Properties

Label 4025.2.a.b.1.1
Level $4025$
Weight $2$
Character 4025.1
Self dual yes
Analytic conductor $32.140$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 805)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4025.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +1.00000 q^{7} -2.00000 q^{9} -5.00000 q^{11} +2.00000 q^{12} -3.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} +5.00000 q^{17} +4.00000 q^{18} +1.00000 q^{21} +10.0000 q^{22} +1.00000 q^{23} +6.00000 q^{26} -5.00000 q^{27} +2.00000 q^{28} +3.00000 q^{29} +6.00000 q^{31} +8.00000 q^{32} -5.00000 q^{33} -10.0000 q^{34} -4.00000 q^{36} +4.00000 q^{37} -3.00000 q^{39} -2.00000 q^{42} +2.00000 q^{43} -10.0000 q^{44} -2.00000 q^{46} +9.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} +5.00000 q^{51} -6.00000 q^{52} +6.00000 q^{53} +10.0000 q^{54} -6.00000 q^{58} -6.00000 q^{59} +10.0000 q^{61} -12.0000 q^{62} -2.00000 q^{63} -8.00000 q^{64} +10.0000 q^{66} -4.00000 q^{67} +10.0000 q^{68} +1.00000 q^{69} -8.00000 q^{71} -10.0000 q^{73} -8.00000 q^{74} -5.00000 q^{77} +6.00000 q^{78} -15.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +2.00000 q^{84} -4.00000 q^{86} +3.00000 q^{87} -10.0000 q^{89} -3.00000 q^{91} +2.00000 q^{92} +6.00000 q^{93} -18.0000 q^{94} +8.00000 q^{96} -7.00000 q^{97} -2.00000 q^{98} +10.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 2.00000 0.577350
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 4.00000 0.942809
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 10.0000 2.13201
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) −5.00000 −0.962250
\(28\) 2.00000 0.377964
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 8.00000 1.41421
\(33\) −5.00000 −0.870388
\(34\) −10.0000 −1.71499
\(35\) 0 0
\(36\) −4.00000 −0.666667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −2.00000 −0.308607
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −10.0000 −1.50756
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) −4.00000 −0.577350
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 5.00000 0.700140
\(52\) −6.00000 −0.832050
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 10.0000 1.36083
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −12.0000 −1.52400
\(63\) −2.00000 −0.251976
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 10.0000 1.23091
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 10.0000 1.21268
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) 6.00000 0.679366
\(79\) −15.0000 −1.68763 −0.843816 0.536633i \(-0.819696\pi\)
−0.843816 + 0.536633i \(0.819696\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 2.00000 0.208514
\(93\) 6.00000 0.622171
\(94\) −18.0000 −1.85656
\(95\) 0 0
\(96\) 8.00000 0.816497
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) −2.00000 −0.202031
\(99\) 10.0000 1.00504
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −10.0000 −0.990148
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) −10.0000 −0.962250
\(109\) −17.0000 −1.62830 −0.814152 0.580651i \(-0.802798\pi\)
−0.814152 + 0.580651i \(0.802798\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) −4.00000 −0.377964
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 6.00000 0.554700
\(118\) 12.0000 1.10469
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −20.0000 −1.81071
\(123\) 0 0
\(124\) 12.0000 1.07763
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) −10.0000 −0.870388
