Properties

Label 3969.2.a.be.1.6
Level $3969$
Weight $2$
Character 3969.1
Self dual yes
Analytic conductor $31.693$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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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: \(6\)
Coefficient field: 6.6.59351616.1
Defining polynomial: \(x^{6} - 12 x^{4} + 21 x^{2} - 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 441)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.820103\) of defining polynomial
Character \(\chi\) \(=\) 3969.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.46050 q^{2} +4.05408 q^{4} +3.65808 q^{5} +5.05408 q^{8} +O(q^{10})\) \(q+2.46050 q^{2} +4.05408 q^{4} +3.65808 q^{5} +5.05408 q^{8} +9.00071 q^{10} +0.406421 q^{11} +0.486796 q^{13} +4.32743 q^{16} -4.85584 q^{17} +1.97351 q^{19} +14.8301 q^{20} +1.00000 q^{22} +4.64766 q^{23} +8.38151 q^{25} +1.19777 q^{26} +7.64766 q^{29} -7.02720 q^{31} +0.539495 q^{32} -11.9478 q^{34} +2.32743 q^{37} +4.85584 q^{38} +18.4882 q^{40} -7.51399 q^{41} -2.32743 q^{43} +1.64766 q^{44} +11.4356 q^{46} +6.31623 q^{47} +20.6228 q^{50} +1.97351 q^{52} -3.56867 q^{53} +1.48672 q^{55} +18.8171 q^{58} -6.11839 q^{59} +8.02712 q^{61} -17.2905 q^{62} -7.32743 q^{64} +1.78074 q^{65} +3.60078 q^{67} -19.6860 q^{68} +8.46050 q^{71} -1.97351 q^{73} +5.72665 q^{74} +8.00079 q^{76} +8.16225 q^{79} +15.8301 q^{80} -18.4882 q^{82} -12.1720 q^{83} -17.7630 q^{85} -5.72665 q^{86} +2.05408 q^{88} -14.8301 q^{89} +18.8420 q^{92} +15.5411 q^{94} +7.21926 q^{95} +9.48751 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} + 6q^{4} + 12q^{8} + O(q^{10}) \) \( 6q + 2q^{2} + 6q^{4} + 12q^{8} + 8q^{11} + 6q^{16} + 6q^{22} + 4q^{23} + 12q^{25} + 22q^{29} + 16q^{32} - 6q^{37} + 6q^{43} - 14q^{44} + 12q^{46} + 56q^{50} + 28q^{53} + 18q^{58} - 24q^{64} - 6q^{65} + 38q^{71} + 36q^{74} - 6q^{79} - 30q^{85} - 36q^{86} - 6q^{88} + 62q^{92} + 60q^{95} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.46050 1.73984 0.869920 0.493193i \(-0.164170\pi\)
0.869920 + 0.493193i \(0.164170\pi\)
\(3\) 0 0
\(4\) 4.05408 2.02704
\(5\) 3.65808 1.63594 0.817970 0.575260i \(-0.195099\pi\)
0.817970 + 0.575260i \(0.195099\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 5.05408 1.78689
\(9\) 0 0
\(10\) 9.00071 2.84628
\(11\) 0.406421 0.122540 0.0612702 0.998121i \(-0.480485\pi\)
0.0612702 + 0.998121i \(0.480485\pi\)
\(12\) 0 0
\(13\) 0.486796 0.135013 0.0675065 0.997719i \(-0.478496\pi\)
0.0675065 + 0.997719i \(0.478496\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.32743 1.08186
\(17\) −4.85584 −1.17771 −0.588857 0.808237i \(-0.700422\pi\)
−0.588857 + 0.808237i \(0.700422\pi\)
\(18\) 0 0
\(19\) 1.97351 0.452755 0.226378 0.974040i \(-0.427312\pi\)
0.226378 + 0.974040i \(0.427312\pi\)
\(20\) 14.8301 3.31612
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 4.64766 0.969105 0.484552 0.874762i \(-0.338982\pi\)
0.484552 + 0.874762i \(0.338982\pi\)
\(24\) 0 0
\(25\) 8.38151 1.67630
\(26\) 1.19777 0.234901
\(27\) 0 0
\(28\) 0 0
\(29\) 7.64766 1.42014 0.710068 0.704133i \(-0.248664\pi\)
0.710068 + 0.704133i \(0.248664\pi\)
\(30\) 0 0
\(31\) −7.02720 −1.26212 −0.631061 0.775733i \(-0.717380\pi\)
−0.631061 + 0.775733i \(0.717380\pi\)
\(32\) 0.539495 0.0953702
\(33\) 0 0
\(34\) −11.9478 −2.04903
\(35\) 0 0
\(36\) 0 0
\(37\) 2.32743 0.382627 0.191314 0.981529i \(-0.438725\pi\)
0.191314 + 0.981529i \(0.438725\pi\)
\(38\) 4.85584 0.787721
\(39\) 0 0
\(40\) 18.4882 2.92324
\(41\) −7.51399 −1.17349 −0.586744 0.809772i \(-0.699591\pi\)
−0.586744 + 0.809772i \(0.699591\pi\)
\(42\) 0 0
\(43\) −2.32743 −0.354930 −0.177465 0.984127i \(-0.556790\pi\)
−0.177465 + 0.984127i \(0.556790\pi\)
\(44\) 1.64766 0.248395
\(45\) 0 0
\(46\) 11.4356 1.68609
\(47\) 6.31623 0.921317 0.460658 0.887578i \(-0.347613\pi\)
0.460658 + 0.887578i \(0.347613\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 20.6228 2.91650
\(51\) 0 0
\(52\) 1.97351 0.273677
\(53\) −3.56867 −0.490195 −0.245097 0.969498i \(-0.578820\pi\)
−0.245097 + 0.969498i \(0.578820\pi\)
\(54\) 0 0
\(55\) 1.48672 0.200469
\(56\) 0 0
\(57\) 0 0
\(58\) 18.8171 2.47081
\(59\) −6.11839 −0.796546 −0.398273 0.917267i \(-0.630390\pi\)
−0.398273 + 0.917267i \(0.630390\pi\)
\(60\) 0 0
\(61\) 8.02712 1.02777 0.513884 0.857860i \(-0.328206\pi\)
0.513884 + 0.857860i \(0.328206\pi\)
\(62\) −17.2905 −2.19589
\(63\) 0 0
\(64\) −7.32743 −0.915929
\(65\) 1.78074 0.220873
\(66\) 0 0
\(67\) 3.60078 0.439905 0.219952 0.975511i \(-0.429410\pi\)
0.219952 + 0.975511i \(0.429410\pi\)
\(68\) −19.6860 −2.38728
