Properties

Label 3822.2.a.bm.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
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 \(\chi\) \(=\) 3822.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.27492 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.27492 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.27492 q^{10} +5.27492 q^{11} +1.00000 q^{12} +1.00000 q^{13} -3.27492 q^{15} +1.00000 q^{16} +7.27492 q^{17} -1.00000 q^{18} -5.27492 q^{19} -3.27492 q^{20} -5.27492 q^{22} -5.27492 q^{23} -1.00000 q^{24} +5.72508 q^{25} -1.00000 q^{26} +1.00000 q^{27} +0.725083 q^{29} +3.27492 q^{30} -8.00000 q^{31} -1.00000 q^{32} +5.27492 q^{33} -7.27492 q^{34} +1.00000 q^{36} +3.27492 q^{37} +5.27492 q^{38} +1.00000 q^{39} +3.27492 q^{40} +8.54983 q^{41} +5.27492 q^{43} +5.27492 q^{44} -3.27492 q^{45} +5.27492 q^{46} +1.00000 q^{48} -5.72508 q^{50} +7.27492 q^{51} +1.00000 q^{52} +10.0000 q^{53} -1.00000 q^{54} -17.2749 q^{55} -5.27492 q^{57} -0.725083 q^{58} -8.00000 q^{59} -3.27492 q^{60} +4.72508 q^{61} +8.00000 q^{62} +1.00000 q^{64} -3.27492 q^{65} -5.27492 q^{66} -2.54983 q^{67} +7.27492 q^{68} -5.27492 q^{69} -1.00000 q^{72} -9.82475 q^{73} -3.27492 q^{74} +5.72508 q^{75} -5.27492 q^{76} -1.00000 q^{78} -2.54983 q^{79} -3.27492 q^{80} +1.00000 q^{81} -8.54983 q^{82} -10.5498 q^{83} -23.8248 q^{85} -5.27492 q^{86} +0.725083 q^{87} -5.27492 q^{88} +14.0000 q^{89} +3.27492 q^{90} -5.27492 q^{92} -8.00000 q^{93} +17.2749 q^{95} -1.00000 q^{96} +15.0997 q^{97} +5.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} - q^{10} + 3 q^{11} + 2 q^{12} + 2 q^{13} + q^{15} + 2 q^{16} + 7 q^{17} - 2 q^{18} - 3 q^{19} + q^{20} - 3 q^{22} - 3 q^{23} - 2 q^{24} + 19 q^{25} - 2 q^{26} + 2 q^{27} + 9 q^{29} - q^{30} - 16 q^{31} - 2 q^{32} + 3 q^{33} - 7 q^{34} + 2 q^{36} - q^{37} + 3 q^{38} + 2 q^{39} - q^{40} + 2 q^{41} + 3 q^{43} + 3 q^{44} + q^{45} + 3 q^{46} + 2 q^{48} - 19 q^{50} + 7 q^{51} + 2 q^{52} + 20 q^{53} - 2 q^{54} - 27 q^{55} - 3 q^{57} - 9 q^{58} - 16 q^{59} + q^{60} + 17 q^{61} + 16 q^{62} + 2 q^{64} + q^{65} - 3 q^{66} + 10 q^{67} + 7 q^{68} - 3 q^{69} - 2 q^{72} + 3 q^{73} + q^{74} + 19 q^{75} - 3 q^{76} - 2 q^{78} + 10 q^{79} + q^{80} + 2 q^{81} - 2 q^{82} - 6 q^{83} - 25 q^{85} - 3 q^{86} + 9 q^{87} - 3 q^{88} + 28 q^{89} - q^{90} - 3 q^{92} - 16 q^{93} + 27 q^{95} - 2 q^{96} + 3 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −3.27492 −1.46459 −0.732294 0.680989i \(-0.761550\pi\)
−0.732294 + 0.680989i \(0.761550\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.27492 1.03562
\(11\) 5.27492 1.59045 0.795224 0.606316i \(-0.207353\pi\)
0.795224 + 0.606316i \(0.207353\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −3.27492 −0.845580
\(16\) 1.00000 0.250000
\(17\) 7.27492 1.76443 0.882213 0.470850i \(-0.156053\pi\)
0.882213 + 0.470850i \(0.156053\pi\)
\(18\) −1.00000 −0.235702
\(19\) −5.27492 −1.21015 −0.605075 0.796169i \(-0.706857\pi\)
−0.605075 + 0.796169i \(0.706857\pi\)
\(20\) −3.27492 −0.732294
\(21\) 0 0
\(22\) −5.27492 −1.12462
\(23\) −5.27492 −1.09990 −0.549948 0.835199i \(-0.685353\pi\)
−0.549948 + 0.835199i \(0.685353\pi\)
\(24\) −1.00000 −0.204124
\(25\) 5.72508 1.14502
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.725083 0.134644 0.0673222 0.997731i \(-0.478554\pi\)
0.0673222 + 0.997731i \(0.478554\pi\)
\(30\) 3.27492 0.597915
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.27492 0.918245
\(34\) −7.27492 −1.24764
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 3.27492 0.538393 0.269197 0.963085i \(-0.413242\pi\)
0.269197 + 0.963085i \(0.413242\pi\)
\(38\) 5.27492 0.855705
\(39\) 1.00000 0.160128
\(40\) 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
\(45\) −3.27492 −0.488196
\(46\) 5.27492 0.777744
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −5.72508 −0.809649
\(51\) 7.27492 1.01869
\(52\) 1.00000 0.138675
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) −1.00000 −0.136083
\(55\) −17.2749 −2.32935
\(56\) 0 0
\(57\) −5.27492 −0.698680
\(58\) −0.725083 −0.0952080
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) −3.27492 −0.422790
\(61\) 4.72508 0.604985 0.302492 0.953152i \(-0.402181\pi\)
