Properties

Label 575.2.a.c.1.1
Level $575$
Weight $2$
Character 575.1
Self dual yes
Analytic conductor $4.591$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.59139811622\)
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 \(\chi\) \(=\) 575.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} +3.00000 q^{18} -5.00000 q^{19} +1.00000 q^{22} +1.00000 q^{23} -1.00000 q^{26} -1.00000 q^{28} -5.00000 q^{29} -2.00000 q^{31} -5.00000 q^{32} +3.00000 q^{36} -4.00000 q^{37} +5.00000 q^{38} -5.00000 q^{41} -9.00000 q^{43} +1.00000 q^{44} -1.00000 q^{46} -6.00000 q^{47} -6.00000 q^{49} -1.00000 q^{52} +2.00000 q^{53} +3.00000 q^{56} +5.00000 q^{58} +8.00000 q^{59} -8.00000 q^{61} +2.00000 q^{62} -3.00000 q^{63} +7.00000 q^{64} +8.00000 q^{67} -10.0000 q^{71} -9.00000 q^{72} -3.00000 q^{73} +4.00000 q^{74} +5.00000 q^{76} -1.00000 q^{77} -3.00000 q^{79} +9.00000 q^{81} +5.00000 q^{82} +3.00000 q^{83} +9.00000 q^{86} -3.00000 q^{88} +10.0000 q^{89} +1.00000 q^{91} -1.00000 q^{92} +6.00000 q^{94} -2.00000 q^{97} +6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 3.00000 1.06066
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 3.00000 0.707107
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 5.00000 0.811107
\(39\) 0 0
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) −9.00000 −1.37249 −0.686244 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) 5.00000 0.656532
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 2.00000 0.254000
\(63\) −3.00000 −0.377964
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) −9.00000 −1.06066
\(73\) −3.00000 −0.351123 −0.175562 0.984468i \(-0.556174\pi\)
−0.175562 + 0.984468i \(0.556174\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −3.00000 −0.337526 −0.168763 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 5.00000 0.552158
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 9.00000 0.970495
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 6.00000 0.606092
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −17.0000 −1.67506 −0.837530 0.546392i \(-0.816001\pi\)
−0.837530 + 0.546392i \(0.816001\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.00000 0.464238
\(117\) −3.00000 −0.277350
\(118\) −8.00000 −0.736460
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 8.00000 0.724286
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) −5.00000 −0.433555
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.0000 0.839181
\(143\) −1.00000 −0.0836242
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) 3.00000 0.248282
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) −15.0000 −1.21666
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 3.00000 0.238667
\(159\) 0 0
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) −9.00000 −0.707107
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) 5.00000 0.390434
\(165\) 0 0
\(166\) −3.00000 −0.232845
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 15.0000 1.14708
\(172\) 9.00000 0.686244
\(173\) 15.0000 1.14043 0.570214 0.821496i \(-0.306860\pi\)
0.570214 + 0.821496i \(0.306860\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 0 0
\(184\) 3.00000 0.221163
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) 0 0
\(191\) 15.0000 1.08536 0.542681 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 6.00000 0.428571
\(197\) −17.0000 −1.21120 −0.605600 0.795769i \(-0.707067\pi\)
−0.605600 + 0.795769i \(0.707067\pi\)
\(198\) −3.00000 −0.213201
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −10.0000 −0.703598
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) 0 0
\(206\) 17.0000 1.18445
\(207\) −3.00000 −0.208514
\(208\) −1.00000 −0.0693375
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) −6.00000 −0.413057 −0.206529 0.978441i \(-0.566217\pi\)
−0.206529 + 0.978441i \(0.566217\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −5.00000 −0.334077
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −15.0000 −0.984798
