Properties

Label 5550.2.a.ba.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 $1$

Related objects

Downloads

Learn more

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

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} -1.00000 q^{12} +7.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{21} +3.00000 q^{22} +3.00000 q^{23} -1.00000 q^{24} +7.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} -10.0000 q^{31} +1.00000 q^{32} -3.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} -1.00000 q^{38} -7.00000 q^{39} -1.00000 q^{42} +4.00000 q^{43} +3.00000 q^{44} +3.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -3.00000 q^{51} +7.00000 q^{52} -9.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} +1.00000 q^{57} +6.00000 q^{58} +6.00000 q^{59} -10.0000 q^{61} -10.0000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} +10.0000 q^{67} +3.00000 q^{68} -3.00000 q^{69} -6.00000 q^{71} +1.00000 q^{72} -11.0000 q^{73} -1.00000 q^{74} -1.00000 q^{76} +3.00000 q^{77} -7.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} -9.00000 q^{83} -1.00000 q^{84} +4.00000 q^{86} -6.00000 q^{87} +3.00000 q^{88} -9.00000 q^{89} +7.00000 q^{91} +3.00000 q^{92} +10.0000 q^{93} +12.0000 q^{94} -1.00000 q^{96} +16.0000 q^{97} -6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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 \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 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
\(13\) 7.00000 1.94145 0.970725 0.240192i \(-0.0772105\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 3.00000 0.639602
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 7.00000 1.37281
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.00000 −0.522233
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) −1.00000 −0.162221
\(39\) −7.00000 −1.12090
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −1.00000 −0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 7.00000 0.970725
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 1.00000 0.132453
\(58\) 6.00000 0.787839
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −10.0000 −1.27000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.00000 −0.369274
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 3.00000 0.363803
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.00000 0.117851
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 3.00000 0.341882
\(78\) −7.00000 −0.792594
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) −6.00000 −0.643268
\(88\) 3.00000 0.319801
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) 7.00000 0.733799
\(92\) 3.00000 0.312772
\(93\) 10.0000 1.03695
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 16.0000 1.62455 0.812277 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) −6.00000 −0.606092
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) −3.00000 −0.297044
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 7.00000 0.686406
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 1.00000 0.0944911
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 7.00000 0.647150
\(118\) 6.00000 0.552345
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −10.0000 −0.898027
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −3.00000 −0.261116
\(133\) −1.00000 −0.0867110
\(134\) 10.0000 0.863868
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) −3.00000 −0.255377
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) −6.00000 −0.503509
\(143\) 21.0000 1.75611
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −11.0000 −0.910366
\(147\) 6.00000 0.494872
\(148\) −1.00000 −0.0821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 23.0000 1.87171 0.935857 0.352381i \(-0.114628\pi\)
0.935857 + 0.352381i \(0.114628\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 3.00000 0.242536
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) −7.00000 −0.560449
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 8.00000 0.636446
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 3.00000 0.236433
\(162\) 1.00000 0.0785674
\(163\) 1.00000 0.0783260 0.0391630 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 36.0000 2.76923
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 4.00000 0.304997
