Properties

Label 882.2.a.n.1.1
Level $882$
Weight $2$
Character 882.1
Self dual yes
Analytic conductor $7.043$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 882.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.82843 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.82843 q^{5} -1.00000 q^{8} +2.82843 q^{10} +2.00000 q^{11} +1.00000 q^{16} +1.41421 q^{17} -7.07107 q^{19} -2.82843 q^{20} -2.00000 q^{22} +4.00000 q^{23} +3.00000 q^{25} -2.00000 q^{29} +8.48528 q^{31} -1.00000 q^{32} -1.41421 q^{34} +10.0000 q^{37} +7.07107 q^{38} +2.82843 q^{40} +9.89949 q^{41} +2.00000 q^{43} +2.00000 q^{44} -4.00000 q^{46} -2.82843 q^{47} -3.00000 q^{50} +2.00000 q^{53} -5.65685 q^{55} +2.00000 q^{58} +1.41421 q^{59} +2.82843 q^{61} -8.48528 q^{62} +1.00000 q^{64} +12.0000 q^{67} +1.41421 q^{68} +12.0000 q^{71} -1.41421 q^{73} -10.0000 q^{74} -7.07107 q^{76} -4.00000 q^{79} -2.82843 q^{80} -9.89949 q^{82} -9.89949 q^{83} -4.00000 q^{85} -2.00000 q^{86} -2.00000 q^{88} +7.07107 q^{89} +4.00000 q^{92} +2.82843 q^{94} +20.0000 q^{95} +9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{11} + 2 q^{16} - 4 q^{22} + 8 q^{23} + 6 q^{25} - 4 q^{29} - 2 q^{32} + 20 q^{37} + 4 q^{43} + 4 q^{44} - 8 q^{46} - 6 q^{50} + 4 q^{53} + 4 q^{58} + 2 q^{64} + 24 q^{67} + 24 q^{71} - 20 q^{74} - 8 q^{79} - 8 q^{85} - 4 q^{86} - 4 q^{88} + 8 q^{92} + 40 q^{95} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.82843 −1.26491 −0.632456 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 2.82843 0.894427
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.41421 0.342997 0.171499 0.985184i \(-0.445139\pi\)
0.171499 + 0.985184i \(0.445139\pi\)
\(18\) 0 0
\(19\) −7.07107 −1.62221 −0.811107 0.584898i \(-0.801135\pi\)
−0.811107 + 0.584898i \(0.801135\pi\)
\(20\) −2.82843 −0.632456
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 8.48528 1.52400 0.762001 0.647576i \(-0.224217\pi\)
0.762001 + 0.647576i \(0.224217\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −1.41421 −0.242536
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 7.07107 1.14708
\(39\) 0 0
\(40\) 2.82843 0.447214
\(41\) 9.89949 1.54604 0.773021 0.634381i \(-0.218745\pi\)
0.773021 + 0.634381i \(0.218745\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −3.00000 −0.424264
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.762770
\(56\) 0 0
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 1.41421 0.184115 0.0920575 0.995754i \(-0.470656\pi\)
0.0920575 + 0.995754i \(0.470656\pi\)
\(60\) 0 0
\(61\) 2.82843 0.362143 0.181071 0.983470i \(-0.442043\pi\)
0.181071 + 0.983470i \(0.442043\pi\)
\(62\) −8.48528 −1.07763
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 1.41421 0.171499
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −1.41421 −0.165521 −0.0827606 0.996569i \(-0.526374\pi\)
−0.0827606 + 0.996569i \(0.526374\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −7.07107 −0.811107
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −2.82843 −0.316228
\(81\) 0 0
\(82\) −9.89949 −1.09322
\(83\) −9.89949 −1.08661 −0.543305 0.839535i \(-0.682827\pi\)
−0.543305 + 0.839535i \(0.682827\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) 7.07107 0.749532 0.374766 0.927119i \(-0.377723\pi\)
0.374766 + 0.927119i \(0.377723\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 2.82843 0.291730
\(95\) 20.0000 2.05196
\(96\) 0 0
