Properties

Label 4560.2.a.bs.1.3
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1772.1
Defining polynomial: \(x^{3} - x^{2} - 12 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.654334\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +4.11309 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +4.11309 q^{7} +1.00000 q^{9} +2.11309 q^{11} +4.80442 q^{13} -1.00000 q^{15} +4.00000 q^{17} -1.00000 q^{19} -4.11309 q^{21} +0.691333 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.11309 q^{29} +2.00000 q^{31} -2.11309 q^{33} +4.11309 q^{35} +3.42176 q^{37} -4.80442 q^{39} +5.42176 q^{41} +2.80442 q^{43} +1.00000 q^{45} -0.691333 q^{47} +9.91751 q^{49} -4.00000 q^{51} -2.69133 q^{53} +2.11309 q^{55} +1.00000 q^{57} -9.53485 q^{59} -2.91751 q^{61} +4.11309 q^{63} +4.80442 q^{65} -4.00000 q^{67} -0.691333 q^{69} -5.60885 q^{71} +6.00000 q^{73} -1.00000 q^{75} +8.69133 q^{77} -15.1437 q^{79} +1.00000 q^{81} -6.22618 q^{83} +4.00000 q^{85} -4.11309 q^{87} -1.42176 q^{89} +19.7610 q^{91} -2.00000 q^{93} -1.00000 q^{95} +15.4218 q^{97} +2.11309 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 3q^{5} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 3q^{5} + 3q^{9} - 6q^{11} + 4q^{13} - 3q^{15} + 12q^{17} - 3q^{19} + 4q^{23} + 3q^{25} - 3q^{27} + 6q^{31} + 6q^{33} - 4q^{37} - 4q^{39} + 2q^{41} - 2q^{43} + 3q^{45} - 4q^{47} + 7q^{49} - 12q^{51} - 10q^{53} - 6q^{55} + 3q^{57} - 2q^{59} + 14q^{61} + 4q^{65} - 12q^{67} - 4q^{69} + 4q^{71} + 18q^{73} - 3q^{75} + 28q^{77} + 2q^{79} + 3q^{81} + 6q^{83} + 12q^{85} + 10q^{89} + 8q^{91} - 6q^{93} - 3q^{95} + 32q^{97} - 6q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.11309 1.55460 0.777301 0.629129i \(-0.216588\pi\)
0.777301 + 0.629129i \(0.216588\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.11309 0.637121 0.318560 0.947903i \(-0.396801\pi\)
0.318560 + 0.947903i \(0.396801\pi\)
\(12\) 0 0
\(13\) 4.80442 1.33251 0.666254 0.745725i \(-0.267897\pi\)
0.666254 + 0.745725i \(0.267897\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −4.11309 −0.897550
\(22\) 0 0
\(23\) 0.691333 0.144153 0.0720764 0.997399i \(-0.477037\pi\)
0.0720764 + 0.997399i \(0.477037\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.11309 0.763782 0.381891 0.924207i \(-0.375273\pi\)
0.381891 + 0.924207i \(0.375273\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) −2.11309 −0.367842
\(34\) 0 0
\(35\) 4.11309 0.695239
\(36\) 0 0
\(37\) 3.42176 0.562533 0.281267 0.959630i \(-0.409245\pi\)
0.281267 + 0.959630i \(0.409245\pi\)
\(38\) 0 0
\(39\) −4.80442 −0.769323
\(40\) 0 0
\(41\) 5.42176 0.846736 0.423368 0.905958i \(-0.360848\pi\)
0.423368 + 0.905958i \(0.360848\pi\)
\(42\) 0 0
\(43\) 2.80442 0.427671 0.213835 0.976870i \(-0.431404\pi\)
0.213835 + 0.976870i \(0.431404\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −0.691333 −0.100841 −0.0504206 0.998728i \(-0.516056\pi\)
−0.0504206 + 0.998728i \(0.516056\pi\)
\(48\) 0 0
\(49\) 9.91751 1.41679
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −2.69133 −0.369683 −0.184842 0.982768i \(-0.559177\pi\)
−0.184842 + 0.982768i \(0.559177\pi\)
\(54\) 0 0
\(55\) 2.11309 0.284929
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −9.53485 −1.24133 −0.620666 0.784075i \(-0.713138\pi\)
−0.620666 + 0.784075i \(0.713138\pi\)
\(60\) 0 0
\(61\) −2.91751 −0.373549 −0.186775 0.982403i \(-0.559803\pi\)
−0.186775 + 0.982403i \(0.559803\pi\)
\(62\) 0 0
\(63\) 4.11309 0.518201
\(64\) 0 0
\(65\) 4.80442 0.595915
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −0.691333 −0.0832267
\(70\) 0 0
\(71\) −5.60885 −0.665648 −0.332824 0.942989i \(-0.608001\pi\)
−0.332824 + 0.942989i \(0.608001\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 8.69133 0.990469
\(78\) 0 0
\(79\) −15.1437 −1.70380 −0.851899 0.523705i \(-0.824549\pi\)
−0.851899 + 0.523705i \(0.824549\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.22618 −0.683412 −0.341706 0.939807i \(-0.611005\pi\)
−0.341706 + 0.939807i \(0.611005\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) −4.11309 −0.440970
