Properties

Label 3822.2.a.bm.1.2
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
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 \(\chi\) \(=\) 3822.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.27492 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.27492 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -4.27492 q^{10} -2.27492 q^{11} +1.00000 q^{12} +1.00000 q^{13} +4.27492 q^{15} +1.00000 q^{16} -0.274917 q^{17} -1.00000 q^{18} +2.27492 q^{19} +4.27492 q^{20} +2.27492 q^{22} +2.27492 q^{23} -1.00000 q^{24} +13.2749 q^{25} -1.00000 q^{26} +1.00000 q^{27} +8.27492 q^{29} -4.27492 q^{30} -8.00000 q^{31} -1.00000 q^{32} -2.27492 q^{33} +0.274917 q^{34} +1.00000 q^{36} -4.27492 q^{37} -2.27492 q^{38} +1.00000 q^{39} -4.27492 q^{40} -6.54983 q^{41} -2.27492 q^{43} -2.27492 q^{44} +4.27492 q^{45} -2.27492 q^{46} +1.00000 q^{48} -13.2749 q^{50} -0.274917 q^{51} +1.00000 q^{52} +10.0000 q^{53} -1.00000 q^{54} -9.72508 q^{55} +2.27492 q^{57} -8.27492 q^{58} -8.00000 q^{59} +4.27492 q^{60} +12.2749 q^{61} +8.00000 q^{62} +1.00000 q^{64} +4.27492 q^{65} +2.27492 q^{66} +12.5498 q^{67} -0.274917 q^{68} +2.27492 q^{69} -1.00000 q^{72} +12.8248 q^{73} +4.27492 q^{74} +13.2749 q^{75} +2.27492 q^{76} -1.00000 q^{78} +12.5498 q^{79} +4.27492 q^{80} +1.00000 q^{81} +6.54983 q^{82} +4.54983 q^{83} -1.17525 q^{85} +2.27492 q^{86} +8.27492 q^{87} +2.27492 q^{88} +14.0000 q^{89} -4.27492 q^{90} +2.27492 q^{92} -8.00000 q^{93} +9.72508 q^{95} -1.00000 q^{96} -15.0997 q^{97} -2.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + q^{5} - 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + q^{5} - 2q^{6} - 2q^{8} + 2q^{9} - q^{10} + 3q^{11} + 2q^{12} + 2q^{13} + q^{15} + 2q^{16} + 7q^{17} - 2q^{18} - 3q^{19} + q^{20} - 3q^{22} - 3q^{23} - 2q^{24} + 19q^{25} - 2q^{26} + 2q^{27} + 9q^{29} - q^{30} - 16q^{31} - 2q^{32} + 3q^{33} - 7q^{34} + 2q^{36} - q^{37} + 3q^{38} + 2q^{39} - q^{40} + 2q^{41} + 3q^{43} + 3q^{44} + q^{45} + 3q^{46} + 2q^{48} - 19q^{50} + 7q^{51} + 2q^{52} + 20q^{53} - 2q^{54} - 27q^{55} - 3q^{57} - 9q^{58} - 16q^{59} + q^{60} + 17q^{61} + 16q^{62} + 2q^{64} + q^{65} - 3q^{66} + 10q^{67} + 7q^{68} - 3q^{69} - 2q^{72} + 3q^{73} + q^{74} + 19q^{75} - 3q^{76} - 2q^{78} + 10q^{79} + q^{80} + 2q^{81} - 2q^{82} - 6q^{83} - 25q^{85} - 3q^{86} + 9q^{87} - 3q^{88} + 28q^{89} - q^{90} - 3q^{92} - 16q^{93} + 27q^{95} - 2q^{96} + 3q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 4.27492 1.91180 0.955901 0.293691i \(-0.0948835\pi\)
0.955901 + 0.293691i \(0.0948835\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −4.27492 −1.35185
\(11\) −2.27492 −0.685913 −0.342957 0.939351i \(-0.611428\pi\)
−0.342957 + 0.939351i \(0.611428\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 4.27492 1.10378
\(16\) 1.00000 0.250000
\(17\) −0.274917 −0.0666772 −0.0333386 0.999444i \(-0.510614\pi\)
−0.0333386 + 0.999444i \(0.510614\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.27492 0.521902 0.260951 0.965352i \(-0.415964\pi\)
0.260951 + 0.965352i \(0.415964\pi\)
\(20\) 4.27492 0.955901
\(21\) 0 0
\(22\) 2.27492 0.485014
\(23\) 2.27492 0.474353 0.237177 0.971467i \(-0.423778\pi\)
0.237177 + 0.971467i \(0.423778\pi\)
\(24\) −1.00000 −0.204124
\(25\) 13.2749 2.65498
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 8.27492 1.53661 0.768307 0.640082i \(-0.221100\pi\)
0.768307 + 0.640082i \(0.221100\pi\)
\(30\) −4.27492 −0.780490
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.27492 −0.396012
\(34\) 0.274917 0.0471479
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.27492 −0.702792 −0.351396 0.936227i \(-0.614293\pi\)
−0.351396 + 0.936227i \(0.614293\pi\)
\(38\) −2.27492 −0.369040
\(39\) 1.00000 0.160128
\(40\) −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
\(45\) 4.27492 0.637267
\(46\) −2.27492 −0.335418
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −13.2749 −1.87736
\(51\) −0.274917 −0.0384961
\(52\) 1.00000 0.138675
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) −1.00000 −0.136083
\(55\) −9.72508 −1.31133
\(56\) 0 0
\(57\) 2.27492 0.301320
\(58\) −8.27492 −1.08655
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 4.27492 0.551889
