Properties

Label 1110.2.a.f.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $0$
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: \(0\)
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 \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} +5.00000 q^{22} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +9.00000 q^{29} +1.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} -5.00000 q^{33} -1.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} -6.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} +9.00000 q^{41} -1.00000 q^{42} +1.00000 q^{43} -5.00000 q^{44} -1.00000 q^{45} +1.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} +2.00000 q^{52} +9.00000 q^{53} -1.00000 q^{54} +5.00000 q^{55} -1.00000 q^{56} +6.00000 q^{57} -9.00000 q^{58} -10.0000 q^{59} -1.00000 q^{60} -5.00000 q^{61} -3.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +5.00000 q^{66} +16.0000 q^{67} +1.00000 q^{68} +1.00000 q^{70} -1.00000 q^{72} +12.0000 q^{73} +1.00000 q^{74} +1.00000 q^{75} +6.00000 q^{76} -5.00000 q^{77} -2.00000 q^{78} -12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.00000 q^{82} +2.00000 q^{83} +1.00000 q^{84} -1.00000 q^{85} -1.00000 q^{86} +9.00000 q^{87} +5.00000 q^{88} -2.00000 q^{89} +1.00000 q^{90} +2.00000 q^{91} +3.00000 q^{93} -6.00000 q^{95} -1.00000 q^{96} +17.0000 q^{97} +6.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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 \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) 5.00000 1.06600
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 1.00000 0.182574
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.00000 −0.870388
\(34\) −1.00000 −0.171499
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) −6.00000 −0.973329
\(39\) 2.00000 0.320256
\(40\) 1.00000 0.158114
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) −1.00000 −0.154303
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) −5.00000 −0.753778
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) −1.00000 −0.141421
\(51\) 1.00000 0.140028
\(52\) 2.00000 0.277350
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −1.00000 −0.136083
\(55\) 5.00000 0.674200
\(56\) −1.00000 −0.133631
\(57\) 6.00000 0.794719
\(58\) −9.00000 −1.18176
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) −1.00000 −0.129099
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) −3.00000 −0.381000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 5.00000 0.615457
\(67\) 16.0000 1.95471 0.977356 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) 1.00000 0.121268
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 1.00000 0.116248
\(75\) 1.00000 0.115470
\(76\) 6.00000 0.688247
\(77\) −5.00000 −0.569803
\(78\) −2.00000 −0.226455
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −9.00000 −0.993884
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 1.00000 0.109109
\(85\) −1.00000 −0.108465
\(86\) −1.00000 −0.107833
\(87\) 9.00000 0.964901
\(88\) 5.00000 0.533002
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 3.00000 0.311086
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) −1.00000 −0.102062
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 6.00000 0.606092
\(99\) −5.00000 −0.502519
\(100\) 1.00000 0.100000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −2.00000 −0.196116
\(105\) −1.00000 −0.0975900
\(106\) −9.00000 −0.874157
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 1.00000 0.0962250
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) −5.00000 −0.476731
\(111\) −1.00000 −0.0949158
\(112\) 1.00000 0.0944911
\(113\) −13.0000 −1.22294 −0.611469 0.791269i \(-0.709421\pi\)
−0.611469 + 0.791269i \(0.709421\pi\)
\(114\) −6.00000 −0.561951
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) 2.00000 0.184900
\(118\) 10.0000 0.920575
\(119\) 1.00000 0.0916698
\(120\) 1.00000 0.0912871
\(121\) 14.0000 1.27273
\(122\) 5.00000 0.452679
\(123\) 9.00000 0.811503
