Properties

Label 550.2.a.f.1.1
Level $550$
Weight $2$
Character 550.1
Self dual yes
Analytic conductor $4.392$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 550.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} +7.00000 q^{17} +2.00000 q^{18} +5.00000 q^{19} -3.00000 q^{21} -1.00000 q^{22} +6.00000 q^{23} -1.00000 q^{24} -6.00000 q^{26} -5.00000 q^{27} -3.00000 q^{28} +5.00000 q^{29} -3.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -7.00000 q^{34} -2.00000 q^{36} -3.00000 q^{37} -5.00000 q^{38} +6.00000 q^{39} +2.00000 q^{41} +3.00000 q^{42} -4.00000 q^{43} +1.00000 q^{44} -6.00000 q^{46} +2.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} +7.00000 q^{51} +6.00000 q^{52} +1.00000 q^{53} +5.00000 q^{54} +3.00000 q^{56} +5.00000 q^{57} -5.00000 q^{58} -10.0000 q^{59} +7.00000 q^{61} +3.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -8.00000 q^{67} +7.00000 q^{68} +6.00000 q^{69} +7.00000 q^{71} +2.00000 q^{72} -14.0000 q^{73} +3.00000 q^{74} +5.00000 q^{76} -3.00000 q^{77} -6.00000 q^{78} +10.0000 q^{79} +1.00000 q^{81} -2.00000 q^{82} +6.00000 q^{83} -3.00000 q^{84} +4.00000 q^{86} +5.00000 q^{87} -1.00000 q^{88} -15.0000 q^{89} -18.0000 q^{91} +6.00000 q^{92} -3.00000 q^{93} -2.00000 q^{94} -1.00000 q^{96} +12.0000 q^{97} -2.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 2.00000 0.471405
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) −1.00000 −0.213201
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) −5.00000 −0.962250
\(28\) −3.00000 −0.566947
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) −7.00000 −1.20049
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) −5.00000 −0.811107
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 3.00000 0.462910
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 7.00000 0.980196
\(52\) 6.00000 0.832050
\(53\) 1.00000 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) 5.00000 0.662266
\(58\) −5.00000 −0.656532
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 3.00000 0.381000
\(63\) 6.00000 0.755929
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 7.00000 0.848875
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 7.00000 0.830747 0.415374 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(72\) 2.00000 0.235702
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) −3.00000 −0.341882
\(78\) −6.00000 −0.679366
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −3.00000 −0.327327
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 5.00000 0.536056
\(88\) −1.00000 −0.106600
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) −18.0000 −1.88691
\(92\) 6.00000 0.625543
\(93\) −3.00000 −0.311086
\(94\) −2.00000 −0.206284
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) −2.00000 −0.202031
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −7.00000 −0.693103
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −5.00000 −0.481125
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) −3.00000 −0.283473
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) −5.00000 −0.468293
\(115\) 0 0
\(116\) 5.00000 0.464238
\(117\) −12.0000 −1.10940
\(118\) 10.0000 0.920575
\(119\) −21.0000 −1.92507
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −7.00000 −0.633750
\(123\) 2.00000 0.180334
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) −6.00000 −0.534522
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 17.0000 1.48530 0.742648 0.669681i \(-0.233569\pi\)
0.742648 + 0.669681i \(0.233569\pi\)
\(132\) 1.00000 0.0870388
\(133\) −15.0000 −1.30066
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −7.00000 −0.600245
