Properties

Label 2898.2.a.f.1.1
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.1406465058\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2898.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{10} -4.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} +2.00000 q^{20} +4.00000 q^{22} -1.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{28} +2.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} -2.00000 q^{35} +6.00000 q^{37} -2.00000 q^{40} +6.00000 q^{41} +12.0000 q^{43} -4.00000 q^{44} +1.00000 q^{46} +12.0000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} -8.00000 q^{55} +1.00000 q^{56} -2.00000 q^{58} +4.00000 q^{59} -10.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +4.00000 q^{67} -6.00000 q^{68} +2.00000 q^{70} +16.0000 q^{71} +2.00000 q^{73} -6.00000 q^{74} +4.00000 q^{77} +8.00000 q^{79} +2.00000 q^{80} -6.00000 q^{82} +16.0000 q^{83} -12.0000 q^{85} -12.0000 q^{86} +4.00000 q^{88} -6.00000 q^{89} -2.00000 q^{91} -1.00000 q^{92} -12.0000 q^{94} -2.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 2.00000 0.223607
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) −12.0000 −1.29399
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 8.00000 0.762770
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) −16.0000 −1.34269
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −4.00000 −0.322329
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) −2.00000 −0.158114
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −16.0000 −1.24184
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 12.0000 0.920358
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −14.0000 −0.985037
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 28.0000 1.92760 0.963800 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 24.0000 1.63679
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) −8.00000 −0.539360
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) −18.0000 −1.18947 −0.594737 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 24.0000 1.56559
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) −6.00000 −0.388922
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 0 0
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 8.00000 0.501965
\(255\) 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
\(259\) −6.00000 −0.372822
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) 24.0000 1.42665 0.713326 0.700832i \(-0.247188\pi\)
0.713326 + 0.700832i \(0.247188\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) −4.00000 −0.234888
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) 34.0000 1.98630 0.993151 0.116841i \(-0.0372769\pi\)
0.993151 + 0.116841i \(0.0372769\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −20.0000 −1.14520
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 16.0000 0.878114
\(333\) 0 0
\(334\) 20.0000 1.09435
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) −12.0000 −0.650791
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 32.0000 1.69838
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −12.0000 −0.623850
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) −18.0000 −0.916176
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) −38.0000 −1.92668 −0.963338 0.268290i \(-0.913542\pi\)
−0.963338 + 0.268290i \(0.913542\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 16.0000 0.805047
\(396\) 0 0
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) −12.0000 −0.592638
\(411\) 0 0
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 32.0000 1.57082
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 0 0
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −28.0000 −1.36302
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −24.0000 −1.15738
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 8.00000 0.381385
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) 4.00000 0.189405
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −24.0000 −1.12638
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 18.0000 0.841085
\(459\) 0 0
\(460\) −2.00000 −0.0932505
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) −24.0000 −1.10704
\(471\) 0 0
\(472\) −4.00000 −0.184115
\(473\) −48.0000 −2.20704
\(474\) 0 0
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) 0 0
\(478\) 8.00000 0.365911
\(479\) −40.0000 −1.82765 −0.913823 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 10.0000 0.452679
\(489\) 0 0
\(490\) −2.00000 −0.0903508
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) 24.0000 1.07117
\(503\) −40.0000 −1.78351 −0.891756 0.452517i \(-0.850526\pi\)
−0.891756 + 0.452517i \(0.850526\pi\)
\(504\) 0 0
\(505\) 28.0000 1.24598
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 0 0
\(517\) −48.0000 −2.11104
\(518\) 6.00000 0.263625
\(519\) 0 0
\(520\) −4.00000 −0.175412
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) 0 0
\(523\) −8.00000 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 12.0000 0.521247
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −20.0000 −0.859074
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) −22.0000 −0.928014
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) −24.0000 −1.00880
\(567\) 0 0
\(568\) −16.0000 −0.671345
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −34.0000 −1.40453
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −8.00000 −0.329355
\(591\) 0 0
\(592\) 6.00000 0.246598
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 2.00000 0.0817861
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 12.0000 0.489083
\(603\) 0 0
\(604\) 0 0
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) 20.0000 0.811775 0.405887 0.913923i \(-0.366962\pi\)
