Properties

Label 1110.2.a.l.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -5.00000 q^{19} +1.00000 q^{20} -3.00000 q^{21} +1.00000 q^{22} +7.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} -2.00000 q^{29} -1.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} -5.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} +4.00000 q^{41} -3.00000 q^{42} -12.0000 q^{43} +1.00000 q^{44} +1.00000 q^{45} +7.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} +1.00000 q^{51} +1.00000 q^{52} -9.00000 q^{53} -1.00000 q^{54} +1.00000 q^{55} +3.00000 q^{56} +5.00000 q^{57} -2.00000 q^{58} +10.0000 q^{59} -1.00000 q^{60} +14.0000 q^{61} +2.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} -1.00000 q^{66} +2.00000 q^{67} -1.00000 q^{68} -7.00000 q^{69} +3.00000 q^{70} +2.00000 q^{71} +1.00000 q^{72} -9.00000 q^{73} +1.00000 q^{74} -1.00000 q^{75} -5.00000 q^{76} +3.00000 q^{77} -1.00000 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} -1.00000 q^{83} -3.00000 q^{84} -1.00000 q^{85} -12.0000 q^{86} +2.00000 q^{87} +1.00000 q^{88} +1.00000 q^{89} +1.00000 q^{90} +3.00000 q^{91} +7.00000 q^{92} -2.00000 q^{93} +12.0000 q^{94} -5.00000 q^{95} -1.00000 q^{96} -8.00000 q^{97} +2.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 3.00000 0.801784
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 1.00000 0.235702
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 1.00000 0.223607
\(21\) −3.00000 −0.654654
\(22\) 1.00000 0.213201
\(23\) 7.00000 1.45960 0.729800 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 3.00000 0.566947
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −1.00000 −0.182574
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) 3.00000 0.507093
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) −5.00000 −0.811107
\(39\) −1.00000 −0.160128
\(40\) 1.00000 0.158114
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) −3.00000 −0.462910
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 1.00000 0.150756
\(45\) 1.00000 0.149071
\(46\) 7.00000 1.03209
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) 1.00000 0.141421
\(51\) 1.00000 0.140028
\(52\) 1.00000 0.138675
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.00000 0.134840
\(56\) 3.00000 0.400892
\(57\) 5.00000 0.662266
\(58\) −2.00000 −0.262613
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) −1.00000 −0.129099
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 2.00000 0.254000
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) −1.00000 −0.123091
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −1.00000 −0.121268
\(69\) −7.00000 −0.842701
\(70\) 3.00000 0.358569
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 1.00000 0.117851
\(73\) −9.00000 −1.05337 −0.526685 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) 1.00000 0.116248
\(75\) −1.00000 −0.115470
\(76\) −5.00000 −0.573539
\(77\) 3.00000 0.341882
\(78\) −1.00000 −0.113228
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) −1.00000 −0.109764 −0.0548821 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) −3.00000 −0.327327
\(85\) −1.00000 −0.108465
\(86\) −12.0000 −1.29399
\(87\) 2.00000 0.214423
\(88\) 1.00000 0.106600
\(89\) 1.00000 0.106000 0.0529999 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 1.00000 0.105409
\(91\) 3.00000 0.314485
\(92\) 7.00000 0.729800
\(93\) −2.00000 −0.207390
\(94\) 12.0000 1.23771
\(95\) −5.00000 −0.512989
\(96\) −1.00000 −0.102062
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 2.00000 0.202031
\(99\) 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 \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 1.00000 0.0980581
\(105\) −3.00000 −0.292770
\(106\) −9.00000 −0.874157
\(107\) −19.0000 −1.83680 −0.918400 0.395654i \(-0.870518\pi\)
−0.918400 + 0.395654i \(0.870518\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −19.0000 −1.81987 −0.909935 0.414751i \(-0.863869\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 1.00000 0.0953463
\(111\) −1.00000 −0.0949158
\(112\) 3.00000 0.283473
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 5.00000 0.468293
\(115\) 7.00000 0.652753
\(116\) −2.00000 −0.185695
\(117\) 1.00000 0.0924500
\(118\) 10.0000 0.920575
\(119\) −3.00000 −0.275010
