Properties

Label 4029.2.a.k.1.15
Level 4029
Weight 2
Character 4029.1
Self dual yes
Analytic conductor 32.172
Analytic rank 0
Dimension 31
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4029.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1717269744\)
Analytic rank: \(0\)
Dimension: \(31\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) = 4029.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.0488355 q^{2} +1.00000 q^{3} -1.99762 q^{4} -0.188126 q^{5} +0.0488355 q^{6} +3.46836 q^{7} -0.195225 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.0488355 q^{2} +1.00000 q^{3} -1.99762 q^{4} -0.188126 q^{5} +0.0488355 q^{6} +3.46836 q^{7} -0.195225 q^{8} +1.00000 q^{9} -0.00918720 q^{10} +3.08722 q^{11} -1.99762 q^{12} -6.69012 q^{13} +0.169379 q^{14} -0.188126 q^{15} +3.98570 q^{16} +1.00000 q^{17} +0.0488355 q^{18} -1.82042 q^{19} +0.375802 q^{20} +3.46836 q^{21} +0.150766 q^{22} +3.48683 q^{23} -0.195225 q^{24} -4.96461 q^{25} -0.326715 q^{26} +1.00000 q^{27} -6.92845 q^{28} -4.01796 q^{29} -0.00918720 q^{30} +10.0379 q^{31} +0.585094 q^{32} +3.08722 q^{33} +0.0488355 q^{34} -0.652488 q^{35} -1.99762 q^{36} +8.47735 q^{37} -0.0889012 q^{38} -6.69012 q^{39} +0.0367269 q^{40} +5.10831 q^{41} +0.169379 q^{42} +10.8185 q^{43} -6.16709 q^{44} -0.188126 q^{45} +0.170281 q^{46} -8.85864 q^{47} +3.98570 q^{48} +5.02954 q^{49} -0.242449 q^{50} +1.00000 q^{51} +13.3643 q^{52} -8.03495 q^{53} +0.0488355 q^{54} -0.580786 q^{55} -0.677113 q^{56} -1.82042 q^{57} -0.196219 q^{58} +12.4261 q^{59} +0.375802 q^{60} +6.94654 q^{61} +0.490206 q^{62} +3.46836 q^{63} -7.94282 q^{64} +1.25858 q^{65} +0.150766 q^{66} -1.50745 q^{67} -1.99762 q^{68} +3.48683 q^{69} -0.0318645 q^{70} -9.99510 q^{71} -0.195225 q^{72} -2.67757 q^{73} +0.413996 q^{74} -4.96461 q^{75} +3.63650 q^{76} +10.7076 q^{77} -0.326715 q^{78} +1.00000 q^{79} -0.749811 q^{80} +1.00000 q^{81} +0.249467 q^{82} -13.1230 q^{83} -6.92845 q^{84} -0.188126 q^{85} +0.528329 q^{86} -4.01796 q^{87} -0.602705 q^{88} +4.05252 q^{89} -0.00918720 q^{90} -23.2038 q^{91} -6.96534 q^{92} +10.0379 q^{93} -0.432616 q^{94} +0.342468 q^{95} +0.585094 q^{96} +8.47463 q^{97} +0.245620 q^{98} +3.08722 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + O(q^{10}) \) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + 5q^{10} + 26q^{11} + 34q^{12} + 7q^{13} + 19q^{14} + 11q^{15} + 40q^{16} + 31q^{17} + 4q^{18} + 32q^{19} + 23q^{20} + 4q^{21} + 2q^{22} + 29q^{23} + 12q^{24} + 32q^{25} + 13q^{26} + 31q^{27} - 13q^{28} + 25q^{29} + 5q^{30} + 22q^{31} + 28q^{32} + 26q^{33} + 4q^{34} + 20q^{35} + 34q^{36} - 4q^{37} + 19q^{38} + 7q^{39} - 3q^{40} + 33q^{41} + 19q^{42} + 6q^{43} + 30q^{44} + 11q^{45} - 11q^{46} + 23q^{47} + 40q^{48} + 31q^{49} + 6q^{50} + 31q^{51} - 7q^{52} + 12q^{53} + 4q^{54} + 40q^{56} + 32q^{57} + 9q^{58} + 27q^{59} + 23q^{60} - 4q^{61} + 25q^{62} + 4q^{63} + 10q^{64} + 54q^{65} + 2q^{66} + 34q^{68} + 29q^{69} - 59q^{70} + 35q^{71} + 12q^{72} + 5q^{73} + 48q^{74} + 32q^{75} + 32q^{76} + 42q^{77} + 13q^{78} + 31q^{79} + 24q^{80} + 31q^{81} + 5q^{82} + 67q^{83} - 13q^{84} + 11q^{85} - 20q^{86} + 25q^{87} - 7q^{88} + 22q^{89} + 5q^{90} + 16q^{91} + 57q^{92} + 22q^{93} + 45q^{94} + 73q^{95} + 28q^{96} - 13q^{97} - 19q^{98} + 26q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0488355 0.0345319 0.0172660 0.999851i \(-0.494504\pi\)
0.0172660 + 0.999851i \(0.494504\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.99762 −0.998808
\(5\) −0.188126 −0.0841323 −0.0420661 0.999115i \(-0.513394\pi\)
−0.0420661 + 0.999115i \(0.513394\pi\)
\(6\) 0.0488355 0.0199370
\(7\) 3.46836 1.31092 0.655459 0.755231i \(-0.272475\pi\)
0.655459 + 0.755231i \(0.272475\pi\)
\(8\) −0.195225 −0.0690226
\(9\) 1.00000 0.333333
\(10\) −0.00918720 −0.00290525
\(11\) 3.08722 0.930833 0.465417 0.885092i \(-0.345905\pi\)
0.465417 + 0.885092i \(0.345905\pi\)
\(12\) −1.99762 −0.576662
\(13\) −6.69012 −1.85551 −0.927753 0.373195i \(-0.878262\pi\)
−0.927753 + 0.373195i \(0.878262\pi\)
\(14\) 0.169379 0.0452685
\(15\) −0.188126 −0.0485738
\(16\) 3.98570 0.996424
\(17\) 1.00000 0.242536
\(18\) 0.0488355 0.0115106
\(19\) −1.82042 −0.417633 −0.208817 0.977955i \(-0.566961\pi\)
−0.208817 + 0.977955i \(0.566961\pi\)
\(20\) 0.375802 0.0840320
\(21\) 3.46836 0.756859
\(22\) 0.150766 0.0321434
\(23\) 3.48683 0.727053 0.363527 0.931584i \(-0.381573\pi\)
0.363527 + 0.931584i \(0.381573\pi\)
\(24\) −0.195225 −0.0398502
\(25\) −4.96461 −0.992922
\(26\) −0.326715 −0.0640742
\(27\) 1.00000 0.192450
\(28\) −6.92845 −1.30935
\(29\) −4.01796 −0.746116 −0.373058 0.927808i \(-0.621691\pi\)
−0.373058 + 0.927808i \(0.621691\pi\)
\(30\) −0.00918720 −0.00167735
\(31\) 10.0379 1.80286 0.901431 0.432923i \(-0.142518\pi\)
0.901431 + 0.432923i \(0.142518\pi\)
\(32\) 0.585094 0.103431
\(33\) 3.08722 0.537417
\(34\) 0.0488355 0.00837522
\(35\) −0.652488 −0.110291
\(36\) −1.99762 −0.332936
\(37\) 8.47735 1.39367 0.696834 0.717232i \(-0.254591\pi\)
0.696834 + 0.717232i \(0.254591\pi\)
\(38\) −0.0889012 −0.0144217
\(39\) −6.69012 −1.07128
\(40\) 0.0367269 0.00580703
\(41\) 5.10831 0.797783 0.398892 0.916998i \(-0.369395\pi\)
0.398892 + 0.916998i \(0.369395\pi\)
\(42\) 0.169379 0.0261358
\(43\) 10.8185 1.64981 0.824906 0.565270i \(-0.191228\pi\)
0.824906 + 0.565270i \(0.191228\pi\)
\(44\) −6.16709 −0.929723
