Properties

Label 7728.2.a.bm.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.12311 q^{11} -3.56155 q^{13} -1.56155 q^{15} -1.12311 q^{17} +5.12311 q^{19} +1.00000 q^{21} -1.00000 q^{23} -2.56155 q^{25} +1.00000 q^{27} +7.56155 q^{29} -3.12311 q^{31} -5.12311 q^{33} -1.56155 q^{35} -1.56155 q^{37} -3.56155 q^{39} +3.56155 q^{41} -6.68466 q^{43} -1.56155 q^{45} -2.43845 q^{47} +1.00000 q^{49} -1.12311 q^{51} +14.2462 q^{53} +8.00000 q^{55} +5.12311 q^{57} -4.87689 q^{59} +0.876894 q^{61} +1.00000 q^{63} +5.56155 q^{65} +1.12311 q^{67} -1.00000 q^{69} +9.36932 q^{71} +9.12311 q^{73} -2.56155 q^{75} -5.12311 q^{77} -14.2462 q^{79} +1.00000 q^{81} +9.12311 q^{83} +1.75379 q^{85} +7.56155 q^{87} +14.0000 q^{89} -3.56155 q^{91} -3.12311 q^{93} -8.00000 q^{95} -12.4384 q^{97} -5.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + q^{5} + 2q^{7} + 2q^{9} - 2q^{11} - 3q^{13} + q^{15} + 6q^{17} + 2q^{19} + 2q^{21} - 2q^{23} - q^{25} + 2q^{27} + 11q^{29} + 2q^{31} - 2q^{33} + q^{35} + q^{37} - 3q^{39} + 3q^{41} - q^{43} + q^{45} - 9q^{47} + 2q^{49} + 6q^{51} + 12q^{53} + 16q^{55} + 2q^{57} - 18q^{59} + 10q^{61} + 2q^{63} + 7q^{65} - 6q^{67} - 2q^{69} - 6q^{71} + 10q^{73} - q^{75} - 2q^{77} - 12q^{79} + 2q^{81} + 10q^{83} + 20q^{85} + 11q^{87} + 28q^{89} - 3q^{91} + 2q^{93} - 16q^{95} - 29q^{97} - 2q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.56155 −0.698348 −0.349174 0.937058i \(-0.613538\pi\)
−0.349174 + 0.937058i \(0.613538\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.12311 −1.54467 −0.772337 0.635213i \(-0.780912\pi\)
−0.772337 + 0.635213i \(0.780912\pi\)
\(12\) 0 0
\(13\) −3.56155 −0.987797 −0.493899 0.869520i \(-0.664429\pi\)
−0.493899 + 0.869520i \(0.664429\pi\)
\(14\) 0 0
\(15\) −1.56155 −0.403191
\(16\) 0 0
\(17\) −1.12311 −0.272393 −0.136197 0.990682i \(-0.543488\pi\)
−0.136197 + 0.990682i \(0.543488\pi\)
\(18\) 0 0
\(19\) 5.12311 1.17532 0.587661 0.809108i \(-0.300049\pi\)
0.587661 + 0.809108i \(0.300049\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −2.56155 −0.512311
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 7.56155 1.40415 0.702073 0.712105i \(-0.252258\pi\)
0.702073 + 0.712105i \(0.252258\pi\)
\(30\) 0 0
\(31\) −3.12311 −0.560926 −0.280463 0.959865i \(-0.590488\pi\)
−0.280463 + 0.959865i \(0.590488\pi\)
\(32\) 0 0
\(33\) −5.12311 −0.891818
\(34\) 0 0
\(35\) −1.56155 −0.263951
\(36\) 0 0
\(37\) −1.56155 −0.256718 −0.128359 0.991728i \(-0.540971\pi\)
−0.128359 + 0.991728i \(0.540971\pi\)
\(38\) 0 0
\(39\) −3.56155 −0.570305
\(40\) 0 0
\(41\) 3.56155 0.556221 0.278111 0.960549i \(-0.410292\pi\)
0.278111 + 0.960549i \(0.410292\pi\)
\(42\) 0 0
\(43\) −6.68466 −1.01940 −0.509700 0.860352i \(-0.670244\pi\)
−0.509700 + 0.860352i \(0.670244\pi\)
\(44\) 0 0
\(45\) −1.56155 −0.232783
\(46\) 0 0
\(47\) −2.43845 −0.355684 −0.177842 0.984059i \(-0.556912\pi\)
−0.177842 + 0.984059i \(0.556912\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.12311 −0.157266
\(52\) 0 0
\(53\) 14.2462 1.95687 0.978434 0.206561i \(-0.0662271\pi\)
0.978434 + 0.206561i \(0.0662271\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 5.12311 0.678572
\(58\) 0 0
\(59\) −4.87689 −0.634918 −0.317459 0.948272i \(-0.602830\pi\)
−0.317459 + 0.948272i \(0.602830\pi\)
\(60\) 0 0
\(61\) 0.876894 0.112275 0.0561374 0.998423i \(-0.482122\pi\)
0.0561374 + 0.998423i \(0.482122\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 5.56155 0.689826
\(66\) 0 0
\(67\) 1.12311 0.137209 0.0686046 0.997644i \(-0.478145\pi\)
0.0686046 + 0.997644i \(0.478145\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 9.36932 1.11193 0.555967 0.831205i \(-0.312348\pi\)
0.555967 + 0.831205i \(0.312348\pi\)
\(72\) 0 0
\(73\) 9.12311 1.06778 0.533889 0.845554i \(-0.320730\pi\)
0.533889 + 0.845554i \(0.320730\pi\)
\(74\) 0 0
\(75\) −2.56155 −0.295783
\(76\) 0 0
\(77\) −5.12311 −0.583832
\(78\) 0 0
\(79\) −14.2462 −1.60282 −0.801412 0.598113i \(-0.795918\pi\)
−0.801412 + 0.598113i \(0.795918\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.12311 1.00139 0.500695 0.865624i \(-0.333078\pi\)
