Properties

Label 4006.2.a.g.1.5
Level 4006
Weight 2
Character 4006.1
Self dual yes
Analytic conductor 31.988
Analytic rank 1
Dimension 40
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4006.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9880710497\)
Analytic rank: \(1\)
Dimension: \(40\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) = 4006.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.39522 q^{3} +1.00000 q^{4} +0.491091 q^{5} +2.39522 q^{6} +3.07111 q^{7} -1.00000 q^{8} +2.73707 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.39522 q^{3} +1.00000 q^{4} +0.491091 q^{5} +2.39522 q^{6} +3.07111 q^{7} -1.00000 q^{8} +2.73707 q^{9} -0.491091 q^{10} +4.17695 q^{11} -2.39522 q^{12} +5.66744 q^{13} -3.07111 q^{14} -1.17627 q^{15} +1.00000 q^{16} -5.34713 q^{17} -2.73707 q^{18} +3.22494 q^{19} +0.491091 q^{20} -7.35598 q^{21} -4.17695 q^{22} -8.14473 q^{23} +2.39522 q^{24} -4.75883 q^{25} -5.66744 q^{26} +0.629784 q^{27} +3.07111 q^{28} +3.25118 q^{29} +1.17627 q^{30} -9.37775 q^{31} -1.00000 q^{32} -10.0047 q^{33} +5.34713 q^{34} +1.50819 q^{35} +2.73707 q^{36} -2.98089 q^{37} -3.22494 q^{38} -13.5748 q^{39} -0.491091 q^{40} -7.26806 q^{41} +7.35598 q^{42} -2.20220 q^{43} +4.17695 q^{44} +1.34415 q^{45} +8.14473 q^{46} -5.87041 q^{47} -2.39522 q^{48} +2.43171 q^{49} +4.75883 q^{50} +12.8075 q^{51} +5.66744 q^{52} -0.642744 q^{53} -0.629784 q^{54} +2.05126 q^{55} -3.07111 q^{56} -7.72443 q^{57} -3.25118 q^{58} -9.99028 q^{59} -1.17627 q^{60} -7.53520 q^{61} +9.37775 q^{62} +8.40583 q^{63} +1.00000 q^{64} +2.78323 q^{65} +10.0047 q^{66} +9.38047 q^{67} -5.34713 q^{68} +19.5084 q^{69} -1.50819 q^{70} -12.1271 q^{71} -2.73707 q^{72} +9.49214 q^{73} +2.98089 q^{74} +11.3984 q^{75} +3.22494 q^{76} +12.8279 q^{77} +13.5748 q^{78} -12.4184 q^{79} +0.491091 q^{80} -9.71967 q^{81} +7.26806 q^{82} +4.34233 q^{83} -7.35598 q^{84} -2.62592 q^{85} +2.20220 q^{86} -7.78728 q^{87} -4.17695 q^{88} -10.1770 q^{89} -1.34415 q^{90} +17.4053 q^{91} -8.14473 q^{92} +22.4617 q^{93} +5.87041 q^{94} +1.58374 q^{95} +2.39522 q^{96} +7.98956 q^{97} -2.43171 q^{98} +11.4326 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + O(q^{10}) \) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + 23q^{10} - 28q^{11} - q^{12} - 12q^{13} - 12q^{14} - 14q^{15} + 40q^{16} - 10q^{17} - 29q^{18} + 3q^{19} - 23q^{20} - 40q^{21} + 28q^{22} - 10q^{23} + q^{24} + 43q^{25} + 12q^{26} - 7q^{27} + 12q^{28} - 37q^{29} + 14q^{30} - 16q^{31} - 40q^{32} - 11q^{33} + 10q^{34} - 24q^{35} + 29q^{36} - 17q^{37} - 3q^{38} - 10q^{39} + 23q^{40} - 58q^{41} + 40q^{42} + 37q^{43} - 28q^{44} - 66q^{45} + 10q^{46} - 34q^{47} - q^{48} + 28q^{49} - 43q^{50} - 7q^{51} - 12q^{52} - 43q^{53} + 7q^{54} + 28q^{55} - 12q^{56} - 25q^{57} + 37q^{58} - 92q^{59} - 14q^{60} - 37q^{61} + 16q^{62} + 14q^{63} + 40q^{64} - 54q^{65} + 11q^{66} - 10q^{67} - 10q^{68} - 49q^{69} + 24q^{70} - 87q^{71} - 29q^{72} + 12q^{73} + 17q^{74} - 31q^{75} + 3q^{76} - 53q^{77} + 10q^{78} + 14q^{79} - 23q^{80} - 4q^{81} + 58q^{82} - 42q^{83} - 40q^{84} - 24q^{85} - 37q^{86} + 24q^{87} + 28q^{88} - 118q^{89} + 66q^{90} - 35q^{91} - 10q^{92} - 22q^{93} + 34q^{94} - 18q^{95} + q^{96} + 2q^{97} - 28q^{98} - 48q^{99} + O(q^{100}) \)

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
\(3\) −2.39522 −1.38288 −0.691440 0.722434i \(-0.743023\pi\)
−0.691440 + 0.722434i \(0.743023\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.491091 0.219622 0.109811 0.993952i \(-0.464975\pi\)
0.109811 + 0.993952i \(0.464975\pi\)
\(6\) 2.39522 0.977843
\(7\) 3.07111 1.16077 0.580385 0.814342i \(-0.302902\pi\)
0.580385 + 0.814342i \(0.302902\pi\)
\(8\) −1.00000 −0.353553
\(9\) 2.73707 0.912355
\(10\) −0.491091 −0.155296
\(11\) 4.17695 1.25940 0.629699 0.776839i \(-0.283178\pi\)
0.629699 + 0.776839i \(0.283178\pi\)
\(12\) −2.39522 −0.691440
\(13\) 5.66744 1.57187 0.785933 0.618312i \(-0.212183\pi\)
0.785933 + 0.618312i \(0.212183\pi\)
\(14\) −3.07111 −0.820789
\(15\) −1.17627 −0.303711
\(16\) 1.00000 0.250000
\(17\) −5.34713 −1.29687 −0.648434 0.761271i \(-0.724576\pi\)
−0.648434 + 0.761271i \(0.724576\pi\)
\(18\) −2.73707 −0.645133
\(19\) 3.22494 0.739852 0.369926 0.929061i \(-0.379383\pi\)
0.369926 + 0.929061i \(0.379383\pi\)
\(20\) 0.491091 0.109811
\(21\) −7.35598 −1.60521
\(22\) −4.17695 −0.890529
\(23\) −8.14473 −1.69829 −0.849147 0.528157i \(-0.822883\pi\)
−0.849147 + 0.528157i \(0.822883\pi\)
\(24\) 2.39522 0.488922
\(25\) −4.75883 −0.951766
\(26\) −5.66744 −1.11148
\(27\) 0.629784 0.121202
\(28\) 3.07111 0.580385
\(29\) 3.25118 0.603729 0.301864 0.953351i \(-0.402391\pi\)
0.301864 + 0.953351i \(0.402391\pi\)
\(30\) 1.17627 0.214756
\(31\) −9.37775 −1.68429 −0.842147 0.539248i \(-0.818708\pi\)
−0.842147 + 0.539248i \(0.818708\pi\)
\(32\) −1.00000 −0.176777
\(33\) −10.0047 −1.74160
\(34\) 5.34713 0.917024
\(35\) 1.50819 0.254931
\(36\) 2.73707 0.456178
\(37\) −2.98089 −0.490056 −0.245028 0.969516i \(-0.578797\pi\)
−0.245028 + 0.969516i \(0.578797\pi\)
\(38\) −3.22494 −0.523155
\(39\) −13.5748 −2.17370
\(40\) −0.491091 −0.0776482
\(41\) −7.26806 −1.13508 −0.567540 0.823346i \(-0.692105\pi\)
−0.567540 + 0.823346i \(0.692105\pi\)
\(42\) 7.35598 1.13505
