Properties

Label 7245.2.a.d.1.1
Level $7245$
Weight $2$
Character 7245.1
Self dual yes
Analytic conductor $57.852$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7245 = 3^{2} \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.8516162644\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 7245.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{10} +5.00000 q^{11} +3.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} +5.00000 q^{17} +2.00000 q^{20} -10.0000 q^{22} +1.00000 q^{23} +1.00000 q^{25} -6.00000 q^{26} -2.00000 q^{28} -3.00000 q^{29} +6.00000 q^{31} +8.00000 q^{32} -10.0000 q^{34} -1.00000 q^{35} -4.00000 q^{37} -2.00000 q^{43} +10.0000 q^{44} -2.00000 q^{46} +9.00000 q^{47} +1.00000 q^{49} -2.00000 q^{50} +6.00000 q^{52} +6.00000 q^{53} +5.00000 q^{55} +6.00000 q^{58} +6.00000 q^{59} +10.0000 q^{61} -12.0000 q^{62} -8.00000 q^{64} +3.00000 q^{65} +4.00000 q^{67} +10.0000 q^{68} +2.00000 q^{70} +8.00000 q^{71} +10.0000 q^{73} +8.00000 q^{74} -5.00000 q^{77} -15.0000 q^{79} -4.00000 q^{80} -12.0000 q^{83} +5.00000 q^{85} +4.00000 q^{86} +10.0000 q^{89} -3.00000 q^{91} +2.00000 q^{92} -18.0000 q^{94} +7.00000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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 \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 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 \(0.292637\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −10.0000 −2.13201
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −10.0000 −1.71499
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 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 \(0.272082\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.00000 −0.282843
\(51\) 0 0
\(52\) 6.00000 0.832050
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 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
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 10.0000 1.21268
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) −15.0000 −1.68763 −0.843816 0.536633i \(-0.819696\pi\)
−0.843816 + 0.536633i \(0.819696\pi\)
\(80\) −4.00000 −0.447214
\(81\) 0 0
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 5.00000 0.542326
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) −18.0000 −1.85656
\(95\) 0 0
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) −2.00000 −0.202031
\(99\) 0 0
\(100\) 2.00000 0.200000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 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 \(0.406338\pi\)
\(108\) 0 0
\(109\) −17.0000 −1.62830 −0.814152 0.580651i \(-0.802798\pi\)
−0.814152 + 0.580651i \(0.802798\pi\)
\(110\) −10.0000 −0.953463
\(111\) 0 0
\(112\) 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
\(115\) 1.00000 0.0932505
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −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
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −6.00000 −0.526235
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) 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 \(0.417494\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) −16.0000 −1.34269
\(143\) 15.0000 1.25436
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) −20.0000 −1.65521
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) −9.00000 −0.732410 −0.366205 0.930534i \(-0.619343\pi\)
−0.366205 + 0.930534i \(0.619343\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 10.0000 0.805823
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 30.0000 2.38667
\(159\) 0 0
\(160\) 8.00000 0.632456
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 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
\(170\) −10.0000 −0.766965
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 3.00000 0.228086 0.114043 0.993476i \(-0.463620\pi\)
0.114043 + 0.993476i \(0.463620\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −20.0000 −1.50756
\(177\) 0 0
\(178\) −20.0000 −1.49906
\(179\) −8.00000 −0.597948 −0.298974 0.954261i \(-0.596644\pi\)
−0.298974 + 0.954261i \(0.596644\pi\)
\(180\) 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
\(183\) 0 0
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 25.0000 1.82818
\(188\) 18.0000 1.31278
\(189\) 0 0
\(190\) 0 0
\(191\) −27.0000 −1.95365 −0.976826 0.214036i \(-0.931339\pi\)
−0.976826 + 0.214036i \(0.931339\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) −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 \(0.359401\pi\)
\(198\) 0 0
