 −0.0821116 0.996623i \(-0.526166\pi\)
−0.0821116 + 0.996623i \(0.526166\pi\)
\(74\) −4.29844 −0.499683
\(75\) −8.70156 −1.00477
\(76\) 5.40312 0.619781
\(77\) 0 0
\(78\) −5.70156 −0.645575
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −3.70156 −0.413847
\(81\) 1.00000 0.111111
\(82\) −0.298438 −0.0329570
\(83\) −13.4031 −1.47118 −0.735592 0.677425i \(-0.763096\pi\)
−0.735592 + 0.677425i \(0.763096\pi\)
\(84\) 0 0
\(85\) 14.8062 1.60596
\(86\) 1.70156 0.183484
\(87\) −0.298438 −0.0319959
\(88\) 4.00000 0.426401
\(89\) 11.4031 1.20873 0.604364 0.796708i \(-0.293427\pi\)
0.604364 + 0.796708i \(0.293427\pi\)
\(90\) 3.70156 0.390179
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −2.00000 −0.207390
\(94\) −11.1047 −1.14536
\(95\) −20.0000 −2.05196
\(96\) 1.00000 0.102062
\(97\) −6.29844 −0.639509 −0.319755 0.947500i \(-0.603601\pi\)
−0.319755 + 0.947500i \(0.603601\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 8.70156 0.870156
\(101\) 3.40312 0.338624 0.169312 0.985563i \(-0.445846\pi\)
0.169312 + 0.985563i \(0.445846\pi\)
\(102\) −4.00000 −0.396059
\(103\) −2.29844 −0.226472 −0.113236 0.993568i \(-0.536122\pi\)
−0.113236 + 0.993568i \(0.536122\pi\)
\(104\) 5.70156 0.559084
\(105\) 0 0
\(106\) 9.40312 0.913312
\(107\) 18.8062 1.81807 0.909034 0.416721i \(-0.136821\pi\)
0.909034 + 0.416721i \(0.136821\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −7.70156 −0.737676 −0.368838 0.929494i \(-0.620244\pi\)
−0.368838 + 0.929494i \(0.620244\pi\)
\(110\) −14.8062 −1.41172
\(111\) −4.29844 −0.407990
\(112\) 0 0
\(113\) 7.10469 0.668353 0.334176 0.942511i \(-0.391542\pi\)
0.334176 + 0.942511i \(0.391542\pi\)
\(114\) 5.40312 0.506049
\(115\) −3.70156 −0.345172
\(116\) 0.298438 0.0277093
\(117\) −5.70156 −0.527110
\(118\) −7.40312 −0.681513
\(119\) 0 0
\(120\) −3.70156 −0.337905
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) −0.298438 −0.0269092
\(124\) 2.00000 0.179605
\(125\) −13.7016 −1.22550
\(126\) 0 0
\(127\) −5.70156 −0.505932 −0.252966 0.967475i \(-0.581406\pi\)
−0.252966 + 0.967475i \(0.581406\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.70156 0.149814
\(130\) −21.1047 −1.85100
\(131\) 7.40312 0.646814 0.323407 0.946260i \(-0.395172\pi\)
0.323407 + 0.946260i \(0.395172\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −14.8062 −1.27907
\(135\) 3.70156 0.318580
\(136\) 4.00000 0.342997
\(137\) −19.7016 −1.68322 −0.841609 0.540087i \(-0.818391\pi\)
−0.841609 + 0.540087i \(0.818391\pi\)
\(138\) 1.00000 0.0851257
\(139\) −17.7016 −1.50143 −0.750713 0.660628i \(-0.770290\pi\)
−0.750713 + 0.660628i \(0.770290\pi\)
\(140\) 0 0
\(141\) −11.1047 −0.935183
\(142\) 7.40312 0.621256
\(143\) 22.8062 1.90715
\(144\) 1.00000 0.0833333
\(145\) −1.10469 −0.0917392
\(146\) 1.40312 0.116123
\(147\) 0 0
\(148\) 4.29844 0.353329
\(149\) −20.2094 −1.65562 −0.827808 0.561011i \(-0.810412\pi\)
−0.827808 + 0.561011i \(0.810412\pi\)
\(150\) 8.70156 0.710480
\(151\) 23.9109 1.94584 0.972922 0.231133i \(-0.0742433\pi\)
0.972922 + 0.231133i \(0.0742433\pi\)
\(152\) −5.40312 −0.438251
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) −7.40312 −0.594633
\(156\) 5.70156 0.456490
\(157\) 1.40312 0.111982 0.0559908 0.998431i \(-0.482168\pi\)
0.0559908 + 0.998431i \(0.482168\pi\)
\(158\) −8.00000 −0.636446
\(159\) 9.40312 0.745716
\(160\) 3.70156 0.292634
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −10.8062 −0.846411 −0.423205 0.906034i \(-0.639095\pi\)
−0.423205 + 0.906034i \(0.639095\pi\)
\(164\) 0.298438 0.0233041
\(165\) −14.8062 −1.15266
\(166\) 13.4031 1.04028
\(167\) 13.4031 1.03716 0.518582 0.855028i \(-0.326460\pi\)
0.518582 + 0.855028i \(0.326460\pi\)
\(168\) 0 0
\(169\) 19.5078 1.50060
\(170\) −14.8062 −1.13559
\(171\) 5.40312 0.413187
\(172\) −1.70156 −0.129743
\(173\) −19.4031 −1.47519 −0.737596 0.675242i \(-0.764039\pi\)
−0.737596 + 0.675242i \(0.764039\pi\)
\(174\) 0.298438 0.0226245
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) −7.40312 −0.556453
\(178\) −11.4031 −0.854700
\(179\) −25.1047 −1.87641 −0.938206 0.346077i \(-0.887514\pi\)
−0.938206 + 0.346077i \(0.887514\pi\)
\(180\) −3.70156 −0.275898
