Properties

Label 507.2.a.l.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

\(f(q)\) \(=\) \(q+2.35690 q^{2} -1.00000 q^{3} +3.55496 q^{4} +3.69202 q^{5} -2.35690 q^{6} -0.801938 q^{7} +3.66487 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.35690 q^{2} -1.00000 q^{3} +3.55496 q^{4} +3.69202 q^{5} -2.35690 q^{6} -0.801938 q^{7} +3.66487 q^{8} +1.00000 q^{9} +8.70171 q^{10} -2.85086 q^{11} -3.55496 q^{12} -1.89008 q^{14} -3.69202 q^{15} +1.52781 q^{16} +2.93900 q^{17} +2.35690 q^{18} -2.44504 q^{19} +13.1250 q^{20} +0.801938 q^{21} -6.71917 q^{22} -7.78986 q^{23} -3.66487 q^{24} +8.63102 q^{25} -1.00000 q^{27} -2.85086 q^{28} +3.85086 q^{29} -8.70171 q^{30} +2.34481 q^{31} -3.72886 q^{32} +2.85086 q^{33} +6.92692 q^{34} -2.96077 q^{35} +3.55496 q^{36} +7.44504 q^{37} -5.76271 q^{38} +13.5308 q^{40} -0.850855 q^{41} +1.89008 q^{42} -1.61596 q^{43} -10.1347 q^{44} +3.69202 q^{45} -18.3599 q^{46} +2.44504 q^{47} -1.52781 q^{48} -6.35690 q^{49} +20.3424 q^{50} -2.93900 q^{51} -9.96077 q^{53} -2.35690 q^{54} -10.5254 q^{55} -2.93900 q^{56} +2.44504 q^{57} +9.07606 q^{58} -5.38404 q^{59} -13.1250 q^{60} -13.2567 q^{61} +5.52648 q^{62} -0.801938 q^{63} -11.8442 q^{64} +6.71917 q^{66} +14.3937 q^{67} +10.4480 q^{68} +7.78986 q^{69} -6.97823 q^{70} +8.12498 q^{71} +3.66487 q^{72} -11.8877 q^{73} +17.5472 q^{74} -8.63102 q^{75} -8.69202 q^{76} +2.28621 q^{77} +5.40581 q^{79} +5.64071 q^{80} +1.00000 q^{81} -2.00538 q^{82} +7.04892 q^{83} +2.85086 q^{84} +10.8509 q^{85} -3.80864 q^{86} -3.85086 q^{87} -10.4480 q^{88} +1.13169 q^{89} +8.70171 q^{90} -27.6926 q^{92} -2.34481 q^{93} +5.76271 q^{94} -9.02715 q^{95} +3.72886 q^{96} -5.94438 q^{97} -14.9825 q^{98} -2.85086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 11q^{4} + 6q^{5} - 3q^{6} + 2q^{7} + 12q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 11q^{4} + 6q^{5} - 3q^{6} + 2q^{7} + 12q^{8} + 3q^{9} - q^{10} + 5q^{11} - 11q^{12} - 5q^{14} - 6q^{15} + 11q^{16} - q^{17} + 3q^{18} - 7q^{19} + 15q^{20} - 2q^{21} - 9q^{22} - 12q^{24} + 11q^{25} - 3q^{27} + 5q^{28} - 2q^{29} + q^{30} - 16q^{31} + 22q^{32} - 5q^{33} - 8q^{34} + 4q^{35} + 11q^{36} + 22q^{37} + 3q^{40} + 11q^{41} + 5q^{42} - 15q^{43} + 16q^{44} + 6q^{45} - 7q^{46} + 7q^{47} - 11q^{48} - 15q^{49} - 3q^{50} + q^{51} - 17q^{53} - 3q^{54} + 3q^{55} + q^{56} + 7q^{57} + 12q^{58} - 6q^{59} - 15q^{60} - 13q^{61} - 2q^{62} + 2q^{63} + 9q^{66} + 11q^{67} - 13q^{68} - 24q^{70} + 12q^{72} + 6q^{73} + 15q^{74} - 11q^{75} - 21q^{76} + 15q^{77} + 3q^{79} - 20q^{80} + 3q^{81} - 3q^{82} + 12q^{83} - 5q^{84} + 19q^{85} - 29q^{86} + 2q^{87} + 13q^{88} + q^{89} - q^{90} + 7q^{92} + 16q^{93} - 21q^{95} - 22q^{96} - 5q^{97} - 29q^{98} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.35690 1.66658 0.833289 0.552838i \(-0.186455\pi\)
0.833289 + 0.552838i \(0.186455\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.55496 1.77748
\(5\) 3.69202 1.65112 0.825561 0.564313i \(-0.190859\pi\)
0.825561 + 0.564313i \(0.190859\pi\)
\(6\) −2.35690 −0.962199
\(7\) −0.801938 −0.303104 −0.151552 0.988449i \(-0.548427\pi\)
−0.151552 + 0.988449i \(0.548427\pi\)
\(8\) 3.66487 1.29573
\(9\) 1.00000 0.333333
\(10\) 8.70171 2.75172
\(11\) −2.85086 −0.859565 −0.429783 0.902932i \(-0.641410\pi\)
−0.429783 + 0.902932i \(0.641410\pi\)
\(12\) −3.55496 −1.02623
\(13\) 0 0
\(14\) −1.89008 −0.505146
\(15\) −3.69202 −0.953276
\(16\) 1.52781 0.381953
\(17\) 2.93900 0.712812 0.356406 0.934331i \(-0.384002\pi\)
0.356406 + 0.934331i \(0.384002\pi\)
\(18\) 2.35690 0.555526
\(19\) −2.44504 −0.560931 −0.280466 0.959864i \(-0.590489\pi\)
−0.280466 + 0.959864i \(0.590489\pi\)
\(20\) 13.1250 2.93484
\(21\) 0.801938 0.174997
\(22\) −6.71917 −1.43253
\(23\) −7.78986 −1.62430 −0.812149 0.583451i \(-0.801702\pi\)
−0.812149 + 0.583451i \(0.801702\pi\)
\(24\) −3.66487 −0.748089
\(25\) 8.63102 1.72620
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −2.85086 −0.538761
\(29\) 3.85086 0.715086 0.357543 0.933897i \(-0.383615\pi\)
0.357543 + 0.933897i \(0.383615\pi\)
\(30\) −8.70171 −1.58871
\(31\) 2.34481 0.421141 0.210571 0.977579i \(-0.432468\pi\)
0.210571 + 0.977579i \(0.432468\pi\)
\(32\) −3.72886 −0.659175
\(33\) 2.85086 0.496270
\(34\) 6.92692 1.18796
\(35\) −2.96077 −0.500462
\(36\) 3.55496 0.592493
\(37\) 7.44504 1.22396 0.611979 0.790874i \(-0.290374\pi\)
0.611979 + 0.790874i \(0.290374\pi\)
\(38\) −5.76271 −0.934835
\(39\) 0 0
\(40\) 13.5308 2.13941
\(41\) −0.850855 −0.132881 −0.0664406 0.997790i \(-0.521164\pi\)
−0.0664406 + 0.997790i \(0.521164\pi\)
\(42\) 1.89008 0.291646
\(43\) −1.61596 −0.246431 −0.123216 0.992380i \(-0.539321\pi\)
−0.123216 + 0.992380i \(0.539321\pi\)
\(44\) −10.1347 −1.52786
\(45\) 3.69202 0.550374
\(46\) −18.3599 −2.70702
\(47\) 2.44504 0.356646 0.178323 0.983972i \(-0.442933\pi\)
0.178323 + 0.983972i \(0.442933\pi\)
\(48\) −1.52781 −0.220521
\(49\) −6.35690 −0.908128
\(50\) 20.3424 2.87685
\(51\) −2.93900 −0.411542
\(52\) 0 0
\(53\) −9.96077 −1.36822 −0.684109 0.729380i \(-0.739809\pi\)
−0.684109 + 0.729380i \(0.739809\pi\)
\(54\) −2.35690 −0.320733
\(55\) −10.5254 −1.41925
\(56\) −2.93900 −0.392741
