Properties

Label 9702.2.a.dg.1.2
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.4708600410\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{8} +1.41421 q^{10} -1.00000 q^{11} -5.65685 q^{13} +1.00000 q^{16} -1.41421 q^{17} +4.24264 q^{19} +1.41421 q^{20} -1.00000 q^{22} -4.00000 q^{23} -3.00000 q^{25} -5.65685 q^{26} +4.24264 q^{31} +1.00000 q^{32} -1.41421 q^{34} -6.00000 q^{37} +4.24264 q^{38} +1.41421 q^{40} -4.24264 q^{41} -4.00000 q^{43} -1.00000 q^{44} -4.00000 q^{46} +1.41421 q^{47} -3.00000 q^{50} -5.65685 q^{52} +6.00000 q^{53} -1.41421 q^{55} +2.82843 q^{59} +5.65685 q^{61} +4.24264 q^{62} +1.00000 q^{64} -8.00000 q^{65} -10.0000 q^{67} -1.41421 q^{68} +4.24264 q^{73} -6.00000 q^{74} +4.24264 q^{76} -6.00000 q^{79} +1.41421 q^{80} -4.24264 q^{82} -15.5563 q^{83} -2.00000 q^{85} -4.00000 q^{86} -1.00000 q^{88} -14.1421 q^{89} -4.00000 q^{92} +1.41421 q^{94} +6.00000 q^{95} +5.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} + 2 q^{16} - 2 q^{22} - 8 q^{23} - 6 q^{25} + 2 q^{32} - 12 q^{37} - 8 q^{43} - 2 q^{44} - 8 q^{46} - 6 q^{50} + 12 q^{53} + 2 q^{64} - 16 q^{65} - 20 q^{67} - 12 q^{74} - 12 q^{79} - 4 q^{85} - 8 q^{86} - 2 q^{88} - 8 q^{92} + 12 q^{95} + O(q^{100}) \)

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
\(5\) 1.41421 0.632456 0.316228 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.41421 0.447214
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.41421 −0.342997 −0.171499 0.985184i \(-0.554861\pi\)
−0.171499 + 0.985184i \(0.554861\pi\)
\(18\) 0 0
\(19\) 4.24264 0.973329 0.486664 0.873589i \(-0.338214\pi\)
0.486664 + 0.873589i \(0.338214\pi\)
\(20\) 1.41421 0.316228
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) −5.65685 −1.10940
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 4.24264 0.762001 0.381000 0.924575i \(-0.375580\pi\)
0.381000 + 0.924575i \(0.375580\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −1.41421 −0.242536
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 4.24264 0.688247
\(39\) 0 0
\(40\) 1.41421 0.223607
\(41\) −4.24264 −0.662589 −0.331295 0.943527i \(-0.607485\pi\)
−0.331295 + 0.943527i \(0.607485\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 1.41421 0.206284 0.103142 0.994667i \(-0.467110\pi\)
0.103142 + 0.994667i \(0.467110\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −3.00000 −0.424264
\(51\) 0 0
\(52\) −5.65685 −0.784465
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −1.41421 −0.190693
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.82843 0.368230 0.184115 0.982905i \(-0.441058\pi\)
0.184115 + 0.982905i \(0.441058\pi\)
\(60\) 0 0
\(61\) 5.65685 0.724286 0.362143 0.932123i \(-0.382045\pi\)
0.362143 + 0.932123i \(0.382045\pi\)
\(62\) 4.24264 0.538816
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −1.41421 −0.171499
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 4.24264 0.496564 0.248282 0.968688i \(-0.420134\pi\)
0.248282 + 0.968688i \(0.420134\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 4.24264 0.486664
\(77\) 0 0
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 1.41421 0.158114
\(81\) 0 0
\(82\) −4.24264 −0.468521
\(83\) −15.5563 −1.70753 −0.853766 0.520658i \(-0.825687\pi\)
−0.853766 + 0.520658i \(0.825687\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −14.1421 −1.49906 −0.749532 0.661968i \(-0.769721\pi\)
−0.749532 + 0.661968i \(0.769721\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 1.41421 0.145865
