Properties

Label 5070.2.a.j.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.00000 q^{11} +1.00000 q^{12} -2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} +3.00000 q^{22} +3.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{28} +3.00000 q^{29} +1.00000 q^{30} +5.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} -6.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -7.00000 q^{37} -2.00000 q^{38} +1.00000 q^{40} +6.00000 q^{41} -2.00000 q^{42} -1.00000 q^{43} -3.00000 q^{44} -1.00000 q^{45} -3.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -6.00000 q^{53} -1.00000 q^{54} +3.00000 q^{55} -2.00000 q^{56} +2.00000 q^{57} -3.00000 q^{58} -9.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -5.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} +8.00000 q^{67} +6.00000 q^{68} +3.00000 q^{69} +2.00000 q^{70} -12.0000 q^{71} -1.00000 q^{72} +14.0000 q^{73} +7.00000 q^{74} +1.00000 q^{75} +2.00000 q^{76} -6.00000 q^{77} +5.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -6.00000 q^{83} +2.00000 q^{84} -6.00000 q^{85} +1.00000 q^{86} +3.00000 q^{87} +3.00000 q^{88} -18.0000 q^{89} +1.00000 q^{90} +3.00000 q^{92} +5.00000 q^{93} +3.00000 q^{94} -2.00000 q^{95} -1.00000 q^{96} +14.0000 q^{97} +3.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −2.00000 −0.534522
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −1.00000 −0.223607
\(21\) 2.00000 0.436436
\(22\) 3.00000 0.639602
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 1.00000 0.182574
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) −6.00000 −1.02899
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −2.00000 −0.308607
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −3.00000 −0.452267
\(45\) −1.00000 −0.149071
\(46\) −3.00000 −0.442326
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 3.00000 0.404520
\(56\) −2.00000 −0.267261
\(57\) 2.00000 0.264906
\(58\) −3.00000 −0.393919
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) −1.00000 −0.129099
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −5.00000 −0.635001
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 6.00000 0.727607
\(69\) 3.00000 0.361158
\(70\) 2.00000 0.239046
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 7.00000 0.813733
\(75\) 1.00000 0.115470
\(76\) 2.00000 0.229416
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 2.00000 0.218218
\(85\) −6.00000 −0.650791
\(86\) 1.00000 0.107833
\(87\) 3.00000 0.321634
\(88\) 3.00000 0.319801
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 3.00000 0.312772
\(93\) 5.00000 0.518476
\(94\) 3.00000 0.309426
\(95\) −2.00000 −0.205196
\(96\) −1.00000 −0.102062
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 3.00000 0.303046
\(99\) −3.00000 −0.301511
\(100\) 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
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 6.00000 0.582772
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 1.00000 0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) −3.00000 −0.286039
\(111\) −7.00000 −0.664411
\(112\) 2.00000 0.188982
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) −2.00000 −0.187317
\(115\) −3.00000 −0.279751
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) 9.00000 0.828517
\(119\) 12.0000 1.10004
\(120\) 1.00000 0.0912871
\(121\) −2.00000 −0.181818
\(122\) −2.00000 −0.181071
\(123\) 6.00000 0.541002
\(124\) 5.00000 0.449013
\(125\) −1.00000 −0.0894427
\(126\) −2.00000 −0.178174
\(127\) 14.0000 1.24230 0.621150 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 9.00000 0.786334 0.393167 0.919467i \(-0.371379\pi\)
0.393167 + 0.919467i \(0.371379\pi\)
\(132\) −3.00000 −0.261116
\(133\) 4.00000 0.346844
\(134\) −8.00000 −0.691095
\(135\) −1.00000 −0.0860663
\(136\) −6.00000 −0.514496
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) −3.00000 −0.255377
