Properties

Label 43.4.a.b.1.3
Level 43
Weight 4
Character 43.1
Self dual yes
Analytic conductor 2.537
Analytic rank 0
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.53708213025\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.299707\) of \(x^{6} - 32 x^{4} - 16 x^{3} + 251 x^{2} + 276 x + 60\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.29971 q^{2} +1.43046 q^{3} -6.31076 q^{4} +20.4116 q^{5} +1.85918 q^{6} +29.9522 q^{7} -18.5998 q^{8} -24.9538 q^{9} +O(q^{10})\) \(q+1.29971 q^{2} +1.43046 q^{3} -6.31076 q^{4} +20.4116 q^{5} +1.85918 q^{6} +29.9522 q^{7} -18.5998 q^{8} -24.9538 q^{9} +26.5291 q^{10} -22.8719 q^{11} -9.02731 q^{12} -44.4397 q^{13} +38.9292 q^{14} +29.1981 q^{15} +26.3118 q^{16} -13.0970 q^{17} -32.4326 q^{18} +5.41527 q^{19} -128.813 q^{20} +42.8456 q^{21} -29.7268 q^{22} -175.226 q^{23} -26.6063 q^{24} +291.634 q^{25} -57.7586 q^{26} -74.3180 q^{27} -189.021 q^{28} +165.972 q^{29} +37.9489 q^{30} -155.581 q^{31} +182.996 q^{32} -32.7174 q^{33} -17.0222 q^{34} +611.374 q^{35} +157.477 q^{36} -95.3991 q^{37} +7.03827 q^{38} -63.5693 q^{39} -379.652 q^{40} +189.928 q^{41} +55.6867 q^{42} -43.0000 q^{43} +144.339 q^{44} -509.347 q^{45} -227.743 q^{46} -37.2349 q^{47} +37.6381 q^{48} +554.137 q^{49} +379.039 q^{50} -18.7348 q^{51} +280.448 q^{52} +559.862 q^{53} -96.5916 q^{54} -466.852 q^{55} -557.106 q^{56} +7.74635 q^{57} +215.715 q^{58} -82.3042 q^{59} -184.262 q^{60} -640.304 q^{61} -202.210 q^{62} -747.422 q^{63} +27.3469 q^{64} -907.085 q^{65} -42.5231 q^{66} -509.592 q^{67} +82.6519 q^{68} -250.655 q^{69} +794.607 q^{70} +792.932 q^{71} +464.135 q^{72} +612.727 q^{73} -123.991 q^{74} +417.171 q^{75} -34.1745 q^{76} -685.065 q^{77} -82.6215 q^{78} +237.047 q^{79} +537.066 q^{80} +567.443 q^{81} +246.850 q^{82} +418.683 q^{83} -270.388 q^{84} -267.330 q^{85} -55.8874 q^{86} +237.417 q^{87} +425.413 q^{88} +113.788 q^{89} -662.002 q^{90} -1331.07 q^{91} +1105.81 q^{92} -222.553 q^{93} -48.3944 q^{94} +110.534 q^{95} +261.769 q^{96} +1649.00 q^{97} +720.216 q^{98} +570.740 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 7q^{3} + 22q^{4} + 43q^{5} - 3q^{6} + 8q^{7} + 54q^{8} + 81q^{9} + O(q^{10}) \) \( 6q + 6q^{2} + 7q^{3} + 22q^{4} + 43q^{5} - 3q^{6} + 8q^{7} + 54q^{8} + 81q^{9} + 57q^{10} - 28q^{11} - 157q^{12} + 56q^{13} - 184q^{14} - 124q^{15} - 54q^{16} + 19q^{17} - 81q^{18} - 75q^{19} + 135q^{20} - 18q^{21} - 504q^{22} + 131q^{23} - 567q^{24} + 105q^{25} + 44q^{26} + 238q^{27} - 404q^{28} + 515q^{29} - 396q^{30} + 237q^{31} + 558q^{32} + 540q^{33} - 107q^{34} + 198q^{35} + 73q^{36} + 269q^{37} + 527q^{38} + 290q^{39} + 613q^{40} + 471q^{41} + 362q^{42} - 258q^{43} - 428q^{44} + 334q^{45} - 67q^{46} + 415q^{47} - 989q^{48} + 350q^{49} + 1335q^{50} - 1241q^{51} - 8q^{52} + 450q^{53} + 402q^{54} - 1732q^{55} - 780q^{56} - 1000q^{57} - 1055q^{58} + 356q^{59} - 2732q^{60} - 1328q^{61} + 1603q^{62} - 2290q^{63} + 466q^{64} - 62q^{65} + 156q^{66} - 632q^{67} + 571q^{68} - 1130q^{69} - 1902q^{70} - 144q^{71} + 567q^{72} + 864q^{73} + 1207q^{74} - 2494q^{75} + 1005q^{76} + 2660q^{77} + 2222q^{78} - 1613q^{79} + 2399q^{80} - 102q^{81} + 1673q^{82} - 682q^{83} + 3758q^{84} + 84q^{85} - 258q^{86} + 449q^{87} - 608q^{88} + 3378q^{89} + 930q^{90} - 3900q^{91} + 3491q^{92} + 1879q^{93} + 3197q^{94} - 79q^{95} - 591q^{96} - 55q^{97} + 2398q^{98} - 1612q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.29971 0.459516 0.229758 0.973248i \(-0.426207\pi\)
0.229758 + 0.973248i \(0.426207\pi\)
\(3\) 1.43046 0.275293 0.137646 0.990481i \(-0.456046\pi\)
0.137646 + 0.990481i \(0.456046\pi\)
\(4\) −6.31076 −0.788845
\(5\) 20.4116 1.82567 0.912835 0.408329i \(-0.133888\pi\)
0.912835 + 0.408329i \(0.133888\pi\)
\(6\) 1.85918 0.126501
\(7\) 29.9522 1.61727 0.808635 0.588311i \(-0.200207\pi\)
0.808635 + 0.588311i \(0.200207\pi\)
\(8\) −18.5998 −0.822003
\(9\) −24.9538 −0.924214
\(10\) 26.5291 0.838924
\(11\) −22.8719 −0.626922 −0.313461 0.949601i \(-0.601488\pi\)
−0.313461 + 0.949601i \(0.601488\pi\)
\(12\) −9.02731 −0.217163
\(13\) −44.4397 −0.948104 −0.474052 0.880497i \(-0.657209\pi\)
−0.474052 + 0.880497i \(0.657209\pi\)
\(14\) 38.9292 0.743161
\(15\) 29.1981 0.502594
\(16\) 26.3118 0.411122
\(17\) −13.0970 −0.186852 −0.0934260 0.995626i \(-0.529782\pi\)
−0.0934260 + 0.995626i \(0.529782\pi\)
\(18\) −32.4326 −0.424691
\(19\) 5.41527 0.0653868 0.0326934 0.999465i \(-0.489592\pi\)
0.0326934 + 0.999465i \(0.489592\pi\)
\(20\) −128.813 −1.44017
\(21\) 42.8456 0.445223
\(22\) −29.7268 −0.288081
\(23\) −175.226 −1.58857 −0.794287 0.607542i \(-0.792155\pi\)
−0.794287 + 0.607542i \(0.792155\pi\)
\(24\) −26.6063 −0.226291
\(25\) 291.634 2.33307
\(26\) −57.7586 −0.435669
\(27\) −74.3180 −0.529722
\(28\) −189.021 −1.27578
\(29\) 165.972 1.06277 0.531383 0.847132i \(-0.321672\pi\)
0.531383 + 0.847132i \(0.321672\pi\)
\(30\) 37.9489 0.230950
\(31\) −155.581 −0.901395 −0.450697 0.892677i \(-0.648825\pi\)
−0.450697 + 0.892677i \(0.648825\pi\)
\(32\) 182.996 1.01092
\(33\) −32.7174 −0.172587
\(34\) −17.0222 −0.0858615
\(35\) 611.374 2.95260
\(36\) 157.477 0.729062
\(37\) −95.3991 −0.423879 −0.211939 0.977283i \(-0.567978\pi\)
−0.211939 + 0.977283i \(0.567978\pi\)
\(38\) 7.03827 0.0300463
\(39\) −63.5693 −0.261006
\(40\) −379.652 −1.50071