\(181\) 16.8062 1.24920 0.624599 0.780945i \(-0.285262\pi\)
0.624599 + 0.780945i \(0.285262\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) −1.00000 −0.0737210
\(185\) −15.9109 −1.16980
\(186\) 2.00000 0.146647
\(187\) 16.0000 1.17004
\(188\) 11.1047 0.809892
\(189\) 0 0
\(190\) 20.0000 1.45095
\(191\) 22.8062 1.65020 0.825101 0.564985i \(-0.191118\pi\)
0.825101 + 0.564985i \(0.191118\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −23.1047 −1.66311 −0.831556 0.555441i \(-0.812549\pi\)
−0.831556 + 0.555441i \(0.812549\pi\)
\(194\) 6.29844 0.452201
\(195\) −21.1047 −1.51134
\(196\) 0 0
\(197\) 22.5078 1.60362 0.801808 0.597582i \(-0.203872\pi\)
0.801808 + 0.597582i \(0.203872\pi\)
\(198\) 4.00000 0.284268
\(199\) 2.29844 0.162932 0.0814660 0.996676i \(-0.474040\pi\)
0.0814660 + 0.996676i \(0.474040\pi\)
\(200\) −8.70156 −0.615293
\(201\) −14.8062 −1.04435
\(202\) −3.40312 −0.239443
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) −1.10469 −0.0771546
\(206\) 2.29844 0.160140
\(207\) 1.00000 0.0695048
\(208\) −5.70156 −0.395332
\(209\) −21.6125 −1.49497
\(210\) 0 0
\(211\) 10.8062 0.743933 0.371966 0.928246i \(-0.378684\pi\)
0.371966 + 0.928246i \(0.378684\pi\)
\(212\) −9.40312 −0.645809
\(213\) 7.40312 0.507254
\(214\) −18.8062 −1.28557
\(215\) 6.29844 0.429550
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 7.70156 0.521616
\(219\) 1.40312 0.0948143
\(220\) 14.8062 0.998237
\(221\) 22.8062 1.53411
\(222\) 4.29844 0.288492
\(223\) 22.0000 1.47323 0.736614 0.676313i \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) 0 0
\(225\) 8.70156 0.580104
\(226\) −7.10469 −0.472597
\(227\) 23.1047 1.53351 0.766756 0.641939i \(-0.221870\pi\)
0.766756 + 0.641939i \(0.221870\pi\)
\(228\) −5.40312 −0.357831
\(229\) 14.5969 0.964589 0.482294 0.876009i \(-0.339804\pi\)
0.482294 + 0.876009i \(0.339804\pi\)
\(230\) 3.70156 0.244074
\(231\) 0 0
\(232\) −0.298438 −0.0195934
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 5.70156 0.372723
\(235\) −41.1047 −2.68137
\(236\) 7.40312 0.481902
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 10.8062 0.698998 0.349499 0.936937i \(-0.386352\pi\)
0.349499 + 0.936937i \(0.386352\pi\)
\(240\) 3.70156 0.238935
\(241\) 15.9109 1.02491 0.512457 0.858713i \(-0.328736\pi\)
0.512457 + 0.858713i \(0.328736\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0.298438 0.0190277
\(247\) −30.8062 −1.96015
\(248\) −2.00000 −0.127000
\(249\) 13.4031 0.849388
\(250\) 13.7016 0.866563
\(251\) −19.7016 −1.24355 −0.621776 0.783195i \(-0.713588\pi\)
−0.621776 + 0.783195i \(0.713588\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 5.70156 0.357748
\(255\) −14.8062 −0.927203
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) −1.70156 −0.105935
\(259\) 0 0
\(260\) 21.1047 1.30886
\(261\) 0.298438 0.0184728
\(262\) −7.40312 −0.457367
\(263\) 9.70156 0.598224 0.299112 0.954218i \(-0.403310\pi\)
0.299112 + 0.954218i \(0.403310\pi\)
\(264\) −4.00000 −0.246183
\(265\) 34.8062 2.13813
\(266\) 0 0
\(267\) −11.4031 −0.697860
\(268\) 14.8062 0.904436
\(269\) 10.2094 0.622476 0.311238 0.950332i \(-0.399256\pi\)
0.311238 + 0.950332i \(0.399256\pi\)
\(270\) −3.70156 −0.225270
\(271\) −1.40312 −0.0852337 −0.0426169 0.999091i \(-0.513569\pi\)
−0.0426169 + 0.999091i \(0.513569\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 19.7016 1.19021
\(275\) −34.8062 −2.09890
\(276\) −1.00000 −0.0601929
\(277\) −21.4031 −1.28599 −0.642995 0.765871i \(-0.722308\pi\)
−0.642995 + 0.765871i \(0.722308\pi\)
\(278\) 17.7016 1.06167
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 14.5078 0.865463 0.432732 0.901523i \(-0.357550\pi\)
0.432732 + 0.901523i \(0.357550\pi\)
\(282\) 11.1047 0.661274
\(283\) −4.80625 −0.285702 −0.142851 0.989744i \(-0.545627\pi\)
−0.142851 + 0.989744i \(0.545627\pi\)
\(284\) −7.40312 −0.439295
\(285\) 20.0000 1.18470
\(286\) −22.8062 −1.34856
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 1.10469 0.0648694
\(291\) 6.29844 0.369221
\(292\) −1.40312 −0.0821116
\(293\) 20.8062 1.21551 0.607757 0.794123i \(-0.292069\pi\)
0.607757 + 0.794123i \(0.292069\pi\)
\(294\) 0 0
\(295\) −27.4031 −1.59547
\(296\) −4.29844 −0.249842
\(297\) 4.00000 0.232104