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) −2.00000 −0.169031
\(141\) −3.00000 −0.252646
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −3.00000 −0.249136
\(146\) −14.0000 −1.15865
\(147\) −3.00000 −0.247436
\(148\) −7.00000 −0.575396
\(149\) −9.00000 −0.737309 −0.368654 0.929567i \(-0.620181\pi\)
−0.368654 + 0.929567i \(0.620181\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −2.00000 −0.162221
\(153\) 6.00000 0.485071
\(154\) 6.00000 0.483494
\(155\) −5.00000 −0.401610
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) −5.00000 −0.397779
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 6.00000 0.472866
\(162\) −1.00000 −0.0785674
\(163\) −13.0000 −1.01824 −0.509119 0.860696i \(-0.670029\pi\)
−0.509119 + 0.860696i \(0.670029\pi\)
\(164\) 6.00000 0.468521
\(165\) 3.00000 0.233550
\(166\) 6.00000 0.465690
\(167\) −9.00000 −0.696441 −0.348220 0.937413i \(-0.613214\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(168\) −2.00000 −0.154303
\(169\) 0 0
\(170\) 6.00000 0.460179
\(171\) 2.00000 0.152944
\(172\) −1.00000 −0.0762493
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) −3.00000 −0.227429
\(175\) 2.00000 0.151186
\(176\) −3.00000 −0.226134
\(177\) −9.00000 −0.676481
\(178\) 18.0000 1.34916
\(179\) −3.00000 −0.224231 −0.112115 0.993695i \(-0.535763\pi\)
−0.112115 + 0.993695i \(0.535763\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) −3.00000 −0.221163
\(185\) 7.00000 0.514650
\(186\) −5.00000 −0.366618
\(187\) −18.0000 −1.31629
\(188\) −3.00000 −0.218797
\(189\) 2.00000 0.145479
\(190\) 2.00000 0.145095
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 24.0000 1.70993 0.854965 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 3.00000 0.213201
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) −6.00000 −0.422159
\(203\) 6.00000 0.421117
\(204\) 6.00000 0.420084
\(205\) −6.00000 −0.419058
\(206\) −14.0000 −0.975426
\(207\) 3.00000 0.208514
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 2.00000 0.138013
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −6.00000 −0.412082
\(213\) −12.0000 −0.822226
\(214\) 6.00000 0.410152
\(215\) 1.00000 0.0681994
\(216\) −1.00000 −0.0680414
\(217\) 10.0000 0.678844
\(218\) −14.0000 −0.948200
\(219\) 14.0000 0.946032
\(220\) 3.00000 0.202260
\(221\) 0 0
\(222\) 7.00000 0.469809
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) −2.00000 −0.133631
\(225\) 1.00000 0.0666667
\(226\) −15.0000 −0.997785
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 2.00000 0.132453
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 3.00000 0.197814
\(231\) −6.00000 −0.394771
\(232\) −3.00000 −0.196960
\(233\) 21.0000 1.37576 0.687878 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) 0 0
\(235\) 3.00000 0.195698
\(236\) −9.00000 −0.585850
\(237\) 5.00000 0.324785
\(238\) −12.0000 −0.777844
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 2.00000 0.128565
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 3.00000 0.191663
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) −5.00000 −0.317500
\(249\) −6.00000 −0.380235
\(250\) 1.00000 0.0632456
\(251\) 15.0000 0.946792 0.473396 0.880850i \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) 2.00000 0.125988
\(253\) −9.00000 −0.565825
\(254\) −14.0000 −0.878438
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) −21.0000 −1.30994 −0.654972 0.755653i \(-0.727320\pi\)
−0.654972 + 0.755653i \(0.727320\pi\)
\(258\) 1.00000 0.0622573
\(259\) −14.0000 −0.869918
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) −9.00000 −0.556022
\(263\) −15.0000 −0.924940 −0.462470 0.886635i \(-0.653037\pi\)
−0.462470 + 0.886635i \(0.653037\pi\)
\(264\) 3.00000 0.184637
\(265\) 6.00000 0.368577
\(266\) −4.00000 −0.245256
\(267\) −18.0000 −1.10158
