Properties

Label 330.2.a.d.1.1
Level $330$
Weight $2$
Character 330.1
Self dual yes
Analytic conductor $2.635$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.63506326670\)
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 \(\chi\) \(=\) 330.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} +1.00000 q^{22} -1.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -10.0000 q^{29} -1.00000 q^{30} +1.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -6.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +1.00000 q^{45} -8.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +1.00000 q^{55} +4.00000 q^{57} -10.0000 q^{58} -4.00000 q^{59} -1.00000 q^{60} -2.00000 q^{61} +1.00000 q^{64} +6.00000 q^{65} -1.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -6.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -12.0000 q^{83} +2.00000 q^{85} +4.00000 q^{86} +10.0000 q^{87} +1.00000 q^{88} -6.00000 q^{89} +1.00000 q^{90} -8.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} +18.0000 q^{97} -7.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) −1.00000 −0.182574
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) −6.00000 −0.960769
\(40\) 1.00000 0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 6.00000 0.832050
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) −10.0000 −1.31306
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) −1.00000 −0.123091
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 6.00000 0.697486
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 4.00000 0.431331
\(87\) 10.0000 1.07211
\(88\) 1.00000 0.106600
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) −7.00000 −0.707107
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −2.00000 −0.198030
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 1.00000 0.0953463
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −10.0000 −0.928477
\(117\) 6.00000 0.554700
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 6.00000 0.526235
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) −1.00000 −0.0860663
\(136\) 2.00000 0.171499
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) −8.00000 −0.671345
\(143\) 6.00000 0.501745
\(144\) 1.00000 0.0833333
\(145\) −10.0000 −0.830455
\(146\) 2.00000 0.165521
\(147\) 7.00000 0.577350
\(148\) 6.00000 0.493197
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −4.00000 −0.324443
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −8.00000 −0.636446
\(159\) 10.0000 0.793052
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 2.00000 0.156174
\(165\) −1.00000 −0.0778499
\(166\) −12.0000 −0.931381
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 2.00000 0.153393
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 10.0000 0.758098
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 4.00000 0.300658
\(178\) −6.00000 −0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 1.00000 0.0745356
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 18.0000 1.29232
\(195\) −6.00000 −0.429669
\(196\) −7.00000 −0.500000
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 1.00000 0.0710669
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) 14.0000 0.985037
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 2.00000 0.139686
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −10.0000 −0.686803
\(213\) 8.00000 0.548151
\(214\) −4.00000 −0.273434
\(215\) 4.00000 0.272798
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) −2.00000 −0.135147
\(220\) 1.00000 0.0674200
\(221\) 12.0000 0.807207
\(222\) −6.00000 −0.402694
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 2.00000 0.133038
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 4.00000 0.264906
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −10.0000 −0.656532
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 6.00000 0.392232
\(235\) −8.00000 −0.521862
\(236\) −4.00000 −0.260378
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) −7.00000 −0.447214
\(246\) −2.00000 −0.127515
\(247\) −24.0000 −1.52708
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 1.00000 0.0632456
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 6.00000 0.372104
\(261\) −10.0000 −0.618984
\(262\) 20.0000 1.23560
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −10.0000 −0.614295
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) −4.00000 −0.244339
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 8.00000 0.476393
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) −8.00000 −0.474713
\(285\) 4.00000 0.236940
\(286\) 6.00000 0.354787
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) −10.0000 −0.587220
\(291\) −18.0000 −1.05518
\(292\) 2.00000 0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 7.00000 0.408248
\(295\) −4.00000 −0.232889
\(296\) 6.00000 0.348743
\(297\) −1.00000 −0.0580259
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) −14.0000 −0.804279
\(304\) −4.00000 −0.229416
\(305\) −2.00000 −0.114520
\(306\) 2.00000 0.114332
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −6.00000 −0.339683
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −34.0000 −1.90963 −0.954815 0.297200i \(-0.903947\pi\)
−0.954815 + 0.297200i \(0.903947\pi\)
\(318\) 10.0000 0.560772
