Properties

Label 507.2.a.k.1.2
Level $507$
Weight $2$
Character 507.1
Self dual yes
Analytic conductor $4.048$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
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 \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.445042 q^{2} +1.00000 q^{3} -1.80194 q^{4} -0.246980 q^{5} +0.445042 q^{6} +1.75302 q^{7} -1.69202 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.445042 q^{2} +1.00000 q^{3} -1.80194 q^{4} -0.246980 q^{5} +0.445042 q^{6} +1.75302 q^{7} -1.69202 q^{8} +1.00000 q^{9} -0.109916 q^{10} +5.65279 q^{11} -1.80194 q^{12} +0.780167 q^{14} -0.246980 q^{15} +2.85086 q^{16} -3.80194 q^{17} +0.445042 q^{18} +5.58211 q^{19} +0.445042 q^{20} +1.75302 q^{21} +2.51573 q^{22} +8.34481 q^{23} -1.69202 q^{24} -4.93900 q^{25} +1.00000 q^{27} -3.15883 q^{28} -5.93900 q^{29} -0.109916 q^{30} +5.26875 q^{31} +4.65279 q^{32} +5.65279 q^{33} -1.69202 q^{34} -0.432960 q^{35} -1.80194 q^{36} +3.19806 q^{37} +2.48427 q^{38} +0.417895 q^{40} +0.445042 q^{41} +0.780167 q^{42} +1.71379 q^{43} -10.1860 q^{44} -0.246980 q^{45} +3.71379 q^{46} -6.73556 q^{47} +2.85086 q^{48} -3.92692 q^{49} -2.19806 q^{50} -3.80194 q^{51} -1.06100 q^{53} +0.445042 q^{54} -1.39612 q^{55} -2.96615 q^{56} +5.58211 q^{57} -2.64310 q^{58} -13.7017 q^{59} +0.445042 q^{60} -8.51573 q^{61} +2.34481 q^{62} +1.75302 q^{63} -3.63102 q^{64} +2.51573 q^{66} +5.96077 q^{67} +6.85086 q^{68} +8.34481 q^{69} -0.192685 q^{70} -5.71917 q^{71} -1.69202 q^{72} +7.35690 q^{73} +1.42327 q^{74} -4.93900 q^{75} -10.0586 q^{76} +9.90946 q^{77} +4.45473 q^{79} -0.704103 q^{80} +1.00000 q^{81} +0.198062 q^{82} -10.1860 q^{83} -3.15883 q^{84} +0.939001 q^{85} +0.762709 q^{86} -5.93900 q^{87} -9.56465 q^{88} +0.137063 q^{89} -0.109916 q^{90} -15.0368 q^{92} +5.26875 q^{93} -2.99761 q^{94} -1.37867 q^{95} +4.65279 q^{96} -13.6896 q^{97} -1.74764 q^{98} +5.65279 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} + 3q^{3} - q^{4} + 4q^{5} + q^{6} + 10q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + q^{2} + 3q^{3} - q^{4} + 4q^{5} + q^{6} + 10q^{7} + 3q^{9} - q^{10} - q^{11} - q^{12} + q^{14} + 4q^{15} - 5q^{16} - 7q^{17} + q^{18} + 11q^{19} + q^{20} + 10q^{21} - 5q^{22} + 2q^{23} - 5q^{25} + 3q^{27} - q^{28} - 8q^{29} - q^{30} + 8q^{31} - 4q^{32} - q^{33} + 18q^{35} - q^{36} + 14q^{37} + 20q^{38} + 7q^{40} + q^{41} + q^{42} - 3q^{43} - 16q^{44} + 4q^{45} + 3q^{46} - 9q^{47} - 5q^{48} + 17q^{49} - 11q^{50} - 7q^{51} - 13q^{53} + q^{54} - 13q^{55} + 7q^{56} + 11q^{57} - 12q^{58} - 14q^{59} + q^{60} - 13q^{61} - 16q^{62} + 10q^{63} + 4q^{64} - 5q^{66} + 5q^{67} + 7q^{68} + 2q^{69} - 8q^{70} - 6q^{71} + 18q^{73} + 7q^{74} - 5q^{75} + q^{76} - 15q^{77} - 9q^{79} - 16q^{80} + 3q^{81} + 5q^{82} - 16q^{83} - q^{84} - 7q^{85} - 15q^{86} - 8q^{87} - 7q^{88} - 5q^{89} - q^{90} - 17q^{92} + 8q^{93} + 32q^{94} + 3q^{95} - 4q^{96} + 5q^{97} - 13q^{98} - q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.445042 0.314692 0.157346 0.987544i \(-0.449706\pi\)
0.157346 + 0.987544i \(0.449706\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.80194 −0.900969
\(5\) −0.246980 −0.110453 −0.0552263 0.998474i \(-0.517588\pi\)
−0.0552263 + 0.998474i \(0.517588\pi\)
\(6\) 0.445042 0.181688
\(7\) 1.75302 0.662579 0.331290 0.943529i \(-0.392516\pi\)
0.331290 + 0.943529i \(0.392516\pi\)
\(8\) −1.69202 −0.598220
\(9\) 1.00000 0.333333
\(10\) −0.109916 −0.0347586
\(11\) 5.65279 1.70438 0.852191 0.523232i \(-0.175274\pi\)
0.852191 + 0.523232i \(0.175274\pi\)
\(12\) −1.80194 −0.520175
\(13\) 0 0
\(14\) 0.780167 0.208509
\(15\) −0.246980 −0.0637699
\(16\) 2.85086 0.712714
\(17\) −3.80194 −0.922105 −0.461053 0.887373i \(-0.652528\pi\)
−0.461053 + 0.887373i \(0.652528\pi\)
\(18\) 0.445042 0.104897
\(19\) 5.58211 1.28062 0.640311 0.768115i \(-0.278805\pi\)
0.640311 + 0.768115i \(0.278805\pi\)
\(20\) 0.445042 0.0995144
\(21\) 1.75302 0.382540
\(22\) 2.51573 0.536355
\(23\) 8.34481 1.74001 0.870007 0.493039i \(-0.164114\pi\)
0.870007 + 0.493039i \(0.164114\pi\)
\(24\) −1.69202 −0.345382
\(25\) −4.93900 −0.987800
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −3.15883 −0.596963
\(29\) −5.93900 −1.10284 −0.551422 0.834226i \(-0.685915\pi\)
−0.551422 + 0.834226i \(0.685915\pi\)
\(30\) −0.109916 −0.0200679
\(31\) 5.26875 0.946295 0.473148 0.880983i \(-0.343118\pi\)
0.473148 + 0.880983i \(0.343118\pi\)
\(32\) 4.65279 0.822505
\(33\) 5.65279 0.984025
\(34\) −1.69202 −0.290179
\(35\) −0.432960 −0.0731836
\(36\) −1.80194 −0.300323
\(37\) 3.19806 0.525758 0.262879 0.964829i \(-0.415328\pi\)
0.262879 + 0.964829i \(0.415328\pi\)
\(38\) 2.48427 0.403002
\(39\) 0 0
\(40\) 0.417895 0.0660750
\(41\) 0.445042 0.0695039 0.0347519 0.999396i \(-0.488936\pi\)
0.0347519 + 0.999396i \(0.488936\pi\)
\(42\) 0.780167 0.120382
\(43\) 1.71379 0.261351 0.130675 0.991425i \(-0.458285\pi\)
0.130675 + 0.991425i \(0.458285\pi\)
\(44\) −10.1860 −1.53559
\(45\) −0.246980 −0.0368175
\(46\) 3.71379 0.547569
\(47\) −6.73556 −0.982483 −0.491241 0.871024i \(-0.663457\pi\)
−0.491241 + 0.871024i \(0.663457\pi\)
\(48\) 2.85086 0.411485
\(49\) −3.92692 −0.560988
\(50\) −2.19806 −0.310853
\(51\) −3.80194 −0.532378
\(52\) 0 0
\(53\) −1.06100 −0.145739 −0.0728697 0.997341i \(-0.523216\pi\)
