Properties

Label 1638.2.a.y.1.1
Level $1638$
Weight $2$
Character 1638.1
Self dual yes
Analytic conductor $13.079$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 1638.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.27492 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.27492 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.27492 q^{10} -5.27492 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +7.27492 q^{17} +5.27492 q^{19} -3.27492 q^{20} -5.27492 q^{22} +5.27492 q^{23} +5.72508 q^{25} -1.00000 q^{26} +1.00000 q^{28} -0.725083 q^{29} +8.00000 q^{31} +1.00000 q^{32} +7.27492 q^{34} -3.27492 q^{35} +3.27492 q^{37} +5.27492 q^{38} -3.27492 q^{40} +8.54983 q^{41} +5.27492 q^{43} -5.27492 q^{44} +5.27492 q^{46} +1.00000 q^{49} +5.72508 q^{50} -1.00000 q^{52} -10.0000 q^{53} +17.2749 q^{55} +1.00000 q^{56} -0.725083 q^{58} -8.00000 q^{59} -4.72508 q^{61} +8.00000 q^{62} +1.00000 q^{64} +3.27492 q^{65} -2.54983 q^{67} +7.27492 q^{68} -3.27492 q^{70} +9.82475 q^{73} +3.27492 q^{74} +5.27492 q^{76} -5.27492 q^{77} -2.54983 q^{79} -3.27492 q^{80} +8.54983 q^{82} -10.5498 q^{83} -23.8248 q^{85} +5.27492 q^{86} -5.27492 q^{88} +14.0000 q^{89} -1.00000 q^{91} +5.27492 q^{92} -17.2749 q^{95} -15.0997 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + q^{5} + 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + q^{5} + 2q^{7} + 2q^{8} + q^{10} - 3q^{11} - 2q^{13} + 2q^{14} + 2q^{16} + 7q^{17} + 3q^{19} + q^{20} - 3q^{22} + 3q^{23} + 19q^{25} - 2q^{26} + 2q^{28} - 9q^{29} + 16q^{31} + 2q^{32} + 7q^{34} + q^{35} - q^{37} + 3q^{38} + q^{40} + 2q^{41} + 3q^{43} - 3q^{44} + 3q^{46} + 2q^{49} + 19q^{50} - 2q^{52} - 20q^{53} + 27q^{55} + 2q^{56} - 9q^{58} - 16q^{59} - 17q^{61} + 16q^{62} + 2q^{64} - q^{65} + 10q^{67} + 7q^{68} + q^{70} - 3q^{73} - q^{74} + 3q^{76} - 3q^{77} + 10q^{79} + q^{80} + 2q^{82} - 6q^{83} - 25q^{85} + 3q^{86} - 3q^{88} + 28q^{89} - 2q^{91} + 3q^{92} - 27q^{95} + 2q^{98} + O(q^{100}) \)

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
\(5\) −3.27492 −1.46459 −0.732294 0.680989i \(-0.761550\pi\)
−0.732294 + 0.680989i \(0.761550\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −3.27492 −1.03562
\(11\) −5.27492 −1.59045 −0.795224 0.606316i \(-0.792647\pi\)
−0.795224 + 0.606316i \(0.792647\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.27492 1.76443 0.882213 0.470850i \(-0.156053\pi\)
0.882213 + 0.470850i \(0.156053\pi\)
\(18\) 0 0
\(19\) 5.27492 1.21015 0.605075 0.796169i \(-0.293143\pi\)
0.605075 + 0.796169i \(0.293143\pi\)
\(20\) −3.27492 −0.732294
\(21\) 0 0
\(22\) −5.27492 −1.12462
\(23\) 5.27492 1.09990 0.549948 0.835199i \(-0.314647\pi\)
0.549948 + 0.835199i \(0.314647\pi\)
\(24\) 0 0
\(25\) 5.72508 1.14502
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −0.725083 −0.134644 −0.0673222 0.997731i \(-0.521446\pi\)
−0.0673222 + 0.997731i \(0.521446\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 7.27492 1.24764
\(35\) −3.27492 −0.553562
\(36\) 0 0
\(37\) 3.27492 0.538393 0.269197 0.963085i \(-0.413242\pi\)
0.269197 + 0.963085i \(0.413242\pi\)
\(38\) 5.27492 0.855705
\(39\) 0 0
\(40\) −3.27492 −0.517810
\(41\) 8.54983 1.33526 0.667630 0.744493i \(-0.267309\pi\)
0.667630 + 0.744493i \(0.267309\pi\)
\(42\) 0 0
\(43\) 5.27492 0.804417 0.402209 0.915548i \(-0.368243\pi\)
0.402209 + 0.915548i \(0.368243\pi\)
\(44\) −5.27492 −0.795224
\(45\) 0 0
\(46\) 5.27492 0.777744
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 5.72508 0.809649
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 17.2749 2.32935
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −0.725083 −0.0952080
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −4.72508 −0.604985 −0.302492 0.953152i \(-0.597819\pi\)
−0.302492 + 0.953152i \(0.597819\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.27492 0.406203
\(66\) 0 0
\(67\) −2.54983 −0.311512 −0.155756 0.987796i \(-0.549781\pi\)
−0.155756 + 0.987796i \(0.549781\pi\)
\(68\) 7.27492 0.882213
\(69\) 0 0
\(70\) −3.27492 −0.391427
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 9.82475 1.14990 0.574950 0.818188i \(-0.305021\pi\)
0.574950 + 0.818188i \(0.305021\pi\)
\(74\) 3.27492 0.380701
\(75\) 0 0
\(76\) 5.27492 0.605075
\(77\) −5.27492 −0.601133
