Properties

Label 418.2.a.g.1.1
Level $418$
Weight $2$
Character 418.1
Self dual yes
Analytic conductor $3.338$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.33774680449\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.14510\) of defining polynomial
Character \(\chi\) \(=\) 418.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.14510 q^{3} +1.00000 q^{4} -1.60147 q^{5} +2.14510 q^{6} +2.89167 q^{7} -1.00000 q^{8} +1.60147 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.14510 q^{3} +1.00000 q^{4} -1.60147 q^{5} +2.14510 q^{6} +2.89167 q^{7} -1.00000 q^{8} +1.60147 q^{9} +1.60147 q^{10} -1.00000 q^{11} -2.14510 q^{12} +4.89167 q^{13} -2.89167 q^{14} +3.43531 q^{15} +1.00000 q^{16} -5.74657 q^{17} -1.60147 q^{18} -1.00000 q^{19} -1.60147 q^{20} -6.20293 q^{21} +1.00000 q^{22} -7.74657 q^{23} +2.14510 q^{24} -2.43531 q^{25} -4.89167 q^{26} +3.00000 q^{27} +2.89167 q^{28} +5.34803 q^{29} -3.43531 q^{30} -7.43531 q^{31} -1.00000 q^{32} +2.14510 q^{33} +5.74657 q^{34} -4.63091 q^{35} +1.60147 q^{36} -7.49314 q^{37} +1.00000 q^{38} -10.4931 q^{39} +1.60147 q^{40} -2.05783 q^{41} +6.20293 q^{42} -10.6887 q^{43} -1.00000 q^{44} -2.56469 q^{45} +7.74657 q^{46} -7.08727 q^{47} -2.14510 q^{48} +1.36176 q^{49} +2.43531 q^{50} +12.3270 q^{51} +4.89167 q^{52} +8.32698 q^{53} -3.00000 q^{54} +1.60147 q^{55} -2.89167 q^{56} +2.14510 q^{57} -5.34803 q^{58} -2.54364 q^{59} +3.43531 q^{60} +13.4931 q^{61} +7.43531 q^{62} +4.63091 q^{63} +1.00000 q^{64} -7.83384 q^{65} -2.14510 q^{66} -10.3848 q^{67} -5.74657 q^{68} +16.6172 q^{69} +4.63091 q^{70} +14.9789 q^{71} -1.60147 q^{72} -3.74657 q^{73} +7.49314 q^{74} +5.22399 q^{75} -1.00000 q^{76} -2.89167 q^{77} +10.4931 q^{78} -8.69607 q^{79} -1.60147 q^{80} -11.2397 q^{81} +2.05783 q^{82} -6.10833 q^{83} -6.20293 q^{84} +9.20293 q^{85} +10.6887 q^{86} -11.4721 q^{87} +1.00000 q^{88} +3.20293 q^{89} +2.56469 q^{90} +14.1451 q^{91} -7.74657 q^{92} +15.9495 q^{93} +7.08727 q^{94} +1.60147 q^{95} +2.14510 q^{96} -6.98627 q^{97} -1.36176 q^{98} -1.60147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{10} - 3 q^{11} + 6 q^{14} - 9 q^{15} + 3 q^{16} - 9 q^{17} - 3 q^{18} - 3 q^{19} - 3 q^{20} - 15 q^{21} + 3 q^{22} - 15 q^{23} + 12 q^{25} + 9 q^{27} - 6 q^{28} + 6 q^{29} + 9 q^{30} - 3 q^{31} - 3 q^{32} + 9 q^{34} + 3 q^{36} - 6 q^{37} + 3 q^{38} - 15 q^{39} + 3 q^{40} - 9 q^{41} + 15 q^{42} - 21 q^{43} - 3 q^{44} - 27 q^{45} + 15 q^{46} - 12 q^{47} + 27 q^{49} - 12 q^{50} + 3 q^{51} - 9 q^{53} - 9 q^{54} + 3 q^{55} + 6 q^{56} - 6 q^{58} - 3 q^{59} - 9 q^{60} + 24 q^{61} + 3 q^{62} + 3 q^{64} - 6 q^{65} - 9 q^{68} + 3 q^{69} + 21 q^{71} - 3 q^{72} - 3 q^{73} + 6 q^{74} + 36 q^{75} - 3 q^{76} + 6 q^{77} + 15 q^{78} - 6 q^{79} - 3 q^{80} - 9 q^{81} + 9 q^{82} - 33 q^{83} - 15 q^{84} + 24 q^{85} + 21 q^{86} + 6 q^{87} + 3 q^{88} + 6 q^{89} + 27 q^{90} + 36 q^{91} - 15 q^{92} + 36 q^{93} + 12 q^{94} + 3 q^{95} + 12 q^{97} - 27 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.14510 −1.23848 −0.619238 0.785204i \(-0.712558\pi\)
−0.619238 + 0.785204i \(0.712558\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.60147 −0.716197 −0.358099 0.933684i \(-0.616575\pi\)
−0.358099 + 0.933684i \(0.616575\pi\)
\(6\) 2.14510 0.875735
\(7\) 2.89167 1.09295 0.546474 0.837476i \(-0.315970\pi\)
0.546474 + 0.837476i \(0.315970\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.60147 0.533822
\(10\) 1.60147 0.506428
\(11\) −1.00000 −0.301511
\(12\) −2.14510 −0.619238
\(13\) 4.89167 1.35671 0.678353 0.734736i \(-0.262694\pi\)
0.678353 + 0.734736i \(0.262694\pi\)
\(14\) −2.89167 −0.772832
\(15\) 3.43531 0.886993
\(16\) 1.00000 0.250000
\(17\) −5.74657 −1.39375 −0.696874 0.717194i \(-0.745426\pi\)
−0.696874 + 0.717194i \(0.745426\pi\)
\(18\) −1.60147 −0.377469
\(19\) −1.00000 −0.229416
\(20\) −1.60147 −0.358099
\(21\) −6.20293 −1.35359
\(22\) 1.00000 0.213201
\(23\) −7.74657 −1.61527 −0.807636 0.589682i \(-0.799253\pi\)
−0.807636 + 0.589682i \(0.799253\pi\)
\(24\) 2.14510 0.437867
\(25\) −2.43531 −0.487062
\(26\) −4.89167 −0.959336
\(27\) 3.00000 0.577350
\(28\) 2.89167 0.546474
\(29\) 5.34803 0.993105 0.496552 0.868007i \(-0.334599\pi\)
0.496552 + 0.868007i \(0.334599\pi\)
\(30\) −3.43531 −0.627199
\(31\) −7.43531 −1.33542 −0.667710 0.744421i \(-0.732726\pi\)
−0.667710 + 0.744421i \(0.732726\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.14510 0.373414
\(34\) 5.74657 0.985528
\(35\) −4.63091 −0.782767
\(36\) 1.60147 0.266911
\(37\) −7.49314 −1.23186 −0.615932 0.787799i \(-0.711220\pi\)
−0.615932 + 0.787799i \(0.711220\pi\)
\(38\) 1.00000 0.162221
\(39\) −10.4931 −1.68025
\(40\) 1.60147 0.253214
\(41\) −2.05783 −0.321379 −0.160689 0.987005i \(-0.551372\pi\)
−0.160689 + 0.987005i \(0.551372\pi\)
\(42\) 6.20293 0.957133
\(43\) −10.6887 −1.63002 −0.815009 0.579449i \(-0.803268\pi\)
−0.815009 + 0.579449i \(0.803268\pi\)
\(44\) −1.00000 −0.150756
\(45\) −2.56469 −0.382322
\(46\) 7.74657 1.14217
\(47\) −7.08727 −1.03379 −0.516893 0.856050i \(-0.672911\pi\)
−0.516893 + 0.856050i \(0.672911\pi\)
\(48\) −2.14510 −0.309619
\(49\) 1.36176 0.194537
\(50\) 2.43531 0.344405
\(51\) 12.3270 1.72612
\(52\) 4.89167 0.678353
