Properties

Label 2310.2.a.b.1.1
Level $2310$
Weight $2$
Character 2310.1
Self dual yes
Analytic conductor $18.445$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2310.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.4454428669\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2310.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} +1.00000 q^{21} -1.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +10.0000 q^{37} -6.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} -1.00000 q^{42} -8.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} +4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} +1.00000 q^{56} +6.00000 q^{58} -12.0000 q^{59} +1.00000 q^{60} +10.0000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -6.00000 q^{65} +1.00000 q^{66} +8.00000 q^{67} +2.00000 q^{68} -1.00000 q^{70} +4.00000 q^{71} -1.00000 q^{72} +10.0000 q^{73} -10.0000 q^{74} -1.00000 q^{75} -1.00000 q^{77} +6.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +12.0000 q^{83} +1.00000 q^{84} -2.00000 q^{85} +8.00000 q^{86} +6.00000 q^{87} -1.00000 q^{88} +6.00000 q^{89} +1.00000 q^{90} -6.00000 q^{91} +8.00000 q^{93} -4.00000 q^{94} +1.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) −1.00000 −0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −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 \(0.688081\pi\)
\(30\) −1.00000 −0.182574
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −2.00000 −0.342997
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −1.00000 −0.154303
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 1.00000 0.150756
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) 6.00000 0.832050
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.00000 −0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 1.00000 0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 8.00000 1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) 1.00000 0.123091
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) −1.00000 −0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −10.0000 −1.16248
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 6.00000 0.679366
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 1.00000 0.109109
\(85\) −2.00000 −0.216930
\(86\) 8.00000 0.862662
\(87\) 6.00000 0.643268
\(88\) −1.00000 −0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 1.00000 0.105409
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 2.00000 0.198030
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −6.00000 −0.588348
\(105\) −1.00000 −0.0975900
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 1.00000 0.0953463
\(111\) −10.0000 −0.949158
\(112\) −1.00000 −0.0944911
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 6.00000 0.554700
\(118\) 12.0000 1.10469
\(119\) −2.00000 −0.183340
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) −10.0000 −0.905357
\(123\) 2.00000 0.180334
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) 6.00000 0.526235
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 1.00000 0.0860663
\(136\) −2.00000 −0.171499
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 1.00000 0.0845154
\(141\) −4.00000 −0.336861
\(142\) −4.00000 −0.335673
\(143\) 6.00000 0.501745
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) −10.0000 −0.827606
\(147\) −1.00000 −0.0824786
\(148\) 10.0000 0.821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 1.00000 0.0816497
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 1.00000 0.0805823
\(155\) 8.00000 0.642575
\(156\) −6.00000 −0.480384
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) −8.00000 −0.636446
\(159\) 6.00000 0.475831
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −2.00000 −0.156174
\(165\) 1.00000 0.0778499
\(166\) −12.0000 −0.931381
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 23.0000 1.76923
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −6.00000 −0.454859
\(175\) −1.00000 −0.0755929
\(176\) 1.00000 0.0753778
\(177\) 12.0000 0.901975
\(178\) −6.00000 −0.449719
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 6.00000 0.444750
