Properties

Label 722.2.a.d.1.1
Level $722$
Weight $2$
Character 722.1
Self dual yes
Analytic conductor $5.765$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.76519902594\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 722.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -2.00000 q^{18} +4.00000 q^{21} +3.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} +5.00000 q^{27} -4.00000 q^{28} -2.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} -6.00000 q^{34} -2.00000 q^{36} +10.0000 q^{37} +2.00000 q^{39} -9.00000 q^{41} +4.00000 q^{42} -4.00000 q^{43} +3.00000 q^{44} -6.00000 q^{46} -1.00000 q^{48} +9.00000 q^{49} -5.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} +5.00000 q^{54} -4.00000 q^{56} +9.00000 q^{59} -4.00000 q^{61} -2.00000 q^{62} +8.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} +7.00000 q^{67} -6.00000 q^{68} +6.00000 q^{69} +6.00000 q^{71} -2.00000 q^{72} -1.00000 q^{73} +10.0000 q^{74} +5.00000 q^{75} -12.0000 q^{77} +2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{81} -9.00000 q^{82} +3.00000 q^{83} +4.00000 q^{84} -4.00000 q^{86} +3.00000 q^{88} -6.00000 q^{89} +8.00000 q^{91} -6.00000 q^{92} +2.00000 q^{93} -1.00000 q^{96} -17.0000 q^{97} +9.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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 \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −2.00000 −0.471405
\(19\) 0 0
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 3.00000 0.639602
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) −2.00000 −0.392232
\(27\) 5.00000 0.962250
\(28\) −4.00000 −0.755929
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.00000 −0.522233
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 4.00000 0.617213
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) −5.00000 −0.707107
\(51\) 6.00000 0.840168
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 0 0
\(58\) 0 0
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) −2.00000 −0.254000
\(63\) 8.00000 1.00791
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.00000 −0.369274
\(67\) 7.00000 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) −6.00000 −0.727607
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −2.00000 −0.235702
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 10.0000 1.16248
\(75\) 5.00000 0.577350
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 2.00000 0.226455
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −9.00000 −0.993884
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) −6.00000 −0.625543
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −17.0000 −1.72609 −0.863044 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) 9.00000 0.909137
\(99\) −6.00000 −0.603023
\(100\) −5.00000 −0.500000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 6.00000 0.594089
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 5.00000 0.481125
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) −4.00000 −0.377964
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 9.00000 0.828517
\(119\) 24.0000 2.20008
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −4.00000 −0.362143
\(123\) 9.00000 0.811503
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 8.00000 0.712697
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 9.00000 0.786334 0.393167 0.919467i \(-0.371379\pi\)
0.393167 + 0.919467i \(0.371379\pi\)
\(132\) −3.00000 −0.261116
\(133\) 0 0
\(134\) 7.00000 0.604708
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 6.00000 0.510754
\(139\) 11.0000 0.933008 0.466504 0.884519i \(-0.345513\pi\)
0.466504 + 0.884519i \(0.345513\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) −6.00000 −0.501745
\(144\) −2.00000 −0.166667
\(145\) 0 0
\(146\) −1.00000 −0.0827606
\(147\) −9.00000 −0.742307
\(148\) 10.0000 0.821995
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 5.00000 0.408248
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) −12.0000 −0.966988
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −16.0000 −1.27694 −0.638470 0.769647i \(-0.720432\pi\)
