Properties

Label 8005.2.a.a.1.1
Level 8005
Weight 2
Character 8005.1
Self dual yes
Analytic conductor 63.920
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 8005 = 5 \cdot 1601 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8005.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9202468180\)
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
Root \(0\) of \(x\)
Character \(\chi\) \(=\) 8005.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -6.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} +6.00000 q^{22} +2.00000 q^{23} -6.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{27} -2.00000 q^{28} +2.00000 q^{29} +2.00000 q^{30} -8.00000 q^{31} -5.00000 q^{32} +12.0000 q^{33} -6.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} +4.00000 q^{39} +3.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} +8.00000 q^{43} +6.00000 q^{44} +1.00000 q^{45} -2.00000 q^{46} +12.0000 q^{47} +2.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -12.0000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -4.00000 q^{54} -6.00000 q^{55} +6.00000 q^{56} +8.00000 q^{57} -2.00000 q^{58} +10.0000 q^{59} +2.00000 q^{60} -2.00000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} -12.0000 q^{66} -8.00000 q^{67} -6.00000 q^{68} -4.00000 q^{69} -2.00000 q^{70} +10.0000 q^{71} +3.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} -2.00000 q^{75} +4.00000 q^{76} -12.0000 q^{77} -4.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} +6.00000 q^{83} +4.00000 q^{84} +6.00000 q^{85} -8.00000 q^{86} -4.00000 q^{87} -18.0000 q^{88} -2.00000 q^{89} -1.00000 q^{90} -4.00000 q^{91} -2.00000 q^{92} +16.0000 q^{93} -12.0000 q^{94} -4.00000 q^{95} +10.0000 q^{96} +10.0000 q^{97} +3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) 2.00000 0.816497
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −2.00000 −0.534522
\(15\) −2.00000 −0.516398
\(16\) −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
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) −4.00000 −0.872872
\(22\) 6.00000 1.27920
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) −6.00000 −1.22474
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 4.00000 0.769800
\(28\) −2.00000 −0.377964
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 2.00000 0.365148
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −5.00000 −0.883883
\(33\) 12.0000 2.08893
\(34\) −6.00000 −1.02899
\(35\) 2.00000 0.338062
\(36\) −1.00000 −0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 4.00000 0.648886
\(39\) 4.00000 0.640513
\(40\) 3.00000 0.474342
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000 0.617213
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 6.00000 0.904534
\(45\) 1.00000 0.149071
\(46\) −2.00000 −0.294884
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 2.00000 0.288675
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) −12.0000 −1.68034
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −4.00000 −0.544331
\(55\) −6.00000 −0.809040
\(56\) 6.00000 0.801784
\(57\) 8.00000 1.05963
\(58\) −2.00000 −0.262613
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 2.00000 0.258199
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000 1.01600
\(63\) 2.00000 0.251976
\(64\) 7.00000 0.875000
\(65\) −2.00000 −0.248069
\(66\) −12.0000 −1.47710
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −6.00000 −0.727607
\(69\) −4.00000 −0.481543
\(70\) −2.00000 −0.239046
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 3.00000 0.353553
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −2.00000 −0.232495
\(75\) −2.00000 −0.230940
\(76\) 4.00000 0.458831
\(77\) −12.0000 −1.36753
\(78\) −4.00000 −0.452911
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 4.00000 0.436436
\(85\) 6.00000 0.650791
\(86\) −8.00000 −0.862662
\(87\) −4.00000 −0.428845
\(88\) −18.0000 −1.91881
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) −1.00000 −0.105409
\(91\) −4.00000 −0.419314
\(92\) −2.00000 −0.208514
