Properties

Label 462.2.a.e.1.1
Level $462$
Weight $2$
Character 462.1
Self dual yes
Analytic conductor $3.689$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.68908857338\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 462.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -4.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} -8.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -6.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +4.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -4.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -2.00000 q^{38} +6.00000 q^{39} -4.00000 q^{40} +12.0000 q^{41} -1.00000 q^{42} +4.00000 q^{43} -1.00000 q^{44} -4.00000 q^{45} -8.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +11.0000 q^{50} +4.00000 q^{51} -6.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} +1.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} +4.00000 q^{60} +10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +24.0000 q^{65} +1.00000 q^{66} +4.00000 q^{67} -4.00000 q^{68} +8.00000 q^{69} -4.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} -6.00000 q^{74} -11.0000 q^{75} -2.00000 q^{76} -1.00000 q^{77} +6.00000 q^{78} -16.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} -14.0000 q^{83} -1.00000 q^{84} +16.0000 q^{85} +4.00000 q^{86} +6.00000 q^{87} -1.00000 q^{88} -14.0000 q^{89} -4.00000 q^{90} -6.00000 q^{91} -8.00000 q^{92} -6.00000 q^{93} +6.00000 q^{94} +8.00000 q^{95} -1.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −4.00000 −1.26491
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 1.00000 0.267261
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −4.00000 −0.894427
\(21\) −1.00000 −0.218218
\(22\) −1.00000 −0.213201
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) −1.00000 −0.204124
\(25\) 11.0000 2.20000
\(26\) −6.00000 −1.17670
\(27\) −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 \(0.688081\pi\)
\(30\) 4.00000 0.730297
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) −4.00000 −0.685994
\(35\) −4.00000 −0.676123
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −2.00000 −0.324443
\(39\) 6.00000 0.960769
\(40\) −4.00000 −0.632456
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) −1.00000 −0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) −4.00000 −0.596285
\(46\) −8.00000 −1.17954
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 11.0000 1.55563
\(51\) 4.00000 0.560112
\(52\) −6.00000 −0.832050
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.00000 0.539360
\(56\) 1.00000 0.133631
\(57\) 2.00000 0.264906
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 4.00000 0.516398
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 6.00000 0.762001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 24.0000 2.97683
\(66\) 1.00000 0.123091
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −4.00000 −0.485071
\(69\) 8.00000 0.963087
\(70\) −4.00000 −0.478091
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −6.00000 −0.697486
\(75\) −11.0000 −1.27017
\(76\) −2.00000 −0.229416
\(77\) −1.00000 −0.113961
\(78\) 6.00000 0.679366
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) −1.00000 −0.109109
\(85\) 16.0000 1.73544
\(86\) 4.00000 0.431331
\(87\) 6.00000 0.643268
\(88\) −1.00000 −0.106600
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) −4.00000 −0.421637
\(91\) −6.00000 −0.628971
\(92\) −8.00000 −0.834058
\(93\) −6.00000 −0.622171
\(94\) 6.00000 0.618853
\(95\) 8.00000 0.820783
\(96\) −1.00000 −0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.00000 −0.100504
\(100\) 11.0000 1.10000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 4.00000 0.396059
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) −6.00000 −0.588348
\(105\) 4.00000 0.390360
