Properties

Label 3450.2.a.x.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +9.00000 q^{29} -10.0000 q^{31} +1.00000 q^{32} +3.00000 q^{34} +1.00000 q^{36} +11.0000 q^{37} +2.00000 q^{38} +2.00000 q^{39} +12.0000 q^{41} -1.00000 q^{42} -10.0000 q^{43} -1.00000 q^{46} -9.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +3.00000 q^{51} +2.00000 q^{52} +1.00000 q^{54} -1.00000 q^{56} +2.00000 q^{57} +9.00000 q^{58} -6.00000 q^{59} +14.0000 q^{61} -10.0000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +14.0000 q^{67} +3.00000 q^{68} -1.00000 q^{69} +3.00000 q^{71} +1.00000 q^{72} +11.0000 q^{73} +11.0000 q^{74} +2.00000 q^{76} +2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{82} -9.00000 q^{83} -1.00000 q^{84} -10.0000 q^{86} +9.00000 q^{87} +3.00000 q^{89} -2.00000 q^{91} -1.00000 q^{92} -10.0000 q^{93} -9.00000 q^{94} +1.00000 q^{96} -10.0000 q^{97} -6.00000 q^{98} +O(q^{100})\)

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
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −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
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 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
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 11.0000 1.80839 0.904194 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 2.00000 0.324443
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) −1.00000 −0.154303
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −9.00000 −1.31278 −0.656392 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 2.00000 0.277350
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 2.00000 0.264906
\(58\) 9.00000 1.18176
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) −10.0000 −1.27000
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 3.00000 0.363803
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 1.00000 0.117851
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 11.0000 1.27872
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 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
\(82\) 12.0000 1.32518
\(83\) −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
\(86\) −10.0000 −1.07833
\(87\) 9.00000 0.964901
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −1.00000 −0.104257
\(93\) −10.0000 −1.03695
\(94\) −9.00000 −0.928279
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −6.00000 −0.606092
\(99\) 0 0
\(100\) 0 0
\(101\) −9.00000 −0.895533 −0.447767 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(102\) 3.00000 0.297044
\(103\) −7.00000 −0.689730 −0.344865 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 11.0000 1.04407
\(112\) −1.00000 −0.0944911
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) 2.00000 0.184900
\(118\) −6.00000 −0.552345
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 14.0000 1.26750
\(123\) 12.0000 1.08200
\(124\) −10.0000 −0.898027
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) −9.00000 −0.757937
\(142\) 3.00000 0.251754
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 11.0000 0.910366
\(147\) −6.00000 −0.494872
\(148\) 11.0000 0.904194
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 2.00000 0.162221
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 1.00000 0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 12.0000 0.937043
\(165\) 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
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) −10.0000 −0.762493
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 9.00000 0.682288
\(175\) 0 0
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) 3.00000 0.224860
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −1.00000 −0.0743294 −0.0371647 0.999309i \(-0.511833\pi\)
−0.0371647 + 0.999309i \(0.511833\pi\)
\(182\) −2.00000 −0.148250
\(183\) 14.0000 1.03491
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −10.0000 −0.733236
\(187\) 0 0
\(188\) −9.00000 −0.656392
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 1.00000 0.0721688