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −2.00000 −0.170251
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) 16.0000 1.34269
\(143\) 15.0000 1.25436
\(144\) 8.00000 0.666667
\(145\) 0 0
\(146\) 20.0000 1.65521
\(147\) 1.00000 0.0824786
\(148\) 8.00000 0.657596
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) −9.00000 −0.732410 −0.366205 0.930534i \(-0.619343\pi\)
−0.366205 + 0.930534i \(0.619343\pi\)
\(152\) 0 0
\(153\) −10.0000 −0.808452
\(154\) 10.0000 0.805823
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 30.0000 2.38667
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) −2.00000 −0.157135
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 24.0000 1.86276
\(167\) 1.00000 0.0773823 0.0386912 0.999251i \(-0.487681\pi\)
0.0386912 + 0.999251i \(0.487681\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 3.00000 0.228086 0.114043 0.993476i \(-0.463620\pi\)
0.114043 + 0.993476i \(0.463620\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 20.0000 1.50756
\(177\) −6.00000 −0.450988
\(178\) 20.0000 1.49906
\(179\) 8.00000 0.597948 0.298974 0.954261i \(-0.403356\pi\)
0.298974 + 0.954261i \(0.403356\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 6.00000 0.444750
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 0 0
\(186\) −12.0000 −0.879883
\(187\) −25.0000 −1.82818
\(188\) 18.0000 1.31278
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 27.0000 1.95365 0.976826 0.214036i \(-0.0686611\pi\)
0.976826 + 0.214036i \(0.0686611\pi\)
\(192\) −8.00000 −0.577350
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) −20.0000 −1.42134
\(199\) −22.0000 −1.55954 −0.779769 0.626067i \(-0.784664\pi\)
−0.779769 + 0.626067i \(0.784664\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) −20.0000 −1.40720
\(203\) 3.00000 0.210559
\(204\) 10.0000 0.700140
\(205\) 0 0
\(206\) 2.00000 0.139347
\(207\) −2.00000 −0.139010
\(208\) 12.0000 0.832050
\(209\) 0 0
\(210\) 0 0
\(211\) −17.0000 −1.17033 −0.585164 0.810915i \(-0.698970\pi\)
−0.585164 + 0.810915i \(0.698970\pi\)
\(212\) 12.0000 0.824163
\(213\) −8.00000 −0.548151
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 34.0000 2.30277
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) −15.0000 −1.00901
\(222\) −8.00000 −0.536925
\(223\) −15.0000 −1.00447 −0.502237 0.864730i \(-0.667490\pi\)
−0.502237 + 0.864730i \(0.667490\pi\)
\(224\) 8.00000 0.534522
\(225\) 0 0
\(226\) 36.0000 2.39468
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 0 0
\(229\) −8.00000 −0.528655 −0.264327 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) −12.0000 −0.784465
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) −15.0000 −0.974355
\(238\) −10.0000 −0.648204
\(239\) 7.00000 0.452792 0.226396 0.974035i \(-0.427306\pi\)
0.226396 + 0.974035i \(0.427306\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −28.0000 −1.79991
\(243\) 16.0000 1.02640
\(244\) 20.0000 1.28037
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) −4.00000 −0.251976
\(253\) −5.00000 −0.314347
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) −4.00000 −0.249029
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 40.0000 2.47121
\(263\) 28.0000 1.72655 0.863277 0.504730i \(-0.168408\pi\)
0.863277 + 0.504730i \(0.168408\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) −8.00000 −0.488678
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) −18.0000 −1.09342 −0.546711 0.837321i \(-0.684120\pi\)