\(57\) 2.44504 0.323854
\(58\) 9.07606 1.19175
\(59\) −5.38404 −0.700943 −0.350471 0.936573i \(-0.613979\pi\)
−0.350471 + 0.936573i \(0.613979\pi\)
\(60\) −13.1250 −1.69443
\(61\) −13.2567 −1.69734 −0.848671 0.528921i \(-0.822597\pi\)
−0.848671 + 0.528921i \(0.822597\pi\)
\(62\) 5.52648 0.701864
\(63\) −0.801938 −0.101035
\(64\) −11.8442 −1.48052
\(65\) 0 0
\(66\) 6.71917 0.827072
\(67\) 14.3937 1.75847 0.879237 0.476384i \(-0.158053\pi\)
0.879237 + 0.476384i \(0.158053\pi\)
\(68\) 10.4480 1.26701
\(69\) 7.78986 0.937788
\(70\) −6.97823 −0.834058
\(71\) 8.12498 0.964258 0.482129 0.876100i \(-0.339864\pi\)
0.482129 + 0.876100i \(0.339864\pi\)
\(72\) 3.66487 0.431910
\(73\) −11.8877 −1.39135 −0.695674 0.718357i \(-0.744894\pi\)
−0.695674 + 0.718357i \(0.744894\pi\)
\(74\) 17.5472 2.03982
\(75\) −8.63102 −0.996625
\(76\) −8.69202 −0.997043
\(77\) 2.28621 0.260538
\(78\) 0 0
\(79\) 5.40581 0.608202 0.304101 0.952640i \(-0.401644\pi\)
0.304101 + 0.952640i \(0.401644\pi\)
\(80\) 5.64071 0.630651
\(81\) 1.00000 0.111111
\(82\) −2.00538 −0.221457
\(83\) 7.04892 0.773719 0.386860 0.922139i \(-0.373560\pi\)
0.386860 + 0.922139i \(0.373560\pi\)
\(84\) 2.85086 0.311054
\(85\) 10.8509 1.17694
\(86\) −3.80864 −0.410696
\(87\) −3.85086 −0.412855
\(88\) −10.4480 −1.11376
\(89\) 1.13169 0.119959 0.0599793 0.998200i \(-0.480897\pi\)
0.0599793 + 0.998200i \(0.480897\pi\)
\(90\) 8.70171 0.917241
\(91\) 0 0
\(92\) −27.6926 −2.88715
\(93\) −2.34481 −0.243146
\(94\) 5.76271 0.594378
\(95\) −9.02715 −0.926166
\(96\) 3.72886 0.380575
\(97\) −5.94438 −0.603560 −0.301780 0.953378i \(-0.597581\pi\)
−0.301780 + 0.953378i \(0.597581\pi\)
\(98\) −14.9825 −1.51347
\(99\) −2.85086 −0.286522
\(100\) 30.6829 3.06829
\(101\) 4.62565 0.460269 0.230134 0.973159i \(-0.426083\pi\)
0.230134 + 0.973159i \(0.426083\pi\)
\(102\) −6.92692 −0.685867
\(103\) 1.20775 0.119003 0.0595016 0.998228i \(-0.481049\pi\)
0.0595016 + 0.998228i \(0.481049\pi\)
\(104\) 0 0
\(105\) 2.96077 0.288942
\(106\) −23.4765 −2.28024
\(107\) −9.52111 −0.920440 −0.460220 0.887805i \(-0.652229\pi\)
−0.460220 + 0.887805i \(0.652229\pi\)
\(108\) −3.55496 −0.342076
\(109\) −1.78448 −0.170922 −0.0854611 0.996342i \(-0.527236\pi\)
−0.0854611 + 0.996342i \(0.527236\pi\)
\(110\) −24.8073 −2.36528
\(111\) −7.44504 −0.706652
\(112\) −1.22521 −0.115771
\(113\) 4.95108 0.465759 0.232879 0.972506i \(-0.425185\pi\)
0.232879 + 0.972506i \(0.425185\pi\)
\(114\) 5.76271 0.539727
\(115\) −28.7603 −2.68191
\(116\) 13.6896 1.27105
\(117\) 0 0
\(118\) −12.6896 −1.16817
\(119\) −2.35690 −0.216056
\(120\) −13.5308 −1.23519
\(121\) −2.87263 −0.261148
\(122\) −31.2446 −2.82875
\(123\) 0.850855 0.0767190
\(124\) 8.33572 0.748569
\(125\) 13.4058 1.19905
\(126\) −1.89008 −0.168382
\(127\) 5.67025 0.503153 0.251577 0.967837i \(-0.419051\pi\)
0.251577 + 0.967837i \(0.419051\pi\)
\(128\) −20.4577 −1.80822
\(129\) 1.61596 0.142277
\(130\) 0 0
\(131\) 18.2228 1.59213 0.796067 0.605208i \(-0.206910\pi\)
0.796067 + 0.605208i \(0.206910\pi\)
\(132\) 10.1347 0.882110
\(133\) 1.96077 0.170020
\(134\) 33.9245 2.93063
\(135\) −3.69202 −0.317759
\(136\) 10.7711 0.923612
\(137\) 9.45042 0.807404 0.403702 0.914891i \(-0.367723\pi\)
0.403702 + 0.914891i \(0.367723\pi\)
\(138\) 18.3599 1.56290
\(139\) 4.01507 0.340553 0.170277 0.985396i \(-0.445534\pi\)
0.170277 + 0.985396i \(0.445534\pi\)
\(140\) −10.5254 −0.889560
\(141\) −2.44504 −0.205910
\(142\) 19.1497 1.60701
\(143\) 0 0
\(144\) 1.52781 0.127318
\(145\) 14.2174 1.18069
\(146\) −28.0180 −2.31879
\(147\) 6.35690 0.524308
\(148\) 26.4668 2.17556
\(149\) 19.4058 1.58979 0.794893 0.606750i \(-0.207527\pi\)
0.794893 + 0.606750i \(0.207527\pi\)
\(150\) −20.3424 −1.66095
\(151\) −12.3623 −1.00603 −0.503014 0.864278i \(-0.667775\pi\)
−0.503014 + 0.864278i \(0.667775\pi\)
\(152\) −8.96077 −0.726815
\(153\) 2.93900 0.237604
\(154\) 5.38835 0.434206
\(155\) 8.65710 0.695355
\(156\) 0 0
\(157\) −18.6775 −1.49063 −0.745315 0.666712i \(-0.767701\pi\)
−0.745315 + 0.666712i \(0.767701\pi\)
\(158\) 12.7409 1.01361
\(159\) 9.96077 0.789941
\(160\) −13.7670 −1.08838
\(161\) 6.24698 0.492331
\(162\) 2.35690 0.185175
\(163\) 12.3394 0.966499 0.483250 0.875483i \(-0.339456\pi\)
0.483250 + 0.875483i \(0.339456\pi\)
\(164\) −3.02475 −0.236194
\(165\) 10.5254 0.819403
\(166\) 16.6136 1.28946
\(167\) 11.4940 0.889429 0.444715 0.895672i \(-0.353305\pi\)
0.444715 + 0.895672i \(0.353305\pi\)
\(168\) 2.93900 0.226749
\(169\) 0 0
\(170\) 25.5743 1.96146
\(171\) −2.44504 −0.186977
\(172\) −5.74466 −0.438026
\(173\) 12.1142 0.921028 0.460514 0.887653i \(-0.347665\pi\)
0.460514 + 0.887653i \(0.347665\pi\)
\(174\) −9.07606 −0.688055
\(175\) −6.92154 −0.523219
\(176\) −4.35557 −0.328313
\(177\) 5.38404 0.404689
\(178\) 2.66727 0.199920
\(179\) −0.538565 −0.0402542 −0.0201271 0.999797i \(-0.506407\pi\)
−0.0201271 + 0.999797i \(0.506407\pi\)
\(180\) 13.1250 0.978278
\(181\) −23.2838 −1.73067 −0.865336 0.501192i \(-0.832895\pi\)
−0.865336 + 0.501192i \(0.832895\pi\)
\(182\) 0 0
\(183\) 13.2567 0.979961
\(184\) −28.5488 −2.10465
\(185\) 27.4873 2.02090
\(186\) −5.52648 −0.405221
\(187\) −8.37867 −0.612709
