Properties

Label 7728.2.a.ch.1.3
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 3 x^{5} - 11 x^{4} + 23 x^{3} + 9 x^{2} - 23 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.0909082\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.535873 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.535873 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.85211 q^{11} +4.64109 q^{13} -0.535873 q^{15} -4.61235 q^{17} +2.85211 q^{19} +1.00000 q^{21} +1.00000 q^{23} -4.71284 q^{25} +1.00000 q^{27} +9.55878 q^{29} +3.22303 q^{31} +2.85211 q^{33} -0.535873 q^{35} -2.23053 q^{37} +4.64109 q^{39} +10.6232 q^{41} -0.687155 q^{43} -0.535873 q^{45} -2.01090 q^{47} +1.00000 q^{49} -4.61235 q^{51} +0.842547 q^{53} -1.52837 q^{55} +2.85211 q^{57} -7.87502 q^{59} -5.76980 q^{61} +1.00000 q^{63} -2.48704 q^{65} +4.55405 q^{67} +1.00000 q^{69} -7.39888 q^{71} -14.9663 q^{73} -4.71284 q^{75} +2.85211 q^{77} +1.54060 q^{79} +1.00000 q^{81} +14.4017 q^{83} +2.47163 q^{85} +9.55878 q^{87} -0.327138 q^{89} +4.64109 q^{91} +3.22303 q^{93} -1.52837 q^{95} +11.8764 q^{97} +2.85211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{3} + 2q^{5} + 6q^{7} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{3} + 2q^{5} + 6q^{7} + 6q^{9} + 3q^{11} + 2q^{15} + 6q^{17} + 3q^{19} + 6q^{21} + 6q^{23} + 10q^{25} + 6q^{27} + 5q^{29} + 4q^{31} + 3q^{33} + 2q^{35} + q^{37} + 12q^{41} + 6q^{43} + 2q^{45} + 6q^{47} + 6q^{49} + 6q^{51} + 10q^{53} - 3q^{55} + 3q^{57} + 14q^{59} + 4q^{61} + 6q^{63} + 27q^{65} - 7q^{67} + 6q^{69} - 7q^{71} + 10q^{73} + 10q^{75} + 3q^{77} - 14q^{79} + 6q^{81} - 14q^{83} + 21q^{85} + 5q^{87} + 25q^{89} + 4q^{93} - 3q^{95} + 11q^{97} + 3q^{99} + O(q^{100}) \)

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
\(5\) −0.535873 −0.239650 −0.119825 0.992795i \(-0.538233\pi\)
−0.119825 + 0.992795i \(0.538233\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.85211 0.859944 0.429972 0.902842i \(-0.358523\pi\)
0.429972 + 0.902842i \(0.358523\pi\)
\(12\) 0 0
\(13\) 4.64109 1.28721 0.643604 0.765359i \(-0.277438\pi\)
0.643604 + 0.765359i \(0.277438\pi\)
\(14\) 0 0
\(15\) −0.535873 −0.138362
\(16\) 0 0
\(17\) −4.61235 −1.11866 −0.559329 0.828946i \(-0.688941\pi\)
−0.559329 + 0.828946i \(0.688941\pi\)
\(18\) 0 0
\(19\) 2.85211 0.654319 0.327160 0.944969i \(-0.393908\pi\)
0.327160 + 0.944969i \(0.393908\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.71284 −0.942568
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 9.55878 1.77502 0.887511 0.460787i \(-0.152433\pi\)
0.887511 + 0.460787i \(0.152433\pi\)
\(30\) 0 0
\(31\) 3.22303 0.578873 0.289436 0.957197i \(-0.406532\pi\)
0.289436 + 0.957197i \(0.406532\pi\)
\(32\) 0 0
\(33\) 2.85211 0.496489
\(34\) 0 0
\(35\) −0.535873 −0.0905790
\(36\) 0 0
\(37\) −2.23053 −0.366697 −0.183349 0.983048i \(-0.558694\pi\)
−0.183349 + 0.983048i \(0.558694\pi\)
\(38\) 0 0
\(39\) 4.64109 0.743170
\(40\) 0 0
\(41\) 10.6232 1.65907 0.829536 0.558453i \(-0.188605\pi\)
0.829536 + 0.558453i \(0.188605\pi\)
\(42\) 0 0
\(43\) −0.687155 −0.104790 −0.0523951 0.998626i \(-0.516686\pi\)
−0.0523951 + 0.998626i \(0.516686\pi\)
\(44\) 0 0
\(45\) −0.535873 −0.0798832
\(46\) 0 0
\(47\) −2.01090 −0.293320 −0.146660 0.989187i \(-0.546852\pi\)
−0.146660 + 0.989187i \(0.546852\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.61235 −0.645858
\(52\) 0 0
\(53\) 0.842547 0.115733 0.0578663 0.998324i \(-0.481570\pi\)
0.0578663 + 0.998324i \(0.481570\pi\)
\(54\) 0 0
\(55\) −1.52837 −0.206085
\(56\) 0 0
\(57\) 2.85211 0.377771
\(58\) 0 0
\(59\) −7.87502 −1.02524 −0.512620 0.858615i \(-0.671325\pi\)
−0.512620 + 0.858615i \(0.671325\pi\)
\(60\) 0 0
\(61\) −5.76980 −0.738747 −0.369374 0.929281i \(-0.620428\pi\)
−0.369374 + 0.929281i \(0.620428\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −2.48704 −0.308479
\(66\) 0 0
\(67\) 4.55405 0.556366 0.278183 0.960528i \(-0.410268\pi\)
0.278183 + 0.960528i \(0.410268\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −7.39888 −0.878086 −0.439043 0.898466i \(-0.644682\pi\)
−0.439043 + 0.898466i \(0.644682\pi\)
\(72\) 0 0
\(73\) −14.9663 −1.75167 −0.875835 0.482610i \(-0.839689\pi\)