−0.546711 + 0.837321i \(0.684120\pi\)
\(272\) −20.0000 −1.21268
\(273\) −3.00000 −0.181568
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 2.00000 0.120386
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 16.0000 0.959616
\(279\) −12.0000 −0.718421
\(280\) 0 0
\(281\) −3.00000 −0.178965 −0.0894825 0.995988i \(-0.528521\pi\)
−0.0894825 + 0.995988i \(0.528521\pi\)
\(282\) −18.0000 −1.07188
\(283\) 25.0000 1.48610 0.743048 0.669238i \(-0.233379\pi\)
0.743048 + 0.669238i \(0.233379\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) −30.0000 −1.77394
\(287\) 0 0
\(288\) −16.0000 −0.942809
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −7.00000 −0.410347
\(292\) −20.0000 −1.17041
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) 0 0
\(297\) 25.0000 1.45065
\(298\) 4.00000 0.231714
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 18.0000 1.03578
\(303\) 10.0000 0.574485
\(304\) 0 0
\(305\) 0 0
\(306\) 20.0000 1.14332
\(307\) −17.0000 −0.970241 −0.485121 0.874447i \(-0.661224\pi\)
−0.485121 + 0.874447i \(0.661224\pi\)
\(308\) −10.0000 −0.569803
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) −20.0000 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) 0 0
\(313\) 15.0000 0.847850 0.423925 0.905697i \(-0.360652\pi\)
0.423925 + 0.905697i \(0.360652\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −30.0000 −1.68763
\(317\) 20.0000 1.12331 0.561656 0.827371i \(-0.310164\pi\)
0.561656 + 0.827371i \(0.310164\pi\)
\(318\) −12.0000 −0.672927
\(319\) −15.0000 −0.839839
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) −2.00000 −0.111456
\(323\) 0 0
\(324\) 2.00000 0.111111
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) −17.0000 −0.940102
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) 24.0000 1.31916 0.659580 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) −24.0000 −1.31717
\(333\) −8.00000 −0.438397
\(334\) −2.00000 −0.109435
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 8.00000 0.435143
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) −30.0000 −1.62459
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 6.00000 0.321634
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 0 0
\(351\) 15.0000 0.800641
\(352\) −40.0000 −2.13201
\(353\) −21.0000 −1.11772 −0.558859 0.829263i \(-0.688761\pi\)
−0.558859 + 0.829263i \(0.688761\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −20.0000 −1.06000
\(357\) 5.00000 0.264628
\(358\) −16.0000 −0.845626
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 28.0000 1.47165
\(363\) 14.0000 0.734809
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) −20.0000 −1.04542
\(367\) 23.0000 1.20059 0.600295 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 12.0000 0.622171
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 50.0000 2.58544
\(375\) 0 0
\(376\) 0 0
\(377\) −9.00000 −0.463524
\(378\) 10.0000 0.514344
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) −54.0000 −2.76288
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 12.0000 0.610784
\(387\) −4.00000 −0.203331
\(388\) −14.0000 −0.710742
\(389\) 31.0000 1.57176 0.785881 0.618378i \(-0.212210\pi\)
0.785881 + 0.618378i \(0.212210\pi\)
\(390\) 0 0
\(391\) 5.00000 0.252861
\(392\) 0 0
\(393\) −20.0000 −1.00887
\(394\) −24.0000 −1.20910
\(395\) 0 0
\(396\) 20.0000 1.00504
\(397\) −5.00000 −0.250943 −0.125471 0.992097i \(-0.540044\pi\)