\(268\) 8.00000 0.488678
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 1.00000 0.0608581
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −9.00000 −0.543710
\(275\) −3.00000 −0.180907
\(276\) 3.00000 0.180579
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) −14.0000 −0.839664
\(279\) 5.00000 0.299342
\(280\) 2.00000 0.119523
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 3.00000 0.178647
\(283\) −31.0000 −1.84276 −0.921379 0.388664i \(-0.872937\pi\)
−0.921379 + 0.388664i \(0.872937\pi\)
\(284\) −12.0000 −0.712069
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 3.00000 0.176166
\(291\) 14.0000 0.820695
\(292\) 14.0000 0.819288
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 3.00000 0.174964
\(295\) 9.00000 0.524000
\(296\) 7.00000 0.406867
\(297\) −3.00000 −0.174078
\(298\) 9.00000 0.521356
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −2.00000 −0.115278
\(302\) −8.00000 −0.460348
\(303\) 6.00000 0.344691
\(304\) 2.00000 0.114708
\(305\) −2.00000 −0.114520
\(306\) −6.00000 −0.342997
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) −6.00000 −0.341882
\(309\) 14.0000 0.796432
\(310\) 5.00000 0.283981
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 13.0000 0.733632
\(315\) −2.00000 −0.112687
\(316\) 5.00000 0.281272
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 6.00000 0.336463
\(319\) −9.00000 −0.503903
\(320\) −1.00000 −0.0559017
\(321\) −6.00000 −0.334887
\(322\) −6.00000 −0.334367
\(323\) 12.0000 0.667698
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 13.0000 0.720003
\(327\) 14.0000 0.774202
\(328\) −6.00000 −0.331295
\(329\) −6.00000 −0.330791
\(330\) −3.00000 −0.165145
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) −6.00000 −0.329293
\(333\) −7.00000 −0.383598
\(334\) 9.00000 0.492458
\(335\) −8.00000 −0.437087
\(336\) 2.00000 0.109109
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 15.0000 0.814688
\(340\) −6.00000 −0.325396
\(341\) −15.0000 −0.812296
\(342\) −2.00000 −0.108148
\(343\) −20.0000 −1.07990
\(344\) 1.00000 0.0539164
\(345\) −3.00000 −0.161515
\(346\) −12.0000 −0.645124
\(347\) −30.0000 −1.61048 −0.805242 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) 3.00000 0.160817
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 9.00000 0.478345
\(355\) 12.0000 0.636894
\(356\) −18.0000 −0.953998
\(357\) 12.0000 0.635107
\(358\) 3.00000 0.158555
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 1.00000 0.0527046
\(361\) −15.0000 −0.789474
\(362\) 16.0000 0.840941
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) −14.0000 −0.732793
\(366\) −2.00000 −0.104542
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 3.00000 0.156386
\(369\) 6.00000 0.312348
\(370\) −7.00000 −0.363913
\(371\) −12.0000 −0.623009
\(372\) 5.00000 0.259238
\(373\) −25.0000 −1.29445 −0.647225 0.762299i \(-0.724071\pi\)
−0.647225 + 0.762299i \(0.724071\pi\)
\(374\) 18.0000 0.930758
\(375\) −1.00000 −0.0516398
\(376\) 3.00000 0.154713
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) 38.0000 1.95193 0.975964 0.217930i \(-0.0699304\pi\)
0.975964 + 0.217930i \(0.0699304\pi\)
\(380\) −2.00000 −0.102598
\(381\) 14.0000 0.717242
\(382\) 12.0000 0.613973
\(383\) −21.0000 −1.07305 −0.536525 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 6.00000 0.305788
\(386\) 4.00000 0.203595
\(387\) −1.00000 −0.0508329
\(388\) 14.0000 0.710742
\(389\) 3.00000 0.152106 0.0760530 0.997104i \(-0.475768\pi\)
0.0760530 + 0.997104i \(0.475768\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 3.00000 0.151523
\(393\) 9.00000 0.453990
\(394\) −24.0000 −1.20910
\(395\) −5.00000 −0.251577
\(396\) −3.00000 −0.150756
\(397\) −31.0000 −1.55585 −0.777923 0.628360i \(-0.783727\pi\)
−0.777923 + 0.628360i \(0.783727\pi\)
\(398\) −8.00000 −0.401004
\(399\) 4.00000 0.200250