\(97\) 9.89949 1.00514 0.502571 0.864536i \(-0.332388\pi\)
0.502571 + 0.864536i \(0.332388\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 3.00000 0.300000
\(101\) −8.48528 −0.844317 −0.422159 0.906522i \(-0.638727\pi\)
−0.422159 + 0.906522i \(0.638727\pi\)
\(102\) 0 0
\(103\) 2.82843 0.278693 0.139347 0.990244i \(-0.455500\pi\)
0.139347 + 0.990244i \(0.455500\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 5.65685 0.539360
\(111\) 0 0
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) −11.3137 −1.05501
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) −1.41421 −0.130189
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −2.82843 −0.256074
\(123\) 0 0
\(124\) 8.48528 0.762001
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −12.7279 −1.11204 −0.556022 0.831168i \(-0.687673\pi\)
−0.556022 + 0.831168i \(0.687673\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −1.41421 −0.121268
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 9.89949 0.839664 0.419832 0.907602i \(-0.362089\pi\)
0.419832 + 0.907602i \(0.362089\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) 0 0
\(145\) 5.65685 0.469776
\(146\) 1.41421 0.117041
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 7.07107 0.573539
\(153\) 0 0
\(154\) 0 0
\(155\) −24.0000 −1.92773
\(156\) 0 0
\(157\) −11.3137 −0.902932 −0.451466 0.892288i \(-0.649099\pi\)
−0.451466 + 0.892288i \(0.649099\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) 2.82843 0.223607
\(161\) 0 0
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 9.89949 0.773021
\(165\) 0 0
\(166\) 9.89949 0.768350
\(167\) 19.7990 1.53209 0.766046 0.642786i \(-0.222221\pi\)
0.766046 + 0.642786i \(0.222221\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) 16.9706 1.29025 0.645124 0.764078i \(-0.276806\pi\)
0.645124 + 0.764078i \(0.276806\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) −7.07107 −0.529999
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) −28.2843 −2.07950
\(186\) 0 0
\(187\) 2.82843 0.206835
\(188\) −2.82843 −0.206284
\(189\) 0 0
\(190\) −20.0000 −1.45095
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) −9.89949 −0.710742
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 8.48528 0.601506 0.300753 0.953702i \(-0.402762\pi\)
0.300753 + 0.953702i \(0.402762\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) 8.48528 0.597022
\(203\) 0 0
\(204\) 0 0
\(205\) −28.0000 −1.95560
\(206\) −2.82843 −0.197066
\(207\) 0 0
\(208\) 0 0
\(209\) −14.1421 −0.978232
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −5.65685 −0.385794
\(216\) 0 0
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) −5.65685 −0.381385
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −12.0000 −0.798228
\(227\) 21.2132 1.40797 0.703985 0.710215i \(-0.251402\pi\)
0.703985 + 0.710215i \(0.251402\pi\)
\(228\) 0 0
\(229\) −16.9706 −1.12145 −0.560723 0.828003i \(-0.689477\pi\)
−0.560723 + 0.828003i \(0.689477\pi\)
\(230\) 11.3137 0.746004
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 1.41421 0.0920575
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −21.2132 −1.36646 −0.683231 0.730202i \(-0.739426\pi\)
−0.683231 + 0.730202i \(0.739426\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) 2.82843 0.181071
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) −8.48528 −0.538816
\(249\) 0 0
\(250\) −5.65685 −0.357771
\(251\) −9.89949 −0.624851 −0.312425 0.949942i \(-0.601141\pi\)