−0.875835 + 0.482610i \(0.839689\pi\)
\(74\) 0 0
\(75\) −4.71284 −0.544192
\(76\) 0 0
\(77\) 2.85211 0.325028
\(78\) 0 0
\(79\) 1.54060 0.173331 0.0866655 0.996237i \(-0.472379\pi\)
0.0866655 + 0.996237i \(0.472379\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.4017 1.58079 0.790394 0.612599i \(-0.209876\pi\)
0.790394 + 0.612599i \(0.209876\pi\)
\(84\) 0 0
\(85\) 2.47163 0.268086
\(86\) 0 0
\(87\) 9.55878 1.02481
\(88\) 0 0
\(89\) −0.327138 −0.0346765 −0.0173383 0.999850i \(-0.505519\pi\)
−0.0173383 + 0.999850i \(0.505519\pi\)
\(90\) 0 0
\(91\) 4.64109 0.486519
\(92\) 0 0
\(93\) 3.22303 0.334212
\(94\) 0 0
\(95\) −1.52837 −0.156807
\(96\) 0 0
\(97\) 11.8764 1.20586 0.602931 0.797794i \(-0.294000\pi\)
0.602931 + 0.797794i \(0.294000\pi\)
\(98\) 0 0
\(99\) 2.85211 0.286648
\(100\) 0 0
\(101\) 14.9468 1.48726 0.743629 0.668592i \(-0.233103\pi\)
0.743629 + 0.668592i \(0.233103\pi\)
\(102\) 0 0
\(103\) −7.29309 −0.718609 −0.359305 0.933220i \(-0.616986\pi\)
−0.359305 + 0.933220i \(0.616986\pi\)
\(104\) 0 0
\(105\) −0.535873 −0.0522958
\(106\) 0 0
\(107\) −0.698547 −0.0675311 −0.0337655 0.999430i \(-0.510750\pi\)
−0.0337655 + 0.999430i \(0.510750\pi\)
\(108\) 0 0
\(109\) 4.22776 0.404946 0.202473 0.979288i \(-0.435102\pi\)
0.202473 + 0.979288i \(0.435102\pi\)
\(110\) 0 0
\(111\) −2.23053 −0.211713
\(112\) 0 0
\(113\) 11.2316 1.05658 0.528292 0.849063i \(-0.322833\pi\)
0.528292 + 0.849063i \(0.322833\pi\)
\(114\) 0 0
\(115\) −0.535873 −0.0499704
\(116\) 0 0
\(117\) 4.64109 0.429069
\(118\) 0 0
\(119\) −4.61235 −0.422813
\(120\) 0 0
\(121\) −2.86546 −0.260496
\(122\) 0 0
\(123\) 10.6232 0.957866
\(124\) 0 0
\(125\) 5.20485 0.465536
\(126\) 0 0
\(127\) −8.77730 −0.778860 −0.389430 0.921056i \(-0.627328\pi\)
−0.389430 + 0.921056i \(0.627328\pi\)
\(128\) 0 0
\(129\) −0.687155 −0.0605007
\(130\) 0 0
\(131\) 8.93720 0.780847 0.390423 0.920635i \(-0.372329\pi\)
0.390423 + 0.920635i \(0.372329\pi\)
\(132\) 0 0
\(133\) 2.85211 0.247309
\(134\) 0 0
\(135\) −0.535873 −0.0461206
\(136\) 0 0
\(137\) −1.55150 −0.132553 −0.0662767 0.997801i \(-0.521112\pi\)
−0.0662767 + 0.997801i \(0.521112\pi\)
\(138\) 0 0
\(139\) −1.71384 −0.145366 −0.0726831 0.997355i \(-0.523156\pi\)
−0.0726831 + 0.997355i \(0.523156\pi\)
\(140\) 0 0
\(141\) −2.01090 −0.169348
\(142\) 0 0
\(143\) 13.2369 1.10693
\(144\) 0 0
\(145\) −5.12229 −0.425383
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 14.8538 1.21687 0.608434 0.793604i \(-0.291798\pi\)
0.608434 + 0.793604i \(0.291798\pi\)
\(150\) 0 0
\(151\) −14.6606 −1.19306 −0.596532 0.802589i \(-0.703455\pi\)
−0.596532 + 0.802589i \(0.703455\pi\)
\(152\) 0 0
\(153\) −4.61235 −0.372886
\(154\) 0 0
\(155\) −1.72713 −0.138727
\(156\) 0 0
\(157\) 13.2998 1.06144 0.530721 0.847547i \(-0.321921\pi\)
0.530721 + 0.847547i \(0.321921\pi\)
\(158\) 0 0
\(159\) 0.842547 0.0668183
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −5.98215 −0.468558 −0.234279 0.972169i \(-0.575273\pi\)
−0.234279 + 0.972169i \(0.575273\pi\)
\(164\) 0 0
\(165\) −1.52837 −0.118983
\(166\) 0 0
\(167\) −16.9590 −1.31233 −0.656163 0.754619i \(-0.727822\pi\)
−0.656163 + 0.754619i \(0.727822\pi\)
\(168\) 0 0
\(169\) 8.53976 0.656905
\(170\) 0 0
\(171\) 2.85211 0.218106
\(172\) 0 0
\(173\) 2.16362 0.164497 0.0822486 0.996612i \(-0.473790\pi\)
0.0822486 + 0.996612i \(0.473790\pi\)
\(174\) 0 0
\(175\) −4.71284 −0.356257
\(176\) 0 0
\(177\) −7.87502 −0.591923
\(178\) 0 0
\(179\) −16.7739 −1.25374 −0.626870 0.779124i \(-0.715664\pi\)
−0.626870 + 0.779124i \(0.715664\pi\)
\(180\) 0 0
\(181\) −18.1059 −1.34580 −0.672900 0.739734i \(-0.734952\pi\)
−0.672900 + 0.739734i \(0.734952\pi\)
\(182\) 0 0
\(183\) −5.76980 −0.426516
\(184\) 0 0
\(185\) 1.19528 0.0878788
\(186\) 0 0
\(187\) −13.1549 −0.961984
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 4.02908 0.291534 0.145767 0.989319i \(-0.453435\pi\)
0.145767 + 0.989319i \(0.453435\pi\)
\(192\) 0 0
\(193\) 24.8327 1.78750 0.893748 0.448570i \(-0.148066\pi\)
0.893748 + 0.448570i \(0.148066\pi\)
\(194\) 0 0
\(195\) −2.48704 −0.178100
\(196\) 0 0
\(197\) 7.87747 0.561246 0.280623 0.959818i \(-0.409459\pi\)
