Properties

Label 1950.2.a.bb.1.1
Level $1950$
Weight $2$
Character 1950.1
Self dual yes
Analytic conductor $15.571$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.5708283941\)
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 \(\chi\) \(=\) 1950.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} +3.00000 q^{21} -3.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} +3.00000 q^{28} +5.00000 q^{29} -3.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} -12.0000 q^{37} +1.00000 q^{39} +2.00000 q^{41} +3.00000 q^{42} +4.00000 q^{43} -3.00000 q^{44} +4.00000 q^{46} +3.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} +3.00000 q^{51} +1.00000 q^{52} +9.00000 q^{53} +1.00000 q^{54} +3.00000 q^{56} +5.00000 q^{58} +15.0000 q^{59} -3.00000 q^{61} -3.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} -7.00000 q^{67} +3.00000 q^{68} +4.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} -16.0000 q^{73} -12.0000 q^{74} -9.00000 q^{77} +1.00000 q^{78} +1.00000 q^{81} +2.00000 q^{82} -1.00000 q^{83} +3.00000 q^{84} +4.00000 q^{86} +5.00000 q^{87} -3.00000 q^{88} +3.00000 q^{91} +4.00000 q^{92} -3.00000 q^{93} +3.00000 q^{94} +1.00000 q^{96} -2.00000 q^{97} +2.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 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
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) 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
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) −3.00000 −0.639602
\(23\) 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
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 3.00000 0.566947
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 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
\(33\) −3.00000 −0.522233
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −12.0000 −1.97279 −0.986394 0.164399i \(-0.947432\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 3.00000 0.462910
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 1.00000 0.138675
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) 5.00000 0.656532
\(59\) 15.0000 1.95283 0.976417 0.215894i \(-0.0692665\pi\)
0.976417 + 0.215894i \(0.0692665\pi\)
\(60\) 0 0
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) −3.00000 −0.381000
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.00000 −0.369274
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 3.00000 0.363803
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) −16.0000 −1.87266 −0.936329 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −12.0000 −1.39497
\(75\) 0 0
\(76\) 0 0
\(77\) −9.00000 −1.02565
\(78\) 1.00000 0.113228
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −1.00000 −0.109764 −0.0548821 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 5.00000 0.536056
\(88\) −3.00000 −0.319801
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) 4.00000 0.417029
\(93\) −3.00000 −0.311086
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 2.00000 0.202031
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 3.00000 0.297044
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) 3.00000 0.283473
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.00000 0.464238
\(117\) 1.00000 0.0924500
\(118\) 15.0000 1.38086
\(119\) 9.00000 0.825029
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −3.00000 −0.271607
\(123\) 2.00000 0.180334
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) −3.00000 −0.261116
\(133\) 0 0
\(134\) −7.00000 −0.604708
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 4.00000 0.340503
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) −8.00000 −0.671345
\(143\) −3.00000 −0.250873
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −16.0000 −1.32417
\(147\) 2.00000 0.164957
\(148\) −12.0000 −0.986394
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 17.0000 1.38344 0.691720 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) −9.00000 −0.725241
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) 0 0
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −1.00000 −0.0776151
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 3.00000 0.231455
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −1.00000 −0.0760286 −0.0380143 0.999277i \(-0.512103\pi\)
−0.0380143 + 0.999277i \(0.512103\pi\)
\(174\) 5.00000 0.379049
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 15.0000 1.12747
\(178\) 0 0
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 3.00000 0.222375
\(183\) −3.00000 −0.221766
