Properties

Label 5520.2.a.by.1.1
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 345)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -4.48929 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -4.48929 q^{7} +1.00000 q^{9} +1.14637 q^{11} +0.853635 q^{13} -1.00000 q^{15} +1.34292 q^{17} +3.83221 q^{19} +4.48929 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.02877 q^{29} -2.19656 q^{31} -1.14637 q^{33} -4.48929 q^{35} -2.48929 q^{37} -0.853635 q^{39} +11.3001 q^{41} -10.6858 q^{43} +1.00000 q^{45} -1.53948 q^{47} +13.1537 q^{49} -1.34292 q^{51} +4.02877 q^{53} +1.14637 q^{55} -3.83221 q^{57} +15.0073 q^{59} -5.83221 q^{61} -4.48929 q^{63} +0.853635 q^{65} +11.5682 q^{67} +1.00000 q^{69} -4.32150 q^{71} -13.1035 q^{73} -1.00000 q^{75} -5.14637 q^{77} -0.585462 q^{79} +1.00000 q^{81} -5.63565 q^{83} +1.34292 q^{85} +8.02877 q^{87} -0.100384 q^{89} -3.83221 q^{91} +2.19656 q^{93} +3.83221 q^{95} +11.5640 q^{97} +1.14637 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} - 6 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 3 q^{5} - 6 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - 3 q^{15} - 2 q^{17} - 2 q^{19} + 6 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 6 q^{29} - 2 q^{31} - 2 q^{33} - 6 q^{35} - 4 q^{39} - 2 q^{41} - 20 q^{43} + 3 q^{45} + 6 q^{47} + 5 q^{49} + 2 q^{51} - 6 q^{53} + 2 q^{55} + 2 q^{57} + 12 q^{59} - 4 q^{61} - 6 q^{63} + 4 q^{65} + 6 q^{67} + 3 q^{69} + 8 q^{71} - 8 q^{73} - 3 q^{75} - 14 q^{77} + 4 q^{79} + 3 q^{81} - 8 q^{83} - 2 q^{85} + 6 q^{87} + 6 q^{89} + 2 q^{91} + 2 q^{93} - 2 q^{95} + 14 q^{97} + 2 q^{99}+O(q^{100}) \) 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
\(7\) −4.48929 −1.69679 −0.848396 0.529362i \(-0.822431\pi\)
−0.848396 + 0.529362i \(0.822431\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.14637 0.345642 0.172821 0.984953i \(-0.444712\pi\)
0.172821 + 0.984953i \(0.444712\pi\)
\(12\) 0 0
\(13\) 0.853635 0.236756 0.118378 0.992969i \(-0.462231\pi\)
0.118378 + 0.992969i \(0.462231\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 1.34292 0.325707 0.162853 0.986650i \(-0.447930\pi\)
0.162853 + 0.986650i \(0.447930\pi\)
\(18\) 0 0
\(19\) 3.83221 0.879170 0.439585 0.898201i \(-0.355126\pi\)
0.439585 + 0.898201i \(0.355126\pi\)
\(20\) 0 0
\(21\) 4.48929 0.979643
\(22\) 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
\(29\) −8.02877 −1.49091 −0.745453 0.666559i \(-0.767767\pi\)
−0.745453 + 0.666559i \(0.767767\pi\)
\(30\) 0 0
\(31\) −2.19656 −0.394513 −0.197257 0.980352i \(-0.563203\pi\)
−0.197257 + 0.980352i \(0.563203\pi\)
\(32\) 0 0
\(33\) −1.14637 −0.199557
\(34\) 0 0
\(35\) −4.48929 −0.758828
\(36\) 0 0
\(37\) −2.48929 −0.409237 −0.204618 0.978842i \(-0.565595\pi\)
−0.204618 + 0.978842i \(0.565595\pi\)
\(38\) 0 0
\(39\) −0.853635 −0.136691
\(40\) 0 0
\(41\) 11.3001 1.76478 0.882388 0.470523i \(-0.155935\pi\)
0.882388 + 0.470523i \(0.155935\pi\)
\(42\) 0 0
\(43\) −10.6858 −1.62958 −0.814788 0.579759i \(-0.803147\pi\)
−0.814788 + 0.579759i \(0.803147\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −1.53948 −0.224556 −0.112278 0.993677i \(-0.535815\pi\)
−0.112278 + 0.993677i \(0.535815\pi\)
\(48\) 0 0
\(49\) 13.1537 1.87910
\(50\) 0 0
\(51\) −1.34292 −0.188047
\(52\) 0 0
\(53\) 4.02877 0.553394 0.276697 0.960957i \(-0.410760\pi\)
0.276697 + 0.960957i \(0.410760\pi\)
\(54\) 0 0
\(55\) 1.14637 0.154576
\(56\) 0 0
\(57\) −3.83221 −0.507589
\(58\) 0 0
\(59\) 15.0073 1.95379 0.976895 0.213720i \(-0.0685579\pi\)
0.976895 + 0.213720i \(0.0685579\pi\)
\(60\) 0 0
\(61\) −5.83221 −0.746738 −0.373369 0.927683i \(-0.621797\pi\)
−0.373369 + 0.927683i \(0.621797\pi\)
\(62\) 0 0
\(63\) −4.48929 −0.565597
\(64\) 0 0
\(65\) 0.853635 0.105880
\(66\) 0 0
\(67\) 11.5682 1.41329 0.706643 0.707570i \(-0.250209\pi\)
0.706643 + 0.707570i \(0.250209\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −4.32150 −0.512868 −0.256434 0.966562i \(-0.582548\pi\)
−0.256434 + 0.966562i \(0.582548\pi\)
\(72\) 0 0
\(73\) −13.1035 −1.53365 −0.766825 0.641856i \(-0.778165\pi\)
−0.766825 + 0.641856i \(0.778165\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −5.14637 −0.586483