\(120\) −1.00000 −0.0912871
\(121\) −10.0000 −0.909091
\(122\) 14.0000 1.26750
\(123\) −4.00000 −0.360668
\(124\) 2.00000 0.179605
\(125\) 1.00000 0.0894427
\(126\) 3.00000 0.267261
\(127\) 13.0000 1.15356 0.576782 0.816898i \(-0.304308\pi\)
0.576782 + 0.816898i \(0.304308\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.0000 1.05654
\(130\) 1.00000 0.0877058
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −15.0000 −1.30066
\(134\) 2.00000 0.172774
\(135\) −1.00000 −0.0860663
\(136\) −1.00000 −0.0857493
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) −7.00000 −0.595880
\(139\) −6.00000 −0.508913 −0.254457 0.967084i \(-0.581897\pi\)
−0.254457 + 0.967084i \(0.581897\pi\)
\(140\) 3.00000 0.253546
\(141\) −12.0000 −1.01058
\(142\) 2.00000 0.167836
\(143\) 1.00000 0.0836242
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) −9.00000 −0.744845
\(147\) −2.00000 −0.164957
\(148\) 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
\(151\) −17.0000 −1.38344 −0.691720 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) −5.00000 −0.405554
\(153\) −1.00000 −0.0808452
\(154\) 3.00000 0.241747
\(155\) 2.00000 0.160644
\(156\) −1.00000 −0.0800641
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) 4.00000 0.318223
\(159\) 9.00000 0.713746
\(160\) 1.00000 0.0790569
\(161\) 21.0000 1.65503
\(162\) 1.00000 0.0785674
\(163\) −9.00000 −0.704934 −0.352467 0.935824i \(-0.614657\pi\)
−0.352467 + 0.935824i \(0.614657\pi\)
\(164\) 4.00000 0.312348
\(165\) −1.00000 −0.0778499
\(166\) −1.00000 −0.0776151
\(167\) −7.00000 −0.541676 −0.270838 0.962625i \(-0.587301\pi\)
−0.270838 + 0.962625i \(0.587301\pi\)
\(168\) −3.00000 −0.231455
\(169\) −12.0000 −0.923077
\(170\) −1.00000 −0.0766965
\(171\) −5.00000 −0.382360
\(172\) −12.0000 −0.914991
\(173\) −23.0000 −1.74866 −0.874329 0.485334i \(-0.838698\pi\)
−0.874329 + 0.485334i \(0.838698\pi\)
\(174\) 2.00000 0.151620
\(175\) 3.00000 0.226779
\(176\) 1.00000 0.0753778
\(177\) −10.0000 −0.751646
\(178\) 1.00000 0.0749532
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 1.00000 0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 3.00000 0.222375
\(183\) −14.0000 −1.03491
\(184\) 7.00000 0.516047
\(185\) 1.00000 0.0735215
\(186\) −2.00000 −0.146647
\(187\) −1.00000 −0.0731272
\(188\) 12.0000 0.875190
\(189\) −3.00000 −0.218218
\(190\) −5.00000 −0.362738
\(191\) 5.00000 0.361787 0.180894 0.983503i \(-0.442101\pi\)
0.180894 + 0.983503i \(0.442101\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −8.00000 −0.574367
\(195\) −1.00000 −0.0716115
\(196\) 2.00000 0.142857
\(197\) 13.0000 0.926212 0.463106 0.886303i \(-0.346735\pi\)
0.463106 + 0.886303i \(0.346735\pi\)
\(198\) 1.00000 0.0710669
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 1.00000 0.0707107
\(201\) −2.00000 −0.141069
\(202\) 6.00000 0.422159
\(203\) −6.00000 −0.421117
\(204\) 1.00000 0.0700140
\(205\) 4.00000 0.279372
\(206\) 8.00000 0.557386
\(207\) 7.00000 0.486534
\(208\) 1.00000 0.0693375
\(209\) −5.00000 −0.345857
\(210\) −3.00000 −0.207020
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) −9.00000 −0.618123
\(213\) −2.00000 −0.137038
\(214\) −19.0000 −1.29881
\(215\) −12.0000 −0.818393
\(216\) −1.00000 −0.0680414
\(217\) 6.00000 0.407307
\(218\) −19.0000 −1.28684
\(219\) 9.00000 0.608164
\(220\) 1.00000 0.0674200
\(221\) −1.00000 −0.0672673
\(222\) −1.00000 −0.0671156
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 3.00000 0.200446
\(225\) 1.00000 0.0666667
\(226\) −2.00000 −0.133038
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 5.00000 0.331133
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 7.00000 0.461566
\(231\) −3.00000 −0.197386
\(232\) −2.00000 −0.131306
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 1.00000 0.0653720
\(235\) 12.0000 0.782794
\(236\) 10.0000 0.650945
\(237\) −4.00000 −0.259828
\(238\) −3.00000 −0.194461
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) −10.0000 −0.642824
\(243\) −1.00000 −0.0641500
\(244\) 14.0000 0.896258
\(245\) 2.00000 0.127775
\(246\) −4.00000 −0.255031
\(247\) −5.00000 −0.318142
\(248\) 2.00000 0.127000
\(249\) 1.00000 0.0633724
\(250\) 1.00000 0.0632456
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 3.00000 0.188982