\(45\) −0.188126 −0.0280441
\(46\) 0.170281 0.0251065
\(47\) −8.85864 −1.29217 −0.646083 0.763267i \(-0.723594\pi\)
−0.646083 + 0.763267i \(0.723594\pi\)
\(48\) 3.98570 0.575286
\(49\) 5.02954 0.718506
\(50\) −0.242449 −0.0342875
\(51\) 1.00000 0.140028
\(52\) 13.3643 1.85329
\(53\) −8.03495 −1.10369 −0.551843 0.833948i \(-0.686075\pi\)
−0.551843 + 0.833948i \(0.686075\pi\)
\(54\) 0.0488355 0.00664567
\(55\) −0.580786 −0.0783131
\(56\) −0.677113 −0.0904830
\(57\) −1.82042 −0.241121
\(58\) −0.196219 −0.0257648
\(59\) 12.4261 1.61774 0.808871 0.587986i \(-0.200079\pi\)
0.808871 + 0.587986i \(0.200079\pi\)
\(60\) 0.375802 0.0485159
\(61\) 6.94654 0.889413 0.444706 0.895676i \(-0.353308\pi\)
0.444706 + 0.895676i \(0.353308\pi\)
\(62\) 0.490206 0.0622563
\(63\) 3.46836 0.436973
\(64\) −7.94282 −0.992852
\(65\) 1.25858 0.156108
\(66\) 0.150766 0.0185580
\(67\) −1.50745 −0.184164 −0.0920820 0.995751i \(-0.529352\pi\)
−0.0920820 + 0.995751i \(0.529352\pi\)
\(68\) −1.99762 −0.242246
\(69\) 3.48683 0.419764
\(70\) −0.0318645 −0.00380854
\(71\) −9.99510 −1.18620 −0.593100 0.805129i \(-0.702096\pi\)
−0.593100 + 0.805129i \(0.702096\pi\)
\(72\) −0.195225 −0.0230075
\(73\) −2.67757 −0.313386 −0.156693 0.987647i \(-0.550083\pi\)
−0.156693 + 0.987647i \(0.550083\pi\)
\(74\) 0.413996 0.0481260
\(75\) −4.96461 −0.573264
\(76\) 3.63650 0.417135
\(77\) 10.7076 1.22025
\(78\) −0.326715 −0.0369932
\(79\) 1.00000 0.112509
\(80\) −0.749811 −0.0838314
\(81\) 1.00000 0.111111
\(82\) 0.249467 0.0275490
\(83\) −13.1230 −1.44044 −0.720221 0.693745i \(-0.755960\pi\)
−0.720221 + 0.693745i \(0.755960\pi\)
\(84\) −6.92845 −0.755956
\(85\) −0.188126 −0.0204051
\(86\) 0.528329 0.0569711
\(87\) −4.01796 −0.430770
\(88\) −0.602705 −0.0642486
\(89\) 4.05252 0.429567 0.214783 0.976662i \(-0.431095\pi\)
0.214783 + 0.976662i \(0.431095\pi\)
\(90\) −0.00918720 −0.000968416 0
\(91\) −23.2038 −2.43242
\(92\) −6.96534 −0.726186
\(93\) 10.0379 1.04088
\(94\) −0.432616 −0.0446209
\(95\) 0.342468 0.0351364
\(96\) 0.585094 0.0597159
\(97\) 8.47463 0.860468 0.430234 0.902717i \(-0.358431\pi\)
0.430234 + 0.902717i \(0.358431\pi\)
\(98\) 0.245620 0.0248114
\(99\) 3.08722 0.310278
\(100\) 9.91738 0.991738
\(101\) 16.5028 1.64209 0.821045 0.570864i \(-0.193391\pi\)
0.821045 + 0.570864i \(0.193391\pi\)
\(102\) 0.0488355 0.00483543
\(103\) −3.65982 −0.360612 −0.180306 0.983611i \(-0.557709\pi\)
−0.180306 + 0.983611i \(0.557709\pi\)
\(104\) 1.30608 0.128072
\(105\) −0.652488 −0.0636763
\(106\) −0.392391 −0.0381123
\(107\) 19.0585 1.84246 0.921228 0.389023i \(-0.127187\pi\)
0.921228 + 0.389023i \(0.127187\pi\)
\(108\) −1.99762 −0.192221
\(109\) −12.7995 −1.22597 −0.612987 0.790093i \(-0.710032\pi\)
−0.612987 + 0.790093i \(0.710032\pi\)
\(110\) −0.0283630 −0.00270430
\(111\) 8.47735 0.804635
\(112\) 13.8238 1.30623
\(113\) 10.3696 0.975487 0.487744 0.872987i \(-0.337820\pi\)
0.487744 + 0.872987i \(0.337820\pi\)
\(114\) −0.0889012 −0.00832636
\(115\) −0.655961 −0.0611687
\(116\) 8.02633 0.745226
\(117\) −6.69012 −0.618502
\(118\) 0.606835 0.0558637
\(119\) 3.46836 0.317944
\(120\) 0.0367269 0.00335269
\(121\) −1.46904 −0.133549
\(122\) 0.339237 0.0307131
\(123\) 5.10831 0.460600
\(124\) −20.0519 −1.80071
\(125\) 1.87460 0.167669
\(126\) 0.169379 0.0150895
\(127\) 3.38064 0.299983 0.149992 0.988687i \(-0.452075\pi\)
0.149992 + 0.988687i \(0.452075\pi\)
\(128\) −1.55808 −0.137716
\(129\) 10.8185 0.952519
\(130\) 0.0614635 0.00539071
\(131\) 14.9439 1.30565 0.652827 0.757507i \(-0.273583\pi\)
0.652827 + 0.757507i \(0.273583\pi\)
\(132\) −6.16709 −0.536776
\(133\) −6.31388 −0.547483
\(134\) −0.0736170 −0.00635954
\(135\) −0.188126 −0.0161913
\(136\) −0.195225 −0.0167404
\(137\) −3.55003 −0.303300 −0.151650 0.988434i \(-0.548459\pi\)
−0.151650 + 0.988434i \(0.548459\pi\)
\(138\) 0.170281 0.0144953
\(139\) −20.5032 −1.73906 −0.869530 0.493880i \(-0.835578\pi\)
−0.869530 + 0.493880i \(0.835578\pi\)
\(140\) 1.30342 0.110159
\(141\) −8.85864 −0.746032
\(142\) −0.488116 −0.0409617
\(143\) −20.6539 −1.72717
\(144\) 3.98570 0.332141
\(145\) 0.755880 0.0627724
\(146\) −0.130760 −0.0108218
\(147\) 5.02954 0.414830
\(148\) −16.9345 −1.39201
\(149\) 5.79237 0.474529 0.237265 0.971445i \(-0.423749\pi\)
0.237265 + 0.971445i \(0.423749\pi\)
\(150\) −0.242449 −0.0197959
\(151\) −7.38646 −0.601102 −0.300551 0.953766i \(-0.597170\pi\)
−0.300551 + 0.953766i \(0.597170\pi\)
\(152\) 0.355393 0.0288261
\(153\) 1.00000 0.0808452
\(154\) 0.522912 0.0421374
\(155\) −1.88839 −0.151679
\(156\) 13.3643 1.07000
\(157\) 12.3260 0.983718 0.491859 0.870675i \(-0.336317\pi\)
0.491859 + 0.870675i \(0.336317\pi\)
\(158\) 0.0488355 0.00388514
\(159\) −8.03495 −0.637213
\(160\) −0.110071 −0.00870189
\(161\) 12.0936 0.953107
\(162\) 0.0488355 0.00383688
\(163\) 13.4096 1.05032 0.525161 0.851003i \(-0.324005\pi\)
0.525161 + 0.851003i \(0.324005\pi\)
\(164\) −10.2044 −0.796832
\(165\) −0.580786 −0.0452141
\(166\) −0.640871 −0.0497412
\(167\) −5.38383 −0.416613 −0.208307 0.978064i \(-0.566795\pi\)
−0.208307 + 0.978064i \(0.566795\pi\)
\(168\) −0.677113 −0.0522404
\(169\) 31.7577 2.44290
\(170\) −0.00918720 −0.000704626 0
\(171\) −1.82042 −0.139211
\(172\) −21.6113 −1.64784
\(173\) 12.1312 0.922320 0.461160 0.887317i \(-0.347433\pi\)
0.461160 + 0.887317i \(0.347433\pi\)
\(174\) −0.196219 −0.0148753
