Properties

Label 847.2.a.d.1.2
Level $847$
Weight $2$
Character 847.1
Self dual yes
Analytic conductor $6.763$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.76332905120\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 847.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.381966 q^{2} -1.61803 q^{3} -1.85410 q^{4} +1.00000 q^{5} +0.618034 q^{6} +1.00000 q^{7} +1.47214 q^{8} -0.381966 q^{9} +O(q^{10})\) \(q-0.381966 q^{2} -1.61803 q^{3} -1.85410 q^{4} +1.00000 q^{5} +0.618034 q^{6} +1.00000 q^{7} +1.47214 q^{8} -0.381966 q^{9} -0.381966 q^{10} +3.00000 q^{12} -1.23607 q^{13} -0.381966 q^{14} -1.61803 q^{15} +3.14590 q^{16} +3.09017 q^{17} +0.145898 q^{18} -1.76393 q^{19} -1.85410 q^{20} -1.61803 q^{21} +5.09017 q^{23} -2.38197 q^{24} -4.00000 q^{25} +0.472136 q^{26} +5.47214 q^{27} -1.85410 q^{28} -4.61803 q^{29} +0.618034 q^{30} -4.23607 q^{31} -4.14590 q^{32} -1.18034 q^{34} +1.00000 q^{35} +0.708204 q^{36} +6.47214 q^{37} +0.673762 q^{38} +2.00000 q^{39} +1.47214 q^{40} -11.1803 q^{41} +0.618034 q^{42} -12.5623 q^{43} -0.381966 q^{45} -1.94427 q^{46} -6.61803 q^{47} -5.09017 q^{48} +1.00000 q^{49} +1.52786 q^{50} -5.00000 q^{51} +2.29180 q^{52} -2.38197 q^{53} -2.09017 q^{54} +1.47214 q^{56} +2.85410 q^{57} +1.76393 q^{58} +11.0902 q^{59} +3.00000 q^{60} -7.61803 q^{61} +1.61803 q^{62} -0.381966 q^{63} -4.70820 q^{64} -1.23607 q^{65} -8.32624 q^{67} -5.72949 q^{68} -8.23607 q^{69} -0.381966 q^{70} -16.0902 q^{71} -0.562306 q^{72} -14.2361 q^{73} -2.47214 q^{74} +6.47214 q^{75} +3.27051 q^{76} -0.763932 q^{78} +6.38197 q^{79} +3.14590 q^{80} -7.70820 q^{81} +4.27051 q^{82} +2.70820 q^{83} +3.00000 q^{84} +3.09017 q^{85} +4.79837 q^{86} +7.47214 q^{87} +6.85410 q^{89} +0.145898 q^{90} -1.23607 q^{91} -9.43769 q^{92} +6.85410 q^{93} +2.52786 q^{94} -1.76393 q^{95} +6.70820 q^{96} +7.00000 q^{97} -0.381966 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - q^{3} + 3 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} - 6 q^{8} - 3 q^{9} + O(q^{10}) \) \( 2 q - 3 q^{2} - q^{3} + 3 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} - 6 q^{8} - 3 q^{9} - 3 q^{10} + 6 q^{12} + 2 q^{13} - 3 q^{14} - q^{15} + 13 q^{16} - 5 q^{17} + 7 q^{18} - 8 q^{19} + 3 q^{20} - q^{21} - q^{23} - 7 q^{24} - 8 q^{25} - 8 q^{26} + 2 q^{27} + 3 q^{28} - 7 q^{29} - q^{30} - 4 q^{31} - 15 q^{32} + 20 q^{34} + 2 q^{35} - 12 q^{36} + 4 q^{37} + 17 q^{38} + 4 q^{39} - 6 q^{40} - q^{42} - 5 q^{43} - 3 q^{45} + 14 q^{46} - 11 q^{47} + q^{48} + 2 q^{49} + 12 q^{50} - 10 q^{51} + 18 q^{52} - 7 q^{53} + 7 q^{54} - 6 q^{56} - q^{57} + 8 q^{58} + 11 q^{59} + 6 q^{60} - 13 q^{61} + q^{62} - 3 q^{63} + 4 q^{64} + 2 q^{65} - q^{67} - 45 q^{68} - 12 q^{69} - 3 q^{70} - 21 q^{71} + 19 q^{72} - 24 q^{73} + 4 q^{74} + 4 q^{75} - 27 q^{76} - 6 q^{78} + 15 q^{79} + 13 q^{80} - 2 q^{81} - 25 q^{82} - 8 q^{83} + 6 q^{84} - 5 q^{85} - 15 q^{86} + 6 q^{87} + 7 q^{89} + 7 q^{90} + 2 q^{91} - 39 q^{92} + 7 q^{93} + 14 q^{94} - 8 q^{95} + 14 q^{97} - 3 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.381966 −0.270091 −0.135045 0.990839i \(-0.543118\pi\)
−0.135045 + 0.990839i \(0.543118\pi\)
\(3\) −1.61803 −0.934172 −0.467086 0.884212i \(-0.654696\pi\)
−0.467086 + 0.884212i \(0.654696\pi\)
\(4\) −1.85410 −0.927051
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0.618034 0.252311
\(7\) 1.00000 0.377964
\(8\) 1.47214 0.520479
\(9\) −0.381966 −0.127322
\(10\) −0.381966 −0.120788
\(11\) 0 0
\(12\) 3.00000 0.866025
\(13\) −1.23607 −0.342824 −0.171412 0.985199i \(-0.554833\pi\)
−0.171412 + 0.985199i \(0.554833\pi\)
\(14\) −0.381966 −0.102085
\(15\) −1.61803 −0.417775
\(16\) 3.14590 0.786475
\(17\) 3.09017 0.749476 0.374738 0.927131i \(-0.377733\pi\)
0.374738 + 0.927131i \(0.377733\pi\)
\(18\) 0.145898 0.0343885
\(19\) −1.76393 −0.404674 −0.202337 0.979316i \(-0.564854\pi\)
−0.202337 + 0.979316i \(0.564854\pi\)
\(20\) −1.85410 −0.414590
\(21\) −1.61803 −0.353084
\(22\) 0 0
\(23\) 5.09017 1.06137 0.530687 0.847568i \(-0.321934\pi\)
0.530687 + 0.847568i \(0.321934\pi\)
\(24\) −2.38197 −0.486217
\(25\) −4.00000 −0.800000
\(26\) 0.472136 0.0925935
\(27\) 5.47214 1.05311
\(28\) −1.85410 −0.350392
\(29\) −4.61803 −0.857547 −0.428774 0.903412i \(-0.641054\pi\)
−0.428774 + 0.903412i \(0.641054\pi\)
\(30\) 0.618034 0.112837
\(31\) −4.23607 −0.760820 −0.380410 0.924818i \(-0.624217\pi\)
−0.380410 + 0.924818i \(0.624217\pi\)
\(32\) −4.14590 −0.732898
\(33\) 0 0
\(34\) −1.18034 −0.202427
\(35\) 1.00000 0.169031
\(36\) 0.708204 0.118034
\(37\) 6.47214 1.06401 0.532006 0.846740i \(-0.321438\pi\)
0.532006 + 0.846740i \(0.321438\pi\)
\(38\) 0.673762 0.109299
\(39\) 2.00000 0.320256
\(40\) 1.47214 0.232765
\(41\) −11.1803 −1.74608 −0.873038 0.487652i \(-0.837853\pi\)
−0.873038 + 0.487652i \(0.837853\pi\)
\(42\) 0.618034 0.0953647
\(43\) −12.5623 −1.91573 −0.957867 0.287213i \(-0.907271\pi\)
−0.957867 + 0.287213i \(0.907271\pi\)
\(44\) 0 0
\(45\) −0.381966 −0.0569401
\(46\) −1.94427 −0.286667
\(47\) −6.61803 −0.965339 −0.482670 0.875802i \(-0.660333\pi\)
−0.482670 + 0.875802i \(0.660333\pi\)
\(48\) −5.09017 −0.734703
\(49\) 1.00000 0.142857
