Properties

Label 1110.2.a.e.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.86339462436\)
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 \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} -7.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} -3.00000 q^{22} -3.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +7.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} +1.00000 q^{30} -10.0000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +3.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} +1.00000 q^{38} -7.00000 q^{39} +1.00000 q^{40} +1.00000 q^{42} -4.00000 q^{43} +3.00000 q^{44} -1.00000 q^{45} +3.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} -3.00000 q^{51} -7.00000 q^{52} +9.00000 q^{53} -1.00000 q^{54} -3.00000 q^{55} +1.00000 q^{56} -1.00000 q^{57} -6.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} +10.0000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +7.00000 q^{65} -3.00000 q^{66} -10.0000 q^{67} -3.00000 q^{68} -3.00000 q^{69} -1.00000 q^{70} -6.00000 q^{71} -1.00000 q^{72} +11.0000 q^{73} -1.00000 q^{74} +1.00000 q^{75} -1.00000 q^{76} -3.00000 q^{77} +7.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +9.00000 q^{83} -1.00000 q^{84} +3.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} -3.00000 q^{88} -9.00000 q^{89} +1.00000 q^{90} +7.00000 q^{91} -3.00000 q^{92} -10.0000 q^{93} +12.0000 q^{94} +1.00000 q^{95} -1.00000 q^{96} -16.0000 q^{97} +6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000 0.288675
\(13\) −7.00000 −1.94145 −0.970725 0.240192i \(-0.922790\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) −3.00000 −0.639602
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 7.00000 1.37281
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 1.00000 0.182574
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.00000 0.522233
\(34\) 3.00000 0.514496
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) 1.00000 0.162221
\(39\) −7.00000 −1.12090
\(40\) 1.00000 0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 1.00000 0.154303
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 3.00000 0.452267
\(45\) −1.00000 −0.149071
\(46\) 3.00000 0.442326
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) −1.00000 −0.141421
\(51\) −3.00000 −0.420084
\(52\) −7.00000 −0.970725
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.00000 −0.404520
\(56\) 1.00000 0.133631
\(57\) −1.00000 −0.132453
\(58\) −6.00000 −0.787839
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 10.0000 1.27000
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 7.00000 0.868243
\(66\) −3.00000 −0.369274
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −3.00000 −0.363803
\(69\) −3.00000 −0.361158
\(70\) −1.00000 −0.119523
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −1.00000 −0.117851
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) −1.00000 −0.116248
\(75\) 1.00000 0.115470
\(76\) −1.00000 −0.114708
\(77\) −3.00000 −0.341882
\(78\) 7.00000 0.792594
\(79\) 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
\(82\) 0 0
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) −1.00000 −0.109109
\(85\) 3.00000 0.325396
\(86\) 4.00000 0.431331
\(87\) 6.00000 0.643268
\(88\) −3.00000 −0.319801
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 1.00000 0.105409
\(91\) 7.00000 0.733799
\(92\) −3.00000 −0.312772
\(93\) −10.0000 −1.03695
\(94\) 12.0000 1.23771
\(95\) 1.00000 0.102598
\(96\) −1.00000 −0.102062
\(97\) −16.0000 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) 6.00000 0.606092
\(99\) 3.00000 0.301511
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 3.00000 0.297044
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 7.00000 0.686406
\(105\) 1.00000 0.0975900
\(106\) −9.00000 −0.874157
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 1.00000 0.0962250
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 3.00000 0.286039
