Properties

Label 2760.2.a.l.1.1
Level $2760$
Weight $2$
Character 2760.1
Self dual yes
Analytic conductor $22.039$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2760.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.0387109579\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.31662\) of defining polynomial
Character \(\chi\) \(=\) 2760.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} -4.31662 q^{11} -2.31662 q^{13} +1.00000 q^{15} -3.31662 q^{17} -1.68338 q^{19} +3.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -0.683375 q^{29} -9.63325 q^{31} +4.31662 q^{33} +3.00000 q^{35} +11.6332 q^{37} +2.31662 q^{39} +3.31662 q^{41} +2.63325 q^{43} -1.00000 q^{45} +10.9499 q^{47} +2.00000 q^{49} +3.31662 q^{51} -9.94987 q^{53} +4.31662 q^{55} +1.68338 q^{57} -11.9499 q^{59} +3.68338 q^{61} -3.00000 q^{63} +2.31662 q^{65} +2.36675 q^{67} -1.00000 q^{69} -7.94987 q^{71} +1.68338 q^{73} -1.00000 q^{75} +12.9499 q^{77} +1.00000 q^{81} +5.31662 q^{83} +3.31662 q^{85} +0.683375 q^{87} +8.63325 q^{89} +6.94987 q^{91} +9.63325 q^{93} +1.68338 q^{95} +12.0000 q^{97} -4.31662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} - 10 q^{19} + 6 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 8 q^{29} - 6 q^{31} + 2 q^{33} + 6 q^{35} + 10 q^{37} - 2 q^{39} - 8 q^{43} - 2 q^{45} + 2 q^{47} + 4 q^{49} + 2 q^{55} + 10 q^{57} - 4 q^{59} + 14 q^{61} - 6 q^{63} - 2 q^{65} + 18 q^{67} - 2 q^{69} + 4 q^{71} + 10 q^{73} - 2 q^{75} + 6 q^{77} + 2 q^{81} + 4 q^{83} + 8 q^{87} + 4 q^{89} - 6 q^{91} + 6 q^{93} + 10 q^{95} + 24 q^{97} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.31662 −1.30151 −0.650756 0.759287i \(-0.725548\pi\)
−0.650756 + 0.759287i \(0.725548\pi\)
\(12\) 0 0
\(13\) −2.31662 −0.642516 −0.321258 0.946992i \(-0.604106\pi\)
−0.321258 + 0.946992i \(0.604106\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −3.31662 −0.804400 −0.402200 0.915552i \(-0.631754\pi\)
−0.402200 + 0.915552i \(0.631754\pi\)
\(18\) 0 0
\(19\) −1.68338 −0.386193 −0.193096 0.981180i \(-0.561853\pi\)
−0.193096 + 0.981180i \(0.561853\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.683375 −0.126900 −0.0634498 0.997985i \(-0.520210\pi\)
−0.0634498 + 0.997985i \(0.520210\pi\)
\(30\) 0 0
\(31\) −9.63325 −1.73018 −0.865091 0.501614i \(-0.832740\pi\)
−0.865091 + 0.501614i \(0.832740\pi\)
\(32\) 0 0
\(33\) 4.31662 0.751428
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) 11.6332 1.91249 0.956247 0.292560i \(-0.0945071\pi\)
0.956247 + 0.292560i \(0.0945071\pi\)
\(38\) 0 0
\(39\) 2.31662 0.370957
\(40\) 0 0
\(41\) 3.31662 0.517970 0.258985 0.965881i \(-0.416612\pi\)
0.258985 + 0.965881i \(0.416612\pi\)
\(42\) 0 0
\(43\) 2.63325 0.401567 0.200783 0.979636i \(-0.435651\pi\)
0.200783 + 0.979636i \(0.435651\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 10.9499 1.59720 0.798602 0.601860i \(-0.205573\pi\)
0.798602 + 0.601860i \(0.205573\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 3.31662 0.464420
\(52\) 0 0
\(53\) −9.94987 −1.36672 −0.683360 0.730081i \(-0.739482\pi\)
−0.683360 + 0.730081i \(0.739482\pi\)
\(54\) 0 0
\(55\) 4.31662 0.582054
\(56\) 0 0
\(57\) 1.68338 0.222968
\(58\) 0 0
\(59\) −11.9499 −1.55574 −0.777871 0.628425i \(-0.783700\pi\)
−0.777871 + 0.628425i \(0.783700\pi\)
\(60\) 0 0
\(61\) 3.68338 0.471608 0.235804 0.971801i \(-0.424228\pi\)
0.235804 + 0.971801i \(0.424228\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) 0 0
\(65\) 2.31662 0.287342
\(66\) 0 0
\(67\) 2.36675 0.289145 0.144572 0.989494i \(-0.453819\pi\)
0.144572 + 0.989494i \(0.453819\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −7.94987 −0.943477 −0.471738 0.881739i \(-0.656373\pi\)
−0.471738 + 0.881739i \(0.656373\pi\)
\(72\) 0 0
\(73\) 1.68338 0.197024 0.0985121 0.995136i \(-0.468592\pi\)
0.0985121 + 0.995136i \(0.468592\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 12.9499 1.47578
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.31662 0.583575 0.291788 0.956483i \(-0.405750\pi\)
