Properties

Label 4368.2.a.bh.1.1
Level $4368$
Weight $2$
Character 4368.1
Self dual yes
Analytic conductor $34.879$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.8786556029\)
Analytic rank: \(1\)
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 1.1
Root \(4.27492\) of defining polynomial
Character \(\chi\) \(=\) 4368.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -4.27492 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -4.27492 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.27492 q^{11} -1.00000 q^{13} -4.27492 q^{15} +0.274917 q^{17} +2.27492 q^{19} -1.00000 q^{21} -2.27492 q^{23} +13.2749 q^{25} +1.00000 q^{27} +8.27492 q^{29} -8.00000 q^{31} +2.27492 q^{33} +4.27492 q^{35} -4.27492 q^{37} -1.00000 q^{39} +6.54983 q^{41} +2.27492 q^{43} -4.27492 q^{45} +1.00000 q^{49} +0.274917 q^{51} +10.0000 q^{53} -9.72508 q^{55} +2.27492 q^{57} -8.00000 q^{59} -12.2749 q^{61} -1.00000 q^{63} +4.27492 q^{65} -12.5498 q^{67} -2.27492 q^{69} -12.8248 q^{73} +13.2749 q^{75} -2.27492 q^{77} -12.5498 q^{79} +1.00000 q^{81} +4.54983 q^{83} -1.17525 q^{85} +8.27492 q^{87} -14.0000 q^{89} +1.00000 q^{91} -8.00000 q^{93} -9.72508 q^{95} +15.0997 q^{97} +2.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - q^{5} - 2q^{7} + 2q^{9} - 3q^{11} - 2q^{13} - q^{15} - 7q^{17} - 3q^{19} - 2q^{21} + 3q^{23} + 19q^{25} + 2q^{27} + 9q^{29} - 16q^{31} - 3q^{33} + q^{35} - q^{37} - 2q^{39} - 2q^{41} - 3q^{43} - q^{45} + 2q^{49} - 7q^{51} + 20q^{53} - 27q^{55} - 3q^{57} - 16q^{59} - 17q^{61} - 2q^{63} + q^{65} - 10q^{67} + 3q^{69} - 3q^{73} + 19q^{75} + 3q^{77} - 10q^{79} + 2q^{81} - 6q^{83} - 25q^{85} + 9q^{87} - 28q^{89} + 2q^{91} - 16q^{93} - 27q^{95} - 3q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −4.27492 −1.91180 −0.955901 0.293691i \(-0.905116\pi\)
−0.955901 + 0.293691i \(0.905116\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 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
\(14\) 0 0
\(15\) −4.27492 −1.10378
\(16\) 0 0
\(17\) 0.274917 0.0666772 0.0333386 0.999444i \(-0.489386\pi\)
0.0333386 + 0.999444i \(0.489386\pi\)
\(18\) 0 0
\(19\) 2.27492 0.521902 0.260951 0.965352i \(-0.415964\pi\)
0.260951 + 0.965352i \(0.415964\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −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
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 8.27492 1.53661 0.768307 0.640082i \(-0.221100\pi\)
0.768307 + 0.640082i \(0.221100\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 2.27492 0.396012
\(34\) 0 0
\(35\) 4.27492 0.722593
\(36\) 0 0
\(37\) −4.27492 −0.702792 −0.351396 0.936227i \(-0.614293\pi\)
−0.351396 + 0.936227i \(0.614293\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 6.54983 1.02291 0.511456 0.859309i \(-0.329106\pi\)
0.511456 + 0.859309i \(0.329106\pi\)
\(42\) 0 0
\(43\) 2.27492 0.346922 0.173461 0.984841i \(-0.444505\pi\)
0.173461 + 0.984841i \(0.444505\pi\)
\(44\) 0 0
\(45\) −4.27492 −0.637267
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.274917 0.0384961
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) −9.72508 −1.31133
\(56\) 0 0
\(57\) 2.27492 0.301320
\(58\) 0 0
\(59\) −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 \(0.787759\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 4.27492 0.530238
\(66\) 0 0
\(67\) −12.5498 −1.53321 −0.766603 0.642121i \(-0.778055\pi\)
