Properties

Label 5550.2.a.bk.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.3169731218\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -1.00000 q^{21} -1.00000 q^{22} +1.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +9.00000 q^{29} +7.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +7.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} -2.00000 q^{38} -2.00000 q^{39} -11.0000 q^{41} -1.00000 q^{42} +11.0000 q^{43} -1.00000 q^{44} -8.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +7.00000 q^{51} -2.00000 q^{52} +11.0000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -2.00000 q^{57} +9.00000 q^{58} -10.0000 q^{59} -1.00000 q^{61} +7.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} +8.00000 q^{67} +7.00000 q^{68} +1.00000 q^{72} -4.00000 q^{73} +1.00000 q^{74} -2.00000 q^{76} +1.00000 q^{77} -2.00000 q^{78} +12.0000 q^{79} +1.00000 q^{81} -11.0000 q^{82} +6.00000 q^{83} -1.00000 q^{84} +11.0000 q^{86} +9.00000 q^{87} -1.00000 q^{88} +6.00000 q^{89} +2.00000 q^{91} +7.00000 q^{93} -8.00000 q^{94} +1.00000 q^{96} +19.0000 q^{97} -6.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) −1.00000 −0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) 7.00000 1.20049
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) −2.00000 −0.324443
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −11.0000 −1.71791 −0.858956 0.512050i \(-0.828886\pi\)
−0.858956 + 0.512050i \(0.828886\pi\)
\(42\) −1.00000 −0.154303
\(43\) 11.0000 1.67748 0.838742 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 7.00000 0.980196
\(52\) −2.00000 −0.277350
\(53\) 11.0000 1.51097 0.755483 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −2.00000 −0.264906
\(58\) 9.00000 1.18176
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 7.00000 0.889001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 7.00000 0.848875
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 1.00000 0.113961
\(78\) −2.00000 −0.226455
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −11.0000 −1.21475
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 11.0000 1.18616
\(87\) 9.00000 0.964901
\(88\) −1.00000 −0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 7.00000 0.725866
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 19.0000 1.92916 0.964579 0.263795i \(-0.0849741\pi\)
0.964579 + 0.263795i \(0.0849741\pi\)
\(98\) −6.00000 −0.606092
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −20.0000 −1.99007 −0.995037 0.0995037i \(-0.968274\pi\)
−0.995037 + 0.0995037i \(0.968274\pi\)
\(102\) 7.00000 0.693103
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 11.0000 1.06841
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 1.00000 0.0962250
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) −1.00000 −0.0944911
\(113\) 13.0000 1.22294 0.611469 0.791269i \(-0.290579\pi\)
0.611469 + 0.791269i \(0.290579\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) −2.00000 −0.184900
\(118\) −10.0000 −0.920575
\(119\) −7.00000 −0.641689
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −1.00000 −0.0905357
\(123\) −11.0000 −0.991837
\(124\) 7.00000 0.628619
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 11.0000 0.968496
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 2.00000 0.173422
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 7.00000 0.600245
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 21.0000 1.78120 0.890598 0.454791i \(-0.150286\pi\)
0.890598 + 0.454791i \(0.150286\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) −6.00000 −0.494872
\(148\) 1.00000 0.0821995
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −6.00000 −0.488273 −0.244137 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) −2.00000 −0.162221
\(153\) 7.00000 0.565916
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −23.0000 −1.83560 −0.917800 0.397043i \(-0.870036\pi\)