\(253\) 7.00000 0.440086
\(254\) 13.0000 0.815693
\(255\) 1.00000 0.0626224
\(256\) 1.00000 0.0625000
\(257\) −13.0000 −0.810918 −0.405459 0.914113i \(-0.632888\pi\)
−0.405459 + 0.914113i \(0.632888\pi\)
\(258\) 12.0000 0.747087
\(259\) 3.00000 0.186411
\(260\) 1.00000 0.0620174
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −9.00000 −0.552866
\(266\) −15.0000 −0.919709
\(267\) −1.00000 −0.0611990
\(268\) 2.00000 0.122169
\(269\) −5.00000 −0.304855 −0.152428 0.988315i \(-0.548709\pi\)
−0.152428 + 0.988315i \(0.548709\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) −1.00000 −0.0606339
\(273\) −3.00000 −0.181568
\(274\) −10.0000 −0.604122
\(275\) 1.00000 0.0603023
\(276\) −7.00000 −0.421350
\(277\) −17.0000 −1.02143 −0.510716 0.859750i \(-0.670619\pi\)
−0.510716 + 0.859750i \(0.670619\pi\)
\(278\) −6.00000 −0.359856
\(279\) 2.00000 0.119737
\(280\) 3.00000 0.179284
\(281\) −11.0000 −0.656205 −0.328102 0.944642i \(-0.606409\pi\)
−0.328102 + 0.944642i \(0.606409\pi\)
\(282\) −12.0000 −0.714590
\(283\) −21.0000 −1.24832 −0.624160 0.781296i \(-0.714559\pi\)
−0.624160 + 0.781296i \(0.714559\pi\)
\(284\) 2.00000 0.118678
\(285\) 5.00000 0.296174
\(286\) 1.00000 0.0591312
\(287\) 12.0000 0.708338
\(288\) 1.00000 0.0589256
\(289\) −16.0000 −0.941176
\(290\) −2.00000 −0.117444
\(291\) 8.00000 0.468968
\(292\) −9.00000 −0.526685
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) −2.00000 −0.116642
\(295\) 10.0000 0.582223
\(296\) 1.00000 0.0581238
\(297\) −1.00000 −0.0580259
\(298\) 6.00000 0.347571
\(299\) 7.00000 0.404820
\(300\) −1.00000 −0.0577350
\(301\) −36.0000 −2.07501
\(302\) −17.0000 −0.978240
\(303\) −6.00000 −0.344691
\(304\) −5.00000 −0.286770
\(305\) 14.0000 0.801638
\(306\) −1.00000 −0.0571662
\(307\) 18.0000 1.02731 0.513657 0.857996i \(-0.328290\pi\)
0.513657 + 0.857996i \(0.328290\pi\)
\(308\) 3.00000 0.170941
\(309\) −8.00000 −0.455104
\(310\) 2.00000 0.113592
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) −8.00000 −0.451466
\(315\) 3.00000 0.169031
\(316\) 4.00000 0.225018
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 9.00000 0.504695
\(319\) −2.00000 −0.111979
\(320\) 1.00000 0.0559017
\(321\) 19.0000 1.06048
\(322\) 21.0000 1.17028
\(323\) 5.00000 0.278207
\(324\) 1.00000 0.0555556
\(325\) 1.00000 0.0554700
\(326\) −9.00000 −0.498464
\(327\) 19.0000 1.05070
\(328\) 4.00000 0.220863
\(329\) 36.0000 1.98474
\(330\) −1.00000 −0.0550482
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −1.00000 −0.0548821
\(333\) 1.00000 0.0547997
\(334\) −7.00000 −0.383023
\(335\) 2.00000 0.109272
\(336\) −3.00000 −0.163663
\(337\) −1.00000 −0.0544735 −0.0272367 0.999629i \(-0.508671\pi\)
−0.0272367 + 0.999629i \(0.508671\pi\)
\(338\) −12.0000 −0.652714
\(339\) 2.00000 0.108625
\(340\) −1.00000 −0.0542326
\(341\) 2.00000 0.108306
\(342\) −5.00000 −0.270369
\(343\) −15.0000 −0.809924
\(344\) −12.0000 −0.646997
\(345\) −7.00000 −0.376867
\(346\) −23.0000 −1.23649
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 2.00000 0.107211
\(349\) −4.00000 −0.214115 −0.107058 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(350\) 3.00000 0.160357
\(351\) −1.00000 −0.0533761
\(352\) 1.00000 0.0533002
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) −10.0000 −0.531494
\(355\) 2.00000 0.106149
\(356\) 1.00000 0.0529999
\(357\) 3.00000 0.158777
\(358\) 12.0000 0.634220
\(359\) 4.00000 0.211112 0.105556 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 1.00000 0.0527046
\(361\) 6.00000 0.315789
\(362\) 2.00000 0.105118
\(363\) 10.0000 0.524864
\(364\) 3.00000 0.157243
\(365\) −9.00000 −0.471082
\(366\) −14.0000 −0.731792
\(367\) 35.0000 1.82699 0.913493 0.406855i \(-0.133375\pi\)
0.913493 + 0.406855i \(0.133375\pi\)
\(368\) 7.00000 0.364900
\(369\) 4.00000 0.208232
\(370\) 1.00000 0.0519875
\(371\) −27.0000 −1.40177
\(372\) −2.00000 −0.103695
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −1.00000 −0.0517088
\(375\) −1.00000 −0.0516398
\(376\) 12.0000 0.618853
\(377\) −2.00000 −0.103005
\(378\) −3.00000 −0.154303
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) −5.00000 −0.256495
\(381\) −13.0000 −0.666010
\(382\) 5.00000 0.255822
\(383\) 13.0000 0.664269 0.332134 0.943232i \(-0.392231\pi\)