\(78\) 0 0
\(79\) −2.54983 −0.286879 −0.143439 0.989659i \(-0.545816\pi\)
−0.143439 + 0.989659i \(0.545816\pi\)
\(80\) −3.27492 −0.366147
\(81\) 0 0
\(82\) 8.54983 0.944171
\(83\) −10.5498 −1.15799 −0.578997 0.815329i \(-0.696556\pi\)
−0.578997 + 0.815329i \(0.696556\pi\)
\(84\) 0 0
\(85\) −23.8248 −2.58416
\(86\) 5.27492 0.568809
\(87\) 0 0
\(88\) −5.27492 −0.562308
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 5.27492 0.549948
\(93\) 0 0
\(94\) 0 0
\(95\) −17.2749 −1.77237
\(96\) 0 0
\(97\) −15.0997 −1.53314 −0.766570 0.642161i \(-0.778038\pi\)
−0.766570 + 0.642161i \(0.778038\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 5.72508 0.572508
\(101\) 12.5498 1.24876 0.624378 0.781123i \(-0.285353\pi\)
0.624378 + 0.781123i \(0.285353\pi\)
\(102\) 0 0
\(103\) 2.72508 0.268510 0.134255 0.990947i \(-0.457136\pi\)
0.134255 + 0.990947i \(0.457136\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) 14.5498 1.40659 0.703293 0.710900i \(-0.251712\pi\)
0.703293 + 0.710900i \(0.251712\pi\)
\(108\) 0 0
\(109\) −7.27492 −0.696811 −0.348405 0.937344i \(-0.613277\pi\)
−0.348405 + 0.937344i \(0.613277\pi\)
\(110\) 17.2749 1.64710
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −17.2749 −1.61089
\(116\) −0.725083 −0.0673222
\(117\) 0 0
\(118\) −8.00000 −0.736460
\(119\) 7.27492 0.666891
\(120\) 0 0
\(121\) 16.8248 1.52952
\(122\) −4.72508 −0.427789
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) −2.37459 −0.212389
\(126\) 0 0
\(127\) −5.45017 −0.483624 −0.241812 0.970323i \(-0.577742\pi\)
−0.241812 + 0.970323i \(0.577742\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 3.27492 0.287229
\(131\) 15.8248 1.38261 0.691307 0.722561i \(-0.257035\pi\)
0.691307 + 0.722561i \(0.257035\pi\)
\(132\) 0 0
\(133\) 5.27492 0.457393
\(134\) −2.54983 −0.220272
\(135\) 0 0
\(136\) 7.27492 0.623819
\(137\) −20.3746 −1.74072 −0.870359 0.492417i \(-0.836113\pi\)
−0.870359 + 0.492417i \(0.836113\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) −3.27492 −0.276781
\(141\) 0 0
\(142\) 0 0
\(143\) 5.27492 0.441111
\(144\) 0 0
\(145\) 2.37459 0.197199
\(146\) 9.82475 0.813102
\(147\) 0 0
\(148\) 3.27492 0.269197
\(149\) −7.45017 −0.610341 −0.305171 0.952298i \(-0.598714\pi\)
−0.305171 + 0.952298i \(0.598714\pi\)
\(150\) 0 0
\(151\) −1.27492 −0.103751 −0.0518756 0.998654i \(-0.516520\pi\)
−0.0518756 + 0.998654i \(0.516520\pi\)
\(152\) 5.27492 0.427852
\(153\) 0 0
\(154\) −5.27492 −0.425065
\(155\) −26.1993 −2.10438
\(156\) 0 0
\(157\) 24.3746 1.94530 0.972652 0.232268i \(-0.0746146\pi\)
0.972652 + 0.232268i \(0.0746146\pi\)
\(158\) −2.54983 −0.202854
\(159\) 0 0
\(160\) −3.27492 −0.258905
\(161\) 5.27492 0.415722
\(162\) 0 0
\(163\) −2.54983 −0.199718 −0.0998592 0.995002i \(-0.531839\pi\)
−0.0998592 + 0.995002i \(0.531839\pi\)
\(164\) 8.54983 0.667630
\(165\) 0 0
\(166\) −10.5498 −0.818826
\(167\) 9.27492 0.717715 0.358857 0.933392i \(-0.383166\pi\)
0.358857 + 0.933392i \(0.383166\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −23.8248 −1.82728
\(171\) 0 0
\(172\) 5.27492 0.402209
\(173\) 23.0997 1.75624 0.878118 0.478445i \(-0.158799\pi\)
0.878118 + 0.478445i \(0.158799\pi\)
\(174\) 0 0
\(175\) 5.72508 0.432776
\(176\) −5.27492 −0.397612
\(177\) 0 0
\(178\) 14.0000 1.04934
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −11.0997 −0.825032 −0.412516 0.910950i \(-0.635350\pi\)
−0.412516 + 0.910950i \(0.635350\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 0 0
\(184\) 5.27492 0.388872
\(185\) −10.7251 −0.788524
\(186\) 0 0
\(187\) −38.3746 −2.80623
\(188\) 0 0
\(189\) 0 0
\(190\) −17.2749 −1.25325
\(191\) 5.27492 0.381680 0.190840 0.981621i \(-0.438879\pi\)
0.190840 + 0.981621i \(0.438879\pi\)
\(192\) 0 0
\(193\) −3.45017 −0.248348 −0.124174 0.992260i \(-0.539628\pi\)
−0.124174 + 0.992260i \(0.539628\pi\)
\(194\) −15.0997 −1.08409
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −12.5498 −0.894139 −0.447069 0.894499i \(-0.647532\pi\)
−0.447069 + 0.894499i \(0.647532\pi\)
\(198\) 0 0
\(199\) −15.8248 −1.12179 −0.560893 0.827888i \(-0.689542\pi\)
−0.560893 + 0.827888i \(0.689542\pi\)
\(200\) 5.72508 0.404824
\(201\) 0 0
\(202\) 12.5498 0.883003
\(203\) −0.725083 −0.0508908
\(204\) 0 0
\(205\) −28.0000 −1.95560
