Properties

Label 418.2.a.h.1.3
Level $418$
Weight $2$
Character 418.1
Self dual yes
Analytic conductor $3.338$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) 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 1.3
Root \(2.39138\) of defining polynomial
Character \(\chi\) \(=\) 418.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.39138 q^{3} +1.00000 q^{4} -0.391382 q^{5} +2.39138 q^{6} -1.71871 q^{7} +1.00000 q^{8} +2.71871 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.39138 q^{3} +1.00000 q^{4} -0.391382 q^{5} +2.39138 q^{6} -1.71871 q^{7} +1.00000 q^{8} +2.71871 q^{9} -0.391382 q^{10} -1.00000 q^{11} +2.39138 q^{12} +3.71871 q^{13} -1.71871 q^{14} -0.935945 q^{15} +1.00000 q^{16} +5.43742 q^{17} +2.71871 q^{18} -1.00000 q^{19} -0.391382 q^{20} -4.11009 q^{21} -1.00000 q^{22} -3.43742 q^{23} +2.39138 q^{24} -4.84682 q^{25} +3.71871 q^{26} -0.672673 q^{27} -1.71871 q^{28} -3.82880 q^{29} -0.935945 q^{30} -3.06406 q^{31} +1.00000 q^{32} -2.39138 q^{33} +5.43742 q^{34} +0.672673 q^{35} +2.71871 q^{36} -4.65465 q^{37} -1.00000 q^{38} +8.89286 q^{39} -0.391382 q^{40} +1.06406 q^{41} -4.11009 q^{42} -3.73673 q^{43} -1.00000 q^{44} -1.06406 q^{45} -3.43742 q^{46} -8.22018 q^{47} +2.39138 q^{48} -4.04604 q^{49} -4.84682 q^{50} +13.0029 q^{51} +3.71871 q^{52} -1.21724 q^{53} -0.672673 q^{54} +0.391382 q^{55} -1.71871 q^{56} -2.39138 q^{57} -3.82880 q^{58} +12.4404 q^{59} -0.935945 q^{60} +2.00000 q^{61} -3.06406 q^{62} -4.67267 q^{63} +1.00000 q^{64} -1.45544 q^{65} -2.39138 q^{66} -1.71871 q^{67} +5.43742 q^{68} -8.22018 q^{69} +0.672673 q^{70} +1.04604 q^{71} +2.71871 q^{72} +13.6576 q^{73} -4.65465 q^{74} -11.5906 q^{75} -1.00000 q^{76} +1.71871 q^{77} +8.89286 q^{78} +13.0029 q^{79} -0.391382 q^{80} -9.76475 q^{81} +1.06406 q^{82} +8.51949 q^{83} -4.11009 q^{84} -2.12811 q^{85} -3.73673 q^{86} -9.15613 q^{87} -1.00000 q^{88} +17.6936 q^{89} -1.06406 q^{90} -6.39138 q^{91} -3.43742 q^{92} -7.32733 q^{93} -8.22018 q^{94} +0.391382 q^{95} +2.39138 q^{96} -4.65465 q^{97} -4.04604 q^{98} -2.71871 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} + 3 q^{8} + 2 q^{9} + 5 q^{10} - 3 q^{11} + q^{12} + 5 q^{13} + q^{14} - 9 q^{15} + 3 q^{16} + 4 q^{17} + 2 q^{18} - 3 q^{19} + 5 q^{20} - 3 q^{22} + 2 q^{23} + q^{24} + 4 q^{25} + 5 q^{26} - 2 q^{27} + q^{28} + 7 q^{29} - 9 q^{30} - 3 q^{31} + 3 q^{32} - q^{33} + 4 q^{34} + 2 q^{35} + 2 q^{36} - 14 q^{37} - 3 q^{38} + 2 q^{39} + 5 q^{40} - 3 q^{41} - 5 q^{43} - 3 q^{44} + 3 q^{45} + 2 q^{46} + q^{48} - 6 q^{49} + 4 q^{50} + 2 q^{51} + 5 q^{52} - 16 q^{53} - 2 q^{54} - 5 q^{55} + q^{56} - q^{57} + 7 q^{58} - 12 q^{59} - 9 q^{60} + 6 q^{61} - 3 q^{62} - 14 q^{63} + 3 q^{64} + 8 q^{65} - q^{66} + q^{67} + 4 q^{68} + 2 q^{70} - 3 q^{71} + 2 q^{72} + 4 q^{73} - 14 q^{74} - 41 q^{75} - 3 q^{76} - q^{77} + 2 q^{78} + 2 q^{79} + 5 q^{80} - 17 q^{81} - 3 q^{82} + 7 q^{83} + 6 q^{85} - 5 q^{86} - 9 q^{87} - 3 q^{88} + 16 q^{89} + 3 q^{90} - 13 q^{91} + 2 q^{92} - 22 q^{93} - 5 q^{95} + q^{96} - 14 q^{97} - 6 q^{98} - 2 q^{99}+O(q^{100}) \) 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
\(3\) 2.39138 1.38067 0.690333 0.723492i \(-0.257464\pi\)
0.690333 + 0.723492i \(0.257464\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.391382 −0.175032 −0.0875158 0.996163i \(-0.527893\pi\)
−0.0875158 + 0.996163i \(0.527893\pi\)
\(6\) 2.39138 0.976278
\(7\) −1.71871 −0.649611 −0.324806 0.945781i \(-0.605299\pi\)
−0.324806 + 0.945781i \(0.605299\pi\)
\(8\) 1.00000 0.353553
\(9\) 2.71871 0.906237
\(10\) −0.391382 −0.123766
\(11\) −1.00000 −0.301511
\(12\) 2.39138 0.690333
\(13\) 3.71871 1.03138 0.515692 0.856774i \(-0.327535\pi\)
0.515692 + 0.856774i \(0.327535\pi\)
\(14\) −1.71871 −0.459344
\(15\) −0.935945 −0.241660
\(16\) 1.00000 0.250000
\(17\) 5.43742 1.31877 0.659384 0.751806i \(-0.270817\pi\)
0.659384 + 0.751806i \(0.270817\pi\)
\(18\) 2.71871 0.640806
\(19\) −1.00000 −0.229416
\(20\) −0.391382 −0.0875158
\(21\) −4.11009 −0.896896
\(22\) −1.00000 −0.213201
\(23\) −3.43742 −0.716751 −0.358376 0.933577i \(-0.616669\pi\)
−0.358376 + 0.933577i \(0.616669\pi\)
\(24\) 2.39138 0.488139
\(25\) −4.84682 −0.969364
\(26\) 3.71871 0.729299
\(27\) −0.672673 −0.129456
\(28\) −1.71871 −0.324806
\(29\) −3.82880 −0.710991 −0.355495 0.934678i \(-0.615688\pi\)
−0.355495 + 0.934678i \(0.615688\pi\)
\(30\) −0.935945 −0.170879
\(31\) −3.06406 −0.550321 −0.275160 0.961398i \(-0.588731\pi\)
−0.275160 + 0.961398i \(0.588731\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.39138 −0.416286
\(34\) 5.43742 0.932510
\(35\) 0.672673 0.113702
\(36\) 2.71871 0.453118
\(37\) −4.65465 −0.765221 −0.382610 0.923910i \(-0.624975\pi\)
−0.382610 + 0.923910i \(0.624975\pi\)
\(38\) −1.00000 −0.162221
\(39\) 8.89286 1.42400
\(40\) −0.391382 −0.0618830
\(41\) 1.06406 0.166177 0.0830887 0.996542i \(-0.473522\pi\)
0.0830887 + 0.996542i \(0.473522\pi\)
\(42\) −4.11009 −0.634201
\(43\) −3.73673 −0.569846 −0.284923 0.958550i \(-0.591968\pi\)
−0.284923 + 0.958550i \(0.591968\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.06406 −0.158620
\(46\) −3.43742 −0.506820
\(47\) −8.22018 −1.19904 −0.599519 0.800361i \(-0.704641\pi\)
−0.599519 + 0.800361i \(0.704641\pi\)
\(48\) 2.39138 0.345166
