Properties

Label 1849.4.a.g.1.17
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $30$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.094531601\)
Analytic rank: \(1\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.830523 q^{2} -7.00735 q^{3} -7.31023 q^{4} -4.56308 q^{5} -5.81976 q^{6} +20.3840 q^{7} -12.7155 q^{8} +22.1029 q^{9} +O(q^{10})\) \(q+0.830523 q^{2} -7.00735 q^{3} -7.31023 q^{4} -4.56308 q^{5} -5.81976 q^{6} +20.3840 q^{7} -12.7155 q^{8} +22.1029 q^{9} -3.78975 q^{10} -14.9524 q^{11} +51.2253 q^{12} -38.4779 q^{13} +16.9294 q^{14} +31.9751 q^{15} +47.9213 q^{16} -56.2723 q^{17} +18.3570 q^{18} +82.5783 q^{19} +33.3572 q^{20} -142.838 q^{21} -12.4183 q^{22} -84.1335 q^{23} +89.1019 q^{24} -104.178 q^{25} -31.9568 q^{26} +34.3154 q^{27} -149.012 q^{28} -144.523 q^{29} +26.5561 q^{30} +136.963 q^{31} +141.524 q^{32} +104.777 q^{33} -46.7354 q^{34} -93.0141 q^{35} -161.578 q^{36} -250.262 q^{37} +68.5831 q^{38} +269.628 q^{39} +58.0219 q^{40} +447.286 q^{41} -118.630 q^{42} +109.306 q^{44} -100.858 q^{45} -69.8748 q^{46} +338.476 q^{47} -335.802 q^{48} +72.5092 q^{49} -86.5224 q^{50} +394.319 q^{51} +281.282 q^{52} +540.674 q^{53} +28.4997 q^{54} +68.2293 q^{55} -259.193 q^{56} -578.655 q^{57} -120.029 q^{58} +48.6306 q^{59} -233.746 q^{60} +710.497 q^{61} +113.751 q^{62} +450.547 q^{63} -265.832 q^{64} +175.578 q^{65} +87.0197 q^{66} +907.541 q^{67} +411.363 q^{68} +589.553 q^{69} -77.2504 q^{70} +856.353 q^{71} -281.050 q^{72} -785.853 q^{73} -207.848 q^{74} +730.013 q^{75} -603.666 q^{76} -304.791 q^{77} +223.932 q^{78} +385.961 q^{79} -218.669 q^{80} -837.239 q^{81} +371.482 q^{82} +959.796 q^{83} +1044.18 q^{84} +256.775 q^{85} +1012.72 q^{87} +190.128 q^{88} -1204.79 q^{89} -83.7646 q^{90} -784.335 q^{91} +615.035 q^{92} -959.746 q^{93} +281.112 q^{94} -376.812 q^{95} -991.706 q^{96} -500.574 q^{97} +60.2205 q^{98} -330.493 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30q - 6q^{2} - 2q^{3} + 114q^{4} - 27q^{5} + 8q^{6} - 48q^{7} - 90q^{8} + 216q^{9} + O(q^{10}) \) \( 30q - 6q^{2} - 2q^{3} + 114q^{4} - 27q^{5} + 8q^{6} - 48q^{7} - 90q^{8} + 216q^{9} - 27q^{10} + 80q^{11} + 36q^{12} - 13q^{13} + 36q^{14} + 16q^{15} + 318q^{16} + 66q^{17} - 80q^{18} - 254q^{19} - 312q^{20} - 548q^{21} - 305q^{22} - 105q^{23} + 123q^{24} + 523q^{25} - 549q^{26} + 10q^{27} - 578q^{28} - 793q^{29} - 1560q^{30} - 359q^{31} - 676q^{32} - 208q^{33} - 1007q^{34} - 514q^{35} + 776q^{36} - 510q^{37} - 2066q^{38} - 898q^{39} - 1248q^{40} - 270q^{41} + 915q^{42} + 3256q^{44} - 807q^{45} - 1960q^{46} + 1421q^{47} + 632q^{48} + 386q^{49} + 141q^{50} - 209q^{51} + 2825q^{52} - 21q^{53} + 2368q^{54} - 2258q^{55} + 2521q^{56} - 1723q^{57} - 347q^{58} + 1752q^{59} + 2711q^{60} - 1759q^{61} - 395q^{62} - 2204q^{63} + 222q^{64} - 1151q^{65} + 160q^{66} - 3001q^{67} + 1921q^{68} - 1660q^{69} - 1597q^{70} - 727q^{71} - 9100q^{72} - 4623q^{73} - 2649q^{74} - 1027q^{75} - 874q^{76} - 3556q^{77} - 4979q^{78} + 546q^{79} - 5809q^{80} - 410q^{81} + 4397q^{82} - 492q^{83} - 10611q^{84} + 1723q^{85} + 5937q^{87} - 3974q^{88} - 5218q^{89} + 10492q^{90} - 1104q^{91} + 1060q^{92} - 1997q^{93} + 2134q^{94} + 6346q^{95} - 11984q^{96} + 2590q^{97} - 6270q^{98} - 2693q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.830523 0.293634 0.146817 0.989164i \(-0.453097\pi\)
0.146817 + 0.989164i \(0.453097\pi\)
\(3\) −7.00735 −1.34856 −0.674282 0.738474i \(-0.735547\pi\)
−0.674282 + 0.738474i \(0.735547\pi\)
\(4\) −7.31023 −0.913779
\(5\) −4.56308 −0.408135 −0.204067 0.978957i \(-0.565416\pi\)
−0.204067 + 0.978957i \(0.565416\pi\)
\(6\) −5.81976 −0.395985
\(7\) 20.3840 1.10063 0.550317 0.834956i \(-0.314507\pi\)
0.550317 + 0.834956i \(0.314507\pi\)
\(8\) −12.7155 −0.561951
\(9\) 22.1029 0.818628
\(10\) −3.78975 −0.119842
\(11\) −14.9524 −0.409848 −0.204924 0.978778i \(-0.565695\pi\)
−0.204924 + 0.978778i \(0.565695\pi\)
\(12\) 51.2253 1.23229
\(13\) −38.4779 −0.820912 −0.410456 0.911880i \(-0.634630\pi\)
−0.410456 + 0.911880i \(0.634630\pi\)
\(14\) 16.9294 0.323184
\(15\) 31.9751 0.550396
\(16\) 47.9213 0.748771
\(17\) −56.2723 −0.802825 −0.401413 0.915897i \(-0.631481\pi\)
−0.401413 + 0.915897i \(0.631481\pi\)
\(18\) 18.3570 0.240377
\(19\) 82.5783 0.997092 0.498546 0.866863i \(-0.333867\pi\)
0.498546 + 0.866863i \(0.333867\pi\)
\(20\) 33.3572 0.372945
\(21\) −142.838 −1.48428
\(22\) −12.4183 −0.120346
\(23\) −84.1335 −0.762741 −0.381371 0.924422i \(-0.624548\pi\)
−0.381371 + 0.924422i \(0.624548\pi\)
\(24\) 89.1019 0.757827
\(25\) −104.178 −0.833426
\(26\) −31.9568 −0.241048
\(27\) 34.3154 0.244593
\(28\) −149.012 −1.00574
\(29\) −144.523 −0.925420 −0.462710 0.886510i \(-0.653123\pi\)
−0.462710 + 0.886510i \(0.653123\pi\)
\(30\) 26.5561 0.161615
\(31\) 136.963 0.793524 0.396762 0.917922i \(-0.370134\pi\)
0.396762 + 0.917922i \(0.370134\pi\)
\(32\) 141.524 0.781816
\(33\) 104.777 0.552707
\(34\) −46.7354 −0.235737
\(35\) −93.0141 −0.449207
\(36\) −161.578 −0.748045
\(37\) −250.262 −1.11197 −0.555984 0.831193i \(-0.687658\pi\)
−0.555984 + 0.831193i \(0.687658\pi\)
\(38\) 68.5831 0.292780
\(39\) 269.628 1.10705
\(40\) 58.0219 0.229352
\(41\) 447.286 1.70377 0.851883 0.523732i \(-0.175461\pi\)