0.280623 + 0.959818i \(0.409459\pi\)
\(198\) 0 0
\(199\) −24.8221 −1.75959 −0.879797 0.475350i \(-0.842321\pi\)
−0.879797 + 0.475350i \(0.842321\pi\)
\(200\) 0 0
\(201\) 4.55405 0.321218
\(202\) 0 0
\(203\) 9.55878 0.670895
\(204\) 0 0
\(205\) −5.69271 −0.397596
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 8.13454 0.562678
\(210\) 0 0
\(211\) 10.0882 0.694503 0.347252 0.937772i \(-0.387115\pi\)
0.347252 + 0.937772i \(0.387115\pi\)
\(212\) 0 0
\(213\) −7.39888 −0.506963
\(214\) 0 0
\(215\) 0.368228 0.0251129
\(216\) 0 0
\(217\) 3.22303 0.218793
\(218\) 0 0
\(219\) −14.9663 −1.01133
\(220\) 0 0
\(221\) −21.4063 −1.43995
\(222\) 0 0
\(223\) −2.46897 −0.165334 −0.0826671 0.996577i \(-0.526344\pi\)
−0.0826671 + 0.996577i \(0.526344\pi\)
\(224\) 0 0
\(225\) −4.71284 −0.314189
\(226\) 0 0
\(227\) 14.5722 0.967193 0.483597 0.875291i \(-0.339330\pi\)
0.483597 + 0.875291i \(0.339330\pi\)
\(228\) 0 0
\(229\) 1.93609 0.127940 0.0639702 0.997952i \(-0.479624\pi\)
0.0639702 + 0.997952i \(0.479624\pi\)
\(230\) 0 0
\(231\) 2.85211 0.187655
\(232\) 0 0
\(233\) 21.5440 1.41140 0.705698 0.708512i \(-0.250633\pi\)
0.705698 + 0.708512i \(0.250633\pi\)
\(234\) 0 0
\(235\) 1.07759 0.0702939
\(236\) 0 0
\(237\) 1.54060 0.100073
\(238\) 0 0
\(239\) −5.95307 −0.385072 −0.192536 0.981290i \(-0.561671\pi\)
−0.192536 + 0.981290i \(0.561671\pi\)
\(240\) 0 0
\(241\) 4.91902 0.316862 0.158431 0.987370i \(-0.449356\pi\)
0.158431 + 0.987370i \(0.449356\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.535873 −0.0342357
\(246\) 0 0
\(247\) 13.2369 0.842245
\(248\) 0 0
\(249\) 14.4017 0.912668
\(250\) 0 0
\(251\) 10.2686 0.648146 0.324073 0.946032i \(-0.394948\pi\)
0.324073 + 0.946032i \(0.394948\pi\)
\(252\) 0 0
\(253\) 2.85211 0.179311
\(254\) 0 0
\(255\) 2.47163 0.154780
\(256\) 0 0
\(257\) 5.81864 0.362957 0.181478 0.983395i \(-0.441912\pi\)
0.181478 + 0.983395i \(0.441912\pi\)
\(258\) 0 0
\(259\) −2.23053 −0.138598
\(260\) 0 0
\(261\) 9.55878 0.591674
\(262\) 0 0
\(263\) −1.50568 −0.0928442 −0.0464221 0.998922i \(-0.514782\pi\)
−0.0464221 + 0.998922i \(0.514782\pi\)
\(264\) 0 0
\(265\) −0.451498 −0.0277353
\(266\) 0 0
\(267\) −0.327138 −0.0200205
\(268\) 0 0
\(269\) −5.41612 −0.330226 −0.165113 0.986275i \(-0.552799\pi\)
−0.165113 + 0.986275i \(0.552799\pi\)
\(270\) 0 0
\(271\) 17.9069 1.08777 0.543883 0.839161i \(-0.316954\pi\)
0.543883 + 0.839161i \(0.316954\pi\)
\(272\) 0 0
\(273\) 4.64109 0.280892
\(274\) 0 0
\(275\) −13.4415 −0.810556
\(276\) 0 0
\(277\) 1.58025 0.0949479 0.0474739 0.998872i \(-0.484883\pi\)
0.0474739 + 0.998872i \(0.484883\pi\)
\(278\) 0 0
\(279\) 3.22303 0.192958
\(280\) 0 0
\(281\) 9.24221 0.551344 0.275672 0.961252i \(-0.411100\pi\)
0.275672 + 0.961252i \(0.411100\pi\)
\(282\) 0 0
\(283\) −1.31768 −0.0783281 −0.0391641 0.999233i \(-0.512469\pi\)
−0.0391641 + 0.999233i \(0.512469\pi\)
\(284\) 0 0
\(285\) −1.52837 −0.0905328
\(286\) 0 0
\(287\) 10.6232 0.627070
\(288\) 0 0
\(289\) 4.27374 0.251396
\(290\) 0 0
\(291\) 11.8764 0.696204
\(292\) 0 0
\(293\) 8.06786 0.471329 0.235665 0.971834i \(-0.424273\pi\)
0.235665 + 0.971834i \(0.424273\pi\)
\(294\) 0 0
\(295\) 4.22001 0.245698
\(296\) 0 0
\(297\) 2.85211 0.165496
\(298\) 0 0
\(299\) 4.64109 0.268401
\(300\) 0 0
\(301\) −0.687155 −0.0396070
\(302\) 0 0
\(303\) 14.9468 0.858669
\(304\) 0 0
\(305\) 3.09188 0.177040
\(306\) 0 0
\(307\) 11.5951 0.661770 0.330885 0.943671i \(-0.392653\pi\)
0.330885 + 0.943671i \(0.392653\pi\)
\(308\) 0 0
\(309\) −7.29309 −0.414889
\(310\) 0 0
\(311\) −16.6910 −0.946461 −0.473231 0.880939i \(-0.656912\pi\)
−0.473231 + 0.880939i \(0.656912\pi\)
\(312\) 0 0
\(313\) 18.0079 1.01787 0.508935 0.860805i \(-0.330039\pi\)
0.508935 + 0.860805i \(0.330039\pi\)
\(314\) 0 0
\(315\) −0.535873 −0.0301930
\(316\) 0 0
\(317\) −20.1974 −1.13440 −0.567198 0.823581i \(-0.691973\pi\)
−0.567198 + 0.823581i \(0.691973\pi\)
\(318\) 0 0
\(319\) 27.2627 1.52642
\(320\) 0 0
\(321\) −0.698547 −0.0389891
\(322\) 0 0
\(323\) −13.1549 −0.731960
\(324\) 0 0
\(325\) −21.8727 −1.21328
\(326\) 0 0
\(327\) 4.22776 0.233795
\(328\) 0 0
\(329\) −2.01090 −0.110864
