Properties

Label 966.2.a.g.1.1
Level $966$
Weight $2$
Character 966.1
Self dual yes
Analytic conductor $7.714$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.71354883526\)
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 \(\chi\) \(=\) 966.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} -4.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -6.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} +1.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} -10.0000 q^{61} -8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} -6.00000 q^{68} +1.00000 q^{69} -2.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -4.00000 q^{77} +2.00000 q^{78} -2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} -1.00000 q^{84} +12.0000 q^{85} -4.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} +2.00000 q^{89} -2.00000 q^{90} -2.00000 q^{91} -1.00000 q^{92} +8.00000 q^{93} -8.00000 q^{94} -8.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −2.00000 −0.447214
\(21\) −1.00000 −0.218218
\(22\) −4.00000 −0.852803
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −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
\(33\) 4.00000 0.696311
\(34\) −6.00000 −1.02899
\(35\) −2.00000 −0.338062
\(36\) 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
\(39\) 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 \(0.655213\pi\)
\(42\) −1.00000 −0.154303
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −4.00000 −0.603023
\(45\) −2.00000 −0.298142
\(46\) −1.00000 −0.147442
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 6.00000 0.840168
\(52\) −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
\(55\) 8.00000 1.07872
\(56\) 1.00000 0.133631
\(57\) −4.00000 −0.529813
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 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
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 4.00000 0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −6.00000 −0.727607
\(69\) 1.00000 0.120386
\(70\) −2.00000 −0.239046
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) −4.00000 −0.455842
\(78\) 2.00000 0.226455
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −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 \(0.728845\pi\)
\(84\) −1.00000 −0.109109
\(85\) 12.0000 1.30158
\(86\) −4.00000 −0.431331
\(87\) 2.00000 0.214423
\(88\) −4.00000 −0.426401
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) −2.00000 −0.210819
\(91\) −2.00000 −0.209657
\(92\) −1.00000 −0.104257
\(93\) 8.00000 0.829561
\(94\) −8.00000 −0.825137
\(95\) −8.00000 −0.820783
\(96\) −1.00000 −0.102062
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 1.00000 0.101015
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 6.00000 0.594089
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) 2.00000 0.195180
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 8.00000 0.762770
\(111\) −6.00000 −0.569495
\(112\) 1.00000 0.0944911
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) −4.00000 −0.374634
\(115\) 2.00000 0.186501
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) 4.00000 0.368230
\(119\) −6.00000 −0.550019
\(120\) 2.00000 0.182574
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) 6.00000 0.541002
\(124\) −8.00000 −0.718421
\(125\) 12.0000 1.07331
\(126\) 1.00000 0.0890871
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 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 \(0.324356\pi\)
\(132\) 4.00000 0.348155
\(133\) 4.00000 0.346844
\(134\) 4.00000 0.345547
\(135\) 2.00000 0.172133
\(136\) −6.00000 −0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 1.00000 0.0851257
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) −2.00000 −0.169031
\(141\) 8.00000 0.673722
\(142\) −8.00000 −0.671345
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) −6.00000 −0.496564
\(147\) −1.00000 −0.0824786
\(148\) 6.00000 0.493197
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 1.00000 0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 4.00000 0.324443
