Properties

Label 155.2.a.c.1.1
Level $155$
Weight $2$
Character 155.1
Self dual yes
Analytic conductor $1.238$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 155.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.23768123133\)
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 \(\chi\) \(=\) 155.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{9} -4.00000 q^{11} +2.00000 q^{12} -6.00000 q^{13} +1.00000 q^{15} +4.00000 q^{16} +5.00000 q^{17} -1.00000 q^{19} +2.00000 q^{20} +8.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} -10.0000 q^{29} -1.00000 q^{31} +4.00000 q^{33} +4.00000 q^{36} +1.00000 q^{37} +6.00000 q^{39} -3.00000 q^{41} -7.00000 q^{43} +8.00000 q^{44} +2.00000 q^{45} -6.00000 q^{47} -4.00000 q^{48} -7.00000 q^{49} -5.00000 q^{51} +12.0000 q^{52} +5.00000 q^{53} +4.00000 q^{55} +1.00000 q^{57} +11.0000 q^{59} -2.00000 q^{60} -12.0000 q^{61} -8.00000 q^{64} +6.00000 q^{65} -2.00000 q^{67} -10.0000 q^{68} -8.00000 q^{69} +9.00000 q^{71} -9.00000 q^{73} -1.00000 q^{75} +2.00000 q^{76} -10.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +9.00000 q^{83} -5.00000 q^{85} +10.0000 q^{87} -16.0000 q^{92} +1.00000 q^{93} +1.00000 q^{95} -14.0000 q^{97} +8.00000 q^{99} +O(q^{100})\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 2.00000 0.577350
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 4.00000 1.00000
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 4.00000 0.666667
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 8.00000 1.20605
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −4.00000 −0.577350
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 12.0000 1.66410
\(53\) 5.00000 0.686803 0.343401 0.939189i \(-0.388421\pi\)
0.343401 + 0.939189i \(0.388421\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 11.0000 1.43208 0.716039 0.698060i \(-0.245953\pi\)
0.716039 + 0.698060i \(0.245953\pi\)
\(60\) −2.00000 −0.258199
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −10.0000 −1.21268
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) −9.00000 −1.05337 −0.526685 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 0 0
\(85\) −5.00000 −0.542326
\(86\) 0 0
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −16.0000 −1.66812
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 8.00000 0.804030
\(100\) −2.00000 −0.200000
\(101\) −7.00000 −0.696526 −0.348263 0.937397i \(-0.613228\pi\)
−0.348263 + 0.937397i \(0.613228\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) −10.0000 −0.962250
\(109\) 15.0000 1.43674 0.718370 0.695662i \(-0.244889\pi\)
0.718370 + 0.695662i \(0.244889\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) 0 0
\(113\) −4.00000 −0.376288 −0.188144 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 20.0000 1.85695
\(117\) 12.0000 1.10940
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 3.00000 0.270501
\(124\) 2.00000 0.179605
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 7.00000 0.616316
\(130\) 0 0
\(131\) −1.00000 −0.0873704 −0.0436852 0.999045i \(-0.513910\pi\)
−0.0436852 + 0.999045i \(0.513910\pi\)
\(132\) −8.00000 −0.696311
\(133\) 0 0
\(134\) 0 0
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 24.0000 2.00698
\(144\) −8.00000 −0.666667
\(145\) 10.0000 0.830455
\(146\) 0 0
\(147\) 7.00000 0.577350
\(148\) −2.00000 −0.164399
\(149\) 1.00000 0.0819232 0.0409616 0.999161i \(-0.486958\pi\)
0.0409616 + 0.999161i \(0.486958\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) −10.0000 −0.808452
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) −12.0000 −0.960769
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) −5.00000 −0.396526