\(53\) 8.32698 1.14380 0.571899 0.820324i \(-0.306207\pi\)
0.571899 + 0.820324i \(0.306207\pi\)
\(54\) −3.00000 −0.408248
\(55\) 1.60147 0.215942
\(56\) −2.89167 −0.386416
\(57\) 2.14510 0.284126
\(58\) −5.34803 −0.702231
\(59\) −2.54364 −0.331153 −0.165577 0.986197i \(-0.552949\pi\)
−0.165577 + 0.986197i \(0.552949\pi\)
\(60\) 3.43531 0.443496
\(61\) 13.4931 1.72762 0.863810 0.503818i \(-0.168072\pi\)
0.863810 + 0.503818i \(0.168072\pi\)
\(62\) 7.43531 0.944285
\(63\) 4.63091 0.583440
\(64\) 1.00000 0.125000
\(65\) −7.83384 −0.971669
\(66\) −2.14510 −0.264044
\(67\) −10.3848 −1.26871 −0.634353 0.773043i \(-0.718733\pi\)
−0.634353 + 0.773043i \(0.718733\pi\)
\(68\) −5.74657 −0.696874
\(69\) 16.6172 2.00047
\(70\) 4.63091 0.553500
\(71\) 14.9789 1.77767 0.888837 0.458224i \(-0.151514\pi\)
0.888837 + 0.458224i \(0.151514\pi\)
\(72\) −1.60147 −0.188735
\(73\) −3.74657 −0.438503 −0.219251 0.975668i \(-0.570361\pi\)
−0.219251 + 0.975668i \(0.570361\pi\)
\(74\) 7.49314 0.871059
\(75\) 5.22399 0.603214
\(76\) −1.00000 −0.114708
\(77\) −2.89167 −0.329536
\(78\) 10.4931 1.18811
\(79\) −8.69607 −0.978384 −0.489192 0.872176i \(-0.662708\pi\)
−0.489192 + 0.872176i \(0.662708\pi\)
\(80\) −1.60147 −0.179049
\(81\) −11.2397 −1.24886
\(82\) 2.05783 0.227249
\(83\) −6.10833 −0.670476 −0.335238 0.942133i \(-0.608817\pi\)
−0.335238 + 0.942133i \(0.608817\pi\)
\(84\) −6.20293 −0.676795
\(85\) 9.20293 0.998198
\(86\) 10.6887 1.15260
\(87\) −11.4721 −1.22994
\(88\) 1.00000 0.106600
\(89\) 3.20293 0.339510 0.169755 0.985486i \(-0.445702\pi\)
0.169755 + 0.985486i \(0.445702\pi\)
\(90\) 2.56469 0.270342
\(91\) 14.1451 1.48281
\(92\) −7.74657 −0.807636
\(93\) 15.9495 1.65389
\(94\) 7.08727 0.730997
\(95\) 1.60147 0.164307
\(96\) 2.14510 0.218934
\(97\) −6.98627 −0.709349 −0.354674 0.934990i \(-0.615408\pi\)
−0.354674 + 0.934990i \(0.615408\pi\)
\(98\) −1.36176 −0.137559
\(99\) −1.60147 −0.160953
\(100\) −2.43531 −0.243531
\(101\) −1.20293 −0.119696 −0.0598481 0.998207i \(-0.519062\pi\)
−0.0598481 + 0.998207i \(0.519062\pi\)
\(102\) −12.3270 −1.22055
\(103\) 16.8779 1.66303 0.831517 0.555500i \(-0.187473\pi\)
0.831517 + 0.555500i \(0.187473\pi\)
\(104\) −4.89167 −0.479668
\(105\) 9.93378 0.969438
\(106\) −8.32698 −0.808788
\(107\) −8.03677 −0.776944 −0.388472 0.921460i \(-0.626997\pi\)
−0.388472 + 0.921460i \(0.626997\pi\)
\(108\) 3.00000 0.288675
\(109\) 11.5299 1.10437 0.552183 0.833723i \(-0.313795\pi\)
0.552183 + 0.833723i \(0.313795\pi\)
\(110\) −1.60147 −0.152694
\(111\) 16.0735 1.52563
\(112\) 2.89167 0.273237
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) −2.14510 −0.200907
\(115\) 12.4059 1.15685
\(116\) 5.34803 0.496552
\(117\) 7.83384 0.724239
\(118\) 2.54364 0.234161
\(119\) −16.6172 −1.52329
\(120\) −3.43531 −0.313599
\(121\) 1.00000 0.0909091
\(122\) −13.4931 −1.22161
\(123\) 4.41425 0.398020
\(124\) −7.43531 −0.667710
\(125\) 11.9074 1.06503
\(126\) −4.63091 −0.412554
\(127\) −21.2765 −1.88798 −0.943991 0.329971i \(-0.892961\pi\)
−0.943991 + 0.329971i \(0.892961\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 22.9284 2.01874
\(130\) 7.83384 0.687073
\(131\) −3.43531 −0.300144 −0.150072 0.988675i \(-0.547951\pi\)
−0.150072 + 0.988675i \(0.547951\pi\)
\(132\) 2.14510 0.186707
\(133\) −2.89167 −0.250740
\(134\) 10.3848 0.897111
\(135\) −4.80440 −0.413497
\(136\) 5.74657 0.492764
\(137\) 13.5877 1.16088 0.580439 0.814303i \(-0.302881\pi\)
0.580439 + 0.814303i \(0.302881\pi\)
\(138\) −16.6172 −1.41455
\(139\) 3.19560 0.271048 0.135524 0.990774i \(-0.456728\pi\)
0.135524 + 0.990774i \(0.456728\pi\)
\(140\) −4.63091 −0.391383
\(141\) 15.2029 1.28032
\(142\) −14.9789 −1.25701
\(143\) −4.89167 −0.409062
\(144\) 1.60147 0.133455
\(145\) −8.56469 −0.711259
\(146\) 3.74657 0.310068
\(147\) −2.92112 −0.240930
\(148\) −7.49314 −0.615932
\(149\) −12.0735 −0.989104 −0.494552 0.869148i \(-0.664668\pi\)
−0.494552 + 0.869148i \(0.664668\pi\)
\(150\) −5.22399 −0.426537
\(151\) −4.91273 −0.399792 −0.199896 0.979817i \(-0.564060\pi\)
−0.199896 + 0.979817i \(0.564060\pi\)
\(152\) 1.00000 0.0811107
\(153\) −9.20293 −0.744013
\(154\) 2.89167 0.233017
\(155\) 11.9074 0.956425
\(156\) −10.4931 −0.840123
\(157\) 8.00733 0.639054 0.319527 0.947577i \(-0.396476\pi\)
0.319527 + 0.947577i \(0.396476\pi\)
\(158\) 8.69607 0.691822
\(159\) −17.8622 −1.41657
\(160\) 1.60147 0.126607
\(161\) −22.4005 −1.76541
\(162\) 11.2397 0.883075
\(163\) 1.88434 0.147593 0.0737966 0.997273i \(-0.476488\pi\)
0.0737966 + 0.997273i \(0.476488\pi\)
\(164\) −2.05783 −0.160689
\(165\) −3.43531 −0.267438
\(166\) 6.10833 0.474098
\(167\) −4.98627 −0.385849 −0.192925 0.981214i \(-0.561797\pi\)
−0.192925 + 0.981214i \(0.561797\pi\)
\(168\) 6.20293 0.478567
\(169\) 10.9284 0.840650
\(170\) −9.20293 −0.705833
\(171\) −1.60147 −0.122467
\(172\) −10.6887 −0.815009
\(173\) 6.51419 0.495265 0.247632 0.968854i \(-0.420347\pi\)
0.247632 + 0.968854i \(0.420347\pi\)
\(174\) 11.4721 0.869696
\(175\) −7.04211 −0.532333
\(176\) −1.00000 −0.0753778
\(177\) 5.45636 0.410125
\(178\) −3.20293 −0.240070
\(179\) −5.55096 −0.414899 −0.207449 0.978246i \(-0.566516\pi\)
−0.207449 + 0.978246i \(0.566516\pi\)
\(180\) −2.56469 −0.191161