−0.766603 + 0.642121i \(0.778055\pi\)
\(68\) 0 0
\(69\) −2.27492 −0.273868
\(70\) 0 0
\(71\) 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 \(0.770193\pi\)
\(74\) 0 0
\(75\) 13.2749 1.53286
\(76\) 0 0
\(77\) −2.27492 −0.259251
\(78\) 0 0
\(79\) −12.5498 −1.41197 −0.705983 0.708228i \(-0.749495\pi\)
−0.705983 + 0.708228i \(0.749495\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 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
\(86\) 0 0
\(87\) 8.27492 0.887164
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 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 \(0.221962\pi\)
\(98\) 0 0
\(99\) 2.27492 0.228638
\(100\) 0 0
\(101\) 2.54983 0.253718 0.126859 0.991921i \(-0.459510\pi\)
0.126859 + 0.991921i \(0.459510\pi\)
\(102\) 0 0
\(103\) −10.2749 −1.01242 −0.506209 0.862411i \(-0.668954\pi\)
−0.506209 + 0.862411i \(0.668954\pi\)
\(104\) 0 0
\(105\) 4.27492 0.417189
\(106\) 0 0
\(107\) −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 \(0.495809\pi\)
\(110\) 0 0
\(111\) −4.27492 −0.405757
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 9.72508 0.906869
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −0.274917 −0.0252016
\(120\) 0 0
\(121\) −5.82475 −0.529523
\(122\) 0 0
\(123\) 6.54983 0.590579
\(124\) 0 0
\(125\) −35.3746 −3.16400
\(126\) 0 0
\(127\) 20.5498 1.82350 0.911751 0.410742i \(-0.134730\pi\)
0.911751 + 0.410742i \(0.134730\pi\)
\(128\) 0 0
\(129\) 2.27492 0.200295
\(130\) 0 0
\(131\) −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
\(134\) 0 0
\(135\) −4.27492 −0.367926
\(136\) 0 0
\(137\) −17.3746 −1.48441 −0.742206 0.670172i \(-0.766220\pi\)
−0.742206 + 0.670172i \(0.766220\pi\)
\(138\) 0 0
\(139\) −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
\(144\) 0 0
\(145\) −35.3746 −2.93770
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 22.5498 1.84735 0.923677 0.383172i \(-0.125168\pi\)
0.923677 + 0.383172i \(0.125168\pi\)
\(150\) 0 0
\(151\) −6.27492 −0.510646 −0.255323 0.966856i \(-0.582182\pi\)
−0.255323 + 0.966856i \(0.582182\pi\)
\(152\) 0 0
\(153\) 0.274917 0.0222257
\(154\) 0 0
\(155\) 34.1993 2.74696
\(156\) 0 0
\(157\) −13.3746 −1.06741 −0.533704 0.845671i \(-0.679200\pi\)
−0.533704 + 0.845671i \(0.679200\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 2.27492 0.179289
\(162\) 0 0
\(163\) −12.5498 −0.982979 −0.491489 0.870884i \(-0.663547\pi\)
−0.491489 + 0.870884i \(0.663547\pi\)
\(164\) 0 0
\(165\) −9.72508 −0.757097
\(166\) 0 0
\(167\) 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
\(170\) 0 0
\(171\) 2.27492 0.173967
\(172\) 0 0
\(173\) 7.09967 0.539778 0.269889 0.962891i \(-0.413013\pi\)
0.269889 + 0.962891i \(0.413013\pi\)
\(174\) 0 0
\(175\) −13.2749 −1.00349
\(176\) 0 0
\(177\) −8.00000 −0.601317
\(178\) 0 0
\(179\) −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 \(0.248770\pi\)
\(182\) 0 0
\(183\) −12.2749 −0.907388
\(184\) 0 0
\(185\) 18.2749 1.34360
\(186\) 0 0
\(187\) 0.625414 0.0457348
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −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 \(0.732688\pi\)
\(194\) 0 0
\(195\) 4.27492 0.306133
\(196\) 0 0
\(197\) −2.54983 −0.181668 −0.0908341 0.995866i \(-0.528953\pi\)
−0.0908341 + 0.995866i \(0.528953\pi\)
\(198\) 0 0
\(199\) −6.82475 −0.483794 −0.241897 0.970302i \(-0.577770\pi\)