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) −6.00000 −0.450988
\(178\) −9.00000 −0.674579
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 7.00000 0.518875
\(183\) 10.0000 0.739221
\(184\) 3.00000 0.221163
\(185\) 0 0
\(186\) 10.0000 0.733236
\(187\) 9.00000 0.658145
\(188\) 12.0000 0.875190
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −3.00000 −0.213741 −0.106871 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) 3.00000 0.213201
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) −10.0000 −0.705346
\(202\) −6.00000 −0.422159
\(203\) 6.00000 0.421117
\(204\) −3.00000 −0.210042
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 3.00000 0.208514
\(208\) 7.00000 0.485363
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) −9.00000 −0.618123
\(213\) 6.00000 0.411113
\(214\) −3.00000 −0.205076
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −10.0000 −0.678844
\(218\) −7.00000 −0.474100
\(219\) 11.0000 0.743311
\(220\) 0 0
\(221\) 21.0000 1.41261
\(222\) 1.00000 0.0671156
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 1.00000 0.0662266
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 6.00000 0.393919
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 7.00000 0.457604
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) −8.00000 −0.519656
\(238\) 3.00000 0.194461
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) −2.00000 −0.128565
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) −7.00000 −0.445399
\(248\) −10.0000 −0.635001
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 1.00000 0.0629941
\(253\) 9.00000 0.565825
\(254\) 7.00000 0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.00000 −0.561405 −0.280702 0.959795i \(-0.590567\pi\)
−0.280702 + 0.959795i \(0.590567\pi\)
\(258\) −4.00000 −0.249029
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) −12.0000 −0.741362
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) −1.00000 −0.0613139
\(267\) 9.00000 0.550791
\(268\) 10.0000 0.610847
\(269\) 9.00000 0.548740 0.274370 0.961624i \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 3.00000 0.181902
\(273\) −7.00000 −0.423659
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) −3.00000 −0.180579
\(277\) −23.0000 −1.38194 −0.690968 0.722885i \(-0.742815\pi\)
−0.690968 + 0.722885i \(0.742815\pi\)
\(278\) 14.0000 0.839664
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 27.0000 1.61068 0.805342 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) −12.0000 −0.714590
\(283\) 13.0000 0.772770 0.386385 0.922338i \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 21.0000 1.24176
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) −11.0000 −0.643726
\(293\) 15.0000 0.876309 0.438155 0.898900i \(-0.355632\pi\)
0.438155 + 0.898900i \(0.355632\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) −3.00000 −0.174078
\(298\) −6.00000 −0.347571
\(299\) 21.0000 1.21446
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 23.0000 1.32350
\(303\) 6.00000 0.344691
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 3.00000 0.171499
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 3.00000 0.170941
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −7.00000 −0.396297
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 9.00000 0.504695
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) 3.00000 0.167444
\(322\) 3.00000 0.167183
\(323\) −3.00000 −0.166924
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 1.00000 0.0553849
\(327\) 7.00000 0.387101
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −9.00000 −0.493939
\(333\) −1.00000 −0.0547997
\(334\) −3.00000 −0.164153
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) 36.0000 1.95814
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) −30.0000 −1.62459
\(342\) −1.00000 −0.0540738
\(343\) −13.0000 −0.701934
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 9.00000 0.483843
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) −6.00000 −0.321634
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) −7.00000 −0.373632
\(352\) 3.00000 0.159901
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) −9.00000 −0.476999
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −22.0000 −1.15629
