Properties

Label 4410.2.a.bk.1.1
Level $4410$
Weight $2$
Character 4410.1
Self dual yes
Analytic conductor $35.214$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.2140272914\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 4410.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} +2.00000 q^{11} +2.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} +6.00000 q^{19} +1.00000 q^{20} +2.00000 q^{22} -4.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -2.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} +2.00000 q^{37} +6.00000 q^{38} +1.00000 q^{40} +10.0000 q^{41} -8.00000 q^{43} +2.00000 q^{44} -4.00000 q^{46} -8.00000 q^{47} +1.00000 q^{50} +2.00000 q^{52} -2.00000 q^{53} +2.00000 q^{55} +4.00000 q^{59} +8.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{67} +2.00000 q^{68} +6.00000 q^{71} -2.00000 q^{73} +2.00000 q^{74} +6.00000 q^{76} -8.00000 q^{79} +1.00000 q^{80} +10.0000 q^{82} +4.00000 q^{83} +2.00000 q^{85} -8.00000 q^{86} +2.00000 q^{88} -10.0000 q^{89} -4.00000 q^{92} -8.00000 q^{94} +6.00000 q^{95} +18.0000 q^{97} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −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
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 10.0000 1.10432
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 8.00000 0.724286
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 20.0000 1.63846 0.819232 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 18.0000 1.29232
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 28.0000 1.87502 0.937509 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) 0 0
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) −26.0000 −1.68180 −0.840900 0.541190i \(-0.817974\pi\)
−0.840900 + 0.541190i \(0.817974\pi\)
\(240\) 0 0
\(241\) −20.0000 −1.28831 −0.644157 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 0 0
\(247\) 12.0000 0.763542
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −22.0000 −1.34136 −0.670682 0.741745i \(-0.733998\pi\)
−0.670682 + 0.741745i \(0.733998\pi\)
\(270\) 0 0
\(271\) 10.0000 0.607457 0.303728 0.952759i \(-0.401768\pi\)
0.303728 + 0.952759i \(0.401768\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −14.0000 −0.845771
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 14.0000 0.839664
\(279\) 0 0
\(280\) 0 0
\(281\) 20.0000 1.19310 0.596550 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) 0 0
\(283\) −8.00000 −0.475551 −0.237775 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 20.0000 1.15857
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.00000 −0.113592
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 8.00000 0.443079
\(327\) 0 0
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) 10.0000 0.528516
\(359\) 34.0000 1.79445 0.897226 0.441572i \(-0.145579\pi\)
0.897226 + 0.441572i \(0.145579\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 20.0000 1.05118
\(363\) 0 0
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 2.00000 0.103975
\(371\) 0 0
\(372\) 0 0
\(373\) −18.0000 −0.932005 −0.466002 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 0 0
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) −6.00000 −0.306987
\(383\) −28.0000 −1.43073 −0.715367 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) 0 0
\(388\) 18.0000 0.913812
\(389\) 36.0000 1.82527 0.912636 0.408773i \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 26.0000 1.30490 0.652451 0.757831i \(-0.273741\pi\)
0.652451 + 0.757831i \(0.273741\pi\)
\(398\) −2.00000 −0.100251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −32.0000 −1.59800 −0.799002 0.601329i \(-0.794638\pi\)
−0.799002 + 0.601329i \(0.794638\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) −8.00000 −0.395575 −0.197787 0.980245i \(-0.563376\pi\)
−0.197787 + 0.980245i \(0.563376\pi\)
\(410\) 10.0000 0.493865
\(411\) 0 0
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 12.0000 0.586939
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) 4.00000 0.190261
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 0 0
\(445\) −10.0000 −0.474045
\(446\) 28.0000 1.32584
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 20.0000 0.941763
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) 30.0000 1.40334 0.701670 0.712502i \(-0.252438\pi\)
0.701670 + 0.712502i \(0.252438\pi\)
\(458\) 16.0000 0.747631
\(459\) 0 0
\(460\) −4.00000 −0.186501
