Properties

Label 1110.2.a.j.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

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} +3.00000 q^{21} -5.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -3.00000 q^{28} -1.00000 q^{29} -1.00000 q^{30} -3.00000 q^{31} +1.00000 q^{32} +5.00000 q^{33} +3.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} -6.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} -7.00000 q^{41} +3.00000 q^{42} +3.00000 q^{43} -5.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} -1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} -3.00000 q^{51} -2.00000 q^{52} +5.00000 q^{53} -1.00000 q^{54} -5.00000 q^{55} -3.00000 q^{56} +6.00000 q^{57} -1.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} +5.00000 q^{61} -3.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +5.00000 q^{66} -4.00000 q^{67} +3.00000 q^{68} +4.00000 q^{69} -3.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} -1.00000 q^{74} -1.00000 q^{75} -6.00000 q^{76} +15.0000 q^{77} +2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -7.00000 q^{82} +6.00000 q^{83} +3.00000 q^{84} +3.00000 q^{85} +3.00000 q^{86} +1.00000 q^{87} -5.00000 q^{88} -18.0000 q^{89} +1.00000 q^{90} +6.00000 q^{91} -4.00000 q^{92} +3.00000 q^{93} -6.00000 q^{95} -1.00000 q^{96} -13.0000 q^{97} +2.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 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 \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −3.00000 −0.801784
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.00000 0.654654
\(22\) −5.00000 −1.06600
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −3.00000 −0.566947
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) −1.00000 −0.182574
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.00000 0.870388
\(34\) 3.00000 0.514496
\(35\) −3.00000 −0.507093
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) −6.00000 −0.973329
\(39\) 2.00000 0.320256
\(40\) 1.00000 0.158114
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 3.00000 0.462910
\(43\) 3.00000 0.457496 0.228748 0.973486i \(-0.426537\pi\)
0.228748 + 0.973486i \(0.426537\pi\)
\(44\) −5.00000 −0.753778
\(45\) 1.00000 0.149071
\(46\) −4.00000 −0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) 1.00000 0.141421
\(51\) −3.00000 −0.420084
\(52\) −2.00000 −0.277350
\(53\) 5.00000 0.686803 0.343401 0.939189i \(-0.388421\pi\)
0.343401 + 0.939189i \(0.388421\pi\)
\(54\) −1.00000 −0.136083
\(55\) −5.00000 −0.674200
\(56\) −3.00000 −0.400892
\(57\) 6.00000 0.794719
\(58\) −1.00000 −0.131306
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −1.00000 −0.129099
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) −3.00000 −0.381000
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 5.00000 0.615457
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 3.00000 0.363803
\(69\) 4.00000 0.481543
\(70\) −3.00000 −0.358569
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −1.00000 −0.116248
\(75\) −1.00000 −0.115470
\(76\) −6.00000 −0.688247
\(77\) 15.0000 1.70941
\(78\) 2.00000 0.226455
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −7.00000 −0.773021
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 3.00000 0.327327
\(85\) 3.00000 0.325396
\(86\) 3.00000 0.323498
\(87\) 1.00000 0.107211
\(88\) −5.00000 −0.533002
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 1.00000 0.105409
\(91\) 6.00000 0.628971
\(92\) −4.00000 −0.417029
\(93\) 3.00000 0.311086
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) −1.00000 −0.102062
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 2.00000 0.202031
\(99\) −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 \(0.703650\pi\)
\(102\) −3.00000 −0.297044
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) −2.00000 −0.196116
\(105\) 3.00000 0.292770
\(106\) 5.00000 0.485643
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) −5.00000 −0.476731
\(111\) 1.00000 0.0949158
\(112\) −3.00000 −0.283473
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 6.00000 0.561951
\(115\) −4.00000 −0.373002
\(116\) −1.00000 −0.0928477
\(117\) −2.00000 −0.184900
\(118\) 6.00000 0.552345
