Properties

Label 5054.2.a.bi.1.7
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
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: \(0\)
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 1.7
Root \(2.04552\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.40760 q^{3} +1.00000 q^{4} +1.76144 q^{5} +2.40760 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.79656 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.40760 q^{3} +1.00000 q^{4} +1.76144 q^{5} +2.40760 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.79656 q^{9} +1.76144 q^{10} +1.18414 q^{11} +2.40760 q^{12} +0.361568 q^{13} -1.00000 q^{14} +4.24085 q^{15} +1.00000 q^{16} +3.54532 q^{17} +2.79656 q^{18} +1.76144 q^{20} -2.40760 q^{21} +1.18414 q^{22} -2.87691 q^{23} +2.40760 q^{24} -1.89733 q^{25} +0.361568 q^{26} -0.489807 q^{27} -1.00000 q^{28} +5.10308 q^{29} +4.24085 q^{30} -1.38152 q^{31} +1.00000 q^{32} +2.85095 q^{33} +3.54532 q^{34} -1.76144 q^{35} +2.79656 q^{36} -3.35526 q^{37} +0.870512 q^{39} +1.76144 q^{40} +10.0120 q^{41} -2.40760 q^{42} +3.47246 q^{43} +1.18414 q^{44} +4.92597 q^{45} -2.87691 q^{46} +11.6169 q^{47} +2.40760 q^{48} +1.00000 q^{49} -1.89733 q^{50} +8.53574 q^{51} +0.361568 q^{52} +12.0168 q^{53} -0.489807 q^{54} +2.08580 q^{55} -1.00000 q^{56} +5.10308 q^{58} -7.72735 q^{59} +4.24085 q^{60} -1.99042 q^{61} -1.38152 q^{62} -2.79656 q^{63} +1.00000 q^{64} +0.636880 q^{65} +2.85095 q^{66} +8.30406 q^{67} +3.54532 q^{68} -6.92645 q^{69} -1.76144 q^{70} -0.226733 q^{71} +2.79656 q^{72} -14.0163 q^{73} -3.35526 q^{74} -4.56803 q^{75} -1.18414 q^{77} +0.870512 q^{78} +5.28063 q^{79} +1.76144 q^{80} -9.56894 q^{81} +10.0120 q^{82} +13.3912 q^{83} -2.40760 q^{84} +6.24487 q^{85} +3.47246 q^{86} +12.2862 q^{87} +1.18414 q^{88} -1.87916 q^{89} +4.92597 q^{90} -0.361568 q^{91} -2.87691 q^{92} -3.32616 q^{93} +11.6169 q^{94} +2.40760 q^{96} -3.99766 q^{97} +1.00000 q^{98} +3.31153 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} + 8 q^{8} + 8 q^{9} + 6 q^{10} + 4 q^{11} + 4 q^{12} + 6 q^{13} - 8 q^{14} + 8 q^{15} + 8 q^{16} + 2 q^{17} + 8 q^{18} + 6 q^{20} - 4 q^{21} + 4 q^{22} + 20 q^{23} + 4 q^{24} - 8 q^{25} + 6 q^{26} + 22 q^{27} - 8 q^{28} + 16 q^{29} + 8 q^{30} + 22 q^{31} + 8 q^{32} - 8 q^{33} + 2 q^{34} - 6 q^{35} + 8 q^{36} - 12 q^{37} + 8 q^{39} + 6 q^{40} + 36 q^{41} - 4 q^{42} + 4 q^{44} - 4 q^{45} + 20 q^{46} + 6 q^{47} + 4 q^{48} + 8 q^{49} - 8 q^{50} - 4 q^{51} + 6 q^{52} + 16 q^{53} + 22 q^{54} + 18 q^{55} - 8 q^{56} + 16 q^{58} + 14 q^{59} + 8 q^{60} + 10 q^{61} + 22 q^{62} - 8 q^{63} + 8 q^{64} + 32 q^{65} - 8 q^{66} - 20 q^{67} + 2 q^{68} + 40 q^{69} - 6 q^{70} + 8 q^{72} - 18 q^{73} - 12 q^{74} + 16 q^{75} - 4 q^{77} + 8 q^{78} + 4 q^{79} + 6 q^{80} + 12 q^{81} + 36 q^{82} + 36 q^{83} - 4 q^{84} - 16 q^{85} + 28 q^{87} + 4 q^{88} + 18 q^{89} - 4 q^{90} - 6 q^{91} + 20 q^{92} + 16 q^{93} + 6 q^{94} + 4 q^{96} + 26 q^{97} + 8 q^{98} + 4 q^{99}+O(q^{100}) \) 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
\(3\) 2.40760 1.39003 0.695015 0.718995i \(-0.255397\pi\)
0.695015 + 0.718995i \(0.255397\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.76144 0.787739 0.393870 0.919166i \(-0.371136\pi\)
0.393870 + 0.919166i \(0.371136\pi\)
\(6\) 2.40760 0.982900
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 2.79656 0.932186
\(10\) 1.76144 0.557016
\(11\) 1.18414 0.357033 0.178516 0.983937i \(-0.442870\pi\)
0.178516 + 0.983937i \(0.442870\pi\)
\(12\) 2.40760 0.695015
\(13\) 0.361568 0.100281 0.0501404 0.998742i \(-0.484033\pi\)
0.0501404 + 0.998742i \(0.484033\pi\)
\(14\) −1.00000 −0.267261
\(15\) 4.24085 1.09498
\(16\) 1.00000 0.250000
\(17\) 3.54532 0.859868 0.429934 0.902860i \(-0.358537\pi\)
0.429934 + 0.902860i \(0.358537\pi\)
\(18\) 2.79656 0.659155
\(19\) 0 0
\(20\) 1.76144 0.393870
\(21\) −2.40760 −0.525382
\(22\) 1.18414 0.252460
\(23\) −2.87691 −0.599877 −0.299938 0.953959i \(-0.596966\pi\)
−0.299938 + 0.953959i \(0.596966\pi\)
\(24\) 2.40760 0.491450
\(25\) −1.89733 −0.379467
\(26\) 0.361568 0.0709093
\(27\) −0.489807 −0.0942634
\(28\) −1.00000 −0.188982
\(29\) 5.10308 0.947618 0.473809 0.880628i \(-0.342879\pi\)
0.473809 + 0.880628i \(0.342879\pi\)
\(30\) 4.24085 0.774269
\(31\) −1.38152 −0.248129 −0.124064 0.992274i \(-0.539593\pi\)
−0.124064 + 0.992274i \(0.539593\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.85095 0.496286
\(34\) 3.54532 0.608018
\(35\) −1.76144 −0.297737
\(36\) 2.79656 0.466093
\(37\) −3.35526 −0.551602 −0.275801 0.961215i \(-0.588943\pi\)
−0.275801 + 0.961215i \(0.588943\pi\)
\(38\) 0 0
\(39\) 0.870512 0.139394
\(40\) 1.76144 0.278508
\(41\) 10.0120 1.56361 0.781803 0.623526i \(-0.214300\pi\)
0.781803 + 0.623526i \(0.214300\pi\)
\(42\) −2.40760 −0.371501
\(43\) 3.47246 0.529545 0.264772 0.964311i \(-0.414703\pi\)
0.264772 + 0.964311i \(0.414703\pi\)
\(44\) 1.18414 0.178516
\(45\) 4.92597 0.734320
\(46\) −2.87691 −0.424177
\(47\) 11.6169 1.69451 0.847253 0.531190i \(-0.178255\pi\)
0.847253 + 0.531190i \(0.178255\pi\)
\(48\) 2.40760 0.347508
\(49\) 1.00000 0.142857
\(50\) −1.89733 −0.268323
\(51\) 8.53574 1.19524
