Properties

Label 1110.2.a.a.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.86339462436\)
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 \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +8.00000 q^{19} -1.00000 q^{20} +4.00000 q^{21} -4.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} -8.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -4.00000 q^{42} +4.00000 q^{43} +4.00000 q^{44} -1.00000 q^{45} -12.0000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{55} +4.00000 q^{56} -8.00000 q^{57} +6.00000 q^{58} +1.00000 q^{60} -6.00000 q^{61} +8.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} +12.0000 q^{67} -2.00000 q^{68} -4.00000 q^{70} -1.00000 q^{72} -6.00000 q^{73} +1.00000 q^{74} -1.00000 q^{75} +8.00000 q^{76} -16.0000 q^{77} +2.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} +4.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} +6.00000 q^{87} -4.00000 q^{88} -14.0000 q^{89} +1.00000 q^{90} -8.00000 q^{91} +8.00000 q^{93} +12.0000 q^{94} -8.00000 q^{95} +1.00000 q^{96} +6.00000 q^{97} -9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 4.00000 1.06904
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) −1.00000 −0.223607
\(21\) 4.00000 0.872872
\(22\) −4.00000 −0.852803
\(23\) 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
\(28\) −4.00000 −0.755929
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −1.00000 −0.182574
\(31\) −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
\(34\) 2.00000 0.342997
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) −8.00000 −1.29777
\(39\) −2.00000 −0.320256
\(40\) 1.00000 0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −4.00000 −0.617213
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 2.00000 0.280056
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) −4.00000 −0.539360
\(56\) 4.00000 0.534522
\(57\) −8.00000 −1.05963
\(58\) 6.00000 0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 1.00000 0.129099
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 8.00000 1.01600
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 4.00000 0.492366
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 1.00000 0.116248
\(75\) −1.00000 −0.115470
\(76\) 8.00000 0.917663
\(77\) −16.0000 −1.82337
\(78\) 2.00000 0.226455
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 4.00000 0.436436
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) 6.00000 0.643268
\(88\) −4.00000 −0.426401
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 1.00000 0.105409
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 12.0000 1.23771
\(95\) −8.00000 −0.820783
\(96\) 1.00000 0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) −9.00000 −0.909137
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −2.00000 −0.198030
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) −4.00000 −0.390360
\(106\) 6.00000 0.582772
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 4.00000 0.381385
\(111\) 1.00000 0.0949158
\(112\) −4.00000 −0.377964
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) −1.00000 −0.0912871
\(121\) 5.00000 0.454545
\(122\) 6.00000 0.543214
\(123\) 6.00000 0.541002
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) 4.00000 0.356348
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 2.00000 0.175412
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) −4.00000 −0.348155
\(133\) −32.0000 −2.77475
\(134\) −12.0000 −1.03664
\(135\) 1.00000 0.0860663
\(136\) 2.00000 0.171499
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 4.00000 0.338062
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) 6.00000 0.496564
\(147\) −9.00000 −0.742307
\(148\) −1.00000 −0.0821995
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 1.00000 0.0816497
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −8.00000 −0.648886
