Properties

Label 722.2.a.b.1.1
Level $722$
Weight $2$
Character 722.1
Self dual yes
Analytic conductor $5.765$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.76519902594\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 722.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +4.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} -3.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} +2.00000 q^{18} -4.00000 q^{20} +3.00000 q^{21} -2.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -1.00000 q^{26} -5.00000 q^{27} +3.00000 q^{28} +5.00000 q^{29} +4.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} -3.00000 q^{34} -12.0000 q^{35} -2.00000 q^{36} +2.00000 q^{37} +1.00000 q^{39} +4.00000 q^{40} +8.00000 q^{41} -3.00000 q^{42} +4.00000 q^{43} +2.00000 q^{44} +8.00000 q^{45} +1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -11.0000 q^{50} +3.00000 q^{51} +1.00000 q^{52} +1.00000 q^{53} +5.00000 q^{54} -8.00000 q^{55} -3.00000 q^{56} -5.00000 q^{58} -15.0000 q^{59} -4.00000 q^{60} +2.00000 q^{61} -8.00000 q^{62} -6.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -2.00000 q^{66} -3.00000 q^{67} +3.00000 q^{68} -1.00000 q^{69} +12.0000 q^{70} -2.00000 q^{71} +2.00000 q^{72} +9.00000 q^{73} -2.00000 q^{74} +11.0000 q^{75} +6.00000 q^{77} -1.00000 q^{78} +10.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} -6.00000 q^{83} +3.00000 q^{84} -12.0000 q^{85} -4.00000 q^{86} +5.00000 q^{87} -2.00000 q^{88} -8.00000 q^{90} +3.00000 q^{91} -1.00000 q^{92} +8.00000 q^{93} -8.00000 q^{94} -1.00000 q^{96} +2.00000 q^{97} -2.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 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
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) 4.00000 1.26491
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −3.00000 −0.801784
\(15\) −4.00000 −1.03280
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 2.00000 0.471405
\(19\) 0 0
\(20\) −4.00000 −0.894427
\(21\) 3.00000 0.654654
\(22\) −2.00000 −0.426401
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) −1.00000 −0.204124
\(25\) 11.0000 2.20000
\(26\) −1.00000 −0.196116
\(27\) −5.00000 −0.962250
\(28\) 3.00000 0.566947
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 4.00000 0.730297
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) −3.00000 −0.514496
\(35\) −12.0000 −2.02837
\(36\) −2.00000 −0.333333
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 4.00000 0.632456
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) −3.00000 −0.462910
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 2.00000 0.301511
\(45\) 8.00000 1.19257
\(46\) 1.00000 0.147442
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) −11.0000 −1.55563
\(51\) 3.00000 0.420084
\(52\) 1.00000 0.138675
\(53\) 1.00000 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 5.00000 0.680414
\(55\) −8.00000 −1.07872
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) −5.00000 −0.656532
\(59\) −15.0000 −1.95283 −0.976417 0.215894i \(-0.930733\pi\)
−0.976417 + 0.215894i \(0.930733\pi\)
\(60\) −4.00000 −0.516398
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −8.00000 −1.01600
\(63\) −6.00000 −0.755929
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) −2.00000 −0.246183
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 3.00000 0.363803
\(69\) −1.00000 −0.120386
\(70\) 12.0000 1.43427
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 2.00000 0.235702
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) −2.00000 −0.232495
\(75\) 11.0000 1.27017
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) −1.00000 −0.113228
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) −8.00000 −0.883452
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 3.00000 0.327327
\(85\) −12.0000 −1.30158
\(86\) −4.00000 −0.431331
\(87\) 5.00000 0.536056
\(88\) −2.00000 −0.213201
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −8.00000 −0.843274
\(91\) 3.00000 0.314485
\(92\) −1.00000 −0.104257
\(93\) 8.00000 0.829561
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −2.00000 −0.202031
\(99\) −4.00000 −0.402015
\(100\) 11.0000 1.10000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −3.00000 −0.297044
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) −1.00000 −0.0980581