\(124\) 3.00000 0.269408
\(125\) −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
\(129\) 1.00000 0.0880451
\(130\) 2.00000 0.175412
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) −5.00000 −0.435194
\(133\) 6.00000 0.520266
\(134\) −16.0000 −1.38219
\(135\) −1.00000 −0.0860663
\(136\) −1.00000 −0.0857493
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) −19.0000 −1.61156 −0.805779 0.592216i \(-0.798253\pi\)
−0.805779 + 0.592216i \(0.798253\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 0 0
\(143\) −10.0000 −0.836242
\(144\) 1.00000 0.0833333
\(145\) −9.00000 −0.747409
\(146\) −12.0000 −0.993127
\(147\) −6.00000 −0.494872
\(148\) −1.00000 −0.0821995
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) −6.00000 −0.486664
\(153\) 1.00000 0.0808452
\(154\) 5.00000 0.402911
\(155\) −3.00000 −0.240966
\(156\) 2.00000 0.160128
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 12.0000 0.954669
\(159\) 9.00000 0.713746
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 1.00000 0.0783260 0.0391630 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 9.00000 0.702782
\(165\) 5.00000 0.389249
\(166\) −2.00000 −0.155230
\(167\) 20.0000 1.54765 0.773823 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 1.00000 0.0766965
\(171\) 6.00000 0.458831
\(172\) 1.00000 0.0762493
\(173\) −7.00000 −0.532200 −0.266100 0.963945i \(-0.585735\pi\)
−0.266100 + 0.963945i \(0.585735\pi\)
\(174\) −9.00000 −0.682288
\(175\) 1.00000 0.0755929
\(176\) −5.00000 −0.376889
\(177\) −10.0000 −0.751646
\(178\) 2.00000 0.149906
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −2.00000 −0.148250
\(183\) −5.00000 −0.369611
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) −3.00000 −0.219971
\(187\) −5.00000 −0.365636
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 6.00000 0.435286
\(191\) 11.0000 0.795932 0.397966 0.917400i \(-0.369716\pi\)
0.397966 + 0.917400i \(0.369716\pi\)
\(192\) 1.00000 0.0721688
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −17.0000 −1.22053
\(195\) −2.00000 −0.143223
\(196\) −6.00000 −0.428571
\(197\) 14.0000 0.997459 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) 5.00000 0.355335
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 16.0000 1.12855
\(202\) 12.0000 0.844317
\(203\) 9.00000 0.631676
\(204\) 1.00000 0.0700140
\(205\) −9.00000 −0.628587
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −30.0000 −2.07514
\(210\) 1.00000 0.0690066
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 9.00000 0.618123
\(213\) 0 0
\(214\) −18.0000 −1.23045
\(215\) −1.00000 −0.0681994
\(216\) −1.00000 −0.0680414
\(217\) 3.00000 0.203653
\(218\) 5.00000 0.338643
\(219\) 12.0000 0.810885
\(220\) 5.00000 0.337100
\(221\) 2.00000 0.134535
\(222\) 1.00000 0.0671156
\(223\) −5.00000 −0.334825 −0.167412 0.985887i \(-0.553541\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 13.0000 0.864747
\(227\) −7.00000 −0.464606 −0.232303 0.972643i \(-0.574626\pi\)
−0.232303 + 0.972643i \(0.574626\pi\)
\(228\) 6.00000 0.397360
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) −9.00000 −0.590879
\(233\) −8.00000 −0.524097 −0.262049 0.965055i \(-0.584398\pi\)
−0.262049 + 0.965055i \(0.584398\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) −12.0000 −0.779484
\(238\) −1.00000 −0.0648204
\(239\) −21.0000 −1.35838 −0.679189 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) −14.0000 −0.899954
\(243\) 1.00000 0.0641500
\(244\) −5.00000 −0.320092
\(245\) 6.00000 0.383326
\(246\) −9.00000 −0.573819
\(247\) 12.0000 0.763542
\(248\) −3.00000 −0.190500
\(249\) 2.00000 0.126745
\(250\) 1.00000 0.0632456
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 0 0
\(255\) −1.00000 −0.0626224
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −1.00000 −0.0622573
\(259\) −1.00000 −0.0621370
\(260\) −2.00000 −0.124035
\(261\) 9.00000 0.557086
\(262\) −10.0000 −0.617802
\(263\) −15.0000 −0.924940 −0.462470 0.886635i \(-0.653037\pi\)
