Properties

Label 3822.2.a.bj.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -0.414214 q^{11} -1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +2.41421 q^{17} -1.00000 q^{18} -1.82843 q^{19} +1.00000 q^{20} +0.414214 q^{22} +2.65685 q^{23} +1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} -8.65685 q^{29} +1.00000 q^{30} +4.24264 q^{31} -1.00000 q^{32} +0.414214 q^{33} -2.41421 q^{34} +1.00000 q^{36} +5.24264 q^{37} +1.82843 q^{38} +1.00000 q^{39} -1.00000 q^{40} +1.17157 q^{41} -7.00000 q^{43} -0.414214 q^{44} +1.00000 q^{45} -2.65685 q^{46} -3.65685 q^{47} -1.00000 q^{48} +4.00000 q^{50} -2.41421 q^{51} -1.00000 q^{52} -2.34315 q^{53} +1.00000 q^{54} -0.414214 q^{55} +1.82843 q^{57} +8.65685 q^{58} +9.89949 q^{59} -1.00000 q^{60} -8.41421 q^{61} -4.24264 q^{62} +1.00000 q^{64} -1.00000 q^{65} -0.414214 q^{66} -1.41421 q^{67} +2.41421 q^{68} -2.65685 q^{69} -9.07107 q^{71} -1.00000 q^{72} -6.17157 q^{73} -5.24264 q^{74} +4.00000 q^{75} -1.82843 q^{76} -1.00000 q^{78} +1.75736 q^{79} +1.00000 q^{80} +1.00000 q^{81} -1.17157 q^{82} +0.343146 q^{83} +2.41421 q^{85} +7.00000 q^{86} +8.65685 q^{87} +0.414214 q^{88} +3.41421 q^{89} -1.00000 q^{90} +2.65685 q^{92} -4.24264 q^{93} +3.65685 q^{94} -1.82843 q^{95} +1.00000 q^{96} -15.3137 q^{97} -0.414214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} - 2 q^{12} - 2 q^{13} - 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} + 2 q^{19} + 2 q^{20} - 2 q^{22} - 6 q^{23} + 2 q^{24} - 8 q^{25} + 2 q^{26} - 2 q^{27} - 6 q^{29} + 2 q^{30} - 2 q^{32} - 2 q^{33} - 2 q^{34} + 2 q^{36} + 2 q^{37} - 2 q^{38} + 2 q^{39} - 2 q^{40} + 8 q^{41} - 14 q^{43} + 2 q^{44} + 2 q^{45} + 6 q^{46} + 4 q^{47} - 2 q^{48} + 8 q^{50} - 2 q^{51} - 2 q^{52} - 16 q^{53} + 2 q^{54} + 2 q^{55} - 2 q^{57} + 6 q^{58} - 2 q^{60} - 14 q^{61} + 2 q^{64} - 2 q^{65} + 2 q^{66} + 2 q^{68} + 6 q^{69} - 4 q^{71} - 2 q^{72} - 18 q^{73} - 2 q^{74} + 8 q^{75} + 2 q^{76} - 2 q^{78} + 12 q^{79} + 2 q^{80} + 2 q^{81} - 8 q^{82} + 12 q^{83} + 2 q^{85} + 14 q^{86} + 6 q^{87} - 2 q^{88} + 4 q^{89} - 2 q^{90} - 6 q^{92} - 4 q^{94} + 2 q^{95} + 2 q^{96} - 8 q^{97} + 2 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −0.414214 −0.124890 −0.0624450 0.998048i \(-0.519890\pi\)
−0.0624450 + 0.998048i \(0.519890\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 2.41421 0.585533 0.292766 0.956184i \(-0.405424\pi\)
0.292766 + 0.956184i \(0.405424\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.82843 −0.419470 −0.209735 0.977758i \(-0.567260\pi\)
−0.209735 + 0.977758i \(0.567260\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 0.414214 0.0883106
\(23\) 2.65685 0.553992 0.276996 0.960871i \(-0.410661\pi\)
0.276996 + 0.960871i \(0.410661\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.00000 −0.800000
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −8.65685 −1.60754 −0.803769 0.594942i \(-0.797175\pi\)
−0.803769 + 0.594942i \(0.797175\pi\)
\(30\) 1.00000 0.182574
\(31\) 4.24264 0.762001 0.381000 0.924575i \(-0.375580\pi\)
0.381000 + 0.924575i \(0.375580\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.414214 0.0721053
\(34\) −2.41421 −0.414034
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 5.24264 0.861885 0.430942 0.902379i \(-0.358181\pi\)
0.430942 + 0.902379i \(0.358181\pi\)
\(38\) 1.82843 0.296610
\(39\) 1.00000 0.160128
\(40\) −1.00000 −0.158114
\(41\) 1.17157 0.182969 0.0914845 0.995807i \(-0.470839\pi\)
0.0914845 + 0.995807i \(0.470839\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) −0.414214 −0.0624450
\(45\) 1.00000 0.149071
\(46\) −2.65685 −0.391732
\(47\) −3.65685 −0.533407 −0.266704 0.963779i \(-0.585934\pi\)
−0.266704 + 0.963779i \(0.585934\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) −2.41421 −0.338058
\(52\) −1.00000 −0.138675
\(53\) −2.34315 −0.321856 −0.160928 0.986966i \(-0.551449\pi\)
−0.160928 + 0.986966i \(0.551449\pi\)
\(54\) 1.00000 0.136083
\(55\) −0.414214 −0.0558525
\(56\) 0 0
\(57\) 1.82843 0.242181
\(58\) 8.65685 1.13670
\(59\) 9.89949 1.28880 0.644402 0.764687i \(-0.277106\pi\)
0.644402 + 0.764687i \(0.277106\pi\)
\(60\) −1.00000 −0.129099
\(61\) −8.41421 −1.07733 −0.538665 0.842520i \(-0.681071\pi\)
−0.538665 + 0.842520i \(0.681071\pi\)
\(62\) −4.24264 −0.538816
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) −0.414214 −0.0509862
\(67\) −1.41421 −0.172774 −0.0863868 0.996262i \(-0.527532\pi\)
−0.0863868 + 0.996262i \(0.527532\pi\)
\(68\) 2.41421 0.292766
\(69\) −2.65685 −0.319848
\(70\) 0 0
\(71\) −9.07107 −1.07654 −0.538269 0.842773i \(-0.680921\pi\)
−0.538269 + 0.842773i \(0.680921\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.17157 −0.722328 −0.361164 0.932502i \(-0.617621\pi\)
−0.361164 + 0.932502i \(0.617621\pi\)
\(74\) −5.24264 −0.609445
\(75\) 4.00000 0.461880
\(76\) −1.82843 −0.209735