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) 5.65685 0.574367 0.287183 0.957876i \(-0.407281\pi\)
0.287183 + 0.957876i \(0.407281\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −3.00000 −0.300000
\(101\) −5.65685 −0.562878 −0.281439 0.959579i \(-0.590812\pi\)
−0.281439 + 0.959579i \(0.590812\pi\)
\(102\) 0 0
\(103\) −7.07107 −0.696733 −0.348367 0.937358i \(-0.613264\pi\)
−0.348367 + 0.937358i \(0.613264\pi\)
\(104\) −5.65685 −0.554700
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) −1.41421 −0.134840
\(111\) 0 0
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) −5.65685 −0.527504
\(116\) 0 0
\(117\) 0 0
\(118\) 2.82843 0.260378
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 5.65685 0.512148
\(123\) 0 0
\(124\) 4.24264 0.381000
\(125\) −11.3137 −1.01193
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −8.00000 −0.701646
\(131\) 15.5563 1.35916 0.679582 0.733599i \(-0.262161\pi\)
0.679582 + 0.733599i \(0.262161\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −10.0000 −0.863868
\(135\) 0 0
\(136\) −1.41421 −0.121268
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 4.24264 0.359856 0.179928 0.983680i \(-0.442414\pi\)
0.179928 + 0.983680i \(0.442414\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.65685 0.473050
\(144\) 0 0
\(145\) 0 0
\(146\) 4.24264 0.351123
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 4.24264 0.344124
\(153\) 0 0
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −18.3848 −1.46726 −0.733632 0.679546i \(-0.762177\pi\)
−0.733632 + 0.679546i \(0.762177\pi\)
\(158\) −6.00000 −0.477334
\(159\) 0 0
\(160\) 1.41421 0.111803
\(161\) 0 0
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −4.24264 −0.331295
\(165\) 0 0
\(166\) −15.5563 −1.20741
\(167\) 19.7990 1.53209 0.766046 0.642786i \(-0.222221\pi\)
0.766046 + 0.642786i \(0.222221\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −2.82843 −0.215041 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −14.1421 −1.06000
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 12.7279 0.946059 0.473029 0.881047i \(-0.343160\pi\)
0.473029 + 0.881047i \(0.343160\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) −8.48528 −0.623850
\(186\) 0 0
\(187\) 1.41421 0.103418
\(188\) 1.41421 0.103142
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 5.65685 0.406138
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 18.3848 1.30326 0.651631 0.758536i \(-0.274085\pi\)
0.651631 + 0.758536i \(0.274085\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −5.65685 −0.398015
\(203\) 0 0
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) −7.07107 −0.492665
\(207\) 0 0
\(208\) −5.65685 −0.392232
\(209\) −4.24264 −0.293470
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −5.65685 −0.385794
\(216\) 0 0
\(217\) 0 0
\(218\) −8.00000 −0.541828
\(219\) 0 0
\(220\) −1.41421 −0.0953463
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) −24.0416 −1.60995 −0.804973 0.593311i \(-0.797820\pi\)
−0.804973 + 0.593311i \(0.797820\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) 7.07107 0.469323 0.234662 0.972077i \(-0.424602\pi\)
0.234662 + 0.972077i \(0.424602\pi\)
\(228\) 0 0
\(229\) −7.07107 −0.467269 −0.233635 0.972324i \(-0.575062\pi\)
−0.233635 + 0.972324i \(0.575062\pi\)
\(230\) −5.65685 −0.373002
\(231\) 0 0
\(232\) 0 0
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) 2.00000 0.130466