0.500695 + 0.865624i \(0.333078\pi\)
\(84\) 0 0
\(85\) 1.75379 0.190225
\(86\) 0 0
\(87\) 7.56155 0.810684
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −3.56155 −0.373352
\(92\) 0 0
\(93\) −3.12311 −0.323851
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −12.4384 −1.26293 −0.631466 0.775403i \(-0.717547\pi\)
−0.631466 + 0.775403i \(0.717547\pi\)
\(98\) 0 0
\(99\) −5.12311 −0.514891
\(100\) 0 0
\(101\) 16.2462 1.61656 0.808279 0.588799i \(-0.200399\pi\)
0.808279 + 0.588799i \(0.200399\pi\)
\(102\) 0 0
\(103\) −18.9309 −1.86531 −0.932657 0.360764i \(-0.882516\pi\)
−0.932657 + 0.360764i \(0.882516\pi\)
\(104\) 0 0
\(105\) −1.56155 −0.152392
\(106\) 0 0
\(107\) 5.12311 0.495269 0.247635 0.968853i \(-0.420347\pi\)
0.247635 + 0.968853i \(0.420347\pi\)
\(108\) 0 0
\(109\) 10.4384 0.999822 0.499911 0.866077i \(-0.333366\pi\)
0.499911 + 0.866077i \(0.333366\pi\)
\(110\) 0 0
\(111\) −1.56155 −0.148216
\(112\) 0 0
\(113\) −2.68466 −0.252551 −0.126276 0.991995i \(-0.540302\pi\)
−0.126276 + 0.991995i \(0.540302\pi\)
\(114\) 0 0
\(115\) 1.56155 0.145616
\(116\) 0 0
\(117\) −3.56155 −0.329266
\(118\) 0 0
\(119\) −1.12311 −0.102955
\(120\) 0 0
\(121\) 15.2462 1.38602
\(122\) 0 0
\(123\) 3.56155 0.321134
\(124\) 0 0
\(125\) 11.8078 1.05612
\(126\) 0 0
\(127\) 16.6847 1.48052 0.740262 0.672318i \(-0.234701\pi\)
0.740262 + 0.672318i \(0.234701\pi\)
\(128\) 0 0
\(129\) −6.68466 −0.588551
\(130\) 0 0
\(131\) −4.87689 −0.426096 −0.213048 0.977042i \(-0.568339\pi\)
−0.213048 + 0.977042i \(0.568339\pi\)
\(132\) 0 0
\(133\) 5.12311 0.444230
\(134\) 0 0
\(135\) −1.56155 −0.134397
\(136\) 0 0
\(137\) −0.438447 −0.0374591 −0.0187295 0.999825i \(-0.505962\pi\)
−0.0187295 + 0.999825i \(0.505962\pi\)
\(138\) 0 0
\(139\) −3.80776 −0.322970 −0.161485 0.986875i \(-0.551628\pi\)
−0.161485 + 0.986875i \(0.551628\pi\)
\(140\) 0 0
\(141\) −2.43845 −0.205354
\(142\) 0 0
\(143\) 18.2462 1.52582
\(144\) 0 0
\(145\) −11.8078 −0.980581
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −6.24621 −0.511710 −0.255855 0.966715i \(-0.582357\pi\)
−0.255855 + 0.966715i \(0.582357\pi\)
\(150\) 0 0
\(151\) −7.31534 −0.595314 −0.297657 0.954673i \(-0.596205\pi\)
−0.297657 + 0.954673i \(0.596205\pi\)
\(152\) 0 0
\(153\) −1.12311 −0.0907977
\(154\) 0 0
\(155\) 4.87689 0.391722
\(156\) 0 0
\(157\) 14.2462 1.13697 0.568486 0.822693i \(-0.307529\pi\)
0.568486 + 0.822693i \(0.307529\pi\)
\(158\) 0 0
\(159\) 14.2462 1.12980
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 0 0
\(165\) 8.00000 0.622799
\(166\) 0 0
\(167\) 14.2462 1.10240 0.551202 0.834372i \(-0.314169\pi\)
0.551202 + 0.834372i \(0.314169\pi\)
\(168\) 0 0
\(169\) −0.315342 −0.0242570
\(170\) 0 0
\(171\) 5.12311 0.391774
\(172\) 0 0
\(173\) 18.4924 1.40595 0.702976 0.711213i \(-0.251854\pi\)
0.702976 + 0.711213i \(0.251854\pi\)
\(174\) 0 0
\(175\) −2.56155 −0.193635
\(176\) 0 0
\(177\) −4.87689 −0.366570
\(178\) 0 0
\(179\) 3.80776 0.284606 0.142303 0.989823i \(-0.454549\pi\)
0.142303 + 0.989823i \(0.454549\pi\)
\(180\) 0 0
\(181\) −15.1231 −1.12409 −0.562046 0.827106i \(-0.689986\pi\)
−0.562046 + 0.827106i \(0.689986\pi\)
\(182\) 0 0
\(183\) 0.876894 0.0648219
\(184\) 0 0
\(185\) 2.43845 0.179278
\(186\) 0 0
\(187\) 5.75379 0.420759
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) 2.68466 0.193246 0.0966230 0.995321i \(-0.469196\pi\)
0.0966230 + 0.995321i \(0.469196\pi\)
\(194\) 0 0
\(195\) 5.56155 0.398271
\(196\) 0 0
\(197\) 24.9309 1.77625 0.888125 0.459601i \(-0.152008\pi\)
0.888125 + 0.459601i \(0.152008\pi\)
\(198\) 0 0
\(199\) 21.1771 1.50120 0.750602 0.660755i \(-0.229764\pi\)
0.750602 + 0.660755i \(0.229764\pi\)
\(200\) 0 0
\(201\) 1.12311 0.0792178
\(202\) 0 0
\(203\) 7.56155 0.530717
\(204\) 0 0
\(205\) −5.56155 −0.388436
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −26.2462 −1.81549
\(210\) 0 0
\(211\) 16.4924 1.13539 0.567693 0.823241i \(-0.307836\pi\)
0.567693 + 0.823241i \(0.307836\pi\)
