Properties

Label 546.2.a.g.1.1
Level $546$
Weight $2$
Character 546.1
Self dual yes
Analytic conductor $4.360$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 546.1

$q$-expansion

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

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 1.00000 0.267261
\(15\) 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 \(0.651732\pi\)
\(20\) 2.00000 0.447214
\(21\) 1.00000 0.218218
\(22\) −4.00000 −0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 2.00000 0.365148
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 6.00000 1.02899
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) −4.00000 −0.648886
\(39\) −1.00000 −0.160128
\(40\) 2.00000 0.316228
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000 0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 6.00000 0.840168
\(52\) −1.00000 −0.138675
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 1.00000 0.136083
\(55\) −8.00000 −1.07872
\(56\) 1.00000 0.133631
\(57\) −4.00000 −0.529813
\(58\) −6.00000 −0.787839
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 2.00000 0.258199
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −8.00000 −1.01600
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) −4.00000 −0.492366
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 10.0000 1.16248
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) −4.00000 −0.455842
\(78\) −1.00000 −0.113228
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 1.00000 0.109109
\(85\) 12.0000 1.30158
\(86\) 4.00000 0.431331
\(87\) −6.00000 −0.643268
\(88\) −4.00000 −0.426401
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 2.00000 0.210819
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 4.00000 0.412568
\(95\) −8.00000 −0.820783
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 1.00000 0.101015
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 6.00000 0.594089
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 2.00000 0.195180
\(106\) 10.0000 0.971286
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 1.00000 0.0962250
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) −8.00000 −0.762770
\(111\) 10.0000 0.949158
\(112\) 1.00000 0.0944911
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) −1.00000 −0.0924500
\(118\) 4.00000 0.368230
\(119\) 6.00000 0.550019
\(120\) 2.00000 0.182574
\(121\) 5.00000 0.454545
\(122\) −6.00000 −0.543214
\(123\) −6.00000 −0.541002
\(124\) −8.00000 −0.718421
\(125\) −12.0000 −1.07331
\(126\) 1.00000 0.0890871
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) −2.00000 −0.175412
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) −4.00000 −0.348155
\(133\) −4.00000 −0.346844
\(134\) −8.00000 −0.691095
\(135\) 2.00000 0.172133
\(136\) 6.00000 0.514496
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 2.00000 0.169031
\(141\) 4.00000 0.336861
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) −12.0000 −0.996546
\(146\) −10.0000 −0.827606
\(147\) 1.00000 0.0824786
\(148\) 10.0000 0.821995
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −4.00000 −0.324443
\(153\) 6.00000 0.485071
\(154\) −4.00000 −0.322329
\(155\) −16.0000 −1.28515
\(156\) −1.00000 −0.0800641
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −8.00000 −0.636446
\(159\) 10.0000 0.793052
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −6.00000 −0.468521
\(165\) −8.00000 −0.622799
\(166\) 4.00000 0.310460
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 1.00000 0.0771517
\(169\) 1.00000 0.0769231
\(170\) 12.0000 0.920358
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −6.00000 −0.454859
\(175\) −1.00000 −0.0755929
\(176\) −4.00000 −0.301511
\(177\) 4.00000 0.300658
\(178\) −6.00000 −0.449719