0.291788 + 0.956483i \(0.405750\pi\)
\(84\) 0 0
\(85\) 3.31662 0.359738
\(86\) 0 0
\(87\) 0.683375 0.0732655
\(88\) 0 0
\(89\) 8.63325 0.915123 0.457561 0.889178i \(-0.348723\pi\)
0.457561 + 0.889178i \(0.348723\pi\)
\(90\) 0 0
\(91\) 6.94987 0.728545
\(92\) 0 0
\(93\) 9.63325 0.998921
\(94\) 0 0
\(95\) 1.68338 0.172711
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 0 0
\(99\) −4.31662 −0.433837
\(100\) 0 0
\(101\) 5.31662 0.529024 0.264512 0.964382i \(-0.414789\pi\)
0.264512 + 0.964382i \(0.414789\pi\)
\(102\) 0 0
\(103\) 18.0000 1.77359 0.886796 0.462160i \(-0.152926\pi\)
0.886796 + 0.462160i \(0.152926\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) −13.9499 −1.34859 −0.674293 0.738464i \(-0.735551\pi\)
−0.674293 + 0.738464i \(0.735551\pi\)
\(108\) 0 0
\(109\) −10.9499 −1.04881 −0.524404 0.851470i \(-0.675712\pi\)
−0.524404 + 0.851470i \(0.675712\pi\)
\(110\) 0 0
\(111\) −11.6332 −1.10418
\(112\) 0 0
\(113\) 15.3166 1.44087 0.720433 0.693524i \(-0.243943\pi\)
0.720433 + 0.693524i \(0.243943\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −2.31662 −0.214172
\(118\) 0 0
\(119\) 9.94987 0.912103
\(120\) 0 0
\(121\) 7.63325 0.693932
\(122\) 0 0
\(123\) −3.31662 −0.299050
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.94987 −0.439230 −0.219615 0.975587i \(-0.570480\pi\)
−0.219615 + 0.975587i \(0.570480\pi\)
\(128\) 0 0
\(129\) −2.63325 −0.231845
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 5.05013 0.437901
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −13.2665 −1.13343 −0.566717 0.823913i \(-0.691787\pi\)
−0.566717 + 0.823913i \(0.691787\pi\)
\(138\) 0 0
\(139\) −1.63325 −0.138530 −0.0692652 0.997598i \(-0.522065\pi\)
−0.0692652 + 0.997598i \(0.522065\pi\)
\(140\) 0 0
\(141\) −10.9499 −0.922146
\(142\) 0 0
\(143\) 10.0000 0.836242
\(144\) 0 0
\(145\) 0.683375 0.0567512
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) 2.31662 0.189785 0.0948926 0.995488i \(-0.469749\pi\)
0.0948926 + 0.995488i \(0.469749\pi\)
\(150\) 0 0
\(151\) −1.36675 −0.111225 −0.0556123 0.998452i \(-0.517711\pi\)
−0.0556123 + 0.998452i \(0.517711\pi\)
\(152\) 0 0
\(153\) −3.31662 −0.268133
\(154\) 0 0
\(155\) 9.63325 0.773761
\(156\) 0 0
\(157\) 6.36675 0.508122 0.254061 0.967188i \(-0.418234\pi\)
0.254061 + 0.967188i \(0.418234\pi\)
\(158\) 0 0
\(159\) 9.94987 0.789076
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 3.26650 0.255852 0.127926 0.991784i \(-0.459168\pi\)
0.127926 + 0.991784i \(0.459168\pi\)
\(164\) 0 0
\(165\) −4.31662 −0.336049
\(166\) 0 0
\(167\) 19.5831 1.51539 0.757694 0.652610i \(-0.226326\pi\)
0.757694 + 0.652610i \(0.226326\pi\)
\(168\) 0 0
\(169\) −7.63325 −0.587173
\(170\) 0 0
\(171\) −1.68338 −0.128731
\(172\) 0 0
\(173\) 4.00000 0.304114 0.152057 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 0 0
\(177\) 11.9499 0.898208
\(178\) 0 0
\(179\) −9.36675 −0.700104 −0.350052 0.936730i \(-0.613836\pi\)
−0.350052 + 0.936730i \(0.613836\pi\)
\(180\) 0 0
\(181\) 20.6332 1.53366 0.766829 0.641852i \(-0.221834\pi\)
0.766829 + 0.641852i \(0.221834\pi\)
\(182\) 0 0
\(183\) −3.68338 −0.272283
\(184\) 0 0
\(185\) −11.6332 −0.855294
\(186\) 0 0
\(187\) 14.3166 1.04694
\(188\) 0 0
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 19.5831 1.41699 0.708493 0.705718i \(-0.249376\pi\)
0.708493 + 0.705718i \(0.249376\pi\)
\(192\) 0 0
\(193\) 12.6332 0.909361 0.454681 0.890655i \(-0.349753\pi\)
0.454681 + 0.890655i \(0.349753\pi\)
\(194\) 0 0
\(195\) −2.31662 −0.165897
\(196\) 0 0
\(197\) −27.8997 −1.98777 −0.993887 0.110399i \(-0.964787\pi\)
−0.993887 + 0.110399i \(0.964787\pi\)
\(198\) 0 0
\(199\) 1.36675 0.0968864 0.0484432 0.998826i \(-0.484574\pi\)
0.0484432 + 0.998826i \(0.484574\pi\)
\(200\) 0 0
\(201\) −2.36675 −0.166938
\(202\) 0 0
\(203\) 2.05013 0.143891
\(204\) 0 0
\(205\) −3.31662 −0.231643
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 7.26650 0.502634
\(210\) 0 0