\(69\) 0 0
\(70\) 0 0
\(71\) 8.46050 1.00408 0.502039 0.864845i \(-0.332584\pi\)
0.502039 + 0.864845i \(0.332584\pi\)
\(72\) 0 0
\(73\) −1.97351 −0.230982 −0.115491 0.993309i \(-0.536844\pi\)
−0.115491 + 0.993309i \(0.536844\pi\)
\(74\) 5.72665 0.665710
\(75\) 0 0
\(76\) 8.00079 0.917754
\(77\) 0 0
\(78\) 0 0
\(79\) 8.16225 0.918325 0.459163 0.888352i \(-0.348150\pi\)
0.459163 + 0.888352i \(0.348150\pi\)
\(80\) 15.8301 1.76986
\(81\) 0 0
\(82\) −18.4882 −2.04168
\(83\) −12.1720 −1.33605 −0.668025 0.744139i \(-0.732860\pi\)
−0.668025 + 0.744139i \(0.732860\pi\)
\(84\) 0 0
\(85\) −17.7630 −1.92667
\(86\) −5.72665 −0.617521
\(87\) 0 0
\(88\) 2.05408 0.218966
\(89\) −14.8301 −1.57199 −0.785996 0.618231i \(-0.787849\pi\)
−0.785996 + 0.618231i \(0.787849\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 18.8420 1.96442
\(93\) 0 0
\(94\) 15.5411 1.60294
\(95\) 7.21926 0.740681
\(96\) 0 0
\(97\) 9.48751 0.963311 0.481655 0.876361i \(-0.340036\pi\)
0.481655 + 0.876361i \(0.340036\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 33.9794 3.39794
\(101\) −8.71176 −0.866852 −0.433426 0.901189i \(-0.642696\pi\)
−0.433426 + 0.901189i \(0.642696\pi\)
\(102\) 0 0
\(103\) −8.02712 −0.790936 −0.395468 0.918480i \(-0.629418\pi\)
−0.395468 + 0.918480i \(0.629418\pi\)
\(104\) 2.46031 0.241253
\(105\) 0 0
\(106\) −8.78074 −0.852861
\(107\) 12.8420 1.24148 0.620742 0.784015i \(-0.286831\pi\)
0.620742 + 0.784015i \(0.286831\pi\)
\(108\) 0 0
\(109\) 2.60078 0.249109 0.124555 0.992213i \(-0.460250\pi\)
0.124555 + 0.992213i \(0.460250\pi\)
\(110\) 3.65808 0.348784
\(111\) 0 0
\(112\) 0 0
\(113\) −13.9502 −1.31232 −0.656162 0.754620i \(-0.727821\pi\)
−0.656162 + 0.754620i \(0.727821\pi\)
\(114\) 0 0
\(115\) 17.0015 1.58540
\(116\) 31.0043 2.87867
\(117\) 0 0
\(118\) −15.0543 −1.38586
\(119\) 0 0
\(120\) 0 0
\(121\) −10.8348 −0.984984
\(122\) 19.7508 1.78815
\(123\) 0 0
\(124\) −28.4889 −2.55837
\(125\) 12.3698 1.10639
\(126\) 0 0
\(127\) −15.5438 −1.37929 −0.689643 0.724149i \(-0.742233\pi\)
−0.689643 + 0.724149i \(0.742233\pi\)
\(128\) −19.1082 −1.68894
\(129\) 0 0
\(130\) 4.38151 0.384284
\(131\) 8.51392 0.743864 0.371932 0.928260i \(-0.378695\pi\)
0.371932 + 0.928260i \(0.378695\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 8.85973 0.765364
\(135\) 0 0
\(136\) −24.5418 −2.10444
\(137\) 0.377242 0.0322300 0.0161150 0.999870i \(-0.494870\pi\)
0.0161150 + 0.999870i \(0.494870\pi\)
\(138\) 0 0
\(139\) 19.0013 1.61167 0.805837 0.592138i \(-0.201716\pi\)
0.805837 + 0.592138i \(0.201716\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 20.8171 1.74693
\(143\) 0.197844 0.0165446
\(144\) 0 0
\(145\) 27.9757 2.32326
\(146\) −4.85584 −0.401872
\(147\) 0 0
\(148\) 9.43560 0.775601
\(149\) −9.70175 −0.794798 −0.397399 0.917646i \(-0.630087\pi\)
−0.397399 + 0.917646i \(0.630087\pi\)
\(150\) 0 0
\(151\) −12.8348 −1.04448 −0.522242 0.852798i \(-0.674904\pi\)
−0.522242 + 0.852798i \(0.674904\pi\)
\(152\) 9.97430 0.809023
\(153\) 0 0
\(154\) 0 0
\(155\) −25.7060 −2.06476
\(156\) 0 0
\(157\) −20.9485 −1.67187 −0.835937 0.548825i \(-0.815075\pi\)
−0.835937 + 0.548825i \(0.815075\pi\)
\(158\) 20.0833 1.59774
\(159\) 0 0
\(160\) 1.97351 0.156020
\(161\) 0 0
\(162\) 0 0
\(163\) −11.1623 −0.874295 −0.437148 0.899390i \(-0.644011\pi\)
−0.437148 + 0.899390i \(0.644011\pi\)
\(164\) −30.4624 −2.37871
\(165\) 0 0
\(166\) −29.9492 −2.32451
\(167\) −3.46023 −0.267761 −0.133880 0.990998i \(-0.542744\pi\)
−0.133880 + 0.990998i \(0.542744\pi\)
\(168\) 0 0
\(169\) −12.7630 −0.981771
\(170\) −43.7060 −3.35210
\(171\) 0 0
\(172\) −9.43560 −0.719458
\(173\) 6.05361 0.460247 0.230124 0.973161i \(-0.426087\pi\)
0.230124 + 0.973161i \(0.426087\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.75876 0.132571
\(177\) 0 0
\(178\) −36.4896 −2.73501
\(179\) 9.13307 0.682638 0.341319 0.939948i \(-0.389126\pi\)
0.341319 + 0.939948i \(0.389126\pi\)
\(180\) 0 0
\(181\) −11.9478 −0.888074 −0.444037 0.896008i \(-0.646454\pi\)
−0.444037 + 0.896008i \(0.646454\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 23.4897 1.73168
\(185\) 8.51392 0.625956
\(186\) 0 0
\(187\) −1.97351 −0.144318
\(188\) 25.6065 1.86755
\(189\) 0 0
\(190\) 17.7630 1.28867
\(191\) 9.14027 0.661367 0.330683 0.943742i \(-0.392721\pi\)
0.330683 + 0.943742i \(0.392721\pi\)
\(192\) 0 0
\(193\) 16.9430 1.21958 0.609792 0.792562i \(-0.291253\pi\)
0.609792 + 0.792562i \(0.291253\pi\)
\(194\) 23.3441 1.67601
\(195\) 0 0
\(196\) 0 0
\(197\) −21.3173 −1.51880 −0.759398 0.650627i \(-0.774506\pi\)
−0.759398 + 0.650627i \(0.774506\pi\)