\(88\) 0 0
\(89\) −1.42176 −0.150706 −0.0753530 0.997157i \(-0.524008\pi\)
−0.0753530 + 0.997157i \(0.524008\pi\)
\(90\) 0 0
\(91\) 19.7610 2.07152
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 15.4218 1.56584 0.782921 0.622121i \(-0.213729\pi\)
0.782921 + 0.622121i \(0.213729\pi\)
\(98\) 0 0
\(99\) 2.11309 0.212374
\(100\) 0 0
\(101\) −10.2262 −1.01754 −0.508772 0.860902i \(-0.669900\pi\)
−0.508772 + 0.860902i \(0.669900\pi\)
\(102\) 0 0
\(103\) −16.4524 −1.62110 −0.810550 0.585670i \(-0.800832\pi\)
−0.810550 + 0.585670i \(0.800832\pi\)
\(104\) 0 0
\(105\) −4.11309 −0.401397
\(106\) 0 0
\(107\) −16.4524 −1.59051 −0.795255 0.606275i \(-0.792663\pi\)
−0.795255 + 0.606275i \(0.792663\pi\)
\(108\) 0 0
\(109\) −7.53485 −0.721708 −0.360854 0.932622i \(-0.617515\pi\)
−0.360854 + 0.932622i \(0.617515\pi\)
\(110\) 0 0
\(111\) −3.42176 −0.324779
\(112\) 0 0
\(113\) −17.5348 −1.64954 −0.824770 0.565469i \(-0.808695\pi\)
−0.824770 + 0.565469i \(0.808695\pi\)
\(114\) 0 0
\(115\) 0.691333 0.0644671
\(116\) 0 0
\(117\) 4.80442 0.444169
\(118\) 0 0
\(119\) 16.4524 1.50819
\(120\) 0 0
\(121\) −6.53485 −0.594077
\(122\) 0 0
\(123\) −5.42176 −0.488863
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 9.83503 0.872718 0.436359 0.899773i \(-0.356268\pi\)
0.436359 + 0.899773i \(0.356268\pi\)
\(128\) 0 0
\(129\) −2.80442 −0.246916
\(130\) 0 0
\(131\) −6.33927 −0.553865 −0.276932 0.960889i \(-0.589318\pi\)
−0.276932 + 0.960889i \(0.589318\pi\)
\(132\) 0 0
\(133\) −4.11309 −0.356650
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 7.30867 0.624422 0.312211 0.950013i \(-0.398930\pi\)
0.312211 + 0.950013i \(0.398930\pi\)
\(138\) 0 0
\(139\) 8.22618 0.697736 0.348868 0.937172i \(-0.386566\pi\)
0.348868 + 0.937172i \(0.386566\pi\)
\(140\) 0 0
\(141\) 0.691333 0.0582207
\(142\) 0 0
\(143\) 10.1522 0.848968
\(144\) 0 0
\(145\) 4.11309 0.341574
\(146\) 0 0
\(147\) −9.91751 −0.817983
\(148\) 0 0
\(149\) −7.38267 −0.604812 −0.302406 0.953179i \(-0.597790\pi\)
−0.302406 + 0.953179i \(0.597790\pi\)
\(150\) 0 0
\(151\) 7.38267 0.600793 0.300396 0.953814i \(-0.402881\pi\)
0.300396 + 0.953814i \(0.402881\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 12.9175 1.03093 0.515465 0.856911i \(-0.327619\pi\)
0.515465 + 0.856911i \(0.327619\pi\)
\(158\) 0 0
\(159\) 2.69133 0.213437
\(160\) 0 0
\(161\) 2.84352 0.224100
\(162\) 0 0
\(163\) 5.42176 0.424665 0.212332 0.977197i \(-0.431894\pi\)
0.212332 + 0.977197i \(0.431894\pi\)
\(164\) 0 0
\(165\) −2.11309 −0.164504
\(166\) 0 0
\(167\) 12.9175 0.999587 0.499794 0.866145i \(-0.333409\pi\)
0.499794 + 0.866145i \(0.333409\pi\)
\(168\) 0 0
\(169\) 10.0825 0.775576
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −13.5348 −1.02904 −0.514518 0.857480i \(-0.672029\pi\)
−0.514518 + 0.857480i \(0.672029\pi\)
\(174\) 0 0
\(175\) 4.11309 0.310920
\(176\) 0 0
\(177\) 9.53485 0.716683
\(178\) 0 0
\(179\) 10.9175 0.816013 0.408007 0.912979i \(-0.366224\pi\)
0.408007 + 0.912979i \(0.366224\pi\)
\(180\) 0 0
\(181\) −17.3699 −1.29109 −0.645546 0.763721i \(-0.723370\pi\)
−0.645546 + 0.763721i \(0.723370\pi\)
\(182\) 0 0
\(183\) 2.91751 0.215669
\(184\) 0 0
\(185\) 3.42176 0.251573
\(186\) 0 0
\(187\) 8.45236 0.618098
\(188\) 0 0
\(189\) −4.11309 −0.299183
\(190\) 0 0
\(191\) −0.730425 −0.0528517 −0.0264258 0.999651i \(-0.508413\pi\)
−0.0264258 + 0.999651i \(0.508413\pi\)
\(192\) 0 0
\(193\) 25.4830 1.83430 0.917152 0.398537i \(-0.130482\pi\)
0.917152 + 0.398537i \(0.130482\pi\)
\(194\) 0 0
\(195\) −4.80442 −0.344052
\(196\) 0 0
\(197\) −21.7610 −1.55041 −0.775205 0.631710i \(-0.782353\pi\)
−0.775205 + 0.631710i \(0.782353\pi\)
\(198\) 0 0
\(199\) −1.60885 −0.114048 −0.0570241 0.998373i \(-0.518161\pi\)
−0.0570241 + 0.998373i \(0.518161\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 16.9175 1.18738
\(204\) 0 0
\(205\) 5.42176 0.378672
\(206\) 0 0
\(207\) 0.691333 0.0480510
\(208\) 0 0
\(209\) −2.11309 −0.146166
\(210\) 0 0