\(175\) −17.2191 −1.30164
\(176\) 12.3047 0.927505
\(177\) 12.4261 0.934004
\(178\) 0.197907 0.0148338
\(179\) 5.40371 0.403892 0.201946 0.979397i \(-0.435273\pi\)
0.201946 + 0.979397i \(0.435273\pi\)
\(180\) 0.375802 0.0280107
\(181\) 21.0681 1.56598 0.782989 0.622035i \(-0.213694\pi\)
0.782989 + 0.622035i \(0.213694\pi\)
\(182\) −1.13317 −0.0839960
\(183\) 6.94654 0.513503
\(184\) −0.680717 −0.0501831
\(185\) −1.59481 −0.117253
\(186\) 0.490206 0.0359437
\(187\) 3.08722 0.225760
\(188\) 17.6962 1.29062
\(189\) 3.46836 0.252286
\(190\) 0.0167246 0.00121333
\(191\) 27.1557 1.96492 0.982458 0.186482i \(-0.0597085\pi\)
0.982458 + 0.186482i \(0.0597085\pi\)
\(192\) −7.94282 −0.573224
\(193\) 9.27823 0.667861 0.333931 0.942598i \(-0.391625\pi\)
0.333931 + 0.942598i \(0.391625\pi\)
\(194\) 0.413863 0.0297136
\(195\) 1.25858 0.0901290
\(196\) −10.0471 −0.717649
\(197\) −20.3967 −1.45321 −0.726603 0.687058i \(-0.758902\pi\)
−0.726603 + 0.687058i \(0.758902\pi\)
\(198\) 0.150766 0.0107145
\(199\) −10.4301 −0.739370 −0.369685 0.929157i \(-0.620534\pi\)
−0.369685 + 0.929157i \(0.620534\pi\)
\(200\) 0.969218 0.0685341
\(201\) −1.50745 −0.106327
\(202\) 0.805922 0.0567045
\(203\) −13.9357 −0.978097
\(204\) −1.99762 −0.139861
\(205\) −0.961003 −0.0671193
\(206\) −0.178729 −0.0124526
\(207\) 3.48683 0.242351
\(208\) −26.6648 −1.84887
\(209\) −5.62005 −0.388747
\(210\) −0.0318645 −0.00219886
\(211\) 4.41022 0.303612 0.151806 0.988410i \(-0.451491\pi\)
0.151806 + 0.988410i \(0.451491\pi\)
\(212\) 16.0507 1.10237
\(213\) −9.99510 −0.684853
\(214\) 0.930732 0.0636235
\(215\) −2.03524 −0.138802
\(216\) −0.195225 −0.0132834
\(217\) 34.8151 2.36340
\(218\) −0.625072 −0.0423352
\(219\) −2.67757 −0.180933
\(220\) 1.16019 0.0782198
\(221\) −6.69012 −0.450026
\(222\) 0.413996 0.0277856
\(223\) −25.4398 −1.70357 −0.851786 0.523890i \(-0.824480\pi\)
−0.851786 + 0.523890i \(0.824480\pi\)
\(224\) 2.02932 0.135590
\(225\) −4.96461 −0.330974
\(226\) 0.506403 0.0336854
\(227\) 0.0735588 0.00488227 0.00244113 0.999997i \(-0.499223\pi\)
0.00244113 + 0.999997i \(0.499223\pi\)
\(228\) 3.63650 0.240833
\(229\) 10.5527 0.697343 0.348672 0.937245i \(-0.386633\pi\)
0.348672 + 0.937245i \(0.386633\pi\)
\(230\) −0.0320342 −0.00211227
\(231\) 10.7076 0.704509
\(232\) 0.784408 0.0514989
\(233\) 18.3410 1.20156 0.600779 0.799415i \(-0.294857\pi\)
0.600779 + 0.799415i \(0.294857\pi\)
\(234\) −0.326715 −0.0213581
\(235\) 1.66654 0.108713
\(236\) −24.8226 −1.61581
\(237\) 1.00000 0.0649570
\(238\) 0.169379 0.0109792
\(239\) 27.2352 1.76170 0.880849 0.473397i \(-0.156972\pi\)
0.880849 + 0.473397i \(0.156972\pi\)
\(240\) −0.749811 −0.0484001
\(241\) 5.06644 0.326358 0.163179 0.986596i \(-0.447825\pi\)
0.163179 + 0.986596i \(0.447825\pi\)
\(242\) −0.0717415 −0.00461172
\(243\) 1.00000 0.0641500
\(244\) −13.8765 −0.888352
\(245\) −0.946185 −0.0604496
\(246\) 0.249467 0.0159054
\(247\) 12.1788 0.774921
\(248\) −1.95966 −0.124438
\(249\) −13.1230 −0.831639
\(250\) 0.0915469 0.00578993
\(251\) 12.4395 0.785173 0.392587 0.919715i \(-0.371580\pi\)
0.392587 + 0.919715i \(0.371580\pi\)
\(252\) −6.92845 −0.436452
\(253\) 10.7646 0.676766
\(254\) 0.165095 0.0103590
\(255\) −0.188126 −0.0117809
\(256\) 15.8095 0.988097
\(257\) 2.31761 0.144569 0.0722843 0.997384i \(-0.476971\pi\)
0.0722843 + 0.997384i \(0.476971\pi\)
\(258\) 0.528329 0.0328923
\(259\) 29.4025 1.82699
\(260\) −2.51416 −0.155922
\(261\) −4.01796 −0.248705
\(262\) 0.729793 0.0450867
\(263\) −14.3401 −0.884250 −0.442125 0.896953i \(-0.645775\pi\)
−0.442125 + 0.896953i \(0.645775\pi\)
\(264\) −0.602705 −0.0370939
\(265\) 1.51158 0.0928555
\(266\) −0.308342 −0.0189056
\(267\) 4.05252 0.248010
\(268\) 3.01130 0.183944
\(269\) −5.00886 −0.305395 −0.152698 0.988273i \(-0.548796\pi\)
−0.152698 + 0.988273i \(0.548796\pi\)
\(270\) −0.00918720 −0.000559115 0
\(271\) −4.90415 −0.297906 −0.148953 0.988844i \(-0.547590\pi\)
−0.148953 + 0.988844i \(0.547590\pi\)
\(272\) 3.98570 0.241668
\(273\) −23.2038 −1.40436
\(274\) −0.173367 −0.0104735
\(275\) −15.3269 −0.924245
\(276\) −6.96534 −0.419264
\(277\) 10.9342 0.656970 0.328485 0.944509i \(-0.393462\pi\)
0.328485 + 0.944509i \(0.393462\pi\)
\(278\) −1.00128 −0.0600531
\(279\) 10.0379 0.600954
\(280\) 0.127382 0.00761254
\(281\) 19.8441 1.18380 0.591899 0.806012i \(-0.298378\pi\)
0.591899 + 0.806012i \(0.298378\pi\)
\(282\) −0.432616 −0.0257619
\(283\) −5.54080 −0.329366 −0.164683 0.986347i \(-0.552660\pi\)
−0.164683 + 0.986347i \(0.552660\pi\)
\(284\) 19.9664 1.18479
\(285\) 0.342468 0.0202860
\(286\) −1.00864 −0.0596424
\(287\) 17.7175 1.04583
\(288\) 0.585094 0.0344770
\(289\) 1.00000 0.0588235
\(290\) 0.0369138 0.00216765
\(291\) 8.47463 0.496792
\(292\) 5.34875 0.313012
\(293\) 14.1163 0.824681 0.412341 0.911030i \(-0.364711\pi\)
0.412341 + 0.911030i \(0.364711\pi\)
\(294\) 0.245620 0.0143249
\(295\) −2.33767 −0.136104
\(296\) −1.65500 −0.0961947
\(297\) 3.08722 0.179139
\(298\) 0.282873 0.0163864
\(299\) −23.3273 −1.34905
\(300\) 9.91738 0.572580
\(301\) 37.5226 2.16277
\(302\) −0.360721 −0.0207572
\(303\) 16.5028 0.948061
\(304\) −7.25565 −0.416140
\(305\) −1.30682 −0.0748283
\(306\) 0.0488355 0.00279174
\(307\) 16.7907 0.958295 0.479147 0.877734i \(-0.340946\pi\)
0.479147 + 0.877734i \(0.340946\pi\)
\(308\) −21.3897 −1.21879
\(309\) −3.65982 −0.208200
