Properties

Label 1849.4.a.i.1.41
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 1.41
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.88871 q^{2} -9.09520 q^{3} +7.12206 q^{4} +19.0468 q^{5} -35.3686 q^{6} -18.7927 q^{7} -3.41407 q^{8} +55.7226 q^{9} +O(q^{10})\) \(q+3.88871 q^{2} -9.09520 q^{3} +7.12206 q^{4} +19.0468 q^{5} -35.3686 q^{6} -18.7927 q^{7} -3.41407 q^{8} +55.7226 q^{9} +74.0673 q^{10} +4.37373 q^{11} -64.7765 q^{12} +45.6452 q^{13} -73.0793 q^{14} -173.234 q^{15} -70.2528 q^{16} -62.8152 q^{17} +216.689 q^{18} -33.1127 q^{19} +135.652 q^{20} +170.923 q^{21} +17.0082 q^{22} -32.0896 q^{23} +31.0516 q^{24} +237.779 q^{25} +177.501 q^{26} -261.237 q^{27} -133.843 q^{28} +291.097 q^{29} -673.657 q^{30} -83.9237 q^{31} -245.880 q^{32} -39.7799 q^{33} -244.270 q^{34} -357.940 q^{35} +396.859 q^{36} +90.1795 q^{37} -128.766 q^{38} -415.152 q^{39} -65.0270 q^{40} -422.014 q^{41} +664.670 q^{42} +31.1499 q^{44} +1061.33 q^{45} -124.787 q^{46} -363.055 q^{47} +638.963 q^{48} +10.1649 q^{49} +924.654 q^{50} +571.316 q^{51} +325.087 q^{52} +387.946 q^{53} -1015.88 q^{54} +83.3054 q^{55} +64.1595 q^{56} +301.166 q^{57} +1131.99 q^{58} -248.120 q^{59} -1233.78 q^{60} -64.5638 q^{61} -326.355 q^{62} -1047.18 q^{63} -394.134 q^{64} +869.393 q^{65} -154.692 q^{66} +80.6113 q^{67} -447.373 q^{68} +291.861 q^{69} -1391.92 q^{70} -936.123 q^{71} -190.241 q^{72} +651.084 q^{73} +350.682 q^{74} -2162.65 q^{75} -235.830 q^{76} -82.1941 q^{77} -1614.40 q^{78} +1114.36 q^{79} -1338.09 q^{80} +871.496 q^{81} -1641.09 q^{82} +26.3884 q^{83} +1217.32 q^{84} -1196.43 q^{85} -2647.58 q^{87} -14.9322 q^{88} +347.849 q^{89} +4127.22 q^{90} -857.795 q^{91} -228.544 q^{92} +763.303 q^{93} -1411.82 q^{94} -630.689 q^{95} +2236.33 q^{96} +291.207 q^{97} +39.5283 q^{98} +243.715 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.88871 1.37487 0.687433 0.726248i \(-0.258737\pi\)
0.687433 + 0.726248i \(0.258737\pi\)
\(3\) −9.09520 −1.75037 −0.875186 0.483787i \(-0.839261\pi\)
−0.875186 + 0.483787i \(0.839261\pi\)
\(4\) 7.12206 0.890257
\(5\) 19.0468 1.70359 0.851797 0.523872i \(-0.175513\pi\)
0.851797 + 0.523872i \(0.175513\pi\)
\(6\) −35.3686 −2.40653
\(7\) −18.7927 −1.01471 −0.507355 0.861737i \(-0.669377\pi\)
−0.507355 + 0.861737i \(0.669377\pi\)
\(8\) −3.41407 −0.150882
\(9\) 55.7226 2.06380
\(10\) 74.0673 2.34221
\(11\) 4.37373 0.119884 0.0599422 0.998202i \(-0.480908\pi\)
0.0599422 + 0.998202i \(0.480908\pi\)
\(12\) −64.7765 −1.55828
\(13\) 45.6452 0.973822 0.486911 0.873452i \(-0.338124\pi\)
0.486911 + 0.873452i \(0.338124\pi\)
\(14\) −73.0793 −1.39509
\(15\) −173.234 −2.98192
\(16\) −70.2528 −1.09770
\(17\) −62.8152 −0.896172 −0.448086 0.893990i \(-0.647894\pi\)
−0.448086 + 0.893990i \(0.647894\pi\)
\(18\) 216.689 2.83745
\(19\) −33.1127 −0.399819 −0.199910 0.979814i \(-0.564065\pi\)
−0.199910 + 0.979814i \(0.564065\pi\)
\(20\) 135.652 1.51664
\(21\) 170.923 1.77612
\(22\) 17.0082 0.164825
\(23\) −32.0896 −0.290919 −0.145459 0.989364i \(-0.546466\pi\)
−0.145459 + 0.989364i \(0.546466\pi\)
\(24\) 31.0516 0.264100
\(25\) 237.779 1.90223
\(26\) 177.501 1.33888
\(27\) −261.237 −1.86204
\(28\) −133.843 −0.903352
\(29\) 291.097 1.86398 0.931988 0.362490i \(-0.118073\pi\)
0.931988 + 0.362490i \(0.118073\pi\)
\(30\) −673.657 −4.09974
\(31\) −83.9237 −0.486231 −0.243115 0.969997i \(-0.578169\pi\)
−0.243115 + 0.969997i \(0.578169\pi\)
\(32\) −245.880 −1.35831
\(33\) −39.7799 −0.209842
\(34\) −244.270 −1.23212
\(35\) −357.940 −1.72865
\(36\) 396.859 1.83731
\(37\) 90.1795 0.400687 0.200343 0.979726i \(-0.435794\pi\)
0.200343 + 0.979726i \(0.435794\pi\)
\(38\) −128.766 −0.549698
\(39\) −415.152 −1.70455
\(40\) −65.0270 −0.257042
\(41\) −422.014 −1.60750 −0.803750 0.594967i \(-0.797165\pi\)
−0.803750 + 0.594967i \(0.797165\pi\)
\(42\) 664.670 2.44192
\(43\) 0 0
\(44\) 31.1499 0.106728
\(45\) 1061.33 3.51588
\(46\) −124.787 −0.399974
\(47\) −363.055 −1.12675 −0.563373 0.826203i \(-0.690497\pi\)
−0.563373 + 0.826203i \(0.690497\pi\)
\(48\) 638.963 1.92138
\(49\) 10.1649 0.0296352
\(50\) 924.654 2.61532
\(51\) 571.316 1.56863
\(52\) 325.087 0.866952
\(53\) 387.946 1.00544 0.502722 0.864448i \(-0.332332\pi\)
0.502722 + 0.864448i \(0.332332\pi\)
\(54\) −1015.88 −2.56006
\(55\) 83.3054 0.204234
\(56\) 64.1595 0.153101
\(57\) 301.166 0.699832
\(58\) 1131.99 2.56272
\(59\) −248.120 −0.547500 −0.273750 0.961801i \(-0.588264\pi\)
−0.273750 + 0.961801i \(0.588264\pi\)
\(60\) −1233.78 −2.65468
\(61\) −64.5638 −0.135517 −0.0677586 0.997702i \(-0.521585\pi\)
−0.0677586 + 0.997702i \(0.521585\pi\)
\(62\) −326.355 −0.668502
\(63\) −1047.18 −2.09416
\(64\) −394.134 −0.769792
\(65\) 869.393 1.65900
\(66\) −154.692 −0.288505
\(67\) 80.6113 0.146989 0.0734943 0.997296i \(-0.476585\pi\)
0.0734943 + 0.997296i \(0.476585\pi\)
\(68\) −447.373 −0.797823
\(69\) 291.861 0.509216
\(70\) −1391.92 −2.37667
\(71\) −936.123 −1.56475 −0.782375 0.622807i \(-0.785992\pi\)
−0.782375 + 0.622807i \(0.785992\pi\)
