Properties

Label 1849.4.a.g.1.5
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 1.5
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.39566 q^{2} -7.85436 q^{3} +11.3218 q^{4} -5.85216 q^{5} +34.5251 q^{6} -12.3980 q^{7} -14.6016 q^{8} +34.6910 q^{9} +O(q^{10})\) \(q-4.39566 q^{2} -7.85436 q^{3} +11.3218 q^{4} -5.85216 q^{5} +34.5251 q^{6} -12.3980 q^{7} -14.6016 q^{8} +34.6910 q^{9} +25.7241 q^{10} +50.9011 q^{11} -88.9257 q^{12} -72.3797 q^{13} +54.4973 q^{14} +45.9650 q^{15} -26.3910 q^{16} +72.1180 q^{17} -152.490 q^{18} -47.7505 q^{19} -66.2570 q^{20} +97.3782 q^{21} -223.744 q^{22} +194.454 q^{23} +114.686 q^{24} -90.7523 q^{25} +318.156 q^{26} -60.4081 q^{27} -140.368 q^{28} -114.570 q^{29} -202.046 q^{30} -108.300 q^{31} +232.818 q^{32} -399.795 q^{33} -317.006 q^{34} +72.5549 q^{35} +392.765 q^{36} -19.4210 q^{37} +209.895 q^{38} +568.496 q^{39} +85.4507 q^{40} -485.191 q^{41} -428.041 q^{42} +576.293 q^{44} -203.017 q^{45} -854.753 q^{46} -20.2520 q^{47} +207.284 q^{48} -189.290 q^{49} +398.916 q^{50} -566.441 q^{51} -819.469 q^{52} +276.032 q^{53} +265.533 q^{54} -297.881 q^{55} +181.030 q^{56} +375.050 q^{57} +503.612 q^{58} -65.3849 q^{59} +520.407 q^{60} +330.113 q^{61} +476.051 q^{62} -430.098 q^{63} -812.262 q^{64} +423.577 q^{65} +1757.36 q^{66} -522.685 q^{67} +816.506 q^{68} -1527.31 q^{69} -318.927 q^{70} -155.769 q^{71} -506.544 q^{72} -543.223 q^{73} +85.3682 q^{74} +712.801 q^{75} -540.622 q^{76} -631.070 q^{77} -2498.91 q^{78} +1153.53 q^{79} +154.444 q^{80} -462.190 q^{81} +2132.73 q^{82} +781.141 q^{83} +1102.50 q^{84} -422.046 q^{85} +899.877 q^{87} -743.236 q^{88} -515.449 q^{89} +892.395 q^{90} +897.361 q^{91} +2201.57 q^{92} +850.630 q^{93} +89.0210 q^{94} +279.443 q^{95} -1828.64 q^{96} +901.767 q^{97} +832.055 q^{98} +1765.81 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30q - 6q^{2} - 2q^{3} + 114q^{4} - 27q^{5} + 8q^{6} - 48q^{7} - 90q^{8} + 216q^{9} + O(q^{10}) \) \( 30q - 6q^{2} - 2q^{3} + 114q^{4} - 27q^{5} + 8q^{6} - 48q^{7} - 90q^{8} + 216q^{9} - 27q^{10} + 80q^{11} + 36q^{12} - 13q^{13} + 36q^{14} + 16q^{15} + 318q^{16} + 66q^{17} - 80q^{18} - 254q^{19} - 312q^{20} - 548q^{21} - 305q^{22} - 105q^{23} + 123q^{24} + 523q^{25} - 549q^{26} + 10q^{27} - 578q^{28} - 793q^{29} - 1560q^{30} - 359q^{31} - 676q^{32} - 208q^{33} - 1007q^{34} - 514q^{35} + 776q^{36} - 510q^{37} - 2066q^{38} - 898q^{39} - 1248q^{40} - 270q^{41} + 915q^{42} + 3256q^{44} - 807q^{45} - 1960q^{46} + 1421q^{47} + 632q^{48} + 386q^{49} + 141q^{50} - 209q^{51} + 2825q^{52} - 21q^{53} + 2368q^{54} - 2258q^{55} + 2521q^{56} - 1723q^{57} - 347q^{58} + 1752q^{59} + 2711q^{60} - 1759q^{61} - 395q^{62} - 2204q^{63} + 222q^{64} - 1151q^{65} + 160q^{66} - 3001q^{67} + 1921q^{68} - 1660q^{69} - 1597q^{70} - 727q^{71} - 9100q^{72} - 4623q^{73} - 2649q^{74} - 1027q^{75} - 874q^{76} - 3556q^{77} - 4979q^{78} + 546q^{79} - 5809q^{80} - 410q^{81} + 4397q^{82} - 492q^{83} - 10611q^{84} + 1723q^{85} + 5937q^{87} - 3974q^{88} - 5218q^{89} + 10492q^{90} - 1104q^{91} + 1060q^{92} - 1997q^{93} + 2134q^{94} + 6346q^{95} - 11984q^{96} + 2590q^{97} - 6270q^{98} - 2693q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.39566 −1.55410 −0.777050 0.629439i \(-0.783285\pi\)
−0.777050 + 0.629439i \(0.783285\pi\)
\(3\) −7.85436 −1.51157 −0.755786 0.654818i \(-0.772745\pi\)
−0.755786 + 0.654818i \(0.772745\pi\)
\(4\) 11.3218 1.41523
\(5\) −5.85216 −0.523433 −0.261716 0.965145i \(-0.584288\pi\)
−0.261716 + 0.965145i \(0.584288\pi\)
\(6\) 34.5251 2.34914
\(7\) −12.3980 −0.669428 −0.334714 0.942320i \(-0.608640\pi\)
−0.334714 + 0.942320i \(0.608640\pi\)
\(8\) −14.6016 −0.645305
\(9\) 34.6910 1.28485
\(10\) 25.7241 0.813467
\(11\) 50.9011 1.39520 0.697602 0.716485i \(-0.254250\pi\)
0.697602 + 0.716485i \(0.254250\pi\)
\(12\) −88.9257 −2.13922
\(13\) −72.3797 −1.54419 −0.772096 0.635506i \(-0.780792\pi\)
−0.772096 + 0.635506i \(0.780792\pi\)
\(14\) 54.4973 1.04036
\(15\) 45.9650 0.791207
\(16\) −26.3910 −0.412359
\(17\) 72.1180 1.02889 0.514446 0.857523i \(-0.327997\pi\)
0.514446 + 0.857523i \(0.327997\pi\)
\(18\) −152.490 −1.99679
\(19\) −47.7505 −0.576564 −0.288282 0.957546i \(-0.593084\pi\)
−0.288282 + 0.957546i \(0.593084\pi\)
\(20\) −66.2570 −0.740776
\(21\) 97.3782 1.01189
\(22\) −223.744 −2.16829
\(23\) 194.454 1.76289 0.881444 0.472289i \(-0.156572\pi\)
0.881444 + 0.472289i \(0.156572\pi\)
\(24\) 114.686 0.975425
\(25\) −90.7523 −0.726018
\(26\) 318.156 2.39983
\(27\) −60.4081 −0.430576
\(28\) −140.368 −0.947392
\(29\) −114.570 −0.733627 −0.366813 0.930295i \(-0.619551\pi\)
−0.366813 + 0.930295i \(0.619551\pi\)
\(30\) −202.046 −1.22961
\(31\) −108.300 −0.627461 −0.313731 0.949512i \(-0.601579\pi\)
−0.313731 + 0.949512i \(0.601579\pi\)
\(32\) 232.818 1.28615
\(33\) −399.795 −2.10895
\(34\) −317.006 −1.59900
\(35\) 72.5549 0.350400
\(36\) 392.765 1.81836
\(37\) −19.4210 −0.0862918 −0.0431459 0.999069i \(-0.513738\pi\)
−0.0431459 + 0.999069i \(0.513738\pi\)
\(38\) 209.895 0.896038
\(39\) 568.496 2.33416
\(40\) 85.4507 0.337774
\(41\) −485.191 −1.84815 −0.924074 0.382214i \(-0.875162\pi\)
−0.924074 + 0.382214i \(0.875162\pi\)
\(42\) −428.041 −1.57258