\(93\) 16.0000 1.65912
\(94\) −12.0000 −1.23771
\(95\) −4.00000 −0.410391
\(96\) 10.0000 1.02062
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 3.00000 0.303046
\(99\) −6.00000 −0.603023
\(100\) −1.00000 −0.100000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 12.0000 1.18818
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −6.00000 −0.588348
\(105\) −4.00000 −0.390360
\(106\) 6.00000 0.582772
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −4.00000 −0.384900
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 6.00000 0.572078
\(111\) −4.00000 −0.379663
\(112\) −2.00000 −0.188982
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −8.00000 −0.749269
\(115\) 2.00000 0.186501
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) −10.0000 −0.920575
\(119\) 12.0000 1.10004
\(120\) −6.00000 −0.547723
\(121\) 25.0000 2.27273
\(122\) 2.00000 0.181071
\(123\) 12.0000 1.08200
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) −2.00000 −0.178174
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 3.00000 0.265165
\(129\) −16.0000 −1.40872
\(130\) 2.00000 0.175412
\(131\) −10.0000 −0.873704 −0.436852 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) −12.0000 −1.04447
\(133\) −8.00000 −0.693688
\(134\) 8.00000 0.691095
\(135\) 4.00000 0.344265
\(136\) 18.0000 1.54349
\(137\) 14.0000 1.19610 0.598050 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) 4.00000 0.340503
\(139\) 18.0000 1.52674 0.763370 0.645961i \(-0.223543\pi\)
0.763370 + 0.645961i \(0.223543\pi\)
\(140\) −2.00000 −0.169031
\(141\) −24.0000 −2.02116
\(142\) −10.0000 −0.839181
\(143\) 12.0000 1.00349
\(144\) −1.00000 −0.0833333
\(145\) 2.00000 0.166091
\(146\) 2.00000 0.165521
\(147\) 6.00000 0.494872
\(148\) −2.00000 −0.164399
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 2.00000 0.163299
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −12.0000 −0.973329
\(153\) 6.00000 0.485071
\(154\) 12.0000 0.966988
\(155\) −8.00000 −0.642575
\(156\) −4.00000 −0.320256
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 8.00000 0.636446
\(159\) 12.0000 0.951662
\(160\) −5.00000 −0.395285
\(161\) 4.00000 0.315244
\(162\) 11.0000 0.864242
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 6.00000 0.468521
\(165\) 12.0000 0.934199
\(166\) −6.00000 −0.465690
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −12.0000 −0.925820
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) −4.00000 −0.305888
\(172\) −8.00000 −0.609994
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 4.00000 0.303239
\(175\) 2.00000 0.151186
\(176\) 6.00000 0.452267
\(177\) −20.0000 −1.50329
\(178\) 2.00000 0.149906
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 4.00000 0.296500
\(183\) 4.00000 0.295689
\(184\) 6.00000 0.442326
\(185\) 2.00000 0.147043
\(186\) −16.0000 −1.17318
\(187\) −36.0000 −2.63258
\(188\) −12.0000 −0.875190
\(189\) 8.00000 0.581914
\(190\) 4.00000 0.290191
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) −14.0000 −1.01036
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) −10.0000 −0.717958
\(195\) 4.00000 0.286446
\(196\) 3.00000 0.214286
\(197\) 26.0000 1.85242 0.926212 0.377004i \(-0.123046\pi\)
0.926212 + 0.377004i \(0.123046\pi\)
\(198\) 6.00000 0.426401
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 3.00000 0.212132
\(201\) 16.0000 1.12855
\(202\) 14.0000 0.985037
\(203\) 4.00000 0.280745
\(204\) 12.0000 0.840168
\(205\) −6.00000 −0.419058
\(206\) 4.00000 0.278693
\(207\) 2.00000 0.139010
\(208\) 2.00000 0.138675
\(209\) 24.0000 1.66011
\(210\) 4.00000 0.276026
\(211\) −14.0000 −0.963800 −0.481900 0.876226i \(-0.660053\pi\)
−0.481900 + 0.876226i \(0.660053\pi\)
\(212\) 6.00000 0.412082
\(213\) −20.0000 −1.37038
\(214\) 6.00000 0.410152
\(215\) 8.00000 0.545595
\(216\) 12.0000 0.816497
\(217\) −16.0000 −1.08615
\(218\) 14.0000 0.948200
\(219\) 4.00000 0.270295
\(220\) 6.00000 0.404520
\(221\) −12.0000 −0.807207
\(222\) 4.00000 0.268462
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −10.0000 −0.668153