\(111\) 1.00000 0.0949158
\(112\) −1.00000 −0.0944911
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 1.00000 0.0936586
\(115\) 3.00000 0.279751
\(116\) 6.00000 0.557086
\(117\) −7.00000 −0.647150
\(118\) −6.00000 −0.552345
\(119\) 3.00000 0.275010
\(120\) 1.00000 0.0912871
\(121\) −2.00000 −0.181818
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) −10.0000 −0.898027
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) −7.00000 −0.613941
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 3.00000 0.261116
\(133\) 1.00000 0.0867110
\(134\) 10.0000 0.863868
\(135\) −1.00000 −0.0860663
\(136\) 3.00000 0.257248
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 3.00000 0.255377
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 1.00000 0.0845154
\(141\) −12.0000 −1.01058
\(142\) 6.00000 0.503509
\(143\) −21.0000 −1.75611
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) −11.0000 −0.910366
\(147\) −6.00000 −0.494872
\(148\) 1.00000 0.0821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 23.0000 1.87171 0.935857 0.352381i \(-0.114628\pi\)
0.935857 + 0.352381i \(0.114628\pi\)
\(152\) 1.00000 0.0811107
\(153\) −3.00000 −0.242536
\(154\) 3.00000 0.241747
\(155\) 10.0000 0.803219
\(156\) −7.00000 −0.560449
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) −8.00000 −0.636446
\(159\) 9.00000 0.713746
\(160\) 1.00000 0.0790569
\(161\) 3.00000 0.236433
\(162\) −1.00000 −0.0785674
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) 0 0
\(165\) −3.00000 −0.233550
\(166\) −9.00000 −0.698535
\(167\) 3.00000 0.232147 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 1.00000 0.0771517
\(169\) 36.0000 2.76923
\(170\) −3.00000 −0.230089
\(171\) −1.00000 −0.0764719
\(172\) −4.00000 −0.304997
\(173\) −9.00000 −0.684257 −0.342129 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) −6.00000 −0.454859
\(175\) −1.00000 −0.0755929
\(176\) 3.00000 0.226134
\(177\) 6.00000 0.450988
\(178\) 9.00000 0.674579
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −7.00000 −0.518875
\(183\) −10.0000 −0.739221
\(184\) 3.00000 0.221163
\(185\) −1.00000 −0.0735215
\(186\) 10.0000 0.733236
\(187\) −9.00000 −0.658145
\(188\) −12.0000 −0.875190
\(189\) −1.00000 −0.0727393
\(190\) −1.00000 −0.0725476
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 1.00000 0.0721688
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 16.0000 1.14873
\(195\) 7.00000 0.501280
\(196\) −6.00000 −0.428571
\(197\) 3.00000 0.213741 0.106871 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) −3.00000 −0.213201
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −10.0000 −0.705346
\(202\) 6.00000 0.422159
\(203\) −6.00000 −0.421117
\(204\) −3.00000 −0.210042
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) −3.00000 −0.208514
\(208\) −7.00000 −0.485363
\(209\) −3.00000 −0.207514
\(210\) −1.00000 −0.0690066
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 9.00000 0.618123
\(213\) −6.00000 −0.411113
\(214\) −3.00000 −0.205076
\(215\) 4.00000 0.272798
\(216\) −1.00000 −0.0680414
\(217\) 10.0000 0.678844
\(218\) 7.00000 0.474100
\(219\) 11.0000 0.743311
\(220\) −3.00000 −0.202260
\(221\) 21.0000 1.41261
\(222\) −1.00000 −0.0671156
\(223\) 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
\(226\) −18.0000 −1.19734
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −3.00000 −0.197814
\(231\) −3.00000 −0.197386
\(232\) −6.00000 −0.393919
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 7.00000 0.457604
\(235\) 12.0000 0.782794
\(236\) 6.00000 0.390567
\(237\) 8.00000 0.519656
\(238\) −3.00000 −0.194461
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 2.00000 0.128565
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) 7.00000 0.445399
\(248\) 10.0000 0.635001
\(249\) 9.00000 0.570352
\(250\) 1.00000 0.0632456
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −9.00000 −0.565825
\(254\) 7.00000 0.439219
\(255\) 3.00000 0.187867
\(256\) 1.00000 0.0625000
