Properties

Label 83.2.a.a.1.1
Level $83$
Weight $2$
Character 83.1
Self dual yes
Analytic conductor $0.663$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 83.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.662758336777\)
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 \(\chi\) \(=\) 83.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} +3.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} +3.00000 q^{8} -2.00000 q^{9} +2.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} +3.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} +5.00000 q^{17} +2.00000 q^{18} +2.00000 q^{19} +2.00000 q^{20} +3.00000 q^{21} -3.00000 q^{22} -4.00000 q^{23} -3.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} +5.00000 q^{27} +3.00000 q^{28} -7.00000 q^{29} -2.00000 q^{30} +5.00000 q^{31} -5.00000 q^{32} -3.00000 q^{33} -5.00000 q^{34} +6.00000 q^{35} +2.00000 q^{36} -11.0000 q^{37} -2.00000 q^{38} +6.00000 q^{39} -6.00000 q^{40} -2.00000 q^{41} -3.00000 q^{42} -8.00000 q^{43} -3.00000 q^{44} +4.00000 q^{45} +4.00000 q^{46} +1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} -5.00000 q^{51} +6.00000 q^{52} +6.00000 q^{53} -5.00000 q^{54} -6.00000 q^{55} -9.00000 q^{56} -2.00000 q^{57} +7.00000 q^{58} +5.00000 q^{59} -2.00000 q^{60} +5.00000 q^{61} -5.00000 q^{62} +6.00000 q^{63} +7.00000 q^{64} +12.0000 q^{65} +3.00000 q^{66} -2.00000 q^{67} -5.00000 q^{68} +4.00000 q^{69} -6.00000 q^{70} +2.00000 q^{71} -6.00000 q^{72} +11.0000 q^{74} +1.00000 q^{75} -2.00000 q^{76} -9.00000 q^{77} -6.00000 q^{78} +14.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -1.00000 q^{83} -3.00000 q^{84} -10.0000 q^{85} +8.00000 q^{86} +7.00000 q^{87} +9.00000 q^{88} -4.00000 q^{90} +18.0000 q^{91} +4.00000 q^{92} -5.00000 q^{93} -4.00000 q^{95} +5.00000 q^{96} -8.00000 q^{97} -2.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) −1.00000 −0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 3.00000 1.06066
\(9\) −2.00000 −0.666667
\(10\) 2.00000 0.632456
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000 0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 3.00000 0.801784
\(15\) 2.00000 0.516398
\(16\) −1.00000 −0.250000
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 2.00000 0.471405
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 2.00000 0.447214
\(21\) 3.00000 0.654654
\(22\) −3.00000 −0.639602
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −3.00000 −0.612372
\(25\) −1.00000 −0.200000
\(26\) 6.00000 1.17670
\(27\) 5.00000 0.962250
\(28\) 3.00000 0.566947
\(29\) −7.00000 −1.29987 −0.649934 0.759991i \(-0.725203\pi\)
−0.649934 + 0.759991i \(0.725203\pi\)
\(30\) −2.00000 −0.365148
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) −5.00000 −0.883883
\(33\) −3.00000 −0.522233
\(34\) −5.00000 −0.857493
\(35\) 6.00000 1.01419
\(36\) 2.00000 0.333333
\(37\) −11.0000 −1.80839 −0.904194 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) −2.00000 −0.324443
\(39\) 6.00000 0.960769
\(40\) −6.00000 −0.948683
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −3.00000 −0.462910
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −3.00000 −0.452267
\(45\) 4.00000 0.596285
\(46\) 4.00000 0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) 1.00000 0.141421
\(51\) −5.00000 −0.700140
\(52\) 6.00000 0.832050
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −5.00000 −0.680414
\(55\) −6.00000 −0.809040
\(56\) −9.00000 −1.20268
\(57\) −2.00000 −0.264906
\(58\) 7.00000 0.919145
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) −2.00000 −0.258199
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) −5.00000 −0.635001
\(63\) 6.00000 0.755929
\(64\) 7.00000 0.875000
\(65\) 12.0000 1.48842
\(66\) 3.00000 0.369274
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −5.00000 −0.606339
\(69\) 4.00000 0.481543
\(70\) −6.00000 −0.717137
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −6.00000 −0.707107
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 11.0000 1.27872
\(75\) 1.00000 0.115470
\(76\) −2.00000 −0.229416
\(77\) −9.00000 −1.02565
\(78\) −6.00000 −0.679366
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −1.00000 −0.109764
\(84\) −3.00000 −0.327327
\(85\) −10.0000 −1.08465
\(86\) 8.00000 0.862662
\(87\) 7.00000 0.750479
\(88\) 9.00000 0.959403
