Properties

Label 1849.4.a.j.1.36
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $50$
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: \(50\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.36
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.06679 q^{2} +2.01332 q^{3} +1.40518 q^{4} +11.9365 q^{5} +6.17442 q^{6} +10.6583 q^{7} -20.2249 q^{8} -22.9465 q^{9} +O(q^{10})\) \(q+3.06679 q^{2} +2.01332 q^{3} +1.40518 q^{4} +11.9365 q^{5} +6.17442 q^{6} +10.6583 q^{7} -20.2249 q^{8} -22.9465 q^{9} +36.6067 q^{10} +65.4818 q^{11} +2.82908 q^{12} -61.8987 q^{13} +32.6868 q^{14} +24.0320 q^{15} -73.2669 q^{16} -22.8412 q^{17} -70.3722 q^{18} -36.6127 q^{19} +16.7730 q^{20} +21.4586 q^{21} +200.819 q^{22} -143.179 q^{23} -40.7192 q^{24} +17.4799 q^{25} -189.830 q^{26} -100.558 q^{27} +14.9769 q^{28} -213.016 q^{29} +73.7010 q^{30} -307.040 q^{31} -62.8949 q^{32} +131.836 q^{33} -70.0492 q^{34} +127.223 q^{35} -32.2441 q^{36} +83.7905 q^{37} -112.284 q^{38} -124.622 q^{39} -241.414 q^{40} +59.1767 q^{41} +65.8090 q^{42} +92.0140 q^{44} -273.901 q^{45} -439.100 q^{46} +201.957 q^{47} -147.510 q^{48} -229.400 q^{49} +53.6072 q^{50} -45.9867 q^{51} -86.9790 q^{52} +406.354 q^{53} -308.391 q^{54} +781.624 q^{55} -215.563 q^{56} -73.7131 q^{57} -653.274 q^{58} +164.777 q^{59} +33.7694 q^{60} +150.450 q^{61} -941.626 q^{62} -244.572 q^{63} +393.250 q^{64} -738.853 q^{65} +404.312 q^{66} +169.033 q^{67} -32.0962 q^{68} -288.265 q^{69} +390.166 q^{70} -1039.96 q^{71} +464.092 q^{72} -374.665 q^{73} +256.968 q^{74} +35.1926 q^{75} -51.4477 q^{76} +697.926 q^{77} -382.188 q^{78} +465.164 q^{79} -874.550 q^{80} +417.101 q^{81} +181.482 q^{82} +1307.62 q^{83} +30.1533 q^{84} -272.644 q^{85} -428.869 q^{87} -1324.36 q^{88} +66.7489 q^{89} -839.997 q^{90} -659.736 q^{91} -201.193 q^{92} -618.169 q^{93} +619.359 q^{94} -437.028 q^{95} -126.628 q^{96} -953.889 q^{97} -703.522 q^{98} -1502.58 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.06679 1.08427 0.542137 0.840290i \(-0.317616\pi\)
0.542137 + 0.840290i \(0.317616\pi\)
\(3\) 2.01332 0.387463 0.193732 0.981055i \(-0.437941\pi\)
0.193732 + 0.981055i \(0.437941\pi\)
\(4\) 1.40518 0.175648
\(5\) 11.9365 1.06763 0.533816 0.845600i \(-0.320757\pi\)
0.533816 + 0.845600i \(0.320757\pi\)
\(6\) 6.17442 0.420116
\(7\) 10.6583 0.575495 0.287748 0.957706i \(-0.407094\pi\)
0.287748 + 0.957706i \(0.407094\pi\)
\(8\) −20.2249 −0.893823
\(9\) −22.9465 −0.849872
\(10\) 36.6067 1.15761
\(11\) 65.4818 1.79487 0.897433 0.441152i \(-0.145430\pi\)
0.897433 + 0.441152i \(0.145430\pi\)
\(12\) 2.82908 0.0680572
\(13\) −61.8987 −1.32058 −0.660292 0.751009i \(-0.729568\pi\)
−0.660292 + 0.751009i \(0.729568\pi\)
\(14\) 32.6868 0.623994
\(15\) 24.0320 0.413669
\(16\) −73.2669 −1.14480
\(17\) −22.8412 −0.325871 −0.162936 0.986637i \(-0.552096\pi\)
−0.162936 + 0.986637i \(0.552096\pi\)
\(18\) −70.3722 −0.921493
\(19\) −36.6127 −0.442081 −0.221041 0.975265i \(-0.570945\pi\)
−0.221041 + 0.975265i \(0.570945\pi\)
\(20\) 16.7730 0.187528
\(21\) 21.4586 0.222983
\(22\) 200.819 1.94612
\(23\) −143.179 −1.29804 −0.649020 0.760772i \(-0.724821\pi\)
−0.649020 + 0.760772i \(0.724821\pi\)
\(24\) −40.7192 −0.346324
\(25\) 17.4799 0.139839
\(26\) −189.830 −1.43187
\(27\) −100.558 −0.716758
\(28\) 14.9769 0.101085
\(29\) −213.016 −1.36400 −0.682001 0.731351i \(-0.738890\pi\)
−0.682001 + 0.731351i \(0.738890\pi\)
\(30\) 73.7010 0.448530
\(31\) −307.040 −1.77890 −0.889452 0.457029i \(-0.848913\pi\)
−0.889452 + 0.457029i \(0.848913\pi\)
\(32\) −62.8949 −0.347449
\(33\) 131.836 0.695445
\(34\) −70.0492 −0.353334
\(35\) 127.223 0.614417
\(36\) −32.2441 −0.149278
\(37\) 83.7905 0.372299 0.186150 0.982521i \(-0.440399\pi\)
0.186150 + 0.982521i \(0.440399\pi\)
\(38\) −112.284 −0.479337
\(39\) −124.622 −0.511678
\(40\) −241.414 −0.954274
\(41\) 59.1767 0.225411 0.112705 0.993628i \(-0.464048\pi\)
0.112705 + 0.993628i \(0.464048\pi\)
\(42\) 65.8090 0.241775
\(43\) 0 0
\(44\) 92.0140 0.315265
\(45\) −273.901 −0.907351
\(46\) −439.100 −1.40743
\(47\) 201.957 0.626775 0.313388 0.949625i \(-0.398536\pi\)
0.313388 + 0.949625i \(0.398536\pi\)
\(48\) −147.510 −0.443567
\(49\) −229.400 −0.668805
\(50\) 53.6072 0.151624
\(51\) −45.9867 −0.126263
\(52\) −86.9790 −0.231958
\(53\) 406.354 1.05315 0.526576 0.850128i \(-0.323476\pi\)
0.526576 + 0.850128i \(0.323476\pi\)
\(54\) −308.391 −0.777161
\(55\) 781.624 1.91626
\(56\) −215.563 −0.514391
\(57\) −73.7131 −0.171290
\(58\) −653.274 −1.47895
\(59\) 164.777 0.363595 0.181798 0.983336i \(-0.441808\pi\)
0.181798 + 0.983336i \(0.441808\pi\)
\(60\) 33.7694 0.0726601
\(61\) 150.450 0.315790 0.157895 0.987456i \(-0.449529\pi\)
0.157895 + 0.987456i \(0.449529\pi\)
\(62\) −941.626 −1.92882
\(63\) −244.572 −0.489097
\(64\) 393.250 0.768067
\(65\) −738.853 −1.40990
\(66\) 404.312 0.754052
\(67\) 169.033 0.308220 0.154110 0.988054i \(-0.450749\pi\)
0.154110 + 0.988054i \(0.450749\pi\)
\(68\) −32.0962 −0.0572387
\(69\) −288.265 −0.502943
\(70\) 390.166 0.666196