\(49\) −4.04604 −0.578005
\(50\) −4.84682 −0.685444
\(51\) 13.0029 1.82078
\(52\) 3.71871 0.515692
\(53\) −1.21724 −0.167200 −0.0836001 0.996499i \(-0.526642\pi\)
−0.0836001 + 0.996499i \(0.526642\pi\)
\(54\) −0.672673 −0.0915392
\(55\) 0.391382 0.0527740
\(56\) −1.71871 −0.229672
\(57\) −2.39138 −0.316746
\(58\) −3.82880 −0.502746
\(59\) 12.4404 1.61960 0.809799 0.586707i \(-0.199576\pi\)
0.809799 + 0.586707i \(0.199576\pi\)
\(60\) −0.935945 −0.120830
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −3.06406 −0.389135
\(63\) −4.67267 −0.588701
\(64\) 1.00000 0.125000
\(65\) −1.45544 −0.180525
\(66\) −2.39138 −0.294359
\(67\) −1.71871 −0.209974 −0.104987 0.994474i \(-0.533480\pi\)
−0.104987 + 0.994474i \(0.533480\pi\)
\(68\) 5.43742 0.659384
\(69\) −8.22018 −0.989594
\(70\) 0.672673 0.0803998
\(71\) 1.04604 0.124142 0.0620709 0.998072i \(-0.480230\pi\)
0.0620709 + 0.998072i \(0.480230\pi\)
\(72\) 2.71871 0.320403
\(73\) 13.6576 1.59850 0.799251 0.600998i \(-0.205230\pi\)
0.799251 + 0.600998i \(0.205230\pi\)
\(74\) −4.65465 −0.541093
\(75\) −11.5906 −1.33837
\(76\) −1.00000 −0.114708
\(77\) 1.71871 0.195865
\(78\) 8.89286 1.00692
\(79\) 13.0029 1.46295 0.731473 0.681870i \(-0.238833\pi\)
0.731473 + 0.681870i \(0.238833\pi\)
\(80\) −0.391382 −0.0437579
\(81\) −9.76475 −1.08497
\(82\) 1.06406 0.117505
\(83\) 8.51949 0.935136 0.467568 0.883957i \(-0.345130\pi\)
0.467568 + 0.883957i \(0.345130\pi\)
\(84\) −4.11009 −0.448448
\(85\) −2.12811 −0.230826
\(86\) −3.73673 −0.402942
\(87\) −9.15613 −0.981640
\(88\) −1.00000 −0.106600
\(89\) 17.6936 1.87552 0.937761 0.347281i \(-0.112895\pi\)
0.937761 + 0.347281i \(0.112895\pi\)
\(90\) −1.06406 −0.112161
\(91\) −6.39138 −0.669999
\(92\) −3.43742 −0.358376
\(93\) −7.32733 −0.759808
\(94\) −8.22018 −0.847847
\(95\) 0.391382 0.0401550
\(96\) 2.39138 0.244069
\(97\) −4.65465 −0.472609 −0.236304 0.971679i \(-0.575936\pi\)
−0.236304 + 0.971679i \(0.575936\pi\)
\(98\) −4.04604 −0.408711
\(99\) −2.71871 −0.273241
\(100\) −4.84682 −0.484682
\(101\) −9.65760 −0.960967 −0.480484 0.877004i \(-0.659539\pi\)
−0.480484 + 0.877004i \(0.659539\pi\)
\(102\) 13.0029 1.28748
\(103\) −6.95396 −0.685194 −0.342597 0.939482i \(-0.611307\pi\)
−0.342597 + 0.939482i \(0.611307\pi\)
\(104\) 3.71871 0.364649
\(105\) 1.60862 0.156985
\(106\) −1.21724 −0.118228
\(107\) 2.65465 0.256635 0.128318 0.991733i \(-0.459042\pi\)
0.128318 + 0.991733i \(0.459042\pi\)
\(108\) −0.672673 −0.0647280
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0.391382 0.0373168
\(111\) −11.1311 −1.05651
\(112\) −1.71871 −0.162403
\(113\) 7.52949 0.708315 0.354158 0.935186i \(-0.384768\pi\)
0.354158 + 0.935186i \(0.384768\pi\)
\(114\) −2.39138 −0.223973
\(115\) 1.34535 0.125454
\(116\) −3.82880 −0.355495
\(117\) 10.1101 0.934678
\(118\) 12.4404 1.14523
\(119\) −9.34535 −0.856686
\(120\) −0.935945 −0.0854397
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) 2.54456 0.229435
\(124\) −3.06406 −0.275160
\(125\) 3.85387 0.344701
\(126\) −4.67267 −0.416275
\(127\) −5.00295 −0.443940 −0.221970 0.975054i \(-0.571249\pi\)
−0.221970 + 0.975054i \(0.571249\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.93594 −0.786766
\(130\) −1.45544 −0.127650
\(131\) −6.50147 −0.568036 −0.284018 0.958819i \(-0.591668\pi\)
−0.284018 + 0.958819i \(0.591668\pi\)
\(132\) −2.39138 −0.208143
\(133\) 1.71871 0.149031
\(134\) −1.71871 −0.148474
\(135\) 0.263272 0.0226589
\(136\) 5.43742 0.466255
\(137\) 15.7187 1.34294 0.671470 0.741032i \(-0.265663\pi\)
0.671470 + 0.741032i \(0.265663\pi\)
\(138\) −8.22018 −0.699749
\(139\) −11.1381 −0.944722 −0.472361 0.881405i \(-0.656598\pi\)
−0.472361 + 0.881405i \(0.656598\pi\)
\(140\) 0.672673 0.0568512
\(141\) −19.6576 −1.65547
\(142\) 1.04604 0.0877815
\(143\) −3.71871 −0.310974
\(144\) 2.71871 0.226559
\(145\) 1.49853 0.124446
\(146\) 13.6576 1.13031
\(147\) −9.67562 −0.798032
\(148\) −4.65465 −0.382610
\(149\) 11.3453 0.929447 0.464723 0.885456i \(-0.346154\pi\)
0.464723 + 0.885456i \(0.346154\pi\)
\(150\) −11.5906 −0.946368
\(151\) −14.6547 −1.19258 −0.596289 0.802770i \(-0.703359\pi\)
−0.596289 + 0.802770i \(0.703359\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 14.7828 1.19512
\(154\) 1.71871 0.138498
\(155\) 1.19922 0.0963234
\(156\) 8.89286 0.711998
\(157\) −10.4835 −0.836671 −0.418335 0.908293i \(-0.637386\pi\)
−0.418335 + 0.908293i \(0.637386\pi\)
\(158\) 13.0029 1.03446
\(159\) −2.91087 −0.230847
\(160\) −0.391382 −0.0309415
\(161\) 5.90793 0.465610
\(162\) −9.76475 −0.767191
\(163\) −3.47346 −0.272062 −0.136031 0.990705i \(-0.543435\pi\)
−0.136031 + 0.990705i \(0.543435\pi\)
\(164\) 1.06406 0.0830887
\(165\) 0.935945 0.0728632
\(166\) 8.51949 0.661241
\(167\) 23.0950 1.78715 0.893573 0.448917i \(-0.148190\pi\)
0.893573 + 0.448917i \(0.148190\pi\)
\(168\) −4.11009 −0.317100
\(169\) 0.828802 0.0637540
\(170\) −2.12811 −0.163219
\(171\) −2.71871 −0.207905
\(172\) −3.73673 −0.284923
\(173\) −18.1771 −1.38198 −0.690990 0.722865i \(-0.742825\pi\)
−0.690990 + 0.722865i \(0.742825\pi\)
\(174\) −9.15613 −0.694124
\(175\) 8.33028 0.629710
\(176\) −1.00000 −0.0753778
\(177\) 29.7497 2.23612
\(178\) 17.6936 1.32619
\(179\) 11.2842 0.843424 0.421712 0.906730i \(-0.361429\pi\)