−0.917800 + 0.397043i \(0.870036\pi\)
\(158\) 12.0000 0.954669
\(159\) 11.0000 0.872357
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −13.0000 −1.01824 −0.509119 0.860696i \(-0.670029\pi\)
−0.509119 + 0.860696i \(0.670029\pi\)
\(164\) −11.0000 −0.858956
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 11.0000 0.838742
\(173\) 3.00000 0.228086 0.114043 0.993476i \(-0.463620\pi\)
0.114043 + 0.993476i \(0.463620\pi\)
\(174\) 9.00000 0.682288
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −10.0000 −0.751646
\(178\) 6.00000 0.449719
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) −26.0000 −1.93256 −0.966282 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 2.00000 0.148250
\(183\) −1.00000 −0.0739221
\(184\) 0 0
\(185\) 0 0
\(186\) 7.00000 0.513265
\(187\) −7.00000 −0.511891
\(188\) −8.00000 −0.583460
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 19.0000 1.37479 0.687396 0.726283i \(-0.258754\pi\)
0.687396 + 0.726283i \(0.258754\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 19.0000 1.36412
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −20.0000 −1.40720
\(203\) −9.00000 −0.631676
\(204\) 7.00000 0.490098
\(205\) 0 0
\(206\) 12.0000 0.836080
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 11.0000 0.755483
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −7.00000 −0.475191
\(218\) 7.00000 0.474100
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −14.0000 −0.941742
\(222\) 1.00000 0.0671156
\(223\) −3.00000 −0.200895 −0.100447 0.994942i \(-0.532027\pi\)
−0.100447 + 0.994942i \(0.532027\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 13.0000 0.864747
\(227\) 23.0000 1.52656 0.763282 0.646066i \(-0.223587\pi\)
0.763282 + 0.646066i \(0.223587\pi\)
\(228\) −2.00000 −0.132453
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 9.00000 0.590879
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) 12.0000 0.779484
\(238\) −7.00000 −0.453743
\(239\) 27.0000 1.74648 0.873242 0.487286i \(-0.162013\pi\)
0.873242 + 0.487286i \(0.162013\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) −10.0000 −0.642824
\(243\) 1.00000 0.0641500
\(244\) −1.00000 −0.0640184
\(245\) 0 0
\(246\) −11.0000 −0.701334
\(247\) 4.00000 0.254514
\(248\) 7.00000 0.444500
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) −16.0000 −1.00991 −0.504956 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 11.0000 0.684830
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) −6.00000 −0.370681
\(263\) −21.0000 −1.29492 −0.647458 0.762101i \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) 6.00000 0.367194
\(268\) 8.00000 0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 7.00000 0.424437
\(273\) 2.00000 0.121046
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 21.0000 1.25950
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −8.00000 −0.476393
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 11.0000 0.649309
\(288\) 1.00000 0.0589256
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) 19.0000 1.11380
\(292\) −4.00000 −0.234082
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) −6.00000 −0.349927
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) −1.00000 −0.0580259
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) −11.0000 −0.634029
\(302\) −6.00000 −0.345261
\(303\) −20.0000 −1.14897
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 7.00000 0.400163
\(307\) −10.0000 −0.570730 −0.285365 0.958419i \(-0.592115\pi\)
−0.285365 + 0.958419i \(0.592115\pi\)
\(308\) 1.00000 0.0569803
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) −9.00000 −0.510343 −0.255172 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(312\) −2.00000 −0.113228
\(313\) 18.0000 1.01742 0.508710 0.860938i \(-0.330123\pi\)
0.508710 + 0.860938i \(0.330123\pi\)
\(314\) −23.0000 −1.29797