\(310\) −0.0922203 −0.00523776
\(311\) 17.2105 0.975920 0.487960 0.872866i \(-0.337741\pi\)
0.487960 + 0.872866i \(0.337741\pi\)
\(312\) 1.30608 0.0739424
\(313\) 11.0304 0.623474 0.311737 0.950168i \(-0.399089\pi\)
0.311737 + 0.950168i \(0.399089\pi\)
\(314\) 0.601944 0.0339697
\(315\) −0.652488 −0.0367635
\(316\) −1.99762 −0.112375
\(317\) −13.9115 −0.781345 −0.390673 0.920530i \(-0.627758\pi\)
−0.390673 + 0.920530i \(0.627758\pi\)
\(318\) −0.392391 −0.0220042
\(319\) −12.4043 −0.694510
\(320\) 1.49425 0.0835309
\(321\) 19.0585 1.06374
\(322\) 0.590596 0.0329126
\(323\) −1.82042 −0.101291
\(324\) −1.99762 −0.110979
\(325\) 33.2138 1.84237
\(326\) 0.654864 0.0362696
\(327\) −12.7995 −0.707817
\(328\) −0.997272 −0.0550651
\(329\) −30.7250 −1.69392
\(330\) −0.0283630 −0.00156133
\(331\) −17.3979 −0.956275 −0.478137 0.878285i \(-0.658688\pi\)
−0.478137 + 0.878285i \(0.658688\pi\)
\(332\) 26.2148 1.43872
\(333\) 8.47735 0.464556
\(334\) −0.262922 −0.0143864
\(335\) 0.283589 0.0154941
\(336\) 13.8238 0.754152
\(337\) −16.4679 −0.897064 −0.448532 0.893767i \(-0.648053\pi\)
−0.448532 + 0.893767i \(0.648053\pi\)
\(338\) 1.55090 0.0843581
\(339\) 10.3696 0.563198
\(340\) 0.375802 0.0203807
\(341\) 30.9893 1.67816
\(342\) −0.0889012 −0.00480722
\(343\) −6.83426 −0.369016
\(344\) −2.11205 −0.113874
\(345\) −0.655961 −0.0353157
\(346\) 0.592434 0.0318495
\(347\) −12.1439 −0.651921 −0.325961 0.945383i \(-0.605688\pi\)
−0.325961 + 0.945383i \(0.605688\pi\)
\(348\) 8.02633 0.430257
\(349\) −24.7722 −1.32603 −0.663013 0.748607i \(-0.730723\pi\)
−0.663013 + 0.748607i \(0.730723\pi\)
\(350\) −0.840901 −0.0449481
\(351\) −6.69012 −0.357092
\(352\) 1.80632 0.0962771
\(353\) −18.9307 −1.00758 −0.503790 0.863826i \(-0.668061\pi\)
−0.503790 + 0.863826i \(0.668061\pi\)
\(354\) 0.606835 0.0322529
\(355\) 1.88033 0.0997977
\(356\) −8.09538 −0.429054
\(357\) 3.46836 0.183565
\(358\) 0.263893 0.0139472
\(359\) 26.0487 1.37480 0.687398 0.726281i \(-0.258753\pi\)
0.687398 + 0.726281i \(0.258753\pi\)
\(360\) 0.0367269 0.00193568
\(361\) −15.6861 −0.825582
\(362\) 1.02887 0.0540762
\(363\) −1.46904 −0.0771048
\(364\) 46.3522 2.42952
\(365\) 0.503719 0.0263659
\(366\) 0.339237 0.0177322
\(367\) −17.6229 −0.919908 −0.459954 0.887943i \(-0.652134\pi\)
−0.459954 + 0.887943i \(0.652134\pi\)
\(368\) 13.8974 0.724454
\(369\) 5.10831 0.265928
\(370\) −0.0778832 −0.00404895
\(371\) −27.8681 −1.44684
\(372\) −20.0519 −1.03964
\(373\) −27.0562 −1.40091 −0.700457 0.713694i \(-0.747021\pi\)
−0.700457 + 0.713694i \(0.747021\pi\)
\(374\) 0.150766 0.00779593
\(375\) 1.87460 0.0968038
\(376\) 1.72943 0.0891887
\(377\) 26.8806 1.38442
\(378\) 0.169379 0.00871193
\(379\) 14.4095 0.740165 0.370083 0.928999i \(-0.379329\pi\)
0.370083 + 0.928999i \(0.379329\pi\)
\(380\) −0.684119 −0.0350945
\(381\) 3.38064 0.173195
\(382\) 1.32616 0.0678523
\(383\) 3.52687 0.180215 0.0901074 0.995932i \(-0.471279\pi\)
0.0901074 + 0.995932i \(0.471279\pi\)
\(384\) −1.55808 −0.0795104
\(385\) −2.01438 −0.102662
\(386\) 0.453107 0.0230625
\(387\) 10.8185 0.549937
\(388\) −16.9290 −0.859442
\(389\) 19.4993 0.988656 0.494328 0.869275i \(-0.335414\pi\)
0.494328 + 0.869275i \(0.335414\pi\)
\(390\) 0.0614635 0.00311233
\(391\) 3.48683 0.176336
\(392\) −0.981895 −0.0495932
\(393\) 14.9439 0.753820
\(394\) −0.996084 −0.0501820
\(395\) −0.188126 −0.00946562
\(396\) −6.16709 −0.309908
\(397\) −10.3140 −0.517646 −0.258823 0.965925i \(-0.583335\pi\)
−0.258823 + 0.965925i \(0.583335\pi\)
\(398\) −0.509359 −0.0255318
\(399\) −6.31388 −0.316089
\(400\) −19.7874 −0.989371
\(401\) 13.9178 0.695024 0.347512 0.937676i \(-0.387027\pi\)
0.347512 + 0.937676i \(0.387027\pi\)
\(402\) −0.0736170 −0.00367168
\(403\) −67.1549 −3.34522
\(404\) −32.9662 −1.64013
\(405\) −0.188126 −0.00934803
\(406\) −0.680558 −0.0337755
\(407\) 26.1715 1.29727
\(408\) −0.195225 −0.00966510
\(409\) −8.13110 −0.402057 −0.201029 0.979585i \(-0.564428\pi\)
−0.201029 + 0.979585i \(0.564428\pi\)
\(410\) −0.0469310 −0.00231776
\(411\) −3.55003 −0.175110
\(412\) 7.31090 0.360182
\(413\) 43.0983 2.12073
\(414\) 0.170281 0.00836885
\(415\) 2.46878 0.121188
\(416\) −3.91435 −0.191917
\(417\) −20.5032 −1.00405
\(418\) −0.274458 −0.0134242
\(419\) −28.7896 −1.40646 −0.703232 0.710961i \(-0.748260\pi\)
−0.703232 + 0.710961i \(0.748260\pi\)
\(420\) 1.30342 0.0636003
\(421\) −6.66404 −0.324786 −0.162393 0.986726i \(-0.551921\pi\)
−0.162393 + 0.986726i \(0.551921\pi\)
\(422\) 0.215375 0.0104843
\(423\) −8.85864 −0.430722
\(424\) 1.56863 0.0761792
\(425\) −4.96461 −0.240819
\(426\) −0.488116 −0.0236493
\(427\) 24.0931 1.16595
\(428\) −38.0716 −1.84026
\(429\) −20.6539 −0.997180
\(430\) −0.0993921 −0.00479311
\(431\) 1.08855 0.0524336 0.0262168 0.999656i \(-0.491654\pi\)
0.0262168 + 0.999656i \(0.491654\pi\)
\(432\) 3.98570 0.191762
\(433\) −20.9678 −1.00765 −0.503823 0.863807i \(-0.668074\pi\)
−0.503823 + 0.863807i \(0.668074\pi\)
\(434\) 1.70021 0.0816129
\(435\) 0.755880 0.0362417
\(436\) 25.5686 1.22451
\(437\) −6.34749 −0.303642
\(438\) −0.130760 −0.00624797
\(439\) −2.44992 −0.116928 −0.0584641 0.998290i \(-0.518620\pi\)
−0.0584641 + 0.998290i \(0.518620\pi\)
\(440\) 0.113384 0.00540538
\(441\) 5.02954 0.239502
\(442\) −0.326715 −0.0155403
\(443\) −6.16931 −0.293113 −0.146556 0.989202i \(-0.546819\pi\)