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −4.00000 −0.421637
\(91\) 18.0000 1.88691
\(92\) 4.00000 0.417029
\(93\) −5.00000 −0.518476
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 5.00000 0.510310
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) −2.00000 −0.202031
\(99\) −6.00000 −0.603023
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 5.00000 0.495074
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −18.0000 −1.76505
\(105\) −6.00000 −0.585540
\(106\) −6.00000 −0.582772
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) −5.00000 −0.481125
\(109\) −17.0000 −1.62830 −0.814152 0.580651i \(-0.802798\pi\)
−0.814152 + 0.580651i \(0.802798\pi\)
\(110\) 6.00000 0.572078
\(111\) 11.0000 1.04407
\(112\) 3.00000 0.283473
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 2.00000 0.187317
\(115\) 8.00000 0.746004
\(116\) 7.00000 0.649934
\(117\) 12.0000 1.10940
\(118\) −5.00000 −0.460287
\(119\) −15.0000 −1.37505
\(120\) 6.00000 0.547723
\(121\) −2.00000 −0.181818
\(122\) −5.00000 −0.452679
\(123\) 2.00000 0.180334
\(124\) −5.00000 −0.449013
\(125\) 12.0000 1.07331
\(126\) −6.00000 −0.534522
\(127\) 13.0000 1.15356 0.576782 0.816898i \(-0.304308\pi\)
0.576782 + 0.816898i \(0.304308\pi\)
\(128\) 3.00000 0.265165
\(129\) 8.00000 0.704361
\(130\) −12.0000 −1.05247
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 3.00000 0.261116
\(133\) −6.00000 −0.520266
\(134\) 2.00000 0.172774
\(135\) −10.0000 −0.860663
\(136\) 15.0000 1.28624
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) −4.00000 −0.340503
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) −6.00000 −0.507093
\(141\) 0 0
\(142\) −2.00000 −0.167836
\(143\) −18.0000 −1.50524
\(144\) 2.00000 0.166667
\(145\) 14.0000 1.16264
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 11.0000 0.904194
\(149\) −16.0000 −1.31077 −0.655386 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 9.00000 0.732410 0.366205 0.930534i \(-0.380657\pi\)
0.366205 + 0.930534i \(0.380657\pi\)
\(152\) 6.00000 0.486664
\(153\) −10.0000 −0.808452
\(154\) 9.00000 0.725241
\(155\) −10.0000 −0.803219
\(156\) −6.00000 −0.480384
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −14.0000 −1.11378
\(159\) −6.00000 −0.475831
\(160\) 10.0000 0.790569
\(161\) 12.0000 0.945732
\(162\) −1.00000 −0.0785674
\(163\) −22.0000 −1.72317 −0.861586 0.507611i \(-0.830529\pi\)
−0.861586 + 0.507611i \(0.830529\pi\)
\(164\) 2.00000 0.156174
\(165\) 6.00000 0.467099
\(166\) 1.00000 0.0776151
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) 9.00000 0.694365
\(169\) 23.0000 1.76923
\(170\) 10.0000 0.766965
\(171\) −4.00000 −0.305888
\(172\) 8.00000 0.609994
\(173\) 21.0000 1.59660 0.798300 0.602260i \(-0.205733\pi\)
0.798300 + 0.602260i \(0.205733\pi\)
\(174\) −7.00000 −0.530669
\(175\) 3.00000 0.226779
\(176\) −3.00000 −0.226134
\(177\) −5.00000 −0.375823
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) −4.00000 −0.298142
\(181\) −26.0000 −1.93256 −0.966282 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) −18.0000 −1.33425
\(183\) −5.00000 −0.369611
\(184\) −12.0000 −0.884652
\(185\) 22.0000 1.61747
\(186\) 5.00000 0.366618
\(187\) 15.0000 1.09691
\(188\) 0 0
\(189\) −15.0000 −1.09109
\(190\) 4.00000 0.290191
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) −7.00000 −0.505181
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 8.00000 0.574367
\(195\) −12.0000 −0.859338
\(196\) −2.00000 −0.142857
\(197\) 11.0000 0.783718 0.391859 0.920025i \(-0.371832\pi\)
0.391859 + 0.920025i \(0.371832\pi\)
\(198\) 6.00000 0.426401
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −3.00000 −0.212132
\(201\) 2.00000 0.141069
\(202\) 6.00000 0.422159
\(203\) 21.0000 1.47391
\(204\) 5.00000 0.350070
\(205\) 4.00000 0.279372
\(206\) 4.00000 0.278693
\(207\) 8.00000 0.556038
\(208\) 6.00000 0.416025
\(209\) 6.00000 0.415029
\(210\) 6.00000 0.414039
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −6.00000 −0.412082
\(213\) −2.00000 −0.137038
\(214\) 18.0000 1.23045
\(215\) 16.0000 1.09119
\(216\) 15.0000 1.02062
\(217\) −15.0000 −1.01827
\(218\) 17.0000 1.15139
\(219\) 0 0
\(220\) 6.00000 0.404520
\(221\) −30.0000 −2.01802
\(222\) −11.0000 −0.738272
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 15.0000 1.00223