−0.241897 + 0.970302i \(0.577770\pi\)
\(200\) 0 0
\(201\) −12.5498 −0.885197
\(202\) 0 0
\(203\) −8.27492 −0.580785
\(204\) 0 0
\(205\) −28.0000 −1.95560
\(206\) 0 0
\(207\) −2.27492 −0.158118
\(208\) 0 0
\(209\) 5.17525 0.357979
\(210\) 0 0
\(211\) −1.17525 −0.0809074 −0.0404537 0.999181i \(-0.512880\pi\)
−0.0404537 + 0.999181i \(0.512880\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −9.72508 −0.663245
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) −12.8248 −0.866616
\(220\) 0 0
\(221\) −0.274917 −0.0184929
\(222\) 0 0
\(223\) −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
\(226\) 0 0
\(227\) 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 \(0.282616\pi\)
\(230\) 0 0
\(231\) −2.27492 −0.149679
\(232\) 0 0
\(233\) 5.45017 0.357052 0.178526 0.983935i \(-0.442867\pi\)
0.178526 + 0.983935i \(0.442867\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −12.5498 −0.815199
\(238\) 0 0
\(239\) −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 \(0.338336\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −4.27492 −0.273114
\(246\) 0 0
\(247\) −2.27492 −0.144750
\(248\) 0 0
\(249\) 4.54983 0.288334
\(250\) 0 0
\(251\) −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
\(254\) 0 0
\(255\) −1.17525 −0.0735969
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 4.27492 0.265630
\(260\) 0 0
\(261\) 8.27492 0.512205
\(262\) 0 0
\(263\) −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
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −28.5498 −1.73428 −0.867139 0.498065i \(-0.834044\pi\)
−0.867139 + 0.498065i \(0.834044\pi\)
\(272\) 0 0
\(273\) 1.00000 0.0605228
\(274\) 0 0
\(275\) 30.1993 1.82109
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) −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
\(286\) 0 0
\(287\) −6.54983 −0.386625
\(288\) 0 0
\(289\) −16.9244 −0.995554
\(290\) 0 0
\(291\) 15.0997 0.885158
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 34.1993 1.99116
\(296\) 0 0
\(297\) 2.27492 0.132004
\(298\) 0 0
\(299\) 2.27492 0.131562
\(300\) 0 0
\(301\) −2.27492 −0.131124
\(302\) 0 0
\(303\) 2.54983 0.146484
\(304\) 0 0
\(305\) 52.4743 3.00467
\(306\) 0 0
\(307\) −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
\(310\) 0 0
\(311\) −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 \(0.785505\pi\)
\(314\) 0 0
\(315\) 4.27492 0.240864
\(316\) 0 0
\(317\) −1.45017 −0.0814494 −0.0407247 0.999170i \(-0.512967\pi\)
−0.0407247 + 0.999170i \(0.512967\pi\)
\(318\) 0 0
\(319\) 18.8248 1.05398
\(320\) 0 0
\(321\) −0.549834 −0.0306888
\(322\) 0 0
\(323\) 0.625414 0.0347990
\(324\) 0 0
\(325\) −13.2749 −0.736360
\(326\) 0 0
\(327\) 0.274917 0.0152030
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 29.6495 1.62968 0.814842 0.579683i \(-0.196824\pi\)
0.814842 + 0.579683i \(0.196824\pi\)
\(332\) 0 0
\(333\) −4.27492 −0.234264
\(334\) 0 0
\(335\) 53.6495 2.93119
\(336\) 0 0
\(337\) 9.37459 0.510666 0.255333 0.966853i \(-0.417815\pi\)
0.255333 + 0.966853i \(0.417815\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −18.1993 −0.985549
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 9.72508 0.523581
\(346\) 0 0
\(347\) −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 \(0.482953\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 27.0997 1.44237 0.721185 0.692743i \(-0.243598\pi\)