\(153\) −2.00000 −0.161690
\(154\) 16.0000 1.28932
\(155\) 8.00000 0.642575
\(156\) −2.00000 −0.160128
\(157\) −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
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −6.00000 −0.468521
\(165\) 4.00000 0.311400
\(166\) −4.00000 −0.310460
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −4.00000 −0.308607
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 8.00000 0.611775
\(172\) 4.00000 0.304997
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) −6.00000 −0.454859
\(175\) −4.00000 −0.302372
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) 14.0000 1.04934
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 8.00000 0.592999
\(183\) 6.00000 0.443533
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) −8.00000 −0.586588
\(187\) −8.00000 −0.585018
\(188\) −12.0000 −0.875190
\(189\) 4.00000 0.290957
\(190\) 8.00000 0.580381
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −6.00000 −0.430775
\(195\) 2.00000 0.143223
\(196\) 9.00000 0.642857
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −4.00000 −0.284268
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −12.0000 −0.846415
\(202\) 10.0000 0.703598
\(203\) 24.0000 1.68447
\(204\) 2.00000 0.140028
\(205\) 6.00000 0.419058
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 32.0000 2.21349
\(210\) 4.00000 0.276026
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −4.00000 −0.272798
\(216\) 1.00000 0.0680414
\(217\) 32.0000 2.17230
\(218\) −2.00000 −0.135457
\(219\) 6.00000 0.405442
\(220\) −4.00000 −0.269680
\(221\) −4.00000 −0.269069
\(222\) −1.00000 −0.0671156
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 4.00000 0.267261
\(225\) 1.00000 0.0666667
\(226\) 18.0000 1.19734
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −8.00000 −0.529813
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) 6.00000 0.393919
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) 12.0000 0.782794
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) −8.00000 −0.518563
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 1.00000 0.0645497
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) −9.00000 −0.574989
\(246\) −6.00000 −0.382546
\(247\) 16.0000 1.01806
\(248\) 8.00000 0.508001
\(249\) −4.00000 −0.253490
\(250\) 1.00000 0.0632456
\(251\) −16.0000 −1.00991 −0.504956 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) −4.00000 −0.251976
\(253\) 0 0
\(254\) −4.00000 −0.250982
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 4.00000 0.249029
\(259\) 4.00000 0.248548
\(260\) −2.00000 −0.124035
\(261\) −6.00000 −0.371391
\(262\) 16.0000 0.988483
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 4.00000 0.246183
\(265\) 6.00000 0.368577
\(266\) 32.0000 1.96205
\(267\) 14.0000 0.856786
\(268\) 12.0000 0.733017
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) −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
\(273\) 8.00000 0.484182
\(274\) 6.00000 0.362473
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 4.00000 0.239904
\(279\) −8.00000 −0.478947
\(280\) −4.00000 −0.239046
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) −12.0000 −0.714590
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 0 0
\(285\) 8.00000 0.473879
\(286\) −8.00000 −0.473050
\(287\) 24.0000 1.41668
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) −6.00000 −0.351726
\(292\) −6.00000 −0.351123
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) −4.00000 −0.232104
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −16.0000 −0.922225
\(302\) 16.0000 0.920697
\(303\) 10.0000 0.574485
\(304\) 8.00000 0.458831
\(305\) 6.00000 0.343559
\(306\) 2.00000 0.114332
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) −16.0000 −0.911685
\(309\) −8.00000 −0.455104
\(310\) −8.00000 −0.454369