\(193\) −25.0000 −1.79954 −0.899770 0.436365i \(-0.856266\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −15.0000 −1.06871 −0.534353 0.845262i \(-0.679445\pi\)
−0.534353 + 0.845262i \(0.679445\pi\)
\(198\) 0 0
\(199\) −1.00000 −0.0708881 −0.0354441 0.999372i \(-0.511285\pi\)
−0.0354441 + 0.999372i \(0.511285\pi\)
\(200\) 0 0
\(201\) 14.0000 0.987484
\(202\) −9.00000 −0.633238
\(203\) −9.00000 −0.631676
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) −7.00000 −0.487713
\(207\) −1.00000 −0.0695048
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 23.0000 1.58339 0.791693 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) 0 0
\(213\) 3.00000 0.205557
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 10.0000 0.678844
\(218\) 11.0000 0.745014
\(219\) 11.0000 0.743311
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 11.0000 0.738272
\(223\) −22.0000 −1.47323 −0.736614 0.676313i \(-0.763577\pi\)
−0.736614 + 0.676313i \(0.763577\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −9.00000 −0.598671
\(227\) −21.0000 −1.39382 −0.696909 0.717159i \(-0.745442\pi\)
−0.696909 + 0.717159i \(0.745442\pi\)
\(228\) 2.00000 0.132453
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 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
\(236\) −6.00000 −0.390567
\(237\) 8.00000 0.519656
\(238\) −3.00000 −0.194461
\(239\) 27.0000 1.74648 0.873242 0.487286i \(-0.162013\pi\)
0.873242 + 0.487286i \(0.162013\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) 4.00000 0.254514
\(248\) −10.0000 −0.635001
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −10.0000 −0.622573
\(259\) −11.0000 −0.683507
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) 12.0000 0.741362
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) 3.00000 0.183597
\(268\) 14.0000 0.855186
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 3.00000 0.181902
\(273\) −2.00000 −0.121046
\(274\) 9.00000 0.543710
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 5.00000 0.299880
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) −9.00000 −0.535942
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 3.00000 0.178017
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 11.0000 0.643726
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) −6.00000 −0.349927
\(295\) 0 0
\(296\) 11.0000 0.639362
\(297\) 0 0
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 10.0000 0.576390
\(302\) 8.00000 0.460348
\(303\) −9.00000 −0.517036
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 3.00000 0.171499
\(307\) 5.00000 0.285365 0.142683 0.989769i \(-0.454427\pi\)
0.142683 + 0.989769i \(0.454427\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) −15.0000 −0.850572 −0.425286 0.905059i \(-0.639826\pi\)
−0.425286 + 0.905059i \(0.639826\pi\)
\(312\) 2.00000 0.113228
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −27.0000 −1.51647 −0.758236 0.651981i \(-0.773938\pi\)
−0.758236 + 0.651981i \(0.773938\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 1.00000 0.0557278
\(323\) 6.00000 0.333849
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) 11.0000 0.608301
\(328\) 12.0000 0.662589
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) 5.00000 0.274825 0.137412 0.990514i \(-0.456121\pi\)
0.137412 + 0.990514i \(0.456121\pi\)
\(332\) −9.00000 −0.493939
\(333\) 11.0000 0.602796
\(334\) −3.00000 −0.164153
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) −28.0000 −1.52526 −0.762629 0.646837i \(-0.776092\pi\)
−0.762629 + 0.646837i \(0.776092\pi\)
\(338\) −9.00000 −0.489535
\(339\) −9.00000 −0.488813
\(340\) 0 0
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) 13.0000 0.701934
\(344\) −10.0000 −0.539164
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −30.0000 −1.61048 −0.805242 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) 9.00000 0.482451
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) 3.00000 0.159000
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −1.00000 −0.0525588
\(363\) −11.0000 −0.577350
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 14.0000 0.731792