\(71\) −1039.96 −1.73832 −0.869162 0.494528i \(-0.835341\pi\)
−0.869162 + 0.494528i \(0.835341\pi\)
\(72\) 464.092 0.759635
\(73\) −374.665 −0.600701 −0.300351 0.953829i \(-0.597104\pi\)
−0.300351 + 0.953829i \(0.597104\pi\)
\(74\) 256.968 0.403674
\(75\) 35.1926 0.0541826
\(76\) −51.4477 −0.0776507
\(77\) 697.926 1.03294
\(78\) −382.188 −0.554799
\(79\) 465.164 0.662468 0.331234 0.943549i \(-0.392535\pi\)
0.331234 + 0.943549i \(0.392535\pi\)
\(80\) −874.550 −1.22222
\(81\) 417.101 0.572155
\(82\) 181.482 0.244407
\(83\) 1307.62 1.72928 0.864641 0.502391i \(-0.167546\pi\)
0.864641 + 0.502391i \(0.167546\pi\)
\(84\) 30.1533 0.0391666
\(85\) −272.644 −0.347911
\(86\) 0 0
\(87\) −428.869 −0.528501
\(88\) −1324.36 −1.60429
\(89\) 66.7489 0.0794986 0.0397493 0.999210i \(-0.487344\pi\)
0.0397493 + 0.999210i \(0.487344\pi\)
\(90\) −839.997 −0.983816
\(91\) −659.736 −0.759990
\(92\) −201.193 −0.227998
\(93\) −618.169 −0.689260
\(94\) 619.359 0.679596
\(95\) −437.028 −0.471980
\(96\) −126.628 −0.134624
\(97\) −953.889 −0.998482 −0.499241 0.866463i \(-0.666388\pi\)
−0.499241 + 0.866463i \(0.666388\pi\)
\(98\) −703.522 −0.725167
\(99\) −1502.58 −1.52541
\(100\) 24.5625 0.0245625
\(101\) −1507.22 −1.48489 −0.742446 0.669905i \(-0.766335\pi\)
−0.742446 + 0.669905i \(0.766335\pi\)
\(102\) −141.031 −0.136904
\(103\) 1071.84 1.02536 0.512679 0.858581i \(-0.328653\pi\)
0.512679 + 0.858581i \(0.328653\pi\)
\(104\) 1251.89 1.18037
\(105\) 256.141 0.238064
\(106\) 1246.20 1.14190
\(107\) 924.397 0.835186 0.417593 0.908634i \(-0.362874\pi\)
0.417593 + 0.908634i \(0.362874\pi\)
\(108\) −141.303 −0.125897
\(109\) −1584.45 −1.39232 −0.696161 0.717886i \(-0.745110\pi\)
−0.696161 + 0.717886i \(0.745110\pi\)
\(110\) 2397.07 2.07775
\(111\) 168.697 0.144252
\(112\) −780.902 −0.658825
\(113\) −1539.52 −1.28164 −0.640822 0.767690i \(-0.721406\pi\)
−0.640822 + 0.767690i \(0.721406\pi\)
\(114\) −226.063 −0.185725
\(115\) −1709.06 −1.38583
\(116\) −299.326 −0.239584
\(117\) 1420.36 1.12233
\(118\) 505.335 0.394236
\(119\) −243.449 −0.187537
\(120\) −486.044 −0.369746
\(121\) 2956.87 2.22154
\(122\) 461.399 0.342403
\(123\) 119.142 0.0873385
\(124\) −431.448 −0.312461
\(125\) −1283.41 −0.918336
\(126\) −750.049 −0.530315
\(127\) −626.337 −0.437625 −0.218813 0.975767i \(-0.570218\pi\)
−0.218813 + 0.975767i \(0.570218\pi\)
\(128\) 1709.17 1.18024
\(129\) 0 0
\(130\) −2265.91 −1.52872
\(131\) 1299.79 0.866894 0.433447 0.901179i \(-0.357297\pi\)
0.433447 + 0.901179i \(0.357297\pi\)
\(132\) 185.254 0.122153
\(133\) −390.230 −0.254416
\(134\) 518.389 0.334194
\(135\) −1200.31 −0.765234
\(136\) 461.962 0.291271
\(137\) −1833.35 −1.14331 −0.571656 0.820493i \(-0.693699\pi\)
−0.571656 + 0.820493i \(0.693699\pi\)
\(138\) −884.048 −0.545327
\(139\) 354.867 0.216543 0.108271 0.994121i \(-0.465468\pi\)
0.108271 + 0.994121i \(0.465468\pi\)
\(140\) 178.772 0.107921
\(141\) 406.604 0.242853
\(142\) −3189.35 −1.88482
\(143\) −4053.24 −2.37027
\(144\) 1681.22 0.972930
\(145\) −2542.66 −1.45625
\(146\) −1149.02 −0.651324
\(147\) −461.856 −0.259138
\(148\) 117.741 0.0653937
\(149\) 3282.46 1.80476 0.902382 0.430938i \(-0.141817\pi\)
0.902382 + 0.430938i \(0.141817\pi\)
\(150\) 107.928 0.0587488
\(151\) −124.151 −0.0669093 −0.0334546 0.999440i \(-0.510651\pi\)
−0.0334546 + 0.999440i \(0.510651\pi\)
\(152\) 740.489 0.395142
\(153\) 524.128 0.276949
\(154\) 2140.39 1.11999
\(155\) −3664.98 −1.89921
\(156\) −175.117 −0.0898753
\(157\) −1119.99 −0.569332 −0.284666 0.958627i \(-0.591883\pi\)
−0.284666 + 0.958627i \(0.591883\pi\)
\(158\) 1426.56 0.718297
\(159\) 818.121 0.408058
\(160\) −750.745 −0.370947
\(161\) −1526.05 −0.747015
\(162\) 1279.16 0.620372
\(163\) −2820.81 −1.35548 −0.677739 0.735303i \(-0.737040\pi\)
−0.677739 + 0.735303i \(0.737040\pi\)
\(164\) 83.1542 0.0395930
\(165\) 1573.66 0.742479
\(166\) 4010.20 1.87501
\(167\) 1945.07 0.901281 0.450640 0.892706i \(-0.351196\pi\)
0.450640 + 0.892706i \(0.351196\pi\)
\(168\) −433.998 −0.199308
\(169\) 1634.44 0.743944
\(170\) −836.142 −0.377231
\(171\) 840.136 0.375712
\(172\) 0 0
\(173\) −1105.62 −0.485889 −0.242945 0.970040i \(-0.578113\pi\)
−0.242945 + 0.970040i \(0.578113\pi\)
\(174\) −1315.25 −0.573039
\(175\) 186.307 0.0804769
\(176\) −4797.65 −2.05475
\(177\) 331.748 0.140880
\(178\) 204.705 0.0861982
\(179\) 1154.35 0.482014 0.241007 0.970523i \(-0.422522\pi\)
0.241007 + 0.970523i \(0.422522\pi\)
\(180\) −384.882 −0.159374
\(181\) −3547.67 −1.45688 −0.728442 0.685108i \(-0.759755\pi\)
−0.728442 + 0.685108i \(0.759755\pi\)
\(182\) −2023.27 −0.824037
\(183\) 302.905 0.122357
\(184\) 2895.78 1.16022
\(185\) 1000.17 0.397479
\(186\) −1895.79 −0.747346
\(187\) −1495.69 −0.584895
\(188\) 283.787 0.110092
\(189\) −1071.78 −0.412491
\(190\) −1340.27 −0.511755
\(191\) −2388.85 −0.904979 −0.452490 0.891770i \(-0.649464\pi\)
−0.452490 + 0.891770i \(0.649464\pi\)
\(192\) 791.738 0.297598
\(193\) 775.397 0.289193 0.144597 0.989491i \(-0.453812\pi\)
0.144597 + 0.989491i \(0.453812\pi\)
\(194\) −2925.37 −1.08263
\(195\) −1487.55 −0.546284