0.851883 + 0.523732i \(0.175461\pi\)
\(42\) −118.630 −0.435835
\(43\) 0 0
\(44\) 109.306 0.374511
\(45\) −100.858 −0.334110
\(46\) −69.8748 −0.223967
\(47\) 338.476 1.05047 0.525233 0.850959i \(-0.323978\pi\)
0.525233 + 0.850959i \(0.323978\pi\)
\(48\) −335.802 −1.00977
\(49\) 72.5092 0.211397
\(50\) −86.5224 −0.244722
\(51\) 394.319 1.08266
\(52\) 281.282 0.750132
\(53\) 540.674 1.40127 0.700634 0.713520i \(-0.252900\pi\)
0.700634 + 0.713520i \(0.252900\pi\)
\(54\) 28.4997 0.0718208
\(55\) 68.2293 0.167273
\(56\) −259.193 −0.618503
\(57\) −578.655 −1.34464
\(58\) −120.029 −0.271735
\(59\) 48.6306 0.107308 0.0536540 0.998560i \(-0.482913\pi\)
0.0536540 + 0.998560i \(0.482913\pi\)
\(60\) −233.746 −0.502940
\(61\) 710.497 1.49131 0.745654 0.666333i \(-0.232137\pi\)
0.745654 + 0.666333i \(0.232137\pi\)
\(62\) 113.751 0.233006
\(63\) 450.547 0.901010
\(64\) −265.832 −0.519203
\(65\) 175.578 0.335043
\(66\) 87.0197 0.162294
\(67\) 907.541 1.65483 0.827417 0.561588i \(-0.189810\pi\)
0.827417 + 0.561588i \(0.189810\pi\)
\(68\) 411.363 0.733605
\(69\) 589.553 1.02861
\(70\) −77.2504 −0.131903
\(71\) 856.353 1.43141 0.715707 0.698401i \(-0.246105\pi\)
0.715707 + 0.698401i \(0.246105\pi\)
\(72\) −281.050 −0.460029
\(73\) −785.853 −1.25996 −0.629980 0.776611i \(-0.716937\pi\)
−0.629980 + 0.776611i \(0.716937\pi\)
\(74\) −207.848 −0.326512
\(75\) 730.013 1.12393
\(76\) −603.666 −0.911122
\(77\) −304.791 −0.451093
\(78\) 223.932 0.325069
\(79\) 385.961 0.549671 0.274836 0.961491i \(-0.411377\pi\)
0.274836 + 0.961491i \(0.411377\pi\)
\(80\) −218.669 −0.305599
\(81\) −837.239 −1.14848
\(82\) 371.482 0.500284
\(83\) 959.796 1.26929 0.634646 0.772803i \(-0.281146\pi\)
0.634646 + 0.772803i \(0.281146\pi\)
\(84\) 1044.18 1.35630
\(85\) 256.775 0.327661
\(86\) 0 0
\(87\) 1012.72 1.24799
\(88\) 190.128 0.230315
\(89\) −1204.79 −1.43491 −0.717457 0.696603i \(-0.754694\pi\)
−0.717457 + 0.696603i \(0.754694\pi\)
\(90\) −83.7646 −0.0981062
\(91\) −784.335 −0.903524
\(92\) 615.035 0.696977
\(93\) −959.746 −1.07012
\(94\) 281.112 0.308453
\(95\) −376.812 −0.406948
\(96\) −991.706 −1.05433
\(97\) −500.574 −0.523976 −0.261988 0.965071i \(-0.584378\pi\)
−0.261988 + 0.965071i \(0.584378\pi\)
\(98\) 60.2205 0.0620734
\(99\) −330.493 −0.335513
\(100\) 761.567 0.761567
\(101\) −1497.80 −1.47561 −0.737807 0.675012i \(-0.764138\pi\)
−0.737807 + 0.675012i \(0.764138\pi\)
\(102\) 327.491 0.317907
\(103\) 2.90953 0.00278334 0.00139167 0.999999i \(-0.499557\pi\)
0.00139167 + 0.999999i \(0.499557\pi\)
\(104\) 489.266 0.461312
\(105\) 651.782 0.605785
\(106\) 449.042 0.411460
\(107\) −1294.15 −1.16926 −0.584629 0.811301i \(-0.698760\pi\)
−0.584629 + 0.811301i \(0.698760\pi\)
\(108\) −250.853 −0.223504
\(109\) 239.337 0.210315 0.105158 0.994456i \(-0.466465\pi\)
0.105158 + 0.994456i \(0.466465\pi\)
\(110\) 56.6660 0.0491172
\(111\) 1753.67 1.49956
\(112\) 976.831 0.824123
\(113\) 18.8930 0.0157283 0.00786416 0.999969i \(-0.497497\pi\)
0.00786416 + 0.999969i \(0.497497\pi\)
\(114\) −480.586 −0.394833
\(115\) 383.908 0.311301
\(116\) 1056.49 0.845630
\(117\) −850.475 −0.672021
\(118\) 40.3889 0.0315093
\(119\) −1147.06 −0.883618
\(120\) −406.580 −0.309296
\(121\) −1107.42 −0.832024
\(122\) 590.084 0.437899
\(123\) −3134.29 −2.29764
\(124\) −1001.23 −0.725105
\(125\) 1045.76 0.748285
\(126\) 374.190 0.264567
\(127\) 1964.13 1.37235 0.686175 0.727436i \(-0.259288\pi\)
0.686175 + 0.727436i \(0.259288\pi\)
\(128\) −1352.97 −0.934272
\(129\) 0 0
\(130\) 145.822 0.0983800
\(131\) 122.057 0.0814059 0.0407030 0.999171i \(-0.487040\pi\)
0.0407030 + 0.999171i \(0.487040\pi\)
\(132\) −765.944 −0.505052
\(133\) 1683.28 1.09743
\(134\) 753.734 0.485916
\(135\) −156.584 −0.0998267
\(136\) 715.530 0.451149
\(137\) −999.830 −0.623513 −0.311756 0.950162i \(-0.600917\pi\)
−0.311756 + 0.950162i \(0.600917\pi\)
\(138\) 489.637 0.302034
\(139\) 79.5799 0.0485603 0.0242801 0.999705i \(-0.492271\pi\)
0.0242801 + 0.999705i \(0.492271\pi\)
\(140\) 679.955 0.410476
\(141\) −2371.82 −1.41662
\(142\) 711.221 0.420312
\(143\) 575.339 0.336449
\(144\) 1059.20 0.612964
\(145\) 659.469 0.377696
\(146\) −652.669 −0.369967
\(147\) −508.097 −0.285083
\(148\) 1829.47 1.01609
\(149\) 235.834 0.129666 0.0648330 0.997896i \(-0.479349\pi\)
0.0648330 + 0.997896i \(0.479349\pi\)
\(150\) 606.293 0.330024
\(151\) −564.405 −0.304176 −0.152088 0.988367i \(-0.548600\pi\)
−0.152088 + 0.988367i \(0.548600\pi\)
\(152\) −1050.02 −0.560317
\(153\) −1243.78 −0.657215
\(154\) −253.136 −0.132456
\(155\) −624.973 −0.323865
\(156\) −1971.04 −1.01160
\(157\) 486.141 0.247123 0.123561 0.992337i \(-0.460568\pi\)
0.123561 + 0.992337i \(0.460568\pi\)
\(158\) 320.550 0.161402
\(159\) −3788.69 −1.88970
\(160\) −645.785 −0.319086
\(161\) −1714.98 −0.839500
\(162\) −695.347 −0.337232
\(163\) −982.776 −0.472251 −0.236126 0.971723i \(-0.575878\pi\)
−0.236126 + 0.971723i \(0.575878\pi\)
\(164\) −3269.77 −1.55687
\(165\) −478.106 −0.225579
\(166\) 797.132 0.372708
\(167\) 2035.17 0.943029 0.471514 0.881858i \(-0.343708\pi\)
0.471514 + 0.881858i \(0.343708\pi\)
\(168\) 1816.26 0.834091
\(169\) −716.450 −0.326104
\(170\) 213.258 0.0962124
\(171\) 1825.22 0.816247
\(172\) 0 0
\(173\) 3600.69 1.58240 0.791201 0.611556i \(-0.209456\pi\)
