Properties

Label 161.2.a.b.1.2
Level 161
Weight 2
Character 161.1
Self dual yes
Analytic conductor 1.286
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 161.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.28559147254\)
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 \(\chi\) \(=\) 161.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -3.23607 q^{5} -0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -3.23607 q^{5} -0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} -2.00000 q^{9} -2.00000 q^{10} +4.47214 q^{11} +1.61803 q^{12} +0.236068 q^{13} -0.618034 q^{14} +3.23607 q^{15} +1.85410 q^{16} -1.23607 q^{18} -7.23607 q^{19} +5.23607 q^{20} +1.00000 q^{21} +2.76393 q^{22} -1.00000 q^{23} +2.23607 q^{24} +5.47214 q^{25} +0.145898 q^{26} +5.00000 q^{27} +1.61803 q^{28} -1.47214 q^{29} +2.00000 q^{30} -9.00000 q^{31} +5.61803 q^{32} -4.47214 q^{33} +3.23607 q^{35} +3.23607 q^{36} -5.70820 q^{37} -4.47214 q^{38} -0.236068 q^{39} +7.23607 q^{40} -2.23607 q^{41} +0.618034 q^{42} +2.47214 q^{43} -7.23607 q^{44} +6.47214 q^{45} -0.618034 q^{46} -3.47214 q^{47} -1.85410 q^{48} +1.00000 q^{49} +3.38197 q^{50} -0.381966 q^{52} +11.2361 q^{53} +3.09017 q^{54} -14.4721 q^{55} +2.23607 q^{56} +7.23607 q^{57} -0.909830 q^{58} -1.52786 q^{59} -5.23607 q^{60} +13.4164 q^{61} -5.56231 q^{62} +2.00000 q^{63} -0.236068 q^{64} -0.763932 q^{65} -2.76393 q^{66} -12.1803 q^{67} +1.00000 q^{69} +2.00000 q^{70} -10.2361 q^{71} +4.47214 q^{72} -6.70820 q^{73} -3.52786 q^{74} -5.47214 q^{75} +11.7082 q^{76} -4.47214 q^{77} -0.145898 q^{78} -7.23607 q^{79} -6.00000 q^{80} +1.00000 q^{81} -1.38197 q^{82} +6.47214 q^{83} -1.61803 q^{84} +1.52786 q^{86} +1.47214 q^{87} -10.0000 q^{88} +8.94427 q^{89} +4.00000 q^{90} -0.236068 q^{91} +1.61803 q^{92} +9.00000 q^{93} -2.14590 q^{94} +23.4164 q^{95} -5.61803 q^{96} +3.70820 q^{97} +0.618034 q^{98} -8.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 2q^{3} - q^{4} - 2q^{5} + q^{6} - 2q^{7} - 4q^{9} + O(q^{10}) \) \( 2q - q^{2} - 2q^{3} - q^{4} - 2q^{5} + q^{6} - 2q^{7} - 4q^{9} - 4q^{10} + q^{12} - 4q^{13} + q^{14} + 2q^{15} - 3q^{16} + 2q^{18} - 10q^{19} + 6q^{20} + 2q^{21} + 10q^{22} - 2q^{23} + 2q^{25} + 7q^{26} + 10q^{27} + q^{28} + 6q^{29} + 4q^{30} - 18q^{31} + 9q^{32} + 2q^{35} + 2q^{36} + 2q^{37} + 4q^{39} + 10q^{40} - q^{42} - 4q^{43} - 10q^{44} + 4q^{45} + q^{46} + 2q^{47} + 3q^{48} + 2q^{49} + 9q^{50} - 3q^{52} + 18q^{53} - 5q^{54} - 20q^{55} + 10q^{57} - 13q^{58} - 12q^{59} - 6q^{60} + 9q^{62} + 4q^{63} + 4q^{64} - 6q^{65} - 10q^{66} - 2q^{67} + 2q^{69} + 4q^{70} - 16q^{71} - 16q^{74} - 2q^{75} + 10q^{76} - 7q^{78} - 10q^{79} - 12q^{80} + 2q^{81} - 5q^{82} + 4q^{83} - q^{84} + 12q^{86} - 6q^{87} - 20q^{88} + 8q^{90} + 4q^{91} + q^{92} + 18q^{93} - 11q^{94} + 20q^{95} - 9q^{96} - 6q^{97} - q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.437016 0.218508 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) −1.61803 −0.809017
\(5\) −3.23607 −1.44721 −0.723607 0.690212i \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) −0.618034 −0.252311
\(7\) −1.00000 −0.377964
\(8\) −2.23607 −0.790569
\(9\) −2.00000 −0.666667
\(10\) −2.00000 −0.632456
\(11\) 4.47214 1.34840 0.674200 0.738549i \(-0.264489\pi\)
0.674200 + 0.738549i \(0.264489\pi\)
\(12\) 1.61803 0.467086
\(13\) 0.236068 0.0654735 0.0327367 0.999464i \(-0.489578\pi\)
0.0327367 + 0.999464i \(0.489578\pi\)
\(14\) −0.618034 −0.165177
\(15\) 3.23607 0.835549
\(16\) 1.85410 0.463525
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.23607 −0.291344
\(19\) −7.23607 −1.66007 −0.830034 0.557713i \(-0.811679\pi\)
−0.830034 + 0.557713i \(0.811679\pi\)
\(20\) 5.23607 1.17082
\(21\) 1.00000 0.218218
\(22\) 2.76393 0.589272
\(23\) −1.00000 −0.208514
\(24\) 2.23607 0.456435
\(25\) 5.47214 1.09443
\(26\) 0.145898 0.0286130
\(27\) 5.00000 0.962250
\(28\) 1.61803 0.305780
\(29\) −1.47214 −0.273369 −0.136684 0.990615i \(-0.543645\pi\)
−0.136684 + 0.990615i \(0.543645\pi\)
\(30\) 2.00000 0.365148
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 5.61803 0.993137
\(33\) −4.47214 −0.778499
\(34\) 0 0
\(35\) 3.23607 0.546995
\(36\) 3.23607 0.539345
\(37\) −5.70820 −0.938423 −0.469211 0.883086i \(-0.655462\pi\)
−0.469211 + 0.883086i \(0.655462\pi\)
\(38\) −4.47214 −0.725476
\(39\) −0.236068 −0.0378011
\(40\) 7.23607 1.14412
\(41\) −2.23607 −0.349215 −0.174608 0.984638i \(-0.555866\pi\)
−0.174608 + 0.984638i \(0.555866\pi\)
\(42\) 0.618034 0.0953647
\(43\) 2.47214 0.376997 0.188499 0.982073i \(-0.439638\pi\)
0.188499 + 0.982073i \(0.439638\pi\)
\(44\) −7.23607 −1.09088
\(45\) 6.47214 0.964809
\(46\) −0.618034 −0.0911241
\(47\) −3.47214 −0.506463 −0.253232 0.967406i \(-0.581493\pi\)
−0.253232 + 0.967406i \(0.581493\pi\)
\(48\) −1.85410 −0.267617
\(49\) 1.00000 0.142857
\(50\) 3.38197 0.478282
\(51\) 0 0
\(52\) −0.381966 −0.0529692
\(53\) 11.2361 1.54339 0.771696 0.635991i \(-0.219409\pi\)
0.771696 + 0.635991i \(0.219409\pi\)
\(54\) 3.09017 0.420519
\(55\) −14.4721 −1.95142
\(56\) 2.23607 0.298807
\(57\) 7.23607 0.958441