\(400\) 1.00000 0.0500000
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) −8.00000 −0.399004
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) −1.00000 −0.0496904
\(406\) −6.00000 −0.297775
\(407\) 21.0000 1.04093
\(408\) −6.00000 −0.297044
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 6.00000 0.296319
\(411\) 9.00000 0.443937
\(412\) 14.0000 0.689730
\(413\) −18.0000 −0.885722
\(414\) −3.00000 −0.147442
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) 14.0000 0.685583
\(418\) 6.00000 0.293470
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) −2.00000 −0.0975900
\(421\) −16.0000 −0.779792 −0.389896 0.920859i \(-0.627489\pi\)
−0.389896 + 0.920859i \(0.627489\pi\)
\(422\) −20.0000 −0.973585
\(423\) −3.00000 −0.145865
\(424\) 6.00000 0.291386
\(425\) 6.00000 0.291043
\(426\) 12.0000 0.581402
\(427\) 4.00000 0.193574
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) −1.00000 −0.0482243
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 1.00000 0.0481125
\(433\) −40.0000 −1.92228 −0.961139 0.276066i \(-0.910969\pi\)
−0.961139 + 0.276066i \(0.910969\pi\)
\(434\) −10.0000 −0.480015
\(435\) −3.00000 −0.143839
\(436\) 14.0000 0.670478
\(437\) 6.00000 0.287019
\(438\) −14.0000 −0.668946
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) −3.00000 −0.143019
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) −7.00000 −0.332205
\(445\) 18.0000 0.853282
\(446\) 10.0000 0.473514
\(447\) −9.00000 −0.425685
\(448\) 2.00000 0.0944911
\(449\) 36.0000 1.69895 0.849473 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −18.0000 −0.847587
\(452\) 15.0000 0.705541
\(453\) 8.00000 0.375873
\(454\) 0 0
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) −14.0000 −0.654177
\(459\) 6.00000 0.280056
\(460\) −3.00000 −0.139876
\(461\) 15.0000 0.698620 0.349310 0.937007i \(-0.386416\pi\)
0.349310 + 0.937007i \(0.386416\pi\)
\(462\) 6.00000 0.279145
\(463\) −34.0000 −1.58011 −0.790057 0.613033i \(-0.789949\pi\)
−0.790057 + 0.613033i \(0.789949\pi\)
\(464\) 3.00000 0.139272
\(465\) −5.00000 −0.231869
\(466\) −21.0000 −0.972806
\(467\) 18.0000 0.832941 0.416470 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) −3.00000 −0.138380
\(471\) −13.0000 −0.599008
\(472\) 9.00000 0.414259
\(473\) 3.00000 0.137940
\(474\) −5.00000 −0.229658
\(475\) 2.00000 0.0917663
\(476\) 12.0000 0.550019
\(477\) −6.00000 −0.274721
\(478\) −24.0000 −1.09773
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −17.0000 −0.774329
\(483\) 6.00000 0.273009
\(484\) −2.00000 −0.0909091
\(485\) −14.0000 −0.635707
\(486\) −1.00000 −0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −13.0000 −0.587880
\(490\) −3.00000 −0.135526
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000 0.270501
\(493\) 18.0000 0.810679
\(494\) 0 0
\(495\) 3.00000 0.134840
\(496\) 5.00000 0.224507
\(497\) −24.0000 −1.07655
\(498\) 6.00000 0.268866
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −9.00000 −0.402090
\(502\) −15.0000 −0.669483
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −2.00000 −0.0890871
\(505\) −6.00000 −0.266996
\(506\) 9.00000 0.400099
\(507\) 0 0
\(508\) 14.0000 0.621150
\(509\) 3.00000 0.132973 0.0664863 0.997787i \(-0.478821\pi\)
0.0664863 + 0.997787i \(0.478821\pi\)
\(510\) 6.00000 0.265684
\(511\) 28.0000 1.23865
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) 21.0000 0.926270
\(515\) −14.0000 −0.616914
\(516\) −1.00000 −0.0440225
\(517\) 9.00000 0.395820
\(518\) 14.0000 0.615125
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −3.00000 −0.131306
\(523\) 11.0000 0.480996 0.240498 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(524\) 9.00000 0.393167
\(525\) 2.00000 0.0872872
\(526\) 15.0000 0.654031
\(527\) 30.0000 1.30682
\(528\) −3.00000 −0.130558
\(529\) −14.0000 −0.608696
\(530\) −6.00000 −0.260623