−0.312425 + 0.949942i \(0.601141\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.7279 −0.793946 −0.396973 0.917830i \(-0.629939\pi\)
−0.396973 + 0.917830i \(0.629939\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 12.7279 0.786334
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) −5.65685 −0.347498
\(266\) 0 0
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) 11.3137 0.689809 0.344904 0.938638i \(-0.387911\pi\)
0.344904 + 0.938638i \(0.387911\pi\)
\(270\) 0 0
\(271\) 22.6274 1.37452 0.687259 0.726413i \(-0.258814\pi\)
0.687259 + 0.726413i \(0.258814\pi\)
\(272\) 1.41421 0.0857493
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 6.00000 0.361814
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −9.89949 −0.593732
\(279\) 0 0
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) −1.41421 −0.0840663 −0.0420331 0.999116i \(-0.513384\pi\)
−0.0420331 + 0.999116i \(0.513384\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) −5.65685 −0.332182
\(291\) 0 0
\(292\) −1.41421 −0.0827606
\(293\) 19.7990 1.15667 0.578335 0.815800i \(-0.303703\pi\)
0.578335 + 0.815800i \(0.303703\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) −7.07107 −0.405554
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) −9.89949 −0.564994 −0.282497 0.959268i \(-0.591163\pi\)
−0.282497 + 0.959268i \(0.591163\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 24.0000 1.36311
\(311\) 11.3137 0.641542 0.320771 0.947157i \(-0.396058\pi\)
0.320771 + 0.947157i \(0.396058\pi\)
\(312\) 0 0
\(313\) 12.7279 0.719425 0.359712 0.933063i \(-0.382875\pi\)
0.359712 + 0.933063i \(0.382875\pi\)
\(314\) 11.3137 0.638470
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −10.0000 −0.561656 −0.280828 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) −2.82843 −0.158114
\(321\) 0 0
\(322\) 0 0
\(323\) −10.0000 −0.556415
\(324\) 0 0
\(325\) 0 0
\(326\) −10.0000 −0.553849
\(327\) 0 0
\(328\) −9.89949 −0.546608
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −9.89949 −0.543305
\(333\) 0 0
\(334\) −19.7990 −1.08335
\(335\) −33.9411 −1.85440
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 13.0000 0.707107
\(339\) 0 0
\(340\) −4.00000 −0.216930
\(341\) 16.9706 0.919007
\(342\) 0 0
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) −16.9706 −0.912343
\(347\) 30.0000 1.61048 0.805242 0.592946i \(-0.202035\pi\)
0.805242 + 0.592946i \(0.202035\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) 1.41421 0.0752710 0.0376355 0.999292i \(-0.488017\pi\)
0.0376355 + 0.999292i \(0.488017\pi\)
\(354\) 0 0
\(355\) −33.9411 −1.80141
\(356\) 7.07107 0.374766
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 31.0000 1.63158
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) 28.2843 1.47643 0.738213 0.674567i \(-0.235670\pi\)
0.738213 + 0.674567i \(0.235670\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 28.2843 1.47043
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −2.82843 −0.146254
\(375\) 0 0
\(376\) 2.82843 0.145865
\(377\) 0 0
\(378\) 0 0
\(379\) −26.0000 −1.33553 −0.667765 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 20.0000 1.02598
\(381\) 0 0
\(382\) −4.00000 −0.204658
\(383\) 36.7696 1.87884 0.939418 0.342773i \(-0.111366\pi\)
0.939418 + 0.342773i \(0.111366\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 16.0000 0.814379
\(387\) 0 0
\(388\) 9.89949 0.502571