0.791201 + 0.611556i \(0.209456\pi\)
\(174\) 841.088 0.366452
\(175\) −2123.57 −0.917298
\(176\) −716.541 −0.306883
\(177\) −340.772 −0.144712
\(178\) −1000.60 −0.421340
\(179\) −1335.29 −0.557564 −0.278782 0.960354i \(-0.589931\pi\)
−0.278782 + 0.960354i \(0.589931\pi\)
\(180\) 737.292 0.305303
\(181\) 1886.27 0.774614 0.387307 0.921951i \(-0.373405\pi\)
0.387307 + 0.921951i \(0.373405\pi\)
\(182\) −651.409 −0.265306
\(183\) −4978.70 −2.01113
\(184\) 1069.80 0.428623
\(185\) 1141.97 0.453832
\(186\) −797.091 −0.314223
\(187\) 841.408 0.329037
\(188\) −2474.34 −0.959893
\(189\) 699.486 0.269207
\(190\) −312.951 −0.119494
\(191\) 144.895 0.0548911 0.0274456 0.999623i \(-0.491263\pi\)
0.0274456 + 0.999623i \(0.491263\pi\)
\(192\) 1862.78 0.700179
\(193\) 3015.69 1.12474 0.562368 0.826887i \(-0.309890\pi\)
0.562368 + 0.826887i \(0.309890\pi\)
\(194\) −415.739 −0.153857
\(195\) −1230.34 −0.451827
\(196\) −530.059 −0.193170
\(197\) −3923.49 −1.41897 −0.709485 0.704720i \(-0.751073\pi\)
−0.709485 + 0.704720i \(0.751073\pi\)
\(198\) −274.482 −0.0985181
\(199\) 2882.17 1.02669 0.513346 0.858182i \(-0.328406\pi\)
0.513346 + 0.858182i \(0.328406\pi\)
\(200\) 1324.68 0.468345
\(201\) −6359.46 −2.23165
\(202\) −1243.96 −0.433290
\(203\) −2945.96 −1.01855
\(204\) −2882.57 −0.989314
\(205\) −2041.01 −0.695366
\(206\) 2.41643 0.000817285 0
\(207\) −1859.60 −0.624401
\(208\) −1843.91 −0.614675
\(209\) −1234.75 −0.408657
\(210\) 541.320 0.177879
\(211\) −3809.94 −1.24307 −0.621533 0.783388i \(-0.713490\pi\)
−0.621533 + 0.783388i \(0.713490\pi\)
\(212\) −3952.45 −1.28045
\(213\) −6000.76 −1.93035
\(214\) −1074.83 −0.343334
\(215\) 0 0
\(216\) −436.337 −0.137449
\(217\) 2791.86 0.873380
\(218\) 198.775 0.0617557
\(219\) 5506.74 1.69914
\(220\) −498.772 −0.152851
\(221\) 2165.24 0.659049
\(222\) 1456.47 0.440322
\(223\) 2160.95 0.648913 0.324456 0.945901i \(-0.394819\pi\)
0.324456 + 0.945901i \(0.394819\pi\)
\(224\) 2884.83 0.860494
\(225\) −2302.65 −0.682266
\(226\) 15.6910 0.00461837
\(227\) −2048.34 −0.598913 −0.299457 0.954110i \(-0.596805\pi\)
−0.299457 + 0.954110i \(0.596805\pi\)
\(228\) 4230.10 1.22871
\(229\) −246.349 −0.0710882 −0.0355441 0.999368i \(-0.511316\pi\)
−0.0355441 + 0.999368i \(0.511316\pi\)
\(230\) 318.845 0.0914087
\(231\) 2135.78 0.608329
\(232\) 1837.68 0.520041
\(233\) −5331.57 −1.49907 −0.749534 0.661965i \(-0.769723\pi\)
−0.749534 + 0.661965i \(0.769723\pi\)
\(234\) −706.339 −0.197328
\(235\) −1544.50 −0.428731
\(236\) −355.501 −0.0980558
\(237\) −2704.57 −0.741268
\(238\) −952.657 −0.259460
\(239\) −3285.36 −0.889172 −0.444586 0.895736i \(-0.646649\pi\)
−0.444586 + 0.895736i \(0.646649\pi\)
\(240\) 1532.29 0.412121
\(241\) −6049.31 −1.61689 −0.808445 0.588571i \(-0.799691\pi\)
−0.808445 + 0.588571i \(0.799691\pi\)
\(242\) −919.741 −0.244311
\(243\) 4940.31 1.30420
\(244\) −5193.90 −1.36273
\(245\) −330.865 −0.0862784
\(246\) −2603.10 −0.674665
\(247\) −3177.44 −0.818525
\(248\) −1741.55 −0.445922
\(249\) −6725.62 −1.71172
\(250\) 868.528 0.219722
\(251\) −3881.73 −0.976147 −0.488073 0.872803i \(-0.662300\pi\)
−0.488073 + 0.872803i \(0.662300\pi\)
\(252\) −3293.61 −0.823324
\(253\) 1258.00 0.312608
\(254\) 1631.26 0.402969
\(255\) −1799.31 −0.441872
\(256\) 1002.98 0.244869
\(257\) 6016.31 1.46026 0.730131 0.683307i \(-0.239459\pi\)
0.730131 + 0.683307i \(0.239459\pi\)
\(258\) 0 0
\(259\) −5101.35 −1.22387
\(260\) −1283.52 −0.306155
\(261\) −3194.38 −0.757575
\(262\) 101.371 0.0239036
\(263\) −795.657 −0.186549 −0.0932743 0.995640i \(-0.529733\pi\)
−0.0932743 + 0.995640i \(0.529733\pi\)
\(264\) −1332.29 −0.310594
\(265\) −2467.14 −0.571906
\(266\) 1398.00 0.322244
\(267\) 8442.37 1.93507
\(268\) −6634.34 −1.51215
\(269\) −6264.61 −1.41993 −0.709963 0.704239i \(-0.751289\pi\)
−0.709963 + 0.704239i \(0.751289\pi\)
\(270\) −130.047 −0.0293125
\(271\) 6986.31 1.56601 0.783004 0.622017i \(-0.213686\pi\)
0.783004 + 0.622017i \(0.213686\pi\)
\(272\) −2696.64 −0.601132
\(273\) 5496.11 1.21846
\(274\) −830.382 −0.183085
\(275\) 1557.72 0.341578
\(276\) −4309.77 −0.939919
\(277\) −78.7787 −0.0170879 −0.00854396 0.999963i \(-0.502720\pi\)
−0.00854396 + 0.999963i \(0.502720\pi\)
\(278\) 66.0930 0.0142590
\(279\) 3027.28 0.649601
\(280\) 1182.72 0.252432
\(281\) −6663.90 −1.41472 −0.707358 0.706856i \(-0.750113\pi\)
−0.707358 + 0.706856i \(0.750113\pi\)
\(282\) −1969.85 −0.415968
\(283\) −1831.66 −0.384738 −0.192369 0.981323i \(-0.561617\pi\)
−0.192369 + 0.981323i \(0.561617\pi\)
\(284\) −6260.14 −1.30800
\(285\) 2640.45 0.548796
\(286\) 477.832 0.0987930
\(287\) 9117.51 1.87522
\(288\) 3128.09 0.640016
\(289\) −1746.43 −0.355471
\(290\) 547.704 0.110905
\(291\) 3507.70 0.706615
\(292\) 5744.77 1.15133
\(293\) −1423.07 −0.283742 −0.141871 0.989885i \(-0.545312\pi\)
−0.141871 + 0.989885i \(0.545312\pi\)
\(294\) −421.986 −0.0837100
\(295\) −221.906 −0.0437961
\(296\) 3182.20 0.624871
\(297\) −513.099 −0.100246
\(298\) 195.865 0.0380744
\(299\) 3237.28 0.626143
\(300\) −5336.57 −1.02702
\(301\) 0 0
\(302\) −468.752 −0.0893166
\(303\) 10495.6 1.98996
\(304\) 3957.26 0.746594
\(305\) −3242.06 −0.608654
\(306\) −1032.99 −0.192981
\(307\) 8324.83 1.54763 0.773816 0.633411i \(-0.218346\pi\)