\(61\) 12.2749 1.57164 0.785821 0.618454i \(-0.212241\pi\)
0.785821 + 0.618454i \(0.212241\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.27492 0.530238
\(66\) 2.27492 0.280023
\(67\) 12.5498 1.53321 0.766603 0.642121i \(-0.221945\pi\)
0.766603 + 0.642121i \(0.221945\pi\)
\(68\) −0.274917 −0.0333386
\(69\) 2.27492 0.273868
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.8248 1.50102 0.750512 0.660857i \(-0.229807\pi\)
0.750512 + 0.660857i \(0.229807\pi\)
\(74\) 4.27492 0.496949
\(75\) 13.2749 1.53286
\(76\) 2.27492 0.260951
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) 12.5498 1.41197 0.705983 0.708228i \(-0.250505\pi\)
0.705983 + 0.708228i \(0.250505\pi\)
\(80\) 4.27492 0.477950
\(81\) 1.00000 0.111111
\(82\) 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
\(87\) 8.27492 0.887164
\(88\) 2.27492 0.242507
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) −4.27492 −0.450616
\(91\) 0 0
\(92\) 2.27492 0.237177
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 9.72508 0.997772
\(96\) −1.00000 −0.102062
\(97\) −15.0997 −1.53314 −0.766570 0.642161i \(-0.778038\pi\)
−0.766570 + 0.642161i \(0.778038\pi\)
\(98\) 0 0
\(99\) −2.27492 −0.228638
\(100\) 13.2749 1.32749
\(101\) −2.54983 −0.253718 −0.126859 0.991921i \(-0.540490\pi\)
−0.126859 + 0.991921i \(0.540490\pi\)
\(102\) 0.274917 0.0272209
\(103\) −10.2749 −1.01242 −0.506209 0.862411i \(-0.668954\pi\)
−0.506209 + 0.862411i \(0.668954\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) 0.549834 0.0531545 0.0265773 0.999647i \(-0.491539\pi\)
0.0265773 + 0.999647i \(0.491539\pi\)
\(108\) 1.00000 0.0962250
\(109\) 0.274917 0.0263323 0.0131661 0.999913i \(-0.495809\pi\)
0.0131661 + 0.999913i \(0.495809\pi\)
\(110\) 9.72508 0.927250
\(111\) −4.27492 −0.405757
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −2.27492 −0.213066
\(115\) 9.72508 0.906869
\(116\) 8.27492 0.768307
\(117\) 1.00000 0.0924500
\(118\) 8.00000 0.736460
\(119\) 0 0
\(120\) −4.27492 −0.390245
\(121\) −5.82475 −0.529523
\(122\) −12.2749 −1.11132
\(123\) −6.54983 −0.590579
\(124\) −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
\(129\) −2.27492 −0.200295
\(130\) −4.27492 −0.374935
\(131\) −6.82475 −0.596281 −0.298141 0.954522i \(-0.596366\pi\)
−0.298141 + 0.954522i \(0.596366\pi\)
\(132\) −2.27492 −0.198006
\(133\) 0 0
\(134\) −12.5498 −1.08414
\(135\) 4.27492 0.367926
\(136\) 0.274917 0.0235740
\(137\) −17.3746 −1.48441 −0.742206 0.670172i \(-0.766220\pi\)
−0.742206 + 0.670172i \(0.766220\pi\)
\(138\) −2.27492 −0.193654
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.27492 −0.190238
\(144\) 1.00000 0.0833333
\(145\) 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 \(-0.125168\pi\)
0.923677 + 0.383172i \(0.125168\pi\)
\(150\) −13.2749 −1.08389
\(151\) 6.27492 0.510646 0.255323 0.966856i \(-0.417818\pi\)
0.255323 + 0.966856i \(0.417818\pi\)
\(152\) −2.27492 −0.184520
\(153\) −0.274917 −0.0222257
\(154\) 0 0
\(155\) −34.1993 −2.74696
\(156\) 1.00000 0.0800641
\(157\) 13.3746 1.06741 0.533704 0.845671i \(-0.320800\pi\)
0.533704 + 0.845671i \(0.320800\pi\)
\(158\) −12.5498 −0.998411
\(159\) 10.0000 0.793052
\(160\) −4.27492 −0.337962
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 12.5498 0.982979 0.491489 0.870884i \(-0.336453\pi\)
0.491489 + 0.870884i \(0.336453\pi\)
\(164\) −6.54983 −0.511456
\(165\) −9.72508 −0.757097
\(166\) −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
\(171\) 2.27492 0.173967
\(172\) −2.27492 −0.173461
\(173\) −7.09967 −0.539778 −0.269889 0.962891i \(-0.586987\pi\)
−0.269889 + 0.962891i \(0.586987\pi\)
\(174\) −8.27492 −0.627320
\(175\) 0 0
\(176\) −2.27492 −0.171478
\(177\) −8.00000 −0.601317
\(178\) −14.0000 −1.04934
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 4.27492 0.318634
\(181\) −19.0997 −1.41967 −0.709834 0.704369i \(-0.751230\pi\)
−0.709834 + 0.704369i \(0.751230\pi\)
\(182\) 0 0
\(183\) 12.2749 0.907388
\(184\) −2.27492 −0.167709
\(185\) −18.2749 −1.34360
\(186\) 8.00000 0.586588
\(187\) 0.625414 0.0457348
\(188\) 0 0
\(189\) 0 0
\(190\) −9.72508 −0.705532
\(191\) 2.27492 0.164607 0.0823036 0.996607i \(-0.473772\pi\)
0.0823036 + 0.996607i \(0.473772\pi\)
\(192\) 1.00000 0.0721688