−0.638470 + 0.769647i \(0.720432\pi\)
\(158\) 4.00000 0.318223
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 24.0000 1.89146
\(162\) 1.00000 0.0785674
\(163\) −19.0000 −1.48819 −0.744097 0.668071i \(-0.767120\pi\)
−0.744097 + 0.668071i \(0.767120\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) 3.00000 0.232845
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 4.00000 0.308607
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 20.0000 1.51186
\(176\) 3.00000 0.226134
\(177\) −9.00000 −0.676481
\(178\) −6.00000 −0.449719
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 8.00000 0.592999
\(183\) 4.00000 0.295689
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 2.00000 0.146647
\(187\) −18.0000 −1.31629
\(188\) 0 0
\(189\) −20.0000 −1.45479
\(190\) 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
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −17.0000 −1.22053
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −6.00000 −0.426401
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) −5.00000 −0.353553
\(201\) −7.00000 −0.493742
\(202\) 0 0
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −2.00000 −0.139347
\(207\) 12.0000 0.834058
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −6.00000 −0.412082
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) 0 0
\(216\) 5.00000 0.340207
\(217\) 8.00000 0.543075
\(218\) 16.0000 1.08366
\(219\) 1.00000 0.0675737
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) −10.0000 −0.671156
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) −4.00000 −0.267261
\(225\) 10.0000 0.666667
\(226\) −15.0000 −0.997785
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) 0 0
\(233\) 3.00000 0.196537 0.0982683 0.995160i \(-0.468670\pi\)
0.0982683 + 0.995160i \(0.468670\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 9.00000 0.585850
\(237\) −4.00000 −0.259828
\(238\) 24.0000 1.55569
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) −2.00000 −0.128565
\(243\) −16.0000 −1.02640
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) 9.00000 0.573819
\(247\) 0 0
\(248\) −2.00000 −0.127000
\(249\) −3.00000 −0.190117
\(250\) 0 0
\(251\) −3.00000 −0.189358 −0.0946792 0.995508i \(-0.530183\pi\)
−0.0946792 + 0.995508i \(0.530183\pi\)
\(252\) 8.00000 0.503953
\(253\) −18.0000 −1.13165
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −3.00000 −0.187135 −0.0935674 0.995613i \(-0.529827\pi\)
−0.0935674 + 0.995613i \(0.529827\pi\)
\(258\) 4.00000 0.249029
\(259\) −40.0000 −2.48548
\(260\) 0 0
\(261\) 0 0
\(262\) 9.00000 0.556022
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 7.00000 0.427593
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −6.00000 −0.363803
\(273\) −8.00000 −0.484182
\(274\) 9.00000 0.543710
\(275\) −15.0000 −0.904534
\(276\) 6.00000 0.361158
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 11.0000 0.659736
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −27.0000 −1.61068 −0.805342 0.592810i \(-0.798019\pi\)
−0.805342 + 0.592810i \(0.798019\pi\)
\(282\) 0 0
\(283\) 5.00000 0.297219 0.148610 0.988896i \(-0.452520\pi\)
0.148610 + 0.988896i \(0.452520\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 36.0000 2.12501
\(288\) −2.00000 −0.117851
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 17.0000 0.996558
\(292\) −1.00000 −0.0585206
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) 15.0000 0.870388
\(298\) 18.0000 1.04271
\(299\) 12.0000 0.693978
\(300\) 5.00000 0.288675
\(301\) 16.0000 0.922225
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 12.0000 0.685994
\(307\) 7.00000 0.399511 0.199756 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) −12.0000 −0.683763
\(309\) 2.00000 0.113776
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 2.00000 0.113228
\(313\) −19.0000 −1.07394 −0.536972 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) −16.0000 −0.902932
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 24.0000 1.33747
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 10.0000 0.554700
\(326\) −19.0000 −1.05231