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) 0 0
\(478\) −26.0000 −1.18921
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) −20.0000 −0.910975
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 18.0000 0.817338
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 8.00000 0.362143
\(489\) 0 0
\(490\) 0 0
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 12.0000 0.539906
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −20.0000 −0.892644
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) 12.0000 0.532414
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −2.00000 −0.0868744
\(531\) 0 0
\(532\) 0 0
\(533\) 20.0000 0.866296
\(534\) 0 0
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) −22.0000 −0.948487
\(539\) 0 0
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 10.0000 0.429537
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −14.0000 −0.598050
\(549\) 0 0
\(550\) 2.00000 0.0852803
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 20.0000 0.843649
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) −8.00000 −0.336265
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) 40.0000 1.67689 0.838444 0.544988i \(-0.183466\pi\)
0.838444 + 0.544988i \(0.183466\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) −40.0000 −1.65098 −0.825488 0.564419i \(-0.809100\pi\)
−0.825488 + 0.564419i \(0.809100\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) 10.0000 0.410651 0.205325 0.978694i \(-0.434175\pi\)
0.205325 + 0.978694i \(0.434175\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 20.0000 0.819232
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) 26.0000 1.06233 0.531166 0.847268i \(-0.321754\pi\)
0.531166 + 0.847268i \(0.321754\pi\)
\(600\) 0 0
\(601\) 4.00000 0.163163 0.0815817 0.996667i \(-0.474003\pi\)
0.0815817 + 0.996667i \(0.474003\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) 8.00000 0.323911
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −18.0000 −0.723481 −0.361741 0.932279i \(-0.617817\pi\)
−0.361741 + 0.932279i \(0.617817\pi\)
\(620\) −2.00000 −0.0803219
\(621\) 0 0
\(622\) −8.00000 −0.320771
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) 6.00000 0.238290
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 0 0
\(643\) −12.0000 −0.473234 −0.236617 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) −46.0000 −1.80012 −0.900060 0.435767i \(-0.856477\pi\)
−0.900060 + 0.435767i \(0.856477\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 10.0000 0.390434
\(657\) 0 0
\(658\) 0 0
\(659\) −34.0000 −1.32445 −0.662226 0.749304i \(-0.730388\pi\)
−0.662226 + 0.749304i \(0.730388\pi\)
\(660\) 0 0
\(661\) 16.0000 0.622328 0.311164 0.950356i \(-0.399281\pi\)
0.311164 + 0.950356i \(0.399281\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) 16.0000 0.617673
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 34.0000 1.30963
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 2.00000 0.0768662 0.0384331 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) 0 0
\(682\) −4.00000 −0.153168
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) −14.0000 −0.534913
\(686\) 0 0
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −46.0000 −1.74992 −0.874961 0.484193i \(-0.839113\pi\)
−0.874961 + 0.484193i \(0.839113\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 14.0000 0.531050
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) 16.0000 0.605609
\(699\) 0 0
\(700\) 0 0
\(701\) −32.0000 −1.20862 −0.604312 0.796748i \(-0.706552\pi\)
−0.604312 + 0.796748i \(0.706552\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) 0 0
\(709\) −42.0000 −1.57734 −0.788672 0.614815i \(-0.789231\pi\)
−0.788672 + 0.614815i \(0.789231\pi\)
\(710\) 6.00000 0.225176
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 10.0000 0.373718
\(717\) 0 0
\(718\) 34.0000 1.26887
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) 20.0000 0.743294
\(725\) 0 0
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2.00000 −0.0740233
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) −4.00000 −0.147643
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 20.0000 0.732743
\(746\) −18.0000 −0.659027
\(747\) 0 0
\(748\) 4.00000 0.146254
\(749\) 0 0
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) 6.00000 0.217643
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) −28.0000 −1.01168