0.302492 + 0.953152i \(0.402181\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.27492 −0.406203
\(66\) −5.27492 −0.649297
\(67\) −2.54983 −0.311512 −0.155756 0.987796i \(-0.549781\pi\)
−0.155756 + 0.987796i \(0.549781\pi\)
\(68\) 7.27492 0.882213
\(69\) −5.27492 −0.635025
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −9.82475 −1.14990 −0.574950 0.818188i \(-0.694979\pi\)
−0.574950 + 0.818188i \(0.694979\pi\)
\(74\) −3.27492 −0.380701
\(75\) 5.72508 0.661076
\(76\) −5.27492 −0.605075
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) −2.54983 −0.286879 −0.143439 0.989659i \(-0.545816\pi\)
−0.143439 + 0.989659i \(0.545816\pi\)
\(80\) −3.27492 −0.366147
\(81\) 1.00000 0.111111
\(82\) −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
\(87\) 0.725083 0.0777370
\(88\) −5.27492 −0.562308
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 3.27492 0.345207
\(91\) 0 0
\(92\) −5.27492 −0.549948
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 17.2749 1.77237
\(96\) −1.00000 −0.102062
\(97\) 15.0997 1.53314 0.766570 0.642161i \(-0.221962\pi\)
0.766570 + 0.642161i \(0.221962\pi\)
\(98\) 0 0
\(99\) 5.27492 0.530149
\(100\) 5.72508 0.572508
\(101\) 12.5498 1.24876 0.624378 0.781123i \(-0.285353\pi\)
0.624378 + 0.781123i \(0.285353\pi\)
\(102\) −7.27492 −0.720324
\(103\) −2.72508 −0.268510 −0.134255 0.990947i \(-0.542864\pi\)
−0.134255 + 0.990947i \(0.542864\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −14.5498 −1.40659 −0.703293 0.710900i \(-0.748288\pi\)
−0.703293 + 0.710900i \(0.748288\pi\)
\(108\) 1.00000 0.0962250
\(109\) −7.27492 −0.696811 −0.348405 0.937344i \(-0.613277\pi\)
−0.348405 + 0.937344i \(0.613277\pi\)
\(110\) 17.2749 1.64710
\(111\) 3.27492 0.310841
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 5.27492 0.494041
\(115\) 17.2749 1.61089
\(116\) 0.725083 0.0673222
\(117\) 1.00000 0.0924500
\(118\) 8.00000 0.736460
\(119\) 0 0
\(120\) 3.27492 0.298958
\(121\) 16.8248 1.52952
\(122\) −4.72508 −0.427789
\(123\) 8.54983 0.770913
\(124\) −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
\(129\) 5.27492 0.464431
\(130\) 3.27492 0.287229
\(131\) 15.8248 1.38261 0.691307 0.722561i \(-0.257035\pi\)
0.691307 + 0.722561i \(0.257035\pi\)
\(132\) 5.27492 0.459123
\(133\) 0 0
\(134\) 2.54983 0.220272
\(135\) −3.27492 −0.281860
\(136\) −7.27492 −0.623819
\(137\) 20.3746 1.74072 0.870359 0.492417i \(-0.163887\pi\)
0.870359 + 0.492417i \(0.163887\pi\)
\(138\) 5.27492 0.449031
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.27492 0.441111
\(144\) 1.00000 0.0833333
\(145\) −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 \(-0.401286\pi\)
0.305171 + 0.952298i \(0.401286\pi\)
\(150\) −5.72508 −0.467451
\(151\) −1.27492 −0.103751 −0.0518756 0.998654i \(-0.516520\pi\)
−0.0518756 + 0.998654i \(0.516520\pi\)
\(152\) 5.27492 0.427852
\(153\) 7.27492 0.588142
\(154\) 0 0
\(155\) 26.1993 2.10438
\(156\) 1.00000 0.0800641
\(157\) −24.3746 −1.94530 −0.972652 0.232268i \(-0.925385\pi\)
−0.972652 + 0.232268i \(0.925385\pi\)
\(158\) 2.54983 0.202854
\(159\) 10.0000 0.793052
\(160\) 3.27492 0.258905
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −2.54983 −0.199718 −0.0998592 0.995002i \(-0.531839\pi\)
−0.0998592 + 0.995002i \(0.531839\pi\)
\(164\) 8.54983 0.667630
\(165\) −17.2749 −1.34485
\(166\) 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
\(171\) −5.27492 −0.403383
\(172\) 5.27492 0.402209
\(173\) 23.0997 1.75624 0.878118 0.478445i \(-0.158799\pi\)
0.878118 + 0.478445i \(0.158799\pi\)
\(174\) −0.725083 −0.0549684
\(175\) 0 0
\(176\) 5.27492 0.397612
\(177\) −8.00000 −0.601317
\(178\) −14.0000 −1.04934
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −3.27492 −0.244098
\(181\) 11.0997 0.825032 0.412516 0.910950i \(-0.364650\pi\)
0.412516 + 0.910950i \(0.364650\pi\)
\(182\) 0 0
\(183\) 4.72508 0.349288
\(184\) 5.27492 0.388872
\(185\) −10.7251 −0.788524
\(186\) 8.00000 0.586588
\(187\) 38.3746 2.80623
\(188\) 0 0
\(189\) 0 0
\(190\) −17.2749 −1.25325
\(191\) −5.27492 −0.381680 −0.190840 0.981621i \(-0.561121\pi\)
−0.190840 + 0.981621i \(0.561121\pi\)
\(192\) 1.00000 0.0721688
\(193\) −3.45017 −0.248348 −0.124174 0.992260i \(-0.539628\pi\)