\(105\) −12.0000 −1.17108
\(106\) −1.00000 −0.0971286
\(107\) 7.00000 0.676716 0.338358 0.941018i \(-0.390129\pi\)
0.338358 + 0.941018i \(0.390129\pi\)
\(108\) −5.00000 −0.481125
\(109\) 15.0000 1.43674 0.718370 0.695662i \(-0.244889\pi\)
0.718370 + 0.695662i \(0.244889\pi\)
\(110\) 8.00000 0.762770
\(111\) 2.00000 0.189832
\(112\) 3.00000 0.283473
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 5.00000 0.464238
\(117\) −2.00000 −0.184900
\(118\) 15.0000 1.38086
\(119\) 9.00000 0.825029
\(120\) 4.00000 0.365148
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) 8.00000 0.721336
\(124\) 8.00000 0.718421
\(125\) −24.0000 −2.14663
\(126\) 6.00000 0.534522
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) 4.00000 0.350823
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 3.00000 0.259161
\(135\) 20.0000 1.72133
\(136\) −3.00000 −0.257248
\(137\) −17.0000 −1.45241 −0.726204 0.687479i \(-0.758717\pi\)
−0.726204 + 0.687479i \(0.758717\pi\)
\(138\) 1.00000 0.0851257
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −12.0000 −1.01419
\(141\) 8.00000 0.673722
\(142\) 2.00000 0.167836
\(143\) 2.00000 0.167248
\(144\) −2.00000 −0.166667
\(145\) −20.0000 −1.66091
\(146\) −9.00000 −0.744845
\(147\) 2.00000 0.164957
\(148\) 2.00000 0.164399
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −11.0000 −0.898146
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) −6.00000 −0.483494
\(155\) −32.0000 −2.57030
\(156\) 1.00000 0.0800641
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −10.0000 −0.795557
\(159\) 1.00000 0.0793052
\(160\) 4.00000 0.316228
\(161\) −3.00000 −0.236433
\(162\) −1.00000 −0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 8.00000 0.624695
\(165\) −8.00000 −0.622799
\(166\) 6.00000 0.465690
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) −3.00000 −0.231455
\(169\) −12.0000 −0.923077
\(170\) 12.0000 0.920358
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −5.00000 −0.379049
\(175\) 33.0000 2.49457
\(176\) 2.00000 0.150756
\(177\) −15.0000 −1.12747
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 8.00000 0.596285
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −3.00000 −0.222375
\(183\) 2.00000 0.147844
\(184\) 1.00000 0.0737210
\(185\) −8.00000 −0.588172
\(186\) −8.00000 −0.586588
\(187\) 6.00000 0.438763
\(188\) 8.00000 0.583460
\(189\) −15.0000 −1.09109
\(190\) 0 0
\(191\) 7.00000 0.506502 0.253251 0.967401i \(-0.418500\pi\)
0.253251 + 0.967401i \(0.418500\pi\)
\(192\) 1.00000 0.0721688
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) −2.00000 −0.143592
\(195\) −4.00000 −0.286446
\(196\) 2.00000 0.142857
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 4.00000 0.284268
\(199\) −25.0000 −1.77220 −0.886102 0.463491i \(-0.846597\pi\)
−0.886102 + 0.463491i \(0.846597\pi\)
\(200\) −11.0000 −0.777817
\(201\) −3.00000 −0.211604
\(202\) −2.00000 −0.140720
\(203\) 15.0000 1.05279
\(204\) 3.00000 0.210042
\(205\) −32.0000 −2.23498
\(206\) −6.00000 −0.418040
\(207\) 2.00000 0.139010
\(208\) 1.00000 0.0693375
\(209\) 0 0
\(210\) 12.0000 0.828079
\(211\) −27.0000 −1.85876 −0.929378 0.369129i \(-0.879656\pi\)
−0.929378 + 0.369129i \(0.879656\pi\)
\(212\) 1.00000 0.0686803
\(213\) −2.00000 −0.137038
\(214\) −7.00000 −0.478510
\(215\) −16.0000 −1.09119
\(216\) 5.00000 0.340207
\(217\) 24.0000 1.62923
\(218\) −15.0000 −1.01593
\(219\) 9.00000 0.608164
\(220\) −8.00000 −0.539360
\(221\) 3.00000 0.201802
\(222\) −2.00000 −0.134231
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) −3.00000 −0.200446
\(225\) −22.0000 −1.46667
\(226\) 14.0000 0.931266
\(227\) 17.0000 1.12833 0.564165 0.825662i \(-0.309198\pi\)
0.564165 + 0.825662i \(0.309198\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −4.00000 −0.263752
\(231\) 6.00000 0.394771
\(232\) −5.00000 −0.328266
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 2.00000 0.130744
\(235\) −32.0000 −2.08745
\(236\) −15.0000 −0.976417
\(237\) 10.0000 0.649570
\(238\) −9.00000 −0.583383
\(239\) 15.0000 0.970269 0.485135 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) −4.00000 −0.258199
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 7.00000 0.449977
\(243\) 16.0000 1.02640
\(244\) 2.00000 0.128037
\(245\) −8.00000 −0.511101