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) 1.75736 0.197718 0.0988592 0.995101i \(-0.468481\pi\)
0.0988592 + 0.995101i \(0.468481\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −1.17157 −0.129379
\(83\) 0.343146 0.0376651 0.0188326 0.999823i \(-0.494005\pi\)
0.0188326 + 0.999823i \(0.494005\pi\)
\(84\) 0 0
\(85\) 2.41421 0.261858
\(86\) 7.00000 0.754829
\(87\) 8.65685 0.928112
\(88\) 0.414214 0.0441553
\(89\) 3.41421 0.361906 0.180953 0.983492i \(-0.442082\pi\)
0.180953 + 0.983492i \(0.442082\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 2.65685 0.276996
\(93\) −4.24264 −0.439941
\(94\) 3.65685 0.377176
\(95\) −1.82843 −0.187593
\(96\) 1.00000 0.102062
\(97\) −15.3137 −1.55487 −0.777436 0.628962i \(-0.783480\pi\)
−0.777436 + 0.628962i \(0.783480\pi\)
\(98\) 0 0
\(99\) −0.414214 −0.0416300
\(100\) −4.00000 −0.400000
\(101\) 13.0711 1.30062 0.650310 0.759669i \(-0.274639\pi\)
0.650310 + 0.759669i \(0.274639\pi\)
\(102\) 2.41421 0.239043
\(103\) 6.07107 0.598200 0.299100 0.954222i \(-0.403314\pi\)
0.299100 + 0.954222i \(0.403314\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 2.34315 0.227586
\(107\) 7.41421 0.716759 0.358380 0.933576i \(-0.383329\pi\)
0.358380 + 0.933576i \(0.383329\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.41421 −0.231240 −0.115620 0.993294i \(-0.536885\pi\)
−0.115620 + 0.993294i \(0.536885\pi\)
\(110\) 0.414214 0.0394937
\(111\) −5.24264 −0.497609
\(112\) 0 0
\(113\) −9.75736 −0.917895 −0.458948 0.888463i \(-0.651773\pi\)
−0.458948 + 0.888463i \(0.651773\pi\)
\(114\) −1.82843 −0.171248
\(115\) 2.65685 0.247753
\(116\) −8.65685 −0.803769
\(117\) −1.00000 −0.0924500
\(118\) −9.89949 −0.911322
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −10.8284 −0.984402
\(122\) 8.41421 0.761787
\(123\) −1.17157 −0.105637
\(124\) 4.24264 0.381000
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −17.0711 −1.51481 −0.757406 0.652944i \(-0.773534\pi\)
−0.757406 + 0.652944i \(0.773534\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.00000 0.616316
\(130\) 1.00000 0.0877058
\(131\) 8.75736 0.765134 0.382567 0.923928i \(-0.375040\pi\)
0.382567 + 0.923928i \(0.375040\pi\)
\(132\) 0.414214 0.0360527
\(133\) 0 0
\(134\) 1.41421 0.122169
\(135\) −1.00000 −0.0860663
\(136\) −2.41421 −0.207017
\(137\) 1.24264 0.106166 0.0530830 0.998590i \(-0.483095\pi\)
0.0530830 + 0.998590i \(0.483095\pi\)
\(138\) 2.65685 0.226166
\(139\) −1.17157 −0.0993715 −0.0496858 0.998765i \(-0.515822\pi\)
−0.0496858 + 0.998765i \(0.515822\pi\)
\(140\) 0 0
\(141\) 3.65685 0.307963
\(142\) 9.07107 0.761227
\(143\) 0.414214 0.0346383
\(144\) 1.00000 0.0833333
\(145\) −8.65685 −0.718913
\(146\) 6.17157 0.510763
\(147\) 0 0
\(148\) 5.24264 0.430942
\(149\) 2.82843 0.231714 0.115857 0.993266i \(-0.463039\pi\)
0.115857 + 0.993266i \(0.463039\pi\)
\(150\) −4.00000 −0.326599
\(151\) −4.75736 −0.387148 −0.193574 0.981086i \(-0.562008\pi\)
−0.193574 + 0.981086i \(0.562008\pi\)
\(152\) 1.82843 0.148305
\(153\) 2.41421 0.195178
\(154\) 0 0
\(155\) 4.24264 0.340777
\(156\) 1.00000 0.0800641
\(157\) 16.8995 1.34873 0.674363 0.738400i \(-0.264418\pi\)
0.674363 + 0.738400i \(0.264418\pi\)
\(158\) −1.75736 −0.139808
\(159\) 2.34315 0.185824
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −0.343146 −0.0268772 −0.0134386 0.999910i \(-0.504278\pi\)
−0.0134386 + 0.999910i \(0.504278\pi\)
\(164\) 1.17157 0.0914845
\(165\) 0.414214 0.0322465
\(166\) −0.343146 −0.0266333
\(167\) 8.17157 0.632335 0.316168 0.948703i \(-0.397604\pi\)
0.316168 + 0.948703i \(0.397604\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −2.41421 −0.185162
\(171\) −1.82843 −0.139823
\(172\) −7.00000 −0.533745
\(173\) 7.75736 0.589781 0.294891 0.955531i \(-0.404717\pi\)
0.294891 + 0.955531i \(0.404717\pi\)
\(174\) −8.65685 −0.656274
\(175\) 0 0
\(176\) −0.414214 −0.0312225
\(177\) −9.89949 −0.744092
\(178\) −3.41421 −0.255906
\(179\) 11.5563 0.863762 0.431881 0.901931i \(-0.357850\pi\)
0.431881 + 0.901931i \(0.357850\pi\)
\(180\) 1.00000 0.0745356
\(181\) −10.1421 −0.753859 −0.376930 0.926242i \(-0.623020\pi\)
−0.376930 + 0.926242i \(0.623020\pi\)
\(182\) 0 0
\(183\) 8.41421 0.621997
\(184\) −2.65685 −0.195866
\(185\) 5.24264 0.385447
\(186\) 4.24264 0.311086
\(187\) −1.00000 −0.0731272
\(188\) −3.65685 −0.266704
\(189\) 0 0
\(190\) 1.82843 0.132648
\(191\) −11.4853 −0.831046 −0.415523 0.909583i \(-0.636401\pi\)
−0.415523 + 0.909583i \(0.636401\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −24.3848 −1.75525 −0.877627 0.479344i \(-0.840875\pi\)
−0.877627 + 0.479344i \(0.840875\pi\)
\(194\) 15.3137 1.09946
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −9.07107 −0.646287 −0.323143 0.946350i \(-0.604740\pi\)
−0.323143 + 0.946350i \(0.604740\pi\)
\(198\) 0.414214 0.0294369
\(199\) −4.07107 −0.288590 −0.144295 0.989535i \(-0.546091\pi\)
