Properties

Label 5054.2.a.bh.1.4
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.19520000000.1
Defining polynomial: \( x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.02989\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.02954 q^{3} +1.00000 q^{4} -1.38232 q^{5} -1.02954 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.94005 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.02954 q^{3} +1.00000 q^{4} -1.38232 q^{5} -1.02954 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.94005 q^{9} -1.38232 q^{10} +0.201562 q^{11} -1.02954 q^{12} +3.61000 q^{13} +1.00000 q^{14} +1.42315 q^{15} +1.00000 q^{16} -3.98785 q^{17} -1.94005 q^{18} -1.38232 q^{20} -1.02954 q^{21} +0.201562 q^{22} -1.95145 q^{23} -1.02954 q^{24} -3.08919 q^{25} +3.61000 q^{26} +5.08597 q^{27} +1.00000 q^{28} +4.85823 q^{29} +1.42315 q^{30} +3.87902 q^{31} +1.00000 q^{32} -0.207516 q^{33} -3.98785 q^{34} -1.38232 q^{35} -1.94005 q^{36} -8.95866 q^{37} -3.71663 q^{39} -1.38232 q^{40} -9.30590 q^{41} -1.02954 q^{42} +5.30315 q^{43} +0.201562 q^{44} +2.68177 q^{45} -1.95145 q^{46} +7.51513 q^{47} -1.02954 q^{48} +1.00000 q^{49} -3.08919 q^{50} +4.10564 q^{51} +3.61000 q^{52} -0.0971934 q^{53} +5.08597 q^{54} -0.278624 q^{55} +1.00000 q^{56} +4.85823 q^{58} -9.25161 q^{59} +1.42315 q^{60} -0.827230 q^{61} +3.87902 q^{62} -1.94005 q^{63} +1.00000 q^{64} -4.99017 q^{65} -0.207516 q^{66} -11.4197 q^{67} -3.98785 q^{68} +2.00909 q^{69} -1.38232 q^{70} +4.03775 q^{71} -1.94005 q^{72} -4.59223 q^{73} -8.95866 q^{74} +3.18044 q^{75} +0.201562 q^{77} -3.71663 q^{78} -13.2028 q^{79} -1.38232 q^{80} +0.583959 q^{81} -9.30590 q^{82} +10.1055 q^{83} -1.02954 q^{84} +5.51247 q^{85} +5.30315 q^{86} -5.00173 q^{87} +0.201562 q^{88} +6.42150 q^{89} +2.68177 q^{90} +3.61000 q^{91} -1.95145 q^{92} -3.99360 q^{93} +7.51513 q^{94} -1.02954 q^{96} -10.9140 q^{97} +1.00000 q^{98} -0.391042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} + 8 q^{8} + 8 q^{9} - 2 q^{10} - 12 q^{11} - 4 q^{12} - 10 q^{13} + 8 q^{14} - 24 q^{15} + 8 q^{16} - 6 q^{17} + 8 q^{18} - 2 q^{20} - 4 q^{21} - 12 q^{22} - 20 q^{23} - 4 q^{24} + 8 q^{25} - 10 q^{26} - 22 q^{27} + 8 q^{28} - 8 q^{29} - 24 q^{30} - 18 q^{31} + 8 q^{32} + 16 q^{33} - 6 q^{34} - 2 q^{35} + 8 q^{36} - 36 q^{37} - 2 q^{40} - 4 q^{41} - 4 q^{42} - 16 q^{43} - 12 q^{44} + 8 q^{45} - 20 q^{46} - 10 q^{47} - 4 q^{48} + 8 q^{49} + 8 q^{50} - 12 q^{51} - 10 q^{52} - 32 q^{53} - 22 q^{54} - 22 q^{55} + 8 q^{56} - 8 q^{58} + 2 q^{59} - 24 q^{60} + 2 q^{61} - 18 q^{62} + 8 q^{63} + 8 q^{64} + 16 q^{66} - 44 q^{67} - 6 q^{68} - 2 q^{70} - 8 q^{71} + 8 q^{72} + 30 q^{73} - 36 q^{74} + 16 q^{75} - 12 q^{77} - 60 q^{79} - 2 q^{80} + 12 q^{81} - 4 q^{82} - 28 q^{83} - 4 q^{84} - 16 q^{85} - 16 q^{86} + 24 q^{87} - 12 q^{88} + 22 q^{89} + 8 q^{90} - 10 q^{91} - 20 q^{92} - 16 q^{93} - 10 q^{94} - 4 q^{96} + 6 q^{97} + 8 q^{98} - 52 q^{99}+O(q^{100}) \) 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
\(3\) −1.02954 −0.594404 −0.297202 0.954815i \(-0.596053\pi\)
−0.297202 + 0.954815i \(0.596053\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.38232 −0.618192 −0.309096 0.951031i \(-0.600026\pi\)
−0.309096 + 0.951031i \(0.600026\pi\)
\(6\) −1.02954 −0.420307
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −1.94005 −0.646684
\(10\) −1.38232 −0.437128
\(11\) 0.201562 0.0607734 0.0303867 0.999538i \(-0.490326\pi\)
0.0303867 + 0.999538i \(0.490326\pi\)
\(12\) −1.02954 −0.297202
\(13\) 3.61000 1.00123 0.500616 0.865669i \(-0.333107\pi\)
0.500616 + 0.865669i \(0.333107\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.42315 0.367456
\(16\) 1.00000 0.250000
\(17\) −3.98785 −0.967195 −0.483597 0.875291i \(-0.660670\pi\)
−0.483597 + 0.875291i \(0.660670\pi\)
\(18\) −1.94005 −0.457275
\(19\) 0 0
\(20\) −1.38232 −0.309096
\(21\) −1.02954 −0.224664
\(22\) 0.201562 0.0429733
\(23\) −1.95145 −0.406905 −0.203453 0.979085i \(-0.565216\pi\)
−0.203453 + 0.979085i \(0.565216\pi\)
\(24\) −1.02954 −0.210153
\(25\) −3.08919 −0.617839
\(26\) 3.61000 0.707978
\(27\) 5.08597 0.978795
\(28\) 1.00000 0.188982
\(29\) 4.85823 0.902150 0.451075 0.892486i \(-0.351041\pi\)
0.451075 + 0.892486i \(0.351041\pi\)
\(30\) 1.42315 0.259830
\(31\) 3.87902 0.696693 0.348347 0.937366i \(-0.386743\pi\)
0.348347 + 0.937366i \(0.386743\pi\)
\(32\) 1.00000 0.176777
\(33\) −0.207516 −0.0361239
\(34\) −3.98785 −0.683910
\(35\) −1.38232 −0.233655
\(36\) −1.94005 −0.323342
\(37\) −8.95866 −1.47279 −0.736397 0.676550i \(-0.763474\pi\)
−0.736397 + 0.676550i \(0.763474\pi\)
\(38\) 0 0
\(39\) −3.71663 −0.595137
\(40\) −1.38232 −0.218564
\(41\) −9.30590 −1.45334 −0.726669 0.686988i \(-0.758932\pi\)
−0.726669 + 0.686988i \(0.758932\pi\)
\(42\) −1.02954 −0.158861
\(43\) 5.30315 0.808722 0.404361 0.914599i \(-0.367494\pi\)
0.404361 + 0.914599i \(0.367494\pi\)
\(44\) 0.201562 0.0303867
\(45\) 2.68177 0.399775
\(46\) −1.95145 −0.287726
\(47\) 7.51513 1.09619 0.548097 0.836414i \(-0.315352\pi\)
0.548097 + 0.836414i \(0.315352\pi\)
\(48\) −1.02954 −0.148601
\(49\) 1.00000 0.142857
\(50\) −3.08919 −0.436878
\(51\) 4.10564 0.574904
\(52\) 3.61000 0.500616