\(233\) −5.00000 −0.327561 −0.163780 0.986497i \(-0.552369\pi\)
−0.163780 + 0.986497i \(0.552369\pi\)
\(234\) 3.00000 0.196116
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 0 0
\(238\) 0 0
\(239\) 10.0000 0.646846 0.323423 0.946254i \(-0.395166\pi\)
0.323423 + 0.946254i \(0.395166\pi\)
\(240\) 0 0
\(241\) −30.0000 −1.93247 −0.966235 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) 10.0000 0.642824
\(243\) 0 0
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 0 0
\(247\) −5.00000 −0.318142
\(248\) −6.00000 −0.381000
\(249\) 0 0
\(250\) 0 0
\(251\) −8.00000 −0.504956 −0.252478 0.967603i \(-0.581245\pi\)
−0.252478 + 0.967603i \(0.581245\pi\)
\(252\) 3.00000 0.188982
\(253\) −1.00000 −0.0628695
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 15.0000 0.928477
\(262\) 6.00000 0.370681
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 5.00000 0.306570
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 3.00000 0.182913 0.0914566 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 0 0
\(271\) −14.0000 −0.850439 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) −33.0000 −1.98278 −0.991389 0.130950i \(-0.958197\pi\)
−0.991389 + 0.130950i \(0.958197\pi\)
\(278\) −20.0000 −1.19952
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 32.0000 1.90220 0.951101 0.308879i \(-0.0999539\pi\)
0.951101 + 0.308879i \(0.0999539\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) 1.00000 0.0591312
\(287\) −5.00000 −0.295141
\(288\) 15.0000 0.883883
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 3.00000 0.175562
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −12.0000 −0.697486
\(297\) 0 0
\(298\) 2.00000 0.115857
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) −9.00000 −0.518751
\(302\) −6.00000 −0.345261
\(303\) 0 0
\(304\) 5.00000 0.286770
\(305\) 0 0
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) 0 0
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) 0 0
\(313\) −12.0000 −0.678280 −0.339140 0.940736i \(-0.610136\pi\)
−0.339140 + 0.940736i \(0.610136\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 3.00000 0.168763
\(317\) −33.0000 −1.85346 −0.926732 0.375722i \(-0.877395\pi\)
−0.926732 + 0.375722i \(0.877395\pi\)
\(318\) 0 0
\(319\) 5.00000 0.279946
\(320\) 0 0
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) 6.00000 0.332309
\(327\) 0 0
\(328\) −15.0000 −0.828236
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 14.0000 0.769510 0.384755 0.923019i \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(332\) −3.00000 −0.164646
\(333\) 12.0000 0.657596
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) −15.0000 −0.811107
\(343\) −13.0000 −0.701934
\(344\) −27.0000 −1.45574
\(345\) 0 0
\(346\) −15.0000 −0.806405
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 0 0
\(349\) −25.0000 −1.33822 −0.669110 0.743164i \(-0.733324\pi\)
−0.669110 + 0.743164i \(0.733324\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) −11.0000 −0.585471 −0.292735 0.956193i \(-0.594566\pi\)
−0.292735 + 0.956193i \(0.594566\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −6.00000 −0.317110
\(359\) 33.0000 1.74167 0.870837 0.491572i \(-0.163578\pi\)
0.870837 + 0.491572i \(0.163578\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 20.0000 1.05118
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) 0 0
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 15.0000 0.780869
\(370\) 0 0
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) 20.0000 1.03556 0.517780 0.855514i \(-0.326758\pi\)
0.517780 + 0.855514i \(0.326758\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −18.0000 −0.928279
\(377\) −5.00000 −0.257513
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −15.0000 −0.767467
\(383\) −19.0000 −0.970855 −0.485427 0.874277i \(-0.661336\pi\)
−0.485427 + 0.874277i \(0.661336\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 27.0000 1.37249
\(388\) 2.00000 0.101535
