Properties

Label 1638.2.a.y.1.2
Level $1638$
Weight $2$
Character 1638.1
Self dual yes
Analytic conductor $13.079$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.27492\) of defining polynomial
Character \(\chi\) \(=\) 1638.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +4.27492 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +4.27492 q^{5} +1.00000 q^{7} +1.00000 q^{8} +4.27492 q^{10} +2.27492 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -0.274917 q^{17} -2.27492 q^{19} +4.27492 q^{20} +2.27492 q^{22} -2.27492 q^{23} +13.2749 q^{25} -1.00000 q^{26} +1.00000 q^{28} -8.27492 q^{29} +8.00000 q^{31} +1.00000 q^{32} -0.274917 q^{34} +4.27492 q^{35} -4.27492 q^{37} -2.27492 q^{38} +4.27492 q^{40} -6.54983 q^{41} -2.27492 q^{43} +2.27492 q^{44} -2.27492 q^{46} +1.00000 q^{49} +13.2749 q^{50} -1.00000 q^{52} -10.0000 q^{53} +9.72508 q^{55} +1.00000 q^{56} -8.27492 q^{58} -8.00000 q^{59} -12.2749 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.27492 q^{65} +12.5498 q^{67} -0.274917 q^{68} +4.27492 q^{70} -12.8248 q^{73} -4.27492 q^{74} -2.27492 q^{76} +2.27492 q^{77} +12.5498 q^{79} +4.27492 q^{80} -6.54983 q^{82} +4.54983 q^{83} -1.17525 q^{85} -2.27492 q^{86} +2.27492 q^{88} +14.0000 q^{89} -1.00000 q^{91} -2.27492 q^{92} -9.72508 q^{95} +15.0997 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + q^{5} + 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + q^{5} + 2q^{7} + 2q^{8} + q^{10} - 3q^{11} - 2q^{13} + 2q^{14} + 2q^{16} + 7q^{17} + 3q^{19} + q^{20} - 3q^{22} + 3q^{23} + 19q^{25} - 2q^{26} + 2q^{28} - 9q^{29} + 16q^{31} + 2q^{32} + 7q^{34} + q^{35} - q^{37} + 3q^{38} + q^{40} + 2q^{41} + 3q^{43} - 3q^{44} + 3q^{46} + 2q^{49} + 19q^{50} - 2q^{52} - 20q^{53} + 27q^{55} + 2q^{56} - 9q^{58} - 16q^{59} - 17q^{61} + 16q^{62} + 2q^{64} - q^{65} + 10q^{67} + 7q^{68} + q^{70} - 3q^{73} - q^{74} + 3q^{76} - 3q^{77} + 10q^{79} + q^{80} + 2q^{82} - 6q^{83} - 25q^{85} + 3q^{86} - 3q^{88} + 28q^{89} - 2q^{91} + 3q^{92} - 27q^{95} + 2q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 4.27492 1.91180 0.955901 0.293691i \(-0.0948835\pi\)
0.955901 + 0.293691i \(0.0948835\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 4.27492 1.35185
\(11\) 2.27492 0.685913 0.342957 0.939351i \(-0.388572\pi\)
0.342957 + 0.939351i \(0.388572\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.274917 −0.0666772 −0.0333386 0.999444i \(-0.510614\pi\)
−0.0333386 + 0.999444i \(0.510614\pi\)
\(18\) 0 0
\(19\) −2.27492 −0.521902 −0.260951 0.965352i \(-0.584036\pi\)
−0.260951 + 0.965352i \(0.584036\pi\)
\(20\) 4.27492 0.955901
\(21\) 0 0
\(22\) 2.27492 0.485014
\(23\) −2.27492 −0.474353 −0.237177 0.971467i \(-0.576222\pi\)
−0.237177 + 0.971467i \(0.576222\pi\)
\(24\) 0 0
\(25\) 13.2749 2.65498
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −8.27492 −1.53661 −0.768307 0.640082i \(-0.778900\pi\)
−0.768307 + 0.640082i \(0.778900\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.274917 −0.0471479
\(35\) 4.27492 0.722593
\(36\) 0 0
\(37\) −4.27492 −0.702792 −0.351396 0.936227i \(-0.614293\pi\)
−0.351396 + 0.936227i \(0.614293\pi\)
\(38\) −2.27492 −0.369040
\(39\) 0 0
\(40\) 4.27492 0.675924
\(41\) −6.54983 −1.02291 −0.511456 0.859309i \(-0.670894\pi\)
−0.511456 + 0.859309i \(0.670894\pi\)
\(42\) 0 0
\(43\) −2.27492 −0.346922 −0.173461 0.984841i \(-0.555495\pi\)
−0.173461 + 0.984841i \(0.555495\pi\)
\(44\) 2.27492 0.342957
\(45\) 0 0
\(46\) −2.27492 −0.335418
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 13.2749 1.87736
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 9.72508 1.31133
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −8.27492 −1.08655
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −12.2749 −1.57164 −0.785821 0.618454i \(-0.787759\pi\)
−0.785821 + 0.618454i \(0.787759\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.27492 −0.530238
\(66\) 0 0
\(67\) 12.5498 1.53321 0.766603 0.642121i \(-0.221945\pi\)
0.766603 + 0.642121i \(0.221945\pi\)
\(68\) −0.274917 −0.0333386
\(69\) 0 0
\(70\) 4.27492 0.510950
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −12.8248 −1.50102 −0.750512 0.660857i \(-0.770193\pi\)
−0.750512 + 0.660857i \(0.770193\pi\)
\(74\) −4.27492 −0.496949
\(75\) 0 0