0.421712 + 0.906730i \(0.361429\pi\)
\(180\) −1.06406 −0.0793100
\(181\) 21.8778 1.62616 0.813082 0.582150i \(-0.197788\pi\)
0.813082 + 0.582150i \(0.197788\pi\)
\(182\) −6.39138 −0.473761
\(183\) 4.78276 0.353552
\(184\) −3.43742 −0.253410
\(185\) 1.82175 0.133938
\(186\) −7.32733 −0.537266
\(187\) −5.43742 −0.397623
\(188\) −8.22018 −0.599519
\(189\) 1.15613 0.0840960
\(190\) 0.391382 0.0283939
\(191\) −7.09502 −0.513378 −0.256689 0.966494i \(-0.582632\pi\)
−0.256689 + 0.966494i \(0.582632\pi\)
\(192\) 2.39138 0.172583
\(193\) 12.6476 0.910394 0.455197 0.890391i \(-0.349569\pi\)
0.455197 + 0.890391i \(0.349569\pi\)
\(194\) −4.65465 −0.334185
\(195\) −3.48051 −0.249244
\(196\) −4.04604 −0.289003
\(197\) 8.43447 0.600931 0.300466 0.953793i \(-0.402858\pi\)
0.300466 + 0.953793i \(0.402858\pi\)
\(198\) −2.71871 −0.193210
\(199\) −2.09207 −0.148303 −0.0741516 0.997247i \(-0.523625\pi\)
−0.0741516 + 0.997247i \(0.523625\pi\)
\(200\) −4.84682 −0.342722
\(201\) −4.11009 −0.289904
\(202\) −9.65760 −0.679507
\(203\) 6.58060 0.461867
\(204\) 13.0029 0.910389
\(205\) −0.416452 −0.0290863
\(206\) −6.95396 −0.484506
\(207\) −9.34535 −0.649546
\(208\) 3.71871 0.257846
\(209\) 1.00000 0.0691714
\(210\) 1.60862 0.111005
\(211\) −6.09207 −0.419396 −0.209698 0.977766i \(-0.567248\pi\)
−0.209698 + 0.977766i \(0.567248\pi\)
\(212\) −1.21724 −0.0836001
\(213\) 2.50147 0.171398
\(214\) 2.65465 0.181468
\(215\) 1.46249 0.0997409
\(216\) −0.672673 −0.0457696
\(217\) 5.26622 0.357494
\(218\) 6.00000 0.406371
\(219\) 32.6606 2.20700
\(220\) 0.391382 0.0263870
\(221\) 20.2202 1.36016
\(222\) −11.1311 −0.747068
\(223\) −18.8748 −1.26395 −0.631976 0.774988i \(-0.717756\pi\)
−0.631976 + 0.774988i \(0.717756\pi\)
\(224\) −1.71871 −0.114836
\(225\) −13.1771 −0.878473
\(226\) 7.52949 0.500854
\(227\) 20.9669 1.39162 0.695811 0.718225i \(-0.255045\pi\)
0.695811 + 0.718225i \(0.255045\pi\)
\(228\) −2.39138 −0.158373
\(229\) 13.8108 0.912642 0.456321 0.889815i \(-0.349167\pi\)
0.456321 + 0.889815i \(0.349167\pi\)
\(230\) 1.34535 0.0887094
\(231\) 4.11009 0.270424
\(232\) −3.82880 −0.251373
\(233\) 4.91087 0.321722 0.160861 0.986977i \(-0.448573\pi\)
0.160861 + 0.986977i \(0.448573\pi\)
\(234\) 10.1101 0.660917
\(235\) 3.21724 0.209869
\(236\) 12.4404 0.809799
\(237\) 31.0950 2.01984
\(238\) −9.34535 −0.605769
\(239\) −14.0490 −0.908753 −0.454377 0.890810i \(-0.650138\pi\)
−0.454377 + 0.890810i \(0.650138\pi\)
\(240\) −0.935945 −0.0604150
\(241\) 22.7397 1.46479 0.732396 0.680879i \(-0.238402\pi\)
0.732396 + 0.680879i \(0.238402\pi\)
\(242\) 1.00000 0.0642824
\(243\) −21.3332 −1.36853
\(244\) 2.00000 0.128037
\(245\) 1.58355 0.101169
\(246\) 2.54456 0.162235
\(247\) −3.71871 −0.236616
\(248\) −3.06406 −0.194568
\(249\) 20.3734 1.29111
\(250\) 3.85387 0.243740
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −4.67267 −0.294351
\(253\) 3.43742 0.216109
\(254\) −5.00295 −0.313913
\(255\) −5.08913 −0.318693
\(256\) 1.00000 0.0625000
\(257\) −19.4433 −1.21284 −0.606420 0.795144i \(-0.707395\pi\)
−0.606420 + 0.795144i \(0.707395\pi\)
\(258\) −8.93594 −0.556328
\(259\) 8.00000 0.497096
\(260\) −1.45544 −0.0902624
\(261\) −10.4094 −0.644326
\(262\) −6.50147 −0.401662
\(263\) 14.6476 0.903210 0.451605 0.892218i \(-0.350852\pi\)
0.451605 + 0.892218i \(0.350852\pi\)
\(264\) −2.39138 −0.147179
\(265\) 0.476404 0.0292653
\(266\) 1.71871 0.105381
\(267\) 42.3123 2.58947
\(268\) −1.71871 −0.104987
\(269\) 8.34829 0.509004 0.254502 0.967072i \(-0.418088\pi\)
0.254502 + 0.967072i \(0.418088\pi\)
\(270\) 0.263272 0.0160222
\(271\) −10.5375 −0.640108 −0.320054 0.947399i \(-0.603701\pi\)
−0.320054 + 0.947399i \(0.603701\pi\)
\(272\) 5.43742 0.329692
\(273\) −15.2842 −0.925044
\(274\) 15.7187 0.949602
\(275\) 4.84682 0.292274
\(276\) −8.22018 −0.494797
\(277\) −22.8807 −1.37477 −0.687385 0.726293i \(-0.741242\pi\)
−0.687385 + 0.726293i \(0.741242\pi\)
\(278\) −11.1381 −0.668020
\(279\) −8.33028 −0.498721
\(280\) 0.672673 0.0401999
\(281\) −4.39138 −0.261968 −0.130984 0.991384i \(-0.541814\pi\)
−0.130984 + 0.991384i \(0.541814\pi\)
\(282\) −19.6576 −1.17059
\(283\) −24.0670 −1.43063 −0.715317 0.698800i \(-0.753718\pi\)
−0.715317 + 0.698800i \(0.753718\pi\)
\(284\) 1.04604 0.0620709
\(285\) 0.935945 0.0554406
\(286\) −3.71871 −0.219892
\(287\) −1.82880 −0.107951
\(288\) 2.71871 0.160202
\(289\) 12.5655 0.739149
\(290\) 1.49853 0.0879965
\(291\) −11.1311 −0.652514
\(292\) 13.6576 0.799251
\(293\) −27.2901 −1.59431 −0.797153 0.603777i \(-0.793662\pi\)
−0.797153 + 0.603777i \(0.793662\pi\)
\(294\) −9.67562 −0.564294
\(295\) −4.86894 −0.283481
\(296\) −4.65465 −0.270546
\(297\) 0.672673 0.0390324
\(298\) 11.3453 0.657218
\(299\) −12.7828 −0.739246
\(300\) −11.5906 −0.669184
\(301\) 6.42235 0.370178
\(302\) −14.6547 −0.843281
\(303\) −23.0950 −1.32677
\(304\) −1.00000 −0.0573539
\(305\) −0.782765 −0.0448210
\(306\) 14.7828 0.845074
\(307\) −9.22313 −0.526392 −0.263196 0.964742i \(-0.584777\pi\)
−0.263196 + 0.964742i \(0.584777\pi\)
\(308\) 1.71871 0.0979326
\(309\) −16.6296 −0.946024
\(310\) 1.19922 0.0681110
\(311\) 6.34829 0.359979 0.179989 0.983669i \(-0.442394\pi\)
0.179989 + 0.983669i \(0.442394\pi\)
\(312\) 8.89286 0.503459
\(313\) 13.3943 0.757092 0.378546 0.925582i \(-0.376424\pi\)