−0.125471 + 0.992097i \(0.540044\pi\)
\(398\) 44.0000 2.20552
\(399\) 0 0
\(400\) 0 0
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) 8.00000 0.399004
\(403\) −18.0000 −0.896644
\(404\) 20.0000 0.995037
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) −20.0000 −0.991363
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) −2.00000 −0.0985329
\(413\) −6.00000 −0.295241
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) −24.0000 −1.17670
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) −37.0000 −1.80327 −0.901635 0.432498i \(-0.857632\pi\)
−0.901635 + 0.432498i \(0.857632\pi\)
\(422\) 34.0000 1.65509
\(423\) −18.0000 −0.875190
\(424\) 0 0
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) 10.0000 0.483934
\(428\) 12.0000 0.580042
\(429\) 15.0000 0.724207
\(430\) 0 0
\(431\) −33.0000 −1.58955 −0.794777 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) 20.0000 0.962250
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −12.0000 −0.576018
\(435\) 0 0
\(436\) −34.0000 −1.62830
\(437\) 0 0
\(438\) 20.0000 0.955637
\(439\) −6.00000 −0.286364 −0.143182 0.989696i \(-0.545733\pi\)
−0.143182 + 0.989696i \(0.545733\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 30.0000 1.42695
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) 30.0000 1.42054
\(447\) −2.00000 −0.0945968
\(448\) −8.00000 −0.377964
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −36.0000 −1.69330
\(453\) −9.00000 −0.422857
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 16.0000 0.747631
\(459\) −25.0000 −1.16690
\(460\) 0 0
\(461\) 32.0000 1.49039 0.745194 0.666847i \(-0.232357\pi\)
0.745194 + 0.666847i \(0.232357\pi\)
\(462\) 10.0000 0.465242
\(463\) 10.0000 0.464739 0.232370 0.972628i \(-0.425352\pi\)
0.232370 + 0.972628i \(0.425352\pi\)
\(464\) −12.0000 −0.557086
\(465\) 0 0
\(466\) 36.0000 1.66767
\(467\) 21.0000 0.971764 0.485882 0.874024i \(-0.338498\pi\)
0.485882 + 0.874024i \(0.338498\pi\)
\(468\) 12.0000 0.554700
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 30.0000 1.37795
\(475\) 0 0
\(476\) 10.0000 0.458349
\(477\) −12.0000 −0.549442
\(478\) −14.0000 −0.640345
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −4.00000 −0.182195
\(483\) 1.00000 0.0455016
\(484\) 28.0000 1.27273
\(485\) 0 0
\(486\) −32.0000 −1.45155
\(487\) −10.0000 −0.453143 −0.226572 0.973995i \(-0.572752\pi\)
−0.226572 + 0.973995i \(0.572752\pi\)
\(488\) 0 0
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −41.0000 −1.85030 −0.925152 0.379597i \(-0.876063\pi\)
−0.925152 + 0.379597i \(0.876063\pi\)
\(492\) 0 0
\(493\) 15.0000 0.675566
\(494\) 0 0
\(495\) 0 0
\(496\) −24.0000 −1.07763
\(497\) −8.00000 −0.358849
\(498\) 24.0000 1.07547
\(499\) 25.0000 1.11915 0.559577 0.828778i \(-0.310964\pi\)
0.559577 + 0.828778i \(0.310964\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 40.0000 1.78529
\(503\) −39.0000 −1.73892 −0.869462 0.494000i \(-0.835534\pi\)
−0.869462 + 0.494000i \(0.835534\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 10.0000 0.444554
\(507\) −4.00000 −0.177646
\(508\) −8.00000 −0.354943
\(509\) 36.0000 1.59567 0.797836 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) 4.00000 0.176432
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) −45.0000 −1.97910
\(518\) −8.00000 −0.351500
\(519\) 3.00000 0.131685
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 12.0000 0.525226