\(319\) −10.0000 −0.559893
\(320\) 1.00000 0.0559017
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) 6.00000 0.332820
\(326\) −20.0000 −1.10770
\(327\) −14.0000 −0.774202
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) −1.00000 −0.0550482
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −12.0000 −0.658586
\(333\) 6.00000 0.328798
\(334\) 24.0000 1.31322
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 23.0000 1.25104
\(339\) −2.00000 −0.108625
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 10.0000 0.536056
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 1.00000 0.0533002
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 4.00000 0.212598
\(355\) −8.00000 −0.424596
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 2.00000 0.104542
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) 0 0
\(373\) −18.0000 −0.932005 −0.466002 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) 2.00000 0.103418
\(375\) −1.00000 −0.0516398
\(376\) −8.00000 −0.412568
\(377\) −60.0000 −3.09016
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) −4.00000 −0.205196
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 4.00000 0.203331
\(388\) 18.0000 0.913812
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −6.00000 −0.303822
\(391\) 0 0
\(392\) −7.00000 −0.353553
\(393\) −20.0000 −1.00887
\(394\) 6.00000 0.302276
\(395\) −8.00000 −0.402524
\(396\) 1.00000 0.0502519
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) −2.00000 −0.0990148
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 2.00000 0.0987730
\(411\) 6.00000 0.295958
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 6.00000 0.294174
\(417\) −4.00000 −0.195881
\(418\) −4.00000 −0.195646
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) −20.0000 −0.973585
\(423\) −8.00000 −0.388973
\(424\) −10.0000 −0.485643
\(425\) 2.00000 0.0970143
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) −6.00000 −0.289683
\(430\) 4.00000 0.192897
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 10.0000 0.479463
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) −2.00000 −0.0955637
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 1.00000 0.0476731
\(441\) −7.00000 −0.333333
\(442\) 12.0000 0.570782
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −6.00000 −0.284747
\(445\) −6.00000 −0.284427
\(446\) 24.0000 1.13643
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 1.00000 0.0471405
\(451\) 2.00000 0.0941763
\(452\) 2.00000 0.0940721
\(453\) 16.0000 0.751746
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −26.0000 −1.21490
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 40.0000 1.85896 0.929479 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) −10.0000 −0.464238
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) 18.0000 0.829396
\(472\) −4.00000 −0.184115
\(473\) 4.00000 0.183920
\(474\) 8.00000 0.367452
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 16.0000 0.731823
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 36.0000 1.64146
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 18.0000 0.817338
\(486\) −1.00000 −0.0453609
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 20.0000 0.904431
\(490\) −7.00000 −0.316228
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) −2.00000 −0.0901670
\(493\) −20.0000 −0.900755
\(494\) −24.0000 −1.07981
\(495\) 1.00000 0.0449467
\(496\) 0 0
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 1.00000 0.0447214
\(501\) −24.0000 −1.07224
\(502\) −4.00000 −0.178529
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) 8.00000 0.354943
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) −14.0000 −0.617514
\(515\) 16.0000 0.705044
\(516\) −4.00000 −0.176090
\(517\) −8.00000 −0.351840
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 6.00000 0.263117
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) −10.0000 −0.437688
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) −1.00000 −0.0435194
\(529\) −23.0000 −1.00000
\(530\) −10.0000 −0.434372
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 6.00000 0.259645
\(535\) −4.00000 −0.172935
\(536\) −4.00000 −0.172774
\(537\) 12.0000 0.517838
\(538\) −2.00000 −0.0862261
\(539\) −7.00000 −0.301511
\(540\) −1.00000 −0.0430331
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −8.00000 −0.343629
\(543\) 10.0000 0.429141
\(544\) 2.00000 0.0857493
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −6.00000 −0.256307
\(549\) −2.00000 −0.0853579
\(550\) 1.00000 0.0426401
\(551\) 40.0000 1.70406
\(552\) 0 0
\(553\) 0 0
\(554\) −18.0000 −0.764747
\(555\) −6.00000 −0.254686
\(556\) 4.00000 0.169638
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) −2.00000 −0.0844401
\(562\) 18.0000 0.759284
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 8.00000 0.336861
\(565\) 2.00000 0.0841406
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 4.00000 0.167542
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 6.00000 0.250873
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −46.0000 −1.91501 −0.957503 0.288425i \(-0.906868\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) −13.0000 −0.540729
\(579\) −10.0000 −0.415586
\(580\) −10.0000 −0.415227
\(581\) 0 0
