Properties

Label 507.2.b.g.337.3
Level $507$
Weight $2$
Character 507.337
Analytic conductor $4.048$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.3
Root \(-0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.2.b.g.337.4

$q$-expansion

\(f(q)\) \(=\) \(q-0.445042i q^{2} +1.00000 q^{3} +1.80194 q^{4} +0.246980i q^{5} -0.445042i q^{6} +1.75302i q^{7} -1.69202i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.445042i q^{2} +1.00000 q^{3} +1.80194 q^{4} +0.246980i q^{5} -0.445042i q^{6} +1.75302i q^{7} -1.69202i q^{8} +1.00000 q^{9} +0.109916 q^{10} +5.65279i q^{11} +1.80194 q^{12} +0.780167 q^{14} +0.246980i q^{15} +2.85086 q^{16} +3.80194 q^{17} -0.445042i q^{18} -5.58211i q^{19} +0.445042i q^{20} +1.75302i q^{21} +2.51573 q^{22} -8.34481 q^{23} -1.69202i q^{24} +4.93900 q^{25} +1.00000 q^{27} +3.15883i q^{28} -5.93900 q^{29} +0.109916 q^{30} -5.26875i q^{31} -4.65279i q^{32} +5.65279i q^{33} -1.69202i q^{34} -0.432960 q^{35} +1.80194 q^{36} +3.19806i q^{37} -2.48427 q^{38} +0.417895 q^{40} -0.445042i q^{41} +0.780167 q^{42} -1.71379 q^{43} +10.1860i q^{44} +0.246980i q^{45} +3.71379i q^{46} -6.73556i q^{47} +2.85086 q^{48} +3.92692 q^{49} -2.19806i q^{50} +3.80194 q^{51} -1.06100 q^{53} -0.445042i q^{54} -1.39612 q^{55} +2.96615 q^{56} -5.58211i q^{57} +2.64310i q^{58} -13.7017i q^{59} +0.445042i q^{60} -8.51573 q^{61} -2.34481 q^{62} +1.75302i q^{63} +3.63102 q^{64} +2.51573 q^{66} -5.96077i q^{67} +6.85086 q^{68} -8.34481 q^{69} +0.192685i q^{70} +5.71917i q^{71} -1.69202i q^{72} +7.35690i q^{73} +1.42327 q^{74} +4.93900 q^{75} -10.0586i q^{76} -9.90946 q^{77} +4.45473 q^{79} +0.704103i q^{80} +1.00000 q^{81} -0.198062 q^{82} +10.1860i q^{83} +3.15883i q^{84} +0.939001i q^{85} +0.762709i q^{86} -5.93900 q^{87} +9.56465 q^{88} +0.137063i q^{89} +0.109916 q^{90} -15.0368 q^{92} -5.26875i q^{93} -2.99761 q^{94} +1.37867 q^{95} -4.65279i q^{96} +13.6896i q^{97} -1.74764i q^{98} +5.65279i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{3} + 2q^{4} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{3} + 2q^{4} + 6q^{9} + 2q^{10} + 2q^{12} + 2q^{14} - 10q^{16} + 14q^{17} - 10q^{22} - 4q^{23} + 10q^{25} + 6q^{27} - 16q^{29} + 2q^{30} + 36q^{35} + 2q^{36} - 40q^{38} + 14q^{40} + 2q^{42} + 6q^{43} - 10q^{48} - 34q^{49} + 14q^{51} - 26q^{53} - 26q^{55} - 14q^{56} - 26q^{61} + 32q^{62} - 8q^{64} - 10q^{66} + 14q^{68} - 4q^{69} + 14q^{74} + 10q^{75} + 30q^{77} - 18q^{79} + 6q^{81} - 10q^{82} - 16q^{87} + 14q^{88} + 2q^{90} - 34q^{92} + 64q^{94} - 6q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.445042i − 0.314692i −0.987544 0.157346i \(-0.949706\pi\)
0.987544 0.157346i \(-0.0502938\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.80194 0.900969
\(5\) 0.246980i 0.110453i 0.998474 + 0.0552263i \(0.0175880\pi\)
−0.998474 + 0.0552263i \(0.982412\pi\)
\(6\) − 0.445042i − 0.181688i
\(7\) 1.75302i 0.662579i 0.943529 + 0.331290i \(0.107484\pi\)
−0.943529 + 0.331290i \(0.892516\pi\)
\(8\) − 1.69202i − 0.598220i
\(9\) 1.00000 0.333333
\(10\) 0.109916 0.0347586
\(11\) 5.65279i 1.70438i 0.523232 + 0.852191i \(0.324726\pi\)
−0.523232 + 0.852191i \(0.675274\pi\)
\(12\) 1.80194 0.520175
\(13\) 0 0
\(14\) 0.780167 0.208509
\(15\) 0.246980i 0.0637699i
\(16\) 2.85086 0.712714
\(17\) 3.80194 0.922105 0.461053 0.887373i \(-0.347472\pi\)
0.461053 + 0.887373i \(0.347472\pi\)
\(18\) − 0.445042i − 0.104897i
\(19\) − 5.58211i − 1.28062i −0.768115 0.640311i \(-0.778805\pi\)
0.768115 0.640311i \(-0.221195\pi\)
\(20\) 0.445042i 0.0995144i
\(21\) 1.75302i 0.382540i
\(22\) 2.51573 0.536355
\(23\) −8.34481 −1.74001 −0.870007 0.493039i \(-0.835886\pi\)
−0.870007 + 0.493039i \(0.835886\pi\)
\(24\) − 1.69202i − 0.345382i
\(25\) 4.93900 0.987800
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 3.15883i 0.596963i
\(29\) −5.93900 −1.10284 −0.551422 0.834226i \(-0.685915\pi\)
−0.551422 + 0.834226i \(0.685915\pi\)
\(30\) 0.109916 0.0200679
\(31\) − 5.26875i − 0.946295i −0.880983 0.473148i \(-0.843118\pi\)
0.880983 0.473148i \(-0.156882\pi\)
\(32\) − 4.65279i − 0.822505i
\(33\) 5.65279i 0.984025i
\(34\) − 1.69202i − 0.290179i
\(35\) −0.432960 −0.0731836
\(36\) 1.80194 0.300323
\(37\) 3.19806i 0.525758i 0.964829 + 0.262879i \(0.0846720\pi\)
−0.964829 + 0.262879i \(0.915328\pi\)
\(38\) −2.48427 −0.403002
\(39\) 0 0
\(40\) 0.417895 0.0660750
\(41\) − 0.445042i − 0.0695039i −0.999396 0.0347519i \(-0.988936\pi\)
0.999396 0.0347519i \(-0.0110641\pi\)
\(42\) 0.780167 0.120382
\(43\) −1.71379 −0.261351 −0.130675 0.991425i \(-0.541715\pi\)
−0.130675 + 0.991425i \(0.541715\pi\)
\(44\) 10.1860i 1.53559i
\(45\) 0.246980i 0.0368175i
\(46\) 3.71379i 0.547569i
\(47\) − 6.73556i − 0.982483i −0.871024 0.491241i \(-0.836543\pi\)
0.871024 0.491241i \(-0.163457\pi\)
\(48\) 2.85086 0.411485
\(49\) 3.92692 0.560988
\(50\) − 2.19806i − 0.310853i
\(51\) 3.80194 0.532378
\(52\) 0 0
\(53\) −1.06100 −0.145739 −0.0728697 0.997341i \(-0.523216\pi\)
−0.0728697 + 0.997341i \(0.523216\pi\)
\(54\) − 0.445042i − 0.0605625i
\(55\) −1.39612 −0.188253
\(56\) 2.96615 0.396368
\(57\) − 5.58211i − 0.739368i
\(58\) 2.64310i 0.347057i
\(59\) − 13.7017i − 1.78381i −0.452222 0.891905i \(-0.649369\pi\)
0.452222 0.891905i \(-0.350631\pi\)
\(60\) 0.445042i 0.0574547i
\(61\) −8.51573 −1.09033 −0.545164 0.838330i \(-0.683533\pi\)
−0.545164 + 0.838330i \(0.683533\pi\)
