Properties

Label 4029.2.a.k.1.2
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 1.2
Character \(\chi\) = 4029.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.56879 q^{2} +1.00000 q^{3} +4.59868 q^{4} +3.14567 q^{5} -2.56879 q^{6} -0.137684 q^{7} -6.67546 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.56879 q^{2} +1.00000 q^{3} +4.59868 q^{4} +3.14567 q^{5} -2.56879 q^{6} -0.137684 q^{7} -6.67546 q^{8} +1.00000 q^{9} -8.08058 q^{10} +0.886329 q^{11} +4.59868 q^{12} +0.153446 q^{13} +0.353681 q^{14} +3.14567 q^{15} +7.95050 q^{16} +1.00000 q^{17} -2.56879 q^{18} +6.70747 q^{19} +14.4660 q^{20} -0.137684 q^{21} -2.27679 q^{22} +7.75524 q^{23} -6.67546 q^{24} +4.89527 q^{25} -0.394171 q^{26} +1.00000 q^{27} -0.633165 q^{28} +4.03660 q^{29} -8.08058 q^{30} -6.23265 q^{31} -7.07224 q^{32} +0.886329 q^{33} -2.56879 q^{34} -0.433109 q^{35} +4.59868 q^{36} +8.71769 q^{37} -17.2301 q^{38} +0.153446 q^{39} -20.9988 q^{40} +4.00646 q^{41} +0.353681 q^{42} +1.85994 q^{43} +4.07594 q^{44} +3.14567 q^{45} -19.9216 q^{46} -8.83188 q^{47} +7.95050 q^{48} -6.98104 q^{49} -12.5749 q^{50} +1.00000 q^{51} +0.705650 q^{52} -1.81352 q^{53} -2.56879 q^{54} +2.78810 q^{55} +0.919105 q^{56} +6.70747 q^{57} -10.3692 q^{58} +1.89769 q^{59} +14.4660 q^{60} -6.28907 q^{61} +16.0104 q^{62} -0.137684 q^{63} +2.26609 q^{64} +0.482691 q^{65} -2.27679 q^{66} -12.1576 q^{67} +4.59868 q^{68} +7.75524 q^{69} +1.11257 q^{70} +7.83419 q^{71} -6.67546 q^{72} +0.0892372 q^{73} -22.3939 q^{74} +4.89527 q^{75} +30.8455 q^{76} -0.122033 q^{77} -0.394171 q^{78} +1.00000 q^{79} +25.0097 q^{80} +1.00000 q^{81} -10.2918 q^{82} +6.29504 q^{83} -0.633165 q^{84} +3.14567 q^{85} -4.77778 q^{86} +4.03660 q^{87} -5.91665 q^{88} -9.08666 q^{89} -8.08058 q^{90} -0.0211271 q^{91} +35.6639 q^{92} -6.23265 q^{93} +22.6872 q^{94} +21.0995 q^{95} -7.07224 q^{96} -16.3940 q^{97} +17.9328 q^{98} +0.886329 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + O(q^{10}) \) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + 5q^{10} + 26q^{11} + 34q^{12} + 7q^{13} + 19q^{14} + 11q^{15} + 40q^{16} + 31q^{17} + 4q^{18} + 32q^{19} + 23q^{20} + 4q^{21} + 2q^{22} + 29q^{23} + 12q^{24} + 32q^{25} + 13q^{26} + 31q^{27} - 13q^{28} + 25q^{29} + 5q^{30} + 22q^{31} + 28q^{32} + 26q^{33} + 4q^{34} + 20q^{35} + 34q^{36} - 4q^{37} + 19q^{38} + 7q^{39} - 3q^{40} + 33q^{41} + 19q^{42} + 6q^{43} + 30q^{44} + 11q^{45} - 11q^{46} + 23q^{47} + 40q^{48} + 31q^{49} + 6q^{50} + 31q^{51} - 7q^{52} + 12q^{53} + 4q^{54} + 40q^{56} + 32q^{57} + 9q^{58} + 27q^{59} + 23q^{60} - 4q^{61} + 25q^{62} + 4q^{63} + 10q^{64} + 54q^{65} + 2q^{66} + 34q^{68} + 29q^{69} - 59q^{70} + 35q^{71} + 12q^{72} + 5q^{73} + 48q^{74} + 32q^{75} + 32q^{76} + 42q^{77} + 13q^{78} + 31q^{79} + 24q^{80} + 31q^{81} + 5q^{82} + 67q^{83} - 13q^{84} + 11q^{85} - 20q^{86} + 25q^{87} - 7q^{88} + 22q^{89} + 5q^{90} + 16q^{91} + 57q^{92} + 22q^{93} + 45q^{94} + 73q^{95} + 28q^{96} - 13q^{97} - 19q^{98} + 26q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.56879 −1.81641 −0.908204 0.418527i \(-0.862546\pi\)
−0.908204 + 0.418527i \(0.862546\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.59868 2.29934
\(5\) 3.14567 1.40679 0.703394 0.710800i \(-0.251667\pi\)
0.703394 + 0.710800i \(0.251667\pi\)
\(6\) −2.56879 −1.04870
\(7\) −0.137684 −0.0520397 −0.0260198 0.999661i \(-0.508283\pi\)
−0.0260198 + 0.999661i \(0.508283\pi\)
\(8\) −6.67546 −2.36013
\(9\) 1.00000 0.333333
\(10\) −8.08058 −2.55530
\(11\) 0.886329 0.267238 0.133619 0.991033i \(-0.457340\pi\)
0.133619 + 0.991033i \(0.457340\pi\)
\(12\) 4.59868 1.32752
\(13\) 0.153446 0.0425583 0.0212791 0.999774i \(-0.493226\pi\)
0.0212791 + 0.999774i \(0.493226\pi\)
\(14\) 0.353681 0.0945254
\(15\) 3.14567 0.812210
\(16\) 7.95050 1.98763
\(17\) 1.00000 0.242536
\(18\) −2.56879 −0.605470
\(19\) 6.70747 1.53880 0.769400 0.638768i \(-0.220555\pi\)
0.769400 + 0.638768i \(0.220555\pi\)
\(20\) 14.4660 3.23468
\(21\) −0.137684 −0.0300451
\(22\) −2.27679 −0.485414
\(23\) 7.75524 1.61708 0.808540 0.588442i \(-0.200258\pi\)
0.808540 + 0.588442i \(0.200258\pi\)
\(24\) −6.67546 −1.36262
\(25\) 4.89527 0.979053
\(26\) −0.394171 −0.0773033
\(27\) 1.00000 0.192450
\(28\) −0.633165 −0.119657
\(29\) 4.03660 0.749577 0.374789 0.927110i \(-0.377715\pi\)
0.374789 + 0.927110i \(0.377715\pi\)
\(30\) −8.08058 −1.47530
\(31\) −6.23265 −1.11942 −0.559709 0.828690i \(-0.689087\pi\)
−0.559709 + 0.828690i \(0.689087\pi\)
\(32\) −7.07224 −1.25021
\(33\) 0.886329 0.154290
\(34\) −2.56879 −0.440544
\(35\) −0.433109 −0.0732088
\(36\) 4.59868 0.766447
\(37\) 8.71769 1.43318 0.716589 0.697495i \(-0.245702\pi\)
0.716589 + 0.697495i \(0.245702\pi\)
\(38\) −17.2301 −2.79509
\(39\) 0.153446 0.0245710
\(40\) −20.9988 −3.32021
\(41\) 4.00646 0.625705 0.312852 0.949802i \(-0.398716\pi\)
0.312852 + 0.949802i \(0.398716\pi\)
\(42\) 0.353681 0.0545742
\(43\) 1.85994 0.283638 0.141819 0.989893i \(-0.454705\pi\)
0.141819 + 0.989893i \(0.454705\pi\)
\(44\) 4.07594 0.614471
\(45\) 3.14567 0.468929
\(46\) −19.9216 −2.93728
\(47\) −8.83188 −1.28826 −0.644131 0.764915i \(-0.722781\pi\)
−0.644131 + 0.764915i \(0.722781\pi\)
\(48\) 7.95050 1.14756
\(49\) −6.98104 −0.997292