0.721185 + 0.692743i \(0.243598\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −0.274917 −0.0145502
\(358\) 0 0
\(359\) 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
\(362\) 0 0
\(363\) −5.82475 −0.305720
\(364\) 0 0
\(365\) 54.8248 2.86966
\(366\) 0 0
\(367\) 2.90033 0.151396 0.0756980 0.997131i \(-0.475881\pi\)
0.0756980 + 0.997131i \(0.475881\pi\)
\(368\) 0 0
\(369\) 6.54983 0.340971
\(370\) 0 0
\(371\) −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 0
\(375\) −35.3746 −1.82674
\(376\) 0 0
\(377\) −8.27492 −0.426180
\(378\) 0 0
\(379\) 33.0997 1.70022 0.850108 0.526609i \(-0.176537\pi\)
0.850108 + 0.526609i \(0.176537\pi\)
\(380\) 0 0
\(381\) 20.5498 1.05280
\(382\) 0 0
\(383\) −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
\(386\) 0 0
\(387\) 2.27492 0.115641
\(388\) 0 0
\(389\) 28.1993 1.42976 0.714882 0.699246i \(-0.246481\pi\)
0.714882 + 0.699246i \(0.246481\pi\)
\(390\) 0 0
\(391\) −0.625414 −0.0316285
\(392\) 0 0
\(393\) −6.82475 −0.344263
\(394\) 0 0
\(395\) 53.6495 2.69940
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) −2.27492 −0.113888
\(400\) 0 0
\(401\) −3.09967 −0.154790 −0.0773950 0.997001i \(-0.524660\pi\)
−0.0773950 + 0.997001i \(0.524660\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 0 0
\(405\) −4.27492 −0.212422
\(406\) 0 0
\(407\) −9.72508 −0.482054
\(408\) 0 0
\(409\) −7.17525 −0.354793 −0.177397 0.984139i \(-0.556768\pi\)
−0.177397 + 0.984139i \(0.556768\pi\)
\(410\) 0 0
\(411\) −17.3746 −0.857025
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) −19.4502 −0.954771
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −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 \(0.475934\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.64950 0.177027
\(426\) 0 0
\(427\) 12.2749 0.594025
\(428\) 0 0
\(429\) −2.27492 −0.109834
\(430\) 0 0
\(431\) 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 \(0.307616\pi\)
\(434\) 0 0
\(435\) −35.3746 −1.69608
\(436\) 0 0
\(437\) −5.17525 −0.247566
\(438\) 0 0
\(439\) 14.8248 0.707547 0.353773 0.935331i \(-0.384898\pi\)
0.353773 + 0.935331i \(0.384898\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −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
\(446\) 0 0
\(447\) 22.5498 1.06657
\(448\) 0 0
\(449\) −0.274917 −0.0129741 −0.00648707 0.999979i \(-0.502065\pi\)
−0.00648707 + 0.999979i \(0.502065\pi\)
\(450\) 0 0
\(451\) 14.9003 0.701629
\(452\) 0 0
\(453\) −6.27492 −0.294821
\(454\) 0 0
\(455\) −4.27492 −0.200411
\(456\) 0 0
\(457\) −18.5498 −0.867725 −0.433862 0.900979i \(-0.642850\pi\)
−0.433862 + 0.900979i \(0.642850\pi\)
\(458\) 0 0
\(459\) 0.274917 0.0128320
\(460\) 0 0
\(461\) −17.9244 −0.834823 −0.417412 0.908717i \(-0.637063\pi\)
−0.417412 + 0.908717i \(0.637063\pi\)
\(462\) 0 0
\(463\) −30.2749 −1.40699 −0.703497 0.710698i \(-0.748379\pi\)
−0.703497 + 0.710698i \(0.748379\pi\)
\(464\) 0 0
\(465\) 34.1993 1.58596
\(466\) 0 0
\(467\) 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
\(471\) −13.3746 −0.616268
\(472\) 0 0
\(473\) 5.17525 0.237958
\(474\) 0 0
\(475\) 30.1993 1.38564
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) 0 0
\(479\) 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
\(482\) 0 0
\(483\) 2.27492 0.103512
\(484\) 0 0
\(485\) −64.5498 −2.93106
\(486\) 0 0