\(43\) 0 0
\(44\) 576.293 1.97453
\(45\) −203.017 −0.672534
\(46\) −854.753 −2.73970
\(47\) −20.2520 −0.0628523 −0.0314262 0.999506i \(-0.510005\pi\)
−0.0314262 + 0.999506i \(0.510005\pi\)
\(48\) 207.284 0.623311
\(49\) −189.290 −0.551866
\(50\) 398.916 1.12830
\(51\) −566.441 −1.55525
\(52\) −819.469 −2.18538
\(53\) 276.032 0.715396 0.357698 0.933837i \(-0.383562\pi\)
0.357698 + 0.933837i \(0.383562\pi\)
\(54\) 265.533 0.669158
\(55\) −297.881 −0.730296
\(56\) 181.030 0.431985
\(57\) 375.050 0.871519
\(58\) 503.612 1.14013
\(59\) −65.3849 −0.144278 −0.0721389 0.997395i \(-0.522982\pi\)
−0.0721389 + 0.997395i \(0.522982\pi\)
\(60\) 520.407 1.11974
\(61\) 330.113 0.692896 0.346448 0.938069i \(-0.387388\pi\)
0.346448 + 0.938069i \(0.387388\pi\)
\(62\) 476.051 0.975138
\(63\) −430.098 −0.860116
\(64\) −812.262 −1.58645
\(65\) 423.577 0.808281
\(66\) 1757.36 3.27752
\(67\) −522.685 −0.953077 −0.476538 0.879154i \(-0.658109\pi\)
−0.476538 + 0.879154i \(0.658109\pi\)
\(68\) 816.506 1.45612
\(69\) −1527.31 −2.66473
\(70\) −318.927 −0.544557
\(71\) −155.769 −0.260371 −0.130185 0.991490i \(-0.541557\pi\)
−0.130185 + 0.991490i \(0.541557\pi\)
\(72\) −506.544 −0.829122
\(73\) −543.223 −0.870951 −0.435476 0.900201i \(-0.643420\pi\)
−0.435476 + 0.900201i \(0.643420\pi\)
\(74\) 85.3682 0.134106
\(75\) 712.801 1.09743
\(76\) −540.622 −0.815969
\(77\) −631.070 −0.933988
\(78\) −2498.91 −3.62752
\(79\) 1153.53 1.64281 0.821407 0.570342i \(-0.193189\pi\)
0.821407 + 0.570342i \(0.193189\pi\)
\(80\) 154.444 0.215842
\(81\) −462.190 −0.634006
\(82\) 2132.73 2.87221
\(83\) 781.141 1.03303 0.516514 0.856279i \(-0.327229\pi\)
0.516514 + 0.856279i \(0.327229\pi\)
\(84\) 1102.50 1.43205
\(85\) −422.046 −0.538556
\(86\) 0 0
\(87\) 899.877 1.10893
\(88\) −743.236 −0.900332
\(89\) −515.449 −0.613904 −0.306952 0.951725i \(-0.599309\pi\)
−0.306952 + 0.951725i \(0.599309\pi\)
\(90\) 892.395 1.04519
\(91\) 897.361 1.03373
\(92\) 2201.57 2.49489
\(93\) 850.630 0.948454
\(94\) 89.0210 0.0976788
\(95\) 279.443 0.301792
\(96\) −1828.64 −1.94411
\(97\) 901.767 0.943924 0.471962 0.881619i \(-0.343546\pi\)
0.471962 + 0.881619i \(0.343546\pi\)
\(98\) 832.055 0.857656
\(99\) 1765.81 1.79263
\(100\) −1027.48 −1.02748
\(101\) −199.561 −0.196605 −0.0983024 0.995157i \(-0.531341\pi\)
−0.0983024 + 0.995157i \(0.531341\pi\)
\(102\) 2489.88 2.41701
\(103\) −881.273 −0.843052 −0.421526 0.906816i \(-0.638505\pi\)
−0.421526 + 0.906816i \(0.638505\pi\)
\(104\) 1056.86 0.996475
\(105\) −569.872 −0.529656
\(106\) −1213.34 −1.11180
\(107\) 1495.28 1.35098 0.675488 0.737371i \(-0.263933\pi\)
0.675488 + 0.737371i \(0.263933\pi\)
\(108\) −683.930 −0.609363
\(109\) −760.930 −0.668660 −0.334330 0.942456i \(-0.608510\pi\)
−0.334330 + 0.942456i \(0.608510\pi\)
\(110\) 1309.38 1.13495
\(111\) 152.540 0.130436
\(112\) 327.195 0.276045
\(113\) 1352.82 1.12621 0.563107 0.826384i \(-0.309606\pi\)
0.563107 + 0.826384i \(0.309606\pi\)
\(114\) −1648.59 −1.35443
\(115\) −1137.97 −0.922753
\(116\) −1297.14 −1.03825
\(117\) −2510.92 −1.98406
\(118\) 287.410 0.224222
\(119\) −894.117 −0.688769
\(120\) −671.161 −0.510570
\(121\) 1259.92 0.946595
\(122\) −1451.06 −1.07683
\(123\) 3810.86 2.79361
\(124\) −1226.16 −0.888000
\(125\) 1262.62 0.903454
\(126\) 1890.57 1.33671
\(127\) 229.260 0.160185 0.0800926 0.996787i \(-0.474478\pi\)
0.0800926 + 0.996787i \(0.474478\pi\)
\(128\) 1707.88 1.17935
\(129\) 0 0
\(130\) −1861.90 −1.25615
\(131\) 1670.22 1.11396 0.556978 0.830527i \(-0.311961\pi\)
0.556978 + 0.830527i \(0.311961\pi\)
\(132\) −4526.41 −2.98465
\(133\) 592.010 0.385968
\(134\) 2297.55 1.48118
\(135\) 353.518 0.225377
\(136\) −1053.04 −0.663949
\(137\) 2120.54 1.32241 0.661203 0.750207i \(-0.270046\pi\)
0.661203 + 0.750207i \(0.270046\pi\)
\(138\) 6713.54 4.14126
\(139\) −546.720 −0.333613 −0.166806 0.985990i \(-0.553346\pi\)
−0.166806 + 0.985990i \(0.553346\pi\)
\(140\) 821.453 0.495896
\(141\) 159.067 0.0950059
\(142\) 684.706 0.404642
\(143\) −3684.20 −2.15446
\(144\) −915.530 −0.529821
\(145\) 670.483 0.384004
\(146\) 2387.82 1.35355
\(147\) 1486.75 0.834186
\(148\) −219.881 −0.122123
\(149\) 3202.06 1.76056 0.880280 0.474455i \(-0.157355\pi\)
0.880280 + 0.474455i \(0.157355\pi\)
\(150\) −3133.23 −1.70552
\(151\) 1219.91 0.657452 0.328726 0.944425i \(-0.393381\pi\)
0.328726 + 0.944425i \(0.393381\pi\)
\(152\) 697.233 0.372060
\(153\) 2501.85 1.32198
\(154\) 2773.97 1.45151
\(155\) 633.790 0.328434
\(156\) 6436.41 3.30337
\(157\) −1934.69 −0.983472 −0.491736 0.870744i \(-0.663637\pi\)
−0.491736 + 0.870744i \(0.663637\pi\)
\(158\) −5070.53 −2.55310
\(159\) −2168.06 −1.08137
\(160\) −1362.49 −0.673214
\(161\) −2410.83 −1.18013
\(162\) 2031.63 0.985309
\(163\) −1091.52 −0.524505 −0.262253 0.964999i \(-0.584465\pi\)
−0.262253 + 0.964999i \(0.584465\pi\)
\(164\) −5493.24 −2.61555
\(165\) 2339.67 1.10390
\(166\) −3433.63 −1.60543
\(167\) 2286.50 1.05949 0.529744 0.848157i \(-0.322288\pi\)
0.529744 + 0.848157i \(0.322288\pi\)
\(168\) −1421.88 −0.652977
\(169\) 3041.81 1.38453
\(170\) 1855.17 0.836970
\(171\) −1656.51 −0.740800
\(172\) 0 0
\(173\) −1171.22 −0.514720 −0.257360 0.966316i \(-0.582853\pi\)