\(196\) −322.350 −0.117474
\(197\) 3582.80 1.29576 0.647878 0.761744i \(-0.275657\pi\)
0.647878 + 0.761744i \(0.275657\pi\)
\(198\) −4608.10 −1.65396
\(199\) 3110.52 1.10804 0.554018 0.832505i \(-0.313094\pi\)
0.554018 + 0.832505i \(0.313094\pi\)
\(200\) −353.529 −0.124992
\(201\) 340.318 0.119424
\(202\) −4622.33 −1.61003
\(203\) −2270.39 −0.784976
\(204\) −64.6198 −0.0221779
\(205\) 706.363 0.240656
\(206\) 3287.11 1.11177
\(207\) 3285.46 1.10317
\(208\) 4535.13 1.51180
\(209\) −2397.47 −0.793476
\(210\) 785.528 0.258127
\(211\) −1101.39 −0.359350 −0.179675 0.983726i \(-0.557505\pi\)
−0.179675 + 0.983726i \(0.557505\pi\)
\(212\) 571.003 0.184984
\(213\) −2093.78 −0.673537
\(214\) 2834.93 0.905569
\(215\) 0 0
\(216\) 2033.78 0.640654
\(217\) −3272.53 −1.02375
\(218\) −4859.18 −1.50966
\(219\) −754.320 −0.232750
\(220\) 1098.33 0.336587
\(221\) 1413.84 0.430341
\(222\) 517.358 0.156409
\(223\) −2413.48 −0.724748 −0.362374 0.932033i \(-0.618034\pi\)
−0.362374 + 0.932033i \(0.618034\pi\)
\(224\) −670.354 −0.199955
\(225\) −401.104 −0.118846
\(226\) −4721.38 −1.38965
\(227\) −2233.25 −0.652979 −0.326489 0.945201i \(-0.605866\pi\)
−0.326489 + 0.945201i \(0.605866\pi\)
\(228\) −103.581 −0.0300868
\(229\) −1359.76 −0.392381 −0.196190 0.980566i \(-0.562857\pi\)
−0.196190 + 0.980566i \(0.562857\pi\)
\(230\) −5241.31 −1.50262
\(231\) 1405.15 0.400225
\(232\) 4308.22 1.21918
\(233\) 6996.98 1.96733 0.983665 0.180012i \(-0.0576135\pi\)
0.983665 + 0.180012i \(0.0576135\pi\)
\(234\) 4355.94 1.21691
\(235\) 2410.66 0.669166
\(236\) 231.542 0.0638648
\(237\) 936.523 0.256682
\(238\) −746.607 −0.203342
\(239\) 4353.01 1.17813 0.589064 0.808086i \(-0.299496\pi\)
0.589064 + 0.808086i \(0.299496\pi\)
\(240\) −1760.75 −0.473566
\(241\) 3059.19 0.817676 0.408838 0.912607i \(-0.365934\pi\)
0.408838 + 0.912607i \(0.365934\pi\)
\(242\) 9068.09 2.40876
\(243\) 3554.83 0.938447
\(244\) 211.411 0.0554679
\(245\) −2738.23 −0.714038
\(246\) 365.382 0.0946988
\(247\) 2266.28 0.583805
\(248\) 6209.85 1.59002
\(249\) 2632.66 0.670033
\(250\) −3935.95 −0.995727
\(251\) 6105.23 1.53529 0.767647 0.640873i \(-0.221427\pi\)
0.767647 + 0.640873i \(0.221427\pi\)
\(252\) −343.668 −0.0859090
\(253\) −9375.63 −2.32980
\(254\) −1920.84 −0.474505
\(255\) −548.920 −0.134803
\(256\) 2095.67 0.511639
\(257\) −1695.01 −0.411409 −0.205704 0.978614i \(-0.565949\pi\)
−0.205704 + 0.978614i \(0.565949\pi\)
\(258\) 0 0
\(259\) 893.066 0.214257
\(260\) −1038.22 −0.247646
\(261\) 4887.98 1.15923
\(262\) 3986.18 0.939950
\(263\) −1328.61 −0.311505 −0.155752 0.987796i \(-0.549780\pi\)
−0.155752 + 0.987796i \(0.549780\pi\)
\(264\) −2666.37 −0.621604
\(265\) 4850.45 1.12438
\(266\) −1196.75 −0.275856
\(267\) 134.387 0.0308028
\(268\) 237.523 0.0541382
\(269\) −4979.49 −1.12864 −0.564321 0.825555i \(-0.690862\pi\)
−0.564321 + 0.825555i \(0.690862\pi\)
\(270\) −3681.11 −0.829723
\(271\) −2747.64 −0.615894 −0.307947 0.951404i \(-0.599642\pi\)
−0.307947 + 0.951404i \(0.599642\pi\)
\(272\) 1673.51 0.373056
\(273\) −1328.26 −0.294468
\(274\) −5622.50 −1.23966
\(275\) 1144.62 0.250993
\(276\) −405.066 −0.0883409
\(277\) −5590.40 −1.21262 −0.606308 0.795230i \(-0.707350\pi\)
−0.606308 + 0.795230i \(0.707350\pi\)
\(278\) 1088.30 0.234792
\(279\) 7045.51 1.51184
\(280\) −2573.07 −0.549180
\(281\) 3044.07 0.646243 0.323121 0.946358i \(-0.395268\pi\)
0.323121 + 0.946358i \(0.395268\pi\)
\(282\) 1246.97 0.263318
\(283\) −3571.65 −0.750221 −0.375111 0.926980i \(-0.622395\pi\)
−0.375111 + 0.926980i \(0.622395\pi\)
\(284\) −1461.34 −0.305333
\(285\) −879.877 −0.182875
\(286\) −12430.4 −2.57002
\(287\) 630.724 0.129723
\(288\) 1443.22 0.295287
\(289\) −4391.28 −0.893808
\(290\) −7797.80 −1.57898
\(291\) −1920.48 −0.386875
\(292\) −526.473 −0.105512
\(293\) −3424.66 −0.682835 −0.341417 0.939912i \(-0.610907\pi\)
−0.341417 + 0.939912i \(0.610907\pi\)
\(294\) −1416.41 −0.280976
\(295\) 1966.86 0.388186
\(296\) −1694.65 −0.332770
\(297\) −6584.74 −1.28648
\(298\) 10066.6 1.95686
\(299\) 8862.59 1.71417
\(300\) 49.4522 0.00951707
\(301\) 0 0
\(302\) −380.746 −0.0725479
\(303\) −3034.52 −0.575342
\(304\) 2682.50 0.506093
\(305\) 1795.85 0.337148
\(306\) 1607.39 0.300288
\(307\) −1249.32 −0.232256 −0.116128 0.993234i \(-0.537048\pi\)
−0.116128 + 0.993234i \(0.537048\pi\)
\(308\) 980.715 0.181433
\(309\) 2157.96 0.397289
\(310\) −11239.7 −2.05927
\(311\) −2947.84 −0.537481 −0.268740 0.963213i \(-0.586607\pi\)
−0.268740 + 0.963213i \(0.586607\pi\)
\(312\) 2520.46 0.457350
\(313\) −8598.81 −1.55282 −0.776411 0.630226i \(-0.782962\pi\)
−0.776411 + 0.630226i \(0.782962\pi\)
\(314\) −3434.78 −0.617311
\(315\) −2919.33 −0.522176
\(316\) 653.641 0.116361
\(317\) −1852.66 −0.328252 −0.164126 0.986439i \(-0.552480\pi\)
−0.164126 + 0.986439i \(0.552480\pi\)
\(318\) 2509.00 0.442446
\(319\) −13948.7 −2.44820
\(320\) 4694.03 0.820013
\(321\) 1861.11 0.323604
\(322\) −4680.07 −0.809969
\(323\) 836.281 0.144062
\(324\) 586.103 0.100498
\(325\) −1081.98 −0.184670
\(326\) −8650.82 −1.46971