\(181\) 1.88434 0.140062 0.0700311 0.997545i \(-0.477690\pi\)
0.0700311 + 0.997545i \(0.477690\pi\)
\(182\) −14.1451 −1.04850
\(183\) −28.9442 −2.13961
\(184\) 7.74657 0.571085
\(185\) 12.0000 0.882258
\(186\) −15.9495 −1.16947
\(187\) 5.74657 0.420231
\(188\) −7.08727 −0.516893
\(189\) 8.67501 0.631014
\(190\) −1.60147 −0.116183
\(191\) 2.44264 0.176743 0.0883715 0.996088i \(-0.471834\pi\)
0.0883715 + 0.996088i \(0.471834\pi\)
\(192\) −2.14510 −0.154809
\(193\) −9.67501 −0.696423 −0.348211 0.937416i \(-0.613211\pi\)
−0.348211 + 0.937416i \(0.613211\pi\)
\(194\) 6.98627 0.501585
\(195\) 16.8044 1.20339
\(196\) 1.36176 0.0972686
\(197\) −17.4931 −1.24633 −0.623167 0.782089i \(-0.714154\pi\)
−0.623167 + 0.782089i \(0.714154\pi\)
\(198\) 1.60147 0.113811
\(199\) −25.2397 −1.78920 −0.894598 0.446873i \(-0.852538\pi\)
−0.894598 + 0.446873i \(0.852538\pi\)
\(200\) 2.43531 0.172202
\(201\) 22.2765 1.57126
\(202\) 1.20293 0.0846379
\(203\) 15.4648 1.08541
\(204\) 12.3270 0.863061
\(205\) 3.29554 0.230171
\(206\) −16.8779 −1.17594
\(207\) −12.4059 −0.862267
\(208\) 4.89167 0.339176
\(209\) 1.00000 0.0691714
\(210\) −9.93378 −0.685496
\(211\) −14.8338 −1.02120 −0.510602 0.859817i \(-0.670577\pi\)
−0.510602 + 0.859817i \(0.670577\pi\)
\(212\) 8.32698 0.571899
\(213\) −32.1314 −2.20161
\(214\) 8.03677 0.549383
\(215\) 17.1176 1.16741
\(216\) −3.00000 −0.204124
\(217\) −21.5005 −1.45955
\(218\) −11.5299 −0.780904
\(219\) 8.03677 0.543075
\(220\) 1.60147 0.107971
\(221\) −28.1103 −1.89090
\(222\) −16.0735 −1.07879
\(223\) 11.4931 0.769637 0.384819 0.922992i \(-0.374264\pi\)
0.384819 + 0.922992i \(0.374264\pi\)
\(224\) −2.89167 −0.193208
\(225\) −3.90006 −0.260004
\(226\) 8.00000 0.532152
\(227\) 8.73284 0.579619 0.289810 0.957084i \(-0.406408\pi\)
0.289810 + 0.957084i \(0.406408\pi\)
\(228\) 2.14510 0.142063
\(229\) 7.84117 0.518159 0.259080 0.965856i \(-0.416581\pi\)
0.259080 + 0.965856i \(0.416581\pi\)
\(230\) −12.4059 −0.818018
\(231\) 6.20293 0.408123
\(232\) −5.34803 −0.351116
\(233\) −12.5069 −0.819352 −0.409676 0.912231i \(-0.634358\pi\)
−0.409676 + 0.912231i \(0.634358\pi\)
\(234\) −7.83384 −0.512114
\(235\) 11.3500 0.740394
\(236\) −2.54364 −0.165577
\(237\) 18.6540 1.21170
\(238\) 16.6172 1.07713
\(239\) 24.6245 1.59283 0.796414 0.604752i \(-0.206728\pi\)
0.796414 + 0.604752i \(0.206728\pi\)
\(240\) 3.43531 0.221748
\(241\) 16.0809 1.03586 0.517930 0.855423i \(-0.326703\pi\)
0.517930 + 0.855423i \(0.326703\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 15.1103 0.969328
\(244\) 13.4931 0.863810
\(245\) −2.18081 −0.139327
\(246\) −4.41425 −0.281443
\(247\) −4.89167 −0.311250
\(248\) 7.43531 0.472143
\(249\) 13.1030 0.830368
\(250\) −11.9074 −0.753089
\(251\) 13.0873 0.826061 0.413031 0.910717i \(-0.364470\pi\)
0.413031 + 0.910717i \(0.364470\pi\)
\(252\) 4.63091 0.291720
\(253\) 7.74657 0.487023
\(254\) 21.2765 1.33500
\(255\) −19.7412 −1.23624
\(256\) 1.00000 0.0625000
\(257\) 0.723522 0.0451320 0.0225660 0.999745i \(-0.492816\pi\)
0.0225660 + 0.999745i \(0.492816\pi\)
\(258\) −22.9284 −1.42746
\(259\) −21.6677 −1.34636
\(260\) −7.83384 −0.485834
\(261\) 8.56469 0.530141
\(262\) 3.43531 0.212234
\(263\) 13.7760 0.849465 0.424733 0.905319i \(-0.360368\pi\)
0.424733 + 0.905319i \(0.360368\pi\)
\(264\) −2.14510 −0.132022
\(265\) −13.3354 −0.819185
\(266\) 2.89167 0.177300
\(267\) −6.87062 −0.420475
\(268\) −10.3848 −0.634353
\(269\) −19.9579 −1.21685 −0.608427 0.793610i \(-0.708199\pi\)
−0.608427 + 0.793610i \(0.708199\pi\)
\(270\) 4.80440 0.292386
\(271\) 28.7486 1.74635 0.873175 0.487406i \(-0.162057\pi\)
0.873175 + 0.487406i \(0.162057\pi\)
\(272\) −5.74657 −0.348437
\(273\) −30.3427 −1.83642
\(274\) −13.5877 −0.820865
\(275\) 2.43531 0.146855
\(276\) 16.6172 1.00024
\(277\) 3.08727 0.185496 0.0927482 0.995690i \(-0.470435\pi\)
0.0927482 + 0.995690i \(0.470435\pi\)
\(278\) −3.19560 −0.191660
\(279\) −11.9074 −0.712877
\(280\) 4.63091 0.276750
\(281\) 9.09460 0.542538 0.271269 0.962504i \(-0.412557\pi\)
0.271269 + 0.962504i \(0.412557\pi\)
\(282\) −15.2029 −0.905321
\(283\) 13.1451 0.781395 0.390698 0.920519i \(-0.372234\pi\)
0.390698 + 0.920519i \(0.372234\pi\)
\(284\) 14.9789 0.888837
\(285\) −3.43531 −0.203490
\(286\) 4.89167 0.289251
\(287\) −5.95056 −0.351251
\(288\) −1.60147 −0.0943673
\(289\) 16.0230 0.942532
\(290\) 8.56469 0.502936
\(291\) 14.9863 0.878511
\(292\) −3.74657 −0.219251
\(293\) 3.10833 0.181591 0.0907953 0.995870i \(-0.471059\pi\)
0.0907953 + 0.995870i \(0.471059\pi\)
\(294\) 2.92112 0.170363
\(295\) 4.07355 0.237171
\(296\) 7.49314 0.435530
\(297\) −3.00000 −0.174078
\(298\) 12.0735 0.699402
\(299\) −37.8937 −2.19145
\(300\) 5.22399 0.301607
\(301\) −30.9083 −1.78153
\(302\) 4.91273 0.282696
\(303\) 2.58041 0.148241
\(304\) −1.00000 −0.0573539
\(305\) −21.6088 −1.23732
\(306\) 9.20293 0.526097
\(307\) 2.18920 0.124944 0.0624722 0.998047i \(-0.480102\pi\)
0.0624722 + 0.998047i \(0.480102\pi\)
\(308\) −2.89167 −0.164768
\(309\) −36.2049 −2.05963
\(310\) −11.9074 −0.676294
\(311\) 3.92112 0.222346 0.111173 0.993801i \(-0.464539\pi\)
0.111173 + 0.993801i \(0.464539\pi\)
\(312\) 10.4931 0.594057
\(313\) −21.7118 −1.22722 −0.613611 0.789608i \(-0.710284\pi\)