\(53\) −0.0971934 −0.0133505 −0.00667527 0.999978i \(-0.502125\pi\)
−0.00667527 + 0.999978i \(0.502125\pi\)
\(54\) 5.08597 0.692113
\(55\) −0.278624 −0.0375696
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 4.85823 0.637917
\(59\) −9.25161 −1.20446 −0.602229 0.798324i \(-0.705720\pi\)
−0.602229 + 0.798324i \(0.705720\pi\)
\(60\) 1.42315 0.183728
\(61\) −0.827230 −0.105916 −0.0529580 0.998597i \(-0.516865\pi\)
−0.0529580 + 0.998597i \(0.516865\pi\)
\(62\) 3.87902 0.492637
\(63\) −1.94005 −0.244424
\(64\) 1.00000 0.125000
\(65\) −4.99017 −0.618954
\(66\) −0.207516 −0.0255435
\(67\) −11.4197 −1.39514 −0.697571 0.716516i \(-0.745736\pi\)
−0.697571 + 0.716516i \(0.745736\pi\)
\(68\) −3.98785 −0.483597
\(69\) 2.00909 0.241866
\(70\) −1.38232 −0.165219
\(71\) 4.03775 0.479193 0.239596 0.970873i \(-0.422985\pi\)
0.239596 + 0.970873i \(0.422985\pi\)
\(72\) −1.94005 −0.228637
\(73\) −4.59223 −0.537480 −0.268740 0.963213i \(-0.586607\pi\)
−0.268740 + 0.963213i \(0.586607\pi\)
\(74\) −8.95866 −1.04142
\(75\) 3.18044 0.367246
\(76\) 0 0
\(77\) 0.201562 0.0229702
\(78\) −3.71663 −0.420825
\(79\) −13.2028 −1.48543 −0.742715 0.669607i \(-0.766462\pi\)
−0.742715 + 0.669607i \(0.766462\pi\)
\(80\) −1.38232 −0.154548
\(81\) 0.583959 0.0648843
\(82\) −9.30590 −1.02766
\(83\) 10.1055 1.10922 0.554610 0.832111i \(-0.312868\pi\)
0.554610 + 0.832111i \(0.312868\pi\)
\(84\) −1.02954 −0.112332
\(85\) 5.51247 0.597912
\(86\) 5.30315 0.571853
\(87\) −5.00173 −0.536242
\(88\) 0.201562 0.0214866
\(89\) 6.42150 0.680677 0.340339 0.940303i \(-0.389458\pi\)
0.340339 + 0.940303i \(0.389458\pi\)
\(90\) 2.68177 0.282683
\(91\) 3.61000 0.378430
\(92\) −1.95145 −0.203453
\(93\) −3.99360 −0.414117
\(94\) 7.51513 0.775127
\(95\) 0 0
\(96\) −1.02954 −0.105077
\(97\) −10.9140 −1.10815 −0.554074 0.832467i \(-0.686928\pi\)
−0.554074 + 0.832467i \(0.686928\pi\)
\(98\) 1.00000 0.101015
\(99\) −0.391042 −0.0393012
\(100\) −3.08919 −0.308919
\(101\) −2.36077 −0.234905 −0.117453 0.993078i \(-0.537473\pi\)
−0.117453 + 0.993078i \(0.537473\pi\)
\(102\) 4.10564 0.406519
\(103\) −7.52998 −0.741951 −0.370975 0.928643i \(-0.620977\pi\)
−0.370975 + 0.928643i \(0.620977\pi\)
\(104\) 3.61000 0.353989
\(105\) 1.42315 0.138885
\(106\) −0.0971934 −0.00944026
\(107\) 3.07601 0.297369 0.148684 0.988885i \(-0.452496\pi\)
0.148684 + 0.988885i \(0.452496\pi\)
\(108\) 5.08597 0.489398
\(109\) −2.31631 −0.221862 −0.110931 0.993828i \(-0.535383\pi\)
−0.110931 + 0.993828i \(0.535383\pi\)
\(110\) −0.278624 −0.0265657
\(111\) 9.22327 0.875434
\(112\) 1.00000 0.0944911
\(113\) −3.65946 −0.344253 −0.172127 0.985075i \(-0.555064\pi\)
−0.172127 + 0.985075i \(0.555064\pi\)
\(114\) 0 0
\(115\) 2.69753 0.251546
\(116\) 4.85823 0.451075
\(117\) −7.00358 −0.647481
\(118\) −9.25161 −0.851680
\(119\) −3.98785 −0.365565
\(120\) 1.42315 0.129915
\(121\) −10.9594 −0.996307
\(122\) −0.827230 −0.0748939
\(123\) 9.58077 0.863869
\(124\) 3.87902 0.348347
\(125\) 11.1818 1.00013
\(126\) −1.94005 −0.172834
\(127\) −3.21620 −0.285392 −0.142696 0.989767i \(-0.545577\pi\)
−0.142696 + 0.989767i \(0.545577\pi\)
\(128\) 1.00000 0.0883883
\(129\) −5.45979 −0.480708
\(130\) −4.99017 −0.437666
\(131\) −6.57612 −0.574559 −0.287279 0.957847i \(-0.592751\pi\)
−0.287279 + 0.957847i \(0.592751\pi\)
\(132\) −0.207516 −0.0180620
\(133\) 0 0
\(134\) −11.4197 −0.986514
\(135\) −7.03043 −0.605083
\(136\) −3.98785 −0.341955
\(137\) −18.4291 −1.57450 −0.787250 0.616634i \(-0.788496\pi\)
−0.787250 + 0.616634i \(0.788496\pi\)
\(138\) 2.00909 0.171025
\(139\) −3.46809 −0.294160 −0.147080 0.989125i \(-0.546987\pi\)
−0.147080 + 0.989125i \(0.546987\pi\)
\(140\) −1.38232 −0.116827
\(141\) −7.73711 −0.651583
\(142\) 4.03775 0.338840
\(143\) 0.727640 0.0608483
\(144\) −1.94005 −0.161671
\(145\) −6.71562 −0.557702
\(146\) −4.59223 −0.380056
\(147\) −1.02954 −0.0849148
\(148\) −8.95866 −0.736397
\(149\) 13.8826 1.13731 0.568653 0.822578i \(-0.307465\pi\)
0.568653 + 0.822578i \(0.307465\pi\)
\(150\) 3.18044 0.259682
\(151\) −19.5806 −1.59345 −0.796725 0.604342i \(-0.793436\pi\)
−0.796725 + 0.604342i \(0.793436\pi\)
\(152\) 0 0
\(153\) 7.73663 0.625469
\(154\) 0.201562 0.0162424
\(155\) −5.36205 −0.430690
\(156\) −3.71663 −0.297568
\(157\) −4.95955 −0.395816 −0.197908 0.980221i \(-0.563415\pi\)
−0.197908 + 0.980221i \(0.563415\pi\)
\(158\) −13.2028 −1.05036
\(159\) 0.100064 0.00793562
\(160\) −1.38232 −0.109282
\(161\) −1.95145 −0.153796
\(162\) 0.583959 0.0458801
\(163\) 10.2851 0.805594 0.402797 0.915289i \(-0.368038\pi\)
0.402797 + 0.915289i \(0.368038\pi\)
\(164\) −9.30590 −0.726669
\(165\) 0.286853 0.0223315
\(166\) 10.1055 0.784336
\(167\) 0.132958 0.0102886 0.00514432 0.999987i \(-0.498363\pi\)
0.00514432 + 0.999987i \(0.498363\pi\)
\(168\) −1.02954 −0.0794306
\(169\) 0.0320700 0.00246692
\(170\) 5.51247 0.422787
\(171\) 0 0
\(172\) 5.30315 0.404361
\(173\) 19.5740 1.48818 0.744091 0.668078i \(-0.232883\pi\)
0.744091 + 0.668078i \(0.232883\pi\)
\(174\) −5.00173 −0.379180
\(175\) −3.08919 −0.233521
\(176\) 0.201562 0.0151933
\(177\) 9.52488 0.715934
\(178\) 6.42150 0.481312
\(179\) −2.05705 −0.153751 −0.0768756 0.997041i \(-0.524494\pi\)