\(58\) −0.909830 −0.119467
\(59\) −1.52786 −0.198911 −0.0994555 0.995042i \(-0.531710\pi\)
−0.0994555 + 0.995042i \(0.531710\pi\)
\(60\) −5.23607 −0.675973
\(61\) 13.4164 1.71780 0.858898 0.512148i \(-0.171150\pi\)
0.858898 + 0.512148i \(0.171150\pi\)
\(62\) −5.56231 −0.706414
\(63\) 2.00000 0.251976
\(64\) −0.236068 −0.0295085
\(65\) −0.763932 −0.0947541
\(66\) −2.76393 −0.340217
\(67\) −12.1803 −1.48807 −0.744033 0.668143i \(-0.767089\pi\)
−0.744033 + 0.668143i \(0.767089\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 2.00000 0.239046
\(71\) −10.2361 −1.21480 −0.607399 0.794397i \(-0.707787\pi\)
−0.607399 + 0.794397i \(0.707787\pi\)
\(72\) 4.47214 0.527046
\(73\) −6.70820 −0.785136 −0.392568 0.919723i \(-0.628413\pi\)
−0.392568 + 0.919723i \(0.628413\pi\)
\(74\) −3.52786 −0.410106
\(75\) −5.47214 −0.631868
\(76\) 11.7082 1.34302
\(77\) −4.47214 −0.509647
\(78\) −0.145898 −0.0165197
\(79\) −7.23607 −0.814121 −0.407061 0.913401i \(-0.633446\pi\)
−0.407061 + 0.913401i \(0.633446\pi\)
\(80\) −6.00000 −0.670820
\(81\) 1.00000 0.111111
\(82\) −1.38197 −0.152613
\(83\) 6.47214 0.710409 0.355205 0.934789i \(-0.384411\pi\)
0.355205 + 0.934789i \(0.384411\pi\)
\(84\) −1.61803 −0.176542
\(85\) 0 0
\(86\) 1.52786 0.164754
\(87\) 1.47214 0.157830
\(88\) −10.0000 −1.06600
\(89\) 8.94427 0.948091 0.474045 0.880500i \(-0.342793\pi\)
0.474045 + 0.880500i \(0.342793\pi\)
\(90\) 4.00000 0.421637
\(91\) −0.236068 −0.0247466
\(92\) 1.61803 0.168692
\(93\) 9.00000 0.933257
\(94\) −2.14590 −0.221332
\(95\) 23.4164 2.40247
\(96\) −5.61803 −0.573388
\(97\) 3.70820 0.376511 0.188256 0.982120i \(-0.439717\pi\)
0.188256 + 0.982120i \(0.439717\pi\)
\(98\) 0.618034 0.0624309
\(99\) −8.94427 −0.898933
\(100\) −8.85410 −0.885410
\(101\) −13.4164 −1.33498 −0.667491 0.744618i \(-0.732632\pi\)
−0.667491 + 0.744618i \(0.732632\pi\)
\(102\) 0 0
\(103\) −15.7082 −1.54778 −0.773888 0.633323i \(-0.781691\pi\)
−0.773888 + 0.633323i \(0.781691\pi\)
\(104\) −0.527864 −0.0517613
\(105\) −3.23607 −0.315808
\(106\) 6.94427 0.674487
\(107\) 11.2361 1.08623 0.543116 0.839658i \(-0.317244\pi\)
0.543116 + 0.839658i \(0.317244\pi\)
\(108\) −8.09017 −0.778477
\(109\) −15.4164 −1.47662 −0.738312 0.674459i \(-0.764377\pi\)
−0.738312 + 0.674459i \(0.764377\pi\)
\(110\) −8.94427 −0.852803
\(111\) 5.70820 0.541799
\(112\) −1.85410 −0.175196
\(113\) 2.47214 0.232559 0.116279 0.993217i \(-0.462903\pi\)
0.116279 + 0.993217i \(0.462903\pi\)
\(114\) 4.47214 0.418854
\(115\) 3.23607 0.301765
\(116\) 2.38197 0.221160
\(117\) −0.472136 −0.0436490
\(118\) −0.944272 −0.0869273
\(119\) 0 0
\(120\) −7.23607 −0.660560
\(121\) 9.00000 0.818182
\(122\) 8.29180 0.750704
\(123\) 2.23607 0.201619
\(124\) 14.5623 1.30773
\(125\) −1.52786 −0.136656
\(126\) 1.23607 0.110118
\(127\) −2.70820 −0.240314 −0.120157 0.992755i \(-0.538340\pi\)
−0.120157 + 0.992755i \(0.538340\pi\)
\(128\) −11.3820 −1.00603
\(129\) −2.47214 −0.217659
\(130\) −0.472136 −0.0414091
\(131\) 3.94427 0.344613 0.172306 0.985043i \(-0.444878\pi\)
0.172306 + 0.985043i \(0.444878\pi\)
\(132\) 7.23607 0.629819
\(133\) 7.23607 0.627447
\(134\) −7.52786 −0.650308
\(135\) −16.1803 −1.39258
\(136\) 0 0
\(137\) 15.7082 1.34204 0.671021 0.741438i \(-0.265856\pi\)
0.671021 + 0.741438i \(0.265856\pi\)
\(138\) 0.618034 0.0526105
\(139\) 2.52786 0.214411 0.107205 0.994237i \(-0.465810\pi\)
0.107205 + 0.994237i \(0.465810\pi\)
\(140\) −5.23607 −0.442529
\(141\) 3.47214 0.292407
\(142\) −6.32624 −0.530886
\(143\) 1.05573 0.0882844
\(144\) −3.70820 −0.309017
\(145\) 4.76393 0.395623
\(146\) −4.14590 −0.343117
\(147\) −1.00000 −0.0824786
\(148\) 9.23607 0.759200
\(149\) 15.2361 1.24819 0.624094 0.781350i \(-0.285468\pi\)
0.624094 + 0.781350i \(0.285468\pi\)
\(150\) −3.38197 −0.276136
\(151\) −15.1803 −1.23536 −0.617679 0.786430i \(-0.711927\pi\)
−0.617679 + 0.786430i \(0.711927\pi\)
\(152\) 16.1803 1.31240
\(153\) 0 0
\(154\) −2.76393 −0.222724
\(155\) 29.1246 2.33935
\(156\) 0.381966 0.0305818
\(157\) 15.4164 1.23036 0.615182 0.788385i \(-0.289083\pi\)
0.615182 + 0.788385i \(0.289083\pi\)
\(158\) −4.47214 −0.355784
\(159\) −11.2361 −0.891078
\(160\) −18.1803 −1.43728
\(161\) 1.00000 0.0788110
\(162\) 0.618034 0.0485573
\(163\) 19.1803 1.50232 0.751160 0.660120i \(-0.229495\pi\)
0.751160 + 0.660120i \(0.229495\pi\)
\(164\) 3.61803 0.282521
\(165\) 14.4721 1.12665
\(166\) 4.00000 0.310460
\(167\) 21.8885 1.69379 0.846893 0.531763i \(-0.178470\pi\)
0.846893 + 0.531763i \(0.178470\pi\)
\(168\) −2.23607 −0.172516
\(169\) −12.9443 −0.995713
\(170\) 0 0
\(171\) 14.4721 1.10671
\(172\) −4.00000 −0.304997
\(173\) −3.52786 −0.268219 −0.134109 0.990967i \(-0.542817\pi\)
−0.134109 + 0.990967i \(0.542817\pi\)
\(174\) 0.909830 0.0689740
\(175\) −5.47214 −0.413655
\(176\) 8.29180 0.625018
\(177\) 1.52786 0.114841
\(178\) 5.52786 0.414331
\(179\) −18.7082 −1.39832 −0.699158 0.714967i \(-0.746442\pi\)
−0.699158 + 0.714967i \(0.746442\pi\)
\(180\) −10.4721 −0.780547
\(181\) −5.05573 −0.375789 −0.187895 0.982189i \(-0.560166\pi\)
−0.187895 + 0.982189i \(0.560166\pi\)
