Properties

Label 507.2.a.l.1.1
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 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.04892 q^{2} -1.00000 q^{3} +2.19806 q^{4} +3.35690 q^{5} +2.04892 q^{6} +2.24698 q^{7} -0.405813 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.04892 q^{2} -1.00000 q^{3} +2.19806 q^{4} +3.35690 q^{5} +2.04892 q^{6} +2.24698 q^{7} -0.405813 q^{8} +1.00000 q^{9} -6.87800 q^{10} +4.93900 q^{11} -2.19806 q^{12} -4.60388 q^{14} -3.35690 q^{15} -3.56465 q^{16} +0.911854 q^{17} -2.04892 q^{18} -3.80194 q^{19} +7.37867 q^{20} -2.24698 q^{21} -10.1196 q^{22} +2.02715 q^{23} +0.405813 q^{24} +6.26875 q^{25} -1.00000 q^{27} +4.93900 q^{28} -3.93900 q^{29} +6.87800 q^{30} -8.82908 q^{31} +8.11529 q^{32} -4.93900 q^{33} -1.86831 q^{34} +7.54288 q^{35} +2.19806 q^{36} +8.80194 q^{37} +7.78986 q^{38} -1.36227 q^{40} +6.93900 q^{41} +4.60388 q^{42} -2.28621 q^{43} +10.8562 q^{44} +3.35690 q^{45} -4.15346 q^{46} +3.80194 q^{47} +3.56465 q^{48} -1.95108 q^{49} -12.8442 q^{50} -0.911854 q^{51} +0.542877 q^{53} +2.04892 q^{54} +16.5797 q^{55} -0.911854 q^{56} +3.80194 q^{57} +8.07069 q^{58} -4.71379 q^{59} -7.37867 q^{60} +3.67994 q^{61} +18.0901 q^{62} +2.24698 q^{63} -9.49827 q^{64} +10.1196 q^{66} -1.52111 q^{67} +2.00431 q^{68} -2.02715 q^{69} -15.4547 q^{70} +2.37867 q^{71} -0.405813 q^{72} +7.41119 q^{73} -18.0344 q^{74} -6.26875 q^{75} -8.35690 q^{76} +11.0978 q^{77} -3.74094 q^{79} -11.9661 q^{80} +1.00000 q^{81} -14.2174 q^{82} +2.30798 q^{83} -4.93900 q^{84} +3.06100 q^{85} +4.68425 q^{86} +3.93900 q^{87} -2.00431 q^{88} -10.0586 q^{89} -6.87800 q^{90} +4.45580 q^{92} +8.82908 q^{93} -7.78986 q^{94} -12.7627 q^{95} -8.11529 q^{96} -16.1293 q^{97} +3.99761 q^{98} +4.93900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 11q^{4} + 6q^{5} - 3q^{6} + 2q^{7} + 12q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 11q^{4} + 6q^{5} - 3q^{6} + 2q^{7} + 12q^{8} + 3q^{9} - q^{10} + 5q^{11} - 11q^{12} - 5q^{14} - 6q^{15} + 11q^{16} - q^{17} + 3q^{18} - 7q^{19} + 15q^{20} - 2q^{21} - 9q^{22} - 12q^{24} + 11q^{25} - 3q^{27} + 5q^{28} - 2q^{29} + q^{30} - 16q^{31} + 22q^{32} - 5q^{33} - 8q^{34} + 4q^{35} + 11q^{36} + 22q^{37} + 3q^{40} + 11q^{41} + 5q^{42} - 15q^{43} + 16q^{44} + 6q^{45} - 7q^{46} + 7q^{47} - 11q^{48} - 15q^{49} - 3q^{50} + q^{51} - 17q^{53} - 3q^{54} + 3q^{55} + q^{56} + 7q^{57} + 12q^{58} - 6q^{59} - 15q^{60} - 13q^{61} - 2q^{62} + 2q^{63} + 9q^{66} + 11q^{67} - 13q^{68} - 24q^{70} + 12q^{72} + 6q^{73} + 15q^{74} - 11q^{75} - 21q^{76} + 15q^{77} + 3q^{79} - 20q^{80} + 3q^{81} - 3q^{82} + 12q^{83} - 5q^{84} + 19q^{85} - 29q^{86} + 2q^{87} + 13q^{88} + q^{89} - q^{90} + 7q^{92} + 16q^{93} - 21q^{95} - 22q^{96} - 5q^{97} - 29q^{98} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.04892 −1.44880 −0.724402 0.689378i \(-0.757884\pi\)
−0.724402 + 0.689378i \(0.757884\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.19806 1.09903
\(5\) 3.35690 1.50125 0.750625 0.660729i \(-0.229753\pi\)
0.750625 + 0.660729i \(0.229753\pi\)
\(6\) 2.04892 0.836467
\(7\) 2.24698 0.849278 0.424639 0.905363i \(-0.360401\pi\)
0.424639 + 0.905363i \(0.360401\pi\)
\(8\) −0.405813 −0.143477
\(9\) 1.00000 0.333333
\(10\) −6.87800 −2.17502
\(11\) 4.93900 1.48916 0.744582 0.667531i \(-0.232649\pi\)
0.744582 + 0.667531i \(0.232649\pi\)
\(12\) −2.19806 −0.634526
\(13\) 0 0
\(14\) −4.60388 −1.23044
\(15\) −3.35690 −0.866747
\(16\) −3.56465 −0.891162
\(17\) 0.911854 0.221157 0.110579 0.993867i \(-0.464730\pi\)
0.110579 + 0.993867i \(0.464730\pi\)
\(18\) −2.04892 −0.482934
\(19\) −3.80194 −0.872224 −0.436112 0.899892i \(-0.643645\pi\)
−0.436112 + 0.899892i \(0.643645\pi\)
\(20\) 7.37867 1.64992
\(21\) −2.24698 −0.490331
\(22\) −10.1196 −2.15751
\(23\) 2.02715 0.422689 0.211345 0.977412i \(-0.432216\pi\)
0.211345 + 0.977412i \(0.432216\pi\)
\(24\) 0.405813 0.0828363
\(25\) 6.26875 1.25375
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 4.93900 0.933383
\(29\) −3.93900 −0.731454 −0.365727 0.930722i \(-0.619180\pi\)
−0.365727 + 0.930722i \(0.619180\pi\)
\(30\) 6.87800 1.25575
\(31\) −8.82908 −1.58575 −0.792875 0.609384i \(-0.791417\pi\)
−0.792875 + 0.609384i \(0.791417\pi\)
\(32\) 8.11529 1.43459
\(33\) −4.93900 −0.859770
\(34\) −1.86831 −0.320413
\(35\) 7.54288 1.27498
\(36\) 2.19806 0.366344
\(37\) 8.80194 1.44703 0.723515 0.690309i \(-0.242525\pi\)
0.723515 + 0.690309i \(0.242525\pi\)
\(38\) 7.78986 1.26368
\(39\) 0 0
\(40\) −1.36227 −0.215394
\(41\) 6.93900 1.08369 0.541845 0.840478i \(-0.317726\pi\)
0.541845 + 0.840478i \(0.317726\pi\)
\(42\) 4.60388 0.710393
\(43\) −2.28621 −0.348643 −0.174322 0.984689i \(-0.555773\pi\)
−0.174322 + 0.984689i \(0.555773\pi\)
\(44\) 10.8562 1.63664
\(45\) 3.35690 0.500416
\(46\) −4.15346 −0.612394
\(47\) 3.80194 0.554570 0.277285 0.960788i \(-0.410565\pi\)
0.277285 + 0.960788i \(0.410565\pi\)
\(48\) 3.56465 0.514512
\(49\) −1.95108 −0.278726
\(50\) −12.8442 −1.81644
\(51\) −0.911854 −0.127685
\(52\) 0 0
\(53\) 0.542877 0.0745698 0.0372849 0.999305i \(-0.488129\pi\)
0.0372849 + 0.999305i \(0.488129\pi\)
\(54\) 2.04892 0.278822
\(55\) 16.5797 2.23561
