Properties

Label 1078.2.a.m.1.1
Level $1078$
Weight $2$
Character 1078.1
Self dual yes
Analytic conductor $8.608$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.60787333789\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1078.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} +2.00000 q^{10} -1.00000 q^{11} +3.00000 q^{12} -7.00000 q^{13} +6.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +6.00000 q^{18} +2.00000 q^{20} -1.00000 q^{22} -8.00000 q^{23} +3.00000 q^{24} -1.00000 q^{25} -7.00000 q^{26} +9.00000 q^{27} -5.00000 q^{29} +6.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} +2.00000 q^{34} +6.00000 q^{36} +4.00000 q^{37} -21.0000 q^{39} +2.00000 q^{40} +4.00000 q^{41} -8.00000 q^{43} -1.00000 q^{44} +12.0000 q^{45} -8.00000 q^{46} +2.00000 q^{47} +3.00000 q^{48} -1.00000 q^{50} +6.00000 q^{51} -7.00000 q^{52} -6.00000 q^{53} +9.00000 q^{54} -2.00000 q^{55} -5.00000 q^{58} +3.00000 q^{59} +6.00000 q^{60} +1.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -14.0000 q^{65} -3.00000 q^{66} +9.00000 q^{67} +2.00000 q^{68} -24.0000 q^{69} -2.00000 q^{71} +6.00000 q^{72} +4.00000 q^{73} +4.00000 q^{74} -3.00000 q^{75} -21.0000 q^{78} +9.00000 q^{79} +2.00000 q^{80} +9.00000 q^{81} +4.00000 q^{82} +6.00000 q^{83} +4.00000 q^{85} -8.00000 q^{86} -15.0000 q^{87} -1.00000 q^{88} +6.00000 q^{89} +12.0000 q^{90} -8.00000 q^{92} +12.0000 q^{93} +2.00000 q^{94} +3.00000 q^{96} +7.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 3.00000 1.22474
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 6.00000 2.00000
\(10\) 2.00000 0.632456
\(11\) −1.00000 −0.301511
\(12\) 3.00000 0.866025
\(13\) −7.00000 −1.94145 −0.970725 0.240192i \(-0.922790\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 6.00000 1.54919
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 6.00000 1.41421
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 3.00000 0.612372
\(25\) −1.00000 −0.200000
\(26\) −7.00000 −1.37281
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 6.00000 1.09545
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.00000 −0.522233
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) −21.0000 −3.36269
\(40\) 2.00000 0.316228
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −1.00000 −0.150756
\(45\) 12.0000 1.78885
\(46\) −8.00000 −1.17954
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 3.00000 0.433013
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 6.00000 0.840168
\(52\) −7.00000 −0.970725
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 9.00000 1.22474
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) −5.00000 −0.656532
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 6.00000 0.774597
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −14.0000 −1.73649
\(66\) −3.00000 −0.369274
\(67\) 9.00000 1.09952 0.549762 0.835321i \(-0.314718\pi\)
0.549762 + 0.835321i \(0.314718\pi\)
\(68\) 2.00000 0.242536
\(69\) −24.0000 −2.88926
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 6.00000 0.707107
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 4.00000 0.464991
\(75\) −3.00000 −0.346410
\(76\) 0 0
\(77\) 0 0
\(78\) −21.0000 −2.37778
\(79\) 9.00000 1.01258 0.506290 0.862364i \(-0.331017\pi\)
0.506290 + 0.862364i \(0.331017\pi\)
\(80\) 2.00000 0.223607
\(81\) 9.00000 1.00000
\(82\) 4.00000 0.441726
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) −8.00000 −0.862662
\(87\) −15.0000 −1.60817
\(88\) −1.00000 −0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 12.0000 1.26491
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 12.0000 1.24434
\(94\) 2.00000 0.206284
\(95\) 0 0
\(96\) 3.00000 0.306186
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) −1.00000 −0.100000
\(101\) −9.00000 −0.895533 −0.447767 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(102\) 6.00000 0.594089
\(103\) 18.0000 1.77359 0.886796 0.462160i \(-0.152926\pi\)