−0.0768756 + 0.997041i \(0.524494\pi\)
\(180\) 2.68177 0.199887
\(181\) −4.47667 −0.332748 −0.166374 0.986063i \(-0.553206\pi\)
−0.166374 + 0.986063i \(0.553206\pi\)
\(182\) 3.61000 0.267591
\(183\) 0.851665 0.0629569
\(184\) −1.95145 −0.143863
\(185\) 12.3837 0.910469
\(186\) −3.99360 −0.292825
\(187\) −0.803800 −0.0587797
\(188\) 7.51513 0.548097
\(189\) 5.08597 0.369950
\(190\) 0 0
\(191\) −3.44067 −0.248958 −0.124479 0.992222i \(-0.539726\pi\)
−0.124479 + 0.992222i \(0.539726\pi\)
\(192\) −1.02954 −0.0743005
\(193\) −25.1998 −1.81392 −0.906959 0.421219i \(-0.861603\pi\)
−0.906959 + 0.421219i \(0.861603\pi\)
\(194\) −10.9140 −0.783579
\(195\) 5.13756 0.367909
\(196\) 1.00000 0.0714286
\(197\) −14.3027 −1.01902 −0.509512 0.860464i \(-0.670174\pi\)
−0.509512 + 0.860464i \(0.670174\pi\)
\(198\) −0.391042 −0.0277901
\(199\) 12.3169 0.873121 0.436560 0.899675i \(-0.356197\pi\)
0.436560 + 0.899675i \(0.356197\pi\)
\(200\) −3.08919 −0.218439
\(201\) 11.7570 0.829278
\(202\) −2.36077 −0.166103
\(203\) 4.85823 0.340981
\(204\) 4.10564 0.287452
\(205\) 12.8637 0.898441
\(206\) −7.52998 −0.524638
\(207\) 3.78591 0.263139
\(208\) 3.61000 0.250308
\(209\) 0 0
\(210\) 1.42315 0.0982066
\(211\) −22.8505 −1.57309 −0.786547 0.617530i \(-0.788133\pi\)
−0.786547 + 0.617530i \(0.788133\pi\)
\(212\) −0.0971934 −0.00667527
\(213\) −4.15702 −0.284834
\(214\) 3.07601 0.210272
\(215\) −7.33064 −0.499946
\(216\) 5.08597 0.346056
\(217\) 3.87902 0.263325
\(218\) −2.31631 −0.156880
\(219\) 4.72788 0.319480
\(220\) −0.278624 −0.0187848
\(221\) −14.3961 −0.968387
\(222\) 9.22327 0.619026
\(223\) 7.72793 0.517501 0.258750 0.965944i \(-0.416689\pi\)
0.258750 + 0.965944i \(0.416689\pi\)
\(224\) 1.00000 0.0668153
\(225\) 5.99320 0.399547
\(226\) −3.65946 −0.243424
\(227\) −25.9981 −1.72556 −0.862779 0.505581i \(-0.831278\pi\)
−0.862779 + 0.505581i \(0.831278\pi\)
\(228\) 0 0
\(229\) −24.1289 −1.59448 −0.797242 0.603660i \(-0.793708\pi\)
−0.797242 + 0.603660i \(0.793708\pi\)
\(230\) 2.69753 0.177870
\(231\) −0.207516 −0.0136536
\(232\) 4.85823 0.318958
\(233\) −27.2074 −1.78242 −0.891208 0.453595i \(-0.850141\pi\)
−0.891208 + 0.453595i \(0.850141\pi\)
\(234\) −7.00358 −0.457838
\(235\) −10.3883 −0.677659
\(236\) −9.25161 −0.602229
\(237\) 13.5928 0.882946
\(238\) −3.98785 −0.258494
\(239\) 26.5413 1.71681 0.858406 0.512971i \(-0.171455\pi\)
0.858406 + 0.512971i \(0.171455\pi\)
\(240\) 1.42315 0.0918639
\(241\) 11.8064 0.760517 0.380258 0.924880i \(-0.375835\pi\)
0.380258 + 0.924880i \(0.375835\pi\)
\(242\) −10.9594 −0.704495
\(243\) −15.8591 −1.01736
\(244\) −0.827230 −0.0529580
\(245\) −1.38232 −0.0883131
\(246\) 9.58077 0.610848
\(247\) 0 0
\(248\) 3.87902 0.246318
\(249\) −10.4040 −0.659324
\(250\) 11.1818 0.707202
\(251\) 11.1809 0.705735 0.352867 0.935673i \(-0.385207\pi\)
0.352867 + 0.935673i \(0.385207\pi\)
\(252\) −1.94005 −0.122212
\(253\) −0.393339 −0.0247290
\(254\) −3.21620 −0.201802
\(255\) −5.67530 −0.355401
\(256\) 1.00000 0.0625000
\(257\) 29.5934 1.84599 0.922993 0.384817i \(-0.125736\pi\)
0.922993 + 0.384817i \(0.125736\pi\)
\(258\) −5.45979 −0.339912
\(259\) −8.95866 −0.556664
\(260\) −4.99017 −0.309477
\(261\) −9.42522 −0.583406
\(262\) −6.57612 −0.406274
\(263\) −24.6167 −1.51793 −0.758966 0.651131i \(-0.774295\pi\)
−0.758966 + 0.651131i \(0.774295\pi\)
\(264\) −0.207516 −0.0127717
\(265\) 0.134352 0.00825320
\(266\) 0 0
\(267\) −6.61117 −0.404597
\(268\) −11.4197 −0.697571
\(269\) 7.74788 0.472397 0.236198 0.971705i \(-0.424098\pi\)
0.236198 + 0.971705i \(0.424098\pi\)
\(270\) −7.03043 −0.427858
\(271\) −11.5762 −0.703204 −0.351602 0.936150i \(-0.614363\pi\)
−0.351602 + 0.936150i \(0.614363\pi\)
\(272\) −3.98785 −0.241799
\(273\) −3.71663 −0.224940
\(274\) −18.4291 −1.11334
\(275\) −0.622666 −0.0375481
\(276\) 2.00909 0.120933
\(277\) 4.90088 0.294465 0.147233 0.989102i \(-0.452963\pi\)
0.147233 + 0.989102i \(0.452963\pi\)
\(278\) −3.46809 −0.208002
\(279\) −7.52551 −0.450541
\(280\) −1.38232 −0.0826094
\(281\) −16.3213 −0.973650 −0.486825 0.873500i \(-0.661845\pi\)
−0.486825 + 0.873500i \(0.661845\pi\)
\(282\) −7.73711 −0.460738
\(283\) −19.1772 −1.13997 −0.569983 0.821656i \(-0.693050\pi\)
−0.569983 + 0.821656i \(0.693050\pi\)
\(284\) 4.03775 0.239596
\(285\) 0 0
\(286\) 0.727640 0.0430262
\(287\) −9.30590 −0.549310
\(288\) −1.94005 −0.114319
\(289\) −1.09709 −0.0645348
\(290\) −6.71562 −0.394355
\(291\) 11.2364 0.658688
\(292\) −4.59223 −0.268740
\(293\) 9.38811 0.548459 0.274230 0.961664i \(-0.411577\pi\)
0.274230 + 0.961664i \(0.411577\pi\)
\(294\) −1.02954 −0.0600439
\(295\) 12.7887 0.744586
\(296\) −8.95866 −0.520711
\(297\) 1.02514 0.0594847
\(298\) 13.8826 0.804196
\(299\) −7.04473 −0.407407
\(300\) 3.18044 0.183623
\(301\) 5.30315 0.305668
\(302\) −19.5806 −1.12674
\(303\) 2.43050 0.139629
\(304\) 0 0
\(305\) 1.14350 0.0654764
\(306\) 7.73663 0.442274
\(307\) −24.5234 −1.39962 −0.699811 0.714328i \(-0.746733\pi\)
−0.699811 + 0.714328i \(0.746733\pi\)
\(308\) 0.201562 0.0114851
\(309\) 7.75240 0.441018
\(310\) −5.36205 −0.304544
\(311\) 11.6440 0.660272 0.330136 0.943933i \(-0.392905\pi\)
