Properties

Label 1122.2.a.e.1.1
Level $1122$
Weight $2$
Character 1122.1
Self dual yes
Analytic conductor $8.959$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1122.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.95921510679\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1122.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} -1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} -6.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} -2.00000 q^{45} +8.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +2.00000 q^{55} +4.00000 q^{57} -2.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} -10.0000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +1.00000 q^{66} +4.00000 q^{67} +1.00000 q^{68} +16.0000 q^{71} +1.00000 q^{72} +2.00000 q^{73} -10.0000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +2.00000 q^{78} -8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} -2.00000 q^{85} +4.00000 q^{86} +2.00000 q^{87} -1.00000 q^{88} +10.0000 q^{89} -2.00000 q^{90} +8.00000 q^{93} +8.00000 q^{94} +8.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} -7.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −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
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −2.00000 −0.447214
\(21\) 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
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 2.00000 0.365148
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) −2.00000 −0.316228
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 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
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) −1.00000 −0.141421
\(51\) −1.00000 −0.140028
\(52\) −2.00000 −0.277350
\(53\) 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
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) −2.00000 −0.262613
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 2.00000 0.258199
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 1.00000 0.123091
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 1.00000 0.121268
\(69\) 0 0
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −10.0000 −1.16248
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −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
\(87\) 2.00000 0.214423
\(88\) −1.00000 −0.106600
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 8.00000 0.825137
\(95\) 8.00000 0.820783
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −7.00000 −0.707107
\(99\) −1.00000 −0.100504
\(100\) −1.00000 −0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 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
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 2.00000 0.190693
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) −10.0000 −0.905357
\(123\) 6.00000 0.541002
\(124\) −8.00000 −0.718421
\(125\) 12.0000 1.07331
\(126\) 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
\(130\) 4.00000 0.350823
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 2.00000 0.172133
\(136\) 1.00000 0.0857493
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 16.0000 1.34269
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 2.00000 0.165521
\(147\) 7.00000 0.577350
\(148\) −10.0000 −0.821995
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 1.00000 0.0816497
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −4.00000 −0.324443
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) 16.0000 1.28515
\(156\) 2.00000 0.160128
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −8.00000 −0.636446
\(159\) −6.00000 −0.475831
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −6.00000 −0.468521
\(165\) −2.00000 −0.155700
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 12.0000 0.901975
\(178\) 10.0000 0.749532
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −2.00000 −0.149071
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 20.0000 1.47043
\(186\) 8.00000 0.586588
\(187\) −1.00000 −0.0731272
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 2.00000 0.143592
\(195\) −4.00000 −0.286446
\(196\) −7.00000 −0.500000
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.00000 −0.282138
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) −1.00000 −0.0700140
\(205\) 12.0000 0.838116
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) −16.0000 −1.09630
\(214\) −12.0000 −0.820303
\(215\) −8.00000 −0.545595
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 6.00000 0.406371
\(219\) −2.00000 −0.135147
\(220\) 2.00000 0.134840
\(221\) −2.00000 −0.134535
\(222\) 10.0000 0.671156
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −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 \(0.369623\pi\)
\(228\) 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
\(232\) −2.00000 −0.131306
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) −16.0000 −1.04372
\(236\) −12.0000 −0.781133
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 2.00000 0.129099
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 14.0000 0.894427
\(246\) 6.00000 0.382546
\(247\) 8.00000 0.509028
\(248\) −8.00000 −0.508001
\(249\) 12.0000 0.760469
\(250\) 12.0000 0.758947
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 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 \(0.480131\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) −2.00000 −0.123797
\(262\) 12.0000 0.741362
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 1.00000 0.0615457
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 4.00000 0.244339