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) −5.00000 −0.280828 −0.140414 0.990093i \(-0.544843\pi\)
−0.140414 + 0.990093i \(0.544843\pi\)
\(318\) 11.0000 0.616849
\(319\) −9.00000 −0.503903
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) −14.0000 −0.778981
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −13.0000 −0.720003
\(327\) 7.00000 0.387101
\(328\) −11.0000 −0.607373
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 6.00000 0.329293
\(333\) 1.00000 0.0547997
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) 13.0000 0.706063
\(340\) 0 0
\(341\) −7.00000 −0.379071
\(342\) −2.00000 −0.108148
\(343\) 13.0000 0.701934
\(344\) 11.0000 0.593080
\(345\) 0 0
\(346\) 3.00000 0.161281
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 9.00000 0.482451
\(349\) −32.0000 −1.71292 −0.856460 0.516213i \(-0.827341\pi\)
−0.856460 + 0.516213i \(0.827341\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −1.00000 −0.0533002
\(353\) 31.0000 1.64996 0.824982 0.565159i \(-0.191185\pi\)
0.824982 + 0.565159i \(0.191185\pi\)
\(354\) −10.0000 −0.531494
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) −7.00000 −0.370479
\(358\) 24.0000 1.26844
\(359\) 14.0000 0.738892 0.369446 0.929252i \(-0.379548\pi\)
0.369446 + 0.929252i \(0.379548\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −26.0000 −1.36653
\(363\) −10.0000 −0.524864
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) −1.00000 −0.0522708
\(367\) −17.0000 −0.887393 −0.443696 0.896177i \(-0.646333\pi\)
−0.443696 + 0.896177i \(0.646333\pi\)
\(368\) 0 0
\(369\) −11.0000 −0.572637
\(370\) 0 0
\(371\) −11.0000 −0.571092
\(372\) 7.00000 0.362933
\(373\) 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
\(376\) −8.00000 −0.412568
\(377\) −18.0000 −0.927047
\(378\) −1.00000 −0.0514344
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 19.0000 0.972125
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 11.0000 0.559161
\(388\) 19.0000 0.964579
\(389\) −21.0000 −1.06474 −0.532371 0.846511i \(-0.678699\pi\)
−0.532371 + 0.846511i \(0.678699\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −6.00000 −0.303046
\(393\) −6.00000 −0.302660
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −4.00000 −0.200502
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 8.00000 0.399004
\(403\) −14.0000 −0.697390
\(404\) −20.0000 −0.995037
\(405\) 0 0
\(406\) −9.00000 −0.446663
\(407\) −1.00000 −0.0495682
\(408\) 7.00000 0.346552
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 12.0000 0.591198
\(413\) 10.0000 0.492068
\(414\) 0 0
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 21.0000 1.02837
\(418\) 2.00000 0.0978232
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) −5.00000 −0.243396
\(423\) −8.00000 −0.388973
\(424\) 11.0000 0.534207
\(425\) 0 0
\(426\) 0 0
\(427\) 1.00000 0.0483934
\(428\) 6.00000 0.290021
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) −1.00000 −0.0481683 −0.0240842 0.999710i \(-0.507667\pi\)
−0.0240842 + 0.999710i \(0.507667\pi\)
\(432\) 1.00000 0.0481125
\(433\) −20.0000 −0.961139 −0.480569 0.876957i \(-0.659570\pi\)
−0.480569 + 0.876957i \(0.659570\pi\)
\(434\) −7.00000 −0.336011
\(435\) 0 0
\(436\) 7.00000 0.335239
\(437\) 0 0
\(438\) −4.00000 −0.191127
\(439\) 1.00000 0.0477274 0.0238637 0.999715i \(-0.492403\pi\)
0.0238637 + 0.999715i \(0.492403\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) −14.0000 −0.665912
\(443\) 30.0000 1.42534 0.712672 0.701498i \(-0.247485\pi\)
0.712672 + 0.701498i \(0.247485\pi\)
\(444\) 1.00000 0.0474579
\(445\) 0 0
\(446\) −3.00000 −0.142054
\(447\) 6.00000 0.283790
\(448\) −1.00000 −0.0472456
\(449\) 8.00000 0.377543 0.188772 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) 0 0
\(451\) 11.0000 0.517970
\(452\) 13.0000 0.611469
\(453\) −6.00000 −0.281905
\(454\) 23.0000 1.07944
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 11.0000 0.514558 0.257279 0.966337i \(-0.417174\pi\)