\(153\) −6.00000 −0.485071
\(154\) −4.00000 −0.322329
\(155\) 16.0000 1.28515
\(156\) 2.00000 0.160128
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) −2.00000 −0.158114
\(161\) −1.00000 −0.0788110
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −6.00000 −0.468521
\(165\) −8.00000 −0.622799
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 12.0000 0.920358
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 2.00000 0.151620
\(175\) −1.00000 −0.0755929
\(176\) −4.00000 −0.301511
\(177\) −4.00000 −0.300658
\(178\) 2.00000 0.149906
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −2.00000 −0.149071
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −2.00000 −0.148250
\(183\) 10.0000 0.739221
\(184\) −1.00000 −0.0737210
\(185\) −12.0000 −0.882258
\(186\) 8.00000 0.586588
\(187\) 24.0000 1.75505
\(188\) −8.00000 −0.583460
\(189\) −1.00000 −0.0727393
\(190\) −8.00000 −0.580381
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 10.0000 0.717958
\(195\) −4.00000 −0.286446
\(196\) 1.00000 0.0714286
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) −4.00000 −0.284268
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.00000 −0.282138
\(202\) 6.00000 0.422159
\(203\) −2.00000 −0.140372
\(204\) 6.00000 0.420084
\(205\) 12.0000 0.838116
\(206\) 8.00000 0.557386
\(207\) −1.00000 −0.0695048
\(208\) −2.00000 −0.138675
\(209\) −16.0000 −1.10674
\(210\) 2.00000 0.138013
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 6.00000 0.412082
\(213\) 8.00000 0.548151
\(214\) 12.0000 0.820303
\(215\) 8.00000 0.545595
\(216\) −1.00000 −0.0680414
\(217\) −8.00000 −0.543075
\(218\) 14.0000 0.948200
\(219\) 6.00000 0.405442
\(220\) 8.00000 0.539360
\(221\) 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 \(0.202926\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.00000 −0.0666667
\(226\) 18.0000 1.19734
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −4.00000 −0.264906
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 2.00000 0.131876
\(231\) 4.00000 0.263181
\(232\) −2.00000 −0.131306
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −2.00000 −0.130744
\(235\) 16.0000 1.04372
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) −6.00000 −0.388922
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 2.00000 0.129099
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) −2.00000 −0.127775
\(246\) 6.00000 0.382546
\(247\) −8.00000 −0.509028
\(248\) −8.00000 −0.508001
\(249\) 12.0000 0.760469
\(250\) 12.0000 0.758947
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 1.00000 0.0629941
\(253\) 4.00000 0.251478
\(254\) −16.0000 −1.00393
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 4.00000 0.249029
\(259\) 6.00000 0.372822
\(260\) 4.00000 0.248069
\(261\) −2.00000 −0.123797
\(262\) 12.0000 0.741362
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 4.00000 0.246183
\(265\) −12.0000 −0.737154
\(266\) 4.00000 0.245256
\(267\) −2.00000 −0.122398
\(268\) 4.00000 0.244339
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 2.00000 0.121716
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) −6.00000 −0.363803
\(273\) 2.00000 0.121046
\(274\) −6.00000 −0.362473
\(275\) 4.00000 0.241209
\(276\) 1.00000 0.0601929
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −12.0000 −0.719712
\(279\) −8.00000 −0.478947
\(280\) −2.00000 −0.119523
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 8.00000 0.476393
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −8.00000 −0.474713
\(285\) 8.00000 0.473879
\(286\) 8.00000 0.473050
\(287\) −6.00000 −0.354169
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 4.00000 0.234888
\(291\) −10.0000 −0.586210
\(292\) −6.00000 −0.351123
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −8.00000 −0.465778
\(296\) 6.00000 0.348743
\(297\) 4.00000 0.232104
\(298\) 6.00000 0.347571
\(299\) 2.00000 0.115663
\(300\) 1.00000 0.0577350
\(301\) −4.00000 −0.230556
\(302\) 8.00000 0.460348
\(303\) −6.00000 −0.344691
\(304\) 4.00000 0.229416
\(305\) 20.0000 1.14520
\(306\) −6.00000 −0.342997