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −44.0000 −1.58668 −0.793340 0.608778i \(-0.791660\pi\)
−0.793340 + 0.608778i \(0.791660\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −6.00000 −0.215945
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 18.0000 0.646162
\(777\) 0 0
\(778\) 36.0000 1.29066
\(779\) 60.0000 2.14972
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) −8.00000 −0.286079
\(783\) 0 0
\(784\) 0 0
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) 0 0
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 26.0000 0.922705
\(795\) 0 0
\(796\) −2.00000 −0.0708881
\(797\) 50.0000 1.77109 0.885545 0.464553i \(-0.153785\pi\)
0.885545 + 0.464553i \(0.153785\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −32.0000 −1.12996
\(803\) −4.00000 −0.141157
\(804\) 0 0
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) −2.00000 −0.0703598
\(809\) 44.0000 1.54696 0.773479 0.633822i \(-0.218515\pi\)
0.773479 + 0.633822i \(0.218515\pi\)
\(810\) 0 0
\(811\) −34.0000 −1.19390 −0.596951 0.802278i \(-0.703621\pi\)
−0.596951 + 0.802278i \(0.703621\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 4.00000 0.140200
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) −48.0000 −1.67931
\(818\) −8.00000 −0.279713
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −52.0000 −1.81261 −0.906303 0.422628i \(-0.861108\pi\)
−0.906303 + 0.422628i \(0.861108\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) 32.0000 1.11275 0.556375 0.830932i \(-0.312192\pi\)
0.556375 + 0.830932i \(0.312192\pi\)
\(828\) 0 0
\(829\) 44.0000 1.52818 0.764092 0.645108i \(-0.223188\pi\)
0.764092 + 0.645108i \(0.223188\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 12.0000 0.415029
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) −2.00000 −0.0686803
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) 2.00000 0.0684787 0.0342393 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) −22.0000 −0.750630 −0.375315 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) −18.0000 −0.613082
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) −22.0000 −0.747590
\(867\) 0 0
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 10.0000 0.338643
\(873\) 0 0
\(874\) −24.0000 −0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) −26.0000 −0.877457
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −10.0000 −0.335201
\(891\) 0 0
\(892\) 28.0000 0.937509
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) 10.0000 0.334263
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 20.0000 0.665927
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) 0 0
\(911\) −14.0000 −0.463841 −0.231920 0.972735i \(-0.574501\pi\)
−0.231920 + 0.972735i \(0.574501\pi\)
\(912\) 0 0
\(913\) 8.00000 0.264761
\(914\) 30.0000 0.992312
\(915\) 0 0
\(916\) 16.0000 0.528655
\(917\) 0 0
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) −4.00000 −0.131876
\(921\) 0 0
\(922\) 18.0000 0.592798
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) −20.0000 −0.657241
\(927\) 0 0
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −26.0000 −0.851658
\(933\) 0 0
\(934\) −28.0000 −0.916188
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) 0 0
\(943\) −40.0000 −1.30258
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 6.00000 0.194666
\(951\) 0 0
\(952\) 0 0
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) 0 0
\(955\) −6.00000 −0.194155
\(956\) −26.0000 −0.840900
\(957\) 0 0
\(958\) −8.00000 −0.258468
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 4.00000 0.128965
\(963\) 0 0
\(964\) −20.0000 −0.644157
\(965\) −6.00000 −0.193147
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 18.0000 0.577945
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) 0 0
\(982\) 18.0000 0.574403
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 0 0
\(994\) 0 0
\(995\) −2.00000 −0.0634043
\(996\) 0 0
\(997\) −54.0000 −1.71020 −0.855099 0.518465i \(-0.826503\pi\)
−0.855099 + 0.518465i \(0.826503\pi\)
\(998\) 4.00000 0.126618
\(999\) 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 4410.2.a.bk.1.1 yes 1
3.2 odd 2 4410.2.a.d.1.1 1
7.6 odd 2 4410.2.a.bb.1.1 yes 1
21.20 even 2 4410.2.a.o.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4410.2.a.d.1.1 1 3.2 odd 2
4410.2.a.o.1.1 yes 1 21.20 even 2
4410.2.a.bb.1.1 yes 1 7.6 odd 2
4410.2.a.bk.1.1 yes 1 1.1 even 1 trivial