\(50\) −12.5749 −1.77836
\(51\) 1.00000 0.140028
\(52\) 0.705650 0.0978560
\(53\) −1.81352 −0.249107 −0.124553 0.992213i \(-0.539750\pi\)
−0.124553 + 0.992213i \(0.539750\pi\)
\(54\) −2.56879 −0.349568
\(55\) 2.78810 0.375947
\(56\) 0.919105 0.122821
\(57\) 6.70747 0.888426
\(58\) −10.3692 −1.36154
\(59\) 1.89769 0.247058 0.123529 0.992341i \(-0.460579\pi\)
0.123529 + 0.992341i \(0.460579\pi\)
\(60\) 14.4660 1.86755
\(61\) −6.28907 −0.805233 −0.402617 0.915369i \(-0.631899\pi\)
−0.402617 + 0.915369i \(0.631899\pi\)
\(62\) 16.0104 2.03332
\(63\) −0.137684 −0.0173466
\(64\) 2.26609 0.283261
\(65\) 0.482691 0.0598705
\(66\) −2.27679 −0.280254
\(67\) −12.1576 −1.48529 −0.742643 0.669688i \(-0.766428\pi\)
−0.742643 + 0.669688i \(0.766428\pi\)
\(68\) 4.59868 0.557672
\(69\) 7.75524 0.933621
\(70\) 1.11257 0.132977
\(71\) 7.83419 0.929747 0.464873 0.885377i \(-0.346100\pi\)
0.464873 + 0.885377i \(0.346100\pi\)
\(72\) −6.67546 −0.786711
\(73\) 0.0892372 0.0104444 0.00522221 0.999986i \(-0.498338\pi\)
0.00522221 + 0.999986i \(0.498338\pi\)
\(74\) −22.3939 −2.60324
\(75\) 4.89527 0.565257
\(76\) 30.8455 3.53822
\(77\) −0.122033 −0.0139070
\(78\) −0.394171 −0.0446311
\(79\) 1.00000 0.112509
\(80\) 25.0097 2.79617
\(81\) 1.00000 0.111111
\(82\) −10.2918 −1.13654
\(83\) 6.29504 0.690970 0.345485 0.938424i \(-0.387714\pi\)
0.345485 + 0.938424i \(0.387714\pi\)
\(84\) −0.633165 −0.0690840
\(85\) 3.14567 0.341196
\(86\) −4.77778 −0.515202
\(87\) 4.03660 0.432769
\(88\) −5.91665 −0.630717
\(89\) −9.08666 −0.963184 −0.481592 0.876396i \(-0.659941\pi\)
−0.481592 + 0.876396i \(0.659941\pi\)
\(90\) −8.08058 −0.851767
\(91\) −0.0211271 −0.00221472
\(92\) 35.6639 3.71822
\(93\) −6.23265 −0.646296
\(94\) 22.6872 2.34001
\(95\) 21.0995 2.16477
\(96\) −7.07224 −0.721807
\(97\) −16.3940 −1.66456 −0.832280 0.554355i \(-0.812965\pi\)
−0.832280 + 0.554355i \(0.812965\pi\)
\(98\) 17.9328 1.81149
\(99\) 0.886329 0.0890794
\(100\) 22.5118 2.25118
\(101\) 13.6640 1.35962 0.679809 0.733389i \(-0.262063\pi\)
0.679809 + 0.733389i \(0.262063\pi\)
\(102\) −2.56879 −0.254348
\(103\) 6.32754 0.623471 0.311736 0.950169i \(-0.399090\pi\)
0.311736 + 0.950169i \(0.399090\pi\)
\(104\) −1.02432 −0.100443
\(105\) −0.433109 −0.0422671
\(106\) 4.65856 0.452479
\(107\) −10.5763 −1.02245 −0.511226 0.859446i \(-0.670809\pi\)
−0.511226 + 0.859446i \(0.670809\pi\)
\(108\) 4.59868 0.442508
\(109\) 4.47043 0.428190 0.214095 0.976813i \(-0.431320\pi\)
0.214095 + 0.976813i \(0.431320\pi\)
\(110\) −7.16204 −0.682874
\(111\) 8.71769 0.827446
\(112\) −1.09466 −0.103435
\(113\) 11.7239 1.10289 0.551447 0.834210i \(-0.314076\pi\)
0.551447 + 0.834210i \(0.314076\pi\)
\(114\) −17.2301 −1.61375
\(115\) 24.3955 2.27489
\(116\) 18.5630 1.72353
\(117\) 0.153446 0.0141861
\(118\) −4.87477 −0.448759
\(119\) −0.137684 −0.0126215
\(120\) −20.9988 −1.91692
\(121\) −10.2144 −0.928584
\(122\) 16.1553 1.46263
\(123\) 4.00646 0.361251
\(124\) −28.6620 −2.57392
\(125\) −0.329457 −0.0294675
\(126\) 0.353681 0.0315085
\(127\) 9.39492 0.833665 0.416832 0.908983i \(-0.363140\pi\)
0.416832 + 0.908983i \(0.363140\pi\)
\(128\) 8.32337 0.735689
\(129\) 1.85994 0.163758
\(130\) −1.23993 −0.108749
\(131\) 17.4924 1.52832 0.764158 0.645029i \(-0.223155\pi\)
0.764158 + 0.645029i \(0.223155\pi\)
\(132\) 4.07594 0.354765
\(133\) −0.923512 −0.0800787
\(134\) 31.2303 2.69789
\(135\) 3.14567 0.270737
\(136\) −6.67546 −0.572416
\(137\) −6.18726 −0.528614 −0.264307 0.964439i \(-0.585143\pi\)
−0.264307 + 0.964439i \(0.585143\pi\)
\(138\) −19.9216 −1.69584
\(139\) −13.4519 −1.14098 −0.570488 0.821306i \(-0.693246\pi\)
−0.570488 + 0.821306i \(0.693246\pi\)
\(140\) −1.99173 −0.168332
\(141\) −8.83188 −0.743778
\(142\) −20.1244 −1.68880
\(143\) 0.136004 0.0113732
\(144\) 7.95050 0.662542
\(145\) 12.6978 1.05450
\(146\) −0.229232 −0.0189713
\(147\) −6.98104 −0.575787
\(148\) 40.0899 3.29537
\(149\) 7.97695 0.653497 0.326749 0.945111i \(-0.394047\pi\)
0.326749 + 0.945111i \(0.394047\pi\)
\(150\) −12.5749 −1.02674
\(151\) 6.34255 0.516150 0.258075 0.966125i \(-0.416912\pi\)
0.258075 + 0.966125i \(0.416912\pi\)
\(152\) −44.7755 −3.63177
\(153\) 1.00000 0.0808452
\(154\) 0.313478 0.0252608
\(155\) −19.6059 −1.57478
\(156\) 0.705650 0.0564972
\(157\) −4.96083 −0.395918 −0.197959 0.980210i \(-0.563431\pi\)
−0.197959 + 0.980210i \(0.563431\pi\)
\(158\) −2.56879 −0.204362
\(159\) −1.81352 −0.143822
\(160\) −22.2470 −1.75878
\(161\) −1.06777 −0.0841523
\(162\) −2.56879 −0.201823
\(163\) −17.8809 −1.40054 −0.700272 0.713877i \(-0.746938\pi\)
−0.700272 + 0.713877i \(0.746938\pi\)
\(164\) 18.4244 1.43871
\(165\) 2.78810 0.217053
\(166\) −16.1706 −1.25508
\(167\) −0.248750 −0.0192489 −0.00962443 0.999954i \(-0.503064\pi\)
−0.00962443 + 0.999954i \(0.503064\pi\)
\(168\) 0.919105 0.0709105
\(169\) −12.9765 −0.998189
\(170\) −8.08058 −0.619752
\(171\) 6.70747 0.512933
\(172\) 8.55325 0.652179
\(173\) −20.4864 −1.55755 −0.778774 0.627304i \(-0.784158\pi\)
−0.778774 + 0.627304i \(0.784158\pi\)
\(174\) −10.3692 −0.786085
\(175\) −0.674000 −0.0509496
\(176\) 7.04676 0.531169
\(177\) 1.89769 0.142639
\(178\) 23.3417 1.74954
\(179\) 20.3027 1.51750 0.758748 0.651384i \(-0.225811\pi\)