\(225\) 2.00000 0.133333
\(226\) −15.0000 −0.997785
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 2.00000 0.132453
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −8.00000 −0.527504
\(231\) 9.00000 0.592157
\(232\) −21.0000 −1.37872
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) −12.0000 −0.784465
\(235\) 0 0
\(236\) −5.00000 −0.325472
\(237\) −14.0000 −0.909398
\(238\) 15.0000 0.972306
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) −2.00000 −0.129099
\(241\) −1.00000 −0.0644157 −0.0322078 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) 2.00000 0.128565
\(243\) −16.0000 −1.02640
\(244\) −5.00000 −0.320092
\(245\) −4.00000 −0.255551
\(246\) −2.00000 −0.127515
\(247\) −12.0000 −0.763542
\(248\) 15.0000 0.952501
\(249\) 1.00000 0.0633724
\(250\) −12.0000 −0.758947
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) −6.00000 −0.377964
\(253\) −12.0000 −0.754434
\(254\) −13.0000 −0.815693
\(255\) 10.0000 0.626224
\(256\) −17.0000 −1.06250
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −8.00000 −0.498058
\(259\) 33.0000 2.05052
\(260\) −12.0000 −0.744208
\(261\) 14.0000 0.866578
\(262\) 4.00000 0.247121
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) −9.00000 −0.553912
\(265\) −12.0000 −0.737154
\(266\) 6.00000 0.367884
\(267\) 0 0
\(268\) 2.00000 0.122169
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 10.0000 0.608581
\(271\) −32.0000 −1.94386 −0.971931 0.235267i \(-0.924404\pi\)
−0.971931 + 0.235267i \(0.924404\pi\)
\(272\) −5.00000 −0.303170
\(273\) −18.0000 −1.08941
\(274\) 14.0000 0.845771
\(275\) −3.00000 −0.180907
\(276\) −4.00000 −0.240772
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 2.00000 0.119952
\(279\) −10.0000 −0.598684
\(280\) 18.0000 1.07571
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 26.0000 1.54554 0.772770 0.634686i \(-0.218871\pi\)
0.772770 + 0.634686i \(0.218871\pi\)
\(284\) −2.00000 −0.118678
\(285\) 4.00000 0.236940
\(286\) 18.0000 1.06436
\(287\) 6.00000 0.354169
\(288\) 10.0000 0.589256
\(289\) 8.00000 0.470588
\(290\) −14.0000 −0.822108
\(291\) 8.00000 0.468968
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 2.00000 0.116642
\(295\) −10.0000 −0.582223
\(296\) −33.0000 −1.91809
\(297\) 15.0000 0.870388
\(298\) 16.0000 0.926855
\(299\) 24.0000 1.38796
\(300\) −1.00000 −0.0577350
\(301\) 24.0000 1.38334
\(302\) −9.00000 −0.517892
\(303\) 6.00000 0.344691
\(304\) −2.00000 −0.114708
\(305\) −10.0000 −0.572598
\(306\) 10.0000 0.571662
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 9.00000 0.512823
\(309\) 4.00000 0.227552
\(310\) 10.0000 0.567962
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 18.0000 1.01905
\(313\) 11.0000 0.621757 0.310878 0.950450i \(-0.399377\pi\)
0.310878 + 0.950450i \(0.399377\pi\)
\(314\) −10.0000 −0.564333
\(315\) −12.0000 −0.676123
\(316\) −14.0000 −0.787562
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) 6.00000 0.336463
\(319\) −21.0000 −1.17577
\(320\) −14.0000 −0.782624
\(321\) 18.0000 1.00466
\(322\) −12.0000 −0.668734
\(323\) 10.0000 0.556415
\(324\) −1.00000 −0.0555556
\(325\) 6.00000 0.332820
\(326\) 22.0000 1.21847
\(327\) 17.0000 0.940102
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) −6.00000 −0.330289
\(331\) −22.0000 −1.20923 −0.604615 0.796518i \(-0.706673\pi\)
−0.604615 + 0.796518i \(0.706673\pi\)
\(332\) 1.00000 0.0548821
\(333\) 22.0000 1.20559
\(334\) −21.0000 −1.14907
\(335\) 4.00000 0.218543
\(336\) −3.00000 −0.163663
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −23.0000 −1.25104
\(339\) −15.0000 −0.814688
\(340\) 10.0000 0.542326
\(341\) 15.0000 0.812296
\(342\) 4.00000 0.216295
\(343\) 15.0000 0.809924
\(344\) −24.0000 −1.29399
\(345\) −8.00000 −0.430706
\(346\) −21.0000 −1.12897
\(347\) 14.0000 0.751559 0.375780 0.926709i \(-0.377375\pi\)
0.375780 + 0.926709i \(0.377375\pi\)
\(348\) −7.00000 −0.375239
\(349\) 23.0000 1.23116 0.615581 0.788074i \(-0.288921\pi\)
0.615581 + 0.788074i \(0.288921\pi\)
\(350\) −3.00000 −0.160357
\(351\) −30.0000 −1.60128
\(352\) −15.0000 −0.799503
\(353\) −1.00000 −0.0532246 −0.0266123 0.999646i \(-0.508472\pi\)
−0.0266123 + 0.999646i \(0.508472\pi\)
\(354\) 5.00000 0.265747
\(355\) −4.00000 −0.212298
\(356\) 0 0
\(357\) 15.0000 0.793884
\(358\) 6.00000 0.317110
