Properties

Label 690.2.a.b.1.1
Level $690$
Weight $2$
Character 690.1
Self dual yes
Analytic conductor $5.510$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.50967773947\)
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 \(\chi\) \(=\) 690.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} -4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} -4.00000 q^{21} +2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +4.00000 q^{28} -4.00000 q^{29} -1.00000 q^{30} -1.00000 q^{32} +2.00000 q^{33} -2.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} +10.0000 q^{37} +1.00000 q^{40} +6.00000 q^{41} +4.00000 q^{42} +2.00000 q^{43} -2.00000 q^{44} -1.00000 q^{45} -1.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} -4.00000 q^{56} +4.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} -14.0000 q^{61} +4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +2.00000 q^{67} +2.00000 q^{68} -1.00000 q^{69} +4.00000 q^{70} -2.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} -10.0000 q^{74} -1.00000 q^{75} -8.00000 q^{77} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +8.00000 q^{83} -4.00000 q^{84} -2.00000 q^{85} -2.00000 q^{86} +4.00000 q^{87} +2.00000 q^{88} -8.00000 q^{89} +1.00000 q^{90} +1.00000 q^{92} -12.0000 q^{94} +1.00000 q^{96} -9.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −4.00000 −1.06904
\(15\) 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
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) −4.00000 −0.872872
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) −1.00000 −0.182574
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) −2.00000 −0.342997
\(35\) −4.00000 −0.676123
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 4.00000 0.617213
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −2.00000 −0.301511
\(45\) −1.00000 −0.149071
\(46\) −1.00000 −0.147442
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.00000 0.269680
\(56\) −4.00000 −0.534522
\(57\) 0 0
\(58\) 4.00000 0.525226
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 1.00000 0.129099
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 2.00000 0.242536
\(69\) −1.00000 −0.120386
\(70\) 4.00000 0.478091
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) −1.00000 −0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −10.0000 −1.16248
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) −4.00000 −0.436436
\(85\) −2.00000 −0.216930
\(86\) −2.00000 −0.215666
\(87\) 4.00000 0.428845
\(88\) 2.00000 0.213201
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −9.00000 −0.909137
\(99\) −2.00000 −0.201008
\(100\) 1.00000 0.100000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 2.00000 0.198030
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) −2.00000 −0.190693
\(111\) −10.0000 −0.949158
\(112\) 4.00000 0.377964
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) 8.00000 0.733359
\(120\) −1.00000 −0.0912871
\(121\) −7.00000 −0.636364
\(122\) 14.0000 1.26750
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) −4.00000 −0.356348
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) 1.00000 0.0860663
\(136\) −2.00000 −0.171499
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 1.00000 0.0851257
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) −4.00000 −0.338062
\(141\) −12.0000 −1.01058
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) −6.00000 −0.496564
\(147\) −9.00000 −0.742307
\(148\) 10.0000 0.821995
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 1.00000 0.0816497
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 8.00000 0.644658
\(155\) 0 0
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) −8.00000 −0.636446
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 4.00000 0.315244
\(162\) −1.00000 −0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 6.00000 0.468521
\(165\) −2.00000 −0.155700
\(166\) −8.00000 −0.620920
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 4.00000 0.308607
\(169\) −13.0000 −1.00000
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −4.00000 −0.303239
\(175\) 4.00000 0.302372
\(176\) −2.00000 −0.150756
\(177\) −12.0000 −0.901975
\(178\) 8.00000 0.599625
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) −1.00000 −0.0737210
\(185\) −10.0000 −0.735215