0.773816 + 0.633411i \(0.218346\pi\)
\(308\) 2228.09 0.412200
\(309\) −20.3881 −0.00375352
\(310\) −519.054 −0.0950977
\(311\) −9795.80 −1.78607 −0.893037 0.449984i \(-0.851430\pi\)
−0.893037 + 0.449984i \(0.851430\pi\)
\(312\) −3428.46 −0.622109
\(313\) −9275.95 −1.67511 −0.837553 0.546357i \(-0.816014\pi\)
−0.837553 + 0.546357i \(0.816014\pi\)
\(314\) 403.751 0.0725636
\(315\) −2055.89 −0.367733
\(316\) −2821.47 −0.502278
\(317\) 7079.71 1.25437 0.627186 0.778870i \(-0.284207\pi\)
0.627186 + 0.778870i \(0.284207\pi\)
\(318\) −3146.59 −0.554881
\(319\) 2160.97 0.379282
\(320\) 1213.01 0.211905
\(321\) 9068.59 1.57682
\(322\) −1424.33 −0.246506
\(323\) −4646.87 −0.800491
\(324\) 6120.41 1.04945
\(325\) 4008.56 0.684169
\(326\) −816.218 −0.138669
\(327\) −1677.12 −0.283624
\(328\) −5687.47 −0.957433
\(329\) 6899.52 1.15618
\(330\) −397.078 −0.0662377
\(331\) −4093.53 −0.679760 −0.339880 0.940469i \(-0.610386\pi\)
−0.339880 + 0.940469i \(0.610386\pi\)
\(332\) −7016.33 −1.15985
\(333\) −5531.52 −0.910287
\(334\) 1690.25 0.276906
\(335\) −4141.19 −0.675395
\(336\) −6844.99 −1.11138
\(337\) −2177.63 −0.351997 −0.175999 0.984390i \(-0.556315\pi\)
−0.175999 + 0.984390i \(0.556315\pi\)
\(338\) −595.028 −0.0957552
\(339\) −132.390 −0.0212107
\(340\) −1877.09 −0.299410
\(341\) −2047.93 −0.325224
\(342\) 1515.89 0.239678
\(343\) −5513.70 −0.867964
\(344\) 0 0
\(345\) −2690.18 −0.419810
\(346\) 2990.46 0.464647
\(347\) −6825.91 −1.05601 −0.528003 0.849243i \(-0.677059\pi\)
−0.528003 + 0.849243i \(0.677059\pi\)
\(348\) −7403.22 −1.14039
\(349\) 6001.37 0.920475 0.460238 0.887796i \(-0.347764\pi\)
0.460238 + 0.887796i \(0.347764\pi\)
\(350\) −1763.68 −0.269350
\(351\) −1320.38 −0.200789
\(352\) −2116.13 −0.320426
\(353\) −5847.25 −0.881636 −0.440818 0.897596i \(-0.645312\pi\)
−0.440818 + 0.897596i \(0.645312\pi\)
\(354\) −283.019 −0.0424923
\(355\) −3907.61 −0.584210
\(356\) 8807.28 1.31119
\(357\) 8037.82 1.19162
\(358\) −1108.99 −0.163720
\(359\) 4930.18 0.724805 0.362403 0.932022i \(-0.381957\pi\)
0.362403 + 0.932022i \(0.381957\pi\)
\(360\) 1282.45 0.187754
\(361\) −39.8311 −0.00580713
\(362\) 1566.59 0.227453
\(363\) 7760.11 1.12204
\(364\) 5733.67 0.825621
\(365\) 3585.91 0.514234
\(366\) −4134.92 −0.590535
\(367\) −3457.00 −0.491700 −0.245850 0.969308i \(-0.579067\pi\)
−0.245850 + 0.969308i \(0.579067\pi\)
\(368\) −4031.79 −0.571118
\(369\) 9886.35 1.39475
\(370\) 948.429 0.133261
\(371\) 11021.1 1.54229
\(372\) 7015.97 0.977852
\(373\) 205.191 0.0284836 0.0142418 0.999899i \(-0.495467\pi\)
0.0142418 + 0.999899i \(0.495467\pi\)
\(374\) 698.809 0.0966164
\(375\) −7328.00 −1.00911
\(376\) −4303.90 −0.590310
\(377\) 5560.93 0.759688
\(378\) 580.939 0.0790484
\(379\) 11728.4 1.58957 0.794785 0.606891i \(-0.207584\pi\)
0.794785 + 0.606891i \(0.207584\pi\)
\(380\) 2754.58 0.371860
\(381\) −13763.4 −1.85070
\(382\) 120.338 0.0161179
\(383\) 4805.06 0.641063 0.320531 0.947238i \(-0.396139\pi\)
0.320531 + 0.947238i \(0.396139\pi\)
\(384\) 9480.73 1.25993
\(385\) 1390.79 0.184107
\(386\) 2504.60 0.330261
\(387\) 0 0
\(388\) 3659.31 0.478798
\(389\) −8640.08 −1.12614 −0.563071 0.826409i \(-0.690380\pi\)
−0.563071 + 0.826409i \(0.690380\pi\)
\(390\) −1021.82 −0.132672
\(391\) 4734.38 0.612348
\(392\) −921.990 −0.118795
\(393\) −855.297 −0.109781
\(394\) −3258.55 −0.416658
\(395\) −1761.17 −0.224340
\(396\) 2415.98 0.306585
\(397\) −1045.09 −0.132120 −0.0660599 0.997816i \(-0.521043\pi\)
−0.0660599 + 0.997816i \(0.521043\pi\)
\(398\) 2393.71 0.301472
\(399\) −11795.3 −1.47996
\(400\) −4992.36 −0.624045
\(401\) −512.318 −0.0638003 −0.0319002 0.999491i \(-0.510156\pi\)
−0.0319002 + 0.999491i \(0.510156\pi\)
\(402\) −5281.68 −0.655289
\(403\) −5270.04 −0.651413
\(404\) 10949.3 1.34838
\(405\) 3820.39 0.468733
\(406\) −2446.68 −0.299081
\(407\) 3742.03 0.455738
\(408\) −5013.97 −0.608403
\(409\) −4472.52 −0.540714 −0.270357 0.962760i \(-0.587142\pi\)
−0.270357 + 0.962760i \(0.587142\pi\)
\(410\) −1695.10 −0.204183
\(411\) 7006.16 0.840848
\(412\) −21.2693 −0.00254336
\(413\) 991.289 0.118107
\(414\) −1544.44 −0.183345
\(415\) −4379.63 −0.518042
\(416\) −5445.54 −0.641802
\(417\) −557.644 −0.0654867
\(418\) −1025.49 −0.119996
\(419\) 1189.05 0.138637 0.0693185 0.997595i \(-0.477918\pi\)
0.0693185 + 0.997595i \(0.477918\pi\)
\(420\) −4764.68 −0.553554
\(421\) −2778.77 −0.321684 −0.160842 0.986980i \(-0.551421\pi\)
−0.160842 + 0.986980i \(0.551421\pi\)
\(422\) −3164.24 −0.365007
\(423\) 7481.33 0.859940
\(424\) −6874.94 −0.787444
\(425\) 5862.35 0.669096
\(426\) −4983.77 −0.566818
\(427\) 14482.8 1.64139
\(428\) 9460.57 1.06844
\(429\) −4031.60 −0.453724
\(430\) 0 0
\(431\) 13056.6 1.45920 0.729599 0.683876i \(-0.239707\pi\)
0.729599 + 0.683876i \(0.239707\pi\)
\(432\) 1644.44 0.183144
\(433\) −1122.53 −0.124585 −0.0622927 0.998058i \(-0.519841\pi\)
−0.0622927 + 0.998058i \(0.519841\pi\)
\(434\) 2318.70 0.256454
\(435\) −4621.13 −0.509348
\(436\) −1749.61 −0.192182
\(437\) −6947.60 −0.760523
\(438\) 4573.48 0.498925
\(439\) 8780.26 0.954576 0.477288 0.878747i \(-0.341620\pi\)
0.477288 + 0.878747i \(0.341620\pi\)
\(440\) −867.569 −0.0939994
\(441\) 1602.67 0.173055
\(442\) 1798.28 0.193519