\(62\) −2.34481 −0.297792
\(63\) 1.75302i 0.220860i
\(64\) 3.63102 0.453878
\(65\) 0 0
\(66\) 2.51573 0.309665
\(67\) − 5.96077i − 0.728224i −0.931355 0.364112i \(-0.881373\pi\)
0.931355 0.364112i \(-0.118627\pi\)
\(68\) 6.85086 0.830788
\(69\) −8.34481 −1.00460
\(70\) 0.192685i 0.0230303i
\(71\) 5.71917i 0.678740i 0.940653 + 0.339370i \(0.110214\pi\)
−0.940653 + 0.339370i \(0.889786\pi\)
\(72\) − 1.69202i − 0.199407i
\(73\) 7.35690i 0.861060i 0.902576 + 0.430530i \(0.141673\pi\)
−0.902576 + 0.430530i \(0.858327\pi\)
\(74\) 1.42327 0.165452
\(75\) 4.93900 0.570307
\(76\) − 10.0586i − 1.15380i
\(77\) −9.90946 −1.12929
\(78\) 0 0
\(79\) 4.45473 0.501196 0.250598 0.968091i \(-0.419373\pi\)
0.250598 + 0.968091i \(0.419373\pi\)
\(80\) 0.704103i 0.0787211i
\(81\) 1.00000 0.111111
\(82\) −0.198062 −0.0218723
\(83\) 10.1860i 1.11806i 0.829149 + 0.559028i \(0.188826\pi\)
−0.829149 + 0.559028i \(0.811174\pi\)
\(84\) 3.15883i 0.344657i
\(85\) 0.939001i 0.101849i
\(86\) 0.762709i 0.0822450i
\(87\) −5.93900 −0.636728
\(88\) 9.56465 1.01959
\(89\) 0.137063i 0.0145287i 0.999974 + 0.00726434i \(0.00231233\pi\)
−0.999974 + 0.00726434i \(0.997688\pi\)
\(90\) 0.109916 0.0115862
\(91\) 0 0
\(92\) −15.0368 −1.56770
\(93\) − 5.26875i − 0.546344i
\(94\) −2.99761 −0.309180
\(95\) 1.37867 0.141448
\(96\) − 4.65279i − 0.474874i
\(97\) 13.6896i 1.38997i 0.719024 + 0.694986i \(0.244589\pi\)
−0.719024 + 0.694986i \(0.755411\pi\)
\(98\) − 1.74764i − 0.176539i
\(99\) 5.65279i 0.568127i
\(100\) 8.89977 0.889977
\(101\) 5.41119 0.538434 0.269217 0.963080i \(-0.413235\pi\)
0.269217 + 0.963080i \(0.413235\pi\)
\(102\) − 1.69202i − 0.167535i
\(103\) −13.7560 −1.35542 −0.677710 0.735330i \(-0.737027\pi\)
−0.677710 + 0.735330i \(0.737027\pi\)
\(104\) 0 0
\(105\) −0.432960 −0.0422526
\(106\) 0.472189i 0.0458630i
\(107\) −12.8170 −1.23907 −0.619533 0.784970i \(-0.712678\pi\)
−0.619533 + 0.784970i \(0.712678\pi\)
\(108\) 1.80194 0.173392
\(109\) − 12.1468i − 1.16345i −0.813386 0.581724i \(-0.802378\pi\)
0.813386 0.581724i \(-0.197622\pi\)
\(110\) 0.621334i 0.0592419i
\(111\) 3.19806i 0.303547i
\(112\) 4.99761i 0.472229i
\(113\) −1.63773 −0.154064 −0.0770322 0.997029i \(-0.524544\pi\)
−0.0770322 + 0.997029i \(0.524544\pi\)
\(114\) −2.48427 −0.232673
\(115\) − 2.06100i − 0.192189i
\(116\) −10.7017 −0.993629
\(117\) 0 0
\(118\) −6.09783 −0.561351
\(119\) 6.66487i 0.610968i
\(120\) 0.417895 0.0381484
\(121\) −20.9541 −1.90492
\(122\) 3.78986i 0.343117i
\(123\) − 0.445042i − 0.0401281i
\(124\) − 9.49396i − 0.852583i
\(125\) 2.45473i 0.219558i
\(126\) 0.780167 0.0695028
\(127\) −10.7995 −0.958305 −0.479152 0.877732i \(-0.659056\pi\)
−0.479152 + 0.877732i \(0.659056\pi\)
\(128\) − 10.9215i − 0.965337i
\(129\) −1.71379 −0.150891
\(130\) 0 0
\(131\) 0.907542 0.0792923 0.0396462 0.999214i \(-0.487377\pi\)
0.0396462 + 0.999214i \(0.487377\pi\)
\(132\) 10.1860i 0.886576i
\(133\) 9.78554 0.848514
\(134\) −2.65279 −0.229166
\(135\) 0.246980i 0.0212566i
\(136\) − 6.43296i − 0.551622i
\(137\) − 9.54825i − 0.815762i −0.913035 0.407881i \(-0.866268\pi\)
0.913035 0.407881i \(-0.133732\pi\)
\(138\) 3.71379i 0.316139i
\(139\) −4.09246 −0.347118 −0.173559 0.984823i \(-0.555527\pi\)
−0.173559 + 0.984823i \(0.555527\pi\)
\(140\) −0.780167 −0.0659362
\(141\) − 6.73556i − 0.567237i
\(142\) 2.54527 0.213594
\(143\) 0 0
\(144\) 2.85086 0.237571
\(145\) − 1.46681i − 0.121812i
\(146\) 3.27413 0.270969
\(147\) 3.92692 0.323887
\(148\) 5.76271i 0.473692i
\(149\) 15.3884i 1.26066i 0.776326 + 0.630332i \(0.217081\pi\)
−0.776326 + 0.630332i \(0.782919\pi\)
\(150\) − 2.19806i − 0.179471i
\(151\) − 3.67456i − 0.299032i −0.988759 0.149516i \(-0.952228\pi\)
0.988759 0.149516i \(-0.0477715\pi\)
\(152\) −9.44504 −0.766094
\(153\) 3.80194 0.307368
\(154\) 4.41013i 0.355378i
\(155\) 1.30127 0.104521
\(156\) 0 0
\(157\) −4.87800 −0.389307 −0.194653 0.980872i \(-0.562358\pi\)
−0.194653 + 0.980872i \(0.562358\pi\)
\(158\) − 1.98254i − 0.157723i
\(159\) −1.06100 −0.0841427
\(160\) 1.14914 0.0908479
\(161\) − 14.6286i − 1.15290i
\(162\) − 0.445042i − 0.0349658i
\(163\) 8.63102i 0.676034i 0.941140 + 0.338017i \(0.109756\pi\)
−0.941140 + 0.338017i \(0.890244\pi\)
\(164\) − 0.801938i − 0.0626208i
\(165\) −1.39612 −0.108688
\(166\) 4.53319 0.351844
\(167\) 9.46980i 0.732795i 0.930458 + 0.366397i \(0.119409\pi\)
−0.930458 + 0.366397i \(0.880591\pi\)
\(168\) 2.96615 0.228843
\(169\) 0 0
\(170\) 0.417895 0.0320511
\(171\) − 5.58211i − 0.426874i
\(172\) −3.08815 −0.235469
\(173\) −4.77479 −0.363021 −0.181510 0.983389i \(-0.558099\pi\)
−0.181510 + 0.983389i \(0.558099\pi\)
\(174\) 2.64310i 0.200373i
\(175\) 8.65817i 0.654496i
\(176\) 16.1153i 1.21474i
\(177\) − 13.7017i − 1.02988i
\(178\) 0.0609989 0.00457206
\(179\) −3.43535 −0.256770 −0.128385 0.991724i \(-0.540979\pi\)
−0.128385 + 0.991724i \(0.540979\pi\)
\(180\) 0.445042i 0.0331715i
\(181\) 13.4862 1.00242 0.501210 0.865326i \(-0.332888\pi\)
0.501210 + 0.865326i \(0.332888\pi\)
\(182\) 0 0
\(183\) −8.51573 −0.629501
\(184\) 14.1196i 1.04091i
\(185\) −0.789856 −0.0580714
\(186\) −2.34481 −0.171930
\(187\) 21.4916i 1.57162i
\(188\) − 12.1371i − 0.885186i
\(189\) 1.75302i 0.127513i
\(190\) − 0.613564i − 0.0445126i
\(191\) −1.30127 −0.0941569 −0.0470784 0.998891i \(-0.514991\pi\)
−0.0470784 + 0.998891i \(0.514991\pi\)
\(192\) 3.63102 0.262046