0.332134 + 0.943232i \(0.392231\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 3.00000 0.152894
\(386\) −22.0000 −1.11977
\(387\) −12.0000 −0.609994
\(388\) −8.00000 −0.406138
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) −1.00000 −0.0506370
\(391\) −7.00000 −0.354005
\(392\) 2.00000 0.101015
\(393\) 0 0
\(394\) 13.0000 0.654931
\(395\) 4.00000 0.201262
\(396\) 1.00000 0.0502519
\(397\) −16.0000 −0.803017 −0.401508 0.915855i \(-0.631514\pi\)
−0.401508 + 0.915855i \(0.631514\pi\)
\(398\) 4.00000 0.200502
\(399\) 15.0000 0.750939
\(400\) 1.00000 0.0500000
\(401\) 25.0000 1.24844 0.624220 0.781248i \(-0.285417\pi\)
0.624220 + 0.781248i \(0.285417\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 2.00000 0.0996271
\(404\) 6.00000 0.298511
\(405\) 1.00000 0.0496904
\(406\) −6.00000 −0.297775
\(407\) 1.00000 0.0495682
\(408\) 1.00000 0.0495074
\(409\) −8.00000 −0.395575 −0.197787 0.980245i \(-0.563376\pi\)
−0.197787 + 0.980245i \(0.563376\pi\)
\(410\) 4.00000 0.197546
\(411\) 10.0000 0.493264
\(412\) 8.00000 0.394132
\(413\) 30.0000 1.47620
\(414\) 7.00000 0.344031
\(415\) −1.00000 −0.0490881
\(416\) 1.00000 0.0490290
\(417\) 6.00000 0.293821
\(418\) −5.00000 −0.244558
\(419\) −1.00000 −0.0488532 −0.0244266 0.999702i \(-0.507776\pi\)
−0.0244266 + 0.999702i \(0.507776\pi\)
\(420\) −3.00000 −0.146385
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −10.0000 −0.486792
\(423\) 12.0000 0.583460
\(424\) −9.00000 −0.437079
\(425\) −1.00000 −0.0485071
\(426\) −2.00000 −0.0969003
\(427\) 42.0000 2.03252
\(428\) −19.0000 −0.918400
\(429\) −1.00000 −0.0482805
\(430\) −12.0000 −0.578691
\(431\) 3.00000 0.144505 0.0722525 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −19.0000 −0.913082 −0.456541 0.889702i \(-0.650912\pi\)
−0.456541 + 0.889702i \(0.650912\pi\)
\(434\) 6.00000 0.288009
\(435\) 2.00000 0.0958927
\(436\) −19.0000 −0.909935
\(437\) −35.0000 −1.67428
\(438\) 9.00000 0.430037
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 1.00000 0.0476731
\(441\) 2.00000 0.0952381
\(442\) −1.00000 −0.0475651
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 1.00000 0.0474045
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 3.00000 0.141737
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 1.00000 0.0471405
\(451\) 4.00000 0.188353
\(452\) −2.00000 −0.0940721
\(453\) 17.0000 0.798730
\(454\) −4.00000 −0.187729
\(455\) 3.00000 0.140642
\(456\) 5.00000 0.234146
\(457\) 24.0000 1.12267 0.561336 0.827588i \(-0.310287\pi\)
0.561336 + 0.827588i \(0.310287\pi\)
\(458\) 26.0000 1.21490
\(459\) 1.00000 0.0466760
\(460\) 7.00000 0.326377
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) −3.00000 −0.139573
\(463\) −36.0000 −1.67306 −0.836531 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) −2.00000 −0.0928477
\(465\) −2.00000 −0.0927478
\(466\) −14.0000 −0.648537
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 1.00000 0.0462250
\(469\) 6.00000 0.277054
\(470\) 12.0000 0.553519
\(471\) 8.00000 0.368621
\(472\) 10.0000 0.460287
\(473\) −12.0000 −0.551761
\(474\) −4.00000 −0.183726
\(475\) −5.00000 −0.229416
\(476\) −3.00000 −0.137505
\(477\) −9.00000 −0.412082
\(478\) 20.0000 0.914779
\(479\) −9.00000 −0.411220 −0.205610 0.978634i \(-0.565918\pi\)
−0.205610 + 0.978634i \(0.565918\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 1.00000 0.0455961
\(482\) −14.0000 −0.637683
\(483\) −21.0000 −0.955533
\(484\) −10.0000 −0.454545
\(485\) −8.00000 −0.363261
\(486\) −1.00000 −0.0453609
\(487\) 6.00000 0.271886 0.135943 0.990717i \(-0.456594\pi\)
0.135943 + 0.990717i \(0.456594\pi\)
\(488\) 14.0000 0.633750
\(489\) 9.00000 0.406994
\(490\) 2.00000 0.0903508
\(491\) −35.0000 −1.57953 −0.789764 0.613411i \(-0.789797\pi\)
−0.789764 + 0.613411i \(0.789797\pi\)
\(492\) −4.00000 −0.180334
\(493\) 2.00000 0.0900755
\(494\) −5.00000 −0.224961
\(495\) 1.00000 0.0449467
\(496\) 2.00000 0.0898027
\(497\) 6.00000 0.269137
\(498\) 1.00000 0.0448111
\(499\) −11.0000 −0.492428 −0.246214 0.969216i \(-0.579187\pi\)
−0.246214 + 0.969216i \(0.579187\pi\)
\(500\) 1.00000 0.0447214
\(501\) 7.00000 0.312737
\(502\) 18.0000 0.803379
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 3.00000 0.133631