\(298\) 20.2094 1.17070
\(299\) −5.70156 −0.329730
\(300\) −8.70156 −0.502385
\(301\) 0 0
\(302\) −23.9109 −1.37592
\(303\) −3.40312 −0.195504
\(304\) 5.40312 0.309890
\(305\) −7.40312 −0.423902
\(306\) 4.00000 0.228665
\(307\) −11.9109 −0.679793 −0.339896 0.940463i \(-0.610392\pi\)
−0.339896 + 0.940463i \(0.610392\pi\)
\(308\) 0 0
\(309\) 2.29844 0.130754
\(310\) 7.40312 0.420469
\(311\) −13.4031 −0.760021 −0.380011 0.924982i \(-0.624080\pi\)
−0.380011 + 0.924982i \(0.624080\pi\)
\(312\) −5.70156 −0.322787
\(313\) 6.20937 0.350974 0.175487 0.984482i \(-0.443850\pi\)
0.175487 + 0.984482i \(0.443850\pi\)
\(314\) −1.40312 −0.0791829
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −8.29844 −0.466087 −0.233043 0.972466i \(-0.574868\pi\)
−0.233043 + 0.972466i \(0.574868\pi\)
\(318\) −9.40312 −0.527301
\(319\) −1.19375 −0.0668373
\(320\) −3.70156 −0.206924
\(321\) −18.8062 −1.04966
\(322\) 0 0
\(323\) −21.6125 −1.20255
\(324\) 1.00000 0.0555556
\(325\) −49.6125 −2.75201
\(326\) 10.8062 0.598503
\(327\) 7.70156 0.425897
\(328\) −0.298438 −0.0164785
\(329\) 0 0
\(330\) 14.8062 0.815057
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −13.4031 −0.735592
\(333\) 4.29844 0.235553
\(334\) −13.4031 −0.733386
\(335\) −54.8062 −2.99439
\(336\) 0 0
\(337\) −4.20937 −0.229299 −0.114650 0.993406i \(-0.536575\pi\)
−0.114650 + 0.993406i \(0.536575\pi\)
\(338\) −19.5078 −1.06109
\(339\) −7.10469 −0.385874
\(340\) 14.8062 0.802982
\(341\) −8.00000 −0.433224
\(342\) −5.40312 −0.292167
\(343\) 0 0
\(344\) 1.70156 0.0917421
\(345\) 3.70156 0.199285
\(346\) 19.4031 1.04312
\(347\) −10.2984 −0.552849 −0.276425 0.961036i \(-0.589150\pi\)
−0.276425 + 0.961036i \(0.589150\pi\)
\(348\) −0.298438 −0.0159979
\(349\) −12.5969 −0.674295 −0.337148 0.941452i \(-0.609462\pi\)
−0.337148 + 0.941452i \(0.609462\pi\)
\(350\) 0 0
\(351\) 5.70156 0.304327
\(352\) 4.00000 0.213201
\(353\) −4.29844 −0.228783 −0.114391 0.993436i \(-0.536492\pi\)
−0.114391 + 0.993436i \(0.536492\pi\)
\(354\) 7.40312 0.393472
\(355\) 27.4031 1.45441
\(356\) 11.4031 0.604364
\(357\) 0 0
\(358\) 25.1047 1.32682
\(359\) −2.89531 −0.152809 −0.0764044 0.997077i \(-0.524344\pi\)
−0.0764044 + 0.997077i \(0.524344\pi\)
\(360\) 3.70156 0.195089
\(361\) 10.1938 0.536513
\(362\) −16.8062 −0.883317
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 5.19375 0.271853
\(366\) 2.00000 0.104542
\(367\) 17.1047 0.892857 0.446429 0.894819i \(-0.352696\pi\)
0.446429 + 0.894819i \(0.352696\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0.298438 0.0155361
\(370\) 15.9109 0.827170
\(371\) 0 0
\(372\) −2.00000 −0.103695
\(373\) −0.806248 −0.0417460 −0.0208730 0.999782i \(-0.506645\pi\)
−0.0208730 + 0.999782i \(0.506645\pi\)
\(374\) −16.0000 −0.827340
\(375\) 13.7016 0.707546
\(376\) −11.1047 −0.572680
\(377\) −1.70156 −0.0876349
\(378\) 0 0
\(379\) 13.7016 0.703802 0.351901 0.936037i \(-0.385535\pi\)
0.351901 + 0.936037i \(0.385535\pi\)
\(380\) −20.0000 −1.02598
\(381\) 5.70156 0.292100
\(382\) −22.8062 −1.16687
\(383\) −29.6125 −1.51313 −0.756564 0.653920i \(-0.773123\pi\)
−0.756564 + 0.653920i \(0.773123\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 23.1047 1.17600
\(387\) −1.70156 −0.0864953
\(388\) −6.29844 −0.319755
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 21.1047 1.06868
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) −7.40312 −0.373438
\(394\) −22.5078 −1.13393
\(395\) −29.6125 −1.48997
\(396\) −4.00000 −0.201008
\(397\) −8.59688 −0.431465 −0.215732 0.976453i \(-0.569214\pi\)
−0.215732 + 0.976453i \(0.569214\pi\)
\(398\) −2.29844 −0.115210
\(399\) 0 0
\(400\) 8.70156 0.435078
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 14.8062 0.738469
\(403\) −11.4031 −0.568030
\(404\) 3.40312 0.169312
\(405\) −3.70156 −0.183932
\(406\) 0 0
\(407\) −17.1938 −0.852263
\(408\) −4.00000 −0.198030
\(409\) −5.40312 −0.267167 −0.133584 0.991038i \(-0.542648\pi\)
−0.133584 + 0.991038i \(0.542648\pi\)
\(410\) 1.10469 0.0545566
\(411\) 19.7016 0.971806
\(412\) −2.29844 −0.113236
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 49.6125 2.43538
\(416\) 5.70156 0.279542
\(417\) 17.7016 0.866849