\(211\) −9.83503 −0.677071 −0.338536 0.940954i \(-0.609932\pi\)
−0.338536 + 0.940954i \(0.609932\pi\)
\(212\) 0 0
\(213\) 5.60885 0.384312
\(214\) 0 0
\(215\) 2.80442 0.191260
\(216\) 0 0
\(217\) 8.22618 0.558430
\(218\) 0 0
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 19.2177 1.29272
\(222\) 0 0
\(223\) −13.8350 −0.926462 −0.463231 0.886238i \(-0.653310\pi\)
−0.463231 + 0.886238i \(0.653310\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −10.3002 −0.683647 −0.341823 0.939764i \(-0.611044\pi\)
−0.341823 + 0.939764i \(0.611044\pi\)
\(228\) 0 0
\(229\) 9.08249 0.600188 0.300094 0.953910i \(-0.402982\pi\)
0.300094 + 0.953910i \(0.402982\pi\)
\(230\) 0 0
\(231\) −8.69133 −0.571848
\(232\) 0 0
\(233\) −22.5264 −1.47575 −0.737875 0.674937i \(-0.764171\pi\)
−0.737875 + 0.674937i \(0.764171\pi\)
\(234\) 0 0
\(235\) −0.691333 −0.0450976
\(236\) 0 0
\(237\) 15.1437 0.983689
\(238\) 0 0
\(239\) 23.7219 1.53444 0.767222 0.641381i \(-0.221638\pi\)
0.767222 + 0.641381i \(0.221638\pi\)
\(240\) 0 0
\(241\) 21.2177 1.36675 0.683376 0.730067i \(-0.260511\pi\)
0.683376 + 0.730067i \(0.260511\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 9.91751 0.633607
\(246\) 0 0
\(247\) −4.80442 −0.305698
\(248\) 0 0
\(249\) 6.22618 0.394568
\(250\) 0 0
\(251\) 28.1743 1.77835 0.889173 0.457571i \(-0.151280\pi\)
0.889173 + 0.457571i \(0.151280\pi\)
\(252\) 0 0
\(253\) 1.46085 0.0918428
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 0 0
\(257\) 12.3002 0.767264 0.383632 0.923486i \(-0.374673\pi\)
0.383632 + 0.923486i \(0.374673\pi\)
\(258\) 0 0
\(259\) 14.0740 0.874516
\(260\) 0 0
\(261\) 4.11309 0.254594
\(262\) 0 0
\(263\) −12.9175 −0.796528 −0.398264 0.917271i \(-0.630387\pi\)
−0.398264 + 0.917271i \(0.630387\pi\)
\(264\) 0 0
\(265\) −2.69133 −0.165327
\(266\) 0 0
\(267\) 1.42176 0.0870102
\(268\) 0 0
\(269\) −6.73042 −0.410361 −0.205181 0.978724i \(-0.565778\pi\)
−0.205181 + 0.978724i \(0.565778\pi\)
\(270\) 0 0
\(271\) 7.77382 0.472226 0.236113 0.971726i \(-0.424126\pi\)
0.236113 + 0.971726i \(0.424126\pi\)
\(272\) 0 0
\(273\) −19.7610 −1.19599
\(274\) 0 0
\(275\) 2.11309 0.127424
\(276\) 0 0
\(277\) 15.3087 0.919809 0.459904 0.887968i \(-0.347884\pi\)
0.459904 + 0.887968i \(0.347884\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 13.4218 0.800675 0.400337 0.916368i \(-0.368893\pi\)
0.400337 + 0.916368i \(0.368893\pi\)
\(282\) 0 0
\(283\) −15.0306 −0.893477 −0.446738 0.894665i \(-0.647415\pi\)
−0.446738 + 0.894665i \(0.647415\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 22.3002 1.31634
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −15.4218 −0.904039
\(292\) 0 0
\(293\) −9.08249 −0.530604 −0.265302 0.964165i \(-0.585472\pi\)
−0.265302 + 0.964165i \(0.585472\pi\)
\(294\) 0 0
\(295\) −9.53485 −0.555140
\(296\) 0 0
\(297\) −2.11309 −0.122614
\(298\) 0 0
\(299\) 3.32146 0.192085
\(300\) 0 0
\(301\) 11.5348 0.664858
\(302\) 0 0
\(303\) 10.2262 0.587479
\(304\) 0 0
\(305\) −2.91751 −0.167056
\(306\) 0 0
\(307\) −32.2262 −1.83925 −0.919623 0.392803i \(-0.871505\pi\)
−0.919623 + 0.392803i \(0.871505\pi\)
\(308\) 0 0
\(309\) 16.4524 0.935942
\(310\) 0 0
\(311\) 16.7304 0.948695 0.474348 0.880338i \(-0.342684\pi\)
0.474348 + 0.880338i \(0.342684\pi\)
\(312\) 0 0
\(313\) −14.6785 −0.829680 −0.414840 0.909894i \(-0.636163\pi\)
−0.414840 + 0.909894i \(0.636163\pi\)
\(314\) 0 0
\(315\) 4.11309 0.231746
\(316\) 0 0
\(317\) −17.9090 −1.00587 −0.502936 0.864324i \(-0.667747\pi\)
−0.502936 + 0.864324i \(0.667747\pi\)
\(318\) 0 0
\(319\) 8.69133 0.486621
\(320\) 0 0
\(321\) 16.4524 0.918281
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) 4.80442 0.266501
\(326\) 0 0
\(327\) 7.53485 0.416678
\(328\) 0 0
\(329\) −2.84352 −0.156768
\(330\) 0 0
\(331\) 1.92600 0.105863 0.0529313 0.998598i \(-0.483144\pi\)
0.0529313 + 0.998598i \(0.483144\pi\)
\(332\) 0 0
\(333\) 3.42176 0.187511
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −17.2568 −0.940037 −0.470019 0.882657i \(-0.655753\pi\)
−0.470019 + 0.882657i \(0.655753\pi\)