0.758748 + 0.651384i \(0.225811\pi\)
\(180\) 14.4660 1.07823
\(181\) 6.39501 0.475337 0.237669 0.971346i \(-0.423617\pi\)
0.237669 + 0.971346i \(0.423617\pi\)
\(182\) 0.0542711 0.00402284
\(183\) −6.28907 −0.464902
\(184\) −51.7698 −3.81652
\(185\) 27.4230 2.01618
\(186\) 16.0104 1.17394
\(187\) 0.886329 0.0648148
\(188\) −40.6150 −2.96215
\(189\) −0.137684 −0.0100150
\(190\) −54.2002 −3.93210
\(191\) 6.78820 0.491177 0.245588 0.969374i \(-0.421019\pi\)
0.245588 + 0.969374i \(0.421019\pi\)
\(192\) 2.26609 0.163541
\(193\) 0.0206068 0.00148331 0.000741653 1.00000i \(-0.499764\pi\)
0.000741653 1.00000i \(0.499764\pi\)
\(194\) 42.1128 3.02352
\(195\) 0.482691 0.0345663
\(196\) −32.1036 −2.29311
\(197\) 5.35940 0.381841 0.190921 0.981605i \(-0.438853\pi\)
0.190921 + 0.981605i \(0.438853\pi\)
\(198\) −2.27679 −0.161805
\(199\) −18.6041 −1.31881 −0.659405 0.751788i \(-0.729192\pi\)
−0.659405 + 0.751788i \(0.729192\pi\)
\(200\) −32.6782 −2.31070
\(201\) −12.1576 −0.857530
\(202\) −35.0999 −2.46962
\(203\) −0.555775 −0.0390078
\(204\) 4.59868 0.321972
\(205\) 12.6030 0.880234
\(206\) −16.2541 −1.13248
\(207\) 7.75524 0.539026
\(208\) 1.21997 0.0845899
\(209\) 5.94502 0.411226
\(210\) 1.11257 0.0767744
\(211\) 12.3382 0.849400 0.424700 0.905334i \(-0.360380\pi\)
0.424700 + 0.905334i \(0.360380\pi\)
\(212\) −8.33981 −0.572781
\(213\) 7.83419 0.536790
\(214\) 27.1684 1.85719
\(215\) 5.85075 0.399018
\(216\) −6.67546 −0.454208
\(217\) 0.858137 0.0582541
\(218\) −11.4836 −0.777767
\(219\) 0.0892372 0.00603009
\(220\) 12.8216 0.864431
\(221\) 0.153446 0.0103219
\(222\) −22.3939 −1.50298
\(223\) 2.52511 0.169094 0.0845469 0.996419i \(-0.473056\pi\)
0.0845469 + 0.996419i \(0.473056\pi\)
\(224\) 0.973735 0.0650604
\(225\) 4.89527 0.326351
\(226\) −30.1163 −2.00331
\(227\) −11.0846 −0.735711 −0.367855 0.929883i \(-0.619908\pi\)
−0.367855 + 0.929883i \(0.619908\pi\)
\(228\) 30.8455 2.04279
\(229\) −9.95351 −0.657746 −0.328873 0.944374i \(-0.606669\pi\)
−0.328873 + 0.944374i \(0.606669\pi\)
\(230\) −62.6668 −4.13213
\(231\) −0.122033 −0.00802920
\(232\) −26.9462 −1.76910
\(233\) −19.2676 −1.26226 −0.631131 0.775676i \(-0.717409\pi\)
−0.631131 + 0.775676i \(0.717409\pi\)
\(234\) −0.394171 −0.0257678
\(235\) −27.7822 −1.81231
\(236\) 8.72687 0.568071
\(237\) 1.00000 0.0649570
\(238\) 0.353681 0.0229258
\(239\) 16.5390 1.06982 0.534911 0.844909i \(-0.320345\pi\)
0.534911 + 0.844909i \(0.320345\pi\)
\(240\) 25.0097 1.61437
\(241\) 26.3974 1.70041 0.850203 0.526454i \(-0.176479\pi\)
0.850203 + 0.526454i \(0.176479\pi\)
\(242\) 26.2387 1.68669
\(243\) 1.00000 0.0641500
\(244\) −28.9214 −1.85150
\(245\) −21.9601 −1.40298
\(246\) −10.2918 −0.656179
\(247\) 1.02924 0.0654887
\(248\) 41.6058 2.64197
\(249\) 6.29504 0.398932
\(250\) 0.846305 0.0535250
\(251\) 1.40438 0.0886437 0.0443219 0.999017i \(-0.485887\pi\)
0.0443219 + 0.999017i \(0.485887\pi\)
\(252\) −0.633165 −0.0398857
\(253\) 6.87369 0.432145
\(254\) −24.1336 −1.51428
\(255\) 3.14567 0.196990
\(256\) −25.9132 −1.61957
\(257\) −11.3803 −0.709881 −0.354941 0.934889i \(-0.615499\pi\)
−0.354941 + 0.934889i \(0.615499\pi\)
\(258\) −4.77778 −0.297452
\(259\) −1.20029 −0.0745822
\(260\) 2.21974 0.137663
\(261\) 4.03660 0.249859
\(262\) −44.9342 −2.77605
\(263\) −18.9971 −1.17141 −0.585705 0.810525i \(-0.699182\pi\)
−0.585705 + 0.810525i \(0.699182\pi\)
\(264\) −5.91665 −0.364145
\(265\) −5.70475 −0.350440
\(266\) 2.37231 0.145456
\(267\) −9.08666 −0.556095
\(268\) −55.9089 −3.41518
\(269\) −8.23267 −0.501955 −0.250977 0.967993i \(-0.580752\pi\)
−0.250977 + 0.967993i \(0.580752\pi\)
\(270\) −8.08058 −0.491768
\(271\) −0.135088 −0.00820603 −0.00410301 0.999992i \(-0.501306\pi\)
−0.00410301 + 0.999992i \(0.501306\pi\)
\(272\) 7.95050 0.482070
\(273\) −0.0211271 −0.00127867
\(274\) 15.8938 0.960178
\(275\) 4.33881 0.261640
\(276\) 35.6639 2.14671
\(277\) −23.2923 −1.39950 −0.699749 0.714388i \(-0.746705\pi\)
−0.699749 + 0.714388i \(0.746705\pi\)
\(278\) 34.5551 2.07248
\(279\) −6.23265 −0.373139
\(280\) 2.89121 0.172783
\(281\) 25.7854 1.53823 0.769113 0.639113i \(-0.220698\pi\)
0.769113 + 0.639113i \(0.220698\pi\)
\(282\) 22.6872 1.35101
\(283\) 18.9333 1.12547 0.562735 0.826637i \(-0.309749\pi\)
0.562735 + 0.826637i \(0.309749\pi\)
\(284\) 36.0269 2.13780
\(285\) 21.0995 1.24983
\(286\) −0.349365 −0.0206584
\(287\) −0.551626 −0.0325615
\(288\) −7.07224 −0.416736
\(289\) 1.00000 0.0588235
\(290\) −32.6180 −1.91540
\(291\) −16.3940 −0.961035
\(292\) 0.410374 0.0240153
\(293\) 7.36864 0.430481 0.215240 0.976561i \(-0.430946\pi\)
0.215240 + 0.976561i \(0.430946\pi\)
\(294\) 17.9328 1.04586
\(295\) 5.96952 0.347559
\(296\) −58.1946 −3.38249
\(297\) 0.886329 0.0514300
\(298\) −20.4911 −1.18702
\(299\) 1.19001 0.0688201
\(300\) 22.5118 1.29972
\(301\) −0.256084 −0.0147604
\(302\) −16.2927 −0.937539
\(303\) 13.6640 0.784976
\(304\) 53.3278 3.05856
\(305\) −19.7834 −1.13279
\(306\) −2.56879 −0.146848
\(307\) −24.2984 −1.38678 −0.693391 0.720561i \(-0.743884\pi\)
−0.693391 + 0.720561i \(0.743884\pi\)
\(308\) −0.561192 −0.0319769
\(309\) 6.32754 0.359961
\(310\) 50.3634 2.86045
\(311\) −6.81598 −0.386499 −0.193249 0.981150i \(-0.561903\pi\)
−0.193249 + 0.981150i \(0.561903\pi\)