\(188\) 8.69202 0.633931
\(189\) 0.801938 0.0583324
\(190\) −21.2760 −1.54353
\(191\) 16.7657 1.21312 0.606561 0.795037i \(-0.292548\pi\)
0.606561 + 0.795037i \(0.292548\pi\)
\(192\) 11.8442 0.854778
\(193\) 25.7439 1.85309 0.926544 0.376186i \(-0.122765\pi\)
0.926544 + 0.376186i \(0.122765\pi\)
\(194\) −14.0103 −1.00588
\(195\) 0 0
\(196\) −22.5985 −1.61418
\(197\) 21.4209 1.52617 0.763087 0.646296i \(-0.223683\pi\)
0.763087 + 0.646296i \(0.223683\pi\)
\(198\) −6.71917 −0.477511
\(199\) 3.52781 0.250080 0.125040 0.992152i \(-0.460094\pi\)
0.125040 + 0.992152i \(0.460094\pi\)
\(200\) 31.6316 2.23669
\(201\) −14.3937 −1.01526
\(202\) 10.9022 0.767074
\(203\) −3.08815 −0.216745
\(204\) −10.4480 −0.731508
\(205\) −3.14138 −0.219403
\(206\) 2.84654 0.198328
\(207\) −7.78986 −0.541432
\(208\) 0 0
\(209\) 6.97046 0.482157
\(210\) 6.97823 0.481544
\(211\) 1.21552 0.0836799 0.0418399 0.999124i \(-0.486678\pi\)
0.0418399 + 0.999124i \(0.486678\pi\)
\(212\) −35.4101 −2.43198
\(213\) −8.12498 −0.556715
\(214\) −22.4403 −1.53398
\(215\) −5.96615 −0.406888
\(216\) −3.66487 −0.249363
\(217\) −1.88040 −0.127650
\(218\) −4.20583 −0.284855
\(219\) 11.8877 0.803296
\(220\) −37.4174 −2.52268
\(221\) 0 0
\(222\) −17.5472 −1.17769
\(223\) −17.3884 −1.16441 −0.582205 0.813042i \(-0.697810\pi\)
−0.582205 + 0.813042i \(0.697810\pi\)
\(224\) 2.99031 0.199799
\(225\) 8.63102 0.575402
\(226\) 11.6692 0.776223
\(227\) 17.4155 1.15591 0.577954 0.816070i \(-0.303851\pi\)
0.577954 + 0.816070i \(0.303851\pi\)
\(228\) 8.69202 0.575643
\(229\) 18.7603 1.23972 0.619858 0.784714i \(-0.287190\pi\)
0.619858 + 0.784714i \(0.287190\pi\)
\(230\) −67.7851 −4.46962
\(231\) −2.28621 −0.150421
\(232\) 14.1129 0.926557
\(233\) 3.95108 0.258844 0.129422 0.991590i \(-0.458688\pi\)
0.129422 + 0.991590i \(0.458688\pi\)
\(234\) 0 0
\(235\) 9.02715 0.588866
\(236\) −19.1400 −1.24591
\(237\) −5.40581 −0.351145
\(238\) −5.55496 −0.360074
\(239\) 0.818331 0.0529334 0.0264667 0.999650i \(-0.491574\pi\)
0.0264667 + 0.999650i \(0.491574\pi\)
\(240\) −5.64071 −0.364106
\(241\) 6.03252 0.388589 0.194295 0.980943i \(-0.437758\pi\)
0.194295 + 0.980943i \(0.437758\pi\)
\(242\) −6.77048 −0.435223
\(243\) −1.00000 −0.0641500
\(244\) −47.1269 −3.01699
\(245\) −23.4698 −1.49943
\(246\) 2.00538 0.127858
\(247\) 0 0
\(248\) 8.59345 0.545685
\(249\) −7.04892 −0.446707
\(250\) 31.5961 1.99831
\(251\) −26.8799 −1.69665 −0.848323 0.529479i \(-0.822387\pi\)
−0.848323 + 0.529479i \(0.822387\pi\)
\(252\) −2.85086 −0.179587
\(253\) 22.2078 1.39619
\(254\) 13.3642 0.838544
\(255\) −10.8509 −0.679507
\(256\) −24.5284 −1.53303
\(257\) −9.05323 −0.564725 −0.282362 0.959308i \(-0.591118\pi\)
−0.282362 + 0.959308i \(0.591118\pi\)
\(258\) 3.80864 0.237116
\(259\) −5.97046 −0.370986
\(260\) 0 0
\(261\) 3.85086 0.238362
\(262\) 42.9493 2.65342
\(263\) 23.1511 1.42756 0.713778 0.700372i \(-0.246983\pi\)
0.713778 + 0.700372i \(0.246983\pi\)
\(264\) 10.4480 0.643032
\(265\) −36.7754 −2.25909
\(266\) 4.62133 0.283352
\(267\) −1.13169 −0.0692581
\(268\) 51.1691 3.12565
\(269\) −2.42088 −0.147604 −0.0738018 0.997273i \(-0.523513\pi\)
−0.0738018 + 0.997273i \(0.523513\pi\)
\(270\) −8.70171 −0.529569
\(271\) −21.4450 −1.30269 −0.651347 0.758780i \(-0.725796\pi\)
−0.651347 + 0.758780i \(0.725796\pi\)
\(272\) 4.49024 0.272261
\(273\) 0 0
\(274\) 22.2737 1.34560
\(275\) −24.6058 −1.48379
\(276\) 27.6926 1.66690
\(277\) −14.8073 −0.889685 −0.444843 0.895609i \(-0.646740\pi\)
−0.444843 + 0.895609i \(0.646740\pi\)
\(278\) 9.46309 0.567558
\(279\) 2.34481 0.140380
\(280\) −10.8509 −0.648463
\(281\) 14.5036 0.865215 0.432608 0.901582i \(-0.357594\pi\)
0.432608 + 0.901582i \(0.357594\pi\)
\(282\) −5.76271 −0.343164
\(283\) 25.6722 1.52605 0.763026 0.646368i \(-0.223713\pi\)
0.763026 + 0.646368i \(0.223713\pi\)
\(284\) 28.8840 1.71395
\(285\) 9.02715 0.534722
\(286\) 0 0
\(287\) 0.682333 0.0402768
\(288\) −3.72886 −0.219725
\(289\) −8.36227 −0.491898
\(290\) 33.5090 1.96772
\(291\) 5.94438 0.348466
\(292\) −42.2602 −2.47309
\(293\) −26.5230 −1.54949 −0.774746 0.632273i \(-0.782122\pi\)
−0.774746 + 0.632273i \(0.782122\pi\)
\(294\) 14.9825 0.873800
\(295\) −19.8780 −1.15734
\(296\) 27.2851 1.58592
\(297\) 2.85086 0.165423
\(298\) 45.7375 2.64950
\(299\) 0 0
\(300\) −30.6829 −1.77148
\(301\) 1.29590 0.0746943
\(302\) −29.1366 −1.67662
\(303\) −4.62565 −0.265736
\(304\) −3.73556 −0.214249
\(305\) −48.9439 −2.80252
\(306\) 6.92692 0.395986
\(307\) −8.24698 −0.470680 −0.235340 0.971913i \(-0.575620\pi\)
−0.235340 + 0.971913i \(0.575620\pi\)
\(308\) 8.12737 0.463100
\(309\) −1.20775 −0.0687066
\(310\) 20.4039 1.15886
\(311\) −14.4179 −0.817564 −0.408782 0.912632i \(-0.634046\pi\)
−0.408782 + 0.912632i \(0.634046\pi\)
\(312\) 0 0
\(313\) 14.2338 0.804544 0.402272 0.915520i \(-0.368221\pi\)
0.402272 + 0.915520i \(0.368221\pi\)
\(314\) −44.0210 −2.48425
\(315\) −2.96077 −0.166821
\(316\) 19.2174 1.08107
\(317\) −6.84415 −0.384406 −0.192203 0.981355i \(-0.561563\pi\)
−0.192203 + 0.981355i \(0.561563\pi\)
\(318\) 23.4765 1.31650
\(319\) −10.9782 −0.614663
\(320\) −43.7289 −2.44452
\(321\) 9.52111 0.531416
\(322\) 14.7235 0.820507