\(193\) − 9.19567i − 0.661919i −0.943645 0.330959i \(-0.892628\pi\)
0.943645 0.330959i \(-0.107372\pi\)
\(194\) 6.09246 0.437413
\(195\) 0 0
\(196\) 7.07606 0.505433
\(197\) − 4.11231i − 0.292990i −0.989211 0.146495i \(-0.953201\pi\)
0.989211 0.146495i \(-0.0467992\pi\)
\(198\) 2.51573 0.178785
\(199\) 24.7724 1.75607 0.878034 0.478598i \(-0.158855\pi\)
0.878034 + 0.478598i \(0.158855\pi\)
\(200\) − 8.35690i − 0.590922i
\(201\) − 5.96077i − 0.420440i
\(202\) − 2.40821i − 0.169441i
\(203\) − 10.4112i − 0.730722i
\(204\) 6.85086 0.479656
\(205\) 0.109916 0.00767688
\(206\) 6.12200i 0.426540i
\(207\) −8.34481 −0.580005
\(208\) 0 0
\(209\) 31.5545 2.18267
\(210\) 0.192685i 0.0132966i
\(211\) −5.93900 −0.408858 −0.204429 0.978881i \(-0.565534\pi\)
−0.204429 + 0.978881i \(0.565534\pi\)
\(212\) −1.91185 −0.131307
\(213\) 5.71917i 0.391871i
\(214\) 5.70410i 0.389924i
\(215\) − 0.423272i − 0.0288669i
\(216\) − 1.69202i − 0.115127i
\(217\) 9.23623 0.626996
\(218\) −5.40581 −0.366128
\(219\) 7.35690i 0.497133i
\(220\) −2.51573 −0.169610
\(221\) 0 0
\(222\) 1.42327 0.0955237
\(223\) − 14.2010i − 0.950972i −0.879723 0.475486i \(-0.842272\pi\)
0.879723 0.475486i \(-0.157728\pi\)
\(224\) 8.15644 0.544975
\(225\) 4.93900 0.329267
\(226\) 0.728857i 0.0484829i
\(227\) − 16.0073i − 1.06244i −0.847233 0.531221i \(-0.821733\pi\)
0.847233 0.531221i \(-0.178267\pi\)
\(228\) − 10.0586i − 0.666147i
\(229\) 1.84117i 0.121668i 0.998148 + 0.0608339i \(0.0193760\pi\)
−0.998148 + 0.0608339i \(0.980624\pi\)
\(230\) −0.917231 −0.0604804
\(231\) −9.90946 −0.651995
\(232\) 10.0489i 0.659744i
\(233\) 23.4252 1.53464 0.767318 0.641267i \(-0.221591\pi\)
0.767318 + 0.641267i \(0.221591\pi\)
\(234\) 0 0
\(235\) 1.66355 0.108518
\(236\) − 24.6896i − 1.60716i
\(237\) 4.45473 0.289366
\(238\) 2.96615 0.192267
\(239\) − 14.6015i − 0.944491i −0.881467 0.472246i \(-0.843444\pi\)
0.881467 0.472246i \(-0.156556\pi\)
\(240\) 0.704103i 0.0454497i
\(241\) 8.63102i 0.555973i 0.960585 + 0.277987i \(0.0896671\pi\)
−0.960585 + 0.277987i \(0.910333\pi\)
\(242\) 9.32544i 0.599462i
\(243\) 1.00000 0.0641500
\(244\) −15.3448 −0.982351
\(245\) 0.969869i 0.0619627i
\(246\) −0.198062 −0.0126280
\(247\) 0 0
\(248\) −8.91484 −0.566093
\(249\) 10.1860i 0.645510i
\(250\) 1.09246 0.0690931
\(251\) 3.80194 0.239976 0.119988 0.992775i \(-0.461714\pi\)
0.119988 + 0.992775i \(0.461714\pi\)
\(252\) 3.15883i 0.198988i
\(253\) − 47.1715i − 2.96565i
\(254\) 4.80625i 0.301571i
\(255\) 0.939001i 0.0588025i
\(256\) 2.40150 0.150094
\(257\) 20.5961 1.28475 0.642375 0.766391i \(-0.277949\pi\)
0.642375 + 0.766391i \(0.277949\pi\)
\(258\) 0.762709i 0.0474842i
\(259\) −5.60627 −0.348357
\(260\) 0 0
\(261\) −5.93900 −0.367615
\(262\) − 0.403894i − 0.0249527i
\(263\) 0.332733 0.0205172 0.0102586 0.999947i \(-0.496735\pi\)
0.0102586 + 0.999947i \(0.496735\pi\)
\(264\) 9.56465 0.588663
\(265\) − 0.262045i − 0.0160973i
\(266\) − 4.35498i − 0.267021i
\(267\) 0.137063i 0.00838814i
\(268\) − 10.7409i − 0.656107i
\(269\) 27.3032 1.66471 0.832353 0.554247i \(-0.186994\pi\)
0.832353 + 0.554247i \(0.186994\pi\)
\(270\) 0.109916 0.00668929
\(271\) 27.9855i 1.70000i 0.526783 + 0.850000i \(0.323398\pi\)
−0.526783 + 0.850000i \(0.676602\pi\)
\(272\) 10.8388 0.657197
\(273\) 0 0
\(274\) −4.24937 −0.256714
\(275\) 27.9191i 1.68359i
\(276\) −15.0368 −0.905111
\(277\) 2.10321 0.126370 0.0631849 0.998002i \(-0.479874\pi\)
0.0631849 + 0.998002i \(0.479874\pi\)
\(278\) 1.82132i 0.109235i
\(279\) − 5.26875i − 0.315432i
\(280\) 0.732578i 0.0437799i
\(281\) 27.2349i 1.62470i 0.583172 + 0.812349i \(0.301811\pi\)
−0.583172 + 0.812349i \(0.698189\pi\)
\(282\) −2.99761 −0.178505
\(283\) −5.28382 −0.314090 −0.157045 0.987591i \(-0.550197\pi\)
−0.157045 + 0.987591i \(0.550197\pi\)
\(284\) 10.3056i 0.611524i
\(285\) 1.37867 0.0816651
\(286\) 0 0
\(287\) 0.780167 0.0460518
\(288\) − 4.65279i − 0.274168i
\(289\) −2.54527 −0.149722
\(290\) −0.652793 −0.0383333
\(291\) 13.6896i 0.802500i
\(292\) 13.2567i 0.775788i
\(293\) 32.6625i 1.90816i 0.299548 + 0.954081i \(0.403164\pi\)
−0.299548 + 0.954081i \(0.596836\pi\)
\(294\) − 1.74764i − 0.101925i
\(295\) 3.38404 0.197027
\(296\) 5.41119 0.314519
\(297\) 5.65279i 0.328008i
\(298\) 6.84846 0.396721
\(299\) 0 0
\(300\) 8.89977 0.513829
\(301\) − 3.00431i − 0.173166i
\(302\) −1.63533 −0.0941029
\(303\) 5.41119 0.310865
\(304\) − 15.9138i − 0.912717i
\(305\) − 2.10321i − 0.120430i
\(306\) − 1.69202i − 0.0967264i
\(307\) − 20.7614i − 1.18491i −0.805602 0.592457i \(-0.798158\pi\)
0.805602 0.592457i \(-0.201842\pi\)
\(308\) −17.8562 −1.01745
\(309\) −13.7560 −0.782552
\(310\) − 0.579121i − 0.0328919i
\(311\) 11.3013 0.640836 0.320418 0.947276i \(-0.396177\pi\)
0.320418 + 0.947276i \(0.396177\pi\)
\(312\) 0 0
\(313\) −4.27173 −0.241453 −0.120726 0.992686i \(-0.538522\pi\)
−0.120726 + 0.992686i \(0.538522\pi\)
\(314\) 2.17092i 0.122512i
\(315\) −0.432960 −0.0243945
\(316\) 8.02715 0.451562
\(317\) − 15.4776i − 0.869307i −0.900598 0.434653i \(-0.856871\pi\)
0.900598 0.434653i \(-0.143129\pi\)
\(318\) 0.472189i 0.0264790i
\(319\) − 33.5719i − 1.87967i
\(320\) 0.896789i 0.0501320i
\(321\) −12.8170 −0.715375
\(322\) −6.51035 −0.362808
\(323\) − 21.2228i − 1.18087i
\(324\) 1.80194 0.100108
\(325\) 0 0
\(326\) 3.84117 0.212743
\(327\) − 12.1468i − 0.671717i
\(328\) −0.753020 −0.0415786
\(329\) 11.8076 0.650973