0.886796 + 0.462160i \(0.152926\pi\)
\(104\) −7.00000 −0.686406
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 9.00000 0.866025
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −2.00000 −0.190693
\(111\) 12.0000 1.13899
\(112\) 0 0
\(113\) 5.00000 0.470360 0.235180 0.971952i \(-0.424432\pi\)
0.235180 + 0.971952i \(0.424432\pi\)
\(114\) 0 0
\(115\) −16.0000 −1.49201
\(116\) −5.00000 −0.464238
\(117\) −42.0000 −3.88290
\(118\) 3.00000 0.276172
\(119\) 0 0
\(120\) 6.00000 0.547723
\(121\) 1.00000 0.0909091
\(122\) 1.00000 0.0905357
\(123\) 12.0000 1.08200
\(124\) 4.00000 0.359211
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −19.0000 −1.68598 −0.842989 0.537931i \(-0.819206\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) 1.00000 0.0883883
\(129\) −24.0000 −2.11308
\(130\) −14.0000 −1.22788
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −3.00000 −0.261116
\(133\) 0 0
\(134\) 9.00000 0.777482
\(135\) 18.0000 1.54919
\(136\) 2.00000 0.171499
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) −24.0000 −2.04302
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) −2.00000 −0.167836
\(143\) 7.00000 0.585369
\(144\) 6.00000 0.500000
\(145\) −10.0000 −0.830455
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −3.00000 −0.244949
\(151\) −3.00000 −0.244137 −0.122068 0.992522i \(-0.538953\pi\)
−0.122068 + 0.992522i \(0.538953\pi\)
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) −21.0000 −1.68135
\(157\) 16.0000 1.27694 0.638470 0.769647i \(-0.279568\pi\)
0.638470 + 0.769647i \(0.279568\pi\)
\(158\) 9.00000 0.716002
\(159\) −18.0000 −1.42749
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 9.00000 0.707107
\(163\) −17.0000 −1.33154 −0.665771 0.746156i \(-0.731897\pi\)
−0.665771 + 0.746156i \(0.731897\pi\)
\(164\) 4.00000 0.312348
\(165\) −6.00000 −0.467099
\(166\) 6.00000 0.465690
\(167\) 19.0000 1.47026 0.735132 0.677924i \(-0.237120\pi\)
0.735132 + 0.677924i \(0.237120\pi\)
\(168\) 0 0
\(169\) 36.0000 2.76923
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 25.0000 1.90071 0.950357 0.311160i \(-0.100718\pi\)
0.950357 + 0.311160i \(0.100718\pi\)
\(174\) −15.0000 −1.13715
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 9.00000 0.676481
\(178\) 6.00000 0.449719
\(179\) 19.0000 1.42013 0.710063 0.704138i \(-0.248666\pi\)
0.710063 + 0.704138i \(0.248666\pi\)
\(180\) 12.0000 0.894427
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) 3.00000 0.221766
\(184\) −8.00000 −0.589768
\(185\) 8.00000 0.588172
\(186\) 12.0000 0.879883
\(187\) −2.00000 −0.146254
\(188\) 2.00000 0.145865
\(189\) 0 0
\(190\) 0 0
\(191\) 2.00000 0.144715 0.0723575 0.997379i \(-0.476948\pi\)
0.0723575 + 0.997379i \(0.476948\pi\)
\(192\) 3.00000 0.216506
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) 7.00000 0.502571
\(195\) −42.0000 −3.00768
\(196\) 0 0
\(197\) 15.0000 1.06871 0.534353 0.845262i \(-0.320555\pi\)
0.534353 + 0.845262i \(0.320555\pi\)
\(198\) −6.00000 −0.426401
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 27.0000 1.90443
\(202\) −9.00000 −0.633238
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 8.00000 0.558744
\(206\) 18.0000 1.25412
\(207\) −48.0000 −3.33623
\(208\) −7.00000 −0.485363
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −6.00000 −0.412082
\(213\) −6.00000 −0.411113
\(214\) 2.00000 0.136717
\(215\) −16.0000 −1.09119
\(216\) 9.00000 0.612372
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 12.0000 0.810885
\(220\) −2.00000 −0.134840
\(221\) −14.0000 −0.941742
\(222\) 12.0000 0.805387
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) −6.00000 −0.400000
\(226\) 5.00000 0.332595
\(227\) −2.00000 −0.132745 −0.0663723 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 0 0
\(229\) −28.0000 −1.85029 −0.925146 0.379611i \(-0.876058\pi\)
−0.925146 + 0.379611i \(0.876058\pi\)
\(230\) −16.0000 −1.05501
\(231\) 0 0
\(232\) −5.00000 −0.328266