\(236\) 2.82843 0.184115
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 1.41421 0.0910975 0.0455488 0.998962i \(-0.485496\pi\)
0.0455488 + 0.998962i \(0.485496\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 5.65685 0.362143
\(245\) 0 0
\(246\) 0 0
\(247\) −24.0000 −1.52708
\(248\) 4.24264 0.269408
\(249\) 0 0
\(250\) −11.3137 −0.715542
\(251\) 19.7990 1.24970 0.624851 0.780744i \(-0.285160\pi\)
0.624851 + 0.780744i \(0.285160\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −31.1127 −1.94076 −0.970378 0.241590i \(-0.922331\pi\)
−0.970378 + 0.241590i \(0.922331\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −8.00000 −0.496139
\(261\) 0 0
\(262\) 15.5563 0.961074
\(263\) −2.00000 −0.123325 −0.0616626 0.998097i \(-0.519640\pi\)
−0.0616626 + 0.998097i \(0.519640\pi\)
\(264\) 0 0
\(265\) 8.48528 0.521247
\(266\) 0 0
\(267\) 0 0
\(268\) −10.0000 −0.610847
\(269\) −12.7279 −0.776035 −0.388018 0.921652i \(-0.626840\pi\)
−0.388018 + 0.921652i \(0.626840\pi\)
\(270\) 0 0
\(271\) −2.82843 −0.171815 −0.0859074 0.996303i \(-0.527379\pi\)
−0.0859074 + 0.996303i \(0.527379\pi\)
\(272\) −1.41421 −0.0857493
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 3.00000 0.180907
\(276\) 0 0
\(277\) −28.0000 −1.68236 −0.841178 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) 4.24264 0.254457
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 15.5563 0.924729 0.462364 0.886690i \(-0.347001\pi\)
0.462364 + 0.886690i \(0.347001\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 5.65685 0.334497
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 0 0
\(291\) 0 0
\(292\) 4.24264 0.248282
\(293\) 31.1127 1.81762 0.908812 0.417207i \(-0.136991\pi\)
0.908812 + 0.417207i \(0.136991\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) −2.00000 −0.115857
\(299\) 22.6274 1.30858
\(300\) 0 0
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) 4.24264 0.243332
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) 7.07107 0.403567 0.201784 0.979430i \(-0.435326\pi\)
0.201784 + 0.979430i \(0.435326\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 6.00000 0.340777
\(311\) 18.3848 1.04251 0.521253 0.853402i \(-0.325465\pi\)
0.521253 + 0.853402i \(0.325465\pi\)
\(312\) 0 0
\(313\) 8.48528 0.479616 0.239808 0.970820i \(-0.422915\pi\)
0.239808 + 0.970820i \(0.422915\pi\)
\(314\) −18.3848 −1.03751
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.41421 0.0790569
\(321\) 0 0
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) 16.9706 0.941357
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) −4.24264 −0.234261
\(329\) 0 0
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) −15.5563 −0.853766
\(333\) 0 0
\(334\) 19.7990 1.08335
\(335\) −14.1421 −0.772667
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 19.0000 1.03346
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) −4.24264 −0.229752
\(342\) 0 0
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −2.82843 −0.152057
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 0 0
\(349\) −25.4558 −1.36262 −0.681310 0.731995i \(-0.738589\pi\)
−0.681310 + 0.731995i \(0.738589\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 25.4558 1.35488 0.677439 0.735579i \(-0.263090\pi\)
0.677439 + 0.735579i \(0.263090\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.1421 −0.749532
\(357\) 0 0
\(358\) 6.00000 0.317110