\(56\) −0.911854 −0.121852
\(57\) 3.80194 0.503579
\(58\) 8.07069 1.05973
\(59\) −4.71379 −0.613683 −0.306842 0.951761i \(-0.599272\pi\)
−0.306842 + 0.951761i \(0.599272\pi\)
\(60\) −7.37867 −0.952582
\(61\) 3.67994 0.471168 0.235584 0.971854i \(-0.424300\pi\)
0.235584 + 0.971854i \(0.424300\pi\)
\(62\) 18.0901 2.29744
\(63\) 2.24698 0.283093
\(64\) −9.49827 −1.18728
\(65\) 0 0
\(66\) 10.1196 1.24564
\(67\) −1.52111 −0.185833 −0.0929164 0.995674i \(-0.529619\pi\)
−0.0929164 + 0.995674i \(0.529619\pi\)
\(68\) 2.00431 0.243059
\(69\) −2.02715 −0.244040
\(70\) −15.4547 −1.84719
\(71\) 2.37867 0.282296 0.141148 0.989989i \(-0.454921\pi\)
0.141148 + 0.989989i \(0.454921\pi\)
\(72\) −0.405813 −0.0478255
\(73\) 7.41119 0.867414 0.433707 0.901054i \(-0.357205\pi\)
0.433707 + 0.901054i \(0.357205\pi\)
\(74\) −18.0344 −2.09646
\(75\) −6.26875 −0.723853
\(76\) −8.35690 −0.958602
\(77\) 11.0978 1.26472
\(78\) 0 0
\(79\) −3.74094 −0.420888 −0.210444 0.977606i \(-0.567491\pi\)
−0.210444 + 0.977606i \(0.567491\pi\)
\(80\) −11.9661 −1.33786
\(81\) 1.00000 0.111111
\(82\) −14.2174 −1.57005
\(83\) 2.30798 0.253334 0.126667 0.991945i \(-0.459572\pi\)
0.126667 + 0.991945i \(0.459572\pi\)
\(84\) −4.93900 −0.538889
\(85\) 3.06100 0.332012
\(86\) 4.68425 0.505116
\(87\) 3.93900 0.422305
\(88\) −2.00431 −0.213660
\(89\) −10.0586 −1.06621 −0.533105 0.846049i \(-0.678975\pi\)
−0.533105 + 0.846049i \(0.678975\pi\)
\(90\) −6.87800 −0.725005
\(91\) 0 0
\(92\) 4.45580 0.464549
\(93\) 8.82908 0.915533
\(94\) −7.78986 −0.803462
\(95\) −12.7627 −1.30943
\(96\) −8.11529 −0.828264
\(97\) −16.1293 −1.63768 −0.818841 0.574021i \(-0.805383\pi\)
−0.818841 + 0.574021i \(0.805383\pi\)
\(98\) 3.99761 0.403819
\(99\) 4.93900 0.496388
\(100\) 13.7791 1.37791
\(101\) −9.94869 −0.989932 −0.494966 0.868912i \(-0.664819\pi\)
−0.494966 + 0.868912i \(0.664819\pi\)
\(102\) 1.86831 0.184991
\(103\) −10.9879 −1.08267 −0.541336 0.840806i \(-0.682081\pi\)
−0.541336 + 0.840806i \(0.682081\pi\)
\(104\) 0 0
\(105\) −7.54288 −0.736109
\(106\) −1.11231 −0.108037
\(107\) −9.87263 −0.954423 −0.477211 0.878789i \(-0.658352\pi\)
−0.477211 + 0.878789i \(0.658352\pi\)
\(108\) −2.19806 −0.211509
\(109\) 20.2446 1.93908 0.969540 0.244934i \(-0.0787662\pi\)
0.969540 + 0.244934i \(0.0787662\pi\)
\(110\) −33.9705 −3.23896
\(111\) −8.80194 −0.835443
\(112\) −8.00969 −0.756844
\(113\) 9.69202 0.911749 0.455874 0.890044i \(-0.349327\pi\)
0.455874 + 0.890044i \(0.349327\pi\)
\(114\) −7.78986 −0.729587
\(115\) 6.80492 0.634562
\(116\) −8.65817 −0.803891
\(117\) 0 0
\(118\) 9.65817 0.889107
\(119\) 2.04892 0.187824
\(120\) 1.36227 0.124358
\(121\) 13.3937 1.21761
\(122\) −7.53989 −0.682630
\(123\) −6.93900 −0.625669
\(124\) −19.4069 −1.74279
\(125\) 4.25906 0.380942
\(126\) −4.60388 −0.410146
\(127\) 13.8116 1.22558 0.612792 0.790244i \(-0.290046\pi\)
0.612792 + 0.790244i \(0.290046\pi\)
\(128\) 3.23059 0.285546
\(129\) 2.28621 0.201289
\(130\) 0 0
\(131\) 2.99462 0.261641 0.130821 0.991406i \(-0.458239\pi\)
0.130821 + 0.991406i \(0.458239\pi\)
\(132\) −10.8562 −0.944914
\(133\) −8.54288 −0.740761
\(134\) 3.11662 0.269235
\(135\) −3.35690 −0.288916
\(136\) −0.370042 −0.0317309
\(137\) 23.0194 1.96668 0.983339 0.181781i \(-0.0581862\pi\)
0.983339 + 0.181781i \(0.0581862\pi\)
\(138\) 4.15346 0.353566
\(139\) 0.982542 0.0833381 0.0416690 0.999131i \(-0.486732\pi\)
0.0416690 + 0.999131i \(0.486732\pi\)
\(140\) 16.5797 1.40124
\(141\) −3.80194 −0.320181
\(142\) −4.87369 −0.408991
\(143\) 0 0
\(144\) −3.56465 −0.297054
\(145\) −13.2228 −1.09810
\(146\) −15.1849 −1.25671
\(147\) 1.95108 0.160923
\(148\) 19.3472 1.59033
\(149\) 10.2591 0.840455 0.420228 0.907419i \(-0.361950\pi\)
0.420228 + 0.907419i \(0.361950\pi\)
\(150\) 12.8442 1.04872
\(151\) −20.1685 −1.64129 −0.820646 0.571438i \(-0.806386\pi\)
−0.820646 + 0.571438i \(0.806386\pi\)
\(152\) 1.54288 0.125144
\(153\) 0.911854 0.0737190
\(154\) −22.7385 −1.83232
\(155\) −29.6383 −2.38061
\(156\) 0 0
\(157\) 10.4383 0.833070 0.416535 0.909120i \(-0.363244\pi\)
0.416535 + 0.909120i \(0.363244\pi\)
\(158\) 7.66487 0.609785
\(159\) −0.542877 −0.0430529
\(160\) 27.2422 2.15368
\(161\) 4.55496 0.358981
\(162\) −2.04892 −0.160978
\(163\) −11.0465 −0.865231 −0.432615 0.901579i \(-0.642409\pi\)
−0.432615 + 0.901579i \(0.642409\pi\)
\(164\) 15.2524 1.19101
\(165\) −16.5797 −1.29073
\(166\) −4.72886 −0.367031
\(167\) 8.10992 0.627564 0.313782 0.949495i \(-0.398404\pi\)
0.313782 + 0.949495i \(0.398404\pi\)
\(168\) 0.911854 0.0703511
\(169\) 0 0
\(170\) −6.27173 −0.481020
\(171\) −3.80194 −0.290741
\(172\) −5.02523 −0.383170
\(173\) −18.0562 −1.37279 −0.686394 0.727230i \(-0.740808\pi\)
−0.686394 + 0.727230i \(0.740808\pi\)
\(174\) −8.07069 −0.611837
\(175\) 14.0858 1.06478
\(176\) −17.6058 −1.32709
\(177\) 4.71379 0.354310
\(178\) 20.6093 1.54473
\(179\) −19.8702 −1.48517 −0.742585 0.669751i \(-0.766401\pi\)
−0.742585 + 0.669751i \(0.766401\pi\)
\(180\) 7.37867 0.549973
\(181\) −10.0828 −0.749446 −0.374723 0.927137i \(-0.622262\pi\)
−0.374723 + 0.927137i \(0.622262\pi\)
\(182\) 0 0
\(183\) −3.67994 −0.272029
\(184\) −0.822643 −0.0606461
\(185\) 29.5472 2.17235