\(43\) −2.20220 −0.335832 −0.167916 0.985801i \(-0.553704\pi\)
−0.167916 + 0.985801i \(0.553704\pi\)
\(44\) 4.17695 0.629699
\(45\) 1.34415 0.200374
\(46\) 8.14473 1.20088
\(47\) −5.87041 −0.856287 −0.428143 0.903711i \(-0.640832\pi\)
−0.428143 + 0.903711i \(0.640832\pi\)
\(48\) −2.39522 −0.345720
\(49\) 2.43171 0.347388
\(50\) 4.75883 0.673000
\(51\) 12.8075 1.79341
\(52\) 5.66744 0.785933
\(53\) −0.642744 −0.0882876 −0.0441438 0.999025i \(-0.514056\pi\)
−0.0441438 + 0.999025i \(0.514056\pi\)
\(54\) −0.629784 −0.0857027
\(55\) 2.05126 0.276592
\(56\) −3.07111 −0.410394
\(57\) −7.72443 −1.02313
\(58\) −3.25118 −0.426901
\(59\) −9.99028 −1.30062 −0.650312 0.759668i \(-0.725362\pi\)
−0.650312 + 0.759668i \(0.725362\pi\)
\(60\) −1.17627 −0.151856
\(61\) −7.53520 −0.964783 −0.482392 0.875956i \(-0.660232\pi\)
−0.482392 + 0.875956i \(0.660232\pi\)
\(62\) 9.37775 1.19098
\(63\) 8.40583 1.05904
\(64\) 1.00000 0.125000
\(65\) 2.78323 0.345217
\(66\) 10.0047 1.23149
\(67\) 9.38047 1.14601 0.573003 0.819553i \(-0.305778\pi\)
0.573003 + 0.819553i \(0.305778\pi\)
\(68\) −5.34713 −0.648434
\(69\) 19.5084 2.34854
\(70\) −1.50819 −0.180264
\(71\) −12.1271 −1.43922 −0.719610 0.694379i \(-0.755679\pi\)
−0.719610 + 0.694379i \(0.755679\pi\)
\(72\) −2.73707 −0.322566
\(73\) 9.49214 1.11097 0.555486 0.831526i \(-0.312532\pi\)
0.555486 + 0.831526i \(0.312532\pi\)
\(74\) 2.98089 0.346522
\(75\) 11.3984 1.31618
\(76\) 3.22494 0.369926
\(77\) 12.8279 1.46187
\(78\) 13.5748 1.53704
\(79\) −12.4184 −1.39717 −0.698587 0.715525i \(-0.746188\pi\)
−0.698587 + 0.715525i \(0.746188\pi\)
\(80\) 0.491091 0.0549056
\(81\) −9.71967 −1.07996
\(82\) 7.26806 0.802623
\(83\) 4.34233 0.476633 0.238316 0.971188i \(-0.423405\pi\)
0.238316 + 0.971188i \(0.423405\pi\)
\(84\) −7.35598 −0.802603
\(85\) −2.62592 −0.284821
\(86\) 2.20220 0.237469
\(87\) −7.78728 −0.834884
\(88\) −4.17695 −0.445264
\(89\) −10.1770 −1.07876 −0.539378 0.842064i \(-0.681340\pi\)
−0.539378 + 0.842064i \(0.681340\pi\)
\(90\) −1.34415 −0.141686
\(91\) 17.4053 1.82457
\(92\) −8.14473 −0.849147
\(93\) 22.4617 2.32917
\(94\) 5.87041 0.605486
\(95\) 1.58374 0.162488
\(96\) 2.39522 0.244461
\(97\) 7.98956 0.811217 0.405608 0.914047i \(-0.367060\pi\)
0.405608 + 0.914047i \(0.367060\pi\)
\(98\) −2.43171 −0.245640
\(99\) 11.4326 1.14902
\(100\) −4.75883 −0.475883
\(101\) −1.68806 −0.167968 −0.0839842 0.996467i \(-0.526765\pi\)
−0.0839842 + 0.996467i \(0.526765\pi\)
\(102\) −12.8075 −1.26813
\(103\) −1.91205 −0.188400 −0.0941998 0.995553i \(-0.530029\pi\)
−0.0941998 + 0.995553i \(0.530029\pi\)
\(104\) −5.66744 −0.555738
\(105\) −3.61245 −0.352539
\(106\) 0.642744 0.0624288
\(107\) 8.77566 0.848376 0.424188 0.905574i \(-0.360560\pi\)
0.424188 + 0.905574i \(0.360560\pi\)
\(108\) 0.629784 0.0606009
\(109\) −14.1944 −1.35957 −0.679786 0.733410i \(-0.737927\pi\)
−0.679786 + 0.733410i \(0.737927\pi\)
\(110\) −2.05126 −0.195580
\(111\) 7.13988 0.677688
\(112\) 3.07111 0.290193
\(113\) 9.20454 0.865891 0.432945 0.901420i \(-0.357474\pi\)
0.432945 + 0.901420i \(0.357474\pi\)
\(114\) 7.72443 0.723460
\(115\) −3.99980 −0.372983
\(116\) 3.25118 0.301864
\(117\) 15.5122 1.43410
\(118\) 9.99028 0.919679
\(119\) −16.4216 −1.50537
\(120\) 1.17627 0.107378
\(121\) 6.44691 0.586082
\(122\) 7.53520 0.682205
\(123\) 17.4086 1.56968
\(124\) −9.37775 −0.842147
\(125\) −4.79247 −0.428652
\(126\) −8.40583 −0.748851
\(127\) −4.22504 −0.374911 −0.187456 0.982273i \(-0.560024\pi\)
−0.187456 + 0.982273i \(0.560024\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.27474 0.464415
\(130\) −2.78323 −0.244105
\(131\) 4.88496 0.426801 0.213401 0.976965i \(-0.431546\pi\)
0.213401 + 0.976965i \(0.431546\pi\)
\(132\) −10.0047 −0.870798
\(133\) 9.90415 0.858798
\(134\) −9.38047 −0.810349
\(135\) 0.309281 0.0266187
\(136\) 5.34713 0.458512
\(137\) −0.798475 −0.0682183 −0.0341092 0.999418i \(-0.510859\pi\)
−0.0341092 + 0.999418i \(0.510859\pi\)
\(138\) −19.5084 −1.66067
\(139\) 9.77048 0.828722 0.414361 0.910113i \(-0.364005\pi\)
0.414361 + 0.910113i \(0.364005\pi\)
\(140\) 1.50819 0.127466
\(141\) 14.0609 1.18414
\(142\) 12.1271 1.01768
\(143\) 23.6726 1.97960
\(144\) 2.73707 0.228089
\(145\) 1.59662 0.132592
\(146\) −9.49214 −0.785575
\(147\) −5.82448 −0.480395
\(148\) −2.98089 −0.245028
\(149\) −21.1919 −1.73610 −0.868052 0.496473i \(-0.834628\pi\)
−0.868052 + 0.496473i \(0.834628\pi\)
\(150\) −11.3984 −0.930678
\(151\) 17.9796 1.46316 0.731578 0.681758i \(-0.238784\pi\)
0.731578 + 0.681758i \(0.238784\pi\)
\(152\) −3.22494 −0.261577
\(153\) −14.6354 −1.18320
\(154\) −12.8279 −1.03370
\(155\) −4.60532 −0.369909
\(156\) −13.5748 −1.08685
\(157\) −11.9540 −0.954031 −0.477015 0.878895i \(-0.658281\pi\)
−0.477015 + 0.878895i \(0.658281\pi\)
\(158\) 12.4184 0.987952
\(159\) 1.53951 0.122091
\(160\) −0.491091 −0.0388241
\(161\) −25.0134 −1.97133
\(162\) 9.71967 0.763649
\(163\) −11.0917 −0.868770 −0.434385 0.900727i \(-0.643034\pi\)
−0.434385 + 0.900727i \(0.643034\pi\)
\(164\) −7.26806 −0.567540
\(165\) −4.91321 −0.382493
\(166\) −4.34233 −0.337030
\(167\) −16.7829 −1.29870 −0.649351 0.760489i \(-0.724959\pi\)
−0.649351 + 0.760489i \(0.724959\pi\)
\(168\) 7.35598 0.567526
\(169\) 19.1199 1.47076
\(170\) 2.62592 0.201399
