Properties

Label 980.2.a.f.1.1
Level $980$
Weight $2$
Character 980.1
Self dual yes
Analytic conductor $7.825$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.82533939809\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 980.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} -1.00000 q^{11} -5.00000 q^{13} -1.00000 q^{15} +1.00000 q^{17} -6.00000 q^{19} -4.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +2.00000 q^{31} -1.00000 q^{33} +8.00000 q^{37} -5.00000 q^{39} -10.0000 q^{41} -2.00000 q^{43} +2.00000 q^{45} -7.00000 q^{47} +1.00000 q^{51} -2.00000 q^{53} +1.00000 q^{55} -6.00000 q^{57} +14.0000 q^{59} -8.00000 q^{61} +5.00000 q^{65} +14.0000 q^{67} -4.00000 q^{69} -10.0000 q^{73} +1.00000 q^{75} -11.0000 q^{79} +1.00000 q^{81} -4.00000 q^{83} -1.00000 q^{85} +3.00000 q^{87} +4.00000 q^{89} +2.00000 q^{93} +6.00000 q^{95} -3.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) −5.00000 −0.800641
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) 14.0000 1.82264 0.911322 0.411693i \(-0.135063\pi\)
0.911322 + 0.411693i \(0.135063\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.00000 0.620174
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −11.0000 −1.23760 −0.618798 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −1.00000 −0.108465
\(86\) 0 0
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −3.00000 −0.304604 −0.152302 0.988334i \(-0.548669\pi\)
−0.152302 + 0.988334i \(0.548669\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) −7.00000 −0.689730 −0.344865 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 10.0000 0.924500
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) −10.0000 −0.901670
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 0 0
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 5.00000 0.430331
\(136\) 0 0
\(137\) 16.0000 1.36697 0.683486 0.729964i \(-0.260463\pi\)
0.683486 + 0.729964i \(0.260463\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) −7.00000 −0.589506
\(142\) 0 0
\(143\) 5.00000 0.418121
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) −3.00000 −0.244137 −0.122068 0.992522i \(-0.538953\pi\)
−0.122068 + 0.992522i \(0.538953\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 0 0
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 12.0000 0.917663
\(172\) 0 0
\(173\) 21.0000 1.59660 0.798300 0.602260i \(-0.205733\pi\)
0.798300 + 0.602260i \(0.205733\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 14.0000 1.05230
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −1.00000 −0.0731272
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 0 0
\(193\) −12.0000 −0.863779 −0.431889 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) 0 0
\(195\) 5.00000 0.358057
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −22.0000 −1.55954 −0.779769 0.626067i \(-0.784664\pi\)
−0.779769 + 0.626067i \(0.784664\pi\)
\(200\) 0 0
\(201\) 14.0000 0.987484
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) 0 0
\(207\) 8.00000 0.556038
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 17.0000 1.17033 0.585164 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) −5.00000 −0.336336
\(222\) 0 0
\(223\) 29.0000 1.94198 0.970992 0.239113i \(-0.0768565\pi\)
0.970992 + 0.239113i \(0.0768565\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) 9.00000 0.597351 0.298675 0.954355i \(-0.403455\pi\)
0.298675 + 0.954355i \(0.403455\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −8.00000 −0.524097 −0.262049 0.965055i \(-0.584398\pi\)
−0.262049 + 0.965055i \(0.584398\pi\)
\(234\) 0 0
\(235\) 7.00000 0.456630
\(236\) 0 0
\(237\) −11.0000 −0.714527
\(238\) 0 0
\(239\) 13.0000 0.840900 0.420450 0.907316i \(-0.361872\pi\)
0.420450 + 0.907316i \(0.361872\pi\)
\(240\) 0 0
\(241\) −16.0000 −1.03065 −0.515325 0.856995i \(-0.672329\pi\)
−0.515325 + 0.856995i \(0.672329\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 30.0000 1.90885
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) −1.00000 −0.0626224