\(186\) −18.0901 −1.32643
\(187\) 4.50365 0.329339
\(188\) 8.35690 0.609489
\(189\) −2.24698 −0.163444
\(190\) 26.1497 1.89710
\(191\) −6.58748 −0.476653 −0.238327 0.971185i \(-0.576599\pi\)
−0.238327 + 0.971185i \(0.576599\pi\)
\(192\) 9.49827 0.685479
\(193\) 10.8672 0.782242 0.391121 0.920339i \(-0.372087\pi\)
0.391121 + 0.920339i \(0.372087\pi\)
\(194\) 33.0476 2.37268
\(195\) 0 0
\(196\) −4.28860 −0.306329
\(197\) 9.24160 0.658437 0.329218 0.944254i \(-0.393215\pi\)
0.329218 + 0.944254i \(0.393215\pi\)
\(198\) −10.1196 −0.719169
\(199\) −1.56465 −0.110915 −0.0554574 0.998461i \(-0.517662\pi\)
−0.0554574 + 0.998461i \(0.517662\pi\)
\(200\) −2.54394 −0.179884
\(201\) 1.52111 0.107291
\(202\) 20.3840 1.43422
\(203\) −8.85086 −0.621208
\(204\) −2.00431 −0.140330
\(205\) 23.2935 1.62689
\(206\) 22.5133 1.56858
\(207\) 2.02715 0.140896
\(208\) 0 0
\(209\) −18.7778 −1.29889
\(210\) 15.4547 1.06648
\(211\) 23.2446 1.60022 0.800112 0.599851i \(-0.204774\pi\)
0.800112 + 0.599851i \(0.204774\pi\)
\(212\) 1.19328 0.0819546
\(213\) −2.37867 −0.162984
\(214\) 20.2282 1.38277
\(215\) −7.67456 −0.523401
\(216\) 0.405813 0.0276121
\(217\) −19.8388 −1.34674
\(218\) −41.4795 −2.80935
\(219\) −7.41119 −0.500802
\(220\) 36.4432 2.45700
\(221\) 0 0
\(222\) 18.0344 1.21039
\(223\) 10.7385 0.719106 0.359553 0.933125i \(-0.382929\pi\)
0.359553 + 0.933125i \(0.382929\pi\)
\(224\) 18.2349 1.21837
\(225\) 6.26875 0.417917
\(226\) −19.8582 −1.32094
\(227\) −6.97584 −0.463003 −0.231501 0.972835i \(-0.574364\pi\)
−0.231501 + 0.972835i \(0.574364\pi\)
\(228\) 8.35690 0.553449
\(229\) −16.8049 −1.11050 −0.555250 0.831683i \(-0.687378\pi\)
−0.555250 + 0.831683i \(0.687378\pi\)
\(230\) −13.9427 −0.919356
\(231\) −11.0978 −0.730184
\(232\) 1.59850 0.104947
\(233\) 8.69202 0.569433 0.284717 0.958612i \(-0.408100\pi\)
0.284717 + 0.958612i \(0.408100\pi\)
\(234\) 0 0
\(235\) 12.7627 0.832547
\(236\) −10.3612 −0.674457
\(237\) 3.74094 0.243000
\(238\) −4.19806 −0.272120
\(239\) −22.9191 −1.48252 −0.741258 0.671220i \(-0.765771\pi\)
−0.741258 + 0.671220i \(0.765771\pi\)
\(240\) 11.9661 0.772412
\(241\) 21.9801 1.41587 0.707933 0.706280i \(-0.249628\pi\)
0.707933 + 0.706280i \(0.249628\pi\)
\(242\) −27.4426 −1.76408
\(243\) −1.00000 −0.0641500
\(244\) 8.08874 0.517828
\(245\) −6.54958 −0.418437
\(246\) 14.2174 0.906471
\(247\) 0 0
\(248\) 3.58296 0.227518
\(249\) −2.30798 −0.146262
\(250\) −8.72646 −0.551910
\(251\) 26.6437 1.68174 0.840868 0.541241i \(-0.182045\pi\)
0.840868 + 0.541241i \(0.182045\pi\)
\(252\) 4.93900 0.311128
\(253\) 10.0121 0.629454
\(254\) −28.2989 −1.77563
\(255\) −3.06100 −0.191687
\(256\) 12.3773 0.773584
\(257\) 23.1444 1.44371 0.721853 0.692047i \(-0.243291\pi\)
0.721853 + 0.692047i \(0.243291\pi\)
\(258\) −4.68425 −0.291629
\(259\) 19.7778 1.22893
\(260\) 0 0
\(261\) −3.93900 −0.243818
\(262\) −6.13574 −0.379067
\(263\) −18.5284 −1.14251 −0.571255 0.820773i \(-0.693543\pi\)
−0.571255 + 0.820773i \(0.693543\pi\)
\(264\) 2.00431 0.123357
\(265\) 1.82238 0.111948
\(266\) 17.5036 1.07322
\(267\) 10.0586 0.615577
\(268\) −3.34349 −0.204236
\(269\) 9.75840 0.594980 0.297490 0.954725i \(-0.403851\pi\)
0.297490 + 0.954725i \(0.403851\pi\)
\(270\) 6.87800 0.418582
\(271\) −22.8019 −1.38512 −0.692560 0.721361i \(-0.743517\pi\)
−0.692560 + 0.721361i \(0.743517\pi\)
\(272\) −3.25044 −0.197087
\(273\) 0 0
\(274\) −47.1648 −2.84933
\(275\) 30.9614 1.86704
\(276\) −4.45580 −0.268207
\(277\) −23.9705 −1.44025 −0.720123 0.693847i \(-0.755914\pi\)
−0.720123 + 0.693847i \(0.755914\pi\)
\(278\) −2.01315 −0.120741
\(279\) −8.82908 −0.528583
\(280\) −3.06100 −0.182930
\(281\) −4.12498 −0.246076 −0.123038 0.992402i \(-0.539264\pi\)
−0.123038 + 0.992402i \(0.539264\pi\)
\(282\) 7.78986 0.463879
\(283\) −15.6558 −0.930639 −0.465320 0.885143i \(-0.654061\pi\)
−0.465320 + 0.885143i \(0.654061\pi\)
\(284\) 5.22846 0.310252
\(285\) 12.7627 0.755998
\(286\) 0 0
\(287\) 15.5918 0.920354
\(288\) 8.11529 0.478198
\(289\) −16.1685 −0.951090
\(290\) 27.0925 1.59092
\(291\) 16.1293 0.945516
\(292\) 16.2903 0.953315
\(293\) 22.5948 1.32000 0.660001 0.751265i \(-0.270556\pi\)
0.660001 + 0.751265i \(0.270556\pi\)
\(294\) −3.99761 −0.233145
\(295\) −15.8237 −0.921292
\(296\) −3.57194 −0.207615
\(297\) −4.93900 −0.286590
\(298\) −21.0200 −1.21765
\(299\) 0 0
\(300\) −13.7791 −0.795537
\(301\) −5.13706 −0.296095
\(302\) 41.3236 2.37791
\(303\) 9.94869 0.571537
\(304\) 13.5526 0.777293
\(305\) 12.3532 0.707341
\(306\) −1.86831 −0.106804
\(307\) −6.55496 −0.374111 −0.187056 0.982349i \(-0.559894\pi\)
−0.187056 + 0.982349i \(0.559894\pi\)
\(308\) 24.3937 1.38996
\(309\) 10.9879 0.625081
\(310\) 60.7265 3.44903
\(311\) −12.0392 −0.682682 −0.341341 0.939940i \(-0.610881\pi\)
−0.341341 + 0.939940i \(0.610881\pi\)
\(312\) 0 0
\(313\) −33.8950 −1.91586 −0.957929 0.287005i \(-0.907340\pi\)
−0.957929 + 0.287005i \(0.907340\pi\)
\(314\) −21.3873 −1.20695
\(315\) 7.54288 0.424993
\(316\) −8.22282 −0.462570
\(317\) −4.49827 −0.252648 −0.126324 0.991989i \(-0.540318\pi\)
−0.126324 + 0.991989i \(0.540318\pi\)
\(318\) 1.11231 0.0623752
\(319\) −19.4547 −1.08926
\(320\) −31.8847 −1.78241