\(119\) −9.00000 −0.825029
\(120\) −1.00000 −0.0912871
\(121\) 14.0000 1.27273
\(122\) 5.00000 0.452679
\(123\) 7.00000 0.631169
\(124\) −3.00000 −0.269408
\(125\) 1.00000 0.0894427
\(126\) −3.00000 −0.267261
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −3.00000 −0.264135
\(130\) −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
\(133\) 18.0000 1.56080
\(134\) −4.00000 −0.345547
\(135\) −1.00000 −0.0860663
\(136\) 3.00000 0.257248
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 4.00000 0.340503
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) −3.00000 −0.253546
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) 10.0000 0.836242
\(144\) 1.00000 0.0833333
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) −1.00000 −0.0821995
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −6.00000 −0.486664
\(153\) 3.00000 0.242536
\(154\) 15.0000 1.20873
\(155\) −3.00000 −0.240966
\(156\) 2.00000 0.160128
\(157\) 11.0000 0.877896 0.438948 0.898513i \(-0.355351\pi\)
0.438948 + 0.898513i \(0.355351\pi\)
\(158\) −4.00000 −0.318223
\(159\) −5.00000 −0.396526
\(160\) 1.00000 0.0790569
\(161\) 12.0000 0.945732
\(162\) 1.00000 0.0785674
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) −7.00000 −0.546608
\(165\) 5.00000 0.389249
\(166\) 6.00000 0.465690
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 3.00000 0.231455
\(169\) −9.00000 −0.692308
\(170\) 3.00000 0.230089
\(171\) −6.00000 −0.458831
\(172\) 3.00000 0.228748
\(173\) 13.0000 0.988372 0.494186 0.869356i \(-0.335466\pi\)
0.494186 + 0.869356i \(0.335466\pi\)
\(174\) 1.00000 0.0758098
\(175\) −3.00000 −0.226779
\(176\) −5.00000 −0.376889
\(177\) −6.00000 −0.450988
\(178\) −18.0000 −1.34916
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 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
\(183\) −5.00000 −0.369611
\(184\) −4.00000 −0.294884
\(185\) −1.00000 −0.0735215
\(186\) 3.00000 0.219971
\(187\) −15.0000 −1.09691
\(188\) 0 0
\(189\) 3.00000 0.218218
\(190\) −6.00000 −0.435286
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) −13.0000 −0.933346
\(195\) 2.00000 0.143223
\(196\) 2.00000 0.142857
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −5.00000 −0.355335
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) −12.0000 −0.844317
\(203\) 3.00000 0.210559
\(204\) −3.00000 −0.210042
\(205\) −7.00000 −0.488901
\(206\) 16.0000 1.11477
\(207\) −4.00000 −0.278019
\(208\) −2.00000 −0.138675
\(209\) 30.0000 2.07514
\(210\) 3.00000 0.207020
\(211\) 19.0000 1.30801 0.654007 0.756489i \(-0.273087\pi\)
0.654007 + 0.756489i \(0.273087\pi\)
\(212\) 5.00000 0.343401
\(213\) 12.0000 0.822226
\(214\) 2.00000 0.136717
\(215\) 3.00000 0.204598
\(216\) −1.00000 −0.0680414
\(217\) 9.00000 0.610960
\(218\) 5.00000 0.338643
\(219\) 0 0
\(220\) −5.00000 −0.337100
\(221\) −6.00000 −0.403604
\(222\) 1.00000 0.0671156
\(223\) 7.00000 0.468755 0.234377 0.972146i \(-0.424695\pi\)
0.234377 + 0.972146i \(0.424695\pi\)
\(224\) −3.00000 −0.200446
\(225\) 1.00000 0.0666667
\(226\) 9.00000 0.598671
\(227\) −21.0000 −1.39382 −0.696909 0.717159i \(-0.745442\pi\)
−0.696909 + 0.717159i \(0.745442\pi\)
\(228\) 6.00000 0.397360
\(229\) 12.0000 0.792982 0.396491 0.918039i \(-0.370228\pi\)
0.396491 + 0.918039i \(0.370228\pi\)
\(230\) −4.00000 −0.263752
\(231\) −15.0000 −0.986928
\(232\) −1.00000 −0.0656532
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 4.00000 0.259828
\(238\) −9.00000 −0.583383
\(239\) 29.0000 1.87585 0.937927 0.346833i \(-0.112743\pi\)
0.937927 + 0.346833i \(0.112743\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 14.0000 0.899954
\(243\) −1.00000 −0.0641500
\(244\) 5.00000 0.320092
\(245\) 2.00000 0.127775
\(246\) 7.00000 0.446304
\(247\) 12.0000 0.763542
\(248\) −3.00000 −0.190500
\(249\) −6.00000 −0.380235
\(250\) 1.00000 0.0632456
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) −3.00000 −0.188982
\(253\) 20.0000 1.25739
\(254\) 8.00000 0.501965
\(255\) −3.00000 −0.187867
\(256\) 1.00000 0.0625000
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) −3.00000 −0.186772
\(259\) 3.00000 0.186411
\(260\) −2.00000 −0.124035
\(261\) −1.00000 −0.0618984
\(262\) 10.0000 0.617802