−0.146556 + 0.989202i \(0.546819\pi\)
\(444\) −16.9345 −0.803675
\(445\) −0.762383 −0.0361404
\(446\) −1.24236 −0.0588276
\(447\) 5.79237 0.273970
\(448\) −27.5486 −1.30155
\(449\) −41.4632 −1.95677 −0.978384 0.206795i \(-0.933697\pi\)
−0.978384 + 0.206795i \(0.933697\pi\)
\(450\) −0.242449 −0.0114292
\(451\) 15.7705 0.742603
\(452\) −20.7144 −0.974324
\(453\) −7.38646 −0.347046
\(454\) 0.00359228 0.000168594 0
\(455\) 4.36522 0.204645
\(456\) 0.355393 0.0166428
\(457\) −37.6028 −1.75898 −0.879492 0.475913i \(-0.842118\pi\)
−0.879492 + 0.475913i \(0.842118\pi\)
\(458\) 0.515347 0.0240806
\(459\) 1.00000 0.0466760
\(460\) 1.31036 0.0610957
\(461\) 22.0115 1.02518 0.512588 0.858635i \(-0.328687\pi\)
0.512588 + 0.858635i \(0.328687\pi\)
\(462\) 0.522912 0.0243281
\(463\) −29.7303 −1.38168 −0.690841 0.723006i \(-0.742760\pi\)
−0.690841 + 0.723006i \(0.742760\pi\)
\(464\) −16.0144 −0.743448
\(465\) −1.88839 −0.0875719
\(466\) 0.895691 0.0414921
\(467\) −20.0102 −0.925961 −0.462981 0.886368i \(-0.653220\pi\)
−0.462981 + 0.886368i \(0.653220\pi\)
\(468\) 13.3643 0.617765
\(469\) −5.22838 −0.241424
\(470\) 0.0813861 0.00375406
\(471\) 12.3260 0.567950
\(472\) −2.42589 −0.111661
\(473\) 33.3993 1.53570
\(474\) 0.0488355 0.00224309
\(475\) 9.03768 0.414677
\(476\) −6.92845 −0.317565
\(477\) −8.03495 −0.367895
\(478\) 1.33004 0.0608348
\(479\) −4.05162 −0.185123 −0.0925616 0.995707i \(-0.529506\pi\)
−0.0925616 + 0.995707i \(0.529506\pi\)
\(480\) −0.110071 −0.00502404
\(481\) −56.7145 −2.58596
\(482\) 0.247422 0.0112698
\(483\) 12.0936 0.550277
\(484\) 2.93458 0.133390
\(485\) −1.59429 −0.0723932
\(486\) 0.0488355 0.00221522
\(487\) −16.7295 −0.758084 −0.379042 0.925379i \(-0.623746\pi\)
−0.379042 + 0.925379i \(0.623746\pi\)
\(488\) −1.35614 −0.0613896
\(489\) 13.4096 0.606403
\(490\) −0.0462074 −0.00208744
\(491\) −42.0166 −1.89618 −0.948092 0.317996i \(-0.896990\pi\)
−0.948092 + 0.317996i \(0.896990\pi\)
\(492\) −10.2044 −0.460051
\(493\) −4.01796 −0.180960
\(494\) 0.594760 0.0267595
\(495\) −0.580786 −0.0261044
\(496\) 40.0081 1.79642
\(497\) −34.6666 −1.55501
\(498\) −0.640871 −0.0287181
\(499\) −29.6144 −1.32572 −0.662861 0.748742i \(-0.730658\pi\)
−0.662861 + 0.748742i \(0.730658\pi\)
\(500\) −3.74472 −0.167469
\(501\) −5.38383 −0.240532
\(502\) 0.607488 0.0271135
\(503\) 14.8661 0.662846 0.331423 0.943482i \(-0.392471\pi\)
0.331423 + 0.943482i \(0.392471\pi\)
\(504\) −0.677113 −0.0301610
\(505\) −3.10460 −0.138153
\(506\) 0.525695 0.0233700
\(507\) 31.7577 1.41041
\(508\) −6.75321 −0.299625
\(509\) 37.4954 1.66195 0.830977 0.556306i \(-0.187782\pi\)
0.830977 + 0.556306i \(0.187782\pi\)
\(510\) −0.00918720 −0.000406816 0
\(511\) −9.28678 −0.410823
\(512\) 3.88823 0.171837
\(513\) −1.82042 −0.0803736
\(514\) 0.113182 0.00499223
\(515\) 0.688505 0.0303391
\(516\) −21.6113 −0.951384
\(517\) −27.3486 −1.20279
\(518\) 1.43589 0.0630893
\(519\) 12.1312 0.532502
\(520\) −0.245707 −0.0107750
\(521\) −23.8055 −1.04294 −0.521468 0.853271i \(-0.674616\pi\)
−0.521468 + 0.853271i \(0.674616\pi\)
\(522\) −0.196219 −0.00858827
\(523\) 40.3639 1.76499 0.882494 0.470323i \(-0.155863\pi\)
0.882494 + 0.470323i \(0.155863\pi\)
\(524\) −29.8522 −1.30410
\(525\) −17.2191 −0.751502
\(526\) −0.700307 −0.0305348
\(527\) 10.0379 0.437258
\(528\) 12.3047 0.535495
\(529\) −10.8420 −0.471393
\(530\) 0.0738187 0.00320648
\(531\) 12.4261 0.539248
\(532\) 12.6127 0.546830
\(533\) −34.1752 −1.48029
\(534\) 0.197907 0.00856427
\(535\) −3.58539 −0.155010
\(536\) 0.294292 0.0127115
\(537\) 5.40371 0.233187
\(538\) −0.244610 −0.0105459
\(539\) 15.5273 0.668809
\(540\) 0.375802 0.0161720
\(541\) −10.8843 −0.467954 −0.233977 0.972242i \(-0.575174\pi\)
−0.233977 + 0.972242i \(0.575174\pi\)
\(542\) −0.239497 −0.0102873
\(543\) 21.0681 0.904118
\(544\) 0.585094 0.0250857
\(545\) 2.40792 0.103144
\(546\) −1.13317 −0.0484951
\(547\) 12.0756 0.516314 0.258157 0.966103i \(-0.416885\pi\)
0.258157 + 0.966103i \(0.416885\pi\)
\(548\) 7.09159 0.302938
\(549\) 6.94654 0.296471
\(550\) −0.748495 −0.0319159
\(551\) 7.31438 0.311603
\(552\) −0.680717 −0.0289732
\(553\) 3.46836 0.147490
\(554\) 0.533975 0.0226864
\(555\) −1.59481 −0.0676958
\(556\) 40.9575 1.73699
\(557\) −18.2844 −0.774737 −0.387368 0.921925i \(-0.626616\pi\)
−0.387368 + 0.921925i \(0.626616\pi\)
\(558\) 0.490206 0.0207521
\(559\) −72.3774 −3.06124
\(560\) −2.60062 −0.109896
\(561\) 3.08722 0.130343
\(562\) 0.969096 0.0408788
\(563\) 18.2598 0.769558 0.384779 0.923009i \(-0.374278\pi\)
0.384779 + 0.923009i \(0.374278\pi\)
\(564\) 17.6962 0.745143
\(565\) −1.95078 −0.0820700
\(566\) −0.270588 −0.0113737
\(567\) 3.46836 0.145658
\(568\) 1.95130 0.0818747
\(569\) −0.690518 −0.0289480 −0.0144740 0.999895i \(-0.504607\pi\)
−0.0144740 + 0.999895i \(0.504607\pi\)
\(570\) 0.0167246 0.000700515 0
\(571\) 22.6322 0.947128 0.473564 0.880759i \(-0.342967\pi\)
0.473564 + 0.880759i \(0.342967\pi\)
\(572\) 41.2586 1.72511
\(573\) 27.1557 1.13445
\(574\) 0.865241 0.0361145
\(575\) −17.3107 −0.721907
\(576\) −7.94282 −0.330951
\(577\) 41.8894 1.74388 0.871939 0.489615i \(-0.162863\pi\)
0.871939 + 0.489615i \(0.162863\pi\)
\(578\) 0.0488355 0.00203129
\(579\) 9.27823 0.385590
\(580\) −1.50996 −0.0626976
\(581\) −45.5155 −1.88830
\(582\) 0.413863 0.0171552
\(583\) −24.8057 −1.02735
\(584\) 0.522730 0.0216307