\(327\) −16.0000 −0.884802
\(328\) −9.00000 −0.496942
\(329\) 0 0
\(330\) 0 0
\(331\) −5.00000 −0.274825 −0.137412 0.990514i \(-0.543879\pi\)
−0.137412 + 0.990514i \(0.543879\pi\)
\(332\) 3.00000 0.164646
\(333\) −20.0000 −1.09599
\(334\) 24.0000 1.31322
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) −11.0000 −0.599208 −0.299604 0.954064i \(-0.596855\pi\)
−0.299604 + 0.954064i \(0.596855\pi\)
\(338\) −9.00000 −0.489535
\(339\) 15.0000 0.814688
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −9.00000 −0.483145 −0.241573 0.970383i \(-0.577663\pi\)
−0.241573 + 0.970383i \(0.577663\pi\)
\(348\) 0 0
\(349\) −4.00000 −0.214115 −0.107058 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(350\) 20.0000 1.06904
\(351\) −10.0000 −0.533761
\(352\) 3.00000 0.159901
\(353\) 3.00000 0.159674 0.0798369 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) −9.00000 −0.478345
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −24.0000 −1.27021
\(358\) −9.00000 −0.475665
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) −2.00000 −0.105118
\(363\) 2.00000 0.104973
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) −6.00000 −0.312772
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 2.00000 0.103695
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) −18.0000 −0.930758
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) −20.0000 −1.02869
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 12.0000 0.613973
\(383\) 36.0000 1.83951 0.919757 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 8.00000 0.406663
\(388\) −17.0000 −0.863044
\(389\) −36.0000 −1.82527 −0.912636 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) 9.00000 0.454569
\(393\) −9.00000 −0.453990
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 27.0000 1.34832 0.674158 0.738587i \(-0.264507\pi\)
0.674158 + 0.738587i \(0.264507\pi\)
\(402\) −7.00000 −0.349128
\(403\) 4.00000 0.199254
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 30.0000 1.48704
\(408\) 6.00000 0.297044
\(409\) −5.00000 −0.247234 −0.123617 0.992330i \(-0.539449\pi\)
−0.123617 + 0.992330i \(0.539449\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) −2.00000 −0.0985329
\(413\) −36.0000 −1.77144
\(414\) 12.0000 0.589768
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −11.0000 −0.538672
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 30.0000 1.45521
\(426\) −6.00000 −0.290701
\(427\) 16.0000 0.774294
\(428\) 0 0
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 5.00000 0.240563
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) 16.0000 0.766261
\(437\) 0 0
\(438\) 1.00000 0.0477818
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) 12.0000 0.570782
\(443\) 9.00000 0.427603 0.213801 0.976877i \(-0.431415\pi\)
0.213801 + 0.976877i \(0.431415\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) −18.0000 −0.851371
\(448\) −4.00000 −0.188982
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 10.0000 0.471405
\(451\) −27.0000 −1.27138
\(452\) −15.0000 −0.705541
\(453\) −10.0000 −0.469841
\(454\) −3.00000 −0.140797
\(455\) 0 0
\(456\) 0 0
\(457\) 5.00000 0.233890 0.116945 0.993138i \(-0.462690\pi\)
0.116945 + 0.993138i \(0.462690\pi\)
\(458\) −16.0000 −0.747631
\(459\) −30.0000 −1.40028
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 12.0000 0.558291
\(463\) −34.0000 −1.58011 −0.790057 0.613033i \(-0.789949\pi\)
−0.790057 + 0.613033i \(0.789949\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 3.00000 0.138972
\(467\) −27.0000 −1.24941 −0.624705 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) 4.00000 0.184900
\(469\) −28.0000 −1.29292
\(470\) 0 0
\(471\) 16.0000 0.737241
\(472\) 9.00000 0.414259
\(473\) −12.0000 −0.551761
\(474\) −4.00000 −0.183726
\(475\) 0 0
\(476\) 24.0000 1.10004
\(477\) 12.0000 0.549442
\(478\) −12.0000 −0.548867
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) −5.00000 −0.227744
\(483\) −24.0000 −1.09204
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) −16.0000 −0.725775
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) −4.00000 −0.181071