\(359\) 33.0000 1.74167 0.870837 0.491572i \(-0.163578\pi\)
0.870837 + 0.491572i \(0.163578\pi\)
\(360\) 12.0000 0.632456
\(361\) −15.0000 −0.789474
\(362\) 26.0000 1.36653
\(363\) 2.00000 0.104973
\(364\) −18.0000 −0.943456
\(365\) 0 0
\(366\) 5.00000 0.261354
\(367\) −14.0000 −0.730794 −0.365397 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) 4.00000 0.208514
\(369\) 4.00000 0.208232
\(370\) −22.0000 −1.14373
\(371\) −18.0000 −0.934513
\(372\) 5.00000 0.259238
\(373\) −11.0000 −0.569558 −0.284779 0.958593i \(-0.591920\pi\)
−0.284779 + 0.958593i \(0.591920\pi\)
\(374\) −15.0000 −0.775632
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) 42.0000 2.16311
\(378\) 15.0000 0.771517
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 4.00000 0.205196
\(381\) −13.0000 −0.666010
\(382\) 3.00000 0.153493
\(383\) −29.0000 −1.48183 −0.740915 0.671598i \(-0.765608\pi\)
−0.740915 + 0.671598i \(0.765608\pi\)
\(384\) −3.00000 −0.153093
\(385\) 18.0000 0.917365
\(386\) −2.00000 −0.101797
\(387\) 16.0000 0.813326
\(388\) 8.00000 0.406138
\(389\) 4.00000 0.202808 0.101404 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) 12.0000 0.607644
\(391\) −20.0000 −1.01144
\(392\) 6.00000 0.303046
\(393\) 4.00000 0.201773
\(394\) −11.0000 −0.554172
\(395\) −28.0000 −1.40883
\(396\) 6.00000 0.301511
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 6.00000 0.300376
\(400\) 1.00000 0.0500000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −2.00000 −0.0997509
\(403\) −30.0000 −1.49441
\(404\) 6.00000 0.298511
\(405\) −2.00000 −0.0993808
\(406\) −21.0000 −1.04221
\(407\) −33.0000 −1.63575
\(408\) −15.0000 −0.742611
\(409\) −21.0000 −1.03838 −0.519192 0.854658i \(-0.673767\pi\)
−0.519192 + 0.854658i \(0.673767\pi\)
\(410\) −4.00000 −0.197546
\(411\) 14.0000 0.690569
\(412\) 4.00000 0.197066
\(413\) −15.0000 −0.738102
\(414\) −8.00000 −0.393179
\(415\) 2.00000 0.0981761
\(416\) 30.0000 1.47087
\(417\) 2.00000 0.0979404
\(418\) −6.00000 −0.293470
\(419\) 27.0000 1.31904 0.659518 0.751689i \(-0.270760\pi\)
0.659518 + 0.751689i \(0.270760\pi\)
\(420\) 6.00000 0.292770
\(421\) −38.0000 −1.85201 −0.926003 0.377515i \(-0.876779\pi\)
−0.926003 + 0.377515i \(0.876779\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) −5.00000 −0.242536
\(426\) 2.00000 0.0969003
\(427\) −15.0000 −0.725901
\(428\) 18.0000 0.870063
\(429\) 18.0000 0.869048
\(430\) −16.0000 −0.771589
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) −5.00000 −0.240563
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 15.0000 0.720023
\(435\) −14.0000 −0.671249
\(436\) 17.0000 0.814152
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) −18.0000 −0.858116
\(441\) −4.00000 −0.190476
\(442\) 30.0000 1.42695
\(443\) 13.0000 0.617649 0.308824 0.951119i \(-0.400064\pi\)
0.308824 + 0.951119i \(0.400064\pi\)
\(444\) −11.0000 −0.522037
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) 16.0000 0.756774
\(448\) −21.0000 −0.992157
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) −2.00000 −0.0942809
\(451\) −6.00000 −0.282529
\(452\) −15.0000 −0.705541
\(453\) −9.00000 −0.422857
\(454\) 28.0000 1.31411
\(455\) −36.0000 −1.68771
\(456\) −6.00000 −0.280976
\(457\) 30.0000 1.40334 0.701670 0.712502i \(-0.252438\pi\)
0.701670 + 0.712502i \(0.252438\pi\)
\(458\) 10.0000 0.467269
\(459\) 25.0000 1.16690
\(460\) −8.00000 −0.373002
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) −9.00000 −0.418718
\(463\) −5.00000 −0.232370 −0.116185 0.993228i \(-0.537067\pi\)
−0.116185 + 0.993228i \(0.537067\pi\)
\(464\) 7.00000 0.324967
\(465\) 10.0000 0.463739
\(466\) −18.0000 −0.833834
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −12.0000 −0.554700
\(469\) 6.00000 0.277054
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 15.0000 0.690431
\(473\) −24.0000 −1.10352
\(474\) 14.0000 0.643041
\(475\) −2.00000 −0.0917663
\(476\) 15.0000 0.687524
\(477\) −12.0000 −0.549442
\(478\) 8.00000 0.365911
\(479\) 21.0000 0.959514 0.479757 0.877401i \(-0.340725\pi\)
0.479757 + 0.877401i \(0.340725\pi\)
\(480\) −10.0000 −0.456435
\(481\) 66.0000 3.00934
\(482\) 1.00000 0.0455488
\(483\) −12.0000 −0.546019
\(484\) 2.00000 0.0909091
\(485\) 16.0000 0.726523
\(486\) 16.0000 0.725775
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 15.0000 0.679018
\(489\) 22.0000 0.994874