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) −6.00000 −0.510754
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) −7.00000 −0.587427
\(143\) 6.00000 0.501745
\(144\) −2.00000 −0.166667
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) 2.00000 0.164957
\(148\) −3.00000 −0.246598
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) −5.00000 −0.405554
\(153\) −14.0000 −1.13183
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) −3.00000 −0.239426 −0.119713 0.992809i \(-0.538197\pi\)
−0.119713 + 0.992809i \(0.538197\pi\)
\(158\) −10.0000 −0.795557
\(159\) 1.00000 0.0793052
\(160\) 0 0
\(161\) −18.0000 −1.41860
\(162\) −1.00000 −0.0785674
\(163\) −19.0000 −1.48819 −0.744097 0.668071i \(-0.767120\pi\)
−0.744097 + 0.668071i \(0.767120\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 3.00000 0.231455
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) −10.0000 −0.764719
\(172\) −4.00000 −0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) −5.00000 −0.379049
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −10.0000 −0.751646
\(178\) 15.0000 1.12430
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 18.0000 1.33425
\(183\) 7.00000 0.517455
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 3.00000 0.219971
\(187\) 7.00000 0.511891
\(188\) 2.00000 0.145865
\(189\) 15.0000 1.09109
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 1.00000 0.0721688
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 2.00000 0.142134
\(199\) −25.0000 −1.77220 −0.886102 0.463491i \(-0.846597\pi\)
−0.886102 + 0.463491i \(0.846597\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) −2.00000 −0.140720
\(203\) −15.0000 −1.05279
\(204\) 7.00000 0.490098
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) −12.0000 −0.834058
\(208\) 6.00000 0.416025
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) −23.0000 −1.58339 −0.791693 0.610920i \(-0.790800\pi\)
−0.791693 + 0.610920i \(0.790800\pi\)
\(212\) 1.00000 0.0686803
\(213\) 7.00000 0.479632
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) 5.00000 0.340207
\(217\) 9.00000 0.610960
\(218\) 10.0000 0.677285
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) 42.0000 2.82523
\(222\) 3.00000 0.201347
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) 3.00000 0.200446
\(225\) 0 0
\(226\) −16.0000 −1.06430
\(227\) 2.00000 0.132745 0.0663723 0.997795i \(-0.478857\pi\)
0.0663723 + 0.997795i \(0.478857\pi\)
\(228\) 5.00000 0.331133
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) −5.00000 −0.328266
\(233\) −9.00000 −0.589610 −0.294805 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) 12.0000 0.784465
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) 10.0000 0.649570
\(238\) 21.0000 1.36123
\(239\) 10.0000 0.646846 0.323423 0.946254i \(-0.395166\pi\)
0.323423 + 0.946254i \(0.395166\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 16.0000 1.02640
\(244\) 7.00000 0.448129
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 30.0000 1.90885
\(248\) 3.00000 0.190500
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 6.00000 0.377964
\(253\) 6.00000 0.377217
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 4.00000 0.249029
\(259\) 9.00000 0.559233
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) −17.0000 −1.05026
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 0 0
\(266\) 15.0000 0.919709
\(267\) −15.0000 −0.917985
\(268\) −8.00000 −0.488678
\(269\) −20.0000 −1.21942 −0.609711 0.792624i \(-0.708714\pi\)
−0.609711 + 0.792624i \(0.708714\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 7.00000 0.424437