−0.144295 + 0.989535i \(0.546091\pi\)
\(200\) 4.00000 0.282843
\(201\) 1.41421 0.0997509
\(202\) −13.0711 −0.919677
\(203\) 0 0
\(204\) −2.41421 −0.169029
\(205\) 1.17157 0.0818262
\(206\) −6.07107 −0.422991
\(207\) 2.65685 0.184664
\(208\) −1.00000 −0.0693375
\(209\) 0.757359 0.0523876
\(210\) 0 0
\(211\) 6.31371 0.434654 0.217327 0.976099i \(-0.430266\pi\)
0.217327 + 0.976099i \(0.430266\pi\)
\(212\) −2.34315 −0.160928
\(213\) 9.07107 0.621539
\(214\) −7.41421 −0.506825
\(215\) −7.00000 −0.477396
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 2.41421 0.163511
\(219\) 6.17157 0.417036
\(220\) −0.414214 −0.0279263
\(221\) −2.41421 −0.162398
\(222\) 5.24264 0.351863
\(223\) −4.34315 −0.290839 −0.145419 0.989370i \(-0.546453\pi\)
−0.145419 + 0.989370i \(0.546453\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 9.75736 0.649050
\(227\) 9.07107 0.602068 0.301034 0.953613i \(-0.402668\pi\)
0.301034 + 0.953613i \(0.402668\pi\)
\(228\) 1.82843 0.121091
\(229\) −4.10051 −0.270969 −0.135485 0.990779i \(-0.543259\pi\)
−0.135485 + 0.990779i \(0.543259\pi\)
\(230\) −2.65685 −0.175188
\(231\) 0 0
\(232\) 8.65685 0.568350
\(233\) −5.17157 −0.338801 −0.169401 0.985547i \(-0.554183\pi\)
−0.169401 + 0.985547i \(0.554183\pi\)
\(234\) 1.00000 0.0653720
\(235\) −3.65685 −0.238547
\(236\) 9.89949 0.644402
\(237\) −1.75736 −0.114153
\(238\) 0 0
\(239\) −10.7279 −0.693932 −0.346966 0.937878i \(-0.612788\pi\)
−0.346966 + 0.937878i \(0.612788\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −23.6569 −1.52387 −0.761936 0.647652i \(-0.775751\pi\)
−0.761936 + 0.647652i \(0.775751\pi\)
\(242\) 10.8284 0.696078
\(243\) −1.00000 −0.0641500
\(244\) −8.41421 −0.538665
\(245\) 0 0
\(246\) 1.17157 0.0746968
\(247\) 1.82843 0.116340
\(248\) −4.24264 −0.269408
\(249\) −0.343146 −0.0217460
\(250\) 9.00000 0.569210
\(251\) 6.41421 0.404862 0.202431 0.979297i \(-0.435116\pi\)
0.202431 + 0.979297i \(0.435116\pi\)
\(252\) 0 0
\(253\) −1.10051 −0.0691882
\(254\) 17.0711 1.07113
\(255\) −2.41421 −0.151184
\(256\) 1.00000 0.0625000
\(257\) 10.8284 0.675459 0.337729 0.941243i \(-0.390341\pi\)
0.337729 + 0.941243i \(0.390341\pi\)
\(258\) −7.00000 −0.435801
\(259\) 0 0
\(260\) −1.00000 −0.0620174
\(261\) −8.65685 −0.535846
\(262\) −8.75736 −0.541031
\(263\) −1.65685 −0.102166 −0.0510830 0.998694i \(-0.516267\pi\)
−0.0510830 + 0.998694i \(0.516267\pi\)
\(264\) −0.414214 −0.0254931
\(265\) −2.34315 −0.143938
\(266\) 0 0
\(267\) −3.41421 −0.208946
\(268\) −1.41421 −0.0863868
\(269\) 8.72792 0.532151 0.266075 0.963952i \(-0.414273\pi\)
0.266075 + 0.963952i \(0.414273\pi\)
\(270\) 1.00000 0.0608581
\(271\) −5.75736 −0.349735 −0.174867 0.984592i \(-0.555950\pi\)
−0.174867 + 0.984592i \(0.555950\pi\)
\(272\) 2.41421 0.146383
\(273\) 0 0
\(274\) −1.24264 −0.0750707
\(275\) 1.65685 0.0999121
\(276\) −2.65685 −0.159924
\(277\) −19.4142 −1.16649 −0.583244 0.812297i \(-0.698217\pi\)
−0.583244 + 0.812297i \(0.698217\pi\)
\(278\) 1.17157 0.0702663
\(279\) 4.24264 0.254000
\(280\) 0 0
\(281\) −21.3137 −1.27147 −0.635735 0.771908i \(-0.719303\pi\)
−0.635735 + 0.771908i \(0.719303\pi\)
\(282\) −3.65685 −0.217763
\(283\) −5.17157 −0.307418 −0.153709 0.988116i \(-0.549122\pi\)
−0.153709 + 0.988116i \(0.549122\pi\)
\(284\) −9.07107 −0.538269
\(285\) 1.82843 0.108307
\(286\) −0.414214 −0.0244930
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −11.1716 −0.657151
\(290\) 8.65685 0.508348
\(291\) 15.3137 0.897705
\(292\) −6.17157 −0.361164
\(293\) −22.1421 −1.29356 −0.646779 0.762678i \(-0.723884\pi\)
−0.646779 + 0.762678i \(0.723884\pi\)
\(294\) 0 0
\(295\) 9.89949 0.576371
\(296\) −5.24264 −0.304722
\(297\) 0.414214 0.0240351
\(298\) −2.82843 −0.163846
\(299\) −2.65685 −0.153650
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) 4.75736 0.273755
\(303\) −13.0711 −0.750913
\(304\) −1.82843 −0.104867
\(305\) −8.41421 −0.481796
\(306\) −2.41421 −0.138011
\(307\) −0.343146 −0.0195844 −0.00979218 0.999952i \(-0.503117\pi\)
−0.00979218 + 0.999952i \(0.503117\pi\)
\(308\) 0 0
\(309\) −6.07107 −0.345371
\(310\) −4.24264 −0.240966
\(311\) −17.7990 −1.00929 −0.504644 0.863328i \(-0.668376\pi\)
−0.504644 + 0.863328i \(0.668376\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −6.92893 −0.391646 −0.195823 0.980639i \(-0.562738\pi\)
−0.195823 + 0.980639i \(0.562738\pi\)
\(314\) −16.8995 −0.953694
\(315\) 0 0
\(316\) 1.75736 0.0988592
\(317\) −30.0416 −1.68731 −0.843653 0.536889i \(-0.819599\pi\)
−0.843653 + 0.536889i \(0.819599\pi\)
\(318\) −2.34315 −0.131397
\(319\) 3.58579 0.200765
\(320\) 1.00000 0.0559017
\(321\) −7.41421 −0.413821
\(322\) 0 0
\(323\) −4.41421 −0.245613
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) 0.343146 0.0190051
\(327\) 2.41421 0.133506
\(328\) −1.17157 −0.0646893
\(329\) 0 0
\(330\) −0.414214 −0.0228017