−0.0728697 + 0.997341i \(0.523216\pi\)
\(54\) 0.445042 0.0605625
\(55\) −1.39612 −0.188253
\(56\) −2.96615 −0.396368
\(57\) 5.58211 0.739368
\(58\) −2.64310 −0.347057
\(59\) −13.7017 −1.78381 −0.891905 0.452222i \(-0.850631\pi\)
−0.891905 + 0.452222i \(0.850631\pi\)
\(60\) 0.445042 0.0574547
\(61\) −8.51573 −1.09033 −0.545164 0.838330i \(-0.683533\pi\)
−0.545164 + 0.838330i \(0.683533\pi\)
\(62\) 2.34481 0.297792
\(63\) 1.75302 0.220860
\(64\) −3.63102 −0.453878
\(65\) 0 0
\(66\) 2.51573 0.309665
\(67\) 5.96077 0.728224 0.364112 0.931355i \(-0.381373\pi\)
0.364112 + 0.931355i \(0.381373\pi\)
\(68\) 6.85086 0.830788
\(69\) 8.34481 1.00460
\(70\) −0.192685 −0.0230303
\(71\) −5.71917 −0.678740 −0.339370 0.940653i \(-0.610214\pi\)
−0.339370 + 0.940653i \(0.610214\pi\)
\(72\) −1.69202 −0.199407
\(73\) 7.35690 0.861060 0.430530 0.902576i \(-0.358327\pi\)
0.430530 + 0.902576i \(0.358327\pi\)
\(74\) 1.42327 0.165452
\(75\) −4.93900 −0.570307
\(76\) −10.0586 −1.15380
\(77\) 9.90946 1.12929
\(78\) 0 0
\(79\) 4.45473 0.501196 0.250598 0.968091i \(-0.419373\pi\)
0.250598 + 0.968091i \(0.419373\pi\)
\(80\) −0.704103 −0.0787211
\(81\) 1.00000 0.111111
\(82\) 0.198062 0.0218723
\(83\) −10.1860 −1.11806 −0.559028 0.829149i \(-0.688826\pi\)
−0.559028 + 0.829149i \(0.688826\pi\)
\(84\) −3.15883 −0.344657
\(85\) 0.939001 0.101849
\(86\) 0.762709 0.0822450
\(87\) −5.93900 −0.636728
\(88\) −9.56465 −1.01959
\(89\) 0.137063 0.0145287 0.00726434 0.999974i \(-0.497688\pi\)
0.00726434 + 0.999974i \(0.497688\pi\)
\(90\) −0.109916 −0.0115862
\(91\) 0 0
\(92\) −15.0368 −1.56770
\(93\) 5.26875 0.546344
\(94\) −2.99761 −0.309180
\(95\) −1.37867 −0.141448
\(96\) 4.65279 0.474874
\(97\) −13.6896 −1.38997 −0.694986 0.719024i \(-0.744589\pi\)
−0.694986 + 0.719024i \(0.744589\pi\)
\(98\) −1.74764 −0.176539
\(99\) 5.65279 0.568127
\(100\) 8.89977 0.889977
\(101\) −5.41119 −0.538434 −0.269217 0.963080i \(-0.586765\pi\)
−0.269217 + 0.963080i \(0.586765\pi\)
\(102\) −1.69202 −0.167535
\(103\) 13.7560 1.35542 0.677710 0.735330i \(-0.262973\pi\)
0.677710 + 0.735330i \(0.262973\pi\)
\(104\) 0 0
\(105\) −0.432960 −0.0422526
\(106\) −0.472189 −0.0458630
\(107\) −12.8170 −1.23907 −0.619533 0.784970i \(-0.712678\pi\)
−0.619533 + 0.784970i \(0.712678\pi\)
\(108\) −1.80194 −0.173392
\(109\) 12.1468 1.16345 0.581724 0.813386i \(-0.302378\pi\)
0.581724 + 0.813386i \(0.302378\pi\)
\(110\) −0.621334 −0.0592419
\(111\) 3.19806 0.303547
\(112\) 4.99761 0.472229
\(113\) −1.63773 −0.154064 −0.0770322 0.997029i \(-0.524544\pi\)
−0.0770322 + 0.997029i \(0.524544\pi\)
\(114\) 2.48427 0.232673
\(115\) −2.06100 −0.192189
\(116\) 10.7017 0.993629
\(117\) 0 0
\(118\) −6.09783 −0.561351
\(119\) −6.66487 −0.610968
\(120\) 0.417895 0.0381484
\(121\) 20.9541 1.90492
\(122\) −3.78986 −0.343117
\(123\) 0.445042 0.0401281
\(124\) −9.49396 −0.852583
\(125\) 2.45473 0.219558
\(126\) 0.780167 0.0695028
\(127\) 10.7995 0.958305 0.479152 0.877732i \(-0.340944\pi\)
0.479152 + 0.877732i \(0.340944\pi\)
\(128\) −10.9215 −0.965337
\(129\) 1.71379 0.150891
\(130\) 0 0
\(131\) 0.907542 0.0792923 0.0396462 0.999214i \(-0.487377\pi\)
0.0396462 + 0.999214i \(0.487377\pi\)
\(132\) −10.1860 −0.886576
\(133\) 9.78554 0.848514
\(134\) 2.65279 0.229166
\(135\) −0.246980 −0.0212566
\(136\) 6.43296 0.551622
\(137\) −9.54825 −0.815762 −0.407881 0.913035i \(-0.633732\pi\)
−0.407881 + 0.913035i \(0.633732\pi\)
\(138\) 3.71379 0.316139
\(139\) −4.09246 −0.347118 −0.173559 0.984823i \(-0.555527\pi\)
−0.173559 + 0.984823i \(0.555527\pi\)
\(140\) 0.780167 0.0659362
\(141\) −6.73556 −0.567237
\(142\) −2.54527 −0.213594
\(143\) 0 0
\(144\) 2.85086 0.237571
\(145\) 1.46681 0.121812
\(146\) 3.27413 0.270969
\(147\) −3.92692 −0.323887
\(148\) −5.76271 −0.473692
\(149\) −15.3884 −1.26066 −0.630332 0.776326i \(-0.717081\pi\)
−0.630332 + 0.776326i \(0.717081\pi\)
\(150\) −2.19806 −0.179471
\(151\) −3.67456 −0.299032 −0.149516 0.988759i \(-0.547772\pi\)
−0.149516 + 0.988759i \(0.547772\pi\)
\(152\) −9.44504 −0.766094
\(153\) −3.80194 −0.307368
\(154\) 4.41013 0.355378
\(155\) −1.30127 −0.104521
\(156\) 0 0
\(157\) −4.87800 −0.389307 −0.194653 0.980872i \(-0.562358\pi\)
−0.194653 + 0.980872i \(0.562358\pi\)
\(158\) 1.98254 0.157723
\(159\) −1.06100 −0.0841427
\(160\) −1.14914 −0.0908479
\(161\) 14.6286 1.15290
\(162\) 0.445042 0.0349658
\(163\) 8.63102 0.676034 0.338017 0.941140i \(-0.390244\pi\)
0.338017 + 0.941140i \(0.390244\pi\)
\(164\) −0.801938 −0.0626208
\(165\) −1.39612 −0.108688
\(166\) −4.53319 −0.351844
\(167\) 9.46980 0.732795 0.366397 0.930458i \(-0.380591\pi\)
0.366397 + 0.930458i \(0.380591\pi\)
\(168\) −2.96615 −0.228843
\(169\) 0 0
\(170\) 0.417895 0.0320511
\(171\) 5.58211 0.426874
\(172\) −3.08815 −0.235469
\(173\) 4.77479 0.363021 0.181510 0.983389i \(-0.441901\pi\)
0.181510 + 0.983389i \(0.441901\pi\)
\(174\) −2.64310 −0.200373
\(175\) −8.65817 −0.654496
\(176\) 16.1153 1.21474
\(177\) −13.7017 −1.02988
\(178\) 0.0609989 0.00457206
\(179\) 3.43535 0.256770 0.128385 0.991724i \(-0.459021\pi\)
0.128385 + 0.991724i \(0.459021\pi\)
\(180\) 0.445042 0.0331715
\(181\) −13.4862 −1.00242 −0.501210 0.865326i \(-0.667112\pi\)
−0.501210 + 0.865326i \(0.667112\pi\)