\(489\) 19.0000 0.859210
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 9.00000 0.405751
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −24.0000 −1.07655
\(498\) −3.00000 −0.134433
\(499\) −25.0000 −1.11915 −0.559577 0.828778i \(-0.689036\pi\)
−0.559577 + 0.828778i \(0.689036\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) −3.00000 −0.133897
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 8.00000 0.356348
\(505\) 0 0
\(506\) −18.0000 −0.800198
\(507\) 9.00000 0.399704
\(508\) −2.00000 −0.0887357
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −3.00000 −0.132324
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) −40.0000 −1.75750
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −9.00000 −0.394297 −0.197149 0.980374i \(-0.563168\pi\)
−0.197149 + 0.980374i \(0.563168\pi\)
\(522\) 0 0
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) 9.00000 0.393167
\(525\) −20.0000 −0.872872
\(526\) −12.0000 −0.523225
\(527\) 12.0000 0.522728
\(528\) −3.00000 −0.130558
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −18.0000 −0.781133
\(532\) 0 0
\(533\) 18.0000 0.779667
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 7.00000 0.302354
\(537\) 9.00000 0.388379
\(538\) −12.0000 −0.517357
\(539\) 27.0000 1.16297
\(540\) 0 0
\(541\) 44.0000 1.89171 0.945854 0.324593i \(-0.105227\pi\)
0.945854 + 0.324593i \(0.105227\pi\)
\(542\) −16.0000 −0.687259
\(543\) 2.00000 0.0858282
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 9.00000 0.384461
\(549\) 8.00000 0.341432
\(550\) −15.0000 −0.639602
\(551\) 0 0
\(552\) 6.00000 0.255377
\(553\) −16.0000 −0.680389
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) 11.0000 0.466504
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 4.00000 0.169334
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) −27.0000 −1.13893
\(563\) 21.0000 0.885044 0.442522 0.896758i \(-0.354084\pi\)
0.442522 + 0.896758i \(0.354084\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 5.00000 0.210166
\(567\) −4.00000 −0.167984
\(568\) 6.00000 0.251754
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −7.00000 −0.292941 −0.146470 0.989215i \(-0.546791\pi\)
−0.146470 + 0.989215i \(0.546791\pi\)
\(572\) −6.00000 −0.250873
\(573\) −12.0000 −0.501307
\(574\) 36.0000 1.50261
\(575\) 30.0000 1.25109
\(576\) −2.00000 −0.0833333
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 19.0000 0.790296
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 17.0000 0.704673
\(583\) −18.0000 −0.745484
\(584\) −1.00000 −0.0413803
\(585\) 0 0
\(586\) −24.0000 −0.991431
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −9.00000 −0.371154
\(589\) 0 0
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 10.0000 0.410997
\(593\) −21.0000 −0.862367 −0.431183 0.902264i \(-0.641904\pi\)
−0.431183 + 0.902264i \(0.641904\pi\)
\(594\) 15.0000 0.615457
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 10.0000 0.409273
\(598\) 12.0000 0.490716
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 5.00000 0.204124
\(601\) 13.0000 0.530281 0.265141 0.964210i \(-0.414582\pi\)
0.265141 + 0.964210i \(0.414582\pi\)
\(602\) 16.0000 0.652111
\(603\) −14.0000 −0.570124
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) −20.0000 −0.811775 −0.405887 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 12.0000 0.485071
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 7.00000 0.282497
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) 3.00000 0.120775 0.0603877 0.998175i \(-0.480766\pi\)
0.0603877 + 0.998175i \(0.480766\pi\)
\(618\) 2.00000 0.0804518
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) −30.0000 −1.20386
\(622\) −30.0000 −1.20289
\(623\) 24.0000 0.961540
\(624\) 2.00000 0.0800641
\(625\) 25.0000 1.00000
\(626\) −19.0000 −0.759393
\(627\) 0 0
\(628\) −16.0000 −0.638470
\(629\) −60.0000 −2.39236
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 4.00000 0.159111
\(633\) 20.0000 0.794929
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) −18.0000 −0.713186