\(78\) 0 0
\(79\) −0.585462 −0.0658696 −0.0329348 0.999458i \(-0.510485\pi\)
−0.0329348 + 0.999458i \(0.510485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.63565 −0.618593 −0.309297 0.950966i \(-0.600094\pi\)
−0.309297 + 0.950966i \(0.600094\pi\)
\(84\) 0 0
\(85\) 1.34292 0.145660
\(86\) 0 0
\(87\) 8.02877 0.860774
\(88\) 0 0
\(89\) −0.100384 −0.0106407 −0.00532035 0.999986i \(-0.501694\pi\)
−0.00532035 + 0.999986i \(0.501694\pi\)
\(90\) 0 0
\(91\) −3.83221 −0.401725
\(92\) 0 0
\(93\) 2.19656 0.227772
\(94\) 0 0
\(95\) 3.83221 0.393177
\(96\) 0 0
\(97\) 11.5640 1.17415 0.587075 0.809532i \(-0.300279\pi\)
0.587075 + 0.809532i \(0.300279\pi\)
\(98\) 0 0
\(99\) 1.14637 0.115214
\(100\) 0 0
\(101\) 11.6932 1.16352 0.581758 0.813362i \(-0.302365\pi\)
0.581758 + 0.813362i \(0.302365\pi\)
\(102\) 0 0
\(103\) 19.7220 1.94326 0.971631 0.236501i \(-0.0760006\pi\)
0.971631 + 0.236501i \(0.0760006\pi\)
\(104\) 0 0
\(105\) 4.48929 0.438110
\(106\) 0 0
\(107\) −5.44331 −0.526224 −0.263112 0.964765i \(-0.584749\pi\)
−0.263112 + 0.964765i \(0.584749\pi\)
\(108\) 0 0
\(109\) −12.5181 −1.19901 −0.599506 0.800370i \(-0.704636\pi\)
−0.599506 + 0.800370i \(0.704636\pi\)
\(110\) 0 0
\(111\) 2.48929 0.236273
\(112\) 0 0
\(113\) −14.1292 −1.32916 −0.664579 0.747218i \(-0.731389\pi\)
−0.664579 + 0.747218i \(0.731389\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 0.853635 0.0789185
\(118\) 0 0
\(119\) −6.02877 −0.552656
\(120\) 0 0
\(121\) −9.68585 −0.880531
\(122\) 0 0
\(123\) −11.3001 −1.01889
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −5.48194 −0.486444 −0.243222 0.969971i \(-0.578204\pi\)
−0.243222 + 0.969971i \(0.578204\pi\)
\(128\) 0 0
\(129\) 10.6858 0.940836
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −17.2039 −1.49177
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 11.7648 1.00514 0.502568 0.864538i \(-0.332389\pi\)
0.502568 + 0.864538i \(0.332389\pi\)
\(138\) 0 0
\(139\) −8.15371 −0.691589 −0.345794 0.938310i \(-0.612391\pi\)
−0.345794 + 0.938310i \(0.612391\pi\)
\(140\) 0 0
\(141\) 1.53948 0.129648
\(142\) 0 0
\(143\) 0.978577 0.0818327
\(144\) 0 0
\(145\) −8.02877 −0.666753
\(146\) 0 0
\(147\) −13.1537 −1.08490
\(148\) 0 0
\(149\) −3.73183 −0.305723 −0.152862 0.988248i \(-0.548849\pi\)
−0.152862 + 0.988248i \(0.548849\pi\)
\(150\) 0 0
\(151\) −16.6430 −1.35439 −0.677194 0.735804i \(-0.736804\pi\)
−0.677194 + 0.735804i \(0.736804\pi\)
\(152\) 0 0
\(153\) 1.34292 0.108569
\(154\) 0 0
\(155\) −2.19656 −0.176432
\(156\) 0 0
\(157\) −13.2327 −1.05608 −0.528041 0.849219i \(-0.677073\pi\)
−0.528041 + 0.849219i \(0.677073\pi\)
\(158\) 0 0
\(159\) −4.02877 −0.319502
\(160\) 0 0
\(161\) 4.48929 0.353806
\(162\) 0 0
\(163\) −16.3931 −1.28401 −0.642004 0.766701i \(-0.721897\pi\)
−0.642004 + 0.766701i \(0.721897\pi\)
\(164\) 0 0
\(165\) −1.14637 −0.0892444
\(166\) 0 0
\(167\) −3.04598 −0.235705 −0.117853 0.993031i \(-0.537601\pi\)
−0.117853 + 0.993031i \(0.537601\pi\)
\(168\) 0 0
\(169\) −12.2713 −0.943947
\(170\) 0 0
\(171\) 3.83221 0.293057
\(172\) 0 0
\(173\) −24.9357 −1.89583 −0.947914 0.318526i \(-0.896812\pi\)
−0.947914 + 0.318526i \(0.896812\pi\)
\(174\) 0 0
\(175\) −4.48929 −0.339358
\(176\) 0 0
\(177\) −15.0073 −1.12802
\(178\) 0 0
\(179\) 6.68585 0.499724 0.249862 0.968282i \(-0.419615\pi\)
0.249862 + 0.968282i \(0.419615\pi\)
\(180\) 0 0
\(181\) −20.6002 −1.53120 −0.765599 0.643318i \(-0.777557\pi\)
−0.765599 + 0.643318i \(0.777557\pi\)
\(182\) 0 0
\(183\) 5.83221 0.431129
\(184\) 0 0
\(185\) −2.48929 −0.183016
\(186\) 0 0
\(187\) 1.53948 0.112578
\(188\) 0 0
\(189\) 4.48929 0.326548
\(190\) 0 0
\(191\) −4.95402 −0.358460 −0.179230 0.983807i \(-0.557361\pi\)
−0.179230 + 0.983807i \(0.557361\pi\)
\(192\) 0 0
\(193\) 6.05754 0.436031 0.218016 0.975945i \(-0.430042\pi\)
0.218016 + 0.975945i \(0.430042\pi\)
\(194\) 0 0
\(195\) −0.853635 −0.0611300
\(196\) 0 0
\(197\) −24.4078 −1.73898 −0.869492 0.493947i \(-0.835554\pi\)
−0.869492 + 0.493947i \(0.835554\pi\)
\(198\) 0 0
\(199\) 3.85677 0.273399 0.136700 0.990613i \(-0.456350\pi\)
0.136700 + 0.990613i \(0.456350\pi\)
\(200\) 0 0
\(201\) −11.5682 −0.815961