\(531\) −9.00000 −0.390567
\(532\) 4.00000 0.173422
\(533\) 0 0
\(534\) 18.0000 0.778936
\(535\) 6.00000 0.259403
\(536\) −8.00000 −0.345547
\(537\) −3.00000 −0.129460
\(538\) −18.0000 −0.776035
\(539\) 9.00000 0.387657
\(540\) −1.00000 −0.0430331
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) −11.0000 −0.472490
\(543\) −16.0000 −0.686626
\(544\) −6.00000 −0.257248
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 9.00000 0.384461
\(549\) 2.00000 0.0853579
\(550\) 3.00000 0.127920
\(551\) 6.00000 0.255609
\(552\) −3.00000 −0.127688
\(553\) 10.0000 0.425243
\(554\) 1.00000 0.0424859
\(555\) 7.00000 0.297133
\(556\) 14.0000 0.593732
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) −5.00000 −0.211667
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) −18.0000 −0.759961
\(562\) −18.0000 −0.759284
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) −3.00000 −0.126323
\(565\) −15.0000 −0.631055
\(566\) 31.0000 1.30303
\(567\) 2.00000 0.0839921
\(568\) 12.0000 0.503509
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 2.00000 0.0837708
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) −12.0000 −0.500870
\(575\) 3.00000 0.125109
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −19.0000 −0.790296
\(579\) −4.00000 −0.166234
\(580\) −3.00000 −0.124568
\(581\) −12.0000 −0.497844
\(582\) −14.0000 −0.580319
\(583\) 18.0000 0.745484
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) −3.00000 −0.123718
\(589\) 10.0000 0.412043
\(590\) −9.00000 −0.370524
\(591\) 24.0000 0.987228
\(592\) −7.00000 −0.287698
\(593\) 27.0000 1.10876 0.554379 0.832265i \(-0.312956\pi\)
0.554379 + 0.832265i \(0.312956\pi\)
\(594\) 3.00000 0.123091
\(595\) −12.0000 −0.491952
\(596\) −9.00000 −0.368654
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 2.00000 0.0815139
\(603\) 8.00000 0.325785
\(604\) 8.00000 0.325515
\(605\) 2.00000 0.0813116
\(606\) −6.00000 −0.243733
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 6.00000 0.243132
\(610\) 2.00000 0.0809776
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) −31.0000 −1.25208 −0.626039 0.779792i \(-0.715325\pi\)
−0.626039 + 0.779792i \(0.715325\pi\)
\(614\) −8.00000 −0.322854
\(615\) −6.00000 −0.241943
\(616\) 6.00000 0.241747
\(617\) 21.0000 0.845428 0.422714 0.906263i \(-0.361077\pi\)
0.422714 + 0.906263i \(0.361077\pi\)
\(618\) −14.0000 −0.563163
\(619\) −46.0000 −1.84890 −0.924448 0.381308i \(-0.875474\pi\)
−0.924448 + 0.381308i \(0.875474\pi\)
\(620\) −5.00000 −0.200805
\(621\) 3.00000 0.120386
\(622\) 12.0000 0.481156
\(623\) −36.0000 −1.44231
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −8.00000 −0.319744
\(627\) −6.00000 −0.239617
\(628\) −13.0000 −0.518756
\(629\) −42.0000 −1.67465
\(630\) 2.00000 0.0796819
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −5.00000 −0.198889
\(633\) 20.0000 0.794929
\(634\) −12.0000 −0.476581
\(635\) −14.0000 −0.555573
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 9.00000 0.356313
\(639\) −12.0000 −0.474713
\(640\) 1.00000 0.0395285
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 6.00000 0.236801
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 6.00000 0.236433
\(645\) 1.00000 0.0393750
\(646\) −12.0000 −0.472134
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 27.0000 1.05984
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) −13.0000 −0.509119
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) −14.0000 −0.547443
\(655\) −9.00000 −0.351659
\(656\) 6.00000 0.234261
\(657\) 14.0000 0.546192
\(658\) 6.00000 0.233904
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 3.00000 0.116775
\(661\) 32.0000 1.24466 0.622328 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) −32.0000 −1.24372
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) −4.00000 −0.155113
\(666\) 7.00000 0.271244