\(199\) −22.0000 −1.55954 −0.779769 0.626067i \(-0.784664\pi\)
−0.779769 + 0.626067i \(0.784664\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 20.0000 1.40720
\(203\) 3.00000 0.210559
\(204\) 0 0
\(205\) 0 0
\(206\) −2.00000 −0.139347
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −2.00000 −0.136399
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 34.0000 2.30277
\(219\) 0 0
\(220\) 10.0000 0.674200
\(221\) 15.0000 1.00901
\(222\) 0 0
\(223\) 15.0000 1.00447 0.502237 0.864730i \(-0.332510\pi\)
0.502237 + 0.864730i \(0.332510\pi\)
\(224\) −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 \(0.585150\pi\)
\(230\) −2.00000 −0.131876
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 10.0000 0.648204
\(239\) −7.00000 −0.452792 −0.226396 0.974035i \(-0.572694\pi\)
−0.226396 + 0.974035i \(0.572694\pi\)
\(240\) 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
\(243\) 0 0
\(244\) 20.0000 1.28037
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −2.00000 −0.126491
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 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 \(0.519869\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) −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
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 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
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 5.00000 0.301511
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 16.0000 0.959616
\(279\) 0 0
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 0 0
\(283\) −25.0000 −1.48610 −0.743048 0.669238i \(-0.766621\pi\)
−0.743048 + 0.669238i \(0.766621\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) −30.0000 −1.77394
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) 20.0000 1.17041
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 0 0
\(297\) 0 0
\(298\) −4.00000 −0.231714
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 18.0000 1.03578
\(303\) 0 0
\(304\) 0 0
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 17.0000 0.970241 0.485121 0.874447i \(-0.338776\pi\)
0.485121 + 0.874447i \(0.338776\pi\)
\(308\) −10.0000 −0.569803
\(309\) 0 0
\(310\) −12.0000 −0.681554
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) 0 0
\(313\) −15.0000 −0.847850 −0.423925 0.905697i \(-0.639348\pi\)
−0.423925 + 0.905697i \(0.639348\pi\)
\(314\) −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 \(0.310164\pi\)
\(318\) 0 0
\(319\) −15.0000 −0.839839
\(320\) −8.00000 −0.447214
\(321\) 0 0
\(322\) 2.00000 0.111456
\(323\) 0 0
\(324\) 0 0
\(325\) 3.00000 0.166410
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) 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
\(333\) 0 0
\(334\) −2.00000 −0.109435
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 8.00000 0.435143
\(339\) 0 0
\(340\) 10.0000 0.542326
\(341\) 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 \(0.604388\pi\)
\(348\) 0 0
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 40.0000 2.13201
\(353\) −21.0000 −1.11772 −0.558859 0.829263i \(-0.688761\pi\)
−0.558859 + 0.829263i \(0.688761\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 20.0000 1.06000
\(357\) 0 0
\(358\) 16.0000 0.845626
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 28.0000 1.47165
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) −23.0000 −1.20059 −0.600295 0.799779i \(-0.704950\pi\)
−0.600295 + 0.799779i \(0.704950\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) −50.0000 −2.58544
\(375\) 0 0
\(376\) 0 0
\(377\) −9.00000 −0.463524
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 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
\(385\) −5.00000 −0.254824
\(386\) −12.0000 −0.610784
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) −31.0000 −1.57176 −0.785881 0.618378i \(-0.787790\pi\)
−0.785881 + 0.618378i \(0.787790\pi\)
\(390\) 0 0
\(391\) 5.00000 0.252861
\(392\) 0 0
\(393\) 0 0
\(394\) −24.0000 −1.20910
\(395\) −15.0000 −0.754732
\(396\) 0 0
\(397\) 5.00000 0.250943 0.125471 0.992097i \(-0.459956\pi\)
0.125471 + 0.992097i \(0.459956\pi\)
\(398\) 44.0000 2.20552
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −3.00000 −0.149813 −0.0749064 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 0 0
\(403\) 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
\(411\) 0 0
\(412\) 2.00000 0.0985329
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 24.0000 1.17670
\(417\) 0 0
\(418\) 0 0