\(41\) 189.928 0.723457 0.361728 0.932284i \(-0.382187\pi\)
0.361728 + 0.932284i \(0.382187\pi\)
\(42\) 55.6867 0.204587
\(43\) −43.0000 −0.152499
\(44\) 144.339 0.494544
\(45\) −509.347 −1.68731
\(46\) −227.743 −0.729975
\(47\) −37.2349 −0.115559 −0.0577794 0.998329i \(-0.518402\pi\)
−0.0577794 + 0.998329i \(0.518402\pi\)
\(48\) 37.6381 0.113179
\(49\) 554.137 1.61556
\(50\) 379.039 1.07208
\(51\) −18.7348 −0.0514390
\(52\) 280.448 0.747907
\(53\) 559.862 1.45100 0.725500 0.688223i \(-0.241609\pi\)
0.725500 + 0.688223i \(0.241609\pi\)
\(54\) −96.5916 −0.243416
\(55\) −466.852 −1.14455
\(56\) −557.106 −1.32940
\(57\) 7.74635 0.0180005
\(58\) 215.715 0.488358
\(59\) −82.3042 −0.181612 −0.0908059 0.995869i \(-0.528944\pi\)
−0.0908059 + 0.995869i \(0.528944\pi\)
\(60\) −184.262 −0.396469
\(61\) −640.304 −1.34398 −0.671988 0.740562i \(-0.734559\pi\)
−0.671988 + 0.740562i \(0.734559\pi\)
\(62\) −202.210 −0.414205
\(63\) −747.422 −1.49470
\(64\) 27.3469 0.0534120
\(65\) −907.085 −1.73092
\(66\) −42.5231 −0.0793065
\(67\) −509.592 −0.929203 −0.464602 0.885520i \(-0.653802\pi\)
−0.464602 + 0.885520i \(0.653802\pi\)
\(68\) 82.6519 0.147397
\(69\) −250.655 −0.437323
\(70\) 794.607 1.35677
\(71\) 792.932 1.32540 0.662702 0.748883i \(-0.269410\pi\)
0.662702 + 0.748883i \(0.269410\pi\)
\(72\) 464.135 0.759706
\(73\) 612.727 0.982387 0.491194 0.871050i \(-0.336561\pi\)
0.491194 + 0.871050i \(0.336561\pi\)
\(74\) −123.991 −0.194779
\(75\) 417.171 0.642277
\(76\) −34.1745 −0.0515800
\(77\) −685.065 −1.01390
\(78\) −82.6215 −0.119936
\(79\) 237.047 0.337594 0.168797 0.985651i \(-0.446012\pi\)
0.168797 + 0.985651i \(0.446012\pi\)
\(80\) 537.066 0.750573
\(81\) 567.443 0.778385
\(82\) 246.850 0.332440
\(83\) 418.683 0.553691 0.276846 0.960914i \(-0.410711\pi\)
0.276846 + 0.960914i \(0.410711\pi\)
\(84\) −270.388 −0.351212
\(85\) −267.330 −0.341130
\(86\) −55.8874 −0.0700755
\(87\) 237.417 0.292572
\(88\) 425.413 0.515331
\(89\) 113.788 0.135523 0.0677615 0.997702i \(-0.478414\pi\)
0.0677615 + 0.997702i \(0.478414\pi\)
\(90\) −662.002 −0.775346
\(91\) −1331.07 −1.53334
\(92\) 1105.81 1.25314
\(93\) −222.553 −0.248147
\(94\) −48.3944 −0.0531011
\(95\) 110.534 0.119375
\(96\) 261.769 0.278299
\(97\) 1649.00 1.72609 0.863045 0.505127i \(-0.168554\pi\)
0.863045 + 0.505127i \(0.168554\pi\)
\(98\) 720.216 0.742376
\(99\) 570.740 0.579410
\(100\) −1840.43 −1.84043
\(101\) −165.065 −0.162619 −0.0813097 0.996689i \(-0.525910\pi\)
−0.0813097 + 0.996689i \(0.525910\pi\)
\(102\) −24.3497 −0.0236370
\(103\) −1665.32 −1.59309 −0.796547 0.604576i \(-0.793342\pi\)
−0.796547 + 0.604576i \(0.793342\pi\)
\(104\) 826.569 0.779344
\(105\) 874.547 0.812829
\(106\) 727.657 0.666757
\(107\) −631.683 −0.570721 −0.285360 0.958420i \(-0.592113\pi\)
−0.285360 + 0.958420i \(0.592113\pi\)
\(108\) 469.003 0.417869
\(109\) 839.368 0.737586 0.368793 0.929512i \(-0.379771\pi\)
0.368793 + 0.929512i \(0.379771\pi\)
\(110\) −606.771 −0.525940
\(111\) −136.465 −0.116691
\(112\) 788.097 0.664895
\(113\) −1080.84 −0.899796 −0.449898 0.893080i \(-0.648540\pi\)
−0.449898 + 0.893080i \(0.648540\pi\)
\(114\) 10.0680 0.00827152
\(115\) −3576.65 −2.90021
\(116\) −1047.41 −0.838358
\(117\) 1108.94 0.876251
\(118\) −106.971 −0.0834535
\(119\) −392.284 −0.302190
\(120\) −543.078 −0.413133
\(121\) −807.876 −0.606969
\(122\) −832.207 −0.617578
\(123\) 271.685 0.199162
\(124\) 981.837 0.711061
\(125\) 3401.26 2.43375
\(126\) −971.429 −0.686840
\(127\) 594.837 0.415616 0.207808 0.978170i \(-0.433367\pi\)
0.207808 + 0.978170i \(0.433367\pi\)
\(128\) −1428.43 −0.986376
\(129\) −61.5099 −0.0419818
\(130\) −1178.95 −0.795387
\(131\) 1987.90 1.32583 0.662916 0.748694i \(-0.269319\pi\)
0.662916 + 0.748694i \(0.269319\pi\)
\(132\) 206.472 0.136144
\(133\) 162.200 0.105748
\(134\) −662.321 −0.426984
\(135\) −1516.95 −0.967098
\(136\) 243.601 0.153593
\(137\) 1717.01 1.07076 0.535379 0.844612i \(-0.320169\pi\)
0.535379 + 0.844612i \(0.320169\pi\)
\(138\) −325.778 −0.200957
\(139\) −1735.09 −1.05876 −0.529381 0.848384i \(-0.677576\pi\)
−0.529381 + 0.848384i \(0.677576\pi\)
\(140\) −3858.23 −2.32914
\(141\) −53.2631 −0.0318125
\(142\) 1030.58 0.609044
\(143\) 1016.42 0.594387
\(144\) −656.579 −0.379964
\(145\) 3387.76 1.94026
\(146\) 796.366 0.451423
\(147\) 792.673 0.444752
\(148\) 602.041 0.334375
\(149\) 456.600 0.251048 0.125524 0.992091i \(-0.459939\pi\)
0.125524 + 0.992091i \(0.459939\pi\)
\(150\) 542.201 0.295137
\(151\) −483.618 −0.260638 −0.130319 0.991472i \(-0.541600\pi\)
−0.130319 + 0.991472i \(0.541600\pi\)
\(152\) −100.723 −0.0537481
\(153\) 326.819 0.172691
\(154\) −890.384 −0.465904
\(155\) −3175.67 −1.64565
\(156\) 401.171 0.205893
\(157\) 2300.64 1.16950 0.584749 0.811214i \(-0.301193\pi\)
0.584749 + 0.811214i \(0.301193\pi\)
\(158\) 308.092 0.155130
\(159\) 800.862 0.399450
\(160\) 3735.24 1.84561
\(161\) −5248.42 −2.56915
\(162\) 737.510 0.357680
\(163\) 314.362 0.151060 0.0755299 0.997144i \(-0.475935\pi\)
0.0755299 + 0.997144i \(0.475935\pi\)
\(164\) −1198.59 −0.570695
\(165\) −667.815 −0.315087
\(166\) 544.165 0.254430
\(167\) −282.441 −0.130874 −0.0654369 0.997857i \(-0.520844\pi\)
−0.0654369 + 0.997857i \(0.520844\pi\)
\(168\) −796.919 −0.365974
\(169\) −222.114 −0.101099
\(170\) −347.451 −0.156755
\(171\) −135.131 −0.0604314