\(487\) 18.1993 0.824691 0.412345 0.911028i \(-0.364710\pi\)
0.412345 + 0.911028i \(0.364710\pi\)
\(488\) 0 0
\(489\) −12.5498 −0.567523
\(490\) 0 0
\(491\) −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
\(494\) 0 0
\(495\) −9.72508 −0.437110
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −25.0997 −1.12362 −0.561808 0.827268i \(-0.689894\pi\)
−0.561808 + 0.827268i \(0.689894\pi\)
\(500\) 0 0
\(501\) 1.72508 0.0770710
\(502\) 0 0
\(503\) 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
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −9.92442 −0.439892 −0.219946 0.975512i \(-0.570588\pi\)
−0.219946 + 0.975512i \(0.570588\pi\)
\(510\) 0 0
\(511\) 12.8248 0.567334
\(512\) 0 0
\(513\) 2.27492 0.100440
\(514\) 0 0
\(515\) 43.9244 1.93554
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 7.09967 0.311641
\(520\) 0 0
\(521\) −4.27492 −0.187288 −0.0936438 0.995606i \(-0.529851\pi\)
−0.0936438 + 0.995606i \(0.529851\pi\)
\(522\) 0 0
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 0 0
\(525\) −13.2749 −0.579365
\(526\) 0 0
\(527\) −2.19934 −0.0958047
\(528\) 0 0
\(529\) −17.8248 −0.774989
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) −6.54983 −0.283705
\(534\) 0 0
\(535\) 2.35050 0.101621
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) 2.27492 0.0979876
\(540\) 0 0
\(541\) 42.4743 1.82611 0.913055 0.407835i \(-0.133716\pi\)
0.913055 + 0.407835i \(0.133716\pi\)
\(542\) 0 0
\(543\) 19.0997 0.819645
\(544\) 0 0
\(545\) −1.17525 −0.0503421
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 0 0
\(549\) −12.2749 −0.523881
\(550\) 0 0
\(551\) 18.8248 0.801961
\(552\) 0 0
\(553\) 12.5498 0.533673
\(554\) 0 0
\(555\) 18.2749 0.775727
\(556\) 0 0
\(557\) −44.7492 −1.89608 −0.948042 0.318146i \(-0.896940\pi\)
−0.948042 + 0.318146i \(0.896940\pi\)
\(558\) 0 0
\(559\) −2.27492 −0.0962187
\(560\) 0 0
\(561\) 0.625414 0.0264050
\(562\) 0 0
\(563\) 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
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 3.09967 0.129945 0.0649724 0.997887i \(-0.479304\pi\)
0.0649724 + 0.997887i \(0.479304\pi\)
\(570\) 0 0
\(571\) 37.0997 1.55257 0.776286 0.630380i \(-0.217101\pi\)
0.776286 + 0.630380i \(0.217101\pi\)
\(572\) 0 0
\(573\) −2.27492 −0.0950360
\(574\) 0 0
\(575\) −30.1993 −1.25940
\(576\) 0 0
\(577\) −8.90033 −0.370526 −0.185263 0.982689i \(-0.559314\pi\)
−0.185263 + 0.982689i \(0.559314\pi\)
\(578\) 0 0
\(579\) −18.5498 −0.770905
\(580\) 0 0
\(581\) −4.54983 −0.188759
\(582\) 0 0
\(583\) 22.7492 0.942174
\(584\) 0 0
\(585\) 4.27492 0.176746
\(586\) 0 0
\(587\) −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
\(590\) 0 0
\(591\) −2.54983 −0.104886
\(592\) 0 0
\(593\) −7.09967 −0.291548 −0.145774 0.989318i \(-0.546567\pi\)
−0.145774 + 0.989318i \(0.546567\pi\)
\(594\) 0 0
\(595\) 1.17525 0.0481805
\(596\) 0 0
\(597\) −6.82475 −0.279318
\(598\) 0 0
\(599\) −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 \(0.339786\pi\)
\(602\) 0 0
\(603\) −12.5498 −0.511069
\(604\) 0 0
\(605\) 24.9003 1.01234
\(606\) 0 0
\(607\) −9.17525 −0.372412 −0.186206 0.982511i \(-0.559619\pi\)
−0.186206 + 0.982511i \(0.559619\pi\)
\(608\) 0 0
\(609\) −8.27492 −0.335317
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 16.2749 0.657338 0.328669 0.944445i \(-0.393400\pi\)
0.328669 + 0.944445i \(0.393400\pi\)
\(614\) 0 0