\(330\) 0.621334i 0.0342033i
\(331\) − 6.06829i − 0.333544i −0.985996 0.166772i \(-0.946666\pi\)
0.985996 0.166772i \(-0.0533343\pi\)
\(332\) 18.3545i 1.00733i
\(333\) 3.19806i 0.175253i
\(334\) 4.21446 0.230605
\(335\) 1.47219 0.0804343
\(336\) 4.99761i 0.272642i
\(337\) −12.1239 −0.660432 −0.330216 0.943905i \(-0.607122\pi\)
−0.330216 + 0.943905i \(0.607122\pi\)
\(338\) 0 0
\(339\) −1.63773 −0.0889491
\(340\) 1.69202i 0.0917627i
\(341\) 29.7832 1.61285
\(342\) −2.48427 −0.134334
\(343\) 19.1551i 1.03428i
\(344\) 2.89977i 0.156345i
\(345\) − 2.06100i − 0.110960i
\(346\) 2.12498i 0.114240i
\(347\) 23.1497 1.24274 0.621371 0.783516i \(-0.286576\pi\)
0.621371 + 0.783516i \(0.286576\pi\)
\(348\) −10.7017 −0.573672
\(349\) − 22.1957i − 1.18811i −0.804426 0.594053i \(-0.797527\pi\)
0.804426 0.594053i \(-0.202473\pi\)
\(350\) 3.85325 0.205965
\(351\) 0 0
\(352\) 26.3013 1.40186
\(353\) 5.07069i 0.269885i 0.990853 + 0.134943i \(0.0430850\pi\)
−0.990853 + 0.134943i \(0.956915\pi\)
\(354\) −6.09783 −0.324096
\(355\) −1.41252 −0.0749687
\(356\) 0.246980i 0.0130899i
\(357\) 6.66487i 0.352743i
\(358\) 1.52888i 0.0808036i
\(359\) − 16.6746i − 0.880050i −0.897986 0.440025i \(-0.854970\pi\)
0.897986 0.440025i \(-0.145030\pi\)
\(360\) 0.417895 0.0220250
\(361\) −12.1599 −0.639995
\(362\) − 6.00192i − 0.315454i
\(363\) −20.9541 −1.09980
\(364\) 0 0
\(365\) −1.81700 −0.0951063
\(366\) 3.78986i 0.198099i
\(367\) −1.17928 −0.0615577 −0.0307789 0.999526i \(-0.509799\pi\)
−0.0307789 + 0.999526i \(0.509799\pi\)
\(368\) −23.7899 −1.24013
\(369\) − 0.445042i − 0.0231680i
\(370\) 0.351519i 0.0182746i
\(371\) − 1.85995i − 0.0965639i
\(372\) − 9.49396i − 0.492239i
\(373\) −30.0925 −1.55813 −0.779064 0.626944i \(-0.784305\pi\)
−0.779064 + 0.626944i \(0.784305\pi\)
\(374\) 9.56465 0.494576
\(375\) 2.45473i 0.126762i
\(376\) −11.3967 −0.587741
\(377\) 0 0
\(378\) 0.780167 0.0401275
\(379\) 19.1631i 0.984345i 0.870498 + 0.492172i \(0.163797\pi\)
−0.870498 + 0.492172i \(0.836203\pi\)
\(380\) 2.48427 0.127440
\(381\) −10.7995 −0.553277
\(382\) 0.579121i 0.0296304i
\(383\) 15.3884i 0.786308i 0.919473 + 0.393154i \(0.128616\pi\)
−0.919473 + 0.393154i \(0.871384\pi\)
\(384\) − 10.9215i − 0.557338i
\(385\) − 2.44743i − 0.124733i
\(386\) −4.09246 −0.208301
\(387\) −1.71379 −0.0871169
\(388\) 24.6679i 1.25232i
\(389\) 24.0315 1.21844 0.609222 0.793000i \(-0.291482\pi\)
0.609222 + 0.793000i \(0.291482\pi\)
\(390\) 0 0
\(391\) −31.7265 −1.60448
\(392\) − 6.64443i − 0.335594i
\(393\) 0.907542 0.0457794
\(394\) −1.83015 −0.0922016
\(395\) 1.10023i 0.0553585i
\(396\) 10.1860i 0.511865i
\(397\) 29.6015i 1.48566i 0.669482 + 0.742828i \(0.266516\pi\)
−0.669482 + 0.742828i \(0.733484\pi\)
\(398\) − 11.0248i − 0.552621i
\(399\) 9.78554 0.489890
\(400\) 14.0804 0.704019
\(401\) − 21.1032i − 1.05384i −0.849914 0.526922i \(-0.823346\pi\)
0.849914 0.526922i \(-0.176654\pi\)
\(402\) −2.65279 −0.132309
\(403\) 0 0
\(404\) 9.75063 0.485112
\(405\) 0.246980i 0.0122725i
\(406\) −4.63342 −0.229953
\(407\) −18.0780 −0.896092
\(408\) − 6.43296i − 0.318479i
\(409\) 36.0224i 1.78119i 0.454796 + 0.890596i \(0.349712\pi\)
−0.454796 + 0.890596i \(0.650288\pi\)
\(410\) − 0.0489173i − 0.00241586i
\(411\) − 9.54825i − 0.470981i
\(412\) −24.7875 −1.22119
\(413\) 24.0194 1.18192
\(414\) 3.71379i 0.182523i
\(415\) −2.51573 −0.123492
\(416\) 0 0
\(417\) −4.09246 −0.200409
\(418\) − 14.0431i − 0.686869i
\(419\) 5.96854 0.291582 0.145791 0.989315i \(-0.453427\pi\)
0.145791 + 0.989315i \(0.453427\pi\)
\(420\) −0.780167 −0.0380683
\(421\) 2.09544i 0.102126i 0.998695 + 0.0510628i \(0.0162609\pi\)
−0.998695 + 0.0510628i \(0.983739\pi\)
\(422\) 2.64310i 0.128664i
\(423\) − 6.73556i − 0.327494i
\(424\) 1.79523i 0.0871842i
\(425\) 18.7778 0.910856
\(426\) 2.54527 0.123319
\(427\) − 14.9282i − 0.722429i
\(428\) −23.0954 −1.11636
\(429\) 0 0
\(430\) −0.188374 −0.00908418
\(431\) 2.88577i 0.139003i 0.997582 + 0.0695014i \(0.0221408\pi\)
−0.997582 + 0.0695014i \(0.977859\pi\)
\(432\) 2.85086 0.137162
\(433\) 12.6485 0.607847 0.303924 0.952696i \(-0.401703\pi\)
0.303924 + 0.952696i \(0.401703\pi\)
\(434\) − 4.11051i − 0.197311i
\(435\) − 1.46681i − 0.0703283i
\(436\) − 21.8877i − 1.04823i
\(437\) 46.5816i 2.22830i
\(438\) 3.27413 0.156444
\(439\) 10.9269 0.521513 0.260757 0.965405i \(-0.416028\pi\)
0.260757 + 0.965405i \(0.416028\pi\)
\(440\) 2.36227i 0.112617i
\(441\) 3.92692 0.186996
\(442\) 0 0
\(443\) −19.2403 −0.914133 −0.457067 0.889433i \(-0.651100\pi\)
−0.457067 + 0.889433i \(0.651100\pi\)
\(444\) 5.76271i 0.273486i
\(445\) −0.0338518 −0.00160473
\(446\) −6.32006 −0.299264
\(447\) 15.3884i 0.727844i
\(448\) 6.36526i 0.300730i
\(449\) − 28.8200i − 1.36010i −0.733166 0.680050i \(-0.761958\pi\)
0.733166 0.680050i \(-0.238042\pi\)
\(450\) − 2.19806i − 0.103618i
\(451\) 2.51573 0.118461
\(452\) −2.95108 −0.138807
\(453\) − 3.67456i − 0.172646i
\(454\) −7.12392 −0.334342
\(455\) 0 0
\(456\) −9.44504 −0.442305
\(457\) 18.0707i 0.845311i 0.906290 + 0.422656i \(0.138902\pi\)
−0.906290 + 0.422656i \(0.861098\pi\)
\(458\) 0.819396 0.0382879
\(459\) 3.80194 0.177459
\(460\) − 3.71379i − 0.173156i
\(461\) 7.56763i 0.352460i 0.984349 + 0.176230i \(0.0563902\pi\)
−0.984349 + 0.176230i \(0.943610\pi\)
\(462\) 4.41013i 0.205178i
\(463\) 35.3551i 1.64309i 0.570143 + 0.821545i \(0.306888\pi\)