\(106\) 2.00000 0.194257
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 4.00000 0.381385
\(111\) 6.00000 0.569495
\(112\) 1.00000 0.0944911
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 2.00000 0.187317
\(115\) 32.0000 2.98402
\(116\) −6.00000 −0.557086
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 4.00000 0.365148
\(121\) 1.00000 0.0909091
\(122\) 10.0000 0.905357
\(123\) −12.0000 −1.08200
\(124\) 6.00000 0.538816
\(125\) −24.0000 −2.14663
\(126\) 1.00000 0.0890871
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 24.0000 2.10494
\(131\) −2.00000 −0.174741 −0.0873704 0.996176i \(-0.527846\pi\)
−0.0873704 + 0.996176i \(0.527846\pi\)
\(132\) 1.00000 0.0870388
\(133\) −2.00000 −0.173422
\(134\) 4.00000 0.345547
\(135\) 4.00000 0.344265
\(136\) −4.00000 −0.342997
\(137\) 14.0000 1.19610 0.598050 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) 8.00000 0.681005
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) −4.00000 −0.338062
\(141\) −6.00000 −0.505291
\(142\) −12.0000 −1.00702
\(143\) 6.00000 0.501745
\(144\) 1.00000 0.0833333
\(145\) 24.0000 1.99309
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) −6.00000 −0.493197
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −11.0000 −0.898146
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −2.00000 −0.162221
\(153\) −4.00000 −0.323381
\(154\) −1.00000 −0.0805823
\(155\) −24.0000 −1.92773
\(156\) 6.00000 0.480384
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) −16.0000 −1.27289
\(159\) −2.00000 −0.158610
\(160\) −4.00000 −0.316228
\(161\) −8.00000 −0.630488
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 12.0000 0.937043
\(165\) −4.00000 −0.311400
\(166\) −14.0000 −1.08661
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 23.0000 1.76923
\(170\) 16.0000 1.22714
\(171\) −2.00000 −0.152944
\(172\) 4.00000 0.304997
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 6.00000 0.454859
\(175\) 11.0000 0.831522
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −4.00000 −0.298142
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) −6.00000 −0.444750
\(183\) −10.0000 −0.739221
\(184\) −8.00000 −0.589768
\(185\) 24.0000 1.76452
\(186\) −6.00000 −0.439941
\(187\) 4.00000 0.292509
\(188\) 6.00000 0.437595
\(189\) −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
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −14.0000 −1.00514
\(195\) −24.0000 −1.71868
\(196\) 1.00000 0.0714286
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 11.0000 0.777817
\(201\) −4.00000 −0.282138
\(202\) 14.0000 0.985037
\(203\) −6.00000 −0.421117
\(204\) 4.00000 0.280056
\(205\) −48.0000 −3.35247
\(206\) −6.00000 −0.418040
\(207\) −8.00000 −0.556038
\(208\) −6.00000 −0.416025
\(209\) 2.00000 0.138343
\(210\) 4.00000 0.276026
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 2.00000 0.137361
\(213\) 12.0000 0.822226
\(214\) −8.00000 −0.546869
\(215\) −16.0000 −1.09119
\(216\) −1.00000 −0.0680414
\(217\) 6.00000 0.407307
\(218\) 6.00000 0.406371
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) 24.0000 1.61441
\(222\) 6.00000 0.402694
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) 1.00000 0.0668153
\(225\) 11.0000 0.733333
\(226\) 14.0000 0.931266
\(227\) 6.00000 0.398234 0.199117 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) 2.00000 0.132453
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 32.0000 2.11002
\(231\) 1.00000 0.0657952
\(232\) −6.00000 −0.393919
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) −6.00000 −0.392232
\(235\) −24.0000 −1.56559
\(236\) 0 0
\(237\) 16.0000 1.03931
\(238\) −4.00000 −0.259281
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 4.00000 0.258199
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) −4.00000 −0.255551
\(246\) −12.0000 −0.765092