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −40.0000 −1.74741
\(525\) 0 0
\(526\) −56.0000 −2.44172
\(527\) 30.0000 1.30682
\(528\) 20.0000 0.870388
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 0 0
\(534\) 20.0000 0.865485
\(535\) 0 0
\(536\) 0 0
\(537\) 8.00000 0.345225
\(538\) −4.00000 −0.172452
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) −9.00000 −0.386940 −0.193470 0.981106i \(-0.561974\pi\)
−0.193470 + 0.981106i \(0.561974\pi\)
\(542\) 36.0000 1.54633
\(543\) −14.0000 −0.600798
\(544\) 40.0000 1.71499
\(545\) 0 0
\(546\) 6.00000 0.256776
\(547\) 38.0000 1.62476 0.812381 0.583127i \(-0.198171\pi\)
0.812381 + 0.583127i \(0.198171\pi\)
\(548\) 12.0000 0.512615
\(549\) −20.0000 −0.853579
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −15.0000 −0.637865
\(554\) −16.0000 −0.679775
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 24.0000 1.01600
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) −25.0000 −1.05550
\(562\) 6.00000 0.253095
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 18.0000 0.757937
\(565\) 0 0
\(566\) −50.0000 −2.10166
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −2.00000 −0.0838444 −0.0419222 0.999121i \(-0.513348\pi\)
−0.0419222 + 0.999121i \(0.513348\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 30.0000 1.25436
\(573\) 27.0000 1.12794
\(574\) 0 0
\(575\) 0 0
\(576\) 16.0000 0.666667
\(577\) −23.0000 −0.957503 −0.478751 0.877951i \(-0.658910\pi\)
−0.478751 + 0.877951i \(0.658910\pi\)
\(578\) −16.0000 −0.665512
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 14.0000 0.580319
\(583\) −30.0000 −1.24247
\(584\) 0 0
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 2.00000 0.0824786
\(589\) 0 0
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) −16.0000 −0.657596
\(593\) −27.0000 −1.10876 −0.554379 0.832265i \(-0.687044\pi\)
−0.554379 + 0.832265i \(0.687044\pi\)
\(594\) −50.0000 −2.05152
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) −22.0000 −0.900400
\(598\) 6.00000 0.245358
\(599\) −43.0000 −1.75693 −0.878466 0.477805i \(-0.841433\pi\)
−0.878466 + 0.477805i \(0.841433\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) −4.00000 −0.163028
\(603\) 8.00000 0.325785
\(604\) −18.0000 −0.732410
\(605\) 0 0
\(606\) −20.0000 −0.812444
\(607\) −33.0000 −1.33943 −0.669714 0.742619i \(-0.733583\pi\)
−0.669714 + 0.742619i \(0.733583\pi\)
\(608\) 0 0
\(609\) 3.00000 0.121566
\(610\) 0 0
\(611\) −27.0000 −1.09230
\(612\) −20.0000 −0.808452
\(613\) −24.0000 −0.969351 −0.484675 0.874694i \(-0.661062\pi\)
−0.484675 + 0.874694i \(0.661062\pi\)
\(614\) 34.0000 1.37213
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 2.00000 0.0804518
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 40.0000 1.60385
\(623\) −10.0000 −0.400642
\(624\) 12.0000 0.480384
\(625\) 0 0
\(626\) −30.0000 −1.19904
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) 35.0000 1.39333 0.696664 0.717398i \(-0.254667\pi\)
0.696664 + 0.717398i \(0.254667\pi\)
\(632\) 0 0
\(633\) −17.0000 −0.675689
\(634\) −40.0000 −1.58860
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) −3.00000 −0.118864
\(638\) 30.0000 1.18771
\(639\) 16.0000 0.632950
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) −12.0000 −0.473602
\(643\) 31.0000 1.22252 0.611260 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(644\) 2.00000 0.0788110
\(645\) 0 0
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) −16.0000 −0.626608