\(312\) −1.02432 −0.0579909
\(313\) 22.8625 1.29226 0.646131 0.763226i \(-0.276386\pi\)
0.646131 + 0.763226i \(0.276386\pi\)
\(314\) 12.7433 0.719148
\(315\) −0.433109 −0.0244029
\(316\) 4.59868 0.258696
\(317\) 10.9591 0.615525 0.307762 0.951463i \(-0.400420\pi\)
0.307762 + 0.951463i \(0.400420\pi\)
\(318\) 4.65856 0.261239
\(319\) 3.57775 0.200316
\(320\) 7.12838 0.398489
\(321\) −10.5763 −0.590313
\(322\) 2.74288 0.152855
\(323\) 6.70747 0.373214
\(324\) 4.59868 0.255482
\(325\) 0.751160 0.0416668
\(326\) 45.9324 2.54396
\(327\) 4.47043 0.247215
\(328\) −26.7450 −1.47675
\(329\) 1.21601 0.0670408
\(330\) −7.16204 −0.394258
\(331\) 24.6789 1.35647 0.678236 0.734844i \(-0.262745\pi\)
0.678236 + 0.734844i \(0.262745\pi\)
\(332\) 28.9489 1.58878
\(333\) 8.71769 0.477726
\(334\) 0.638987 0.0349638
\(335\) −38.2438 −2.08948
\(336\) −1.09466 −0.0597185
\(337\) 18.1589 0.989181 0.494590 0.869126i \(-0.335318\pi\)
0.494590 + 0.869126i \(0.335318\pi\)
\(338\) 33.3338 1.81312
\(339\) 11.7239 0.636756
\(340\) 14.4660 0.784526
\(341\) −5.52418 −0.299151
\(342\) −17.2301 −0.931696
\(343\) 1.92497 0.103938
\(344\) −12.4159 −0.669422
\(345\) 24.3955 1.31341
\(346\) 52.6251 2.82914
\(347\) 4.94297 0.265353 0.132676 0.991159i \(-0.457643\pi\)
0.132676 + 0.991159i \(0.457643\pi\)
\(348\) 18.5630 0.995082
\(349\) −10.9480 −0.586032 −0.293016 0.956108i \(-0.594659\pi\)
−0.293016 + 0.956108i \(0.594659\pi\)
\(350\) 1.73137 0.0925454
\(351\) 0.153446 0.00819035
\(352\) −6.26833 −0.334103
\(353\) −6.40755 −0.341039 −0.170520 0.985354i \(-0.554545\pi\)
−0.170520 + 0.985354i \(0.554545\pi\)
\(354\) −4.87477 −0.259091
\(355\) 24.6438 1.30796
\(356\) −41.7866 −2.21469
\(357\) −0.137684 −0.00728702
\(358\) −52.1534 −2.75639
\(359\) −26.0161 −1.37308 −0.686538 0.727094i \(-0.740870\pi\)
−0.686538 + 0.727094i \(0.740870\pi\)
\(360\) −20.9988 −1.10674
\(361\) 25.9902 1.36790
\(362\) −16.4274 −0.863407
\(363\) −10.2144 −0.536118
\(364\) −0.0971567 −0.00509240
\(365\) 0.280711 0.0146931
\(366\) 16.1553 0.844451
\(367\) 7.02332 0.366614 0.183307 0.983056i \(-0.441320\pi\)
0.183307 + 0.983056i \(0.441320\pi\)
\(368\) 61.6580 3.21415
\(369\) 4.00646 0.208568
\(370\) −70.4439 −3.66221
\(371\) 0.249693 0.0129634
\(372\) −28.6620 −1.48605
\(373\) −20.5593 −1.06452 −0.532260 0.846581i \(-0.678657\pi\)
−0.532260 + 0.846581i \(0.678657\pi\)
\(374\) −2.27679 −0.117730
\(375\) −0.329457 −0.0170131
\(376\) 58.9569 3.04047
\(377\) 0.619400 0.0319007
\(378\) 0.353681 0.0181914
\(379\) −28.3640 −1.45696 −0.728480 0.685067i \(-0.759773\pi\)
−0.728480 + 0.685067i \(0.759773\pi\)
\(380\) 97.0300 4.97753
\(381\) 9.39492 0.481317
\(382\) −17.4375 −0.892178
\(383\) −0.877959 −0.0448616 −0.0224308 0.999748i \(-0.507141\pi\)
−0.0224308 + 0.999748i \(0.507141\pi\)
\(384\) 8.32337 0.424750
\(385\) −0.383877 −0.0195642
\(386\) −0.0529344 −0.00269429
\(387\) 1.85994 0.0945458
\(388\) −75.3909 −3.82739
\(389\) 18.5261 0.939313 0.469657 0.882849i \(-0.344378\pi\)
0.469657 + 0.882849i \(0.344378\pi\)
\(390\) −1.23993 −0.0627864
\(391\) 7.75524 0.392199
\(392\) 46.6017 2.35374
\(393\) 17.4924 0.882374
\(394\) −13.7672 −0.693580
\(395\) 3.14567 0.158276
\(396\) 4.07594 0.204824
\(397\) 7.78973 0.390955 0.195478 0.980708i \(-0.437374\pi\)
0.195478 + 0.980708i \(0.437374\pi\)
\(398\) 47.7900 2.39550
\(399\) −0.923512 −0.0462334
\(400\) 38.9198 1.94599
\(401\) 30.5551 1.52585 0.762923 0.646489i \(-0.223763\pi\)
0.762923 + 0.646489i \(0.223763\pi\)
\(402\) 31.2303 1.55762
\(403\) −0.956376 −0.0476405
\(404\) 62.8363 3.12622
\(405\) 3.14567 0.156310
\(406\) 1.42767 0.0708541
\(407\) 7.72673 0.383000
\(408\) −6.67546 −0.330485
\(409\) 39.4597 1.95116 0.975579 0.219651i \(-0.0704918\pi\)
0.975579 + 0.219651i \(0.0704918\pi\)
\(410\) −32.3745 −1.59886
\(411\) −6.18726 −0.305195
\(412\) 29.0983 1.43357
\(413\) −0.261282 −0.0128568
\(414\) −19.9216 −0.979092
\(415\) 19.8021 0.972049
\(416\) −1.08521 −0.0532067
\(417\) −13.4519 −0.658742
\(418\) −15.2715 −0.746954
\(419\) −3.10743 −0.151808 −0.0759039 0.997115i \(-0.524184\pi\)
−0.0759039 + 0.997115i \(0.524184\pi\)
\(420\) −1.99173 −0.0971865
\(421\) −11.1773 −0.544748 −0.272374 0.962192i \(-0.587809\pi\)
−0.272374 + 0.962192i \(0.587809\pi\)
\(422\) −31.6943 −1.54286
\(423\) −8.83188 −0.429421
\(424\) 12.1061 0.587925
\(425\) 4.89527 0.237455
\(426\) −20.1244 −0.975029
\(427\) 0.865905 0.0419041
\(428\) −48.6372 −2.35097
\(429\) 0.136004 0.00656632
\(430\) −15.0294 −0.724780
\(431\) 7.74831 0.373223 0.186611 0.982434i \(-0.440249\pi\)
0.186611 + 0.982434i \(0.440249\pi\)
\(432\) 7.95050 0.382519
\(433\) −12.6430 −0.607584 −0.303792 0.952738i \(-0.598253\pi\)
−0.303792 + 0.952738i \(0.598253\pi\)
\(434\) −2.20437 −0.105813
\(435\) 12.6978 0.608814
\(436\) 20.5581 0.984554
\(437\) 52.0180 2.48836
\(438\) −0.229232 −0.0109531
\(439\) 24.0391 1.14733 0.573663 0.819092i \(-0.305522\pi\)
0.573663 + 0.819092i \(0.305522\pi\)
\(440\) −18.6119 −0.887286
\(441\) −6.98104 −0.332431
\(442\) −0.394171 −0.0187488
\(443\) −8.61339 −0.409235 −0.204617 0.978842i \(-0.565595\pi\)
−0.204617 + 0.978842i \(0.565595\pi\)
\(444\) 40.0899 1.90258
\(445\) −28.5837 −1.35500
\(446\) −6.48648 −0.307144