\(338\) 0 0
\(339\) 17.5348 0.952362
\(340\) 0 0
\(341\) 4.22618 0.228861
\(342\) 0 0
\(343\) 12.0000 0.647939
\(344\) 0 0
\(345\) −0.691333 −0.0372201
\(346\) 0 0
\(347\) 22.6785 1.21745 0.608724 0.793382i \(-0.291682\pi\)
0.608724 + 0.793382i \(0.291682\pi\)
\(348\) 0 0
\(349\) 4.61733 0.247160 0.123580 0.992335i \(-0.460562\pi\)
0.123580 + 0.992335i \(0.460562\pi\)
\(350\) 0 0
\(351\) −4.80442 −0.256441
\(352\) 0 0
\(353\) −23.7610 −1.26467 −0.632336 0.774694i \(-0.717904\pi\)
−0.632336 + 0.774694i \(0.717904\pi\)
\(354\) 0 0
\(355\) −5.60885 −0.297687
\(356\) 0 0
\(357\) −16.4524 −0.870751
\(358\) 0 0
\(359\) 12.5042 0.659949 0.329974 0.943990i \(-0.392960\pi\)
0.329974 + 0.943990i \(0.392960\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 6.53485 0.342991
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 6.73042 0.351325 0.175663 0.984450i \(-0.443793\pi\)
0.175663 + 0.984450i \(0.443793\pi\)
\(368\) 0 0
\(369\) 5.42176 0.282245
\(370\) 0 0
\(371\) −11.0697 −0.574710
\(372\) 0 0
\(373\) 8.80442 0.455876 0.227938 0.973676i \(-0.426802\pi\)
0.227938 + 0.973676i \(0.426802\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 19.7610 1.01774
\(378\) 0 0
\(379\) 31.7610 1.63145 0.815727 0.578437i \(-0.196337\pi\)
0.815727 + 0.578437i \(0.196337\pi\)
\(380\) 0 0
\(381\) −9.83503 −0.503864
\(382\) 0 0
\(383\) −23.0697 −1.17881 −0.589403 0.807839i \(-0.700637\pi\)
−0.589403 + 0.807839i \(0.700637\pi\)
\(384\) 0 0
\(385\) 8.69133 0.442951
\(386\) 0 0
\(387\) 2.80442 0.142557
\(388\) 0 0
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 2.76533 0.139849
\(392\) 0 0
\(393\) 6.33927 0.319774
\(394\) 0 0
\(395\) −15.1437 −0.761962
\(396\) 0 0
\(397\) 38.5264 1.93358 0.966791 0.255567i \(-0.0822622\pi\)
0.966791 + 0.255567i \(0.0822622\pi\)
\(398\) 0 0
\(399\) 4.11309 0.205912
\(400\) 0 0
\(401\) 22.1003 1.10364 0.551818 0.833964i \(-0.313934\pi\)
0.551818 + 0.833964i \(0.313934\pi\)
\(402\) 0 0
\(403\) 9.60885 0.478651
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 7.23048 0.358402
\(408\) 0 0
\(409\) 30.2262 1.49459 0.747294 0.664493i \(-0.231353\pi\)
0.747294 + 0.664493i \(0.231353\pi\)
\(410\) 0 0
\(411\) −7.30867 −0.360510
\(412\) 0 0
\(413\) −39.2177 −1.92978
\(414\) 0 0
\(415\) −6.22618 −0.305631
\(416\) 0 0
\(417\) −8.22618 −0.402838
\(418\) 0 0
\(419\) −11.7219 −0.572654 −0.286327 0.958132i \(-0.592434\pi\)
−0.286327 + 0.958132i \(0.592434\pi\)
\(420\) 0 0
\(421\) −30.9787 −1.50981 −0.754905 0.655834i \(-0.772317\pi\)
−0.754905 + 0.655834i \(0.772317\pi\)
\(422\) 0 0
\(423\) −0.691333 −0.0336138
\(424\) 0 0
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) −12.0000 −0.580721
\(428\) 0 0
\(429\) −10.1522 −0.490152
\(430\) 0 0
\(431\) 19.2959 0.929450 0.464725 0.885455i \(-0.346153\pi\)
0.464725 + 0.885455i \(0.346153\pi\)
\(432\) 0 0
\(433\) −23.4218 −1.12558 −0.562789 0.826601i \(-0.690272\pi\)
−0.562789 + 0.826601i \(0.690272\pi\)
\(434\) 0 0
\(435\) −4.11309 −0.197208
\(436\) 0 0
\(437\) −0.691333 −0.0330709
\(438\) 0 0
\(439\) 8.85630 0.422688 0.211344 0.977412i \(-0.432216\pi\)
0.211344 + 0.977412i \(0.432216\pi\)
\(440\) 0 0
\(441\) 9.91751 0.472263
\(442\) 0 0
\(443\) 7.23467 0.343729 0.171865 0.985121i \(-0.445021\pi\)
0.171865 + 0.985121i \(0.445021\pi\)
\(444\) 0 0
\(445\) −1.42176 −0.0673978
\(446\) 0 0
\(447\) 7.38267 0.349188
\(448\) 0 0
\(449\) −1.19558 −0.0564227 −0.0282114 0.999602i \(-0.508981\pi\)
−0.0282114 + 0.999602i \(0.508981\pi\)
\(450\) 0 0
\(451\) 11.4567 0.539473
\(452\) 0 0
\(453\) −7.38267 −0.346868
\(454\) 0 0
\(455\) 19.7610 0.926411
\(456\) 0 0
\(457\) 16.9915 0.794829 0.397415 0.917639i \(-0.369907\pi\)
0.397415 + 0.917639i \(0.369907\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −16.6173 −0.773946 −0.386973 0.922091i \(-0.626479\pi\)
−0.386973 + 0.922091i \(0.626479\pi\)
\(462\) 0 0
\(463\) 12.1131 0.562943 0.281472 0.959570i \(-0.409178\pi\)
0.281472 + 0.959570i \(0.409178\pi\)
\(464\) 0 0
\(465\) −2.00000 −0.0927478
\(466\) 0 0