\(198\) 0 0
\(199\) −9.97430 −0.707060 −0.353530 0.935423i \(-0.615019\pi\)
−0.353530 + 0.935423i \(0.615019\pi\)
\(200\) 42.3609 2.99537
\(201\) 0 0
\(202\) −21.4353 −1.50818
\(203\) 0 0
\(204\) 0 0
\(205\) −27.4868 −1.91976
\(206\) −19.7508 −1.37610
\(207\) 0 0
\(208\) 2.10658 0.146065
\(209\) 0.802077 0.0554808
\(210\) 0 0
\(211\) 4.89183 0.336768 0.168384 0.985722i \(-0.446145\pi\)
0.168384 + 0.985722i \(0.446145\pi\)
\(212\) −14.4677 −0.993646
\(213\) 0 0
\(214\) 31.5979 2.15998
\(215\) −8.51392 −0.580644
\(216\) 0 0
\(217\) 0 0
\(218\) 6.39922 0.433410
\(219\) 0 0
\(220\) 6.02728 0.406359
\(221\) −2.36381 −0.159007
\(222\) 0 0
\(223\) 23.4088 1.56757 0.783786 0.621031i \(-0.213286\pi\)
0.783786 + 0.621031i \(0.213286\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −34.3245 −2.28323
\(227\) 6.11839 0.406092 0.203046 0.979169i \(-0.434916\pi\)
0.203046 + 0.979169i \(0.434916\pi\)
\(228\) 0 0
\(229\) −1.46039 −0.0965052 −0.0482526 0.998835i \(-0.515365\pi\)
−0.0482526 + 0.998835i \(0.515365\pi\)
\(230\) 41.8323 2.75834
\(231\) 0 0
\(232\) 38.6519 2.53762
\(233\) −13.2484 −0.867934 −0.433967 0.900929i \(-0.642887\pi\)
−0.433967 + 0.900929i \(0.642887\pi\)
\(234\) 0 0
\(235\) 23.1052 1.50722
\(236\) −24.8045 −1.61463
\(237\) 0 0
\(238\) 0 0
\(239\) 19.3887 1.25415 0.627076 0.778958i \(-0.284252\pi\)
0.627076 + 0.778958i \(0.284252\pi\)
\(240\) 0 0
\(241\) 5.05368 0.325536 0.162768 0.986664i \(-0.447958\pi\)
0.162768 + 0.986664i \(0.447958\pi\)
\(242\) −26.6591 −1.71371
\(243\) 0 0
\(244\) 32.5426 2.08333
\(245\) 0 0
\(246\) 0 0
\(247\) 0.960699 0.0611278
\(248\) −35.5161 −2.25527
\(249\) 0 0
\(250\) 30.4360 1.92494
\(251\) 15.0928 0.952647 0.476324 0.879270i \(-0.341969\pi\)
0.476324 + 0.879270i \(0.341969\pi\)
\(252\) 0 0
\(253\) 1.88891 0.118755
\(254\) −38.2455 −2.39974
\(255\) 0 0
\(256\) −32.3609 −2.02256
\(257\) 7.71184 0.481051 0.240526 0.970643i \(-0.422680\pi\)
0.240526 + 0.970643i \(0.422680\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 7.21926 0.447720
\(261\) 0 0
\(262\) 20.9485 1.29420
\(263\) 4.21206 0.259727 0.129864 0.991532i \(-0.458546\pi\)
0.129864 + 0.991532i \(0.458546\pi\)
\(264\) 0 0
\(265\) −13.0545 −0.801930
\(266\) 0 0
\(267\) 0 0
\(268\) 14.5979 0.891706
\(269\) 20.7507 1.26519 0.632596 0.774482i \(-0.281989\pi\)
0.632596 + 0.774482i \(0.281989\pi\)
\(270\) 0 0
\(271\) −28.4889 −1.73057 −0.865287 0.501276i \(-0.832864\pi\)
−0.865287 + 0.501276i \(0.832864\pi\)
\(272\) −21.0133 −1.27412
\(273\) 0 0
\(274\) 0.928207 0.0560750
\(275\) 3.40642 0.205415
\(276\) 0 0
\(277\) 17.1623 1.03118 0.515590 0.856835i \(-0.327573\pi\)
0.515590 + 0.856835i \(0.327573\pi\)
\(278\) 46.7529 2.80405
\(279\) 0 0
\(280\) 0 0
\(281\) 9.44280 0.563310 0.281655 0.959516i \(-0.409117\pi\)
0.281655 + 0.959516i \(0.409117\pi\)
\(282\) 0 0
\(283\) 16.8684 1.00272 0.501362 0.865237i \(-0.332832\pi\)
0.501362 + 0.865237i \(0.332832\pi\)
\(284\) 34.2996 2.03531
\(285\) 0 0
\(286\) 0.486796 0.0287849
\(287\) 0 0
\(288\) 0 0
\(289\) 6.57918 0.387011
\(290\) 68.8344 4.04210
\(291\) 0 0
\(292\) −8.00079 −0.468211
\(293\) −3.72286 −0.217492 −0.108746 0.994070i \(-0.534683\pi\)
−0.108746 + 0.994070i \(0.534683\pi\)
\(294\) 0 0
\(295\) −22.3815 −1.30310
\(296\) 11.7630 0.683712
\(297\) 0 0
\(298\) −23.8712 −1.38282
\(299\) 2.26247 0.130842
\(300\) 0 0
\(301\) 0 0
\(302\) −31.5801 −1.81723
\(303\) 0 0
\(304\) 8.54024 0.489817
\(305\) 29.3638 1.68137
\(306\) 0 0
\(307\) 30.5691 1.74467 0.872335 0.488908i \(-0.162605\pi\)
0.872335 + 0.488908i \(0.162605\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −63.2498 −3.59235
\(311\) −10.4348 −0.591702 −0.295851 0.955234i \(-0.595603\pi\)
−0.295851 + 0.955234i \(0.595603\pi\)
\(312\) 0 0
\(313\) −0.619860 −0.0350366 −0.0175183 0.999847i \(-0.505577\pi\)
−0.0175183 + 0.999847i \(0.505577\pi\)
\(314\) −51.5440 −2.90879
\(315\) 0 0
\(316\) 33.0905 1.86148
\(317\) 10.2484 0.575610 0.287805 0.957689i \(-0.407075\pi\)
0.287805 + 0.957689i \(0.407075\pi\)
\(318\) 0 0
\(319\) 3.10817 0.174024
\(320\) −26.8043 −1.49841
\(321\) 0 0
\(322\) 0 0
\(323\) −9.58307 −0.533216
\(324\) 0 0
\(325\) 4.08009 0.226323
\(326\) −27.4648 −1.52113
\(327\) 0 0
\(328\) −37.9764 −2.09689
\(329\) 0 0
\(330\) 0 0
\(331\) −20.3638 −1.11930 −0.559648 0.828730i \(-0.689064\pi\)
−0.559648 + 0.828730i \(0.689064\pi\)
\(332\) −49.3463 −2.70823
\(333\) 0 0
\(334\) −8.51392 −0.465861
\(335\) 13.1719 0.719658
\(336\) 0 0
\(337\) −5.71187 −0.311145 −0.155573 0.987824i \(-0.549722\pi\)
−0.155573 + 0.987824i \(0.549722\pi\)
