Properties

Label 4800.2.a.be.1.1
Level $4800$
Weight $2$
Character 4800.1
Self dual yes
Analytic conductor $38.328$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.3281929702\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} +2.00000 q^{17} -5.00000 q^{19} -3.00000 q^{21} -6.00000 q^{23} -1.00000 q^{27} -10.0000 q^{29} +3.00000 q^{31} -2.00000 q^{33} -2.00000 q^{37} +1.00000 q^{39} -8.00000 q^{41} +1.00000 q^{43} -2.00000 q^{47} +2.00000 q^{49} -2.00000 q^{51} +4.00000 q^{53} +5.00000 q^{57} -10.0000 q^{59} -7.00000 q^{61} +3.00000 q^{63} -3.00000 q^{67} +6.00000 q^{69} +8.00000 q^{71} -14.0000 q^{73} +6.00000 q^{77} +1.00000 q^{81} +6.00000 q^{83} +10.0000 q^{87} -3.00000 q^{91} -3.00000 q^{93} +17.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.00000 0.662266
\(58\) 0 0
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 0 0
\(63\) 3.00000 0.377964
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 0 0
\(93\) −3.00000 −0.311086
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) −4.00000 −0.376288 −0.188144 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −15.0000 −1.30066
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −7.00000 −0.569652 −0.284826 0.958579i \(-0.591936\pi\)
−0.284826 + 0.958579i \(0.591936\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 13.0000 1.03751 0.518756 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) 0 0
\(159\) −4.00000 −0.317221
\(160\) 0 0
\(161\) −18.0000 −1.41860
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −5.00000 −0.382360
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 10.0000 0.751646
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −17.0000 −1.26360 −0.631800 0.775131i \(-0.717684\pi\)
−0.631800 + 0.775131i \(0.717684\pi\)
\(182\) 0 0
\(183\) 7.00000 0.517455
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) −22.0000 −1.59186 −0.795932 0.605386i \(-0.793019\pi\)
−0.795932 + 0.605386i \(0.793019\pi\)
\(192\) 0 0
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 5.00000 0.354441 0.177220 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) 0 0
\(201\) 3.00000 0.211604
\(202\) 0 0
\(203\) −30.0000 −2.10559
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) −10.0000 −0.691714
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 0 0
\(213\) −8.00000 −0.548151
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 9.00000 0.610960
\(218\) 0 0
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) 19.0000 1.27233 0.636167 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) 15.0000 0.991228 0.495614 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −23.0000 −1.48156 −0.740780 0.671748i \(-0.765544\pi\)
−0.740780 + 0.671748i \(0.765544\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.00000 0.318142
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) 0 0
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 3.00000 0.181568
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.00000 0.180253 0.0901263 0.995930i \(-0.471273\pi\)
0.0901263 + 0.995930i \(0.471273\pi\)
\(278\) 0 0
\(279\) 3.00000 0.179605
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) −9.00000 −0.534994 −0.267497 0.963559i \(-0.586197\pi\)
−0.267497 + 0.963559i \(0.586197\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −17.0000 −0.996558
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) 0 0
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 3.00000 0.172917
\(302\) 0 0
\(303\) 12.0000 0.689382
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7.00000 0.399511 0.199756 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 11.0000 0.621757 0.310878 0.950450i \(-0.399377\pi\)
0.310878 + 0.950450i \(0.399377\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.00000 0.449325 0.224662 0.974437i \(-0.427872\pi\)
0.224662 + 0.974437i \(0.427872\pi\)
\(318\) 0 0
\(319\) −20.0000 −1.11979
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −10.0000 −0.556415
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 5.00000 0.276501
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) 0 0
\(339\) 4.00000 0.217250
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −6.00000 −0.317554
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −27.0000 −1.40939 −0.704694 0.709511i \(-0.748916\pi\)
−0.704694 + 0.709511i \(0.748916\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 29.0000 1.50156 0.750782 0.660551i \(-0.229677\pi\)