\(76\) −2.27492 −0.260951
\(77\) 2.27492 0.259251
\(78\) 0 0
\(79\) 12.5498 1.41197 0.705983 0.708228i \(-0.250505\pi\)
0.705983 + 0.708228i \(0.250505\pi\)
\(80\) 4.27492 0.477950
\(81\) 0 0
\(82\) −6.54983 −0.723308
\(83\) 4.54983 0.499409 0.249705 0.968322i \(-0.419666\pi\)
0.249705 + 0.968322i \(0.419666\pi\)
\(84\) 0 0
\(85\) −1.17525 −0.127474
\(86\) −2.27492 −0.245311
\(87\) 0 0
\(88\) 2.27492 0.242507
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) −2.27492 −0.237177
\(93\) 0 0
\(94\) 0 0
\(95\) −9.72508 −0.997772
\(96\) 0 0
\(97\) 15.0997 1.53314 0.766570 0.642161i \(-0.221962\pi\)
0.766570 + 0.642161i \(0.221962\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 13.2749 1.32749
\(101\) −2.54983 −0.253718 −0.126859 0.991921i \(-0.540490\pi\)
−0.126859 + 0.991921i \(0.540490\pi\)
\(102\) 0 0
\(103\) 10.2749 1.01242 0.506209 0.862411i \(-0.331046\pi\)
0.506209 + 0.862411i \(0.331046\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −0.549834 −0.0531545 −0.0265773 0.999647i \(-0.508461\pi\)
−0.0265773 + 0.999647i \(0.508461\pi\)
\(108\) 0 0
\(109\) 0.274917 0.0263323 0.0131661 0.999913i \(-0.495809\pi\)
0.0131661 + 0.999913i \(0.495809\pi\)
\(110\) 9.72508 0.927250
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −9.72508 −0.906869
\(116\) −8.27492 −0.768307
\(117\) 0 0
\(118\) −8.00000 −0.736460
\(119\) −0.274917 −0.0252016
\(120\) 0 0
\(121\) −5.82475 −0.529523
\(122\) −12.2749 −1.11132
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) 35.3746 3.16400
\(126\) 0 0
\(127\) −20.5498 −1.82350 −0.911751 0.410742i \(-0.865270\pi\)
−0.911751 + 0.410742i \(0.865270\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −4.27492 −0.374935
\(131\) −6.82475 −0.596281 −0.298141 0.954522i \(-0.596366\pi\)
−0.298141 + 0.954522i \(0.596366\pi\)
\(132\) 0 0
\(133\) −2.27492 −0.197260
\(134\) 12.5498 1.08414
\(135\) 0 0
\(136\) −0.274917 −0.0235740
\(137\) 17.3746 1.48441 0.742206 0.670172i \(-0.233780\pi\)
0.742206 + 0.670172i \(0.233780\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 4.27492 0.361296
\(141\) 0 0
\(142\) 0 0
\(143\) −2.27492 −0.190238
\(144\) 0 0
\(145\) −35.3746 −2.93770
\(146\) −12.8248 −1.06138
\(147\) 0 0
\(148\) −4.27492 −0.351396
\(149\) −22.5498 −1.84735 −0.923677 0.383172i \(-0.874832\pi\)
−0.923677 + 0.383172i \(0.874832\pi\)
\(150\) 0 0
\(151\) 6.27492 0.510646 0.255323 0.966856i \(-0.417818\pi\)
0.255323 + 0.966856i \(0.417818\pi\)
\(152\) −2.27492 −0.184520
\(153\) 0 0
\(154\) 2.27492 0.183318
\(155\) 34.1993 2.74696
\(156\) 0 0
\(157\) −13.3746 −1.06741 −0.533704 0.845671i \(-0.679200\pi\)
−0.533704 + 0.845671i \(0.679200\pi\)
\(158\) 12.5498 0.998411
\(159\) 0 0
\(160\) 4.27492 0.337962
\(161\) −2.27492 −0.179289
\(162\) 0 0
\(163\) 12.5498 0.982979 0.491489 0.870884i \(-0.336453\pi\)
0.491489 + 0.870884i \(0.336453\pi\)
\(164\) −6.54983 −0.511456
\(165\) 0 0
\(166\) 4.54983 0.353136
\(167\) 1.72508 0.133491 0.0667455 0.997770i \(-0.478738\pi\)
0.0667455 + 0.997770i \(0.478738\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −1.17525 −0.0901374
\(171\) 0 0
\(172\) −2.27492 −0.173461
\(173\) −7.09967 −0.539778 −0.269889 0.962891i \(-0.586987\pi\)
−0.269889 + 0.962891i \(0.586987\pi\)
\(174\) 0 0
\(175\) 13.2749 1.00349
\(176\) 2.27492 0.171478
\(177\) 0 0
\(178\) 14.0000 1.04934
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 19.0997 1.41967 0.709834 0.704369i \(-0.248770\pi\)
0.709834 + 0.704369i \(0.248770\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 0 0
\(184\) −2.27492 −0.167709
\(185\) −18.2749 −1.34360
\(186\) 0 0
\(187\) −0.625414 −0.0457348
\(188\) 0 0
\(189\) 0 0
\(190\) −9.72508 −0.705532
\(191\) −2.27492 −0.164607 −0.0823036 0.996607i \(-0.526228\pi\)
−0.0823036 + 0.996607i \(0.526228\pi\)
\(192\) 0 0
\(193\) −18.5498 −1.33525 −0.667623 0.744499i \(-0.732688\pi\)
−0.667623 + 0.744499i \(0.732688\pi\)
\(194\) 15.0997 1.08409
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 2.54983 0.181668 0.0908341 0.995866i \(-0.471047\pi\)
0.0908341 + 0.995866i \(0.471047\pi\)
\(198\) 0 0
\(199\) 6.82475 0.483794 0.241897 0.970302i \(-0.422230\pi\)
0.241897 + 0.970302i \(0.422230\pi\)
\(200\) 13.2749 0.938678
\(201\) 0 0
\(202\) −2.54983 −0.179406
\(203\) −8.27492 −0.580785