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −4.00000 −0.227921
\(309\) −8.00000 −0.455104
\(310\) 16.0000 0.908739
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 2.00000 0.113228
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 22.0000 1.24153
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −6.00000 −0.336463
\(319\) 8.00000 0.447914
\(320\) −2.00000 −0.111803
\(321\) −12.0000 −0.669775
\(322\) −1.00000 −0.0557278
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 4.00000 0.221540
\(327\) −14.0000 −0.774202
\(328\) −6.00000 −0.331295
\(329\) −8.00000 −0.441054
\(330\) −8.00000 −0.440386
\(331\) −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
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) −1.00000 −0.0545545
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) −9.00000 −0.489535
\(339\) −18.0000 −0.977626
\(340\) 12.0000 0.650791
\(341\) 32.0000 1.73290
\(342\) 4.00000 0.216295
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) −2.00000 −0.107676
\(346\) −2.00000 −0.107521
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 2.00000 0.107211
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 2.00000 0.106752
\(352\) −4.00000 −0.213201
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) −4.00000 −0.212598
\(355\) 16.0000 0.849192
\(356\) 2.00000 0.106000
\(357\) 6.00000 0.317554
\(358\) −12.0000 −0.634220
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) −2.00000 −0.105409
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) −5.00000 −0.262432
\(364\) −2.00000 −0.104828
\(365\) 12.0000 0.628109
\(366\) 10.0000 0.522708
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −6.00000 −0.312348
\(370\) −12.0000 −0.623850
\(371\) 6.00000 0.311504
\(372\) 8.00000 0.414781
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 24.0000 1.24101
\(375\) −12.0000 −0.619677
\(376\) −8.00000 −0.412568
\(377\) 4.00000 0.206010
\(378\) −1.00000 −0.0514344
\(379\) −36.0000 −1.84920 −0.924598 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) −8.00000 −0.410391
\(381\) 16.0000 0.819705
\(382\) −16.0000 −0.818631
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 8.00000 0.407718
\(386\) 2.00000 0.101797
\(387\) −4.00000 −0.203331
\(388\) 10.0000 0.507673
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −4.00000 −0.202548
\(391\) 6.00000 0.303433
\(392\) 1.00000 0.0505076
\(393\) −12.0000 −0.605320
\(394\) −26.0000 −1.30986
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −24.0000 −1.20301
\(399\) −4.00000 −0.200250
\(400\) −1.00000 −0.0500000
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) −4.00000 −0.199502
\(403\) 16.0000 0.797017
\(404\) 6.00000 0.298511
\(405\) −2.00000 −0.0993808
\(406\) −2.00000 −0.0992583
\(407\) −24.0000 −1.18964
\(408\) 6.00000 0.297044
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 12.0000 0.592638
\(411\) 6.00000 0.295958
\(412\) 8.00000 0.394132
\(413\) 4.00000 0.196827
\(414\) −1.00000 −0.0491473
\(415\) 24.0000 1.17811
\(416\) −2.00000 −0.0980581
\(417\) 12.0000 0.587643
\(418\) −16.0000 −0.782586
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 2.00000 0.0975900
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 4.00000 0.194717
\(423\) −8.00000 −0.388973
\(424\) 6.00000 0.291386
\(425\) 6.00000 0.291043
\(426\) 8.00000 0.387601
\(427\) −10.0000 −0.483934
\(428\) 12.0000 0.580042
\(429\) −8.00000 −0.386244
\(430\) 8.00000 0.385794
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) −8.00000 −0.384012
\(435\) −4.00000 −0.191785
\(436\) 14.0000 0.670478
\(437\) −4.00000 −0.191346
\(438\) 6.00000 0.286691
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 8.00000 0.381385
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) −6.00000 −0.284747
\(445\) −4.00000 −0.189618
\(446\) 24.0000 1.13643
\(447\) −6.00000 −0.283790
\(448\) 1.00000 0.0472456
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 24.0000 1.13012
\(452\) 18.0000 0.846649
\(453\) −8.00000 −0.375873
\(454\) −12.0000 −0.563188
\(455\) 4.00000 0.187523