\(490\) 4.00000 0.180702
\(491\) −10.0000 −0.451294 −0.225647 0.974209i \(-0.572450\pi\)
−0.225647 + 0.974209i \(0.572450\pi\)
\(492\) −2.00000 −0.0901670
\(493\) −35.0000 −1.57632
\(494\) 12.0000 0.539906
\(495\) 12.0000 0.539360
\(496\) −5.00000 −0.224507
\(497\) −6.00000 −0.269137
\(498\) −1.00000 −0.0448111
\(499\) −9.00000 −0.402895 −0.201448 0.979499i \(-0.564565\pi\)
−0.201448 + 0.979499i \(0.564565\pi\)
\(500\) −12.0000 −0.536656
\(501\) −21.0000 −0.938211
\(502\) −8.00000 −0.357057
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 18.0000 0.801784
\(505\) 12.0000 0.533993
\(506\) 12.0000 0.533465
\(507\) −23.0000 −1.02147
\(508\) −13.0000 −0.576782
\(509\) −33.0000 −1.46270 −0.731350 0.682003i \(-0.761109\pi\)
−0.731350 + 0.682003i \(0.761109\pi\)
\(510\) −10.0000 −0.442807
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) 10.0000 0.441511
\(514\) 6.00000 0.264649
\(515\) 8.00000 0.352522
\(516\) −8.00000 −0.352180
\(517\) 0 0
\(518\) −33.0000 −1.44994
\(519\) −21.0000 −0.921798
\(520\) 36.0000 1.57870
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) −14.0000 −0.612763
\(523\) 29.0000 1.26808 0.634041 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 4.00000 0.174741
\(525\) −3.00000 −0.130931
\(526\) 6.00000 0.261612
\(527\) 25.0000 1.08902
\(528\) 3.00000 0.130558
\(529\) −7.00000 −0.304348
\(530\) 12.0000 0.521247
\(531\) −10.0000 −0.433963
\(532\) 6.00000 0.260133
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 36.0000 1.55642
\(536\) −6.00000 −0.259161
\(537\) 6.00000 0.258919
\(538\) −10.0000 −0.431131
\(539\) 6.00000 0.258438
\(540\) 10.0000 0.430331
\(541\) −26.0000 −1.11783 −0.558914 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) 32.0000 1.37452
\(543\) 26.0000 1.11577
\(544\) −25.0000 −1.07187
\(545\) 34.0000 1.45640
\(546\) 18.0000 0.770329
\(547\) 13.0000 0.555840 0.277920 0.960604i \(-0.410355\pi\)
0.277920 + 0.960604i \(0.410355\pi\)
\(548\) 14.0000 0.598050
\(549\) −10.0000 −0.426790
\(550\) 3.00000 0.127920
\(551\) −14.0000 −0.596420
\(552\) 12.0000 0.510754
\(553\) −42.0000 −1.78602
\(554\) −10.0000 −0.424859
\(555\) −22.0000 −0.933848
\(556\) 2.00000 0.0848189
\(557\) −26.0000 −1.10166 −0.550828 0.834619i \(-0.685688\pi\)
−0.550828 + 0.834619i \(0.685688\pi\)
\(558\) 10.0000 0.423334
\(559\) 48.0000 2.03018
\(560\) −6.00000 −0.253546
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) 21.0000 0.885044 0.442522 0.896758i \(-0.354084\pi\)
0.442522 + 0.896758i \(0.354084\pi\)
\(564\) 0 0
\(565\) −30.0000 −1.26211
\(566\) −26.0000 −1.09286
\(567\) −3.00000 −0.125988
\(568\) 6.00000 0.251754
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) −4.00000 −0.167542
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 18.0000 0.752618
\(573\) 3.00000 0.125327
\(574\) −6.00000 −0.250435
\(575\) 4.00000 0.166812
\(576\) −14.0000 −0.583333
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) −8.00000 −0.332756
\(579\) −2.00000 −0.0831172
\(580\) −14.0000 −0.581318
\(581\) 3.00000 0.124461
\(582\) −8.00000 −0.331611
\(583\) 18.0000 0.745484
\(584\) 0 0
\(585\) −24.0000 −0.992278
\(586\) 6.00000 0.247858
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 2.00000 0.0824786
\(589\) 10.0000 0.412043
\(590\) 10.0000 0.411693
\(591\) −11.0000 −0.452480
\(592\) 11.0000 0.452097
\(593\) −15.0000 −0.615976 −0.307988 0.951390i \(-0.599656\pi\)
−0.307988 + 0.951390i \(0.599656\pi\)
\(594\) −15.0000 −0.615457
\(595\) 30.0000 1.22988
\(596\) 16.0000 0.655386
\(597\) 0 0
\(598\) −24.0000 −0.981433
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 3.00000 0.122474
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) −24.0000 −0.978167
\(603\) 4.00000 0.162893
\(604\) −9.00000 −0.366205
\(605\) 4.00000 0.162623
\(606\) −6.00000 −0.243733
\(607\) 24.0000 0.974130 0.487065 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) −10.0000 −0.405554
\(609\) −21.0000 −0.850963
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) 10.0000 0.404226
\(613\) −8.00000 −0.323117 −0.161558 0.986863i \(-0.551652\pi\)
−0.161558 + 0.986863i \(0.551652\pi\)
\(614\) −20.0000 −0.807134
\(615\) −4.00000 −0.161296
\(616\) −27.0000 −1.08786
\(617\) 11.0000 0.442843 0.221422 0.975178i \(-0.428930\pi\)
0.221422 + 0.975178i \(0.428930\pi\)
\(618\) −4.00000 −0.160904