−0.613611 + 0.789608i \(0.710284\pi\)
\(314\) −8.00733 −0.451880
\(315\) −7.41625 −0.417858
\(316\) −8.69607 −0.489192
\(317\) 6.90739 0.387958 0.193979 0.981006i \(-0.437861\pi\)
0.193979 + 0.981006i \(0.437861\pi\)
\(318\) 17.8622 1.00166
\(319\) −5.34803 −0.299432
\(320\) −1.60147 −0.0895246
\(321\) 17.2397 0.962226
\(322\) 22.4005 1.24833
\(323\) 5.74657 0.319748
\(324\) −11.2397 −0.624428
\(325\) −11.9127 −0.660799
\(326\) −1.88434 −0.104364
\(327\) −24.7328 −1.36773
\(328\) 2.05783 0.113625
\(329\) −20.4941 −1.12987
\(330\) 3.43531 0.189107
\(331\) 32.1030 1.76454 0.882270 0.470744i \(-0.156014\pi\)
0.882270 + 0.470744i \(0.156014\pi\)
\(332\) −6.10833 −0.335238
\(333\) −12.0000 −0.657596
\(334\) 4.98627 0.272837
\(335\) 16.6309 0.908644
\(336\) −6.20293 −0.338398
\(337\) −1.55096 −0.0844864 −0.0422432 0.999107i \(-0.513450\pi\)
−0.0422432 + 0.999107i \(0.513450\pi\)
\(338\) −10.9284 −0.594429
\(339\) 17.1608 0.932048
\(340\) 9.20293 0.499099
\(341\) 7.43531 0.402645
\(342\) 1.60147 0.0865973
\(343\) −16.3039 −0.880330
\(344\) 10.6887 0.576298
\(345\) −26.6118 −1.43273
\(346\) −6.51419 −0.350205
\(347\) −0.506864 −0.0272099 −0.0136049 0.999907i \(-0.504331\pi\)
−0.0136049 + 0.999907i \(0.504331\pi\)
\(348\) −11.4721 −0.614968
\(349\) 10.8117 0.578738 0.289369 0.957218i \(-0.406554\pi\)
0.289369 + 0.957218i \(0.406554\pi\)
\(350\) 7.04211 0.376417
\(351\) 14.6750 0.783294
\(352\) 1.00000 0.0533002
\(353\) 27.0966 1.44221 0.721103 0.692828i \(-0.243635\pi\)
0.721103 + 0.692828i \(0.243635\pi\)
\(354\) −5.45636 −0.290002
\(355\) −23.9883 −1.27316
\(356\) 3.20293 0.169755
\(357\) 35.6456 1.88656
\(358\) 5.55096 0.293378
\(359\) −20.7486 −1.09507 −0.547534 0.836784i \(-0.684433\pi\)
−0.547534 + 0.836784i \(0.684433\pi\)
\(360\) 2.56469 0.135171
\(361\) 1.00000 0.0526316
\(362\) −1.88434 −0.0990389
\(363\) −2.14510 −0.112589
\(364\) 14.1451 0.741405
\(365\) 6.00000 0.314054
\(366\) 28.9442 1.51294
\(367\) 25.6402 1.33841 0.669205 0.743078i \(-0.266635\pi\)
0.669205 + 0.743078i \(0.266635\pi\)
\(368\) −7.74657 −0.403818
\(369\) −3.29554 −0.171559
\(370\) −12.0000 −0.623850
\(371\) 24.0789 1.25011
\(372\) 15.9495 0.826943
\(373\) 25.2049 1.30506 0.652531 0.757762i \(-0.273707\pi\)
0.652531 + 0.757762i \(0.273707\pi\)
\(374\) −5.74657 −0.297148
\(375\) −25.5426 −1.31901
\(376\) 7.08727 0.365498
\(377\) 26.1608 1.34735
\(378\) −8.67501 −0.446195
\(379\) 3.47208 0.178349 0.0891744 0.996016i \(-0.471577\pi\)
0.0891744 + 0.996016i \(0.471577\pi\)
\(380\) 1.60147 0.0821534
\(381\) 45.6402 2.33822
\(382\) −2.44264 −0.124976
\(383\) 23.1176 1.18126 0.590628 0.806944i \(-0.298880\pi\)
0.590628 + 0.806944i \(0.298880\pi\)
\(384\) 2.14510 0.109467
\(385\) 4.63091 0.236013
\(386\) 9.67501 0.492445
\(387\) −17.1176 −0.870139
\(388\) −6.98627 −0.354674
\(389\) −26.7623 −1.35690 −0.678451 0.734646i \(-0.737348\pi\)
−0.678451 + 0.734646i \(0.737348\pi\)
\(390\) −16.8044 −0.850924
\(391\) 44.5162 2.25128
\(392\) −1.36176 −0.0687793
\(393\) 7.36909 0.371721
\(394\) 17.4931 0.881291
\(395\) 13.9265 0.700716
\(396\) −1.60147 −0.0804767
\(397\) −31.1030 −1.56101 −0.780507 0.625147i \(-0.785039\pi\)
−0.780507 + 0.625147i \(0.785039\pi\)
\(398\) 25.2397 1.26515
\(399\) 6.20293 0.310535
\(400\) −2.43531 −0.121765
\(401\) 29.3500 1.46567 0.732835 0.680406i \(-0.238197\pi\)
0.732835 + 0.680406i \(0.238197\pi\)
\(402\) −22.2765 −1.11105
\(403\) −36.3711 −1.81177
\(404\) −1.20293 −0.0598481
\(405\) 18.0000 0.894427
\(406\) −15.4648 −0.767503
\(407\) 7.49314 0.371421
\(408\) −12.3270 −0.610276
\(409\) 8.58774 0.424636 0.212318 0.977201i \(-0.431899\pi\)
0.212318 + 0.977201i \(0.431899\pi\)
\(410\) −3.29554 −0.162755
\(411\) −29.1471 −1.43772
\(412\) 16.8779 0.831517
\(413\) −7.35536 −0.361934
\(414\) 12.4059 0.609715
\(415\) 9.78228 0.480193
\(416\) −4.89167 −0.239834
\(417\) −6.85490 −0.335686
\(418\) −1.00000 −0.0489116
\(419\) 7.59414 0.370998 0.185499 0.982644i \(-0.440610\pi\)
0.185499 + 0.982644i \(0.440610\pi\)
\(420\) 9.93378 0.484719
\(421\) 8.76030 0.426951 0.213475 0.976948i \(-0.431522\pi\)
0.213475 + 0.976948i \(0.431522\pi\)
\(422\) 14.8338 0.722100
\(423\) −11.3500 −0.551857
\(424\) −8.32698 −0.404394
\(425\) 13.9947 0.678841
\(426\) 32.1314 1.55677
\(427\) 39.0177 1.88820
\(428\) −8.03677 −0.388472
\(429\) 10.4931 0.506613
\(430\) −17.1176 −0.825486
\(431\) −33.6677 −1.62172 −0.810858 0.585243i \(-0.800999\pi\)
−0.810858 + 0.585243i \(0.800999\pi\)
\(432\) 3.00000 0.144338
\(433\) 30.5383 1.46758 0.733789 0.679378i \(-0.237750\pi\)
0.733789 + 0.679378i \(0.237750\pi\)
\(434\) 21.5005 1.03206
\(435\) 18.3721 0.880877
\(436\) 11.5299 0.552183
\(437\) 7.74657 0.370569
\(438\) −8.03677 −0.384012
\(439\) −12.5069 −0.596920 −0.298460 0.954422i \(-0.596473\pi\)
−0.298460 + 0.954422i \(0.596473\pi\)
\(440\) −1.60147 −0.0763469
\(441\) 2.18081 0.103848
\(442\) 28.1103 1.33707
\(443\) −17.6677 −0.839417 −0.419709 0.907659i \(-0.637868\pi\)
−0.419709 + 0.907659i \(0.637868\pi\)
\(444\) 16.0735 0.762817
\(445\) −5.12938 −0.243156
\(446\) −11.4931 −0.544216
\(447\) 25.8990 1.22498
\(448\) 2.89167 0.136619
\(449\) −17.7098 −0.835777 −0.417888 0.908498i \(-0.637230\pi\)
−0.417888 + 0.908498i \(0.637230\pi\)