\(443\) −15938.5 −1.70939 −0.854694 0.519133i \(-0.826255\pi\)
−0.854694 + 0.519133i \(0.826255\pi\)
\(444\) −12819.8 −1.37027
\(445\) 5497.55 0.585638
\(446\) 1794.71 0.190543
\(447\) −1652.57 −0.174863
\(448\) −5418.73 −0.571453
\(449\) −12696.6 −1.33450 −0.667248 0.744835i \(-0.732528\pi\)
−0.667248 + 0.744835i \(0.732528\pi\)
\(450\) −1912.40 −0.200337
\(451\) −6688.03 −0.698286
\(452\) −138.112 −0.0143722
\(453\) 3954.98 0.410202
\(454\) −1701.20 −0.175861
\(455\) 3578.99 0.368760
\(456\) 7357.88 0.755624
\(457\) 11229.2 1.14941 0.574704 0.818362i \(-0.305117\pi\)
0.574704 + 0.818362i \(0.305117\pi\)
\(458\) −204.598 −0.0208739
\(459\) −1931.01 −0.196365
\(460\) −2806.46 −0.284460
\(461\) 2948.65 0.297900 0.148950 0.988845i \(-0.452411\pi\)
0.148950 + 0.988845i \(0.452411\pi\)
\(462\) 1773.81 0.178626
\(463\) −15736.8 −1.57959 −0.789797 0.613368i \(-0.789814\pi\)
−0.789797 + 0.613368i \(0.789814\pi\)
\(464\) −6925.72 −0.692928
\(465\) 4379.40 0.436753
\(466\) −4427.99 −0.440178
\(467\) 14900.1 1.47643 0.738215 0.674566i \(-0.235669\pi\)
0.738215 + 0.674566i \(0.235669\pi\)
\(468\) 6217.17 0.614079
\(469\) 18499.4 1.82137
\(470\) −1282.74 −0.125890
\(471\) −3406.56 −0.333261
\(472\) −618.363 −0.0603018
\(473\) 0 0
\(474\) −2246.20 −0.217662
\(475\) −8602.86 −0.831003
\(476\) 8385.25 0.807431
\(477\) 11950.5 1.14712
\(478\) −2728.56 −0.261091
\(479\) 6419.60 0.612357 0.306178 0.951974i \(-0.400950\pi\)
0.306178 + 0.951974i \(0.400950\pi\)
\(480\) 4525.24 0.430308
\(481\) 9629.55 0.912827
\(482\) −5024.09 −0.474774
\(483\) 12017.5 1.13212
\(484\) 8095.53 0.760286
\(485\) 2284.16 0.213853
\(486\) 4103.04 0.382959
\(487\) 14002.5 1.30290 0.651450 0.758691i \(-0.274161\pi\)
0.651450 + 0.758691i \(0.274161\pi\)
\(488\) −9034.32 −0.838042
\(489\) 6886.65 0.636861
\(490\) −274.791 −0.0253343
\(491\) 4910.67 0.451356 0.225678 0.974202i \(-0.427540\pi\)
0.225678 + 0.974202i \(0.427540\pi\)
\(492\) 22912.4 2.09953
\(493\) 8132.62 0.742951
\(494\) −2638.94 −0.240347
\(495\) 1508.07 0.136935
\(496\) 6563.44 0.594168
\(497\) 17455.9 1.57546
\(498\) −5585.78 −0.502620
\(499\) 957.649 0.0859123 0.0429562 0.999077i \(-0.486322\pi\)
0.0429562 + 0.999077i \(0.486322\pi\)
\(500\) −7644.75 −0.683767
\(501\) −14261.1 −1.27174
\(502\) −3223.87 −0.286630
\(503\) −12247.8 −1.08569 −0.542843 0.839834i \(-0.682652\pi\)
−0.542843 + 0.839834i \(0.682652\pi\)
\(504\) −5728.93 −0.506323
\(505\) 6834.60 0.602249
\(506\) 1044.80 0.0917925
\(507\) 5020.42 0.439772
\(508\) −14358.3 −1.25403
\(509\) 9063.40 0.789250 0.394625 0.918842i \(-0.370875\pi\)
0.394625 + 0.918842i \(0.370875\pi\)
\(510\) −1494.37 −0.129749
\(511\) −16018.9 −1.38676
\(512\) 11656.8 1.00617
\(513\) 2833.71 0.243881
\(514\) 4996.69 0.428783
\(515\) −13.2764 −0.00113598
\(516\) 0 0
\(517\) −5061.05 −0.430531
\(518\) −4236.79 −0.359370
\(519\) −25231.3 −2.13397
\(520\) −2232.56 −0.188278
\(521\) −14421.2 −1.21267 −0.606336 0.795208i \(-0.707362\pi\)
−0.606336 + 0.795208i \(0.707362\pi\)
\(522\) −2653.00 −0.222450
\(523\) −5350.14 −0.447314 −0.223657 0.974668i \(-0.571800\pi\)
−0.223657 + 0.974668i \(0.571800\pi\)
\(524\) −892.266 −0.0743870
\(525\) 14880.6 1.23704
\(526\) −660.811 −0.0547771
\(527\) −7707.21 −0.637061
\(528\) 5021.05 0.413851
\(529\) −5088.55 −0.418226
\(530\) −2049.02 −0.167931
\(531\) 1074.88 0.0878453
\(532\) −12305.2 −1.00281
\(533\) −17210.6 −1.39864
\(534\) 7011.59 0.568204
\(535\) 5905.34 0.477215
\(536\) −11539.8 −0.929935
\(537\) 9356.82 0.751912
\(538\) −5202.91 −0.416939
\(539\) −1084.19 −0.0866407
\(540\) 1144.67 0.0912196
\(541\) −5131.45 −0.407797 −0.203898 0.978992i \(-0.565361\pi\)
−0.203898 + 0.978992i \(0.565361\pi\)
\(542\) 5802.29 0.459833
\(543\) −13217.7 −1.04462
\(544\) −7963.86 −0.627662
\(545\) −1092.12 −0.0858369
\(546\) 4564.65 0.357782
\(547\) 16592.7 1.29699 0.648494 0.761220i \(-0.275399\pi\)
0.648494 + 0.761220i \(0.275399\pi\)
\(548\) 7308.99 0.569753
\(549\) 15704.1 1.22083
\(550\) 1293.72 0.100299
\(551\) −11934.4 −0.922729
\(552\) −7496.46 −0.578026
\(553\) 7867.45 0.604987
\(554\) −65.4275 −0.00501760
\(555\) −8002.15 −0.612023
\(556\) −581.748 −0.0443734
\(557\) 4232.85 0.321996 0.160998 0.986955i \(-0.448529\pi\)
0.160998 + 0.986955i \(0.448529\pi\)
\(558\) 2514.23 0.190745
\(559\) 0 0
\(560\) −4457.36 −0.336353
\(561\) −5896.04 −0.443727
\(562\) −5534.53 −0.415409
\(563\) 2958.31 0.221453 0.110726 0.993851i \(-0.464682\pi\)
0.110726 + 0.993851i \(0.464682\pi\)
\(564\) 17338.6 1.29448
\(565\) −86.2101 −0.00641927
\(566\) −1521.23 −0.112972
\(567\) −17066.3 −1.26405
\(568\) −10889.0 −0.804384
\(569\) 5234.91 0.385692 0.192846 0.981229i \(-0.438228\pi\)
0.192846 + 0.981229i \(0.438228\pi\)
\(570\) 2192.95 0.161145
\(571\) 13989.0 1.02525 0.512627 0.858611i \(-0.328672\pi\)
0.512627 + 0.858611i \(0.328672\pi\)
\(572\) −4205.86 −0.307440
\(573\) −1015.33 −0.0740242
\(574\) 7572.30 0.550630
\(575\) 8764.88 0.635688
\(576\) −5875.67 −0.425034
\(577\) 23239.0 1.67670 0.838348 0.545135i \(-0.183522\pi\)
0.838348 + 0.545135i \(0.183522\pi\)
\(578\) −1450.45 −0.104379
\(579\) −21132.0 −1.51678
\(580\) −4820.87 −0.345131
\(581\) 19564.5 1.39703
\(582\) 2913.22 0.207486
\(583\) −8084.39 −0.574308
\(584\) 9992.51 0.708036
\(585\) 3880.79 0.274275