\(456\) −4.00000 −0.187317
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 14.0000 0.654177
\(459\) 6.00000 0.280056
\(460\) 2.00000 0.0932505
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 4.00000 0.186097
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) −2.00000 −0.0928477
\(465\) −16.0000 −0.741982
\(466\) 10.0000 0.463241
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 4.00000 0.184703
\(470\) 16.0000 0.738025
\(471\) −22.0000 −1.01371
\(472\) 4.00000 0.184115
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) −6.00000 −0.275010
\(477\) 6.00000 0.274721
\(478\) −16.0000 −0.731823
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 2.00000 0.0912871
\(481\) −12.0000 −0.547153
\(482\) −22.0000 −1.00207
\(483\) 1.00000 0.0455016
\(484\) 5.00000 0.227273
\(485\) −20.0000 −0.908153
\(486\) −1.00000 −0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) −10.0000 −0.452679
\(489\) −4.00000 −0.180886
\(490\) −2.00000 −0.0903508
\(491\) −4.00000 −0.180517 −0.0902587 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) 6.00000 0.270501
\(493\) 12.0000 0.540453
\(494\) −8.00000 −0.359937
\(495\) 8.00000 0.359573
\(496\) −8.00000 −0.359211
\(497\) −8.00000 −0.358849
\(498\) 12.0000 0.537733
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 1.00000 0.0445435
\(505\) −12.0000 −0.533993
\(506\) 4.00000 0.177822
\(507\) 9.00000 0.399704
\(508\) −16.0000 −0.709885
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) −12.0000 −0.531369
\(511\) −6.00000 −0.265424
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 18.0000 0.793946
\(515\) −16.0000 −0.705044
\(516\) 4.00000 0.176090
\(517\) 32.0000 1.40736
\(518\) 6.00000 0.263625
\(519\) 2.00000 0.0877903
\(520\) 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
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 12.0000 0.524222
\(525\) 1.00000 0.0436436
\(526\) −24.0000 −1.04645
\(527\) 48.0000 2.09091
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) −12.0000 −0.521247
\(531\) 4.00000 0.173585
\(532\) 4.00000 0.173422
\(533\) 12.0000 0.519778
\(534\) −2.00000 −0.0865485
\(535\) −24.0000 −1.03761
\(536\) 4.00000 0.172774
\(537\) 12.0000 0.517838
\(538\) −2.00000 −0.0862261
\(539\) −4.00000 −0.172292
\(540\) 2.00000 0.0860663
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 24.0000 1.03089
\(543\) 2.00000 0.0858282
\(544\) −6.00000 −0.257248
\(545\) −28.0000 −1.19939
\(546\) 2.00000 0.0855921
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −6.00000 −0.256307
\(549\) −10.0000 −0.426790
\(550\) 4.00000 0.170561
\(551\) −8.00000 −0.340811
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 22.0000 0.934690
\(555\) 12.0000 0.509372
\(556\) −12.0000 −0.508913
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) −8.00000 −0.338667
\(559\) 8.00000 0.338364
\(560\) −2.00000 −0.0845154
\(561\) −24.0000 −1.01328
\(562\) 10.0000 0.421825
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 8.00000 0.336861
\(565\) −36.0000 −1.51453
\(566\) −20.0000 −0.840663
\(567\) 1.00000 0.0419961
\(568\) −8.00000 −0.335673
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 8.00000 0.335083
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 8.00000 0.334497
\(573\) 16.0000 0.668410
\(574\) −6.00000 −0.250435
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 19.0000 0.790296
\(579\) −2.00000 −0.0831172
\(580\) 4.00000 0.166091
\(581\) −12.0000 −0.497844
\(582\) −10.0000 −0.414513
\(583\) −24.0000 −0.993978
\(584\) −6.00000 −0.248282
\(585\) 4.00000 0.165380
\(586\) −2.00000 −0.0826192
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −32.0000 −1.31854
\(590\) −8.00000 −0.329355
\(591\) 26.0000 1.06950
\(592\) 6.00000 0.246598
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 4.00000 0.164122
\(595\) 12.0000 0.491952
\(596\) 6.00000 0.245770
\(597\) 24.0000 0.982255
\(598\) 2.00000 0.0817861
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 1.00000 0.0408248
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) −4.00000 −0.163028