\(505\) 6.00000 0.266996
\(506\) 7.00000 0.311188
\(507\) 12.0000 0.532939
\(508\) 13.0000 0.576782
\(509\) −11.0000 −0.487566 −0.243783 0.969830i \(-0.578389\pi\)
−0.243783 + 0.969830i \(0.578389\pi\)
\(510\) 1.00000 0.0442807
\(511\) −27.0000 −1.19441
\(512\) 1.00000 0.0441942
\(513\) 5.00000 0.220755
\(514\) −13.0000 −0.573405
\(515\) 8.00000 0.352522
\(516\) 12.0000 0.528271
\(517\) 12.0000 0.527759
\(518\) 3.00000 0.131812
\(519\) 23.0000 1.00959
\(520\) 1.00000 0.0438529
\(521\) 36.0000 1.57719 0.788594 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) −3.00000 −0.130931
\(526\) 8.00000 0.348817
\(527\) −2.00000 −0.0871214
\(528\) −1.00000 −0.0435194
\(529\) 26.0000 1.13043
\(530\) −9.00000 −0.390935
\(531\) 10.0000 0.433963
\(532\) −15.0000 −0.650332
\(533\) 4.00000 0.173259
\(534\) −1.00000 −0.0432742
\(535\) −19.0000 −0.821442
\(536\) 2.00000 0.0863868
\(537\) −12.0000 −0.517838
\(538\) −5.00000 −0.215565
\(539\) 2.00000 0.0861461
\(540\) −1.00000 −0.0430331
\(541\) 3.00000 0.128980 0.0644900 0.997918i \(-0.479458\pi\)
0.0644900 + 0.997918i \(0.479458\pi\)
\(542\) −28.0000 −1.20270
\(543\) −2.00000 −0.0858282
\(544\) −1.00000 −0.0428746
\(545\) −19.0000 −0.813871
\(546\) −3.00000 −0.128388
\(547\) −13.0000 −0.555840 −0.277920 0.960604i \(-0.589645\pi\)
−0.277920 + 0.960604i \(0.589645\pi\)
\(548\) −10.0000 −0.427179
\(549\) 14.0000 0.597505
\(550\) 1.00000 0.0426401
\(551\) 10.0000 0.426014
\(552\) −7.00000 −0.297940
\(553\) 12.0000 0.510292
\(554\) −17.0000 −0.722261
\(555\) −1.00000 −0.0424476
\(556\) −6.00000 −0.254457
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 2.00000 0.0846668
\(559\) −12.0000 −0.507546
\(560\) 3.00000 0.126773
\(561\) 1.00000 0.0422200
\(562\) −11.0000 −0.464007
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) −12.0000 −0.505291
\(565\) −2.00000 −0.0841406
\(566\) −21.0000 −0.882696
\(567\) 3.00000 0.125988
\(568\) 2.00000 0.0839181
\(569\) −1.00000 −0.0419222 −0.0209611 0.999780i \(-0.506673\pi\)
−0.0209611 + 0.999780i \(0.506673\pi\)
\(570\) 5.00000 0.209427
\(571\) 18.0000 0.753277 0.376638 0.926360i \(-0.377080\pi\)
0.376638 + 0.926360i \(0.377080\pi\)
\(572\) 1.00000 0.0418121
\(573\) −5.00000 −0.208878
\(574\) 12.0000 0.500870
\(575\) 7.00000 0.291920
\(576\) 1.00000 0.0416667
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) −16.0000 −0.665512
\(579\) 22.0000 0.914289
\(580\) −2.00000 −0.0830455
\(581\) −3.00000 −0.124461
\(582\) 8.00000 0.331611
\(583\) −9.00000 −0.372742
\(584\) −9.00000 −0.372423
\(585\) 1.00000 0.0413449
\(586\) −9.00000 −0.371787
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −10.0000 −0.412043
\(590\) 10.0000 0.411693
\(591\) −13.0000 −0.534749
\(592\) 1.00000 0.0410997
\(593\) −8.00000 −0.328521 −0.164260 0.986417i \(-0.552524\pi\)
−0.164260 + 0.986417i \(0.552524\pi\)
\(594\) −1.00000 −0.0410305
\(595\) −3.00000 −0.122988
\(596\) 6.00000 0.245770
\(597\) −4.00000 −0.163709
\(598\) 7.00000 0.286251
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) −36.0000 −1.46725
\(603\) 2.00000 0.0814463
\(604\) −17.0000 −0.691720
\(605\) −10.0000 −0.406558
\(606\) −6.00000 −0.243733
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) −5.00000 −0.202777
\(609\) 6.00000 0.243132
\(610\) 14.0000 0.566843
\(611\) 12.0000 0.485468
\(612\) −1.00000 −0.0404226
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 18.0000 0.726421
\(615\) −4.00000 −0.161296
\(616\) 3.00000 0.120873
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) −8.00000 −0.321807
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 2.00000 0.0803219
\(621\) −7.00000 −0.280900
\(622\) 24.0000 0.962312
\(623\) 3.00000 0.120192
\(624\) −1.00000 −0.0400320
\(625\) 1.00000 0.0400000
\(626\) 8.00000 0.319744
\(627\) 5.00000 0.199681
\(628\) −8.00000 −0.319235
\(629\) −1.00000 −0.0398726
\(630\) 3.00000 0.119523
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 4.00000 0.159111
\(633\) 10.0000 0.397464
\(634\) −6.00000 −0.238290
\(635\) 13.0000 0.515889
\(636\) 9.00000 0.356873
\(637\) 2.00000 0.0792429
\(638\) −2.00000 −0.0791808
\(639\) 2.00000 0.0791188
\(640\) 1.00000 0.0395285
\(641\) 44.0000 1.73790 0.868948 0.494904i \(-0.164797\pi\)