0.257279 + 0.966337i \(0.417174\pi\)
\(458\) −16.0000 −0.747631
\(459\) 7.00000 0.326732
\(460\) 0 0
\(461\) −13.0000 −0.605470 −0.302735 0.953075i \(-0.597900\pi\)
−0.302735 + 0.953075i \(0.597900\pi\)
\(462\) 1.00000 0.0465242
\(463\) −2.00000 −0.0929479 −0.0464739 0.998920i \(-0.514798\pi\)
−0.0464739 + 0.998920i \(0.514798\pi\)
\(464\) 9.00000 0.417815
\(465\) 0 0
\(466\) −24.0000 −1.11178
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −23.0000 −1.05978
\(472\) −10.0000 −0.460287
\(473\) −11.0000 −0.505781
\(474\) 12.0000 0.551178
\(475\) 0 0
\(476\) −7.00000 −0.320844
\(477\) 11.0000 0.503655
\(478\) 27.0000 1.23495
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 6.00000 0.273293
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −18.0000 −0.815658 −0.407829 0.913058i \(-0.633714\pi\)
−0.407829 + 0.913058i \(0.633714\pi\)
\(488\) −1.00000 −0.0452679
\(489\) −13.0000 −0.587880
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −11.0000 −0.495918
\(493\) 63.0000 2.83738
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) 7.00000 0.314309
\(497\) 0 0
\(498\) 6.00000 0.268866
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) −16.0000 −0.714115
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 16.0000 0.709885
\(509\) 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
\(513\) −2.00000 −0.0883022
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) 11.0000 0.484248
\(517\) 8.00000 0.351840
\(518\) −1.00000 −0.0439375
\(519\) 3.00000 0.131685
\(520\) 0 0
\(521\) 15.0000 0.657162 0.328581 0.944476i \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) 9.00000 0.393919
\(523\) 24.0000 1.04945 0.524723 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −21.0000 −0.915644
\(527\) 49.0000 2.13447
\(528\) −1.00000 −0.0435194
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 2.00000 0.0867110
\(533\) 22.0000 0.952926
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 24.0000 1.03568
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 8.00000 0.343629
\(543\) −26.0000 −1.11577
\(544\) 7.00000 0.300123
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) 11.0000 0.470326 0.235163 0.971956i \(-0.424438\pi\)
0.235163 + 0.971956i \(0.424438\pi\)
\(548\) 6.00000 0.256307
\(549\) −1.00000 −0.0426790
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) −12.0000 −0.510292
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) 21.0000 0.890598
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 7.00000 0.296334
\(559\) −22.0000 −0.930501
\(560\) 0 0
\(561\) −7.00000 −0.295540
\(562\) 6.00000 0.253095
\(563\) −13.0000 −0.547885 −0.273942 0.961746i \(-0.588328\pi\)
−0.273942 + 0.961746i \(0.588328\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) −12.0000 −0.504398
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 11.0000 0.460336 0.230168 0.973151i \(-0.426072\pi\)
0.230168 + 0.973151i \(0.426072\pi\)
\(572\) 2.00000 0.0836242
\(573\) 19.0000 0.793736
\(574\) 11.0000 0.459131
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 32.0000 1.33102
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) 19.0000 0.787575
\(583\) −11.0000 −0.455573
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) −43.0000 −1.77480 −0.887400 0.461000i \(-0.847491\pi\)
−0.887400 + 0.461000i \(0.847491\pi\)
\(588\) −6.00000 −0.247436
\(589\) −14.0000 −0.576860
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 1.00000 0.0410997
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) −23.0000 −0.938190 −0.469095 0.883148i \(-0.655420\pi\)
−0.469095 + 0.883148i \(0.655420\pi\)
\(602\) −11.0000 −0.448327
\(603\) 8.00000 0.325785
\(604\) −6.00000 −0.244137
\(605\) 0 0
\(606\) −20.0000 −0.812444
\(607\) 10.0000 0.405887 0.202944 0.979190i \(-0.434949\pi\)
0.202944 + 0.979190i \(0.434949\pi\)