\(257\) 9.00000 0.561405 0.280702 0.959795i \(-0.409433\pi\)
0.280702 + 0.959795i \(0.409433\pi\)
\(258\) 4.00000 0.249029
\(259\) −1.00000 −0.0621370
\(260\) 7.00000 0.434122
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −3.00000 −0.184637
\(265\) −9.00000 −0.552866
\(266\) −1.00000 −0.0613139
\(267\) −9.00000 −0.550791
\(268\) −10.0000 −0.610847
\(269\) 9.00000 0.548740 0.274370 0.961624i \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) 1.00000 0.0608581
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −3.00000 −0.181902
\(273\) 7.00000 0.423659
\(274\) 18.0000 1.08742
\(275\) 3.00000 0.180907
\(276\) −3.00000 −0.180579
\(277\) 23.0000 1.38194 0.690968 0.722885i \(-0.257185\pi\)
0.690968 + 0.722885i \(0.257185\pi\)
\(278\) −14.0000 −0.839664
\(279\) −10.0000 −0.598684
\(280\) −1.00000 −0.0597614
\(281\) 27.0000 1.61068 0.805342 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) 12.0000 0.714590
\(283\) −13.0000 −0.772770 −0.386385 0.922338i \(-0.626276\pi\)
−0.386385 + 0.922338i \(0.626276\pi\)
\(284\) −6.00000 −0.356034
\(285\) 1.00000 0.0592349
\(286\) 21.0000 1.24176
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −8.00000 −0.470588
\(290\) 6.00000 0.352332
\(291\) −16.0000 −0.937937
\(292\) 11.0000 0.643726
\(293\) −15.0000 −0.876309 −0.438155 0.898900i \(-0.644368\pi\)
−0.438155 + 0.898900i \(0.644368\pi\)
\(294\) 6.00000 0.349927
\(295\) −6.00000 −0.349334
\(296\) −1.00000 −0.0581238
\(297\) 3.00000 0.174078
\(298\) 6.00000 0.347571
\(299\) 21.0000 1.21446
\(300\) 1.00000 0.0577350
\(301\) 4.00000 0.230556
\(302\) −23.0000 −1.32350
\(303\) −6.00000 −0.344691
\(304\) −1.00000 −0.0573539
\(305\) 10.0000 0.572598
\(306\) 3.00000 0.171499
\(307\) −22.0000 −1.25561 −0.627803 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) −3.00000 −0.170941
\(309\) −4.00000 −0.227552
\(310\) −10.0000 −0.567962
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 7.00000 0.396297
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 4.00000 0.225733
\(315\) 1.00000 0.0563436
\(316\) 8.00000 0.450035
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −9.00000 −0.504695
\(319\) 18.0000 1.00781
\(320\) −1.00000 −0.0559017
\(321\) 3.00000 0.167444
\(322\) −3.00000 −0.167183
\(323\) 3.00000 0.166924
\(324\) 1.00000 0.0555556
\(325\) −7.00000 −0.388290
\(326\) 1.00000 0.0553849
\(327\) −7.00000 −0.387101
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 3.00000 0.165145
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 9.00000 0.493939
\(333\) 1.00000 0.0547997
\(334\) −3.00000 −0.164153
\(335\) 10.0000 0.546358
\(336\) −1.00000 −0.0545545
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) −36.0000 −1.95814
\(339\) 18.0000 0.977626
\(340\) 3.00000 0.162698
\(341\) −30.0000 −1.62459
\(342\) 1.00000 0.0540738
\(343\) 13.0000 0.701934
\(344\) 4.00000 0.215666
\(345\) 3.00000 0.161515
\(346\) 9.00000 0.483843
\(347\) 36.0000 1.93258 0.966291 0.257454i \(-0.0828835\pi\)
0.966291 + 0.257454i \(0.0828835\pi\)
\(348\) 6.00000 0.321634
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 1.00000 0.0534522
\(351\) −7.00000 −0.373632
\(352\) −3.00000 −0.159901
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) −6.00000 −0.318896
\(355\) 6.00000 0.318447
\(356\) −9.00000 −0.476999
\(357\) 3.00000 0.158777
\(358\) 0 0
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 1.00000 0.0527046
\(361\) −18.0000 −0.947368
\(362\) 22.0000 1.15629
\(363\) −2.00000 −0.104973
\(364\) 7.00000 0.366900
\(365\) −11.0000 −0.575766
\(366\) 10.0000 0.522708
\(367\) 23.0000 1.20059 0.600295 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) −3.00000 −0.156386
\(369\) 0 0
\(370\) 1.00000 0.0519875
\(371\) −9.00000 −0.467257
\(372\) −10.0000 −0.518476
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 9.00000 0.465379
\(375\) −1.00000 −0.0516398
\(376\) 12.0000 0.618853
\(377\) −42.0000 −2.16311
\(378\) 1.00000 0.0514344