\(171\) 8.82688 0.675008
\(172\) −2.20220 −0.167916
\(173\) −9.99615 −0.759993 −0.379996 0.924988i \(-0.624075\pi\)
−0.379996 + 0.924988i \(0.624075\pi\)
\(174\) 7.78728 0.590352
\(175\) −14.6149 −1.10478
\(176\) 4.17695 0.314849
\(177\) 23.9289 1.79861
\(178\) 10.1770 0.762795
\(179\) −10.9214 −0.816304 −0.408152 0.912914i \(-0.633827\pi\)
−0.408152 + 0.912914i \(0.633827\pi\)
\(180\) 1.34415 0.100187
\(181\) 3.90061 0.289930 0.144965 0.989437i \(-0.453693\pi\)
0.144965 + 0.989437i \(0.453693\pi\)
\(182\) −17.4053 −1.29017
\(183\) 18.0484 1.33418
\(184\) 8.14473 0.600438
\(185\) −1.46389 −0.107627
\(186\) −22.4617 −1.64698
\(187\) −22.3347 −1.63327
\(188\) −5.87041 −0.428143
\(189\) 1.93413 0.140688
\(190\) −1.58374 −0.114896
\(191\) 2.97007 0.214907 0.107453 0.994210i \(-0.465730\pi\)
0.107453 + 0.994210i \(0.465730\pi\)
\(192\) −2.39522 −0.172860
\(193\) 7.73876 0.557048 0.278524 0.960429i \(-0.410155\pi\)
0.278524 + 0.960429i \(0.410155\pi\)
\(194\) −7.98956 −0.573617
\(195\) −6.66643 −0.477393
\(196\) 2.43171 0.173694
\(197\) 0.354426 0.0252518 0.0126259 0.999920i \(-0.495981\pi\)
0.0126259 + 0.999920i \(0.495981\pi\)
\(198\) −11.4326 −0.812479
\(199\) 14.1460 1.00278 0.501390 0.865221i \(-0.332822\pi\)
0.501390 + 0.865221i \(0.332822\pi\)
\(200\) 4.75883 0.336500
\(201\) −22.4683 −1.58479
\(202\) 1.68806 0.118772
\(203\) 9.98472 0.700790
\(204\) 12.8075 0.896706
\(205\) −3.56928 −0.249289
\(206\) 1.91205 0.133219
\(207\) −22.2927 −1.54945
\(208\) 5.66744 0.392966
\(209\) 13.4704 0.931768
\(210\) 3.61245 0.249283
\(211\) 24.6919 1.69986 0.849931 0.526894i \(-0.176644\pi\)
0.849931 + 0.526894i \(0.176644\pi\)
\(212\) −0.642744 −0.0441438
\(213\) 29.0470 1.99027
\(214\) −8.77566 −0.599892
\(215\) −1.08148 −0.0737563
\(216\) −0.629784 −0.0428513
\(217\) −28.8001 −1.95508
\(218\) 14.1944 0.961363
\(219\) −22.7357 −1.53634
\(220\) 2.05126 0.138296
\(221\) −30.3045 −2.03850
\(222\) −7.13988 −0.479198
\(223\) −27.1716 −1.81955 −0.909774 0.415105i \(-0.863745\pi\)
−0.909774 + 0.415105i \(0.863745\pi\)
\(224\) −3.07111 −0.205197
\(225\) −13.0252 −0.868349
\(226\) −9.20454 −0.612277
\(227\) 11.4716 0.761400 0.380700 0.924699i \(-0.375683\pi\)
0.380700 + 0.924699i \(0.375683\pi\)
\(228\) −7.72443 −0.511563
\(229\) −6.94051 −0.458642 −0.229321 0.973351i \(-0.573651\pi\)
−0.229321 + 0.973351i \(0.573651\pi\)
\(230\) 3.99980 0.263739
\(231\) −30.7255 −2.02159
\(232\) −3.25118 −0.213450
\(233\) −16.4108 −1.07511 −0.537553 0.843230i \(-0.680651\pi\)
−0.537553 + 0.843230i \(0.680651\pi\)
\(234\) −15.5122 −1.01406
\(235\) −2.88290 −0.188060
\(236\) −9.99028 −0.650312
\(237\) 29.7447 1.93212
\(238\) 16.4216 1.06445
\(239\) 22.3045 1.44276 0.721379 0.692540i \(-0.243508\pi\)
0.721379 + 0.692540i \(0.243508\pi\)
\(240\) −1.17627 −0.0759278
\(241\) −22.4473 −1.44596 −0.722979 0.690870i \(-0.757228\pi\)
−0.722979 + 0.690870i \(0.757228\pi\)
\(242\) −6.44691 −0.414423
\(243\) 21.3914 1.37226
\(244\) −7.53520 −0.482392
\(245\) 1.19419 0.0762941
\(246\) −17.4086 −1.10993
\(247\) 18.2772 1.16295
\(248\) 9.37775 0.595488
\(249\) −10.4008 −0.659125
\(250\) 4.79247 0.303102
\(251\) 20.4423 1.29031 0.645153 0.764053i \(-0.276793\pi\)
0.645153 + 0.764053i \(0.276793\pi\)
\(252\) 8.40583 0.529518
\(253\) −34.0201 −2.13883
\(254\) 4.22504 0.265102
\(255\) 6.28966 0.393874
\(256\) 1.00000 0.0625000
\(257\) −12.3192 −0.768450 −0.384225 0.923239i \(-0.625531\pi\)
−0.384225 + 0.923239i \(0.625531\pi\)
\(258\) −5.27474 −0.328391
\(259\) −9.15465 −0.568842
\(260\) 2.78323 0.172608
\(261\) 8.89869 0.550815
\(262\) −4.88496 −0.301794
\(263\) 17.1786 1.05928 0.529640 0.848223i \(-0.322327\pi\)
0.529640 + 0.848223i \(0.322327\pi\)
\(264\) 10.0047 0.615747
\(265\) −0.315645 −0.0193899
\(266\) −9.90415 −0.607262
\(267\) 24.3760 1.49179
\(268\) 9.38047 0.573003
\(269\) −0.264663 −0.0161368 −0.00806840 0.999967i \(-0.502568\pi\)
−0.00806840 + 0.999967i \(0.502568\pi\)
\(270\) −0.309281 −0.0188222
\(271\) −9.31291 −0.565719 −0.282859 0.959161i \(-0.591283\pi\)
−0.282859 + 0.959161i \(0.591283\pi\)
\(272\) −5.34713 −0.324217
\(273\) −41.6896 −2.52317
\(274\) 0.798475 0.0482376
\(275\) −19.8774 −1.19865
\(276\) 19.5084 1.17427
\(277\) −12.0706 −0.725254 −0.362627 0.931934i \(-0.618120\pi\)
−0.362627 + 0.931934i \(0.618120\pi\)
\(278\) −9.77048 −0.585995
\(279\) −25.6675 −1.53667
\(280\) −1.50819 −0.0901318
\(281\) −20.0764 −1.19766 −0.598828 0.800878i \(-0.704367\pi\)
−0.598828 + 0.800878i \(0.704367\pi\)
\(282\) −14.0609 −0.837314
\(283\) 33.1312 1.96944 0.984722 0.174133i \(-0.0557124\pi\)
0.984722 + 0.174133i \(0.0557124\pi\)
\(284\) −12.1271 −0.719610
\(285\) −3.79340 −0.224701
\(286\) −23.6726 −1.39979
\(287\) −22.3210 −1.31757
\(288\) −2.73707 −0.161283
\(289\) 11.5917 0.681868
\(290\) −1.59662 −0.0937569
\(291\) −19.1367 −1.12182
\(292\) 9.49214 0.555486
\(293\) 23.7516 1.38758 0.693792 0.720176i \(-0.255939\pi\)
0.693792 + 0.720176i \(0.255939\pi\)
\(294\) 5.82448 0.339691
\(295\) −4.90613 −0.285646
\(296\) 2.98089 0.173261
\(297\) 2.63057 0.152641
\(298\) 21.1919 1.22761
\(299\) −46.1598 −2.66949
\(300\) 11.3984 0.658089
\(301\) −6.76319 −0.389824
\(302\) −17.9796 −1.03461
\(303\) 4.04328 0.232280
\(304\) 3.22494 0.184963
\(305\) −3.70046 −0.211888
