Properties

Label 2450.2.a.bg.1.1
Level $2450$
Weight $2$
Character 2450.1
Self dual yes
Analytic conductor $19.563$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5633484952\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 350)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} -5.00000 q^{11} +3.00000 q^{12} +6.00000 q^{13} +1.00000 q^{16} +1.00000 q^{17} +6.00000 q^{18} +3.00000 q^{19} -5.00000 q^{22} +3.00000 q^{24} +6.00000 q^{26} +9.00000 q^{27} -6.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -15.0000 q^{33} +1.00000 q^{34} +6.00000 q^{36} +8.00000 q^{37} +3.00000 q^{38} +18.0000 q^{39} -11.0000 q^{41} -8.00000 q^{43} -5.00000 q^{44} -2.00000 q^{47} +3.00000 q^{48} +3.00000 q^{51} +6.00000 q^{52} +4.00000 q^{53} +9.00000 q^{54} +9.00000 q^{57} -6.00000 q^{58} -4.00000 q^{59} +2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -15.0000 q^{66} +9.00000 q^{67} +1.00000 q^{68} -10.0000 q^{71} +6.00000 q^{72} +7.00000 q^{73} +8.00000 q^{74} +3.00000 q^{76} +18.0000 q^{78} -2.00000 q^{79} +9.00000 q^{81} -11.0000 q^{82} -11.0000 q^{83} -8.00000 q^{86} -18.0000 q^{87} -5.00000 q^{88} +11.0000 q^{89} +12.0000 q^{93} -2.00000 q^{94} +3.00000 q^{96} +10.0000 q^{97} -30.0000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 3.00000 1.22474
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 3.00000 0.866025
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 6.00000 1.41421
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.00000 −1.06600
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) −15.0000 −2.61116
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 3.00000 0.486664
\(39\) 18.0000 2.88231
\(40\) 0 0
\(41\) −11.0000 −1.71791 −0.858956 0.512050i \(-0.828886\pi\)
−0.858956 + 0.512050i \(0.828886\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) 0 0
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 3.00000 0.433013
\(49\) 0 0
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 6.00000 0.832050
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 9.00000 1.22474
\(55\) 0 0
\(56\) 0 0
\(57\) 9.00000 1.19208
\(58\) −6.00000 −0.787839
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −15.0000 −1.84637
\(67\) 9.00000 1.09952 0.549762 0.835321i \(-0.314718\pi\)
0.549762 + 0.835321i \(0.314718\pi\)
\(68\) 1.00000 0.121268
\(69\) 0 0
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 6.00000 0.707107
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 3.00000 0.344124
\(77\) 0 0
\(78\) 18.0000 2.03810
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −11.0000 −1.21475
\(83\) −11.0000 −1.20741 −0.603703 0.797209i \(-0.706309\pi\)
−0.603703 + 0.797209i \(0.706309\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) −18.0000 −1.92980
\(88\) −5.00000 −0.533002
\(89\) 11.0000 1.16600 0.582999 0.812473i \(-0.301879\pi\)
0.582999 + 0.812473i \(0.301879\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 12.0000 1.24434
\(94\) −2.00000 −0.206284
\(95\) 0 0
\(96\) 3.00000 0.306186
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) −30.0000 −3.01511
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 3.00000 0.297044
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 9.00000 0.866025
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 24.0000 2.27798
\(112\) 0 0
\(113\) −1.00000 −0.0940721 −0.0470360 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) 9.00000 0.842927
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 36.0000 3.32820
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 2.00000 0.181071
\(123\) −33.0000 −2.97551
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) −14.0000 −1.24230 −0.621150 0.783692i \(-0.713334\pi\)
−0.621150 + 0.783692i \(0.713334\pi\)
\(128\) 1.00000 0.0883883
\(129\) −24.0000 −2.11308
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) −15.0000 −1.30558
\(133\) 0 0
\(134\) 9.00000 0.777482