\(603\) 4.00000 0.162893
\(604\) 8.00000 0.325515
\(605\) −10.0000 −0.406558
\(606\) −6.00000 −0.243733
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 4.00000 0.162221
\(609\) 2.00000 0.0810441
\(610\) 20.0000 0.809776
\(611\) 16.0000 0.647291
\(612\) −6.00000 −0.242536
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 12.0000 0.484281
\(615\) −12.0000 −0.483887
\(616\) −4.00000 −0.161165
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) −8.00000 −0.321807
\(619\) 12.0000 0.482321 0.241160 0.970485i \(-0.422472\pi\)
0.241160 + 0.970485i \(0.422472\pi\)
\(620\) 16.0000 0.642575
\(621\) 1.00000 0.0401286
\(622\) −16.0000 −0.641542
\(623\) 2.00000 0.0801283
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) −14.0000 −0.559553
\(627\) 16.0000 0.638978
\(628\) 22.0000 0.877896
\(629\) −36.0000 −1.43541
\(630\) −2.00000 −0.0796819
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) −2.00000 −0.0794301
\(635\) 32.0000 1.26988
\(636\) −6.00000 −0.237915
\(637\) −2.00000 −0.0792429
\(638\) 8.00000 0.316723
\(639\) −8.00000 −0.316475
\(640\) −2.00000 −0.0790569
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −12.0000 −0.473602
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −8.00000 −0.315000
\(646\) −24.0000 −0.944267
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 1.00000 0.0392837
\(649\) −16.0000 −0.628055
\(650\) 2.00000 0.0784465
\(651\) 8.00000 0.313545
\(652\) 4.00000 0.156652
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −14.0000 −0.547443
\(655\) −24.0000 −0.937758
\(656\) −6.00000 −0.234261
\(657\) −6.00000 −0.234082
\(658\) −8.00000 −0.311872
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) −8.00000 −0.311400
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) −4.00000 −0.155464
\(663\) −12.0000 −0.466041
\(664\) −12.0000 −0.465690
\(665\) −8.00000 −0.310227
\(666\) 6.00000 0.232495
\(667\) 2.00000 0.0774403
\(668\) 0 0
\(669\) −24.0000 −0.927894
\(670\) −8.00000 −0.309067
\(671\) 40.0000 1.54418
\(672\) −1.00000 −0.0385758
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) −30.0000 −1.15556
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 14.0000 0.538064 0.269032 0.963131i \(-0.413296\pi\)
0.269032 + 0.963131i \(0.413296\pi\)
\(678\) −18.0000 −0.691286
\(679\) 10.0000 0.383765
\(680\) 12.0000 0.460179
\(681\) 12.0000 0.459841
\(682\) 32.0000 1.22534
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 4.00000 0.152944
\(685\) 12.0000 0.458496
\(686\) 1.00000 0.0381802
\(687\) −14.0000 −0.534133
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) −2.00000 −0.0761387
\(691\) −52.0000 −1.97817 −0.989087 0.147335i \(-0.952930\pi\)
−0.989087 + 0.147335i \(0.952930\pi\)
\(692\) −2.00000 −0.0760286
\(693\) −4.00000 −0.151947
\(694\) −36.0000 −1.36654
\(695\) 24.0000 0.910372
\(696\) 2.00000 0.0758098
\(697\) 36.0000 1.36360
\(698\) −2.00000 −0.0757011
\(699\) −10.0000 −0.378235
\(700\) −1.00000 −0.0377964
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 2.00000 0.0754851
\(703\) 24.0000 0.905177
\(704\) −4.00000 −0.150756
\(705\) −16.0000 −0.602595
\(706\) −30.0000 −1.12906
\(707\) 6.00000 0.225653
\(708\) −4.00000 −0.150329
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 16.0000 0.600469
\(711\) 0 0
\(712\) 2.00000 0.0749532
\(713\) 8.00000 0.299602
\(714\) 6.00000 0.224544
\(715\) −16.0000 −0.598366
\(716\) −12.0000 −0.448461
\(717\) 16.0000 0.597531
\(718\) −24.0000 −0.895672
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 8.00000 0.297936
\(722\) −3.00000 −0.111648
\(723\) 22.0000 0.818189
\(724\) −2.00000 −0.0743294
\(725\) 2.00000 0.0742781
\(726\) −5.00000 −0.185567
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 12.0000 0.444140
\(731\) 24.0000 0.887672
\(732\) 10.0000 0.369611
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 0 0
\(735\) 2.00000 0.0737711
\(736\) −1.00000 −0.0368605
\(737\) −16.0000 −0.589368
\(738\) −6.00000 −0.220863
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) −12.0000 −0.441129
\(741\) 8.00000 0.293887