\(52\) 0.361568 0.0501404
\(53\) 12.0168 1.65063 0.825316 0.564672i \(-0.190997\pi\)
0.825316 + 0.564672i \(0.190997\pi\)
\(54\) −0.489807 −0.0666543
\(55\) 2.08580 0.281249
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 5.10308 0.670067
\(59\) −7.72735 −1.00602 −0.503008 0.864282i \(-0.667773\pi\)
−0.503008 + 0.864282i \(0.667773\pi\)
\(60\) 4.24085 0.547491
\(61\) −1.99042 −0.254848 −0.127424 0.991848i \(-0.540671\pi\)
−0.127424 + 0.991848i \(0.540671\pi\)
\(62\) −1.38152 −0.175453
\(63\) −2.79656 −0.352333
\(64\) 1.00000 0.125000
\(65\) 0.636880 0.0789952
\(66\) 2.85095 0.350927
\(67\) 8.30406 1.01450 0.507251 0.861798i \(-0.330662\pi\)
0.507251 + 0.861798i \(0.330662\pi\)
\(68\) 3.54532 0.429934
\(69\) −6.92645 −0.833847
\(70\) −1.76144 −0.210532
\(71\) −0.226733 −0.0269083 −0.0134541 0.999909i \(-0.504283\pi\)
−0.0134541 + 0.999909i \(0.504283\pi\)
\(72\) 2.79656 0.329578
\(73\) −14.0163 −1.64048 −0.820242 0.572017i \(-0.806161\pi\)
−0.820242 + 0.572017i \(0.806161\pi\)
\(74\) −3.35526 −0.390041
\(75\) −4.56803 −0.527470
\(76\) 0 0
\(77\) −1.18414 −0.134946
\(78\) 0.870512 0.0985661
\(79\) 5.28063 0.594118 0.297059 0.954859i \(-0.403994\pi\)
0.297059 + 0.954859i \(0.403994\pi\)
\(80\) 1.76144 0.196935
\(81\) −9.56894 −1.06322
\(82\) 10.0120 1.10564
\(83\) 13.3912 1.46988 0.734939 0.678133i \(-0.237211\pi\)
0.734939 + 0.678133i \(0.237211\pi\)
\(84\) −2.40760 −0.262691
\(85\) 6.24487 0.677351
\(86\) 3.47246 0.374445
\(87\) 12.2862 1.31722
\(88\) 1.18414 0.126230
\(89\) −1.87916 −0.199191 −0.0995954 0.995028i \(-0.531755\pi\)
−0.0995954 + 0.995028i \(0.531755\pi\)
\(90\) 4.92597 0.519242
\(91\) −0.361568 −0.0379026
\(92\) −2.87691 −0.299938
\(93\) −3.32616 −0.344906
\(94\) 11.6169 1.19820
\(95\) 0 0
\(96\) 2.40760 0.245725
\(97\) −3.99766 −0.405901 −0.202950 0.979189i \(-0.565053\pi\)
−0.202950 + 0.979189i \(0.565053\pi\)
\(98\) 1.00000 0.101015
\(99\) 3.31153 0.332821
\(100\) −1.89733 −0.189733
\(101\) −18.6641 −1.85715 −0.928575 0.371145i \(-0.878965\pi\)
−0.928575 + 0.371145i \(0.878965\pi\)
\(102\) 8.53574 0.845164
\(103\) −2.35245 −0.231793 −0.115897 0.993261i \(-0.536974\pi\)
−0.115897 + 0.993261i \(0.536974\pi\)
\(104\) 0.361568 0.0354546
\(105\) −4.24085 −0.413864
\(106\) 12.0168 1.16717
\(107\) −3.11852 −0.301479 −0.150739 0.988574i \(-0.548165\pi\)
−0.150739 + 0.988574i \(0.548165\pi\)
\(108\) −0.489807 −0.0471317
\(109\) −14.1835 −1.35853 −0.679266 0.733892i \(-0.737702\pi\)
−0.679266 + 0.733892i \(0.737702\pi\)
\(110\) 2.08580 0.198873
\(111\) −8.07815 −0.766744
\(112\) −1.00000 −0.0944911
\(113\) −3.82085 −0.359436 −0.179718 0.983718i \(-0.557518\pi\)
−0.179718 + 0.983718i \(0.557518\pi\)
\(114\) 0 0
\(115\) −5.06749 −0.472546
\(116\) 5.10308 0.473809
\(117\) 1.01115 0.0934804
\(118\) −7.72735 −0.711360
\(119\) −3.54532 −0.324999
\(120\) 4.24085 0.387135
\(121\) −9.59781 −0.872528
\(122\) −1.99042 −0.180205
\(123\) 24.1048 2.17346
\(124\) −1.38152 −0.124064
\(125\) −12.1492 −1.08666
\(126\) −2.79656 −0.249137
\(127\) −19.3815 −1.71983 −0.859917 0.510434i \(-0.829485\pi\)
−0.859917 + 0.510434i \(0.829485\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.36030 0.736084
\(130\) 0.636880 0.0558580
\(131\) −4.99933 −0.436794 −0.218397 0.975860i \(-0.570083\pi\)
−0.218397 + 0.975860i \(0.570083\pi\)
\(132\) 2.85095 0.248143
\(133\) 0 0
\(134\) 8.30406 0.717362
\(135\) −0.862765 −0.0742550
\(136\) 3.54532 0.304009
\(137\) 20.8876 1.78455 0.892276 0.451491i \(-0.149108\pi\)
0.892276 + 0.451491i \(0.149108\pi\)
\(138\) −6.92645 −0.589619
\(139\) −13.2393 −1.12294 −0.561472 0.827496i \(-0.689765\pi\)
−0.561472 + 0.827496i \(0.689765\pi\)
\(140\) −1.76144 −0.148869
\(141\) 27.9690 2.35541
\(142\) −0.226733 −0.0190270
\(143\) 0.428148 0.0358035
\(144\) 2.79656 0.233047
\(145\) 8.98876 0.746476
\(146\) −14.0163 −1.16000
\(147\) 2.40760 0.198576
\(148\) −3.35526 −0.275801
\(149\) 3.08216 0.252500 0.126250 0.991998i \(-0.459706\pi\)
0.126250 + 0.991998i \(0.459706\pi\)
\(150\) −4.56803 −0.372978
\(151\) −6.35364 −0.517052 −0.258526 0.966004i \(-0.583237\pi\)
−0.258526 + 0.966004i \(0.583237\pi\)
\(152\) 0 0
\(153\) 9.91471 0.801557
\(154\) −1.18414 −0.0954210
\(155\) −2.43347 −0.195461
\(156\) 0.870512 0.0696968
\(157\) 4.20918 0.335929 0.167965 0.985793i \(-0.446281\pi\)
0.167965 + 0.985793i \(0.446281\pi\)
\(158\) 5.28063 0.420105
\(159\) 28.9316 2.29443
\(160\) 1.76144 0.139254
\(161\) 2.87691 0.226732
\(162\) −9.56894 −0.751807
\(163\) 7.15678 0.560562 0.280281 0.959918i \(-0.409572\pi\)
0.280281 + 0.959918i \(0.409572\pi\)
\(164\) 10.0120 0.781803
\(165\) 5.02177 0.390944
\(166\) 13.3912 1.03936
\(167\) 22.1964 1.71761 0.858806 0.512300i \(-0.171207\pi\)
0.858806 + 0.512300i \(0.171207\pi\)
\(168\) −2.40760 −0.185751
\(169\) −12.8693 −0.989944
\(170\) 6.24487 0.478960
\(171\) 0 0
\(172\) 3.47246 0.264772
\(173\) 10.8227 0.822835 0.411418 0.911447i \(-0.365034\pi\)
0.411418 + 0.911447i \(0.365034\pi\)
\(174\) 12.2862 0.931414
\(175\) 1.89733 0.143425
\(176\) 1.18414 0.0892581
\(177\) −18.6044 −1.39839
\(178\) −1.87916 −0.140849
\(179\) −11.7586 −0.878880 −0.439440 0.898272i \(-0.644823\pi\)