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) −10.0000 −0.518476
\(373\) −13.0000 −0.673114 −0.336557 0.941663i \(-0.609263\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) 18.0000 0.927047
\(378\) −1.00000 −0.0514344
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) −18.0000 −0.920960
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −25.0000 −1.27247
\(387\) −10.0000 −0.508329
\(388\) −10.0000 −0.507673
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) −6.00000 −0.303046
\(393\) 12.0000 0.605320
\(394\) −15.0000 −0.755689
\(395\) 0 0
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) −1.00000 −0.0501255
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 14.0000 0.698257
\(403\) −20.0000 −0.996271
\(404\) −9.00000 −0.447767
\(405\) 0 0
\(406\) −9.00000 −0.446663
\(407\) 0 0
\(408\) 3.00000 0.148522
\(409\) 17.0000 0.840596 0.420298 0.907386i \(-0.361926\pi\)
0.420298 + 0.907386i \(0.361926\pi\)
\(410\) 0 0
\(411\) 9.00000 0.443937
\(412\) −7.00000 −0.344865
\(413\) 6.00000 0.295241
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 5.00000 0.244851
\(418\) 0 0
\(419\) 15.0000 0.732798 0.366399 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 23.0000 1.11962
\(423\) −9.00000 −0.437595
\(424\) 0 0
\(425\) 0 0
\(426\) 3.00000 0.145350
\(427\) −14.0000 −0.677507
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 10.0000 0.480015
\(435\) 0 0
\(436\) 11.0000 0.526804
\(437\) −2.00000 −0.0956730
\(438\) 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
\(442\) 6.00000 0.285391
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 11.0000 0.522037
\(445\) 0 0
\(446\) −22.0000 −1.04173
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −9.00000 −0.423324
\(453\) 8.00000 0.375873
\(454\) −21.0000 −0.985579
\(455\) 0 0
\(456\) 2.00000 0.0936586
\(457\) 32.0000 1.49690 0.748448 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 2.00000 0.0934539
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) 3.00000 0.139724 0.0698620 0.997557i \(-0.477744\pi\)
0.0698620 + 0.997557i \(0.477744\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 9.00000 0.417815
\(465\) 0 0
\(466\) −24.0000 −1.11178
\(467\) −33.0000 −1.52706 −0.763529 0.645774i \(-0.776535\pi\)
−0.763529 + 0.645774i \(0.776535\pi\)
\(468\) 2.00000 0.0924500
\(469\) −14.0000 −0.646460
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) −6.00000 −0.276172
\(473\) 0 0
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) −3.00000 −0.137505
\(477\) 0 0
\(478\) 27.0000 1.23495
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) 22.0000 1.00311
\(482\) 8.00000 0.364390
\(483\) 1.00000 0.0455016
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) 14.0000 0.633750
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 12.0000 0.541002
\(493\) 27.0000 1.21602
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) −3.00000 −0.134568
\(498\) −9.00000 −0.403300
\(499\) −19.0000 −0.850557 −0.425278 0.905063i \(-0.639824\pi\)
−0.425278 + 0.905063i \(0.639824\pi\)
\(500\) 0 0
\(501\) −3.00000 −0.134030
\(502\) 3.00000 0.133897
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) −27.0000 −1.19675 −0.598377 0.801215i \(-0.704187\pi\)
−0.598377 + 0.801215i \(0.704187\pi\)
\(510\) 0 0
\(511\) −11.0000 −0.486611
\(512\) 1.00000 0.0441942
\(513\) 2.00000 0.0883022
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −10.0000 −0.440225
\(517\) 0 0
\(518\) −11.0000 −0.483312
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −39.0000 −1.70862 −0.854311 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) 9.00000 0.393919
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −18.0000 −0.784837
\(527\) −30.0000 −1.30682
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) −2.00000 −0.0867110
\(533\) 24.0000 1.03956
\(534\) 3.00000 0.129823
\(535\) 0 0
\(536\) 14.0000 0.604708
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) 0 0
\(541\) 44.0000 1.89171 0.945854 0.324593i \(-0.105227\pi\)
0.945854 + 0.324593i \(0.105227\pi\)
\(542\) −28.0000 −1.20270
\(543\) −1.00000 −0.0429141