\(418\) 21.6125 1.05710
\(419\) 0.209373 0.0102285 0.00511426 0.999987i \(-0.498372\pi\)
0.00511426 + 0.999987i \(0.498372\pi\)
\(420\) 0 0
\(421\) −15.7016 −0.765247 −0.382624 0.923904i \(-0.624979\pi\)
−0.382624 + 0.923904i \(0.624979\pi\)
\(422\) −10.8062 −0.526040
\(423\) 11.1047 0.539928
\(424\) 9.40312 0.456656
\(425\) −34.8062 −1.68835
\(426\) −7.40312 −0.358683
\(427\) 0 0
\(428\) 18.8062 0.909034
\(429\) −22.8062 −1.10110
\(430\) −6.29844 −0.303738
\(431\) −3.91093 −0.188383 −0.0941916 0.995554i \(-0.530027\pi\)
−0.0941916 + 0.995554i \(0.530027\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −31.3141 −1.50486 −0.752429 0.658674i \(-0.771118\pi\)
−0.752429 + 0.658674i \(0.771118\pi\)
\(434\) 0 0
\(435\) 1.10469 0.0529657
\(436\) −7.70156 −0.368838
\(437\) 5.40312 0.258466
\(438\) −1.40312 −0.0670439
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) −14.8062 −0.705860
\(441\) 0 0
\(442\) −22.8062 −1.08478
\(443\) −13.7016 −0.650981 −0.325490 0.945545i \(-0.605529\pi\)
−0.325490 + 0.945545i \(0.605529\pi\)
\(444\) −4.29844 −0.203995
\(445\) −42.2094 −2.00092
\(446\) −22.0000 −1.04173
\(447\) 20.2094 0.955871
\(448\) 0 0
\(449\) −38.4187 −1.81309 −0.906546 0.422106i \(-0.861291\pi\)
−0.906546 + 0.422106i \(0.861291\pi\)
\(450\) −8.70156 −0.410196
\(451\) −1.19375 −0.0562116
\(452\) 7.10469 0.334176
\(453\) −23.9109 −1.12343
\(454\) −23.1047 −1.08436
\(455\) 0 0
\(456\) 5.40312 0.253024
\(457\) 28.2094 1.31958 0.659789 0.751451i \(-0.270645\pi\)
0.659789 + 0.751451i \(0.270645\pi\)
\(458\) −14.5969 −0.682067
\(459\) 4.00000 0.186704
\(460\) −3.70156 −0.172586
\(461\) 2.20937 0.102901 0.0514504 0.998676i \(-0.483616\pi\)
0.0514504 + 0.998676i \(0.483616\pi\)
\(462\) 0 0
\(463\) −22.2984 −1.03630 −0.518148 0.855291i \(-0.673378\pi\)
−0.518148 + 0.855291i \(0.673378\pi\)
\(464\) 0.298438 0.0138546
\(465\) 7.40312 0.343312
\(466\) 26.0000 1.20443
\(467\) 3.70156 0.171288 0.0856439 0.996326i \(-0.472705\pi\)
0.0856439 + 0.996326i \(0.472705\pi\)
\(468\) −5.70156 −0.263555
\(469\) 0 0
\(470\) 41.1047 1.89602
\(471\) −1.40312 −0.0646526
\(472\) −7.40312 −0.340756
\(473\) 6.80625 0.312952
\(474\) 8.00000 0.367452
\(475\) 47.0156 2.15722
\(476\) 0 0
\(477\) −9.40312 −0.430539
\(478\) −10.8062 −0.494266
\(479\) −16.5969 −0.758331 −0.379165 0.925329i \(-0.623789\pi\)
−0.379165 + 0.925329i \(0.623789\pi\)
\(480\) −3.70156 −0.168952
\(481\) −24.5078 −1.11746
\(482\) −15.9109 −0.724723
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 23.3141 1.05864
\(486\) 1.00000 0.0453609
\(487\) −6.29844 −0.285409 −0.142705 0.989765i \(-0.545580\pi\)
−0.142705 + 0.989765i \(0.545580\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 10.8062 0.488675
\(490\) 0 0
\(491\) −9.61250 −0.433806 −0.216903 0.976193i \(-0.569596\pi\)
−0.216903 + 0.976193i \(0.569596\pi\)
\(492\) −0.298438 −0.0134546
\(493\) −1.19375 −0.0537639
\(494\) 30.8062 1.38604
\(495\) 14.8062 0.665491
\(496\) 2.00000 0.0898027
\(497\) 0 0
\(498\) −13.4031 −0.600608
\(499\) −23.4031 −1.04767 −0.523834 0.851820i \(-0.675499\pi\)
−0.523834 + 0.851820i \(0.675499\pi\)
\(500\) −13.7016 −0.612752
\(501\) −13.4031 −0.598807
\(502\) 19.7016 0.879324
\(503\) −4.59688 −0.204965 −0.102482 0.994735i \(-0.532678\pi\)
−0.102482 + 0.994735i \(0.532678\pi\)
\(504\) 0 0
\(505\) −12.5969 −0.560554
\(506\) 4.00000 0.177822
\(507\) −19.5078 −0.866372
\(508\) −5.70156 −0.252966
\(509\) −37.6125 −1.66714 −0.833572 0.552410i \(-0.813708\pi\)
−0.833572 + 0.552410i \(0.813708\pi\)
\(510\) 14.8062 0.655632
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −5.40312 −0.238554
\(514\) 2.00000 0.0882162
\(515\) 8.50781 0.374899
\(516\) 1.70156 0.0749071
\(517\) −44.4187 −1.95353
\(518\) 0 0
\(519\) 19.4031 0.851703
\(520\) −21.1047 −0.925502
\(521\) 20.5969 0.902366 0.451183 0.892432i \(-0.351002\pi\)
0.451183 + 0.892432i \(0.351002\pi\)
\(522\) −0.298438 −0.0130623
\(523\) −20.8062 −0.909794 −0.454897 0.890544i \(-0.650324\pi\)
−0.454897 + 0.890544i \(0.650324\pi\)
\(524\) 7.40312 0.323407
\(525\) 0 0
\(526\) −9.70156 −0.423008
\(527\) −8.00000 −0.348485
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) −34.8062 −1.51189