\(182\) −0.145898 −0.0108147
\(183\) −13.4164 −0.991769
\(184\) 2.23607 0.164845
\(185\) 18.4721 1.35810
\(186\) 5.56231 0.407848
\(187\) 0 0
\(188\) 5.61803 0.409737
\(189\) −5.00000 −0.363696
\(190\) 14.4721 1.04992
\(191\) −24.1803 −1.74963 −0.874814 0.484459i \(-0.839016\pi\)
−0.874814 + 0.484459i \(0.839016\pi\)
\(192\) 0.236068 0.0170367
\(193\) 4.41641 0.317900 0.158950 0.987287i \(-0.449189\pi\)
0.158950 + 0.987287i \(0.449189\pi\)
\(194\) 2.29180 0.164541
\(195\) 0.763932 0.0547063
\(196\) −1.61803 −0.115574
\(197\) −21.4721 −1.52983 −0.764913 0.644133i \(-0.777218\pi\)
−0.764913 + 0.644133i \(0.777218\pi\)
\(198\) −5.52786 −0.392848
\(199\) 15.8885 1.12631 0.563155 0.826352i \(-0.309588\pi\)
0.563155 + 0.826352i \(0.309588\pi\)
\(200\) −12.2361 −0.865221
\(201\) 12.1803 0.859135
\(202\) −8.29180 −0.583409
\(203\) 1.47214 0.103324
\(204\) 0 0
\(205\) 7.23607 0.505389
\(206\) −9.70820 −0.676403
\(207\) 2.00000 0.139010
\(208\) 0.437694 0.0303486
\(209\) −32.3607 −2.23844
\(210\) −2.00000 −0.138013
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −18.1803 −1.24863
\(213\) 10.2361 0.701364
\(214\) 6.94427 0.474701
\(215\) −8.00000 −0.545595
\(216\) −11.1803 −0.760726
\(217\) 9.00000 0.610960
\(218\) −9.52786 −0.645308
\(219\) 6.70820 0.453298
\(220\) 23.4164 1.57873
\(221\) 0 0
\(222\) 3.52786 0.236775
\(223\) 3.41641 0.228780 0.114390 0.993436i \(-0.463509\pi\)
0.114390 + 0.993436i \(0.463509\pi\)
\(224\) −5.61803 −0.375371
\(225\) −10.9443 −0.729618
\(226\) 1.52786 0.101632
\(227\) −5.88854 −0.390836 −0.195418 0.980720i \(-0.562606\pi\)
−0.195418 + 0.980720i \(0.562606\pi\)
\(228\) −11.7082 −0.775395
\(229\) 14.7639 0.975628 0.487814 0.872948i \(-0.337794\pi\)
0.487814 + 0.872948i \(0.337794\pi\)
\(230\) 2.00000 0.131876
\(231\) 4.47214 0.294245
\(232\) 3.29180 0.216117
\(233\) −11.4721 −0.751565 −0.375782 0.926708i \(-0.622626\pi\)
−0.375782 + 0.926708i \(0.622626\pi\)
\(234\) −0.291796 −0.0190753
\(235\) 11.2361 0.732960
\(236\) 2.47214 0.160922
\(237\) 7.23607 0.470033
\(238\) 0 0
\(239\) −15.7639 −1.01968 −0.509842 0.860268i \(-0.670296\pi\)
−0.509842 + 0.860268i \(0.670296\pi\)
\(240\) 6.00000 0.387298
\(241\) −21.4164 −1.37955 −0.689776 0.724023i \(-0.742291\pi\)
−0.689776 + 0.724023i \(0.742291\pi\)
\(242\) 5.56231 0.357559
\(243\) −16.0000 −1.02640
\(244\) −21.7082 −1.38973
\(245\) −3.23607 −0.206745
\(246\) 1.38197 0.0881109
\(247\) −1.70820 −0.108690
\(248\) 20.1246 1.27791
\(249\) −6.47214 −0.410155
\(250\) −0.944272 −0.0597210
\(251\) 8.29180 0.523374 0.261687 0.965153i \(-0.415721\pi\)
0.261687 + 0.965153i \(0.415721\pi\)
\(252\) −3.23607 −0.203853
\(253\) −4.47214 −0.281161
\(254\) −1.67376 −0.105021
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) 4.23607 0.264239 0.132119 0.991234i \(-0.457822\pi\)
0.132119 + 0.991234i \(0.457822\pi\)
\(258\) −1.52786 −0.0951207
\(259\) 5.70820 0.354691
\(260\) 1.23607 0.0766577
\(261\) 2.94427 0.182246
\(262\) 2.43769 0.150601
\(263\) −26.9443 −1.66145 −0.830727 0.556679i \(-0.812075\pi\)
−0.830727 + 0.556679i \(0.812075\pi\)
\(264\) 10.0000 0.615457
\(265\) −36.3607 −2.23362
\(266\) 4.47214 0.274204
\(267\) −8.94427 −0.547381
\(268\) 19.7082 1.20387
\(269\) −9.18034 −0.559735 −0.279868 0.960039i \(-0.590291\pi\)
−0.279868 + 0.960039i \(0.590291\pi\)
\(270\) −10.0000 −0.608581
\(271\) −16.9443 −1.02929 −0.514646 0.857403i \(-0.672076\pi\)
−0.514646 + 0.857403i \(0.672076\pi\)
\(272\) 0 0
\(273\) 0.236068 0.0142875
\(274\) 9.70820 0.586494
\(275\) 24.4721 1.47573
\(276\) −1.61803 −0.0973942
\(277\) 20.4164 1.22670 0.613352 0.789810i \(-0.289821\pi\)
0.613352 + 0.789810i \(0.289821\pi\)
\(278\) 1.56231 0.0937009
\(279\) 18.0000 1.07763
\(280\) −7.23607 −0.432438
\(281\) 3.70820 0.221213 0.110606 0.993864i \(-0.464721\pi\)
0.110606 + 0.993864i \(0.464721\pi\)
\(282\) 2.14590 0.127786
\(283\) 18.9443 1.12612 0.563060 0.826416i \(-0.309624\pi\)
0.563060 + 0.826416i \(0.309624\pi\)
\(284\) 16.5623 0.982792
\(285\) −23.4164 −1.38707
\(286\) 0.652476 0.0385817
\(287\) 2.23607 0.131991
\(288\) −11.2361 −0.662092
\(289\) −17.0000 −1.00000
\(290\) 2.94427 0.172894
\(291\) −3.70820 −0.217379
\(292\) 10.8541 0.635188
\(293\) −15.7082 −0.917683 −0.458842 0.888518i \(-0.651735\pi\)
−0.458842 + 0.888518i \(0.651735\pi\)
\(294\) −0.618034 −0.0360445
\(295\) 4.94427 0.287867
\(296\) 12.7639 0.741888
\(297\) 22.3607 1.29750
\(298\) 9.41641 0.545478
\(299\) −0.236068 −0.0136522
\(300\) 8.85410 0.511192
\(301\) −2.47214 −0.142492
\(302\) −9.38197 −0.539871
\(303\) 13.4164 0.770752
\(304\) −13.4164 −0.769484
\(305\) −43.4164 −2.48602
\(306\) 0 0
\(307\) −11.4164 −0.651569 −0.325784 0.945444i \(-0.605628\pi\)
−0.325784 + 0.945444i \(0.605628\pi\)
\(308\) 7.23607 0.412313
\(309\) 15.7082 0.893609
\(310\) 18.0000 1.02233
\(311\) −2.88854 −0.163794 −0.0818971 0.996641i \(-0.526098\pi\)
−0.0818971 + 0.996641i \(0.526098\pi\)
\(312\) 0.527864 0.0298844
\(313\) 2.76393 0.156227 0.0781133 0.996944i \(-0.475110\pi\)
0.0781133 + 0.996944i \(0.475110\pi\)
\(314\) 9.52786 0.537688