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) 11.0000 0.933008 0.466504 0.884519i \(-0.345513\pi\)
0.466504 + 0.884519i \(0.345513\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) −10.0000 −0.839181
\(143\) −30.0000 −2.50873
\(144\) 6.00000 0.500000
\(145\) 0 0
\(146\) 7.00000 0.579324
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 3.00000 0.243332
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 18.0000 1.44115
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) −2.00000 −0.159111
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 9.00000 0.707107
\(163\) −19.0000 −1.48819 −0.744097 0.668071i \(-0.767120\pi\)
−0.744097 + 0.668071i \(0.767120\pi\)
\(164\) −11.0000 −0.858956
\(165\) 0 0
\(166\) −11.0000 −0.853766
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 18.0000 1.37649
\(172\) −8.00000 −0.609994
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) −18.0000 −1.36458
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) −12.0000 −0.901975
\(178\) 11.0000 0.824485
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 0 0
\(185\) 0 0
\(186\) 12.0000 0.879883
\(187\) −5.00000 −0.365636
\(188\) −2.00000 −0.145865
\(189\) 0 0
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 3.00000 0.216506
\(193\) −19.0000 −1.36765 −0.683825 0.729646i \(-0.739685\pi\)
−0.683825 + 0.729646i \(0.739685\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 0 0
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) −30.0000 −2.13201
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 27.0000 1.90443
\(202\) 0 0
\(203\) 0 0
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) −15.0000 −1.03757
\(210\) 0 0
\(211\) 1.00000 0.0688428 0.0344214 0.999407i \(-0.489041\pi\)
0.0344214 + 0.999407i \(0.489041\pi\)
\(212\) 4.00000 0.274721
\(213\) −30.0000 −2.05557
\(214\) 3.00000 0.205076
\(215\) 0 0
\(216\) 9.00000 0.612372
\(217\) 0 0
\(218\) −18.0000 −1.21911
\(219\) 21.0000 1.41905
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 24.0000 1.61077
\(223\) −22.0000 −1.47323 −0.736614 0.676313i \(-0.763577\pi\)
−0.736614 + 0.676313i \(0.763577\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1.00000 −0.0665190
\(227\) 28.0000 1.85843 0.929213 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) 9.00000 0.596040
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 36.0000 2.35339
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) −6.00000 −0.389742
\(238\) 0 0
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) 5.00000 0.322078 0.161039 0.986948i \(-0.448515\pi\)
0.161039 + 0.986948i \(0.448515\pi\)
\(242\) 14.0000 0.899954
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −33.0000 −2.10400
\(247\) 18.0000 1.14531
\(248\) 4.00000 0.254000
\(249\) −33.0000 −2.09129
\(250\) 0 0
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −14.0000 −0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) −24.0000 −1.49417
\(259\) 0 0
\(260\) 0 0
\(261\) −36.0000 −2.22834
\(262\) −8.00000 −0.494242
\(263\) −10.0000 −0.616626 −0.308313 0.951285i \(-0.599764\pi\)
−0.308313 + 0.951285i \(0.599764\pi\)
\(264\) −15.0000 −0.923186
\(265\) 0 0
\(266\) 0 0
\(267\) 33.0000 2.01957
\(268\) 9.00000 0.549762
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 6.00000 0.364474 0.182237 0.983255i \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) −3.00000 −0.181237
\(275\) 0 0
\(276\) 0 0
\(277\) −30.0000 −1.80253 −0.901263 0.433273i \(-0.857359\pi\)
−0.901263 + 0.433273i \(0.857359\pi\)
\(278\) 11.0000 0.659736
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) 14.0000 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) −6.00000 −0.357295
\(283\) 13.0000 0.772770 0.386385 0.922338i \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) −30.0000 −1.77394
\(287\) 0 0
\(288\) 6.00000 0.353553
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 30.0000 1.75863
\(292\) 7.00000 0.409644