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 4.00000 0.244796
\(268\) 0 0
\(269\) −8.00000 −0.487769 −0.243884 0.969804i \(-0.578422\pi\)
−0.243884 + 0.969804i \(0.578422\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 20.0000 1.20168 0.600842 0.799368i \(-0.294832\pi\)
0.600842 + 0.799368i \(0.294832\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −13.0000 −0.775515 −0.387757 0.921761i \(-0.626750\pi\)
−0.387757 + 0.921761i \(0.626750\pi\)
\(282\) 0 0
\(283\) −1.00000 −0.0594438 −0.0297219 0.999558i \(-0.509462\pi\)
−0.0297219 + 0.999558i \(0.509462\pi\)
\(284\) 0 0
\(285\) 6.00000 0.355409
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −3.00000 −0.175863
\(292\) 0 0
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) −14.0000 −0.815112
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) 20.0000 1.15663
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −10.0000 −0.574485
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) 23.0000 1.31268 0.656340 0.754466i \(-0.272104\pi\)
0.656340 + 0.754466i \(0.272104\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) 14.0000 0.793867 0.396934 0.917847i \(-0.370074\pi\)
0.396934 + 0.917847i \(0.370074\pi\)
\(312\) 0 0
\(313\) −25.0000 −1.41308 −0.706542 0.707671i \(-0.749746\pi\)
−0.706542 + 0.707671i \(0.749746\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −3.00000 −0.167968
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) −5.00000 −0.277350
\(326\) 0 0
\(327\) −11.0000 −0.608301
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) −16.0000 −0.876795
\(334\) 0 0
\(335\) −14.0000 −0.764902
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 0 0
\(339\) 16.0000 0.869001
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) 8.00000 0.429463 0.214731 0.976673i \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 25.0000 1.33440
\(352\) 0 0
\(353\) 9.00000 0.479022 0.239511 0.970894i \(-0.423013\pi\)
0.239511 + 0.970894i \(0.423013\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) −23.0000 −1.20059 −0.600295 0.799779i \(-0.704950\pi\)
−0.600295 + 0.799779i \(0.704950\pi\)
\(368\) 0 0
\(369\) 20.0000 1.04116
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 12.0000 0.621336 0.310668 0.950518i \(-0.399447\pi\)
0.310668 + 0.950518i \(0.399447\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −15.0000 −0.772539
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −6.00000 −0.307389
\(382\) 0 0
\(383\) −32.0000 −1.63512 −0.817562 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) −15.0000 −0.760530 −0.380265 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 11.0000 0.553470
\(396\) 0 0
\(397\) −23.0000 −1.15434 −0.577168 0.816625i \(-0.695842\pi\)
−0.577168 + 0.816625i \(0.695842\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11.0000 −0.549314 −0.274657 0.961542i \(-0.588564\pi\)
−0.274657 + 0.961542i \(0.588564\pi\)
\(402\) 0 0
\(403\) −10.0000 −0.498135
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 16.0000 0.789222
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 16.0000 0.783523
\(418\) 0 0
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) 14.0000 0.680703
\(424\) 0 0
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 5.00000 0.241402
\(430\) 0 0
\(431\) −33.0000 −1.58955 −0.794777 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) −3.00000 −0.143839
\(436\) 0 0
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −26.0000 −1.23530 −0.617649 0.786454i \(-0.711915\pi\)
−0.617649 + 0.786454i \(0.711915\pi\)
\(444\) 0 0
\(445\) −4.00000 −0.189618
\(446\) 0 0
\(447\) −22.0000 −1.04056
\(448\) 0 0
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 0 0
\(453\) −3.00000 −0.140952
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 0 0
\(459\) −5.00000 −0.233380
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) −2.00000 −0.0927478
\(466\) 0 0
\(467\) 19.0000 0.879215 0.439608 0.898190i \(-0.355118\pi\)
0.439608 + 0.898190i \(0.355118\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 0 0
\(473\) 2.00000 0.0919601