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) 0 0
\(394\) 2.00000 0.100759
\(395\) 11.3137 0.569254
\(396\) 0 0
\(397\) 22.6274 1.13564 0.567819 0.823154i \(-0.307787\pi\)
0.567819 + 0.823154i \(0.307787\pi\)
\(398\) −8.48528 −0.425329
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −8.48528 −0.422159
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) 38.1838 1.88807 0.944033 0.329851i \(-0.106999\pi\)
0.944033 + 0.329851i \(0.106999\pi\)
\(410\) 28.0000 1.38282
\(411\) 0 0
\(412\) 2.82843 0.139347
\(413\) 0 0
\(414\) 0 0
\(415\) 28.0000 1.37447
\(416\) 0 0
\(417\) 0 0
\(418\) 14.1421 0.691714
\(419\) 9.89949 0.483622 0.241811 0.970323i \(-0.422259\pi\)
0.241811 + 0.970323i \(0.422259\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) 4.24264 0.205798
\(426\) 0 0
\(427\) 0 0
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 5.65685 0.272798
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −29.6985 −1.42722 −0.713609 0.700544i \(-0.752941\pi\)
−0.713609 + 0.700544i \(0.752941\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) −28.2843 −1.35302
\(438\) 0 0
\(439\) −16.9706 −0.809961 −0.404980 0.914325i \(-0.632722\pi\)
−0.404980 + 0.914325i \(0.632722\pi\)
\(440\) 5.65685 0.269680
\(441\) 0 0
\(442\) 0 0
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 0 0
\(445\) −20.0000 −0.948091
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 19.7990 0.932298
\(452\) 12.0000 0.564433
\(453\) 0 0
\(454\) −21.2132 −0.995585
\(455\) 0 0
\(456\) 0 0
\(457\) 24.0000 1.12267 0.561336 0.827588i \(-0.310287\pi\)
0.561336 + 0.827588i \(0.310287\pi\)
\(458\) 16.9706 0.792982
\(459\) 0 0
\(460\) −11.3137 −0.527504
\(461\) −39.5980 −1.84426 −0.922131 0.386878i \(-0.873553\pi\)
−0.922131 + 0.386878i \(0.873553\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) −32.5269 −1.50517 −0.752583 0.658497i \(-0.771192\pi\)
−0.752583 + 0.658497i \(0.771192\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) −1.41421 −0.0650945
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) −21.2132 −0.973329
\(476\) 0 0
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) 31.1127 1.42158 0.710788 0.703407i \(-0.248339\pi\)
0.710788 + 0.703407i \(0.248339\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 21.2132 0.966235
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −28.0000 −1.27141
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) −2.82843 −0.128037
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −2.82843 −0.127386
\(494\) 0 0
\(495\) 0 0
\(496\) 8.48528 0.381000
\(497\) 0 0
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 5.65685 0.252982
\(501\) 0 0
\(502\) 9.89949 0.441836
\(503\) −39.5980 −1.76559 −0.882793 0.469762i \(-0.844340\pi\)
−0.882793 + 0.469762i \(0.844340\pi\)
\(504\) 0 0
\(505\) 24.0000 1.06799
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) 16.0000 0.709885
\(509\) −22.6274 −1.00294 −0.501471 0.865174i \(-0.667208\pi\)
−0.501471 + 0.865174i \(0.667208\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 12.7279 0.561405
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) −5.65685 −0.248788
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.41421 0.0619578 0.0309789 0.999520i \(-0.490138\pi\)
0.0309789 + 0.999520i \(0.490138\pi\)
\(522\) 0 0
\(523\) 12.7279 0.556553 0.278277 0.960501i \(-0.410237\pi\)
0.278277 + 0.960501i \(0.410237\pi\)
\(524\) −12.7279 −0.556022