−0.462470 + 0.886635i \(0.653037\pi\)
\(264\) 5.00000 0.307729
\(265\) −9.00000 −0.552866
\(266\) −6.00000 −0.367884
\(267\) −2.00000 −0.122398
\(268\) 16.0000 0.977356
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 1.00000 0.0608581
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 1.00000 0.0606339
\(273\) 2.00000 0.121046
\(274\) 14.0000 0.845771
\(275\) −5.00000 −0.301511
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 19.0000 1.13954
\(279\) 3.00000 0.179605
\(280\) 1.00000 0.0597614
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 0 0
\(285\) −6.00000 −0.355409
\(286\) 10.0000 0.591312
\(287\) 9.00000 0.531253
\(288\) −1.00000 −0.0589256
\(289\) −16.0000 −0.941176
\(290\) 9.00000 0.528498
\(291\) 17.0000 0.996558
\(292\) 12.0000 0.702247
\(293\) 19.0000 1.10999 0.554996 0.831853i \(-0.312720\pi\)
0.554996 + 0.831853i \(0.312720\pi\)
\(294\) 6.00000 0.349927
\(295\) 10.0000 0.582223
\(296\) 1.00000 0.0581238
\(297\) −5.00000 −0.290129
\(298\) −14.0000 −0.810998
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 1.00000 0.0576390
\(302\) −18.0000 −1.03578
\(303\) −12.0000 −0.689382
\(304\) 6.00000 0.344124
\(305\) 5.00000 0.286299
\(306\) −1.00000 −0.0571662
\(307\) −30.0000 −1.71219 −0.856095 0.516818i \(-0.827116\pi\)
−0.856095 + 0.516818i \(0.827116\pi\)
\(308\) −5.00000 −0.284901
\(309\) −4.00000 −0.227552
\(310\) 3.00000 0.170389
\(311\) −25.0000 −1.41762 −0.708810 0.705399i \(-0.750768\pi\)
−0.708810 + 0.705399i \(0.750768\pi\)
\(312\) −2.00000 −0.113228
\(313\) −18.0000 −1.01742 −0.508710 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) −7.00000 −0.395033
\(315\) −1.00000 −0.0563436
\(316\) −12.0000 −0.675053
\(317\) −7.00000 −0.393159 −0.196580 0.980488i \(-0.562983\pi\)
−0.196580 + 0.980488i \(0.562983\pi\)
\(318\) −9.00000 −0.504695
\(319\) −45.0000 −2.51952
\(320\) −1.00000 −0.0559017
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −1.00000 −0.0553849
\(327\) −5.00000 −0.276501
\(328\) −9.00000 −0.496942
\(329\) 0 0
\(330\) −5.00000 −0.275241
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 2.00000 0.109764
\(333\) −1.00000 −0.0547997
\(334\) −20.0000 −1.09435
\(335\) −16.0000 −0.874173
\(336\) 1.00000 0.0545545
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 9.00000 0.489535
\(339\) −13.0000 −0.706063
\(340\) −1.00000 −0.0542326
\(341\) −15.0000 −0.812296
\(342\) −6.00000 −0.324443
\(343\) −13.0000 −0.701934
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) 7.00000 0.376322
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 9.00000 0.482451
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 2.00000 0.106752
\(352\) 5.00000 0.266501
\(353\) 1.00000 0.0532246 0.0266123 0.999646i \(-0.491528\pi\)
0.0266123 + 0.999646i \(0.491528\pi\)
\(354\) 10.0000 0.531494
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 1.00000 0.0529256
\(358\) 16.0000 0.845626
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 1.00000 0.0527046
\(361\) 17.0000 0.894737
\(362\) −14.0000 −0.735824
\(363\) 14.0000 0.734809
\(364\) 2.00000 0.104828
\(365\) −12.0000 −0.628109
\(366\) 5.00000 0.261354
\(367\) −23.0000 −1.20059 −0.600295 0.799779i \(-0.704950\pi\)
−0.600295 + 0.799779i \(0.704950\pi\)
\(368\) 0 0
\(369\) 9.00000 0.468521
\(370\) −1.00000 −0.0519875
\(371\) 9.00000 0.467257
\(372\) 3.00000 0.155543
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 5.00000 0.258544
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 18.0000 0.927047
\(378\) −1.00000 −0.0514344
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −6.00000 −0.307794
\(381\) 0 0
\(382\) −11.0000 −0.562809
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 5.00000 0.254824
\(386\) 2.00000 0.101797
\(387\) 1.00000 0.0508329
\(388\) 17.0000 0.863044
\(389\) 19.0000 0.963338 0.481669 0.876353i \(-0.340031\pi\)
0.481669 + 0.876353i \(0.340031\pi\)
\(390\) 2.00000 0.101274
\(391\) 0 0
\(392\) 6.00000 0.303046
\(393\) 10.0000 0.504433
