Properties

Label 1849.4.a.i.1.15
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 1.15
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 \(-0.682384\pi\)
−0.542137 + 0.840290i \(0.682384\pi\)
\(3\) −2.01332 −0.387463 −0.193732 0.981055i \(-0.562059\pi\)
−0.193732 + 0.981055i \(0.562059\pi\)
\(4\) 1.40518 0.175648
\(5\) −11.9365 −1.06763 −0.533816 0.845600i \(-0.679243\pi\)
−0.533816 + 0.845600i \(0.679243\pi\)
\(6\) 6.17442 0.420116
\(7\) −10.6583 −0.575495 −0.287748 0.957706i \(-0.592906\pi\)
−0.287748 + 0.957706i \(0.592906\pi\)
\(8\) 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 \(-0.429055\pi\)
0.221041 + 0.975265i \(0.429055\pi\)
\(20\) −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 \(-0.261110\pi\)
0.682001 + 0.731351i \(0.261110\pi\)
\(30\) −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 \(-0.559601\pi\)
−0.186150 + 0.982521i \(0.559601\pi\)
\(38\) −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 \(-0.550471\pi\)
−0.157895 + 0.987456i \(0.550471\pi\)
\(62\) 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 \(-0.164659\pi\)
0.869162 + 0.494528i \(0.164659\pi\)
\(72\) −464.092 −0.759635
\(73\) 374.665 0.600701 0.300351 0.953829i \(-0.402896\pi\)
0.300351 + 0.953829i \(0.402896\pi\)
\(74\) 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 \(-0.512656\pi\)
−0.0397493 + 0.999210i \(0.512656\pi\)
\(90\) −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 \(-0.278594\pi\)
0.640822 + 0.767690i \(0.278594\pi\)
\(114\) 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 \(-0.642703\pi\)
−0.433447 + 0.901179i \(0.642703\pi\)
\(132\) −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 \(-0.306301\pi\)
0.571656 + 0.820493i \(0.306301\pi\)
\(138\) −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 \(-0.858183\pi\)
−0.902382 + 0.430938i \(0.858183\pi\)
\(150\) 107.928 0.0587488
\(151\) 124.151 0.0669093 0.0334546 0.999440i \(-0.489349\pi\)
0.0334546 + 0.999440i \(0.489349\pi\)
\(152\) 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 \(-0.408117\pi\)
0.284666 + 0.958627i \(0.408117\pi\)
\(158\) −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 \(-0.262960\pi\)
0.677739 + 0.735303i \(0.262960\pi\)
\(164\) 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 \(-0.577478\pi\)
−0.241007 + 0.970523i \(0.577478\pi\)
\(180\) 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 \(-0.350536\pi\)
0.452490 + 0.891770i \(0.350536\pi\)
\(192\) −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 \(-0.686906\pi\)
−0.554018 + 0.832505i \(0.686906\pi\)
\(200\) 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 \(-0.442495\pi\)
0.179675 + 0.983726i \(0.442495\pi\)
\(212\) 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 \(-0.381966\pi\)
0.362374 + 0.932033i \(0.381966\pi\)
\(224\) −670.354 −0.199955
\(225\) −401.104 −0.118846
\(226\) −4721.38 −1.38965
\(227\) 2233.25 0.652979 0.326489 0.945201i \(-0.394134\pi\)
0.326489 + 0.945201i \(0.394134\pi\)
\(228\) −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 \(-0.942386\pi\)
−0.983665 + 0.180012i \(0.942386\pi\)
\(234\) −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 \(-0.634066\pi\)
−0.408838 + 0.912607i \(0.634066\pi\)
\(242\) −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 \(-0.434051\pi\)
0.205704 + 0.978614i \(0.434051\pi\)
\(258\) 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 \(-0.450220\pi\)
0.155752 + 0.987796i \(0.450220\pi\)
\(264\) −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 \(-0.292650\pi\)