−0.257360 + 0.966316i \(0.582853\pi\)
\(174\) −3955.55 −1.72339
\(175\) 1125.14 0.486017
\(176\) −1343.33 −0.575325
\(177\) 513.557 0.218087
\(178\) 2265.74 0.954069
\(179\) −931.735 −0.389057 −0.194528 0.980897i \(-0.562318\pi\)
−0.194528 + 0.980897i \(0.562318\pi\)
\(180\) −2298.52 −0.951788
\(181\) −305.691 −0.125535 −0.0627675 0.998028i \(-0.519993\pi\)
−0.0627675 + 0.998028i \(0.519993\pi\)
\(182\) −3944.49 −1.60651
\(183\) −2592.83 −1.04736
\(184\) −2839.33 −1.13760
\(185\) 113.655 0.0451680
\(186\) −3739.08 −1.47399
\(187\) 3670.88 1.43552
\(188\) −229.290 −0.0889503
\(189\) 748.938 0.288239
\(190\) −1228.34 −0.469016
\(191\) −302.996 −0.114786 −0.0573928 0.998352i \(-0.518279\pi\)
−0.0573928 + 0.998352i \(0.518279\pi\)
\(192\) 6379.80 2.39803
\(193\) 3593.56 1.34026 0.670130 0.742244i \(-0.266239\pi\)
0.670130 + 0.742244i \(0.266239\pi\)
\(194\) −3963.86 −1.46695
\(195\) −3326.93 −1.22178
\(196\) −2143.11 −0.781017
\(197\) −2255.81 −0.815838 −0.407919 0.913018i \(-0.633745\pi\)
−0.407919 + 0.913018i \(0.633745\pi\)
\(198\) −7761.90 −2.78593
\(199\) 533.234 0.189950 0.0949748 0.995480i \(-0.469723\pi\)
0.0949748 + 0.995480i \(0.469723\pi\)
\(200\) 1325.13 0.468503
\(201\) 4105.36 1.44065
\(202\) 877.203 0.305543
\(203\) 1420.44 0.491110
\(204\) −6413.14 −2.20103
\(205\) 2839.41 0.967381
\(206\) 3873.78 1.31019
\(207\) 6745.80 2.26505
\(208\) 1910.17 0.636762
\(209\) −2430.55 −0.804425
\(210\) 2504.96 0.823138
\(211\) 4938.57 1.61130 0.805651 0.592390i \(-0.201816\pi\)
0.805651 + 0.592390i \(0.201816\pi\)
\(212\) 3125.19 1.01245
\(213\) 1223.46 0.393570
\(214\) −6572.75 −2.09955
\(215\) 0 0
\(216\) 882.054 0.277853
\(217\) 1342.70 0.420040
\(218\) 3344.79 1.03916
\(219\) 4266.67 1.31651
\(220\) −3372.55 −1.03353
\(221\) −5219.87 −1.58881
\(222\) −670.513 −0.202711
\(223\) 1407.81 0.422752 0.211376 0.977405i \(-0.432206\pi\)
0.211376 + 0.977405i \(0.432206\pi\)
\(224\) −2886.48 −0.860986
\(225\) −3148.29 −0.932827
\(226\) −5946.52 −1.75025
\(227\) 297.127 0.0868766 0.0434383 0.999056i \(-0.486169\pi\)
0.0434383 + 0.999056i \(0.486169\pi\)
\(228\) 4246.25 1.23340
\(229\) 3885.00 1.12108 0.560542 0.828126i \(-0.310593\pi\)
0.560542 + 0.828126i \(0.310593\pi\)
\(230\) 5002.15 1.43405
\(231\) 4956.65 1.41179
\(232\) 1672.91 0.473413
\(233\) 5538.50 1.55725 0.778625 0.627489i \(-0.215917\pi\)
0.778625 + 0.627489i \(0.215917\pi\)
\(234\) 11037.2 3.08343
\(235\) 118.518 0.0328990
\(236\) −740.276 −0.204186
\(237\) −9060.25 −2.48323
\(238\) 3930.23 1.07042
\(239\) −2719.79 −0.736103 −0.368052 0.929805i \(-0.619975\pi\)
−0.368052 + 0.929805i \(0.619975\pi\)
\(240\) −1213.06 −0.326261
\(241\) −1477.37 −0.394878 −0.197439 0.980315i \(-0.563262\pi\)
−0.197439 + 0.980315i \(0.563262\pi\)
\(242\) −5538.17 −1.47110
\(243\) 5261.23 1.38892
\(244\) 3737.48 0.980605
\(245\) 1107.76 0.288865
\(246\) −16751.3 −4.34155
\(247\) 3456.16 0.890326
\(248\) 1581.35 0.404904
\(249\) −6135.36 −1.56150
\(250\) −5550.03 −1.40406
\(251\) −3405.07 −0.856280 −0.428140 0.903712i \(-0.640831\pi\)
−0.428140 + 0.903712i \(0.640831\pi\)
\(252\) −4869.50 −1.21726
\(253\) 9897.90 2.45959
\(254\) −1007.75 −0.248944
\(255\) 3314.90 0.814067
\(256\) −1009.17 −0.246378
\(257\) −2338.70 −0.567642 −0.283821 0.958877i \(-0.591602\pi\)
−0.283821 + 0.958877i \(0.591602\pi\)
\(258\) 0 0
\(259\) 240.781 0.0577661
\(260\) 4795.66 1.14390
\(261\) −3974.56 −0.942602
\(262\) −7341.74 −1.73120
\(263\) 2652.29 0.621854 0.310927 0.950434i \(-0.399361\pi\)
0.310927 + 0.950434i \(0.399361\pi\)
\(264\) 5837.65 1.36092
\(265\) −1615.38 −0.374462
\(266\) −2602.27 −0.599833
\(267\) 4048.52 0.927961
\(268\) −5917.75 −1.34882
\(269\) −3751.84 −0.850387 −0.425193 0.905103i \(-0.639794\pi\)
−0.425193 + 0.905103i \(0.639794\pi\)
\(270\) −1553.94 −0.350259
\(271\) −6834.16 −1.53190 −0.765952 0.642898i \(-0.777732\pi\)
−0.765952 + 0.642898i \(0.777732\pi\)
\(272\) −1903.26 −0.424273
\(273\) −7048.20 −1.56255
\(274\) −9321.16 −2.05515
\(275\) −4619.39 −1.01294
\(276\) −17291.9 −3.77120
\(277\) −3328.70 −0.722029 −0.361015 0.932560i \(-0.617570\pi\)
−0.361015 + 0.932560i \(0.617570\pi\)
\(278\) 2403.20 0.518468
\(279\) −3757.05 −0.806195
\(280\) −1059.42 −0.226115
\(281\) −273.014 −0.0579596 −0.0289798 0.999580i \(-0.509226\pi\)
−0.0289798 + 0.999580i \(0.509226\pi\)
\(282\) −699.203 −0.147649
\(283\) 4738.49 0.995314 0.497657 0.867374i \(-0.334194\pi\)
0.497657 + 0.867374i \(0.334194\pi\)
\(284\) −1763.58 −0.368484
\(285\) −2194.85 −0.456181
\(286\) 16194.5 3.34825
\(287\) 6015.38 1.23720
\(288\) 8076.71 1.65252
\(289\) 287.999 0.0586199
\(290\) −2947.22 −0.596781
\(291\) −7082.81 −1.42681
\(292\) −6150.27 −1.23259
\(293\) 6636.13 1.32316 0.661581 0.749873i \(-0.269886\pi\)
0.661581 + 0.749873i \(0.269886\pi\)
\(294\) −6535.26 −1.29641
\(295\) 382.643 0.0755198
\(296\) 283.578 0.0556845
\(297\) −3074.84 −0.600741
\(298\) −14075.2 −2.73609
\(299\) −14074.5 −2.72224
\(300\) 8070.21 1.55311
\(301\) 0 0
\(302\) −5362.33 −1.02175
\(303\) 1567.43 0.297182
\(304\) 1260.18 0.237751
\(305\) −1931.87 −0.362684
\(306\) −10997.3 −2.05448
\(307\) 6964.04 1.29465 0.647327 0.762212i \(-0.275887\pi\)
0.647327 + 0.762212i \(0.275887\pi\)