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) −10.0000 −0.735215
\(186\) −8.00000 −0.586588
\(187\) 2.00000 0.146254
\(188\) 4.00000 0.291730
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −2.00000 −0.143592
\(195\) 6.00000 0.429669
\(196\) 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
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −8.00000 −0.564276
\(202\) 6.00000 0.422159
\(203\) 6.00000 0.421117
\(204\) −2.00000 −0.140028
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) 0 0
\(210\) 1.00000 0.0690066
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −6.00000 −0.412082
\(213\) −4.00000 −0.274075
\(214\) −12.0000 −0.820303
\(215\) 8.00000 0.545595
\(216\) 1.00000 0.0680414
\(217\) 8.00000 0.543075
\(218\) 10.0000 0.677285
\(219\) −10.0000 −0.675737
\(220\) −1.00000 −0.0674200
\(221\) 12.0000 0.807207
\(222\) 10.0000 0.671156
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) −14.0000 −0.931266
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 6.00000 0.393919
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) −6.00000 −0.392232
\(235\) −4.00000 −0.260931
\(236\) −12.0000 −0.781133
\(237\) −8.00000 −0.519656
\(238\) 2.00000 0.129641
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 1.00000 0.0645497
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) −1.00000 −0.0638877
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) −12.0000 −0.760469
\(250\) 1.00000 0.0632456
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 0 0
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −8.00000 −0.498058
\(259\) −10.0000 −0.621370
\(260\) −6.00000 −0.372104
\(261\) −6.00000 −0.371391
\(262\) −4.00000 −0.247121
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 1.00000 0.0615457
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 8.00000 0.488678
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 2.00000 0.121268
\(273\) 6.00000 0.363137
\(274\) 18.0000 1.08742
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) −1.00000 −0.0597614
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 4.00000 0.238197
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 2.00000 0.118056
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) −2.00000 −0.117242
\(292\) 10.0000 0.585206
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 1.00000 0.0583212
\(295\) 12.0000 0.698667
\(296\) −10.0000 −0.581238
\(297\) −1.00000 −0.0580259
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 8.00000 0.461112
\(302\) −16.0000 −0.920697
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) −2.00000 −0.114332
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 6.00000 0.339683
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −22.0000 −1.24153
\(315\) 1.00000 0.0563436
\(316\) 8.00000 0.450035
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −6.00000 −0.336463
\(319\) −6.00000 −0.335936
\(320\) −1.00000 −0.0559017
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 6.00000 0.332820
\(326\) 8.00000 0.443079
\(327\) 10.0000 0.553001
\(328\) 2.00000 0.110432
\(329\) −4.00000 −0.220527
\(330\) −1.00000 −0.0550482
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 12.0000 0.658586
\(333\) 10.0000 0.547997
\(334\) 20.0000 1.09435
\(335\) −8.00000 −0.437087
\(336\) 1.00000 0.0545545
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) −23.0000 −1.25104
\(339\) −14.0000 −0.760376
\(340\) −2.00000 −0.108465
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 6.00000 0.321634
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 1.00000 0.0534522
\(351\) −6.00000 −0.320256
\(352\) −1.00000 −0.0533002
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) −12.0000 −0.637793
\(355\) −4.00000 −0.212298
\(356\) 6.00000 0.317999
\(357\) 2.00000 0.105851
\(358\) 4.00000 0.211407
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 1.00000 0.0527046
\(361\) −19.0000 −1.00000
\(362\) −10.0000 −0.525588
\(363\) −1.00000 −0.0524864
\(364\) −6.00000 −0.314485
\(365\) −10.0000 −0.523424
\(366\) 10.0000 0.522708
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 10.0000 0.519875
\(371\) 6.00000 0.311504
\(372\) 8.00000 0.414781