\(447\) 7.97695 0.377297
\(448\) −0.312005 −0.0147408
\(449\) −41.1657 −1.94273 −0.971365 0.237592i \(-0.923642\pi\)
−0.971365 + 0.237592i \(0.923642\pi\)
\(450\) −12.5749 −0.592787
\(451\) 3.55104 0.167212
\(452\) 53.9146 2.53593
\(453\) 6.34255 0.297999
\(454\) 28.4740 1.33635
\(455\) −0.0664589 −0.00311564
\(456\) −44.7755 −2.09680
\(457\) −39.5093 −1.84817 −0.924084 0.382190i \(-0.875170\pi\)
−0.924084 + 0.382190i \(0.875170\pi\)
\(458\) 25.5685 1.19474
\(459\) 1.00000 0.0466760
\(460\) 112.187 5.23074
\(461\) 28.8203 1.34229 0.671147 0.741324i \(-0.265802\pi\)
0.671147 + 0.741324i \(0.265802\pi\)
\(462\) 0.313478 0.0145843
\(463\) 21.6787 1.00749 0.503747 0.863851i \(-0.331954\pi\)
0.503747 + 0.863851i \(0.331954\pi\)
\(464\) 32.0930 1.48988
\(465\) −19.6059 −0.909201
\(466\) 49.4944 2.29278
\(467\) −7.24370 −0.335198 −0.167599 0.985855i \(-0.553601\pi\)
−0.167599 + 0.985855i \(0.553601\pi\)
\(468\) 0.705650 0.0326187
\(469\) 1.67391 0.0772938
\(470\) 71.3667 3.29190
\(471\) −4.96083 −0.228583
\(472\) −12.6680 −0.583090
\(473\) 1.64851 0.0757988
\(474\) −2.56879 −0.117988
\(475\) 32.8349 1.50657
\(476\) −0.633165 −0.0290211
\(477\) −1.81352 −0.0830355
\(478\) −42.4853 −1.94323
\(479\) −13.3885 −0.611737 −0.305869 0.952074i \(-0.598947\pi\)
−0.305869 + 0.952074i \(0.598947\pi\)
\(480\) −22.2470 −1.01543
\(481\) 1.33770 0.0609936
\(482\) −67.8094 −3.08863
\(483\) −1.06777 −0.0485854
\(484\) −46.9729 −2.13513
\(485\) −51.5702 −2.34168
\(486\) −2.56879 −0.116523
\(487\) 8.61232 0.390261 0.195131 0.980777i \(-0.437487\pi\)
0.195131 + 0.980777i \(0.437487\pi\)
\(488\) 41.9825 1.90046
\(489\) −17.8809 −0.808604
\(490\) 56.4108 2.54838
\(491\) 26.6586 1.20308 0.601542 0.798841i \(-0.294553\pi\)
0.601542 + 0.798841i \(0.294553\pi\)
\(492\) 18.4244 0.830638
\(493\) 4.03660 0.181799
\(494\) −2.64389 −0.118954
\(495\) 2.78810 0.125316
\(496\) −49.5527 −2.22498
\(497\) −1.07864 −0.0483837
\(498\) −16.1706 −0.724623
\(499\) −6.29777 −0.281927 −0.140963 0.990015i \(-0.545020\pi\)
−0.140963 + 0.990015i \(0.545020\pi\)
\(500\) −1.51507 −0.0677558
\(501\) −0.248750 −0.0111133
\(502\) −3.60756 −0.161013
\(503\) 17.6609 0.787460 0.393730 0.919226i \(-0.371185\pi\)
0.393730 + 0.919226i \(0.371185\pi\)
\(504\) 0.919105 0.0409402
\(505\) 42.9825 1.91269
\(506\) −17.6571 −0.784952
\(507\) −12.9765 −0.576305
\(508\) 43.2043 1.91688
\(509\) −5.36610 −0.237848 −0.118924 0.992903i \(-0.537944\pi\)
−0.118924 + 0.992903i \(0.537944\pi\)
\(510\) −8.08058 −0.357814
\(511\) −0.0122865 −0.000543525 0
\(512\) 49.9187 2.20612
\(513\) 6.70747 0.296142
\(514\) 29.2335 1.28943
\(515\) 19.9044 0.877092
\(516\) 8.55325 0.376536
\(517\) −7.82795 −0.344273
\(518\) 3.08328 0.135472
\(519\) −20.4864 −0.899251
\(520\) −3.22219 −0.141302
\(521\) −6.54100 −0.286566 −0.143283 0.989682i \(-0.545766\pi\)
−0.143283 + 0.989682i \(0.545766\pi\)
\(522\) −10.3692 −0.453846
\(523\) 24.4974 1.07120 0.535599 0.844473i \(-0.320086\pi\)
0.535599 + 0.844473i \(0.320086\pi\)
\(524\) 80.4418 3.51412
\(525\) −0.674000 −0.0294158
\(526\) 48.7995 2.12776
\(527\) −6.23265 −0.271499
\(528\) 7.04676 0.306671
\(529\) 37.1437 1.61494
\(530\) 14.6543 0.636543
\(531\) 1.89769 0.0823528
\(532\) −4.24694 −0.184128
\(533\) 0.614776 0.0266289
\(534\) 23.3417 1.01009
\(535\) −33.2697 −1.43837
\(536\) 81.1575 3.50547
\(537\) 20.3027 0.876127
\(538\) 21.1480 0.911755
\(539\) −6.18750 −0.266514
\(540\) 14.4660 0.622515
\(541\) −42.9711 −1.84747 −0.923737 0.383028i \(-0.874881\pi\)
−0.923737 + 0.383028i \(0.874881\pi\)
\(542\) 0.347013 0.0149055
\(543\) 6.39501 0.274436
\(544\) −7.07224 −0.303220
\(545\) 14.0625 0.602372
\(546\) 0.0542711 0.00232259
\(547\) 9.48682 0.405627 0.202814 0.979217i \(-0.434991\pi\)
0.202814 + 0.979217i \(0.434991\pi\)
\(548\) −28.4532 −1.21546
\(549\) −6.28907 −0.268411
\(550\) −11.1455 −0.475246
\(551\) 27.0754 1.15345
\(552\) −51.7698 −2.20347
\(553\) −0.137684 −0.00585492
\(554\) 59.8330 2.54206
\(555\) 27.4230 1.16404
\(556\) −61.8610 −2.62349
\(557\) −16.5982 −0.703290 −0.351645 0.936133i \(-0.614378\pi\)
−0.351645 + 0.936133i \(0.614378\pi\)
\(558\) 16.0104 0.677773
\(559\) 0.285400 0.0120711
\(560\) −3.44344 −0.145512
\(561\) 0.886329 0.0374208
\(562\) −66.2372 −2.79405
\(563\) 12.6343 0.532471 0.266236 0.963908i \(-0.414220\pi\)
0.266236 + 0.963908i \(0.414220\pi\)
\(564\) −40.6150 −1.71020
\(565\) 36.8796 1.55154
\(566\) −48.6357 −2.04431
\(567\) −0.137684 −0.00578219
\(568\) −52.2968 −2.19433
\(569\) 24.9192 1.04467 0.522334 0.852741i \(-0.325062\pi\)
0.522334 + 0.852741i \(0.325062\pi\)
\(570\) −54.2002 −2.27020
\(571\) 6.28106 0.262854 0.131427 0.991326i \(-0.458044\pi\)
0.131427 + 0.991326i \(0.458044\pi\)
\(572\) 0.625437 0.0261509
\(573\) 6.78820 0.283581
\(574\) 1.41701 0.0591449
\(575\) 37.9640 1.58321
\(576\) 2.26609 0.0944204
\(577\) −33.9537 −1.41351 −0.706756 0.707457i \(-0.749842\pi\)
−0.706756 + 0.707457i \(0.749842\pi\)
\(578\) −2.56879 −0.106848
\(579\) 0.0206068 0.000856388 0
\(580\) 58.3932 2.42465
\(581\) −0.866727 −0.0359579
\(582\) 42.1128 1.74563
\(583\) −1.60738 −0.0665708
\(584\) −0.595700 −0.0246502
\(585\) 0.482691 0.0199568
\(586\) −18.9285 −0.781929