\(327\) −3190.01 −0.539474
\(328\) −1196.84 −0.201477
\(329\) 2152.52 0.360706
\(330\) 4826.07 0.805050
\(331\) 316.494 0.0525562 0.0262781 0.999655i \(-0.491634\pi\)
0.0262781 + 0.999655i \(0.491634\pi\)
\(332\) 1837.45 0.303745
\(333\) −1922.70 −0.316407
\(334\) 5965.11 0.977234
\(335\) 2017.67 0.329065
\(336\) −1572.21 −0.255270
\(337\) −2975.67 −0.480994 −0.240497 0.970650i \(-0.577310\pi\)
−0.240497 + 0.970650i \(0.577310\pi\)
\(338\) 5012.49 0.806638
\(339\) −3099.54 −0.496590
\(340\) −383.116 −0.0611099
\(341\) −20105.5 −3.19289
\(342\) 2576.52 0.407375
\(343\) −6100.83 −0.960389
\(344\) 0 0
\(345\) −3440.88 −0.536958
\(346\) −3390.71 −0.526837
\(347\) 3588.41 0.555147 0.277573 0.960704i \(-0.410470\pi\)
0.277573 + 0.960704i \(0.410470\pi\)
\(348\) −602.640 −0.0928301
\(349\) 10696.8 1.64065 0.820325 0.571898i \(-0.193793\pi\)
0.820325 + 0.571898i \(0.193793\pi\)
\(350\) 571.362 0.0872589
\(351\) 6224.43 0.946539
\(352\) −4118.47 −0.623623
\(353\) −7340.86 −1.10684 −0.553420 0.832902i \(-0.686678\pi\)
−0.553420 + 0.832902i \(0.686678\pi\)
\(354\) 1017.40 0.152752
\(355\) −12413.5 −1.85589
\(356\) 93.7946 0.0139638
\(357\) −490.141 −0.0726639
\(358\) 3540.16 0.522635
\(359\) 8011.90 1.17786 0.588931 0.808184i \(-0.299549\pi\)
0.588931 + 0.808184i \(0.299549\pi\)
\(360\) 5539.63 0.811011
\(361\) −5518.51 −0.804564
\(362\) −10879.9 −1.57966
\(363\) 5953.12 0.860766
\(364\) −927.050 −0.133491
\(365\) −4472.18 −0.641328
\(366\) 928.944 0.132669
\(367\) −8926.05 −1.26958 −0.634790 0.772685i \(-0.718913\pi\)
−0.634790 + 0.772685i \(0.718913\pi\)
\(368\) 10490.3 1.48599
\(369\) −1357.90 −0.191570
\(370\) 3067.29 0.430976
\(371\) 4331.05 0.606084
\(372\) −868.642 −0.121067
\(373\) −3809.92 −0.528874 −0.264437 0.964403i \(-0.585186\pi\)
−0.264437 + 0.964403i \(0.585186\pi\)
\(374\) −4586.95 −0.634186
\(375\) −2583.92 −0.355821
\(376\) −4084.56 −0.560226
\(377\) 13185.4 1.80128
\(378\) −3286.93 −0.447253
\(379\) 2451.19 0.332214 0.166107 0.986108i \(-0.446880\pi\)
0.166107 + 0.986108i \(0.446880\pi\)
\(380\) −614.105 −0.0829024
\(381\) −1261.02 −0.169564
\(382\) −7326.09 −0.981244
\(383\) 3346.93 0.446528 0.223264 0.974758i \(-0.428329\pi\)
0.223264 + 0.974758i \(0.428329\pi\)
\(384\) 3441.11 0.457301
\(385\) 8330.80 1.10280
\(386\) 2377.98 0.313564
\(387\) 0 0
\(388\) −1340.39 −0.175381
\(389\) 5274.35 0.687455 0.343727 0.939069i \(-0.388310\pi\)
0.343727 + 0.939069i \(0.388310\pi\)
\(390\) −4561.99 −0.592321
\(391\) 3270.39 0.422994
\(392\) 4639.60 0.597793
\(393\) 2616.89 0.335890
\(394\) 10987.7 1.40495
\(395\) 5552.42 0.707273
\(396\) −2111.40 −0.267935
\(397\) −9500.83 −1.20109 −0.600545 0.799591i \(-0.705050\pi\)
−0.600545 + 0.799591i \(0.705050\pi\)
\(398\) 9539.32 1.20141
\(399\) −785.658 −0.0985767
\(400\) −1280.70 −0.160087
\(401\) −14475.4 −1.80266 −0.901330 0.433132i \(-0.857408\pi\)
−0.901330 + 0.433132i \(0.857408\pi\)
\(402\) 1043.68 0.129488
\(403\) 19005.4 2.34919
\(404\) −2117.92 −0.260819
\(405\) 4978.72 0.610851
\(406\) −6962.81 −0.851129
\(407\) 5486.76 0.668227
\(408\) 930.077 0.112857
\(409\) 12284.3 1.48513 0.742567 0.669772i \(-0.233608\pi\)
0.742567 + 0.669772i \(0.233608\pi\)
\(410\) 2166.26 0.260937
\(411\) −3691.12 −0.442992
\(412\) 1506.14 0.180102
\(413\) 1756.24 0.209247
\(414\) 10075.8 1.19613
\(415\) 15608.4 1.84624
\(416\) 3893.11 0.458835
\(417\) 714.462 0.0839025
\(418\) −7352.53 −0.860345
\(419\) 2550.74 0.297403 0.148701 0.988882i \(-0.452491\pi\)
0.148701 + 0.988882i \(0.452491\pi\)
\(420\) 359.925 0.0418155
\(421\) −4680.41 −0.541827 −0.270913 0.962604i \(-0.587326\pi\)
−0.270913 + 0.962604i \(0.587326\pi\)
\(422\) −3377.73 −0.389634
\(423\) −4634.21 −0.532679
\(424\) −8218.47 −0.941331
\(425\) −399.263 −0.0455696
\(426\) −6421.18 −0.730298
\(427\) 1603.55 0.181736
\(428\) 1298.95 0.146699
\(429\) −8160.46 −0.918394
\(430\) 0 0
\(431\) 6700.14 0.748803 0.374402 0.927267i \(-0.377848\pi\)
0.374402 + 0.927267i \(0.377848\pi\)
\(432\) 7367.60 0.820541
\(433\) −11197.9 −1.24281 −0.621407 0.783488i \(-0.713438\pi\)
−0.621407 + 0.783488i \(0.713438\pi\)
\(434\) −10036.2 −1.11002
\(435\) −5119.19 −0.564245
\(436\) −2226.45 −0.244559
\(437\) 5242.18 0.573838
\(438\) −2313.34 −0.252364
\(439\) 13944.1 1.51598 0.757992 0.652264i \(-0.226181\pi\)
0.757992 + 0.652264i \(0.226181\pi\)
\(440\) −15808.3 −1.71279
\(441\) 5263.94 0.568399
\(442\) 4335.95 0.466607
\(443\) 2589.51 0.277723 0.138862 0.990312i \(-0.455656\pi\)
0.138862 + 0.990312i \(0.455656\pi\)
\(444\) 237.050 0.0253377
\(445\) 796.748 0.0848753
\(446\) −7401.64 −0.785825
\(447\) 6608.64 0.699280
\(448\) 4191.39 0.442019
\(449\) 13244.7 1.39211 0.696056 0.717987i \(-0.254936\pi\)
0.696056 + 0.717987i \(0.254936\pi\)
\(450\) −1230.10 −0.128861
\(451\) 3875.00 0.404582
\(452\) −2163.31 −0.225118
\(453\) −249.957 −0.0259249
\(454\) −6848.91 −0.708007
\(455\) −7874.93 −0.811390
\(456\) 1490.84 0.153103
\(457\) 7940.24 0.812754 0.406377 0.913705i \(-0.366792\pi\)
0.406377 + 0.913705i \(0.366792\pi\)
\(458\) −4170.08 −0.425448