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 2.00000 0.121716
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 20.0000 1.19952
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) −8.00000 −0.476393
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 16.0000 0.949425
\(285\) −8.00000 −0.473879
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 4.00000 0.234888
\(291\) −2.00000 −0.117242
\(292\) 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
\(295\) 24.0000 1.39733
\(296\) −10.0000 −0.581238
\(297\) 1.00000 0.0580259
\(298\) 22.0000 1.27443
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) −6.00000 −0.344691
\(304\) −4.00000 −0.229416
\(305\) 20.0000 1.14520
\(306\) 1.00000 0.0571662
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 16.0000 0.908739
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 2.00000 0.113228
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) −6.00000 −0.336463
\(319\) 2.00000 0.111979
\(320\) −2.00000 −0.111803
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −4.00000 −0.221540
\(327\) −6.00000 −0.331801
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) −2.00000 −0.110096
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −12.0000 −0.658586
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) 6.00000 0.325875
\(340\) −2.00000 −0.108465
\(341\) 8.00000 0.433224
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 2.00000 0.107211
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −1.00000 −0.0533002
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 12.0000 0.637793
\(355\) −32.0000 −1.69838
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) −2.00000 −0.105409
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) 10.0000 0.522708
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 20.0000 1.03975
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −1.00000 −0.0517088
\(375\) −12.0000 −0.619677
\(376\) 8.00000 0.412568
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 8.00000 0.410391
\(381\) −8.00000 −0.409852
\(382\) 8.00000 0.409316
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) 4.00000 0.203331
\(388\) 2.00000 0.101535
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −4.00000 −0.202548
\(391\) 0 0
\(392\) −7.00000 −0.353553
\(393\) −12.0000 −0.605320
\(394\) −26.0000 −1.30986
\(395\) 16.0000 0.805047
\(396\) −1.00000 −0.0502519
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) −4.00000 −0.199502
\(403\) 16.0000 0.797017
\(404\) 6.00000 0.298511
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 10.0000 0.495682
\(408\) −1.00000 −0.0495074
\(409\) −38.0000 −1.87898 −0.939490 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 12.0000 0.592638
\(411\) 6.00000 0.295958
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) −2.00000 −0.0980581
\(417\) −20.0000 −0.979404
\(418\) 4.00000 0.195646
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) −4.00000 −0.194717
\(423\) 8.00000 0.388973
\(424\) 6.00000 0.291386
\(425\) −1.00000 −0.0485071
\(426\) −16.0000 −0.775203
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) −2.00000 −0.0965609
\(430\) −8.00000 −0.385794
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) −4.00000 −0.191785
\(436\) 6.00000 0.287348
\(437\) 0 0
\(438\) −2.00000 −0.0955637
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 2.00000 0.0953463
\(441\) −7.00000 −0.333333
\(442\) −2.00000 −0.0951303
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 10.0000 0.474579
\(445\) −20.0000 −0.948091
\(446\) 0 0
\(447\) −22.0000 −1.04056
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 6.00000 0.282529
\(452\) −6.00000 −0.282216
\(453\) −16.0000 −0.751746
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −26.0000 −1.21490
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −2.00000 −0.0928477
\(465\) −16.0000 −0.741982
\(466\) −6.00000 −0.277945
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) −16.0000 −0.738025
\(471\) 2.00000 0.0921551
\(472\) −12.0000 −0.552345
\(473\) −4.00000 −0.183920
\(474\) 8.00000 0.367452
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 16.0000 0.731823
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 2.00000 0.0912871
\(481\) 20.0000 0.911922
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −4.00000 −0.181631
\(486\) −1.00000 −0.0453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −10.0000 −0.452679
\(489\) 4.00000 0.180886
\(490\) 14.0000 0.632456
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 6.00000 0.270501
\(493\) −2.00000 −0.0900755
\(494\) 8.00000 0.359937
\(495\) 2.00000 0.0898933
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 44.0000 1.96971 0.984855 0.173379i \(-0.0554684\pi\)
0.984855 + 0.173379i \(0.0554684\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 20.0000 0.892644
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 8.00000 0.354943
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 2.00000 0.0885615
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) 2.00000 0.0882162
\(515\) 16.0000 0.705044
\(516\) −4.00000 −0.176090
\(517\) −8.00000 −0.351840
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 4.00000 0.175412
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −8.00000 −0.348485
\(528\) 1.00000 0.0435194
\(529\) −23.0000 −1.00000
\(530\) −12.0000 −0.521247
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) −10.0000 −0.432742
\(535\) 24.0000 1.03761
\(536\) 4.00000 0.172774
\(537\) 4.00000 0.172613
\(538\) −26.0000 −1.12094
\(539\) 7.00000 0.301511