\(323\) −7.18598 −0.399839
\(324\) 3.55496 0.197498
\(325\) 0 0
\(326\) 29.0828 1.61075
\(327\) 1.78448 0.0986819
\(328\) −3.11828 −0.172178
\(329\) −1.96077 −0.108101
\(330\) 24.8073 1.36560
\(331\) 9.44265 0.519015 0.259507 0.965741i \(-0.416440\pi\)
0.259507 + 0.965741i \(0.416440\pi\)
\(332\) 25.0586 1.37527
\(333\) 7.44504 0.407986
\(334\) 27.0901 1.48230
\(335\) 53.1420 2.90346
\(336\) 1.22521 0.0668406
\(337\) 2.64310 0.143979 0.0719895 0.997405i \(-0.477065\pi\)
0.0719895 + 0.997405i \(0.477065\pi\)
\(338\) 0 0
\(339\) −4.95108 −0.268906
\(340\) 38.5743 2.09199
\(341\) −6.68473 −0.361998
\(342\) −5.76271 −0.311612
\(343\) 10.7114 0.578361
\(344\) −5.92228 −0.319308
\(345\) 28.7603 1.54840
\(346\) 28.5520 1.53496
\(347\) −10.1588 −0.545355 −0.272677 0.962106i \(-0.587909\pi\)
−0.272677 + 0.962106i \(0.587909\pi\)
\(348\) −13.6896 −0.733841
\(349\) 10.4397 0.558822 0.279411 0.960172i \(-0.409861\pi\)
0.279411 + 0.960172i \(0.409861\pi\)
\(350\) −16.3134 −0.871986
\(351\) 0 0
\(352\) 10.6304 0.566604
\(353\) 18.2911 0.973538 0.486769 0.873531i \(-0.338175\pi\)
0.486769 + 0.873531i \(0.338175\pi\)
\(354\) 12.6896 0.674446
\(355\) 29.9976 1.59211
\(356\) 4.02310 0.213224
\(357\) 2.35690 0.124740
\(358\) −1.26934 −0.0670867
\(359\) −15.2731 −0.806081 −0.403041 0.915182i \(-0.632047\pi\)
−0.403041 + 0.915182i \(0.632047\pi\)
\(360\) 13.5308 0.713136
\(361\) −13.0218 −0.685356
\(362\) −54.8775 −2.88430
\(363\) 2.87263 0.150774
\(364\) 0 0
\(365\) −43.8896 −2.29729
\(366\) 31.2446 1.63318
\(367\) −22.2717 −1.16258 −0.581288 0.813698i \(-0.697451\pi\)
−0.581288 + 0.813698i \(0.697451\pi\)
\(368\) −11.9014 −0.620405
\(369\) −0.850855 −0.0442937
\(370\) 64.7846 3.36799
\(371\) 7.98792 0.414712
\(372\) −8.33572 −0.432187
\(373\) 4.12631 0.213652 0.106826 0.994278i \(-0.465931\pi\)
0.106826 + 0.994278i \(0.465931\pi\)
\(374\) −19.7476 −1.02113
\(375\) −13.4058 −0.692273
\(376\) 8.96077 0.462116
\(377\) 0 0
\(378\) 1.89008 0.0972154
\(379\) −10.7071 −0.549986 −0.274993 0.961446i \(-0.588676\pi\)
−0.274993 + 0.961446i \(0.588676\pi\)
\(380\) −32.0911 −1.64624
\(381\) −5.67025 −0.290496
\(382\) 39.5150 2.02176
\(383\) 6.52648 0.333488 0.166744 0.986000i \(-0.446675\pi\)
0.166744 + 0.986000i \(0.446675\pi\)
\(384\) 20.4577 1.04398
\(385\) 8.44073 0.430179
\(386\) 60.6757 3.08831
\(387\) −1.61596 −0.0821437
\(388\) −21.1320 −1.07282
\(389\) −11.7922 −0.597891 −0.298945 0.954270i \(-0.596635\pi\)
−0.298945 + 0.954270i \(0.596635\pi\)
\(390\) 0 0
\(391\) −22.8944 −1.15782
\(392\) −23.2972 −1.17669
\(393\) −18.2228 −0.919219
\(394\) 50.4868 2.54349
\(395\) 19.9584 1.00422
\(396\) −10.1347 −0.509286
\(397\) −12.5429 −0.629509 −0.314754 0.949173i \(-0.601922\pi\)
−0.314754 + 0.949173i \(0.601922\pi\)
\(398\) 8.31468 0.416777
\(399\) −1.96077 −0.0981613
\(400\) 13.1866 0.659329
\(401\) 17.8702 0.892397 0.446198 0.894934i \(-0.352778\pi\)
0.446198 + 0.894934i \(0.352778\pi\)
\(402\) −33.9245 −1.69200
\(403\) 0 0
\(404\) 16.4440 0.818118
\(405\) 3.69202 0.183458
\(406\) −7.27844 −0.361223
\(407\) −21.2247 −1.05207
\(408\) −10.7711 −0.533247
\(409\) −27.9119 −1.38015 −0.690076 0.723737i \(-0.742423\pi\)
−0.690076 + 0.723737i \(0.742423\pi\)
\(410\) −7.40389 −0.365652
\(411\) −9.45042 −0.466155
\(412\) 4.29350 0.211526
\(413\) 4.31767 0.212459
\(414\) −18.3599 −0.902339
\(415\) 26.0248 1.27750
\(416\) 0 0
\(417\) −4.01507 −0.196619
\(418\) 16.4286 0.803551
\(419\) −16.4034 −0.801360 −0.400680 0.916218i \(-0.631226\pi\)
−0.400680 + 0.916218i \(0.631226\pi\)
\(420\) 10.5254 0.513588
\(421\) −3.03684 −0.148006 −0.0740032 0.997258i \(-0.523577\pi\)
−0.0740032 + 0.997258i \(0.523577\pi\)
\(422\) 2.86486 0.139459
\(423\) 2.44504 0.118882
\(424\) −36.5050 −1.77284
\(425\) 25.3666 1.23046
\(426\) −19.1497 −0.927808
\(427\) 10.6310 0.514471
\(428\) −33.8471 −1.63606
\(429\) 0 0
\(430\) −14.0616 −0.678110
\(431\) −3.33811 −0.160791 −0.0803955 0.996763i \(-0.525618\pi\)
−0.0803955 + 0.996763i \(0.525618\pi\)
\(432\) −1.52781 −0.0735068
\(433\) 11.9028 0.572010 0.286005 0.958228i \(-0.407673\pi\)
0.286005 + 0.958228i \(0.407673\pi\)
\(434\) −4.43190 −0.212738
\(435\) −14.2174 −0.681674
\(436\) −6.34375 −0.303810
\(437\) 19.0465 0.911119
\(438\) 28.0180 1.33875
\(439\) 3.71810 0.177455 0.0887277 0.996056i \(-0.471720\pi\)
0.0887277 + 0.996056i \(0.471720\pi\)
\(440\) −38.5743 −1.83896
\(441\) −6.35690 −0.302709
\(442\) 0 0
\(443\) 1.45712 0.0692300 0.0346150 0.999401i \(-0.488979\pi\)
0.0346150 + 0.999401i \(0.488979\pi\)
\(444\) −26.4668 −1.25606
\(445\) 4.17821 0.198066
\(446\) −40.9825 −1.94058
\(447\) −19.4058 −0.917863
\(448\) 9.49827 0.448751
\(449\) −12.1274 −0.572326 −0.286163 0.958181i \(-0.592380\pi\)
−0.286163 + 0.958181i \(0.592380\pi\)
\(450\) 20.3424 0.958951
\(451\) 2.42566 0.114220
\(452\) 17.6009 0.827876
\(453\) 12.3623 0.580830
\(454\) 41.0465 1.92641
\(455\) 0 0
\(456\) 8.96077 0.419627
\(457\) −3.44803 −0.161292 −0.0806459 0.996743i \(-0.525698\pi\)
−0.0806459 + 0.996743i \(0.525698\pi\)
\(458\) 44.2161 2.06608
\(459\) −2.93900 −0.137181
\(460\) −102.242 −4.76704
\(461\) −6.75600 −0.314658 −0.157329 0.987546i \(-0.550288\pi\)