\(246\) −8.00000 −0.510061
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) −6.00000 −0.380235
\(250\) 24.0000 1.51789
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) −6.00000 −0.377964
\(253\) −2.00000 −0.125739
\(254\) 18.0000 1.12942
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) −4.00000 −0.249029
\(259\) 6.00000 0.372822
\(260\) −4.00000 −0.248069
\(261\) −10.0000 −0.618984
\(262\) −12.0000 −0.741362
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) −2.00000 −0.123091
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 0 0
\(268\) −3.00000 −0.183254
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) −20.0000 −1.21716
\(271\) 7.00000 0.425220 0.212610 0.977137i \(-0.431804\pi\)
0.212610 + 0.977137i \(0.431804\pi\)
\(272\) 3.00000 0.181902
\(273\) 3.00000 0.181568
\(274\) 17.0000 1.02701
\(275\) 22.0000 1.32665
\(276\) −1.00000 −0.0601929
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 0 0
\(279\) −16.0000 −0.957895
\(280\) 12.0000 0.717137
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) −8.00000 −0.476393
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 24.0000 1.41668
\(288\) 2.00000 0.117851
\(289\) −8.00000 −0.470588
\(290\) 20.0000 1.17444
\(291\) 2.00000 0.117242
\(292\) 9.00000 0.526685
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) −2.00000 −0.116642
\(295\) 60.0000 3.49334
\(296\) −2.00000 −0.116248
\(297\) −10.0000 −0.580259
\(298\) 0 0
\(299\) −1.00000 −0.0578315
\(300\) 11.0000 0.635085
\(301\) 12.0000 0.691669
\(302\) 2.00000 0.115087
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) 6.00000 0.342997
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 6.00000 0.341882
\(309\) 6.00000 0.341328
\(310\) 32.0000 1.81748
\(311\) 7.00000 0.396934 0.198467 0.980108i \(-0.436404\pi\)
0.198467 + 0.980108i \(0.436404\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 29.0000 1.63918 0.819588 0.572953i \(-0.194202\pi\)
0.819588 + 0.572953i \(0.194202\pi\)
\(314\) 2.00000 0.112867
\(315\) 24.0000 1.35225
\(316\) 10.0000 0.562544
\(317\) 27.0000 1.51647 0.758236 0.651981i \(-0.226062\pi\)
0.758236 + 0.651981i \(0.226062\pi\)
\(318\) −1.00000 −0.0560772
\(319\) 10.0000 0.559893
\(320\) −4.00000 −0.223607
\(321\) 7.00000 0.390702
\(322\) 3.00000 0.167183
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 11.0000 0.610170
\(326\) 16.0000 0.886158
\(327\) 15.0000 0.829502
\(328\) −8.00000 −0.441726
\(329\) 24.0000 1.32316
\(330\) 8.00000 0.440386
\(331\) −17.0000 −0.934405 −0.467202 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) −6.00000 −0.329293
\(333\) −4.00000 −0.219199
\(334\) −12.0000 −0.656611
\(335\) 12.0000 0.655630
\(336\) 3.00000 0.163663
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) 12.0000 0.652714
\(339\) −14.0000 −0.760376
\(340\) −12.0000 −0.650791
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) −4.00000 −0.215666
\(345\) 4.00000 0.215353
\(346\) −6.00000 −0.322562
\(347\) −2.00000 −0.107366 −0.0536828 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) 5.00000 0.268028
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) −33.0000 −1.76392
\(351\) −5.00000 −0.266880
\(352\) −2.00000 −0.106600
\(353\) 9.00000 0.479022 0.239511 0.970894i \(-0.423013\pi\)
0.239511 + 0.970894i \(0.423013\pi\)
\(354\) 15.0000 0.797241
\(355\) 8.00000 0.424596
\(356\) 0 0
\(357\) 9.00000 0.476331
\(358\) 0 0
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) −8.00000 −0.421637
\(361\) 0 0
\(362\) 22.0000 1.15629
\(363\) −7.00000 −0.367405
\(364\) 3.00000 0.157243
\(365\) −36.0000 −1.88433
\(366\) −2.00000 −0.104542
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −16.0000 −0.832927
\(370\) 8.00000 0.415900
\(371\) 3.00000 0.155752
\(372\) 8.00000 0.414781
\(373\) −29.0000 −1.50156 −0.750782 0.660551i \(-0.770323\pi\)
−0.750782 + 0.660551i \(0.770323\pi\)
\(374\) −6.00000 −0.310253
\(375\) −24.0000 −1.23935
\(376\) −8.00000 −0.412568
\(377\) 5.00000 0.257513
\(378\) 15.0000 0.771517
\(379\) −15.0000 −0.770498 −0.385249 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) −7.00000 −0.358151
\(383\) 26.0000 1.32854 0.664269 0.747494i \(-0.268743\pi\)
0.664269 + 0.747494i \(0.268743\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −24.0000 −1.22315