\(540\) 2.00000 0.0860663
\(541\) −26.0000 −1.11783 −0.558914 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) 24.0000 1.03089
\(543\) 10.0000 0.429141
\(544\) 1.00000 0.0428746
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −6.00000 −0.256307
\(549\) −10.0000 −0.426790
\(550\) 1.00000 0.0426401
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) −20.0000 −0.848953
\(556\) 20.0000 0.848189
\(557\) 46.0000 1.94908 0.974541 0.224208i \(-0.0719796\pi\)
0.974541 + 0.224208i \(0.0719796\pi\)
\(558\) −8.00000 −0.338667
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 1.00000 0.0422200
\(562\) −6.00000 −0.253095
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) −8.00000 −0.336861
\(565\) 12.0000 0.504844
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) 16.0000 0.671345
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) −8.00000 −0.335083
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) 2.00000 0.0836242
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 1.00000 0.0415945
\(579\) 6.00000 0.249351
\(580\) 4.00000 0.166091
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) −6.00000 −0.248495
\(584\) 2.00000 0.0827606
\(585\) 4.00000 0.165380
\(586\) 6.00000 0.247858
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 7.00000 0.288675
\(589\) 32.0000 1.31854
\(590\) 24.0000 0.988064
\(591\) 26.0000 1.06950
\(592\) −10.0000 −0.410997
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 1.00000 0.0408248
\(601\) 18.0000 0.734235 0.367118 0.930175i \(-0.380345\pi\)
0.367118 + 0.930175i \(0.380345\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 16.0000 0.651031
\(605\) −2.00000 −0.0813116
\(606\) −6.00000 −0.243733
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 20.0000 0.809776
\(611\) −16.0000 −0.647291
\(612\) 1.00000 0.0404226
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) −4.00000 −0.161427
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 8.00000 0.321807
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 16.0000 0.642575
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) −6.00000 −0.239808
\(627\) −4.00000 −0.159745
\(628\) −2.00000 −0.0798087
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −8.00000 −0.318223
\(633\) 4.00000 0.158986
\(634\) 22.0000 0.873732
\(635\) −16.0000 −0.634941
\(636\) −6.00000 −0.237915
\(637\) 14.0000 0.554700
\(638\) 2.00000 0.0791808
\(639\) 16.0000 0.632950
\(640\) −2.00000 −0.0790569
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 12.0000 0.473602
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) −4.00000 −0.157378
\(647\) −16.0000 −0.629025 −0.314512 0.949253i \(-0.601841\pi\)
−0.314512 + 0.949253i \(0.601841\pi\)
\(648\) 1.00000 0.0392837
\(649\) 12.0000 0.471041
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) −6.00000 −0.234619
\(655\) −24.0000 −0.937758
\(656\) −6.00000 −0.234261
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 12.0000 0.466393
\(663\) 2.00000 0.0776736
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) −22.0000 −0.847408
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) −2.00000 −0.0766965
\(681\) −12.0000 −0.459841
\(682\) 8.00000 0.306336
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) −4.00000 −0.152944
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 26.0000 0.991962
\(688\) 4.00000 0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −40.0000 −1.51729
\(696\) 2.00000 0.0758098
\(697\) −6.00000 −0.227266
\(698\) −18.0000 −0.681310
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 2.00000 0.0754851
\(703\) 40.0000 1.50863
\(704\) −1.00000 −0.0376889
\(705\) 16.0000 0.602595
\(706\) 2.00000 0.0752710
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) −32.0000 −1.20094
\(711\) −8.00000 −0.300023
\(712\) 10.0000 0.374766
\(713\) 0 0
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) −4.00000 −0.149487
\(717\) −16.0000 −0.597531
\(718\) −8.00000 −0.298557
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) −10.0000 −0.371904
\(724\) −10.0000 −0.371647
\(725\) 2.00000 0.0742781
\(726\) −1.00000 −0.0371135
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −4.00000 −0.148047
\(731\) 4.00000 0.147945
\(732\) 10.0000 0.369611
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −8.00000 −0.295285
\(735\) −14.0000 −0.516398
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) −6.00000 −0.220863
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 20.0000 0.735215
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 8.00000 0.293294
\(745\) −44.0000 −1.61204
\(746\) −26.0000 −0.951928
\(747\) −12.0000 −0.439057
\(748\) −1.00000 −0.0365636
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 8.00000 0.291730
\(753\) −20.0000 −0.728841
\(754\) 4.00000 0.145671
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −12.0000 −0.435860
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −8.00000 −0.289809
\(763\) 0 0
\(764\) 8.00000 0.289430
\(765\) −2.00000 −0.0723102
\(766\) −24.0000 −0.867155
\(767\) 24.0000 0.866590
\(768\) −1.00000 −0.0360844
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) −6.00000 −0.215945
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 4.00000 0.143777
\(775\) 8.00000 0.287368
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 24.0000 0.859889
\(780\) −4.00000 −0.143223
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) −7.00000 −0.250000
\(785\) 4.00000 0.142766
\(786\) −12.0000 −0.428026
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −26.0000 −0.926212