\(172\) 271.363 0.120298
\(173\) 1774.10 0.779667 0.389834 0.920885i \(-0.372532\pi\)
0.389834 + 0.920885i \(0.372532\pi\)
\(174\) 308.572 0.134441
\(175\) 8735.09 3.77320
\(176\) −601.801 −0.257741
\(177\) −117.733 −0.0499964
\(178\) 147.892 0.0622750
\(179\) 890.113 0.371677 0.185839 0.982580i \(-0.440500\pi\)
0.185839 + 0.982580i \(0.440500\pi\)
\(180\) 3214.37 1.33103
\(181\) −2883.82 −1.18427 −0.592135 0.805838i \(-0.701715\pi\)
−0.592135 + 0.805838i \(0.701715\pi\)
\(182\) −1730.00 −0.704594
\(183\) −915.931 −0.369987
\(184\) 3259.17 1.30581
\(185\) −1947.25 −0.773863
\(186\) −289.254 −0.114028
\(187\) 299.553 0.117142
\(188\) 234.980 0.0911580
\(189\) −2225.99 −0.856704
\(190\) 143.662 0.0548546
\(191\) 1120.48 0.424478 0.212239 0.977218i \(-0.431924\pi\)
0.212239 + 0.977218i \(0.431924\pi\)
\(192\) 39.1188 0.0147039
\(193\) −2777.72 −1.03598 −0.517992 0.855385i \(-0.673320\pi\)
−0.517992 + 0.855385i \(0.673320\pi\)
\(194\) 2143.22 0.793166
\(195\) −1297.55 −0.476511
\(196\) −3497.03 −1.27443
\(197\) −1266.30 −0.457971 −0.228985 0.973430i \(-0.573541\pi\)
−0.228985 + 0.973430i \(0.573541\pi\)
\(198\) 741.795 0.266248
\(199\) −5113.40 −1.82150 −0.910752 0.412955i \(-0.864497\pi\)
−0.910752 + 0.412955i \(0.864497\pi\)
\(200\) −5424.33 −1.91779
\(201\) −728.953 −0.255803
\(202\) −214.536 −0.0747262
\(203\) 4971.23 1.71878
\(204\) 118.231 0.0405774
\(205\) 3876.73 1.32079
\(206\) −2164.43 −0.732052
\(207\) 4372.56 1.46818
\(208\) −1169.29 −0.389786
\(209\) −123.858 −0.0409924
\(210\) 1136.66 0.373508
\(211\) −1318.35 −0.430138 −0.215069 0.976599i \(-0.568998\pi\)
−0.215069 + 0.976599i \(0.568998\pi\)
\(212\) −3533.15 −1.14461
\(213\) 1134.26 0.364874
\(214\) −821.003 −0.262255
\(215\) −877.699 −0.278412
\(216\) 1382.30 0.435433
\(217\) −4660.01 −1.45780
\(218\) 1090.93 0.338932
\(219\) 876.483 0.270444
\(220\) 2946.19 0.902874
\(221\) 582.026 0.177155
\(222\) −177.365 −0.0536213
\(223\) −812.504 −0.243988 −0.121994 0.992531i \(-0.538929\pi\)
−0.121994 + 0.992531i \(0.538929\pi\)
\(224\) 5481.14 1.63493
\(225\) −7277.36 −2.15626
\(226\) −1404.78 −0.413470
\(227\) 1890.51 0.552763 0.276382 0.961048i \(-0.410865\pi\)
0.276382 + 0.961048i \(0.410865\pi\)
\(228\) −48.8853 −0.0141996
\(229\) 1368.43 0.394884 0.197442 0.980315i \(-0.436737\pi\)
0.197442 + 0.980315i \(0.436737\pi\)
\(230\) −4648.60 −1.33269
\(231\) −979.960 −0.279120
\(232\) −3087.05 −0.873597
\(233\) −3535.33 −0.994023 −0.497012 0.867744i \(-0.665569\pi\)
−0.497012 + 0.867744i \(0.665569\pi\)
\(234\) 1441.29 0.402651
\(235\) −760.023 −0.210972
\(236\) 519.402 0.143264
\(237\) 339.088 0.0929371
\(238\) −509.854 −0.138861
\(239\) 1515.24 0.410095 0.205047 0.978752i \(-0.434265\pi\)
0.205047 + 0.978752i \(0.434265\pi\)
\(240\) 768.253 0.206627
\(241\) −6717.46 −1.79548 −0.897738 0.440530i \(-0.854791\pi\)
−0.897738 + 0.440530i \(0.854791\pi\)
\(242\) −1050.00 −0.278912
\(243\) 2818.29 0.744006
\(244\) 4040.80 1.06019
\(245\) 11310.8 2.94948
\(246\) 353.110 0.0915183
\(247\) −240.653 −0.0619934
\(248\) 2893.78 0.740949
\(249\) 598.910 0.152427
\(250\) 4420.65 1.11835
\(251\) −2291.38 −0.576217 −0.288109 0.957598i \(-0.593026\pi\)
−0.288109 + 0.957598i \(0.593026\pi\)
\(252\) 4716.80 1.17909
\(253\) 4007.76 0.995912
\(254\) 773.114 0.190982
\(255\) −382.406 −0.0939107
\(256\) −2075.31 −0.506668
\(257\) −2236.18 −0.542760 −0.271380 0.962472i \(-0.587480\pi\)
−0.271380 + 0.962472i \(0.587480\pi\)
\(258\) −79.9449 −0.0192913
\(259\) −2857.42 −0.685526
\(260\) 5724.40 1.36543
\(261\) −4141.63 −0.982224
\(262\) 2583.69 0.609241
\(263\) −6393.87 −1.49910 −0.749550 0.661948i \(-0.769730\pi\)
−0.749550 + 0.661948i \(0.769730\pi\)
\(264\) 608.538 0.141867
\(265\) 11427.7 2.64905
\(266\) 210.812 0.0485929
\(267\) 162.770 0.0373085
\(268\) 3215.92 0.732997
\(269\) −2688.95 −0.609473 −0.304736 0.952437i \(-0.598568\pi\)
−0.304736 + 0.952437i \(0.598568\pi\)
\(270\) −1971.59 −0.444397
\(271\) −1057.95 −0.237143 −0.118572 0.992945i \(-0.537832\pi\)
−0.118572 + 0.992945i \(0.537832\pi\)
\(272\) −344.605 −0.0768189
\(273\) −1904.04 −0.422117
\(274\) 2231.61 0.492030
\(275\) −6670.22 −1.46265
\(276\) 1581.82 0.344980
\(277\) 1904.17 0.413035 0.206518 0.978443i \(-0.433787\pi\)
0.206518 + 0.978443i \(0.433787\pi\)
\(278\) −2255.10 −0.486518
\(279\) 3882.34 0.833081
\(280\) −11371.4 −2.42705
\(281\) 6520.37 1.38424 0.692122 0.721781i \(-0.256676\pi\)
0.692122 + 0.721781i \(0.256676\pi\)
\(282\) −69.2264 −0.0146183
\(283\) 130.376 0.0273854 0.0136927 0.999906i \(-0.495641\pi\)
0.0136927 + 0.999906i \(0.495641\pi\)
\(284\) −5004.00 −1.04554
\(285\) 158.115 0.0328630
\(286\) 1321.05 0.273130
\(287\) 5688.76 1.17002
\(288\) −4566.44 −0.934306
\(289\) −4741.47 −0.965086
\(290\) 4403.09 0.891581
\(291\) 2358.84 0.475180
\(292\) −3866.77 −0.774951
\(293\) −321.150 −0.0640333 −0.0320166 0.999487i \(-0.510193\pi\)
−0.0320166 + 0.999487i \(0.510193\pi\)
\(294\) 1030.24 0.204371
\(295\) −1679.96 −0.331563
\(296\) 1774.40 0.348430
\(297\) 1699.79 0.332094
\(298\) 593.446 0.115361
\(299\) 7787.00 1.50613
\(300\) −2632.67 −0.506657
\(301\) −1287.95 −0.246631
\(302\) −628.562 −0.119767
\(303\) −236.119 −0.0447679
\(304\) 142.486 0.0268819
\(305\) −13069.6 −2.45366
\(306\) 424.769 0.0793544
\(307\) 3944.87 0.733374 0.366687 0.930344i \(-0.380492\pi\)