\(202\) 0 0
\(203\) 36.0435 2.52976
\(204\) 0 0
\(205\) 11.3001 0.789232
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 4.39312 0.303878
\(210\) 0 0
\(211\) 0.824865 0.0567861 0.0283930 0.999597i \(-0.490961\pi\)
0.0283930 + 0.999597i \(0.490961\pi\)
\(212\) 0 0
\(213\) 4.32150 0.296104
\(214\) 0 0
\(215\) −10.6858 −0.728769
\(216\) 0 0
\(217\) 9.86098 0.669407
\(218\) 0 0
\(219\) 13.1035 0.885454
\(220\) 0 0
\(221\) 1.14637 0.0771129
\(222\) 0 0
\(223\) −5.56404 −0.372596 −0.186298 0.982493i \(-0.559649\pi\)
−0.186298 + 0.982493i \(0.559649\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −16.5855 −1.10082 −0.550408 0.834896i \(-0.685528\pi\)
−0.550408 + 0.834896i \(0.685528\pi\)
\(228\) 0 0
\(229\) 6.64300 0.438982 0.219491 0.975615i \(-0.429560\pi\)
0.219491 + 0.975615i \(0.429560\pi\)
\(230\) 0 0
\(231\) 5.14637 0.338606
\(232\) 0 0
\(233\) −14.3931 −0.942924 −0.471462 0.881886i \(-0.656274\pi\)
−0.471462 + 0.881886i \(0.656274\pi\)
\(234\) 0 0
\(235\) −1.53948 −0.100425
\(236\) 0 0
\(237\) 0.585462 0.0380298
\(238\) 0 0
\(239\) 13.1077 0.847869 0.423934 0.905693i \(-0.360649\pi\)
0.423934 + 0.905693i \(0.360649\pi\)
\(240\) 0 0
\(241\) 12.5181 0.806359 0.403179 0.915121i \(-0.367905\pi\)
0.403179 + 0.915121i \(0.367905\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 13.1537 0.840360
\(246\) 0 0
\(247\) 3.27131 0.208148
\(248\) 0 0
\(249\) 5.63565 0.357145
\(250\) 0 0
\(251\) 26.3503 1.66321 0.831607 0.555364i \(-0.187421\pi\)
0.831607 + 0.555364i \(0.187421\pi\)
\(252\) 0 0
\(253\) −1.14637 −0.0720714
\(254\) 0 0
\(255\) −1.34292 −0.0840971
\(256\) 0 0
\(257\) −27.0116 −1.68493 −0.842467 0.538747i \(-0.818898\pi\)
−0.842467 + 0.538747i \(0.818898\pi\)
\(258\) 0 0
\(259\) 11.1751 0.694389
\(260\) 0 0
\(261\) −8.02877 −0.496968
\(262\) 0 0
\(263\) −21.8855 −1.34952 −0.674760 0.738037i \(-0.735753\pi\)
−0.674760 + 0.738037i \(0.735753\pi\)
\(264\) 0 0
\(265\) 4.02877 0.247485
\(266\) 0 0
\(267\) 0.100384 0.00614341
\(268\) 0 0
\(269\) 3.69319 0.225178 0.112589 0.993642i \(-0.464086\pi\)
0.112589 + 0.993642i \(0.464086\pi\)
\(270\) 0 0
\(271\) −0.288520 −0.0175264 −0.00876318 0.999962i \(-0.502789\pi\)
−0.00876318 + 0.999962i \(0.502789\pi\)
\(272\) 0 0
\(273\) 3.83221 0.231936
\(274\) 0 0
\(275\) 1.14637 0.0691284
\(276\) 0 0
\(277\) 21.3288 1.28153 0.640763 0.767739i \(-0.278618\pi\)
0.640763 + 0.767739i \(0.278618\pi\)
\(278\) 0 0
\(279\) −2.19656 −0.131504
\(280\) 0 0
\(281\) −26.6676 −1.59085 −0.795427 0.606050i \(-0.792753\pi\)
−0.795427 + 0.606050i \(0.792753\pi\)
\(282\) 0 0
\(283\) −25.9185 −1.54070 −0.770348 0.637624i \(-0.779918\pi\)
−0.770348 + 0.637624i \(0.779918\pi\)
\(284\) 0 0
\(285\) −3.83221 −0.227001
\(286\) 0 0
\(287\) −50.7293 −2.99446
\(288\) 0 0
\(289\) −15.1966 −0.893915
\(290\) 0 0
\(291\) −11.5640 −0.677896
\(292\) 0 0
\(293\) 13.6503 0.797462 0.398731 0.917068i \(-0.369451\pi\)
0.398731 + 0.917068i \(0.369451\pi\)
\(294\) 0 0
\(295\) 15.0073 0.873761
\(296\) 0 0
\(297\) −1.14637 −0.0665189
\(298\) 0 0
\(299\) −0.853635 −0.0493670
\(300\) 0 0
\(301\) 47.9718 2.76505
\(302\) 0 0
\(303\) −11.6932 −0.671756
\(304\) 0 0
\(305\) −5.83221 −0.333951
\(306\) 0 0
\(307\) 9.78937 0.558709 0.279354 0.960188i \(-0.409880\pi\)
0.279354 + 0.960188i \(0.409880\pi\)
\(308\) 0 0
\(309\) −19.7220 −1.12194
\(310\) 0 0
\(311\) 3.32885 0.188762 0.0943808 0.995536i \(-0.469913\pi\)
0.0943808 + 0.995536i \(0.469913\pi\)
\(312\) 0 0
\(313\) −18.8824 −1.06730 −0.533648 0.845707i \(-0.679179\pi\)
−0.533648 + 0.845707i \(0.679179\pi\)
\(314\) 0 0
\(315\) −4.48929 −0.252943
\(316\) 0 0
\(317\) −8.06740 −0.453111 −0.226555 0.973998i \(-0.572746\pi\)
−0.226555 + 0.973998i \(0.572746\pi\)
\(318\) 0 0
\(319\) −9.20390 −0.515320
\(320\) 0 0
\(321\) 5.44331 0.303816
\(322\) 0 0
\(323\) 5.14637 0.286351
\(324\) 0 0
\(325\) 0.853635 0.0473511
\(326\) 0 0
\(327\) 12.5181 0.692250
\(328\) 0 0
\(329\) 6.91117 0.381025
\(330\) 0 0
\(331\) 26.6388 1.46420 0.732100 0.681197i \(-0.238540\pi\)
0.732100 + 0.681197i \(0.238540\pi\)
\(332\) 0 0
\(333\) −2.48929 −0.136412
\(334\) 0 0
\(335\) 11.5682 0.632041
\(336\) 0 0