\(467\) −37.6701 −1.74316 −0.871581 0.490251i \(-0.836905\pi\)
−0.871581 + 0.490251i \(0.836905\pi\)
\(468\) 0 0
\(469\) −16.4524 −0.759700
\(470\) 0 0
\(471\) −12.9175 −0.595208
\(472\) 0 0
\(473\) 5.92600 0.272478
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −2.69133 −0.123228
\(478\) 0 0
\(479\) 34.1131 1.55867 0.779333 0.626609i \(-0.215558\pi\)
0.779333 + 0.626609i \(0.215558\pi\)
\(480\) 0 0
\(481\) 16.4396 0.749580
\(482\) 0 0
\(483\) −2.84352 −0.129384
\(484\) 0 0
\(485\) 15.4218 0.700266
\(486\) 0 0
\(487\) 38.5136 1.74522 0.872608 0.488421i \(-0.162427\pi\)
0.872608 + 0.488421i \(0.162427\pi\)
\(488\) 0 0
\(489\) −5.42176 −0.245180
\(490\) 0 0
\(491\) −3.49576 −0.157761 −0.0788806 0.996884i \(-0.525135\pi\)
−0.0788806 + 0.996884i \(0.525135\pi\)
\(492\) 0 0
\(493\) 16.4524 0.740977
\(494\) 0 0
\(495\) 2.11309 0.0949764
\(496\) 0 0
\(497\) −23.0697 −1.03482
\(498\) 0 0
\(499\) 16.2262 0.726384 0.363192 0.931714i \(-0.381687\pi\)
0.363192 + 0.931714i \(0.381687\pi\)
\(500\) 0 0
\(501\) −12.9175 −0.577112
\(502\) 0 0
\(503\) 33.3699 1.48789 0.743945 0.668241i \(-0.232953\pi\)
0.743945 + 0.668241i \(0.232953\pi\)
\(504\) 0 0
\(505\) −10.2262 −0.455059
\(506\) 0 0
\(507\) −10.0825 −0.447779
\(508\) 0 0
\(509\) 19.5570 0.866847 0.433424 0.901190i \(-0.357305\pi\)
0.433424 + 0.901190i \(0.357305\pi\)
\(510\) 0 0
\(511\) 24.6785 1.09171
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) −16.4524 −0.724978
\(516\) 0 0
\(517\) −1.46085 −0.0642481
\(518\) 0 0
\(519\) 13.5348 0.594114
\(520\) 0 0
\(521\) 4.63945 0.203258 0.101629 0.994822i \(-0.467595\pi\)
0.101629 + 0.994822i \(0.467595\pi\)
\(522\) 0 0
\(523\) 35.8962 1.56963 0.784816 0.619728i \(-0.212757\pi\)
0.784816 + 0.619728i \(0.212757\pi\)
\(524\) 0 0
\(525\) −4.11309 −0.179510
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −22.5221 −0.979220
\(530\) 0 0
\(531\) −9.53485 −0.413777
\(532\) 0 0
\(533\) 26.0484 1.12828
\(534\) 0 0
\(535\) −16.4524 −0.711298
\(536\) 0 0
\(537\) −10.9175 −0.471126
\(538\) 0 0
\(539\) 20.9566 0.902665
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) 17.3699 0.745413
\(544\) 0 0
\(545\) −7.53485 −0.322757
\(546\) 0 0
\(547\) −24.2262 −1.03584 −0.517918 0.855430i \(-0.673293\pi\)
−0.517918 + 0.855430i \(0.673293\pi\)
\(548\) 0 0
\(549\) −2.91751 −0.124516
\(550\) 0 0
\(551\) −4.11309 −0.175224
\(552\) 0 0
\(553\) −62.2874 −2.64873
\(554\) 0 0
\(555\) −3.42176 −0.145246
\(556\) 0 0
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) 13.4736 0.569874
\(560\) 0 0
\(561\) −8.45236 −0.356859
\(562\) 0 0
\(563\) −44.9175 −1.89305 −0.946524 0.322634i \(-0.895432\pi\)
−0.946524 + 0.322634i \(0.895432\pi\)
\(564\) 0 0
\(565\) −17.5348 −0.737697
\(566\) 0 0
\(567\) 4.11309 0.172734
\(568\) 0 0
\(569\) −17.1956 −0.720876 −0.360438 0.932783i \(-0.617373\pi\)
−0.360438 + 0.932783i \(0.617373\pi\)
\(570\) 0 0
\(571\) 1.15648 0.0483974 0.0241987 0.999707i \(-0.492297\pi\)
0.0241987 + 0.999707i \(0.492297\pi\)
\(572\) 0 0
\(573\) 0.730425 0.0305139
\(574\) 0 0
\(575\) 0.691333 0.0288306
\(576\) 0 0
\(577\) −36.5136 −1.52008 −0.760040 0.649876i \(-0.774821\pi\)
−0.760040 + 0.649876i \(0.774821\pi\)
\(578\) 0 0
\(579\) −25.4830 −1.05904
\(580\) 0 0
\(581\) −25.6088 −1.06243
\(582\) 0 0
\(583\) −5.68703 −0.235533
\(584\) 0 0
\(585\) 4.80442 0.198638
\(586\) 0 0
\(587\) −4.61733 −0.190578 −0.0952889 0.995450i \(-0.530377\pi\)
−0.0952889 + 0.995450i \(0.530377\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 21.7610 0.895129
\(592\) 0 0
\(593\) 25.5961 1.05110 0.525552 0.850761i \(-0.323859\pi\)
0.525552 + 0.850761i \(0.323859\pi\)
\(594\) 0 0
\(595\) 16.4524 0.674481
\(596\) 0 0
\(597\) 1.60885 0.0658457
\(598\) 0 0
\(599\) −45.3571 −1.85324 −0.926620 0.375999i \(-0.877300\pi\)
−0.926620 + 0.375999i \(0.877300\pi\)
\(600\) 0 0
\(601\) 15.6088 0.636698 0.318349 0.947974i \(-0.396872\pi\)
0.318349 + 0.947974i \(0.396872\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) −6.53485 −0.265679
\(606\) 0 0