\(273\) −18.0000 −1.08941
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 20.0000 1.19952
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −2.00000 −0.119098
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 7.00000 0.415374
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) −6.00000 −0.354169
\(288\) 2.00000 0.117851
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) −14.0000 −0.819288
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) 3.00000 0.174371
\(297\) −5.00000 −0.290129
\(298\) −15.0000 −0.868927
\(299\) 36.0000 2.08193
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) −2.00000 −0.115087
\(303\) 2.00000 0.114897
\(304\) 5.00000 0.286770
\(305\) 0 0
\(306\) 14.0000 0.800327
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) −3.00000 −0.170941
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −3.00000 −0.170114 −0.0850572 0.996376i \(-0.527107\pi\)
−0.0850572 + 0.996376i \(0.527107\pi\)
\(312\) −6.00000 −0.339683
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 3.00000 0.169300
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 7.00000 0.393159 0.196580 0.980488i \(-0.437017\pi\)
0.196580 + 0.980488i \(0.437017\pi\)
\(318\) −1.00000 −0.0560772
\(319\) 5.00000 0.279946
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 18.0000 1.00310
\(323\) 35.0000 1.94745
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 19.0000 1.05231
\(327\) −10.0000 −0.553001
\(328\) −2.00000 −0.110432
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 6.00000 0.329293
\(333\) 6.00000 0.328798
\(334\) 3.00000 0.164153
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) 17.0000 0.926049 0.463025 0.886345i \(-0.346764\pi\)
0.463025 + 0.886345i \(0.346764\pi\)
\(338\) −23.0000 −1.25104
\(339\) 16.0000 0.869001
\(340\) 0 0
\(341\) −3.00000 −0.162459
\(342\) 10.0000 0.540738
\(343\) 15.0000 0.809924
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 5.00000 0.268028
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) −30.0000 −1.60128
\(352\) −1.00000 −0.0533002
\(353\) −34.0000 −1.80964 −0.904819 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 10.0000 0.531494
\(355\) 0 0
\(356\) −15.0000 −0.794998
\(357\) −21.0000 −1.11144
\(358\) 0 0
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) −2.00000 −0.105118
\(363\) 1.00000 0.0524864
\(364\) −18.0000 −0.943456
\(365\) 0 0
\(366\) −7.00000 −0.365896
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 6.00000 0.312772
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) −3.00000 −0.155752
\(372\) −3.00000 −0.155543
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) −7.00000 −0.361961
\(375\) 0 0
\(376\) −2.00000 −0.103142
\(377\) 30.0000 1.54508
\(378\) −15.0000 −0.771517
\(379\) −30.0000 −1.54100 −0.770498 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) −12.0000 −0.613973
\(383\) −34.0000 −1.73732 −0.868659 0.495410i \(-0.835018\pi\)
−0.868659 + 0.495410i \(0.835018\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −11.0000 −0.559885
\(387\) 8.00000 0.406663
\(388\) 12.0000 0.609208
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 42.0000 2.12403
\(392\) −2.00000 −0.101015
\(393\) 17.0000 0.857537
\(394\) −12.0000 −0.604551
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 25.0000 1.25314
\(399\) −15.0000 −0.750939
\(400\) 0 0
\(401\) −13.0000 −0.649189 −0.324595 0.945853i \(-0.605228\pi\)
−0.324595 + 0.945853i \(0.605228\pi\)
\(402\) 8.00000 0.399004
\(403\) −18.0000 −0.896644
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 15.0000 0.744438
\(407\) −3.00000 −0.148704
\(408\) −7.00000 −0.346552