\(263\) 5.00000 0.308313 0.154157 0.988046i \(-0.450734\pi\)
0.154157 + 0.988046i \(0.450734\pi\)
\(264\) 5.00000 0.307729
\(265\) 5.00000 0.307148
\(266\) 18.0000 1.10365
\(267\) 18.0000 1.10158
\(268\) −4.00000 −0.244339
\(269\) −16.0000 −0.975537 −0.487769 0.872973i \(-0.662189\pi\)
−0.487769 + 0.872973i \(0.662189\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 3.00000 0.181902
\(273\) −6.00000 −0.363137
\(274\) −10.0000 −0.604122
\(275\) −5.00000 −0.301511
\(276\) 4.00000 0.240772
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) 5.00000 0.299880
\(279\) −3.00000 −0.179605
\(280\) −3.00000 −0.179284
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −12.0000 −0.712069
\(285\) 6.00000 0.355409
\(286\) 10.0000 0.591312
\(287\) 21.0000 1.23959
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) −1.00000 −0.0587220
\(291\) 13.0000 0.762073
\(292\) 0 0
\(293\) 7.00000 0.408944 0.204472 0.978872i \(-0.434452\pi\)
0.204472 + 0.978872i \(0.434452\pi\)
\(294\) −2.00000 −0.116642
\(295\) 6.00000 0.349334
\(296\) −1.00000 −0.0581238
\(297\) 5.00000 0.290129
\(298\) −18.0000 −1.04271
\(299\) 8.00000 0.462652
\(300\) −1.00000 −0.0577350
\(301\) −9.00000 −0.518751
\(302\) 10.0000 0.575435
\(303\) 12.0000 0.689382
\(304\) −6.00000 −0.344124
\(305\) 5.00000 0.286299
\(306\) 3.00000 0.171499
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 15.0000 0.854704
\(309\) −16.0000 −0.910208
\(310\) −3.00000 −0.170389
\(311\) −31.0000 −1.75785 −0.878924 0.476961i \(-0.841738\pi\)
−0.878924 + 0.476961i \(0.841738\pi\)
\(312\) 2.00000 0.113228
\(313\) −30.0000 −1.69570 −0.847850 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) 11.0000 0.620766
\(315\) −3.00000 −0.169031
\(316\) −4.00000 −0.225018
\(317\) −27.0000 −1.51647 −0.758236 0.651981i \(-0.773938\pi\)
−0.758236 + 0.651981i \(0.773938\pi\)
\(318\) −5.00000 −0.280386
\(319\) 5.00000 0.279946
\(320\) 1.00000 0.0559017
\(321\) −2.00000 −0.111629
\(322\) 12.0000 0.668734
\(323\) −18.0000 −1.00155
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) 11.0000 0.609234
\(327\) −5.00000 −0.276501
\(328\) −7.00000 −0.386510
\(329\) 0 0
\(330\) 5.00000 0.275241
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 6.00000 0.329293
\(333\) −1.00000 −0.0547997
\(334\) −24.0000 −1.31322
\(335\) −4.00000 −0.218543
\(336\) 3.00000 0.163663
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) −9.00000 −0.489535
\(339\) −9.00000 −0.488813
\(340\) 3.00000 0.162698
\(341\) 15.0000 0.812296
\(342\) −6.00000 −0.324443
\(343\) 15.0000 0.809924
\(344\) 3.00000 0.161749
\(345\) 4.00000 0.215353
\(346\) 13.0000 0.698884
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 1.00000 0.0536056
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) −3.00000 −0.160357
\(351\) 2.00000 0.106752
\(352\) −5.00000 −0.266501
\(353\) −37.0000 −1.96931 −0.984656 0.174509i \(-0.944166\pi\)
−0.984656 + 0.174509i \(0.944166\pi\)
\(354\) −6.00000 −0.318896
\(355\) −12.0000 −0.636894
\(356\) −18.0000 −0.953998
\(357\) 9.00000 0.476331
\(358\) 12.0000 0.634220
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 1.00000 0.0527046
\(361\) 17.0000 0.894737
\(362\) 10.0000 0.525588
\(363\) −14.0000 −0.734809
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) −5.00000 −0.261354
\(367\) −3.00000 −0.156599 −0.0782994 0.996930i \(-0.524949\pi\)
−0.0782994 + 0.996930i \(0.524949\pi\)
\(368\) −4.00000 −0.208514
\(369\) −7.00000 −0.364405
\(370\) −1.00000 −0.0519875
\(371\) −15.0000 −0.778761
\(372\) 3.00000 0.155543
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) −15.0000 −0.775632
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 2.00000 0.103005
\(378\) 3.00000 0.154303
\(379\) 32.0000 1.64373 0.821865 0.569683i \(-0.192934\pi\)
0.821865 + 0.569683i \(0.192934\pi\)
\(380\) −6.00000 −0.307794
\(381\) −8.00000 −0.409852
\(382\) −3.00000 −0.153493
\(383\) −20.0000 −1.02195 −0.510976 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 15.0000 0.764471
\(386\) 18.0000 0.916176
\(387\) 3.00000 0.152499
\(388\) −13.0000 −0.659975
\(389\) −3.00000 −0.152106 −0.0760530 0.997104i \(-0.524232\pi\)
−0.0760530 + 0.997104i \(0.524232\pi\)