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 34.0000 1.32951
\(655\) 0 0
\(656\) 0 0
\(657\) 20.0000 0.780274
\(658\) −18.0000 −0.701713
\(659\) −21.0000 −0.818044 −0.409022 0.912525i \(-0.634130\pi\)
−0.409022 + 0.912525i \(0.634130\pi\)
\(660\) 0 0
\(661\) 40.0000 1.55582 0.777910 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) −48.0000 −1.86557
\(663\) −15.0000 −0.582552
\(664\) 0 0
\(665\) 0 0
\(666\) 16.0000 0.619987
\(667\) 3.00000 0.116160
\(668\) 2.00000 0.0773823
\(669\) −15.0000 −0.579934
\(670\) 0 0
\(671\) −50.0000 −1.93023
\(672\) 8.00000 0.308607
\(673\) −44.0000 −1.69608 −0.848038 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) 4.00000 0.154074
\(675\) 0 0
\(676\) −8.00000 −0.307692
\(677\) 39.0000 1.49889 0.749446 0.662066i \(-0.230320\pi\)
0.749446 + 0.662066i \(0.230320\pi\)
\(678\) 36.0000 1.38257
\(679\) −7.00000 −0.268635
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 60.0000 2.29752
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.00000 −0.0763604
\(687\) −8.00000 −0.305219
\(688\) −8.00000 −0.304997
\(689\) −18.0000 −0.685745
\(690\) 0 0
\(691\) 26.0000 0.989087 0.494543 0.869153i \(-0.335335\pi\)
0.494543 + 0.869153i \(0.335335\pi\)
\(692\) 6.00000 0.228086
\(693\) 10.0000 0.379869
\(694\) 24.0000 0.911028
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −56.0000 −2.11963
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 45.0000 1.69963 0.849813 0.527084i \(-0.176715\pi\)
0.849813 + 0.527084i \(0.176715\pi\)
\(702\) −30.0000 −1.13228
\(703\) 0 0
\(704\) 40.0000 1.50756
\(705\) 0 0
\(706\) 42.0000 1.58069
\(707\) 10.0000 0.376089
\(708\) −12.0000 −0.450988
\(709\) −51.0000 −1.91535 −0.957673 0.287860i \(-0.907056\pi\)
−0.957673 + 0.287860i \(0.907056\pi\)
\(710\) 0 0
\(711\) 30.0000 1.12509
\(712\) 0 0
\(713\) 6.00000 0.224702
\(714\) −10.0000 −0.374241
\(715\) 0 0
\(716\) 16.0000 0.597948
\(717\) 7.00000 0.261420
\(718\) −64.0000 −2.38846
\(719\) 26.0000 0.969636 0.484818 0.874615i \(-0.338886\pi\)
0.484818 + 0.874615i \(0.338886\pi\)
\(720\) 0 0
\(721\) −1.00000 −0.0372419
\(722\) 38.0000 1.41421
\(723\) 2.00000 0.0743808
\(724\) −28.0000 −1.04061
\(725\) 0 0
\(726\) −28.0000 −1.03918
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 10.0000 0.369863
\(732\) 20.0000 0.739221
\(733\) −29.0000 −1.07114 −0.535570 0.844491i \(-0.679903\pi\)
−0.535570 + 0.844491i \(0.679903\pi\)
\(734\) −46.0000 −1.69789
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) 20.0000 0.736709
\(738\) 0 0
\(739\) −43.0000 −1.58178 −0.790890 0.611958i \(-0.790382\pi\)
−0.790890 + 0.611958i \(0.790382\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) 30.0000 1.10059 0.550297 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 52.0000 1.90386
\(747\) 24.0000 0.878114
\(748\) −50.0000 −1.82818
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 53.0000 1.93400 0.966999 0.254781i \(-0.0820034\pi\)
0.966999 + 0.254781i \(0.0820034\pi\)
\(752\) −36.0000 −1.31278
\(753\) −20.0000 −0.728841
\(754\) 18.0000 0.655521
\(755\) 0 0
\(756\) −10.0000 −0.363696
\(757\) −8.00000 −0.290765 −0.145382 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) 40.0000 1.45287
\(759\) −5.00000 −0.181489
\(760\) 0 0
\(761\) −36.0000 −1.30500 −0.652499 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) 8.00000 0.289809
\(763\) −17.0000 −0.615441
\(764\) 54.0000 1.95365
\(765\) 0 0