\(337\) −8.54262 −0.465346 −0.232673 0.972555i \(-0.574747\pi\)
−0.232673 + 0.972555i \(0.574747\pi\)
\(338\) 0 0
\(339\) 14.1292 0.767390
\(340\) 0 0
\(341\) −2.51806 −0.136360
\(342\) 0 0
\(343\) −27.6258 −1.49165
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) 16.2499 0.872340 0.436170 0.899864i \(-0.356335\pi\)
0.436170 + 0.899864i \(0.356335\pi\)
\(348\) 0 0
\(349\) 23.1898 1.24132 0.620662 0.784079i \(-0.286864\pi\)
0.620662 + 0.784079i \(0.286864\pi\)
\(350\) 0 0
\(351\) −0.853635 −0.0455636
\(352\) 0 0
\(353\) −7.48194 −0.398224 −0.199112 0.979977i \(-0.563806\pi\)
−0.199112 + 0.979977i \(0.563806\pi\)
\(354\) 0 0
\(355\) −4.32150 −0.229361
\(356\) 0 0
\(357\) 6.02877 0.319076
\(358\) 0 0
\(359\) −1.87506 −0.0989617 −0.0494809 0.998775i \(-0.515757\pi\)
−0.0494809 + 0.998775i \(0.515757\pi\)
\(360\) 0 0
\(361\) −4.31415 −0.227061
\(362\) 0 0
\(363\) 9.68585 0.508375
\(364\) 0 0
\(365\) −13.1035 −0.685870
\(366\) 0 0
\(367\) 4.68164 0.244379 0.122190 0.992507i \(-0.461008\pi\)
0.122190 + 0.992507i \(0.461008\pi\)
\(368\) 0 0
\(369\) 11.3001 0.588259
\(370\) 0 0
\(371\) −18.0863 −0.938994
\(372\) 0 0
\(373\) 18.6430 0.965298 0.482649 0.875814i \(-0.339675\pi\)
0.482649 + 0.875814i \(0.339675\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −6.85363 −0.352980
\(378\) 0 0
\(379\) −29.8715 −1.53439 −0.767197 0.641412i \(-0.778349\pi\)
−0.767197 + 0.641412i \(0.778349\pi\)
\(380\) 0 0
\(381\) 5.48194 0.280848
\(382\) 0 0
\(383\) −20.5714 −1.05115 −0.525574 0.850748i \(-0.676150\pi\)
−0.525574 + 0.850748i \(0.676150\pi\)
\(384\) 0 0
\(385\) −5.14637 −0.262283
\(386\) 0 0
\(387\) −10.6858 −0.543192
\(388\) 0 0
\(389\) −2.33558 −0.118418 −0.0592092 0.998246i \(-0.518858\pi\)
−0.0592092 + 0.998246i \(0.518858\pi\)
\(390\) 0 0
\(391\) −1.34292 −0.0679145
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.585462 −0.0294578
\(396\) 0 0
\(397\) 29.1365 1.46232 0.731160 0.682207i \(-0.238980\pi\)
0.731160 + 0.682207i \(0.238980\pi\)
\(398\) 0 0
\(399\) 17.2039 0.861272
\(400\) 0 0
\(401\) −14.8353 −0.740842 −0.370421 0.928864i \(-0.620787\pi\)
−0.370421 + 0.928864i \(0.620787\pi\)
\(402\) 0 0
\(403\) −1.87506 −0.0934033
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −2.85363 −0.141449
\(408\) 0 0
\(409\) −38.0189 −1.87991 −0.939957 0.341293i \(-0.889135\pi\)
−0.939957 + 0.341293i \(0.889135\pi\)
\(410\) 0 0
\(411\) −11.7648 −0.580315
\(412\) 0 0
\(413\) −67.3723 −3.31517
\(414\) 0 0
\(415\) −5.63565 −0.276643
\(416\) 0 0
\(417\) 8.15371 0.399289
\(418\) 0 0
\(419\) −21.8469 −1.06729 −0.533646 0.845708i \(-0.679178\pi\)
−0.533646 + 0.845708i \(0.679178\pi\)
\(420\) 0 0
\(421\) 21.4391 1.04488 0.522439 0.852677i \(-0.325022\pi\)
0.522439 + 0.852677i \(0.325022\pi\)
\(422\) 0 0
\(423\) −1.53948 −0.0748521
\(424\) 0 0
\(425\) 1.34292 0.0651413
\(426\) 0 0
\(427\) 26.1825 1.26706
\(428\) 0 0
\(429\) −0.978577 −0.0472461
\(430\) 0 0
\(431\) −29.8223 −1.43649 −0.718246 0.695789i \(-0.755055\pi\)
−0.718246 + 0.695789i \(0.755055\pi\)
\(432\) 0 0
\(433\) −0.925249 −0.0444647 −0.0222323 0.999753i \(-0.507077\pi\)
−0.0222323 + 0.999753i \(0.507077\pi\)
\(434\) 0 0
\(435\) 8.02877 0.384950
\(436\) 0 0
\(437\) −3.83221 −0.183320
\(438\) 0 0
\(439\) 21.6791 1.03469 0.517344 0.855778i \(-0.326921\pi\)
0.517344 + 0.855778i \(0.326921\pi\)
\(440\) 0 0
\(441\) 13.1537 0.626367
\(442\) 0 0
\(443\) −5.20390 −0.247245 −0.123622 0.992329i \(-0.539451\pi\)
−0.123622 + 0.992329i \(0.539451\pi\)
\(444\) 0 0
\(445\) −0.100384 −0.00475867
\(446\) 0 0
\(447\) 3.73183 0.176509
\(448\) 0 0
\(449\) −10.8578 −0.512413 −0.256207 0.966622i \(-0.582473\pi\)
−0.256207 + 0.966622i \(0.582473\pi\)
\(450\) 0 0
\(451\) 12.9540 0.609981
\(452\) 0 0
\(453\) 16.6430 0.781956
\(454\) 0 0
\(455\) −3.83221 −0.179657
\(456\) 0 0
\(457\) 27.2755 1.27589 0.637947 0.770080i \(-0.279784\pi\)
0.637947 + 0.770080i \(0.279784\pi\)
\(458\) 0 0
\(459\) −1.34292 −0.0626823
\(460\) 0 0
\(461\) 20.2070 0.941136 0.470568 0.882364i \(-0.344049\pi\)
0.470568 + 0.882364i \(0.344049\pi\)
\(462\) 0 0
\(463\) −8.03298 −0.373324 −0.186662 0.982424i \(-0.559767\pi\)
−0.186662 + 0.982424i \(0.559767\pi\)