\(586\) −1181.89 −0.0833165
\(587\) −6240.67 −0.438807 −0.219404 0.975634i \(-0.570411\pi\)
−0.219404 + 0.975634i \(0.570411\pi\)
\(588\) 3714.31 0.260502
\(589\) 11310.1 0.791217
\(590\) −184.298 −0.0128600
\(591\) 27493.3 1.91357
\(592\) −11992.9 −0.832609
\(593\) −11379.5 −0.788029 −0.394014 0.919104i \(-0.628914\pi\)
−0.394014 + 0.919104i \(0.628914\pi\)
\(594\) −426.141 −0.0294356
\(595\) 5234.12 0.360635
\(596\) −1724.00 −0.118486
\(597\) −20196.4 −1.38456
\(598\) 2688.64 0.183857
\(599\) −23163.8 −1.58005 −0.790024 0.613076i \(-0.789932\pi\)
−0.790024 + 0.613076i \(0.789932\pi\)
\(600\) −9282.49 −0.631593
\(601\) −4928.18 −0.334484 −0.167242 0.985916i \(-0.553486\pi\)
−0.167242 + 0.985916i \(0.553486\pi\)
\(602\) 0 0
\(603\) 20059.3 1.35469
\(604\) 4125.93 0.277950
\(605\) 5053.27 0.339578
\(606\) 8716.86 0.584320
\(607\) 7495.38 0.501199 0.250600 0.968091i \(-0.419372\pi\)
0.250600 + 0.968091i \(0.419372\pi\)
\(608\) 11686.8 0.779542
\(609\) 20643.3 1.37358
\(610\) −2692.60 −0.178722
\(611\) −13023.9 −0.862339
\(612\) 9092.34 0.600549
\(613\) −7140.15 −0.470453 −0.235227 0.971941i \(-0.575583\pi\)
−0.235227 + 0.971941i \(0.575583\pi\)
\(614\) 6913.96 0.454438
\(615\) 14302.0 0.937746
\(616\) 3875.57 0.253492
\(617\) −14180.3 −0.925249 −0.462624 0.886554i \(-0.653092\pi\)
−0.462624 + 0.886554i \(0.653092\pi\)
\(618\) −16.9328 −0.00110216
\(619\) 10072.1 0.654007 0.327003 0.945023i \(-0.393961\pi\)
0.327003 + 0.945023i \(0.393961\pi\)
\(620\) 4568.70 0.295941
\(621\) −2887.07 −0.186561
\(622\) −8135.64 −0.524452
\(623\) −24558.5 −1.57932
\(624\) 12920.9 0.828929
\(625\) 8250.39 0.528025
\(626\) −7703.89 −0.491868
\(627\) 8652.30 0.551100
\(628\) −3553.80 −0.225815
\(629\) 14082.8 0.892716
\(630\) −1707.46 −0.107979
\(631\) −10425.8 −0.657755 −0.328877 0.944373i \(-0.606670\pi\)
−0.328877 + 0.944373i \(0.606670\pi\)
\(632\) −4907.69 −0.308888
\(633\) 26697.6 1.67636
\(634\) 5879.86 0.368327
\(635\) −8962.50 −0.560104
\(636\) 27696.2 1.72677
\(637\) −2790.00 −0.173538
\(638\) 1794.73 0.111370
\(639\) 18927.9 1.17179
\(640\) 6173.72 0.381309
\(641\) 11982.7 0.738359 0.369180 0.929358i \(-0.379639\pi\)
0.369180 + 0.929358i \(0.379639\pi\)
\(642\) 7531.67 0.463009
\(643\) 3919.88 0.240412 0.120206 0.992749i \(-0.461644\pi\)
0.120206 + 0.992749i \(0.461644\pi\)
\(644\) 12536.9 0.767117
\(645\) 0 0
\(646\) −3859.33 −0.235052
\(647\) −179.894 −0.0109310 −0.00546549 0.999985i \(-0.501740\pi\)
−0.00546549 + 0.999985i \(0.501740\pi\)
\(648\) 10645.9 0.645388
\(649\) −727.147 −0.0439800
\(650\) 3329.20 0.200896
\(651\) −19563.5 −1.17781
\(652\) 7184.32 0.431533
\(653\) 24418.3 1.46334 0.731670 0.681659i \(-0.238741\pi\)
0.731670 + 0.681659i \(0.238741\pi\)
\(654\) −1392.89 −0.0832816
\(655\) −556.957 −0.0332246
\(656\) 21434.6 1.27573
\(657\) −17369.7 −1.03144
\(658\) 5730.21 0.339494
\(659\) 12541.3 0.741336 0.370668 0.928765i \(-0.379129\pi\)
0.370668 + 0.928765i \(0.379129\pi\)
\(660\) 3495.07 0.206129
\(661\) −19977.8 −1.17556 −0.587782 0.809019i \(-0.699999\pi\)
−0.587782 + 0.809019i \(0.699999\pi\)
\(662\) −3399.77 −0.199601
\(663\) −15172.6 −0.888770
\(664\) −12204.3 −0.713280
\(665\) −7680.94 −0.447901
\(666\) −4594.06 −0.267291
\(667\) 12159.2 0.705856
\(668\) −14877.5 −0.861720
\(669\) −15142.5 −0.875101
\(670\) −3439.35 −0.198319
\(671\) −10623.7 −0.611210
\(672\) −20215.0 −1.16043
\(673\) −4879.78 −0.279497 −0.139749 0.990187i \(-0.544629\pi\)
−0.139749 + 0.990187i \(0.544629\pi\)
\(674\) −1808.57 −0.103358
\(675\) −3574.92 −0.203850
\(676\) 5237.42 0.297987
\(677\) −25770.9 −1.46301 −0.731505 0.681836i \(-0.761182\pi\)
−0.731505 + 0.681836i \(0.761182\pi\)
\(678\) −109.953 −0.00622817
\(679\) −10203.7 −0.576706
\(680\) −3265.02 −0.184129
\(681\) 14353.5 0.807674
\(682\) −1700.85 −0.0954970
\(683\) −17463.2 −0.978345 −0.489172 0.872187i \(-0.662701\pi\)
−0.489172 + 0.872187i \(0.662701\pi\)
\(684\) −13342.8 −0.745869
\(685\) 4562.31 0.254477
\(686\) −4579.25 −0.254864
\(687\) 1726.25 0.0958670
\(688\) 0 0
\(689\) −20804.0 −1.15032
\(690\) −2234.26 −0.123271
\(691\) 12899.2 0.710140 0.355070 0.934840i \(-0.384457\pi\)
0.355070 + 0.934840i \(0.384457\pi\)
\(692\) −26321.9 −1.44597
\(693\) −6736.78 −0.369277
\(694\) −5669.07 −0.310079
\(695\) −363.130 −0.0198191
\(696\) −12877.3 −0.701309
\(697\) −25169.8 −1.36783
\(698\) 4984.27 0.270283
\(699\) 37360.2 2.02159
\(700\) 15523.8 0.838207
\(701\) −27155.9 −1.46314 −0.731572 0.681764i \(-0.761213\pi\)
−0.731572 + 0.681764i \(0.761213\pi\)
\(702\) −1096.61 −0.0589585
\(703\) −20666.2 −1.10873
\(704\) 3974.84 0.212795
\(705\) 10822.8 0.578172
\(706\) −4856.27 −0.258879
\(707\) −30531.3 −1.62411
\(708\) 2491.12 0.132235
\(709\) 32347.2 1.71343 0.856717 0.515787i \(-0.172500\pi\)
0.856717 + 0.515787i \(0.172500\pi\)
\(710\) −3245.36 −0.171544
\(711\) 8530.88 0.449976
\(712\) 15319.5 0.806351
\(713\) −11523.2 −0.605253
\(714\) 6675.60 0.349899
\(715\) −2625.32 −0.137317
\(716\) 9761.25 0.509490
\(717\) 23021.6 1.19911
\(718\) 4094.63 0.212828
\(719\) 2462.26 0.127715 0.0638574 0.997959i \(-0.479660\pi\)
0.0638574 + 0.997959i \(0.479660\pi\)
\(720\) −4833.23 −0.250172
\(721\) 59.3080 0.00306344
\(722\) −33.0807 −0.00170517