−0.124174 + 0.992260i \(0.539628\pi\)
\(194\) −15.0997 −1.08409
\(195\) −3.27492 −0.234522
\(196\) 0 0
\(197\) 12.5498 0.894139 0.447069 0.894499i \(-0.352468\pi\)
0.447069 + 0.894499i \(0.352468\pi\)
\(198\) −5.27492 −0.374872
\(199\) 15.8248 1.12179 0.560893 0.827888i \(-0.310458\pi\)
0.560893 + 0.827888i \(0.310458\pi\)
\(200\) −5.72508 −0.404824
\(201\) −2.54983 −0.179851
\(202\) −12.5498 −0.883003
\(203\) 0 0
\(204\) 7.27492 0.509346
\(205\) −28.0000 −1.95560
\(206\) 2.72508 0.189866
\(207\) −5.27492 −0.366632
\(208\) 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
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 7.27492 0.492720
\(219\) −9.82475 −0.663895
\(220\) −17.2749 −1.16467
\(221\) 7.27492 0.489364
\(222\) −3.27492 −0.219798
\(223\) 23.6495 1.58369 0.791844 0.610723i \(-0.209121\pi\)
0.791844 + 0.610723i \(0.209121\pi\)
\(224\) 0 0
\(225\) 5.72508 0.381672
\(226\) 6.00000 0.399114
\(227\) 5.45017 0.361740 0.180870 0.983507i \(-0.442109\pi\)
0.180870 + 0.983507i \(0.442109\pi\)
\(228\) −5.27492 −0.349340
\(229\) 11.0997 0.733487 0.366743 0.930322i \(-0.380473\pi\)
0.366743 + 0.930322i \(0.380473\pi\)
\(230\) −17.2749 −1.13907
\(231\) 0 0
\(232\) −0.725083 −0.0476040
\(233\) 20.5498 1.34626 0.673132 0.739522i \(-0.264948\pi\)
0.673132 + 0.739522i \(0.264948\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) −2.54983 −0.165630
\(238\) 0 0
\(239\) −5.09967 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(240\) −3.27492 −0.211395
\(241\) 15.0997 0.972655 0.486328 0.873777i \(-0.338336\pi\)
0.486328 + 0.873777i \(0.338336\pi\)
\(242\) −16.8248 −1.08154
\(243\) 1.00000 0.0641500
\(244\) 4.72508 0.302492
\(245\) 0 0
\(246\) −8.54983 −0.545118
\(247\) −5.27492 −0.335635
\(248\) 8.00000 0.508001
\(249\) −10.5498 −0.668569
\(250\) 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
\(255\) −23.8248 −1.49196
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) −5.27492 −0.328402
\(259\) 0 0
\(260\) −3.27492 −0.203102
\(261\) 0.725083 0.0448815
\(262\) −15.8248 −0.977656
\(263\) 28.0000 1.72655 0.863277 0.504730i \(-0.168408\pi\)
0.863277 + 0.504730i \(0.168408\pi\)
\(264\) −5.27492 −0.324649
\(265\) −32.7492 −2.01177
\(266\) 0 0
\(267\) 14.0000 0.856786
\(268\) −2.54983 −0.155756
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 3.27492 0.199305
\(271\) −13.4502 −0.817039 −0.408520 0.912750i \(-0.633955\pi\)
−0.408520 + 0.912750i \(0.633955\pi\)
\(272\) 7.27492 0.441107
\(273\) 0 0
\(274\) −20.3746 −1.23087
\(275\) 30.1993 1.82109
\(276\) −5.27492 −0.317513
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 4.00000 0.239904
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −15.0997 −0.900771 −0.450385 0.892834i \(-0.648713\pi\)
−0.450385 + 0.892834i \(0.648713\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 17.2749 1.02328
\(286\) −5.27492 −0.311912
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 35.9244 2.11320
\(290\) 2.37459 0.139440
\(291\) 15.0997 0.885158
\(292\) −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
\(297\) 5.27492 0.306082
\(298\) −7.45017 −0.431577
\(299\) −5.27492 −0.305056
\(300\) 5.72508 0.330538
\(301\) 0 0
\(302\) 1.27492 0.0733632
\(303\) 12.5498 0.720969
\(304\) −5.27492 −0.302537
\(305\) −15.4743 −0.886053
\(306\) −7.27492 −0.415879
\(307\) 9.09967 0.519346 0.259673 0.965697i \(-0.416385\pi\)
0.259673 + 0.965697i \(0.416385\pi\)
\(308\) 0 0
\(309\) −2.72508 −0.155025
\(310\) −26.1993 −1.48802
\(311\) −18.5498 −1.05186 −0.525932 0.850526i \(-0.676283\pi\)
−0.525932 + 0.850526i \(0.676283\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −17.6495 −0.997609 −0.498804 0.866715i \(-0.666227\pi\)
−0.498804 + 0.866715i \(0.666227\pi\)
\(314\) 24.3746 1.37554
\(315\) 0 0
\(316\) −2.54983 −0.143439
\(317\) −16.5498 −0.929531 −0.464766 0.885434i \(-0.653861\pi\)
−0.464766 + 0.885434i \(0.653861\pi\)
\(318\) −10.0000 −0.560772
\(319\) 3.82475 0.214145
\(320\) −3.27492 −0.183073
\(321\) −14.5498 −0.812093
\(322\) 0 0
\(323\) −38.3746 −2.13522
\(324\) 1.00000 0.0555556
\(325\) 5.72508 0.317570
\(326\) 2.54983 0.141222
\(327\) −7.27492 −0.402304
\(328\) −8.54983 −0.472086
\(329\) 0 0
\(330\) 17.2749 0.950953