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 2.00000 0.149071
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −1.00000 −0.0741249
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) 20.0000 1.47043
\(186\) −8.00000 −0.586588
\(187\) −24.0000 −1.75505
\(188\) 4.00000 0.291730
\(189\) 1.00000 0.0727393
\(190\) −8.00000 −0.580381
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) −2.00000 −0.143592
\(195\) −2.00000 −0.143223
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −4.00000 −0.284268
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −8.00000 −0.564276
\(202\) 10.0000 0.703598
\(203\) −6.00000 −0.421117
\(204\) 6.00000 0.420084
\(205\) −12.0000 −0.838116
\(206\) −12.0000 −0.836080
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 16.0000 1.10674
\(210\) 2.00000 0.138013
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 10.0000 0.686803
\(213\) 0 0
\(214\) 16.0000 1.09374
\(215\) 8.00000 0.545595
\(216\) 1.00000 0.0680414
\(217\) −8.00000 −0.543075
\(218\) −6.00000 −0.406371
\(219\) −10.0000 −0.675737
\(220\) −8.00000 −0.539360
\(221\) −6.00000 −0.403604
\(222\) 10.0000 0.671156
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.00000 −0.0666667
\(226\) 2.00000 0.133038
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −4.00000 −0.264906
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) −6.00000 −0.393919
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 8.00000 0.521862
\(236\) 4.00000 0.260378
\(237\) −8.00000 −0.519656
\(238\) 6.00000 0.388922
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 2.00000 0.129099
\(241\) 30.0000 1.93247 0.966235 0.257663i \(-0.0829523\pi\)
0.966235 + 0.257663i \(0.0829523\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) −6.00000 −0.384111
\(245\) 2.00000 0.127775
\(246\) −6.00000 −0.382546
\(247\) 4.00000 0.254514
\(248\) −8.00000 −0.508001
\(249\) 4.00000 0.253490
\(250\) −12.0000 −0.758947
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) 12.0000 0.751469
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 4.00000 0.249029
\(259\) 10.0000 0.621370
\(260\) −2.00000 −0.124035
\(261\) −6.00000 −0.371391
\(262\) −20.0000 −1.23560
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) −4.00000 −0.246183
\(265\) 20.0000 1.22859
\(266\) −4.00000 −0.245256
\(267\) −6.00000 −0.367194
\(268\) −8.00000 −0.488678
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 2.00000 0.121716
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 6.00000 0.363803
\(273\) −1.00000 −0.0605228
\(274\) 10.0000 0.604122
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −12.0000 −0.719712
\(279\) −8.00000 −0.478947
\(280\) 2.00000 0.119523
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 4.00000 0.238197
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 4.00000 0.236525
\(287\) −6.00000 −0.354169
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −12.0000 −0.704664
\(291\) −2.00000 −0.117242
\(292\) −10.0000 −0.585206
\(293\) 2.00000 0.116841 0.0584206 0.998292i \(-0.481394\pi\)
0.0584206 + 0.998292i \(0.481394\pi\)
\(294\) 1.00000 0.0583212
\(295\) 8.00000 0.465778
\(296\) 10.0000 0.581238
\(297\) −4.00000 −0.232104
\(298\) 14.0000 0.810998
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 4.00000 0.230556
\(302\) 16.0000 0.920697
\(303\) 10.0000 0.574485
\(304\) −4.00000 −0.229416
\(305\) −12.0000 −0.687118
\(306\) 6.00000 0.342997
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) −4.00000 −0.227921
\(309\) −12.0000 −0.682656
\(310\) −16.0000 −0.908739
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 18.0000 1.01580
\(315\) 2.00000 0.112687
\(316\) −8.00000 −0.450035
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 10.0000 0.560772
\(319\) 24.0000 1.34374