0.378546 + 0.925582i \(0.376424\pi\)
\(314\) −10.4835 −0.591616
\(315\) 1.82880 0.103041
\(316\) 13.0029 0.731473
\(317\) 31.0029 1.74130 0.870650 0.491904i \(-0.163699\pi\)
0.870650 + 0.491904i \(0.163699\pi\)
\(318\) −2.91087 −0.163234
\(319\) 3.82880 0.214372
\(320\) −0.391382 −0.0218789
\(321\) 6.34829 0.354327
\(322\) 5.90793 0.329236
\(323\) −5.43742 −0.302546
\(324\) −9.76475 −0.542486
\(325\) −18.0239 −0.999787
\(326\) −3.47346 −0.192377
\(327\) 14.3483 0.793462
\(328\) 1.06406 0.0587526
\(329\) 14.1281 0.778908
\(330\) 0.935945 0.0515221
\(331\) −16.2993 −0.895891 −0.447946 0.894061i \(-0.647844\pi\)
−0.447946 + 0.894061i \(0.647844\pi\)
\(332\) 8.51949 0.467568
\(333\) −12.6547 −0.693471
\(334\) 23.0950 1.26370
\(335\) 0.672673 0.0367520
\(336\) −4.11009 −0.224224
\(337\) −3.19217 −0.173888 −0.0869442 0.996213i \(-0.527710\pi\)
−0.0869442 + 0.996213i \(0.527710\pi\)
\(338\) 0.828802 0.0450809
\(339\) 18.0059 0.977946
\(340\) −2.12811 −0.115413
\(341\) 3.06406 0.165928
\(342\) −2.71871 −0.147011
\(343\) 18.9849 1.02509
\(344\) −3.73673 −0.201471
\(345\) 3.21724 0.173210
\(346\) −18.1771 −0.977207
\(347\) 24.6966 1.32578 0.662891 0.748716i \(-0.269329\pi\)
0.662891 + 0.748716i \(0.269329\pi\)
\(348\) −9.15613 −0.490820
\(349\) −4.43447 −0.237372 −0.118686 0.992932i \(-0.537868\pi\)
−0.118686 + 0.992932i \(0.537868\pi\)
\(350\) 8.33028 0.445272
\(351\) −2.50147 −0.133519
\(352\) −1.00000 −0.0533002
\(353\) 25.3152 1.34739 0.673696 0.739008i \(-0.264706\pi\)
0.673696 + 0.739008i \(0.264706\pi\)
\(354\) 29.7497 1.58118
\(355\) −0.409400 −0.0217287
\(356\) 17.6936 0.937761
\(357\) −22.3483 −1.18280
\(358\) 11.2842 0.596391
\(359\) −13.9749 −0.737569 −0.368784 0.929515i \(-0.620226\pi\)
−0.368784 + 0.929515i \(0.620226\pi\)
\(360\) −1.06406 −0.0560806
\(361\) 1.00000 0.0526316
\(362\) 21.8778 1.14987
\(363\) 2.39138 0.125515
\(364\) −6.39138 −0.334999
\(365\) −5.34535 −0.279788
\(366\) 4.78276 0.249999
\(367\) −25.3453 −1.32302 −0.661508 0.749938i \(-0.730083\pi\)
−0.661508 + 0.749938i \(0.730083\pi\)
\(368\) −3.43742 −0.179188
\(369\) 2.89286 0.150596
\(370\) 1.82175 0.0947083
\(371\) 2.09207 0.108615
\(372\) −7.32733 −0.379904
\(373\) 8.95396 0.463619 0.231809 0.972761i \(-0.425535\pi\)
0.231809 + 0.972761i \(0.425535\pi\)
\(374\) −5.43742 −0.281162
\(375\) 9.21608 0.475916
\(376\) −8.22018 −0.423924
\(377\) −14.2382 −0.733305
\(378\) 1.15613 0.0594649
\(379\) −25.8527 −1.32796 −0.663982 0.747748i \(-0.731135\pi\)
−0.663982 + 0.747748i \(0.731135\pi\)
\(380\) 0.391382 0.0200775
\(381\) −11.9640 −0.612932
\(382\) −7.09502 −0.363013
\(383\) −20.6296 −1.05412 −0.527061 0.849827i \(-0.676706\pi\)
−0.527061 + 0.849827i \(0.676706\pi\)
\(384\) 2.39138 0.122035
\(385\) −0.672673 −0.0342826
\(386\) 12.6476 0.643746
\(387\) −10.1591 −0.516415
\(388\) −4.65465 −0.236304
\(389\) 15.6086 0.791388 0.395694 0.918382i \(-0.370504\pi\)
0.395694 + 0.918382i \(0.370504\pi\)
\(390\) −3.48051 −0.176242
\(391\) −18.6907 −0.945229
\(392\) −4.04604 −0.204356
\(393\) −15.5475 −0.784268
\(394\) 8.43447 0.424922
\(395\) −5.08913 −0.256062
\(396\) −2.71871 −0.136620
\(397\) −34.2512 −1.71902 −0.859508 0.511122i \(-0.829230\pi\)
−0.859508 + 0.511122i \(0.829230\pi\)
\(398\) −2.09207 −0.104866
\(399\) 4.11009 0.205762
\(400\) −4.84682 −0.242341
\(401\) 4.09207 0.204348 0.102174 0.994767i \(-0.467420\pi\)
0.102174 + 0.994767i \(0.467420\pi\)
\(402\) −4.11009 −0.204993
\(403\) −11.3943 −0.567592
\(404\) −9.65760 −0.480484
\(405\) 3.82175 0.189904
\(406\) 6.58060 0.326590
\(407\) 4.65465 0.230723
\(408\) 13.0029 0.643742
\(409\) 2.48346 0.122799 0.0613995 0.998113i \(-0.480444\pi\)
0.0613995 + 0.998113i \(0.480444\pi\)
\(410\) −0.416452 −0.0205671
\(411\) 37.5894 1.85415
\(412\) −6.95396 −0.342597
\(413\) −21.3814 −1.05211
\(414\) −9.34535 −0.459299
\(415\) −3.33438 −0.163678
\(416\) 3.71871 0.182325
\(417\) −26.6355 −1.30435
\(418\) 1.00000 0.0489116
\(419\) 35.0950 1.71450 0.857252 0.514897i \(-0.172170\pi\)
0.857252 + 0.514897i \(0.172170\pi\)
\(420\) 1.60862 0.0784925
\(421\) −12.3483 −0.601819 −0.300910 0.953653i \(-0.597290\pi\)
−0.300910 + 0.953653i \(0.597290\pi\)
\(422\) −6.09207 −0.296558
\(423\) −22.3483 −1.08661
\(424\) −1.21724 −0.0591142
\(425\) −26.3542 −1.27837
\(426\) 2.50147 0.121197
\(427\) −3.43742 −0.166348
\(428\) 2.65465 0.128318
\(429\) −8.89286 −0.429351
\(430\) 1.46249 0.0705275
\(431\) −6.94691 −0.334621 −0.167310 0.985904i \(-0.553508\pi\)
−0.167310 + 0.985904i \(0.553508\pi\)
\(432\) −0.672673 −0.0323640
\(433\) −13.2533 −0.636912 −0.318456 0.947938i \(-0.603164\pi\)
−0.318456 + 0.947938i \(0.603164\pi\)
\(434\) 5.26622 0.252787
\(435\) 3.58355 0.171818
\(436\) 6.00000 0.287348
\(437\) 3.43742 0.164434
\(438\) 32.6606 1.56058
\(439\) 15.3152 0.730955 0.365477 0.930820i \(-0.380906\pi\)
0.365477 + 0.930820i \(0.380906\pi\)
\(440\) 0.391382 0.0186584
\(441\) −11.0000 −0.523810
\(442\) 20.2202 0.961776
\(443\) −20.8807 −0.992074 −0.496037 0.868301i \(-0.665212\pi\)
−0.496037 + 0.868301i \(0.665212\pi\)
\(444\) −11.1311 −0.528257
\(445\) −6.92498 −0.328275
\(446\) −18.8748 −0.893750
\(447\) 27.1311 1.28326
\(448\) −1.71871 −0.0812014
\(449\) −28.7887 −1.35862 −0.679310 0.733851i \(-0.737721\pi\)