\(766\) 32.0000 1.15621
\(767\) 18.0000 0.649942
\(768\) 16.0000 0.577350
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) −12.0000 −0.431889
\(773\) 15.0000 0.539513 0.269756 0.962929i \(-0.413057\pi\)
0.269756 + 0.962929i \(0.413057\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) 0 0
\(777\) 4.00000 0.143499
\(778\) −62.0000 −2.22281
\(779\) 0 0
\(780\) 0 0
\(781\) 40.0000 1.43131
\(782\) −10.0000 −0.357599
\(783\) −15.0000 −0.536056
\(784\) −4.00000 −0.142857
\(785\) 0 0
\(786\) 40.0000 1.42675
\(787\) −37.0000 −1.31891 −0.659454 0.751745i \(-0.729212\pi\)
−0.659454 + 0.751745i \(0.729212\pi\)
\(788\) 24.0000 0.854965
\(789\) 28.0000 0.996826
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) −30.0000 −1.06533
\(794\) 10.0000 0.354887
\(795\) 0 0
\(796\) −44.0000 −1.55954
\(797\) −51.0000 −1.80651 −0.903256 0.429101i \(-0.858830\pi\)
−0.903256 + 0.429101i \(0.858830\pi\)
\(798\) 0 0
\(799\) 45.0000 1.59199
\(800\) 0 0
\(801\) 20.0000 0.706665
\(802\) −6.00000 −0.211867
\(803\) 50.0000 1.76446
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 36.0000 1.26805
\(807\) 2.00000 0.0704033
\(808\) 0 0
\(809\) 9.00000 0.316423 0.158212 0.987405i \(-0.449427\pi\)
0.158212 + 0.987405i \(0.449427\pi\)
\(810\) 0 0
\(811\) 22.0000 0.772524 0.386262 0.922389i \(-0.373766\pi\)
0.386262 + 0.922389i \(0.373766\pi\)
\(812\) 6.00000 0.210559
\(813\) −18.0000 −0.631288
\(814\) 40.0000 1.40200
\(815\) 0 0
\(816\) −20.0000 −0.700140
\(817\) 0 0
\(818\) 12.0000 0.419570
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) −19.0000 −0.663105 −0.331552 0.943437i \(-0.607572\pi\)
−0.331552 + 0.943437i \(0.607572\pi\)
\(822\) −12.0000 −0.418548
\(823\) −42.0000 −1.46403 −0.732014 0.681290i \(-0.761419\pi\)
−0.732014 + 0.681290i \(0.761419\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) −4.00000 −0.139010
\(829\) 16.0000 0.555703 0.277851 0.960624i \(-0.410378\pi\)
0.277851 + 0.960624i \(0.410378\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 24.0000 0.832050
\(833\) 5.00000 0.173240
\(834\) 16.0000 0.554035
\(835\) 0 0
\(836\) 0 0
\(837\) −30.0000 −1.03695
\(838\) 36.0000 1.24360
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 74.0000 2.55021
\(843\) −3.00000 −0.103325
\(844\) −34.0000 −1.17033
\(845\) 0 0
\(846\) 36.0000 1.23771
\(847\) 14.0000 0.481046
\(848\) −24.0000 −0.824163
\(849\) 25.0000 0.857998
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) −16.0000 −0.548151
\(853\) 2.00000 0.0684787 0.0342393 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) −20.0000 −0.684386
\(855\) 0 0
\(856\) 0 0
\(857\) −46.0000 −1.57133 −0.785665 0.618652i \(-0.787679\pi\)
−0.785665 + 0.618652i \(0.787679\pi\)
\(858\) −30.0000 −1.02418
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 66.0000 2.24797
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) −40.0000 −1.36083
\(865\) 0 0
\(866\) 68.0000 2.31073
\(867\) 8.00000 0.271694
\(868\) 12.0000 0.407307
\(869\) 75.0000 2.54420
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 0 0
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) 0 0
\(876\) −20.0000 −0.675737
\(877\) −8.00000 −0.270141 −0.135070 0.990836i \(-0.543126\pi\)
−0.135070 + 0.990836i \(0.543126\pi\)
\(878\) 12.0000 0.404980
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) −34.0000 −1.14549 −0.572745 0.819734i \(-0.694121\pi\)
−0.572745 + 0.819734i \(0.694121\pi\)
\(882\) 4.00000 0.134687