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) −8.00000 −0.529813
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −2.00000 −0.131876
\(231\) 24.0000 1.57908
\(232\) 6.00000 0.393919
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 2.00000 0.130744
\(235\) 12.0000 0.782794
\(236\) −10.0000 −0.650945
\(237\) 16.0000 1.03931
\(238\) −12.0000 −0.777844
\(239\) 22.0000 1.42306 0.711531 0.702655i \(-0.248002\pi\)
0.711531 + 0.702655i \(0.248002\pi\)
\(240\) 2.00000 0.129099
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −25.0000 −1.60706
\(243\) 10.0000 0.641500
\(244\) 2.00000 0.128037
\(245\) −3.00000 −0.191663
\(246\) −12.0000 −0.765092
\(247\) 8.00000 0.509028
\(248\) −24.0000 −1.52400
\(249\) −12.0000 −0.760469
\(250\) −1.00000 −0.0632456
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) −2.00000 −0.125988
\(253\) −12.0000 −0.754434
\(254\) 6.00000 0.376473
\(255\) −12.0000 −0.751469
\(256\) −17.0000 −1.06250
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 16.0000 0.996116
\(259\) 4.00000 0.248548
\(260\) 2.00000 0.124035
\(261\) 2.00000 0.123797
\(262\) 10.0000 0.617802
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 36.0000 2.21565
\(265\) −6.00000 −0.368577
\(266\) 8.00000 0.490511
\(267\) 4.00000 0.244796
\(268\) 8.00000 0.488678
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) −4.00000 −0.243432
\(271\) 6.00000 0.364474 0.182237 0.983255i \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(272\) −6.00000 −0.363803
\(273\) 8.00000 0.484182
\(274\) −14.0000 −0.845771
\(275\) −6.00000 −0.361814
\(276\) 4.00000 0.240772
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −18.0000 −1.07957
\(279\) −8.00000 −0.478947
\(280\) 6.00000 0.358569
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 24.0000 1.42918
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) −10.0000 −0.593391
\(285\) 8.00000 0.473879
\(286\) −12.0000 −0.709575
\(287\) −12.0000 −0.708338
\(288\) −5.00000 −0.294628
\(289\) 19.0000 1.11765
\(290\) −2.00000 −0.117444
\(291\) −20.0000 −1.17242
\(292\) 2.00000 0.117041
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) −6.00000 −0.349927
\(295\) 10.0000 0.582223
\(296\) 6.00000 0.348743
\(297\) −24.0000 −1.39262
\(298\) −10.0000 −0.579284
\(299\) −4.00000 −0.231326
\(300\) 2.00000 0.115470
\(301\) 16.0000 0.922225
\(302\) 8.00000 0.460348
\(303\) 28.0000 1.60856
\(304\) 4.00000 0.229416
\(305\) −2.00000 −0.114520
\(306\) −6.00000 −0.342997
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 12.0000 0.683763
\(309\) 8.00000 0.455104
\(310\) 8.00000 0.454369
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) 12.0000 0.679366
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −18.0000 −1.01580
\(315\) 2.00000 0.112687
\(316\) 8.00000 0.450035
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −12.0000 −0.672927
\(319\) −12.0000 −0.671871
\(320\) 7.00000 0.391312
\(321\) 12.0000 0.669775
\(322\) −4.00000 −0.222911
\(323\) −24.0000 −1.33540
\(324\) 11.0000 0.611111
\(325\) −2.00000 −0.110940
\(326\) −4.00000 −0.221540
\(327\) 28.0000 1.54840
\(328\) −18.0000 −0.993884
\(329\) 24.0000 1.32316
\(330\) −12.0000 −0.660578
\(331\) 6.00000 0.329790 0.164895 0.986311i \(-0.447272\pi\)
0.164895 + 0.986311i \(0.447272\pi\)
\(332\) −6.00000 −0.329293
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 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
\(339\) −12.0000 −0.651751
\(340\) −6.00000 −0.325396
\(341\) 48.0000 2.59935
\(342\) 4.00000 0.216295
\(343\) −20.0000 −1.07990
\(344\) 24.0000 1.29399
\(345\) −4.00000 −0.215353
\(346\) −6.00000 −0.322562
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 4.00000 0.214423
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) −2.00000 −0.106904
\(351\) −8.00000 −0.427008
\(352\) 30.0000 1.59901
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 20.0000 1.06299
\(355\) 10.0000 0.530745
\(356\) 2.00000 0.106000
\(357\) −24.0000 −1.27021
\(358\) 16.0000 0.845626