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 39.0000 1.54041 0.770204 0.637798i \(-0.220155\pi\)
0.770204 + 0.637798i \(0.220155\pi\)
\(642\) 0 0
\(643\) −43.0000 −1.69575 −0.847877 0.530193i \(-0.822120\pi\)
−0.847877 + 0.530193i \(0.822120\pi\)
\(644\) 24.0000 0.945732
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 1.00000 0.0392837
\(649\) 27.0000 1.05984
\(650\) 10.0000 0.392232
\(651\) −8.00000 −0.313545
\(652\) −19.0000 −0.744097
\(653\) −12.0000 −0.469596 −0.234798 0.972044i \(-0.575443\pi\)
−0.234798 + 0.972044i \(0.575443\pi\)
\(654\) −16.0000 −0.625650
\(655\) 0 0
\(656\) −9.00000 −0.351391
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 40.0000 1.55582 0.777910 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) −5.00000 −0.194331
\(663\) −12.0000 −0.466041
\(664\) 3.00000 0.116423
\(665\) 0 0
\(666\) −20.0000 −0.774984
\(667\) 0 0
\(668\) 24.0000 0.928588
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 4.00000 0.154303
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −11.0000 −0.423704
\(675\) −25.0000 −0.962250
\(676\) −9.00000 −0.346154
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 15.0000 0.576072
\(679\) 68.0000 2.60960
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) −6.00000 −0.229752
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 16.0000 0.610438
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) −6.00000 −0.228086
\(693\) 24.0000 0.911685
\(694\) −9.00000 −0.341635
\(695\) 0 0
\(696\) 0 0
\(697\) 54.0000 2.04540
\(698\) −4.00000 −0.151402
\(699\) −3.00000 −0.113470
\(700\) 20.0000 0.755929
\(701\) −24.0000 −0.906467 −0.453234 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(702\) −10.0000 −0.377426
\(703\) 0 0
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 3.00000 0.112906
\(707\) 0 0
\(708\) −9.00000 −0.338241
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −6.00000 −0.224860
\(713\) 12.0000 0.449404
\(714\) −24.0000 −0.898177
\(715\) 0 0
\(716\) −9.00000 −0.336346
\(717\) 12.0000 0.448148
\(718\) −6.00000 −0.223918
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 5.00000 0.185952
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 8.00000 0.296500
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 4.00000 0.147844
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −22.0000 −0.812035
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 21.0000 0.773545
\(738\) 18.0000 0.662589
\(739\) 35.0000 1.28750 0.643748 0.765238i \(-0.277379\pi\)
0.643748 + 0.765238i \(0.277379\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) 18.0000 0.660356 0.330178 0.943919i \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 2.00000 0.0733236
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) −6.00000 −0.219529
\(748\) −18.0000 −0.658145
\(749\) 0 0
\(750\) 0 0
\(751\) −38.0000 −1.38664 −0.693320 0.720630i \(-0.743853\pi\)
−0.693320 + 0.720630i \(0.743853\pi\)
\(752\) 0 0
\(753\) 3.00000 0.109326
\(754\) 0 0
\(755\) 0 0
\(756\) −20.0000 −0.727393
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 28.0000 1.01701
\(759\) 18.0000 0.653359
\(760\) 0 0
\(761\) −39.0000 −1.41375 −0.706874 0.707339i \(-0.749895\pi\)
−0.706874 + 0.707339i \(0.749895\pi\)
\(762\) 2.00000 0.0724524
\(763\) −64.0000 −2.31696
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) −18.0000 −0.649942
\(768\) −1.00000 −0.0360844
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 3.00000 0.108042
\(772\) −2.00000 −0.0719816
\(773\) 48.0000 1.72644 0.863220 0.504828i \(-0.168444\pi\)
0.863220 + 0.504828i \(0.168444\pi\)
\(774\) 8.00000 0.287554
\(775\) 10.0000 0.359211
\(776\) −17.0000 −0.610264
\(777\) 40.0000 1.43499
\(778\) −36.0000 −1.29066
\(779\) 0 0
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 36.0000 1.28736
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) −9.00000 −0.321019
\(787\) 7.00000 0.249523 0.124762 0.992187i \(-0.460183\pi\)
0.124762 + 0.992187i \(0.460183\pi\)
\(788\) 18.0000 0.641223
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) 60.0000 2.13335
\(792\) −6.00000 −0.213201
\(793\) 8.00000 0.284088