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 1.00000 0.0512989
\(381\) −7.00000 −0.358621
\(382\) −3.00000 −0.153493
\(383\) −9.00000 −0.459879 −0.229939 0.973205i \(-0.573853\pi\)
−0.229939 + 0.973205i \(0.573853\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 3.00000 0.152894
\(386\) 10.0000 0.508987
\(387\) −4.00000 −0.203331
\(388\) −16.0000 −0.812277
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) −7.00000 −0.354459
\(391\) 9.00000 0.455150
\(392\) 6.00000 0.303046
\(393\) −12.0000 −0.605320
\(394\) −3.00000 −0.151138
\(395\) −8.00000 −0.402524
\(396\) 3.00000 0.150756
\(397\) −4.00000 −0.200754 −0.100377 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) −20.0000 −1.00251
\(399\) 1.00000 0.0500626
\(400\) 1.00000 0.0500000
\(401\) 15.0000 0.749064 0.374532 0.927214i \(-0.377803\pi\)
0.374532 + 0.927214i \(0.377803\pi\)
\(402\) 10.0000 0.498755
\(403\) 70.0000 3.48695
\(404\) −6.00000 −0.298511
\(405\) −1.00000 −0.0496904
\(406\) 6.00000 0.297775
\(407\) 3.00000 0.148704
\(408\) 3.00000 0.148522
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) −4.00000 −0.197066
\(413\) −6.00000 −0.295241
\(414\) 3.00000 0.147442
\(415\) −9.00000 −0.441793
\(416\) 7.00000 0.343203
\(417\) 14.0000 0.685583
\(418\) 3.00000 0.146735
\(419\) −27.0000 −1.31904 −0.659518 0.751689i \(-0.729240\pi\)
−0.659518 + 0.751689i \(0.729240\pi\)
\(420\) 1.00000 0.0487950
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −2.00000 −0.0973585
\(423\) −12.0000 −0.583460
\(424\) −9.00000 −0.437079
\(425\) −3.00000 −0.145521
\(426\) 6.00000 0.290701
\(427\) 10.0000 0.483934
\(428\) 3.00000 0.145010
\(429\) −21.0000 −1.01389
\(430\) −4.00000 −0.192897
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 1.00000 0.0481125
\(433\) −31.0000 −1.48976 −0.744882 0.667196i \(-0.767494\pi\)
−0.744882 + 0.667196i \(0.767494\pi\)
\(434\) −10.0000 −0.480015
\(435\) −6.00000 −0.287678
\(436\) −7.00000 −0.335239
\(437\) 3.00000 0.143509
\(438\) −11.0000 −0.525600
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 3.00000 0.143019
\(441\) −6.00000 −0.285714
\(442\) −21.0000 −0.998868
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 1.00000 0.0474579
\(445\) 9.00000 0.426641
\(446\) −8.00000 −0.378811
\(447\) −6.00000 −0.283790
\(448\) −1.00000 −0.0472456
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) 18.0000 0.846649
\(453\) 23.0000 1.08063
\(454\) −24.0000 −1.12638
\(455\) −7.00000 −0.328165
\(456\) 1.00000 0.0468293
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 10.0000 0.467269
\(459\) −3.00000 −0.140028
\(460\) 3.00000 0.139876
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 3.00000 0.139573
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 6.00000 0.278543
\(465\) 10.0000 0.463739
\(466\) −18.0000 −0.833834
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) −7.00000 −0.323575
\(469\) 10.0000 0.461757
\(470\) −12.0000 −0.553519
\(471\) −4.00000 −0.184310
\(472\) −6.00000 −0.276172
\(473\) −12.0000 −0.551761
\(474\) −8.00000 −0.367452
\(475\) −1.00000 −0.0458831
\(476\) 3.00000 0.137505
\(477\) 9.00000 0.412082
\(478\) −12.0000 −0.548867
\(479\) −39.0000 −1.78196 −0.890978 0.454047i \(-0.849980\pi\)
−0.890978 + 0.454047i \(0.849980\pi\)
\(480\) 1.00000 0.0456435
\(481\) −7.00000 −0.319173
\(482\) −26.0000 −1.18427
\(483\) 3.00000 0.136505
\(484\) −2.00000 −0.0909091
\(485\) 16.0000 0.726523
\(486\) −1.00000 −0.0453609
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) 10.0000 0.452679
\(489\) −1.00000 −0.0452216
\(490\) −6.00000 −0.271052
\(491\) −9.00000 −0.406164 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) −18.0000 −0.810679
\(494\) −7.00000 −0.314945
\(495\) −3.00000 −0.134840
\(496\) −10.0000 −0.449013
\(497\) 6.00000 0.269137
\(498\) −9.00000 −0.403300
\(499\) 41.0000 1.83541 0.917706 0.397260i \(-0.130039\pi\)
0.917706 + 0.397260i \(0.130039\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 3.00000 0.134030