\(50\) 1.52786 0.216073
\(51\) −5.00000 −0.700140
\(52\) 2.29180 0.317815
\(53\) −2.38197 −0.327188 −0.163594 0.986528i \(-0.552309\pi\)
−0.163594 + 0.986528i \(0.552309\pi\)
\(54\) −2.09017 −0.284436
\(55\) 0 0
\(56\) 1.47214 0.196722
\(57\) 2.85410 0.378035
\(58\) 1.76393 0.231616
\(59\) 11.0902 1.44382 0.721909 0.691988i \(-0.243265\pi\)
0.721909 + 0.691988i \(0.243265\pi\)
\(60\) 3.00000 0.387298
\(61\) −7.61803 −0.975389 −0.487695 0.873014i \(-0.662162\pi\)
−0.487695 + 0.873014i \(0.662162\pi\)
\(62\) 1.61803 0.205491
\(63\) −0.381966 −0.0481232
\(64\) −4.70820 −0.588525
\(65\) −1.23607 −0.153315
\(66\) 0 0
\(67\) −8.32624 −1.01721 −0.508606 0.860999i \(-0.669839\pi\)
−0.508606 + 0.860999i \(0.669839\pi\)
\(68\) −5.72949 −0.694803
\(69\) −8.23607 −0.991506
\(70\) −0.381966 −0.0456537
\(71\) −16.0902 −1.90955 −0.954776 0.297326i \(-0.903905\pi\)
−0.954776 + 0.297326i \(0.903905\pi\)
\(72\) −0.562306 −0.0662684
\(73\) −14.2361 −1.66621 −0.833103 0.553118i \(-0.813438\pi\)
−0.833103 + 0.553118i \(0.813438\pi\)
\(74\) −2.47214 −0.287380
\(75\) 6.47214 0.747338
\(76\) 3.27051 0.375153
\(77\) 0 0
\(78\) −0.763932 −0.0864983
\(79\) 6.38197 0.718027 0.359014 0.933332i \(-0.383113\pi\)
0.359014 + 0.933332i \(0.383113\pi\)
\(80\) 3.14590 0.351722
\(81\) −7.70820 −0.856467
\(82\) 4.27051 0.471599
\(83\) 2.70820 0.297264 0.148632 0.988893i \(-0.452513\pi\)
0.148632 + 0.988893i \(0.452513\pi\)
\(84\) 3.00000 0.327327
\(85\) 3.09017 0.335176
\(86\) 4.79837 0.517422
\(87\) 7.47214 0.801097
\(88\) 0 0
\(89\) 6.85410 0.726533 0.363267 0.931685i \(-0.381661\pi\)
0.363267 + 0.931685i \(0.381661\pi\)
\(90\) 0.145898 0.0153790
\(91\) −1.23607 −0.129575
\(92\) −9.43769 −0.983948
\(93\) 6.85410 0.710737
\(94\) 2.52786 0.260729
\(95\) −1.76393 −0.180976
\(96\) 6.70820 0.684653
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) −0.381966 −0.0385844
\(99\) 0 0
\(100\) 7.41641 0.741641
\(101\) −14.7984 −1.47249 −0.736247 0.676713i \(-0.763404\pi\)
−0.736247 + 0.676713i \(0.763404\pi\)
\(102\) 1.90983 0.189101
\(103\) 8.85410 0.872421 0.436210 0.899845i \(-0.356320\pi\)
0.436210 + 0.899845i \(0.356320\pi\)
\(104\) −1.81966 −0.178432
\(105\) −1.61803 −0.157904
\(106\) 0.909830 0.0883705
\(107\) 2.70820 0.261812 0.130906 0.991395i \(-0.458211\pi\)
0.130906 + 0.991395i \(0.458211\pi\)
\(108\) −10.1459 −0.976289
\(109\) −1.52786 −0.146343 −0.0731714 0.997319i \(-0.523312\pi\)
−0.0731714 + 0.997319i \(0.523312\pi\)
\(110\) 0 0
\(111\) −10.4721 −0.993971
\(112\) 3.14590 0.297259
\(113\) 0.145898 0.0137249 0.00686247 0.999976i \(-0.497816\pi\)
0.00686247 + 0.999976i \(0.497816\pi\)
\(114\) −1.09017 −0.102104
\(115\) 5.09017 0.474661
\(116\) 8.56231 0.794990
\(117\) 0.472136 0.0436490
\(118\) −4.23607 −0.389962
\(119\) 3.09017 0.283275
\(120\) −2.38197 −0.217443
\(121\) 0 0
\(122\) 2.90983 0.263444
\(123\) 18.0902 1.63114
\(124\) 7.85410 0.705319
\(125\) −9.00000 −0.804984
\(126\) 0.145898 0.0129976
\(127\) 2.94427 0.261262 0.130631 0.991431i \(-0.458300\pi\)
0.130631 + 0.991431i \(0.458300\pi\)
\(128\) 10.0902 0.891853
\(129\) 20.3262 1.78963
\(130\) 0.472136 0.0414091
\(131\) 18.9443 1.65517 0.827584 0.561341i \(-0.189715\pi\)
0.827584 + 0.561341i \(0.189715\pi\)
\(132\) 0 0
\(133\) −1.76393 −0.152952
\(134\) 3.18034 0.274740
\(135\) 5.47214 0.470966
\(136\) 4.54915 0.390086
\(137\) −15.3262 −1.30941 −0.654704 0.755885i \(-0.727207\pi\)
−0.654704 + 0.755885i \(0.727207\pi\)
\(138\) 3.14590 0.267797
\(139\) 11.9443 1.01310 0.506550 0.862211i \(-0.330921\pi\)
0.506550 + 0.862211i \(0.330921\pi\)
\(140\) −1.85410 −0.156700
\(141\) 10.7082 0.901793
\(142\) 6.14590 0.515752
\(143\) 0 0
\(144\) −1.20163 −0.100136
\(145\) −4.61803 −0.383507
\(146\) 5.43769 0.450027
\(147\) −1.61803 −0.133453
\(148\) −12.0000 −0.986394
\(149\) −8.85410 −0.725356 −0.362678 0.931914i \(-0.618138\pi\)
−0.362678 + 0.931914i \(0.618138\pi\)
\(150\) −2.47214 −0.201849
\(151\) 0.0557281 0.00453509 0.00226754 0.999997i \(-0.499278\pi\)
0.00226754 + 0.999997i \(0.499278\pi\)
\(152\) −2.59675 −0.210624
\(153\) −1.18034 −0.0954248
\(154\) 0 0
\(155\) −4.23607 −0.340249
\(156\) −3.70820 −0.296894
\(157\) 19.8885 1.58728 0.793639 0.608389i \(-0.208184\pi\)
0.793639 + 0.608389i \(0.208184\pi\)
\(158\) −2.43769 −0.193933
\(159\) 3.85410 0.305650
\(160\) −4.14590 −0.327762
\(161\) 5.09017 0.401162
\(162\) 2.94427 0.231324
\(163\) −8.70820 −0.682079 −0.341040 0.940049i \(-0.610779\pi\)
−0.341040 + 0.940049i \(0.610779\pi\)
\(164\) 20.7295 1.61870
\(165\) 0 0
\(166\) −1.03444 −0.0802883
\(167\) 6.47214 0.500829 0.250414 0.968139i \(-0.419433\pi\)
0.250414 + 0.968139i \(0.419433\pi\)
\(168\) −2.38197 −0.183773
\(169\) −11.4721 −0.882472
\(170\) −1.18034 −0.0905279
\(171\) 0.673762 0.0515239
\(172\) 23.2918 1.77598
\(173\) −15.3820 −1.16947 −0.584735 0.811225i \(-0.698801\pi\)
−0.584735 + 0.811225i \(0.698801\pi\)
\(174\) −2.85410 −0.216369
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) −17.9443 −1.34877
\(178\) −2.61803 −0.196230
\(179\) −3.76393 −0.281329 −0.140665 0.990057i \(-0.544924\pi\)
−0.140665 + 0.990057i \(0.544924\pi\)
\(180\) 0.708204 0.0527864