\(72\) −190.241 −0.311390
\(73\) 651.084 1.04389 0.521943 0.852980i \(-0.325207\pi\)
0.521943 + 0.852980i \(0.325207\pi\)
\(74\) 350.682 0.550891
\(75\) −2162.65 −3.32962
\(76\) −235.830 −0.355942
\(77\) −82.1941 −0.121648
\(78\) −1614.40 −2.34353
\(79\) 1114.36 1.58703 0.793516 0.608549i \(-0.208248\pi\)
0.793516 + 0.608549i \(0.208248\pi\)
\(80\) −1338.09 −1.87003
\(81\) 871.496 1.19547
\(82\) −1641.09 −2.21010
\(83\) 26.3884 0.0348976 0.0174488 0.999848i \(-0.494446\pi\)
0.0174488 + 0.999848i \(0.494446\pi\)
\(84\) 1217.32 1.58120
\(85\) −1196.43 −1.52671
\(86\) 0 0
\(87\) −2647.58 −3.26265
\(88\) −14.9322 −0.0180884
\(89\) 347.849 0.414292 0.207146 0.978310i \(-0.433583\pi\)
0.207146 + 0.978310i \(0.433583\pi\)
\(90\) 4127.22 4.83386
\(91\) −857.795 −0.988147
\(92\) −228.544 −0.258993
\(93\) 763.303 0.851084
\(94\) −1411.82 −1.54912
\(95\) −630.689 −0.681130
\(96\) 2236.33 2.37754
\(97\) 291.207 0.304821 0.152410 0.988317i \(-0.451296\pi\)
0.152410 + 0.988317i \(0.451296\pi\)
\(98\) 39.5283 0.0407445
\(99\) 243.715 0.247417
\(100\) 1693.48 1.69348
\(101\) −1176.57 −1.15913 −0.579567 0.814924i \(-0.696778\pi\)
−0.579567 + 0.814924i \(0.696778\pi\)
\(102\) 2221.68 2.15666
\(103\) 457.000 0.437181 0.218590 0.975817i \(-0.429854\pi\)
0.218590 + 0.975817i \(0.429854\pi\)
\(104\) −155.836 −0.146932
\(105\) 3255.53 3.02579
\(106\) 1508.61 1.38235
\(107\) −731.873 −0.661242 −0.330621 0.943764i \(-0.607258\pi\)
−0.330621 + 0.943764i \(0.607258\pi\)
\(108\) −1860.55 −1.65770
\(109\) −119.276 −0.104813 −0.0524063 0.998626i \(-0.516689\pi\)
−0.0524063 + 0.998626i \(0.516689\pi\)
\(110\) 323.950 0.280795
\(111\) −820.200 −0.701351
\(112\) 1320.24 1.11385
\(113\) −1149.02 −0.956554 −0.478277 0.878209i \(-0.658739\pi\)
−0.478277 + 0.878209i \(0.658739\pi\)
\(114\) 1171.15 0.962176
\(115\) −611.202 −0.495608
\(116\) 2073.21 1.65942
\(117\) 2543.47 2.00977
\(118\) −964.868 −0.752740
\(119\) 1180.47 0.909354
\(120\) 591.433 0.449919
\(121\) −1311.87 −0.985628
\(122\) −251.070 −0.186318
\(123\) 3838.30 2.81372
\(124\) −597.710 −0.432870
\(125\) 2148.08 1.53704
\(126\) −4072.17 −2.87918
\(127\) −1354.82 −0.946621 −0.473310 0.880896i \(-0.656941\pi\)
−0.473310 + 0.880896i \(0.656941\pi\)
\(128\) 434.370 0.299947
\(129\) 0 0
\(130\) 3380.81 2.28090
\(131\) −986.023 −0.657628 −0.328814 0.944395i \(-0.606649\pi\)
−0.328814 + 0.944395i \(0.606649\pi\)
\(132\) −283.315 −0.186813
\(133\) 622.276 0.405701
\(134\) 313.474 0.202090
\(135\) −4975.73 −3.17217
\(136\) 214.456 0.135216
\(137\) −876.858 −0.546825 −0.273412 0.961897i \(-0.588152\pi\)
−0.273412 + 0.961897i \(0.588152\pi\)
\(138\) 1134.96 0.700104
\(139\) −1371.22 −0.836731 −0.418365 0.908279i \(-0.637397\pi\)
−0.418365 + 0.908279i \(0.637397\pi\)
\(140\) −2549.27 −1.53895
\(141\) 3302.06 1.97222
\(142\) −3640.31 −2.15132
\(143\) 199.639 0.116746
\(144\) −3914.67 −2.26543
\(145\) 5544.45 3.17546
\(146\) 2531.88 1.43520
\(147\) −92.4516 −0.0518727
\(148\) 642.263 0.356714
\(149\) −2597.36 −1.42808 −0.714040 0.700105i \(-0.753137\pi\)
−0.714040 + 0.700105i \(0.753137\pi\)
\(150\) −8409.91 −4.57778
\(151\) −285.155 −0.153679 −0.0768397 0.997043i \(-0.524483\pi\)
−0.0768397 + 0.997043i \(0.524483\pi\)
\(152\) 113.049 0.0603256
\(153\) −3500.22 −1.84952
\(154\) −319.629 −0.167250
\(155\) −1598.48 −0.828340
\(156\) −2956.73 −1.51749
\(157\) −1326.53 −0.674323 −0.337161 0.941447i \(-0.609467\pi\)
−0.337161 + 0.941447i \(0.609467\pi\)
\(158\) 4333.43 2.18196
\(159\) −3528.45 −1.75990
\(160\) −4683.22 −2.31401
\(161\) 603.049 0.295198
\(162\) 3388.99 1.64361
\(163\) 182.755 0.0878191 0.0439095 0.999036i \(-0.486019\pi\)
0.0439095 + 0.999036i \(0.486019\pi\)
\(164\) −3005.61 −1.43109
\(165\) −757.679 −0.357486
\(166\) 102.617 0.0479795
\(167\) −2822.36 −1.30779 −0.653894 0.756586i \(-0.726866\pi\)
−0.653894 + 0.756586i \(0.726866\pi\)
\(168\) −583.544 −0.267984
\(169\) −113.520 −0.0516704
\(170\) −4652.55 −2.09903
\(171\) −1845.12 −0.825147
\(172\) 0 0
\(173\) −2917.04 −1.28196 −0.640978 0.767559i \(-0.721471\pi\)
−0.640978 + 0.767559i \(0.721471\pi\)
\(174\) −10295.7 −4.48571
\(175\) −4468.51 −1.93022
\(176\) −307.267 −0.131597
\(177\) 2256.70 0.958329
\(178\) 1352.68 0.569595
\(179\) 2240.14 0.935397 0.467699 0.883888i \(-0.345083\pi\)
0.467699 + 0.883888i \(0.345083\pi\)
\(180\) 7558.89 3.13003
\(181\) −4100.39 −1.68387 −0.841934 0.539581i \(-0.818583\pi\)
−0.841934 + 0.539581i \(0.818583\pi\)
\(182\) −3335.71 −1.35857
\(183\) 587.221 0.237205
\(184\) 109.556 0.0438944
\(185\) 1717.63 0.682608
\(186\) 2968.26 1.17013
\(187\) −274.737 −0.107437
\(188\) −2585.70 −1.00309
\(189\) 4909.35 1.88943
\(190\) −2452.57 −0.936463
\(191\) −3114.34 −1.17982 −0.589910 0.807469i \(-0.700837\pi\)
−0.589910 + 0.807469i \(0.700837\pi\)
\(192\) 3584.72 1.34742
\(193\) 1369.02 0.510590 0.255295 0.966863i \(-0.417827\pi\)
0.255295 + 0.966863i \(0.417827\pi\)
\(194\) 1132.42 0.419088
\(195\) −7907.30 −2.90386
\(196\) 72.3949 0.0263830