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −3.00000 −0.219971
\(187\) −9.00000 −0.658145
\(188\) 3.00000 0.218797
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 1.00000 0.0721688
\(193\) −26.0000 −1.87152 −0.935760 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −3.00000 −0.213201
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) −3.00000 −0.211079
\(203\) 15.0000 1.05279
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) −6.00000 −0.418040
\(207\) 4.00000 0.278019
\(208\) 1.00000 0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 9.00000 0.618123
\(213\) −8.00000 −0.548151
\(214\) −2.00000 −0.136717
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −9.00000 −0.610960
\(218\) 10.0000 0.677285
\(219\) −16.0000 −1.08118
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) −12.0000 −0.805387
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 3.00000 0.200446
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) 13.0000 0.862840 0.431420 0.902151i \(-0.358013\pi\)
0.431420 + 0.902151i \(0.358013\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −9.00000 −0.592157
\(232\) 5.00000 0.328266
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) 15.0000 0.976417
\(237\) 0 0
\(238\) 9.00000 0.583383
\(239\) −5.00000 −0.323423 −0.161712 0.986838i \(-0.551701\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) −2.00000 −0.128565
\(243\) 1.00000 0.0641500
\(244\) −3.00000 −0.192055
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 0 0
\(248\) −3.00000 −0.190500
\(249\) −1.00000 −0.0633724
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 3.00000 0.188982
\(253\) −12.0000 −0.754434
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −27.0000 −1.68421 −0.842107 0.539311i \(-0.818685\pi\)
−0.842107 + 0.539311i \(0.818685\pi\)
\(258\) 4.00000 0.249029
\(259\) −36.0000 −2.23693
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) −18.0000 −1.11204
\(263\) −26.0000 −1.60323 −0.801614 0.597841i \(-0.796025\pi\)
−0.801614 + 0.597841i \(0.796025\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −7.00000 −0.427593
\(269\) 5.00000 0.304855 0.152428 0.988315i \(-0.451291\pi\)
0.152428 + 0.988315i \(0.451291\pi\)
\(270\) 0 0
\(271\) −3.00000 −0.182237 −0.0911185 0.995840i \(-0.529044\pi\)
−0.0911185 + 0.995840i \(0.529044\pi\)
\(272\) 3.00000 0.181902
\(273\) 3.00000 0.181568
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 10.0000 0.599760
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 3.00000 0.178647
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −3.00000 −0.177394
\(287\) 6.00000 0.354169
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) −16.0000 −0.936329
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) −12.0000 −0.697486
\(297\) −3.00000 −0.174078
\(298\) 10.0000 0.579284
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 17.0000 0.978240
\(303\) −3.00000 −0.172345
\(304\) 0 0
\(305\) 0 0
\(306\) 3.00000 0.171499
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) −9.00000 −0.512823
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) 2.00000 0.113410 0.0567048 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) 1.00000 0.0566139
\(313\) 19.0000 1.07394 0.536972 0.843600i \(-0.319568\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) −7.00000 −0.395033
\(315\) 0 0
\(316\) 0 0
\(317\) 28.0000 1.57264 0.786318 0.617822i \(-0.211985\pi\)
0.786318 + 0.617822i \(0.211985\pi\)
\(318\) 9.00000 0.504695
\(319\) −15.0000 −0.839839
\(320\) 0 0
\(321\) −2.00000 −0.111629
\(322\) 12.0000 0.668734
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 10.0000 0.553001
\(328\) 2.00000 0.110432
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −1.00000 −0.0548821
\(333\) −12.0000 −0.657596
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 3.00000 0.163663
\(337\) −27.0000 −1.47078 −0.735392 0.677642i \(-0.763002\pi\)
−0.735392 + 0.677642i \(0.763002\pi\)
\(338\) 1.00000 0.0543928
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 9.00000 0.487377
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −1.00000 −0.0537603
\(347\) 18.0000 0.966291 0.483145 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 5.00000 0.268028
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) −3.00000 −0.159901
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 15.0000 0.797241
\(355\) 0 0
\(356\) 0 0
\(357\) 9.00000 0.476331
\(358\) 10.0000 0.528516
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 17.0000 0.893500
\(363\) −2.00000 −0.104973