−0.679310 + 0.733851i \(0.737721\pi\)
\(450\) −13.1771 −0.621174
\(451\) −1.06406 −0.0501044
\(452\) 7.52949 0.354158
\(453\) −35.0449 −1.64655
\(454\) 20.9669 0.984026
\(455\) 2.50147 0.117271
\(456\) −2.39138 −0.111987
\(457\) −32.7526 −1.53210 −0.766052 0.642779i \(-0.777781\pi\)
−0.766052 + 0.642779i \(0.777781\pi\)
\(458\) 13.8108 0.645336
\(459\) −3.65760 −0.170722
\(460\) 1.34535 0.0627271
\(461\) 20.2261 0.942023 0.471011 0.882127i \(-0.343889\pi\)
0.471011 + 0.882127i \(0.343889\pi\)
\(462\) 4.11009 0.191219
\(463\) −18.6907 −0.868630 −0.434315 0.900761i \(-0.643010\pi\)
−0.434315 + 0.900761i \(0.643010\pi\)
\(464\) −3.82880 −0.177748
\(465\) 2.86779 0.132990
\(466\) 4.91087 0.227492
\(467\) 34.5685 1.59964 0.799819 0.600241i \(-0.204929\pi\)
0.799819 + 0.600241i \(0.204929\pi\)
\(468\) 10.1101 0.467339
\(469\) 2.95396 0.136401
\(470\) 3.21724 0.148400
\(471\) −25.0700 −1.15516
\(472\) 12.4404 0.572614
\(473\) 3.73673 0.171815
\(474\) 31.0950 1.42824
\(475\) 4.84682 0.222387
\(476\) −9.34535 −0.428343
\(477\) −3.30931 −0.151523
\(478\) −14.0490 −0.642586
\(479\) −17.7187 −0.809589 −0.404794 0.914408i \(-0.632657\pi\)
−0.404794 + 0.914408i \(0.632657\pi\)
\(480\) −0.935945 −0.0427198
\(481\) −17.3093 −0.789237
\(482\) 22.7397 1.03576
\(483\) 14.1281 0.642851
\(484\) 1.00000 0.0454545
\(485\) 1.82175 0.0827214
\(486\) −21.3332 −0.967695
\(487\) 37.7426 1.71028 0.855141 0.518396i \(-0.173471\pi\)
0.855141 + 0.518396i \(0.173471\pi\)
\(488\) 2.00000 0.0905357
\(489\) −8.30636 −0.375627
\(490\) 1.58355 0.0715374
\(491\) 0.373364 0.0168497 0.00842485 0.999965i \(-0.497318\pi\)
0.00842485 + 0.999965i \(0.497318\pi\)
\(492\) 2.54456 0.114718
\(493\) −20.8188 −0.937632
\(494\) −3.71871 −0.167313
\(495\) 1.06406 0.0478257
\(496\) −3.06406 −0.137580
\(497\) −1.79783 −0.0806439
\(498\) 20.3734 0.912952
\(499\) −14.8388 −0.664276 −0.332138 0.943231i \(-0.607770\pi\)
−0.332138 + 0.943231i \(0.607770\pi\)
\(500\) 3.85387 0.172350
\(501\) 55.2290 2.46745
\(502\) −12.0000 −0.535586
\(503\) 33.2662 1.48327 0.741634 0.670805i \(-0.234051\pi\)
0.741634 + 0.670805i \(0.234051\pi\)
\(504\) −4.67267 −0.208137
\(505\) 3.77982 0.168200
\(506\) 3.43742 0.152812
\(507\) 1.98198 0.0880229
\(508\) −5.00295 −0.221970
\(509\) −8.47051 −0.375449 −0.187724 0.982222i \(-0.560111\pi\)
−0.187724 + 0.982222i \(0.560111\pi\)
\(510\) −5.08913 −0.225350
\(511\) −23.4735 −1.03840
\(512\) 1.00000 0.0441942
\(513\) 0.672673 0.0296992
\(514\) −19.4433 −0.857608
\(515\) 2.72166 0.119931
\(516\) −8.93594 −0.393383
\(517\) 8.22018 0.361523
\(518\) 8.00000 0.351500
\(519\) −43.4684 −1.90805
\(520\) −1.45544 −0.0638252
\(521\) −37.3152 −1.63481 −0.817404 0.576064i \(-0.804588\pi\)
−0.817404 + 0.576064i \(0.804588\pi\)
\(522\) −10.4094 −0.455607
\(523\) −31.6576 −1.38429 −0.692145 0.721758i \(-0.743334\pi\)
−0.692145 + 0.721758i \(0.743334\pi\)
\(524\) −6.50147 −0.284018
\(525\) 19.9209 0.869418
\(526\) 14.6476 0.638666
\(527\) −16.6606 −0.725745
\(528\) −2.39138 −0.104072
\(529\) −11.1841 −0.486267
\(530\) 0.476404 0.0206937
\(531\) 33.8217 1.46774
\(532\) 1.71871 0.0745155
\(533\) 3.95691 0.171393
\(534\) 42.3123 1.83103
\(535\) −1.03899 −0.0449192
\(536\) −1.71871 −0.0742370
\(537\) 26.9849 1.16449
\(538\) 8.34829 0.359921
\(539\) 4.04604 0.174275
\(540\) 0.263272 0.0113294
\(541\) 28.0921 1.20777 0.603886 0.797070i \(-0.293618\pi\)
0.603886 + 0.797070i \(0.293618\pi\)
\(542\) −10.5375 −0.452625
\(543\) 52.3182 2.24519
\(544\) 5.43742 0.233127
\(545\) −2.34829 −0.100590
\(546\) −15.2842 −0.654105
\(547\) 24.1841 1.03404 0.517020 0.855973i \(-0.327041\pi\)
0.517020 + 0.855973i \(0.327041\pi\)
\(548\) 15.7187 0.671470
\(549\) 5.43742 0.232063
\(550\) 4.84682 0.206669
\(551\) 3.82880 0.163112
\(552\) −8.22018 −0.349874
\(553\) −22.3483 −0.950346
\(554\) −22.8807 −0.972109
\(555\) 4.35650 0.184923
\(556\) −11.1381 −0.472361
\(557\) 39.9699 1.69358 0.846789 0.531929i \(-0.178533\pi\)
0.846789 + 0.531929i \(0.178533\pi\)
\(558\) −8.33028 −0.352649
\(559\) −13.8958 −0.587730
\(560\) 0.672673 0.0284256
\(561\) −13.0029 −0.548985
\(562\) −4.39138 −0.185239
\(563\) 16.4764 0.694398 0.347199 0.937792i \(-0.387133\pi\)
0.347199 + 0.937792i \(0.387133\pi\)
\(564\) −19.6576 −0.827734
\(565\) −2.94691 −0.123977
\(566\) −24.0670 −1.01161
\(567\) 16.7828 0.704810
\(568\) 1.04604 0.0438907
\(569\) −24.9418 −1.04562 −0.522808 0.852450i \(-0.675116\pi\)
−0.522808 + 0.852450i \(0.675116\pi\)
\(570\) 0.935945 0.0392024
\(571\) −8.50737 −0.356022 −0.178011 0.984028i \(-0.556966\pi\)
−0.178011 + 0.984028i \(0.556966\pi\)
\(572\) −3.71871 −0.155487
\(573\) −16.9669 −0.708803
\(574\) −1.82880 −0.0763327
\(575\) 16.6606 0.694793
\(576\) 2.71871 0.113280
\(577\) 31.8527 1.32605 0.663023 0.748599i \(-0.269273\pi\)
0.663023 + 0.748599i \(0.269273\pi\)
\(578\) 12.5655 0.522657
\(579\) 30.2453 1.25695
\(580\) 1.49853 0.0622229
\(581\) −14.6425 −0.607475
\(582\) −11.1311 −0.461397
\(583\) 1.21724 0.0504127
\(584\) 13.6576 0.565156
\(585\) −3.95691 −0.163598
\(586\) −27.2901 −1.12735
\(587\) 2.43447 0.100481 0.0502407 0.998737i \(-0.484001\pi\)
0.0502407 + 0.998737i \(0.484001\pi\)
\(588\) −9.67562 −0.399016
\(589\) 3.06406 0.126252
\(590\) −4.86894 −0.200451