\(211\) 9.63325 0.663180 0.331590 0.943424i \(-0.392415\pi\)
0.331590 + 0.943424i \(0.392415\pi\)
\(212\) 0 0
\(213\) 7.94987 0.544716
\(214\) 0 0
\(215\) −2.63325 −0.179586
\(216\) 0 0
\(217\) 28.8997 1.96184
\(218\) 0 0
\(219\) −1.68338 −0.113752
\(220\) 0 0
\(221\) 7.68338 0.516840
\(222\) 0 0
\(223\) −3.26650 −0.218741 −0.109370 0.994001i \(-0.534883\pi\)
−0.109370 + 0.994001i \(0.534883\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 0 0
\(229\) 16.6332 1.09916 0.549578 0.835442i \(-0.314789\pi\)
0.549578 + 0.835442i \(0.314789\pi\)
\(230\) 0 0
\(231\) −12.9499 −0.852039
\(232\) 0 0
\(233\) 8.00000 0.524097 0.262049 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 0 0
\(235\) −10.9499 −0.714291
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.94987 0.255496 0.127748 0.991807i \(-0.459225\pi\)
0.127748 + 0.991807i \(0.459225\pi\)
\(240\) 0 0
\(241\) 20.2164 1.30225 0.651126 0.758970i \(-0.274297\pi\)
0.651126 + 0.758970i \(0.274297\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 3.89975 0.248135
\(248\) 0 0
\(249\) −5.31662 −0.336927
\(250\) 0 0
\(251\) −3.26650 −0.206180 −0.103090 0.994672i \(-0.532873\pi\)
−0.103090 + 0.994672i \(0.532873\pi\)
\(252\) 0 0
\(253\) −4.31662 −0.271384
\(254\) 0 0
\(255\) −3.31662 −0.207695
\(256\) 0 0
\(257\) −18.9499 −1.18206 −0.591030 0.806649i \(-0.701279\pi\)
−0.591030 + 0.806649i \(0.701279\pi\)
\(258\) 0 0
\(259\) −34.8997 −2.16856
\(260\) 0 0
\(261\) −0.683375 −0.0422999
\(262\) 0 0
\(263\) −4.58312 −0.282608 −0.141304 0.989966i \(-0.545129\pi\)
−0.141304 + 0.989966i \(0.545129\pi\)
\(264\) 0 0
\(265\) 9.94987 0.611216
\(266\) 0 0
\(267\) −8.63325 −0.528346
\(268\) 0 0
\(269\) 10.5831 0.645264 0.322632 0.946524i \(-0.395432\pi\)
0.322632 + 0.946524i \(0.395432\pi\)
\(270\) 0 0
\(271\) −2.26650 −0.137680 −0.0688400 0.997628i \(-0.521930\pi\)
−0.0688400 + 0.997628i \(0.521930\pi\)
\(272\) 0 0
\(273\) −6.94987 −0.420626
\(274\) 0 0
\(275\) −4.31662 −0.260302
\(276\) 0 0
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 0 0
\(279\) −9.63325 −0.576728
\(280\) 0 0
\(281\) −16.9499 −1.01114 −0.505572 0.862784i \(-0.668719\pi\)
−0.505572 + 0.862784i \(0.668719\pi\)
\(282\) 0 0
\(283\) 7.63325 0.453750 0.226875 0.973924i \(-0.427149\pi\)
0.226875 + 0.973924i \(0.427149\pi\)
\(284\) 0 0
\(285\) −1.68338 −0.0997145
\(286\) 0 0
\(287\) −9.94987 −0.587323
\(288\) 0 0
\(289\) −6.00000 −0.352941
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) 0 0
\(293\) 11.9499 0.698119 0.349060 0.937101i \(-0.386501\pi\)
0.349060 + 0.937101i \(0.386501\pi\)
\(294\) 0 0
\(295\) 11.9499 0.695749
\(296\) 0 0
\(297\) 4.31662 0.250476
\(298\) 0 0
\(299\) −2.31662 −0.133974
\(300\) 0 0
\(301\) −7.89975 −0.455334
\(302\) 0 0
\(303\) −5.31662 −0.305432
\(304\) 0 0
\(305\) −3.68338 −0.210909
\(306\) 0 0
\(307\) −5.05013 −0.288226 −0.144113 0.989561i \(-0.546033\pi\)
−0.144113 + 0.989561i \(0.546033\pi\)
\(308\) 0 0
\(309\) −18.0000 −1.02398
\(310\) 0 0
\(311\) 29.2665 1.65955 0.829775 0.558097i \(-0.188468\pi\)
0.829775 + 0.558097i \(0.188468\pi\)
\(312\) 0 0
\(313\) 20.3668 1.15120 0.575598 0.817733i \(-0.304769\pi\)
0.575598 + 0.817733i \(0.304769\pi\)
\(314\) 0 0
\(315\) 3.00000 0.169031
\(316\) 0 0
\(317\) −9.68338 −0.543873 −0.271936 0.962315i \(-0.587664\pi\)
−0.271936 + 0.962315i \(0.587664\pi\)
\(318\) 0 0
\(319\) 2.94987 0.165161
\(320\) 0 0
\(321\) 13.9499 0.778606
\(322\) 0 0
\(323\) 5.58312 0.310653
\(324\) 0 0
\(325\) −2.31662 −0.128503
\(326\) 0 0
\(327\) 10.9499 0.605529
\(328\) 0 0
\(329\) −32.8496 −1.81106
\(330\) 0 0
\(331\) −31.5330 −1.73321 −0.866605 0.498994i \(-0.833703\pi\)
−0.866605 + 0.498994i \(0.833703\pi\)
\(332\) 0 0
\(333\) 11.6332 0.637498
\(334\) 0 0
\(335\) −2.36675 −0.129309
\(336\) 0 0
\(337\) 31.2665 1.70319 0.851597 0.524196i \(-0.175634\pi\)
0.851597 + 0.524196i \(0.175634\pi\)
\(338\) 0 0
\(339\) −15.3166 −0.831885
\(340\) 0 0
\(341\) 41.5831 2.25185
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) −11.3668 −0.610199 −0.305100 0.952320i \(-0.598690\pi\)