\(206\) 2.72508 0.189866
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) −27.8248 −1.92468
\(210\) 0 0
\(211\) 23.8248 1.64016 0.820082 0.572246i \(-0.193928\pi\)
0.820082 + 0.572246i \(0.193928\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) 14.5498 0.994606
\(215\) −17.2749 −1.17814
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) −7.27492 −0.492720
\(219\) 0 0
\(220\) 17.2749 1.16467
\(221\) −7.27492 −0.489364
\(222\) 0 0
\(223\) −23.6495 −1.58369 −0.791844 0.610723i \(-0.790879\pi\)
−0.791844 + 0.610723i \(0.790879\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 5.45017 0.361740 0.180870 0.983507i \(-0.442109\pi\)
0.180870 + 0.983507i \(0.442109\pi\)
\(228\) 0 0
\(229\) −11.0997 −0.733487 −0.366743 0.930322i \(-0.619527\pi\)
−0.366743 + 0.930322i \(0.619527\pi\)
\(230\) −17.2749 −1.13907
\(231\) 0 0
\(232\) −0.725083 −0.0476040
\(233\) −20.5498 −1.34626 −0.673132 0.739522i \(-0.735052\pi\)
−0.673132 + 0.739522i \(0.735052\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 0 0
\(238\) 7.27492 0.471563
\(239\) 5.09967 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(240\) 0 0
\(241\) −15.0997 −0.972655 −0.486328 0.873777i \(-0.661664\pi\)
−0.486328 + 0.873777i \(0.661664\pi\)
\(242\) 16.8248 1.08154
\(243\) 0 0
\(244\) −4.72508 −0.302492
\(245\) −3.27492 −0.209227
\(246\) 0 0
\(247\) −5.27492 −0.335635
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) −2.37459 −0.150182
\(251\) 28.9244 1.82569 0.912847 0.408303i \(-0.133879\pi\)
0.912847 + 0.408303i \(0.133879\pi\)
\(252\) 0 0
\(253\) −27.8248 −1.74933
\(254\) −5.45017 −0.341974
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) 3.27492 0.203493
\(260\) 3.27492 0.203102
\(261\) 0 0
\(262\) 15.8248 0.977656
\(263\) −28.0000 −1.72655 −0.863277 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) 0 0
\(265\) 32.7492 2.01177
\(266\) 5.27492 0.323426
\(267\) 0 0
\(268\) −2.54983 −0.155756
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 13.4502 0.817039 0.408520 0.912750i \(-0.366045\pi\)
0.408520 + 0.912750i \(0.366045\pi\)
\(272\) 7.27492 0.441107
\(273\) 0 0
\(274\) −20.3746 −1.23087
\(275\) −30.1993 −1.82109
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) −3.27492 −0.195714
\(281\) 15.0997 0.900771 0.450385 0.892834i \(-0.351287\pi\)
0.450385 + 0.892834i \(0.351287\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 5.27492 0.311912
\(287\) 8.54983 0.504681
\(288\) 0 0
\(289\) 35.9244 2.11320
\(290\) 2.37459 0.139440
\(291\) 0 0
\(292\) 9.82475 0.574950
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 26.1993 1.52538
\(296\) 3.27492 0.190351
\(297\) 0 0
\(298\) −7.45017 −0.431577
\(299\) −5.27492 −0.305056
\(300\) 0 0
\(301\) 5.27492 0.304041
\(302\) −1.27492 −0.0733632
\(303\) 0 0
\(304\) 5.27492 0.302537
\(305\) 15.4743 0.886053
\(306\) 0 0
\(307\) −9.09967 −0.519346 −0.259673 0.965697i \(-0.583615\pi\)
−0.259673 + 0.965697i \(0.583615\pi\)
\(308\) −5.27492 −0.300566
\(309\) 0 0
\(310\) −26.1993 −1.48802
\(311\) −18.5498 −1.05186 −0.525932 0.850526i \(-0.676283\pi\)
−0.525932 + 0.850526i \(0.676283\pi\)
\(312\) 0 0
\(313\) 17.6495 0.997609 0.498804 0.866715i \(-0.333773\pi\)
0.498804 + 0.866715i \(0.333773\pi\)
\(314\) 24.3746 1.37554
\(315\) 0 0
\(316\) −2.54983 −0.143439
\(317\) 16.5498 0.929531 0.464766 0.885434i \(-0.346139\pi\)
0.464766 + 0.885434i \(0.346139\pi\)
\(318\) 0 0
\(319\) 3.82475 0.214145
\(320\) −3.27492 −0.183073
\(321\) 0 0
\(322\) 5.27492 0.293960
\(323\) 38.3746 2.13522
\(324\) 0 0
\(325\) −5.72508 −0.317570
\(326\) −2.54983 −0.141222
\(327\) 0 0
\(328\) 8.54983 0.472086
\(329\) 0 0
\(330\) 0 0
\(331\) 15.6495 0.860174 0.430087 0.902787i \(-0.358483\pi\)
0.430087 + 0.902787i \(0.358483\pi\)
\(332\) −10.5498 −0.578997
\(333\) 0 0
\(334\) 9.27492 0.507501
\(335\) 8.35050 0.456236
\(336\) 0 0
\(337\) −28.3746 −1.54566 −0.772831 0.634612i \(-0.781160\pi\)
−0.772831 + 0.634612i \(0.781160\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) −23.8248 −1.29208
\(341\) −42.1993 −2.28522
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 5.27492 0.284404
\(345\) 0 0
\(346\) 23.0997 1.24185
\(347\) 25.0997 1.34742 0.673710 0.738995i \(-0.264700\pi\)