\(306\) 14.6354 0.836652
\(307\) 9.38157 0.535435 0.267717 0.963498i \(-0.413731\pi\)
0.267717 + 0.963498i \(0.413731\pi\)
\(308\) 12.8279 0.730936
\(309\) 4.57977 0.260534
\(310\) 4.60532 0.261565
\(311\) −5.26368 −0.298476 −0.149238 0.988801i \(-0.547682\pi\)
−0.149238 + 0.988801i \(0.547682\pi\)
\(312\) 13.5748 0.768519
\(313\) 24.8871 1.40670 0.703350 0.710844i \(-0.251687\pi\)
0.703350 + 0.710844i \(0.251687\pi\)
\(314\) 11.9540 0.674602
\(315\) 4.12802 0.232588
\(316\) −12.4184 −0.698587
\(317\) 16.8876 0.948502 0.474251 0.880390i \(-0.342719\pi\)
0.474251 + 0.880390i \(0.342719\pi\)
\(318\) −1.53951 −0.0863315
\(319\) 13.5800 0.760334
\(320\) 0.491091 0.0274528
\(321\) −21.0196 −1.17320
\(322\) 25.0134 1.39394
\(323\) −17.2442 −0.959491
\(324\) −9.71967 −0.539982
\(325\) −26.9704 −1.49605
\(326\) 11.0917 0.614313
\(327\) 33.9986 1.88012
\(328\) 7.26806 0.401311
\(329\) −18.0287 −0.993952
\(330\) 4.91321 0.270464
\(331\) 22.3758 1.22988 0.614941 0.788573i \(-0.289180\pi\)
0.614941 + 0.788573i \(0.289180\pi\)
\(332\) 4.34233 0.238316
\(333\) −8.15890 −0.447105
\(334\) 16.7829 0.918321
\(335\) 4.60666 0.251689
\(336\) −7.35598 −0.401301
\(337\) 21.8971 1.19281 0.596406 0.802683i \(-0.296595\pi\)
0.596406 + 0.802683i \(0.296595\pi\)
\(338\) −19.1199 −1.03999
\(339\) −22.0469 −1.19742
\(340\) −2.62592 −0.142411
\(341\) −39.1704 −2.12120
\(342\) −8.82688 −0.477303
\(343\) −14.0297 −0.757533
\(344\) 2.20220 0.118735
\(345\) 9.58039 0.515791
\(346\) 9.99615 0.537396
\(347\) −12.6063 −0.676744 −0.338372 0.941012i \(-0.609876\pi\)
−0.338372 + 0.941012i \(0.609876\pi\)
\(348\) −7.78728 −0.417442
\(349\) −28.4651 −1.52370 −0.761852 0.647752i \(-0.775709\pi\)
−0.761852 + 0.647752i \(0.775709\pi\)
\(350\) 14.6149 0.781199
\(351\) 3.56926 0.190513
\(352\) −4.17695 −0.222632
\(353\) 8.69186 0.462621 0.231311 0.972880i \(-0.425699\pi\)
0.231311 + 0.972880i \(0.425699\pi\)
\(354\) −23.9289 −1.27181
\(355\) −5.95549 −0.316085
\(356\) −10.1770 −0.539378
\(357\) 39.3333 2.08174
\(358\) 10.9214 0.577214
\(359\) 12.1148 0.639394 0.319697 0.947520i \(-0.396419\pi\)
0.319697 + 0.947520i \(0.396419\pi\)
\(360\) −1.34415 −0.0708428
\(361\) −8.59976 −0.452619
\(362\) −3.90061 −0.205012
\(363\) −15.4417 −0.810481
\(364\) 17.4053 0.912287
\(365\) 4.66150 0.243994
\(366\) −18.0484 −0.943407
\(367\) −32.7058 −1.70723 −0.853615 0.520904i \(-0.825595\pi\)
−0.853615 + 0.520904i \(0.825595\pi\)
\(368\) −8.14473 −0.424573
\(369\) −19.8932 −1.03560
\(370\) 1.46389 0.0761039
\(371\) −1.97394 −0.102482
\(372\) 22.4617 1.16459
\(373\) −14.0290 −0.726394 −0.363197 0.931712i \(-0.618315\pi\)
−0.363197 + 0.931712i \(0.618315\pi\)
\(374\) 22.3347 1.15490
\(375\) 11.4790 0.592773
\(376\) 5.87041 0.302743
\(377\) 18.4259 0.948980
\(378\) −1.93413 −0.0994811
\(379\) −7.19822 −0.369748 −0.184874 0.982762i \(-0.559188\pi\)
−0.184874 + 0.982762i \(0.559188\pi\)
\(380\) 1.58374 0.0812441
\(381\) 10.1199 0.518457
\(382\) −2.97007 −0.151962
\(383\) 6.09068 0.311219 0.155610 0.987819i \(-0.450266\pi\)
0.155610 + 0.987819i \(0.450266\pi\)
\(384\) 2.39522 0.122230
\(385\) 6.29965 0.321060
\(386\) −7.73876 −0.393893
\(387\) −6.02756 −0.306398
\(388\) 7.98956 0.405608
\(389\) −16.9965 −0.861758 −0.430879 0.902410i \(-0.641796\pi\)
−0.430879 + 0.902410i \(0.641796\pi\)
\(390\) 6.66643 0.337568
\(391\) 43.5509 2.20246
\(392\) −2.43171 −0.122820
\(393\) −11.7005 −0.590214
\(394\) −0.354426 −0.0178557
\(395\) −6.09854 −0.306851
\(396\) 11.4326 0.574509
\(397\) −32.9086 −1.65164 −0.825818 0.563937i \(-0.809286\pi\)
−0.825818 + 0.563937i \(0.809286\pi\)
\(398\) −14.1460 −0.709072
\(399\) −23.7226 −1.18761
\(400\) −4.75883 −0.237942
\(401\) −24.4612 −1.22153 −0.610767 0.791811i \(-0.709139\pi\)
−0.610767 + 0.791811i \(0.709139\pi\)
\(402\) 22.4683 1.12061
\(403\) −53.1478 −2.64748
\(404\) −1.68806 −0.0839842
\(405\) −4.77324 −0.237184
\(406\) −9.98472 −0.495533
\(407\) −12.4510 −0.617175
\(408\) −12.8075 −0.634067
\(409\) 2.74577 0.135769 0.0678847 0.997693i \(-0.478375\pi\)
0.0678847 + 0.997693i \(0.478375\pi\)
\(410\) 3.56928 0.176274
\(411\) 1.91252 0.0943377
\(412\) −1.91205 −0.0941998
\(413\) −30.6812 −1.50972
\(414\) 22.2927 1.09562
\(415\) 2.13248 0.104679
\(416\) −5.66744 −0.277869
\(417\) −23.4024 −1.14602
\(418\) −13.4704 −0.658860
\(419\) −12.3308 −0.602398 −0.301199 0.953561i \(-0.597387\pi\)
−0.301199 + 0.953561i \(0.597387\pi\)
\(420\) −3.61245 −0.176270
\(421\) −2.41067 −0.117489 −0.0587446 0.998273i \(-0.518710\pi\)
−0.0587446 + 0.998273i \(0.518710\pi\)
\(422\) −24.6919 −1.20198
\(423\) −16.0677 −0.781238
\(424\) 0.642744 0.0312144
\(425\) 25.4461 1.23432
\(426\) −29.0470 −1.40733
\(427\) −23.1414 −1.11989
\(428\) 8.77566 0.424188
\(429\) −56.7011 −2.73755
\(430\) 1.08148 0.0521535
\(431\) −12.0286 −0.579396 −0.289698 0.957118i \(-0.593555\pi\)
−0.289698 + 0.957118i \(0.593555\pi\)
\(432\) 0.629784 0.0303005
\(433\) 27.1227 1.30343 0.651716 0.758463i \(-0.274049\pi\)
0.651716 + 0.758463i \(0.274049\pi\)
\(434\) 28.8001 1.38245
\(435\) −3.82426 −0.183359
\(436\) −14.1944 −0.679786
\(437\) −26.2663 −1.25649
\(438\) 22.7357 1.08636
\(439\) −28.9910 −1.38366 −0.691832 0.722059i \(-0.743196\pi\)
−0.691832 + 0.722059i \(0.743196\pi\)