0.868948 + 0.494904i \(0.164797\pi\)
\(642\) 19.0000 0.749870
\(643\) −7.00000 −0.276053 −0.138027 0.990429i \(-0.544076\pi\)
−0.138027 + 0.990429i \(0.544076\pi\)
\(644\) 21.0000 0.827516
\(645\) 12.0000 0.472500
\(646\) 5.00000 0.196722
\(647\) −17.0000 −0.668339 −0.334169 0.942513i \(-0.608456\pi\)
−0.334169 + 0.942513i \(0.608456\pi\)
\(648\) 1.00000 0.0392837
\(649\) 10.0000 0.392534
\(650\) 1.00000 0.0392232
\(651\) −6.00000 −0.235159
\(652\) −9.00000 −0.352467
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) 19.0000 0.742959
\(655\) 0 0
\(656\) 4.00000 0.156174
\(657\) −9.00000 −0.351123
\(658\) 36.0000 1.40343
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) −1.00000 −0.0389249
\(661\) 7.00000 0.272268 0.136134 0.990690i \(-0.456532\pi\)
0.136134 + 0.990690i \(0.456532\pi\)
\(662\) 12.0000 0.466393
\(663\) 1.00000 0.0388368
\(664\) −1.00000 −0.0388075
\(665\) −15.0000 −0.581675
\(666\) 1.00000 0.0387492
\(667\) −14.0000 −0.542082
\(668\) −7.00000 −0.270838
\(669\) 0 0
\(670\) 2.00000 0.0772667
\(671\) 14.0000 0.540464
\(672\) −3.00000 −0.115728
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) −1.00000 −0.0385186
\(675\) −1.00000 −0.0384900
\(676\) −12.0000 −0.461538
\(677\) −3.00000 −0.115299 −0.0576497 0.998337i \(-0.518361\pi\)
−0.0576497 + 0.998337i \(0.518361\pi\)
\(678\) 2.00000 0.0768095
\(679\) −24.0000 −0.921035
\(680\) −1.00000 −0.0383482
\(681\) 4.00000 0.153280
\(682\) 2.00000 0.0765840
\(683\) 40.0000 1.53056 0.765279 0.643699i \(-0.222601\pi\)
0.765279 + 0.643699i \(0.222601\pi\)
\(684\) −5.00000 −0.191180
\(685\) −10.0000 −0.382080
\(686\) −15.0000 −0.572703
\(687\) −26.0000 −0.991962
\(688\) −12.0000 −0.457496
\(689\) −9.00000 −0.342873
\(690\) −7.00000 −0.266485
\(691\) −2.00000 −0.0760836 −0.0380418 0.999276i \(-0.512112\pi\)
−0.0380418 + 0.999276i \(0.512112\pi\)
\(692\) −23.0000 −0.874329
\(693\) 3.00000 0.113961
\(694\) 24.0000 0.911028
\(695\) −6.00000 −0.227593
\(696\) 2.00000 0.0758098
\(697\) −4.00000 −0.151511
\(698\) −4.00000 −0.151402
\(699\) 14.0000 0.529529
\(700\) 3.00000 0.113389
\(701\) −40.0000 −1.51078 −0.755390 0.655276i \(-0.772552\pi\)
−0.755390 + 0.655276i \(0.772552\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −5.00000 −0.188579
\(704\) 1.00000 0.0376889
\(705\) −12.0000 −0.451946
\(706\) 14.0000 0.526897
\(707\) 18.0000 0.676960
\(708\) −10.0000 −0.375823
\(709\) 27.0000 1.01401 0.507003 0.861944i \(-0.330753\pi\)
0.507003 + 0.861944i \(0.330753\pi\)
\(710\) 2.00000 0.0750587
\(711\) 4.00000 0.150012
\(712\) 1.00000 0.0374766
\(713\) 14.0000 0.524304
\(714\) 3.00000 0.112272
\(715\) 1.00000 0.0373979
\(716\) 12.0000 0.448461
\(717\) −20.0000 −0.746914
\(718\) 4.00000 0.149279
\(719\) 10.0000 0.372937 0.186469 0.982461i \(-0.440296\pi\)
0.186469 + 0.982461i \(0.440296\pi\)
\(720\) 1.00000 0.0372678
\(721\) 24.0000 0.893807
\(722\) 6.00000 0.223297
\(723\) 14.0000 0.520666
\(724\) 2.00000 0.0743294
\(725\) −2.00000 −0.0742781
\(726\) 10.0000 0.371135
\(727\) 30.0000 1.11264 0.556319 0.830969i \(-0.312213\pi\)
0.556319 + 0.830969i \(0.312213\pi\)
\(728\) 3.00000 0.111187
\(729\) 1.00000 0.0370370
\(730\) −9.00000 −0.333105
\(731\) 12.0000 0.443836
\(732\) −14.0000 −0.517455
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 35.0000 1.29187
\(735\) −2.00000 −0.0737711
\(736\) 7.00000 0.258023
\(737\) 2.00000 0.0736709
\(738\) 4.00000 0.147242
\(739\) 54.0000 1.98642 0.993211 0.116326i \(-0.0371118\pi\)
0.993211 + 0.116326i \(0.0371118\pi\)
\(740\) 1.00000 0.0367607
\(741\) 5.00000 0.183680
\(742\) −27.0000 −0.991201
\(743\) 14.0000 0.513610 0.256805 0.966463i \(-0.417330\pi\)
0.256805 + 0.966463i \(0.417330\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 6.00000 0.219823
\(746\) 10.0000 0.366126
\(747\) −1.00000 −0.0365881
\(748\) −1.00000 −0.0365636
\(749\) −57.0000 −2.08273
\(750\) −1.00000 −0.0365148
\(751\) 44.0000 1.60558 0.802791 0.596260i \(-0.203347\pi\)
0.802791 + 0.596260i \(0.203347\pi\)
\(752\) 12.0000 0.437595
\(753\) −18.0000 −0.655956
\(754\) −2.00000 −0.0728357
\(755\) −17.0000 −0.618693
\(756\) −3.00000 −0.109109
\(757\) 43.0000 1.56286 0.781431 0.623992i \(-0.214490\pi\)
0.781431 + 0.623992i \(0.214490\pi\)
\(758\) −2.00000 −0.0726433