\(667\) 9.00000 0.348481
\(668\) −9.00000 −0.348220
\(669\) −10.0000 −0.386622
\(670\) 8.00000 0.309067
\(671\) −6.00000 −0.231627
\(672\) −2.00000 −0.0771517
\(673\) −4.00000 −0.154189 −0.0770943 0.997024i \(-0.524564\pi\)
−0.0770943 + 0.997024i \(0.524564\pi\)
\(674\) −14.0000 −0.539260
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −36.0000 −1.38359 −0.691796 0.722093i \(-0.743180\pi\)
−0.691796 + 0.722093i \(0.743180\pi\)
\(678\) −15.0000 −0.576072
\(679\) 28.0000 1.07454
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) 15.0000 0.574380
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 2.00000 0.0764719
\(685\) −9.00000 −0.343872
\(686\) 20.0000 0.763604
\(687\) 14.0000 0.534133
\(688\) −1.00000 −0.0381246
\(689\) 0 0
\(690\) 3.00000 0.114208
\(691\) −46.0000 −1.74992 −0.874961 0.484193i \(-0.839113\pi\)
−0.874961 + 0.484193i \(0.839113\pi\)
\(692\) 12.0000 0.456172
\(693\) −6.00000 −0.227921
\(694\) 30.0000 1.13878
\(695\) −14.0000 −0.531050
\(696\) −3.00000 −0.113715
\(697\) 36.0000 1.36360
\(698\) −8.00000 −0.302804
\(699\) 21.0000 0.794293
\(700\) 2.00000 0.0755929
\(701\) −21.0000 −0.793159 −0.396580 0.918000i \(-0.629803\pi\)
−0.396580 + 0.918000i \(0.629803\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) −3.00000 −0.113067
\(705\) 3.00000 0.112987
\(706\) −30.0000 −1.12906
\(707\) 12.0000 0.451306
\(708\) −9.00000 −0.338241
\(709\) 32.0000 1.20179 0.600893 0.799330i \(-0.294812\pi\)
0.600893 + 0.799330i \(0.294812\pi\)
\(710\) −12.0000 −0.450352
\(711\) 5.00000 0.187515
\(712\) 18.0000 0.674579
\(713\) 15.0000 0.561754
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) −3.00000 −0.112115
\(717\) 24.0000 0.896296
\(718\) −24.0000 −0.895672
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 28.0000 1.04277
\(722\) 15.0000 0.558242
\(723\) 17.0000 0.632237
\(724\) −16.0000 −0.594635
\(725\) 3.00000 0.111417
\(726\) 2.00000 0.0742270
\(727\) −4.00000 −0.148352 −0.0741759 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 14.0000 0.518163
\(731\) −6.00000 −0.221918
\(732\) 2.00000 0.0739221
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −20.0000 −0.738213
\(735\) 3.00000 0.110657
\(736\) −3.00000 −0.110581
\(737\) −24.0000 −0.884051
\(738\) −6.00000 −0.220863
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 7.00000 0.257325
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) −9.00000 −0.330178 −0.165089 0.986279i \(-0.552791\pi\)
−0.165089 + 0.986279i \(0.552791\pi\)
\(744\) −5.00000 −0.183309
\(745\) 9.00000 0.329734
\(746\) 25.0000 0.915315
\(747\) −6.00000 −0.219529
\(748\) −18.0000 −0.658145
\(749\) −12.0000 −0.438470
\(750\) 1.00000 0.0365148
\(751\) 41.0000 1.49611 0.748056 0.663636i \(-0.230988\pi\)
0.748056 + 0.663636i \(0.230988\pi\)
\(752\) −3.00000 −0.109399
\(753\) 15.0000 0.546630
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) 2.00000 0.0727393
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −38.0000 −1.38022
\(759\) −9.00000 −0.326679
\(760\) 2.00000 0.0725476
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) −14.0000 −0.507166
\(763\) 28.0000 1.01367
\(764\) −12.0000 −0.434145
\(765\) −6.00000 −0.216930
\(766\) 21.0000 0.758761
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −13.0000 −0.468792 −0.234396 0.972141i \(-0.575311\pi\)
−0.234396 + 0.972141i \(0.575311\pi\)
\(770\) −6.00000 −0.216225
\(771\) −21.0000 −0.756297
\(772\) −4.00000 −0.143963
\(773\) −48.0000 −1.72644 −0.863220 0.504828i \(-0.831556\pi\)
−0.863220 + 0.504828i \(0.831556\pi\)
\(774\) 1.00000 0.0359443
\(775\) 5.00000 0.179605
\(776\) −14.0000 −0.502571
\(777\) −14.0000 −0.502247
\(778\) −3.00000 −0.107555
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) −18.0000 −0.643679
\(783\) 3.00000 0.107211
\(784\) −3.00000 −0.107143
\(785\) 13.0000 0.463990
\(786\) −9.00000 −0.321019