\(591\) 20.1700 0.829685
\(592\) −4.65465 −0.191305
\(593\) −18.1841 −0.746733 −0.373367 0.927684i \(-0.621797\pi\)
−0.373367 + 0.927684i \(0.621797\pi\)
\(594\) 0.672673 0.0276001
\(595\) 3.65760 0.149947
\(596\) 11.3453 0.464723
\(597\) −5.00295 −0.204757
\(598\) −12.7828 −0.522726
\(599\) 4.33534 0.177137 0.0885687 0.996070i \(-0.471771\pi\)
0.0885687 + 0.996070i \(0.471771\pi\)
\(600\) −11.5906 −0.473184
\(601\) 16.2453 0.662658 0.331329 0.943515i \(-0.392503\pi\)
0.331329 + 0.943515i \(0.392503\pi\)
\(602\) 6.42235 0.261755
\(603\) −4.67267 −0.190286
\(604\) −14.6547 −0.596289
\(605\) −0.391382 −0.0159120
\(606\) −23.0950 −0.938171
\(607\) 41.6995 1.69253 0.846266 0.532761i \(-0.178845\pi\)
0.846266 + 0.532761i \(0.178845\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 15.7367 0.637684
\(610\) −0.782765 −0.0316932
\(611\) −30.5685 −1.23667
\(612\) 14.7828 0.597558
\(613\) 48.1900 1.94638 0.973189 0.230008i \(-0.0738751\pi\)
0.973189 + 0.230008i \(0.0738751\pi\)
\(614\) −9.22313 −0.372215
\(615\) −0.995897 −0.0401584
\(616\) 1.71871 0.0692488
\(617\) 15.7187 0.632811 0.316406 0.948624i \(-0.397524\pi\)
0.316406 + 0.948624i \(0.397524\pi\)
\(618\) −16.6296 −0.668940
\(619\) 44.6606 1.79506 0.897530 0.440954i \(-0.145360\pi\)
0.897530 + 0.440954i \(0.145360\pi\)
\(620\) 1.19922 0.0481617
\(621\) 2.31226 0.0927877
\(622\) 6.34829 0.254543
\(623\) −30.4102 −1.21836
\(624\) 8.89286 0.355999
\(625\) 22.7258 0.909030
\(626\) 13.3943 0.535345
\(627\) 2.39138 0.0955026
\(628\) −10.4835 −0.418335
\(629\) −25.3093 −1.00915
\(630\) 1.82880 0.0728612
\(631\) 2.38433 0.0949187 0.0474593 0.998873i \(-0.484888\pi\)
0.0474593 + 0.998873i \(0.484888\pi\)
\(632\) 13.0029 0.517230
\(633\) −14.5685 −0.579045
\(634\) 31.0029 1.23128
\(635\) 1.95807 0.0777035
\(636\) −2.91087 −0.115424
\(637\) −15.0460 −0.596146
\(638\) 3.82880 0.151584
\(639\) 2.84387 0.112502
\(640\) −0.391382 −0.0154707
\(641\) −45.6936 −1.80479 −0.902395 0.430910i \(-0.858193\pi\)
−0.902395 + 0.430910i \(0.858193\pi\)
\(642\) 6.34829 0.250547
\(643\) 42.2621 1.66666 0.833328 0.552779i \(-0.186433\pi\)
0.833328 + 0.552779i \(0.186433\pi\)
\(644\) 5.90793 0.232805
\(645\) 3.49737 0.137709
\(646\) −5.43742 −0.213932
\(647\) −38.7166 −1.52211 −0.761053 0.648690i \(-0.775317\pi\)
−0.761053 + 0.648690i \(0.775317\pi\)
\(648\) −9.76475 −0.383595
\(649\) −12.4404 −0.488327
\(650\) −18.0239 −0.706956
\(651\) 12.5935 0.493580
\(652\) −3.47346 −0.136031
\(653\) 38.6915 1.51412 0.757058 0.653348i \(-0.226636\pi\)
0.757058 + 0.653348i \(0.226636\pi\)
\(654\) 14.3483 0.561063
\(655\) 2.54456 0.0994243
\(656\) 1.06406 0.0415444
\(657\) 37.1311 1.44862
\(658\) 14.1281 0.550771
\(659\) 12.7467 0.496542 0.248271 0.968691i \(-0.420138\pi\)
0.248271 + 0.968691i \(0.420138\pi\)
\(660\) 0.935945 0.0364316
\(661\) −21.1671 −0.823305 −0.411652 0.911341i \(-0.635048\pi\)
−0.411652 + 0.911341i \(0.635048\pi\)
\(662\) −16.2993 −0.633491
\(663\) 48.3542 1.87792
\(664\) 8.51949 0.330620
\(665\) −0.672673 −0.0260851
\(666\) −12.6547 −0.490358
\(667\) 13.1612 0.509604
\(668\) 23.0950 0.893573
\(669\) −45.1370 −1.74510
\(670\) 0.672673 0.0259876
\(671\) −2.00000 −0.0772091
\(672\) −4.11009 −0.158550
\(673\) 2.25115 0.0867755 0.0433878 0.999058i \(-0.486185\pi\)
0.0433878 + 0.999058i \(0.486185\pi\)
\(674\) −3.19217 −0.122958
\(675\) 3.26032 0.125490
\(676\) 0.828802 0.0318770
\(677\) −9.57848 −0.368131 −0.184065 0.982914i \(-0.558926\pi\)
−0.184065 + 0.982914i \(0.558926\pi\)
\(678\) 18.0059 0.691512
\(679\) 8.00000 0.307012
\(680\) −2.12811 −0.0816093
\(681\) 50.1399 1.92137
\(682\) 3.06406 0.117329
\(683\) −32.0118 −1.22490 −0.612449 0.790510i \(-0.709815\pi\)
−0.612449 + 0.790510i \(0.709815\pi\)
\(684\) −2.71871 −0.103952
\(685\) −6.15203 −0.235057
\(686\) 18.9849 0.724848
\(687\) 33.0269 1.26005
\(688\) −3.73673 −0.142461
\(689\) −4.52654 −0.172448
\(690\) 3.21724 0.122478
\(691\) −25.9699 −0.987940 −0.493970 0.869479i \(-0.664455\pi\)
−0.493970 + 0.869479i \(0.664455\pi\)
\(692\) −18.1771 −0.690990
\(693\) 4.67267 0.177500
\(694\) 24.6966 0.937470
\(695\) 4.35926 0.165356
\(696\) −9.15613 −0.347062
\(697\) 5.78571 0.219150
\(698\) −4.43447 −0.167847
\(699\) 11.7438 0.444191
\(700\) 8.33028 0.314855
\(701\) −21.0950 −0.796748 −0.398374 0.917223i \(-0.630425\pi\)
−0.398374 + 0.917223i \(0.630425\pi\)
\(702\) −2.50147 −0.0944121
\(703\) 4.65465 0.175554
\(704\) −1.00000 −0.0376889
\(705\) 7.69364 0.289759
\(706\) 25.3152 0.952750
\(707\) 16.5986 0.624255
\(708\) 29.7497 1.11806
\(709\) 29.1981 1.09656 0.548278 0.836296i \(-0.315284\pi\)
0.548278 + 0.836296i \(0.315284\pi\)
\(710\) −0.409400 −0.0153645
\(711\) 35.3512 1.32578
\(712\) 17.6936 0.663097
\(713\) 10.5324 0.394443
\(714\) −22.3483 −0.836364
\(715\) 1.45544 0.0544303
\(716\) 11.2842 0.421712
\(717\) −33.5965 −1.25468
\(718\) −13.9749 −0.521540
\(719\) −13.9079 −0.518678 −0.259339 0.965786i \(-0.583505\pi\)
−0.259339 + 0.965786i \(0.583505\pi\)
\(720\) −1.06406 −0.0396550
\(721\) 11.9518 0.445110
\(722\) 1.00000 0.0372161
\(723\) 54.3793 2.02239
\(724\) 21.8778 0.813082
\(725\) 18.5575 0.689209
\(726\) 2.39138 0.0887525
\(727\) −42.0059 −1.55791 −0.778956 0.627078i \(-0.784251\pi\)
−0.778956 + 0.627078i \(0.784251\pi\)