\(615\) −28.0000 −1.12907
\(616\) 0 0
\(617\) −20.8248 −0.838373 −0.419186 0.907900i \(-0.637685\pi\)
−0.419186 + 0.907900i \(0.637685\pi\)
\(618\) 0 0
\(619\) −29.7251 −1.19475 −0.597376 0.801961i \(-0.703790\pi\)
−0.597376 + 0.801961i \(0.703790\pi\)
\(620\) 0 0
\(621\) −2.27492 −0.0912893
\(622\) 0 0
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) 84.8488 3.39395
\(626\) 0 0
\(627\) 5.17525 0.206680
\(628\) 0 0
\(629\) −1.17525 −0.0468602
\(630\) 0 0
\(631\) 10.8248 0.430927 0.215463 0.976512i \(-0.430874\pi\)
0.215463 + 0.976512i \(0.430874\pi\)
\(632\) 0 0
\(633\) −1.17525 −0.0467119
\(634\) 0 0
\(635\) −87.8488 −3.48617
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −15.0997 −0.596401 −0.298201 0.954503i \(-0.596386\pi\)
−0.298201 + 0.954503i \(0.596386\pi\)
\(642\) 0 0
\(643\) 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 0
\(647\) 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
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 0 0
\(653\) −5.37459 −0.210324 −0.105162 0.994455i \(-0.533536\pi\)
−0.105162 + 0.994455i \(0.533536\pi\)
\(654\) 0 0
\(655\) 29.1752 1.13997
\(656\) 0 0
\(657\) −12.8248 −0.500341
\(658\) 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 \(0.785691\pi\)
\(662\) 0 0
\(663\) −0.274917 −0.0106769
\(664\) 0 0
\(665\) 9.72508 0.377123
\(666\) 0 0
\(667\) −18.8248 −0.728897
\(668\) 0 0
\(669\) −21.6495 −0.837018
\(670\) 0 0
\(671\) −27.9244 −1.07801
\(672\) 0 0
\(673\) 16.2749 0.627352 0.313676 0.949530i \(-0.398439\pi\)
0.313676 + 0.949530i \(0.398439\pi\)
\(674\) 0 0
\(675\) 13.2749 0.510952
\(676\) 0 0
\(677\) 16.1993 0.622591 0.311296 0.950313i \(-0.399237\pi\)
0.311296 + 0.950313i \(0.399237\pi\)
\(678\) 0 0
\(679\) −15.0997 −0.579472
\(680\) 0 0
\(681\) 20.5498 0.787471
\(682\) 0 0
\(683\) −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
\(686\) 0 0
\(687\) 19.0997 0.728698
\(688\) 0 0
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) −2.27492 −0.0864170
\(694\) 0 0
\(695\) 17.0997 0.648627
\(696\) 0 0
\(697\) 1.80066 0.0682049
\(698\) 0 0
\(699\) 5.45017 0.206144
\(700\) 0 0
\(701\) −15.0997 −0.570307 −0.285153 0.958482i \(-0.592045\pi\)
−0.285153 + 0.958482i \(0.592045\pi\)
\(702\) 0 0
\(703\) −9.72508 −0.366788
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −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
\(711\) −12.5498 −0.470656
\(712\) 0 0
\(713\) 18.1993 0.681571
\(714\) 0 0
\(715\) 9.72508 0.363697
\(716\) 0 0
\(717\) −25.0997 −0.937364
\(718\) 0 0
\(719\) 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
\(722\) 0 0
\(723\) 15.0997 0.561563
\(724\) 0 0
\(725\) 109.849 4.07968
\(726\) 0 0
\(727\) −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
\(730\) 0 0
\(731\) 0.625414 0.0231318
\(732\) 0 0
\(733\) 14.5498 0.537410 0.268705 0.963222i \(-0.413404\pi\)
0.268705 + 0.963222i \(0.413404\pi\)
\(734\) 0 0
\(735\) −4.27492 −0.157683
\(736\) 0 0
\(737\) −28.5498 −1.05165
\(738\) 0 0
\(739\) −37.6495 −1.38496 −0.692480 0.721437i \(-0.743482\pi\)
−0.692480 + 0.721437i \(0.743482\pi\)
\(740\) 0 0
\(741\) −2.27492 −0.0835712
\(742\) 0 0
\(743\) −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
\(746\) 0 0
\(747\) 4.54983 0.166470
\(748\) 0 0
\(749\) 0.549834 0.0200905
\(750\) 0 0
\(751\) −10.1993 −0.372179 −0.186090 0.982533i \(-0.559581\pi\)