\(363\) 2.00000 0.104973
\(364\) 7.00000 0.366900
\(365\) 0 0
\(366\) 10.0000 0.522708
\(367\) −23.0000 −1.20059 −0.600295 0.799779i \(-0.704950\pi\)
−0.600295 + 0.799779i \(0.704950\pi\)
\(368\) 3.00000 0.156386
\(369\) 0 0
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 10.0000 0.518476
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 42.0000 2.16311
\(378\) −1.00000 −0.0514344
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) −7.00000 −0.358621
\(382\) 3.00000 0.153493
\(383\) 9.00000 0.459879 0.229939 0.973205i \(-0.426147\pi\)
0.229939 + 0.973205i \(0.426147\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 4.00000 0.203331
\(388\) 16.0000 0.812277
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 9.00000 0.455150
\(392\) −6.00000 −0.303046
\(393\) 12.0000 0.605320
\(394\) −3.00000 −0.151138
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) 4.00000 0.200754 0.100377 0.994949i \(-0.467995\pi\)
0.100377 + 0.994949i \(0.467995\pi\)
\(398\) 20.0000 1.00251
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) 15.0000 0.749064 0.374532 0.927214i \(-0.377803\pi\)
0.374532 + 0.927214i \(0.377803\pi\)
\(402\) −10.0000 −0.498755
\(403\) −70.0000 −3.48695
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) −3.00000 −0.148704
\(408\) −3.00000 −0.148522
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 4.00000 0.197066
\(413\) 6.00000 0.295241
\(414\) 3.00000 0.147442
\(415\) 0 0
\(416\) 7.00000 0.343203
\(417\) −14.0000 −0.685583
\(418\) −3.00000 −0.146735
\(419\) −27.0000 −1.31904 −0.659518 0.751689i \(-0.729240\pi\)
−0.659518 + 0.751689i \(0.729240\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 2.00000 0.0973585
\(423\) 12.0000 0.583460
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) 6.00000 0.290701
\(427\) −10.0000 −0.483934
\(428\) −3.00000 −0.145010
\(429\) −21.0000 −1.01389
\(430\) 0 0
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 31.0000 1.48976 0.744882 0.667196i \(-0.232506\pi\)
0.744882 + 0.667196i \(0.232506\pi\)
\(434\) −10.0000 −0.480015
\(435\) 0 0
\(436\) −7.00000 −0.335239
\(437\) −3.00000 −0.143509
\(438\) 11.0000 0.525600
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 21.0000 0.998868
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 1.00000 0.0474579
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 6.00000 0.283790
\(448\) 1.00000 0.0472456
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −18.0000 −0.846649
\(453\) −23.0000 −1.08063
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −10.0000 −0.467269
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) −3.00000 −0.139573
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 7.00000 0.323575
\(469\) 10.0000 0.461757
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 6.00000 0.276172
\(473\) 12.0000 0.551761
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 3.00000 0.137505
\(477\) −9.00000 −0.412082
\(478\) 12.0000 0.548867
\(479\) −39.0000 −1.78196 −0.890978 0.454047i \(-0.849980\pi\)
−0.890978 + 0.454047i \(0.849980\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) 26.0000 1.18427
\(483\) −3.00000 −0.136505
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) −10.0000 −0.452679
\(489\) −1.00000 −0.0452216
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) 18.0000 0.810679
\(494\) −7.00000 −0.314945
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) −6.00000 −0.269137
\(498\) 9.00000 0.403300
\(499\) 41.0000 1.83541 0.917706 0.397260i \(-0.130039\pi\)
0.917706 + 0.397260i \(0.130039\pi\)
\(500\) 0 0
\(501\) 3.00000 0.134030
\(502\) 6.00000 0.267793
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 9.00000 0.400099
\(507\) −36.0000 −1.59882
\(508\) 7.00000 0.310575
\(509\) −33.0000 −1.46270 −0.731350 0.682003i \(-0.761109\pi\)
−0.731350 + 0.682003i \(0.761109\pi\)
\(510\) 0 0
\(511\) −11.0000 −0.486611
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) −9.00000 −0.396973
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 36.0000 1.58328
\(518\) −1.00000 −0.0439375
\(519\) −9.00000 −0.395056
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 6.00000 0.262613
\(523\) −44.0000 −1.92399 −0.961993 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) −30.0000 −1.30682