\(315\) −6.47214 −0.364664
\(316\) 11.7082 0.658638
\(317\) 31.3050 1.75826 0.879131 0.476581i \(-0.158124\pi\)
0.879131 + 0.476581i \(0.158124\pi\)
\(318\) −6.94427 −0.389415
\(319\) −6.58359 −0.368610
\(320\) 0.763932 0.0427051
\(321\) −11.2361 −0.627136
\(322\) 0.618034 0.0344417
\(323\) 0 0
\(324\) −1.61803 −0.0898908
\(325\) 1.29180 0.0716560
\(326\) 11.8541 0.656538
\(327\) 15.4164 0.852529
\(328\) 5.00000 0.276079
\(329\) 3.47214 0.191425
\(330\) 8.94427 0.492366
\(331\) 0.708204 0.0389264 0.0194632 0.999811i \(-0.493804\pi\)
0.0194632 + 0.999811i \(0.493804\pi\)
\(332\) −10.4721 −0.574733
\(333\) 11.4164 0.625615
\(334\) 13.5279 0.740212
\(335\) 39.4164 2.15355
\(336\) 1.85410 0.101150
\(337\) −17.5967 −0.958556 −0.479278 0.877663i \(-0.659101\pi\)
−0.479278 + 0.877663i \(0.659101\pi\)
\(338\) −8.00000 −0.435143
\(339\) −2.47214 −0.134268
\(340\) 0 0
\(341\) −40.2492 −2.17962
\(342\) 8.94427 0.483651
\(343\) −1.00000 −0.0539949
\(344\) −5.52786 −0.298042
\(345\) −3.23607 −0.174224
\(346\) −2.18034 −0.117216
\(347\) −5.88854 −0.316114 −0.158057 0.987430i \(-0.550523\pi\)
−0.158057 + 0.987430i \(0.550523\pi\)
\(348\) −2.38197 −0.127687
\(349\) 1.29180 0.0691483 0.0345741 0.999402i \(-0.488993\pi\)
0.0345741 + 0.999402i \(0.488993\pi\)
\(350\) −3.38197 −0.180774
\(351\) 1.18034 0.0630019
\(352\) 25.1246 1.33915
\(353\) −1.76393 −0.0938846 −0.0469423 0.998898i \(-0.514948\pi\)
−0.0469423 + 0.998898i \(0.514948\pi\)
\(354\) 0.944272 0.0501875
\(355\) 33.1246 1.75807
\(356\) −14.4721 −0.767022
\(357\) 0 0
\(358\) −11.5623 −0.611087
\(359\) 4.76393 0.251431 0.125715 0.992066i \(-0.459877\pi\)
0.125715 + 0.992066i \(0.459877\pi\)
\(360\) −14.4721 −0.762749
\(361\) 33.3607 1.75583
\(362\) −3.12461 −0.164226
\(363\) −9.00000 −0.472377
\(364\) 0.381966 0.0200205
\(365\) 21.7082 1.13626
\(366\) −8.29180 −0.433419
\(367\) −4.58359 −0.239262 −0.119631 0.992818i \(-0.538171\pi\)
−0.119631 + 0.992818i \(0.538171\pi\)
\(368\) −1.85410 −0.0966517
\(369\) 4.47214 0.232810
\(370\) 11.4164 0.593511
\(371\) −11.2361 −0.583348
\(372\) −14.5623 −0.755020
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 1.52786 0.0788986
\(376\) 7.76393 0.400394
\(377\) −0.347524 −0.0178984
\(378\) −3.09017 −0.158941
\(379\) 8.29180 0.425921 0.212960 0.977061i \(-0.431689\pi\)
0.212960 + 0.977061i \(0.431689\pi\)
\(380\) −37.8885 −1.94364
\(381\) 2.70820 0.138745
\(382\) −14.9443 −0.764615
\(383\) −15.7082 −0.802652 −0.401326 0.915935i \(-0.631451\pi\)
−0.401326 + 0.915935i \(0.631451\pi\)
\(384\) 11.3820 0.580834
\(385\) 14.4721 0.737568
\(386\) 2.72949 0.138927
\(387\) −4.94427 −0.251331
\(388\) −6.00000 −0.304604
\(389\) −7.41641 −0.376027 −0.188013 0.982166i \(-0.560205\pi\)
−0.188013 + 0.982166i \(0.560205\pi\)
\(390\) 0.472136 0.0239075
\(391\) 0 0
\(392\) −2.23607 −0.112938
\(393\) −3.94427 −0.198962
\(394\) −13.2705 −0.668559
\(395\) 23.4164 1.17821
\(396\) 14.4721 0.727252
\(397\) −25.6525 −1.28746 −0.643730 0.765252i \(-0.722614\pi\)
−0.643730 + 0.765252i \(0.722614\pi\)
\(398\) 9.81966 0.492215
\(399\) −7.23607 −0.362257
\(400\) 10.1459 0.507295
\(401\) −19.8885 −0.993186 −0.496593 0.867983i \(-0.665416\pi\)
−0.496593 + 0.867983i \(0.665416\pi\)
\(402\) 7.52786 0.375456
\(403\) −2.12461 −0.105834
\(404\) 21.7082 1.08002
\(405\) −3.23607 −0.160802
\(406\) 0.909830 0.0451541
\(407\) −25.5279 −1.26537
\(408\) 0 0
\(409\) 4.12461 0.203949 0.101974 0.994787i \(-0.467484\pi\)
0.101974 + 0.994787i \(0.467484\pi\)
\(410\) 4.47214 0.220863
\(411\) −15.7082 −0.774829
\(412\) 25.4164 1.25218
\(413\) 1.52786 0.0751813
\(414\) 1.23607 0.0607494
\(415\) −20.9443 −1.02811
\(416\) 1.32624 0.0650242
\(417\) −2.52786 −0.123790
\(418\) −20.0000 −0.978232
\(419\) 26.4721 1.29325 0.646624 0.762809i \(-0.276180\pi\)
0.646624 + 0.762809i \(0.276180\pi\)
\(420\) 5.23607 0.255494
\(421\) −8.58359 −0.418339 −0.209169 0.977879i \(-0.567076\pi\)
−0.209169 + 0.977879i \(0.567076\pi\)
\(422\) −7.41641 −0.361025
\(423\) 6.94427 0.337642
\(424\) −25.1246 −1.22016
\(425\) 0 0
\(426\) 6.32624 0.306507
\(427\) −13.4164 −0.649265
\(428\) −18.1803 −0.878780
\(429\) −1.05573 −0.0509710
\(430\) −4.94427 −0.238434
\(431\) 18.1803 0.875716 0.437858 0.899044i \(-0.355737\pi\)
0.437858 + 0.899044i \(0.355737\pi\)
\(432\) 9.27051 0.446028
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 5.56231 0.266999
\(435\) −4.76393 −0.228413
\(436\) 24.9443 1.19461
\(437\) 7.23607 0.346148
\(438\) 4.14590 0.198099
\(439\) 30.3050 1.44638 0.723188 0.690651i \(-0.242676\pi\)
0.723188 + 0.690651i \(0.242676\pi\)
\(440\) 32.3607 1.54273
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) −7.18034 −0.341148 −0.170574 0.985345i \(-0.554562\pi\)
−0.170574 + 0.985345i \(0.554562\pi\)
\(444\) −9.23607 −0.438324
\(445\) −28.9443 −1.37209
\(446\) 2.11146 0.0999803
\(447\) −15.2361 −0.720641
\(448\) 0.236068 0.0111532
\(449\) 32.8328 1.54948 0.774738 0.632282i \(-0.217882\pi\)
0.774738 + 0.632282i \(0.217882\pi\)
\(450\) −6.76393 −0.318855
\(451\) −10.0000 −0.470882
\(452\) −4.00000 −0.188144
