Properties

Label 1050.2.a.m.1.1
Level $1050$
Weight $2$
Character 1050.1
Self dual yes
Analytic conductor $8.384$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1050.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +8.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -1.00000 q^{21} -2.00000 q^{22} -1.00000 q^{24} +2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +6.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +8.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} -2.00000 q^{38} -2.00000 q^{39} +6.00000 q^{41} -1.00000 q^{42} +8.00000 q^{43} -2.00000 q^{44} +4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -8.00000 q^{51} +2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} -8.00000 q^{59} +10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -12.0000 q^{67} +8.00000 q^{68} -14.0000 q^{71} +1.00000 q^{72} -10.0000 q^{73} +8.00000 q^{74} -2.00000 q^{76} -2.00000 q^{77} -2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +16.0000 q^{83} -1.00000 q^{84} +8.00000 q^{86} +6.00000 q^{87} -2.00000 q^{88} +10.0000 q^{89} +2.00000 q^{91} -6.00000 q^{93} +4.00000 q^{94} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −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
\(17\) 8.00000 1.94029 0.970143 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) 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
\(22\) −2.00000 −0.426401
\(23\) 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
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) 8.00000 1.37199
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −2.00000 −0.324443
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −1.00000 −0.154303
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 2.00000 0.277350
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 2.00000 0.264906
\(58\) −6.00000 −0.787839
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 6.00000 0.762001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 8.00000 0.970143
\(69\) 0 0
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) −2.00000 −0.227921
\(78\) −2.00000 −0.226455
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 6.00000 0.643268
\(88\) −2.00000 −0.213201
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −6.00000 −0.622171
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 1.00000 0.101015
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −8.00000 −0.792118
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 1.00000 0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) −8.00000 −0.736460
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 10.0000 0.905357
\(123\) −6.00000 −0.541002
\(124\) 6.00000 0.538816
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 2.00000 0.174078
\(133\) −2.00000 −0.173422
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 8.00000 0.685994
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) −14.0000 −1.17485
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) −1.00000 −0.0824786
\(148\) 8.00000 0.657596
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 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
\(153\) 8.00000 0.646762
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 4.00000 0.318223
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 8.00000 0.609994
\(173\) −16.0000 −1.21646 −0.608229 0.793762i \(-0.708120\pi\)
−0.608229 + 0.793762i \(0.708120\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 8.00000 0.601317
\(178\) 10.0000 0.749532
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 2.00000 0.148250
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) −16.0000 −1.17004
\(188\) 4.00000 0.291730
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 14.0000 1.01300 0.506502 0.862239i \(-0.330938\pi\)
0.506502 + 0.862239i \(0.330938\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −2.00000 −0.142134
\(199\) −26.0000 −1.84309 −0.921546 0.388270i \(-0.873073\pi\)
−0.921546 + 0.388270i \(0.873073\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) −10.0000 −0.703598
\(203\) −6.00000 −0.421117
\(204\) −8.00000 −0.560112
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −2.00000 −0.137361
\(213\) 14.0000 0.959264
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 6.00000 0.407307
\(218\) −6.00000 −0.406371
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 16.0000 1.07628
\(222\) −8.00000 −0.536925
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 2.00000 0.132453