\(182\) 0 0
\(183\) −8.51573 −0.629501
\(184\) −14.1196 −1.04091
\(185\) −0.789856 −0.0580714
\(186\) 2.34481 0.171930
\(187\) −21.4916 −1.57162
\(188\) 12.1371 0.885186
\(189\) 1.75302 0.127513
\(190\) −0.613564 −0.0445126
\(191\) −1.30127 −0.0941569 −0.0470784 0.998891i \(-0.514991\pi\)
−0.0470784 + 0.998891i \(0.514991\pi\)
\(192\) −3.63102 −0.262046
\(193\) −9.19567 −0.661919 −0.330959 0.943645i \(-0.607372\pi\)
−0.330959 + 0.943645i \(0.607372\pi\)
\(194\) −6.09246 −0.437413
\(195\) 0 0
\(196\) 7.07606 0.505433
\(197\) 4.11231 0.292990 0.146495 0.989211i \(-0.453201\pi\)
0.146495 + 0.989211i \(0.453201\pi\)
\(198\) 2.51573 0.178785
\(199\) −24.7724 −1.75607 −0.878034 0.478598i \(-0.841145\pi\)
−0.878034 + 0.478598i \(0.841145\pi\)
\(200\) 8.35690 0.590922
\(201\) 5.96077 0.420440
\(202\) −2.40821 −0.169441
\(203\) −10.4112 −0.730722
\(204\) 6.85086 0.479656
\(205\) −0.109916 −0.00767688
\(206\) 6.12200 0.426540
\(207\) 8.34481 0.580005
\(208\) 0 0
\(209\) 31.5545 2.18267
\(210\) −0.192685 −0.0132966
\(211\) −5.93900 −0.408858 −0.204429 0.978881i \(-0.565534\pi\)
−0.204429 + 0.978881i \(0.565534\pi\)
\(212\) 1.91185 0.131307
\(213\) −5.71917 −0.391871
\(214\) −5.70410 −0.389924
\(215\) −0.423272 −0.0288669
\(216\) −1.69202 −0.115127
\(217\) 9.23623 0.626996
\(218\) 5.40581 0.366128
\(219\) 7.35690 0.497133
\(220\) 2.51573 0.169610
\(221\) 0 0
\(222\) 1.42327 0.0955237
\(223\) 14.2010 0.950972 0.475486 0.879723i \(-0.342272\pi\)
0.475486 + 0.879723i \(0.342272\pi\)
\(224\) 8.15644 0.544975
\(225\) −4.93900 −0.329267
\(226\) −0.728857 −0.0484829
\(227\) 16.0073 1.06244 0.531221 0.847233i \(-0.321733\pi\)
0.531221 + 0.847233i \(0.321733\pi\)
\(228\) −10.0586 −0.666147
\(229\) 1.84117 0.121668 0.0608339 0.998148i \(-0.480624\pi\)
0.0608339 + 0.998148i \(0.480624\pi\)
\(230\) −0.917231 −0.0604804
\(231\) 9.90946 0.651995
\(232\) 10.0489 0.659744
\(233\) −23.4252 −1.53464 −0.767318 0.641267i \(-0.778409\pi\)
−0.767318 + 0.641267i \(0.778409\pi\)
\(234\) 0 0
\(235\) 1.66355 0.108518
\(236\) 24.6896 1.60716
\(237\) 4.45473 0.289366
\(238\) −2.96615 −0.192267
\(239\) 14.6015 0.944491 0.472246 0.881467i \(-0.343444\pi\)
0.472246 + 0.881467i \(0.343444\pi\)
\(240\) −0.704103 −0.0454497
\(241\) 8.63102 0.555973 0.277987 0.960585i \(-0.410333\pi\)
0.277987 + 0.960585i \(0.410333\pi\)
\(242\) 9.32544 0.599462
\(243\) 1.00000 0.0641500
\(244\) 15.3448 0.982351
\(245\) 0.969869 0.0619627
\(246\) 0.198062 0.0126280
\(247\) 0 0
\(248\) −8.91484 −0.566093
\(249\) −10.1860 −0.645510
\(250\) 1.09246 0.0690931
\(251\) −3.80194 −0.239976 −0.119988 0.992775i \(-0.538286\pi\)
−0.119988 + 0.992775i \(0.538286\pi\)
\(252\) −3.15883 −0.198988
\(253\) 47.1715 2.96565
\(254\) 4.80625 0.301571
\(255\) 0.939001 0.0588025
\(256\) 2.40150 0.150094
\(257\) −20.5961 −1.28475 −0.642375 0.766391i \(-0.722051\pi\)
−0.642375 + 0.766391i \(0.722051\pi\)
\(258\) 0.762709 0.0474842
\(259\) 5.60627 0.348357
\(260\) 0 0
\(261\) −5.93900 −0.367615
\(262\) 0.403894 0.0249527
\(263\) 0.332733 0.0205172 0.0102586 0.999947i \(-0.496735\pi\)
0.0102586 + 0.999947i \(0.496735\pi\)
\(264\) −9.56465 −0.588663
\(265\) 0.262045 0.0160973
\(266\) 4.35498 0.267021
\(267\) 0.137063 0.00838814
\(268\) −10.7409 −0.656107
\(269\) 27.3032 1.66471 0.832353 0.554247i \(-0.186994\pi\)
0.832353 + 0.554247i \(0.186994\pi\)
\(270\) −0.109916 −0.00668929
\(271\) 27.9855 1.70000 0.850000 0.526783i \(-0.176602\pi\)
0.850000 + 0.526783i \(0.176602\pi\)
\(272\) −10.8388 −0.657197
\(273\) 0 0
\(274\) −4.24937 −0.256714
\(275\) −27.9191 −1.68359
\(276\) −15.0368 −0.905111
\(277\) −2.10321 −0.126370 −0.0631849 0.998002i \(-0.520126\pi\)
−0.0631849 + 0.998002i \(0.520126\pi\)
\(278\) −1.82132 −0.109235
\(279\) 5.26875 0.315432
\(280\) 0.732578 0.0437799
\(281\) 27.2349 1.62470 0.812349 0.583172i \(-0.198189\pi\)
0.812349 + 0.583172i \(0.198189\pi\)
\(282\) −2.99761 −0.178505
\(283\) 5.28382 0.314090 0.157045 0.987591i \(-0.449803\pi\)
0.157045 + 0.987591i \(0.449803\pi\)
\(284\) 10.3056 0.611524
\(285\) −1.37867 −0.0816651
\(286\) 0 0
\(287\) 0.780167 0.0460518
\(288\) 4.65279 0.274168
\(289\) −2.54527 −0.149722
\(290\) 0.652793 0.0383333
\(291\) −13.6896 −0.802500
\(292\) −13.2567 −0.775788
\(293\) 32.6625 1.90816 0.954081 0.299548i \(-0.0968358\pi\)
0.954081 + 0.299548i \(0.0968358\pi\)
\(294\) −1.74764 −0.101925
\(295\) 3.38404 0.197027
\(296\) −5.41119 −0.314519
\(297\) 5.65279 0.328008
\(298\) −6.84846 −0.396721
\(299\) 0 0
\(300\) 8.89977 0.513829
\(301\) 3.00431 0.173166
\(302\) −1.63533 −0.0941029
\(303\) −5.41119 −0.310865
\(304\) 15.9138 0.912717
\(305\) 2.10321 0.120430
\(306\) −1.69202 −0.0967264
\(307\) −20.7614 −1.18491 −0.592457 0.805602i \(-0.701842\pi\)
−0.592457 + 0.805602i \(0.701842\pi\)
\(308\) −17.8562 −1.01745
\(309\) 13.7560 0.782552
\(310\) −0.579121 −0.0328919
\(311\) −11.3013 −0.640836 −0.320418 0.947276i \(-0.603823\pi\)
−0.320418 + 0.947276i \(0.603823\pi\)
\(312\) 0 0
\(313\) −4.27173 −0.241453 −0.120726 0.992686i \(-0.538522\pi\)
−0.120726 + 0.992686i \(0.538522\pi\)
\(314\) −2.17092 −0.122512
\(315\) −0.432960 −0.0243945
\(316\) −8.02715 −0.451562
\(317\) 15.4776 0.869307 0.434653 0.900598i \(-0.356871\pi\)
0.434653 + 0.900598i \(0.356871\pi\)
