Properties

Label 1078.2.a.g.1.1
Level $1078$
Weight $2$
Character 1078.1
Self dual yes
Analytic conductor $8.608$
Analytic rank $1$
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: \(1\)
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 \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −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 \(-0.0772105\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) 6.00000 1.54919
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 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 \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 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 \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 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 \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) −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 \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 6.00000 0.774597
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) −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 \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 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 \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 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 \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −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 \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) −1.00000 −0.100000
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) 6.00000 0.594089
\(103\) −18.0000 −1.77359 −0.886796 0.462160i \(-0.847074\pi\)
−0.886796 + 0.462160i \(0.847074\pi\)
\(104\) 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 \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 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 \(-0.720432\pi\)
−0.638470 + 0.769647i \(0.720432\pi\)
\(158\) 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 \(-0.762880\pi\)
−0.735132 + 0.677924i \(0.762880\pi\)
\(168\) 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 \(-0.899282\pi\)
−0.950357 + 0.311160i \(0.899282\pi\)
\(174\) 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 \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 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 \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −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 \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) −6.00000 −0.400000
\(226\) 5.00000 0.332595
\(227\) 2.00000 0.132745 0.0663723 0.997795i \(-0.478857\pi\)
0.0663723 + 0.997795i \(0.478857\pi\)
\(228\) 0 0
\(229\) 28.0000 1.85029 0.925146 0.379611i \(-0.123942\pi\)
0.925146 + 0.379611i \(0.123942\pi\)
\(230\) 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 \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 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 \(-0.529827\pi\)
−0.0935674 + 0.995613i \(0.529827\pi\)
\(258\) 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 \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 18.0000 1.09545
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) −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 \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 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 \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) 54.0000 3.07195
\(310\) 8.00000 0.454369
\(311\) 10.0000 0.567048 0.283524 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) −21.0000 −1.18889
\(313\) −1.00000 −0.0565233 −0.0282617 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) −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 \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −63.0000 −3.36269
\(352\) −1.00000 −0.0533002
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 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 \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −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 \(-0.731257\pi\)
−0.664269 + 0.747494i \(0.731257\pi\)
\(384\) −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 \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) −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 \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 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 \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 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 \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 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 \(-0.461929\pi\)
0.119318 + 0.992856i \(0.461929\pi\)
\(440\) 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 \(-0.716436\pi\)
−0.628758 + 0.777601i \(0.716436\pi\)
\(462\) 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 \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 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 \(-0.492727\pi\)
0.0228456 + 0.999739i \(0.492727\pi\)
\(480\) 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 \(-0.521305\pi\)
−0.0668817 + 0.997761i \(0.521305\pi\)
\(504\) 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 \(-0.337885\pi\)
0.487566 + 0.873086i \(0.337885\pi\)
\(510\) −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 \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) −30.0000 −1.31306
\(523\) 10.0000 0.437269 0.218635 0.975807i \(-0.429840\pi\)
0.218635 + 0.975807i \(0.429840\pi\)
\(524\) 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 \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 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 \(-0.147135\pi\)
0.895057 + 0.445952i \(0.147135\pi\)
\(578\) −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 \(-0.688126\pi\)
−0.557205 + 0.830375i \(0.688126\pi\)
\(588\) 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 \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 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 \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 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 \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 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 \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 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 \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) 0 0
\(645\) −48.0000 −1.89000
\(646\) 0 0
\(647\) −30.0000 −1.17942 −0.589711 0.807614i \(-0.700758\pi\)
−0.589711 + 0.807614i \(0.700758\pi\)
\(648\) 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 \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 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 \(-0.586704\pi\)
−0.269032 + 0.963131i \(0.586704\pi\)
\(678\) −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 \(-0.715999\pi\)
−0.627690 + 0.778464i \(0.715999\pi\)
\(692\) −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 \(-0.249336\pi\)
0.708580 + 0.705630i \(0.249336\pi\)
\(720\) −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 \(-0.366238\pi\)
0.407967 + 0.912996i \(0.366238\pi\)
\(728\) 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 \(-0.385879\pi\)
0.350891 + 0.936416i \(0.385879\pi\)
\(734\) −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 \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 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 \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 9.00000 0.324127
\(772\) 8.00000 0.287926
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) −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 \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) 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 \(-0.846215\pi\)
−0.885545 + 0.464553i \(0.846215\pi\)
\(798\) 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 \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) 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 \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 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 \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) 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 \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 2.00000 0.0683586
\(857\) −44.0000 −1.50301 −0.751506 0.659727i \(-0.770672\pi\)
−0.751506 + 0.659727i \(0.770672\pi\)
\(858\) 21.0000 0.716928
\(859\) −13.0000 −0.443554 −0.221777 0.975097i \(-0.571186\pi\)
−0.221777 + 0.975097i \(0.571186\pi\)
\(860\) 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 \(-0.615095\pi\)
−0.353754 + 0.935339i \(0.615095\pi\)
\(882\) 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 \(-0.258347\pi\)
0.688323 + 0.725405i \(0.258347\pi\)
\(888\) −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 \(-0.579138\pi\)
−0.246067 + 0.969253i \(0.579138\pi\)
\(930\) −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 \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 0 0
\(939\) 3.00000 0.0979013
\(940\) 4.00000 0.130466
\(941\) 11.0000 0.358590 0.179295 0.983795i \(-0.442618\pi\)
0.179295 + 0.983795i \(0.442618\pi\)
\(942\) 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 \(-0.728545\pi\)
−0.657876 + 0.753126i \(0.728545\pi\)
\(972\) 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 \(-0.407324\pi\)
0.287055 + 0.957914i \(0.407324\pi\)
\(984\) 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 \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) 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.g.1.1 1
3.2 odd 2 9702.2.a.y.1.1 1
4.3 odd 2 8624.2.a.be.1.1 1
7.2 even 3 1078.2.e.f.67.1 2
7.3 odd 6 154.2.e.a.23.1 2
7.4 even 3 1078.2.e.f.177.1 2
7.5 odd 6 154.2.e.a.67.1 yes 2
7.6 odd 2 1078.2.a.m.1.1 1
21.5 even 6 1386.2.k.o.991.1 2
21.17 even 6 1386.2.k.o.793.1 2
21.20 even 2 9702.2.a.i.1.1 1
28.3 even 6 1232.2.q.e.177.1 2
28.19 even 6 1232.2.q.e.529.1 2
28.27 even 2 8624.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.a.23.1 2 7.3 odd 6
154.2.e.a.67.1 yes 2 7.5 odd 6
1078.2.a.g.1.1 1 1.1 even 1 trivial
1078.2.a.m.1.1 1 7.6 odd 2
1078.2.e.f.67.1 2 7.2 even 3
1078.2.e.f.177.1 2 7.4 even 3
1232.2.q.e.177.1 2 28.3 even 6
1232.2.q.e.529.1 2 28.19 even 6
1386.2.k.o.793.1 2 21.17 even 6
1386.2.k.o.991.1 2 21.5 even 6
8624.2.a.b.1.1 1 28.27 even 2
8624.2.a.be.1.1 1 4.3 odd 2
9702.2.a.i.1.1 1 21.20 even 2
9702.2.a.y.1.1 1 3.2 odd 2