0.750782 + 0.660551i \(0.229677\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 10.0000 0.515026
\(378\) 0 0
\(379\) 25.0000 1.28416 0.642082 0.766636i \(-0.278071\pi\)
0.642082 + 0.766636i \(0.278071\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.00000 0.0508329
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −7.00000 −0.351320 −0.175660 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) 0 0
\(399\) 15.0000 0.750939
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) −3.00000 −0.149441
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 0 0
\(413\) −30.0000 −1.47620
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) −2.00000 −0.0972433
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −21.0000 −1.01626
\(428\) 0 0
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) −29.0000 −1.39365 −0.696826 0.717241i \(-0.745405\pi\)
−0.696826 + 0.717241i \(0.745405\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 30.0000 1.43509
\(438\) 0 0
\(439\) 35.0000 1.67046 0.835229 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) −16.0000 −0.753411
\(452\) 0 0
\(453\) 7.00000 0.328889
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 0 0
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −38.0000 −1.75843 −0.879215 0.476425i \(-0.841932\pi\)
−0.879215 + 0.476425i \(0.841932\pi\)
\(468\) 0 0
\(469\) −9.00000 −0.415581
\(470\) 0 0
\(471\) −13.0000 −0.599008
\(472\) 0 0
\(473\) 2.00000 0.0919601
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) 0 0
\(479\) −30.0000 −1.37073 −0.685367 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 0 0
\(483\) 18.0000 0.819028
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 13.0000 0.589086 0.294543 0.955638i \(-0.404833\pi\)
0.294543 + 0.955638i \(0.404833\pi\)
\(488\) 0 0
\(489\) −11.0000 −0.497437
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) −20.0000 −0.900755
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) −5.00000 −0.223831 −0.111915 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 0 0
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) −42.0000 −1.85797
\(512\) 0 0
\(513\) 5.00000 0.220755
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −4.00000 −0.175920
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) 31.0000 1.35554 0.677768 0.735276i \(-0.262948\pi\)
0.677768 + 0.735276i \(0.262948\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.00000 0.261364
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 10.0000 0.431532
\(538\) 0 0
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) 3.00000 0.128980 0.0644900 0.997918i \(-0.479458\pi\)
0.0644900 + 0.997918i \(0.479458\pi\)
\(542\) 0 0
\(543\) 17.0000 0.729540
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 0 0
\(549\) −7.00000 −0.298753
\(550\) 0 0
\(551\) 50.0000 2.13007
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −42.0000 −1.77960 −0.889799 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) 6.00000 0.252870 0.126435 0.991975i \(-0.459647\pi\)
0.126435 + 0.991975i \(0.459647\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.00000 0.125988
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −13.0000 −0.544033 −0.272017 0.962293i \(-0.587691\pi\)
−0.272017 + 0.962293i \(0.587691\pi\)
\(572\) 0 0
\(573\) 22.0000 0.919063
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −13.0000 −0.541197 −0.270599 0.962692i \(-0.587222\pi\)
−0.270599 + 0.962692i \(0.587222\pi\)
\(578\) 0 0
\(579\) −11.0000 −0.457144
\(580\) 0 0
\(581\) 18.0000 0.746766
\(582\) 0 0
\(583\) 8.00000 0.331326
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −15.0000 −0.618064
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) 16.0000 0.657041 0.328521 0.944497i \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5.00000 −0.204636
\(598\) 0 0
\(599\) 20.0000 0.817178 0.408589 0.912719i \(-0.366021\pi\)
0.408589 + 0.912719i \(0.366021\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) 0 0
\(603\) −3.00000 −0.122169
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 0 0
\(609\) 30.0000 1.21566
\(610\) 0 0
\(611\) 2.00000 0.0809113
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 0 0
\(615\) 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
\(619\) −25.0000 −1.00483 −0.502417 0.864625i \(-0.667556\pi\)