\(528\) −3.00000 −0.130558
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) −1.00000 −0.0433555
\(533\) 0 0
\(534\) 9.00000 0.389468
\(535\) 0 0
\(536\) 10.0000 0.431934
\(537\) 0 0
\(538\) 9.00000 0.388018
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 20.0000 0.859074
\(543\) 22.0000 0.944110
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) −7.00000 −0.299572
\(547\) 37.0000 1.58201 0.791003 0.611812i \(-0.209559\pi\)
0.791003 + 0.611812i \(0.209559\pi\)
\(548\) 18.0000 0.768922
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) −3.00000 −0.127688
\(553\) 8.00000 0.340195
\(554\) −23.0000 −0.977176
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) −10.0000 −0.423334
\(559\) 28.0000 1.18427
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 27.0000 1.13893
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) 13.0000 0.546431
\(567\) 1.00000 0.0419961
\(568\) −6.00000 −0.251754
\(569\) −39.0000 −1.63497 −0.817483 0.575953i \(-0.804631\pi\)
−0.817483 + 0.575953i \(0.804631\pi\)
\(570\) 0 0
\(571\) 14.0000 0.585882 0.292941 0.956131i \(-0.405366\pi\)
0.292941 + 0.956131i \(0.405366\pi\)
\(572\) 21.0000 0.878054
\(573\) −3.00000 −0.125327
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) −8.00000 −0.332756
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) −9.00000 −0.373383
\(582\) −16.0000 −0.663221
\(583\) −27.0000 −1.11823
\(584\) −11.0000 −0.455183
\(585\) 0 0
\(586\) 15.0000 0.619644
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 6.00000 0.247436
\(589\) 10.0000 0.412043
\(590\) 0 0
\(591\) 3.00000 0.123404
\(592\) −1.00000 −0.0410997
\(593\) 36.0000 1.47834 0.739171 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) −3.00000 −0.123091
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −20.0000 −0.818546
\(598\) 21.0000 0.858754
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 4.00000 0.163028
\(603\) 10.0000 0.407231
\(604\) 23.0000 0.935857
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 84.0000 3.39828
\(612\) 3.00000 0.121268
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 22.0000 0.887848
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) −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
\(621\) −3.00000 −0.120386
\(622\) −24.0000 −0.962312
\(623\) −9.00000 −0.360577
\(624\) −7.00000 −0.280224
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) 3.00000 0.119808
\(628\) 4.00000 0.159617
\(629\) −3.00000 −0.119618
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 8.00000 0.318223
\(633\) −2.00000 −0.0794929
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 9.00000 0.356873
\(637\) −42.0000 −1.66410
\(638\) 18.0000 0.712627
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 3.00000 0.118401
\(643\) 31.0000 1.22252 0.611260 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(644\) 3.00000 0.118217
\(645\) 0 0
\(646\) −3.00000 −0.118033
\(647\) −45.0000 −1.76913 −0.884566 0.466415i \(-0.845546\pi\)
−0.884566 + 0.466415i \(0.845546\pi\)
\(648\) 1.00000 0.0392837
\(649\) 18.0000 0.706562
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) 1.00000 0.0391630
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 7.00000 0.273722
\(655\) 0 0
\(656\) 0 0
\(657\) −11.0000 −0.429151
\(658\) 12.0000 0.467809
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) −28.0000 −1.08825
\(663\) −21.0000 −0.815572
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) 18.0000 0.696963
\(668\) −3.00000 −0.116073
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) −30.0000 −1.15814
\(672\) −1.00000 −0.0385758
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 13.0000 0.500741
\(675\) 0 0
\(676\) 36.0000 1.38462
\(677\) −3.00000 −0.115299 −0.0576497 0.998337i \(-0.518361\pi\)
−0.0576497 + 0.998337i \(0.518361\pi\)
\(678\) 18.0000 0.691286
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) −30.0000 −1.14876
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) 10.0000 0.381524
\(688\) 4.00000 0.152499
\(689\) −63.0000 −2.40011
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 9.00000 0.342129
\(693\) 3.00000 0.113961
\(694\) −36.0000 −1.36654
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) 0 0
\(698\) 8.00000 0.302804