\(193\) −18.5498 −1.33525 −0.667623 0.744499i \(-0.732688\pi\)
−0.667623 + 0.744499i \(0.732688\pi\)
\(194\) 15.0997 1.08409
\(195\) 4.27492 0.306133
\(196\) 0 0
\(197\) −2.54983 −0.181668 −0.0908341 0.995866i \(-0.528953\pi\)
−0.0908341 + 0.995866i \(0.528953\pi\)
\(198\) 2.27492 0.161671
\(199\) −6.82475 −0.483794 −0.241897 0.970302i \(-0.577770\pi\)
−0.241897 + 0.970302i \(0.577770\pi\)
\(200\) −13.2749 −0.938678
\(201\) 12.5498 0.885197
\(202\) 2.54983 0.179406
\(203\) 0 0
\(204\) −0.274917 −0.0192481
\(205\) −28.0000 −1.95560
\(206\) 10.2749 0.715887
\(207\) 2.27492 0.158118
\(208\) 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
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −0.274917 −0.0186197
\(219\) 12.8248 0.866616
\(220\) −9.72508 −0.655665
\(221\) −0.274917 −0.0184929
\(222\) 4.27492 0.286914
\(223\) −21.6495 −1.44976 −0.724879 0.688876i \(-0.758104\pi\)
−0.724879 + 0.688876i \(0.758104\pi\)
\(224\) 0 0
\(225\) 13.2749 0.884994
\(226\) 6.00000 0.399114
\(227\) 20.5498 1.36394 0.681970 0.731380i \(-0.261123\pi\)
0.681970 + 0.731380i \(0.261123\pi\)
\(228\) 2.27492 0.150660
\(229\) −19.0997 −1.26214 −0.631071 0.775725i \(-0.717384\pi\)
−0.631071 + 0.775725i \(0.717384\pi\)
\(230\) −9.72508 −0.641253
\(231\) 0 0
\(232\) −8.27492 −0.543275
\(233\) 5.45017 0.357052 0.178526 0.983935i \(-0.442867\pi\)
0.178526 + 0.983935i \(0.442867\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 12.5498 0.815199
\(238\) 0 0
\(239\) 25.0997 1.62356 0.811781 0.583962i \(-0.198498\pi\)
0.811781 + 0.583962i \(0.198498\pi\)
\(240\) 4.27492 0.275945
\(241\) −15.0997 −0.972655 −0.486328 0.873777i \(-0.661664\pi\)
−0.486328 + 0.873777i \(0.661664\pi\)
\(242\) 5.82475 0.374429
\(243\) 1.00000 0.0641500
\(244\) 12.2749 0.785821
\(245\) 0 0
\(246\) 6.54983 0.417602
\(247\) 2.27492 0.144750
\(248\) 8.00000 0.508001
\(249\) 4.54983 0.288334
\(250\) −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
\(255\) −1.17525 −0.0735969
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 2.27492 0.141630
\(259\) 0 0
\(260\) 4.27492 0.265119
\(261\) 8.27492 0.512205
\(262\) 6.82475 0.421635
\(263\) 28.0000 1.72655 0.863277 0.504730i \(-0.168408\pi\)
0.863277 + 0.504730i \(0.168408\pi\)
\(264\) 2.27492 0.140011
\(265\) 42.7492 2.62606
\(266\) 0 0
\(267\) 14.0000 0.856786
\(268\) 12.5498 0.766603
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) −4.27492 −0.260163
\(271\) −28.5498 −1.73428 −0.867139 0.498065i \(-0.834044\pi\)
−0.867139 + 0.498065i \(0.834044\pi\)
\(272\) −0.274917 −0.0166693
\(273\) 0 0
\(274\) 17.3746 1.04964
\(275\) −30.1993 −1.82109
\(276\) 2.27492 0.136934
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 4.00000 0.239904
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 15.0997 0.900771 0.450385 0.892834i \(-0.351287\pi\)
0.450385 + 0.892834i \(0.351287\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 9.72508 0.576064
\(286\) 2.27492 0.134519
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −16.9244 −0.995554
\(290\) −35.3746 −2.07727
\(291\) −15.0997 −0.885158
\(292\) 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
\(297\) −2.27492 −0.132004
\(298\) −22.5498 −1.30628
\(299\) 2.27492 0.131562
\(300\) 13.2749 0.766428
\(301\) 0 0
\(302\) −6.27492 −0.361081
\(303\) −2.54983 −0.146484
\(304\) 2.27492 0.130475
\(305\) 52.4743 3.00467
\(306\) 0.274917 0.0157160
\(307\) −21.0997 −1.20422 −0.602111 0.798412i \(-0.705673\pi\)
−0.602111 + 0.798412i \(0.705673\pi\)
\(308\) 0 0
\(309\) −10.2749 −0.584520
\(310\) 34.1993 1.94239
\(311\) −3.45017 −0.195641 −0.0978205 0.995204i \(-0.531187\pi\)
−0.0978205 + 0.995204i \(0.531187\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 27.6495 1.56284 0.781421 0.624004i \(-0.214495\pi\)
0.781421 + 0.624004i \(0.214495\pi\)
\(314\) −13.3746 −0.754772
\(315\) 0 0
\(316\) 12.5498 0.705983
\(317\) −1.45017 −0.0814494 −0.0407247 0.999170i \(-0.512967\pi\)
−0.0407247 + 0.999170i \(0.512967\pi\)
\(318\) −10.0000 −0.560772
\(319\) −18.8248 −1.05398
\(320\) 4.27492 0.238975
\(321\) 0.549834 0.0306888
\(322\) 0 0
\(323\) −0.625414 −0.0347990
\(324\) 1.00000 0.0555556
\(325\) 13.2749 0.736360
\(326\) −12.5498 −0.695071
\(327\) 0.274917 0.0152030
\(328\) 6.54983 0.361654
\(329\) 0 0