\(330\) 0 0
\(331\) −11.2080 −0.616045 −0.308022 0.951379i \(-0.599667\pi\)
−0.308022 + 0.951379i \(0.599667\pi\)
\(332\) 0 0
\(333\) −2.23053 −0.122232
\(334\) 0 0
\(335\) −2.44039 −0.133333
\(336\) 0 0
\(337\) 15.1177 0.823512 0.411756 0.911294i \(-0.364916\pi\)
0.411756 + 0.911294i \(0.364916\pi\)
\(338\) 0 0
\(339\) 11.2316 0.610019
\(340\) 0 0
\(341\) 9.19244 0.497798
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −0.535873 −0.0288504
\(346\) 0 0
\(347\) −15.4640 −0.830151 −0.415075 0.909787i \(-0.636245\pi\)
−0.415075 + 0.909787i \(0.636245\pi\)
\(348\) 0 0
\(349\) 1.29276 0.0691997 0.0345998 0.999401i \(-0.488984\pi\)
0.0345998 + 0.999401i \(0.488984\pi\)
\(350\) 0 0
\(351\) 4.64109 0.247723
\(352\) 0 0
\(353\) 25.9277 1.37999 0.689997 0.723812i \(-0.257612\pi\)
0.689997 + 0.723812i \(0.257612\pi\)
\(354\) 0 0
\(355\) 3.96486 0.210433
\(356\) 0 0
\(357\) −4.61235 −0.244111
\(358\) 0 0
\(359\) −22.6668 −1.19631 −0.598153 0.801382i \(-0.704099\pi\)
−0.598153 + 0.801382i \(0.704099\pi\)
\(360\) 0 0
\(361\) −10.8655 −0.571866
\(362\) 0 0
\(363\) −2.86546 −0.150397
\(364\) 0 0
\(365\) 8.02002 0.419787
\(366\) 0 0
\(367\) 3.14797 0.164323 0.0821613 0.996619i \(-0.473818\pi\)
0.0821613 + 0.996619i \(0.473818\pi\)
\(368\) 0 0
\(369\) 10.6232 0.553024
\(370\) 0 0
\(371\) 0.842547 0.0437428
\(372\) 0 0
\(373\) 2.29387 0.118772 0.0593860 0.998235i \(-0.481086\pi\)
0.0593860 + 0.998235i \(0.481086\pi\)
\(374\) 0 0
\(375\) 5.20485 0.268777
\(376\) 0 0
\(377\) 44.3632 2.28482
\(378\) 0 0
\(379\) 27.9754 1.43700 0.718501 0.695526i \(-0.244829\pi\)
0.718501 + 0.695526i \(0.244829\pi\)
\(380\) 0 0
\(381\) −8.77730 −0.449675
\(382\) 0 0
\(383\) 25.6133 1.30878 0.654390 0.756157i \(-0.272926\pi\)
0.654390 + 0.756157i \(0.272926\pi\)
\(384\) 0 0
\(385\) −1.52837 −0.0778929
\(386\) 0 0
\(387\) −0.687155 −0.0349301
\(388\) 0 0
\(389\) 23.5377 1.19341 0.596704 0.802461i \(-0.296477\pi\)
0.596704 + 0.802461i \(0.296477\pi\)
\(390\) 0 0
\(391\) −4.61235 −0.233256
\(392\) 0 0
\(393\) 8.93720 0.450822
\(394\) 0 0
\(395\) −0.825566 −0.0415387
\(396\) 0 0
\(397\) −14.8211 −0.743851 −0.371926 0.928263i \(-0.621302\pi\)
−0.371926 + 0.928263i \(0.621302\pi\)
\(398\) 0 0
\(399\) 2.85211 0.142784
\(400\) 0 0
\(401\) 14.2657 0.712395 0.356197 0.934411i \(-0.384073\pi\)
0.356197 + 0.934411i \(0.384073\pi\)
\(402\) 0 0
\(403\) 14.9584 0.745130
\(404\) 0 0
\(405\) −0.535873 −0.0266277
\(406\) 0 0
\(407\) −6.36173 −0.315339
\(408\) 0 0
\(409\) −0.437603 −0.0216381 −0.0108190 0.999941i \(-0.503444\pi\)
−0.0108190 + 0.999941i \(0.503444\pi\)
\(410\) 0 0
\(411\) −1.55150 −0.0765298
\(412\) 0 0
\(413\) −7.87502 −0.387504
\(414\) 0 0
\(415\) −7.71746 −0.378835
\(416\) 0 0
\(417\) −1.71384 −0.0839272
\(418\) 0 0
\(419\) −12.9168 −0.631026 −0.315513 0.948921i \(-0.602177\pi\)
−0.315513 + 0.948921i \(0.602177\pi\)
\(420\) 0 0
\(421\) −33.2522 −1.62061 −0.810306 0.586006i \(-0.800699\pi\)
−0.810306 + 0.586006i \(0.800699\pi\)
\(422\) 0 0
\(423\) −2.01090 −0.0977732
\(424\) 0 0
\(425\) 21.7373 1.05441
\(426\) 0 0
\(427\) −5.76980 −0.279220
\(428\) 0 0
\(429\) 13.2369 0.639085
\(430\) 0 0
\(431\) −37.9235 −1.82671 −0.913354 0.407166i \(-0.866517\pi\)
−0.913354 + 0.407166i \(0.866517\pi\)
\(432\) 0 0
\(433\) 24.9694 1.19995 0.599976 0.800018i \(-0.295177\pi\)
0.599976 + 0.800018i \(0.295177\pi\)
\(434\) 0 0
\(435\) −5.12229 −0.245595
\(436\) 0 0
\(437\) 2.85211 0.136435
\(438\) 0 0
\(439\) 21.9241 1.04638 0.523190 0.852216i \(-0.324742\pi\)
0.523190 + 0.852216i \(0.324742\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 10.7672 0.511567 0.255784 0.966734i \(-0.417667\pi\)
0.255784 + 0.966734i \(0.417667\pi\)
\(444\) 0 0
\(445\) 0.175304 0.00831022
\(446\) 0 0
\(447\) 14.8538 0.702560
\(448\) 0 0
\(449\) −16.6801 −0.787181 −0.393591 0.919286i \(-0.628767\pi\)
−0.393591 + 0.919286i \(0.628767\pi\)
\(450\) 0 0
\(451\) 30.2987 1.42671
\(452\) 0 0
\(453\) −14.6606 −0.688816
\(454\) 0 0
\(455\) −2.48704 −0.116594
\(456\) 0 0
\(457\) 39.8951 1.86621 0.933107 0.359600i \(-0.117087\pi\)
0.933107 + 0.359600i \(0.117087\pi\)
\(458\) 0 0
\(459\) −4.61235 −0.215286