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) −42.0000 −2.74563
\(235\) 4.00000 0.260931
\(236\) 3.00000 0.195283
\(237\) 27.0000 1.75384
\(238\) 0 0
\(239\) −5.00000 −0.323423 −0.161712 0.986838i \(-0.551701\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(240\) 6.00000 0.387298
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 1.00000 0.0640184
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 18.0000 1.14070
\(250\) −12.0000 −0.758947
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) −19.0000 −1.19217
\(255\) 12.0000 0.751469
\(256\) 1.00000 0.0625000
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) −24.0000 −1.49417
\(259\) 0 0
\(260\) −14.0000 −0.868243
\(261\) −30.0000 −1.85695
\(262\) 0 0
\(263\) 27.0000 1.66489 0.832446 0.554107i \(-0.186940\pi\)
0.832446 + 0.554107i \(0.186940\pi\)
\(264\) −3.00000 −0.184637
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 18.0000 1.10158
\(268\) 9.00000 0.549762
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 18.0000 1.09545
\(271\) −11.0000 −0.668202 −0.334101 0.942537i \(-0.608433\pi\)
−0.334101 + 0.942537i \(0.608433\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 3.00000 0.181237
\(275\) 1.00000 0.0603023
\(276\) −24.0000 −1.44463
\(277\) −9.00000 −0.540758 −0.270379 0.962754i \(-0.587149\pi\)
−0.270379 + 0.962754i \(0.587149\pi\)
\(278\) −4.00000 −0.239904
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) −28.0000 −1.67034 −0.835170 0.549992i \(-0.814631\pi\)
−0.835170 + 0.549992i \(0.814631\pi\)
\(282\) 6.00000 0.357295
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 7.00000 0.413919
\(287\) 0 0
\(288\) 6.00000 0.353553
\(289\) −13.0000 −0.764706
\(290\) −10.0000 −0.587220
\(291\) 21.0000 1.23104
\(292\) 4.00000 0.234082
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 4.00000 0.232495
\(297\) −9.00000 −0.522233
\(298\) −10.0000 −0.579284
\(299\) 56.0000 3.23856
\(300\) −3.00000 −0.173205
\(301\) 0 0
\(302\) −3.00000 −0.172631
\(303\) −27.0000 −1.55111
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 12.0000 0.685994
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 0 0
\(309\) 54.0000 3.07195
\(310\) 8.00000 0.454369
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) −21.0000 −1.18889
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 16.0000 0.902932
\(315\) 0 0
\(316\) 9.00000 0.506290
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) −18.0000 −1.00939
\(319\) 5.00000 0.279946
\(320\) 2.00000 0.111803
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) 0 0
\(324\) 9.00000 0.500000
\(325\) 7.00000 0.388290
\(326\) −17.0000 −0.941543
\(327\) −6.00000 −0.331801
\(328\) 4.00000 0.220863
\(329\) 0 0
\(330\) −6.00000 −0.330289
\(331\) 13.0000 0.714545 0.357272 0.934000i \(-0.383707\pi\)
0.357272 + 0.934000i \(0.383707\pi\)
\(332\) 6.00000 0.329293
\(333\) 24.0000 1.31519
\(334\) 19.0000 1.03963
\(335\) 18.0000 0.983445
\(336\) 0 0
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) 36.0000 1.95814
\(339\) 15.0000 0.814688
\(340\) 4.00000 0.216930
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) −8.00000 −0.431331
\(345\) −48.0000 −2.58423
\(346\) 25.0000 1.34401
\(347\) −22.0000 −1.18102 −0.590511 0.807030i \(-0.701074\pi\)
−0.590511 + 0.807030i \(0.701074\pi\)
\(348\) −15.0000 −0.804084
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −63.0000 −3.36269
\(352\) −1.00000 −0.0533002
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 9.00000 0.478345
\(355\) −4.00000 −0.212298
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 19.0000 1.00418
\(359\) −19.0000 −1.00278 −0.501391 0.865221i \(-0.667178\pi\)
−0.501391 + 0.865221i \(0.667178\pi\)
\(360\) 12.0000 0.632456
\(361\) −19.0000 −1.00000
\(362\) −22.0000 −1.15629
\(363\) 3.00000 0.157459
\(364\) 0 0
\(365\) 8.00000 0.418739
\(366\) 3.00000 0.156813
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) −8.00000 −0.417029
\(369\) 24.0000 1.24939