\(318\) −0.472189 −0.0264790
\(319\) −33.5719 −1.87967
\(320\) 0.896789 0.0501320
\(321\) −12.8170 −0.715375
\(322\) 6.51035 0.362808
\(323\) −21.2228 −1.18087
\(324\) −1.80194 −0.100108
\(325\) 0 0
\(326\) 3.84117 0.212743
\(327\) 12.1468 0.671717
\(328\) −0.753020 −0.0415786
\(329\) −11.8076 −0.650973
\(330\) −0.621334 −0.0342033
\(331\) 6.06829 0.333544 0.166772 0.985996i \(-0.446666\pi\)
0.166772 + 0.985996i \(0.446666\pi\)
\(332\) 18.3545 1.00733
\(333\) 3.19806 0.175253
\(334\) 4.21446 0.230605
\(335\) −1.47219 −0.0804343
\(336\) 4.99761 0.272642
\(337\) 12.1239 0.660432 0.330216 0.943905i \(-0.392878\pi\)
0.330216 + 0.943905i \(0.392878\pi\)
\(338\) 0 0
\(339\) −1.63773 −0.0889491
\(340\) −1.69202 −0.0917627
\(341\) 29.7832 1.61285
\(342\) 2.48427 0.134334
\(343\) −19.1551 −1.03428
\(344\) −2.89977 −0.156345
\(345\) −2.06100 −0.110960
\(346\) 2.12498 0.114240
\(347\) 23.1497 1.24274 0.621371 0.783516i \(-0.286576\pi\)
0.621371 + 0.783516i \(0.286576\pi\)
\(348\) 10.7017 0.573672
\(349\) −22.1957 −1.18811 −0.594053 0.804426i \(-0.702473\pi\)
−0.594053 + 0.804426i \(0.702473\pi\)
\(350\) −3.85325 −0.205965
\(351\) 0 0
\(352\) 26.3013 1.40186
\(353\) −5.07069 −0.269885 −0.134943 0.990853i \(-0.543085\pi\)
−0.134943 + 0.990853i \(0.543085\pi\)
\(354\) −6.09783 −0.324096
\(355\) 1.41252 0.0749687
\(356\) −0.246980 −0.0130899
\(357\) −6.66487 −0.352743
\(358\) 1.52888 0.0808036
\(359\) −16.6746 −0.880050 −0.440025 0.897986i \(-0.645030\pi\)
−0.440025 + 0.897986i \(0.645030\pi\)
\(360\) 0.417895 0.0220250
\(361\) 12.1599 0.639995
\(362\) −6.00192 −0.315454
\(363\) 20.9541 1.09980
\(364\) 0 0
\(365\) −1.81700 −0.0951063
\(366\) −3.78986 −0.198099
\(367\) −1.17928 −0.0615577 −0.0307789 0.999526i \(-0.509799\pi\)
−0.0307789 + 0.999526i \(0.509799\pi\)
\(368\) 23.7899 1.24013
\(369\) 0.445042 0.0231680
\(370\) −0.351519 −0.0182746
\(371\) −1.85995 −0.0965639
\(372\) −9.49396 −0.492239
\(373\) −30.0925 −1.55813 −0.779064 0.626944i \(-0.784305\pi\)
−0.779064 + 0.626944i \(0.784305\pi\)
\(374\) −9.56465 −0.494576
\(375\) 2.45473 0.126762
\(376\) 11.3967 0.587741
\(377\) 0 0
\(378\) 0.780167 0.0401275
\(379\) −19.1631 −0.984345 −0.492172 0.870498i \(-0.663797\pi\)
−0.492172 + 0.870498i \(0.663797\pi\)
\(380\) 2.48427 0.127440
\(381\) 10.7995 0.553277
\(382\) −0.579121 −0.0296304
\(383\) −15.3884 −0.786308 −0.393154 0.919473i \(-0.628616\pi\)
−0.393154 + 0.919473i \(0.628616\pi\)
\(384\) −10.9215 −0.557338
\(385\) −2.44743 −0.124733
\(386\) −4.09246 −0.208301
\(387\) 1.71379 0.0871169
\(388\) 24.6679 1.25232
\(389\) −24.0315 −1.21844 −0.609222 0.793000i \(-0.708518\pi\)
−0.609222 + 0.793000i \(0.708518\pi\)
\(390\) 0 0
\(391\) −31.7265 −1.60448
\(392\) 6.64443 0.335594
\(393\) 0.907542 0.0457794
\(394\) 1.83015 0.0922016
\(395\) −1.10023 −0.0553585
\(396\) −10.1860 −0.511865
\(397\) 29.6015 1.48566 0.742828 0.669482i \(-0.233484\pi\)
0.742828 + 0.669482i \(0.233484\pi\)
\(398\) −11.0248 −0.552621
\(399\) 9.78554 0.489890
\(400\) −14.0804 −0.704019
\(401\) −21.1032 −1.05384 −0.526922 0.849914i \(-0.676654\pi\)
−0.526922 + 0.849914i \(0.676654\pi\)
\(402\) 2.65279 0.132309
\(403\) 0 0
\(404\) 9.75063 0.485112
\(405\) −0.246980 −0.0122725
\(406\) −4.63342 −0.229953
\(407\) 18.0780 0.896092
\(408\) 6.43296 0.318479
\(409\) −36.0224 −1.78119 −0.890596 0.454796i \(-0.849712\pi\)
−0.890596 + 0.454796i \(0.849712\pi\)
\(410\) −0.0489173 −0.00241586
\(411\) −9.54825 −0.470981
\(412\) −24.7875 −1.22119
\(413\) −24.0194 −1.18192
\(414\) 3.71379 0.182523
\(415\) 2.51573 0.123492
\(416\) 0 0
\(417\) −4.09246 −0.200409
\(418\) 14.0431 0.686869
\(419\) 5.96854 0.291582 0.145791 0.989315i \(-0.453427\pi\)
0.145791 + 0.989315i \(0.453427\pi\)
\(420\) 0.780167 0.0380683
\(421\) −2.09544 −0.102126 −0.0510628 0.998695i \(-0.516261\pi\)
−0.0510628 + 0.998695i \(0.516261\pi\)
\(422\) −2.64310 −0.128664
\(423\) −6.73556 −0.327494
\(424\) 1.79523 0.0871842
\(425\) 18.7778 0.910856
\(426\) −2.54527 −0.123319
\(427\) −14.9282 −0.722429
\(428\) 23.0954 1.11636
\(429\) 0 0
\(430\) −0.188374 −0.00908418
\(431\) −2.88577 −0.139003 −0.0695014 0.997582i \(-0.522141\pi\)
−0.0695014 + 0.997582i \(0.522141\pi\)
\(432\) 2.85086 0.137162
\(433\) −12.6485 −0.607847 −0.303924 0.952696i \(-0.598297\pi\)
−0.303924 + 0.952696i \(0.598297\pi\)
\(434\) 4.11051 0.197311
\(435\) 1.46681 0.0703283
\(436\) −21.8877 −1.04823
\(437\) 46.5816 2.22830
\(438\) 3.27413 0.156444
\(439\) −10.9269 −0.521513 −0.260757 0.965405i \(-0.583972\pi\)
−0.260757 + 0.965405i \(0.583972\pi\)
\(440\) 2.36227 0.112617
\(441\) −3.92692 −0.186996
\(442\) 0 0
\(443\) −19.2403 −0.914133 −0.457067 0.889433i \(-0.651100\pi\)
−0.457067 + 0.889433i \(0.651100\pi\)
\(444\) −5.76271 −0.273486
\(445\) −0.0338518 −0.00160473
\(446\) 6.32006 0.299264
\(447\) −15.3884 −0.727844
\(448\) −6.36526 −0.300730
\(449\) −28.8200 −1.36010 −0.680050 0.733166i \(-0.738042\pi\)
−0.680050 + 0.733166i \(0.738042\pi\)
\(450\) −2.19806 −0.103618
\(451\) 2.51573 0.118461
\(452\) 2.95108 0.138807
\(453\) −3.67456 −0.172646
\(454\) 7.12392 0.334342
\(455\) 0 0
\(456\) −9.44504 −0.442305
\(457\) −18.0707 −0.845311 −0.422656 0.906290i \(-0.638902\pi\)
−0.422656 + 0.906290i \(0.638902\pi\)