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.00000 0.136223
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) 0 0
\(493\) 3.00000 0.135113
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −5.00000 −0.223831 −0.111915 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 0 0
\(501\) 21.0000 0.938211
\(502\) 0 0
\(503\) −9.00000 −0.401290 −0.200645 0.979664i \(-0.564304\pi\)
−0.200645 + 0.979664i \(0.564304\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 0 0
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 30.0000 1.32453
\(514\) 0 0
\(515\) 7.00000 0.308457
\(516\) 0 0
\(517\) 7.00000 0.307860
\(518\) 0 0
\(519\) 21.0000 0.921798
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.00000 0.0871214
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −28.0000 −1.21510
\(532\) 0 0
\(533\) 50.0000 2.16574
\(534\) 0 0
\(535\) −18.0000 −0.778208
\(536\) 0 0
\(537\) −20.0000 −0.863064
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) 11.0000 0.471188
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 0 0
\(549\) 16.0000 0.682863
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −8.00000 −0.339581
\(556\) 0 0
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) −1.00000 −0.0422200
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) −16.0000 −0.673125
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) 3.00000 0.125327
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 0 0
\(579\) −12.0000 −0.498703
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2.00000 0.0828315
\(584\) 0 0
\(585\) −10.0000 −0.413449
\(586\) 0 0
\(587\) −44.0000 −1.81607 −0.908037 0.418890i \(-0.862419\pi\)
−0.908037 + 0.418890i \(0.862419\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) 23.0000 0.944497 0.472248 0.881466i \(-0.343443\pi\)
0.472248 + 0.881466i \(0.343443\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −22.0000 −0.900400
\(598\) 0 0
\(599\) 35.0000 1.43006 0.715031 0.699093i \(-0.246413\pi\)
0.715031 + 0.699093i \(0.246413\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) −28.0000 −1.14025
\(604\) 0 0
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) −29.0000 −1.17707 −0.588537 0.808470i \(-0.700296\pi\)
−0.588537 + 0.808470i \(0.700296\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 35.0000 1.41595
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 10.0000 0.403239
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 20.0000 0.802572
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.00000 0.239617
\(628\) 0 0
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) 31.0000 1.23409 0.617045 0.786928i \(-0.288330\pi\)
0.617045 + 0.786928i \(0.288330\pi\)
\(632\) 0 0
\(633\) 17.0000 0.675689
\(634\) 0 0
\(635\) 6.00000 0.238103
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −15.0000 −0.591542 −0.295771 0.955259i \(-0.595577\pi\)
−0.295771 + 0.955259i \(0.595577\pi\)
\(644\) 0 0
\(645\) 2.00000 0.0787499
\(646\) 0 0
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 0 0
\(649\) −14.0000 −0.549548
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 0 0
\(657\) 20.0000 0.780274
\(658\) 0 0
\(659\) −37.0000 −1.44132 −0.720658 0.693291i \(-0.756160\pi\)
−0.720658 + 0.693291i \(0.756160\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) 0 0
\(663\) −5.00000 −0.194184
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) 0 0
\(669\) 29.0000 1.12120
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 4.00000 0.154189 0.0770943 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) 0 0
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) 31.0000 1.19143 0.595713 0.803197i \(-0.296869\pi\)
0.595713 + 0.803197i \(0.296869\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 9.00000 0.344881
\(682\) 0 0
\(683\) 48.0000 1.83667 0.918334 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) 0 0
\(685\) −16.0000 −0.611329
\(686\) 0 0
\(687\) 2.00000 0.0763048
\(688\) 0 0
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.0000 −0.606915
\(696\) 0 0
\(697\) −10.0000 −0.378777