−0.157329 + 0.987546i \(0.550288\pi\)
\(462\) −5.38835 −0.250689
\(463\) −7.45175 −0.346312 −0.173156 0.984894i \(-0.555396\pi\)
−0.173156 + 0.984894i \(0.555396\pi\)
\(464\) 5.88338 0.273129
\(465\) −8.65710 −0.401464
\(466\) 9.31229 0.431384
\(467\) 32.6098 1.50900 0.754502 0.656298i \(-0.227879\pi\)
0.754502 + 0.656298i \(0.227879\pi\)
\(468\) 0 0
\(469\) −11.5429 −0.533001
\(470\) 21.2760 0.981391
\(471\) 18.6775 0.860616
\(472\) −19.7318 −0.908232
\(473\) 4.60686 0.211824
\(474\) −12.7409 −0.585211
\(475\) −21.1032 −0.968282
\(476\) −8.37867 −0.384036
\(477\) −9.96077 −0.456072
\(478\) 1.92872 0.0882177
\(479\) 2.82908 0.129264 0.0646321 0.997909i \(-0.479413\pi\)
0.0646321 + 0.997909i \(0.479413\pi\)
\(480\) 13.7670 0.628376
\(481\) 0 0
\(482\) 14.2180 0.647614
\(483\) −6.24698 −0.284247
\(484\) −10.2121 −0.464185
\(485\) −21.9468 −0.996552
\(486\) −2.35690 −0.106911
\(487\) −41.2935 −1.87119 −0.935594 0.353079i \(-0.885135\pi\)
−0.935594 + 0.353079i \(0.885135\pi\)
\(488\) −48.5840 −2.19930
\(489\) −12.3394 −0.558009
\(490\) −55.3159 −2.49892
\(491\) −34.6698 −1.56463 −0.782313 0.622886i \(-0.785960\pi\)
−0.782313 + 0.622886i \(0.785960\pi\)
\(492\) 3.02475 0.136366
\(493\) 11.3177 0.509722
\(494\) 0 0
\(495\) −10.5254 −0.473082
\(496\) 3.58243 0.160856
\(497\) −6.51573 −0.292270
\(498\) −16.6136 −0.744472
\(499\) 17.9409 0.803146 0.401573 0.915827i \(-0.368464\pi\)
0.401573 + 0.915827i \(0.368464\pi\)
\(500\) 47.6571 2.13129
\(501\) −11.4940 −0.513512
\(502\) −63.3532 −2.82759
\(503\) −26.1812 −1.16736 −0.583681 0.811983i \(-0.698388\pi\)
−0.583681 + 0.811983i \(0.698388\pi\)
\(504\) −2.93900 −0.130914
\(505\) 17.0780 0.759960
\(506\) 52.3414 2.32686
\(507\) 0 0
\(508\) 20.1575 0.894345
\(509\) −5.50604 −0.244051 −0.122025 0.992527i \(-0.538939\pi\)
−0.122025 + 0.992527i \(0.538939\pi\)
\(510\) −25.5743 −1.13245
\(511\) 9.53319 0.421723
\(512\) −16.8955 −0.746681
\(513\) 2.44504 0.107951
\(514\) −21.3375 −0.941158
\(515\) 4.45904 0.196489
\(516\) 5.74466 0.252895
\(517\) −6.97046 −0.306560
\(518\) −14.0718 −0.618277
\(519\) −12.1142 −0.531756
\(520\) 0 0
\(521\) −26.7211 −1.17067 −0.585336 0.810791i \(-0.699037\pi\)
−0.585336 + 0.810791i \(0.699037\pi\)
\(522\) 9.07606 0.397249
\(523\) 36.5230 1.59704 0.798520 0.601968i \(-0.205617\pi\)
0.798520 + 0.601968i \(0.205617\pi\)
\(524\) 64.7813 2.82999
\(525\) 6.92154 0.302081
\(526\) 54.5646 2.37913
\(527\) 6.89141 0.300195
\(528\) 4.35557 0.189552
\(529\) 37.6819 1.63834
\(530\) −86.6757 −3.76495
\(531\) −5.38404 −0.233648
\(532\) 6.97046 0.302208
\(533\) 0 0
\(534\) −2.66727 −0.115424
\(535\) −35.1521 −1.51976
\(536\) 52.7512 2.27851
\(537\) 0.538565 0.0232408
\(538\) −5.70576 −0.245993
\(539\) 18.1226 0.780595
\(540\) −13.1250 −0.564809
\(541\) 18.4655 0.793893 0.396947 0.917842i \(-0.370070\pi\)
0.396947 + 0.917842i \(0.370070\pi\)
\(542\) −50.5437 −2.17104
\(543\) 23.2838 0.999204
\(544\) −10.9591 −0.469868
\(545\) −6.58834 −0.282213
\(546\) 0 0
\(547\) 39.8471 1.70374 0.851870 0.523753i \(-0.175468\pi\)
0.851870 + 0.523753i \(0.175468\pi\)
\(548\) 33.5958 1.43514
\(549\) −13.2567 −0.565781
\(550\) −57.9933 −2.47284
\(551\) −9.41550 −0.401114
\(552\) 28.5488 1.21512
\(553\) −4.33513 −0.184348
\(554\) −34.8993 −1.48273
\(555\) −27.4873 −1.16677
\(556\) 14.2734 0.605327
\(557\) 9.20477 0.390018 0.195009 0.980801i \(-0.437526\pi\)
0.195009 + 0.980801i \(0.437526\pi\)
\(558\) 5.52648 0.233955
\(559\) 0 0
\(560\) −4.52350 −0.191153
\(561\) 8.37867 0.353748
\(562\) 34.1836 1.44195
\(563\) −0.975246 −0.0411017 −0.0205509 0.999789i \(-0.506542\pi\)
−0.0205509 + 0.999789i \(0.506542\pi\)
\(564\) −8.69202 −0.366000
\(565\) 18.2795 0.769024
\(566\) 60.5066 2.54328
\(567\) −0.801938 −0.0336782
\(568\) 29.7770 1.24942
\(569\) −16.8944 −0.708250 −0.354125 0.935198i \(-0.615221\pi\)
−0.354125 + 0.935198i \(0.615221\pi\)
\(570\) 21.2760 0.891155
\(571\) −44.3226 −1.85484 −0.927421 0.374019i \(-0.877979\pi\)
−0.927421 + 0.374019i \(0.877979\pi\)
\(572\) 0 0
\(573\) −16.7657 −0.700397
\(574\) 1.60819 0.0671244
\(575\) −67.2344 −2.80387
\(576\) −11.8442 −0.493506
\(577\) 3.56704 0.148498 0.0742489 0.997240i \(-0.476344\pi\)
0.0742489 + 0.997240i \(0.476344\pi\)
\(578\) −19.7090 −0.819787
\(579\) −25.7439 −1.06988
\(580\) 50.5424 2.09866
\(581\) −5.65279 −0.234517
\(582\) 14.0103 0.580745
\(583\) 28.3967 1.17607
\(584\) −43.5669 −1.80281
\(585\) 0 0
\(586\) −62.5120 −2.58235
\(587\) −16.1172 −0.665229 −0.332614 0.943063i \(-0.607931\pi\)
−0.332614 + 0.943063i \(0.607931\pi\)
\(588\) 22.5985 0.931946
\(589\) −5.73317 −0.236231
\(590\) −46.8504 −1.92880
\(591\) −21.4209 −0.881137
\(592\) 11.3746 0.467494
\(593\) −42.8611 −1.76010 −0.880048 0.474885i \(-0.842490\pi\)
−0.880048 + 0.474885i \(0.842490\pi\)
\(594\) 6.71917 0.275691
\(595\) −8.70171 −0.356735
\(596\) 68.9869 2.82581
\(597\) −3.52781 −0.144384
\(598\) 0 0
\(599\) 40.9420 1.67284 0.836422 0.548086i \(-0.184643\pi\)
0.836422 + 0.548086i \(0.184643\pi\)
\(600\) −31.6316 −1.29136
\(601\) 1.18705 0.0484206 0.0242103 0.999707i \(-0.492293\pi\)
0.0242103 + 0.999707i \(0.492293\pi\)
\(602\) 3.05429 0.124484