−0.439440 + 0.898272i \(0.644823\pi\)
\(180\) 4.92597 0.367160
\(181\) 1.62634 0.120885 0.0604426 0.998172i \(-0.480749\pi\)
0.0604426 + 0.998172i \(0.480749\pi\)
\(182\) −0.361568 −0.0268012
\(183\) −4.79215 −0.354246
\(184\) −2.87691 −0.212088
\(185\) −5.91009 −0.434519
\(186\) −3.32616 −0.243886
\(187\) 4.19817 0.307001
\(188\) 11.6169 0.847253
\(189\) 0.489807 0.0356282
\(190\) 0 0
\(191\) 15.1136 1.09358 0.546792 0.837268i \(-0.315849\pi\)
0.546792 + 0.837268i \(0.315849\pi\)
\(192\) 2.40760 0.173754
\(193\) −21.0053 −1.51200 −0.755998 0.654574i \(-0.772848\pi\)
−0.755998 + 0.654574i \(0.772848\pi\)
\(194\) −3.99766 −0.287015
\(195\) 1.53335 0.109806
\(196\) 1.00000 0.0714286
\(197\) −24.5580 −1.74969 −0.874844 0.484405i \(-0.839036\pi\)
−0.874844 + 0.484405i \(0.839036\pi\)
\(198\) 3.31153 0.235340
\(199\) 5.50367 0.390145 0.195073 0.980789i \(-0.437506\pi\)
0.195073 + 0.980789i \(0.437506\pi\)
\(200\) −1.89733 −0.134162
\(201\) 19.9929 1.41019
\(202\) −18.6641 −1.31320
\(203\) −5.10308 −0.358166
\(204\) 8.53574 0.597621
\(205\) 17.6355 1.23171
\(206\) −2.35245 −0.163903
\(207\) −8.04544 −0.559197
\(208\) 0.361568 0.0250702
\(209\) 0 0
\(210\) −4.24085 −0.292646
\(211\) 4.53017 0.311870 0.155935 0.987767i \(-0.450161\pi\)
0.155935 + 0.987767i \(0.450161\pi\)
\(212\) 12.0168 0.825316
\(213\) −0.545884 −0.0374033
\(214\) −3.11852 −0.213178
\(215\) 6.11652 0.417143
\(216\) −0.489807 −0.0333271
\(217\) 1.38152 0.0937838
\(218\) −14.1835 −0.960628
\(219\) −33.7457 −2.28032
\(220\) 2.08580 0.140624
\(221\) 1.28188 0.0862283
\(222\) −8.07815 −0.542170
\(223\) 10.0211 0.671066 0.335533 0.942029i \(-0.391084\pi\)
0.335533 + 0.942029i \(0.391084\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −5.30600 −0.353734
\(226\) −3.82085 −0.254159
\(227\) −7.43478 −0.493463 −0.246732 0.969084i \(-0.579357\pi\)
−0.246732 + 0.969084i \(0.579357\pi\)
\(228\) 0 0
\(229\) 3.96142 0.261778 0.130889 0.991397i \(-0.458217\pi\)
0.130889 + 0.991397i \(0.458217\pi\)
\(230\) −5.06749 −0.334141
\(231\) −2.85095 −0.187579
\(232\) 5.10308 0.335034
\(233\) −14.3859 −0.942452 −0.471226 0.882012i \(-0.656188\pi\)
−0.471226 + 0.882012i \(0.656188\pi\)
\(234\) 1.01115 0.0661007
\(235\) 20.4625 1.33483
\(236\) −7.72735 −0.503008
\(237\) 12.7137 0.825842
\(238\) −3.54532 −0.229809
\(239\) −7.12027 −0.460572 −0.230286 0.973123i \(-0.573966\pi\)
−0.230286 + 0.973123i \(0.573966\pi\)
\(240\) 4.24085 0.273746
\(241\) −14.9816 −0.965051 −0.482525 0.875882i \(-0.660280\pi\)
−0.482525 + 0.875882i \(0.660280\pi\)
\(242\) −9.59781 −0.616970
\(243\) −21.5688 −1.38364
\(244\) −1.99042 −0.127424
\(245\) 1.76144 0.112534
\(246\) 24.1048 1.53687
\(247\) 0 0
\(248\) −1.38152 −0.0877267
\(249\) 32.2408 2.04318
\(250\) −12.1492 −0.768385
\(251\) −28.2614 −1.78385 −0.891923 0.452187i \(-0.850644\pi\)
−0.891923 + 0.452187i \(0.850644\pi\)
\(252\) −2.79656 −0.176167
\(253\) −3.40667 −0.214175
\(254\) −19.3815 −1.21611
\(255\) 15.0352 0.941540
\(256\) 1.00000 0.0625000
\(257\) −11.5197 −0.718576 −0.359288 0.933227i \(-0.616980\pi\)
−0.359288 + 0.933227i \(0.616980\pi\)
\(258\) 8.36030 0.520490
\(259\) 3.35526 0.208486
\(260\) 0.636880 0.0394976
\(261\) 14.2711 0.883357
\(262\) −4.99933 −0.308860
\(263\) 26.6852 1.64548 0.822740 0.568418i \(-0.192445\pi\)
0.822740 + 0.568418i \(0.192445\pi\)
\(264\) 2.85095 0.175464
\(265\) 21.1668 1.30027
\(266\) 0 0
\(267\) −4.52428 −0.276881
\(268\) 8.30406 0.507251
\(269\) −2.07333 −0.126413 −0.0632065 0.998000i \(-0.520133\pi\)
−0.0632065 + 0.998000i \(0.520133\pi\)
\(270\) −0.862765 −0.0525062
\(271\) −23.0985 −1.40314 −0.701568 0.712602i \(-0.747516\pi\)
−0.701568 + 0.712602i \(0.747516\pi\)
\(272\) 3.54532 0.214967
\(273\) −0.870512 −0.0526858
\(274\) 20.8876 1.26187
\(275\) −2.24671 −0.135482
\(276\) −6.92645 −0.416923
\(277\) 20.3780 1.22440 0.612198 0.790705i \(-0.290286\pi\)
0.612198 + 0.790705i \(0.290286\pi\)
\(278\) −13.2393 −0.794042
\(279\) −3.86351 −0.231302
\(280\) −1.76144 −0.105266
\(281\) 15.6633 0.934394 0.467197 0.884153i \(-0.345264\pi\)
0.467197 + 0.884153i \(0.345264\pi\)
\(282\) 27.9690 1.66553
\(283\) 2.37228 0.141017 0.0705086 0.997511i \(-0.477538\pi\)
0.0705086 + 0.997511i \(0.477538\pi\)
\(284\) −0.226733 −0.0134541
\(285\) 0 0
\(286\) 0.428148 0.0253169
\(287\) −10.0120 −0.590987
\(288\) 2.79656 0.164789
\(289\) −4.43067 −0.260628
\(290\) 8.98876 0.527838
\(291\) −9.62478 −0.564215
\(292\) −14.0163 −0.820242
\(293\) −0.586563 −0.0342674 −0.0171337 0.999853i \(-0.505454\pi\)
−0.0171337 + 0.999853i \(0.505454\pi\)
\(294\) 2.40760 0.140414
\(295\) −13.6113 −0.792478
\(296\) −3.35526 −0.195021
\(297\) −0.580001 −0.0336551
\(298\) 3.08216 0.178545
\(299\) −1.04020 −0.0601561
\(300\) −4.56803 −0.263735
\(301\) −3.47246 −0.200149
\(302\) −6.35364 −0.365611
\(303\) −44.9358 −2.58150
\(304\) 0 0
\(305\) −3.50601 −0.200754
\(306\) 9.91471 0.566786
\(307\) 19.9202 1.13690 0.568452 0.822717i \(-0.307543\pi\)
0.568452 + 0.822717i \(0.307543\pi\)
\(308\) −1.18414 −0.0674728
\(309\) −5.66376 −0.322200
\(310\) −2.43347 −0.138212
\(311\) −14.6345 −0.829849 −0.414925 0.909856i \(-0.636192\pi\)