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 3.00000 0.158114
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) −50.0000 −2.62432
\(364\) 4.00000 0.209657
\(365\) −2.00000 −0.104685
\(366\) −4.00000 −0.209083
\(367\) 34.0000 1.77479 0.887393 0.461014i \(-0.152514\pi\)
0.887393 + 0.461014i \(0.152514\pi\)
\(368\) −2.00000 −0.104257
\(369\) −6.00000 −0.312348
\(370\) −2.00000 −0.103975
\(371\) −12.0000 −0.623009
\(372\) −16.0000 −0.829561
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 36.0000 1.86152
\(375\) −2.00000 −0.103280
\(376\) 36.0000 1.85656
\(377\) −4.00000 −0.206010
\(378\) −8.00000 −0.411476
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) 4.00000 0.205196
\(381\) 12.0000 0.614779
\(382\) 10.0000 0.511645
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) −6.00000 −0.306186
\(385\) −12.0000 −0.611577
\(386\) −22.0000 −1.11977
\(387\) 8.00000 0.406663
\(388\) −10.0000 −0.507673
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) −4.00000 −0.202548
\(391\) 12.0000 0.606866
\(392\) −9.00000 −0.454569
\(393\) 20.0000 1.00887
\(394\) −26.0000 −1.30986
\(395\) −8.00000 −0.402524
\(396\) 6.00000 0.301511
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) −4.00000 −0.200502
\(399\) 16.0000 0.801002
\(400\) −1.00000 −0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) −16.0000 −0.798007
\(403\) 16.0000 0.797017
\(404\) 14.0000 0.696526
\(405\) −11.0000 −0.546594
\(406\) −4.00000 −0.198517
\(407\) −12.0000 −0.594818
\(408\) −36.0000 −1.78227
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 6.00000 0.296319
\(411\) −28.0000 −1.38114
\(412\) 4.00000 0.197066
\(413\) 20.0000 0.984136
\(414\) −2.00000 −0.0982946
\(415\) 6.00000 0.294528
\(416\) 10.0000 0.490290
\(417\) −36.0000 −1.76293
\(418\) −24.0000 −1.17388
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 4.00000 0.195180
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) 14.0000 0.681509
\(423\) 12.0000 0.583460
\(424\) −18.0000 −0.874157
\(425\) 6.00000 0.291043
\(426\) 20.0000 0.969003
\(427\) −4.00000 −0.193574
\(428\) 6.00000 0.290021
\(429\) −24.0000 −1.15873
\(430\) −8.00000 −0.385794
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) −4.00000 −0.192450
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 16.0000 0.768025
\(435\) −4.00000 −0.191785
\(436\) 14.0000 0.670478
\(437\) −8.00000 −0.382692
\(438\) −4.00000 −0.191127
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) −18.0000 −0.858116
\(441\) −3.00000 −0.142857
\(442\) 12.0000 0.570782
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 4.00000 0.189832
\(445\) −2.00000 −0.0948091
\(446\) 2.00000 0.0947027
\(447\) −20.0000 −0.945968
\(448\) 14.0000 0.661438
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 36.0000 1.69517
\(452\) −6.00000 −0.282216
\(453\) 16.0000 0.751746
\(454\) 24.0000 1.12638
\(455\) −4.00000 −0.187523
\(456\) 24.0000 1.12390
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) −10.0000 −0.467269
\(459\) 24.0000 1.12022
\(460\) −2.00000 −0.0932505
\(461\) 34.0000 1.58354 0.791769 0.610821i \(-0.209160\pi\)
0.791769 + 0.610821i \(0.209160\pi\)
\(462\) −24.0000 −1.11658
\(463\) 18.0000 0.836531 0.418265 0.908325i \(-0.362638\pi\)
0.418265 + 0.908325i \(0.362638\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 16.0000 0.741982
\(466\) −18.0000 −0.833834
\(467\) −30.0000 −1.38823 −0.694117 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 2.00000 0.0924500
\(469\) −16.0000 −0.738811
\(470\) −12.0000 −0.553519
\(471\) −36.0000 −1.65879
\(472\) 30.0000 1.38086
\(473\) −48.0000 −2.20704
\(474\) −16.0000 −0.734904
\(475\) −4.00000 −0.183533
\(476\) −12.0000 −0.550019
\(477\) −6.00000 −0.274721
\(478\) −22.0000 −1.00626
\(479\) −14.0000 −0.639676 −0.319838 0.947472i \(-0.603629\pi\)
−0.319838 + 0.947472i \(0.603629\pi\)
\(480\) 10.0000 0.456435
\(481\) −4.00000 −0.182384
\(482\) −10.0000 −0.455488
\(483\) −8.00000 −0.364013
\(484\) −25.0000 −1.13636
\(485\) 10.0000 0.454077
\(486\) −10.0000 −0.453609