\(364\) 3.00000 0.157243
\(365\) 0 0
\(366\) −3.00000 −0.156813
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 4.00000 0.208514
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 27.0000 1.40177
\(372\) −3.00000 −0.155543
\(373\) 29.0000 1.50156 0.750782 0.660551i \(-0.229677\pi\)
0.750782 + 0.660551i \(0.229677\pi\)
\(374\) −9.00000 −0.465379
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) 5.00000 0.257513
\(378\) 3.00000 0.154303
\(379\) 15.0000 0.770498 0.385249 0.922813i \(-0.374116\pi\)
0.385249 + 0.922813i \(0.374116\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) −18.0000 −0.920960
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −26.0000 −1.32337
\(387\) 4.00000 0.203331
\(388\) −2.00000 −0.101535
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 2.00000 0.101015
\(393\) −18.0000 −0.907980
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) −3.00000 −0.150756
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) −7.00000 −0.349128
\(403\) −3.00000 −0.149441
\(404\) −3.00000 −0.149256
\(405\) 0 0
\(406\) 15.0000 0.744438
\(407\) 36.0000 1.78445
\(408\) 3.00000 0.148522
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) −6.00000 −0.295599
\(413\) 45.0000 2.21431
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 10.0000 0.489702
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −28.0000 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) 12.0000 0.584151
\(423\) 3.00000 0.145865
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) −9.00000 −0.435541
\(428\) −2.00000 −0.0966736
\(429\) −3.00000 −0.144841
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) −9.00000 −0.432014
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 0 0
\(438\) −16.0000 −0.764510
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 3.00000 0.142695
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −12.0000 −0.569495
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 10.0000 0.472984
\(448\) 3.00000 0.141737
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 14.0000 0.658505
\(453\) 17.0000 0.798730
\(454\) 13.0000 0.610120
\(455\) 0 0
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −10.0000 −0.467269
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) −38.0000 −1.76984 −0.884918 0.465746i \(-0.845786\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) −9.00000 −0.418718
\(463\) 9.00000 0.418265 0.209133 0.977887i \(-0.432936\pi\)
0.209133 + 0.977887i \(0.432936\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −32.0000 −1.48078 −0.740392 0.672176i \(-0.765360\pi\)
−0.740392 + 0.672176i \(0.765360\pi\)
\(468\) 1.00000 0.0462250
\(469\) −21.0000 −0.969690
\(470\) 0 0
\(471\) −7.00000 −0.322543
\(472\) 15.0000 0.690431
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 0 0
\(476\) 9.00000 0.412514
\(477\) 9.00000 0.412082
\(478\) −5.00000 −0.228695
\(479\) 15.0000 0.685367 0.342684 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 22.0000 1.00207
\(483\) 12.0000 0.546019
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 43.0000 1.94852 0.974258 0.225436i \(-0.0723806\pi\)
0.974258 + 0.225436i \(0.0723806\pi\)
\(488\) −3.00000 −0.135804
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 2.00000 0.0902587 0.0451294 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) 2.00000 0.0901670
\(493\) 15.0000 0.675566
\(494\) 0 0
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) −24.0000 −1.07655
\(498\) −1.00000 −0.0448111
\(499\) 5.00000 0.223831 0.111915 0.993718i \(-0.464301\pi\)
0.111915 + 0.993718i \(0.464301\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 2.00000 0.0892644
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) −12.0000 −0.533465
\(507\) 1.00000 0.0444116
\(508\) −12.0000 −0.532414
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) −48.0000 −2.12339
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −27.0000 −1.19092
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) −9.00000 −0.395820
\(518\) −36.0000 −1.58175
\(519\) −1.00000 −0.0438951
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 5.00000 0.218844
\(523\) 14.0000 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) −18.0000 −0.786334
\(525\) 0 0
\(526\) −26.0000 −1.13365
\(527\) −9.00000 −0.392046
\(528\) −3.00000 −0.130558
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 15.0000 0.650945
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) 0 0
\(536\) −7.00000 −0.302354
\(537\) 10.0000 0.431532
\(538\) 5.00000 0.215565