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 2.00000 0.113228
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 22.0000 1.24153
\(315\) 4.00000 0.225374
\(316\) −8.00000 −0.450035
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) −6.00000 −0.336463
\(319\) −24.0000 −1.34374
\(320\) −1.00000 −0.0559017
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 12.0000 0.664619
\(327\) −2.00000 −0.110600
\(328\) 6.00000 0.331295
\(329\) 48.0000 2.64633
\(330\) −4.00000 −0.220193
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 4.00000 0.219529
\(333\) −1.00000 −0.0547997
\(334\) 8.00000 0.437741
\(335\) −12.0000 −0.655630
\(336\) 4.00000 0.218218
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 9.00000 0.489535
\(339\) 18.0000 0.977626
\(340\) 2.00000 0.108465
\(341\) −32.0000 −1.73290
\(342\) −8.00000 −0.432590
\(343\) −8.00000 −0.431959
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 6.00000 0.321634
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 4.00000 0.213809
\(351\) −2.00000 −0.106752
\(352\) −4.00000 −0.213201
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) −8.00000 −0.423405
\(358\) −24.0000 −1.26844
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 1.00000 0.0527046
\(361\) 45.0000 2.36842
\(362\) −14.0000 −0.735824
\(363\) −5.00000 −0.262432
\(364\) −8.00000 −0.419314
\(365\) 6.00000 0.314054
\(366\) −6.00000 −0.313625
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) −1.00000 −0.0519875
\(371\) 24.0000 1.24602
\(372\) 8.00000 0.414781
\(373\) 34.0000 1.76045 0.880227 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) 8.00000 0.413670
\(375\) 1.00000 0.0516398
\(376\) 12.0000 0.618853
\(377\) −12.0000 −0.618031
\(378\) −4.00000 −0.205738
\(379\) −36.0000 −1.84920 −0.924598 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) −8.00000 −0.410391
\(381\) −4.00000 −0.204926
\(382\) −24.0000 −1.22795
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 1.00000 0.0510310
\(385\) 16.0000 0.815436
\(386\) −14.0000 −0.712581
\(387\) 4.00000 0.203331
\(388\) 6.00000 0.304604
\(389\) 34.0000 1.72387 0.861934 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) −9.00000 −0.454569
\(393\) 16.0000 0.807093
\(394\) −2.00000 −0.100759
\(395\) 8.00000 0.402524
\(396\) 4.00000 0.201008
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −16.0000 −0.802008
\(399\) 32.0000 1.60200
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 12.0000 0.598506
\(403\) −16.0000 −0.797017
\(404\) −10.0000 −0.497519
\(405\) −1.00000 −0.0496904
\(406\) −24.0000 −1.19110
\(407\) −4.00000 −0.198273
\(408\) −2.00000 −0.0990148
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) −6.00000 −0.296319
\(411\) 6.00000 0.295958
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) −2.00000 −0.0980581
\(417\) 4.00000 0.195881
\(418\) −32.0000 −1.56517
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) −4.00000 −0.195180
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 20.0000 0.973585
\(423\) −12.0000 −0.583460
\(424\) 6.00000 0.291386
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 24.0000 1.16144
\(428\) 4.00000 0.193347
\(429\) −8.00000 −0.386244
\(430\) 4.00000 0.192897
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) −32.0000 −1.53605
\(435\) −6.00000 −0.287678
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 4.00000 0.190693
\(441\) 9.00000 0.428571
\(442\) 4.00000 0.190261
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 1.00000 0.0474579
\(445\) 14.0000 0.663664
\(446\) 12.0000 0.568216
\(447\) −6.00000 −0.283790
\(448\) −4.00000 −0.188982
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −24.0000 −1.13012
\(452\) −18.0000 −0.846649
\(453\) 16.0000 0.751746
\(454\) −12.0000 −0.563188
\(455\) 8.00000 0.375046
\(456\) 8.00000 0.374634
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) −14.0000 −0.654177