\(386\) −6.00000 −0.305392
\(387\) −8.00000 −0.406663
\(388\) 2.00000 0.101535
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 4.00000 0.202548
\(391\) −3.00000 −0.151717
\(392\) −2.00000 −0.101015
\(393\) 12.0000 0.605320
\(394\) −8.00000 −0.403034
\(395\) −40.0000 −2.01262
\(396\) −4.00000 −0.201008
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 25.0000 1.25314
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) 8.00000 0.399501 0.199750 0.979847i \(-0.435987\pi\)
0.199750 + 0.979847i \(0.435987\pi\)
\(402\) 3.00000 0.149626
\(403\) 8.00000 0.398508
\(404\) 2.00000 0.0995037
\(405\) −4.00000 −0.198762
\(406\) −15.0000 −0.744438
\(407\) 4.00000 0.198273
\(408\) −3.00000 −0.148522
\(409\) 20.0000 0.988936 0.494468 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 32.0000 1.58037
\(411\) −17.0000 −0.838548
\(412\) 6.00000 0.295599
\(413\) −45.0000 −2.21431
\(414\) −2.00000 −0.0982946
\(415\) 24.0000 1.17811
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −12.0000 −0.585540
\(421\) 13.0000 0.633581 0.316791 0.948495i \(-0.397395\pi\)
0.316791 + 0.948495i \(0.397395\pi\)
\(422\) 27.0000 1.31434
\(423\) −16.0000 −0.777947
\(424\) −1.00000 −0.0485643
\(425\) 33.0000 1.60074
\(426\) 2.00000 0.0969003
\(427\) 6.00000 0.290360
\(428\) 7.00000 0.338358
\(429\) 2.00000 0.0965609
\(430\) 16.0000 0.771589
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) −5.00000 −0.240563
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) −24.0000 −1.15204
\(435\) −20.0000 −0.958927
\(436\) 15.0000 0.718370
\(437\) 0 0
\(438\) −9.00000 −0.430037
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 8.00000 0.381385
\(441\) −4.00000 −0.190476
\(442\) −3.00000 −0.142695
\(443\) −26.0000 −1.23530 −0.617649 0.786454i \(-0.711915\pi\)
−0.617649 + 0.786454i \(0.711915\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) 3.00000 0.141737
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 22.0000 1.03709
\(451\) 16.0000 0.753411
\(452\) −14.0000 −0.658505
\(453\) −2.00000 −0.0939682
\(454\) −17.0000 −0.797850
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) −7.00000 −0.327446 −0.163723 0.986506i \(-0.552350\pi\)
−0.163723 + 0.986506i \(0.552350\pi\)
\(458\) 10.0000 0.467269
\(459\) −15.0000 −0.700140
\(460\) 4.00000 0.186501
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) −6.00000 −0.279145
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 5.00000 0.232119
\(465\) −32.0000 −1.48396
\(466\) 6.00000 0.277945
\(467\) −2.00000 −0.0925490 −0.0462745 0.998929i \(-0.514735\pi\)
−0.0462745 + 0.998929i \(0.514735\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −9.00000 −0.415581
\(470\) 32.0000 1.47605
\(471\) −2.00000 −0.0921551
\(472\) 15.0000 0.690431
\(473\) 8.00000 0.367840
\(474\) −10.0000 −0.459315
\(475\) 0 0
\(476\) 9.00000 0.412514
\(477\) −2.00000 −0.0915737
\(478\) −15.0000 −0.686084
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 4.00000 0.182574
\(481\) 2.00000 0.0911922
\(482\) −8.00000 −0.364390
\(483\) −3.00000 −0.136505
\(484\) −7.00000 −0.318182
\(485\) −8.00000 −0.363261
\(486\) −16.0000 −0.725775
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −16.0000 −0.723545
\(490\) 8.00000 0.361403
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 8.00000 0.360668
\(493\) 15.0000 0.675566
\(494\) 0 0
\(495\) 16.0000 0.719147
\(496\) 8.00000 0.359211
\(497\) −6.00000 −0.269137
\(498\) 6.00000 0.268866
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) −24.0000 −1.07331
\(501\) 12.0000 0.536120
\(502\) −2.00000 −0.0892644
\(503\) 39.0000 1.73892 0.869462 0.494000i \(-0.164466\pi\)
0.869462 + 0.494000i \(0.164466\pi\)
\(504\) 6.00000 0.267261
\(505\) −8.00000 −0.355995
\(506\) 2.00000 0.0889108
\(507\) −12.0000 −0.532939
\(508\) −18.0000 −0.798621
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 12.0000 0.531369
\(511\) 27.0000 1.19441
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 8.00000 0.352865
\(515\) −24.0000 −1.05757
\(516\) 4.00000 0.176090
\(517\) 16.0000 0.703679
\(518\) −6.00000 −0.263625
\(519\) 6.00000 0.263371
\(520\) 4.00000 0.175412
\(521\) 28.0000 1.22670 0.613351 0.789810i \(-0.289821\pi\)
0.613351 + 0.789810i \(0.289821\pi\)
\(522\) 10.0000 0.437688
\(523\) −29.0000 −1.26808 −0.634041 0.773300i \(-0.718605\pi\)