\(181\) −7.41641 −0.551257 −0.275629 0.961264i \(-0.588886\pi\)
−0.275629 + 0.961264i \(0.588886\pi\)
\(182\) 0.472136 0.0349970
\(183\) 12.3262 0.911182
\(184\) 7.49342 0.552422
\(185\) 6.47214 0.475841
\(186\) −2.61803 −0.191964
\(187\) 0 0
\(188\) 12.2705 0.894919
\(189\) 5.47214 0.398039
\(190\) 0.673762 0.0488798
\(191\) 15.7639 1.14064 0.570319 0.821423i \(-0.306820\pi\)
0.570319 + 0.821423i \(0.306820\pi\)
\(192\) 7.61803 0.549784
\(193\) −15.3262 −1.10321 −0.551603 0.834107i \(-0.685984\pi\)
−0.551603 + 0.834107i \(0.685984\pi\)
\(194\) −2.67376 −0.191965
\(195\) 2.00000 0.143223
\(196\) −1.85410 −0.132436
\(197\) −4.29180 −0.305778 −0.152889 0.988243i \(-0.548858\pi\)
−0.152889 + 0.988243i \(0.548858\pi\)
\(198\) 0 0
\(199\) −8.23607 −0.583839 −0.291920 0.956443i \(-0.594294\pi\)
−0.291920 + 0.956443i \(0.594294\pi\)
\(200\) −5.88854 −0.416383
\(201\) 13.4721 0.950251
\(202\) 5.65248 0.397707
\(203\) −4.61803 −0.324122
\(204\) 9.27051 0.649066
\(205\) −11.1803 −0.780869
\(206\) −3.38197 −0.235633
\(207\) −1.94427 −0.135136
\(208\) −3.88854 −0.269622
\(209\) 0 0
\(210\) 0.618034 0.0426484
\(211\) 15.6180 1.07519 0.537595 0.843203i \(-0.319333\pi\)
0.537595 + 0.843203i \(0.319333\pi\)
\(212\) 4.41641 0.303320
\(213\) 26.0344 1.78385
\(214\) −1.03444 −0.0707130
\(215\) −12.5623 −0.856742
\(216\) 8.05573 0.548123
\(217\) −4.23607 −0.287563
\(218\) 0.583592 0.0395258
\(219\) 23.0344 1.55652
\(220\) 0 0
\(221\) −3.81966 −0.256938
\(222\) 4.00000 0.268462
\(223\) 2.03444 0.136236 0.0681182 0.997677i \(-0.478301\pi\)
0.0681182 + 0.997677i \(0.478301\pi\)
\(224\) −4.14590 −0.277009
\(225\) 1.52786 0.101858
\(226\) −0.0557281 −0.00370698
\(227\) −16.0344 −1.06424 −0.532122 0.846668i \(-0.678605\pi\)
−0.532122 + 0.846668i \(0.678605\pi\)
\(228\) −5.29180 −0.350458
\(229\) 6.76393 0.446973 0.223487 0.974707i \(-0.428256\pi\)
0.223487 + 0.974707i \(0.428256\pi\)
\(230\) −1.94427 −0.128201
\(231\) 0 0
\(232\) −6.79837 −0.446335
\(233\) −29.4164 −1.92713 −0.963566 0.267469i \(-0.913813\pi\)
−0.963566 + 0.267469i \(0.913813\pi\)
\(234\) −0.180340 −0.0117892
\(235\) −6.61803 −0.431713
\(236\) −20.5623 −1.33849
\(237\) −10.3262 −0.670761
\(238\) −1.18034 −0.0765101
\(239\) −17.0902 −1.10547 −0.552736 0.833357i \(-0.686416\pi\)
−0.552736 + 0.833357i \(0.686416\pi\)
\(240\) −5.09017 −0.328569
\(241\) 16.2705 1.04808 0.524038 0.851695i \(-0.324425\pi\)
0.524038 + 0.851695i \(0.324425\pi\)
\(242\) 0 0
\(243\) −3.94427 −0.253025
\(244\) 14.1246 0.904236
\(245\) 1.00000 0.0638877
\(246\) −6.90983 −0.440555
\(247\) 2.18034 0.138732
\(248\) −6.23607 −0.395991
\(249\) −4.38197 −0.277696
\(250\) 3.43769 0.217419
\(251\) 23.0000 1.45175 0.725874 0.687828i \(-0.241436\pi\)
0.725874 + 0.687828i \(0.241436\pi\)
\(252\) 0.708204 0.0446127
\(253\) 0 0
\(254\) −1.12461 −0.0705644
\(255\) −5.00000 −0.313112
\(256\) 5.56231 0.347644
\(257\) −8.56231 −0.534102 −0.267051 0.963682i \(-0.586049\pi\)
−0.267051 + 0.963682i \(0.586049\pi\)
\(258\) −7.76393 −0.483361
\(259\) 6.47214 0.402159
\(260\) 2.29180 0.142131
\(261\) 1.76393 0.109185
\(262\) −7.23607 −0.447046
\(263\) −17.1246 −1.05595 −0.527974 0.849260i \(-0.677048\pi\)
−0.527974 + 0.849260i \(0.677048\pi\)
\(264\) 0 0
\(265\) −2.38197 −0.146323
\(266\) 0.673762 0.0413110
\(267\) −11.0902 −0.678707
\(268\) 15.4377 0.943007
\(269\) 16.8541 1.02761 0.513806 0.857906i \(-0.328235\pi\)
0.513806 + 0.857906i \(0.328235\pi\)
\(270\) −2.09017 −0.127204
\(271\) −4.79837 −0.291480 −0.145740 0.989323i \(-0.546556\pi\)
−0.145740 + 0.989323i \(0.546556\pi\)
\(272\) 9.72136 0.589444
\(273\) 2.00000 0.121046
\(274\) 5.85410 0.353659
\(275\) 0 0
\(276\) 15.2705 0.919177
\(277\) 23.0344 1.38401 0.692003 0.721895i \(-0.256729\pi\)
0.692003 + 0.721895i \(0.256729\pi\)
\(278\) −4.56231 −0.273629
\(279\) 1.61803 0.0968692
\(280\) 1.47214 0.0879770
\(281\) −25.1803 −1.50213 −0.751067 0.660226i \(-0.770460\pi\)
−0.751067 + 0.660226i \(0.770460\pi\)
\(282\) −4.09017 −0.243566
\(283\) 11.9443 0.710013 0.355007 0.934864i \(-0.384479\pi\)
0.355007 + 0.934864i \(0.384479\pi\)
\(284\) 29.8328 1.77025
\(285\) 2.85410 0.169062
\(286\) 0 0
\(287\) −11.1803 −0.659955
\(288\) 1.58359 0.0933141
\(289\) −7.45085 −0.438285
\(290\) 1.76393 0.103582
\(291\) −11.3262 −0.663956
\(292\) 26.3951 1.54466
\(293\) −11.0000 −0.642627 −0.321313 0.946973i \(-0.604124\pi\)
−0.321313 + 0.946973i \(0.604124\pi\)
\(294\) 0.618034 0.0360445
\(295\) 11.0902 0.645695
\(296\) 9.52786 0.553796
\(297\) 0 0
\(298\) 3.38197 0.195912
\(299\) −6.29180 −0.363864
\(300\) −12.0000 −0.692820
\(301\) −12.5623 −0.724079
\(302\) −0.0212862 −0.00122489
\(303\) 23.9443 1.37556
\(304\) −5.54915 −0.318266
\(305\) −7.61803 −0.436207
\(306\) 0.450850 0.0257734
\(307\) −23.1803 −1.32297 −0.661486 0.749958i \(-0.730074\pi\)
−0.661486 + 0.749958i \(0.730074\pi\)
\(308\) 0 0
\(309\) −14.3262 −0.814991
\(310\) 1.61803 0.0918982
\(311\) 9.00000 0.510343 0.255172 0.966896i \(-0.417868\pi\)
0.255172 + 0.966896i \(0.417868\pi\)
\(312\) 2.94427 0.166687
\(313\) 18.3262 1.03586 0.517930 0.855423i \(-0.326703\pi\)
0.517930 + 0.855423i \(0.326703\pi\)
\(314\) −7.59675 −0.428709