\(458\) 0.819396 0.0382879
\(459\) −3.80194 −0.177459
\(460\) 3.71379 0.173156
\(461\) −7.56763 −0.352460 −0.176230 0.984349i \(-0.556390\pi\)
−0.176230 + 0.984349i \(0.556390\pi\)
\(462\) 4.41013 0.205178
\(463\) 35.3551 1.64309 0.821545 0.570143i \(-0.193112\pi\)
0.821545 + 0.570143i \(0.193112\pi\)
\(464\) −16.9312 −0.786013
\(465\) −1.30127 −0.0603451
\(466\) −10.4252 −0.482938
\(467\) 13.0000 0.601568 0.300784 0.953692i \(-0.402752\pi\)
0.300784 + 0.953692i \(0.402752\pi\)
\(468\) 0 0
\(469\) 10.4494 0.482506
\(470\) 0.740348 0.0341497
\(471\) −4.87800 −0.224766
\(472\) 23.1836 1.06711
\(473\) 9.68771 0.445441
\(474\) 1.98254 0.0910612
\(475\) −27.5700 −1.26500
\(476\) 12.0097 0.550463
\(477\) −1.06100 −0.0485798
\(478\) 6.49827 0.297224
\(479\) −25.5265 −1.16633 −0.583167 0.812352i \(-0.698187\pi\)
−0.583167 + 0.812352i \(0.698187\pi\)
\(480\) −1.14914 −0.0524510
\(481\) 0 0
\(482\) 3.84117 0.174960
\(483\) 14.6286 0.665626
\(484\) −37.7579 −1.71627
\(485\) 3.38106 0.153526
\(486\) 0.445042 0.0201875
\(487\) 16.0073 0.725360 0.362680 0.931914i \(-0.381862\pi\)
0.362680 + 0.931914i \(0.381862\pi\)
\(488\) 14.4088 0.652256
\(489\) 8.63102 0.390308
\(490\) 0.431632 0.0194992
\(491\) 20.7385 0.935917 0.467959 0.883750i \(-0.344990\pi\)
0.467959 + 0.883750i \(0.344990\pi\)
\(492\) −0.801938 −0.0361541
\(493\) 22.5797 1.01694
\(494\) 0 0
\(495\) −1.39612 −0.0627511
\(496\) 15.0204 0.674438
\(497\) −10.0258 −0.449719
\(498\) −4.53319 −0.203137
\(499\) −8.06770 −0.361160 −0.180580 0.983560i \(-0.557797\pi\)
−0.180580 + 0.983560i \(0.557797\pi\)
\(500\) −4.42327 −0.197815
\(501\) 9.46980 0.423079
\(502\) −1.69202 −0.0755186
\(503\) −30.2422 −1.34843 −0.674216 0.738534i \(-0.735518\pi\)
−0.674216 + 0.738534i \(0.735518\pi\)
\(504\) −2.96615 −0.132123
\(505\) 1.33645 0.0594714
\(506\) 20.9933 0.933266
\(507\) 0 0
\(508\) −19.4601 −0.863403
\(509\) 16.0495 0.711382 0.355691 0.934604i \(-0.384246\pi\)
0.355691 + 0.934604i \(0.384246\pi\)
\(510\) 0.417895 0.0185047
\(511\) 12.8968 0.570520
\(512\) 22.9119 1.01257
\(513\) 5.58211 0.246456
\(514\) −9.16613 −0.404301
\(515\) −3.39745 −0.149710
\(516\) −3.08815 −0.135948
\(517\) −38.0747 −1.67452
\(518\) 2.49502 0.109625
\(519\) 4.77479 0.209590
\(520\) 0 0
\(521\) −2.69309 −0.117986 −0.0589931 0.998258i \(-0.518789\pi\)
−0.0589931 + 0.998258i \(0.518789\pi\)
\(522\) −2.64310 −0.115686
\(523\) 35.3957 1.54774 0.773872 0.633342i \(-0.218317\pi\)
0.773872 + 0.633342i \(0.218317\pi\)
\(524\) −1.63533 −0.0714399
\(525\) −8.65817 −0.377874
\(526\) 0.148080 0.00645659
\(527\) −20.0315 −0.872584
\(528\) 16.1153 0.701328
\(529\) 46.6359 2.02765
\(530\) 0.116621 0.00506569
\(531\) −13.7017 −0.594604
\(532\) −17.6329 −0.764485
\(533\) 0 0
\(534\) 0.0609989 0.00263968
\(535\) 3.16554 0.136858
\(536\) −10.0858 −0.435638
\(537\) 3.43535 0.148246
\(538\) 12.1511 0.523870
\(539\) −22.1981 −0.956138
\(540\) 0.445042 0.0191516
\(541\) −34.7338 −1.49332 −0.746660 0.665205i \(-0.768344\pi\)
−0.746660 + 0.665205i \(0.768344\pi\)
\(542\) 12.4547 0.534976
\(543\) −13.4862 −0.578748
\(544\) −17.6896 −0.758437
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) −26.1183 −1.11674 −0.558368 0.829593i \(-0.688572\pi\)
−0.558368 + 0.829593i \(0.688572\pi\)
\(548\) 17.2054 0.734976
\(549\) −8.51573 −0.363442
\(550\) −12.4252 −0.529812
\(551\) −33.1521 −1.41233
\(552\) −14.1196 −0.600970
\(553\) 7.80923 0.332082
\(554\) −0.936017 −0.0397676
\(555\) −0.789856 −0.0335275
\(556\) 7.37435 0.312742
\(557\) 24.7748 1.04974 0.524871 0.851182i \(-0.324114\pi\)
0.524871 + 0.851182i \(0.324114\pi\)
\(558\) 2.34481 0.0992639
\(559\) 0 0
\(560\) −1.23431 −0.0521590
\(561\) −21.4916 −0.907375
\(562\) 12.1207 0.511280
\(563\) 5.26098 0.221724 0.110862 0.993836i \(-0.464639\pi\)
0.110862 + 0.993836i \(0.464639\pi\)
\(564\) 12.1371 0.511063
\(565\) 0.404485 0.0170168
\(566\) 2.35152 0.0988417
\(567\) 1.75302 0.0736199
\(568\) 9.67696 0.406036
\(569\) −33.7458 −1.41470 −0.707350 0.706864i \(-0.750109\pi\)
−0.707350 + 0.706864i \(0.750109\pi\)
\(570\) −0.613564 −0.0256994
\(571\) 23.0887 0.966234 0.483117 0.875556i \(-0.339505\pi\)
0.483117 + 0.875556i \(0.339505\pi\)
\(572\) 0 0
\(573\) −1.30127 −0.0543615
\(574\) 0.347207 0.0144921
\(575\) −41.2150 −1.71879
\(576\) −3.63102 −0.151293
\(577\) 3.57002 0.148622 0.0743110 0.997235i \(-0.476324\pi\)
0.0743110 + 0.997235i \(0.476324\pi\)
\(578\) −1.13275 −0.0471162
\(579\) −9.19567 −0.382159
\(580\) −2.64310 −0.109749
\(581\) −17.8562 −0.740801
\(582\) −6.09246 −0.252541
\(583\) −5.99761 −0.248396
\(584\) −12.4480 −0.515103
\(585\) 0 0
\(586\) 14.5362 0.600484
\(587\) −11.4625 −0.473108 −0.236554 0.971618i \(-0.576018\pi\)
−0.236554 + 0.971618i \(0.576018\pi\)
\(588\) 7.07606 0.291812
\(589\) 29.4107 1.21185
\(590\) 1.50604 0.0620027
\(591\) 4.11231 0.169158
\(592\) 9.11721 0.374715
\(593\) −21.8538 −0.897430 −0.448715 0.893675i \(-0.648118\pi\)
−0.448715 + 0.893675i \(0.648118\pi\)
\(594\) 2.51573 0.103222
\(595\) 1.64609 0.0674830
\(596\) 27.7289 1.13582
\(597\) −24.7724 −1.01387
\(598\) 0 0
\(599\) 27.0573 1.10553 0.552765 0.833337i \(-0.313573\pi\)
0.552765 + 0.833337i \(0.313573\pi\)
\(600\) 8.35690 0.341169
\(601\) 10.8780 0.443723 0.221861 0.975078i \(-0.428787\pi\)