\(460\) 0 0
\(461\) −4.14238 −0.192930 −0.0964649 0.995336i \(-0.530754\pi\)
−0.0964649 + 0.995336i \(0.530754\pi\)
\(462\) 0 0
\(463\) 25.8453 1.20113 0.600566 0.799575i \(-0.294942\pi\)
0.600566 + 0.799575i \(0.294942\pi\)
\(464\) 0 0
\(465\) −1.72713 −0.0800939
\(466\) 0 0
\(467\) 12.6526 0.585491 0.292745 0.956190i \(-0.405431\pi\)
0.292745 + 0.956190i \(0.405431\pi\)
\(468\) 0 0
\(469\) 4.55405 0.210287
\(470\) 0 0
\(471\) 13.2998 0.612824
\(472\) 0 0
\(473\) −1.95984 −0.0901137
\(474\) 0 0
\(475\) −13.4415 −0.616741
\(476\) 0 0
\(477\) 0.842547 0.0385776
\(478\) 0 0
\(479\) 12.3166 0.562758 0.281379 0.959597i \(-0.409208\pi\)
0.281379 + 0.959597i \(0.409208\pi\)
\(480\) 0 0
\(481\) −10.3521 −0.472016
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −6.36421 −0.288984
\(486\) 0 0
\(487\) −26.6188 −1.20622 −0.603108 0.797660i \(-0.706071\pi\)
−0.603108 + 0.797660i \(0.706071\pi\)
\(488\) 0 0
\(489\) −5.98215 −0.270522
\(490\) 0 0
\(491\) −26.5310 −1.19733 −0.598664 0.801000i \(-0.704301\pi\)
−0.598664 + 0.801000i \(0.704301\pi\)
\(492\) 0 0
\(493\) −44.0884 −1.98564
\(494\) 0 0
\(495\) −1.52837 −0.0686951
\(496\) 0 0
\(497\) −7.39888 −0.331885
\(498\) 0 0
\(499\) −24.3625 −1.09062 −0.545308 0.838236i \(-0.683587\pi\)
−0.545308 + 0.838236i \(0.683587\pi\)
\(500\) 0 0
\(501\) −16.9590 −0.757672
\(502\) 0 0
\(503\) 29.8898 1.33272 0.666359 0.745631i \(-0.267852\pi\)
0.666359 + 0.745631i \(0.267852\pi\)
\(504\) 0 0
\(505\) −8.00957 −0.356421
\(506\) 0 0
\(507\) 8.53976 0.379264
\(508\) 0 0
\(509\) −18.9450 −0.839722 −0.419861 0.907588i \(-0.637921\pi\)
−0.419861 + 0.907588i \(0.637921\pi\)
\(510\) 0 0
\(511\) −14.9663 −0.662069
\(512\) 0 0
\(513\) 2.85211 0.125924
\(514\) 0 0
\(515\) 3.90817 0.172214
\(516\) 0 0
\(517\) −5.73531 −0.252239
\(518\) 0 0
\(519\) 2.16362 0.0949726
\(520\) 0 0
\(521\) 8.65650 0.379248 0.189624 0.981857i \(-0.439273\pi\)
0.189624 + 0.981857i \(0.439273\pi\)
\(522\) 0 0
\(523\) 9.94399 0.434820 0.217410 0.976080i \(-0.430239\pi\)
0.217410 + 0.976080i \(0.430239\pi\)
\(524\) 0 0
\(525\) −4.71284 −0.205685
\(526\) 0 0
\(527\) −14.8657 −0.647561
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −7.87502 −0.341747
\(532\) 0 0
\(533\) 49.3035 2.13557
\(534\) 0 0
\(535\) 0.374332 0.0161838
\(536\) 0 0
\(537\) −16.7739 −0.723848
\(538\) 0 0
\(539\) 2.85211 0.122849
\(540\) 0 0
\(541\) 26.0542 1.12016 0.560079 0.828439i \(-0.310771\pi\)
0.560079 + 0.828439i \(0.310771\pi\)
\(542\) 0 0
\(543\) −18.1059 −0.776998
\(544\) 0 0
\(545\) −2.26554 −0.0970450
\(546\) 0 0
\(547\) −6.71844 −0.287260 −0.143630 0.989631i \(-0.545877\pi\)
−0.143630 + 0.989631i \(0.545877\pi\)
\(548\) 0 0
\(549\) −5.76980 −0.246249
\(550\) 0 0
\(551\) 27.2627 1.16143
\(552\) 0 0
\(553\) 1.54060 0.0655130
\(554\) 0 0
\(555\) 1.19528 0.0507369
\(556\) 0 0
\(557\) −0.232042 −0.00983195 −0.00491597 0.999988i \(-0.501565\pi\)
−0.00491597 + 0.999988i \(0.501565\pi\)
\(558\) 0 0
\(559\) −3.18915 −0.134887
\(560\) 0 0
\(561\) −13.1549 −0.555402
\(562\) 0 0
\(563\) 33.6859 1.41969 0.709846 0.704357i \(-0.248765\pi\)
0.709846 + 0.704357i \(0.248765\pi\)
\(564\) 0 0
\(565\) −6.01873 −0.253210
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −3.31826 −0.139109 −0.0695543 0.997578i \(-0.522158\pi\)
−0.0695543 + 0.997578i \(0.522158\pi\)
\(570\) 0 0
\(571\) 9.02468 0.377671 0.188836 0.982009i \(-0.439529\pi\)
0.188836 + 0.982009i \(0.439529\pi\)
\(572\) 0 0
\(573\) 4.02908 0.168317
\(574\) 0 0
\(575\) −4.71284 −0.196539
\(576\) 0 0
\(577\) −42.1352 −1.75411 −0.877056 0.480388i \(-0.840496\pi\)
−0.877056 + 0.480388i \(0.840496\pi\)
\(578\) 0 0
\(579\) 24.8327 1.03201
\(580\) 0 0
\(581\) 14.4017 0.597482
\(582\) 0 0
\(583\) 2.40304 0.0995237
\(584\) 0 0
\(585\) −2.48704 −0.102826
\(586\) 0 0
\(587\) −39.2874 −1.62156 −0.810781 0.585349i \(-0.800957\pi\)
−0.810781 + 0.585349i \(0.800957\pi\)
\(588\) 0 0
\(589\) 9.19244 0.378768
\(590\) 0 0
\(591\) 7.87747 0.324036
\(592\) 0 0
\(593\) −14.7695 −0.606509 −0.303255 0.952910i \(-0.598073\pi\)
−0.303255 + 0.952910i \(0.598073\pi\)
\(594\) 0 0
\(595\) 2.47163 0.101327
\(596\) 0 0
\(597\) −24.8221 −1.01590