\(204\) 0 0
\(205\) −28.0000 −1.95560
\(206\) 10.2749 0.715887
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) −5.17525 −0.357979
\(210\) 0 0
\(211\) 1.17525 0.0809074 0.0404537 0.999181i \(-0.487120\pi\)
0.0404537 + 0.999181i \(0.487120\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) −0.549834 −0.0375859
\(215\) −9.72508 −0.663245
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0.274917 0.0186197
\(219\) 0 0
\(220\) 9.72508 0.655665
\(221\) 0.274917 0.0184929
\(222\) 0 0
\(223\) 21.6495 1.44976 0.724879 0.688876i \(-0.241896\pi\)
0.724879 + 0.688876i \(0.241896\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 20.5498 1.36394 0.681970 0.731380i \(-0.261123\pi\)
0.681970 + 0.731380i \(0.261123\pi\)
\(228\) 0 0
\(229\) 19.0997 1.26214 0.631071 0.775725i \(-0.282616\pi\)
0.631071 + 0.775725i \(0.282616\pi\)
\(230\) −9.72508 −0.641253
\(231\) 0 0
\(232\) −8.27492 −0.543275
\(233\) −5.45017 −0.357052 −0.178526 0.983935i \(-0.557133\pi\)
−0.178526 + 0.983935i \(0.557133\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 0 0
\(238\) −0.274917 −0.0178202
\(239\) −25.0997 −1.62356 −0.811781 0.583962i \(-0.801502\pi\)
−0.811781 + 0.583962i \(0.801502\pi\)
\(240\) 0 0
\(241\) 15.0997 0.972655 0.486328 0.873777i \(-0.338336\pi\)
0.486328 + 0.873777i \(0.338336\pi\)
\(242\) −5.82475 −0.374429
\(243\) 0 0
\(244\) −12.2749 −0.785821
\(245\) 4.27492 0.273114
\(246\) 0 0
\(247\) 2.27492 0.144750
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) 35.3746 2.23729
\(251\) −23.9244 −1.51010 −0.755048 0.655669i \(-0.772386\pi\)
−0.755048 + 0.655669i \(0.772386\pi\)
\(252\) 0 0
\(253\) −5.17525 −0.325365
\(254\) −20.5498 −1.28941
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) −4.27492 −0.265630
\(260\) −4.27492 −0.265119
\(261\) 0 0
\(262\) −6.82475 −0.421635
\(263\) −28.0000 −1.72655 −0.863277 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) 0 0
\(265\) −42.7492 −2.62606
\(266\) −2.27492 −0.139484
\(267\) 0 0
\(268\) 12.5498 0.766603
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 28.5498 1.73428 0.867139 0.498065i \(-0.165956\pi\)
0.867139 + 0.498065i \(0.165956\pi\)
\(272\) −0.274917 −0.0166693
\(273\) 0 0
\(274\) 17.3746 1.04964
\(275\) 30.1993 1.82109
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 4.27492 0.255475
\(281\) −15.0997 −0.900771 −0.450385 0.892834i \(-0.648713\pi\)
−0.450385 + 0.892834i \(0.648713\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −2.27492 −0.134519
\(287\) −6.54983 −0.386625
\(288\) 0 0
\(289\) −16.9244 −0.995554
\(290\) −35.3746 −2.07727
\(291\) 0 0
\(292\) −12.8248 −0.750512
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −34.1993 −1.99116
\(296\) −4.27492 −0.248475
\(297\) 0 0
\(298\) −22.5498 −1.30628
\(299\) 2.27492 0.131562
\(300\) 0 0
\(301\) −2.27492 −0.131124
\(302\) 6.27492 0.361081
\(303\) 0 0
\(304\) −2.27492 −0.130475
\(305\) −52.4743 −3.00467
\(306\) 0 0
\(307\) 21.0997 1.20422 0.602111 0.798412i \(-0.294327\pi\)
0.602111 + 0.798412i \(0.294327\pi\)
\(308\) 2.27492 0.129625
\(309\) 0 0
\(310\) 34.1993 1.94239
\(311\) −3.45017 −0.195641 −0.0978205 0.995204i \(-0.531187\pi\)
−0.0978205 + 0.995204i \(0.531187\pi\)
\(312\) 0 0
\(313\) −27.6495 −1.56284 −0.781421 0.624004i \(-0.785505\pi\)
−0.781421 + 0.624004i \(0.785505\pi\)
\(314\) −13.3746 −0.754772
\(315\) 0 0
\(316\) 12.5498 0.705983
\(317\) 1.45017 0.0814494 0.0407247 0.999170i \(-0.487033\pi\)
0.0407247 + 0.999170i \(0.487033\pi\)
\(318\) 0 0
\(319\) −18.8248 −1.05398
\(320\) 4.27492 0.238975
\(321\) 0 0
\(322\) −2.27492 −0.126776
\(323\) 0.625414 0.0347990
\(324\) 0 0
\(325\) −13.2749 −0.736360
\(326\) 12.5498 0.695071
\(327\) 0 0
\(328\) −6.54983 −0.361654
\(329\) 0 0
\(330\) 0 0
\(331\) −29.6495 −1.62968 −0.814842 0.579683i \(-0.803176\pi\)
−0.814842 + 0.579683i \(0.803176\pi\)
\(332\) 4.54983 0.249705
\(333\) 0 0
\(334\) 1.72508 0.0943923
\(335\) 53.6495 2.93119
\(336\) 0 0
\(337\) 9.37459 0.510666 0.255333 0.966853i \(-0.417815\pi\)
0.255333 + 0.966853i \(0.417815\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) −1.17525 −0.0637368
\(341\) 18.1993 0.985549
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −2.27492 −0.122655
\(345\) 0 0
\(346\) −7.09967 −0.381681