\(394\) −14.0000 −0.705310
\(395\) 12.0000 0.603786
\(396\) −5.00000 −0.251259
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 12.0000 0.601506
\(399\) 6.00000 0.300376
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −16.0000 −0.798007
\(403\) 6.00000 0.298881
\(404\) −12.0000 −0.597022
\(405\) −1.00000 −0.0496904
\(406\) −9.00000 −0.446663
\(407\) 5.00000 0.247841
\(408\) −1.00000 −0.0495074
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 9.00000 0.444478
\(411\) −14.0000 −0.690569
\(412\) −4.00000 −0.197066
\(413\) −10.0000 −0.492068
\(414\) 0 0
\(415\) −2.00000 −0.0981761
\(416\) −2.00000 −0.0980581
\(417\) −19.0000 −0.930434
\(418\) 30.0000 1.46735
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 5.00000 0.243396
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) −5.00000 −0.241967
\(428\) 18.0000 0.870063
\(429\) −10.0000 −0.482805
\(430\) 1.00000 0.0482243
\(431\) 15.0000 0.722525 0.361262 0.932464i \(-0.382346\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(432\) 1.00000 0.0481125
\(433\) 36.0000 1.73005 0.865025 0.501729i \(-0.167303\pi\)
0.865025 + 0.501729i \(0.167303\pi\)
\(434\) −3.00000 −0.144005
\(435\) −9.00000 −0.431517
\(436\) −5.00000 −0.239457
\(437\) 0 0
\(438\) −12.0000 −0.573382
\(439\) −27.0000 −1.28864 −0.644320 0.764756i \(-0.722859\pi\)
−0.644320 + 0.764756i \(0.722859\pi\)
\(440\) −5.00000 −0.238366
\(441\) −6.00000 −0.285714
\(442\) −2.00000 −0.0951303
\(443\) 2.00000 0.0950229 0.0475114 0.998871i \(-0.484871\pi\)
0.0475114 + 0.998871i \(0.484871\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 2.00000 0.0948091
\(446\) 5.00000 0.236757
\(447\) 14.0000 0.662177
\(448\) 1.00000 0.0472456
\(449\) 24.0000 1.13263 0.566315 0.824189i \(-0.308369\pi\)
0.566315 + 0.824189i \(0.308369\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −45.0000 −2.11897
\(452\) −13.0000 −0.611469
\(453\) 18.0000 0.845714
\(454\) 7.00000 0.328526
\(455\) −2.00000 −0.0937614
\(456\) −6.00000 −0.280976
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) 0 0
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) 35.0000 1.63011 0.815056 0.579382i \(-0.196706\pi\)
0.815056 + 0.579382i \(0.196706\pi\)
\(462\) 5.00000 0.232621
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) 9.00000 0.417815
\(465\) −3.00000 −0.139122
\(466\) 8.00000 0.370593
\(467\) −21.0000 −0.971764 −0.485882 0.874024i \(-0.661502\pi\)
−0.485882 + 0.874024i \(0.661502\pi\)
\(468\) 2.00000 0.0924500
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 7.00000 0.322543
\(472\) 10.0000 0.460287
\(473\) −5.00000 −0.229900
\(474\) 12.0000 0.551178
\(475\) 6.00000 0.275299
\(476\) 1.00000 0.0458349
\(477\) 9.00000 0.412082
\(478\) 21.0000 0.960518
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 1.00000 0.0456435
\(481\) −2.00000 −0.0911922
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) −17.0000 −0.771930
\(486\) −1.00000 −0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 5.00000 0.226339
\(489\) 1.00000 0.0452216
\(490\) −6.00000 −0.271052
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 9.00000 0.405751
\(493\) 9.00000 0.405340
\(494\) −12.0000 −0.539906
\(495\) 5.00000 0.224733
\(496\) 3.00000 0.134704
\(497\) 0 0
\(498\) −2.00000 −0.0896221
\(499\) −44.0000 −1.96971 −0.984855 0.173379i \(-0.944532\pi\)
−0.984855 + 0.173379i \(0.944532\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 20.0000 0.893534
\(502\) 24.0000 1.07117
\(503\) 4.00000 0.178351 0.0891756 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 1.00000 0.0442807
\(511\) 12.0000 0.530849
\(512\) −1.00000 −0.0441942
\(513\) 6.00000 0.264906
\(514\) 18.0000 0.793946
\(515\) 4.00000 0.176261
\(516\) 1.00000 0.0440225
\(517\) 0 0
\(518\) 1.00000 0.0439375
\(519\) −7.00000 −0.307266
\(520\) 2.00000 0.0877058
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) −9.00000 −0.393919
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 10.0000 0.436852
\(525\) 1.00000 0.0436436