\(450\) 3.90006 0.183851
\(451\) 2.05783 0.0968994
\(452\) −8.00000 −0.376288
\(453\) 10.5383 0.495133
\(454\) −8.73284 −0.409853
\(455\) −22.6529 −1.06198
\(456\) −2.14510 −0.100454
\(457\) −24.8064 −1.16039 −0.580197 0.814476i \(-0.697024\pi\)
−0.580197 + 0.814476i \(0.697024\pi\)
\(458\) −7.84117 −0.366394
\(459\) −17.2397 −0.804681
\(460\) 12.4059 0.578426
\(461\) −2.98627 −0.139085 −0.0695423 0.997579i \(-0.522154\pi\)
−0.0695423 + 0.997579i \(0.522154\pi\)
\(462\) −6.20293 −0.288586
\(463\) −39.5667 −1.83882 −0.919410 0.393301i \(-0.871333\pi\)
−0.919410 + 0.393301i \(0.871333\pi\)
\(464\) 5.34803 0.248276
\(465\) −25.5426 −1.18451
\(466\) 12.5069 0.579369
\(467\) 4.46475 0.206604 0.103302 0.994650i \(-0.467059\pi\)
0.103302 + 0.994650i \(0.467059\pi\)
\(468\) 7.83384 0.362119
\(469\) −30.0294 −1.38663
\(470\) −11.3500 −0.523538
\(471\) −17.1765 −0.791453
\(472\) 2.54364 0.117080
\(473\) 10.6887 0.491469
\(474\) −18.6540 −0.856805
\(475\) 2.43531 0.111740
\(476\) −16.6172 −0.761647
\(477\) 13.3354 0.610585
\(478\) −24.6245 −1.12630
\(479\) −20.8863 −0.954321 −0.477161 0.878816i \(-0.658334\pi\)
−0.477161 + 0.878816i \(0.658334\pi\)
\(480\) −3.43531 −0.156800
\(481\) −36.6540 −1.67128
\(482\) −16.0809 −0.732464
\(483\) 48.0514 2.18642
\(484\) 1.00000 0.0454545
\(485\) 11.1883 0.508033
\(486\) −15.1103 −0.685418
\(487\) 15.2838 0.692575 0.346288 0.938128i \(-0.387442\pi\)
0.346288 + 0.938128i \(0.387442\pi\)
\(488\) −13.4931 −0.610806
\(489\) −4.04211 −0.182791
\(490\) 2.18081 0.0985191
\(491\) 3.31965 0.149814 0.0749069 0.997191i \(-0.476134\pi\)
0.0749069 + 0.997191i \(0.476134\pi\)
\(492\) 4.41425 0.199010
\(493\) −30.7328 −1.38414
\(494\) 4.89167 0.220087
\(495\) 2.56469 0.115274
\(496\) −7.43531 −0.333855
\(497\) 43.3142 1.94291
\(498\) −13.1030 −0.587159
\(499\) −2.07355 −0.0928247 −0.0464124 0.998922i \(-0.514779\pi\)
−0.0464124 + 0.998922i \(0.514779\pi\)
\(500\) 11.9074 0.532515
\(501\) 10.6961 0.477865
\(502\) −13.0873 −0.584114
\(503\) −1.11672 −0.0497921 −0.0248960 0.999690i \(-0.507925\pi\)
−0.0248960 + 0.999690i \(0.507925\pi\)
\(504\) −4.63091 −0.206277
\(505\) 1.92645 0.0857260
\(506\) −7.74657 −0.344377
\(507\) −23.4426 −1.04112
\(508\) −21.2765 −0.943991
\(509\) −38.2481 −1.69532 −0.847659 0.530542i \(-0.821988\pi\)
−0.847659 + 0.530542i \(0.821988\pi\)
\(510\) 19.7412 0.874156
\(511\) −10.8338 −0.479261
\(512\) −1.00000 −0.0441942
\(513\) −3.00000 −0.132453
\(514\) −0.723522 −0.0319132
\(515\) −27.0294 −1.19106
\(516\) 22.9284 1.00937
\(517\) 7.08727 0.311698
\(518\) 21.6677 0.952023
\(519\) −13.9736 −0.613373
\(520\) 7.83384 0.343537
\(521\) 10.8853 0.476892 0.238446 0.971156i \(-0.423362\pi\)
0.238446 + 0.971156i \(0.423362\pi\)
\(522\) −8.56469 −0.374866
\(523\) −15.7191 −0.687349 −0.343674 0.939089i \(-0.611672\pi\)
−0.343674 + 0.939089i \(0.611672\pi\)
\(524\) −3.43531 −0.150072
\(525\) 15.1060 0.659282
\(526\) −13.7760 −0.600663
\(527\) 42.7275 1.86124
\(528\) 2.14510 0.0933536
\(529\) 37.0093 1.60910
\(530\) 13.3354 0.579251
\(531\) −4.07355 −0.176777
\(532\) −2.89167 −0.125370
\(533\) −10.0662 −0.436016
\(534\) 6.87062 0.297321
\(535\) 12.8706 0.556445
\(536\) 10.3848 0.448555
\(537\) 11.9074 0.513842
\(538\) 19.9579 0.860446
\(539\) −1.36176 −0.0586552
\(540\) −4.80440 −0.206748
\(541\) −26.4373 −1.13663 −0.568314 0.822812i \(-0.692404\pi\)
−0.568314 + 0.822812i \(0.692404\pi\)
\(542\) −28.7486 −1.23486
\(543\) −4.04211 −0.173464
\(544\) 5.74657 0.246382
\(545\) −18.4648 −0.790943
\(546\) 30.3427 1.29855
\(547\) −24.9588 −1.06716 −0.533581 0.845749i \(-0.679154\pi\)
−0.533581 + 0.845749i \(0.679154\pi\)
\(548\) 13.5877 0.580439
\(549\) 21.6088 0.922241
\(550\) −2.43531 −0.103842
\(551\) −5.34803 −0.227834
\(552\) −16.6172 −0.707274
\(553\) −25.1462 −1.06932
\(554\) −3.08727 −0.131166
\(555\) −25.7412 −1.09265
\(556\) 3.19560 0.135524
\(557\) 8.30486 0.351888 0.175944 0.984400i \(-0.443702\pi\)
0.175944 + 0.984400i \(0.443702\pi\)
\(558\) 11.9074 0.504080
\(559\) −52.2858 −2.21145
\(560\) −4.63091 −0.195692
\(561\) −12.3270 −0.520445
\(562\) −9.09460 −0.383633
\(563\) 41.6402 1.75493 0.877463 0.479644i \(-0.159234\pi\)
0.877463 + 0.479644i \(0.159234\pi\)
\(564\) 15.2029 0.640159
\(565\) 12.8117 0.538993
\(566\) −13.1451 −0.552530
\(567\) −32.5015 −1.36494
\(568\) −14.9789 −0.628503
\(569\) −14.9138 −0.625219 −0.312609 0.949882i \(-0.601203\pi\)
−0.312609 + 0.949882i \(0.601203\pi\)
\(570\) 3.43531 0.143889
\(571\) −23.0020 −0.962603 −0.481302 0.876555i \(-0.659836\pi\)
−0.481302 + 0.876555i \(0.659836\pi\)
\(572\) −4.89167 −0.204531
\(573\) −5.23970 −0.218892
\(574\) 5.95056 0.248372
\(575\) 18.8653 0.786737
\(576\) 1.60147 0.0667277
\(577\) −33.6613 −1.40134 −0.700669 0.713487i \(-0.747115\pi\)
−0.700669 + 0.713487i \(0.747115\pi\)
\(578\) −16.0230 −0.666471
\(579\) 20.7539 0.862502
\(580\) −8.56469 −0.355629
\(581\) −17.6633 −0.732796
\(582\) −14.9863 −0.621201
\(583\) −8.32698 −0.344868
\(584\) 3.74657 0.155034
\(585\) −12.5456 −0.518698
\(586\) −3.10833 −0.128404
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) −2.92112 −0.120465
\(589\) 7.43531 0.306367
\(590\) −4.07355 −0.167705
\(591\) 37.5246 1.54355
\(592\) −7.49314 −0.307966