\(321\) 9.87263 0.551036
\(322\) −9.33273 −0.520093
\(323\) −3.46681 −0.192899
\(324\) 2.19806 0.122115
\(325\) 0 0
\(326\) 22.6334 1.25355
\(327\) −20.2446 −1.11953
\(328\) −2.81594 −0.155484
\(329\) 8.54288 0.470984
\(330\) 33.9705 1.87001
\(331\) −11.2131 −0.616329 −0.308165 0.951333i \(-0.599715\pi\)
−0.308165 + 0.951333i \(0.599715\pi\)
\(332\) 5.07308 0.278421
\(333\) 8.80194 0.482343
\(334\) −16.6165 −0.909217
\(335\) −5.10620 −0.278981
\(336\) 8.00969 0.436964
\(337\) 7.04892 0.383979 0.191989 0.981397i \(-0.438506\pi\)
0.191989 + 0.981397i \(0.438506\pi\)
\(338\) 0 0
\(339\) −9.69202 −0.526398
\(340\) 6.72827 0.364891
\(341\) −43.6069 −2.36144
\(342\) 7.78986 0.421227
\(343\) −20.1129 −1.08599
\(344\) 0.927774 0.0500222
\(345\) −6.80492 −0.366365
\(346\) 36.9957 1.98890
\(347\) −2.70410 −0.145164 −0.0725819 0.997362i \(-0.523124\pi\)
−0.0725819 + 0.997362i \(0.523124\pi\)
\(348\) 8.65817 0.464127
\(349\) −0.415502 −0.0222413 −0.0111207 0.999938i \(-0.503540\pi\)
−0.0111207 + 0.999938i \(0.503540\pi\)
\(350\) −28.8605 −1.54266
\(351\) 0 0
\(352\) 40.0814 2.13635
\(353\) −32.1672 −1.71209 −0.856044 0.516904i \(-0.827084\pi\)
−0.856044 + 0.516904i \(0.827084\pi\)
\(354\) −9.65817 −0.513326
\(355\) 7.98493 0.423796
\(356\) −22.1094 −1.17180
\(357\) −2.04892 −0.108440
\(358\) 40.7125 2.15172
\(359\) 22.3521 1.17970 0.589850 0.807513i \(-0.299187\pi\)
0.589850 + 0.807513i \(0.299187\pi\)
\(360\) −1.36227 −0.0717981
\(361\) −4.54527 −0.239225
\(362\) 20.6588 1.08580
\(363\) −13.3937 −0.702989
\(364\) 0 0
\(365\) 24.8786 1.30221
\(366\) 7.53989 0.394117
\(367\) −2.30260 −0.120195 −0.0600974 0.998193i \(-0.519141\pi\)
−0.0600974 + 0.998193i \(0.519141\pi\)
\(368\) −7.22606 −0.376685
\(369\) 6.93900 0.361230
\(370\) −60.5397 −3.14731
\(371\) 1.21983 0.0633305
\(372\) 19.4069 1.00620
\(373\) −19.2760 −0.998076 −0.499038 0.866580i \(-0.666313\pi\)
−0.499038 + 0.866580i \(0.666313\pi\)
\(374\) −9.22760 −0.477148
\(375\) −4.25906 −0.219937
\(376\) −1.54288 −0.0795678
\(377\) 0 0
\(378\) 4.60388 0.236798
\(379\) −7.33944 −0.377002 −0.188501 0.982073i \(-0.560363\pi\)
−0.188501 + 0.982073i \(0.560363\pi\)
\(380\) −28.0532 −1.43910
\(381\) −13.8116 −0.707591
\(382\) 13.4972 0.690577
\(383\) 19.0901 0.975457 0.487728 0.872995i \(-0.337826\pi\)
0.487728 + 0.872995i \(0.337826\pi\)
\(384\) −3.23059 −0.164860
\(385\) 37.2543 1.89865
\(386\) −22.2661 −1.13331
\(387\) −2.28621 −0.116214
\(388\) −35.4532 −1.79986
\(389\) −23.9879 −1.21624 −0.608118 0.793847i \(-0.708075\pi\)
−0.608118 + 0.793847i \(0.708075\pi\)
\(390\) 0 0
\(391\) 1.84846 0.0934807
\(392\) 0.791775 0.0399907
\(393\) −2.99462 −0.151059
\(394\) −18.9353 −0.953946
\(395\) −12.5579 −0.631859
\(396\) 10.8562 0.545546
\(397\) −4.41789 −0.221728 −0.110864 0.993836i \(-0.535362\pi\)
−0.110864 + 0.993836i \(0.535362\pi\)
\(398\) 3.20583 0.160694
\(399\) 8.54288 0.427679
\(400\) −22.3459 −1.11729
\(401\) −20.4088 −1.01917 −0.509583 0.860421i \(-0.670200\pi\)
−0.509583 + 0.860421i \(0.670200\pi\)
\(402\) −3.11662 −0.155443
\(403\) 0 0
\(404\) −21.8678 −1.08797
\(405\) 3.35690 0.166805
\(406\) 18.1347 0.900009
\(407\) 43.4728 2.15487
\(408\) 0.370042 0.0183198
\(409\) −22.1491 −1.09520 −0.547602 0.836739i \(-0.684459\pi\)
−0.547602 + 0.836739i \(0.684459\pi\)
\(410\) −47.7265 −2.35704
\(411\) −23.0194 −1.13546
\(412\) −24.1521 −1.18989
\(413\) −10.5918 −0.521188
\(414\) −4.15346 −0.204131
\(415\) 7.74764 0.380317
\(416\) 0 0
\(417\) −0.982542 −0.0481153
\(418\) 38.4741 1.88183
\(419\) 14.7560 0.720878 0.360439 0.932783i \(-0.382627\pi\)
0.360439 + 0.932783i \(0.382627\pi\)
\(420\) −16.5797 −0.809007
\(421\) 8.47219 0.412909 0.206455 0.978456i \(-0.433807\pi\)
0.206455 + 0.978456i \(0.433807\pi\)
\(422\) −47.6262 −2.31841
\(423\) 3.80194 0.184857
\(424\) −0.220306 −0.0106990
\(425\) 5.71618 0.277276
\(426\) 4.87369 0.236131
\(427\) 8.26875 0.400153
\(428\) −21.7006 −1.04894
\(429\) 0 0
\(430\) 15.7245 0.758305
\(431\) 2.39181 0.115210 0.0576048 0.998339i \(-0.481654\pi\)
0.0576048 + 0.998339i \(0.481654\pi\)
\(432\) 3.56465 0.171504
\(433\) −10.4286 −0.501169 −0.250584 0.968095i \(-0.580623\pi\)
−0.250584 + 0.968095i \(0.580623\pi\)
\(434\) 40.6480 1.95117
\(435\) 13.2228 0.633986
\(436\) 44.4989 2.13111
\(437\) −7.70709 −0.368680
\(438\) 15.1849 0.725563
\(439\) −32.5502 −1.55353 −0.776767 0.629787i \(-0.783142\pi\)
−0.776767 + 0.629787i \(0.783142\pi\)
\(440\) −6.72827 −0.320758
\(441\) −1.95108 −0.0929087
\(442\) 0 0
\(443\) 9.58211 0.455260 0.227630 0.973748i \(-0.426902\pi\)
0.227630 + 0.973748i \(0.426902\pi\)
\(444\) −19.3472 −0.918178
\(445\) −33.7657 −1.60065
\(446\) −22.0024 −1.04184
\(447\) −10.2591 −0.485237
\(448\) −21.3424 −1.00833
\(449\) −28.3937 −1.33998 −0.669992 0.742369i \(-0.733702\pi\)
−0.669992 + 0.742369i \(0.733702\pi\)
\(450\) −12.8442 −0.605479
\(451\) 34.2717 1.61379
\(452\) 21.3037 1.00204
\(453\) 20.1685 0.947600
\(454\) 14.2929 0.670800
\(455\) 0 0
\(456\) −1.54288 −0.0722518
\(457\) 4.99569 0.233688 0.116844 0.993150i \(-0.462722\pi\)
0.116844 + 0.993150i \(0.462722\pi\)
\(458\) 34.4319 1.60890
\(459\) −0.911854 −0.0425617
\(460\) 14.9576 0.697404