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) −6.00000 −0.271607
\(489\) −8.00000 −0.361773
\(490\) 3.00000 0.135526
\(491\) 14.0000 0.631811 0.315906 0.948791i \(-0.397692\pi\)
0.315906 + 0.948791i \(0.397692\pi\)
\(492\) −12.0000 −0.541002
\(493\) 12.0000 0.540453
\(494\) −8.00000 −0.359937
\(495\) −6.00000 −0.269680
\(496\) 8.00000 0.359211
\(497\) 20.0000 0.897123
\(498\) 12.0000 0.537733
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 2.00000 0.0892644
\(503\) 4.00000 0.178351 0.0891756 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) 6.00000 0.267261
\(505\) −14.0000 −0.622992
\(506\) 12.0000 0.533465
\(507\) 18.0000 0.799408
\(508\) 6.00000 0.266207
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 12.0000 0.531369
\(511\) −4.00000 −0.176950
\(512\) 11.0000 0.486136
\(513\) −16.0000 −0.706417
\(514\) 18.0000 0.793946
\(515\) −4.00000 −0.176261
\(516\) 16.0000 0.704361
\(517\) −72.0000 −3.16656
\(518\) −4.00000 −0.175750
\(519\) −12.0000 −0.526742
\(520\) −6.00000 −0.263117
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −6.00000 −0.262362 −0.131181 0.991358i \(-0.541877\pi\)
−0.131181 + 0.991358i \(0.541877\pi\)
\(524\) 10.0000 0.436852
\(525\) −4.00000 −0.174574
\(526\) 12.0000 0.523225
\(527\) −48.0000 −2.09091
\(528\) −12.0000 −0.522233
\(529\) −19.0000 −0.826087
\(530\) 6.00000 0.260623
\(531\) 10.0000 0.433963
\(532\) 8.00000 0.346844
\(533\) 12.0000 0.519778
\(534\) −4.00000 −0.173097
\(535\) −6.00000 −0.259403
\(536\) −24.0000 −1.03664
\(537\) 32.0000 1.38090
\(538\) −14.0000 −0.603583
\(539\) 18.0000 0.775315
\(540\) −4.00000 −0.172133
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) −6.00000 −0.257722
\(543\) −20.0000 −0.858282
\(544\) −30.0000 −1.28624
\(545\) −14.0000 −0.599694
\(546\) −8.00000 −0.342368
\(547\) −38.0000 −1.62476 −0.812381 0.583127i \(-0.801829\pi\)
−0.812381 + 0.583127i \(0.801829\pi\)
\(548\) −14.0000 −0.598050
\(549\) −2.00000 −0.0853579
\(550\) 6.00000 0.255841
\(551\) −8.00000 −0.340811
\(552\) −12.0000 −0.510754
\(553\) −16.0000 −0.680389
\(554\) 10.0000 0.424859
\(555\) −4.00000 −0.169791
\(556\) −18.0000 −0.763370
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 8.00000 0.338667
\(559\) −16.0000 −0.676728
\(560\) −2.00000 −0.0845154
\(561\) 72.0000 3.03984
\(562\) −30.0000 −1.26547
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 24.0000 1.01058
\(565\) 6.00000 0.252422
\(566\) −14.0000 −0.588464
\(567\) −22.0000 −0.923913
\(568\) 30.0000 1.25877
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) −8.00000 −0.335083
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) −12.0000 −0.501745
\(573\) 20.0000 0.835512
\(574\) 12.0000 0.500870
\(575\) 2.00000 0.0834058
\(576\) 7.00000 0.291667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −19.0000 −0.790296
\(579\) −44.0000 −1.82858
\(580\) −2.00000 −0.0830455
\(581\) 12.0000 0.497844
\(582\) 20.0000 0.829027
\(583\) 36.0000 1.49097
\(584\) −6.00000 −0.248282
\(585\) −2.00000 −0.0826898
\(586\) 26.0000 1.07405
\(587\) −30.0000 −1.23823 −0.619116 0.785299i \(-0.712509\pi\)
−0.619116 + 0.785299i \(0.712509\pi\)
\(588\) −6.00000 −0.247436
\(589\) 32.0000 1.31854
\(590\) −10.0000 −0.411693
\(591\) −52.0000 −2.13899
\(592\) −2.00000 −0.0821995
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 24.0000 0.984732
\(595\) 12.0000 0.491952
\(596\) −10.0000 −0.409616
\(597\) −8.00000 −0.327418
\(598\) 4.00000 0.163572
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) −6.00000 −0.244949
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) −16.0000 −0.652111
\(603\) −8.00000 −0.325785
\(604\) 8.00000 0.325515
\(605\) 25.0000 1.01639
\(606\) −28.0000 −1.13742
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 20.0000 0.811107
\(609\) −8.00000 −0.324176
\(610\) 2.00000 0.0809776
\(611\) −24.0000 −0.970936
\(612\) −6.00000 −0.242536
\(613\) 22.0000 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) 32.0000 1.29141
\(615\) 12.0000 0.483887
\(616\) −36.0000 −1.45048