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −18.0000 −0.909137
\(393\) 0 0
\(394\) 17.0000 0.856448
\(395\) 0 0
\(396\) −3.00000 −0.150756
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 7.00000 0.350878
\(399\) 0 0
\(400\) 0 0
\(401\) 32.0000 1.59800 0.799002 0.601329i \(-0.205362\pi\)
0.799002 + 0.601329i \(0.205362\pi\)
\(402\) 0 0
\(403\) −2.00000 −0.0996271
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 5.00000 0.248146
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) 1.00000 0.0494468 0.0247234 0.999694i \(-0.492129\pi\)
0.0247234 + 0.999694i \(0.492129\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 17.0000 0.837530
\(413\) 8.00000 0.393654
\(414\) 3.00000 0.147442
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) −5.00000 −0.244558
\(419\) 9.00000 0.439679 0.219839 0.975536i \(-0.429447\pi\)
0.219839 + 0.975536i \(0.429447\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 6.00000 0.292075
\(423\) 18.0000 0.875190
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) 32.0000 1.53782 0.768911 0.639356i \(-0.220799\pi\)
0.768911 + 0.639356i \(0.220799\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) −5.00000 −0.239182
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) 0 0
\(443\) −14.0000 −0.665160 −0.332580 0.943075i \(-0.607919\pi\)
−0.332580 + 0.943075i \(0.607919\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) 0 0
\(448\) 7.00000 0.330719
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 5.00000 0.235441
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) 22.0000 1.02799
\(459\) 0 0
\(460\) 0 0
\(461\) 7.00000 0.326023 0.163011 0.986624i \(-0.447879\pi\)
0.163011 + 0.986624i \(0.447879\pi\)
\(462\) 0 0
\(463\) −28.0000 −1.30127 −0.650635 0.759390i \(-0.725497\pi\)
−0.650635 + 0.759390i \(0.725497\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) 5.00000 0.231621
\(467\) 33.0000 1.52706 0.763529 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) 3.00000 0.138675
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 24.0000 1.10469
\(473\) 9.00000 0.413820
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −10.0000 −0.457389
\(479\) −9.00000 −0.411220 −0.205610 0.978634i \(-0.565918\pi\)
−0.205610 + 0.978634i \(0.565918\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 30.0000 1.36646
\(483\) 0 0
\(484\) 10.0000 0.454545
\(485\) 0 0
\(486\) 0 0
\(487\) 14.0000 0.634401 0.317200 0.948359i \(-0.397257\pi\)
0.317200 + 0.948359i \(0.397257\pi\)
\(488\) −24.0000 −1.08643
\(489\) 0 0
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 5.00000 0.224961
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −10.0000 −0.448561
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 8.00000 0.357057
\(503\) −21.0000 −0.936344 −0.468172 0.883637i \(-0.655087\pi\)
−0.468172 + 0.883637i \(0.655087\pi\)
\(504\) −9.00000 −0.400892
\(505\) 0 0
\(506\) 1.00000 0.0444554
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) −3.00000 −0.132712
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) 4.00000 0.175750
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −15.0000 −0.656532
\(523\) 5.00000 0.218635 0.109317 0.994007i \(-0.465134\pi\)
0.109317 + 0.994007i \(0.465134\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −24.0000 −1.04151
\(532\) 5.00000 0.216777
\(533\) −5.00000 −0.216574
\(534\) 0 0
\(535\) 0 0
\(536\) 24.0000 1.03664
\(537\) 0 0
\(538\) −3.00000 −0.129339
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 31.0000 1.33279 0.666397 0.745597i \(-0.267836\pi\)
0.666397 + 0.745597i \(0.267836\pi\)
\(542\) 14.0000 0.601351
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −18.0000 −0.769624 −0.384812 0.922995i \(-0.625734\pi\)
−0.384812 + 0.922995i \(0.625734\pi\)
\(548\) −18.0000 −0.768922
\(549\) 24.0000 1.02430
\(550\) 0 0
\(551\) 25.0000 1.06504
\(552\) 0 0
\(553\) −3.00000 −0.127573
\(554\) 33.0000 1.40204
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) −16.0000 −0.677942 −0.338971 0.940797i \(-0.610079\pi\)