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) −3.00000 −0.128861
\(543\) 17.0000 0.729540
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 3.00000 0.128388
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) −12.0000 −0.512615
\(549\) −3.00000 −0.128037
\(550\) 0 0
\(551\) 0 0
\(552\) 4.00000 0.170251
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 10.0000 0.424094
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −3.00000 −0.127000
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) −18.0000 −0.759284
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 3.00000 0.126323
\(565\) 0 0
\(566\) −6.00000 −0.252199
\(567\) 3.00000 0.125988
\(568\) −8.00000 −0.335673
\(569\) 15.0000 0.628833 0.314416 0.949285i \(-0.398191\pi\)
0.314416 + 0.949285i \(0.398191\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) −3.00000 −0.125436
\(573\) −18.0000 −0.751961
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) −8.00000 −0.332756
\(579\) −26.0000 −1.08052
\(580\) 0 0
\(581\) −3.00000 −0.124461
\(582\) −2.00000 −0.0829027
\(583\) −27.0000 −1.11823
\(584\) −16.0000 −0.662085
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) 33.0000 1.36206 0.681028 0.732257i \(-0.261533\pi\)
0.681028 + 0.732257i \(0.261533\pi\)
\(588\) 2.00000 0.0824786
\(589\) 0 0
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) −12.0000 −0.493197
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) −3.00000 −0.123091
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −10.0000 −0.409273
\(598\) 4.00000 0.163572
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −3.00000 −0.122373 −0.0611863 0.998126i \(-0.519488\pi\)
−0.0611863 + 0.998126i \(0.519488\pi\)
\(602\) 12.0000 0.489083
\(603\) −7.00000 −0.285062
\(604\) 17.0000 0.691720
\(605\) 0 0
\(606\) −3.00000 −0.121867
\(607\) 18.0000 0.730597 0.365299 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) 0 0
\(609\) 15.0000 0.607831
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) 3.00000 0.121268
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) −9.00000 −0.362620
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) −6.00000 −0.241355
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 2.00000 0.0801927
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) 19.0000 0.759393
\(627\) 0 0
\(628\) −7.00000 −0.279330
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) 12.0000 0.477712 0.238856 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 28.0000 1.11202
\(635\) 0 0
\(636\) 9.00000 0.356873
\(637\) 2.00000 0.0792429
\(638\) −15.0000 −0.593856
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 17.0000 0.671460 0.335730 0.941958i \(-0.391017\pi\)
0.335730 + 0.941958i \(0.391017\pi\)
\(642\) −2.00000 −0.0789337
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) 0 0
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 1.00000 0.0392837
\(649\) −45.0000 −1.76640
\(650\) 0 0
\(651\) −9.00000 −0.352738
\(652\) 4.00000 0.156652
\(653\) 39.0000 1.52619 0.763094 0.646288i \(-0.223679\pi\)
0.763094 + 0.646288i \(0.223679\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) −16.0000 −0.624219
\(658\) 9.00000 0.350857
\(659\) 40.0000 1.55818 0.779089 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 0 0
\(661\) 42.0000 1.63361 0.816805 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) −28.0000 −1.08825
\(663\) 3.00000 0.116510
\(664\) −1.00000 −0.0388075
\(665\) 0 0
\(666\) −12.0000 −0.464991
\(667\) 20.0000 0.774403
\(668\) −12.0000 −0.464294
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 9.00000 0.347441
\(672\) 3.00000 0.115728
\(673\) −51.0000 −1.96591 −0.982953 0.183858i \(-0.941141\pi\)
−0.982953 + 0.183858i \(0.941141\pi\)
\(674\) −27.0000 −1.04000
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 14.0000 0.537667
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) 13.0000 0.498161
\(682\) 9.00000 0.344628
\(683\) −21.0000 −0.803543 −0.401771 0.915740i \(-0.631605\pi\)
−0.401771 + 0.915740i \(0.631605\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) −10.0000 −0.381524
\(688\) 4.00000 0.152499
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) −1.00000 −0.0380143
\(693\) −9.00000 −0.341882
\(694\) 18.0000 0.683271
\(695\) 0 0
\(696\) 5.00000 0.189525
\(697\) 6.00000 0.227266
\(698\) −20.0000 −0.757011
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 27.0000 1.01978 0.509888 0.860241i \(-0.329687\pi\)
0.509888 + 0.860241i \(0.329687\pi\)
\(702\) 1.00000 0.0377426
\(703\) 0 0