\(728\) −6.39138 −0.236880
\(729\) −21.7217 −0.804506
\(730\) −5.34535 −0.197840
\(731\) −20.3182 −0.751494
\(732\) 4.78276 0.176776
\(733\) 36.0780 1.33257 0.666285 0.745697i \(-0.267883\pi\)
0.666285 + 0.745697i \(0.267883\pi\)
\(734\) −25.3453 −0.935514
\(735\) 3.78687 0.139681
\(736\) −3.43742 −0.126705
\(737\) 1.71871 0.0633095
\(738\) 2.89286 0.106488
\(739\) 49.9988 1.83924 0.919619 0.392812i \(-0.128498\pi\)
0.919619 + 0.392812i \(0.128498\pi\)
\(740\) 1.82175 0.0669689
\(741\) −8.89286 −0.326687
\(742\) 2.09207 0.0768025
\(743\) 0.220184 0.00807777 0.00403889 0.999992i \(-0.498714\pi\)
0.00403889 + 0.999992i \(0.498714\pi\)
\(744\) −7.32733 −0.268633
\(745\) −4.44037 −0.162683
\(746\) 8.95396 0.327828
\(747\) 23.1620 0.847454
\(748\) −5.43742 −0.198812
\(749\) −4.56258 −0.166713
\(750\) 9.21608 0.336524
\(751\) −23.1311 −0.844064 −0.422032 0.906581i \(-0.638683\pi\)
−0.422032 + 0.906581i \(0.638683\pi\)
\(752\) −8.22018 −0.299759
\(753\) −28.6966 −1.04576
\(754\) −14.2382 −0.518525
\(755\) 5.73557 0.208739
\(756\) 1.15613 0.0420480
\(757\) 37.4304 1.36043 0.680215 0.733013i \(-0.261886\pi\)
0.680215 + 0.733013i \(0.261886\pi\)
\(758\) −25.8527 −0.939013
\(759\) 8.22018 0.298374
\(760\) 0.391382 0.0141969
\(761\) −33.9499 −1.23068 −0.615341 0.788261i \(-0.710982\pi\)
−0.615341 + 0.788261i \(0.710982\pi\)
\(762\) −11.9640 −0.433409
\(763\) −10.3123 −0.373329
\(764\) −7.09502 −0.256689
\(765\) −5.78571 −0.209183
\(766\) −20.6296 −0.745377
\(767\) 46.2621 1.67043
\(768\) 2.39138 0.0862916
\(769\) −44.7025 −1.61201 −0.806006 0.591907i \(-0.798375\pi\)
−0.806006 + 0.591907i \(0.798375\pi\)
\(770\) −0.672673 −0.0242414
\(771\) −46.4964 −1.67453
\(772\) 12.6476 0.455197
\(773\) −44.8748 −1.61404 −0.807018 0.590527i \(-0.798920\pi\)
−0.807018 + 0.590527i \(0.798920\pi\)
\(774\) −10.1591 −0.365161
\(775\) 14.8509 0.533461
\(776\) −4.65465 −0.167092
\(777\) 19.1311 0.686323
\(778\) 15.6086 0.559596
\(779\) −1.06406 −0.0381237
\(780\) −3.48051 −0.124622
\(781\) −1.04604 −0.0374301
\(782\) −18.6907 −0.668378
\(783\) 2.57553 0.0920419
\(784\) −4.04604 −0.144501
\(785\) 4.10304 0.146444
\(786\) −15.5475 −0.554561
\(787\) 11.8418 0.422113 0.211056 0.977474i \(-0.432310\pi\)
0.211056 + 0.977474i \(0.432310\pi\)
\(788\) 8.43447 0.300466
\(789\) 35.0280 1.24703
\(790\) −5.08913 −0.181063
\(791\) −12.9410 −0.460129
\(792\) −2.71871 −0.0966051
\(793\) 7.43742 0.264111
\(794\) −34.2512 −1.21553
\(795\) 1.13927 0.0404056
\(796\) −2.09207 −0.0741516
\(797\) 0.128110 0.00453789 0.00226895 0.999997i \(-0.499278\pi\)
0.00226895 + 0.999997i \(0.499278\pi\)
\(798\) 4.11009 0.145496
\(799\) −44.6966 −1.58125
\(800\) −4.84682 −0.171361
\(801\) 48.1039 1.69967
\(802\) 4.09207 0.144496
\(803\) −13.6576 −0.481966
\(804\) −4.11009 −0.144952
\(805\) −2.31226 −0.0814964
\(806\) −11.3943 −0.401348
\(807\) 19.9640 0.702765
\(808\) −9.65760 −0.339753
\(809\) 17.5354 0.616512 0.308256 0.951304i \(-0.400255\pi\)
0.308256 + 0.951304i \(0.400255\pi\)
\(810\) 3.82175 0.134283
\(811\) −39.3512 −1.38181 −0.690905 0.722946i \(-0.742788\pi\)
−0.690905 + 0.722946i \(0.742788\pi\)
\(812\) 6.58060 0.230934
\(813\) −25.1992 −0.883775
\(814\) 4.65465 0.163146
\(815\) 1.35945 0.0476194
\(816\) 13.0029 0.455194
\(817\) 3.73673 0.130732
\(818\) 2.48346 0.0868320
\(819\) −17.3763 −0.607178
\(820\) −0.416452 −0.0145431
\(821\) 14.0980 0.492023 0.246011 0.969267i \(-0.420880\pi\)
0.246011 + 0.969267i \(0.420880\pi\)
\(822\) 37.5894 1.31108
\(823\) −1.12516 −0.0392207 −0.0196103 0.999808i \(-0.506243\pi\)
−0.0196103 + 0.999808i \(0.506243\pi\)
\(824\) −6.95396 −0.242253
\(825\) 11.5906 0.403533
\(826\) −21.3814 −0.743953
\(827\) −33.8217 −1.17610 −0.588049 0.808825i \(-0.700104\pi\)
−0.588049 + 0.808825i \(0.700104\pi\)
\(828\) −9.34535 −0.324773
\(829\) −36.1900 −1.25693 −0.628466 0.777837i \(-0.716317\pi\)
−0.628466 + 0.777837i \(0.716317\pi\)
\(830\) −3.33438 −0.115738
\(831\) −54.7166 −1.89810
\(832\) 3.71871 0.128923
\(833\) −22.0000 −0.762255
\(834\) −26.6355 −0.922311
\(835\) −9.03899 −0.312807
\(836\) 1.00000 0.0345857
\(837\) 2.06111 0.0712423
\(838\) 35.0950 1.21234
\(839\) −31.8347 −1.09906 −0.549528 0.835475i \(-0.685192\pi\)
−0.549528 + 0.835475i \(0.685192\pi\)
\(840\) 1.60862 0.0555026
\(841\) −14.3403 −0.494492
\(842\) −12.3483 −0.425550
\(843\) −10.5015 −0.361690
\(844\) −6.09207 −0.209698
\(845\) −0.324378 −0.0111590
\(846\) −22.3483 −0.768350
\(847\) −1.71871 −0.0590556
\(848\) −1.21724 −0.0418000
\(849\) −57.5534 −1.97523
\(850\) −26.3542 −0.903941
\(851\) 16.0000 0.548473
\(852\) 2.50147 0.0856991
\(853\) −31.1009 −1.06488 −0.532438 0.846469i \(-0.678724\pi\)
−0.532438 + 0.846469i \(0.678724\pi\)
\(854\) −3.43742 −0.117626
\(855\) 1.06406 0.0363899
\(856\) 2.65465 0.0907342
\(857\) −35.4985 −1.21261 −0.606303 0.795234i \(-0.707348\pi\)
−0.606303 + 0.795234i \(0.707348\pi\)
\(858\) −8.89286 −0.303597
\(859\) −5.63760 −0.192352 −0.0961762 0.995364i \(-0.530661\pi\)
−0.0961762 + 0.995364i \(0.530661\pi\)
\(860\) 1.46249 0.0498705
\(861\) −4.37336 −0.149044
\(862\) −6.94691 −0.236613
\(863\) 31.9949 1.08912 0.544560 0.838722i \(-0.316697\pi\)
0.544560 + 0.838722i \(0.316697\pi\)
\(864\) −0.672673 −0.0228848
\(865\) 7.11420 0.241890
