Properties

Label 5550.2.a.t.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.3169731218\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5550.1

$q$-expansion

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

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
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −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 \(0.410544\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 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
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 5.00000 1.06600
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −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 \(0.529597\pi\)
\(30\) 0 0
\(31\) −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
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) 6.00000 0.973329
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −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 \(-0.573463\pi\)
−0.228748 + 0.973486i \(0.573463\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) −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
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 2.00000 0.277350
\(53\) −5.00000 −0.686803 −0.343401 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −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 \(0.372279\pi\)
\(60\) 0 0
\(61\) 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
\(65\) 0 0
\(66\) 5.00000 0.615457
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −3.00000 −0.363803
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −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
\(75\) 0 0
\(76\) −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 \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 7.00000 0.773021
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) 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 \(0.903076\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 4.00000 0.417029
\(93\) −3.00000 −0.311086
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 13.0000 1.31995 0.659975 0.751288i \(-0.270567\pi\)
0.659975 + 0.751288i \(0.270567\pi\)
\(98\) −2.00000 −0.202031
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) −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 \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 5.00000 0.485643
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 1.00000 0.0962250
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 3.00000 0.283473
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 6.00000 0.561951
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) 2.00000 0.184900
\(118\) −6.00000 −0.552345
\(119\) −9.00000 −0.825029
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −5.00000 −0.452679
\(123\) −7.00000 −0.631169
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) −3.00000 −0.267261
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −3.00000 −0.264135
\(130\) 0 0
\(131\) 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
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −4.00000 −0.340503
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) −10.0000 −0.836242
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 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 \(0.763904\pi\)
\(150\) 0 0
\(151\) 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
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −11.0000 −0.877896 −0.438948 0.898513i \(-0.644649\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 4.00000 0.318223
\(159\) −5.00000 −0.396526
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) −1.00000 −0.0785674
\(163\) −11.0000 −0.861586 −0.430793 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) −7.00000 −0.546608
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) −3.00000 −0.231455
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) −3.00000 −0.228748
\(173\) −13.0000 −0.988372 −0.494186 0.869356i \(-0.664534\pi\)
−0.494186 + 0.869356i \(0.664534\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) −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 \(0.351972\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) 3.00000 0.219971
\(187\) 15.0000 1.09691
\(188\) 0 0
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) −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 \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) −13.0000 −0.933346
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 5.00000 0.355335
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 12.0000 0.844317
\(203\) −3.00000 −0.210559
\(204\) −3.00000 −0.210042
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 4.00000 0.278019
\(208\) 2.00000 0.138675
\(209\) 30.0000 2.07514
\(210\) 0 0
\(211\) 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
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −9.00000 −0.610960
\(218\) −5.00000 −0.338643