\(587\) 33.7214 1.39183 0.695916 0.718124i \(-0.254999\pi\)
0.695916 + 0.718124i \(0.254999\pi\)
\(588\) −32.1036 −1.32393
\(589\) −41.8053 −1.72256
\(590\) −15.3344 −0.631309
\(591\) 5.35940 0.220456
\(592\) 69.3100 2.84862
\(593\) −31.6351 −1.29910 −0.649549 0.760320i \(-0.725042\pi\)
−0.649549 + 0.760320i \(0.725042\pi\)
\(594\) −2.27679 −0.0934179
\(595\) −0.433109 −0.0177558
\(596\) 36.6834 1.50261
\(597\) −18.6041 −0.761415
\(598\) −3.05689 −0.125005
\(599\) −10.0084 −0.408931 −0.204465 0.978874i \(-0.565546\pi\)
−0.204465 + 0.978874i \(0.565546\pi\)
\(600\) −32.6782 −1.33408
\(601\) 39.2087 1.59936 0.799679 0.600428i \(-0.205003\pi\)
0.799679 + 0.600428i \(0.205003\pi\)
\(602\) 0.657825 0.0268109
\(603\) −12.1576 −0.495095
\(604\) 29.1674 1.18680
\(605\) −32.1312 −1.30632
\(606\) −35.0999 −1.42584
\(607\) −1.91002 −0.0775251 −0.0387626 0.999248i \(-0.512342\pi\)
−0.0387626 + 0.999248i \(0.512342\pi\)
\(608\) −47.4368 −1.92382
\(609\) −0.555775 −0.0225211
\(610\) 50.8193 2.05761
\(611\) −1.35522 −0.0548262
\(612\) 4.59868 0.185891
\(613\) −10.1587 −0.410305 −0.205152 0.978730i \(-0.565769\pi\)
−0.205152 + 0.978730i \(0.565769\pi\)
\(614\) 62.4175 2.51896
\(615\) 12.6030 0.508203
\(616\) 0.814629 0.0328223
\(617\) 6.58019 0.264909 0.132454 0.991189i \(-0.457714\pi\)
0.132454 + 0.991189i \(0.457714\pi\)
\(618\) −16.2541 −0.653837
\(619\) −20.5201 −0.824771 −0.412386 0.911009i \(-0.635304\pi\)
−0.412386 + 0.911009i \(0.635304\pi\)
\(620\) −90.1612 −3.62096
\(621\) 7.75524 0.311207
\(622\) 17.5088 0.702040
\(623\) 1.25109 0.0501238
\(624\) 1.21997 0.0488380
\(625\) −25.5127 −1.02051
\(626\) −58.7289 −2.34728
\(627\) 5.94502 0.237421
\(628\) −22.8133 −0.910349
\(629\) 8.71769 0.347597
\(630\) 1.11257 0.0443257
\(631\) 22.7532 0.905792 0.452896 0.891563i \(-0.350391\pi\)
0.452896 + 0.891563i \(0.350391\pi\)
\(632\) −6.67546 −0.265536
\(633\) 12.3382 0.490401
\(634\) −28.1516 −1.11804
\(635\) 29.5534 1.17279
\(636\) −8.33981 −0.330695
\(637\) −1.07121 −0.0424430
\(638\) −9.19049 −0.363855
\(639\) 7.83419 0.309916
\(640\) 26.1826 1.03496
\(641\) 34.8738 1.37743 0.688715 0.725032i \(-0.258175\pi\)
0.688715 + 0.725032i \(0.258175\pi\)
\(642\) 27.1684 1.07225
\(643\) 36.3233 1.43245 0.716225 0.697870i \(-0.245868\pi\)
0.716225 + 0.697870i \(0.245868\pi\)
\(644\) −4.91035 −0.193495
\(645\) 5.85075 0.230373
\(646\) −17.2301 −0.677909
\(647\) 13.9056 0.546687 0.273343 0.961917i \(-0.411870\pi\)
0.273343 + 0.961917i \(0.411870\pi\)
\(648\) −6.67546 −0.262237
\(649\) 1.68198 0.0660234
\(650\) −1.92957 −0.0756840
\(651\) 0.858137 0.0336330
\(652\) −82.2287 −3.22032
\(653\) −37.4461 −1.46538 −0.732690 0.680562i \(-0.761736\pi\)
−0.732690 + 0.680562i \(0.761736\pi\)
\(654\) −11.4836 −0.449044
\(655\) 55.0253 2.15002
\(656\) 31.8534 1.24367
\(657\) 0.0892372 0.00348148
\(658\) −3.12367 −0.121773
\(659\) −28.9304 −1.12697 −0.563485 0.826126i \(-0.690540\pi\)
−0.563485 + 0.826126i \(0.690540\pi\)
\(660\) 12.8216 0.499080
\(661\) 21.5071 0.836529 0.418264 0.908325i \(-0.362639\pi\)
0.418264 + 0.908325i \(0.362639\pi\)
\(662\) −63.3948 −2.46391
\(663\) 0.153446 0.00595935
\(664\) −42.0223 −1.63078
\(665\) −2.90507 −0.112654
\(666\) −22.3939 −0.867746
\(667\) 31.3048 1.21213
\(668\) −1.14392 −0.0442597
\(669\) 2.52511 0.0976264
\(670\) 98.2403 3.79535
\(671\) −5.57418 −0.215189
\(672\) 0.973735 0.0375626
\(673\) −28.6479 −1.10430 −0.552148 0.833746i \(-0.686192\pi\)
−0.552148 + 0.833746i \(0.686192\pi\)
\(674\) −46.6465 −1.79676
\(675\) 4.89527 0.188419
\(676\) −59.6746 −2.29518
\(677\) −5.18671 −0.199341 −0.0996707 0.995020i \(-0.531779\pi\)
−0.0996707 + 0.995020i \(0.531779\pi\)
\(678\) −30.1163 −1.15661
\(679\) 2.25720 0.0866232
\(680\) −20.9988 −0.805269
\(681\) −11.0846 −0.424763
\(682\) 14.1904 0.543380
\(683\) 3.48378 0.133303 0.0666516 0.997776i \(-0.478768\pi\)
0.0666516 + 0.997776i \(0.478768\pi\)
\(684\) 30.8455 1.17941
\(685\) −19.4631 −0.743647
\(686\) −4.94484 −0.188795
\(687\) −9.95351 −0.379750
\(688\) 14.7874 0.563765
\(689\) −0.278278 −0.0106016
\(690\) −62.6668 −2.38568
\(691\) −13.1216 −0.499169 −0.249584 0.968353i \(-0.580294\pi\)
−0.249584 + 0.968353i \(0.580294\pi\)
\(692\) −94.2102 −3.58133
\(693\) −0.122033 −0.00463566
\(694\) −12.6975 −0.481989
\(695\) −42.3153 −1.60511
\(696\) −26.9462 −1.02139
\(697\) 4.00646 0.151756
\(698\) 28.1231 1.06447
\(699\) −19.2676 −0.728767
\(700\) −3.09951 −0.117151
\(701\) −33.0949 −1.24998 −0.624988 0.780634i \(-0.714896\pi\)
−0.624988 + 0.780634i \(0.714896\pi\)
\(702\) −0.394171 −0.0148770
\(703\) 58.4736 2.20537
\(704\) 2.00850 0.0756982
\(705\) −27.7822 −1.04634
\(706\) 16.4596 0.619467
\(707\) −1.88131 −0.0707541
\(708\) 8.72687 0.327976
\(709\) 6.41526 0.240930 0.120465 0.992718i \(-0.461561\pi\)
0.120465 + 0.992718i \(0.461561\pi\)
\(710\) −63.3047 −2.37578
\(711\) 1.00000 0.0375029
\(712\) 60.6577 2.27324
\(713\) −48.3357 −1.81019
\(714\) 0.353681 0.0132362
\(715\) 0.427823 0.0159997
\(716\) 93.3657 3.48924
\(717\) 16.5390 0.617662
\(718\) 66.8298 2.49407
\(719\) 13.4629 0.502081 0.251040 0.967977i \(-0.419227\pi\)
0.251040 + 0.967977i \(0.419227\pi\)
\(720\) 25.0097 0.932056
\(721\) −0.871202 −0.0324452
\(722\) −66.7633 −2.48467
\(723\) 26.3974 0.981730