\(247\) 12.0000 0.763542
\(248\) 6.00000 0.381000
\(249\) 14.0000 0.887214
\(250\) −24.0000 −1.51789
\(251\) −16.0000 −1.00991 −0.504956 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 1.00000 0.0629941
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) −16.0000 −1.00196
\(256\) 1.00000 0.0625000
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) −4.00000 −0.249029
\(259\) −6.00000 −0.372822
\(260\) 24.0000 1.48842
\(261\) −6.00000 −0.371391
\(262\) −2.00000 −0.123560
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 1.00000 0.0615457
\(265\) −8.00000 −0.491436
\(266\) −2.00000 −0.122628
\(267\) 14.0000 0.856786
\(268\) 4.00000 0.244339
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 4.00000 0.243432
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) −4.00000 −0.242536
\(273\) 6.00000 0.363137
\(274\) 14.0000 0.845771
\(275\) −11.0000 −0.663325
\(276\) 8.00000 0.481543
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 2.00000 0.119952
\(279\) 6.00000 0.359211
\(280\) −4.00000 −0.239046
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) −6.00000 −0.357295
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −12.0000 −0.712069
\(285\) −8.00000 −0.473879
\(286\) 6.00000 0.354787
\(287\) 12.0000 0.708338
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 24.0000 1.40933
\(291\) 14.0000 0.820695
\(292\) 0 0
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 1.00000 0.0580259
\(298\) −10.0000 −0.579284
\(299\) 48.0000 2.77591
\(300\) −11.0000 −0.635085
\(301\) 4.00000 0.230556
\(302\) −8.00000 −0.460348
\(303\) −14.0000 −0.804279
\(304\) −2.00000 −0.114708
\(305\) −40.0000 −2.29039
\(306\) −4.00000 −0.228665
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 6.00000 0.341328
\(310\) −24.0000 −1.36311
\(311\) 10.0000 0.567048 0.283524 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) 6.00000 0.339683
\(313\) −34.0000 −1.92179 −0.960897 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) −4.00000 −0.225733
\(315\) −4.00000 −0.225374
\(316\) −16.0000 −0.900070
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −2.00000 −0.112154
\(319\) 6.00000 0.335936
\(320\) −4.00000 −0.223607
\(321\) 8.00000 0.446516
\(322\) −8.00000 −0.445823
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) −66.0000 −3.66102
\(326\) 4.00000 0.221540
\(327\) −6.00000 −0.331801
\(328\) 12.0000 0.662589
\(329\) 6.00000 0.330791
\(330\) −4.00000 −0.220193
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −14.0000 −0.768350
\(333\) −6.00000 −0.328798
\(334\) −12.0000 −0.656611
\(335\) −16.0000 −0.874173
\(336\) −1.00000 −0.0545545
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 23.0000 1.25104
\(339\) −14.0000 −0.760376
\(340\) 16.0000 0.867722
\(341\) −6.00000 −0.324918
\(342\) −2.00000 −0.108148
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) −32.0000 −1.72282
\(346\) 2.00000 0.107521
\(347\) 8.00000 0.429463 0.214731 0.976673i \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(348\) 6.00000 0.321634
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 11.0000 0.587975
\(351\) 6.00000 0.320256
\(352\) −1.00000 −0.0533002
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 48.0000 2.54758
\(356\) −14.0000 −0.741999
\(357\) 4.00000 0.211702
\(358\) −4.00000 −0.211407
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −4.00000 −0.210819
\(361\) −15.0000 −0.789474
\(362\) −8.00000 −0.420471
\(363\) −1.00000 −0.0524864
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) −34.0000 −1.77479 −0.887393 0.461014i \(-0.847486\pi\)
−0.887393 + 0.461014i \(0.847486\pi\)
\(368\) −8.00000 −0.417029
\(369\) 12.0000 0.624695
\(370\) 24.0000 1.24770
\(371\) 2.00000 0.103835
\(372\) −6.00000 −0.311086
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 4.00000 0.206835