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 12.7279 0.668965
\(363\) 0 0
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) −21.2132 −1.10732 −0.553660 0.832743i \(-0.686769\pi\)
−0.553660 + 0.832743i \(0.686769\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) −8.48528 −0.441129
\(371\) 0 0
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 1.41421 0.0731272
\(375\) 0 0
\(376\) 1.41421 0.0729325
\(377\) 0 0
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) 18.3848 0.939418 0.469709 0.882821i \(-0.344359\pi\)
0.469709 + 0.882821i \(0.344359\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) 5.65685 0.287183
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.48528 −0.426941
\(396\) 0 0
\(397\) 29.6985 1.49052 0.745262 0.666772i \(-0.232324\pi\)
0.745262 + 0.666772i \(0.232324\pi\)
\(398\) 18.3848 0.921546
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) −24.0000 −1.19553
\(404\) −5.65685 −0.281439
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) 4.24264 0.209785 0.104893 0.994484i \(-0.466550\pi\)
0.104893 + 0.994484i \(0.466550\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) −7.07107 −0.348367
\(413\) 0 0
\(414\) 0 0
\(415\) −22.0000 −1.07994
\(416\) −5.65685 −0.277350
\(417\) 0 0
\(418\) −4.24264 −0.207514
\(419\) −11.3137 −0.552711 −0.276355 0.961056i \(-0.589127\pi\)
−0.276355 + 0.961056i \(0.589127\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 4.24264 0.205798
\(426\) 0 0
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) −5.65685 −0.272798
\(431\) 26.0000 1.25238 0.626188 0.779672i \(-0.284614\pi\)
0.626188 + 0.779672i \(0.284614\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) −16.9706 −0.811812
\(438\) 0 0
\(439\) 19.7990 0.944954 0.472477 0.881343i \(-0.343360\pi\)
0.472477 + 0.881343i \(0.343360\pi\)
\(440\) −1.41421 −0.0674200
\(441\) 0 0
\(442\) 8.00000 0.380521
\(443\) −10.0000 −0.475114 −0.237557 0.971374i \(-0.576347\pi\)
−0.237557 + 0.971374i \(0.576347\pi\)
\(444\) 0 0
\(445\) −20.0000 −0.948091
\(446\) −24.0416 −1.13840
\(447\) 0 0
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 4.24264 0.199778
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) 7.07107 0.331862
\(455\) 0 0
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −7.07107 −0.330409
\(459\) 0 0
\(460\) −5.65685 −0.263752
\(461\) 28.2843 1.31733 0.658665 0.752436i \(-0.271121\pi\)
0.658665 + 0.752436i \(0.271121\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 2.00000 0.0922531
\(471\) 0 0
\(472\) 2.82843 0.130189
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) −12.7279 −0.583997
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5.65685 −0.258468 −0.129234 0.991614i \(-0.541252\pi\)
−0.129234 + 0.991614i \(0.541252\pi\)
\(480\) 0 0
\(481\) 33.9411 1.54758
\(482\) 1.41421 0.0644157
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) 5.65685 0.256074
\(489\) 0 0
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −24.0000 −1.07981
\(495\) 0 0
\(496\) 4.24264 0.190500
\(497\) 0 0
\(498\) 0 0
\(499\) −22.0000 −0.984855 −0.492428 0.870353i \(-0.663890\pi\)
−0.492428 + 0.870353i \(0.663890\pi\)
\(500\) −11.3137 −0.505964
\(501\) 0 0
\(502\) 19.7990 0.883672
\(503\) 19.7990 0.882793 0.441397 0.897312i \(-0.354483\pi\)
0.441397 + 0.897312i \(0.354483\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 4.00000 0.177822
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 26.8701 1.19099 0.595497 0.803357i \(-0.296955\pi\)