\(619\) 23.0000 0.924448 0.462224 0.886763i \(-0.347052\pi\)
0.462224 + 0.886763i \(0.347052\pi\)
\(620\) 10.0000 0.401610
\(621\) −20.0000 −0.802572
\(622\) 0 0
\(623\) 0 0
\(624\) −6.00000 −0.240192
\(625\) −19.0000 −0.760000
\(626\) −11.0000 −0.439648
\(627\) −6.00000 −0.239617
\(628\) −10.0000 −0.399043
\(629\) −55.0000 −2.19299
\(630\) 12.0000 0.478091
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) 42.0000 1.67067
\(633\) 8.00000 0.317971
\(634\) −3.00000 −0.119145
\(635\) −26.0000 −1.03178
\(636\) 6.00000 0.237915
\(637\) −12.0000 −0.475457
\(638\) 21.0000 0.831398
\(639\) −4.00000 −0.158238
\(640\) −6.00000 −0.237171
\(641\) 8.00000 0.315981 0.157991 0.987441i \(-0.449498\pi\)
0.157991 + 0.987441i \(0.449498\pi\)
\(642\) −18.0000 −0.710403
\(643\) 8.00000 0.315489 0.157745 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) −12.0000 −0.472866
\(645\) −16.0000 −0.629999
\(646\) −10.0000 −0.393445
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) 3.00000 0.117851
\(649\) 15.0000 0.588802
\(650\) −6.00000 −0.235339
\(651\) 15.0000 0.587896
\(652\) 22.0000 0.861586
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) −17.0000 −0.664753
\(655\) 8.00000 0.312586
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) −6.00000 −0.233550
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 22.0000 0.855054
\(663\) 30.0000 1.16510
\(664\) −3.00000 −0.116423
\(665\) 12.0000 0.465340
\(666\) −22.0000 −0.852483
\(667\) 28.0000 1.08416
\(668\) −21.0000 −0.812514
\(669\) 16.0000 0.618596
\(670\) −4.00000 −0.154533
\(671\) 15.0000 0.579069
\(672\) −15.0000 −0.578638
\(673\) −23.0000 −0.886585 −0.443292 0.896377i \(-0.646190\pi\)
−0.443292 + 0.896377i \(0.646190\pi\)
\(674\) −2.00000 −0.0770371
\(675\) −5.00000 −0.192450
\(676\) −23.0000 −0.884615
\(677\) −8.00000 −0.307465 −0.153732 0.988113i \(-0.549129\pi\)
−0.153732 + 0.988113i \(0.549129\pi\)
\(678\) 15.0000 0.576072
\(679\) 24.0000 0.921035
\(680\) −30.0000 −1.15045
\(681\) 28.0000 1.07296
\(682\) −15.0000 −0.574380
\(683\) 28.0000 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) 4.00000 0.152944
\(685\) 28.0000 1.06983
\(686\) −15.0000 −0.572703
\(687\) 10.0000 0.381524
\(688\) 8.00000 0.304997
\(689\) −36.0000 −1.37149
\(690\) 8.00000 0.304555
\(691\) −35.0000 −1.33146 −0.665731 0.746191i \(-0.731880\pi\)
−0.665731 + 0.746191i \(0.731880\pi\)
\(692\) −21.0000 −0.798300
\(693\) 18.0000 0.683763
\(694\) −14.0000 −0.531433
\(695\) 4.00000 0.151729
\(696\) 21.0000 0.796003
\(697\) −10.0000 −0.378777
\(698\) −23.0000 −0.870563
\(699\) −18.0000 −0.680823
\(700\) −3.00000 −0.113389
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 30.0000 1.13228
\(703\) −22.0000 −0.829746
\(704\) 21.0000 0.791467
\(705\) 0 0
\(706\) 1.00000 0.0376355
\(707\) 18.0000 0.676960
\(708\) 5.00000 0.187912
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 4.00000 0.150117
\(711\) −28.0000 −1.05008
\(712\) 0 0
\(713\) −20.0000 −0.749006
\(714\) −15.0000 −0.561361
\(715\) 36.0000 1.34632
\(716\) 6.00000 0.224231
\(717\) 8.00000 0.298765
\(718\) −33.0000 −1.23155
\(719\) 44.0000 1.64092 0.820462 0.571702i \(-0.193717\pi\)
0.820462 + 0.571702i \(0.193717\pi\)
\(720\) −4.00000 −0.149071
\(721\) 12.0000 0.446903
\(722\) 15.0000 0.558242
\(723\) 1.00000 0.0371904
\(724\) 26.0000 0.966282
\(725\) 7.00000 0.259973
\(726\) −2.00000 −0.0742270
\(727\) 23.0000 0.853023 0.426511 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) 54.0000 2.00137
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −40.0000 −1.47945
\(732\) 5.00000 0.184805
\(733\) −1.00000 −0.0369358 −0.0184679 0.999829i \(-0.505879\pi\)
−0.0184679 + 0.999829i \(0.505879\pi\)
\(734\) 14.0000 0.516749
\(735\) 4.00000 0.147542
\(736\) 20.0000 0.737210
\(737\) −6.00000 −0.221013
\(738\) −4.00000 −0.147242
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) −22.0000 −0.808736
\(741\) 12.0000 0.440831
\(742\) 18.0000 0.660801
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) −15.0000 −0.549927
\(745\) 32.0000 1.17239
\(746\) 11.0000 0.402739
\(747\) 2.00000 0.0731762
\(748\) −15.0000 −0.548454
\(749\) 54.0000 1.97312
\(750\) 12.0000 0.438178
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) −8.00000 −0.291536
\(754\) −42.0000 −1.52955
\(755\) −18.0000 −0.655087
\(756\) 15.0000 0.545545