−0.570143 + 0.821545i \(0.693112\pi\)
\(464\) −16.9312 −0.786013
\(465\) 1.30127 0.0603451
\(466\) − 10.4252i − 0.482938i
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) 10.4494 0.482506
\(470\) − 0.740348i − 0.0341497i
\(471\) −4.87800 −0.224766
\(472\) −23.1836 −1.06711
\(473\) − 9.68771i − 0.445441i
\(474\) − 1.98254i − 0.0910612i
\(475\) − 27.5700i − 1.26500i
\(476\) 12.0097i 0.550463i
\(477\) −1.06100 −0.0485798
\(478\) −6.49827 −0.297224
\(479\) − 25.5265i − 1.16633i −0.812352 0.583167i \(-0.801813\pi\)
0.812352 0.583167i \(-0.198187\pi\)
\(480\) 1.14914 0.0524510
\(481\) 0 0
\(482\) 3.84117 0.174960
\(483\) − 14.6286i − 0.665626i
\(484\) −37.7579 −1.71627
\(485\) −3.38106 −0.153526
\(486\) − 0.445042i − 0.0201875i
\(487\) − 16.0073i − 0.725360i −0.931914 0.362680i \(-0.881862\pi\)
0.931914 0.362680i \(-0.118138\pi\)
\(488\) 14.4088i 0.652256i
\(489\) 8.63102i 0.390308i
\(490\) 0.431632 0.0194992
\(491\) −20.7385 −0.935917 −0.467959 0.883750i \(-0.655010\pi\)
−0.467959 + 0.883750i \(0.655010\pi\)
\(492\) − 0.801938i − 0.0361541i
\(493\) −22.5797 −1.01694
\(494\) 0 0
\(495\) −1.39612 −0.0627511
\(496\) − 15.0204i − 0.674438i
\(497\) −10.0258 −0.449719
\(498\) 4.53319 0.203137
\(499\) 8.06770i 0.361160i 0.983560 + 0.180580i \(0.0577975\pi\)
−0.983560 + 0.180580i \(0.942203\pi\)
\(500\) 4.42327i 0.197815i
\(501\) 9.46980i 0.423079i
\(502\) − 1.69202i − 0.0755186i
\(503\) −30.2422 −1.34843 −0.674216 0.738534i \(-0.735518\pi\)
−0.674216 + 0.738534i \(0.735518\pi\)
\(504\) 2.96615 0.132123
\(505\) 1.33645i 0.0594714i
\(506\) −20.9933 −0.933266
\(507\) 0 0
\(508\) −19.4601 −0.863403
\(509\) − 16.0495i − 0.711382i −0.934604 0.355691i \(-0.884246\pi\)
0.934604 0.355691i \(-0.115754\pi\)
\(510\) 0.417895 0.0185047
\(511\) −12.8968 −0.570520
\(512\) − 22.9119i − 1.01257i
\(513\) − 5.58211i − 0.246456i
\(514\) − 9.16613i − 0.404301i
\(515\) − 3.39745i − 0.149710i
\(516\) −3.08815 −0.135948
\(517\) 38.0747 1.67452
\(518\) 2.49502i 0.109625i
\(519\) −4.77479 −0.209590
\(520\) 0 0
\(521\) −2.69309 −0.117986 −0.0589931 0.998258i \(-0.518789\pi\)
−0.0589931 + 0.998258i \(0.518789\pi\)
\(522\) 2.64310i 0.115686i
\(523\) 35.3957 1.54774 0.773872 0.633342i \(-0.218317\pi\)
0.773872 + 0.633342i \(0.218317\pi\)
\(524\) 1.63533 0.0714399
\(525\) 8.65817i 0.377874i
\(526\) − 0.148080i − 0.00645659i
\(527\) − 20.0315i − 0.872584i
\(528\) 16.1153i 0.701328i
\(529\) 46.6359 2.02765
\(530\) −0.116621 −0.00506569
\(531\) − 13.7017i − 0.594604i
\(532\) 17.6329 0.764485
\(533\) 0 0
\(534\) 0.0609989 0.00263968
\(535\) − 3.16554i − 0.136858i
\(536\) −10.0858 −0.435638
\(537\) −3.43535 −0.148246
\(538\) − 12.1511i − 0.523870i
\(539\) 22.1981i 0.956138i
\(540\) 0.445042i 0.0191516i
\(541\) − 34.7338i − 1.49332i −0.665205 0.746660i \(-0.731656\pi\)
0.665205 0.746660i \(-0.268344\pi\)
\(542\) 12.4547 0.534976
\(543\) 13.4862 0.578748
\(544\) − 17.6896i − 0.758437i
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) −26.1183 −1.11674 −0.558368 0.829593i \(-0.688572\pi\)
−0.558368 + 0.829593i \(0.688572\pi\)
\(548\) − 17.2054i − 0.734976i
\(549\) −8.51573 −0.363442
\(550\) 12.4252 0.529812
\(551\) 33.1521i 1.41233i
\(552\) 14.1196i 0.600970i
\(553\) 7.80923i 0.332082i
\(554\) − 0.936017i − 0.0397676i
\(555\) −0.789856 −0.0335275
\(556\) −7.37435 −0.312742
\(557\) 24.7748i 1.04974i 0.851182 + 0.524871i \(0.175886\pi\)
−0.851182 + 0.524871i \(0.824114\pi\)
\(558\) −2.34481 −0.0992639
\(559\) 0 0
\(560\) −1.23431 −0.0521590
\(561\) 21.4916i 0.907375i
\(562\) 12.1207 0.511280
\(563\) −5.26098 −0.221724 −0.110862 0.993836i \(-0.535361\pi\)
−0.110862 + 0.993836i \(0.535361\pi\)
\(564\) − 12.1371i − 0.511063i
\(565\) − 0.404485i − 0.0170168i
\(566\) 2.35152i 0.0988417i
\(567\) 1.75302i 0.0736199i
\(568\) 9.67696 0.406036
\(569\) 33.7458 1.41470 0.707350 0.706864i \(-0.249891\pi\)
0.707350 + 0.706864i \(0.249891\pi\)
\(570\) − 0.613564i − 0.0256994i
\(571\) −23.0887 −0.966234 −0.483117 0.875556i \(-0.660495\pi\)
−0.483117 + 0.875556i \(0.660495\pi\)
\(572\) 0 0
\(573\) −1.30127 −0.0543615
\(574\) − 0.347207i − 0.0144921i
\(575\) −41.2150 −1.71879
\(576\) 3.63102 0.151293
\(577\) − 3.57002i − 0.148622i −0.997235 0.0743110i \(-0.976324\pi\)
0.997235 0.0743110i \(-0.0236758\pi\)
\(578\) 1.13275i 0.0471162i
\(579\) − 9.19567i − 0.382159i
\(580\) − 2.64310i − 0.109749i
\(581\) −17.8562 −0.740801
\(582\) 6.09246 0.252541
\(583\) − 5.99761i − 0.248396i
\(584\) 12.4480 0.515103
\(585\) 0 0
\(586\) 14.5362 0.600484
\(587\) 11.4625i 0.473108i 0.971618 + 0.236554i \(0.0760180\pi\)
−0.971618 + 0.236554i \(0.923982\pi\)
\(588\) 7.07606 0.291812
\(589\) −29.4107 −1.21185
\(590\) − 1.50604i − 0.0620027i
\(591\) − 4.11231i − 0.169158i
\(592\) 9.11721i 0.374715i
\(593\) − 21.8538i − 0.897430i −0.893675 0.448715i \(-0.851882\pi\)
0.893675 0.448715i \(-0.148118\pi\)
\(594\) 2.51573 0.103222
\(595\) −1.64609 −0.0674830
\(596\) 27.7289i 1.13582i
\(597\) 24.7724 1.01387
\(598\) 0 0
\(599\) 27.0573 1.10553 0.552765 0.833337i \(-0.313573\pi\)
0.552765 + 0.833337i \(0.313573\pi\)
\(600\) − 8.35690i − 0.341169i
\(601\) 10.8780 0.443723 0.221861 0.975078i \(-0.428787\pi\)
0.221861 + 0.975078i \(0.428787\pi\)
\(602\) −1.33704 −0.0544939
\(603\) − 5.96077i − 0.242741i
\(604\) − 6.62133i − 0.269418i
\(605\) − 5.17523i − 0.210403i
\(606\) − 2.40821i − 0.0978267i