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) −48.0000 −1.81293 −0.906467 0.422276i \(-0.861231\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) −7.00000 −0.264198
\(703\) 1.00000 0.0377157
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) −6.00000 −0.225653
\(708\) −6.00000 −0.225494
\(709\) −25.0000 −0.938895 −0.469447 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −9.00000 −0.337289
\(713\) −30.0000 −1.12351
\(714\) −3.00000 −0.112272
\(715\) 0 0
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) −12.0000 −0.447836
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) −18.0000 −0.669891
\(723\) −26.0000 −0.966950
\(724\) −22.0000 −0.817624
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) −2.00000 −0.0741759 −0.0370879 0.999312i \(-0.511808\pi\)
−0.0370879 + 0.999312i \(0.511808\pi\)
\(728\) 7.00000 0.259437
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 10.0000 0.369611
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) −23.0000 −0.848945
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) 30.0000 1.10506
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 0 0
\(741\) 7.00000 0.257151
\(742\) −9.00000 −0.330400
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 10.0000 0.366618
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) −9.00000 −0.329293
\(748\) 9.00000 0.329073
\(749\) −3.00000 −0.109618
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 12.0000 0.437595
\(753\) −6.00000 −0.218652
\(754\) 42.0000 1.52955
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) 13.0000 0.472493 0.236247 0.971693i \(-0.424083\pi\)
0.236247 + 0.971693i \(0.424083\pi\)
\(758\) 26.0000 0.944363
\(759\) −9.00000 −0.326679
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) −7.00000 −0.253583
\(763\) −7.00000 −0.253417
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) 9.00000 0.325183
\(767\) 42.0000 1.51653
\(768\) −1.00000 −0.0360844
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 9.00000 0.324127
\(772\) 10.0000 0.359908
\(773\) −15.0000 −0.539513 −0.269756 0.962929i \(-0.586943\pi\)
−0.269756 + 0.962929i \(0.586943\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 16.0000 0.574367
\(777\) 1.00000 0.0358748
\(778\) 18.0000 0.645331
\(779\) 0 0
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 9.00000 0.321839
\(783\) −6.00000 −0.214423
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) −2.00000 −0.0712923 −0.0356462 0.999364i \(-0.511349\pi\)
−0.0356462 + 0.999364i \(0.511349\pi\)
\(788\) −3.00000 −0.106871
\(789\) 0 0
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 3.00000 0.106600
\(793\) −70.0000 −2.48577
\(794\) 4.00000 0.141955
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 1.00000 0.0353996
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) −9.00000 −0.317999
\(802\) 15.0000 0.529668
\(803\) −33.0000 −1.16454
\(804\) −10.0000 −0.352673
\(805\) 0 0
\(806\) −70.0000 −2.46564
\(807\) −9.00000 −0.316815
\(808\) −6.00000 −0.211079
\(809\) −21.0000 −0.738321 −0.369160 0.929366i \(-0.620355\pi\)
−0.369160 + 0.929366i \(0.620355\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 6.00000 0.210559
\(813\) −20.0000 −0.701431
\(814\) −3.00000 −0.105150
\(815\) 0 0
\(816\) −3.00000 −0.105021
\(817\) −4.00000 −0.139942
\(818\) −4.00000 −0.139857
\(819\) 7.00000 0.244600
\(820\) 0 0
\(821\) −9.00000 −0.314102 −0.157051 0.987590i \(-0.550199\pi\)
−0.157051 + 0.987590i \(0.550199\pi\)
\(822\) −18.0000 −0.627822
\(823\) 7.00000 0.244005 0.122002 0.992530i \(-0.461068\pi\)
0.122002 + 0.992530i \(0.461068\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 3.00000 0.104257
\(829\) 29.0000 1.00721 0.503606 0.863934i \(-0.332006\pi\)
0.503606 + 0.863934i \(0.332006\pi\)
\(830\) 0 0
\(831\) 23.0000 0.797861
\(832\) 7.00000 0.242681
\(833\) −18.0000 −0.623663
\(834\) −14.0000 −0.484780
\(835\) 0 0
\(836\) −3.00000 −0.103757
\(837\) 10.0000 0.345651
\(838\) −27.0000 −0.932700
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) −27.0000 −0.929929
\(844\) 2.00000 0.0688428
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) −2.00000 −0.0687208
\(848\) −9.00000 −0.309061
\(849\) −13.0000 −0.446159
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) 6.00000 0.205557