\(723\) 42389.7 2.18048
\(724\) −13789.0 −0.707826
\(725\) 15056.1 0.771269
\(726\) 6444.95 0.329469
\(727\) −34210.8 −1.74526 −0.872632 0.488379i \(-0.837588\pi\)
−0.872632 + 0.488379i \(0.837588\pi\)
\(728\) 9973.22 0.507736
\(729\) −12013.0 −0.610325
\(730\) 2978.18 0.150997
\(731\) 0 0
\(732\) 36395.4 1.83772
\(733\) 5710.38 0.287746 0.143873 0.989596i \(-0.454044\pi\)
0.143873 + 0.989596i \(0.454044\pi\)
\(734\) −2871.12 −0.144380
\(735\) 2318.49 0.116352
\(736\) −11906.9 −0.596323
\(737\) −13570.0 −0.678231
\(738\) 8210.84 0.409546
\(739\) −27827.8 −1.38520 −0.692601 0.721321i \(-0.743535\pi\)
−0.692601 + 0.721321i \(0.743535\pi\)
\(740\) −8348.04 −0.414703
\(741\) 22265.4 1.10383
\(742\) 9153.29 0.452868
\(743\) −4999.54 −0.246858 −0.123429 0.992353i \(-0.539389\pi\)
−0.123429 + 0.992353i \(0.539389\pi\)
\(744\) 12203.7 0.601354
\(745\) −1076.13 −0.0529212
\(746\) 170.416 0.00836375
\(747\) 21214.3 1.03908
\(748\) −6150.89 −0.300667
\(749\) −26380.1 −1.28693
\(750\) −6086.08 −0.296309
\(751\) −3252.06 −0.158015 −0.0790077 0.996874i \(-0.525175\pi\)
−0.0790077 + 0.996874i \(0.525175\pi\)
\(752\) 16220.2 0.786558
\(753\) 27200.7 1.31640
\(754\) 4618.48 0.223071
\(755\) 2575.43 0.124145
\(756\) −5113.41 −0.245996
\(757\) 32449.8 1.55800 0.779000 0.627024i \(-0.215727\pi\)
0.779000 + 0.627024i \(0.215727\pi\)
\(758\) 9740.70 0.466752
\(759\) −8815.26 −0.421573
\(760\) 4791.35 0.228685
\(761\) −1370.12 −0.0652650 −0.0326325 0.999467i \(-0.510389\pi\)
−0.0326325 + 0.999467i \(0.510389\pi\)
\(762\) −11430.8 −0.543430
\(763\) 4878.66 0.231480
\(764\) −1059.21 −0.0501583
\(765\) 5675.49 0.268232
\(766\) 3990.71 0.188238
\(767\) −1871.21 −0.0880904
\(768\) −7028.25 −0.330222
\(769\) 7651.00 0.358780 0.179390 0.983778i \(-0.442588\pi\)
0.179390 + 0.983778i \(0.442588\pi\)
\(770\) 1155.08 0.0540601
\(771\) −42158.4 −1.96926
\(772\) −22045.4 −1.02776
\(773\) 13408.1 0.623873 0.311937 0.950103i \(-0.399022\pi\)
0.311937 + 0.950103i \(0.399022\pi\)
\(774\) 0 0
\(775\) −14268.5 −0.661344
\(776\) 6365.05 0.294449
\(777\) 35746.9 1.65047
\(778\) −7175.78 −0.330674
\(779\) 36936.1 1.69881
\(780\) 8994.04 0.412870
\(781\) −12804.6 −0.586662
\(782\) 3932.01 0.179806
\(783\) −4959.35 −0.226351
\(784\) 3474.74 0.158288
\(785\) −2218.30 −0.100859
\(786\) −710.343 −0.0322355
\(787\) −18848.1 −0.853700 −0.426850 0.904323i \(-0.640377\pi\)
−0.426850 + 0.904323i \(0.640377\pi\)
\(788\) 28681.6 1.29663
\(789\) 5575.44 0.251573
\(790\) −1462.70 −0.0658739
\(791\) 385.115 0.0173111
\(792\) 4202.38 0.188542
\(793\) −27338.4 −1.22423
\(794\) −867.971 −0.0387949
\(795\) 17288.1 0.771253
\(796\) −21069.3 −0.938169
\(797\) 32490.5 1.44401 0.722004 0.691889i \(-0.243221\pi\)
0.722004 + 0.691889i \(0.243221\pi\)
\(798\) −9796.29 −0.434567
\(799\) −19046.8 −0.843340
\(800\) −14743.7 −0.651586
\(801\) −26629.4 −1.17466
\(802\) −425.492 −0.0187340
\(803\) 11750.4 0.516393
\(804\) 46489.1 2.03924
\(805\) 7825.60 0.342629
\(806\) −4376.89 −0.191277
\(807\) 43898.3 1.91486
\(808\) 19045.3 0.829222
\(809\) 14885.6 0.646908 0.323454 0.946244i \(-0.395156\pi\)
0.323454 + 0.946244i \(0.395156\pi\)
\(810\) 3172.93 0.137636
\(811\) 23932.3 1.03622 0.518112 0.855313i \(-0.326635\pi\)
0.518112 + 0.855313i \(0.326635\pi\)
\(812\) 21535.6 0.930729
\(813\) −48955.5 −2.11186
\(814\) 3107.84 0.133820
\(815\) 4484.49 0.192742
\(816\) 18896.3 0.810666
\(817\) 0 0
\(818\) −3714.53 −0.158772
\(819\) −17336.1 −0.739650
\(820\) 14920.2 0.635411
\(821\) −35623.3 −1.51433 −0.757164 0.653225i \(-0.773416\pi\)
−0.757164 + 0.653225i \(0.773416\pi\)
\(822\) 5818.78 0.246902
\(823\) −409.139 −0.0173289 −0.00866446 0.999962i \(-0.502758\pi\)
−0.00866446 + 0.999962i \(0.502758\pi\)
\(824\) −36.9961 −0.00156410
\(825\) −10915.5 −0.460641
\(826\) 823.288 0.0346802
\(827\) 21237.5 0.892988 0.446494 0.894787i \(-0.352672\pi\)
0.446494 + 0.894787i \(0.352672\pi\)
\(828\) 13594.1 0.570564
\(829\) −1728.67 −0.0724236 −0.0362118 0.999344i \(-0.511529\pi\)
−0.0362118 + 0.999344i \(0.511529\pi\)
\(830\) −3637.38 −0.152115
\(831\) 552.030 0.0230442
\(832\) 10228.7 0.426220
\(833\) −4080.26 −0.169715
\(834\) −463.136 −0.0192291
\(835\) −9286.63 −0.384883
\(836\) 9026.29 0.373422
\(837\) 4699.93 0.194090
\(838\) 987.534 0.0407086
\(839\) −41221.3 −1.69621 −0.848104 0.529830i \(-0.822256\pi\)
−0.848104 + 0.529830i \(0.822256\pi\)
\(840\) −8287.74 −0.340422
\(841\) −3502.19 −0.143597
\(842\) −2307.83 −0.0944575
\(843\) 46696.3 1.90784
\(844\) 27851.5 1.13589
\(845\) 3269.22 0.133094
\(846\) 6213.41 0.252508
\(847\) −22573.8 −0.915755
\(848\) 25909.8 1.04923
\(849\) 12835.1 0.518844
\(850\) 4868.81 0.196469
\(851\) 21055.4 0.848143
\(852\) 43867.0 1.76392
\(853\) −14786.9 −0.593547 −0.296773 0.954948i \(-0.595911\pi\)
−0.296773 + 0.954948i \(0.595911\pi\)
\(854\) 12028.3 0.481967
\(855\) −8328.64 −0.333139
\(856\) 16455.8 0.657066
\(857\) −20068.7 −0.799921 −0.399961 0.916532i \(-0.630976\pi\)
−0.399961 + 0.916532i \(0.630976\pi\)
\(858\) −3348.34 −0.133229
\(859\) −37526.7 −1.49056 −0.745282 0.666750i \(-0.767685\pi\)
−0.745282 + 0.666750i \(0.767685\pi\)
\(860\) 0 0
\(861\) −63889.5 −2.52886
\(862\) 10843.8 0.428470
\(863\) 28185.4 1.11175 0.555875 0.831266i \(-0.312383\pi\)