\(330\) 9.72508 0.535348
\(331\) −29.6495 −1.62968 −0.814842 0.579683i \(-0.803176\pi\)
−0.814842 + 0.579683i \(0.803176\pi\)
\(332\) 4.54983 0.249705
\(333\) −4.27492 −0.234264
\(334\) −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
\(339\) −6.00000 −0.325875
\(340\) −1.17525 −0.0637368
\(341\) 18.1993 0.985549
\(342\) −2.27492 −0.123013
\(343\) 0 0
\(344\) 2.27492 0.122655
\(345\) 9.72508 0.523581
\(346\) 7.09967 0.381681
\(347\) 5.09967 0.273765 0.136882 0.990587i \(-0.456292\pi\)
0.136882 + 0.990587i \(0.456292\pi\)
\(348\) 8.27492 0.443582
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 2.27492 0.121253
\(353\) −27.0997 −1.44237 −0.721185 0.692743i \(-0.756402\pi\)
−0.721185 + 0.692743i \(0.756402\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) −4.27492 −0.225308
\(361\) −13.8248 −0.727619
\(362\) 19.0997 1.00386
\(363\) −5.82475 −0.305720
\(364\) 0 0
\(365\) 54.8248 2.86966
\(366\) −12.2749 −0.641620
\(367\) 2.90033 0.151396 0.0756980 0.997131i \(-0.475881\pi\)
0.0756980 + 0.997131i \(0.475881\pi\)
\(368\) 2.27492 0.118588
\(369\) −6.54983 −0.340971
\(370\) 18.2749 0.950068
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) 26.5498 1.37470 0.687349 0.726327i \(-0.258774\pi\)
0.687349 + 0.726327i \(0.258774\pi\)
\(374\) −0.625414 −0.0323394
\(375\) 35.3746 1.82674
\(376\) 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
\(381\) −20.5498 −1.05280
\(382\) −2.27492 −0.116395
\(383\) −30.2749 −1.54698 −0.773488 0.633811i \(-0.781490\pi\)
−0.773488 + 0.633811i \(0.781490\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 18.5498 0.944162
\(387\) −2.27492 −0.115641
\(388\) −15.0997 −0.766570
\(389\) 28.1993 1.42976 0.714882 0.699246i \(-0.246481\pi\)
0.714882 + 0.699246i \(0.246481\pi\)
\(390\) −4.27492 −0.216469
\(391\) −0.625414 −0.0316285
\(392\) 0 0
\(393\) −6.82475 −0.344263
\(394\) 2.54983 0.128459
\(395\) 53.6495 2.69940
\(396\) −2.27492 −0.114319
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 6.82475 0.342094
\(399\) 0 0
\(400\) 13.2749 0.663746
\(401\) −3.09967 −0.154790 −0.0773950 0.997001i \(-0.524660\pi\)
−0.0773950 + 0.997001i \(0.524660\pi\)
\(402\) −12.5498 −0.625929
\(403\) −8.00000 −0.398508
\(404\) −2.54983 −0.126859
\(405\) 4.27492 0.212422
\(406\) 0 0
\(407\) 9.72508 0.482054
\(408\) 0.274917 0.0136104
\(409\) 7.17525 0.354793 0.177397 0.984139i \(-0.443232\pi\)
0.177397 + 0.984139i \(0.443232\pi\)
\(410\) 28.0000 1.38282
\(411\) −17.3746 −0.857025
\(412\) −10.2749 −0.506209
\(413\) 0 0
\(414\) −2.27492 −0.111806
\(415\) 19.4502 0.954771
\(416\) −1.00000 −0.0490290
\(417\) −4.00000 −0.195881
\(418\) 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
\(427\) 0 0
\(428\) 0.549834 0.0265773
\(429\) −2.27492 −0.109834
\(430\) 9.72508 0.468985
\(431\) −11.4502 −0.551535 −0.275768 0.961224i \(-0.588932\pi\)
−0.275768 + 0.961224i \(0.588932\pi\)
\(432\) 1.00000 0.0481125
\(433\) −23.6495 −1.13652 −0.568261 0.822848i \(-0.692384\pi\)
−0.568261 + 0.822848i \(0.692384\pi\)
\(434\) 0 0
\(435\) 35.3746 1.69608
\(436\) 0.274917 0.0131661
\(437\) 5.17525 0.247566
\(438\) −12.8248 −0.612790
\(439\) 14.8248 0.707547 0.353773 0.935331i \(-0.384898\pi\)
0.353773 + 0.935331i \(0.384898\pi\)
\(440\) 9.72508 0.463625
\(441\) 0 0
\(442\) 0.274917 0.0130765
\(443\) 7.45017 0.353968 0.176984 0.984214i \(-0.443366\pi\)
0.176984 + 0.984214i \(0.443366\pi\)
\(444\) −4.27492 −0.202879
\(445\) 59.8488 2.83711
\(446\) 21.6495 1.02513
\(447\) 22.5498 1.06657
\(448\) 0 0
\(449\) −0.274917 −0.0129741 −0.00648707 0.999979i \(-0.502065\pi\)
−0.00648707 + 0.999979i \(0.502065\pi\)
\(450\) −13.2749 −0.625786
\(451\) 14.9003 0.701629
\(452\) −6.00000 −0.282216
\(453\) 6.27492 0.294821
\(454\) −20.5498 −0.964452
\(455\) 0 0
\(456\) −2.27492 −0.106533
\(457\) −18.5498 −0.867725 −0.433862 0.900979i \(-0.642850\pi\)
−0.433862 + 0.900979i \(0.642850\pi\)
\(458\) 19.0997 0.892469
\(459\) −0.274917 −0.0128320
\(460\) 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
\(465\) −34.1993 −1.58596
\(466\) −5.45017 −0.252474
\(467\) 26.2749 1.21586 0.607929 0.793991i \(-0.292000\pi\)
0.607929 + 0.793991i \(0.292000\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) 13.3746 0.616268