−0.502417 + 0.864625i \(0.667556\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 10.0000 0.399362
\(628\) 0 0
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 23.0000 0.915616 0.457808 0.889051i \(-0.348635\pi\)
0.457808 + 0.889051i \(0.348635\pi\)
\(632\) 0 0
\(633\) 13.0000 0.516704
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 0 0
\(643\) 36.0000 1.41970 0.709851 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) −9.00000 −0.352738
\(652\) 0 0
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) −40.0000 −1.55818 −0.779089 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) 2.00000 0.0776736
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 60.0000 2.32321
\(668\) 0 0
\(669\) −19.0000 −0.734582
\(670\) 0 0
\(671\) −14.0000 −0.540464
\(672\) 0 0
\(673\) 6.00000 0.231283 0.115642 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) 51.0000 1.95720
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −15.0000 −0.572286
\(688\) 0 0
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 6.00000 0.227921
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) 0 0
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) 10.0000 0.377157
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −36.0000 −1.35392
\(708\) 0 0
\(709\) −25.0000 −0.938895 −0.469447 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −18.0000 −0.674105
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 20.0000 0.746914
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) 23.0000 0.855379
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 43.0000 1.59478 0.797391 0.603463i \(-0.206213\pi\)
0.797391 + 0.603463i \(0.206213\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.00000 0.0739727
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.00000 −0.221013
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −5.00000 −0.183680
\(742\) 0 0
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) 36.0000 1.31541
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 23.0000 0.835949 0.417975 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) 0 0
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) −15.0000 −0.543036
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10.0000 0.361079
\(768\) 0 0
\(769\) 35.0000 1.26213 0.631066 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 0 0
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 6.00000 0.215249
\(778\) 0 0
\(779\) 40.0000 1.43315
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) 0 0
\(783\) 10.0000 0.357371
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 7.00000 0.249523 0.124762 0.992187i \(-0.460183\pi\)
0.124762 + 0.992187i \(0.460183\pi\)
\(788\) 0 0
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 7.00000 0.248577
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −52.0000 −1.84193 −0.920967 0.389640i \(-0.872599\pi\)
−0.920967 + 0.389640i \(0.872599\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −28.0000 −0.988099
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −10.0000 −0.352017
\(808\) 0 0
\(809\) −20.0000 −0.703163 −0.351581 0.936157i \(-0.614356\pi\)
−0.351581 + 0.936157i \(0.614356\pi\)
\(810\) 0 0
\(811\) 27.0000 0.948098 0.474049 0.880498i \(-0.342792\pi\)
0.474049 + 0.880498i \(0.342792\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −5.00000 −0.174928
\(818\) 0 0
\(819\) −3.00000 −0.104828
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) −41.0000 −1.42917 −0.714585 0.699549i \(-0.753384\pi\)
−0.714585 + 0.699549i \(0.753384\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) −3.00000 −0.104069
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −3.00000 −0.103695
\(838\) 0 0
\(839\) −10.0000 −0.345238 −0.172619 0.984989i \(-0.555223\pi\)
−0.172619 + 0.984989i \(0.555223\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 18.0000 0.619953
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −21.0000 −0.721569
\(848\) 0 0
\(849\) 9.00000 0.308879
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) −51.0000 −1.74621 −0.873103 0.487535i \(-0.837896\pi\)
−0.873103 + 0.487535i \(0.837896\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −28.0000 −0.956462 −0.478231 0.878234i \(-0.658722\pi\)
−0.478231 + 0.878234i \(0.658722\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) 0 0
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 3.00000 0.101651
\(872\) 0 0
\(873\) 17.0000 0.575363