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) −9.00000 −0.338480
\(708\) 15.0000 0.563735
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −12.0000 −0.449404
\(714\) 9.00000 0.336817
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) −5.00000 −0.186728
\(718\) −15.0000 −0.559795
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) −19.0000 −0.707107
\(723\) 22.0000 0.818189
\(724\) 17.0000 0.631800
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) −42.0000 −1.55769 −0.778847 0.627214i \(-0.784195\pi\)
−0.778847 + 0.627214i \(0.784195\pi\)
\(728\) 3.00000 0.111187
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) −3.00000 −0.110883
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 21.0000 0.773545
\(738\) 2.00000 0.0736210
\(739\) 5.00000 0.183928 0.0919640 0.995762i \(-0.470686\pi\)
0.0919640 + 0.995762i \(0.470686\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 27.0000 0.991201
\(743\) −21.0000 −0.770415 −0.385208 0.922830i \(-0.625870\pi\)
−0.385208 + 0.922830i \(0.625870\pi\)
\(744\) −3.00000 −0.109985
\(745\) 0 0
\(746\) 29.0000 1.06177
\(747\) −1.00000 −0.0365881
\(748\) −9.00000 −0.329073
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) 12.0000 0.437886 0.218943 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 3.00000 0.109399
\(753\) 2.00000 0.0728841
\(754\) 5.00000 0.182089
\(755\) 0 0
\(756\) 3.00000 0.109109
\(757\) 43.0000 1.56286 0.781431 0.623992i \(-0.214490\pi\)
0.781431 + 0.623992i \(0.214490\pi\)
\(758\) 15.0000 0.544825
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) −28.0000 −1.01500 −0.507500 0.861652i \(-0.669430\pi\)
−0.507500 + 0.861652i \(0.669430\pi\)
\(762\) −12.0000 −0.434714
\(763\) 30.0000 1.08607
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 15.0000 0.541619
\(768\) 1.00000 0.0360844
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) −27.0000 −0.972381
\(772\) −26.0000 −0.935760
\(773\) 34.0000 1.22290 0.611448 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) −36.0000 −1.29149
\(778\) −30.0000 −1.07555
\(779\) 0 0
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 12.0000 0.429119
\(783\) 5.00000 0.178685
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) −18.0000 −0.642039
\(787\) 3.00000 0.106938 0.0534692 0.998569i \(-0.482972\pi\)
0.0534692 + 0.998569i \(0.482972\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −26.0000 −0.925625
\(790\) 0 0
\(791\) 42.0000 1.49335
\(792\) −3.00000 −0.106600
\(793\) −3.00000 −0.106533
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) 33.0000 1.16892 0.584460 0.811423i \(-0.301306\pi\)
0.584460 + 0.811423i \(0.301306\pi\)
\(798\) 0 0
\(799\) 9.00000 0.318397
\(800\) 0 0
\(801\) 0 0
\(802\) 12.0000 0.423735
\(803\) 48.0000 1.69388
\(804\) −7.00000 −0.246871
\(805\) 0 0
\(806\) −3.00000 −0.105670
\(807\) 5.00000 0.176008
\(808\) −3.00000 −0.105540
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) −43.0000 −1.50993 −0.754967 0.655763i \(-0.772347\pi\)
−0.754967 + 0.655763i \(0.772347\pi\)
\(812\) 15.0000 0.526397
\(813\) −3.00000 −0.105215
\(814\) 36.0000 1.26180
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) 0 0
\(818\) −10.0000 −0.349642
\(819\) 3.00000 0.104828
\(820\) 0 0
\(821\) −28.0000 −0.977207 −0.488603 0.872506i \(-0.662493\pi\)
−0.488603 + 0.872506i \(0.662493\pi\)
\(822\) −12.0000 −0.418548
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) −6.00000 −0.209020
\(825\) 0 0
\(826\) 45.0000 1.56575
\(827\) 3.00000 0.104320 0.0521601 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) 4.00000 0.139010
\(829\) 15.0000 0.520972 0.260486 0.965478i \(-0.416117\pi\)
0.260486 + 0.965478i \(0.416117\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 1.00000 0.0346688
\(833\) 6.00000 0.207888
\(834\) 10.0000 0.346272
\(835\) 0 0
\(836\) 0 0
\(837\) −3.00000 −0.103695
\(838\) 30.0000 1.03633
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −28.0000 −0.964944
\(843\) −18.0000 −0.619953
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) −6.00000 −0.206162
\(848\) 9.00000 0.309061
\(849\) −6.00000 −0.205919
\(850\) 0 0
\(851\) −48.0000 −1.64542
\(852\) −8.00000 −0.274075
\(853\) −16.0000 −0.547830 −0.273915 0.961754i \(-0.588319\pi\)
−0.273915 + 0.961754i \(0.588319\pi\)
\(854\) −9.00000 −0.307974
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) −3.00000 −0.102418
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 12.0000 0.408722
\(863\) 39.0000 1.32758 0.663788 0.747921i \(-0.268948\pi\)