\(742\) 6.00000 0.220267
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 8.00000 0.293294
\(745\) −12.0000 −0.439646
\(746\) 6.00000 0.219676
\(747\) −12.0000 −0.439057
\(748\) 24.0000 0.877527
\(749\) 12.0000 0.438470
\(750\) −12.0000 −0.438178
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −8.00000 −0.291730
\(753\) −12.0000 −0.437304
\(754\) 4.00000 0.145671
\(755\) −16.0000 −0.582300
\(756\) −1.00000 −0.0363696
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −36.0000 −1.30758
\(759\) −4.00000 −0.145191
\(760\) −8.00000 −0.290191
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 16.0000 0.579619
\(763\) 14.0000 0.506834
\(764\) −16.0000 −0.578860
\(765\) 12.0000 0.433861
\(766\) 16.0000 0.578103
\(767\) −8.00000 −0.288863
\(768\) −1.00000 −0.0360844
\(769\) −38.0000 −1.37032 −0.685158 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(770\) 8.00000 0.288300
\(771\) −18.0000 −0.648254
\(772\) 2.00000 0.0719816
\(773\) 46.0000 1.65451 0.827253 0.561830i \(-0.189903\pi\)
0.827253 + 0.561830i \(0.189903\pi\)
\(774\) −4.00000 −0.143777
\(775\) 8.00000 0.287368
\(776\) 10.0000 0.358979
\(777\) −6.00000 −0.215249
\(778\) 6.00000 0.215110
\(779\) −24.0000 −0.859889
\(780\) −4.00000 −0.143223
\(781\) 32.0000 1.14505
\(782\) 6.00000 0.214560
\(783\) 2.00000 0.0714742
\(784\) 1.00000 0.0357143
\(785\) −44.0000 −1.57043
\(786\) −12.0000 −0.428026
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) −26.0000 −0.926212
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) −4.00000 −0.142134
\(793\) 20.0000 0.710221
\(794\) −2.00000 −0.0709773
\(795\) 12.0000 0.425596
\(796\) −24.0000 −0.850657
\(797\) 54.0000 1.91278 0.956389 0.292096i \(-0.0943526\pi\)
0.956389 + 0.292096i \(0.0943526\pi\)
\(798\) −4.00000 −0.141598
\(799\) 48.0000 1.69812
\(800\) −1.00000 −0.0353553
\(801\) 2.00000 0.0706665
\(802\) 34.0000 1.20058
\(803\) 24.0000 0.846942
\(804\) −4.00000 −0.141069
\(805\) 2.00000 0.0704907
\(806\) 16.0000 0.563576
\(807\) 2.00000 0.0704033
\(808\) 6.00000 0.211079
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 52.0000 1.82597 0.912983 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) −2.00000 −0.0701862
\(813\) −24.0000 −0.841717
\(814\) −24.0000 −0.841200
\(815\) −8.00000 −0.280228
\(816\) 6.00000 0.210042
\(817\) −16.0000 −0.559769
\(818\) 10.0000 0.349642
\(819\) −2.00000 −0.0698857
\(820\) 12.0000 0.419058
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 6.00000 0.209274
\(823\) 56.0000 1.95204 0.976019 0.217687i \(-0.0698512\pi\)
0.976019 + 0.217687i \(0.0698512\pi\)
\(824\) 8.00000 0.278693
\(825\) −4.00000 −0.139262
\(826\) 4.00000 0.139178
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 24.0000 0.833052
\(831\) −22.0000 −0.763172
\(832\) −2.00000 −0.0693375
\(833\) −6.00000 −0.207888
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 8.00000 0.276520
\(838\) −12.0000 −0.414533
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 2.00000 0.0690066
\(841\) −25.0000 −0.862069
\(842\) −10.0000 −0.344623
\(843\) −10.0000 −0.344418
\(844\) 4.00000 0.137686
\(845\) 18.0000 0.619219
\(846\) −8.00000 −0.275046
\(847\) 5.00000 0.171802
\(848\) 6.00000 0.206041
\(849\) 20.0000 0.686398
\(850\) 6.00000 0.205798
\(851\) −6.00000 −0.205677
\(852\) 8.00000 0.274075
\(853\) 38.0000 1.30110 0.650548 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) −10.0000 −0.342193
\(855\) −8.00000 −0.273594
\(856\) 12.0000 0.410152
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) −8.00000 −0.273115
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 8.00000 0.272798
\(861\) 6.00000 0.204479
\(862\) 0 0
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 4.00000 0.136004
\(866\) 26.0000 0.883516
\(867\) −19.0000 −0.645274
\(868\) −8.00000 −0.271538
\(869\) 0 0
\(870\) −4.00000 −0.135613
\(871\) −8.00000 −0.271070
\(872\) 14.0000 0.474100
\(873\) 10.0000 0.338449
\(874\) −4.00000 −0.135302
\(875\) 12.0000 0.405674
\(876\) 6.00000 0.202721
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) −16.0000 −0.539974