0.595497 + 0.803357i \(0.296955\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −31.1127 −1.37232
\(515\) −10.0000 −0.440653
\(516\) 0 0
\(517\) −1.41421 −0.0621970
\(518\) 0 0
\(519\) 0 0
\(520\) −8.00000 −0.350823
\(521\) 19.7990 0.867409 0.433705 0.901055i \(-0.357206\pi\)
0.433705 + 0.901055i \(0.357206\pi\)
\(522\) 0 0
\(523\) −41.0122 −1.79334 −0.896669 0.442702i \(-0.854020\pi\)
−0.896669 + 0.442702i \(0.854020\pi\)
\(524\) 15.5563 0.679582
\(525\) 0 0
\(526\) −2.00000 −0.0872041
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 8.48528 0.368577
\(531\) 0 0
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) −5.65685 −0.244567
\(536\) −10.0000 −0.431934
\(537\) 0 0
\(538\) −12.7279 −0.548740
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −2.82843 −0.121491
\(543\) 0 0
\(544\) −1.41421 −0.0606339
\(545\) −11.3137 −0.484626
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) 3.00000 0.127920
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −28.0000 −1.18961
\(555\) 0 0
\(556\) 4.24264 0.179928
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) 22.6274 0.957038
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −15.5563 −0.655622 −0.327811 0.944743i \(-0.606311\pi\)
−0.327811 + 0.944743i \(0.606311\pi\)
\(564\) 0 0
\(565\) −25.4558 −1.07094
\(566\) 15.5563 0.653882
\(567\) 0 0
\(568\) 0 0
\(569\) 14.0000 0.586911 0.293455 0.955973i \(-0.405195\pi\)
0.293455 + 0.955973i \(0.405195\pi\)
\(570\) 0 0
\(571\) −24.0000 −1.00437 −0.502184 0.864761i \(-0.667470\pi\)
−0.502184 + 0.864761i \(0.667470\pi\)
\(572\) 5.65685 0.236525
\(573\) 0 0
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) −8.48528 −0.353247 −0.176623 0.984278i \(-0.556517\pi\)
−0.176623 + 0.984278i \(0.556517\pi\)
\(578\) −15.0000 −0.623918
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −6.00000 −0.248495
\(584\) 4.24264 0.175562
\(585\) 0 0
\(586\) 31.1127 1.28525
\(587\) −31.1127 −1.28416 −0.642079 0.766638i \(-0.721928\pi\)
−0.642079 + 0.766638i \(0.721928\pi\)
\(588\) 0 0
\(589\) 18.0000 0.741677
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) 7.07107 0.290374 0.145187 0.989404i \(-0.453622\pi\)
0.145187 + 0.989404i \(0.453622\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −2.00000 −0.0819232
\(597\) 0 0
\(598\) 22.6274 0.925304
\(599\) −20.0000 −0.817178 −0.408589 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 0 0
\(601\) 43.8406 1.78830 0.894148 0.447771i \(-0.147782\pi\)
0.894148 + 0.447771i \(0.147782\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) 1.41421 0.0574960
\(606\) 0 0
\(607\) −16.9706 −0.688814 −0.344407 0.938820i \(-0.611920\pi\)
−0.344407 + 0.938820i \(0.611920\pi\)
\(608\) 4.24264 0.172062
\(609\) 0 0
\(610\) 8.00000 0.323911
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 7.07107 0.285365
\(615\) 0 0
\(616\) 0 0
\(617\) 28.0000 1.12724 0.563619 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) 0 0
\(619\) −11.3137 −0.454736 −0.227368 0.973809i \(-0.573012\pi\)
−0.227368 + 0.973809i \(0.573012\pi\)
\(620\) 6.00000 0.240966
\(621\) 0 0
\(622\) 18.3848 0.737162
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 8.48528 0.339140
\(627\) 0 0
\(628\) −18.3848 −0.733632
\(629\) 8.48528 0.338330
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) −6.00000 −0.238667
\(633\) 0 0
\(634\) −22.0000 −0.873732
\(635\) 2.82843 0.112243