\(308\) −7144.86 −1.32181
\(309\) 6921.84 1.27434
\(310\) −2785.92 −0.510419
\(311\) −1532.20 −0.279366 −0.139683 0.990196i \(-0.544608\pi\)
−0.139683 + 0.990196i \(0.544608\pi\)
\(312\) −8300.94 −1.50624
\(313\) −7971.39 −1.43952 −0.719760 0.694223i \(-0.755748\pi\)
−0.719760 + 0.694223i \(0.755748\pi\)
\(314\) 8504.24 1.52841
\(315\) 2517.00 0.450213
\(316\) 13060.1 2.32496
\(317\) −3644.45 −0.645719 −0.322860 0.946447i \(-0.604644\pi\)
−0.322860 + 0.946447i \(0.604644\pi\)
\(318\) 9530.05 1.68056
\(319\) −5831.75 −1.02356
\(320\) 4753.49 0.830400
\(321\) −11744.5 −2.04210
\(322\) 10597.2 1.83403
\(323\) −3443.67 −0.593222
\(324\) −5232.84 −0.897263
\(325\) 6568.62 1.12111
\(326\) 4797.94 0.815133
\(327\) 5976.62 1.01073
\(328\) 7084.55 1.19262
\(329\) 251.084 0.0420751
\(330\) −10284.4 −1.71556
\(331\) −5229.57 −0.868409 −0.434204 0.900814i \(-0.642970\pi\)
−0.434204 + 0.900814i \(0.642970\pi\)
\(332\) 8843.93 1.46197
\(333\) −673.735 −0.110872
\(334\) −10050.7 −1.64655
\(335\) 3058.83 0.498872
\(336\) −2569.91 −0.417261
\(337\) −4729.01 −0.764408 −0.382204 0.924078i \(-0.624835\pi\)
−0.382204 + 0.924078i \(0.624835\pi\)
\(338\) −13370.8 −2.15170
\(339\) −10625.5 −1.70236
\(340\) −4778.32 −0.762179
\(341\) −5512.60 −0.875437
\(342\) 7281.47 1.15128
\(343\) 6599.32 1.03886
\(344\) 0 0
\(345\) 8938.06 1.39481
\(346\) 5148.31 0.799927
\(347\) −10045.8 −1.55415 −0.777074 0.629409i \(-0.783297\pi\)
−0.777074 + 0.629409i \(0.783297\pi\)
\(348\) 10188.2 1.56939
\(349\) −2985.23 −0.457867 −0.228933 0.973442i \(-0.573524\pi\)
−0.228933 + 0.973442i \(0.573524\pi\)
\(350\) −4945.75 −0.755319
\(351\) 4372.32 0.664892
\(352\) 11850.7 1.79444
\(353\) 4066.23 0.613098 0.306549 0.951855i \(-0.400826\pi\)
0.306549 + 0.951855i \(0.400826\pi\)
\(354\) −2257.42 −0.338928
\(355\) 911.582 0.136287
\(356\) −5835.82 −0.868814
\(357\) 7022.72 1.04112
\(358\) 4095.59 0.604633
\(359\) −5734.13 −0.842996 −0.421498 0.906829i \(-0.638496\pi\)
−0.421498 + 0.906829i \(0.638496\pi\)
\(360\) 2964.37 0.433989
\(361\) −4578.89 −0.667574
\(362\) 1343.71 0.195094
\(363\) −9895.85 −1.43085
\(364\) 10159.8 1.46296
\(365\) 3179.02 0.455884
\(366\) 11397.2 1.62771
\(367\) −7460.30 −1.06110 −0.530551 0.847653i \(-0.678015\pi\)
−0.530551 + 0.847653i \(0.678015\pi\)
\(368\) −5131.83 −0.726943
\(369\) −16831.8 −2.37460
\(370\) −499.588 −0.0701955
\(371\) −3422.24 −0.478906
\(372\) 9630.67 1.34228
\(373\) −5646.39 −0.783805 −0.391902 0.920007i \(-0.628183\pi\)
−0.391902 + 0.920007i \(0.628183\pi\)
\(374\) −16135.9 −2.23093
\(375\) −9917.04 −1.36564
\(376\) 295.711 0.0405589
\(377\) 8292.56 1.13286
\(378\) −3292.08 −0.447953
\(379\) −3098.26 −0.419912 −0.209956 0.977711i \(-0.567332\pi\)
−0.209956 + 0.977711i \(0.567332\pi\)
\(380\) 3163.81 0.427105
\(381\) −1800.69 −0.242132
\(382\) 1331.87 0.178388
\(383\) −4625.15 −0.617060 −0.308530 0.951215i \(-0.599837\pi\)
−0.308530 + 0.951215i \(0.599837\pi\)
\(384\) −13414.3 −1.78267
\(385\) 3693.12 0.488880
\(386\) −15796.1 −2.08290
\(387\) 0 0
\(388\) 10209.6 1.33587
\(389\) 13403.3 1.74698 0.873488 0.486845i \(-0.161852\pi\)
0.873488 + 0.486845i \(0.161852\pi\)
\(390\) 14624.0 1.89876
\(391\) 14023.6 1.81382
\(392\) 2763.94 0.356122
\(393\) −13118.6 −1.68383
\(394\) 9915.79 1.26789
\(395\) −6750.64 −0.859903
\(396\) 19992.2 2.53698
\(397\) 8906.42 1.12595 0.562973 0.826475i \(-0.309657\pi\)
0.562973 + 0.826475i \(0.309657\pi\)
\(398\) −2343.92 −0.295201
\(399\) −4649.86 −0.583419
\(400\) 2395.04 0.299380
\(401\) −10.3507 −0.00128900 −0.000644499 1.00000i \(-0.500205\pi\)
−0.000644499 1.00000i \(0.500205\pi\)
\(402\) −18045.8 −2.23891
\(403\) 7838.73 0.968921
\(404\) −2259.40 −0.278240
\(405\) 2704.81 0.331860
\(406\) −6243.77 −0.763234
\(407\) −988.551 −0.120395
\(408\) 8270.93 1.00361
\(409\) 7522.58 0.909456 0.454728 0.890630i \(-0.349736\pi\)
0.454728 + 0.890630i \(0.349736\pi\)
\(410\) −12481.1 −1.50341
\(411\) −16655.5 −1.99891
\(412\) −9977.61 −1.19311
\(413\) 810.641 0.0965836
\(414\) −29652.2 −3.52012
\(415\) −4571.36 −0.540721
\(416\) −16851.3 −1.98607
\(417\) 4294.14 0.504280
\(418\) 10683.9 1.25016
\(419\) 109.458 0.0127622 0.00638110 0.999980i \(-0.497969\pi\)
0.00638110 + 0.999980i \(0.497969\pi\)
\(420\) −6451.99 −0.749583
\(421\) 8920.89 1.03273 0.516363 0.856370i \(-0.327286\pi\)
0.516363 + 0.856370i \(0.327286\pi\)
\(422\) −21708.3 −2.50413
\(423\) −702.563 −0.0807560
\(424\) −4030.51 −0.461648
\(425\) −6544.87 −0.746995
\(426\) −5377.93 −0.611647
\(427\) −4092.73 −0.463844
\(428\) 16929.3 1.91194
\(429\) 28937.1 3.25663
\(430\) 0 0
\(431\) −16280.0 −1.81944 −0.909720 0.415223i \(-0.863703\pi\)
−0.909720 + 0.415223i \(0.863703\pi\)
\(432\) 1594.23 0.177552
\(433\) −12363.2 −1.37214 −0.686070 0.727535i \(-0.740666\pi\)
−0.686070 + 0.727535i \(0.740666\pi\)
\(434\) −5902.07 −0.652784
\(435\) −5266.22 −0.580450
\(436\) −8615.12 −0.946306
\(437\) −9285.27 −1.01642
\(438\) −18754.8 −2.04598
\(439\) −15673.1 −1.70396 −0.851979 0.523576i \(-0.824598\pi\)
−0.851979 + 0.523576i \(0.824598\pi\)
\(440\) 4349.53 0.471263
\(441\) −6566.67 −0.709067
\(442\) 22944.8 2.46917
\(443\) 8049.79 0.863335 0.431667 0.902033i \(-0.357925\pi\)