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) −2.00000 −0.103418
\(375\) 1.00000 0.0516398
\(376\) −4.00000 −0.206284
\(377\) −36.0000 −1.85409
\(378\) −1.00000 −0.0514344
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 1.00000 0.0510310
\(385\) 1.00000 0.0509647
\(386\) −10.0000 −0.508987
\(387\) −8.00000 −0.406663
\(388\) 2.00000 0.101535
\(389\) −22.0000 −1.11544 −0.557722 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) −6.00000 −0.303822
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) −4.00000 −0.201773
\(394\) −10.0000 −0.503793
\(395\) −8.00000 −0.402524
\(396\) 1.00000 0.0502519
\(397\) 30.0000 1.50566 0.752828 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) −24.0000 −1.20301
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 8.00000 0.399004
\(403\) −48.0000 −2.39105
\(404\) −6.00000 −0.298511
\(405\) −1.00000 −0.0496904
\(406\) −6.00000 −0.297775
\(407\) 10.0000 0.495682
\(408\) 2.00000 0.0990148
\(409\) −38.0000 −1.87898 −0.939490 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 18.0000 0.887875
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) −6.00000 −0.294174
\(417\) 0 0
\(418\) 0 0
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) 12.0000 0.584151
\(423\) 4.00000 0.194487
\(424\) 6.00000 0.291386
\(425\) 2.00000 0.0970143
\(426\) 4.00000 0.193801
\(427\) −10.0000 −0.483934
\(428\) 12.0000 0.580042
\(429\) −6.00000 −0.289683
\(430\) −8.00000 −0.385794
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) −8.00000 −0.384012
\(435\) −6.00000 −0.287678
\(436\) −10.0000 −0.478913
\(437\) 0 0
\(438\) 10.0000 0.477818
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 1.00000 0.0476731
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) −28.0000 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) −10.0000 −0.474579
\(445\) −6.00000 −0.284427
\(446\) −8.00000 −0.378811
\(447\) 6.00000 0.283790
\(448\) −1.00000 −0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −2.00000 −0.0941763
\(452\) 14.0000 0.658505
\(453\) −16.0000 −0.751746
\(454\) −4.00000 −0.187729
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 14.0000 0.654177
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) −1.00000 −0.0465242
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) −6.00000 −0.278543
\(465\) −8.00000 −0.370991
\(466\) 18.0000 0.833834
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 6.00000 0.277350
\(469\) −8.00000 −0.369406
\(470\) 4.00000 0.184506
\(471\) −22.0000 −1.01371
\(472\) 12.0000 0.552345
\(473\) −8.00000 −0.367840
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) −6.00000 −0.274721
\(478\) 20.0000 0.914779
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 60.0000 2.73576
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −2.00000 −0.0908153
\(486\) 1.00000 0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −10.0000 −0.452679
\(489\) 8.00000 0.361773
\(490\) 1.00000 0.0451754
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) 2.00000 0.0901670
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) −8.00000 −0.359211
\(497\) −4.00000 −0.179425
\(498\) 12.0000 0.537733
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 20.0000 0.893534
\(502\) −28.0000 −1.24970
\(503\) −20.0000 −0.891756 −0.445878 0.895094i \(-0.647108\pi\)
−0.445878 + 0.895094i \(0.647108\pi\)
\(504\) 1.00000 0.0445435
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) 0 0
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) −2.00000 −0.0885615
\(511\) −10.0000 −0.442374
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 4.00000 0.175920
\(518\) 10.0000 0.439375
\(519\) −18.0000 −0.790112
\(520\) 6.00000 0.263117
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) 6.00000 0.262613
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 4.00000 0.174741
\(525\) 1.00000 0.0436436
\(526\) 16.0000 0.697633
\(527\) −16.0000 −0.696971
\(528\) −1.00000 −0.0435194
\(529\) −23.0000 −1.00000
\(530\) −6.00000 −0.260623
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 6.00000 0.259645
\(535\) −12.0000 −0.518805
\(536\) −8.00000 −0.345547
\(537\) 4.00000 0.172613