\(453\) 15.1803 0.713235
\(454\) −3.63932 −0.170802
\(455\) 0.763932 0.0358137
\(456\) −16.1803 −0.757714
\(457\) 14.4721 0.676978 0.338489 0.940970i \(-0.390084\pi\)
0.338489 + 0.940970i \(0.390084\pi\)
\(458\) 9.12461 0.426365
\(459\) 0 0
\(460\) −5.23607 −0.244133
\(461\) −25.7639 −1.19995 −0.599973 0.800020i \(-0.704822\pi\)
−0.599973 + 0.800020i \(0.704822\pi\)
\(462\) 2.76393 0.128590
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) −2.72949 −0.126713
\(465\) −29.1246 −1.35062
\(466\) −7.09017 −0.328446
\(467\) −11.5279 −0.533446 −0.266723 0.963773i \(-0.585941\pi\)
−0.266723 + 0.963773i \(0.585941\pi\)
\(468\) 0.763932 0.0353128
\(469\) 12.1803 0.562436
\(470\) 6.94427 0.320315
\(471\) −15.4164 −0.710351
\(472\) 3.41641 0.157253
\(473\) 11.0557 0.508343
\(474\) 4.47214 0.205412
\(475\) −39.5967 −1.81682
\(476\) 0 0
\(477\) −22.4721 −1.02893
\(478\) −9.74265 −0.445618
\(479\) −23.1246 −1.05659 −0.528295 0.849061i \(-0.677169\pi\)
−0.528295 + 0.849061i \(0.677169\pi\)
\(480\) 18.1803 0.829815
\(481\) −1.34752 −0.0614418
\(482\) −13.2361 −0.602886
\(483\) −1.00000 −0.0455016
\(484\) −14.5623 −0.661923
\(485\) −12.0000 −0.544892
\(486\) −9.88854 −0.448553
\(487\) 20.1246 0.911933 0.455967 0.889997i \(-0.349294\pi\)
0.455967 + 0.889997i \(0.349294\pi\)
\(488\) −30.0000 −1.35804
\(489\) −19.1803 −0.867365
\(490\) −2.00000 −0.0903508
\(491\) 11.7639 0.530899 0.265449 0.964125i \(-0.414480\pi\)
0.265449 + 0.964125i \(0.414480\pi\)
\(492\) −3.61803 −0.163114
\(493\) 0 0
\(494\) −1.05573 −0.0474995
\(495\) 28.9443 1.30095
\(496\) −16.6869 −0.749265
\(497\) 10.2361 0.459150
\(498\) −4.00000 −0.179244
\(499\) −36.7082 −1.64328 −0.821642 0.570003i \(-0.806942\pi\)
−0.821642 + 0.570003i \(0.806942\pi\)
\(500\) 2.47214 0.110557
\(501\) −21.8885 −0.977908
\(502\) 5.12461 0.228723
\(503\) −15.5967 −0.695425 −0.347712 0.937601i \(-0.613041\pi\)
−0.347712 + 0.937601i \(0.613041\pi\)
\(504\) −4.47214 −0.199205
\(505\) 43.4164 1.93200
\(506\) −2.76393 −0.122872
\(507\) 12.9443 0.574875
\(508\) 4.38197 0.194418
\(509\) −12.8197 −0.568221 −0.284111 0.958791i \(-0.591698\pi\)
−0.284111 + 0.958791i \(0.591698\pi\)
\(510\) 0 0
\(511\) 6.70820 0.296753
\(512\) 18.7082 0.826794
\(513\) −36.1803 −1.59740
\(514\) 2.61803 0.115477
\(515\) 50.8328 2.23996
\(516\) 4.00000 0.176090
\(517\) −15.5279 −0.682915
\(518\) 3.52786 0.155005
\(519\) 3.52786 0.154856
\(520\) 1.70820 0.0749097
\(521\) 21.3050 0.933387 0.466693 0.884419i \(-0.345445\pi\)
0.466693 + 0.884419i \(0.345445\pi\)
\(522\) 1.81966 0.0796444
\(523\) −7.70820 −0.337056 −0.168528 0.985697i \(-0.553901\pi\)
−0.168528 + 0.985697i \(0.553901\pi\)
\(524\) −6.38197 −0.278797
\(525\) 5.47214 0.238824
\(526\) −16.6525 −0.726082
\(527\) 0 0
\(528\) −8.29180 −0.360854
\(529\) 1.00000 0.0434783
\(530\) −22.4721 −0.976127
\(531\) 3.05573 0.132607
\(532\) −11.7082 −0.507615
\(533\) −0.527864 −0.0228643
\(534\) −5.52786 −0.239214
\(535\) −36.3607 −1.57201
\(536\) 27.2361 1.17642
\(537\) 18.7082 0.807319
\(538\) −5.67376 −0.244613
\(539\) 4.47214 0.192629
\(540\) 26.1803 1.12662
\(541\) 21.0000 0.902861 0.451430 0.892306i \(-0.350914\pi\)
0.451430 + 0.892306i \(0.350914\pi\)
\(542\) −10.4721 −0.449817
\(543\) 5.05573 0.216962
\(544\) 0 0
\(545\) 49.8885 2.13699
\(546\) 0.145898 0.00624386
\(547\) 43.1803 1.84626 0.923129 0.384490i \(-0.125623\pi\)
0.923129 + 0.384490i \(0.125623\pi\)
\(548\) −25.4164 −1.08574
\(549\) −26.8328 −1.14520
\(550\) 15.1246 0.644916
\(551\) 10.6525 0.453811
\(552\) −2.23607 −0.0951734
\(553\) 7.23607 0.307709
\(554\) 12.6180 0.536089
\(555\) −18.4721 −0.784099
\(556\) −4.09017 −0.173462
\(557\) −26.7639 −1.13402 −0.567012 0.823709i \(-0.691901\pi\)
−0.567012 + 0.823709i \(0.691901\pi\)
\(558\) 11.1246 0.470942
\(559\) 0.583592 0.0246833
\(560\) 6.00000 0.253546
\(561\) 0 0
\(562\) 2.29180 0.0966736
\(563\) −9.59675 −0.404455 −0.202227 0.979339i \(-0.564818\pi\)
−0.202227 + 0.979339i \(0.564818\pi\)
\(564\) −5.61803 −0.236562
\(565\) −8.00000 −0.336563
\(566\) 11.7082 0.492133
\(567\) −1.00000 −0.0419961
\(568\) 22.8885 0.960382
\(569\) −8.18034 −0.342938 −0.171469 0.985190i \(-0.554851\pi\)
−0.171469 + 0.985190i \(0.554851\pi\)
\(570\) −14.4721 −0.606171
\(571\) 16.2918 0.681790 0.340895 0.940101i \(-0.389270\pi\)
0.340895 + 0.940101i \(0.389270\pi\)
\(572\) −1.70820 −0.0714236
\(573\) 24.1803 1.01015
\(574\) 1.38197 0.0576821
\(575\) −5.47214 −0.228204
\(576\) 0.472136 0.0196723
\(577\) −15.2918 −0.636606 −0.318303 0.947989i \(-0.603113\pi\)
−0.318303 + 0.947989i \(0.603113\pi\)
\(578\) −10.5066 −0.437016
\(579\) −4.41641 −0.183540
\(580\) −7.70820 −0.320066
\(581\) −6.47214 −0.268509
\(582\) −2.29180 −0.0949980
\(583\) 50.2492 2.08111
\(584\) 15.0000 0.620704
\(585\) 1.52786 0.0631694
\(586\) −9.70820 −0.401042
\(587\) 20.8885 0.862162 0.431081 0.902313i \(-0.358132\pi\)
0.431081 + 0.902313i \(0.358132\pi\)
\(588\) 1.61803 0.0667266
\(589\) 65.1246 2.68341
\(590\) 3.05573 0.125802
\(591\) 21.4721 0.883246
\(592\) −10.5836 −0.434983
\(593\) 34.3607 1.41102 0.705512 0.708698i \(-0.250717\pi\)