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) −1.00000 −0.0671156
\(223\) −7.00000 −0.468755 −0.234377 0.972146i \(-0.575305\pi\)
−0.234377 + 0.972146i \(0.575305\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) 9.00000 0.598671
\(227\) 21.0000 1.39382 0.696909 0.717159i \(-0.254558\pi\)
0.696909 + 0.717159i \(0.254558\pi\)
\(228\) −6.00000 −0.397360
\(229\) 12.0000 0.792982 0.396491 0.918039i \(-0.370228\pi\)
0.396491 + 0.918039i \(0.370228\pi\)
\(230\) 0 0
\(231\) −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 \(0.112743\pi\)
\(240\) 0 0
\(241\) −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
\(245\) 0 0
\(246\) 7.00000 0.446304
\(247\) −12.0000 −0.763542
\(248\) 3.00000 0.190500
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −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
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −26.0000 −1.62184 −0.810918 0.585160i \(-0.801032\pi\)
−0.810918 + 0.585160i \(0.801032\pi\)
\(258\) 3.00000 0.186772
\(259\) 3.00000 0.186411
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) −10.0000 −0.617802
\(263\) −5.00000 −0.308313 −0.154157 0.988046i \(-0.549266\pi\)
−0.154157 + 0.988046i \(0.549266\pi\)
\(264\) 5.00000 0.307729
\(265\) 0 0
\(266\) 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 \(0.662189\pi\)
\(270\) 0 0
\(271\) 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
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 16.0000 0.961347 0.480673 0.876900i \(-0.340392\pi\)
0.480673 + 0.876900i \(0.340392\pi\)
\(278\) −5.00000 −0.299880
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) 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 \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 10.0000 0.591312
\(287\) −21.0000 −1.23959
\(288\) −1.00000 −0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 13.0000 0.762073
\(292\) 0 0
\(293\) −7.00000 −0.408944 −0.204472 0.978872i \(-0.565548\pi\)
−0.204472 + 0.978872i \(0.565548\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) −5.00000 −0.290129
\(298\) 18.0000 1.04271
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) −9.00000 −0.518751
\(302\) −10.0000 −0.575435
\(303\) −12.0000 −0.689382
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) 3.00000 0.171499
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) −15.0000 −0.854704
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −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 \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) 11.0000 0.620766
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 27.0000 1.51647 0.758236 0.651981i \(-0.226062\pi\)
0.758236 + 0.651981i \(0.226062\pi\)
\(318\) 5.00000 0.280386
\(319\) 5.00000 0.279946
\(320\) 0 0
\(321\) −2.00000 −0.111629
\(322\) −12.0000 −0.668734
\(323\) 18.0000 1.00155
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 11.0000 0.609234
\(327\) 5.00000 0.276501
\(328\) 7.00000 0.386510
\(329\) 0 0
\(330\) 0 0
\(331\) −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
\(335\) 0 0
\(336\) 3.00000 0.163663
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 9.00000 0.489535
\(339\) −9.00000 −0.488813
\(340\) 0 0
\(341\) 15.0000 0.812296
\(342\) 6.00000 0.324443
\(343\) −15.0000 −0.809924
\(344\) 3.00000 0.161749
\(345\) 0 0
\(346\) 13.0000 0.698884
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) −1.00000 −0.0536056
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 5.00000 0.266501
\(353\) 37.0000 1.96931 0.984656 0.174509i \(-0.0558337\pi\)
0.984656 + 0.174509i \(0.0558337\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) −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 \(0.790788\pi\)
\(360\) 0 0
\(361\) 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 \(-0.475051\pi\)
0.0782994 + 0.996930i \(0.475051\pi\)
\(368\) 4.00000 0.208514
\(369\) −7.00000 −0.364405
\(370\) 0 0
\(371\) −15.0000 −0.778761
\(372\) −3.00000 −0.155543
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −15.0000 −0.775632
\(375\) 0 0
\(376\) 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 \(0.192934\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 3.00000 0.153493
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 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 \(0.524232\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) −2.00000 −0.101015
\(393\) 10.0000 0.504433
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 4.00000 0.200502
\(399\) −18.0000 −0.901127
\(400\) 0 0
\(401\) −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
\(405\) 0 0
\(406\) 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 \(0.452607\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) −16.0000 −0.788263
\(413\) 18.0000 0.885722
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) −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 \(0.594693\pi\)
\(420\) 0 0
\(421\) 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