\(538\) −18.0000 −0.776035
\(539\) 1.00000 0.0430730
\(540\) 1.00000 0.0430331
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −24.0000 −1.03089
\(543\) −10.0000 −0.429141
\(544\) −2.00000 −0.0857493
\(545\) 10.0000 0.428353
\(546\) −6.00000 −0.256776
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) −18.0000 −0.768922
\(549\) 10.0000 0.426790
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 6.00000 0.254916
\(555\) 10.0000 0.424476
\(556\) 0 0
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 8.00000 0.338667
\(559\) −48.0000 −2.03018
\(560\) 1.00000 0.0422577
\(561\) −2.00000 −0.0844401
\(562\) −18.0000 −0.759284
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) −4.00000 −0.168430
\(565\) −14.0000 −0.588984
\(566\) −4.00000 −0.168133
\(567\) −1.00000 −0.0419961
\(568\) −4.00000 −0.167836
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 6.00000 0.250873
\(573\) −12.0000 −0.501307
\(574\) −2.00000 −0.0834784
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 13.0000 0.540729
\(579\) −10.0000 −0.415586
\(580\) 6.00000 0.249136
\(581\) −12.0000 −0.497844
\(582\) 2.00000 0.0829027
\(583\) −6.00000 −0.248495
\(584\) −10.0000 −0.413803
\(585\) −6.00000 −0.248069
\(586\) −18.0000 −0.743573
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) −12.0000 −0.494032
\(591\) −10.0000 −0.411345
\(592\) 10.0000 0.410997
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 1.00000 0.0410305
\(595\) 2.00000 0.0819920
\(596\) −6.00000 −0.245770
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 1.00000 0.0408248
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −8.00000 −0.326056
\(603\) 8.00000 0.325785
\(604\) 16.0000 0.651031
\(605\) −1.00000 −0.0406558
\(606\) −6.00000 −0.243733
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 10.0000 0.404888
\(611\) 24.0000 0.970936
\(612\) 2.00000 0.0808452
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 4.00000 0.161427
\(615\) −2.00000 −0.0806478
\(616\) 1.00000 0.0402911
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −48.0000 −1.92928 −0.964641 0.263566i \(-0.915101\pi\)
−0.964641 + 0.263566i \(0.915101\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) −6.00000 −0.240385
\(624\) −6.00000 −0.240192
\(625\) 1.00000 0.0400000
\(626\) −26.0000 −1.03917
\(627\) 0 0
\(628\) 22.0000 0.877896
\(629\) 20.0000 0.797452
\(630\) −1.00000 −0.0398410
\(631\) −24.0000 −0.955425 −0.477712 0.878516i \(-0.658534\pi\)
−0.477712 + 0.878516i \(0.658534\pi\)
\(632\) −8.00000 −0.318223
\(633\) 12.0000 0.476957
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 6.00000 0.237729
\(638\) 6.00000 0.237542
\(639\) 4.00000 0.158238
\(640\) 1.00000 0.0395285
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 12.0000 0.473602
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −12.0000 −0.471041
\(650\) −6.00000 −0.235339
\(651\) −8.00000 −0.313545
\(652\) −8.00000 −0.313304
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) −10.0000 −0.391031
\(655\) −4.00000 −0.156293
\(656\) −2.00000 −0.0780869
\(657\) 10.0000 0.390137
\(658\) 4.00000 0.155936
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 1.00000 0.0389249
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) −20.0000 −0.777322
\(663\) −12.0000 −0.466041
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 0 0
\(668\) −20.0000 −0.773823
\(669\) −8.00000 −0.309298
\(670\) 8.00000 0.309067
\(671\) 10.0000 0.386046
\(672\) −1.00000 −0.0385758
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 6.00000 0.231111
\(675\) −1.00000 −0.0384900
\(676\) 23.0000 0.884615
\(677\) 10.0000 0.384331 0.192166 0.981363i \(-0.438449\pi\)
0.192166 + 0.981363i \(0.438449\pi\)
\(678\) 14.0000 0.537667
\(679\) −2.00000 −0.0767530
\(680\) 2.00000 0.0766965
\(681\) −4.00000 −0.153280
\(682\) 8.00000 0.306336
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 1.00000 0.0381802
\(687\) 14.0000 0.534133
\(688\) −8.00000 −0.304997
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 18.0000 0.684257
\(693\) −1.00000 −0.0379869
\(694\) 28.0000 1.06287
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) −4.00000 −0.151511