\(794\) −10.0000 −0.354887
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −5.00000 −0.176777
\(801\) 12.0000 0.423999
\(802\) 27.0000 0.953403
\(803\) −3.00000 −0.105868
\(804\) −7.00000 −0.246871
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 12.0000 0.422420
\(808\) 0 0
\(809\) 45.0000 1.58212 0.791058 0.611741i \(-0.209531\pi\)
0.791058 + 0.611741i \(0.209531\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 30.0000 1.05150
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 0 0
\(818\) −5.00000 −0.174821
\(819\) −16.0000 −0.559085
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) −9.00000 −0.313911
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) −2.00000 −0.0696733
\(825\) 15.0000 0.522233
\(826\) −36.0000 −1.25260
\(827\) 39.0000 1.35616 0.678081 0.734987i \(-0.262812\pi\)
0.678081 + 0.734987i \(0.262812\pi\)
\(828\) 12.0000 0.417029
\(829\) −44.0000 −1.52818 −0.764092 0.645108i \(-0.776812\pi\)
−0.764092 + 0.645108i \(0.776812\pi\)
\(830\) 0 0
\(831\) −8.00000 −0.277517
\(832\) −2.00000 −0.0693375
\(833\) −54.0000 −1.87099
\(834\) −11.0000 −0.380899
\(835\) 0 0
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) −12.0000 −0.414533
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 10.0000 0.344623
\(843\) 27.0000 0.929929
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) 8.00000 0.274883
\(848\) −6.00000 −0.206041
\(849\) −5.00000 −0.171600
\(850\) 30.0000 1.02899
\(851\) −60.0000 −2.05677
\(852\) −6.00000 −0.205557
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) 16.0000 0.547509
\(855\) 0 0
\(856\) 0 0
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 6.00000 0.204837
\(859\) −43.0000 −1.46714 −0.733571 0.679613i \(-0.762148\pi\)
−0.733571 + 0.679613i \(0.762148\pi\)
\(860\) 0 0
\(861\) −36.0000 −1.22688
\(862\) 30.0000 1.02180
\(863\) −18.0000 −0.612727 −0.306364 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) −26.0000 −0.883516
\(867\) −19.0000 −0.645274
\(868\) 8.00000 0.271538
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) −14.0000 −0.474372
\(872\) 16.0000 0.541828
\(873\) 34.0000 1.15073
\(874\) 0 0
\(875\) 0 0
\(876\) 1.00000 0.0337869
\(877\) −20.0000 −0.675352 −0.337676 0.941262i \(-0.609641\pi\)
−0.337676 + 0.941262i \(0.609641\pi\)
\(878\) −14.0000 −0.472477
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) −18.0000 −0.606092
\(883\) −19.0000 −0.639401 −0.319700 0.947519i \(-0.603582\pi\)
−0.319700 + 0.947519i \(0.603582\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 9.00000 0.302361
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −10.0000 −0.335578
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) −14.0000 −0.468755
\(893\) 0 0
\(894\) −18.0000 −0.602010
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) −12.0000 −0.400668
\(898\) −9.00000 −0.300334
\(899\) 0 0
\(900\) 10.0000 0.333333
\(901\) 36.0000 1.19933
\(902\) −27.0000 −0.899002
\(903\) −16.0000 −0.532447
\(904\) −15.0000 −0.498893
\(905\) 0 0
\(906\) −10.0000 −0.332228
\(907\) −17.0000 −0.564476 −0.282238 0.959344i \(-0.591077\pi\)
−0.282238 + 0.959344i \(0.591077\pi\)
\(908\) −3.00000 −0.0995585
\(909\) 0 0
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 9.00000 0.297857
\(914\) 5.00000 0.165385
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) −36.0000 −1.18882
\(918\) −30.0000 −0.990148
\(919\) −10.0000 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) 6.00000 0.197599
\(923\) −12.0000 −0.394985
\(924\) 12.0000 0.394771
\(925\) −50.0000 −1.64399
\(926\) −34.0000 −1.11731
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) −3.00000 −0.0984268 −0.0492134 0.998788i \(-0.515671\pi\)
−0.0492134 + 0.998788i \(0.515671\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 3.00000 0.0982683
\(933\) 30.0000 0.982156
\(934\) −27.0000 −0.883467
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 35.0000 1.14340 0.571700 0.820463i \(-0.306284\pi\)
0.571700 + 0.820463i \(0.306284\pi\)
\(938\) −28.0000 −0.914232
\(939\) 19.0000 0.620042
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 16.0000 0.521308