−0.338971 + 0.940797i \(0.610079\pi\)
\(558\) −6.00000 −0.254000
\(559\) −9.00000 −0.380659
\(560\) 0 0
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) −41.0000 −1.72794 −0.863972 0.503540i \(-0.832031\pi\)
−0.863972 + 0.503540i \(0.832031\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −32.0000 −1.34506
\(567\) 9.00000 0.377964
\(568\) −30.0000 −1.25877
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 1.00000 0.0418121
\(573\) 0 0
\(574\) 5.00000 0.208696
\(575\) 0 0
\(576\) −21.0000 −0.875000
\(577\) −31.0000 −1.29055 −0.645273 0.763952i \(-0.723257\pi\)
−0.645273 + 0.763952i \(0.723257\pi\)
\(578\) 17.0000 0.707107
\(579\) 0 0
\(580\) 0 0
\(581\) 3.00000 0.124461
\(582\) 0 0
\(583\) −2.00000 −0.0828315
\(584\) −9.00000 −0.372423
\(585\) 0 0
\(586\) −24.0000 −0.991431
\(587\) 8.00000 0.330195 0.165098 0.986277i \(-0.447206\pi\)
0.165098 + 0.986277i \(0.447206\pi\)
\(588\) 0 0
\(589\) 10.0000 0.412043
\(590\) 0 0
\(591\) 0 0
\(592\) 4.00000 0.164399
\(593\) −25.0000 −1.02663 −0.513313 0.858201i \(-0.671582\pi\)
−0.513313 + 0.858201i \(0.671582\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.00000 0.0819232
\(597\) 0 0
\(598\) −1.00000 −0.0408930
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −30.0000 −1.22373 −0.611863 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 9.00000 0.366813
\(603\) −24.0000 −0.977356
\(604\) −6.00000 −0.244137
\(605\) 0 0
\(606\) 0 0
\(607\) 2.00000 0.0811775 0.0405887 0.999176i \(-0.487077\pi\)
0.0405887 + 0.999176i \(0.487077\pi\)
\(608\) 25.0000 1.01388
\(609\) 0 0
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 28.0000 1.13091 0.565455 0.824779i \(-0.308701\pi\)
0.565455 + 0.824779i \(0.308701\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) 28.0000 1.12724 0.563619 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) 0 0
\(619\) 36.0000 1.44696 0.723481 0.690344i \(-0.242541\pi\)
0.723481 + 0.690344i \(0.242541\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −28.0000 −1.12270
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) 0 0
\(626\) 12.0000 0.479616
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 0 0
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) −9.00000 −0.358001
\(633\) 0 0
\(634\) 33.0000 1.31060
\(635\) 0 0
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) −5.00000 −0.197952
\(639\) 30.0000 1.18678
\(640\) 0 0
\(641\) −16.0000 −0.631962 −0.315981 0.948766i \(-0.602334\pi\)
−0.315981 + 0.948766i \(0.602334\pi\)
\(642\) 0 0
\(643\) 31.0000 1.22252 0.611260 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) 0 0
\(647\) −38.0000 −1.49393 −0.746967 0.664861i \(-0.768491\pi\)
−0.746967 + 0.664861i \(0.768491\pi\)
\(648\) 27.0000 1.06066
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 6.00000 0.234978
\(653\) −27.0000 −1.05659 −0.528296 0.849060i \(-0.677169\pi\)
−0.528296 + 0.849060i \(0.677169\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 5.00000 0.195217
\(657\) 9.00000 0.351123
\(658\) 6.00000 0.233904
\(659\) −3.00000 −0.116863 −0.0584317 0.998291i \(-0.518610\pi\)
−0.0584317 + 0.998291i \(0.518610\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −14.0000 −0.544125
\(663\) 0 0
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) −12.0000 −0.464991
\(667\) −5.00000 −0.193601
\(668\) −18.0000 −0.696441
\(669\) 0 0
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) −12.0000 −0.461197 −0.230599 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) −2.00000 −0.0765840
\(683\) 2.00000 0.0765279 0.0382639 0.999268i \(-0.487817\pi\)
0.0382639 + 0.999268i \(0.487817\pi\)
\(684\) −15.0000 −0.573539
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) 0 0
\(688\) 9.00000 0.343122
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −15.0000 −0.570214
\(693\) 3.00000 0.113961
\(694\) 24.0000 0.911028
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 25.0000 0.946264
\(699\) 0 0
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 0 0
