Properties

Label 7728.2.a.cg.1.1
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: \(x^{6} - x^{5} - 13 x^{4} + 7 x^{3} + 31 x^{2} - 17 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 1.1
Root \(0.174924\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -4.39875 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -4.39875 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.51973 q^{11} -5.51804 q^{13} +4.39875 q^{15} +3.11930 q^{17} +5.05863 q^{19} -1.00000 q^{21} -1.00000 q^{23} +14.3490 q^{25} -1.00000 q^{27} -6.66406 q^{29} -1.11930 q^{31} +3.51973 q^{33} -4.39875 q^{35} -6.66406 q^{37} +5.51804 q^{39} -10.6034 q^{41} -6.81835 q^{43} -4.39875 q^{45} -3.43403 q^{47} +1.00000 q^{49} -3.11930 q^{51} -9.54307 q^{53} +15.4824 q^{55} -5.05863 q^{57} -11.5431 q^{59} +13.9627 q^{61} +1.00000 q^{63} +24.2725 q^{65} -6.11258 q^{67} +1.00000 q^{69} -8.02334 q^{71} -2.95022 q^{73} -14.3490 q^{75} -3.51973 q^{77} -13.2171 q^{79} +1.00000 q^{81} +9.91679 q^{83} -13.7210 q^{85} +6.66406 q^{87} -4.31305 q^{89} -5.51804 q^{91} +1.11930 q^{93} -22.2516 q^{95} +16.2030 q^{97} -3.51973 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} + O(q^{10}) \) \( 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} - 5 q^{11} + 2 q^{13} + 10 q^{17} - 3 q^{19} - 6 q^{21} - 6 q^{23} + 18 q^{25} - 6 q^{27} + 3 q^{29} + 2 q^{31} + 5 q^{33} + 3 q^{37} - 2 q^{39} + 4 q^{41} - 6 q^{43} + 2 q^{47} + 6 q^{49} - 10 q^{51} - 4 q^{53} + 15 q^{55} + 3 q^{57} - 16 q^{59} + 22 q^{61} + 6 q^{63} + 35 q^{65} - 9 q^{67} + 6 q^{69} - 11 q^{71} + 24 q^{73} - 18 q^{75} - 5 q^{77} - 18 q^{79} + 6 q^{81} - 2 q^{83} + 13 q^{85} - 3 q^{87} + 7 q^{89} + 2 q^{91} - 2 q^{93} - 5 q^{95} + 37 q^{97} - 5 q^{99} + O(q^{100}) \)

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
\(5\) −4.39875 −1.96718 −0.983590 0.180421i \(-0.942254\pi\)
−0.983590 + 0.180421i \(0.942254\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.51973 −1.06124 −0.530619 0.847610i \(-0.678041\pi\)
−0.530619 + 0.847610i \(0.678041\pi\)
\(12\) 0 0
\(13\) −5.51804 −1.53043 −0.765215 0.643775i \(-0.777367\pi\)
−0.765215 + 0.643775i \(0.777367\pi\)
\(14\) 0 0
\(15\) 4.39875 1.13575
\(16\) 0 0
\(17\) 3.11930 0.756541 0.378270 0.925695i \(-0.376519\pi\)
0.378270 + 0.925695i \(0.376519\pi\)
\(18\) 0 0
\(19\) 5.05863 1.16053 0.580264 0.814428i \(-0.302949\pi\)
0.580264 + 0.814428i \(0.302949\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 14.3490 2.86979
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.66406 −1.23748 −0.618742 0.785594i \(-0.712357\pi\)
−0.618742 + 0.785594i \(0.712357\pi\)
\(30\) 0 0
\(31\) −1.11930 −0.201032 −0.100516 0.994935i \(-0.532049\pi\)
−0.100516 + 0.994935i \(0.532049\pi\)
\(32\) 0 0
\(33\) 3.51973 0.612706
\(34\) 0 0
\(35\) −4.39875 −0.743524
\(36\) 0 0
\(37\) −6.66406 −1.09556 −0.547782 0.836621i \(-0.684528\pi\)
−0.547782 + 0.836621i \(0.684528\pi\)
\(38\) 0 0
\(39\) 5.51804 0.883594
\(40\) 0 0
\(41\) −10.6034 −1.65597 −0.827985 0.560750i \(-0.810513\pi\)
−0.827985 + 0.560750i \(0.810513\pi\)
\(42\) 0 0
\(43\) −6.81835 −1.03979 −0.519894 0.854231i \(-0.674029\pi\)
−0.519894 + 0.854231i \(0.674029\pi\)
\(44\) 0 0
\(45\) −4.39875 −0.655726
\(46\) 0 0
\(47\) −3.43403 −0.500905 −0.250453 0.968129i \(-0.580579\pi\)
−0.250453 + 0.968129i \(0.580579\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.11930 −0.436789
\(52\) 0 0
\(53\) −9.54307 −1.31084 −0.655421 0.755264i \(-0.727509\pi\)
−0.655421 + 0.755264i \(0.727509\pi\)
\(54\) 0 0
\(55\) 15.4824 2.08765
\(56\) 0 0
\(57\) −5.05863 −0.670032
\(58\) 0 0
\(59\) −11.5431 −1.50278 −0.751390 0.659858i \(-0.770616\pi\)
−0.751390 + 0.659858i \(0.770616\pi\)
\(60\) 0 0
\(61\) 13.9627 1.78774 0.893868 0.448329i \(-0.147981\pi\)
0.893868 + 0.448329i \(0.147981\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 24.2725 3.01063
\(66\) 0 0
\(67\) −6.11258 −0.746771 −0.373385 0.927676i \(-0.621803\pi\)
−0.373385 + 0.927676i \(0.621803\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −8.02334 −0.952195 −0.476098 0.879392i \(-0.657949\pi\)
−0.476098 + 0.879392i \(0.657949\pi\)
\(72\) 0 0