\(197\) −4821.42 −1.74372 −0.871858 0.489758i \(-0.837085\pi\)
−0.871858 + 0.489758i \(0.837085\pi\)
\(198\) 947.738 0.340166
\(199\) 3469.11 1.23577 0.617887 0.786267i \(-0.287989\pi\)
0.617887 + 0.786267i \(0.287989\pi\)
\(200\) −811.795 −0.287013
\(201\) −733.175 −0.257285
\(202\) −4575.32 −1.59366
\(203\) −5470.49 −1.89139
\(204\) 4068.95 1.39649
\(205\) −8038.00 −2.73853
\(206\) 1777.14 0.601065
\(207\) −1788.11 −0.600398
\(208\) −3206.70 −1.06896
\(209\) −144.826 −0.0479321
\(210\) 12659.8 4.16005
\(211\) 949.908 0.309926 0.154963 0.987920i \(-0.450474\pi\)
0.154963 + 0.987920i \(0.450474\pi\)
\(212\) 2762.97 0.895103
\(213\) 8514.22 2.73889
\(214\) −2846.04 −0.909119
\(215\) 0 0
\(216\) 891.883 0.280949
\(217\) 1577.15 0.493383
\(218\) −463.830 −0.144103
\(219\) −5921.74 −1.82719
\(220\) 593.305 0.181821
\(221\) −2867.21 −0.872712
\(222\) −3189.52 −0.964264
\(223\) 5128.19 1.53995 0.769976 0.638073i \(-0.220268\pi\)
0.769976 + 0.638073i \(0.220268\pi\)
\(224\) 4620.74 1.37829
\(225\) 13249.7 3.92583
\(226\) −4468.20 −1.31513
\(227\) −3818.68 −1.11654 −0.558270 0.829659i \(-0.688535\pi\)
−0.558270 + 0.829659i \(0.688535\pi\)
\(228\) 2144.92 0.623031
\(229\) −2587.09 −0.746548 −0.373274 0.927721i \(-0.621765\pi\)
−0.373274 + 0.927721i \(0.621765\pi\)
\(230\) −2376.79 −0.681394
\(231\) 747.571 0.212929
\(232\) −993.825 −0.281240
\(233\) −3660.55 −1.02923 −0.514615 0.857421i \(-0.672065\pi\)
−0.514615 + 0.857421i \(0.672065\pi\)
\(234\) 9890.80 2.76317
\(235\) −6915.03 −1.91952
\(236\) −1767.13 −0.487416
\(237\) −10135.3 −2.77789
\(238\) 4590.49 1.25024
\(239\) 840.112 0.227374 0.113687 0.993517i \(-0.463734\pi\)
0.113687 + 0.993517i \(0.463734\pi\)
\(240\) 12170.2 3.27326
\(241\) 1237.68 0.330813 0.165406 0.986226i \(-0.447106\pi\)
0.165406 + 0.986226i \(0.447106\pi\)
\(242\) −5101.48 −1.35511
\(243\) −873.016 −0.230469
\(244\) −459.827 −0.120645
\(245\) 193.608 0.0504864
\(246\) 14926.0 3.86849
\(247\) −1511.43 −0.389353
\(248\) 286.522 0.0733635
\(249\) −240.007 −0.0610837
\(250\) 8353.26 2.11323
\(251\) −4783.43 −1.20290 −0.601449 0.798911i \(-0.705410\pi\)
−0.601449 + 0.798911i \(0.705410\pi\)
\(252\) −7458.05 −1.86434
\(253\) −140.351 −0.0348766
\(254\) −5268.50 −1.30148
\(255\) 10881.7 2.67232
\(256\) 4842.20 1.18218
\(257\) −3254.28 −0.789869 −0.394935 0.918709i \(-0.629233\pi\)
−0.394935 + 0.918709i \(0.629233\pi\)
\(258\) 0 0
\(259\) −1694.71 −0.406581
\(260\) 6191.86 1.47693
\(261\) 16220.7 3.84687
\(262\) −3834.36 −0.904150
\(263\) 4499.66 1.05498 0.527492 0.849560i \(-0.323132\pi\)
0.527492 + 0.849560i \(0.323132\pi\)
\(264\) 135.811 0.0316614
\(265\) 7389.12 1.71287
\(266\) 2419.85 0.557784
\(267\) −3163.76 −0.725164
\(268\) 574.118 0.130858
\(269\) 4396.09 0.996409 0.498205 0.867059i \(-0.333993\pi\)
0.498205 + 0.867059i \(0.333993\pi\)
\(270\) −19349.2 −4.36131
\(271\) 489.737 0.109776 0.0548882 0.998493i \(-0.482520\pi\)
0.0548882 + 0.998493i \(0.482520\pi\)
\(272\) 4412.94 0.983728
\(273\) 7801.81 1.72962
\(274\) −3409.84 −0.751811
\(275\) 1039.98 0.228048
\(276\) 2078.65 0.453333
\(277\) 2576.01 0.558762 0.279381 0.960180i \(-0.409871\pi\)
0.279381 + 0.960180i \(0.409871\pi\)
\(278\) −5332.28 −1.15039
\(279\) −4676.45 −1.00348
\(280\) 1222.03 0.260823
\(281\) 6339.36 1.34582 0.672908 0.739726i \(-0.265045\pi\)
0.672908 + 0.739726i \(0.265045\pi\)
\(282\) 12840.7 2.71154
\(283\) 3492.92 0.733684 0.366842 0.930283i \(-0.380439\pi\)
0.366842 + 0.930283i \(0.380439\pi\)
\(284\) −6667.12 −1.39303
\(285\) 5736.24 1.19223
\(286\) 776.340 0.160510
\(287\) 7930.77 1.63115
\(288\) −13701.1 −2.80327
\(289\) −967.251 −0.196876
\(290\) 21560.8 4.36583
\(291\) −2648.59 −0.533550
\(292\) 4637.06 0.929327
\(293\) −6042.69 −1.20484 −0.602419 0.798180i \(-0.705796\pi\)
−0.602419 + 0.798180i \(0.705796\pi\)
\(294\) −359.517 −0.0713180
\(295\) −4725.89 −0.932719
\(296\) −307.879 −0.0604565
\(297\) −1142.58 −0.223230
\(298\) −10100.4 −1.96342
\(299\) −1464.73 −0.283303
\(300\) −15402.5 −2.96421
\(301\) 0 0
\(302\) −1108.89 −0.211289
\(303\) 10701.1 2.02892
\(304\) 2326.26 0.438882
\(305\) −1229.73 −0.230866
\(306\) −13611.4 −2.54284
\(307\) 7899.54 1.46857 0.734285 0.678842i \(-0.237518\pi\)
0.734285 + 0.678842i \(0.237518\pi\)
\(308\) −585.391 −0.108298
\(309\) −4156.51 −0.765228
\(310\) −6216.01 −1.13886
\(311\) −10690.9 −1.94927 −0.974636 0.223798i \(-0.928154\pi\)
−0.974636 + 0.223798i \(0.928154\pi\)
\(312\) 1417.36 0.257186
\(313\) 1211.51 0.218782 0.109391 0.993999i \(-0.465110\pi\)
0.109391 + 0.993999i \(0.465110\pi\)
\(314\) −5158.49 −0.927103
\(315\) −19945.3 −3.56759
\(316\) 7936.55 1.41287
\(317\) −6471.09 −1.14654 −0.573269 0.819367i \(-0.694325\pi\)
−0.573269 + 0.819367i \(0.694325\pi\)
\(318\) −13721.1 −2.41963
\(319\) 1273.18 0.223462
\(320\) −7506.97 −1.31141
\(321\) 6656.53 1.15742
\(322\) 2345.08 0.405858
\(323\) 2079.98 0.358307
\(324\) 6206.84 1.06427
\(325\) 10853.5 1.85244
\(326\) 710.682 0.120739
\(327\) 1084.84 0.183461
\(328\) 1440.79 0.242543