−0.414925 + 0.909856i \(0.636192\pi\)
\(312\) 0.870512 0.0492831
\(313\) −26.9261 −1.52195 −0.760975 0.648781i \(-0.775279\pi\)
−0.760975 + 0.648781i \(0.775279\pi\)
\(314\) 4.20918 0.237538
\(315\) −4.92597 −0.277547
\(316\) 5.28063 0.297059
\(317\) 30.5772 1.71739 0.858694 0.512489i \(-0.171276\pi\)
0.858694 + 0.512489i \(0.171276\pi\)
\(318\) 28.9316 1.62241
\(319\) 6.04278 0.338331
\(320\) 1.76144 0.0984674
\(321\) −7.50816 −0.419065
\(322\) 2.87691 0.160324
\(323\) 0 0
\(324\) −9.56894 −0.531608
\(325\) −0.686015 −0.0380533
\(326\) 7.15678 0.396377
\(327\) −34.1483 −1.88840
\(328\) 10.0120 0.552818
\(329\) −11.6169 −0.640463
\(330\) 5.02177 0.276439
\(331\) 20.3542 1.11877 0.559384 0.828909i \(-0.311038\pi\)
0.559384 + 0.828909i \(0.311038\pi\)
\(332\) 13.3912 0.734939
\(333\) −9.38319 −0.514196
\(334\) 22.1964 1.21454
\(335\) 14.6271 0.799164
\(336\) −2.40760 −0.131346
\(337\) −14.8090 −0.806700 −0.403350 0.915046i \(-0.632154\pi\)
−0.403350 + 0.915046i \(0.632154\pi\)
\(338\) −12.8693 −0.699996
\(339\) −9.19910 −0.499627
\(340\) 6.24487 0.338676
\(341\) −1.63592 −0.0885900
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 3.47246 0.187222
\(345\) −12.2005 −0.656854
\(346\) 10.8227 0.581832
\(347\) −19.0467 −1.02248 −0.511241 0.859438i \(-0.670814\pi\)
−0.511241 + 0.859438i \(0.670814\pi\)
\(348\) 12.2862 0.658609
\(349\) −21.1239 −1.13074 −0.565368 0.824839i \(-0.691266\pi\)
−0.565368 + 0.824839i \(0.691266\pi\)
\(350\) 1.89733 0.101417
\(351\) −0.177098 −0.00945282
\(352\) 1.18414 0.0631150
\(353\) 2.24313 0.119390 0.0596949 0.998217i \(-0.480987\pi\)
0.0596949 + 0.998217i \(0.480987\pi\)
\(354\) −18.6044 −0.988813
\(355\) −0.399377 −0.0211967
\(356\) −1.87916 −0.0995954
\(357\) −8.53574 −0.451759
\(358\) −11.7586 −0.621462
\(359\) 3.31981 0.175213 0.0876066 0.996155i \(-0.472078\pi\)
0.0876066 + 0.996155i \(0.472078\pi\)
\(360\) 4.92597 0.259621
\(361\) 0 0
\(362\) 1.62634 0.0854787
\(363\) −23.1077 −1.21284
\(364\) −0.361568 −0.0189513
\(365\) −24.6888 −1.29227
\(366\) −4.79215 −0.250490
\(367\) −18.3886 −0.959875 −0.479938 0.877303i \(-0.659341\pi\)
−0.479938 + 0.877303i \(0.659341\pi\)
\(368\) −2.87691 −0.149969
\(369\) 27.9990 1.45757
\(370\) −5.91009 −0.307251
\(371\) −12.0168 −0.623880
\(372\) −3.32616 −0.172453
\(373\) −3.33060 −0.172452 −0.0862259 0.996276i \(-0.527481\pi\)
−0.0862259 + 0.996276i \(0.527481\pi\)
\(374\) 4.19817 0.217082
\(375\) −29.2505 −1.51049
\(376\) 11.6169 0.599098
\(377\) 1.84511 0.0950280
\(378\) 0.489807 0.0251929
\(379\) 30.8785 1.58612 0.793061 0.609142i \(-0.208486\pi\)
0.793061 + 0.609142i \(0.208486\pi\)
\(380\) 0 0
\(381\) −46.6631 −2.39062
\(382\) 15.1136 0.773281
\(383\) −4.18785 −0.213989 −0.106995 0.994260i \(-0.534123\pi\)
−0.106995 + 0.994260i \(0.534123\pi\)
\(384\) 2.40760 0.122863
\(385\) −2.08580 −0.106302
\(386\) −21.0053 −1.06914
\(387\) 9.71093 0.493634
\(388\) −3.99766 −0.202950
\(389\) 13.7229 0.695779 0.347890 0.937536i \(-0.386898\pi\)
0.347890 + 0.937536i \(0.386898\pi\)
\(390\) 1.53335 0.0776444
\(391\) −10.1996 −0.515814
\(392\) 1.00000 0.0505076
\(393\) −12.0364 −0.607157
\(394\) −24.5580 −1.23722
\(395\) 9.30151 0.468010
\(396\) 3.31153 0.166410
\(397\) −12.7243 −0.638614 −0.319307 0.947651i \(-0.603450\pi\)
−0.319307 + 0.947651i \(0.603450\pi\)
\(398\) 5.50367 0.275874
\(399\) 0 0
\(400\) −1.89733 −0.0948667
\(401\) 27.5174 1.37415 0.687077 0.726585i \(-0.258894\pi\)
0.687077 + 0.726585i \(0.258894\pi\)
\(402\) 19.9929 0.997155
\(403\) −0.499514 −0.0248826
\(404\) −18.6641 −0.928575
\(405\) −16.8551 −0.837536
\(406\) −5.10308 −0.253262
\(407\) −3.97311 −0.196940
\(408\) 8.53574 0.422582
\(409\) 30.1566 1.49115 0.745574 0.666423i \(-0.232175\pi\)
0.745574 + 0.666423i \(0.232175\pi\)
\(410\) 17.6355 0.870953
\(411\) 50.2892 2.48058
\(412\) −2.35245 −0.115897
\(413\) 7.72735 0.380238
\(414\) −8.04544 −0.395412
\(415\) 23.5878 1.15788
\(416\) 0.361568 0.0177273
\(417\) −31.8750 −1.56093
\(418\) 0 0
\(419\) 25.2631 1.23418 0.617092 0.786891i \(-0.288311\pi\)
0.617092 + 0.786891i \(0.288311\pi\)
\(420\) −4.24085 −0.206932
\(421\) −38.0387 −1.85389 −0.926945 0.375197i \(-0.877575\pi\)
−0.926945 + 0.375197i \(0.877575\pi\)
\(422\) 4.53017 0.220525
\(423\) 32.4875 1.57959
\(424\) 12.0168 0.583586
\(425\) −6.72666 −0.326291
\(426\) −0.545884 −0.0264482
\(427\) 1.99042 0.0963234
\(428\) −3.11852 −0.150739
\(429\) 1.03081 0.0497680
\(430\) 6.11652 0.294965
\(431\) 18.9095 0.910837 0.455419 0.890277i \(-0.349490\pi\)
0.455419 + 0.890277i \(0.349490\pi\)
\(432\) −0.489807 −0.0235658
\(433\) −12.6102 −0.606009 −0.303005 0.952989i \(-0.597990\pi\)
−0.303005 + 0.952989i \(0.597990\pi\)
\(434\) 1.38152 0.0663152
\(435\) 21.6414 1.03763
\(436\) −14.1835 −0.679266
\(437\) 0 0
\(438\) −33.7457 −1.61243
\(439\) 33.9256 1.61918 0.809591 0.586994i \(-0.199689\pi\)
0.809591 + 0.586994i \(0.199689\pi\)
\(440\) 2.08580 0.0994364
\(441\) 2.79656 0.133169
\(442\) 1.28188 0.0609726
\(443\) −4.69792 −0.223205 −0.111602 0.993753i \(-0.535598\pi\)
−0.111602 + 0.993753i \(0.535598\pi\)
\(444\) −8.07815 −0.383372
\(445\) −3.31003 −0.156910