0.366687 + 0.930344i \(0.380492\pi\)
\(308\) 4323.28 0.799811
\(309\) −2382.18 −0.438567
\(310\) −4127.44 −0.756202
\(311\) 7046.04 1.28471 0.642354 0.766408i \(-0.277958\pi\)
0.642354 + 0.766408i \(0.277958\pi\)
\(312\) 1182.38 0.214548
\(313\) −230.283 −0.0415858 −0.0207929 0.999784i \(-0.506619\pi\)
−0.0207929 + 0.999784i \(0.506619\pi\)
\(314\) 2990.16 0.537403
\(315\) −15256.1 −2.72883
\(316\) −1495.95 −0.266309
\(317\) 6574.02 1.16477 0.582387 0.812911i \(-0.302119\pi\)
0.582387 + 0.812911i \(0.302119\pi\)
\(318\) 1040.89 0.183553
\(319\) −3796.10 −0.666271
\(320\) 558.195 0.0975126
\(321\) −903.600 −0.157115
\(322\) −6821.41 −1.18057
\(323\) −70.9237 −0.0122177
\(324\) −3581.00 −0.614025
\(325\) −12960.1 −2.21199
\(326\) 408.579 0.0694144
\(327\) 1200.68 0.203052
\(328\) −3532.62 −0.594683
\(329\) −1115.27 −0.186890
\(330\) −867.964 −0.144787
\(331\) 8326.42 1.38266 0.691331 0.722538i \(-0.257025\pi\)
0.691331 + 0.722538i \(0.257025\pi\)
\(332\) −2642.21 −0.436777
\(333\) 2380.57 0.391755
\(334\) −367.091 −0.0601386
\(335\) −10401.6 −1.69642
\(336\) 1127.34 0.183041
\(337\) 9021.52 1.45826 0.729130 0.684375i \(-0.239925\pi\)
0.729130 + 0.684375i \(0.239925\pi\)
\(338\) −288.684 −0.0464566
\(339\) −1546.10 −0.247707
\(340\) 1687.06 0.269099
\(341\) 3558.44 0.565104
\(342\) −175.631 −0.0277692
\(343\) 6324.03 0.995526
\(344\) 799.791 0.125354
\(345\) −5116.27 −0.798408
\(346\) 2305.81 0.358270
\(347\) 133.107 0.0205925 0.0102962 0.999947i \(-0.496723\pi\)
0.0102962 + 0.999947i \(0.496723\pi\)
\(348\) −1498.28 −0.230794
\(349\) −1322.67 −0.202868 −0.101434 0.994842i \(-0.532343\pi\)
−0.101434 + 0.994842i \(0.532343\pi\)
\(350\) 11353.1 1.73385
\(351\) 3302.67 0.502232
\(352\) −4185.47 −0.633768
\(353\) 12515.7 1.88709 0.943543 0.331251i \(-0.107471\pi\)
0.943543 + 0.331251i \(0.107471\pi\)
\(354\) −153.019 −0.0229741
\(355\) 16185.0 2.41975
\(356\) −718.091 −0.106907
\(357\) −561.148 −0.0831907
\(358\) 1156.89 0.170792
\(359\) −12654.2 −1.86035 −0.930174 0.367118i \(-0.880344\pi\)
−0.930174 + 0.367118i \(0.880344\pi\)
\(360\) 9473.75 1.38697
\(361\) −6829.67 −0.995725
\(362\) −3748.13 −0.544191
\(363\) −1155.64 −0.167094
\(364\) 8400.05 1.20957
\(365\) 12506.7 1.79351
\(366\) −1190.44 −0.170015
\(367\) −13307.8 −1.89281 −0.946407 0.322976i \(-0.895317\pi\)
−0.946407 + 0.322976i \(0.895317\pi\)
\(368\) −4610.52 −0.653098
\(369\) −4739.41 −0.668629
\(370\) −2530.86 −0.355602
\(371\) 16769.1 2.34666
\(372\) 1404.48 0.195750
\(373\) −6552.61 −0.909602 −0.454801 0.890593i \(-0.650290\pi\)
−0.454801 + 0.890593i \(0.650290\pi\)
\(374\) 389.331 0.0538284
\(375\) 4865.38 0.669993
\(376\) 692.561 0.0949896
\(377\) −7375.74 −1.00761
\(378\) −2893.14 −0.393669
\(379\) 11236.0 1.52284 0.761418 0.648262i \(-0.224504\pi\)
0.761418 + 0.648262i \(0.224504\pi\)
\(380\) −697.556 −0.0941681
\(381\) 850.893 0.114416
\(382\) 1456.30 0.195054
\(383\) 6353.36 0.847628 0.423814 0.905749i \(-0.360691\pi\)
0.423814 + 0.905749i \(0.360691\pi\)
\(384\) −2043.31 −0.271542
\(385\) −13983.3 −1.85105
\(386\) −3610.23 −0.476051
\(387\) 1073.01 0.140941
\(388\) −10406.5 −1.36162
\(389\) 3788.91 0.493844 0.246922 0.969035i \(-0.420581\pi\)
0.246922 + 0.969035i \(0.420581\pi\)
\(390\) −1686.44 −0.218964
\(391\) 2294.94 0.296828
\(392\) −10306.8 −1.32799
\(393\) 2843.62 0.364992
\(394\) −1645.82 −0.210445
\(395\) 4838.52 0.616335
\(396\) −3601.81 −0.457065
\(397\) −2601.66 −0.328901 −0.164451 0.986385i \(-0.552585\pi\)
−0.164451 + 0.986385i \(0.552585\pi\)
\(398\) −6645.92 −0.837010
\(399\) 232.021 0.0291117
\(400\) 7673.41 0.959176
\(401\) −1698.36 −0.211501 −0.105750 0.994393i \(-0.533724\pi\)
−0.105750 + 0.994393i \(0.533724\pi\)
\(402\) −947.426 −0.117546
\(403\) 6913.99 0.854616
\(404\) 1041.68 0.128281
\(405\) 11582.4 1.42107
\(406\) 6461.15 0.789807
\(407\) 2181.96 0.265739
\(408\) 348.463 0.0422830
\(409\) −5511.51 −0.666324 −0.333162 0.942870i \(-0.608116\pi\)
−0.333162 + 0.942870i \(0.608116\pi\)
\(410\) 5038.61 0.606925
\(411\) 2456.12 0.294772
\(412\) 10509.4 1.25670
\(413\) −2465.20 −0.293715
\(414\) 5683.05 0.674653
\(415\) 8545.99 1.01086
\(416\) −8132.29 −0.958457
\(417\) −2481.98 −0.291470
\(418\) −160.979 −0.0188367
\(419\) −6137.26 −0.715573 −0.357786 0.933803i \(-0.616468\pi\)
−0.357786 + 0.933803i \(0.616468\pi\)
\(420\) −5519.06 −0.641197
\(421\) 9168.60 1.06140 0.530701 0.847559i \(-0.321929\pi\)
0.530701 + 0.847559i \(0.321929\pi\)
\(422\) −1713.47 −0.197655
\(423\) 929.150 0.106801
\(424\) −10413.3 −1.19273
\(425\) −3819.52 −0.435939
\(426\) 1474.21 0.167666
\(427\) −19178.5 −2.17357
\(428\) 3986.40 0.450210
\(429\) 1453.95 0.163630
\(430\) −1140.75 −0.127935
\(431\) 12535.8 1.40099 0.700494 0.713658i \(-0.252963\pi\)
0.700494 + 0.713658i \(0.252963\pi\)
\(432\) −1955.44 −0.217780
\(433\) 7163.93 0.795095 0.397548 0.917582i \(-0.369861\pi\)
0.397548 + 0.917582i \(0.369861\pi\)
\(434\) −6056.65 −0.669881
\(435\) 4846.06 0.534140
\(436\) −5297.05 −0.581841
\(437\) −948.898 −0.103872
\(438\) 1139.17 0.124273
\(439\) −2988.31 −0.324884 −0.162442 0.986718i \(-0.551937\pi\)
−0.162442 + 0.986718i \(0.551937\pi\)
\(440\) 8683.36 0.940825
\(441\) −13827.8 −1.49312
\(442\) 756.463 0.0814056
\(443\) −11257.1 −1.20732 −0.603660 0.797242i \(-0.706292\pi\)