\(461\) 1.35258 0.0629961 0.0314981 0.999504i \(-0.489972\pi\)
0.0314981 + 0.999504i \(0.489972\pi\)
\(462\) 22.7385 1.05789
\(463\) −3.36467 −0.156369 −0.0781846 0.996939i \(-0.524912\pi\)
−0.0781846 + 0.996939i \(0.524912\pi\)
\(464\) 14.0411 0.651844
\(465\) 29.6383 1.37444
\(466\) −17.8092 −0.824997
\(467\) 6.91079 0.319793 0.159897 0.987134i \(-0.448884\pi\)
0.159897 + 0.987134i \(0.448884\pi\)
\(468\) 0 0
\(469\) −3.41789 −0.157824
\(470\) −26.1497 −1.20620
\(471\) −10.4383 −0.480973
\(472\) 1.91292 0.0880492
\(473\) −11.2916 −0.519188
\(474\) −7.66487 −0.352059
\(475\) −23.8334 −1.09355
\(476\) 4.50365 0.206424
\(477\) 0.542877 0.0248566
\(478\) 46.9594 2.14787
\(479\) 3.51573 0.160638 0.0803189 0.996769i \(-0.474406\pi\)
0.0803189 + 0.996769i \(0.474406\pi\)
\(480\) −27.2422 −1.24343
\(481\) 0 0
\(482\) −45.0355 −2.05131
\(483\) −4.55496 −0.207258
\(484\) 29.4403 1.33819
\(485\) −54.1444 −2.45857
\(486\) 2.04892 0.0929408
\(487\) −12.8479 −0.582193 −0.291096 0.956694i \(-0.594020\pi\)
−0.291096 + 0.956694i \(0.594020\pi\)
\(488\) −1.49337 −0.0676016
\(489\) 11.0465 0.499541
\(490\) 13.4196 0.606234
\(491\) 28.6708 1.29390 0.646948 0.762534i \(-0.276045\pi\)
0.646948 + 0.762534i \(0.276045\pi\)
\(492\) −15.2524 −0.687629
\(493\) −3.59179 −0.161766
\(494\) 0 0
\(495\) 16.5797 0.745203
\(496\) 31.4726 1.41316
\(497\) 5.34481 0.239748
\(498\) 4.72886 0.211905
\(499\) −33.5555 −1.50215 −0.751076 0.660215i \(-0.770465\pi\)
−0.751076 + 0.660215i \(0.770465\pi\)
\(500\) 9.36168 0.418667
\(501\) −8.10992 −0.362324
\(502\) −54.5907 −2.43650
\(503\) 21.5633 0.961461 0.480730 0.876868i \(-0.340372\pi\)
0.480730 + 0.876868i \(0.340372\pi\)
\(504\) −0.911854 −0.0406172
\(505\) −33.3967 −1.48613
\(506\) −20.5139 −0.911955
\(507\) 0 0
\(508\) 30.3588 1.34695
\(509\) −8.89008 −0.394046 −0.197023 0.980399i \(-0.563127\pi\)
−0.197023 + 0.980399i \(0.563127\pi\)
\(510\) 6.27173 0.277717
\(511\) 16.6528 0.736676
\(512\) −31.8213 −1.40632
\(513\) 3.80194 0.167860
\(514\) −47.4209 −2.09165
\(515\) −36.8853 −1.62536
\(516\) 5.02523 0.221223
\(517\) 18.7778 0.825846
\(518\) −40.5230 −1.78048
\(519\) 18.0562 0.792580
\(520\) 0 0
\(521\) 19.3478 0.847642 0.423821 0.905746i \(-0.360688\pi\)
0.423821 + 0.905746i \(0.360688\pi\)
\(522\) 8.07069 0.353244
\(523\) −12.5948 −0.550731 −0.275366 0.961340i \(-0.588799\pi\)
−0.275366 + 0.961340i \(0.588799\pi\)
\(524\) 6.58237 0.287552
\(525\) −14.0858 −0.614753
\(526\) 37.9632 1.65527
\(527\) −8.05084 −0.350700
\(528\) 17.6058 0.766194
\(529\) −18.8907 −0.821334
\(530\) −3.73391 −0.162191
\(531\) −4.71379 −0.204561
\(532\) −18.7778 −0.814120
\(533\) 0 0
\(534\) −20.6093 −0.891850
\(535\) −33.1414 −1.43283
\(536\) 0.617285 0.0266627
\(537\) 19.8702 0.857464
\(538\) −19.9941 −0.862009
\(539\) −9.63640 −0.415069
\(540\) −7.37867 −0.317527
\(541\) 29.0019 1.24689 0.623445 0.781867i \(-0.285733\pi\)
0.623445 + 0.781867i \(0.285733\pi\)
\(542\) 46.7193 2.00677
\(543\) 10.0828 0.432693
\(544\) 7.39996 0.317271
\(545\) 67.9590 2.91104
\(546\) 0 0
\(547\) 27.7006 1.18439 0.592197 0.805793i \(-0.298261\pi\)
0.592197 + 0.805793i \(0.298261\pi\)
\(548\) 50.5980 2.16144
\(549\) 3.67994 0.157056
\(550\) −63.4373 −2.70497
\(551\) 14.9758 0.637992
\(552\) 0.822643 0.0350140
\(553\) −8.40581 −0.357452
\(554\) 49.1135 2.08663
\(555\) −29.5472 −1.25421
\(556\) 2.15969 0.0915912
\(557\) 6.80971 0.288537 0.144268 0.989539i \(-0.453917\pi\)
0.144268 + 0.989539i \(0.453917\pi\)
\(558\) 18.0901 0.765814
\(559\) 0 0
\(560\) −26.8877 −1.13621
\(561\) −4.50365 −0.190144
\(562\) 8.45175 0.356515
\(563\) −19.2524 −0.811390 −0.405695 0.914008i \(-0.632970\pi\)
−0.405695 + 0.914008i \(0.632970\pi\)
\(564\) −8.35690 −0.351889
\(565\) 32.5351 1.36876
\(566\) 32.0774 1.34831
\(567\) 2.24698 0.0943643
\(568\) −0.965294 −0.0405028
\(569\) 7.84846 0.329025 0.164512 0.986375i \(-0.447395\pi\)
0.164512 + 0.986375i \(0.447395\pi\)
\(570\) −26.1497 −1.09529
\(571\) 29.8568 1.24947 0.624735 0.780837i \(-0.285207\pi\)
0.624735 + 0.780837i \(0.285207\pi\)
\(572\) 0 0
\(573\) 6.58748 0.275196
\(574\) −31.9463 −1.33341
\(575\) 12.7077 0.529947
\(576\) −9.49827 −0.395761
\(577\) 8.97823 0.373769 0.186884 0.982382i \(-0.440161\pi\)
0.186884 + 0.982382i \(0.440161\pi\)
\(578\) 33.1280 1.37794
\(579\) −10.8672 −0.451627
\(580\) −29.0646 −1.20684
\(581\) 5.18598 0.215151
\(582\) −33.0476 −1.36987
\(583\) 2.68127 0.111047
\(584\) −3.00756 −0.124454
\(585\) 0 0
\(586\) −46.2948 −1.91242
\(587\) 23.8538 0.984553 0.492277 0.870439i \(-0.336165\pi\)
0.492277 + 0.870439i \(0.336165\pi\)
\(588\) 4.28860 0.176859
\(589\) 33.5676 1.38313
\(590\) 32.4215 1.33477
\(591\) −9.24160 −0.380149
\(592\) −31.3758 −1.28954
\(593\) 11.9866 0.492230 0.246115 0.969241i \(-0.420846\pi\)
0.246115 + 0.969241i \(0.420846\pi\)
\(594\) 10.1196 0.415212
\(595\) 6.87800 0.281971
\(596\) 22.5501 0.923686
\(597\) 1.56465 0.0640367
\(598\) 0 0
\(599\) 29.1142 1.18958 0.594788 0.803883i \(-0.297236\pi\)
0.594788 + 0.803883i \(0.297236\pi\)
\(600\) 2.54394 0.103856
\(601\) 37.1366 1.51483 0.757417 0.652932i \(-0.226461\pi\)
0.757417 + 0.652932i \(0.226461\pi\)
\(602\) 10.5254 0.428984