\(585\) 1.25858 0.0520360
\(586\) 0.689375 0.0284778
\(587\) −39.0445 −1.61154 −0.805769 0.592230i \(-0.798248\pi\)
−0.805769 + 0.592230i \(0.798248\pi\)
\(588\) −10.0471 −0.414335
\(589\) −18.2732 −0.752935
\(590\) −0.114161 −0.00469994
\(591\) −20.3967 −0.839009
\(592\) 33.7882 1.38868
\(593\) 40.1747 1.64978 0.824888 0.565296i \(-0.191238\pi\)
0.824888 + 0.565296i \(0.191238\pi\)
\(594\) 0.150766 0.00618601
\(595\) −0.652488 −0.0267494
\(596\) −11.5709 −0.473963
\(597\) −10.4301 −0.426875
\(598\) −1.13920 −0.0465853
\(599\) −4.51721 −0.184568 −0.0922841 0.995733i \(-0.529417\pi\)
−0.0922841 + 0.995733i \(0.529417\pi\)
\(600\) 0.969218 0.0395682
\(601\) 5.24339 0.213882 0.106941 0.994265i \(-0.465894\pi\)
0.106941 + 0.994265i \(0.465894\pi\)
\(602\) 1.83244 0.0746845
\(603\) −1.50745 −0.0613880
\(604\) 14.7553 0.600385
\(605\) 0.276365 0.0112358
\(606\) 0.805922 0.0327384
\(607\) −32.4421 −1.31679 −0.658393 0.752674i \(-0.728763\pi\)
−0.658393 + 0.752674i \(0.728763\pi\)
\(608\) −1.06512 −0.0431963
\(609\) −13.9357 −0.564704
\(610\) −0.0638192 −0.00258396
\(611\) 59.2654 2.39762
\(612\) −1.99762 −0.0807488
\(613\) −19.4452 −0.785383 −0.392692 0.919670i \(-0.628456\pi\)
−0.392692 + 0.919670i \(0.628456\pi\)
\(614\) 0.819981 0.0330917
\(615\) −0.961003 −0.0387514
\(616\) −2.09040 −0.0842246
\(617\) 30.8778 1.24309 0.621546 0.783377i \(-0.286505\pi\)
0.621546 + 0.783377i \(0.286505\pi\)
\(618\) −0.178729 −0.00718953
\(619\) 1.50085 0.0603244 0.0301622 0.999545i \(-0.490398\pi\)
0.0301622 + 0.999545i \(0.490398\pi\)
\(620\) 3.77227 0.151498
\(621\) 3.48683 0.139921
\(622\) 0.840485 0.0337004
\(623\) 14.0556 0.563127
\(624\) −26.6648 −1.06745
\(625\) 24.4704 0.978815
\(626\) 0.538674 0.0215298
\(627\) −5.62005 −0.224443
\(628\) −24.6225 −0.982545
\(629\) 8.47735 0.338014
\(630\) −0.0318645 −0.00126951
\(631\) −20.0363 −0.797634 −0.398817 0.917031i \(-0.630579\pi\)
−0.398817 + 0.917031i \(0.630579\pi\)
\(632\) −0.195225 −0.00776565
\(633\) 4.41022 0.175290
\(634\) −0.679373 −0.0269813
\(635\) −0.635984 −0.0252383
\(636\) 16.0507 0.636453
\(637\) −33.6483 −1.33319
\(638\) −0.605772 −0.0239827
\(639\) −9.99510 −0.395400
\(640\) 0.293115 0.0115864
\(641\) −46.6685 −1.84329 −0.921646 0.388031i \(-0.873155\pi\)
−0.921646 + 0.388031i \(0.873155\pi\)
\(642\) 0.930732 0.0367331
\(643\) 18.7704 0.740231 0.370115 0.928986i \(-0.379318\pi\)
0.370115 + 0.928986i \(0.379318\pi\)
\(644\) −24.1583 −0.951971
\(645\) −2.03524 −0.0801376
\(646\) −0.0889012 −0.00349777
\(647\) −36.9010 −1.45073 −0.725364 0.688366i \(-0.758328\pi\)
−0.725364 + 0.688366i \(0.758328\pi\)
\(648\) −0.195225 −0.00766918
\(649\) 38.3622 1.50585
\(650\) 1.62201 0.0636206
\(651\) 34.8151 1.36451
\(652\) −26.7872 −1.04907
\(653\) −19.9016 −0.778808 −0.389404 0.921067i \(-0.627319\pi\)
−0.389404 + 0.921067i \(0.627319\pi\)
\(654\) −0.625072 −0.0244423
\(655\) −2.81133 −0.109848
\(656\) 20.3602 0.794931
\(657\) −2.67757 −0.104462
\(658\) −1.50047 −0.0584944
\(659\) 2.98589 0.116314 0.0581568 0.998307i \(-0.481478\pi\)
0.0581568 + 0.998307i \(0.481478\pi\)
\(660\) 1.16019 0.0451602
\(661\) 25.4871 0.991334 0.495667 0.868513i \(-0.334924\pi\)
0.495667 + 0.868513i \(0.334924\pi\)
\(662\) −0.849635 −0.0330220
\(663\) −6.69012 −0.259823
\(664\) 2.56195 0.0994231
\(665\) 1.18780 0.0460610
\(666\) 0.413996 0.0160420
\(667\) −14.0099 −0.542466
\(668\) 10.7548 0.416116
\(669\) −25.4398 −0.983558
\(670\) 0.0138492 0.000535042 0
\(671\) 21.4455 0.827895
\(672\) 2.02932 0.0782827
\(673\) −5.43037 −0.209325 −0.104663 0.994508i \(-0.533376\pi\)
−0.104663 + 0.994508i \(0.533376\pi\)
\(674\) −0.804218 −0.0309773
\(675\) −4.96461 −0.191088
\(676\) −63.4397 −2.43999
\(677\) −10.6352 −0.408742 −0.204371 0.978893i \(-0.565515\pi\)
−0.204371 + 0.978893i \(0.565515\pi\)
\(678\) 0.506403 0.0194483
\(679\) 29.3931 1.12800
\(680\) 0.0367269 0.00140841
\(681\) 0.0735588 0.00281878
\(682\) 1.51338 0.0579502
\(683\) 4.50761 0.172479 0.0862395 0.996274i \(-0.472515\pi\)
0.0862395 + 0.996274i \(0.472515\pi\)
\(684\) 3.63650 0.139045
\(685\) 0.667851 0.0255173
\(686\) −0.333755 −0.0127428
\(687\) 10.5527 0.402611
\(688\) 43.1194 1.64391
\(689\) 53.7548 2.04789
\(690\) −0.0320342 −0.00121952
\(691\) 7.65022 0.291028 0.145514 0.989356i \(-0.453516\pi\)
0.145514 + 0.989356i \(0.453516\pi\)
\(692\) −24.2335 −0.921220
\(693\) 10.7076 0.406749
\(694\) −0.593055 −0.0225121
\(695\) 3.85718 0.146311
\(696\) 0.784408 0.0297329
\(697\) 5.10831 0.193491
\(698\) −1.20976 −0.0457902
\(699\) 18.3410 0.693720
\(700\) 34.3971 1.30009
\(701\) 22.7818 0.860458 0.430229 0.902720i \(-0.358433\pi\)
0.430229 + 0.902720i \(0.358433\pi\)
\(702\) −0.326715 −0.0123311
\(703\) −15.4324 −0.582042
\(704\) −24.5213 −0.924180
\(705\) 1.66654 0.0627654
\(706\) −0.924490 −0.0347937
\(707\) 57.2377 2.15265
\(708\) −24.8226 −0.932890
\(709\) −4.36980 −0.164111 −0.0820557 0.996628i \(-0.526149\pi\)
−0.0820557 + 0.996628i \(0.526149\pi\)
\(710\) 0.0918270 0.00344621
\(711\) 1.00000 0.0375029
\(712\) −0.791156 −0.0296498
\(713\) 35.0005 1.31078
\(714\) 0.169379 0.00633886
\(715\) 3.88553 0.145311
\(716\) −10.7945 −0.403411
\(717\) 27.2352 1.01712
\(718\) 1.27210 0.0474743
\(719\) 15.3235 0.571471 0.285736 0.958308i \(-0.407762\pi\)
0.285736 + 0.958308i \(0.407762\pi\)
\(720\) −0.749811 −0.0279438
\(721\) −12.6936 −0.472733