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 12.0000 0.875190
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 2.00000 0.142134
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −2.00000 −0.141069
\(202\) 12.0000 0.844317
\(203\) −16.0000 −1.12298
\(204\) −2.00000 −0.140028
\(205\) −6.00000 −0.419058
\(206\) −4.00000 −0.278693
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 0 0
\(210\) −4.00000 −0.276026
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 6.00000 0.412082
\(213\) 2.00000 0.137038
\(214\) 12.0000 0.820303
\(215\) −2.00000 −0.136399
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −14.0000 −0.948200
\(219\) −6.00000 −0.405442
\(220\) 2.00000 0.134840
\(221\) 0 0
\(222\) 10.0000 0.671156
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −4.00000 −0.267261
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 1.00000 0.0659380
\(231\) 8.00000 0.526361
\(232\) 4.00000 0.262613
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 12.0000 0.781133
\(237\) −8.00000 −0.519656
\(238\) −8.00000 −0.518563
\(239\) 22.0000 1.42306 0.711531 0.702655i \(-0.248002\pi\)
0.711531 + 0.702655i \(0.248002\pi\)
\(240\) 1.00000 0.0645497
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) −14.0000 −0.896258
\(245\) −9.00000 −0.574989
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 0 0
\(249\) −8.00000 −0.506979
\(250\) 1.00000 0.0632456
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 4.00000 0.251976
\(253\) −2.00000 −0.125739
\(254\) −6.00000 −0.376473
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 2.00000 0.124515
\(259\) 40.0000 2.48548
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) 0 0
\(263\) −32.0000 −1.97320 −0.986602 0.163144i \(-0.947836\pi\)
−0.986602 + 0.163144i \(0.947836\pi\)
\(264\) −2.00000 −0.123091
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) 2.00000 0.122169
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) −2.00000 −0.120605
\(276\) −1.00000 −0.0601929
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) 32.0000 1.90896 0.954480 0.298275i \(-0.0964112\pi\)
0.954480 + 0.298275i \(0.0964112\pi\)
\(282\) 12.0000 0.714590
\(283\) −10.0000 −0.594438 −0.297219 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) −4.00000 −0.234888
\(291\) 0 0
\(292\) 6.00000 0.351123
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 9.00000 0.524891
\(295\) −12.0000 −0.698667
\(296\) −10.0000 −0.581238
\(297\) 2.00000 0.116052
\(298\) 14.0000 0.810998
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 8.00000 0.461112
\(302\) 12.0000 0.690522
\(303\) 12.0000 0.689382
\(304\) 0 0
\(305\) 14.0000 0.801638
\(306\) −2.00000 −0.114332
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) −8.00000 −0.455842
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −2.00000 −0.113410 −0.0567048 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) 0 0
\(313\) 20.0000 1.13047 0.565233 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) 22.0000 1.24153
\(315\) −4.00000 −0.225374
\(316\) 8.00000 0.450035
\(317\) −10.0000 −0.561656 −0.280828 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 6.00000 0.336463
\(319\) 8.00000 0.447914
\(320\) −1.00000 −0.0559017
\(321\) 12.0000 0.669775
\(322\) −4.00000 −0.222911
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) −14.0000 −0.774202
\(328\) −6.00000 −0.331295
\(329\) 48.0000 2.64633
\(330\) 2.00000 0.110096
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 8.00000 0.439057
\(333\) 10.0000 0.547997
\(334\) 0 0
\(335\) −2.00000 −0.109272
\(336\) −4.00000 −0.218218
\(337\) −4.00000 −0.217894 −0.108947 0.994048i \(-0.534748\pi\)
−0.108947 + 0.994048i \(0.534748\pi\)
\(338\) 13.0000 0.707107
\(339\) −6.00000 −0.325875
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) −2.00000 −0.107833
\(345\) 1.00000 0.0538382
\(346\) 6.00000 0.322562
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 4.00000 0.214423
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) −34.0000 −1.80964 −0.904819 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 12.0000 0.637793
\(355\) 2.00000 0.106149
\(356\) −8.00000 −0.423999
\(357\) −8.00000 −0.423405
\(358\) −4.00000 −0.211407
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 1.00000 0.0527046
\(361\) −19.0000 −1.00000