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) −45.0000 −2.61116
\(298\) 12.0000 0.695141
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 3.00000 0.172062
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −13.0000 −0.741949 −0.370975 0.928643i \(-0.620976\pi\)
−0.370975 + 0.928643i \(0.620976\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 18.0000 1.01905
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) 4.00000 0.224662 0.112331 0.993671i \(-0.464168\pi\)
0.112331 + 0.993671i \(0.464168\pi\)
\(318\) 12.0000 0.672927
\(319\) 30.0000 1.67968
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −19.0000 −1.05231
\(327\) −54.0000 −2.98621
\(328\) −11.0000 −0.607373
\(329\) 0 0
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) −11.0000 −0.603703
\(333\) 48.0000 2.63038
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) −29.0000 −1.57973 −0.789865 0.613280i \(-0.789850\pi\)
−0.789865 + 0.613280i \(0.789850\pi\)
\(338\) 23.0000 1.25104
\(339\) −3.00000 −0.162938
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) 18.0000 0.973329
\(343\) 0 0
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) 19.0000 1.01997 0.509987 0.860182i \(-0.329650\pi\)
0.509987 + 0.860182i \(0.329650\pi\)
\(348\) −18.0000 −0.964901
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) 54.0000 2.88231
\(352\) −5.00000 −0.266501
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 11.0000 0.582999
\(357\) 0 0
\(358\) 3.00000 0.158555
\(359\) 26.0000 1.37223 0.686114 0.727494i \(-0.259315\pi\)
0.686114 + 0.727494i \(0.259315\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) −10.0000 −0.525588
\(363\) 42.0000 2.20443
\(364\) 0 0
\(365\) 0 0
\(366\) 6.00000 0.313625
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) −66.0000 −3.43582
\(370\) 0 0
\(371\) 0 0
\(372\) 12.0000 0.622171
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) −5.00000 −0.258544
\(375\) 0 0
\(376\) −2.00000 −0.103142
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) 9.00000 0.462299 0.231149 0.972918i \(-0.425751\pi\)
0.231149 + 0.972918i \(0.425751\pi\)
\(380\) 0 0
\(381\) −42.0000 −2.15173
\(382\) −6.00000 −0.306987
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −19.0000 −0.967075
\(387\) −48.0000 −2.43998
\(388\) 10.0000 0.507673
\(389\) 8.00000 0.405616 0.202808 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −24.0000 −1.21064
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) −30.0000 −1.50756
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) 10.0000 0.501255
\(399\) 0 0
\(400\) 0 0
\(401\) 37.0000 1.84769 0.923846 0.382765i \(-0.125028\pi\)
0.923846 + 0.382765i \(0.125028\pi\)
\(402\) 27.0000 1.34664
\(403\) 24.0000 1.19553
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −40.0000 −1.98273
\(408\) 3.00000 0.148522
\(409\) 21.0000 1.03838 0.519192 0.854658i \(-0.326233\pi\)
0.519192 + 0.854658i \(0.326233\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 33.0000 1.61602
\(418\) −15.0000 −0.733674
\(419\) 39.0000 1.90527 0.952637 0.304109i \(-0.0983586\pi\)
0.952637 + 0.304109i \(0.0983586\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 1.00000 0.0486792
\(423\) −12.0000 −0.583460
\(424\) 4.00000 0.194257
\(425\) 0 0
\(426\) −30.0000 −1.45350
\(427\) 0 0
\(428\) 3.00000 0.145010
\(429\) −90.0000 −4.34524
\(430\) 0 0
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) 9.00000 0.433013
\(433\) 1.00000 0.0480569 0.0240285 0.999711i \(-0.492351\pi\)
0.0240285 + 0.999711i \(0.492351\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) 0 0
\(438\) 21.0000 1.00342
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6.00000 0.285391
\(443\) 37.0000 1.75792 0.878962 0.476893i \(-0.158237\pi\)
0.878962 + 0.476893i \(0.158237\pi\)
\(444\) 24.0000 1.13899
\(445\) 0 0
\(446\) −22.0000 −1.04173
\(447\) 36.0000 1.70274
\(448\) 0 0
\(449\) 33.0000 1.55737 0.778683 0.627417i \(-0.215888\pi\)