\(531\) 7.40312 0.321268
\(532\) 0 0
\(533\) −1.70156 −0.0737028
\(534\) 11.4031 0.493461
\(535\) −69.6125 −3.00961
\(536\) −14.8062 −0.639533
\(537\) 25.1047 1.08335
\(538\) −10.2094 −0.440157
\(539\) 0 0
\(540\) 3.70156 0.159290
\(541\) 33.4031 1.43611 0.718056 0.695985i \(-0.245032\pi\)
0.718056 + 0.695985i \(0.245032\pi\)
\(542\) 1.40312 0.0602693
\(543\) −16.8062 −0.721225
\(544\) 4.00000 0.171499
\(545\) 28.5078 1.22114
\(546\) 0 0
\(547\) 29.0156 1.24062 0.620309 0.784357i \(-0.287007\pi\)
0.620309 + 0.784357i \(0.287007\pi\)
\(548\) −19.7016 −0.841609
\(549\) 2.00000 0.0853579
\(550\) 34.8062 1.48414
\(551\) 1.61250 0.0686947
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 21.4031 0.909332
\(555\) 15.9109 0.675382
\(556\) −17.7016 −0.750713
\(557\) −37.4031 −1.58482 −0.792411 0.609988i \(-0.791174\pi\)
−0.792411 + 0.609988i \(0.791174\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 9.70156 0.410332
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) −14.5078 −0.611975
\(563\) 3.10469 0.130847 0.0654235 0.997858i \(-0.479160\pi\)
0.0654235 + 0.997858i \(0.479160\pi\)
\(564\) −11.1047 −0.467592
\(565\) −26.2984 −1.10638
\(566\) 4.80625 0.202022
\(567\) 0 0
\(568\) 7.40312 0.310628
\(569\) 31.7016 1.32900 0.664499 0.747289i \(-0.268645\pi\)
0.664499 + 0.747289i \(0.268645\pi\)
\(570\) −20.0000 −0.837708
\(571\) −14.8062 −0.619622 −0.309811 0.950798i \(-0.600266\pi\)
−0.309811 + 0.950798i \(0.600266\pi\)
\(572\) 22.8062 0.953577
\(573\) −22.8062 −0.952745
\(574\) 0 0
\(575\) 8.70156 0.362880
\(576\) 1.00000 0.0416667
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 1.00000 0.0415945
\(579\) 23.1047 0.960198
\(580\) −1.10469 −0.0458696
\(581\) 0 0
\(582\) −6.29844 −0.261079
\(583\) 37.6125 1.55775
\(584\) 1.40312 0.0580617
\(585\) 21.1047 0.872571
\(586\) −20.8062 −0.859498
\(587\) −14.8062 −0.611119 −0.305560 0.952173i \(-0.598844\pi\)
−0.305560 + 0.952173i \(0.598844\pi\)
\(588\) 0 0
\(589\) 10.8062 0.445264
\(590\) 27.4031 1.12817
\(591\) −22.5078 −0.925848
\(592\) 4.29844 0.176665
\(593\) −43.7016 −1.79461 −0.897304 0.441413i \(-0.854477\pi\)
−0.897304 + 0.441413i \(0.854477\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −20.2094 −0.827808
\(597\) −2.29844 −0.0940688
\(598\) 5.70156 0.233154
\(599\) 13.6125 0.556192 0.278096 0.960553i \(-0.410297\pi\)
0.278096 + 0.960553i \(0.410297\pi\)
\(600\) 8.70156 0.355240
\(601\) 13.4031 0.546725 0.273362 0.961911i \(-0.411864\pi\)
0.273362 + 0.961911i \(0.411864\pi\)
\(602\) 0 0
\(603\) 14.8062 0.602957
\(604\) 23.9109 0.972922
\(605\) −18.5078 −0.752450
\(606\) 3.40312 0.138242
\(607\) 39.6125 1.60782 0.803911 0.594750i \(-0.202749\pi\)
0.803911 + 0.594750i \(0.202749\pi\)
\(608\) −5.40312 −0.219126
\(609\) 0 0
\(610\) 7.40312 0.299744
\(611\) −63.3141 −2.56141
\(612\) −4.00000 −0.161690
\(613\) 3.70156 0.149505 0.0747523 0.997202i \(-0.476183\pi\)
0.0747523 + 0.997202i \(0.476183\pi\)
\(614\) 11.9109 0.480686
\(615\) 1.10469 0.0445453
\(616\) 0 0
\(617\) −19.1938 −0.772711 −0.386356 0.922350i \(-0.626266\pi\)
−0.386356 + 0.922350i \(0.626266\pi\)
\(618\) −2.29844 −0.0924567
\(619\) 7.19375 0.289141 0.144571 0.989494i \(-0.453820\pi\)
0.144571 + 0.989494i \(0.453820\pi\)
\(620\) −7.40312 −0.297317
\(621\) −1.00000 −0.0401286
\(622\) 13.4031 0.537416
\(623\) 0 0
\(624\) 5.70156 0.228245
\(625\) 7.20937 0.288375
\(626\) −6.20937 −0.248176
\(627\) 21.6125 0.863120
\(628\) 1.40312 0.0559908
\(629\) −17.1938 −0.685560
\(630\) 0 0
\(631\) −29.6125 −1.17885 −0.589427 0.807821i \(-0.700647\pi\)
−0.589427 + 0.807821i \(0.700647\pi\)
\(632\) −8.00000 −0.318223
\(633\) −10.8062 −0.429510
\(634\) 8.29844 0.329573
\(635\) 21.1047 0.837514
\(636\) 9.40312 0.372858
\(637\) 0 0
\(638\) 1.19375 0.0472611
\(639\) −7.40312 −0.292863
\(640\) 3.70156 0.146317
\(641\) −20.7172 −0.818280 −0.409140 0.912472i \(-0.634171\pi\)
−0.409140 + 0.912472i \(0.634171\pi\)
\(642\) 18.8062 0.742223
\(643\) −0.806248 −0.0317953 −0.0158977 0.999874i \(-0.505061\pi\)
−0.0158977 + 0.999874i \(0.505061\pi\)
\(644\) 0 0
\(645\) −6.29844 −0.248001
\(646\) 21.6125 0.850332
\(647\) 6.59688 0.259350 0.129675 0.991557i \(-0.458607\pi\)