\(338\) −31.4035 −1.70812
\(339\) 0 0
\(340\) −72.0128 −3.90544
\(341\) −2.85600 −0.154661
\(342\) 0 0
\(343\) 0 0
\(344\) −11.7630 −0.634220
\(345\) 0 0
\(346\) 14.8949 0.800756
\(347\) 8.88132 0.476774 0.238387 0.971170i \(-0.423381\pi\)
0.238387 + 0.971170i \(0.423381\pi\)
\(348\) 0 0
\(349\) −20.9749 −1.12276 −0.561379 0.827559i \(-0.689729\pi\)
−0.561379 + 0.827559i \(0.689729\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.219262 0.0116867
\(353\) 14.7654 0.785881 0.392941 0.919564i \(-0.371458\pi\)
0.392941 + 0.919564i \(0.371458\pi\)
\(354\) 0 0
\(355\) 30.9492 1.64261
\(356\) −60.1227 −3.18649
\(357\) 0 0
\(358\) 22.4720 1.18768
\(359\) 7.21206 0.380638 0.190319 0.981722i \(-0.439048\pi\)
0.190319 + 0.981722i \(0.439048\pi\)
\(360\) 0 0
\(361\) −15.1052 −0.795013
\(362\) −29.3977 −1.54511
\(363\) 0 0
\(364\) 0 0
\(365\) −7.21926 −0.377873
\(366\) 0 0
\(367\) 10.9742 0.572850 0.286425 0.958103i \(-0.407533\pi\)
0.286425 + 0.958103i \(0.407533\pi\)
\(368\) 20.1124 1.04843
\(369\) 0 0
\(370\) 20.9485 1.08906
\(371\) 0 0
\(372\) 0 0
\(373\) −0.543767 −0.0281552 −0.0140776 0.999901i \(-0.504481\pi\)
−0.0140776 + 0.999901i \(0.504481\pi\)
\(374\) −4.85584 −0.251090
\(375\) 0 0
\(376\) 31.9228 1.64629
\(377\) 3.72286 0.191737
\(378\) 0 0
\(379\) −22.6912 −1.16557 −0.582785 0.812626i \(-0.698037\pi\)
−0.582785 + 0.812626i \(0.698037\pi\)
\(380\) 29.2675 1.50139
\(381\) 0 0
\(382\) 22.4897 1.15067
\(383\) −35.7139 −1.82489 −0.912447 0.409194i \(-0.865810\pi\)
−0.912447 + 0.409194i \(0.865810\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 41.6883 2.12188
\(387\) 0 0
\(388\) 38.4632 1.95267
\(389\) 38.6591 1.96010 0.980048 0.198761i \(-0.0636919\pi\)
0.980048 + 0.198761i \(0.0636919\pi\)
\(390\) 0 0
\(391\) −22.5683 −1.14133
\(392\) 0 0
\(393\) 0 0
\(394\) −52.4513 −2.64246
\(395\) 29.8581 1.50233
\(396\) 0 0
\(397\) 11.9478 0.599644 0.299822 0.953995i \(-0.403073\pi\)
0.299822 + 0.953995i \(0.403073\pi\)
\(398\) −24.5418 −1.23017
\(399\) 0 0
\(400\) 36.2704 1.81352
\(401\) 32.3566 1.61581 0.807906 0.589311i \(-0.200601\pi\)
0.807906 + 0.589311i \(0.200601\pi\)
\(402\) 0 0
\(403\) −3.42082 −0.170403
\(404\) −35.3182 −1.75715
\(405\) 0 0
\(406\) 0 0
\(407\) 0.945916 0.0468873
\(408\) 0 0
\(409\) −18.9750 −0.938254 −0.469127 0.883131i \(-0.655431\pi\)
−0.469127 + 0.883131i \(0.655431\pi\)
\(410\) −67.6313 −3.34007
\(411\) 0 0
\(412\) −32.5426 −1.60326
\(413\) 0 0
\(414\) 0 0
\(415\) −44.5261 −2.18570
\(416\) 0.262624 0.0128762
\(417\) 0 0
\(418\) 1.97351 0.0965277
\(419\) 17.2905 0.844694 0.422347 0.906434i \(-0.361206\pi\)
0.422347 + 0.906434i \(0.361206\pi\)
\(420\) 0 0
\(421\) 18.6008 0.906546 0.453273 0.891372i \(-0.350256\pi\)
0.453273 + 0.891372i \(0.350256\pi\)
\(422\) 12.0364 0.585922
\(423\) 0 0
\(424\) −18.0364 −0.875924
\(425\) −40.6993 −1.97421
\(426\) 0 0
\(427\) 0 0
\(428\) 52.0626 2.51654
\(429\) 0 0
\(430\) −20.9485 −1.01023
\(431\) −15.8784 −0.764835 −0.382418 0.923990i \(-0.624908\pi\)
−0.382418 + 0.923990i \(0.624908\pi\)
\(432\) 0 0
\(433\) −40.4367 −1.94326 −0.971631 0.236501i \(-0.923999\pi\)
−0.971631 + 0.236501i \(0.923999\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10.5438 0.504955
\(437\) 9.17223 0.438767
\(438\) 0 0
\(439\) −12.4609 −0.594728 −0.297364 0.954764i \(-0.596108\pi\)
−0.297364 + 0.954764i \(0.596108\pi\)
\(440\) 7.51399 0.358216
\(441\) 0 0
\(442\) −5.81616 −0.276646
\(443\) −8.23073 −0.391054 −0.195527 0.980698i \(-0.562642\pi\)
−0.195527 + 0.980698i \(0.562642\pi\)
\(444\) 0 0
\(445\) −54.2498 −2.57169
\(446\) 57.5976 2.72732
\(447\) 0 0
\(448\) 0 0
\(449\) 5.64474 0.266392 0.133196 0.991090i \(-0.457476\pi\)
0.133196 + 0.991090i \(0.457476\pi\)
\(450\) 0 0
\(451\) −3.05384 −0.143800
\(452\) −56.5552 −2.66013
\(453\) 0 0
\(454\) 15.0543 0.706534
\(455\) 0 0
\(456\) 0 0
\(457\) 5.06887 0.237112 0.118556 0.992947i \(-0.462174\pi\)
0.118556 + 0.992947i \(0.462174\pi\)
\(458\) −3.59330 −0.167904
\(459\) 0 0
\(460\) 68.9255 3.21367
\(461\) 7.77662 0.362193 0.181097 0.983465i \(-0.442035\pi\)
0.181097 + 0.983465i \(0.442035\pi\)
\(462\) 0 0
\(463\) −9.17996 −0.426629 −0.213314 0.976984i \(-0.568426\pi\)
−0.213314 + 0.976984i \(0.568426\pi\)
\(464\) 33.0947 1.53638
\(465\) 0 0
\(466\) −32.5979 −1.51007
\(467\) 13.7654 0.636989 0.318494 0.947925i \(-0.396823\pi\)
0.318494 + 0.947925i \(0.396823\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 56.8506 2.62232
\(471\) 0 0
\(472\) −30.9228 −1.42334
\(473\) −0.945916 −0.0434933
\(474\) 0 0
\(475\) 16.5410 0.758955
\(476\) 0 0
\(477\) 0 0
\(478\) 47.7060 2.18202