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) −2.00000 −0.0855921
\(547\) −1.00000 −0.0427569 −0.0213785 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) 9.00000 0.384461
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) 18.0000 0.766826
\(552\) −1.00000 −0.0425628
\(553\) −8.00000 −0.340195
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) 5.00000 0.212047
\(557\) −42.0000 −1.77960 −0.889799 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) −10.0000 −0.423334
\(559\) −20.0000 −0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) 3.00000 0.126547
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −9.00000 −0.378968
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) −1.00000 −0.0419961
\(568\) 3.00000 0.125877
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) 26.0000 1.08807 0.544033 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(572\) 0 0
\(573\) −18.0000 −0.751961
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) −8.00000 −0.332756
\(579\) −25.0000 −1.03896
\(580\) 0 0
\(581\) 9.00000 0.373383
\(582\) −10.0000 −0.414513
\(583\) 0 0
\(584\) 11.0000 0.455183
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −6.00000 −0.247436
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) −15.0000 −0.617018
\(592\) 11.0000 0.452097
\(593\) 36.0000 1.47834 0.739171 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.00000 −0.0409273
\(598\) −2.00000 −0.0817861
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 10.0000 0.407570
\(603\) 14.0000 0.570124
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) −9.00000 −0.365600
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 2.00000 0.0811107
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) −18.0000 −0.728202
\(612\) 3.00000 0.121268
\(613\) −19.0000 −0.767403 −0.383701 0.923457i \(-0.625351\pi\)
−0.383701 + 0.923457i \(0.625351\pi\)
\(614\) 5.00000 0.201784
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) −7.00000 −0.281581
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −15.0000 −0.601445
\(623\) −3.00000 −0.120192
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 33.0000 1.31580
\(630\) 0 0
\(631\) 5.00000 0.199047 0.0995234 0.995035i \(-0.468268\pi\)
0.0995234 + 0.995035i \(0.468268\pi\)
\(632\) 8.00000 0.318223
\(633\) 23.0000 0.914168
\(634\) −27.0000 −1.07231
\(635\) 0 0
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) 3.00000 0.118678
\(640\) 0 0
\(641\) −21.0000 −0.829450 −0.414725 0.909947i \(-0.636122\pi\)
−0.414725 + 0.909947i \(0.636122\pi\)
\(642\) 12.0000 0.473602
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) 21.0000 0.825595 0.412798 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) −16.0000 −0.626608
\(653\) 9.00000 0.352197 0.176099 0.984373i \(-0.443652\pi\)
0.176099 + 0.984373i \(0.443652\pi\)
\(654\) 11.0000 0.430134
\(655\) 0 0
\(656\) 12.0000 0.468521
\(657\) 11.0000 0.429151
\(658\) 9.00000 0.350857
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 0 0
\(661\) −31.0000 −1.20576 −0.602880 0.797832i \(-0.705980\pi\)
−0.602880 + 0.797832i \(0.705980\pi\)
\(662\) 5.00000 0.194331
\(663\) 6.00000 0.233021
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) 11.0000 0.426241
\(667\) −9.00000 −0.348481
\(668\) −3.00000 −0.116073
\(669\) −22.0000 −0.850569
\(670\) 0 0
\(671\) 0 0
\(672\) −1.00000 −0.0385758
\(673\) −7.00000 −0.269830 −0.134915 0.990857i \(-0.543076\pi\)
−0.134915 + 0.990857i \(0.543076\pi\)
\(674\) −28.0000 −1.07852
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) −9.00000 −0.345643
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −21.0000 −0.804722
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) 2.00000 0.0763048
\(688\) −10.0000 −0.381246
\(689\) 0 0
\(690\) 0 0
\(691\) 11.0000 0.418460 0.209230 0.977866i \(-0.432904\pi\)
0.209230 + 0.977866i \(0.432904\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −30.0000 −1.13878
\(695\) 0 0
\(696\) 9.00000 0.341144
\(697\) 36.0000 1.36360
\(698\) −10.0000 −0.378506