0.778683 + 0.627417i \(0.215888\pi\)
\(450\) 0 0
\(451\) 55.0000 2.58985
\(452\) −1.00000 −0.0470360
\(453\) 24.0000 1.12762
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) 9.00000 0.421464
\(457\) 25.0000 1.16945 0.584725 0.811231i \(-0.301202\pi\)
0.584725 + 0.811231i \(0.301202\pi\)
\(458\) 14.0000 0.654177
\(459\) 9.00000 0.420084
\(460\) 0 0
\(461\) 38.0000 1.76984 0.884918 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) 36.0000 1.66410
\(469\) 0 0
\(470\) 0 0
\(471\) −12.0000 −0.552931
\(472\) −4.00000 −0.184115
\(473\) 40.0000 1.83920
\(474\) −6.00000 −0.275589
\(475\) 0 0
\(476\) 0 0
\(477\) 24.0000 1.09888
\(478\) 4.00000 0.182956
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 48.0000 2.18861
\(482\) 5.00000 0.227744
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) 0 0
\(487\) −34.0000 −1.54069 −0.770344 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) 2.00000 0.0905357
\(489\) −57.0000 −2.57763
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −33.0000 −1.48775
\(493\) −6.00000 −0.270226
\(494\) 18.0000 0.809858
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) −33.0000 −1.47877
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) −36.0000 −1.60836
\(502\) −27.0000 −1.20507
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 69.0000 3.06440
\(508\) −14.0000 −0.621150
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 27.0000 1.19208
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) −24.0000 −1.05654
\(517\) 10.0000 0.439799
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −11.0000 −0.481919 −0.240959 0.970535i \(-0.577462\pi\)
−0.240959 + 0.970535i \(0.577462\pi\)
\(522\) −36.0000 −1.57568
\(523\) −13.0000 −0.568450 −0.284225 0.958758i \(-0.591736\pi\)
−0.284225 + 0.958758i \(0.591736\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) −10.0000 −0.436021
\(527\) 4.00000 0.174243
\(528\) −15.0000 −0.652791
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −24.0000 −1.04151
\(532\) 0 0
\(533\) −66.0000 −2.85878
\(534\) 33.0000 1.42805
\(535\) 0 0
\(536\) 9.00000 0.388741
\(537\) 9.00000 0.388379
\(538\) −18.0000 −0.776035
\(539\) 0 0
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) 6.00000 0.257722
\(543\) −30.0000 −1.28742
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) 0 0
\(547\) 27.0000 1.15444 0.577218 0.816590i \(-0.304138\pi\)
0.577218 + 0.816590i \(0.304138\pi\)
\(548\) −3.00000 −0.128154
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) 0 0
\(554\) −30.0000 −1.27458
\(555\) 0 0
\(556\) 11.0000 0.466504
\(557\) 4.00000 0.169485 0.0847427 0.996403i \(-0.472993\pi\)
0.0847427 + 0.996403i \(0.472993\pi\)
\(558\) 24.0000 1.01600
\(559\) −48.0000 −2.03018
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 14.0000 0.590554
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) 13.0000 0.546431
\(567\) 0 0
\(568\) −10.0000 −0.419591
\(569\) 21.0000 0.880366 0.440183 0.897908i \(-0.354914\pi\)
0.440183 + 0.897908i \(0.354914\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −30.0000 −1.25436
\(573\) −18.0000 −0.751961
\(574\) 0 0
\(575\) 0 0
\(576\) 6.00000 0.250000
\(577\) −13.0000 −0.541197 −0.270599 0.962692i \(-0.587222\pi\)
−0.270599 + 0.962692i \(0.587222\pi\)
\(578\) −16.0000 −0.665512
\(579\) −57.0000 −2.36884
\(580\) 0 0
\(581\) 0 0
\(582\) 30.0000 1.24354
\(583\) −20.0000 −0.828315
\(584\) 7.00000 0.289662
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) 13.0000 0.536567 0.268284 0.963340i \(-0.413544\pi\)
0.268284 + 0.963340i \(0.413544\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) −66.0000 −2.71488
\(592\) 8.00000 0.328798
\(593\) 39.0000 1.60154 0.800769 0.598973i \(-0.204424\pi\)
0.800769 + 0.598973i \(0.204424\pi\)
\(594\) −45.0000 −1.84637
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) 30.0000 1.22782
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −21.0000 −0.856608 −0.428304 0.903635i \(-0.640889\pi\)