−0.305100 + 0.952320i \(0.598690\pi\)
\(348\) 0 0
\(349\) −21.6332 −1.15800 −0.579001 0.815327i \(-0.696557\pi\)
−0.579001 + 0.815327i \(0.696557\pi\)
\(350\) 0 0
\(351\) 2.31662 0.123652
\(352\) 0 0
\(353\) −20.2164 −1.07601 −0.538004 0.842942i \(-0.680822\pi\)
−0.538004 + 0.842942i \(0.680822\pi\)
\(354\) 0 0
\(355\) 7.94987 0.421936
\(356\) 0 0
\(357\) −9.94987 −0.526603
\(358\) 0 0
\(359\) −28.8496 −1.52262 −0.761312 0.648385i \(-0.775445\pi\)
−0.761312 + 0.648385i \(0.775445\pi\)
\(360\) 0 0
\(361\) −16.1662 −0.850855
\(362\) 0 0
\(363\) −7.63325 −0.400642
\(364\) 0 0
\(365\) −1.68338 −0.0881119
\(366\) 0 0
\(367\) −17.5330 −0.915215 −0.457608 0.889154i \(-0.651294\pi\)
−0.457608 + 0.889154i \(0.651294\pi\)
\(368\) 0 0
\(369\) 3.31662 0.172657
\(370\) 0 0
\(371\) 29.8496 1.54972
\(372\) 0 0
\(373\) 29.8997 1.54815 0.774075 0.633094i \(-0.218215\pi\)
0.774075 + 0.633094i \(0.218215\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 1.58312 0.0815350
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 4.94987 0.253590
\(382\) 0 0
\(383\) 19.3166 0.987033 0.493517 0.869736i \(-0.335711\pi\)
0.493517 + 0.869736i \(0.335711\pi\)
\(384\) 0 0
\(385\) −12.9499 −0.659987
\(386\) 0 0
\(387\) 2.63325 0.133856
\(388\) 0 0
\(389\) −23.8997 −1.21177 −0.605883 0.795554i \(-0.707180\pi\)
−0.605883 + 0.795554i \(0.707180\pi\)
\(390\) 0 0
\(391\) −3.31662 −0.167729
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 12.0000 0.602263 0.301131 0.953583i \(-0.402636\pi\)
0.301131 + 0.953583i \(0.402636\pi\)
\(398\) 0 0
\(399\) −5.05013 −0.252823
\(400\) 0 0
\(401\) −19.8997 −0.993746 −0.496873 0.867823i \(-0.665519\pi\)
−0.496873 + 0.867823i \(0.665519\pi\)
\(402\) 0 0
\(403\) 22.3166 1.11167
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −50.2164 −2.48913
\(408\) 0 0
\(409\) 0.366750 0.0181346 0.00906732 0.999959i \(-0.497114\pi\)
0.00906732 + 0.999959i \(0.497114\pi\)
\(410\) 0 0
\(411\) 13.2665 0.654388
\(412\) 0 0
\(413\) 35.8496 1.76404
\(414\) 0 0
\(415\) −5.31662 −0.260983
\(416\) 0 0
\(417\) 1.63325 0.0799806
\(418\) 0 0
\(419\) −11.6834 −0.570770 −0.285385 0.958413i \(-0.592122\pi\)
−0.285385 + 0.958413i \(0.592122\pi\)
\(420\) 0 0
\(421\) 23.5831 1.14937 0.574686 0.818374i \(-0.305124\pi\)
0.574686 + 0.818374i \(0.305124\pi\)
\(422\) 0 0
\(423\) 10.9499 0.532401
\(424\) 0 0
\(425\) −3.31662 −0.160880
\(426\) 0 0
\(427\) −11.0501 −0.534753
\(428\) 0 0
\(429\) −10.0000 −0.482805
\(430\) 0 0
\(431\) −16.6332 −0.801195 −0.400598 0.916254i \(-0.631198\pi\)
−0.400598 + 0.916254i \(0.631198\pi\)
\(432\) 0 0
\(433\) −10.3668 −0.498194 −0.249097 0.968479i \(-0.580134\pi\)
−0.249097 + 0.968479i \(0.580134\pi\)
\(434\) 0 0
\(435\) −0.683375 −0.0327653
\(436\) 0 0
\(437\) −1.68338 −0.0805268
\(438\) 0 0
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) −21.6834 −1.03021 −0.515104 0.857128i \(-0.672247\pi\)
−0.515104 + 0.857128i \(0.672247\pi\)
\(444\) 0 0
\(445\) −8.63325 −0.409255
\(446\) 0 0
\(447\) −2.31662 −0.109573
\(448\) 0 0
\(449\) −0.683375 −0.0322505 −0.0161252 0.999870i \(-0.505133\pi\)
−0.0161252 + 0.999870i \(0.505133\pi\)
\(450\) 0 0
\(451\) −14.3166 −0.674144
\(452\) 0 0
\(453\) 1.36675 0.0642155
\(454\) 0 0
\(455\) −6.94987 −0.325815
\(456\) 0 0
\(457\) 12.3668 0.578492 0.289246 0.957255i \(-0.406595\pi\)
0.289246 + 0.957255i \(0.406595\pi\)
\(458\) 0 0
\(459\) 3.31662 0.154807
\(460\) 0 0
\(461\) −10.6332 −0.495240 −0.247620 0.968857i \(-0.579648\pi\)
−0.247620 + 0.968857i \(0.579648\pi\)
\(462\) 0 0
\(463\) −8.21637 −0.381847 −0.190924 0.981605i \(-0.561148\pi\)
−0.190924 + 0.981605i \(0.561148\pi\)
\(464\) 0 0
\(465\) −9.63325 −0.446731
\(466\) 0 0
\(467\) −29.9499 −1.38591 −0.692957 0.720978i \(-0.743693\pi\)
−0.692957 + 0.720978i \(0.743693\pi\)
\(468\) 0 0
\(469\) −7.10025 −0.327859
\(470\) 0 0
\(471\) −6.36675 −0.293364
\(472\) 0 0
\(473\) −11.3668 −0.522644
\(474\) 0 0
\(475\) −1.68338 −0.0772386
\(476\) 0 0