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 24.0000 1.87983 0.939913 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(164\) 6.00000 0.468521
\(165\) −4.00000 −0.311400
\(166\) 0 0
\(167\) −15.0000 −1.16073 −0.580367 0.814355i \(-0.697091\pi\)
−0.580367 + 0.814355i \(0.697091\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 14.0000 1.06749
\(173\) 4.00000 0.304114 0.152057 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −16.0000 −1.20605
\(177\) −11.0000 −0.826811
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −4.00000 −0.298142
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) 12.0000 0.887066
\(184\) 0 0
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −20.0000 −1.46254
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 8.00000 0.577350
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) −6.00000 −0.429669
\(196\) 14.0000 1.00000
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) 0 0
\(203\) 0 0
\(204\) 10.0000 0.700140
\(205\) 3.00000 0.209529
\(206\) 0 0
\(207\) −16.0000 −1.11208
\(208\) −24.0000 −1.66410
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −10.0000 −0.686803
\(213\) −9.00000 −0.616670
\(214\) 0 0
\(215\) 7.00000 0.477396
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 9.00000 0.608164
\(220\) −8.00000 −0.539360
\(221\) −30.0000 −2.01802
\(222\) 0 0
\(223\) −17.0000 −1.13840 −0.569202 0.822198i \(-0.692748\pi\)
−0.569202 + 0.822198i \(0.692748\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) −10.0000 −0.663723 −0.331862 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(228\) −2.00000 −0.132453
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0000 1.04819 0.524097 0.851658i \(-0.324403\pi\)
0.524097 + 0.851658i \(0.324403\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) −22.0000 −1.43208
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 4.00000 0.258199
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 24.0000 1.53644
\(245\) 7.00000 0.447214
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 0 0
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) −22.0000 −1.38863 −0.694314 0.719672i \(-0.744292\pi\)
−0.694314 + 0.719672i \(0.744292\pi\)
\(252\) 0 0
\(253\) −32.0000 −2.01182
\(254\) 0 0
\(255\) 5.00000 0.313112
\(256\) 16.0000 1.00000
\(257\) 10.0000 0.623783 0.311891 0.950118i \(-0.399037\pi\)
0.311891 + 0.950118i \(0.399037\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −12.0000 −0.744208
\(261\) 20.0000 1.23797
\(262\) 0 0
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 0 0
\(265\) −5.00000 −0.307148
\(266\) 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
\(271\) −14.0000 −0.850439 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 20.0000 1.21268
\(273\) 0 0
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) 16.0000 0.963087
\(277\) 9.00000 0.540758 0.270379 0.962754i \(-0.412851\pi\)
0.270379 + 0.962754i \(0.412851\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 7.00000 0.417585 0.208792 0.977960i \(-0.433047\pi\)
0.208792 + 0.977960i \(0.433047\pi\)
\(282\) 0 0
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) −18.0000 −1.06810
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 18.0000 1.05337
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −11.0000 −0.640445
\(296\) 0 0
\(297\) −20.0000 −1.16052
\(298\) 0 0
\(299\) −48.0000 −2.77591
\(300\) 2.00000 0.115470
\(301\) 0 0
\(302\) 0 0
\(303\) 7.00000 0.402139
\(304\) −4.00000 −0.229416
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −7.00000 −0.396934 −0.198467 0.980108i \(-0.563596\pi\)
−0.198467 + 0.980108i \(0.563596\pi\)
\(312\) 0 0
\(313\) 29.0000 1.63918 0.819588 0.572953i \(-0.194202\pi\)