0.673710 + 0.738995i \(0.264700\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 5.72508 0.306019
\(351\) 0 0
\(352\) −5.27492 −0.281154
\(353\) 3.09967 0.164979 0.0824894 0.996592i \(-0.473713\pi\)
0.0824894 + 0.996592i \(0.473713\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 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
\(361\) 8.82475 0.464461
\(362\) −11.0997 −0.583386
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) −32.1752 −1.68413
\(366\) 0 0
\(367\) −33.0997 −1.72779 −0.863894 0.503673i \(-0.831982\pi\)
−0.863894 + 0.503673i \(0.831982\pi\)
\(368\) 5.27492 0.274974
\(369\) 0 0
\(370\) −10.7251 −0.557571
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) 11.4502 0.592867 0.296434 0.955053i \(-0.404203\pi\)
0.296434 + 0.955053i \(0.404203\pi\)
\(374\) −38.3746 −1.98430
\(375\) 0 0
\(376\) 0 0
\(377\) 0.725083 0.0373437
\(378\) 0 0
\(379\) −2.90033 −0.148980 −0.0744900 0.997222i \(-0.523733\pi\)
−0.0744900 + 0.997222i \(0.523733\pi\)
\(380\) −17.2749 −0.886185
\(381\) 0 0
\(382\) 5.27492 0.269888
\(383\) −22.7251 −1.16120 −0.580599 0.814190i \(-0.697181\pi\)
−0.580599 + 0.814190i \(0.697181\pi\)
\(384\) 0 0
\(385\) 17.2749 0.880411
\(386\) −3.45017 −0.175609
\(387\) 0 0
\(388\) −15.0997 −0.766570
\(389\) 32.1993 1.63257 0.816286 0.577649i \(-0.196030\pi\)
0.816286 + 0.577649i \(0.196030\pi\)
\(390\) 0 0
\(391\) 38.3746 1.94069
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −12.5498 −0.632252
\(395\) 8.35050 0.420159
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −15.8248 −0.793223
\(399\) 0 0
\(400\) 5.72508 0.286254
\(401\) −27.0997 −1.35329 −0.676646 0.736308i \(-0.736567\pi\)
−0.676646 + 0.736308i \(0.736567\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 12.5498 0.624378
\(405\) 0 0
\(406\) −0.725083 −0.0359853
\(407\) −17.2749 −0.856286
\(408\) 0 0
\(409\) −29.8248 −1.47474 −0.737370 0.675490i \(-0.763932\pi\)
−0.737370 + 0.675490i \(0.763932\pi\)
\(410\) −28.0000 −1.38282
\(411\) 0 0
\(412\) 2.72508 0.134255
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 34.5498 1.69598
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) −27.8248 −1.36095
\(419\) 5.27492 0.257697 0.128848 0.991664i \(-0.458872\pi\)
0.128848 + 0.991664i \(0.458872\pi\)
\(420\) 0 0
\(421\) −27.0997 −1.32076 −0.660379 0.750933i \(-0.729604\pi\)
−0.660379 + 0.750933i \(0.729604\pi\)
\(422\) 23.8248 1.15977
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) 41.6495 2.02030
\(426\) 0 0
\(427\) −4.72508 −0.228663
\(428\) 14.5498 0.703293
\(429\) 0 0
\(430\) −17.2749 −0.833070
\(431\) 26.5498 1.27886 0.639430 0.768849i \(-0.279170\pi\)
0.639430 + 0.768849i \(0.279170\pi\)
\(432\) 0 0
\(433\) −21.6495 −1.04041 −0.520204 0.854042i \(-0.674144\pi\)
−0.520204 + 0.854042i \(0.674144\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −7.27492 −0.348405
\(437\) 27.8248 1.33104
\(438\) 0 0
\(439\) 7.82475 0.373455 0.186728 0.982412i \(-0.440212\pi\)
0.186728 + 0.982412i \(0.440212\pi\)
\(440\) 17.2749 0.823549
\(441\) 0 0
\(442\) −7.27492 −0.346033
\(443\) −22.5498 −1.07137 −0.535687 0.844416i \(-0.679947\pi\)
−0.535687 + 0.844416i \(0.679947\pi\)
\(444\) 0 0
\(445\) −45.8488 −2.17344
\(446\) −23.6495 −1.11984
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −7.27492 −0.343325 −0.171662 0.985156i \(-0.554914\pi\)
−0.171662 + 0.985156i \(0.554914\pi\)
\(450\) 0 0
\(451\) −45.0997 −2.12366
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 5.45017 0.255789
\(455\) 3.27492 0.153530
\(456\) 0 0
\(457\) −3.45017 −0.161392 −0.0806960 0.996739i \(-0.525714\pi\)
−0.0806960 + 0.996739i \(0.525714\pi\)
\(458\) −11.0997 −0.518653
\(459\) 0 0
\(460\) −17.2749 −0.805447
\(461\) −34.9244 −1.62659 −0.813296 0.581850i \(-0.802329\pi\)
−0.813296 + 0.581850i \(0.802329\pi\)
\(462\) 0 0
\(463\) 22.7251 1.05612 0.528062 0.849206i \(-0.322919\pi\)
0.528062 + 0.849206i \(0.322919\pi\)
\(464\) −0.725083 −0.0336611
\(465\) 0 0
\(466\) −20.5498 −0.951953
\(467\) 18.7251 0.866493 0.433247 0.901275i \(-0.357368\pi\)
0.433247 + 0.901275i \(0.357368\pi\)
\(468\) 0 0
\(469\) −2.54983 −0.117740
\(470\) 0 0
\(471\) 0 0
\(472\) −8.00000 −0.368230
\(473\) −27.8248 −1.27938
\(474\) 0 0
\(475\) 30.1993 1.38564
\(476\) 7.27492 0.333445
\(477\) 0 0