\(502\) −6.00000 −0.267793
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 1.00000 0.0445435
\(505\) 6.00000 0.266996
\(506\) 9.00000 0.400099
\(507\) 36.0000 1.59882
\(508\) −7.00000 −0.310575
\(509\) −33.0000 −1.46270 −0.731350 0.682003i \(-0.761109\pi\)
−0.731350 + 0.682003i \(0.761109\pi\)
\(510\) −3.00000 −0.132842
\(511\) −11.0000 −0.486611
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −9.00000 −0.396973
\(515\) 4.00000 0.176261
\(516\) −4.00000 −0.176090
\(517\) −36.0000 −1.58328
\(518\) 1.00000 0.0439375
\(519\) −9.00000 −0.395056
\(520\) −7.00000 −0.306970
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) −6.00000 −0.262613
\(523\) 44.0000 1.92399 0.961993 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) −12.0000 −0.524222
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 30.0000 1.30682
\(528\) 3.00000 0.130558
\(529\) −14.0000 −0.608696
\(530\) 9.00000 0.390935
\(531\) 6.00000 0.260378
\(532\) 1.00000 0.0433555
\(533\) 0 0
\(534\) 9.00000 0.389468
\(535\) −3.00000 −0.129701
\(536\) 10.0000 0.431934
\(537\) 0 0
\(538\) −9.00000 −0.388018
\(539\) −18.0000 −0.775315
\(540\) −1.00000 −0.0430331
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) −20.0000 −0.859074
\(543\) −22.0000 −0.944110
\(544\) 3.00000 0.128624
\(545\) 7.00000 0.299847
\(546\) −7.00000 −0.299572
\(547\) −37.0000 −1.58201 −0.791003 0.611812i \(-0.790441\pi\)
−0.791003 + 0.611812i \(0.790441\pi\)
\(548\) −18.0000 −0.768922
\(549\) −10.0000 −0.426790
\(550\) −3.00000 −0.127920
\(551\) −6.00000 −0.255609
\(552\) 3.00000 0.127688
\(553\) −8.00000 −0.340195
\(554\) −23.0000 −0.977176
\(555\) −1.00000 −0.0424476
\(556\) 14.0000 0.593732
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 10.0000 0.423334
\(559\) 28.0000 1.18427
\(560\) 1.00000 0.0422577
\(561\) −9.00000 −0.379980
\(562\) −27.0000 −1.13893
\(563\) −18.0000 −0.758610 −0.379305 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) −12.0000 −0.505291
\(565\) −18.0000 −0.757266
\(566\) 13.0000 0.546431
\(567\) −1.00000 −0.0419961
\(568\) 6.00000 0.251754
\(569\) −39.0000 −1.63497 −0.817483 0.575953i \(-0.804631\pi\)
−0.817483 + 0.575953i \(0.804631\pi\)
\(570\) −1.00000 −0.0418854
\(571\) 14.0000 0.585882 0.292941 0.956131i \(-0.405366\pi\)
0.292941 + 0.956131i \(0.405366\pi\)
\(572\) −21.0000 −0.878054
\(573\) 3.00000 0.125327
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 8.00000 0.332756
\(579\) −10.0000 −0.415586
\(580\) −6.00000 −0.249136
\(581\) −9.00000 −0.373383
\(582\) 16.0000 0.663221
\(583\) 27.0000 1.11823
\(584\) −11.0000 −0.455183
\(585\) 7.00000 0.289414
\(586\) 15.0000 0.619644
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) −6.00000 −0.247436
\(589\) 10.0000 0.412043
\(590\) 6.00000 0.247016
\(591\) 3.00000 0.123404
\(592\) 1.00000 0.0410997
\(593\) −36.0000 −1.47834 −0.739171 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) −3.00000 −0.123091
\(595\) −3.00000 −0.122988
\(596\) −6.00000 −0.245770
\(597\) 20.0000 0.818546
\(598\) −21.0000 −0.858754
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) −4.00000 −0.163028
\(603\) −10.0000 −0.407231
\(604\) 23.0000 0.935857
\(605\) 2.00000 0.0813116
\(606\) 6.00000 0.243733
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 1.00000 0.0405554
\(609\) −6.00000 −0.243132
\(610\) −10.0000 −0.404888
\(611\) 84.0000 3.39828
\(612\) −3.00000 −0.121268
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 22.0000 0.887848
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 4.00000 0.160904
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 10.0000 0.401610
\(621\) −3.00000 −0.120386
\(622\) 24.0000 0.962312
\(623\) 9.00000 0.360577
\(624\) −7.00000 −0.280224
\(625\) 1.00000 0.0400000
\(626\) −8.00000 −0.319744
\(627\) −3.00000 −0.119808
\(628\) −4.00000 −0.159617
\(629\) −3.00000 −0.119618
\(630\) −1.00000 −0.0398410
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) −8.00000 −0.318223