\(603\) −1.52111 −0.0619442
\(604\) −44.3317 −1.80383
\(605\) 44.9614 1.82794
\(606\) −20.3840 −0.828045
\(607\) −14.2325 −0.577680 −0.288840 0.957377i \(-0.593269\pi\)
−0.288840 + 0.957377i \(0.593269\pi\)
\(608\) −30.8538 −1.25129
\(609\) 8.85086 0.358655
\(610\) −25.3106 −1.02480
\(611\) 0 0
\(612\) 2.00431 0.0810195
\(613\) 30.3139 1.22437 0.612184 0.790715i \(-0.290291\pi\)
0.612184 + 0.790715i \(0.290291\pi\)
\(614\) 13.4306 0.542014
\(615\) −23.2935 −0.939285
\(616\) −4.50365 −0.181457
\(617\) −24.9638 −1.00500 −0.502501 0.864576i \(-0.667587\pi\)
−0.502501 + 0.864576i \(0.667587\pi\)
\(618\) −22.5133 −0.905619
\(619\) 35.6122 1.43138 0.715688 0.698420i \(-0.246113\pi\)
0.715688 + 0.698420i \(0.246113\pi\)
\(620\) −65.1469 −2.61636
\(621\) −2.02715 −0.0813466
\(622\) 24.6674 0.989072
\(623\) −22.6015 −0.905509
\(624\) 0 0
\(625\) −17.0465 −0.681861
\(626\) 69.4480 2.77570
\(627\) 18.7778 0.749912
\(628\) 22.9441 0.915570
\(629\) 8.02608 0.320021
\(630\) −15.4547 −0.615731
\(631\) 23.8829 0.950763 0.475382 0.879780i \(-0.342310\pi\)
0.475382 + 0.879780i \(0.342310\pi\)
\(632\) 1.51812 0.0603877
\(633\) −23.2446 −0.923889
\(634\) 9.21659 0.366037
\(635\) 46.3642 1.83991
\(636\) −1.19328 −0.0473165
\(637\) 0 0
\(638\) 39.8611 1.57812
\(639\) 2.37867 0.0940986
\(640\) 10.8447 0.428676
\(641\) −24.6577 −0.973920 −0.486960 0.873424i \(-0.661894\pi\)
−0.486960 + 0.873424i \(0.661894\pi\)
\(642\) −20.2282 −0.798343
\(643\) 47.8165 1.88570 0.942850 0.333218i \(-0.108134\pi\)
0.942850 + 0.333218i \(0.108134\pi\)
\(644\) 10.0121 0.394531
\(645\) 7.67456 0.302186
\(646\) 7.10321 0.279472
\(647\) −5.38942 −0.211880 −0.105940 0.994373i \(-0.533785\pi\)
−0.105940 + 0.994373i \(0.533785\pi\)
\(648\) −0.405813 −0.0159418
\(649\) −23.2814 −0.913876
\(650\) 0 0
\(651\) 19.8388 0.777543
\(652\) −24.2809 −0.950915
\(653\) −5.03790 −0.197148 −0.0985741 0.995130i \(-0.531428\pi\)
−0.0985741 + 0.995130i \(0.531428\pi\)
\(654\) 41.4795 1.62198
\(655\) 10.0526 0.392789
\(656\) −24.7351 −0.965743
\(657\) 7.41119 0.289138
\(658\) −17.5036 −0.682363
\(659\) −43.9812 −1.71326 −0.856632 0.515927i \(-0.827447\pi\)
−0.856632 + 0.515927i \(0.827447\pi\)
\(660\) −36.4432 −1.41855
\(661\) −38.2194 −1.48656 −0.743280 0.668980i \(-0.766731\pi\)
−0.743280 + 0.668980i \(0.766731\pi\)
\(662\) 22.9748 0.892940
\(663\) 0 0
\(664\) −0.936608 −0.0363474
\(665\) −28.6775 −1.11207
\(666\) −18.0344 −0.698820
\(667\) −7.98493 −0.309178
\(668\) 17.8261 0.689713
\(669\) −10.7385 −0.415176
\(670\) 10.4622 0.404189
\(671\) 18.1752 0.701647
\(672\) −18.2349 −0.703426
\(673\) −6.66487 −0.256912 −0.128456 0.991715i \(-0.541002\pi\)
−0.128456 + 0.991715i \(0.541002\pi\)
\(674\) −14.4426 −0.556310
\(675\) −6.26875 −0.241284
\(676\) 0 0
\(677\) 4.80194 0.184553 0.0922767 0.995733i \(-0.470586\pi\)
0.0922767 + 0.995733i \(0.470586\pi\)
\(678\) 19.8582 0.762648
\(679\) −36.2422 −1.39085
\(680\) −1.24219 −0.0476360
\(681\) 6.97584 0.267315
\(682\) 89.3469 3.42127
\(683\) 11.2591 0.430816 0.215408 0.976524i \(-0.430892\pi\)
0.215408 + 0.976524i \(0.430892\pi\)
\(684\) −8.35690 −0.319534
\(685\) 77.2737 2.95247
\(686\) 41.2097 1.57339
\(687\) 16.8049 0.641148
\(688\) 8.14952 0.310698
\(689\) 0 0
\(690\) 13.9427 0.530790
\(691\) −24.7144 −0.940179 −0.470090 0.882619i \(-0.655778\pi\)
−0.470090 + 0.882619i \(0.655778\pi\)
\(692\) −39.6887 −1.50874
\(693\) 11.0978 0.421572
\(694\) 5.54048 0.210314
\(695\) 3.29829 0.125111
\(696\) −1.59850 −0.0605909
\(697\) 6.32736 0.239666
\(698\) 0.851329 0.0322233
\(699\) −8.69202 −0.328762
\(700\) 30.9614 1.17023
\(701\) 25.8920 0.977927 0.488964 0.872304i \(-0.337375\pi\)
0.488964 + 0.872304i \(0.337375\pi\)
\(702\) 0 0
\(703\) −33.4644 −1.26213
\(704\) −46.9120 −1.76806
\(705\) −12.7627 −0.480671
\(706\) 65.9079 2.48048
\(707\) −22.3545 −0.840728
\(708\) 10.3612 0.389398
\(709\) 0.0851621 0.00319833 0.00159916 0.999999i \(-0.499491\pi\)
0.00159916 + 0.999999i \(0.499491\pi\)
\(710\) −16.3605 −0.613998
\(711\) −3.74094 −0.140296
\(712\) 4.08192 0.152976
\(713\) −17.8979 −0.670280
\(714\) 4.19806 0.157109
\(715\) 0 0
\(716\) −43.6760 −1.63225
\(717\) 22.9191 0.855931
\(718\) −45.7976 −1.70915
\(719\) 27.0508 1.00883 0.504413 0.863463i \(-0.331709\pi\)
0.504413 + 0.863463i \(0.331709\pi\)
\(720\) −11.9661 −0.445952
\(721\) −24.6896 −0.919490
\(722\) 9.31288 0.346590
\(723\) −21.9801 −0.817451
\(724\) −22.1626 −0.823665
\(725\) −24.6926 −0.917061
\(726\) 27.4426 1.01849
\(727\) −47.1584 −1.74901 −0.874503 0.485019i \(-0.838813\pi\)
−0.874503 + 0.485019i \(0.838813\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −50.9742 −1.88664
\(731\) −2.08469 −0.0771050
\(732\) −8.08874 −0.298968
\(733\) −9.68186 −0.357608 −0.178804 0.983885i \(-0.557223\pi\)
−0.178804 + 0.983885i \(0.557223\pi\)
\(734\) 4.71784 0.174139
\(735\) 6.54958 0.241585
\(736\) 16.4509 0.606388
\(737\) −7.51275 −0.276736
\(738\) −14.2174 −0.523351
\(739\) 36.3139 1.33583 0.667915 0.744238i \(-0.267187\pi\)
0.667915 + 0.744238i \(0.267187\pi\)
\(740\) 64.9466 2.38748
\(741\) 0 0
\(742\) −2.49934 −0.0917535
\(743\) −41.7536 −1.53179 −0.765896 0.642965i \(-0.777704\pi\)