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −8.00000 −0.321807
\(619\) −42.0000 −1.68812 −0.844061 0.536247i \(-0.819842\pi\)
−0.844061 + 0.536247i \(0.819842\pi\)
\(620\) 8.00000 0.321288
\(621\) 8.00000 0.321029
\(622\) 10.0000 0.400963
\(623\) −4.00000 −0.160257
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) −48.0000 −1.91694
\(628\) −18.0000 −0.718278
\(629\) 12.0000 0.478471
\(630\) −2.00000 −0.0796819
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) −24.0000 −0.954669
\(633\) 28.0000 1.11290
\(634\) 2.00000 0.0794301
\(635\) −6.00000 −0.238103
\(636\) −12.0000 −0.475831
\(637\) 6.00000 0.237729
\(638\) 12.0000 0.475085
\(639\) 10.0000 0.395594
\(640\) 3.00000 0.118585
\(641\) 34.0000 1.34292 0.671460 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) −12.0000 −0.473602
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) −4.00000 −0.157622
\(645\) −16.0000 −0.629999
\(646\) 24.0000 0.944267
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) −33.0000 −1.29636
\(649\) −60.0000 −2.35521
\(650\) 2.00000 0.0784465
\(651\) 32.0000 1.25418
\(652\) −4.00000 −0.156652
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) −28.0000 −1.09489
\(655\) −10.0000 −0.390732
\(656\) 6.00000 0.234261
\(657\) −2.00000 −0.0780274
\(658\) −24.0000 −0.935617
\(659\) −14.0000 −0.545363 −0.272681 0.962104i \(-0.587910\pi\)
−0.272681 + 0.962104i \(0.587910\pi\)
\(660\) −12.0000 −0.467099
\(661\) −34.0000 −1.32245 −0.661223 0.750189i \(-0.729962\pi\)
−0.661223 + 0.750189i \(0.729962\pi\)
\(662\) −6.00000 −0.233197
\(663\) 24.0000 0.932083
\(664\) 18.0000 0.698535
\(665\) −8.00000 −0.310227
\(666\) −2.00000 −0.0774984
\(667\) 4.00000 0.154881
\(668\) 0 0
\(669\) 4.00000 0.154649
\(670\) 8.00000 0.309067
\(671\) 12.0000 0.463255
\(672\) 20.0000 0.771517
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 4.00000 0.153960
\(676\) 9.00000 0.346154
\(677\) 14.0000 0.538064 0.269032 0.963131i \(-0.413296\pi\)
0.269032 + 0.963131i \(0.413296\pi\)
\(678\) 12.0000 0.460857
\(679\) 20.0000 0.767530
\(680\) 18.0000 0.690268
\(681\) 48.0000 1.83936
\(682\) −48.0000 −1.83801
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 4.00000 0.152944
\(685\) 14.0000 0.534913
\(686\) 20.0000 0.763604
\(687\) −20.0000 −0.763048
\(688\) −8.00000 −0.304997
\(689\) 12.0000 0.457164
\(690\) 4.00000 0.152277
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −6.00000 −0.228086
\(693\) −12.0000 −0.455842
\(694\) 8.00000 0.303676
\(695\) 18.0000 0.682779
\(696\) −12.0000 −0.454859
\(697\) −36.0000 −1.36360
\(698\) −22.0000 −0.832712
\(699\) −36.0000 −1.36165
\(700\) −2.00000 −0.0755929
\(701\) −46.0000 −1.73740 −0.868698 0.495342i \(-0.835043\pi\)
−0.868698 + 0.495342i \(0.835043\pi\)
\(702\) 8.00000 0.301941
\(703\) −8.00000 −0.301726
\(704\) −42.0000 −1.58293
\(705\) −24.0000 −0.903892
\(706\) −10.0000 −0.376355
\(707\) −28.0000 −1.05305
\(708\) 20.0000 0.751646
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) −10.0000 −0.375293
\(711\) −8.00000 −0.300023
\(712\) −6.00000 −0.224860
\(713\) −16.0000 −0.599205
\(714\) 24.0000 0.898177
\(715\) 12.0000 0.448775
\(716\) 16.0000 0.597948
\(717\) −44.0000 −1.64321
\(718\) 36.0000 1.34351
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −8.00000 −0.297936
\(722\) 3.00000 0.111648
\(723\) −20.0000 −0.743808
\(724\) −10.0000 −0.371647
\(725\) 2.00000 0.0742781
\(726\) 50.0000 1.85567
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) −12.0000 −0.444750
\(729\) 13.0000 0.481481
\(730\) 2.00000 0.0740233
\(731\) 48.0000 1.77534
\(732\) −4.00000 −0.147844
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) −34.0000 −1.25496
\(735\) 6.00000 0.221313
\(736\) −10.0000 −0.368605
\(737\) 48.0000 1.76810
\(738\) 6.00000 0.220863
\(739\) 14.0000 0.514998 0.257499 0.966279i \(-0.417102\pi\)
0.257499 + 0.966279i \(0.417102\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −16.0000 −0.587775
\(742\) 12.0000 0.440534
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 48.0000 1.75977