\(375\) 24.0000 1.23935
\(376\) 6.00000 0.309426
\(377\) 36.0000 1.85409
\(378\) −1.00000 −0.0514344
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) 0 0
\(383\) −22.0000 −1.12415 −0.562074 0.827087i \(-0.689996\pi\)
−0.562074 + 0.827087i \(0.689996\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 4.00000 0.203859
\(386\) −14.0000 −0.712581
\(387\) 4.00000 0.203331
\(388\) −14.0000 −0.710742
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −24.0000 −1.21529
\(391\) 32.0000 1.61831
\(392\) 1.00000 0.0505076
\(393\) 2.00000 0.100887
\(394\) 10.0000 0.503793
\(395\) 64.0000 3.22019
\(396\) −1.00000 −0.0502519
\(397\) −20.0000 −1.00377 −0.501886 0.864934i \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) 2.00000 0.100251
\(399\) 2.00000 0.100125
\(400\) 11.0000 0.550000
\(401\) −26.0000 −1.29838 −0.649189 0.760627i \(-0.724892\pi\)
−0.649189 + 0.760627i \(0.724892\pi\)
\(402\) −4.00000 −0.199502
\(403\) −36.0000 −1.79329
\(404\) 14.0000 0.696526
\(405\) −4.00000 −0.198762
\(406\) −6.00000 −0.297775
\(407\) 6.00000 0.297409
\(408\) 4.00000 0.198030
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) −48.0000 −2.37055
\(411\) −14.0000 −0.690569
\(412\) −6.00000 −0.295599
\(413\) 0 0
\(414\) −8.00000 −0.393179
\(415\) 56.0000 2.74893
\(416\) −6.00000 −0.294174
\(417\) −2.00000 −0.0979404
\(418\) 2.00000 0.0978232
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 4.00000 0.195180
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) −4.00000 −0.194717
\(423\) 6.00000 0.291730
\(424\) 2.00000 0.0971286
\(425\) −44.0000 −2.13431
\(426\) 12.0000 0.581402
\(427\) 10.0000 0.483934
\(428\) −8.00000 −0.386695
\(429\) −6.00000 −0.289683
\(430\) −16.0000 −0.771589
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 6.00000 0.288009
\(435\) −24.0000 −1.15071
\(436\) 6.00000 0.287348
\(437\) 16.0000 0.765384
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 4.00000 0.190693
\(441\) 1.00000 0.0476190
\(442\) 24.0000 1.14156
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 6.00000 0.284747
\(445\) 56.0000 2.65465
\(446\) 6.00000 0.284108
\(447\) 10.0000 0.472984
\(448\) 1.00000 0.0472456
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 11.0000 0.518545
\(451\) −12.0000 −0.565058
\(452\) 14.0000 0.658505
\(453\) 8.00000 0.375873
\(454\) 6.00000 0.281594
\(455\) 24.0000 1.12514
\(456\) 2.00000 0.0936586
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 8.00000 0.373815
\(459\) 4.00000 0.186704
\(460\) 32.0000 1.49201
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 1.00000 0.0465242
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −6.00000 −0.278543
\(465\) 24.0000 1.11297
\(466\) −14.0000 −0.648537
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) −6.00000 −0.277350
\(469\) 4.00000 0.184703
\(470\) −24.0000 −1.10704
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 16.0000 0.734904
\(475\) −22.0000 −1.00943
\(476\) −4.00000 −0.183340
\(477\) 2.00000 0.0915737
\(478\) −16.0000 −0.731823
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 4.00000 0.182574
\(481\) 36.0000 1.64146
\(482\) −8.00000 −0.364390
\(483\) 8.00000 0.364013
\(484\) 1.00000 0.0454545
\(485\) 56.0000 2.54283
\(486\) −1.00000 −0.0453609
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 10.0000 0.452679
\(489\) −4.00000 −0.180886
\(490\) −4.00000 −0.180702
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) −12.0000 −0.541002
\(493\) 24.0000 1.08091
\(494\) 12.0000 0.539906
\(495\) 4.00000 0.179787
\(496\) 6.00000 0.269408
\(497\) −12.0000 −0.538274
\(498\) 14.0000 0.627355
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −24.0000 −1.07331
\(501\) 12.0000 0.536120
\(502\) −16.0000 −0.714115
\(503\) 20.0000 0.891756 0.445878 0.895094i \(-0.352892\pi\)