0.705512 + 0.708698i \(0.250717\pi\)
\(594\) 13.8197 0.567028
\(595\) 0 0
\(596\) −24.6525 −1.00980
\(597\) −15.8885 −0.650275
\(598\) −0.145898 −0.00596621
\(599\) −2.47214 −0.101009 −0.0505044 0.998724i \(-0.516083\pi\)
−0.0505044 + 0.998724i \(0.516083\pi\)
\(600\) 12.2361 0.499535
\(601\) 22.2361 0.907028 0.453514 0.891249i \(-0.350170\pi\)
0.453514 + 0.891249i \(0.350170\pi\)
\(602\) −1.52786 −0.0622711
\(603\) 24.3607 0.992044
\(604\) 24.5623 0.999426
\(605\) −29.1246 −1.18408
\(606\) 8.29180 0.336831
\(607\) −33.3050 −1.35181 −0.675903 0.736990i \(-0.736246\pi\)
−0.675903 + 0.736990i \(0.736246\pi\)
\(608\) −40.6525 −1.64868
\(609\) −1.47214 −0.0596540
\(610\) −26.8328 −1.08643
\(611\) −0.819660 −0.0331599
\(612\) 0 0
\(613\) −22.1803 −0.895855 −0.447928 0.894070i \(-0.647838\pi\)
−0.447928 + 0.894070i \(0.647838\pi\)
\(614\) −7.05573 −0.284746
\(615\) −7.23607 −0.291786
\(616\) 10.0000 0.402911
\(617\) 35.2361 1.41855 0.709275 0.704932i \(-0.249022\pi\)
0.709275 + 0.704932i \(0.249022\pi\)
\(618\) 9.70820 0.390521
\(619\) −47.4853 −1.90860 −0.954298 0.298858i \(-0.903394\pi\)
−0.954298 + 0.298858i \(0.903394\pi\)
\(620\) −47.1246 −1.89257
\(621\) −5.00000 −0.200643
\(622\) −1.78522 −0.0715807
\(623\) −8.94427 −0.358345
\(624\) −0.437694 −0.0175218
\(625\) −22.4164 −0.896656
\(626\) 1.70820 0.0682736
\(627\) 32.3607 1.29236
\(628\) −24.9443 −0.995385
\(629\) 0 0
\(630\) −4.00000 −0.159364
\(631\) 3.41641 0.136005 0.0680025 0.997685i \(-0.478337\pi\)
0.0680025 + 0.997685i \(0.478337\pi\)
\(632\) 16.1803 0.643619
\(633\) 12.0000 0.476957
\(634\) 19.3475 0.768388
\(635\) 8.76393 0.347786
\(636\) 18.1803 0.720897
\(637\) 0.236068 0.00935335
\(638\) −4.06888 −0.161089
\(639\) 20.4721 0.809865
\(640\) 36.8328 1.45594
\(641\) 7.05573 0.278685 0.139342 0.990244i \(-0.455501\pi\)
0.139342 + 0.990244i \(0.455501\pi\)
\(642\) −6.94427 −0.274069
\(643\) −17.7082 −0.698343 −0.349172 0.937059i \(-0.613537\pi\)
−0.349172 + 0.937059i \(0.613537\pi\)
\(644\) −1.61803 −0.0637595
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) −18.5279 −0.728405 −0.364203 0.931320i \(-0.618658\pi\)
−0.364203 + 0.931320i \(0.618658\pi\)
\(648\) −2.23607 −0.0878410
\(649\) −6.83282 −0.268211
\(650\) 0.798374 0.0313148
\(651\) −9.00000 −0.352738
\(652\) −31.0344 −1.21540
\(653\) 24.5279 0.959849 0.479925 0.877310i \(-0.340664\pi\)
0.479925 + 0.877310i \(0.340664\pi\)
\(654\) 9.52786 0.372569
\(655\) −12.7639 −0.498728
\(656\) −4.14590 −0.161870
\(657\) 13.4164 0.523424
\(658\) 2.14590 0.0836558
\(659\) 6.76393 0.263485 0.131743 0.991284i \(-0.457943\pi\)
0.131743 + 0.991284i \(0.457943\pi\)
\(660\) −23.4164 −0.911482
\(661\) −41.7082 −1.62226 −0.811131 0.584865i \(-0.801147\pi\)
−0.811131 + 0.584865i \(0.801147\pi\)
\(662\) 0.437694 0.0170115
\(663\) 0 0
\(664\) −14.4721 −0.561628
\(665\) −23.4164 −0.908049
\(666\) 7.05573 0.273404
\(667\) 1.47214 0.0570013
\(668\) −35.4164 −1.37030
\(669\) −3.41641 −0.132086
\(670\) 24.3607 0.941135
\(671\) 60.0000 2.31627
\(672\) 5.61803 0.216720
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −10.8754 −0.418904
\(675\) 27.3607 1.05311
\(676\) 20.9443 0.805549
\(677\) 23.2361 0.893035 0.446517 0.894775i \(-0.352664\pi\)
0.446517 + 0.894775i \(0.352664\pi\)
\(678\) −1.52786 −0.0586773
\(679\) −3.70820 −0.142308
\(680\) 0 0
\(681\) 5.88854 0.225649
\(682\) −24.8754 −0.952528
\(683\) −5.18034 −0.198220 −0.0991101 0.995076i \(-0.531600\pi\)
−0.0991101 + 0.995076i \(0.531600\pi\)
\(684\) −23.4164 −0.895349
\(685\) −50.8328 −1.94222
\(686\) −0.618034 −0.0235966
\(687\) −14.7639 −0.563279
\(688\) 4.58359 0.174748
\(689\) 2.65248 0.101051
\(690\) −2.00000 −0.0761387
\(691\) 42.8328 1.62944 0.814719 0.579857i \(-0.196891\pi\)
0.814719 + 0.579857i \(0.196891\pi\)
\(692\) 5.70820 0.216993
\(693\) 8.94427 0.339765
\(694\) −3.63932 −0.138147
\(695\) −8.18034 −0.310298
\(696\) −3.29180 −0.124775
\(697\) 0 0
\(698\) 0.798374 0.0302189
\(699\) 11.4721 0.433916
\(700\) 8.85410 0.334654
\(701\) −22.7639 −0.859782 −0.429891 0.902881i \(-0.641448\pi\)
−0.429891 + 0.902881i \(0.641448\pi\)
\(702\) 0.729490 0.0275328
\(703\) 41.3050 1.55785
\(704\) −1.05573 −0.0397892
\(705\) −11.2361 −0.423175
\(706\) −1.09017 −0.0410291
\(707\) 13.4164 0.504576
\(708\) −2.47214 −0.0929086
\(709\) 34.2492 1.28626 0.643128 0.765758i \(-0.277636\pi\)
0.643128 + 0.765758i \(0.277636\pi\)
\(710\) 20.4721 0.768306
\(711\) 14.4721 0.542748
\(712\) −20.0000 −0.749532
\(713\) 9.00000 0.337053
\(714\) 0 0
\(715\) −3.41641 −0.127766
\(716\) 30.2705 1.13126
\(717\) 15.7639 0.588715
\(718\) 2.94427 0.109879
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 12.0000 0.447214
\(721\) 15.7082 0.585004
\(722\) 20.6180 0.767324
\(723\) 21.4164 0.796485
\(724\) 8.18034 0.304020
\(725\) −8.05573 −0.299182
\(726\) −5.56231 −0.206437
\(727\) 3.23607 0.120019 0.0600096 0.998198i \(-0.480887\pi\)
0.0600096 + 0.998198i \(0.480887\pi\)
\(728\) 0.527864 0.0195639
\(729\) 13.0000 0.481481
\(730\) 13.4164 0.496564
\(731\) 0 0
\(732\) 21.7082 0.802358