−0.634041 + 0.773300i \(0.718605\pi\)
\(524\) 12.0000 0.524222
\(525\) 33.0000 1.44024
\(526\) −24.0000 −1.04645
\(527\) 24.0000 1.04546
\(528\) 2.00000 0.0870388
\(529\) −22.0000 −0.956522
\(530\) 4.00000 0.173749
\(531\) 30.0000 1.30189
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) −28.0000 −1.21055
\(536\) 3.00000 0.129580
\(537\) 0 0
\(538\) 30.0000 1.29339
\(539\) 4.00000 0.172292
\(540\) 20.0000 0.860663
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) −7.00000 −0.300676
\(543\) −22.0000 −0.944110
\(544\) −3.00000 −0.128624
\(545\) −60.0000 −2.57012
\(546\) −3.00000 −0.128388
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −17.0000 −0.726204
\(549\) −4.00000 −0.170716
\(550\) −22.0000 −0.938083
\(551\) 0 0
\(552\) 1.00000 0.0425628
\(553\) 30.0000 1.27573
\(554\) −28.0000 −1.18961
\(555\) −8.00000 −0.339581
\(556\) 0 0
\(557\) 28.0000 1.18640 0.593199 0.805056i \(-0.297865\pi\)
0.593199 + 0.805056i \(0.297865\pi\)
\(558\) 16.0000 0.677334
\(559\) 4.00000 0.169182
\(560\) −12.0000 −0.507093
\(561\) 6.00000 0.253320
\(562\) −8.00000 −0.337460
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 8.00000 0.336861
\(565\) 56.0000 2.35594
\(566\) 6.00000 0.252199
\(567\) 3.00000 0.125988
\(568\) 2.00000 0.0839181
\(569\) −40.0000 −1.67689 −0.838444 0.544988i \(-0.816534\pi\)
−0.838444 + 0.544988i \(0.816534\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 2.00000 0.0836242
\(573\) 7.00000 0.292429
\(574\) −24.0000 −1.00174
\(575\) −11.0000 −0.458732
\(576\) −2.00000 −0.0833333
\(577\) −37.0000 −1.54033 −0.770165 0.637845i \(-0.779826\pi\)
−0.770165 + 0.637845i \(0.779826\pi\)
\(578\) 8.00000 0.332756
\(579\) 6.00000 0.249351
\(580\) −20.0000 −0.830455
\(581\) −18.0000 −0.746766
\(582\) −2.00000 −0.0829027
\(583\) 2.00000 0.0828315
\(584\) −9.00000 −0.372423
\(585\) 8.00000 0.330759
\(586\) 9.00000 0.371787
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 2.00000 0.0824786
\(589\) 0 0
\(590\) −60.0000 −2.47016
\(591\) 8.00000 0.329076
\(592\) 2.00000 0.0821995
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 10.0000 0.410305
\(595\) −36.0000 −1.47586
\(596\) 0 0
\(597\) −25.0000 −1.02318
\(598\) 1.00000 0.0408930
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −11.0000 −0.449073
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) −12.0000 −0.489083
\(603\) 6.00000 0.244339
\(604\) −2.00000 −0.0813788
\(605\) 28.0000 1.13836
\(606\) −2.00000 −0.0812444
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 0 0
\(609\) 15.0000 0.607831
\(610\) 8.00000 0.323911
\(611\) 8.00000 0.323645
\(612\) −6.00000 −0.242536
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) −12.0000 −0.484281
\(615\) −32.0000 −1.29036
\(616\) −6.00000 −0.241747
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) −6.00000 −0.241355
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) −32.0000 −1.28515
\(621\) 5.00000 0.200643
\(622\) −7.00000 −0.280674
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 41.0000 1.64000
\(626\) −29.0000 −1.15907
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) 6.00000 0.239236
\(630\) −24.0000 −0.956183
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) −10.0000 −0.397779
\(633\) −27.0000 −1.07315
\(634\) −27.0000 −1.07231
\(635\) 72.0000 2.85723
\(636\) 1.00000 0.0396526
\(637\) 2.00000 0.0792429
\(638\) −10.0000 −0.395904
\(639\) 4.00000 0.158238
\(640\) 4.00000 0.158114
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) −7.00000 −0.276268
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) −3.00000 −0.118217
\(645\) −16.0000 −0.629999
\(646\) 0 0
\(647\) 23.0000 0.904223 0.452112 0.891961i \(-0.350671\pi\)
0.452112 + 0.891961i \(0.350671\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −30.0000 −1.17760
\(650\) −11.0000 −0.431455
\(651\) 24.0000 0.940634
\(652\) −16.0000 −0.626608
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) −15.0000 −0.586546
\(655\) −48.0000 −1.87552
\(656\) 8.00000 0.312348
\(657\) −18.0000 −0.702247
\(658\) −24.0000 −0.935617
\(659\) −5.00000 −0.194772 −0.0973862 0.995247i \(-0.531048\pi\)
−0.0973862 + 0.995247i \(0.531048\pi\)
\(660\) −8.00000 −0.311400
\(661\) 23.0000 0.894596 0.447298 0.894385i \(-0.352386\pi\)
0.447298 + 0.894385i \(0.352386\pi\)