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) −6.00000 −0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) −4.00000 −0.259828
\(238\) 8.00000 0.518563
\(239\) −22.0000 −1.42306 −0.711531 0.702655i \(-0.751998\pi\)
−0.711531 + 0.702655i \(0.751998\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −7.00000 −0.449977
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −4.00000 −0.254514
\(248\) 6.00000 0.381000
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −20.0000 −1.24757 −0.623783 0.781598i \(-0.714405\pi\)
−0.623783 + 0.781598i \(0.714405\pi\)
\(258\) −8.00000 −0.498058
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) −20.0000 −1.23560
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) −10.0000 −0.611990
\(268\) −12.0000 −0.733017
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 8.00000 0.485071
\(273\) −2.00000 −0.121046
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −2.00000 −0.119952
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) −4.00000 −0.238197
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −14.0000 −0.830747
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 6.00000 0.354169
\(288\) 1.00000 0.0589256
\(289\) 47.0000 2.76471
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) −10.0000 −0.585206
\(293\) −16.0000 −0.934730 −0.467365 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) 2.00000 0.116052
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 8.00000 0.460348
\(303\) 10.0000 0.574485
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 8.00000 0.457330
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) −2.00000 −0.113228
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 2.00000 0.112154
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 0 0
\(327\) 6.00000 0.331801
\(328\) 6.00000 0.331295
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 16.0000 0.878114
\(333\) 8.00000 0.438397
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) −9.00000 −0.489535
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) −2.00000 −0.108148
\(343\) 1.00000 0.0539949
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −16.0000 −0.860165
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 6.00000 0.321634
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −2.00000 −0.106600
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) −8.00000 −0.423405
\(358\) −2.00000 −0.105703
\(359\) 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
\(362\) 22.0000 1.15629
\(363\) 7.00000 0.367405
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) −6.00000 −0.311086
\(373\) 36.0000 1.86401 0.932005 0.362446i \(-0.118058\pi\)
0.932005 + 0.362446i \(0.118058\pi\)
\(374\) −16.0000 −0.827340
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) −12.0000 −0.618031
\(378\) −1.00000 −0.0514344
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 14.0000 0.716302
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 0 0
\(387\) 8.00000 0.406663
\(388\) 10.0000 0.507673
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 20.0000 1.00887
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −26.0000 −1.30326
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 12.0000 0.598506
\(403\) 12.0000 0.597763
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) −16.0000 −0.793091
\(408\) −8.00000 −0.396059
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 2.00000 0.0979404
\(418\) 4.00000 0.195646
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −8.00000 −0.389434
\(423\) 4.00000 0.194487
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) 14.0000 0.678302
\(427\) 10.0000 0.483934
\(428\) −12.0000 −0.580042
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) −6.00000 −0.289010 −0.144505 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 6.00000 0.288009
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) 10.0000 0.477818
\(439\) 6.00000 0.286364 0.143182 0.989696i \(-0.454267\pi\)
0.143182 + 0.989696i \(0.454267\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 16.0000 0.761042
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 18.0000 0.851371
\(448\) 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
\(451\) −12.0000 −0.565058
\(452\) 6.00000 0.282216
\(453\) −8.00000 −0.375873
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) 2.00000 0.0936586
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 26.0000 1.21490
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 2.00000 0.0930484
\(463\) −12.0000 −0.557687 −0.278844 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 16.0000 0.740392 0.370196 0.928954i \(-0.379291\pi\)