\(479\) −8.71176 −0.398050 −0.199025 0.979994i \(-0.563778\pi\)
−0.199025 + 0.979994i \(0.563778\pi\)
\(480\) 0 0
\(481\) 1.13298 0.0516597
\(482\) 12.4346 0.566381
\(483\) 0 0
\(484\) −43.9253 −1.99660
\(485\) 34.7060 1.57592
\(486\) 0 0
\(487\) −18.0364 −0.817306 −0.408653 0.912690i \(-0.634001\pi\)
−0.408653 + 0.912690i \(0.634001\pi\)
\(488\) 40.5697 1.83651
\(489\) 0 0
\(490\) 0 0
\(491\) 2.04689 0.0923747 0.0461874 0.998933i \(-0.485293\pi\)
0.0461874 + 0.998933i \(0.485293\pi\)
\(492\) 0 0
\(493\) −37.1358 −1.67251
\(494\) 2.36381 0.106353
\(495\) 0 0
\(496\) −30.4097 −1.36544
\(497\) 0 0
\(498\) 0 0
\(499\) −39.0875 −1.74980 −0.874899 0.484305i \(-0.839072\pi\)
−0.874899 + 0.484305i \(0.839072\pi\)
\(500\) 50.1484 2.24270
\(501\) 0 0
\(502\) 37.1358 1.65745
\(503\) −5.11846 −0.228221 −0.114111 0.993468i \(-0.536402\pi\)
−0.114111 + 0.993468i \(0.536402\pi\)
\(504\) 0 0
\(505\) −31.8683 −1.41812
\(506\) 4.64766 0.206614
\(507\) 0 0
\(508\) −63.0157 −2.79587
\(509\) 29.5272 1.30877 0.654386 0.756161i \(-0.272927\pi\)
0.654386 + 0.756161i \(0.272927\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −41.4078 −1.82998
\(513\) 0 0
\(514\) 18.9750 0.836952
\(515\) −29.3638 −1.29392
\(516\) 0 0
\(517\) 2.56705 0.112899
\(518\) 0 0
\(519\) 0 0
\(520\) 9.00000 0.394676
\(521\) −1.06470 −0.0466454 −0.0233227 0.999728i \(-0.507425\pi\)
−0.0233227 + 0.999728i \(0.507425\pi\)
\(522\) 0 0
\(523\) −13.3819 −0.585149 −0.292574 0.956243i \(-0.594512\pi\)
−0.292574 + 0.956243i \(0.594512\pi\)
\(524\) 34.5161 1.50784
\(525\) 0 0
\(526\) 10.3638 0.451883
\(527\) 34.1230 1.48642
\(528\) 0 0
\(529\) −1.39922 −0.0608358
\(530\) −32.1206 −1.39523
\(531\) 0 0
\(532\) 0 0
\(533\) −3.65779 −0.158436
\(534\) 0 0
\(535\) 46.9771 2.03100
\(536\) 18.1986 0.786061
\(537\) 0 0
\(538\) 51.0572 2.20123
\(539\) 0 0
\(540\) 0 0
\(541\) −34.0875 −1.46554 −0.732769 0.680478i \(-0.761772\pi\)
−0.732769 + 0.680478i \(0.761772\pi\)
\(542\) −70.0970 −3.01092
\(543\) 0 0
\(544\) −2.61970 −0.112319
\(545\) 9.51384 0.407528
\(546\) 0 0
\(547\) −5.94299 −0.254104 −0.127052 0.991896i \(-0.540551\pi\)
−0.127052 + 0.991896i \(0.540551\pi\)
\(548\) 1.52937 0.0653316
\(549\) 0 0
\(550\) 8.38151 0.357389
\(551\) 15.0928 0.642974
\(552\) 0 0
\(553\) 0 0
\(554\) 42.2278 1.79409
\(555\) 0 0
\(556\) 77.0331 3.26693
\(557\) −30.0803 −1.27454 −0.637272 0.770639i \(-0.719937\pi\)
−0.637272 + 0.770639i \(0.719937\pi\)
\(558\) 0 0
\(559\) −1.13298 −0.0479202
\(560\) 0 0
\(561\) 0 0
\(562\) 23.2340 0.980069
\(563\) −19.6212 −0.826935 −0.413468 0.910519i \(-0.635683\pi\)
−0.413468 + 0.910519i \(0.635683\pi\)
\(564\) 0 0
\(565\) −51.0308 −2.14688
\(566\) 41.5049 1.74458
\(567\) 0 0
\(568\) 42.7601 1.79417
\(569\) −1.37432 −0.0576144 −0.0288072 0.999585i \(-0.509171\pi\)
−0.0288072 + 0.999585i \(0.509171\pi\)
\(570\) 0 0
\(571\) 17.3815 0.727394 0.363697 0.931517i \(-0.381514\pi\)
0.363697 + 0.931517i \(0.381514\pi\)
\(572\) 0.802077 0.0335365
\(573\) 0 0
\(574\) 0 0
\(575\) 38.9545 1.62451
\(576\) 0 0
\(577\) −27.0548 −1.12631 −0.563153 0.826353i \(-0.690412\pi\)
−0.563153 + 0.826353i \(0.690412\pi\)
\(578\) 16.1881 0.673337
\(579\) 0 0
\(580\) 113.416 4.70934
\(581\) 0 0
\(582\) 0 0
\(583\) −1.45038 −0.0600687
\(584\) −9.97430 −0.412740
\(585\) 0 0
\(586\) −9.16010 −0.378400
\(587\) 7.51399 0.310136 0.155068 0.987904i \(-0.450440\pi\)
0.155068 + 0.987904i \(0.450440\pi\)
\(588\) 0 0
\(589\) −13.8683 −0.571432
\(590\) −55.0698 −2.26719
\(591\) 0 0
\(592\) 10.0718 0.413948
\(593\) 35.5808 1.46113 0.730565 0.682843i \(-0.239257\pi\)
0.730565 + 0.682843i \(0.239257\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −39.3317 −1.61109
\(597\) 0 0
\(598\) 5.56681 0.227644
\(599\) −11.4821 −0.469146 −0.234573 0.972099i \(-0.575369\pi\)
−0.234573 + 0.972099i \(0.575369\pi\)
\(600\) 0 0
\(601\) −0.380061 −0.0155030 −0.00775150 0.999970i \(-0.502467\pi\)
−0.00775150 + 0.999970i \(0.502467\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −52.0335 −2.11721
\(605\) −39.6346 −1.61138
\(606\) 0 0
\(607\) −18.5409 −0.752551 −0.376275 0.926508i \(-0.622795\pi\)
−0.376275 + 0.926508i \(0.622795\pi\)
\(608\) 1.06470 0.0431793
\(609\) 0 0
\(610\) 72.2498 2.92531
\(611\) 3.07472 0.124390
\(612\) 0 0
\(613\) 7.32451 0.295834 0.147917 0.989000i \(-0.452743\pi\)
0.147917 + 0.989000i \(0.452743\pi\)
\(614\) 75.2154 3.03545
\(615\) 0 0
\(616\) 0 0
\(617\) −25.4854 −1.02600 −0.513002 0.858387i \(-0.671467\pi\)
−0.513002 + 0.858387i \(0.671467\pi\)
\(618\) 0 0
\(619\) 32.8963 1.32222 0.661108 0.750291i \(-0.270087\pi\)