\(459\) 2296.88 0.233571
\(460\) −2401.54 −0.243418
\(461\) 5251.72 0.530579 0.265290 0.964169i \(-0.414532\pi\)
0.265290 + 0.964169i \(0.414532\pi\)
\(462\) 4309.29 0.433953
\(463\) 9045.26 0.907924 0.453962 0.891021i \(-0.350010\pi\)
0.453962 + 0.891021i \(0.350010\pi\)
\(464\) 15607.0 1.56150
\(465\) −7378.78 −0.735876
\(466\) 21458.3 2.13312
\(467\) 11982.6 1.18734 0.593671 0.804708i \(-0.297678\pi\)
0.593671 + 0.804708i \(0.297678\pi\)
\(468\) 1995.87 0.197135
\(469\) 1801.61 0.177379
\(470\) 7392.97 0.725558
\(471\) −2254.90 −0.220595
\(472\) −3332.59 −0.324989
\(473\) 0 0
\(474\) 2872.12 0.278314
\(475\) −639.988 −0.0618203
\(476\) −342.091 −0.0329406
\(477\) −9324.43 −0.895044
\(478\) 13349.8 1.27741
\(479\) 5589.14 0.533140 0.266570 0.963816i \(-0.414110\pi\)
0.266570 + 0.963816i \(0.414110\pi\)
\(480\) −1511.49 −0.143729
\(481\) −5186.52 −0.491653
\(482\) 9381.89 0.886584
\(483\) −3072.42 −0.289441
\(484\) 4154.95 0.390209
\(485\) −11386.1 −1.06601
\(486\) 10901.9 1.01753
\(487\) 9311.43 0.866409 0.433205 0.901296i \(-0.357383\pi\)
0.433205 + 0.901296i \(0.357383\pi\)
\(488\) −3042.84 −0.282260
\(489\) −5679.19 −0.525198
\(490\) −8397.58 −0.774212
\(491\) −12911.8 −1.18677 −0.593383 0.804920i \(-0.702208\pi\)
−0.593383 + 0.804920i \(0.702208\pi\)
\(492\) 167.416 0.0153408
\(493\) 4865.55 0.444489
\(494\) 6950.20 0.633005
\(495\) −17935.6 −1.62857
\(496\) 22495.9 2.03648
\(497\) −11084.3 −1.00040
\(498\) 8073.82 0.726499
\(499\) 6845.56 0.614127 0.307064 0.951689i \(-0.400654\pi\)
0.307064 + 0.951689i \(0.400654\pi\)
\(500\) −1803.43 −0.161304
\(501\) 3916.04 0.349213
\(502\) 18723.4 1.66468
\(503\) −5914.10 −0.524248 −0.262124 0.965034i \(-0.584423\pi\)
−0.262124 + 0.965034i \(0.584423\pi\)
\(504\) 4946.44 0.437166
\(505\) −17990.9 −1.58532
\(506\) −28753.1 −2.52614
\(507\) 3290.66 0.288251
\(508\) −880.118 −0.0768680
\(509\) −5224.84 −0.454984 −0.227492 0.973780i \(-0.573053\pi\)
−0.227492 + 0.973780i \(0.573053\pi\)
\(510\) −1683.42 −0.146163
\(511\) −3993.30 −0.345701
\(512\) −7246.41 −0.625487
\(513\) 3681.72 0.316865
\(514\) −5198.25 −0.446079
\(515\) 12794.0 1.09471
\(516\) 0 0
\(517\) 13224.5 1.12498
\(518\) 2738.84 0.232313
\(519\) −2225.97 −0.188264
\(520\) 14943.2 1.26020
\(521\) −1983.41 −0.166785 −0.0833923 0.996517i \(-0.526575\pi\)
−0.0833923 + 0.996517i \(0.526575\pi\)
\(522\) 14990.4 1.25692
\(523\) 588.084 0.0491685 0.0245843 0.999698i \(-0.492174\pi\)
0.0245843 + 0.999698i \(0.492174\pi\)
\(524\) 1826.44 0.152268
\(525\) 375.095 0.0311818
\(526\) −4074.57 −0.337756
\(527\) 7013.17 0.579694
\(528\) −9659.21 −0.796142
\(529\) 8333.24 0.684905
\(530\) 14875.3 1.21913
\(531\) −3781.06 −0.309009
\(532\) −548.346 −0.0446876
\(533\) −3662.96 −0.297674
\(534\) 412.136 0.0333986
\(535\) 11034.1 0.891671
\(536\) −3418.68 −0.275494
\(537\) 2324.08 0.186763
\(538\) −15271.0 −1.22376
\(539\) −15021.5 −1.20042
\(540\) −1686.66 −0.134412
\(541\) −14115.0 −1.12173 −0.560863 0.827909i \(-0.689530\pi\)
−0.560863 + 0.827909i \(0.689530\pi\)
\(542\) −8426.42 −0.667797
\(543\) −7142.58 −0.564489
\(544\) 1436.60 0.113224
\(545\) −18912.8 −1.48649
\(546\) −4073.49 −0.319284
\(547\) 9946.81 0.777504 0.388752 0.921342i \(-0.372906\pi\)
0.388752 + 0.921342i \(0.372906\pi\)
\(548\) −2576.20 −0.200820
\(549\) −3452.32 −0.268381
\(550\) 3510.30 0.272145
\(551\) 7799.09 0.602999
\(552\) 5830.13 0.449542
\(553\) 4957.86 0.381247
\(554\) −17144.6 −1.31481
\(555\) 2013.65 0.154009
\(556\) 498.654 0.0380353
\(557\) −12763.4 −0.970918 −0.485459 0.874259i \(-0.661347\pi\)
−0.485459 + 0.874259i \(0.661347\pi\)
\(558\) 21607.1 1.63925
\(559\) 0 0
\(560\) −9321.24 −0.703383
\(561\) −3011.29 −0.226626
\(562\) 9335.53 0.700704
\(563\) 2108.25 0.157819 0.0789096 0.996882i \(-0.474856\pi\)
0.0789096 + 0.996882i \(0.474856\pi\)
\(564\) 571.353 0.0426566
\(565\) −18376.5 −1.36832
\(566\) −10953.5 −0.813445
\(567\) 4445.59 0.329272
\(568\) 21033.2 1.55375
\(569\) 6912.66 0.509304 0.254652 0.967033i \(-0.418039\pi\)
0.254652 + 0.967033i \(0.418039\pi\)
\(570\) −2698.39 −0.198287
\(571\) 24459.8 1.79266 0.896330 0.443387i \(-0.146223\pi\)
0.896330 + 0.443387i \(0.146223\pi\)
\(572\) −5695.55 −0.416334
\(573\) −4809.51 −0.350646
\(574\) 1934.30 0.140655
\(575\) −2502.76 −0.181517
\(576\) −9023.73 −0.652758
\(577\) 22269.4 1.60674 0.803370 0.595480i \(-0.203038\pi\)
0.803370 + 0.595480i \(0.203038\pi\)
\(578\) −13467.1 −0.969132
\(579\) 1561.12 0.112052
\(580\) −3572.91 −0.255788
\(581\) 13937.1 0.995193
\(582\) −5889.71 −0.419478
\(583\) 26608.8 1.89027
\(584\) 7577.55 0.536920
\(585\) 16954.1 1.19823
\(586\) −10502.7 −0.740379
\(587\) 23935.8 1.68303 0.841514 0.540235i \(-0.181664\pi\)
0.841514 + 0.540235i \(0.181664\pi\)
\(588\) −648.993 −0.0455170
\(589\) 11241.6 0.786419
\(590\) 6031.93 0.420900
\(591\) 7213.32 0.502058
\(592\) −6139.07 −0.426207
\(593\) −25229.9 −1.74717 −0.873583 0.486675i \(-0.838209\pi\)
−0.873583 + 0.486675i \(0.838209\pi\)
\(594\) −20194.0 −1.39490
\(595\) −2905.93 −0.200221
\(596\) 4612.46 0.317003