0.606308 + 0.795230i \(0.292650\pi\)
\(278\) −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 \(-0.217038\pi\)
0.776411 + 0.630226i \(0.217038\pi\)
\(314\) −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 \(-0.508366\pi\)
−0.0262781 + 0.999655i \(0.508366\pi\)
\(332\) 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 \(-0.589530\pi\)
−0.277573 + 0.960704i \(0.589530\pi\)
\(348\) −602.640 −0.0928301
\(349\) −10696.8 −1.64065 −0.820325 0.571898i \(-0.806207\pi\)
−0.820325 + 0.571898i \(0.806207\pi\)
\(350\) 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 \(-0.414814\pi\)
0.264437 + 0.964403i \(0.414814\pi\)
\(374\) 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 \(-0.571671\pi\)
−0.223264 + 0.974758i \(0.571671\pi\)
\(384\) 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 \(-0.611690\pi\)
−0.343727 + 0.939069i \(0.611690\pi\)
\(390\) 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 \(-0.766392\pi\)
−0.742567 + 0.669772i \(0.766392\pi\)
\(410\) 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 \(-0.547509\pi\)
−0.148701 + 0.988882i \(0.547509\pi\)
\(420\) −359.925 −0.0418155
\(421\) 4680.41 0.541827 0.270913 0.962604i \(-0.412674\pi\)
0.270913 + 0.962604i \(0.412674\pi\)
\(422\) −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 \(-0.286562\pi\)
0.621407 + 0.783488i \(0.286562\pi\)
\(434\) −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 \(-0.745064\pi\)
−0.696056 + 0.717987i \(0.745064\pi\)
\(450\) 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 \(-0.633208\pi\)
−0.406377 + 0.913705i \(0.633208\pi\)
\(458\) 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 \(-0.649990\pi\)
−0.453962 + 0.891021i \(0.649990\pi\)
\(464\) −15607.0 −1.56150
\(465\) −7378.78 −0.735876
\(466\) 21458.3 2.13312
\(467\) −11982.6 −1.18734 −0.593671 0.804708i \(-0.702322\pi\)
−0.593671 + 0.804708i \(0.702322\pi\)
\(468\) 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 \(-0.297792\pi\)
0.593383 + 0.804920i \(0.297792\pi\)
\(492\) −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 \(-0.599346\pi\)
−0.307064 + 0.951689i \(0.599346\pi\)
\(500\) 1803.43 0.161304
\(501\) −3916.04 −0.349213
\(502\) −18723.4 −1.66468
\(503\) 5914.10 0.524248 0.262124 0.965034i \(-0.415577\pi\)
0.262124 + 0.965034i \(0.415577\pi\)
\(504\) 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 \(-0.473425\pi\)
0.0833923 + 0.996517i \(0.473425\pi\)
\(522\) 14990.4 1.25692
\(523\) −588.084 −0.0491685 −0.0245843 0.999698i \(-0.507826\pi\)
−0.0245843 + 0.999698i \(0.507826\pi\)
\(524\) −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 \(-0.853777\pi\)
−0.896330 + 0.443387i \(0.853777\pi\)
\(572\) −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 \(-0.796962\pi\)
−0.803370 + 0.595480i \(0.796962\pi\)
\(578\) 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 \(-0.818336\pi\)
−0.841514 + 0.540235i \(0.818336\pi\)
\(588\) 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 \(-0.161791\pi\)
0.873583 + 0.486675i \(0.161791\pi\)
\(594\) −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 \(-0.284598\pi\)
0.626228 + 0.779640i \(0.284598\pi\)
\(602\) 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 \(-0.147825\pi\)
0.894088 + 0.447892i \(0.147825\pi\)
\(608\) 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 \(-0.399688\pi\)
0.309950 + 0.950753i \(0.399688\pi\)
\(632\) 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 \(-0.446753\pi\)
0.166502 + 0.986041i \(0.446753\pi\)