\(73\) −2.95022 −0.345297 −0.172649 0.984983i \(-0.555233\pi\)
−0.172649 + 0.984983i \(0.555233\pi\)
\(74\) 0 0
\(75\) −14.3490 −1.65688
\(76\) 0 0
\(77\) −3.51973 −0.401111
\(78\) 0 0
\(79\) −13.2171 −1.48704 −0.743519 0.668714i \(-0.766845\pi\)
−0.743519 + 0.668714i \(0.766845\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.91679 1.08851 0.544255 0.838920i \(-0.316813\pi\)
0.544255 + 0.838920i \(0.316813\pi\)
\(84\) 0 0
\(85\) −13.7210 −1.48825
\(86\) 0 0
\(87\) 6.66406 0.714462
\(88\) 0 0
\(89\) −4.31305 −0.457182 −0.228591 0.973523i \(-0.573412\pi\)
−0.228591 + 0.973523i \(0.573412\pi\)
\(90\) 0 0
\(91\) −5.51804 −0.578448
\(92\) 0 0
\(93\) 1.11930 0.116066
\(94\) 0 0
\(95\) −22.2516 −2.28297
\(96\) 0 0
\(97\) 16.2030 1.64516 0.822580 0.568649i \(-0.192534\pi\)
0.822580 + 0.568649i \(0.192534\pi\)
\(98\) 0 0
\(99\) −3.51973 −0.353746
\(100\) 0 0
\(101\) 1.44528 0.143811 0.0719053 0.997411i \(-0.477092\pi\)
0.0719053 + 0.997411i \(0.477092\pi\)
\(102\) 0 0
\(103\) −10.9024 −1.07424 −0.537121 0.843505i \(-0.680488\pi\)
−0.537121 + 0.843505i \(0.680488\pi\)
\(104\) 0 0
\(105\) 4.39875 0.429274
\(106\) 0 0
\(107\) −8.77415 −0.848229 −0.424115 0.905609i \(-0.639415\pi\)
−0.424115 + 0.905609i \(0.639415\pi\)
\(108\) 0 0
\(109\) −7.47654 −0.716123 −0.358061 0.933698i \(-0.616562\pi\)
−0.358061 + 0.933698i \(0.616562\pi\)
\(110\) 0 0
\(111\) 6.66406 0.632524
\(112\) 0 0
\(113\) 6.90759 0.649811 0.324906 0.945746i \(-0.394667\pi\)
0.324906 + 0.945746i \(0.394667\pi\)
\(114\) 0 0
\(115\) 4.39875 0.410185
\(116\) 0 0
\(117\) −5.51804 −0.510143
\(118\) 0 0
\(119\) 3.11930 0.285946
\(120\) 0 0
\(121\) 1.38851 0.126228
\(122\) 0 0
\(123\) 10.6034 0.956075
\(124\) 0 0
\(125\) −41.1237 −3.67822
\(126\) 0 0
\(127\) −17.5269 −1.55526 −0.777630 0.628722i \(-0.783578\pi\)
−0.777630 + 0.628722i \(0.783578\pi\)
\(128\) 0 0
\(129\) 6.81835 0.600322
\(130\) 0 0
\(131\) 9.00752 0.786990 0.393495 0.919327i \(-0.371266\pi\)
0.393495 + 0.919327i \(0.371266\pi\)
\(132\) 0 0
\(133\) 5.05863 0.438639
\(134\) 0 0
\(135\) 4.39875 0.378584
\(136\) 0 0
\(137\) −2.72473 −0.232789 −0.116395 0.993203i \(-0.537134\pi\)
−0.116395 + 0.993203i \(0.537134\pi\)
\(138\) 0 0
\(139\) 5.43821 0.461263 0.230631 0.973041i \(-0.425921\pi\)
0.230631 + 0.973041i \(0.425921\pi\)
\(140\) 0 0
\(141\) 3.43403 0.289198
\(142\) 0 0
\(143\) 19.4220 1.62415
\(144\) 0 0
\(145\) 29.3135 2.43435
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 12.9788 1.06326 0.531632 0.846975i \(-0.321579\pi\)
0.531632 + 0.846975i \(0.321579\pi\)
\(150\) 0 0
\(151\) −22.7393 −1.85050 −0.925249 0.379359i \(-0.876144\pi\)
−0.925249 + 0.379359i \(0.876144\pi\)
\(152\) 0 0
\(153\) 3.11930 0.252180
\(154\) 0 0
\(155\) 4.92350 0.395465
\(156\) 0 0
\(157\) 13.9393 1.11248 0.556240 0.831022i \(-0.312244\pi\)
0.556240 + 0.831022i \(0.312244\pi\)
\(158\) 0 0
\(159\) 9.54307 0.756815
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 1.82337 0.142818 0.0714088 0.997447i \(-0.477251\pi\)
0.0714088 + 0.997447i \(0.477251\pi\)
\(164\) 0 0
\(165\) −15.4824 −1.20530
\(166\) 0 0
\(167\) 9.05863 0.700978 0.350489 0.936567i \(-0.386015\pi\)
0.350489 + 0.936567i \(0.386015\pi\)
\(168\) 0 0
\(169\) 17.4488 1.34222
\(170\) 0 0
\(171\) 5.05863 0.386843
\(172\) 0 0
\(173\) 8.71766 0.662791 0.331396 0.943492i \(-0.392481\pi\)
0.331396 + 0.943492i \(0.392481\pi\)
\(174\) 0 0
\(175\) 14.3490 1.08468
\(176\) 0 0
\(177\) 11.5431 0.867630
\(178\) 0 0
\(179\) −1.59788 −0.119431 −0.0597155 0.998215i \(-0.519019\pi\)
−0.0597155 + 0.998215i \(0.519019\pi\)
\(180\) 0 0
\(181\) 16.7083 1.24191 0.620957 0.783844i \(-0.286744\pi\)
0.620957 + 0.783844i \(0.286744\pi\)
\(182\) 0 0
\(183\) −13.9627 −1.03215
\(184\) 0 0
\(185\) 29.3135 2.15517
\(186\) 0 0
\(187\) −10.9791 −0.802870
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 9.78971 0.708359 0.354179 0.935177i \(-0.384760\pi\)
0.354179 + 0.935177i \(0.384760\pi\)
\(192\) 0 0
\(193\) 6.13373 0.441516 0.220758 0.975329i \(-0.429147\pi\)
0.220758 + 0.975329i \(0.429147\pi\)
\(194\) 0 0
\(195\) −24.2725 −1.73819
\(196\) 0 0