\(390\) 2.00000 0.101274
\(391\) −12.0000 −0.606866
\(392\) 2.00000 0.101015
\(393\) −10.0000 −0.504433
\(394\) −2.00000 −0.100759
\(395\) −4.00000 −0.201262
\(396\) −5.00000 −0.251259
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −4.00000 −0.200502
\(399\) −18.0000 −0.901127
\(400\) 1.00000 0.0500000
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 4.00000 0.199502
\(403\) 6.00000 0.298881
\(404\) −12.0000 −0.597022
\(405\) 1.00000 0.0496904
\(406\) 3.00000 0.148888
\(407\) 5.00000 0.247841
\(408\) −3.00000 −0.148522
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) −7.00000 −0.345705
\(411\) 10.0000 0.493264
\(412\) 16.0000 0.788263
\(413\) −18.0000 −0.885722
\(414\) −4.00000 −0.196589
\(415\) 6.00000 0.294528
\(416\) −2.00000 −0.0980581
\(417\) −5.00000 −0.244851
\(418\) 30.0000 1.46735
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 3.00000 0.146385
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 19.0000 0.924906
\(423\) 0 0
\(424\) 5.00000 0.242821
\(425\) 3.00000 0.145521
\(426\) 12.0000 0.581402
\(427\) −15.0000 −0.725901
\(428\) 2.00000 0.0966736
\(429\) −10.0000 −0.482805
\(430\) 3.00000 0.144673
\(431\) −7.00000 −0.337178 −0.168589 0.985686i \(-0.553921\pi\)
−0.168589 + 0.985686i \(0.553921\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −8.00000 −0.384455 −0.192228 0.981350i \(-0.561571\pi\)
−0.192228 + 0.981350i \(0.561571\pi\)
\(434\) 9.00000 0.432014
\(435\) 1.00000 0.0479463
\(436\) 5.00000 0.239457
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) −29.0000 −1.38409 −0.692047 0.721852i \(-0.743291\pi\)
−0.692047 + 0.721852i \(0.743291\pi\)
\(440\) −5.00000 −0.238366
\(441\) 2.00000 0.0952381
\(442\) −6.00000 −0.285391
\(443\) 14.0000 0.665160 0.332580 0.943075i \(-0.392081\pi\)
0.332580 + 0.943075i \(0.392081\pi\)
\(444\) 1.00000 0.0474579
\(445\) −18.0000 −0.853282
\(446\) 7.00000 0.331460
\(447\) 18.0000 0.851371
\(448\) −3.00000 −0.141737
\(449\) −36.0000 −1.69895 −0.849473 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 1.00000 0.0471405
\(451\) 35.0000 1.64809
\(452\) 9.00000 0.423324
\(453\) −10.0000 −0.469841
\(454\) −21.0000 −0.985579
\(455\) 6.00000 0.281284
\(456\) 6.00000 0.280976
\(457\) −5.00000 −0.233890 −0.116945 0.993138i \(-0.537310\pi\)
−0.116945 + 0.993138i \(0.537310\pi\)
\(458\) 12.0000 0.560723
\(459\) −3.00000 −0.140028
\(460\) −4.00000 −0.186501
\(461\) 21.0000 0.978068 0.489034 0.872265i \(-0.337349\pi\)
0.489034 + 0.872265i \(0.337349\pi\)
\(462\) −15.0000 −0.697863
\(463\) 34.0000 1.58011 0.790057 0.613033i \(-0.210051\pi\)
0.790057 + 0.613033i \(0.210051\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 3.00000 0.139122
\(466\) 0 0
\(467\) −15.0000 −0.694117 −0.347059 0.937843i \(-0.612820\pi\)
−0.347059 + 0.937843i \(0.612820\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) −11.0000 −0.506853
\(472\) 6.00000 0.276172
\(473\) −15.0000 −0.689701
\(474\) 4.00000 0.183726
\(475\) −6.00000 −0.275299
\(476\) −9.00000 −0.412514
\(477\) 5.00000 0.228934
\(478\) 29.0000 1.32643
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 2.00000 0.0911922
\(482\) −10.0000 −0.455488
\(483\) −12.0000 −0.546019
\(484\) 14.0000 0.636364
\(485\) −13.0000 −0.590300
\(486\) −1.00000 −0.0453609
\(487\) −6.00000 −0.271886 −0.135943 0.990717i \(-0.543406\pi\)
−0.135943 + 0.990717i \(0.543406\pi\)
\(488\) 5.00000 0.226339
\(489\) −11.0000 −0.497437
\(490\) 2.00000 0.0903508
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) 7.00000 0.315584
\(493\) −3.00000 −0.135113
\(494\) 12.0000 0.539906
\(495\) −5.00000 −0.224733
\(496\) −3.00000 −0.134704
\(497\) 36.0000 1.61482
\(498\) −6.00000 −0.268866
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 1.00000 0.0447214
\(501\) 24.0000 1.07224
\(502\) −24.0000 −1.07117
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) −3.00000 −0.133631
\(505\) −12.0000 −0.533993
\(506\) 20.0000 0.889108
\(507\) 9.00000 0.399704
\(508\) 8.00000 0.354943
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) −3.00000 −0.132842
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 6.00000 0.264906
\(514\) 26.0000 1.14681
\(515\) 16.0000 0.705044