0.445878 + 0.895094i \(0.352892\pi\)
\(504\) 1.00000 0.0445435
\(505\) −56.0000 −2.49197
\(506\) 8.00000 0.355643
\(507\) −23.0000 −1.02147
\(508\) 0 0
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) −16.0000 −0.708492
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 2.00000 0.0883022
\(514\) 26.0000 1.14681
\(515\) 24.0000 1.05757
\(516\) −4.00000 −0.176090
\(517\) −6.00000 −0.263880
\(518\) −6.00000 −0.263625
\(519\) −2.00000 −0.0877903
\(520\) 24.0000 1.05247
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −6.00000 −0.262613
\(523\) −22.0000 −0.961993 −0.480996 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) −2.00000 −0.0873704
\(525\) −11.0000 −0.480079
\(526\) 0 0
\(527\) −24.0000 −1.04546
\(528\) 1.00000 0.0435194
\(529\) 41.0000 1.78261
\(530\) −8.00000 −0.347498
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) −72.0000 −3.11867
\(534\) 14.0000 0.605839
\(535\) 32.0000 1.38348
\(536\) 4.00000 0.172774
\(537\) 4.00000 0.172613
\(538\) −4.00000 −0.172452
\(539\) −1.00000 −0.0430730
\(540\) 4.00000 0.172133
\(541\) −14.0000 −0.601907 −0.300954 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) −4.00000 −0.171815
\(543\) 8.00000 0.343313
\(544\) −4.00000 −0.171499
\(545\) −24.0000 −1.02805
\(546\) 6.00000 0.256776
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 14.0000 0.598050
\(549\) 10.0000 0.426790
\(550\) −11.0000 −0.469042
\(551\) 12.0000 0.511217
\(552\) 8.00000 0.340503
\(553\) −16.0000 −0.680389
\(554\) −10.0000 −0.424859
\(555\) −24.0000 −1.01874
\(556\) 2.00000 0.0848189
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 6.00000 0.254000
\(559\) −24.0000 −1.01509
\(560\) −4.00000 −0.169031
\(561\) −4.00000 −0.168880
\(562\) 26.0000 1.09674
\(563\) −26.0000 −1.09577 −0.547885 0.836554i \(-0.684567\pi\)
−0.547885 + 0.836554i \(0.684567\pi\)
\(564\) −6.00000 −0.252646
\(565\) −56.0000 −2.35594
\(566\) −14.0000 −0.588464
\(567\) 1.00000 0.0419961
\(568\) −12.0000 −0.503509
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) −8.00000 −0.335083
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 6.00000 0.250873
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) −88.0000 −3.66985
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 14.0000 0.581820
\(580\) 24.0000 0.996546
\(581\) −14.0000 −0.580818
\(582\) 14.0000 0.580319
\(583\) −2.00000 −0.0828315
\(584\) 0 0
\(585\) 24.0000 0.992278
\(586\) −14.0000 −0.578335
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) −6.00000 −0.246598
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 1.00000 0.0410305
\(595\) 16.0000 0.655936
\(596\) −10.0000 −0.409616
\(597\) −2.00000 −0.0818546
\(598\) 48.0000 1.96287
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) −11.0000 −0.449073
\(601\) 4.00000 0.163163 0.0815817 0.996667i \(-0.474003\pi\)
0.0815817 + 0.996667i \(0.474003\pi\)
\(602\) 4.00000 0.163028
\(603\) 4.00000 0.162893
\(604\) −8.00000 −0.325515
\(605\) −4.00000 −0.162623
\(606\) −14.0000 −0.568711
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 6.00000 0.243132
\(610\) −40.0000 −1.61955
\(611\) −36.0000 −1.45640
\(612\) −4.00000 −0.161690
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) −14.0000 −0.564994
\(615\) 48.0000 1.93555
\(616\) −1.00000 −0.0402911
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 6.00000 0.241355
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) −24.0000 −0.963863
\(621\) 8.00000 0.321029
\(622\) 10.0000 0.400963
\(623\) −14.0000 −0.560898
\(624\) 6.00000 0.240192
\(625\) 41.0000 1.64000
\(626\) −34.0000 −1.35891
\(627\) −2.00000 −0.0798723
\(628\) −4.00000 −0.159617
\(629\) 24.0000 0.956943
\(630\) −4.00000 −0.159364
\(631\) 28.0000 1.11466 0.557331 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) −16.0000 −0.636446
\(633\) 4.00000 0.158986