\(787\) 35.0000 1.24762 0.623808 0.781578i \(-0.285585\pi\)
0.623808 + 0.781578i \(0.285585\pi\)
\(788\) 24.0000 0.854965
\(789\) −15.0000 −0.534014
\(790\) 5.00000 0.177892
\(791\) 30.0000 1.06668
\(792\) 3.00000 0.106600
\(793\) 0 0
\(794\) 31.0000 1.10015
\(795\) 6.00000 0.212798
\(796\) 8.00000 0.283552
\(797\) −24.0000 −0.850124 −0.425062 0.905164i \(-0.639748\pi\)
−0.425062 + 0.905164i \(0.639748\pi\)
\(798\) −4.00000 −0.141598
\(799\) −18.0000 −0.636794
\(800\) −1.00000 −0.0353553
\(801\) −18.0000 −0.635999
\(802\) 12.0000 0.423735
\(803\) −42.0000 −1.48215
\(804\) 8.00000 0.282138
\(805\) −6.00000 −0.211472
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) −6.00000 −0.211079
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 1.00000 0.0351364
\(811\) 32.0000 1.12367 0.561836 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 6.00000 0.210559
\(813\) 11.0000 0.385787
\(814\) −21.0000 −0.736050
\(815\) 13.0000 0.455370
\(816\) 6.00000 0.210042
\(817\) −2.00000 −0.0699711
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 27.0000 0.942306 0.471153 0.882051i \(-0.343838\pi\)
0.471153 + 0.882051i \(0.343838\pi\)
\(822\) −9.00000 −0.313911
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) −14.0000 −0.487713
\(825\) −3.00000 −0.104447
\(826\) 18.0000 0.626300
\(827\) 6.00000 0.208640 0.104320 0.994544i \(-0.466733\pi\)
0.104320 + 0.994544i \(0.466733\pi\)
\(828\) 3.00000 0.104257
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) −6.00000 −0.208263
\(831\) −1.00000 −0.0346896
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) −14.0000 −0.484780
\(835\) 9.00000 0.311458
\(836\) −6.00000 −0.207514
\(837\) 5.00000 0.172825
\(838\) 12.0000 0.414533
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 2.00000 0.0690066
\(841\) −20.0000 −0.689655
\(842\) 16.0000 0.551396
\(843\) 18.0000 0.619953
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) −4.00000 −0.137442
\(848\) −6.00000 −0.206041
\(849\) −31.0000 −1.06392
\(850\) −6.00000 −0.205798
\(851\) −21.0000 −0.719871
\(852\) −12.0000 −0.411113
\(853\) −19.0000 −0.650548 −0.325274 0.945620i \(-0.605456\pi\)
−0.325274 + 0.945620i \(0.605456\pi\)
\(854\) −4.00000 −0.136877
\(855\) −2.00000 −0.0683986
\(856\) 6.00000 0.205076
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) −46.0000 −1.56950 −0.784750 0.619813i \(-0.787209\pi\)
−0.784750 + 0.619813i \(0.787209\pi\)
\(860\) 1.00000 0.0340997
\(861\) 12.0000 0.408959
\(862\) −12.0000 −0.408722
\(863\) −45.0000 −1.53182 −0.765909 0.642949i \(-0.777711\pi\)
−0.765909 + 0.642949i \(0.777711\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −12.0000 −0.408012
\(866\) 40.0000 1.35926
\(867\) 19.0000 0.645274
\(868\) 10.0000 0.339422
\(869\) −15.0000 −0.508840
\(870\) 3.00000 0.101710
\(871\) 0 0
\(872\) −14.0000 −0.474100
\(873\) 14.0000 0.473828
\(874\) −6.00000 −0.202953
\(875\) −2.00000 −0.0676123
\(876\) 14.0000 0.473016
\(877\) 23.0000 0.776655 0.388327 0.921521i \(-0.373053\pi\)
0.388327 + 0.921521i \(0.373053\pi\)
\(878\) 4.00000 0.134993
\(879\) −30.0000 −1.01187
\(880\) 3.00000 0.101130
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 3.00000 0.101015
\(883\) −19.0000 −0.639401 −0.319700 0.947519i \(-0.603582\pi\)
−0.319700 + 0.947519i \(0.603582\pi\)
\(884\) 0 0
\(885\) 9.00000 0.302532
\(886\) 36.0000 1.20944
\(887\) −57.0000 −1.91387 −0.956936 0.290298i \(-0.906246\pi\)
−0.956936 + 0.290298i \(0.906246\pi\)
\(888\) 7.00000 0.234905
\(889\) 28.0000 0.939090
\(890\) −18.0000 −0.603361
\(891\) −3.00000 −0.100504
\(892\) −10.0000 −0.334825
\(893\) −6.00000 −0.200782
\(894\) 9.00000 0.301005
\(895\) 3.00000 0.100279
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −36.0000 −1.20134
\(899\) 15.0000 0.500278
\(900\) 1.00000 0.0333333
\(901\) −36.0000 −1.19933
\(902\) 18.0000 0.599334