\(212\) 0 0
\(213\) 9.36932 0.641975
\(214\) 0 0
\(215\) 10.4384 0.711896
\(216\) 0 0
\(217\) −3.12311 −0.212010
\(218\) 0 0
\(219\) 9.12311 0.616482
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) −3.12311 −0.209139 −0.104569 0.994518i \(-0.533346\pi\)
−0.104569 + 0.994518i \(0.533346\pi\)
\(224\) 0 0
\(225\) −2.56155 −0.170770
\(226\) 0 0
\(227\) 6.19224 0.410993 0.205497 0.978658i \(-0.434119\pi\)
0.205497 + 0.978658i \(0.434119\pi\)
\(228\) 0 0
\(229\) 2.24621 0.148434 0.0742169 0.997242i \(-0.476354\pi\)
0.0742169 + 0.997242i \(0.476354\pi\)
\(230\) 0 0
\(231\) −5.12311 −0.337076
\(232\) 0 0
\(233\) −18.4924 −1.21148 −0.605739 0.795663i \(-0.707123\pi\)
−0.605739 + 0.795663i \(0.707123\pi\)
\(234\) 0 0
\(235\) 3.80776 0.248391
\(236\) 0 0
\(237\) −14.2462 −0.925391
\(238\) 0 0
\(239\) −4.49242 −0.290591 −0.145295 0.989388i \(-0.546413\pi\)
−0.145295 + 0.989388i \(0.546413\pi\)
\(240\) 0 0
\(241\) −20.0540 −1.29179 −0.645895 0.763426i \(-0.723516\pi\)
−0.645895 + 0.763426i \(0.723516\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −1.56155 −0.0997639
\(246\) 0 0
\(247\) −18.2462 −1.16098
\(248\) 0 0
\(249\) 9.12311 0.578153
\(250\) 0 0
\(251\) 0.438447 0.0276745 0.0138373 0.999904i \(-0.495595\pi\)
0.0138373 + 0.999904i \(0.495595\pi\)
\(252\) 0 0
\(253\) 5.12311 0.322087
\(254\) 0 0
\(255\) 1.75379 0.109827
\(256\) 0 0
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) −1.56155 −0.0970302
\(260\) 0 0
\(261\) 7.56155 0.468048
\(262\) 0 0
\(263\) −22.9309 −1.41398 −0.706989 0.707225i \(-0.749947\pi\)
−0.706989 + 0.707225i \(0.749947\pi\)
\(264\) 0 0
\(265\) −22.2462 −1.36657
\(266\) 0 0
\(267\) 14.0000 0.856786
\(268\) 0 0
\(269\) 12.2462 0.746665 0.373332 0.927698i \(-0.378215\pi\)
0.373332 + 0.927698i \(0.378215\pi\)
\(270\) 0 0
\(271\) −14.2462 −0.865396 −0.432698 0.901539i \(-0.642438\pi\)
−0.432698 + 0.901539i \(0.642438\pi\)
\(272\) 0 0
\(273\) −3.56155 −0.215555
\(274\) 0 0
\(275\) 13.1231 0.791353
\(276\) 0 0
\(277\) 13.1231 0.788491 0.394245 0.919005i \(-0.371006\pi\)
0.394245 + 0.919005i \(0.371006\pi\)
\(278\) 0 0
\(279\) −3.12311 −0.186975
\(280\) 0 0
\(281\) 4.93087 0.294151 0.147076 0.989125i \(-0.453014\pi\)
0.147076 + 0.989125i \(0.453014\pi\)
\(282\) 0 0
\(283\) 10.4924 0.623710 0.311855 0.950130i \(-0.399050\pi\)
0.311855 + 0.950130i \(0.399050\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) 3.56155 0.210232
\(288\) 0 0
\(289\) −15.7386 −0.925802
\(290\) 0 0
\(291\) −12.4384 −0.729155
\(292\) 0 0
\(293\) 27.6155 1.61332 0.806658 0.591018i \(-0.201274\pi\)
0.806658 + 0.591018i \(0.201274\pi\)
\(294\) 0 0
\(295\) 7.61553 0.443393
\(296\) 0 0
\(297\) −5.12311 −0.297273
\(298\) 0 0
\(299\) 3.56155 0.205970
\(300\) 0 0
\(301\) −6.68466 −0.385297
\(302\) 0 0
\(303\) 16.2462 0.933320
\(304\) 0 0
\(305\) −1.36932 −0.0784069
\(306\) 0 0
\(307\) −2.93087 −0.167274 −0.0836368 0.996496i \(-0.526654\pi\)
−0.0836368 + 0.996496i \(0.526654\pi\)
\(308\) 0 0
\(309\) −18.9309 −1.07694
\(310\) 0 0
\(311\) −18.7386 −1.06257 −0.531285 0.847193i \(-0.678291\pi\)
−0.531285 + 0.847193i \(0.678291\pi\)
\(312\) 0 0
\(313\) 4.24621 0.240010 0.120005 0.992773i \(-0.461709\pi\)
0.120005 + 0.992773i \(0.461709\pi\)
\(314\) 0 0
\(315\) −1.56155 −0.0879835
\(316\) 0 0
\(317\) −17.8078 −1.00018 −0.500092 0.865972i \(-0.666700\pi\)
−0.500092 + 0.865972i \(0.666700\pi\)
\(318\) 0 0
\(319\) −38.7386 −2.16895
\(320\) 0 0
\(321\) 5.12311 0.285944
\(322\) 0 0
\(323\) −5.75379 −0.320149
\(324\) 0 0
\(325\) 9.12311 0.506059
\(326\) 0 0
\(327\) 10.4384 0.577247
\(328\) 0 0
\(329\) −2.43845 −0.134436
\(330\) 0 0
\(331\) 24.4924 1.34623 0.673113 0.739540i \(-0.264957\pi\)
0.673113 + 0.739540i \(0.264957\pi\)
\(332\) 0 0
\(333\) −1.56155 −0.0855726
\(334\) 0 0
\(335\) −1.75379 −0.0958197
\(336\) 0 0
\(337\) 27.3693 1.49090 0.745451 0.666561i \(-0.232234\pi\)
0.745451 + 0.666561i \(0.232234\pi\)
\(338\) 0 0
\(339\) −2.68466 −0.145811
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 1.56155 0.0840712
\(346\) 0 0