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) −6.00000 −0.237729
\(638\) 6.00000 0.237542
\(639\) −12.0000 −0.474713
\(640\) −4.00000 −0.158114
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 8.00000 0.315735
\(643\) 32.0000 1.26196 0.630978 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) −8.00000 −0.315244
\(645\) 16.0000 0.629999
\(646\) 8.00000 0.314756
\(647\) 26.0000 1.02217 0.511083 0.859532i \(-0.329245\pi\)
0.511083 + 0.859532i \(0.329245\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −66.0000 −2.58873
\(651\) −6.00000 −0.235159
\(652\) 4.00000 0.156652
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) −6.00000 −0.234619
\(655\) 8.00000 0.312586
\(656\) 12.0000 0.468521
\(657\) 0 0
\(658\) 6.00000 0.233904
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) −4.00000 −0.155700
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 20.0000 0.777322
\(663\) −24.0000 −0.932083
\(664\) −14.0000 −0.543305
\(665\) 8.00000 0.310227
\(666\) −6.00000 −0.232495
\(667\) 48.0000 1.85857
\(668\) −12.0000 −0.464294
\(669\) −6.00000 −0.231973
\(670\) −16.0000 −0.618134
\(671\) −10.0000 −0.386046
\(672\) −1.00000 −0.0385758
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 6.00000 0.231111
\(675\) −11.0000 −0.423390
\(676\) 23.0000 0.884615
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) −14.0000 −0.537667
\(679\) −14.0000 −0.537271
\(680\) 16.0000 0.613572
\(681\) −6.00000 −0.229920
\(682\) −6.00000 −0.229752
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −56.0000 −2.13965
\(686\) 1.00000 0.0381802
\(687\) −8.00000 −0.305219
\(688\) 4.00000 0.152499
\(689\) −12.0000 −0.457164
\(690\) −32.0000 −1.21822
\(691\) 24.0000 0.913003 0.456502 0.889723i \(-0.349102\pi\)
0.456502 + 0.889723i \(0.349102\pi\)
\(692\) 2.00000 0.0760286
\(693\) −1.00000 −0.0379869
\(694\) 8.00000 0.303676
\(695\) −8.00000 −0.303457
\(696\) 6.00000 0.227429
\(697\) −48.0000 −1.81813
\(698\) 6.00000 0.227103
\(699\) 14.0000 0.529529
\(700\) 11.0000 0.415761
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 6.00000 0.226455
\(703\) 12.0000 0.452589
\(704\) −1.00000 −0.0376889
\(705\) 24.0000 0.903892
\(706\) −6.00000 −0.225813
\(707\) 14.0000 0.526524
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 48.0000 1.80141
\(711\) −16.0000 −0.600047
\(712\) −14.0000 −0.524672
\(713\) −48.0000 −1.79761
\(714\) 4.00000 0.149696
\(715\) −24.0000 −0.897549
\(716\) −4.00000 −0.149487
\(717\) 16.0000 0.597531
\(718\) 24.0000 0.895672
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) −4.00000 −0.149071
\(721\) −6.00000 −0.223452
\(722\) −15.0000 −0.558242
\(723\) 8.00000 0.297523
\(724\) −8.00000 −0.297318
\(725\) −66.0000 −2.45118
\(726\) −1.00000 −0.0371135
\(727\) −42.0000 −1.55769 −0.778847 0.627214i \(-0.784195\pi\)
−0.778847 + 0.627214i \(0.784195\pi\)
\(728\) −6.00000 −0.222375
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) −10.0000 −0.369611
\(733\) 50.0000 1.84679 0.923396 0.383849i \(-0.125402\pi\)
0.923396 + 0.383849i \(0.125402\pi\)
\(734\) −34.0000 −1.25496
\(735\) 4.00000 0.147542
\(736\) −8.00000 −0.294884
\(737\) −4.00000 −0.147342
\(738\) 12.0000 0.441726
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 24.0000 0.882258
\(741\) −12.0000 −0.440831
\(742\) 2.00000 0.0734223
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) −6.00000 −0.219971
\(745\) 40.0000 1.46549
\(746\) −14.0000 −0.512576
\(747\) −14.0000 −0.512233
\(748\) 4.00000 0.146254
\(749\) −8.00000 −0.292314
\(750\) 24.0000 0.876356
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) 6.00000 0.218797
\(753\) 16.0000 0.583072
\(754\) 36.0000 1.31104
\(755\) 32.0000 1.16460
\(756\) −1.00000 −0.0363696
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 20.0000 0.726433
\(759\) −8.00000 −0.290382