\(362\) 6.00000 0.315353
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) −14.0000 −0.731792
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 1.00000 0.0521286
\(369\) 6.00000 0.312348
\(370\) 10.0000 0.519875
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 4.00000 0.206835
\(375\) 1.00000 0.0516398
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 4.00000 0.205738
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) −6.00000 −0.307389
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 8.00000 0.407718
\(386\) 14.0000 0.712581
\(387\) 2.00000 0.101666
\(388\) 0 0
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) −8.00000 −0.402524
\(396\) −2.00000 −0.100504
\(397\) 32.0000 1.60603 0.803017 0.595956i \(-0.203227\pi\)
0.803017 + 0.595956i \(0.203227\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 2.00000 0.0997509
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) −1.00000 −0.0496904
\(406\) 16.0000 0.794067
\(407\) −20.0000 −0.991363
\(408\) 2.00000 0.0990148
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 6.00000 0.296319
\(411\) 2.00000 0.0986527
\(412\) 4.00000 0.197066
\(413\) 48.0000 2.36193
\(414\) −1.00000 −0.0491473
\(415\) −8.00000 −0.392705
\(416\) 0 0
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 4.00000 0.195180
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) −4.00000 −0.194717
\(423\) 12.0000 0.583460
\(424\) −6.00000 −0.291386
\(425\) 2.00000 0.0970143
\(426\) −2.00000 −0.0969003
\(427\) −56.0000 −2.71003
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 2.00000 0.0964486
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 24.0000 1.15337 0.576683 0.816968i \(-0.304347\pi\)
0.576683 + 0.816968i \(0.304347\pi\)
\(434\) 0 0
\(435\) −4.00000 −0.191785
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) 6.00000 0.286691
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) −10.0000 −0.474579
\(445\) 8.00000 0.379236
\(446\) −2.00000 −0.0947027
\(447\) 14.0000 0.662177
\(448\) 4.00000 0.188982
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −12.0000 −0.565058
\(452\) 6.00000 0.282216
\(453\) 12.0000 0.563809
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) −24.0000 −1.12267 −0.561336 0.827588i \(-0.689713\pi\)
−0.561336 + 0.827588i \(0.689713\pi\)
\(458\) 26.0000 1.21490
\(459\) −2.00000 −0.0933520
\(460\) −1.00000 −0.0466252
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) −8.00000 −0.372194
\(463\) 18.0000 0.836531 0.418265 0.908325i \(-0.362638\pi\)
0.418265 + 0.908325i \(0.362638\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 24.0000 1.11059 0.555294 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 12.0000 0.553519
\(471\) 22.0000 1.01371
\(472\) −12.0000 −0.552345
\(473\) −4.00000 −0.183920
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) 6.00000 0.274721
\(478\) −22.0000 −1.00626
\(479\) −40.0000 −1.82765 −0.913823 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 14.0000 0.637683
\(483\) −4.00000 −0.182006
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 18.0000 0.815658 0.407829 0.913058i \(-0.366286\pi\)
0.407829 + 0.913058i \(0.366286\pi\)
\(488\) 14.0000 0.633750
\(489\) 16.0000 0.723545
\(490\) 9.00000 0.406579
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) −6.00000 −0.270501
\(493\) −8.00000 −0.360302
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 8.00000 0.358489
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −18.0000 −0.803379
\(503\) −8.00000 −0.356702 −0.178351 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) −4.00000 −0.178174
\(505\) 12.0000 0.533993
\(506\) 2.00000 0.0889108
\(507\) 13.0000 0.577350
\(508\) 6.00000 0.266207
\(509\) 28.0000 1.24108 0.620539 0.784176i \(-0.286914\pi\)
0.620539 + 0.784176i \(0.286914\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 24.0000 1.06170
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) −4.00000 −0.176261
\(516\) −2.00000 −0.0880451
\(517\) −24.0000 −1.05552
\(518\) −40.0000 −1.75750
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 32.0000 1.40195 0.700973 0.713188i \(-0.252749\pi\)
0.700973 + 0.713188i \(0.252749\pi\)
\(522\) 4.00000 0.175075
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) 32.0000 1.39527
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) 6.00000 0.260623