0.370196 + 0.928954i \(0.379291\pi\)
\(468\) 2.00000 0.0924500
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) −8.00000 −0.368230
\(473\) −16.0000 −0.735681
\(474\) −4.00000 −0.183726
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) −2.00000 −0.0915737
\(478\) −22.0000 −1.00626
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 36.0000 1.63132 0.815658 0.578535i \(-0.196375\pi\)
0.815658 + 0.578535i \(0.196375\pi\)
\(488\) 10.0000 0.452679
\(489\) 0 0
\(490\) 0 0
\(491\) −42.0000 −1.89543 −0.947717 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) −6.00000 −0.270501
\(493\) −48.0000 −2.16181
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) −14.0000 −0.627986
\(498\) −16.0000 −0.716977
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 12.0000 0.532414
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 1.00000 0.0441942
\(513\) 2.00000 0.0883022
\(514\) −20.0000 −0.882162
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) −8.00000 −0.351840
\(518\) 8.00000 0.351500
\(519\) 16.0000 0.702322
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −6.00000 −0.262613
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) −20.0000 −0.873704
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 48.0000 2.09091
\(528\) 2.00000 0.0870388
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) −2.00000 −0.0867110
\(533\) 12.0000 0.519778
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 2.00000 0.0863064
\(538\) 14.0000 0.603583
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 2.00000 0.0859074
\(543\) −22.0000 −0.944110
\(544\) 8.00000 0.342997
\(545\) 0 0
\(546\) −2.00000 −0.0855921
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −14.0000 −0.598050
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) −2.00000 −0.0848189
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 6.00000 0.254000
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 16.0000 0.675521
\(562\) 22.0000 0.928014
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) 1.00000 0.0419961
\(568\) −14.0000 −0.587427
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) −4.00000 −0.167248
\(573\) −14.0000 −0.584858
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 47.0000 1.95494
\(579\) 0 0
\(580\) 0 0
\(581\) 16.0000 0.663792
\(582\) −10.0000 −0.414513
\(583\) 4.00000 0.165663
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) −16.0000 −0.660954
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 8.00000 0.328798
\(593\) −32.0000 −1.31408 −0.657041 0.753855i \(-0.728192\pi\)
−0.657041 + 0.753855i \(0.728192\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) 26.0000 1.06411
\(598\) 0 0
\(599\) 14.0000 0.572024 0.286012 0.958226i \(-0.407670\pi\)
0.286012 + 0.958226i \(0.407670\pi\)
\(600\) 0 0
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 8.00000 0.326056
\(603\) −12.0000 −0.488678
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 10.0000 0.406222
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 8.00000 0.323381
\(613\) 32.0000 1.29247 0.646234 0.763139i \(-0.276343\pi\)
0.646234 + 0.763139i \(0.276343\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −18.0000 −0.723481 −0.361741 0.932279i \(-0.617817\pi\)
−0.361741 + 0.932279i \(0.617817\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −8.00000 −0.320771
\(623\) 10.0000 0.400642
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) −4.00000 −0.159745
\(628\) −10.0000 −0.399043
\(629\) 64.0000 2.55185
\(630\) 0 0
\(631\) 12.0000 0.477712 0.238856 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) 4.00000 0.159111
\(633\) 8.00000 0.317971
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) 2.00000 0.0792429
\(638\) 12.0000 0.475085
\(639\) −14.0000 −0.553831
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 12.0000 0.473602
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) 16.0000 0.629025 0.314512 0.949253i \(-0.398159\pi\)
0.314512 + 0.949253i \(0.398159\pi\)
\(648\) 1.00000 0.0392837
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 0 0
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 6.00000 0.234619
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) −10.0000 −0.390137
\(658\) 4.00000 0.155936
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 0 0
\(663\) −16.0000 −0.621389
\(664\) 16.0000 0.620920
\(665\) 0 0
\(666\) 8.00000 0.309994
\(667\) 0 0
\(668\) −8.00000 −0.309529
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −20.0000 −0.772091