0.129675 + 0.991557i \(0.458607\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −29.6125 −1.16239
\(650\) 49.6125 1.94596
\(651\) 0 0
\(652\) −10.8062 −0.423205
\(653\) 7.10469 0.278028 0.139014 0.990290i \(-0.455607\pi\)
0.139014 + 0.990290i \(0.455607\pi\)
\(654\) −7.70156 −0.301155
\(655\) −27.4031 −1.07073
\(656\) 0.298438 0.0116520
\(657\) −1.40312 −0.0547411
\(658\) 0 0
\(659\) 8.00000 0.311636 0.155818 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) −14.8062 −0.576332
\(661\) −42.4187 −1.64990 −0.824949 0.565207i \(-0.808796\pi\)
−0.824949 + 0.565207i \(0.808796\pi\)
\(662\) 20.0000 0.777322
\(663\) −22.8062 −0.885721
\(664\) 13.4031 0.520142
\(665\) 0 0
\(666\) −4.29844 −0.166561
\(667\) 0.298438 0.0115556
\(668\) 13.4031 0.518582
\(669\) −22.0000 −0.850569
\(670\) 54.8062 2.11735
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 7.70156 0.296873 0.148437 0.988922i \(-0.452576\pi\)
0.148437 + 0.988922i \(0.452576\pi\)
\(674\) 4.20937 0.162139
\(675\) −8.70156 −0.334923
\(676\) 19.5078 0.750300
\(677\) 50.4187 1.93775 0.968875 0.247551i \(-0.0796257\pi\)
0.968875 + 0.247551i \(0.0796257\pi\)
\(678\) 7.10469 0.272854
\(679\) 0 0
\(680\) −14.8062 −0.567794
\(681\) −23.1047 −0.885374
\(682\) 8.00000 0.306336
\(683\) 33.6125 1.28615 0.643073 0.765805i \(-0.277659\pi\)
0.643073 + 0.765805i \(0.277659\pi\)
\(684\) 5.40312 0.206594
\(685\) 72.9266 2.78638
\(686\) 0 0
\(687\) −14.5969 −0.556906
\(688\) −1.70156 −0.0648714
\(689\) 53.6125 2.04247
\(690\) −3.70156 −0.140916
\(691\) 0.507811 0.0193180 0.00965901 0.999953i \(-0.496925\pi\)
0.00965901 + 0.999953i \(0.496925\pi\)
\(692\) −19.4031 −0.737596
\(693\) 0 0
\(694\) 10.2984 0.390923
\(695\) 65.5234 2.48545
\(696\) 0.298438 0.0113123
\(697\) −1.19375 −0.0452166
\(698\) 12.5969 0.476799
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) 47.0156 1.77576 0.887878 0.460079i \(-0.152179\pi\)
0.887878 + 0.460079i \(0.152179\pi\)
\(702\) −5.70156 −0.215192
\(703\) 23.2250 0.875947
\(704\) −4.00000 −0.150756
\(705\) 41.1047 1.54809
\(706\) 4.29844 0.161774
\(707\) 0 0
\(708\) −7.40312 −0.278226
\(709\) 23.1938 0.871060 0.435530 0.900174i \(-0.356561\pi\)
0.435530 + 0.900174i \(0.356561\pi\)
\(710\) −27.4031 −1.02842
\(711\) 8.00000 0.300023
\(712\) −11.4031 −0.427350
\(713\) 2.00000 0.0749006
\(714\) 0 0
\(715\) −84.4187 −3.15708
\(716\) −25.1047 −0.938206
\(717\) −10.8062 −0.403567
\(718\) 2.89531 0.108052
\(719\) 5.91093 0.220441 0.110220 0.993907i \(-0.464844\pi\)
0.110220 + 0.993907i \(0.464844\pi\)
\(720\) −3.70156 −0.137949
\(721\) 0 0
\(722\) −10.1938 −0.379372
\(723\) −15.9109 −0.591734
\(724\) 16.8062 0.624599
\(725\) 2.59688 0.0964455
\(726\) 5.00000 0.185567
\(727\) 12.0000 0.445055 0.222528 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −5.19375 −0.192229
\(731\) 6.80625 0.251738
\(732\) −2.00000 −0.0739221
\(733\) −52.2094 −1.92840 −0.964199 0.265181i \(-0.914568\pi\)
−0.964199 + 0.265181i \(0.914568\pi\)
\(734\) −17.1047 −0.631345
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −59.2250 −2.18158
\(738\) −0.298438 −0.0109857
\(739\) −29.0156 −1.06736 −0.533678 0.845687i \(-0.679191\pi\)
−0.533678 + 0.845687i \(0.679191\pi\)
\(740\) −15.9109 −0.584898
\(741\) 30.8062 1.13170
\(742\) 0 0
\(743\) 32.0000 1.17397 0.586983 0.809599i \(-0.300316\pi\)
0.586983 + 0.809599i \(0.300316\pi\)
\(744\) 2.00000 0.0733236
\(745\) 74.8062 2.74069
\(746\) 0.806248 0.0295189
\(747\) −13.4031 −0.490395
\(748\) 16.0000 0.585018
\(749\) 0 0
\(750\) −13.7016 −0.500310
\(751\) 29.6125 1.08058 0.540288 0.841480i \(-0.318315\pi\)
0.540288 + 0.841480i \(0.318315\pi\)
\(752\) 11.1047 0.404946
\(753\) 19.7016 0.717965
\(754\) 1.70156 0.0619672
\(755\) −88.5078 −3.22113
\(756\) 0 0
\(757\) −26.4187 −0.960206 −0.480103 0.877212i \(-0.659401\pi\)
−0.480103 + 0.877212i \(0.659401\pi\)
\(758\) −13.7016 −0.497663
\(759\) 4.00000 0.145191
\(760\) 20.0000 0.725476
\(761\) 35.6125 1.29095 0.645476 0.763781i \(-0.276659\pi\)
0.645476 + 0.763781i \(0.276659\pi\)
\(762\) −5.70156 −0.206546
\(763\) 0 0
\(764\) 22.8062 0.825101
\(765\) 14.8062 0.535321
\(766\) 29.6125 1.06994
\(767\) −42.2094 −1.52409
\(768\) −1.00000 −0.0360844