\(464\) 0 0
\(465\) 2.19656 0.101863
\(466\) 0 0
\(467\) 19.4580 0.900409 0.450204 0.892926i \(-0.351351\pi\)
0.450204 + 0.892926i \(0.351351\pi\)
\(468\) 0 0
\(469\) −51.9332 −2.39805
\(470\) 0 0
\(471\) 13.2327 0.609729
\(472\) 0 0
\(473\) −12.2499 −0.563250
\(474\) 0 0
\(475\) 3.83221 0.175834
\(476\) 0 0
\(477\) 4.02877 0.184465
\(478\) 0 0
\(479\) −28.9540 −1.32294 −0.661471 0.749970i \(-0.730068\pi\)
−0.661471 + 0.749970i \(0.730068\pi\)
\(480\) 0 0
\(481\) −2.12494 −0.0968890
\(482\) 0 0
\(483\) −4.48929 −0.204270
\(484\) 0 0
\(485\) 11.5640 0.525096
\(486\) 0 0
\(487\) −20.8108 −0.943027 −0.471513 0.881859i \(-0.656292\pi\)
−0.471513 + 0.881859i \(0.656292\pi\)
\(488\) 0 0
\(489\) 16.3931 0.741322
\(490\) 0 0
\(491\) −35.1140 −1.58467 −0.792336 0.610084i \(-0.791135\pi\)
−0.792336 + 0.610084i \(0.791135\pi\)
\(492\) 0 0
\(493\) −10.7820 −0.485598
\(494\) 0 0
\(495\) 1.14637 0.0515253
\(496\) 0 0
\(497\) 19.4005 0.870230
\(498\) 0 0
\(499\) 7.31836 0.327615 0.163807 0.986492i \(-0.447622\pi\)
0.163807 + 0.986492i \(0.447622\pi\)
\(500\) 0 0
\(501\) 3.04598 0.136084
\(502\) 0 0
\(503\) 22.0006 0.980959 0.490479 0.871453i \(-0.336822\pi\)
0.490479 + 0.871453i \(0.336822\pi\)
\(504\) 0 0
\(505\) 11.6932 0.520340
\(506\) 0 0
\(507\) 12.2713 0.544988
\(508\) 0 0
\(509\) −35.4783 −1.57255 −0.786275 0.617877i \(-0.787993\pi\)
−0.786275 + 0.617877i \(0.787993\pi\)
\(510\) 0 0
\(511\) 58.8255 2.60229
\(512\) 0 0
\(513\) −3.83221 −0.169196
\(514\) 0 0
\(515\) 19.7220 0.869053
\(516\) 0 0
\(517\) −1.76481 −0.0776161
\(518\) 0 0
\(519\) 24.9357 1.09456
\(520\) 0 0
\(521\) 21.4966 0.941785 0.470892 0.882191i \(-0.343932\pi\)
0.470892 + 0.882191i \(0.343932\pi\)
\(522\) 0 0
\(523\) 11.3288 0.495376 0.247688 0.968840i \(-0.420329\pi\)
0.247688 + 0.968840i \(0.420329\pi\)
\(524\) 0 0
\(525\) 4.48929 0.195929
\(526\) 0 0
\(527\) −2.94981 −0.128496
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 15.0073 0.651263
\(532\) 0 0
\(533\) 9.64614 0.417821
\(534\) 0 0
\(535\) −5.44331 −0.235335
\(536\) 0 0
\(537\) −6.68585 −0.288516
\(538\) 0 0
\(539\) 15.0790 0.649497
\(540\) 0 0
\(541\) −7.67912 −0.330151 −0.165075 0.986281i \(-0.552787\pi\)
−0.165075 + 0.986281i \(0.552787\pi\)
\(542\) 0 0
\(543\) 20.6002 0.884037
\(544\) 0 0
\(545\) −12.5181 −0.536215
\(546\) 0 0
\(547\) −31.5296 −1.34811 −0.674054 0.738682i \(-0.735449\pi\)
−0.674054 + 0.738682i \(0.735449\pi\)
\(548\) 0 0
\(549\) −5.83221 −0.248913
\(550\) 0 0
\(551\) −30.7679 −1.31076
\(552\) 0 0
\(553\) 2.62831 0.111767
\(554\) 0 0
\(555\) 2.48929 0.105664
\(556\) 0 0
\(557\) −10.1292 −0.429186 −0.214593 0.976704i \(-0.568843\pi\)
−0.214593 + 0.976704i \(0.568843\pi\)
\(558\) 0 0
\(559\) −9.12181 −0.385811
\(560\) 0 0
\(561\) −1.53948 −0.0649969
\(562\) 0 0
\(563\) −40.7146 −1.71592 −0.857958 0.513719i \(-0.828267\pi\)
−0.857958 + 0.513719i \(0.828267\pi\)
\(564\) 0 0
\(565\) −14.1292 −0.594418
\(566\) 0 0
\(567\) −4.48929 −0.188532
\(568\) 0 0
\(569\) 1.66442 0.0697763 0.0348881 0.999391i \(-0.488893\pi\)
0.0348881 + 0.999391i \(0.488893\pi\)
\(570\) 0 0
\(571\) 41.8469 1.75124 0.875619 0.483002i \(-0.160454\pi\)
0.875619 + 0.483002i \(0.160454\pi\)
\(572\) 0 0
\(573\) 4.95402 0.206957
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 21.6069 0.899506 0.449753 0.893153i \(-0.351512\pi\)
0.449753 + 0.893153i \(0.351512\pi\)
\(578\) 0 0
\(579\) −6.05754 −0.251743
\(580\) 0 0
\(581\) 25.3001 1.04962
\(582\) 0 0
\(583\) 4.61844 0.191276
\(584\) 0 0
\(585\) 0.853635 0.0352934
\(586\) 0 0
\(587\) −0.393115 −0.0162256 −0.00811280 0.999967i \(-0.502582\pi\)
−0.00811280 + 0.999967i \(0.502582\pi\)
\(588\) 0 0
\(589\) −8.41767 −0.346844
\(590\) 0 0
\(591\) 24.4078 1.00400
\(592\) 0 0
\(593\) −8.31729 −0.341550 −0.170775 0.985310i \(-0.554627\pi\)
−0.170775 + 0.985310i \(0.554627\pi\)
\(594\) 0 0
\(595\) −6.02877 −0.247155
\(596\) 0 0
\(597\) −3.85677 −0.157847
\(598\) 0 0
\(599\) −9.03612 −0.369206 −0.184603 0.982813i \(-0.559100\pi\)
−0.184603 + 0.982813i \(0.559100\pi\)
\(600\) 0 0
\(601\) −27.6602 −1.12828 −0.564142 0.825678i \(-0.690793\pi\)
−0.564142 + 0.825678i \(0.690793\pi\)
\(602\) 0 0