\(472\) 8.00000 0.368230
\(473\) 5.17525 0.237958
\(474\) −12.5498 −0.576433
\(475\) 30.1993 1.38564
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) −25.0997 −1.14803
\(479\) 23.3746 1.06801 0.534006 0.845481i \(-0.320686\pi\)
0.534006 + 0.845481i \(0.320686\pi\)
\(480\) −4.27492 −0.195122
\(481\) −4.27492 −0.194919
\(482\) 15.0997 0.687771
\(483\) 0 0
\(484\) −5.82475 −0.264761
\(485\) −64.5498 −2.93106
\(486\) −1.00000 −0.0453609
\(487\) −18.1993 −0.824691 −0.412345 0.911028i \(-0.635290\pi\)
−0.412345 + 0.911028i \(0.635290\pi\)
\(488\) −12.2749 −0.555659
\(489\) 12.5498 0.567523
\(490\) 0 0
\(491\) 8.54983 0.385849 0.192924 0.981214i \(-0.438203\pi\)
0.192924 + 0.981214i \(0.438203\pi\)
\(492\) −6.54983 −0.295289
\(493\) −2.27492 −0.102457
\(494\) −2.27492 −0.102353
\(495\) −9.72508 −0.437110
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) −4.54983 −0.203883
\(499\) 25.0997 1.12362 0.561808 0.827268i \(-0.310106\pi\)
0.561808 + 0.827268i \(0.310106\pi\)
\(500\) 35.3746 1.58200
\(501\) 1.72508 0.0770710
\(502\) 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
\(507\) 1.00000 0.0444116
\(508\) −20.5498 −0.911751
\(509\) 9.92442 0.439892 0.219946 0.975512i \(-0.429412\pi\)
0.219946 + 0.975512i \(0.429412\pi\)
\(510\) 1.17525 0.0520409
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 2.27492 0.100440
\(514\) −14.0000 −0.617514
\(515\) −43.9244 −1.93554
\(516\) −2.27492 −0.100148
\(517\) 0 0
\(518\) 0 0
\(519\) −7.09967 −0.311641
\(520\) −4.27492 −0.187468
\(521\) 4.27492 0.187288 0.0936438 0.995606i \(-0.470149\pi\)
0.0936438 + 0.995606i \(0.470149\pi\)
\(522\) −8.27492 −0.362183
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) −6.82475 −0.298141
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) 2.19934 0.0958047
\(528\) −2.27492 −0.0990031
\(529\) −17.8248 −0.774989
\(530\) −42.7492 −1.85691
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) −6.54983 −0.283705
\(534\) −14.0000 −0.605839
\(535\) 2.35050 0.101621
\(536\) −12.5498 −0.542070
\(537\) 12.0000 0.517838
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) 4.27492 0.183963
\(541\) 42.4743 1.82611 0.913055 0.407835i \(-0.133716\pi\)
0.913055 + 0.407835i \(0.133716\pi\)
\(542\) 28.5498 1.22632
\(543\) −19.0997 −0.819645
\(544\) 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
\(549\) 12.2749 0.523881
\(550\) 30.1993 1.28770
\(551\) 18.8248 0.801961
\(552\) −2.27492 −0.0968269
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) −18.2749 −0.775727
\(556\) −4.00000 −0.169638
\(557\) −44.7492 −1.89608 −0.948042 0.318146i \(-0.896940\pi\)
−0.948042 + 0.318146i \(0.896940\pi\)
\(558\) 8.00000 0.338667
\(559\) −2.27492 −0.0962187
\(560\) 0 0
\(561\) 0.625414 0.0264050
\(562\) −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 \(-0.479304\pi\)
0.0649724 + 0.997887i \(0.479304\pi\)
\(570\) −9.72508 −0.407339
\(571\) −37.0997 −1.55257 −0.776286 0.630380i \(-0.782899\pi\)
−0.776286 + 0.630380i \(0.782899\pi\)
\(572\) −2.27492 −0.0951191
\(573\) 2.27492 0.0950360
\(574\) 0 0
\(575\) 30.1993 1.25940
\(576\) 1.00000 0.0416667
\(577\) 8.90033 0.370526 0.185263 0.982689i \(-0.440686\pi\)
0.185263 + 0.982689i \(0.440686\pi\)
\(578\) 16.9244 0.703963
\(579\) −18.5498 −0.770905
\(580\) 35.3746 1.46885
\(581\) 0 0
\(582\) 15.0997 0.625901
\(583\) −22.7492 −0.942174
\(584\) −12.8248 −0.530692
\(585\) 4.27492 0.176746
\(586\) −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
\(591\) −2.54983 −0.104886
\(592\) −4.27492 −0.175698
\(593\) 7.09967 0.291548 0.145774 0.989318i \(-0.453433\pi\)
0.145774 + 0.989318i \(0.453433\pi\)
\(594\) 2.27492 0.0933410
\(595\) 0 0
\(596\) 22.5498 0.923677
\(597\) −6.82475 −0.279318
\(598\) −2.27492 −0.0930283
\(599\) 21.7251 0.887663 0.443831 0.896110i \(-0.353619\pi\)
0.443831 + 0.896110i \(0.353619\pi\)
\(600\) −13.2749 −0.541946
\(601\) −23.6495 −0.964683 −0.482342 0.875983i \(-0.660214\pi\)
−0.482342 + 0.875983i \(0.660214\pi\)
\(602\) 0 0
\(603\) 12.5498 0.511069
\(604\) 6.27492 0.255323
\(605\) −24.9003 −1.01234
\(606\) 2.54983 0.103580
\(607\) −9.17525 −0.372412 −0.186206 0.982511i \(-0.559619\pi\)
−0.186206 + 0.982511i \(0.559619\pi\)
\(608\) −2.27492 −0.0922601
\(609\) 0 0