\(672\) −1.00000 −0.0385758
\(673\) 12.0000 0.462566 0.231283 0.972887i \(-0.425708\pi\)
0.231283 + 0.972887i \(0.425708\pi\)
\(674\) 32.0000 1.23259
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) −6.00000 −0.230429
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) −12.0000 −0.459504
\(683\) 28.0000 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) −26.0000 −0.991962
\(688\) 8.00000 0.304997
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) −16.0000 −0.608229
\(693\) −2.00000 −0.0759737
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 48.0000 1.81813
\(698\) 18.0000 0.681310
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −16.0000 −0.603451
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 0 0
\(707\) −10.0000 −0.376089
\(708\) 8.00000 0.300658
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 10.0000 0.374766
\(713\) 0 0
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) −2.00000 −0.0747435
\(717\) 22.0000 0.821605
\(718\) 14.0000 0.522475
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −15.0000 −0.558242
\(723\) 10.0000 0.371904
\(724\) 22.0000 0.817624
\(725\) 0 0
\(726\) 7.00000 0.259794
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 64.0000 2.36713
\(732\) −10.0000 −0.369611
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) 0 0
\(737\) 24.0000 0.884051
\(738\) 6.00000 0.220863
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) −2.00000 −0.0734223
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) −6.00000 −0.219971
\(745\) 0 0
\(746\) 36.0000 1.31805
\(747\) 16.0000 0.585409
\(748\) −16.0000 −0.585018
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) −4.00000 −0.145382 −0.0726912 0.997354i \(-0.523159\pi\)
−0.0726912 + 0.997354i \(0.523159\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) −12.0000 −0.434714
\(763\) −6.00000 −0.217215
\(764\) 14.0000 0.506502
\(765\) 0 0
\(766\) 0 0
\(767\) −16.0000 −0.577727
\(768\) −1.00000 −0.0360844
\(769\) −42.0000 −1.51456 −0.757279 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(770\) 0 0
\(771\) 20.0000 0.720282
\(772\) 0 0
\(773\) −32.0000 −1.15096 −0.575480 0.817816i \(-0.695185\pi\)
−0.575480 + 0.817816i \(0.695185\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) −8.00000 −0.286998
\(778\) 6.00000 0.215110
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 28.0000 1.00192
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 20.0000 0.713376
\(787\) −44.0000 −1.56843 −0.784215 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) −6.00000 −0.213741
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) −2.00000 −0.0710669
\(793\) 20.0000 0.710221
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −26.0000 −0.921546
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 2.00000 0.0707992
\(799\) 32.0000 1.13208
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 18.0000 0.635602
\(803\) 20.0000 0.705785
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 12.0000 0.422682
\(807\) −14.0000 −0.492823
\(808\) −10.0000 −0.351799
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 14.0000 0.491606 0.245803 0.969320i \(-0.420948\pi\)
0.245803 + 0.969320i \(0.420948\pi\)
\(812\) −6.00000 −0.210559
\(813\) −2.00000 −0.0701431
\(814\) −16.0000 −0.560800
\(815\) 0 0
\(816\) −8.00000 −0.280056
\(817\) −16.0000 −0.559769
\(818\) 10.0000 0.349642
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 14.0000 0.488306
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 2.00000 0.0693375
\(833\) 8.00000 0.277184
\(834\) 2.00000 0.0692543
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) −6.00000 −0.207390
\(838\) 12.0000 0.414533
\(839\) 44.0000 1.51905 0.759524 0.650479i \(-0.225432\pi\)
0.759524 + 0.650479i \(0.225432\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) −22.0000 −0.757720
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) −7.00000 −0.240523
\(848\) −2.00000 −0.0686803
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 0 0
\(852\) 14.0000 0.479632
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 20.0000 0.683187 0.341593 0.939848i \(-0.389033\pi\)
0.341593 + 0.939848i \(0.389033\pi\)
\(858\) 4.00000 0.136558
\(859\) −6.00000 −0.204717 −0.102359 0.994748i \(-0.532639\pi\)
−0.102359 + 0.994748i \(0.532639\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) −6.00000 −0.204361
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) −47.0000 −1.59620
\(868\) 6.00000 0.203653
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) −6.00000 −0.203186