\(745\) 10.0000 0.366372
\(746\) −2.00000 −0.0732252
\(747\) 6.00000 0.219529
\(748\) 36.0000 1.31629
\(749\) −12.0000 −0.438470
\(750\) 2.00000 0.0730297
\(751\) 38.0000 1.38664 0.693320 0.720630i \(-0.256147\pi\)
0.693320 + 0.720630i \(0.256147\pi\)
\(752\) −12.0000 −0.437595
\(753\) 4.00000 0.145768
\(754\) 4.00000 0.145671
\(755\) −8.00000 −0.291150
\(756\) −8.00000 −0.290957
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 34.0000 1.23494
\(759\) 24.0000 0.871145
\(760\) −12.0000 −0.435286
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) −12.0000 −0.434714
\(763\) −28.0000 −1.01367
\(764\) 10.0000 0.361787
\(765\) 6.00000 0.216930
\(766\) 16.0000 0.578103
\(767\) −20.0000 −0.722158
\(768\) 34.0000 1.22687
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 12.0000 0.432450
\(771\) 36.0000 1.29651
\(772\) −22.0000 −0.791797
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −8.00000 −0.287554
\(775\) −8.00000 −0.287368
\(776\) 30.0000 1.07694
\(777\) −8.00000 −0.286998
\(778\) 2.00000 0.0717035
\(779\) 24.0000 0.859889
\(780\) −4.00000 −0.143223
\(781\) −60.0000 −2.14697
\(782\) −12.0000 −0.429119
\(783\) 8.00000 0.285897
\(784\) 3.00000 0.107143
\(785\) 18.0000 0.642448
\(786\) −20.0000 −0.713376
\(787\) −38.0000 −1.35455 −0.677277 0.735728i \(-0.736840\pi\)
−0.677277 + 0.735728i \(0.736840\pi\)
\(788\) −26.0000 −0.926212
\(789\) 24.0000 0.854423
\(790\) 8.00000 0.284627
\(791\) 12.0000 0.426671
\(792\) −18.0000 −0.639602
\(793\) 4.00000 0.142044
\(794\) 18.0000 0.638796
\(795\) 12.0000 0.425596
\(796\) −4.00000 −0.141776
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) −16.0000 −0.566394
\(799\) 72.0000 2.54718
\(800\) −5.00000 −0.176777
\(801\) −2.00000 −0.0706665
\(802\) 30.0000 1.05934
\(803\) 12.0000 0.423471
\(804\) −16.0000 −0.564276
\(805\) 4.00000 0.140981
\(806\) −16.0000 −0.563576
\(807\) −28.0000 −0.985647
\(808\) −42.0000 −1.47755
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 11.0000 0.386501
\(811\) 36.0000 1.26413 0.632065 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) −4.00000 −0.140372
\(813\) −12.0000 −0.420858
\(814\) 12.0000 0.420600
\(815\) 4.00000 0.140114
\(816\) 12.0000 0.420084
\(817\) −32.0000 −1.11954
\(818\) −2.00000 −0.0699284
\(819\) −4.00000 −0.139771
\(820\) 6.00000 0.209529
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 28.0000 0.976612
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −12.0000 −0.418040
\(825\) 12.0000 0.417786
\(826\) −20.0000 −0.695889
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −2.00000 −0.0695048
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) −6.00000 −0.208263
\(831\) 20.0000 0.693792
\(832\) −14.0000 −0.485363
\(833\) −18.0000 −0.623663
\(834\) 36.0000 1.24658
\(835\) 0 0
\(836\) −24.0000 −0.830057
\(837\) −32.0000 −1.10608
\(838\) 6.00000 0.207267
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) −12.0000 −0.414039
\(841\) −25.0000 −0.862069
\(842\) −18.0000 −0.620321
\(843\) −60.0000 −2.06651
\(844\) 14.0000 0.481900
\(845\) −9.00000 −0.309609
\(846\) −12.0000 −0.412568
\(847\) 50.0000 1.71802
\(848\) 6.00000 0.206041
\(849\) −28.0000 −0.960958
\(850\) −6.00000 −0.205798
\(851\) 4.00000 0.137118
\(852\) 20.0000 0.685189
\(853\) 42.0000 1.43805 0.719026 0.694983i \(-0.244588\pi\)
0.719026 + 0.694983i \(0.244588\pi\)
\(854\) 4.00000 0.136877
\(855\) −4.00000 −0.136797
\(856\) −18.0000 −0.615227
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 24.0000 0.819346
\(859\) 22.0000 0.750630 0.375315 0.926897i \(-0.377534\pi\)
0.375315 + 0.926897i \(0.377534\pi\)
\(860\) −8.00000 −0.272798
\(861\) 24.0000 0.817918
\(862\) 18.0000 0.613082
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −20.0000 −0.680414
\(865\) 6.00000 0.204006
\(866\) 26.0000 0.883516
\(867\) −38.0000 −1.29055
\(868\) 16.0000 0.543075
\(869\) 48.0000 1.62829
\(870\) 4.00000 0.135613
\(871\) 16.0000 0.542139
\(872\) −42.0000 −1.42230
\(873\) 10.0000 0.338449
\(874\) 8.00000 0.270604
\(875\) 2.00000 0.0676123