\(597\) 6262.48 0.429324
\(598\) 27179.7 1.85863
\(599\) −11758.7 −0.802086 −0.401043 0.916059i \(-0.631352\pi\)
−0.401043 + 0.916059i \(0.631352\pi\)
\(600\) −711.768 −0.0484297
\(601\) −18453.3 −1.25246 −0.626228 0.779640i \(-0.715402\pi\)
−0.626228 + 0.779640i \(0.715402\pi\)
\(602\) 0 0
\(603\) −3878.73 −0.261947
\(604\) −174.456 −0.0117525
\(605\) 35294.7 2.37179
\(606\) −9306.22 −0.623828
\(607\) −26742.0 −1.78818 −0.894088 0.447892i \(-0.852175\pi\)
−0.894088 + 0.447892i \(0.852175\pi\)
\(608\) 2302.76 0.153600
\(609\) −4571.02 −0.304150
\(610\) 5507.49 0.365560
\(611\) −12500.9 −0.827710
\(612\) 736.496 0.0486456
\(613\) −2389.43 −0.157436 −0.0787179 0.996897i \(-0.525083\pi\)
−0.0787179 + 0.996897i \(0.525083\pi\)
\(614\) −3831.40 −0.251829
\(615\) 1422.13 0.0932454
\(616\) −14115.5 −0.923262
\(617\) 16915.3 1.10371 0.551853 0.833942i \(-0.313921\pi\)
0.551853 + 0.833942i \(0.313921\pi\)
\(618\) 6618.01 0.430769
\(619\) 18741.2 1.21692 0.608460 0.793584i \(-0.291787\pi\)
0.608460 + 0.793584i \(0.291787\pi\)
\(620\) −5149.97 −0.333593
\(621\) 14397.8 0.930380
\(622\) −9040.39 −0.582776
\(623\) 711.432 0.0457511
\(624\) 9130.65 0.585767
\(625\) −17504.4 −1.12028
\(626\) −26370.7 −1.68368
\(627\) −4826.87 −0.307443
\(628\) −1573.79 −0.100002
\(629\) −1913.88 −0.121322
\(630\) −8952.96 −0.566182
\(631\) −9825.75 −0.619900 −0.309950 0.950753i \(-0.600312\pi\)
−0.309950 + 0.950753i \(0.600312\pi\)
\(632\) −9407.89 −0.592129
\(633\) −2217.45 −0.139235
\(634\) −5681.73 −0.355915
\(635\) −7476.26 −0.467223
\(636\) 1149.61 0.0716746
\(637\) 14199.6 0.883214
\(638\) −42777.6 −2.65452
\(639\) 23863.6 1.47735
\(640\) 20401.5 1.26007
\(641\) −5404.26 −0.333004 −0.166502 0.986041i \(-0.553247\pi\)
−0.166502 + 0.986041i \(0.553247\pi\)
\(642\) 5707.62 0.350875
\(643\) 24177.5 1.48284 0.741422 0.671039i \(-0.234152\pi\)
0.741422 + 0.671039i \(0.234152\pi\)
\(644\) −2144.38 −0.131212
\(645\) 0 0
\(646\) 2564.70 0.156202
\(647\) 9145.13 0.555691 0.277846 0.960626i \(-0.410380\pi\)
0.277846 + 0.960626i \(0.410380\pi\)
\(648\) −8435.82 −0.511405
\(649\) 10789.9 0.652604
\(650\) −3318.21 −0.200232
\(651\) −6588.65 −0.396666
\(652\) −3963.76 −0.238087
\(653\) 161.209 0.00966091 0.00483046 0.999988i \(-0.498462\pi\)
0.00483046 + 0.999988i \(0.498462\pi\)
\(654\) −9783.08 −0.584937
\(655\) 15514.9 0.925524
\(656\) −4335.70 −0.258050
\(657\) 8597.26 0.510519
\(658\) 6601.33 0.391104
\(659\) −8406.91 −0.496945 −0.248472 0.968639i \(-0.579929\pi\)
−0.248472 + 0.968639i \(0.579929\pi\)
\(660\) 2211.28 0.130415
\(661\) −17911.7 −1.05399 −0.526994 0.849869i \(-0.676681\pi\)
−0.526994 + 0.849869i \(0.676681\pi\)
\(662\) 970.621 0.0569853
\(663\) 2846.52 0.166741
\(664\) −26446.6 −1.54567
\(665\) −4657.98 −0.271622
\(666\) −5896.52 −0.343071
\(667\) 30499.4 1.77053
\(668\) 2733.18 0.158308
\(669\) −4859.11 −0.280813
\(670\) 6187.75 0.356797
\(671\) 9851.77 0.566801
\(672\) −1349.64 −0.0774753
\(673\) −11202.5 −0.641642 −0.320821 0.947140i \(-0.603959\pi\)
−0.320821 + 0.947140i \(0.603959\pi\)
\(674\) −9125.74 −0.521529
\(675\) −1757.75 −0.100231
\(676\) 2296.70 0.130672
\(677\) 29615.8 1.68128 0.840640 0.541594i \(-0.182179\pi\)
0.840640 + 0.541594i \(0.182179\pi\)
\(678\) −9505.64 −0.538439
\(679\) −10166.9 −0.574622
\(680\) 5514.20 0.310971
\(681\) −4496.25 −0.253005
\(682\) −61659.4 −3.46197
\(683\) 24045.9 1.34713 0.673566 0.739127i \(-0.264762\pi\)
0.673566 + 0.739127i \(0.264762\pi\)
\(684\) 1180.55 0.0659931
\(685\) −21883.8 −1.22064
\(686\) −18709.9 −1.04132
\(687\) −2737.62 −0.152033
\(688\) 0 0
\(689\) −25152.8 −1.39078
\(690\) −10552.4 −0.582209
\(691\) −21794.1 −1.19983 −0.599917 0.800062i \(-0.704800\pi\)
−0.599917 + 0.800062i \(0.704800\pi\)
\(692\) −1553.60 −0.0853455
\(693\) −16015.0 −0.877864
\(694\) 11004.9 0.601931
\(695\) 4235.87 0.231188
\(696\) 8673.83 0.472386
\(697\) −1351.67 −0.0734550
\(698\) 32804.8 1.77891
\(699\) 14087.2 0.762268
\(700\) 261.795 0.0141356
\(701\) −25915.8 −1.39633 −0.698164 0.715937i \(-0.746000\pi\)
−0.698164 + 0.715937i \(0.746000\pi\)
\(702\) 19089.0 1.02631
\(703\) −3067.80 −0.164587
\(704\) 25750.7 1.37858
\(705\) 4853.42 0.259277
\(706\) −22512.9 −1.20012
\(707\) −16064.5 −0.854549
\(708\) 466.167 0.0247453
\(709\) −26358.5 −1.39621 −0.698106 0.715995i \(-0.745973\pi\)
−0.698106 + 0.715995i \(0.745973\pi\)
\(710\) −38069.6 −2.01229
\(711\) −10673.9 −0.563013
\(712\) −1349.99 −0.0710576
\(713\) 43961.7 2.30909
\(714\) −1503.16 −0.0787875
\(715\) −48381.5 −2.53058
\(716\) 1622.08 0.0846648
\(717\) 8764.00 0.456482
\(718\) 24570.8 1.27712
\(719\) 522.060 0.0270787 0.0135393 0.999908i \(-0.495690\pi\)
0.0135393 + 0.999908i \(0.495690\pi\)
\(720\) 20067.9 1.03873
\(721\) 11424.0 0.590088
\(722\) −16924.1 −0.872367
\(723\) 6159.13 0.316819
\(724\) −4985.12 −0.255899
\(725\) −3723.50 −0.190741
\(726\) 18257.0 0.933305
\(727\) −8443.21 −0.430731 −0.215366 0.976534i \(-0.569094\pi\)
−0.215366 + 0.976534i \(0.569094\pi\)
\(728\) 13343.1 0.679296
\(729\) −4104.71 −0.208541
\(730\) −13715.2 −0.695375