0.221861 + 0.975078i \(0.428787\pi\)
\(602\) 1.33704 0.0544939
\(603\) 5.96077 0.242741
\(604\) 6.62133 0.269418
\(605\) −5.17523 −0.210403
\(606\) −2.40821 −0.0978267
\(607\) −29.6359 −1.20289 −0.601443 0.798916i \(-0.705407\pi\)
−0.601443 + 0.798916i \(0.705407\pi\)
\(608\) 25.9724 1.05332
\(609\) −10.4112 −0.421883
\(610\) 0.936017 0.0378982
\(611\) 0 0
\(612\) 6.85086 0.276929
\(613\) 10.2343 0.413360 0.206680 0.978409i \(-0.433734\pi\)
0.206680 + 0.978409i \(0.433734\pi\)
\(614\) −9.23968 −0.372883
\(615\) −0.109916 −0.00443225
\(616\) −16.7670 −0.675563
\(617\) 26.2828 1.05810 0.529052 0.848590i \(-0.322548\pi\)
0.529052 + 0.848590i \(0.322548\pi\)
\(618\) 6.12200 0.246263
\(619\) 29.0834 1.16896 0.584479 0.811408i \(-0.301299\pi\)
0.584479 + 0.811408i \(0.301299\pi\)
\(620\) 2.34481 0.0941700
\(621\) 8.34481 0.334866
\(622\) −5.02954 −0.201666
\(623\) 0.240275 0.00962641
\(624\) 0 0
\(625\) 24.0887 0.963549
\(626\) −1.90110 −0.0759833
\(627\) 31.5545 1.26016
\(628\) 8.78986 0.350753
\(629\) −12.1588 −0.484804
\(630\) −0.192685 −0.00767677
\(631\) 25.4480 1.01307 0.506535 0.862219i \(-0.330926\pi\)
0.506535 + 0.862219i \(0.330926\pi\)
\(632\) −7.53750 −0.299826
\(633\) −5.93900 −0.236054
\(634\) 6.88816 0.273564
\(635\) −2.66727 −0.105847
\(636\) 1.91185 0.0758099
\(637\) 0 0
\(638\) −14.9409 −0.591517
\(639\) −5.71917 −0.226247
\(640\) 2.69740 0.106624
\(641\) −26.7409 −1.05620 −0.528102 0.849181i \(-0.677096\pi\)
−0.528102 + 0.849181i \(0.677096\pi\)
\(642\) −5.70410 −0.225123
\(643\) −32.9614 −1.29987 −0.649935 0.759990i \(-0.725204\pi\)
−0.649935 + 0.759990i \(0.725204\pi\)
\(644\) −26.3599 −1.03872
\(645\) −0.423272 −0.0166663
\(646\) −9.44504 −0.371610
\(647\) 34.4946 1.35612 0.678060 0.735006i \(-0.262821\pi\)
0.678060 + 0.735006i \(0.262821\pi\)
\(648\) −1.69202 −0.0664689
\(649\) −77.4529 −3.04029
\(650\) 0 0
\(651\) 9.23623 0.361996
\(652\) −15.5526 −0.609085
\(653\) 36.1517 1.41472 0.707362 0.706852i \(-0.249885\pi\)
0.707362 + 0.706852i \(0.249885\pi\)
\(654\) 5.40581 0.211384
\(655\) −0.224144 −0.00875805
\(656\) 1.26875 0.0495364
\(657\) 7.35690 0.287020
\(658\) −5.25487 −0.204856
\(659\) −6.81700 −0.265553 −0.132776 0.991146i \(-0.542389\pi\)
−0.132776 + 0.991146i \(0.542389\pi\)
\(660\) 2.51573 0.0979246
\(661\) −10.8944 −0.423743 −0.211871 0.977298i \(-0.567956\pi\)
−0.211871 + 0.977298i \(0.567956\pi\)
\(662\) 2.70065 0.104964
\(663\) 0 0
\(664\) 17.2349 0.668844
\(665\) −2.41683 −0.0937206
\(666\) 1.42327 0.0551507
\(667\) −49.5599 −1.91897
\(668\) −17.0640 −0.660225
\(669\) 14.2010 0.549044
\(670\) −0.655186 −0.0253120
\(671\) −48.1377 −1.85833
\(672\) 8.15644 0.314642
\(673\) 20.7385 0.799412 0.399706 0.916643i \(-0.369112\pi\)
0.399706 + 0.916643i \(0.369112\pi\)
\(674\) 5.39565 0.207833
\(675\) −4.93900 −0.190102
\(676\) 0 0
\(677\) −25.5786 −0.983067 −0.491534 0.870859i \(-0.663564\pi\)
−0.491534 + 0.870859i \(0.663564\pi\)
\(678\) −0.728857 −0.0279916
\(679\) −23.9982 −0.920966
\(680\) −1.58881 −0.0609281
\(681\) 16.0073 0.613401
\(682\) 13.2547 0.507551
\(683\) 21.6310 0.827688 0.413844 0.910348i \(-0.364186\pi\)
0.413844 + 0.910348i \(0.364186\pi\)
\(684\) −10.0586 −0.384600
\(685\) 2.35822 0.0901031
\(686\) −8.52483 −0.325479
\(687\) 1.84117 0.0702449
\(688\) 4.88577 0.186268
\(689\) 0 0
\(690\) −0.917231 −0.0349184
\(691\) 2.62996 0.100048 0.0500242 0.998748i \(-0.484070\pi\)
0.0500242 + 0.998748i \(0.484070\pi\)
\(692\) −8.60388 −0.327070
\(693\) 9.90946 0.376429
\(694\) 10.3026 0.391081
\(695\) 1.01075 0.0383401
\(696\) 10.0489 0.380903
\(697\) −1.69202 −0.0640899
\(698\) −9.87800 −0.373888
\(699\) −23.4252 −0.886022
\(700\) 15.6015 0.589681
\(701\) 40.0925 1.51427 0.757136 0.653258i \(-0.226598\pi\)
0.757136 + 0.653258i \(0.226598\pi\)
\(702\) 0 0
\(703\) 17.8519 0.673298
\(704\) −20.5254 −0.773581
\(705\) 1.66355 0.0626528
\(706\) −2.25667 −0.0849308
\(707\) −9.48593 −0.356755
\(708\) 24.6896 0.927893
\(709\) 23.2097 0.871657 0.435829 0.900030i \(-0.356455\pi\)
0.435829 + 0.900030i \(0.356455\pi\)
\(710\) 0.628630 0.0235921
\(711\) 4.45473 0.167065
\(712\) −0.231914 −0.00869135
\(713\) 43.9667 1.64657
\(714\) −2.96615 −0.111005
\(715\) 0 0
\(716\) −6.19029 −0.231342
\(717\) 14.6015 0.545302
\(718\) −7.42088 −0.276945
\(719\) −26.0146 −0.970181 −0.485090 0.874464i \(-0.661213\pi\)
−0.485090 + 0.874464i \(0.661213\pi\)
\(720\) −0.704103 −0.0262404
\(721\) 24.1146 0.898073
\(722\) 5.41166 0.201401
\(723\) 8.63102 0.320991
\(724\) 24.3013 0.903150
\(725\) 29.3327 1.08939
\(726\) 9.32544 0.346099
\(727\) −16.5472 −0.613701 −0.306851 0.951758i \(-0.599275\pi\)
−0.306851 + 0.951758i \(0.599275\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −0.808643 −0.0299292
\(731\) −6.51573 −0.240993
\(732\) 15.3448 0.567161
\(733\) 18.8750 0.697165 0.348582 0.937278i \(-0.386663\pi\)
0.348582 + 0.937278i \(0.386663\pi\)
\(734\) −0.524827 −0.0193717
\(735\) 0.969869 0.0357742
\(736\) 38.8267 1.43117
\(737\) 33.6950 1.24117
\(738\) 0.198062 0.00729077
\(739\) 47.3239 1.74084 0.870419 0.492312i \(-0.163848\pi\)
0.870419 + 0.492312i \(0.163848\pi\)
\(740\) 1.42327 0.0523205
\(741\) 0 0
\(742\) −0.827757 −0.0303879
\(743\) −8.88769 −0.326058 −0.163029 0.986621i \(-0.552126\pi\)