−0.603660 + 0.797242i \(0.706292\pi\)
\(444\) 861.198 0.0920510
\(445\) 2322.60 0.247420
\(446\) −1056.02 −0.112116
\(447\) 653.150 0.0691117
\(448\) 819.102 0.0863816
\(449\) −4960.39 −0.521371 −0.260685 0.965424i \(-0.583948\pi\)
−0.260685 + 0.965424i \(0.583948\pi\)
\(450\) −9458.44 −0.990834
\(451\) −4344.01 −0.453551
\(452\) 6820.92 0.709799
\(453\) −691.798 −0.0717517
\(454\) 2457.10 0.254004
\(455\) −27169.2 −2.79937
\(456\) −144.081 −0.0147965
\(457\) −4911.00 −0.502685 −0.251343 0.967898i \(-0.580872\pi\)
−0.251343 + 0.967898i \(0.580872\pi\)
\(458\) 1778.56 0.181455
\(459\) 973.341 0.0989797
\(460\) 22571.4 2.28782
\(461\) −3089.54 −0.312135 −0.156068 0.987746i \(-0.549882\pi\)
−0.156068 + 0.987746i \(0.549882\pi\)
\(462\) −1273.66 −0.128260
\(463\) 2821.14 0.283174 0.141587 0.989926i \(-0.454779\pi\)
0.141587 + 0.989926i \(0.454779\pi\)
\(464\) 4367.02 0.436926
\(465\) −4542.67 −0.453035
\(466\) −4594.90 −0.456770
\(467\) 483.354 0.0478950 0.0239475 0.999713i \(-0.492377\pi\)
0.0239475 + 0.999713i \(0.492377\pi\)
\(468\) −6998.24 −0.691226
\(469\) −15263.4 −1.50277
\(470\) −987.808 −0.0969451
\(471\) 3290.99 0.321955
\(472\) 1530.84 0.149285
\(473\) 983.492 0.0956047
\(474\) 440.715 0.0427061
\(475\) 1579.28 0.152552
\(476\) 2475.61 0.238381
\(477\) −13970.7 −1.34103
\(478\) 1969.37 0.188445
\(479\) −5570.85 −0.531396 −0.265698 0.964056i \(-0.585602\pi\)
−0.265698 + 0.964056i \(0.585602\pi\)
\(480\) 5343.13 0.508082
\(481\) 4239.51 0.401881
\(482\) −8730.73 −0.825050
\(483\) −7507.68 −0.707269
\(484\) 5098.31 0.478805
\(485\) 33658.8 3.15127
\(486\) 3662.95 0.341883
\(487\) 453.859 0.0422306 0.0211153 0.999777i \(-0.493278\pi\)
0.0211153 + 0.999777i \(0.493278\pi\)
\(488\) 11909.5 1.10475
\(489\) 449.684 0.0415857
\(490\) 14700.8 1.35533
\(491\) 2787.17 0.256178 0.128089 0.991763i \(-0.459116\pi\)
0.128089 + 0.991763i \(0.459116\pi\)
\(492\) −1714.54 −0.157108
\(493\) −2173.73 −0.198580
\(494\) −312.778 −0.0284870
\(495\) 11649.7 1.05781
\(496\) −4093.62 −0.370583
\(497\) 23750.1 2.14354
\(498\) 778.408 0.0700428
\(499\) −325.118 −0.0291669 −0.0145834 0.999894i \(-0.504642\pi\)
−0.0145834 + 0.999894i \(0.504642\pi\)
\(500\) −21464.6 −1.91985
\(501\) −404.022 −0.0360286
\(502\) −2978.12 −0.264781
\(503\) −11441.8 −1.01424 −0.507120 0.861875i \(-0.669290\pi\)
−0.507120 + 0.861875i \(0.669290\pi\)
\(504\) 13901.9 1.22865
\(505\) −3369.24 −0.296889
\(506\) 5208.91 0.457637
\(507\) −317.727 −0.0278318
\(508\) −3753.88 −0.327857
\(509\) −8398.29 −0.731332 −0.365666 0.930746i \(-0.619159\pi\)
−0.365666 + 0.930746i \(0.619159\pi\)
\(510\) −497.016 −0.0431534
\(511\) 18352.5 1.58878
\(512\) 8730.11 0.753554
\(513\) −402.452 −0.0346368
\(514\) −2906.39 −0.249407
\(515\) −33991.8 −2.90846
\(516\) 388.174 0.0331171
\(517\) 851.632 0.0724463
\(518\) −3713.81 −0.315010
\(519\) 2537.79 0.214637
\(520\) 16871.6 1.42282
\(521\) 9762.74 0.820947 0.410474 0.911873i \(-0.365363\pi\)
0.410474 + 0.911873i \(0.365363\pi\)
\(522\) −5382.90 −0.451347
\(523\) −14406.0 −1.20445 −0.602227 0.798325i \(-0.705720\pi\)
−0.602227 + 0.798325i \(0.705720\pi\)
\(524\) −12545.2 −1.04588
\(525\) 12495.2 1.03874
\(526\) −8310.16 −0.688860
\(527\) 2037.65 0.168427
\(528\) −860.854 −0.0709543
\(529\) 18537.3 1.52357
\(530\) 14852.6 1.21728
\(531\) 2053.80 0.167848
\(532\) −1023.60 −0.0834188
\(533\) −8440.33 −0.685912
\(534\) 211.554 0.0171439
\(535\) −12893.7 −1.04195
\(536\) 9478.32 0.763808
\(537\) 1273.27 0.102320
\(538\) −3494.85 −0.280062
\(539\) −12674.2 −1.01283
\(540\) 9573.10 0.762890
\(541\) 22048.2 1.75217 0.876087 0.482153i \(-0.160145\pi\)
0.876087 + 0.482153i \(0.160145\pi\)
\(542\) −1375.02 −0.108971
\(543\) −4125.21 −0.326021
\(544\) −2396.70 −0.188892
\(545\) 17132.8 1.34659
\(546\) −2474.70 −0.193970
\(547\) 6165.00 0.481895 0.240947 0.970538i \(-0.422542\pi\)
0.240947 + 0.970538i \(0.422542\pi\)
\(548\) −10835.6 −0.844662
\(549\) 15978.0 1.24212
\(550\) −8669.33 −0.672112
\(551\) 898.784 0.0694909
\(552\) 4662.13 0.359481
\(553\) 7100.10 0.545980
\(554\) 2474.87 0.189796
\(555\) −2785.47 −0.213039
\(556\) 10949.7 0.835200
\(557\) −12519.9 −0.952400 −0.476200 0.879337i \(-0.657986\pi\)
−0.476200 + 0.879337i \(0.657986\pi\)
\(558\) 5045.91 0.382814
\(559\) 1910.91 0.144584
\(560\) 16086.3 1.21388
\(561\) 428.499 0.0322482
\(562\) 8474.57 0.636082
\(563\) 21124.7 1.58135 0.790674 0.612237i \(-0.209730\pi\)
0.790674 + 0.612237i \(0.209730\pi\)
\(564\) 336.131 0.0250951
\(565\) −22061.7 −1.64273
\(566\) 169.451 0.0125840
\(567\) 16996.2 1.25886
\(568\) −14748.4 −1.08949
\(569\) −22782.7 −1.67856 −0.839279 0.543700i \(-0.817023\pi\)
−0.839279 + 0.543700i \(0.817023\pi\)
\(570\) 205.504 0.0151011
\(571\) −12990.7 −0.952092 −0.476046 0.879420i \(-0.657930\pi\)
−0.476046 + 0.879420i \(0.657930\pi\)
\(572\) −6414.39 −0.468879
\(573\) 1602.81 0.116856
\(574\) 7393.72 0.537645
\(575\) −51101.9 −3.70626
\(576\) −682.409 −0.0493641
\(577\) −25759.9 −1.85858 −0.929289 0.369353i \(-0.879579\pi\)
−0.929289 + 0.369353i \(0.879579\pi\)
\(578\) −6162.52 −0.443473
\(579\) −3973.43 −0.285199
\(580\) −21379.3 −1.53057
\(581\) 12540.5 0.895468
\(582\) 3065.80 0.218353
\(583\) −12805.1 −0.909663
\(584\) −11396.6 −0.807525
\(585\) 22635.2 1.59974
\(586\) −417.400 −0.0294243