\(347\) −5.09967 −0.273765 −0.136882 0.990587i \(-0.543708\pi\)
−0.136882 + 0.990587i \(0.543708\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 13.2749 0.709574
\(351\) 0 0
\(352\) 2.27492 0.121253
\(353\) −27.0997 −1.44237 −0.721185 0.692743i \(-0.756402\pi\)
−0.721185 + 0.692743i \(0.756402\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) −13.8248 −0.727619
\(362\) 19.0997 1.00386
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) −54.8248 −2.86966
\(366\) 0 0
\(367\) −2.90033 −0.151396 −0.0756980 0.997131i \(-0.524119\pi\)
−0.0756980 + 0.997131i \(0.524119\pi\)
\(368\) −2.27492 −0.118588
\(369\) 0 0
\(370\) −18.2749 −0.950068
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) 26.5498 1.37470 0.687349 0.726327i \(-0.258774\pi\)
0.687349 + 0.726327i \(0.258774\pi\)
\(374\) −0.625414 −0.0323394
\(375\) 0 0
\(376\) 0 0
\(377\) 8.27492 0.426180
\(378\) 0 0
\(379\) −33.0997 −1.70022 −0.850108 0.526609i \(-0.823463\pi\)
−0.850108 + 0.526609i \(0.823463\pi\)
\(380\) −9.72508 −0.498886
\(381\) 0 0
\(382\) −2.27492 −0.116395
\(383\) −30.2749 −1.54698 −0.773488 0.633811i \(-0.781490\pi\)
−0.773488 + 0.633811i \(0.781490\pi\)
\(384\) 0 0
\(385\) 9.72508 0.495636
\(386\) −18.5498 −0.944162
\(387\) 0 0
\(388\) 15.0997 0.766570
\(389\) −28.1993 −1.42976 −0.714882 0.699246i \(-0.753519\pi\)
−0.714882 + 0.699246i \(0.753519\pi\)
\(390\) 0 0
\(391\) 0.625414 0.0316285
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 2.54983 0.128459
\(395\) 53.6495 2.69940
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 6.82475 0.342094
\(399\) 0 0
\(400\) 13.2749 0.663746
\(401\) 3.09967 0.154790 0.0773950 0.997001i \(-0.475340\pi\)
0.0773950 + 0.997001i \(0.475340\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) −2.54983 −0.126859
\(405\) 0 0
\(406\) −8.27492 −0.410677
\(407\) −9.72508 −0.482054
\(408\) 0 0
\(409\) −7.17525 −0.354793 −0.177397 0.984139i \(-0.556768\pi\)
−0.177397 + 0.984139i \(0.556768\pi\)
\(410\) −28.0000 −1.38282
\(411\) 0 0
\(412\) 10.2749 0.506209
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 19.4502 0.954771
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) −5.17525 −0.253130
\(419\) −2.27492 −0.111137 −0.0555685 0.998455i \(-0.517697\pi\)
−0.0555685 + 0.998455i \(0.517697\pi\)
\(420\) 0 0
\(421\) 3.09967 0.151069 0.0755343 0.997143i \(-0.475934\pi\)
0.0755343 + 0.997143i \(0.475934\pi\)
\(422\) 1.17525 0.0572102
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) −3.64950 −0.177027
\(426\) 0 0
\(427\) −12.2749 −0.594025
\(428\) −0.549834 −0.0265773
\(429\) 0 0
\(430\) −9.72508 −0.468985
\(431\) 11.4502 0.551535 0.275768 0.961224i \(-0.411068\pi\)
0.275768 + 0.961224i \(0.411068\pi\)
\(432\) 0 0
\(433\) 23.6495 1.13652 0.568261 0.822848i \(-0.307616\pi\)
0.568261 + 0.822848i \(0.307616\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) 0.274917 0.0131661
\(437\) 5.17525 0.247566
\(438\) 0 0
\(439\) −14.8248 −0.707547 −0.353773 0.935331i \(-0.615102\pi\)
−0.353773 + 0.935331i \(0.615102\pi\)
\(440\) 9.72508 0.463625
\(441\) 0 0
\(442\) 0.274917 0.0130765
\(443\) −7.45017 −0.353968 −0.176984 0.984214i \(-0.556634\pi\)
−0.176984 + 0.984214i \(0.556634\pi\)
\(444\) 0 0
\(445\) 59.8488 2.83711
\(446\) 21.6495 1.02513
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 0.274917 0.0129741 0.00648707 0.999979i \(-0.497935\pi\)
0.00648707 + 0.999979i \(0.497935\pi\)
\(450\) 0 0
\(451\) −14.9003 −0.701629
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 20.5498 0.964452
\(455\) −4.27492 −0.200411
\(456\) 0 0
\(457\) −18.5498 −0.867725 −0.433862 0.900979i \(-0.642850\pi\)
−0.433862 + 0.900979i \(0.642850\pi\)
\(458\) 19.0997 0.892469
\(459\) 0 0
\(460\) −9.72508 −0.453434
\(461\) 17.9244 0.834823 0.417412 0.908717i \(-0.362937\pi\)
0.417412 + 0.908717i \(0.362937\pi\)
\(462\) 0 0
\(463\) 30.2749 1.40699 0.703497 0.710698i \(-0.251621\pi\)
0.703497 + 0.710698i \(0.251621\pi\)
\(464\) −8.27492 −0.384153
\(465\) 0 0
\(466\) −5.45017 −0.252474
\(467\) 26.2749 1.21586 0.607929 0.793991i \(-0.292000\pi\)
0.607929 + 0.793991i \(0.292000\pi\)
\(468\) 0 0
\(469\) 12.5498 0.579498
\(470\) 0 0
\(471\) 0 0
\(472\) −8.00000 −0.368230
\(473\) −5.17525 −0.237958
\(474\) 0 0
\(475\) −30.1993 −1.38564
\(476\) −0.274917 −0.0126008