\(733\) 12.9443 0.478108 0.239054 0.971006i \(-0.423163\pi\)
0.239054 + 0.971006i \(0.423163\pi\)
\(734\) −2.83282 −0.104561
\(735\) 3.23607 0.119364
\(736\) −5.61803 −0.207083
\(737\) −54.4721 −2.00651
\(738\) 2.76393 0.101742
\(739\) 8.70820 0.320336 0.160168 0.987090i \(-0.448796\pi\)
0.160168 + 0.987090i \(0.448796\pi\)
\(740\) −29.8885 −1.09872
\(741\) 1.70820 0.0627524
\(742\) −6.94427 −0.254932
\(743\) −25.5279 −0.936527 −0.468263 0.883589i \(-0.655120\pi\)
−0.468263 + 0.883589i \(0.655120\pi\)
\(744\) −20.1246 −0.737804
\(745\) −49.3050 −1.80639
\(746\) 16.0689 0.588324
\(747\) −12.9443 −0.473606
\(748\) 0 0
\(749\) −11.2361 −0.410557
\(750\) 0.944272 0.0344799
\(751\) −8.58359 −0.313220 −0.156610 0.987661i \(-0.550057\pi\)
−0.156610 + 0.987661i \(0.550057\pi\)
\(752\) −6.43769 −0.234759
\(753\) −8.29180 −0.302170
\(754\) −0.214782 −0.00782189
\(755\) 49.1246 1.78783
\(756\) 8.09017 0.294237
\(757\) −28.8328 −1.04795 −0.523973 0.851735i \(-0.675551\pi\)
−0.523973 + 0.851735i \(0.675551\pi\)
\(758\) 5.12461 0.186134
\(759\) 4.47214 0.162328
\(760\) −52.3607 −1.89932
\(761\) −31.6525 −1.14740 −0.573701 0.819065i \(-0.694493\pi\)
−0.573701 + 0.819065i \(0.694493\pi\)
\(762\) 1.67376 0.0606340
\(763\) 15.4164 0.558111
\(764\) 39.1246 1.41548
\(765\) 0 0
\(766\) −9.70820 −0.350772
\(767\) −0.360680 −0.0130234
\(768\) 6.56231 0.236797
\(769\) −43.2361 −1.55913 −0.779566 0.626320i \(-0.784560\pi\)
−0.779566 + 0.626320i \(0.784560\pi\)
\(770\) 8.94427 0.322329
\(771\) −4.23607 −0.152558
\(772\) −7.14590 −0.257186
\(773\) 40.6525 1.46217 0.731084 0.682288i \(-0.239015\pi\)
0.731084 + 0.682288i \(0.239015\pi\)
\(774\) −3.05573 −0.109836
\(775\) −49.2492 −1.76908
\(776\) −8.29180 −0.297658
\(777\) −5.70820 −0.204781
\(778\) −4.58359 −0.164330
\(779\) 16.1803 0.579721
\(780\) −1.23607 −0.0442583
\(781\) −45.7771 −1.63803
\(782\) 0 0
\(783\) −7.36068 −0.263049
\(784\) 1.85410 0.0662179
\(785\) −49.8885 −1.78060
\(786\) −2.43769 −0.0869497
\(787\) −0.360680 −0.0128568 −0.00642842 0.999979i \(-0.502046\pi\)
−0.00642842 + 0.999979i \(0.502046\pi\)
\(788\) 34.7426 1.23766
\(789\) 26.9443 0.959241
\(790\) 14.4721 0.514895
\(791\) −2.47214 −0.0878990
\(792\) 20.0000 0.710669
\(793\) 3.16718 0.112470
\(794\) −15.8541 −0.562641
\(795\) 36.3607 1.28958
\(796\) −25.7082 −0.911203
\(797\) −36.7639 −1.30225 −0.651123 0.758973i \(-0.725702\pi\)
−0.651123 + 0.758973i \(0.725702\pi\)
\(798\) −4.47214 −0.158312
\(799\) 0 0
\(800\) 30.7426 1.08692
\(801\) −17.8885 −0.632061
\(802\) −12.2918 −0.434038
\(803\) −30.0000 −1.05868
\(804\) −19.7082 −0.695055
\(805\) −3.23607 −0.114056
\(806\) −1.31308 −0.0462514
\(807\) 9.18034 0.323163
\(808\) 30.0000 1.05540
\(809\) −32.8328 −1.15434 −0.577170 0.816624i \(-0.695843\pi\)
−0.577170 + 0.816624i \(0.695843\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −45.3607 −1.59283 −0.796414 0.604751i \(-0.793273\pi\)
−0.796414 + 0.604751i \(0.793273\pi\)
\(812\) −2.38197 −0.0835906
\(813\) 16.9443 0.594262
\(814\) −15.7771 −0.552987
\(815\) −62.0689 −2.17418
\(816\) 0 0
\(817\) −17.8885 −0.625841
\(818\) 2.54915 0.0891289
\(819\) 0.472136 0.0164978
\(820\) −11.7082 −0.408868
\(821\) 22.5836 0.788173 0.394086 0.919073i \(-0.371061\pi\)
0.394086 + 0.919073i \(0.371061\pi\)
\(822\) −9.70820 −0.338612
\(823\) 25.5410 0.890304 0.445152 0.895455i \(-0.353150\pi\)
0.445152 + 0.895455i \(0.353150\pi\)
\(824\) 35.1246 1.22362
\(825\) −24.4721 −0.852010
\(826\) 0.944272 0.0328554
\(827\) −17.3050 −0.601752 −0.300876 0.953663i \(-0.597279\pi\)
−0.300876 + 0.953663i \(0.597279\pi\)
\(828\) −3.23607 −0.112461
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) −12.9443 −0.449302
\(831\) −20.4164 −0.708237
\(832\) −0.0557281 −0.00193202
\(833\) 0 0
\(834\) −1.56231 −0.0540982
\(835\) −70.8328 −2.45127
\(836\) 52.3607 1.81093
\(837\) −45.0000 −1.55543
\(838\) 16.3607 0.565170
\(839\) 20.0689 0.692855 0.346427 0.938077i \(-0.387395\pi\)
0.346427 + 0.938077i \(0.387395\pi\)
\(840\) 7.23607 0.249668
\(841\) −26.8328 −0.925270
\(842\) −5.30495 −0.182821
\(843\) −3.70820 −0.127717
\(844\) 19.4164 0.668340
\(845\) 41.8885 1.44101
\(846\) 4.29180 0.147555
\(847\) −9.00000 −0.309244
\(848\) 20.8328 0.715402
\(849\) −18.9443 −0.650166
\(850\) 0 0
\(851\) 5.70820 0.195675
\(852\) −16.5623 −0.567415
\(853\) 44.8328 1.53505 0.767523 0.641021i \(-0.221489\pi\)
0.767523 + 0.641021i \(0.221489\pi\)
\(854\) −8.29180 −0.283739
\(855\) −46.8328 −1.60165
\(856\) −25.1246 −0.858742
\(857\) 9.54102 0.325915 0.162958 0.986633i \(-0.447897\pi\)
0.162958 + 0.986633i \(0.447897\pi\)
\(858\) −0.652476 −0.0222752
\(859\) 11.5836 0.395227 0.197614 0.980280i \(-0.436681\pi\)
0.197614 + 0.980280i \(0.436681\pi\)
\(860\) 12.9443 0.441396
\(861\) −2.23607 −0.0762050
\(862\) 11.2361 0.382702
\(863\) 2.23607 0.0761166 0.0380583 0.999276i \(-0.487883\pi\)
0.0380583 + 0.999276i \(0.487883\pi\)
\(864\) 28.0902 0.955647
\(865\) 11.4164 0.388170
\(866\) 8.65248 0.294023
\(867\) 17.0000 0.577350
\(868\) −14.5623 −0.494277
\(869\) −32.3607 −1.09776