\(760\) 8.00000 0.290191
\(761\) 24.0000 0.869999 0.435000 0.900431i \(-0.356748\pi\)
0.435000 + 0.900431i \(0.356748\pi\)
\(762\) 0 0
\(763\) 6.00000 0.217215
\(764\) 0 0
\(765\) 16.0000 0.578481
\(766\) −22.0000 −0.794892
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 28.0000 1.00971 0.504853 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(770\) 4.00000 0.144150
\(771\) −26.0000 −0.936367
\(772\) −14.0000 −0.503871
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 4.00000 0.143777
\(775\) 66.0000 2.37079
\(776\) −14.0000 −0.502571
\(777\) 6.00000 0.215249
\(778\) 6.00000 0.215110
\(779\) −24.0000 −0.859889
\(780\) −24.0000 −0.859338
\(781\) 12.0000 0.429394
\(782\) 32.0000 1.14432
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) 16.0000 0.571064
\(786\) 2.00000 0.0713376
\(787\) −50.0000 −1.78231 −0.891154 0.453701i \(-0.850103\pi\)
−0.891154 + 0.453701i \(0.850103\pi\)
\(788\) 10.0000 0.356235
\(789\) 0 0
\(790\) 64.0000 2.27702
\(791\) 14.0000 0.497783
\(792\) −1.00000 −0.0355335
\(793\) −60.0000 −2.13066
\(794\) −20.0000 −0.709773
\(795\) 8.00000 0.283731
\(796\) 2.00000 0.0708881
\(797\) 24.0000 0.850124 0.425062 0.905164i \(-0.360252\pi\)
0.425062 + 0.905164i \(0.360252\pi\)
\(798\) 2.00000 0.0707992
\(799\) −24.0000 −0.849059
\(800\) 11.0000 0.388909
\(801\) −14.0000 −0.494666
\(802\) −26.0000 −0.918092
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 32.0000 1.12785
\(806\) −36.0000 −1.26805
\(807\) 4.00000 0.140807
\(808\) 14.0000 0.492518
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) −4.00000 −0.140546
\(811\) 26.0000 0.912983 0.456492 0.889728i \(-0.349106\pi\)
0.456492 + 0.889728i \(0.349106\pi\)
\(812\) −6.00000 −0.210559
\(813\) 4.00000 0.140286
\(814\) 6.00000 0.210300
\(815\) −16.0000 −0.560456
\(816\) 4.00000 0.140028
\(817\) −8.00000 −0.279885
\(818\) 4.00000 0.139857
\(819\) −6.00000 −0.209657
\(820\) −48.0000 −1.67623
\(821\) −26.0000 −0.907406 −0.453703 0.891153i \(-0.649897\pi\)
−0.453703 + 0.891153i \(0.649897\pi\)
\(822\) −14.0000 −0.488306
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) −6.00000 −0.209020
\(825\) 11.0000 0.382971
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) −8.00000 −0.278019
\(829\) 16.0000 0.555703 0.277851 0.960624i \(-0.410378\pi\)
0.277851 + 0.960624i \(0.410378\pi\)
\(830\) 56.0000 1.94379
\(831\) 10.0000 0.346896
\(832\) −6.00000 −0.208013
\(833\) −4.00000 −0.138592
\(834\) −2.00000 −0.0692543
\(835\) 48.0000 1.66111
\(836\) 2.00000 0.0691714
\(837\) −6.00000 −0.207390
\(838\) 12.0000 0.414533
\(839\) −54.0000 −1.86429 −0.932144 0.362089i \(-0.882064\pi\)
−0.932144 + 0.362089i \(0.882064\pi\)
\(840\) 4.00000 0.138013
\(841\) 7.00000 0.241379
\(842\) 38.0000 1.30957
\(843\) −26.0000 −0.895488
\(844\) −4.00000 −0.137686
\(845\) −92.0000 −3.16490
\(846\) 6.00000 0.206284
\(847\) 1.00000 0.0343604
\(848\) 2.00000 0.0686803
\(849\) 14.0000 0.480479
\(850\) −44.0000 −1.50919
\(851\) 48.0000 1.64542
\(852\) 12.0000 0.411113
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 10.0000 0.342193
\(855\) 8.00000 0.273594
\(856\) −8.00000 −0.273434
\(857\) −40.0000 −1.36637 −0.683187 0.730243i \(-0.739407\pi\)
−0.683187 + 0.730243i \(0.739407\pi\)
\(858\) −6.00000 −0.204837
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) −16.0000 −0.545595
\(861\) −12.0000 −0.408959
\(862\) −24.0000 −0.817443
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −8.00000 −0.272008
\(866\) 26.0000 0.883516
\(867\) 1.00000 0.0339618
\(868\) 6.00000 0.203653
\(869\) 16.0000 0.542763
\(870\) −24.0000 −0.813676
\(871\) −24.0000 −0.813209
\(872\) 6.00000 0.203186
\(873\) −14.0000 −0.473828
\(874\) 16.0000 0.541208
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) −20.0000 −0.674967
\(879\) 14.0000 0.472208