\(724\) 29.4086 1.09296
\(725\) 19.7602 0.733876
\(726\) 26.2387 0.973810
\(727\) −24.5117 −0.909090 −0.454545 0.890724i \(-0.650198\pi\)
−0.454545 + 0.890724i \(0.650198\pi\)
\(728\) 0.141033 0.00522704
\(729\) 1.00000 0.0370370
\(730\) −0.721088 −0.0266887
\(731\) 1.85994 0.0687922
\(732\) −28.9214 −1.06897
\(733\) 46.4494 1.71565 0.857824 0.513944i \(-0.171816\pi\)
0.857824 + 0.513944i \(0.171816\pi\)
\(734\) −18.0414 −0.665921
\(735\) −21.9601 −0.810010
\(736\) −54.8469 −2.02168
\(737\) −10.7756 −0.396925
\(738\) −10.2918 −0.378845
\(739\) −25.7210 −0.946163 −0.473081 0.881019i \(-0.656858\pi\)
−0.473081 + 0.881019i \(0.656858\pi\)
\(740\) 126.110 4.63588
\(741\) 1.02924 0.0378099
\(742\) −0.641410 −0.0235469
\(743\) −40.1614 −1.47338 −0.736689 0.676232i \(-0.763612\pi\)
−0.736689 + 0.676232i \(0.763612\pi\)
\(744\) 41.6058 1.52534
\(745\) 25.0929 0.919332
\(746\) 52.8125 1.93360
\(747\) 6.29504 0.230323
\(748\) 4.07594 0.149031
\(749\) 1.45619 0.0532081
\(750\) 0.846305 0.0309027
\(751\) −34.8498 −1.27169 −0.635844 0.771818i \(-0.719348\pi\)
−0.635844 + 0.771818i \(0.719348\pi\)
\(752\) −70.2179 −2.56058
\(753\) 1.40438 0.0511785
\(754\) −1.59111 −0.0579448
\(755\) 19.9516 0.726113
\(756\) −0.633165 −0.0230280
\(757\) 23.6413 0.859259 0.429629 0.903005i \(-0.358644\pi\)
0.429629 + 0.903005i \(0.358644\pi\)
\(758\) 72.8611 2.64643
\(759\) 6.87369 0.249499
\(760\) −140.849 −5.10913
\(761\) −23.9851 −0.869458 −0.434729 0.900561i \(-0.643156\pi\)
−0.434729 + 0.900561i \(0.643156\pi\)
\(762\) −24.1336 −0.874267
\(763\) −0.615507 −0.0222829
\(764\) 31.2168 1.12938
\(765\) 3.14567 0.113732
\(766\) 2.25529 0.0814870
\(767\) 0.291193 0.0105144
\(768\) −25.9132 −0.935061
\(769\) 5.09074 0.183577 0.0917884 0.995779i \(-0.470742\pi\)
0.0917884 + 0.995779i \(0.470742\pi\)
\(770\) 0.986100 0.0355366
\(771\) −11.3803 −0.409850
\(772\) 0.0947639 0.00341063
\(773\) −39.5891 −1.42392 −0.711960 0.702220i \(-0.752192\pi\)
−0.711960 + 0.702220i \(0.752192\pi\)
\(774\) −4.77778 −0.171734
\(775\) −30.5105 −1.09597
\(776\) 109.438 3.92858
\(777\) −1.20029 −0.0430600
\(778\) −47.5898 −1.70618
\(779\) 26.8732 0.962834
\(780\) 2.21974 0.0794796
\(781\) 6.94366 0.248464
\(782\) −19.9216 −0.712394
\(783\) 4.03660 0.144256
\(784\) −55.5028 −1.98224
\(785\) −15.6052 −0.556972
\(786\) −44.9342 −1.60275
\(787\) 13.0925 0.466697 0.233348 0.972393i \(-0.425032\pi\)
0.233348 + 0.972393i \(0.425032\pi\)
\(788\) 24.6461 0.877983
\(789\) −18.9971 −0.676313
\(790\) −8.08058 −0.287494
\(791\) −1.61420 −0.0573943
\(792\) −5.91665 −0.210239
\(793\) −0.965034 −0.0342693
\(794\) −20.0102 −0.710134
\(795\) −5.70475 −0.202327
\(796\) −85.5543 −3.03239
\(797\) −18.4655 −0.654082 −0.327041 0.945010i \(-0.606052\pi\)
−0.327041 + 0.945010i \(0.606052\pi\)
\(798\) 2.37231 0.0839788
\(799\) −8.83188 −0.312449
\(800\) −34.6205 −1.22402
\(801\) −9.08666 −0.321061
\(802\) −78.4895 −2.77156
\(803\) 0.0790935 0.00279115
\(804\) −55.9089 −1.97175
\(805\) −3.35887 −0.118384
\(806\) 2.45673 0.0865346
\(807\) −8.23267 −0.289804
\(808\) −91.2135 −3.20888
\(809\) −49.5903 −1.74350 −0.871751 0.489949i \(-0.837015\pi\)
−0.871751 + 0.489949i \(0.837015\pi\)
\(810\) −8.08058 −0.283922
\(811\) 17.5640 0.616756 0.308378 0.951264i \(-0.400214\pi\)
0.308378 + 0.951264i \(0.400214\pi\)
\(812\) −2.55583 −0.0896921
\(813\) −0.135088 −0.00473775
\(814\) −19.8484 −0.695684
\(815\) −56.2476 −1.97027
\(816\) 7.95050 0.278323
\(817\) 12.4755 0.436461
\(818\) −101.364 −3.54410
\(819\) −0.0211271 −0.000738240 0
\(820\) 57.9573 2.02396
\(821\) −37.4516 −1.30707 −0.653535 0.756896i \(-0.726715\pi\)
−0.653535 + 0.756896i \(0.726715\pi\)
\(822\) 15.8938 0.554359
\(823\) −0.551602 −0.0192276 −0.00961382 0.999954i \(-0.503060\pi\)
−0.00961382 + 0.999954i \(0.503060\pi\)
\(824\) −42.2393 −1.47147
\(825\) 4.33881 0.151058
\(826\) 0.671178 0.0233533
\(827\) −23.1491 −0.804971 −0.402486 0.915426i \(-0.631854\pi\)
−0.402486 + 0.915426i \(0.631854\pi\)
\(828\) 35.6639 1.23941
\(829\) 25.0048 0.868454 0.434227 0.900803i \(-0.357022\pi\)
0.434227 + 0.900803i \(0.357022\pi\)
\(830\) −50.8675 −1.76564
\(831\) −23.2923 −0.808001
\(832\) 0.347723 0.0120551
\(833\) −6.98104 −0.241879
\(834\) 34.5551 1.19655
\(835\) −0.782487 −0.0270791
\(836\) 27.3393 0.945548
\(837\) −6.23265 −0.215432
\(838\) 7.98232 0.275745
\(839\) 11.0268 0.380687 0.190343 0.981718i \(-0.439040\pi\)
0.190343 + 0.981718i \(0.439040\pi\)
\(840\) 2.89121 0.0997561
\(841\) −12.7059 −0.438134
\(842\) 28.7121 0.989484
\(843\) 25.7854 0.888095
\(844\) 56.7396 1.95306
\(845\) −40.8197 −1.40424
\(846\) 22.6872 0.780003
\(847\) 1.40636 0.0483232
\(848\) −14.4184 −0.495131
\(849\) 18.9333 0.649790
\(850\) −12.5749 −0.431316
\(851\) 67.6077 2.31756
\(852\) 36.0269 1.23426
\(853\) 39.0814 1.33812 0.669061 0.743207i \(-0.266696\pi\)
0.669061 + 0.743207i \(0.266696\pi\)
\(854\) −2.22433 −0.0761149
\(855\) 21.0995 0.721588
\(856\) 70.6019 2.41312
\(857\) −42.7467 −1.46020 −0.730099 0.683341i \(-0.760526\pi\)
−0.730099 + 0.683341i \(0.760526\pi\)
\(858\) −0.349365 −0.0119271
\(859\) 42.2640 1.44203 0.721014 0.692920i \(-0.243676\pi\)
0.721014 + 0.692920i \(0.243676\pi\)
\(860\) 26.9057 0.917478