\(610\) −52.4743 −2.12462
\(611\) 0 0
\(612\) −0.274917 −0.0111129
\(613\) 16.2749 0.657338 0.328669 0.944445i \(-0.393400\pi\)
0.328669 + 0.944445i \(0.393400\pi\)
\(614\) 21.0997 0.851513
\(615\) −28.0000 −1.12907
\(616\) 0 0
\(617\) −20.8248 −0.838373 −0.419186 0.907900i \(-0.637685\pi\)
−0.419186 + 0.907900i \(0.637685\pi\)
\(618\) 10.2749 0.413318
\(619\) −29.7251 −1.19475 −0.597376 0.801961i \(-0.703790\pi\)
−0.597376 + 0.801961i \(0.703790\pi\)
\(620\) −34.1993 −1.37348
\(621\) 2.27492 0.0912893
\(622\) 3.45017 0.138339
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 84.8488 3.39395
\(626\) −27.6495 −1.10510
\(627\) −5.17525 −0.206680
\(628\) 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
\(633\) 1.17525 0.0467119
\(634\) 1.45017 0.0575934
\(635\) −87.8488 −3.48617
\(636\) 10.0000 0.396526
\(637\) 0 0
\(638\) 18.8248 0.745279
\(639\) 0 0
\(640\) −4.27492 −0.168981
\(641\) −15.0997 −0.596401 −0.298201 0.954503i \(-0.596386\pi\)
−0.298201 + 0.954503i \(0.596386\pi\)
\(642\) −0.549834 −0.0217002
\(643\) 23.9244 0.943487 0.471744 0.881736i \(-0.343625\pi\)
0.471744 + 0.881736i \(0.343625\pi\)
\(644\) 0 0
\(645\) −9.72508 −0.382925
\(646\) 0.625414 0.0246066
\(647\) 18.1993 0.715490 0.357745 0.933819i \(-0.383546\pi\)
0.357745 + 0.933819i \(0.383546\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 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 \(-0.533536\pi\)
−0.105162 + 0.994455i \(0.533536\pi\)
\(654\) −0.274917 −0.0107501
\(655\) −29.1752 −1.13997
\(656\) −6.54983 −0.255728
\(657\) 12.8248 0.500341
\(658\) 0 0
\(659\) −7.45017 −0.290217 −0.145109 0.989416i \(-0.546353\pi\)
−0.145109 + 0.989416i \(0.546353\pi\)
\(660\) −9.72508 −0.378548
\(661\) 40.1993 1.56357 0.781787 0.623546i \(-0.214309\pi\)
0.781787 + 0.623546i \(0.214309\pi\)
\(662\) 29.6495 1.15236
\(663\) −0.274917 −0.0106769
\(664\) −4.54983 −0.176568
\(665\) 0 0
\(666\) 4.27492 0.165650
\(667\) 18.8248 0.728897
\(668\) 1.72508 0.0667455
\(669\) −21.6495 −0.837018
\(670\) −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
\(675\) 13.2749 0.510952
\(676\) 1.00000 0.0384615
\(677\) −16.1993 −0.622591 −0.311296 0.950313i \(-0.600763\pi\)
−0.311296 + 0.950313i \(0.600763\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 1.17525 0.0450687
\(681\) 20.5498 0.787471
\(682\) −18.1993 −0.696889
\(683\) 3.37459 0.129125 0.0645625 0.997914i \(-0.479435\pi\)
0.0645625 + 0.997914i \(0.479435\pi\)
\(684\) 2.27492 0.0869836
\(685\) −74.2749 −2.83790
\(686\) 0 0
\(687\) −19.0997 −0.728698
\(688\) −2.27492 −0.0867304
\(689\) 10.0000 0.380970
\(690\) −9.72508 −0.370228
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −7.09967 −0.269889
\(693\) 0 0
\(694\) −5.09967 −0.193581
\(695\) −17.0997 −0.648627
\(696\) −8.27492 −0.313660
\(697\) 1.80066 0.0682049
\(698\) 2.00000 0.0757011
\(699\) 5.45017 0.206144
\(700\) 0 0
\(701\) −15.0997 −0.570307 −0.285153 0.958482i \(-0.592045\pi\)
−0.285153 + 0.958482i \(0.592045\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −9.72508 −0.366788
\(704\) −2.27492 −0.0857392
\(705\) 0 0
\(706\) 27.0997 1.01991
\(707\) 0 0
\(708\) −8.00000 −0.300658
\(709\) −23.0997 −0.867526 −0.433763 0.901027i \(-0.642815\pi\)
−0.433763 + 0.901027i \(0.642815\pi\)
\(710\) 0 0
\(711\) 12.5498 0.470656
\(712\) −14.0000 −0.524672
\(713\) −18.1993 −0.681571
\(714\) 0 0
\(715\) −9.72508 −0.363697
\(716\) 12.0000 0.448461
\(717\) 25.0997 0.937364
\(718\) 32.0000 1.19423
\(719\) 29.6495 1.10574 0.552870 0.833268i \(-0.313533\pi\)
0.552870 + 0.833268i \(0.313533\pi\)
\(720\) 4.27492 0.159317
\(721\) 0 0
\(722\) 13.8248 0.514504
\(723\) −15.0997 −0.561563
\(724\) −19.0997 −0.709834
\(725\) 109.849 4.07968
\(726\) 5.82475 0.216177
\(727\) −35.3746 −1.31197 −0.655985 0.754774i \(-0.727747\pi\)
−0.655985 + 0.754774i \(0.727747\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −54.8248 −2.02916
\(731\) 0.625414 0.0231318
\(732\) 12.2749 0.453694
\(733\) −14.5498 −0.537410 −0.268705 0.963222i \(-0.586596\pi\)
−0.268705 + 0.963222i \(0.586596\pi\)
\(734\) −2.90033 −0.107053
\(735\) 0 0
\(736\) −2.27492 −0.0838546
\(737\) −28.5498 −1.05165
\(738\) 6.54983 0.241103
\(739\) 37.6495 1.38496 0.692480 0.721437i \(-0.256518\pi\)