0.405887 + 0.913923i \(0.366962\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 20.0000 0.809776
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) −36.0000 −1.43541
\(630\) 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
\(634\) −10.0000 −0.397151
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 8.00000 0.316723
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 24.0000 0.946468 0.473234 0.880937i \(-0.343087\pi\)
0.473234 + 0.880937i \(0.343087\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 0 0
\(647\) −36.0000 −1.41531 −0.707653 0.706560i \(-0.750246\pi\)
−0.707653 + 0.706560i \(0.750246\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 12.0000 0.467809
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) −28.0000 −1.08825
\(663\) 0 0
\(664\) −16.0000 −0.620920
\(665\) 0 0
\(666\) 0 0
\(667\) −2.00000 −0.0774403
\(668\) −20.0000 −0.773823
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 12.0000 0.460179
\(681\) 0 0
\(682\) 16.0000 0.612672
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) −20.0000 −0.764161
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 12.0000 0.457496
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 6.00000 0.227103
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) −32.0000 −1.20094
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) −16.0000 −0.597115
\(719\) −28.0000 −1.04422 −0.522112 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 2.00000 0.0741249
\(729\) 0 0
\(730\) −4.00000 −0.148047
\(731\) −72.0000 −2.66302
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 24.0000 0.885856
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) −22.0000 −0.805477
\(747\) 0 0
\(748\) 24.0000 0.877527
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) −8.00000 −0.288300
\(771\) 0 0
\(772\) 18.0000 0.647834
\(773\) 26.0000 0.935155 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 38.0000 1.36237
\(779\) 0 0
\(780\) 0 0
\(781\) −64.0000 −2.29010
\(782\) −6.00000 −0.214560
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −36.0000 −1.28490
\(786\) 0 0
\(787\) −40.0000 −1.42585 −0.712923 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) −16.0000 −0.569254
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) −34.0000 −1.20661
\(795\) 0 0
\(796\) 0 0
\(797\) 34.0000 1.20434 0.602171 0.798367i \(-0.294303\pi\)
0.602171 + 0.798367i \(0.294303\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) −8.00000 −0.282314
\(804\) 0 0
\(805\) 2.00000 0.0704907
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) −14.0000 −0.492518
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) 24.0000 0.841200
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 0 0
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 12.0000 0.419058
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) −32.0000 −1.11074
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −40.0000 −1.38426
\(836\) 0 0
\(837\) 0 0
\(838\) 16.0000 0.552711
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) 28.0000 0.963800
\(845\) −18.0000 −0.619219
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −6.00000 −0.205798
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) −38.0000 −1.30110 −0.650548 0.759465i \(-0.725461\pi\)
−0.650548 + 0.759465i \(0.725461\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) −12.0000 −0.409435 −0.204717 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) 24.0000 0.818393
\(861\) 0 0
\(862\) 16.0000 0.544962
\(863\) 8.00000 0.272323 0.136162 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) −26.0000 −0.877958 −0.438979 0.898497i \(-0.644660\pi\)
−0.438979 + 0.898497i \(0.644660\pi\)
\(878\) −20.0000 −0.674967
\(879\) 0 0
\(880\) −8.00000 −0.269680
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) 44.0000 1.47738 0.738688 0.674048i \(-0.235446\pi\)
0.738688 + 0.674048i \(0.235446\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 12.0000 0.402241
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) 0 0
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 24.0000 0.799113
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) −4.00000 −0.132964
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 24.0000 0.796468
\(909\) 0 0
\(910\) 4.00000 0.132599
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) −64.0000 −2.11809
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) −18.0000 −0.594737
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 2.00000 0.0659380
\(921\) 0 0
\(922\) −22.0000 −0.724531
\(923\) 32.0000 1.05329
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) 46.0000 1.50921 0.754606 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) 0 0
\(934\) 8.00000 0.261768
\(935\) 48.0000 1.56977
\(936\) 0 0
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) 24.0000 0.782794
\(941\) −38.0000 −1.23876 −0.619382 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) −6.00000 −0.194461
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) 40.0000 1.29234
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) 36.0000 1.15888
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 2.00000 0.0638877
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) −56.0000 −1.77890 −0.889449 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) 0 0
\(997\) −6.00000 −0.190022 −0.0950110 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) 36.0000 1.13956
\(999\) 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 2898.2.a.f.1.1 1
3.2 odd 2 966.2.a.j.1.1 1
12.11 even 2 7728.2.a.b.1.1 1
21.20 even 2 6762.2.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.j.1.1 1 3.2 odd 2
2898.2.a.f.1.1 1 1.1 even 1 trivial
6762.2.a.bc.1.1 1 21.20 even 2
7728.2.a.b.1.1 1 12.11 even 2