\(731\) 0 0
\(732\) 425.637 0.0214918
\(733\) −9103.47 −0.458723 −0.229362 0.973341i \(-0.573664\pi\)
−0.229362 + 0.973341i \(0.573664\pi\)
\(734\) −27374.3 −1.37657
\(735\) −5512.94 −0.276664
\(736\) 9005.24 0.451002
\(737\) 11068.6 0.553212
\(738\) −4164.39 −0.207715
\(739\) −1195.87 −0.0595275 −0.0297638 0.999557i \(-0.509476\pi\)
−0.0297638 + 0.999557i \(0.509476\pi\)
\(740\) 1405.42 0.0698164
\(741\) 4562.75 0.226203
\(742\) 13282.4 0.657160
\(743\) 18356.6 0.906379 0.453189 0.891414i \(-0.350286\pi\)
0.453189 + 0.891414i \(0.350286\pi\)
\(744\) 12502.4 0.616076
\(745\) 39181.1 1.92682
\(746\) −11684.2 −0.573444
\(747\) −30005.4 −1.46967
\(748\) −2101.72 −0.102736
\(749\) 9852.52 0.480645
\(750\) −7924.33 −0.385808
\(751\) −37294.3 −1.81210 −0.906052 0.423167i \(-0.860918\pi\)
−0.906052 + 0.423167i \(0.860918\pi\)
\(752\) −14796.8 −0.717530
\(753\) 12291.8 0.594870
\(754\) 40436.8 1.95308
\(755\) −1481.93 −0.0714345
\(756\) −1506.05 −0.0724532
\(757\) −14854.1 −0.713187 −0.356593 0.934260i \(-0.616062\pi\)
−0.356593 + 0.934260i \(0.616062\pi\)
\(758\) 7517.27 0.360211
\(759\) −18876.1 −0.902714
\(760\) 8838.84 0.421867
\(761\) 35725.3 1.70176 0.850881 0.525358i \(-0.176069\pi\)
0.850881 + 0.525358i \(0.176069\pi\)
\(762\) −3867.27 −0.183853
\(763\) −16887.6 −0.801275
\(764\) −3356.77 −0.158958
\(765\) 6256.25 0.295680
\(766\) 10264.3 0.484158
\(767\) −10199.5 −0.480158
\(768\) 4219.26 0.198241
\(769\) 6836.30 0.320576 0.160288 0.987070i \(-0.448758\pi\)
0.160288 + 0.987070i \(0.448758\pi\)
\(770\) 25548.8 1.19573
\(771\) −3412.60 −0.159406
\(772\) 1089.58 0.0507962
\(773\) 16076.5 0.748034 0.374017 0.927422i \(-0.377980\pi\)
0.374017 + 0.927422i \(0.377980\pi\)
\(774\) 0 0
\(775\) −5367.03 −0.248761
\(776\) 19292.3 0.892466
\(777\) 1798.03 0.0830166
\(778\) 16175.3 0.745389
\(779\) −2166.62 −0.0996499
\(780\) −2090.28 −0.0959538
\(781\) −68098.7 −3.12006
\(782\) 10029.6 0.458641
\(783\) 21420.5 0.977659
\(784\) 16807.4 0.765645
\(785\) −13368.8 −0.607837
\(786\) 8025.45 0.364196
\(787\) −23639.9 −1.07074 −0.535370 0.844618i \(-0.679828\pi\)
−0.535370 + 0.844618i \(0.679828\pi\)
\(788\) 5034.49 0.227597
\(789\) −2674.92 −0.120697
\(790\) 17028.1 0.766877
\(791\) −16408.7 −0.737580
\(792\) 30389.6 1.36344
\(793\) −9312.68 −0.417028
\(794\) −29137.0 −1.30231
\(795\) 9765.49 0.435656
\(796\) 4370.86 0.194624
\(797\) 25100.7 1.11557 0.557786 0.829985i \(-0.311651\pi\)
0.557786 + 0.829985i \(0.311651\pi\)
\(798\) −2409.45 −0.106884
\(799\) −4612.95 −0.204248
\(800\) −1099.40 −0.0485870
\(801\) −1531.66 −0.0675636
\(802\) −44393.0 −1.95458
\(803\) −24533.7 −1.07818
\(804\) 478.210 0.0209766
\(805\) −18215.7 −0.797538
\(806\) 58285.4 2.54717
\(807\) −10025.3 −0.437308
\(808\) 30483.4 1.32723
\(809\) 13162.4 0.572021 0.286011 0.958226i \(-0.407671\pi\)
0.286011 + 0.958226i \(0.407671\pi\)
\(810\) 15268.7 0.662329
\(811\) −25843.3 −1.11897 −0.559484 0.828841i \(-0.689000\pi\)
−0.559484 + 0.828841i \(0.689000\pi\)
\(812\) −3190.32 −0.137880
\(813\) −5531.87 −0.238636
\(814\) 16826.7 0.724541
\(815\) −33670.6 −1.44715
\(816\) 3369.31 0.144546
\(817\) 0 0
\(818\) 37673.4 1.61029
\(819\) 15138.7 0.645894
\(820\) 992.570 0.0422708
\(821\) 20649.7 0.877805 0.438903 0.898535i \(-0.355367\pi\)
0.438903 + 0.898535i \(0.355367\pi\)
\(822\) −11319.9 −0.480324
\(823\) −33853.1 −1.43384 −0.716918 0.697158i \(-0.754448\pi\)
−0.716918 + 0.697158i \(0.754448\pi\)
\(824\) −21677.9 −0.916488
\(825\) 2304.48 0.0972505
\(826\) 5386.03 0.226881
\(827\) 12153.3 0.511019 0.255509 0.966807i \(-0.417757\pi\)
0.255509 + 0.966807i \(0.417757\pi\)
\(828\) 4616.68 0.193769
\(829\) −4628.02 −0.193894 −0.0969468 0.995290i \(-0.530908\pi\)
−0.0969468 + 0.995290i \(0.530908\pi\)
\(830\) 47867.8 2.00182
\(831\) −11255.3 −0.469844
\(832\) −24341.7 −1.01430
\(833\) 5239.79 0.217945
\(834\) 2191.10 0.0909732
\(835\) 23217.3 0.962236
\(836\) −3368.89 −0.139372
\(837\) 30875.4 1.27504
\(838\) 7822.57 0.322466
\(839\) 29113.1 1.19797 0.598984 0.800761i \(-0.295571\pi\)
0.598984 + 0.800761i \(0.295571\pi\)
\(840\) −5180.42 −0.212787
\(841\) 20986.7 0.860500
\(842\) −14353.8 −0.587488
\(843\) 6128.69 0.250395
\(844\) −1547.66 −0.0631192
\(845\) 19509.5 0.794259
\(846\) −14212.1 −0.577569
\(847\) 31515.3 1.27849
\(848\) −29772.3 −1.20564
\(849\) −7190.87 −0.290683
\(850\) −1224.45 −0.0494099
\(851\) −11997.0 −0.483259
\(852\) −2942.15 −0.118305
\(853\) −2836.73 −0.113866 −0.0569331 0.998378i \(-0.518132\pi\)
−0.0569331 + 0.998378i \(0.518132\pi\)
\(854\) 4917.74 0.197051
\(855\) 10028.3 0.401123
\(856\) −18695.8 −0.746508
\(857\) −2323.85 −0.0926269 −0.0463134 0.998927i \(-0.514747\pi\)
−0.0463134 + 0.998927i \(0.514747\pi\)
\(858\) −25026.4 −0.995789
\(859\) −20417.4 −0.810981 −0.405490 0.914099i \(-0.632899\pi\)
−0.405490 + 0.914099i \(0.632899\pi\)
\(860\) 0 0
\(861\) 1269.85 0.0502629
\(862\) 20547.9 0.811907
\(863\) 11193.8 0.441532 0.220766 0.975327i \(-0.429144\pi\)
0.220766 + 0.975327i \(0.429144\pi\)
\(864\) 6324.61 0.249037