\(320\) 2.00000 0.111803
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) 1.00000 0.0554700
\(326\) 16.0000 0.886158
\(327\) −6.00000 −0.331801
\(328\) −6.00000 −0.331295
\(329\) 4.00000 0.220527
\(330\) −8.00000 −0.440386
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 4.00000 0.219529
\(333\) 10.0000 0.547997
\(334\) −12.0000 −0.656611
\(335\) −16.0000 −0.874173
\(336\) 1.00000 0.0545545
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) 1.00000 0.0543928
\(339\) 2.00000 0.108625
\(340\) 12.0000 0.650791
\(341\) 32.0000 1.73290
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 8.00000 0.429463 0.214731 0.976673i \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(348\) −6.00000 −0.321634
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −1.00000 −0.0533761
\(352\) −4.00000 −0.213201
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 6.00000 0.317554
\(358\) 24.0000 1.26844
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 2.00000 0.105409
\(361\) −3.00000 −0.157895
\(362\) −22.0000 −1.15629
\(363\) 5.00000 0.262432
\(364\) −1.00000 −0.0524142
\(365\) −20.0000 −1.04685
\(366\) −6.00000 −0.313625
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 20.0000 1.03975
\(371\) 10.0000 0.519174
\(372\) −8.00000 −0.414781
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −24.0000 −1.24101
\(375\) −12.0000 −0.619677
\(376\) 4.00000 0.206284
\(377\) 6.00000 0.309016
\(378\) 1.00000 0.0514344
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −8.00000 −0.410391
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 1.00000 0.0510310
\(385\) −8.00000 −0.407718
\(386\) −6.00000 −0.305392
\(387\) 4.00000 0.203331
\(388\) −2.00000 −0.101535
\(389\) 34.0000 1.72387 0.861934 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) −20.0000 −1.00887
\(394\) 6.00000 0.302276
\(395\) −16.0000 −0.805047
\(396\) −4.00000 −0.201008
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 20.0000 1.00251
\(399\) −4.00000 −0.200250
\(400\) −1.00000 −0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) −8.00000 −0.399004
\(403\) 8.00000 0.398508
\(404\) 10.0000 0.497519
\(405\) 2.00000 0.0993808
\(406\) −6.00000 −0.297775
\(407\) −40.0000 −1.98273
\(408\) 6.00000 0.297044
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) −12.0000 −0.592638
\(411\) 10.0000 0.493264
\(412\) −12.0000 −0.591198
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) −1.00000 −0.0490290
\(417\) −12.0000 −0.587643
\(418\) 16.0000 0.782586
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 2.00000 0.0975900
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) 12.0000 0.584151
\(423\) 4.00000 0.194487
\(424\) 10.0000 0.485643
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) 16.0000 0.773389
\(429\) 4.00000 0.193122
\(430\) 8.00000 0.385794
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 1.00000 0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) −8.00000 −0.384012
\(435\) −12.0000 −0.575356
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) −10.0000 −0.477818
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) −8.00000 −0.381385
\(441\) 1.00000 0.0476190
\(442\) −6.00000 −0.285391
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 10.0000 0.474579
\(445\) −12.0000 −0.568855
\(446\) −8.00000 −0.378811
\(447\) 14.0000 0.662177
\(448\) 1.00000 0.0472456
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 24.0000 1.13012
\(452\) 2.00000 0.0940721
\(453\) 16.0000 0.751746
\(454\) −12.0000 −0.563188
\(455\) −2.00000 −0.0937614
\(456\) −4.00000 −0.187317
\(457\) −14.0000 −0.654892 −0.327446 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) −6.00000 −0.280362
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) −4.00000 −0.186097