\(329\) 6822.78 1.14332
\(330\) −2946.39 −0.491496
\(331\) −1395.83 −0.231788 −0.115894 0.993262i \(-0.536973\pi\)
−0.115894 + 0.993262i \(0.536973\pi\)
\(332\) 187.939 0.0310678
\(333\) 5025.03 0.826937
\(334\) −10975.3 −1.79803
\(335\) 1535.38 0.250409
\(336\) −12007.8 −1.94964
\(337\) 11548.8 1.86678 0.933391 0.358860i \(-0.116835\pi\)
0.933391 + 0.358860i \(0.116835\pi\)
\(338\) −441.446 −0.0710399
\(339\) 10450.6 1.67432
\(340\) −8521.02 −1.35917
\(341\) −367.060 −0.0582915
\(342\) −7175.15 −1.13447
\(343\) 6254.86 0.984638
\(344\) 0 0
\(345\) 5559.00 0.867498
\(346\) −11343.5 −1.76252
\(347\) −3970.96 −0.614330 −0.307165 0.951656i \(-0.599380\pi\)
−0.307165 + 0.951656i \(0.599380\pi\)
\(348\) −18856.2 −2.90460
\(349\) 1461.92 0.224226 0.112113 0.993695i \(-0.464238\pi\)
0.112113 + 0.993695i \(0.464238\pi\)
\(350\) −17376.7 −2.65379
\(351\) −11924.2 −1.81330
\(352\) −1075.41 −0.162840
\(353\) −4652.97 −0.701565 −0.350783 0.936457i \(-0.614084\pi\)
−0.350783 + 0.936457i \(0.614084\pi\)
\(354\) 8775.66 1.31757
\(355\) −17830.1 −2.66570
\(356\) 2477.40 0.368826
\(357\) −10736.6 −1.59171
\(358\) 8711.26 1.28605
\(359\) 84.9556 0.0124897 0.00624483 0.999981i \(-0.498012\pi\)
0.00624483 + 0.999981i \(0.498012\pi\)
\(360\) −3623.47 −0.530483
\(361\) −5762.55 −0.840144
\(362\) −15945.2 −2.31509
\(363\) 11931.7 1.72521
\(364\) −6109.26 −0.879704
\(365\) 12401.1 1.77836
\(366\) 2283.53 0.326126
\(367\) −5121.53 −0.728452 −0.364226 0.931311i \(-0.618666\pi\)
−0.364226 + 0.931311i \(0.618666\pi\)
\(368\) 2254.38 0.319341
\(369\) −23515.7 −3.31756
\(370\) 6679.35 0.938495
\(371\) −7290.55 −1.02023
\(372\) 5436.29 0.757683
\(373\) 898.596 0.124739 0.0623693 0.998053i \(-0.480134\pi\)
0.0623693 + 0.998053i \(0.480134\pi\)
\(374\) −1068.37 −0.147712
\(375\) −19537.2 −2.69039
\(376\) 1239.50 0.170006
\(377\) 13287.2 1.81518
\(378\) 19091.0 2.59772
\(379\) 9099.47 1.23327 0.616634 0.787250i \(-0.288496\pi\)
0.616634 + 0.787250i \(0.288496\pi\)
\(380\) −4491.81 −0.606381
\(381\) 12322.3 1.65694
\(382\) −12110.8 −1.62209
\(383\) 6069.35 0.809738 0.404869 0.914375i \(-0.367317\pi\)
0.404869 + 0.914375i \(0.367317\pi\)
\(384\) −3950.68 −0.525018
\(385\) −1565.53 −0.207239
\(386\) 5323.70 0.701993
\(387\) 0 0
\(388\) 2073.99 0.271369
\(389\) −8283.52 −1.07967 −0.539834 0.841771i \(-0.681513\pi\)
−0.539834 + 0.841771i \(0.681513\pi\)
\(390\) −30749.2 −3.99242
\(391\) 2015.71 0.260713
\(392\) −34.7036 −0.00447142
\(393\) 8968.08 1.15109
\(394\) −18749.1 −2.39738
\(395\) 21225.0 2.70366
\(396\) 1735.75 0.220265
\(397\) 5831.90 0.737266 0.368633 0.929575i \(-0.379826\pi\)
0.368633 + 0.929575i \(0.379826\pi\)
\(398\) 13490.4 1.69902
\(399\) −5659.72 −0.710127
\(400\) −16704.7 −2.08808
\(401\) −14041.3 −1.74860 −0.874299 0.485389i \(-0.838678\pi\)
−0.874299 + 0.485389i \(0.838678\pi\)
\(402\) −2851.10 −0.353732
\(403\) −3830.71 −0.473502
\(404\) −8379.56 −1.03193
\(405\) 16599.2 2.03659
\(406\) −21273.1 −2.60041
\(407\) 394.421 0.0480361
\(408\) −1950.51 −0.236679
\(409\) −2456.40 −0.296971 −0.148486 0.988915i \(-0.547440\pi\)
−0.148486 + 0.988915i \(0.547440\pi\)
\(410\) −31257.4 −3.76511
\(411\) 7975.19 0.957147
\(412\) 3254.78 0.389203
\(413\) 4662.85 0.555554
\(414\) −6953.45 −0.825467
\(415\) 502.613 0.0594513
\(416\) −11223.2 −1.32275
\(417\) 12471.5 1.46459
\(418\) −563.186 −0.0659003
\(419\) −9412.48 −1.09745 −0.548723 0.836004i \(-0.684886\pi\)
−0.548723 + 0.836004i \(0.684886\pi\)
\(420\) 23186.1 2.69373
\(421\) 7475.80 0.865435 0.432718 0.901529i \(-0.357555\pi\)
0.432718 + 0.901529i \(0.357555\pi\)
\(422\) 3693.92 0.426107
\(423\) −20230.4 −2.32538
\(424\) −1324.48 −0.151703
\(425\) −14936.2 −1.70473
\(426\) 33109.3 3.76561
\(427\) 1213.33 0.137511
\(428\) −5212.44 −0.588675
\(429\) −1815.76 −0.204349
\(430\) 0 0
\(431\) 12819.7 1.43273 0.716363 0.697728i \(-0.245806\pi\)
0.716363 + 0.697728i \(0.245806\pi\)
\(432\) 18352.7 2.04396
\(433\) 8929.20 0.991017 0.495508 0.868603i \(-0.334982\pi\)
0.495508 + 0.868603i \(0.334982\pi\)
\(434\) 6133.09 0.678335
\(435\) −50427.9 −5.55823
\(436\) −849.490 −0.0933101
\(437\) 1062.57 0.116315
\(438\) −23027.9 −2.51214
\(439\) −9033.45 −0.982103 −0.491051 0.871131i \(-0.663387\pi\)
−0.491051 + 0.871131i \(0.663387\pi\)
\(440\) −284.410 −0.0308153
\(441\) 566.414 0.0611612
\(442\) −11149.7 −1.19986
\(443\) 7257.25 0.778335 0.389167 0.921167i \(-0.372763\pi\)
0.389167 + 0.921167i \(0.372763\pi\)
\(444\) −5841.51 −0.624382
\(445\) 6625.40 0.705785
\(446\) 19942.1 2.11723
\(447\) 23623.5 2.49967
\(448\) 7406.83 0.781115
\(449\) −11843.1 −1.24479 −0.622394 0.782704i \(-0.713840\pi\)
−0.622394 + 0.782704i \(0.713840\pi\)
\(450\) 51524.1 5.39749
\(451\) −1845.77 −0.192714
\(452\) −8183.38 −0.851579
\(453\) 2593.54 0.268996
\(454\) −14849.7 −1.53509
\(455\) −16338.2 −1.68340
\(456\) −1028.20 −0.105592
\(457\) −6136.43 −0.628119 −0.314059 0.949403i \(-0.601689\pi\)
−0.314059 + 0.949403i \(0.601689\pi\)
\(458\) −10060.4 −1.02640
\(459\) 16409.7 1.66871