\(331\) 15.6495 0.860174 0.430087 0.902787i \(-0.358483\pi\)
0.430087 + 0.902787i \(0.358483\pi\)
\(332\) −10.5498 −0.578997
\(333\) 3.27492 0.179464
\(334\) −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
\(339\) −6.00000 −0.325875
\(340\) −23.8248 −1.29208
\(341\) −42.1993 −2.28522
\(342\) 5.27492 0.285235
\(343\) 0 0
\(344\) −5.27492 −0.284404
\(345\) 17.2749 0.930050
\(346\) −23.0997 −1.24185
\(347\) −25.0997 −1.34742 −0.673710 0.738995i \(-0.735300\pi\)
−0.673710 + 0.738995i \(0.735300\pi\)
\(348\) 0.725083 0.0388685
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) −5.27492 −0.281154
\(353\) 3.09967 0.164979 0.0824894 0.996592i \(-0.473713\pi\)
0.0824894 + 0.996592i \(0.473713\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 3.27492 0.172603
\(361\) 8.82475 0.464461
\(362\) −11.0997 −0.583386
\(363\) 16.8248 0.883070
\(364\) 0 0
\(365\) 32.1752 1.68413
\(366\) −4.72508 −0.246984
\(367\) 33.0997 1.72779 0.863894 0.503673i \(-0.168018\pi\)
0.863894 + 0.503673i \(0.168018\pi\)
\(368\) −5.27492 −0.274974
\(369\) 8.54983 0.445087
\(370\) 10.7251 0.557571
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) 11.4502 0.592867 0.296434 0.955053i \(-0.404203\pi\)
0.296434 + 0.955053i \(0.404203\pi\)
\(374\) −38.3746 −1.98430
\(375\) −2.37459 −0.122623
\(376\) 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
\(381\) −5.45017 −0.279220
\(382\) 5.27492 0.269888
\(383\) −22.7251 −1.16120 −0.580599 0.814190i \(-0.697181\pi\)
−0.580599 + 0.814190i \(0.697181\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 3.45017 0.175609
\(387\) 5.27492 0.268139
\(388\) 15.0997 0.766570
\(389\) −32.1993 −1.63257 −0.816286 0.577649i \(-0.803970\pi\)
−0.816286 + 0.577649i \(0.803970\pi\)
\(390\) 3.27492 0.165832
\(391\) −38.3746 −1.94069
\(392\) 0 0
\(393\) 15.8248 0.798253
\(394\) −12.5498 −0.632252
\(395\) 8.35050 0.420159
\(396\) 5.27492 0.265075
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −15.8248 −0.793223
\(399\) 0 0
\(400\) 5.72508 0.286254
\(401\) 27.0997 1.35329 0.676646 0.736308i \(-0.263433\pi\)
0.676646 + 0.736308i \(0.263433\pi\)
\(402\) 2.54983 0.127174
\(403\) −8.00000 −0.398508
\(404\) 12.5498 0.624378
\(405\) −3.27492 −0.162732
\(406\) 0 0
\(407\) 17.2749 0.856286
\(408\) −7.27492 −0.360162
\(409\) 29.8248 1.47474 0.737370 0.675490i \(-0.236068\pi\)
0.737370 + 0.675490i \(0.236068\pi\)
\(410\) 28.0000 1.38282
\(411\) 20.3746 1.00500
\(412\) −2.72508 −0.134255
\(413\) 0 0
\(414\) 5.27492 0.259248
\(415\) 34.5498 1.69598
\(416\) −1.00000 −0.0490290
\(417\) −4.00000 −0.195881
\(418\) 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
\(427\) 0 0
\(428\) −14.5498 −0.703293
\(429\) 5.27492 0.254675
\(430\) 17.2749 0.833070
\(431\) −26.5498 −1.27886 −0.639430 0.768849i \(-0.720830\pi\)
−0.639430 + 0.768849i \(0.720830\pi\)
\(432\) 1.00000 0.0481125
\(433\) 21.6495 1.04041 0.520204 0.854042i \(-0.325856\pi\)
0.520204 + 0.854042i \(0.325856\pi\)
\(434\) 0 0
\(435\) −2.37459 −0.113853
\(436\) −7.27492 −0.348405
\(437\) 27.8248 1.33104
\(438\) 9.82475 0.469445
\(439\) −7.82475 −0.373455 −0.186728 0.982412i \(-0.559788\pi\)
−0.186728 + 0.982412i \(0.559788\pi\)
\(440\) 17.2749 0.823549
\(441\) 0 0
\(442\) −7.27492 −0.346033
\(443\) 22.5498 1.07137 0.535687 0.844416i \(-0.320053\pi\)
0.535687 + 0.844416i \(0.320053\pi\)
\(444\) 3.27492 0.155421
\(445\) −45.8488 −2.17344
\(446\) −23.6495 −1.11984
\(447\) 7.45017 0.352381
\(448\) 0 0
\(449\) 7.27492 0.343325 0.171662 0.985156i \(-0.445086\pi\)
0.171662 + 0.985156i \(0.445086\pi\)
\(450\) −5.72508 −0.269883
\(451\) 45.0997 2.12366
\(452\) −6.00000 −0.282216
\(453\) −1.27492 −0.0599008
\(454\) −5.45017 −0.255789
\(455\) 0 0
\(456\) 5.27492 0.247021
\(457\) −3.45017 −0.161392 −0.0806960 0.996739i \(-0.525714\pi\)
−0.0806960 + 0.996739i \(0.525714\pi\)
\(458\) −11.0997 −0.518653
\(459\) 7.27492 0.339564
\(460\) 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
\(465\) 26.1993 1.21497
\(466\) −20.5498 −0.951953
\(467\) 18.7251 0.866493 0.433247 0.901275i \(-0.357368\pi\)
0.433247 + 0.901275i \(0.357368\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) −24.3746 −1.12312
\(472\) 8.00000 0.368230
\(473\) 27.8248 1.27938