0.431667 + 0.902033i \(0.357925\pi\)
\(444\) 1727.03 0.184597
\(445\) 3016.49 0.321338
\(446\) −6188.23 −0.656999
\(447\) −25150.2 −2.66121
\(448\) 10070.4 1.06201
\(449\) −7655.72 −0.804668 −0.402334 0.915493i \(-0.631801\pi\)
−0.402334 + 0.915493i \(0.631801\pi\)
\(450\) 13838.8 1.44971
\(451\) −24696.7 −2.57854
\(452\) 15316.3 1.59385
\(453\) −9581.65 −0.993787
\(454\) −1306.07 −0.135015
\(455\) −5251.50 −0.541086
\(456\) −5476.32 −0.562395
\(457\) 5935.62 0.607563 0.303782 0.952742i \(-0.401751\pi\)
0.303782 + 0.952742i \(0.401751\pi\)
\(458\) −17077.1 −1.74228
\(459\) −4356.51 −0.443016
\(460\) −12883.9 −1.30591
\(461\) 14649.8 1.48007 0.740033 0.672571i \(-0.234810\pi\)
0.740033 + 0.672571i \(0.234810\pi\)
\(462\) −21787.8 −2.19407
\(463\) 8724.18 0.875695 0.437848 0.899049i \(-0.355741\pi\)
0.437848 + 0.899049i \(0.355741\pi\)
\(464\) 3023.62 0.302518
\(465\) −4978.02 −0.496452
\(466\) −24345.4 −2.42012
\(467\) −211.747 −0.0209818 −0.0104909 0.999945i \(-0.503339\pi\)
−0.0104909 + 0.999945i \(0.503339\pi\)
\(468\) −28428.2 −2.80790
\(469\) 6480.24 0.638016
\(470\) −520.964 −0.0511283
\(471\) 15195.8 1.48659
\(472\) 954.724 0.0931032
\(473\) 0 0
\(474\) 39825.8 3.85919
\(475\) 4333.47 0.418596
\(476\) −10123.0 −0.974765
\(477\) 9575.85 0.919178
\(478\) 11955.3 1.14398
\(479\) 8169.97 0.779323 0.389661 0.920958i \(-0.372592\pi\)
0.389661 + 0.920958i \(0.372592\pi\)
\(480\) 10701.5 1.01761
\(481\) 1405.69 0.133251
\(482\) 6494.00 0.613679
\(483\) 18935.6 1.78385
\(484\) 14264.6 1.33965
\(485\) −5277.28 −0.494080
\(486\) −23126.6 −2.15852
\(487\) 8646.83 0.804570 0.402285 0.915515i \(-0.368216\pi\)
0.402285 + 0.915515i \(0.368216\pi\)
\(488\) −4820.17 −0.447129
\(489\) 8573.18 0.792828
\(490\) −4869.32 −0.448925
\(491\) 7773.64 0.714500 0.357250 0.934009i \(-0.383714\pi\)
0.357250 + 0.934009i \(0.383714\pi\)
\(492\) 43145.9 3.95359
\(493\) −8262.58 −0.754823
\(494\) −15192.1 −1.38366
\(495\) −10333.8 −0.938322
\(496\) 2858.15 0.258739
\(497\) 1931.22 0.174299
\(498\) 26969.0 2.42672
\(499\) 17480.8 1.56823 0.784114 0.620617i \(-0.213118\pi\)
0.784114 + 0.620617i \(0.213118\pi\)
\(500\) 14295.1 1.27859
\(501\) −17959.0 −1.60149
\(502\) 14967.5 1.33075
\(503\) −10065.2 −0.892218 −0.446109 0.894979i \(-0.647191\pi\)
−0.446109 + 0.894979i \(0.647191\pi\)
\(504\) 6280.12 0.555037
\(505\) 1167.86 0.102909
\(506\) −43507.8 −3.82245
\(507\) −23891.5 −2.09282
\(508\) 2595.64 0.226698
\(509\) 5106.23 0.444656 0.222328 0.974972i \(-0.428634\pi\)
0.222328 + 0.974972i \(0.428634\pi\)
\(510\) −14571.2 −1.26514
\(511\) 6734.86 0.583039
\(512\) −9227.10 −0.796454
\(513\) 2884.52 0.248255
\(514\) 10280.1 0.882172
\(515\) 5157.35 0.441281
\(516\) 0 0
\(517\) −1030.85 −0.0876919
\(518\) −1058.39 −0.0897743
\(519\) 9199.23 0.778037
\(520\) −6184.89 −0.521588
\(521\) −7651.54 −0.643416 −0.321708 0.946839i \(-0.604257\pi\)
−0.321708 + 0.946839i \(0.604257\pi\)
\(522\) 17470.8 1.46490
\(523\) −3318.37 −0.277442 −0.138721 0.990331i \(-0.544299\pi\)
−0.138721 + 0.990331i \(0.544299\pi\)
\(524\) 18910.0 1.57650
\(525\) −8837.29 −0.734650
\(526\) −11658.6 −0.966423
\(527\) −7810.39 −0.645590
\(528\) 10551.0 0.869646
\(529\) 25645.3 2.10777
\(530\) 7100.68 0.581951
\(531\) −2268.27 −0.185376
\(532\) 6702.62 0.546232
\(533\) 35117.9 2.85390
\(534\) −17795.9 −1.44214
\(535\) −8750.63 −0.707145
\(536\) 7632.03 0.615025
\(537\) 7318.18 0.588087
\(538\) 16491.8 1.32159
\(539\) −9635.07 −0.769966
\(540\) 4002.46 0.318960
\(541\) 3001.86 0.238558 0.119279 0.992861i \(-0.461942\pi\)
0.119279 + 0.992861i \(0.461942\pi\)
\(542\) 30040.7 2.38073
\(543\) 2401.01 0.189755
\(544\) 16790.4 1.32331
\(545\) 4453.08 0.349998
\(546\) 30981.5 2.42836
\(547\) −7159.56 −0.559636 −0.279818 0.960053i \(-0.590274\pi\)
−0.279818 + 0.960053i \(0.590274\pi\)
\(548\) 24008.3 1.87151
\(549\) 11452.0 0.890269
\(550\) 20305.3 1.57422
\(551\) 5470.79 0.422983
\(552\) 22301.2 1.71957
\(553\) −14301.4 −1.09975
\(554\) 14631.8 1.12211
\(555\) −892.687 −0.0682747
\(556\) −6189.87 −0.472138
\(557\) −8290.11 −0.630634 −0.315317 0.948986i \(-0.602111\pi\)
−0.315317 + 0.948986i \(0.602111\pi\)
\(558\) 16514.7 1.25291
\(559\) 0 0
\(560\) −1914.79 −0.144491
\(561\) −28832.4 −2.16989
\(562\) 1200.08 0.0900751
\(563\) 11830.8 0.885630 0.442815 0.896613i \(-0.353980\pi\)
0.442815 + 0.896613i \(0.353980\pi\)
\(564\) 1800.92 0.134455
\(565\) −7916.89 −0.589498
\(566\) −20828.8 −1.54682
\(567\) 5730.23 0.424421
\(568\) 2274.47 0.168019
\(569\) −661.208 −0.0487158 −0.0243579 0.999703i \(-0.507754\pi\)
−0.0243579 + 0.999703i \(0.507754\pi\)
\(570\) 9647.81 0.708951
\(571\) 13866.7 1.01629 0.508146 0.861271i \(-0.330331\pi\)
0.508146 + 0.861271i \(0.330331\pi\)
\(572\) −41711.9 −3.04906
\(573\) 2379.84 0.173507
\(574\) −26441.6 −1.92273
\(575\) −17647.1 −1.27989
\(576\) −28178.2 −2.03835
\(577\) 15498.6 1.11822 0.559112 0.829092i \(-0.311142\pi\)
0.559112 + 0.829092i \(0.311142\pi\)
\(578\) −1265.95 −0.0911011
\(579\) −28225.1 −2.02590
\(580\) 7591.09 0.543453
\(581\) −9684.56 −0.691538
\(582\) 31133.6 2.21740
\(583\) 14050.3 0.998123
\(584\) 7931.91 0.562029
\(585\) 14694.3 1.03852
\(586\) −29170.2 −2.05633