0.661108 + 0.750291i \(0.270087\pi\)
\(620\) −104.214 −4.18535
\(621\) 0 0
\(622\) −25.6748 −1.02947
\(623\) 0 0
\(624\) 0 0
\(625\) 3.34221 0.133689
\(626\) −1.52517 −0.0609580
\(627\) 0 0
\(628\) −84.9271 −3.38896
\(629\) −11.3016 −0.450626
\(630\) 0 0
\(631\) 29.8683 1.18904 0.594519 0.804082i \(-0.297343\pi\)
0.594519 + 0.804082i \(0.297343\pi\)
\(632\) 41.2527 1.64094
\(633\) 0 0
\(634\) 25.2163 1.00147
\(635\) −56.8603 −2.25643
\(636\) 0 0
\(637\) 0 0
\(638\) 7.64766 0.302774
\(639\) 0 0
\(640\) −69.8991 −2.76301
\(641\) 11.4605 0.452663 0.226331 0.974050i \(-0.427327\pi\)
0.226331 + 0.974050i \(0.427327\pi\)
\(642\) 0 0
\(643\) −17.3816 −0.685462 −0.342731 0.939434i \(-0.611352\pi\)
−0.342731 + 0.939434i \(0.611352\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −23.5792 −0.927711
\(647\) 25.3439 0.996372 0.498186 0.867070i \(-0.334000\pi\)
0.498186 + 0.867070i \(0.334000\pi\)
\(648\) 0 0
\(649\) −2.48664 −0.0976091
\(650\) 10.0391 0.393765
\(651\) 0 0
\(652\) −45.2527 −1.77223
\(653\) 14.0833 0.551121 0.275560 0.961284i \(-0.411137\pi\)
0.275560 + 0.961284i \(0.411137\pi\)
\(654\) 0 0
\(655\) 31.1445 1.21692
\(656\) −32.5163 −1.26955
\(657\) 0 0
\(658\) 0 0
\(659\) 38.1708 1.48692 0.743462 0.668779i \(-0.233183\pi\)
0.743462 + 0.668779i \(0.233183\pi\)
\(660\) 0 0
\(661\) −0.353732 −0.0137586 −0.00687930 0.999976i \(-0.502190\pi\)
−0.00687930 + 0.999976i \(0.502190\pi\)
\(662\) −50.1052 −1.94740
\(663\) 0 0
\(664\) −61.5183 −2.38737
\(665\) 0 0
\(666\) 0 0
\(667\) 35.5438 1.37626
\(668\) −14.0281 −0.542762
\(669\) 0 0
\(670\) 32.4096 1.25209
\(671\) 3.26239 0.125943
\(672\) 0 0
\(673\) −21.1111 −0.813773 −0.406886 0.913479i \(-0.633386\pi\)
−0.406886 + 0.913479i \(0.633386\pi\)
\(674\) −14.0541 −0.541343
\(675\) 0 0
\(676\) −51.7424 −1.99009
\(677\) 21.1464 0.812721 0.406361 0.913713i \(-0.366798\pi\)
0.406361 + 0.913713i \(0.366798\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −89.7758 −3.44275
\(681\) 0 0
\(682\) −7.02720 −0.269085
\(683\) 34.7716 1.33050 0.665249 0.746622i \(-0.268326\pi\)
0.665249 + 0.746622i \(0.268326\pi\)
\(684\) 0 0
\(685\) 1.37998 0.0527264
\(686\) 0 0
\(687\) 0 0
\(688\) −10.0718 −0.383984
\(689\) −1.73722 −0.0661827
\(690\) 0 0
\(691\) 34.6492 1.31812 0.659059 0.752091i \(-0.270955\pi\)
0.659059 + 0.752091i \(0.270955\pi\)
\(692\) 24.5418 0.932940
\(693\) 0 0
\(694\) 21.8525 0.829511
\(695\) 69.5083 2.63660
\(696\) 0 0
\(697\) 36.4868 1.38203
\(698\) −51.6087 −1.95342
\(699\) 0 0
\(700\) 0 0
\(701\) −48.6050 −1.83579 −0.917894 0.396826i \(-0.870111\pi\)
−0.917894 + 0.396826i \(0.870111\pi\)
\(702\) 0 0
\(703\) 4.59322 0.173236
\(704\) −2.97802 −0.112238
\(705\) 0 0
\(706\) 36.3303 1.36731
\(707\) 0 0
\(708\) 0 0
\(709\) 4.10817 0.154286 0.0771428 0.997020i \(-0.475420\pi\)
0.0771428 + 0.997020i \(0.475420\pi\)
\(710\) 76.1506 2.85788
\(711\) 0 0
\(712\) −74.9528 −2.80898
\(713\) −32.6601 −1.22313
\(714\) 0 0
\(715\) 0.723729 0.0270659
\(716\) 37.0263 1.38374
\(717\) 0 0
\(718\) 17.7453 0.662249
\(719\) 48.2816 1.80060 0.900299 0.435271i \(-0.143347\pi\)
0.900299 + 0.435271i \(0.143347\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −37.1665 −1.38319
\(723\) 0 0
\(724\) −48.4375 −1.80016
\(725\) 64.0990 2.38058
\(726\) 0 0
\(727\) −41.0302 −1.52173 −0.760863 0.648913i \(-0.775224\pi\)
−0.760863 + 0.648913i \(0.775224\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −17.7630 −0.657439
\(731\) 11.3016 0.418006
\(732\) 0 0
\(733\) 30.5428 1.12812 0.564062 0.825733i \(-0.309238\pi\)
0.564062 + 0.825733i \(0.309238\pi\)
\(734\) 27.0021 0.996667
\(735\) 0 0
\(736\) 2.50739 0.0924237
\(737\) 1.46343 0.0539061
\(738\) 0 0
\(739\) 23.8200 0.876234 0.438117 0.898918i \(-0.355646\pi\)
0.438117 + 0.898918i \(0.355646\pi\)
\(740\) 34.5161 1.26884
\(741\) 0 0
\(742\) 0 0
\(743\) 10.5218 0.386007 0.193003 0.981198i \(-0.438177\pi\)
0.193003 + 0.981198i \(0.438177\pi\)
\(744\) 0 0
\(745\) −35.4897 −1.30024
\(746\) −1.33794 −0.0489855
\(747\) 0 0
\(748\) −8.00079 −0.292538
\(749\) 0 0
\(750\) 0 0
\(751\) −10.2704 −0.374773 −0.187386 0.982286i \(-0.560002\pi\)
−0.187386 + 0.982286i \(0.560002\pi\)
\(752\) 27.3330 0.996734
\(753\) 0 0
\(754\) 9.16010 0.333591
\(755\) −46.9507 −1.70871
\(756\) 0 0
\(757\) 8.03930 0.292193 0.146097 0.989270i \(-0.453329\pi\)
0.146097 + 0.989270i \(0.453329\pi\)
\(758\) −55.8319 −2.02791
\(759\) 0 0
\(760\) 36.4868 1.32351
\(761\) 27.6604 1.00269 0.501345 0.865247i \(-0.332839\pi\)
0.501345 + 0.865247i \(0.332839\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 37.0554 1.34062