\(197\) 24.9157 1.77517 0.887584 0.460646i \(-0.152382\pi\)
0.887584 + 0.460646i \(0.152382\pi\)
\(198\) 0 0
\(199\) −17.9499 −1.27243 −0.636216 0.771511i \(-0.719501\pi\)
−0.636216 + 0.771511i \(0.719501\pi\)
\(200\) 0 0
\(201\) 6.11258 0.431148
\(202\) 0 0
\(203\) −6.66406 −0.467725
\(204\) 0 0
\(205\) 46.6416 3.25759
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −17.8050 −1.23160
\(210\) 0 0
\(211\) 3.51973 0.242308 0.121154 0.992634i \(-0.461340\pi\)
0.121154 + 0.992634i \(0.461340\pi\)
\(212\) 0 0
\(213\) 8.02334 0.549750
\(214\) 0 0
\(215\) 29.9922 2.04545
\(216\) 0 0
\(217\) −1.11930 −0.0759829
\(218\) 0 0
\(219\) 2.95022 0.199357
\(220\) 0 0
\(221\) −17.2124 −1.15783
\(222\) 0 0
\(223\) 15.1709 1.01592 0.507960 0.861381i \(-0.330400\pi\)
0.507960 + 0.861381i \(0.330400\pi\)
\(224\) 0 0
\(225\) 14.3490 0.956598
\(226\) 0 0
\(227\) 17.6039 1.16841 0.584206 0.811605i \(-0.301406\pi\)
0.584206 + 0.811605i \(0.301406\pi\)
\(228\) 0 0
\(229\) −14.4746 −0.956510 −0.478255 0.878221i \(-0.658731\pi\)
−0.478255 + 0.878221i \(0.658731\pi\)
\(230\) 0 0
\(231\) 3.51973 0.231581
\(232\) 0 0
\(233\) −26.5474 −1.73918 −0.869588 0.493779i \(-0.835615\pi\)
−0.869588 + 0.493779i \(0.835615\pi\)
\(234\) 0 0
\(235\) 15.1054 0.985370
\(236\) 0 0
\(237\) 13.2171 0.858542
\(238\) 0 0
\(239\) −4.35118 −0.281454 −0.140727 0.990048i \(-0.544944\pi\)
−0.140727 + 0.990048i \(0.544944\pi\)
\(240\) 0 0
\(241\) 20.0505 1.29157 0.645784 0.763520i \(-0.276531\pi\)
0.645784 + 0.763520i \(0.276531\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −4.39875 −0.281026
\(246\) 0 0
\(247\) −27.9137 −1.77611
\(248\) 0 0
\(249\) −9.91679 −0.628451
\(250\) 0 0
\(251\) −14.1426 −0.892672 −0.446336 0.894866i \(-0.647271\pi\)
−0.446336 + 0.894866i \(0.647271\pi\)
\(252\) 0 0
\(253\) 3.51973 0.221284
\(254\) 0 0
\(255\) 13.7210 0.859242
\(256\) 0 0
\(257\) −22.8292 −1.42404 −0.712022 0.702157i \(-0.752221\pi\)
−0.712022 + 0.702157i \(0.752221\pi\)
\(258\) 0 0
\(259\) −6.66406 −0.414084
\(260\) 0 0
\(261\) −6.66406 −0.412495
\(262\) 0 0
\(263\) −7.95143 −0.490306 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(264\) 0 0
\(265\) 41.9776 2.57866
\(266\) 0 0
\(267\) 4.31305 0.263954
\(268\) 0 0
\(269\) 2.07445 0.126482 0.0632408 0.997998i \(-0.479856\pi\)
0.0632408 + 0.997998i \(0.479856\pi\)
\(270\) 0 0
\(271\) 18.7177 1.13702 0.568508 0.822678i \(-0.307521\pi\)
0.568508 + 0.822678i \(0.307521\pi\)
\(272\) 0 0
\(273\) 5.51804 0.333967
\(274\) 0 0
\(275\) −50.5045 −3.04554
\(276\) 0 0
\(277\) 5.07342 0.304832 0.152416 0.988316i \(-0.451295\pi\)
0.152416 + 0.988316i \(0.451295\pi\)
\(278\) 0 0
\(279\) −1.11930 −0.0670106
\(280\) 0 0
\(281\) 13.6688 0.815410 0.407705 0.913114i \(-0.366329\pi\)
0.407705 + 0.913114i \(0.366329\pi\)
\(282\) 0 0
\(283\) −28.8011 −1.71205 −0.856024 0.516936i \(-0.827073\pi\)
−0.856024 + 0.516936i \(0.827073\pi\)
\(284\) 0 0
\(285\) 22.2516 1.31807
\(286\) 0 0
\(287\) −10.6034 −0.625898
\(288\) 0 0
\(289\) −7.26998 −0.427646
\(290\) 0 0
\(291\) −16.2030 −0.949834
\(292\) 0 0
\(293\) 2.03833 0.119081 0.0595403 0.998226i \(-0.481037\pi\)
0.0595403 + 0.998226i \(0.481037\pi\)
\(294\) 0 0
\(295\) 50.7750 2.95624
\(296\) 0 0
\(297\) 3.51973 0.204235
\(298\) 0 0
\(299\) 5.51804 0.319117
\(300\) 0 0
\(301\) −6.81835 −0.393003
\(302\) 0 0
\(303\) −1.44528 −0.0830290
\(304\) 0 0
\(305\) −61.4182 −3.51680
\(306\) 0 0
\(307\) 5.85313 0.334056 0.167028 0.985952i \(-0.446583\pi\)
0.167028 + 0.985952i \(0.446583\pi\)
\(308\) 0 0
\(309\) 10.9024 0.620213
\(310\) 0 0
\(311\) 0.298169 0.0169076 0.00845381 0.999964i \(-0.497309\pi\)
0.00845381 + 0.999964i \(0.497309\pi\)
\(312\) 0 0
\(313\) −23.7991 −1.34521 −0.672603 0.740004i \(-0.734824\pi\)
−0.672603 + 0.740004i \(0.734824\pi\)
\(314\) 0 0
\(315\) −4.39875 −0.247841
\(316\) 0 0
\(317\) 2.90759 0.163306 0.0816532 0.996661i \(-0.473980\pi\)
0.0816532 + 0.996661i \(0.473980\pi\)
\(318\) 0 0
\(319\) 23.4557 1.31327
\(320\) 0 0
\(321\) 8.77415 0.489725
\(322\) 0 0
\(323\) 15.7794 0.877987
\(324\) 0 0
\(325\) −79.1782 −4.39202
\(326\) 0 0
\(327\) 7.47654 0.413454
\(328\) 0 0
\(329\) −3.43403 −0.189324