\(703\) 20.0000 0.754314
\(704\) −7.00000 −0.263822
\(705\) 0 0
\(706\) 11.0000 0.413990
\(707\) 10.0000 0.376089
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) 9.00000 0.337526
\(712\) 30.0000 1.12430
\(713\) −2.00000 −0.0749006
\(714\) 0 0
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) 0 0
\(718\) −33.0000 −1.23155
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) −17.0000 −0.633113
\(722\) −6.00000 −0.223297
\(723\) 0 0
\(724\) 20.0000 0.743294
\(725\) 0 0
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 3.00000 0.111187
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −17.0000 −0.627481
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) −8.00000 −0.294684
\(738\) −15.0000 −0.552158
\(739\) −22.0000 −0.809283 −0.404642 0.914475i \(-0.632604\pi\)
−0.404642 + 0.914475i \(0.632604\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.00000 −0.0734223
\(743\) −11.0000 −0.403551 −0.201775 0.979432i \(-0.564671\pi\)
−0.201775 + 0.979432i \(0.564671\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −20.0000 −0.732252
\(747\) −9.00000 −0.329293
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 23.0000 0.839282 0.419641 0.907690i \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 5.00000 0.182089
\(755\) 0 0
\(756\) 0 0
\(757\) 40.0000 1.45382 0.726912 0.686730i \(-0.240955\pi\)
0.726912 + 0.686730i \(0.240955\pi\)
\(758\) 12.0000 0.435860
\(759\) 0 0
\(760\) 0 0
\(761\) −39.0000 −1.41375 −0.706874 0.707339i \(-0.749895\pi\)
−0.706874 + 0.707339i \(0.749895\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) −15.0000 −0.542681
\(765\) 0 0
\(766\) 19.0000 0.686498
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −52.0000 −1.87517 −0.937584 0.347759i \(-0.886943\pi\)
−0.937584 + 0.347759i \(0.886943\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) 20.0000 0.719350 0.359675 0.933078i \(-0.382888\pi\)
0.359675 + 0.933078i \(0.382888\pi\)
\(774\) −27.0000 −0.970495
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) 14.0000 0.501924
\(779\) 25.0000 0.895718
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) 0 0
\(783\) 0 0
\(784\) 6.00000 0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) −43.0000 −1.53278 −0.766392 0.642373i \(-0.777950\pi\)
−0.766392 + 0.642373i \(0.777950\pi\)
\(788\) 17.0000 0.605600
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 9.00000 0.319801
\(793\) −8.00000 −0.284088
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) 7.00000 0.248108
\(797\) 8.00000 0.283375 0.141687 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) −32.0000 −1.12996
\(803\) 3.00000 0.105868
\(804\) 0 0
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) 0 0
\(808\) 30.0000 1.05540
\(809\) −15.0000 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 5.00000 0.175466
\(813\) 0 0
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) 0 0
\(817\) 45.0000 1.57435
\(818\) −1.00000 −0.0349642
\(819\) −3.00000 −0.104828
\(820\) 0 0
\(821\) 1.00000 0.0349002 0.0174501 0.999848i \(-0.494445\pi\)
0.0174501 + 0.999848i \(0.494445\pi\)
\(822\) 0 0
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) −51.0000 −1.77667
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 37.0000 1.28662 0.643308 0.765607i \(-0.277561\pi\)
0.643308 + 0.765607i \(0.277561\pi\)
\(828\) 3.00000 0.104257
\(829\) 15.0000 0.520972 0.260486 0.965478i \(-0.416117\pi\)
0.260486 + 0.965478i \(0.416117\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 7.00000 0.242681
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −5.00000 −0.172929
\(837\) 0 0
\(838\) −9.00000 −0.310900
\(839\) −9.00000 −0.310715 −0.155357 0.987858i \(-0.549653\pi\)
−0.155357 + 0.987858i \(0.549653\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −22.0000 −0.758170
\(843\) 0 0
\(844\) 6.00000 0.206529
\(845\) 0 0
\(846\) −18.0000 −0.618853
\(847\) −10.0000 −0.343604
\(848\) −2.00000 −0.0686803
\(849\) 0 0
\(850\) 0 0