−0.428304 + 0.903635i \(0.640889\pi\)
\(602\) 0 0
\(603\) 54.0000 2.19905
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 3.00000 0.121666
\(609\) 0 0
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 6.00000 0.242536
\(613\) 18.0000 0.727013 0.363507 0.931592i \(-0.381579\pi\)
0.363507 + 0.931592i \(0.381579\pi\)
\(614\) −13.0000 −0.524637
\(615\) 0 0
\(616\) 0 0
\(617\) 14.0000 0.563619 0.281809 0.959470i \(-0.409065\pi\)
0.281809 + 0.959470i \(0.409065\pi\)
\(618\) −12.0000 −0.482711
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −6.00000 −0.240578
\(623\) 0 0
\(624\) 18.0000 0.720577
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) −45.0000 −1.79713
\(628\) −4.00000 −0.159617
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −2.00000 −0.0795557
\(633\) 3.00000 0.119239
\(634\) 4.00000 0.158860
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) 30.0000 1.18771
\(639\) −60.0000 −2.37356
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 9.00000 0.355202
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 3.00000 0.118033
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 9.00000 0.353553
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) −19.0000 −0.744097
\(653\) −28.0000 −1.09572 −0.547862 0.836569i \(-0.684558\pi\)
−0.547862 + 0.836569i \(0.684558\pi\)
\(654\) −54.0000 −2.11157
\(655\) 0 0
\(656\) −11.0000 −0.429478
\(657\) 42.0000 1.63858
\(658\) 0 0
\(659\) 1.00000 0.0389545 0.0194772 0.999810i \(-0.493800\pi\)
0.0194772 + 0.999810i \(0.493800\pi\)
\(660\) 0 0
\(661\) 50.0000 1.94477 0.972387 0.233373i \(-0.0749763\pi\)
0.972387 + 0.233373i \(0.0749763\pi\)
\(662\) −17.0000 −0.660724
\(663\) 18.0000 0.699062
\(664\) −11.0000 −0.426883
\(665\) 0 0
\(666\) 48.0000 1.85996
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) −66.0000 −2.55171
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −29.0000 −1.11704
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 48.0000 1.84479 0.922395 0.386248i \(-0.126229\pi\)
0.922395 + 0.386248i \(0.126229\pi\)
\(678\) −3.00000 −0.115214
\(679\) 0 0
\(680\) 0 0
\(681\) 84.0000 3.21889
\(682\) −20.0000 −0.765840
\(683\) −13.0000 −0.497431 −0.248716 0.968577i \(-0.580008\pi\)
−0.248716 + 0.968577i \(0.580008\pi\)
\(684\) 18.0000 0.688247
\(685\) 0 0
\(686\) 0 0
\(687\) 42.0000 1.60240
\(688\) −8.00000 −0.304997
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) −49.0000 −1.86405 −0.932024 0.362397i \(-0.881959\pi\)
−0.932024 + 0.362397i \(0.881959\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 0 0
\(694\) 19.0000 0.721230
\(695\) 0 0
\(696\) −18.0000 −0.682288
\(697\) −11.0000 −0.416655
\(698\) 8.00000 0.302804
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) −32.0000 −1.20862 −0.604312 0.796748i \(-0.706552\pi\)
−0.604312 + 0.796748i \(0.706552\pi\)
\(702\) 54.0000 2.03810
\(703\) 24.0000 0.905177
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) −12.0000 −0.450988
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 11.0000 0.412242
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 3.00000 0.112115
\(717\) 12.0000 0.448148
\(718\) 26.0000 0.970311
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −10.0000 −0.372161
\(723\) 15.0000 0.557856
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 42.0000 1.55877
\(727\) −6.00000 −0.222528 −0.111264 0.993791i \(-0.535490\pi\)
−0.111264 + 0.993791i \(0.535490\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 6.00000 0.221766
\(733\) −40.0000 −1.47743 −0.738717 0.674016i \(-0.764568\pi\)
−0.738717 + 0.674016i \(0.764568\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) −45.0000 −1.65760
\(738\) −66.0000 −2.42949
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 54.0000 1.98374
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 12.0000 0.439941
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) −66.0000 −2.41481