\(477\) −9.94987 −0.455573
\(478\) 0 0
\(479\) 18.3166 0.836908 0.418454 0.908238i \(-0.362572\pi\)
0.418454 + 0.908238i \(0.362572\pi\)
\(480\) 0 0
\(481\) −26.9499 −1.22881
\(482\) 0 0
\(483\) 3.00000 0.136505
\(484\) 0 0
\(485\) −12.0000 −0.544892
\(486\) 0 0
\(487\) 13.5831 0.615510 0.307755 0.951466i \(-0.400422\pi\)
0.307755 + 0.951466i \(0.400422\pi\)
\(488\) 0 0
\(489\) −3.26650 −0.147716
\(490\) 0 0
\(491\) 9.94987 0.449032 0.224516 0.974470i \(-0.427920\pi\)
0.224516 + 0.974470i \(0.427920\pi\)
\(492\) 0 0
\(493\) 2.26650 0.102078
\(494\) 0 0
\(495\) 4.31662 0.194018
\(496\) 0 0
\(497\) 23.8496 1.06980
\(498\) 0 0
\(499\) 18.2665 0.817721 0.408860 0.912597i \(-0.365926\pi\)
0.408860 + 0.912597i \(0.365926\pi\)
\(500\) 0 0
\(501\) −19.5831 −0.874909
\(502\) 0 0
\(503\) 37.8496 1.68763 0.843816 0.536633i \(-0.180304\pi\)
0.843816 + 0.536633i \(0.180304\pi\)
\(504\) 0 0
\(505\) −5.31662 −0.236587
\(506\) 0 0
\(507\) 7.63325 0.339005
\(508\) 0 0
\(509\) 8.00000 0.354594 0.177297 0.984157i \(-0.443265\pi\)
0.177297 + 0.984157i \(0.443265\pi\)
\(510\) 0 0
\(511\) −5.05013 −0.223404
\(512\) 0 0
\(513\) 1.68338 0.0743228
\(514\) 0 0
\(515\) −18.0000 −0.793175
\(516\) 0 0
\(517\) −47.2665 −2.07878
\(518\) 0 0
\(519\) −4.00000 −0.175581
\(520\) 0 0
\(521\) −0.316625 −0.0138716 −0.00693579 0.999976i \(-0.502208\pi\)
−0.00693579 + 0.999976i \(0.502208\pi\)
\(522\) 0 0
\(523\) −22.5330 −0.985299 −0.492650 0.870228i \(-0.663972\pi\)
−0.492650 + 0.870228i \(0.663972\pi\)
\(524\) 0 0
\(525\) 3.00000 0.130931
\(526\) 0 0
\(527\) 31.9499 1.39176
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −11.9499 −0.518580
\(532\) 0 0
\(533\) −7.68338 −0.332804
\(534\) 0 0
\(535\) 13.9499 0.603106
\(536\) 0 0
\(537\) 9.36675 0.404205
\(538\) 0 0
\(539\) −8.63325 −0.371860
\(540\) 0 0
\(541\) 21.2665 0.914318 0.457159 0.889385i \(-0.348867\pi\)
0.457159 + 0.889385i \(0.348867\pi\)
\(542\) 0 0
\(543\) −20.6332 −0.885457
\(544\) 0 0
\(545\) 10.9499 0.469041
\(546\) 0 0
\(547\) −25.2665 −1.08032 −0.540159 0.841563i \(-0.681636\pi\)
−0.540159 + 0.841563i \(0.681636\pi\)
\(548\) 0 0
\(549\) 3.68338 0.157203
\(550\) 0 0
\(551\) 1.15038 0.0490077
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 11.6332 0.493804
\(556\) 0 0
\(557\) −25.9499 −1.09953 −0.549766 0.835319i \(-0.685283\pi\)
−0.549766 + 0.835319i \(0.685283\pi\)
\(558\) 0 0
\(559\) −6.10025 −0.258013
\(560\) 0 0
\(561\) −14.3166 −0.604448
\(562\) 0 0
\(563\) 36.6834 1.54602 0.773010 0.634394i \(-0.218750\pi\)
0.773010 + 0.634394i \(0.218750\pi\)
\(564\) 0 0
\(565\) −15.3166 −0.644375
\(566\) 0 0
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) 3.89975 0.163486 0.0817430 0.996653i \(-0.473951\pi\)
0.0817430 + 0.996653i \(0.473951\pi\)
\(570\) 0 0
\(571\) 24.9499 1.04412 0.522060 0.852909i \(-0.325164\pi\)
0.522060 + 0.852909i \(0.325164\pi\)
\(572\) 0 0
\(573\) −19.5831 −0.818097
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −9.26650 −0.385769 −0.192885 0.981221i \(-0.561784\pi\)
−0.192885 + 0.981221i \(0.561784\pi\)
\(578\) 0 0
\(579\) −12.6332 −0.525020
\(580\) 0 0
\(581\) −15.9499 −0.661712
\(582\) 0 0
\(583\) 42.9499 1.77880
\(584\) 0 0
\(585\) 2.31662 0.0957806
\(586\) 0 0
\(587\) 2.00000 0.0825488 0.0412744 0.999148i \(-0.486858\pi\)
0.0412744 + 0.999148i \(0.486858\pi\)
\(588\) 0 0
\(589\) 16.2164 0.668184
\(590\) 0 0
\(591\) 27.8997 1.14764
\(592\) 0 0
\(593\) −22.3166 −0.916434 −0.458217 0.888840i \(-0.651512\pi\)
−0.458217 + 0.888840i \(0.651512\pi\)
\(594\) 0 0
\(595\) −9.94987 −0.407905
\(596\) 0 0
\(597\) −1.36675 −0.0559374
\(598\) 0 0
\(599\) 28.6332 1.16992 0.584961 0.811061i \(-0.301110\pi\)
0.584961 + 0.811061i \(0.301110\pi\)
\(600\) 0 0
\(601\) 9.63325 0.392948 0.196474 0.980509i \(-0.437051\pi\)
0.196474 + 0.980509i \(0.437051\pi\)
\(602\) 0 0
\(603\) 2.36675 0.0963815
\(604\) 0 0
\(605\) −7.63325 −0.310336
\(606\) 0 0
\(607\) −29.5831 −1.20074 −0.600371 0.799722i \(-0.704980\pi\)