\(460\) −4353.02 −0.441218
\(461\) 1927.73 0.194758 0.0973788 0.995247i \(-0.468954\pi\)
0.0973788 + 0.995247i \(0.468954\pi\)
\(462\) 2907.09 0.292749
\(463\) 10705.1 1.07453 0.537264 0.843414i \(-0.319458\pi\)
0.537264 + 0.843414i \(0.319458\pi\)
\(464\) −20450.3 −2.04609
\(465\) 14538.5 1.44990
\(466\) −14234.8 −1.41505
\(467\) 13219.7 1.30992 0.654962 0.755661i \(-0.272684\pi\)
0.654962 + 0.755661i \(0.272684\pi\)
\(468\) 18114.7 1.78921
\(469\) −1514.90 −0.149151
\(470\) −26890.5 −2.63908
\(471\) 12065.1 1.18032
\(472\) 847.101 0.0826080
\(473\) 0 0
\(474\) −39413.4 −3.81923
\(475\) −7873.51 −0.760550
\(476\) 8407.34 0.809559
\(477\) 21617.4 2.07503
\(478\) 3266.95 0.312608
\(479\) −2249.15 −0.214543 −0.107271 0.994230i \(-0.534211\pi\)
−0.107271 + 0.994230i \(0.534211\pi\)
\(480\) 42594.8 4.05037
\(481\) 4116.26 0.390198
\(482\) 4812.97 0.454823
\(483\) −5484.85 −0.516706
\(484\) −9343.21 −0.877462
\(485\) 5546.56 0.519291
\(486\) −3394.90 −0.316864
\(487\) 17135.5 1.59443 0.797213 0.603698i \(-0.206307\pi\)
0.797213 + 0.603698i \(0.206307\pi\)
\(488\) 220.425 0.0204471
\(489\) −1662.20 −0.153716
\(490\) 752.886 0.0694121
\(491\) 17052.6 1.56736 0.783680 0.621165i \(-0.213340\pi\)
0.783680 + 0.621165i \(0.213340\pi\)
\(492\) 27336.6 2.50494
\(493\) −18285.3 −1.67044
\(494\) −5877.52 −0.535308
\(495\) 4641.99 0.421499
\(496\) 5895.88 0.533735
\(497\) 17592.3 1.58777
\(498\) −933.319 −0.0839820
\(499\) −7199.58 −0.645887 −0.322944 0.946418i \(-0.604672\pi\)
−0.322944 + 0.946418i \(0.604672\pi\)
\(500\) 15298.8 1.36836
\(501\) 25669.9 2.28911
\(502\) −18601.4 −1.65382
\(503\) 17120.4 1.51761 0.758807 0.651316i \(-0.225783\pi\)
0.758807 + 0.651316i \(0.225783\pi\)
\(504\) 3575.14 0.315971
\(505\) −22409.8 −1.97470
\(506\) −545.784 −0.0479507
\(507\) 1032.49 0.0904424
\(508\) −9649.10 −0.842735
\(509\) 12743.3 1.10970 0.554848 0.831952i \(-0.312776\pi\)
0.554848 + 0.831952i \(0.312776\pi\)
\(510\) 42315.9 3.67408
\(511\) −12235.6 −1.05924
\(512\) 15355.0 1.32539
\(513\) 8650.27 0.744481
\(514\) −12655.0 −1.08596
\(515\) 8704.38 0.744778
\(516\) 0 0
\(517\) −1587.91 −0.135079
\(518\) −6590.25 −0.558994
\(519\) 26531.1 2.24390
\(520\) −2968.17 −0.250313
\(521\) 15646.4 1.31570 0.657851 0.753148i \(-0.271465\pi\)
0.657851 + 0.753148i \(0.271465\pi\)
\(522\) 63077.4 5.28893
\(523\) −11556.5 −0.966213 −0.483106 0.875562i \(-0.660492\pi\)
−0.483106 + 0.875562i \(0.660492\pi\)
\(524\) −7022.51 −0.585458
\(525\) 40642.0 3.37859
\(526\) 17497.9 1.45046
\(527\) 5271.69 0.435746
\(528\) 2794.65 0.230344
\(529\) −11137.3 −0.915366
\(530\) 28734.1 2.35496
\(531\) −13825.9 −1.12993
\(532\) 4431.88 0.361178
\(533\) −19262.9 −1.56542
\(534\) −12302.9 −0.997003
\(535\) −13939.8 −1.12649
\(536\) −275.213 −0.0221779
\(537\) −20374.5 −1.63729
\(538\) 17095.1 1.36993
\(539\) 44.4584 0.00355280
\(540\) −35437.4 −2.82404
\(541\) 4750.49 0.377523 0.188761 0.982023i \(-0.439553\pi\)
0.188761 + 0.982023i \(0.439553\pi\)
\(542\) 1904.44 0.150928
\(543\) 37293.9 2.94739
\(544\) 15445.0 1.21728
\(545\) −2271.82 −0.178558
\(546\) 30339.0 2.37800
\(547\) −3938.22 −0.307836 −0.153918 0.988084i \(-0.549189\pi\)
−0.153918 + 0.988084i \(0.549189\pi\)
\(548\) −6245.03 −0.486815
\(549\) −3597.66 −0.279680
\(550\) 4044.19 0.313536
\(551\) −9638.99 −0.745254
\(552\) −996.433 −0.0768315
\(553\) −20941.9 −1.61038
\(554\) 10017.3 0.768223
\(555\) −15622.2 −1.19482
\(556\) −9765.92 −0.744905
\(557\) −8504.39 −0.646934 −0.323467 0.946239i \(-0.604849\pi\)
−0.323467 + 0.946239i \(0.604849\pi\)
\(558\) −18185.3 −1.37965
\(559\) 0 0
\(560\) 25146.3 1.89754
\(561\) 2498.78 0.188055
\(562\) 24651.9 1.85032
\(563\) −18079.0 −1.35335 −0.676676 0.736281i \(-0.736580\pi\)
−0.676676 + 0.736281i \(0.736580\pi\)
\(564\) 23517.4 1.75579
\(565\) −21885.1 −1.62958
\(566\) 13582.9 1.00872
\(567\) −16377.7 −1.21305
\(568\) 3195.99 0.236093
\(569\) 18686.5 1.37677 0.688383 0.725347i \(-0.258321\pi\)
0.688383 + 0.725347i \(0.258321\pi\)
\(570\) 22306.6 1.63916
\(571\) −5322.93 −0.390119 −0.195059 0.980791i \(-0.562490\pi\)
−0.195059 + 0.980791i \(0.562490\pi\)
\(572\) 1421.84 0.103934
\(573\) 28325.5 2.06512
\(574\) 30840.5 2.24261
\(575\) −7630.23 −0.553396
\(576\) −21962.1 −1.58870
\(577\) 8955.95 0.646171 0.323086 0.946370i \(-0.395280\pi\)
0.323086 + 0.946370i \(0.395280\pi\)
\(578\) −3761.36 −0.270678
\(579\) −12451.5 −0.893722
\(580\) 39487.9 2.82697
\(581\) −495.908 −0.0354109
\(582\) −10299.6 −0.733559
\(583\) 1696.77 0.120537
\(584\) −2222.85 −0.157504
\(585\) 48444.8 3.42384
\(586\) −23498.2 −1.65649
\(587\) 15496.4 1.08962 0.544810 0.838560i \(-0.316602\pi\)
0.544810 + 0.838560i \(0.316602\pi\)
\(588\) −658.446 −0.0461800
\(589\) 2778.94 0.194404
\(590\) −18377.6 −1.28236
\(591\) 43851.8 3.05215
\(592\) −6335.36 −0.439834
\(593\) 4288.84 0.297001 0.148500 0.988912i \(-0.452555\pi\)
0.148500 + 0.988912i \(0.452555\pi\)
\(594\) −4443.17 −0.306911
\(595\) 22484.1 1.54917
\(596\) −18498.5 −1.27136