\(331\) −10.1005 −0.555174 −0.277587 0.960701i \(-0.589535\pi\)
−0.277587 + 0.960701i \(0.589535\pi\)
\(332\) 0.343146 0.0188326
\(333\) 5.24264 0.287295
\(334\) −8.17157 −0.447129
\(335\) −1.41421 −0.0772667
\(336\) 0 0
\(337\) −0.514719 −0.0280385 −0.0140193 0.999902i \(-0.504463\pi\)
−0.0140193 + 0.999902i \(0.504463\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 9.75736 0.529947
\(340\) 2.41421 0.130929
\(341\) −1.75736 −0.0951663
\(342\) 1.82843 0.0988700
\(343\) 0 0
\(344\) 7.00000 0.377415
\(345\) −2.65685 −0.143040
\(346\) −7.75736 −0.417038
\(347\) −25.4142 −1.36431 −0.682153 0.731209i \(-0.738956\pi\)
−0.682153 + 0.731209i \(0.738956\pi\)
\(348\) 8.65685 0.464056
\(349\) −10.7279 −0.574253 −0.287126 0.957893i \(-0.592700\pi\)
−0.287126 + 0.957893i \(0.592700\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0.414214 0.0220777
\(353\) −31.4558 −1.67423 −0.837113 0.547030i \(-0.815758\pi\)
−0.837113 + 0.547030i \(0.815758\pi\)
\(354\) 9.89949 0.526152
\(355\) −9.07107 −0.481442
\(356\) 3.41421 0.180953
\(357\) 0 0
\(358\) −11.5563 −0.610772
\(359\) 21.4558 1.13240 0.566198 0.824269i \(-0.308414\pi\)
0.566198 + 0.824269i \(0.308414\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −15.6569 −0.824045
\(362\) 10.1421 0.533059
\(363\) 10.8284 0.568345
\(364\) 0 0
\(365\) −6.17157 −0.323035
\(366\) −8.41421 −0.439818
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 2.65685 0.138498
\(369\) 1.17157 0.0609896
\(370\) −5.24264 −0.272552
\(371\) 0 0
\(372\) −4.24264 −0.219971
\(373\) −4.24264 −0.219676 −0.109838 0.993950i \(-0.535033\pi\)
−0.109838 + 0.993950i \(0.535033\pi\)
\(374\) 1.00000 0.0517088
\(375\) 9.00000 0.464758
\(376\) 3.65685 0.188588
\(377\) 8.65685 0.445851
\(378\) 0 0
\(379\) 12.3848 0.636163 0.318082 0.948063i \(-0.396961\pi\)
0.318082 + 0.948063i \(0.396961\pi\)
\(380\) −1.82843 −0.0937963
\(381\) 17.0711 0.874577
\(382\) 11.4853 0.587638
\(383\) −22.6569 −1.15771 −0.578856 0.815430i \(-0.696501\pi\)
−0.578856 + 0.815430i \(0.696501\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 24.3848 1.24115
\(387\) −7.00000 −0.355830
\(388\) −15.3137 −0.777436
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) −1.00000 −0.0506370
\(391\) 6.41421 0.324381
\(392\) 0 0
\(393\) −8.75736 −0.441750
\(394\) 9.07107 0.456994
\(395\) 1.75736 0.0884223
\(396\) −0.414214 −0.0208150
\(397\) −29.3137 −1.47121 −0.735606 0.677409i \(-0.763103\pi\)
−0.735606 + 0.677409i \(0.763103\pi\)
\(398\) 4.07107 0.204064
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 29.4558 1.47095 0.735477 0.677549i \(-0.236958\pi\)
0.735477 + 0.677549i \(0.236958\pi\)
\(402\) −1.41421 −0.0705346
\(403\) −4.24264 −0.211341
\(404\) 13.0711 0.650310
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −2.17157 −0.107641
\(408\) 2.41421 0.119521
\(409\) −0.0294373 −0.00145558 −0.000727789 1.00000i \(-0.500232\pi\)
−0.000727789 1.00000i \(0.500232\pi\)
\(410\) −1.17157 −0.0578599
\(411\) −1.24264 −0.0612949
\(412\) 6.07107 0.299100
\(413\) 0 0
\(414\) −2.65685 −0.130577
\(415\) 0.343146 0.0168444
\(416\) 1.00000 0.0490290
\(417\) 1.17157 0.0573722
\(418\) −0.757359 −0.0370437
\(419\) 20.5563 1.00424 0.502122 0.864797i \(-0.332553\pi\)
0.502122 + 0.864797i \(0.332553\pi\)
\(420\) 0 0
\(421\) 29.1127 1.41887 0.709433 0.704773i \(-0.248951\pi\)
0.709433 + 0.704773i \(0.248951\pi\)
\(422\) −6.31371 −0.307347
\(423\) −3.65685 −0.177802
\(424\) 2.34315 0.113793
\(425\) −9.65685 −0.468426
\(426\) −9.07107 −0.439495
\(427\) 0 0
\(428\) 7.41421 0.358380
\(429\) −0.414214 −0.0199984
\(430\) 7.00000 0.337570
\(431\) 20.8284 1.00327 0.501635 0.865079i \(-0.332732\pi\)
0.501635 + 0.865079i \(0.332732\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 18.6274 0.895177 0.447588 0.894240i \(-0.352283\pi\)
0.447588 + 0.894240i \(0.352283\pi\)
\(434\) 0 0
\(435\) 8.65685 0.415064
\(436\) −2.41421 −0.115620
\(437\) −4.85786 −0.232383
\(438\) −6.17157 −0.294889
\(439\) −4.21320 −0.201085 −0.100543 0.994933i \(-0.532058\pi\)
−0.100543 + 0.994933i \(0.532058\pi\)
\(440\) 0.414214 0.0197469
\(441\) 0 0
\(442\) 2.41421 0.114832
\(443\) −4.82843 −0.229405 −0.114703 0.993400i \(-0.536592\pi\)
−0.114703 + 0.993400i \(0.536592\pi\)
\(444\) −5.24264 −0.248805
\(445\) 3.41421 0.161849
\(446\) 4.34315 0.205654
\(447\) −2.82843 −0.133780
\(448\) 0 0
\(449\) −12.5563 −0.592571 −0.296285 0.955099i \(-0.595748\pi\)
−0.296285 + 0.955099i \(0.595748\pi\)
\(450\) 4.00000 0.188562
\(451\) −0.485281 −0.0228510
\(452\) −9.75736 −0.458948
\(453\) 4.75736 0.223520
\(454\) −9.07107 −0.425726
\(455\) 0 0
\(456\) −1.82843 −0.0856239
\(457\) −13.7574 −0.643542 −0.321771 0.946818i \(-0.604278\pi\)
−0.321771 + 0.946818i \(0.604278\pi\)
\(458\) 4.10051 0.191604
\(459\) −2.41421 −0.112686
\(460\) 2.65685 0.123876