\(440\) −2.05126 −0.0977900
\(441\) 6.65576 0.316941
\(442\) 30.3045 1.44144
\(443\) 11.4191 0.542539 0.271269 0.962503i \(-0.412557\pi\)
0.271269 + 0.962503i \(0.412557\pi\)
\(444\) 7.13988 0.338844
\(445\) −4.99781 −0.236919
\(446\) 27.1716 1.28661
\(447\) 50.7591 2.40082
\(448\) 3.07111 0.145096
\(449\) −16.5317 −0.780181 −0.390091 0.920776i \(-0.627556\pi\)
−0.390091 + 0.920776i \(0.627556\pi\)
\(450\) 13.0252 0.614015
\(451\) −30.3583 −1.42952
\(452\) 9.20454 0.432945
\(453\) −43.0650 −2.02337
\(454\) −11.4716 −0.538391
\(455\) 8.54760 0.400718
\(456\) 7.72443 0.361730
\(457\) −26.9824 −1.26218 −0.631092 0.775708i \(-0.717393\pi\)
−0.631092 + 0.775708i \(0.717393\pi\)
\(458\) 6.94051 0.324309
\(459\) −3.36753 −0.157183
\(460\) −3.99980 −0.186492
\(461\) 16.6346 0.774751 0.387375 0.921922i \(-0.373382\pi\)
0.387375 + 0.921922i \(0.373382\pi\)
\(462\) 30.7255 1.42948
\(463\) −26.3819 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(464\) 3.25118 0.150932
\(465\) 11.0308 0.511539
\(466\) 16.4108 0.760215
\(467\) 20.9194 0.968034 0.484017 0.875058i \(-0.339177\pi\)
0.484017 + 0.875058i \(0.339177\pi\)
\(468\) 15.5122 0.717050
\(469\) 28.8084 1.33025
\(470\) 2.88290 0.132978
\(471\) 28.6324 1.31931
\(472\) 9.99028 0.459840
\(473\) −9.19847 −0.422946
\(474\) −29.7447 −1.36622
\(475\) −15.3469 −0.704166
\(476\) −16.4216 −0.752683
\(477\) −1.75923 −0.0805497
\(478\) −22.3045 −1.02018
\(479\) 17.9881 0.821898 0.410949 0.911658i \(-0.365197\pi\)
0.410949 + 0.911658i \(0.365197\pi\)
\(480\) 1.17627 0.0536891
\(481\) −16.8940 −0.770301
\(482\) 22.4473 1.02245
\(483\) 59.9124 2.72611
\(484\) 6.44691 0.293041
\(485\) 3.92360 0.178161
\(486\) −21.3914 −0.970332
\(487\) 38.3070 1.73585 0.867927 0.496692i \(-0.165452\pi\)
0.867927 + 0.496692i \(0.165452\pi\)
\(488\) 7.53520 0.341102
\(489\) 26.5671 1.20140
\(490\) −1.19419 −0.0539481
\(491\) 36.3842 1.64200 0.820998 0.570931i \(-0.193418\pi\)
0.820998 + 0.570931i \(0.193418\pi\)
\(492\) 17.4086 0.784840
\(493\) −17.3845 −0.782956
\(494\) −18.2772 −0.822329
\(495\) 5.61444 0.252350
\(496\) −9.37775 −0.421073
\(497\) −37.2436 −1.67060
\(498\) 10.4008 0.466072
\(499\) 24.6930 1.10541 0.552705 0.833377i \(-0.313596\pi\)
0.552705 + 0.833377i \(0.313596\pi\)
\(500\) −4.79247 −0.214326
\(501\) 40.1988 1.79595
\(502\) −20.4423 −0.912385
\(503\) −21.1450 −0.942809 −0.471404 0.881917i \(-0.656253\pi\)
−0.471404 + 0.881917i \(0.656253\pi\)
\(504\) −8.40583 −0.374425
\(505\) −0.828992 −0.0368896
\(506\) 34.0201 1.51238
\(507\) −45.7963 −2.03389
\(508\) −4.22504 −0.187456
\(509\) −12.3985 −0.549553 −0.274776 0.961508i \(-0.588604\pi\)
−0.274776 + 0.961508i \(0.588604\pi\)
\(510\) −6.28966 −0.278511
\(511\) 29.1514 1.28958
\(512\) −1.00000 −0.0441942
\(513\) 2.03101 0.0896715
\(514\) 12.3192 0.543376
\(515\) −0.938989 −0.0413768
\(516\) 5.27474 0.232208
\(517\) −24.5204 −1.07841
\(518\) 9.15465 0.402232
\(519\) 23.9429 1.05098
\(520\) −2.78323 −0.122053
\(521\) −7.01193 −0.307198 −0.153599 0.988133i \(-0.549086\pi\)
−0.153599 + 0.988133i \(0.549086\pi\)
\(522\) −8.89869 −0.389485
\(523\) 29.6863 1.29809 0.649045 0.760750i \(-0.275169\pi\)
0.649045 + 0.760750i \(0.275169\pi\)
\(524\) 4.88496 0.213401
\(525\) 35.0058 1.52778
\(526\) −17.1786 −0.749024
\(527\) 50.1440 2.18431
\(528\) −10.0047 −0.435399
\(529\) 43.3367 1.88420
\(530\) 0.315645 0.0137108
\(531\) −27.3440 −1.18663
\(532\) 9.90415 0.429399
\(533\) −41.1913 −1.78419
\(534\) −24.3760 −1.05485
\(535\) 4.30965 0.186322
\(536\) −9.38047 −0.405175
\(537\) 26.1591 1.12885
\(538\) 0.264663 0.0114104
\(539\) 10.1571 0.437499
\(540\) 0.309281 0.0133093
\(541\) −0.402620 −0.0173100 −0.00865499 0.999963i \(-0.502755\pi\)
−0.00865499 + 0.999963i \(0.502755\pi\)
\(542\) 9.31291 0.400024
\(543\) −9.34281 −0.400938
\(544\) 5.34713 0.229256
\(545\) −6.97071 −0.298593
\(546\) 41.6896 1.78415
\(547\) 25.9079 1.10774 0.553871 0.832602i \(-0.313150\pi\)
0.553871 + 0.832602i \(0.313150\pi\)
\(548\) −0.798475 −0.0341092
\(549\) −20.6243 −0.880225
\(550\) 19.8774 0.847575
\(551\) 10.4849 0.446670
\(552\) −19.5084 −0.830333
\(553\) −38.1381 −1.62180
\(554\) 12.0706 0.512832
\(555\) 3.50633 0.148835
\(556\) 9.77048 0.414361
\(557\) −21.2299 −0.899538 −0.449769 0.893145i \(-0.648494\pi\)
−0.449769 + 0.893145i \(0.648494\pi\)
\(558\) 25.6675 1.08659
\(559\) −12.4808 −0.527883
\(560\) 1.50819 0.0637328
\(561\) 53.4964 2.25862
\(562\) 20.0764 0.846870
\(563\) −20.0942 −0.846872 −0.423436 0.905926i \(-0.639176\pi\)
−0.423436 + 0.905926i \(0.639176\pi\)
\(564\) 14.0609 0.592071
\(565\) 4.52027 0.190169
\(566\) −33.1312 −1.39261
\(567\) −29.8502 −1.25359
\(568\) 12.1271 0.508841
\(569\) 18.0634 0.757258 0.378629 0.925549i \(-0.376396\pi\)
0.378629 + 0.925549i \(0.376396\pi\)
\(570\) 3.79340 0.158888
\(571\) −16.1702 −0.676702 −0.338351 0.941020i \(-0.609869\pi\)
−0.338351 + 0.941020i \(0.609869\pi\)
\(572\) 23.6726 0.989802
\(573\) −7.11396 −0.297190
\(574\) 22.3210 0.931661
\(575\) 38.7594 1.61638
\(576\) 2.73707 0.114044
\(577\) 44.0917 1.83556 0.917781 0.397088i \(-0.129979\pi\)
0.917781 + 0.397088i \(0.129979\pi\)
\(578\) −11.5917 −0.482153
\(579\) −18.5360 −0.770330
\(580\) 1.59662 0.0662962
\(581\) 13.3358 0.553261