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −38.0000 −1.76984 −0.884918 0.465746i \(-0.845786\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) −16.0000 −0.744387
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −6.00000 −0.278543
\(465\) −8.00000 −0.370991
\(466\) 6.00000 0.277945
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 2.00000 0.0924500
\(469\) −48.0000 −2.21643
\(470\) −12.0000 −0.553519
\(471\) 22.0000 1.01371
\(472\) 0 0
\(473\) 16.0000 0.735681
\(474\) −8.00000 −0.367452
\(475\) 8.00000 0.367065
\(476\) 8.00000 0.366679
\(477\) −6.00000 −0.274721
\(478\) 8.00000 0.365911
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −2.00000 −0.0911922
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −6.00000 −0.272446
\(486\) 1.00000 0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 6.00000 0.271607
\(489\) 12.0000 0.542659
\(490\) 9.00000 0.406579
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 6.00000 0.270501
\(493\) 12.0000 0.540453
\(494\) −16.0000 −0.719874
\(495\) −4.00000 −0.179787
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 8.00000 0.357414
\(502\) 16.0000 0.714115
\(503\) 8.00000 0.356702 0.178351 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) 4.00000 0.178174
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 4.00000 0.177471
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 2.00000 0.0885615
\(511\) 24.0000 1.06170
\(512\) −1.00000 −0.0441942
\(513\) −8.00000 −0.353209
\(514\) −14.0000 −0.617514
\(515\) −8.00000 −0.352522
\(516\) −4.00000 −0.176090
\(517\) −48.0000 −2.11104
\(518\) −4.00000 −0.175750
\(519\) −2.00000 −0.0877903
\(520\) 2.00000 0.0877058
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 6.00000 0.262613
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −16.0000 −0.698963
\(525\) 4.00000 0.174574
\(526\) 4.00000 0.174408
\(527\) 16.0000 0.696971
\(528\) −4.00000 −0.174078
\(529\) −23.0000 −1.00000
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) −32.0000 −1.38738
\(533\) −12.0000 −0.519778
\(534\) −14.0000 −0.605839
\(535\) −4.00000 −0.172935
\(536\) −12.0000 −0.518321
\(537\) −24.0000 −1.03568
\(538\) −30.0000 −1.29339
\(539\) 36.0000 1.55063
\(540\) 1.00000 0.0430331
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) −8.00000 −0.343629
\(543\) −14.0000 −0.600798
\(544\) 2.00000 0.0857493
\(545\) −2.00000 −0.0856706
\(546\) −8.00000 −0.342368
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −6.00000 −0.256307
\(549\) −6.00000 −0.256074
\(550\) −4.00000 −0.170561
\(551\) −48.0000 −2.04487
\(552\) 0 0
\(553\) 32.0000 1.36078
\(554\) 22.0000 0.934690
\(555\) −1.00000 −0.0424476
\(556\) −4.00000 −0.169638
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 8.00000 0.338667
\(559\) 8.00000 0.338364
\(560\) 4.00000 0.169031
\(561\) 8.00000 0.337760
\(562\) 14.0000 0.590554
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 12.0000 0.505291
\(565\) 18.0000 0.757266
\(566\) −12.0000 −0.504398
\(567\) −4.00000 −0.167984
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) −8.00000 −0.335083
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 8.00000 0.334497
\(573\) −24.0000 −1.00261
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 13.0000 0.540729
\(579\) −14.0000 −0.581820
\(580\) 6.00000 0.249136
\(581\) −16.0000 −0.663792
\(582\) 6.00000 0.248708
\(583\) −24.0000 −0.993978
\(584\) 6.00000 0.248282
\(585\) −2.00000 −0.0826898
\(586\) 14.0000 0.578335
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −9.00000 −0.371154
\(589\) −64.0000 −2.63707
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) −1.00000 −0.0410997
\(593\) 2.00000 0.0821302 0.0410651 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 4.00000 0.164122
\(595\) −8.00000 −0.327968
\(596\) 6.00000 0.245770