\(461\) −6.51472 −0.303421 −0.151710 0.988425i \(-0.548478\pi\)
−0.151710 + 0.988425i \(0.548478\pi\)
\(462\) 0 0
\(463\) −21.8701 −1.01639 −0.508194 0.861243i \(-0.669687\pi\)
−0.508194 + 0.861243i \(0.669687\pi\)
\(464\) −8.65685 −0.401884
\(465\) −4.24264 −0.196748
\(466\) 5.17157 0.239568
\(467\) 28.8995 1.33731 0.668655 0.743573i \(-0.266871\pi\)
0.668655 + 0.743573i \(0.266871\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 3.65685 0.168678
\(471\) −16.8995 −0.778688
\(472\) −9.89949 −0.455661
\(473\) 2.89949 0.133319
\(474\) 1.75736 0.0807182
\(475\) 7.31371 0.335576
\(476\) 0 0
\(477\) −2.34315 −0.107285
\(478\) 10.7279 0.490684
\(479\) 7.48528 0.342011 0.171006 0.985270i \(-0.445298\pi\)
0.171006 + 0.985270i \(0.445298\pi\)
\(480\) 1.00000 0.0456435
\(481\) −5.24264 −0.239044
\(482\) 23.6569 1.07754
\(483\) 0 0
\(484\) −10.8284 −0.492201
\(485\) −15.3137 −0.695360
\(486\) 1.00000 0.0453609
\(487\) −41.5980 −1.88498 −0.942492 0.334228i \(-0.891524\pi\)
−0.942492 + 0.334228i \(0.891524\pi\)
\(488\) 8.41421 0.380894
\(489\) 0.343146 0.0155176
\(490\) 0 0
\(491\) −38.7696 −1.74965 −0.874823 0.484443i \(-0.839022\pi\)
−0.874823 + 0.484443i \(0.839022\pi\)
\(492\) −1.17157 −0.0528186
\(493\) −20.8995 −0.941266
\(494\) −1.82843 −0.0822648
\(495\) −0.414214 −0.0186175
\(496\) 4.24264 0.190500
\(497\) 0 0
\(498\) 0.343146 0.0153767
\(499\) −2.72792 −0.122119 −0.0610593 0.998134i \(-0.519448\pi\)
−0.0610593 + 0.998134i \(0.519448\pi\)
\(500\) −9.00000 −0.402492
\(501\) −8.17157 −0.365079
\(502\) −6.41421 −0.286280
\(503\) 3.07107 0.136932 0.0684661 0.997653i \(-0.478190\pi\)
0.0684661 + 0.997653i \(0.478190\pi\)
\(504\) 0 0
\(505\) 13.0711 0.581655
\(506\) 1.10051 0.0489234
\(507\) −1.00000 −0.0444116
\(508\) −17.0711 −0.757406
\(509\) 23.1421 1.02576 0.512879 0.858461i \(-0.328579\pi\)
0.512879 + 0.858461i \(0.328579\pi\)
\(510\) 2.41421 0.106903
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 1.82843 0.0807270
\(514\) −10.8284 −0.477621
\(515\) 6.07107 0.267523
\(516\) 7.00000 0.308158
\(517\) 1.51472 0.0666172
\(518\) 0 0
\(519\) −7.75736 −0.340510
\(520\) 1.00000 0.0438529
\(521\) 20.5563 0.900590 0.450295 0.892880i \(-0.351319\pi\)
0.450295 + 0.892880i \(0.351319\pi\)
\(522\) 8.65685 0.378900
\(523\) −30.7279 −1.34364 −0.671819 0.740715i \(-0.734487\pi\)
−0.671819 + 0.740715i \(0.734487\pi\)
\(524\) 8.75736 0.382567
\(525\) 0 0
\(526\) 1.65685 0.0722423
\(527\) 10.2426 0.446176
\(528\) 0.414214 0.0180263
\(529\) −15.9411 −0.693092
\(530\) 2.34315 0.101780
\(531\) 9.89949 0.429601
\(532\) 0 0
\(533\) −1.17157 −0.0507465
\(534\) 3.41421 0.147747
\(535\) 7.41421 0.320544
\(536\) 1.41421 0.0610847
\(537\) −11.5563 −0.498693
\(538\) −8.72792 −0.376287
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) 11.3848 0.489470 0.244735 0.969590i \(-0.421299\pi\)
0.244735 + 0.969590i \(0.421299\pi\)
\(542\) 5.75736 0.247300
\(543\) 10.1421 0.435241
\(544\) −2.41421 −0.103509
\(545\) −2.41421 −0.103414
\(546\) 0 0
\(547\) 13.7990 0.590002 0.295001 0.955497i \(-0.404680\pi\)
0.295001 + 0.955497i \(0.404680\pi\)
\(548\) 1.24264 0.0530830
\(549\) −8.41421 −0.359110
\(550\) −1.65685 −0.0706485
\(551\) 15.8284 0.674314
\(552\) 2.65685 0.113083
\(553\) 0 0
\(554\) 19.4142 0.824831
\(555\) −5.24264 −0.222538
\(556\) −1.17157 −0.0496858
\(557\) 8.48528 0.359533 0.179766 0.983709i \(-0.442466\pi\)
0.179766 + 0.983709i \(0.442466\pi\)
\(558\) −4.24264 −0.179605
\(559\) 7.00000 0.296068
\(560\) 0 0
\(561\) 1.00000 0.0422200
\(562\) 21.3137 0.899065
\(563\) 5.58579 0.235413 0.117706 0.993048i \(-0.462446\pi\)
0.117706 + 0.993048i \(0.462446\pi\)
\(564\) 3.65685 0.153981
\(565\) −9.75736 −0.410495
\(566\) 5.17157 0.217377
\(567\) 0 0
\(568\) 9.07107 0.380614
\(569\) −1.51472 −0.0635003 −0.0317502 0.999496i \(-0.510108\pi\)
−0.0317502 + 0.999496i \(0.510108\pi\)
\(570\) −1.82843 −0.0765844
\(571\) −26.2843 −1.09996 −0.549981 0.835177i \(-0.685365\pi\)
−0.549981 + 0.835177i \(0.685365\pi\)
\(572\) 0.414214 0.0173191
\(573\) 11.4853 0.479805
\(574\) 0 0
\(575\) −10.6274 −0.443194
\(576\) 1.00000 0.0416667
\(577\) 26.8284 1.11688 0.558441 0.829544i \(-0.311400\pi\)
0.558441 + 0.829544i \(0.311400\pi\)
\(578\) 11.1716 0.464676
\(579\) 24.3848 1.01340
\(580\) −8.65685 −0.359456
\(581\) 0 0
\(582\) −15.3137 −0.634774
\(583\) 0.970563 0.0401966
\(584\) 6.17157 0.255382
\(585\) −1.00000 −0.0413449
\(586\) 22.1421 0.914683
\(587\) 25.1127 1.03651 0.518256 0.855226i \(-0.326581\pi\)
0.518256 + 0.855226i \(0.326581\pi\)
\(588\) 0 0
\(589\) −7.75736 −0.319636
\(590\) −9.89949 −0.407556
\(591\) 9.07107 0.373134
\(592\) 5.24264 0.215471
\(593\) 4.58579 0.188316 0.0941578 0.995557i \(-0.469984\pi\)
0.0941578 + 0.995557i \(0.469984\pi\)
\(594\) −0.414214 −0.0169954
\(595\) 0 0
\(596\) 2.82843 0.115857
\(597\) 4.07107 0.166618
\(598\) 2.65685 0.108647