−0.600371 + 0.799722i \(0.704980\pi\)
\(608\) 0 0
\(609\) −2.05013 −0.0830753
\(610\) 0 0
\(611\) −25.3668 −1.02623
\(612\) 0 0
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 0 0
\(615\) 3.31662 0.133739
\(616\) 0 0
\(617\) 31.9499 1.28625 0.643127 0.765760i \(-0.277637\pi\)
0.643127 + 0.765760i \(0.277637\pi\)
\(618\) 0 0
\(619\) 43.7995 1.76045 0.880225 0.474556i \(-0.157391\pi\)
0.880225 + 0.474556i \(0.157391\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −25.8997 −1.03765
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −7.26650 −0.290196
\(628\) 0 0
\(629\) −38.5831 −1.53841
\(630\) 0 0
\(631\) 16.9499 0.674764 0.337382 0.941368i \(-0.390459\pi\)
0.337382 + 0.941368i \(0.390459\pi\)
\(632\) 0 0
\(633\) −9.63325 −0.382887
\(634\) 0 0
\(635\) 4.94987 0.196430
\(636\) 0 0
\(637\) −4.63325 −0.183576
\(638\) 0 0
\(639\) −7.94987 −0.314492
\(640\) 0 0
\(641\) −9.58312 −0.378511 −0.189255 0.981928i \(-0.560607\pi\)
−0.189255 + 0.981928i \(0.560607\pi\)
\(642\) 0 0
\(643\) −18.2665 −0.720360 −0.360180 0.932883i \(-0.617285\pi\)
−0.360180 + 0.932883i \(0.617285\pi\)
\(644\) 0 0
\(645\) 2.63325 0.103684
\(646\) 0 0
\(647\) 17.5831 0.691264 0.345632 0.938370i \(-0.387665\pi\)
0.345632 + 0.938370i \(0.387665\pi\)
\(648\) 0 0
\(649\) 51.5831 2.02481
\(650\) 0 0
\(651\) −28.8997 −1.13267
\(652\) 0 0
\(653\) −50.2164 −1.96512 −0.982559 0.185950i \(-0.940464\pi\)
−0.982559 + 0.185950i \(0.940464\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.68338 0.0656747
\(658\) 0 0
\(659\) −37.5831 −1.46403 −0.732015 0.681288i \(-0.761420\pi\)
−0.732015 + 0.681288i \(0.761420\pi\)
\(660\) 0 0
\(661\) −34.6332 −1.34708 −0.673539 0.739152i \(-0.735226\pi\)
−0.673539 + 0.739152i \(0.735226\pi\)
\(662\) 0 0
\(663\) −7.68338 −0.298398
\(664\) 0 0
\(665\) −5.05013 −0.195835
\(666\) 0 0
\(667\) −0.683375 −0.0264604
\(668\) 0 0
\(669\) 3.26650 0.126290
\(670\) 0 0
\(671\) −15.8997 −0.613803
\(672\) 0 0
\(673\) −3.05013 −0.117574 −0.0587869 0.998271i \(-0.518723\pi\)
−0.0587869 + 0.998271i \(0.518723\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −26.0501 −1.00119 −0.500594 0.865682i \(-0.666885\pi\)
−0.500594 + 0.865682i \(0.666885\pi\)
\(678\) 0 0
\(679\) −36.0000 −1.38155
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) 0 0
\(683\) 13.6834 0.523580 0.261790 0.965125i \(-0.415687\pi\)
0.261790 + 0.965125i \(0.415687\pi\)
\(684\) 0 0
\(685\) 13.2665 0.506887
\(686\) 0 0
\(687\) −16.6332 −0.634598
\(688\) 0 0
\(689\) 23.0501 0.878140
\(690\) 0 0
\(691\) 10.6332 0.404508 0.202254 0.979333i \(-0.435173\pi\)
0.202254 + 0.979333i \(0.435173\pi\)
\(692\) 0 0
\(693\) 12.9499 0.491925
\(694\) 0 0
\(695\) 1.63325 0.0619527
\(696\) 0 0
\(697\) −11.0000 −0.416655
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) −10.3166 −0.389654 −0.194827 0.980838i \(-0.562414\pi\)
−0.194827 + 0.980838i \(0.562414\pi\)
\(702\) 0 0
\(703\) −19.5831 −0.738592
\(704\) 0 0
\(705\) 10.9499 0.412396
\(706\) 0 0
\(707\) −15.9499 −0.599857
\(708\) 0 0
\(709\) −46.8496 −1.75947 −0.879737 0.475460i \(-0.842282\pi\)
−0.879737 + 0.475460i \(0.842282\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −9.63325 −0.360768
\(714\) 0 0
\(715\) −10.0000 −0.373979
\(716\) 0 0
\(717\) −3.94987 −0.147511
\(718\) 0 0
\(719\) −37.8496 −1.41155 −0.705776 0.708435i \(-0.749402\pi\)
−0.705776 + 0.708435i \(0.749402\pi\)
\(720\) 0 0
\(721\) −54.0000 −2.01107
\(722\) 0 0
\(723\) −20.2164 −0.751855
\(724\) 0 0
\(725\) −0.683375 −0.0253799
\(726\) 0 0
\(727\) 10.3668 0.384481 0.192241 0.981348i \(-0.438425\pi\)
0.192241 + 0.981348i \(0.438425\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.73350 −0.323020
\(732\) 0 0
\(733\) −42.2665 −1.56115 −0.780574 0.625063i \(-0.785073\pi\)
−0.780574 + 0.625063i \(0.785073\pi\)
\(734\) 0 0
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) −10.2164 −0.376325
\(738\) 0 0
\(739\) −37.0000 −1.36107 −0.680534 0.732717i \(-0.738252\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(740\) 0 0