\(662\) 17.0000 0.660724
\(663\) 3.00000 0.116510
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) −5.00000 −0.193601
\(668\) 12.0000 0.464294
\(669\) −14.0000 −0.541271
\(670\) −12.0000 −0.463600
\(671\) 4.00000 0.154418
\(672\) −3.00000 −0.115728
\(673\) −44.0000 −1.69608 −0.848038 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) −32.0000 −1.23259
\(675\) −55.0000 −2.11695
\(676\) −12.0000 −0.461538
\(677\) −13.0000 −0.499631 −0.249815 0.968294i \(-0.580370\pi\)
−0.249815 + 0.968294i \(0.580370\pi\)
\(678\) 14.0000 0.537667
\(679\) 6.00000 0.230259
\(680\) 12.0000 0.460179
\(681\) 17.0000 0.651441
\(682\) −16.0000 −0.612672
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) 68.0000 2.59815
\(686\) 15.0000 0.572703
\(687\) −10.0000 −0.381524
\(688\) 4.00000 0.152499
\(689\) 1.00000 0.0380970
\(690\) −4.00000 −0.152277
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) 6.00000 0.228086
\(693\) −12.0000 −0.455842
\(694\) 2.00000 0.0759190
\(695\) 0 0
\(696\) −5.00000 −0.189525
\(697\) 24.0000 0.909065
\(698\) −10.0000 −0.378506
\(699\) −6.00000 −0.226941
\(700\) 33.0000 1.24728
\(701\) −28.0000 −1.05755 −0.528773 0.848763i \(-0.677348\pi\)
−0.528773 + 0.848763i \(0.677348\pi\)
\(702\) 5.00000 0.188713
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) −32.0000 −1.20519
\(706\) −9.00000 −0.338719
\(707\) 6.00000 0.225653
\(708\) −15.0000 −0.563735
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) −8.00000 −0.300235
\(711\) −20.0000 −0.750059
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) −9.00000 −0.336817
\(715\) −8.00000 −0.299183
\(716\) 0 0
\(717\) 15.0000 0.560185
\(718\) 15.0000 0.559795
\(719\) −5.00000 −0.186469 −0.0932343 0.995644i \(-0.529721\pi\)
−0.0932343 + 0.995644i \(0.529721\pi\)
\(720\) 8.00000 0.298142
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) 8.00000 0.297523
\(724\) −22.0000 −0.817624
\(725\) 55.0000 2.04265
\(726\) 7.00000 0.259794
\(727\) −17.0000 −0.630495 −0.315248 0.949009i \(-0.602088\pi\)
−0.315248 + 0.949009i \(0.602088\pi\)
\(728\) −3.00000 −0.111187
\(729\) 13.0000 0.481481
\(730\) 36.0000 1.33242
\(731\) 12.0000 0.443836
\(732\) 2.00000 0.0739221
\(733\) −36.0000 −1.32969 −0.664845 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) −28.0000 −1.03350
\(735\) −8.00000 −0.295084
\(736\) 1.00000 0.0368605
\(737\) −6.00000 −0.221013
\(738\) 16.0000 0.588968
\(739\) −40.0000 −1.47142 −0.735712 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) −3.00000 −0.110133
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 29.0000 1.06177
\(747\) 12.0000 0.439057
\(748\) 6.00000 0.219382
\(749\) 21.0000 0.767323
\(750\) 24.0000 0.876356
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 8.00000 0.291730
\(753\) 2.00000 0.0728841
\(754\) −5.00000 −0.182089
\(755\) 8.00000 0.291150
\(756\) −15.0000 −0.545545
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 15.0000 0.544825
\(759\) −2.00000 −0.0725954
\(760\) 0 0
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) 18.0000 0.652071
\(763\) 45.0000 1.62911
\(764\) 7.00000 0.253251
\(765\) 24.0000 0.867722
\(766\) −26.0000 −0.939418
\(767\) −15.0000 −0.541619
\(768\) 1.00000 0.0360844
\(769\) −35.0000 −1.26213 −0.631066 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) 24.0000 0.864900
\(771\) −8.00000 −0.288113
\(772\) 6.00000 0.215945
\(773\) −9.00000 −0.323708 −0.161854 0.986815i \(-0.551747\pi\)
−0.161854 + 0.986815i \(0.551747\pi\)
\(774\) 8.00000 0.287554
\(775\) 88.0000 3.16105
\(776\) −2.00000 −0.0717958
\(777\) 6.00000 0.215249
\(778\) 30.0000 1.07555
\(779\) 0 0
\(780\) −4.00000 −0.143223
\(781\) −4.00000 −0.143131
\(782\) 3.00000 0.107280
\(783\) −25.0000 −0.893427
\(784\) 2.00000 0.0714286
\(785\) 8.00000 0.285532
\(786\) −12.0000 −0.428026
\(787\) 17.0000 0.605985 0.302992 0.952993i \(-0.402014\pi\)
0.302992 + 0.952993i \(0.402014\pi\)
\(788\) 8.00000 0.284988
\(789\) 24.0000 0.854423
\(790\) 40.0000 1.42314
\(791\) −42.0000 −1.49335
\(792\) 4.00000 0.142134
\(793\) 2.00000 0.0710221
\(794\) −8.00000 −0.283909
\(795\) −4.00000 −0.141865
\(796\) −25.0000 −0.886102
\(797\) −3.00000 −0.106265 −0.0531327 0.998587i \(-0.516921\pi\)
−0.0531327 + 0.998587i \(0.516921\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) −11.0000 −0.388909