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) 37.0000 1.26686 0.633428 0.773802i \(-0.281647\pi\)
0.633428 + 0.773802i \(0.281647\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) −26.0000 −0.888143 −0.444072 0.895991i \(-0.646466\pi\)
−0.444072 + 0.895991i \(0.646466\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8.00000 0.272481
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −32.0000 −1.08740
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) 3.00000 0.101768
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −12.0000 −0.406371
\(873\) 6.00000 0.203069
\(874\) 5.00000 0.169128
\(875\) 0 0
\(876\) 0 0
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) −20.0000 −0.674967
\(879\) 0 0
\(880\) 0 0
\(881\) −20.0000 −0.673817 −0.336909 0.941537i \(-0.609381\pi\)
−0.336909 + 0.941537i \(0.609381\pi\)
\(882\) −18.0000 −0.606092
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 14.0000 0.470339
\(887\) 28.0000 0.940148 0.470074 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) −2.00000 −0.0669650
\(893\) 30.0000 1.00391
\(894\) 0 0
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) 10.0000 0.333519
\(900\) 0 0
\(901\) 0 0
\(902\) −5.00000 −0.166482
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) 0 0
\(907\) 17.0000 0.564476 0.282238 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(908\) 20.0000 0.663723
\(909\) −30.0000 −0.995037
\(910\) 0 0
\(911\) −55.0000 −1.82223 −0.911116 0.412151i \(-0.864778\pi\)
−0.911116 + 0.412151i \(0.864778\pi\)
\(912\) 0 0
\(913\) −3.00000 −0.0992855
\(914\) 38.0000 1.25693
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −7.00000 −0.230533
\(923\) −10.0000 −0.329154
\(924\) 0 0
\(925\) 0 0
\(926\) 28.0000 0.920137
\(927\) 51.0000 1.67506
\(928\) 25.0000 0.820665
\(929\) −45.0000 −1.47640 −0.738201 0.674581i \(-0.764324\pi\)
−0.738201 + 0.674581i \(0.764324\pi\)
\(930\) 0 0
\(931\) 30.0000 0.983210
\(932\) 5.00000 0.163780
\(933\) 0 0
\(934\) −33.0000 −1.07979
\(935\) 0 0
\(936\) −9.00000 −0.294174
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) −5.00000 −0.162822
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) −9.00000 −0.292615
\(947\) −22.0000 −0.714904 −0.357452 0.933932i \(-0.616354\pi\)
−0.357452 + 0.933932i \(0.616354\pi\)
\(948\) 0 0
\(949\) −3.00000 −0.0973841
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −28.0000 −0.907009 −0.453504 0.891254i \(-0.649826\pi\)
−0.453504 + 0.891254i \(0.649826\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −10.0000 −0.323423
\(957\) 0 0
\(958\) 9.00000 0.290777
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 4.00000 0.128965
\(963\) −36.0000 −1.16008
\(964\) 30.0000 0.966235
\(965\) 0 0
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) −30.0000 −0.964237
\(969\) 0 0
\(970\) 0 0
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) 20.0000 0.641171
\(974\) −14.0000 −0.448589
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) −24.0000 −0.765871
\(983\) 39.0000 1.24391 0.621953 0.783054i \(-0.286339\pi\)
0.621953 + 0.783054i \(0.286339\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 5.00000 0.159071
\(989\) −9.00000 −0.286183
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) 10.0000 0.317500
\(993\) 0 0
\(994\) 10.0000 0.317181
\(995\) 0 0
\(996\) 0 0
\(997\) −37.0000 −1.17180 −0.585901 0.810383i \(-0.699259\pi\)
−0.585901 + 0.810383i \(0.699259\pi\)
\(998\) −10.0000 −0.316544
\(999\) 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 575.2.a.c.1.1 1
3.2 odd 2 5175.2.a.u.1.1 1
4.3 odd 2 9200.2.a.r.1.1 1
5.2 odd 4 575.2.b.b.24.1 2
5.3 odd 4 575.2.b.b.24.2 2
5.4 even 2 575.2.a.d.1.1 yes 1
15.14 odd 2 5175.2.a.e.1.1 1
20.19 odd 2 9200.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
575.2.a.c.1.1 1 1.1 even 1 trivial
575.2.a.d.1.1 yes 1 5.4 even 2
575.2.b.b.24.1 2 5.2 odd 4
575.2.b.b.24.2 2 5.3 odd 4
5175.2.a.e.1.1 1 15.14 odd 2
5175.2.a.u.1.1 1 3.2 odd 2
9200.2.a.r.1.1 1 4.3 odd 2
9200.2.a.u.1.1 1 20.19 odd 2