\(370\) 8.00000 0.415900
\(371\) 0 0
\(372\) 12.0000 0.622171
\(373\) 11.0000 0.569558 0.284779 0.958593i \(-0.408080\pi\)
0.284779 + 0.958593i \(0.408080\pi\)
\(374\) −2.00000 −0.103418
\(375\) −36.0000 −1.85903
\(376\) 2.00000 0.103142
\(377\) 35.0000 1.80259
\(378\) 0 0
\(379\) 1.00000 0.0513665 0.0256833 0.999670i \(-0.491824\pi\)
0.0256833 + 0.999670i \(0.491824\pi\)
\(380\) 0 0
\(381\) −57.0000 −2.92020
\(382\) 2.00000 0.102329
\(383\) 26.0000 1.32854 0.664269 0.747494i \(-0.268743\pi\)
0.664269 + 0.747494i \(0.268743\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 8.00000 0.407189
\(387\) −48.0000 −2.43998
\(388\) 7.00000 0.355371
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) −42.0000 −2.12675
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 15.0000 0.755689
\(395\) 18.0000 0.905678
\(396\) −6.00000 −0.301511
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −9.00000 −0.449439 −0.224719 0.974424i \(-0.572147\pi\)
−0.224719 + 0.974424i \(0.572147\pi\)
\(402\) 27.0000 1.34664
\(403\) −28.0000 −1.39478
\(404\) −9.00000 −0.447767
\(405\) 18.0000 0.894427
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 6.00000 0.297044
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 8.00000 0.395092
\(411\) 9.00000 0.443937
\(412\) 18.0000 0.886796
\(413\) 0 0
\(414\) −48.0000 −2.35907
\(415\) 12.0000 0.589057
\(416\) −7.00000 −0.343203
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) −20.0000 −0.973585
\(423\) 12.0000 0.583460
\(424\) −6.00000 −0.291386
\(425\) −2.00000 −0.0970143
\(426\) −6.00000 −0.290701
\(427\) 0 0
\(428\) 2.00000 0.0966736
\(429\) 21.0000 1.01389
\(430\) −16.0000 −0.771589
\(431\) 25.0000 1.20421 0.602104 0.798418i \(-0.294329\pi\)
0.602104 + 0.798418i \(0.294329\pi\)
\(432\) 9.00000 0.433013
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) −30.0000 −1.43839
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) 12.0000 0.573382
\(439\) −5.00000 −0.238637 −0.119318 0.992856i \(-0.538071\pi\)
−0.119318 + 0.992856i \(0.538071\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) −14.0000 −0.665912
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 12.0000 0.569495
\(445\) 12.0000 0.568855
\(446\) 4.00000 0.189405
\(447\) −30.0000 −1.41895
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) −6.00000 −0.282843
\(451\) −4.00000 −0.188353
\(452\) 5.00000 0.235180
\(453\) −9.00000 −0.422857
\(454\) −2.00000 −0.0938647
\(455\) 0 0
\(456\) 0 0
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) −28.0000 −1.30835
\(459\) 18.0000 0.840168
\(460\) −16.0000 −0.746004
\(461\) 27.0000 1.25752 0.628758 0.777601i \(-0.283564\pi\)
0.628758 + 0.777601i \(0.283564\pi\)
\(462\) 0 0
\(463\) −2.00000 −0.0929479 −0.0464739 0.998920i \(-0.514798\pi\)
−0.0464739 + 0.998920i \(0.514798\pi\)
\(464\) −5.00000 −0.232119
\(465\) 24.0000 1.11297
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −42.0000 −1.94145
\(469\) 0 0
\(470\) 4.00000 0.184506
\(471\) 48.0000 2.21172
\(472\) 3.00000 0.138086
\(473\) 8.00000 0.367840
\(474\) 27.0000 1.24015
\(475\) 0 0
\(476\) 0 0
\(477\) −36.0000 −1.64833
\(478\) −5.00000 −0.228695
\(479\) −1.00000 −0.0456912 −0.0228456 0.999739i \(-0.507273\pi\)
−0.0228456 + 0.999739i \(0.507273\pi\)
\(480\) 6.00000 0.273861
\(481\) −28.0000 −1.27669
\(482\) 0 0
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 1.00000 0.0452679
\(489\) −51.0000 −2.30630
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 12.0000 0.541002
\(493\) −10.0000 −0.450377
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 18.0000 0.806599
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) −12.0000 −0.536656
\(501\) 57.0000 2.54657
\(502\) −24.0000 −1.07117
\(503\) 3.00000 0.133763 0.0668817 0.997761i \(-0.478695\pi\)
0.0668817 + 0.997761i \(0.478695\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 8.00000 0.355643
\(507\) 108.000 4.79645
\(508\) −19.0000 −0.842989