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −6.00000 −0.278543
\(465\) −16.0000 −0.741982
\(466\) 18.0000 0.833834
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −8.00000 −0.369406
\(470\) 8.00000 0.369012
\(471\) 18.0000 0.829396
\(472\) 4.00000 0.184115
\(473\) −16.0000 −0.735681
\(474\) −8.00000 −0.367452
\(475\) 4.00000 0.183533
\(476\) 6.00000 0.275010
\(477\) 10.0000 0.457869
\(478\) −24.0000 −1.09773
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 2.00000 0.0912871
\(481\) −10.0000 −0.455961
\(482\) 30.0000 1.36646
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −4.00000 −0.181631
\(486\) 1.00000 0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) −6.00000 −0.271607
\(489\) 16.0000 0.723545
\(490\) 2.00000 0.0903508
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) −6.00000 −0.270501
\(493\) −36.0000 −1.62136
\(494\) 4.00000 0.179969
\(495\) −8.00000 −0.359573
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) −12.0000 −0.536656
\(501\) −12.0000 −0.536120
\(502\) −4.00000 −0.178529
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 1.00000 0.0445435
\(505\) 20.0000 0.889988
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) −16.0000 −0.709885
\(509\) −14.0000 −0.620539 −0.310270 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 12.0000 0.531369
\(511\) −10.0000 −0.442374
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 6.00000 0.264649
\(515\) −24.0000 −1.05757
\(516\) 4.00000 0.176090
\(517\) −16.0000 −0.703679
\(518\) 10.0000 0.439375
\(519\) 18.0000 0.790112
\(520\) −2.00000 −0.0877058
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) −6.00000 −0.262613
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −20.0000 −0.873704
\(525\) −1.00000 −0.0436436
\(526\) 24.0000 1.04645
\(527\) −48.0000 −2.09091
\(528\) −4.00000 −0.174078
\(529\) −23.0000 −1.00000
\(530\) 20.0000 0.868744
\(531\) 4.00000 0.173585
\(532\) −4.00000 −0.173422
\(533\) 6.00000 0.259889
\(534\) −6.00000 −0.259645
\(535\) 32.0000 1.38348
\(536\) −8.00000 −0.345547
\(537\) 24.0000 1.03568
\(538\) 18.0000 0.776035
\(539\) −4.00000 −0.172292
\(540\) 2.00000 0.0860663
\(541\) −46.0000 −1.97769 −0.988847 0.148933i \(-0.952416\pi\)
−0.988847 + 0.148933i \(0.952416\pi\)
\(542\) 8.00000 0.343629
\(543\) −22.0000 −0.944110
\(544\) 6.00000 0.257248
\(545\) −12.0000 −0.514024
\(546\) −1.00000 −0.0427960
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 10.0000 0.427179
\(549\) −6.00000 −0.256074
\(550\) 4.00000 0.170561
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) −10.0000 −0.424859
\(555\) 20.0000 0.848953
\(556\) −12.0000 −0.508913
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) −8.00000 −0.338667
\(559\) −4.00000 −0.169182
\(560\) 2.00000 0.0845154
\(561\) −24.0000 −1.01328
\(562\) 10.0000 0.421825
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 4.00000 0.168430
\(565\) 4.00000 0.168281
\(566\) 28.0000 1.17693
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) −8.00000 −0.335083
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 19.0000 0.790296
\(579\) −6.00000 −0.249351
\(580\) −12.0000 −0.498273
\(581\) 4.00000 0.165948
\(582\) −2.00000 −0.0829027
\(583\) −40.0000 −1.65663
\(584\) −10.0000 −0.413803
\(585\) −2.00000 −0.0826898
\(586\) 2.00000 0.0826192
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 1.00000 0.0412393
\(589\) 32.0000 1.31854
\(590\) 8.00000 0.329355
\(591\) 6.00000 0.246807
\(592\) 10.0000 0.410997
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) −4.00000 −0.164122
\(595\) 12.0000 0.491952
\(596\) 14.0000 0.573462
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 4.00000 0.163028
\(603\) −8.00000 −0.325785
\(604\) 16.0000 0.651031
\(605\) 10.0000 0.406558
\(606\) 10.0000 0.406222