\(769\) −15.9109 −0.573763 −0.286881 0.957966i \(-0.592619\pi\)
−0.286881 + 0.957966i \(0.592619\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) −23.1047 −0.831556
\(773\) 7.10469 0.255538 0.127769 0.991804i \(-0.459218\pi\)
0.127769 + 0.991804i \(0.459218\pi\)
\(774\) 1.70156 0.0611614
\(775\) 17.4031 0.625139
\(776\) 6.29844 0.226101
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 1.61250 0.0577737
\(780\) −21.1047 −0.755669
\(781\) 29.6125 1.05962
\(782\) 4.00000 0.143040
\(783\) −0.298438 −0.0106653
\(784\) 0 0
\(785\) −5.19375 −0.185373
\(786\) 7.40312 0.264061
\(787\) 18.5969 0.662907 0.331454 0.943472i \(-0.392461\pi\)
0.331454 + 0.943472i \(0.392461\pi\)
\(788\) 22.5078 0.801808
\(789\) −9.70156 −0.345385
\(790\) 29.6125 1.05357
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) −11.4031 −0.404937
\(794\) 8.59688 0.305092
\(795\) −34.8062 −1.23445
\(796\) 2.29844 0.0814660
\(797\) −15.7016 −0.556178 −0.278089 0.960555i \(-0.589701\pi\)
−0.278089 + 0.960555i \(0.589701\pi\)
\(798\) 0 0
\(799\) −44.4187 −1.57142
\(800\) −8.70156 −0.307647
\(801\) 11.4031 0.402910
\(802\) −30.0000 −1.05934
\(803\) 5.61250 0.198061
\(804\) −14.8062 −0.522176
\(805\) 0 0
\(806\) 11.4031 0.401658
\(807\) −10.2094 −0.359387
\(808\) −3.40312 −0.119721
\(809\) 31.0156 1.09045 0.545226 0.838289i \(-0.316444\pi\)
0.545226 + 0.838289i \(0.316444\pi\)
\(810\) 3.70156 0.130060
\(811\) −52.5078 −1.84380 −0.921899 0.387430i \(-0.873363\pi\)
−0.921899 + 0.387430i \(0.873363\pi\)
\(812\) 0 0
\(813\) 1.40312 0.0492097
\(814\) 17.1938 0.602641
\(815\) 40.0000 1.40114
\(816\) 4.00000 0.140028
\(817\) −9.19375 −0.321649
\(818\) 5.40312 0.188916
\(819\) 0 0
\(820\) −1.10469 −0.0385773
\(821\) 27.6125 0.963683 0.481841 0.876258i \(-0.339968\pi\)
0.481841 + 0.876258i \(0.339968\pi\)
\(822\) −19.7016 −0.687171
\(823\) −13.7016 −0.477606 −0.238803 0.971068i \(-0.576755\pi\)
−0.238803 + 0.971068i \(0.576755\pi\)
\(824\) 2.29844 0.0800699
\(825\) 34.8062 1.21180
\(826\) 0 0
\(827\) −47.4031 −1.64837 −0.824184 0.566322i \(-0.808366\pi\)
−0.824184 + 0.566322i \(0.808366\pi\)
\(828\) 1.00000 0.0347524
\(829\) −55.4031 −1.92423 −0.962115 0.272644i \(-0.912102\pi\)
−0.962115 + 0.272644i \(0.912102\pi\)
\(830\) −49.6125 −1.72207
\(831\) 21.4031 0.742466
\(832\) −5.70156 −0.197666
\(833\) 0 0
\(834\) −17.7016 −0.612955
\(835\) −49.6125 −1.71691
\(836\) −21.6125 −0.747484
\(837\) −2.00000 −0.0691301
\(838\) −0.209373 −0.00723266
\(839\) −42.2094 −1.45723 −0.728615 0.684924i \(-0.759835\pi\)
−0.728615 + 0.684924i \(0.759835\pi\)
\(840\) 0 0
\(841\) −28.9109 −0.996929
\(842\) 15.7016 0.541112
\(843\) −14.5078 −0.499676
\(844\) 10.8062 0.371966
\(845\) −72.2094 −2.48408
\(846\) −11.1047 −0.381787
\(847\) 0 0
\(848\) −9.40312 −0.322905
\(849\) 4.80625 0.164950
\(850\) 34.8062 1.19384
\(851\) 4.29844 0.147349
\(852\) 7.40312 0.253627
\(853\) −9.10469 −0.311739 −0.155869 0.987778i \(-0.549818\pi\)
−0.155869 + 0.987778i \(0.549818\pi\)
\(854\) 0 0
\(855\) −20.0000 −0.683986
\(856\) −18.8062 −0.642784
\(857\) 0.298438 0.0101944 0.00509722 0.999987i \(-0.498377\pi\)
0.00509722 + 0.999987i \(0.498377\pi\)
\(858\) 22.8062 0.778592
\(859\) 15.4922 0.528587 0.264293 0.964442i \(-0.414861\pi\)
0.264293 + 0.964442i \(0.414861\pi\)
\(860\) 6.29844 0.214775
\(861\) 0 0
\(862\) 3.91093 0.133207
\(863\) −6.38750 −0.217433 −0.108717 0.994073i \(-0.534674\pi\)
−0.108717 + 0.994073i \(0.534674\pi\)
\(864\) 1.00000 0.0340207
\(865\) 71.8219 2.44202
\(866\) 31.3141 1.06410
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) −1.10469 −0.0374524
\(871\) −84.4187 −2.86042
\(872\) 7.70156 0.260808
\(873\) −6.29844 −0.213170
\(874\) −5.40312 −0.182763
\(875\) 0 0
\(876\) 1.40312 0.0474072
\(877\) −17.4031 −0.587662 −0.293831 0.955857i \(-0.594930\pi\)
−0.293831 + 0.955857i \(0.594930\pi\)
\(878\) −22.0000 −0.742464
\(879\) −20.8062 −0.701777
\(880\) 14.8062 0.499119
\(881\) −6.20937 −0.209199 −0.104600 0.994514i \(-0.533356\pi\)
−0.104600 + 0.994514i \(0.533356\pi\)
\(882\) 0 0
\(883\) −24.5969 −0.827751 −0.413875 0.910334i \(-0.635825\pi\)
−0.413875 + 0.910334i \(0.635825\pi\)
\(884\) 22.8062 0.767057
\(885\) 27.4031 0.921146