\(633\) 2.00000 0.0794929
\(634\) 18.0000 0.714871
\(635\) 7.00000 0.277787
\(636\) 9.00000 0.356873
\(637\) 42.0000 1.66410
\(638\) −18.0000 −0.712627
\(639\) −6.00000 −0.237356
\(640\) 1.00000 0.0395285
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) −3.00000 −0.118401
\(643\) −31.0000 −1.22252 −0.611260 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) 3.00000 0.118217
\(645\) 4.00000 0.157500
\(646\) −3.00000 −0.118033
\(647\) 45.0000 1.76913 0.884566 0.466415i \(-0.154454\pi\)
0.884566 + 0.466415i \(0.154454\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 18.0000 0.706562
\(650\) 7.00000 0.274563
\(651\) 10.0000 0.391931
\(652\) −1.00000 −0.0391630
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 7.00000 0.273722
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) 11.0000 0.429151
\(658\) −12.0000 −0.467809
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) −3.00000 −0.116775
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 28.0000 1.08825
\(663\) 21.0000 0.815572
\(664\) −9.00000 −0.349268
\(665\) −1.00000 −0.0387783
\(666\) −1.00000 −0.0387492
\(667\) −18.0000 −0.696963
\(668\) 3.00000 0.116073
\(669\) 8.00000 0.309298
\(670\) −10.0000 −0.386334
\(671\) −30.0000 −1.15814
\(672\) 1.00000 0.0385758
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 13.0000 0.500741
\(675\) 1.00000 0.0384900
\(676\) 36.0000 1.38462
\(677\) 3.00000 0.115299 0.0576497 0.998337i \(-0.481639\pi\)
0.0576497 + 0.998337i \(0.481639\pi\)
\(678\) −18.0000 −0.691286
\(679\) 16.0000 0.614024
\(680\) −3.00000 −0.115045
\(681\) 24.0000 0.919682
\(682\) 30.0000 1.14876
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 18.0000 0.687745
\(686\) −13.0000 −0.496342
\(687\) −10.0000 −0.381524
\(688\) −4.00000 −0.152499
\(689\) −63.0000 −2.40011
\(690\) −3.00000 −0.114208
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) −9.00000 −0.342129
\(693\) −3.00000 −0.113961
\(694\) −36.0000 −1.36654
\(695\) −14.0000 −0.531050
\(696\) −6.00000 −0.227429
\(697\) 0 0
\(698\) −8.00000 −0.302804
\(699\) 18.0000 0.680823
\(700\) −1.00000 −0.0377964
\(701\) −48.0000 −1.81293 −0.906467 0.422276i \(-0.861231\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) 7.00000 0.264198
\(703\) −1.00000 −0.0377157
\(704\) 3.00000 0.113067
\(705\) 12.0000 0.451946
\(706\) −18.0000 −0.677439
\(707\) 6.00000 0.225653
\(708\) 6.00000 0.225494
\(709\) −25.0000 −0.938895 −0.469447 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) −6.00000 −0.225176
\(711\) 8.00000 0.300023
\(712\) 9.00000 0.337289
\(713\) 30.0000 1.12351
\(714\) −3.00000 −0.112272
\(715\) 21.0000 0.785355
\(716\) 0 0
\(717\) 12.0000 0.448148
\(718\) 12.0000 0.447836
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 4.00000 0.148968
\(722\) 18.0000 0.669891
\(723\) 26.0000 0.966950
\(724\) −22.0000 −0.817624
\(725\) 6.00000 0.222834
\(726\) 2.00000 0.0742270
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) −7.00000 −0.259437
\(729\) 1.00000 0.0370370
\(730\) 11.0000 0.407128
\(731\) 12.0000 0.443836
\(732\) −10.0000 −0.369611
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −23.0000 −0.848945
\(735\) 6.00000 0.221313
\(736\) 3.00000 0.110581
\(737\) −30.0000 −1.10506
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 7.00000 0.257151
\(742\) 9.00000 0.330400
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 10.0000 0.366618
\(745\) 6.00000 0.219823
\(746\) −14.0000 −0.512576
\(747\) 9.00000 0.329293
\(748\) −9.00000 −0.329073
\(749\) −3.00000 −0.109618
\(750\) 1.00000 0.0365148
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −12.0000 −0.437595
\(753\) 6.00000 0.218652
\(754\) 42.0000 1.52955
\(755\) −23.0000 −0.837056
\(756\) −1.00000 −0.0363696
\(757\) −13.0000 −0.472493 −0.236247 0.971693i \(-0.575917\pi\)
−0.236247 + 0.971693i \(0.575917\pi\)
\(758\) −26.0000 −0.944363
\(759\) −9.00000 −0.326679
\(760\) −1.00000 −0.0362738
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 7.00000 0.253583