\(477\) 0 0
\(478\) −25.0997 −1.14803
\(479\) 23.3746 1.06801 0.534006 0.845481i \(-0.320686\pi\)
0.534006 + 0.845481i \(0.320686\pi\)
\(480\) 0 0
\(481\) 4.27492 0.194919
\(482\) 15.0997 0.687771
\(483\) 0 0
\(484\) −5.82475 −0.264761
\(485\) 64.5498 2.93106
\(486\) 0 0
\(487\) −18.1993 −0.824691 −0.412345 0.911028i \(-0.635290\pi\)
−0.412345 + 0.911028i \(0.635290\pi\)
\(488\) −12.2749 −0.555659
\(489\) 0 0
\(490\) 4.27492 0.193121
\(491\) −8.54983 −0.385849 −0.192924 0.981214i \(-0.561797\pi\)
−0.192924 + 0.981214i \(0.561797\pi\)
\(492\) 0 0
\(493\) 2.27492 0.102457
\(494\) 2.27492 0.102353
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) 25.0997 1.12362 0.561808 0.827268i \(-0.310106\pi\)
0.561808 + 0.827268i \(0.310106\pi\)
\(500\) 35.3746 1.58200
\(501\) 0 0
\(502\) −23.9244 −1.06780
\(503\) 3.45017 0.153835 0.0769176 0.997037i \(-0.475492\pi\)
0.0769176 + 0.997037i \(0.475492\pi\)
\(504\) 0 0
\(505\) −10.9003 −0.485058
\(506\) −5.17525 −0.230068
\(507\) 0 0
\(508\) −20.5498 −0.911751
\(509\) 9.92442 0.439892 0.219946 0.975512i \(-0.429412\pi\)
0.219946 + 0.975512i \(0.429412\pi\)
\(510\) 0 0
\(511\) −12.8248 −0.567334
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) 43.9244 1.93554
\(516\) 0 0
\(517\) 0 0
\(518\) −4.27492 −0.187829
\(519\) 0 0
\(520\) −4.27492 −0.187468
\(521\) 4.27492 0.187288 0.0936438 0.995606i \(-0.470149\pi\)
0.0936438 + 0.995606i \(0.470149\pi\)
\(522\) 0 0
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) −6.82475 −0.298141
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) −2.19934 −0.0958047
\(528\) 0 0
\(529\) −17.8248 −0.774989
\(530\) −42.7492 −1.85691
\(531\) 0 0
\(532\) −2.27492 −0.0986302
\(533\) 6.54983 0.283705
\(534\) 0 0
\(535\) −2.35050 −0.101621
\(536\) 12.5498 0.542070
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) 2.27492 0.0979876
\(540\) 0 0
\(541\) 42.4743 1.82611 0.913055 0.407835i \(-0.133716\pi\)
0.913055 + 0.407835i \(0.133716\pi\)
\(542\) 28.5498 1.22632
\(543\) 0 0
\(544\) −0.274917 −0.0117870
\(545\) 1.17525 0.0503421
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 17.3746 0.742206
\(549\) 0 0
\(550\) 30.1993 1.28770
\(551\) 18.8248 0.801961
\(552\) 0 0
\(553\) 12.5498 0.533673
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 44.7492 1.89608 0.948042 0.318146i \(-0.103060\pi\)
0.948042 + 0.318146i \(0.103060\pi\)
\(558\) 0 0
\(559\) 2.27492 0.0962187
\(560\) 4.27492 0.180648
\(561\) 0 0
\(562\) −15.0997 −0.636941
\(563\) 35.3746 1.49086 0.745431 0.666583i \(-0.232244\pi\)
0.745431 + 0.666583i \(0.232244\pi\)
\(564\) 0 0
\(565\) 25.6495 1.07908
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 0 0
\(569\) −3.09967 −0.129945 −0.0649724 0.997887i \(-0.520696\pi\)
−0.0649724 + 0.997887i \(0.520696\pi\)
\(570\) 0 0
\(571\) −37.0997 −1.55257 −0.776286 0.630380i \(-0.782899\pi\)
−0.776286 + 0.630380i \(0.782899\pi\)
\(572\) −2.27492 −0.0951191
\(573\) 0 0
\(574\) −6.54983 −0.273385
\(575\) −30.1993 −1.25940
\(576\) 0 0
\(577\) −8.90033 −0.370526 −0.185263 0.982689i \(-0.559314\pi\)
−0.185263 + 0.982689i \(0.559314\pi\)
\(578\) −16.9244 −0.703963
\(579\) 0 0
\(580\) −35.3746 −1.46885
\(581\) 4.54983 0.188759
\(582\) 0 0
\(583\) −22.7492 −0.942174
\(584\) −12.8248 −0.530692
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −46.7492 −1.92954 −0.964772 0.263086i \(-0.915260\pi\)
−0.964772 + 0.263086i \(0.915260\pi\)
\(588\) 0 0
\(589\) −18.1993 −0.749891
\(590\) −34.1993 −1.40796
\(591\) 0 0
\(592\) −4.27492 −0.175698
\(593\) 7.09967 0.291548 0.145774 0.989318i \(-0.453433\pi\)
0.145774 + 0.989318i \(0.453433\pi\)
\(594\) 0 0
\(595\) −1.17525 −0.0481805
\(596\) −22.5498 −0.923677
\(597\) 0 0
\(598\) 2.27492 0.0930283
\(599\) −21.7251 −0.887663 −0.443831 0.896110i \(-0.646381\pi\)
−0.443831 + 0.896110i \(0.646381\pi\)
\(600\) 0 0
\(601\) 23.6495 0.964683 0.482342 0.875983i \(-0.339786\pi\)
0.482342 + 0.875983i \(0.339786\pi\)
\(602\) −2.27492 −0.0927187
\(603\) 0 0
\(604\) 6.27492 0.255323
\(605\) −24.9003 −1.01234
\(606\) 0 0
\(607\) 9.17525 0.372412 0.186206 0.982511i \(-0.440381\pi\)
0.186206 + 0.982511i \(0.440381\pi\)
\(608\) −2.27492 −0.0922601
\(609\) 0 0
\(610\) −52.4743 −2.12462
\(611\) 0 0
\(612\) 0 0
\(613\) 16.2749 0.657338 0.328669 0.944445i \(-0.393400\pi\)