\(886\) 13.7016 0.460313
\(887\) 28.2094 0.947178 0.473589 0.880746i \(-0.342958\pi\)
0.473589 + 0.880746i \(0.342958\pi\)
\(888\) 4.29844 0.144246
\(889\) 0 0
\(890\) 42.2094 1.41486
\(891\) −4.00000 −0.134005
\(892\) 22.0000 0.736614
\(893\) 60.0000 2.00782
\(894\) −20.2094 −0.675903
\(895\) 92.9266 3.10619
\(896\) 0 0
\(897\) 5.70156 0.190370
\(898\) 38.4187 1.28205
\(899\) 0.596876 0.0199069
\(900\) 8.70156 0.290052
\(901\) 37.6125 1.25305
\(902\) 1.19375 0.0397476
\(903\) 0 0
\(904\) −7.10469 −0.236298
\(905\) −62.2094 −2.06791
\(906\) 23.9109 0.794388
\(907\) 20.5078 0.680951 0.340475 0.940253i \(-0.389412\pi\)
0.340475 + 0.940253i \(0.389412\pi\)
\(908\) 23.1047 0.766756
\(909\) 3.40312 0.112875
\(910\) 0 0
\(911\) 53.1047 1.75944 0.879718 0.475495i \(-0.157731\pi\)
0.879718 + 0.475495i \(0.157731\pi\)
\(912\) −5.40312 −0.178915
\(913\) 53.6125 1.77431
\(914\) −28.2094 −0.933083
\(915\) 7.40312 0.244740
\(916\) 14.5969 0.482294
\(917\) 0 0
\(918\) −4.00000 −0.132020
\(919\) −52.4187 −1.72913 −0.864567 0.502517i \(-0.832407\pi\)
−0.864567 + 0.502517i \(0.832407\pi\)
\(920\) 3.70156 0.122037
\(921\) 11.9109 0.392479
\(922\) −2.20937 −0.0727618
\(923\) 42.2094 1.38934
\(924\) 0 0
\(925\) 37.4031 1.22981
\(926\) 22.2984 0.732772
\(927\) −2.29844 −0.0754906
\(928\) −0.298438 −0.00979670
\(929\) 45.9109 1.50629 0.753144 0.657855i \(-0.228536\pi\)
0.753144 + 0.657855i \(0.228536\pi\)
\(930\) −7.40312 −0.242758
\(931\) 0 0
\(932\) −26.0000 −0.851658
\(933\) 13.4031 0.438799
\(934\) −3.70156 −0.121119
\(935\) −59.2250 −1.93686
\(936\) 5.70156 0.186361
\(937\) −25.1047 −0.820134 −0.410067 0.912055i \(-0.634495\pi\)
−0.410067 + 0.912055i \(0.634495\pi\)
\(938\) 0 0
\(939\) −6.20937 −0.202635
\(940\) −41.1047 −1.34069
\(941\) −25.3141 −0.825215 −0.412607 0.910909i \(-0.635382\pi\)
−0.412607 + 0.910909i \(0.635382\pi\)
\(942\) 1.40312 0.0457163
\(943\) 0.298438 0.00971847
\(944\) 7.40312 0.240951
\(945\) 0 0
\(946\) −6.80625 −0.221290
\(947\) 15.9109 0.517036 0.258518 0.966006i \(-0.416766\pi\)
0.258518 + 0.966006i \(0.416766\pi\)
\(948\) −8.00000 −0.259828
\(949\) 8.00000 0.259691
\(950\) −47.0156 −1.52539
\(951\) 8.29844 0.269095
\(952\) 0 0
\(953\) 61.2250 1.98327 0.991636 0.129066i \(-0.0411978\pi\)
0.991636 + 0.129066i \(0.0411978\pi\)
\(954\) 9.40312 0.304437
\(955\) −84.4187 −2.73173
\(956\) 10.8062 0.349499
\(957\) 1.19375 0.0385885
\(958\) 16.5969 0.536221
\(959\) 0 0
\(960\) 3.70156 0.119467
\(961\) −27.0000 −0.870968
\(962\) 24.5078 0.790164
\(963\) 18.8062 0.606023
\(964\) 15.9109 0.512457
\(965\) 85.5234 2.75310
\(966\) 0 0
\(967\) −14.8062 −0.476137 −0.238068 0.971248i \(-0.576514\pi\)
−0.238068 + 0.971248i \(0.576514\pi\)
\(968\) −5.00000 −0.160706
\(969\) 21.6125 0.694293
\(970\) −23.3141 −0.748569
\(971\) −49.8219 −1.59886 −0.799430 0.600759i \(-0.794865\pi\)
−0.799430 + 0.600759i \(0.794865\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 6.29844 0.201815
\(975\) 49.6125 1.58887
\(976\) 2.00000 0.0640184
\(977\) 16.7172 0.534830 0.267415 0.963581i \(-0.413831\pi\)
0.267415 + 0.963581i \(0.413831\pi\)
\(978\) −10.8062 −0.345546
\(979\) −45.6125 −1.45778
\(980\) 0 0
\(981\) −7.70156 −0.245892
\(982\) 9.61250 0.306747
\(983\) −42.2094 −1.34627 −0.673135 0.739520i \(-0.735053\pi\)
−0.673135 + 0.739520i \(0.735053\pi\)
\(984\) 0.298438 0.00951385
\(985\) −83.3141 −2.65461
\(986\) 1.19375 0.0380168
\(987\) 0 0
\(988\) −30.8062 −0.980077
\(989\) −1.70156 −0.0541065
\(990\) −14.8062 −0.470573
\(991\) 28.0000 0.889449 0.444725 0.895667i \(-0.353302\pi\)
0.444725 + 0.895667i \(0.353302\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) −8.50781 −0.269716
\(996\) 13.4031 0.424694
\(997\) 23.4031 0.741184 0.370592 0.928796i \(-0.379155\pi\)
0.370592 + 0.928796i \(0.379155\pi\)
\(998\) 23.4031 0.740813
\(999\) −4.29844 −0.135997
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 6762.2.a.bo.1.1 2
7.6 odd 2 966.2.a.n.1.2 2
21.20 even 2 2898.2.a.bb.1.1 2
28.27 even 2 7728.2.a.bc.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.n.1.2 2 7.6 odd 2
2898.2.a.bb.1.1 2 21.20 even 2
6762.2.a.bo.1.1 2 1.1 even 1 trivial
7728.2.a.bc.1.2 2 28.27 even 2