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) −48.0000 −1.62084 −0.810422 0.585846i \(-0.800762\pi\)
−0.810422 + 0.585846i \(0.800762\pi\)
\(878\) 6.00000 0.202490
\(879\) 16.0000 0.539667
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 1.00000 0.0336718
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 16.0000 0.538138
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) −8.00000 −0.268462
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) −16.0000 −0.535720
\(893\) −8.00000 −0.267710
\(894\) 18.0000 0.602010
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) −36.0000 −1.20067
\(900\) 0 0
\(901\) −16.0000 −0.533037
\(902\) −12.0000 −0.399556
\(903\) −8.00000 −0.266223
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) −8.00000 −0.265489
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 6.00000 0.198789 0.0993944 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) 2.00000 0.0662266
\(913\) −32.0000 −1.05905
\(914\) −28.0000 −0.926158
\(915\) 0 0
\(916\) 26.0000 0.859064
\(917\) −20.0000 −0.660458
\(918\) −8.00000 −0.264039
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) 18.0000 0.592798
\(923\) −28.0000 −0.921631
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) −12.0000 −0.394344
\(927\) 0 0
\(928\) −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
\(931\) −2.00000 −0.0655474
\(932\) 6.00000 0.196537
\(933\) 8.00000 0.261908
\(934\) 16.0000 0.523536
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) −12.0000 −0.391814
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 10.0000 0.325818
\(943\) 0 0
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −4.00000 −0.129914
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 8.00000 0.259281
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) −22.0000 −0.711531
\(957\) −12.0000 −0.387905
\(958\) 8.00000 0.258468
\(959\) −14.0000 −0.452084
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 16.0000 0.515861
\(963\) −12.0000 −0.386695
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −24.0000 −0.771788 −0.385894 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) −7.00000 −0.224989
\(969\) 16.0000 0.513994
\(970\) 0 0
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −2.00000 −0.0641171
\(974\) 36.0000 1.15351
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) −42.0000 −1.34027
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −48.0000 −1.52863
\(987\) −4.00000 −0.127321
\(988\) −4.00000 −0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) −60.0000 −1.90596 −0.952981 0.303029i \(-0.902002\pi\)
−0.952981 + 0.303029i \(0.902002\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) −14.0000 −0.444053
\(995\) 0 0
\(996\) −16.0000 −0.506979
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) −28.0000 −0.886325
\(999\) −8.00000 −0.253109
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 1050.2.a.m.1.1 1
3.2 odd 2 3150.2.a.q.1.1 1
4.3 odd 2 8400.2.a.ca.1.1 1
5.2 odd 4 210.2.g.a.169.2 yes 2
5.3 odd 4 210.2.g.a.169.1 2
5.4 even 2 1050.2.a.g.1.1 1
7.6 odd 2 7350.2.a.co.1.1 1
15.2 even 4 630.2.g.d.379.1 2
15.8 even 4 630.2.g.d.379.2 2
15.14 odd 2 3150.2.a.be.1.1 1
20.3 even 4 1680.2.t.d.1009.2 2
20.7 even 4 1680.2.t.d.1009.1 2
20.19 odd 2 8400.2.a.bd.1.1 1
35.2 odd 12 1470.2.n.g.949.2 4
35.3 even 12 1470.2.n.c.79.2 4
35.12 even 12 1470.2.n.c.949.2 4
35.13 even 4 1470.2.g.e.589.1 2
35.17 even 12 1470.2.n.c.79.1 4
35.18 odd 12 1470.2.n.g.79.2 4
35.23 odd 12 1470.2.n.g.949.1 4
35.27 even 4 1470.2.g.e.589.2 2
35.32 odd 12 1470.2.n.g.79.1 4
35.33 even 12 1470.2.n.c.949.1 4
35.34 odd 2 7350.2.a.g.1.1 1
60.23 odd 4 5040.2.t.k.1009.2 2
60.47 odd 4 5040.2.t.k.1009.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.g.a.169.1 2 5.3 odd 4
210.2.g.a.169.2 yes 2 5.2 odd 4
630.2.g.d.379.1 2 15.2 even 4
630.2.g.d.379.2 2 15.8 even 4
1050.2.a.g.1.1 1 5.4 even 2
1050.2.a.m.1.1 1 1.1 even 1 trivial
1470.2.g.e.589.1 2 35.13 even 4
1470.2.g.e.589.2 2 35.27 even 4
1470.2.n.c.79.1 4 35.17 even 12
1470.2.n.c.79.2 4 35.3 even 12
1470.2.n.c.949.1 4 35.33 even 12
1470.2.n.c.949.2 4 35.12 even 12
1470.2.n.g.79.1 4 35.32 odd 12
1470.2.n.g.79.2 4 35.18 odd 12
1470.2.n.g.949.1 4 35.23 odd 12
1470.2.n.g.949.2 4 35.2 odd 12
1680.2.t.d.1009.1 2 20.7 even 4
1680.2.t.d.1009.2 2 20.3 even 4
3150.2.a.q.1.1 1 3.2 odd 2
3150.2.a.be.1.1 1 15.14 odd 2
5040.2.t.k.1009.1 2 60.47 odd 4
5040.2.t.k.1009.2 2 60.23 odd 4
7350.2.a.g.1.1 1 35.34 odd 2
7350.2.a.co.1.1 1 7.6 odd 2
8400.2.a.bd.1.1 1 20.19 odd 2
8400.2.a.ca.1.1 1 4.3 odd 2