\(597\) −31552.3 −2.16306
\(598\) −5695.92 −0.389504
\(599\) 9242.19 0.630427 0.315213 0.949021i \(-0.397924\pi\)
0.315213 + 0.949021i \(0.397924\pi\)
\(600\) 7383.44 0.502379
\(601\) 2151.32 0.146014 0.0730068 0.997331i \(-0.476741\pi\)
0.0730068 + 0.997331i \(0.476741\pi\)
\(602\) 0 0
\(603\) 4491.87 0.303355
\(604\) −2030.89 −0.136814
\(605\) −24986.9 −1.67911
\(606\) 41613.4 2.78949
\(607\) −3764.19 −0.251703 −0.125851 0.992049i \(-0.540166\pi\)
−0.125851 + 0.992049i \(0.540166\pi\)
\(608\) 8141.75 0.543078
\(609\) 49755.1 3.31064
\(610\) −4782.07 −0.317410
\(611\) −16571.7 −1.09725
\(612\) −24928.8 −1.64655
\(613\) −22176.0 −1.46114 −0.730571 0.682837i \(-0.760746\pi\)
−0.730571 + 0.682837i \(0.760746\pi\)
\(614\) 30719.0 2.01909
\(615\) 73107.2 4.79344
\(616\) 280.616 0.0183545
\(617\) −8804.57 −0.574487 −0.287244 0.957858i \(-0.592739\pi\)
−0.287244 + 0.957858i \(0.592739\pi\)
\(618\) −16163.5 −1.05209
\(619\) −11451.5 −0.743579 −0.371789 0.928317i \(-0.621256\pi\)
−0.371789 + 0.928317i \(0.621256\pi\)
\(620\) −11384.4 −0.737435
\(621\) 8382.99 0.541704
\(622\) −41573.7 −2.67999
\(623\) −6537.02 −0.420386
\(624\) 29165.5 1.87108
\(625\) 11191.6 0.716262
\(626\) 4711.23 0.300796
\(627\) 1317.22 0.0838990
\(628\) −9447.62 −0.600320
\(629\) −5664.64 −0.359084
\(630\) −77561.6 −4.90496
\(631\) −1972.40 −0.124437 −0.0622187 0.998063i \(-0.519818\pi\)
−0.0622187 + 0.998063i \(0.519818\pi\)
\(632\) −3804.51 −0.239455
\(633\) −8639.60 −0.542486
\(634\) −25164.2 −1.57634
\(635\) −25804.9 −1.61266
\(636\) −25129.8 −1.56676
\(637\) 463.978 0.0288594
\(638\) 4951.02 0.307230
\(639\) −52163.2 −3.22933
\(640\) 8273.34 0.510988
\(641\) −4829.62 −0.297595 −0.148798 0.988868i \(-0.547540\pi\)
−0.148798 + 0.988868i \(0.547540\pi\)
\(642\) 25885.3 1.59130
\(643\) 7867.97 0.482554 0.241277 0.970456i \(-0.422434\pi\)
0.241277 + 0.970456i \(0.422434\pi\)
\(644\) 4294.95 0.262802
\(645\) 0 0
\(646\) 8088.43 0.492624
\(647\) −378.715 −0.0230121 −0.0115060 0.999934i \(-0.503663\pi\)
−0.0115060 + 0.999934i \(0.503663\pi\)
\(648\) −2975.35 −0.180375
\(649\) −1085.21 −0.0656368
\(650\) 42206.0 2.54685
\(651\) −14344.5 −0.863603
\(652\) 1301.59 0.0781815
\(653\) 4432.62 0.265639 0.132819 0.991140i \(-0.457597\pi\)
0.132819 + 0.991140i \(0.457597\pi\)
\(654\) 4218.62 0.252234
\(655\) −18780.6 −1.12033
\(656\) 29647.6 1.76455
\(657\) 36280.1 2.15437
\(658\) 26531.8 1.57191
\(659\) 24352.2 1.43950 0.719748 0.694236i \(-0.244257\pi\)
0.719748 + 0.694236i \(0.244257\pi\)
\(660\) −5396.23 −0.318254
\(661\) −7875.21 −0.463404 −0.231702 0.972787i \(-0.574429\pi\)
−0.231702 + 0.972787i \(0.574429\pi\)
\(662\) −5427.98 −0.318678
\(663\) 26077.8 1.52757
\(664\) −90.0918 −0.00526542
\(665\) 11852.3 0.691149
\(666\) 19540.9 1.13693
\(667\) −9341.16 −0.542266
\(668\) −20101.0 −1.16427
\(669\) −46641.9 −2.69549
\(670\) 5970.66 0.344279
\(671\) −282.385 −0.0162464
\(672\) −42026.6 −2.41252
\(673\) −3508.40 −0.200949 −0.100475 0.994940i \(-0.532036\pi\)
−0.100475 + 0.994940i \(0.532036\pi\)
\(674\) 44910.1 2.56658
\(675\) −62116.9 −3.54204
\(676\) −808.495 −0.0459999
\(677\) 18304.0 1.03912 0.519558 0.854435i \(-0.326097\pi\)
0.519558 + 0.854435i \(0.326097\pi\)
\(678\) 40639.1 2.30197
\(679\) −5472.57 −0.309305
\(680\) 4084.68 0.230354
\(681\) 34731.6 1.95436
\(682\) −1427.39 −0.0801430
\(683\) 14143.7 0.792378 0.396189 0.918169i \(-0.370332\pi\)
0.396189 + 0.918169i \(0.370332\pi\)
\(684\) −13141.1 −0.734593
\(685\) −16701.3 −0.931568
\(686\) 24323.3 1.35375
\(687\) 23530.0 1.30674
\(688\) 0 0
\(689\) 17707.9 0.979123
\(690\) 21617.3 1.19269
\(691\) −2755.08 −0.151676 −0.0758380 0.997120i \(-0.524163\pi\)
−0.0758380 + 0.997120i \(0.524163\pi\)
\(692\) −20775.3 −1.14127
\(693\) −4580.07 −0.251057
\(694\) −15441.9 −0.844621
\(695\) −26117.4 −1.42545
\(696\) 9039.03 0.492275
\(697\) 26508.9 1.44060
\(698\) 5684.98 0.308281
\(699\) 33293.4 1.80153
\(700\) −31825.0 −1.71839
\(701\) −27986.8 −1.50792 −0.753958 0.656923i \(-0.771858\pi\)
−0.753958 + 0.656923i \(0.771858\pi\)
\(702\) −46369.8 −2.49304
\(703\) −2986.08 −0.160202
\(704\) −1723.83 −0.0922861
\(705\) 62893.5 3.35987
\(706\) −18094.0 −0.964559
\(707\) 22110.8 1.17618
\(708\) 16072.4 0.853159
\(709\) 18906.6 1.00148 0.500741 0.865597i \(-0.333061\pi\)
0.500741 + 0.865597i \(0.333061\pi\)
\(710\) −69336.1 −3.66498
\(711\) 62095.1 3.27532
\(712\) −1187.58 −0.0625092
\(713\) 2693.08 0.141454
\(714\) −41751.4 −2.18838
\(715\) 3802.49 0.198888
\(716\) 15954.4 0.832744
\(717\) −7640.98 −0.397988
\(718\) 330.368 0.0171716
\(719\) 25788.9 1.33764 0.668820 0.743424i \(-0.266800\pi\)
0.668820 + 0.743424i \(0.266800\pi\)
\(720\) −74561.7 −3.85938
\(721\) −8588.26 −0.443611
\(722\) −22408.9 −1.15509
\(723\) −11256.9 −0.579045
\(724\) −29203.2 −1.49907
\(725\) 69216.8 3.54572
\(726\) 46399.0 2.37194
\(727\) −15335.9 −0.782364 −0.391182 0.920313i \(-0.627934\pi\)
−0.391182 + 0.920313i \(0.627934\pi\)
\(728\) 2928.57 0.149094
\(729\) −15590.1 −0.792061
\(730\) 48224.1 2.44500