0.328669 + 0.944445i \(0.393400\pi\)
\(614\) 21.0997 0.851513
\(615\) 0 0
\(616\) 2.27492 0.0916590
\(617\) 20.8248 0.838373 0.419186 0.907900i \(-0.362315\pi\)
0.419186 + 0.907900i \(0.362315\pi\)
\(618\) 0 0
\(619\) 29.7251 1.19475 0.597376 0.801961i \(-0.296210\pi\)
0.597376 + 0.801961i \(0.296210\pi\)
\(620\) 34.1993 1.37348
\(621\) 0 0
\(622\) −3.45017 −0.138339
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) 84.8488 3.39395
\(626\) −27.6495 −1.10510
\(627\) 0 0
\(628\) −13.3746 −0.533704
\(629\) 1.17525 0.0468602
\(630\) 0 0
\(631\) −10.8248 −0.430927 −0.215463 0.976512i \(-0.569126\pi\)
−0.215463 + 0.976512i \(0.569126\pi\)
\(632\) 12.5498 0.499206
\(633\) 0 0
\(634\) 1.45017 0.0575934
\(635\) −87.8488 −3.48617
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) −18.8248 −0.745279
\(639\) 0 0
\(640\) 4.27492 0.168981
\(641\) 15.0997 0.596401 0.298201 0.954503i \(-0.403614\pi\)
0.298201 + 0.954503i \(0.403614\pi\)
\(642\) 0 0
\(643\) −23.9244 −0.943487 −0.471744 0.881736i \(-0.656375\pi\)
−0.471744 + 0.881736i \(0.656375\pi\)
\(644\) −2.27492 −0.0896443
\(645\) 0 0
\(646\) 0.625414 0.0246066
\(647\) 18.1993 0.715490 0.357745 0.933819i \(-0.383546\pi\)
0.357745 + 0.933819i \(0.383546\pi\)
\(648\) 0 0
\(649\) −18.1993 −0.714386
\(650\) −13.2749 −0.520685
\(651\) 0 0
\(652\) 12.5498 0.491489
\(653\) 5.37459 0.210324 0.105162 0.994455i \(-0.466464\pi\)
0.105162 + 0.994455i \(0.466464\pi\)
\(654\) 0 0
\(655\) −29.1752 −1.13997
\(656\) −6.54983 −0.255728
\(657\) 0 0
\(658\) 0 0
\(659\) 7.45017 0.290217 0.145109 0.989416i \(-0.453647\pi\)
0.145109 + 0.989416i \(0.453647\pi\)
\(660\) 0 0
\(661\) −40.1993 −1.56357 −0.781787 0.623546i \(-0.785691\pi\)
−0.781787 + 0.623546i \(0.785691\pi\)
\(662\) −29.6495 −1.15236
\(663\) 0 0
\(664\) 4.54983 0.176568
\(665\) −9.72508 −0.377123
\(666\) 0 0
\(667\) 18.8248 0.728897
\(668\) 1.72508 0.0667455
\(669\) 0 0
\(670\) 53.6495 2.07266
\(671\) −27.9244 −1.07801
\(672\) 0 0
\(673\) 16.2749 0.627352 0.313676 0.949530i \(-0.398439\pi\)
0.313676 + 0.949530i \(0.398439\pi\)
\(674\) 9.37459 0.361096
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −16.1993 −0.622591 −0.311296 0.950313i \(-0.600763\pi\)
−0.311296 + 0.950313i \(0.600763\pi\)
\(678\) 0 0
\(679\) 15.0997 0.579472
\(680\) −1.17525 −0.0450687
\(681\) 0 0
\(682\) 18.1993 0.696889
\(683\) −3.37459 −0.129125 −0.0645625 0.997914i \(-0.520565\pi\)
−0.0645625 + 0.997914i \(0.520565\pi\)
\(684\) 0 0
\(685\) 74.2749 2.83790
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −2.27492 −0.0867304
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −7.09967 −0.269889
\(693\) 0 0
\(694\) −5.09967 −0.193581
\(695\) 17.0997 0.648627
\(696\) 0 0
\(697\) 1.80066 0.0682049
\(698\) 2.00000 0.0757011
\(699\) 0 0
\(700\) 13.2749 0.501745
\(701\) 15.0997 0.570307 0.285153 0.958482i \(-0.407955\pi\)
0.285153 + 0.958482i \(0.407955\pi\)
\(702\) 0 0
\(703\) 9.72508 0.366788
\(704\) 2.27492 0.0857392
\(705\) 0 0
\(706\) −27.0997 −1.01991
\(707\) −2.54983 −0.0958964
\(708\) 0 0
\(709\) −23.0997 −0.867526 −0.433763 0.901027i \(-0.642815\pi\)
−0.433763 + 0.901027i \(0.642815\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 14.0000 0.524672
\(713\) −18.1993 −0.681571
\(714\) 0 0
\(715\) −9.72508 −0.363697
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 32.0000 1.19423
\(719\) 29.6495 1.10574 0.552870 0.833268i \(-0.313533\pi\)
0.552870 + 0.833268i \(0.313533\pi\)
\(720\) 0 0
\(721\) 10.2749 0.382658
\(722\) −13.8248 −0.514504
\(723\) 0 0
\(724\) 19.0997 0.709834
\(725\) −109.849 −4.07968
\(726\) 0 0
\(727\) 35.3746 1.31197 0.655985 0.754774i \(-0.272253\pi\)
0.655985 + 0.754774i \(0.272253\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 0 0
\(730\) −54.8248 −2.02916
\(731\) 0.625414 0.0231318
\(732\) 0 0
\(733\) 14.5498 0.537410 0.268705 0.963222i \(-0.413404\pi\)
0.268705 + 0.963222i \(0.413404\pi\)
\(734\) −2.90033 −0.107053
\(735\) 0 0
\(736\) −2.27492 −0.0838546
\(737\) 28.5498 1.05165
\(738\) 0 0
\(739\) 37.6495 1.38496 0.692480 0.721437i \(-0.256518\pi\)
0.692480 + 0.721437i \(0.256518\pi\)
\(740\) −18.2749 −0.671799
\(741\) 0 0
\(742\) −10.0000 −0.367112
\(743\) −11.4502 −0.420066 −0.210033 0.977694i \(-0.567357\pi\)