\(741\) −3.89975 −0.143261
\(742\) 0 0
\(743\) 47.7995 1.75359 0.876797 0.480861i \(-0.159676\pi\)
0.876797 + 0.480861i \(0.159676\pi\)
\(744\) 0 0
\(745\) −2.31662 −0.0848746
\(746\) 0 0
\(747\) 5.31662 0.194525
\(748\) 0 0
\(749\) 41.8496 1.52915
\(750\) 0 0
\(751\) 48.8496 1.78255 0.891274 0.453465i \(-0.149812\pi\)
0.891274 + 0.453465i \(0.149812\pi\)
\(752\) 0 0
\(753\) 3.26650 0.119038
\(754\) 0 0
\(755\) 1.36675 0.0497411
\(756\) 0 0
\(757\) 29.5330 1.07340 0.536698 0.843775i \(-0.319672\pi\)
0.536698 + 0.843775i \(0.319672\pi\)
\(758\) 0 0
\(759\) 4.31662 0.156684
\(760\) 0 0
\(761\) 38.5831 1.39864 0.699319 0.714810i \(-0.253487\pi\)
0.699319 + 0.714810i \(0.253487\pi\)
\(762\) 0 0
\(763\) 32.8496 1.18924
\(764\) 0 0
\(765\) 3.31662 0.119913
\(766\) 0 0
\(767\) 27.6834 0.999589
\(768\) 0 0
\(769\) −21.5831 −0.778307 −0.389154 0.921173i \(-0.627232\pi\)
−0.389154 + 0.921173i \(0.627232\pi\)
\(770\) 0 0
\(771\) 18.9499 0.682463
\(772\) 0 0
\(773\) 21.1662 0.761297 0.380649 0.924720i \(-0.375701\pi\)
0.380649 + 0.924720i \(0.375701\pi\)
\(774\) 0 0
\(775\) −9.63325 −0.346037
\(776\) 0 0
\(777\) 34.8997 1.25202
\(778\) 0 0
\(779\) −5.58312 −0.200036
\(780\) 0 0
\(781\) 34.3166 1.22795
\(782\) 0 0
\(783\) 0.683375 0.0244218
\(784\) 0 0
\(785\) −6.36675 −0.227239
\(786\) 0 0
\(787\) 53.5330 1.90825 0.954123 0.299416i \(-0.0967918\pi\)
0.954123 + 0.299416i \(0.0967918\pi\)
\(788\) 0 0
\(789\) 4.58312 0.163164
\(790\) 0 0
\(791\) −45.9499 −1.63379
\(792\) 0 0
\(793\) −8.53300 −0.303016
\(794\) 0 0
\(795\) −9.94987 −0.352886
\(796\) 0 0
\(797\) −41.8496 −1.48239 −0.741195 0.671290i \(-0.765741\pi\)
−0.741195 + 0.671290i \(0.765741\pi\)
\(798\) 0 0
\(799\) −36.3166 −1.28479
\(800\) 0 0
\(801\) 8.63325 0.305041
\(802\) 0 0
\(803\) −7.26650 −0.256429
\(804\) 0 0
\(805\) 3.00000 0.105736
\(806\) 0 0
\(807\) −10.5831 −0.372544
\(808\) 0 0
\(809\) −4.78363 −0.168183 −0.0840917 0.996458i \(-0.526799\pi\)
−0.0840917 + 0.996458i \(0.526799\pi\)
\(810\) 0 0
\(811\) −42.1662 −1.48066 −0.740329 0.672245i \(-0.765330\pi\)
−0.740329 + 0.672245i \(0.765330\pi\)
\(812\) 0 0
\(813\) 2.26650 0.0794896
\(814\) 0 0
\(815\) −3.26650 −0.114420
\(816\) 0 0
\(817\) −4.43275 −0.155082
\(818\) 0 0
\(819\) 6.94987 0.242848
\(820\) 0 0
\(821\) 29.2665 1.02141 0.510704 0.859757i \(-0.329385\pi\)
0.510704 + 0.859757i \(0.329385\pi\)
\(822\) 0 0
\(823\) 53.2665 1.85675 0.928377 0.371641i \(-0.121205\pi\)
0.928377 + 0.371641i \(0.121205\pi\)
\(824\) 0 0
\(825\) 4.31662 0.150286
\(826\) 0 0
\(827\) 18.5831 0.646199 0.323099 0.946365i \(-0.395275\pi\)
0.323099 + 0.946365i \(0.395275\pi\)
\(828\) 0 0
\(829\) 19.0000 0.659897 0.329949 0.943999i \(-0.392969\pi\)
0.329949 + 0.943999i \(0.392969\pi\)
\(830\) 0 0
\(831\) −18.0000 −0.624413
\(832\) 0 0
\(833\) −6.63325 −0.229828
\(834\) 0 0
\(835\) −19.5831 −0.677702
\(836\) 0 0
\(837\) 9.63325 0.332974
\(838\) 0 0
\(839\) 7.26650 0.250867 0.125434 0.992102i \(-0.459968\pi\)
0.125434 + 0.992102i \(0.459968\pi\)
\(840\) 0 0
\(841\) −28.5330 −0.983896
\(842\) 0 0
\(843\) 16.9499 0.583785
\(844\) 0 0
\(845\) 7.63325 0.262592
\(846\) 0 0
\(847\) −22.8997 −0.786845
\(848\) 0 0
\(849\) −7.63325 −0.261972
\(850\) 0 0
\(851\) 11.6332 0.398783
\(852\) 0 0
\(853\) 26.6332 0.911905 0.455953 0.890004i \(-0.349299\pi\)
0.455953 + 0.890004i \(0.349299\pi\)
\(854\) 0 0
\(855\) 1.68338 0.0575702
\(856\) 0 0
\(857\) 24.5330 0.838031 0.419016 0.907979i \(-0.362375\pi\)
0.419016 + 0.907979i \(0.362375\pi\)
\(858\) 0 0
\(859\) 20.3668 0.694905 0.347452 0.937698i \(-0.387047\pi\)
0.347452 + 0.937698i \(0.387047\pi\)
\(860\) 0 0
\(861\) 9.94987 0.339091
\(862\) 0 0
\(863\) −1.36675 −0.0465247 −0.0232624 0.999729i \(-0.507405\pi\)
−0.0232624 + 0.999729i \(0.507405\pi\)
\(864\) 0 0
\(865\) −4.00000 −0.136004
\(866\) 0 0
\(867\) 6.00000 0.203771
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −5.48287 −0.185780
\(872\) 0 0