0.555875 + 0.831266i \(0.312383\pi\)
\(864\) 4856.44 0.191226
\(865\) −16430.3 −0.645833
\(866\) −932.290 −0.0365826
\(867\) 12237.9 0.479376
\(868\) −20409.1 −0.798076
\(869\) −5771.07 −0.225282
\(870\) −3837.96 −0.149562
\(871\) −34920.3 −1.35847
\(872\) −3043.29 −0.118187
\(873\) −11064.2 −0.428941
\(874\) −5770.14 −0.223316
\(875\) 21316.8 0.823588
\(876\) −40255.6 −1.55264
\(877\) 17560.4 0.676136 0.338068 0.941122i \(-0.390227\pi\)
0.338068 + 0.941122i \(0.390227\pi\)
\(878\) 7292.20 0.280296
\(879\) 9971.94 0.382645
\(880\) 3269.64 0.125249
\(881\) −35647.1 −1.36320 −0.681601 0.731724i \(-0.738716\pi\)
−0.681601 + 0.731724i \(0.738716\pi\)
\(882\) 1331.05 0.0508150
\(883\) −50715.7 −1.93286 −0.966431 0.256925i \(-0.917291\pi\)
−0.966431 + 0.256925i \(0.917291\pi\)
\(884\) −15828.4 −0.602225
\(885\) 1554.97 0.0590619
\(886\) −13237.3 −0.501935
\(887\) −14869.8 −0.562884 −0.281442 0.959578i \(-0.590813\pi\)
−0.281442 + 0.959578i \(0.590813\pi\)
\(888\) −22298.8 −0.842679
\(889\) 40036.9 1.51046
\(890\) 4565.84 0.171963
\(891\) 12518.8 0.470701
\(892\) −15797.0 −0.592963
\(893\) 27950.8 1.04741
\(894\) −1372.50 −0.0513458
\(895\) 6093.02 0.227561
\(896\) −27579.0 −1.02829
\(897\) −22684.8 −0.844395
\(898\) −10544.8 −0.391854
\(899\) −19794.2 −0.734343
\(900\) 16832.9 0.623440
\(901\) −30424.9 −1.12497
\(902\) −5554.56 −0.205041
\(903\) 0 0
\(904\) −240.233 −0.00883854
\(905\) −8607.19 −0.316147
\(906\) 3284.71 0.120449
\(907\) −38314.2 −1.40265 −0.701324 0.712843i \(-0.747407\pi\)
−0.701324 + 0.712843i \(0.747407\pi\)
\(908\) 14973.9 0.547274
\(909\) −33105.8 −1.20798
\(910\) 2972.43 0.108280
\(911\) 41494.7 1.50909 0.754545 0.656248i \(-0.227858\pi\)
0.754545 + 0.656248i \(0.227858\pi\)
\(912\) −27729.9 −1.00683
\(913\) −14351.3 −0.520217
\(914\) 9326.09 0.337505
\(915\) 22718.2 0.820810
\(916\) 1800.87 0.0649589
\(917\) 2488.02 0.0895982
\(918\) −1603.74 −0.0576595
\(919\) −22998.3 −0.825511 −0.412756 0.910842i \(-0.635434\pi\)
−0.412756 + 0.910842i \(0.635434\pi\)
\(920\) −4881.59 −0.174936
\(921\) −58335.0 −2.08708
\(922\) 2448.92 0.0874738
\(923\) −32950.7 −1.17506
\(924\) −15613.0 −0.555878
\(925\) 26071.8 0.926743
\(926\) −13069.8 −0.463823
\(927\) 64.3091 0.00227852
\(928\) −20453.4 −0.723508
\(929\) −43145.2 −1.52373 −0.761867 0.647733i \(-0.775717\pi\)
−0.761867 + 0.647733i \(0.775717\pi\)
\(930\) 3637.19 0.128245
\(931\) 5987.68 0.210782
\(932\) 38975.0 1.36982
\(933\) 68642.6 2.40864
\(934\) 12374.8 0.433530
\(935\) −3839.42 −0.134291
\(936\) 10814.2 0.377643
\(937\) 19389.3 0.676009 0.338005 0.941144i \(-0.390248\pi\)
0.338005 + 0.941144i \(0.390248\pi\)
\(938\) 15364.1 0.534816
\(939\) 64999.8 2.25899
\(940\) 11290.6 0.391766
\(941\) 27492.5 0.952424 0.476212 0.879330i \(-0.342009\pi\)
0.476212 + 0.879330i \(0.342009\pi\)
\(942\) −2829.22 −0.0978568
\(943\) −37631.8 −1.29953
\(944\) 2330.45 0.0803491
\(945\) −3191.82 −0.109873
\(946\) 0 0
\(947\) −36198.3 −1.24212 −0.621059 0.783764i \(-0.713297\pi\)
−0.621059 + 0.783764i \(0.713297\pi\)
\(948\) 19771.0 0.677355
\(949\) 30238.0 1.03432
\(950\) −7144.87 −0.244011
\(951\) −49610.0 −1.69160
\(952\) 14585.4 0.496550
\(953\) −29531.3 −1.00379 −0.501895 0.864929i \(-0.667363\pi\)
−0.501895 + 0.864929i \(0.667363\pi\)
\(954\) 9925.15 0.336833
\(955\) −661.166 −0.0224030
\(956\) 24016.7 0.812507
\(957\) −15142.7 −0.511486
\(958\) 5331.62 0.179809
\(959\) −20380.6 −0.686260
\(960\) −8500.01 −0.285767
\(961\) −11032.2 −0.370320
\(962\) 7997.57 0.268037
\(963\) −28604.6 −0.957187
\(964\) 44221.9 1.47748
\(965\) −13760.9 −0.459044
\(966\) 9980.78 0.332429
\(967\) 9561.17 0.317959 0.158980 0.987282i \(-0.449180\pi\)
0.158980 + 0.987282i \(0.449180\pi\)
\(968\) 14081.5 0.467557
\(969\) 32562.2 1.07951
\(970\) 1897.05 0.0627944
\(971\) 4107.09 0.135739 0.0678696 0.997694i \(-0.478380\pi\)
0.0678696 + 0.997694i \(0.478380\pi\)
\(972\) −36114.8 −1.19175
\(973\) 1622.16 0.0534472
\(974\) 11629.4 0.382576
\(975\) −28089.4 −0.922647
\(976\) 34048.0 1.11665
\(977\) 28818.7 0.943698 0.471849 0.881679i \(-0.343587\pi\)
0.471849 + 0.881679i \(0.343587\pi\)
\(978\) 5719.52 0.187004
\(979\) 18014.5 0.588097
\(980\) 2418.70 0.0788394
\(981\) 5290.06 0.172170
\(982\) 4078.43 0.132533
\(983\) 9721.35 0.315425 0.157713 0.987485i \(-0.449588\pi\)
0.157713 + 0.987485i \(0.449588\pi\)
\(984\) 39854.1 1.29116
\(985\) 17903.2 0.579131
\(986\) 6754.33 0.218156
\(987\) −48347.3 −1.55918
\(988\) 23227.8 0.747951
\(989\) 0 0
\(990\) 1252.48 0.0402087
\(991\) −35015.2 −1.12240 −0.561198 0.827682i \(-0.689659\pi\)
−0.561198 + 0.827682i \(0.689659\pi\)
\(992\) 19383.5 0.620390
\(993\) 28684.8 0.916700
\(994\) 14497.6 0.462610
\(995\) −13151.6 −0.419028
\(996\) 49165.9 1.56414
\(997\) −13390.4 −0.425355 −0.212677 0.977122i \(-0.568218\pi\)
−0.212677 + 0.977122i \(0.568218\pi\)
\(998\) 795.350 0.0252268
\(999\) −8587.83 −0.271979
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 1849.4.a.g.1.17 30
43.8 odd 14 43.4.e.a.21.7 60
43.27 odd 14 43.4.e.a.41.7 yes 60
43.42 odd 2 1849.4.a.h.1.14 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.e.a.21.7 60 43.8 odd 14
43.4.e.a.41.7 yes 60 43.27 odd 14
1849.4.a.g.1.17 30 1.1 even 1 trivial
1849.4.a.h.1.14 30 43.42 odd 2