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 0 0
\(534\) −8.00000 −0.346194
\(535\) 12.0000 0.518805
\(536\) −2.00000 −0.0863868
\(537\) −4.00000 −0.172613
\(538\) −24.0000 −1.03471
\(539\) −18.0000 −0.775315
\(540\) 1.00000 0.0430331
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) −8.00000 −0.343629
\(543\) 6.00000 0.257485
\(544\) −2.00000 −0.0857493
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −14.0000 −0.597505
\(550\) 2.00000 0.0852803
\(551\) 0 0
\(552\) 1.00000 0.0425628
\(553\) 32.0000 1.36078
\(554\) −8.00000 −0.339887
\(555\) 10.0000 0.424476
\(556\) 20.0000 0.848189
\(557\) 26.0000 1.10166 0.550828 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −4.00000 −0.169031
\(561\) 4.00000 0.168880
\(562\) −32.0000 −1.34984
\(563\) 8.00000 0.337160 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(564\) −12.0000 −0.505291
\(565\) −6.00000 −0.252422
\(566\) 10.0000 0.420331
\(567\) 4.00000 0.167984
\(568\) 2.00000 0.0839181
\(569\) −12.0000 −0.503066 −0.251533 0.967849i \(-0.580935\pi\)
−0.251533 + 0.967849i \(0.580935\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 13.0000 0.540729
\(579\) 14.0000 0.581820
\(580\) 4.00000 0.166091
\(581\) 32.0000 1.32758
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) −4.00000 −0.165098 −0.0825488 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) −9.00000 −0.371154
\(589\) 0 0
\(590\) 12.0000 0.494032
\(591\) 6.00000 0.246807
\(592\) 10.0000 0.410997
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) −2.00000 −0.0820610
\(595\) −8.00000 −0.327968
\(596\) −14.0000 −0.573462
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 1.00000 0.0408248
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −8.00000 −0.326056
\(603\) 2.00000 0.0814463
\(604\) −12.0000 −0.488273
\(605\) 7.00000 0.284590
\(606\) −12.0000 −0.487467
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 0 0
\(609\) 16.0000 0.648353
\(610\) −14.0000 −0.566843
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −4.00000 −0.161427
\(615\) 6.00000 0.241943
\(616\) 8.00000 0.322329
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 4.00000 0.160904
\(619\) −24.0000 −0.964641 −0.482321 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 2.00000 0.0801927
\(623\) −32.0000 −1.28205
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −20.0000 −0.799361
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) 20.0000 0.797452
\(630\) 4.00000 0.159364
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −8.00000 −0.318223
\(633\) −4.00000 −0.158986
\(634\) 10.0000 0.397151
\(635\) −6.00000 −0.238103
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) −8.00000 −0.316723
\(639\) −2.00000 −0.0791188
\(640\) 1.00000 0.0395285
\(641\) −8.00000 −0.315981 −0.157991 0.987441i \(-0.550502\pi\)
−0.157991 + 0.987441i \(0.550502\pi\)
\(642\) −12.0000 −0.473602
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 4.00000 0.157622
\(645\) 2.00000 0.0787499
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 6.00000 0.234082
\(658\) −48.0000 −1.87123
\(659\) −18.0000 −0.701180 −0.350590 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 6.00000 0.233373 0.116686 0.993169i \(-0.462773\pi\)
0.116686 + 0.993169i \(0.462773\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) −4.00000 −0.154881
\(668\) 0 0
\(669\) −2.00000 −0.0773245
\(670\) 2.00000 0.0772667
\(671\) 28.0000 1.08093
\(672\) 4.00000 0.154303
\(673\) −18.0000 −0.693849 −0.346925 0.937893i \(-0.612774\pi\)
−0.346925 + 0.937893i \(0.612774\pi\)
\(674\) 4.00000 0.154074
\(675\) −1.00000 −0.0384900
\(676\) −13.0000 −0.500000
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) −8.00000 −0.305441
\(687\) 26.0000 0.991962
\(688\) 2.00000 0.0762493
\(689\) 0 0
\(690\) −1.00000 −0.0380693
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −6.00000 −0.228086
\(693\) −8.00000 −0.303895
\(694\) −4.00000 −0.151838
\(695\) −20.0000 −0.758643
\(696\) −4.00000 −0.151620
\(697\) 12.0000 0.454532
\(698\) 30.0000 1.13552
\(699\) −10.0000 −0.378235
\(700\) 4.00000 0.151186
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 12.0000 0.451946
\(706\) 34.0000 1.27961
\(707\) −48.0000 −1.80523
\(708\) −12.0000 −0.450988