\(587\) −4369.06 −0.307207 −0.153604 0.988133i \(-0.549088\pi\)
−0.153604 + 0.988133i \(0.549088\pi\)
\(588\) −5002.37 −0.350840
\(589\) −842.515 −0.0589393
\(590\) −2183.46 −0.152358
\(591\) −1811.40 −0.126076
\(592\) −2510.12 −0.174266
\(593\) 3241.12 0.224447 0.112223 0.993683i \(-0.464203\pi\)
0.112223 + 0.993683i \(0.464203\pi\)
\(594\) 2209.23 0.152603
\(595\) −8007.15 −0.551699
\(596\) −2881.49 −0.198038
\(597\) −7314.53 −0.501447
\(598\) 10120.8 0.692092
\(599\) −4294.37 −0.292927 −0.146463 0.989216i \(-0.546789\pi\)
−0.146463 + 0.989216i \(0.546789\pi\)
\(600\) −7759.31 −0.527954
\(601\) 1277.55 0.0867093 0.0433546 0.999060i \(-0.486195\pi\)
0.0433546 + 0.999060i \(0.486195\pi\)
\(602\) −1673.95 −0.113331
\(603\) 12716.3 0.858782
\(604\) 3052.00 0.205603
\(605\) −16490.0 −1.10813
\(606\) −306.886 −0.0205716
\(607\) 17305.3 1.15717 0.578584 0.815623i \(-0.303606\pi\)
0.578584 + 0.815623i \(0.303606\pi\)
\(608\) 990.973 0.0661008
\(609\) 7111.17 0.473168
\(610\) −16986.7 −1.12749
\(611\) 1654.71 0.109562
\(612\) −2062.48 −0.136227
\(613\) −12812.9 −0.844225 −0.422112 0.906544i \(-0.638711\pi\)
−0.422112 + 0.906544i \(0.638711\pi\)
\(614\) 5127.18 0.336997
\(615\) 5545.52 0.363605
\(616\) 12742.1 0.833430
\(617\) −3963.52 −0.258615 −0.129307 0.991605i \(-0.541275\pi\)
−0.129307 + 0.991605i \(0.541275\pi\)
\(618\) −3096.13 −0.201529
\(619\) 26937.7 1.74914 0.874570 0.484899i \(-0.161144\pi\)
0.874570 + 0.484899i \(0.161144\pi\)
\(620\) 20040.9 1.29816
\(621\) 13022.5 0.841503
\(622\) 9157.79 0.590344
\(623\) 3408.22 0.219177
\(624\) −1672.62 −0.107305
\(625\) 32971.0 2.11015
\(626\) −299.300 −0.0191093
\(627\) −177.174 −0.0112849
\(628\) −14518.8 −0.922554
\(629\) 1249.44 0.0792026
\(630\) −19828.4 −1.25394
\(631\) −2401.37 −0.151501 −0.0757504 0.997127i \(-0.524135\pi\)
−0.0757504 + 0.997127i \(0.524135\pi\)
\(632\) −4409.03 −0.277503
\(633\) −1885.85 −0.118414
\(634\) 8544.30 0.535232
\(635\) 12141.6 0.758778
\(636\) −5054.05 −0.315104
\(637\) −24625.7 −1.53172
\(638\) −4933.81 −0.306162
\(639\) −19786.6 −1.22496
\(640\) −29156.5 −1.80080
\(641\) −22872.8 −1.40940 −0.704698 0.709508i \(-0.748917\pi\)
−0.704698 + 0.709508i \(0.748917\pi\)
\(642\) −1174.42 −0.0721970
\(643\) 27051.9 1.65913 0.829566 0.558409i \(-0.188588\pi\)
0.829566 + 0.558409i \(0.188588\pi\)
\(644\) 33121.5 2.02666
\(645\) −1255.52 −0.0766448
\(646\) −92.1801 −0.00561420
\(647\) 12947.9 0.786760 0.393380 0.919376i \(-0.371306\pi\)
0.393380 + 0.919376i \(0.371306\pi\)
\(648\) −10554.3 −0.639835
\(649\) 1882.45 0.113856
\(650\) −16844.4 −1.01645
\(651\) −6665.97 −0.401321
\(652\) −1983.87 −0.119163
\(653\) 16218.4 0.971934 0.485967 0.873977i \(-0.338467\pi\)
0.485967 + 0.873977i \(0.338467\pi\)
\(654\) 1560.54 0.0933056
\(655\) 40576.3 2.42053
\(656\) 4997.34 0.297429
\(657\) −15289.8 −0.907936
\(658\) −1449.52 −0.0858788
\(659\) −26252.1 −1.55180 −0.775901 0.630855i \(-0.782704\pi\)
−0.775901 + 0.630855i \(0.782704\pi\)
\(660\) 4214.42 0.248555
\(661\) 24649.9 1.45049 0.725243 0.688493i \(-0.241727\pi\)
0.725243 + 0.688493i \(0.241727\pi\)
\(662\) 10821.9 0.635355
\(663\) 832.566 0.0487695
\(664\) −7787.41 −0.455136
\(665\) 3310.75 0.193061
\(666\) 3094.04 0.180018
\(667\) −29082.7 −1.68828
\(668\) 1782.42 0.103239
\(669\) −1162.26 −0.0671681
\(670\) −13519.0 −0.779531
\(671\) 14645.0 0.842567
\(672\) 7840.57 0.450084
\(673\) 4424.70 0.253432 0.126716 0.991939i \(-0.459556\pi\)
0.126716 + 0.991939i \(0.459556\pi\)
\(674\) 11725.3 0.670094
\(675\) −21673.6 −1.23588
\(676\) 1401.71 0.0797514
\(677\) 18153.6 1.03057 0.515287 0.857018i \(-0.327685\pi\)
0.515287 + 0.857018i \(0.327685\pi\)
\(678\) −2009.48 −0.113825
\(679\) 49391.3 2.79155
\(680\) 4972.29 0.280410
\(681\) 2704.30 0.152172
\(682\) 4624.93 0.259674
\(683\) 19887.7 1.11417 0.557087 0.830454i \(-0.311919\pi\)
0.557087 + 0.830454i \(0.311919\pi\)
\(684\) 852.782 0.0476710
\(685\) 35046.9 1.95485
\(686\) 8219.39 0.457460
\(687\) 1957.49 0.108709
\(688\) −1131.41 −0.0626955
\(689\) −24880.1 −1.37570
\(690\) −6649.65 −0.366881
\(691\) 8550.10 0.470711 0.235355 0.971909i \(-0.424375\pi\)
0.235355 + 0.971909i \(0.424375\pi\)
\(692\) −11195.9 −0.615037
\(693\) 17095.0 0.937062
\(694\) 173.001 0.00946256
\(695\) −35415.9 −1.93295
\(696\) −4415.91 −0.240495
\(697\) −2487.48 −0.135179
\(698\) −1719.09 −0.0932212
\(699\) −5057.17 −0.273648
\(700\) −55125.1 −2.97647
\(701\) −19280.9 −1.03884 −0.519421 0.854518i \(-0.673852\pi\)
−0.519421 + 0.854518i \(0.673852\pi\)
\(702\) 4292.50 0.230783
\(703\) −516.612 −0.0277161
\(704\) −625.477 −0.0334851
\(705\) −1087.19 −0.0580791
\(706\) 16266.7 0.867146
\(707\) −4944.06 −0.262999
\(708\) 742.986 0.0394394
\(709\) 19384.3 1.02679 0.513394 0.858153i \(-0.328388\pi\)
0.513394 + 0.858153i \(0.328388\pi\)
\(710\) 21035.8 1.11191
\(711\) −5915.23 −0.312009
\(712\) −2116.44 −0.111400
\(713\) 27261.9 1.43193
\(714\) −729.328 −0.0382275
\(715\) 20746.8 1.08515
\(716\) −5617.29 −0.293196
\(717\) 2167.49 0.112896
\(718\) −16446.8 −0.854860
\(719\) −11746.0 −0.609253 −0.304627 0.952472i \(-0.598532\pi\)
−0.304627 + 0.952472i \(0.598532\pi\)
\(720\) −13401.8 −0.693690
\(721\) −49880.0 −2.57646
\(722\) −8876.58 −0.457551
\(723\) −9609.08 −0.494282
\(724\) 18199.1 0.934206
\(725\) 48403.0 2.47951
\(726\) −1501.99 −0.0767825