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) 15.0000 0.725901
\(428\) −2.00000 −0.0966736
\(429\) −10.0000 −0.482805
\(430\) 0 0
\(431\) −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 \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) 9.00000 0.432014
\(435\) 0 0
\(436\) 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 \(0.743291\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 6.00000 0.285391
\(443\) −14.0000 −0.665160 −0.332580 0.943075i \(-0.607919\pi\)
−0.332580 + 0.943075i \(0.607919\pi\)
\(444\) 1.00000 0.0474579
\(445\) 0 0
\(446\) 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 \(0.823080\pi\)
\(450\) 0 0
\(451\) 35.0000 1.64809
\(452\) −9.00000 −0.423324
\(453\) 10.0000 0.469841
\(454\) −21.0000 −0.985579
\(455\) 0 0
\(456\) 6.00000 0.280976
\(457\) 5.00000 0.233890 0.116945 0.993138i \(-0.462690\pi\)
0.116945 + 0.993138i \(0.462690\pi\)
\(458\) −12.0000 −0.560723
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) 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 \(-0.789949\pi\)
−0.790057 + 0.613033i \(0.789949\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) 0 0
\(467\) 15.0000 0.694117 0.347059 0.937843i \(-0.387180\pi\)
0.347059 + 0.937843i \(0.387180\pi\)
\(468\) 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
\(475\) 0 0
\(476\) −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 \(0.619112\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 10.0000 0.455488
\(483\) 12.0000 0.546019
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 6.00000 0.271886 0.135943 0.990717i \(-0.456594\pi\)
0.135943 + 0.990717i \(0.456594\pi\)
\(488\) −5.00000 −0.226339
\(489\) −11.0000 −0.497437
\(490\) 0 0
\(491\) 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
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) −36.0000 −1.61482
\(498\) 6.00000 0.268866
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) 24.0000 1.07117
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) 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 \(0.321482\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −6.00000 −0.264906
\(514\) 26.0000 1.14681
\(515\) 0 0
\(516\) −3.00000 −0.132068
\(517\) 0 0
\(518\) −3.00000 −0.131812
\(519\) −13.0000 −0.570637
\(520\) 0 0
\(521\) 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 \(-0.161164\pi\)
0.874539 + 0.484955i \(0.161164\pi\)
\(524\) 10.0000 0.436852
\(525\) 0 0
\(526\) 5.00000 0.218010
\(527\) 9.00000 0.392046
\(528\) −5.00000 −0.217597
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) −18.0000 −0.780399
\(533\) −14.0000 −0.606407
\(534\) 18.0000 0.778936
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 12.0000 0.517838
\(538\) 16.0000 0.689809
\(539\) −10.0000 −0.430730
\(540\) 0 0
\(541\) −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
\(545\) 0 0
\(546\) −6.00000 −0.256776
\(547\) −19.0000 −0.812381 −0.406191 0.913788i \(-0.633143\pi\)
−0.406191 + 0.913788i \(0.633143\pi\)
\(548\) 10.0000 0.427179
\(549\) 5.00000 0.213395
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) −4.00000 −0.170251
\(553\) −12.0000 −0.510292
\(554\) −16.0000 −0.679775
\(555\) 0 0
\(556\) 5.00000 0.212047
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 3.00000 0.127000
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) 15.0000 0.633300
\(562\) −2.00000 −0.0843649
\(563\) 17.0000 0.716465 0.358232 0.933632i \(-0.383380\pi\)
0.358232 + 0.933632i \(0.383380\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 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 \(0.0854179\pi\)
\(570\) 0 0
\(571\) −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
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 8.00000 0.332756
\(579\) −18.0000 −0.748054
\(580\) 0 0
\(581\) −18.0000 −0.746766
\(582\) −13.0000 −0.538867
\(583\) 25.0000 1.03539
\(584\) 0 0
\(585\) 0 0
\(586\) 7.00000 0.289167
\(587\) −1.00000 −0.0412744 −0.0206372 0.999787i \(-0.506569\pi\)
−0.0206372 + 0.999787i \(0.506569\pi\)
\(588\) 2.00000 0.0824786
\(589\) 18.0000 0.741677
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 1.00000 0.0410997
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 5.00000 0.205152
\(595\) 0 0
\(596\) −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 \(0.726802\pi\)
\(600\) 0 0
\(601\) −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
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) 6.00000 0.243332
\(609\) −3.00000 −0.121566
\(610\) 0 0
\(611\) 0 0
\(612\) −3.00000 −0.121268
\(613\) 43.0000 1.73675 0.868377 0.495905i \(-0.165164\pi\)
0.868377 + 0.495905i \(0.165164\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 15.0000 0.604367
\(617\) −34.0000 −1.36879 −0.684394 0.729112i \(-0.739933\pi\)
−0.684394 + 0.729112i \(0.739933\pi\)
\(618\) 16.0000 0.643614
\(619\) −27.0000 −1.08522 −0.542611 0.839984i \(-0.682564\pi\)