\(801\) 0 0
\(802\) −8.00000 −0.282490
\(803\) 18.0000 0.635206
\(804\) −3.00000 −0.105802
\(805\) 12.0000 0.422944
\(806\) −8.00000 −0.281788
\(807\) −30.0000 −1.05605
\(808\) −2.00000 −0.0703598
\(809\) −15.0000 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 4.00000 0.140546
\(811\) 3.00000 0.105344 0.0526721 0.998612i \(-0.483226\pi\)
0.0526721 + 0.998612i \(0.483226\pi\)
\(812\) 15.0000 0.526397
\(813\) 7.00000 0.245501
\(814\) −4.00000 −0.140200
\(815\) 64.0000 2.24182
\(816\) 3.00000 0.105021
\(817\) 0 0
\(818\) −20.0000 −0.699284
\(819\) −6.00000 −0.209657
\(820\) −32.0000 −1.11749
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 17.0000 0.592943
\(823\) 29.0000 1.01088 0.505438 0.862863i \(-0.331331\pi\)
0.505438 + 0.862863i \(0.331331\pi\)
\(824\) −6.00000 −0.209020
\(825\) 22.0000 0.765942
\(826\) 45.0000 1.56575
\(827\) −23.0000 −0.799788 −0.399894 0.916561i \(-0.630953\pi\)
−0.399894 + 0.916561i \(0.630953\pi\)
\(828\) 2.00000 0.0695048
\(829\) 15.0000 0.520972 0.260486 0.965478i \(-0.416117\pi\)
0.260486 + 0.965478i \(0.416117\pi\)
\(830\) −24.0000 −0.833052
\(831\) 28.0000 0.971309
\(832\) 1.00000 0.0346688
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) −48.0000 −1.66111
\(836\) 0 0
\(837\) −40.0000 −1.38260
\(838\) 0 0
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 12.0000 0.414039
\(841\) −4.00000 −0.137931
\(842\) −13.0000 −0.448010
\(843\) 8.00000 0.275535
\(844\) −27.0000 −0.929378
\(845\) 48.0000 1.65125
\(846\) 16.0000 0.550091
\(847\) −21.0000 −0.721569
\(848\) 1.00000 0.0343401
\(849\) −6.00000 −0.205919
\(850\) −33.0000 −1.13189
\(851\) −2.00000 −0.0685591
\(852\) −2.00000 −0.0685189
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) −7.00000 −0.239255
\(857\) 12.0000 0.409912 0.204956 0.978771i \(-0.434295\pi\)
0.204956 + 0.978771i \(0.434295\pi\)
\(858\) −2.00000 −0.0682789
\(859\) −50.0000 −1.70598 −0.852989 0.521929i \(-0.825213\pi\)
−0.852989 + 0.521929i \(0.825213\pi\)
\(860\) −16.0000 −0.545595
\(861\) 24.0000 0.817918
\(862\) −18.0000 −0.613082
\(863\) −54.0000 −1.83818 −0.919091 0.394046i \(-0.871075\pi\)
−0.919091 + 0.394046i \(0.871075\pi\)
\(864\) 5.00000 0.170103
\(865\) −24.0000 −0.816024
\(866\) 14.0000 0.475739
\(867\) −8.00000 −0.271694
\(868\) 24.0000 0.814613
\(869\) 20.0000 0.678454
\(870\) 20.0000 0.678064
\(871\) −3.00000 −0.101651
\(872\) −15.0000 −0.507964
\(873\) −4.00000 −0.135379
\(874\) 0 0
\(875\) −72.0000 −2.43404
\(876\) 9.00000 0.304082
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) 20.0000 0.674967
\(879\) −9.00000 −0.303562
\(880\) −8.00000 −0.269680
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 4.00000 0.134687
\(883\) 34.0000 1.14419 0.572096 0.820187i \(-0.306131\pi\)
0.572096 + 0.820187i \(0.306131\pi\)
\(884\) 3.00000 0.100901
\(885\) 60.0000 2.01688
\(886\) 26.0000 0.873487
\(887\) 2.00000 0.0671534 0.0335767 0.999436i \(-0.489310\pi\)
0.0335767 + 0.999436i \(0.489310\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −54.0000 −1.81110
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −14.0000 −0.468755
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −3.00000 −0.100223
\(897\) −1.00000 −0.0333890
\(898\) 10.0000 0.333704
\(899\) 40.0000 1.33407
\(900\) −22.0000 −0.733333
\(901\) 3.00000 0.0999445
\(902\) −16.0000 −0.532742
\(903\) 12.0000 0.399335
\(904\) 14.0000 0.465633
\(905\) 88.0000 2.92522
\(906\) 2.00000 0.0664455
\(907\) −53.0000 −1.75984 −0.879918 0.475125i \(-0.842403\pi\)
−0.879918 + 0.475125i \(0.842403\pi\)
\(908\) 17.0000 0.564165
\(909\) −4.00000 −0.132672
\(910\) 12.0000 0.397796
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 7.00000 0.231539
\(915\) −8.00000 −0.264472
\(916\) −10.0000 −0.330409
\(917\) 36.0000 1.18882
\(918\) 15.0000 0.495074
\(919\) 5.00000 0.164935 0.0824674 0.996594i \(-0.473720\pi\)
0.0824674 + 0.996594i \(0.473720\pi\)
\(920\) −4.00000 −0.131876
\(921\) 12.0000 0.395413
\(922\) 28.0000 0.922131
\(923\) −2.00000 −0.0658308
\(924\) 6.00000 0.197386
\(925\) 22.0000 0.723356
\(926\) −4.00000 −0.131448
\(927\) −12.0000 −0.394132
\(928\) −5.00000 −0.164133
\(929\) −55.0000 −1.80449 −0.902246 0.431222i \(-0.858082\pi\)
−0.902246 + 0.431222i \(0.858082\pi\)
\(930\) 32.0000 1.04932