\(608\) −2.00000 −0.0811107
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) 7.00000 0.282958
\(613\) −33.0000 −1.33286 −0.666429 0.745569i \(-0.732178\pi\)
−0.666429 + 0.745569i \(0.732178\pi\)
\(614\) −10.0000 −0.403567
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 12.0000 0.482711
\(619\) −19.0000 −0.763674 −0.381837 0.924230i \(-0.624709\pi\)
−0.381837 + 0.924230i \(0.624709\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −9.00000 −0.360867
\(623\) −6.00000 −0.240385
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 18.0000 0.719425
\(627\) 2.00000 0.0798723
\(628\) −23.0000 −0.917800
\(629\) 7.00000 0.279108
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 12.0000 0.477334
\(633\) −5.00000 −0.198732
\(634\) −5.00000 −0.198575
\(635\) 0 0
\(636\) 11.0000 0.436178
\(637\) 12.0000 0.475457
\(638\) −9.00000 −0.356313
\(639\) 0 0
\(640\) 0 0
\(641\) −27.0000 −1.06644 −0.533218 0.845978i \(-0.679017\pi\)
−0.533218 + 0.845978i \(0.679017\pi\)
\(642\) 6.00000 0.236801
\(643\) 7.00000 0.276053 0.138027 0.990429i \(-0.455924\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −14.0000 −0.550823
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 1.00000 0.0392837
\(649\) 10.0000 0.392534
\(650\) 0 0
\(651\) −7.00000 −0.274352
\(652\) −13.0000 −0.509119
\(653\) −4.00000 −0.156532 −0.0782660 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 7.00000 0.273722
\(655\) 0 0
\(656\) −11.0000 −0.429478
\(657\) −4.00000 −0.156055
\(658\) 8.00000 0.311872
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) 41.0000 1.59472 0.797358 0.603507i \(-0.206231\pi\)
0.797358 + 0.603507i \(0.206231\pi\)
\(662\) −8.00000 −0.310929
\(663\) −14.0000 −0.543715
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) −3.00000 −0.115987
\(670\) 0 0
\(671\) 1.00000 0.0386046
\(672\) −1.00000 −0.0385758
\(673\) −4.00000 −0.154189 −0.0770943 0.997024i \(-0.524564\pi\)
−0.0770943 + 0.997024i \(0.524564\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 13.0000 0.499262
\(679\) −19.0000 −0.729153
\(680\) 0 0
\(681\) 23.0000 0.881362
\(682\) −7.00000 −0.268044
\(683\) 15.0000 0.573959 0.286980 0.957937i \(-0.407349\pi\)
0.286980 + 0.957937i \(0.407349\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) −16.0000 −0.610438
\(688\) 11.0000 0.419371
\(689\) −22.0000 −0.838133
\(690\) 0 0
\(691\) −33.0000 −1.25538 −0.627690 0.778464i \(-0.715999\pi\)
−0.627690 + 0.778464i \(0.715999\pi\)
\(692\) 3.00000 0.114043
\(693\) 1.00000 0.0379869
\(694\) −28.0000 −1.06287
\(695\) 0 0
\(696\) 9.00000 0.341144
\(697\) −77.0000 −2.91658
\(698\) −32.0000 −1.21122
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −2.00000 −0.0754314
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 31.0000 1.16670
\(707\) 20.0000 0.752177
\(708\) −10.0000 −0.375823
\(709\) 35.0000 1.31445 0.657226 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) −7.00000 −0.261968
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) 27.0000 1.00833
\(718\) 14.0000 0.522475
\(719\) 26.0000 0.969636 0.484818 0.874615i \(-0.338886\pi\)
0.484818 + 0.874615i \(0.338886\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) −15.0000 −0.558242
\(723\) 6.00000 0.223142
\(724\) −26.0000 −0.966282
\(725\) 0 0
\(726\) −10.0000 −0.371135
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 77.0000 2.84795
\(732\) −1.00000 −0.0369611
\(733\) 15.0000 0.554038 0.277019 0.960864i \(-0.410654\pi\)
0.277019 + 0.960864i \(0.410654\pi\)
\(734\) −17.0000 −0.627481
\(735\) 0 0
\(736\) 0 0
\(737\) −8.00000 −0.294684
\(738\) −11.0000 −0.404916
\(739\) −31.0000 −1.14035 −0.570177 0.821522i \(-0.693125\pi\)
−0.570177 + 0.821522i \(0.693125\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) −11.0000 −0.403823
\(743\) −21.0000 −0.770415 −0.385208 0.922830i \(-0.625870\pi\)
−0.385208 + 0.922830i \(0.625870\pi\)
\(744\) 7.00000 0.256632
\(745\) 0 0