−0.542611 + 0.839984i \(0.682564\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 31.0000 1.24299
\(623\) −54.0000 −2.16346
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −30.0000 −1.19904
\(627\) 30.0000 1.19808
\(628\) −11.0000 −0.438948
\(629\) −3.00000 −0.119618
\(630\) 0 0
\(631\) 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
\(635\) 0 0
\(636\) −5.00000 −0.198263
\(637\) 4.00000 0.158486
\(638\) −5.00000 −0.197952
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −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 \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) −18.0000 −0.708201
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −30.0000 −1.17760
\(650\) 0 0
\(651\) −9.00000 −0.352738
\(652\) −11.0000 −0.430793
\(653\) −16.0000 −0.626128 −0.313064 0.949732i \(-0.601356\pi\)
−0.313064 + 0.949732i \(0.601356\pi\)
\(654\) −5.00000 −0.195515
\(655\) 0 0
\(656\) −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 \(0.399123\pi\)
\(660\) 0 0
\(661\) −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
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) −4.00000 −0.154881
\(668\) 24.0000 0.928588
\(669\) −7.00000 −0.270636
\(670\) 0 0
\(671\) −25.0000 −0.965114
\(672\) −3.00000 −0.115728
\(673\) −48.0000 −1.85026 −0.925132 0.379646i \(-0.876046\pi\)
−0.925132 + 0.379646i \(0.876046\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −10.0000 −0.384331 −0.192166 0.981363i \(-0.561551\pi\)
−0.192166 + 0.981363i \(0.561551\pi\)
\(678\) 9.00000 0.345643
\(679\) 39.0000 1.49668
\(680\) 0 0
\(681\) 21.0000 0.804722
\(682\) −15.0000 −0.574380
\(683\) 29.0000 1.10965 0.554827 0.831966i \(-0.312784\pi\)
0.554827 + 0.831966i \(0.312784\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) 15.0000 0.572703
\(687\) 12.0000 0.457829
\(688\) −3.00000 −0.114374
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) −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
\(695\) 0 0
\(696\) 1.00000 0.0379049
\(697\) 21.0000 0.795432
\(698\) 20.0000 0.757011
\(699\) 0 0
\(700\) 0 0
\(701\) −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 \(0.128094\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 18.0000 0.674579
\(713\) −12.0000 −0.449404
\(714\) 9.00000 0.336817
\(715\) 0 0
\(716\) 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 \(0.608954\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) −17.0000 −0.632674
\(723\) −10.0000 −0.371904
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) −14.0000 −0.519589
\(727\) 22.0000 0.815935 0.407967 0.912996i \(-0.366238\pi\)
0.407967 + 0.912996i \(0.366238\pi\)
\(728\) −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 \(-0.109072\pi\)
0.941864 + 0.335994i \(0.109072\pi\)
\(734\) −3.00000 −0.110732
\(735\) 0 0
\(736\) −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 \(0.494145\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 15.0000 0.550667
\(743\) 3.00000 0.110059 0.0550297 0.998485i \(-0.482475\pi\)
0.0550297 + 0.998485i \(0.482475\pi\)
\(744\) 3.00000 0.109985
\(745\) 0 0
\(746\) 26.0000 0.951928
\(747\) −6.00000 −0.219529
\(748\) 15.0000 0.548454
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) −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
\(755\) 0 0
\(756\) 3.00000 0.109109
\(757\) −30.0000 −1.09037 −0.545184 0.838316i \(-0.683540\pi\)
−0.545184 + 0.838316i \(0.683540\pi\)
\(758\) −32.0000 −1.16229
\(759\) −20.0000 −0.725954
\(760\) 0 0
\(761\) 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
\(765\) 0 0
\(766\) −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 \(0.259733\pi\)
\(770\) 0 0
\(771\) −26.0000 −0.936367
\(772\) −18.0000 −0.647834
\(773\) 17.0000 0.611448 0.305724 0.952120i \(-0.401102\pi\)
0.305724 + 0.952120i \(0.401102\pi\)
\(774\) 3.00000 0.107833
\(775\) 0 0
\(776\) −13.0000 −0.466673
\(777\) 3.00000 0.107624
\(778\) 3.00000 0.107555
\(779\) 42.0000 1.50481
\(780\) 0 0
\(781\) 60.0000 2.14697
\(782\) 12.0000 0.429119
\(783\) −1.00000 −0.0357371
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) −10.0000 −0.356688
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 2.00000 0.0712470
\(789\) −5.00000 −0.178005
\(790\) 0 0
\(791\) −27.0000 −0.960009
\(792\) 5.00000 0.177667
\(793\) 10.0000 0.355110
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −40.0000 −1.41687 −0.708436 0.705775i \(-0.750599\pi\)
−0.708436 + 0.705775i \(0.750599\pi\)
\(798\) 18.0000 0.637193
\(799\) 0 0
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) 2.00000 0.0706225
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 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 \(0.567655\pi\)
\(810\) 0 0
\(811\) 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
\(815\) 0 0
\(816\) −3.00000 −0.105021
\(817\) 18.0000 0.629740
\(818\) −6.00000 −0.209785
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 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 \(-0.930149\pi\)