\(347\) −14.9309 −0.801531 −0.400766 0.916181i \(-0.631256\pi\)
−0.400766 + 0.916181i \(0.631256\pi\)
\(348\) 0 0
\(349\) −16.2462 −0.869640 −0.434820 0.900517i \(-0.643188\pi\)
−0.434820 + 0.900517i \(0.643188\pi\)
\(350\) 0 0
\(351\) −3.56155 −0.190102
\(352\) 0 0
\(353\) −33.8078 −1.79941 −0.899703 0.436503i \(-0.856217\pi\)
−0.899703 + 0.436503i \(0.856217\pi\)
\(354\) 0 0
\(355\) −14.6307 −0.776516
\(356\) 0 0
\(357\) −1.12311 −0.0594411
\(358\) 0 0
\(359\) 6.43845 0.339808 0.169904 0.985461i \(-0.445654\pi\)
0.169904 + 0.985461i \(0.445654\pi\)
\(360\) 0 0
\(361\) 7.24621 0.381380
\(362\) 0 0
\(363\) 15.2462 0.800219
\(364\) 0 0
\(365\) −14.2462 −0.745681
\(366\) 0 0
\(367\) 18.4384 0.962479 0.481240 0.876589i \(-0.340187\pi\)
0.481240 + 0.876589i \(0.340187\pi\)
\(368\) 0 0
\(369\) 3.56155 0.185407
\(370\) 0 0
\(371\) 14.2462 0.739626
\(372\) 0 0
\(373\) 15.1231 0.783045 0.391522 0.920169i \(-0.371949\pi\)
0.391522 + 0.920169i \(0.371949\pi\)
\(374\) 0 0
\(375\) 11.8078 0.609750
\(376\) 0 0
\(377\) −26.9309 −1.38701
\(378\) 0 0
\(379\) −20.0540 −1.03010 −0.515052 0.857159i \(-0.672227\pi\)
−0.515052 + 0.857159i \(0.672227\pi\)
\(380\) 0 0
\(381\) 16.6847 0.854781
\(382\) 0 0
\(383\) 6.24621 0.319166 0.159583 0.987184i \(-0.448985\pi\)
0.159583 + 0.987184i \(0.448985\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) 0 0
\(387\) −6.68466 −0.339800
\(388\) 0 0
\(389\) −29.3693 −1.48908 −0.744542 0.667576i \(-0.767332\pi\)
−0.744542 + 0.667576i \(0.767332\pi\)
\(390\) 0 0
\(391\) 1.12311 0.0567979
\(392\) 0 0
\(393\) −4.87689 −0.246007
\(394\) 0 0
\(395\) 22.2462 1.11933
\(396\) 0 0
\(397\) −7.75379 −0.389152 −0.194576 0.980887i \(-0.562333\pi\)
−0.194576 + 0.980887i \(0.562333\pi\)
\(398\) 0 0
\(399\) 5.12311 0.256476
\(400\) 0 0
\(401\) 16.2462 0.811297 0.405649 0.914029i \(-0.367046\pi\)
0.405649 + 0.914029i \(0.367046\pi\)
\(402\) 0 0
\(403\) 11.1231 0.554081
\(404\) 0 0
\(405\) −1.56155 −0.0775942
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 33.6155 1.66218 0.831090 0.556137i \(-0.187717\pi\)
0.831090 + 0.556137i \(0.187717\pi\)
\(410\) 0 0
\(411\) −0.438447 −0.0216270
\(412\) 0 0
\(413\) −4.87689 −0.239976
\(414\) 0 0
\(415\) −14.2462 −0.699319
\(416\) 0 0
\(417\) −3.80776 −0.186467
\(418\) 0 0
\(419\) −23.8617 −1.16572 −0.582861 0.812572i \(-0.698067\pi\)
−0.582861 + 0.812572i \(0.698067\pi\)
\(420\) 0 0
\(421\) 33.1771 1.61695 0.808476 0.588529i \(-0.200293\pi\)
0.808476 + 0.588529i \(0.200293\pi\)
\(422\) 0 0
\(423\) −2.43845 −0.118561
\(424\) 0 0
\(425\) 2.87689 0.139550
\(426\) 0 0
\(427\) 0.876894 0.0424359
\(428\) 0 0
\(429\) 18.2462 0.880935
\(430\) 0 0
\(431\) 3.31534 0.159694 0.0798472 0.996807i \(-0.474557\pi\)
0.0798472 + 0.996807i \(0.474557\pi\)
\(432\) 0 0
\(433\) 14.6847 0.705700 0.352850 0.935680i \(-0.385213\pi\)
0.352850 + 0.935680i \(0.385213\pi\)
\(434\) 0 0
\(435\) −11.8078 −0.566139
\(436\) 0 0
\(437\) −5.12311 −0.245071
\(438\) 0 0
\(439\) 23.6155 1.12711 0.563554 0.826079i \(-0.309434\pi\)
0.563554 + 0.826079i \(0.309434\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −9.06913 −0.430887 −0.215444 0.976516i \(-0.569120\pi\)
−0.215444 + 0.976516i \(0.569120\pi\)
\(444\) 0 0
\(445\) −21.8617 −1.03635
\(446\) 0 0
\(447\) −6.24621 −0.295436
\(448\) 0 0
\(449\) −0.246211 −0.0116194 −0.00580971 0.999983i \(-0.501849\pi\)
−0.00580971 + 0.999983i \(0.501849\pi\)
\(450\) 0 0
\(451\) −18.2462 −0.859181
\(452\) 0 0
\(453\) −7.31534 −0.343705
\(454\) 0 0
\(455\) 5.56155 0.260730
\(456\) 0 0
\(457\) 7.36932 0.344722 0.172361 0.985034i \(-0.444860\pi\)
0.172361 + 0.985034i \(0.444860\pi\)
\(458\) 0 0
\(459\) −1.12311 −0.0524221
\(460\) 0 0
\(461\) 40.7386 1.89739 0.948694 0.316197i \(-0.102406\pi\)
0.948694 + 0.316197i \(0.102406\pi\)
\(462\) 0 0
\(463\) 13.5616 0.630259 0.315129 0.949049i \(-0.397952\pi\)
0.315129 + 0.949049i \(0.397952\pi\)
\(464\) 0 0
\(465\) 4.87689 0.226161
\(466\) 0 0
\(467\) −2.68466 −0.124231 −0.0621156 0.998069i \(-0.519785\pi\)