\(731\) 0 0
\(732\) 4182.22 0.211174
\(733\) 36179.3 1.82308 0.911538 0.411217i \(-0.134896\pi\)
0.911538 + 0.411217i \(0.134896\pi\)
\(734\) −19916.2 −1.00152
\(735\) −1760.90 −0.0883700
\(736\) 7890.18 0.395157
\(737\) 352.572 0.0176216
\(738\) −91445.7 −4.56120
\(739\) −26665.5 −1.32735 −0.663673 0.748023i \(-0.731003\pi\)
−0.663673 + 0.748023i \(0.731003\pi\)
\(740\) 12233.0 0.607697
\(741\) 13746.8 0.681512
\(742\) −28350.8 −1.40268
\(743\) 18545.2 0.915692 0.457846 0.889032i \(-0.348621\pi\)
0.457846 + 0.889032i \(0.348621\pi\)
\(744\) −2605.97 −0.128413
\(745\) −49471.3 −2.43287
\(746\) 3494.38 0.171499
\(747\) 1470.43 0.0720216
\(748\) −1956.69 −0.0956466
\(749\) 13753.9 0.670968
\(750\) −75974.5 −3.69893
\(751\) −19902.5 −0.967048 −0.483524 0.875331i \(-0.660643\pi\)
−0.483524 + 0.875331i \(0.660643\pi\)
\(752\) 25505.6 1.23683
\(753\) 43506.2 2.10552
\(754\) 51669.9 2.49563
\(755\) −5431.28 −0.261808
\(756\) 34964.7 1.68208
\(757\) 34693.7 1.66574 0.832869 0.553470i \(-0.186697\pi\)
0.832869 + 0.553470i \(0.186697\pi\)
\(758\) 35385.2 1.69558
\(759\) 1276.52 0.0610471
\(760\) 2153.22 0.102770
\(761\) −6333.20 −0.301680 −0.150840 0.988558i \(-0.548198\pi\)
−0.150840 + 0.988558i \(0.548198\pi\)
\(762\) 47918.0 2.27807
\(763\) 2241.52 0.106354
\(764\) −22180.5 −1.05034
\(765\) −66668.0 −3.15083
\(766\) 23602.0 1.11328
\(767\) −11325.5 −0.533168
\(768\) −44040.8 −2.06925
\(769\) −7479.29 −0.350729 −0.175364 0.984504i \(-0.556110\pi\)
−0.175364 + 0.984504i \(0.556110\pi\)
\(770\) −6087.90 −0.284925
\(771\) 29598.3 1.38256
\(772\) 9750.20 0.454556
\(773\) 18737.7 0.871860 0.435930 0.899981i \(-0.356420\pi\)
0.435930 + 0.899981i \(0.356420\pi\)
\(774\) 0 0
\(775\) −19955.3 −0.924925
\(776\) −994.202 −0.0459920
\(777\) 15413.8 0.711667
\(778\) −32212.2 −1.48440
\(779\) 13974.0 0.642710
\(780\) −56316.2 −2.58518
\(781\) −4094.35 −0.187589
\(782\) 7838.51 0.358446
\(783\) −76045.3 −3.47080
\(784\) −714.111 −0.0325306
\(785\) −25266.1 −1.14877
\(786\) 34874.2 1.58260
\(787\) −29807.9 −1.35011 −0.675054 0.737768i \(-0.735880\pi\)
−0.675054 + 0.737768i \(0.735880\pi\)
\(788\) −34338.4 −1.55236
\(789\) −40925.3 −1.84661
\(790\) 82537.8 3.71717
\(791\) 21593.1 0.970624
\(792\) −832.062 −0.0373308
\(793\) −2947.03 −0.131970
\(794\) 22678.6 1.01364
\(795\) −67205.5 −2.99815
\(796\) 24707.2 1.10016
\(797\) −23158.3 −1.02924 −0.514622 0.857417i \(-0.672068\pi\)
−0.514622 + 0.857417i \(0.672068\pi\)
\(798\) −22009.0 −0.976329
\(799\) 22805.4 1.00976
\(800\) −58465.2 −2.58382
\(801\) 19383.1 0.855015
\(802\) −54602.4 −2.40409
\(803\) 2847.67 0.125146
\(804\) −5221.71 −0.229049
\(805\) 11486.1 0.502898
\(806\) −14896.5 −0.651002
\(807\) −39983.3 −1.74409
\(808\) 4016.88 0.174893
\(809\) −18381.5 −0.798836 −0.399418 0.916769i \(-0.630788\pi\)
−0.399418 + 0.916769i \(0.630788\pi\)
\(810\) 64549.4 2.80004
\(811\) −32529.5 −1.40847 −0.704233 0.709969i \(-0.748709\pi\)
−0.704233 + 0.709969i \(0.748709\pi\)
\(812\) −38961.1 −1.68383
\(813\) −4454.25 −0.192149
\(814\) 1533.79 0.0660432
\(815\) 3480.90 0.149608
\(816\) −40136.6 −1.72189
\(817\) 0 0
\(818\) −9552.23 −0.408296
\(819\) −47798.5 −2.03934
\(820\) −57247.1 −2.43799
\(821\) 24119.8 1.02532 0.512660 0.858592i \(-0.328660\pi\)
0.512660 + 0.858592i \(0.328660\pi\)
\(822\) 31013.2 1.31595
\(823\) −22679.7 −0.960588 −0.480294 0.877108i \(-0.659470\pi\)
−0.480294 + 0.877108i \(0.659470\pi\)
\(824\) −1560.23 −0.0659627
\(825\) −9458.84 −0.399169
\(826\) 18132.5 0.763812
\(827\) 7335.57 0.308443 0.154222 0.988036i \(-0.450713\pi\)
0.154222 + 0.988036i \(0.450713\pi\)
\(828\) −12735.0 −0.534509
\(829\) −2389.09 −0.100092 −0.0500461 0.998747i \(-0.515937\pi\)
−0.0500461 + 0.998747i \(0.515937\pi\)
\(830\) 1954.52 0.0817376
\(831\) −23429.3 −0.978042
\(832\) −17990.3 −0.749641
\(833\) −638.509 −0.0265583
\(834\) 48498.2 2.01361
\(835\) −53756.8 −2.22794
\(836\) −1031.46 −0.0426719
\(837\) 21924.0 0.905383
\(838\) −36602.4 −1.50884
\(839\) 6893.41 0.283656 0.141828 0.989891i \(-0.454702\pi\)
0.141828 + 0.989891i \(0.454702\pi\)
\(840\) −11114.6 −0.456537
\(841\) 60348.3 2.47440
\(842\) 29071.2 1.18986
\(843\) −57657.7 −2.35568
\(844\) 6765.30 0.275914
\(845\) −2162.19 −0.0880254
\(846\) −78670.0 −3.19708
\(847\) 24653.6 1.00013
\(848\) −27254.3 −1.10367
\(849\) −31768.8 −1.28422
\(850\) −58082.4 −2.34377
\(851\) −2893.82 −0.116567
\(852\) 60638.7 2.43832
\(853\) −6006.16 −0.241087 −0.120543 0.992708i \(-0.538464\pi\)
−0.120543 + 0.992708i \(0.538464\pi\)
\(854\) 4718.28 0.189059
\(855\) −35143.6 −1.40572
\(856\) 2498.67 0.0997695
\(857\) 9759.72 0.389015 0.194507 0.980901i \(-0.437689\pi\)
0.194507 + 0.980901i \(0.437689\pi\)
\(858\) −7060.96 −0.280953
\(859\) −12026.8 −0.477705 −0.238853 0.971056i \(-0.576771\pi\)
−0.238853 + 0.971056i \(0.576771\pi\)
\(860\) 0 0
\(861\) −72131.9 −2.85511
\(862\) 49852.2 1.96981
\(863\) −36262.9 −1.43036 −0.715182 0.698938i \(-0.753656\pi\)
−0.715182 + 0.698938i \(0.753656\pi\)