0.692480 + 0.721437i \(0.256518\pi\)
\(740\) −18.2749 −0.671799
\(741\) 2.27492 0.0835712
\(742\) 0 0
\(743\) 11.4502 0.420066 0.210033 0.977694i \(-0.432643\pi\)
0.210033 + 0.977694i \(0.432643\pi\)
\(744\) 8.00000 0.293294
\(745\) 96.3987 3.53177
\(746\) −26.5498 −0.972059
\(747\) 4.54983 0.166470
\(748\) 0.625414 0.0228674
\(749\) 0 0
\(750\) −35.3746 −1.29170
\(751\) 10.1993 0.372179 0.186090 0.982533i \(-0.440419\pi\)
0.186090 + 0.982533i \(0.440419\pi\)
\(752\) 0 0
\(753\) −23.9244 −0.871854
\(754\) −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
\(759\) −5.17525 −0.187850
\(760\) −9.72508 −0.352766
\(761\) −38.5498 −1.39743 −0.698715 0.715400i \(-0.746245\pi\)
−0.698715 + 0.715400i \(0.746245\pi\)
\(762\) 20.5498 0.744442
\(763\) 0 0
\(764\) 2.27492 0.0823036
\(765\) −1.17525 −0.0424912
\(766\) 30.2749 1.09388
\(767\) −8.00000 −0.288863
\(768\) 1.00000 0.0360844
\(769\) −32.8248 −1.18369 −0.591845 0.806051i \(-0.701600\pi\)
−0.591845 + 0.806051i \(0.701600\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) −18.5498 −0.667623
\(773\) 9.92442 0.356957 0.178478 0.983944i \(-0.442883\pi\)
0.178478 + 0.983944i \(0.442883\pi\)
\(774\) 2.27492 0.0817702
\(775\) −106.199 −3.81479
\(776\) 15.0997 0.542047
\(777\) 0 0
\(778\) −28.1993 −1.01100
\(779\) −14.9003 −0.533860
\(780\) 4.27492 0.153067
\(781\) 0 0
\(782\) 0.625414 0.0223648
\(783\) 8.27492 0.295721
\(784\) 0 0
\(785\) 57.1752 2.04067
\(786\) 6.82475 0.243431
\(787\) 10.2749 0.366261 0.183131 0.983089i \(-0.441377\pi\)
0.183131 + 0.983089i \(0.441377\pi\)
\(788\) −2.54983 −0.0908341
\(789\) 28.0000 0.996826
\(790\) −53.6495 −1.90876
\(791\) 0 0
\(792\) 2.27492 0.0808357
\(793\) 12.2749 0.435895
\(794\) −14.0000 −0.496841
\(795\) 42.7492 1.51616
\(796\) −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
\(801\) 14.0000 0.494666
\(802\) 3.09967 0.109453
\(803\) −29.1752 −1.02957
\(804\) 12.5498 0.442599
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) −6.00000 −0.211210
\(808\) 2.54983 0.0897029
\(809\) −56.1993 −1.97586 −0.987932 0.154890i \(-0.950498\pi\)
−0.987932 + 0.154890i \(0.950498\pi\)
\(810\) −4.27492 −0.150205
\(811\) 11.3746 0.399416 0.199708 0.979855i \(-0.436001\pi\)
0.199708 + 0.979855i \(0.436001\pi\)
\(812\) 0 0
\(813\) −28.5498 −1.00129
\(814\) −9.72508 −0.340864
\(815\) 53.6495 1.87926
\(816\) −0.274917 −0.00962403
\(817\) −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 \(-0.313755\pi\)
0.552288 + 0.833654i \(0.313755\pi\)
\(822\) 17.3746 0.606008
\(823\) −25.0997 −0.874919 −0.437460 0.899238i \(-0.644122\pi\)
−0.437460 + 0.899238i \(0.644122\pi\)
\(824\) 10.2749 0.357944
\(825\) −30.1993 −1.05141
\(826\) 0 0
\(827\) −26.2749 −0.913668 −0.456834 0.889552i \(-0.651017\pi\)
−0.456834 + 0.889552i \(0.651017\pi\)
\(828\) 2.27492 0.0790588
\(829\) −11.7251 −0.407229 −0.203614 0.979051i \(-0.565269\pi\)
−0.203614 + 0.979051i \(0.565269\pi\)
\(830\) −19.4502 −0.675125
\(831\) −10.0000 −0.346896
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) 4.00000 0.138509
\(835\) 7.37459 0.255208
\(836\) −5.17525 −0.178990
\(837\) −8.00000 −0.276520
\(838\) 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
\(843\) 15.0997 0.520060
\(844\) 1.17525 0.0404537
\(845\) 4.27492 0.147062
\(846\) 0 0
\(847\) 0 0
\(848\) 10.0000 0.343401
\(849\) 4.00000 0.137280
\(850\) 3.64950 0.125177
\(851\) −9.72508 −0.333372
\(852\) 0 0
\(853\) 27.6495 0.946701 0.473350 0.880874i \(-0.343044\pi\)
0.473350 + 0.880874i \(0.343044\pi\)
\(854\) 0 0
\(855\) 9.72508 0.332591
\(856\) −0.549834 −0.0187930
\(857\) −19.0997 −0.652432 −0.326216 0.945295i \(-0.605774\pi\)
−0.326216 + 0.945295i \(0.605774\pi\)
\(858\) 2.27492 0.0776644
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −9.72508 −0.331623
\(861\) 0 0
\(862\) 11.4502 0.389994
\(863\) 29.6495 1.00928 0.504640 0.863330i \(-0.331625\pi\)
0.504640 + 0.863330i \(0.331625\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −30.3505 −1.03195
\(866\) 23.6495 0.803643
\(867\) −16.9244 −0.574783
\(868\) 0 0
\(869\) −28.5498 −0.968487
\(870\) −35.3746 −1.19931
\(871\) 12.5498 0.425235
\(872\) −0.274917 −0.00930987
\(873\) −15.0997 −0.511046
\(874\) −5.17525 −0.175055
\(875\) 0 0
\(876\) 12.8248 0.433308