−0.163029 + 0.986621i \(0.552126\pi\)
\(744\) −8.91484 −0.326834
\(745\) 3.80061 0.139244
\(746\) −13.3924 −0.490331
\(747\) −10.1860 −0.372686
\(748\) 38.7265 1.41598
\(749\) −22.4685 −0.820980
\(750\) 1.09246 0.0398909
\(751\) 0.710808 0.0259377 0.0129689 0.999916i \(-0.495872\pi\)
0.0129689 + 0.999916i \(0.495872\pi\)
\(752\) −19.2021 −0.700229
\(753\) −3.80194 −0.138550
\(754\) 0 0
\(755\) 0.907542 0.0330288
\(756\) −3.15883 −0.114886
\(757\) 9.78554 0.355662 0.177831 0.984061i \(-0.443092\pi\)
0.177831 + 0.984061i \(0.443092\pi\)
\(758\) −8.52840 −0.309766
\(759\) 47.1715 1.71222
\(760\) 2.33273 0.0846171
\(761\) −18.8810 −0.684435 −0.342218 0.939621i \(-0.611178\pi\)
−0.342218 + 0.939621i \(0.611178\pi\)
\(762\) 4.80625 0.174112
\(763\) 21.2935 0.770877
\(764\) 2.34481 0.0848324
\(765\) 0.939001 0.0339497
\(766\) −6.84846 −0.247445
\(767\) 0 0
\(768\) 2.40150 0.0866567
\(769\) −12.4349 −0.448413 −0.224207 0.974542i \(-0.571979\pi\)
−0.224207 + 0.974542i \(0.571979\pi\)
\(770\) −1.08921 −0.0392524
\(771\) −20.5961 −0.741751
\(772\) 16.5700 0.596368
\(773\) 45.6746 1.64280 0.821400 0.570353i \(-0.193193\pi\)
0.821400 + 0.570353i \(0.193193\pi\)
\(774\) 0.762709 0.0274150
\(775\) −26.0224 −0.934751
\(776\) 23.1631 0.831508
\(777\) 5.60627 0.201124
\(778\) −10.6950 −0.383435
\(779\) 2.48427 0.0890082
\(780\) 0 0
\(781\) −32.3293 −1.15683
\(782\) −14.1196 −0.504916
\(783\) −5.93900 −0.212243
\(784\) −11.1951 −0.399824
\(785\) 1.20477 0.0430000
\(786\) 0.403894 0.0144064
\(787\) −4.51871 −0.161075 −0.0805374 0.996752i \(-0.525664\pi\)
−0.0805374 + 0.996752i \(0.525664\pi\)
\(788\) −7.41013 −0.263975
\(789\) 0.332733 0.0118456
\(790\) −0.489647 −0.0174209
\(791\) −2.87097 −0.102080
\(792\) −9.56465 −0.339865
\(793\) 0 0
\(794\) 13.1739 0.467524
\(795\) 0.262045 0.00929378
\(796\) 44.6383 1.58216
\(797\) −28.7391 −1.01799 −0.508996 0.860769i \(-0.669983\pi\)
−0.508996 + 0.860769i \(0.669983\pi\)
\(798\) 4.35498 0.154165
\(799\) 25.6082 0.905953
\(800\) −22.9801 −0.812471
\(801\) 0.137063 0.00484289
\(802\) −9.39181 −0.331636
\(803\) 41.5870 1.46757
\(804\) −10.7409 −0.378804
\(805\) −3.61297 −0.127341
\(806\) 0 0
\(807\) 27.3032 0.961118
\(808\) 9.15585 0.322102
\(809\) 5.42891 0.190870 0.0954352 0.995436i \(-0.469576\pi\)
0.0954352 + 0.995436i \(0.469576\pi\)
\(810\) −0.109916 −0.00386206
\(811\) 0.629104 0.0220908 0.0110454 0.999939i \(-0.496484\pi\)
0.0110454 + 0.999939i \(0.496484\pi\)
\(812\) 18.7603 0.658358
\(813\) 27.9855 0.981495
\(814\) 8.04546 0.281993
\(815\) −2.13169 −0.0746697
\(816\) −10.8388 −0.379433
\(817\) 9.56657 0.334692
\(818\) −16.0315 −0.560527
\(819\) 0 0
\(820\) 0.198062 0.00691663
\(821\) 36.2640 1.26562 0.632811 0.774307i \(-0.281901\pi\)
0.632811 + 0.774307i \(0.281901\pi\)
\(822\) −4.24937 −0.148214
\(823\) 41.7396 1.45495 0.727476 0.686133i \(-0.240693\pi\)
0.727476 + 0.686133i \(0.240693\pi\)
\(824\) −23.2755 −0.810839
\(825\) −27.9191 −0.972020
\(826\) −10.6896 −0.371940
\(827\) 38.1997 1.32833 0.664167 0.747584i \(-0.268786\pi\)
0.664167 + 0.747584i \(0.268786\pi\)
\(828\) −15.0368 −0.522566
\(829\) 15.9788 0.554967 0.277484 0.960730i \(-0.410500\pi\)
0.277484 + 0.960730i \(0.410500\pi\)
\(830\) 1.11960 0.0388621
\(831\) −2.10321 −0.0729596
\(832\) 0 0
\(833\) 14.9299 0.517290
\(834\) −1.82132 −0.0630670
\(835\) −2.33885 −0.0809391
\(836\) −56.8592 −1.96652
\(837\) 5.26875 0.182115
\(838\) 2.65625 0.0917587
\(839\) −4.63879 −0.160149 −0.0800744 0.996789i \(-0.525516\pi\)
−0.0800744 + 0.996789i \(0.525516\pi\)
\(840\) 0.732578 0.0252763
\(841\) 6.27173 0.216267
\(842\) −0.932559 −0.0321381
\(843\) 27.2349 0.938020
\(844\) 10.7017 0.368368
\(845\) 0 0
\(846\) −2.99761 −0.103060
\(847\) 36.7329 1.26216
\(848\) −3.02475 −0.103870
\(849\) 5.28382 0.181340
\(850\) 8.35690 0.286639
\(851\) 26.6872 0.914827
\(852\) 10.3056 0.353064
\(853\) 18.3884 0.629605 0.314803 0.949157i \(-0.398062\pi\)
0.314803 + 0.949157i \(0.398062\pi\)
\(854\) −6.64370 −0.227343
\(855\) −1.37867 −0.0471494
\(856\) 21.6866 0.741234
\(857\) 28.1849 0.962778 0.481389 0.876507i \(-0.340132\pi\)
0.481389 + 0.876507i \(0.340132\pi\)
\(858\) 0 0
\(859\) 33.3957 1.13944 0.569722 0.821837i \(-0.307051\pi\)
0.569722 + 0.821837i \(0.307051\pi\)
\(860\) 0.762709 0.0260082
\(861\) 0.780167 0.0265880
\(862\) −1.28429 −0.0437431
\(863\) −43.0640 −1.46592 −0.732958 0.680274i \(-0.761861\pi\)
−0.732958 + 0.680274i \(0.761861\pi\)
\(864\) 4.65279 0.158291
\(865\) −1.17928 −0.0400966
\(866\) −5.62910 −0.191285
\(867\) −2.54527 −0.0864419
\(868\) −16.6431 −0.564904
\(869\) 25.1817 0.854230
\(870\) 0.652793 0.0221317
\(871\) 0 0
\(872\) −20.5526 −0.695998
\(873\) −13.6896 −0.463324
\(874\) 20.7308 0.701229
\(875\) 4.30319 0.145474
\(876\) −13.2567 −0.447901
\(877\) −30.1702 −1.01877 −0.509387 0.860537i \(-0.670128\pi\)
−0.509387 + 0.860537i \(0.670128\pi\)
\(878\) −4.86294 −0.164116
\(879\) 32.6625 1.10168
\(880\) −3.98015 −0.134171
\(881\) −3.56273 −0.120031 −0.0600157 0.998197i \(-0.519115\pi\)
−0.0600157 + 0.998197i \(0.519115\pi\)
\(882\) −1.74764 −0.0588462
\(883\) −10.2088 −0.343554 −0.171777 0.985136i \(-0.554951\pi\)
−0.171777 + 0.985136i \(0.554951\pi\)
\(884\) 0 0
\(885\) 3.38404 0.113753