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 5.65685 0.245718
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −11.3137 −0.489134
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) −11.3137 −0.487769
\(539\) 0 0
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) −22.6274 −0.971931
\(543\) 0 0
\(544\) −1.41421 −0.0606339
\(545\) 5.65685 0.242313
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) −12.0000 −0.512615
\(549\) 0 0
\(550\) −6.00000 −0.255841
\(551\) 14.1421 0.602475
\(552\) 0 0
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 9.89949 0.419832
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 16.0000 0.674919
\(563\) 1.41421 0.0596020 0.0298010 0.999556i \(-0.490513\pi\)
0.0298010 + 0.999556i \(0.490513\pi\)
\(564\) 0 0
\(565\) −33.9411 −1.42791
\(566\) 1.41421 0.0594438
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −2.00000 −0.0836974 −0.0418487 0.999124i \(-0.513325\pi\)
−0.0418487 + 0.999124i \(0.513325\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) −21.2132 −0.883117 −0.441559 0.897232i \(-0.645574\pi\)
−0.441559 + 0.897232i \(0.645574\pi\)
\(578\) 15.0000 0.623918
\(579\) 0 0
\(580\) 5.65685 0.234888
\(581\) 0 0
\(582\) 0 0
\(583\) 4.00000 0.165663
\(584\) 1.41421 0.0585206
\(585\) 0 0
\(586\) −19.7990 −0.817889
\(587\) 29.6985 1.22579 0.612894 0.790165i \(-0.290005\pi\)
0.612894 + 0.790165i \(0.290005\pi\)
\(588\) 0 0
\(589\) −60.0000 −2.47226
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) 10.0000 0.410997
\(593\) 7.07107 0.290374 0.145187 0.989404i \(-0.453622\pi\)
0.145187 + 0.989404i \(0.453622\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) 29.6985 1.21143 0.605713 0.795683i \(-0.292888\pi\)
0.605713 + 0.795683i \(0.292888\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) 19.7990 0.804943
\(606\) 0 0
\(607\) −16.9706 −0.688814 −0.344407 0.938820i \(-0.611920\pi\)
−0.344407 + 0.938820i \(0.611920\pi\)
\(608\) 7.07107 0.286770
\(609\) 0 0
\(610\) 8.00000 0.323911
\(611\) 0 0
\(612\) 0 0
\(613\) −30.0000 −1.21169 −0.605844 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) 9.89949 0.399511
\(615\) 0 0
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 0 0
\(619\) 18.3848 0.738947 0.369473 0.929241i \(-0.379538\pi\)
0.369473 + 0.929241i \(0.379538\pi\)
\(620\) −24.0000 −0.963863
\(621\) 0 0
\(622\) −11.3137 −0.453638
\(623\) 0 0
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) −12.7279 −0.508710
\(627\) 0 0
\(628\) −11.3137 −0.451466
\(629\) 14.1421 0.563884
\(630\) 0 0
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) 10.0000 0.397151
\(635\) −45.2548 −1.79588
\(636\) 0 0
\(637\) 0 0
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) 2.82843 0.111803
\(641\) −26.0000 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) 0 0
\(643\) 9.89949 0.390398 0.195199 0.980764i \(-0.437465\pi\)
0.195199 + 0.980764i \(0.437465\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 10.0000 0.393445
\(647\) −8.48528 −0.333591 −0.166795 0.985992i \(-0.553342\pi\)
−0.166795 + 0.985992i \(0.553342\pi\)
\(648\) 0 0
\(649\) 2.82843 0.111025
\(650\) 0 0
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 36.0000 1.40664
\(656\) 9.89949 0.386510
\(657\) 0 0
\(658\) 0 0
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) 8.48528 0.330039 0.165020 0.986290i \(-0.447231\pi\)
0.165020 + 0.986290i \(0.447231\pi\)