\(315\) −0.381966 −0.0215213
\(316\) −11.8328 −0.665648
\(317\) −4.43769 −0.249246 −0.124623 0.992204i \(-0.539772\pi\)
−0.124623 + 0.992204i \(0.539772\pi\)
\(318\) −1.47214 −0.0825533
\(319\) 0 0
\(320\) −4.70820 −0.263197
\(321\) −4.38197 −0.244577
\(322\) −1.94427 −0.108350
\(323\) −5.45085 −0.303293
\(324\) 14.2918 0.793989
\(325\) 4.94427 0.274259
\(326\) 3.32624 0.184223
\(327\) 2.47214 0.136709
\(328\) −16.4590 −0.908795
\(329\) −6.61803 −0.364864
\(330\) 0 0
\(331\) 6.18034 0.339702 0.169851 0.985470i \(-0.445671\pi\)
0.169851 + 0.985470i \(0.445671\pi\)
\(332\) −5.02129 −0.275579
\(333\) −2.47214 −0.135472
\(334\) −2.47214 −0.135269
\(335\) −8.32624 −0.454911
\(336\) −5.09017 −0.277692
\(337\) 25.9443 1.41327 0.706637 0.707576i \(-0.250211\pi\)
0.706637 + 0.707576i \(0.250211\pi\)
\(338\) 4.38197 0.238348
\(339\) −0.236068 −0.0128215
\(340\) −5.72949 −0.310725
\(341\) 0 0
\(342\) −0.257354 −0.0139161
\(343\) 1.00000 0.0539949
\(344\) −18.4934 −0.997099
\(345\) −8.23607 −0.443415
\(346\) 5.87539 0.315863
\(347\) 2.34752 0.126022 0.0630108 0.998013i \(-0.479930\pi\)
0.0630108 + 0.998013i \(0.479930\pi\)
\(348\) −13.8541 −0.742658
\(349\) 32.7984 1.75566 0.877828 0.478975i \(-0.158992\pi\)
0.877828 + 0.478975i \(0.158992\pi\)
\(350\) 1.52786 0.0816678
\(351\) −6.76393 −0.361032
\(352\) 0 0
\(353\) 24.0902 1.28219 0.641095 0.767461i \(-0.278480\pi\)
0.641095 + 0.767461i \(0.278480\pi\)
\(354\) 6.85410 0.364291
\(355\) −16.0902 −0.853978
\(356\) −12.7082 −0.673533
\(357\) −5.00000 −0.264628
\(358\) 1.43769 0.0759845
\(359\) −15.6180 −0.824288 −0.412144 0.911119i \(-0.635220\pi\)
−0.412144 + 0.911119i \(0.635220\pi\)
\(360\) −0.562306 −0.0296361
\(361\) −15.8885 −0.836239
\(362\) 2.83282 0.148889
\(363\) 0 0
\(364\) 2.29180 0.120123
\(365\) −14.2361 −0.745150
\(366\) −4.70820 −0.246102
\(367\) 32.8328 1.71386 0.856930 0.515434i \(-0.172369\pi\)
0.856930 + 0.515434i \(0.172369\pi\)
\(368\) 16.0132 0.834743
\(369\) 4.27051 0.222314
\(370\) −2.47214 −0.128520
\(371\) −2.38197 −0.123666
\(372\) −12.7082 −0.658890
\(373\) −15.4377 −0.799334 −0.399667 0.916661i \(-0.630874\pi\)
−0.399667 + 0.916661i \(0.630874\pi\)
\(374\) 0 0
\(375\) 14.5623 0.751994
\(376\) −9.74265 −0.502439
\(377\) 5.70820 0.293987
\(378\) −2.09017 −0.107507
\(379\) 32.0689 1.64727 0.823634 0.567122i \(-0.191943\pi\)
0.823634 + 0.567122i \(0.191943\pi\)
\(380\) 3.27051 0.167774
\(381\) −4.76393 −0.244064
\(382\) −6.02129 −0.308076
\(383\) −34.5967 −1.76781 −0.883906 0.467665i \(-0.845095\pi\)
−0.883906 + 0.467665i \(0.845095\pi\)
\(384\) −16.3262 −0.833145
\(385\) 0 0
\(386\) 5.85410 0.297966
\(387\) 4.79837 0.243915
\(388\) −12.9787 −0.658894
\(389\) −34.1803 −1.73301 −0.866506 0.499167i \(-0.833640\pi\)
−0.866506 + 0.499167i \(0.833640\pi\)
\(390\) −0.763932 −0.0386832
\(391\) 15.7295 0.795475
\(392\) 1.47214 0.0743541
\(393\) −30.6525 −1.54621
\(394\) 1.63932 0.0825878
\(395\) 6.38197 0.321112
\(396\) 0 0
\(397\) −0.819660 −0.0411376 −0.0205688 0.999788i \(-0.506548\pi\)
−0.0205688 + 0.999788i \(0.506548\pi\)
\(398\) 3.14590 0.157690
\(399\) 2.85410 0.142884
\(400\) −12.5836 −0.629180
\(401\) −17.5279 −0.875300 −0.437650 0.899145i \(-0.644189\pi\)
−0.437650 + 0.899145i \(0.644189\pi\)
\(402\) −5.14590 −0.256654
\(403\) 5.23607 0.260827
\(404\) 27.4377 1.36508
\(405\) −7.70820 −0.383024
\(406\) 1.76393 0.0875425
\(407\) 0 0
\(408\) −7.36068 −0.364408
\(409\) −17.7082 −0.875614 −0.437807 0.899069i \(-0.644245\pi\)
−0.437807 + 0.899069i \(0.644245\pi\)
\(410\) 4.27051 0.210905
\(411\) 24.7984 1.22321
\(412\) −16.4164 −0.808778
\(413\) 11.0902 0.545712
\(414\) 0.742646 0.0364990
\(415\) 2.70820 0.132941
\(416\) 5.12461 0.251255
\(417\) −19.3262 −0.946410
\(418\) 0 0
\(419\) 29.8885 1.46015 0.730075 0.683367i \(-0.239485\pi\)
0.730075 + 0.683367i \(0.239485\pi\)
\(420\) 3.00000 0.146385
\(421\) 0.763932 0.0372318 0.0186159 0.999827i \(-0.494074\pi\)
0.0186159 + 0.999827i \(0.494074\pi\)
\(422\) −5.96556 −0.290399
\(423\) 2.52786 0.122909
\(424\) −3.50658 −0.170294
\(425\) −12.3607 −0.599581
\(426\) −9.94427 −0.481802
\(427\) −7.61803 −0.368663
\(428\) −5.02129 −0.242713
\(429\) 0 0
\(430\) 4.79837 0.231398
\(431\) −13.8541 −0.667329 −0.333664 0.942692i \(-0.608285\pi\)
−0.333664 + 0.942692i \(0.608285\pi\)
\(432\) 17.2148 0.828247
\(433\) −8.96556 −0.430857 −0.215429 0.976520i \(-0.569115\pi\)
−0.215429 + 0.976520i \(0.569115\pi\)
\(434\) 1.61803 0.0776681
\(435\) 7.47214 0.358261
\(436\) 2.83282 0.135667
\(437\) −8.97871 −0.429510
\(438\) −8.79837 −0.420403
\(439\) 3.38197 0.161412 0.0807062 0.996738i \(-0.474282\pi\)
0.0807062 + 0.996738i \(0.474282\pi\)
\(440\) 0 0
\(441\) −0.381966 −0.0181889
\(442\) 1.45898 0.0693966
\(443\) 3.09017 0.146818 0.0734092 0.997302i \(-0.476612\pi\)
0.0734092 + 0.997302i \(0.476612\pi\)
\(444\) 19.4164 0.921462
\(445\) 6.85410 0.324916
\(446\) −0.777088 −0.0367962
\(447\) 14.3262 0.677608
\(448\) −4.70820 −0.222442
\(449\) 12.3262 0.581711 0.290856 0.956767i \(-0.406060\pi\)
0.290856 + 0.956767i \(0.406060\pi\)
\(450\) −0.583592 −0.0275108
\(451\) 0 0
\(452\) −0.270510 −0.0127237
\(453\) −0.0901699 −0.00423655