\(526\) 15.0000 0.654031
\(527\) 3.00000 0.130682
\(528\) −5.00000 −0.217597
\(529\) −23.0000 −1.00000
\(530\) 9.00000 0.390935
\(531\) −10.0000 −0.433963
\(532\) 6.00000 0.260133
\(533\) 18.0000 0.779667
\(534\) 2.00000 0.0865485
\(535\) −18.0000 −0.778208
\(536\) −16.0000 −0.691095
\(537\) −16.0000 −0.690451
\(538\) −24.0000 −1.03471
\(539\) 30.0000 1.29219
\(540\) −1.00000 −0.0430331
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 16.0000 0.687259
\(543\) 14.0000 0.600798
\(544\) −1.00000 −0.0428746
\(545\) 5.00000 0.214176
\(546\) −2.00000 −0.0855921
\(547\) 17.0000 0.726868 0.363434 0.931620i \(-0.381604\pi\)
0.363434 + 0.931620i \(0.381604\pi\)
\(548\) −14.0000 −0.598050
\(549\) −5.00000 −0.213395
\(550\) 5.00000 0.213201
\(551\) 54.0000 2.30048
\(552\) 0 0
\(553\) −12.0000 −0.510292
\(554\) 8.00000 0.339887
\(555\) 1.00000 0.0424476
\(556\) −19.0000 −0.805779
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −3.00000 −0.127000
\(559\) 2.00000 0.0845910
\(560\) −1.00000 −0.0422577
\(561\) −5.00000 −0.211100
\(562\) 2.00000 0.0843649
\(563\) 13.0000 0.547885 0.273942 0.961746i \(-0.411672\pi\)
0.273942 + 0.961746i \(0.411672\pi\)
\(564\) 0 0
\(565\) 13.0000 0.546914
\(566\) 28.0000 1.17693
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 38.0000 1.59304 0.796521 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 6.00000 0.251312
\(571\) 35.0000 1.46470 0.732352 0.680926i \(-0.238422\pi\)
0.732352 + 0.680926i \(0.238422\pi\)
\(572\) −10.0000 −0.418121
\(573\) 11.0000 0.459532
\(574\) −9.00000 −0.375653
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 16.0000 0.665512
\(579\) −2.00000 −0.0831172
\(580\) −9.00000 −0.373705
\(581\) 2.00000 0.0829740
\(582\) −17.0000 −0.704673
\(583\) −45.0000 −1.86371
\(584\) −12.0000 −0.496564
\(585\) −2.00000 −0.0826898
\(586\) −19.0000 −0.784883
\(587\) 3.00000 0.123823 0.0619116 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) −6.00000 −0.247436
\(589\) 18.0000 0.741677
\(590\) −10.0000 −0.411693
\(591\) 14.0000 0.575883
\(592\) −1.00000 −0.0410997
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 5.00000 0.205152
\(595\) −1.00000 −0.0409960
\(596\) 14.0000 0.573462
\(597\) −12.0000 −0.491127
\(598\) 0 0
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −7.00000 −0.285536 −0.142768 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) −1.00000 −0.0407570
\(603\) 16.0000 0.651570
\(604\) 18.0000 0.732410
\(605\) −14.0000 −0.569181
\(606\) 12.0000 0.487467
\(607\) 6.00000 0.243532 0.121766 0.992559i \(-0.461144\pi\)
0.121766 + 0.992559i \(0.461144\pi\)
\(608\) −6.00000 −0.243332
\(609\) 9.00000 0.364698
\(610\) −5.00000 −0.202444
\(611\) 0 0
\(612\) 1.00000 0.0404226
\(613\) −23.0000 −0.928961 −0.464481 0.885583i \(-0.653759\pi\)
−0.464481 + 0.885583i \(0.653759\pi\)
\(614\) 30.0000 1.21070
\(615\) −9.00000 −0.362915
\(616\) 5.00000 0.201456
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 4.00000 0.160904
\(619\) −11.0000 −0.442127 −0.221064 0.975259i \(-0.570953\pi\)
−0.221064 + 0.975259i \(0.570953\pi\)
\(620\) −3.00000 −0.120483
\(621\) 0 0
\(622\) 25.0000 1.00241
\(623\) −2.00000 −0.0801283
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 18.0000 0.719425
\(627\) −30.0000 −1.19808
\(628\) 7.00000 0.279330
\(629\) −1.00000 −0.0398726
\(630\) 1.00000 0.0398410
\(631\) −13.0000 −0.517522 −0.258761 0.965941i \(-0.583314\pi\)
−0.258761 + 0.965941i \(0.583314\pi\)
\(632\) 12.0000 0.477334
\(633\) −5.00000 −0.198732
\(634\) 7.00000 0.278006
\(635\) 0 0
\(636\) 9.00000 0.356873
\(637\) −12.0000 −0.475457
\(638\) 45.0000 1.78157
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −15.0000 −0.592464 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) −18.0000 −0.710403
\(643\) −19.0000 −0.749287 −0.374643 0.927169i \(-0.622235\pi\)
−0.374643 + 0.927169i \(0.622235\pi\)
\(644\) 0 0
\(645\) −1.00000 −0.0393750
\(646\) −6.00000 −0.236067
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 50.0000 1.96267