\(587\) 21748.6 1.52923 0.764617 0.644485i \(-0.222928\pi\)
0.764617 + 0.644485i \(0.222928\pi\)
\(588\) 16832.8 1.18056
\(589\) 5171.39 0.361772
\(590\) −1681.97 −0.117365
\(591\) 17718.0 1.23320
\(592\) 512.540 0.0355832
\(593\) 22274.2 1.54248 0.771240 0.636545i \(-0.219637\pi\)
0.771240 + 0.636545i \(0.219637\pi\)
\(594\) 13515.9 0.933612
\(595\) 5232.51 0.360524
\(596\) 36253.2 2.49159
\(597\) −4188.22 −0.287123
\(598\) 61866.7 4.23063
\(599\) 22572.5 1.53971 0.769856 0.638217i \(-0.220328\pi\)
0.769856 + 0.638217i \(0.220328\pi\)
\(600\) −10408.0 −0.708176
\(601\) −22982.2 −1.55984 −0.779918 0.625881i \(-0.784739\pi\)
−0.779918 + 0.625881i \(0.784739\pi\)
\(602\) 0 0
\(603\) −18132.5 −1.22456
\(604\) 13811.7 0.930444
\(605\) −7373.24 −0.495479
\(606\) −6889.87 −0.461851
\(607\) −9708.39 −0.649178 −0.324589 0.945855i \(-0.605226\pi\)
−0.324589 + 0.945855i \(0.605226\pi\)
\(608\) −11117.2 −0.741549
\(609\) −11156.7 −0.742349
\(610\) 8491.86 0.563648
\(611\) 1465.83 0.0970561
\(612\) 28325.4 1.87090
\(613\) 13924.3 0.917450 0.458725 0.888578i \(-0.348306\pi\)
0.458725 + 0.888578i \(0.348306\pi\)
\(614\) −30611.5 −2.01202
\(615\) −22301.8 −1.46227
\(616\) 9214.62 0.602707
\(617\) 7924.09 0.517037 0.258519 0.966006i \(-0.416766\pi\)
0.258519 + 0.966006i \(0.416766\pi\)
\(618\) −30426.0 −1.98044
\(619\) −7645.23 −0.496426 −0.248213 0.968705i \(-0.579843\pi\)
−0.248213 + 0.968705i \(0.579843\pi\)
\(620\) 7175.66 0.464808
\(621\) −11746.6 −0.759057
\(622\) 6735.02 0.434163
\(623\) 6390.53 0.410965
\(624\) −15003.2 −0.962512
\(625\) 3955.01 0.253121
\(626\) 35039.5 2.23716
\(627\) 19090.4 1.21595
\(628\) −21904.2 −1.39184
\(629\) −1400.60 −0.0887850
\(630\) −11063.9 −0.699676
\(631\) −10857.9 −0.685020 −0.342510 0.939514i \(-0.611277\pi\)
−0.342510 + 0.939514i \(0.611277\pi\)
\(632\) −16843.4 −1.06012
\(633\) −38789.3 −2.43560
\(634\) 16019.8 1.00351
\(635\) −1341.66 −0.0838461
\(636\) −24546.4 −1.53039
\(637\) 13700.8 0.852188
\(638\) 25634.4 1.59071
\(639\) −5403.77 −0.334538
\(640\) −9994.79 −0.617310
\(641\) −9993.21 −0.615769 −0.307885 0.951424i \(-0.599621\pi\)
−0.307885 + 0.951424i \(0.599621\pi\)
\(642\) 51624.8 3.17363
\(643\) 4952.30 0.303732 0.151866 0.988401i \(-0.451472\pi\)
0.151866 + 0.988401i \(0.451472\pi\)
\(644\) −27295.0 −1.67015
\(645\) 0 0
\(646\) 15137.2 0.921927
\(647\) −12682.7 −0.770644 −0.385322 0.922782i \(-0.625910\pi\)
−0.385322 + 0.922782i \(0.625910\pi\)
\(648\) 6748.71 0.409127
\(649\) −3328.16 −0.201297
\(650\) −28873.4 −1.74232
\(651\) −10546.1 −0.634921
\(652\) −12358.0 −0.742294
\(653\) −599.654 −0.0359361 −0.0179681 0.999839i \(-0.505720\pi\)
−0.0179681 + 0.999839i \(0.505720\pi\)
\(654\) −26271.2 −1.57077
\(655\) −9774.42 −0.583081
\(656\) 12804.7 0.762100
\(657\) −18845.0 −1.11904
\(658\) −1103.68 −0.0653889
\(659\) −19082.8 −1.12801 −0.564007 0.825770i \(-0.690741\pi\)
−0.564007 + 0.825770i \(0.690741\pi\)
\(660\) 26489.3 1.56226
\(661\) 30565.8 1.79860 0.899298 0.437336i \(-0.144078\pi\)
0.899298 + 0.437336i \(0.144078\pi\)
\(662\) 22987.4 1.34959
\(663\) 40998.8 2.40160
\(664\) −11405.9 −0.666618
\(665\) −3464.53 −0.202028
\(666\) 2961.51 0.172307
\(667\) −22278.6 −1.29330
\(668\) 25887.3 1.49942
\(669\) −11057.4 −0.639020
\(670\) −13445.6 −0.775296
\(671\) 16803.1 0.966732
\(672\) 22671.4 1.30144
\(673\) 16705.8 0.956850 0.478425 0.878128i \(-0.341208\pi\)
0.478425 + 0.878128i \(0.341208\pi\)
\(674\) 20787.1 1.18797
\(675\) 5482.17 0.312606
\(676\) 34438.9 1.95943
\(677\) 11786.3 0.669105 0.334553 0.942377i \(-0.391415\pi\)
0.334553 + 0.942377i \(0.391415\pi\)
\(678\) 46706.1 2.64563
\(679\) −11180.1 −0.631889
\(680\) 6162.53 0.347533
\(681\) −2333.74 −0.131320
\(682\) 24231.5 1.36052
\(683\) −2424.38 −0.135822 −0.0679110 0.997691i \(-0.521633\pi\)
−0.0679110 + 0.997691i \(0.521633\pi\)
\(684\) −18754.7 −1.04840
\(685\) −12409.7 −0.692191
\(686\) −29008.4 −1.61450
\(687\) −30514.2 −1.69460
\(688\) 0 0
\(689\) −19979.1 −1.10471
\(690\) −39288.7 −2.16767
\(691\) −3786.01 −0.208432 −0.104216 0.994555i \(-0.533233\pi\)
−0.104216 + 0.994555i \(0.533233\pi\)
\(692\) −13260.4 −0.728446
\(693\) −21892.5 −1.20004
\(694\) 44158.1 2.41530
\(695\) 3199.49 0.174624
\(696\) −13139.6 −0.715598
\(697\) −34991.0 −1.90155
\(698\) 13122.0 0.711571
\(699\) −43501.4 −2.35390
\(700\) 12738.7 0.687824
\(701\) 2332.79 0.125689 0.0628447 0.998023i \(-0.479983\pi\)
0.0628447 + 0.998023i \(0.479983\pi\)
\(702\) −19219.2 −1.03331
\(703\) 927.364 0.0497527
\(704\) −41345.0 −2.21342
\(705\) −930.883 −0.0497292
\(706\) −17873.8 −0.952815
\(707\) 2474.15 0.131613
\(708\) 5814.40 0.308642
\(709\) −33435.9 −1.77110 −0.885550 0.464544i \(-0.846218\pi\)
−0.885550 + 0.464544i \(0.846218\pi\)
\(710\) −4007.00 −0.211803
\(711\) 40017.2 2.11077
\(712\) 7526.37 0.396155
\(713\) −21059.4 −1.10614
\(714\) −30869.5 −1.61801
\(715\) 21560.5 1.12772
\(716\) −10548.9 −0.550603
\(717\) 21362.2 1.11267
\(718\) 25205.3 1.31010
\(719\) 21181.7 1.09867 0.549334 0.835603i \(-0.314881\pi\)
0.549334 + 0.835603i \(0.314881\pi\)
\(720\) 5357.82 0.277325
\(721\) 10926.0 0.564363
\(722\) 20127.2 1.03748
\(723\) 11603.8 0.596886
\(724\) −3460.98 −0.177661