\(454\) 6.12461 0.287442
\(455\) −1.23607 −0.0579478
\(456\) 4.20163 0.196759
\(457\) −0.145898 −0.00682482 −0.00341241 0.999994i \(-0.501086\pi\)
−0.00341241 + 0.999994i \(0.501086\pi\)
\(458\) −2.58359 −0.120723
\(459\) 16.9098 0.789283
\(460\) −9.43769 −0.440035
\(461\) −18.5066 −0.861937 −0.430969 0.902367i \(-0.641828\pi\)
−0.430969 + 0.902367i \(0.641828\pi\)
\(462\) 0 0
\(463\) 22.2361 1.03340 0.516699 0.856167i \(-0.327161\pi\)
0.516699 + 0.856167i \(0.327161\pi\)
\(464\) −14.5279 −0.674439
\(465\) 6.85410 0.317851
\(466\) 11.2361 0.520501
\(467\) −4.50658 −0.208540 −0.104270 0.994549i \(-0.533251\pi\)
−0.104270 + 0.994549i \(0.533251\pi\)
\(468\) −0.875388 −0.0404648
\(469\) −8.32624 −0.384470
\(470\) 2.52786 0.116602
\(471\) −32.1803 −1.48279
\(472\) 16.3262 0.751476
\(473\) 0 0
\(474\) 3.94427 0.181166
\(475\) 7.05573 0.323739
\(476\) −5.72949 −0.262611
\(477\) 0.909830 0.0416583
\(478\) 6.52786 0.298578
\(479\) 21.7639 0.994419 0.497210 0.867630i \(-0.334358\pi\)
0.497210 + 0.867630i \(0.334358\pi\)
\(480\) 6.70820 0.306186
\(481\) −8.00000 −0.364769
\(482\) −6.21478 −0.283076
\(483\) −8.23607 −0.374754
\(484\) 0 0
\(485\) 7.00000 0.317854
\(486\) 1.50658 0.0683398
\(487\) −32.5967 −1.47710 −0.738550 0.674199i \(-0.764489\pi\)
−0.738550 + 0.674199i \(0.764489\pi\)
\(488\) −11.2148 −0.507669
\(489\) 14.0902 0.637180
\(490\) −0.381966 −0.0172555
\(491\) 22.1459 0.999430 0.499715 0.866190i \(-0.333438\pi\)
0.499715 + 0.866190i \(0.333438\pi\)
\(492\) −33.5410 −1.51215
\(493\) −14.2705 −0.642711
\(494\) −0.832816 −0.0374702
\(495\) 0 0
\(496\) −13.3262 −0.598366
\(497\) −16.0902 −0.721743
\(498\) 1.67376 0.0750031
\(499\) −10.0902 −0.451698 −0.225849 0.974162i \(-0.572516\pi\)
−0.225849 + 0.974162i \(0.572516\pi\)
\(500\) 16.6869 0.746262
\(501\) −10.4721 −0.467861
\(502\) −8.78522 −0.392103
\(503\) −34.8328 −1.55312 −0.776559 0.630044i \(-0.783037\pi\)
−0.776559 + 0.630044i \(0.783037\pi\)
\(504\) −0.562306 −0.0250471
\(505\) −14.7984 −0.658519
\(506\) 0 0
\(507\) 18.5623 0.824381
\(508\) −5.45898 −0.242203
\(509\) −10.7426 −0.476159 −0.238080 0.971246i \(-0.576518\pi\)
−0.238080 + 0.971246i \(0.576518\pi\)
\(510\) 1.90983 0.0845687
\(511\) −14.2361 −0.629767
\(512\) −22.3050 −0.985749
\(513\) −9.65248 −0.426167
\(514\) 3.27051 0.144256
\(515\) 8.85410 0.390158
\(516\) −37.6869 −1.65907
\(517\) 0 0
\(518\) −2.47214 −0.108619
\(519\) 24.8885 1.09249
\(520\) −1.81966 −0.0797974
\(521\) −12.2361 −0.536072 −0.268036 0.963409i \(-0.586375\pi\)
−0.268036 + 0.963409i \(0.586375\pi\)
\(522\) −0.673762 −0.0294898
\(523\) 16.5623 0.724219 0.362110 0.932136i \(-0.382057\pi\)
0.362110 + 0.932136i \(0.382057\pi\)
\(524\) −35.1246 −1.53443
\(525\) 6.47214 0.282467
\(526\) 6.54102 0.285202
\(527\) −13.0902 −0.570217
\(528\) 0 0
\(529\) 2.90983 0.126514
\(530\) 0.909830 0.0395205
\(531\) −4.23607 −0.183830
\(532\) 3.27051 0.141795
\(533\) 13.8197 0.598596
\(534\) 4.23607 0.183313
\(535\) 2.70820 0.117086
\(536\) −12.2574 −0.529437
\(537\) 6.09017 0.262810
\(538\) −6.43769 −0.277549
\(539\) 0 0
\(540\) −10.1459 −0.436610
\(541\) 0.708204 0.0304481 0.0152240 0.999884i \(-0.495154\pi\)
0.0152240 + 0.999884i \(0.495154\pi\)
\(542\) 1.83282 0.0787262
\(543\) 12.0000 0.514969
\(544\) −12.8115 −0.549290
\(545\) −1.52786 −0.0654465
\(546\) −0.763932 −0.0326933
\(547\) −1.72949 −0.0739477 −0.0369738 0.999316i \(-0.511772\pi\)
−0.0369738 + 0.999316i \(0.511772\pi\)
\(548\) 28.4164 1.21389
\(549\) 2.90983 0.124189
\(550\) 0 0
\(551\) 8.14590 0.347027
\(552\) −12.1246 −0.516058
\(553\) 6.38197 0.271389
\(554\) −8.79837 −0.373807
\(555\) −10.4721 −0.444517
\(556\) −22.1459 −0.939195
\(557\) −9.76393 −0.413711 −0.206856 0.978371i \(-0.566323\pi\)
−0.206856 + 0.978371i \(0.566323\pi\)
\(558\) −0.618034 −0.0261635
\(559\) 15.5279 0.656759
\(560\) 3.14590 0.132938
\(561\) 0 0
\(562\) 9.61803 0.405712
\(563\) −31.9787 −1.34774 −0.673871 0.738849i \(-0.735370\pi\)
−0.673871 + 0.738849i \(0.735370\pi\)
\(564\) −19.8541 −0.836009
\(565\) 0.145898 0.00613798
\(566\) −4.56231 −0.191768
\(567\) −7.70820 −0.323714
\(568\) −23.6869 −0.993881
\(569\) 37.3050 1.56390 0.781952 0.623338i \(-0.214224\pi\)
0.781952 + 0.623338i \(0.214224\pi\)
\(570\) −1.09017 −0.0456622
\(571\) 34.3050 1.43562 0.717809 0.696240i \(-0.245145\pi\)
0.717809 + 0.696240i \(0.245145\pi\)
\(572\) 0 0
\(573\) −25.5066 −1.06555
\(574\) 4.27051 0.178248
\(575\) −20.3607 −0.849099
\(576\) 1.79837 0.0749322
\(577\) 9.65248 0.401838 0.200919 0.979608i \(-0.435607\pi\)
0.200919 + 0.979608i \(0.435607\pi\)
\(578\) 2.84597 0.118377
\(579\) 24.7984 1.03059
\(580\) 8.56231 0.355530
\(581\) 2.70820 0.112355
\(582\) 4.32624 0.179328
\(583\) 0 0
\(584\) −20.9574 −0.867225
\(585\) 0.472136 0.0195204
\(586\) 4.20163 0.173568
\(587\) 1.00000 0.0412744 0.0206372 0.999787i \(-0.493431\pi\)
0.0206372 + 0.999787i \(0.493431\pi\)
\(588\) 3.00000 0.123718
\(589\) 7.47214 0.307884
\(590\) −4.23607 −0.174396
\(591\) 6.94427 0.285649
\(592\) 20.3607 0.836819
\(593\) 11.1246 0.456833 0.228417 0.973564i \(-0.426645\pi\)
0.228417 + 0.973564i \(0.426645\pi\)
\(594\) 0 0
\(595\) 3.09017 0.126685