\(599\) 37.4853 1.53161 0.765804 0.643075i \(-0.222341\pi\)
0.765804 + 0.643075i \(0.222341\pi\)
\(600\) −4.00000 −0.163299
\(601\) 21.4558 0.875202 0.437601 0.899169i \(-0.355828\pi\)
0.437601 + 0.899169i \(0.355828\pi\)
\(602\) 0 0
\(603\) −1.41421 −0.0575912
\(604\) −4.75736 −0.193574
\(605\) −10.8284 −0.440238
\(606\) 13.0711 0.530976
\(607\) −41.1838 −1.67160 −0.835799 0.549036i \(-0.814995\pi\)
−0.835799 + 0.549036i \(0.814995\pi\)
\(608\) 1.82843 0.0741525
\(609\) 0 0
\(610\) 8.41421 0.340682
\(611\) 3.65685 0.147940
\(612\) 2.41421 0.0975888
\(613\) −8.07107 −0.325987 −0.162994 0.986627i \(-0.552115\pi\)
−0.162994 + 0.986627i \(0.552115\pi\)
\(614\) 0.343146 0.0138482
\(615\) −1.17157 −0.0472424
\(616\) 0 0
\(617\) −28.4142 −1.14391 −0.571957 0.820284i \(-0.693815\pi\)
−0.571957 + 0.820284i \(0.693815\pi\)
\(618\) 6.07107 0.244214
\(619\) −35.2843 −1.41819 −0.709097 0.705111i \(-0.750897\pi\)
−0.709097 + 0.705111i \(0.750897\pi\)
\(620\) 4.24264 0.170389
\(621\) −2.65685 −0.106616
\(622\) 17.7990 0.713674
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 11.0000 0.440000
\(626\) 6.92893 0.276936
\(627\) −0.757359 −0.0302460
\(628\) 16.8995 0.674363
\(629\) 12.6569 0.504662
\(630\) 0 0
\(631\) 37.3848 1.48826 0.744132 0.668032i \(-0.232863\pi\)
0.744132 + 0.668032i \(0.232863\pi\)
\(632\) −1.75736 −0.0699040
\(633\) −6.31371 −0.250947
\(634\) 30.0416 1.19311
\(635\) −17.0711 −0.677445
\(636\) 2.34315 0.0929118
\(637\) 0 0
\(638\) −3.58579 −0.141963
\(639\) −9.07107 −0.358846
\(640\) −1.00000 −0.0395285
\(641\) 22.5269 0.889760 0.444880 0.895590i \(-0.353246\pi\)
0.444880 + 0.895590i \(0.353246\pi\)
\(642\) 7.41421 0.292616
\(643\) 38.3137 1.51095 0.755473 0.655180i \(-0.227407\pi\)
0.755473 + 0.655180i \(0.227407\pi\)
\(644\) 0 0
\(645\) 7.00000 0.275625
\(646\) 4.41421 0.173675
\(647\) −44.9706 −1.76798 −0.883988 0.467510i \(-0.845151\pi\)
−0.883988 + 0.467510i \(0.845151\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −4.10051 −0.160959
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) −0.343146 −0.0134386
\(653\) −4.51472 −0.176675 −0.0883373 0.996091i \(-0.528155\pi\)
−0.0883373 + 0.996091i \(0.528155\pi\)
\(654\) −2.41421 −0.0944032
\(655\) 8.75736 0.342178
\(656\) 1.17157 0.0457422
\(657\) −6.17157 −0.240776
\(658\) 0 0
\(659\) 39.4558 1.53698 0.768491 0.639861i \(-0.221008\pi\)
0.768491 + 0.639861i \(0.221008\pi\)
\(660\) 0.414214 0.0161232
\(661\) 1.02944 0.0400405 0.0200202 0.999800i \(-0.493627\pi\)
0.0200202 + 0.999800i \(0.493627\pi\)
\(662\) 10.1005 0.392567
\(663\) 2.41421 0.0937603
\(664\) −0.343146 −0.0133166
\(665\) 0 0
\(666\) −5.24264 −0.203148
\(667\) −23.0000 −0.890564
\(668\) 8.17157 0.316168
\(669\) 4.34315 0.167916
\(670\) 1.41421 0.0546358
\(671\) 3.48528 0.134548
\(672\) 0 0
\(673\) −24.1127 −0.929476 −0.464738 0.885448i \(-0.653852\pi\)
−0.464738 + 0.885448i \(0.653852\pi\)
\(674\) 0.514719 0.0198262
\(675\) 4.00000 0.153960
\(676\) 1.00000 0.0384615
\(677\) −34.7696 −1.33630 −0.668151 0.744025i \(-0.732914\pi\)
−0.668151 + 0.744025i \(0.732914\pi\)
\(678\) −9.75736 −0.374729
\(679\) 0 0
\(680\) −2.41421 −0.0925809
\(681\) −9.07107 −0.347604
\(682\) 1.75736 0.0672928
\(683\) 26.4142 1.01071 0.505356 0.862911i \(-0.331361\pi\)
0.505356 + 0.862911i \(0.331361\pi\)
\(684\) −1.82843 −0.0699117
\(685\) 1.24264 0.0474789
\(686\) 0 0
\(687\) 4.10051 0.156444
\(688\) −7.00000 −0.266872
\(689\) 2.34315 0.0892667
\(690\) 2.65685 0.101145
\(691\) −28.8284 −1.09669 −0.548343 0.836254i \(-0.684741\pi\)
−0.548343 + 0.836254i \(0.684741\pi\)
\(692\) 7.75736 0.294891
\(693\) 0 0
\(694\) 25.4142 0.964710
\(695\) −1.17157 −0.0444403
\(696\) −8.65685 −0.328137
\(697\) 2.82843 0.107134
\(698\) 10.7279 0.406058
\(699\) 5.17157 0.195607
\(700\) 0 0
\(701\) 29.1716 1.10180 0.550898 0.834573i \(-0.314285\pi\)
0.550898 + 0.834573i \(0.314285\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −9.58579 −0.361535
\(704\) −0.414214 −0.0156113
\(705\) 3.65685 0.137725
\(706\) 31.4558 1.18386
\(707\) 0 0
\(708\) −9.89949 −0.372046
\(709\) 35.1716 1.32090 0.660448 0.750872i \(-0.270366\pi\)
0.660448 + 0.750872i \(0.270366\pi\)
\(710\) 9.07107 0.340431
\(711\) 1.75736 0.0659061
\(712\) −3.41421 −0.127953
\(713\) 11.2721 0.422143
\(714\) 0 0
\(715\) 0.414214 0.0154907
\(716\) 11.5563 0.431881
\(717\) 10.7279 0.400642
\(718\) −21.4558 −0.800725
\(719\) 13.4142 0.500266 0.250133 0.968212i \(-0.419526\pi\)
0.250133 + 0.968212i \(0.419526\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) 15.6569 0.582688
\(723\) 23.6569 0.879808
\(724\) −10.1421 −0.376930
\(725\) 34.6274 1.28603
\(726\) −10.8284 −0.401881
\(727\) 50.2132 1.86230 0.931152 0.364630i \(-0.118805\pi\)
0.931152 + 0.364630i \(0.118805\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.17157 0.228420
\(731\) −16.8995 −0.625050