\(330\) 0 0
\(331\) 13.3354 0.732977 0.366489 0.930423i \(-0.380560\pi\)
0.366489 + 0.930423i \(0.380560\pi\)
\(332\) 0 0
\(333\) −6.66406 −0.365188
\(334\) 0 0
\(335\) 26.8877 1.46903
\(336\) 0 0
\(337\) 27.5716 1.50192 0.750962 0.660346i \(-0.229590\pi\)
0.750962 + 0.660346i \(0.229590\pi\)
\(338\) 0 0
\(339\) −6.90759 −0.375169
\(340\) 0 0
\(341\) 3.93963 0.213343
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −4.39875 −0.236821
\(346\) 0 0
\(347\) 27.1016 1.45489 0.727445 0.686166i \(-0.240707\pi\)
0.727445 + 0.686166i \(0.240707\pi\)
\(348\) 0 0
\(349\) −30.8549 −1.65162 −0.825812 0.563946i \(-0.809283\pi\)
−0.825812 + 0.563946i \(0.809283\pi\)
\(350\) 0 0
\(351\) 5.51804 0.294531
\(352\) 0 0
\(353\) −6.13344 −0.326450 −0.163225 0.986589i \(-0.552190\pi\)
−0.163225 + 0.986589i \(0.552190\pi\)
\(354\) 0 0
\(355\) 35.2926 1.87314
\(356\) 0 0
\(357\) −3.11930 −0.165091
\(358\) 0 0
\(359\) −24.6084 −1.29878 −0.649390 0.760456i \(-0.724976\pi\)
−0.649390 + 0.760456i \(0.724976\pi\)
\(360\) 0 0
\(361\) 6.58971 0.346827
\(362\) 0 0
\(363\) −1.38851 −0.0728776
\(364\) 0 0
\(365\) 12.9773 0.679261
\(366\) 0 0
\(367\) 31.3384 1.63585 0.817924 0.575326i \(-0.195125\pi\)
0.817924 + 0.575326i \(0.195125\pi\)
\(368\) 0 0
\(369\) −10.6034 −0.551990
\(370\) 0 0
\(371\) −9.54307 −0.495452
\(372\) 0 0
\(373\) 27.7860 1.43871 0.719353 0.694644i \(-0.244438\pi\)
0.719353 + 0.694644i \(0.244438\pi\)
\(374\) 0 0
\(375\) 41.1237 2.12362
\(376\) 0 0
\(377\) 36.7726 1.89388
\(378\) 0 0
\(379\) 18.1449 0.932041 0.466020 0.884774i \(-0.345687\pi\)
0.466020 + 0.884774i \(0.345687\pi\)
\(380\) 0 0
\(381\) 17.5269 0.897930
\(382\) 0 0
\(383\) −9.59881 −0.490476 −0.245238 0.969463i \(-0.578866\pi\)
−0.245238 + 0.969463i \(0.578866\pi\)
\(384\) 0 0
\(385\) 15.4824 0.789056
\(386\) 0 0
\(387\) −6.81835 −0.346596
\(388\) 0 0
\(389\) −19.9645 −1.01224 −0.506121 0.862463i \(-0.668921\pi\)
−0.506121 + 0.862463i \(0.668921\pi\)
\(390\) 0 0
\(391\) −3.11930 −0.157750
\(392\) 0 0
\(393\) −9.00752 −0.454369
\(394\) 0 0
\(395\) 58.1386 2.92527
\(396\) 0 0
\(397\) 15.1473 0.760222 0.380111 0.924941i \(-0.375886\pi\)
0.380111 + 0.924941i \(0.375886\pi\)
\(398\) 0 0
\(399\) −5.05863 −0.253248
\(400\) 0 0
\(401\) −23.6837 −1.18271 −0.591353 0.806413i \(-0.701406\pi\)
−0.591353 + 0.806413i \(0.701406\pi\)
\(402\) 0 0
\(403\) 6.17633 0.307665
\(404\) 0 0
\(405\) −4.39875 −0.218575
\(406\) 0 0
\(407\) 23.4557 1.16266
\(408\) 0 0
\(409\) 24.2817 1.20065 0.600327 0.799755i \(-0.295037\pi\)
0.600327 + 0.799755i \(0.295037\pi\)
\(410\) 0 0
\(411\) 2.72473 0.134401
\(412\) 0 0
\(413\) −11.5431 −0.567997
\(414\) 0 0
\(415\) −43.6214 −2.14129
\(416\) 0 0
\(417\) −5.43821 −0.266310
\(418\) 0 0
\(419\) −23.0431 −1.12573 −0.562864 0.826549i \(-0.690301\pi\)
−0.562864 + 0.826549i \(0.690301\pi\)
\(420\) 0 0
\(421\) 1.77888 0.0866975 0.0433487 0.999060i \(-0.486197\pi\)
0.0433487 + 0.999060i \(0.486197\pi\)
\(422\) 0 0
\(423\) −3.43403 −0.166968
\(424\) 0 0
\(425\) 44.7587 2.17112
\(426\) 0 0
\(427\) 13.9627 0.675701
\(428\) 0 0
\(429\) −19.4220 −0.937704
\(430\) 0 0
\(431\) −0.691068 −0.0332876 −0.0166438 0.999861i \(-0.505298\pi\)
−0.0166438 + 0.999861i \(0.505298\pi\)
\(432\) 0 0
\(433\) −31.0340 −1.49140 −0.745699 0.666283i \(-0.767884\pi\)
−0.745699 + 0.666283i \(0.767884\pi\)
\(434\) 0 0
\(435\) −29.3135 −1.40547
\(436\) 0 0
\(437\) −5.05863 −0.241987
\(438\) 0 0
\(439\) −10.3974 −0.496239 −0.248119 0.968729i \(-0.579813\pi\)
−0.248119 + 0.968729i \(0.579813\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −9.20500 −0.437343 −0.218671 0.975799i \(-0.570172\pi\)
−0.218671 + 0.975799i \(0.570172\pi\)
\(444\) 0 0
\(445\) 18.9720 0.899359
\(446\) 0 0
\(447\) −12.9788 −0.613876
\(448\) 0 0
\(449\) 20.2127 0.953896 0.476948 0.878931i \(-0.341743\pi\)
0.476948 + 0.878931i \(0.341743\pi\)
\(450\) 0 0
\(451\) 37.3211 1.75738
\(452\) 0 0
\(453\) 22.7393 1.06839
\(454\) 0 0
\(455\) 24.2725 1.13791
\(456\) 0 0
\(457\) −31.8183 −1.48840 −0.744199 0.667958i \(-0.767169\pi\)
−0.744199 + 0.667958i \(0.767169\pi\)
\(458\) 0 0
\(459\) −3.11930 −0.145596
\(460\) 0 0