\(603\) 11.5682 0.471096
\(604\) 0 0
\(605\) −9.68585 −0.393786
\(606\) 0 0
\(607\) −29.6461 −1.20330 −0.601650 0.798760i \(-0.705490\pi\)
−0.601650 + 0.798760i \(0.705490\pi\)
\(608\) 0 0
\(609\) −36.0435 −1.46055
\(610\) 0 0
\(611\) −1.31415 −0.0531650
\(612\) 0 0
\(613\) 39.4868 1.59486 0.797428 0.603414i \(-0.206193\pi\)
0.797428 + 0.603414i \(0.206193\pi\)
\(614\) 0 0
\(615\) −11.3001 −0.455663
\(616\) 0 0
\(617\) 16.8641 0.678924 0.339462 0.940620i \(-0.389755\pi\)
0.339462 + 0.940620i \(0.389755\pi\)
\(618\) 0 0
\(619\) −2.74338 −0.110266 −0.0551330 0.998479i \(-0.517558\pi\)
−0.0551330 + 0.998479i \(0.517558\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 0.450654 0.0180551
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −4.39312 −0.175444
\(628\) 0 0
\(629\) −3.34292 −0.133291
\(630\) 0 0
\(631\) 9.93260 0.395410 0.197705 0.980262i \(-0.436651\pi\)
0.197705 + 0.980262i \(0.436651\pi\)
\(632\) 0 0
\(633\) −0.824865 −0.0327855
\(634\) 0 0
\(635\) −5.48194 −0.217544
\(636\) 0 0
\(637\) 11.2285 0.444888
\(638\) 0 0
\(639\) −4.32150 −0.170956
\(640\) 0 0
\(641\) −10.6184 −0.419403 −0.209702 0.977765i \(-0.567249\pi\)
−0.209702 + 0.977765i \(0.567249\pi\)
\(642\) 0 0
\(643\) −20.5959 −0.812225 −0.406112 0.913823i \(-0.633116\pi\)
−0.406112 + 0.913823i \(0.633116\pi\)
\(644\) 0 0
\(645\) 10.6858 0.420755
\(646\) 0 0
\(647\) 6.71883 0.264144 0.132072 0.991240i \(-0.457837\pi\)
0.132072 + 0.991240i \(0.457837\pi\)
\(648\) 0 0
\(649\) 17.2039 0.675312
\(650\) 0 0
\(651\) −9.86098 −0.386482
\(652\) 0 0
\(653\) −4.07583 −0.159499 −0.0797497 0.996815i \(-0.525412\pi\)
−0.0797497 + 0.996815i \(0.525412\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −13.1035 −0.511217
\(658\) 0 0
\(659\) 29.0607 1.13204 0.566022 0.824390i \(-0.308482\pi\)
0.566022 + 0.824390i \(0.308482\pi\)
\(660\) 0 0
\(661\) 31.2369 1.21497 0.607487 0.794330i \(-0.292178\pi\)
0.607487 + 0.794330i \(0.292178\pi\)
\(662\) 0 0
\(663\) −1.14637 −0.0445211
\(664\) 0 0
\(665\) −17.2039 −0.667139
\(666\) 0 0
\(667\) 8.02877 0.310875
\(668\) 0 0
\(669\) 5.56404 0.215118
\(670\) 0 0
\(671\) −6.68585 −0.258104
\(672\) 0 0
\(673\) 9.74652 0.375701 0.187850 0.982198i \(-0.439848\pi\)
0.187850 + 0.982198i \(0.439848\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 6.80031 0.261357 0.130679 0.991425i \(-0.458284\pi\)
0.130679 + 0.991425i \(0.458284\pi\)
\(678\) 0 0
\(679\) −51.9143 −1.99229
\(680\) 0 0
\(681\) 16.5855 0.635556
\(682\) 0 0
\(683\) 41.5029 1.58806 0.794032 0.607876i \(-0.207978\pi\)
0.794032 + 0.607876i \(0.207978\pi\)
\(684\) 0 0
\(685\) 11.7648 0.449510
\(686\) 0 0
\(687\) −6.64300 −0.253446
\(688\) 0 0
\(689\) 3.43910 0.131019
\(690\) 0 0
\(691\) 31.2713 1.18962 0.594808 0.803868i \(-0.297228\pi\)
0.594808 + 0.803868i \(0.297228\pi\)
\(692\) 0 0
\(693\) −5.14637 −0.195494
\(694\) 0 0
\(695\) −8.15371 −0.309288
\(696\) 0 0
\(697\) 15.1751 0.574799
\(698\) 0 0
\(699\) 14.3931 0.544398
\(700\) 0 0
\(701\) −6.02456 −0.227544 −0.113772 0.993507i \(-0.536293\pi\)
−0.113772 + 0.993507i \(0.536293\pi\)
\(702\) 0 0
\(703\) −9.53948 −0.359788
\(704\) 0 0
\(705\) 1.53948 0.0579802
\(706\) 0 0
\(707\) −52.4941 −1.97424
\(708\) 0 0
\(709\) 37.6974 1.41576 0.707878 0.706335i \(-0.249653\pi\)
0.707878 + 0.706335i \(0.249653\pi\)
\(710\) 0 0
\(711\) −0.585462 −0.0219565
\(712\) 0 0
\(713\) 2.19656 0.0822617
\(714\) 0 0
\(715\) 0.978577 0.0365967
\(716\) 0 0
\(717\) −13.1077 −0.489517
\(718\) 0 0
\(719\) −19.8708 −0.741058 −0.370529 0.928821i \(-0.620824\pi\)
−0.370529 + 0.928821i \(0.620824\pi\)
\(720\) 0 0
\(721\) −88.5376 −3.29731
\(722\) 0 0
\(723\) −12.5181 −0.465552
\(724\) 0 0
\(725\) −8.02877 −0.298181
\(726\) 0 0
\(727\) −15.8757 −0.588796 −0.294398 0.955683i \(-0.595119\pi\)
−0.294398 + 0.955683i \(0.595119\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −14.3503 −0.530764
\(732\) 0 0
\(733\) 37.1751 1.37309 0.686547 0.727085i \(-0.259125\pi\)
0.686547 + 0.727085i \(0.259125\pi\)
\(734\) 0 0
\(735\) −13.1537 −0.485182
\(736\) 0 0
\(737\) 13.2614 0.488492
\(738\) 0 0
\(739\) −44.0189 −1.61926 −0.809631 0.586940i \(-0.800333\pi\)