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) −27.8997 −0.942108 −0.471054 0.882104i \(-0.656126\pi\)
−0.471054 + 0.882104i \(0.656126\pi\)
\(878\) 0 0
\(879\) −11.9499 −0.403059
\(880\) 0 0
\(881\) −8.41688 −0.283572 −0.141786 0.989897i \(-0.545284\pi\)
−0.141786 + 0.989897i \(0.545284\pi\)
\(882\) 0 0
\(883\) −11.6834 −0.393177 −0.196588 0.980486i \(-0.562986\pi\)
−0.196588 + 0.980486i \(0.562986\pi\)
\(884\) 0 0
\(885\) −11.9499 −0.401691
\(886\) 0 0
\(887\) 33.3668 1.12035 0.560173 0.828376i \(-0.310735\pi\)
0.560173 + 0.828376i \(0.310735\pi\)
\(888\) 0 0
\(889\) 14.8496 0.498040
\(890\) 0 0
\(891\) −4.31662 −0.144612
\(892\) 0 0
\(893\) −18.4327 −0.616828
\(894\) 0 0
\(895\) 9.36675 0.313096
\(896\) 0 0
\(897\) 2.31662 0.0773499
\(898\) 0 0
\(899\) 6.58312 0.219559
\(900\) 0 0
\(901\) 33.0000 1.09939
\(902\) 0 0
\(903\) 7.89975 0.262887
\(904\) 0 0
\(905\) −20.6332 −0.685872
\(906\) 0 0
\(907\) 7.63325 0.253458 0.126729 0.991937i \(-0.459552\pi\)
0.126729 + 0.991937i \(0.459552\pi\)
\(908\) 0 0
\(909\) 5.31662 0.176341
\(910\) 0 0
\(911\) 47.7995 1.58367 0.791834 0.610736i \(-0.209126\pi\)
0.791834 + 0.610736i \(0.209126\pi\)
\(912\) 0 0
\(913\) −22.9499 −0.759530
\(914\) 0 0
\(915\) 3.68338 0.121769
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −17.2665 −0.569569 −0.284785 0.958592i \(-0.591922\pi\)
−0.284785 + 0.958592i \(0.591922\pi\)
\(920\) 0 0
\(921\) 5.05013 0.166407
\(922\) 0 0
\(923\) 18.4169 0.606199
\(924\) 0 0
\(925\) 11.6332 0.382499
\(926\) 0 0
\(927\) 18.0000 0.591198
\(928\) 0 0
\(929\) −17.3166 −0.568140 −0.284070 0.958804i \(-0.591685\pi\)
−0.284070 + 0.958804i \(0.591685\pi\)
\(930\) 0 0
\(931\) −3.36675 −0.110341
\(932\) 0 0
\(933\) −29.2665 −0.958142
\(934\) 0 0
\(935\) −14.3166 −0.468204
\(936\) 0 0
\(937\) 40.0000 1.30674 0.653372 0.757037i \(-0.273354\pi\)
0.653372 + 0.757037i \(0.273354\pi\)
\(938\) 0 0
\(939\) −20.3668 −0.664644
\(940\) 0 0
\(941\) 4.41688 0.143986 0.0719930 0.997405i \(-0.477064\pi\)
0.0719930 + 0.997405i \(0.477064\pi\)
\(942\) 0 0
\(943\) 3.31662 0.108004
\(944\) 0 0
\(945\) −3.00000 −0.0975900
\(946\) 0 0
\(947\) 0.633250 0.0205778 0.0102889 0.999947i \(-0.496725\pi\)
0.0102889 + 0.999947i \(0.496725\pi\)
\(948\) 0 0
\(949\) −3.89975 −0.126591
\(950\) 0 0
\(951\) 9.68338 0.314005
\(952\) 0 0
\(953\) 0.532998 0.0172655 0.00863275 0.999963i \(-0.497252\pi\)
0.00863275 + 0.999963i \(0.497252\pi\)
\(954\) 0 0
\(955\) −19.5831 −0.633695
\(956\) 0 0
\(957\) −2.94987 −0.0953559
\(958\) 0 0
\(959\) 39.7995 1.28519
\(960\) 0 0
\(961\) 61.7995 1.99353
\(962\) 0 0
\(963\) −13.9499 −0.449528
\(964\) 0 0
\(965\) −12.6332 −0.406679
\(966\) 0 0
\(967\) −0.949874 −0.0305459 −0.0152730 0.999883i \(-0.504862\pi\)
−0.0152730 + 0.999883i \(0.504862\pi\)
\(968\) 0 0
\(969\) −5.58312 −0.179356
\(970\) 0 0
\(971\) 8.63325 0.277054 0.138527 0.990359i \(-0.455763\pi\)
0.138527 + 0.990359i \(0.455763\pi\)
\(972\) 0 0
\(973\) 4.89975 0.157079
\(974\) 0 0
\(975\) 2.31662 0.0741914
\(976\) 0 0
\(977\) 13.8496 0.443089 0.221544 0.975150i \(-0.428890\pi\)
0.221544 + 0.975150i \(0.428890\pi\)
\(978\) 0 0
\(979\) −37.2665 −1.19104
\(980\) 0 0
\(981\) −10.9499 −0.349603
\(982\) 0 0
\(983\) −18.4829 −0.589512 −0.294756 0.955573i \(-0.595238\pi\)
−0.294756 + 0.955573i \(0.595238\pi\)
\(984\) 0 0
\(985\) 27.8997 0.888960
\(986\) 0 0
\(987\) 32.8496 1.04562
\(988\) 0 0
\(989\) 2.63325 0.0837325
\(990\) 0 0
\(991\) 52.1662 1.65712 0.828558 0.559904i \(-0.189162\pi\)
0.828558 + 0.559904i \(0.189162\pi\)
\(992\) 0 0
\(993\) 31.5330 1.00067
\(994\) 0 0
\(995\) −1.36675 −0.0433289
\(996\) 0 0
\(997\) 24.6332 0.780143 0.390071 0.920785i \(-0.372450\pi\)
0.390071 + 0.920785i \(0.372450\pi\)
\(998\) 0 0
\(999\) −11.6332 −0.368060
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 2760.2.a.l.1.1 2
3.2 odd 2 8280.2.a.bc.1.2 2
4.3 odd 2 5520.2.a.bo.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.l.1.1 2 1.1 even 1 trivial
5520.2.a.bo.1.2 2 4.3 odd 2
8280.2.a.bc.1.2 2 3.2 odd 2