\(598\) 0 0
\(599\) 4.45606 0.182070 0.0910349 0.995848i \(-0.470983\pi\)
0.0910349 + 0.995848i \(0.470983\pi\)
\(600\) 0 0
\(601\) −47.4750 −1.93654 −0.968272 0.249899i \(-0.919602\pi\)
−0.968272 + 0.249899i \(0.919602\pi\)
\(602\) 0 0
\(603\) 4.55405 0.185455
\(604\) 0 0
\(605\) 1.53552 0.0624278
\(606\) 0 0
\(607\) 22.1967 0.900934 0.450467 0.892793i \(-0.351257\pi\)
0.450467 + 0.892793i \(0.351257\pi\)
\(608\) 0 0
\(609\) 9.55878 0.387341
\(610\) 0 0
\(611\) −9.33277 −0.377563
\(612\) 0 0
\(613\) −40.8295 −1.64909 −0.824545 0.565797i \(-0.808569\pi\)
−0.824545 + 0.565797i \(0.808569\pi\)
\(614\) 0 0
\(615\) −5.69271 −0.229552
\(616\) 0 0
\(617\) −37.0083 −1.48990 −0.744949 0.667121i \(-0.767526\pi\)
−0.744949 + 0.667121i \(0.767526\pi\)
\(618\) 0 0
\(619\) −14.8684 −0.597611 −0.298806 0.954314i \(-0.596588\pi\)
−0.298806 + 0.954314i \(0.596588\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −0.327138 −0.0131065
\(624\) 0 0
\(625\) 20.7751 0.831003
\(626\) 0 0
\(627\) 8.13454 0.324862
\(628\) 0 0
\(629\) 10.2880 0.410209
\(630\) 0 0
\(631\) −12.4650 −0.496222 −0.248111 0.968732i \(-0.579810\pi\)
−0.248111 + 0.968732i \(0.579810\pi\)
\(632\) 0 0
\(633\) 10.0882 0.400972
\(634\) 0 0
\(635\) 4.70352 0.186653
\(636\) 0 0
\(637\) 4.64109 0.183887
\(638\) 0 0
\(639\) −7.39888 −0.292695
\(640\) 0 0
\(641\) 10.0656 0.397568 0.198784 0.980043i \(-0.436301\pi\)
0.198784 + 0.980043i \(0.436301\pi\)
\(642\) 0 0
\(643\) 8.66680 0.341785 0.170893 0.985290i \(-0.445335\pi\)
0.170893 + 0.985290i \(0.445335\pi\)
\(644\) 0 0
\(645\) 0.368228 0.0144990
\(646\) 0 0
\(647\) 29.1473 1.14590 0.572949 0.819591i \(-0.305799\pi\)
0.572949 + 0.819591i \(0.305799\pi\)
\(648\) 0 0
\(649\) −22.4604 −0.881650
\(650\) 0 0
\(651\) 3.22303 0.126320
\(652\) 0 0
\(653\) 20.0418 0.784294 0.392147 0.919902i \(-0.371732\pi\)
0.392147 + 0.919902i \(0.371732\pi\)
\(654\) 0 0
\(655\) −4.78920 −0.187130
\(656\) 0 0
\(657\) −14.9663 −0.583890
\(658\) 0 0
\(659\) 42.9537 1.67324 0.836619 0.547785i \(-0.184529\pi\)
0.836619 + 0.547785i \(0.184529\pi\)
\(660\) 0 0
\(661\) −25.8875 −1.00691 −0.503453 0.864022i \(-0.667937\pi\)
−0.503453 + 0.864022i \(0.667937\pi\)
\(662\) 0 0
\(663\) −21.4063 −0.831353
\(664\) 0 0
\(665\) −1.52837 −0.0592676
\(666\) 0 0
\(667\) 9.55878 0.370118
\(668\) 0 0
\(669\) −2.46897 −0.0954557
\(670\) 0 0
\(671\) −16.4561 −0.635281
\(672\) 0 0
\(673\) 25.6833 0.990019 0.495009 0.868888i \(-0.335165\pi\)
0.495009 + 0.868888i \(0.335165\pi\)
\(674\) 0 0
\(675\) −4.71284 −0.181397
\(676\) 0 0
\(677\) 33.5087 1.28784 0.643921 0.765092i \(-0.277306\pi\)
0.643921 + 0.765092i \(0.277306\pi\)
\(678\) 0 0
\(679\) 11.8764 0.455773
\(680\) 0 0
\(681\) 14.5722 0.558409
\(682\) 0 0
\(683\) 2.51401 0.0961960 0.0480980 0.998843i \(-0.484684\pi\)
0.0480980 + 0.998843i \(0.484684\pi\)
\(684\) 0 0
\(685\) 0.831406 0.0317664
\(686\) 0 0
\(687\) 1.93609 0.0738664
\(688\) 0 0
\(689\) 3.91034 0.148972
\(690\) 0 0
\(691\) −26.9417 −1.02491 −0.512454 0.858714i \(-0.671264\pi\)
−0.512454 + 0.858714i \(0.671264\pi\)
\(692\) 0 0
\(693\) 2.85211 0.108343
\(694\) 0 0
\(695\) 0.918401 0.0348369
\(696\) 0 0
\(697\) −48.9981 −1.85593
\(698\) 0 0
\(699\) 21.5440 0.814870
\(700\) 0 0
\(701\) −4.90071 −0.185097 −0.0925486 0.995708i \(-0.529501\pi\)
−0.0925486 + 0.995708i \(0.529501\pi\)
\(702\) 0 0
\(703\) −6.36173 −0.239937
\(704\) 0 0
\(705\) 1.07759 0.0405842
\(706\) 0 0
\(707\) 14.9468 0.562131
\(708\) 0 0
\(709\) 14.6930 0.551807 0.275904 0.961185i \(-0.411023\pi\)
0.275904 + 0.961185i \(0.411023\pi\)
\(710\) 0 0
\(711\) 1.54060 0.0577770
\(712\) 0 0
\(713\) 3.22303 0.120703
\(714\) 0 0
\(715\) −7.09331 −0.265275
\(716\) 0 0
\(717\) −5.95307 −0.222321
\(718\) 0 0
\(719\) 5.43643 0.202744 0.101372 0.994849i \(-0.467677\pi\)
0.101372 + 0.994849i \(0.467677\pi\)
\(720\) 0 0
\(721\) −7.29309 −0.271609
\(722\) 0 0
\(723\) 4.91902 0.182940
\(724\) 0 0
\(725\) −45.0490 −1.67308
\(726\) 0 0
\(727\) −1.64622 −0.0610549 −0.0305274 0.999534i \(-0.509719\pi\)
−0.0305274 + 0.999534i \(0.509719\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 3.16940 0.117224
\(732\) 0 0