\(765\) 0 0
\(766\) −87.8742 −3.17502
\(767\) −2.97841 −0.107544
\(768\) 0 0
\(769\) −33.9226 −1.22328 −0.611640 0.791136i \(-0.709490\pi\)
−0.611640 + 0.791136i \(0.709490\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 68.6883 2.47215
\(773\) −28.5956 −1.02851 −0.514256 0.857637i \(-0.671932\pi\)
−0.514256 + 0.857637i \(0.671932\pi\)
\(774\) 0 0
\(775\) −58.8986 −2.11570
\(776\) 47.9507 1.72133
\(777\) 0 0
\(778\) 95.1210 3.41025
\(779\) −14.8290 −0.531303
\(780\) 0 0
\(781\) 3.43852 0.123040
\(782\) −55.5294 −1.98573
\(783\) 0 0
\(784\) 0 0
\(785\) −76.6313 −2.73509
\(786\) 0 0
\(787\) 46.0035 1.63985 0.819923 0.572473i \(-0.194016\pi\)
0.819923 + 0.572473i \(0.194016\pi\)
\(788\) −86.4222 −3.07866
\(789\) 0 0
\(790\) 73.4661 2.61381
\(791\) 0 0
\(792\) 0 0
\(793\) 3.90757 0.138762
\(794\) 29.3977 1.04328
\(795\) 0 0
\(796\) −40.4367 −1.43324
\(797\) −36.9117 −1.30748 −0.653739 0.756720i \(-0.726801\pi\)
−0.653739 + 0.756720i \(0.726801\pi\)
\(798\) 0 0
\(799\) −30.6706 −1.08505
\(800\) 4.52179 0.159869
\(801\) 0 0
\(802\) 79.6136 2.81125
\(803\) −0.802077 −0.0283047
\(804\) 0 0
\(805\) 0 0
\(806\) −8.41693 −0.296474
\(807\) 0 0
\(808\) −44.0300 −1.54897
\(809\) 47.6887 1.67665 0.838323 0.545174i \(-0.183537\pi\)
0.838323 + 0.545174i \(0.183537\pi\)
\(810\) 0 0
\(811\) −6.02728 −0.211646 −0.105823 0.994385i \(-0.533748\pi\)
−0.105823 + 0.994385i \(0.533748\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 2.32743 0.0815764
\(815\) −40.8324 −1.43030
\(816\) 0 0
\(817\) −4.59322 −0.160696
\(818\) −46.6881 −1.63241
\(819\) 0 0
\(820\) −111.434 −3.89143
\(821\) −39.2642 −1.37033 −0.685165 0.728388i \(-0.740270\pi\)
−0.685165 + 0.728388i \(0.740270\pi\)
\(822\) 0 0
\(823\) 22.7630 0.793469 0.396735 0.917933i \(-0.370143\pi\)
0.396735 + 0.917933i \(0.370143\pi\)
\(824\) −40.5697 −1.41331
\(825\) 0 0
\(826\) 0 0
\(827\) −28.9286 −1.00595 −0.502973 0.864302i \(-0.667760\pi\)
−0.502973 + 0.864302i \(0.667760\pi\)
\(828\) 0 0
\(829\) 33.9767 1.18006 0.590029 0.807382i \(-0.299116\pi\)
0.590029 + 0.807382i \(0.299116\pi\)
\(830\) −109.557 −3.80276
\(831\) 0 0
\(832\) −3.56697 −0.123662
\(833\) 0 0
\(834\) 0 0
\(835\) −12.6578 −0.438041
\(836\) 3.25169 0.112462
\(837\) 0 0
\(838\) 42.5432 1.46963
\(839\) −38.0411 −1.31333 −0.656663 0.754184i \(-0.728033\pi\)
−0.656663 + 0.754184i \(0.728033\pi\)
\(840\) 0 0
\(841\) 29.4868 1.01678
\(842\) 45.7673 1.57725
\(843\) 0 0
\(844\) 19.8319 0.682642
\(845\) −46.6881 −1.60612
\(846\) 0 0
\(847\) 0 0
\(848\) −15.4432 −0.530321
\(849\) 0 0
\(850\) −100.141 −3.43480
\(851\) 10.8171 0.370806
\(852\) 0 0
\(853\) −9.81491 −0.336056 −0.168028 0.985782i \(-0.553740\pi\)
−0.168028 + 0.985782i \(0.553740\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 64.9046 2.21840
\(857\) −26.7360 −0.913285 −0.456642 0.889650i \(-0.650948\pi\)
−0.456642 + 0.889650i \(0.650948\pi\)
\(858\) 0 0
\(859\) 18.4882 0.630810 0.315405 0.948957i \(-0.397860\pi\)
0.315405 + 0.948957i \(0.397860\pi\)
\(860\) −34.5161 −1.17699
\(861\) 0 0
\(862\) −39.0689 −1.33069
\(863\) −9.14786 −0.311397 −0.155698 0.987805i \(-0.549763\pi\)
−0.155698 + 0.987805i \(0.549763\pi\)
\(864\) 0 0
\(865\) 22.1445 0.752937
\(866\) −99.4946 −3.38097
\(867\) 0 0
\(868\) 0 0
\(869\) 3.31731 0.112532
\(870\) 0 0
\(871\) 1.75285 0.0593929
\(872\) 13.1445 0.445130
\(873\) 0 0
\(874\) 22.5683 0.763385
\(875\) 0 0
\(876\) 0 0
\(877\) 22.6883 0.766130 0.383065 0.923721i \(-0.374869\pi\)
0.383065 + 0.923721i \(0.374869\pi\)
\(878\) −30.6602 −1.03473
\(879\) 0 0
\(880\) 6.43367 0.216879
\(881\) −15.3696 −0.517815 −0.258907 0.965902i \(-0.583362\pi\)
−0.258907 + 0.965902i \(0.583362\pi\)
\(882\) 0 0
\(883\) 29.6372 0.997370 0.498685 0.866783i \(-0.333817\pi\)
0.498685 + 0.866783i \(0.333817\pi\)
\(884\) −9.58307 −0.322313
\(885\) 0 0
\(886\) −20.2518 −0.680371
\(887\) 18.7650 0.630069 0.315034 0.949080i \(-0.397984\pi\)
0.315034 + 0.949080i \(0.397984\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −133.482 −4.47432
\(891\) 0 0
\(892\) 94.9014 3.17753
\(893\) 12.4652 0.417131
\(894\) 0 0
\(895\) 33.4095 1.11676
\(896\) 0 0
\(897\) 0 0
\(898\) 13.8889 0.463479
\(899\) −53.7416 −1.79238
\(900\) 0 0
\(901\) 17.3289 0.577310
\(902\) −7.51399 −0.250189
\(903\) 0 0
\(904\) −70.5054 −2.34498
\(905\) −43.7060 −1.45284
\(906\) 0 0
\(907\) −31.1082 −1.03293 −0.516465 0.856308i \(-0.672752\pi\)
−0.516465 + 0.856308i \(0.672752\pi\)
\(908\) 24.8045 0.823165
\(909\) 0 0
\(910\) 0 0
\(911\) 17.4753 0.578982 0.289491 0.957181i \(-0.406514\pi\)
0.289491 + 0.957181i \(0.406514\pi\)