\(866\) −13.2533 −0.450364
\(867\) 30.0490 1.02052
\(868\) 5.26622 0.178747
\(869\) −13.0029 −0.441095
\(870\) 3.58355 0.121494
\(871\) −6.39138 −0.216564
\(872\) 6.00000 0.203186
\(873\) −12.6547 −0.428295
\(874\) 3.43742 0.116272
\(875\) −6.62369 −0.223921
\(876\) 32.6606 1.10350
\(877\) −3.23019 −0.109076 −0.0545378 0.998512i \(-0.517369\pi\)
−0.0545378 + 0.998512i \(0.517369\pi\)
\(878\) 15.3152 0.516863
\(879\) −65.2612 −2.20120
\(880\) 0.391382 0.0131935
\(881\) −24.7217 −0.832894 −0.416447 0.909160i \(-0.636725\pi\)
−0.416447 + 0.909160i \(0.636725\pi\)
\(882\) −11.0000 −0.370389
\(883\) 11.9138 0.400932 0.200466 0.979701i \(-0.435754\pi\)
0.200466 + 0.979701i \(0.435754\pi\)
\(884\) 20.2202 0.680078
\(885\) −11.6435 −0.391392
\(886\) −20.8807 −0.701502
\(887\) −12.1841 −0.409104 −0.204552 0.978856i \(-0.565574\pi\)
−0.204552 + 0.978856i \(0.565574\pi\)
\(888\) −11.1311 −0.373534
\(889\) 8.59862 0.288388
\(890\) −6.92498 −0.232126
\(891\) 9.76475 0.327131
\(892\) −18.8748 −0.631976
\(893\) 8.22018 0.275078
\(894\) 27.1311 0.907398
\(895\) −4.41645 −0.147626
\(896\) −1.71871 −0.0574181
\(897\) −30.5685 −1.02065
\(898\) −28.7887 −0.960690
\(899\) 11.7317 0.391273
\(900\) −13.1771 −0.439237
\(901\) −6.61862 −0.220498
\(902\) −1.06406 −0.0354292
\(903\) 15.3583 0.511092
\(904\) 7.52949 0.250427
\(905\) −8.56258 −0.284630
\(906\) −35.0449 −1.16429
\(907\) 45.3211 1.50486 0.752431 0.658671i \(-0.228881\pi\)
0.752431 + 0.658671i \(0.228881\pi\)
\(908\) 20.9669 0.695811
\(909\) −26.2562 −0.870864
\(910\) 2.50147 0.0829231
\(911\) −13.6376 −0.451834 −0.225917 0.974147i \(-0.572538\pi\)
−0.225917 + 0.974147i \(0.572538\pi\)
\(912\) −2.39138 −0.0791866
\(913\) −8.51949 −0.281954
\(914\) −32.7526 −1.08336
\(915\) −1.87189 −0.0618828
\(916\) 13.8108 0.456321
\(917\) 11.1741 0.369003
\(918\) −3.65760 −0.120719
\(919\) 39.5283 1.30392 0.651960 0.758254i \(-0.273947\pi\)
0.651960 + 0.758254i \(0.273947\pi\)
\(920\) 1.34535 0.0443547
\(921\) −22.0560 −0.726771
\(922\) 20.2261 0.666111
\(923\) 3.88991 0.128038
\(924\) 4.11009 0.135212
\(925\) 22.5603 0.741777
\(926\) −18.6907 −0.614214
\(927\) −18.9058 −0.620948
\(928\) −3.82880 −0.125687
\(929\) 6.89991 0.226379 0.113189 0.993573i \(-0.463893\pi\)
0.113189 + 0.993573i \(0.463893\pi\)
\(930\) 2.86779 0.0940384
\(931\) 4.04604 0.132604
\(932\) 4.91087 0.160861
\(933\) 15.1812 0.497010
\(934\) 34.5685 1.13112
\(935\) 2.12811 0.0695966
\(936\) 10.1101 0.330459
\(937\) −21.3873 −0.698692 −0.349346 0.936994i \(-0.613596\pi\)
−0.349346 + 0.936994i \(0.613596\pi\)
\(938\) 2.95396 0.0964503
\(939\) 32.0310 1.04529
\(940\) 3.21724 0.104935
\(941\) −27.5655 −0.898611 −0.449305 0.893378i \(-0.648328\pi\)
−0.449305 + 0.893378i \(0.648328\pi\)
\(942\) −25.0700 −0.816823
\(943\) −3.65760 −0.119108
\(944\) 12.4404 0.404899
\(945\) −0.452489 −0.0147195
\(946\) 3.73673 0.121491
\(947\) −9.22313 −0.299712 −0.149856 0.988708i \(-0.547881\pi\)
−0.149856 + 0.988708i \(0.547881\pi\)
\(948\) 31.0950 1.00992
\(949\) 50.7887 1.64867
\(950\) 4.84682 0.157252
\(951\) 74.1399 2.40415
\(952\) −9.34535 −0.302884
\(953\) 22.6966 0.735215 0.367607 0.929981i \(-0.380177\pi\)
0.367607 + 0.929981i \(0.380177\pi\)
\(954\) −3.30931 −0.107143
\(955\) 2.77687 0.0898573
\(956\) −14.0490 −0.454377
\(957\) 9.15613 0.295976
\(958\) −17.7187 −0.572466
\(959\) −27.0159 −0.872389
\(960\) −0.935945 −0.0302075
\(961\) −21.6116 −0.697147
\(962\) −17.3093 −0.558075
\(963\) 7.21724 0.232572
\(964\) 22.7397 0.732396
\(965\) −4.95005 −0.159348
\(966\) 14.1281 0.454564
\(967\) −48.6966 −1.56598 −0.782988 0.622036i \(-0.786306\pi\)
−0.782988 + 0.622036i \(0.786306\pi\)
\(968\) 1.00000 0.0321412
\(969\) −13.0029 −0.417715
\(970\) 1.82175 0.0584929
\(971\) −19.5165 −0.626316 −0.313158 0.949701i \(-0.601387\pi\)
−0.313158 + 0.949701i \(0.601387\pi\)
\(972\) −21.3332 −0.684264
\(973\) 19.1432 0.613702
\(974\) 37.7426 1.20935
\(975\) −43.1021 −1.38037
\(976\) 2.00000 0.0640184
\(977\) 37.3512 1.19497 0.597486 0.801879i \(-0.296166\pi\)
0.597486 + 0.801879i \(0.296166\pi\)
\(978\) −8.30636 −0.265608
\(979\) −17.6936 −0.565491
\(980\) 1.58355 0.0505846
\(981\) 16.3123 0.520810
\(982\) 0.373364 0.0119145
\(983\) −45.8888 −1.46362 −0.731812 0.681507i \(-0.761325\pi\)
−0.731812 + 0.681507i \(0.761325\pi\)
\(984\) 2.54456 0.0811177
\(985\) −3.30110 −0.105182
\(986\) −20.8188 −0.663006
\(987\) 33.7857 1.07541
\(988\) −3.71871 −0.118308
\(989\) 12.8447 0.408438
\(990\) 1.06406 0.0338179
\(991\) −30.9418 −0.982900 −0.491450 0.870906i \(-0.663533\pi\)
−0.491450 + 0.870906i \(0.663533\pi\)
\(992\) −3.06406 −0.0972838
\(993\) −38.9779 −1.23693
\(994\) −1.79783 −0.0570238
\(995\) 0.818801 0.0259577
\(996\) 20.3734 0.645555
\(997\) 49.8418 1.57850 0.789252 0.614069i \(-0.210469\pi\)
0.789252 + 0.614069i \(0.210469\pi\)
\(998\) −14.8388 −0.469714
\(999\) 3.13106 0.0990623
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.h.1.3 3
3.2 odd 2 3762.2.a.bd.1.3 3
4.3 odd 2 3344.2.a.p.1.1 3
11.10 odd 2 4598.2.a.bm.1.3 3
19.18 odd 2 7942.2.a.bc.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.h.1.3 3 1.1 even 1 trivial
3344.2.a.p.1.1 3 4.3 odd 2
3762.2.a.bd.1.3 3 3.2 odd 2
4598.2.a.bm.1.3 3 11.10 odd 2
7942.2.a.bc.1.1 3 19.18 odd 2