−0.809631 + 0.586940i \(0.800333\pi\)
\(740\) 0 0
\(741\) −3.27131 −0.120175
\(742\) 0 0
\(743\) −51.5296 −1.89044 −0.945219 0.326437i \(-0.894152\pi\)
−0.945219 + 0.326437i \(0.894152\pi\)
\(744\) 0 0
\(745\) −3.73183 −0.136724
\(746\) 0 0
\(747\) −5.63565 −0.206198
\(748\) 0 0
\(749\) 24.4366 0.892893
\(750\) 0 0
\(751\) 3.91790 0.142966 0.0714832 0.997442i \(-0.477227\pi\)
0.0714832 + 0.997442i \(0.477227\pi\)
\(752\) 0 0
\(753\) −26.3503 −0.960257
\(754\) 0 0
\(755\) −16.6430 −0.605701
\(756\) 0 0
\(757\) 25.7606 0.936285 0.468142 0.883653i \(-0.344923\pi\)
0.468142 + 0.883653i \(0.344923\pi\)
\(758\) 0 0
\(759\) 1.14637 0.0416104
\(760\) 0 0
\(761\) −15.7507 −0.570964 −0.285482 0.958384i \(-0.592154\pi\)
−0.285482 + 0.958384i \(0.592154\pi\)
\(762\) 0 0
\(763\) 56.1972 2.03447
\(764\) 0 0
\(765\) 1.34292 0.0485535
\(766\) 0 0
\(767\) 12.8108 0.462571
\(768\) 0 0
\(769\) −19.0031 −0.685271 −0.342635 0.939468i \(-0.611320\pi\)
−0.342635 + 0.939468i \(0.611320\pi\)
\(770\) 0 0
\(771\) 27.0116 0.972797
\(772\) 0 0
\(773\) −41.7795 −1.50270 −0.751352 0.659902i \(-0.770598\pi\)
−0.751352 + 0.659902i \(0.770598\pi\)
\(774\) 0 0
\(775\) −2.19656 −0.0789027
\(776\) 0 0
\(777\) −11.1751 −0.400906
\(778\) 0 0
\(779\) 43.3043 1.55154
\(780\) 0 0
\(781\) −4.95402 −0.177269
\(782\) 0 0
\(783\) 8.02877 0.286925
\(784\) 0 0
\(785\) −13.2327 −0.472294
\(786\) 0 0
\(787\) 21.8610 0.779260 0.389630 0.920972i \(-0.372603\pi\)
0.389630 + 0.920972i \(0.372603\pi\)
\(788\) 0 0
\(789\) 21.8855 0.779146
\(790\) 0 0
\(791\) 63.4298 2.25531
\(792\) 0 0
\(793\) −4.97858 −0.176794
\(794\) 0 0
\(795\) −4.02877 −0.142886
\(796\) 0 0
\(797\) −27.8280 −0.985718 −0.492859 0.870109i \(-0.664048\pi\)
−0.492859 + 0.870109i \(0.664048\pi\)
\(798\) 0 0
\(799\) −2.06740 −0.0731395
\(800\) 0 0
\(801\) −0.100384 −0.00354690
\(802\) 0 0
\(803\) −15.0214 −0.530095
\(804\) 0 0
\(805\) 4.48929 0.158227
\(806\) 0 0
\(807\) −3.69319 −0.130007
\(808\) 0 0
\(809\) −21.0565 −0.740306 −0.370153 0.928971i \(-0.620695\pi\)
−0.370153 + 0.928971i \(0.620695\pi\)
\(810\) 0 0
\(811\) 16.8249 0.590801 0.295400 0.955374i \(-0.404547\pi\)
0.295400 + 0.955374i \(0.404547\pi\)
\(812\) 0 0
\(813\) 0.288520 0.0101188
\(814\) 0 0
\(815\) −16.3931 −0.574226
\(816\) 0 0
\(817\) −40.9504 −1.43267
\(818\) 0 0
\(819\) −3.83221 −0.133908
\(820\) 0 0
\(821\) −2.59388 −0.0905272 −0.0452636 0.998975i \(-0.514413\pi\)
−0.0452636 + 0.998975i \(0.514413\pi\)
\(822\) 0 0
\(823\) −0.786230 −0.0274063 −0.0137031 0.999906i \(-0.504362\pi\)
−0.0137031 + 0.999906i \(0.504362\pi\)
\(824\) 0 0
\(825\) −1.14637 −0.0399113
\(826\) 0 0
\(827\) −10.2211 −0.355423 −0.177712 0.984083i \(-0.556869\pi\)
−0.177712 + 0.984083i \(0.556869\pi\)
\(828\) 0 0
\(829\) −8.97437 −0.311693 −0.155846 0.987781i \(-0.549810\pi\)
−0.155846 + 0.987781i \(0.549810\pi\)
\(830\) 0 0
\(831\) −21.3288 −0.739889
\(832\) 0 0
\(833\) 17.6644 0.612036
\(834\) 0 0
\(835\) −3.04598 −0.105411
\(836\) 0 0
\(837\) 2.19656 0.0759241
\(838\) 0 0
\(839\) −19.6069 −0.676905 −0.338452 0.940984i \(-0.609903\pi\)
−0.338452 + 0.940984i \(0.609903\pi\)
\(840\) 0 0
\(841\) 35.4611 1.22280
\(842\) 0 0
\(843\) 26.6676 0.918480
\(844\) 0 0
\(845\) −12.2713 −0.422146
\(846\) 0 0
\(847\) 43.4826 1.49408
\(848\) 0 0
\(849\) 25.9185 0.889521
\(850\) 0 0
\(851\) 2.48929 0.0853317
\(852\) 0 0
\(853\) −14.3650 −0.491847 −0.245923 0.969289i \(-0.579091\pi\)
−0.245923 + 0.969289i \(0.579091\pi\)
\(854\) 0 0
\(855\) 3.83221 0.131059
\(856\) 0 0
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) 4.85425 0.165625 0.0828125 0.996565i \(-0.473610\pi\)
0.0828125 + 0.996565i \(0.473610\pi\)
\(860\) 0 0
\(861\) 50.7293 1.72885
\(862\) 0 0
\(863\) −41.8139 −1.42336 −0.711681 0.702503i \(-0.752066\pi\)
−0.711681 + 0.702503i \(0.752066\pi\)
\(864\) 0 0
\(865\) −24.9357 −0.847840
\(866\) 0 0
\(867\) 15.1966 0.516102
\(868\) 0 0
\(869\) −0.671153 −0.0227673
\(870\) 0 0
\(871\) 9.87506 0.334604
\(872\) 0 0
\(873\) 11.5640 0.391383
\(874\) 0 0
\(875\) −4.48929 −0.151766
\(876\) 0 0
\(877\) −1.27973 −0.0432134 −0.0216067 0.999767i \(-0.506878\pi\)
−0.0216067 + 0.999767i \(0.506878\pi\)
\(878\) 0 0