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) −4.00000 −0.197066
\(413\) 30.0000 1.47620
\(414\) 12.0000 0.589768
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) −20.0000 −0.979404
\(418\) −5.00000 −0.244558
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) 23.0000 1.11962
\(423\) −4.00000 −0.194487
\(424\) −1.00000 −0.0485643
\(425\) 0 0
\(426\) −7.00000 −0.339151
\(427\) −21.0000 −1.01626
\(428\) −8.00000 −0.386695
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) −5.00000 −0.240563
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) −9.00000 −0.432014
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 30.0000 1.43509
\(438\) 14.0000 0.668946
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) −42.0000 −1.99774
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −3.00000 −0.142374
\(445\) 0 0
\(446\) −6.00000 −0.284108
\(447\) 15.0000 0.709476
\(448\) −3.00000 −0.141737
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) 16.0000 0.752577
\(453\) 2.00000 0.0939682
\(454\) −2.00000 −0.0938647
\(455\) 0 0
\(456\) −5.00000 −0.234146
\(457\) −3.00000 −0.140334 −0.0701670 0.997535i \(-0.522353\pi\)
−0.0701670 + 0.997535i \(0.522353\pi\)
\(458\) −10.0000 −0.467269
\(459\) −35.0000 −1.63366
\(460\) 0 0
\(461\) 27.0000 1.25752 0.628758 0.777601i \(-0.283564\pi\)
0.628758 + 0.777601i \(0.283564\pi\)
\(462\) 3.00000 0.139573
\(463\) −34.0000 −1.58011 −0.790057 0.613033i \(-0.789949\pi\)
−0.790057 + 0.613033i \(0.789949\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) 9.00000 0.416917
\(467\) −23.0000 −1.06431 −0.532157 0.846646i \(-0.678618\pi\)
−0.532157 + 0.846646i \(0.678618\pi\)
\(468\) −12.0000 −0.554700
\(469\) 24.0000 1.10822
\(470\) 0 0
\(471\) −3.00000 −0.138233
\(472\) 10.0000 0.460287
\(473\) −4.00000 −0.183920
\(474\) −10.0000 −0.459315
\(475\) 0 0
\(476\) −21.0000 −0.962533
\(477\) −2.00000 −0.0915737
\(478\) −10.0000 −0.457389
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −18.0000 −0.820729
\(482\) 18.0000 0.819878
\(483\) −18.0000 −0.819028
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −16.0000 −0.725775
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) −7.00000 −0.316875
\(489\) −19.0000 −0.859210
\(490\) 0 0
\(491\) −3.00000 −0.135388 −0.0676941 0.997706i \(-0.521564\pi\)
−0.0676941 + 0.997706i \(0.521564\pi\)
\(492\) 2.00000 0.0901670
\(493\) 35.0000 1.57632
\(494\) −30.0000 −1.34976
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) −21.0000 −0.941979
\(498\) −6.00000 −0.268866
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) −3.00000 −0.134030
\(502\) −2.00000 −0.0892644
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −6.00000 −0.267261
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) 23.0000 1.02147
\(508\) −8.00000 −0.354943
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 0 0
\(511\) 42.0000 1.85797
\(512\) −1.00000 −0.0441942
\(513\) −25.0000 −1.10378
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 2.00000 0.0879599
\(518\) −9.00000 −0.395437
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 10.0000 0.437688
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 17.0000 0.742648
\(525\) 0 0
\(526\) 9.00000 0.392419
\(527\) −21.0000 −0.914774
\(528\) 1.00000 0.0435194
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 20.0000 0.867926
\(532\) −15.0000 −0.650332
\(533\) 12.0000 0.519778
\(534\) 15.0000 0.649113
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 20.0000 0.862261
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −23.0000 −0.988847 −0.494424 0.869221i \(-0.664621\pi\)