\(725\) 10397.5 0.532626
\(726\) 43498.8 2.22368
\(727\) −29543.9 −1.50718 −0.753592 0.657343i \(-0.771680\pi\)
−0.753592 + 0.657343i \(0.771680\pi\)
\(728\) −13102.9 −0.667068
\(729\) −28844.5 −1.46545
\(730\) −13973.9 −0.708490
\(731\) 0 0
\(732\) −29355.5 −1.48226
\(733\) −17766.9 −0.895273 −0.447637 0.894216i \(-0.647734\pi\)
−0.447637 + 0.894216i \(0.647734\pi\)
\(734\) 32793.0 1.64906
\(735\) −8700.72 −0.436640
\(736\) 45272.4 2.26734
\(737\) −26605.2 −1.32974
\(738\) 73986.7 3.69036
\(739\) −21245.7 −1.05756 −0.528778 0.848760i \(-0.677350\pi\)
−0.528778 + 0.848760i \(0.677350\pi\)
\(740\) 1286.78 0.0639229
\(741\) −27146.0 −1.34579
\(742\) 15043.0 0.744268
\(743\) −397.687 −0.0196362 −0.00981812 0.999952i \(-0.503125\pi\)
−0.00981812 + 0.999952i \(0.503125\pi\)
\(744\) −12420.5 −0.612042
\(745\) −18739.0 −0.921534
\(746\) 24819.6 1.21811
\(747\) 27098.6 1.32729
\(748\) 41561.0 2.03158
\(749\) −18538.5 −0.904381
\(750\) 43591.9 2.12234
\(751\) 4939.05 0.239985 0.119992 0.992775i \(-0.461713\pi\)
0.119992 + 0.992775i \(0.461713\pi\)
\(752\) 534.470 0.0259177
\(753\) 26744.7 1.29433
\(754\) −36451.3 −1.76058
\(755\) −7139.13 −0.344132
\(756\) 8479.34 0.407924
\(757\) −32519.8 −1.56136 −0.780682 0.624928i \(-0.785128\pi\)
−0.780682 + 0.624928i \(0.785128\pi\)
\(758\) 13618.9 0.652585
\(759\) −77741.7 −3.71785
\(760\) −4080.31 −0.194748
\(761\) 2808.28 0.133771 0.0668856 0.997761i \(-0.478694\pi\)
0.0668856 + 0.997761i \(0.478694\pi\)
\(762\) 7915.22 0.376297
\(763\) 9434.00 0.447619
\(764\) −3430.47 −0.162448
\(765\) −14641.2 −0.691965
\(766\) 20330.6 0.958973
\(767\) 4732.54 0.222793
\(768\) 7926.35 0.372419
\(769\) −2452.64 −0.115012 −0.0575061 0.998345i \(-0.518315\pi\)
−0.0575061 + 0.998345i \(0.518315\pi\)
\(770\) −16233.7 −0.759769
\(771\) 18369.0 0.858032
\(772\) 40685.6 1.89677
\(773\) −15873.4 −0.738584 −0.369292 0.929313i \(-0.620400\pi\)
−0.369292 + 0.929313i \(0.620400\pi\)
\(774\) 0 0
\(775\) 9828.49 0.455548
\(776\) −13167.2 −0.609118
\(777\) −1891.18 −0.0873177
\(778\) −58916.3 −2.71498
\(779\) 23168.1 1.06558
\(780\) −37666.9 −1.72909
\(781\) −7928.79 −0.363271
\(782\) −61643.0 −2.81886
\(783\) 6920.98 0.315882
\(784\) 4995.55 0.227567
\(785\) 11322.1 0.514781
\(786\) 57664.7 2.61683
\(787\) −5695.97 −0.257992 −0.128996 0.991645i \(-0.541175\pi\)
−0.128996 + 0.991645i \(0.541175\pi\)
\(788\) −25539.9 −1.15460
\(789\) −20832.1 −0.939977
\(790\) 29673.5 1.33638
\(791\) −16772.2 −0.753919
\(792\) −25783.6 −1.15679
\(793\) −23893.5 −1.06996
\(794\) −39149.6 −1.74983
\(795\) 12687.8 0.566026
\(796\) 6037.18 0.268822
\(797\) −2203.96 −0.0979528 −0.0489764 0.998800i \(-0.515596\pi\)
−0.0489764 + 0.998800i \(0.515596\pi\)
\(798\) 20439.2 0.906691
\(799\) −1460.53 −0.0646683
\(800\) −21128.8 −0.933770
\(801\) −17881.5 −0.788777
\(802\) 45.4980 0.00200323
\(803\) −27650.6 −1.21515
\(804\) 46480.1 2.03884
\(805\) 14108.6 0.617717
\(806\) −34456.4 −1.50580
\(807\) 29468.4 1.28542
\(808\) 2913.91 0.126870
\(809\) −247.927 −0.0107746 −0.00538730 0.999985i \(-0.501715\pi\)
−0.00538730 + 0.999985i \(0.501715\pi\)
\(810\) −11889.4 −0.515743
\(811\) 16460.2 0.712694 0.356347 0.934354i \(-0.384022\pi\)
0.356347 + 0.934354i \(0.384022\pi\)
\(812\) 16082.0 0.695032
\(813\) 53678.0 2.31558
\(814\) 4345.33 0.187105
\(815\) 6387.74 0.274543
\(816\) 14948.9 0.641320
\(817\) 0 0
\(818\) −33066.7 −1.41339
\(819\) 31130.4 1.32818
\(820\) 32147.3 1.36906
\(821\) 26642.8 1.13257 0.566286 0.824209i \(-0.308380\pi\)
0.566286 + 0.824209i \(0.308380\pi\)
\(822\) 73211.7 3.10651
\(823\) 13805.3 0.584716 0.292358 0.956309i \(-0.405560\pi\)
0.292358 + 0.956309i \(0.405560\pi\)
\(824\) 12868.0 0.544026
\(825\) 36282.3 1.53114
\(826\) −3563.30 −0.150101
\(827\) −42308.6 −1.77898 −0.889488 0.456959i \(-0.848939\pi\)
−0.889488 + 0.456959i \(0.848939\pi\)
\(828\) 76374.7 3.20556
\(829\) −1206.36 −0.0505412 −0.0252706 0.999681i \(-0.508045\pi\)
−0.0252706 + 0.999681i \(0.508045\pi\)
\(830\) 20094.1 0.840334
\(831\) 26144.8 1.09140
\(832\) 58791.3 2.44978
\(833\) −13651.2 −0.567811
\(834\) −18875.6 −0.783702
\(835\) −13381.0 −0.554571
\(836\) −27518.3 −1.13844
\(837\) 6542.21 0.270170
\(838\) −481.139 −0.0198338
\(839\) −23105.7 −0.950773 −0.475386 0.879777i \(-0.657692\pi\)
−0.475386 + 0.879777i \(0.657692\pi\)
\(840\) 8321.04 0.341789
\(841\) −11262.6 −0.461792
\(842\) −39213.2 −1.60496
\(843\) 2144.35 0.0876102
\(844\) 55913.5 2.28036
\(845\) −17801.2 −0.724709
\(846\) 3088.23 0.125503
\(847\) −15620.4 −0.633677
\(848\) −7284.77 −0.295000
\(849\) −37217.8 −1.50449
\(850\) 28769.0 1.16090
\(851\) −3776.49 −0.152123
\(852\) 13851.8 0.556990
\(853\) −16448.8 −0.660253 −0.330126 0.943937i \(-0.607091\pi\)
−0.330126 + 0.943937i \(0.607091\pi\)
\(854\) 17990.3 0.720860
\(855\) 9694.18 0.387759
\(856\) −21833.5 −0.871791
\(857\) 35801.6 1.42702 0.713512 0.700643i \(-0.247103\pi\)
0.713512 + 0.700643i \(0.247103\pi\)
\(858\) −127197. −5.06113
\(859\) 10727.9 0.426113 0.213057 0.977040i \(-0.431658\pi\)
0.213057 + 0.977040i \(0.431658\pi\)
\(860\) 0 0
\(861\) −47247.0 −1.87012
\(862\) 71561.2 2.82759
\(863\) −16939.7 −0.668172 −0.334086 0.942543i \(-0.608428\pi\)