\(662\) −10.0000 −0.388661
\(663\) 0 0
\(664\) 9.89949 0.384175
\(665\) 0 0
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) 19.7990 0.766046
\(669\) 0 0
\(670\) 33.9411 1.31126
\(671\) 5.65685 0.218380
\(672\) 0 0
\(673\) −12.0000 −0.462566 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 16.9706 0.652232 0.326116 0.945330i \(-0.394260\pi\)
0.326116 + 0.945330i \(0.394260\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) 0 0
\(682\) −16.9706 −0.649836
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 33.9411 1.29682
\(686\) 0 0
\(687\) 0 0
\(688\) 2.00000 0.0762493
\(689\) 0 0
\(690\) 0 0
\(691\) 12.7279 0.484193 0.242096 0.970252i \(-0.422165\pi\)
0.242096 + 0.970252i \(0.422165\pi\)
\(692\) 16.9706 0.645124
\(693\) 0 0
\(694\) −30.0000 −1.13878
\(695\) −28.0000 −1.06210
\(696\) 0 0
\(697\) 14.0000 0.530288
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) −70.7107 −2.66690
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −1.41421 −0.0532246
\(707\) 0 0
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 33.9411 1.27379
\(711\) 0 0
\(712\) −7.07107 −0.264999
\(713\) 33.9411 1.27111
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) −32.0000 −1.19423
\(719\) −2.82843 −0.105483 −0.0527413 0.998608i \(-0.516796\pi\)
−0.0527413 + 0.998608i \(0.516796\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −31.0000 −1.15370
\(723\) 0 0
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −19.7990 −0.734304 −0.367152 0.930161i \(-0.619667\pi\)
−0.367152 + 0.930161i \(0.619667\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −4.00000 −0.148047
\(731\) 2.82843 0.104613
\(732\) 0 0
\(733\) 42.4264 1.56706 0.783528 0.621357i \(-0.213418\pi\)
0.783528 + 0.621357i \(0.213418\pi\)
\(734\) −28.2843 −1.04399
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 24.0000 0.884051
\(738\) 0 0
\(739\) −30.0000 −1.10357 −0.551784 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) −28.2843 −1.03975
\(741\) 0 0
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) 28.2843 1.03626
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) 2.82843 0.103418
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −2.82843 −0.103142
\(753\) 0 0
\(754\) 0 0
\(755\) 45.2548 1.64699
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 26.0000 0.944363
\(759\) 0 0
\(760\) −20.0000 −0.725476
\(761\) 7.07107 0.256326 0.128163 0.991753i \(-0.459092\pi\)
0.128163 + 0.991753i \(0.459092\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 4.00000 0.144715
\(765\) 0 0
\(766\) −36.7696 −1.32854
\(767\) 0 0
\(768\) 0 0
\(769\) 29.6985 1.07095 0.535477 0.844550i \(-0.320132\pi\)
0.535477 + 0.844550i \(0.320132\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −16.0000 −0.575853
\(773\) −48.0833 −1.72943 −0.864717 0.502259i \(-0.832502\pi\)
−0.864717 + 0.502259i \(0.832502\pi\)
\(774\) 0 0
\(775\) 25.4558 0.914401
\(776\) −9.89949 −0.355371
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) −70.0000 −2.50801
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) −5.65685 −0.202289
\(783\) 0 0
\(784\) 0 0
\(785\) 32.0000 1.14213
\(786\) 0 0
\(787\) −1.41421 −0.0504113 −0.0252056 0.999682i \(-0.508024\pi\)
−0.0252056 + 0.999682i \(0.508024\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) −11.3137 −0.402524
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −22.6274 −0.803017
\(795\) 0 0
\(796\) 8.48528 0.300753