−0.765896 + 0.642965i \(0.777704\pi\)
\(744\) −3.58296 −0.131358
\(745\) 34.4386 1.26173
\(746\) 39.4950 1.44602
\(747\) 2.30798 0.0844445
\(748\) 9.89930 0.361954
\(749\) −22.1836 −0.810571
\(750\) 8.72646 0.318645
\(751\) −22.1927 −0.809823 −0.404911 0.914356i \(-0.632698\pi\)
−0.404911 + 0.914356i \(0.632698\pi\)
\(752\) −13.5526 −0.494211
\(753\) −26.6437 −0.970950
\(754\) 0 0
\(755\) −67.7036 −2.46399
\(756\) −4.93900 −0.179630
\(757\) 9.23729 0.335735 0.167868 0.985810i \(-0.446312\pi\)
0.167868 + 0.985810i \(0.446312\pi\)
\(758\) 15.0379 0.546201
\(759\) −10.0121 −0.363416
\(760\) 5.17928 0.187872
\(761\) −7.50173 −0.271937 −0.135969 0.990713i \(-0.543415\pi\)
−0.135969 + 0.990713i \(0.543415\pi\)
\(762\) 28.2989 1.02516
\(763\) 45.4892 1.64682
\(764\) −14.4797 −0.523857
\(765\) 3.06100 0.110671
\(766\) −39.1140 −1.41325
\(767\) 0 0
\(768\) −12.3773 −0.446629
\(769\) 5.14005 0.185355 0.0926774 0.995696i \(-0.470457\pi\)
0.0926774 + 0.995696i \(0.470457\pi\)
\(770\) −76.3309 −2.75078
\(771\) −23.1444 −0.833524
\(772\) 23.8869 0.859708
\(773\) −8.50173 −0.305786 −0.152893 0.988243i \(-0.548859\pi\)
−0.152893 + 0.988243i \(0.548859\pi\)
\(774\) 4.68425 0.168372
\(775\) −55.3473 −1.98813
\(776\) 6.54548 0.234969
\(777\) −19.7778 −0.709524
\(778\) 49.1493 1.76209
\(779\) −26.3817 −0.945221
\(780\) 0 0
\(781\) 11.7482 0.420385
\(782\) −3.78735 −0.135435
\(783\) 3.93900 0.140768
\(784\) 6.95492 0.248390
\(785\) 35.0404 1.25065
\(786\) 6.13574 0.218854
\(787\) −7.78554 −0.277525 −0.138762 0.990326i \(-0.544312\pi\)
−0.138762 + 0.990326i \(0.544312\pi\)
\(788\) 20.3136 0.723643
\(789\) 18.5284 0.659629
\(790\) 25.7302 0.915439
\(791\) 21.7778 0.774329
\(792\) −2.00431 −0.0712201
\(793\) 0 0
\(794\) 9.05190 0.321240
\(795\) −1.82238 −0.0646332
\(796\) −3.43919 −0.121899
\(797\) 21.3840 0.757462 0.378731 0.925507i \(-0.376361\pi\)
0.378731 + 0.925507i \(0.376361\pi\)
\(798\) −17.5036 −0.619622
\(799\) 3.46681 0.122647
\(800\) 50.8727 1.79862
\(801\) −10.0586 −0.355403
\(802\) 41.8159 1.47657
\(803\) 36.6039 1.29172
\(804\) 3.34349 0.117916
\(805\) 15.2905 0.538920
\(806\) 0 0
\(807\) −9.75840 −0.343512
\(808\) 4.03731 0.142032
\(809\) 2.28621 0.0803788 0.0401894 0.999192i \(-0.487204\pi\)
0.0401894 + 0.999192i \(0.487204\pi\)
\(810\) −6.87800 −0.241668
\(811\) 17.8079 0.625320 0.312660 0.949865i \(-0.398780\pi\)
0.312660 + 0.949865i \(0.398780\pi\)
\(812\) −19.4547 −0.682727
\(813\) 22.8019 0.799699
\(814\) −89.0721 −3.12198
\(815\) −37.0820 −1.29893
\(816\) 3.25044 0.113788
\(817\) 8.69202 0.304095
\(818\) 45.3818 1.58674
\(819\) 0 0
\(820\) 51.2006 1.78800
\(821\) 13.9054 0.485302 0.242651 0.970114i \(-0.421983\pi\)
0.242651 + 0.970114i \(0.421983\pi\)
\(822\) 47.1648 1.64506
\(823\) −7.24831 −0.252660 −0.126330 0.991988i \(-0.540320\pi\)
−0.126330 + 0.991988i \(0.540320\pi\)
\(824\) 4.45904 0.155338
\(825\) −30.9614 −1.07794
\(826\) 21.7017 0.755099
\(827\) 54.1191 1.88191 0.940953 0.338536i \(-0.109932\pi\)
0.940953 + 0.338536i \(0.109932\pi\)
\(828\) 4.45580 0.154850
\(829\) −5.79178 −0.201157 −0.100578 0.994929i \(-0.532069\pi\)
−0.100578 + 0.994929i \(0.532069\pi\)
\(830\) −15.8743 −0.551004
\(831\) 23.9705 0.831526
\(832\) 0 0
\(833\) −1.77910 −0.0616422
\(834\) 2.01315 0.0697096
\(835\) 27.2241 0.942130
\(836\) −41.2747 −1.42752
\(837\) 8.82908 0.305178
\(838\) −30.2338 −1.04441
\(839\) −5.29350 −0.182752 −0.0913760 0.995816i \(-0.529127\pi\)
−0.0913760 + 0.995816i \(0.529127\pi\)
\(840\) 3.06100 0.105614
\(841\) −13.4843 −0.464975
\(842\) −17.3588 −0.598224
\(843\) 4.12498 0.142072
\(844\) 51.0930 1.75870
\(845\) 0 0
\(846\) −7.78986 −0.267821
\(847\) 30.0954 1.03409
\(848\) −1.93516 −0.0664538
\(849\) 15.6558 0.537305
\(850\) −11.7120 −0.401718
\(851\) 17.8428 0.611644
\(852\) −5.22846 −0.179124
\(853\) −13.5961 −0.465522 −0.232761 0.972534i \(-0.574776\pi\)
−0.232761 + 0.972534i \(0.574776\pi\)
\(854\) −16.9420 −0.579743
\(855\) −12.7627 −0.436475
\(856\) 4.00644 0.136937
\(857\) −23.8323 −0.814097 −0.407048 0.913407i \(-0.633442\pi\)
−0.407048 + 0.913407i \(0.633442\pi\)
\(858\) 0 0
\(859\) −26.9861 −0.920754 −0.460377 0.887723i \(-0.652286\pi\)
−0.460377 + 0.887723i \(0.652286\pi\)
\(860\) −16.8692 −0.575234
\(861\) −15.5918 −0.531367
\(862\) −4.90063 −0.166916
\(863\) −27.0291 −0.920080 −0.460040 0.887898i \(-0.652165\pi\)
−0.460040 + 0.887898i \(0.652165\pi\)
\(864\) −8.11529 −0.276088
\(865\) −60.6128 −2.06090
\(866\) 21.3674 0.726095
\(867\) 16.1685 0.549112
\(868\) −43.6069 −1.48011
\(869\) −18.4765 −0.626772
\(870\) −27.0925 −0.918520
\(871\) 0 0
\(872\) −8.21552 −0.278213
\(873\) −16.1293 −0.545894
\(874\) 15.7912 0.534145
\(875\) 9.57002 0.323526
\(876\) −16.2903 −0.550397
\(877\) 40.8780 1.38035 0.690176 0.723642i \(-0.257533\pi\)
0.690176 + 0.723642i \(0.257533\pi\)
\(878\) 66.6926 2.25077
\(879\) −22.5948 −0.762103
\(880\) −59.1008 −1.99229
\(881\) −22.4964 −0.757921 −0.378961 0.925413i \(-0.623718\pi\)
−0.378961 + 0.925413i \(0.623718\pi\)
\(882\) 3.99761 0.134606
\(883\) −4.16315 −0.140101 −0.0700505 0.997543i \(-0.522316\pi\)
−0.0700505 + 0.997543i \(0.522316\pi\)
\(884\) 0 0
\(885\) 15.8237 0.531908