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) −29.0000 −1.09531 −0.547657 0.836703i \(-0.684480\pi\)
−0.547657 + 0.836703i \(0.684480\pi\)
\(702\) 0 0
\(703\) −48.0000 −1.81035
\(704\) 0 0
\(705\) 7.00000 0.263635
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 27.0000 1.01401 0.507003 0.861944i \(-0.330753\pi\)
0.507003 + 0.861944i \(0.330753\pi\)
\(710\) 0 0
\(711\) 22.0000 0.825064
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) −5.00000 −0.186989
\(716\) 0 0
\(717\) 13.0000 0.485494
\(718\) 0 0
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −16.0000 −0.595046
\(724\) 0 0
\(725\) 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
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −2.00000 −0.0739727
\(732\) 0 0
\(733\) −33.0000 −1.21888 −0.609441 0.792831i \(-0.708606\pi\)
−0.609441 + 0.792831i \(0.708606\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −14.0000 −0.515697
\(738\) 0 0
\(739\) 3.00000 0.110357 0.0551784 0.998477i \(-0.482427\pi\)
0.0551784 + 0.998477i \(0.482427\pi\)
\(740\) 0 0
\(741\) 30.0000 1.10208
\(742\) 0 0
\(743\) −18.0000 −0.660356 −0.330178 0.943919i \(-0.607109\pi\)
−0.330178 + 0.943919i \(0.607109\pi\)
\(744\) 0 0
\(745\) 22.0000 0.806018
\(746\) 0 0
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 13.0000 0.474377 0.237188 0.971464i \(-0.423774\pi\)
0.237188 + 0.971464i \(0.423774\pi\)
\(752\) 0 0
\(753\) −4.00000 −0.145768
\(754\) 0 0
\(755\) 3.00000 0.109181
\(756\) 0 0
\(757\) 16.0000 0.581530 0.290765 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) 0 0
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) 0 0
\(767\) −70.0000 −2.52755
\(768\) 0 0
\(769\) 50.0000 1.80305 0.901523 0.432731i \(-0.142450\pi\)
0.901523 + 0.432731i \(0.142450\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) −21.0000 −0.755318 −0.377659 0.925945i \(-0.623271\pi\)
−0.377659 + 0.925945i \(0.623271\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 60.0000 2.14972
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −15.0000 −0.536056
\(784\) 0 0
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) −3.00000 −0.106938 −0.0534692 0.998569i \(-0.517028\pi\)
−0.0534692 + 0.998569i \(0.517028\pi\)
\(788\) 0 0
\(789\) 18.0000 0.640817
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) 0 0
\(795\) 2.00000 0.0709327
\(796\) 0 0
\(797\) 37.0000 1.31061 0.655304 0.755366i \(-0.272541\pi\)
0.655304 + 0.755366i \(0.272541\pi\)
\(798\) 0 0
\(799\) −7.00000 −0.247642
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) 0 0
\(803\) 10.0000 0.352892
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −8.00000 −0.281613
\(808\) 0 0
\(809\) 5.00000 0.175791 0.0878953 0.996130i \(-0.471986\pi\)
0.0878953 + 0.996130i \(0.471986\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 12.0000 0.419827
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.00000 −0.104701 −0.0523504 0.998629i \(-0.516671\pi\)
−0.0523504 + 0.998629i \(0.516671\pi\)
\(822\) 0 0
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) 32.0000 1.11275 0.556375 0.830932i \(-0.312192\pi\)
0.556375 + 0.830932i \(0.312192\pi\)
\(828\) 0 0
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) 20.0000 0.693792
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −21.0000 −0.726735
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −13.0000 −0.447744
\(844\) 0 0
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −1.00000 −0.0343199
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 0 0
\(855\) −12.0000 −0.410391
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 52.0000 1.77422 0.887109 0.461561i \(-0.152710\pi\)
0.887109 + 0.461561i \(0.152710\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) 0 0
\(865\) −21.0000 −0.714021
\(866\) 0 0
\(867\) −16.0000 −0.543388
\(868\) 0 0
\(869\) 11.0000 0.373149
\(870\) 0 0
\(871\) −70.0000 −2.37186
\(872\) 0 0
\(873\) 6.00000 0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −54.0000 −1.82345 −0.911725 0.410801i \(-0.865249\pi\)