\(607\) −29.6359 −1.20289 −0.601443 0.798916i \(-0.705407\pi\)
−0.601443 + 0.798916i \(0.705407\pi\)
\(608\) −25.9724 −1.05332
\(609\) − 10.4112i − 0.421883i
\(610\) −0.936017 −0.0378982
\(611\) 0 0
\(612\) 6.85086 0.276929
\(613\) − 10.2343i − 0.413360i −0.978409 0.206680i \(-0.933734\pi\)
0.978409 0.206680i \(-0.0662659\pi\)
\(614\) −9.23968 −0.372883
\(615\) 0.109916 0.00443225
\(616\) 16.7670i 0.675563i
\(617\) − 26.2828i − 1.05810i −0.848590 0.529052i \(-0.822548\pi\)
0.848590 0.529052i \(-0.177452\pi\)
\(618\) 6.12200i 0.246263i
\(619\) 29.0834i 1.16896i 0.811408 + 0.584479i \(0.198701\pi\)
−0.811408 + 0.584479i \(0.801299\pi\)
\(620\) 2.34481 0.0941700
\(621\) −8.34481 −0.334866
\(622\) − 5.02954i − 0.201666i
\(623\) −0.240275 −0.00962641
\(624\) 0 0
\(625\) 24.0887 0.963549
\(626\) 1.90110i 0.0759833i
\(627\) 31.5545 1.26016
\(628\) −8.78986 −0.350753
\(629\) 12.1588i 0.484804i
\(630\) 0.192685i 0.00767677i
\(631\) 25.4480i 1.01307i 0.862219 + 0.506535i \(0.169074\pi\)
−0.862219 + 0.506535i \(0.830926\pi\)
\(632\) − 7.53750i − 0.299826i
\(633\) −5.93900 −0.236054
\(634\) −6.88816 −0.273564
\(635\) − 2.66727i − 0.105847i
\(636\) −1.91185 −0.0758099
\(637\) 0 0
\(638\) −14.9409 −0.591517
\(639\) 5.71917i 0.226247i
\(640\) 2.69740 0.106624
\(641\) 26.7409 1.05620 0.528102 0.849181i \(-0.322904\pi\)
0.528102 + 0.849181i \(0.322904\pi\)
\(642\) 5.70410i 0.225123i
\(643\) 32.9614i 1.29987i 0.759990 + 0.649935i \(0.225204\pi\)
−0.759990 + 0.649935i \(0.774796\pi\)
\(644\) − 26.3599i − 1.03872i
\(645\) − 0.423272i − 0.0166663i
\(646\) −9.44504 −0.371610
\(647\) −34.4946 −1.35612 −0.678060 0.735006i \(-0.737179\pi\)
−0.678060 + 0.735006i \(0.737179\pi\)
\(648\) − 1.69202i − 0.0664689i
\(649\) 77.4529 3.04029
\(650\) 0 0
\(651\) 9.23623 0.361996
\(652\) 15.5526i 0.609085i
\(653\) 36.1517 1.41472 0.707362 0.706852i \(-0.249885\pi\)
0.707362 + 0.706852i \(0.249885\pi\)
\(654\) −5.40581 −0.211384
\(655\) 0.224144i 0.00875805i
\(656\) − 1.26875i − 0.0495364i
\(657\) 7.35690i 0.287020i
\(658\) − 5.25487i − 0.204856i
\(659\) −6.81700 −0.265553 −0.132776 0.991146i \(-0.542389\pi\)
−0.132776 + 0.991146i \(0.542389\pi\)
\(660\) −2.51573 −0.0979246
\(661\) − 10.8944i − 0.423743i −0.977298 0.211871i \(-0.932044\pi\)
0.977298 0.211871i \(-0.0679558\pi\)
\(662\) −2.70065 −0.104964
\(663\) 0 0
\(664\) 17.2349 0.668844
\(665\) 2.41683i 0.0937206i
\(666\) 1.42327 0.0551507
\(667\) 49.5599 1.91897
\(668\) 17.0640i 0.660225i
\(669\) − 14.2010i − 0.549044i
\(670\) − 0.655186i − 0.0253120i
\(671\) − 48.1377i − 1.85833i
\(672\) 8.15644 0.314642
\(673\) −20.7385 −0.799412 −0.399706 0.916643i \(-0.630888\pi\)
−0.399706 + 0.916643i \(0.630888\pi\)
\(674\) 5.39565i 0.207833i
\(675\) 4.93900 0.190102
\(676\) 0 0
\(677\) −25.5786 −0.983067 −0.491534 0.870859i \(-0.663564\pi\)
−0.491534 + 0.870859i \(0.663564\pi\)
\(678\) 0.728857i 0.0279916i
\(679\) −23.9982 −0.920966
\(680\) 1.58881 0.0609281
\(681\) − 16.0073i − 0.613401i
\(682\) − 13.2547i − 0.507551i
\(683\) 21.6310i 0.827688i 0.910348 + 0.413844i \(0.135814\pi\)
−0.910348 + 0.413844i \(0.864186\pi\)
\(684\) − 10.0586i − 0.384600i
\(685\) 2.35822 0.0901031
\(686\) 8.52483 0.325479
\(687\) 1.84117i 0.0702449i
\(688\) −4.88577 −0.186268
\(689\) 0 0
\(690\) −0.917231 −0.0349184
\(691\) − 2.62996i − 0.100048i −0.998748 0.0500242i \(-0.984070\pi\)
0.998748 0.0500242i \(-0.0159298\pi\)
\(692\) −8.60388 −0.327070
\(693\) −9.90946 −0.376429
\(694\) − 10.3026i − 0.391081i
\(695\) − 1.01075i − 0.0383401i
\(696\) 10.0489i 0.380903i
\(697\) − 1.69202i − 0.0640899i
\(698\) −9.87800 −0.373888
\(699\) 23.4252 0.886022
\(700\) 15.6015i 0.589681i
\(701\) −40.0925 −1.51427 −0.757136 0.653258i \(-0.773402\pi\)
−0.757136 + 0.653258i \(0.773402\pi\)
\(702\) 0 0
\(703\) 17.8519 0.673298
\(704\) 20.5254i 0.773581i
\(705\) 1.66355 0.0626528
\(706\) 2.25667 0.0849308
\(707\) 9.48593i 0.356755i
\(708\) − 24.6896i − 0.927893i
\(709\) 23.2097i 0.871657i 0.900030 + 0.435829i \(0.143545\pi\)
−0.900030 + 0.435829i \(0.856455\pi\)
\(710\) 0.628630i 0.0235921i
\(711\) 4.45473 0.167065
\(712\) 0.231914 0.00869135
\(713\) 43.9667i 1.64657i
\(714\) 2.96615 0.111005
\(715\) 0 0
\(716\) −6.19029 −0.231342
\(717\) − 14.6015i − 0.545302i
\(718\) −7.42088 −0.276945
\(719\) 26.0146 0.970181 0.485090 0.874464i \(-0.338787\pi\)
0.485090 + 0.874464i \(0.338787\pi\)
\(720\) 0.704103i 0.0262404i
\(721\) − 24.1146i − 0.898073i
\(722\) 5.41166i 0.201401i
\(723\) 8.63102i 0.320991i
\(724\) 24.3013 0.903150
\(725\) −29.3327 −1.08939
\(726\) 9.32544i 0.346099i
\(727\) 16.5472 0.613701 0.306851 0.951758i \(-0.400725\pi\)
0.306851 + 0.951758i \(0.400725\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0.808643i 0.0299292i
\(731\) −6.51573 −0.240993
\(732\) −15.3448 −0.567161
\(733\) − 18.8750i − 0.697165i −0.937278 0.348582i \(-0.886663\pi\)
0.937278 0.348582i \(-0.113337\pi\)
\(734\) 0.524827i 0.0193717i
\(735\) 0.969869i 0.0357742i
\(736\) 38.8267i 1.43117i
\(737\) 33.6950 1.24117
\(738\) −0.198062 −0.00729077
\(739\) 47.3239i 1.74084i 0.492312 + 0.870419i \(0.336152\pi\)
−0.492312 + 0.870419i \(0.663848\pi\)
\(740\) −1.42327 −0.0523205
\(741\) 0 0
\(742\) −0.827757 −0.0303879
\(743\) 8.88769i 0.326058i 0.986621 + 0.163029i \(0.0521264\pi\)
−0.986621 + 0.163029i \(0.947874\pi\)
\(744\) −8.91484 −0.326834
\(745\) −3.80061 −0.139244