\(699\) −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
\(703\) 22.0000 0.829746
\(704\) 0 0
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 9.00000 0.338480
\(708\) −6.00000 −0.225494
\(709\) −19.0000 −0.713560 −0.356780 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 3.00000 0.112430
\(713\) 10.0000 0.374503
\(714\) −3.00000 −0.112272
\(715\) 0 0
\(716\) 0 0
\(717\) 27.0000 1.00833
\(718\) 36.0000 1.34351
\(719\) −15.0000 −0.559406 −0.279703 0.960087i \(-0.590236\pi\)
−0.279703 + 0.960087i \(0.590236\pi\)
\(720\) 0 0
\(721\) 7.00000 0.260694
\(722\) −15.0000 −0.558242
\(723\) 8.00000 0.297523
\(724\) −1.00000 −0.0371647
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −30.0000 −1.10959
\(732\) 14.0000 0.517455
\(733\) 29.0000 1.07114 0.535570 0.844491i \(-0.320097\pi\)
0.535570 + 0.844491i \(0.320097\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) 12.0000 0.441726
\(739\) −1.00000 −0.0367856 −0.0183928 0.999831i \(-0.505855\pi\)
−0.0183928 + 0.999831i \(0.505855\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) −30.0000 −1.10059 −0.550297 0.834969i \(-0.685485\pi\)
−0.550297 + 0.834969i \(0.685485\pi\)
\(744\) −10.0000 −0.366618
\(745\) 0 0
\(746\) −13.0000 −0.475964
\(747\) −9.00000 −0.329293
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −13.0000 −0.474377 −0.237188 0.971464i \(-0.576226\pi\)
−0.237188 + 0.971464i \(0.576226\pi\)
\(752\) −9.00000 −0.328196
\(753\) 3.00000 0.109326
\(754\) 18.0000 0.655521
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) −13.0000 −0.472493 −0.236247 0.971693i \(-0.575917\pi\)
−0.236247 + 0.971693i \(0.575917\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 8.00000 0.289809
\(763\) −11.0000 −0.398227
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) −12.0000 −0.433295
\(768\) 1.00000 0.0360844
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) −25.0000 −0.899770
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) −10.0000 −0.359443
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) −11.0000 −0.394623
\(778\) −30.0000 −1.07555
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) −3.00000 −0.107280
\(783\) 9.00000 0.321634
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −15.0000 −0.534353
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) 28.0000 0.994309
\(794\) 8.00000 0.283909
\(795\) 0 0
\(796\) −1.00000 −0.0354441
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) −2.00000 −0.0707992
\(799\) −27.0000 −0.955191
\(800\) 0 0
\(801\) 3.00000 0.106000
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) 14.0000 0.493742
\(805\) 0 0
\(806\) −20.0000 −0.704470
\(807\) 6.00000 0.211210
\(808\) −9.00000 −0.316619
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −52.0000 −1.82597 −0.912983 0.407997i \(-0.866228\pi\)
−0.912983 + 0.407997i \(0.866228\pi\)
\(812\) −9.00000 −0.315838
\(813\) −28.0000 −0.982003
\(814\) 0 0
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) −20.0000 −0.699711
\(818\) 17.0000 0.594391
\(819\) −2.00000 −0.0698857
\(820\) 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
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −7.00000 −0.243857
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) −33.0000 −1.14752 −0.573761 0.819023i \(-0.694516\pi\)
−0.573761 + 0.819023i \(0.694516\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 2.00000 0.0693375
\(833\) −18.0000 −0.623663
\(834\) 5.00000 0.173136
\(835\) 0 0
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) 15.0000 0.518166
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 26.0000 0.896019
\(843\) 3.00000 0.103325
\(844\) 23.0000 0.791693
\(845\) 0 0
\(846\) −9.00000 −0.309426
\(847\) 11.0000 0.377964
\(848\) 0 0
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) −11.0000 −0.377075
\(852\) 3.00000 0.102778
\(853\) 44.0000 1.50653 0.753266 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) −14.0000 −0.479070
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 30.0000 1.02180
\(863\) 27.0000 0.919091 0.459545 0.888154i \(-0.348012\pi\)