\(478\) 5.09967 0.233253
\(479\) −14.3746 −0.656792 −0.328396 0.944540i \(-0.606508\pi\)
−0.328396 + 0.944540i \(0.606508\pi\)
\(480\) 0 0
\(481\) −3.27492 −0.149323
\(482\) −15.0997 −0.687771
\(483\) 0 0
\(484\) 16.8248 0.764761
\(485\) 49.4502 2.24542
\(486\) 0 0
\(487\) 42.1993 1.91223 0.956117 0.292984i \(-0.0946484\pi\)
0.956117 + 0.292984i \(0.0946484\pi\)
\(488\) −4.72508 −0.213894
\(489\) 0 0
\(490\) −3.27492 −0.147946
\(491\) 6.54983 0.295590 0.147795 0.989018i \(-0.452782\pi\)
0.147795 + 0.989018i \(0.452782\pi\)
\(492\) 0 0
\(493\) −5.27492 −0.237570
\(494\) −5.27492 −0.237330
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) −5.09967 −0.228293 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(500\) −2.37459 −0.106195
\(501\) 0 0
\(502\) 28.9244 1.29096
\(503\) 18.5498 0.827096 0.413548 0.910482i \(-0.364289\pi\)
0.413548 + 0.910482i \(0.364289\pi\)
\(504\) 0 0
\(505\) −41.0997 −1.82891
\(506\) −27.8248 −1.23696
\(507\) 0 0
\(508\) −5.45017 −0.241812
\(509\) −42.9244 −1.90259 −0.951296 0.308280i \(-0.900247\pi\)
−0.951296 + 0.308280i \(0.900247\pi\)
\(510\) 0 0
\(511\) 9.82475 0.434621
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) −8.92442 −0.393257
\(516\) 0 0
\(517\) 0 0
\(518\) 3.27492 0.143892
\(519\) 0 0
\(520\) 3.27492 0.143615
\(521\) −3.27492 −0.143477 −0.0717384 0.997423i \(-0.522855\pi\)
−0.0717384 + 0.997423i \(0.522855\pi\)
\(522\) 0 0
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) 15.8248 0.691307
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) 58.1993 2.53520
\(528\) 0 0
\(529\) 4.82475 0.209772
\(530\) 32.7492 1.42253
\(531\) 0 0
\(532\) 5.27492 0.228697
\(533\) −8.54983 −0.370334
\(534\) 0 0
\(535\) −47.6495 −2.06007
\(536\) −2.54983 −0.110136
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) −5.27492 −0.227207
\(540\) 0 0
\(541\) −25.4743 −1.09522 −0.547612 0.836732i \(-0.684463\pi\)
−0.547612 + 0.836732i \(0.684463\pi\)
\(542\) 13.4502 0.577734
\(543\) 0 0
\(544\) 7.27492 0.311910
\(545\) 23.8248 1.02054
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −20.3746 −0.870359
\(549\) 0 0
\(550\) −30.1993 −1.28770
\(551\) −3.82475 −0.162940
\(552\) 0 0
\(553\) −2.54983 −0.108430
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) −30.7492 −1.30288 −0.651442 0.758698i \(-0.725836\pi\)
−0.651442 + 0.758698i \(0.725836\pi\)
\(558\) 0 0
\(559\) −5.27492 −0.223105
\(560\) −3.27492 −0.138391
\(561\) 0 0
\(562\) 15.0997 0.636941
\(563\) −2.37459 −0.100077 −0.0500384 0.998747i \(-0.515934\pi\)
−0.0500384 + 0.998747i \(0.515934\pi\)
\(564\) 0 0
\(565\) −19.6495 −0.826661
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 0 0
\(569\) 27.0997 1.13608 0.568039 0.823002i \(-0.307702\pi\)
0.568039 + 0.823002i \(0.307702\pi\)
\(570\) 0 0
\(571\) −6.90033 −0.288770 −0.144385 0.989522i \(-0.546120\pi\)
−0.144385 + 0.989522i \(0.546120\pi\)
\(572\) 5.27492 0.220555
\(573\) 0 0
\(574\) 8.54983 0.356863
\(575\) 30.1993 1.25940
\(576\) 0 0
\(577\) −39.0997 −1.62774 −0.813870 0.581047i \(-0.802643\pi\)
−0.813870 + 0.581047i \(0.802643\pi\)
\(578\) 35.9244 1.49426
\(579\) 0 0
\(580\) 2.37459 0.0985993
\(581\) −10.5498 −0.437681
\(582\) 0 0
\(583\) 52.7492 2.18465
\(584\) 9.82475 0.406551
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 28.7492 1.18661 0.593303 0.804979i \(-0.297824\pi\)
0.593303 + 0.804979i \(0.297824\pi\)
\(588\) 0 0
\(589\) 42.1993 1.73879
\(590\) 26.1993 1.07861
\(591\) 0 0
\(592\) 3.27492 0.134598
\(593\) −23.0997 −0.948590 −0.474295 0.880366i \(-0.657297\pi\)
−0.474295 + 0.880366i \(0.657297\pi\)
\(594\) 0 0
\(595\) −23.8248 −0.976720
\(596\) −7.45017 −0.305171
\(597\) 0 0
\(598\) −5.27492 −0.215707
\(599\) −29.2749 −1.19614 −0.598070 0.801444i \(-0.704066\pi\)
−0.598070 + 0.801444i \(0.704066\pi\)
\(600\) 0 0
\(601\) −21.6495 −0.883102 −0.441551 0.897236i \(-0.645572\pi\)
−0.441551 + 0.897236i \(0.645572\pi\)
\(602\) 5.27492 0.214990
\(603\) 0 0
\(604\) −1.27492 −0.0518756
\(605\) −55.0997 −2.24012
\(606\) 0 0
\(607\) 31.8248 1.29173 0.645863 0.763453i \(-0.276498\pi\)
0.645863 + 0.763453i \(0.276498\pi\)
\(608\) 5.27492 0.213926
\(609\) 0 0
\(610\) 15.4743 0.626534
\(611\) 0 0
\(612\) 0 0
\(613\) 8.72508 0.352403 0.176201 0.984354i \(-0.443619\pi\)