\(419\) 18.0000 0.879358 0.439679 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 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
\(423\) 0 0
\(424\) 0 0
\(425\) 5.00000 0.242536
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 33.0000 1.58955 0.794777 0.606902i \(-0.207588\pi\)
0.794777 + 0.606902i \(0.207588\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) −34.0000 −1.62830
\(437\) 0 0
\(438\) 0 0
\(439\) −6.00000 −0.286364 −0.143182 0.989696i \(-0.545733\pi\)
−0.143182 + 0.989696i \(0.545733\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −30.0000 −1.42695
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) −30.0000 −1.42054
\(447\) 0 0
\(448\) 8.00000 0.377964
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −36.0000 −1.69330
\(453\) 0 0
\(454\) −6.00000 −0.281594
\(455\) −3.00000 −0.140642
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 16.0000 0.747631
\(459\) 0 0
\(460\) 2.00000 0.0932505
\(461\) −32.0000 −1.49039 −0.745194 0.666847i \(-0.767643\pi\)
−0.745194 + 0.666847i \(0.767643\pi\)
\(462\) 0 0
\(463\) −10.0000 −0.464739 −0.232370 0.972628i \(-0.574648\pi\)
−0.232370 + 0.972628i \(0.574648\pi\)
\(464\) 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 \(0.338498\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) −18.0000 −0.830278
\(471\) 0 0
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) 0 0
\(476\) −10.0000 −0.458349
\(477\) 0 0
\(478\) 14.0000 0.640345
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −4.00000 −0.182195
\(483\) 0 0
\(484\) 28.0000 1.27273
\(485\) 7.00000 0.317854
\(486\) 0 0
\(487\) 10.0000 0.453143 0.226572 0.973995i \(-0.427248\pi\)
0.226572 + 0.973995i \(0.427248\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −2.00000 −0.0903508
\(491\) 41.0000 1.85030 0.925152 0.379597i \(-0.123937\pi\)
0.925152 + 0.379597i \(0.123937\pi\)
\(492\) 0 0
\(493\) −15.0000 −0.675566
\(494\) 0 0
\(495\) 0 0
\(496\) −24.0000 −1.07763
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 25.0000 1.11915 0.559577 0.828778i \(-0.310964\pi\)
0.559577 + 0.828778i \(0.310964\pi\)
\(500\) 2.00000 0.0894427
\(501\) 0 0
\(502\) −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
\(505\) −10.0000 −0.444994
\(506\) −10.0000 −0.444554
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −36.0000 −1.59567 −0.797836 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) 4.00000 0.176432
\(515\) 1.00000 0.0440653
\(516\) 0 0
\(517\) 45.0000 1.97910
\(518\) −8.00000 −0.351500
\(519\) 0 0
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 40.0000 1.74741
\(525\) 0 0
\(526\) −56.0000 −2.44172
\(527\) 30.0000 1.30682
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −12.0000 −0.521247
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) 0 0
\(538\) 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
\(543\) 0 0
\(544\) 40.0000 1.71499
\(545\) −17.0000 −0.728200
\(546\) 0 0
\(547\) −38.0000 −1.62476 −0.812381 0.583127i \(-0.801829\pi\)
−0.812381 + 0.583127i \(0.801829\pi\)
\(548\) 12.0000 0.512615
\(549\) 0 0
\(550\) −10.0000 −0.426401
\(551\) 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 \(0.404147\pi\)
\(558\) 0 0
\(559\) −6.00000 −0.253773
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) 50.0000 2.10166
\(567\) 0 0
\(568\) 0 0
\(569\) 2.00000 0.0838444 0.0419222 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 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
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 23.0000 0.957503 0.478751 0.877951i \(-0.341090\pi\)
0.478751 + 0.877951i \(0.341090\pi\)
\(578\) −16.0000 −0.665512
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 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 \(0.420343\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −12.0000 −0.494032
\(591\) 0 0
\(592\) 16.0000 0.657596
\(593\) −27.0000 −1.10876 −0.554379 0.832265i \(-0.687044\pi\)
−0.554379 + 0.832265i \(0.687044\pi\)
\(594\) 0 0
\(595\) −5.00000 −0.204980
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) −6.00000 −0.245358
\(599\) 43.0000 1.75693 0.878466 0.477805i \(-0.158567\pi\)
0.878466 + 0.477805i \(0.158567\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) −18.0000 −0.732410
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) 33.0000 1.33943 0.669714 0.742619i \(-0.266417\pi\)
0.669714 + 0.742619i \(0.266417\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −20.0000 −0.809776
\(611\) 27.0000 1.09230
\(612\) 0 0