\(903\) −2.00000 −0.0665558
\(904\) −15.0000 −0.498893
\(905\) 16.0000 0.531858
\(906\) −8.00000 −0.265782
\(907\) 17.0000 0.564476 0.282238 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) 2.00000 0.0662266
\(913\) 18.0000 0.595713
\(914\) −2.00000 −0.0661541
\(915\) −2.00000 −0.0661180
\(916\) 14.0000 0.462573
\(917\) 18.0000 0.594412
\(918\) −6.00000 −0.198030
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 3.00000 0.0989071
\(921\) 8.00000 0.263609
\(922\) −15.0000 −0.493999
\(923\) 0 0
\(924\) −6.00000 −0.197386
\(925\) −7.00000 −0.230159
\(926\) 34.0000 1.11731
\(927\) 14.0000 0.459820
\(928\) −3.00000 −0.0984798
\(929\) −12.0000 −0.393707 −0.196854 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) 5.00000 0.163956
\(931\) −6.00000 −0.196642
\(932\) 21.0000 0.687878
\(933\) −12.0000 −0.392862
\(934\) −18.0000 −0.588978
\(935\) 18.0000 0.588663
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −16.0000 −0.522419
\(939\) 8.00000 0.261070
\(940\) 3.00000 0.0978492
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 13.0000 0.423563
\(943\) 18.0000 0.586161
\(944\) −9.00000 −0.292925
\(945\) −2.00000 −0.0650600
\(946\) −3.00000 −0.0975384
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 5.00000 0.162392
\(949\) 0 0
\(950\) −2.00000 −0.0648886
\(951\) 12.0000 0.389127
\(952\) −12.0000 −0.388922
\(953\) 9.00000 0.291539 0.145769 0.989319i \(-0.453434\pi\)
0.145769 + 0.989319i \(0.453434\pi\)
\(954\) 6.00000 0.194257
\(955\) 12.0000 0.388311
\(956\) 24.0000 0.776215
\(957\) −9.00000 −0.290929
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) −1.00000 −0.0322749
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 17.0000 0.547533
\(965\) 4.00000 0.128765
\(966\) −6.00000 −0.193047
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 2.00000 0.0642824
\(969\) 12.0000 0.385496
\(970\) 14.0000 0.449513
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 1.00000 0.0320750
\(973\) 28.0000 0.897639
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −57.0000 −1.82359 −0.911796 0.410644i \(-0.865304\pi\)
−0.911796 + 0.410644i \(0.865304\pi\)
\(978\) 13.0000 0.415694
\(979\) 54.0000 1.72585
\(980\) 3.00000 0.0958315
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) 9.00000 0.287055 0.143528 0.989646i \(-0.454155\pi\)
0.143528 + 0.989646i \(0.454155\pi\)
\(984\) −6.00000 −0.191273
\(985\) −24.0000 −0.764704
\(986\) −18.0000 −0.573237
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) −3.00000 −0.0953945
\(990\) −3.00000 −0.0953463
\(991\) −25.0000 −0.794151 −0.397076 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) −5.00000 −0.158750
\(993\) 32.0000 1.01549
\(994\) 24.0000 0.761234
\(995\) −8.00000 −0.253617
\(996\) −6.00000 −0.190117
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) −32.0000 −1.01294
\(999\) −7.00000 −0.221470
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 5070.2.a.j.1.1 1
13.3 even 3 390.2.i.c.61.1 2
13.5 odd 4 5070.2.b.j.1351.2 2
13.8 odd 4 5070.2.b.j.1351.1 2
13.9 even 3 390.2.i.c.211.1 yes 2
13.12 even 2 5070.2.a.v.1.1 1
39.29 odd 6 1170.2.i.d.451.1 2
39.35 odd 6 1170.2.i.d.991.1 2
65.3 odd 12 1950.2.z.k.1699.1 4
65.9 even 6 1950.2.i.n.601.1 2
65.22 odd 12 1950.2.z.k.1849.1 4
65.29 even 6 1950.2.i.n.451.1 2
65.42 odd 12 1950.2.z.k.1699.2 4
65.48 odd 12 1950.2.z.k.1849.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.c.61.1 2 13.3 even 3
390.2.i.c.211.1 yes 2 13.9 even 3
1170.2.i.d.451.1 2 39.29 odd 6
1170.2.i.d.991.1 2 39.35 odd 6
1950.2.i.n.451.1 2 65.29 even 6
1950.2.i.n.601.1 2 65.9 even 6
1950.2.z.k.1699.1 4 65.3 odd 12
1950.2.z.k.1699.2 4 65.42 odd 12
1950.2.z.k.1849.1 4 65.22 odd 12
1950.2.z.k.1849.2 4 65.48 odd 12
5070.2.a.j.1.1 1 1.1 even 1 trivial
5070.2.a.v.1.1 1 13.12 even 2
5070.2.b.j.1351.1 2 13.8 odd 4
5070.2.b.j.1351.2 2 13.5 odd 4