\(461\) 30.9379 1.44092 0.720460 0.693496i \(-0.243930\pi\)
0.720460 + 0.693496i \(0.243930\pi\)
\(462\) 0 0
\(463\) 41.5016 1.92874 0.964371 0.264553i \(-0.0852245\pi\)
0.964371 + 0.264553i \(0.0852245\pi\)
\(464\) 0 0
\(465\) −4.92350 −0.228322
\(466\) 0 0
\(467\) −36.9392 −1.70934 −0.854672 0.519169i \(-0.826242\pi\)
−0.854672 + 0.519169i \(0.826242\pi\)
\(468\) 0 0
\(469\) −6.11258 −0.282253
\(470\) 0 0
\(471\) −13.9393 −0.642290
\(472\) 0 0
\(473\) 23.9987 1.10346
\(474\) 0 0
\(475\) 72.5861 3.33048
\(476\) 0 0
\(477\) −9.54307 −0.436947
\(478\) 0 0
\(479\) −2.97512 −0.135937 −0.0679684 0.997687i \(-0.521652\pi\)
−0.0679684 + 0.997687i \(0.521652\pi\)
\(480\) 0 0
\(481\) 36.7726 1.67668
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −71.2727 −3.23633
\(486\) 0 0
\(487\) −18.3542 −0.831707 −0.415854 0.909432i \(-0.636517\pi\)
−0.415854 + 0.909432i \(0.636517\pi\)
\(488\) 0 0
\(489\) −1.82337 −0.0824558
\(490\) 0 0
\(491\) −8.65281 −0.390496 −0.195248 0.980754i \(-0.562551\pi\)
−0.195248 + 0.980754i \(0.562551\pi\)
\(492\) 0 0
\(493\) −20.7872 −0.936207
\(494\) 0 0
\(495\) 15.4824 0.695882
\(496\) 0 0
\(497\) −8.02334 −0.359896
\(498\) 0 0
\(499\) −0.624797 −0.0279698 −0.0139849 0.999902i \(-0.504452\pi\)
−0.0139849 + 0.999902i \(0.504452\pi\)
\(500\) 0 0
\(501\) −9.05863 −0.404710
\(502\) 0 0
\(503\) 0.197295 0.00879695 0.00439848 0.999990i \(-0.498600\pi\)
0.00439848 + 0.999990i \(0.498600\pi\)
\(504\) 0 0
\(505\) −6.35741 −0.282901
\(506\) 0 0
\(507\) −17.4488 −0.774929
\(508\) 0 0
\(509\) −14.8988 −0.660378 −0.330189 0.943915i \(-0.607113\pi\)
−0.330189 + 0.943915i \(0.607113\pi\)
\(510\) 0 0
\(511\) −2.95022 −0.130510
\(512\) 0 0
\(513\) −5.05863 −0.223344
\(514\) 0 0
\(515\) 47.9567 2.11322
\(516\) 0 0
\(517\) 12.0869 0.531580
\(518\) 0 0
\(519\) −8.71766 −0.382663
\(520\) 0 0
\(521\) 16.7786 0.735085 0.367542 0.930007i \(-0.380199\pi\)
0.367542 + 0.930007i \(0.380199\pi\)
\(522\) 0 0
\(523\) −18.8854 −0.825800 −0.412900 0.910776i \(-0.635484\pi\)
−0.412900 + 0.910776i \(0.635484\pi\)
\(524\) 0 0
\(525\) −14.3490 −0.626240
\(526\) 0 0
\(527\) −3.49142 −0.152089
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −11.5431 −0.500927
\(532\) 0 0
\(533\) 58.5100 2.53435
\(534\) 0 0
\(535\) 38.5953 1.66862
\(536\) 0 0
\(537\) 1.59788 0.0689536
\(538\) 0 0
\(539\) −3.51973 −0.151606
\(540\) 0 0
\(541\) 35.0510 1.50696 0.753479 0.657472i \(-0.228374\pi\)
0.753479 + 0.657472i \(0.228374\pi\)
\(542\) 0 0
\(543\) −16.7083 −0.717020
\(544\) 0 0
\(545\) 32.8874 1.40874
\(546\) 0 0
\(547\) 17.8488 0.763161 0.381580 0.924336i \(-0.375380\pi\)
0.381580 + 0.924336i \(0.375380\pi\)
\(548\) 0 0
\(549\) 13.9627 0.595912
\(550\) 0 0
\(551\) −33.7110 −1.43614
\(552\) 0 0
\(553\) −13.2171 −0.562048
\(554\) 0 0
\(555\) −29.3135 −1.24429
\(556\) 0 0
\(557\) 8.11388 0.343796 0.171898 0.985115i \(-0.445010\pi\)
0.171898 + 0.985115i \(0.445010\pi\)
\(558\) 0 0
\(559\) 37.6239 1.59132
\(560\) 0 0
\(561\) 10.9791 0.463537
\(562\) 0 0
\(563\) 36.4963 1.53813 0.769067 0.639168i \(-0.220721\pi\)
0.769067 + 0.639168i \(0.220721\pi\)
\(564\) 0 0
\(565\) −30.3847 −1.27829
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 7.04084 0.295167 0.147584 0.989050i \(-0.452850\pi\)
0.147584 + 0.989050i \(0.452850\pi\)
\(570\) 0 0
\(571\) 8.90016 0.372460 0.186230 0.982506i \(-0.440373\pi\)
0.186230 + 0.982506i \(0.440373\pi\)
\(572\) 0 0
\(573\) −9.78971 −0.408971
\(574\) 0 0
\(575\) −14.3490 −0.598393
\(576\) 0 0
\(577\) 11.5829 0.482203 0.241102 0.970500i \(-0.422491\pi\)
0.241102 + 0.970500i \(0.422491\pi\)
\(578\) 0 0
\(579\) −6.13373 −0.254909
\(580\) 0 0
\(581\) 9.91679 0.411418
\(582\) 0 0
\(583\) 33.5890 1.39112
\(584\) 0 0
\(585\) 24.2725 1.00354
\(586\) 0 0
\(587\) 38.7702 1.60022 0.800109 0.599855i \(-0.204775\pi\)
0.800109 + 0.599855i \(0.204775\pi\)
\(588\) 0 0
\(589\) −5.66211 −0.233303
\(590\) 0 0
\(591\) −24.9157 −1.02489
\(592\) 0 0
\(593\) −29.8912 −1.22748 −0.613742 0.789506i \(-0.710337\pi\)
−0.613742 + 0.789506i \(0.710337\pi\)
\(594\) 0 0
\(595\) −13.7210 −0.562506
\(596\) 0 0
\(597\) 17.9499 0.734639