\(613\) 24.0000 0.969351 0.484675 0.874694i \(-0.338938\pi\)
0.484675 + 0.874694i \(0.338938\pi\)
\(614\) −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 \(0.422344\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 12.0000 0.481932
\(621\) 0 0
\(622\) −40.0000 −1.60385
\(623\) −10.0000 −0.400642
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 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
\(633\) 0 0
\(634\) −40.0000 −1.58860
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 30.0000 1.18771
\(639\) 0 0
\(640\) 0 0
\(641\) −26.0000 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) 0 0
\(643\) −31.0000 −1.22252 −0.611260 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) −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
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) 20.0000 0.781465
\(656\) 0 0
\(657\) 0 0
\(658\) 18.0000 0.701713
\(659\) 21.0000 0.818044 0.409022 0.912525i \(-0.365870\pi\)
0.409022 + 0.912525i \(0.365870\pi\)
\(660\) 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
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.00000 −0.116160
\(668\) 2.00000 0.0773823
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) 50.0000 1.93023
\(672\) 0 0
\(673\) 44.0000 1.69608 0.848038 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) −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 \(0.230320\pi\)
\(678\) 0 0
\(679\) −7.00000 −0.268635
\(680\) 0 0
\(681\) 0 0
\(682\) −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
\(685\) 6.00000 0.229248
\(686\) 2.00000 0.0763604
\(687\) 0 0
\(688\) 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
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) −8.00000 −0.303457
\(696\) 0 0
\(697\) 0 0
\(698\) −56.0000 −2.11963
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) −45.0000 −1.69963 −0.849813 0.527084i \(-0.823285\pi\)
−0.849813 + 0.527084i \(0.823285\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −40.0000 −1.50756
\(705\) 0 0
\(706\) 42.0000 1.58069
\(707\) 10.0000 0.376089
\(708\) 0 0
\(709\) −51.0000 −1.91535 −0.957673 0.287860i \(-0.907056\pi\)
−0.957673 + 0.287860i \(0.907056\pi\)
\(710\) −16.0000 −0.600469
\(711\) 0 0
\(712\) 0 0
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 15.0000 0.560968
\(716\) −16.0000 −0.597948
\(717\) 0 0
\(718\) 64.0000 2.38846
\(719\) −26.0000 −0.969636 −0.484818 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(720\) 0 0
\(721\) −1.00000 −0.0372419
\(722\) 38.0000 1.41421
\(723\) 0 0
\(724\) −28.0000 −1.04061
\(725\) −3.00000 −0.111417
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −20.0000 −0.740233
\(731\) −10.0000 −0.369863
\(732\) 0 0
\(733\) 29.0000 1.07114 0.535570 0.844491i \(-0.320097\pi\)
0.535570 + 0.844491i \(0.320097\pi\)
\(734\) 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 \(0.790382\pi\)
\(740\) −8.00000 −0.294086
\(741\) 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
\(745\) 2.00000 0.0732743
\(746\) −52.0000 −1.90386
\(747\) 0 0
\(748\) 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
\(753\) 0 0
\(754\) 18.0000 0.655521
\(755\) −9.00000 −0.327544
\(756\) 0 0
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) 40.0000 1.45287
\(759\) 0 0
\(760\) 0 0
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) 0 0
\(763\) 17.0000 0.615441
\(764\) −54.0000 −1.95365
\(765\) 0 0
\(766\) 32.0000 1.15621
\(767\) 18.0000 0.649942
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 10.0000 0.360375
\(771\) 0 0
\(772\) 12.0000 0.431889
\(773\) 15.0000 0.539513 0.269756 0.962929i \(-0.413057\pi\)
0.269756 + 0.962929i \(0.413057\pi\)
\(774\) 0 0
\(775\) 6.00000 0.215526
\(776\) 0 0
\(777\) 0 0
\(778\) 62.0000 2.22281
\(779\) 0 0
\(780\) 0 0
\(781\) 40.0000 1.43131
\(782\) −10.0000 −0.357599
\(783\) 0 0
\(784\) −4.00000 −0.142857
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) 37.0000 1.31891 0.659454 0.751745i \(-0.270788\pi\)
0.659454 + 0.751745i \(0.270788\pi\)
\(788\) 24.0000 0.854965
\(789\) 0 0
\(790\) 30.0000 1.06735
\(791\) 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
\(800\) 8.00000 0.282843
\(801\) 0 0
\(802\) 6.00000 0.211867
\(803\) 50.0000 1.76446
\(804\) 0 0
\(805\) −1.00000 −0.0352454
\(806\) −36.0000 −1.26805
\(807\) 0 0
\(808\) 0 0
\(809\) −9.00000 −0.316423 −0.158212 0.987405i \(-0.550573\pi\)
−0.158212 + 0.987405i \(0.550573\pi\)
\(810\) 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
\(813\) 0 0
\(814\) 40.0000 1.40200