\(516\) −3.00000 −0.132068
\(517\) 0 0
\(518\) 3.00000 0.131812
\(519\) −13.0000 −0.570637
\(520\) −2.00000 −0.0877058
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) −1.00000 −0.0437688
\(523\) −40.0000 −1.74908 −0.874539 0.484955i \(-0.838836\pi\)
−0.874539 + 0.484955i \(0.838836\pi\)
\(524\) 10.0000 0.436852
\(525\) 3.00000 0.130931
\(526\) 5.00000 0.218010
\(527\) −9.00000 −0.392046
\(528\) 5.00000 0.217597
\(529\) −7.00000 −0.304348
\(530\) 5.00000 0.217186
\(531\) 6.00000 0.260378
\(532\) 18.0000 0.780399
\(533\) 14.0000 0.606407
\(534\) 18.0000 0.778936
\(535\) 2.00000 0.0864675
\(536\) −4.00000 −0.172774
\(537\) −12.0000 −0.517838
\(538\) −16.0000 −0.689809
\(539\) −10.0000 −0.430730
\(540\) −1.00000 −0.0430331
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 8.00000 0.343629
\(543\) −10.0000 −0.429141
\(544\) 3.00000 0.128624
\(545\) 5.00000 0.214176
\(546\) −6.00000 −0.256776
\(547\) 19.0000 0.812381 0.406191 0.913788i \(-0.366857\pi\)
0.406191 + 0.913788i \(0.366857\pi\)
\(548\) −10.0000 −0.427179
\(549\) 5.00000 0.213395
\(550\) −5.00000 −0.213201
\(551\) 6.00000 0.255609
\(552\) 4.00000 0.170251
\(553\) 12.0000 0.510292
\(554\) −16.0000 −0.679775
\(555\) 1.00000 0.0424476
\(556\) 5.00000 0.212047
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) −3.00000 −0.127000
\(559\) −6.00000 −0.253773
\(560\) −3.00000 −0.126773
\(561\) 15.0000 0.633300
\(562\) 2.00000 0.0843649
\(563\) −17.0000 −0.716465 −0.358232 0.933632i \(-0.616620\pi\)
−0.358232 + 0.933632i \(0.616620\pi\)
\(564\) 0 0
\(565\) 9.00000 0.378633
\(566\) 4.00000 0.168133
\(567\) −3.00000 −0.125988
\(568\) −12.0000 −0.503509
\(569\) 46.0000 1.92842 0.964210 0.265139i \(-0.0854179\pi\)
0.964210 + 0.265139i \(0.0854179\pi\)
\(570\) 6.00000 0.251312
\(571\) −21.0000 −0.878823 −0.439411 0.898286i \(-0.644813\pi\)
−0.439411 + 0.898286i \(0.644813\pi\)
\(572\) 10.0000 0.418121
\(573\) 3.00000 0.125327
\(574\) 21.0000 0.876523
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −8.00000 −0.332756
\(579\) −18.0000 −0.748054
\(580\) −1.00000 −0.0415227
\(581\) −18.0000 −0.746766
\(582\) 13.0000 0.538867
\(583\) −25.0000 −1.03539
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 7.00000 0.289167
\(587\) 1.00000 0.0412744 0.0206372 0.999787i \(-0.493431\pi\)
0.0206372 + 0.999787i \(0.493431\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 18.0000 0.741677
\(590\) 6.00000 0.247016
\(591\) 2.00000 0.0822690
\(592\) −1.00000 −0.0410997
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 5.00000 0.205152
\(595\) −9.00000 −0.368964
\(596\) −18.0000 −0.737309
\(597\) 4.00000 0.163709
\(598\) 8.00000 0.327144
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −31.0000 −1.26452 −0.632258 0.774758i \(-0.717872\pi\)
−0.632258 + 0.774758i \(0.717872\pi\)
\(602\) −9.00000 −0.366813
\(603\) −4.00000 −0.162893
\(604\) 10.0000 0.406894
\(605\) 14.0000 0.569181
\(606\) 12.0000 0.487467
\(607\) −34.0000 −1.38002 −0.690009 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) −6.00000 −0.243332
\(609\) −3.00000 −0.121566
\(610\) 5.00000 0.202444
\(611\) 0 0
\(612\) 3.00000 0.121268
\(613\) −43.0000 −1.73675 −0.868377 0.495905i \(-0.834836\pi\)
−0.868377 + 0.495905i \(0.834836\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 7.00000 0.282267
\(616\) 15.0000 0.604367
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) −16.0000 −0.643614
\(619\) −27.0000 −1.08522 −0.542611 0.839984i \(-0.682564\pi\)
−0.542611 + 0.839984i \(0.682564\pi\)
\(620\) −3.00000 −0.120483
\(621\) 4.00000 0.160514
\(622\) −31.0000 −1.24299
\(623\) 54.0000 2.16346
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −30.0000 −1.19904
\(627\) −30.0000 −1.19808
\(628\) 11.0000 0.438948
\(629\) −3.00000 −0.119618
\(630\) −3.00000 −0.119523
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) −4.00000 −0.159111
\(633\) −19.0000 −0.755182
\(634\) −27.0000 −1.07231
\(635\) 8.00000 0.317470
\(636\) −5.00000 −0.198263
\(637\) −4.00000 −0.158486
\(638\) 5.00000 0.197952
\(639\) −12.0000 −0.474713
\(640\) 1.00000 0.0395285
\(641\) −23.0000 −0.908445 −0.454223 0.890888i \(-0.650083\pi\)
−0.454223 + 0.890888i \(0.650083\pi\)