−0.494424 + 0.869221i \(0.664621\pi\)
\(542\) 8.00000 0.343629
\(543\) 2.00000 0.0858282
\(544\) −7.00000 −0.300123
\(545\) 0 0
\(546\) 18.0000 0.770329
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 12.0000 0.512615
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 25.0000 1.06504
\(552\) −6.00000 −0.255377
\(553\) −30.0000 −1.27573
\(554\) −12.0000 −0.509831
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −6.00000 −0.254000
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 7.00000 0.295540
\(562\) 18.0000 0.759284
\(563\) 6.00000 0.252870 0.126435 0.991975i \(-0.459647\pi\)
0.126435 + 0.991975i \(0.459647\pi\)
\(564\) 2.00000 0.0842152
\(565\) 0 0
\(566\) −6.00000 −0.252199
\(567\) −3.00000 −0.125988
\(568\) −7.00000 −0.293713
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 27.0000 1.12991 0.564957 0.825120i \(-0.308893\pi\)
0.564957 + 0.825120i \(0.308893\pi\)
\(572\) 6.00000 0.250873
\(573\) 12.0000 0.501307
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) −2.00000 −0.0833333
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) −32.0000 −1.33102
\(579\) 11.0000 0.457144
\(580\) 0 0
\(581\) −18.0000 −0.746766
\(582\) −12.0000 −0.497416
\(583\) 1.00000 0.0414158
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 27.0000 1.11441 0.557205 0.830375i \(-0.311874\pi\)
0.557205 + 0.830375i \(0.311874\pi\)
\(588\) 2.00000 0.0824786
\(589\) −15.0000 −0.618064
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) −3.00000 −0.123299
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 5.00000 0.205152
\(595\) 0 0
\(596\) 15.0000 0.614424
\(597\) −25.0000 −1.02318
\(598\) −36.0000 −1.47215
\(599\) 45.0000 1.83865 0.919325 0.393499i \(-0.128735\pi\)
0.919325 + 0.393499i \(0.128735\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) −12.0000 −0.489083
\(603\) 16.0000 0.651570
\(604\) 2.00000 0.0813788
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) 47.0000 1.90767 0.953836 0.300329i \(-0.0970966\pi\)
0.953836 + 0.300329i \(0.0970966\pi\)
\(608\) −5.00000 −0.202777
\(609\) −15.0000 −0.607831
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) −14.0000 −0.565916
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −8.00000 −0.322068 −0.161034 0.986949i \(-0.551483\pi\)
−0.161034 + 0.986949i \(0.551483\pi\)
\(618\) 4.00000 0.160904
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) −30.0000 −1.20386
\(622\) 3.00000 0.120289
\(623\) 45.0000 1.80289
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) 5.00000 0.199681
\(628\) −3.00000 −0.119713
\(629\) −21.0000 −0.837325
\(630\) 0 0
\(631\) −33.0000 −1.31371 −0.656855 0.754017i \(-0.728113\pi\)
−0.656855 + 0.754017i \(0.728113\pi\)
\(632\) −10.0000 −0.397779
\(633\) −23.0000 −0.914168
\(634\) −7.00000 −0.278006
\(635\) 0 0
\(636\) 1.00000 0.0396526
\(637\) 12.0000 0.475457
\(638\) −5.00000 −0.197952
\(639\) −14.0000 −0.553831
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 8.00000 0.315735
\(643\) −19.0000 −0.749287 −0.374643 0.927169i \(-0.622235\pi\)
−0.374643 + 0.927169i \(0.622235\pi\)
\(644\) −18.0000 −0.709299
\(645\) 0 0
\(646\) −35.0000 −1.37706
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −10.0000 −0.392534
\(650\) 0 0
\(651\) 9.00000 0.352738
\(652\) −19.0000 −0.744097
\(653\) 31.0000 1.21312 0.606562 0.795036i \(-0.292548\pi\)
0.606562 + 0.795036i \(0.292548\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 28.0000 1.09238
\(658\) 6.00000 0.233904
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 28.0000 1.08825
\(663\) 42.0000 1.63114
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 30.0000 1.16160