\(446\) 10.0211 0.474515
\(447\) 7.42061 0.350983
\(448\) −1.00000 −0.0472456
\(449\) 25.3251 1.19517 0.597583 0.801807i \(-0.296128\pi\)
0.597583 + 0.801807i \(0.296128\pi\)
\(450\) −5.30600 −0.250127
\(451\) 11.8556 0.558258
\(452\) −3.82085 −0.179718
\(453\) −15.2970 −0.718718
\(454\) −7.43478 −0.348931
\(455\) −0.636880 −0.0298574
\(456\) 0 0
\(457\) −40.3466 −1.88734 −0.943668 0.330894i \(-0.892650\pi\)
−0.943668 + 0.330894i \(0.892650\pi\)
\(458\) 3.96142 0.185105
\(459\) −1.73652 −0.0810540
\(460\) −5.06749 −0.236273
\(461\) 8.28479 0.385861 0.192931 0.981212i \(-0.438201\pi\)
0.192931 + 0.981212i \(0.438201\pi\)
\(462\) −2.85095 −0.132638
\(463\) −26.7426 −1.24283 −0.621416 0.783481i \(-0.713442\pi\)
−0.621416 + 0.783481i \(0.713442\pi\)
\(464\) 5.10308 0.236905
\(465\) −5.85882 −0.271696
\(466\) −14.3859 −0.666414
\(467\) 7.29907 0.337761 0.168880 0.985637i \(-0.445985\pi\)
0.168880 + 0.985637i \(0.445985\pi\)
\(468\) 1.01115 0.0467402
\(469\) −8.30406 −0.383446
\(470\) 20.4625 0.943866
\(471\) 10.1340 0.466952
\(472\) −7.72735 −0.355680
\(473\) 4.11189 0.189065
\(474\) 12.7137 0.583959
\(475\) 0 0
\(476\) −3.54532 −0.162500
\(477\) 33.6056 1.53870
\(478\) −7.12027 −0.325673
\(479\) −27.1804 −1.24191 −0.620953 0.783848i \(-0.713254\pi\)
−0.620953 + 0.783848i \(0.713254\pi\)
\(480\) 4.24085 0.193567
\(481\) −1.21316 −0.0553151
\(482\) −14.9816 −0.682394
\(483\) 6.92645 0.315165
\(484\) −9.59781 −0.436264
\(485\) −7.04163 −0.319744
\(486\) −21.5688 −0.978380
\(487\) 6.88190 0.311849 0.155924 0.987769i \(-0.450164\pi\)
0.155924 + 0.987769i \(0.450164\pi\)
\(488\) −1.99042 −0.0901023
\(489\) 17.2307 0.779199
\(490\) 1.76144 0.0795737
\(491\) −9.47678 −0.427681 −0.213841 0.976869i \(-0.568597\pi\)
−0.213841 + 0.976869i \(0.568597\pi\)
\(492\) 24.1048 1.08673
\(493\) 18.0921 0.814826
\(494\) 0 0
\(495\) 5.83305 0.262176
\(496\) −1.38152 −0.0620322
\(497\) 0.226733 0.0101704
\(498\) 32.2408 1.44474
\(499\) −5.58793 −0.250150 −0.125075 0.992147i \(-0.539917\pi\)
−0.125075 + 0.992147i \(0.539917\pi\)
\(500\) −12.1492 −0.543330
\(501\) 53.4403 2.38753
\(502\) −28.2614 −1.26137
\(503\) −9.98556 −0.445234 −0.222617 0.974906i \(-0.571460\pi\)
−0.222617 + 0.974906i \(0.571460\pi\)
\(504\) −2.79656 −0.124569
\(505\) −32.8757 −1.46295
\(506\) −3.40667 −0.151445
\(507\) −30.9841 −1.37605
\(508\) −19.3815 −0.859917
\(509\) −35.5061 −1.57378 −0.786890 0.617093i \(-0.788310\pi\)
−0.786890 + 0.617093i \(0.788310\pi\)
\(510\) 15.0352 0.665769
\(511\) 14.0163 0.620044
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −11.5197 −0.508110
\(515\) −4.14369 −0.182593
\(516\) 8.36030 0.368042
\(517\) 13.7561 0.604993
\(518\) 3.35526 0.147422
\(519\) 26.0568 1.14377
\(520\) 0.636880 0.0279290
\(521\) 4.29772 0.188287 0.0941433 0.995559i \(-0.469989\pi\)
0.0941433 + 0.995559i \(0.469989\pi\)
\(522\) 14.2711 0.624627
\(523\) 31.1762 1.36324 0.681620 0.731707i \(-0.261276\pi\)
0.681620 + 0.731707i \(0.261276\pi\)
\(524\) −4.99933 −0.218397
\(525\) 4.56803 0.199365
\(526\) 26.6852 1.16353
\(527\) −4.89794 −0.213358
\(528\) 2.85095 0.124072
\(529\) −14.7234 −0.640148
\(530\) 21.1668 0.919428
\(531\) −21.6100 −0.937794
\(532\) 0 0
\(533\) 3.62000 0.156800
\(534\) −4.52428 −0.195785
\(535\) −5.49308 −0.237487
\(536\) 8.30406 0.358681
\(537\) −28.3101 −1.22167
\(538\) −2.07333 −0.0893874
\(539\) 1.18414 0.0510047
\(540\) −0.862765 −0.0371275
\(541\) −5.51854 −0.237261 −0.118630 0.992938i \(-0.537850\pi\)
−0.118630 + 0.992938i \(0.537850\pi\)
\(542\) −23.0985 −0.992167
\(543\) 3.91559 0.168034
\(544\) 3.54532 0.152005
\(545\) −24.9834 −1.07017
\(546\) −0.870512 −0.0372545
\(547\) 8.52743 0.364607 0.182303 0.983242i \(-0.441645\pi\)
0.182303 + 0.983242i \(0.441645\pi\)
\(548\) 20.8876 0.892276
\(549\) −5.56634 −0.237565
\(550\) −2.24671 −0.0958002
\(551\) 0 0
\(552\) −6.92645 −0.294809
\(553\) −5.28063 −0.224555
\(554\) 20.3780 0.865778
\(555\) −14.2292 −0.603994
\(556\) −13.2393 −0.561472
\(557\) 22.8183 0.966843 0.483422 0.875388i \(-0.339394\pi\)
0.483422 + 0.875388i \(0.339394\pi\)
\(558\) −3.86351 −0.163555
\(559\) 1.25553 0.0531032
\(560\) −1.76144 −0.0744344
\(561\) 10.1075 0.426740
\(562\) 15.6633 0.660716
\(563\) 10.7974 0.455056 0.227528 0.973772i \(-0.426936\pi\)
0.227528 + 0.973772i \(0.426936\pi\)
\(564\) 27.9690 1.17771
\(565\) −6.73020 −0.283142
\(566\) 2.37228 0.0997142
\(567\) 9.56894 0.401858
\(568\) −0.226733 −0.00951351
\(569\) 5.10510 0.214017 0.107008 0.994258i \(-0.465873\pi\)
0.107008 + 0.994258i \(0.465873\pi\)
\(570\) 0 0
\(571\) 4.62174 0.193414 0.0967070 0.995313i \(-0.469169\pi\)
0.0967070 + 0.995313i \(0.469169\pi\)
\(572\) 0.428148 0.0179018
\(573\) 36.3877 1.52012
\(574\) −10.0120 −0.417891
\(575\) 5.45845 0.227633
\(576\) 2.79656 0.116523
\(577\) −32.4426 −1.35060 −0.675302 0.737541i \(-0.735987\pi\)
−0.675302 + 0.737541i \(0.735987\pi\)
\(578\) −4.43067 −0.184292
\(579\) −50.5725 −2.10172
\(580\) 8.98876 0.373238
\(581\) −13.3912 −0.555562
\(582\) −9.62478 −0.398960
\(583\) 14.2296 0.589329
\(584\) −14.0163 −0.579998
\(585\) 1.78107 0.0736382
\(586\) −0.586563 −0.0242307