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) −2.00000 −0.0750587
\(711\) 8.00000 0.300023
\(712\) 8.00000 0.299813
\(713\) 0 0
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) −22.0000 −0.821605
\(718\) 0 0
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 16.0000 0.595871
\(722\) 19.0000 0.707107
\(723\) 14.0000 0.520666
\(724\) −6.00000 −0.222988
\(725\) −4.00000 −0.148556
\(726\) −7.00000 −0.259794
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.00000 0.222070
\(731\) 4.00000 0.147945
\(732\) 14.0000 0.517455
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) −4.00000 −0.147643
\(735\) 9.00000 0.331970
\(736\) −1.00000 −0.0368605
\(737\) −4.00000 −0.147342
\(738\) −6.00000 −0.220863
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) −10.0000 −0.367607
\(741\) 0 0
\(742\) −24.0000 −0.881068
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) 22.0000 0.805477
\(747\) 8.00000 0.292705
\(748\) −4.00000 −0.146254
\(749\) −48.0000 −1.75388
\(750\) −1.00000 −0.0365148
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 12.0000 0.437595
\(753\) −18.0000 −0.655956
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) −4.00000 −0.145479
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −4.00000 −0.145287
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) 2.00000 0.0724999 0.0362500 0.999343i \(-0.488459\pi\)
0.0362500 + 0.999343i \(0.488459\pi\)
\(762\) 6.00000 0.217357
\(763\) 56.0000 2.02734
\(764\) 0 0
\(765\) −2.00000 −0.0723102
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 6.00000 0.216366 0.108183 0.994131i \(-0.465497\pi\)
0.108183 + 0.994131i \(0.465497\pi\)
\(770\) −8.00000 −0.288300
\(771\) −2.00000 −0.0720282
\(772\) −14.0000 −0.503871
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) 0 0
\(777\) −40.0000 −1.43499
\(778\) −22.0000 −0.788738
\(779\) 0 0
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) −2.00000 −0.0715199
\(783\) 4.00000 0.142948
\(784\) 9.00000 0.321429
\(785\) 22.0000 0.785214
\(786\) 0 0
\(787\) −34.0000 −1.21197 −0.605985 0.795476i \(-0.707221\pi\)
−0.605985 + 0.795476i \(0.707221\pi\)
\(788\) −6.00000 −0.213741
\(789\) 32.0000 1.13923
\(790\) 8.00000 0.284627
\(791\) 24.0000 0.853342
\(792\) 2.00000 0.0710669
\(793\) 0 0
\(794\) −32.0000 −1.13564
\(795\) 6.00000 0.212798
\(796\) 16.0000 0.567105
\(797\) 50.0000 1.77109 0.885545 0.464553i \(-0.153785\pi\)
0.885545 + 0.464553i \(0.153785\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) −1.00000 −0.0353553
\(801\) −8.00000 −0.282666
\(802\) 12.0000 0.423735
\(803\) −12.0000 −0.423471
\(804\) −2.00000 −0.0705346
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) −24.0000 −0.844840
\(808\) 12.0000 0.422159
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 1.00000 0.0351364
\(811\) 52.0000 1.82597 0.912983 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) −16.0000 −0.561490
\(813\) −8.00000 −0.280572
\(814\) 20.0000 0.701000
\(815\) 16.0000 0.560456
\(816\) −2.00000 −0.0700140
\(817\) 0 0
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) −16.0000 −0.558404 −0.279202 0.960232i \(-0.590070\pi\)
−0.279202 + 0.960232i \(0.590070\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −10.0000 −0.348578 −0.174289 0.984695i \(-0.555763\pi\)
−0.174289 + 0.984695i \(0.555763\pi\)
\(824\) −4.00000 −0.139347
\(825\) 2.00000 0.0696311
\(826\) −48.0000 −1.67013
\(827\) 44.0000 1.53003 0.765015 0.644013i \(-0.222732\pi\)
0.765015 + 0.644013i \(0.222732\pi\)
\(828\) 1.00000 0.0347524
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 8.00000 0.277684
\(831\) −8.00000 −0.277517
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 26.0000 0.898155
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) −4.00000 −0.138013
\(841\) −13.0000 −0.448276
\(842\) 6.00000 0.206774
\(843\) −32.0000 −1.10214
\(844\) 4.00000 0.137686
\(845\) 13.0000 0.447214
\(846\) −12.0000 −0.412568
\(847\) −28.0000 −0.962091
\(848\) 6.00000 0.206041
\(849\) 10.0000 0.343199
\(850\) −2.00000 −0.0685994
\(851\) 10.0000 0.342796
\(852\) 2.00000 0.0685189
\(853\) 56.0000 1.91740 0.958702 0.284413i \(-0.0917988\pi\)
0.958702 + 0.284413i \(0.0917988\pi\)
\(854\) 56.0000 1.91628
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) −2.00000 −0.0681994
\(861\) −24.0000 −0.817918
\(862\) −24.0000 −0.817443