0.819588 + 0.572953i \(0.194202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 20.0000 1.12509
\(317\) 32.0000 1.79730 0.898650 0.438667i \(-0.144549\pi\)
0.898650 + 0.438667i \(0.144549\pi\)
\(318\) 0 0
\(319\) 40.0000 2.23957
\(320\) 8.00000 0.447214
\(321\) 2.00000 0.111629
\(322\) 0 0
\(323\) −5.00000 −0.278207
\(324\) −2.00000 −0.111111
\(325\) −6.00000 −0.332820
\(326\) 0 0
\(327\) −15.0000 −0.829502
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −18.0000 −0.987878
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) −19.0000 −1.03500 −0.517498 0.855684i \(-0.673136\pi\)
−0.517498 + 0.855684i \(0.673136\pi\)
\(338\) 0 0
\(339\) 4.00000 0.217250
\(340\) 10.0000 0.542326
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 8.00000 0.430706
\(346\) 0 0
\(347\) 8.00000 0.429463 0.214731 0.976673i \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(348\) −20.0000 −1.07211
\(349\) −29.0000 −1.55233 −0.776167 0.630527i \(-0.782839\pi\)
−0.776167 + 0.630527i \(0.782839\pi\)
\(350\) 0 0
\(351\) −30.0000 −1.60128
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −9.00000 −0.477670
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 9.00000 0.471082
\(366\) 0 0
\(367\) 25.0000 1.30499 0.652495 0.757793i \(-0.273722\pi\)
0.652495 + 0.757793i \(0.273722\pi\)
\(368\) 32.0000 1.66812
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) −2.00000 −0.103695
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 60.0000 3.09016
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) −2.00000 −0.102598
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) 1.00000 0.0510976 0.0255488 0.999674i \(-0.491867\pi\)
0.0255488 + 0.999674i \(0.491867\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 14.0000 0.711660
\(388\) 28.0000 1.42148
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 40.0000 2.02289
\(392\) 0 0
\(393\) 1.00000 0.0504433
\(394\) 0 0
\(395\) 10.0000 0.503155
\(396\) −16.0000 −0.804030
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −16.0000 −0.799002 −0.399501 0.916733i \(-0.630817\pi\)
−0.399501 + 0.916733i \(0.630817\pi\)
\(402\) 0 0
\(403\) 6.00000 0.298881
\(404\) 14.0000 0.696526
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) −3.00000 −0.147979
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) 0 0
\(415\) −9.00000 −0.441793
\(416\) 0 0
\(417\) −2.00000 −0.0979404
\(418\) 0 0
\(419\) −15.0000 −0.732798 −0.366399 0.930458i \(-0.619409\pi\)
−0.366399 + 0.930458i \(0.619409\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 0 0
\(423\) 12.0000 0.583460
\(424\) 0 0
\(425\) 5.00000 0.242536
\(426\) 0 0
\(427\) 0 0
\(428\) 4.00000 0.193347
\(429\) −24.0000 −1.15873
\(430\) 0 0
\(431\) −33.0000 −1.58955 −0.794777 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) 20.0000 0.962250
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) −10.0000 −0.479463
\(436\) −30.0000 −1.43674
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) −35.0000 −1.67046 −0.835229 0.549902i \(-0.814665\pi\)
−0.835229 + 0.549902i \(0.814665\pi\)
\(440\) 0 0
\(441\) 14.0000 0.666667
\(442\) 0 0
\(443\) −34.0000 −1.61539 −0.807694 0.589601i \(-0.799285\pi\)
−0.807694 + 0.589601i \(0.799285\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 0 0
\(447\) −1.00000 −0.0472984
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 8.00000 0.376288
\(453\) −10.0000 −0.469841
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) 0 0
\(459\) 25.0000 1.16690
\(460\) 16.0000 0.746004
\(461\) 4.00000 0.186299 0.0931493 0.995652i \(-0.470307\pi\)
0.0931493 + 0.995652i \(0.470307\pi\)
\(462\) 0 0