\(668\) −3.00000 −0.116073
\(669\) 6.00000 0.231973
\(670\) 0 0
\(671\) 7.00000 0.270232
\(672\) 3.00000 0.115728
\(673\) −29.0000 −1.11787 −0.558934 0.829212i \(-0.688789\pi\)
−0.558934 + 0.829212i \(0.688789\pi\)
\(674\) −17.0000 −0.654816
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −28.0000 −1.07613 −0.538064 0.842904i \(-0.680844\pi\)
−0.538064 + 0.842904i \(0.680844\pi\)
\(678\) −16.0000 −0.614476
\(679\) −36.0000 −1.38155
\(680\) 0 0
\(681\) 2.00000 0.0766402
\(682\) 3.00000 0.114876
\(683\) 31.0000 1.18618 0.593091 0.805135i \(-0.297907\pi\)
0.593091 + 0.805135i \(0.297907\pi\)
\(684\) −10.0000 −0.382360
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) 10.0000 0.381524
\(688\) −4.00000 −0.152499
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −38.0000 −1.44559 −0.722794 0.691063i \(-0.757142\pi\)
−0.722794 + 0.691063i \(0.757142\pi\)
\(692\) −14.0000 −0.532200
\(693\) 6.00000 0.227921
\(694\) 18.0000 0.683271
\(695\) 0 0
\(696\) −5.00000 −0.189525
\(697\) 14.0000 0.530288
\(698\) −30.0000 −1.13552
\(699\) −9.00000 −0.340411
\(700\) 0 0
\(701\) 7.00000 0.264386 0.132193 0.991224i \(-0.457798\pi\)
0.132193 + 0.991224i \(0.457798\pi\)
\(702\) 30.0000 1.13228
\(703\) −15.0000 −0.565736
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 34.0000 1.27961
\(707\) −6.00000 −0.225653
\(708\) −10.0000 −0.375823
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) −20.0000 −0.750059
\(712\) 15.0000 0.562149
\(713\) −18.0000 −0.674105
\(714\) 21.0000 0.785905
\(715\) 0 0
\(716\) 0 0
\(717\) 10.0000 0.373457
\(718\) 20.0000 0.746393
\(719\) 25.0000 0.932343 0.466171 0.884694i \(-0.345633\pi\)
0.466171 + 0.884694i \(0.345633\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) −6.00000 −0.223297
\(723\) −18.0000 −0.669427
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) −1.00000 −0.0371135
\(727\) 22.0000 0.815935 0.407967 0.912996i \(-0.366238\pi\)
0.407967 + 0.912996i \(0.366238\pi\)
\(728\) 18.0000 0.667124
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −28.0000 −1.03562
\(732\) 7.00000 0.258727
\(733\) −24.0000 −0.886460 −0.443230 0.896408i \(-0.646168\pi\)
−0.443230 + 0.896408i \(0.646168\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −8.00000 −0.294684
\(738\) 4.00000 0.147242
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) 30.0000 1.10208
\(742\) 3.00000 0.110133
\(743\) 21.0000 0.770415 0.385208 0.922830i \(-0.374130\pi\)
0.385208 + 0.922830i \(0.374130\pi\)
\(744\) 3.00000 0.109985
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) −12.0000 −0.439057
\(748\) 7.00000 0.255945
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) 17.0000 0.620339 0.310169 0.950681i \(-0.399614\pi\)
0.310169 + 0.950681i \(0.399614\pi\)
\(752\) 2.00000 0.0729325
\(753\) 2.00000 0.0728841
\(754\) −30.0000 −1.09254
\(755\) 0 0
\(756\) 15.0000 0.545545
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 30.0000 1.08965
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 8.00000 0.289809
\(763\) 30.0000 1.08607
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 34.0000 1.22847
\(767\) −60.0000 −2.16647
\(768\) 1.00000 0.0360844
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) 11.0000 0.395899
\(773\) −19.0000 −0.683383 −0.341691 0.939812i \(-0.611000\pi\)
−0.341691 + 0.939812i \(0.611000\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) −12.0000 −0.430775
\(777\) 9.00000 0.322873
\(778\) 30.0000 1.07555
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) 7.00000 0.250480
\(782\) −42.0000 −1.50192
\(783\) −25.0000 −0.893427
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) −17.0000 −0.606370