\(853\) 19.0000 0.650548 0.325274 0.945620i \(-0.394544\pi\)
0.325274 + 0.945620i \(0.394544\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) −3.00000 −0.102538
\(857\) 9.00000 0.307434 0.153717 0.988115i \(-0.450876\pi\)
0.153717 + 0.988115i \(0.450876\pi\)
\(858\) −21.0000 −0.716928
\(859\) 5.00000 0.170598 0.0852989 0.996355i \(-0.472815\pi\)
0.0852989 + 0.996355i \(0.472815\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −3.00000 −0.102180
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 31.0000 1.05342
\(867\) 8.00000 0.271694
\(868\) −10.0000 −0.339422
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 70.0000 2.37186
\(872\) −7.00000 −0.237050
\(873\) 16.0000 0.541518
\(874\) −3.00000 −0.101477
\(875\) 0 0
\(876\) 11.0000 0.371656
\(877\) −32.0000 −1.08056 −0.540282 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) 20.0000 0.674967
\(879\) −15.0000 −0.505937
\(880\) 0 0
\(881\) −12.0000 −0.404290 −0.202145 0.979356i \(-0.564791\pi\)
−0.202145 + 0.979356i \(0.564791\pi\)
\(882\) −6.00000 −0.202031
\(883\) −35.0000 −1.17784 −0.588922 0.808190i \(-0.700447\pi\)
−0.588922 + 0.808190i \(0.700447\pi\)
\(884\) 21.0000 0.706306
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 18.0000 0.604381 0.302190 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) 1.00000 0.0335578
\(889\) 7.00000 0.234772
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) −8.00000 −0.267860
\(893\) −12.0000 −0.401565
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) −21.0000 −0.701170
\(898\) −30.0000 −1.00111
\(899\) −60.0000 −2.00111
\(900\) 0 0
\(901\) −27.0000 −0.899500
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) −23.0000 −0.764124
\(907\) 37.0000 1.22856 0.614282 0.789086i \(-0.289446\pi\)
0.614282 + 0.789086i \(0.289446\pi\)
\(908\) −24.0000 −0.796468
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 1.00000 0.0331133
\(913\) −27.0000 −0.893570
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) −12.0000 −0.396275
\(918\) −3.00000 −0.0990148
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) −22.0000 −0.724925
\(922\) 18.0000 0.592798
\(923\) −42.0000 −1.38245
\(924\) −3.00000 −0.0986928
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) 4.00000 0.131377
\(928\) 6.00000 0.196960
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) −18.0000 −0.589610
\(933\) 24.0000 0.785725
\(934\) 0 0
\(935\) 0 0
\(936\) 7.00000 0.228802
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 10.0000 0.326512
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −4.00000 −0.130327
\(943\) 0 0
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) −8.00000 −0.259828
\(949\) −77.0000 −2.49953
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 3.00000 0.0972306
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −9.00000 −0.291386
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) −18.0000 −0.581857
\(958\) −39.0000 −1.26003
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) −7.00000 −0.225689
\(963\) −3.00000 −0.0966736
\(964\) 26.0000 0.837404
\(965\) 0 0
\(966\) −3.00000 −0.0965234
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 3.00000 0.0963739
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 14.0000 0.448819
\(974\) 22.0000 0.704925
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −9.00000 −0.287936 −0.143968 0.989582i \(-0.545986\pi\)
−0.143968 + 0.989582i \(0.545986\pi\)
\(978\) −1.00000 −0.0319765
\(979\) −27.0000 −0.862924
\(980\) 0 0
\(981\) −7.00000 −0.223493
\(982\) −9.00000 −0.287202
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 18.0000 0.573237
\(987\) −12.0000 −0.381964
\(988\) −7.00000 −0.222700
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) −10.0000 −0.317500
\(993\) 28.0000 0.888553
\(994\) −6.00000 −0.190308
\(995\) 0 0
\(996\) 9.00000 0.285176
\(997\) 1.00000 0.0316703 0.0158352 0.999875i \(-0.494959\pi\)
0.0158352 + 0.999875i \(0.494959\pi\)
\(998\) 41.0000 1.29783
\(999\) 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.ba.1.1 1
5.4 even 2 1110.2.a.e.1.1 1
15.14 odd 2 3330.2.a.u.1.1 1
20.19 odd 2 8880.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.e.1.1 1 5.4 even 2
3330.2.a.u.1.1 1 15.14 odd 2
5550.2.a.ba.1.1 1 1.1 even 1 trivial
8880.2.a.f.1.1 1 20.19 odd 2