−0.186090 + 0.982533i \(0.559581\pi\)
\(752\) 0 0
\(753\) −23.9244 −0.871854
\(754\) 0 0
\(755\) 26.8248 0.976253
\(756\) 0 0
\(757\) −35.0997 −1.27572 −0.637860 0.770153i \(-0.720180\pi\)
−0.637860 + 0.770153i \(0.720180\pi\)
\(758\) 0 0
\(759\) −5.17525 −0.187850
\(760\) 0 0
\(761\) 38.5498 1.39743 0.698715 0.715400i \(-0.253755\pi\)
0.698715 + 0.715400i \(0.253755\pi\)
\(762\) 0 0
\(763\) −0.274917 −0.00995267
\(764\) 0 0
\(765\) −1.17525 −0.0424912
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 32.8248 1.18369 0.591845 0.806051i \(-0.298400\pi\)
0.591845 + 0.806051i \(0.298400\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 0 0
\(773\) −9.92442 −0.356957 −0.178478 0.983944i \(-0.557117\pi\)
−0.178478 + 0.983944i \(0.557117\pi\)
\(774\) 0 0
\(775\) −106.199 −3.81479
\(776\) 0 0
\(777\) 4.27492 0.153362
\(778\) 0 0
\(779\) 14.9003 0.533860
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 8.27492 0.295721
\(784\) 0 0
\(785\) 57.1752 2.04067
\(786\) 0 0
\(787\) 10.2749 0.366261 0.183131 0.983089i \(-0.441377\pi\)
0.183131 + 0.983089i \(0.441377\pi\)
\(788\) 0 0
\(789\) −28.0000 −0.996826
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 12.2749 0.435895
\(794\) 0 0
\(795\) −42.7492 −1.51616
\(796\) 0 0
\(797\) −15.6495 −0.554334 −0.277167 0.960822i \(-0.589396\pi\)
−0.277167 + 0.960822i \(0.589396\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 0 0
\(803\) −29.1752 −1.02957
\(804\) 0 0
\(805\) −9.72508 −0.342764
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) −56.1993 −1.97586 −0.987932 0.154890i \(-0.950498\pi\)
−0.987932 + 0.154890i \(0.950498\pi\)
\(810\) 0 0
\(811\) 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
\(814\) 0 0
\(815\) 53.6495 1.87926
\(816\) 0 0
\(817\) 5.17525 0.181059
\(818\) 0 0
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) 31.6495 1.10458 0.552288 0.833654i \(-0.313755\pi\)
0.552288 + 0.833654i \(0.313755\pi\)
\(822\) 0 0
\(823\) 25.0997 0.874919 0.437460 0.899238i \(-0.355878\pi\)
0.437460 + 0.899238i \(0.355878\pi\)
\(824\) 0 0
\(825\) 30.1993 1.05141
\(826\) 0 0
\(827\) 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 \(0.434731\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 0.274917 0.00952532
\(834\) 0 0
\(835\) −7.37459 −0.255208
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 0 0
\(839\) −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
\(842\) 0 0
\(843\) 15.0997 0.520060
\(844\) 0 0
\(845\) −4.27492 −0.147062
\(846\) 0 0
\(847\) 5.82475 0.200141
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 9.72508 0.333372
\(852\) 0 0
\(853\) −27.6495 −0.946701 −0.473350 0.880874i \(-0.656956\pi\)
−0.473350 + 0.880874i \(0.656956\pi\)
\(854\) 0 0
\(855\) −9.72508 −0.332591
\(856\) 0 0
\(857\) 19.0997 0.652432 0.326216 0.945295i \(-0.394226\pi\)
0.326216 + 0.945295i \(0.394226\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) −6.54983 −0.223218
\(862\) 0 0
\(863\) −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
\(866\) 0 0
\(867\) −16.9244 −0.574783
\(868\) 0 0
\(869\) −28.5498 −0.968487
\(870\) 0 0
\(871\) 12.5498 0.425235
\(872\) 0 0
\(873\) 15.0997 0.511046
\(874\) 0 0
\(875\) 35.3746 1.19588
\(876\) 0 0
\(877\) 51.0997 1.72551 0.862757 0.505619i \(-0.168736\pi\)
0.862757 + 0.505619i \(0.168736\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −31.7251 −1.06885 −0.534423 0.845217i \(-0.679471\pi\)