\(582\) 19.1367 0.793243
\(583\) −2.68471 −0.111189
\(584\) −9.49214 −0.392788
\(585\) 7.61788 0.314960
\(586\) −23.7516 −0.981170
\(587\) 2.45312 0.101251 0.0506256 0.998718i \(-0.483878\pi\)
0.0506256 + 0.998718i \(0.483878\pi\)
\(588\) −5.82448 −0.240198
\(589\) −30.2427 −1.24613
\(590\) 4.90613 0.201982
\(591\) −0.848927 −0.0349202
\(592\) −2.98089 −0.122514
\(593\) −38.4514 −1.57901 −0.789505 0.613744i \(-0.789663\pi\)
−0.789505 + 0.613744i \(0.789663\pi\)
\(594\) −2.63057 −0.107934
\(595\) −8.06450 −0.330612
\(596\) −21.1919 −0.868052
\(597\) −33.8826 −1.38672
\(598\) 46.1598 1.88761
\(599\) 28.2670 1.15496 0.577479 0.816406i \(-0.304037\pi\)
0.577479 + 0.816406i \(0.304037\pi\)
\(600\) −11.3984 −0.465339
\(601\) −35.8944 −1.46416 −0.732082 0.681217i \(-0.761451\pi\)
−0.732082 + 0.681217i \(0.761451\pi\)
\(602\) 6.76319 0.275647
\(603\) 25.6750 1.04557
\(604\) 17.9796 0.731578
\(605\) 3.16601 0.128717
\(606\) −4.04328 −0.164247
\(607\) −10.0515 −0.407976 −0.203988 0.978973i \(-0.565390\pi\)
−0.203988 + 0.978973i \(0.565390\pi\)
\(608\) −3.22494 −0.130789
\(609\) −23.9156 −0.969108
\(610\) 3.70046 0.149827
\(611\) −33.2702 −1.34597
\(612\) −14.6354 −0.591602
\(613\) 13.7202 0.554154 0.277077 0.960848i \(-0.410634\pi\)
0.277077 + 0.960848i \(0.410634\pi\)
\(614\) −9.38157 −0.378609
\(615\) 8.54919 0.344737
\(616\) −12.8279 −0.516850
\(617\) 4.83317 0.194576 0.0972881 0.995256i \(-0.468983\pi\)
0.0972881 + 0.995256i \(0.468983\pi\)
\(618\) −4.57977 −0.184225
\(619\) 9.03192 0.363024 0.181512 0.983389i \(-0.441901\pi\)
0.181512 + 0.983389i \(0.441901\pi\)
\(620\) −4.60532 −0.184954
\(621\) −5.12942 −0.205836
\(622\) 5.26368 0.211054
\(623\) −31.2546 −1.25219
\(624\) −13.5748 −0.543425
\(625\) 21.4406 0.857625
\(626\) −24.8871 −0.994687
\(627\) −32.2646 −1.28852
\(628\) −11.9540 −0.477015
\(629\) 15.9392 0.635538
\(630\) −4.12802 −0.164464
\(631\) 3.48726 0.138826 0.0694129 0.997588i \(-0.477887\pi\)
0.0694129 + 0.997588i \(0.477887\pi\)
\(632\) 12.4184 0.493976
\(633\) −59.1425 −2.35070
\(634\) −16.8876 −0.670692
\(635\) −2.07488 −0.0823390
\(636\) 1.53951 0.0610456
\(637\) 13.7816 0.546047
\(638\) −13.5800 −0.537638
\(639\) −33.1926 −1.31308
\(640\) −0.491091 −0.0194121
\(641\) 10.2789 0.405992 0.202996 0.979180i \(-0.434932\pi\)
0.202996 + 0.979180i \(0.434932\pi\)
\(642\) 21.0196 0.829578
\(643\) −6.19060 −0.244133 −0.122067 0.992522i \(-0.538952\pi\)
−0.122067 + 0.992522i \(0.538952\pi\)
\(644\) −25.0134 −0.985665
\(645\) 2.59038 0.101996
\(646\) 17.2442 0.678463
\(647\) 20.5889 0.809433 0.404717 0.914442i \(-0.367370\pi\)
0.404717 + 0.914442i \(0.367370\pi\)
\(648\) 9.71967 0.381825
\(649\) −41.7289 −1.63800
\(650\) 26.9704 1.05787
\(651\) 68.9825 2.70364
\(652\) −11.0917 −0.434385
\(653\) 27.1456 1.06229 0.531145 0.847281i \(-0.321762\pi\)
0.531145 + 0.847281i \(0.321762\pi\)
\(654\) −33.9986 −1.32945
\(655\) 2.39896 0.0937351
\(656\) −7.26806 −0.283770
\(657\) 25.9806 1.01360
\(658\) 18.0287 0.702831
\(659\) 30.8015 1.19986 0.599929 0.800053i \(-0.295196\pi\)
0.599929 + 0.800053i \(0.295196\pi\)
\(660\) −4.91321 −0.191247
\(661\) 23.5729 0.916881 0.458441 0.888725i \(-0.348408\pi\)
0.458441 + 0.888725i \(0.348408\pi\)
\(662\) −22.3758 −0.869659
\(663\) 72.5859 2.81900
\(664\) −4.34233 −0.168515
\(665\) 4.86383 0.188611
\(666\) 8.15890 0.316151
\(667\) −26.4800 −1.02531
\(668\) −16.7829 −0.649351
\(669\) 65.0820 2.51621
\(670\) −4.60666 −0.177971
\(671\) −31.4741 −1.21505
\(672\) 7.35598 0.283763
\(673\) 8.13234 0.313479 0.156739 0.987640i \(-0.449902\pi\)
0.156739 + 0.987640i \(0.449902\pi\)
\(674\) −21.8971 −0.843445
\(675\) −2.99703 −0.115356
\(676\) 19.1199 0.735381
\(677\) 51.4609 1.97780 0.988901 0.148578i \(-0.0474696\pi\)
0.988901 + 0.148578i \(0.0474696\pi\)
\(678\) 22.0469 0.846705
\(679\) 24.5368 0.941637
\(680\) 2.62592 0.100700
\(681\) −27.4771 −1.05292
\(682\) 39.1704 1.49991
\(683\) −2.92145 −0.111786 −0.0558930 0.998437i \(-0.517801\pi\)
−0.0558930 + 0.998437i \(0.517801\pi\)
\(684\) 8.82688 0.337504
\(685\) −0.392124 −0.0149823
\(686\) 14.0297 0.535657
\(687\) 16.6240 0.634246
\(688\) −2.20220 −0.0839580
\(689\) −3.64271 −0.138776
\(690\) −9.58039 −0.364719
\(691\) 35.1316 1.33647 0.668234 0.743951i \(-0.267050\pi\)
0.668234 + 0.743951i \(0.267050\pi\)
\(692\) −9.99615 −0.379996
\(693\) 35.1107 1.33375
\(694\) 12.6063 0.478530
\(695\) 4.79819 0.182006
\(696\) 7.78728 0.295176
\(697\) 38.8632 1.47205
\(698\) 28.4651 1.07742
\(699\) 39.3074 1.48674
\(700\) −14.6149 −0.552391
\(701\) 9.01787 0.340600 0.170300 0.985392i \(-0.445526\pi\)
0.170300 + 0.985392i \(0.445526\pi\)
\(702\) −3.56926 −0.134713
\(703\) −9.61320 −0.362569
\(704\) 4.17695 0.157425
\(705\) 6.90518 0.260064
\(706\) −8.69186 −0.327123
\(707\) −5.18422 −0.194973
\(708\) 23.9289 0.899303
\(709\) 51.0066 1.91559 0.957796 0.287450i \(-0.0928074\pi\)
0.957796 + 0.287450i \(0.0928074\pi\)
\(710\) 5.95549 0.223506
\(711\) −33.9899 −1.27472
\(712\) 10.1770 0.381398
\(713\) 76.3793 2.86043
\(714\) −39.3333 −1.47201
\(715\) 11.6254 0.434765
\(716\) −10.9214 −0.408152
\(717\) −53.4242 −1.99516
\(718\) −12.1148 −0.452120
\(719\) −40.9282 −1.52637 −0.763183 0.646183i \(-0.776364\pi\)
−0.763183 + 0.646183i \(0.776364\pi\)