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 1.41421 0.0559017
\(641\) 44.0000 1.73790 0.868948 0.494904i \(-0.164797\pi\)
0.868948 + 0.494904i \(0.164797\pi\)
\(642\) 0 0
\(643\) −36.7696 −1.45005 −0.725025 0.688723i \(-0.758172\pi\)
−0.725025 + 0.688723i \(0.758172\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) 7.07107 0.277992 0.138996 0.990293i \(-0.455612\pi\)
0.138996 + 0.990293i \(0.455612\pi\)
\(648\) 0 0
\(649\) −2.82843 −0.111025
\(650\) 16.9706 0.665640
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) 22.0000 0.859611
\(656\) −4.24264 −0.165647
\(657\) 0 0
\(658\) 0 0
\(659\) 32.0000 1.24654 0.623272 0.782006i \(-0.285803\pi\)
0.623272 + 0.782006i \(0.285803\pi\)
\(660\) 0 0
\(661\) −29.6985 −1.15514 −0.577569 0.816342i \(-0.695998\pi\)
−0.577569 + 0.816342i \(0.695998\pi\)
\(662\) 2.00000 0.0777322
\(663\) 0 0
\(664\) −15.5563 −0.603703
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 19.7990 0.766046
\(669\) 0 0
\(670\) −14.1421 −0.546358
\(671\) −5.65685 −0.218380
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) 19.0000 0.730769
\(677\) −2.82843 −0.108705 −0.0543526 0.998522i \(-0.517310\pi\)
−0.0543526 + 0.998522i \(0.517310\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) −4.24264 −0.162459
\(683\) −46.0000 −1.76014 −0.880071 0.474843i \(-0.842505\pi\)
−0.880071 + 0.474843i \(0.842505\pi\)
\(684\) 0 0
\(685\) 8.48528 0.324206
\(686\) 0 0
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −33.9411 −1.29305
\(690\) 0 0
\(691\) −25.4558 −0.968386 −0.484193 0.874961i \(-0.660887\pi\)
−0.484193 + 0.874961i \(0.660887\pi\)
\(692\) −2.82843 −0.107521
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) 6.00000 0.227593
\(696\) 0 0
\(697\) 6.00000 0.227266
\(698\) −25.4558 −0.963518
\(699\) 0 0
\(700\) 0 0
\(701\) 46.0000 1.73740 0.868698 0.495342i \(-0.164957\pi\)
0.868698 + 0.495342i \(0.164957\pi\)
\(702\) 0 0
\(703\) −25.4558 −0.960085
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 25.4558 0.958043
\(707\) 0 0
\(708\) 0 0
\(709\) −14.0000 −0.525781 −0.262891 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −14.1421 −0.529999
\(713\) −16.9706 −0.635553
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 6.00000 0.224231
\(717\) 0 0
\(718\) −30.0000 −1.11959
\(719\) −21.2132 −0.791119 −0.395559 0.918440i \(-0.629449\pi\)
−0.395559 + 0.918440i \(0.629449\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) 12.7279 0.473029
\(725\) 0 0
\(726\) 0 0
\(727\) −18.3848 −0.681854 −0.340927 0.940090i \(-0.610741\pi\)
−0.340927 + 0.940090i \(0.610741\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 6.00000 0.222070
\(731\) 5.65685 0.209226
\(732\) 0 0
\(733\) −33.9411 −1.25364 −0.626822 0.779162i \(-0.715645\pi\)
−0.626822 + 0.779162i \(0.715645\pi\)
\(734\) −21.2132 −0.782994
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 10.0000 0.368355
\(738\) 0 0
\(739\) −24.0000 −0.882854 −0.441427 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) −8.48528 −0.311925
\(741\) 0 0
\(742\) 0 0
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) −2.82843 −0.103626
\(746\) −26.0000 −0.951928
\(747\) 0 0
\(748\) 1.41421 0.0517088
\(749\) 0 0
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 1.41421 0.0515711
\(753\) 0 0
\(754\) 0 0
\(755\) −22.6274 −0.823496
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 6.00000 0.217643
\(761\) −7.07107 −0.256326 −0.128163 0.991753i \(-0.540908\pi\)