\(943\) 54.0000 1.75848
\(944\) 9.00000 0.292925
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 60.0000 1.94974 0.974869 0.222779i \(-0.0715128\pi\)
0.974869 + 0.222779i \(0.0715128\pi\)
\(948\) −4.00000 −0.129914
\(949\) 2.00000 0.0649227
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 24.0000 0.777844
\(953\) 15.0000 0.485898 0.242949 0.970039i \(-0.421885\pi\)
0.242949 + 0.970039i \(0.421885\pi\)
\(954\) 12.0000 0.388514
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 36.0000 1.16311
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −20.0000 −0.644826
\(963\) 0 0
\(964\) −5.00000 −0.161039
\(965\) 0 0
\(966\) −24.0000 −0.772187
\(967\) −34.0000 −1.09337 −0.546683 0.837340i \(-0.684110\pi\)
−0.546683 + 0.837340i \(0.684110\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) 0 0
\(971\) −21.0000 −0.673922 −0.336961 0.941519i \(-0.609399\pi\)
−0.336961 + 0.941519i \(0.609399\pi\)
\(972\) −16.0000 −0.513200
\(973\) −44.0000 −1.41058
\(974\) −2.00000 −0.0640841
\(975\) −10.0000 −0.320256
\(976\) −4.00000 −0.128037
\(977\) −33.0000 −1.05576 −0.527882 0.849318i \(-0.677014\pi\)
−0.527882 + 0.849318i \(0.677014\pi\)
\(978\) 19.0000 0.607553
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) −32.0000 −1.02168
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 9.00000 0.286910
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 5.00000 0.158670
\(994\) −24.0000 −0.761234
\(995\) 0 0
\(996\) −3.00000 −0.0950586
\(997\) −4.00000 −0.126681 −0.0633406 0.997992i \(-0.520175\pi\)
−0.0633406 + 0.997992i \(0.520175\pi\)
\(998\) −25.0000 −0.791361
\(999\) 50.0000 1.58193
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 722.2.a.d.1.1 1
3.2 odd 2 6498.2.a.e.1.1 1
4.3 odd 2 5776.2.a.n.1.1 1
19.2 odd 18 722.2.e.j.99.1 6
19.3 odd 18 722.2.e.j.389.1 6
19.4 even 9 722.2.e.i.415.1 6
19.5 even 9 722.2.e.i.595.1 6
19.6 even 9 722.2.e.i.245.1 6
19.7 even 3 722.2.c.b.429.1 2
19.8 odd 6 38.2.c.a.7.1 2
19.9 even 9 722.2.e.i.423.1 6
19.10 odd 18 722.2.e.j.423.1 6
19.11 even 3 722.2.c.b.653.1 2
19.12 odd 6 38.2.c.a.11.1 yes 2
19.13 odd 18 722.2.e.j.245.1 6
19.14 odd 18 722.2.e.j.595.1 6
19.15 odd 18 722.2.e.j.415.1 6
19.16 even 9 722.2.e.i.389.1 6
19.17 even 9 722.2.e.i.99.1 6
19.18 odd 2 722.2.a.c.1.1 1
57.8 even 6 342.2.g.b.235.1 2
57.50 even 6 342.2.g.b.163.1 2
57.56 even 2 6498.2.a.s.1.1 1
76.27 even 6 304.2.i.c.273.1 2
76.31 even 6 304.2.i.c.49.1 2
76.75 even 2 5776.2.a.g.1.1 1
95.8 even 12 950.2.j.e.349.2 4
95.12 even 12 950.2.j.e.49.2 4
95.27 even 12 950.2.j.e.349.1 4
95.69 odd 6 950.2.e.d.201.1 2
95.84 odd 6 950.2.e.d.501.1 2
95.88 even 12 950.2.j.e.49.1 4
152.27 even 6 1216.2.i.d.577.1 2
152.69 odd 6 1216.2.i.h.961.1 2
152.107 even 6 1216.2.i.d.961.1 2
152.141 odd 6 1216.2.i.h.577.1 2
228.107 odd 6 2736.2.s.m.1873.1 2
228.179 odd 6 2736.2.s.m.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.c.a.7.1 2 19.8 odd 6
38.2.c.a.11.1 yes 2 19.12 odd 6
304.2.i.c.49.1 2 76.31 even 6
304.2.i.c.273.1 2 76.27 even 6
342.2.g.b.163.1 2 57.50 even 6
342.2.g.b.235.1 2 57.8 even 6
722.2.a.c.1.1 1 19.18 odd 2
722.2.a.d.1.1 1 1.1 even 1 trivial
722.2.c.b.429.1 2 19.7 even 3
722.2.c.b.653.1 2 19.11 even 3
722.2.e.i.99.1 6 19.17 even 9
722.2.e.i.245.1 6 19.6 even 9
722.2.e.i.389.1 6 19.16 even 9
722.2.e.i.415.1 6 19.4 even 9
722.2.e.i.423.1 6 19.9 even 9
722.2.e.i.595.1 6 19.5 even 9
722.2.e.j.99.1 6 19.2 odd 18
722.2.e.j.245.1 6 19.13 odd 18
722.2.e.j.389.1 6 19.3 odd 18
722.2.e.j.415.1 6 19.15 odd 18
722.2.e.j.423.1 6 19.10 odd 18
722.2.e.j.595.1 6 19.14 odd 18
950.2.e.d.201.1 2 95.69 odd 6
950.2.e.d.501.1 2 95.84 odd 6
950.2.j.e.49.1 4 95.88 even 12
950.2.j.e.49.2 4 95.12 even 12
950.2.j.e.349.1 4 95.27 even 12
950.2.j.e.349.2 4 95.8 even 12
1216.2.i.d.577.1 2 152.27 even 6
1216.2.i.d.961.1 2 152.107 even 6
1216.2.i.h.577.1 2 152.141 odd 6
1216.2.i.h.961.1 2 152.69 odd 6
2736.2.s.m.577.1 2 228.179 odd 6
2736.2.s.m.1873.1 2 228.107 odd 6
5776.2.a.g.1.1 1 76.75 even 2
5776.2.a.n.1.1 1 4.3 odd 2
6498.2.a.e.1.1 1 3.2 odd 2
6498.2.a.s.1.1 1 57.56 even 2