\(720\) 1.34415 0.0500934
\(721\) −5.87211 −0.218689
\(722\) 8.59976 0.320050
\(723\) 53.7661 1.99958
\(724\) 3.90061 0.144965
\(725\) −15.4718 −0.574608
\(726\) 15.4417 0.573097
\(727\) 23.3071 0.864413 0.432206 0.901775i \(-0.357735\pi\)
0.432206 + 0.901775i \(0.357735\pi\)
\(728\) −17.4053 −0.645085
\(729\) −22.0780 −0.817702
\(730\) −4.66150 −0.172530
\(731\) 11.7754 0.435530
\(732\) 18.0484 0.667089
\(733\) −32.1927 −1.18906 −0.594532 0.804072i \(-0.702663\pi\)
−0.594532 + 0.804072i \(0.702663\pi\)
\(734\) 32.7058 1.20719
\(735\) −2.86035 −0.105506
\(736\) 8.14473 0.300219
\(737\) 39.1817 1.44328
\(738\) 19.8932 0.732277
\(739\) −37.3545 −1.37411 −0.687053 0.726607i \(-0.741096\pi\)
−0.687053 + 0.726607i \(0.741096\pi\)
\(740\) −1.46389 −0.0538136
\(741\) −43.7778 −1.60822
\(742\) 1.97394 0.0724655
\(743\) −10.6076 −0.389156 −0.194578 0.980887i \(-0.562334\pi\)
−0.194578 + 0.980887i \(0.562334\pi\)
\(744\) −22.4617 −0.823488
\(745\) −10.4071 −0.381287
\(746\) 14.0290 0.513638
\(747\) 11.8852 0.434858
\(748\) −22.3347 −0.816636
\(749\) 26.9510 0.984769
\(750\) −11.4790 −0.419154
\(751\) −5.39998 −0.197048 −0.0985240 0.995135i \(-0.531412\pi\)
−0.0985240 + 0.995135i \(0.531412\pi\)
\(752\) −5.87041 −0.214072
\(753\) −48.9638 −1.78434
\(754\) −18.4259 −0.671030
\(755\) 8.82959 0.321342
\(756\) 1.93413 0.0703438
\(757\) 40.6092 1.47597 0.737983 0.674819i \(-0.235778\pi\)
0.737983 + 0.674819i \(0.235778\pi\)
\(758\) 7.19822 0.261451
\(759\) 81.4856 2.95774
\(760\) −1.58374 −0.0574482
\(761\) 2.77011 0.100417 0.0502083 0.998739i \(-0.484011\pi\)
0.0502083 + 0.998739i \(0.484011\pi\)
\(762\) −10.1199 −0.366605
\(763\) −43.5924 −1.57815
\(764\) 2.97007 0.107453
\(765\) −7.18733 −0.259858
\(766\) −6.09068 −0.220065
\(767\) −56.6193 −2.04440
\(768\) −2.39522 −0.0864300
\(769\) −17.8199 −0.642603 −0.321301 0.946977i \(-0.604120\pi\)
−0.321301 + 0.946977i \(0.604120\pi\)
\(770\) −6.29965 −0.227023
\(771\) 29.5071 1.06267
\(772\) 7.73876 0.278524
\(773\) −23.3260 −0.838977 −0.419489 0.907761i \(-0.637791\pi\)
−0.419489 + 0.907761i \(0.637791\pi\)
\(774\) 6.02756 0.216656
\(775\) 44.6271 1.60305
\(776\) −7.98956 −0.286808
\(777\) 21.9274 0.786640
\(778\) 16.9965 0.609355
\(779\) −23.4391 −0.839792
\(780\) −6.66643 −0.238697
\(781\) −50.6542 −1.81255
\(782\) −43.5509 −1.55738
\(783\) 2.04754 0.0731730
\(784\) 2.43171 0.0868469
\(785\) −5.87048 −0.209526
\(786\) 11.7005 0.417345
\(787\) 20.2693 0.722523 0.361262 0.932464i \(-0.382346\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(788\) 0.354426 0.0126259
\(789\) −41.1466 −1.46486
\(790\) 6.09854 0.216976
\(791\) 28.2682 1.00510
\(792\) −11.4326 −0.406239
\(793\) −42.7053 −1.51651
\(794\) 32.9086 1.16788
\(795\) 0.756039 0.0268139
\(796\) 14.1460 0.501390
\(797\) 12.3513 0.437504 0.218752 0.975780i \(-0.429801\pi\)
0.218752 + 0.975780i \(0.429801\pi\)
\(798\) 23.7226 0.839770
\(799\) 31.3898 1.11049
\(800\) 4.75883 0.168250
\(801\) −27.8550 −0.984208
\(802\) 24.4612 0.863754
\(803\) 39.6482 1.39915
\(804\) −22.4683 −0.792394
\(805\) −12.2838 −0.432948
\(806\) 53.1478 1.87205
\(807\) 0.633926 0.0223152
\(808\) 1.68806 0.0593858
\(809\) 19.0325 0.669148 0.334574 0.942369i \(-0.391408\pi\)
0.334574 + 0.942369i \(0.391408\pi\)
\(810\) 4.77324 0.167714
\(811\) −27.9661 −0.982022 −0.491011 0.871153i \(-0.663373\pi\)
−0.491011 + 0.871153i \(0.663373\pi\)
\(812\) 9.98472 0.350395
\(813\) 22.3064 0.782321
\(814\) 12.4510 0.436409
\(815\) −5.44704 −0.190801
\(816\) 12.8075 0.448353
\(817\) −7.10196 −0.248466
\(818\) −2.74577 −0.0960035
\(819\) 47.6396 1.66466
\(820\) −3.56928 −0.124645
\(821\) 25.4108 0.886844 0.443422 0.896313i \(-0.353764\pi\)
0.443422 + 0.896313i \(0.353764\pi\)
\(822\) −1.91252 −0.0667068
\(823\) 35.1045 1.22367 0.611833 0.790987i \(-0.290432\pi\)
0.611833 + 0.790987i \(0.290432\pi\)
\(824\) 1.91205 0.0666093
\(825\) 47.6107 1.65759
\(826\) 30.6812 1.06754
\(827\) −33.0004 −1.14754 −0.573768 0.819018i \(-0.694519\pi\)
−0.573768 + 0.819018i \(0.694519\pi\)
\(828\) −22.2927 −0.774724
\(829\) 9.61320 0.333880 0.166940 0.985967i \(-0.446611\pi\)
0.166940 + 0.985967i \(0.446611\pi\)
\(830\) −2.13248 −0.0740194
\(831\) 28.9118 1.00294
\(832\) 5.66744 0.196483
\(833\) −13.0027 −0.450516
\(834\) 23.4024 0.810360
\(835\) −8.24194 −0.285224
\(836\) 13.4704 0.465884
\(837\) −5.90595 −0.204140
\(838\) 12.3308 0.425959
\(839\) −46.8043 −1.61587 −0.807933 0.589275i \(-0.799414\pi\)
−0.807933 + 0.589275i \(0.799414\pi\)
\(840\) 3.61245 0.124641
\(841\) −18.4298 −0.635512
\(842\) 2.41067 0.0830773
\(843\) 48.0873 1.65621
\(844\) 24.6919 0.849931
\(845\) 9.38960 0.323012
\(846\) 16.0677 0.552419
\(847\) 19.7992 0.680307
\(848\) −0.642744 −0.0220719
\(849\) −79.3564 −2.72350
\(850\) −25.4461 −0.872793
\(851\) 24.2786 0.832258
\(852\) 29.0470 0.995133
\(853\) −23.6958 −0.811328 −0.405664 0.914022i \(-0.632960\pi\)
−0.405664 + 0.914022i \(0.632960\pi\)
\(854\) 23.1414 0.791883
\(855\) 4.33480 0.148247
\(856\) −8.77566 −0.299946
\(857\) 33.1710 1.13310 0.566550 0.824027i \(-0.308278\pi\)
0.566550 + 0.824027i \(0.308278\pi\)
\(858\) 56.7011 1.93574
\(859\) −2.65757 −0.0906751 −0.0453375 0.998972i \(-0.514436\pi\)
−0.0453375 + 0.998972i \(0.514436\pi\)