\(603\) 14.3937 0.586158
\(604\) −43.9474 −1.78819
\(605\) −10.6058 −0.431187
\(606\) −10.9022 −0.442870
\(607\) 19.9922 0.811460 0.405730 0.913993i \(-0.367017\pi\)
0.405730 + 0.913993i \(0.367017\pi\)
\(608\) 9.11721 0.369752
\(609\) 3.08815 0.125138
\(610\) −115.356 −4.67062
\(611\) 0 0
\(612\) 10.4480 0.422336
\(613\) −33.3618 −1.34747 −0.673735 0.738973i \(-0.735311\pi\)
−0.673735 + 0.738973i \(0.735311\pi\)
\(614\) −19.4373 −0.784424
\(615\) 3.14138 0.126672
\(616\) 8.37867 0.337586
\(617\) 11.6233 0.467935 0.233967 0.972244i \(-0.424829\pi\)
0.233967 + 0.972244i \(0.424829\pi\)
\(618\) −2.84654 −0.114505
\(619\) −16.5381 −0.664722 −0.332361 0.943152i \(-0.607845\pi\)
−0.332361 + 0.943152i \(0.607845\pi\)
\(620\) 30.7756 1.23598
\(621\) 7.78986 0.312596
\(622\) −33.9815 −1.36253
\(623\) −0.907542 −0.0363599
\(624\) 0 0
\(625\) 6.33944 0.253577
\(626\) 33.5477 1.34083
\(627\) −6.97046 −0.278373
\(628\) −66.3979 −2.64956
\(629\) 21.8810 0.872452
\(630\) −6.97823 −0.278019
\(631\) −36.4416 −1.45072 −0.725358 0.688372i \(-0.758326\pi\)
−0.725358 + 0.688372i \(0.758326\pi\)
\(632\) 19.8116 0.788064
\(633\) −1.21552 −0.0483126
\(634\) −16.1309 −0.640642
\(635\) 20.9347 0.830768
\(636\) 35.4101 1.40410
\(637\) 0 0
\(638\) −25.8745 −1.02438
\(639\) 8.12498 0.321419
\(640\) −75.5303 −2.98560
\(641\) 27.2067 1.07460 0.537300 0.843391i \(-0.319444\pi\)
0.537300 + 0.843391i \(0.319444\pi\)
\(642\) 22.4403 0.885646
\(643\) −5.06962 −0.199926 −0.0999632 0.994991i \(-0.531873\pi\)
−0.0999632 + 0.994991i \(0.531873\pi\)
\(644\) 22.2078 0.875108
\(645\) 5.96615 0.234917
\(646\) −16.9366 −0.666362
\(647\) 19.3207 0.759573 0.379787 0.925074i \(-0.375997\pi\)
0.379787 + 0.925074i \(0.375997\pi\)
\(648\) 3.66487 0.143970
\(649\) 15.3491 0.602506
\(650\) 0 0
\(651\) 1.88040 0.0736985
\(652\) 43.8662 1.71793
\(653\) 35.2355 1.37887 0.689436 0.724347i \(-0.257859\pi\)
0.689436 + 0.724347i \(0.257859\pi\)
\(654\) 4.20583 0.164461
\(655\) 67.2790 2.62881
\(656\) −1.29995 −0.0507544
\(657\) −11.8877 −0.463783
\(658\) −4.62133 −0.180158
\(659\) −4.36168 −0.169907 −0.0849535 0.996385i \(-0.527074\pi\)
−0.0849535 + 0.996385i \(0.527074\pi\)
\(660\) 37.4174 1.45647
\(661\) −15.4709 −0.601747 −0.300873 0.953664i \(-0.597278\pi\)
−0.300873 + 0.953664i \(0.597278\pi\)
\(662\) 22.2553 0.864978
\(663\) 0 0
\(664\) 25.8334 1.00253
\(665\) 7.23921 0.280725
\(666\) 17.5472 0.679940
\(667\) −29.9976 −1.16151
\(668\) 40.8605 1.58094
\(669\) 17.3884 0.672273
\(670\) 125.250 4.83883
\(671\) 37.7928 1.45898
\(672\) −2.99031 −0.115354
\(673\) −11.7409 −0.452580 −0.226290 0.974060i \(-0.572660\pi\)
−0.226290 + 0.974060i \(0.572660\pi\)
\(674\) 6.22952 0.239952
\(675\) −8.63102 −0.332208
\(676\) 0 0
\(677\) 3.44504 0.132404 0.0662019 0.997806i \(-0.478912\pi\)
0.0662019 + 0.997806i \(0.478912\pi\)
\(678\) −11.6692 −0.448152
\(679\) 4.76702 0.182941
\(680\) 39.7670 1.52500
\(681\) −17.4155 −0.667363
\(682\) −15.7552 −0.603298
\(683\) 20.4058 0.780807 0.390403 0.920644i \(-0.372336\pi\)
0.390403 + 0.920644i \(0.372336\pi\)
\(684\) −8.69202 −0.332348
\(685\) 34.8911 1.33312
\(686\) 25.2457 0.963883
\(687\) −18.7603 −0.715751
\(688\) −2.46888 −0.0941251
\(689\) 0 0
\(690\) 67.7851 2.58053
\(691\) 27.8039 1.05771 0.528854 0.848713i \(-0.322622\pi\)
0.528854 + 0.848713i \(0.322622\pi\)
\(692\) 43.0656 1.63711
\(693\) 2.28621 0.0868459
\(694\) −23.9433 −0.908876
\(695\) 14.8237 0.562295
\(696\) −14.1129 −0.534948
\(697\) −2.50066 −0.0947194
\(698\) 24.6052 0.931321
\(699\) −3.95108 −0.149444
\(700\) −24.6058 −0.930012
\(701\) 11.9715 0.452158 0.226079 0.974109i \(-0.427409\pi\)
0.226079 + 0.974109i \(0.427409\pi\)
\(702\) 0 0
\(703\) −18.2034 −0.686556
\(704\) 33.7660 1.27260
\(705\) −9.02715 −0.339982
\(706\) 43.1102 1.62248
\(707\) −3.70948 −0.139509
\(708\) 19.1400 0.719327
\(709\) 32.2664 1.21179 0.605894 0.795545i \(-0.292815\pi\)
0.605894 + 0.795545i \(0.292815\pi\)
\(710\) 70.7012 2.65337
\(711\) 5.40581 0.202734
\(712\) 4.14749 0.155434
\(713\) −18.2658 −0.684058
\(714\) 5.55496 0.207889
\(715\) 0 0
\(716\) −1.91457 −0.0715510
\(717\) −0.818331 −0.0305611
\(718\) −35.9970 −1.34340
\(719\) 12.1086 0.451574 0.225787 0.974177i \(-0.427505\pi\)
0.225787 + 0.974177i \(0.427505\pi\)
\(720\) 5.64071 0.210217
\(721\) −0.968541 −0.0360704
\(722\) −30.6910 −1.14220
\(723\) −6.03252 −0.224352
\(724\) −82.7730 −3.07623
\(725\) 33.2368 1.23438
\(726\) 6.77048 0.251276
\(727\) −16.6200 −0.616402 −0.308201 0.951321i \(-0.599727\pi\)
−0.308201 + 0.951321i \(0.599727\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −103.443 −3.82861
\(731\) −4.74930 −0.175659
\(732\) 47.1269 1.74186
\(733\) 17.7912 0.657132 0.328566 0.944481i \(-0.393435\pi\)
0.328566 + 0.944481i \(0.393435\pi\)
\(734\) −52.4922 −1.93752
\(735\) 23.4698 0.865696
\(736\) 29.0473 1.07070
\(737\) −41.0344 −1.51152
\(738\) −2.00538 −0.0738189
\(739\) −27.3618 −1.00652 −0.503260 0.864135i \(-0.667866\pi\)
−0.503260 + 0.864135i \(0.667866\pi\)
\(740\) 97.7160 3.59211
\(741\) 0 0
\(742\) 18.8267 0.691150
\(743\) 8.38596 0.307651 0.153826 0.988098i \(-0.450841\pi\)
0.153826 + 0.988098i \(0.450841\pi\)