\(879\) −13.6503 −0.460415
\(880\) 0 0
\(881\) 52.9687 1.78456 0.892281 0.451481i \(-0.149104\pi\)
0.892281 + 0.451481i \(0.149104\pi\)
\(882\) 0 0
\(883\) 17.5479 0.590534 0.295267 0.955415i \(-0.404591\pi\)
0.295267 + 0.955415i \(0.404591\pi\)
\(884\) 0 0
\(885\) −15.0073 −0.504466
\(886\) 0 0
\(887\) 11.2713 0.378453 0.189227 0.981933i \(-0.439402\pi\)
0.189227 + 0.981933i \(0.439402\pi\)
\(888\) 0 0
\(889\) 24.6100 0.825394
\(890\) 0 0
\(891\) 1.14637 0.0384047
\(892\) 0 0
\(893\) −5.89962 −0.197423
\(894\) 0 0
\(895\) 6.68585 0.223483
\(896\) 0 0
\(897\) 0.853635 0.0285020
\(898\) 0 0
\(899\) 17.6357 0.588182
\(900\) 0 0
\(901\) 5.41033 0.180244
\(902\) 0 0
\(903\) −47.9718 −1.59640
\(904\) 0 0
\(905\) −20.6002 −0.684772
\(906\) 0 0
\(907\) −27.2045 −0.903311 −0.451656 0.892192i \(-0.649166\pi\)
−0.451656 + 0.892192i \(0.649166\pi\)
\(908\) 0 0
\(909\) 11.6932 0.387839
\(910\) 0 0
\(911\) 3.41454 0.113129 0.0565643 0.998399i \(-0.481985\pi\)
0.0565643 + 0.998399i \(0.481985\pi\)
\(912\) 0 0
\(913\) −6.46052 −0.213812
\(914\) 0 0
\(915\) 5.83221 0.192807
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −20.4998 −0.676225 −0.338113 0.941106i \(-0.609788\pi\)
−0.338113 + 0.941106i \(0.609788\pi\)
\(920\) 0 0
\(921\) −9.78937 −0.322571
\(922\) 0 0
\(923\) −3.68898 −0.121424
\(924\) 0 0
\(925\) −2.48929 −0.0818473
\(926\) 0 0
\(927\) 19.7220 0.647754
\(928\) 0 0
\(929\) 38.2640 1.25540 0.627700 0.778455i \(-0.283996\pi\)
0.627700 + 0.778455i \(0.283996\pi\)
\(930\) 0 0
\(931\) 50.4078 1.65205
\(932\) 0 0
\(933\) −3.32885 −0.108982
\(934\) 0 0
\(935\) 1.53948 0.0503464
\(936\) 0 0
\(937\) 35.6791 1.16559 0.582793 0.812621i \(-0.301960\pi\)
0.582793 + 0.812621i \(0.301960\pi\)
\(938\) 0 0
\(939\) 18.8824 0.616204
\(940\) 0 0
\(941\) 15.2321 0.496551 0.248275 0.968689i \(-0.420136\pi\)
0.248275 + 0.968689i \(0.420136\pi\)
\(942\) 0 0
\(943\) −11.3001 −0.367981
\(944\) 0 0
\(945\) 4.48929 0.146037
\(946\) 0 0
\(947\) −6.29273 −0.204486 −0.102243 0.994759i \(-0.532602\pi\)
−0.102243 + 0.994759i \(0.532602\pi\)
\(948\) 0 0
\(949\) −11.1856 −0.363100
\(950\) 0 0
\(951\) 8.06740 0.261604
\(952\) 0 0
\(953\) 20.3418 0.658937 0.329469 0.944167i \(-0.393130\pi\)
0.329469 + 0.944167i \(0.393130\pi\)
\(954\) 0 0
\(955\) −4.95402 −0.160308
\(956\) 0 0
\(957\) 9.20390 0.297520
\(958\) 0 0
\(959\) −52.8156 −1.70551
\(960\) 0 0
\(961\) −26.1751 −0.844359
\(962\) 0 0
\(963\) −5.44331 −0.175408
\(964\) 0 0
\(965\) 6.05754 0.194999
\(966\) 0 0
\(967\) 37.6461 1.21062 0.605309 0.795991i \(-0.293050\pi\)
0.605309 + 0.795991i \(0.293050\pi\)
\(968\) 0 0
\(969\) −5.14637 −0.165325
\(970\) 0 0
\(971\) 29.6216 0.950602 0.475301 0.879823i \(-0.342339\pi\)
0.475301 + 0.879823i \(0.342339\pi\)
\(972\) 0 0
\(973\) 36.6044 1.17348
\(974\) 0 0
\(975\) −0.853635 −0.0273382
\(976\) 0 0
\(977\) 21.0158 0.672354 0.336177 0.941799i \(-0.390866\pi\)
0.336177 + 0.941799i \(0.390866\pi\)
\(978\) 0 0
\(979\) −0.115077 −0.00367788
\(980\) 0 0
\(981\) −12.5181 −0.399671
\(982\) 0 0
\(983\) 27.5647 0.879176 0.439588 0.898200i \(-0.355124\pi\)
0.439588 + 0.898200i \(0.355124\pi\)
\(984\) 0 0
\(985\) −24.4078 −0.777697
\(986\) 0 0
\(987\) −6.91117 −0.219985
\(988\) 0 0
\(989\) 10.6858 0.339790
\(990\) 0 0
\(991\) 20.1537 0.640204 0.320102 0.947383i \(-0.396283\pi\)
0.320102 + 0.947383i \(0.396283\pi\)
\(992\) 0 0
\(993\) −26.6388 −0.845356
\(994\) 0 0
\(995\) 3.85677 0.122268
\(996\) 0 0
\(997\) 14.9295 0.472821 0.236410 0.971653i \(-0.424029\pi\)
0.236410 + 0.971653i \(0.424029\pi\)
\(998\) 0 0
\(999\) 2.48929 0.0787576
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 5520.2.a.by.1.1 3
4.3 odd 2 345.2.a.j.1.1 3
12.11 even 2 1035.2.a.n.1.3 3
20.3 even 4 1725.2.b.u.1174.6 6
20.7 even 4 1725.2.b.u.1174.1 6
20.19 odd 2 1725.2.a.bi.1.3 3
60.59 even 2 5175.2.a.br.1.1 3
92.91 even 2 7935.2.a.u.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.a.j.1.1 3 4.3 odd 2
1035.2.a.n.1.3 3 12.11 even 2
1725.2.a.bi.1.3 3 20.19 odd 2
1725.2.b.u.1174.1 6 20.7 even 4
1725.2.b.u.1174.6 6 20.3 even 4
5175.2.a.br.1.1 3 60.59 even 2
5520.2.a.by.1.1 3 1.1 even 1 trivial
7935.2.a.u.1.1 3 92.91 even 2