\(746\) 6.00000 0.219676
\(747\) 6.00000 0.219529
\(748\) −7.00000 −0.255945
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) −8.00000 −0.291730
\(753\) −16.0000 −0.583072
\(754\) −18.0000 −0.655521
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) −43.0000 −1.55875 −0.779374 0.626559i \(-0.784463\pi\)
−0.779374 + 0.626559i \(0.784463\pi\)
\(762\) 16.0000 0.579619
\(763\) −7.00000 −0.253417
\(764\) 19.0000 0.687396
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 20.0000 0.722158
\(768\) 1.00000 0.0360844
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) −14.0000 −0.503871
\(773\) 9.00000 0.323708 0.161854 0.986815i \(-0.448253\pi\)
0.161854 + 0.986815i \(0.448253\pi\)
\(774\) 11.0000 0.395387
\(775\) 0 0
\(776\) 19.0000 0.682060
\(777\) −1.00000 −0.0358748
\(778\) −21.0000 −0.752886
\(779\) 22.0000 0.788232
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 9.00000 0.321634
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) −6.00000 −0.214013
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 18.0000 0.641223
\(789\) −21.0000 −0.747620
\(790\) 0 0
\(791\) −13.0000 −0.462227
\(792\) −1.00000 −0.0355335
\(793\) 2.00000 0.0710221
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 48.0000 1.70025 0.850124 0.526583i \(-0.176527\pi\)
0.850124 + 0.526583i \(0.176527\pi\)
\(798\) 2.00000 0.0707992
\(799\) −56.0000 −1.98114
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) −6.00000 −0.211867
\(803\) 4.00000 0.141157
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −14.0000 −0.493129
\(807\) 0 0
\(808\) −20.0000 −0.703598
\(809\) −24.0000 −0.843795 −0.421898 0.906644i \(-0.638636\pi\)
−0.421898 + 0.906644i \(0.638636\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) −9.00000 −0.315838
\(813\) 8.00000 0.280572
\(814\) −1.00000 −0.0350500
\(815\) 0 0
\(816\) 7.00000 0.245049
\(817\) −22.0000 −0.769683
\(818\) −18.0000 −0.629355
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 6.00000 0.209274
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) 10.0000 0.347945
\(827\) −21.0000 −0.730242 −0.365121 0.930960i \(-0.618972\pi\)
−0.365121 + 0.930960i \(0.618972\pi\)
\(828\) 0 0
\(829\) 19.0000 0.659897 0.329949 0.943999i \(-0.392969\pi\)
0.329949 + 0.943999i \(0.392969\pi\)
\(830\) 0 0
\(831\) −8.00000 −0.277517
\(832\) −2.00000 −0.0693375
\(833\) −42.0000 −1.45521
\(834\) 21.0000 0.727171
\(835\) 0 0
\(836\) 2.00000 0.0691714
\(837\) 7.00000 0.241955
\(838\) 28.0000 0.967244
\(839\) −26.0000 −0.897620 −0.448810 0.893627i \(-0.648152\pi\)
−0.448810 + 0.893627i \(0.648152\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) −2.00000 −0.0689246
\(843\) 6.00000 0.206651
\(844\) −5.00000 −0.172107
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 10.0000 0.343604
\(848\) 11.0000 0.377742
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 50.0000 1.71197 0.855984 0.517003i \(-0.172952\pi\)
0.855984 + 0.517003i \(0.172952\pi\)
\(854\) 1.00000 0.0342193
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) −31.0000 −1.05894 −0.529470 0.848329i \(-0.677609\pi\)
−0.529470 + 0.848329i \(0.677609\pi\)
\(858\) 2.00000 0.0682789
\(859\) 18.0000 0.614152 0.307076 0.951685i \(-0.400649\pi\)
0.307076 + 0.951685i \(0.400649\pi\)
\(860\) 0 0
\(861\) 11.0000 0.374879
\(862\) −1.00000 −0.0340601
\(863\) −27.0000 −0.919091 −0.459545 0.888154i \(-0.651988\pi\)
−0.459545 + 0.888154i \(0.651988\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −20.0000 −0.679628
\(867\) 32.0000 1.08678
\(868\) −7.00000 −0.237595
\(869\) −12.0000 −0.407072
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 7.00000 0.237050
\(873\) 19.0000 0.643053
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) −23.0000 −0.776655 −0.388327 0.921521i \(-0.626947\pi\)
−0.388327 + 0.921521i \(0.626947\pi\)
\(878\) 1.00000 0.0337484