\(746\) 13.3924i 0.490331i
\(747\) 10.1860i 0.372686i
\(748\) 38.7265i 1.41598i
\(749\) − 22.4685i − 0.820980i
\(750\) 1.09246 0.0398909
\(751\) −0.710808 −0.0259377 −0.0129689 0.999916i \(-0.504128\pi\)
−0.0129689 + 0.999916i \(0.504128\pi\)
\(752\) − 19.2021i − 0.700229i
\(753\) 3.80194 0.138550
\(754\) 0 0
\(755\) 0.907542 0.0330288
\(756\) 3.15883i 0.114886i
\(757\) 9.78554 0.355662 0.177831 0.984061i \(-0.443092\pi\)
0.177831 + 0.984061i \(0.443092\pi\)
\(758\) 8.52840 0.309766
\(759\) − 47.1715i − 1.71222i
\(760\) − 2.33273i − 0.0846171i
\(761\) − 18.8810i − 0.684435i −0.939621 0.342218i \(-0.888822\pi\)
0.939621 0.342218i \(-0.111178\pi\)
\(762\) 4.80625i 0.174112i
\(763\) 21.2935 0.770877
\(764\) −2.34481 −0.0848324
\(765\) 0.939001i 0.0339497i
\(766\) 6.84846 0.247445
\(767\) 0 0
\(768\) 2.40150 0.0866567
\(769\) 12.4349i 0.448413i 0.974542 + 0.224207i \(0.0719790\pi\)
−0.974542 + 0.224207i \(0.928021\pi\)
\(770\) −1.08921 −0.0392524
\(771\) 20.5961 0.741751
\(772\) − 16.5700i − 0.596368i
\(773\) − 45.6746i − 1.64280i −0.570353 0.821400i \(-0.693193\pi\)
0.570353 0.821400i \(-0.306807\pi\)
\(774\) 0.762709i 0.0274150i
\(775\) − 26.0224i − 0.934751i
\(776\) 23.1631 0.831508
\(777\) −5.60627 −0.201124
\(778\) − 10.6950i − 0.383435i
\(779\) −2.48427 −0.0890082
\(780\) 0 0
\(781\) −32.3293 −1.15683
\(782\) 14.1196i 0.504916i
\(783\) −5.93900 −0.212243
\(784\) 11.1951 0.399824
\(785\) − 1.20477i − 0.0430000i
\(786\) − 0.403894i − 0.0144064i
\(787\) − 4.51871i − 0.161075i −0.996752 0.0805374i \(-0.974336\pi\)
0.996752 0.0805374i \(-0.0256636\pi\)
\(788\) − 7.41013i − 0.263975i
\(789\) 0.332733 0.0118456
\(790\) 0.489647 0.0174209
\(791\) − 2.87097i − 0.102080i
\(792\) 9.56465 0.339865
\(793\) 0 0
\(794\) 13.1739 0.467524
\(795\) − 0.262045i − 0.00929378i
\(796\) 44.6383 1.58216
\(797\) 28.7391 1.01799 0.508996 0.860769i \(-0.330017\pi\)
0.508996 + 0.860769i \(0.330017\pi\)
\(798\) − 4.35498i − 0.154165i
\(799\) − 25.6082i − 0.905953i
\(800\) − 22.9801i − 0.812471i
\(801\) 0.137063i 0.00484289i
\(802\) −9.39181 −0.331636
\(803\) −41.5870 −1.46757
\(804\) − 10.7409i − 0.378804i
\(805\) 3.61297 0.127341
\(806\) 0 0
\(807\) 27.3032 0.961118
\(808\) − 9.15585i − 0.322102i
\(809\) 5.42891 0.190870 0.0954352 0.995436i \(-0.469576\pi\)
0.0954352 + 0.995436i \(0.469576\pi\)
\(810\) 0.109916 0.00386206
\(811\) − 0.629104i − 0.0220908i −0.999939 0.0110454i \(-0.996484\pi\)
0.999939 0.0110454i \(-0.00351593\pi\)
\(812\) − 18.7603i − 0.658358i
\(813\) 27.9855i 0.981495i
\(814\) 8.04546i 0.281993i
\(815\) −2.13169 −0.0746697
\(816\) 10.8388 0.379433
\(817\) 9.56657i 0.334692i
\(818\) 16.0315 0.560527
\(819\) 0 0
\(820\) 0.198062 0.00691663
\(821\) − 36.2640i − 1.26562i −0.774307 0.632811i \(-0.781901\pi\)
0.774307 0.632811i \(-0.218099\pi\)
\(822\) −4.24937 −0.148214
\(823\) −41.7396 −1.45495 −0.727476 0.686133i \(-0.759307\pi\)
−0.727476 + 0.686133i \(0.759307\pi\)
\(824\) 23.2755i 0.810839i
\(825\) 27.9191i 0.972020i
\(826\) − 10.6896i − 0.371940i
\(827\) 38.1997i 1.32833i 0.747584 + 0.664167i \(0.231214\pi\)
−0.747584 + 0.664167i \(0.768786\pi\)
\(828\) −15.0368 −0.522566
\(829\) −15.9788 −0.554967 −0.277484 0.960730i \(-0.589500\pi\)
−0.277484 + 0.960730i \(0.589500\pi\)
\(830\) 1.11960i 0.0388621i
\(831\) 2.10321 0.0729596
\(832\) 0 0
\(833\) 14.9299 0.517290
\(834\) 1.82132i 0.0630670i
\(835\) −2.33885 −0.0809391
\(836\) 56.8592 1.96652
\(837\) − 5.26875i − 0.182115i
\(838\) − 2.65625i − 0.0917587i
\(839\) − 4.63879i − 0.160149i −0.996789 0.0800744i \(-0.974484\pi\)
0.996789 0.0800744i \(-0.0255158\pi\)
\(840\) 0.732578i 0.0252763i
\(841\) 6.27173 0.216267
\(842\) 0.932559 0.0321381
\(843\) 27.2349i 0.938020i
\(844\) −10.7017 −0.368368
\(845\) 0 0
\(846\) −2.99761 −0.103060
\(847\) − 36.7329i − 1.26216i
\(848\) −3.02475 −0.103870
\(849\) −5.28382 −0.181340
\(850\) − 8.35690i − 0.286639i
\(851\) − 26.6872i − 0.914827i
\(852\) 10.3056i 0.353064i
\(853\) 18.3884i 0.629605i 0.949157 + 0.314803i \(0.101938\pi\)
−0.949157 + 0.314803i \(0.898062\pi\)
\(854\) −6.64370 −0.227343
\(855\) 1.37867 0.0471494
\(856\) 21.6866i 0.741234i
\(857\) −28.1849 −0.962778 −0.481389 0.876507i \(-0.659868\pi\)
−0.481389 + 0.876507i \(0.659868\pi\)
\(858\) 0 0
\(859\) 33.3957 1.13944 0.569722 0.821837i \(-0.307051\pi\)
0.569722 + 0.821837i \(0.307051\pi\)
\(860\) − 0.762709i − 0.0260082i
\(861\) 0.780167 0.0265880
\(862\) 1.28429 0.0437431
\(863\) 43.0640i 1.46592i 0.680274 + 0.732958i \(0.261861\pi\)
−0.680274 + 0.732958i \(0.738139\pi\)
\(864\) − 4.65279i − 0.158291i
\(865\) − 1.17928i − 0.0400966i
\(866\) − 5.62910i − 0.191285i
\(867\) −2.54527 −0.0864419
\(868\) 16.6431 0.564904
\(869\) 25.1817i 0.854230i
\(870\) −0.652793 −0.0221317
\(871\) 0 0
\(872\) −20.5526 −0.695998
\(873\) 13.6896i 0.463324i
\(874\) 20.7308 0.701229
\(875\) −4.30319 −0.145474
\(876\) 13.2567i 0.447901i
\(877\) 30.1702i 1.01877i 0.860537 + 0.509387i \(0.170128\pi\)
−0.860537 + 0.509387i \(0.829872\pi\)
\(878\) − 4.86294i − 0.164116i
\(879\) 32.6625i 1.10168i
\(880\) −3.98015 −0.134171
\(881\) 3.56273 0.120031 0.0600157 0.998197i \(-0.480885\pi\)
0.0600157 + 0.998197i \(0.480885\pi\)
\(882\) − 1.74764i − 0.0588462i
\(883\) 10.2088 0.343554 0.171777 0.985136i \(-0.445049\pi\)
0.171777 + 0.985136i \(0.445049\pi\)
\(884\) 0 0
\(885\) 3.38404 0.113753
\(886\) 8.56273i 0.287670i