\(596\) 16.4164 0.672442
\(597\) 13.3262 0.545407
\(598\) 2.40325 0.0982763
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 9.52786 0.388973
\(601\) 13.7082 0.559169 0.279585 0.960121i \(-0.409803\pi\)
0.279585 + 0.960121i \(0.409803\pi\)
\(602\) 4.79837 0.195567
\(603\) 3.18034 0.129513
\(604\) −0.103326 −0.00420426
\(605\) 0 0
\(606\) −9.14590 −0.371527
\(607\) 20.2016 0.819959 0.409979 0.912095i \(-0.365536\pi\)
0.409979 + 0.912095i \(0.365536\pi\)
\(608\) 7.31308 0.296585
\(609\) 7.47214 0.302786
\(610\) 2.90983 0.117816
\(611\) 8.18034 0.330941
\(612\) 2.18847 0.0884637
\(613\) −0.888544 −0.0358879 −0.0179440 0.999839i \(-0.505712\pi\)
−0.0179440 + 0.999839i \(0.505712\pi\)
\(614\) 8.85410 0.357322
\(615\) 18.0902 0.729466
\(616\) 0 0
\(617\) 9.41641 0.379090 0.189545 0.981872i \(-0.439299\pi\)
0.189545 + 0.981872i \(0.439299\pi\)
\(618\) 5.47214 0.220122
\(619\) −39.7082 −1.59601 −0.798004 0.602653i \(-0.794111\pi\)
−0.798004 + 0.602653i \(0.794111\pi\)
\(620\) 7.85410 0.315428
\(621\) 27.8541 1.11775
\(622\) −3.43769 −0.137839
\(623\) 6.85410 0.274604
\(624\) 6.29180 0.251873
\(625\) 11.0000 0.440000
\(626\) −7.00000 −0.279776
\(627\) 0 0
\(628\) −36.8754 −1.47149
\(629\) 20.0000 0.797452
\(630\) 0.145898 0.00581272
\(631\) 17.8328 0.709913 0.354957 0.934883i \(-0.384496\pi\)
0.354957 + 0.934883i \(0.384496\pi\)
\(632\) 9.39512 0.373718
\(633\) −25.2705 −1.00441
\(634\) 1.69505 0.0673190
\(635\) 2.94427 0.116840
\(636\) −7.14590 −0.283353
\(637\) −1.23607 −0.0489748
\(638\) 0 0
\(639\) 6.14590 0.243128
\(640\) 10.0902 0.398849
\(641\) 31.6525 1.25020 0.625099 0.780546i \(-0.285059\pi\)
0.625099 + 0.780546i \(0.285059\pi\)
\(642\) 1.67376 0.0660581
\(643\) 1.58359 0.0624508 0.0312254 0.999512i \(-0.490059\pi\)
0.0312254 + 0.999512i \(0.490059\pi\)
\(644\) −9.43769 −0.371897
\(645\) 20.3262 0.800345
\(646\) 2.08204 0.0819167
\(647\) 6.18034 0.242974 0.121487 0.992593i \(-0.461234\pi\)
0.121487 + 0.992593i \(0.461234\pi\)
\(648\) −11.3475 −0.445773
\(649\) 0 0
\(650\) −1.88854 −0.0740748
\(651\) 6.85410 0.268633
\(652\) 16.1459 0.632322
\(653\) 18.2361 0.713632 0.356816 0.934175i \(-0.383862\pi\)
0.356816 + 0.934175i \(0.383862\pi\)
\(654\) −0.944272 −0.0369240
\(655\) 18.9443 0.740214
\(656\) −35.1722 −1.37324
\(657\) 5.43769 0.212145
\(658\) 2.52786 0.0985464
\(659\) 31.4721 1.22598 0.612990 0.790091i \(-0.289967\pi\)
0.612990 + 0.790091i \(0.289967\pi\)
\(660\) 0 0
\(661\) −34.5623 −1.34432 −0.672159 0.740407i \(-0.734633\pi\)
−0.672159 + 0.740407i \(0.734633\pi\)
\(662\) −2.36068 −0.0917504
\(663\) 6.18034 0.240025
\(664\) 3.98684 0.154720
\(665\) −1.76393 −0.0684023
\(666\) 0.944272 0.0365898
\(667\) −23.5066 −0.910178
\(668\) −12.0000 −0.464294
\(669\) −3.29180 −0.127268
\(670\) 3.18034 0.122867
\(671\) 0 0
\(672\) 6.70820 0.258775
\(673\) −6.58359 −0.253779 −0.126889 0.991917i \(-0.540499\pi\)
−0.126889 + 0.991917i \(0.540499\pi\)
\(674\) −9.90983 −0.381712
\(675\) −21.8885 −0.842490
\(676\) 21.2705 0.818097
\(677\) −36.2705 −1.39399 −0.696994 0.717077i \(-0.745480\pi\)
−0.696994 + 0.717077i \(0.745480\pi\)
\(678\) 0.0901699 0.00346296
\(679\) 7.00000 0.268635
\(680\) 4.54915 0.174452
\(681\) 25.9443 0.994187
\(682\) 0 0
\(683\) 0.493422 0.0188803 0.00944014 0.999955i \(-0.496995\pi\)
0.00944014 + 0.999955i \(0.496995\pi\)
\(684\) −1.24922 −0.0477653
\(685\) −15.3262 −0.585585
\(686\) −0.381966 −0.0145835
\(687\) −10.9443 −0.417550
\(688\) −39.5197 −1.50668
\(689\) 2.94427 0.112168
\(690\) 3.14590 0.119762
\(691\) 18.5410 0.705334 0.352667 0.935749i \(-0.385275\pi\)
0.352667 + 0.935749i \(0.385275\pi\)
\(692\) 28.5197 1.08416
\(693\) 0 0
\(694\) −0.896674 −0.0340373
\(695\) 11.9443 0.453072
\(696\) 11.0000 0.416954
\(697\) −34.5492 −1.30864
\(698\) −12.5279 −0.474187
\(699\) 47.5967 1.80027
\(700\) 7.41641 0.280314
\(701\) 18.5066 0.698984 0.349492 0.936939i \(-0.386354\pi\)
0.349492 + 0.936939i \(0.386354\pi\)
\(702\) 2.58359 0.0975114
\(703\) −11.4164 −0.430578
\(704\) 0 0
\(705\) 10.7082 0.403294
\(706\) −9.20163 −0.346308
\(707\) −14.7984 −0.556550
\(708\) 33.2705 1.25038
\(709\) −7.03444 −0.264184 −0.132092 0.991237i \(-0.542169\pi\)
−0.132092 + 0.991237i \(0.542169\pi\)
\(710\) 6.14590 0.230651
\(711\) −2.43769 −0.0914207
\(712\) 10.0902 0.378145
\(713\) −21.5623 −0.807515
\(714\) 1.90983 0.0714736
\(715\) 0 0
\(716\) 6.97871 0.260807
\(717\) 27.6525 1.03270
\(718\) 5.96556 0.222633
\(719\) −2.88854 −0.107725 −0.0538623 0.998548i \(-0.517153\pi\)
−0.0538623 + 0.998548i \(0.517153\pi\)
\(720\) −1.20163 −0.0447820
\(721\) 8.85410 0.329744
\(722\) 6.06888 0.225860
\(723\) −26.3262 −0.979083
\(724\) 13.7508 0.511044
\(725\) 18.4721 0.686038
\(726\) 0 0
\(727\) −12.4508 −0.461776 −0.230888 0.972980i \(-0.574163\pi\)
−0.230888 + 0.972980i \(0.574163\pi\)
\(728\) −1.81966 −0.0674411
\(729\) 29.5066 1.09284
\(730\) 5.43769 0.201258
\(731\) −38.8197 −1.43580
\(732\) −22.8541 −0.844712
\(733\) 52.9443 1.95554 0.977771 0.209677i \(-0.0672413\pi\)
0.977771 + 0.209677i \(0.0672413\pi\)
\(734\) −12.5410 −0.462897
\(735\) −1.61803 −0.0596821
\(736\) −21.1033 −0.777879
\(737\) 0 0