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 12.0000 0.427482
\(789\) −9.00000 −0.320408
\(790\) 0 0
\(791\) −48.0000 −1.70668
\(792\) 2.00000 0.0710669
\(793\) 42.0000 1.49146
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −25.0000 −0.886102
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 15.0000 0.530994
\(799\) 14.0000 0.495284
\(800\) 0 0
\(801\) 30.0000 1.06000
\(802\) 13.0000 0.459046
\(803\) −14.0000 −0.494049
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 18.0000 0.634023
\(807\) −20.0000 −0.704033
\(808\) −2.00000 −0.0703598
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 7.00000 0.245803 0.122902 0.992419i \(-0.460780\pi\)
0.122902 + 0.992419i \(0.460780\pi\)
\(812\) −15.0000 −0.526397
\(813\) −8.00000 −0.280572
\(814\) 3.00000 0.105150
\(815\) 0 0
\(816\) 7.00000 0.245049
\(817\) −20.0000 −0.699711
\(818\) 20.0000 0.699284
\(819\) 36.0000 1.25794
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) −12.0000 −0.418548
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) −30.0000 −1.04383
\(827\) −8.00000 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) −12.0000 −0.417029
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 0 0
\(831\) 12.0000 0.416275
\(832\) 6.00000 0.208013
\(833\) 14.0000 0.485071
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 5.00000 0.172929
\(837\) 15.0000 0.518476
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −32.0000 −1.10279
\(843\) −18.0000 −0.619953
\(844\) −23.0000 −0.791693
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) −3.00000 −0.103081
\(848\) 1.00000 0.0343401
\(849\) 6.00000 0.205919
\(850\) 0 0
\(851\) −18.0000 −0.617032
\(852\) 7.00000 0.239816
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 21.0000 0.718605
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) 7.00000 0.239115 0.119558 0.992827i \(-0.461852\pi\)
0.119558 + 0.992827i \(0.461852\pi\)
\(858\) −6.00000 −0.204837
\(859\) 30.0000 1.02359 0.511793 0.859109i \(-0.328981\pi\)
0.511793 + 0.859109i \(0.328981\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 8.00000 0.272481
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) −16.0000 −0.543702
\(867\) 32.0000 1.08678
\(868\) 9.00000 0.305480
\(869\) 10.0000 0.339227
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 10.0000 0.338643
\(873\) −24.0000 −0.812277
\(874\) −30.0000 −1.01477
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) −20.0000 −0.674967
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 4.00000 0.134687
\(883\) −9.00000 −0.302874 −0.151437 0.988467i \(-0.548390\pi\)
−0.151437 + 0.988467i \(0.548390\pi\)
\(884\) 42.0000 1.41261
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 3.00000 0.100673
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 6.00000 0.200895
\(893\) 10.0000 0.334637
\(894\) −15.0000 −0.501675
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 36.0000 1.20201
\(898\) 30.0000 1.00111
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) 7.00000 0.233204
\(902\) −2.00000 −0.0665927
\(903\) 12.0000 0.399335
\(904\) −16.0000 −0.532152
\(905\) 0 0
\(906\) −2.00000 −0.0664455
\(907\) 57.0000 1.89265 0.946327 0.323211i \(-0.104762\pi\)
0.946327 + 0.323211i \(0.104762\pi\)
\(908\) 2.00000 0.0663723
\(909\) −4.00000 −0.132672
\(910\) 0 0
\(911\) 27.0000 0.894550 0.447275 0.894397i \(-0.352395\pi\)
0.447275 + 0.894397i \(0.352395\pi\)
\(912\) 5.00000 0.165567
\(913\) 6.00000 0.198571
\(914\) 3.00000 0.0992312
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) −51.0000 −1.68417
\(918\) 35.0000 1.15517