\(879\) 2.00000 0.0674583
\(880\) 8.00000 0.269680
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 1.00000 0.0336718
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 12.0000 0.403604
\(885\) 8.00000 0.268917
\(886\) −20.0000 −0.671913
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) −6.00000 −0.201347
\(889\) −16.0000 −0.536623
\(890\) −4.00000 −0.134080
\(891\) −4.00000 −0.134005
\(892\) 24.0000 0.803579
\(893\) −32.0000 −1.07084
\(894\) −6.00000 −0.200670
\(895\) 24.0000 0.802232
\(896\) 1.00000 0.0334077
\(897\) −2.00000 −0.0667781
\(898\) 2.00000 0.0667409
\(899\) 16.0000 0.533630
\(900\) −1.00000 −0.0333333
\(901\) −36.0000 −1.19933
\(902\) 24.0000 0.799113
\(903\) 4.00000 0.133112
\(904\) 18.0000 0.598671
\(905\) 4.00000 0.132964
\(906\) −8.00000 −0.265782
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) −12.0000 −0.398234
\(909\) 6.00000 0.199007
\(910\) 4.00000 0.132599
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) −4.00000 −0.132453
\(913\) 48.0000 1.58857
\(914\) −22.0000 −0.727695
\(915\) −20.0000 −0.661180
\(916\) 14.0000 0.462573
\(917\) 12.0000 0.396275
\(918\) 6.00000 0.198030
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 2.00000 0.0659380
\(921\) −12.0000 −0.395413
\(922\) 30.0000 0.987997
\(923\) 16.0000 0.526646
\(924\) 4.00000 0.131590
\(925\) −6.00000 −0.197279
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) −2.00000 −0.0656532
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) −16.0000 −0.524661
\(931\) 4.00000 0.131095
\(932\) 10.0000 0.327561
\(933\) 16.0000 0.523816
\(934\) 4.00000 0.130884
\(935\) −48.0000 −1.56977
\(936\) −2.00000 −0.0653720
\(937\) 18.0000 0.588034 0.294017 0.955800i \(-0.405008\pi\)
0.294017 + 0.955800i \(0.405008\pi\)
\(938\) 4.00000 0.130605
\(939\) 14.0000 0.456873
\(940\) 16.0000 0.521862
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) −22.0000 −0.716799
\(943\) 6.00000 0.195387
\(944\) 4.00000 0.130189
\(945\) 2.00000 0.0650600
\(946\) 16.0000 0.520205
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) −4.00000 −0.129777
\(951\) 2.00000 0.0648544
\(952\) −6.00000 −0.194461
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 6.00000 0.194257
\(955\) 32.0000 1.03550
\(956\) −16.0000 −0.517477
\(957\) −8.00000 −0.258603
\(958\) −32.0000 −1.03387
\(959\) −6.00000 −0.193750
\(960\) 2.00000 0.0645497
\(961\) 33.0000 1.06452
\(962\) −12.0000 −0.386896
\(963\) 12.0000 0.386695
\(964\) −22.0000 −0.708572
\(965\) −4.00000 −0.128765
\(966\) 1.00000 0.0321745
\(967\) 40.0000 1.28631 0.643157 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 5.00000 0.160706
\(969\) 24.0000 0.770991
\(970\) −20.0000 −0.642161
\(971\) 44.0000 1.41203 0.706014 0.708198i \(-0.250492\pi\)
0.706014 + 0.708198i \(0.250492\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −12.0000 −0.384702
\(974\) 8.00000 0.256337
\(975\) −2.00000 −0.0640513
\(976\) −10.0000 −0.320092
\(977\) −62.0000 −1.98356 −0.991778 0.127971i \(-0.959153\pi\)
−0.991778 + 0.127971i \(0.959153\pi\)
\(978\) −4.00000 −0.127906
\(979\) −8.00000 −0.255681
\(980\) −2.00000 −0.0638877
\(981\) 14.0000 0.446986
\(982\) −4.00000 −0.127645
\(983\) 40.0000 1.27580 0.637901 0.770118i \(-0.279803\pi\)
0.637901 + 0.770118i \(0.279803\pi\)
\(984\) 6.00000 0.191273
\(985\) 52.0000 1.65686
\(986\) 12.0000 0.382158
\(987\) 8.00000 0.254643
\(988\) −8.00000 −0.254514
\(989\) 4.00000 0.127193
\(990\) 8.00000 0.254257
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) −8.00000 −0.254000
\(993\) 4.00000 0.126936
\(994\) −8.00000 −0.253745
\(995\) 48.0000 1.52170
\(996\) 12.0000 0.380235
\(997\) −58.0000 −1.83688 −0.918439 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) −28.0000 −0.886325
\(999\) −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 966.2.a.g.1.1 1
3.2 odd 2 2898.2.a.i.1.1 1
4.3 odd 2 7728.2.a.o.1.1 1
7.6 odd 2 6762.2.a.bl.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.g.1.1 1 1.1 even 1 trivial
2898.2.a.i.1.1 1 3.2 odd 2
6762.2.a.bl.1.1 1 7.6 odd 2
7728.2.a.o.1.1 1 4.3 odd 2