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 1.00000 0.0408248
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 16.0000 0.652111
\(603\) 12.0000 0.488678
\(604\) −16.0000 −0.651031
\(605\) −5.00000 −0.203279
\(606\) −10.0000 −0.406222
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) −8.00000 −0.324443
\(609\) −24.0000 −0.972529
\(610\) −6.00000 −0.242933
\(611\) −24.0000 −0.970936
\(612\) −2.00000 −0.0808452
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) −20.0000 −0.807134
\(615\) −6.00000 −0.241943
\(616\) 16.0000 0.644658
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 8.00000 0.321807
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) 56.0000 2.24359
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) −22.0000 −0.879297
\(627\) −32.0000 −1.27796
\(628\) −22.0000 −0.877896
\(629\) 2.00000 0.0797452
\(630\) −4.00000 −0.159364
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 8.00000 0.318223
\(633\) 20.0000 0.794929
\(634\) 30.0000 1.19145
\(635\) −4.00000 −0.158735
\(636\) 6.00000 0.237915
\(637\) 18.0000 0.713186
\(638\) 24.0000 0.950169
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 4.00000 0.157867
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) 16.0000 0.629512
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) −32.0000 −1.25418
\(652\) −12.0000 −0.469956
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 2.00000 0.0782062
\(655\) 16.0000 0.625172
\(656\) −6.00000 −0.234261
\(657\) −6.00000 −0.234082
\(658\) −48.0000 −1.87123
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 4.00000 0.155700
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 8.00000 0.310929
\(663\) 4.00000 0.155347
\(664\) −4.00000 −0.155230
\(665\) 32.0000 1.24091
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) −8.00000 −0.309529
\(669\) 12.0000 0.463947
\(670\) 12.0000 0.463600
\(671\) −24.0000 −0.926510
\(672\) −4.00000 −0.154303
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) −2.00000 −0.0770371
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) −18.0000 −0.691286
\(679\) −24.0000 −0.921035
\(680\) −2.00000 −0.0766965
\(681\) −12.0000 −0.459841
\(682\) 32.0000 1.22534
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 8.00000 0.305888
\(685\) 6.00000 0.229248
\(686\) 8.00000 0.305441
\(687\) −14.0000 −0.534133
\(688\) 4.00000 0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 2.00000 0.0760286
\(693\) −16.0000 −0.607790
\(694\) 12.0000 0.455514
\(695\) 4.00000 0.151729
\(696\) −6.00000 −0.227429
\(697\) 12.0000 0.454532
\(698\) 2.00000 0.0757011
\(699\) 6.00000 0.226941
\(700\) −4.00000 −0.151186
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 2.00000 0.0754851
\(703\) −8.00000 −0.301726
\(704\) 4.00000 0.150756
\(705\) −12.0000 −0.451946
\(706\) 26.0000 0.978523
\(707\) 40.0000 1.50435
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 14.0000 0.524672
\(713\) 0 0
\(714\) 8.00000 0.299392
\(715\) −8.00000 −0.299183
\(716\) 24.0000 0.896922
\(717\) 8.00000 0.298765
\(718\) −24.0000 −0.895672
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −32.0000 −1.19174
\(722\) −45.0000 −1.67473
\(723\) 14.0000 0.520666
\(724\) 14.0000 0.520306
\(725\) −6.00000 −0.222834
\(726\) 5.00000 0.185567
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 8.00000 0.296500
\(729\) 1.00000 0.0370370
\(730\) −6.00000 −0.222070
\(731\) −8.00000 −0.295891
\(732\) 6.00000 0.221766
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 28.0000 1.03350
\(735\) 9.00000 0.331970
\(736\) 0 0
\(737\) 48.0000 1.76810
\(738\) 6.00000 0.220863
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 1.00000 0.0367607
\(741\) −16.0000 −0.587775
\(742\) −24.0000 −0.881068
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) −8.00000 −0.293294
\(745\) −6.00000 −0.219823
\(746\) −34.0000 −1.24483
\(747\) 4.00000 0.146352