\(650\) −2.00000 −0.0784465
\(651\) 3.00000 0.117579
\(652\) 1.00000 0.0391630
\(653\) −12.0000 −0.469596 −0.234798 0.972044i \(-0.575443\pi\)
−0.234798 + 0.972044i \(0.575443\pi\)
\(654\) 5.00000 0.195515
\(655\) −10.0000 −0.390732
\(656\) 9.00000 0.351391
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) −32.0000 −1.24654 −0.623272 0.782006i \(-0.714197\pi\)
−0.623272 + 0.782006i \(0.714197\pi\)
\(660\) 5.00000 0.194625
\(661\) −11.0000 −0.427850 −0.213925 0.976850i \(-0.568625\pi\)
−0.213925 + 0.976850i \(0.568625\pi\)
\(662\) 8.00000 0.310929
\(663\) 2.00000 0.0776736
\(664\) −2.00000 −0.0776151
\(665\) −6.00000 −0.232670
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) 20.0000 0.773823
\(669\) −5.00000 −0.193311
\(670\) 16.0000 0.618134
\(671\) 25.0000 0.965114
\(672\) −1.00000 −0.0385758
\(673\) −44.0000 −1.69608 −0.848038 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) −6.00000 −0.231111
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 13.0000 0.499262
\(679\) 17.0000 0.652400
\(680\) 1.00000 0.0383482
\(681\) −7.00000 −0.268241
\(682\) 15.0000 0.574380
\(683\) 33.0000 1.26271 0.631355 0.775494i \(-0.282499\pi\)
0.631355 + 0.775494i \(0.282499\pi\)
\(684\) 6.00000 0.229416
\(685\) 14.0000 0.534913
\(686\) 13.0000 0.496342
\(687\) 0 0
\(688\) 1.00000 0.0381246
\(689\) 18.0000 0.685745
\(690\) 0 0
\(691\) −1.00000 −0.0380418 −0.0190209 0.999819i \(-0.506055\pi\)
−0.0190209 + 0.999819i \(0.506055\pi\)
\(692\) −7.00000 −0.266100
\(693\) −5.00000 −0.189934
\(694\) −12.0000 −0.455514
\(695\) 19.0000 0.720711
\(696\) −9.00000 −0.341144
\(697\) 9.00000 0.340899
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 1.00000 0.0377964
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −6.00000 −0.226294
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −1.00000 −0.0376355
\(707\) −12.0000 −0.451306
\(708\) −10.0000 −0.375823
\(709\) −17.0000 −0.638448 −0.319224 0.947679i \(-0.603422\pi\)
−0.319224 + 0.947679i \(0.603422\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 2.00000 0.0749532
\(713\) 0 0
\(714\) −1.00000 −0.0374241
\(715\) 10.0000 0.373979
\(716\) −16.0000 −0.597948
\(717\) −21.0000 −0.784259
\(718\) −6.00000 −0.223918
\(719\) 10.0000 0.372937 0.186469 0.982461i \(-0.440296\pi\)
0.186469 + 0.982461i \(0.440296\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −4.00000 −0.148968
\(722\) −17.0000 −0.632674
\(723\) −18.0000 −0.669427
\(724\) 14.0000 0.520306
\(725\) 9.00000 0.334252
\(726\) −14.0000 −0.519589
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 12.0000 0.444140
\(731\) 1.00000 0.0369863
\(732\) −5.00000 −0.184805
\(733\) −47.0000 −1.73598 −0.867992 0.496578i \(-0.834590\pi\)
−0.867992 + 0.496578i \(0.834590\pi\)
\(734\) 23.0000 0.848945
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) −80.0000 −2.94684
\(738\) −9.00000 −0.331295
\(739\) −7.00000 −0.257499 −0.128750 0.991677i \(-0.541096\pi\)
−0.128750 + 0.991677i \(0.541096\pi\)
\(740\) 1.00000 0.0367607
\(741\) 12.0000 0.440831
\(742\) −9.00000 −0.330400
\(743\) 1.00000 0.0366864 0.0183432 0.999832i \(-0.494161\pi\)
0.0183432 + 0.999832i \(0.494161\pi\)
\(744\) −3.00000 −0.109985
\(745\) −14.0000 −0.512920
\(746\) −26.0000 −0.951928
\(747\) 2.00000 0.0731762
\(748\) −5.00000 −0.182818
\(749\) 18.0000 0.657706
\(750\) 1.00000 0.0365148
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) −24.0000 −0.874609
\(754\) −18.0000 −0.655521
\(755\) −18.0000 −0.655087
\(756\) 1.00000 0.0363696
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 6.00000 0.217643
\(761\) 33.0000 1.19625 0.598125 0.801403i \(-0.295913\pi\)
0.598125 + 0.801403i \(0.295913\pi\)
\(762\) 0 0
\(763\) −5.00000 −0.181012
\(764\) 11.0000 0.397966
\(765\) −1.00000 −0.0361551
\(766\) 0 0
\(767\) −20.0000 −0.722158
\(768\) 1.00000 0.0360844
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) −5.00000 −0.180187
\(771\) −18.0000 −0.648254
\(772\) −2.00000 −0.0719816