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) −3.00000 −0.106066
\(801\) 0 0
\(802\) −18.0000 −0.635602
\(803\) −2.82843 −0.0998130
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 8.48528 0.298511
\(809\) 16.0000 0.562530 0.281265 0.959630i \(-0.409246\pi\)
0.281265 + 0.959630i \(0.409246\pi\)
\(810\) 0 0
\(811\) −29.6985 −1.04285 −0.521427 0.853296i \(-0.674600\pi\)
−0.521427 + 0.853296i \(0.674600\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −20.0000 −0.701000
\(815\) −28.2843 −0.990755
\(816\) 0 0
\(817\) −14.1421 −0.494771
\(818\) −38.1838 −1.33506
\(819\) 0 0
\(820\) −28.0000 −0.977802
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) −2.82843 −0.0985329
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −31.1127 −1.08059 −0.540294 0.841476i \(-0.681687\pi\)
−0.540294 + 0.841476i \(0.681687\pi\)
\(830\) −28.0000 −0.971894
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −56.0000 −1.93796
\(836\) −14.1421 −0.489116
\(837\) 0 0
\(838\) −9.89949 −0.341972
\(839\) −19.7990 −0.683537 −0.341769 0.939784i \(-0.611026\pi\)
−0.341769 + 0.939784i \(0.611026\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −30.0000 −1.03387
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) 36.7696 1.26491
\(846\) 0 0
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) −4.24264 −0.145521
\(851\) 40.0000 1.37118
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) −18.3848 −0.628012 −0.314006 0.949421i \(-0.601671\pi\)
−0.314006 + 0.949421i \(0.601671\pi\)
\(858\) 0 0
\(859\) −26.8701 −0.916795 −0.458397 0.888747i \(-0.651576\pi\)
−0.458397 + 0.888747i \(0.651576\pi\)
\(860\) −5.65685 −0.192897
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) −48.0000 −1.63205
\(866\) 29.6985 1.00920
\(867\) 0 0
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 0 0
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) 28.2843 0.956730
\(875\) 0 0
\(876\) 0 0
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) 16.9706 0.572729
\(879\) 0 0
\(880\) −5.65685 −0.190693
\(881\) −29.6985 −1.00057 −0.500284 0.865862i \(-0.666771\pi\)
−0.500284 + 0.865862i \(0.666771\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 36.7696 1.23460 0.617300 0.786728i \(-0.288226\pi\)
0.617300 + 0.786728i \(0.288226\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 20.0000 0.670402
\(891\) 0 0
\(892\) 0 0
\(893\) 20.0000 0.669274
\(894\) 0 0
\(895\) 33.9411 1.13453
\(896\) 0 0
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) −16.9706 −0.566000
\(900\) 0 0
\(901\) 2.82843 0.0942286
\(902\) −19.7990 −0.659234
\(903\) 0 0
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 21.2132 0.703985
\(909\) 0 0
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) −19.7990 −0.655251
\(914\) −24.0000 −0.793849
\(915\) 0 0
\(916\) −16.9706 −0.560723
\(917\) 0 0
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 11.3137 0.373002
\(921\) 0 0
\(922\) 39.5980 1.30409
\(923\) 0 0
\(924\) 0 0
\(925\) 30.0000 0.986394
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) −32.5269 −1.06717 −0.533587 0.845745i \(-0.679156\pi\)
−0.533587 + 0.845745i \(0.679156\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −24.0000 −0.786146
\(933\) 0 0
\(934\) 32.5269 1.06431
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) 9.89949 0.323402 0.161701 0.986840i \(-0.448302\pi\)