\(860\) −1.08148 −0.0368781
\(861\) 53.4637 1.82204
\(862\) 12.0286 0.409695
\(863\) −36.7289 −1.25027 −0.625133 0.780518i \(-0.714955\pi\)
−0.625133 + 0.780518i \(0.714955\pi\)
\(864\) −0.629784 −0.0214257
\(865\) −4.90901 −0.166911
\(866\) −27.1227 −0.921666
\(867\) −27.7648 −0.942941
\(868\) −28.8001 −0.977539
\(869\) −51.8709 −1.75960
\(870\) 3.82426 0.129655
\(871\) 53.1633 1.80137
\(872\) 14.1944 0.480681
\(873\) 21.8680 0.740118
\(874\) 26.2663 0.888470
\(875\) −14.7182 −0.497566
\(876\) −22.7357 −0.768170
\(877\) 0.989371 0.0334087 0.0167043 0.999860i \(-0.494683\pi\)
0.0167043 + 0.999860i \(0.494683\pi\)
\(878\) 28.9910 0.978398
\(879\) −56.8903 −1.91886
\(880\) 2.05126 0.0691480
\(881\) −27.2205 −0.917082 −0.458541 0.888673i \(-0.651628\pi\)
−0.458541 + 0.888673i \(0.651628\pi\)
\(882\) −6.65576 −0.224111
\(883\) 24.4715 0.823530 0.411765 0.911290i \(-0.364912\pi\)
0.411765 + 0.911290i \(0.364912\pi\)
\(884\) −30.3045 −1.01925
\(885\) 11.7512 0.395014
\(886\) −11.4191 −0.383633
\(887\) −24.1615 −0.811263 −0.405631 0.914037i \(-0.632948\pi\)
−0.405631 + 0.914037i \(0.632948\pi\)
\(888\) −7.13988 −0.239599
\(889\) −12.9756 −0.435186
\(890\) 4.99781 0.167527
\(891\) −40.5986 −1.36010
\(892\) −27.1716 −0.909774
\(893\) −18.9317 −0.633526
\(894\) −50.7591 −1.69764
\(895\) −5.36340 −0.179279
\(896\) −3.07111 −0.102599
\(897\) 110.563 3.69158
\(898\) 16.5317 0.551672
\(899\) −30.4887 −1.01686
\(900\) −13.0252 −0.434174
\(901\) 3.43683 0.114497
\(902\) 30.3583 1.01082
\(903\) 16.1993 0.539080
\(904\) −9.20454 −0.306139
\(905\) 1.91555 0.0636752
\(906\) 43.0650 1.43074
\(907\) 47.7732 1.58628 0.793142 0.609037i \(-0.208444\pi\)
0.793142 + 0.609037i \(0.208444\pi\)
\(908\) 11.4716 0.380700
\(909\) −4.62034 −0.153247
\(910\) −8.54760 −0.283350
\(911\) −40.6459 −1.34666 −0.673330 0.739342i \(-0.735137\pi\)
−0.673330 + 0.739342i \(0.735137\pi\)
\(912\) −7.72443 −0.255782
\(913\) 18.1377 0.600270
\(914\) 26.9824 0.892499
\(915\) 8.86342 0.293016
\(916\) −6.94051 −0.229321
\(917\) 15.0023 0.495418
\(918\) 3.36753 0.111145
\(919\) 16.1240 0.531881 0.265940 0.963989i \(-0.414318\pi\)
0.265940 + 0.963989i \(0.414318\pi\)
\(920\) 3.99980 0.131870
\(921\) −22.4709 −0.740441
\(922\) −16.6346 −0.547831
\(923\) −68.7295 −2.26226
\(924\) −30.7255 −1.01080
\(925\) 14.1856 0.466418
\(926\) 26.3819 0.866964
\(927\) −5.23340 −0.171887
\(928\) −3.25118 −0.106725
\(929\) −3.68430 −0.120878 −0.0604390 0.998172i \(-0.519250\pi\)
−0.0604390 + 0.998172i \(0.519250\pi\)
\(930\) −11.0308 −0.361713
\(931\) 7.84214 0.257016
\(932\) −16.4108 −0.537553
\(933\) 12.6076 0.412756
\(934\) −20.9194 −0.684504
\(935\) −10.9683 −0.358703
\(936\) −15.5122 −0.507031
\(937\) 30.4101 0.993456 0.496728 0.867906i \(-0.334535\pi\)
0.496728 + 0.867906i \(0.334535\pi\)
\(938\) −28.8084 −0.940629
\(939\) −59.6099 −1.94530
\(940\) −2.88290 −0.0940299
\(941\) 52.1075 1.69866 0.849329 0.527865i \(-0.177007\pi\)
0.849329 + 0.527865i \(0.177007\pi\)
\(942\) −28.6324 −0.932893
\(943\) 59.1964 1.92770
\(944\) −9.99028 −0.325156
\(945\) 0.949835 0.0308981
\(946\) 9.19847 0.299068
\(947\) 2.95686 0.0960852 0.0480426 0.998845i \(-0.484702\pi\)
0.0480426 + 0.998845i \(0.484702\pi\)
\(948\) 29.7447 0.966062
\(949\) 53.7962 1.74630
\(950\) 15.3469 0.497921
\(951\) −40.4495 −1.31166
\(952\) 16.4216 0.532227
\(953\) −25.0877 −0.812669 −0.406334 0.913724i \(-0.633193\pi\)
−0.406334 + 0.913724i \(0.633193\pi\)
\(954\) 1.75923 0.0569572
\(955\) 1.45857 0.0471983
\(956\) 22.3045 0.721379
\(957\) −32.5271 −1.05145
\(958\) −17.9881 −0.581170
\(959\) −2.45220 −0.0791858
\(960\) −1.17627 −0.0379639
\(961\) 56.9422 1.83684
\(962\) 16.8940 0.544685
\(963\) 24.0196 0.774020
\(964\) −22.4473 −0.722979
\(965\) 3.80043 0.122340
\(966\) −59.9124 −1.92765
\(967\) 3.29962 0.106109 0.0530543 0.998592i \(-0.483104\pi\)
0.0530543 + 0.998592i \(0.483104\pi\)
\(968\) −6.44691 −0.207211
\(969\) 41.3035 1.32686
\(970\) −3.92360 −0.125979
\(971\) −3.20895 −0.102980 −0.0514900 0.998674i \(-0.516397\pi\)
−0.0514900 + 0.998674i \(0.516397\pi\)
\(972\) 21.3914 0.686128
\(973\) 30.0062 0.961955
\(974\) −38.3070 −1.22743
\(975\) 64.6000 2.06885
\(976\) −7.53520 −0.241196
\(977\) −52.1630 −1.66884 −0.834421 0.551127i \(-0.814198\pi\)
−0.834421 + 0.551127i \(0.814198\pi\)
\(978\) −26.5671 −0.849521
\(979\) −42.5086 −1.35858
\(980\) 1.19419 0.0381471
\(981\) −38.8509 −1.24041
\(982\) −36.3842 −1.16107
\(983\) 23.5438 0.750930 0.375465 0.926837i \(-0.377483\pi\)
0.375465 + 0.926837i \(0.377483\pi\)
\(984\) −17.4086 −0.554965
\(985\) 0.174055 0.00554586
\(986\) 17.3845 0.553634
\(987\) 43.1826 1.37452
\(988\) 18.2772 0.581474
\(989\) 17.9363 0.570342
\(990\) −5.61444 −0.178438
\(991\) 20.4687 0.650211 0.325105 0.945678i \(-0.394600\pi\)
0.325105 + 0.945678i \(0.394600\pi\)
\(992\) 9.37775 0.297744
\(993\) −53.5948 −1.70078
\(994\) 37.2436 1.18129
\(995\) 6.94694 0.220233
\(996\) −10.4008 −0.329563
\(997\) 28.5292 0.903530 0.451765 0.892137i \(-0.350795\pi\)
0.451765 + 0.892137i \(0.350795\pi\)
\(998\) −24.6930 −0.781642
\(999\) −1.87732 −0.0593957
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 4006.2.a.g.1.5 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4006.2.a.g.1.5 40 1.1 even 1 trivial