−0.0621156 + 0.998069i \(0.519785\pi\)
\(468\) 0 0
\(469\) 1.12311 0.0518602
\(470\) 0 0
\(471\) 14.2462 0.656431
\(472\) 0 0
\(473\) 34.2462 1.57464
\(474\) 0 0
\(475\) −13.1231 −0.602129
\(476\) 0 0
\(477\) 14.2462 0.652289
\(478\) 0 0
\(479\) −31.6155 −1.44455 −0.722275 0.691606i \(-0.756904\pi\)
−0.722275 + 0.691606i \(0.756904\pi\)
\(480\) 0 0
\(481\) 5.56155 0.253585
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 19.4233 0.881966
\(486\) 0 0
\(487\) 5.56155 0.252018 0.126009 0.992029i \(-0.459783\pi\)
0.126009 + 0.992029i \(0.459783\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) 26.7386 1.20670 0.603349 0.797477i \(-0.293833\pi\)
0.603349 + 0.797477i \(0.293833\pi\)
\(492\) 0 0
\(493\) −8.49242 −0.382479
\(494\) 0 0
\(495\) 8.00000 0.359573
\(496\) 0 0
\(497\) 9.36932 0.420271
\(498\) 0 0
\(499\) −41.3693 −1.85194 −0.925972 0.377591i \(-0.876753\pi\)
−0.925972 + 0.377591i \(0.876753\pi\)
\(500\) 0 0
\(501\) 14.2462 0.636474
\(502\) 0 0
\(503\) 32.1080 1.43162 0.715811 0.698294i \(-0.246057\pi\)
0.715811 + 0.698294i \(0.246057\pi\)
\(504\) 0 0
\(505\) −25.3693 −1.12892
\(506\) 0 0
\(507\) −0.315342 −0.0140048
\(508\) 0 0
\(509\) −3.36932 −0.149342 −0.0746712 0.997208i \(-0.523791\pi\)
−0.0746712 + 0.997208i \(0.523791\pi\)
\(510\) 0 0
\(511\) 9.12311 0.403582
\(512\) 0 0
\(513\) 5.12311 0.226191
\(514\) 0 0
\(515\) 29.5616 1.30264
\(516\) 0 0
\(517\) 12.4924 0.549416
\(518\) 0 0
\(519\) 18.4924 0.811727
\(520\) 0 0
\(521\) 0.246211 0.0107867 0.00539336 0.999985i \(-0.498283\pi\)
0.00539336 + 0.999985i \(0.498283\pi\)
\(522\) 0 0
\(523\) −14.0000 −0.612177 −0.306089 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) 0 0
\(525\) −2.56155 −0.111795
\(526\) 0 0
\(527\) 3.50758 0.152792
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −4.87689 −0.211639
\(532\) 0 0
\(533\) −12.6847 −0.549434
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) 0 0
\(537\) 3.80776 0.164317
\(538\) 0 0
\(539\) −5.12311 −0.220668
\(540\) 0 0
\(541\) −15.3693 −0.660779 −0.330389 0.943845i \(-0.607180\pi\)
−0.330389 + 0.943845i \(0.607180\pi\)
\(542\) 0 0
\(543\) −15.1231 −0.648995
\(544\) 0 0
\(545\) −16.3002 −0.698223
\(546\) 0 0
\(547\) 3.61553 0.154589 0.0772944 0.997008i \(-0.475372\pi\)
0.0772944 + 0.997008i \(0.475372\pi\)
\(548\) 0 0
\(549\) 0.876894 0.0374249
\(550\) 0 0
\(551\) 38.7386 1.65032
\(552\) 0 0
\(553\) −14.2462 −0.605811
\(554\) 0 0
\(555\) 2.43845 0.103506
\(556\) 0 0
\(557\) −28.9848 −1.22813 −0.614064 0.789257i \(-0.710466\pi\)
−0.614064 + 0.789257i \(0.710466\pi\)
\(558\) 0 0
\(559\) 23.8078 1.00696
\(560\) 0 0
\(561\) 5.75379 0.242925
\(562\) 0 0
\(563\) −19.5616 −0.824421 −0.412211 0.911089i \(-0.635243\pi\)
−0.412211 + 0.911089i \(0.635243\pi\)
\(564\) 0 0
\(565\) 4.19224 0.176369
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −44.5464 −1.86748 −0.933741 0.357949i \(-0.883476\pi\)
−0.933741 + 0.357949i \(0.883476\pi\)
\(570\) 0 0
\(571\) −43.3693 −1.81495 −0.907475 0.420107i \(-0.861993\pi\)
−0.907475 + 0.420107i \(0.861993\pi\)
\(572\) 0 0
\(573\) 20.0000 0.835512
\(574\) 0 0
\(575\) 2.56155 0.106824
\(576\) 0 0
\(577\) −19.7538 −0.822361 −0.411180 0.911554i \(-0.634883\pi\)
−0.411180 + 0.911554i \(0.634883\pi\)
\(578\) 0 0
\(579\) 2.68466 0.111571
\(580\) 0 0
\(581\) 9.12311 0.378490
\(582\) 0 0
\(583\) −72.9848 −3.02272
\(584\) 0 0
\(585\) 5.56155 0.229942
\(586\) 0 0
\(587\) 24.9848 1.03123 0.515617 0.856819i \(-0.327563\pi\)
0.515617 + 0.856819i \(0.327563\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 24.9309 1.02552
\(592\) 0 0
\(593\) 30.3002 1.24428 0.622140 0.782906i \(-0.286264\pi\)
0.622140 + 0.782906i \(0.286264\pi\)
\(594\) 0 0
\(595\) 1.75379 0.0718983
\(596\) 0 0
\(597\) 21.1771 0.866720
\(598\) 0 0
\(599\) 10.7386 0.438769 0.219384 0.975639i \(-0.429595\pi\)
0.219384 + 0.975639i \(0.429595\pi\)
\(600\) 0 0
\(601\) −3.36932 −0.137437 −0.0687187 0.997636i \(-0.521891\pi\)
−0.0687187 + 0.997636i \(0.521891\pi\)
\(602\) 0 0
\(603\) 1.12311 0.0457364
\(604\) 0 0