\(732\) 8.41421 0.310998
\(733\) 40.7279 1.50432 0.752160 0.658980i \(-0.229012\pi\)
0.752160 + 0.658980i \(0.229012\pi\)
\(734\) −4.00000 −0.147643
\(735\) 0 0
\(736\) −2.65685 −0.0979329
\(737\) 0.585786 0.0215777
\(738\) −1.17157 −0.0431262
\(739\) −22.2843 −0.819740 −0.409870 0.912144i \(-0.634426\pi\)
−0.409870 + 0.912144i \(0.634426\pi\)
\(740\) 5.24264 0.192723
\(741\) −1.82843 −0.0671689
\(742\) 0 0
\(743\) 7.37258 0.270474 0.135237 0.990813i \(-0.456820\pi\)
0.135237 + 0.990813i \(0.456820\pi\)
\(744\) 4.24264 0.155543
\(745\) 2.82843 0.103626
\(746\) 4.24264 0.155334
\(747\) 0.343146 0.0125550
\(748\) −1.00000 −0.0365636
\(749\) 0 0
\(750\) −9.00000 −0.328634
\(751\) 47.5563 1.73536 0.867678 0.497127i \(-0.165612\pi\)
0.867678 + 0.497127i \(0.165612\pi\)
\(752\) −3.65685 −0.133352
\(753\) −6.41421 −0.233747
\(754\) −8.65685 −0.315264
\(755\) −4.75736 −0.173138
\(756\) 0 0
\(757\) 9.07107 0.329694 0.164847 0.986319i \(-0.447287\pi\)
0.164847 + 0.986319i \(0.447287\pi\)
\(758\) −12.3848 −0.449835
\(759\) 1.10051 0.0399458
\(760\) 1.82843 0.0663240
\(761\) 1.17157 0.0424695 0.0212347 0.999775i \(-0.493240\pi\)
0.0212347 + 0.999775i \(0.493240\pi\)
\(762\) −17.0711 −0.618420
\(763\) 0 0
\(764\) −11.4853 −0.415523
\(765\) 2.41421 0.0872861
\(766\) 22.6569 0.818625
\(767\) −9.89949 −0.357450
\(768\) −1.00000 −0.0360844
\(769\) −29.1421 −1.05089 −0.525446 0.850827i \(-0.676102\pi\)
−0.525446 + 0.850827i \(0.676102\pi\)
\(770\) 0 0
\(771\) −10.8284 −0.389976
\(772\) −24.3848 −0.877627
\(773\) −41.9706 −1.50958 −0.754788 0.655969i \(-0.772260\pi\)
−0.754788 + 0.655969i \(0.772260\pi\)
\(774\) 7.00000 0.251610
\(775\) −16.9706 −0.609601
\(776\) 15.3137 0.549730
\(777\) 0 0
\(778\) 8.00000 0.286814
\(779\) −2.14214 −0.0767500
\(780\) 1.00000 0.0358057
\(781\) 3.75736 0.134449
\(782\) −6.41421 −0.229372
\(783\) 8.65685 0.309371
\(784\) 0 0
\(785\) 16.8995 0.603169
\(786\) 8.75736 0.312365
\(787\) −15.8284 −0.564222 −0.282111 0.959382i \(-0.591035\pi\)
−0.282111 + 0.959382i \(0.591035\pi\)
\(788\) −9.07107 −0.323143
\(789\) 1.65685 0.0589856
\(790\) −1.75736 −0.0625240
\(791\) 0 0
\(792\) 0.414214 0.0147184
\(793\) 8.41421 0.298797
\(794\) 29.3137 1.04030
\(795\) 2.34315 0.0831028
\(796\) −4.07107 −0.144295
\(797\) −18.7279 −0.663377 −0.331689 0.943389i \(-0.607618\pi\)
−0.331689 + 0.943389i \(0.607618\pi\)
\(798\) 0 0
\(799\) −8.82843 −0.312327
\(800\) 4.00000 0.141421
\(801\) 3.41421 0.120635
\(802\) −29.4558 −1.04012
\(803\) 2.55635 0.0902116
\(804\) 1.41421 0.0498755
\(805\) 0 0
\(806\) 4.24264 0.149441
\(807\) −8.72792 −0.307237
\(808\) −13.0711 −0.459839
\(809\) 17.5563 0.617248 0.308624 0.951184i \(-0.400131\pi\)
0.308624 + 0.951184i \(0.400131\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 32.6569 1.14674 0.573369 0.819298i \(-0.305636\pi\)
0.573369 + 0.819298i \(0.305636\pi\)
\(812\) 0 0
\(813\) 5.75736 0.201919
\(814\) 2.17157 0.0761136
\(815\) −0.343146 −0.0120199
\(816\) −2.41421 −0.0845144
\(817\) 12.7990 0.447780
\(818\) 0.0294373 0.00102925
\(819\) 0 0
\(820\) 1.17157 0.0409131
\(821\) 11.5147 0.401866 0.200933 0.979605i \(-0.435603\pi\)
0.200933 + 0.979605i \(0.435603\pi\)
\(822\) 1.24264 0.0433421
\(823\) 44.5269 1.55211 0.776055 0.630665i \(-0.217218\pi\)
0.776055 + 0.630665i \(0.217218\pi\)
\(824\) −6.07107 −0.211496
\(825\) −1.65685 −0.0576843
\(826\) 0 0
\(827\) −37.5269 −1.30494 −0.652469 0.757815i \(-0.726267\pi\)
−0.652469 + 0.757815i \(0.726267\pi\)
\(828\) 2.65685 0.0923321
\(829\) 13.5269 0.469809 0.234905 0.972018i \(-0.424522\pi\)
0.234905 + 0.972018i \(0.424522\pi\)
\(830\) −0.343146 −0.0119108
\(831\) 19.4142 0.673472
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −1.17157 −0.0405683
\(835\) 8.17157 0.282789
\(836\) 0.757359 0.0261938
\(837\) −4.24264 −0.146647
\(838\) −20.5563 −0.710107
\(839\) −11.3726 −0.392625 −0.196313 0.980541i \(-0.562897\pi\)
−0.196313 + 0.980541i \(0.562897\pi\)
\(840\) 0 0
\(841\) 45.9411 1.58418
\(842\) −29.1127 −1.00329
\(843\) 21.3137 0.734083
\(844\) 6.31371 0.217327
\(845\) 1.00000 0.0344010
\(846\) 3.65685 0.125725
\(847\) 0 0
\(848\) −2.34315 −0.0804640
\(849\) 5.17157 0.177488
\(850\) 9.65685 0.331227
\(851\) 13.9289 0.477478
\(852\) 9.07107 0.310770
\(853\) −22.3431 −0.765015 −0.382507 0.923952i \(-0.624939\pi\)
−0.382507 + 0.923952i \(0.624939\pi\)
\(854\) 0 0
\(855\) −1.82843 −0.0625309
\(856\) −7.41421 −0.253413
\(857\) 39.1716 1.33808 0.669038 0.743228i \(-0.266706\pi\)
0.669038 + 0.743228i \(0.266706\pi\)
\(858\) 0.414214 0.0141410
\(859\) 15.0294 0.512798 0.256399 0.966571i \(-0.417464\pi\)
0.256399 + 0.966571i \(0.417464\pi\)
\(860\) −7.00000 −0.238698
\(861\) 0 0
\(862\) −20.8284 −0.709419
\(863\) 22.9706 0.781927 0.390964 0.920406i \(-0.372142\pi\)
0.390964 + 0.920406i \(0.372142\pi\)
\(864\) 1.00000 0.0340207