−0.128163 + 0.991753i \(0.540908\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) 18.3848 0.664269
\(767\) −16.0000 −0.577727
\(768\) 0 0
\(769\) −26.8701 −0.968959 −0.484480 0.874803i \(-0.660991\pi\)
−0.484480 + 0.874803i \(0.660991\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) −9.89949 −0.356060 −0.178030 0.984025i \(-0.556972\pi\)
−0.178030 + 0.984025i \(0.556972\pi\)
\(774\) 0 0
\(775\) −12.7279 −0.457200
\(776\) 5.65685 0.203069
\(777\) 0 0
\(778\) 14.0000 0.501924
\(779\) −18.0000 −0.644917
\(780\) 0 0
\(781\) 0 0
\(782\) 5.65685 0.202289
\(783\) 0 0
\(784\) 0 0
\(785\) −26.0000 −0.927980
\(786\) 0 0
\(787\) 21.2132 0.756169 0.378085 0.925771i \(-0.376583\pi\)
0.378085 + 0.925771i \(0.376583\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) −8.48528 −0.301893
\(791\) 0 0
\(792\) 0 0
\(793\) −32.0000 −1.13635
\(794\) 29.6985 1.05396
\(795\) 0 0
\(796\) 18.3848 0.651631
\(797\) 18.3848 0.651222 0.325611 0.945504i \(-0.394430\pi\)
0.325611 + 0.945504i \(0.394430\pi\)
\(798\) 0 0
\(799\) −2.00000 −0.0707549
\(800\) −3.00000 −0.106066
\(801\) 0 0
\(802\) −2.00000 −0.0706225
\(803\) −4.24264 −0.149720
\(804\) 0 0
\(805\) 0 0
\(806\) −24.0000 −0.845364
\(807\) 0 0
\(808\) −5.65685 −0.199007
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) 4.24264 0.148979 0.0744896 0.997222i \(-0.476267\pi\)
0.0744896 + 0.997222i \(0.476267\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 6.00000 0.210300
\(815\) −16.9706 −0.594453
\(816\) 0 0
\(817\) −16.9706 −0.593725
\(818\) 4.24264 0.148340
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) 0 0
\(823\) −48.0000 −1.67317 −0.836587 0.547833i \(-0.815453\pi\)
−0.836587 + 0.547833i \(0.815453\pi\)
\(824\) −7.07107 −0.246332
\(825\) 0 0
\(826\) 0 0
\(827\) 52.0000 1.80822 0.904109 0.427303i \(-0.140536\pi\)
0.904109 + 0.427303i \(0.140536\pi\)
\(828\) 0 0
\(829\) −21.2132 −0.736765 −0.368383 0.929674i \(-0.620088\pi\)
−0.368383 + 0.929674i \(0.620088\pi\)
\(830\) −22.0000 −0.763631
\(831\) 0 0
\(832\) −5.65685 −0.196116
\(833\) 0 0
\(834\) 0 0
\(835\) 28.0000 0.968980
\(836\) −4.24264 −0.146735
\(837\) 0 0
\(838\) −11.3137 −0.390826
\(839\) −15.5563 −0.537065 −0.268532 0.963271i \(-0.586539\pi\)
−0.268532 + 0.963271i \(0.586539\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 22.0000 0.758170
\(843\) 0 0
\(844\) −8.00000 −0.275371
\(845\) 26.8701 0.924358
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 4.24264 0.145521
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) 33.9411 1.16212 0.581061 0.813860i \(-0.302638\pi\)
0.581061 + 0.813860i \(0.302638\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 1.41421 0.0483086 0.0241543 0.999708i \(-0.492311\pi\)
0.0241543 + 0.999708i \(0.492311\pi\)
\(858\) 0 0
\(859\) 19.7990 0.675533 0.337766 0.941230i \(-0.390329\pi\)
0.337766 + 0.941230i \(0.390329\pi\)
\(860\) −5.65685 −0.192897
\(861\) 0 0
\(862\) 26.0000 0.885564
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) −4.00000 −0.136004
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6.00000 0.203536
\(870\) 0 0
\(871\) 56.5685 1.91675
\(872\) −8.00000 −0.270914
\(873\) 0 0
\(874\) −16.9706 −0.574038
\(875\) 0 0
\(876\) 0 0
\(877\) 4.00000 0.135070 0.0675352 0.997717i \(-0.478487\pi\)
0.0675352 + 0.997717i \(0.478487\pi\)
\(878\) 19.7990 0.668184
\(879\) 0 0
\(880\) −1.41421 −0.0476731
\(881\) −45.2548 −1.52467 −0.762337 0.647180i \(-0.775948\pi\)