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) −27.0000 −0.889198
\(923\) 42.0000 1.38245
\(924\) −3.00000 −0.0986928
\(925\) 0 0
\(926\) 34.0000 1.11731
\(927\) 8.00000 0.262754
\(928\) −5.00000 −0.164133
\(929\) 35.0000 1.14831 0.574156 0.818746i \(-0.305330\pi\)
0.574156 + 0.818746i \(0.305330\pi\)
\(930\) 0 0
\(931\) 10.0000 0.327737
\(932\) −9.00000 −0.294805
\(933\) −3.00000 −0.0982156
\(934\) 23.0000 0.752583
\(935\) 0 0
\(936\) 12.0000 0.392232
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) −24.0000 −0.783628
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 17.0000 0.554184 0.277092 0.960843i \(-0.410629\pi\)
0.277092 + 0.960843i \(0.410629\pi\)
\(942\) 3.00000 0.0977453
\(943\) 12.0000 0.390774
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 27.0000 0.877382 0.438691 0.898638i \(-0.355442\pi\)
0.438691 + 0.898638i \(0.355442\pi\)
\(948\) 10.0000 0.324785
\(949\) −84.0000 −2.72676
\(950\) 0 0
\(951\) 7.00000 0.226991
\(952\) 21.0000 0.680614
\(953\) −39.0000 −1.26333 −0.631667 0.775240i \(-0.717629\pi\)
−0.631667 + 0.775240i \(0.717629\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) 10.0000 0.323423
\(957\) 5.00000 0.161627
\(958\) 0 0
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 18.0000 0.580343
\(963\) 16.0000 0.515593
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) 18.0000 0.579141
\(967\) 27.0000 0.868261 0.434131 0.900850i \(-0.357056\pi\)
0.434131 + 0.900850i \(0.357056\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 35.0000 1.12436
\(970\) 0 0
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 16.0000 0.513200
\(973\) 60.0000 1.92351
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) 7.00000 0.224065
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 19.0000 0.607553
\(979\) −15.0000 −0.479402
\(980\) 0 0
\(981\) 20.0000 0.638551
\(982\) 3.00000 0.0957338
\(983\) −54.0000 −1.72233 −0.861166 0.508323i \(-0.830265\pi\)
−0.861166 + 0.508323i \(0.830265\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) −35.0000 −1.11463
\(987\) −6.00000 −0.190982
\(988\) 30.0000 0.954427
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 3.00000 0.0952501
\(993\) −28.0000 −0.888553
\(994\) 21.0000 0.666080
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) 32.0000 1.01345 0.506725 0.862108i \(-0.330856\pi\)
0.506725 + 0.862108i \(0.330856\pi\)
\(998\) −20.0000 −0.633089
\(999\) 15.0000 0.474579
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 550.2.a.f.1.1 1
3.2 odd 2 4950.2.a.bc.1.1 1
4.3 odd 2 4400.2.a.l.1.1 1
5.2 odd 4 550.2.b.a.199.1 2
5.3 odd 4 550.2.b.a.199.2 2
5.4 even 2 110.2.a.b.1.1 1
11.10 odd 2 6050.2.a.bj.1.1 1
15.2 even 4 4950.2.c.m.199.2 2
15.8 even 4 4950.2.c.m.199.1 2
15.14 odd 2 990.2.a.d.1.1 1
20.3 even 4 4400.2.b.i.4049.1 2
20.7 even 4 4400.2.b.i.4049.2 2
20.19 odd 2 880.2.a.i.1.1 1
35.34 odd 2 5390.2.a.bf.1.1 1
40.19 odd 2 3520.2.a.h.1.1 1
40.29 even 2 3520.2.a.y.1.1 1
55.54 odd 2 1210.2.a.b.1.1 1
60.59 even 2 7920.2.a.d.1.1 1
220.219 even 2 9680.2.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.b.1.1 1 5.4 even 2
550.2.a.f.1.1 1 1.1 even 1 trivial
550.2.b.a.199.1 2 5.2 odd 4
550.2.b.a.199.2 2 5.3 odd 4
880.2.a.i.1.1 1 20.19 odd 2
990.2.a.d.1.1 1 15.14 odd 2
1210.2.a.b.1.1 1 55.54 odd 2
3520.2.a.h.1.1 1 40.19 odd 2
3520.2.a.y.1.1 1 40.29 even 2
4400.2.a.l.1.1 1 4.3 odd 2
4400.2.b.i.4049.1 2 20.3 even 4
4400.2.b.i.4049.2 2 20.7 even 4
4950.2.a.bc.1.1 1 3.2 odd 2
4950.2.c.m.199.1 2 15.8 even 4
4950.2.c.m.199.2 2 15.2 even 4
5390.2.a.bf.1.1 1 35.34 odd 2
6050.2.a.bj.1.1 1 11.10 odd 2
7920.2.a.d.1.1 1 60.59 even 2
9680.2.a.x.1.1 1 220.219 even 2