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) −4.00000 −0.162221
\(609\) −6.00000 −0.243132
\(610\) −12.0000 −0.485866
\(611\) −4.00000 −0.161823
\(612\) 6.00000 0.242536
\(613\) −30.0000 −1.21169 −0.605844 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) −12.0000 −0.484281
\(615\) −12.0000 −0.483887
\(616\) −4.00000 −0.161165
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −12.0000 −0.482711
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) −16.0000 −0.642575
\(621\) 0 0
\(622\) −16.0000 −0.641542
\(623\) −6.00000 −0.240385
\(624\) −1.00000 −0.0400320
\(625\) −19.0000 −0.760000
\(626\) 10.0000 0.399680
\(627\) 16.0000 0.638978
\(628\) 18.0000 0.718278
\(629\) 60.0000 2.39236
\(630\) 2.00000 0.0796819
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −8.00000 −0.318223
\(633\) 12.0000 0.476957
\(634\) −18.0000 −0.714871
\(635\) −32.0000 −1.26988
\(636\) 10.0000 0.396526
\(637\) −1.00000 −0.0396214
\(638\) 24.0000 0.950169
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 16.0000 0.631470
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) −24.0000 −0.944267
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 1.00000 0.0392837
\(649\) −16.0000 −0.628055
\(650\) 1.00000 0.0392232
\(651\) −8.00000 −0.313545
\(652\) 16.0000 0.626608
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) −6.00000 −0.234619
\(655\) −40.0000 −1.56293
\(656\) −6.00000 −0.234261
\(657\) −10.0000 −0.390137
\(658\) 4.00000 0.155936
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) −8.00000 −0.311400
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) −8.00000 −0.310929
\(663\) −6.00000 −0.233021
\(664\) 4.00000 0.155230
\(665\) −8.00000 −0.310227
\(666\) 10.0000 0.387492
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) −8.00000 −0.309298
\(670\) −16.0000 −0.618134
\(671\) 24.0000 0.926510
\(672\) 1.00000 0.0385758
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) 34.0000 1.30963
\(675\) −1.00000 −0.0384900
\(676\) 1.00000 0.0384615
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 2.00000 0.0768095
\(679\) −2.00000 −0.0767530
\(680\) 12.0000 0.460179
\(681\) −12.0000 −0.459841
\(682\) 32.0000 1.22534
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −4.00000 −0.152944
\(685\) 20.0000 0.764161
\(686\) 1.00000 0.0381802
\(687\) −6.00000 −0.228914
\(688\) 4.00000 0.152499
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 18.0000 0.684257
\(693\) −4.00000 −0.151947
\(694\) 8.00000 0.303676
\(695\) −24.0000 −0.910372
\(696\) −6.00000 −0.227429
\(697\) −36.0000 −1.36360
\(698\) 2.00000 0.0757011
\(699\) 18.0000 0.680823
\(700\) −1.00000 −0.0377964
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −40.0000 −1.50863
\(704\) −4.00000 −0.150756
\(705\) 8.00000 0.301297
\(706\) −6.00000 −0.225813
\(707\) 10.0000 0.376089
\(708\) 4.00000 0.150329
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 6.00000 0.224544
\(715\) 8.00000 0.299183
\(716\) 24.0000 0.896922
\(717\) −24.0000 −0.896296
\(718\) −16.0000 −0.597115
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 2.00000 0.0745356
\(721\) −12.0000 −0.446903
\(722\) −3.00000 −0.111648
\(723\) 30.0000 1.11571
\(724\) −22.0000 −0.817624
\(725\) 6.00000 0.222834
\(726\) 5.00000 0.185567
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 1.00000 0.0370370
\(730\) −20.0000 −0.740233
\(731\) 24.0000 0.887672
\(732\) −6.00000 −0.221766
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) −28.0000 −1.03350
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) 32.0000 1.17874
\(738\) −6.00000 −0.220863
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 20.0000 0.735215
\(741\) 4.00000 0.146944
\(742\) 10.0000 0.367112
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) −8.00000 −0.293294