\(474\) 2.54983 0.117118
\(475\) −30.1993 −1.38564
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) 5.09967 0.233253
\(479\) −14.3746 −0.656792 −0.328396 0.944540i \(-0.606508\pi\)
−0.328396 + 0.944540i \(0.606508\pi\)
\(480\) 3.27492 0.149479
\(481\) 3.27492 0.149323
\(482\) −15.0997 −0.687771
\(483\) 0 0
\(484\) 16.8248 0.764761
\(485\) −49.4502 −2.24542
\(486\) −1.00000 −0.0453609
\(487\) 42.1993 1.91223 0.956117 0.292984i \(-0.0946484\pi\)
0.956117 + 0.292984i \(0.0946484\pi\)
\(488\) −4.72508 −0.213894
\(489\) −2.54983 −0.115307
\(490\) 0 0
\(491\) −6.54983 −0.295590 −0.147795 0.989018i \(-0.547218\pi\)
−0.147795 + 0.989018i \(0.547218\pi\)
\(492\) 8.54983 0.385456
\(493\) 5.27492 0.237570
\(494\) 5.27492 0.237330
\(495\) −17.2749 −0.776450
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 10.5498 0.472749
\(499\) −5.09967 −0.228293 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(500\) −2.37459 −0.106195
\(501\) 9.27492 0.414373
\(502\) −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
\(507\) 1.00000 0.0444116
\(508\) −5.45017 −0.241812
\(509\) −42.9244 −1.90259 −0.951296 0.308280i \(-0.900247\pi\)
−0.951296 + 0.308280i \(0.900247\pi\)
\(510\) 23.8248 1.05498
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −5.27492 −0.232893
\(514\) −14.0000 −0.617514
\(515\) 8.92442 0.393257
\(516\) 5.27492 0.232215
\(517\) 0 0
\(518\) 0 0
\(519\) 23.0997 1.01396
\(520\) 3.27492 0.143615
\(521\) −3.27492 −0.143477 −0.0717384 0.997423i \(-0.522855\pi\)
−0.0717384 + 0.997423i \(0.522855\pi\)
\(522\) −0.725083 −0.0317360
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 15.8248 0.691307
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) −58.1993 −2.53520
\(528\) 5.27492 0.229561
\(529\) 4.82475 0.209772
\(530\) 32.7492 1.42253
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 8.54983 0.370334
\(534\) −14.0000 −0.605839
\(535\) 47.6495 2.06007
\(536\) 2.54983 0.110136
\(537\) 12.0000 0.517838
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) −3.27492 −0.140930
\(541\) −25.4743 −1.09522 −0.547612 0.836732i \(-0.684463\pi\)
−0.547612 + 0.836732i \(0.684463\pi\)
\(542\) 13.4502 0.577734
\(543\) 11.0997 0.476332
\(544\) −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
\(549\) 4.72508 0.201662
\(550\) −30.1993 −1.28770
\(551\) −3.82475 −0.162940
\(552\) 5.27492 0.224515
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) −10.7251 −0.455254
\(556\) −4.00000 −0.169638
\(557\) 30.7492 1.30288 0.651442 0.758698i \(-0.274164\pi\)
0.651442 + 0.758698i \(0.274164\pi\)
\(558\) 8.00000 0.338667
\(559\) 5.27492 0.223105
\(560\) 0 0
\(561\) 38.3746 1.62018
\(562\) 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 \(-0.692298\pi\)
−0.568039 + 0.823002i \(0.692298\pi\)
\(570\) −17.2749 −0.723567
\(571\) −6.90033 −0.288770 −0.144385 0.989522i \(-0.546120\pi\)
−0.144385 + 0.989522i \(0.546120\pi\)
\(572\) 5.27492 0.220555
\(573\) −5.27492 −0.220363
\(574\) 0 0
\(575\) −30.1993 −1.25940
\(576\) 1.00000 0.0416667
\(577\) 39.0997 1.62774 0.813870 0.581047i \(-0.197357\pi\)
0.813870 + 0.581047i \(0.197357\pi\)
\(578\) −35.9244 −1.49426
\(579\) −3.45017 −0.143384
\(580\) −2.37459 −0.0985993
\(581\) 0 0
\(582\) −15.0997 −0.625901
\(583\) 52.7492 2.18465
\(584\) 9.82475 0.406551
\(585\) −3.27492 −0.135401
\(586\) −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
\(591\) 12.5498 0.516231
\(592\) 3.27492 0.134598
\(593\) −23.0997 −0.948590 −0.474295 0.880366i \(-0.657297\pi\)
−0.474295 + 0.880366i \(0.657297\pi\)
\(594\) −5.27492 −0.216432
\(595\) 0 0
\(596\) 7.45017 0.305171
\(597\) 15.8248 0.647664
\(598\) 5.27492 0.215707
\(599\) 29.2749 1.19614 0.598070 0.801444i \(-0.295934\pi\)
0.598070 + 0.801444i \(0.295934\pi\)
\(600\) −5.72508 −0.233726
\(601\) 21.6495 0.883102 0.441551 0.897236i \(-0.354428\pi\)
0.441551 + 0.897236i \(0.354428\pi\)
\(602\) 0 0
\(603\) −2.54983 −0.103837
\(604\) −1.27492 −0.0518756
\(605\) −55.0997 −2.24012
\(606\) −12.5498 −0.509802
\(607\) −31.8248 −1.29173 −0.645863 0.763453i \(-0.723502\pi\)
−0.645863 + 0.763453i \(0.723502\pi\)
\(608\) 5.27492 0.213926
\(609\) 0 0
\(610\) 15.4743 0.626534
\(611\) 0 0
\(612\) 7.27492 0.294071