\(748\) −5.00000 −0.182818
\(749\) 0 0
\(750\) 0 0
\(751\) 50.0000 1.82453 0.912263 0.409605i \(-0.134333\pi\)
0.912263 + 0.409605i \(0.134333\pi\)
\(752\) −2.00000 −0.0729325
\(753\) −81.0000 −2.95180
\(754\) −36.0000 −1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 9.00000 0.326895
\(759\) 0 0
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) −42.0000 −1.52150
\(763\) 0 0
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) −24.0000 −0.866590
\(768\) 3.00000 0.108253
\(769\) −19.0000 −0.685158 −0.342579 0.939489i \(-0.611300\pi\)
−0.342579 + 0.939489i \(0.611300\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) −19.0000 −0.683825
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) −48.0000 −1.72532
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) 8.00000 0.286814
\(779\) −33.0000 −1.18235
\(780\) 0 0
\(781\) 50.0000 1.78914
\(782\) 0 0
\(783\) −54.0000 −1.92980
\(784\) 0 0
\(785\) 0 0
\(786\) −24.0000 −0.856052
\(787\) −52.0000 −1.85360 −0.926800 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) −22.0000 −0.783718
\(789\) −30.0000 −1.06803
\(790\) 0 0
\(791\) 0 0
\(792\) −30.0000 −1.06600
\(793\) 12.0000 0.426132
\(794\) −10.0000 −0.354887
\(795\) 0 0
\(796\) 10.0000 0.354441
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) −2.00000 −0.0707549
\(800\) 0 0
\(801\) 66.0000 2.33200
\(802\) 37.0000 1.30652
\(803\) −35.0000 −1.23512
\(804\) 27.0000 0.952217
\(805\) 0 0
\(806\) 24.0000 0.845364
\(807\) −54.0000 −1.90089
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 18.0000 0.631288
\(814\) −40.0000 −1.40200
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) −24.0000 −0.839654
\(818\) 21.0000 0.734248
\(819\) 0 0
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) −9.00000 −0.313911
\(823\) 10.0000 0.348578 0.174289 0.984695i \(-0.444237\pi\)
0.174289 + 0.984695i \(0.444237\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) 41.0000 1.42571 0.712855 0.701312i \(-0.247402\pi\)
0.712855 + 0.701312i \(0.247402\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) −90.0000 −3.12207
\(832\) 6.00000 0.208013
\(833\) 0 0
\(834\) 33.0000 1.14270
\(835\) 0 0
\(836\) −15.0000 −0.518786
\(837\) 36.0000 1.24434
\(838\) 39.0000 1.34723
\(839\) −2.00000 −0.0690477 −0.0345238 0.999404i \(-0.510991\pi\)
−0.0345238 + 0.999404i \(0.510991\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 20.0000 0.689246
\(843\) 42.0000 1.44656
\(844\) 1.00000 0.0344214
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) 4.00000 0.137361
\(849\) 39.0000 1.33848
\(850\) 0 0
\(851\) 0 0
\(852\) −30.0000 −1.02778
\(853\) 34.0000 1.16414 0.582069 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) −3.00000 −0.102478 −0.0512390 0.998686i \(-0.516317\pi\)
−0.0512390 + 0.998686i \(0.516317\pi\)
\(858\) −90.0000 −3.07255
\(859\) 51.0000 1.74010 0.870049 0.492966i \(-0.164087\pi\)
0.870049 + 0.492966i \(0.164087\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −36.0000 −1.22616
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 9.00000 0.306186
\(865\) 0 0
\(866\) 1.00000 0.0339814
\(867\) −48.0000 −1.63017
\(868\) 0 0
\(869\) 10.0000 0.339227
\(870\) 0 0
\(871\) 54.0000 1.82972
\(872\) −18.0000 −0.609557
\(873\) 60.0000 2.03069
\(874\) 0 0
\(875\) 0 0
\(876\) 21.0000 0.709524
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) −28.0000 −0.944954
\(879\) 42.0000 1.41662
\(880\) 0 0
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 0 0
\(883\) 15.0000 0.504790 0.252395 0.967624i \(-0.418782\pi\)
0.252395 + 0.967624i \(0.418782\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) 37.0000 1.24304
\(887\) 34.0000 1.14161 0.570804 0.821086i \(-0.306632\pi\)
0.570804 + 0.821086i \(0.306632\pi\)
\(888\) 24.0000 0.805387
\(889\) 0 0
\(890\) 0 0
\(891\) −45.0000 −1.50756
\(892\) −22.0000 −0.736614
\(893\) −6.00000 −0.200782
\(894\) 36.0000 1.20402
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 33.0000 1.10122