\(745\) 28.0000 1.02584
\(746\) 14.0000 0.512576
\(747\) 4.00000 0.146352
\(748\) −24.0000 −0.877527
\(749\) 16.0000 0.584627
\(750\) −12.0000 −0.438178
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 4.00000 0.145865
\(753\) −4.00000 −0.145768
\(754\) 6.00000 0.218507
\(755\) 32.0000 1.16460
\(756\) 1.00000 0.0363696
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) −16.0000 −0.579619
\(763\) −6.00000 −0.217215
\(764\) 0 0
\(765\) 12.0000 0.433861
\(766\) 4.00000 0.144526
\(767\) −4.00000 −0.144432
\(768\) 1.00000 0.0360844
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) −8.00000 −0.288300
\(771\) 6.00000 0.216085
\(772\) −6.00000 −0.215945
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 4.00000 0.143777
\(775\) 8.00000 0.287368
\(776\) −2.00000 −0.0717958
\(777\) 10.0000 0.358748
\(778\) 34.0000 1.21896
\(779\) 24.0000 0.859889
\(780\) −2.00000 −0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 1.00000 0.0357143
\(785\) 36.0000 1.28490
\(786\) −20.0000 −0.713376
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) 6.00000 0.213741
\(789\) 24.0000 0.854423
\(790\) −16.0000 −0.569254
\(791\) 2.00000 0.0711118
\(792\) −4.00000 −0.142134
\(793\) 6.00000 0.213066
\(794\) 18.0000 0.638796
\(795\) 20.0000 0.709327
\(796\) 20.0000 0.708881
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) −4.00000 −0.141598
\(799\) 24.0000 0.849059
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) −30.0000 −1.05934
\(803\) 40.0000 1.41157
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 18.0000 0.633630
\(808\) 10.0000 0.351799
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 2.00000 0.0702728
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −6.00000 −0.210559
\(813\) 8.00000 0.280572
\(814\) −40.0000 −1.40200
\(815\) 32.0000 1.12091
\(816\) 6.00000 0.210042
\(817\) −16.0000 −0.559769
\(818\) 22.0000 0.769212
\(819\) −1.00000 −0.0349428
\(820\) −12.0000 −0.419058
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 10.0000 0.348790
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) −12.0000 −0.418040
\(825\) 4.00000 0.139262
\(826\) 4.00000 0.139178
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 8.00000 0.277684
\(831\) −10.0000 −0.346896
\(832\) −1.00000 −0.0346688
\(833\) 6.00000 0.207888
\(834\) −12.0000 −0.415526
\(835\) −24.0000 −0.830554
\(836\) 16.0000 0.553372
\(837\) −8.00000 −0.276520
\(838\) 28.0000 0.967244
\(839\) −4.00000 −0.138095 −0.0690477 0.997613i \(-0.521996\pi\)
−0.0690477 + 0.997613i \(0.521996\pi\)
\(840\) 2.00000 0.0690066
\(841\) 7.00000 0.241379
\(842\) −30.0000 −1.03387
\(843\) 10.0000 0.344418
\(844\) 12.0000 0.413057
\(845\) 2.00000 0.0688021
\(846\) 4.00000 0.137523
\(847\) 5.00000 0.171802
\(848\) 10.0000 0.343401
\(849\) 28.0000 0.960958
\(850\) −6.00000 −0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) −6.00000 −0.205316
\(855\) −8.00000 −0.273594
\(856\) 16.0000 0.546869
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 4.00000 0.136558
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 8.00000 0.272798
\(861\) −6.00000 −0.204479
\(862\) −32.0000 −1.08992
\(863\) 56.0000 1.90626 0.953131 0.302558i \(-0.0978405\pi\)
0.953131 + 0.302558i \(0.0978405\pi\)
\(864\) 1.00000 0.0340207
\(865\) 36.0000 1.22404
\(866\) 34.0000 1.15537
\(867\) 19.0000 0.645274
\(868\) −8.00000 −0.271538
\(869\) 32.0000 1.08553
\(870\) −12.0000 −0.406838
\(871\) 8.00000 0.271070
\(872\) −6.00000 −0.203186
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) −10.0000 −0.337869
\(877\) −6.00000 −0.202606 −0.101303 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) −20.0000 −0.674967
\(879\) 2.00000 0.0674583
\(880\) −8.00000 −0.269680