−0.976019 + 0.217687i \(0.930149\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) −18.0000 −0.626300
\(827\) −23.0000 −0.799788 −0.399894 0.916561i \(-0.630953\pi\)
−0.399894 + 0.916561i \(0.630953\pi\)
\(828\) 4.00000 0.139010
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 0 0
\(831\) 16.0000 0.555034
\(832\) 2.00000 0.0693375
\(833\) −6.00000 −0.207888
\(834\) −5.00000 −0.173136
\(835\) 0 0
\(836\) 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 \(0.648152\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −2.00000 −0.0689246
\(843\) 2.00000 0.0688837
\(844\) 19.0000 0.654007
\(845\) 0 0
\(846\) 0 0
\(847\) 42.0000 1.44314
\(848\) −5.00000 −0.171701
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) −12.0000 −0.411113
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) −15.0000 −0.513289
\(855\) 0 0
\(856\) 2.00000 0.0683586
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 10.0000 0.341394
\(859\) −26.0000 −0.887109 −0.443554 0.896248i \(-0.646283\pi\)
−0.443554 + 0.896248i \(0.646283\pi\)
\(860\) 0 0
\(861\) −21.0000 −0.715678
\(862\) 7.00000 0.238421
\(863\) −27.0000 −0.919091 −0.459545 0.888154i \(-0.651988\pi\)
−0.459545 + 0.888154i \(0.651988\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −8.00000 −0.271851
\(867\) −8.00000 −0.271694
\(868\) −9.00000 −0.305480
\(869\) 20.0000 0.678454
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −5.00000 −0.169321
\(873\) 13.0000 0.439983
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) −27.0000 −0.911725 −0.455863 0.890050i \(-0.650669\pi\)
−0.455863 + 0.890050i \(0.650669\pi\)
\(878\) 29.0000 0.978703
\(879\) −7.00000 −0.236104
\(880\) 0 0
\(881\) −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 \(-0.483925\pi\)
0.0504790 + 0.998725i \(0.483925\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) 14.0000 0.470339
\(887\) 7.00000 0.235037 0.117518 0.993071i \(-0.462506\pi\)
0.117518 + 0.993071i \(0.462506\pi\)
\(888\) −1.00000 −0.0335578
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) −7.00000 −0.234377
\(893\) 0 0
\(894\) 18.0000 0.602010
\(895\) 0 0
\(896\) −3.00000 −0.100223
\(897\) 8.00000 0.267112
\(898\) 36.0000 1.20134
\(899\) 3.00000 0.100056
\(900\) 0 0
\(901\) 15.0000 0.499722
\(902\) −35.0000 −1.16537
\(903\) −9.00000 −0.299501
\(904\) 9.00000 0.299336
\(905\) 0 0
\(906\) −10.0000 −0.332228
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 21.0000 0.696909
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −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
\(915\) 0 0
\(916\) 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 \(0.542123\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) −21.0000 −0.691598
\(923\) −24.0000 −0.789970
\(924\) −15.0000 −0.493464
\(925\) 0 0
\(926\) 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 \(0.505222\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 0 0
\(933\) −31.0000 −1.01489
\(934\) −15.0000 −0.490815
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 40.0000 1.30674 0.653372 0.757037i \(-0.273354\pi\)
0.653372 + 0.757037i \(0.273354\pi\)
\(938\) −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
\(945\) 0 0
\(946\) −15.0000 −0.487692
\(947\) 29.0000 0.942373 0.471187 0.882034i \(-0.343826\pi\)
0.471187 + 0.882034i \(0.343826\pi\)
\(948\) −4.00000 −0.129914
\(949\) 0 0
\(950\) 0 0
\(951\) 27.0000 0.875535
\(952\) 9.00000 0.291692
\(953\) 40.0000 1.29573 0.647864 0.761756i \(-0.275663\pi\)
0.647864 + 0.761756i \(0.275663\pi\)
\(954\) 5.00000 0.161881
\(955\) 0 0
\(956\) 29.0000 0.937927
\(957\) 5.00000 0.161627
\(958\) 16.0000 0.516937
\(959\) 30.0000 0.968751
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −2.00000 −0.0644826
\(963\) −2.00000 −0.0644491
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −14.0000 −0.449977
\(969\) 18.0000 0.578243
\(970\) 0 0
\(971\) −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
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) 25.0000 0.799821 0.399910 0.916554i \(-0.369041\pi\)
0.399910 + 0.916554i \(0.369041\pi\)
\(978\) 11.0000 0.351741
\(979\) 90.0000 2.87641
\(980\) 0 0
\(981\) 5.00000 0.159638
\(982\) −4.00000 −0.127645
\(983\) 29.0000 0.924956 0.462478 0.886631i \(-0.346960\pi\)
0.462478 + 0.886631i \(0.346960\pi\)
\(984\) 7.00000 0.223152
\(985\) 0 0
\(986\) −3.00000 −0.0955395
\(987\) 0 0
\(988\) −12.0000 −0.381771
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 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
\(995\) 0 0
\(996\) −6.00000 −0.190117
\(997\) 18.0000 0.570066 0.285033 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) 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 5550.2.a.t.1.1 1
5.4 even 2 1110.2.a.j.1.1 1
15.14 odd 2 3330.2.a.b.1.1 1
20.19 odd 2 8880.2.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.j.1.1 1 5.4 even 2
3330.2.a.b.1.1 1 15.14 odd 2
5550.2.a.t.1.1 1 1.1 even 1 trivial
8880.2.a.bc.1.1 1 20.19 odd 2