\(727\) 10042.8 0.512333 0.256167 0.966633i \(-0.417540\pi\)
0.256167 + 0.966633i \(0.417540\pi\)
\(728\) 24757.6 1.26041
\(729\) −11289.5 −0.573566
\(730\) 16255.1 0.824148
\(731\) 563.170 0.0284947
\(732\) 5780.22 0.291862
\(733\) −23541.9 −1.18627 −0.593136 0.805102i \(-0.702111\pi\)
−0.593136 + 0.805102i \(0.702111\pi\)
\(734\) −17296.3 −0.869778
\(735\) 16179.7 0.811970
\(736\) −32065.7 −1.60592
\(737\) 11655.3 0.582538
\(738\) −6159.85 −0.307245
\(739\) −39871.6 −1.98471 −0.992354 0.123422i \(-0.960613\pi\)
−0.992354 + 0.123422i \(0.960613\pi\)
\(740\) 12288.6 0.610458
\(741\) −344.245 −0.0170664
\(742\) 21795.0 1.07833
\(743\) −11387.7 −0.562281 −0.281141 0.959667i \(-0.590713\pi\)
−0.281141 + 0.959667i \(0.590713\pi\)
\(744\) 4139.45 0.203978
\(745\) 9319.94 0.458331
\(746\) −8516.48 −0.417977
\(747\) −10447.7 −0.511729
\(748\) −1890.41 −0.0924066
\(749\) −18920.3 −0.923009
\(750\) 6323.57 0.307872
\(751\) −26156.9 −1.27094 −0.635471 0.772125i \(-0.719194\pi\)
−0.635471 + 0.772125i \(0.719194\pi\)
\(752\) −979.716 −0.0475087
\(753\) −3277.73 −0.158629
\(754\) −9586.31 −0.463014
\(755\) −9871.43 −0.475839
\(756\) 14047.7 0.675806
\(757\) 9147.11 0.439178 0.219589 0.975593i \(-0.429528\pi\)
0.219589 + 0.975593i \(0.429528\pi\)
\(758\) 14603.5 0.699767
\(759\) 5732.95 0.274167
\(760\) −2055.92 −0.0981263
\(761\) −41329.2 −1.96870 −0.984350 0.176225i \(-0.943611\pi\)
−0.984350 + 0.176225i \(0.943611\pi\)
\(762\) 1105.91 0.0525761
\(763\) 25140.9 1.19287
\(764\) −7071.10 −0.334847
\(765\) 6670.90 0.315277
\(766\) 8257.51 0.389499
\(767\) 3657.57 0.172187
\(768\) −2968.66 −0.139482
\(769\) −3722.69 −0.174569 −0.0872846 0.996183i \(-0.527819\pi\)
−0.0872846 + 0.996183i \(0.527819\pi\)
\(770\) −18174.2 −0.850586
\(771\) −3198.78 −0.149418
\(772\) 17529.6 0.817231
\(773\) 34945.5 1.62600 0.813002 0.582261i \(-0.197832\pi\)
0.813002 + 0.582261i \(0.197832\pi\)
\(774\) 1394.60 0.0647648
\(775\) −45372.8 −2.10302
\(776\) −30671.1 −1.41885
\(777\) −4087.43 −0.188721
\(778\) 4924.48 0.226929
\(779\) 1028.51 0.0473045
\(780\) 8188.54 0.375893
\(781\) −18135.9 −0.830925
\(782\) 2982.74 0.136397
\(783\) −12334.7 −0.562971
\(784\) 14580.3 0.664192
\(785\) 46959.8 2.13512
\(786\) 3695.88 0.167720
\(787\) −26135.3 −1.18377 −0.591883 0.806024i \(-0.701615\pi\)
−0.591883 + 0.806024i \(0.701615\pi\)
\(788\) 7991.33 0.361268
\(789\) −9146.20 −0.412691
\(790\) 6288.66 0.283216
\(791\) −32373.6 −1.45521
\(792\) −10615.7 −0.476276
\(793\) 28454.9 1.27423
\(794\) −3381.40 −0.151135
\(795\) 16346.9 0.729263
\(796\) 32269.4 1.43688
\(797\) 20345.9 0.904250 0.452125 0.891955i \(-0.350666\pi\)
0.452125 + 0.891955i \(0.350666\pi\)
\(798\) 301.559 0.0133773
\(799\) 487.664 0.0215924
\(800\) 53367.8 2.35855
\(801\) −2839.45 −0.125252
\(802\) −2207.37 −0.0971881
\(803\) −14014.2 −0.615880
\(804\) 4600.25 0.201789
\(805\) −107129. −4.69042
\(806\) 8986.16 0.392710
\(807\) −3846.44 −0.167783
\(808\) 3070.17 0.133674
\(809\) −21515.4 −0.935034 −0.467517 0.883984i \(-0.654851\pi\)
−0.467517 + 0.883984i \(0.654851\pi\)
\(810\) 15053.8 0.653006
\(811\) −19685.9 −0.852363 −0.426181 0.904638i \(-0.640142\pi\)
−0.426181 + 0.904638i \(0.640142\pi\)
\(812\) −31372.3 −1.35585
\(813\) −1513.36 −0.0652838
\(814\) 2835.91 0.122111
\(815\) 6416.64 0.275785
\(816\) −492.945 −0.0211477
\(817\) −232.857 −0.00997139
\(818\) −7163.35 −0.306187
\(819\) 33215.2 1.41713
\(820\) −24465.1 −1.04190
\(821\) 23501.4 0.999029 0.499515 0.866305i \(-0.333512\pi\)
0.499515 + 0.866305i \(0.333512\pi\)
\(822\) 3192.23 0.135452
\(823\) 25153.6 1.06537 0.532684 0.846314i \(-0.321183\pi\)
0.532684 + 0.846314i \(0.321183\pi\)
\(824\) 30974.6 1.30953
\(825\) −9541.51 −0.402658
\(826\) −3204.03 −0.134967
\(827\) 31043.6 1.30531 0.652655 0.757655i \(-0.273655\pi\)
0.652655 + 0.757655i \(0.273655\pi\)
\(828\) −27594.2 −1.15817
\(829\) 16479.2 0.690404 0.345202 0.938528i \(-0.387810\pi\)
0.345202 + 0.938528i \(0.387810\pi\)
\(830\) 11107.3 0.464505
\(831\) 2723.85 0.113706
\(832\) −1215.29 −0.0506401
\(833\) −7257.52 −0.301871
\(834\) −3225.84 −0.133935
\(835\) −5765.08 −0.238933
\(836\) 781.636 0.0323366
\(837\) 11562.5 0.477489
\(838\) −7976.65 −0.328817
\(839\) 10409.3 0.428329 0.214164 0.976798i \(-0.431297\pi\)
0.214164 + 0.976798i \(0.431297\pi\)
\(840\) −16266.4 −0.668148
\(841\) 3157.71 0.129473
\(842\) 11916.5 0.487731
\(843\) 9327.15 0.381072
\(844\) 8319.80 0.339312
\(845\) −4533.71 −0.184573
\(846\) 1207.62 0.0490768
\(847\) −24197.7 −0.981633
\(848\) 14731.0 0.596537
\(849\) 186.499 0.00753901
\(850\) −4964.26 −0.200321
\(851\) 16716.4 0.673363
\(852\) −7158.04 −0.287829
\(853\) 1701.91 0.0683147 0.0341573 0.999416i \(-0.489125\pi\)
0.0341573 + 0.999416i \(0.489125\pi\)
\(854\) −24926.5 −0.998790
\(855\) −2758.25 −0.110328
\(856\) 11749.2 0.469134
\(857\) −13987.3 −0.557524 −0.278762 0.960360i \(-0.589924\pi\)
−0.278762 + 0.960360i \(0.589924\pi\)
\(858\) 1889.71 0.0751908
\(859\) −845.375 −0.0335784 −0.0167892 0.999859i \(-0.505344\pi\)
−0.0167892 + 0.999859i \(0.505344\pi\)
\(860\) 5538.95 0.219624
\(861\) 8137.56 0.322099
\(862\) 16292.8 0.643776
\(863\) −16970.3 −0.669381 −0.334691 0.942328i \(-0.608632\pi\)
−0.334691 + 0.942328i \(0.608632\pi\)
\(864\) −13599.9 −0.535507
\(865\) 36212.3 1.42342