\(773\) 27.0000 0.971123 0.485561 0.874203i \(-0.338615\pi\)
0.485561 + 0.874203i \(0.338615\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 3.00000 0.107763
\(776\) −17.0000 −0.610264
\(777\) −1.00000 −0.0358748
\(778\) −19.0000 −0.681183
\(779\) 54.0000 1.93475
\(780\) −2.00000 −0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) 9.00000 0.321634
\(784\) −6.00000 −0.214286
\(785\) −7.00000 −0.249841
\(786\) −10.0000 −0.356688
\(787\) 36.0000 1.28326 0.641631 0.767014i \(-0.278258\pi\)
0.641631 + 0.767014i \(0.278258\pi\)
\(788\) 14.0000 0.498729
\(789\) −15.0000 −0.534014
\(790\) −12.0000 −0.426941
\(791\) −13.0000 −0.462227
\(792\) 5.00000 0.177667
\(793\) −10.0000 −0.355110
\(794\) −18.0000 −0.638796
\(795\) −9.00000 −0.319197
\(796\) −12.0000 −0.425329
\(797\) 40.0000 1.41687 0.708436 0.705775i \(-0.249401\pi\)
0.708436 + 0.705775i \(0.249401\pi\)
\(798\) −6.00000 −0.212398
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −2.00000 −0.0706665
\(802\) 6.00000 0.211867
\(803\) −60.0000 −2.11735
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) −6.00000 −0.211341
\(807\) 24.0000 0.844840
\(808\) 12.0000 0.422159
\(809\) −24.0000 −0.843795 −0.421898 0.906644i \(-0.638636\pi\)
−0.421898 + 0.906644i \(0.638636\pi\)
\(810\) 1.00000 0.0351364
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 9.00000 0.315838
\(813\) −16.0000 −0.561144
\(814\) −5.00000 −0.175250
\(815\) −1.00000 −0.0350285
\(816\) 1.00000 0.0350070
\(817\) 6.00000 0.209913
\(818\) −6.00000 −0.209785
\(819\) 2.00000 0.0698857
\(820\) −9.00000 −0.314294
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 14.0000 0.488306
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 4.00000 0.139347
\(825\) −5.00000 −0.174078
\(826\) 10.0000 0.347945
\(827\) −3.00000 −0.104320 −0.0521601 0.998639i \(-0.516611\pi\)
−0.0521601 + 0.998639i \(0.516611\pi\)
\(828\) 0 0
\(829\) −17.0000 −0.590434 −0.295217 0.955430i \(-0.595392\pi\)
−0.295217 + 0.955430i \(0.595392\pi\)
\(830\) 2.00000 0.0694210
\(831\) −8.00000 −0.277517
\(832\) 2.00000 0.0693375
\(833\) −6.00000 −0.207888
\(834\) 19.0000 0.657916
\(835\) −20.0000 −0.692129
\(836\) −30.0000 −1.03757
\(837\) 3.00000 0.103695
\(838\) −20.0000 −0.690889
\(839\) 54.0000 1.86429 0.932144 0.362089i \(-0.117936\pi\)
0.932144 + 0.362089i \(0.117936\pi\)
\(840\) 1.00000 0.0345033
\(841\) 52.0000 1.79310
\(842\) −30.0000 −1.03387
\(843\) −2.00000 −0.0688837
\(844\) −5.00000 −0.172107
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 9.00000 0.309061
\(849\) −28.0000 −0.960958
\(850\) −1.00000 −0.0342997
\(851\) 0 0
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 5.00000 0.171096
\(855\) −6.00000 −0.205196
\(856\) −18.0000 −0.615227
\(857\) −33.0000 −1.12726 −0.563629 0.826028i \(-0.690595\pi\)
−0.563629 + 0.826028i \(0.690595\pi\)
\(858\) 10.0000 0.341394
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) −1.00000 −0.0340997
\(861\) 9.00000 0.306719
\(862\) −15.0000 −0.510902
\(863\) 39.0000 1.32758 0.663788 0.747921i \(-0.268948\pi\)
0.663788 + 0.747921i \(0.268948\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 7.00000 0.238007
\(866\) −36.0000 −1.22333
\(867\) −16.0000 −0.543388
\(868\) 3.00000 0.101827
\(869\) 60.0000 2.03536
\(870\) 9.00000 0.305129
\(871\) 32.0000 1.08428
\(872\) 5.00000 0.169321
\(873\) 17.0000 0.575363
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 12.0000 0.405442
\(877\) 47.0000 1.58708 0.793539 0.608520i \(-0.208236\pi\)
0.793539 + 0.608520i \(0.208236\pi\)
\(878\) 27.0000 0.911206
\(879\) 19.0000 0.640854
\(880\) 5.00000 0.168550
\(881\) 41.0000 1.38133 0.690663 0.723177i \(-0.257319\pi\)
0.690663 + 0.723177i \(0.257319\pi\)
\(882\) 6.00000 0.202031
\(883\) −9.00000 −0.302874 −0.151437 0.988467i \(-0.548390\pi\)
−0.151437 + 0.988467i \(0.548390\pi\)
\(884\) 2.00000 0.0672673
\(885\) 10.0000 0.336146
\(886\) −2.00000 −0.0671913
\(887\) −35.0000 −1.17518 −0.587592 0.809157i \(-0.699924\pi\)