\(883\) 48.0000 1.61533 0.807664 0.589643i \(-0.200731\pi\)
0.807664 + 0.589643i \(0.200731\pi\)
\(884\) −30.0000 −1.00901
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 16.0000 0.537227 0.268614 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) −30.0000 −1.00447
\(893\) 0 0
\(894\) 4.00000 0.133780
\(895\) 0 0
\(896\) 0 0
\(897\) −3.00000 −0.100167
\(898\) 18.0000 0.600668
\(899\) 18.0000 0.600334
\(900\) 0 0
\(901\) 30.0000 0.999445
\(902\) 0 0
\(903\) 2.00000 0.0665558
\(904\) 0 0
\(905\) 0 0
\(906\) 18.0000 0.598010
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 6.00000 0.199117
\(909\) −20.0000 −0.663358
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 0 0
\(913\) 60.0000 1.98571
\(914\) 0 0
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) −20.0000 −0.660458
\(918\) 50.0000 1.65025
\(919\) 37.0000 1.22052 0.610259 0.792202i \(-0.291065\pi\)
0.610259 + 0.792202i \(0.291065\pi\)
\(920\) 0 0
\(921\) −17.0000 −0.560169
\(922\) −64.0000 −2.10773
\(923\) 24.0000 0.789970
\(924\) −10.0000 −0.328976
\(925\) 0 0
\(926\) −20.0000 −0.657241
\(927\) 2.00000 0.0656886
\(928\) 24.0000 0.787839
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −36.0000 −1.17922
\(933\) −20.0000 −0.654771
\(934\) −42.0000 −1.37428
\(935\) 0 0
\(936\) 0 0
\(937\) −3.00000 −0.0980057 −0.0490029 0.998799i \(-0.515604\pi\)
−0.0490029 + 0.998799i \(0.515604\pi\)
\(938\) 8.00000 0.261209
\(939\) 15.0000 0.489506
\(940\) 0 0
\(941\) 20.0000 0.651981 0.325991 0.945373i \(-0.394302\pi\)
0.325991 + 0.945373i \(0.394302\pi\)
\(942\) 4.00000 0.130327
\(943\) 0 0
\(944\) 24.0000 0.781133
\(945\) 0 0
\(946\) 20.0000 0.650256
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −30.0000 −0.974355
\(949\) 30.0000 0.973841
\(950\) 0 0
\(951\) 20.0000 0.648544
\(952\) 0 0
\(953\) −50.0000 −1.61966 −0.809829 0.586665i \(-0.800440\pi\)
−0.809829 + 0.586665i \(0.800440\pi\)
\(954\) 24.0000 0.777029
\(955\) 0 0
\(956\) 14.0000 0.452792
\(957\) −15.0000 −0.484881
\(958\) 36.0000 1.16311
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 24.0000 0.773791
\(963\) −12.0000 −0.386695
\(964\) 4.00000 0.128831
\(965\) 0 0
\(966\) −2.00000 −0.0643489
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 32.0000 1.02640
\(973\) −8.00000 −0.256468
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) −40.0000 −1.28037
\(977\) 48.0000 1.53566 0.767828 0.640656i \(-0.221338\pi\)
0.767828 + 0.640656i \(0.221338\pi\)
\(978\) 16.0000 0.511624
\(979\) 50.0000 1.59801
\(980\) 0 0
\(981\) 34.0000 1.08554
\(982\) 82.0000 2.61673
\(983\) 13.0000 0.414636 0.207318 0.978274i \(-0.433527\pi\)
0.207318 + 0.978274i \(0.433527\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −30.0000 −0.955395
\(987\) 9.00000 0.286473
\(988\) 0 0
\(989\) 2.00000 0.0635963
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 48.0000 1.52400
\(993\) 24.0000 0.761617
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) −24.0000 −0.760469
\(997\) 51.0000 1.61519 0.807593 0.589740i \(-0.200770\pi\)
0.807593 + 0.589740i \(0.200770\pi\)
\(998\) −50.0000 −1.58272
\(999\) −20.0000 −0.632772
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 4025.2.a.b.1.1 1
5.4 even 2 805.2.a.c.1.1 1
15.14 odd 2 7245.2.a.d.1.1 1
35.34 odd 2 5635.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
805.2.a.c.1.1 1 5.4 even 2
4025.2.a.b.1.1 1 1.1 even 1 trivial
5635.2.a.k.1.1 1 35.34 odd 2
7245.2.a.d.1.1 1 15.14 odd 2