\(698\) 30.0000 1.13552
\(699\) 18.0000 0.680823
\(700\) −1.00000 −0.0377964
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 6.00000 0.226455
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) 4.00000 0.150649
\(706\) −18.0000 −0.677439
\(707\) 6.00000 0.225653
\(708\) 12.0000 0.450988
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 4.00000 0.150117
\(711\) 8.00000 0.300023
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) −2.00000 −0.0748481
\(715\) −6.00000 −0.224387
\(716\) −4.00000 −0.149487
\(717\) 20.0000 0.746914
\(718\) −20.0000 −0.746393
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) −18.0000 −0.669427
\(724\) 10.0000 0.371647
\(725\) −6.00000 −0.222834
\(726\) 1.00000 0.0371135
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) −16.0000 −0.591781
\(732\) −10.0000 −0.369611
\(733\) −10.0000 −0.369358 −0.184679 0.982799i \(-0.559125\pi\)
−0.184679 + 0.982799i \(0.559125\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 2.00000 0.0736210
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) −10.0000 −0.367607
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) −8.00000 −0.293294
\(745\) 6.00000 0.219823
\(746\) −2.00000 −0.0732252
\(747\) 12.0000 0.439057
\(748\) 2.00000 0.0731272
\(749\) −12.0000 −0.438470
\(750\) −1.00000 −0.0365148
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 4.00000 0.145865
\(753\) −28.0000 −1.02038
\(754\) 36.0000 1.31104
\(755\) −16.0000 −0.582300
\(756\) 1.00000 0.0363696
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) 10.0000 0.362024
\(764\) 12.0000 0.434145
\(765\) −2.00000 −0.0723102
\(766\) −4.00000 −0.144526
\(767\) −72.0000 −2.59977
\(768\) −1.00000 −0.0360844
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) −1.00000 −0.0360375
\(771\) −18.0000 −0.648254
\(772\) 10.0000 0.359908
\(773\) 50.0000 1.79838 0.899188 0.437564i \(-0.144158\pi\)
0.899188 + 0.437564i \(0.144158\pi\)
\(774\) 8.00000 0.287554
\(775\) −8.00000 −0.287368
\(776\) −2.00000 −0.0717958
\(777\) 10.0000 0.358748
\(778\) 22.0000 0.788738
\(779\) 0 0
\(780\) 6.00000 0.214834
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) −22.0000 −0.785214
\(786\) 4.00000 0.142675
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 10.0000 0.356235
\(789\) 16.0000 0.569615
\(790\) 8.00000 0.284627
\(791\) −14.0000 −0.497783
\(792\) −1.00000 −0.0355335
\(793\) 60.0000 2.13066
\(794\) −30.0000 −1.06466
\(795\) −6.00000 −0.212798
\(796\) 24.0000 0.850657
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) −1.00000 −0.0353553
\(801\) 6.00000 0.212000
\(802\) 30.0000 1.05934
\(803\) 10.0000 0.352892
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 48.0000 1.69073
\(807\) −18.0000 −0.633630
\(808\) 6.00000 0.211079
\(809\) 34.0000 1.19538 0.597688 0.801729i \(-0.296086\pi\)
0.597688 + 0.801729i \(0.296086\pi\)
\(810\) 1.00000 0.0351364
\(811\) −32.0000 −1.12367 −0.561836 0.827249i \(-0.689905\pi\)
−0.561836 + 0.827249i \(0.689905\pi\)
\(812\) 6.00000 0.210559
\(813\) −24.0000 −0.841717
\(814\) −10.0000 −0.350500
\(815\) 8.00000 0.280228
\(816\) −2.00000 −0.0700140
\(817\) 0 0
\(818\) 38.0000 1.32864
\(819\) −6.00000 −0.209657
\(820\) 2.00000 0.0698430
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) −18.0000 −0.627822
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) −12.0000 −0.417533
\(827\) −44.0000 −1.53003 −0.765015 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 0 0
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 12.0000 0.416526
\(831\) 6.00000 0.208138
\(832\) 6.00000 0.208013
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 20.0000 0.692129
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) −28.0000 −0.967244
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 1.00000 0.0345033
\(841\) 7.00000 0.241379
\(842\) −38.0000 −1.30957
\(843\) −18.0000 −0.619953
\(844\) −12.0000 −0.413057
\(845\) −23.0000 −0.791224
\(846\) −4.00000 −0.137523
\(847\) −1.00000 −0.0343604
\(848\) −6.00000 −0.206041
\(849\) −4.00000 −0.137280
\(850\) −2.00000 −0.0685994
\(851\) 0 0
\(852\) −4.00000 −0.137038