\(613\) 8.72508 0.352403 0.176201 0.984354i \(-0.443619\pi\)
0.176201 + 0.984354i \(0.443619\pi\)
\(614\) −9.09967 −0.367233
\(615\) −28.0000 −1.12907
\(616\) 0 0
\(617\) 1.82475 0.0734617 0.0367309 0.999325i \(-0.488306\pi\)
0.0367309 + 0.999325i \(0.488306\pi\)
\(618\) 2.72508 0.109619
\(619\) −37.2749 −1.49821 −0.749103 0.662454i \(-0.769515\pi\)
−0.749103 + 0.662454i \(0.769515\pi\)
\(620\) 26.1993 1.05219
\(621\) −5.27492 −0.211675
\(622\) 18.5498 0.743781
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) −20.8488 −0.833954
\(626\) 17.6495 0.705416
\(627\) −27.8248 −1.11121
\(628\) −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
\(633\) 23.8248 0.946949
\(634\) 16.5498 0.657278
\(635\) 17.8488 0.708310
\(636\) 10.0000 0.396526
\(637\) 0 0
\(638\) −3.82475 −0.151423
\(639\) 0 0
\(640\) 3.27492 0.129452
\(641\) 15.0997 0.596401 0.298201 0.954503i \(-0.403614\pi\)
0.298201 + 0.954503i \(0.403614\pi\)
\(642\) 14.5498 0.574236
\(643\) −28.9244 −1.14067 −0.570334 0.821413i \(-0.693186\pi\)
−0.570334 + 0.821413i \(0.693186\pi\)
\(644\) 0 0
\(645\) −17.2749 −0.680199
\(646\) 38.3746 1.50983
\(647\) −42.1993 −1.65903 −0.829514 0.558487i \(-0.811382\pi\)
−0.829514 + 0.558487i \(0.811382\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −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 \(-0.281635\pi\)
0.633458 + 0.773777i \(0.281635\pi\)
\(654\) 7.27492 0.284472
\(655\) −51.8248 −2.02496
\(656\) 8.54983 0.333815
\(657\) −9.82475 −0.383300
\(658\) 0 0
\(659\) −22.5498 −0.878417 −0.439208 0.898385i \(-0.644741\pi\)
−0.439208 + 0.898385i \(0.644741\pi\)
\(660\) −17.2749 −0.672425
\(661\) −20.1993 −0.785663 −0.392832 0.919610i \(-0.628504\pi\)
−0.392832 + 0.919610i \(0.628504\pi\)
\(662\) −15.6495 −0.608235
\(663\) 7.27492 0.282534
\(664\) 10.5498 0.409413
\(665\) 0 0
\(666\) −3.27492 −0.126900
\(667\) −3.82475 −0.148095
\(668\) 9.27492 0.358857
\(669\) 23.6495 0.914343
\(670\) −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
\(675\) 5.72508 0.220359
\(676\) 1.00000 0.0384615
\(677\) 44.1993 1.69872 0.849359 0.527815i \(-0.176989\pi\)
0.849359 + 0.527815i \(0.176989\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 23.8248 0.913638
\(681\) 5.45017 0.208851
\(682\) 42.1993 1.61590
\(683\) −34.3746 −1.31531 −0.657653 0.753321i \(-0.728451\pi\)
−0.657653 + 0.753321i \(0.728451\pi\)
\(684\) −5.27492 −0.201692
\(685\) −66.7251 −2.54943
\(686\) 0 0
\(687\) 11.0997 0.423479
\(688\) 5.27492 0.201104
\(689\) 10.0000 0.380970
\(690\) −17.2749 −0.657645
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 23.0997 0.878118
\(693\) 0 0
\(694\) 25.0997 0.952770
\(695\) 13.0997 0.496899
\(696\) −0.725083 −0.0274842
\(697\) 62.1993 2.35597
\(698\) 2.00000 0.0757011
\(699\) 20.5498 0.777266
\(700\) 0 0
\(701\) 15.0997 0.570307 0.285153 0.958482i \(-0.407955\pi\)
0.285153 + 0.958482i \(0.407955\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −17.2749 −0.651536
\(704\) 5.27492 0.198806
\(705\) 0 0
\(706\) −3.09967 −0.116658
\(707\) 0 0
\(708\) −8.00000 −0.300658
\(709\) 7.09967 0.266634 0.133317 0.991073i \(-0.457437\pi\)
0.133317 + 0.991073i \(0.457437\pi\)
\(710\) 0 0
\(711\) −2.54983 −0.0956263
\(712\) −14.0000 −0.524672
\(713\) 42.1993 1.58038
\(714\) 0 0
\(715\) −17.2749 −0.646045
\(716\) 12.0000 0.448461
\(717\) −5.09967 −0.190451
\(718\) 32.0000 1.19423
\(719\) −15.6495 −0.583628 −0.291814 0.956475i \(-0.594259\pi\)
−0.291814 + 0.956475i \(0.594259\pi\)
\(720\) −3.27492 −0.122049
\(721\) 0 0
\(722\) −8.82475 −0.328423
\(723\) 15.0997 0.561563
\(724\) 11.0997 0.412516
\(725\) 4.15116 0.154170
\(726\) −16.8248 −0.624425
\(727\) 2.37459 0.0880685 0.0440343 0.999030i \(-0.485979\pi\)
0.0440343 + 0.999030i \(0.485979\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −32.1752 −1.19086
\(731\) 38.3746 1.41934
\(732\) 4.72508 0.174644
\(733\) 0.549834 0.0203086 0.0101543 0.999948i \(-0.496768\pi\)
0.0101543 + 0.999948i \(0.496768\pi\)
\(734\) −33.0997 −1.22173
\(735\) 0 0
\(736\) 5.27492 0.194436
\(737\) −13.4502 −0.495443
\(738\) −8.54983 −0.314724
\(739\) −7.64950 −0.281392 −0.140696 0.990053i \(-0.544934\pi\)
−0.140696 + 0.990053i \(0.544934\pi\)
\(740\) −10.7251 −0.394262