\(605\) −23.8078 −0.967923
\(606\) 0 0
\(607\) 47.6155 1.93265 0.966327 0.257316i \(-0.0828381\pi\)
0.966327 + 0.257316i \(0.0828381\pi\)
\(608\) 0 0
\(609\) 7.56155 0.306410
\(610\) 0 0
\(611\) 8.68466 0.351344
\(612\) 0 0
\(613\) 3.80776 0.153794 0.0768971 0.997039i \(-0.475499\pi\)
0.0768971 + 0.997039i \(0.475499\pi\)
\(614\) 0 0
\(615\) −5.56155 −0.224263
\(616\) 0 0
\(617\) 31.7538 1.27836 0.639180 0.769057i \(-0.279274\pi\)
0.639180 + 0.769057i \(0.279274\pi\)
\(618\) 0 0
\(619\) −8.24621 −0.331443 −0.165722 0.986173i \(-0.552995\pi\)
−0.165722 + 0.986173i \(0.552995\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) −5.63068 −0.225227
\(626\) 0 0
\(627\) −26.2462 −1.04817
\(628\) 0 0
\(629\) 1.75379 0.0699281
\(630\) 0 0
\(631\) −8.49242 −0.338078 −0.169039 0.985609i \(-0.554066\pi\)
−0.169039 + 0.985609i \(0.554066\pi\)
\(632\) 0 0
\(633\) 16.4924 0.655515
\(634\) 0 0
\(635\) −26.0540 −1.03392
\(636\) 0 0
\(637\) −3.56155 −0.141114
\(638\) 0 0
\(639\) 9.36932 0.370644
\(640\) 0 0
\(641\) −13.3153 −0.525924 −0.262962 0.964806i \(-0.584699\pi\)
−0.262962 + 0.964806i \(0.584699\pi\)
\(642\) 0 0
\(643\) 38.0000 1.49857 0.749287 0.662246i \(-0.230396\pi\)
0.749287 + 0.662246i \(0.230396\pi\)
\(644\) 0 0
\(645\) 10.4384 0.411013
\(646\) 0 0
\(647\) 3.50758 0.137897 0.0689486 0.997620i \(-0.478036\pi\)
0.0689486 + 0.997620i \(0.478036\pi\)
\(648\) 0 0
\(649\) 24.9848 0.980741
\(650\) 0 0
\(651\) −3.12311 −0.122404
\(652\) 0 0
\(653\) 36.5464 1.43017 0.715086 0.699037i \(-0.246388\pi\)
0.715086 + 0.699037i \(0.246388\pi\)
\(654\) 0 0
\(655\) 7.61553 0.297563
\(656\) 0 0
\(657\) 9.12311 0.355926
\(658\) 0 0
\(659\) −5.61553 −0.218750 −0.109375 0.994001i \(-0.534885\pi\)
−0.109375 + 0.994001i \(0.534885\pi\)
\(660\) 0 0
\(661\) 32.8769 1.27876 0.639381 0.768890i \(-0.279190\pi\)
0.639381 + 0.768890i \(0.279190\pi\)
\(662\) 0 0
\(663\) 4.00000 0.155347
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) −7.56155 −0.292784
\(668\) 0 0
\(669\) −3.12311 −0.120746
\(670\) 0 0
\(671\) −4.49242 −0.173428
\(672\) 0 0
\(673\) 22.6847 0.874429 0.437215 0.899357i \(-0.355965\pi\)
0.437215 + 0.899357i \(0.355965\pi\)
\(674\) 0 0
\(675\) −2.56155 −0.0985942
\(676\) 0 0
\(677\) 2.63068 0.101105 0.0505527 0.998721i \(-0.483902\pi\)
0.0505527 + 0.998721i \(0.483902\pi\)
\(678\) 0 0
\(679\) −12.4384 −0.477344
\(680\) 0 0
\(681\) 6.19224 0.237287
\(682\) 0 0
\(683\) 7.50758 0.287269 0.143635 0.989631i \(-0.454121\pi\)
0.143635 + 0.989631i \(0.454121\pi\)
\(684\) 0 0
\(685\) 0.684658 0.0261595
\(686\) 0 0
\(687\) 2.24621 0.0856983
\(688\) 0 0
\(689\) −50.7386 −1.93299
\(690\) 0 0
\(691\) 50.0540 1.90414 0.952071 0.305876i \(-0.0989492\pi\)
0.952071 + 0.305876i \(0.0989492\pi\)
\(692\) 0 0
\(693\) −5.12311 −0.194611
\(694\) 0 0
\(695\) 5.94602 0.225546
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 0 0
\(699\) −18.4924 −0.699448
\(700\) 0 0
\(701\) 34.2462 1.29346 0.646731 0.762718i \(-0.276136\pi\)
0.646731 + 0.762718i \(0.276136\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 3.80776 0.143409
\(706\) 0 0
\(707\) 16.2462 0.611002
\(708\) 0 0
\(709\) 10.6307 0.399244 0.199622 0.979873i \(-0.436029\pi\)
0.199622 + 0.979873i \(0.436029\pi\)
\(710\) 0 0
\(711\) −14.2462 −0.534275
\(712\) 0 0
\(713\) 3.12311 0.116961
\(714\) 0 0
\(715\) −28.4924 −1.06556
\(716\) 0 0
\(717\) −4.49242 −0.167773
\(718\) 0 0
\(719\) −10.4384 −0.389288 −0.194644 0.980874i \(-0.562355\pi\)
−0.194644 + 0.980874i \(0.562355\pi\)
\(720\) 0 0
\(721\) −18.9309 −0.705022
\(722\) 0 0
\(723\) −20.0540 −0.745815
\(724\) 0 0
\(725\) −19.3693 −0.719358
\(726\) 0 0
\(727\) 1.75379 0.0650444 0.0325222 0.999471i \(-0.489646\pi\)
0.0325222 + 0.999471i \(0.489646\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 7.50758 0.277678
\(732\) 0 0
\(733\) 25.7538 0.951238 0.475619 0.879651i \(-0.342224\pi\)
0.475619 + 0.879651i \(0.342224\pi\)
\(734\) 0 0
\(735\) −1.56155 −0.0575987
\(736\) 0 0
\(737\) −5.75379 −0.211944
\(738\) 0 0