\(877\) 51.0997 1.72551 0.862757 0.505619i \(-0.168736\pi\)
0.862757 + 0.505619i \(0.168736\pi\)
\(878\) −14.8248 −0.500311
\(879\) 6.00000 0.202375
\(880\) −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
\(885\) −34.1993 −1.14960
\(886\) −7.45017 −0.250293
\(887\) 33.0997 1.11138 0.555689 0.831390i \(-0.312455\pi\)
0.555689 + 0.831390i \(0.312455\pi\)
\(888\) 4.27492 0.143457
\(889\) 0 0
\(890\) −59.8488 −2.00614
\(891\) −2.27492 −0.0762126
\(892\) −21.6495 −0.724879
\(893\) 0 0
\(894\) −22.5498 −0.754179
\(895\) 51.2990 1.71474
\(896\) 0 0
\(897\) 2.27492 0.0759573
\(898\) 0.274917 0.00917411
\(899\) −66.1993 −2.20787
\(900\) 13.2749 0.442497
\(901\) −2.74917 −0.0915882
\(902\) −14.9003 −0.496127
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −81.6495 −2.71412
\(906\) −6.27492 −0.208470
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 20.5498 0.681970
\(909\) −2.54983 −0.0845727
\(910\) 0 0
\(911\) −52.4743 −1.73855 −0.869275 0.494329i \(-0.835414\pi\)
−0.869275 + 0.494329i \(0.835414\pi\)
\(912\) 2.27492 0.0753300
\(913\) −10.3505 −0.342551
\(914\) 18.5498 0.613574
\(915\) 52.4743 1.73475
\(916\) −19.0997 −0.631071
\(917\) 0 0
\(918\) 0.274917 0.00907362
\(919\) −12.5498 −0.413981 −0.206990 0.978343i \(-0.566367\pi\)
−0.206990 + 0.978343i \(0.566367\pi\)
\(920\) −9.72508 −0.320626
\(921\) −21.0997 −0.695258
\(922\) −17.9244 −0.590309
\(923\) 0 0
\(924\) 0 0
\(925\) −56.7492 −1.86590
\(926\) −30.2749 −0.994896
\(927\) −10.2749 −0.337473
\(928\) −8.27492 −0.271637
\(929\) −14.5498 −0.477365 −0.238682 0.971098i \(-0.576715\pi\)
−0.238682 + 0.971098i \(0.576715\pi\)
\(930\) 34.1993 1.12144
\(931\) 0 0
\(932\) 5.45017 0.178526
\(933\) −3.45017 −0.112953
\(934\) −26.2749 −0.859742
\(935\) 2.67359 0.0874358
\(936\) −1.00000 −0.0326860
\(937\) −16.9003 −0.552110 −0.276055 0.961142i \(-0.589027\pi\)
−0.276055 + 0.961142i \(0.589027\pi\)
\(938\) 0 0
\(939\) 27.6495 0.902307
\(940\) 0 0
\(941\) −51.0997 −1.66580 −0.832901 0.553422i \(-0.813322\pi\)
−0.832901 + 0.553422i \(0.813322\pi\)
\(942\) −13.3746 −0.435768
\(943\) −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 \(-0.590023\pi\)
−0.279060 + 0.960274i \(0.590023\pi\)
\(948\) 12.5498 0.407600
\(949\) 12.8248 0.416309
\(950\) −30.1993 −0.979796
\(951\) −1.45017 −0.0470248
\(952\) 0 0
\(953\) 31.6495 1.02523 0.512614 0.858619i \(-0.328677\pi\)
0.512614 + 0.858619i \(0.328677\pi\)
\(954\) −10.0000 −0.323762
\(955\) 9.72508 0.314696
\(956\) 25.0997 0.811781
\(957\) −18.8248 −0.608518
\(958\) −23.3746 −0.755199
\(959\) 0 0
\(960\) 4.27492 0.137972
\(961\) 33.0000 1.06452
\(962\) 4.27492 0.137829
\(963\) 0.549834 0.0177182
\(964\) −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
\(969\) −0.625414 −0.0200912
\(970\) 64.5498 2.07257
\(971\) 29.0997 0.933853 0.466926 0.884296i \(-0.345361\pi\)
0.466926 + 0.884296i \(0.345361\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 18.1993 0.583144
\(975\) 13.2749 0.425138
\(976\) 12.2749 0.392911
\(977\) −13.9244 −0.445482 −0.222741 0.974878i \(-0.571500\pi\)
−0.222741 + 0.974878i \(0.571500\pi\)
\(978\) −12.5498 −0.401299
\(979\) −31.8488 −1.01789
\(980\) 0 0
\(981\) 0.274917 0.00877743
\(982\) −8.54983 −0.272836
\(983\) −61.0241 −1.94637 −0.973183 0.230032i \(-0.926117\pi\)
−0.973183 + 0.230032i \(0.926117\pi\)
\(984\) 6.54983 0.208801
\(985\) −10.9003 −0.347313
\(986\) 2.27492 0.0724481
\(987\) 0 0
\(988\) 2.27492 0.0723748
\(989\) −5.17525 −0.164563
\(990\) 9.72508 0.309083
\(991\) −46.7492 −1.48504 −0.742518 0.669826i \(-0.766369\pi\)
−0.742518 + 0.669826i \(0.766369\pi\)
\(992\) 8.00000 0.254000
\(993\) −29.6495 −0.940899
\(994\) 0 0
\(995\) −29.1752 −0.924918
\(996\) 4.54983 0.144167
\(997\) 12.9003 0.408558 0.204279 0.978913i \(-0.434515\pi\)
0.204279 + 0.978913i \(0.434515\pi\)
\(998\) −25.0997 −0.794516
\(999\) −4.27492 −0.135252
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 3822.2.a.bm.1.2 2
7.6 odd 2 546.2.a.h.1.1 2
21.20 even 2 1638.2.a.y.1.2 2
28.27 even 2 4368.2.a.bh.1.1 2
91.90 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 7.6 odd 2
1638.2.a.y.1.2 2 21.20 even 2
3822.2.a.bm.1.2 2 1.1 even 1 trivial
4368.2.a.bh.1.1 2 28.27 even 2
7098.2.a.bu.1.2 2 91.90 odd 2