\(598\) 0 0
\(599\) −30.6861 −1.25380 −0.626899 0.779100i \(-0.715676\pi\)
−0.626899 + 0.779100i \(0.715676\pi\)
\(600\) 0 0
\(601\) 25.4323 1.03741 0.518703 0.854954i \(-0.326415\pi\)
0.518703 + 0.854954i \(0.326415\pi\)
\(602\) 0 0
\(603\) −6.11258 −0.248924
\(604\) 0 0
\(605\) −6.10768 −0.248313
\(606\) 0 0
\(607\) 1.11462 0.0452412 0.0226206 0.999744i \(-0.492799\pi\)
0.0226206 + 0.999744i \(0.492799\pi\)
\(608\) 0 0
\(609\) 6.66406 0.270041
\(610\) 0 0
\(611\) 18.9491 0.766600
\(612\) 0 0
\(613\) 23.7284 0.958382 0.479191 0.877711i \(-0.340930\pi\)
0.479191 + 0.877711i \(0.340930\pi\)
\(614\) 0 0
\(615\) −46.6416 −1.88077
\(616\) 0 0
\(617\) 4.48212 0.180444 0.0902218 0.995922i \(-0.471242\pi\)
0.0902218 + 0.995922i \(0.471242\pi\)
\(618\) 0 0
\(619\) −45.2828 −1.82007 −0.910035 0.414531i \(-0.863946\pi\)
−0.910035 + 0.414531i \(0.863946\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −4.31305 −0.172799
\(624\) 0 0
\(625\) 109.148 4.36592
\(626\) 0 0
\(627\) 17.8050 0.711064
\(628\) 0 0
\(629\) −20.7872 −0.828839
\(630\) 0 0
\(631\) 41.5050 1.65229 0.826145 0.563458i \(-0.190529\pi\)
0.826145 + 0.563458i \(0.190529\pi\)
\(632\) 0 0
\(633\) −3.51973 −0.139897
\(634\) 0 0
\(635\) 77.0963 3.05948
\(636\) 0 0
\(637\) −5.51804 −0.218633
\(638\) 0 0
\(639\) −8.02334 −0.317398
\(640\) 0 0
\(641\) −23.0886 −0.911946 −0.455973 0.889994i \(-0.650709\pi\)
−0.455973 + 0.889994i \(0.650709\pi\)
\(642\) 0 0
\(643\) 25.0216 0.986755 0.493378 0.869815i \(-0.335762\pi\)
0.493378 + 0.869815i \(0.335762\pi\)
\(644\) 0 0
\(645\) −29.9922 −1.18094
\(646\) 0 0
\(647\) 3.50932 0.137966 0.0689828 0.997618i \(-0.478025\pi\)
0.0689828 + 0.997618i \(0.478025\pi\)
\(648\) 0 0
\(649\) 40.6285 1.59481
\(650\) 0 0
\(651\) 1.11930 0.0438687
\(652\) 0 0
\(653\) −11.2514 −0.440300 −0.220150 0.975466i \(-0.570655\pi\)
−0.220150 + 0.975466i \(0.570655\pi\)
\(654\) 0 0
\(655\) −39.6218 −1.54815
\(656\) 0 0
\(657\) −2.95022 −0.115099
\(658\) 0 0
\(659\) −25.9614 −1.01131 −0.505656 0.862735i \(-0.668750\pi\)
−0.505656 + 0.862735i \(0.668750\pi\)
\(660\) 0 0
\(661\) −1.56641 −0.0609264 −0.0304632 0.999536i \(-0.509698\pi\)
−0.0304632 + 0.999536i \(0.509698\pi\)
\(662\) 0 0
\(663\) 17.2124 0.668475
\(664\) 0 0
\(665\) −22.2516 −0.862881
\(666\) 0 0
\(667\) 6.66406 0.258033
\(668\) 0 0
\(669\) −15.1709 −0.586541
\(670\) 0 0
\(671\) −49.1448 −1.89722
\(672\) 0 0
\(673\) 6.85541 0.264257 0.132128 0.991233i \(-0.457819\pi\)
0.132128 + 0.991233i \(0.457819\pi\)
\(674\) 0 0
\(675\) −14.3490 −0.552292
\(676\) 0 0
\(677\) −0.443792 −0.0170563 −0.00852817 0.999964i \(-0.502715\pi\)
−0.00852817 + 0.999964i \(0.502715\pi\)
\(678\) 0 0
\(679\) 16.2030 0.621812
\(680\) 0 0
\(681\) −17.6039 −0.674583
\(682\) 0 0
\(683\) 19.7794 0.756836 0.378418 0.925635i \(-0.376468\pi\)
0.378418 + 0.925635i \(0.376468\pi\)
\(684\) 0 0
\(685\) 11.9854 0.457938
\(686\) 0 0
\(687\) 14.4746 0.552241
\(688\) 0 0
\(689\) 52.6591 2.00615
\(690\) 0 0
\(691\) 12.4590 0.473963 0.236981 0.971514i \(-0.423842\pi\)
0.236981 + 0.971514i \(0.423842\pi\)
\(692\) 0 0
\(693\) −3.51973 −0.133704
\(694\) 0 0
\(695\) −23.9213 −0.907386
\(696\) 0 0
\(697\) −33.0751 −1.25281
\(698\) 0 0
\(699\) 26.5474 1.00411
\(700\) 0 0
\(701\) 14.2983 0.540038 0.270019 0.962855i \(-0.412970\pi\)
0.270019 + 0.962855i \(0.412970\pi\)
\(702\) 0 0
\(703\) −33.7110 −1.27143
\(704\) 0 0
\(705\) −15.1054 −0.568904
\(706\) 0 0
\(707\) 1.44528 0.0543553
\(708\) 0 0
\(709\) 3.46444 0.130110 0.0650550 0.997882i \(-0.479278\pi\)
0.0650550 + 0.997882i \(0.479278\pi\)
\(710\) 0 0
\(711\) −13.2171 −0.495680
\(712\) 0 0
\(713\) 1.11930 0.0419180
\(714\) 0 0
\(715\) −85.4326 −3.19500
\(716\) 0 0
\(717\) 4.35118 0.162498
\(718\) 0 0
\(719\) 25.8379 0.963593 0.481796 0.876283i \(-0.339985\pi\)
0.481796 + 0.876283i \(0.339985\pi\)
\(720\) 0 0
\(721\) −10.9024 −0.406025
\(722\) 0 0
\(723\) −20.0505 −0.745687
\(724\) 0 0
\(725\) −95.6223 −3.55132
\(726\) 0 0
\(727\) −22.7739 −0.844637 −0.422319 0.906447i \(-0.638784\pi\)
−0.422319 + 0.906447i \(0.638784\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −21.2684 −0.786642