\(509\) −22.0000 −0.975133 −0.487566 0.873086i \(-0.662115\pi\)
−0.487566 + 0.873086i \(0.662115\pi\)
\(510\) 12.0000 0.531369
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 3.00000 0.132324
\(515\) 36.0000 1.58635
\(516\) −24.0000 −1.05654
\(517\) −2.00000 −0.0879599
\(518\) 0 0
\(519\) 75.0000 3.29213
\(520\) −14.0000 −0.613941
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) −30.0000 −1.31306
\(523\) −10.0000 −0.437269 −0.218635 0.975807i \(-0.570160\pi\)
−0.218635 + 0.975807i \(0.570160\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 27.0000 1.17726
\(527\) 8.00000 0.348485
\(528\) −3.00000 −0.130558
\(529\) 41.0000 1.78261
\(530\) −12.0000 −0.521247
\(531\) 18.0000 0.781133
\(532\) 0 0
\(533\) −28.0000 −1.21281
\(534\) 18.0000 0.778936
\(535\) 4.00000 0.172935
\(536\) 9.00000 0.388741
\(537\) 57.0000 2.45973
\(538\) −24.0000 −1.03471
\(539\) 0 0
\(540\) 18.0000 0.774597
\(541\) 5.00000 0.214967 0.107483 0.994207i \(-0.465721\pi\)
0.107483 + 0.994207i \(0.465721\pi\)
\(542\) −11.0000 −0.472490
\(543\) −66.0000 −2.83233
\(544\) 2.00000 0.0857493
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) 30.0000 1.28271 0.641354 0.767245i \(-0.278373\pi\)
0.641354 + 0.767245i \(0.278373\pi\)
\(548\) 3.00000 0.128154
\(549\) 6.00000 0.256074
\(550\) 1.00000 0.0426401
\(551\) 0 0
\(552\) −24.0000 −1.02151
\(553\) 0 0
\(554\) −9.00000 −0.382373
\(555\) 24.0000 1.01874
\(556\) −4.00000 −0.169638
\(557\) 26.0000 1.10166 0.550828 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) 24.0000 1.01600
\(559\) 56.0000 2.36855
\(560\) 0 0
\(561\) −6.00000 −0.253320
\(562\) −28.0000 −1.18111
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 6.00000 0.252646
\(565\) 10.0000 0.420703
\(566\) 0 0
\(567\) 0 0
\(568\) −2.00000 −0.0839181
\(569\) −12.0000 −0.503066 −0.251533 0.967849i \(-0.580935\pi\)
−0.251533 + 0.967849i \(0.580935\pi\)
\(570\) 0 0
\(571\) 22.0000 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) 7.00000 0.292685
\(573\) 6.00000 0.250654
\(574\) 0 0
\(575\) 8.00000 0.333623
\(576\) 6.00000 0.250000
\(577\) −43.0000 −1.79011 −0.895057 0.445952i \(-0.852865\pi\)
−0.895057 + 0.445952i \(0.852865\pi\)
\(578\) −13.0000 −0.540729
\(579\) 24.0000 0.997406
\(580\) −10.0000 −0.415227
\(581\) 0 0
\(582\) 21.0000 0.870478
\(583\) 6.00000 0.248495
\(584\) 4.00000 0.165521
\(585\) −84.0000 −3.47297
\(586\) −18.0000 −0.743573
\(587\) 27.0000 1.11441 0.557205 0.830375i \(-0.311874\pi\)
0.557205 + 0.830375i \(0.311874\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 6.00000 0.247016
\(591\) 45.0000 1.85105
\(592\) 4.00000 0.164399
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) −9.00000 −0.369274
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 12.0000 0.491127
\(598\) 56.0000 2.29001
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) −3.00000 −0.122474
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 54.0000 2.19905
\(604\) −3.00000 −0.122068
\(605\) 2.00000 0.0813116
\(606\) −27.0000 −1.09680
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) −14.0000 −0.566379
\(612\) 12.0000 0.485071
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 2.00000 0.0807134
\(615\) 24.0000 0.967773
\(616\) 0 0
\(617\) −15.0000 −0.603877 −0.301939 0.953327i \(-0.597634\pi\)
−0.301939 + 0.953327i \(0.597634\pi\)
\(618\) 54.0000 2.17220
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 8.00000 0.321288
\(621\) −72.0000 −2.88926
\(622\) −10.0000 −0.400963
\(623\) 0 0
\(624\) −21.0000 −0.840673
\(625\) −19.0000 −0.760000
\(626\) 1.00000 0.0399680
\(627\) 0 0
\(628\) 16.0000 0.638470
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) −18.0000 −0.716569 −0.358284 0.933613i \(-0.616638\pi\)
−0.358284 + 0.933613i \(0.616638\pi\)
\(632\) 9.00000 0.358001
\(633\) −60.0000 −2.38479
\(634\) −12.0000 −0.476581
\(635\) −38.0000 −1.50798
\(636\) −18.0000 −0.713746
\(637\) 0 0
\(638\) 5.00000 0.197952