0.176201 + 0.984354i \(0.443619\pi\)
\(614\) −9.09967 −0.367233
\(615\) 0 0
\(616\) −5.27492 −0.212532
\(617\) −1.82475 −0.0734617 −0.0367309 0.999325i \(-0.511694\pi\)
−0.0367309 + 0.999325i \(0.511694\pi\)
\(618\) 0 0
\(619\) 37.2749 1.49821 0.749103 0.662454i \(-0.230485\pi\)
0.749103 + 0.662454i \(0.230485\pi\)
\(620\) −26.1993 −1.05219
\(621\) 0 0
\(622\) −18.5498 −0.743781
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) −20.8488 −0.833954
\(626\) 17.6495 0.705416
\(627\) 0 0
\(628\) 24.3746 0.972652
\(629\) 23.8248 0.949955
\(630\) 0 0
\(631\) 11.8248 0.470736 0.235368 0.971906i \(-0.424370\pi\)
0.235368 + 0.971906i \(0.424370\pi\)
\(632\) −2.54983 −0.101427
\(633\) 0 0
\(634\) 16.5498 0.657278
\(635\) 17.8488 0.708310
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 3.82475 0.151423
\(639\) 0 0
\(640\) −3.27492 −0.129452
\(641\) −15.0997 −0.596401 −0.298201 0.954503i \(-0.596386\pi\)
−0.298201 + 0.954503i \(0.596386\pi\)
\(642\) 0 0
\(643\) 28.9244 1.14067 0.570334 0.821413i \(-0.306814\pi\)
0.570334 + 0.821413i \(0.306814\pi\)
\(644\) 5.27492 0.207861
\(645\) 0 0
\(646\) 38.3746 1.50983
\(647\) −42.1993 −1.65903 −0.829514 0.558487i \(-0.811382\pi\)
−0.829514 + 0.558487i \(0.811382\pi\)
\(648\) 0 0
\(649\) 42.1993 1.65647
\(650\) −5.72508 −0.224556
\(651\) 0 0
\(652\) −2.54983 −0.0998592
\(653\) −32.3746 −1.26692 −0.633458 0.773777i \(-0.718365\pi\)
−0.633458 + 0.773777i \(0.718365\pi\)
\(654\) 0 0
\(655\) −51.8248 −2.02496
\(656\) 8.54983 0.333815
\(657\) 0 0
\(658\) 0 0
\(659\) 22.5498 0.878417 0.439208 0.898385i \(-0.355259\pi\)
0.439208 + 0.898385i \(0.355259\pi\)
\(660\) 0 0
\(661\) 20.1993 0.785663 0.392832 0.919610i \(-0.371496\pi\)
0.392832 + 0.919610i \(0.371496\pi\)
\(662\) 15.6495 0.608235
\(663\) 0 0
\(664\) −10.5498 −0.409413
\(665\) −17.2749 −0.669893
\(666\) 0 0
\(667\) −3.82475 −0.148095
\(668\) 9.27492 0.358857
\(669\) 0 0
\(670\) 8.35050 0.322608
\(671\) 24.9244 0.962197
\(672\) 0 0
\(673\) 8.72508 0.336327 0.168164 0.985759i \(-0.446216\pi\)
0.168164 + 0.985759i \(0.446216\pi\)
\(674\) −28.3746 −1.09295
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 44.1993 1.69872 0.849359 0.527815i \(-0.176989\pi\)
0.849359 + 0.527815i \(0.176989\pi\)
\(678\) 0 0
\(679\) −15.0997 −0.579472
\(680\) −23.8248 −0.913638
\(681\) 0 0
\(682\) −42.1993 −1.61590
\(683\) 34.3746 1.31531 0.657653 0.753321i \(-0.271549\pi\)
0.657653 + 0.753321i \(0.271549\pi\)
\(684\) 0 0
\(685\) 66.7251 2.54943
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 5.27492 0.201104
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 23.0997 0.878118
\(693\) 0 0
\(694\) 25.0997 0.952770
\(695\) −13.0997 −0.496899
\(696\) 0 0
\(697\) 62.1993 2.35597
\(698\) 2.00000 0.0757011
\(699\) 0 0
\(700\) 5.72508 0.216388
\(701\) −15.0997 −0.570307 −0.285153 0.958482i \(-0.592045\pi\)
−0.285153 + 0.958482i \(0.592045\pi\)
\(702\) 0 0
\(703\) 17.2749 0.651536
\(704\) −5.27492 −0.198806
\(705\) 0 0
\(706\) 3.09967 0.116658
\(707\) 12.5498 0.471985
\(708\) 0 0
\(709\) 7.09967 0.266634 0.133317 0.991073i \(-0.457437\pi\)
0.133317 + 0.991073i \(0.457437\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 14.0000 0.524672
\(713\) 42.1993 1.58038
\(714\) 0 0
\(715\) −17.2749 −0.646045
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 32.0000 1.19423
\(719\) −15.6495 −0.583628 −0.291814 0.956475i \(-0.594259\pi\)
−0.291814 + 0.956475i \(0.594259\pi\)
\(720\) 0 0
\(721\) 2.72508 0.101487
\(722\) 8.82475 0.328423
\(723\) 0 0
\(724\) −11.0997 −0.412516
\(725\) −4.15116 −0.154170
\(726\) 0 0
\(727\) −2.37459 −0.0880685 −0.0440343 0.999030i \(-0.514021\pi\)
−0.0440343 + 0.999030i \(0.514021\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 0 0
\(730\) −32.1752 −1.19086
\(731\) 38.3746 1.41934
\(732\) 0 0
\(733\) −0.549834 −0.0203086 −0.0101543 0.999948i \(-0.503232\pi\)
−0.0101543 + 0.999948i \(0.503232\pi\)
\(734\) −33.0997 −1.22173
\(735\) 0 0
\(736\) 5.27492 0.194436
\(737\) 13.4502 0.495443
\(738\) 0 0
\(739\) −7.64950 −0.281392 −0.140696 0.990053i \(-0.544934\pi\)
−0.140696 + 0.990053i \(0.544934\pi\)
\(740\) −10.7251 −0.394262
\(741\) 0 0
\(742\) −10.0000 −0.367112
\(743\) −26.5498 −0.974019 −0.487009 0.873397i \(-0.661912\pi\)