\(733\) −38.3270 −1.41564 −0.707820 0.706393i \(-0.750321\pi\)
−0.707820 + 0.706393i \(0.750321\pi\)
\(734\) 0 0
\(735\) −0.535873 −0.0197660
\(736\) 0 0
\(737\) 12.9887 0.478444
\(738\) 0 0
\(739\) −44.0327 −1.61977 −0.809884 0.586591i \(-0.800470\pi\)
−0.809884 + 0.586591i \(0.800470\pi\)
\(740\) 0 0
\(741\) 13.2369 0.486271
\(742\) 0 0
\(743\) 41.0103 1.50452 0.752260 0.658866i \(-0.228964\pi\)
0.752260 + 0.658866i \(0.228964\pi\)
\(744\) 0 0
\(745\) −7.95973 −0.291622
\(746\) 0 0
\(747\) 14.4017 0.526929
\(748\) 0 0
\(749\) −0.698547 −0.0255244
\(750\) 0 0
\(751\) −37.5508 −1.37025 −0.685125 0.728426i \(-0.740252\pi\)
−0.685125 + 0.728426i \(0.740252\pi\)
\(752\) 0 0
\(753\) 10.2686 0.374207
\(754\) 0 0
\(755\) 7.85622 0.285917
\(756\) 0 0
\(757\) 5.12915 0.186422 0.0932110 0.995646i \(-0.470287\pi\)
0.0932110 + 0.995646i \(0.470287\pi\)
\(758\) 0 0
\(759\) 2.85211 0.103525
\(760\) 0 0
\(761\) 4.05700 0.147066 0.0735331 0.997293i \(-0.476573\pi\)
0.0735331 + 0.997293i \(0.476573\pi\)
\(762\) 0 0
\(763\) 4.22776 0.153055
\(764\) 0 0
\(765\) 2.47163 0.0893620
\(766\) 0 0
\(767\) −36.5487 −1.31970
\(768\) 0 0
\(769\) −11.5331 −0.415894 −0.207947 0.978140i \(-0.566678\pi\)
−0.207947 + 0.978140i \(0.566678\pi\)
\(770\) 0 0
\(771\) 5.81864 0.209553
\(772\) 0 0
\(773\) −4.58380 −0.164868 −0.0824340 0.996597i \(-0.526269\pi\)
−0.0824340 + 0.996597i \(0.526269\pi\)
\(774\) 0 0
\(775\) −15.1896 −0.545627
\(776\) 0 0
\(777\) −2.23053 −0.0800199
\(778\) 0 0
\(779\) 30.2987 1.08556
\(780\) 0 0
\(781\) −21.1024 −0.755105
\(782\) 0 0
\(783\) 9.55878 0.341603
\(784\) 0 0
\(785\) −7.12702 −0.254374
\(786\) 0 0
\(787\) 2.26946 0.0808977 0.0404488 0.999182i \(-0.487121\pi\)
0.0404488 + 0.999182i \(0.487121\pi\)
\(788\) 0 0
\(789\) −1.50568 −0.0536036
\(790\) 0 0
\(791\) 11.2316 0.399351
\(792\) 0 0
\(793\) −26.7782 −0.950921
\(794\) 0 0
\(795\) −0.451498 −0.0160130
\(796\) 0 0
\(797\) −43.3260 −1.53468 −0.767342 0.641238i \(-0.778421\pi\)
−0.767342 + 0.641238i \(0.778421\pi\)
\(798\) 0 0
\(799\) 9.27496 0.328124
\(800\) 0 0
\(801\) −0.327138 −0.0115588
\(802\) 0 0
\(803\) −42.6855 −1.50634
\(804\) 0 0
\(805\) −0.535873 −0.0188870
\(806\) 0 0
\(807\) −5.41612 −0.190656
\(808\) 0 0
\(809\) −24.9986 −0.878906 −0.439453 0.898266i \(-0.644828\pi\)
−0.439453 + 0.898266i \(0.644828\pi\)
\(810\) 0 0
\(811\) 41.8959 1.47116 0.735582 0.677435i \(-0.236909\pi\)
0.735582 + 0.677435i \(0.236909\pi\)
\(812\) 0 0
\(813\) 17.9069 0.628022
\(814\) 0 0
\(815\) 3.20567 0.112290
\(816\) 0 0
\(817\) −1.95984 −0.0685663
\(818\) 0 0
\(819\) 4.64109 0.162173
\(820\) 0 0
\(821\) −15.9857 −0.557903 −0.278952 0.960305i \(-0.589987\pi\)
−0.278952 + 0.960305i \(0.589987\pi\)
\(822\) 0 0
\(823\) 10.3882 0.362111 0.181055 0.983473i \(-0.442049\pi\)
0.181055 + 0.983473i \(0.442049\pi\)
\(824\) 0 0
\(825\) −13.4415 −0.467975
\(826\) 0 0
\(827\) 10.2074 0.354946 0.177473 0.984126i \(-0.443208\pi\)
0.177473 + 0.984126i \(0.443208\pi\)
\(828\) 0 0
\(829\) −11.9033 −0.413420 −0.206710 0.978402i \(-0.566276\pi\)
−0.206710 + 0.978402i \(0.566276\pi\)
\(830\) 0 0
\(831\) 1.58025 0.0548182
\(832\) 0 0
\(833\) −4.61235 −0.159808
\(834\) 0 0
\(835\) 9.08787 0.314499
\(836\) 0 0
\(837\) 3.22303 0.111404
\(838\) 0 0
\(839\) −50.3792 −1.73928 −0.869641 0.493685i \(-0.835650\pi\)
−0.869641 + 0.493685i \(0.835650\pi\)
\(840\) 0 0
\(841\) 62.3703 2.15070
\(842\) 0 0
\(843\) 9.24221 0.318319
\(844\) 0 0
\(845\) −4.57623 −0.157427
\(846\) 0 0
\(847\) −2.86546 −0.0984582
\(848\) 0 0
\(849\) −1.31768 −0.0452228
\(850\) 0 0
\(851\) −2.23053 −0.0764616
\(852\) 0 0
\(853\) 15.7938 0.540771 0.270385 0.962752i \(-0.412849\pi\)
0.270385 + 0.962752i \(0.412849\pi\)
\(854\) 0 0
\(855\) −1.52837 −0.0522691
\(856\) 0 0
\(857\) 1.13726 0.0388479 0.0194240 0.999811i \(-0.493817\pi\)
0.0194240 + 0.999811i \(0.493817\pi\)
\(858\) 0 0
\(859\) −24.2237 −0.826503 −0.413252 0.910617i \(-0.635607\pi\)
−0.413252 + 0.910617i \(0.635607\pi\)
\(860\) 0 0
\(861\) 10.6232 0.362039
\(862\) 0 0
\(863\) 34.1994 1.16416 0.582081 0.813131i \(-0.302239\pi\)
0.582081 + 0.813131i \(0.302239\pi\)
\(864\) 0 0
\(865\) −1.15943 −0.0394217