0.330136 + 0.943933i \(0.392905\pi\)
\(312\) −3.71663 −0.210413
\(313\) −9.14724 −0.517033 −0.258516 0.966007i \(-0.583234\pi\)
−0.258516 + 0.966007i \(0.583234\pi\)
\(314\) −4.95955 −0.279884
\(315\) 2.68177 0.151101
\(316\) −13.2028 −0.742715
\(317\) 9.31007 0.522906 0.261453 0.965216i \(-0.415798\pi\)
0.261453 + 0.965216i \(0.415798\pi\)
\(318\) 0.100064 0.00561133
\(319\) 0.979237 0.0548267
\(320\) −1.38232 −0.0772740
\(321\) −3.16687 −0.176757
\(322\) −1.95145 −0.108750
\(323\) 0 0
\(324\) 0.583959 0.0324422
\(325\) −11.1520 −0.618601
\(326\) 10.2851 0.569641
\(327\) 2.38472 0.131876
\(328\) −9.30590 −0.513832
\(329\) 7.51513 0.414323
\(330\) 0.286853 0.0157908
\(331\) −26.6779 −1.46635 −0.733174 0.680041i \(-0.761962\pi\)
−0.733174 + 0.680041i \(0.761962\pi\)
\(332\) 10.1055 0.554610
\(333\) 17.3803 0.952432
\(334\) 0.132958 0.00727516
\(335\) 15.7857 0.862465
\(336\) −1.02954 −0.0561659
\(337\) −4.47140 −0.243573 −0.121786 0.992556i \(-0.538862\pi\)
−0.121786 + 0.992556i \(0.538862\pi\)
\(338\) 0.0320700 0.00174438
\(339\) 3.76755 0.204625
\(340\) 5.51247 0.298956
\(341\) 0.781866 0.0423404
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 5.30315 0.285927
\(345\) −2.77720 −0.149520
\(346\) 19.5740 1.05230
\(347\) 3.89738 0.209222 0.104611 0.994513i \(-0.466640\pi\)
0.104611 + 0.994513i \(0.466640\pi\)
\(348\) −5.00173 −0.268121
\(349\) 13.5634 0.726031 0.363015 0.931783i \(-0.381747\pi\)
0.363015 + 0.931783i \(0.381747\pi\)
\(350\) −3.08919 −0.165124
\(351\) 18.3603 0.980002
\(352\) 0.201562 0.0107433
\(353\) 27.5213 1.46481 0.732405 0.680869i \(-0.238398\pi\)
0.732405 + 0.680869i \(0.238398\pi\)
\(354\) 9.52488 0.506242
\(355\) −5.58146 −0.296233
\(356\) 6.42150 0.340339
\(357\) 4.10564 0.217293
\(358\) −2.05705 −0.108718
\(359\) 16.7795 0.885590 0.442795 0.896623i \(-0.353987\pi\)
0.442795 + 0.896623i \(0.353987\pi\)
\(360\) 2.68177 0.141342
\(361\) 0 0
\(362\) −4.47667 −0.235288
\(363\) 11.2831 0.592208
\(364\) 3.61000 0.189215
\(365\) 6.34793 0.332266
\(366\) 0.851665 0.0445172
\(367\) 28.5583 1.49073 0.745365 0.666656i \(-0.232275\pi\)
0.745365 + 0.666656i \(0.232275\pi\)
\(368\) −1.95145 −0.101726
\(369\) 18.0539 0.939850
\(370\) 12.3837 0.643799
\(371\) −0.0971934 −0.00504603
\(372\) −3.99360 −0.207059
\(373\) −0.530273 −0.0274565 −0.0137282 0.999906i \(-0.504370\pi\)
−0.0137282 + 0.999906i \(0.504370\pi\)
\(374\) −0.803800 −0.0415635
\(375\) −11.5121 −0.594484
\(376\) 7.51513 0.387563
\(377\) 17.5382 0.903263
\(378\) 5.08597 0.261594
\(379\) 0.442763 0.0227432 0.0113716 0.999935i \(-0.496380\pi\)
0.0113716 + 0.999935i \(0.496380\pi\)
\(380\) 0 0
\(381\) 3.31120 0.169638
\(382\) −3.44067 −0.176040
\(383\) −3.06625 −0.156678 −0.0783390 0.996927i \(-0.524962\pi\)
−0.0783390 + 0.996927i \(0.524962\pi\)
\(384\) −1.02954 −0.0525384
\(385\) −0.278624 −0.0142000
\(386\) −25.1998 −1.28263
\(387\) −10.2884 −0.522988
\(388\) −10.9140 −0.554074
\(389\) −14.5166 −0.736021 −0.368010 0.929822i \(-0.619961\pi\)
−0.368010 + 0.929822i \(0.619961\pi\)
\(390\) 5.13756 0.260151
\(391\) 7.78208 0.393557
\(392\) 1.00000 0.0505076
\(393\) 6.77037 0.341520
\(394\) −14.3027 −0.720559
\(395\) 18.2505 0.918281
\(396\) −0.391042 −0.0196506
\(397\) 27.0449 1.35735 0.678674 0.734440i \(-0.262555\pi\)
0.678674 + 0.734440i \(0.262555\pi\)
\(398\) 12.3169 0.617390
\(399\) 0 0
\(400\) −3.08919 −0.154460
\(401\) −1.61913 −0.0808556 −0.0404278 0.999182i \(-0.512872\pi\)
−0.0404278 + 0.999182i \(0.512872\pi\)
\(402\) 11.7570 0.586388
\(403\) 14.0033 0.697552
\(404\) −2.36077 −0.117453
\(405\) −0.807217 −0.0401110
\(406\) 4.85823 0.241110
\(407\) −1.80573 −0.0895067
\(408\) 4.10564 0.203259
\(409\) 6.10992 0.302116 0.151058 0.988525i \(-0.451732\pi\)
0.151058 + 0.988525i \(0.451732\pi\)
\(410\) 12.8637 0.635294
\(411\) 18.9734 0.935889
\(412\) −7.52998 −0.370975
\(413\) −9.25161 −0.455242
\(414\) 3.78591 0.186068
\(415\) −13.9690 −0.685710
\(416\) 3.61000 0.176995
\(417\) 3.57053 0.174850
\(418\) 0 0
\(419\) 1.92152 0.0938723 0.0469362 0.998898i \(-0.485054\pi\)
0.0469362 + 0.998898i \(0.485054\pi\)
\(420\) 1.42315 0.0694426
\(421\) 31.5836 1.53929 0.769645 0.638472i \(-0.220433\pi\)
0.769645 + 0.638472i \(0.220433\pi\)
\(422\) −22.8505 −1.11235
\(423\) −14.5798 −0.708892
\(424\) −0.0971934 −0.00472013
\(425\) 12.3192 0.597570
\(426\) −4.15702 −0.201408
\(427\) −0.827230 −0.0400325
\(428\) 3.07601 0.148684
\(429\) −0.749132 −0.0361685
\(430\) −7.33064 −0.353515
\(431\) 24.5430 1.18219 0.591097 0.806601i \(-0.298695\pi\)
0.591097 + 0.806601i \(0.298695\pi\)
\(432\) 5.08597 0.244699
\(433\) −35.6753 −1.71444 −0.857222 0.514947i \(-0.827811\pi\)
−0.857222 + 0.514947i \(0.827811\pi\)
\(434\) 3.87902 0.186199
\(435\) 6.91399 0.331500
\(436\) −2.31631 −0.110931
\(437\) 0 0
\(438\) 4.72788 0.225907
\(439\) 10.6745 0.509466 0.254733 0.967011i \(-0.418012\pi\)
0.254733 + 0.967011i \(0.418012\pi\)
\(440\) −0.278624 −0.0132829
\(441\) −1.94005 −0.0923834
\(442\) −14.3961 −0.684753
\(443\) −28.1790 −1.33882 −0.669412 0.742891i \(-0.733454\pi\)
−0.669412 + 0.742891i \(0.733454\pi\)
\(444\) 9.22327 0.437717
\(445\) −8.87656 −0.420789
\(446\) 7.72793 0.365928