−0.534423 + 0.845217i \(0.679471\pi\)
\(882\) 0 0
\(883\) −31.9244 −1.07434 −0.537171 0.843473i \(-0.680507\pi\)
−0.537171 + 0.843473i \(0.680507\pi\)
\(884\) 0 0
\(885\) 34.1993 1.14960
\(886\) 0 0
\(887\) 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
\(890\) 0 0
\(891\) 2.27492 0.0762126
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 51.2990 1.71474
\(896\) 0 0
\(897\) 2.27492 0.0759573
\(898\) 0 0
\(899\) −66.1993 −2.20787
\(900\) 0 0
\(901\) 2.74917 0.0915882
\(902\) 0 0
\(903\) −2.27492 −0.0757045
\(904\) 0 0
\(905\) −81.6495 −2.71412
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 0 0
\(909\) 2.54983 0.0845727
\(910\) 0 0
\(911\) 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
\(914\) 0 0
\(915\) 52.4743 1.73475
\(916\) 0 0
\(917\) 6.82475 0.225373
\(918\) 0 0
\(919\) 12.5498 0.413981 0.206990 0.978343i \(-0.433633\pi\)
0.206990 + 0.978343i \(0.433633\pi\)
\(920\) 0 0
\(921\) −21.0997 −0.695258
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −56.7492 −1.86590
\(926\) 0 0
\(927\) −10.2749 −0.337473
\(928\) 0 0
\(929\) 14.5498 0.477365 0.238682 0.971098i \(-0.423285\pi\)
0.238682 + 0.971098i \(0.423285\pi\)
\(930\) 0 0
\(931\) 2.27492 0.0745574
\(932\) 0 0
\(933\) −3.45017 −0.112953
\(934\) 0 0
\(935\) −2.67359 −0.0874358
\(936\) 0 0
\(937\) 16.9003 0.552110 0.276055 0.961142i \(-0.410973\pi\)
0.276055 + 0.961142i \(0.410973\pi\)
\(938\) 0 0
\(939\) −27.6495 −0.902307
\(940\) 0 0
\(941\) 51.0997 1.66580 0.832901 0.553422i \(-0.186678\pi\)
0.832901 + 0.553422i \(0.186678\pi\)
\(942\) 0 0
\(943\) −14.9003 −0.485222
\(944\) 0 0
\(945\) 4.27492 0.139063
\(946\) 0 0
\(947\) 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
\(950\) 0 0
\(951\) −1.45017 −0.0470248
\(952\) 0 0
\(953\) 31.6495 1.02523 0.512614 0.858619i \(-0.328677\pi\)
0.512614 + 0.858619i \(0.328677\pi\)
\(954\) 0 0
\(955\) 9.72508 0.314696
\(956\) 0 0
\(957\) 18.8248 0.608518
\(958\) 0 0
\(959\) 17.3746 0.561055
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −0.549834 −0.0177182
\(964\) 0 0
\(965\) 79.2990 2.55273
\(966\) 0 0
\(967\) −42.8248 −1.37715 −0.688576 0.725165i \(-0.741764\pi\)
−0.688576 + 0.725165i \(0.741764\pi\)
\(968\) 0 0
\(969\) 0.625414 0.0200912
\(970\) 0 0
\(971\) 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
\(974\) 0 0
\(975\) −13.2749 −0.425138
\(976\) 0 0
\(977\) −13.9244 −0.445482 −0.222741 0.974878i \(-0.571500\pi\)
−0.222741 + 0.974878i \(0.571500\pi\)
\(978\) 0 0
\(979\) −31.8488 −1.01789
\(980\) 0 0
\(981\) 0.274917 0.00877743
\(982\) 0 0
\(983\) −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
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −5.17525 −0.164563
\(990\) 0 0
\(991\) 46.7492 1.48504 0.742518 0.669826i \(-0.233631\pi\)
0.742518 + 0.669826i \(0.233631\pi\)
\(992\) 0 0
\(993\) 29.6495 0.940899
\(994\) 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 \(0.565485\pi\)
\(998\) 0 0
\(999\) −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 4368.2.a.bh.1.1 2
4.3 odd 2 546.2.a.h.1.1 2
12.11 even 2 1638.2.a.y.1.2 2
28.27 even 2 3822.2.a.bm.1.2 2
52.51 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 4.3 odd 2
1638.2.a.y.1.2 2 12.11 even 2
3822.2.a.bm.1.2 2 28.27 even 2
4368.2.a.bh.1.1 2 1.1 even 1 trivial
7098.2.a.bu.1.2 2 52.51 odd 2