−0.334086 + 0.942543i \(0.608428\pi\)
\(864\) −14064.1 −0.553786
\(865\) 6854.19 0.269421
\(866\) 54344.4 2.13244
\(867\) −2262.05 −0.0886082
\(868\) 15201.9 0.594452
\(869\) 58715.9 2.29206
\(870\) 23148.5 0.902078
\(871\) 37831.8 1.47173
\(872\) 11110.8 0.431489
\(873\) 31283.2 1.21280
\(874\) 40814.9 1.57961
\(875\) −15653.9 −0.604797
\(876\) 48306.5 1.86316
\(877\) −9358.04 −0.360318 −0.180159 0.983638i \(-0.557661\pi\)
−0.180159 + 0.983638i \(0.557661\pi\)
\(878\) 68893.7 2.64812
\(879\) −52122.6 −2.00006
\(880\) 7861.37 0.301144
\(881\) 11704.6 0.447603 0.223801 0.974635i \(-0.428153\pi\)
0.223801 + 0.974635i \(0.428153\pi\)
\(882\) 28864.8 1.10196
\(883\) 3294.49 0.125559 0.0627794 0.998027i \(-0.480004\pi\)
0.0627794 + 0.998027i \(0.480004\pi\)
\(884\) −59098.4 −2.24852
\(885\) −3005.42 −0.114154
\(886\) −35384.2 −1.34171
\(887\) −19593.1 −0.741683 −0.370842 0.928696i \(-0.620931\pi\)
−0.370842 + 0.928696i \(0.620931\pi\)
\(888\) −2227.32 −0.0841712
\(889\) −2842.36 −0.107232
\(890\) −13259.5 −0.499391
\(891\) −23526.0 −0.884568
\(892\) 15938.9 0.598290
\(893\) 967.044 0.0362384
\(894\) 110552. 4.13579
\(895\) 5452.66 0.203645
\(896\) −21174.3 −0.789490
\(897\) 110546. 4.11486
\(898\) 33652.0 1.25053
\(899\) 12408.0 0.460322
\(900\) −35644.4 −1.32016
\(901\) 19906.9 0.736065
\(902\) 108558. 4.00731
\(903\) 0 0
\(904\) −19753.3 −0.726752
\(905\) 1788.95 0.0657092
\(906\) 42117.7 1.54444
\(907\) −33443.1 −1.22432 −0.612162 0.790733i \(-0.709700\pi\)
−0.612162 + 0.790733i \(0.709700\pi\)
\(908\) 3364.02 0.122950
\(909\) −6922.98 −0.252608
\(910\) 23083.8 0.840901
\(911\) 31893.5 1.15991 0.579955 0.814649i \(-0.303070\pi\)
0.579955 + 0.814649i \(0.303070\pi\)
\(912\) −9897.93 −0.359379
\(913\) 39760.9 1.44129
\(914\) −26090.9 −0.944214
\(915\) 15173.6 0.548224
\(916\) 43985.3 1.58659
\(917\) −20707.4 −0.745713
\(918\) 19149.7 0.688492
\(919\) 15843.1 0.568678 0.284339 0.958724i \(-0.408226\pi\)
0.284339 + 0.958724i \(0.408226\pi\)
\(920\) 16616.2 0.595457
\(921\) −54698.1 −1.95696
\(922\) −64395.6 −2.30017
\(923\) 11274.5 0.402063
\(924\) 56118.3 1.99801
\(925\) 1762.50 0.0626494
\(926\) −38348.5 −1.36092
\(927\) −30572.3 −1.08320
\(928\) −26674.1 −0.943555
\(929\) 5624.10 0.198623 0.0993114 0.995056i \(-0.468336\pi\)
0.0993114 + 0.995056i \(0.468336\pi\)
\(930\) 21881.7 0.771536
\(931\) 9038.70 0.318186
\(932\) 62705.9 2.20386
\(933\) 12034.4 0.422283
\(934\) 930.767 0.0326077
\(935\) −21482.6 −0.751396
\(936\) 36663.5 1.28032
\(937\) −24982.7 −0.871024 −0.435512 0.900183i \(-0.643433\pi\)
−0.435512 + 0.900183i \(0.643433\pi\)
\(938\) −28484.9 −0.991541
\(939\) 62610.2 2.17594
\(940\) 1341.84 0.0465595
\(941\) −2409.74 −0.0834806 −0.0417403 0.999128i \(-0.513290\pi\)
−0.0417403 + 0.999128i \(0.513290\pi\)
\(942\) −66795.4 −2.31031
\(943\) −94347.2 −3.25808
\(944\) 1725.57 0.0594943
\(945\) −4382.90 −0.150874
\(946\) 0 0
\(947\) 13276.4 0.455569 0.227785 0.973712i \(-0.426852\pi\)
0.227785 + 0.973712i \(0.426852\pi\)
\(948\) −102579. −3.51434
\(949\) 39318.3 1.34492
\(950\) −19048.4 −0.650540
\(951\) 28624.9 0.976051
\(952\) 13055.5 0.444466
\(953\) 16222.6 0.551418 0.275709 0.961241i \(-0.411087\pi\)
0.275709 + 0.961241i \(0.411087\pi\)
\(954\) −42092.2 −1.42849
\(955\) 1773.18 0.0600825
\(956\) −30793.0 −1.04175
\(957\) 45804.7 1.54718
\(958\) −35912.4 −1.21115
\(959\) −26290.4 −0.885255
\(960\) −37335.6 −1.25521
\(961\) −18062.1 −0.606292
\(962\) −6178.92 −0.207086
\(963\) 51872.9 1.73581
\(964\) −16726.5 −0.558842
\(965\) −21030.1 −0.701535
\(966\) −83234.3 −2.77228
\(967\) 22602.3 0.751647 0.375823 0.926691i \(-0.377360\pi\)
0.375823 + 0.926691i \(0.377360\pi\)
\(968\) −18396.8 −0.610842
\(969\) 27047.8 0.896699
\(970\) 23197.1 0.767851
\(971\) 8215.32 0.271516 0.135758 0.990742i \(-0.456653\pi\)
0.135758 + 0.990742i \(0.456653\pi\)
\(972\) 59566.7 1.96564
\(973\) 6778.22 0.223330
\(974\) −38008.5 −1.25038
\(975\) −51592.3 −1.69464
\(976\) −8712.01 −0.285722
\(977\) 13701.3 0.448662 0.224331 0.974513i \(-0.427980\pi\)
0.224331 + 0.974513i \(0.427980\pi\)
\(978\) −37684.8 −1.23213
\(979\) −26236.9 −0.856522
\(980\) 12541.8 0.408810
\(981\) −26397.5 −0.859129
\(982\) −34170.3 −1.11040
\(983\) 35295.3 1.14521 0.572606 0.819830i \(-0.305932\pi\)
0.572606 + 0.819830i \(0.305932\pi\)
\(984\) −55644.6 −1.80273
\(985\) 13201.4 0.427036
\(986\) 36319.5 1.17307
\(987\) −1972.10 −0.0635996
\(988\) 39130.1 1.26001
\(989\) 0 0
\(990\) 45423.8 1.45825
\(991\) −10086.0 −0.323302 −0.161651 0.986848i \(-0.551682\pi\)
−0.161651 + 0.986848i \(0.551682\pi\)
\(992\) −25214.3 −0.807011
\(993\) 41075.0 1.31266
\(994\) −8488.96 −0.270879
\(995\) −3120.57 −0.0994259
\(996\) −69463.5 −2.20987
\(997\) −14690.8 −0.466662 −0.233331 0.972397i \(-0.574963\pi\)
−0.233331 + 0.972397i \(0.574963\pi\)
\(998\) −76839.4 −2.43718
\(999\) 1173.19 0.0371552
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.1.5 30
43.8 odd 14 43.4.e.a.21.2 60
43.27 odd 14 43.4.e.a.41.2 yes 60
43.42 odd 2 1849.4.a.h.1.26 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.e.a.21.2 60 43.8 odd 14
43.4.e.a.41.2 yes 60 43.27 odd 14
1849.4.a.g.1.5 30 1.1 even 1 trivial
1849.4.a.h.1.26 30 43.42 odd 2