\(593\) −30.9588 −1.27133 −0.635663 0.771967i \(-0.719273\pi\)
−0.635663 + 0.771967i \(0.719273\pi\)
\(594\) 3.00000 0.123091
\(595\) 26.6118 1.09098
\(596\) −12.0735 −0.494552
\(597\) 54.1418 2.21587
\(598\) 37.8937 1.54959
\(599\) 3.93378 0.160730 0.0803650 0.996766i \(-0.474391\pi\)
0.0803650 + 0.996766i \(0.474391\pi\)
\(600\) −5.22399 −0.213268
\(601\) 23.1451 0.944108 0.472054 0.881570i \(-0.343513\pi\)
0.472054 + 0.881570i \(0.343513\pi\)
\(602\) 30.9083 1.25973
\(603\) −16.6309 −0.677263
\(604\) −4.91273 −0.199896
\(605\) −1.60147 −0.0651088
\(606\) −2.58041 −0.104822
\(607\) −1.78334 −0.0723836 −0.0361918 0.999345i \(-0.511523\pi\)
−0.0361918 + 0.999345i \(0.511523\pi\)
\(608\) 1.00000 0.0405554
\(609\) −33.1735 −1.34426
\(610\) 21.6088 0.874914
\(611\) −34.6686 −1.40254
\(612\) −9.20293 −0.372006
\(613\) −33.0598 −1.33527 −0.667637 0.744487i \(-0.732694\pi\)
−0.667637 + 0.744487i \(0.732694\pi\)
\(614\) −2.18920 −0.0883491
\(615\) −7.06927 −0.285061
\(616\) 2.89167 0.116509
\(617\) −21.8412 −0.879292 −0.439646 0.898171i \(-0.644896\pi\)
−0.439646 + 0.898171i \(0.644896\pi\)
\(618\) 36.2049 1.45638
\(619\) −46.2206 −1.85776 −0.928882 0.370375i \(-0.879229\pi\)
−0.928882 + 0.370375i \(0.879229\pi\)
\(620\) 11.9074 0.478212
\(621\) −23.2397 −0.932577
\(622\) −3.92112 −0.157222
\(623\) 9.26182 0.371067
\(624\) −10.4931 −0.420062
\(625\) −6.89273 −0.275709
\(626\) 21.7118 0.867778
\(627\) −2.14510 −0.0856671
\(628\) 8.00733 0.319527
\(629\) 43.0598 1.71691
\(630\) 7.41625 0.295470
\(631\) 0.317659 0.0126458 0.00632291 0.999980i \(-0.497987\pi\)
0.00632291 + 0.999980i \(0.497987\pi\)
\(632\) 8.69607 0.345911
\(633\) 31.8201 1.26474
\(634\) −6.90739 −0.274327
\(635\) 34.0735 1.35217
\(636\) −17.8622 −0.708283
\(637\) 6.66129 0.263930
\(638\) 5.34803 0.211731
\(639\) 23.9883 0.948961
\(640\) 1.60147 0.0633035
\(641\) −41.5246 −1.64012 −0.820061 0.572276i \(-0.806061\pi\)
−0.820061 + 0.572276i \(0.806061\pi\)
\(642\) −17.2397 −0.680397
\(643\) 7.26182 0.286378 0.143189 0.989695i \(-0.454264\pi\)
0.143189 + 0.989695i \(0.454264\pi\)
\(644\) −22.4005 −0.882704
\(645\) −36.7191 −1.44581
\(646\) −5.74657 −0.226096
\(647\) 29.0819 1.14333 0.571664 0.820487i \(-0.306298\pi\)
0.571664 + 0.820487i \(0.306298\pi\)
\(648\) 11.2397 0.441537
\(649\) 2.54364 0.0998465
\(650\) 11.9127 0.467256
\(651\) 46.1207 1.80761
\(652\) 1.88434 0.0737966
\(653\) 12.6529 0.495146 0.247573 0.968869i \(-0.420367\pi\)
0.247573 + 0.968869i \(0.420367\pi\)
\(654\) 24.7328 0.967131
\(655\) 5.50153 0.214962
\(656\) −2.05783 −0.0803447
\(657\) −6.00000 −0.234082
\(658\) 20.4941 0.798942
\(659\) 49.4878 1.92777 0.963886 0.266317i \(-0.0858068\pi\)
0.963886 + 0.266317i \(0.0858068\pi\)
\(660\) −3.43531 −0.133719
\(661\) 4.94950 0.192513 0.0962566 0.995357i \(-0.469313\pi\)
0.0962566 + 0.995357i \(0.469313\pi\)
\(662\) −32.1030 −1.24772
\(663\) 60.2995 2.34184
\(664\) 6.10833 0.237049
\(665\) 4.63091 0.179579
\(666\) 12.0000 0.464991
\(667\) −41.4289 −1.60413
\(668\) −4.98627 −0.192925
\(669\) −24.6540 −0.953177
\(670\) −16.6309 −0.642508
\(671\) −13.4931 −0.520897
\(672\) 6.20293 0.239283
\(673\) 21.9001 0.844185 0.422093 0.906553i \(-0.361296\pi\)
0.422093 + 0.906553i \(0.361296\pi\)
\(674\) 1.55096 0.0597409
\(675\) −7.30592 −0.281205
\(676\) 10.9284 0.420325
\(677\) 26.4353 1.01599 0.507996 0.861360i \(-0.330387\pi\)
0.507996 + 0.861360i \(0.330387\pi\)
\(678\) −17.1608 −0.659057
\(679\) −20.2020 −0.775282
\(680\) −9.20293 −0.352916
\(681\) −18.7328 −0.717844
\(682\) −7.43531 −0.284713
\(683\) 9.82545 0.375960 0.187980 0.982173i \(-0.439806\pi\)
0.187980 + 0.982173i \(0.439806\pi\)
\(684\) −1.60147 −0.0612336
\(685\) −21.7603 −0.831418
\(686\) 16.3039 0.622487
\(687\) −16.8201 −0.641727
\(688\) −10.6887 −0.407504
\(689\) 40.7328 1.55180
\(690\) 26.6118 1.01310
\(691\) −19.8255 −0.754196 −0.377098 0.926173i \(-0.623078\pi\)
−0.377098 + 0.926173i \(0.623078\pi\)
\(692\) 6.51419 0.247632
\(693\) −4.63091 −0.175914
\(694\) 0.506864 0.0192403
\(695\) −5.11765 −0.194123
\(696\) 11.4721 0.434848
\(697\) 11.8255 0.447921
\(698\) −10.8117 −0.409230
\(699\) 26.8285 1.01475
\(700\) −7.04211 −0.266167
\(701\) −10.7550 −0.406209 −0.203105 0.979157i \(-0.565103\pi\)
−0.203105 + 0.979157i \(0.565103\pi\)
\(702\) −14.6750 −0.553873
\(703\) 7.49314 0.282609
\(704\) −1.00000 −0.0376889
\(705\) −24.3470 −0.916960
\(706\) −27.0966 −1.01979
\(707\) −3.47848 −0.130822
\(708\) 5.45636 0.205063
\(709\) −1.82651 −0.0685962 −0.0342981 0.999412i \(-0.510920\pi\)
−0.0342981 + 0.999412i \(0.510920\pi\)
\(710\) 23.9883 0.900264
\(711\) −13.9265 −0.522283
\(712\) −3.20293 −0.120035
\(713\) 57.5981 2.15707
\(714\) −35.6456 −1.33400
\(715\) 7.83384 0.292969
\(716\) −5.55096 −0.207449
\(717\) −52.8221 −1.97268
\(718\) 20.7486 0.774329
\(719\) −4.65929 −0.173762 −0.0868812 0.996219i \(-0.527690\pi\)
−0.0868812 + 0.996219i \(0.527690\pi\)
\(720\) −2.56469 −0.0955804
\(721\) 48.8055 1.81761
\(722\) −1.00000 −0.0372161
\(723\) −34.4951 −1.28289
\(724\) 1.88434 0.0700311
\(725\) −13.0241 −0.483703
\(726\) 2.14510 0.0796122
\(727\) 19.9358 0.739377 0.369688 0.929156i \(-0.379464\pi\)
0.369688 + 0.929156i \(0.379464\pi\)