\(763\) 7.00000 0.253417
\(764\) 3.00000 0.108536
\(765\) 3.00000 0.108465
\(766\) 9.00000 0.325183
\(767\) −42.0000 −1.51653
\(768\) 1.00000 0.0360844
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) −3.00000 −0.108112
\(771\) 9.00000 0.324127
\(772\) −10.0000 −0.359908
\(773\) 15.0000 0.539513 0.269756 0.962929i \(-0.413057\pi\)
0.269756 + 0.962929i \(0.413057\pi\)
\(774\) 4.00000 0.143777
\(775\) −10.0000 −0.359211
\(776\) 16.0000 0.574367
\(777\) −1.00000 −0.0358748
\(778\) −18.0000 −0.645331
\(779\) 0 0
\(780\) 7.00000 0.250640
\(781\) −18.0000 −0.644091
\(782\) −9.00000 −0.321839
\(783\) 6.00000 0.214423
\(784\) −6.00000 −0.214286
\(785\) 4.00000 0.142766
\(786\) 12.0000 0.428026
\(787\) 2.00000 0.0712923 0.0356462 0.999364i \(-0.488651\pi\)
0.0356462 + 0.999364i \(0.488651\pi\)
\(788\) 3.00000 0.106871
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) −18.0000 −0.640006
\(792\) −3.00000 −0.106600
\(793\) 70.0000 2.48577
\(794\) 4.00000 0.141955
\(795\) −9.00000 −0.319197
\(796\) 20.0000 0.708881
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) −1.00000 −0.0353996
\(799\) 36.0000 1.27359
\(800\) −1.00000 −0.0353553
\(801\) −9.00000 −0.317999
\(802\) −15.0000 −0.529668
\(803\) 33.0000 1.16454
\(804\) −10.0000 −0.352673
\(805\) −3.00000 −0.105736
\(806\) −70.0000 −2.46564
\(807\) 9.00000 0.316815
\(808\) 6.00000 0.211079
\(809\) −21.0000 −0.738321 −0.369160 0.929366i \(-0.620355\pi\)
−0.369160 + 0.929366i \(0.620355\pi\)
\(810\) 1.00000 0.0351364
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) −6.00000 −0.210559
\(813\) 20.0000 0.701431
\(814\) −3.00000 −0.105150
\(815\) 1.00000 0.0350285
\(816\) −3.00000 −0.105021
\(817\) 4.00000 0.139942
\(818\) 4.00000 0.139857
\(819\) 7.00000 0.244600
\(820\) 0 0
\(821\) −9.00000 −0.314102 −0.157051 0.987590i \(-0.550199\pi\)
−0.157051 + 0.987590i \(0.550199\pi\)
\(822\) 18.0000 0.627822
\(823\) −7.00000 −0.244005 −0.122002 0.992530i \(-0.538932\pi\)
−0.122002 + 0.992530i \(0.538932\pi\)
\(824\) 4.00000 0.139347
\(825\) 3.00000 0.104447
\(826\) 6.00000 0.208767
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −3.00000 −0.104257
\(829\) 29.0000 1.00721 0.503606 0.863934i \(-0.332006\pi\)
0.503606 + 0.863934i \(0.332006\pi\)
\(830\) 9.00000 0.312395
\(831\) 23.0000 0.797861
\(832\) −7.00000 −0.242681
\(833\) 18.0000 0.623663
\(834\) −14.0000 −0.484780
\(835\) −3.00000 −0.103819
\(836\) −3.00000 −0.103757
\(837\) −10.0000 −0.345651
\(838\) 27.0000 0.932700
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) 27.0000 0.929929
\(844\) 2.00000 0.0688428
\(845\) −36.0000 −1.23844
\(846\) 12.0000 0.412568
\(847\) 2.00000 0.0687208
\(848\) 9.00000 0.309061
\(849\) −13.0000 −0.446159
\(850\) 3.00000 0.102899
\(851\) −3.00000 −0.102839
\(852\) −6.00000 −0.205557
\(853\) −19.0000 −0.650548 −0.325274 0.945620i \(-0.605456\pi\)
−0.325274 + 0.945620i \(0.605456\pi\)
\(854\) −10.0000 −0.342193
\(855\) 1.00000 0.0341993
\(856\) −3.00000 −0.102538
\(857\) −9.00000 −0.307434 −0.153717 0.988115i \(-0.549124\pi\)
−0.153717 + 0.988115i \(0.549124\pi\)
\(858\) 21.0000 0.716928
\(859\) 5.00000 0.170598 0.0852989 0.996355i \(-0.472815\pi\)
0.0852989 + 0.996355i \(0.472815\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 3.00000 0.102180
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 9.00000 0.306009
\(866\) 31.0000 1.05342
\(867\) −8.00000 −0.271694
\(868\) 10.0000 0.339422
\(869\) 24.0000 0.814144
\(870\) 6.00000 0.203419
\(871\) 70.0000 2.37186
\(872\) 7.00000 0.237050
\(873\) −16.0000 −0.541518
\(874\) −3.00000 −0.101477
\(875\) 1.00000 0.0338062
\(876\) 11.0000 0.371656
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) −20.0000 −0.674967
\(879\) −15.0000 −0.505937
\(880\) −3.00000 −0.101130
\(881\) −12.0000 −0.404290 −0.202145 0.979356i \(-0.564791\pi\)
−0.202145 + 0.979356i \(0.564791\pi\)
\(882\) 6.00000 0.202031
\(883\) 35.0000 1.17784 0.588922 0.808190i \(-0.299553\pi\)