\(738\) −1.63119 −0.0600449
\(739\) −11.3820 −0.418692 −0.209346 0.977842i \(-0.567134\pi\)
−0.209346 + 0.977842i \(0.567134\pi\)
\(740\) −12.0000 −0.441129
\(741\) −3.52786 −0.129599
\(742\) 0.909830 0.0334009
\(743\) −14.9098 −0.546989 −0.273494 0.961874i \(-0.588179\pi\)
−0.273494 + 0.961874i \(0.588179\pi\)
\(744\) 10.0902 0.369924
\(745\) −8.85410 −0.324389
\(746\) 5.89667 0.215893
\(747\) −1.03444 −0.0378482
\(748\) 0 0
\(749\) 2.70820 0.0989556
\(750\) −5.56231 −0.203107
\(751\) 41.3050 1.50724 0.753620 0.657311i \(-0.228306\pi\)
0.753620 + 0.657311i \(0.228306\pi\)
\(752\) −20.8197 −0.759215
\(753\) −37.2148 −1.35618
\(754\) −2.18034 −0.0794033
\(755\) 0.0557281 0.00202815
\(756\) −10.1459 −0.369003
\(757\) −14.7639 −0.536604 −0.268302 0.963335i \(-0.586463\pi\)
−0.268302 + 0.963335i \(0.586463\pi\)
\(758\) −12.2492 −0.444912
\(759\) 0 0
\(760\) −2.59675 −0.0941939
\(761\) 15.3050 0.554804 0.277402 0.960754i \(-0.410527\pi\)
0.277402 + 0.960754i \(0.410527\pi\)
\(762\) 1.81966 0.0659193
\(763\) −1.52786 −0.0553124
\(764\) −29.2279 −1.05743
\(765\) −1.18034 −0.0426753
\(766\) 13.2148 0.477469
\(767\) −13.7082 −0.494975
\(768\) −9.00000 −0.324760
\(769\) 48.5623 1.75120 0.875601 0.483035i \(-0.160466\pi\)
0.875601 + 0.483035i \(0.160466\pi\)
\(770\) 0 0
\(771\) 13.8541 0.498943
\(772\) 28.4164 1.02273
\(773\) 40.2361 1.44719 0.723595 0.690224i \(-0.242488\pi\)
0.723595 + 0.690224i \(0.242488\pi\)
\(774\) −1.83282 −0.0658792
\(775\) 16.9443 0.608656
\(776\) 10.3050 0.369926
\(777\) −10.4721 −0.375686
\(778\) 13.0557 0.468071
\(779\) 19.7214 0.706591
\(780\) −3.70820 −0.132775
\(781\) 0 0
\(782\) −6.00813 −0.214850
\(783\) −25.2705 −0.903094
\(784\) 3.14590 0.112354
\(785\) 19.8885 0.709853
\(786\) 11.7082 0.417618
\(787\) 20.5623 0.732967 0.366484 0.930425i \(-0.380562\pi\)
0.366484 + 0.930425i \(0.380562\pi\)
\(788\) 7.95743 0.283472
\(789\) 27.7082 0.986438
\(790\) −2.43769 −0.0867293
\(791\) 0.145898 0.00518754
\(792\) 0 0
\(793\) 9.41641 0.334386
\(794\) 0.313082 0.0111109
\(795\) 3.85410 0.136691
\(796\) 15.2705 0.541249
\(797\) 32.2361 1.14186 0.570930 0.820999i \(-0.306583\pi\)
0.570930 + 0.820999i \(0.306583\pi\)
\(798\) −1.09017 −0.0385916
\(799\) −20.4508 −0.723499
\(800\) 16.5836 0.586319
\(801\) −2.61803 −0.0925037
\(802\) 6.69505 0.236410
\(803\) 0 0
\(804\) −24.9787 −0.880931
\(805\) 5.09017 0.179405
\(806\) −2.00000 −0.0704470
\(807\) −27.2705 −0.959967
\(808\) −21.7852 −0.766401
\(809\) 32.5066 1.14287 0.571435 0.820647i \(-0.306387\pi\)
0.571435 + 0.820647i \(0.306387\pi\)
\(810\) 2.94427 0.103451
\(811\) 1.16718 0.0409854 0.0204927 0.999790i \(-0.493477\pi\)
0.0204927 + 0.999790i \(0.493477\pi\)
\(812\) 8.56231 0.300478
\(813\) 7.76393 0.272293
\(814\) 0 0
\(815\) −8.70820 −0.305035
\(816\) −15.7295 −0.550642
\(817\) 22.1591 0.775247
\(818\) 6.76393 0.236495
\(819\) 0.472136 0.0164978
\(820\) 20.7295 0.723905
\(821\) 56.6656 1.97764 0.988822 0.149100i \(-0.0476377\pi\)
0.988822 + 0.149100i \(0.0476377\pi\)
\(822\) −9.47214 −0.330379
\(823\) −31.8885 −1.11156 −0.555782 0.831328i \(-0.687581\pi\)
−0.555782 + 0.831328i \(0.687581\pi\)
\(824\) 13.0344 0.454076
\(825\) 0 0
\(826\) −4.23607 −0.147392
\(827\) −19.5967 −0.681446 −0.340723 0.940164i \(-0.610672\pi\)
−0.340723 + 0.940164i \(0.610672\pi\)
\(828\) 3.60488 0.125278
\(829\) −47.9787 −1.66637 −0.833185 0.552995i \(-0.813485\pi\)
−0.833185 + 0.552995i \(0.813485\pi\)
\(830\) −1.03444 −0.0359060
\(831\) −37.2705 −1.29290
\(832\) 5.81966 0.201760
\(833\) 3.09017 0.107068
\(834\) 7.38197 0.255617
\(835\) 6.47214 0.223978
\(836\) 0 0
\(837\) −23.1803 −0.801230
\(838\) −11.4164 −0.394373
\(839\) 30.5967 1.05632 0.528159 0.849146i \(-0.322883\pi\)
0.528159 + 0.849146i \(0.322883\pi\)
\(840\) −2.38197 −0.0821856
\(841\) −7.67376 −0.264612
\(842\) −0.291796 −0.0100560
\(843\) 40.7426 1.40325
\(844\) −28.9574 −0.996756
\(845\) −11.4721 −0.394653
\(846\) −0.965558 −0.0331966
\(847\) 0 0
\(848\) −7.49342 −0.257325
\(849\) −19.3262 −0.663275
\(850\) 4.72136 0.161941
\(851\) 32.9443 1.12932
\(852\) −48.2705 −1.65372
\(853\) −34.9098 −1.19529 −0.597645 0.801761i \(-0.703897\pi\)
−0.597645 + 0.801761i \(0.703897\pi\)
\(854\) 2.90983 0.0995723
\(855\) 0.673762 0.0230422
\(856\) 3.98684 0.136268
\(857\) −15.0902 −0.515470 −0.257735 0.966216i \(-0.582976\pi\)
−0.257735 + 0.966216i \(0.582976\pi\)
\(858\) 0 0
\(859\) 21.5623 0.735696 0.367848 0.929886i \(-0.380095\pi\)
0.367848 + 0.929886i \(0.380095\pi\)
\(860\) 23.2918 0.794244
\(861\) 18.0902 0.616511
\(862\) 5.29180 0.180239
\(863\) −47.9443 −1.63204 −0.816021 0.578022i \(-0.803825\pi\)
−0.816021 + 0.578022i \(0.803825\pi\)
\(864\) −22.6869 −0.771825
\(865\) −15.3820 −0.523003
\(866\) 3.42454 0.116371
\(867\) 12.0557 0.409434
\(868\) 7.85410 0.266586
\(869\) 0 0
\(870\) −2.85410 −0.0967631
\(871\) 10.2918 0.348724
\(872\) −2.24922 −0.0761683
\(873\) −2.67376 −0.0904931
\(874\) 3.42956 0.116007
\(875\) −9.00000 −0.304256
\(876\) −42.7082 −1.44298
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) −1.29180 −0.0435960
\(879\) 17.7984 0.600324
\(880\) 0 0
\(881\) 14.1459 0.476587 0.238294 0.971193i \(-0.423412\pi\)