\(759\) −7.00000 −0.254084
\(760\) −5.00000 −0.181369
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) −13.0000 −0.470940
\(763\) −57.0000 −2.06354
\(764\) 5.00000 0.180894
\(765\) −1.00000 −0.0361551
\(766\) 13.0000 0.469709
\(767\) 10.0000 0.361079
\(768\) −1.00000 −0.0360844
\(769\) 38.0000 1.37032 0.685158 0.728395i \(-0.259733\pi\)
0.685158 + 0.728395i \(0.259733\pi\)
\(770\) 3.00000 0.108112
\(771\) 13.0000 0.468184
\(772\) −22.0000 −0.791797
\(773\) 9.00000 0.323708 0.161854 0.986815i \(-0.448253\pi\)
0.161854 + 0.986815i \(0.448253\pi\)
\(774\) −12.0000 −0.431331
\(775\) 2.00000 0.0718421
\(776\) −8.00000 −0.287183
\(777\) −3.00000 −0.107624
\(778\) 14.0000 0.501924
\(779\) −20.0000 −0.716574
\(780\) −1.00000 −0.0358057
\(781\) 2.00000 0.0715656
\(782\) −7.00000 −0.250319
\(783\) 2.00000 0.0714742
\(784\) 2.00000 0.0714286
\(785\) −8.00000 −0.285532
\(786\) 0 0
\(787\) 14.0000 0.499046 0.249523 0.968369i \(-0.419726\pi\)
0.249523 + 0.968369i \(0.419726\pi\)
\(788\) 13.0000 0.463106
\(789\) −8.00000 −0.284808
\(790\) 4.00000 0.142314
\(791\) −6.00000 −0.213335
\(792\) 1.00000 0.0355335
\(793\) 14.0000 0.497155
\(794\) −16.0000 −0.567819
\(795\) 9.00000 0.319197
\(796\) 4.00000 0.141776
\(797\) −46.0000 −1.62940 −0.814702 0.579880i \(-0.803099\pi\)
−0.814702 + 0.579880i \(0.803099\pi\)
\(798\) 15.0000 0.530994
\(799\) −12.0000 −0.424529
\(800\) 1.00000 0.0353553
\(801\) 1.00000 0.0353333
\(802\) 25.0000 0.882781
\(803\) −9.00000 −0.317603
\(804\) −2.00000 −0.0705346
\(805\) 21.0000 0.740153
\(806\) 2.00000 0.0704470
\(807\) 5.00000 0.176008
\(808\) 6.00000 0.211079
\(809\) 5.00000 0.175791 0.0878953 0.996130i \(-0.471986\pi\)
0.0878953 + 0.996130i \(0.471986\pi\)
\(810\) 1.00000 0.0351364
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) −6.00000 −0.210559
\(813\) 28.0000 0.982003
\(814\) 1.00000 0.0350500
\(815\) −9.00000 −0.315256
\(816\) 1.00000 0.0350070
\(817\) 60.0000 2.09913
\(818\) −8.00000 −0.279713
\(819\) 3.00000 0.104828
\(820\) 4.00000 0.139686
\(821\) −3.00000 −0.104701 −0.0523504 0.998629i \(-0.516671\pi\)
−0.0523504 + 0.998629i \(0.516671\pi\)
\(822\) 10.0000 0.348790
\(823\) 13.0000 0.453152 0.226576 0.973994i \(-0.427247\pi\)
0.226576 + 0.973994i \(0.427247\pi\)
\(824\) 8.00000 0.278693
\(825\) −1.00000 −0.0348155
\(826\) 30.0000 1.04383
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 7.00000 0.243267
\(829\) −7.00000 −0.243120 −0.121560 0.992584i \(-0.538790\pi\)
−0.121560 + 0.992584i \(0.538790\pi\)
\(830\) −1.00000 −0.0347105
\(831\) 17.0000 0.589723
\(832\) 1.00000 0.0346688
\(833\) −2.00000 −0.0692959
\(834\) 6.00000 0.207763
\(835\) −7.00000 −0.242245
\(836\) −5.00000 −0.172929
\(837\) −2.00000 −0.0691301
\(838\) −1.00000 −0.0345444
\(839\) −54.0000 −1.86429 −0.932144 0.362089i \(-0.882064\pi\)
−0.932144 + 0.362089i \(0.882064\pi\)
\(840\) −3.00000 −0.103510
\(841\) −25.0000 −0.862069
\(842\) 22.0000 0.758170
\(843\) 11.0000 0.378860
\(844\) −10.0000 −0.344214
\(845\) −12.0000 −0.412813
\(846\) 12.0000 0.412568
\(847\) −30.0000 −1.03081
\(848\) −9.00000 −0.309061
\(849\) 21.0000 0.720718
\(850\) −1.00000 −0.0342997
\(851\) 7.00000 0.239957
\(852\) −2.00000 −0.0685189
\(853\) 21.0000 0.719026 0.359513 0.933140i \(-0.382943\pi\)
0.359513 + 0.933140i \(0.382943\pi\)
\(854\) 42.0000 1.43721
\(855\) −5.00000 −0.170996
\(856\) −19.0000 −0.649407
\(857\) −27.0000 −0.922302 −0.461151 0.887322i \(-0.652563\pi\)
−0.461151 + 0.887322i \(0.652563\pi\)
\(858\) −1.00000 −0.0341394
\(859\) −39.0000 −1.33066 −0.665331 0.746548i \(-0.731710\pi\)
−0.665331 + 0.746548i \(0.731710\pi\)
\(860\) −12.0000 −0.409197
\(861\) −12.0000 −0.408959
\(862\) 3.00000 0.102180
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −23.0000 −0.782023
\(866\) −19.0000 −0.645646
\(867\) 16.0000 0.543388
\(868\) 6.00000 0.203653
\(869\) 4.00000 0.135691
\(870\) 2.00000 0.0678064
\(871\) 2.00000 0.0677674
\(872\) −19.0000 −0.643421
\(873\) −8.00000 −0.270759
\(874\) −35.0000 −1.18389
\(875\) 3.00000 0.101419
\(876\) 9.00000 0.304082
\(877\) 12.0000 0.405211 0.202606 0.979260i \(-0.435059\pi\)
0.202606 + 0.979260i \(0.435059\pi\)
\(878\) 24.0000 0.809961
\(879\) 9.00000 0.303562
\(880\) 1.00000 0.0337100