\(728\) −14.1451 −0.524252
\(729\) 1.30592 0.0483676
\(730\) −6.00000 −0.222070
\(731\) 61.4236 2.27183
\(732\) −28.9442 −1.06981
\(733\) 13.7687 0.508558 0.254279 0.967131i \(-0.418162\pi\)
0.254279 + 0.967131i \(0.418162\pi\)
\(734\) −25.6402 −0.946398
\(735\) 4.67807 0.172553
\(736\) 7.74657 0.285542
\(737\) 10.3848 0.382529
\(738\) 3.29554 0.121311
\(739\) 22.4867 0.827188 0.413594 0.910461i \(-0.364273\pi\)
0.413594 + 0.910461i \(0.364273\pi\)
\(740\) 12.0000 0.441129
\(741\) 10.4931 0.385475
\(742\) −24.0789 −0.883964
\(743\) −50.6540 −1.85831 −0.929157 0.369686i \(-0.879465\pi\)
−0.929157 + 0.369686i \(0.879465\pi\)
\(744\) −15.9495 −0.584737
\(745\) 19.3354 0.708393
\(746\) −25.2049 −0.922818
\(747\) −9.78228 −0.357915
\(748\) 5.74657 0.210115
\(749\) −23.2397 −0.849160
\(750\) 25.5426 0.932683
\(751\) −38.9127 −1.41995 −0.709973 0.704229i \(-0.751293\pi\)
−0.709973 + 0.704229i \(0.751293\pi\)
\(752\) −7.08727 −0.258446
\(753\) −28.0735 −1.02306
\(754\) −26.1608 −0.952721
\(755\) 7.86756 0.286330
\(756\) 8.67501 0.315507
\(757\) −20.5877 −0.748274 −0.374137 0.927373i \(-0.622061\pi\)
−0.374137 + 0.927373i \(0.622061\pi\)
\(758\) −3.47208 −0.126112
\(759\) −16.6172 −0.603166
\(760\) −1.60147 −0.0580913
\(761\) 10.6025 0.384341 0.192171 0.981362i \(-0.438447\pi\)
0.192171 + 0.981362i \(0.438447\pi\)
\(762\) −45.6402 −1.65337
\(763\) 33.3407 1.20701
\(764\) 2.44264 0.0883715
\(765\) 14.7382 0.532860
\(766\) −23.1176 −0.835275
\(767\) −12.4426 −0.449278
\(768\) −2.14510 −0.0774047
\(769\) −8.77495 −0.316433 −0.158216 0.987404i \(-0.550574\pi\)
−0.158216 + 0.987404i \(0.550574\pi\)
\(770\) −4.63091 −0.166886
\(771\) −1.55203 −0.0558949
\(772\) −9.67501 −0.348211
\(773\) −46.8653 −1.68563 −0.842813 0.538206i \(-0.819102\pi\)
−0.842813 + 0.538206i \(0.819102\pi\)
\(774\) 17.1176 0.615281
\(775\) 18.1073 0.650432
\(776\) 6.98627 0.250793
\(777\) 46.4794 1.66744
\(778\) 26.7623 0.959474
\(779\) 2.05783 0.0737294
\(780\) 16.8044 0.601694
\(781\) −14.9789 −0.535989
\(782\) −44.5162 −1.59190
\(783\) 16.0441 0.573369
\(784\) 1.36176 0.0486343
\(785\) −12.8235 −0.457689
\(786\) −7.36909 −0.262847
\(787\) −27.4143 −0.977213 −0.488606 0.872504i \(-0.662495\pi\)
−0.488606 + 0.872504i \(0.662495\pi\)
\(788\) −17.4931 −0.623167
\(789\) −29.5510 −1.05204
\(790\) −13.9265 −0.495481
\(791\) −23.1334 −0.822528
\(792\) 1.60147 0.0569056
\(793\) 66.0040 2.34387
\(794\) 31.1030 1.10380
\(795\) 28.6057 1.01454
\(796\) −25.2397 −0.894598
\(797\) −9.16616 −0.324682 −0.162341 0.986735i \(-0.551904\pi\)
−0.162341 + 0.986735i \(0.551904\pi\)
\(798\) −6.20293 −0.219581
\(799\) 40.7275 1.44084
\(800\) 2.43531 0.0861011
\(801\) 5.12938 0.181238
\(802\) −29.3500 −1.03639
\(803\) 3.74657 0.132214
\(804\) 22.2765 0.785631
\(805\) 35.8737 1.26438
\(806\) 36.3711 1.28112
\(807\) 42.8117 1.50704
\(808\) 1.20293 0.0423190
\(809\) 48.0829 1.69050 0.845252 0.534368i \(-0.179450\pi\)
0.845252 + 0.534368i \(0.179450\pi\)
\(810\) −18.0000 −0.632456
\(811\) −2.67395 −0.0938951 −0.0469475 0.998897i \(-0.514949\pi\)
−0.0469475 + 0.998897i \(0.514949\pi\)
\(812\) 15.4648 0.542706
\(813\) −61.6686 −2.16281
\(814\) −7.49314 −0.262634
\(815\) −3.01771 −0.105706
\(816\) 12.3270 0.431531
\(817\) 10.6887 0.373952
\(818\) −8.58774 −0.300263
\(819\) 22.6529 0.791556
\(820\) 3.29554 0.115085
\(821\) 29.5078 1.02983 0.514915 0.857242i \(-0.327824\pi\)
0.514915 + 0.857242i \(0.327824\pi\)
\(822\) 29.1471 1.01662
\(823\) −2.39654 −0.0835382 −0.0417691 0.999127i \(-0.513299\pi\)
−0.0417691 + 0.999127i \(0.513299\pi\)
\(824\) −16.8779 −0.587971
\(825\) −5.22399 −0.181876
\(826\) 7.35536 0.255926
\(827\) 7.78868 0.270839 0.135419 0.990788i \(-0.456762\pi\)
0.135419 + 0.990788i \(0.456762\pi\)
\(828\) −12.4059 −0.431134
\(829\) 13.2818 0.461296 0.230648 0.973037i \(-0.425915\pi\)
0.230648 + 0.973037i \(0.425915\pi\)
\(830\) −9.78228 −0.339548
\(831\) −6.62252 −0.229733
\(832\) 4.89167 0.169588
\(833\) −7.82545 −0.271136
\(834\) 6.85490 0.237366
\(835\) 7.98534 0.276344
\(836\) 1.00000 0.0345857
\(837\) −22.3059 −0.771006
\(838\) −7.59414 −0.262335
\(839\) −33.9063 −1.17058 −0.585288 0.810825i \(-0.699019\pi\)
−0.585288 + 0.810825i \(0.699019\pi\)
\(840\) −9.93378 −0.342748
\(841\) −0.398534 −0.0137426
\(842\) −8.76030 −0.301900
\(843\) −19.5089 −0.671921
\(844\) −14.8338 −0.510602
\(845\) −17.5015 −0.602071
\(846\) 11.3500 0.390222
\(847\) 2.89167 0.0993590
\(848\) 8.32698 0.285950
\(849\) −28.1976 −0.967739
\(850\) −13.9947 −0.480013
\(851\) 58.0461 1.98979
\(852\) −32.1314 −1.10080
\(853\) 40.0735 1.37209 0.686046 0.727558i \(-0.259345\pi\)
0.686046 + 0.727558i \(0.259345\pi\)
\(854\) −39.0177 −1.33516
\(855\) 2.56469 0.0877106
\(856\) 8.03677 0.274691
\(857\) 0.665693 0.0227396 0.0113698 0.999935i \(-0.496381\pi\)
0.0113698 + 0.999935i \(0.496381\pi\)
\(858\) −10.4931 −0.358230
\(859\) 13.8990 0.474228 0.237114 0.971482i \(-0.423799\pi\)
0.237114 + 0.971482i \(0.423799\pi\)
\(860\) 17.1176 0.583707
\(861\) 12.7646 0.435015
\(862\) 33.6677 1.14673
\(863\) −40.3941 −1.37503 −0.687516 0.726169i \(-0.741299\pi\)
−0.687516 + 0.726169i \(0.741299\pi\)
\(864\) −3.00000 −0.102062
\(865\) −10.4323 −0.354707
\(866\) −30.5383 −1.03773