\(870\) −2.94427 −0.0998202
\(871\) −2.87539 −0.0974288
\(872\) 34.4721 1.16737
\(873\) −7.41641 −0.251007
\(874\) 4.47214 0.151272
\(875\) 1.52786 0.0516512
\(876\) −10.8541 −0.366726
\(877\) −42.0000 −1.41824 −0.709120 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) 18.7295 0.632090
\(879\) 15.7082 0.529825
\(880\) −26.8328 −0.904534
\(881\) 25.5967 0.862376 0.431188 0.902262i \(-0.358095\pi\)
0.431188 + 0.902262i \(0.358095\pi\)
\(882\) −1.23607 −0.0416206
\(883\) 9.88854 0.332776 0.166388 0.986060i \(-0.446790\pi\)
0.166388 + 0.986060i \(0.446790\pi\)
\(884\) 0 0
\(885\) −4.94427 −0.166200
\(886\) −4.43769 −0.149087
\(887\) 16.8885 0.567062 0.283531 0.958963i \(-0.408494\pi\)
0.283531 + 0.958963i \(0.408494\pi\)
\(888\) −12.7639 −0.428330
\(889\) 2.70820 0.0908302
\(890\) −17.8885 −0.599625
\(891\) 4.47214 0.149822
\(892\) −5.52786 −0.185087
\(893\) 25.1246 0.840763
\(894\) −9.41641 −0.314932
\(895\) 60.5410 2.02366
\(896\) 11.3820 0.380245
\(897\) 0.236068 0.00788208
\(898\) 20.2918 0.677146
\(899\) 13.2492 0.441886
\(900\) 17.7082 0.590273
\(901\) 0 0
\(902\) −6.18034 −0.205783
\(903\) 2.47214 0.0822675
\(904\) −5.52786 −0.183854
\(905\) 16.3607 0.543847
\(906\) 9.38197 0.311695
\(907\) 12.5410 0.416418 0.208209 0.978084i \(-0.433237\pi\)
0.208209 + 0.978084i \(0.433237\pi\)
\(908\) 9.52786 0.316193
\(909\) 26.8328 0.889988
\(910\) 0.472136 0.0156512
\(911\) −21.7771 −0.721507 −0.360754 0.932661i \(-0.617480\pi\)
−0.360754 + 0.932661i \(0.617480\pi\)
\(912\) 13.4164 0.444262
\(913\) 28.9443 0.957916
\(914\) 8.94427 0.295850
\(915\) 43.4164 1.43530
\(916\) −23.8885 −0.789300
\(917\) −3.94427 −0.130251
\(918\) 0 0
\(919\) 3.70820 0.122322 0.0611612 0.998128i \(-0.480520\pi\)
0.0611612 + 0.998128i \(0.480520\pi\)
\(920\) −7.23607 −0.238566
\(921\) 11.4164 0.376183
\(922\) −15.9230 −0.524396
\(923\) −2.41641 −0.0795370
\(924\) −7.23607 −0.238049
\(925\) −31.2361 −1.02704
\(926\) 0 0
\(927\) 31.4164 1.03185
\(928\) −8.27051 −0.271493
\(929\) 51.6525 1.69466 0.847331 0.531065i \(-0.178208\pi\)
0.847331 + 0.531065i \(0.178208\pi\)
\(930\) −18.0000 −0.590243
\(931\) −7.23607 −0.237153
\(932\) 18.5623 0.608029
\(933\) 2.88854 0.0945667
\(934\) −7.12461 −0.233124
\(935\) 0 0
\(936\) 1.05573 0.0345076
\(937\) −57.1246 −1.86618 −0.933090 0.359643i \(-0.882898\pi\)
−0.933090 + 0.359643i \(0.882898\pi\)
\(938\) 7.52786 0.245793
\(939\) −2.76393 −0.0901975
\(940\) −18.1803 −0.592977
\(941\) 53.0132 1.72818 0.864090 0.503338i \(-0.167895\pi\)
0.864090 + 0.503338i \(0.167895\pi\)
\(942\) −9.52786 −0.310435
\(943\) 2.23607 0.0728164
\(944\) −2.83282 −0.0922003
\(945\) 16.1803 0.526346
\(946\) 6.83282 0.222154
\(947\) −22.5967 −0.734296 −0.367148 0.930163i \(-0.619666\pi\)
−0.367148 + 0.930163i \(0.619666\pi\)
\(948\) −11.7082 −0.380265
\(949\) −1.58359 −0.0514056
\(950\) −24.4721 −0.793981
\(951\) −31.3050 −1.01513
\(952\) 0 0
\(953\) 12.1115 0.392329 0.196164 0.980571i \(-0.437151\pi\)
0.196164 + 0.980571i \(0.437151\pi\)
\(954\) −13.8885 −0.449658
\(955\) 78.2492 2.53209
\(956\) 25.5066 0.824942
\(957\) 6.58359 0.212817
\(958\) −14.2918 −0.461747
\(959\) −15.7082 −0.507244
\(960\) −0.763932 −0.0246558
\(961\) 50.0000 1.61290
\(962\) −0.832816 −0.0268511
\(963\) −22.4721 −0.724154
\(964\) 34.6525 1.11608
\(965\) −14.2918 −0.460069
\(966\) −0.618034 −0.0198849
\(967\) −31.0689 −0.999108 −0.499554 0.866283i \(-0.666503\pi\)
−0.499554 + 0.866283i \(0.666503\pi\)
\(968\) −20.1246 −0.646830
\(969\) 0 0
\(970\) −7.41641 −0.238127
\(971\) 33.7082 1.08175 0.540874 0.841104i \(-0.318094\pi\)
0.540874 + 0.841104i \(0.318094\pi\)
\(972\) 25.8885 0.830375
\(973\) −2.52786 −0.0810396
\(974\) 12.4377 0.398529
\(975\) −1.29180 −0.0413706
\(976\) 24.8754 0.796242
\(977\) −18.6525 −0.596746 −0.298373 0.954449i \(-0.596444\pi\)
−0.298373 + 0.954449i \(0.596444\pi\)
\(978\) −11.8541 −0.379052
\(979\) 40.0000 1.27841
\(980\) 5.23607 0.167260
\(981\) 30.8328 0.984416
\(982\) 7.27051 0.232011
\(983\) −4.18034 −0.133332 −0.0666661 0.997775i \(-0.521236\pi\)
−0.0666661 + 0.997775i \(0.521236\pi\)
\(984\) −5.00000 −0.159394
\(985\) 69.4853 2.21399
\(986\) 0 0
\(987\) −3.47214 −0.110519
\(988\) 2.76393 0.0879324
\(989\) −2.47214 −0.0786094
\(990\) 17.8885 0.568535
\(991\) −48.9443 −1.55477 −0.777383 0.629028i \(-0.783453\pi\)
−0.777383 + 0.629028i \(0.783453\pi\)
\(992\) −50.5623 −1.60535
\(993\) −0.708204 −0.0224742
\(994\) 6.32624 0.200656
\(995\) −51.4164 −1.63001
\(996\) 10.4721 0.331822
\(997\) −27.3050 −0.864756 −0.432378 0.901692i \(-0.642325\pi\)
−0.432378 + 0.901692i \(0.642325\pi\)
\(998\) −22.6869 −0.718142
\(999\) −28.5410 −0.902998
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 161.2.a.b.1.2 2
3.2 odd 2 1449.2.a.i.1.1 2
4.3 odd 2 2576.2.a.s.1.1 2
5.4 even 2 4025.2.a.i.1.1 2
7.6 odd 2 1127.2.a.d.1.2 2
23.22 odd 2 3703.2.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
161.2.a.b.1.2 2 1.1 even 1 trivial
1127.2.a.d.1.2 2 7.6 odd 2
1449.2.a.i.1.1 2 3.2 odd 2
2576.2.a.s.1.1 2 4.3 odd 2
3703.2.a.b.1.2 2 23.22 odd 2
4025.2.a.i.1.1 2 5.4 even 2