\(582\) −18.0000 −0.746124
\(583\) −10.0000 −0.414158
\(584\) 2.00000 0.0827606
\(585\) 6.00000 0.248069
\(586\) 6.00000 0.247858
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 7.00000 0.288675
\(589\) 0 0
\(590\) −4.00000 −0.164677
\(591\) −6.00000 −0.246807
\(592\) 6.00000 0.246598
\(593\) −46.0000 −1.88899 −0.944497 0.328521i \(-0.893450\pi\)
−0.944497 + 0.328521i \(0.893450\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −16.0000 −0.651031
\(605\) 1.00000 0.0406558
\(606\) −14.0000 −0.568711
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) −48.0000 −1.94187
\(612\) 2.00000 0.0808452
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −20.0000 −0.807134
\(615\) −2.00000 −0.0806478
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) −16.0000 −0.643614
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) 0 0
\(624\) −6.00000 −0.240192
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) 4.00000 0.159745
\(628\) −18.0000 −0.718278
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −24.0000 −0.955425 −0.477712 0.878516i \(-0.658534\pi\)
−0.477712 + 0.878516i \(0.658534\pi\)
\(632\) −8.00000 −0.318223
\(633\) 20.0000 0.794929
\(634\) −34.0000 −1.35031
\(635\) 8.00000 0.317470
\(636\) 10.0000 0.396526
\(637\) −42.0000 −1.66410
\(638\) −10.0000 −0.395904
\(639\) −8.00000 −0.316475
\(640\) 1.00000 0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 4.00000 0.157867
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) −8.00000 −0.314756
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 1.00000 0.0392837
\(649\) −4.00000 −0.157014
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) −20.0000 −0.783260
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −14.0000 −0.547443
\(655\) 20.0000 0.781465
\(656\) 2.00000 0.0780869
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) −1.00000 −0.0389249
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −4.00000 −0.155464
\(663\) −12.0000 −0.466041
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 0 0
\(668\) 24.0000 0.928588
\(669\) −24.0000 −0.927894
\(670\) −4.00000 −0.154533
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 10.0000 0.385186
\(675\) −1.00000 −0.0384900
\(676\) 23.0000 0.884615
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) −2.00000 −0.0768095
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) −4.00000 −0.152944
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) 26.0000 0.991962
\(688\) 4.00000 0.152499
\(689\) −60.0000 −2.28582
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 4.00000 0.151729
\(696\) 10.0000 0.379049
\(697\) 4.00000 0.151511
\(698\) 30.0000 1.13552
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −26.0000 −0.982006 −0.491003 0.871158i \(-0.663370\pi\)
−0.491003 + 0.871158i \(0.663370\pi\)
\(702\) −6.00000 −0.226455
\(703\) −24.0000 −0.905177
\(704\) 1.00000 0.0376889
\(705\) 8.00000 0.301297
\(706\) −14.0000 −0.526897
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) −8.00000 −0.300235
\(711\) −8.00000 −0.300023
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) −12.0000 −0.448461
\(717\) −16.0000 −0.597531
\(718\) −8.00000 −0.298557
\(719\) −32.0000 −1.19340 −0.596699 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) −18.0000 −0.669427
\(724\) −10.0000 −0.371647
\(725\) −10.0000 −0.371391
\(726\) −1.00000 −0.0371135
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) 8.00000 0.295891
\(732\) 2.00000 0.0739221
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) −8.00000 −0.295285
\(735\) 7.00000 0.258199
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) 2.00000 0.0736210
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 6.00000 0.220564
\(741\) 24.0000 0.881662
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) −18.0000 −0.659027
\(747\) −12.0000 −0.439057
\(748\) 2.00000 0.0731272
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −8.00000 −0.291730
\(753\) 4.00000 0.145768
\(754\) −60.0000 −2.18507
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 12.0000 0.435860
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) −8.00000 −0.289809
\(763\) 0 0
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) −8.00000 −0.289052
\(767\) −24.0000 −0.866590
\(768\) −1.00000 −0.0360844
\(769\) −46.0000 −1.65880 −0.829401 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 10.0000 0.359908
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 18.0000 0.646162
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −8.00000 −0.286630
\(780\) −6.00000 −0.214834
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) 10.0000 0.357371
\(784\) −7.00000 −0.250000
\(785\) −18.0000 −0.642448
\(786\) −20.0000 −0.713376
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 6.00000 0.213741
\(789\) −24.0000 −0.854423
\(790\) −8.00000 −0.284627
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) −12.0000 −0.426132
\(794\) −2.00000 −0.0709773
\(795\) 10.0000 0.354663
\(796\) 8.00000 0.283552
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) 2.00000 0.0706225
\(803\) 2.00000 0.0705785
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) 2.00000 0.0704033
\(808\) 14.0000 0.492518
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 1.00000 0.0351364