\(732\) 0 0
\(733\) 44.3651 1.63866 0.819331 0.573321i \(-0.194345\pi\)
0.819331 + 0.573321i \(0.194345\pi\)
\(734\) 0 0
\(735\) 4.39875 0.162250
\(736\) 0 0
\(737\) 21.5146 0.792502
\(738\) 0 0
\(739\) −20.6877 −0.761011 −0.380505 0.924779i \(-0.624250\pi\)
−0.380505 + 0.924779i \(0.624250\pi\)
\(740\) 0 0
\(741\) 27.9137 1.02544
\(742\) 0 0
\(743\) 21.2056 0.777959 0.388980 0.921246i \(-0.372828\pi\)
0.388980 + 0.921246i \(0.372828\pi\)
\(744\) 0 0
\(745\) −57.0904 −2.09163
\(746\) 0 0
\(747\) 9.91679 0.362836
\(748\) 0 0
\(749\) −8.77415 −0.320601
\(750\) 0 0
\(751\) 35.6869 1.30223 0.651117 0.758977i \(-0.274301\pi\)
0.651117 + 0.758977i \(0.274301\pi\)
\(752\) 0 0
\(753\) 14.1426 0.515384
\(754\) 0 0
\(755\) 100.024 3.64026
\(756\) 0 0
\(757\) −25.1596 −0.914440 −0.457220 0.889354i \(-0.651155\pi\)
−0.457220 + 0.889354i \(0.651155\pi\)
\(758\) 0 0
\(759\) −3.51973 −0.127758
\(760\) 0 0
\(761\) 35.4096 1.28360 0.641799 0.766873i \(-0.278189\pi\)
0.641799 + 0.766873i \(0.278189\pi\)
\(762\) 0 0
\(763\) −7.47654 −0.270669
\(764\) 0 0
\(765\) −13.7210 −0.496084
\(766\) 0 0
\(767\) 63.6952 2.29990
\(768\) 0 0
\(769\) 8.97083 0.323496 0.161748 0.986832i \(-0.448287\pi\)
0.161748 + 0.986832i \(0.448287\pi\)
\(770\) 0 0
\(771\) 22.8292 0.822173
\(772\) 0 0
\(773\) −15.2725 −0.549312 −0.274656 0.961543i \(-0.588564\pi\)
−0.274656 + 0.961543i \(0.588564\pi\)
\(774\) 0 0
\(775\) −16.0608 −0.576920
\(776\) 0 0
\(777\) 6.66406 0.239072
\(778\) 0 0
\(779\) −53.6386 −1.92180
\(780\) 0 0
\(781\) 28.2400 1.01051
\(782\) 0 0
\(783\) 6.66406 0.238154
\(784\) 0 0
\(785\) −61.3156 −2.18845
\(786\) 0 0
\(787\) −4.07137 −0.145129 −0.0725644 0.997364i \(-0.523118\pi\)
−0.0725644 + 0.997364i \(0.523118\pi\)
\(788\) 0 0
\(789\) 7.95143 0.283078
\(790\) 0 0
\(791\) 6.90759 0.245605
\(792\) 0 0
\(793\) −77.0466 −2.73601
\(794\) 0 0
\(795\) −41.9776 −1.48879
\(796\) 0 0
\(797\) −2.06935 −0.0733002 −0.0366501 0.999328i \(-0.511669\pi\)
−0.0366501 + 0.999328i \(0.511669\pi\)
\(798\) 0 0
\(799\) −10.7118 −0.378955
\(800\) 0 0
\(801\) −4.31305 −0.152394
\(802\) 0 0
\(803\) 10.3840 0.366443
\(804\) 0 0
\(805\) 4.39875 0.155035
\(806\) 0 0
\(807\) −2.07445 −0.0730242
\(808\) 0 0
\(809\) −3.73806 −0.131423 −0.0657117 0.997839i \(-0.520932\pi\)
−0.0657117 + 0.997839i \(0.520932\pi\)
\(810\) 0 0
\(811\) −43.7178 −1.53514 −0.767571 0.640965i \(-0.778535\pi\)
−0.767571 + 0.640965i \(0.778535\pi\)
\(812\) 0 0
\(813\) −18.7177 −0.656457
\(814\) 0 0
\(815\) −8.02056 −0.280948
\(816\) 0 0
\(817\) −34.4915 −1.20670
\(818\) 0 0
\(819\) −5.51804 −0.192816
\(820\) 0 0
\(821\) 36.7480 1.28251 0.641257 0.767326i \(-0.278413\pi\)
0.641257 + 0.767326i \(0.278413\pi\)
\(822\) 0 0
\(823\) −25.5698 −0.891306 −0.445653 0.895206i \(-0.647028\pi\)
−0.445653 + 0.895206i \(0.647028\pi\)
\(824\) 0 0
\(825\) 50.5045 1.75834
\(826\) 0 0
\(827\) −2.15043 −0.0747776 −0.0373888 0.999301i \(-0.511904\pi\)
−0.0373888 + 0.999301i \(0.511904\pi\)
\(828\) 0 0
\(829\) −54.4745 −1.89198 −0.945989 0.324199i \(-0.894905\pi\)
−0.945989 + 0.324199i \(0.894905\pi\)
\(830\) 0 0
\(831\) −5.07342 −0.175995
\(832\) 0 0
\(833\) 3.11930 0.108077
\(834\) 0 0
\(835\) −39.8466 −1.37895
\(836\) 0 0
\(837\) 1.11930 0.0386886
\(838\) 0 0
\(839\) −35.4836 −1.22503 −0.612514 0.790460i \(-0.709842\pi\)
−0.612514 + 0.790460i \(0.709842\pi\)
\(840\) 0 0
\(841\) 15.4097 0.531367
\(842\) 0 0
\(843\) −13.6688 −0.470777
\(844\) 0 0
\(845\) −76.7529 −2.64038
\(846\) 0 0
\(847\) 1.38851 0.0477096
\(848\) 0 0
\(849\) 28.8011 0.988452
\(850\) 0 0
\(851\) 6.66406 0.228441
\(852\) 0 0
\(853\) 8.84772 0.302940 0.151470 0.988462i \(-0.451599\pi\)
0.151470 + 0.988462i \(0.451599\pi\)
\(854\) 0 0
\(855\) −22.2516 −0.760989
\(856\) 0 0
\(857\) 24.5225 0.837673 0.418836 0.908062i \(-0.362438\pi\)
0.418836 + 0.908062i \(0.362438\pi\)
\(858\) 0 0
\(859\) −6.52138 −0.222507 −0.111253 0.993792i \(-0.535487\pi\)
−0.111253 + 0.993792i \(0.535487\pi\)
\(860\) 0 0
\(861\) 10.6034 0.361362
\(862\) 0 0
\(863\) −3.21523 −0.109448 −0.0547238 0.998502i \(-0.517428\pi\)
−0.0547238 + 0.998502i \(0.517428\pi\)
\(864\) 0 0