\(722\) −0.766037 −0.0285089
\(723\) 5.06644 0.188423
\(724\) −42.0859 −1.56411
\(725\) 19.9476 0.740835
\(726\) −0.0717415 −0.00266258
\(727\) −45.1511 −1.67456 −0.837280 0.546774i \(-0.815856\pi\)
−0.837280 + 0.546774i \(0.815856\pi\)
\(728\) 4.52997 0.167892
\(729\) 1.00000 0.0370370
\(730\) 0.0245994 0.000910463 0
\(731\) 10.8185 0.400138
\(732\) −13.8765 −0.512890
\(733\) 36.4108 1.34486 0.672431 0.740160i \(-0.265250\pi\)
0.672431 + 0.740160i \(0.265250\pi\)
\(734\) −0.860623 −0.0317662
\(735\) −0.946185 −0.0349006
\(736\) 2.04012 0.0751999
\(737\) −4.65383 −0.171426
\(738\) 0.249467 0.00918299
\(739\) 26.4373 0.972511 0.486256 0.873817i \(-0.338362\pi\)
0.486256 + 0.873817i \(0.338362\pi\)
\(740\) 3.18581 0.117113
\(741\) 12.1788 0.447401
\(742\) −1.36095 −0.0499622
\(743\) 25.0440 0.918777 0.459388 0.888235i \(-0.348069\pi\)
0.459388 + 0.888235i \(0.348069\pi\)
\(744\) −1.95966 −0.0718445
\(745\) −1.08969 −0.0399232
\(746\) −1.32130 −0.0483763
\(747\) −13.1230 −0.480147
\(748\) −6.16709 −0.225491
\(749\) 66.1018 2.41531
\(750\) 0.0915469 0.00334282
\(751\) −33.2486 −1.21326 −0.606630 0.794985i \(-0.707479\pi\)
−0.606630 + 0.794985i \(0.707479\pi\)
\(752\) −35.3079 −1.28755
\(753\) 12.4395 0.453320
\(754\) 1.31273 0.0478068
\(755\) 1.38958 0.0505721
\(756\) −6.92845 −0.251985
\(757\) −0.252350 −0.00917182 −0.00458591 0.999989i \(-0.501460\pi\)
−0.00458591 + 0.999989i \(0.501460\pi\)
\(758\) 0.703694 0.0255593
\(759\) 10.7646 0.390731
\(760\) −0.0668584 −0.00242521
\(761\) −25.2599 −0.915671 −0.457835 0.889037i \(-0.651375\pi\)
−0.457835 + 0.889037i \(0.651375\pi\)
\(762\) 0.165095 0.00598077
\(763\) −44.3935 −1.60715
\(764\) −54.2466 −1.96257
\(765\) −0.188126 −0.00680169
\(766\) 0.172236 0.00622316
\(767\) −83.1322 −3.00173
\(768\) 15.8095 0.570478
\(769\) −13.6542 −0.492385 −0.246192 0.969221i \(-0.579179\pi\)
−0.246192 + 0.969221i \(0.579179\pi\)
\(770\) −0.0983730 −0.00354512
\(771\) 2.31761 0.0834667
\(772\) −18.5343 −0.667065
\(773\) −23.8219 −0.856813 −0.428406 0.903586i \(-0.640925\pi\)
−0.428406 + 0.903586i \(0.640925\pi\)
\(774\) 0.528329 0.0189904
\(775\) −49.8343 −1.79010
\(776\) −1.65446 −0.0593918
\(777\) 29.4025 1.05481
\(778\) 0.952260 0.0341402
\(779\) −9.29927 −0.333181
\(780\) −2.51416 −0.0900215
\(781\) −30.8571 −1.10415
\(782\) 0.170281 0.00608923
\(783\) −4.01796 −0.143590
\(784\) 20.0462 0.715937
\(785\) −2.31883 −0.0827625
\(786\) 0.729793 0.0260308
\(787\) −5.27091 −0.187888 −0.0939438 0.995577i \(-0.529947\pi\)
−0.0939438 + 0.995577i \(0.529947\pi\)
\(788\) 40.7448 1.45147
\(789\) −14.3401 −0.510522
\(790\) −0.00918720 −0.000326866 0
\(791\) 35.9654 1.27878
\(792\) −0.602705 −0.0214162
\(793\) −46.4732 −1.65031
\(794\) −0.503690 −0.0178753
\(795\) 1.51158 0.0536102
\(796\) 20.8353 0.738488
\(797\) 19.8387 0.702722 0.351361 0.936240i \(-0.385719\pi\)
0.351361 + 0.936240i \(0.385719\pi\)
\(798\) −0.308342 −0.0109152
\(799\) −8.85864 −0.313396
\(800\) −2.90476 −0.102699
\(801\) 4.05252 0.143189
\(802\) 0.679684 0.0240005
\(803\) −8.26626 −0.291710
\(804\) 3.01130 0.106200
\(805\) −2.27511 −0.0801871
\(806\) −3.27954 −0.115517
\(807\) −5.00886 −0.176320
\(808\) −3.22177 −0.113341
\(809\) 31.8772 1.12074 0.560371 0.828242i \(-0.310659\pi\)
0.560371 + 0.828242i \(0.310659\pi\)
\(810\) −0.00918720 −0.000322805 0
\(811\) −25.3291 −0.889425 −0.444713 0.895673i \(-0.646694\pi\)
−0.444713 + 0.895673i \(0.646694\pi\)
\(812\) 27.8382 0.976930
\(813\) −4.90415 −0.171996
\(814\) 1.27810 0.0447973
\(815\) −2.52269 −0.0883659
\(816\) 3.98570 0.139527
\(817\) −19.6943 −0.689016
\(818\) −0.397086 −0.0138838
\(819\) −23.2038 −0.810805
\(820\) 1.91971 0.0670393
\(821\) −18.9958 −0.662958 −0.331479 0.943463i \(-0.607548\pi\)
−0.331479 + 0.943463i \(0.607548\pi\)
\(822\) −0.173367 −0.00604688
\(823\) 9.08785 0.316782 0.158391 0.987376i \(-0.449369\pi\)
0.158391 + 0.987376i \(0.449369\pi\)
\(824\) 0.714489 0.0248904
\(825\) −15.3269 −0.533613
\(826\) 2.10473 0.0732328
\(827\) 27.4456 0.954376 0.477188 0.878801i \(-0.341656\pi\)
0.477188 + 0.878801i \(0.341656\pi\)
\(828\) −6.96534 −0.242062
\(829\) 53.2375 1.84901 0.924507 0.381164i \(-0.124477\pi\)
0.924507 + 0.381164i \(0.124477\pi\)
\(830\) 0.120564 0.00418484
\(831\) 10.9342 0.379302
\(832\) 53.1384 1.84224
\(833\) 5.02954 0.174263
\(834\) −1.00128 −0.0346716
\(835\) 1.01284 0.0350506
\(836\) 11.2267 0.388283
\(837\) 10.0379 0.346961
\(838\) −1.40595 −0.0485679
\(839\) 6.57324 0.226934 0.113467 0.993542i \(-0.463804\pi\)
0.113467 + 0.993542i \(0.463804\pi\)
\(840\) 0.127382 0.00439510
\(841\) −12.8560 −0.443311
\(842\) −0.325442 −0.0112155
\(843\) 19.8441 0.683467
\(844\) −8.80991 −0.303250
\(845\) −5.97444 −0.205527
\(846\) −0.432616 −0.0148736
\(847\) −5.09518 −0.175072
\(848\) −32.0249 −1.09974
\(849\) −5.54080 −0.190160
\(850\) −0.242449 −0.00831594
\(851\) 29.5591 1.01327
\(852\) 19.9664 0.684036
\(853\) −25.0968 −0.859297 −0.429648 0.902996i \(-0.641363\pi\)
−0.429648 + 0.902996i \(0.641363\pi\)
\(854\) 1.17660 0.0402624
\(855\) 0.342468 0.0117121
\(856\) −3.72071 −0.127171
\(857\) −31.7374 −1.08413 −0.542065 0.840336i \(-0.682357\pi\)
−0.542065 + 0.840336i \(0.682357\pi\)
\(858\) −1.00864 −0.0344345
\(859\) −43.1041 −1.47069 −0.735347 0.677691i \(-0.762981\pi\)
−0.735347 + 0.677691i \(0.762981\pi\)
\(860\) 4.06563 0.138637