\(866\) 9311.01 0.365359
\(867\) −6782.50 −0.265681
\(868\) 29408.2 1.14998
\(869\) −5421.72 −0.211645
\(870\) 6298.46 0.245446
\(871\) 22646.1 0.880981
\(872\) −15612.1 −0.606297
\(873\) −41148.8 −1.59528
\(874\) −1233.29 −0.0477307
\(875\) 101876. 3.93602
\(876\) −5531.28 −0.213339
\(877\) −3402.23 −0.130998 −0.0654989 0.997853i \(-0.520864\pi\)
−0.0654989 + 0.997853i \(0.520864\pi\)
\(878\) −3883.92 −0.149289
\(879\) −459.393 −0.0176279
\(880\) −12283.7 −0.470550
\(881\) −9753.40 −0.372986 −0.186493 0.982456i \(-0.559712\pi\)
−0.186493 + 0.982456i \(0.559712\pi\)
\(882\) −17972.1 −0.686114
\(883\) 42672.3 1.62632 0.813159 0.582042i \(-0.197746\pi\)
0.813159 + 0.582042i \(0.197746\pi\)
\(884\) −3673.03 −0.139748
\(885\) −2403.12 −0.0912769
\(886\) −14631.0 −0.554783
\(887\) 13027.3 0.493139 0.246570 0.969125i \(-0.420697\pi\)
0.246570 + 0.969125i \(0.420697\pi\)
\(888\) 2538.22 0.0959202
\(889\) 17816.7 0.672164
\(890\) 3018.71 0.113694
\(891\) −12978.5 −0.487987
\(892\) 5127.52 0.192469
\(893\) −201.637 −0.00755601
\(894\) 848.903 0.0317579
\(895\) 18168.6 0.678560
\(896\) −42784.5 −1.59524
\(897\) 11139.0 0.414628
\(898\) −6447.06 −0.239578
\(899\) −25822.1 −0.957972
\(900\) 45925.7 1.70095
\(901\) −7332.50 −0.271122
\(902\) −5645.94 −0.208414
\(903\) −1842.36 −0.0678958
\(904\) 20103.4 0.739634
\(905\) −58863.5 −2.16209
\(906\) −899.135 −0.0329711
\(907\) −2184.13 −0.0799591 −0.0399796 0.999200i \(-0.512729\pi\)
−0.0399796 + 0.999200i \(0.512729\pi\)
\(908\) −11930.5 −0.436045
\(909\) 4118.99 0.150295
\(910\) −35312.1 −1.28636
\(911\) 15349.5 0.558236 0.279118 0.960257i \(-0.409958\pi\)
0.279118 + 0.960257i \(0.409958\pi\)
\(912\) 203.820 0.00740040
\(913\) −9576.07 −0.347121
\(914\) −6382.87 −0.230992
\(915\) −18695.6 −0.675474
\(916\) −8635.84 −0.311502
\(917\) 59542.2 2.14423
\(918\) 1265.06 0.0454827
\(919\) −50672.7 −1.81887 −0.909433 0.415850i \(-0.863484\pi\)
−0.909433 + 0.415850i \(0.863484\pi\)
\(920\) 66525.0 2.38398
\(921\) 5642.99 0.201892
\(922\) −4015.50 −0.143431
\(923\) −35237.6 −1.25662
\(924\) 6184.30 0.220182
\(925\) −27821.6 −0.988940
\(926\) 3666.66 0.130123
\(927\) 41556.0 1.47236
\(928\) 30372.2 1.07437
\(929\) −40538.7 −1.43168 −0.715841 0.698263i \(-0.753957\pi\)
−0.715841 + 0.698263i \(0.753957\pi\)
\(930\) −5904.15 −0.208177
\(931\) 3000.80 0.105636
\(932\) 22310.7 0.784131
\(933\) 10079.1 0.353671
\(934\) 628.219 0.0220085
\(935\) 6114.36 0.213862
\(936\) −20626.0 −0.720281
\(937\) 13492.4 0.470413 0.235206 0.971945i \(-0.424423\pi\)
0.235206 + 0.971945i \(0.424423\pi\)
\(938\) −19838.0 −0.690548
\(939\) −329.411 −0.0114483
\(940\) 4796.33 0.166424
\(941\) 22597.2 0.782835 0.391417 0.920213i \(-0.371985\pi\)
0.391417 + 0.920213i \(0.371985\pi\)
\(942\) 4277.32 0.147943
\(943\) −33280.3 −1.14926
\(944\) −2165.57 −0.0746645
\(945\) −45436.0 −1.56406
\(946\) 1278.25 0.0439319
\(947\) 21796.6 0.747933 0.373967 0.927442i \(-0.377997\pi\)
0.373967 + 0.927442i \(0.377997\pi\)
\(948\) −2139.90 −0.0733130
\(949\) −27229.4 −0.931405
\(950\) 2052.60 0.0701000
\(951\) 9403.89 0.320654
\(952\) 7296.40 0.248401
\(953\) −40073.5 −1.36213 −0.681064 0.732224i \(-0.738482\pi\)
−0.681064 + 0.732224i \(0.738482\pi\)
\(954\) −18157.8 −0.616226
\(955\) 22870.9 0.774957
\(956\) −9562.31 −0.323501
\(957\) −5430.18 −0.183420
\(958\) −7240.47 −0.244185
\(959\) 51428.2 1.73170
\(960\) 798.477 0.0268445
\(961\) −5585.45 −0.187488
\(962\) 5510.12 0.184671
\(963\) 15762.9 0.527468
\(964\) 42392.3 1.41635
\(965\) −56697.8 −1.89137
\(966\) −9757.78 −0.325001
\(967\) −20836.9 −0.692935 −0.346468 0.938062i \(-0.612619\pi\)
−0.346468 + 0.938062i \(0.612619\pi\)
\(968\) 15026.3 0.498930
\(969\) −101.454 −0.00336343
\(970\) 43746.6 1.44806
\(971\) −5902.12 −0.195065 −0.0975325 0.995232i \(-0.531095\pi\)
−0.0975325 + 0.995232i \(0.531095\pi\)
\(972\) −17785.6 −0.586906
\(973\) −51969.7 −1.71230
\(974\) 589.884 0.0194056
\(975\) −18539.0 −0.608946
\(976\) −16847.5 −0.552538
\(977\) 17199.6 0.563218 0.281609 0.959529i \(-0.409132\pi\)
0.281609 + 0.959529i \(0.409132\pi\)
\(978\) 584.457 0.0191093
\(979\) −2602.56 −0.0849623
\(980\) −71379.9 −2.32668
\(981\) −20945.4 −0.681687
\(982\) 3622.51 0.117718
\(983\) 8184.53 0.265561 0.132780 0.991145i \(-0.457610\pi\)
0.132780 + 0.991145i \(0.457610\pi\)
\(984\) −5053.28 −0.163712
\(985\) −25847.3 −0.836104
\(986\) −2825.22 −0.0912507
\(987\) −1595.35 −0.0514494
\(988\) 1518.70 0.0489032
\(989\) 7534.73 0.242255
\(990\) 15141.2 0.486081
\(991\) −43473.7 −1.39353 −0.696765 0.717300i \(-0.745378\pi\)
−0.696765 + 0.717300i \(0.745378\pi\)
\(992\) −28470.8 −0.911238
\(993\) 11910.6 0.380637
\(994\) 30868.2 0.984989
\(995\) −104373. −3.32546
\(996\) −3779.58 −0.120242
\(997\) 42253.7 1.34221 0.671107 0.741361i \(-0.265819\pi\)
0.671107 + 0.741361i \(0.265819\pi\)
\(998\) −422.558 −0.0134026
\(999\) 7089.87 0.224538
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 43.4.a.b.1.3 6
3.2 odd 2 387.4.a.h.1.4 6
4.3 odd 2 688.4.a.i.1.4 6
5.4 even 2 1075.4.a.b.1.4 6
7.6 odd 2 2107.4.a.c.1.3 6
43.42 odd 2 1849.4.a.c.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.a.b.1.3 6 1.1 even 1 trivial
387.4.a.h.1.4 6 3.2 odd 2
688.4.a.i.1.4 6 4.3 odd 2
1075.4.a.b.1.4 6 5.4 even 2
1849.4.a.c.1.4 6 43.42 odd 2
2107.4.a.c.1.3 6 7.6 odd 2