\(865\) −38.3468 −1.30383
\(866\) 0 0
\(867\) 7.26998 0.246902
\(868\) 0 0
\(869\) 46.5206 1.57810
\(870\) 0 0
\(871\) 33.7295 1.14288
\(872\) 0 0
\(873\) 16.2030 0.548387
\(874\) 0 0
\(875\) −41.1237 −1.39024
\(876\) 0 0
\(877\) 12.8366 0.433462 0.216731 0.976231i \(-0.430460\pi\)
0.216731 + 0.976231i \(0.430460\pi\)
\(878\) 0 0
\(879\) −2.03833 −0.0687512
\(880\) 0 0
\(881\) 34.4082 1.15924 0.579621 0.814886i \(-0.303200\pi\)
0.579621 + 0.814886i \(0.303200\pi\)
\(882\) 0 0
\(883\) 14.9055 0.501612 0.250806 0.968037i \(-0.419304\pi\)
0.250806 + 0.968037i \(0.419304\pi\)
\(884\) 0 0
\(885\) −50.7750 −1.70678
\(886\) 0 0
\(887\) −16.3556 −0.549169 −0.274584 0.961563i \(-0.588540\pi\)
−0.274584 + 0.961563i \(0.588540\pi\)
\(888\) 0 0
\(889\) −17.5269 −0.587833
\(890\) 0 0
\(891\) −3.51973 −0.117915
\(892\) 0 0
\(893\) −17.3715 −0.581315
\(894\) 0 0
\(895\) 7.02866 0.234942
\(896\) 0 0
\(897\) −5.51804 −0.184242
\(898\) 0 0
\(899\) 7.45906 0.248774
\(900\) 0 0
\(901\) −29.7677 −0.991705
\(902\) 0 0
\(903\) 6.81835 0.226900
\(904\) 0 0
\(905\) −73.4954 −2.44307
\(906\) 0 0
\(907\) −32.6282 −1.08340 −0.541700 0.840572i \(-0.682219\pi\)
−0.541700 + 0.840572i \(0.682219\pi\)
\(908\) 0 0
\(909\) 1.44528 0.0479368
\(910\) 0 0
\(911\) 11.9478 0.395847 0.197923 0.980217i \(-0.436580\pi\)
0.197923 + 0.980217i \(0.436580\pi\)
\(912\) 0 0
\(913\) −34.9044 −1.15517
\(914\) 0 0
\(915\) 61.4182 2.03042
\(916\) 0 0
\(917\) 9.00752 0.297454
\(918\) 0 0
\(919\) −29.0429 −0.958037 −0.479019 0.877805i \(-0.659007\pi\)
−0.479019 + 0.877805i \(0.659007\pi\)
\(920\) 0 0
\(921\) −5.85313 −0.192867
\(922\) 0 0
\(923\) 44.2731 1.45727
\(924\) 0 0
\(925\) −95.6223 −3.14404
\(926\) 0 0
\(927\) −10.9024 −0.358080
\(928\) 0 0
\(929\) 38.3200 1.25724 0.628619 0.777713i \(-0.283620\pi\)
0.628619 + 0.777713i \(0.283620\pi\)
\(930\) 0 0
\(931\) 5.05863 0.165790
\(932\) 0 0
\(933\) −0.298169 −0.00976161
\(934\) 0 0
\(935\) 48.2942 1.57939
\(936\) 0 0
\(937\) 39.5517 1.29210 0.646049 0.763296i \(-0.276420\pi\)
0.646049 + 0.763296i \(0.276420\pi\)
\(938\) 0 0
\(939\) 23.7991 0.776655
\(940\) 0 0
\(941\) 45.1917 1.47321 0.736604 0.676324i \(-0.236428\pi\)
0.736604 + 0.676324i \(0.236428\pi\)
\(942\) 0 0
\(943\) 10.6034 0.345294
\(944\) 0 0
\(945\) 4.39875 0.143091
\(946\) 0 0
\(947\) 43.3517 1.40874 0.704370 0.709833i \(-0.251230\pi\)
0.704370 + 0.709833i \(0.251230\pi\)
\(948\) 0 0
\(949\) 16.2794 0.528453
\(950\) 0 0
\(951\) −2.90759 −0.0942849
\(952\) 0 0
\(953\) −22.7725 −0.737673 −0.368836 0.929494i \(-0.620244\pi\)
−0.368836 + 0.929494i \(0.620244\pi\)
\(954\) 0 0
\(955\) −43.0625 −1.39347
\(956\) 0 0
\(957\) −23.4557 −0.758215
\(958\) 0 0
\(959\) −2.72473 −0.0879860
\(960\) 0 0
\(961\) −29.7472 −0.959586
\(962\) 0 0
\(963\) −8.77415 −0.282743
\(964\) 0 0
\(965\) −26.9807 −0.868540
\(966\) 0 0
\(967\) −6.09956 −0.196149 −0.0980743 0.995179i \(-0.531268\pi\)
−0.0980743 + 0.995179i \(0.531268\pi\)
\(968\) 0 0
\(969\) −15.7794 −0.506906
\(970\) 0 0
\(971\) 29.6337 0.950991 0.475495 0.879718i \(-0.342269\pi\)
0.475495 + 0.879718i \(0.342269\pi\)
\(972\) 0 0
\(973\) 5.43821 0.174341
\(974\) 0 0
\(975\) 79.1782 2.53573
\(976\) 0 0
\(977\) 5.51346 0.176391 0.0881956 0.996103i \(-0.471890\pi\)
0.0881956 + 0.996103i \(0.471890\pi\)
\(978\) 0 0
\(979\) 15.1808 0.485179
\(980\) 0 0
\(981\) −7.47654 −0.238708
\(982\) 0 0
\(983\) 6.35637 0.202737 0.101368 0.994849i \(-0.467678\pi\)
0.101368 + 0.994849i \(0.467678\pi\)
\(984\) 0 0
\(985\) −109.598 −3.49207
\(986\) 0 0
\(987\) 3.43403 0.109306
\(988\) 0 0
\(989\) 6.81835 0.216811
\(990\) 0 0
\(991\) 5.45383 0.173247 0.0866233 0.996241i \(-0.472392\pi\)
0.0866233 + 0.996241i \(0.472392\pi\)
\(992\) 0 0
\(993\) −13.3354 −0.423185
\(994\) 0 0
\(995\) 78.9569 2.50310
\(996\) 0 0
\(997\) −27.2701 −0.863651 −0.431826 0.901957i \(-0.642130\pi\)
−0.431826 + 0.901957i \(0.642130\pi\)
\(998\) 0 0
\(999\) 6.66406 0.210841
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.cg.1.1 6
4.3 odd 2 3864.2.a.x.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.x.1.1 6 4.3 odd 2
7728.2.a.cg.1.1 6 1.1 even 1 trivial