\(587\) −7.36820 −0.304118 −0.152059 0.988371i \(-0.548590\pi\)
−0.152059 + 0.988371i \(0.548590\pi\)
\(588\) 2.40760 0.0992879
\(589\) 0 0
\(590\) −13.6113 −0.560367
\(591\) −59.1261 −2.43212
\(592\) −3.35526 −0.137900
\(593\) 3.98443 0.163621 0.0818105 0.996648i \(-0.473930\pi\)
0.0818105 + 0.996648i \(0.473930\pi\)
\(594\) −0.580001 −0.0237977
\(595\) −6.24487 −0.256015
\(596\) 3.08216 0.126250
\(597\) 13.2507 0.542314
\(598\) −1.04020 −0.0425368
\(599\) 4.19582 0.171436 0.0857182 0.996319i \(-0.472682\pi\)
0.0857182 + 0.996319i \(0.472682\pi\)
\(600\) −4.56803 −0.186489
\(601\) 30.5991 1.24816 0.624081 0.781360i \(-0.285474\pi\)
0.624081 + 0.781360i \(0.285474\pi\)
\(602\) −3.47246 −0.141527
\(603\) 23.2228 0.945706
\(604\) −6.35364 −0.258526
\(605\) −16.9059 −0.687324
\(606\) −44.9358 −1.82539
\(607\) 42.8583 1.73957 0.869783 0.493435i \(-0.164259\pi\)
0.869783 + 0.493435i \(0.164259\pi\)
\(608\) 0 0
\(609\) −12.2862 −0.497862
\(610\) −3.50601 −0.141954
\(611\) 4.20031 0.169926
\(612\) 9.91471 0.400778
\(613\) 24.8978 1.00561 0.502806 0.864399i \(-0.332301\pi\)
0.502806 + 0.864399i \(0.332301\pi\)
\(614\) 19.9202 0.803912
\(615\) 42.4592 1.71212
\(616\) −1.18414 −0.0477105
\(617\) 13.1675 0.530102 0.265051 0.964234i \(-0.414611\pi\)
0.265051 + 0.964234i \(0.414611\pi\)
\(618\) −5.66376 −0.227830
\(619\) −5.34109 −0.214676 −0.107338 0.994223i \(-0.534233\pi\)
−0.107338 + 0.994223i \(0.534233\pi\)
\(620\) −2.43347 −0.0977303
\(621\) 1.40913 0.0565464
\(622\) −14.6345 −0.586792
\(623\) 1.87916 0.0752870
\(624\) 0.870512 0.0348484
\(625\) −11.9135 −0.476538
\(626\) −26.9261 −1.07618
\(627\) 0 0
\(628\) 4.20918 0.167965
\(629\) −11.8955 −0.474305
\(630\) −4.92597 −0.196255
\(631\) −49.2604 −1.96103 −0.980513 0.196453i \(-0.937058\pi\)
−0.980513 + 0.196453i \(0.937058\pi\)
\(632\) 5.28063 0.210052
\(633\) 10.9069 0.433509
\(634\) 30.5772 1.21438
\(635\) −34.1394 −1.35478
\(636\) 28.9316 1.14721
\(637\) 0.361568 0.0143258
\(638\) 6.04278 0.239236
\(639\) −0.634073 −0.0250835
\(640\) 1.76144 0.0696270
\(641\) −34.2166 −1.35147 −0.675737 0.737143i \(-0.736174\pi\)
−0.675737 + 0.737143i \(0.736174\pi\)
\(642\) −7.50816 −0.296324
\(643\) −44.9211 −1.77152 −0.885758 0.464148i \(-0.846361\pi\)
−0.885758 + 0.464148i \(0.846361\pi\)
\(644\) 2.87691 0.113366
\(645\) 14.7262 0.579842
\(646\) 0 0
\(647\) 19.1593 0.753230 0.376615 0.926370i \(-0.377088\pi\)
0.376615 + 0.926370i \(0.377088\pi\)
\(648\) −9.56894 −0.375903
\(649\) −9.15029 −0.359180
\(650\) −0.686015 −0.0269077
\(651\) 3.32616 0.130362
\(652\) 7.15678 0.280281
\(653\) 26.2846 1.02860 0.514299 0.857611i \(-0.328052\pi\)
0.514299 + 0.857611i \(0.328052\pi\)
\(654\) −34.1483 −1.33530
\(655\) −8.80602 −0.344079
\(656\) 10.0120 0.390901
\(657\) −39.1974 −1.52924
\(658\) −11.6169 −0.452876
\(659\) −37.2268 −1.45015 −0.725074 0.688671i \(-0.758195\pi\)
−0.725074 + 0.688671i \(0.758195\pi\)
\(660\) 5.02177 0.195472
\(661\) 10.9786 0.427018 0.213509 0.976941i \(-0.431511\pi\)
0.213509 + 0.976941i \(0.431511\pi\)
\(662\) 20.3542 0.791089
\(663\) 3.08625 0.119860
\(664\) 13.3912 0.519681
\(665\) 0 0
\(666\) −9.38319 −0.363591
\(667\) −14.6811 −0.568454
\(668\) 22.1964 0.858806
\(669\) 24.1270 0.932802
\(670\) 14.6271 0.565094
\(671\) −2.35695 −0.0909889
\(672\) −2.40760 −0.0928754
\(673\) 7.09081 0.273331 0.136665 0.990617i \(-0.456362\pi\)
0.136665 + 0.990617i \(0.456362\pi\)
\(674\) −14.8090 −0.570423
\(675\) 0.929327 0.0357698
\(676\) −12.8693 −0.494972
\(677\) −4.66569 −0.179317 −0.0896584 0.995973i \(-0.528578\pi\)
−0.0896584 + 0.995973i \(0.528578\pi\)
\(678\) −9.19910 −0.353289
\(679\) 3.99766 0.153416
\(680\) 6.24487 0.239480
\(681\) −17.9000 −0.685930
\(682\) −1.63592 −0.0626426
\(683\) −33.2650 −1.27285 −0.636425 0.771338i \(-0.719588\pi\)
−0.636425 + 0.771338i \(0.719588\pi\)
\(684\) 0 0
\(685\) 36.7923 1.40576
\(686\) −1.00000 −0.0381802
\(687\) 9.53753 0.363880
\(688\) 3.47246 0.132386
\(689\) 4.34488 0.165527
\(690\) −12.2005 −0.464466
\(691\) 27.5209 1.04694 0.523472 0.852043i \(-0.324637\pi\)
0.523472 + 0.852043i \(0.324637\pi\)
\(692\) 10.8227 0.411418
\(693\) −3.31153 −0.125794
\(694\) −19.0467 −0.723004
\(695\) −23.3203 −0.884588
\(696\) 12.2862 0.465707
\(697\) 35.4956 1.34449
\(698\) −21.1239 −0.799550
\(699\) −34.6356 −1.31004
\(700\) 1.89733 0.0717125
\(701\) 6.21171 0.234613 0.117306 0.993096i \(-0.462574\pi\)
0.117306 + 0.993096i \(0.462574\pi\)
\(702\) −0.177098 −0.00668415
\(703\) 0 0
\(704\) 1.18414 0.0446291
\(705\) 49.2657 1.85545
\(706\) 2.24313 0.0844213
\(707\) 18.6641 0.701937
\(708\) −18.6044 −0.699196
\(709\) −1.17736 −0.0442167 −0.0221083 0.999756i \(-0.507038\pi\)
−0.0221083 + 0.999756i \(0.507038\pi\)
\(710\) −0.399377 −0.0149883
\(711\) 14.7676 0.553828
\(712\) −1.87916 −0.0704246
\(713\) 3.97451 0.148847
\(714\) −8.53574 −0.319442
\(715\) 0.754157 0.0282039
\(716\) −11.7586 −0.439440
\(717\) −17.1428 −0.640209
\(718\) 3.31981 0.123894
\(719\) 37.2872 1.39058 0.695289 0.718730i \(-0.255276\pi\)
0.695289 + 0.718730i \(0.255276\pi\)
\(720\) 4.92597 0.183580
\(721\) 2.35245 0.0876097
\(722\) 0 0
\(723\) −36.0698 −1.34145