\(744\) −8.59345 −0.315051
\(745\) 71.6467 2.62493
\(746\) 9.72528 0.356068
\(747\) 7.04892 0.257906
\(748\) −29.7858 −1.08908
\(749\) 7.63533 0.278989
\(750\) −31.5961 −1.15373
\(751\) −38.7778 −1.41502 −0.707511 0.706703i \(-0.750182\pi\)
−0.707511 + 0.706703i \(0.750182\pi\)
\(752\) 3.73556 0.136222
\(753\) 26.8799 0.979559
\(754\) 0 0
\(755\) −45.6418 −1.66107
\(756\) 2.85086 0.103685
\(757\) 12.9729 0.471506 0.235753 0.971813i \(-0.424244\pi\)
0.235753 + 0.971813i \(0.424244\pi\)
\(758\) −25.2355 −0.916594
\(759\) −22.2078 −0.806090
\(760\) −33.0834 −1.20006
\(761\) −5.15585 −0.186899 −0.0934497 0.995624i \(-0.529789\pi\)
−0.0934497 + 0.995624i \(0.529789\pi\)
\(762\) −13.3642 −0.484134
\(763\) 1.43104 0.0518072
\(764\) 59.6013 2.15630
\(765\) 10.8509 0.392313
\(766\) 15.3822 0.555783
\(767\) 0 0
\(768\) 24.5284 0.885092
\(769\) −35.5013 −1.28021 −0.640104 0.768288i \(-0.721109\pi\)
−0.640104 + 0.768288i \(0.721109\pi\)
\(770\) 19.8939 0.716927
\(771\) 9.05323 0.326044
\(772\) 91.5186 3.29383
\(773\) −6.15585 −0.221411 −0.110705 0.993853i \(-0.535311\pi\)
−0.110705 + 0.993853i \(0.535311\pi\)
\(774\) −3.80864 −0.136899
\(775\) 20.2381 0.726976
\(776\) −21.7854 −0.782050
\(777\) 5.97046 0.214189
\(778\) −27.7931 −0.996431
\(779\) 2.08038 0.0745372
\(780\) 0 0
\(781\) −23.1631 −0.828843
\(782\) −53.9597 −1.92960
\(783\) −3.85086 −0.137618
\(784\) −9.71214 −0.346862
\(785\) −68.9579 −2.46121
\(786\) −42.9493 −1.53195
\(787\) 14.2107 0.506558 0.253279 0.967393i \(-0.418491\pi\)
0.253279 + 0.967393i \(0.418491\pi\)
\(788\) 76.1503 2.71274
\(789\) −23.1511 −0.824200
\(790\) 47.0398 1.67360
\(791\) −3.97046 −0.141173
\(792\) −10.4480 −0.371254
\(793\) 0 0
\(794\) −29.5623 −1.04913
\(795\) 36.7754 1.30429
\(796\) 12.5412 0.444512
\(797\) 11.9022 0.421596 0.210798 0.977530i \(-0.432394\pi\)
0.210798 + 0.977530i \(0.432394\pi\)
\(798\) −4.62133 −0.163593
\(799\) 7.18598 0.254222
\(800\) −32.1839 −1.13787
\(801\) 1.13169 0.0399862
\(802\) 42.1183 1.48725
\(803\) 33.8901 1.19596
\(804\) −51.1691 −1.80460
\(805\) 23.0640 0.812899
\(806\) 0 0
\(807\) 2.42088 0.0852190
\(808\) 16.9524 0.596384
\(809\) 1.61596 0.0568140 0.0284070 0.999596i \(-0.490957\pi\)
0.0284070 + 0.999596i \(0.490957\pi\)
\(810\) 8.70171 0.305747
\(811\) −51.9657 −1.82476 −0.912381 0.409342i \(-0.865758\pi\)
−0.912381 + 0.409342i \(0.865758\pi\)
\(812\) −10.9782 −0.385260
\(813\) 21.4450 0.752110
\(814\) −50.0245 −1.75336
\(815\) 45.5575 1.59581
\(816\) −4.49024 −0.157190
\(817\) 3.95108 0.138231
\(818\) −65.7853 −2.30013
\(819\) 0 0
\(820\) −11.1675 −0.389985
\(821\) 54.8327 1.91367 0.956837 0.290627i \(-0.0938638\pi\)
0.956837 + 0.290627i \(0.0938638\pi\)
\(822\) −22.2737 −0.776883
\(823\) −46.8514 −1.63314 −0.816569 0.577247i \(-0.804127\pi\)
−0.816569 + 0.577247i \(0.804127\pi\)
\(824\) 4.42626 0.154196
\(825\) 24.6058 0.856664
\(826\) 10.1763 0.354078
\(827\) 21.2021 0.737270 0.368635 0.929574i \(-0.379825\pi\)
0.368635 + 0.929574i \(0.379825\pi\)
\(828\) −27.6926 −0.962385
\(829\) 18.2972 0.635489 0.317744 0.948176i \(-0.397075\pi\)
0.317744 + 0.948176i \(0.397075\pi\)
\(830\) 61.3376 2.12906
\(831\) 14.8073 0.513660
\(832\) 0 0
\(833\) −18.6829 −0.647325
\(834\) −9.46309 −0.327680
\(835\) 42.4359 1.46856
\(836\) 24.7797 0.857024
\(837\) −2.34481 −0.0810486
\(838\) −38.6612 −1.33553
\(839\) 21.1414 0.729881 0.364941 0.931031i \(-0.381089\pi\)
0.364941 + 0.931031i \(0.381089\pi\)
\(840\) 10.8509 0.374390
\(841\) −14.1709 −0.488652
\(842\) −7.15751 −0.246664
\(843\) −14.5036 −0.499532
\(844\) 4.32113 0.148739
\(845\) 0 0
\(846\) 5.76271 0.198126
\(847\) 2.30367 0.0791549
\(848\) −15.2182 −0.522594
\(849\) −25.6722 −0.881067
\(850\) 59.7864 2.05066
\(851\) −57.9958 −1.98807
\(852\) −28.8840 −0.989549
\(853\) −7.13036 −0.244139 −0.122069 0.992522i \(-0.538953\pi\)
−0.122069 + 0.992522i \(0.538953\pi\)
\(854\) 25.0562 0.857406
\(855\) −9.02715 −0.308722
\(856\) −34.8937 −1.19264
\(857\) −44.7741 −1.52945 −0.764726 0.644355i \(-0.777126\pi\)
−0.764726 + 0.644355i \(0.777126\pi\)
\(858\) 0 0
\(859\) 57.3782 1.95772 0.978859 0.204535i \(-0.0655681\pi\)
0.978859 + 0.204535i \(0.0655681\pi\)
\(860\) −21.2094 −0.723235
\(861\) −0.682333 −0.0232538
\(862\) −7.86758 −0.267971
\(863\) −6.67563 −0.227241 −0.113621 0.993524i \(-0.536245\pi\)
−0.113621 + 0.993524i \(0.536245\pi\)
\(864\) 3.72886 0.126858
\(865\) 44.7260 1.52073
\(866\) 28.0536 0.953299
\(867\) 8.36227 0.283998
\(868\) −6.68473 −0.226894
\(869\) −15.4112 −0.522789
\(870\) −33.5090 −1.13606
\(871\) 0 0
\(872\) −6.53989 −0.221469
\(873\) −5.94438 −0.201187
\(874\) 44.8907 1.51845
\(875\) −10.7506 −0.363438
\(876\) 42.2602 1.42784
\(877\) 25.2983 0.854263 0.427131 0.904190i \(-0.359524\pi\)
0.427131 + 0.904190i \(0.359524\pi\)
\(878\) 8.76318 0.295743
\(879\) 26.5230 0.894599
\(880\) −16.0809 −0.542085
\(881\) −35.3787 −1.19194 −0.595969 0.803008i \(-0.703232\pi\)
−0.595969 + 0.803008i \(0.703232\pi\)
\(882\) −14.9825 −0.504488
\(883\) −11.5851 −0.389869 −0.194935 0.980816i \(-0.562449\pi\)
−0.194935 + 0.980816i \(0.562449\pi\)
\(884\) 0 0
\(885\) 19.8780 0.668192
\(886\) 3.43429 0.115377