\(607\) −39.8962 −1.61934 −0.809669 0.586887i \(-0.800353\pi\)
−0.809669 + 0.586887i \(0.800353\pi\)
\(608\) 0 0
\(609\) −16.9175 −0.685532
\(610\) 0 0
\(611\) −3.32146 −0.134372
\(612\) 0 0
\(613\) 11.6828 0.471866 0.235933 0.971769i \(-0.424185\pi\)
0.235933 + 0.971769i \(0.424185\pi\)
\(614\) 0 0
\(615\) −5.42176 −0.218626
\(616\) 0 0
\(617\) −26.2874 −1.05829 −0.529145 0.848531i \(-0.677487\pi\)
−0.529145 + 0.848531i \(0.677487\pi\)
\(618\) 0 0
\(619\) −23.2959 −0.936340 −0.468170 0.883638i \(-0.655087\pi\)
−0.468170 + 0.883638i \(0.655087\pi\)
\(620\) 0 0
\(621\) −0.691333 −0.0277422
\(622\) 0 0
\(623\) −5.84782 −0.234288
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 2.11309 0.0843887
\(628\) 0 0
\(629\) 13.6870 0.545738
\(630\) 0 0
\(631\) 21.2347 0.845339 0.422669 0.906284i \(-0.361093\pi\)
0.422669 + 0.906284i \(0.361093\pi\)
\(632\) 0 0
\(633\) 9.83503 0.390907
\(634\) 0 0
\(635\) 9.83503 0.390291
\(636\) 0 0
\(637\) 47.6479 1.88788
\(638\) 0 0
\(639\) −5.60885 −0.221883
\(640\) 0 0
\(641\) −39.7091 −1.56842 −0.784209 0.620497i \(-0.786931\pi\)
−0.784209 + 0.620497i \(0.786931\pi\)
\(642\) 0 0
\(643\) −26.8044 −1.05706 −0.528532 0.848914i \(-0.677257\pi\)
−0.528532 + 0.848914i \(0.677257\pi\)
\(644\) 0 0
\(645\) −2.80442 −0.110424
\(646\) 0 0
\(647\) −10.3002 −0.404942 −0.202471 0.979288i \(-0.564897\pi\)
−0.202471 + 0.979288i \(0.564897\pi\)
\(648\) 0 0
\(649\) −20.1480 −0.790878
\(650\) 0 0
\(651\) −8.22618 −0.322409
\(652\) 0 0
\(653\) −16.8563 −0.659638 −0.329819 0.944044i \(-0.606988\pi\)
−0.329819 + 0.944044i \(0.606988\pi\)
\(654\) 0 0
\(655\) −6.33927 −0.247696
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 1.90903 0.0743651 0.0371826 0.999308i \(-0.488162\pi\)
0.0371826 + 0.999308i \(0.488162\pi\)
\(660\) 0 0
\(661\) 2.30018 0.0894666 0.0447333 0.998999i \(-0.485756\pi\)
0.0447333 + 0.998999i \(0.485756\pi\)
\(662\) 0 0
\(663\) −19.2177 −0.746353
\(664\) 0 0
\(665\) −4.11309 −0.159499
\(666\) 0 0
\(667\) 2.84352 0.110101
\(668\) 0 0
\(669\) 13.8350 0.534893
\(670\) 0 0
\(671\) −6.16497 −0.237996
\(672\) 0 0
\(673\) −32.4745 −1.25180 −0.625900 0.779904i \(-0.715268\pi\)
−0.625900 + 0.779904i \(0.715268\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −8.07400 −0.310309 −0.155154 0.987890i \(-0.549588\pi\)
−0.155154 + 0.987890i \(0.549588\pi\)
\(678\) 0 0
\(679\) 63.4311 2.43426
\(680\) 0 0
\(681\) 10.3002 0.394704
\(682\) 0 0
\(683\) −18.6173 −0.712372 −0.356186 0.934415i \(-0.615923\pi\)
−0.356186 + 0.934415i \(0.615923\pi\)
\(684\) 0 0
\(685\) 7.30867 0.279250
\(686\) 0 0
\(687\) −9.08249 −0.346518
\(688\) 0 0
\(689\) −12.9303 −0.492605
\(690\) 0 0
\(691\) 25.6088 0.974206 0.487103 0.873344i \(-0.338054\pi\)
0.487103 + 0.873344i \(0.338054\pi\)
\(692\) 0 0
\(693\) 8.69133 0.330156
\(694\) 0 0
\(695\) 8.22618 0.312037
\(696\) 0 0
\(697\) 21.6870 0.821455
\(698\) 0 0
\(699\) 22.5264 0.852025
\(700\) 0 0
\(701\) 0.765332 0.0289062 0.0144531 0.999896i \(-0.495399\pi\)
0.0144531 + 0.999896i \(0.495399\pi\)
\(702\) 0 0
\(703\) −3.42176 −0.129054
\(704\) 0 0
\(705\) 0.691333 0.0260371
\(706\) 0 0
\(707\) −42.0612 −1.58187
\(708\) 0 0
\(709\) 35.8222 1.34533 0.672666 0.739946i \(-0.265149\pi\)
0.672666 + 0.739946i \(0.265149\pi\)
\(710\) 0 0
\(711\) −15.1437 −0.567933
\(712\) 0 0
\(713\) 1.38267 0.0517812
\(714\) 0 0
\(715\) 10.1522 0.379670
\(716\) 0 0
\(717\) −23.7219 −0.885912
\(718\) 0 0
\(719\) 22.7916 0.849985 0.424992 0.905197i \(-0.360277\pi\)
0.424992 + 0.905197i \(0.360277\pi\)
\(720\) 0 0
\(721\) −67.6701 −2.52016
\(722\) 0 0
\(723\) −21.2177 −0.789095
\(724\) 0 0
\(725\) 4.11309 0.152756
\(726\) 0 0
\(727\) −35.6351 −1.32163 −0.660817 0.750547i \(-0.729790\pi\)
−0.660817 + 0.750547i \(0.729790\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 11.2177 0.414901
\(732\) 0 0
\(733\) 20.6913 0.764252 0.382126 0.924110i \(-0.375192\pi\)
0.382126 + 0.924110i \(0.375192\pi\)
\(734\) 0 0
\(735\) −9.91751 −0.365813
\(736\) 0 0
\(737\) −8.45236 −0.311347
\(738\) 0 0