\(864\) 64233.1 2.52923
\(865\) −55560.2 −2.18393
\(866\) 34723.1 1.36252
\(867\) 8797.34 0.344606
\(868\) 11232.6 0.439237
\(869\) 4873.92 0.190260
\(870\) −196099. −7.64182
\(871\) 3679.51 0.143141
\(872\) 407.217 0.0158143
\(873\) 16226.8 0.629089
\(874\) 4132.03 0.159918
\(875\) −40368.2 −1.55965
\(876\) −42175.0 −1.62667
\(877\) −3560.34 −0.137086 −0.0685429 0.997648i \(-0.521835\pi\)
−0.0685429 + 0.997648i \(0.521835\pi\)
\(878\) −35128.5 −1.35026
\(879\) 54959.4 2.10891
\(880\) −5852.43 −0.224188
\(881\) 45565.8 1.74251 0.871255 0.490831i \(-0.163307\pi\)
0.871255 + 0.490831i \(0.163307\pi\)
\(882\) 2202.62 0.0840884
\(883\) −12993.7 −0.495214 −0.247607 0.968861i \(-0.579644\pi\)
−0.247607 + 0.968861i \(0.579644\pi\)
\(884\) −20420.4 −0.776938
\(885\) 42982.9 1.63260
\(886\) 28221.3 1.07011
\(887\) −33676.4 −1.27480 −0.637398 0.770535i \(-0.719989\pi\)
−0.637398 + 0.770535i \(0.719989\pi\)
\(888\) 2800.22 0.105821
\(889\) 25460.7 0.960545
\(890\) 25764.3 0.970360
\(891\) 3811.69 0.143318
\(892\) 36523.3 1.37095
\(893\) 12021.7 0.450495
\(894\) 91864.9 3.43671
\(895\) 42667.5 1.59354
\(896\) −8162.97 −0.304359
\(897\) 13322.0 0.495886
\(898\) −46054.3 −1.71142
\(899\) −24429.9 −0.906322
\(900\) 94364.9 3.49500
\(901\) −24368.9 −0.901050
\(902\) −7177.68 −0.264956
\(903\) 0 0
\(904\) 3922.83 0.144327
\(905\) −78099.3 −2.86863
\(906\) 10085.5 0.369834
\(907\) 22376.4 0.819180 0.409590 0.912270i \(-0.365672\pi\)
0.409590 + 0.912270i \(0.365672\pi\)
\(908\) −27196.8 −0.994008
\(909\) −65561.2 −2.39222
\(910\) −63534.6 −2.31445
\(911\) −52260.5 −1.90062 −0.950312 0.311298i \(-0.899236\pi\)
−0.950312 + 0.311298i \(0.899236\pi\)
\(912\) −21157.8 −0.768206
\(913\) 115.416 0.00418368
\(914\) −23862.8 −0.863579
\(915\) 11184.7 0.404102
\(916\) −18425.4 −0.664619
\(917\) 18530.0 0.667301
\(918\) 63812.5 2.29425
\(919\) −30391.1 −1.09087 −0.545436 0.838153i \(-0.683636\pi\)
−0.545436 + 0.838153i \(0.683636\pi\)
\(920\) 2086.69 0.0747783
\(921\) −71847.9 −2.57054
\(922\) 7496.37 0.267766
\(923\) −42729.5 −1.52379
\(924\) 5324.24 0.189561
\(925\) 21442.8 0.762201
\(926\) 41628.9 1.47733
\(927\) 25465.2 0.902253
\(928\) −71574.8 −2.53185
\(929\) 9900.90 0.349664 0.174832 0.984598i \(-0.444062\pi\)
0.174832 + 0.984598i \(0.444062\pi\)
\(930\) 56535.8 1.99342
\(931\) −336.587 −0.0118487
\(932\) −26070.6 −0.916279
\(933\) 97235.5 3.41195
\(934\) 51407.6 1.80097
\(935\) −5232.84 −0.183029
\(936\) −8683.57 −0.303239
\(937\) 24469.5 0.853131 0.426566 0.904457i \(-0.359723\pi\)
0.426566 + 0.904457i \(0.359723\pi\)
\(938\) −5891.01 −0.205062
\(939\) −11019.0 −0.382950
\(940\) −49249.2 −1.70886
\(941\) 8819.85 0.305546 0.152773 0.988261i \(-0.451180\pi\)
0.152773 + 0.988261i \(0.451180\pi\)
\(942\) 46917.5 1.62278
\(943\) 13542.2 0.467652
\(944\) 17431.1 0.600991
\(945\) 93507.3 3.21883
\(946\) 0 0
\(947\) 46001.3 1.57850 0.789250 0.614072i \(-0.210469\pi\)
0.789250 + 0.614072i \(0.210469\pi\)
\(948\) −72184.5 −2.47304
\(949\) 29718.9 1.01656
\(950\) −30617.8 −1.04565
\(951\) 58855.8 2.00687
\(952\) −4030.19 −0.137205
\(953\) −12737.3 −0.432950 −0.216475 0.976288i \(-0.569456\pi\)
−0.216475 + 0.976288i \(0.569456\pi\)
\(954\) 84063.6 2.85289
\(955\) −59318.1 −2.00994
\(956\) 5983.32 0.202421
\(957\) −11579.8 −0.391141
\(958\) −8746.27 −0.294968
\(959\) 16478.5 0.554868
\(960\) 68277.3 2.29546
\(961\) −22747.8 −0.763580
\(962\) 16006.9 0.536470
\(963\) −40781.9 −1.36467
\(964\) 8814.81 0.294508
\(965\) 26075.3 0.869838
\(966\) −21329.0 −0.710402
\(967\) 28175.8 0.936992 0.468496 0.883466i \(-0.344796\pi\)
0.468496 + 0.883466i \(0.344796\pi\)
\(968\) 4478.82 0.148714
\(969\) −18917.8 −0.627170
\(970\) 21568.9 0.713956
\(971\) 14012.5 0.463113 0.231557 0.972821i \(-0.425618\pi\)
0.231557 + 0.972821i \(0.425618\pi\)
\(972\) −6217.67 −0.205177
\(973\) 25768.9 0.849039
\(974\) 66635.2 2.19212
\(975\) −98714.5 −3.24245
\(976\) 4535.79 0.148757
\(977\) 25344.3 0.829924 0.414962 0.909839i \(-0.363795\pi\)
0.414962 + 0.909839i \(0.363795\pi\)
\(978\) −6463.80 −0.211339
\(979\) 1521.40 0.0496671
\(980\) 1378.89 0.0449459
\(981\) −6646.37 −0.216312
\(982\) 66312.6 2.15491
\(983\) 31795.3 1.03165 0.515825 0.856694i \(-0.327486\pi\)
0.515825 + 0.856694i \(0.327486\pi\)
\(984\) −13104.2 −0.424540
\(985\) −91832.5 −2.97059
\(986\) −71106.2 −2.29663
\(987\) −62054.5 −2.00123
\(988\) −10764.5 −0.346624
\(989\) 0 0
\(990\) 18051.3 0.579505
\(991\) 57046.1 1.82859 0.914294 0.405051i \(-0.132746\pi\)
0.914294 + 0.405051i \(0.132746\pi\)
\(992\) 20635.2 0.660451
\(993\) 12695.4 0.405715
\(994\) 68411.2 2.18297
\(995\) 66075.4 2.10526
\(996\) −1709.35 −0.0543802
\(997\) 14197.2 0.450984 0.225492 0.974245i \(-0.427601\pi\)
0.225492 + 0.974245i \(0.427601\pi\)
\(998\) −27997.1 −0.888008
\(999\) −23558.3 −0.746096
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.1.41 50
43.42 odd 2 1849.4.a.j.1.10 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.41 50 1.1 even 1 trivial
1849.4.a.j.1.10 yes 50 43.42 odd 2