\(789\) −24.0000 −0.854423
\(790\) 16.0000 0.569254
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) 20.0000 0.710221
\(794\) 14.0000 0.496841
\(795\) 12.0000 0.425596
\(796\) 16.0000 0.567105
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) −1.00000 −0.0353553
\(801\) 10.0000 0.353333
\(802\) 10.0000 0.353112
\(803\) −2.00000 −0.0705785
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 26.0000 0.915243
\(808\) 6.00000 0.211079
\(809\) −22.0000 −0.773479 −0.386739 0.922189i \(-0.626399\pi\)
−0.386739 + 0.922189i \(0.626399\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) −24.0000 −0.841717
\(814\) 10.0000 0.350500
\(815\) 8.00000 0.280228
\(816\) −1.00000 −0.0350070
\(817\) −16.0000 −0.559769
\(818\) −38.0000 −1.32864
\(819\) 0 0
\(820\) 12.0000 0.419058
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 6.00000 0.209274
\(823\) −48.0000 −1.67317 −0.836587 0.547833i \(-0.815453\pi\)
−0.836587 + 0.547833i \(0.815453\pi\)
\(824\) −8.00000 −0.278693
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 24.0000 0.833052
\(831\) 2.00000 0.0693792
\(832\) −2.00000 −0.0693375
\(833\) −7.00000 −0.242536
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) 8.00000 0.276520
\(838\) 28.0000 0.967244
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 6.00000 0.206774
\(843\) 6.00000 0.206651
\(844\) −4.00000 −0.137686
\(845\) 18.0000 0.619219
\(846\) 8.00000 0.275046
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 28.0000 0.960958
\(850\) −1.00000 −0.0342997
\(851\) 0 0
\(852\) −16.0000 −0.548151
\(853\) −34.0000 −1.16414 −0.582069 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) −12.0000 −0.410152
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) −2.00000 −0.0682789
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 36.0000 1.22404
\(866\) −14.0000 −0.475739
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) −4.00000 −0.135613
\(871\) −8.00000 −0.271070
\(872\) 6.00000 0.203186
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) 6.00000 0.202606 0.101303 0.994856i \(-0.467699\pi\)
0.101303 + 0.994856i \(0.467699\pi\)
\(878\) −16.0000 −0.539974
\(879\) −6.00000 −0.202375
\(880\) 2.00000 0.0674200
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) −7.00000 −0.235702
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) −2.00000 −0.0672673
\(885\) −24.0000 −0.806751
\(886\) 4.00000 0.134383
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 10.0000 0.335578
\(889\) 0 0
\(890\) −20.0000 −0.670402
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) −32.0000 −1.07084
\(894\) −22.0000 −0.735790
\(895\) 8.00000 0.267411
\(896\) 0 0
\(897\) 0 0
\(898\) 10.0000 0.333704
\(899\) 16.0000 0.533630
\(900\) −1.00000 −0.0333333
\(901\) 6.00000 0.199889
\(902\) 6.00000 0.199778
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 20.0000 0.664822
\(906\) −16.0000 −0.531564
\(907\) 36.0000 1.19536 0.597680 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) 12.0000 0.398234
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 4.00000 0.132453
\(913\) 12.0000 0.397142
\(914\) −22.0000 −0.727695
\(915\) −20.0000 −0.661180
\(916\) −26.0000 −0.859064
\(917\) 0 0
\(918\) −1.00000 −0.0330049
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 14.0000 0.461065
\(923\) −32.0000 −1.05329
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) 16.0000 0.525793
\(927\) −8.00000 −0.262754
\(928\) −2.00000 −0.0656532
\(929\) 10.0000 0.328089 0.164045 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) −16.0000 −0.524661
\(931\) 28.0000 0.917663
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) 2.00000 0.0654070
\(936\) −2.00000 −0.0653720
\(937\) 58.0000 1.89478 0.947389 0.320085i \(-0.103712\pi\)
0.947389 + 0.320085i \(0.103712\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) −16.0000 −0.521862
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 2.00000 0.0651635
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 8.00000 0.259828
\(949\) −4.00000 −0.129845
\(950\) 4.00000 0.129777
\(951\) −22.0000 −0.713399
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 6.00000 0.194257
\(955\) −16.0000 −0.517748
\(956\) 16.0000 0.517477
\(957\) −2.00000 −0.0646508
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) 33.0000 1.06452
\(962\) 20.0000 0.644826
\(963\) −12.0000 −0.386695
\(964\) 10.0000 0.322078
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 1.00000 0.0321412
\(969\) 4.00000 0.128499
\(970\) −4.00000 −0.128432
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 32.0000 1.02535
\(975\) −2.00000 −0.0640513
\(976\) −10.0000 −0.320092
\(977\) 34.0000 1.08776 0.543878 0.839164i \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) 4.00000 0.127906
\(979\) −10.0000 −0.319601
\(980\) 14.0000 0.447214
\(981\) 6.00000 0.191565
\(982\) −20.0000 −0.638226
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 6.00000 0.191273
\(985\) 52.0000 1.65686
\(986\) −2.00000 −0.0636930
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) 2.00000 0.0635642
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) −8.00000 −0.254000
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) −32.0000 −1.01447
\(996\) 12.0000 0.380235
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 44.0000 1.39280
\(999\) 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 1122.2.a.e.1.1 1
3.2 odd 2 3366.2.a.k.1.1 1
4.3 odd 2 8976.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1122.2.a.e.1.1 1 1.1 even 1 trivial
3366.2.a.k.1.1 1 3.2 odd 2
8976.2.a.w.1.1 1 4.3 odd 2