−0.210033 + 0.977694i \(0.567357\pi\)
\(744\) 0 0
\(745\) −96.3987 −3.53177
\(746\) 26.5498 0.972059
\(747\) 0 0
\(748\) −0.625414 −0.0228674
\(749\) −0.549834 −0.0200905
\(750\) 0 0
\(751\) 10.1993 0.372179 0.186090 0.982533i \(-0.440419\pi\)
0.186090 + 0.982533i \(0.440419\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 8.27492 0.301355
\(755\) 26.8248 0.976253
\(756\) 0 0
\(757\) −35.0997 −1.27572 −0.637860 0.770153i \(-0.720180\pi\)
−0.637860 + 0.770153i \(0.720180\pi\)
\(758\) −33.0997 −1.20223
\(759\) 0 0
\(760\) −9.72508 −0.352766
\(761\) −38.5498 −1.39743 −0.698715 0.715400i \(-0.746245\pi\)
−0.698715 + 0.715400i \(0.746245\pi\)
\(762\) 0 0
\(763\) 0.274917 0.00995267
\(764\) −2.27492 −0.0823036
\(765\) 0 0
\(766\) −30.2749 −1.09388
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 32.8248 1.18369 0.591845 0.806051i \(-0.298400\pi\)
0.591845 + 0.806051i \(0.298400\pi\)
\(770\) 9.72508 0.350468
\(771\) 0 0
\(772\) −18.5498 −0.667623
\(773\) 9.92442 0.356957 0.178478 0.983944i \(-0.442883\pi\)
0.178478 + 0.983944i \(0.442883\pi\)
\(774\) 0 0
\(775\) 106.199 3.81479
\(776\) 15.0997 0.542047
\(777\) 0 0
\(778\) −28.1993 −1.01100
\(779\) 14.9003 0.533860
\(780\) 0 0
\(781\) 0 0
\(782\) 0.625414 0.0223648
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −57.1752 −2.04067
\(786\) 0 0
\(787\) −10.2749 −0.366261 −0.183131 0.983089i \(-0.558623\pi\)
−0.183131 + 0.983089i \(0.558623\pi\)
\(788\) 2.54983 0.0908341
\(789\) 0 0
\(790\) 53.6495 1.90876
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 12.2749 0.435895
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 6.82475 0.241897
\(797\) 15.6495 0.554334 0.277167 0.960822i \(-0.410604\pi\)
0.277167 + 0.960822i \(0.410604\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 13.2749 0.469339
\(801\) 0 0
\(802\) 3.09967 0.109453
\(803\) −29.1752 −1.02957
\(804\) 0 0
\(805\) −9.72508 −0.342764
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) −2.54983 −0.0897029
\(809\) 56.1993 1.97586 0.987932 0.154890i \(-0.0495023\pi\)
0.987932 + 0.154890i \(0.0495023\pi\)
\(810\) 0 0
\(811\) −11.3746 −0.399416 −0.199708 0.979855i \(-0.563999\pi\)
−0.199708 + 0.979855i \(0.563999\pi\)
\(812\) −8.27492 −0.290393
\(813\) 0 0
\(814\) −9.72508 −0.340864
\(815\) 53.6495 1.87926
\(816\) 0 0
\(817\) 5.17525 0.181059
\(818\) −7.17525 −0.250877
\(819\) 0 0
\(820\) −28.0000 −0.977802
\(821\) −31.6495 −1.10458 −0.552288 0.833654i \(-0.686245\pi\)
−0.552288 + 0.833654i \(0.686245\pi\)
\(822\) 0 0
\(823\) −25.0997 −0.874919 −0.437460 0.899238i \(-0.644122\pi\)
−0.437460 + 0.899238i \(0.644122\pi\)
\(824\) 10.2749 0.357944
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 26.2749 0.913668 0.456834 0.889552i \(-0.348983\pi\)
0.456834 + 0.889552i \(0.348983\pi\)
\(828\) 0 0
\(829\) 11.7251 0.407229 0.203614 0.979051i \(-0.434731\pi\)
0.203614 + 0.979051i \(0.434731\pi\)
\(830\) 19.4502 0.675125
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) −0.274917 −0.00952532
\(834\) 0 0
\(835\) 7.37459 0.255208
\(836\) −5.17525 −0.178990
\(837\) 0 0
\(838\) −2.27492 −0.0785857
\(839\) −22.9003 −0.790607 −0.395304 0.918551i \(-0.629361\pi\)
−0.395304 + 0.918551i \(0.629361\pi\)
\(840\) 0 0
\(841\) 39.4743 1.36118
\(842\) 3.09967 0.106822
\(843\) 0 0
\(844\) 1.17525 0.0404537
\(845\) 4.27492 0.147062
\(846\) 0 0
\(847\) −5.82475 −0.200141
\(848\) −10.0000 −0.343401
\(849\) 0 0
\(850\) −3.64950 −0.125177
\(851\) 9.72508 0.333372
\(852\) 0 0
\(853\) −27.6495 −0.946701 −0.473350 0.880874i \(-0.656956\pi\)
−0.473350 + 0.880874i \(0.656956\pi\)
\(854\) −12.2749 −0.420039
\(855\) 0 0
\(856\) −0.549834 −0.0187930
\(857\) −19.0997 −0.652432 −0.326216 0.945295i \(-0.605774\pi\)
−0.326216 + 0.945295i \(0.605774\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) −9.72508 −0.331623
\(861\) 0 0
\(862\) 11.4502 0.389994
\(863\) −29.6495 −1.00928 −0.504640 0.863330i \(-0.668375\pi\)
−0.504640 + 0.863330i \(0.668375\pi\)
\(864\) 0 0
\(865\) −30.3505 −1.03195
\(866\) 23.6495 0.803643
\(867\) 0 0
\(868\) 8.00000 0.271538
\(869\) 28.5498 0.968487
\(870\) 0 0
\(871\) −12.5498 −0.425235
\(872\) 0.274917 0.00930987
\(873\) 0 0
\(874\) 5.17525 0.175055
\(875\) 35.3746 1.19588
\(876\) 0 0