\(867\) −34.3711 −1.16730
\(868\) −21.5005 −0.729773
\(869\) 8.69607 0.294994
\(870\) −18.3721 −0.622874
\(871\) −50.7991 −1.72126
\(872\) −11.5299 −0.390452
\(873\) −11.1883 −0.378666
\(874\) −7.74657 −0.262032
\(875\) 34.4323 1.16402
\(876\) 8.03677 0.271537
\(877\) 15.6108 0.527139 0.263569 0.964640i \(-0.415100\pi\)
0.263569 + 0.964640i \(0.415100\pi\)
\(878\) 12.5069 0.422086
\(879\) −6.66769 −0.224895
\(880\) 1.60147 0.0539854
\(881\) 19.9843 0.673288 0.336644 0.941632i \(-0.390708\pi\)
0.336644 + 0.941632i \(0.390708\pi\)
\(882\) −2.18081 −0.0734318
\(883\) 20.2167 0.680345 0.340172 0.940363i \(-0.389515\pi\)
0.340172 + 0.940363i \(0.389515\pi\)
\(884\) −28.1103 −0.945452
\(885\) −8.73818 −0.293731
\(886\) 17.6677 0.593557
\(887\) 20.1471 0.676473 0.338237 0.941061i \(-0.390170\pi\)
0.338237 + 0.941061i \(0.390170\pi\)
\(888\) −16.0735 −0.539393
\(889\) −61.5246 −2.06347
\(890\) 5.12938 0.171937
\(891\) 11.2397 0.376544
\(892\) 11.4931 0.384819
\(893\) 7.08727 0.237167
\(894\) −25.8990 −0.866192
\(895\) 8.88968 0.297149
\(896\) −2.89167 −0.0966039
\(897\) 81.2858 2.71405
\(898\) 17.7098 0.590984
\(899\) −39.7643 −1.32621
\(900\) −3.90006 −0.130002
\(901\) −47.8516 −1.59417
\(902\) −2.05783 −0.0685182
\(903\) 66.3015 2.20638
\(904\) 8.00000 0.266076
\(905\) −3.01771 −0.100312
\(906\) −10.5383 −0.350112
\(907\) 4.57109 0.151781 0.0758903 0.997116i \(-0.475820\pi\)
0.0758903 + 0.997116i \(0.475820\pi\)
\(908\) 8.73284 0.289810
\(909\) −1.92645 −0.0638964
\(910\) 22.6529 0.750936
\(911\) 51.6402 1.71092 0.855459 0.517871i \(-0.173275\pi\)
0.855459 + 0.517871i \(0.173275\pi\)
\(912\) 2.14510 0.0710314
\(913\) 6.10833 0.202156
\(914\) 24.8064 0.820522
\(915\) 46.3531 1.53239
\(916\) 7.84117 0.259080
\(917\) −9.93378 −0.328042
\(918\) 17.2397 0.568995
\(919\) −48.8412 −1.61112 −0.805561 0.592513i \(-0.798136\pi\)
−0.805561 + 0.592513i \(0.798136\pi\)
\(920\) −12.4059 −0.409009
\(921\) −4.69607 −0.154741
\(922\) 2.98627 0.0983477
\(923\) 73.2721 2.41178
\(924\) 6.20293 0.204061
\(925\) 18.2481 0.599994
\(926\) 39.5667 1.30024
\(927\) 27.0294 0.887763
\(928\) −5.34803 −0.175558
\(929\) −57.8819 −1.89904 −0.949522 0.313700i \(-0.898431\pi\)
−0.949522 + 0.313700i \(0.898431\pi\)
\(930\) 25.5426 0.837574
\(931\) −1.36176 −0.0446299
\(932\) −12.5069 −0.409676
\(933\) −8.41120 −0.275370
\(934\) −4.46475 −0.146091
\(935\) −9.20293 −0.300968
\(936\) −7.83384 −0.256057
\(937\) −57.6496 −1.88333 −0.941664 0.336553i \(-0.890739\pi\)
−0.941664 + 0.336553i \(0.890739\pi\)
\(938\) 30.0294 0.980496
\(939\) 46.5740 1.51989
\(940\) 11.3500 0.370197
\(941\) −15.0966 −0.492135 −0.246067 0.969253i \(-0.579138\pi\)
−0.246067 + 0.969253i \(0.579138\pi\)
\(942\) 17.1765 0.559642
\(943\) 15.9411 0.519114
\(944\) −2.54364 −0.0827883
\(945\) −13.8927 −0.451931
\(946\) −10.6887 −0.347521
\(947\) 33.3775 1.08462 0.542311 0.840178i \(-0.317549\pi\)
0.542311 + 0.840178i \(0.317549\pi\)
\(948\) 18.6540 0.605852
\(949\) −18.3270 −0.594919
\(950\) −2.43531 −0.0790118
\(951\) −14.8171 −0.480476
\(952\) 16.6172 0.538566
\(953\) −12.4794 −0.404248 −0.202124 0.979360i \(-0.564784\pi\)
−0.202124 + 0.979360i \(0.564784\pi\)
\(954\) −13.3354 −0.431749
\(955\) −3.91180 −0.126583
\(956\) 24.6245 0.796414
\(957\) 11.4721 0.370840
\(958\) 20.8863 0.674807
\(959\) 39.2913 1.26878
\(960\) 3.43531 0.110874
\(961\) 24.2838 0.783349
\(962\) 36.6540 1.18177
\(963\) −12.8706 −0.414750
\(964\) 16.0809 0.517930
\(965\) 15.4942 0.498776
\(966\) −48.0514 −1.54603
\(967\) −12.5804 −0.404559 −0.202279 0.979328i \(-0.564835\pi\)
−0.202279 + 0.979328i \(0.564835\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −12.3270 −0.396000
\(970\) −11.1883 −0.359234
\(971\) −1.79067 −0.0574653 −0.0287327 0.999587i \(-0.509147\pi\)
−0.0287327 + 0.999587i \(0.509147\pi\)
\(972\) 15.1103 0.484664
\(973\) 9.24063 0.296241
\(974\) −15.2838 −0.489725
\(975\) 25.5540 0.818384
\(976\) 13.4931 0.431905
\(977\) 6.40586 0.204942 0.102471 0.994736i \(-0.467325\pi\)
0.102471 + 0.994736i \(0.467325\pi\)
\(978\) 4.04211 0.129252
\(979\) −3.20293 −0.102366
\(980\) −2.18081 −0.0696635
\(981\) 18.4648 0.589534
\(982\) −3.31965 −0.105934
\(983\) −20.5647 −0.655912 −0.327956 0.944693i \(-0.606360\pi\)
−0.327956 + 0.944693i \(0.606360\pi\)
\(984\) −4.41425 −0.140721
\(985\) 28.0147 0.892621
\(986\) 30.7328 0.978733
\(987\) 43.9619 1.39932
\(988\) −4.89167 −0.155625
\(989\) 82.8011 2.63292
\(990\) −2.56469 −0.0815113
\(991\) −29.0166 −0.921744 −0.460872 0.887467i \(-0.652463\pi\)
−0.460872 + 0.887467i \(0.652463\pi\)
\(992\) 7.43531 0.236071
\(993\) −68.8642 −2.18534
\(994\) −43.3142 −1.37384
\(995\) 40.4205 1.28142
\(996\) 13.1030 0.415184
\(997\) 42.3638 1.34167 0.670837 0.741605i \(-0.265935\pi\)
0.670837 + 0.741605i \(0.265935\pi\)
\(998\) 2.07355 0.0656370
\(999\) −22.4794 −0.711217
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 418.2.a.g.1.1 3
3.2 odd 2 3762.2.a.bg.1.2 3
4.3 odd 2 3344.2.a.q.1.3 3
11.10 odd 2 4598.2.a.bo.1.1 3
19.18 odd 2 7942.2.a.bi.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.1 3 1.1 even 1 trivial
3344.2.a.q.1.3 3 4.3 odd 2
3762.2.a.bg.1.2 3 3.2 odd 2
4598.2.a.bo.1.1 3 11.10 odd 2
7942.2.a.bi.1.3 3 19.18 odd 2