\(447\) −14.2926 −0.676019
\(448\) 1.00000 0.0472456
\(449\) −1.02365 −0.0483091 −0.0241545 0.999708i \(-0.507689\pi\)
−0.0241545 + 0.999708i \(0.507689\pi\)
\(450\) 5.99320 0.282522
\(451\) −1.87572 −0.0883242
\(452\) −3.65946 −0.172127
\(453\) 20.1590 0.947153
\(454\) −25.9981 −1.22015
\(455\) −4.99017 −0.233943
\(456\) 0 0
\(457\) 30.0439 1.40539 0.702697 0.711489i \(-0.251979\pi\)
0.702697 + 0.711489i \(0.251979\pi\)
\(458\) −24.1289 −1.12747
\(459\) −20.2821 −0.946685
\(460\) 2.69753 0.125773
\(461\) −11.9442 −0.556298 −0.278149 0.960538i \(-0.589721\pi\)
−0.278149 + 0.960538i \(0.589721\pi\)
\(462\) −0.207516 −0.00965452
\(463\) −27.3252 −1.26991 −0.634954 0.772550i \(-0.718981\pi\)
−0.634954 + 0.772550i \(0.718981\pi\)
\(464\) 4.85823 0.225538
\(465\) 5.52043 0.256004
\(466\) −27.2074 −1.26036
\(467\) −7.73250 −0.357818 −0.178909 0.983866i \(-0.557257\pi\)
−0.178909 + 0.983866i \(0.557257\pi\)
\(468\) −7.00358 −0.323741
\(469\) −11.4197 −0.527314
\(470\) −10.3883 −0.479177
\(471\) 5.10605 0.235274
\(472\) −9.25161 −0.425840
\(473\) 1.06892 0.0491488
\(474\) 13.5928 0.624337
\(475\) 0 0
\(476\) −3.98785 −0.182783
\(477\) 0.188560 0.00863358
\(478\) 26.5413 1.21397
\(479\) −1.52899 −0.0698614 −0.0349307 0.999390i \(-0.511121\pi\)
−0.0349307 + 0.999390i \(0.511121\pi\)
\(480\) 1.42315 0.0649576
\(481\) −32.3407 −1.47461
\(482\) 11.8064 0.537767
\(483\) 2.00909 0.0914168
\(484\) −10.9594 −0.498153
\(485\) 15.0866 0.685048
\(486\) −15.8591 −0.719384
\(487\) 14.6532 0.663999 0.332000 0.943280i \(-0.392277\pi\)
0.332000 + 0.943280i \(0.392277\pi\)
\(488\) −0.827230 −0.0374469
\(489\) −10.5889 −0.478848
\(490\) −1.38232 −0.0624468
\(491\) 19.4632 0.878362 0.439181 0.898399i \(-0.355269\pi\)
0.439181 + 0.898399i \(0.355269\pi\)
\(492\) 9.58077 0.431935
\(493\) −19.3739 −0.872555
\(494\) 0 0
\(495\) 0.540544 0.0242957
\(496\) 3.87902 0.174173
\(497\) 4.03775 0.181118
\(498\) −10.4040 −0.466213
\(499\) 8.21283 0.367657 0.183828 0.982958i \(-0.441151\pi\)
0.183828 + 0.982958i \(0.441151\pi\)
\(500\) 11.1818 0.500067
\(501\) −0.136886 −0.00611560
\(502\) 11.1809 0.499030
\(503\) 0.969968 0.0432488 0.0216244 0.999766i \(-0.493116\pi\)
0.0216244 + 0.999766i \(0.493116\pi\)
\(504\) −1.94005 −0.0864168
\(505\) 3.26333 0.145216
\(506\) −0.393339 −0.0174861
\(507\) −0.0330173 −0.00146635
\(508\) −3.21620 −0.142696
\(509\) −10.3220 −0.457516 −0.228758 0.973483i \(-0.573466\pi\)
−0.228758 + 0.973483i \(0.573466\pi\)
\(510\) −5.67530 −0.251306
\(511\) −4.59223 −0.203148
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 29.5934 1.30531
\(515\) 10.4088 0.458668
\(516\) −5.45979 −0.240354
\(517\) 1.51477 0.0666195
\(518\) −8.95866 −0.393621
\(519\) −20.1521 −0.884581
\(520\) −4.99017 −0.218833
\(521\) 6.77463 0.296802 0.148401 0.988927i \(-0.452587\pi\)
0.148401 + 0.988927i \(0.452587\pi\)
\(522\) −9.42522 −0.412531
\(523\) 41.6053 1.81927 0.909636 0.415405i \(-0.136360\pi\)
0.909636 + 0.415405i \(0.136360\pi\)
\(524\) −6.57612 −0.287279
\(525\) 3.18044 0.138806
\(526\) −24.6167 −1.07334
\(527\) −15.4689 −0.673838
\(528\) −0.207516 −0.00903098
\(529\) −19.1918 −0.834428
\(530\) 0.134352 0.00583589
\(531\) 17.9486 0.778903
\(532\) 0 0
\(533\) −33.5943 −1.45513
\(534\) −6.61117 −0.286093
\(535\) −4.25202 −0.183831
\(536\) −11.4197 −0.493257
\(537\) 2.11781 0.0913903
\(538\) 7.74788 0.334035
\(539\) 0.201562 0.00868191
\(540\) −7.03043 −0.302542
\(541\) −40.8132 −1.75470 −0.877348 0.479855i \(-0.840689\pi\)
−0.877348 + 0.479855i \(0.840689\pi\)
\(542\) −11.5762 −0.497241
\(543\) 4.60890 0.197787
\(544\) −3.98785 −0.170977
\(545\) 3.20187 0.137153
\(546\) −3.71663 −0.159057
\(547\) −27.9469 −1.19492 −0.597461 0.801898i \(-0.703824\pi\)
−0.597461 + 0.801898i \(0.703824\pi\)
\(548\) −18.4291 −0.787250
\(549\) 1.60487 0.0684942
\(550\) −0.622666 −0.0265506
\(551\) 0 0
\(552\) 2.00909 0.0855126
\(553\) −13.2028 −0.561440
\(554\) 4.90088 0.208218
\(555\) −12.7495 −0.541186
\(556\) −3.46809 −0.147080
\(557\) 22.0059 0.932421 0.466211 0.884674i \(-0.345619\pi\)
0.466211 + 0.884674i \(0.345619\pi\)
\(558\) −7.52551 −0.318580
\(559\) 19.1443 0.809719
\(560\) −1.38232 −0.0584136
\(561\) 0.827542 0.0349389
\(562\) −16.3213 −0.688475
\(563\) 43.5246 1.83434 0.917171 0.398493i \(-0.130467\pi\)
0.917171 + 0.398493i \(0.130467\pi\)
\(564\) −7.73711 −0.325791
\(565\) 5.05854 0.212814
\(566\) −19.1772 −0.806078
\(567\) 0.583959 0.0245240
\(568\) 4.03775 0.169420
\(569\) −27.9700 −1.17256 −0.586281 0.810108i \(-0.699409\pi\)
−0.586281 + 0.810108i \(0.699409\pi\)
\(570\) 0 0
\(571\) 29.3563 1.22852 0.614261 0.789103i \(-0.289454\pi\)
0.614261 + 0.789103i \(0.289454\pi\)
\(572\) 0.727640 0.0304241
\(573\) 3.54230 0.147982
\(574\) −9.30590 −0.388421
\(575\) 6.02841 0.251402
\(576\) −1.94005 −0.0808355
\(577\) −5.18369 −0.215800 −0.107900 0.994162i \(-0.534413\pi\)
−0.107900 + 0.994162i \(0.534413\pi\)
\(578\) −1.09709 −0.0456330
\(579\) 25.9441 1.07820
\(580\) −6.71562 −0.278851
\(581\) 10.1055 0.419245
\(582\) 11.2364 0.465762
\(583\) −0.0195905 −0.000811358 0
\(584\) −4.59223 −0.190028
\(585\) 9.68118 0.400268
\(586\) 9.38811 0.387819