\(877\) 51.0997 1.72551 0.862757 0.505619i \(-0.168736\pi\)
0.862757 + 0.505619i \(0.168736\pi\)
\(878\) −14.8248 −0.500311
\(879\) 0 0
\(880\) 9.72508 0.327832
\(881\) 31.7251 1.06885 0.534423 0.845217i \(-0.320529\pi\)
0.534423 + 0.845217i \(0.320529\pi\)
\(882\) 0 0
\(883\) 31.9244 1.07434 0.537171 0.843473i \(-0.319493\pi\)
0.537171 + 0.843473i \(0.319493\pi\)
\(884\) 0.274917 0.00924647
\(885\) 0 0
\(886\) −7.45017 −0.250293
\(887\) 33.0997 1.11138 0.555689 0.831390i \(-0.312455\pi\)
0.555689 + 0.831390i \(0.312455\pi\)
\(888\) 0 0
\(889\) −20.5498 −0.689219
\(890\) 59.8488 2.00614
\(891\) 0 0
\(892\) 21.6495 0.724879
\(893\) 0 0
\(894\) 0 0
\(895\) −51.2990 −1.71474
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 0.274917 0.00917411
\(899\) −66.1993 −2.20787
\(900\) 0 0
\(901\) 2.74917 0.0915882
\(902\) −14.9003 −0.496127
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 81.6495 2.71412
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 20.5498 0.681970
\(909\) 0 0
\(910\) −4.27492 −0.141712
\(911\) 52.4743 1.73855 0.869275 0.494329i \(-0.164586\pi\)
0.869275 + 0.494329i \(0.164586\pi\)
\(912\) 0 0
\(913\) 10.3505 0.342551
\(914\) −18.5498 −0.613574
\(915\) 0 0
\(916\) 19.0997 0.631071
\(917\) −6.82475 −0.225373
\(918\) 0 0
\(919\) −12.5498 −0.413981 −0.206990 0.978343i \(-0.566367\pi\)
−0.206990 + 0.978343i \(0.566367\pi\)
\(920\) −9.72508 −0.320626
\(921\) 0 0
\(922\) 17.9244 0.590309
\(923\) 0 0
\(924\) 0 0
\(925\) −56.7492 −1.86590
\(926\) 30.2749 0.994896
\(927\) 0 0
\(928\) −8.27492 −0.271637
\(929\) −14.5498 −0.477365 −0.238682 0.971098i \(-0.576715\pi\)
−0.238682 + 0.971098i \(0.576715\pi\)
\(930\) 0 0
\(931\) −2.27492 −0.0745574
\(932\) −5.45017 −0.178526
\(933\) 0 0
\(934\) 26.2749 0.859742
\(935\) −2.67359 −0.0874358
\(936\) 0 0
\(937\) 16.9003 0.552110 0.276055 0.961142i \(-0.410973\pi\)
0.276055 + 0.961142i \(0.410973\pi\)
\(938\) 12.5498 0.409767
\(939\) 0 0
\(940\) 0 0
\(941\) −51.0997 −1.66580 −0.832901 0.553422i \(-0.813322\pi\)
−0.832901 + 0.553422i \(0.813322\pi\)
\(942\) 0 0
\(943\) 14.9003 0.485222
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) −5.17525 −0.168262
\(947\) 17.1752 0.558121 0.279060 0.960274i \(-0.409977\pi\)
0.279060 + 0.960274i \(0.409977\pi\)
\(948\) 0 0
\(949\) 12.8248 0.416309
\(950\) −30.1993 −0.979796
\(951\) 0 0
\(952\) −0.274917 −0.00891012
\(953\) −31.6495 −1.02523 −0.512614 0.858619i \(-0.671323\pi\)
−0.512614 + 0.858619i \(0.671323\pi\)
\(954\) 0 0
\(955\) −9.72508 −0.314696
\(956\) −25.0997 −0.811781
\(957\) 0 0
\(958\) 23.3746 0.755199
\(959\) 17.3746 0.561055
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 4.27492 0.137829
\(963\) 0 0
\(964\) 15.0997 0.486328
\(965\) −79.2990 −2.55273
\(966\) 0 0
\(967\) 42.8248 1.37715 0.688576 0.725165i \(-0.258236\pi\)
0.688576 + 0.725165i \(0.258236\pi\)
\(968\) −5.82475 −0.187215
\(969\) 0 0
\(970\) 64.5498 2.07257
\(971\) 29.0997 0.933853 0.466926 0.884296i \(-0.345361\pi\)
0.466926 + 0.884296i \(0.345361\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) −18.1993 −0.583144
\(975\) 0 0
\(976\) −12.2749 −0.392911
\(977\) 13.9244 0.445482 0.222741 0.974878i \(-0.428500\pi\)
0.222741 + 0.974878i \(0.428500\pi\)
\(978\) 0 0
\(979\) 31.8488 1.01789
\(980\) 4.27492 0.136557
\(981\) 0 0
\(982\) −8.54983 −0.272836
\(983\) −61.0241 −1.94637 −0.973183 0.230032i \(-0.926117\pi\)
−0.973183 + 0.230032i \(0.926117\pi\)
\(984\) 0 0
\(985\) 10.9003 0.347313
\(986\) 2.27492 0.0724481
\(987\) 0 0
\(988\) 2.27492 0.0723748
\(989\) 5.17525 0.164563
\(990\) 0 0
\(991\) −46.7492 −1.48504 −0.742518 0.669826i \(-0.766369\pi\)
−0.742518 + 0.669826i \(0.766369\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) 0 0
\(995\) 29.1752 0.924918
\(996\) 0 0
\(997\) −12.9003 −0.408558 −0.204279 0.978913i \(-0.565485\pi\)
−0.204279 + 0.978913i \(0.565485\pi\)
\(998\) 25.0997 0.794516
\(999\) 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 1638.2.a.y.1.2 2
3.2 odd 2 546.2.a.h.1.1 2
12.11 even 2 4368.2.a.bh.1.1 2
21.20 even 2 3822.2.a.bm.1.2 2
39.38 odd 2 7098.2.a.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.h.1.1 2 3.2 odd 2
1638.2.a.y.1.2 2 1.1 even 1 trivial
3822.2.a.bm.1.2 2 21.20 even 2
4368.2.a.bh.1.1 2 12.11 even 2
7098.2.a.bu.1.2 2 39.38 odd 2