−0.587592 + 0.809157i \(0.699924\pi\)
\(888\) 1.00000 0.0335578
\(889\) 0 0
\(890\) −2.00000 −0.0670402
\(891\) −5.00000 −0.167506
\(892\) −5.00000 −0.167412
\(893\) 0 0
\(894\) −14.0000 −0.468230
\(895\) 16.0000 0.534821
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −24.0000 −0.800890
\(899\) 27.0000 0.900500
\(900\) 1.00000 0.0333333
\(901\) 9.00000 0.299833
\(902\) 45.0000 1.49834
\(903\) 1.00000 0.0332779
\(904\) 13.0000 0.432374
\(905\) −14.0000 −0.465376
\(906\) −18.0000 −0.598010
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) −7.00000 −0.232303
\(909\) −12.0000 −0.398015
\(910\) 2.00000 0.0662994
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 6.00000 0.198680
\(913\) −10.0000 −0.330952
\(914\) −1.00000 −0.0330771
\(915\) 5.00000 0.165295
\(916\) 0 0
\(917\) 10.0000 0.330229
\(918\) −1.00000 −0.0330049
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 0 0
\(921\) −30.0000 −0.988534
\(922\) −35.0000 −1.15266
\(923\) 0 0
\(924\) −5.00000 −0.164488
\(925\) −1.00000 −0.0328798
\(926\) −26.0000 −0.854413
\(927\) −4.00000 −0.131377
\(928\) −9.00000 −0.295439
\(929\) −1.00000 −0.0328089 −0.0164045 0.999865i \(-0.505222\pi\)
−0.0164045 + 0.999865i \(0.505222\pi\)
\(930\) 3.00000 0.0983739
\(931\) −36.0000 −1.17985
\(932\) −8.00000 −0.262049
\(933\) −25.0000 −0.818463
\(934\) 21.0000 0.687141
\(935\) 5.00000 0.163517
\(936\) −2.00000 −0.0653720
\(937\) 56.0000 1.82944 0.914720 0.404088i \(-0.132411\pi\)
0.914720 + 0.404088i \(0.132411\pi\)
\(938\) −16.0000 −0.522419
\(939\) −18.0000 −0.587408
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) −7.00000 −0.228072
\(943\) 0 0
\(944\) −10.0000 −0.325472
\(945\) −1.00000 −0.0325300
\(946\) 5.00000 0.162564
\(947\) 1.00000 0.0324956 0.0162478 0.999868i \(-0.494828\pi\)
0.0162478 + 0.999868i \(0.494828\pi\)
\(948\) −12.0000 −0.389742
\(949\) 24.0000 0.779073
\(950\) −6.00000 −0.194666
\(951\) −7.00000 −0.226991
\(952\) −1.00000 −0.0324102
\(953\) −4.00000 −0.129573 −0.0647864 0.997899i \(-0.520637\pi\)
−0.0647864 + 0.997899i \(0.520637\pi\)
\(954\) −9.00000 −0.291386
\(955\) −11.0000 −0.355952
\(956\) −21.0000 −0.679189
\(957\) −45.0000 −1.45464
\(958\) 24.0000 0.775405
\(959\) −14.0000 −0.452084
\(960\) −1.00000 −0.0322749
\(961\) −22.0000 −0.709677
\(962\) 2.00000 0.0644826
\(963\) 18.0000 0.580042
\(964\) −18.0000 −0.579741
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) 36.0000 1.15768 0.578841 0.815440i \(-0.303505\pi\)
0.578841 + 0.815440i \(0.303505\pi\)
\(968\) −14.0000 −0.449977
\(969\) 6.00000 0.192748
\(970\) 17.0000 0.545837
\(971\) 29.0000 0.930654 0.465327 0.885139i \(-0.345937\pi\)
0.465327 + 0.885139i \(0.345937\pi\)
\(972\) 1.00000 0.0320750
\(973\) −19.0000 −0.609112
\(974\) −2.00000 −0.0640841
\(975\) 2.00000 0.0640513
\(976\) −5.00000 −0.160046
\(977\) −27.0000 −0.863807 −0.431903 0.901920i \(-0.642158\pi\)
−0.431903 + 0.901920i \(0.642158\pi\)
\(978\) −1.00000 −0.0319765
\(979\) 10.0000 0.319601
\(980\) 6.00000 0.191663
\(981\) −5.00000 −0.159638
\(982\) 36.0000 1.14881
\(983\) −49.0000 −1.56286 −0.781429 0.623995i \(-0.785509\pi\)
−0.781429 + 0.623995i \(0.785509\pi\)
\(984\) −9.00000 −0.286910
\(985\) −14.0000 −0.446077
\(986\) −9.00000 −0.286618
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) 0 0
\(990\) −5.00000 −0.158910
\(991\) −31.0000 −0.984747 −0.492374 0.870384i \(-0.663871\pi\)
−0.492374 + 0.870384i \(0.663871\pi\)
\(992\) −3.00000 −0.0952501
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) 12.0000 0.380426
\(996\) 2.00000 0.0633724
\(997\) 6.00000 0.190022 0.0950110 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) 44.0000 1.39280
\(999\) −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.f.1.1 1
3.2 odd 2 3330.2.a.y.1.1 1
4.3 odd 2 8880.2.a.d.1.1 1
5.4 even 2 5550.2.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.f.1.1 1 1.1 even 1 trivial
3330.2.a.y.1.1 1 3.2 odd 2
5550.2.a.y.1.1 1 5.4 even 2
8880.2.a.d.1.1 1 4.3 odd 2