\(739\) −23.6155 −0.868711 −0.434356 0.900741i \(-0.643024\pi\)
−0.434356 + 0.900741i \(0.643024\pi\)
\(740\) 0 0
\(741\) −18.2462 −0.670291
\(742\) 0 0
\(743\) 24.9848 0.916605 0.458303 0.888796i \(-0.348458\pi\)
0.458303 + 0.888796i \(0.348458\pi\)
\(744\) 0 0
\(745\) 9.75379 0.357351
\(746\) 0 0
\(747\) 9.12311 0.333797
\(748\) 0 0
\(749\) 5.12311 0.187194
\(750\) 0 0
\(751\) −24.4924 −0.893741 −0.446871 0.894599i \(-0.647462\pi\)
−0.446871 + 0.894599i \(0.647462\pi\)
\(752\) 0 0
\(753\) 0.438447 0.0159779
\(754\) 0 0
\(755\) 11.4233 0.415736
\(756\) 0 0
\(757\) −25.3693 −0.922064 −0.461032 0.887384i \(-0.652521\pi\)
−0.461032 + 0.887384i \(0.652521\pi\)
\(758\) 0 0
\(759\) 5.12311 0.185957
\(760\) 0 0
\(761\) −3.26137 −0.118224 −0.0591122 0.998251i \(-0.518827\pi\)
−0.0591122 + 0.998251i \(0.518827\pi\)
\(762\) 0 0
\(763\) 10.4384 0.377897
\(764\) 0 0
\(765\) 1.75379 0.0634084
\(766\) 0 0
\(767\) 17.3693 0.627170
\(768\) 0 0
\(769\) −47.1771 −1.70125 −0.850625 0.525774i \(-0.823776\pi\)
−0.850625 + 0.525774i \(0.823776\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 0 0
\(773\) −32.3002 −1.16176 −0.580878 0.813990i \(-0.697291\pi\)
−0.580878 + 0.813990i \(0.697291\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) −1.56155 −0.0560204
\(778\) 0 0
\(779\) 18.2462 0.653738
\(780\) 0 0
\(781\) −48.0000 −1.71758
\(782\) 0 0
\(783\) 7.56155 0.270228
\(784\) 0 0
\(785\) −22.2462 −0.794001
\(786\) 0 0
\(787\) 10.8769 0.387719 0.193860 0.981029i \(-0.437899\pi\)
0.193860 + 0.981029i \(0.437899\pi\)
\(788\) 0 0
\(789\) −22.9309 −0.816361
\(790\) 0 0
\(791\) −2.68466 −0.0954555
\(792\) 0 0
\(793\) −3.12311 −0.110905
\(794\) 0 0
\(795\) −22.2462 −0.788992
\(796\) 0 0
\(797\) 27.4233 0.971383 0.485691 0.874130i \(-0.338568\pi\)
0.485691 + 0.874130i \(0.338568\pi\)
\(798\) 0 0
\(799\) 2.73863 0.0968859
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) 0 0
\(803\) −46.7386 −1.64937
\(804\) 0 0
\(805\) 1.56155 0.0550375
\(806\) 0 0
\(807\) 12.2462 0.431087
\(808\) 0 0
\(809\) 23.8617 0.838934 0.419467 0.907771i \(-0.362217\pi\)
0.419467 + 0.907771i \(0.362217\pi\)
\(810\) 0 0
\(811\) 24.6847 0.866796 0.433398 0.901203i \(-0.357315\pi\)
0.433398 + 0.901203i \(0.357315\pi\)
\(812\) 0 0
\(813\) −14.2462 −0.499636
\(814\) 0 0
\(815\) −12.4924 −0.437590
\(816\) 0 0
\(817\) −34.2462 −1.19812
\(818\) 0 0
\(819\) −3.56155 −0.124451
\(820\) 0 0
\(821\) −55.4773 −1.93617 −0.968085 0.250622i \(-0.919365\pi\)
−0.968085 + 0.250622i \(0.919365\pi\)
\(822\) 0 0
\(823\) −23.3153 −0.812722 −0.406361 0.913713i \(-0.633202\pi\)
−0.406361 + 0.913713i \(0.633202\pi\)
\(824\) 0 0
\(825\) 13.1231 0.456888
\(826\) 0 0
\(827\) −48.7386 −1.69481 −0.847404 0.530948i \(-0.821836\pi\)
−0.847404 + 0.530948i \(0.821836\pi\)
\(828\) 0 0
\(829\) 24.7386 0.859208 0.429604 0.903017i \(-0.358653\pi\)
0.429604 + 0.903017i \(0.358653\pi\)
\(830\) 0 0
\(831\) 13.1231 0.455235
\(832\) 0 0
\(833\) −1.12311 −0.0389133
\(834\) 0 0
\(835\) −22.2462 −0.769862
\(836\) 0 0
\(837\) −3.12311 −0.107950
\(838\) 0 0
\(839\) −7.61553 −0.262917 −0.131459 0.991322i \(-0.541966\pi\)
−0.131459 + 0.991322i \(0.541966\pi\)
\(840\) 0 0
\(841\) 28.1771 0.971623
\(842\) 0 0
\(843\) 4.93087 0.169828
\(844\) 0 0
\(845\) 0.492423 0.0169398
\(846\) 0 0
\(847\) 15.2462 0.523866
\(848\) 0 0
\(849\) 10.4924 0.360099
\(850\) 0 0
\(851\) 1.56155 0.0535293
\(852\) 0 0
\(853\) −0.438447 −0.0150121 −0.00750607 0.999972i \(-0.502389\pi\)
−0.00750607 + 0.999972i \(0.502389\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 0 0
\(857\) 1.31534 0.0449312 0.0224656 0.999748i \(-0.492848\pi\)
0.0224656 + 0.999748i \(0.492848\pi\)
\(858\) 0 0
\(859\) 33.1771 1.13199 0.565994 0.824410i \(-0.308493\pi\)
0.565994 + 0.824410i \(0.308493\pi\)
\(860\) 0 0
\(861\) 3.56155 0.121377
\(862\) 0 0
\(863\) −10.7386 −0.365547 −0.182774 0.983155i \(-0.558508\pi\)
−0.182774 + 0.983155i \(0.558508\pi\)
\(864\) 0 0
\(865\) −28.8769 −0.981844
\(866\) 0 0
\(867\) −15.7386 −0.534512
\(868\) 0 0
\(869\) 72.9848 2.47584