\(887\) −9.33645 −0.313487 −0.156744 0.987639i \(-0.550100\pi\)
−0.156744 + 0.987639i \(0.550100\pi\)
\(888\) 5.41119 0.181588
\(889\) − 18.9318i − 0.634953i
\(890\) 0.0150655i 0 0.000504996i
\(891\) 5.65279i 0.189376i
\(892\) − 25.5894i − 0.856797i
\(893\) −37.5986 −1.25819
\(894\) 6.84846 0.229047
\(895\) − 0.848462i − 0.0283610i
\(896\) 19.1457 0.639613
\(897\) 0 0
\(898\) −12.8261 −0.428013
\(899\) 31.2911i 1.04362i
\(900\) 8.89977 0.296659
\(901\) −4.03385 −0.134387
\(902\) − 1.11960i − 0.0372788i
\(903\) − 3.00431i − 0.0999772i
\(904\) 2.77107i 0.0921644i
\(905\) 3.33081i 0.110720i
\(906\) −1.63533 −0.0543303
\(907\) 41.0804 1.36405 0.682026 0.731328i \(-0.261099\pi\)
0.682026 + 0.731328i \(0.261099\pi\)
\(908\) − 28.8442i − 0.957227i
\(909\) 5.41119 0.179478
\(910\) 0 0
\(911\) −18.9705 −0.628519 −0.314260 0.949337i \(-0.601756\pi\)
−0.314260 + 0.949337i \(0.601756\pi\)
\(912\) − 15.9138i − 0.526958i
\(913\) −57.5792 −1.90559
\(914\) 8.04221 0.266013
\(915\) − 2.10321i − 0.0695300i
\(916\) 3.31767i 0.109619i
\(917\) 1.59094i 0.0525375i
\(918\) − 1.69202i − 0.0558450i
\(919\) 29.0019 0.956685 0.478343 0.878173i \(-0.341238\pi\)
0.478343 + 0.878173i \(0.341238\pi\)
\(920\) −3.48725 −0.114971
\(921\) − 20.7614i − 0.684111i
\(922\) 3.36791 0.110916
\(923\) 0 0
\(924\) −17.8562 −0.587427
\(925\) 15.7952i 0.519344i
\(926\) 15.7345 0.517068
\(927\) −13.7560 −0.451806
\(928\) 27.6329i 0.907096i
\(929\) 50.7211i 1.66410i 0.554697 + 0.832052i \(0.312834\pi\)
−0.554697 + 0.832052i \(0.687166\pi\)
\(930\) − 0.579121i − 0.0189901i
\(931\) − 21.9205i − 0.718415i
\(932\) 42.2107 1.38266
\(933\) 11.3013 0.369987
\(934\) 5.78554i 0.189309i
\(935\) −5.30798 −0.173589
\(936\) 0 0
\(937\) 51.3051 1.67606 0.838032 0.545620i \(-0.183706\pi\)
0.838032 + 0.545620i \(0.183706\pi\)
\(938\) − 4.65040i − 0.151841i
\(939\) −4.27173 −0.139403
\(940\) 2.99761 0.0977712
\(941\) 34.7036i 1.13131i 0.824643 + 0.565653i \(0.191376\pi\)
−0.824643 + 0.565653i \(0.808624\pi\)
\(942\) 2.17092i 0.0707322i
\(943\) 3.71379i 0.120938i
\(944\) − 39.0616i − 1.27135i
\(945\) −0.432960 −0.0140842
\(946\) −4.31144 −0.140177
\(947\) − 13.0127i − 0.422855i −0.977394 0.211428i \(-0.932189\pi\)
0.977394 0.211428i \(-0.0678112\pi\)
\(948\) 8.02715 0.260710
\(949\) 0 0
\(950\) −12.2698 −0.398085
\(951\) − 15.4776i − 0.501894i
\(952\) 11.2771 0.365493
\(953\) −26.0151 −0.842711 −0.421355 0.906896i \(-0.638445\pi\)
−0.421355 + 0.906896i \(0.638445\pi\)
\(954\) 0.472189i 0.0152877i
\(955\) − 0.321388i − 0.0103999i
\(956\) − 26.3110i − 0.850957i
\(957\) − 33.5719i − 1.08523i
\(958\) −11.3604 −0.367036
\(959\) 16.7383 0.540507
\(960\) 0.896789i 0.0289437i
\(961\) 3.24027 0.104525
\(962\) 0 0
\(963\) −12.8170 −0.413022
\(964\) 15.5526i 0.500914i
\(965\) 2.27114 0.0731107
\(966\) −6.51035 −0.209467
\(967\) − 36.6644i − 1.17905i −0.807751 0.589524i \(-0.799315\pi\)
0.807751 0.589524i \(-0.200685\pi\)
\(968\) 35.4547i 1.13956i
\(969\) − 21.2228i − 0.681775i
\(970\) 1.50471i 0.0483134i
\(971\) −37.8465 −1.21455 −0.607277 0.794490i \(-0.707738\pi\)
−0.607277 + 0.794490i \(0.707738\pi\)
\(972\) 1.80194 0.0577972
\(973\) − 7.17416i − 0.229993i
\(974\) −7.12392 −0.228265
\(975\) 0 0
\(976\) −24.2771 −0.777091
\(977\) − 28.8998i − 0.924586i −0.886727 0.462293i \(-0.847027\pi\)
0.886727 0.462293i \(-0.152973\pi\)
\(978\) 3.84117 0.122827
\(979\) −0.774791 −0.0247624
\(980\) 1.74764i 0.0558264i
\(981\) − 12.1468i − 0.387816i
\(982\) 9.22952i 0.294526i
\(983\) − 19.3991i − 0.618735i −0.950942 0.309368i \(-0.899883\pi\)
0.950942 0.309368i \(-0.100117\pi\)
\(984\) −0.753020 −0.0240054
\(985\) 1.01566 0.0323615
\(986\) 10.0489i 0.320023i
\(987\) 11.8076 0.375839
\(988\) 0 0
\(989\) 14.3013 0.454754
\(990\) 0.621334i 0.0197473i
\(991\) −5.18300 −0.164643 −0.0823217 0.996606i \(-0.526233\pi\)
−0.0823217 + 0.996606i \(0.526233\pi\)
\(992\) −24.5144 −0.778333
\(993\) − 6.06829i − 0.192572i
\(994\) 4.46191i 0.141523i
\(995\) 6.11828i 0.193962i
\(996\) 18.3545i 0.581585i
\(997\) −49.3642 −1.56338 −0.781690 0.623667i \(-0.785642\pi\)
−0.781690 + 0.623667i \(0.785642\pi\)
\(998\) 3.59047 0.113654
\(999\) 3.19806i 0.101182i
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.b.g.337.3 6
3.2 odd 2 1521.2.b.m.1351.4 6
13.2 odd 12 507.2.e.k.22.2 6
13.3 even 3 507.2.j.h.316.4 12
13.4 even 6 507.2.j.h.361.4 12
13.5 odd 4 507.2.a.j.1.2 3
13.6 odd 12 507.2.e.k.484.2 6
13.7 odd 12 507.2.e.j.484.2 6
13.8 odd 4 507.2.a.k.1.2 yes 3
13.9 even 3 507.2.j.h.361.3 12
13.10 even 6 507.2.j.h.316.3 12
13.11 odd 12 507.2.e.j.22.2 6
13.12 even 2 inner 507.2.b.g.337.4 6
39.5 even 4 1521.2.a.q.1.2 3
39.8 even 4 1521.2.a.p.1.2 3
39.38 odd 2 1521.2.b.m.1351.3 6
52.31 even 4 8112.2.a.by.1.3 3
52.47 even 4 8112.2.a.cf.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.2 3 13.5 odd 4
507.2.a.k.1.2 yes 3 13.8 odd 4
507.2.b.g.337.3 6 1.1 even 1 trivial
507.2.b.g.337.4 6 13.12 even 2 inner
507.2.e.j.22.2 6 13.11 odd 12
507.2.e.j.484.2 6 13.7 odd 12
507.2.e.k.22.2 6 13.2 odd 12
507.2.e.k.484.2 6 13.6 odd 12
507.2.j.h.316.3 12 13.10 even 6
507.2.j.h.316.4 12 13.3 even 3
507.2.j.h.361.3 12 13.9 even 3
507.2.j.h.361.4 12 13.4 even 6
1521.2.a.p.1.2 3 39.8 even 4
1521.2.a.q.1.2 3 39.5 even 4
1521.2.b.m.1351.3 6 39.38 odd 2
1521.2.b.m.1351.4 6 3.2 odd 2
8112.2.a.by.1.3 3 52.31 even 4
8112.2.a.cf.1.1 3 52.47 even 4