\(463\) −12.0000 −0.557687 −0.278844 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) −40.0000 −1.85695
\(465\) −1.00000 −0.0463739
\(466\) 0 0
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) −24.0000 −1.10940
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 28.0000 1.28744
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 0 0
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) −19.0000 −0.860972 −0.430486 0.902597i \(-0.641658\pi\)
−0.430486 + 0.902597i \(0.641658\pi\)
\(488\) 0 0
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) −4.00000 −0.180517 −0.0902587 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) −6.00000 −0.270501
\(493\) −50.0000 −2.25189
\(494\) 0 0
\(495\) −8.00000 −0.359573
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) 34.0000 1.52205 0.761025 0.648723i \(-0.224697\pi\)
0.761025 + 0.648723i \(0.224697\pi\)
\(500\) 2.00000 0.0894427
\(501\) 15.0000 0.670151
\(502\) 0 0
\(503\) −22.0000 −0.980932 −0.490466 0.871460i \(-0.663173\pi\)
−0.490466 + 0.871460i \(0.663173\pi\)
\(504\) 0 0
\(505\) 7.00000 0.311496
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) 16.0000 0.709885
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −5.00000 −0.220755
\(514\) 0 0
\(515\) −8.00000 −0.352522
\(516\) −14.0000 −0.616316
\(517\) 24.0000 1.05552
\(518\) 0 0
\(519\) −4.00000 −0.175581
\(520\) 0 0
\(521\) 19.0000 0.832405 0.416203 0.909272i \(-0.363361\pi\)
0.416203 + 0.909272i \(0.363361\pi\)
\(522\) 0 0
\(523\) 11.0000 0.480996 0.240498 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(524\) 2.00000 0.0873704
\(525\) 0 0
\(526\) 0 0
\(527\) −5.00000 −0.217803
\(528\) 16.0000 0.696311
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) −22.0000 −0.954719
\(532\) 0 0
\(533\) 18.0000 0.779667
\(534\) 0 0
\(535\) 2.00000 0.0864675
\(536\) 0 0
\(537\) 4.00000 0.172613
\(538\) 0 0
\(539\) 28.0000 1.20605
\(540\) 10.0000 0.430331
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 0 0
\(543\) 18.0000 0.772454
\(544\) 0 0
\(545\) −15.0000 −0.642529
\(546\) 0 0
\(547\) 40.0000 1.71028 0.855138 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) −6.00000 −0.256307
\(549\) 24.0000 1.02430
\(550\) 0 0
\(551\) 10.0000 0.426014
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1.00000 0.0424476
\(556\) −4.00000 −0.169638
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 42.0000 1.77641
\(560\) 0 0
\(561\) 20.0000 0.844401
\(562\) 0 0
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) −12.0000 −0.505291
\(565\) 4.00000 0.168281
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) −26.0000 −1.08807 −0.544033 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) −48.0000 −2.00698
\(573\) −4.00000 −0.167102
\(574\) 0 0
\(575\) 8.00000 0.333623
\(576\) 16.0000 0.666667
\(577\) 6.00000 0.249783 0.124892 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) 0 0
\(579\) −14.0000 −0.581820
\(580\) −20.0000 −0.830455
\(581\) 0 0
\(582\) 0 0
\(583\) −20.0000 −0.828315
\(584\) 0 0
\(585\) −12.0000 −0.496139
\(586\) 0 0
\(587\) −8.00000 −0.330195 −0.165098 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) −14.0000 −0.577350
\(589\) 1.00000 0.0412043
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 4.00000 0.164399
\(593\) −4.00000 −0.164260 −0.0821302 0.996622i \(-0.526172\pi\)
−0.0821302 + 0.996622i \(0.526172\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −2.00000 −0.0819232
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) −20.0000 −0.813788
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) −42.0000 −1.70473 −0.852364 0.522949i \(-0.824832\pi\)
−0.852364 + 0.522949i \(0.824832\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 36.0000 1.45640