\(639\) −12.0000 −0.474713
\(640\) 2.00000 0.0790569
\(641\) −27.0000 −1.06644 −0.533218 0.845978i \(-0.679017\pi\)
−0.533218 + 0.845978i \(0.679017\pi\)
\(642\) 6.00000 0.236801
\(643\) 31.0000 1.22252 0.611260 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(644\) 0 0
\(645\) −48.0000 −1.89000
\(646\) 0 0
\(647\) 30.0000 1.17942 0.589711 0.807614i \(-0.299242\pi\)
0.589711 + 0.807614i \(0.299242\pi\)
\(648\) 9.00000 0.353553
\(649\) −3.00000 −0.117760
\(650\) 7.00000 0.274563
\(651\) 0 0
\(652\) −17.0000 −0.665771
\(653\) 40.0000 1.56532 0.782660 0.622449i \(-0.213862\pi\)
0.782660 + 0.622449i \(0.213862\pi\)
\(654\) −6.00000 −0.234619
\(655\) 0 0
\(656\) 4.00000 0.156174
\(657\) 24.0000 0.936329
\(658\) 0 0
\(659\) 28.0000 1.09073 0.545363 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) −6.00000 −0.233550
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 13.0000 0.505259
\(663\) −42.0000 −1.63114
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 24.0000 0.929981
\(667\) 40.0000 1.54881
\(668\) 19.0000 0.735132
\(669\) 12.0000 0.463947
\(670\) 18.0000 0.695401
\(671\) −1.00000 −0.0386046
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −12.0000 −0.462223
\(675\) −9.00000 −0.346410
\(676\) 36.0000 1.38462
\(677\) 14.0000 0.538064 0.269032 0.963131i \(-0.413296\pi\)
0.269032 + 0.963131i \(0.413296\pi\)
\(678\) 15.0000 0.576072
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) −6.00000 −0.229920
\(682\) −4.00000 −0.153168
\(683\) 21.0000 0.803543 0.401771 0.915740i \(-0.368395\pi\)
0.401771 + 0.915740i \(0.368395\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 0 0
\(687\) −84.0000 −3.20480
\(688\) −8.00000 −0.304997
\(689\) 42.0000 1.60007
\(690\) −48.0000 −1.82733
\(691\) 33.0000 1.25538 0.627690 0.778464i \(-0.284001\pi\)
0.627690 + 0.778464i \(0.284001\pi\)
\(692\) 25.0000 0.950357
\(693\) 0 0
\(694\) −22.0000 −0.835109
\(695\) −8.00000 −0.303457
\(696\) −15.0000 −0.568574
\(697\) 8.00000 0.303022
\(698\) 2.00000 0.0757011
\(699\) 0 0
\(700\) 0 0
\(701\) −27.0000 −1.01978 −0.509888 0.860241i \(-0.670313\pi\)
−0.509888 + 0.860241i \(0.670313\pi\)
\(702\) −63.0000 −2.37778
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) 12.0000 0.451946
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) 9.00000 0.338241
\(709\) −48.0000 −1.80268 −0.901339 0.433114i \(-0.857415\pi\)
−0.901339 + 0.433114i \(0.857415\pi\)
\(710\) −4.00000 −0.150117
\(711\) 54.0000 2.02516
\(712\) 6.00000 0.224860
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) 14.0000 0.523570
\(716\) 19.0000 0.710063
\(717\) −15.0000 −0.560185
\(718\) −19.0000 −0.709074
\(719\) −38.0000 −1.41716 −0.708580 0.705630i \(-0.750664\pi\)
−0.708580 + 0.705630i \(0.750664\pi\)
\(720\) 12.0000 0.447214
\(721\) 0 0
\(722\) −19.0000 −0.707107
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) 5.00000 0.185695
\(726\) 3.00000 0.111340
\(727\) −22.0000 −0.815935 −0.407967 0.912996i \(-0.633762\pi\)
−0.407967 + 0.912996i \(0.633762\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 8.00000 0.296093
\(731\) −16.0000 −0.591781
\(732\) 3.00000 0.110883
\(733\) −19.0000 −0.701781 −0.350891 0.936416i \(-0.614121\pi\)
−0.350891 + 0.936416i \(0.614121\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) −9.00000 −0.331519
\(738\) 24.0000 0.883452
\(739\) 42.0000 1.54499 0.772497 0.635018i \(-0.219007\pi\)
0.772497 + 0.635018i \(0.219007\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 12.0000 0.439941
\(745\) −20.0000 −0.732743
\(746\) 11.0000 0.402739
\(747\) 36.0000 1.31717
\(748\) −2.00000 −0.0731272
\(749\) 0 0
\(750\) −36.0000 −1.31453
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) 2.00000 0.0729325
\(753\) −72.0000 −2.62383
\(754\) 35.0000 1.27462
\(755\) −6.00000 −0.218362
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 1.00000 0.0363216
\(759\) 24.0000 0.871145
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −57.0000 −2.06489