\(853\) −2.00000 −0.0684787 −0.0342393 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −46.0000 −1.57133 −0.785665 0.618652i \(-0.787679\pi\)
−0.785665 + 0.618652i \(0.787679\pi\)
\(858\) 6.00000 0.204837
\(859\) −48.0000 −1.63774 −0.818869 0.573980i \(-0.805399\pi\)
−0.818869 + 0.573980i \(0.805399\pi\)
\(860\) 8.00000 0.272798
\(861\) −2.00000 −0.0681598
\(862\) 12.0000 0.408722
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 1.00000 0.0340207
\(865\) −18.0000 −0.612018
\(866\) 30.0000 1.01944
\(867\) 13.0000 0.441503
\(868\) 8.00000 0.271538
\(869\) 8.00000 0.271381
\(870\) 6.00000 0.203419
\(871\) 48.0000 1.62642
\(872\) 10.0000 0.338643
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) −10.0000 −0.337869
\(877\) 50.0000 1.68838 0.844190 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) 32.0000 1.07995
\(879\) −18.0000 −0.607125
\(880\) −1.00000 −0.0337100
\(881\) 54.0000 1.81931 0.909653 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 56.0000 1.88455 0.942275 0.334840i \(-0.108682\pi\)
0.942275 + 0.334840i \(0.108682\pi\)
\(884\) 12.0000 0.403604
\(885\) −12.0000 −0.403376
\(886\) 28.0000 0.940678
\(887\) 28.0000 0.940148 0.470074 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) 10.0000 0.335578
\(889\) 0 0
\(890\) 6.00000 0.201120
\(891\) 1.00000 0.0335013
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 4.00000 0.133705
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) 48.0000 1.60089
\(900\) 1.00000 0.0333333
\(901\) −12.0000 −0.399778
\(902\) 2.00000 0.0665927
\(903\) −8.00000 −0.266223
\(904\) −14.0000 −0.465633
\(905\) −10.0000 −0.332411
\(906\) 16.0000 0.531564
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 4.00000 0.132745
\(909\) −6.00000 −0.199007
\(910\) −6.00000 −0.198898
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 6.00000 0.198462
\(915\) 10.0000 0.330590
\(916\) −14.0000 −0.462573
\(917\) −4.00000 −0.132092
\(918\) 2.00000 0.0660098
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) −18.0000 −0.592798
\(923\) 24.0000 0.789970
\(924\) 1.00000 0.0328976
\(925\) 10.0000 0.328798
\(926\) −24.0000 −0.788689
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 8.00000 0.262330
\(931\) 0 0
\(932\) −18.0000 −0.589610
\(933\) 24.0000 0.785725
\(934\) −28.0000 −0.916188
\(935\) −2.00000 −0.0654070
\(936\) −6.00000 −0.196116
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 8.00000 0.261209
\(939\) −26.0000 −0.848478
\(940\) −4.00000 −0.130466
\(941\) −14.0000 −0.456387 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) 22.0000 0.716799
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) −1.00000 −0.0325300
\(946\) 8.00000 0.260102
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) −8.00000 −0.259828
\(949\) 60.0000 1.94768
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 2.00000 0.0648204
\(953\) −2.00000 −0.0647864 −0.0323932 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 6.00000 0.194257
\(955\) −12.0000 −0.388311
\(956\) −20.0000 −0.646846
\(957\) 6.00000 0.193952
\(958\) 8.00000 0.258468
\(959\) 18.0000 0.581250
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) −60.0000 −1.93448
\(963\) 12.0000 0.386695
\(964\) 18.0000 0.579741
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) −6.00000 −0.192154
\(976\) 10.0000 0.320092
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) −8.00000 −0.255812
\(979\) 6.00000 0.191761
\(980\) −1.00000 −0.0319438
\(981\) −10.0000 −0.319275
\(982\) −4.00000 −0.127645
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) −2.00000 −0.0637577
\(985\) −10.0000 −0.318626
\(986\) 12.0000 0.382158
\(987\) 4.00000 0.127321
\(988\) 0 0
\(989\) 0 0
\(990\) 1.00000 0.0317821
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 8.00000 0.254000
\(993\) −20.0000 −0.634681
\(994\) 4.00000 0.126872
\(995\) −24.0000 −0.760851
\(996\) −12.0000 −0.380235
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) −12.0000 −0.379853
\(999\) −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 2310.2.a.b.1.1 1
3.2 odd 2 6930.2.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.b.1.1 1 1.1 even 1 trivial
6930.2.a.bc.1.1 1 3.2 odd 2