\(612\) 20.0000 0.808452
\(613\) −5.00000 −0.201948 −0.100974 0.994889i \(-0.532196\pi\)
−0.100974 + 0.994889i \(0.532196\pi\)
\(614\) 0 0
\(615\) −3.00000 −0.120972
\(616\) 0 0
\(617\) 16.0000 0.644136 0.322068 0.946717i \(-0.395622\pi\)
0.322068 + 0.946717i \(0.395622\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) −2.00000 −0.0803219
\(621\) 40.0000 1.60514
\(622\) 0 0
\(623\) 0 0
\(624\) 24.0000 0.960769
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −4.00000 −0.159745
\(628\) 0 0
\(629\) 5.00000 0.199363
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 10.0000 0.396526
\(637\) 42.0000 1.66410
\(638\) 0 0
\(639\) −18.0000 −0.712069
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) 7.00000 0.276053 0.138027 0.990429i \(-0.455924\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) 0 0
\(645\) −7.00000 −0.275625
\(646\) 0 0
\(647\) −21.0000 −0.825595 −0.412798 0.910823i \(-0.635448\pi\)
−0.412798 + 0.910823i \(0.635448\pi\)
\(648\) 0 0
\(649\) −44.0000 −1.72715
\(650\) 0 0
\(651\) 0 0
\(652\) −48.0000 −1.87983
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 1.00000 0.0390732
\(656\) −12.0000 −0.468521
\(657\) 18.0000 0.702247
\(658\) 0 0
\(659\) 39.0000 1.51922 0.759612 0.650376i \(-0.225389\pi\)
0.759612 + 0.650376i \(0.225389\pi\)
\(660\) 8.00000 0.311400
\(661\) 29.0000 1.12797 0.563985 0.825785i \(-0.309268\pi\)
0.563985 + 0.825785i \(0.309268\pi\)
\(662\) 0 0
\(663\) 30.0000 1.16510
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −80.0000 −3.09761
\(668\) 30.0000 1.16073
\(669\) 17.0000 0.657258
\(670\) 0 0
\(671\) 48.0000 1.85302
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) −46.0000 −1.76923
\(677\) 27.0000 1.03769 0.518847 0.854867i \(-0.326361\pi\)
0.518847 + 0.854867i \(0.326361\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 10.0000 0.383201
\(682\) 0 0
\(683\) 48.0000 1.83667 0.918334 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) −4.00000 −0.152944
\(685\) −3.00000 −0.114624
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) −28.0000 −1.06749
\(689\) −30.0000 −1.14291
\(690\) 0 0
\(691\) −17.0000 −0.646710 −0.323355 0.946278i \(-0.604811\pi\)
−0.323355 + 0.946278i \(0.604811\pi\)
\(692\) −8.00000 −0.304114
\(693\) 0 0
\(694\) 0 0
\(695\) −2.00000 −0.0758643
\(696\) 0 0
\(697\) −15.0000 −0.568166
\(698\) 0 0
\(699\) −16.0000 −0.605176
\(700\) 0 0
\(701\) −26.0000 −0.982006 −0.491003 0.871158i \(-0.663370\pi\)
−0.491003 + 0.871158i \(0.663370\pi\)
\(702\) 0 0
\(703\) −1.00000 −0.0377157
\(704\) 32.0000 1.20605
\(705\) −6.00000 −0.225973
\(706\) 0 0
\(707\) 0 0
\(708\) 22.0000 0.826811
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) −24.0000 −0.897549
\(716\) 8.00000 0.298974
\(717\) −6.00000 −0.224074
\(718\) 0 0
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 8.00000 0.298142
\(721\) 0 0
\(722\) 0 0
\(723\) −4.00000 −0.148762
\(724\) 36.0000 1.33793
\(725\) −10.0000 −0.371391
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −35.0000 −1.29452
\(732\) −24.0000 −0.887066
\(733\) −40.0000 −1.47743 −0.738717 0.674016i \(-0.764568\pi\)
−0.738717 + 0.674016i \(0.764568\pi\)
\(734\) 0 0
\(735\) −7.00000 −0.258199
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 2.00000 0.0735712 0.0367856 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) 2.00000 0.0735215
\(741\) −6.00000 −0.220416
\(742\) 0 0
\(743\) −15.0000 −0.550297 −0.275148 0.961402i \(-0.588727\pi\)
−0.275148 + 0.961402i \(0.588727\pi\)
\(744\) 0 0
\(745\) −1.00000 −0.0366372
\(746\) 0 0
\(747\) −18.0000 −0.658586
\(748\) 40.0000 1.46254
\(749\) 0 0
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) −24.0000 −0.875190