\(886\) −19.6329 −0.659582
\(887\) −26.3002 −0.883075 −0.441537 0.897243i \(-0.645567\pi\)
−0.441537 + 0.897243i \(0.645567\pi\)
\(888\) 3.57194 0.119867
\(889\) 31.0344 1.04086
\(890\) 69.1831 2.31902
\(891\) 4.93900 0.165463
\(892\) 23.6040 0.790320
\(893\) −14.4547 −0.483709
\(894\) 21.0200 0.703013
\(895\) −66.7023 −2.22961
\(896\) 7.25906 0.242508
\(897\) 0 0
\(898\) 58.1764 1.94137
\(899\) 34.7778 1.15990
\(900\) 13.7791 0.459303
\(901\) 0.495024 0.0164916
\(902\) −70.2199 −2.33807
\(903\) 5.13706 0.170951
\(904\) −3.93315 −0.130815
\(905\) −33.8468 −1.12511
\(906\) −41.3236 −1.37289
\(907\) −57.9114 −1.92292 −0.961458 0.274952i \(-0.911338\pi\)
−0.961458 + 0.274952i \(0.911338\pi\)
\(908\) −15.3333 −0.508854
\(909\) −9.94869 −0.329977
\(910\) 0 0
\(911\) −0.286799 −0.00950208 −0.00475104 0.999989i \(-0.501512\pi\)
−0.00475104 + 0.999989i \(0.501512\pi\)
\(912\) −13.5526 −0.448770
\(913\) 11.3991 0.377255
\(914\) −10.2358 −0.338569
\(915\) −12.3532 −0.408383
\(916\) −36.9383 −1.22047
\(917\) 6.72886 0.222206
\(918\) 1.86831 0.0616635
\(919\) −31.1239 −1.02668 −0.513342 0.858184i \(-0.671593\pi\)
−0.513342 + 0.858184i \(0.671593\pi\)
\(920\) −2.76153 −0.0910449
\(921\) 6.55496 0.215993
\(922\) −2.77133 −0.0912690
\(923\) 0 0
\(924\) −24.3937 −0.802495
\(925\) 55.1771 1.81421
\(926\) 6.89392 0.226548
\(927\) −10.9879 −0.360891
\(928\) −31.9661 −1.04934
\(929\) 7.62671 0.250224 0.125112 0.992143i \(-0.460071\pi\)
0.125112 + 0.992143i \(0.460071\pi\)
\(930\) −60.7265 −1.99130
\(931\) 7.41789 0.243112
\(932\) 19.1056 0.625825
\(933\) 12.0392 0.394147
\(934\) −14.1596 −0.463317
\(935\) 15.1183 0.494421
\(936\) 0 0
\(937\) −5.67324 −0.185337 −0.0926683 0.995697i \(-0.529540\pi\)
−0.0926683 + 0.995697i \(0.529540\pi\)
\(938\) 7.00298 0.228656
\(939\) 33.8950 1.10612
\(940\) 28.0532 0.914995
\(941\) 41.5394 1.35415 0.677073 0.735916i \(-0.263248\pi\)
0.677073 + 0.735916i \(0.263248\pi\)
\(942\) 21.3873 0.696836
\(943\) 14.0664 0.458064
\(944\) 16.8030 0.546891
\(945\) −7.54288 −0.245370
\(946\) 23.1355 0.752201
\(947\) −47.3110 −1.53740 −0.768700 0.639610i \(-0.779096\pi\)
−0.768700 + 0.639610i \(0.779096\pi\)
\(948\) 8.22282 0.267065
\(949\) 0 0
\(950\) 48.8327 1.58434
\(951\) 4.49827 0.145866
\(952\) −0.831478 −0.0269483
\(953\) −34.3435 −1.11249 −0.556247 0.831017i \(-0.687759\pi\)
−0.556247 + 0.831017i \(0.687759\pi\)
\(954\) −1.11231 −0.0360123
\(955\) −22.1135 −0.715576
\(956\) −50.3777 −1.62933
\(957\) 19.4547 0.628882
\(958\) −7.20344 −0.232733
\(959\) 51.7241 1.67026
\(960\) 31.8847 1.02907
\(961\) 46.9527 1.51460
\(962\) 0 0
\(963\) −9.87263 −0.318141
\(964\) 48.3137 1.55608
\(965\) 36.4802 1.17434
\(966\) 9.33273 0.300276
\(967\) 48.5096 1.55996 0.779982 0.625802i \(-0.215228\pi\)
0.779982 + 0.625802i \(0.215228\pi\)
\(968\) −5.43535 −0.174699
\(969\) 3.46681 0.111370
\(970\) 110.937 3.56198
\(971\) 41.4650 1.33068 0.665338 0.746542i \(-0.268288\pi\)
0.665338 + 0.746542i \(0.268288\pi\)
\(972\) −2.19806 −0.0705029
\(973\) 2.20775 0.0707772
\(974\) 26.3242 0.843483
\(975\) 0 0
\(976\) −13.1177 −0.419887
\(977\) −2.09677 −0.0670816 −0.0335408 0.999437i \(-0.510678\pi\)
−0.0335408 + 0.999437i \(0.510678\pi\)
\(978\) −22.6334 −0.723737
\(979\) −49.6795 −1.58776
\(980\) −14.3964 −0.459876
\(981\) 20.2446 0.646360
\(982\) −58.7442 −1.87460
\(983\) 25.2336 0.804826 0.402413 0.915458i \(-0.368172\pi\)
0.402413 + 0.915458i \(0.368172\pi\)
\(984\) 2.81594 0.0897688
\(985\) 31.0231 0.988478
\(986\) 7.35929 0.234367
\(987\) −8.54288 −0.271923
\(988\) 0 0
\(989\) −4.63448 −0.147368
\(990\) −33.9705 −1.07965
\(991\) 12.0489 0.382746 0.191373 0.981517i \(-0.438706\pi\)
0.191373 + 0.981517i \(0.438706\pi\)
\(992\) −71.6506 −2.27491
\(993\) 11.2131 0.355838
\(994\) −10.9511 −0.347347
\(995\) −5.25236 −0.166511
\(996\) −5.07308 −0.160747
\(997\) −1.43403 −0.0454160 −0.0227080 0.999742i \(-0.507229\pi\)
−0.0227080 + 0.999742i \(0.507229\pi\)
\(998\) 68.7525 2.17632
\(999\) −8.80194 −0.278481
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.1.1 yes 3
3.2 odd 2 1521.2.a.n.1.3 3
4.3 odd 2 8112.2.a.cp.1.2 3
13.2 odd 12 507.2.j.i.316.3 12
13.3 even 3 507.2.e.i.22.3 6
13.4 even 6 507.2.e.l.484.1 6
13.5 odd 4 507.2.b.f.337.4 6
13.6 odd 12 507.2.j.i.361.4 12
13.7 odd 12 507.2.j.i.361.3 12
13.8 odd 4 507.2.b.f.337.3 6
13.9 even 3 507.2.e.i.484.3 6
13.10 even 6 507.2.e.l.22.1 6
13.11 odd 12 507.2.j.i.316.4 12
13.12 even 2 507.2.a.i.1.3 3
39.5 even 4 1521.2.b.k.1351.3 6
39.8 even 4 1521.2.b.k.1351.4 6
39.38 odd 2 1521.2.a.s.1.1 3
52.51 odd 2 8112.2.a.cg.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.3 3 13.12 even 2
507.2.a.l.1.1 yes 3 1.1 even 1 trivial
507.2.b.f.337.3 6 13.8 odd 4
507.2.b.f.337.4 6 13.5 odd 4
507.2.e.i.22.3 6 13.3 even 3
507.2.e.i.484.3 6 13.9 even 3
507.2.e.l.22.1 6 13.10 even 6
507.2.e.l.484.1 6 13.4 even 6
507.2.j.i.316.3 12 13.2 odd 12
507.2.j.i.316.4 12 13.11 odd 12
507.2.j.i.361.3 12 13.7 odd 12
507.2.j.i.361.4 12 13.6 odd 12
1521.2.a.n.1.3 3 3.2 odd 2
1521.2.a.s.1.1 3 39.38 odd 2
1521.2.b.k.1351.3 6 39.5 even 4
1521.2.b.k.1351.4 6 39.8 even 4
8112.2.a.cg.1.2 3 52.51 odd 2
8112.2.a.cp.1.2 3 4.3 odd 2