\(757\) 23.0000 0.835949 0.417975 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) −16.0000 −0.581146
\(759\) 12.0000 0.435572
\(760\) −12.0000 −0.435286
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 13.0000 0.470940
\(763\) 51.0000 1.84632
\(764\) 3.00000 0.108536
\(765\) 20.0000 0.723102
\(766\) 29.0000 1.04781
\(767\) −30.0000 −1.08324
\(768\) 17.0000 0.613435
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) −18.0000 −0.648675
\(771\) 6.00000 0.216085
\(772\) −2.00000 −0.0719816
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) −16.0000 −0.575108
\(775\) −5.00000 −0.179605
\(776\) −24.0000 −0.861550
\(777\) −33.0000 −1.18387
\(778\) −4.00000 −0.143407
\(779\) −4.00000 −0.143315
\(780\) 12.0000 0.429669
\(781\) 6.00000 0.214697
\(782\) 20.0000 0.715199
\(783\) −35.0000 −1.25080
\(784\) −2.00000 −0.0714286
\(785\) −20.0000 −0.713831
\(786\) −4.00000 −0.142675
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) −11.0000 −0.391859
\(789\) 6.00000 0.213606
\(790\) 28.0000 0.996195
\(791\) −45.0000 −1.60002
\(792\) −18.0000 −0.639602
\(793\) −30.0000 −1.06533
\(794\) −2.00000 −0.0709773
\(795\) 12.0000 0.425596
\(796\) 0 0
\(797\) −24.0000 −0.850124 −0.425062 0.905164i \(-0.639748\pi\)
−0.425062 + 0.905164i \(0.639748\pi\)
\(798\) −6.00000 −0.212398
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) −2.00000 −0.0705346
\(805\) −24.0000 −0.845889
\(806\) 30.0000 1.05670
\(807\) −10.0000 −0.352017
\(808\) −18.0000 −0.633238
\(809\) −28.0000 −0.984428 −0.492214 0.870474i \(-0.663812\pi\)
−0.492214 + 0.870474i \(0.663812\pi\)
\(810\) 2.00000 0.0702728
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) −21.0000 −0.736956
\(813\) 32.0000 1.12229
\(814\) 33.0000 1.15665
\(815\) 44.0000 1.54125
\(816\) 5.00000 0.175035
\(817\) −16.0000 −0.559769
\(818\) 21.0000 0.734248
\(819\) −36.0000 −1.25794
\(820\) −4.00000 −0.139686
\(821\) −32.0000 −1.11681 −0.558404 0.829569i \(-0.688586\pi\)
−0.558404 + 0.829569i \(0.688586\pi\)
\(822\) −14.0000 −0.488306
\(823\) 36.0000 1.25488 0.627441 0.778664i \(-0.284103\pi\)
0.627441 + 0.778664i \(0.284103\pi\)
\(824\) −12.0000 −0.418040
\(825\) 3.00000 0.104447
\(826\) 15.0000 0.521917
\(827\) −32.0000 −1.11275 −0.556375 0.830932i \(-0.687808\pi\)
−0.556375 + 0.830932i \(0.687808\pi\)
\(828\) −8.00000 −0.278019
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) −2.00000 −0.0694210
\(831\) −10.0000 −0.346896
\(832\) −42.0000 −1.45609
\(833\) 10.0000 0.346479
\(834\) −2.00000 −0.0692543
\(835\) −42.0000 −1.45347
\(836\) −6.00000 −0.207514
\(837\) 25.0000 0.864126
\(838\) −27.0000 −0.932700
\(839\) −45.0000 −1.55357 −0.776786 0.629764i \(-0.783151\pi\)
−0.776786 + 0.629764i \(0.783151\pi\)
\(840\) −18.0000 −0.621059
\(841\) 20.0000 0.689655
\(842\) 38.0000 1.30957
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) −46.0000 −1.58245
\(846\) 0 0
\(847\) 6.00000 0.206162
\(848\) −6.00000 −0.206041
\(849\) −26.0000 −0.892318
\(850\) 5.00000 0.171499
\(851\) 44.0000 1.50830
\(852\) 2.00000 0.0685189
\(853\) 1.00000 0.0342393 0.0171197 0.999853i \(-0.494550\pi\)
0.0171197 + 0.999853i \(0.494550\pi\)
\(854\) 15.0000 0.513289
\(855\) 8.00000 0.273594
\(856\) −54.0000 −1.84568
\(857\) −49.0000 −1.67381 −0.836904 0.547350i \(-0.815637\pi\)
−0.836904 + 0.547350i \(0.815637\pi\)
\(858\) −18.0000 −0.614510
\(859\) 41.0000 1.39890 0.699451 0.714681i \(-0.253428\pi\)
0.699451 + 0.714681i \(0.253428\pi\)
\(860\) −16.0000 −0.545595
\(861\) −6.00000 −0.204479
\(862\) −21.0000 −0.715263
\(863\) 37.0000 1.25949 0.629747 0.776800i \(-0.283158\pi\)
0.629747 + 0.776800i \(0.283158\pi\)
\(864\) −25.0000 −0.850517
\(865\) −42.0000 −1.42804
\(866\) 16.0000 0.543702
\(867\) −8.00000 −0.271694
\(868\) 15.0000 0.509133
\(869\) 42.0000 1.42475
\(870\) 14.0000 0.474644
\(871\) 12.0000 0.406604
\(872\) −51.0000 −1.72708
\(873\) 16.0000 0.541518
\(874\) 8.00000 0.270604
\(875\) −36.0000 −1.21702
\(876\) 0 0
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 8.00000 0.269987
\(879\) 6.00000 0.202375
\(880\) 6.00000 0.202260
\(881\) 21.0000 0.707508 0.353754 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) 4.00000 0.134687
\(883\) −46.0000 −1.54802 −0.774012 0.633171i \(-0.781753\pi\)
−0.774012 + 0.633171i \(0.781753\pi\)
\(884\) 30.0000 1.00901