\(865\) −13197.2 −0.518751
\(866\) −34341.7 −1.34755
\(867\) −8841.04 −0.346318
\(868\) −4598.51 −0.179820
\(869\) 30459.8 1.18904
\(870\) −15699.5 −0.611795
\(871\) −10462.9 −0.407030
\(872\) 32045.4 1.24449
\(873\) 21888.5 0.848582
\(874\) 16076.6 0.622198
\(875\) −13679.0 −0.528498
\(876\) −1059.96 −0.0408820
\(877\) −14906.8 −0.573966 −0.286983 0.957936i \(-0.592652\pi\)
−0.286983 + 0.957936i \(0.592652\pi\)
\(878\) 42763.7 1.64374
\(879\) −6894.93 −0.264574
\(880\) −57267.2 −2.19372
\(881\) 32192.4 1.23109 0.615544 0.788102i \(-0.288936\pi\)
0.615544 + 0.788102i \(0.288936\pi\)
\(882\) 16143.4 0.616300
\(883\) −15047.5 −0.573486 −0.286743 0.958008i \(-0.592573\pi\)
−0.286743 + 0.958008i \(0.592573\pi\)
\(884\) 1986.71 0.0755885
\(885\) 3959.91 0.150408
\(886\) 7941.48 0.301128
\(887\) −45336.4 −1.71618 −0.858088 0.513502i \(-0.828348\pi\)
−0.858088 + 0.513502i \(0.828348\pi\)
\(888\) −3411.88 −0.128936
\(889\) −6675.70 −0.251851
\(890\) 2443.46 0.0920280
\(891\) 27312.5 1.02694
\(892\) −3391.39 −0.127301
\(893\) −7394.20 −0.277086
\(894\) 20267.3 0.758210
\(895\) 13778.9 0.514614
\(896\) 18216.9 0.679224
\(897\) 17843.2 0.664178
\(898\) 40618.8 1.50943
\(899\) 65404.4 2.42643
\(900\) −563.624 −0.0208750
\(901\) −9281.64 −0.343192
\(902\) 11883.8 0.438678
\(903\) 0 0
\(904\) 31136.6 1.14556
\(905\) −42346.7 −1.55542
\(906\) −766.564 −0.0281097
\(907\) 1995.61 0.0730573 0.0365287 0.999333i \(-0.488370\pi\)
0.0365287 + 0.999333i \(0.488370\pi\)
\(908\) −3138.13 −0.114694
\(909\) 34585.5 1.26197
\(910\) −24150.7 −0.879769
\(911\) 14639.1 0.532398 0.266199 0.963918i \(-0.414232\pi\)
0.266199 + 0.963918i \(0.414232\pi\)
\(912\) 5400.74 0.196092
\(913\) 85625.6 3.10383
\(914\) 24351.0 0.881248
\(915\) 3615.62 0.130632
\(916\) −1910.71 −0.0689209
\(917\) 13853.6 0.498893
\(918\) 7044.04 0.253255
\(919\) −3884.35 −0.139427 −0.0697133 0.997567i \(-0.522208\pi\)
−0.0697133 + 0.997567i \(0.522208\pi\)
\(920\) 34565.5 1.23869
\(921\) −2515.28 −0.0899906
\(922\) 16105.9 0.575293
\(923\) 64372.4 2.29560
\(924\) 1974.49 0.0702988
\(925\) 1464.65 0.0520621
\(926\) 27739.9 0.984437
\(927\) −24595.1 −0.871423
\(928\) 13397.6 0.473920
\(929\) −11634.6 −0.410892 −0.205446 0.978668i \(-0.565865\pi\)
−0.205446 + 0.978668i \(0.565865\pi\)
\(930\) −22629.1 −0.797891
\(931\) 8398.97 0.295666
\(932\) 9832.05 0.345557
\(933\) −5934.94 −0.208254
\(934\) 36748.1 1.28740
\(935\) −17853.3 −0.624453
\(936\) −28726.6 −1.00316
\(937\) 16593.3 0.578527 0.289263 0.957250i \(-0.406590\pi\)
0.289263 + 0.957250i \(0.406590\pi\)
\(938\) 5525.16 0.192327
\(939\) −17312.2 −0.601662
\(940\) 3387.42 0.117538
\(941\) 20113.2 0.696780 0.348390 0.937350i \(-0.386728\pi\)
0.348390 + 0.937350i \(0.386728\pi\)
\(942\) −6915.30 −0.239185
\(943\) −8472.87 −0.292592
\(944\) −12072.7 −0.416242
\(945\) −12793.3 −0.440389
\(946\) 0 0
\(947\) −2781.93 −0.0954600 −0.0477300 0.998860i \(-0.515199\pi\)
−0.0477300 + 0.998860i \(0.515199\pi\)
\(948\) 1315.99 0.0450857
\(949\) 23191.2 0.793277
\(950\) −1962.71 −0.0670301
\(951\) −3730.01 −0.127186
\(952\) 4923.74 0.167625
\(953\) −5315.57 −0.180680 −0.0903400 0.995911i \(-0.528795\pi\)
−0.0903400 + 0.995911i \(0.528795\pi\)
\(954\) −28596.0 −0.970472
\(955\) −28514.5 −0.966185
\(956\) 6116.78 0.206936
\(957\) −28083.1 −0.948588
\(958\) 17140.7 0.578070
\(959\) −19540.4 −0.657971
\(960\) 9450.58 0.317725
\(961\) 64482.5 2.16450
\(962\) −15906.0 −0.533086
\(963\) −21211.7 −0.709801
\(964\) 4298.73 0.143623
\(965\) 9255.52 0.308752
\(966\) −9422.47 −0.313833
\(967\) −28577.3 −0.950344 −0.475172 0.879893i \(-0.657614\pi\)
−0.475172 + 0.879893i \(0.657614\pi\)
\(968\) −59802.4 −1.98566
\(969\) 1683.70 0.0558186
\(970\) −34918.7 −1.15585
\(971\) 28460.3 0.940612 0.470306 0.882503i \(-0.344144\pi\)
0.470306 + 0.882503i \(0.344144\pi\)
\(972\) 4995.19 0.164836
\(973\) 3782.29 0.124619
\(974\) 28556.2 0.939424
\(975\) −2178.38 −0.0715527
\(976\) −11023.0 −0.361515
\(977\) 1661.03 0.0543919 0.0271960 0.999630i \(-0.491342\pi\)
0.0271960 + 0.999630i \(0.491342\pi\)
\(978\) −17416.9 −0.569458
\(979\) 4370.84 0.142689
\(980\) −3847.72 −0.125419
\(981\) 36357.7 1.18330
\(982\) −39597.8 −1.28678
\(983\) −24756.7 −0.803271 −0.401636 0.915800i \(-0.631558\pi\)
−0.401636 + 0.915800i \(0.631558\pi\)
\(984\) −2409.63 −0.0780651
\(985\) 42766.1 1.38339
\(986\) 14921.6 0.481948
\(987\) 4333.71 0.139760
\(988\) 3184.54 0.102544
\(989\) 0 0
\(990\) −55004.6 −1.76582
\(991\) −20329.5 −0.651654 −0.325827 0.945429i \(-0.605643\pi\)
−0.325827 + 0.945429i \(0.605643\pi\)
\(992\) 19311.3 0.618077
\(993\) 637.204 0.0203636
\(994\) −33993.1 −1.08470
\(995\) 37128.8 1.18298
\(996\) 3699.38 0.117690
\(997\) −51774.9 −1.64466 −0.822330 0.569011i \(-0.807326\pi\)
−0.822330 + 0.569011i \(0.807326\pi\)
\(998\) 20993.9 0.665882
\(999\) −8425.84 −0.266848
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.j.1.36 yes 50
43.42 odd 2 1849.4.a.i.1.15 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.15 50 43.42 odd 2
1849.4.a.j.1.36 yes 50 1.1 even 1 trivial