\(881\) 36.0000 1.21287 0.606435 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(882\) 2.00000 0.0673435
\(883\) −37.0000 −1.24515 −0.622575 0.782560i \(-0.713913\pi\)
−0.622575 + 0.782560i \(0.713913\pi\)
\(884\) −1.00000 −0.0336336
\(885\) −10.0000 −0.336146
\(886\) −36.0000 −1.20944
\(887\) 18.0000 0.604381 0.302190 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) −1.00000 −0.0335578
\(889\) 39.0000 1.30802
\(890\) 1.00000 0.0335201
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) −60.0000 −2.00782
\(894\) −6.00000 −0.200670
\(895\) 12.0000 0.401116
\(896\) 3.00000 0.100223
\(897\) −7.00000 −0.233723
\(898\) −18.0000 −0.600668
\(899\) −4.00000 −0.133407
\(900\) 1.00000 0.0333333
\(901\) 9.00000 0.299833
\(902\) 4.00000 0.133185
\(903\) 36.0000 1.19800
\(904\) −2.00000 −0.0665190
\(905\) 2.00000 0.0664822
\(906\) 17.0000 0.564787
\(907\) 27.0000 0.896520 0.448260 0.893903i \(-0.352044\pi\)
0.448260 + 0.893903i \(0.352044\pi\)
\(908\) −4.00000 −0.132745
\(909\) 6.00000 0.199007
\(910\) 3.00000 0.0994490
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 5.00000 0.165567
\(913\) −1.00000 −0.0330952
\(914\) 24.0000 0.793849
\(915\) −14.0000 −0.462826
\(916\) 26.0000 0.859064
\(917\) 0 0
\(918\) 1.00000 0.0330049
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 7.00000 0.230783
\(921\) −18.0000 −0.593120
\(922\) 6.00000 0.197599
\(923\) 2.00000 0.0658308
\(924\) −3.00000 −0.0986928
\(925\) 1.00000 0.0328798
\(926\) −36.0000 −1.18303
\(927\) 8.00000 0.262754
\(928\) −2.00000 −0.0656532
\(929\) 46.0000 1.50921 0.754606 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) −2.00000 −0.0655826
\(931\) −10.0000 −0.327737
\(932\) −14.0000 −0.458585
\(933\) −24.0000 −0.785725
\(934\) −12.0000 −0.392652
\(935\) −1.00000 −0.0327035
\(936\) 1.00000 0.0326860
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 6.00000 0.195907
\(939\) −8.00000 −0.261070
\(940\) 12.0000 0.391397
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 8.00000 0.260654
\(943\) 28.0000 0.911805
\(944\) 10.0000 0.325472
\(945\) −3.00000 −0.0975900
\(946\) −12.0000 −0.390154
\(947\) −44.0000 −1.42981 −0.714904 0.699223i \(-0.753530\pi\)
−0.714904 + 0.699223i \(0.753530\pi\)
\(948\) −4.00000 −0.129914
\(949\) −9.00000 −0.292152
\(950\) −5.00000 −0.162221
\(951\) 6.00000 0.194563
\(952\) −3.00000 −0.0972306
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) −9.00000 −0.291386
\(955\) 5.00000 0.161796
\(956\) 20.0000 0.646846
\(957\) 2.00000 0.0646508
\(958\) −9.00000 −0.290777
\(959\) −30.0000 −0.968751
\(960\) −1.00000 −0.0322749
\(961\) −27.0000 −0.870968
\(962\) 1.00000 0.0322413
\(963\) −19.0000 −0.612266
\(964\) −14.0000 −0.450910
\(965\) −22.0000 −0.708205
\(966\) −21.0000 −0.675664
\(967\) −58.0000 −1.86515 −0.932577 0.360971i \(-0.882445\pi\)
−0.932577 + 0.360971i \(0.882445\pi\)
\(968\) −10.0000 −0.321412
\(969\) −5.00000 −0.160623
\(970\) −8.00000 −0.256865
\(971\) −4.00000 −0.128366 −0.0641831 0.997938i \(-0.520444\pi\)
−0.0641831 + 0.997938i \(0.520444\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −18.0000 −0.577054
\(974\) 6.00000 0.192252
\(975\) −1.00000 −0.0320256
\(976\) 14.0000 0.448129
\(977\) −61.0000 −1.95156 −0.975781 0.218748i \(-0.929803\pi\)
−0.975781 + 0.218748i \(0.929803\pi\)
\(978\) 9.00000 0.287788
\(979\) 1.00000 0.0319601
\(980\) 2.00000 0.0638877
\(981\) −19.0000 −0.606623
\(982\) −35.0000 −1.11689
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) −4.00000 −0.127515
\(985\) 13.0000 0.414214
\(986\) 2.00000 0.0636930
\(987\) −36.0000 −1.14589
\(988\) −5.00000 −0.159071
\(989\) −84.0000 −2.67104
\(990\) 1.00000 0.0317821
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 2.00000 0.0635001
\(993\) −12.0000 −0.380808
\(994\) 6.00000 0.190308
\(995\) 4.00000 0.126809
\(996\) 1.00000 0.0316862
\(997\) 39.0000 1.23514 0.617571 0.786515i \(-0.288117\pi\)
0.617571 + 0.786515i \(0.288117\pi\)
\(998\) −11.0000 −0.348199
\(999\) −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.l.1.1 1
3.2 odd 2 3330.2.a.e.1.1 1
4.3 odd 2 8880.2.a.y.1.1 1
5.4 even 2 5550.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.l.1.1 1 1.1 even 1 trivial
3330.2.a.e.1.1 1 3.2 odd 2
5550.2.a.m.1.1 1 5.4 even 2
8880.2.a.y.1.1 1 4.3 odd 2