\(885\) 10.0000 0.336146
\(886\) −13.0000 −0.436744
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 33.0000 1.10741
\(889\) −39.0000 −1.30802
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) −16.0000 −0.535120
\(895\) 12.0000 0.401116
\(896\) −9.00000 −0.300669
\(897\) −24.0000 −0.801337
\(898\) 2.00000 0.0667409
\(899\) −35.0000 −1.16732
\(900\) −2.00000 −0.0666667
\(901\) 30.0000 0.999445
\(902\) 6.00000 0.199778
\(903\) −24.0000 −0.798670
\(904\) 45.0000 1.49668
\(905\) 52.0000 1.72854
\(906\) 9.00000 0.299005
\(907\) −47.0000 −1.56061 −0.780305 0.625400i \(-0.784936\pi\)
−0.780305 + 0.625400i \(0.784936\pi\)
\(908\) 28.0000 0.929213
\(909\) 12.0000 0.398015
\(910\) 36.0000 1.19339
\(911\) 3.00000 0.0993944 0.0496972 0.998764i \(-0.484174\pi\)
0.0496972 + 0.998764i \(0.484174\pi\)
\(912\) 2.00000 0.0662266
\(913\) −3.00000 −0.0992855
\(914\) −30.0000 −0.992312
\(915\) 10.0000 0.330590
\(916\) 10.0000 0.330409
\(917\) 12.0000 0.396275
\(918\) −25.0000 −0.825123
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 24.0000 0.791257
\(921\) −20.0000 −0.659022
\(922\) −20.0000 −0.658665
\(923\) −12.0000 −0.394985
\(924\) −9.00000 −0.296078
\(925\) 11.0000 0.361678
\(926\) 5.00000 0.164310
\(927\) 8.00000 0.262754
\(928\) 35.0000 1.14893
\(929\) −35.0000 −1.14831 −0.574156 0.818746i \(-0.694670\pi\)
−0.574156 + 0.818746i \(0.694670\pi\)
\(930\) −10.0000 −0.327913
\(931\) 4.00000 0.131095
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) −30.0000 −0.981105
\(936\) 36.0000 1.17670
\(937\) 12.0000 0.392023 0.196011 0.980602i \(-0.437201\pi\)
0.196011 + 0.980602i \(0.437201\pi\)
\(938\) −6.00000 −0.195907
\(939\) −11.0000 −0.358971
\(940\) 0 0
\(941\) −25.0000 −0.814977 −0.407488 0.913210i \(-0.633595\pi\)
−0.407488 + 0.913210i \(0.633595\pi\)
\(942\) 10.0000 0.325818
\(943\) 8.00000 0.260516
\(944\) −5.00000 −0.162736
\(945\) 30.0000 0.975900
\(946\) 24.0000 0.780307
\(947\) 32.0000 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 14.0000 0.454699
\(949\) 0 0
\(950\) 2.00000 0.0648886
\(951\) −3.00000 −0.0972817
\(952\) −45.0000 −1.45846
\(953\) −27.0000 −0.874616 −0.437308 0.899312i \(-0.644068\pi\)
−0.437308 + 0.899312i \(0.644068\pi\)
\(954\) 12.0000 0.388514
\(955\) 6.00000 0.194155
\(956\) 8.00000 0.258738
\(957\) 21.0000 0.678834
\(958\) −21.0000 −0.678479
\(959\) 42.0000 1.35625
\(960\) 14.0000 0.451848
\(961\) −6.00000 −0.193548
\(962\) −66.0000 −2.12793
\(963\) 36.0000 1.16008
\(964\) 1.00000 0.0322078
\(965\) −4.00000 −0.128765
\(966\) 12.0000 0.386094
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) −6.00000 −0.192847
\(969\) −10.0000 −0.321246
\(970\) −16.0000 −0.513729
\(971\) −22.0000 −0.706014 −0.353007 0.935621i \(-0.614841\pi\)
−0.353007 + 0.935621i \(0.614841\pi\)
\(972\) 16.0000 0.513200
\(973\) 6.00000 0.192351
\(974\) 16.0000 0.512673
\(975\) −6.00000 −0.192154
\(976\) −5.00000 −0.160046
\(977\) −19.0000 −0.607864 −0.303932 0.952694i \(-0.598300\pi\)
−0.303932 + 0.952694i \(0.598300\pi\)
\(978\) −22.0000 −0.703482
\(979\) 0 0
\(980\) 4.00000 0.127775
\(981\) 34.0000 1.08554
\(982\) 10.0000 0.319113
\(983\) −44.0000 −1.40338 −0.701691 0.712481i \(-0.747571\pi\)
−0.701691 + 0.712481i \(0.747571\pi\)
\(984\) 6.00000 0.191273
\(985\) −22.0000 −0.700978
\(986\) 35.0000 1.11463
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) 32.0000 1.01754
\(990\) −12.0000 −0.381385
\(991\) 1.00000 0.0317660 0.0158830 0.999874i \(-0.494944\pi\)
0.0158830 + 0.999874i \(0.494944\pi\)
\(992\) −25.0000 −0.793751
\(993\) 22.0000 0.698149
\(994\) 6.00000 0.190308
\(995\) 0 0
\(996\) −1.00000 −0.0316862
\(997\) −9.00000 −0.285033 −0.142516 0.989792i \(-0.545519\pi\)
−0.142516 + 0.989792i \(0.545519\pi\)
\(998\) 9.00000 0.284890
\(999\) −55.0000 −1.74012
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 83.2.a.a.1.1 1
3.2 odd 2 747.2.a.d.1.1 1
4.3 odd 2 1328.2.a.c.1.1 1
5.4 even 2 2075.2.a.d.1.1 1
7.6 odd 2 4067.2.a.a.1.1 1
8.3 odd 2 5312.2.a.h.1.1 1
8.5 even 2 5312.2.a.l.1.1 1
83.82 odd 2 6889.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
83.2.a.a.1.1 1 1.1 even 1 trivial
747.2.a.d.1.1 1 3.2 odd 2
1328.2.a.c.1.1 1 4.3 odd 2
2075.2.a.d.1.1 1 5.4 even 2
4067.2.a.a.1.1 1 7.6 odd 2
5312.2.a.h.1.1 1 8.3 odd 2
5312.2.a.l.1.1 1 8.5 even 2
6889.2.a.a.1.1 1 83.82 odd 2