−0.911725 + 0.410801i \(0.865249\pi\)
\(878\) 0 0
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) 16.0000 0.539054 0.269527 0.962993i \(-0.413133\pi\)
0.269527 + 0.962993i \(0.413133\pi\)
\(882\) 0 0
\(883\) −56.0000 −1.88455 −0.942275 0.334840i \(-0.891318\pi\)
−0.942275 + 0.334840i \(0.891318\pi\)
\(884\) 0 0
\(885\) −14.0000 −0.470605
\(886\) 0 0
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 42.0000 1.40548
\(894\) 0 0
\(895\) 20.0000 0.668526
\(896\) 0 0
\(897\) 20.0000 0.667781
\(898\) 0 0
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) −2.00000 −0.0666297
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) 16.0000 0.531271 0.265636 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) 0 0
\(909\) 20.0000 0.663358
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) 0 0
\(915\) 8.00000 0.264472
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 29.0000 0.956622 0.478311 0.878191i \(-0.341249\pi\)
0.478311 + 0.878191i \(0.341249\pi\)
\(920\) 0 0
\(921\) 23.0000 0.757876
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 0 0
\(927\) 14.0000 0.459820
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 14.0000 0.458339
\(934\) 0 0
\(935\) 1.00000 0.0327035
\(936\) 0 0
\(937\) 9.00000 0.294017 0.147009 0.989135i \(-0.453036\pi\)
0.147009 + 0.989135i \(0.453036\pi\)
\(938\) 0 0
\(939\) −25.0000 −0.815844
\(940\) 0 0
\(941\) 40.0000 1.30396 0.651981 0.758235i \(-0.273938\pi\)
0.651981 + 0.758235i \(0.273938\pi\)
\(942\) 0 0
\(943\) 40.0000 1.30258
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −54.0000 −1.75476 −0.877382 0.479792i \(-0.840712\pi\)
−0.877382 + 0.479792i \(0.840712\pi\)
\(948\) 0 0
\(949\) 50.0000 1.62307
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 50.0000 1.61966 0.809829 0.586665i \(-0.199560\pi\)
0.809829 + 0.586665i \(0.199560\pi\)
\(954\) 0 0
\(955\) −3.00000 −0.0970777
\(956\) 0 0
\(957\) −3.00000 −0.0969762
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −36.0000 −1.16008
\(964\) 0 0
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) 14.0000 0.450210 0.225105 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) 0 0
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) 2.00000 0.0641831 0.0320915 0.999485i \(-0.489783\pi\)
0.0320915 + 0.999485i \(0.489783\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −5.00000 −0.160128
\(976\) 0 0
\(977\) −54.0000 −1.72761 −0.863807 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) 0 0
\(979\) −4.00000 −0.127841
\(980\) 0 0
\(981\) 22.0000 0.702406
\(982\) 0 0
\(983\) −53.0000 −1.69044 −0.845219 0.534421i \(-0.820530\pi\)
−0.845219 + 0.534421i \(0.820530\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 12.0000 0.381193 0.190596 0.981669i \(-0.438958\pi\)
0.190596 + 0.981669i \(0.438958\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) 22.0000 0.697447
\(996\) 0 0
\(997\) −3.00000 −0.0950110 −0.0475055 0.998871i \(-0.515127\pi\)
−0.0475055 + 0.998871i \(0.515127\pi\)
\(998\) 0 0
\(999\) −40.0000 −1.26554
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 980.2.a.f.1.1 yes 1
3.2 odd 2 8820.2.a.v.1.1 1
4.3 odd 2 3920.2.a.n.1.1 1
5.2 odd 4 4900.2.e.k.2549.1 2
5.3 odd 4 4900.2.e.k.2549.2 2
5.4 even 2 4900.2.a.h.1.1 1
7.2 even 3 980.2.i.e.361.1 2
7.3 odd 6 980.2.i.g.961.1 2
7.4 even 3 980.2.i.e.961.1 2
7.5 odd 6 980.2.i.g.361.1 2
7.6 odd 2 980.2.a.d.1.1 1
21.20 even 2 8820.2.a.i.1.1 1
28.27 even 2 3920.2.a.z.1.1 1
35.13 even 4 4900.2.e.j.2549.1 2
35.27 even 4 4900.2.e.j.2549.2 2
35.34 odd 2 4900.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
980.2.a.d.1.1 1 7.6 odd 2
980.2.a.f.1.1 yes 1 1.1 even 1 trivial
980.2.i.e.361.1 2 7.2 even 3
980.2.i.e.961.1 2 7.4 even 3
980.2.i.g.361.1 2 7.5 odd 6
980.2.i.g.961.1 2 7.3 odd 6
3920.2.a.n.1.1 1 4.3 odd 2
3920.2.a.z.1.1 1 28.27 even 2
4900.2.a.h.1.1 1 5.4 even 2
4900.2.a.o.1.1 1 35.34 odd 2
4900.2.e.j.2549.1 2 35.13 even 4
4900.2.e.j.2549.2 2 35.27 even 4
4900.2.e.k.2549.1 2 5.2 odd 4
4900.2.e.k.2549.2 2 5.3 odd 4
8820.2.a.i.1.1 1 21.20 even 2
8820.2.a.v.1.1 1 3.2 odd 2