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) −12.4384 −0.420978
\(874\) 0 0
\(875\) 11.8078 0.399175
\(876\) 0 0
\(877\) −35.8617 −1.21096 −0.605482 0.795859i \(-0.707020\pi\)
−0.605482 + 0.795859i \(0.707020\pi\)
\(878\) 0 0
\(879\) 27.6155 0.931449
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 8.87689 0.298731 0.149366 0.988782i \(-0.452277\pi\)
0.149366 + 0.988782i \(0.452277\pi\)
\(884\) 0 0
\(885\) 7.61553 0.255993
\(886\) 0 0
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) 16.6847 0.559585
\(890\) 0 0
\(891\) −5.12311 −0.171630
\(892\) 0 0
\(893\) −12.4924 −0.418043
\(894\) 0 0
\(895\) −5.94602 −0.198754
\(896\) 0 0
\(897\) 3.56155 0.118917
\(898\) 0 0
\(899\) −23.6155 −0.787622
\(900\) 0 0
\(901\) −16.0000 −0.533037
\(902\) 0 0
\(903\) −6.68466 −0.222452
\(904\) 0 0
\(905\) 23.6155 0.785007
\(906\) 0 0
\(907\) −52.5464 −1.74477 −0.872387 0.488815i \(-0.837429\pi\)
−0.872387 + 0.488815i \(0.837429\pi\)
\(908\) 0 0
\(909\) 16.2462 0.538853
\(910\) 0 0
\(911\) 1.94602 0.0644747 0.0322373 0.999480i \(-0.489737\pi\)
0.0322373 + 0.999480i \(0.489737\pi\)
\(912\) 0 0
\(913\) −46.7386 −1.54682
\(914\) 0 0
\(915\) −1.36932 −0.0452682
\(916\) 0 0
\(917\) −4.87689 −0.161049
\(918\) 0 0
\(919\) 50.7386 1.67371 0.836857 0.547422i \(-0.184391\pi\)
0.836857 + 0.547422i \(0.184391\pi\)
\(920\) 0 0
\(921\) −2.93087 −0.0965754
\(922\) 0 0
\(923\) −33.3693 −1.09836
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) 0 0
\(927\) −18.9309 −0.621771
\(928\) 0 0
\(929\) −36.4384 −1.19551 −0.597753 0.801680i \(-0.703940\pi\)
−0.597753 + 0.801680i \(0.703940\pi\)
\(930\) 0 0
\(931\) 5.12311 0.167903
\(932\) 0 0
\(933\) −18.7386 −0.613475
\(934\) 0 0
\(935\) −8.98485 −0.293836
\(936\) 0 0
\(937\) −27.1771 −0.887837 −0.443918 0.896067i \(-0.646412\pi\)
−0.443918 + 0.896067i \(0.646412\pi\)
\(938\) 0 0
\(939\) 4.24621 0.138570
\(940\) 0 0
\(941\) 1.94602 0.0634386 0.0317193 0.999497i \(-0.489902\pi\)
0.0317193 + 0.999497i \(0.489902\pi\)
\(942\) 0 0
\(943\) −3.56155 −0.115980
\(944\) 0 0
\(945\) −1.56155 −0.0507973
\(946\) 0 0
\(947\) 54.9309 1.78501 0.892507 0.451034i \(-0.148945\pi\)
0.892507 + 0.451034i \(0.148945\pi\)
\(948\) 0 0
\(949\) −32.4924 −1.05475
\(950\) 0 0
\(951\) −17.8078 −0.577456
\(952\) 0 0
\(953\) 50.0000 1.61966 0.809829 0.586665i \(-0.199560\pi\)
0.809829 + 0.586665i \(0.199560\pi\)
\(954\) 0 0
\(955\) −31.2311 −1.01061
\(956\) 0 0
\(957\) −38.7386 −1.25224
\(958\) 0 0
\(959\) −0.438447 −0.0141582
\(960\) 0 0
\(961\) −21.2462 −0.685362
\(962\) 0 0
\(963\) 5.12311 0.165090
\(964\) 0 0
\(965\) −4.19224 −0.134953
\(966\) 0 0
\(967\) 9.75379 0.313661 0.156830 0.987626i \(-0.449872\pi\)
0.156830 + 0.987626i \(0.449872\pi\)
\(968\) 0 0
\(969\) −5.75379 −0.184838
\(970\) 0 0
\(971\) 1.61553 0.0518448 0.0259224 0.999664i \(-0.491748\pi\)
0.0259224 + 0.999664i \(0.491748\pi\)
\(972\) 0 0
\(973\) −3.80776 −0.122071
\(974\) 0 0
\(975\) 9.12311 0.292173
\(976\) 0 0
\(977\) −26.1922 −0.837964 −0.418982 0.907995i \(-0.637613\pi\)
−0.418982 + 0.907995i \(0.637613\pi\)
\(978\) 0 0
\(979\) −71.7235 −2.29229
\(980\) 0 0
\(981\) 10.4384 0.333274
\(982\) 0 0
\(983\) −33.8617 −1.08002 −0.540011 0.841658i \(-0.681580\pi\)
−0.540011 + 0.841658i \(0.681580\pi\)
\(984\) 0 0
\(985\) −38.9309 −1.24044
\(986\) 0 0
\(987\) −2.43845 −0.0776166
\(988\) 0 0
\(989\) 6.68466 0.212560
\(990\) 0 0
\(991\) 6.24621 0.198417 0.0992087 0.995067i \(-0.468369\pi\)
0.0992087 + 0.995067i \(0.468369\pi\)
\(992\) 0 0
\(993\) 24.4924 0.777244
\(994\) 0 0
\(995\) −33.0691 −1.04836
\(996\) 0 0
\(997\) 51.4773 1.63030 0.815151 0.579249i \(-0.196654\pi\)
0.815151 + 0.579249i \(0.196654\pi\)
\(998\) 0 0
\(999\) −1.56155 −0.0494053
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 7728.2.a.bm.1.1 2
4.3 odd 2 966.2.a.l.1.1 2
12.11 even 2 2898.2.a.ba.1.2 2
28.27 even 2 6762.2.a.bw.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.l.1.1 2 4.3 odd 2
2898.2.a.ba.1.2 2 12.11 even 2
6762.2.a.bw.1.2 2 28.27 even 2
7728.2.a.bm.1.1 2 1.1 even 1 trivial