\(642\) 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 \(-0.589620\pi\)
−0.277846 + 0.960626i \(0.589620\pi\)
\(648\) 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 \(-0.501538\pi\)
−0.00483046 + 0.999988i \(0.501538\pi\)
\(654\) −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 \(-0.396041\pi\)
0.320821 + 0.947140i \(0.396041\pi\)
\(674\) 9125.74 0.521529
\(675\) 1757.75 0.100231
\(676\) 2296.70 0.130672
\(677\) −29615.8 −1.68128 −0.840640 0.541594i \(-0.817821\pi\)
−0.840640 + 0.541594i \(0.817821\pi\)
\(678\) 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 \(-0.295200\pi\)
0.599917 + 0.800062i \(0.295200\pi\)
\(692\) −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 \(-0.430906\pi\)
0.215366 + 0.976534i \(0.430906\pi\)
\(728\) 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 \(-0.426336\pi\)
0.229362 + 0.973341i \(0.426336\pi\)
\(734\) 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 \(-0.490524\pi\)
0.0297638 + 0.999557i \(0.490524\pi\)
\(740\) 1405.42 0.0698164
\(741\) 4562.75 0.226203
\(742\) 13282.4 0.657160
\(743\) −18356.6 −0.906379 −0.453189 0.891414i \(-0.649714\pi\)
−0.453189 + 0.891414i \(0.649714\pi\)
\(744\) 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 \(-0.139082\pi\)
0.906052 + 0.423167i \(0.139082\pi\)
\(752\) −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 \(-0.383938\pi\)
0.356593 + 0.934260i \(0.383938\pi\)
\(758\) −7517.27 −0.360211
\(759\) 18876.1 0.902714
\(760\) −8838.84 −0.421867
\(761\) −35725.3 −1.70176 −0.850881 0.525358i \(-0.823931\pi\)
−0.850881 + 0.525358i \(0.823931\pi\)
\(762\) −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 \(-0.622020\pi\)
−0.374017 + 0.927422i \(0.622020\pi\)
\(774\) 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 \(-0.311000\pi\)
0.559484 + 0.828841i \(0.311000\pi\)
\(812\) −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 \(-0.469092\pi\)
0.0969468 + 0.995290i \(0.469092\pi\)
\(830\) 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 \(-0.704429\pi\)
−0.598984 + 0.800761i \(0.704429\pi\)
\(840\) −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 \(-0.367101\pi\)
0.405490 + 0.914099i \(0.367101\pi\)
\(860\) 0 0
\(861\) 1269.85 0.0502629
\(862\) −20547.9 −0.811907
\(863\) −11193.8 −0.441532 −0.220766 0.975327i \(-0.570856\pi\)
−0.220766 + 0.975327i \(0.570856\pi\)
\(864\) 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 \(-0.171652\pi\)
0.858088 + 0.513502i \(0.171652\pi\)
\(888\) 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 \(-0.585768\pi\)
−0.266199 + 0.963918i \(0.585768\pi\)
\(912\) 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 \(-0.434135\pi\)
0.205446 + 0.978668i \(0.434135\pi\)
\(930\) 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 \(-0.593410\pi\)
−0.289263 + 0.957250i \(0.593410\pi\)
\(938\) 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 \(-0.471205\pi\)
0.0903400 + 0.995911i \(0.471205\pi\)
\(954\) 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 \(-0.368442\pi\)
0.401636 + 0.915800i \(0.368442\pi\)
\(984\) −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 \(-0.394357\pi\)
0.325827 + 0.945429i \(0.394357\pi\)
\(992\) −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 \(-0.192674\pi\)
0.822330 + 0.569011i \(0.192674\pi\)
\(998\) 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.i.1.15 50
43.42 odd 2 1849.4.a.j.1.36 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.15 50 1.1 even 1 trivial
1849.4.a.j.1.36 yes 50 43.42 odd 2