\(753\) 22.0000 0.801725
\(754\) 0 0
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) −45.0000 −1.63555 −0.817776 0.575536i \(-0.804793\pi\)
−0.817776 + 0.575536i \(0.804793\pi\)
\(758\) 0 0
\(759\) 32.0000 1.16153
\(760\) 0 0
\(761\) −44.0000 −1.59500 −0.797499 0.603320i \(-0.793844\pi\)
−0.797499 + 0.603320i \(0.793844\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) 10.0000 0.361551
\(766\) 0 0
\(767\) −66.0000 −2.38312
\(768\) −16.0000 −0.577350
\(769\) −29.0000 −1.04577 −0.522883 0.852404i \(-0.675144\pi\)
−0.522883 + 0.852404i \(0.675144\pi\)
\(770\) 0 0
\(771\) −10.0000 −0.360141
\(772\) −28.0000 −1.00774
\(773\) 2.00000 0.0719350 0.0359675 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.00000 0.107486
\(780\) 12.0000 0.429669
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) −50.0000 −1.78685
\(784\) −28.0000 −1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 44.0000 1.56744
\(789\) 9.00000 0.320408
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 72.0000 2.55679
\(794\) 0 0
\(795\) 5.00000 0.177332
\(796\) 16.0000 0.567105
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) −30.0000 −1.06132
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 36.0000 1.27041
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) −22.0000 −0.773479 −0.386739 0.922189i \(-0.626399\pi\)
−0.386739 + 0.922189i \(0.626399\pi\)
\(810\) 0 0
\(811\) −3.00000 −0.105344 −0.0526721 0.998612i \(-0.516774\pi\)
−0.0526721 + 0.998612i \(0.516774\pi\)
\(812\) 0 0
\(813\) 14.0000 0.491001
\(814\) 0 0
\(815\) −24.0000 −0.840683
\(816\) −20.0000 −0.700140
\(817\) 7.00000 0.244899
\(818\) 0 0
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 56.0000 1.95441 0.977207 0.212290i \(-0.0680921\pi\)
0.977207 + 0.212290i \(0.0680921\pi\)
\(822\) 0 0
\(823\) −23.0000 −0.801730 −0.400865 0.916137i \(-0.631290\pi\)
−0.400865 + 0.916137i \(0.631290\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 37.0000 1.28662 0.643308 0.765607i \(-0.277561\pi\)
0.643308 + 0.765607i \(0.277561\pi\)
\(828\) 32.0000 1.11208
\(829\) −46.0000 −1.59765 −0.798823 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 0 0
\(831\) −9.00000 −0.312207
\(832\) 48.0000 1.66410
\(833\) −35.0000 −1.21268
\(834\) 0 0
\(835\) 15.0000 0.519096
\(836\) −8.00000 −0.276686
\(837\) −5.00000 −0.172825
\(838\) 0 0
\(839\) −35.0000 −1.20833 −0.604167 0.796858i \(-0.706494\pi\)
−0.604167 + 0.796858i \(0.706494\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) −7.00000 −0.241093
\(844\) −40.0000 −1.37686
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) 0 0
\(848\) 20.0000 0.686803
\(849\) 6.00000 0.205919
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 18.0000 0.616670
\(853\) −32.0000 −1.09566 −0.547830 0.836590i \(-0.684546\pi\)
−0.547830 + 0.836590i \(0.684546\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) 0 0
\(857\) 36.0000 1.22974 0.614868 0.788630i \(-0.289209\pi\)
0.614868 + 0.788630i \(0.289209\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) −14.0000 −0.477396
\(861\) 0 0
\(862\) 0 0
\(863\) −45.0000 −1.53182 −0.765909 0.642949i \(-0.777711\pi\)
−0.765909 + 0.642949i \(0.777711\pi\)
\(864\) 0 0
\(865\) −4.00000 −0.136004
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 40.0000 1.35691
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 0 0
\(873\) 28.0000 0.947656
\(874\) 0 0
\(875\) 0 0
\(876\) −18.0000 −0.608164
\(877\) −44.0000 −1.48577 −0.742887 0.669417i \(-0.766544\pi\)
−0.742887 + 0.669417i \(0.766544\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 16.0000 0.539360
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −11.0000 −0.370179 −0.185090 0.982722i \(-0.559258\pi\)