\(724\) 1.62634 0.0604426
\(725\) −9.68225 −0.359590
\(726\) −23.1077 −0.857608
\(727\) 17.8760 0.662983 0.331492 0.943458i \(-0.392448\pi\)
0.331492 + 0.943458i \(0.392448\pi\)
\(728\) −0.361568 −0.0134006
\(729\) −23.2223 −0.860085
\(730\) −24.6888 −0.913775
\(731\) 12.3110 0.455338
\(732\) −4.79215 −0.177123
\(733\) 7.02123 0.259335 0.129667 0.991558i \(-0.458609\pi\)
0.129667 + 0.991558i \(0.458609\pi\)
\(734\) −18.3886 −0.678734
\(735\) 4.24085 0.156426
\(736\) −2.87691 −0.106044
\(737\) 9.83320 0.362211
\(738\) 27.9990 1.03066
\(739\) −46.3549 −1.70519 −0.852596 0.522570i \(-0.824973\pi\)
−0.852596 + 0.522570i \(0.824973\pi\)
\(740\) −5.91009 −0.217259
\(741\) 0 0
\(742\) −12.0168 −0.441150
\(743\) 11.0899 0.406850 0.203425 0.979091i \(-0.434793\pi\)
0.203425 + 0.979091i \(0.434793\pi\)
\(744\) −3.32616 −0.121943
\(745\) 5.42903 0.198904
\(746\) −3.33060 −0.121942
\(747\) 37.4494 1.37020
\(748\) 4.19817 0.153500
\(749\) 3.11852 0.113948
\(750\) −29.2505 −1.06808
\(751\) 16.6367 0.607080 0.303540 0.952819i \(-0.401831\pi\)
0.303540 + 0.952819i \(0.401831\pi\)
\(752\) 11.6169 0.423626
\(753\) −68.0424 −2.47960
\(754\) 1.84511 0.0671949
\(755\) −11.1915 −0.407302
\(756\) 0.489807 0.0178141
\(757\) −23.9144 −0.869185 −0.434592 0.900627i \(-0.643108\pi\)
−0.434592 + 0.900627i \(0.643108\pi\)
\(758\) 30.8785 1.12156
\(759\) −8.20191 −0.297711
\(760\) 0 0
\(761\) 50.6040 1.83439 0.917196 0.398435i \(-0.130447\pi\)
0.917196 + 0.398435i \(0.130447\pi\)
\(762\) −46.6631 −1.69042
\(763\) 14.1835 0.513477
\(764\) 15.1136 0.546792
\(765\) 17.4641 0.631418
\(766\) −4.18785 −0.151313
\(767\) −2.79396 −0.100884
\(768\) 2.40760 0.0868769
\(769\) −8.66228 −0.312370 −0.156185 0.987728i \(-0.549920\pi\)
−0.156185 + 0.987728i \(0.549920\pi\)
\(770\) −2.08580 −0.0751669
\(771\) −27.7348 −0.998843
\(772\) −21.0053 −0.755998
\(773\) 28.4345 1.02272 0.511358 0.859368i \(-0.329142\pi\)
0.511358 + 0.859368i \(0.329142\pi\)
\(774\) 9.71093 0.349052
\(775\) 2.62121 0.0941566
\(776\) −3.99766 −0.143508
\(777\) 8.07815 0.289802
\(778\) 13.7229 0.491990
\(779\) 0 0
\(780\) 1.53335 0.0549029
\(781\) −0.268485 −0.00960713
\(782\) −10.1996 −0.364736
\(783\) −2.49952 −0.0893257
\(784\) 1.00000 0.0357143
\(785\) 7.41422 0.264625
\(786\) −12.0364 −0.429325
\(787\) 7.80328 0.278157 0.139078 0.990281i \(-0.455586\pi\)
0.139078 + 0.990281i \(0.455586\pi\)
\(788\) −24.5580 −0.874844
\(789\) 64.2474 2.28727
\(790\) 9.30151 0.330933
\(791\) 3.82085 0.135854
\(792\) 3.31153 0.117670
\(793\) −0.719673 −0.0255564
\(794\) −12.7243 −0.451568
\(795\) 50.9613 1.80741
\(796\) 5.50367 0.195073
\(797\) 22.7219 0.804849 0.402425 0.915453i \(-0.368168\pi\)
0.402425 + 0.915453i \(0.368168\pi\)
\(798\) 0 0
\(799\) 41.1858 1.45705
\(800\) −1.89733 −0.0670809
\(801\) −5.25519 −0.185683
\(802\) 27.5174 0.971673
\(803\) −16.5973 −0.585706
\(804\) 19.9929 0.705095
\(805\) 5.06749 0.178606
\(806\) −0.499514 −0.0175946
\(807\) −4.99175 −0.175718
\(808\) −18.6641 −0.656602
\(809\) −4.48771 −0.157779 −0.0788897 0.996883i \(-0.525137\pi\)
−0.0788897 + 0.996883i \(0.525137\pi\)
\(810\) −16.8551 −0.592228
\(811\) −17.5040 −0.614648 −0.307324 0.951605i \(-0.599434\pi\)
−0.307324 + 0.951605i \(0.599434\pi\)
\(812\) −5.10308 −0.179083
\(813\) −55.6121 −1.95040
\(814\) −3.97311 −0.139258
\(815\) 12.6062 0.441577
\(816\) 8.53574 0.298811
\(817\) 0 0
\(818\) 30.1566 1.05440
\(819\) −1.01115 −0.0353323
\(820\) 17.6355 0.615857
\(821\) 9.51661 0.332132 0.166066 0.986115i \(-0.446893\pi\)
0.166066 + 0.986115i \(0.446893\pi\)
\(822\) 50.2892 1.75404
\(823\) 31.8532 1.11033 0.555167 0.831739i \(-0.312654\pi\)
0.555167 + 0.831739i \(0.312654\pi\)
\(824\) −2.35245 −0.0819513
\(825\) −5.40920 −0.188324
\(826\) 7.72735 0.268869
\(827\) 10.9457 0.380620 0.190310 0.981724i \(-0.439051\pi\)
0.190310 + 0.981724i \(0.439051\pi\)
\(828\) −8.04544 −0.279598
\(829\) −30.2531 −1.05073 −0.525366 0.850876i \(-0.676072\pi\)
−0.525366 + 0.850876i \(0.676072\pi\)
\(830\) 23.5878 0.818746
\(831\) 49.0622 1.70195
\(832\) 0.361568 0.0125351
\(833\) 3.54532 0.122838
\(834\) −31.8750 −1.10374
\(835\) 39.0977 1.35303
\(836\) 0 0
\(837\) 0.676679 0.0233894
\(838\) 25.2631 0.872699
\(839\) 38.9311 1.34405 0.672025 0.740528i \(-0.265425\pi\)
0.672025 + 0.740528i \(0.265425\pi\)
\(840\) −4.24085 −0.146323
\(841\) −2.95857 −0.102020
\(842\) −38.0387 −1.31090
\(843\) 37.7110 1.29884
\(844\) 4.53017 0.155935
\(845\) −22.6684 −0.779818
\(846\) 32.4875 1.11694
\(847\) 9.59781 0.329784
\(848\) 12.0168 0.412658
\(849\) 5.71150 0.196018
\(850\) −6.72666 −0.230723
\(851\) 9.65278 0.330893
\(852\) −0.545884 −0.0187017
\(853\) −40.9671 −1.40269 −0.701344 0.712823i \(-0.747416\pi\)
−0.701344 + 0.712823i \(0.747416\pi\)
\(854\) 1.99042 0.0681109
\(855\) 0 0
\(856\) −3.11852 −0.106589
\(857\) −13.0672 −0.446367 −0.223184 0.974776i \(-0.571645\pi\)
−0.223184 + 0.974776i \(0.571645\pi\)
\(858\) 1.03081 0.0351913
\(859\) 14.8518 0.506735 0.253368 0.967370i \(-0.418462\pi\)
0.253368 + 0.967370i \(0.418462\pi\)
\(860\) 6.11652 0.208572
\(861\) −24.1048 −0.821491
\(862\) 18.9095 0.644059