\(763\) 0 0
\(764\) 2.00000 0.0723575
\(765\) 24.0000 0.867722
\(766\) 26.0000 0.939418
\(767\) −21.0000 −0.758266
\(768\) 3.00000 0.108253
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 9.00000 0.324127
\(772\) 8.00000 0.287926
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) −48.0000 −1.72532
\(775\) −4.00000 −0.143684
\(776\) 7.00000 0.251285
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) 0 0
\(780\) −42.0000 −1.50384
\(781\) 2.00000 0.0715656
\(782\) −16.0000 −0.572159
\(783\) −45.0000 −1.60817
\(784\) 0 0
\(785\) 32.0000 1.14213
\(786\) 0 0
\(787\) −40.0000 −1.42585 −0.712923 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) 15.0000 0.534353
\(789\) 81.0000 2.88368
\(790\) 18.0000 0.640411
\(791\) 0 0
\(792\) −6.00000 −0.213201
\(793\) −7.00000 −0.248577
\(794\) 6.00000 0.212932
\(795\) −36.0000 −1.27679
\(796\) 4.00000 0.141776
\(797\) 50.0000 1.77109 0.885545 0.464553i \(-0.153785\pi\)
0.885545 + 0.464553i \(0.153785\pi\)
\(798\) 0 0
\(799\) 4.00000 0.141510
\(800\) −1.00000 −0.0353553
\(801\) 36.0000 1.27200
\(802\) −9.00000 −0.317801
\(803\) −4.00000 −0.141157
\(804\) 27.0000 0.952217
\(805\) 0 0
\(806\) −28.0000 −0.986258
\(807\) −72.0000 −2.53452
\(808\) −9.00000 −0.316619
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 18.0000 0.632456
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 0 0
\(813\) −33.0000 −1.15736
\(814\) −4.00000 −0.140200
\(815\) −34.0000 −1.19097
\(816\) 6.00000 0.210042
\(817\) 0 0
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) 8.00000 0.279372
\(821\) −45.0000 −1.57051 −0.785255 0.619172i \(-0.787468\pi\)
−0.785255 + 0.619172i \(0.787468\pi\)
\(822\) 9.00000 0.313911
\(823\) 26.0000 0.906303 0.453152 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) 18.0000 0.627060
\(825\) 3.00000 0.104447
\(826\) 0 0
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) −48.0000 −1.66812
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 12.0000 0.416526
\(831\) −27.0000 −0.936620
\(832\) −7.00000 −0.242681
\(833\) 0 0
\(834\) −12.0000 −0.415526
\(835\) 38.0000 1.31504
\(836\) 0 0
\(837\) 36.0000 1.24434
\(838\) 16.0000 0.552711
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −20.0000 −0.689246
\(843\) −84.0000 −2.89311
\(844\) −20.0000 −0.688428
\(845\) 72.0000 2.47688
\(846\) 12.0000 0.412568
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) −32.0000 −1.09695
\(852\) −6.00000 −0.205557
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 2.00000 0.0683586
\(857\) 44.0000 1.50301 0.751506 0.659727i \(-0.229328\pi\)
0.751506 + 0.659727i \(0.229328\pi\)
\(858\) 21.0000 0.716928
\(859\) 13.0000 0.443554 0.221777 0.975097i \(-0.428814\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(860\) −16.0000 −0.545595
\(861\) 0 0
\(862\) 25.0000 0.851503
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 9.00000 0.306186
\(865\) 50.0000 1.70005
\(866\) −34.0000 −1.15537
\(867\) −39.0000 −1.32451
\(868\) 0 0
\(869\) −9.00000 −0.305304
\(870\) −30.0000 −1.01710
\(871\) −63.0000 −2.13467
\(872\) −2.00000 −0.0677285
\(873\) 42.0000 1.42148
\(874\) 0 0
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) −29.0000 −0.979260 −0.489630 0.871930i \(-0.662868\pi\)
−0.489630 + 0.871930i \(0.662868\pi\)
\(878\) −5.00000 −0.168742
\(879\) −54.0000 −1.82137
\(880\) −2.00000 −0.0674200
\(881\) 21.0000 0.707508 0.353754 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) 0 0
\(883\) −29.0000 −0.975928 −0.487964 0.872864i \(-0.662260\pi\)
−0.487964 + 0.872864i \(0.662260\pi\)
\(884\) −14.0000 −0.470871
\(885\) 18.0000 0.605063
\(886\) 36.0000 1.20944
\(887\) −41.0000 −1.37665 −0.688323 0.725405i \(-0.741653\pi\)
−0.688323 + 0.725405i \(0.741653\pi\)
\(888\) 12.0000 0.402694
\(889\) 0 0
\(890\) 12.0000 0.402241
\(891\) −9.00000 −0.301511
\(892\) 4.00000 0.133930
\(893\) 0 0
\(894\) −30.0000 −1.00335
\(895\) 38.0000 1.27020
\(896\) 0 0