0.238294 + 0.971193i \(0.423412\pi\)
\(882\) 0.145898 0.00491264
\(883\) 31.6312 1.06447 0.532237 0.846595i \(-0.321351\pi\)
0.532237 + 0.846595i \(0.321351\pi\)
\(884\) 7.08204 0.238195
\(885\) −17.9443 −0.603190
\(886\) −1.18034 −0.0396543
\(887\) 57.2148 1.92108 0.960542 0.278134i \(-0.0897161\pi\)
0.960542 + 0.278134i \(0.0897161\pi\)
\(888\) −15.4164 −0.517341
\(889\) 2.94427 0.0987477
\(890\) −2.61803 −0.0877567
\(891\) 0 0
\(892\) −3.77206 −0.126298
\(893\) 11.6738 0.390648
\(894\) −5.47214 −0.183016
\(895\) −3.76393 −0.125814
\(896\) 10.0902 0.337089
\(897\) 10.1803 0.339912
\(898\) −4.70820 −0.157115
\(899\) 19.5623 0.652439
\(900\) −2.83282 −0.0944272
\(901\) −7.36068 −0.245220
\(902\) 0 0
\(903\) 20.3262 0.676415
\(904\) 0.214782 0.00714353
\(905\) −7.41641 −0.246530
\(906\) 0.0344419 0.00114425
\(907\) −15.7639 −0.523433 −0.261716 0.965145i \(-0.584289\pi\)
−0.261716 + 0.965145i \(0.584289\pi\)
\(908\) 29.7295 0.986608
\(909\) 5.65248 0.187481
\(910\) 0.472136 0.0156512
\(911\) −45.1803 −1.49689 −0.748446 0.663196i \(-0.769200\pi\)
−0.748446 + 0.663196i \(0.769200\pi\)
\(912\) 8.97871 0.297315
\(913\) 0 0
\(914\) 0.0557281 0.00184332
\(915\) 12.3262 0.407493
\(916\) −12.5410 −0.414367
\(917\) 18.9443 0.625595
\(918\) −6.45898 −0.213178
\(919\) −14.8885 −0.491128 −0.245564 0.969380i \(-0.578973\pi\)
−0.245564 + 0.969380i \(0.578973\pi\)
\(920\) 7.49342 0.247051
\(921\) 37.5066 1.23588
\(922\) 7.06888 0.232801
\(923\) 19.8885 0.654639
\(924\) 0 0
\(925\) −25.8885 −0.851210
\(926\) −8.49342 −0.279111
\(927\) −3.38197 −0.111078
\(928\) 19.1459 0.628495
\(929\) 24.0000 0.787414 0.393707 0.919236i \(-0.371192\pi\)
0.393707 + 0.919236i \(0.371192\pi\)
\(930\) −2.61803 −0.0858487
\(931\) −1.76393 −0.0578105
\(932\) 54.5410 1.78655
\(933\) −14.5623 −0.476748
\(934\) 1.72136 0.0563246
\(935\) 0 0
\(936\) 0.695048 0.0227184
\(937\) −51.7214 −1.68966 −0.844832 0.535032i \(-0.820299\pi\)
−0.844832 + 0.535032i \(0.820299\pi\)
\(938\) 3.18034 0.103842
\(939\) −29.6525 −0.967672
\(940\) 12.2705 0.400220
\(941\) 27.0689 0.882420 0.441210 0.897404i \(-0.354549\pi\)
0.441210 + 0.897404i \(0.354549\pi\)
\(942\) 12.2918 0.400488
\(943\) −56.9098 −1.85324
\(944\) 34.8885 1.13553
\(945\) 5.47214 0.178009
\(946\) 0 0
\(947\) 6.29180 0.204456 0.102228 0.994761i \(-0.467403\pi\)
0.102228 + 0.994761i \(0.467403\pi\)
\(948\) 19.1459 0.621830
\(949\) 17.5967 0.571215
\(950\) −2.69505 −0.0874389
\(951\) 7.18034 0.232838
\(952\) 4.54915 0.147439
\(953\) −6.23607 −0.202006 −0.101003 0.994886i \(-0.532205\pi\)
−0.101003 + 0.994886i \(0.532205\pi\)
\(954\) −0.347524 −0.0112515
\(955\) 15.7639 0.510109
\(956\) 31.6869 1.02483
\(957\) 0 0
\(958\) −8.31308 −0.268583
\(959\) −15.3262 −0.494910
\(960\) 7.61803 0.245871
\(961\) −13.0557 −0.421153
\(962\) 3.05573 0.0985206
\(963\) −1.03444 −0.0333344
\(964\) −30.1672 −0.971620
\(965\) −15.3262 −0.493369
\(966\) 3.14590 0.101218
\(967\) 27.4508 0.882760 0.441380 0.897320i \(-0.354489\pi\)
0.441380 + 0.897320i \(0.354489\pi\)
\(968\) 0 0
\(969\) 8.81966 0.283328
\(970\) −2.67376 −0.0858493
\(971\) −28.2492 −0.906561 −0.453280 0.891368i \(-0.649746\pi\)
−0.453280 + 0.891368i \(0.649746\pi\)
\(972\) 7.31308 0.234567
\(973\) 11.9443 0.382916
\(974\) 12.4508 0.398951
\(975\) −8.00000 −0.256205
\(976\) −23.9656 −0.767119
\(977\) −10.8197 −0.346152 −0.173076 0.984909i \(-0.555371\pi\)
−0.173076 + 0.984909i \(0.555371\pi\)
\(978\) −5.38197 −0.172096
\(979\) 0 0
\(980\) −1.85410 −0.0592271
\(981\) 0.583592 0.0186327
\(982\) −8.45898 −0.269937
\(983\) −12.3820 −0.394923 −0.197462 0.980311i \(-0.563270\pi\)
−0.197462 + 0.980311i \(0.563270\pi\)
\(984\) 26.6312 0.848971
\(985\) −4.29180 −0.136748
\(986\) 5.45085 0.173590
\(987\) 10.7082 0.340846
\(988\) −4.04257 −0.128611
\(989\) −63.9443 −2.03331
\(990\) 0 0
\(991\) −0.729490 −0.0231730 −0.0115865 0.999933i \(-0.503688\pi\)
−0.0115865 + 0.999933i \(0.503688\pi\)
\(992\) 17.5623 0.557604
\(993\) −10.0000 −0.317340
\(994\) 6.14590 0.194936
\(995\) −8.23607 −0.261101
\(996\) 8.12461 0.257438
\(997\) −55.8673 −1.76933 −0.884667 0.466224i \(-0.845614\pi\)
−0.884667 + 0.466224i \(0.845614\pi\)
\(998\) 3.85410 0.121999
\(999\) 35.4164 1.12053
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 847.2.a.d.1.2 2
3.2 odd 2 7623.2.a.bx.1.1 2
7.6 odd 2 5929.2.a.i.1.2 2
11.2 odd 10 847.2.f.j.323.1 4
11.3 even 5 847.2.f.l.372.1 4
11.4 even 5 847.2.f.l.148.1 4
11.5 even 5 847.2.f.d.729.1 4
11.6 odd 10 847.2.f.j.729.1 4
11.7 odd 10 847.2.f.c.148.1 4
11.8 odd 10 847.2.f.c.372.1 4
11.9 even 5 847.2.f.d.323.1 4
11.10 odd 2 847.2.a.h.1.1 yes 2
33.32 even 2 7623.2.a.t.1.2 2
77.76 even 2 5929.2.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.d.1.2 2 1.1 even 1 trivial
847.2.a.h.1.1 yes 2 11.10 odd 2
847.2.f.c.148.1 4 11.7 odd 10
847.2.f.c.372.1 4 11.8 odd 10
847.2.f.d.323.1 4 11.9 even 5
847.2.f.d.729.1 4 11.5 even 5
847.2.f.j.323.1 4 11.2 odd 10
847.2.f.j.729.1 4 11.6 odd 10
847.2.f.l.148.1 4 11.4 even 5
847.2.f.l.372.1 4 11.3 even 5
5929.2.a.i.1.2 2 7.6 odd 2
5929.2.a.s.1.1 2 77.76 even 2
7623.2.a.t.1.2 2 33.32 even 2
7623.2.a.bx.1.1 2 3.2 odd 2