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −27.0000 −0.911725 −0.455863 0.890050i \(-0.650669\pi\)
−0.455863 + 0.890050i \(0.650669\pi\)
\(878\) 0 0
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 32.0000 1.07811 0.539054 0.842271i \(-0.318782\pi\)
0.539054 + 0.842271i \(0.318782\pi\)
\(882\) 0 0
\(883\) 41.0000 1.37976 0.689880 0.723924i \(-0.257663\pi\)
0.689880 + 0.723924i \(0.257663\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 18.0000 0.604381 0.302190 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) 0 0
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) 10.0000 0.334637
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) 0 0
\(899\) −30.0000 −1.00056
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) −3.00000 −0.0998337
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 0 0
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) 58.0000 1.92163 0.960813 0.277198i \(-0.0894057\pi\)
0.960813 + 0.277198i \(0.0894057\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 36.0000 1.18882
\(918\) 0 0
\(919\) −55.0000 −1.81428 −0.907141 0.420826i \(-0.861740\pi\)
−0.907141 + 0.420826i \(0.861740\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) 0 0
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) −50.0000 −1.64045 −0.820223 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) −10.0000 −0.327737
\(932\) 0 0
\(933\) −18.0000 −0.589294
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −33.0000 −1.07806 −0.539032 0.842286i \(-0.681210\pi\)
−0.539032 + 0.842286i \(0.681210\pi\)
\(938\) 0 0
\(939\) −11.0000 −0.358971
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 0 0
\(943\) 48.0000 1.56310
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −18.0000 −0.584921 −0.292461 0.956278i \(-0.594474\pi\)
−0.292461 + 0.956278i \(0.594474\pi\)
\(948\) 0 0
\(949\) 14.0000 0.454459
\(950\) 0 0
\(951\) −8.00000 −0.259418
\(952\) 0 0
\(953\) 56.0000 1.81402 0.907009 0.421111i \(-0.138360\pi\)
0.907009 + 0.421111i \(0.138360\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 20.0000 0.646508
\(958\) 0 0
\(959\) −54.0000 −1.74375
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 10.0000 0.321246
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 0 0
\(973\) 60.0000 1.92351
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −5.00000 −0.159638
\(982\) 0 0
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) −17.0000 −0.540023 −0.270011 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(992\) 0 0
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 0 0
\(999\) 2.00000 0.0632772
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 4800.2.a.be.1.1 1
4.3 odd 2 4800.2.a.bq.1.1 1
5.2 odd 4 4800.2.f.y.3649.2 2
5.3 odd 4 4800.2.f.y.3649.1 2
5.4 even 2 4800.2.a.br.1.1 1
8.3 odd 2 75.2.a.c.1.1 yes 1
8.5 even 2 1200.2.a.p.1.1 1
20.3 even 4 4800.2.f.l.3649.2 2
20.7 even 4 4800.2.f.l.3649.1 2
20.19 odd 2 4800.2.a.bb.1.1 1
24.5 odd 2 3600.2.a.bk.1.1 1
24.11 even 2 225.2.a.a.1.1 1
40.3 even 4 75.2.b.a.49.1 2
40.13 odd 4 1200.2.f.d.49.2 2
40.19 odd 2 75.2.a.a.1.1 1
40.27 even 4 75.2.b.a.49.2 2
40.29 even 2 1200.2.a.c.1.1 1
40.37 odd 4 1200.2.f.d.49.1 2
56.27 even 2 3675.2.a.q.1.1 1
88.43 even 2 9075.2.a.a.1.1 1
120.29 odd 2 3600.2.a.j.1.1 1
120.53 even 4 3600.2.f.p.2449.1 2
120.59 even 2 225.2.a.e.1.1 1
120.77 even 4 3600.2.f.p.2449.2 2
120.83 odd 4 225.2.b.a.199.2 2
120.107 odd 4 225.2.b.a.199.1 2
280.139 even 2 3675.2.a.b.1.1 1
440.219 even 2 9075.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.a.a.1.1 1 40.19 odd 2
75.2.a.c.1.1 yes 1 8.3 odd 2
75.2.b.a.49.1 2 40.3 even 4
75.2.b.a.49.2 2 40.27 even 4
225.2.a.a.1.1 1 24.11 even 2
225.2.a.e.1.1 1 120.59 even 2
225.2.b.a.199.1 2 120.107 odd 4
225.2.b.a.199.2 2 120.83 odd 4
1200.2.a.c.1.1 1 40.29 even 2
1200.2.a.p.1.1 1 8.5 even 2
1200.2.f.d.49.1 2 40.37 odd 4
1200.2.f.d.49.2 2 40.13 odd 4
3600.2.a.j.1.1 1 120.29 odd 2
3600.2.a.bk.1.1 1 24.5 odd 2
3600.2.f.p.2449.1 2 120.53 even 4
3600.2.f.p.2449.2 2 120.77 even 4
3675.2.a.b.1.1 1 280.139 even 2
3675.2.a.q.1.1 1 56.27 even 2
4800.2.a.bb.1.1 1 20.19 odd 2
4800.2.a.be.1.1 1 1.1 even 1 trivial
4800.2.a.bq.1.1 1 4.3 odd 2
4800.2.a.br.1.1 1 5.4 even 2
4800.2.f.l.3649.1 2 20.7 even 4
4800.2.f.l.3649.2 2 20.3 even 4
4800.2.f.y.3649.1 2 5.3 odd 4
4800.2.f.y.3649.2 2 5.2 odd 4
9075.2.a.a.1.1 1 88.43 even 2
9075.2.a.s.1.1 1 440.219 even 2