\(748\) −8.00000 −0.292509
\(749\) −16.0000 −0.584627
\(750\) −1.00000 −0.0365148
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) −12.0000 −0.437595
\(753\) 16.0000 0.583072
\(754\) 12.0000 0.437014
\(755\) 16.0000 0.582300
\(756\) 4.00000 0.145479
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 36.0000 1.30758
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 4.00000 0.144905
\(763\) −8.00000 −0.289619
\(764\) 24.0000 0.868290
\(765\) 2.00000 0.0723102
\(766\) −24.0000 −0.867155
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) −16.0000 −0.576600
\(771\) −14.0000 −0.504198
\(772\) 14.0000 0.503871
\(773\) −54.0000 −1.94225 −0.971123 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) −4.00000 −0.143777
\(775\) −8.00000 −0.287368
\(776\) −6.00000 −0.215387
\(777\) −4.00000 −0.143499
\(778\) −34.0000 −1.21896
\(779\) −48.0000 −1.71978
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 9.00000 0.321429
\(785\) 22.0000 0.785214
\(786\) −16.0000 −0.570701
\(787\) −52.0000 −1.85360 −0.926800 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) 2.00000 0.0712470
\(789\) 4.00000 0.142404
\(790\) −8.00000 −0.284627
\(791\) 72.0000 2.56003
\(792\) −4.00000 −0.142134
\(793\) −12.0000 −0.426132
\(794\) 14.0000 0.496841
\(795\) −6.00000 −0.212798
\(796\) 16.0000 0.567105
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) −32.0000 −1.13279
\(799\) 24.0000 0.849059
\(800\) −1.00000 −0.0353553
\(801\) −14.0000 −0.494666
\(802\) 6.00000 0.211867
\(803\) −24.0000 −0.846942
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) −30.0000 −1.05605
\(808\) 10.0000 0.351799
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 1.00000 0.0351364
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 24.0000 0.842235
\(813\) −8.00000 −0.280572
\(814\) 4.00000 0.140200
\(815\) 12.0000 0.420342
\(816\) 2.00000 0.0700140
\(817\) 32.0000 1.11954
\(818\) −2.00000 −0.0699284
\(819\) −8.00000 −0.279543
\(820\) 6.00000 0.209529
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −6.00000 −0.209274
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) −8.00000 −0.278693
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) −44.0000 −1.53003 −0.765015 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 0 0
\(829\) −46.0000 −1.59765 −0.798823 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 4.00000 0.138842
\(831\) 22.0000 0.763172
\(832\) 2.00000 0.0693375
\(833\) −18.0000 −0.623663
\(834\) −4.00000 −0.138509
\(835\) 8.00000 0.276851
\(836\) 32.0000 1.10674
\(837\) 8.00000 0.276520
\(838\) 12.0000 0.414533
\(839\) −56.0000 −1.93333 −0.966667 0.256036i \(-0.917584\pi\)
−0.966667 + 0.256036i \(0.917584\pi\)
\(840\) 4.00000 0.138013
\(841\) 7.00000 0.241379
\(842\) 22.0000 0.758170
\(843\) 14.0000 0.482186
\(844\) −20.0000 −0.688428
\(845\) 9.00000 0.309609
\(846\) 12.0000 0.412568
\(847\) −20.0000 −0.687208
\(848\) −6.00000 −0.206041
\(849\) −12.0000 −0.411839
\(850\) 2.00000 0.0685994
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) −24.0000 −0.821263
\(855\) −8.00000 −0.273594
\(856\) −4.00000 −0.136717
\(857\) 46.0000 1.57133 0.785665 0.618652i \(-0.212321\pi\)
0.785665 + 0.618652i \(0.212321\pi\)
\(858\) 8.00000 0.273115
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) −4.00000 −0.136399
\(861\) −24.0000 −0.817918
\(862\) −24.0000 −0.817443
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 1.00000 0.0340207
\(865\) −2.00000 −0.0680020
\(866\) 30.0000 1.01944
\(867\) 13.0000 0.441503
\(868\) 32.0000 1.08615
\(869\) −32.0000 −1.08553
\(870\) 6.00000 0.203419
\(871\) 24.0000 0.813209
\(872\) −2.00000 −0.0677285
\(873\) 6.00000 0.203069
\(874\) 0 0
\(875\) 4.00000 0.135225
\(876\) 6.00000 0.202721
\(877\) 58.0000 1.95852 0.979260 0.202606i \(-0.0649409\pi\)
0.979260 + 0.202606i \(0.0649409\pi\)
\(878\) −16.0000 −0.539974