−0.185090 + 0.982722i \(0.559258\pi\)
\(884\) 60.0000 2.01802
\(885\) 11.0000 0.369761
\(886\) 0 0
\(887\) 18.0000 0.604381 0.302190 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 34.0000 1.13840
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) 48.0000 1.60267
\(898\) 0 0
\(899\) 10.0000 0.333519
\(900\) 4.00000 0.133333
\(901\) 25.0000 0.832871
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18.0000 0.598340
\(906\) 0 0
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) 20.0000 0.663723
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 4.00000 0.132453
\(913\) −36.0000 −1.19143
\(914\) 0 0
\(915\) −12.0000 −0.396708
\(916\) 28.0000 0.925146
\(917\) 0 0
\(918\) 0 0
\(919\) −17.0000 −0.560778 −0.280389 0.959886i \(-0.590464\pi\)
−0.280389 + 0.959886i \(0.590464\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) −54.0000 −1.77743
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 0 0
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) 48.0000 1.57483 0.787414 0.616424i \(-0.211419\pi\)
0.787414 + 0.616424i \(0.211419\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) −32.0000 −1.04819
\(933\) 7.00000 0.229170
\(934\) 0 0
\(935\) 20.0000 0.654070
\(936\) 0 0
\(937\) 28.0000 0.914720 0.457360 0.889282i \(-0.348795\pi\)
0.457360 + 0.889282i \(0.348795\pi\)
\(938\) 0 0
\(939\) −29.0000 −0.946379
\(940\) −12.0000 −0.391397
\(941\) 54.0000 1.76035 0.880175 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 44.0000 1.43208
\(945\) 0 0
\(946\) 0 0
\(947\) 55.0000 1.78726 0.893630 0.448805i \(-0.148150\pi\)
0.893630 + 0.448805i \(0.148150\pi\)
\(948\) −20.0000 −0.649570
\(949\) 54.0000 1.75291
\(950\) 0 0
\(951\) −32.0000 −1.03767
\(952\) 0 0
\(953\) −9.00000 −0.291539 −0.145769 0.989319i \(-0.546566\pi\)
−0.145769 + 0.989319i \(0.546566\pi\)
\(954\) 0 0
\(955\) −4.00000 −0.129437
\(956\) −12.0000 −0.388108
\(957\) −40.0000 −1.29302
\(958\) 0 0
\(959\) 0 0
\(960\) −8.00000 −0.258199
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) −8.00000 −0.257663
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 5.00000 0.160623
\(970\) 0 0
\(971\) 33.0000 1.05902 0.529510 0.848304i \(-0.322376\pi\)
0.529510 + 0.848304i \(0.322376\pi\)
\(972\) 32.0000 1.02640
\(973\) 0 0
\(974\) 0 0
\(975\) 6.00000 0.192154
\(976\) −48.0000 −1.53644
\(977\) −44.0000 −1.40768 −0.703842 0.710356i \(-0.748534\pi\)
−0.703842 + 0.710356i \(0.748534\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −14.0000 −0.447214
\(981\) −30.0000 −0.957826
\(982\) 0 0
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 0 0
\(985\) 22.0000 0.700978
\(986\) 0 0
\(987\) 0 0
\(988\) −12.0000 −0.381771
\(989\) −56.0000 −1.78070
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 0 0
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 18.0000 0.570352
\(997\) −28.0000 −0.886769 −0.443384 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) 0 0
\(999\) 5.00000 0.158193
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 155.2.a.c.1.1 1
3.2 odd 2 1395.2.a.b.1.1 1
4.3 odd 2 2480.2.a.k.1.1 1
5.2 odd 4 775.2.b.c.249.2 2
5.3 odd 4 775.2.b.c.249.1 2
5.4 even 2 775.2.a.a.1.1 1
7.6 odd 2 7595.2.a.h.1.1 1
8.3 odd 2 9920.2.a.l.1.1 1
8.5 even 2 9920.2.a.ba.1.1 1
15.14 odd 2 6975.2.a.l.1.1 1
31.30 odd 2 4805.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
155.2.a.c.1.1 1 1.1 even 1 trivial
775.2.a.a.1.1 1 5.4 even 2
775.2.b.c.249.1 2 5.3 odd 4
775.2.b.c.249.2 2 5.2 odd 4
1395.2.a.b.1.1 1 3.2 odd 2
2480.2.a.k.1.1 1 4.3 odd 2
4805.2.a.e.1.1 1 31.30 odd 2
6975.2.a.l.1.1 1 15.14 odd 2
7595.2.a.h.1.1 1 7.6 odd 2
9920.2.a.l.1.1 1 8.3 odd 2
9920.2.a.ba.1.1 1 8.5 even 2