0.588922 + 0.808190i \(0.299553\pi\)
\(884\) 21.0000 0.706306
\(885\) −6.00000 −0.201688
\(886\) 12.0000 0.403148
\(887\) −18.0000 −0.604381 −0.302190 0.953248i \(-0.597718\pi\)
−0.302190 + 0.953248i \(0.597718\pi\)
\(888\) −1.00000 −0.0335578
\(889\) 7.00000 0.234772
\(890\) −9.00000 −0.301681
\(891\) 3.00000 0.100504
\(892\) 8.00000 0.267860
\(893\) 12.0000 0.401565
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 21.0000 0.701170
\(898\) 30.0000 1.00111
\(899\) −60.0000 −2.00111
\(900\) 1.00000 0.0333333
\(901\) −27.0000 −0.899500
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) −18.0000 −0.598671
\(905\) 22.0000 0.731305
\(906\) −23.0000 −0.764124
\(907\) −37.0000 −1.22856 −0.614282 0.789086i \(-0.710554\pi\)
−0.614282 + 0.789086i \(0.710554\pi\)
\(908\) 24.0000 0.796468
\(909\) −6.00000 −0.199007
\(910\) 7.00000 0.232048
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 27.0000 0.893570
\(914\) −8.00000 −0.264616
\(915\) 10.0000 0.330590
\(916\) −10.0000 −0.330409
\(917\) 12.0000 0.396275
\(918\) 3.00000 0.0990148
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) −3.00000 −0.0989071
\(921\) −22.0000 −0.724925
\(922\) −18.0000 −0.592798
\(923\) 42.0000 1.38245
\(924\) −3.00000 −0.0986928
\(925\) 1.00000 0.0328798
\(926\) 4.00000 0.131448
\(927\) −4.00000 −0.131377
\(928\) −6.00000 −0.196960
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) −10.0000 −0.327913
\(931\) 6.00000 0.196642
\(932\) 18.0000 0.589610
\(933\) −24.0000 −0.785725
\(934\) 0 0
\(935\) 9.00000 0.294331
\(936\) 7.00000 0.228802
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) −10.0000 −0.326512
\(939\) 8.00000 0.261070
\(940\) 12.0000 0.391397
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 4.00000 0.130327
\(943\) 0 0
\(944\) 6.00000 0.195283
\(945\) 1.00000 0.0325300
\(946\) 12.0000 0.390154
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 8.00000 0.259828
\(949\) −77.0000 −2.49953
\(950\) 1.00000 0.0324443
\(951\) −18.0000 −0.583690
\(952\) −3.00000 −0.0972306
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −9.00000 −0.291386
\(955\) −3.00000 −0.0970777
\(956\) 12.0000 0.388108
\(957\) 18.0000 0.581857
\(958\) 39.0000 1.26003
\(959\) 18.0000 0.581250
\(960\) −1.00000 −0.0322749
\(961\) 69.0000 2.22581
\(962\) 7.00000 0.225689
\(963\) 3.00000 0.0966736
\(964\) 26.0000 0.837404
\(965\) 10.0000 0.321911
\(966\) −3.00000 −0.0965234
\(967\) −22.0000 −0.707472 −0.353736 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 2.00000 0.0642824
\(969\) 3.00000 0.0963739
\(970\) −16.0000 −0.513729
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000 0.0320750
\(973\) −14.0000 −0.448819
\(974\) 22.0000 0.704925
\(975\) −7.00000 −0.224179
\(976\) −10.0000 −0.320092
\(977\) 9.00000 0.287936 0.143968 0.989582i \(-0.454014\pi\)
0.143968 + 0.989582i \(0.454014\pi\)
\(978\) 1.00000 0.0319765
\(979\) −27.0000 −0.862924
\(980\) 6.00000 0.191663
\(981\) −7.00000 −0.223493
\(982\) 9.00000 0.287202
\(983\) 42.0000 1.33959 0.669796 0.742545i \(-0.266382\pi\)
0.669796 + 0.742545i \(0.266382\pi\)
\(984\) 0 0
\(985\) −3.00000 −0.0955879
\(986\) 18.0000 0.573237
\(987\) 12.0000 0.381964
\(988\) 7.00000 0.222700
\(989\) 12.0000 0.381578
\(990\) 3.00000 0.0953463
\(991\) −52.0000 −1.65183 −0.825917 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 10.0000 0.317500
\(993\) −28.0000 −0.888553
\(994\) −6.00000 −0.190308
\(995\) −20.0000 −0.634043
\(996\) 9.00000 0.285176
\(997\) −1.00000 −0.0316703 −0.0158352 0.999875i \(-0.505041\pi\)
−0.0158352 + 0.999875i \(0.505041\pi\)
\(998\) −41.0000 −1.29783
\(999\) 1.00000 0.0316386
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 1110.2.a.e.1.1 1
3.2 odd 2 3330.2.a.u.1.1 1
4.3 odd 2 8880.2.a.f.1.1 1
5.4 even 2 5550.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.e.1.1 1 1.1 even 1 trivial
3330.2.a.u.1.1 1 3.2 odd 2
5550.2.a.ba.1.1 1 5.4 even 2
8880.2.a.f.1.1 1 4.3 odd 2