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) 0 0
\(818\) 12.0000 0.419570
\(819\) 0 0
\(820\) 0 0
\(821\) 19.0000 0.663105 0.331552 0.943437i \(-0.392428\pi\)
0.331552 + 0.943437i \(0.392428\pi\)
\(822\) 0 0
\(823\) 42.0000 1.46403 0.732014 0.681290i \(-0.238581\pi\)
0.732014 + 0.681290i \(0.238581\pi\)
\(824\) 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 \(0.284723\pi\)
\(828\) 0 0
\(829\) 16.0000 0.555703 0.277851 0.960624i \(-0.410378\pi\)
0.277851 + 0.960624i \(0.410378\pi\)
\(830\) 24.0000 0.833052
\(831\) 0 0
\(832\) −24.0000 −0.832050
\(833\) 5.00000 0.173240
\(834\) 0 0
\(835\) 1.00000 0.0346064
\(836\) 0 0
\(837\) 0 0
\(838\) −36.0000 −1.24360
\(839\) −36.0000 −1.24286 −0.621429 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 74.0000 2.55021
\(843\) 0 0
\(844\) −34.0000 −1.17033
\(845\) −4.00000 −0.137604
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) −24.0000 −0.824163
\(849\) 0 0
\(850\) −10.0000 −0.342997
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) −2.00000 −0.0684787 −0.0342393 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) 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 \(0.787679\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −66.0000 −2.24797
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) 0 0
\(865\) 3.00000 0.102003
\(866\) −68.0000 −2.31073
\(867\) 0 0
\(868\) −12.0000 −0.407307
\(869\) −75.0000 −2.54420
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 8.00000 0.270141 0.135070 0.990836i \(-0.456874\pi\)
0.135070 + 0.990836i \(0.456874\pi\)
\(878\) 12.0000 0.404980
\(879\) 0 0
\(880\) −20.0000 −0.674200
\(881\) 34.0000 1.14549 0.572745 0.819734i \(-0.305879\pi\)
0.572745 + 0.819734i \(0.305879\pi\)
\(882\) 0 0
\(883\) −48.0000 −1.61533 −0.807664 0.589643i \(-0.799269\pi\)
−0.807664 + 0.589643i \(0.799269\pi\)
\(884\) 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
\(890\) −20.0000 −0.670402
\(891\) 0 0
\(892\) 30.0000 1.00447
\(893\) 0 0
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) −18.0000 −0.600334
\(900\) 0 0
\(901\) 30.0000 0.999445
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) 6.00000 0.199117
\(909\) 0 0
\(910\) 6.00000 0.198898
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) −60.0000 −1.98571
\(914\) 0 0
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) −20.0000 −0.660458
\(918\) 0 0
\(919\) 37.0000 1.22052 0.610259 0.792202i \(-0.291065\pi\)
0.610259 + 0.792202i \(0.291065\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 64.0000 2.10773
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 20.0000 0.657241
\(927\) 0 0
\(928\) −24.0000 −0.787839
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −36.0000 −1.17922
\(933\) 0 0
\(934\) −42.0000 −1.37428
\(935\) 25.0000 0.817587
\(936\) 0 0
\(937\) 3.00000 0.0980057 0.0490029 0.998799i \(-0.484396\pi\)
0.0490029 + 0.998799i \(0.484396\pi\)
\(938\) 8.00000 0.261209
\(939\) 0 0
\(940\) 18.0000 0.587095
\(941\) −20.0000 −0.651981 −0.325991 0.945373i \(-0.605698\pi\)
−0.325991 + 0.945373i \(0.605698\pi\)
\(942\) 0 0
\(943\) 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 \(0.562462\pi\)
\(948\) 0 0
\(949\) 30.0000 0.973841
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −50.0000 −1.61966 −0.809829 0.586665i \(-0.800440\pi\)
−0.809829 + 0.586665i \(0.800440\pi\)
\(954\) 0 0
\(955\) −27.0000 −0.873699
\(956\) −14.0000 −0.452792
\(957\) 0 0
\(958\) −36.0000 −1.16311
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 24.0000 0.773791
\(963\) 0 0
\(964\) 4.00000 0.128831
\(965\) 6.00000 0.193147
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 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 \(0.221338\pi\)
\(978\) 0 0
\(979\) 50.0000 1.59801
\(980\) 2.00000 0.0638877
\(981\) 0 0
\(982\) −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
\(985\) 12.0000 0.382352
\(986\) 30.0000 0.955395
\(987\) 0 0
\(988\) 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
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) −22.0000 −0.697447
\(996\) 0 0
\(997\) −51.0000 −1.61519 −0.807593 0.589740i \(-0.799230\pi\)
−0.807593 + 0.589740i \(0.799230\pi\)
\(998\) −50.0000 −1.58272
\(999\) 0 0
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 7245.2.a.d.1.1 1
3.2 odd 2 805.2.a.c.1.1 1
15.14 odd 2 4025.2.a.b.1.1 1
21.20 even 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 3.2 odd 2
4025.2.a.b.1.1 1 15.14 odd 2
5635.2.a.k.1.1 1 21.20 even 2
7245.2.a.d.1.1 1 1.1 even 1 trivial