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 1.00000 0.0340207
\(865\) 6.00000 0.204006
\(866\) −24.0000 −0.815553
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 4.00000 0.135613
\(871\) 0 0
\(872\) −14.0000 −0.474100
\(873\) 0 0
\(874\) 0 0
\(875\) −4.00000 −0.135225
\(876\) −6.00000 −0.202721
\(877\) −56.0000 −1.89099 −0.945493 0.325643i \(-0.894419\pi\)
−0.945493 + 0.325643i \(0.894419\pi\)
\(878\) −20.0000 −0.674967
\(879\) −14.0000 −0.472208
\(880\) 2.00000 0.0674200
\(881\) 40.0000 1.34763 0.673817 0.738898i \(-0.264654\pi\)
0.673817 + 0.738898i \(0.264654\pi\)
\(882\) −9.00000 −0.303046
\(883\) −48.0000 −1.61533 −0.807664 0.589643i \(-0.799269\pi\)
−0.807664 + 0.589643i \(0.799269\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) 20.0000 0.671913
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 10.0000 0.335578
\(889\) 24.0000 0.804934
\(890\) −8.00000 −0.268161
\(891\) −2.00000 −0.0670025
\(892\) 2.00000 0.0669650
\(893\) 0 0
\(894\) −14.0000 −0.468230
\(895\) −4.00000 −0.133705
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) 34.0000 1.13459
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) 12.0000 0.399556
\(903\) −8.00000 −0.266223
\(904\) −6.00000 −0.199557
\(905\) 6.00000 0.199447
\(906\) −12.0000 −0.398673
\(907\) 42.0000 1.39459 0.697294 0.716786i \(-0.254387\pi\)
0.697294 + 0.716786i \(0.254387\pi\)
\(908\) 12.0000 0.398234
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 24.0000 0.793849
\(915\) −14.0000 −0.462826
\(916\) −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 \(0.676980\pi\)
\(920\) 1.00000 0.0329690
\(921\) −4.00000 −0.131804
\(922\) −20.0000 −0.658665
\(923\) 0 0
\(924\) 8.00000 0.263181
\(925\) 10.0000 0.328798
\(926\) −18.0000 −0.591517
\(927\) 4.00000 0.131377
\(928\) 4.00000 0.131306
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) 2.00000 0.0654771
\(934\) −24.0000 −0.785304
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) 48.0000 1.56809 0.784046 0.620703i \(-0.213153\pi\)
0.784046 + 0.620703i \(0.213153\pi\)
\(938\) −8.00000 −0.261209
\(939\) −20.0000 −0.652675
\(940\) −12.0000 −0.391397
\(941\) 26.0000 0.847576 0.423788 0.905761i \(-0.360700\pi\)
0.423788 + 0.905761i \(0.360700\pi\)
\(942\) −22.0000 −0.716799
\(943\) 6.00000 0.195387
\(944\) 12.0000 0.390567
\(945\) 4.00000 0.130120
\(946\) 4.00000 0.130051
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) −8.00000 −0.259828
\(949\) 0 0
\(950\) 0 0
\(951\) 10.0000 0.324272
\(952\) −8.00000 −0.259281
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 22.0000 0.711531
\(957\) −8.00000 −0.258603
\(958\) 40.0000 1.29234
\(959\) −8.00000 −0.258333
\(960\) 1.00000 0.0322749
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) −14.0000 −0.450910
\(965\) 14.0000 0.450676
\(966\) 4.00000 0.128698
\(967\) 58.0000 1.86515 0.932577 0.360971i \(-0.117555\pi\)
0.932577 + 0.360971i \(0.117555\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 0 0
\(971\) 50.0000 1.60458 0.802288 0.596937i \(-0.203616\pi\)
0.802288 + 0.596937i \(0.203616\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 80.0000 2.56468
\(974\) −18.0000 −0.576757
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −16.0000 −0.511624
\(979\) 16.0000 0.511362
\(980\) −9.00000 −0.287494
\(981\) 14.0000 0.446986
\(982\) 24.0000 0.765871
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 6.00000 0.191273
\(985\) 6.00000 0.191176
\(986\) 8.00000 0.254772
\(987\) −48.0000 −1.52786
\(988\) 0 0
\(989\) 2.00000 0.0635963
\(990\) −2.00000 −0.0635642
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 0 0
\(993\) 28.0000 0.888553
\(994\) 8.00000 0.253745
\(995\) −16.0000 −0.507234
\(996\) −8.00000 −0.253490
\(997\) 40.0000 1.26681 0.633406 0.773819i \(-0.281656\pi\)
0.633406 + 0.773819i \(0.281656\pi\)
\(998\) 28.0000 0.886325
\(999\) −10.0000 −0.316386
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 690.2.a.b.1.1 1
3.2 odd 2 2070.2.a.s.1.1 1
4.3 odd 2 5520.2.a.r.1.1 1
5.2 odd 4 3450.2.d.n.2899.1 2
5.3 odd 4 3450.2.d.n.2899.2 2
5.4 even 2 3450.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.b.1.1 1 1.1 even 1 trivial
2070.2.a.s.1.1 1 3.2 odd 2
3450.2.a.t.1.1 1 5.4 even 2
3450.2.d.n.2899.1 2 5.2 odd 4
3450.2.d.n.2899.2 2 5.3 odd 4
5520.2.a.r.1.1 1 4.3 odd 2