−0.762337 + 0.647180i \(0.775948\pi\)
\(882\) 0 0
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) −10.0000 −0.335957
\(887\) −45.2548 −1.51951 −0.759754 0.650210i \(-0.774681\pi\)
−0.759754 + 0.650210i \(0.774681\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −20.0000 −0.670402
\(891\) 0 0
\(892\) −24.0416 −0.804973
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) 8.48528 0.283632
\(896\) 0 0
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) 0 0
\(900\) 0 0
\(901\) −8.48528 −0.282686
\(902\) 4.24264 0.141264
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 18.0000 0.598340
\(906\) 0 0
\(907\) 6.00000 0.199227 0.0996134 0.995026i \(-0.468239\pi\)
0.0996134 + 0.995026i \(0.468239\pi\)
\(908\) 7.07107 0.234662
\(909\) 0 0
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) 15.5563 0.514840
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) −7.07107 −0.233635
\(917\) 0 0
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −5.65685 −0.186501
\(921\) 0 0
\(922\) 28.2843 0.931493
\(923\) 0 0
\(924\) 0 0
\(925\) 18.0000 0.591836
\(926\) 24.0000 0.788689
\(927\) 0 0
\(928\) 0 0
\(929\) −2.82843 −0.0927977 −0.0463988 0.998923i \(-0.514775\pi\)
−0.0463988 + 0.998923i \(0.514775\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 14.0000 0.458585
\(933\) 0 0
\(934\) 0 0
\(935\) 2.00000 0.0654070
\(936\) 0 0
\(937\) 15.5563 0.508204 0.254102 0.967177i \(-0.418220\pi\)
0.254102 + 0.967177i \(0.418220\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 2.00000 0.0652328
\(941\) 25.4558 0.829837 0.414918 0.909859i \(-0.363810\pi\)
0.414918 + 0.909859i \(0.363810\pi\)
\(942\) 0 0
\(943\) 16.9706 0.552638
\(944\) 2.82843 0.0920575
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 0 0
\(949\) −24.0000 −0.779073
\(950\) −12.7279 −0.412948
\(951\) 0 0
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) −22.6274 −0.732206
\(956\) 0 0
\(957\) 0 0
\(958\) −5.65685 −0.182765
\(959\) 0 0
\(960\) 0 0
\(961\) −13.0000 −0.419355
\(962\) 33.9411 1.09431
\(963\) 0 0
\(964\) 1.41421 0.0455488
\(965\) −2.82843 −0.0910503
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 8.00000 0.256865
\(971\) −22.6274 −0.726148 −0.363074 0.931760i \(-0.618273\pi\)
−0.363074 + 0.931760i \(0.618273\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −4.00000 −0.128168
\(975\) 0 0
\(976\) 5.65685 0.181071
\(977\) 48.0000 1.53566 0.767828 0.640656i \(-0.221338\pi\)
0.767828 + 0.640656i \(0.221338\pi\)
\(978\) 0 0
\(979\) 14.1421 0.451985
\(980\) 0 0
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) −9.89949 −0.315745 −0.157872 0.987460i \(-0.550463\pi\)
−0.157872 + 0.987460i \(0.550463\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 36.0000 1.14358 0.571789 0.820401i \(-0.306250\pi\)
0.571789 + 0.820401i \(0.306250\pi\)
\(992\) 4.24264 0.134704
\(993\) 0 0
\(994\) 0 0
\(995\) 26.0000 0.824255
\(996\) 0 0
\(997\) −2.82843 −0.0895772 −0.0447886 0.998996i \(-0.514261\pi\)
−0.0447886 + 0.998996i \(0.514261\pi\)
\(998\) −22.0000 −0.696398
\(999\) 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 9702.2.a.dg.1.2 yes 2
3.2 odd 2 9702.2.a.ct.1.1 2
7.6 odd 2 inner 9702.2.a.dg.1.1 yes 2
21.20 even 2 9702.2.a.ct.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9702.2.a.ct.1.1 2 3.2 odd 2
9702.2.a.ct.1.2 yes 2 21.20 even 2
9702.2.a.dg.1.1 yes 2 7.6 odd 2 inner
9702.2.a.dg.1.2 yes 2 1.1 even 1 trivial