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 55.0000 1.83130
\(903\) 0 0
\(904\) −1.00000 −0.0332595
\(905\) 0 0
\(906\) 24.0000 0.797347
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 28.0000 0.929213
\(909\) 0 0
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 9.00000 0.298020
\(913\) 55.0000 1.82023
\(914\) 25.0000 0.826927
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) 9.00000 0.297044
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) −39.0000 −1.28509
\(922\) 38.0000 1.25146
\(923\) −60.0000 −1.97492
\(924\) 0 0
\(925\) 0 0
\(926\) 8.00000 0.262896
\(927\) −24.0000 −0.788263
\(928\) −6.00000 −0.196960
\(929\) −46.0000 −1.50921 −0.754606 0.656179i \(-0.772172\pi\)
−0.754606 + 0.656179i \(0.772172\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) −18.0000 −0.589294
\(934\) 4.00000 0.130884
\(935\) 0 0
\(936\) 36.0000 1.17670
\(937\) 7.00000 0.228680 0.114340 0.993442i \(-0.463525\pi\)
0.114340 + 0.993442i \(0.463525\pi\)
\(938\) 0 0
\(939\) 30.0000 0.979013
\(940\) 0 0
\(941\) −56.0000 −1.82555 −0.912774 0.408465i \(-0.866064\pi\)
−0.912774 + 0.408465i \(0.866064\pi\)
\(942\) −12.0000 −0.390981
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 40.0000 1.30051
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) −6.00000 −0.194871
\(949\) 42.0000 1.36338
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) 9.00000 0.291539 0.145769 0.989319i \(-0.453434\pi\)
0.145769 + 0.989319i \(0.453434\pi\)
\(954\) 24.0000 0.777029
\(955\) 0 0
\(956\) 4.00000 0.129369
\(957\) 90.0000 2.90929
\(958\) 6.00000 0.193851
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 48.0000 1.54758
\(963\) 18.0000 0.580042
\(964\) 5.00000 0.161039
\(965\) 0 0
\(966\) 0 0
\(967\) −2.00000 −0.0643157 −0.0321578 0.999483i \(-0.510238\pi\)
−0.0321578 + 0.999483i \(0.510238\pi\)
\(968\) 14.0000 0.449977
\(969\) 9.00000 0.289122
\(970\) 0 0
\(971\) 51.0000 1.63667 0.818334 0.574743i \(-0.194898\pi\)
0.818334 + 0.574743i \(0.194898\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −34.0000 −1.08943
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 21.0000 0.671850 0.335925 0.941889i \(-0.390951\pi\)
0.335925 + 0.941889i \(0.390951\pi\)
\(978\) −57.0000 −1.82266
\(979\) −55.0000 −1.75781
\(980\) 0 0
\(981\) −108.000 −3.44817
\(982\) −12.0000 −0.382935
\(983\) 4.00000 0.127580 0.0637901 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) −33.0000 −1.05200
\(985\) 0 0
\(986\) −6.00000 −0.191079
\(987\) 0 0
\(988\) 18.0000 0.572656
\(989\) 0 0
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 4.00000 0.127000
\(993\) −51.0000 −1.61844
\(994\) 0 0
\(995\) 0 0
\(996\) −33.0000 −1.04565
\(997\) 58.0000 1.83688 0.918439 0.395562i \(-0.129450\pi\)
0.918439 + 0.395562i \(0.129450\pi\)
\(998\) 36.0000 1.13956
\(999\) 72.0000 2.27798
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 2450.2.a.bg.1.1 1
5.2 odd 4 2450.2.c.r.99.2 2
5.3 odd 4 2450.2.c.r.99.1 2
5.4 even 2 2450.2.a.a.1.1 1
7.6 odd 2 350.2.a.d.1.1 yes 1
21.20 even 2 3150.2.a.j.1.1 1
28.27 even 2 2800.2.a.bg.1.1 1
35.13 even 4 350.2.c.a.99.1 2
35.27 even 4 350.2.c.a.99.2 2
35.34 odd 2 350.2.a.c.1.1 1
105.62 odd 4 3150.2.g.v.2899.1 2
105.83 odd 4 3150.2.g.v.2899.2 2
105.104 even 2 3150.2.a.bq.1.1 1
140.27 odd 4 2800.2.g.a.449.1 2
140.83 odd 4 2800.2.g.a.449.2 2
140.139 even 2 2800.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
350.2.a.c.1.1 1 35.34 odd 2
350.2.a.d.1.1 yes 1 7.6 odd 2
350.2.c.a.99.1 2 35.13 even 4
350.2.c.a.99.2 2 35.27 even 4
2450.2.a.a.1.1 1 5.4 even 2
2450.2.a.bg.1.1 1 1.1 even 1 trivial
2450.2.c.r.99.1 2 5.3 odd 4
2450.2.c.r.99.2 2 5.2 odd 4
2800.2.a.b.1.1 1 140.139 even 2
2800.2.a.bg.1.1 1 28.27 even 2
2800.2.g.a.449.1 2 140.27 odd 4
2800.2.g.a.449.2 2 140.83 odd 4
3150.2.a.j.1.1 1 21.20 even 2
3150.2.a.bq.1.1 1 105.104 even 2
3150.2.g.v.2899.1 2 105.62 odd 4
3150.2.g.v.2899.2 2 105.83 odd 4