\(879\) 14.0000 0.472208
\(880\) −4.00000 −0.134840
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) −9.00000 −0.303046
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) −52.0000 −1.74599 −0.872995 0.487730i \(-0.837825\pi\)
−0.872995 + 0.487730i \(0.837825\pi\)
\(888\) −1.00000 −0.0335578
\(889\) −16.0000 −0.536623
\(890\) −14.0000 −0.469281
\(891\) 4.00000 0.134005
\(892\) −12.0000 −0.401790
\(893\) −96.0000 −3.21252
\(894\) 6.00000 0.200670
\(895\) −24.0000 −0.802232
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) 22.0000 0.734150
\(899\) 48.0000 1.60089
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) 24.0000 0.799113
\(903\) 16.0000 0.532447
\(904\) 18.0000 0.598671
\(905\) −14.0000 −0.465376
\(906\) −16.0000 −0.531564
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 12.0000 0.398234
\(909\) −10.0000 −0.331679
\(910\) −8.00000 −0.265197
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) −8.00000 −0.264906
\(913\) 16.0000 0.529523
\(914\) −14.0000 −0.463079
\(915\) −6.00000 −0.198354
\(916\) 14.0000 0.462573
\(917\) 64.0000 2.11347
\(918\) −2.00000 −0.0660098
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 38.0000 1.25146
\(923\) 0 0
\(924\) 16.0000 0.526361
\(925\) −1.00000 −0.0328798
\(926\) 8.00000 0.262896
\(927\) 8.00000 0.262754
\(928\) 6.00000 0.196960
\(929\) −46.0000 −1.50921 −0.754606 0.656179i \(-0.772172\pi\)
−0.754606 + 0.656179i \(0.772172\pi\)
\(930\) 8.00000 0.262330
\(931\) 72.0000 2.35970
\(932\) −6.00000 −0.196537
\(933\) 24.0000 0.785725
\(934\) 12.0000 0.392652
\(935\) 8.00000 0.261628
\(936\) −2.00000 −0.0653720
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 48.0000 1.56726
\(939\) −22.0000 −0.717943
\(940\) 12.0000 0.391397
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) −22.0000 −0.716799
\(943\) 0 0
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) −16.0000 −0.520205
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 8.00000 0.259828
\(949\) −12.0000 −0.389536
\(950\) −8.00000 −0.259554
\(951\) 30.0000 0.972817
\(952\) −8.00000 −0.259281
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 6.00000 0.194257
\(955\) −24.0000 −0.776622
\(956\) −8.00000 −0.258738
\(957\) 24.0000 0.775810
\(958\) −24.0000 −0.775405
\(959\) 24.0000 0.775000
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) 2.00000 0.0644826
\(963\) 4.00000 0.128898
\(964\) −14.0000 −0.450910
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 48.0000 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) −5.00000 −0.160706
\(969\) 16.0000 0.513994
\(970\) 6.00000 0.192648
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 16.0000 0.512936
\(974\) −8.00000 −0.256337
\(975\) −2.00000 −0.0640513
\(976\) −6.00000 −0.192055
\(977\) −50.0000 −1.59964 −0.799821 0.600239i \(-0.795072\pi\)
−0.799821 + 0.600239i \(0.795072\pi\)
\(978\) −12.0000 −0.383718
\(979\) −56.0000 −1.78977
\(980\) −9.00000 −0.287494
\(981\) 2.00000 0.0638551
\(982\) 12.0000 0.382935
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) −6.00000 −0.191273
\(985\) −2.00000 −0.0637253
\(986\) −12.0000 −0.382158
\(987\) −48.0000 −1.52786
\(988\) 16.0000 0.509028
\(989\) 0 0
\(990\) 4.00000 0.127128
\(991\) −48.0000 −1.52477 −0.762385 0.647124i \(-0.775972\pi\)
−0.762385 + 0.647124i \(0.775972\pi\)
\(992\) 8.00000 0.254000
\(993\) 8.00000 0.253872
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) −4.00000 −0.126745
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) −32.0000 −1.01294
\(999\) 1.00000 0.0316386
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 1110.2.a.a.1.1 1
3.2 odd 2 3330.2.a.s.1.1 1
4.3 odd 2 8880.2.a.w.1.1 1
5.4 even 2 5550.2.a.br.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.a.1.1 1 1.1 even 1 trivial
3330.2.a.s.1.1 1 3.2 odd 2
5550.2.a.br.1.1 1 5.4 even 2
8880.2.a.w.1.1 1 4.3 odd 2