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 7.00000 0.229170
\(934\) 2.00000 0.0654420
\(935\) −24.0000 −0.784884
\(936\) 2.00000 0.0653720
\(937\) −7.00000 −0.228680 −0.114340 0.993442i \(-0.536475\pi\)
−0.114340 + 0.993442i \(0.536475\pi\)
\(938\) 9.00000 0.293860
\(939\) 29.0000 0.946379
\(940\) −32.0000 −1.04372
\(941\) −7.00000 −0.228193 −0.114097 0.993470i \(-0.536397\pi\)
−0.114097 + 0.993470i \(0.536397\pi\)
\(942\) 2.00000 0.0651635
\(943\) −8.00000 −0.260516
\(944\) −15.0000 −0.488208
\(945\) 60.0000 1.95180
\(946\) −8.00000 −0.260102
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 10.0000 0.324785
\(949\) 9.00000 0.292152
\(950\) 0 0
\(951\) 27.0000 0.875535
\(952\) −9.00000 −0.291692
\(953\) 46.0000 1.49009 0.745043 0.667016i \(-0.232429\pi\)
0.745043 + 0.667016i \(0.232429\pi\)
\(954\) 2.00000 0.0647524
\(955\) −28.0000 −0.906059
\(956\) 15.0000 0.485135
\(957\) 10.0000 0.323254
\(958\) 20.0000 0.646171
\(959\) −51.0000 −1.64688
\(960\) −4.00000 −0.129099
\(961\) 33.0000 1.06452
\(962\) −2.00000 −0.0644826
\(963\) −14.0000 −0.451144
\(964\) 8.00000 0.257663
\(965\) −24.0000 −0.772587
\(966\) 3.00000 0.0965234
\(967\) 48.0000 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 8.00000 0.256865
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 16.0000 0.513200
\(973\) 0 0
\(974\) −2.00000 −0.0640841
\(975\) 11.0000 0.352282
\(976\) 2.00000 0.0640184
\(977\) −8.00000 −0.255943 −0.127971 0.991778i \(-0.540847\pi\)
−0.127971 + 0.991778i \(0.540847\pi\)
\(978\) 16.0000 0.511624
\(979\) 0 0
\(980\) −8.00000 −0.255551
\(981\) −30.0000 −0.957826
\(982\) 28.0000 0.893516
\(983\) 6.00000 0.191370 0.0956851 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) −8.00000 −0.255031
\(985\) −32.0000 −1.01960
\(986\) −15.0000 −0.477697
\(987\) 24.0000 0.763928
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) −16.0000 −0.508513
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −8.00000 −0.254000
\(993\) −17.0000 −0.539479
\(994\) 6.00000 0.190308
\(995\) 100.000 3.17021
\(996\) −6.00000 −0.190117
\(997\) 28.0000 0.886769 0.443384 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) −40.0000 −1.26618
\(999\) −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 722.2.a.b.1.1 1
3.2 odd 2 6498.2.a.y.1.1 1
4.3 odd 2 5776.2.a.d.1.1 1
19.2 odd 18 722.2.e.c.99.1 6
19.3 odd 18 722.2.e.c.389.1 6
19.4 even 9 722.2.e.d.415.1 6
19.5 even 9 722.2.e.d.595.1 6
19.6 even 9 722.2.e.d.245.1 6
19.7 even 3 722.2.c.f.429.1 2
19.8 odd 6 722.2.c.d.653.1 2
19.9 even 9 722.2.e.d.423.1 6
19.10 odd 18 722.2.e.c.423.1 6
19.11 even 3 722.2.c.f.653.1 2
19.12 odd 6 722.2.c.d.429.1 2
19.13 odd 18 722.2.e.c.245.1 6
19.14 odd 18 722.2.e.c.595.1 6
19.15 odd 18 722.2.e.c.415.1 6
19.16 even 9 722.2.e.d.389.1 6
19.17 even 9 722.2.e.d.99.1 6
19.18 odd 2 38.2.a.b.1.1 1
57.56 even 2 342.2.a.d.1.1 1
76.75 even 2 304.2.a.d.1.1 1
95.18 even 4 950.2.b.c.799.1 2
95.37 even 4 950.2.b.c.799.2 2
95.94 odd 2 950.2.a.b.1.1 1
133.132 even 2 1862.2.a.f.1.1 1
152.37 odd 2 1216.2.a.n.1.1 1
152.75 even 2 1216.2.a.g.1.1 1
209.208 even 2 4598.2.a.a.1.1 1
228.227 odd 2 2736.2.a.w.1.1 1
247.246 odd 2 6422.2.a.b.1.1 1
285.284 even 2 8550.2.a.u.1.1 1
380.379 even 2 7600.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.a.b.1.1 1 19.18 odd 2
304.2.a.d.1.1 1 76.75 even 2
342.2.a.d.1.1 1 57.56 even 2
722.2.a.b.1.1 1 1.1 even 1 trivial
722.2.c.d.429.1 2 19.12 odd 6
722.2.c.d.653.1 2 19.8 odd 6
722.2.c.f.429.1 2 19.7 even 3
722.2.c.f.653.1 2 19.11 even 3
722.2.e.c.99.1 6 19.2 odd 18
722.2.e.c.245.1 6 19.13 odd 18
722.2.e.c.389.1 6 19.3 odd 18
722.2.e.c.415.1 6 19.15 odd 18
722.2.e.c.423.1 6 19.10 odd 18
722.2.e.c.595.1 6 19.14 odd 18
722.2.e.d.99.1 6 19.17 even 9
722.2.e.d.245.1 6 19.6 even 9
722.2.e.d.389.1 6 19.16 even 9
722.2.e.d.415.1 6 19.4 even 9
722.2.e.d.423.1 6 19.9 even 9
722.2.e.d.595.1 6 19.5 even 9
950.2.a.b.1.1 1 95.94 odd 2
950.2.b.c.799.1 2 95.18 even 4
950.2.b.c.799.2 2 95.37 even 4
1216.2.a.g.1.1 1 152.75 even 2
1216.2.a.n.1.1 1 152.37 odd 2
1862.2.a.f.1.1 1 133.132 even 2
2736.2.a.w.1.1 1 228.227 odd 2
4598.2.a.a.1.1 1 209.208 even 2
5776.2.a.d.1.1 1 4.3 odd 2
6422.2.a.b.1.1 1 247.246 odd 2
6498.2.a.y.1.1 1 3.2 odd 2
7600.2.a.h.1.1 1 380.379 even 2
8550.2.a.u.1.1 1 285.284 even 2