\(861\) 17.7175 0.603809
\(862\) 0.0531599 0.00181063
\(863\) 35.8401 1.22001 0.610006 0.792397i \(-0.291167\pi\)
0.610006 + 0.792397i \(0.291167\pi\)
\(864\) 0.585094 0.0199053
\(865\) −2.28219 −0.0775969
\(866\) −1.02397 −0.0347960
\(867\) 1.00000 0.0339618
\(868\) −69.5472 −2.36059
\(869\) 3.08722 0.104727
\(870\) 0.0369138 0.00125149
\(871\) 10.0850 0.341718
\(872\) 2.49880 0.0846200
\(873\) 8.47463 0.286823
\(874\) −0.309983 −0.0104853
\(875\) 6.50178 0.219800
\(876\) 5.34875 0.180718
\(877\) 36.3503 1.22746 0.613730 0.789516i \(-0.289668\pi\)
0.613730 + 0.789516i \(0.289668\pi\)
\(878\) −0.119643 −0.00403775
\(879\) 14.1163 0.476130
\(880\) −2.31484 −0.0780331
\(881\) −49.9058 −1.68137 −0.840684 0.541526i \(-0.817847\pi\)
−0.840684 + 0.541526i \(0.817847\pi\)
\(882\) 0.245620 0.00827046
\(883\) 1.84497 0.0620882 0.0310441 0.999518i \(-0.490117\pi\)
0.0310441 + 0.999518i \(0.490117\pi\)
\(884\) 13.3643 0.449490
\(885\) −2.33767 −0.0785799
\(886\) −0.301281 −0.0101217
\(887\) 7.83251 0.262990 0.131495 0.991317i \(-0.458022\pi\)
0.131495 + 0.991317i \(0.458022\pi\)
\(888\) −1.65500 −0.0555380
\(889\) 11.7253 0.393253
\(890\) −0.0372314 −0.00124800
\(891\) 3.08722 0.103426
\(892\) 50.8188 1.70154
\(893\) 16.1265 0.539651
\(894\) 0.282873 0.00946069
\(895\) −1.01658 −0.0339804
\(896\) −5.40399 −0.180535
\(897\) −23.3273 −0.778876
\(898\) −2.02487 −0.0675710
\(899\) −40.3319 −1.34514
\(900\) 9.91738 0.330579
\(901\) −8.03495 −0.267683
\(902\) 0.770160 0.0256435
\(903\) 37.5226 1.24867
\(904\) −2.02440 −0.0673307
\(905\) −3.96344 −0.131749
\(906\) −0.360721 −0.0119842
\(907\) 9.38676 0.311682 0.155841 0.987782i \(-0.450191\pi\)
0.155841 + 0.987782i \(0.450191\pi\)
\(908\) −0.146942 −0.00487645
\(909\) 16.5028 0.547363
\(910\) 0.213178 0.00706677
\(911\) −33.2686 −1.10224 −0.551118 0.834427i \(-0.685799\pi\)
−0.551118 + 0.834427i \(0.685799\pi\)
\(912\) −7.25565 −0.240258
\(913\) −40.5138 −1.34081
\(914\) −1.83635 −0.0607411
\(915\) −1.30682 −0.0432022
\(916\) −21.0803 −0.696512
\(917\) 51.8309 1.71161
\(918\) 0.0488355 0.00161181
\(919\) 4.42083 0.145830 0.0729148 0.997338i \(-0.476770\pi\)
0.0729148 + 0.997338i \(0.476770\pi\)
\(920\) 0.128060 0.00422202
\(921\) 16.7907 0.553272
\(922\) 1.07494 0.0354013
\(923\) 66.8684 2.20100
\(924\) −21.3897 −0.703669
\(925\) −42.0868 −1.38380
\(926\) −1.45189 −0.0477121
\(927\) −3.65982 −0.120204
\(928\) −2.35088 −0.0771716
\(929\) 13.9434 0.457468 0.228734 0.973489i \(-0.426541\pi\)
0.228734 + 0.973489i \(0.426541\pi\)
\(930\) −0.0922203 −0.00302402
\(931\) −9.15589 −0.300072
\(932\) −36.6382 −1.20013
\(933\) 17.2105 0.563448
\(934\) −0.977207 −0.0319752
\(935\) −0.580786 −0.0189937
\(936\) 1.30608 0.0426906
\(937\) 9.60780 0.313873 0.156937 0.987609i \(-0.449838\pi\)
0.156937 + 0.987609i \(0.449838\pi\)
\(938\) −0.255330 −0.00833683
\(939\) 11.0304 0.359963
\(940\) −3.32910 −0.108583
\(941\) −27.7472 −0.904532 −0.452266 0.891883i \(-0.649384\pi\)
−0.452266 + 0.891883i \(0.649384\pi\)
\(942\) 0.601944 0.0196124
\(943\) 17.8118 0.580031
\(944\) 49.5267 1.61196
\(945\) −0.652488 −0.0212254
\(946\) 1.63107 0.0530306
\(947\) −46.9156 −1.52455 −0.762276 0.647252i \(-0.775918\pi\)
−0.762276 + 0.647252i \(0.775918\pi\)
\(948\) −1.99762 −0.0648795
\(949\) 17.9133 0.581489
\(950\) 0.441360 0.0143196
\(951\) −13.9115 −0.451110
\(952\) −0.677113 −0.0219454
\(953\) 23.8600 0.772902 0.386451 0.922310i \(-0.373701\pi\)
0.386451 + 0.922310i \(0.373701\pi\)
\(954\) −0.392391 −0.0127041
\(955\) −5.10868 −0.165313
\(956\) −54.4054 −1.75960
\(957\) −12.4043 −0.400975
\(958\) −0.197863 −0.00639266
\(959\) −12.3128 −0.397601
\(960\) 1.49425 0.0482266
\(961\) 69.7597 2.25031
\(962\) −2.76968 −0.0892981
\(963\) 19.0585 0.614152
\(964\) −10.1208 −0.325969
\(965\) −1.74547 −0.0561887
\(966\) 0.590596 0.0190021
\(967\) 11.8915 0.382405 0.191203 0.981551i \(-0.438761\pi\)
0.191203 + 0.981551i \(0.438761\pi\)
\(968\) 0.286795 0.00921793
\(969\) −1.82042 −0.0584804
\(970\) −0.0778581 −0.00249987
\(971\) 56.1500 1.80194 0.900969 0.433883i \(-0.142857\pi\)
0.900969 + 0.433883i \(0.142857\pi\)
\(972\) −1.99762 −0.0640735
\(973\) −71.1126 −2.27977
\(974\) −0.816992 −0.0261781
\(975\) 33.2138 1.06369
\(976\) 27.6868 0.886232
\(977\) 37.7889 1.20897 0.604486 0.796615i \(-0.293378\pi\)
0.604486 + 0.796615i \(0.293378\pi\)
\(978\) 0.654864 0.0209403
\(979\) 12.5111 0.399855
\(980\) 1.89011 0.0603775
\(981\) −12.7995 −0.408658
\(982\) −2.05190 −0.0654789
\(983\) −45.3739 −1.44720 −0.723601 0.690218i \(-0.757514\pi\)
−0.723601 + 0.690218i \(0.757514\pi\)
\(984\) −0.997272 −0.0317919
\(985\) 3.83714 0.122262
\(986\) −0.196219 −0.00624888
\(987\) −30.7250 −0.977987
\(988\) −24.3286 −0.773997
\(989\) 37.7224 1.19950
\(990\) −0.0283630 −0.000901434 0
\(991\) 1.53256 0.0486834 0.0243417 0.999704i \(-0.492251\pi\)
0.0243417 + 0.999704i \(0.492251\pi\)
\(992\) 5.87313 0.186472
\(993\) −17.3979 −0.552105
\(994\) −1.69296 −0.0536975
\(995\) 1.96217 0.0622049
\(996\) 26.2148 0.830648
\(997\) 28.8584 0.913953 0.456977 0.889479i \(-0.348932\pi\)
0.456977 + 0.889479i \(0.348932\pi\)
\(998\) −1.44623 −0.0457797
\(999\) 8.47735 0.268212
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 4029.2.a.k.1.15 31
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4029.2.a.k.1.15 31 1.1 even 1 trivial