\(587\) −15.2575 −0.629744 −0.314872 0.949134i \(-0.601962\pi\)
−0.314872 + 0.949134i \(0.601962\pi\)
\(588\) −1.02954 −0.0424574
\(589\) 0 0
\(590\) 12.7887 0.526502
\(591\) 14.7252 0.605712
\(592\) −8.95866 −0.368199
\(593\) 41.0813 1.68701 0.843504 0.537123i \(-0.180489\pi\)
0.843504 + 0.537123i \(0.180489\pi\)
\(594\) 1.02514 0.0420620
\(595\) 5.51247 0.225989
\(596\) 13.8826 0.568653
\(597\) −12.6807 −0.518986
\(598\) −7.04473 −0.288080
\(599\) −25.9404 −1.05990 −0.529948 0.848030i \(-0.677789\pi\)
−0.529948 + 0.848030i \(0.677789\pi\)
\(600\) 3.18044 0.129841
\(601\) −21.2064 −0.865026 −0.432513 0.901628i \(-0.642373\pi\)
−0.432513 + 0.901628i \(0.642373\pi\)
\(602\) 5.30315 0.216140
\(603\) 22.1549 0.902216
\(604\) −19.5806 −0.796725
\(605\) 15.1493 0.615909
\(606\) 2.43050 0.0987323
\(607\) −3.45922 −0.140405 −0.0702027 0.997533i \(-0.522365\pi\)
−0.0702027 + 0.997533i \(0.522365\pi\)
\(608\) 0 0
\(609\) −5.00173 −0.202680
\(610\) 1.14350 0.0462988
\(611\) 27.1296 1.09755
\(612\) 7.73663 0.312735
\(613\) 40.4587 1.63411 0.817056 0.576558i \(-0.195604\pi\)
0.817056 + 0.576558i \(0.195604\pi\)
\(614\) −24.5234 −0.989682
\(615\) −13.2437 −0.534037
\(616\) 0.201562 0.00812118
\(617\) −6.54854 −0.263634 −0.131817 0.991274i \(-0.542081\pi\)
−0.131817 + 0.991274i \(0.542081\pi\)
\(618\) 7.75240 0.311847
\(619\) 1.12076 0.0450470 0.0225235 0.999746i \(-0.492830\pi\)
0.0225235 + 0.999746i \(0.492830\pi\)
\(620\) −5.36205 −0.215345
\(621\) −9.92501 −0.398277
\(622\) 11.6440 0.466883
\(623\) 6.42150 0.257272
\(624\) −3.71663 −0.148784
\(625\) −0.0109054 −0.000436216 0
\(626\) −9.14724 −0.365597
\(627\) 0 0
\(628\) −4.95955 −0.197908
\(629\) 35.7257 1.42448
\(630\) 2.68177 0.106844
\(631\) 46.4791 1.85030 0.925152 0.379598i \(-0.123938\pi\)
0.925152 + 0.379598i \(0.123938\pi\)
\(632\) −13.2028 −0.525179
\(633\) 23.5255 0.935054
\(634\) 9.31007 0.369750
\(635\) 4.44581 0.176427
\(636\) 0.100064 0.00396781
\(637\) 3.61000 0.143033
\(638\) 0.979237 0.0387683
\(639\) −7.83345 −0.309886
\(640\) −1.38232 −0.0546410
\(641\) −9.98955 −0.394563 −0.197282 0.980347i \(-0.563211\pi\)
−0.197282 + 0.980347i \(0.563211\pi\)
\(642\) −3.16687 −0.124986
\(643\) −18.7882 −0.740936 −0.370468 0.928845i \(-0.620803\pi\)
−0.370468 + 0.928845i \(0.620803\pi\)
\(644\) −1.95145 −0.0768979
\(645\) 7.54717 0.297170
\(646\) 0 0
\(647\) −6.40876 −0.251954 −0.125977 0.992033i \(-0.540207\pi\)
−0.125977 + 0.992033i \(0.540207\pi\)
\(648\) 0.583959 0.0229401
\(649\) −1.86478 −0.0731989
\(650\) −11.1520 −0.437417
\(651\) −3.99360 −0.156522
\(652\) 10.2851 0.402797
\(653\) 38.4509 1.50470 0.752351 0.658763i \(-0.228920\pi\)
0.752351 + 0.658763i \(0.228920\pi\)
\(654\) 2.38472 0.0932501
\(655\) 9.09030 0.355187
\(656\) −9.30590 −0.363334
\(657\) 8.90917 0.347580
\(658\) 7.51513 0.292970
\(659\) −18.6345 −0.725897 −0.362948 0.931809i \(-0.618230\pi\)
−0.362948 + 0.931809i \(0.618230\pi\)
\(660\) 0.286853 0.0111658
\(661\) −20.2296 −0.786840 −0.393420 0.919359i \(-0.628708\pi\)
−0.393420 + 0.919359i \(0.628708\pi\)
\(662\) −26.6779 −1.03687
\(663\) 14.8213 0.575613
\(664\) 10.1055 0.392168
\(665\) 0 0
\(666\) 17.3803 0.673471
\(667\) −9.48059 −0.367090
\(668\) 0.132958 0.00514432
\(669\) −7.95620 −0.307604
\(670\) 15.7857 0.609855
\(671\) −0.166739 −0.00643687
\(672\) −1.02954 −0.0397153
\(673\) 27.1339 1.04593 0.522967 0.852353i \(-0.324825\pi\)
0.522967 + 0.852353i \(0.324825\pi\)
\(674\) −4.47140 −0.172232
\(675\) −15.7115 −0.604738
\(676\) 0.0320700 0.00123346
\(677\) 23.1322 0.889041 0.444521 0.895769i \(-0.353374\pi\)
0.444521 + 0.895769i \(0.353374\pi\)
\(678\) 3.76755 0.144692
\(679\) −10.9140 −0.418841
\(680\) 5.51247 0.211394
\(681\) 26.7661 1.02568
\(682\) 0.781866 0.0299392
\(683\) −24.3905 −0.933277 −0.466639 0.884448i \(-0.654535\pi\)
−0.466639 + 0.884448i \(0.654535\pi\)
\(684\) 0 0
\(685\) 25.4748 0.973343
\(686\) 1.00000 0.0381802
\(687\) 24.8416 0.947767
\(688\) 5.30315 0.202181
\(689\) −0.350868 −0.0133670
\(690\) −2.77720 −0.105726
\(691\) −39.6683 −1.50905 −0.754527 0.656269i \(-0.772134\pi\)
−0.754527 + 0.656269i \(0.772134\pi\)
\(692\) 19.5740 0.744091
\(693\) −0.391042 −0.0148544
\(694\) 3.89738 0.147942
\(695\) 4.79401 0.181847
\(696\) −5.00173 −0.189590
\(697\) 37.1105 1.40566
\(698\) 13.5634 0.513381
\(699\) 28.0110 1.05947
\(700\) −3.08919 −0.116761
\(701\) 46.0304 1.73854 0.869272 0.494334i \(-0.164588\pi\)
0.869272 + 0.494334i \(0.164588\pi\)
\(702\) 18.3603 0.692966
\(703\) 0 0
\(704\) 0.201562 0.00759667
\(705\) 10.6952 0.402803
\(706\) 27.5213 1.03578
\(707\) −2.36077 −0.0887858
\(708\) 9.52488 0.357967
\(709\) 41.5265 1.55956 0.779780 0.626053i \(-0.215331\pi\)
0.779780 + 0.626053i \(0.215331\pi\)
\(710\) −5.58146 −0.209468
\(711\) 25.6141 0.960604
\(712\) 6.42150 0.240656
\(713\) −7.56972 −0.283488
\(714\) 4.10564 0.153650
\(715\) −1.00583 −0.0376159
\(716\) −2.05705 −0.0768756
\(717\) −27.3252 −1.02048
\(718\) 16.7795 0.626207
\(719\) −2.36816 −0.0883174 −0.0441587 0.999025i \(-0.514061\pi\)
−0.0441587 + 0.999025i \(0.514061\pi\)
\(720\) 2.68177 0.0999437
\(721\) −7.52998 −0.280431
\(722\) 0 0
\(723\) −12.1551 −0.452054
\(724\) −4.47667 −0.166374