\(887\) −27.2892 −0.916281 −0.458141 0.888880i \(-0.651484\pi\)
−0.458141 + 0.888880i \(0.651484\pi\)
\(888\) −27.2851 −0.915629
\(889\) −4.54719 −0.152508
\(890\) 9.84761 0.330093
\(891\) −2.85086 −0.0955072
\(892\) −61.8149 −2.06972
\(893\) −5.97823 −0.200054
\(894\) −45.7375 −1.52969
\(895\) −1.98839 −0.0664646
\(896\) 16.4058 0.548080
\(897\) 0 0
\(898\) −28.5830 −0.953826
\(899\) 9.02954 0.301152
\(900\) 30.6829 1.02276
\(901\) −29.2747 −0.975282
\(902\) 5.71704 0.190357
\(903\) −1.29590 −0.0431247
\(904\) 18.1451 0.603497
\(905\) −85.9643 −2.85755
\(906\) 29.1366 0.967998
\(907\) −30.4219 −1.01014 −0.505072 0.863077i \(-0.668534\pi\)
−0.505072 + 0.863077i \(0.668534\pi\)
\(908\) 61.9114 2.05460
\(909\) 4.62565 0.153423
\(910\) 0 0
\(911\) 53.5719 1.77492 0.887459 0.460887i \(-0.152469\pi\)
0.887459 + 0.460887i \(0.152469\pi\)
\(912\) 3.73556 0.123697
\(913\) −20.0954 −0.665062
\(914\) −8.12664 −0.268805
\(915\) 48.9439 1.61804
\(916\) 66.6921 2.20357
\(917\) −14.6136 −0.482582
\(918\) −6.92692 −0.228622
\(919\) −36.1672 −1.19305 −0.596523 0.802596i \(-0.703451\pi\)
−0.596523 + 0.802596i \(0.703451\pi\)
\(920\) −105.403 −3.47503
\(921\) 8.24698 0.271747
\(922\) −15.9232 −0.524403
\(923\) 0 0
\(924\) −8.12737 −0.267371
\(925\) 64.2583 2.11280
\(926\) −17.5630 −0.577156
\(927\) 1.20775 0.0396677
\(928\) −14.3593 −0.471367
\(929\) −13.3478 −0.437927 −0.218964 0.975733i \(-0.570268\pi\)
−0.218964 + 0.975733i \(0.570268\pi\)
\(930\) −20.4039 −0.669070
\(931\) 15.5429 0.509397
\(932\) 14.0459 0.460090
\(933\) 14.4179 0.472021
\(934\) 76.8580 2.51487
\(935\) −30.9342 −1.01166
\(936\) 0 0
\(937\) 38.6872 1.26386 0.631928 0.775027i \(-0.282264\pi\)
0.631928 + 0.775027i \(0.282264\pi\)
\(938\) −27.2054 −0.888286
\(939\) −14.2338 −0.464504
\(940\) 32.0911 1.04670
\(941\) 35.7275 1.16468 0.582342 0.812944i \(-0.302136\pi\)
0.582342 + 0.812944i \(0.302136\pi\)
\(942\) 44.0210 1.43428
\(943\) 6.62804 0.215839
\(944\) −8.22580 −0.267727
\(945\) 2.96077 0.0963139
\(946\) 10.8579 0.353020
\(947\) −17.8436 −0.579838 −0.289919 0.957051i \(-0.593628\pi\)
−0.289919 + 0.957051i \(0.593628\pi\)
\(948\) −19.2174 −0.624153
\(949\) 0 0
\(950\) −49.7381 −1.61372
\(951\) 6.84415 0.221937
\(952\) −8.63773 −0.279950
\(953\) 20.1691 0.653342 0.326671 0.945138i \(-0.394073\pi\)
0.326671 + 0.945138i \(0.394073\pi\)
\(954\) −23.4765 −0.760080
\(955\) 61.8993 2.00301
\(956\) 2.90913 0.0940881
\(957\) 10.9782 0.354876
\(958\) 6.66786 0.215429
\(959\) −7.57865 −0.244727
\(960\) 43.7289 1.41134
\(961\) −25.5018 −0.822640
\(962\) 0 0
\(963\) −9.52111 −0.306813
\(964\) 21.4454 0.690709
\(965\) 95.0471 3.05967
\(966\) −14.7235 −0.473720
\(967\) −32.7894 −1.05444 −0.527218 0.849730i \(-0.676765\pi\)
−0.527218 + 0.849730i \(0.676765\pi\)
\(968\) −10.5278 −0.338377
\(969\) 7.18598 0.230847
\(970\) −51.7263 −1.66083
\(971\) −26.9845 −0.865973 −0.432986 0.901401i \(-0.642540\pi\)
−0.432986 + 0.901401i \(0.642540\pi\)
\(972\) −3.55496 −0.114025
\(973\) −3.21983 −0.103223
\(974\) −97.3245 −3.11848
\(975\) 0 0
\(976\) −20.2537 −0.648305
\(977\) −16.9571 −0.542504 −0.271252 0.962508i \(-0.587438\pi\)
−0.271252 + 0.962508i \(0.587438\pi\)
\(978\) −29.0828 −0.929964
\(979\) −3.22627 −0.103112
\(980\) −83.4341 −2.66521
\(981\) −1.78448 −0.0569740
\(982\) −81.7131 −2.60757
\(983\) −32.6631 −1.04179 −0.520895 0.853621i \(-0.674402\pi\)
−0.520895 + 0.853621i \(0.674402\pi\)
\(984\) 3.11828 0.0994070
\(985\) 79.0863 2.51990
\(986\) 26.6746 0.849491
\(987\) 1.96077 0.0624120
\(988\) 0 0
\(989\) 12.5881 0.400277
\(990\) −24.8073 −0.788428
\(991\) 7.64310 0.242791 0.121396 0.992604i \(-0.461263\pi\)
0.121396 + 0.992604i \(0.461263\pi\)
\(992\) −8.74348 −0.277606
\(993\) −9.44265 −0.299653
\(994\) −15.3569 −0.487091
\(995\) 13.0248 0.412912
\(996\) −25.0586 −0.794012
\(997\) 36.1256 1.14411 0.572054 0.820216i \(-0.306147\pi\)
0.572054 + 0.820216i \(0.306147\pi\)
\(998\) 42.2849 1.33850
\(999\) −7.44504 −0.235551
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.l.1.2 yes 3
3.2 odd 2 1521.2.a.n.1.2 3
4.3 odd 2 8112.2.a.cp.1.3 3
13.2 odd 12 507.2.j.i.316.5 12
13.3 even 3 507.2.e.i.22.2 6
13.4 even 6 507.2.e.l.484.2 6
13.5 odd 4 507.2.b.f.337.2 6
13.6 odd 12 507.2.j.i.361.2 12
13.7 odd 12 507.2.j.i.361.5 12
13.8 odd 4 507.2.b.f.337.5 6
13.9 even 3 507.2.e.i.484.2 6
13.10 even 6 507.2.e.l.22.2 6
13.11 odd 12 507.2.j.i.316.2 12
13.12 even 2 507.2.a.i.1.2 3
39.5 even 4 1521.2.b.k.1351.5 6
39.8 even 4 1521.2.b.k.1351.2 6
39.38 odd 2 1521.2.a.s.1.2 3
52.51 odd 2 8112.2.a.cg.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.2 3 13.12 even 2
507.2.a.l.1.2 yes 3 1.1 even 1 trivial
507.2.b.f.337.2 6 13.5 odd 4
507.2.b.f.337.5 6 13.8 odd 4
507.2.e.i.22.2 6 13.3 even 3
507.2.e.i.484.2 6 13.9 even 3
507.2.e.l.22.2 6 13.10 even 6
507.2.e.l.484.2 6 13.4 even 6
507.2.j.i.316.2 12 13.11 odd 12
507.2.j.i.316.5 12 13.2 odd 12
507.2.j.i.361.2 12 13.6 odd 12
507.2.j.i.361.5 12 13.7 odd 12
1521.2.a.n.1.2 3 3.2 odd 2
1521.2.a.s.1.2 3 39.38 odd 2
1521.2.b.k.1351.2 6 39.8 even 4
1521.2.b.k.1351.5 6 39.5 even 4
8112.2.a.cg.1.1 3 52.51 odd 2
8112.2.a.cp.1.3 3 4.3 odd 2