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 1.00000 0.0336718
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) −6.00000 −0.201802
\(885\) 8.00000 0.268917
\(886\) 24.0000 0.806296
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 10.0000 0.335578
\(889\) −16.0000 −0.536623
\(890\) −12.0000 −0.402241
\(891\) −4.00000 −0.134005
\(892\) −8.00000 −0.267860
\(893\) −16.0000 −0.535420
\(894\) 14.0000 0.468230
\(895\) 48.0000 1.60446
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 2.00000 0.0667409
\(899\) 48.0000 1.60089
\(900\) −1.00000 −0.0333333
\(901\) 60.0000 1.99889
\(902\) 24.0000 0.799113
\(903\) 4.00000 0.133112
\(904\) 2.00000 0.0665190
\(905\) −44.0000 −1.46261
\(906\) 16.0000 0.531564
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −12.0000 −0.398234
\(909\) 10.0000 0.331679
\(910\) −2.00000 −0.0662994
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −4.00000 −0.132453
\(913\) −16.0000 −0.529523
\(914\) −14.0000 −0.463079
\(915\) −12.0000 −0.396708
\(916\) −6.00000 −0.198246
\(917\) −20.0000 −0.660458
\(918\) 6.00000 0.198030
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) −14.0000 −0.461065
\(923\) 0 0
\(924\) −4.00000 −0.131590
\(925\) −10.0000 −0.328798
\(926\) −24.0000 −0.788689
\(927\) −12.0000 −0.394132
\(928\) −6.00000 −0.196960
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) −16.0000 −0.524661
\(931\) −4.00000 −0.131095
\(932\) 18.0000 0.589610
\(933\) −16.0000 −0.523816
\(934\) 12.0000 0.392652
\(935\) −48.0000 −1.56977
\(936\) −1.00000 −0.0326860
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) −8.00000 −0.261209
\(939\) 10.0000 0.326338
\(940\) 8.00000 0.260931
\(941\) −54.0000 −1.76035 −0.880175 0.474650i \(-0.842575\pi\)
−0.880175 + 0.474650i \(0.842575\pi\)
\(942\) 18.0000 0.586472
\(943\) 0 0
\(944\) 4.00000 0.130189
\(945\) 2.00000 0.0650600
\(946\) −16.0000 −0.520205
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −8.00000 −0.259828
\(949\) 10.0000 0.324614
\(950\) 4.00000 0.129777
\(951\) −18.0000 −0.583690
\(952\) 6.00000 0.194461
\(953\) −14.0000 −0.453504 −0.226752 0.973952i \(-0.572811\pi\)
−0.226752 + 0.973952i \(0.572811\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 24.0000 0.775810
\(958\) −4.00000 −0.129234
\(959\) 10.0000 0.322917
\(960\) 2.00000 0.0645497
\(961\) 33.0000 1.06452
\(962\) −10.0000 −0.322413
\(963\) 16.0000 0.515593
\(964\) 30.0000 0.966235
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 5.00000 0.160706
\(969\) −24.0000 −0.770991
\(970\) −4.00000 −0.128432
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) 1.00000 0.0320750
\(973\) −12.0000 −0.384702
\(974\) 8.00000 0.256337
\(975\) 1.00000 0.0320256
\(976\) −6.00000 −0.192055
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 16.0000 0.511624
\(979\) 24.0000 0.767043
\(980\) 2.00000 0.0638877
\(981\) −6.00000 −0.191565
\(982\) −32.0000 −1.02116
\(983\) 4.00000 0.127580 0.0637901 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) −6.00000 −0.191273
\(985\) 12.0000 0.382352
\(986\) −36.0000 −1.14647
\(987\) 4.00000 0.127321
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) −8.00000 −0.254257
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) −8.00000 −0.254000
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) 40.0000 1.26809
\(996\) 4.00000 0.126745
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) 32.0000 1.01294
\(999\) 10.0000 0.316386
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 546.2.a.g.1.1 1
3.2 odd 2 1638.2.a.d.1.1 1
4.3 odd 2 4368.2.a.k.1.1 1
7.6 odd 2 3822.2.a.t.1.1 1
13.12 even 2 7098.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.g.1.1 1 1.1 even 1 trivial
1638.2.a.d.1.1 1 3.2 odd 2
3822.2.a.t.1.1 1 7.6 odd 2
4368.2.a.k.1.1 1 4.3 odd 2
7098.2.a.j.1.1 1 13.12 even 2