\(725\) −15.0080 −0.557384
\(726\) 11.2831 0.418755
\(727\) 50.4470 1.87098 0.935489 0.353357i \(-0.114960\pi\)
0.935489 + 0.353357i \(0.114960\pi\)
\(728\) 3.61000 0.133795
\(729\) 14.5757 0.539840
\(730\) 6.34793 0.234947
\(731\) −21.1481 −0.782192
\(732\) 0.851665 0.0314784
\(733\) −5.61817 −0.207512 −0.103756 0.994603i \(-0.533086\pi\)
−0.103756 + 0.994603i \(0.533086\pi\)
\(734\) 28.5583 1.05411
\(735\) 1.42315 0.0524937
\(736\) −1.95145 −0.0719314
\(737\) −2.30179 −0.0847875
\(738\) 18.0539 0.664574
\(739\) 21.1345 0.777443 0.388722 0.921355i \(-0.372917\pi\)
0.388722 + 0.921355i \(0.372917\pi\)
\(740\) 12.3837 0.455235
\(741\) 0 0
\(742\) −0.0971934 −0.00356808
\(743\) −5.74269 −0.210679 −0.105340 0.994436i \(-0.533593\pi\)
−0.105340 + 0.994436i \(0.533593\pi\)
\(744\) −3.99360 −0.146413
\(745\) −19.1902 −0.703073
\(746\) −0.530273 −0.0194147
\(747\) −19.6051 −0.717314
\(748\) −0.803800 −0.0293898
\(749\) 3.07601 0.112395
\(750\) −11.5121 −0.420364
\(751\) −35.9013 −1.31006 −0.655029 0.755604i \(-0.727344\pi\)
−0.655029 + 0.755604i \(0.727344\pi\)
\(752\) 7.51513 0.274049
\(753\) −11.5112 −0.419491
\(754\) 17.5382 0.638703
\(755\) 27.0667 0.985058
\(756\) 5.08597 0.184975
\(757\) 18.3088 0.665445 0.332722 0.943025i \(-0.392033\pi\)
0.332722 + 0.943025i \(0.392033\pi\)
\(758\) 0.442763 0.0160819
\(759\) 0.404957 0.0146990
\(760\) 0 0
\(761\) −48.7667 −1.76779 −0.883896 0.467684i \(-0.845089\pi\)
−0.883896 + 0.467684i \(0.845089\pi\)
\(762\) 3.31120 0.119952
\(763\) −2.31631 −0.0838559
\(764\) −3.44067 −0.124479
\(765\) −10.6945 −0.386660
\(766\) −3.06625 −0.110788
\(767\) −33.3983 −1.20594
\(768\) −1.02954 −0.0371502
\(769\) 44.8502 1.61734 0.808671 0.588262i \(-0.200188\pi\)
0.808671 + 0.588262i \(0.200188\pi\)
\(770\) −0.278624 −0.0100409
\(771\) −30.4675 −1.09726
\(772\) −25.1998 −0.906959
\(773\) −40.8558 −1.46948 −0.734740 0.678349i \(-0.762696\pi\)
−0.734740 + 0.678349i \(0.762696\pi\)
\(774\) −10.2884 −0.369808
\(775\) −11.9831 −0.430444
\(776\) −10.9140 −0.391790
\(777\) 9.22327 0.330883
\(778\) −14.5166 −0.520445
\(779\) 0 0
\(780\) 5.13756 0.183954
\(781\) 0.813859 0.0291222
\(782\) 7.78208 0.278287
\(783\) 24.7088 0.883021
\(784\) 1.00000 0.0357143
\(785\) 6.85569 0.244690
\(786\) 6.77037 0.241491
\(787\) 3.83803 0.136811 0.0684056 0.997658i \(-0.478209\pi\)
0.0684056 + 0.997658i \(0.478209\pi\)
\(788\) −14.3027 −0.509512
\(789\) 25.3438 0.902264
\(790\) 18.2505 0.649323
\(791\) −3.65946 −0.130115
\(792\) −0.391042 −0.0138951
\(793\) −2.98630 −0.106047
\(794\) 27.0449 0.959789
\(795\) −0.138321 −0.00490573
\(796\) 12.3169 0.436560
\(797\) 21.6035 0.765235 0.382617 0.923907i \(-0.375023\pi\)
0.382617 + 0.923907i \(0.375023\pi\)
\(798\) 0 0
\(799\) −29.9692 −1.06023
\(800\) −3.08919 −0.109220
\(801\) −12.4580 −0.440183
\(802\) −1.61913 −0.0571735
\(803\) −0.925622 −0.0326645
\(804\) 11.7570 0.414639
\(805\) 2.69753 0.0950753
\(806\) 14.0033 0.493244
\(807\) −7.97674 −0.280794
\(808\) −2.36077 −0.0830515
\(809\) −25.6743 −0.902660 −0.451330 0.892357i \(-0.649050\pi\)
−0.451330 + 0.892357i \(0.649050\pi\)
\(810\) −0.807217 −0.0283627
\(811\) −49.7923 −1.74844 −0.874222 0.485526i \(-0.838628\pi\)
−0.874222 + 0.485526i \(0.838628\pi\)
\(812\) 4.85823 0.170490
\(813\) 11.9181 0.417987
\(814\) −1.80573 −0.0632908
\(815\) −14.2173 −0.498011
\(816\) 4.10564 0.143726
\(817\) 0 0
\(818\) 6.10992 0.213628
\(819\) −7.00358 −0.244725
\(820\) 12.8637 0.449221
\(821\) −28.5028 −0.994756 −0.497378 0.867534i \(-0.665704\pi\)
−0.497378 + 0.867534i \(0.665704\pi\)
\(822\) 18.9734 0.661774
\(823\) 18.0495 0.629165 0.314582 0.949230i \(-0.398136\pi\)
0.314582 + 0.949230i \(0.398136\pi\)
\(824\) −7.52998 −0.262319
\(825\) 0.641058 0.0223188
\(826\) −9.25161 −0.321905
\(827\) −26.1255 −0.908472 −0.454236 0.890881i \(-0.650088\pi\)
−0.454236 + 0.890881i \(0.650088\pi\)
\(828\) 3.78591 0.131570
\(829\) 27.1011 0.941261 0.470630 0.882330i \(-0.344027\pi\)
0.470630 + 0.882330i \(0.344027\pi\)
\(830\) −13.9690 −0.484870
\(831\) −5.04564 −0.175031
\(832\) 3.61000 0.125154
\(833\) −3.98785 −0.138171
\(834\) 3.57053 0.123637
\(835\) −0.183791 −0.00636035
\(836\) 0 0
\(837\) 19.7286 0.681920
\(838\) 1.92152 0.0663777
\(839\) −37.2618 −1.28642 −0.643210 0.765690i \(-0.722398\pi\)
−0.643210 + 0.765690i \(0.722398\pi\)
\(840\) 1.42315 0.0491033
\(841\) −5.39761 −0.186124
\(842\) 31.5836 1.08844
\(843\) 16.8034 0.578741
\(844\) −22.8505 −0.786547
\(845\) −0.0443310 −0.00152503
\(846\) −14.5798 −0.501262
\(847\) −10.9594 −0.376568
\(848\) −0.0971934 −0.00333764
\(849\) 19.7437 0.677601
\(850\) 12.3192 0.422546
\(851\) 17.4824 0.599288
\(852\) −4.15702 −0.142417
\(853\) −3.05187 −0.104494 −0.0522471 0.998634i \(-0.516638\pi\)
−0.0522471 + 0.998634i \(0.516638\pi\)
\(854\) −0.827230 −0.0283072
\(855\) 0 0
\(856\) 3.07601 0.105136
\(857\) 54.9863 1.87830 0.939148 0.343513i \(-0.111617\pi\)
0.939148 + 0.343513i \(0.111617\pi\)
\(858\) −0.749132 −0.0255750
\(859\) 48.8210 1.66575 0.832876 0.553460i \(-0.186693\pi\)
0.832876 + 0.553460i \(0.186693\pi\)
\(860\) −7.33064 −0.249973
\(861\) 9.58077 0.326512
\(862\) 24.5430 0.835937