−0.487009 + 0.873397i \(0.661912\pi\)
\(744\) 0 0
\(745\) 24.3987 0.893898
\(746\) 11.4502 0.419220
\(747\) 0 0
\(748\) −38.3746 −1.40311
\(749\) 14.5498 0.531639
\(750\) 0 0
\(751\) −50.1993 −1.83180 −0.915900 0.401407i \(-0.868521\pi\)
−0.915900 + 0.401407i \(0.868521\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0.725083 0.0264060
\(755\) 4.17525 0.151953
\(756\) 0 0
\(757\) −4.90033 −0.178106 −0.0890528 0.996027i \(-0.528384\pi\)
−0.0890528 + 0.996027i \(0.528384\pi\)
\(758\) −2.90033 −0.105345
\(759\) 0 0
\(760\) −17.2749 −0.626627
\(761\) −23.4502 −0.850068 −0.425034 0.905177i \(-0.639738\pi\)
−0.425034 + 0.905177i \(0.639738\pi\)
\(762\) 0 0
\(763\) −7.27492 −0.263370
\(764\) 5.27492 0.190840
\(765\) 0 0
\(766\) −22.7251 −0.821091
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 10.1752 0.366929 0.183464 0.983026i \(-0.441269\pi\)
0.183464 + 0.983026i \(0.441269\pi\)
\(770\) 17.2749 0.622545
\(771\) 0 0
\(772\) −3.45017 −0.124174
\(773\) −42.9244 −1.54388 −0.771942 0.635693i \(-0.780714\pi\)
−0.771942 + 0.635693i \(0.780714\pi\)
\(774\) 0 0
\(775\) 45.8007 1.64521
\(776\) −15.0997 −0.542047
\(777\) 0 0
\(778\) 32.1993 1.15440
\(779\) 45.0997 1.61586
\(780\) 0 0
\(781\) 0 0
\(782\) 38.3746 1.37227
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −79.8248 −2.84907
\(786\) 0 0
\(787\) −2.72508 −0.0971387 −0.0485694 0.998820i \(-0.515466\pi\)
−0.0485694 + 0.998820i \(0.515466\pi\)
\(788\) −12.5498 −0.447069
\(789\) 0 0
\(790\) 8.35050 0.297097
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 4.72508 0.167793
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −15.8248 −0.560893
\(797\) −29.6495 −1.05024 −0.525120 0.851028i \(-0.675979\pi\)
−0.525120 + 0.851028i \(0.675979\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 5.72508 0.202412
\(801\) 0 0
\(802\) −27.0997 −0.956923
\(803\) −51.8248 −1.82886
\(804\) 0 0
\(805\) −17.2749 −0.608861
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) 12.5498 0.441502
\(809\) −4.19934 −0.147641 −0.0738204 0.997272i \(-0.523519\pi\)
−0.0738204 + 0.997272i \(0.523519\pi\)
\(810\) 0 0
\(811\) 26.3746 0.926137 0.463068 0.886322i \(-0.346748\pi\)
0.463068 + 0.886322i \(0.346748\pi\)
\(812\) −0.725083 −0.0254454
\(813\) 0 0
\(814\) −17.2749 −0.605486
\(815\) 8.35050 0.292505
\(816\) 0 0
\(817\) 27.8248 0.973465
\(818\) −29.8248 −1.04280
\(819\) 0 0
\(820\) −28.0000 −0.977802
\(821\) 13.6495 0.476371 0.238185 0.971220i \(-0.423447\pi\)
0.238185 + 0.971220i \(0.423447\pi\)
\(822\) 0 0
\(823\) 5.09967 0.177763 0.0888816 0.996042i \(-0.471671\pi\)
0.0888816 + 0.996042i \(0.471671\pi\)
\(824\) 2.72508 0.0949328
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 18.7251 0.651135 0.325567 0.945519i \(-0.394445\pi\)
0.325567 + 0.945519i \(0.394445\pi\)
\(828\) 0 0
\(829\) 19.2749 0.669446 0.334723 0.942317i \(-0.391357\pi\)
0.334723 + 0.942317i \(0.391357\pi\)
\(830\) 34.5498 1.19924
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 7.27492 0.252061
\(834\) 0 0
\(835\) −30.3746 −1.05116
\(836\) −27.8248 −0.962339
\(837\) 0 0
\(838\) 5.27492 0.182219
\(839\) −53.0997 −1.83320 −0.916602 0.399801i \(-0.869079\pi\)
−0.916602 + 0.399801i \(0.869079\pi\)
\(840\) 0 0
\(841\) −28.4743 −0.981871
\(842\) −27.0997 −0.933916
\(843\) 0 0
\(844\) 23.8248 0.820082
\(845\) −3.27492 −0.112661
\(846\) 0 0
\(847\) 16.8248 0.578105
\(848\) −10.0000 −0.343401
\(849\) 0 0
\(850\) 41.6495 1.42857
\(851\) 17.2749 0.592177
\(852\) 0 0
\(853\) 17.6495 0.604307 0.302154 0.953259i \(-0.402294\pi\)
0.302154 + 0.953259i \(0.402294\pi\)
\(854\) −4.72508 −0.161689
\(855\) 0 0
\(856\) 14.5498 0.497303
\(857\) 11.0997 0.379157 0.189579 0.981866i \(-0.439288\pi\)
0.189579 + 0.981866i \(0.439288\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) −17.2749 −0.589070
\(861\) 0 0
\(862\) 26.5498 0.904291
\(863\) 15.6495 0.532715 0.266358 0.963874i \(-0.414180\pi\)
0.266358 + 0.963874i \(0.414180\pi\)
\(864\) 0 0
\(865\) −75.6495 −2.57216
\(866\) −21.6495 −0.735680
\(867\) 0 0
\(868\) 8.00000 0.271538
\(869\) 13.4502 0.456266
\(870\) 0 0
\(871\) 2.54983 0.0863978
\(872\) −7.27492 −0.246360
\(873\) 0 0
\(874\) 27.8248 0.941186
\(875\) −2.37459 −0.0802757
\(876\) 0 0
\(877\) 20.9003 0.705754 0.352877 0.935670i \(-0.385203\pi\)