\(642\) −2.00000 −0.0789337
\(643\) 31.0000 1.22252 0.611260 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(644\) 12.0000 0.472866
\(645\) −3.00000 −0.118125
\(646\) −18.0000 −0.708201
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 1.00000 0.0392837
\(649\) −30.0000 −1.17760
\(650\) −2.00000 −0.0784465
\(651\) −9.00000 −0.352738
\(652\) 11.0000 0.430793
\(653\) 16.0000 0.626128 0.313064 0.949732i \(-0.398644\pi\)
0.313064 + 0.949732i \(0.398644\pi\)
\(654\) −5.00000 −0.195515
\(655\) 10.0000 0.390732
\(656\) −7.00000 −0.273304
\(657\) 0 0
\(658\) 0 0
\(659\) 16.0000 0.623272 0.311636 0.950202i \(-0.399123\pi\)
0.311636 + 0.950202i \(0.399123\pi\)
\(660\) 5.00000 0.194625
\(661\) −29.0000 −1.12797 −0.563985 0.825785i \(-0.690732\pi\)
−0.563985 + 0.825785i \(0.690732\pi\)
\(662\) −20.0000 −0.777322
\(663\) 6.00000 0.233021
\(664\) 6.00000 0.232845
\(665\) 18.0000 0.698010
\(666\) −1.00000 −0.0387492
\(667\) 4.00000 0.154881
\(668\) −24.0000 −0.928588
\(669\) −7.00000 −0.270636
\(670\) −4.00000 −0.154533
\(671\) −25.0000 −0.965114
\(672\) 3.00000 0.115728
\(673\) 48.0000 1.85026 0.925132 0.379646i \(-0.123954\pi\)
0.925132 + 0.379646i \(0.123954\pi\)
\(674\) −6.00000 −0.231111
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 10.0000 0.384331 0.192166 0.981363i \(-0.438449\pi\)
0.192166 + 0.981363i \(0.438449\pi\)
\(678\) −9.00000 −0.345643
\(679\) 39.0000 1.49668
\(680\) 3.00000 0.115045
\(681\) 21.0000 0.804722
\(682\) 15.0000 0.574380
\(683\) −29.0000 −1.10965 −0.554827 0.831966i \(-0.687216\pi\)
−0.554827 + 0.831966i \(0.687216\pi\)
\(684\) −6.00000 −0.229416
\(685\) −10.0000 −0.382080
\(686\) 15.0000 0.572703
\(687\) −12.0000 −0.457829
\(688\) 3.00000 0.114374
\(689\) −10.0000 −0.380970
\(690\) 4.00000 0.152277
\(691\) −33.0000 −1.25538 −0.627690 0.778464i \(-0.715999\pi\)
−0.627690 + 0.778464i \(0.715999\pi\)
\(692\) 13.0000 0.494186
\(693\) 15.0000 0.569803
\(694\) −20.0000 −0.759190
\(695\) 5.00000 0.189661
\(696\) 1.00000 0.0379049
\(697\) −21.0000 −0.795432
\(698\) −20.0000 −0.757011
\(699\) 0 0
\(700\) −3.00000 −0.113389
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 2.00000 0.0754851
\(703\) 6.00000 0.226294
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −37.0000 −1.39251
\(707\) 36.0000 1.35392
\(708\) −6.00000 −0.225494
\(709\) 49.0000 1.84023 0.920117 0.391644i \(-0.128094\pi\)
0.920117 + 0.391644i \(0.128094\pi\)
\(710\) −12.0000 −0.450352
\(711\) −4.00000 −0.150012
\(712\) −18.0000 −0.674579
\(713\) 12.0000 0.449404
\(714\) 9.00000 0.336817
\(715\) 10.0000 0.373979
\(716\) 12.0000 0.448461
\(717\) −29.0000 −1.08302
\(718\) −30.0000 −1.11959
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 1.00000 0.0372678
\(721\) −48.0000 −1.78761
\(722\) 17.0000 0.632674
\(723\) 10.0000 0.371904
\(724\) 10.0000 0.371647
\(725\) −1.00000 −0.0371391
\(726\) −14.0000 −0.519589
\(727\) −22.0000 −0.815935 −0.407967 0.912996i \(-0.633762\pi\)
−0.407967 + 0.912996i \(0.633762\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 9.00000 0.332877
\(732\) −5.00000 −0.184805
\(733\) −51.0000 −1.88373 −0.941864 0.335994i \(-0.890928\pi\)
−0.941864 + 0.335994i \(0.890928\pi\)
\(734\) −3.00000 −0.110732
\(735\) −2.00000 −0.0737711
\(736\) −4.00000 −0.147442
\(737\) 20.0000 0.736709
\(738\) −7.00000 −0.257674
\(739\) 1.00000 0.0367856 0.0183928 0.999831i \(-0.494145\pi\)
0.0183928 + 0.999831i \(0.494145\pi\)
\(740\) −1.00000 −0.0367607
\(741\) −12.0000 −0.440831
\(742\) −15.0000 −0.550667
\(743\) −3.00000 −0.110059 −0.0550297 0.998485i \(-0.517525\pi\)
−0.0550297 + 0.998485i \(0.517525\pi\)
\(744\) 3.00000 0.109985
\(745\) −18.0000 −0.659469
\(746\) 26.0000 0.951928
\(747\) 6.00000 0.219529
\(748\) −15.0000 −0.548454
\(749\) −6.00000 −0.219235
\(750\) −1.00000 −0.0365148
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 0 0
\(753\) 24.0000 0.874609
\(754\) 2.00000 0.0728357
\(755\) 10.0000 0.363937
\(756\) 3.00000 0.109109
\(757\) 30.0000 1.09037 0.545184 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) 32.0000 1.16229
\(759\) −20.0000 −0.725954
\(760\) −6.00000 −0.217643
\(761\) 25.0000 0.906249 0.453125 0.891447i \(-0.350309\pi\)