0.459545 + 0.888154i \(0.348012\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) −8.00000 −0.271694
\(868\) 10.0000 0.339422
\(869\) 0 0
\(870\) 0 0
\(871\) 28.0000 0.948744
\(872\) 11.0000 0.372507
\(873\) −10.0000 −0.338449
\(874\) −2.00000 −0.0676510
\(875\) 0 0
\(876\) 11.0000 0.371656
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 20.0000 0.674967
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) −6.00000 −0.202031
\(883\) 11.0000 0.370179 0.185090 0.982722i \(-0.440742\pi\)
0.185090 + 0.982722i \(0.440742\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) 0 0
\(887\) 39.0000 1.30949 0.654746 0.755849i \(-0.272776\pi\)
0.654746 + 0.755849i \(0.272776\pi\)
\(888\) 11.0000 0.369136
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) −22.0000 −0.736614
\(893\) −18.0000 −0.602347
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) −2.00000 −0.0667781
\(898\) −6.00000 −0.200223
\(899\) −90.0000 −3.00167
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 10.0000 0.332779
\(904\) −9.00000 −0.299336
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) 38.0000 1.26177 0.630885 0.775877i \(-0.282692\pi\)
0.630885 + 0.775877i \(0.282692\pi\)
\(908\) −21.0000 −0.696909
\(909\) −9.00000 −0.298511
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 2.00000 0.0662266
\(913\) 0 0
\(914\) 32.0000 1.05847
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) −12.0000 −0.396275
\(918\) 3.00000 0.0990148
\(919\) −31.0000 −1.02260 −0.511298 0.859404i \(-0.670835\pi\)
−0.511298 + 0.859404i \(0.670835\pi\)
\(920\) 0 0
\(921\) 5.00000 0.164756
\(922\) 3.00000 0.0987997
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) 0 0
\(926\) 8.00000 0.262896
\(927\) −7.00000 −0.229910
\(928\) 9.00000 0.295439
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) −24.0000 −0.786146
\(933\) −15.0000 −0.491078
\(934\) −33.0000 −1.07979
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 50.0000 1.63343 0.816714 0.577042i \(-0.195793\pi\)
0.816714 + 0.577042i \(0.195793\pi\)
\(938\) −14.0000 −0.457116
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 14.0000 0.456145
\(943\) −12.0000 −0.390774
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 8.00000 0.259828
\(949\) 22.0000 0.714150
\(950\) 0 0
\(951\) −27.0000 −0.875535
\(952\) −3.00000 −0.0972306
\(953\) 15.0000 0.485898 0.242949 0.970039i \(-0.421885\pi\)
0.242949 + 0.970039i \(0.421885\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 27.0000 0.873242
\(957\) 0 0
\(958\) −18.0000 −0.581554
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 22.0000 0.709308
\(963\) 12.0000 0.386695
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) 1.00000 0.0321745
\(967\) 2.00000 0.0643157 0.0321578 0.999483i \(-0.489762\pi\)
0.0321578 + 0.999483i \(0.489762\pi\)
\(968\) −11.0000 −0.353553
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) −21.0000 −0.673922 −0.336961 0.941519i \(-0.609399\pi\)
−0.336961 + 0.941519i \(0.609399\pi\)
\(972\) 1.00000 0.0320750
\(973\) −5.00000 −0.160293
\(974\) −28.0000 −0.897178
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) 21.0000 0.671850 0.335925 0.941889i \(-0.390951\pi\)
0.335925 + 0.941889i \(0.390951\pi\)
\(978\) −16.0000 −0.511624
\(979\) 0 0
\(980\) 0 0
\(981\) 11.0000 0.351203
\(982\) −30.0000 −0.957338
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 12.0000 0.382546
\(985\) 0 0
\(986\) 27.0000 0.859855
\(987\) 9.00000 0.286473
\(988\) 4.00000 0.127257
\(989\) 10.0000 0.317982
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) −10.0000 −0.317500
\(993\) 5.00000 0.158670
\(994\) −3.00000 −0.0951542
\(995\) 0 0
\(996\) −9.00000 −0.285176
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −19.0000 −0.601434
\(999\) 11.0000 0.348025
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 3450.2.a.x.1.1 yes 1
5.2 odd 4 3450.2.d.p.2899.2 2
5.3 odd 4 3450.2.d.p.2899.1 2
5.4 even 2 3450.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.f.1.1 1 5.4 even 2
3450.2.a.x.1.1 yes 1 1.1 even 1 trivial
3450.2.d.p.2899.1 2 5.3 odd 4
3450.2.d.p.2899.2 2 5.2 odd 4