\(739\) 23.2177 0.854077 0.427038 0.904234i \(-0.359557\pi\)
0.427038 + 0.904234i \(0.359557\pi\)
\(740\) 0 0
\(741\) 4.80442 0.176495
\(742\) 0 0
\(743\) −7.06970 −0.259362 −0.129681 0.991556i \(-0.541395\pi\)
−0.129681 + 0.991556i \(0.541395\pi\)
\(744\) 0 0
\(745\) −7.38267 −0.270480
\(746\) 0 0
\(747\) −6.22618 −0.227804
\(748\) 0 0
\(749\) −67.6701 −2.47261
\(750\) 0 0
\(751\) 8.61733 0.314451 0.157225 0.987563i \(-0.449745\pi\)
0.157225 + 0.987563i \(0.449745\pi\)
\(752\) 0 0
\(753\) −28.1743 −1.02673
\(754\) 0 0
\(755\) 7.38267 0.268683
\(756\) 0 0
\(757\) 47.9872 1.74412 0.872062 0.489395i \(-0.162782\pi\)
0.872062 + 0.489395i \(0.162782\pi\)
\(758\) 0 0
\(759\) −1.46085 −0.0530255
\(760\) 0 0
\(761\) −23.0527 −0.835661 −0.417830 0.908525i \(-0.637209\pi\)
−0.417830 + 0.908525i \(0.637209\pi\)
\(762\) 0 0
\(763\) −30.9915 −1.12197
\(764\) 0 0
\(765\) 4.00000 0.144620
\(766\) 0 0
\(767\) −45.8094 −1.65408
\(768\) 0 0
\(769\) 37.0697 1.33677 0.668384 0.743817i \(-0.266986\pi\)
0.668384 + 0.743817i \(0.266986\pi\)
\(770\) 0 0
\(771\) −12.3002 −0.442980
\(772\) 0 0
\(773\) −25.3087 −0.910289 −0.455145 0.890417i \(-0.650412\pi\)
−0.455145 + 0.890417i \(0.650412\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) −14.0740 −0.504902
\(778\) 0 0
\(779\) −5.42176 −0.194255
\(780\) 0 0
\(781\) −11.8520 −0.424098
\(782\) 0 0
\(783\) −4.11309 −0.146990
\(784\) 0 0
\(785\) 12.9175 0.461046
\(786\) 0 0
\(787\) 10.9915 0.391805 0.195903 0.980623i \(-0.437236\pi\)
0.195903 + 0.980623i \(0.437236\pi\)
\(788\) 0 0
\(789\) 12.9175 0.459876
\(790\) 0 0
\(791\) −72.1224 −2.56438
\(792\) 0 0
\(793\) −14.0170 −0.497757
\(794\) 0 0
\(795\) 2.69133 0.0954517
\(796\) 0 0
\(797\) −7.69982 −0.272742 −0.136371 0.990658i \(-0.543544\pi\)
−0.136371 + 0.990658i \(0.543544\pi\)
\(798\) 0 0
\(799\) −2.76533 −0.0978304
\(800\) 0 0
\(801\) −1.42176 −0.0502353
\(802\) 0 0
\(803\) 12.6785 0.447416
\(804\) 0 0
\(805\) 2.84352 0.100221
\(806\) 0 0
\(807\) 6.73042 0.236922
\(808\) 0 0
\(809\) 22.4524 0.789383 0.394692 0.918814i \(-0.370851\pi\)
0.394692 + 0.918814i \(0.370851\pi\)
\(810\) 0 0
\(811\) −53.5051 −1.87882 −0.939409 0.342799i \(-0.888625\pi\)
−0.939409 + 0.342799i \(0.888625\pi\)
\(812\) 0 0
\(813\) −7.77382 −0.272640
\(814\) 0 0
\(815\) 5.42176 0.189916
\(816\) 0 0
\(817\) −2.80442 −0.0981144
\(818\) 0 0
\(819\) 19.7610 0.690506
\(820\) 0 0
\(821\) 1.62582 0.0567415 0.0283708 0.999597i \(-0.490968\pi\)
0.0283708 + 0.999597i \(0.490968\pi\)
\(822\) 0 0
\(823\) 29.4958 1.02816 0.514079 0.857743i \(-0.328134\pi\)
0.514079 + 0.857743i \(0.328134\pi\)
\(824\) 0 0
\(825\) −2.11309 −0.0735684
\(826\) 0 0
\(827\) 4.91751 0.170999 0.0854994 0.996338i \(-0.472751\pi\)
0.0854994 + 0.996338i \(0.472751\pi\)
\(828\) 0 0
\(829\) −16.9175 −0.587570 −0.293785 0.955872i \(-0.594915\pi\)
−0.293785 + 0.955872i \(0.594915\pi\)
\(830\) 0 0
\(831\) −15.3087 −0.531052
\(832\) 0 0
\(833\) 39.6701 1.37449
\(834\) 0 0
\(835\) 12.9175 0.447029
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) 10.8435 0.374360 0.187180 0.982326i \(-0.440065\pi\)
0.187180 + 0.982326i \(0.440065\pi\)
\(840\) 0 0
\(841\) −12.0825 −0.416637
\(842\) 0 0
\(843\) −13.4218 −0.462270
\(844\) 0 0
\(845\) 10.0825 0.346848
\(846\) 0 0
\(847\) −26.8784 −0.923554
\(848\) 0 0
\(849\) 15.0306 0.515849
\(850\) 0 0
\(851\) 2.36557 0.0810908
\(852\) 0 0
\(853\) 28.9175 0.990117 0.495058 0.868860i \(-0.335147\pi\)
0.495058 + 0.868860i \(0.335147\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) 27.2917 0.932266 0.466133 0.884715i \(-0.345647\pi\)
0.466133 + 0.884715i \(0.345647\pi\)
\(858\) 0 0
\(859\) 10.6173 0.362259 0.181129 0.983459i \(-0.442025\pi\)
0.181129 + 0.983459i \(0.442025\pi\)
\(860\) 0 0
\(861\) −22.3002 −0.759988
\(862\) 0 0
\(863\) 6.61733 0.225257 0.112628 0.993637i \(-0.464073\pi\)
0.112628 + 0.993637i \(0.464073\pi\)
\(864\) 0 0
\(865\) −13.5348 −0.460199
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) −19.2177 −0.651167
\(872\) 0 0
\(873\) 15.4218 0.521947
\(874\) 0 0