\(866\) 0 0
\(867\) 4.27374 0.145144
\(868\) 0 0
\(869\) 4.39397 0.149055
\(870\) 0 0
\(871\) 21.1358 0.716159
\(872\) 0 0
\(873\) 11.8764 0.401954
\(874\) 0 0
\(875\) 5.20485 0.175956
\(876\) 0 0
\(877\) 4.17323 0.140920 0.0704600 0.997515i \(-0.477553\pi\)
0.0704600 + 0.997515i \(0.477553\pi\)
\(878\) 0 0
\(879\) 8.06786 0.272122
\(880\) 0 0
\(881\) 32.8322 1.10615 0.553073 0.833133i \(-0.313455\pi\)
0.553073 + 0.833133i \(0.313455\pi\)
\(882\) 0 0
\(883\) 4.67210 0.157229 0.0786144 0.996905i \(-0.474950\pi\)
0.0786144 + 0.996905i \(0.474950\pi\)
\(884\) 0 0
\(885\) 4.22001 0.141854
\(886\) 0 0
\(887\) 47.4119 1.59194 0.795968 0.605339i \(-0.206962\pi\)
0.795968 + 0.605339i \(0.206962\pi\)
\(888\) 0 0
\(889\) −8.77730 −0.294381
\(890\) 0 0
\(891\) 2.85211 0.0955494
\(892\) 0 0
\(893\) −5.73531 −0.191925
\(894\) 0 0
\(895\) 8.98868 0.300458
\(896\) 0 0
\(897\) 4.64109 0.154962
\(898\) 0 0
\(899\) 30.8082 1.02751
\(900\) 0 0
\(901\) −3.88612 −0.129465
\(902\) 0 0
\(903\) −0.687155 −0.0228671
\(904\) 0 0
\(905\) 9.70245 0.322520
\(906\) 0 0
\(907\) −51.1953 −1.69991 −0.849955 0.526855i \(-0.823371\pi\)
−0.849955 + 0.526855i \(0.823371\pi\)
\(908\) 0 0
\(909\) 14.9468 0.495753
\(910\) 0 0
\(911\) −44.1658 −1.46328 −0.731638 0.681693i \(-0.761244\pi\)
−0.731638 + 0.681693i \(0.761244\pi\)
\(912\) 0 0
\(913\) 41.0751 1.35939
\(914\) 0 0
\(915\) 3.09188 0.102214
\(916\) 0 0
\(917\) 8.93720 0.295132
\(918\) 0 0
\(919\) −8.67207 −0.286065 −0.143033 0.989718i \(-0.545685\pi\)
−0.143033 + 0.989718i \(0.545685\pi\)
\(920\) 0 0
\(921\) 11.5951 0.382073
\(922\) 0 0
\(923\) −34.3389 −1.13028
\(924\) 0 0
\(925\) 10.5121 0.345637
\(926\) 0 0
\(927\) −7.29309 −0.239536
\(928\) 0 0
\(929\) 6.08501 0.199643 0.0998213 0.995005i \(-0.468173\pi\)
0.0998213 + 0.995005i \(0.468173\pi\)
\(930\) 0 0
\(931\) 2.85211 0.0934742
\(932\) 0 0
\(933\) −16.6910 −0.546440
\(934\) 0 0
\(935\) 7.04937 0.230539
\(936\) 0 0
\(937\) −30.7574 −1.00480 −0.502400 0.864635i \(-0.667549\pi\)
−0.502400 + 0.864635i \(0.667549\pi\)
\(938\) 0 0
\(939\) 18.0079 0.587667
\(940\) 0 0
\(941\) 17.2274 0.561597 0.280799 0.959767i \(-0.409401\pi\)
0.280799 + 0.959767i \(0.409401\pi\)
\(942\) 0 0
\(943\) 10.6232 0.345940
\(944\) 0 0
\(945\) −0.535873 −0.0174319
\(946\) 0 0
\(947\) 27.0945 0.880453 0.440226 0.897887i \(-0.354898\pi\)
0.440226 + 0.897887i \(0.354898\pi\)
\(948\) 0 0
\(949\) −69.4599 −2.25476
\(950\) 0 0
\(951\) −20.1974 −0.654944
\(952\) 0 0
\(953\) −7.18512 −0.232749 −0.116374 0.993205i \(-0.537127\pi\)
−0.116374 + 0.993205i \(0.537127\pi\)
\(954\) 0 0
\(955\) −2.15907 −0.0698660
\(956\) 0 0
\(957\) 27.2627 0.881279
\(958\) 0 0
\(959\) −1.55150 −0.0501005
\(960\) 0 0
\(961\) −20.6121 −0.664906
\(962\) 0 0
\(963\) −0.698547 −0.0225104
\(964\) 0 0
\(965\) −13.3072 −0.428373
\(966\) 0 0
\(967\) −8.81557 −0.283490 −0.141745 0.989903i \(-0.545271\pi\)
−0.141745 + 0.989903i \(0.545271\pi\)
\(968\) 0 0
\(969\) −13.1549 −0.422597
\(970\) 0 0
\(971\) −49.3898 −1.58500 −0.792498 0.609875i \(-0.791220\pi\)
−0.792498 + 0.609875i \(0.791220\pi\)
\(972\) 0 0
\(973\) −1.71384 −0.0549433
\(974\) 0 0
\(975\) −21.8727 −0.700488
\(976\) 0 0
\(977\) 27.9017 0.892653 0.446327 0.894870i \(-0.352732\pi\)
0.446327 + 0.894870i \(0.352732\pi\)
\(978\) 0 0
\(979\) −0.933034 −0.0298199
\(980\) 0 0
\(981\) 4.22776 0.134982
\(982\) 0 0
\(983\) 47.0421 1.50041 0.750205 0.661205i \(-0.229955\pi\)
0.750205 + 0.661205i \(0.229955\pi\)
\(984\) 0 0
\(985\) −4.22132 −0.134502
\(986\) 0 0
\(987\) −2.01090 −0.0640076
\(988\) 0 0
\(989\) −0.687155 −0.0218503
\(990\) 0 0
\(991\) −0.452567 −0.0143763 −0.00718814 0.999974i \(-0.502288\pi\)
−0.00718814 + 0.999974i \(0.502288\pi\)
\(992\) 0 0
\(993\) −11.2080 −0.355674
\(994\) 0 0
\(995\) 13.3015 0.421686
\(996\) 0 0
\(997\) 5.38153 0.170435 0.0852173 0.996362i \(-0.472842\pi\)
0.0852173 + 0.996362i \(0.472842\pi\)
\(998\) 0 0
\(999\) −2.23053 −0.0705709
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7728.2.a.ch.1.3 6
4.3 odd 2 3864.2.a.w.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.w.1.3 6 4.3 odd 2
7728.2.a.ch.1.3 6 1.1 even 1 trivial