\(741\) −5.27492 −0.193779
\(742\) 0 0
\(743\) 26.5498 0.974019 0.487009 0.873397i \(-0.338088\pi\)
0.487009 + 0.873397i \(0.338088\pi\)
\(744\) 8.00000 0.293294
\(745\) −24.3987 −0.893898
\(746\) −11.4502 −0.419220
\(747\) −10.5498 −0.385998
\(748\) 38.3746 1.40311
\(749\) 0 0
\(750\) 2.37459 0.0867076
\(751\) −50.1993 −1.83180 −0.915900 0.401407i \(-0.868521\pi\)
−0.915900 + 0.401407i \(0.868521\pi\)
\(752\) 0 0
\(753\) 28.9244 1.05406
\(754\) −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
\(759\) −27.8248 −1.00997
\(760\) −17.2749 −0.626627
\(761\) −23.4502 −0.850068 −0.425034 0.905177i \(-0.639738\pi\)
−0.425034 + 0.905177i \(0.639738\pi\)
\(762\) 5.45017 0.197439
\(763\) 0 0
\(764\) −5.27492 −0.190840
\(765\) −23.8248 −0.861386
\(766\) 22.7251 0.821091
\(767\) −8.00000 −0.288863
\(768\) 1.00000 0.0360844
\(769\) −10.1752 −0.366929 −0.183464 0.983026i \(-0.558731\pi\)
−0.183464 + 0.983026i \(0.558731\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) −3.45017 −0.124174
\(773\) −42.9244 −1.54388 −0.771942 0.635693i \(-0.780714\pi\)
−0.771942 + 0.635693i \(0.780714\pi\)
\(774\) −5.27492 −0.189603
\(775\) −45.8007 −1.64521
\(776\) −15.0997 −0.542047
\(777\) 0 0
\(778\) 32.1993 1.15440
\(779\) −45.0997 −1.61586
\(780\) −3.27492 −0.117261
\(781\) 0 0
\(782\) 38.3746 1.37227
\(783\) 0.725083 0.0259123
\(784\) 0 0
\(785\) 79.8248 2.84907
\(786\) −15.8248 −0.564450
\(787\) 2.72508 0.0971387 0.0485694 0.998820i \(-0.484534\pi\)
0.0485694 + 0.998820i \(0.484534\pi\)
\(788\) 12.5498 0.447069
\(789\) 28.0000 0.996826
\(790\) −8.35050 −0.297097
\(791\) 0 0
\(792\) −5.27492 −0.187436
\(793\) 4.72508 0.167793
\(794\) −14.0000 −0.496841
\(795\) −32.7492 −1.16149
\(796\) 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
\(801\) 14.0000 0.494666
\(802\) −27.0997 −0.956923
\(803\) −51.8248 −1.82886
\(804\) −2.54983 −0.0899257
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) −6.00000 −0.211210
\(808\) −12.5498 −0.441502
\(809\) 4.19934 0.147641 0.0738204 0.997272i \(-0.476481\pi\)
0.0738204 + 0.997272i \(0.476481\pi\)
\(810\) 3.27492 0.115069
\(811\) −26.3746 −0.926137 −0.463068 0.886322i \(-0.653252\pi\)
−0.463068 + 0.886322i \(0.653252\pi\)
\(812\) 0 0
\(813\) −13.4502 −0.471718
\(814\) −17.2749 −0.605486
\(815\) 8.35050 0.292505
\(816\) 7.27492 0.254673
\(817\) −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 \(-0.576553\pi\)
−0.238185 + 0.971220i \(0.576553\pi\)
\(822\) −20.3746 −0.710645
\(823\) 5.09967 0.177763 0.0888816 0.996042i \(-0.471671\pi\)
0.0888816 + 0.996042i \(0.471671\pi\)
\(824\) 2.72508 0.0949328
\(825\) 30.1993 1.05141
\(826\) 0 0
\(827\) −18.7251 −0.651135 −0.325567 0.945519i \(-0.605555\pi\)
−0.325567 + 0.945519i \(0.605555\pi\)
\(828\) −5.27492 −0.183316
\(829\) −19.2749 −0.669446 −0.334723 0.942317i \(-0.608643\pi\)
−0.334723 + 0.942317i \(0.608643\pi\)
\(830\) −34.5498 −1.19924
\(831\) −10.0000 −0.346896
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) 4.00000 0.138509
\(835\) −30.3746 −1.05116
\(836\) −27.8248 −0.962339
\(837\) −8.00000 −0.276520
\(838\) −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
\(843\) −15.0997 −0.520060
\(844\) 23.8248 0.820082
\(845\) −3.27492 −0.112661
\(846\) 0 0
\(847\) 0 0
\(848\) 10.0000 0.343401
\(849\) 4.00000 0.137280
\(850\) −41.6495 −1.42857
\(851\) −17.2749 −0.592177
\(852\) 0 0
\(853\) −17.6495 −0.604307 −0.302154 0.953259i \(-0.597706\pi\)
−0.302154 + 0.953259i \(0.597706\pi\)
\(854\) 0 0
\(855\) 17.2749 0.590790
\(856\) 14.5498 0.497303
\(857\) 11.0997 0.379157 0.189579 0.981866i \(-0.439288\pi\)
0.189579 + 0.981866i \(0.439288\pi\)
\(858\) −5.27492 −0.180083
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −17.2749 −0.589070
\(861\) 0 0
\(862\) 26.5498 0.904291
\(863\) −15.6495 −0.532715 −0.266358 0.963874i \(-0.585820\pi\)
−0.266358 + 0.963874i \(0.585820\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −75.6495 −2.57216
\(866\) −21.6495 −0.735680
\(867\) 35.9244 1.22006
\(868\) 0 0
\(869\) −13.4502 −0.456266
\(870\) 2.37459 0.0805060
\(871\) −2.54983 −0.0863978
\(872\) 7.27492 0.246360
\(873\) 15.0997 0.511046
\(874\) −27.8248 −0.941186
\(875\) 0 0
\(876\) −9.82475 −0.331948