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 0 0
\(813\) 8.00000 0.280572
\(814\) 6.00000 0.210300
\(815\) −20.0000 −0.700569
\(816\) −2.00000 −0.0700140
\(817\) −16.0000 −0.559769
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 6.00000 0.209274
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 16.0000 0.557386
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) 44.0000 1.53003 0.765015 0.644013i \(-0.222732\pi\)
0.765015 + 0.644013i \(0.222732\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) −12.0000 −0.416526
\(831\) 18.0000 0.624413
\(832\) 6.00000 0.208013
\(833\) −14.0000 −0.485071
\(834\) −4.00000 −0.138509
\(835\) 24.0000 0.830554
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 38.0000 1.30957
\(843\) −18.0000 −0.619953
\(844\) −20.0000 −0.688428
\(845\) 23.0000 0.791224
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) −10.0000 −0.343401
\(849\) 12.0000 0.411839
\(850\) 2.00000 0.0685994
\(851\) 0 0
\(852\) 8.00000 0.274075
\(853\) −50.0000 −1.71197 −0.855984 0.517003i \(-0.827048\pi\)
−0.855984 + 0.517003i \(0.827048\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) −4.00000 −0.136717
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) −6.00000 −0.204837
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 16.0000 0.544962
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 14.0000 0.476014
\(866\) 2.00000 0.0679628
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 10.0000 0.339032
\(871\) −24.0000 −0.813209
\(872\) 14.0000 0.474100
\(873\) 18.0000 0.609208
\(874\) 0 0
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) −58.0000 −1.95852 −0.979260 0.202606i \(-0.935059\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) 32.0000 1.07995
\(879\) −6.00000 −0.202375
\(880\) 1.00000 0.0337100
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) −7.00000 −0.235702
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 12.0000 0.403604
\(885\) 4.00000 0.134459
\(886\) 4.00000 0.134383
\(887\) 56.0000 1.88030 0.940148 0.340766i \(-0.110687\pi\)
0.940148 + 0.340766i \(0.110687\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) −6.00000 −0.201120
\(891\) 1.00000 0.0335013
\(892\) 24.0000 0.803579
\(893\) 32.0000 1.07084
\(894\) 18.0000 0.602010
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −20.0000 −0.666297
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) −10.0000 −0.332411
\(906\) 16.0000 0.531564
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 4.00000 0.132745
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 4.00000 0.132453
\(913\) −12.0000 −0.397142
\(914\) 18.0000 0.595387
\(915\) 2.00000 0.0661180
\(916\) −26.0000 −0.859064
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 6.00000 0.197599
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 40.0000 1.31448
\(927\) 16.0000 0.525509
\(928\) −10.0000 −0.328266
\(929\) 50.0000 1.64045 0.820223 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 0 0
\(931\) 28.0000 0.917663
\(932\) −6.00000 −0.196537
\(933\) −24.0000 −0.785725
\(934\) −4.00000 −0.130884
\(935\) 2.00000 0.0654070
\(936\) 6.00000 0.196116
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) −8.00000 −0.260931
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 18.0000 0.586472
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 8.00000 0.259828
\(949\) 12.0000 0.389536
\(950\) −4.00000 −0.129777
\(951\) 34.0000 1.10253
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 10.0000 0.323254
\(958\) 0 0
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) −31.0000 −1.00000
\(962\) 36.0000 1.16069
\(963\) −4.00000 −0.128898
\(964\) 18.0000 0.579741
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) 48.0000 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) 1.00000 0.0321412
\(969\) 8.00000 0.256997
\(970\) 18.0000 0.577945
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 0 0
\(975\) −6.00000 −0.192154
\(976\) −2.00000 −0.0640184
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 20.0000 0.639529
\(979\) −6.00000 −0.191761
\(980\) −7.00000 −0.223607
\(981\) 14.0000 0.446986
\(982\) −36.0000 −1.14881
\(983\) −48.0000 −1.53096 −0.765481 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 6.00000 0.191176
\(986\) −20.0000 −0.636930
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) 0 0
\(990\) 1.00000 0.0317821
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 12.0000 0.380235
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) 20.0000 0.633089
\(999\) −6.00000 −0.189832
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 330.2.a.d.1.1 1
3.2 odd 2 990.2.a.b.1.1 1
4.3 odd 2 2640.2.a.t.1.1 1
5.2 odd 4 1650.2.c.g.199.2 2
5.3 odd 4 1650.2.c.g.199.1 2
5.4 even 2 1650.2.a.h.1.1 1
11.10 odd 2 3630.2.a.f.1.1 1
12.11 even 2 7920.2.a.m.1.1 1
15.2 even 4 4950.2.c.j.199.1 2
15.8 even 4 4950.2.c.j.199.2 2
15.14 odd 2 4950.2.a.bg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
330.2.a.d.1.1 1 1.1 even 1 trivial
990.2.a.b.1.1 1 3.2 odd 2
1650.2.a.h.1.1 1 5.4 even 2
1650.2.c.g.199.1 2 5.3 odd 4
1650.2.c.g.199.2 2 5.2 odd 4
2640.2.a.t.1.1 1 4.3 odd 2
3630.2.a.f.1.1 1 11.10 odd 2
4950.2.a.bg.1.1 1 15.14 odd 2
4950.2.c.j.199.1 2 15.2 even 4
4950.2.c.j.199.2 2 15.8 even 4
7920.2.a.m.1.1 1 12.11 even 2