\(865\) 7.75736 0.263758
\(866\) −18.6274 −0.632985
\(867\) 11.1716 0.379407
\(868\) 0 0
\(869\) −0.727922 −0.0246931
\(870\) −8.65685 −0.293495
\(871\) 1.41421 0.0479188
\(872\) 2.41421 0.0817556
\(873\) −15.3137 −0.518291
\(874\) 4.85786 0.164320
\(875\) 0 0
\(876\) 6.17157 0.208518
\(877\) 8.48528 0.286528 0.143264 0.989685i \(-0.454240\pi\)
0.143264 + 0.989685i \(0.454240\pi\)
\(878\) 4.21320 0.142189
\(879\) 22.1421 0.746836
\(880\) −0.414214 −0.0139631
\(881\) −4.61522 −0.155491 −0.0777454 0.996973i \(-0.524772\pi\)
−0.0777454 + 0.996973i \(0.524772\pi\)
\(882\) 0 0
\(883\) 55.9706 1.88356 0.941780 0.336231i \(-0.109152\pi\)
0.941780 + 0.336231i \(0.109152\pi\)
\(884\) −2.41421 −0.0811988
\(885\) −9.89949 −0.332768
\(886\) 4.82843 0.162214
\(887\) −3.11270 −0.104514 −0.0522571 0.998634i \(-0.516642\pi\)
−0.0522571 + 0.998634i \(0.516642\pi\)
\(888\) 5.24264 0.175932
\(889\) 0 0
\(890\) −3.41421 −0.114445
\(891\) −0.414214 −0.0138767
\(892\) −4.34315 −0.145419
\(893\) 6.68629 0.223748
\(894\) 2.82843 0.0945968
\(895\) 11.5563 0.386286
\(896\) 0 0
\(897\) 2.65685 0.0887098
\(898\) 12.5563 0.419011
\(899\) −36.7279 −1.22494
\(900\) −4.00000 −0.133333
\(901\) −5.65685 −0.188457
\(902\) 0.485281 0.0161581
\(903\) 0 0
\(904\) 9.75736 0.324525
\(905\) −10.1421 −0.337136
\(906\) −4.75736 −0.158053
\(907\) 56.8284 1.88696 0.943478 0.331434i \(-0.107532\pi\)
0.943478 + 0.331434i \(0.107532\pi\)
\(908\) 9.07107 0.301034
\(909\) 13.0711 0.433540
\(910\) 0 0
\(911\) 23.9706 0.794180 0.397090 0.917780i \(-0.370020\pi\)
0.397090 + 0.917780i \(0.370020\pi\)
\(912\) 1.82843 0.0605453
\(913\) −0.142136 −0.00470400
\(914\) 13.7574 0.455053
\(915\) 8.41421 0.278165
\(916\) −4.10051 −0.135485
\(917\) 0 0
\(918\) 2.41421 0.0796809
\(919\) 16.1421 0.532480 0.266240 0.963907i \(-0.414219\pi\)
0.266240 + 0.963907i \(0.414219\pi\)
\(920\) −2.65685 −0.0875939
\(921\) 0.343146 0.0113070
\(922\) 6.51472 0.214551
\(923\) 9.07107 0.298578
\(924\) 0 0
\(925\) −20.9706 −0.689508
\(926\) 21.8701 0.718695
\(927\) 6.07107 0.199400
\(928\) 8.65685 0.284175
\(929\) −30.1421 −0.988931 −0.494466 0.869197i \(-0.664636\pi\)
−0.494466 + 0.869197i \(0.664636\pi\)
\(930\) 4.24264 0.139122
\(931\) 0 0
\(932\) −5.17157 −0.169401
\(933\) 17.7990 0.582713
\(934\) −28.8995 −0.945620
\(935\) −1.00000 −0.0327035
\(936\) 1.00000 0.0326860
\(937\) −17.6985 −0.578184 −0.289092 0.957301i \(-0.593353\pi\)
−0.289092 + 0.957301i \(0.593353\pi\)
\(938\) 0 0
\(939\) 6.92893 0.226117
\(940\) −3.65685 −0.119273
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 16.8995 0.550615
\(943\) 3.11270 0.101363
\(944\) 9.89949 0.322201
\(945\) 0 0
\(946\) −2.89949 −0.0942707
\(947\) 12.7574 0.414558 0.207279 0.978282i \(-0.433539\pi\)
0.207279 + 0.978282i \(0.433539\pi\)
\(948\) −1.75736 −0.0570764
\(949\) 6.17157 0.200338
\(950\) −7.31371 −0.237288
\(951\) 30.0416 0.974167
\(952\) 0 0
\(953\) −20.2843 −0.657072 −0.328536 0.944491i \(-0.606555\pi\)
−0.328536 + 0.944491i \(0.606555\pi\)
\(954\) 2.34315 0.0758621
\(955\) −11.4853 −0.371655
\(956\) −10.7279 −0.346966
\(957\) −3.58579 −0.115912
\(958\) −7.48528 −0.241838
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) −13.0000 −0.419355
\(962\) 5.24264 0.169030
\(963\) 7.41421 0.238920
\(964\) −23.6569 −0.761936
\(965\) −24.3848 −0.784974
\(966\) 0 0
\(967\) −4.41421 −0.141952 −0.0709758 0.997478i \(-0.522611\pi\)
−0.0709758 + 0.997478i \(0.522611\pi\)
\(968\) 10.8284 0.348039
\(969\) 4.41421 0.141805
\(970\) 15.3137 0.491694
\(971\) −48.4853 −1.55597 −0.777983 0.628285i \(-0.783757\pi\)
−0.777983 + 0.628285i \(0.783757\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 41.5980 1.33289
\(975\) −4.00000 −0.128103
\(976\) −8.41421 −0.269332
\(977\) 5.87006 0.187800 0.0938999 0.995582i \(-0.470067\pi\)
0.0938999 + 0.995582i \(0.470067\pi\)
\(978\) −0.343146 −0.0109726
\(979\) −1.41421 −0.0451985
\(980\) 0 0
\(981\) −2.41421 −0.0770799
\(982\) 38.7696 1.23719
\(983\) −36.3137 −1.15823 −0.579114 0.815247i \(-0.696601\pi\)
−0.579114 + 0.815247i \(0.696601\pi\)
\(984\) 1.17157 0.0373484
\(985\) −9.07107 −0.289028
\(986\) 20.8995 0.665576
\(987\) 0 0
\(988\) 1.82843 0.0581700
\(989\) −18.5980 −0.591381
\(990\) 0.414214 0.0131646
\(991\) 14.6863 0.466525 0.233263 0.972414i \(-0.425060\pi\)
0.233263 + 0.972414i \(0.425060\pi\)
\(992\) −4.24264 −0.134704
\(993\) 10.1005 0.320530
\(994\) 0 0
\(995\) −4.07107 −0.129062
\(996\) −0.343146 −0.0108730
\(997\) −1.71573 −0.0543377 −0.0271688 0.999631i \(-0.508649\pi\)
−0.0271688 + 0.999631i \(0.508649\pi\)
\(998\) 2.72792 0.0863509
\(999\) −5.24264 −0.165870
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 3822.2.a.bj.1.1 2
7.6 odd 2 3822.2.a.bl.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3822.2.a.bj.1.1 2 1.1 even 1 trivial
3822.2.a.bl.1.1 yes 2 7.6 odd 2