0.161701 + 0.986840i \(0.448302\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) 31.1127 1.01424 0.507122 0.861874i \(-0.330709\pi\)
0.507122 + 0.861874i \(0.330709\pi\)
\(942\) 0 0
\(943\) 39.5980 1.28949
\(944\) 1.41421 0.0460287
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 21.2132 0.688247
\(951\) 0 0
\(952\) 0 0
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 0 0
\(955\) −11.3137 −0.366103
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) −31.1127 −1.00521
\(959\) 0 0
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) 0 0
\(963\) 0 0
\(964\) −21.2132 −0.683231
\(965\) 45.2548 1.45680
\(966\) 0 0
\(967\) −12.0000 −0.385894 −0.192947 0.981209i \(-0.561805\pi\)
−0.192947 + 0.981209i \(0.561805\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 28.0000 0.899026
\(971\) −32.5269 −1.04384 −0.521919 0.852995i \(-0.674784\pi\)
−0.521919 + 0.852995i \(0.674784\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) 2.82843 0.0905357
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 0 0
\(979\) 14.1421 0.451985
\(980\) 0 0
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) −48.0833 −1.53362 −0.766809 0.641875i \(-0.778157\pi\)
−0.766809 + 0.641875i \(0.778157\pi\)
\(984\) 0 0
\(985\) 5.65685 0.180242
\(986\) 2.82843 0.0900755
\(987\) 0 0
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −8.48528 −0.269408
\(993\) 0 0
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) 0 0
\(997\) −31.1127 −0.985349 −0.492675 0.870214i \(-0.663981\pi\)
−0.492675 + 0.870214i \(0.663981\pi\)
\(998\) 4.00000 0.126618
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 882.2.a.n.1.1 2
3.2 odd 2 98.2.a.b.1.1 2
4.3 odd 2 7056.2.a.cl.1.1 2
7.2 even 3 882.2.g.l.361.2 4
7.3 odd 6 882.2.g.l.667.1 4
7.4 even 3 882.2.g.l.667.2 4
7.5 odd 6 882.2.g.l.361.1 4
7.6 odd 2 inner 882.2.a.n.1.2 2
12.11 even 2 784.2.a.l.1.2 2
15.2 even 4 2450.2.c.v.99.4 4
15.8 even 4 2450.2.c.v.99.1 4
15.14 odd 2 2450.2.a.bj.1.2 2
21.2 odd 6 98.2.c.c.67.2 4
21.5 even 6 98.2.c.c.67.1 4
21.11 odd 6 98.2.c.c.79.2 4
21.17 even 6 98.2.c.c.79.1 4
21.20 even 2 98.2.a.b.1.2 yes 2
24.5 odd 2 3136.2.a.bn.1.2 2
24.11 even 2 3136.2.a.bm.1.1 2
28.27 even 2 7056.2.a.cl.1.2 2
84.11 even 6 784.2.i.m.177.1 4
84.23 even 6 784.2.i.m.753.1 4
84.47 odd 6 784.2.i.m.753.2 4
84.59 odd 6 784.2.i.m.177.2 4
84.83 odd 2 784.2.a.l.1.1 2
105.62 odd 4 2450.2.c.v.99.3 4
105.83 odd 4 2450.2.c.v.99.2 4
105.104 even 2 2450.2.a.bj.1.1 2
168.83 odd 2 3136.2.a.bm.1.2 2
168.125 even 2 3136.2.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.2.a.b.1.1 2 3.2 odd 2
98.2.a.b.1.2 yes 2 21.20 even 2
98.2.c.c.67.1 4 21.5 even 6
98.2.c.c.67.2 4 21.2 odd 6
98.2.c.c.79.1 4 21.17 even 6
98.2.c.c.79.2 4 21.11 odd 6
784.2.a.l.1.1 2 84.83 odd 2
784.2.a.l.1.2 2 12.11 even 2
784.2.i.m.177.1 4 84.11 even 6
784.2.i.m.177.2 4 84.59 odd 6
784.2.i.m.753.1 4 84.23 even 6
784.2.i.m.753.2 4 84.47 odd 6
882.2.a.n.1.1 2 1.1 even 1 trivial
882.2.a.n.1.2 2 7.6 odd 2 inner
882.2.g.l.361.1 4 7.5 odd 6
882.2.g.l.361.2 4 7.2 even 3
882.2.g.l.667.1 4 7.3 odd 6
882.2.g.l.667.2 4 7.4 even 3
2450.2.a.bj.1.1 2 105.104 even 2
2450.2.a.bj.1.2 2 15.14 odd 2
2450.2.c.v.99.1 4 15.8 even 4
2450.2.c.v.99.2 4 105.83 odd 4
2450.2.c.v.99.3 4 105.62 odd 4
2450.2.c.v.99.4 4 15.2 even 4
3136.2.a.bm.1.1 2 24.11 even 2
3136.2.a.bm.1.2 2 168.83 odd 2
3136.2.a.bn.1.1 2 168.125 even 2
3136.2.a.bn.1.2 2 24.5 odd 2
7056.2.a.cl.1.1 2 4.3 odd 2
7056.2.a.cl.1.2 2 28.27 even 2