\(876\) −4.00000 −0.135147
\(877\) −58.0000 −1.95852 −0.979260 0.202606i \(-0.935059\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) −8.00000 −0.269987
\(879\) 52.0000 1.75392
\(880\) 6.00000 0.202260
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 3.00000 0.101015
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 12.0000 0.403604
\(885\) −20.0000 −0.672293
\(886\) −16.0000 −0.537531
\(887\) 58.0000 1.94745 0.973725 0.227728i \(-0.0731298\pi\)
0.973725 + 0.227728i \(0.0731298\pi\)
\(888\) −12.0000 −0.402694
\(889\) −12.0000 −0.402467
\(890\) 2.00000 0.0670402
\(891\) 66.0000 2.21108
\(892\) 2.00000 0.0669650
\(893\) −48.0000 −1.60626
\(894\) 20.0000 0.668900
\(895\) −16.0000 −0.534821
\(896\) 6.00000 0.200446
\(897\) 8.00000 0.267112
\(898\) −22.0000 −0.734150
\(899\) −16.0000 −0.533630
\(900\) −1.00000 −0.0333333
\(901\) −36.0000 −1.19933
\(902\) −36.0000 −1.19867
\(903\) −32.0000 −1.06489
\(904\) 18.0000 0.598671
\(905\) 10.0000 0.332411
\(906\) −16.0000 −0.531564
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) 24.0000 0.796468
\(909\) −14.0000 −0.464351
\(910\) 4.00000 0.132599
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −8.00000 −0.264906
\(913\) −36.0000 −1.19143
\(914\) −22.0000 −0.727695
\(915\) 4.00000 0.132236
\(916\) −10.0000 −0.330409
\(917\) −20.0000 −0.660458
\(918\) −24.0000 −0.792118
\(919\) −46.0000 −1.51740 −0.758700 0.651440i \(-0.774165\pi\)
−0.758700 + 0.651440i \(0.774165\pi\)
\(920\) 6.00000 0.197814
\(921\) 64.0000 2.10887
\(922\) −34.0000 −1.11973
\(923\) −20.0000 −0.658308
\(924\) −24.0000 −0.789542
\(925\) 2.00000 0.0657596
\(926\) −18.0000 −0.591517
\(927\) −4.00000 −0.131377
\(928\) −10.0000 −0.328266
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) −16.0000 −0.524661
\(931\) 12.0000 0.393284
\(932\) −18.0000 −0.589610
\(933\) 20.0000 0.654771
\(934\) 30.0000 0.981630
\(935\) −36.0000 −1.17733
\(936\) −6.00000 −0.196116
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 16.0000 0.522419
\(939\) −12.0000 −0.391605
\(940\) −12.0000 −0.391397
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 36.0000 1.17294
\(943\) −12.0000 −0.390774
\(944\) −10.0000 −0.325472
\(945\) 8.00000 0.260240
\(946\) 48.0000 1.56061
\(947\) −38.0000 −1.23483 −0.617417 0.786636i \(-0.711821\pi\)
−0.617417 + 0.786636i \(0.711821\pi\)
\(948\) −16.0000 −0.519656
\(949\) 4.00000 0.129845
\(950\) 4.00000 0.129777
\(951\) 4.00000 0.129709
\(952\) 36.0000 1.16677
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 6.00000 0.194257
\(955\) −10.0000 −0.323592
\(956\) −22.0000 −0.711531
\(957\) 24.0000 0.775810
\(958\) 14.0000 0.452319
\(959\) 28.0000 0.904167
\(960\) −14.0000 −0.451848
\(961\) 33.0000 1.06452
\(962\) 4.00000 0.128965
\(963\) −6.00000 −0.193347
\(964\) −10.0000 −0.322078
\(965\) 22.0000 0.708205
\(966\) 8.00000 0.257396
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 75.0000 2.41059
\(969\) 48.0000 1.54198
\(970\) −10.0000 −0.321081
\(971\) −38.0000 −1.21948 −0.609739 0.792602i \(-0.708726\pi\)
−0.609739 + 0.792602i \(0.708726\pi\)
\(972\) −10.0000 −0.320750
\(973\) 36.0000 1.15411
\(974\) 4.00000 0.128168
\(975\) 4.00000 0.128103
\(976\) 2.00000 0.0640184
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 8.00000 0.255812
\(979\) 12.0000 0.383522
\(980\) 3.00000 0.0958315
\(981\) −14.0000 −0.446986
\(982\) −14.0000 −0.446758
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 36.0000 1.14764
\(985\) 26.0000 0.828429
\(986\) −12.0000 −0.382158
\(987\) −48.0000 −1.52786
\(988\) −8.00000 −0.254514
\(989\) 16.0000 0.508770
\(990\) 6.00000 0.190693
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 40.0000 1.27000
\(993\) −12.0000 −0.380808
\(994\) −20.0000 −0.634361
\(995\) 4.00000 0.126809
\(996\) 12.0000 0.380235
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) 20.0000 0.633089
\(999\) 8.00000 0.253109
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 8005.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8005.2.a.a.1.1 1 1.1 even 1 trivial