0.352877 + 0.935670i \(0.385203\pi\)
\(878\) 7.82475 0.264073
\(879\) 0 0
\(880\) 17.2749 0.582337
\(881\) 39.2749 1.32321 0.661603 0.749854i \(-0.269877\pi\)
0.661603 + 0.749854i \(0.269877\pi\)
\(882\) 0 0
\(883\) −20.9244 −0.704163 −0.352081 0.935969i \(-0.614526\pi\)
−0.352081 + 0.935969i \(0.614526\pi\)
\(884\) −7.27492 −0.244682
\(885\) 0 0
\(886\) −22.5498 −0.757577
\(887\) 2.90033 0.0973836 0.0486918 0.998814i \(-0.484495\pi\)
0.0486918 + 0.998814i \(0.484495\pi\)
\(888\) 0 0
\(889\) −5.45017 −0.182793
\(890\) −45.8488 −1.53686
\(891\) 0 0
\(892\) −23.6495 −0.791844
\(893\) 0 0
\(894\) 0 0
\(895\) 39.2990 1.31362
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −7.27492 −0.242767
\(899\) −5.80066 −0.193463
\(900\) 0 0
\(901\) −72.7492 −2.42363
\(902\) −45.0997 −1.50165
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 36.3505 1.20833
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 5.45017 0.180870
\(909\) 0 0
\(910\) 3.27492 0.108562
\(911\) −15.4743 −0.512685 −0.256342 0.966586i \(-0.582517\pi\)
−0.256342 + 0.966586i \(0.582517\pi\)
\(912\) 0 0
\(913\) 55.6495 1.84173
\(914\) −3.45017 −0.114121
\(915\) 0 0
\(916\) −11.0997 −0.366743
\(917\) 15.8248 0.522579
\(918\) 0 0
\(919\) 2.54983 0.0841113 0.0420556 0.999115i \(-0.486609\pi\)
0.0420556 + 0.999115i \(0.486609\pi\)
\(920\) −17.2749 −0.569537
\(921\) 0 0
\(922\) −34.9244 −1.15017
\(923\) 0 0
\(924\) 0 0
\(925\) 18.7492 0.616469
\(926\) 22.7251 0.746793
\(927\) 0 0
\(928\) −0.725083 −0.0238020
\(929\) 0.549834 0.0180395 0.00901974 0.999959i \(-0.497129\pi\)
0.00901974 + 0.999959i \(0.497129\pi\)
\(930\) 0 0
\(931\) 5.27492 0.172878
\(932\) −20.5498 −0.673132
\(933\) 0 0
\(934\) 18.7251 0.612703
\(935\) 125.674 4.10997
\(936\) 0 0
\(937\) 47.0997 1.53868 0.769340 0.638840i \(-0.220585\pi\)
0.769340 + 0.638840i \(0.220585\pi\)
\(938\) −2.54983 −0.0832550
\(939\) 0 0
\(940\) 0 0
\(941\) −20.9003 −0.681331 −0.340666 0.940185i \(-0.610652\pi\)
−0.340666 + 0.940185i \(0.610652\pi\)
\(942\) 0 0
\(943\) 45.0997 1.46865
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) −27.8248 −0.904661
\(947\) 39.8248 1.29413 0.647065 0.762435i \(-0.275996\pi\)
0.647065 + 0.762435i \(0.275996\pi\)
\(948\) 0 0
\(949\) −9.82475 −0.318925
\(950\) 30.1993 0.979796
\(951\) 0 0
\(952\) 7.27492 0.235781
\(953\) 13.6495 0.442151 0.221075 0.975257i \(-0.429043\pi\)
0.221075 + 0.975257i \(0.429043\pi\)
\(954\) 0 0
\(955\) −17.2749 −0.559003
\(956\) 5.09967 0.164935
\(957\) 0 0
\(958\) −14.3746 −0.464422
\(959\) −20.3746 −0.657930
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −3.27492 −0.105588
\(963\) 0 0
\(964\) −15.0997 −0.486328
\(965\) 11.2990 0.363728
\(966\) 0 0
\(967\) 20.1752 0.648792 0.324396 0.945921i \(-0.394839\pi\)
0.324396 + 0.945921i \(0.394839\pi\)
\(968\) 16.8248 0.540768
\(969\) 0 0
\(970\) 49.4502 1.58775
\(971\) −1.09967 −0.0352901 −0.0176450 0.999844i \(-0.505617\pi\)
−0.0176450 + 0.999844i \(0.505617\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) 42.1993 1.35215
\(975\) 0 0
\(976\) −4.72508 −0.151246
\(977\) −38.9244 −1.24530 −0.622651 0.782499i \(-0.713944\pi\)
−0.622651 + 0.782499i \(0.713944\pi\)
\(978\) 0 0
\(979\) −73.8488 −2.36022
\(980\) −3.27492 −0.104613
\(981\) 0 0
\(982\) 6.54983 0.209014
\(983\) 22.0241 0.702459 0.351230 0.936289i \(-0.385764\pi\)
0.351230 + 0.936289i \(0.385764\pi\)
\(984\) 0 0
\(985\) 41.0997 1.30954
\(986\) −5.27492 −0.167988
\(987\) 0 0
\(988\) −5.27492 −0.167817
\(989\) 27.8248 0.884776
\(990\) 0 0
\(991\) 28.7492 0.913248 0.456624 0.889660i \(-0.349059\pi\)
0.456624 + 0.889660i \(0.349059\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) 0 0
\(995\) 51.8248 1.64296
\(996\) 0 0
\(997\) −43.0997 −1.36498 −0.682490 0.730895i \(-0.739103\pi\)
−0.682490 + 0.730895i \(0.739103\pi\)
\(998\) −5.09967 −0.161427
\(999\) 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 1638.2.a.y.1.1 2
3.2 odd 2 546.2.a.h.1.2 2
12.11 even 2 4368.2.a.bh.1.2 2
21.20 even 2 3822.2.a.bm.1.1 2
39.38 odd 2 7098.2.a.bu.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.h.1.2 2 3.2 odd 2
1638.2.a.y.1.1 2 1.1 even 1 trivial
3822.2.a.bm.1.1 2 21.20 even 2
4368.2.a.bh.1.2 2 12.11 even 2
7098.2.a.bu.1.1 2 39.38 odd 2