\(897\) 168.000 5.60936
\(898\) −6.00000 −0.200223
\(899\) −20.0000 −0.667037
\(900\) −6.00000 −0.200000
\(901\) −12.0000 −0.399778
\(902\) −4.00000 −0.133185
\(903\) 0 0
\(904\) 5.00000 0.166298
\(905\) −44.0000 −1.46261
\(906\) −9.00000 −0.299005
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −2.00000 −0.0663723
\(909\) −54.0000 −1.79107
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 0 0
\(913\) −6.00000 −0.198571
\(914\) 14.0000 0.463079
\(915\) 6.00000 0.198354
\(916\) −28.0000 −0.925146
\(917\) 0 0
\(918\) 18.0000 0.594089
\(919\) 36.0000 1.18753 0.593765 0.804638i \(-0.297641\pi\)
0.593765 + 0.804638i \(0.297641\pi\)
\(920\) −16.0000 −0.527504
\(921\) 6.00000 0.197707
\(922\) 27.0000 0.889198
\(923\) 14.0000 0.460816
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −2.00000 −0.0657241
\(927\) 108.000 3.54719
\(928\) −5.00000 −0.164133
\(929\) 15.0000 0.492134 0.246067 0.969253i \(-0.420862\pi\)
0.246067 + 0.969253i \(0.420862\pi\)
\(930\) 24.0000 0.786991
\(931\) 0 0
\(932\) 0 0
\(933\) −30.0000 −0.982156
\(934\) 12.0000 0.392652
\(935\) −4.00000 −0.130814
\(936\) −42.0000 −1.37281
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 0 0
\(939\) 3.00000 0.0979013
\(940\) 4.00000 0.130466
\(941\) −11.0000 −0.358590 −0.179295 0.983795i \(-0.557382\pi\)
−0.179295 + 0.983795i \(0.557382\pi\)
\(942\) 48.0000 1.56392
\(943\) −32.0000 −1.04206
\(944\) 3.00000 0.0976417
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 27.0000 0.876919
\(949\) −28.0000 −0.908918
\(950\) 0 0
\(951\) −36.0000 −1.16738
\(952\) 0 0
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) −36.0000 −1.16554
\(955\) 4.00000 0.129437
\(956\) −5.00000 −0.161712
\(957\) 15.0000 0.484881
\(958\) −1.00000 −0.0323085
\(959\) 0 0
\(960\) 6.00000 0.193649
\(961\) −15.0000 −0.483871
\(962\) −28.0000 −0.902756
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) 16.0000 0.515058
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 14.0000 0.449513
\(971\) 41.0000 1.31575 0.657876 0.753126i \(-0.271455\pi\)
0.657876 + 0.753126i \(0.271455\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −40.0000 −1.28168
\(975\) 21.0000 0.672538
\(976\) 1.00000 0.0320092
\(977\) −46.0000 −1.47167 −0.735835 0.677161i \(-0.763210\pi\)
−0.735835 + 0.677161i \(0.763210\pi\)
\(978\) −51.0000 −1.63080
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) −12.0000 −0.383131
\(982\) 30.0000 0.957338
\(983\) −18.0000 −0.574111 −0.287055 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) 12.0000 0.382546
\(985\) 30.0000 0.955879
\(986\) −10.0000 −0.318465
\(987\) 0 0
\(988\) 0 0
\(989\) 64.0000 2.03508
\(990\) −12.0000 −0.381385
\(991\) 10.0000 0.317660 0.158830 0.987306i \(-0.449228\pi\)
0.158830 + 0.987306i \(0.449228\pi\)
\(992\) 4.00000 0.127000
\(993\) 39.0000 1.23763
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 18.0000 0.570352
\(997\) −6.00000 −0.190022 −0.0950110 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) 20.0000 0.633089
\(999\) 36.0000 1.13899
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 1078.2.a.m.1.1 1
3.2 odd 2 9702.2.a.i.1.1 1
4.3 odd 2 8624.2.a.b.1.1 1
7.2 even 3 154.2.e.a.67.1 yes 2
7.3 odd 6 1078.2.e.f.177.1 2
7.4 even 3 154.2.e.a.23.1 2
7.5 odd 6 1078.2.e.f.67.1 2
7.6 odd 2 1078.2.a.g.1.1 1
21.2 odd 6 1386.2.k.o.991.1 2
21.11 odd 6 1386.2.k.o.793.1 2
21.20 even 2 9702.2.a.y.1.1 1
28.11 odd 6 1232.2.q.e.177.1 2
28.23 odd 6 1232.2.q.e.529.1 2
28.27 even 2 8624.2.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.a.23.1 2 7.4 even 3
154.2.e.a.67.1 yes 2 7.2 even 3
1078.2.a.g.1.1 1 7.6 odd 2
1078.2.a.m.1.1 1 1.1 even 1 trivial
1078.2.e.f.67.1 2 7.5 odd 6
1078.2.e.f.177.1 2 7.3 odd 6
1232.2.q.e.177.1 2 28.11 odd 6
1232.2.q.e.529.1 2 28.23 odd 6
1386.2.k.o.793.1 2 21.11 odd 6
1386.2.k.o.991.1 2 21.2 odd 6
8624.2.a.b.1.1 1 4.3 odd 2
8624.2.a.be.1.1 1 28.27 even 2
9702.2.a.i.1.1 1 3.2 odd 2
9702.2.a.y.1.1 1 21.20 even 2