\(863\) 14.1254 0.480833 0.240416 0.970670i \(-0.422716\pi\)
0.240416 + 0.970670i \(0.422716\pi\)
\(864\) 5.08597 0.173028
\(865\) −27.0575 −0.919982
\(866\) −35.6753 −1.21229
\(867\) 1.12950 0.0383597
\(868\) 3.87902 0.131663
\(869\) −2.66119 −0.0902746
\(870\) 6.91399 0.234406
\(871\) −41.2252 −1.39686
\(872\) −2.31631 −0.0784400
\(873\) 21.1737 0.716622
\(874\) 0 0
\(875\) 11.1818 0.378015
\(876\) 4.72788 0.159740
\(877\) 13.0738 0.441472 0.220736 0.975334i \(-0.429154\pi\)
0.220736 + 0.975334i \(0.429154\pi\)
\(878\) 10.6745 0.360247
\(879\) −9.66541 −0.326006
\(880\) −0.278624 −0.00939240
\(881\) 55.8320 1.88103 0.940514 0.339754i \(-0.110344\pi\)
0.940514 + 0.339754i \(0.110344\pi\)
\(882\) −1.94005 −0.0653250
\(883\) −5.50018 −0.185096 −0.0925478 0.995708i \(-0.529501\pi\)
−0.0925478 + 0.995708i \(0.529501\pi\)
\(884\) −14.3961 −0.484193
\(885\) −13.1664 −0.442585
\(886\) −28.1790 −0.946692
\(887\) −42.0908 −1.41327 −0.706636 0.707578i \(-0.749788\pi\)
−0.706636 + 0.707578i \(0.749788\pi\)
\(888\) 9.22327 0.309513
\(889\) −3.21620 −0.107868
\(890\) −8.87656 −0.297543
\(891\) 0.117704 0.00394324
\(892\) 7.72793 0.258750
\(893\) 0 0
\(894\) −14.2926 −0.478017
\(895\) 2.84350 0.0950477
\(896\) 1.00000 0.0334077
\(897\) 7.25281 0.242164
\(898\) −1.02365 −0.0341597
\(899\) 18.8452 0.628522
\(900\) 5.99320 0.199773
\(901\) 0.387592 0.0129126
\(902\) −1.87572 −0.0624546
\(903\) −5.45979 −0.181690
\(904\) −3.65946 −0.121712
\(905\) 6.18818 0.205702
\(906\) 20.1590 0.669738
\(907\) −24.9876 −0.829700 −0.414850 0.909890i \(-0.636166\pi\)
−0.414850 + 0.909890i \(0.636166\pi\)
\(908\) −25.9981 −0.862779
\(909\) 4.58001 0.151909
\(910\) −4.99017 −0.165422
\(911\) 57.0687 1.89077 0.945385 0.325955i \(-0.105686\pi\)
0.945385 + 0.325955i \(0.105686\pi\)
\(912\) 0 0
\(913\) 2.03688 0.0674110
\(914\) 30.0439 0.993763
\(915\) −1.17727 −0.0389194
\(916\) −24.1289 −0.797242
\(917\) −6.57612 −0.217163
\(918\) −20.2821 −0.669408
\(919\) −2.77343 −0.0914872 −0.0457436 0.998953i \(-0.514566\pi\)
−0.0457436 + 0.998953i \(0.514566\pi\)
\(920\) 2.69753 0.0889348
\(921\) 25.2477 0.831941
\(922\) −11.9442 −0.393362
\(923\) 14.5763 0.479783
\(924\) −0.207516 −0.00682678
\(925\) 27.6750 0.909949
\(926\) −27.3252 −0.897960
\(927\) 14.6085 0.479808
\(928\) 4.85823 0.159479
\(929\) −12.7799 −0.419295 −0.209647 0.977777i \(-0.567232\pi\)
−0.209647 + 0.977777i \(0.567232\pi\)
\(930\) 5.52043 0.181022
\(931\) 0 0
\(932\) −27.2074 −0.891208
\(933\) −11.9880 −0.392468
\(934\) −7.73250 −0.253015
\(935\) 1.11111 0.0363371
\(936\) −7.00358 −0.228919
\(937\) 19.8246 0.647640 0.323820 0.946119i \(-0.395033\pi\)
0.323820 + 0.946119i \(0.395033\pi\)
\(938\) −11.4197 −0.372867
\(939\) 9.41743 0.307326
\(940\) −10.3883 −0.338829
\(941\) −25.2379 −0.822732 −0.411366 0.911470i \(-0.634948\pi\)
−0.411366 + 0.911470i \(0.634948\pi\)
\(942\) 5.10605 0.166364
\(943\) 18.1600 0.591371
\(944\) −9.25161 −0.301114
\(945\) −7.03043 −0.228700
\(946\) 1.06892 0.0347534
\(947\) −9.02597 −0.293305 −0.146652 0.989188i \(-0.546850\pi\)
−0.146652 + 0.989188i \(0.546850\pi\)
\(948\) 13.5928 0.441473
\(949\) −16.5779 −0.538143
\(950\) 0 0
\(951\) −9.58507 −0.310817
\(952\) −3.98785 −0.129247
\(953\) −23.2965 −0.754647 −0.377323 0.926082i \(-0.623155\pi\)
−0.377323 + 0.926082i \(0.623155\pi\)
\(954\) 0.188560 0.00610487
\(955\) 4.75610 0.153904
\(956\) 26.5413 0.858406
\(957\) −1.00816 −0.0325892
\(958\) −1.52899 −0.0493995
\(959\) −18.4291 −0.595105
\(960\) 1.42315 0.0459319
\(961\) −15.9532 −0.514618
\(962\) −32.3407 −1.04271
\(963\) −5.96762 −0.192304
\(964\) 11.8064 0.380258
\(965\) 34.8341 1.12135
\(966\) 2.00909 0.0646414
\(967\) −23.9018 −0.768631 −0.384316 0.923202i \(-0.625563\pi\)
−0.384316 + 0.923202i \(0.625563\pi\)
\(968\) −10.9594 −0.352248
\(969\) 0 0
\(970\) 15.0866 0.484402
\(971\) 58.0700 1.86356 0.931778 0.363029i \(-0.118258\pi\)
0.931778 + 0.363029i \(0.118258\pi\)
\(972\) −15.8591 −0.508681
\(973\) −3.46809 −0.111182
\(974\) 14.6532 0.469518
\(975\) 11.4814 0.367699
\(976\) −0.827230 −0.0264790
\(977\) 45.3835 1.45195 0.725974 0.687722i \(-0.241389\pi\)
0.725974 + 0.687722i \(0.241389\pi\)
\(978\) −10.5889 −0.338597
\(979\) 1.29433 0.0413671
\(980\) −1.38232 −0.0441566
\(981\) 4.49376 0.143475
\(982\) 19.4632 0.621096
\(983\) 46.3756 1.47915 0.739576 0.673073i \(-0.235026\pi\)
0.739576 + 0.673073i \(0.235026\pi\)
\(984\) 9.58077 0.305424
\(985\) 19.7709 0.629952
\(986\) −19.3739 −0.616990
\(987\) −7.73711 −0.246275
\(988\) 0 0
\(989\) −10.3488 −0.329073
\(990\) 0.540544 0.0171796
\(991\) 39.5347 1.25586 0.627931 0.778269i \(-0.283902\pi\)
0.627931 + 0.778269i \(0.283902\pi\)
\(992\) 3.87902 0.123159
\(993\) 27.4659 0.871603
\(994\) 4.03775 0.128070
\(995\) −17.0259 −0.539756
\(996\) −10.4040 −0.329662
\(997\) −51.7281 −1.63825 −0.819123 0.573618i \(-0.805539\pi\)
−0.819123 + 0.573618i \(0.805539\pi\)
\(998\) 8.21283 0.259972
\(999\) −45.5635 −1.44156
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 5054.2.a.bh.1.4 yes 8
19.18 odd 2 5054.2.a.bg.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.bg.1.5 8 19.18 odd 2
5054.2.a.bh.1.4 yes 8 1.1 even 1 trivial