\(886\) −8.56273 −0.287670
\(887\) −9.33645 −0.313487 −0.156744 0.987639i \(-0.550100\pi\)
−0.156744 + 0.987639i \(0.550100\pi\)
\(888\) −5.41119 −0.181588
\(889\) 18.9318 0.634953
\(890\) −0.0150655 −0.000504996 0
\(891\) 5.65279 0.189376
\(892\) −25.5894 −0.856797
\(893\) −37.5986 −1.25819
\(894\) −6.84846 −0.229047
\(895\) −0.848462 −0.0283610
\(896\) −19.1457 −0.639613
\(897\) 0 0
\(898\) −12.8261 −0.428013
\(899\) −31.2911 −1.04362
\(900\) 8.89977 0.296659
\(901\) 4.03385 0.134387
\(902\) 1.11960 0.0372788
\(903\) 3.00431 0.0999772
\(904\) 2.77107 0.0921644
\(905\) 3.33081 0.110720
\(906\) −1.63533 −0.0543303
\(907\) −41.0804 −1.36405 −0.682026 0.731328i \(-0.738901\pi\)
−0.682026 + 0.731328i \(0.738901\pi\)
\(908\) −28.8442 −0.957227
\(909\) −5.41119 −0.179478
\(910\) 0 0
\(911\) −18.9705 −0.628519 −0.314260 0.949337i \(-0.601756\pi\)
−0.314260 + 0.949337i \(0.601756\pi\)
\(912\) 15.9138 0.526958
\(913\) −57.5792 −1.90559
\(914\) −8.04221 −0.266013
\(915\) 2.10321 0.0695300
\(916\) −3.31767 −0.109619
\(917\) 1.59094 0.0525375
\(918\) −1.69202 −0.0558450
\(919\) 29.0019 0.956685 0.478343 0.878173i \(-0.341238\pi\)
0.478343 + 0.878173i \(0.341238\pi\)
\(920\) 3.48725 0.114971
\(921\) −20.7614 −0.684111
\(922\) −3.36791 −0.110916
\(923\) 0 0
\(924\) −17.8562 −0.587427
\(925\) −15.7952 −0.519344
\(926\) 15.7345 0.517068
\(927\) 13.7560 0.451806
\(928\) −27.6329 −0.907096
\(929\) −50.7211 −1.66410 −0.832052 0.554697i \(-0.812834\pi\)
−0.832052 + 0.554697i \(0.812834\pi\)
\(930\) −0.579121 −0.0189901
\(931\) −21.9205 −0.718415
\(932\) 42.2107 1.38266
\(933\) −11.3013 −0.369987
\(934\) 5.78554 0.189309
\(935\) 5.30798 0.173589
\(936\) 0 0
\(937\) 51.3051 1.67606 0.838032 0.545620i \(-0.183706\pi\)
0.838032 + 0.545620i \(0.183706\pi\)
\(938\) 4.65040 0.151841
\(939\) −4.27173 −0.139403
\(940\) −2.99761 −0.0977712
\(941\) −34.7036 −1.13131 −0.565653 0.824643i \(-0.691376\pi\)
−0.565653 + 0.824643i \(0.691376\pi\)
\(942\) −2.17092 −0.0707322
\(943\) 3.71379 0.120938
\(944\) −39.0616 −1.27135
\(945\) −0.432960 −0.0140842
\(946\) 4.31144 0.140177
\(947\) −13.0127 −0.422855 −0.211428 0.977394i \(-0.567811\pi\)
−0.211428 + 0.977394i \(0.567811\pi\)
\(948\) −8.02715 −0.260710
\(949\) 0 0
\(950\) −12.2698 −0.398085
\(951\) 15.4776 0.501894
\(952\) 11.2771 0.365493
\(953\) 26.0151 0.842711 0.421355 0.906896i \(-0.361555\pi\)
0.421355 + 0.906896i \(0.361555\pi\)
\(954\) −0.472189 −0.0152877
\(955\) 0.321388 0.0103999
\(956\) −26.3110 −0.850957
\(957\) −33.5719 −1.08523
\(958\) −11.3604 −0.367036
\(959\) −16.7383 −0.540507
\(960\) 0.896789 0.0289437
\(961\) −3.24027 −0.104525
\(962\) 0 0
\(963\) −12.8170 −0.413022
\(964\) −15.5526 −0.500914
\(965\) 2.27114 0.0731107
\(966\) 6.51035 0.209467
\(967\) 36.6644 1.17905 0.589524 0.807751i \(-0.299315\pi\)
0.589524 + 0.807751i \(0.299315\pi\)
\(968\) −35.4547 −1.13956
\(969\) −21.2228 −0.681775
\(970\) 1.50471 0.0483134
\(971\) −37.8465 −1.21455 −0.607277 0.794490i \(-0.707738\pi\)
−0.607277 + 0.794490i \(0.707738\pi\)
\(972\) −1.80194 −0.0577972
\(973\) −7.17416 −0.229993
\(974\) 7.12392 0.228265
\(975\) 0 0
\(976\) −24.2771 −0.777091
\(977\) 28.8998 0.924586 0.462293 0.886727i \(-0.347027\pi\)
0.462293 + 0.886727i \(0.347027\pi\)
\(978\) 3.84117 0.122827
\(979\) 0.774791 0.0247624
\(980\) −1.74764 −0.0558264
\(981\) 12.1468 0.387816
\(982\) 9.22952 0.294526
\(983\) −19.3991 −0.618735 −0.309368 0.950942i \(-0.600117\pi\)
−0.309368 + 0.950942i \(0.600117\pi\)
\(984\) −0.753020 −0.0240054
\(985\) −1.01566 −0.0323615
\(986\) 10.0489 0.320023
\(987\) −11.8076 −0.375839
\(988\) 0 0
\(989\) 14.3013 0.454754
\(990\) −0.621334 −0.0197473
\(991\) −5.18300 −0.164643 −0.0823217 0.996606i \(-0.526233\pi\)
−0.0823217 + 0.996606i \(0.526233\pi\)
\(992\) 24.5144 0.778333
\(993\) 6.06829 0.192572
\(994\) −4.46191 −0.141523
\(995\) 6.11828 0.193962
\(996\) 18.3545 0.581585
\(997\) −49.3642 −1.56338 −0.781690 0.623667i \(-0.785642\pi\)
−0.781690 + 0.623667i \(0.785642\pi\)
\(998\) −3.59047 −0.113654
\(999\) 3.19806 0.101182
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 507.2.a.k.1.2 yes 3
3.2 odd 2 1521.2.a.p.1.2 3
4.3 odd 2 8112.2.a.cf.1.1 3
13.2 odd 12 507.2.j.h.316.4 12
13.3 even 3 507.2.e.j.22.2 6
13.4 even 6 507.2.e.k.484.2 6
13.5 odd 4 507.2.b.g.337.3 6
13.6 odd 12 507.2.j.h.361.3 12
13.7 odd 12 507.2.j.h.361.4 12
13.8 odd 4 507.2.b.g.337.4 6
13.9 even 3 507.2.e.j.484.2 6
13.10 even 6 507.2.e.k.22.2 6
13.11 odd 12 507.2.j.h.316.3 12
13.12 even 2 507.2.a.j.1.2 3
39.5 even 4 1521.2.b.m.1351.4 6
39.8 even 4 1521.2.b.m.1351.3 6
39.38 odd 2 1521.2.a.q.1.2 3
52.51 odd 2 8112.2.a.by.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.2 3 13.12 even 2
507.2.a.k.1.2 yes 3 1.1 even 1 trivial
507.2.b.g.337.3 6 13.5 odd 4
507.2.b.g.337.4 6 13.8 odd 4
507.2.e.j.22.2 6 13.3 even 3
507.2.e.j.484.2 6 13.9 even 3
507.2.e.k.22.2 6 13.10 even 6
507.2.e.k.484.2 6 13.4 even 6
507.2.j.h.316.3 12 13.11 odd 12
507.2.j.h.316.4 12 13.2 odd 12
507.2.j.h.361.3 12 13.6 odd 12
507.2.j.h.361.4 12 13.7 odd 12
1521.2.a.p.1.2 3 3.2 odd 2
1521.2.a.q.1.2 3 39.38 odd 2
1521.2.b.m.1351.3 6 39.8 even 4
1521.2.b.m.1351.4 6 39.5 even 4
8112.2.a.by.1.3 3 52.51 odd 2
8112.2.a.cf.1.1 3 4.3 odd 2