0.663788 + 0.747921i \(0.268948\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −26.0000 −0.883516
\(867\) −8.00000 −0.271694
\(868\) −9.00000 −0.305480
\(869\) 0 0
\(870\) 0 0
\(871\) −7.00000 −0.237186
\(872\) 10.0000 0.338643
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) −16.0000 −0.540590
\(877\) 8.00000 0.270141 0.135070 0.990836i \(-0.456874\pi\)
0.135070 + 0.990836i \(0.456874\pi\)
\(878\) 0 0
\(879\) −26.0000 −0.876958
\(880\) 0 0
\(881\) 7.00000 0.235836 0.117918 0.993023i \(-0.462378\pi\)
0.117918 + 0.993023i \(0.462378\pi\)
\(882\) 2.00000 0.0673435
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 3.00000 0.100901
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) −12.0000 −0.402694
\(889\) −36.0000 −1.20740
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 4.00000 0.133556
\(898\) −20.0000 −0.667409
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) 27.0000 0.899500
\(902\) −6.00000 −0.199778
\(903\) 12.0000 0.399335
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 17.0000 0.564787
\(907\) 8.00000 0.265636 0.132818 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) 13.0000 0.431420
\(909\) −3.00000 −0.0995037
\(910\) 0 0
\(911\) 52.0000 1.72284 0.861418 0.507896i \(-0.169577\pi\)
0.861418 + 0.507896i \(0.169577\pi\)
\(912\) 0 0
\(913\) 3.00000 0.0992855
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) −54.0000 −1.78324
\(918\) 3.00000 0.0990148
\(919\) 50.0000 1.64935 0.824674 0.565608i \(-0.191359\pi\)
0.824674 + 0.565608i \(0.191359\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) −38.0000 −1.25146
\(923\) −8.00000 −0.263323
\(924\) −9.00000 −0.296078
\(925\) 0 0
\(926\) 9.00000 0.295758
\(927\) −6.00000 −0.197066
\(928\) 5.00000 0.164133
\(929\) −10.0000 −0.328089 −0.164045 0.986453i \(-0.552454\pi\)
−0.164045 + 0.986453i \(0.552454\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 2.00000 0.0654771
\(934\) −32.0000 −1.04707
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) 13.0000 0.424691 0.212346 0.977195i \(-0.431890\pi\)
0.212346 + 0.977195i \(0.431890\pi\)
\(938\) −21.0000 −0.685674
\(939\) 19.0000 0.620042
\(940\) 0 0
\(941\) 52.0000 1.69515 0.847576 0.530674i \(-0.178061\pi\)
0.847576 + 0.530674i \(0.178061\pi\)
\(942\) −7.00000 −0.228072
\(943\) 8.00000 0.260516
\(944\) 15.0000 0.488208
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) −57.0000 −1.85225 −0.926126 0.377215i \(-0.876882\pi\)
−0.926126 + 0.377215i \(0.876882\pi\)
\(948\) 0 0
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) 28.0000 0.907962
\(952\) 9.00000 0.291692
\(953\) 29.0000 0.939402 0.469701 0.882826i \(-0.344362\pi\)
0.469701 + 0.882826i \(0.344362\pi\)
\(954\) 9.00000 0.291386
\(955\) 0 0
\(956\) −5.00000 −0.161712
\(957\) −15.0000 −0.484881
\(958\) 15.0000 0.484628
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −12.0000 −0.386896
\(963\) −2.00000 −0.0644491
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) 12.0000 0.386094
\(967\) −37.0000 −1.18984 −0.594920 0.803785i \(-0.702816\pi\)
−0.594920 + 0.803785i \(0.702816\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 1.00000 0.0320750
\(973\) 30.0000 0.961756
\(974\) 43.0000 1.37781
\(975\) 0 0
\(976\) −3.00000 −0.0960277
\(977\) −52.0000 −1.66363 −0.831814 0.555055i \(-0.812697\pi\)
−0.831814 + 0.555055i \(0.812697\pi\)
\(978\) 4.00000 0.127906
\(979\) 0 0
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 2.00000 0.0638226
\(983\) 29.0000 0.924956 0.462478 0.886631i \(-0.346960\pi\)
0.462478 + 0.886631i \(0.346960\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) 15.0000 0.477697
\(987\) 9.00000 0.286473
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −3.00000 −0.0952501
\(993\) −28.0000 −0.888553
\(994\) −24.0000 −0.761234
\(995\) 0 0
\(996\) −1.00000 −0.0316862
\(997\) −17.0000 −0.538395 −0.269198 0.963085i \(-0.586759\pi\)
−0.269198 + 0.963085i \(0.586759\pi\)
\(998\) 5.00000 0.158272
\(999\) −12.0000 −0.379663
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 1950.2.a.bb.1.1 yes 1
3.2 odd 2 5850.2.a.w.1.1 1
5.2 odd 4 1950.2.e.j.1249.2 2
5.3 odd 4 1950.2.e.j.1249.1 2
5.4 even 2 1950.2.a.a.1.1 1
15.2 even 4 5850.2.e.z.5149.1 2
15.8 even 4 5850.2.e.z.5149.2 2
15.14 odd 2 5850.2.a.bf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.a.1.1 1 5.4 even 2
1950.2.a.bb.1.1 yes 1 1.1 even 1 trivial
1950.2.e.j.1249.1 2 5.3 odd 4
1950.2.e.j.1249.2 2 5.2 odd 4
5850.2.a.w.1.1 1 3.2 odd 2
5850.2.a.bf.1.1 1 15.14 odd 2
5850.2.e.z.5149.1 2 15.2 even 4
5850.2.e.z.5149.2 2 15.8 even 4