\(877\) 20.9003 0.705754 0.352877 0.935670i \(-0.385203\pi\)
0.352877 + 0.935670i \(0.385203\pi\)
\(878\) 7.82475 0.264073
\(879\) 6.00000 0.202375
\(880\) −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
\(885\) 26.1993 0.880681
\(886\) −22.5498 −0.757577
\(887\) 2.90033 0.0973836 0.0486918 0.998814i \(-0.484495\pi\)
0.0486918 + 0.998814i \(0.484495\pi\)
\(888\) −3.27492 −0.109899
\(889\) 0 0
\(890\) 45.8488 1.53686
\(891\) 5.27492 0.176716
\(892\) 23.6495 0.791844
\(893\) 0 0
\(894\) −7.45017 −0.249171
\(895\) −39.2990 −1.31362
\(896\) 0 0
\(897\) −5.27492 −0.176124
\(898\) −7.27492 −0.242767
\(899\) −5.80066 −0.193463
\(900\) 5.72508 0.190836
\(901\) 72.7492 2.42363
\(902\) −45.0997 −1.50165
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −36.3505 −1.20833
\(906\) 1.27492 0.0423563
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 5.45017 0.180870
\(909\) 12.5498 0.416252
\(910\) 0 0
\(911\) 15.4743 0.512685 0.256342 0.966586i \(-0.417483\pi\)
0.256342 + 0.966586i \(0.417483\pi\)
\(912\) −5.27492 −0.174670
\(913\) −55.6495 −1.84173
\(914\) 3.45017 0.114121
\(915\) −15.4743 −0.511563
\(916\) 11.0997 0.366743
\(917\) 0 0
\(918\) −7.27492 −0.240108
\(919\) 2.54983 0.0841113 0.0420556 0.999115i \(-0.486609\pi\)
0.0420556 + 0.999115i \(0.486609\pi\)
\(920\) −17.2749 −0.569537
\(921\) 9.09967 0.299844
\(922\) 34.9244 1.15017
\(923\) 0 0
\(924\) 0 0
\(925\) 18.7492 0.616469
\(926\) −22.7251 −0.746793
\(927\) −2.72508 −0.0895035
\(928\) −0.725083 −0.0238020
\(929\) 0.549834 0.0180395 0.00901974 0.999959i \(-0.497129\pi\)
0.00901974 + 0.999959i \(0.497129\pi\)
\(930\) −26.1993 −0.859110
\(931\) 0 0
\(932\) 20.5498 0.673132
\(933\) −18.5498 −0.607294
\(934\) −18.7251 −0.612703
\(935\) −125.674 −4.10997
\(936\) −1.00000 −0.0326860
\(937\) −47.0997 −1.53868 −0.769340 0.638840i \(-0.779415\pi\)
−0.769340 + 0.638840i \(0.779415\pi\)
\(938\) 0 0
\(939\) −17.6495 −0.575970
\(940\) 0 0
\(941\) −20.9003 −0.681331 −0.340666 0.940185i \(-0.610652\pi\)
−0.340666 + 0.940185i \(0.610652\pi\)
\(942\) 24.3746 0.794167
\(943\) −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 \(-0.724004\pi\)
−0.647065 + 0.762435i \(0.724004\pi\)
\(948\) −2.54983 −0.0828148
\(949\) −9.82475 −0.318925
\(950\) 30.1993 0.979796
\(951\) −16.5498 −0.536665
\(952\) 0 0
\(953\) −13.6495 −0.442151 −0.221075 0.975257i \(-0.570957\pi\)
−0.221075 + 0.975257i \(0.570957\pi\)
\(954\) −10.0000 −0.323762
\(955\) 17.2749 0.559003
\(956\) −5.09967 −0.164935
\(957\) 3.82475 0.123637
\(958\) 14.3746 0.464422
\(959\) 0 0
\(960\) −3.27492 −0.105697
\(961\) 33.0000 1.06452
\(962\) −3.27492 −0.105588
\(963\) −14.5498 −0.468862
\(964\) 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
\(969\) −38.3746 −1.23277
\(970\) 49.4502 1.58775
\(971\) −1.09967 −0.0352901 −0.0176450 0.999844i \(-0.505617\pi\)
−0.0176450 + 0.999844i \(0.505617\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −42.1993 −1.35215
\(975\) 5.72508 0.183349
\(976\) 4.72508 0.151246
\(977\) 38.9244 1.24530 0.622651 0.782499i \(-0.286056\pi\)
0.622651 + 0.782499i \(0.286056\pi\)
\(978\) 2.54983 0.0815347
\(979\) 73.8488 2.36022
\(980\) 0 0
\(981\) −7.27492 −0.232270
\(982\) 6.54983 0.209014
\(983\) 22.0241 0.702459 0.351230 0.936289i \(-0.385764\pi\)
0.351230 + 0.936289i \(0.385764\pi\)
\(984\) −8.54983 −0.272559
\(985\) −41.0997 −1.30954
\(986\) −5.27492 −0.167988
\(987\) 0 0
\(988\) −5.27492 −0.167817
\(989\) −27.8248 −0.884776
\(990\) 17.2749 0.549033
\(991\) 28.7492 0.913248 0.456624 0.889660i \(-0.349059\pi\)
0.456624 + 0.889660i \(0.349059\pi\)
\(992\) 8.00000 0.254000
\(993\) 15.6495 0.496622
\(994\) 0 0
\(995\) −51.8248 −1.64296
\(996\) −10.5498 −0.334284
\(997\) 43.0997 1.36498 0.682490 0.730895i \(-0.260897\pi\)
0.682490 + 0.730895i \(0.260897\pi\)
\(998\) 5.09967 0.161427
\(999\) 3.27492 0.103614
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 3822.2.a.bm.1.1 2
7.6 odd 2 546.2.a.h.1.2 2
21.20 even 2 1638.2.a.y.1.1 2
28.27 even 2 4368.2.a.bh.1.2 2
91.90 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 7.6 odd 2
1638.2.a.y.1.1 2 21.20 even 2
3822.2.a.bm.1.1 2 1.1 even 1 trivial
4368.2.a.bh.1.2 2 28.27 even 2
7098.2.a.bu.1.1 2 91.90 odd 2