\(875\) 4.11309 0.139048
\(876\) 0 0
\(877\) 22.4133 0.756842 0.378421 0.925634i \(-0.376467\pi\)
0.378421 + 0.925634i \(0.376467\pi\)
\(878\) 0 0
\(879\) 9.08249 0.306345
\(880\) 0 0
\(881\) −12.5392 −0.422455 −0.211227 0.977437i \(-0.567746\pi\)
−0.211227 + 0.977437i \(0.567746\pi\)
\(882\) 0 0
\(883\) −7.96091 −0.267906 −0.133953 0.990988i \(-0.542767\pi\)
−0.133953 + 0.990988i \(0.542767\pi\)
\(884\) 0 0
\(885\) 9.53485 0.320510
\(886\) 0 0
\(887\) 5.68703 0.190952 0.0954759 0.995432i \(-0.469563\pi\)
0.0954759 + 0.995432i \(0.469563\pi\)
\(888\) 0 0
\(889\) 40.4524 1.35673
\(890\) 0 0
\(891\) 2.11309 0.0707912
\(892\) 0 0
\(893\) 0.691333 0.0231346
\(894\) 0 0
\(895\) 10.9175 0.364932
\(896\) 0 0
\(897\) −3.32146 −0.110900
\(898\) 0 0
\(899\) 8.22618 0.274359
\(900\) 0 0
\(901\) −10.7653 −0.358645
\(902\) 0 0
\(903\) −11.5348 −0.383856
\(904\) 0 0
\(905\) −17.3699 −0.577394
\(906\) 0 0
\(907\) −17.6088 −0.584692 −0.292346 0.956313i \(-0.594436\pi\)
−0.292346 + 0.956313i \(0.594436\pi\)
\(908\) 0 0
\(909\) −10.2262 −0.339181
\(910\) 0 0
\(911\) 23.2177 0.769237 0.384618 0.923076i \(-0.374333\pi\)
0.384618 + 0.923076i \(0.374333\pi\)
\(912\) 0 0
\(913\) −13.1565 −0.435416
\(914\) 0 0
\(915\) 2.91751 0.0964500
\(916\) 0 0
\(917\) −26.0740 −0.861039
\(918\) 0 0
\(919\) −7.06970 −0.233208 −0.116604 0.993179i \(-0.537201\pi\)
−0.116604 + 0.993179i \(0.537201\pi\)
\(920\) 0 0
\(921\) 32.2262 1.06189
\(922\) 0 0
\(923\) −26.9473 −0.886980
\(924\) 0 0
\(925\) 3.42176 0.112507
\(926\) 0 0
\(927\) −16.4524 −0.540366
\(928\) 0 0
\(929\) −52.9659 −1.73776 −0.868878 0.495026i \(-0.835158\pi\)
−0.868878 + 0.495026i \(0.835158\pi\)
\(930\) 0 0
\(931\) −9.91751 −0.325033
\(932\) 0 0
\(933\) −16.7304 −0.547730
\(934\) 0 0
\(935\) 8.45236 0.276422
\(936\) 0 0
\(937\) 3.75685 0.122731 0.0613654 0.998115i \(-0.480455\pi\)
0.0613654 + 0.998115i \(0.480455\pi\)
\(938\) 0 0
\(939\) 14.6785 0.479016
\(940\) 0 0
\(941\) −13.0178 −0.424369 −0.212184 0.977230i \(-0.568058\pi\)
−0.212184 + 0.977230i \(0.568058\pi\)
\(942\) 0 0
\(943\) 3.74824 0.122059
\(944\) 0 0
\(945\) −4.11309 −0.133799
\(946\) 0 0
\(947\) −20.3912 −0.662623 −0.331312 0.943521i \(-0.607491\pi\)
−0.331312 + 0.943521i \(0.607491\pi\)
\(948\) 0 0
\(949\) 28.8265 0.935749
\(950\) 0 0
\(951\) 17.9090 0.580740
\(952\) 0 0
\(953\) −48.2746 −1.56377 −0.781884 0.623424i \(-0.785741\pi\)
−0.781884 + 0.623424i \(0.785741\pi\)
\(954\) 0 0
\(955\) −0.730425 −0.0236360
\(956\) 0 0
\(957\) −8.69133 −0.280951
\(958\) 0 0
\(959\) 30.0612 0.970727
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −16.4524 −0.530170
\(964\) 0 0
\(965\) 25.4830 0.820326
\(966\) 0 0
\(967\) 28.9396 0.930636 0.465318 0.885144i \(-0.345940\pi\)
0.465318 + 0.885144i \(0.345940\pi\)
\(968\) 0 0
\(969\) 4.00000 0.128499
\(970\) 0 0
\(971\) 3.92600 0.125991 0.0629957 0.998014i \(-0.479935\pi\)
0.0629957 + 0.998014i \(0.479935\pi\)
\(972\) 0 0
\(973\) 33.8350 1.08470
\(974\) 0 0
\(975\) −4.80442 −0.153865
\(976\) 0 0
\(977\) −9.30867 −0.297811 −0.148905 0.988851i \(-0.547575\pi\)
−0.148905 + 0.988851i \(0.547575\pi\)
\(978\) 0 0
\(979\) −3.00430 −0.0960179
\(980\) 0 0
\(981\) −7.53485 −0.240569
\(982\) 0 0
\(983\) 13.8478 0.441677 0.220838 0.975310i \(-0.429121\pi\)
0.220838 + 0.975310i \(0.429121\pi\)
\(984\) 0 0
\(985\) −21.7610 −0.693364
\(986\) 0 0
\(987\) 2.84352 0.0905101
\(988\) 0 0
\(989\) 1.93879 0.0616500
\(990\) 0 0
\(991\) −30.5433 −0.970241 −0.485121 0.874447i \(-0.661224\pi\)
−0.485121 + 0.874447i \(0.661224\pi\)
\(992\) 0 0
\(993\) −1.92600 −0.0611198
\(994\) 0 0
\(995\) −1.60885 −0.0510039
\(996\) 0 0
\(997\) −6.75254 −0.213855 −0.106928 0.994267i \(-0.534101\pi\)
−0.106928 + 0.994267i \(0.534101\pi\)
\(998\) 0 0
\(999\) −3.42176 −0.108260
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 4560.2.a.bs.1.3 3
4.3 odd 2 2280.2.a.u.1.1 3
12.11 even 2 6840.2.a.bh.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.u.1.1 3 4.3 odd 2
4560.2.a.bs.1.3 3 1.1 even 1 trivial
6840.2.a.bh.1.1 3 12.11 even 2