\(863\) −56.6176 −1.92729 −0.963643 0.267193i \(-0.913904\pi\)
−0.963643 + 0.267193i \(0.913904\pi\)
\(864\) −0.489807 −0.0166636
\(865\) 19.0635 0.648180
\(866\) −12.6102 −0.428513
\(867\) −10.6673 −0.362281
\(868\) 1.38152 0.0468919
\(869\) 6.25303 0.212119
\(870\) 21.6414 0.733712
\(871\) 3.00248 0.101735
\(872\) −14.1835 −0.480314
\(873\) −11.1797 −0.378375
\(874\) 0 0
\(875\) 12.1492 0.410719
\(876\) −33.7457 −1.14016
\(877\) −29.6812 −1.00226 −0.501131 0.865372i \(-0.667082\pi\)
−0.501131 + 0.865372i \(0.667082\pi\)
\(878\) 33.9256 1.14494
\(879\) −1.41221 −0.0476327
\(880\) 2.08580 0.0703122
\(881\) −21.9013 −0.737875 −0.368937 0.929454i \(-0.620278\pi\)
−0.368937 + 0.929454i \(0.620278\pi\)
\(882\) 2.79656 0.0941650
\(883\) −22.8216 −0.768008 −0.384004 0.923331i \(-0.625455\pi\)
−0.384004 + 0.923331i \(0.625455\pi\)
\(884\) 1.28188 0.0431141
\(885\) −32.7705 −1.10157
\(886\) −4.69792 −0.157830
\(887\) 19.8209 0.665519 0.332760 0.943012i \(-0.392020\pi\)
0.332760 + 0.943012i \(0.392020\pi\)
\(888\) −8.07815 −0.271085
\(889\) 19.3815 0.650036
\(890\) −3.31003 −0.110952
\(891\) −11.3310 −0.379602
\(892\) 10.0211 0.335533
\(893\) 0 0
\(894\) 7.42061 0.248182
\(895\) −20.7121 −0.692329
\(896\) −1.00000 −0.0334077
\(897\) −2.50438 −0.0836189
\(898\) 25.3251 0.845109
\(899\) −7.05002 −0.235131
\(900\) −5.30600 −0.176867
\(901\) 42.6034 1.41932
\(902\) 11.8556 0.394748
\(903\) −8.36030 −0.278213
\(904\) −3.82085 −0.127080
\(905\) 2.86470 0.0952260
\(906\) −15.2970 −0.508210
\(907\) −51.2543 −1.70187 −0.850936 0.525270i \(-0.823965\pi\)
−0.850936 + 0.525270i \(0.823965\pi\)
\(908\) −7.43478 −0.246732
\(909\) −52.1953 −1.73121
\(910\) −0.636880 −0.0211124
\(911\) −31.5849 −1.04646 −0.523228 0.852193i \(-0.675272\pi\)
−0.523228 + 0.852193i \(0.675272\pi\)
\(912\) 0 0
\(913\) 15.8571 0.524795
\(914\) −40.3466 −1.33455
\(915\) −8.44108 −0.279054
\(916\) 3.96142 0.130889
\(917\) 4.99933 0.165092
\(918\) −1.73652 −0.0573138
\(919\) −9.70848 −0.320253 −0.160127 0.987096i \(-0.551190\pi\)
−0.160127 + 0.987096i \(0.551190\pi\)
\(920\) −5.06749 −0.167070
\(921\) 47.9599 1.58033
\(922\) 8.28479 0.272845
\(923\) −0.0819794 −0.00269839
\(924\) −2.85095 −0.0937893
\(925\) 6.36605 0.209315
\(926\) −26.7426 −0.878815
\(927\) −6.57875 −0.216075
\(928\) 5.10308 0.167517
\(929\) −11.8253 −0.387977 −0.193988 0.981004i \(-0.562142\pi\)
−0.193988 + 0.981004i \(0.562142\pi\)
\(930\) −5.85882 −0.192118
\(931\) 0 0
\(932\) −14.3859 −0.471226
\(933\) −35.2342 −1.15352
\(934\) 7.29907 0.238833
\(935\) 7.39482 0.241837
\(936\) 1.01115 0.0330503
\(937\) 39.1239 1.27812 0.639062 0.769155i \(-0.279323\pi\)
0.639062 + 0.769155i \(0.279323\pi\)
\(938\) −8.30406 −0.271137
\(939\) −64.8273 −2.11556
\(940\) 20.4625 0.667414
\(941\) 16.5733 0.540274 0.270137 0.962822i \(-0.412931\pi\)
0.270137 + 0.962822i \(0.412931\pi\)
\(942\) 10.1340 0.330185
\(943\) −28.8035 −0.937970
\(944\) −7.72735 −0.251504
\(945\) 0.862765 0.0280657
\(946\) 4.11189 0.133689
\(947\) 27.0746 0.879805 0.439902 0.898046i \(-0.355013\pi\)
0.439902 + 0.898046i \(0.355013\pi\)
\(948\) 12.7137 0.412921
\(949\) −5.06784 −0.164509
\(950\) 0 0
\(951\) 73.6178 2.38722
\(952\) −3.54532 −0.114905
\(953\) −23.7483 −0.769284 −0.384642 0.923066i \(-0.625675\pi\)
−0.384642 + 0.923066i \(0.625675\pi\)
\(954\) 33.6056 1.08802
\(955\) 26.6218 0.861460
\(956\) −7.12027 −0.230286
\(957\) 14.5486 0.470290
\(958\) −27.1804 −0.878160
\(959\) −20.8876 −0.674497
\(960\) 4.24085 0.136873
\(961\) −29.0914 −0.938432
\(962\) −1.21316 −0.0391137
\(963\) −8.72113 −0.281034
\(964\) −14.9816 −0.482525
\(965\) −36.9996 −1.19106
\(966\) 6.92645 0.222855
\(967\) 43.7978 1.40844 0.704221 0.709981i \(-0.251297\pi\)
0.704221 + 0.709981i \(0.251297\pi\)
\(968\) −9.59781 −0.308485
\(969\) 0 0
\(970\) −7.04163 −0.226093
\(971\) 17.3224 0.555901 0.277951 0.960595i \(-0.410345\pi\)
0.277951 + 0.960595i \(0.410345\pi\)
\(972\) −21.5688 −0.691819
\(973\) 13.2393 0.424433
\(974\) 6.88190 0.220510
\(975\) −1.65165 −0.0528952
\(976\) −1.99042 −0.0637119
\(977\) −55.2674 −1.76816 −0.884080 0.467336i \(-0.845214\pi\)
−0.884080 + 0.467336i \(0.845214\pi\)
\(978\) 17.2307 0.550977
\(979\) −2.22520 −0.0711176
\(980\) 1.76144 0.0562671
\(981\) −39.6650 −1.26641
\(982\) −9.47678 −0.302416
\(983\) −15.1410 −0.482922 −0.241461 0.970411i \(-0.577627\pi\)
−0.241461 + 0.970411i \(0.577627\pi\)
\(984\) 24.1048 0.768434
\(985\) −43.2575 −1.37830
\(986\) 18.0921 0.576169
\(987\) −27.9690 −0.890263
\(988\) 0 0
\(989\) −9.98994 −0.317662
\(990\) 5.83305 0.185386
\(991\) −32.4171 −1.02976 −0.514882 0.857261i \(-0.672164\pi\)
−0.514882 + 0.857261i \(0.672164\pi\)
\(992\) −1.38152 −0.0438634
\(993\) 49.0049 1.55512
\(994\) 0.226733 0.00719154
\(995\) 9.69439 0.307333
\(996\) 32.2408 1.02159
\(997\) 49.4268 1.56536 0.782681 0.622423i \(-0.213852\pi\)
0.782681 + 0.622423i \(0.213852\pi\)
\(998\) −5.58793 −0.176883
\(999\) 1.64343 0.0519959
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.bi.1.7 yes 8
19.18 odd 2 5054.2.a.bf.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.bf.1.2 8 19.18 odd 2
5054.2.a.bi.1.7 yes 8 1.1 even 1 trivial