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) 21.0000 0.707508 0.353754 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) −6.00000 −0.202031
\(883\) −19.0000 −0.639401 −0.319700 0.947519i \(-0.603582\pi\)
−0.319700 + 0.947519i \(0.603582\pi\)
\(884\) −14.0000 −0.470871
\(885\) 0 0
\(886\) 30.0000 1.00787
\(887\) 47.0000 1.57811 0.789053 0.614325i \(-0.210572\pi\)
0.789053 + 0.614325i \(0.210572\pi\)
\(888\) 1.00000 0.0335578
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) −3.00000 −0.100447
\(893\) 16.0000 0.535420
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 8.00000 0.266963
\(899\) 63.0000 2.10117
\(900\) 0 0
\(901\) 77.0000 2.56524
\(902\) 11.0000 0.366260
\(903\) −11.0000 −0.366057
\(904\) 13.0000 0.432374
\(905\) 0 0
\(906\) −6.00000 −0.199337
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 23.0000 0.763282
\(909\) −20.0000 −0.663358
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) −2.00000 −0.0662266
\(913\) −6.00000 −0.198571
\(914\) 11.0000 0.363848
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) 6.00000 0.198137
\(918\) 7.00000 0.231034
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −10.0000 −0.329511
\(922\) −13.0000 −0.428132
\(923\) 0 0
\(924\) 1.00000 0.0328976
\(925\) 0 0
\(926\) −2.00000 −0.0657241
\(927\) 12.0000 0.394132
\(928\) 9.00000 0.295439
\(929\) −21.0000 −0.688988 −0.344494 0.938789i \(-0.611949\pi\)
−0.344494 + 0.938789i \(0.611949\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) −24.0000 −0.786146
\(933\) −9.00000 −0.294647
\(934\) −3.00000 −0.0981630
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −32.0000 −1.04539 −0.522697 0.852518i \(-0.675074\pi\)
−0.522697 + 0.852518i \(0.675074\pi\)
\(938\) −8.00000 −0.261209
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) −23.0000 −0.749380
\(943\) 0 0
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) −11.0000 −0.357641
\(947\) 47.0000 1.52729 0.763647 0.645634i \(-0.223407\pi\)
0.763647 + 0.645634i \(0.223407\pi\)
\(948\) 12.0000 0.389742
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) −5.00000 −0.162136
\(952\) −7.00000 −0.226871
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 11.0000 0.356138
\(955\) 0 0
\(956\) 27.0000 0.873242
\(957\) −9.00000 −0.290929
\(958\) 24.0000 0.775405
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) −2.00000 −0.0644826
\(963\) 6.00000 0.193347
\(964\) 6.00000 0.193247
\(965\) 0 0
\(966\) 0 0
\(967\) −28.0000 −0.900419 −0.450210 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) −10.0000 −0.321412
\(969\) −14.0000 −0.449745
\(970\) 0 0
\(971\) 49.0000 1.57248 0.786242 0.617918i \(-0.212024\pi\)
0.786242 + 0.617918i \(0.212024\pi\)
\(972\) 1.00000 0.0320750
\(973\) −21.0000 −0.673229
\(974\) −18.0000 −0.576757
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) −5.00000 −0.159964 −0.0799821 0.996796i \(-0.525486\pi\)
−0.0799821 + 0.996796i \(0.525486\pi\)
\(978\) −13.0000 −0.415694
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 7.00000 0.223493
\(982\) −12.0000 −0.382935
\(983\) −51.0000 −1.62665 −0.813324 0.581811i \(-0.802344\pi\)
−0.813324 + 0.581811i \(0.802344\pi\)
\(984\) −11.0000 −0.350667
\(985\) 0 0
\(986\) 63.0000 2.00633
\(987\) 8.00000 0.254643
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) 37.0000 1.17534 0.587672 0.809099i \(-0.300045\pi\)
0.587672 + 0.809099i \(0.300045\pi\)
\(992\) 7.00000 0.222250
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) 18.0000 0.570066 0.285033 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) −28.0000 −0.886325
\(999\) 1.00000 0.0316386
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 5550.2.a.bk.1.1 1
5.4 even 2 1110.2.a.b.1.1 1
15.14 odd 2 3330.2.a.x.1.1 1
20.19 odd 2 8880.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.b.1.1 1 5.4 even 2
3330.2.a.x.1.1 1 15.14 odd 2
5550.2.a.bk.1.1 1 1.1 even 1 trivial
8880.2.a.s.1.1 1 20.19 odd 2