\(880\) 4.00000 0.134840
\(881\) −50.0000 −1.68454 −0.842271 0.539054i \(-0.818782\pi\)
−0.842271 + 0.539054i \(0.818782\pi\)
\(882\) 1.00000 0.0336718
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 28.0000 0.940678
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 6.00000 0.201347
\(889\) 0 0
\(890\) 56.0000 1.87712
\(891\) −1.00000 −0.0335013
\(892\) 6.00000 0.200895
\(893\) −12.0000 −0.401565
\(894\) 10.0000 0.334450
\(895\) 16.0000 0.534821
\(896\) 1.00000 0.0334077
\(897\) −48.0000 −1.60267
\(898\) 14.0000 0.467186
\(899\) −36.0000 −1.20067
\(900\) 11.0000 0.366667
\(901\) −8.00000 −0.266519
\(902\) −12.0000 −0.399556
\(903\) −4.00000 −0.133112
\(904\) 14.0000 0.465633
\(905\) 32.0000 1.06372
\(906\) 8.00000 0.265782
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) 6.00000 0.199117
\(909\) 14.0000 0.464351
\(910\) 24.0000 0.795592
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 2.00000 0.0662266
\(913\) 14.0000 0.463332
\(914\) 10.0000 0.330771
\(915\) 40.0000 1.32236
\(916\) 8.00000 0.264327
\(917\) −2.00000 −0.0660458
\(918\) 4.00000 0.132020
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 32.0000 1.05501
\(921\) 14.0000 0.461316
\(922\) 14.0000 0.461065
\(923\) 72.0000 2.36991
\(924\) 1.00000 0.0328976
\(925\) −66.0000 −2.17007
\(926\) −24.0000 −0.788689
\(927\) −6.00000 −0.197066
\(928\) −6.00000 −0.196960
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 24.0000 0.786991
\(931\) −2.00000 −0.0655474
\(932\) −14.0000 −0.458585
\(933\) −10.0000 −0.327385
\(934\) 36.0000 1.17796
\(935\) −16.0000 −0.523256
\(936\) −6.00000 −0.196116
\(937\) −32.0000 −1.04539 −0.522697 0.852518i \(-0.675074\pi\)
−0.522697 + 0.852518i \(0.675074\pi\)
\(938\) 4.00000 0.130605
\(939\) 34.0000 1.10955
\(940\) −24.0000 −0.782794
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 4.00000 0.130327
\(943\) −96.0000 −3.12619
\(944\) 0 0
\(945\) 4.00000 0.130120
\(946\) −4.00000 −0.130051
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 16.0000 0.519656
\(949\) 0 0
\(950\) −22.0000 −0.713774
\(951\) 6.00000 0.194563
\(952\) −4.00000 −0.129641
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) −6.00000 −0.193952
\(958\) 24.0000 0.775405
\(959\) 14.0000 0.452084
\(960\) 4.00000 0.129099
\(961\) 5.00000 0.161290
\(962\) 36.0000 1.16069
\(963\) −8.00000 −0.257796
\(964\) −8.00000 −0.257663
\(965\) 56.0000 1.80270
\(966\) 8.00000 0.257396
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 1.00000 0.0321412
\(969\) −8.00000 −0.256997
\(970\) 56.0000 1.79805
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 2.00000 0.0641171
\(974\) 20.0000 0.640841
\(975\) 66.0000 2.11369
\(976\) 10.0000 0.320092
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) −4.00000 −0.127906
\(979\) 14.0000 0.447442
\(980\) −4.00000 −0.127775
\(981\) 6.00000 0.191565
\(982\) −28.0000 −0.893516
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) −12.0000 −0.382546
\(985\) −40.0000 −1.27451
\(986\) 24.0000 0.764316
\(987\) −6.00000 −0.190982
\(988\) 12.0000 0.381771
\(989\) −32.0000 −1.01754
\(990\) 4.00000 0.127128
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 6.00000 0.190500
\(993\) −20.0000 −0.634681
\(994\) −12.0000 −0.380617
\(995\) −8.00000 −0.253617
\(996\) 14.0000 0.443607
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) −28.0000 −0.886325
\(999\) 6.00000 0.189832
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 462.2.a.e.1.1 1
3.2 odd 2 1386.2.a.e.1.1 1
4.3 odd 2 3696.2.a.p.1.1 1
7.6 odd 2 3234.2.a.v.1.1 1
11.10 odd 2 5082.2.a.a.1.1 1
21.20 even 2 9702.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.e.1.1 1 1.1 even 1 trivial
1386.2.a.e.1.1 1 3.2 odd 2
3234.2.a.v.1.1 1 7.6 odd 2
3696.2.a.p.1.1 1 4.3 odd 2
5082.2.a.a.1.1 1 11.10 odd 2
9702.2.a.b.1.1 1 21.20 even 2