0.453125 + 0.891447i \(0.350309\pi\)
\(762\) −8.00000 −0.289809
\(763\) −15.0000 −0.543036
\(764\) −3.00000 −0.108536
\(765\) 3.00000 0.108465
\(766\) −20.0000 −0.722629
\(767\) −12.0000 −0.433295
\(768\) −1.00000 −0.0360844
\(769\) 38.0000 1.37032 0.685158 0.728395i \(-0.259733\pi\)
0.685158 + 0.728395i \(0.259733\pi\)
\(770\) 15.0000 0.540562
\(771\) −26.0000 −0.936367
\(772\) 18.0000 0.647834
\(773\) −17.0000 −0.611448 −0.305724 0.952120i \(-0.598898\pi\)
−0.305724 + 0.952120i \(0.598898\pi\)
\(774\) 3.00000 0.107833
\(775\) −3.00000 −0.107763
\(776\) −13.0000 −0.466673
\(777\) −3.00000 −0.107624
\(778\) −3.00000 −0.107555
\(779\) 42.0000 1.50481
\(780\) 2.00000 0.0716115
\(781\) 60.0000 2.14697
\(782\) −12.0000 −0.429119
\(783\) 1.00000 0.0357371
\(784\) 2.00000 0.0714286
\(785\) 11.0000 0.392607
\(786\) −10.0000 −0.356688
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −5.00000 −0.178005
\(790\) −4.00000 −0.142314
\(791\) −27.0000 −0.960009
\(792\) −5.00000 −0.177667
\(793\) −10.0000 −0.355110
\(794\) 18.0000 0.638796
\(795\) −5.00000 −0.177332
\(796\) −4.00000 −0.141776
\(797\) 40.0000 1.41687 0.708436 0.705775i \(-0.249401\pi\)
0.708436 + 0.705775i \(0.249401\pi\)
\(798\) −18.0000 −0.637193
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −18.0000 −0.635999
\(802\) −2.00000 −0.0706225
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 12.0000 0.422944
\(806\) 6.00000 0.211341
\(807\) 16.0000 0.563227
\(808\) −12.0000 −0.422159
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 1.00000 0.0351364
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 3.00000 0.105279
\(813\) −8.00000 −0.280572
\(814\) 5.00000 0.175250
\(815\) 11.0000 0.385313
\(816\) −3.00000 −0.105021
\(817\) −18.0000 −0.629740
\(818\) 6.00000 0.209785
\(819\) 6.00000 0.209657
\(820\) −7.00000 −0.244451
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 10.0000 0.348790
\(823\) 56.0000 1.95204 0.976019 0.217687i \(-0.0698512\pi\)
0.976019 + 0.217687i \(0.0698512\pi\)
\(824\) 16.0000 0.557386
\(825\) 5.00000 0.174078
\(826\) −18.0000 −0.626300
\(827\) 23.0000 0.799788 0.399894 0.916561i \(-0.369047\pi\)
0.399894 + 0.916561i \(0.369047\pi\)
\(828\) −4.00000 −0.139010
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 6.00000 0.208263
\(831\) 16.0000 0.555034
\(832\) −2.00000 −0.0693375
\(833\) 6.00000 0.207888
\(834\) −5.00000 −0.173136
\(835\) −24.0000 −0.830554
\(836\) 30.0000 1.03757
\(837\) 3.00000 0.103695
\(838\) −12.0000 −0.414533
\(839\) −26.0000 −0.897620 −0.448810 0.893627i \(-0.648152\pi\)
−0.448810 + 0.893627i \(0.648152\pi\)
\(840\) 3.00000 0.103510
\(841\) −28.0000 −0.965517
\(842\) 2.00000 0.0689246
\(843\) −2.00000 −0.0688837
\(844\) 19.0000 0.654007
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −42.0000 −1.44314
\(848\) 5.00000 0.171701
\(849\) −4.00000 −0.137280
\(850\) 3.00000 0.102899
\(851\) 4.00000 0.137118
\(852\) 12.0000 0.411113
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) −15.0000 −0.513289
\(855\) −6.00000 −0.205196
\(856\) 2.00000 0.0683586
\(857\) −3.00000 −0.102478 −0.0512390 0.998686i \(-0.516317\pi\)
−0.0512390 + 0.998686i \(0.516317\pi\)
\(858\) −10.0000 −0.341394
\(859\) −26.0000 −0.887109 −0.443554 0.896248i \(-0.646283\pi\)
−0.443554 + 0.896248i \(0.646283\pi\)
\(860\) 3.00000 0.102299
\(861\) −21.0000 −0.715678
\(862\) −7.00000 −0.238421
\(863\) 27.0000 0.919091 0.459545 0.888154i \(-0.348012\pi\)
0.459545 + 0.888154i \(0.348012\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 13.0000 0.442013
\(866\) −8.00000 −0.271851
\(867\) 8.00000 0.271694
\(868\) 9.00000 0.305480
\(869\) 20.0000 0.678454
\(870\) 1.00000 0.0339032
\(871\) 8.00000 0.271070
\(872\) 5.00000 0.169321
\(873\) −13.0000 −0.439983
\(874\) 24.0000 0.811812
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) 27.0000 0.911725 0.455863 0.890050i \(-0.349331\pi\)
0.455863 + 0.890050i \(0.349331\pi\)
\(878\) −29.0000 −0.978703
\(879\) −7.00000 −0.236104
\(880\) −5.00000 −0.168550
\(881\) −39.0000 −1.31394 −0.656972 0.753915i \(-0.728163\pi\)
−0.656972 + 0.753915i \(0.728163\pi\)
\(882\) 2.00000 0.0673435
\(883\) −3.00000 −0.100958 −0.0504790 0.998725i \(-0.516075\pi\)