\(912\) 0 0
\(913\) −4.94695 −0.163720
\(914\) 12.4720 0.412536
\(915\) 0 0
\(916\) −5.92054 −0.195620
\(917\) 0 0
\(918\) 0 0
\(919\) 42.1986 1.39200 0.696002 0.718040i \(-0.254960\pi\)
0.696002 + 0.718040i \(0.254960\pi\)
\(920\) 85.9270 2.83293
\(921\) 0 0
\(922\) 19.1344 0.630158
\(923\) 4.11854 0.135564
\(924\) 0 0
\(925\) 19.5074 0.641399
\(926\) −22.5873 −0.742266
\(927\) 0 0
\(928\) 4.12588 0.135439
\(929\) −48.4111 −1.58832 −0.794159 0.607710i \(-0.792088\pi\)
−0.794159 + 0.607710i \(0.792088\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −53.7103 −1.75934
\(933\) 0 0
\(934\) 33.8699 1.10826
\(935\) −7.21926 −0.236095
\(936\) 0 0
\(937\) 22.6750 0.740762 0.370381 0.928880i \(-0.379227\pi\)
0.370381 + 0.928880i \(0.379227\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 93.6706 3.05520
\(941\) 49.9505 1.62834 0.814170 0.580627i \(-0.197192\pi\)
0.814170 + 0.580627i \(0.197192\pi\)
\(942\) 0 0
\(943\) −34.9225 −1.13723
\(944\) −26.4769 −0.861749
\(945\) 0 0
\(946\) −2.32743 −0.0756713
\(947\) 17.7123 0.575571 0.287786 0.957695i \(-0.407081\pi\)
0.287786 + 0.957695i \(0.407081\pi\)
\(948\) 0 0
\(949\) −0.960699 −0.0311856
\(950\) 40.6993 1.32046
\(951\) 0 0
\(952\) 0 0
\(953\) 38.0229 1.23168 0.615842 0.787870i \(-0.288816\pi\)
0.615842 + 0.787870i \(0.288816\pi\)
\(954\) 0 0
\(955\) 33.4358 1.08196
\(956\) 78.6035 2.54222
\(957\) 0 0
\(958\) −21.4353 −0.692544
\(959\) 0 0
\(960\) 0 0
\(961\) 18.3815 0.592952
\(962\) 2.78771 0.0898795
\(963\) 0 0
\(964\) 20.4881 0.659876
\(965\) 61.9787 1.99517
\(966\) 0 0
\(967\) −31.0128 −0.997305 −0.498652 0.866802i \(-0.666172\pi\)
−0.498652 + 0.866802i \(0.666172\pi\)
\(968\) −54.7601 −1.76006
\(969\) 0 0
\(970\) 85.3943 2.74185
\(971\) 14.5675 0.467494 0.233747 0.972297i \(-0.424901\pi\)
0.233747 + 0.972297i \(0.424901\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −44.3786 −1.42198
\(975\) 0 0
\(976\) 34.7368 1.11190
\(977\) −47.8797 −1.53181 −0.765904 0.642955i \(-0.777708\pi\)
−0.765904 + 0.642955i \(0.777708\pi\)
\(978\) 0 0
\(979\) −6.02728 −0.192633
\(980\) 0 0
\(981\) 0 0
\(982\) 5.03638 0.160717
\(983\) 44.4257 1.41696 0.708479 0.705732i \(-0.249382\pi\)
0.708479 + 0.705732i \(0.249382\pi\)
\(984\) 0 0
\(985\) −77.9803 −2.48466
\(986\) −91.3729 −2.90991
\(987\) 0 0
\(988\) 3.89476 0.123909
\(989\) −10.8171 −0.343964
\(990\) 0 0
\(991\) −41.6156 −1.32196 −0.660981 0.750403i \(-0.729860\pi\)
−0.660981 + 0.750403i \(0.729860\pi\)
\(992\) −3.79114 −0.120369
\(993\) 0 0
\(994\) 0 0
\(995\) −36.4868 −1.15671
\(996\) 0 0
\(997\) −13.0281 −0.412606 −0.206303 0.978488i \(-0.566143\pi\)
−0.206303 + 0.978488i \(0.566143\pi\)
\(998\) −96.1751 −3.04437
\(999\) 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.be.1.6 6
3.2 odd 2 3969.2.a.bd.1.1 6
7.6 odd 2 inner 3969.2.a.be.1.5 6
9.2 odd 6 1323.2.f.g.442.6 12
9.4 even 3 441.2.f.g.295.1 yes 12
9.5 odd 6 1323.2.f.g.883.6 12
9.7 even 3 441.2.f.g.148.1 12
21.20 even 2 3969.2.a.bd.1.2 6
63.2 odd 6 1323.2.g.g.361.5 12
63.4 even 3 441.2.g.g.79.2 12
63.5 even 6 1323.2.h.g.802.1 12
63.11 odd 6 1323.2.h.g.226.2 12
63.13 odd 6 441.2.f.g.295.2 yes 12
63.16 even 3 441.2.g.g.67.2 12
63.20 even 6 1323.2.f.g.442.5 12
63.23 odd 6 1323.2.h.g.802.2 12
63.25 even 3 441.2.h.g.373.6 12
63.31 odd 6 441.2.g.g.79.1 12
63.32 odd 6 1323.2.g.g.667.5 12
63.34 odd 6 441.2.f.g.148.2 yes 12
63.38 even 6 1323.2.h.g.226.1 12
63.40 odd 6 441.2.h.g.214.5 12
63.41 even 6 1323.2.f.g.883.5 12
63.47 even 6 1323.2.g.g.361.6 12
63.52 odd 6 441.2.h.g.373.5 12
63.58 even 3 441.2.h.g.214.6 12
63.59 even 6 1323.2.g.g.667.6 12
63.61 odd 6 441.2.g.g.67.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.f.g.148.1 12 9.7 even 3
441.2.f.g.148.2 yes 12 63.34 odd 6
441.2.f.g.295.1 yes 12 9.4 even 3
441.2.f.g.295.2 yes 12 63.13 odd 6
441.2.g.g.67.1 12 63.61 odd 6
441.2.g.g.67.2 12 63.16 even 3
441.2.g.g.79.1 12 63.31 odd 6
441.2.g.g.79.2 12 63.4 even 3
441.2.h.g.214.5 12 63.40 odd 6
441.2.h.g.214.6 12 63.58 even 3
441.2.h.g.373.5 12 63.52 odd 6
441.2.h.g.373.6 12 63.25 even 3
1323.2.f.g.442.5 12 63.20 even 6
1323.2.f.g.442.6 12 9.2 odd 6
1323.2.f.g.883.5 12 63.41 even 6
1323.2.f.g.883.6 12 9.5 odd 6
1323.2.g.g.361.5 12 63.2 odd 6
1323.2.g.g.361.6 12 63.47 even 6
1323.2.g.g.667.5 12 63.32 odd 6
1323.2.g.g.667.6 12 63.59 even 6
1323.2.h.g.226.1 12 63.38 even 6
1323.2.h.g.226.2 12 63.11 odd 6
1323.2.h.g.802.1 12 63.5 even 6
1323.2.h.g.802.2 12 63.23 odd 6
3969.2.a.bd.1.1 6 3.2 odd 2
3969.2.a.bd.1.2 6 21.20 even 2
3969.2.a.be.1.5 6 7.6 odd 2 inner
3969.2.a.be.1.6 6 1.1 even 1 trivial