\(861\) −0.551626 −0.0187994
\(862\) −19.9038 −0.677925
\(863\) −10.1925 −0.346958 −0.173479 0.984838i \(-0.555501\pi\)
−0.173479 + 0.984838i \(0.555501\pi\)
\(864\) −7.07224 −0.240602
\(865\) −64.4434 −2.19114
\(866\) 32.4772 1.10362
\(867\) 1.00000 0.0339618
\(868\) 3.94630 0.133946
\(869\) 0.886329 0.0300666
\(870\) −32.6180 −1.10585
\(871\) −1.86553 −0.0632112
\(872\) −29.8422 −1.01058
\(873\) −16.3940 −0.554854
\(874\) −133.623 −4.51988
\(875\) 0.0453610 0.00153348
\(876\) 0.410374 0.0138652
\(877\) −29.6015 −0.999571 −0.499786 0.866149i \(-0.666588\pi\)
−0.499786 + 0.866149i \(0.666588\pi\)
\(878\) −61.7515 −2.08401
\(879\) 7.36864 0.248538
\(880\) 22.1668 0.747243
\(881\) 18.6139 0.627117 0.313559 0.949569i \(-0.398479\pi\)
0.313559 + 0.949569i \(0.398479\pi\)
\(882\) 17.9328 0.603830
\(883\) −20.1737 −0.678899 −0.339449 0.940624i \(-0.610241\pi\)
−0.339449 + 0.940624i \(0.610241\pi\)
\(884\) 0.705650 0.0237336
\(885\) 5.96952 0.200663
\(886\) 22.1260 0.743337
\(887\) 32.5535 1.09304 0.546521 0.837446i \(-0.315952\pi\)
0.546521 + 0.837446i \(0.315952\pi\)
\(888\) −58.1946 −1.95288
\(889\) −1.29353 −0.0433837
\(890\) 73.4254 2.46123
\(891\) 0.886329 0.0296931
\(892\) 11.6122 0.388804
\(893\) −59.2396 −1.98238
\(894\) −20.4911 −0.685325
\(895\) 63.8657 2.13480
\(896\) −1.14600 −0.0382850
\(897\) 1.19001 0.0397333
\(898\) 105.746 3.52879
\(899\) −25.1587 −0.839090
\(900\) 22.5118 0.750392
\(901\) −1.81352 −0.0604172
\(902\) −9.12188 −0.303725
\(903\) −0.256084 −0.00852193
\(904\) −78.2626 −2.60298
\(905\) 20.1166 0.668699
\(906\) −16.2927 −0.541288
\(907\) −48.4169 −1.60766 −0.803829 0.594860i \(-0.797207\pi\)
−0.803829 + 0.594860i \(0.797207\pi\)
\(908\) −50.9745 −1.69165
\(909\) 13.6640 0.453206
\(910\) 0.170719 0.00565928
\(911\) 25.5200 0.845517 0.422758 0.906242i \(-0.361062\pi\)
0.422758 + 0.906242i \(0.361062\pi\)
\(912\) 53.3278 1.76586
\(913\) 5.57947 0.184654
\(914\) 101.491 3.35703
\(915\) −19.7834 −0.654018
\(916\) −45.7730 −1.51238
\(917\) −2.40842 −0.0795331
\(918\) −2.56879 −0.0847827
\(919\) 45.6377 1.50545 0.752724 0.658337i \(-0.228740\pi\)
0.752724 + 0.658337i \(0.228740\pi\)
\(920\) −162.851 −5.36904
\(921\) −24.2984 −0.800659
\(922\) −74.0332 −2.43815
\(923\) 1.20213 0.0395684
\(924\) −0.561192 −0.0184619
\(925\) 42.6754 1.40316
\(926\) −55.6880 −1.83002
\(927\) 6.32754 0.207824
\(928\) −28.5478 −0.937127
\(929\) 28.5673 0.937262 0.468631 0.883394i \(-0.344747\pi\)
0.468631 + 0.883394i \(0.344747\pi\)
\(930\) 50.3634 1.65148
\(931\) −46.8252 −1.53463
\(932\) −88.6055 −2.90237
\(933\) −6.81598 −0.223145
\(934\) 18.6075 0.608857
\(935\) 2.78810 0.0911806
\(936\) −1.02432 −0.0334811
\(937\) −28.4968 −0.930951 −0.465476 0.885061i \(-0.654117\pi\)
−0.465476 + 0.885061i \(0.654117\pi\)
\(938\) −4.29991 −0.140397
\(939\) 22.8625 0.746088
\(940\) −127.762 −4.16712
\(941\) −17.4581 −0.569119 −0.284559 0.958658i \(-0.591847\pi\)
−0.284559 + 0.958658i \(0.591847\pi\)
\(942\) 12.7433 0.415200
\(943\) 31.0711 1.01181
\(944\) 15.0876 0.491059
\(945\) −0.433109 −0.0140890
\(946\) −4.23469 −0.137682
\(947\) −45.4816 −1.47795 −0.738976 0.673731i \(-0.764690\pi\)
−0.738976 + 0.673731i \(0.764690\pi\)
\(948\) 4.59868 0.149358
\(949\) 0.0136931 0.000444497 0
\(950\) −84.3459 −2.73654
\(951\) 10.9591 0.355373
\(952\) 0.919105 0.0297884
\(953\) 40.1735 1.30135 0.650674 0.759357i \(-0.274486\pi\)
0.650674 + 0.759357i \(0.274486\pi\)
\(954\) 4.65856 0.150826
\(955\) 21.3535 0.690982
\(956\) 76.0577 2.45988
\(957\) 3.57775 0.115652
\(958\) 34.3923 1.11116
\(959\) 0.851888 0.0275089
\(960\) 7.12838 0.230067
\(961\) 7.84593 0.253094
\(962\) −3.43626 −0.110789
\(963\) −10.5763 −0.340817
\(964\) 121.393 3.90981
\(965\) 0.0648221 0.00208670
\(966\) 2.74288 0.0882509
\(967\) 31.9226 1.02656 0.513281 0.858221i \(-0.328430\pi\)
0.513281 + 0.858221i \(0.328430\pi\)
\(968\) 68.1860 2.19158
\(969\) 6.70747 0.215475
\(970\) 132.473 4.25346
\(971\) −14.9843 −0.480869 −0.240434 0.970665i \(-0.577290\pi\)
−0.240434 + 0.970665i \(0.577290\pi\)
\(972\) 4.59868 0.147503
\(973\) 1.85211 0.0593760
\(974\) −22.1232 −0.708874
\(975\) 0.751160 0.0240564
\(976\) −50.0013 −1.60050
\(977\) −8.88467 −0.284246 −0.142123 0.989849i \(-0.545393\pi\)
−0.142123 + 0.989849i \(0.545393\pi\)
\(978\) 45.9324 1.46876
\(979\) −8.05377 −0.257399
\(980\) −100.987 −3.22592
\(981\) 4.47043 0.142730
\(982\) −68.4803 −2.18529
\(983\) 33.4795 1.06783 0.533915 0.845538i \(-0.320720\pi\)
0.533915 + 0.845538i \(0.320720\pi\)
\(984\) −26.7450 −0.852600
\(985\) 16.8589 0.537170
\(986\) −10.3692 −0.330222
\(987\) 1.21601 0.0387060
\(988\) 4.73313 0.150581
\(989\) 14.4242 0.458664
\(990\) −7.16204 −0.227625
\(991\) −52.5256 −1.66853 −0.834265 0.551364i \(-0.814108\pi\)
−0.834265 + 0.551364i \(0.814108\pi\)
\(992\) 44.0788 1.39950
\(993\) 24.6789 0.783160
\(994\) 2.77081 0.0878847
\(995\) −58.5224 −1.85529
\(996\) 28.9489 0.917280
\(997\) 10.5707 0.334776 0.167388 0.985891i \(-0.446467\pi\)
0.167388 + 0.985891i \(0.446467\pi\)
\(998\) 16.1776 0.512094
\(999\) 8.71769 0.275815
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.1.2 31
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4029.2.a.k.1.2 31 1.1 even 1 trivial