−0.0504790 + 0.998725i \(0.516075\pi\)
\(884\) −6.00000 −0.201802
\(885\) −6.00000 −0.201688
\(886\) 14.0000 0.470339
\(887\) −7.00000 −0.235037 −0.117518 0.993071i \(-0.537494\pi\)
−0.117518 + 0.993071i \(0.537494\pi\)
\(888\) 1.00000 0.0335578
\(889\) −24.0000 −0.804934
\(890\) −18.0000 −0.603361
\(891\) −5.00000 −0.167506
\(892\) 7.00000 0.234377
\(893\) 0 0
\(894\) 18.0000 0.602010
\(895\) 12.0000 0.401116
\(896\) −3.00000 −0.100223
\(897\) −8.00000 −0.267112
\(898\) −36.0000 −1.20134
\(899\) 3.00000 0.100056
\(900\) 1.00000 0.0333333
\(901\) 15.0000 0.499722
\(902\) 35.0000 1.16537
\(903\) 9.00000 0.299501
\(904\) 9.00000 0.299336
\(905\) 10.0000 0.332411
\(906\) −10.0000 −0.332228
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −21.0000 −0.696909
\(909\) −12.0000 −0.398015
\(910\) 6.00000 0.198898
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 6.00000 0.198680
\(913\) −30.0000 −0.992855
\(914\) −5.00000 −0.165385
\(915\) −5.00000 −0.165295
\(916\) 12.0000 0.396491
\(917\) −30.0000 −0.990687
\(918\) −3.00000 −0.0990148
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) −4.00000 −0.131876
\(921\) 2.00000 0.0659022
\(922\) 21.0000 0.691598
\(923\) 24.0000 0.789970
\(924\) −15.0000 −0.493464
\(925\) −1.00000 −0.0328798
\(926\) 34.0000 1.11731
\(927\) 16.0000 0.525509
\(928\) −1.00000 −0.0328266
\(929\) −1.00000 −0.0328089 −0.0164045 0.999865i \(-0.505222\pi\)
−0.0164045 + 0.999865i \(0.505222\pi\)
\(930\) 3.00000 0.0983739
\(931\) −12.0000 −0.393284
\(932\) 0 0
\(933\) 31.0000 1.01489
\(934\) −15.0000 −0.490815
\(935\) −15.0000 −0.490552
\(936\) −2.00000 −0.0653720
\(937\) −40.0000 −1.30674 −0.653372 0.757037i \(-0.726646\pi\)
−0.653372 + 0.757037i \(0.726646\pi\)
\(938\) 12.0000 0.391814
\(939\) 30.0000 0.979013
\(940\) 0 0
\(941\) 46.0000 1.49956 0.749779 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) −11.0000 −0.358399
\(943\) 28.0000 0.911805
\(944\) 6.00000 0.195283
\(945\) 3.00000 0.0975900
\(946\) −15.0000 −0.487692
\(947\) −29.0000 −0.942373 −0.471187 0.882034i \(-0.656174\pi\)
−0.471187 + 0.882034i \(0.656174\pi\)
\(948\) 4.00000 0.129914
\(949\) 0 0
\(950\) −6.00000 −0.194666
\(951\) 27.0000 0.875535
\(952\) −9.00000 −0.291692
\(953\) −40.0000 −1.29573 −0.647864 0.761756i \(-0.724337\pi\)
−0.647864 + 0.761756i \(0.724337\pi\)
\(954\) 5.00000 0.161881
\(955\) −3.00000 −0.0970777
\(956\) 29.0000 0.937927
\(957\) −5.00000 −0.161627
\(958\) −16.0000 −0.516937
\(959\) 30.0000 0.968751
\(960\) −1.00000 −0.0322749
\(961\) −22.0000 −0.709677
\(962\) 2.00000 0.0644826
\(963\) 2.00000 0.0644491
\(964\) −10.0000 −0.322078
\(965\) 18.0000 0.579441
\(966\) −12.0000 −0.386094
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 14.0000 0.449977
\(969\) 18.0000 0.578243
\(970\) −13.0000 −0.417405
\(971\) −35.0000 −1.12320 −0.561602 0.827408i \(-0.689815\pi\)
−0.561602 + 0.827408i \(0.689815\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −15.0000 −0.480878
\(974\) −6.00000 −0.192252
\(975\) 2.00000 0.0640513
\(976\) 5.00000 0.160046
\(977\) −25.0000 −0.799821 −0.399910 0.916554i \(-0.630959\pi\)
−0.399910 + 0.916554i \(0.630959\pi\)
\(978\) −11.0000 −0.351741
\(979\) 90.0000 2.87641
\(980\) 2.00000 0.0638877
\(981\) 5.00000 0.159638
\(982\) 4.00000 0.127645
\(983\) −29.0000 −0.924956 −0.462478 0.886631i \(-0.653040\pi\)
−0.462478 + 0.886631i \(0.653040\pi\)
\(984\) 7.00000 0.223152
\(985\) −2.00000 −0.0637253
\(986\) −3.00000 −0.0955395
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) −12.0000 −0.381578
\(990\) −5.00000 −0.158910
\(991\) 7.00000 0.222362 0.111181 0.993800i \(-0.464537\pi\)
0.111181 + 0.993800i \(0.464537\pi\)
\(992\) −3.00000 −0.0952501
\(993\) 20.0000 0.634681
\(994\) 36.0000 1.14185
\(995\) −4.00000 −0.126809
\(996\) −6.00000 −0.190117
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) 0 0
\(999\) 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.j.1.1 1
3.2 odd 2 3330.2.a.b.1.1 1
4.3 odd 2 8880.2.a.bc.1.1 1
5.4 even 2 5550.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.j.1.1 1 1.1 even 1 trivial
3330.2.a.b.1.1 1 3.2 odd 2
5550.2.a.t.1.1 1 5.4 even 2
8880.2.a.bc.1.1 1 4.3 odd 2